117694972 Fluid Mechanics and Pipe Flow

FLUID MECHANICS AND PIPE FLOW:TURBULENCE, SIMULATION

AND DYNAMICS

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FLUID MECHANICS AND PIPE FLOW:TURBULENCE, SIMULATION

AND DYNAMICS

DONALD MATOSAND

CRISTIAN VALERIOEDITORS

Nova Science Publishers, Inc.New York

Copyright © 2009 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com

NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Fluid mechanics and pipe flow : turbulence, simulation, and dynamics / editors, Donald Matos and Cristian Valerio. p. cm. Includes bibliographical references and index. ISBN 978-1-61668-990-2 (E-Book) 1. Fluid mechanics. 2. Pipe--Fluid dynamics. I. Matos, Donald. II. Valerio, Cristian. TA357.F5787 2009 620.1'06--dc22 2009017666

Published by Nova Science Publishers, Inc. New York

CONTENTS

Preface vii

Chapter 1 Solute Transport, Dispersion, and Separation in NanofluidicChannels

1

Xiangchun Xuan

Chapter 2 H2O in the Mantle: From Fluid to High-Pressure HydrousSilicates

27

N.R. Khisina, R. Wirth and S. Matsyuk

Chapter 3 On the Numerical Simulation of Turbulence Modulation in Two-Phase Flows

41

K. Mohanarangam and J.Y. Tu

Chapter 4 A Review of Population Balance Modelling for Multiphase Flows:Approaches, Applications and Future Aspects

117

Sherman C.P. Cheung, G.H. Yeoh and J.Y. Tu

Chapter 5 Numerical Analysis of Heat Transfer and Fluid Flow for Three-Dimensional Horizontal Annuli with Open Ends

171

Chun-Lang Yeh

Chapter 6 Convective Heat Transfer in the Thermal Entrance Regionof Parallel Flow Double-Pipe Heat Exchangersfor Non-Newtonian Fluids

205

Ryoichi Chiba, Masaaki Izumi and Yoshihiro Sugano

Chapter 7 Numerical Simulation of Turbulent Pipe Flow 231M. Ould-Rouis and A.A. Feiz

Chapter 8 Pipe Flow Analysis of Uranium Nuclear Heating with ConjugateHeat Transfer

269

G.H. Yeoh and M.K.M. Ho

Chapter 9 First and Second Law Thermodynamics Analysis of Pipe Flow 317Ahmet Z. Sahin

Contentsvi

Chapter 10 Single-Phase Incompressible Fluid Flowin Mini- and Micro-channels

343

Lixin Cheng

Chapter 11 Experimental Study of Pulsating Turbulent Flowthrough a Divergent Tube

365

Masaru Sumida

Chapter 12 Solution of an Airfoil Design Inverse Problem for a Viscous FlowUsing a Contractive Operator

379

Jan Šimák and Jaroslav Pelant

Chapter 13 Some Free Boundary Problems in Potential Flow Regime Usingthe Level Set Method

399

M. Garzon, N. Bobillo-Ares and J.A. Sethian

Chapter 14 A New Approach for Polydispersed Turbulent Two-Phase Flows:The Case of Deposition in Pipe-Flows

441

S. Chibbaro

Index 455

PREFACE

Fluid mechanics is the study of how fluids move and the forces that develop as a result.Fluids include liquids and gases and fluid flow can be either laminar or turbulent. This bookpresents a level set based methodology that will avoid problems in potential flow models withmoving boundaries. A review of the state-of-the-art population balance modelling techniquesthat have been adopted to describe the nature of dispersed phase in multiphase problems ispresented as well. Recent works that are aimed at putting forward the main ideas behind anew theoretical approach to turbulent wall-bounded flows are examined, including a state-of-the-art review on single-phase incompressible fluid flow.

Recent breakthrough in nanofabrication has stimulated the interest of solute separation innanofluidic channels. Since the hydraulic radius of nanochannels is comparable to thethickness of electric double layers, the enormous electric fields inherent to the latter generatetransverse electromigrations causing charge-dependent solute distributions over the channelcross-section. As a consequence, the non-uniform fluid flow through nanochannels yieldscharge-dependent solute speeds enabling the separation of solutes by charge alone.In Chapter 1 we develop a theoretical model of solute transport, dispersion and separation inelectroosmotic and pressure-driven flows through nanofluidic channels. This model providesa basis for the optimization of solute separation in nanochannels in terms of selectivity andresolution as traditionally defined.

As presented in Chapter 2, infrared spectroscopic data show that nominally anhydrousolivine (Mg,Fe)2SiO4 contains traces of H2O, up to several hundred wt. ppm of H2O (Miller etal., 1987; Bell et al., 2004; Koch-Muller et al., 2006; Matsyuk & Langer, 2004) and thereforeolivine is suggested to be a water carrier in the mantle (Thompson, 1992). Protonation ofolivine during its crystallization from a hydrous melt resulted in the appearance of intrinsicOH-defects (Libowitsky & Beran, 1995). Mantle olivine nodules from kimberlites wereinvestigated with FTIR and TEM methods (Khisina et al., 2001, 2002, 2008). The results arethe following: (1) Water content in xenoliths is lower than water content in xenocrysts. Fromthese data we concluded that kimberlite magma had been saturated by H2O, whereas adjacentmantle rocks had been crystallized from water-depleted melts. (2) Extrinsic water in olivine isrepresented by high-pressure phases, 10Å-Phase Mg3Si4O10(OH)2.nH2O and hydrous olivinen(Mg,Fe)2SiO4.(H2MgSiO4), both of which belong to the group of Dense HydrousMagnesium Silicates (DHMS), which were synthesized in laboratory high-pressureexperiments (Prewitt & Downs, 1999). The DHMS were regarded as possible mineral carriersfor H2O in the mantle; however, they were not found in natural material until quite recently.

Donald Matos and Cristian Valerioviii

Our observations demonstrate the first finding of the 10Å-Phase and hydrous olivine as amantle substance. (3) 10Å-Phase, which occurred as either nanoinclusions or narrow veins inolivine, is a ubiquitous nano-mineral of kimberlite and closely related to olivine. (4) There aretwo different mechanisms of the 10Å-Phase formation: (a) purification of olivine from OH-bearing defects resulting in transformation of olivine to the 10 Å-Phase with the liberation ofwater fluid; and (b) replacement of olivine for the 10Å-Phase due to hydrous metasomatismin the mantle in the presence of H2O fluid.

With the increase of computational power, computational modelling of two-phase flowproblems using computational fluid dynamics (CFD) techniques is gradually becomingattractive in the engineering field. The major aim of Chapter 3 is to investigate the TurbulenceModulation (TM) of dilute two phase flows. Various density regimes of the two-phase flowshave been investigated in this paper, namely the dilute Gas-Particle (GP) flow, Liquid-Particle (LP) flow and also the Liquid-Air (LA) flows. While the density is quite high for thedispersed phase flow for the gas-particle flow, the density ratio is almost the same for theliquid particle flow, while for the liquid-air flow the density is quite high for the carrier phaseflow. The study of all these density regimes gives a clear picture of how the carrier phasebehaves in the presence of the dispersed phases, which ultimately leads to better design andsafety of many two-phase flow equipments and processes. In order to carry out this approach,an Eulerian-Eulerian Two-Fluid model, with additional source terms to account for thepresence of the dispersed phase in the turbulence equations has been employed for particulateflows, whereas Population Balance (PB) have been employed to study the bubbly flows. Forthe dilute gas-particle flows, particle-turbulence interaction over a backward-facing stepgeometry was numerically investigated. Two different particle classes with same Stokesnumber and varied particle Reynolds number are considered in this study. A detailed studyinto the turbulent behaviour of dilute particulate flow under the influence of two carrierphases namely gas and liquid was also been carried out behind a sudden expansion geometry.The major endeavour of the study is to ascertain the response of the particles within thecarrier (gas or liquid) phase. The main aim prompting the current study is the densitydifference between the carrier and the dispersed phase. While the ratio is quite high in termsof the dispersed phase for the gas-particle flows, the ratio is far more less in terms of theliquid-particle flows. Numerical simulations were carried out for both these classes of flowsand their results were validated against their respective sets of experimental data. For theLiquid-Air flows the phenomenon of drag reduction by the injection of micro-bubbles intoturbulent boundary layer has been investigated using an Eulerian-Eulerian two-fluid model.Two variants namely the Inhomogeneous and MUSIG (MUltiple-SIze-Group) based onPopulation balance models are investigated. The simulated results were benchmarked againstthe experimental findings and also against other numerical studies explaining the variousaspects of drag reduction. For the two Reynolds number cases considered, the buoyancy withthe plate on the bottom configuration is investigated, as from the experiments it is seen thatbuoyancy seem to play a role in the drag reduction. The under predictions of the MUSIGmodel at low flow rates was investigated and reported, their predictions seem to fair betterwith the decrease of the break-up tendency among the micro-bubbles.

Population balance modelling is of significant importance in many scientific andindustrial instances such as: fluidizations, precipitation, particles formation in aerosols,bubbly and droplet flows and so on. In population balance modelling, the solution of thepopulation balance equation (PBE) records the number of entities in dispersed phase that

Preface ix

always governs the overall behaviour of the practical system under consideration. For themajority of cases, the solution evolves dynamically according to the “birth” and “death”processes of which it is tightly coupled with the system operation condition. Theimplementation of PBE in conjunction with the Computational Fluid Dynamics (CFD) isthereby becoming ever a crucial consideration in multiphase flow simulations. Nevertheless,the inherent integrodifferential form of the PBE poses tremendous difficulties on its solutionprocedures where analytical solutions are rare and impossible to be achieved. In Chapter 4,we present a review of the state-of-the-art population balance modelling techniques that havebeen adopted to describe the phenomenological nature of dispersed phase in multiphaseproblems. The main focus of the review can be broadly classified into three categories: (i)Numerical approaches or solution algorithms of the PBE; (ii) Applications of the PBE inpractical gas-liquid multiphase problems and (iii) Possible aspects of the future developmentin population balance modelling. For the first category, details of solution algorithms basedon both method of moment (MOM) and discrete class method (CM) that have been proposedin the literature are provided. Advantages and drawbacks of both approaches are alsodiscussed from the theoretical and practical viewpoints. For the second category, applicationsof existing population balance models in practical multiphase problems that have beenproposed in the literature are summarized. Selected existing mathematical closures formodelling the “birth” and “death” rate of bubbles in gas-liquid flows are introduced.Particular attention is devoted to assess the capability of some selected models in predictingbubbly flow conditions through detail validation studies against experimental data. Thesestudies demonstrate that good agreement can be achieved by the present model by comparingthe predicted results against measured data with regards to the radial distribution of voidfraction, Sauter mean bubble diameter, interfacial area concentration and liquid axial velocity.Finally, weaknesses and limitations of the existing models are revealed are suggestions forfurther development are discussed. Emerging topics for future population balance studies areprovided as to complete the aspect of population balance modelling.

Study of the heat transfer and fluid flow inside concentric or eccentric annuli can beapplied in many engineering fields, e.g. solar energy collection, fire protection, undergroundconduit, heat dissipation for electrical equipment, etc. In the past few decades, these studieswere concentrated in two-dimensional research and were mostly devoted to the investigationof the effects of convective heat transfer. However, in practical situation, this problem shouldbe three-dimensional, except for the vertical concentric annuli which could be modeled astwo-dimensional (axisymmetric). In addition, the effects of heat conduction and radiationshould not be neglected unless the outer cylinder is adiabatic and the temperature of the flowfield is sufficiently low. As the author knows, none of the open literature is devoted to theinvestigation of the conjugated heat transfer of convection, conduction and radiation for thisproblem. The author has worked in industrial piping design area and is experienced in thisfield. The author has also employed three-dimensional body-fitted coordinate systemassociated with zonal grid method to analyze the natural convective heat transfer and fluidflow inside three-dimensional horizontal concentric or eccentric annuli with open ends.Owing to its broad application in practical engineering problems, Chapter 5 is devoted to adetailed discussion of the simulation method for the heat transfer and fluid flow inside three-dimensional horizontal concentric or eccentric annuli with open ends. Two illustrativeproblems are exhibited to demonstrate its practical applications.

Donald Matos and Cristian Valeriox

In Chapter 6, the conjugated Graetz problem in parallel flow double-pipe heat exchangersis analytically solved by an integral transform method—Vodicka’s method—and an analyticalsolution to the fluid temperatures varying along the radial and axial directions is obtained in acompletely explicit form. Since the present study focuses on the range of a sufficiently largePéclet number, heat conduction along the axial direction is considered to be negligible. Animportant feature of the analytical method presented is that it permits arbitrary velocitydistributions of the fluids as long as they are hydrodynamically fully developed. Numericalcalculations are performed for the case in which a Newtonian fluid flows in the annulus of thedouble pipe, whereas a non-Newtonian fluid obeying a simple power law flows through theinner pipe. The numerical results demonstrate the effects of the thermal conductivity ratio ofthe fluids, Péclet number ratio and power-law index on the temperature distributions in thefluids and the amount of exchanged heat between the two fluids.

Many experimental and numerical studies have been devoted to turbulent pipe flows dueto the number of applications in which theses flows govern heat or mass transfer processes:heat exchangers, agricultural spraying machines, gasoline engines, and gas turbines forexamples. The simplest case of non-rotating pipe has been extensively studied experimentallyand numerically. Most of pipe flow numerical simulations have studied stability andtransition. Some Direct Numerical Simulations (DNS) have been performed, with a 3-Dspectral code, or using mixed finite difference and spectral methods. There is few DNS of theturbulent rotating pipe flow in the literature. Investigations devoted to Large EddySimulations (LES) of turbulence pipe flow are very limited. With DNS and LES, one canderive more turbulence statistics and determine a well-resolved flow field which is aprerequisite for correct predictions of heat transfer. However, the turbulent pipe flows havenot been so deeply studied through DNS and LES as the plane-channel flows, due to thepeculiar numerical difficulties associated with the cylindrical coordinate system used for thenumerical simulation of the pipe flows.

Chapter 7 presents Direct Numerical Simulations and Large Eddy Simulations of fullydeveloped turbulent pipe flow in non-rotating and rotating cases. The governing equations arediscretized on a staggered mesh in cylindrical coordinates. The numerical integration isperformed by a finite difference scheme, second-order accurate in space and time. The timeadvancement employs a fractional step method. The aim of this study is to investigate theeffects of the Reynolds number and of the rotation number on the turbulent flowcharacteristics. The mean velocity profiles and many turbulence statistics are compared tonumerical and experimental data available in the literature, and reasonably good agreement isobtained. In particular, the results show that the axial velocity profile gradually approaches alaminar shape when increasing the rotation rate, due to the stability effect caused by thecentrifugal force. Consequently, the friction factor decreases. The rotation of the wall haslarge effects on the root mean square (rms), these effects being more pronounced for thestreamwise rms velocity. The influence of rotation is to reduce the Reynolds stress component⟨Vr'Vz'⟩ and to increase the two other stresses ⟨Vr'Vθ'⟩ and ⟨Vθ'Vz'⟩. The effect of the Reynoldsnumber on the rms of the axial velocity (⟨Vz'2⟩1/2) and the distributions of ⟨Vr'Vz'⟩ is evident,and it increases with an increase in the Reynolds number. On the other hand, the ⟨Vr'Vθ'⟩-profiles appear to be nearly independent of the Reynolds number. The present DNS and LESpredictions will be helpful for developing more accurate turbulence models for heat transferand fluid flow in pipe flows.

Preface xi

The field of computational fluid dynamics (CFD) has evolved from an academic curiosityto a tool of practical importance. Applications of CFD have become increasingly important innuclear engineering and science, where exacting standards of safety and reliability areparamount. The newly-commissioned Open Pool Australian Light-water (OPAL) researchreactor at the Australian Nuclear Science and Technology Organisation (ANSTO) has beendesigned to irradiate uranium targets to produce molybdenum medical isotopes for diagnosisand radiotherapy. During the irradiation process, a vast amount of power is generated whichrequires efficient heat removal. The preferred method is by light-water forced convectioncooling—essentially a study of complex pipe flows with coupled conjugate heat transfer.Feasibility investigation on the use of computational fluid dynamics methodologies intovarious pipe flow configurations for a variety of molybdenum targets and pipe geometries aredetailed in Chapter 8. Such an undertaking has been met with a number of significantmodeling challenges: firstly, the complexity of the geometry that needed to be modeled.Herein, challenges in grid generation are addressed by the creation of purpose-built body-fitted and/or unstructured meshes to map the intricacies within the geometry in order toensure numerical accuracy as well as computational efficiency in the solution of the predictedresult. Secondly, various parts of the irradiation rig that are required to be specified ascomposite solid materials are defined to attain the correct heat transfer characteristics.Thirdly, the use of an appropriate turbulence model is deemed to be necessary for the correctdescription of the fluid and heat flow through the irradiation targets, since the heat removal isforced convection and the flow regime is fully turbulent, which further adds to the complexityof the solution. As complicated as the computational fluid dynamics modeling is, numericalmodeling has significantly reduced the cost and lead time in the molybdenum-target designprocess, and such an approach would not have been possible without the continualimprovement of computational power and hardware. This chapter also addresses theimportance of experimental modeling to evaluate the design and numerical results of thevelocity and flow paths generated by the numerical models. Predicted results have been foundto agree well with experimental observations of pipe flows through transparent models andexperimental measurements via the Laser Doppler Velocimetry instrument.

In Chapter 9, the entropy generation for during fluid flow in a pipe is investigated. Thetemperature dependence of the viscosity is taken into consideration in the analysis. Laminarand turbulent flow cases are treated separately. Two types of thermal boundary conditions areconsidered; uniform heat flux and constant wall temperature. In addition, various cross-sectional pipe geometries were compared from the point of view of entropy generation andpumping power requirement in order to determine the possible optimum pipe geometry whichminimizes the exergy losses.

Chapter 10 aims to present a state-of-the-art review on single-phase incompressible fluidflow in mini- and micro-channels. First, classification of mini- and micro-channels isdiscussed. Then, conventional theories on laminar, laminar to turbulent transition andturbulent fluid flow in macro-channels (conventional channels) are summarized. Next, a briefreview of the available studies on single-phase incompressible fluid flow in mini- and micro-channels is presented. Some experimental results on single phase laminar, laminar toturbulent transition and turbulent flows are presented. The deviations from the conventionalfriction factor correlations for single-phase incompressible fluid flow in mini and micro-channels are discussed. The effect factors on mini- and micro-channel single-phase fluid floware analyzed. Especially, the surface roughness effect is focused on. According to this review,

Donald Matos and Cristian Valerioxii

the future research needs have been identified. So far, no systematic agreed knowledge ofsingle-phase fluid flow in mini- and micro-channels has yet been achieved. Therefore, effortsshould be made to contribute to systematic theories for microscale fluid flow through verycareful experiments.

In Chapter 11, an experimental investigation was conducted of pulsating turbulent flow ina conically divergent tube with a total divergence angle of 12°. The experiments were carriedout under the conditions of Womersley numbers of α =10∼40, mean Reynolds number of Reta

=20000 and oscillatory Reynolds number of Reos =10000 (the flow rate ratio of η = 0.5).Time-dependent wall static pressure and axial velocity were measured at several longitudinalstations and the distributions were illustrated for representative phases within one cycle. Therise between the pressures at the inlet and the exit of the divergent tube does not become toolarge when the flow rate increases, it being moderately high in the decelerative phase. Theprofiles of the phase-averaged velocity and the turbulence intensity in the cross section arevery different from those for steady flow. Also, they show complex changes along the tubeaxis in both the accelerative and decelerative phases.

Chapter 12 deals with a numerical method for a solution of an airfoil design inverseproblem. The presented method is intended for a design of an airfoil based on a prescribedpressure distribution along a mean camber line, especially for modifying existing airfoils. Themain idea of this method is a coupling of a direct and approximate inverse operator. The goalis to find a pseudo-distribution corresponding to the desired airfoil with respect to theapproximate inversion. This is done in an iterative way. The direct operator represents asolution of a flow around an airfoil, described by a system of the Navier-Stokes equations inthe case of a laminar flow and by the k−ω model in the case of a turbulent flow. There is arelative freedom of choosing the model describing the flow. The system of PDEs is solved byan implicit finite volume method. The approximate inverse operator is based on a thin airfoiltheory for a potential flow, equipped with some corrections according to the model used. Theairfoil is constructed using a mean camber line and a thickness function. The so far developedmethod has several restrictions. It is applicable to a subsonic pressure distribution satisfying acertain condition for the position of a stagnation point. Numerical results are presented.

Recent advances in the field of fluid mechanics with moving fronts are linked to the useof Level SetMethods, a versatile mathematical technique to follow free boundaries whichundergo topological changes. A challenging class of problems in this context are those relatedto the solution of a partial differential equation posed on a moving domain, in which theboundary condition for the PDE solver has to be obtained from a partial differential equationdefined on the front. This is the case of potential flow models with moving boundaries.Moreover, the fluid front may carry some material substance which diffuses in the front and isadvected by the front velocity, as for example the use of surfactants to lower surface tension.We present a Level Set based methodology to embed this partial differential equationsdefined on the front in a complete Eulerian framework, fully avoiding the tracking of fluidparticles and its known limitations. To show the advantages of this approach in the field ofFluid Mechanics we present in Chapter 13 one particular application: the numericalapproximation of a potential flow model to simulate the evolution and breaking of a solitarywave propagating over a slopping bottom and compare the level set based algorithm withprevious front tracking models.

Preface xiii

Chapter 14 is basically a review of recent works that is aimed at putting forward the mainideas behind a new theoretical approach to turbulent wall-bounded flows, notably pipe-flows,in which complex physics is involved, such as combustion or particle transport. Pipe flowsare ubiquitous in industrial applications and have been studied intensively in the last century,both from a theoretical and experimental point of view. The result of such a strong effort is agood comprehension of the physics underlying the dynamics of these flows and theproposition of reliable models for simple turbulent pipe-flows at large Reynolds numberNevertheless, the advancing of engineering frontiers casts a growing demand for modelssuitable for the study of more complex flows. For instance, the motion and the interactionwith walls of aerosol particles, the presence of roughness on walls and the possibility of dragreduction through the introduction of few complex molecules in the flow constitute someinteresting examples of pipe-flows with some new complex physics involved. A goodmodeling approach to these flows is yet to come and, in the commentary, we support the ideathat a new angle of attack is needed with respect to present methods. In this article, weanalyze which are the fundamental features of complex two-phase flows and we point out thatthere are two key elements to be taken into account by a suitable theoretical model: 1) Theseflows exhibit chaotic patterns; 2) The presence of instantaneous coherent structures radicallychange the flow properties. From a methodological point of view, two main theoreticalapproaches have been considered so far: the solution of equations based on first principles(for example, the Navier-Stokes equations for a single phase fluid) or Eulerian models basedon constitutive relations. In analogy with the language of statistical physics, we consider theformer as a microscopic approach and the later as a macroscopic one. We discuss why weconsider both approaches unsatisfying with regard to the description of general complexturbulent flows, like two-phase flows. Hence, we argue that a significant breakthrough can beobtained by choosing a new approach based upon two main ideas: 1) The approach has to bemesoscopic (in the middle between the microscopic and the macroscopic) and statistical; 2)Some geometrical features of turbulence have to be introduced in the statistical model. Wepresent the main characteristics of a stochastic model which respects the conditions expressedby the point 1) and a method to fulfill the point 2). These arguments are backed up with somerecent numerical results of deposition onto walls in turbulent pipe-flows. Finally, someperspectives are also given.

In: Fluid Mechanics and Pipe Flow ISBN: 978-1-60741-037-9Editors: D. Matos and C. Valerio, pp. 1-26 © 2009 Nova Science Publishers, Inc.

Chapter 1

SOLUTE TRANSPORT, DISPERSION, AND SEPARATIONIN NANOFLUIDIC CHANNELS

Xiangchun Xuan*

Department of Mechanical Engineering, Clemson University,Clemson, SC 29634-0921, USA

Abstract

Recent breakthrough in nanofabrication has stimulated the interest of solute separation innanofluidic channels. Since the hydraulic radius of nanochannels is comparable to thethickness of electric double layers, the enormous electric fields inherent to the latter generatetransverse electromigrations causing charge-dependent solute distributions over the channelcross-section. As a consequence, the non-uniform fluid flow through nanochannels yieldscharge-dependent solute speeds enabling the separation of solutes by charge alone. In thischapter we develop a theoretical model of solute transport, dispersion and separation inelectroosmotic and pressure-driven flows through nanofluidic channels. This model provides abasis for the optimization of solute separation in nanochannels in terms of selectivity andresolution as traditionally defined.

1. Introduction

Solute transport and separation in micro-columns (e.g., micro capillaries and chip-basedmicrochannels) have been a focus of research and development in electrophoresis andchromatography communities for many years. Recent breakthrough in nanofabrication hasinitiated the study of these topics among others in nanofluidic channels [1-3]. Since thehydraulic radius of nanochannels is comparable to the thickness of electric double layers(EDL), the enormous electric fields inherent to the latter generate transverse electromigrationscausing charge-dependent solute distributions over the channel cross-section [4-7]. As aconsequence, the non-uniform fluid flow in nanochannels yields charge-dependent solutespeeds enabling the separation of solutes by charge alone [8,9]. Such charge-based solute

* E-mail address: [email protected]. Tel: (864) 656-5630. Fax: (864) 656-7299

Xiangchun Xuan2

separation was first proposed and implemented by Pennathur and Santiago [10] and Garcia etal. [11] in electroosmotic flow through nanoscale channels, termed nanochannelelectrophoresis. As a matter of fact, this separation may also happen in pressure-driven flowalong nanoscale channels, termed here as nanochannel chromatography for comparison,which was first demonstrated theoretically by Griffiths and Nilson [12] and Xuan and Li [13],and later experimentally verified by Liu’s group [14].

So far, a number of theoretical studies have been conducted on the transport [4-6,9-13,15,19], dispersion [7,9,12,15-19] and separation [4,9-13,15,19] of solutes in free solutionsthrough nanofluidic channels. This chapter combines and unites the works from ourselves inthis area [6,13,15,17-19], and is aimed to develop a general analytical model of solutetransport, dispersion and separation in nanochannels. It is important to note that this modelapplies only to point-like solutes. For those with a finite size, one must consider thehydrodynamic and electrostatic interactions among solutes, electric field, and flow fluid, andas well the Steric interactions between solutes and channel walls etc [20].

2. Nomenclature

a channel half-heightBi defined function, = exp(−ziΨ)cb bulk concentration of the background electrolyteci concentration of solute species iCi bulk concentration of solute species ici,0 concentration of solute species i at the channel centerlineCi,0 initial concentration of solute species i at the channel centerlineDi solute diffusion coefficient

iD′ effective solute diffusion coefficientE axial electric fieldEst streaming potential fieldF Faraday’s constanthi reduced theoretical plate heightj electric current densityKi hydrodynamic dispersionL channel length P hydrodynamic pressure drop per unit channel lengthPe Peclet numberrji solute selectivityR Universal gas constantRji resolutiont time coordinateT absolute temperatureui axial solute speed

iu mean solute speedvi solute mobilityWi half width of the initially injected solute zone

Solute Transport, Dispersion, and Separation in Nanofluidic Channels 3

x streamwise or longitudinal coordinateXi the central location of the injected solute zoney transverse coordinatezi valence of ionsZ electrokinetic “figure of merit”

Greek Symbols

β non-dimensional product of fluid properties, = λbμ/εRTε permittivityγ apparent viscosity ratioκ reciprocal of Debye lengthχi dispersion coefficientλb molar conductivity of the background electrolyteμ fluid viscosityψ electrical double layer potentialΨ non-dimensional EDL potentialΨ0 EDL potential at the channel centerρe net charge densityσb bulk electric conductivity of background electrolyteσi standard deviation of solute peak distributionσt standard deviation of solute peak distribution in the time domainζ zeta potentialζ* non-dimensional zeta potential

Subscripts

e electroosmosis relatedp pressure-driven relatedi solute species i

3. Fluid Flow in Nanochannels

Given the fact that the width (in micrometers) of state-of-the-art nanofluidic channels isusually much larger than their depth (in nanometers) [1-3], we consider the solute transport influid flow through a long straight nanoslit, see Figure 1 for the schematic. The flow may beelectric field-driven, i.e., electroosmotic, or pressure-driven. For simplicity, the electrolytesolution is assumed symmetric with unit-charge, e.g., KCl. As the time scale for fluid flow (inthe order of nanoseconds) is far less than that of solute transport (typically of tens ofseconds), we assume a steady-state, fully-developed incompressible fluid motion, which in aslit channel is governed by

Xiangchun Xuan4

x

y

u + viziFE a u

Solute zone

Figure 1. Schematic of solute transport in a slit nanochannel (only the top half is illustrated due to thesymmetry).

2

2 0ed u P Edy

μ ρ+ + = (3-1)

where μ is the fluid viscosity, u the axial fluid velocity, y the transverse coordinate originatingfrom the channel axis, P the pressure drop per unit channel length, and E the axial electricfield either externally applied in electroosmotic flow or internally induced in pressure-drivenflow (i.e., the so-called streaming potential field) [21-24]. The net charge density, ρe, issolved from the Poisson equation [25]

2

2eddyψρ ε= − (3-2)

where ε is the fluid permittivity and ψ is the EDL potential. Invoking the no-slip condition forEq. (3-1) and the zeta potential condition for Eq. (3-2) on the channel wall (i.e., y = a), onecan easily obtain

p eu u u= + (3-3)

2 2

212pa yu P

aμ⎛ ⎞

= −⎜ ⎟⎝ ⎠

(3-4)

1eu Eεζμ ζ ∗

⎛ ⎞Ψ= − −⎜ ⎟

⎝ ⎠(3-5)

where up is the pressure-driven fluid velocity, ue the electroosmotic fluid velocity, a the half-height of the channel, and Ψ = Fψ/RT and ζ* = Fζ/RT the dimensionless forms of the EDLand wall zeta potentials with F the Faraday’s constant, R the universal gas constant and T theabsolute fluid temperature. It is noted that the contribution of charged solutes to the netcharge density ρe has been neglected. This is reasonable as long as the solute concentration ismuch lower than the ionic concentration of the background electrolyte, which is fulfilled intypical solute separations. Under such a condition, it is also safe to assume a uniform zetapotential on the channel wall.


The non-dimensional EDL potential in Eq. (3-5), Ψ, may be solved from the Poisson-Boltzmann equation [25]

( )2

22 sinhd

dyκΨ

= Ψ (3-6)

where 22 bF c RTκ ε= is the inverse of the so-called Debye screening length with cb the

bulk concentration of the background electrolyte. We recognize that the assumed Boltzmanndistribution of electrolyte ions in Poisson-Boltzmann equation might be questionable innanoscale channels, especially in those with strong EDL overlapping [26,27]. However, thisequation has been successfully used to explain the experimentally measured electricconductance and streaming current in variable nanofluidic channels [28-32], and is thus stillemployed here.

For the case of a small magnitude of ζ (e.g., |ζ| < 25 mV or |ζ*| < 1) which is actuallydesirable for sensitive solute separations in nanochannels as demonstrated by Griffiths andNilson [12], one may use the Debye-Huckel approximation to simplify Eq. (3-6) as [21,25]

22

2

ddy

κΨ= Ψ (3-7)

It is then straightforward to obtain

( )( )

* coshcosh

ya

κζ

κΨ = (3-8)

where κa may be viewed as the normalized channel half-height. It is important to note that fora given fluid and channel combination, the wall zeta potential will in general vary with κa[28,31,32]. One option to address this is to use a surface-charge based potential parameter forscaling instead of zeta potential [16]. In this work and other studies [6-14], the zeta potentialis used directly, because it may be readily determined through experiment and provides adirect measure of the electroosmotic mobility.

The area-averaged fluid velocity u may be written in terms of the Poiseuille and

electroosmotic components

ep uuu += (3-9)

Pau p μ3

2

= (3-10)

Xiangchun Xuan6

( )tanh1e

au E

aκεζ

μ κ⎡ ⎤

= − −⎢ ⎥⎣ ⎦

(3-11)

where ( ) ( )0

ad y a= ∫ signifies an area-averaged quantity.

3.1. Electroosmotic Flow

For electroosmotic flow, no pressure gradient is present, and so the fluid motion in Eq.(3-3) is described by

0pu = and ( )( )

cosh1

coshe

yu E

aκεζ

μ κ⎡ ⎤

= − −⎢ ⎥⎣ ⎦

(3-12)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Elec

troo

smot

ic v

eloc

ity p

rofil

e

y/a

κa = 1

2

510

20

Figure 2. Radial profile of the normalized electroosmotic fluid velocity, ue/UHS = 1 − cosh(κy)/cosh(κa),at different κa values. All symbols are referred to the nomenclature.

Figure 2 shows the profile of electroosmotic velocity normalized by the so-called Helmholtz-Smoluchowski velocity UHS = −εζE/μ [25], i.e., ue/UHS = 1 − cosh(κy)/cosh(κa), in a slitchannel with different κa values. When κa > 10, the curves are almost plug-like except nearthe channel wall and the bulk velocity is equal to UHS. These are the typical features ofelectroosmotic flow when there is little or zero EDL overlapping. The profiles at κa < 5become essentially parabolic resembling the traditional pressure-driven flow. Moreover, themaximum velocity along the channel centerline is significantly lower than UHS and decreaseswith κa, indicating a vanishingly small electroosmotic mobility when κa approaches 0.


3.2. Pressure-Driven Flow

For pressure-driven flow, the downstream accumulation of counter-ions results in thedevelopment of a streaming potential field [21-24]. This induced electric field, Est, can bedetermined from the condition of zero electric current though the channel. If an equalmobility for the positive and negative ions of the electrolyte is assumed, the electric currentdensity, j, in pressure-driven flow is given as [21-24]

( )coshe b stj u Eρ σ= + Ψ (3-13)

where σb = cbλb is the bulk conductivity of the electrolyte with λb being the molarconductivity. Referring to Eqs. (3-2) and (3-3), one may rewrite the last equation as

( ) ( )2

2 coshp e b b std RTj u u c Edy F

ε λΨ= − + + Ψ (3-14)

Note that the surface conductance of the outer diffusion layer in the EDL has been consideredthrough the cosine hyperbolic function in Eq. (3-14) (which reduces to 1 at Ψ = 0). Thecontribution of the inner Stern layer conductance [33] to the electric current is, however,ignored. Readers may be referred to Davidson and Xuan [34] for a discussion of this issue inelectrokinetic streaming effects.

Integrating j in Eq. (3-14) over the channel cross-section and using the zero electriccurrent condition in a steady-state pressure-driven flow yield

( )*

1*2

2 3st

b

gE Pc F g g

ζβ ζ

=+

(3-15)

( )a

agκκtanh11 −= ,

( )( )aa

agκκ

κ22 cosh1tanh

−= and ( )∫ ⎟⎠⎞

⎜⎝⎛Ψ=

a

aydg

03 cosh (3-16)

Therefore, the fluid motion in pressure-driven flow is characterized as

2 2

212pa yu P

aμ⎛ ⎞

= −⎜ ⎟⎝ ⎠

and ( )( )

cosh1

coshe st

yu E

aκεζ

μ κ⎡ ⎤

= − −⎢ ⎥⎣ ⎦

(3-17)

Area-averaging the two velocity components in the last equation and combining themwith Eq. (3-15) lead to

Zuu

p

e −= (3-18)

Xiangchun Xuan8

where Z is previously termed electrokinetic “figure of merit” as it gauges the efficiency ofelectrokinetic energy conversion [35,36], and defined as

( ) ( )21

2 22 3

3gZa g gκ β ζ ∗

=+

(3-19)

RTb

εμλβ = (3-20)

where κ2 = 2F2cb/εRT has been invoked during the derivation, and β is a non-dimensionalproduct of fluid properties whose reciprocal was termed Levine number by Griffiths andNilson [37]. Apparently, Z depends on three non-dimensional parameters, β, κa and ζ*,among which β spans in the range of 2 ≤ β ≤ 10 and ζ* spans in the rage of −8 ≤ ζ* ≤ 0 [33]for typical aqueous solutions. Moreover, Z is unconditionally positive and less than unity dueto the entropy generation in non-equilibrium electrokinetic flow [38]. The curves of Z at ζ* =−1 (or ζ ≈ −25 mV) and β = 2 and 10, respectively, are displayed in Figure 3 as a function ofκa. One can see that Z achieves the maximum at around κa = 2, indicating that nanochannelswith a strong EDL overlapping are the necessary conditions for efficient electrokinetic energyconversion. For more information about Z and its function in electrokinetic energyconversion, the reader is referred to Xuan and Li [36].

0.00

0.03

0.06

0.09

0.12

0.15

0.1 1 10 100

Z

κa

β = 2

β = 10

Figure 3. The electrokinetic “figure of merit” as a function of κa at ζ* = −1 and β = 2 and 10,respectively.


Based on Eq. (3-18), the effects of streaming potential on the average fluid speed, i.e.,u in Eq. (3-9), in an otherwise pure pressure-driven flow, are characterized by

1p

uZ

u= − (3-21)

This equation also provides a measure of the so-called electro-viscous effects inmicro/nanochannels [39]. If the concept of apparent viscosity is employed to characterizesuch retardation effects, the apparent viscosity ratio γ is readily derived as

( )1 1 Zγ = − (3-22)

For more information on this topic, the reader is referred to Li [39] and Xuan [40].

4. Solute Transport in Nanochannels

Solute transport in nanochannels is governed by the Nernst-Planck equation in theabsence of chemical reactions, which under the assumption of fully-developed fluid flow iswritten as

2 2

2 2i i i i

i i i i i ic c c cu D D v z F ct x x y y y

ψ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂+ = + + ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

(4-1)

i e p i iu u u v z FE= + + (4-2)

where ci is the concentration of solute species i, t the time coordinate, ui the local solute speed(a combination of electroosmosis, pressure-driven motion, and electrophoresis), x the axialcoordinate originating from the channel inlet (see Figure 1), Di the molecular diffusioncoefficient, vi the solute mobility, and zi the solute charge number. Note that the product viziFrepresents the solute electrophoretic mobility. Integrating Eq. (4.1) over the channel cross-section eliminates the last two terms on the right hand side due to the impermeable wallconditions

2

2i i i i

i

c u c cD

t x x∂ ∂ ∂

+ =∂ ∂ ∂

(4-3)

where again ... indicates the area-average over the channel cross-section as defined earlier.

Since the time scale for transverse solute diffusion in nanochannels (characterized bya2/Di, which is about 100 μs when a = 100 nm and Di = 1×10−10 m2/s) is much shorter than

Xiangchun Xuan10

that for longitudinal solute transport (typically of tens of seconds), it is reasonable to assumethat solute species are at a quasi-steady equilibrium in the y direction, i.e.,

2

20 ii i i i

cD v z F cy y y

ψ⎛ ⎞∂ ∂ ∂= + ⎜ ⎟∂ ∂ ∂⎝ ⎠

(4-4)

Integrating Eq. (4.4) twice and using the Nernst-Einstein relation [33], vi = Di/RT, one obtains

( ) ( ) ( ),0 0, , , expi i ic x y t c x t z= − Ψ −Ψ⎡ ⎤⎣ ⎦ (4-5)

where ci,0 is the solute concentration at the channel centerline where the local EDL potentialis defined as Ψ0 (non-zero in the presence of EDL overlapping). Substituting Eq. (4.5) intoEq. (4.3) and considering the hydrodynamic dispersion due to the velocity non-uniformityover the channel cross-section [41,42], one may obtain

2,0 ,0 ,0

2i i i

i i

c c cu D

t x x∂ ∂ ∂

′+ =∂ ∂ ∂

(4-6)

i ip ie i iu u u v z FE= + + (4-7)

p iip

i

u Bu

B= and e i

iei

u Bu

B= (4-8)

where iu is the mean solute speed (i.e., zone velocity) with ipu and ieu being its components

due to pressure-driven and electroosmotic flows, iD′ the effective diffusion coefficient whichis a combination of molecule diffusion and hydrodynamic dispersion and will be addressed inthe next section, and Bi = exp(−ziΨ) the like-Boltzmann distribution of solutes in the cross-stream direction. It is the dependence of iu on the charge number zi that enables the charged-based solute separation in nanochannels.

For an initially uniform concentration Ci,0 of solute species i along the channel axis, aclosed-form solution to Eq. (4-6) is given by [5]

,0,0 erf erf

2 2 2i i i i i i

ii i

C W x X u t x X u tcD t D t

⎡ ⎤⎛ ⎞ ⎛ ⎞− + + − −= +⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟′ ′⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

(4-9)

where erf denotes the error function, and Wi is the half width of the initial solute zone with itscenter being located at Xi. As such, the electrokinetic transport of solute species innanochannels is described by


( ) ( ), , erf erf exp2 2 2

i i i i i ii i

i i

C W x X u t x X u tc x y t zD t D t

⎡ ⎤⎛ ⎞ ⎛ ⎞− + + − −= + − Ψ⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟′ ′⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

(4-10)

where ( ),0 0expi i iC C z= Ψ is the bulk solute concentration at zero potential outside the slit

nanochannel, refer to Eq. (4-5).

zv = +2

zv = +1

zv = 0

zv = −1

zv = −2

Cmax 0.8 Cmax 0.6 Cmax 0.4 Cmax 0.2 Cmax

Figure 4. Transport of solutes with zi = [+2, −2] through a 100 nm deep channel in nanochannelchromatography. Other parameters are referred to the text. Reprinted with permission from [13].

Figure 4 illustrates the transport of an initially Wi = 1 μm wide plug of solutes with zi =[+2, −2] (from top to bottom) through a 100 nm deep (i.e., a = 50 nm) channel 5 s after apressure gradient P = 1×108 Pam-1 was imposed. Note that only the top half of the channel isshown due to symmetry. The ionic concentration of the background electrolyte is cb = 1 mM,corresponding to κa ≈ 5. The other two non-dimensional parameters are assumed to be β = 4and ζ* = −2 (or ζ = −50 mV), both of which are typical to aqueous solutions as indicatedabove. As to the validity of the Debye-Huckel approximation at ζ* = −2, we have recentlydemonstrated using numerical simulation the fairly good accuracy of Eq. (3-7) in predictingthe solute migration velocity [5]. As shown, positive solutes are concentrated to near thenegatively charged wall due to the solute-wall electrostatic interactions [9,11], or in essencethe transverse electromigration in response to the induced EDL field [5,10,12]. Moreover, thehigher the charge number is, the closer the solutes are to the walls. As the fluid velocity nearno-slip walls is slower than its average, positive solutes move slower than neutral solutes thatare still uniformly distributed over the channel cross-section. Conversely, negative solutes arerepelled by the negatively changed walls and concentrated to the region close to the channelcenter. Hence, they move faster than neutral solutes as seen in Figure 4.

As the fluid velocity profile is available in Eq. (3-12) for electroosmotic flow and in Eq.(3-17) for pressure-driven flow, the mean speed of solutes, iu , is readily obtained from Eq.

(4-7). Figure 5 compares the mean speed of solutes with zi = [−2, +2] in electroosmotic flowwith an electric field of 4 kV/m. One can see that iu of all five solutes decreases when κagets smaller. This reduction may be explained by the overall lower electroosmotic velocity ata smaller κa, as demonstrated in Figure 2. When κa > 100, negatively charged solutes moveslower than positive ones due to their opposite electrophoresis to fluid electroosmosis

Xiangchun Xuan12

(identical to the curve with zi = 0). When κa gets smaller than 100, however, negativelysolutes start moving faster than positively solutes as the latter ones are concentrated to theEDL region within which the fluid has a slower speed than the bulk as explained above. Therelative magnitude of iu between the solutes of like charges is, however, a complex functionof both the charge number zi, which determines the velocity component due to fluid flow, i.e.,

ip ieu u+ in Eq. (4-7), and the solute mobility vi, which determines the velocity component

due to solute electrophoresis, i.e., the most right term in Eq. (4-7). When κa further decreasesto less than 1, the EDL potential becomes nearly flat due to the strong EDL overlapping (seeFigure 2), and so the order of iu for the three solutes at large κa (i.e., microchannelelectrophoresis) is recovered.

Figure 5. Comparison of the mean speeds of solutes with zi = [−2, +2] as a function of κa innanochannel electrophoresis. The solute diffusivity is assumed to be constant, Di = 5×10−11 m2/s. Otherparameters are referred to the text.

Combining Eqs. (4-7) and (4-8) provides a measure of the streaming potential effects onthe solute mean speed in pressure-driven flow,

1 e i i i i sti

ip p i

u B v z F B Euu u B

+= + (4-11)

It is obvious that the last equation is reduced to Eq. (3-21) for neutral solutes, i.e., zi = 0 andthus Bi = 1. Figure 6 shows the ratio i ipu u as a function of κa. As expected, streaming


potential effects reduce the solute speed due to the induced electroosmotic backflow. Thisreduction varies with the solute charge zi and attains the extreme at about κa = 3, where thefigure of merit Z (refer to Figure 3) approaches its maximum indicating the largest streamingpotential effects. In both the high and low limits of κa, streaming potential effects becomenegligible, i.e., Z → 0, see Figure 3. Accordingly, i ipu u reduces to 1 for all solutes at large

κa while varying with zi at small κa because of the finite solute mobility [15].

0.7

0.8

0.9

1

1.1

0.1 1 10 100

u i/u

ip

κa

zi = +2

+1

0

−2

−1

Figure 6. Effects of streaming potential on the solute mean speed in nanochannel chromatography.

5. Solute Dispersion in Nanochannels

As up and ue vary over the channel cross-section (refer to Eqs. (3-4) and (3-5), and Figure2), they both contribute to the spreading of solutes along the flow direction, which is termedhydrodynamic dispersion or Taylor dispersion [41,42]. The general formula for calculatingthis dispersion is given by [43,44]

( )2

12 0

y

i i i i

ii i

B B u u dyaKD B

− ⎡ ⎤′−⎢ ⎥⎣ ⎦=

∫(5-1)

Referring to Eqs. (4-2) and (4-7), one may then rewrite the last equation as

( )2 2 2

i ip p ipe p e ie ei

aK F u F u u F uD

= + + (5-2)

Xiangchun Xuan14

( )2

11

0

y

ip i i p ip iF B B u u dy B −− ∗ ∗⎡ ⎤′= −⎢ ⎥⎣ ⎦∫ (5-3a)

( )2

11

0

y

ie i i e ie iF B B u u dy B −− ∗ ∗⎡ ⎤′= −⎢ ⎥⎣ ⎦∫ (5-3b)

( ) ( ) 11

0 02

y y

ipe i p ip i e ie i iF B u u B dy u u B dy B −− ∗ ∗ ∗ ∗⎡ ⎤ ⎡ ⎤′ ′= − −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦∫ ∫ (5-3c)

where m m mu u u∗ = and im im mu u u∗ = (m = p and e). Note that the three terms, Fip, Fipe

and Fie in Eq. (5-2) represent the contributions to dispersion due to the pressure-driven flow,the coupling between pressure-driven and electroosmotic flows, and the electroosmotic flow,respectively.

Hydrodynamic dispersion is often expressed in terms of a non-dimensional dispersioncoefficient χi [45],

iiii DPeK 2χ= (5-4)

where the Peclet number Pei in this case may be defined with respect to the mean solutespeed, i.e., iii DauPe = [12,19], or to the area-averaged fluid velocity, i.e.,

( ) iepi DauuPe += [15-18,45]. Using the solute speed-based Peclet number, χi

becomes dependent on the solute diffusivity Di which complicates the analysis. This isbecause the solute mobility vi in iu [see Eq. (4-7)] is coupled to Di via the Nernst-Einstein

relation, vi = Di/RT. Such dependence doesn’t occur if χi is defined using the fluid velocity-based Peclet number. Here, we employ the latter definition in keeping with the dispersionstudies of neutral solutes in the literature [45]. As such, the dispersion coefficient χi may beeasily obtained from Eq. (5-2) as

( )

2 2

2ip p ipe p e ie e

i

p e

F u F u u F u

u uχ

+ +=

+(5-5)

It is important to note that χi is independent of the solute speed or the driving force of theflow while Ki (in the unit of Di) not. Instead, χi is primarily determined by the flow type(pressure- or electric field-driven), channel structure (including shape and size) and solutecharge number.


5.1. Electroosmotic Flow

In electroosmotic flow, the hydrodynamic dispersion in Eq. (5-2) is reduced to

22e

i iei

a uK F

D= (5-6)

Accordingly, the dispersion coefficient in Eq. (5.5) is simplified as

i ieFχ = (5-7)

0.0001

0.001

0.01

0.1

0.1 1 10 100

χ ifo

r nan

ocha

nnel

elec

trop

hore

sis

κa

zi = 0

+1

+2

−1−2

Figure 7. Illustration of dispersion coefficient χi of solutes with zi = [−2, +2] in nanochannelelectrophoresis as a function of κa.

Figure 7 shows χi of solutes with zi = [−2, +2] in nanochannel electrophoresis as afunction of κa. We see that in the entire range of κa, χi of positive solutes is larger than thatof neutral ones while χi of negative solutes is smaller than the latter. This is because positivesolutes are concentrated to near the channel walls where the velocity gradients are large whilenegative ones are concentrated to the channel centerline where the velocity gradients aresmall (refer to Figure 4). Moreover, the higher the charge number zi, the larger is χi forpositive solutes and the smaller for negative ones. In the low limit of κa (i.e., the narrowestchannel), the EDL potential is essentially uniform over the channel cross-section (refer toFigure 2), and so is the solute distribution regardless of the charge number. As a consequence,χi of all charged solutes approach that of neutral solutes, i.e., 2/105. Note that this value isequal to the dispersion coefficient of neutral solutes in a pure pressure-driven flow indicatingthe resemblance between pressure-driven and electroosmotic flows in very small

Xiangchun Xuan16

nanochannels. This aspect will be revisited shortly. In the high limit of κa (i.e., the widestchannel), the EDL thickness is so thin compared to the channel height that the solutedistribution becomes once again uniform across the channel (the EDL potential is, here,uniformly zero while equal to the wall zeta potential in the low limit of κa). Therefore, thehydrodynamic dispersion in electroosmotic flow, or the so-called electrokinetic dispersion[46], decreases with the square of κa and ultimately converges to zero [47,48].

5.2. Pressure-Driven Flow

In pressure-driven flow with consideration of streaming potential, the hydrodynamicdispersion is obtained from Eq. (5-2) as

( )22

22 31p

i ip i ii

a uK F Z Z

Dδ δ= − + (5-8)

2i ipe ipF Fδ = and 3i ie ipF Fδ = (5-9)

during which Eqs. (3-17) and (3-18) have been invoked and Z is the electrokinetic “figure ofmerit” as defined in Eq. (3-19). It is apparent that the streaming potential inducedelectroosmotic backflow produces two additional dispersions in pressure-driven flow: one isthe electrokinetic dispersion due to electroosmotic flow itself, the term with δi3 in Eq. (5-8),which tends to increase the total dispersion, and the other is due to the coupling of pressure-driven and electroosmotic flows, the term with δi2 in Eq. (5-8), which tends to decrease thetotal dispersion. The latter phenomenon has been employed previously to reduce thehydrodynamic dispersion in capillary electrophoresis where a pressure-driven backflow isintentionally introduced to partially compensate the non-uniformity in electroosmotic velocityprofile [49,50]. If streaming potential effects are ignored, i.e., for a pure pressure-driven flowwith Z = 0, Eq. (5-8) reduces to

22p

ip ipi

a uK F

D= (5-10)

such that2

2 31ii i

ip

K Z ZK

δ δ= − + (5-11)

Similarly, the dispersion coefficient in Eq. (5-5) for real pressure-driven flow is obtainedas

( )

22 3

2

11i i

i ipZ ZF

Zδ δχ − +

=−

(5-12)


where Fip = χip is the dispersion coefficient for a pure pressure-driven flow by analogy to Eq.(5-7) in a pure electroosmotic flow. We thus have

( )

222 3

2

11

i i i i

ip ip

Z Z KKZ

χ δ δ γχ

− += =

−(5-13)

Therefore, χi/χip differs from Ki/Kip by only the square of the apparent viscosity ratio γ, seethe definition in Eq. (3-22). As γ is independent of the solute charge number zi, it is expectedthat the variation of χi/χip with respect to zi will be identical to that of Ki/Kip.

0.001

0.01

0.1

1

0.1 1 10 100

χ i fo

r nan

ocha

nnel

chro

mat

ogra

phy

κa

zi = 0

+1

+2

−1

−2

Figure 8. Dispersion coefficient χi of solutes with zi = [−2, +2] in nanochannel chromatography as afunction of κa.

Figure 8 shows χi of solutes with zi = [−2, +2] as a function of κa in nanochannelchromatography. Due to the same reason as stated above for nanochannel electrophoresis, χi

of positive solutes is larger than that of neutral ones while χi of negative solutes is thesmallest. In both the high and low limits of κa, the flow-induced streaming potential isnegligible, see Eq. (3-18) and Figure 3. Hence, the electroosmotic back flow and the inducedsolute electrophoresis vanish, yielding χi = 2/105 regardless of the solute charge [51]. It isimportant to note that χi of neutral solutes in pressure-driven flow is not uniformly 2/105 asaccepted in the literature. Due to the effects of flow-induced streaming potential, χi isincreased by the electroosmotic backflow [i.e., δi3Z2 term in Eq. (5-12)] even though thecoupled dispersion term [i.e., −δi2Z term in Eq. (5-12)] drops for neutral solutes [17].

Xiangchun Xuan18

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

0.1 1 10 100

Ki/K

iporχ i

/χip

κa

zi

zi

Ki/Kip

χi/χip

.

Figure 9. Effects of streaming potential on the ratio of solute dispersion, Ki/Kip, and the ratio ofdispersion coefficient, χi/χip, in nanochannel chromatography as a function of κa. Adapted withpermission from [18].

Figure 9 displays the effects of streaming potential on the ratio of solute dispersion,Ki/Kip, and the ratio of dispersion coefficient, χi/χip, in nanochannel chromatography as afunction of κa. In all cases, Ki/Kip is less than 1 indicating that streaming potential effectsresult in a decrease in hydrodynamic dispersion. This reduction, as a consequence of theinduced electroosmotic backflow, gets larger (i.e., Ki/Kip deviates further away from 1) whenthe solute charge zi increases. The optimum κa at which Ki/Kip achieves its extreme increasesslightly with zi. In contrast to the decrease in solute dispersion, the dispersion coefficient isincreased by the effects of streaming potential, i.e., χi/χip > 1. These dissimilar trends stemfrom the dependence of γ on κa, see Eqs. (3-19), (3-22) and (5-13). As streaming potentialeffects increase (or in other words, the electrokinetic “figure of merit” Z increases), theelectroosmotic backflow increases causing a decrease in Ki/Kip (and Ki/Kip < 1) while anincrease in γ (and γ > 1). The net result is the observed variation of χi/χip with respect to κa.The increase in χi/χip is more sensitive to zi than the decrease in Ki/Kip. However, the trendthat χi/χip varies with respect to zi is consistent with Ki/Kip as pointed out earlier. In addition,χi/χip attains a maximum at a larger value of κa than that at which Ki/Kip is minimized.

5.3. Neutral Solutes

For neutral solutes (zi = 0), closed-form formulae are available for the functions Fm (m =ip, ipe, ie) defined in Eq. (5-3) which in turn determine δi2 and δi3 through Eq. (5-9).Specifically, we find [17]


2 105ipF = (5-14)

( ) ( ) ( )2 42 2 6 1

1 15ipeFa a

ω ωω ωκ κ

⎡ ⎤−⎛ ⎞= − +⎢ ⎥⎜ ⎟− ⎝ ⎠⎢ ⎥⎣ ⎦(5-15)

( ) ( ) ( ) ( ) ( )

2

2 2 2 22 2

1 2 3 131 2 2 coshieF

a a a aωω κ ω κ ω κ κ

⎡ ⎤= + − −⎢ ⎥

− ⎢ ⎥⎣ ⎦(5-16)

( )tanh aaκ

ωκ

= (5-17)

Figure 10 compares the magnitude of δi2 and δi3 [see their definitions in Eq. (5-9)] forneutral solutes as a function of κa. As shown, δi2 is always larger than δi3. In the low limit ofκa, δi2 approaches 2 while δi3 approaches 1, and the square bracketed terms in Eq. (5-2) thusreduces to (<up> + <ue>)2 reflecting the similarity of pressure-driven and electroosmotic flowprofiles in very narrow nanochannels. In the high limit of κa, both δi2 and δi3 approach zerobecause the streaming potential is negligible and the electroosmotic velocity profile becomesessentially plug-like. Note that Eq. (5-14) gives the well-known hydrodynamic dispersioncoefficient of neutral solutes in a pure pressure-driven flow between two parallel plates [51].Moreover, Eq. (5-16) is identical to that derived by Griffiths and Nilson [47,48] which givesthe electrokinetic dispersion coefficient of neutral solutes in a pure electroosmotic flowbetween two parallel plates.

0

0.5

1

1.5

2

0.1 1 10 100 1000

δ

κa

δi2

δi3

Figure 10. Plot of δi2 and δ i3 for neutral solutes as a function of κa. Adapted with permission from [17].

Xiangchun Xuan20

6. Solute Separation in Nanochannels

Solute separation is typically characterized by retention, selectivity, plate height (or platenumber), and resolution [43], of which retention and plate height are related to only one typeof solutes. In contrast, selectivity and resolution are both dependent on two types of solutespecies, and thus provide a direct measure of the separation performance of solutes. As plateheight is involved in the definition of resolution, see Eq. (6-8), it will still be consideredbelow along with the selectivity and resolution for a comprehensive understanding of soluteseparation in nanofluidic channels.

In order to emphasize the advantage of electrophoresis and chromatography innanochannels over those taking place in micro-columns, we focus on the solutes with asimilar electrophoretic mobility, or specifically, viziF = constant. This is equivalent toassuming a constant charge-to-size ratio or a constant product, Dz = Dizi, of solute chargeand diffusivity because solute size is inversely proportional to its diffusivity via the Nernst-Einstein relation [33]. Such solutes are unable to be separated in free solutions throughpressure-driven or electroosmotic microchannel flows. A typical value of the solute charge-diffusivity product, Dz = 1×10−10 m2/s, was selected in the following demonstrations whilethe solute charge number zi may be varied from −4 to +4. The ratio of channel length to half-height was fixed at L/a = 104 for convenience even though we recognize that fixing thechannel length might be a wiser option when the channel height is varied.

6.1. Selectivity

Selectivity, rji, is defined as the ratio of the mean speeds of solutes i and j

jiji uur = (6-1)

and should be larger than 1 as traditionally defined [43]. A larger rji indicates a betterseparation. Figure 11 compares the selectivity, rji, of (a) positive and (b) negative solutes innanochannel chromatography (solid lines) and nanochannel electrophoresis (dashed lines). Itis important to note that the indices of rji, which indicate the charge values of the two solutesto be separated, are switched between positive and negative solutes in order that rji > 1 astraditional defined [43]. Specifically, we use r21, r32, and r43 for positive solutes (or moregenerally, solutes with ziζ* < 0) as those with higher charges are concentrated in a region ofsmaller fluid speed (i.e., closer to the channel wall) and thus move slower. Note that thesolute electrophoretic mobility has been assumed to remain unvaried. In contrast, negativesolutes (or solutes with ziζ* > 0) with higher charges appear predominantly in the region oflarger fluid speed (closer to the channel center) and thus move faster. Therefore, we need touse r12, r23, and r34 for negative solutes. This index switch also applies to the resolution, Rji,which will be illustrated in Figure 12.


1

2

3

4

0.1 1 10 100

Sele

ctiv

ity, r

ji

κa

(a)

r21

r32

r43

r21

r32

r43

1.0

1.1

1.2

0.1 1 10 100

Sele

ctiv

ity, r

ji

κa

(b)

r12

r23

r34

r12

r23

r34

Figure 11: Selectivity, rji, of (a) positive and (b) negative solutes in nanochannel chromatography (solidlines) and nanochannel electrophoresis (dashed lines). Reprinted with permission from [19].

One can see in Figure 11a that the selectivity, rji, of positive solutes in nanochannelchromatography is always greater than that of the same pair of solutes in nanochannelelectrophoresis. This discrepancy gets larger when the solute charge number zi increases.Meanwhile, the optimal κa value at which rji is maximized increases for both chromatographyand electrophoresis though it is always smaller in the former case. The discrepancy between

Xiangchun Xuan22

these two optimal κa also increases with the rise of zi. For negative solutes, Figure 11b showsa significantly lower rji than positive solutes in nanochannel chromatography. Moreover, rji

decreases when the solute charge number increases. The optimal κa at which rji is maximizedis also smaller than that for positive solutes, and decreases (but only slightly) with zi. Allthese results apply equally to rji of negative solutes in nanochannel electrophoresis except ataround κa = 0.6 where rji varies rapidly with κa. Within this region of κa, the electrophoreticvelocity of negative solutes is close to the fluid electroosmotic velocity [more accurately, ieuin Eq. (4-7)] while in the opposite direction. Therefore, the real solute speed is essentially sosmall that even a trivial difference in the solute speed (essentially the difference in ieu asthe solute electrophoretic velocity is constant due to the fixed charge-to-size ratio) could yielda large rji.

It is, however, important to note that the speed of negative solutes could be reversed innanochannel electrophoresis when κa is less than a threshold value (e.g., κa = 0.6 in Figure11b). In other words, solutes migrate to the anode side instead of the cathode side along withthe electrolyte solution. In such circumstances, it is very likely that only one of the two solutespecies migrates toward the detector no matter the detector is placed in the cathode or theanode side of the channel. Another consequence is that the maximum rji in nanochannelelectrophoresis might be achieved with a fairly long analysis time, which makes theseparation practically meaningless. We therefore expect that solutes with a constantelectrophoretic mobility can be better separated in nanochannel chromatography than innanochannel electrophoresis. Moreover, solutes with ziζ* < 0 can be separated more easilythan can those with ziζ* > 0.

6.2. Plate Height

Plate height, Hi, is the spatial variance of the solute peak distribution, 2iσ , divided by the

migration distance, L, within a time period of ti. It is often expressed in the followingdimensionless form of a reduced plate height, hi [42,45]

i

iiiiii ua

DaL

tDaLa

Hh′

=′

===222σ

(6-2)

( )21i i i i i iD D K D Peχ′ = + = + (6-3)

where iD′ is the effective diffusion coefficient due to a combination of hydrodynamicdispersion Ki [see Eq. (5-4)] and molecular diffusion Di. Note that other sources of dispersionsuch as injection and detection (refer to [46,52,53] for detail) have been neglected forsimplicity.

Following Griffiths and Nilson’s analysis [12], we may combine Eqs. (6-2) and (6-3) torewrite the reduced plate height as


( )iiii PePeh χ+= 12 (6-4)

Therefore, hi attains its minimum

iih χ4min, = at ioptiPe χ1, = (6-5)

In other words, there exists an optimal value for the mean solute speed, iiopti aDu χ=, ,

and thus an optimal electric field in nanochannel electrophoresis or an optimal pressuregradient in nanochannel chromatography, at which the separation efficiency is maximized. Ashi is a function of solely the dispersion coefficient χi that has been demonstrated in Figures 7and 8 for nanochannel electrophoresis and chromatography, respectively, its variations withrespect to zi and κa are not repeated here for brevity.

6.3. Resolution

Resolution, Rji, can be defined in two different ways: the one introduced by Giddings[54], i.e., Eq. (6-6), and the one adopted by Huber [55] and Kenndler et al. [56-58], i.e., Eq.(6-7),

( )jtit

ijji

ttR

,,2 σσ +−

= (6-6)

it

ijji

ttR

,σ−

= (6-7)

where t is the migration time as defined in Eq. (6-2) and σt is the standard deviation of solutepeak distribution in the time domain. Consistent with the solute selectivity rji, a larger valueof Rji indicates a better separation. Substituting ii uLt = , jj uLt = and iiit uσσ =, into

the last equation leads to

( ) ( )1 1ji ji jii i

L aLR r rhσ

= − = − (6-8)

Referring back to Eq. (6-5), it is straightforward to obtain

( ),max,min

1ji jii

L aR r

h= − at ioptiPe χ1, = (6-9)

Xiangchun Xuan24

because the selectivity rji is independent of the solute Peclet number. Therefore, when theplate height of one type of solute is minimized, the separation resolution of this solute fromanother type of solute may reach the maximum value.

Figure 12 compares the maximum resolution, Rji,max, of positive and negative solutes (aslabeled) in nanochannel chromatography (solid lines) and nanochannel electrophoresis(dashed lines). The indices of Rji,max are assigned following those of the selectivity, rji, inFigure 11, to ensure Rji,max > 0. One can see that Rji,max of positive solutes in chromatographyis larger than that of negative ones throughout the range of κa. In electrophoresis, the formeralso yield a better resolution if κa > 1. When κa < 1, Rji,max of negative solutes increases andreaches the extremes at κa = 0.6 due to the sudden rise in the selectivity (refer to Figure 11b)as explained above. Within the same range of κa, Rji,max of positive solutes continuesdecreasing when κa decreases and thus becomes smaller than that of negative solutes.Interestingly, chromatography and electrophoresis offer a comparable resolution for bothtypes of solutes in nanoscale channels if κa > 1.

1

10

100

0.1 1 10 100

Max

imum

reso

lutio

n, R

ji,m

ax

κa

R32 R43R21

R21

R32

R43

R34

R12

R23Negative solutes

Positive solutes

Figure 12. Maximum resolution, Rji,max, of positive and negative solutes in nanochannelchromatography (solid lines) and nanochannel electrophoresis (dashed lines). Reprinted withpermission from [19].

It is also noted in Figure 12 that the optimum channel size for both separation methodsappears to be 1 < κa < 10. In other words, the best channel half-height for solute separation innanochannels will be 10 nm < a < 100 nm if 1 mM electrolyte solutions are used. In thiscontext, the optimum Peclet number to achieve the maximum resolution in a channel of κa =5 (or a ≈ 50 nm) will be Pei,opt = O(4) because hi,min = O(1). Although this Peclet number(corresponding to the mean solute speed of the order of 8 mm/s) seems a little too high incurrent nanofluidics, it indicates that large fluid flows are preferred in both nanochannel


chromatography and nanochannel electrophoresis for high throughputs and separationefficiencies.

7. Conclusion

We have developed an analytical model to study the transport, dispersion and separationof solutes (both charged and non-charged) in electroosmotic and pressure-driven flowsthrough nanoscale slit channels. This model explains why solutes can be separated by chargein nanochannels, and provides compact formulas for calculating the migration speed andhydrodynamic dispersion of solutes. It also presents a simple approach to optimizing theseparation performance in nanochannels, which has been applied particularly to solutes with asimilar electrophoretic mobility. In addition, we would like to point out that the model or theapproach developed in this work can be readily extended to one-dimensional round nanotubes[12,15-17,19] and to even two-dimensional rectangular nanochannels [7,59,60].

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[24] Li, D. Colloid. Surf. A 2001, 191, 35-57.[25] Hunter, R. J. Zeta potential in colloid science, principles and applications, Academic

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275-290.[46] Ghosal, S., Annu. Rev. Fluid Mech. 2006, 38, 309-338.[47] Griffiths, S. K., Nilson, R. H. Anal. Chem. 1999, 71, 5522-5529.[48] Griffiths, S. K., and Nilson, R. N., Anal. Chem. 2000, 72, 4767-4777.[49] Datta, R. Biotechnol. Prog. 1990, 6, 485-493.[50] Datta, R.; and Kotamarthi, V. R AICHE J. 1990, 36. 916-926.[51] Wooding, R. A., J. Fluid. Mech. 1960, 7, 501-515.[52] Gas, B.; Stedry, M.; and Kenndler, E. Electrophoresis 1997, 18, 2123-2133.[53] Gas, B.; Kenndler, E. Electrophoresis 2000, 21, 3888-3897.[54] Giddings, J. C. Sep. Sci. 1969, 4, 181-189.[55] Huber, J. F. K. Fresenius' Z. Anal. Chem. 1975, 277, 341-347.[56] Kenndler, E. J Cap. Elec. 1996, 3, 191-198.[57] Schwer, C.; and Kenndler, E. Chromatographia 1992, 33, 331-335.[58] Kenndler, E.; and Fridel, W. J Chromatography 1992, 608, 161-170.[59] Dutta, D. Electrophoresis 2007, 28, 4552-4560.[60] Dutta, D. Anal. Chem. 2008, 80, 4723-4730.


Chapter 2

H2O IN THE MANTLE: FROM FLUIDTO HIGH-PRESSURE HYDROUS SILICATES

N.R. Khisina1,*, R. Wirth2 and S. Matsyuk3

1Institute of Geochemistry and Analytical Chemistry of Russian Academy of Sciences,Kosygin st. 19, 119991 Moscow, Russia

2GeoForschungZentrum Potsdam, Germany3Institute of Geochemistry, Mineralogy and Ore Formation, National Academy of

Sciences of Ukraine, Paladin Ave., 34, 03680 Kiev-142, Ukraine

Abstract

Infrared spectroscopic data show that nominally anhydrous olivine (Mg,Fe)2SiO4 containstraces of H2O, up to several hundred wt. ppm of H2O (Miller et al., 1987; Bell et al., 2004;Koch-Muller et al., 2006; Matsyuk & Langer, 2004) and therefore olivine is suggested to be awater carrier in the mantle (Thompson, 1992). Protonation of olivine during its crystallizationfrom a hydrous melt resulted in the appearance of intrinsic OH-defects (Libowitsky & Beran,1995). Mantle olivine nodules from kimberlites were investigated with FTIR and TEMmethods (Khisina et al., 2001, 2002, 2008). The results are the following: (1) Water content inxenoliths is lower than water content in xenocrysts. From these data we concluded thatkimberlite magma had been saturated by H2O, whereas adjacent mantle rocks had beencrystallized from water-depleted melts. (2) Extrinsic water in olivine is represented by high-pressure phases, 10Å-Phase Mg3Si4O10(OH)2.nH2O and hydrous olivinen(Mg,Fe)2SiO4.(H2MgSiO4), both of which belong to the group of Dense Hydrous MagnesiumSilicates (DHMS), which were synthesized in laboratory high-pressure experiments (Prewitt& Downs, 1999). The DHMS were regarded as possible mineral carriers for H2O in themantle; however, they were not found in natural material until quite recently. Ourobservations demonstrate the first finding of the 10Å-Phase and hydrous olivine as a mantlesubstance. (3) 10Å-Phase, which occurred as either nanoinclusions or narrow veins in olivine,is a ubiquitous nano-mineral of kimberlite and closely related to olivine. (4) There are twodifferent mechanisms of the 10Å-Phase formation: (a) purification of olivine from OH-bearing defects resulting in transformation of olivine to the 10 Å-Phase with the liberation ofwater fluid; and (b) replacement of olivine for the 10Å-Phase due to hydrous metasomatism inthe mantle in the presence of H2O fluid.

* E-mail address: [email protected]

N.R. Khisina, R. Wirth and S. Matsyuk28

Introduction

The presence of water in the mantle, either as H2O molecules or OH- groups, has been thesubject of long-term interest in geochemistry and geophysics because of the dramatic H2Oinfluence on the melting and the physical properties of mantle rocks. Nowdays the concept of“wet” and heterogeneous mantle is universally accepted among petrologists (Kamenetsky etal., 2004; Sobolev & Chaussidon, 1996; Katayama et al., 2005). However, which mineralscould serve as deep water storage in the mantle is yet to be a widely-discussed topic. Theproblem is that the mantle material available for a direct investigation is very limited andrestricted by mantle nodules trapped by kimberlitic or basaltic magma from the depth. Amongthe presumed candidates considered for water storage in the mantle were the so-called DHMSphases (dense hydrous magnesium silicates) synthesized in laboratory experiments at P-Tconditions of the mantle (Prewitt & Downs, 1999); however, DHMS phases have not yet beenfound as macroscopic minerals in mantle material. The main mineral of the mantle is olivine(Mg,Fe)2SiO4, which is stable at mantle pressures up to ~ 15 GPa; with increasing pressurethe olivine ( α-phase) transforms to higher density structure of wadsleite (β-phase). The P-T-conditions of α-β transition in olivine correspond to the depth of ~ 400 km; according togeophysical data, the mantle has a discontinuity at this depth, specified as a boundarybetween the upper mantle and transition zone. Infrared spectroscopic data show that olivinecontains traces of H2O up to several hundreds wt. ppm of H2O (Bell et al., 2003; Koch-Mülleret al., 2006; Kurosawa et al., 1997; Matsyuk & Langer, 2004; Miller et al., 1987); therefore,nominally anhydrous olivine is considered as water storage in the mantle (Thompson, 1992).

The highest water content, such as about 400 wt. ppm of H2O, was registered for olivinesamples from mantle peridotite nodules in kimberlites. Water in olivine occurs as either OH-or H2O. There are two modes of “water” occurrence in olivine: intrinsic and extrinsic. Anintrinsic mode represents the OH- incorporated into the olivine structure and is considered awater-derived defect complex either associated with a metal vacancy vMe, 2OH- or by avacancy at the Si site vSi,4OH- (Beran & Putnis, 1983; Beran & Libovitzky, 2006; Lemaireet al., 2004). Extrinsic water is possessed by inclusions, either solid or fluid, and occurs asOH- or H2O. Recent TEM investigations of olivine nodules from Udachnaya kimberlite(Yakuyia) revealed the nanoinclusions of hydrous magnesium silicates represented by high-pressure phases, 10Å-Phase and hydrous olivine, partially replaced by low-pressureserpentine + talc assemblage (Wirth & Khisina, 1998; Khisina et al., 2001; Khisina & Wirth,2002, 2008a, 2008b). 10Å-Phase and hydrous olivine belong to the group of DHMS phases.This first finding of DHMS phases in mantle material (Wirth & Khisina, 1998; Khisina et al.,2001; Khisina & Wirth, 2002, 2008a, 2008b) provides direct evidence of the DHMSoccurrence in the mantle and specifies them as nanominerals of non-magmatic origin, closelyrelated to olivine.

The amount of water incorporated into the olivine structure depends on the P-Tconditions as well as on the chemical environment and olivine composition (Fe/Mg ratio inolivine), and increases with increasing water activity, oxygen fugacity, pressure andtemperature (Kohlstedt et al., 1996; Bai & Kohlstedt, 1993; Zhao et al., 2004). Diffusion ofhydrogen in olivine is very fast (Kohlstedt & Mackwell, 1998); therefore, due to interactionof olivine and surrounding melt the initial water content in olivine can be changed underchanges of either P, or T, or fO2 or fH2O during a post-crystallization stage. Olivine remained

H2O in the Mantle: From Fluid to High-Pressure Hydrous Silicates 29

in a host melt can loose water due to decompression; olivine trapped by a foreign melt canbecome either deprotonated or secondarily protonated, depending on whether the foreign meltis lower or higher by water concentration (by water activity) in comparison to the host melt,correspondingly. Experiments by Peslier and Luhr (2006) and Mosenfelder et al. (2006a,2006b) demonstrate rapid processes of deprotonation and secondary protonation of olivine.

FTIR and TEM data provide the information about H2O content and a mode of wateroccurrence in olivine. Here we suggest a way to reconstruct the P-history of the kimberliteprocess and elucidate the water behavior at different stages of the kimberlite process. Thecollected data on the H2O occurrence in mantle olivine nodules represented by xenoliths,xenocrysts and phenocrysts from Yakutian kimberlites (Udachnaya, Obnazennaya, Mir,Kievlyanka, Slyudyanka, Vtorogodnitza and Bazovayua pipes) are used here as a guide fortracing H2O behavior from fluid to high-pressure DHMS phases in the mantle. We show herethat nominally anhydrous olivine is a carrier of water in the mantle and can be used asindicator of P-f(H2O) regime in the mantle.

Samples and Collected Data

Sample Description

Typical kimberlites сontain xenocrysts and xenoliths of mantle and crustal originembedded into a fine- to coarse-grained groundmass of crystallized kimberlite melt (SobolevV.S. et al, 1972; Sobolev N.V., 1974; Pokhilenko et al., 1993; Ukhanov et al., 1988; Matsyuket al., 1995).

We collected the mantle olivine samples represented by xenoliths, xenocrysts andphenocrysts from kimberlite pipes of Yakutian kimberlite province (Udachnaya,Obnazennaya, Mir, Slyudyanka, Vtorogodniza and Kievlyanka). Olivine samples areclassified as xenoliths, xenocrysts and phenocrysts on the base of petrographic examination.Phenocrysts are olivine single grains disintegrated from groundmass of kimberlite rock.Xenoliths are fragments of adjacent mantle and crustal rocks trapped by kimberlite magmaand transported from the depths during kimberlite eruption. Xenocrysts are olivine single-grains disintegrated from adjacent rocks, either mantle or crustal, and lifted from the depthsduring kimberlite eruption. Xenoliths are of several cm in size. Xenocrysts are less than 1 cmin size and comparable by size with phenocrysts.

The samples were studied with optical microscopy, FTIR and TEM.

H2O Content in the Olivine Samples

The highest H2O content in the mantle olivine samples from Yakutian kimberlites wasmeasured as 400–420 wt. ppm of H2O (Koch-Muller et al., 2006; Matsyuk & Langer, 2004).H2O contents in xenoliths from Udachnaya and Obnazennaya pipes vary between 14 and 246wt. ppm (Table 1).

Present FTIR study on the H2O content in xenoliths from Udachnaya and Obnazennaya,together with previous Infrared spectroscopic data on xenocrystic and phenocrystic olivinesamples (Matsyuk and Langer, 2004; Koch-Muller et al., 2006) show the wide variation of


the H2O content in the olivine samples, such as between 0–3 and 400 wt. ppm.of H2O. Theresults are shown in histograms (Figure 1).

Summarized FTIR data on the H2O content in the all studied mantle olivine samplesincluding xenoliths, xenocrysts and phenocrysts from Yakutian kimberlites (Matsyuk andLanger, 2004; Koch-Müller et al., 2006; present study) are represented at histogram (Figure1a). The same data is plotted individually for phenocrysts (Figure 1b) and xenoliths togetherwith xenocrysts (Figure 1c). For comparison, the H2O contents in mantle-derived olivinemegacrysts from the Monastery kimberlite, South Africa (Bell et al., 2004) are plotted on thehistogram in Figure 1d. The most frequent values of H2O in olivine samples from Monasterykimberlite are 150–200 wt. ppm of H2O.

Table 1. Water contents in olivine from mantle xenoliths

Xenolith sample Locality Xenolith description H2O content,wt. ppm

Ob-152 Obnazennaya pipe Garnet lerzolite 246

Ob-105 ″ Garnet lerzolite 14–65

Ob-174 ″ Garnet lerzolite 143(15)

Ob-312 ″ Garnet lerzolite 45

U-47/76 Udachnaya pipe 9.5 cm in sizemegacrystalline . diamond-bearingxenolith. Harzburgite-dunite

18

9206 ″ 5.2 cm in size coarse-grainedxenolith. Garnet-free harzburgite

29

0 100 200 300 400H2O, wt. ppm

0

4

8

12

0 100 200 300 400H2O, wt. ppm

0

4

8

12

(A) (B)

Figure 1. Continued on next page.


0 100 200 300 400H2O, wt. ppm

0

4

8

12

0 50 100 150 200 250 300 H2O, wt. ppmÐ2Ùá öåþ ççüÐ2Ùá öåþ ççü

0

2

4

6

8

10

(C) (D)

Figure 1. Histograms of the H2O contents in mantle olivine samples from kimberlites. a – c: The FTIRdata for Yakutiyan olivine samples from Udachnaya, Obnazennaya, Mir, Vtorogodnitza, Kievlyankaand Sluydyanka kimberlites (present data; Matsyuk & Langer, 2004; Koch-Müller et al., 2006);a – xenoliths, xenocrysts and phenocrysts all together; b – phenocrysts; c – xenoliths and xenocrysts;d – Xenoliths from Monastery kimberlite, South Africa (Bell et al., 2003).

Extrinsic H2O in Olivine Samples

TEM examination of the olivine samples (Khisina et al., 2001; Khisina & Wirth, 2002;Khisina et al., 2008) revealed the OH- segregation resulted in nano-heterogeneity of severaltypes: (i) nanoinclusions; (ii) lamellar precipitates; (iii) veins developed along healedmicrocracks. All kinds of heterogeneity are associated with deformation slip bands in thesamples understudy (Khisina et al., 2008).

(i) Nanoinclusions were observed in both xenoliths and xenocrysts. They are several tens ofnanometers in size and have a shape of pseudohexagonal negative crystals. Thenanoinclusions are often arranged in arrays along [100], [011], [101] and [-101]crystallographic directions of the olivine host. The phase constituents ofnanoinclusions were identified from TEM data (Wirth & Khisina, 1998; Khisina etal., 2001; Khisina & Wirth, 2002; Khisina et al., 2008) as 10Å-PhaseMg3Si4O10(OH)2

.nH2O, where n = 0.65, 1.0, and 2.0, and hydrous olivine(MgH2SiO4).n(Mg2SiO4), both of them represent high-pressure DHMS phases (Bauer& Sclar, 1981; Khisina & Wirth, 2002; Churakov et al., 2003). 10Å-Phase is atypical constituent of nanoinclusions in xenocrysts, while hydrous olivine is morecommon for nanoinclusions in xenoliths. Both 10Å-Phase and hydrous olivine arereplaced often by a low-pressure assemblage of serpentine Mg3Si2O5(OH)4 + talcMg3Si4O10(OH)2. High-pressure phases in nanoinclusions as well as theirreplacement products are strictly aligned relative to the crystallographic directions ofthe olivine matrics, with aol ahy c10Å ctc cserp (hy is abbreviation of hydrous


Figure 2. Array of nanoinclusions in olivine xenocryst from Udachnaya kimberlite. Nanoinclusions arecomposed of the 10Å-Phase in the middle part of the inclusions, and filled by H2O fluid in the regionsbordered the adjacent olivine (white areas of nanoinclusions).

Figure 3. Healed microcrack filled by 10Å-Phase in olivine xenolite sample from Udachnayakimberlite.

olivine), which is indicative of the topotaxic character of these intergrowths. Acharacteristic feature of nanoinclusions in xenocrystic olivine samples is the presenceof voids unfilled with solid material (Figure 2). Solid phase fills the equatorial areaof the inclusions parallel to the (100) plane of the olivine host, and voids innanoinclusions are observed at the polar areas bordering the olivine matrix (Figure


2). The observations led to the conclusion that these voids are not artifacts and werenot produced during sample preparation. According to TEM observations, thenanoinclusions are not connected with the surface of the crystals through channels,which could have served as pathways for the transport of H2O from the externalmedium during the formation of inclusions.

(ii) (100) Lamellar precipitates of hydrous olivine, several nanometers in thickness wereobserved in xenolithic olivine (Khisina et al., 2001; Khisina & Wirth, 2002;Churakov et al., 2003).

Veins filled by 10Å-Phase + talc, were observed in xenolithic olivine (Figure 3). Theseveins were developed along (100) healed microcracks.

Discussion

Olivine as Water Storage in the Mantle

Histogram of the all H2O content measurements for olivine samples, as summarized forxenoliths, xenocrysts and phenocrysts together (Figure 1a) reveals two maximums, at 0–50wt. ppm of H2O and 200–250 wt. ppm of H2O. The same maximums are pronounced at thehistogram represented the H2O contents in xenoliths and xenocrysts together (Figure 1c),whereas only one maximum at 200–250 wt. ppm of H2O is observed at the histogramrepresented the H2O data for phenocrystic olivine (Figure 1b). Histograms at Figure 1b,cshow that the maximum between 0–50 wt. ppm of H2O corresponds to the most frequent H2Ocontents in xenoliths and xenocrysts, while the maximum between 200–250 wt. ppm of H2Ocorresponds to the most frequent H2O values in phenocrysts. We suggest that the mostfrequent H2O contents correspond to the initial H2O contents incorporated by the olivinesamples during crystallization. Hence, the initial H2O content in olivine derived from adjacentand kimberlite rocks has been different. We conclude that compared to xenolithic andxenocrystic samples, the olivine phenocrysts have incorporated more amount of water duringcrystallization. According to the experimental data on the P-f(H2O)–dependence of the OH-solubility in olivine (Kohlstedt et al, 1996), this could mean that the trapped fragments ofadjacent and kimberlite rocks have been crystallized either at different depths or fromdifferent melts. On the basis of the data by Kohlstedt (1996) the pressure conditions of olivinecrystallization from water-saturated melt can be estimated as < 1 GPa for xenocrysts andxenoliths and as 4–4.5 GPa for phenocrysts. The pressure of 4–4.5 GPa estimated as thepressure of crystallization of phenocrysts corresponds to the depth of 120–135 km that isconsistent with the upper mantle depths and is in good agreement with estimations of thedepths of kimberlite formation as between 120 and 230 km (Pochilenko et al., 1993). Thepressure < 1 GPa estimated as the pressure of crystallization of mantle olivine samplesrepresented by xenoliths and xenocrysts, is less than the lowest pressures in the mantle andcorresponds to the depth of <30 km that is consistent with crustal depths; this is notcompatible with petrographic identification of these samples as derived from mantle rocks.Therefore the difference between mantle xenoliths (and xenocrysts) on the one hand, andphenocrysts on the other hand, in respect of initial water contents can not be explained bydifferent depths of their crystallization. Consequently, this led us to the conclusion that the


mantle adjacent rocks and the kimberlites have been crystallized from different melts, water-saturated and water depleted, respectively.

Post-Crystallization H2O Behavior in Olivine

Post-crystallization H2O behavior in the olivine samples is pronounced mostly inxenoliths and xenocrysts and manifests itself in the following processes.

(i) Secondary protonation of olivine samples derived from adjacent mantle rocks. Secondaryprotonation of the olivine derived from adjacent mantle rocks could take place at thecontact of adjacent rocks and the parent kimberlite magma chamber. Interaction ofadjacent rocks with water-saturated parent kimberlite melt resulted in H2Oenrichment up to 150–200 wt. ppm in olivine (Figure 1).

(ii) Internal re-distribution of H2O within the olivine host resulted in nucleation ofnanoinclusions consisted of high-pressure DHMS phases represented by10Å-Phaseand hydrous olivine, with the H2O fluid separation from the solid inside of thenanoinclusions.

(iii) Subsequent replacement of DHMS by low-pressure assemblage of serpentine + talc innanoinclusions.

(iv) Olivine hydration due to interaction of adjacent rocks with the H2O fluid collected inkimberlite magma chamber; this resulted in the olivine replacement by 10Å-Phasealong microcracks;

(v) Subsequent dehydration of the 10Å-Phase developed along micrcracks in olivine.

H2O Fluid in Kimberlite Melt

During a crystallization of early generation of olivine from the parent water-saturatedkimberlite melt in the magma chamber the separation of H2O fluid from the melt occurredthat led to the internal pressure increase in the kimberlite magma chamber. The bulk H2Ocontents in the olivine phenocrystic samples vary between150 and 300 wt. ppm about themost frequent content as 200–250 wt. ppm of H2O. This variations may indicate that thepressure in kimberlite magma chamber has been oscillated with time by magnitude betweenat least of 3 GPa and 5.2 GPa because of the alternating processes of pressure increasing dueto the H2O fluid separation and accumulation in the chamber followed then by decompressionduring chamber expansion and H2O fluid liberation. The pressure oscillation with time tookplace in kimberlite chamber prior to explosion, and resulted in the deformation of olivine andsubsequent appearance of DHMS nanoinclusions associated with deformation slip bands inolivine. The alteration of adjacent rocks took place at the contact with the kimberlite magmachamber before the explosion and eruption of kimberlite, and resulted in (i) secondaryprotonation of olivine due to interaction with water-saturated kimberlite melt and (ii)metasomatic hydration of olivine resulted in the olivine replacement by the 10Å-Phase due tointeraction of the olivine with H2O fluid. The reaction of hydration of olivine may be writtenas

4(Mg,Fe)2SiO4 + 3H2O(fluid) = Mg3Si4O14H6 + 5(Fe,Mg)O (1)


The kimberlite explosion happened because the internal pressure increased as the H2Ofluid separated from the melt inside of the kimberlite magma chamber. This event resultedwhen the pressure corresponded to the highest H2O content in olivine phenocrysts measuredas 400–420 wt. ppm (Figure 1) attained at least 6.5 GPa, as it may be estimated from thepressure dependence on the H2O content in olivine (Kohlstedt et al., 1996). The subsequenteruption and rapid lifting of the mantle material from the depth of 120-135 km wasaccompanied by destruction of adjacent rocks through the flow path. The pressure increasinginside the chamber prior to explosion followed then by rapid decompression during eruption,when the 10Å-Phase substituted for talc Mg3Si4O10(OH)2 (low-pressure phase) with H2O fluidliberation:

Mg3Si4O14H6 = Mg3Si4O10(OH)2 + H2O (fluid)↑ (2)

According to Chinnery et al. (1999), dehydration of 10Å-Phase (reaction 2) proceeds atpressures lower than 3.5–5.2 GPa at T <700°C.

OH-Bearing Nanoinclusions and Intracrystalline H2O Fluid

The nucleation of OH-bearing nanoinclusions could be initiated by deformation ofolivine that is evident from the similar arrangement of arrays of nanoinclusions and theoptically visible slip bands (Khisina et al., 2008), both along common crystallographicdirections in the olivine host. The slip bands in olivine could be produced by the internalpressure progressively increasing in the kimberlite magma chamber due to separation of theH2O fluid from the magma melt. Slip bands are zones of high density of dislocations in acrystal and, consequently, contain relatively high concentration of OH-bearing point defects.Nanoinclusions composed of DHMS phases could be nucleated within the former slip bandsunder subsequent decompression and cooling. Further, the DHMS phases were substituted bya low-pressure serpentine + talc assemblage.

The question is what was a driving force for the exsolution of high-pressure hydroussilicate phases from the host olivine that occurs within slip bands. If it is assumed that theprocess of self-purification of olivine from water took a place under decompression, then howto explain that this process resulted in creation of high-pressure phases? Are they inequilibrium with the olivine host?

In order to explain the mechanism and reactions of formation and further transformationof the primary high-pressure phases, 10Å-Phase and hydrous olivine, to the low-pressureserpentine + talc assemblage, we need to understand the nature of voids in the nanoinclusions(Figure 2), the source of H2O, and the mechanism of formation of nanoinclusions themselves.

TEM observations show the topotaxic relationships of the olivine/10Å-Phase intergrowth,the exact Mg/Si stoichiometric ratio and the lack of channels connecting the inclusions withthe grain surface. Therefore we propose that no fluid infiltration occurred, and thenanoinclusions observed in olivine samples might have formed due to internal isochemicalprocess within a crystal. Consequently, a crystal should have been water saturated prior to thenucleation of inclusions. We suggest that nanoinclusions in olivine have been formed at thepost-crystallization stage under the pressure oscillation in the kimberlite chamber and theformation of the nanoinclusions proceed through the diffusion and segregation of intrinsicOH-bearing point defects, i.e., without any infiltration from outside. Diffusion experiments


have shown that hydrogen can rapidly diffuse through olivine at temperatures >800°C(Kohlstedt & Mackwell, 1998). The diffusion process could result in an aggregation of theOH- and associated metal vacancies within the OH-saturated slip bands of the olivine host.We consider the formation of OH-bearing nanoinclusions in olivine as a process of self-purification of olivine from the OH-bearing defects by the coupled diffusion mechanism Mg2+

vMg,2H+.With the assumption that the cation composition of the inclusion corresponds tothe stoichiometry of the 10Å Phase (Mg/Si = 3:4), the reaction of the nanoinclusion formationby unmixing of OH-bearing olivine solid solution is the following one:

OH-olivine (s.s.) = [stoichiometric olivine](matrix) + [10Å-Phase + H2O(fluid)](inclusion) (3a)

or5y[Mg2-xvxSiO4H2x] = (5y-4yx)[Mg2SiO4](matrix) + yx[Mg3Si4O14H6 + 2H2O](inclusion) (3b)

Taking into account the variations of the 10Å-Phase chemical composition with respectto H2O content (Bauer & Sclar, 1981; Chinnery et al., 1999), the bracketed part of the right-side of the reaction (3b) should be more correctly presented as Mg3Si4O10(OH)2

.nH2O + (4-n)H2O, where n is 0.65; 1.0; 2.0. Reaction (3) occurs in a closed volume limited by the size ofthe inclusion. The plausibility of the proposed model for the nanoinclusion formation can bechecked by the criterion of volume conservation, i.e., in reaction (3) the volume of reactantsmust be equal to the volume of products. The volume of the inclusion can be expressed as afunction of the molar volume of olivine as 4yx[Vu.c.(Ol)/Z].N, and the volume occupied by the10Å-Phase formed in reaction (3) is yx[Vu.c.(10Å-Phase)/Z].N, where N is Avogadro’s number,Vu.c. is a unit cell volume and Z is a number of chemical formula units per unit cell. Thedifference of these volumes is ΔV = 52yxN under normal P-T-conditions. With theassumption that the volume ΔV is filled with the H2O fluid released owing to reaction (3), wecalculated the density of the fluid by the equation ρ = M/Vmol, where Vmol is the molarvolume per one molecule of H2O in the inclusion, which equals ΔV/2yx in accordance with

Figure 4. Fluid inclusion in olivine xenolith (Bullfontain kimberlite, South Africa). 10Å-Phase isobserved as a shell of the inclusion at the contact between the inclusion and olivine matrix.


reaction (3), and M is the formula weight of H2O (M=18). For n of 0.65, 1.0 and 2.0 in thechemical formula of the 10Å-Phase, the number of formula units of H2O fluid is 3.35, 3.0,and 2.0, and the calculated fluid density ρ is approximately 1.9, 1.7, and 1.1 g/cm3,respectively. The value ρ=1.1 g/cm3 corresponding to the highest H2O content in the 10Å-Phase (n=2) is most consistent with the density of water under normal P-T-conditions (ρH2O= 1.0 g/cm3), which allows consideration of the proposed model of inclusion formation asrather realistic at n=2.

It should be mentioned here that the 10Å-Phase together with H2O fluid was alsoobserved in the shell-rim of fluid inclusion occurring in the mantle olivine sample fromBullfontain kimberlite (South Africa) (Figure 4). In this case, the 10Å-Phase was most likelycreated due to recrystallization of the trapped fluid inclusion.

Conclusion

The collected data show that xenoliths, xenocrysts and phenocrysts from kimberliterepresent specimens of the so-called “wet” olivine; xenoliths and xenocrysts have a numberof common features, such as frequently occurring deformation slip bands and the OH-bearingnanoinclusions that are filled by high- and low-pressure hydrous magnesium silicates togetherwith the H2O fluid. The observations show that the 10Å-Phase is closely related to olivine inboth kimberlite and adjacent rocks and, therefore, the 10Å-Phase could be consideredubiquitous nanomineral of kimberlites marking at about 4–5 GPa a certain stage of thekimberlite process.

On the other hand, xenoliths and xenocrysts differ from phenocrysts by the water bulkcontent, as well as by the mechanism of the 10Å-Phase formation. On the basis of collectedFTIR data, the xenoliths and xenocrysts on one hand, and phenocrysts on the other hand, areconsidered crystallized from different melts, such as water-depleted magma and water-saturated kimberlite melt, correspondingly.

The pressure regime inside the kimberlite magma chamber was controlled by H2O fluid,with pressure oscillation with time between of at least 3 GPa and 5.2 GPa due to alternatingprocesses of pressure increasing during the H2O fluid separation and accumulation in thechamber followed then by decompression due to chamber expansion and H2O fluid liberation.The released H2O fluid participates in metasomatic hydration of the olivine-bearing adjacentmantle rocks, resulting in the olivine replacement for the 10Å-Phase.

Reviewed by

Prof. O. LukaninInstitute of Geochemictry and Analytical Chemistry,Russian Academy of Sciences, Kosygin st. 19,119991 Moscow Russia


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[8] Kamenetsky M.B., Sobolev A.V., and Kamenetsky V.S. (2004) Kimberlite melts rich inalkali chlorides and carbonates: a potent metasomatic agent in the mantle. Geology, 32,10, 845 – 848.

[9] Katayama I., Karato S-i., and Brandon M. Evidence of high water content in the deepupper mantle inferred from deformation microstructures. (2005): Geology, 33, 613-616.

[10] Khisina, N.R., and Wirth, R. (2002): Hydrous olivine (Mg1-yFe2+y)2-xvxSiO4H2x – a new

DHMS phase of variable composition observed as sized-sized precipitations in mantleolivine: Phys. Chem. Minerals, 29, 98-11.

[11] Khisina, N.R., Wirth, R., Andrut, M., and Ukhanov, A.V. (2001): Extrinsic and intrinsicmode of hydrogen occurrence in natural olivines: FTIR and TEM investigation. Phys.Chem. Minerals, 28, 291-301.

[12] Khisina N.R., and Wirth R. (2008) Nanoinclusions of high-pressure hydrous silicate,Mg3Si4O10(OH)2

.nH2O (10Å-Phase), in mantle olivine: mechanisms of formation andtransformation. Geochemistry International, 46, 4, 319 – 327.

[13] Khisina N.R., Wirth R., Matsyuk S., and Koch-Müller M. (2008) Microstructures andOH-bearing nano-inclusions in “wet” olivine xenocrysts from Udachnaya kimberlite.Eur. J. Mineral. (in press),

[14] Koch-Müller, M., Matsyuk, S.S., Rhede, D., Wirth, R., and Khisina, N. (2006):Hydroxyl in mantle olivine xenocrysts from the Udachnaya kimberlite pipe. Phys.Chem. Minerals, 33, 276-287.

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[16] Kurosawa, M., Yurimoto, H., and Sueno, S. (1997): Patterns in the hydrogen and traceelement compositions of mantle olivines. Phys. Chem. Minerals, 24, 385-395.

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[19] Matsyuk, S.S., and Langer, K. (2004): Hydroxyl in olivines from mantle xenoliths inkimberlites of the Siberian platform. Contrib. Mineral. Petrol., 147, 413-437.

[20] Miller, G.H., Rossman, G.R., and Harlow, G.E. (1987): The natural occurrence ofhydroxide in olivine. Phys. Chem. Minerals, 14, 461-472.

[21] Peslier, A.N., and Luhr, J.F. (2006): Hydrogen loss from olivines in mantle xenolithsfrom Simcoe (USA) and Mexico: mafic alkalic magma ascent rates and water budget ofthe sub-continental lithosphere. Earth and Planetary Science Letters, 242, 302-319.

[22] Pokhilenko, N.P., Sobolev, N.V., Boyd, F.R., Pearson, D.G., and Shimizu, N. (1993):Megacrystalline pyrope peridotites in the lithosphere of the Siberian Platform:mineralogy, geochemical peculiarities and the problem of their origin. Geol. Geophys.,34/1, 50-62. (in Russian).

[23] Prewitt, C.T., and Downs, R.T. (1999): High-pressure crystal chemistry. in J. Hemley(Ed), Rev. Mineral., vol. 37, pp. 301-312.

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[25] Sobolev, N.V. (1974): Deep-seated inclusions in kimberlites and the problem of thecomposition of the upper mantle. Nauka, Novosibirsk. 264 pp. (in Russian).

[26] Sobolev A.V., and Chaussidon M. (1996) H2O concentrations in primary melts fromsupra-subduction zones and mid-ocean ridges: Implications for H2O storage andrecycling in the mantle. Earth and Planetary Sci. Lett., 137, 45 – 55.

[27] Thompson, A.B. (1992): Water in the Earth upper mantle. Nature, 358, 295-302.[28] Ukhanov, A.V., Ryabchikov, I.D., and Kharkiv, A.D. (1988): Lithospheric mantle of the

Yakutiya kimberlite province . Nauka, Moscow, 286 pp.[29] Wirth, R., and Khisina, N.R. (1998): OH-bearing crystalline inclusions in olivine from

kimberlitic peridotite (Udachnaya-East, Yakutiya). Suppl. EOS Trans., 79,45, T32B-17.[30] Zhao, Y.H., Ginsberg, S.B., and Kohlstedt, D.L. (2004): Solubility of hydrogen in

olivine: dependence on temperature and iron content. Contrib. Mineral. Petrol., 147,155-161.


Chapter 3

ON THE NUMERICAL SIMULATION OF TURBULENCEMODULATION IN TWO-PHASE FLOWS

K. Mohanarangam and J.Y. Tu*

School of Aerospace, Mechanical and Manufacturing EngineeringRMIT University, Vic. 3083, Australia

Abstract

With the increase of computational power, computational modelling of two-phase flowproblems using computational fluid dynamics (CFD) techniques is gradually becomingattractive in the engineering field. The major aim of this book chapter is to investigate theTurbulence Modulation (TM) of dilute two phase flows. Various density regimes of the two-phase flows have been investigated in this paper, namely the dilute Gas-Particle (GP) flow,Liquid-Particle (LP) flow and also the Liquid-Air (LA) flows. While the density is quite highfor the dispersed phase flow for the gas-particle flow, the density ratio is almost the same forthe liquid particle flow, while for the liquid-air flow the density is quite high for the carrierphase flow. The study of all these density regimes gives a clear picture of how the carrierphase behaves in the presence of the dispersed phases, which ultimately leads to better designand safety of many two-phase flow equipments and processes. In order to carry out thisapproach, an Eulerian-Eulerian Two-Fluid model, with additional source terms to account forthe presence of the dispersed phase in the turbulence equations has been employed forparticulate flows, whereas Population Balance (PB) have been employed to study the bubblyflows. For the dilute gas-particle flows, particle-turbulence interaction over a backward-facingstep geometry was numerically investigated. Two different particle classes with same Stokesnumber and varied particle Reynolds number are considered in this study. A detailed studyinto the turbulent behaviour of dilute particulate flow under the influence of two carrier phasesnamely gas and liquid was also been carried out behind a sudden expansion geometry. Themajor endeavour of the study is to ascertain the response of the particles within the carrier(gas or liquid) phase. The main aim prompting the current study is the density differencebetween the carrier and the dispersed phase. While the ratio is quite high in terms of thedispersed phase for the gas-particle flows, the ratio is far more less in terms of the liquid-particle flows. Numerical simulations were carried out for both these classes of flows and

* E-mail address: [email protected]. Tel: +61-3-99256191. Fax: +61-3-99256108. Mailing address: Prof

Jiyuan Tu, School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, PO Box 71,Bundoora Vic 3083, AUSTRALIA

K. Mohanarangam and J.Y. Tu42

their results were validated against their respective sets of experimental data. For the Liquid-Air flows the phenomenon of drag reduction by the injection of micro-bubbles into turbulentboundary layer has been investigated using an Eulerian-Eulerian two-fluid model. Twovariants namely the Inhomogeneous and MUSIG (MUltiple-SIze-Group) based on Populationbalance models are investigated. The simulated results were benchmarked against theexperimental findings and also against other numerical studies explaining the various aspectsof drag reduction. For the two Reynolds number cases considered, the buoyancy with the plateon the bottom configuration is investigated, as from the experiments it is seen that buoyancyseem to play a role in the drag reduction. The under predictions of the MUSIG model at lowflow rates was investigated and reported, their predictions seem to fair better with the decreaseof the break-up tendency among the micro-bubbles.

Introduction

The basics of two-phase flows have received interdisciplinary attention from chemical,mechanical and industrial engineers for decades as they play a major role in a variety ofindustrial and design processes. Among these dilute particulate flows are encountered in avariety of industrial (Kolaitis & Founti, 2002) and natural processes irrespective of theircarrier phase being gas or liquid. Particles constantly interact with the gas in industriesthrough sand blasting equipments (Qianpu et al., 2005), pneumatic transport equipments (Gilet al., 2002), and also play a major role in the safe operation of the power plants (Tu, 1999),gas turbine engines (Awatef et al., 2005) and helicopters. In chemical industries they appearas reactants and catalysts, thereby controlling the order and the fate of the chemical reaction.There are even found in nature as dust dispersed in the room and as pollen released from theplants carried away by the wind. Lately there has been continued interest in these classes offlows in Bio-medical applications, to study the dust deposition patterns in realistic humannasal airway (Inthavong et al., 2006) and also to aid better delivery of the medications into thehuman nose. In industrial and engineering applications a better understanding of the physicsof these flows will not only lead to better operating efficiency by improved equipmentdesigns but also increase the longevity accompanied with lower maintenance costs and betteroperational improvements.

Flows with particles amid liquid are an important class of two-phase flows classifiedunder as slurry flows, whose flow systems are representative of many mineral processingoperations and also provide useful operation correlations for such processes. They form animportant class of flows encompassing pneumatic conveying system, turbines andmachineries operating in particulate-laden environments. These flows provide a useful tool inthe simulation of sprays in industrial and natural processes, as they have comparable phase-density ratios. Comparable densities are of particular interest, since all the effects ofinterphase momentum transfer are important (Parthasarathy and Faeth, 1987). They also serveas a good test of methods to predict particle motion in turbulent environments (Parthasarathyand Faeth, 1987) as they exhibit high relative turbulence intensities for particle motion, whichinfluence particle drag properties (Clift et al., 1978).

Over the past few decades there have also been inter disciplinary research to study themechanism of drag reduction with a mission to adopt them as tangible practice for a widerange of applications using micro-bubbles. While surfactant and polymer injections along theturbulent boundary layer, contribute a major portion towards this endeavour. Nonetheless,they are limited in terms of realistic applications due to their inherent ability to cause

On the Numerical Simulation of Turbulence Modulation in Two-Phase Flows 43

environmental pollution and there by causing the destruction of the oceans flora and fauna.Drag reduction by the injection of micro-bubbles along the boundary layer seems to be arealistic approach because fixing the dispersed phase as air for drag reduction devices maysound quite pragmatic; due to its wide spread availability unlike polymers or surfactants.Although there may be some overhead losses in placing them along the boundary layer, theyare still undoubtedly the cheapest source, with no requirement of large storage space unlikethe use of other low density inert (environment friendly) gases to form micro-bubbles.

With the increase in the computational power and efficiency, computational fluiddynamics (CFD) have taken a centre point in offering effective solutions to not only singlephase flows, but also for the simulation of wide range of two-phase flows viz., gas-particle,liquid-particle and liquid-gas flows. They not only offer a cheap solution, but also helpscientists and engineers probe into places prohibitive by experimental methods or hostileenvironments unsuitable for human life forms. Numerical simulation of these two-phaseflows lay broadly into two major categories namely the Eulerian-Eulerain two-fluid modelapproach and the Eulerian-Lagrangian particle tracking approach.

In the Lagrangian particle tracking approach, each and every particle is tracked, therebyproviding a detailed behaviour of their trajectories, velocities, bounce back angles and otherparameters. Although they encompass a great deal of information within the domain, theyprove to be rather cumbersome for multi-dimensional problems for the same reason being notable to track rather large number of particles given the computational power such as to obtaina good statistical information of the dispersed phase. Besides this, there is the problem ofrepresenting turbulent interactions between two phases (two-way coupling), whichnecessitates the need to fully understand the interactions of particles with individual vortices(Fessler and Eaton, 1997).

Eulerian-Eulerian approach constituted by Anderson and Jackson (1967), Ishii (1975),regard the carrier and the dispersed phases as two interacting fluids with momentum andenergy exchange between them. One major advantage of using the Eulerian approach is thatthe well-proven numerical procedures for single-phase flows can be directly extended to thesecondary phase with the effects of turbulent interactions between the two fluids, lately thereare considerations of extending the Eulerian two-fluid model by adopting the large eddysimulation (LES) approach (Pandya and Mashayek, 2002). Shirolkar et al (1996) in theirpaper stated that Eulerian models have problems to account for the particle history effects asthey do not re-trace the motion of individual particles together; they also suffer fromcontinuum assumption problems with respect to particles, as the particles equilibrate withneither local fluid nor each other when flowing through the flow field, in addition, crossingtrajectories become more pronounced as particle inertia increases and Eulerian methods maybecome less accurate with increasing Stokes number, so a priori and rudimentary Lagrangiancalculations should always be performed to check its validity.

The above argument may be true for dispersed phase flows that contain solid particlesthat does not undergo any shape changes, the same was used to simulate the bubbly flows nottime long, however with the advances in CFD in relation to bubble dynamics, it is envisionedthat the constant bubble size may only be valid for problems where the dispersed phase doesnot undergo deformation. Directly adopting the model for flow problems, wherein dispersedphase undergoes constant change in the shape, may introduce substantial error into the finalpredictions. The MUSIG model was thereby developed to circumvent the above short-coming(Lo, 1996). Theoretically speaking, the model works in a manner in which it resolves the


bubble size mechanistically while working within a range specified groups for the dispersedphase. However, each group possess an individual equation to be solved, which increases theturn around time for each simulation, there by making the parametric study of the problemcomputationally expensive.

Numerical modelling of these classes of flows poses a problem, considering the diversityof two-phase flows, wherein the addition of a dispersed phase to a flow greatly increases theparameter space of the problem. In addition to the complexity involved in tracking thedispersed phase, their mere presence can drastically change the characteristics of the flowitself (Fessler and Eaton, 1995). The turbulent dispersion of these dispersed phaseparticles/bubbles within the carrier phase can be studied by two orders, one by presuming thatthe presence of the dispersed phase does not have any effect on the turbulent carrier flow fieldwhich is considered to be ‘one way coupling’ and the second by considering a feedback of thedispersed phase into the carrier flow field, in addition to the dispersed phase getting affectedby the carrier flow field, this is considered to be ‘two-way coupling’, which is most of thetimes the precise phenomena taking place in the real world.

Turbulence Modulation (TM) which re-defines the carrier phase both at the velocity andat the turbulence level in the presence of dispersed phase is crucial in the design ofengineering applications. However, this study is paralysed owing to the complexities of theflows and limitations of the instruments. A nearly homogenous flow like liquid-particle flowcan circumvent this problem, wherein all turbulence properties are attributed due to therelative motion of the particles; thereby any change felt due to the dispersed phase on thecarrier phase is a direct result of only the TM phenomenon (Parthasarathy and Faeth, 1990).This phenomenon have been exploited by many experimental researchers (Parthasarathy andFaeth, 1987; Parthasarathy and Faeth, 1990; Alajbegovic et al, 1994; Rashidi et al, 1990; Satoand Hishida, 1996; Ishima et al, 2007; Borowsky and Wei, 2007; Righetti and Romano, 2007)to not only investigate, study and understand the basic features of TM but also to aid in thebetter formulation of numerical models. A number of previous studies have examined particleresponse for gas-particle flows in a sudden expansion flow experimentally (Ruck andMakiola, 1988; Hishida and Maeda, 1999; Fessler and Eaton, 1997). Whereas for the liquid-particle flows, although there have been studies in channel flow geometries, but theirpublication is limited for a sudden expansion geometry except for Founti & Klipfel (1998).

Reynolds-averaged Navier-Stokes (RANS) equations are one of the well knownnumerical approaches to predict the Turbulence Modulation (TM) of the carrier phase inmany industrial flow applications. They stem out as a consequence of time average to yieldthe constitutive mean flow equations, however the turbulent stresses which arise as a directconsequence should be modeled with some degree of approximation. Specification of theturbulent eddy viscosity serves as the turbulent closure and the k-ε turbulence formulationspecifies this by solving two additional transport equations for turbulent kinetic energy andthe eddy dissipation. This being the case for single phase flows, for dilute non-reactingparticulate flows, these pose a problem in the sense the momentum of the carrier phaseundergoes a phenomenal change due to the presence of particles, this is also reflected in termsof TM of the carrier phase.

The major aim of this article is to investigate the various density regimes of the two-phase flows and to study the effects of the dispersed phase onto the carrier phase at thevelocity and at the turbulence level (Turbulence Modulation), for the dilute gas-particle flow,liquid-particle flow and also the air-liquid flow. While the density is quite high for the


dispersed phase flow for the gas-particle flow, the density ratio is almost the same for theliquid particle flow, while for the air-liquid flow the density is quite high for the carrier phaseflow. The study of all these density regimes gives a clear picture of how the carrier phasebehaves in the presence of the dispersed phases, which ultimately leads to better design andsafety of many two-phase flow equipments.

Conservation Equations

3.1. Gas-Particle and Liquid-Particle Flows

The modified Eulerian two-fluid model developed by Tu and Fletcher (1995) and Tu(1997) is employed for the simulation of gas-particle and liquid-particle flows, whichbasically considers the carrier and the dispersed phases as two interpenetrating continua.Hereby, a two way coupling is achieved between the dispersed and the carrier phases.

The underlying assumptions employed in the current study are:

1) The particulate phase is dilute and consists of mono disperse spherical particles.2) For such a dilute flow, the gas volume fraction is approximated by unity.3) The viscous stress and the pressure of the particulate phase are negligible.4) The flow field is isothermal.

3.1.1. Governing Equations for Carrier Phase Modeling

The governing equations in Cartesian form for steady, mean turbulent gas flow areobtained by Favre averaging the instantaneous continuity and momentum equations

0 = )u(x

igg

iρ

∂∂ (3.1)

Diig

jgg

j

ig

jg gl

ji

gig

jgg

j

Fuu(x

)ux

(x

+ x

p- = )uu(

x−

∂∂

−∂∂

∂∂

∂

∂

∂∂ )''ρνρρ (3.2)

Eq. (3.1) and (3.2) respectively are the continuity and momentum equation of the carriergas phase, where gggg panduu ',,ρ are the bulk density, mean velocity, fluctuating velocity

and mean pressure of the gas phase, respectively. νgl is the laminar viscosity of the gas phase. FDi

is the Favre-averaged aerodynamic drag force due to the slip velocity between the two phasesand is given by

t)u-u( f

=Fp

ip

ig

pDi ρ (3.3)

where the correction factor f is selected according to Schuh et al (1989)


⎪⎪⎩

⎪⎪⎨

⎧

<

≤<+

≤<+

=

pp

ppp

pp

f

Re2500Re0167.0

2500Re200Re0135.0Re914.0

200Re0Re15.01282.0

687.0

(3.4)

with the particle response or relaxation time given by )18/(dt 2pp glgs νρρ= , wherein dp is

the diameter of the particle.

3.1.2. Governing Equations for Particulate Phase Modeling

After Favre averaging, the steady form of the governing equations for the particulatephase is

0 = )u(x

ipp

iρ

∂∂ (3.5)

WMiDiGiip

jpp

j

ip

jpp

j

F+F+F+uu(x

-=)uu(x

)''ρρ∂∂

∂∂ (3.6)

where ppp uandu ',ρ are the bulk density, mean and fluctuating velocity of the particulate

phase, respectively. In equation (3.6), there are three additional terms representing the gravityforce, aerodynamic drag force, and the wall-momentum transfer force due to particle-wallcollisions, respectively. The gravity force is gF pρ=Gi

, where g is the gravitational acceleration.

3.1.3. Turbulence Modeling for Carrier Phase

For the carrier gas phase, which uses an eddy-viscosity model, the Reynolds stresses aregiven by

ijggi

jg

j

ig

gtgj

gigg k

32)

xu

xu

('u'u δρ∂∂

∂∂

νρρ ++−= (3.7)

where νgt is the turbulent or ‘eddy’ viscosity of the gas phase, which is computed byν εμgt C= ( / )k g g

2 . The kinetic energy of the turbulence, kg and its dissipation rate, εg is

governed by separate transport equations. The RNG theory, models the kg and εg transportequations (3.8) & (3.9) respectively by taking into account the particulate turbulencemodulation, in which α is the inverse Prandtl number.

kggkgg

gtgjg SP

kku +−+= ερ

∂∂

ναρ∂∂ρ

∂∂ )

x(

x)(

x jg

jg

j

(3.8)


εεε ρερε

∂∂ε

ναρ∂∂ερ

∂∂ S+R)CPC(

k)

x(

x)(

x g2kg1gj

gj

gj

gggg

gtgjgu −−+= (3.9)

The rate of strain term R in the εg-equation is expressed as

( )R

Ckg

=−

+μη η η

βη

ε30

3

21

1g , η

ε

∂

∂

∂

∂= = +

⎛

⎝⎜⎜

⎞

⎠⎟⎟

k ux

ux

g gi

j

gj

igij ijS S( ) ,/2 1

22 1 2 (3.10)

where β = 0.015, η0 = 4.38. The major endeavour of including this term is to take intoaccount the effects of rapid strain rate along with the streamline curvature, which in manycases the standard k-ε turbulence model fails to predict. The constants in the turbulenttransport equations are given by α = 1.3929, Cμ = 0.0845, Cε1 = 1.42 and Cε2 = 1.68 as per theRNG theory (Yakhot & Orszag, 1986).

For the confined two-phase flow, the effects of the particulate phase on the turbulence ofthe gas phase are taken into account through the additional terms Sk and Sε in the kg and εg

equations which arise from the correlation term given by

)(2)''( gpgpp

Diigk kk

tfFuS −−=−= ρ (3.11)

in the kg equation and

)(t

f2xx

'F'u2S gpgp

pjj

Diig

gl εερνε −−=∂∂∂∂

−= (3.12)

in the εg equation, where kgp and εgp will be presented in the next following section discussingthe particulate turbulence modeling.

3.1.4. Turbulence Modeling for the Dispersed Phase

The transport equation governing the particulate turbulent fluctuating energy can bewritten as follows:

gpkpj

p

p

ptp

jp

jpp

j

IPxk

xku

x−+= )()(

∂∂

σν

ρ∂∂ρ

∂∂ (3.13)

The turbulence production Pkp of the particulate phase is given by

k

ip

k

kp

ptpijpk

ip

i

jp

j

ip

ptpkp xu

xu

kxu

xu

xu

P∂∂

∂∂

νδρ∂∂

∂∂

∂∂

νρ )(32)( +−+= (3.14)

and the turbulence interaction between two phases Igp is given by


)(2gppp

pgp kk

tfI −= ρ (3.15)

Here ipui

gugpk ''2

1= is the turbulence kinetic energy interaction between two phases.

The transport equation for the gas-particle covariance ip

uig

u '' can be again derived (Tu,

1997), to obtain the transport equation governing the gas-particle correlation which is givenby

gpgppgpj

gp

p

pt

g

gtp

jgp

jp

jgp

j

IIPxk

xkuu

x−−++=+ ερ

∂∂

σν

σν

ρ∂∂ρ

∂∂ )(])([ (3.16)

where the turbulence production by the mean velocity gradients of two phases is

))(31

32)( (

j

ip

j

ig

k

kp

ptk

kp

gtijpgpijpi

jp

ptj

ig

gtpgp xu

xu

xu

xu

kxu

xu

P∂∂

∂∂

∂∂

ν∂∂

νδρδρ∂∂

ν∂∂

νρ ++−−+= (3.17)

The interaction term between the two phases takes the form

]222)1[(2 pggpp

pgp kmkkm

tfII −−+= ρ (3.18)

here m is the mass ratio of the particle to the gas, m=ρp/ρg. The dissipation term due to thegas viscous effect is modeled by

)exp(g

gpggp k

tBε

εε ε−= (3.19)

where Bε=0.4.The turbulent eddy viscosity of the particulate phase, νpt, is defined in a similar way as

the gas phase as:

ν pt p pt pt pk t l k= =23

23

(3.20)

The turbulent characteristic length of the particulate phase is modeled by),min( sptpt Dll ′= where ptl′ is given by

)](|'||'|exp[)cos1(

22

pgg

rgp

gtpt kksign

uuB

ll −−+=′ θ (3.21)

where θ is the angle between the velocity of the particle and the velocity of the gas to accountfor the crossing trajectories effect (Huang et al, 1993). Bgp is an experimentally determined


constant, which takes a value of 0.01. Ds is the characteristic length of the system andprovides a limit to the characteristic length of the particulate phase.

The relative fluctuating velocity is given by

pgr uuu ''' −= (3.22)

and

)2(32'''2'|'|

22pgpgppggr kkkuuuuu +−=+−= (3.23)

The solution process of the above stated numerical formulation can be depicted in theform of a flow chart as shown in figure 3.1.

No

Initial flowsetup

Solve gas-phase momentumequations

Solve pressure correction forthe gas phase

Solve particle-phasemomentum equations

Solve particle concentrationequation

Solve turbulence equations forthe gas phase (kg, eg)

Solve turbulence equations forthe particle phase (kp)

Solve turbulence equations forgas-particle coupling (kgp, egp)

Update primitive variables(concentration, viscosity)

Two-way coupling terms

Stop

A

A

B

B

Overallconvergence

Yes

Figure 3.1. Solution procedure for Eulerian two-fluid model.

3.2. Liquid-Air Flows (Micro-bubble)

3.2.1. Inhomogeneous Two-Fluid Model

3.2.1.1. Mass Conservation

Numerical simulations presented in this paper are based on the two-fluid model usingEulerian-Eulerian approach. The liquid phase is treated as continuum while the gas phase(bubbles) is considered as dispersed phase (ANSYS, 2006). Under isothermal flow condition,


with no interfacial mass transfer, the continuity equation of the two-phases with reference toIshii (1975) and Drew and Lahey (1979) can be written as:

( ) ( ) 0=⋅∇+∂

∂iii

ii uαρtαρ

(3.24)

where α, ρ and u is the void fraction, density and velocity of each phase. The subscripts i = lor g denotes the liquid or gas phase.

3.2.1.2. Momentum Conservation

The momentum equation for the two-phase can be expressed as follow:

( ) ( ) ( )( )[ ] lg

T

iieiiiiiiiii

iii FuuμαgραPαuuαρt

uαρ+∇+∇⋅∇++∇−=⋅∇+

∂∂ (3.25)

On the right hand side of Eq. (3.25), Flg represents the total interfacial force calculatedwith averaged variables, g is the gravity acceleration vector and P is the pressure. The termFlg represents the inter-phase momentum transfer between gas and liquid due to the drag forceresulted from shear and drag which is modelled according to Ishii and Zuber (1979) as:

( )lglglifD uuuuρaCFF −−=−=81

gllg

where DC is the drag coefficient which can be evaluated by correlation of several distinctReynolds number regions for individual bubbles proposed by Ishii and Zuber (1979).

3.2.1.3. Interfacial Area Density

In Eq. (3.25), interfacial momentum transfer due to the drag force is directly dependenton the contact surface area between the two phases and is characterized by the interfacial areaper unit volume between gas and liquid phase, named as the interfacial area density aif. Basedon the particle model, assuming that liquid phase is continuous and the gas phase is dispersed,the interfacial area per unit volume is then calculated based on the Sauter mean bubblediameter dg given by

g

gif d

a*6α

= where

⎪⎩

⎪⎨

⎧

>−

−

<

=)(),

11

max(

)(),max(

maxminmaxmax

maxmin*

αααααα

ααααα

gg

gg

g if

if


The non-dimensional inter-phase transfer coefficients can be correlated in terms of theparticle Reynolds number and is given by

l

glgl dUU

μ

ρ −=lgRe

where µl is the viscosity of the liquid phase.

3.2.2. MUSIG Model

To account for non-uniform bubble size distribution, the MUSIG model employs multiplediscrete bubble size groups to represent the population balance of bubbles. Assuming eachbubble class travel at the same mean algebraic velocity, individual number density of bubbleclass i based on Kumar and Ramkrishna (1996a) can be expressed as:

( )ij

jigi Rnu

tn

⎟⎟⎠

⎞⎜⎜⎝

⎛=⋅∇+

∂∂ ∑ (3.26)

where ( )ij jR∑ represents the net change in the number density distribution due to

coalescence and break-up processes. The discrete bubble class between bubble volumes ivand 1+iv is represented by the centre point of a fixed non-uniform volume distributed grid

interval. The interaction term ( ) ( )BCBCij j DDPPR −−+=∑ contains the source rate of

CP , BP , CD and BD , which are, respectively, the production rates due to coalescence andbreak-up and the death rate due to coalescence and break-up of bubbles.

3.2.2.1. MUSIG Break-up Rate

The production and death rate of bubbles due to the turbulent induced breakage isformulated as:

( ) jij

N

ijB nvvΩP :

1∑

+=

= iiB nΩD = with ∑=

=N

kkii ΩΩ

1(3.27)

Here, the break-up rate of bubbles of volume jv into volume iv is modelled according

to the model developed by Luo and Svendsen (1996). The model is developed based on theassumption of bubble binary break-up under isotropic turbulence situation. The majordifference is the daughter size distribution which has been taken account using a stochasticbreakage volume fraction fBV. By incorporating the increase coefficient of surface area, cf =


[ 32 /BVf +(1-fBV)2/3-1], into the breakage efficient, the break-up rate of bubbles can be obtained

as:

( )( )

( ) dξdβρc

ξξ

dεCF

nαvvΩ

l

f

ξj

Bjg

ij

min ⎟⎟⎠

⎞⎜⎜⎝

⎛−×

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

− ∫ 3/113/53/2

1

3/11

23/1

2

12exp1

1:

ξεσ

where jd/λξ = is the size ratio between an eddy and a particle in the inertial sub-range

and consequently jminmin d/λξ = and C and β are determined, respectively, from

fundamental consideration of drops or bubbles breakage in turbulent dispersion systems to be0.923 and 2.0.

3.2.2.2. MUSIG Coalescence Rate

The number density of individual bubble groups governed by coalescence can beexpressed as:

∑∑= =

=i

k

i

ljiijjkiC nnP

1 121 χη

)/()( 11 −− −−+ iiikj ννννν if ikji νννν <+<−1

=jkiη )/()( 11 iikji ννννν −+− ++ if 1+<+< ikji νννν

0 otherwise

∑=

=N

jjiijC nnD

1χ

From the physical point of view, bubble coalescence occurs via collision of two bubbleswhich may be caused by wake entrainment, random turbulence and buoyancy. However, onlyturbulence random collision is considered in the present study as all bubbles are assumed tobe spherical (wake entrainment becomes negligible). Furthermore, as all bubbles travel at thesame velocity in the MUSIG model, buoyancy effect is also eliminated. The coalescence rateconsidering turbulent collision taken from Prince and Blanch (1990) can be expressed as:

[ ] ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−++=

ij

ijtjtijiCij

tuuddF

τπχ exp4

5.0222

where ijτ is the contact time for two bubbles given by 3/13/2 /)2/( εijd and ijt is the time

required for two bubbles to coalesce having diameter di and dj estimated to be)/ln(]16/)2/[( 0

5.03flij hhd σρ . The equivalent diameter dij is calculated as suggested by


Chesters and Hoffman (1982): 1)/2/2(( −+= jiij ddd . According to Prince and Blanch

(1990), for air-water systems, experiments have determined the initial film thickness ho andcritical film thickness hf at which rupture occurs as 4101 −× and 8101 −× m respectively. Theturbulent velocity ut in the inertial subrange of isotropic turbulence (Rotta, 1972) is given

by: 3/13/12 dut ε= .

Numerical Procedure

All the transport equations are discretized using a finite volume formulation in ageneralized coordinate space, with metric information expressed in terms of area vectors. Theequations are solved on a nonstaggered grid system, wherein all primitive variables are storedat the centroids of the mass control volumes. Third-order QUICK scheme is used toapproximate the convective terms, while second-order accurate central difference scheme isadopted for the diffusion terms. The velocity correction is realized to satisfy continuitythrough SIMPLE algorithm, which couples velocity and pressure. At the inlet boundary theparticulate phase velocity is taken to be the same as the gas velocity. The concentration of theparticulate phase is set to be uniform at the inlet. At the outlet the zero streamwise gradientsare used for all variables. The wall boundary conditions are based on the model of Tu andFletcher (1995).

All the governing equations for both the carrier and dispersed phases are solvedsequentially at each iteration, the solution process is started by solving the momentumequations for the gas phase followed by the pressure-correction through the continuityequation, turbulence equations for the gas phase, are solved in succession. While the solutionprocess for the particle phase starts by the solution of momentum equations followed by theconcentration then gas-particle turbulence interaction to reflect the two-way coupling, theprocess ends by the solution of turbulence equation for the particulate phase. At each globaliteration, each equation is iterated, typically 3 to 5 times, using a strongly implicit procedure(SIP).The above solution process is marched towards a steady state and is repeated until aconverged solution is obtained.

Numerical Predictions

Gas –Particle Flow

4.1. Code Verification

In this section the code is validated for mean streamwise velocities and fluctuations forboth the carrier and dispersed phases against the benchmark experimental data of Fessler andEaton (1995). This task is undertaken to verify the fact that these two classes of particles,which share the same Stokes number but varied particle Reynolds number can be handled bythe code. Figure 4.1 show the backward facing step geometry used in this study, which issimilar to the one used in the experiments of Fessler and Eaton (1995), which has got a stepheight (h) of 26.7mm. As the span wise z-direction perpendicular to the paper is much larger


than the y-direction used in the experiments, the flow is considered to be essentially two-dimensional. The backward facing step has an expansion ratio of 5:3. The Reynolds numberover the step works out to be 18,400 calculated based on the centerline velocity and stepheight (h). The independency of grid over the converged solution was checked by refining themesh system through doubling the number of grid points along the streamwise and the lateraldirections. Simulations results revealed that the difference of the reattachment length betweenthe two mesh schemes is less than 3%.

xy H

h

H=40mm h=26.7mm

x=35h

Figure 4.1. Backward facing step geometry.

4.1.1. Mean Streamwise Velocities

The mean streamwise velocities for the gas phase have been shown in Figure 4.2 forvarious stations along the backward-facing step geometry, it can be generally seen that thereis fairly good agreement with experimental findings of Fessler and Eaton (1995). This is thenfollowed by the streamwise velocities for the two classes of particles considered in this study,whose properties are tabulated in Table 1. The broad varying characteristics of differentparticle sizes and material properties can be unified by a single dimensionless parameter; it isalso used to quantify the particles responsivity to fluid motions and this non-dimensionalparameter is the Stokes number (St) and is given by the ratio of particle relaxation time totime that of the appropriate fluid time scale, St = tp/ts. In choosing the appropriate fluid timescale, the reattachment length has not been considered as it varies due to addition of particlesand is not constant in this study, rather a constant length scale of five step heights, which is inaccordance to the reattachment length is used. The resulting time scale is given by ts=5h/Uo,in accordance with the experiments of Fessler and Eaton (1995).

Table 4.1 Properties of the dispersed phase particles

Nominal Diameter(μm) 150 70

Material Glass Copper

Density(kg/m3) 2500 8800

Stokes Number (St) 7.4 7.1

Particle Reynolds number (Rep) 9.0 4.0


0.00

0.50

1.00

1.50

2.00

2.50

Distance along the step (x/h)

Hei

ght (

y/h)

x/h=1x/h= x/h=9x/h= x/h= x/h=

0

x/h=1

u/Ub 1.5

Figure 4.2. Mean streamwise gas velocities.

0.00

0.50

1.00

1.50

2.00

2.50


Hei

ght (

y/h) x/h=1x/h= x/h=5 x/h= x/h=9

0 u/Ub1.5

Figure 4.3.a. Streamwise mean velocity for 70μm copper particles.

Experimental Numerical



The significance of the Stokes number is that a particle with a small Stokes number(St<<1) are found to be in near velocity equilibrium with the surrounding carrier fluid, thereby making them extremely responsive to fluid velocity fluctuations, in fact Laser Dopplervelocimetry (LDV) takes advantage of this to deduce local fluid velocity from the measuredvelocity of these very small Stokes number particles, however on the other hand for a largerStokes number (St>>1) particles are found be no longer in equilibrium with the surroundingfluid phase as they are unresponsive to fluid velocity fluctuations and they will passunaffected through eddies and other flow structures.

Figures 4.3a & 4.3b shows the mean streamwise velocity profiles for the two particleclasses considered in this study, it can be seen that there is generally a fairly good agreementwith the experimental results. From the particle mean velocity graphs of the carrier anddispersed phases it can be inferred, that the particle streamwise velocity at the first stationx/h=2 is lower than the corresponding gas velocities, this is in lines with the fully developedchannel flow reaching the step as described in the experiments of Kulick et al (1994), whereinthe particles at the channel centerline have lower streamwise velocities than that of the fluidas a result of cross-stream mixing. However the gas velocity lags behind the particlevelocities aft of the sudden expansion as the particles inertia makes them slower to respond tothe adverse pressure gradient than the fluid.

0.00

0.50

1.00

1.50

2.00

2.50


Hei

ght (

y/h) x/h= x/h= x/h=7 x/h=9 x/h=1

0 u/Ub 1.5

Figure 4.3.b. Streamwise mean velocity for 150μm glass particles.

4.1.2. Mean Streamwise Fluctuations

The code has been further validated for mean streamwise fluctuations and Figure 4.4shows the mean streamwise fluctuations for the gas phase against the experimental data. It isseen that there is a general under prediction of the simulated data with the experimental



results and this is more pronounced towards the lower wall for a height of up to y/h≤2,however the pattern of the simulated results have been found to be in tune with itsexperimental counterpart.

0.00

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1.00

1.50

2.00

2.50


Hei

ght (

y/h)

x/h= x/h=x/h x/h x/h=1

0 u'/Ub 0.2

Figure 4.4. Fluctuating streamwise gas velocities particles.

0.00

0.50

1.00

1.50

2.00

2.50


Hei

ght (

y/h)

x/h=2 x/h=5 x/h=7 x/h=9 x/h=12

0 u'/Ub 0.3

Figure 4.5.a. Fluctuating streamwise particle velocities for 70μm copper particles.



0.00

0.50

1.00

1.50

2.00

2.50


Hei

ght (

y/h)

0 u'/Ub 0.3

x/h= x/h=5 x/h=7 x/h=9 x/h=1

Figure 4.5.b. Fluctuating streamwise particle velocities for 150μm glass particles.

Figures 4.5a & 4.5b show the streamwise fluctuating particle velocities for the twodifferent classes of particles considered, there has been a minor under-prediction until stationsy/h ≤ 1, over all a fairly good agreement can be observed. It can be also seen that for y/h >1.5, the particle fluctuating velocities are considerably larger than those of the fluid. Thisagain is in accordance with channel flow inlet conditions, where the particles have higherfluctuating velocities than those of the fluid owing to cross-stream mixing. All theexperimental results used for comparison of particle fluctuating velocities correspond tomaximum mass loadings of particles as reported in the experiments of Fessler & Eaton(1999).

4.2. Results and Discussion

4.2.1. Turbulence Modulation (TM)

The plot depicted in the following sections to represent the Turbulent Modulation (TM)of the carrier gas phase is given by the ratio of the laden flow r.m.s streamwise velocity to theunladen r.m.s streamwise velocity. These plots signify that any turbulence modulation felt inthe carrier phase is reflected as an exit of the ratio from unity.

4.2.1.1. Analysis of Experimental Data

Plots 4.6a and 4.6b depict the experimental data as obtained from the experiments ofFessler and Eaton (1995). Figures 4.6a&b represent the mean streamwise particle velocities



and fluctuations respectively for the two sets of dispersed phase particles i.e., copper andglass particles. It can be well seen that the mean velocities and to a similar extent thefluctuations for these two classes seem to behave analogous to each other. However fromfigures 4.7a-c, which shows the experimental TM for the carrier phase in the presence of thedispersed phase at the same mass loading of 40% seem to behave in contrary to the abovefindings. In all these stations considered in this study the glass particle seems to attenuate thecarrier phase more than copper particles. It is also be seen at the station x/h=2, the attenuation

0.00

0.50

1.00

1.50

2.00

2.50


Hei

ght (

y/h)

x/h=2 x/h=5 x/h=7 x/h=9

0 u/Ub 1.5

Figure 4.6.a. Experimental mean streamwise particle velocity.

of the turbulence for glass particle is totally opposite in relation to copper particle for locationy/h>1.25. At station x/h=7 for certain regions along the height of the step the turbulenceattenuation for the two particle classes seems to be in phase, whereas before and after thissmall region of unison the glass particle seem to attenuate more than the copper particles. Atthe station x/h=14, it can be clearly seen, that there is a uniform degree of difference inattenuation all along the step, this is more attributed to the uniform distribution of theparticles seen along the step. The maximum turbulence attenuation can be seen for the 150μmparticles, which up to 35% as reported by Fessler and Eaton (1999).

From the above analysis, it is quite clear that particles with the same Stokes numbermodulate the carrier phase turbulence in a totally different fashion. This makes us concludethat although Stokes number can be generalized to account for mean values of velocity andfluctuations, it cannot be generalized when it comes to TM, in which case something morethan Stokes number is required to define one’s particle response to surrounding carrier phaseturbulence either to attenuate or enhance it.

70μm copper 150μm glass


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1.00

1.50

2.00

2.50


Hei

ght (

y/h)

x/h= x/h= x/h=7 x/h=

0 u'/Ub 0.3

Figure 4.6.b. Experimental fluctuating streamwise particle velocity.

0.60

0.70

0.80

0.90

1.00

1.10

0.00 0.50 1.00 1.50 2.00 2.50Height (y/h)

Turb

ulen

ce M

odul

atio

n (T

M)

x/h=2

Figure 4.7.a. Experimental Turbulence Modulation at x/h=2.



0.60

0.70

0.80

0.90

1.00

1.10

0.00 0.50 1.00 1.50 2.00 2.50Height (y/h)

Turb

ulen

ce M

odul

atio

n (T

M)

x/h=

Figure 4.7.b. Experimental Turbulence Modulation at x/h=7.

0.60

0.70

0.80

0.90

1.00

1.10

0.00 0.50 1.00 1.50 2.00 2.50Height (y/h)

Turb

ulen

ce M

odul

atio

n (T

M)

x/h=14

Figure 4.7.c. Experimental Turbulence Modulation at x/h=14.



4.2.2. TM & (Particle Number Density) PND Results

In this section we have tried to derive an understanding for the Turbulence Modulation(TM) of the carrier gas phase in the presence of particles using our turbulent formulation,along with its corresponding PND results for the two classes we have considered in thisstudy. The simulated results of the above two parameters are plotted along side theexperimental findings of Fessler and Eaton (1995). The experimental values for themodulation are plotted with error bars, as significant scatter are apparent in these plots,thereby making any variations on the order of ±5% insignificant (Fessler & Eaton, 1995).

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10.00

12.00

0.00 0.50 1.00 1.50 2.00 2.50Height (y/h)

Par

ticle

Num

ber d

ensi

ty (P

ND

)

0.00

0.20

0.40

0.60

0.80

1.00

1.20

Turb

ulen

ce m

odifi

catio

n (T

M)

Figure 4.8.a. Turbulence Modulation & Particle Number Density for 70μm copper particles at x/h=2.

The plots 4.8a-c shows the combined numerical and experimental results of TM(secondary axis) and PND (primary axis) for three sections along the step viz. x/H=2, 7 &14for copper particles. It can be noted at station x/h=2 just aft of the step, for y/h<1 there existvery few particles, but however the experimental TM shows a wavy behavior in comparisonto the simulated results which seem to behave in unison with the PND, however aft of thissection, the simulated values compare well with the experimental data. In the middle of thestep at station x/h=7, there is generally a good comparison of the experimental and simulatedvalues for both the TM and PND. At the exit (x/h=14) however, there seems to be generalunder prediction of TM for y/h>1.5, while the PND seem to vary from under predicting toover predicting the experimental data. Figures 4.9a-c shows plots of the glass particles for thesame three sections along the step. At section x/h=2, the simulated values seem to overpredict the experimental data for y/h>1, however this distinct behavior of maximumattenuation as reported by Fessler and Eaton (1995) occurs here, this under prediction is notquite in terms with the PND results which seem to show a uniform distribution. A fairly good

Experimental-TM Numerical-TM Experimental-PND Numerical-PND


agreement is with both turbulence modulation and PND can be seen for stations x/h=7 and14.

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10.00

12.00

0.00 0.50 1.00 1.50 2.00 2.50Height (y/h)

Part

icle

Num

ber d

ensi

ty (P

ND

)

0.00

0.20

0.40

0.60

0.80

1.00

1.20

Turb

ulen

ce m

odifi

catio

n (T

M)

`

Figure 4.8.b. Turbulence Modulation & Particle Number Density for 70μm copper particles at x/h=7.

It is generally seen that for the two classes of particles considered, there existconsiderably more particles in the region y/h>1 for locations x/h=2 and 7, and the particlesexhibit a uniform distribution by the time it reaches the location x/h=14 due to its uniformspreading action, but despite a uniform distribution the turbulence attenuation is still small fory/h<1 at the x/h=14, and this is attributed to the non-effectiveness of the particles to largescale vortices which are reported to exist downstream of the single phase backward facingstep (Le et al., 1997). From the PND results for the two classes of particles, both simulatedand experimental plots show that there exits very few particles in the region y/h<1 before there-attachment point and a significant spreading has taken place by, at locations x/h=9 and 14,considering this behavior along side with the turbulence attenuation one can state that lack ofturbulence modulation is not simple due to the absence of particles but clearly a difference onresponse of the turbulence in a specific region due to the presence of particles (Fessler &Eaton, 1995) and this has been explicitly seen in our simulated values using our turbulenceformulation mentioned in the previous section.



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6.00

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10.00

12.00

0.00 0.50 1.00 1.50 2.00 2.50Height (y/h)

Part

icle

Num

ber d

ensi

ty (P

ND

)

0.00

0.20

0.40

0.60

0.80

1.00

1.20

Turb

ulen

ce m

odifi

catio

n (T

M)

Figure 4.8.c. Turbulence Modulation & Particle Number Density for 70μm copper particles at x/h=14.

4.2.3. Effect of Particle Reynolds Number on TM

In this section, we have tried to explain the phenomenon of the carrier phase TM byparticles in relation to particle Reynolds number. Looking back at Table 4.1, it is seen that forthe glass particles the Rep is almost 125% more than for the copper particles and alsoconsidering the size of the particles which is around 114% more for the glass particles incomparison to the copper particles. The presented results on the degree of carrier phaseturbulence attenuation also seem to throw some interesting pattern especially in regards tocopper and glass particles, which almost share the same Stokes number. Based on thissimilarity one would expect that their response to the turbulence be the same, although thisfact seems to be quite in lines with the particle mean and fluctuating velocities but theirbehavior in regards to turbulence modulation paint a totally different picture, which makes usconclude that the turbulence attenuation has a direct link of the particles onto the turbulencerather a derived effect on the mean flow modifications (Fessler and Eaton, 1999). And thisattenuation cannot follow suit with increasing loading and particle size as there will be anincrease in the turbulence as reported by Hetsroni (1989), it is also interesting to note fromthe same author that apart from the Stokes number, particle Reynolds number plays animportant role in the systematic behavior of the particles, the same has been reported from theexperimental findings of Fessler and Eaton (1995), this is in conformity to our simulatedresults wherein the two classes of copper and glass behave differently although their particleStokes number remain the same.



0.00

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4.00

6.00

8.00

10.00

12.00

0.00 0.50 1.00 1.50 2.00 2.50Height(y/h)

Part

icle

Num

ber d

ensi

ty(P

ND

)

0.00

0.20

0.40

0.60

0.80

1.00

1.20

Turb

ulen

ce M

odifi

catio

n(TM

)

Figure 4.9.a. Turbulence Modulation & Particle Number Density for 150μm glass particles at x/h=2.

It is known from previous studies that there is a physical increase in the carrier phaseturbulence and this is attributed to the wake formation behind these large particles similar tothe ones behind the cylinders encountered in single phase flows. Experimental studies fromEaton et al (1999) state that for Rep around 10, the modulation is caused due to the strongasymmetric wake distortion extending many particle diameters downstream, whereas forRep=1, the flow distortion is highly localized to a nearly symmetric region extending only afew particle diameters. Taking this into consideration one would expect a significantenhancement of the carrier phase turbulence for the glass particles but in contrast significantattenuation have taken place, the maximum attenuation is noted at location x/h=2 and theattenuation of glass particles is roughly about 44% more than its counterpart copper particles.This behavior of the glass particle to attenuate rather than enhance the turbulence of thecarrier phase can be explained by the fact that, due to strong asymmetric wake, the distortioncaused by this class of particles is dissipated more easily within the carrier gas phase, this isin sharp contrast to the parallel experiments run by Sato et al (2000) using water as the carrierphase fluid, although similar Stokes number were achieved, much higher Rep was realized,this in total showed significant vortices behind the particles and also an enhancement in thecarrier phase turbulence, which is primarily caused due to the different density ratios betweenthe carrier and dispersed phases, which makes one to justify that Stokes number alone will notbe able to describe the parameter space of the dispersed two-phase flow problems, Rep shouldalso be taken into consideration.



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0.00 0.50 1.00 1.50 2.00 2.50Height(y/h)

Part

icle

Num

ber d

ensi

ty

(PN

D)

0.00

0.20

0.40

0.60

0.80

1.00

1.20

Turb

ulen

ce M

odifi

catio

n(TM

)

Figure 4.9.b. Turbulence Modulation & Particle Number Density for 150μm glass particles at x/h=7.

0.00

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4.00

6.00

8.00

10.00

12.00

0.00 0.50 1.00 1.50 2.00 2.50Height(y/h)

Part

icle

Num

ber

dens

ity(P

ND

)

0.00

0.20

0.40

0.60

0.80

1.00

1.20

Turb

ulen

ce M

odifi

catio

n(TM

)

Figure 4.9.c. Turbulence Modulation & Particle Number Density for 150μm glass particles at x/h=14.

Liquid–Particle Flow

This section describes the numerical investigation of the turbulent gas-particle flow overa backward facing step, using the Eulerian-Eulerian model as described in the previoussection. The numerical procedure is almost the same as of section 3.1. However, two



backward-facing step geometries are utilized in this study one to validate the GP flow and theother for the LP flow. Figure 5.1a shows the backward facing step geometry, which is similarto the one used in the experiments of Fessler and Eaton (1995), comprising of a step height(h) of 26.7mm. As the span wise z-direction perpendicular to the paper is much larger thanthe y-direction used in the experiments, the flow is considered to be essentially two-dimensional. The backward facing step has an expansion ratio of 5:3. The Reynolds numberover the step works out to be 18,400 calculated based on the centerline velocity and stepheight (h). The experimental set up of Founti and Klipfel (1998) consisted of a pipe flow witha sudden expansion ratio of 1:2, with a step height of 25.5mm, as depicted in figure 5.1b,working at a Reynolds number of 28,000. The summary of the flow conditions along with theproperties of the dispersed phase particles used in this study are summarized in Table 5.1.

xy H

h

H=40mm h=26.7mm

x=35h

Figure 5.1.a. Backward facing step geometry (Fessler & Eaton; 1995).

yx H

hH=25.5mm h=25.5mm

x=35h

Figure 5.1.b. Backward facing step geometry (Founti & Klipfel; 1998).

Table 5.1. Flow properties of carrier and dispersed phases for LP & GP flows

Parameters Gas-Particle (GP) flowFessler & Eaton (1995)

Liquid-Particle (LP) flowFounti & Klipfel (1998)

Reynolds number (Reh) 18,700 28,000Geometry BFS Pipe (BFS)Continuous Phase Air Diesel OilMass loading 20% 15%Particle Density 2500 2500Particle Diameter 150 micron 450 micronPhase-density ratio 2137:1 3.0:1


5.1. Analysis of Experimental Data

In this section, the experimental data for mean velocities and fluctuations are analysed, inorder to understand the particle behaviour in relation to its carrier phase namely the gas andthe liquid. For this purpose, three sections were selected aft of the sudden expansion one nearthe step, another almost at the middle and the other farther away from the step nearing the exitof the geometry.

An important, dimensionless scaling parameter in defining on how the fluid-particlebehave with the flow field is the Stokes number (St), which is given by the ratio of theparticle relaxation time to a time characteristic of the fluid motion, i.e., St = tp/ts. Thisdetermines the kinetic equilibrium of the particles with the surrounding liquid. In choosing,the fluid time scale ts, amidst the complexity of having two different geometries withdifferent expansion ratios and also with the re-attachment length varying with the additionof the particles, the fluid time scales were determined by ts=5h/Uo in lieu with theexperimental conditions of Fessler and Eaton (1997). A small stokes number (St << 1)signifies that the particles are in near velocity equilibrium with the carrier fluid. For largerstokes number (St >> 1) particles are no longer in equilibrium with the surrounding fluidphase, which will be exemplified in the later sections. Based on the above definition, theStokes number for the GP and the LP flow examined in our study worked out to be 14.2and 0.59 respectively.

Figures 5.2a-c shows the mean streamwise velocities of the liquid and particle phaseflows as presented from the experiments of Founti & Klipfel (1998). It can be seen at sectionx/h=0.7, the particles seem to exhibit a higher negative velocity for a section of y/h<1, afterthis height the particles seem to surpass the liquid velocities for the section y/h>1, while for asmall region at the proximity of the step they seem to exhibit a homogeneous behaviour. Atsection x/h=7.8, which is almost the middle section, the liquid phase have a higher negativevelocity than that of the particulate phase, this behaviour seems to follow for a height of up toy/h=1, after which the both the phases seem to behave in unison. The final section is x/h=15.7, herein again the liquid seems to exhibit a higher velocity than that of the particles, withunified flow at the top.

Figures 5.3a-c shows the mean velocities of the GP flow as obtained from theexperiments of Fessler and Eaton (1999). At section x/h=2, just aft of the step the particlevelocities seem to lag behind the gas velocities, whereas at section x/h=7 in the middle ofthe geometry, the particle velocities seem to ‘catch up’ with that of the gas phase and theirvelocities are more or less the same, however at the exit of the backward-facing stepgeometry (x/h=14), a clear marked difference in velocities is observed, wherein theparticles seem to overtake the gas due to its inertial. From the two sets of experimentalresults outlined above, one could observe that at near the inlet sections, where the re-circulation is quite predominant for both the cases, particles seem to lag behind the gas forthe GP flows, where as the particles seem to exhibit a higher inertial with respect to the LPflows, with particles leading throughout the height of the step. Overall, with respect to themagnitude of the mean velocities, the particle seem to exhibit more or less a change in thepattern from ‘lead’ to ‘lag’, as we proceed along the step for LP flows, whereas theparticles seem to exhibit the opposite pattern from ‘lag’ to ‘lead’ for the GP flows along thestep. The presence of a clear shear layer just aft of the step is quite prominent for the


particles at sections all along the step for LP flow, which in single phase is the major sourceof turbulence generation due to shear.

In order to understand this behaviour better we now turn to the mean streamwisefluctuations for these two kinds of flows. Figures 5.4a-c shows the fluctuating velocities ofLP flows and it can be seen that at the entry the particles seem to ‘lag’ behind the liquid for aheight of y/h>1, while the showing an increase with respect to liquid in the lower part. At themiddle section considered, for a height of about y/h>1 they exhibit a homogenous flowbehaviour, whereas at the lower part the particles again seem to exceed its liquid counterpart.Whereas near the exit, both the continuous and dispersed phases have almost the same patternprompting to the fact that the particle ‘catch up’ with the liquid phase, mimicking ahomogenous flow pattern.

Figures 5.5a-c show the fluctuation results for the GP flow, at the entry section x/h=2, itcan be seen that the particulate phase has a higher fluctuation in comparison to the gas phase,while at the middle section x/h=7, the gas phase seem to catch up with the particulate phase,while at the exit at x/h=14, it is observed that both the dispersed and the continuous phaseseem to fluctuate in unison. From the above experimental results of the fluctuation, it can beascertained that the particle ‘lag’ behind with respect to the continuous phase in terms of LPflows, whereas they ‘lead’ in terms of the gas-particle flows.

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0.75

1.00

1.25

1.50

1.75

2.00

-0.20 0.00 0.20 0.40 0.60 0.80 1.00

x/h=0.7

y/h

u/Uo

Figure 5.2.a. Experimental mean streamwise velocities at x/h=0.7 for liquid-particle flow.

Liquid Particle


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0.50

0.75

1.00

1.25

1.50

1.75

2.00

-0.20 0.00 0.20 0.40 0.60 0.80 1.00

x/h=7.8

y/h

u/Uo

Figure 5.2.b. Experimental mean streamwise velocities at x/h=7.8 for liquid-particle flow.

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0.75

1.00

1.25

1.50

1.75

2.00

-0.20 0.00 0.20 0.40 0.60 0.80 1.00

y/h

u/Uo

x/h=15.7

Figure 5.2.c. Experimental mean streamwise velocities at x/h=15.7 for liquid-particle flow.

Liquid Particle


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1.00

1.50

2.00

2.50

-0.25 0.00 0.25 0.50 0.75 1.00

x/h=2

y/h

u/Uo

Figure 5.3.a. Experimental mean streamwise velocities at x/h=2 for gas-particle flow.

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1.00

1.50

2.00

2.50

0.00 0.25 0.50 0.75 1.00

x/h=7

y/h

u/Uo

Figure 5.3.b. Experimental mean streamwise velocities at x/h=7 for gas-particle flow.

Gas Particle


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2.00

2.50

0.00 0.20 0.40 0.60 0.80 1.00

x/h=14y/

h

u/Uo

Figure 5.3.c. Experimental mean streamwise velocities at x/h=14 for gas-particle flow.

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0.75

1.00

1.25

1.50

1.75

2.00

0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16u'/Uo

y/h

x/h=0.7

Figure 5.4.a. Experimental mean fluctuating velocities at x/h=0.7 for liquid-particle flow.

Liquid Particle

Gas Particle


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0.75

1.00

1.25

1.50

1.75

2.00

0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16u'/Uo

y/h

x/h=7.8

Figure 5.4.b. Experimental mean fluctuating velocities at x/h=7.8 for liquid-particle flow.

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0.75

1.00

1.25

1.50

1.75

2.00

0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16u'/Uo

y/h

x/h=15.7

Figure 5.4.c. Experimental mean fluctuating velocities at x/h=15.7 for liquid-particle flow.

Liquid Particle


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1.50

2.00

2.50

0.00 0.05 0.10 0.15 0.20

x/h=2y/

h

u'/Uo

Figure 5.5.a. Experimental mean fluctuating velocities at x/h=2 for gas-particle flow.

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1.00

1.50

2.00

2.50

0.00 0.05 0.10 0.15 0.20

x/h=7

y/h

u'/Uo

Figure 5.5.b. Experimental mean fluctuating velocities at x/h=7 for gas-particle flow.

Gas Particle


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1.00

1.50

2.00

2.50

0.00 0.05 0.10 0.15 0.20

x/h=14y/

h

u'/Uo

Figure 5.5.c. Experimental mean fluctuating velocities at x/h=14 for gas-particle flow.

5.2. Numerical Code Validation

In this section the code is validated for mean streamwise velocities and fluctuations forboth the carrier and dispersed phases against the benchmark experimental data of Fessler andEaton (1995) for GP flow and the experimental data of Founti and Klipfel (1998) for the LPflow. This task is undertaken to verify the fact that particulate flows with two varied carrierphases can be handled by the code.

The ability of the numerical code to replicate the experimental results of GP (Fesslerand Eaton, 1999) and LP (Founti and Klipfel, 1998) flows are tested. Figure 5.6a shows thenumerical findings of single phase (Diesel oil) mean velocities against the experimentaldata, although the overall behaviour is replicated numerically there have been some underprediction for a height of y/h>1 for mid-section of the geometry, while a minor overprediction is felt along the entire height at section x/h=15.7. Figure 5.6b shows thefluctuating liquid velocities along the step compared against the experimental findings,there have been some minor under prediction for a height of y/h<1 at some sections, themajority show a good comparison with the experimental data. Figure 5.6c and figure 5.6ddepicts the experimental and numerical comparison of particle mean and fluctuatingvelocities and it can be seen that overall numerical results have a good agreement with theexperimental data.

Gas Particle


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1.00

1.50

2.00


Hei

ght (

y/h)

x/h=0 x/h=5 x/h=7 x/h=11 x/h=15

0 1.5u/U

Figure 5.6.a. Axial liquid velocities along the step for LP flows.

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1.00

1.50

2.00


Hei

ght (

y/h)

u'/Uo0 0.

x/h=0.7 x/h=5.9 x/h=7. x/h=11. x/h=15.

Figure 5.6.b. Fluctuating axial liquid velocities along the step for LP flows.



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1.00

1.50

2.00


Hei

ght (

y/h)

u/Uo0 1.5

x/h=0.7 x/h=5.9 x/h=7.8 x/h=11.8 x/h=15.7

Figure 5.6.c. Axial particle velocities along the step for LP flows.

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1.00

1.50

2.00


Hei

ght (

y/h)

u'/U0 0.3

x/h=0. x/h=5. x/h=7. x/h=11. x/h=15.

Figure 5.6.d. Fluctuating axial particle velocities along the step for LP flows.



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1.00

1.50

2.00

2.50


Hei

ght (

y/h)

0 u/U o 1.5

x/h=2 x/h=5 x/h=7 x/h=9 x/h=14

Figure 5.7.a. Streamwise gas velocities along the step for GP flows.

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1.00

1.50

2.00

2.50


Hei

ght (

y/h)

x/h= x/h=x/h= x/h x/h=1

0 u'/Uo 0.2

Figure 5.7.b. Fluctuating streamwise gas velocities along the step for GP flows.



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1.00

1.50

2.00

2.50


Hei

ght (

y/h) x/h= x/h= x/h= x/h=9 x/h=1

4

0 u/Uo 1.5

Figure 5.7.c. Streamwise mean velocity for 150μm glass particles.

0.00

0.50

1.00

1.50

2.00

2.50


Hei

ght (

y/h)

0 u'/Uo 0.3

x/h=2 x/h=5 x/h=7 x/h=9 x/h=1

Figure 5.7.d. Fluctuating streamwise particle velocities for 150μm glass particles.



With the LP flow showing satisfactory agreement, the code is further validated tosubstantiate the numerical findings of GP flows against the experimental data of Fessler andEaton (1999). Figure 5.7a shows the numerical comparison against the experimental findingsand it can be seen that a fairly good agreement have been obtained between the experimentaland its numerical counterpart, minor under prediction have been observed within a height ofy/h<1 for the first two sections considered along the step. Figure 5.7b shows the meanstreamwise fluctuating velocities along the step and it can be there is a general underprediction along the length of the step with some minor over prediction for a height of y/h<1at section x/h=2. But however, the trend as seen from the numerical simulation is in lines withthe experimental data. Figure 5.7c shows the comparison of the mean streamwise particlevelocities for the 150µm glass particle against its experimental counterpart, it can be seen thatthere is a good agreement between the experimental and the numerical findings all along thestep. The simulated streamwise fluctuating velocities are compared against the experimentalfindings in figure 5.7d and it can be seen that a fairly good agreement is felt along varioussections of the backward-facing step geometry.

From the comparisons of the GP flows, it is worth while to note, that mean velocities ofthe particles are lower at x/h=2 at the entry of the step than the gas phase, this is similar to thefully developed channel flow before the step, as reported in the experiments of Kulick et al.(1994), wherein the particles at the channel centreline exhibit lower streamwise velocitiesthan that of the fluid as a result of cross-stream mixing. However, the gas velocity lags behindthe particles aft of the step as particle inertia is slower to respond to the adverse pressuregradient than that of the fluid. At the fluctuation level, the particles exhibit higher values thanthe gas which again is attributed to the cross-stream mixing (Kulick et al., 1994).


5.4.1. Particle Response- Mean Velocity Level

With the numerical code apt to replicate the experimental findings of the GP and LP flows,we now embark on a mission to study the response of the particles to the surrounding carrierphases. In order to proceed with this endeavour, four different Stokes numbers, by invariablychanging the particle response time have been chosen for both the GP and the LP flows. For thisnumerical experiment the step geometry of Fessler and Eaton (1995) has been adopted, whilethe inlet velocity for the carrier and the particulate phase corresponds to that of the experimentalconditions of Founti and Klipfel (1998). The four different Stokes number chosen correspondsto 0.05, 0.5, 2.0 and 6.0. Particles with Stokes number 0.05 acts as tracers to the carrier phaseand are widely used by experimentalists for PIV/LDA study. The Stokes number of 0.5corresponds to unveil realistically the response of particles for St≤1. Stokes number of 2.0 waschosen such as to fall within the range of overshoot phenomena (Chein and Chung, 1987;Hishida et al., 1992; Ishima et al., 1993), wherein the particles is said to disperse more readilythan the fluid, while higher Stokes number of 6.0 is to study the independent responsitivity ofthe particles in relation to the two different carrier phases namely the air and diesel oil. In orderto represent the results more qualitatively 12 interrogation points made up of a matrix of threesections (x/h=2, 7 & 14) along the length of the step and four sections along the height of thestep (y/h=0.5, 1.0, 1.5, 2.0) have been considered.


up/Uo

y/h

0 0.2 0.4 0.6 0.80.00

0.50

1.00

1.50

2.00

2.50

St=0.05St=0.50St=2.00St=6.00

Figure 5.8.a. Mean streamwise particle velocities for varying Stokes number for LP Flows.

TKP/Uo2

y/h

0 0.005 0.01 0.015 0.02 0.025 0.030.00

0.50

1.00

1.50

2.00

2.50 St=0.05St=0.50St=2.00St=6.00

Figure 5.8.b. Fluctuating streamwise particle velocities for varying Stokes number for GP flows.

Figure 5.8a shows the mean streamwise particle velocities along a section for the LP flows forvarying Stokes number, the monitoring stations along the height of the step has been depictedwith the help lines running with constant y/h values. Figure 5.8b also shows the monitoring


stations of particle fluctuation for the GP flows along varying Stokes number. From both thefigures shown here, the mean velocities as well as the fluctuations increase with acorresponding increase in Stokes number for LP and GP flow respectively.

0.20

0.40

0.60

0.80

1.00

1.20

0.00 1.00 2.00 3.00 4.00 5.00 6.00

x/h=2; y/h=0.5

St

up/U

o

Figure 5.9.a. Mean streamwise particle velocities for varying Stokes number along the height of thestep for x/h=2 & y/h=0.5.

0.20

0.40

0.60

0.80

1.00

1.20

0.00 1.00 2.00 3.00 4.00 5.00 6.00

x/h=2; y/h=1.0

St

up/U

o

Figure 5.9.b. Mean streamwise particle velocities for varying Stokes number along the height of thestep for x/h=2 & y/h=1.0.


0.20

0.40

0.60

0.80

1.00

1.20

0.00 1.00 2.00 3.00 4.00 5.00 6.00

x/h=2; y/h=1.5

St

up/U

o

Figure 5.9.c. Mean streamwise particle velocities for varying Stokes number along the height of thestep for x/h=2 & & y/h=1.5.

0.20

0.40

0.60

0.80

1.00

1.20

0.00 1.00 2.00 3.00 4.00 5.00 6.00

x/h=2; y/h=2.0

St

up/U

o

Figure 5.9.d. Mean streamwise particle velocities for varying Stokes number along the height of thestep for x/h=2 & y/h=2.0.

In this part of the results and discussion section the response of the particles to the meanflow with two different carrier mediums are investigated in relation to varying Stokesnumber. Three sections along the step and four along the height of the step have beeninvestigated. Figures 5.9a-d shows the plot of the normalized mean particle velocities forx/h=2 section along varying Stokes number across the step height. The solid lines with circlesin the figure depict the particle behaviour in GP flow, while the broken lines with squaresdepict again the particle behaviour but in LP flow. It can be seen from the figures that themean particle velocities increase with a corresponding increase in its Stokes number for both


the carrier phases with the maximum particulate velocity occurring for the LP flows as oneprogresses along the height. There is along a steady increase in the particulate velocities alongthe height of the step for up to y/h=1.5 after which there is a meagre drop for y/h=2.0 wherein the wall boundary conditions try to retard the flow. The particle velocities for the GP tendto overtake its counterpart LP flow at section y/h=2.0 but however not for the maximumStokes number considered in our study.

0.30

0.40

0.50

0.60

0.70

0.80

0.00 1.00 2.00 3.00 4.00 5.00 6.00

x/h=7; y/h=0.5

St

up/U

o


0.30

0.40

0.50

0.60

0.70

0.80

0.00 1.00 2.00 3.00 4.00 5.00 6.00

x/h=7; y/h=1.0

St

up/U

o


u p/U

o


0.30

0.40

0.50

0.60

0.70

0.80

0.00 1.00 2.00 3.00 4.00 5.00 6.00

x/h=7; y/h=1.5

St

up/U

o

Figure 5.10.c. Mean streamwise particle velocities for varying Stokes number along the height of thestep for x/h=7 & y/h=1.5.

0.30

0.40

0.50

0.60

0.70

0.80

0.00 1.00 2.00 3.00 4.00 5.00 6.00

x/h=7; y/h=2.0

St

up/U

o


Figures 5.10a-d shows the mean particulate velocities at section x/h=7 along the height ofthe step geometry, it can also be seen here that the velocities keep increasing for a height ofabout y/h=1.5 after which there is a small drop due to the wall interference. However in thissection of the step geometry, it can seen that the particulate velocities for the GP flows mostof the time exceed than that of the particles in the LP except for section y/h=0.5. This isattributed to the fact that the GP flow particles are free to move in a less restricted


environment like air while particles in the LP flow move in a highly viscous environment,which fundamentally restricts it motion.

0.40

0.50

0.60

0.70

0.00 1.00 2.00 3.00 4.00 5.00 6.00

x/h=14; y/h=0.5

St

up/U

o


0.40

0.50

0.60

0.70

0.00 1.00 2.00 3.00 4.00 5.00 6.00

x/h=14; y/h=1.0

St

up/U

o



0.40

0.50

0.60

0.70

0.00 1.00 2.00 3.00 4.00 5.00 6.00

x/h=14; y/h=1.5

St

up/U

o

Figure 5.11.c. Mean streamwise particle velocities for varying Stokes number along the height of thestep for x/h=14 & y/h=1.5.

0.40

0.50

0.60

0.70

0.00 1.00 2.00 3.00 4.00 5.00 6.00

x/h=14; y/h=2.0

St

up/U

o


Figures 5.11a-d shows the plot of particle velocities for the section x/h=14 along the step,here also it can be seen that similar trend corresponding to two previous sections along thestep is felt, with the increase in the particle velocities along the height and also with theparticles in GP flow exceeding than that of the LP flow. For all the three x/h sectionsconsidered along the length of the step it can be seen that there is decrease in the particlevelocities, which is attributed to the fact that particle losing their momentum as they travelalong the step of the geometry.


5.4.2. Particle Response-Turbulence Level

In this part of the results and discussion section, particles turbulent behaviour isinvestigated for two variants of carrier phase flows namely the GP and the LP flows. Hereagain three sections along the length of the step and four along its height have beenconsidered. Figures 5.12a-d shows the turbulent kinetic energy of the particles normalized byits free stream velocity at section x/h=2 for four different heights and it can be seen that forall sections considered there has been an increase in the particle kinetic energy with asubsequent increase in the Stokes number for the GP flows but however for the LP flowsthere has been a decrease with subsequent increase in Stokes number. It can also be seenalong the height of the step a decrease in the particulate kinetic energy is pronounced for bothGP as well as LP flows.

Figures 5.13a-d shows the particulate turbulent kinetic energy for the section x/h=7 of thestep. While there have been a steady increase in the kinetic energy with a rise in the Stokesnumber for the GP flows, they seem to work in the opposite fashion for the LP flows whichshow a decrease with increase in Stokes number. It can also be seen that the magnitude of thekinetic energy is a fold less than at the section x/h=2 and that also a decrease is felt with theincrease in the height of the step for the corresponding Stokes number. Figures 5.14a-d alsoshows a similar pattern for the section x/h=14 at the farther end near the exit of the geometry.The magnitude of the particulate turbulent kinetic energy is lesser than the previous section ofx/h=7 and x/h=2. However, the turbulent kinetic energy of the particles seems to decreasewith the subsequent increase in the Stokes number for LP flows.

0.000

0.010

0.020

0.030

0.040

0.050

0.00 1.00 2.00 3.00 4.00 5.00 6.00

x/h=2; y/h=0.5

TKP/

Uo2

Figure 5.12.a. Fluctuating streamwise particle velocities for varying Stokes number along the height ofthe step for x/h=2 & y/h=0.5.

St


0.000

0.010

0.020

0.030

0.040

0.050

0.00 1.00 2.00 3.00 4.00 5.00 6.00

x/h=2; y/h=1.0

TKP/

Uo2

Figure 5.12.b. Fluctuating streamwise particle velocities for varying Stokes number along the height ofthe step for x/h=2 & y/h=1.0.

0.000

0.010

0.020

0.030

0.040

0.050

0.00 1.00 2.00 3.00 4.00 5.00 6.00

x/h=2; y/h=1.5

TKP/

Uo2

Figure 5.12.c. Fluctuating streamwise particle velocities for varying Stokes number along the height ofthe step for x/h=2 & y/h=1.5.

St

St


0.000

0.010

0.020

0.030

0.040

0.050

0.00 1.00 2.00 3.00 4.00 5.00 6.00

x/h=2; y/h=2.0

TKP/

Uo2

Figure 5.12.d. Fluctuating streamwise particle velocities for varying Stokes number along the height ofthe step for x/h=2 & y/h=2.0.

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.00 1.00 2.00 3.00 4.00 5.00 6.00

x/h=7; y/h=0.5

TKP/

Uo2


St

St


0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.00 1.00 2.00 3.00 4.00 5.00 6.00

x/h=7; y/h=1.0

TKP/

Uo2


0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.00 1.00 2.00 3.00 4.00 5.00 6.00

x/h=7; y/h=1.5

TKP/

Uo2


St

St


0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.00 1.00 2.00 3.00 4.00 5.00 6.00

x/h=7; y/h=2.0

St

TKP/

Uo2


0.000

0.005

0.010

0.015

0.020

0.00 1.00 2.00 3.00 4.00 5.00 6.00

x/h=14; y/h=0.5

TKP/

Uo2


St

St


0.000

0.005

0.010

0.015

0.020

0.00 1.00 2.00 3.00 4.00 5.00 6.00

x/h=14; y/h=1.0

TKP/

Uo2


0.000

0.005

0.010

0.015

0.020

0.00 1.00 2.00 3.00 4.00 5.00 6.00

x/h=14; y/h=1.5

TKP/

Uo2


St

St


0.000

0.005

0.010

0.015

0.020

0.00 1.00 2.00 3.00 4.00 5.00 6.00

x/h=14; y/h=2.0

TKP/

Uo2


5.4.3. Summary of Particulate Responsitivity

The above two sections which outlined the particle response at the mean velocity leveland at the turbulence level shows that the mean velocity of the particles increase with asubsequent increase in the Stokes number for both the carrier phases namely the gas and theliquid. The mean particulate velocity not only increase with the Stokes number but is alsohigher than its corresponding carrier phase velocities for the three Stokes number viz St=0.5,2.0 & 6.0 considered in our study. This is quite in lines with the recent experimental data ofIshima et al (2007) and the phenomenon is explained with the help of the particle terminalvelocity which gives a rough approximation as a percentage of how much the particle velocityexceeds the carrier phase velocity. The other reason for the particle velocity to lead the carrierphase is the attribute of the particulate phase to respond slowly to the adverse pressuregradient dominant in shear flow geometries like back-ward facing step, in lieu to the carrierphase.

The particle turbulent kinetic energy plots for both the GP and the LP flows depict theresponse of the particles at the turbulence level, across these plots it can be summarized thatwhile there is a steady increase in the particulate turbulence for the GP flows with successiveincrease in Stokes number, with some sections showing even a 100% increase between theminimum and the maximum Stokes number considered. However, for the LP flows, themagnitude of the increase in the particulate turbulence across the increasing Stokes number isnot as characteristic as its counterpart. Across the same sections for LP flows the majority ofthe trend shows a decrease after which they more of less remain a constant. Previous studiesof liquid-particle flows in vertical channel (Ishima et al., 2007; Borowsky & Wei, 2007)

St


shows that with the increase in the Stokes number there is usually a corresponding increase inthe particulate turbulence, however the flow considered in this study is a shear flow geometry,which basically depicts a totally different flow feature unlike the simple channel geometry.

In these lines even the Turbulence Modulation (TM) of the shear flow geometry for thewell established gas-particle flows using the same set of experimental data employed in thisstudy does not seem to well correspond with the models been employed and formulated forthe vertical channel flows (Mohanrangam & Tu, 2007). Thereby, given the complexity of theproblem much deeper understanding and experimental data may be required to ascertain thesame. The readers are also advised that the two sets of experimental data (Fessler & Eaton,1997; Founti & Klipfel, 1998) used in his study were not alike in all respects within the flowfield with parity only in carrier phases. However, for the current set of results obtained theparticle response to the turbulence for the LP flow in comparison to the GP flow may beexplained in terms of the carrier phase employed to study the particle response. Firstly, thedensity and the viscosity used to study LP flow are approximately 709 and 257 times higherthan that of the GP flow, which basically prohibits the fluctuating motion of the particle.Another rationale being, in regions of strong mean velocity gradient, the streamwise particlefluctuating velocities are determined more by the mean gradient than by the actual responseof the particles to turbulent fluctuations. In the absence of the same the particle velocityfluctuations tend to be lower than the fluid velocity fluctuations as noted from theexperiments of Fessler and Eaton (1995), it was also stated by the same authors that in theirexperiments wall-normal fluctuating particle velocities were lower that fluid, which wasattributed to the large inertia of the particles making them unresponsive to many of the fluidmotions. From these conclusions the eventual decrease in the particle fluctuation is more orless attributed to the decrease in the velocity gradient with a corresponding increase in Stokesnumber. The cross-stream mixing, which attributes towards higher particle fluctuatingvelocities in GP flow may be prohibitive in LP flow considering the elevated density andviscosity of LP flows.

Air-Liquid Flows

In the modelling of micro-bubble laden Air_liquid flows, two sets of governing equationsfor momentum were solved. The generic CFD code ANSYS CFX 11 (ANSYS, 2006) wasemployed as a platform for two-fluid flow computation. The built-in Inhomogeneous andMUSIG models had been adopted for our numerical simulations. Figure 6.1a shows theschematic diagram of the numerical model used in our computations. Numerical simulationswere performed with a velocity inlet and a pressure outlet, on the left and right side of the 2Dcomputational domain respectively. The top wall is modelled as a friction free boundarycondition, wherein the height of the computational domain reflects only half the height of theoriginal test section. The bottom part of the domain has been divided into three distinctsections, section 1 & 3 were modelled as no-slip walls for liquid and free-slip for micro-bubbles, emulating the experimental boundary conditions. The section 2 is specified as theinlet boundary condition for our gas inlet imitating the experimental conditions of gasinjection thought the porous plate.


Traction-free opening

VelocityInlet

x

y 0.057m

Pressureoutlet

0.280m 0.178m 0.254m

Section 1 Section 2 Section 3

Figure 6.1.a. Schematic diagram of the numerical model.

A uniform liquid velocity was specified at the inlet of the test section, different gas flowrates were specified along the section 2 of the computational domain, the free streamvelocities and the gas injection rates used in the simulations are summarized in Table 1. Anarea permeability of 0.3 which lies in line with the sintered metal used in the experiments andalso employed in the numerical work of Kunz et al. (2003) is used all along section 2 for gasinjection purposes. At the outlet, a relative averaged static pressure of zero was specified. Forall flow conditions, reliable convergence were achieved within 2500 iterations when the RMS(root mean square) pressure residual dropped below 1.0×10-7. A fixed physical time scale of0.002s is adopted for all steady state simulations.

Table 6.1. Input boundary conditions for the computational model

Case Air flow rate Qa(m3/s) Water free streamvelocity (m/s)

ReL based on the totalplate length

Q0-V9.6 (Cfo) 0 9.6 7.66 x 106

Q1-V9.6 0.001 9.6 7.66 x 106

Q2-V9.6 0.0015 9.6 7.66 x 106

Q3-V9.6 0.002 9.6 7.66 x 106

Q4-V9.6 0.0025 9.6 7.66 x 106

Q5-V9.6 0.003 9.6 7.66 x 106

Q0-14.2(Cfo) 0 14.2 1.13 x 107

Q1-V14.2 0.001 14.2 1.13 x 107

Q2-V14.2 0.0015 14.2 1.13 x 107

Q3-V14.2 0.002 14.2 1.13 x 107

Q4-V14.2 0.0025 14.2 1.13 x 107

Q5-V14.2 0.003 14.2 1.13 x 107

Section 1 Section 2 Section 3

Figure 6.1.b. Computational grid used for computations.


In handling turbulent micro-bubble flow, unlike single phase fluid flow problem, nostandard turbulence model has been customized for two-phase (liquid-air) flow. Nevertheless,numerical investigation revealed that standard k-ε model predicted an unrealistically high gasvoid fraction peak close to wall (Frank et al., 2004, Cheung et al., 2006). The k-ω based ShearStress Transport (SST) model by Menter (1994) provided more realistic prediction of voidfraction close to wall.

0.00

5.00

10.00

15.00

20.00

25.00

30.00

1 10 100 1000 10000 100000

U+=y+; U+ = (1/k)Ln(Ey+)Numerical

y+

U+

Figure 6.2. Comparison of simulated boundary layer velocity profile with the standard law of curves forU∞=14.2m/s.

The SST model is a hybrid version of the k-ε and k-ω models with a specific blendingfunction. Instead of using empirical wall function to bridge the wall and the far-awayturbulent flow, the k-ω model solves the two turbulence scalars right up to the wall boundary.This approach eliminates errors arising from empirical wall function and thus provides betterprediction at the near wall region. The SST model is thereby employed in the present study.Moreover, to account for the effect of bubbles on liquid turbulence, the Sato’s bubble-inducedturbulent viscosity model (Sato et al., 1981) has been adopted as well.

Figure 6.1b shows the mesh distribution within the computational model, wherein a non-uniform orthogonal mesh with 151x101 grid points was spanned over the whole computationdomain. Wall-normal clustering was used in order to resolve the boundary layer and theheight of the two closest cells next to the walls was designed to be y+≤1, the height of whichcan be approximated from the logarithmic law


κυκ

κυκ

τ

τ

τ

1ln1

1ln1

−+⎟⎟⎠

⎞⎜⎜⎝

⎛=

−+⎟⎠⎞

⎜⎝⎛=

∞

∞

∞

BUuLU

BLu

uU

Using the relation2/1

2⎟⎟⎠

⎞⎜⎜⎝

⎛=∞

fCuU

τ

(3.28)

and substituting it to the above equation, we get

κκ1

2Reln12

2/12/1

−+⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟

⎟⎠

⎞⎜⎜⎝

⎛∞ B

CC

f

f

(3.29)

The skin friction co-efficient (Cf) can be solved from the above equation (3.29) bysubstituting values of B=5.0 and κ=0.41. uτ can be solved from the equation (3.28) and thedistance of the first grid point from the wall can be solved from the equation

τ

υuyy first

+

=


In order to better understand the drag reduction phenomenon and to better comprehendthe various mechanisms behind it, a prognostic approach has been carried out in throughoutthis section starting from the single phase and then to the air-liquid micro-bubble flows,wherein at each juncture our numerical outcomes have been verified and validated againstwell established experimental and numerical findings.

Before investigating the physical phenomenon of the drag reduction, it is very crucial toascertain adequate grid spacing has been generated to resolve the inner, buffer and the outerboundary layer. Figure 6.2 shows the single phase boundary layer velocity profile of theoutlet compared against the standard law of the wall curves for the maximum free streamvelocity of 14.2 m/s. The excellent agreement with the standard curve clearly confirmed thatsufficient mesh resolution has been spanned throughout the boundary layer to capture theassociated velocity gradient. The 9.6 m/s case had the similar y+ values and hence been notshown here for brevity.


6.1.1. Experimental Validation (Inhomogeneous Model)

With the mesh spacing sufficiently fine enough to resolve the turbulent boundary layerattention is then focussed on the micro-bubbles drag reduction mechanism. Figure 6.3 showsthe comparison of the predicted skin friction ratios of the two-fluid inhomogeneous modelagainst the experimental data of Madavan et al. (1984) along various gas injection rates (Q1-Q5) for both the freestream velocities of 9.6 and 14.2m/. Herein, Cf & Cfo are the skin-frictionco-efficients with and without the gas injection respectively. The skin-friction co-efficientthroughout our numerical study have been obtained by averaging out the entire flat plate of‘section 3’. It can be observed from the figure that satisfactory agreement was obtained usingthe inhomogeneous two-fluid model for both the free stream velocities considered in ourstudy.

0.50

0.60

0.70

0.80

0.90

1.00

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035

14.2m/s (Expt)14.2m/s (Num) 9.6m/s (Expt) 9.6m/s (Num)

Cf/C

fo

Qa

Figure 6.3. Comparison of skin friction co-efficient with the experimental findings for U∞=14.2m/s &9.6m/s.

Readers are advised that these results have been compared treating that the experimentalerrors encountered during various process of measurements are nil (error percentages of theexperimental data). However, hypothetical they may seem, the numerical predictions were inremarkable agreement with the measurements. It can be seen that there is always an increasein the drag reduction (DR) or a decrease in the Cf/Cfo ratio with a corresponding increase inthe gas flow rates for both the free stream velocities considered in our simulation, whichagain is congruent to the previous micro-bubble studies. In all the simulations presented,buoyancy was included as a necessary force, as one could find a marked change from theexperimental results with the plate on ‘Top’ and ‘Bottom’ configuration.


0.000

0.005

0.010

0.015

0.020

0.025

6.00 8.00 10.00 12.00 14.00 16.00

Streamwise liquid velocity (m/s)

Y (m

)

Q4-V14.2

Q1-V14.2Q2-V14.2Q3-V14.2

Q5-V14.2

Figure 6.4.a. Velocity profiles for varying gas injection rates for free stream velocity U∞ = 14.2m/s.

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1 10 100 1000

Q4-V14.2

Q1-V14.2Q2-V14.2Q3-V14.2

Q5-V14.2

ul/u

ul

y+

Figure 6.4.b. Change in the mean flow velocity for the carrier phase along the boundary layer for U∞ =14.2m/s.


6.1.2. Investigation of Mechanisms of Drag Reduction

With the skin friction co-efficients showing fairly good comparison for the two-fluidinhomogeneous model, it can be further investigated to study the various mechanisms of dragreduction. To begin with the mean streamwise velocities of the carrier phase are scrutinised.Figure 6.4a shows the mean streamwise liquid velocity profiles along varying gas injectionrates for the high Reynolds number case of 14.2m/s. As depicted, a clearly marked change inthe velocity profile can be seen with a subsequent increase in the gas flow injection rates,which is certainly in relation to the large amount of micro-bubbles present along the boundarylayer as shown from the void fraction profiles in 6.7b. The streamwise velocities reveal thatwith the increase in the gas flow rates, there is a subsequent and a gradual increase in thestreamwise velocities in the outer layer. In contrast, a close investigation of the velocitieswithin the buffer and inner layer shows an opposite phenomenon. In figure 6.4b, the velocityprofiles of the liquid phase for the micro-bubbles laden flows have been normalized with thecorresponding single phase liquid velocities along the length of the boundary layer, where byany velocity change felt in the carrier liquid phase is reflected as an exit of the ratio fromunity, from the figure it can be revealed, that there is a marked decrease in the meanstreamwise velocities with a subsequent increase of the gas injection rates. It can also be seen,that the flow undergoes a maximum reduction in the mean velocity of about 40% for thehighest gas flow rate, while it is quite nominal and about 13% for the lowest gas injectionrate, this trend keeps increasing until a y+ value of 100, aftermath of which there is a spike forthe largest of the three gas flow rates and then a downward trend follows.

0.00

0.01

0.02

0.03

0.04

0.05

0.06

1 10 100 1000

Nor

mal

liqu

id v

eloc

ity (m

/s)

y+

Q4-V14.2

Q1-V14.2Q2-V14.2Q3-V14.2

Q5-V14.2

Q0-V14.2

Figure 6.4.c. Liquid normal velocity for the carrier phase along the boundary layer for U∞ = 14.2m/s.


These findings reported above are in lines with the DNS findings of Ferrante andElghobashi (2004), wherein the presence of micro-bubbles in the turbulent boundary layerresults in a local positive divergence of the fluid velocity, 0U >•∇ , creating a positivemean velocity normal to (and away from) the wall which in turn, reduces the meanstreamwise velocity and displaces the quasi-streamwise longitudinal vortical structures awayfrom the wall. The shifting of the vortical structures away from the wall indicates that the‘sweep’ and ‘ejection’ events (Robinson, 1991), which are located respectively at thedownward and upward sides of these longitudinal vortical structures, are moved farther awayfrom the wall, thereby reducing the intensity of wall streaks along the wall and consequentlydecreasing the skin-friction. It was also reported that there is shift with respect to the locationof peak Reynolds stress production away from the wall, thus reducing the production rate ofturbulence kinetic energy and enstrophy.

Figure 6.4.d. Air void fraction contour plot for Q5-V14.2.

Figure 6.4c shows the plot of water normal velocity through varying gas injection rates, itcan be seen that there is generally an increasing trend in the velocities and then a decreasewhich is followed by a maximum peak, and this is in direct relation to the loss incurred by theflow along the streamwise direction, across varying gas injection rates. There is also a suddenspike in the normal velocities within a y+ range of 60-120. However, the onset of the increaseand the occurrence of the maximum differ in accordance to the gas injection rates. For the

Outlet

Water flow U∞

x

y


three higher injection rates (Q3-Q5) the location of the start of sudden increase and theoccurrence of maximum seems to occur more or less in unison, but their magnitude ofmaximum normal velocities differ wide apart. While for the lower gas injection rates (Q1 &Q2) the location and the magnitude are more distinct and separated wide apart. It can also beseen that the unladen wall normal velocity is quite smaller in lieu with the laden normalvelocities. This can be further confirmed from the contour plot of the air void fraction alongthe boundary layer as shown in figure 6.4d, for the highest air flow rate considered in ourstudy, where there is a small layer of water, which is immediately followed on the top by athick layer of air and then followed again by water, herein due to the inherent presence of themicro-bubbles in the middle section, caused an upward shift in the water normal velocities.

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

40.00

1 10 100 1000 10000 100000y+

U+

U+=y+; U+ = (1/k)ln(Ey+)

Q4-V14.2

Q1-V14.2Q2-V14.2Q3-V14.2

Q5-V14.2

Figure 6.5. Change in the boundary layer for varying gas flow rates for U∞ = 14.2m/s.

Figure 6.5 shows the plot of non-dimensional streamwise velocity profiles along theboundary layer for varying gas flow rates at the middle of ‘section 3’. The presence of themicro bubbles can be felt for a y+≥10, where in there is a gradual thickening of the viscouszone with an upward shift of the logarithmic region, while the inner layer seems more or lessunaltered. With these findings, it can be ascertained that the important aspect in achievingdrag reductions is the accumulation of the micro bubbles within a critical zone in the bufferlayer. This is in lieu with the experimental findings of Villafuerte & Hassan (2006), wherebyhigh drag reductions were reported due the accumulation of the micro bubbles within a rangeof 15 ≥ y+ ≤ 30.

6.1.3. Turbulence Modulation (TM)

The plot depicted in the figure 6.6 demonstrates the turbulent modulation (TM) of theliquid phase in the presence of the micro-bubbles and is given by the ratio of the micro-


bubble laden flow r.m.s streamwise velocity to the unladen r.m.s streamwise velocity. Theseplots signify that any TM felt in the carrier liquid phase is reflected as an exit of the ratiofrom unity. It can be seen from the plot, across various gas injection rates a markedattenuation is felt up to a distance along the boundary layer and then a subsequent increase,which is attributed towards the turbulence enhancement of the liquid phase. It is alsoworthwhile to note that the flow has a tendency to attenuate more for higher gas flow rates.On the other hand there is a turbulence augmentation effect pronounced more in the outerlayer of the boundary. The marked attenuation felt for a small distance from the wall isattributed to the presence of a thin lining of liquid all along the wall (as explained above).However, in order to explain the augmentation of the turbulence felt within the boundary, the“bubble-repelling” and the “bubble-rising” events observed from the experiments of Y.Muraiet al. (2006) is used. From their findings, it is outlined that the vertical rise velocity of thebubble or the “bubble-rising” event towards the wall is only about 5% of the streamwisevelocity, after which the bubble reaches equilibrium with its surroundings and starts itsjourney back through the “bubble-repelling” event away from the wall, but however thisdownward journey accounts for 25% of the streamwise velocity.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

2.00

1.00 10.00 100.00 1000.00y+

u'/u

' l

Q4-V14.2

Q1-V14.2Q2-V14.2Q3-V14.2

Q5-V14.2

Figure 6.6. Turbulence Modulation (TM) along the boundary layer for U∞ = 14.2m/s.

Although the aforementioned experimental observations refer to individual bubblemotion which can only be tracked numerically using Lagrangian approach, the phenomenonof turbulence augmentation in the carrier phase is taken care in our simulation through theSATO (Sato et al; 1981) model, which accounts for the additional viscosity generated throughthe bubble slip velocity, wherein the vortices are formed behind the bubbles by their motion,thereby causing an increase in the turbulence levels in the outer layer of the boundary. It can


also be seen that this turbulence enhancement is more pronounced in the outer layer of theboundary, while most part of the inner and the buffer layer experiences attenuation.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.00 500.00 1000.00 1500.00 2000.00

Volu

me

fract

ion

of a

ir

Q4-V9.6

Q1-V9.6

Q2-V9.6

Q3-V9.6

Q5-V9.6

y+

Figure 6.7.a. Volume fraction of air along the outlet for U∞ = 9.6m/s.

Figure 6.7a shows the void fraction profiles for the dispersed phase for the freestreamvelocity of 9.6m/s, along the outlet plane of the geometry, it can be seen that there is a sharpincrease in the void fraction for a y+ value of about 200 for the maximum flow rate, with themaximum occurring there, where as from figure 6.7b, it can be seen that the void fractionprofiles for the dispersed phase occurring in a slightly different pattern with the maximumoccurring at a distance around y+ =150. This occurrence of maximum void fraction can bebest used to explain the degree of drag reduction between the two Reynolds numbersconsidered in our study. For the 9.6m/s case where the maximum void fraction occurs at adistance higher than that of the 14.2m/s, a higher degree of drag reduction is seen and this isattributed to the fact that the bubbles can rise and distribute themselves within the boundarylayer there by expanding the boundary layer (buffer and outer layer) and thus causing agreater drag reduction, but however for the high Reynolds number 14.2m/s case the bubbleshave lower residence time to re-distribute themselves within the boundary layer there bycausing a smaller drag reduction and consequently a higher Cf/Cfo value (Kodama et al. 2000).From figures 6.7a and 6.7b, it can be seen that with the increase in the gas flow rates the voidfraction of the injected air increases for all the cases and later flatten out due to the dispersionand also the washing down effect of the continuous phase. From the close examination of thegraphs it can be seen that there exists a thin film of liquid covering all over the wallirrespective of the injected gas flow rates, these finding are in accordance to the experimentalfindings of Murai et al. (2007), who later concluded that this liquid film thickness graduallybecomes thinner from the front to the rear part of the bubble such that the skin friction variesalong the coordinate. From the work of Tinse et al. (2003) it can be deduced that thinner this


liquid film becomes the lower the shear stress the film has. This has also been verified fromour numerical findings that the skin friction ratio decreases with the increase in the gas flowdue to the inherent thinning of the liquid film adjacent to the wall.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.00 500.00 1000.00 1500.00 2000.00 2500.00

Q4-V14.2

Q1-V14.2Q2-V14.2Q3-V14.2

Q5-V14.2

y+

Volu

me

fract

ion

Figure 6.7.b. Volume fraction of air along the outlet for U∞ = 14.2m/s.

6.1.3. Effect of Bubble Coalescence and Break-up in Drag Reduction

In the light of the aforementioned results, one should notice that a prescribed bubblediameter has been specified throughout all numerical simulations for Inhomogeneous two-fluid model. Although encouraging results have been presented in previous sections, the valueof bubble diameter (i.e. 500 µm) unfortunately at best served as a fair engineering estimationwhich is calibrated against experimental data based on trail-and-error without solid physicalinterpretations. As discussed before, the constant diameter assumption may introducenumerical errors if the bubble coalescence and break-up become dominant in the problem,especially when the air injection rate is considerably high. In attempting to overcome thisproblem, the MUSIG model is introduced into the simulation allowing bubble diameter to beevaluated mechanistically using the coalescence and breakage kernels.

Figure 6.8a shows the numerical comparison against its experimental counterpart for afree stream velocity of 9.6m/s, the MUSIG model in addition to Inhomogeneous model isused for comparison. Herein the MUSIG model had been specified 10 groups of bubbles,diameters ranging from 100µm-1000µm. As depicted in the figure, at high flow rates (i.e. Q4and Q5), MUSIG model gives the best agreement while the inhomogeneous model tends toslightly over-predict the skin friction coefficients. However, using the default coefficients for


break-up and coalescence models serious under-prediction has been observed for the MUSIGmodel for low gas flow rates (i.e. Q1-Q3). Figure 6.8b shows the similar comparison ofexperimental data with higher free-stream velocity of 14.2 m/s. Analogy to the previousresult, predictions of the MUSIG model appear marginally superior to the inhomogeneous athigh gas injection rates (i.e. Q4 & Q5), while considerably under predictions have shown forlower gas injection rates.

0.50

0.60

0.70

0.80

0.90

1.00

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040

Expt

Two-Fluid-Inhomogenous

MUSIG

Cf/C

fo

Flow rate of Air (Qa)

9.6m/s

Figure 6.8.a. Comparison of computed skin-friction co-efficient Inhomogeneous & MUSIG modelsU∞= 9.6m/s.

U=14.2m/s

0.5

0.6

0.7

0.8

0.9

1.0

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035

Expt

Two-Fluid-Inhomogenous

MUSIG


Cf/C

fo

Figure 6.8.b. Comparison of computed plate drag co-efficient Inhomogeneous & MUSIG models U∞=14.2m/s.


One possible reason attributed to the under-prediction of the MUSIG model for low gasinjection rates could be the over-estimation of bubble break-up rate which sequentiallyintroduced more small bubbles into calculations. These additional small bubbles were therebydispersed with the boundary layer caused greater drag reduction on the surface. It should beemphasized that default model parameters of the MUSIG have been adopted directly in theabove numerical investigation. These parameters were calibrated with bubbly flow conditionwhere isotopic turbulence was assumed. At high air injection rate, such assumption may bequite close the physical behaviour as higher turbulence modulation has been introduced bythe presence of bubbles. However, it may become invalid for low flow rates. In essence, it iswell known that these model parameters may vary from cases to cases which should be re-calibrated for particular flow condition.

0.50

0.60

0.70

0.80

0.90

1.00

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040

Expt

MUSIG Breakup co-efficient=1.0

MUSIG Breakup co-efficient=0.05

Cf/C

fo


9.6m/s

Figure 6.9.a. Comparison of skin friction co-efficients with different break-up co-efficients for U∞=9.6m/s.

Based on the above argument, also serves as a confirmation of the above observations,another set of simulations have been carried out with the break up coefficient deliberatelydecreased 20 times to minimize the resultant bubble break-up rate. Figure 6.9a shows thecorresponding predictions of the skin-friction coefficient plots for the free stream velocity of9.6m/s. Compared with the default MUSIG model, the predicted results were generally insatisfactory agreement with measurements across varying flow rates, while a considerablyimprovements have been obtained for the lower air injection rates. Similar observations canalso be found for the freestream velocity of 14.2m/s showing in Figure 6.9b.

The effect of the reduced break-up rate can be exemplified by a closer visualization of thepredicted bubble size distributions of the two numerical results. Figure 6.10a shows thepredicted bubble size distribution at the outlet obtained from the default and the re-calibratedMUSIG models. With the default break-up coefficient, in both free-stream velocities, therelatively high volume fraction of small bubble clearly demonstrated that the default model


tends to create additional small bubble via the dominating break-up mechanism. In contrast,by limiting the break-up rate, bubble coalescence overcomes the break-up mode formingrelatively higher volume fraction for the larger bubbles.

0.5

0.6

0.7

0.8

0.9

1.0

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040

Expt

MUSIG breakup co-efficient=1.0



Cf/C

fo

14.2 m/s

Figure 6.9.b. Comparison of skin friction co-efficients with different break-up co-efficients for U∞=14.2m/s.

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

6.0%

7.0%

8.0%

100 200 300 400 500 600 700 800 900 1000



d (µm)

Volu

me

fract

ion

Figure 6.10.a. Bubble diameter distribution function for Q1-V9.6.


In the present study, the deduction of break-up coefficient arose only as an engineeringestimation. In fact, as the air injection rates are considerably low, interactions betweenbubbles are relatively insignificant compared with that for the high injection rates. It isthereby unsurprising to re-calibrate the model constants for obtaining a “better” comparison.Although revealing the short-comings of the current kernels is certainly one of the findings,this drawback of the model should be circumvented by refining the model assumption and themechanism which unfortunately is left far beyond the focus of this paper.

Directing back to the theme of current work, one could easily state that good predictionsof the skin-friction coefficients can be obtained by specifying a proper bubble size for eachsimulation. Nevertheless, one should also be reminded that bubble sizes may changesignificantly which is impossible to be represented by a fixed average value. This problem isfurther exacerbated; if rigorous bubble interactions are involved at high air injection rate. Inpractical micro-bubble problems, it is more easily to acquire the range of bubble size ratherthan exact bubble diameter. The MUSIG model which tailored to resolve the bubble sizedistribution mechanistically within a given range of bubble size appeared as the bestcandidate to resolve the physics embedded in micro-bubble drag reduction problems.

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

6.0%

7.0%

8.0%

9.0%

10.0%

100 200 300 400 500 600 700 800 900 1000



d (µm)

Volu

me

fract

ion

Figure 6.10.b. Bubble diameter distribution function for Q2-V14.2.

Conclusion

In this article outlined above, a lot of work was undertaken numerically to study thebehavior of two-phase turbulent flows of varying density regimes viz., Gas-Particle, Liquid-Particle and Liquid-Air flows. In addition to the carrier and the dispersed phases mean andturbulent behavior, Turbulence Modulation (TM) has also been investigated. It is given by the


change in the carrier phase amidst the dispersed phase or the effect of the dispersed phase onto the carrier phase at the turbulence level.

For the Gas-Particle flow, the particle-turbulence two-phase flow interaction has beensuccessfully investigated with the Eulerian two-fluid model. The numerical code has beenvalidated against the experimental results of Fessler and Eaton (1999) for mean velocity andthe fluctuating velocities for both carrier and the dispersed phases. Two classes of particlessharing the same Stokes number, but different particle Reynolds number has also beeninvestigated in this study. The majority of the results agree well with the experimental data;however there have been some minor discrepancies felt at the proximity of the experimentalresults.

The Turbulence Modulation (TM) of the carrier phase for these two classes of particleshave been studied along the three sections that is near the inlet (x/h=2), in the mid-section(x/h=7) and just aft of the exit (x/h=14). It can be concluded that even though the 70μmcopper and 150μm glass particles share the same Stokes number, their behavior seems to bequite different, which suggests that Stokes number alone does not characterize the particlebehavior, thereby making particle Reynolds number an important parameter in classifying theway the particles behave.

Particles response to turbulent GP (Gas-Particle) and LP (Liquid-Particle) flow, behind aturbulent backward-facing step geometry have also been successfully analysed and simulatednumerically using an Eulerian two-fluid model. A significant amount of work was undertakento provide an in-depth understanding of the particle response, amidst turbulent flowconditions for two different carrier phases namely the gas and the liquid (diesel oil). Fromthe two sets of experimental data, at the mean velocity level, the particles seem to ‘lead’ andlater ‘catch up’ with the carrier phase for the LP flow, whereas they ‘lag’ behind and later‘lead’ for the GP flow. While at the turbulence level, the particles seem to ‘lag’ and then‘catch up’ for the LP flow while they ‘lag’ and phenomenally ‘lead’ for the GP flow. Thedetailed study and also the numerical diagnosis undertaken for turbulent particulate flowswith two different carrier phases, to study the particle response both at the mean velocity andat the turbulence level, behind a shear flow sudden expansion geometry is quite unique andone of its kind, as there is no current published work dealing with the analysis and numericalvalidation of the same.

Numerically the code was validated against the benchmark experimental data of Fesslerand Eaton (1997) for GP and the experimental data of Founti & Klipfel (1998) for the LPflows. The numerical results revealed good agreement with the experimental data. From therethe code was further used to investigate Stokes number effect on the two different carrierphases both at the mean velocity and at the turbulence level, for this exercise the experimentalgeometry of the GP flow and the inlet conditions of the LP flow were used. In order topresent the results in a more methodical manner, 12 points consisting of a matrix of threesections along the length of the step and four along the height of the step were used to studythe Stokes number effect on the two types of flows.

At the mean velocity level the particles seem to move faster that the carrier phases forboth the GP and the LP flow. However, at the particle fluctuation level, although the GP flowshow an escalation with the increase in the Stokes number, the same feature seem to be absentin the LP flow, wherein the particle fluctuation seem to decrease and almost flatten out withthe increase in its Stokes number. The main reason for this behaviour is the difference in thephysical characteristics of the carrier phase namely the liquid, which is far denser than the


gas, this eventually changes the cross-stream and the mean gradient behaviour, which isshown to cause elevated particle fluctuations in the GP flow.

For the Liquid-Air flows, the turbulent micro-bubble laden flow has been investigatedwith the help of two numerical models namely the two-fluid Inhomogeneous and MUSIGmodels, for two different free-stream velocities. Inhomogeneous model, which uses a fixedbubble diameter, shows a very good comparison of the skin-friction co-efficients with theexperiment. This model is further probed to study the various physical phenomenon’s causingthe drag reduction along the boundary layer, firstly it was observed that there is drop in themean streamwise water velocities with a subsequent increase in the normal along varying gasinjection rates. Secondly, the presence of the micro-bubbles caused turbulence attenuation forsome distance along the boundary layer and later an augmentation was felt due to theshedding of the vortices behind the bubbles. Thirdly, the peak of the void fractions seem todiffer in relation to the degree of drag reduction along the two free-stream velocitiesconsidered in our study.

However, with respect to the drag reduction caused due to the presence of micro-bubblesin the turbulent boundary layer MUSIG model seem to show good predictions for higher gasflow rates while under predicting for lower gas flow rates. This poor prediction of the modelat low flow rates was investigated to be the dominating break-up phenomenon that was takingplace within the flow. This was done in order to represent the actual flow condition, where bygroups of bubbles of varying bubble sizes are found within the boundary layer. Therebyallowing the MUSIG model to resolve the bubble size distribution mechanistically within agiven range of bubble size and feeding it back to the Inhomogeneous model appeared as thebest candidate to resolve the hidden physics in micro-bubble induced drag reductionproblems.

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Chapter 4

A REVIEW OF POPULATION BALANCE MODELLINGFOR MULTIPHASE FLOWS: APPROACHES,

APPLICATIONS AND FUTURE ASPECTS

Sherman C.P. Cheung1, G.H. Yeoh2 and J.Y. Tu1*

1School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University,Victoria 3083, Australia

2Australian Nuclear Science and Technology Organisation (ANSTO), PMB 1, Menai,NSW 2234, Australia

Abstract

Population balance modelling is of significant importance in many scientific and industrialinstances such as: fluidizations, precipitation, particles formation in aerosols, bubbly anddroplet flows and so on. In population balance modelling, the solution of the populationbalance equation (PBE) records the number of entities in dispersed phase that always governsthe overall behaviour of the practical system under consideration. For the majority of cases,the solution evolves dynamically according to the “birth” and “death” processes of which it istightly coupled with the system operation condition. The implementation of PBE inconjunction with the Computational Fluid Dynamics (CFD) is thereby becoming ever acrucial consideration in multiphase flow simulations. Nevertheless, the inherentintegrodifferential form of the PBE poses tremendous difficulties on its solution procedureswhere analytical solutions are rare and impossible to be achieved. In this article, we present areview of the state-of-the-art population balance modelling techniques that have been adoptedto describe the phenomenological nature of dispersed phase in multiphase problems. The mainfocus of the review can be broadly classified into three categories: (i) Numerical approachesor solution algorithms of the PBE; (ii) Applications of the PBE in practical gas-liquidmultiphase problems and (iii) Possible aspects of the future development in populationbalance modelling. For the first category, details of solution algorithms based on both methodof moment (MOM) and discrete class method (CM) that have been proposed in the literatureare provided. Advantages and drawbacks of both approaches are also discussed from thetheoretical and practical viewpoints. For the second category, applications of existing

* E-mail address: [email protected]. Phone no.:+61-3-9925 6191. Fax no.:+61-3-9925 6108. Corresponding

Author: Prof. Jiyuan Tu, SAMME, RMIT University, Bundoora, Melbourne, Victoria 3083, Australia

Sherman C.P. Cheung, G.H. Yeoh and J.Y. Tu118

population balance models in practical multiphase problems that have been proposed in theliterature are summarized. Selected existing mathematical closures for modelling the “birth”and “death” rate of bubbles in gas-liquid flows are introduced. Particular attention is devotedto assess the capability of some selected models in predicting bubbly flow conditions throughdetail validation studies against experimental data. These studies demonstrate that goodagreement can be achieved by the present model by comparing the predicted results againstmeasured data with regards to the radial distribution of void fraction, Sauter mean bubblediameter, interfacial area concentration and liquid axial velocity. Finally, weaknesses andlimitations of the existing models are revealed are suggestions for further development arediscussed. Emerging topics for future population balance studies are provided as to completethe aspect of population balance modelling.

Keywords: Population balance; Computational Fluid Dynamics; bubbly flow.

1. Introduction

Particles embedded within flow structures are featured in a wide diversity of industrialsystems, such as gas-solid dispersion in combustors, catalytic reactions in fluidized beds,liquid-liquid dispersion in stirring tanks, microbial processes in bioreactors, and gas-liquidheat and mass transfer in bubble column reactors. In most cases, these particles (regardlesswhether they are inherently presence within the system or deliberately introduce into thesystem) are often the dominant factor affecting the behaviour of the systems. Such mountingindustrial interests have certainly stimulated numerous scientific and engineering studiesattempting to synthesize the behaviour of the population of particles and its dynamicalevolution subject to the system environments in which has resulted in a widely adoptedconcept known as Population Balance.

The population balance of any system is a record for the number of particles, which maybe solid particles, liquid nuclei, bubbles or, variables (in mathematical terms) whose presenceor occurrence governs the overall behaviour of the system under consideration. In most of thesystems under concern, the record of these particles is dynamically depended on the “birth”and “death” processes that terminate existing particles and create new particles within a finiteor defined space. Mathematically, dependent variables of these particles may exist in twodifferent coordinates: namely “internal” and “external” coordinates (Ramkrishna andMahoney, 2002). The external coordinates refer to the spatial location of each particle whichis governed by its motion due to convection and diffusion flow behaviour. On the other hand,internal coordinates concerns the internal properties of particles such as: size, surface area,composition and so on. Figure 1 shows an example of the internal and external coordinatesinvolve in the population balance for gas-solid particles flows.

From a modelling perspective such as demonstrated in Figure 1, enormous challengesremain in fully resolving the associated nucleation, growth and agglomeration processes ofparticles within the internal coordinates and flow motions of external coordinates which aresubjected to interfacial momentum transfer and turbulence modulation between gas and solidphases. Owing to the significant advancement of computer hardware and increasingcomputing power over the past decades, Direct Numerical Simulations (DNS), which attemptto resolve the whole spectrum of possible turbulent length scales in the flow, provide thepropensity of describing the complex flow structures within the external coordinates (Biswaset al., 2005; Lu et al., 2006). Nevertheless, practical multiphase flows that are encountered in

A Review of Population Balance Modelling for Multiphase Flows 119

natural and technological systems generally contain millions of particles that aresimultaneously varying along the internal coordinates. Hence, the feasibility of DNS inresolving such flows is still far beyond the capacity of existing computer resources. Thepopulation balance approach, which records the number of particles as an averaged function,has shown to be a more promising way in handling the flow complexity because of itscomparatively lower computational requirements. It is envisaged that the next stages ofmultiphase flow modelling in research and in practice would most probably concentrate onthe development of more definitive efficient algorithms for solving the population balanceequation (PBE).

Figure 1. An exmple of the internal and external coordinates of population balance for gas-solid particleflows.

The development of population balance model has a long standing history. Back to theend of 18th century, the Boltzmann equation, devised by Ludwig Boltzmann, could beregarded as the first population balance equation which can be expressed in terms ofstatistical distribution of molecules or particles in a state space. Nonetheless, the derivation ofa generic population balance concept was actually initiated from the middle of 19th century. In1960s, Hulburt and Katz (1964) and Pandolph and Larson (1964), based on the statisticalmechanics and continuum mechanical framework respectively, presented the populationbalance concept to solve particle size variation due to nucleation, growth and agglomerationprocesses. A series of research development were thereafter presented by Fredrickson et al.(1967), Ramkrishna and Borwanker (1973) and Ramkrishna (1979, 1985) where the treatmentof population balance equations were successfully generalized with various internalcoordinates. A number of textbooks mainly concerning the population balance of


aerocolloidal systems have also been published (Hidy and Brock, 1970; Pandis and Seinfeld,1998; Friedlander, 2000). The flexibility and capability of population balance in solvingpractical engineering problems has not been fully exposed, until recently, where Ramkrishna(2000) wrote a textbook focusing on the generic issues of population balance for variousapplications.

Although the concept of population balance has been formulated over many decades,implementation of population balance modelling was only realized until very recent times.Such dramatic breakthrough was made possible by the rapid development of computationalfluid dynamics (CFD) and in-situ experimental measuring techniques. The availability ofCFD softwares certainly facilitated a solid foundation in obtaining useful PBE solutions. Withthe field information provided by the CFD framework, external variables of the PBE can beeasily acquired by decoupling the equation from external coordinates which can then enabledsolution algorithms to be developed within internal coordinates. The capacity to measureparticle sizes or other population balance variables from experiment is also of significantimportance. These experimental data not only allow the knowledge of particle sizes and theirevolution within systems to be realized but also provide a scientific basis for modelcalibrations and validations.

In view of current developments of the state-of-the-art, this paper aims to further exploitthe methodology of population balance modelling for multiphase flows from several aspects.First, it attempts to elucidate the implementation of population balance in conjunction withCFD techniques in handling practical industrial problems. Second, it seeks to reviews thesolution algorithms of the PBE that have been proposed in literatures and discuss theadvantages and drawbacks of each from the theoretical and practical viewpoints. Third, itdemonstrates the model’s capability in predicting gas-liquid bubbly flows with or withoutheat and mass transfer through rigorous validation studies against various experimental data.Finally, it seeks to outline the fundamental weakness and limitations of current modeldevelopment. Particular focus will be centred on the possible directions for furtherdevelopment.

2. Population Balance Approaches

2.1. Population Balance Equation

The foundation development of the PBE stems from the consideration of the Boltzmanequation. Such equation is generally expressed in an integrodifferential form describing theparticle size distribution (PSD) as follow:

( ) ( ) ∫ ′′′−′′−=⋅∇+∂

∂ ξξξξξξξξξξξ

0),(),(),(

21),,(),,(,, dtftfatrftru

ttrf

∫∞

′′′′−−0

),(),(),( ξξξξξξ dtfatf

∫∞

′′′′+ξ

ξξξξξγ dstfpb ),()/()()( ),()( tfb ξξ− (1)


where ),,( trf ξ is the particle size distribution dependent on the internal space vectorξ ,whose components could be characteristics dimensions, surface area, volume and so on. r andt are the external variables representing the spatial position vector and physical time inexternal coordinate respectively. ),,( tru ξ is velocity vector in external space. On the RHS,the first and second terms denote birth and death rate of particle of space vectorξ due tomerging processes, such as: coalescence or agglomeration processes; the third and fourthterms account for the birth and death rate caused by the breakage processes respectively.

),( ξξ ′a is the coalescence or agglomeration rate between particles of sizeξ andξ ′ .Conversely, )(ξb is the breakage rate of particles of sizeξ . )(ξγ ′ is the number offragments/daughter particles generated from the breakage of a particle of sizeξ ′ and

)/( ξξ ′p represents the probability density function for a particle of sizeξ to be generated bybreakage of a particle of sizeξ ′ .

Owing to the complex phenomenological nature of particle dynamics, analytical solutionsonly exist in very few cases of which coalescence and breakage kernels are substantiallysimplified (Scott, 1968; McCoy and Madras, 2003). Driven by practical interest, numericalapproaches have been developed to solve the PBEs. The most common methods are MonteCarlo methods, Method of Moments and Class Methods. Theoretical speaking, Monte Carlomethods, which solve the PBE based on statistical ensemble approach (Domilovskii et al.,1979; Liffman, 1992; Debry et al., 2003; Maisels et al., 2004), are attractive in contrast toother methods. The main advantage of the method is the flexibility and accuracy to trackparticle changes in multidimensional systems. Nonetheless, as the accuracy of the MonteCarlo method is directly proportion to number of simulations particles, extensivecomputational time is normally required. Furthermore, incorporating the method intoconventional CFD program is also not straightforward which greatly degraded itsapplicability for industrial problems. Because of their relevance in CFD applications,numerical approaches developed for Method of Moments and Class Methods are thendiscussed.

2.2. Method of Moments (MOM) Approach

The method of moments (MOM), first introduced by Hulburt and Katz (1964), has beenconsidered as one of the many promising approaches in viably attaining practical solutions tothe PBE. The basic idea behind MOM centres in the transformation of the problem intolower-order of moments of the size distribution. The moments of the particle size distributionare defined as:

ξξξ dtftm kk ∫∞

=0

)( ),()( (2)

From the above equation, the first few moments give important statistical descriptions onthe population which can be related directly to some physical quantities. In the case spacevectorξ represents the volume of particle, the zero order moment (k = 0) represents the total


number density of population and the fraction moment, k = 1/3 and k = 2/3 gives informationon the mean diameter and mean surface area respectively.

The primary advantage of MOM is its numerical economy which condenses the problemsubstantially by tracking the evolution of limited number of moments (Frenklach, 2002). Thisbecomes a critical value in modeling complex industrial systems when particle dynamics iscoupled with already time-consuming calculations of turbulence multiphase flows. Anothersignificance of the MOM is that it does not suffer from truncation errors in the PSDapproximation. Mathematically, the transformation from the PSD space to the space ofmoments is rigorous. Unfortunately, throughout the transformation process, fractionmoments, representing mean diameter or surface area, are normally involved posing seriousclosure problem (Frenklach and Harris, 1987). In order to overcome the closure problem, inthe early development of MOM, Frencklach and his co-workers (Frenklach and Wang, 1991;Markatou et al., 1993; Frenklach and Wang, 1994) proposed an interpolative scheme todetermine the fraction moment from integer moments – namely Method of moments withinterpolative closure (MOMIC).

2.2.1. Quadrature Method of Moments

Another different approach for computing the moment is to approximate the integrals inEq. (1) using numerical quadrature scheme – the quadrature method of moment (QMOM) assuggested by McGraw (1997). In the QMOM, instead of space transformation, Gaussianquadrature closure is adopted to approximate the PSD by a finite set of Dirac’s delta functionsas follow:

)(),(1

i

M

ii xNtf −≈∑

=

ξδξ (3)

where Ni represents the number density or weight of the ith class consists of all particles perunit volume with a pivot size or abscissa, ix . A graphical representation of the QMOM inapproximating the PSD is depicted in Figure 2.

Although the numerical quadrature approach suffers from truncation errors, itsuccessfully eliminates the problem of fraction moment which special closure is usuallyrequired. The closure of the method is then brought down to solving 2M unknowns, ix and Ni.A number of approaches in the specific evaluation of the quadrature abscissas and weightshave been proposed. McGraw (1997) first introduced the product-difference (PD) algorithmformulated by Gordon (1968) for solving monovariate problem. Nonetheless, as point out byDorao et al. (2006, 2008), the PD algorithm is a numerical ill-conditioned method forcomputing the Gauss quadrature rule (Lambin and Gaspard, 1982). Comprehensive derivationof the PD algorithm can be found in Bove (2005). In general, the computation of thequadrature rule is unstable and sensitive to small errors, especially if large number ofmoments is used. Later, McGraw and Wright (2003) derived the Jacobian MatrixTransformation (JMT) for multi-component population which avoids the instability inducedby the PD algorithm. Very recently, Grosch et al. (2007) proposed a generalized frameworkfor various QMOM approaches and evaluated different QMOM formulations in terms of


numerical quadrature and dynamics simulation. Several studies have also been carried outvalidating the method against different gas-solid particle problems (Barrett and Webb, 1998;Marchisio et al., 2003a,b,c). Encouraging results obtained thus far clearly demonstrated itsusefulness in solving monovariate problems and its potential fusing within ComputationalFluid Dynamics (CFD) simulations. One of the main limitations of the QMOM is thatmoments are adopted to represent the PSD, each moment is “convected” in the same phasevelocity which is apparently non-physical, especially for gas-liquid flow where bubble couldbe deformed and travel in different trajectory.

Figure 2. Graphical presentations of the Quadrature Method of Moments (QMOM).

2.2.2. Direct Quadrature Method of Moments (DQMOM)

With the aim to solve multi-dimensional problems, Marchisio and Fox (2005) extendedthe method by developing the direct quadrature method of moment (DQMOM) where thequadrature abscissas and weights are formulated as transport equations. The main idea of themethod is to keep track the primitive variables appearing in the quadrature approximation,instead of moments of the PSD. As a result, the evaluation of the abscissas and weights aresolved obtained using matrix operations. Substitute Eq. (3) into Eq. (1) and after somemathematical manipulations, transport equations for weights and abscissas are given by:

( ) ( ) iii atrNtrut

trN=⋅∇+

∂∂ ),(),(,

(4)

( ) ( ) iii btrtru

ttr

=⋅∇+∂

∂ ),(),(, ζζ(5)


where iii xN=ζ is the weighted abscissas and the terms ia and ib are related to the “birth” and“death” rate of population which forms 2M linear equations of which unknowns can beevaluated via matrix inversion:

dA =α (6)

where the 2M×2M coefficient matrix A= ]AA[ 21 is given by:

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−

−

−=

−− 12

2

121

211

)1(2)1(2

01

01

A

MM

M

M xM

x

xM

x (7)

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

=

−− 22221

12

)12(

2

)2(

210

10

A

MM

M

M xM

x

xxM

x (8)

The 2M vector of unknownsα is defined by:

[ ] ⎥⎦

⎤⎢⎣

⎡==

ba

bbaa TMM 11α (9)

and the source on the RHS is:

[ ]TMSSd 120 −= (10)

The source term for the kth moment kS is defined by:

ξξξ dtrStrS kk ),,(),(

0∫∞

= (11)

One attractive feature of the DQMOM is that it permits weights and abscissas to bevaried within the state space according to PSD evolution. Furthermore, different travellingvelocity can be also incorporated into transport equations allowing the flexibility to solvepoly-dispersed flows where weights and abscissas travel indifferent flow fields (Ervin andTryggvason, 1997; Bothe et al., 2006). In summary, the MOM represents a rather sound


mathematical approach and an elegant tool of solving the PBE with limited computationalburden. Such approach with no doubt is an emerging technique for solving PBE, due to theconsiderably short development history, thorough validation studies comparing modelpredictions against experimental data are however outstanding. It can be concluded that theDQMOM or other moment methods still require further assessments and validations forvarious multiphase flow problems.

2.3. Class Method (CM) Approach

Instead of inferring the PSD to derivative variables (i.e. moments), the class method(CM) which directly simulate its main characteristic using primitive variable (i.e. particlenumber density) has received greater attention due to its rather straightforwardimplementation within CFD software packages. In the method of discrete classes, thecontinuous size range of particles is discretized into a series number of discrete size classes.For each class, a scalar (number density of particles) equation is solved to accommodate thepopulation changes caused by intra/inter-group particle coalescence and breakage. Theparticle size distribution is thereby approximated as follow:

)(),(1

i

M

ii xNtf −≈∑

=

ξδξ (12)

The expression is exactly the same with QMOM in Eq. (3), however, the groups (orabscissas) of class methods are fixed and aligned continuously in the state space. A graphicalrepresentation of the CM in approximating the PSD is depicted in Figure 3.

Figure 3. Graphical presentations of Class Methods (CM).


2.3.1. Average Quantities Approach

The simplest approach in class methods is adopted an averaged quantity to represent theoverall changes of the particle population. Kocamustafaogullari and Ishii (1995) first derivedan interfacial area concentration (IAC) transport equation for tracking the interfacial areabetween gas and liquid phase in bubbly flow problems. They concluded that the governingfactor of the interfacial transfer mechanisms is strongly dominated by the interfacial areaconcentration. Modelling of the interfacial area concentration is therefore essential. Inaddition, as the population of particle is represented by a single average scalar, such averagequantity approach requires very limited computational time in solving the PBE, whichprovides an attractive feature for practical engineering problems.

Following their study, Ishii and his co-workers preformed extended the capability of their(IAC) transport model to simulate different bubbly flow regime in different flow conditions(Wu et al, 1998; Hibiki and Ishii, 2002; Fu et al., 2002a,b; Sun et al., 2004a,b). A series ofexperimental studies covering a wide range of flow conditions have been carried out in orderto provide a solid foundation for their model development and calibration. Recently, similarmodelling approach has been also adopted by Yao and Morel (2004) with the attempt ofbetter improving the bubble coalescence and breakage kernels. On the other hand, equivalentto the formulation of the interfacial area transport equation, an Average Bubble NumberDensity (ABND) equation has been proposed very recently in our previous studies (Yeoh andTu, 2006; Cheung et al., 2007).

2.3.2. MUltiple SIze Group (MUSIG) model

Besides the proposed the average quantity approach, a more sophisticated model, namelyhomogeneous MUltiple-SIze-Group (MUSIG) model which first introduced by Lo (1996) isbecoming widely adopted. Research studies based on the by Pochorecki et al. (2001), Olmoset al. (2001), Frank et al. (2004), Yeoh and Tu (2005) and Cheung et al. (2007) typified theapplication of MUSIG model in bubbly flow simulations. In the MUSIG model, thecontinuous particle size distribution (PSD) function is approximated by M number sizefractions; the mass conversation of each size fractions are balanced by the inter-fraction masstransfer due to the mechanisms of particle coalescence/agglomeration and breakageprocesses. The overall PSD evolution can then be explicitly resolved via source terms withinthe transport equations.

Snayal et al. (2005) examined and compared the CM and QMOM in a two-dimensionalbubbly column simulation; both methods were found to yield very similar results. The CMsolution has been found to be independent of the resolution of the internal coordinate ifsufficient number of classes were adopted. Computationally speaking, as the number oftransport equations depends on the number of group adopted, the MUSIG model requiresmore computational time and resources than the MOM to achieve stable and accuratenumerical predictions. For typical bubbly flow simulations, the model requires around 10groups to yield accurate results.


Figure 4. Schemetic diagram of the homogeneous and imhomogeneous MUSIG models.

Nonetheless, unlike the QMOM, CM provides the feasibility of accounting differentbubble shapes and travelling gas velocities. The inhomogeneous MUSIG model developed byKrepper et al. (2005), which consisted of sub-dividing the dispersed phase into N number ofvelocity fields, demonstrated the practicability of such an extension. This flexibilityrepresents a robust feature for multiphase flows modelling, especially for bubbly flowsimulations where bubbles may deform into different shapes. Figure 4 shows the concept ofthe inhomogeneous MUSIG in comparison to homogeneous MUSIG. Useful information onthe implementation and application of the inhomogeneous MUSIG model can be found in Shiet al. (2004) and Krepper et al. (2007). In spite of the sacrifices being made to computationalefficiency, the extra computational effort will rapidly diminish due to foreseeableadvancement of computer technology; the class method should therefore suffice as thepreferred approach in tackling more complex multiphase flows.

The formulation of the MUSIG model originates from the discretised PSE is given by:

( ) ( ) iphiiii RRnu

tn )(−=⋅∇+∂∂ ∑ (13)

To ensure overall mass conservation for all poly-dispersed vapour phases, the abovebubble number density equation for the inhomogeneous MUSIG model can be re-expressed interms of the volume fraction and size fraction of the bubble size class i, ],1[ jMi∈ , of

velocity group j, [ ]Nj ,1∈ according to:

( ) iphiijjijgijj RmSufαρ

tfαρ

)(, −=⋅∇+∂

∂(14)

with additional relations and constraints:

∑∑∑= ==

==N

j

M

iij

N

jjg

j

1 11ααα ; ∑

=

=jM

iij

1αα


1=+ lg αα ; ∑=

=jM

iif

11 (15)

where mi is the mass fraction of the particular size group I and phR is the mass transfer rate

due to phase changes which will be discussed in the coming sections.On the right hand side of Eq. (14) , the term ( ) ( )BCBCiiij DDPPRmS −−+== ∑,

represents the net mass transfer rate of the bubble class i resulting from the source of CP ,

BP , CD and BD , which are the production rates due to coalescence and breakage and thedeath rate due to coalescence and breakage of particles respectively. They can be formulatedas:

∑∑= =

=i

k

i

ljiijjkiC nnP

1 121 χη

)/()( 11 −− −−+ iiikj ννννν if ikji νννν <+<−1

=jkiη )/()( 11 iikji ννννν −+− ++ if 1+<+< ikji νννν0 otherwise

∑=

=N

jjiijC nnD

1χ

( ) jij

N

ijB nvvΩP :

1∑

+=

=

iiB nΩD = with ∑=

=N

kkii ΩΩ

1(16)

with

∑=

=jM

iijj SS

1, and 0

1=∑

=

N

jjS (17)

3. Population Balance Modelling for Bubbly Flows

In the previous section, approaches of population balance modelling have been reviewed.One should now realize the broad application of population balance modelling in practicalengineering problems. In essence, population balance has been applied in numerousmultiphase flow systems. For example: predicting the soot formation rate from incomplete


combustion processes (Frenklach and Wang, 1994; Zucca et al., 2007); solving the turbulentreacting flow in fluidized bed (Lakatos et al., 2008; Khan et al., 2007; Fan et al., 2004);evaluating the bubble side distribution and interfacial area in bubble column (Sha et al., 2006;Borel et al., 2006; Jia et al., 2007). In the following sections, to exemplify the capability ofpopulation balance modelling in conjunction with computational fluid dynamics framework,numerical simulations based on the MUSIG model of the CM were performed to predict theevolution of bubble size in both isothermal and heat and mass transfer conditions. Predictionswere validated against experimental data measured by Hibiki et al. (2001), Yun et al. (1997)and Lee et al. (2002). Before presenting the methodology of the numerical study,phenomenological discussion and background of bubbly flows is firstly provided below.

Figure 5. Schematic of the physical phenomenon embedded in (a) isothermal bubbly flows and (b)subcooled boiling flows.

3.1. Isothermal Bubbly Flows

For bubbly flows, the dynamic evolution of the PSD is governed predominantly by thebubble mechanistic behaviours such as bubble coalescence and bubble breakage. Figure 5aillustrates the physical characteristic of a typical isothermal bubbly flow. Owing to thepositive lift force created by the lateral velocity gradient, small bubbles (under 5.5 mm forair-water flows) being injected at the bottom have a tendency to migrate towards the channelwalls thereby increasing the bubble number density near the wall region. The net effect of


bubble coalescence encourages the formation of larger bubbles downstream. Larger bubbles(above 5.5mm), driven by the negative lift force, will move towards to the centre core of thechannel of which they will further coalesce with other bubbles to yield distorted/cap bubbles.These mechanisms strongly govern the distribution of bubble size and void fraction of the gasphase within the bulk liquid; appropriate sub-models (or kernels) in describing the bubblecoalescence and bubble breakage are essential to the proper modelling of bubbly flows. ThePrince and Blanch (1990) coalescence and breakage kernels are widely applied for the bubblyflow simulations. Various models focusing towards modifying the coalescence frequency anddaughter bubble distribution due to breakage have also been proposed such as those fromChester et al. (1982), Luo and Svendsen (1996), Lehr et al. (2001), Hagesaether et al. (2002),Wang et al. (2003, 2005, 2006) and Andersson and Andersson (2006). Chen et al. (2005)compared various kernels using a two-dimensional two-fluid model and showed that althoughthe mechanistic considerations were different in each model, the predicted results were found

Figure 6. Schematic drawings illustrating (a) the mechanism of bubble departing, sliding and lifting offfrom a vertical heated surface and (b) the area of influence due to bubble growth and sliding referring toEq. (26).


to be rather similar across the broad range of models. A number of research studies have alsobeen performed applying the aforementioned kernels to simulate size distribution evolution inbubbly turbulent pipe flows (Yeoh and Tu, 2004; Sha et al., 2006; Borel et al., 2006; Jia et al.,2007). All these works were nonetheless limited in solving isothermal bubbly turbulent co-flow problems with low superficial gas velocity where there was no significant formation ofcap/slug bubbles. It should be noted that the adoption of the above kernels remain debatablefor modelling the bubble dynamics beyond bubbly flow regime, especially for highsuperficial gas velocity and flows involving complex inter-phase exchange of heat and masstransfer.

Subcooled boiling flow belongs to another special category of bubbly turbulent pipe flowwhich embraces the complex dynamic interactions of bubble coalescence and bubblebreakage in the bulk flow as well as the presence of heat and mass transfer occurring in thevicinity of the heated wall due to nucleation and condensation. Heterogeneous bubblenucleation occurs naturally within small pits and cavities on the heated surface designated asnucleation sites as stipulated in Figure 5b, which is in contrast to having the bubbles beinginjected externally at the bottom of the flow as illustrated in Figure 5a for isothermal bubblyturbulent pipe flow. These nucleation sites, activated by external heat, act as a continuoussource generating relatively small bubbles along the vertical wall of the channel. Thepresence of a heated wall represents the fundamental difference between isothermal bubblyand subcooled boiling flows, where the former has a constant bubble injection rate to governthe overall void fraction of gas phase, but the latter has a variable bubble nucleation ratewhich is subjected mainly to the heat transfer and phase changing phenomena. Furthermore,experimental observations have confirmed that the vapour bubbles, driving by external forces,have a tendency to travel a short distance away from the nucleation sites, gradually increasingin size, before lifting off into the bulk subcooled liquid (Klausner et al., 1993). Figure 6shows a schematic illustration of the bubble motion and its area of influence on the heatersurface. Such bubble motion not only alters the mode of heat transfer on the surface, but alsogoverns the departure and lift-off diameter of bubbles, which in turn also influences thebubble distribution in the bulk liquid. In isothermal flow, coalescence prevails in the channelcore where the turbulent dissipation rate is relatively low. Conversely, bubbles tend todecrease in size for subcooled boiling flow as a result of increasing condensation away fromthe heated walls since the temperature in the bulk liquid remains below the saturationtemperature limit. This subcooling effect is a well-known phenomenon confirmed by variousexperiments (Gopinath et al., 2002). Subject to this effect, such flows will certainly yield abroader range of bubble sizes and possibly even amplifying greater dynamical changes of thebubble size distribution when compared to isothermal bubbly turbulent pipe flow. Given thechallenging task of modelling the sophisticated phenomena, empirical equations (Anglart andNylund, 1996) were inevitably employed to determine the bubble diameter in the gas phase.In order to make progress, a modified version of the MUSIG model incorporating nucleationat the heated wall and condensation in the subcooled liquid were developed to better resolvethe problem (Yeoh and Tu, 2005; Yeoh and Tu, 2006). Although some empirical equationswere still retained to determine the bubble nucleation and detachment, the potential ofadopting population balance approach in subcooled boiling flow demonstrated considerablesuccess in aptly predicting the bubble Sauter diameter distribution of the gas bubbles forvertical boiling flows.


4. Mathematical Models

Referring back to the formulation of PBE, one should notice that the left-hand side of theequation denotes the time and spatial variations of the PSD which depends on the externalvariables. By incorporating the PBE within CFD solver, external variables can be obtained. Inthis section, governing equations of the two fluid model and its associated model for handlinginterfacial momentum and mass transfer are introduced.

4.1. Two-Fluid Model

The three-dimensional two-fluid model solves the ensemble-averaged of mass,momentum and energy transport equations governing each phase. Denoting the liquid as thecontinuum phase (αl) and the vapour (i.e. bubbles) as disperse phase (αg), these equations canbe written as:

Continuity Equation of Liquid Phase

( ) lgΓ=⋅∇+∂

∂lll

ll uαρtαρ

(18)

Continuity Equation of Vapour Phase

( ) lgΓ−=⋅∇+∂

∂iigigg

igg fSufαρt

fαρ (19)

Momentum Equation of Liquid Phase

( ) gPuuαρt

uαρlllllll

lll ραα +∇−=⋅∇+∂

∂ (20)

lggllg )()])(([ Fuuuu lgT

llell +Γ−Γ+∇+∇∇+ μα

Momentum Equation of Vapour Phase

( ) gPuuαρt

uαρggggggg

ggg ραα +∇−=⋅∇+∂

∂ (21)

glglT

ggegg Fuuuu +Γ−Γ+∇+∇∇+ )()])(([ lgglμα

Energy Equation of Liquid Phase

( ) )()]([ lggl gllellllll

lll HHTHuαρtHαρ

Γ−Γ+∇∇=⋅∇+∂

∂ λα (22)


Energy Equation of Vapour Phase

( ) )()]([ lggl glgegggggg

ggg HHTHuαρtHαρ

Γ−Γ+∇∇=⋅∇+∂

∂λα (23)

On the right-hand side of equation (4), Si represents the additional source terms due tocoalescence and breakage. For isothermal bubbly turbulent pipe flows, it should be noted thatthe mass transfer rate lgΓ and glΓ are essentially zero. The total interfacial force lgF appearing

in equation (5) is formulated according to appropriate consideration of different sub-forcesaffecting the interface between each phase. For the liquid phase, the total interfacial force isgiven by:

dispersionlg

nlubricatiolg

liftlg

draglglg FFFFF +++= (24)

The sub-forces appearing on the right hand side of equation (9) are: drag force, lift force,wall lubrication force and turbulent dispersion force. More detail descriptions of these sub-forces can be found in Anglart and Nylund (1996). Note that for the gas phase, Fgl = - Flg.

The interfacial mass transfer rate due to condensation in the bulk subcooled liquid inequation (4) can be expressed by:

fg

lsatif

hTTah )(

lg

−=Γ (25)

Here, h indicates the inter-phase heat transfer coefficient which is correlated in terms ofthe Nusselt number (Tu and Yeoh, 2002). The wall generation rate for the vapour is modelledin a mechanistic manner by considering the total mass of bubbles detaching from the heatedsurface as:

)( lsatplfg

egl TTCh

Q−+

=Γ (26)

Here, Qe refers as the heat transfer rate due to evaporation. For subcooled boiling flows,the wall nucleation rate is accounted in equation (4) as a specified boundary conditionapportioned to the discrete bubble class based on the size of the bubble lift-off diameter,which is evaluated from the improved wall heat partition model. The term lgΓif represents the

mass transfer due to condensation. The gas void fraction along with the scalar size fraction fi

are related to the number density of the discrete bubble ith class ni (similarly to the jth classnj) as iiig nf να = .


4.1.1. Turbulence Modelling for Two-Fluid Model

In handling bubble induced turbulent flow, unlike single phase fluid flow problem, nostandard turbulence model is tailored for multiphase flow. For simplicity, the standard k-εmodel has been employed with encouraging results in early studies (Schwarz and Turner,1988; Davidson, 1990). Nonetheless, based on our previous study (Cheung et al, 2006), theMenter’s (1994) k-ω based Shear Stress Transport (SST) model were found superior to thestandard k-ε model. Similar observations have been also reported by Frank et al. (2004).Based on their bubbly flow validation study, they discovered that standard k-ε modelpredicted an unrealistically high gas void fraction peak close to wall. Interestingly, they alsofound that the two turbulence models behaved very similar by reducing the inlet gas voidfraction to a negligible value. This could be attributed to a more realistic prediction ofturbulent dissipation close to wall provided by the k-ω formulation. It revealed that furtherdevelopment should be focused on multiphase flow turbulence modelling in order to betterunderstand or improve the existing models.

The SST model is a hybrid version of the k-ε and k-ω models with a specific blendingfunction. Instead of using empirical wall function to bridge the wall and the far-awayturbulent flow, it solves the two turbulence scalars (i.e. k and ω) explicitly down to the wallboundary. The ensemble-averaged transport equations of the SST model are given as:

( ) llllkllk

ltllllll

lll kρPkαkuαρt

kαρ ωβασμ

μ ′−+⎟⎟⎠

⎞⎜⎜⎝

⎛∇+⋅∇=⋅∇+

∂∂

,3

, )(

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛∇+⋅∇=⋅∇+

∂∂

llt

lllllllll αuαρ

tαρ ω

σμ

μωω

ω

)(3

,

23,3

211

1)1(2 lllkl

ll

j

l

j

l

ll ρP

kxxkF ωβωγαω

ωσαρ

ω

−+∂∂

∂∂

−− (27)

where 3kσ , 3ωσ , 3γ and 3β are the model constants which are evaluated based on the blending

function F1. The shear induced turbulent viscosity lts,μ is given by:

),max( 21

1, SFa

ka

l

llts ω

ρμ = , ijij SSS 2= (28)

The success of SST model hinges on the use of blending functions of F1 and F2 which

govern the crossover point between the k-ω and k-ε models. The blending functions are givenby:

)tanh( 411 Φ=F ,

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛=Φ + 2

221

4,500,09.0

maxminn

ll

nll

l

nl

l

dDk

ddk

ωωσρ

ωρμ

ω


)tanh( 222 Φ=F , ⎟

⎟⎠

⎞⎜⎜⎝

⎛=Φ 22

500,09.0

maxnll

l

nl

l

ddk

ωρμ

ω(29)

Here, default values of model constants were adopted. More detail descriptions of thesemodel constants can be found in Menter (1994). In addition, to account the effect of bubbleson liquid turbulence, the Sato’s bubble-induced turbulent viscosity model was also employed(Sato et al., 1981). The turbulent viscosity of liquid phase is therefore given by:

ltdltslt ,,, μμμ += (30)

and the particle induced turbulence can be expressed as:

lgSglpltd UUDC −= αρμ μ, (31)

For the gas phase, dispersed phase zero equation model was adopted and the turbulentviscosity of gas phase can be obtained as:

g

lt

l

ggt σ

μρρ

μ ,, = (32)

where gσ is the turbulent Prandtl number of the gas phase.

4.2. Coalescence and Breakage Kernels

4.2.1. Bubble Coalescence and Breakage Kernels

Bubble breakage rate of volume jv into volume iv is modelled according to Luo and

Svendsen (1996), which is based on the assumption of bubble binary breakage under isotropicturbulence situation. The daughter size distribution is accounted using a stochastic breakagevolume fraction BVf . Denoting the increase coefficient of surface area as cf = [ 3/2

BVf +(1-

BVf )2/3-1], the breakage rate can be obtained as:

( )( )

( ) ξξε

σ

ξ

ddβρc

ξξ

dεCF

nαvvΩ

l

f

jB

jg

ij⎟⎟⎠

⎞⎜⎜⎝

⎛−×

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

− ∫ 3/113/53/2

1

3/11

23/1

2

12exp1

1:

min

(33)

where jd/λξ = is the size ratio between an eddy and a particle in the inertial sub-range and

consequently jd/minmin λξ = and C and β are determined from fundamental consideration of


drops or bubbles breakage in turbulent dispersion systems to be 0.923 and 2.0. FB is thebreakage calibration factor.

Bubble coalescence occurs via collision of two bubbles which may be caused by wakeentrainment, turbulence random collision and buoyancy. Only turbulence random collision isconsidered in the present study as all bubbles are assumed to be of spherical shape (wakeentrainment becomes negligible). The coalescence rate considering the turbulent collisiontaken from Prince and Blanch (1990) can be expressed as:

[ ] ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−++=

ij

ijtjtijiCij

tuuddF

τπχ exp4

5.0222 (34)

where ijτ is the contact time for two bubbles given by 3/13/2 /)2/( εijd and ijt is the time

required for two bubbles to coalesce having diameter di and dj estimated to be)/ln(]16/)2/[( 0

5.03flij hhd σρ . The equivalent diameter dij is calculated as suggested by

Chesters and Hoffman (1982): 1)/2/2(( −+= jiij ddd . According to Prince and Blanch

(1990), experiments have determined the initial film thickness ho = 4101 −× m and critical

film thickness hf = 8101 −× m at which rupture for air-water systems. The turbulent velocityut in the inertial sub-range of isotropic turbulence (Rotta, 1972) which is given

by: 3/13/12 dut ε= . FC is the coalescence calibration factor.

4.2.2. Bubble Source and Sink Due to Phase Change in Subcooled Boiling Flows

The term iphR )( in Eq. (14) constitutes the essential formulation of the source/sink rate

for the phase change processes associated with subcooled boiling flow. Considering thecondensation of bubbles, the bubble condensation rate in a control volume for each bubbleclass can be determined from:

dtdRA

Vn

BB

iCOND −=φ (35)

The bubble condensation velocity (Gopinath et al., 2002) is obtained from:

fgg

lsat

hTTh

dtdR

ρ)( −

= (36)

Substituting Eq. (21) into Eq. (20), and given that the bubble surface are AB and volumeVB which are based on Sauter mean bubble diameter, Eq. (20) can be rearranged to yield:


ifg

lsatif

igCONDiph n

hTTha

R⎥⎥⎦

⎤

⎢⎢⎣

⎡ −−==

)(1)(αρ

φ (37)

At the heated surface, bubbles at the nucleation sites are formed through the evaporationprocesses. The bubble nucleation rate from these sites can be expressed as:

C

HWN A

Nafξφ = (38)

where Na , f, Hξ and AC is the active nucleation site density, the bubble generation frequencyfrom the nucleation sites, the heated perimeter, and the cross-sectional area of the boilingchannel, respectively. Since the bubble nucleation process is taken to occur only at the heatedsurface, this heated wall nucleation rate is treated as a specified boundary condition to Eq.(14) apportioned to the discrete bubble class i, based on the bubble lift-off diameterdetermined from the mechanistic wall heat partition model.

4.3. Mechanistic Wall Heat Partition Model

To determine the bubble generation frequency and lift-off diameter for the boundarycondition of the MUSIG model, an improved wall heat partition model based on amechanistic approach is proposed (Yeoh et al., 2008) and briefly discussed in this section.With the presence of convective force or buoyancy force acting upon a vertically orientatedboiling flow as depicted in Figure 6a, vapour bubble departs from its nucleation site, slidesalong the heating surface and continues to grow downstream until it lifts off from the surface(Klausner et al., 1993). The motion of the travelling bubble affects the heat transfer at theheated wall according to two mechanisms: (i) the latent heat transfer due to micro-layerevaporation and (ii) transient conduction as the disrupted thermal boundary layer reformsduring the waiting period (i.e. incipience of the next bubble at the same nucleation site).

4.3.1. Transient Conduction Due to Bubble Motion

Transient conduction occurs in regions at the point of inception and in regions beingswept by sliding bubbles. For a stationary bubble, the heat flux is given by:

ftD

KNRTTtCk

Q wd

aflsw

pllltc ⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

4)(2

2ππρ

)1(4

)(22

ftD

NRTTtCk

wd

aflsw

plll −⎟⎟⎠

⎞⎜⎜⎝

⎛−+

ππρ

(39)

where Dd is the bubble departure diameter, Ts is the temperature of the heater surface and Tl isthe temperature of the liquid. Eq. (24) indicates that some fraction of the nucleation sites will


undergo transient conduction while the remaining will be in the growth period. For a slidingbubble, the heat flux due to transient conduction that takes place during the sliding phase andthe area occupied by the sliding bubble at any instant of time is given by:

fKDtlNRTTtCk

Q wsaflsw

pllltcsl )(2 −=

πρ

)1(4

)(22

ftDftNRTTtCk

wslaflsw

plll −⎟⎟⎠

⎞⎜⎜⎝

⎛−+

ππρ

(40)

where the average bubble diameter D is given by ( ) 2ld DDD += and Dl is the bubblelift-off diameter. In this study, a value of 1.8 is assumed for the area of influence, K (Juddand Hwang, 1976). The reduction factor Rf appearing in equations (24) and (25) depicts theratio of the actual number of bubbles lifting off per unit area of the heater surface to thenumber of active nucleation sites per unit area, viz., ( )slR sf /1= where ls is the sliding

distance and s is the spacing between nucleation sites. In the present study, the spacing

between nucleation sites is approximated as aNs 1= (Basu et al., 2005). The active

nucleation site density, Na, is expressed by ( ) 805.1]185[ satwa TTN −= (Končar et al., 2004).The factor Rf is obtained alongside with the sliding distance evaluated from the force balancemodel described below.

4.3.2. Force Convection for Single Phase Component

Forced convection always prevails at all times in areas of the heater surface that are notinfluenced by the stationary and sliding bubbles (see also in Figure 6b). The fraction of theheater area for stationary and sliding bubbles is given by:

)1(44

1122

ftD

NftD

KNRA wd

awd

afq −⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−=−

ππ

⎥⎥⎦

⎤−⎟⎟

⎠

⎞⎜⎜⎝

⎛++ )1(

4

2

ftD

ftNfKDtlN wd

slawsaπ

(41)

The heat flux due to forced convection can be obtained according to the definition oflocal Stanton number St for turbulent convection is:

))(1( lwqlpllc TTAuCStQ −−= ρ (42)

where ul is the adjacent liquid velocity.


4.3.3. Latent Heat Due to Vapour Evaporation Processes

The heat flux attributed to vapour generation is given by the energy carried away by thebubbles lifting off from the heated surface. It also represents the energy of vaporizationwhereby the bubble size of Dl is produced, which is expressed as:

fggl

afe hD

fNRQ ρπ

⎟⎟⎠

⎞⎜⎜⎝

⎛=

6

3

(43)

The total wall heat flux Qw is the combination of the following heat flux components: Qw

= Qe + Qtc + Qtcsl + Qc.

Mechanistic Approach for Bubble Frequency Evaluation

The bubble nucleation rate in Eq. (23) requires the knowledge of the bubble frequency(f). Within the wall partition model, the bubble frequency is determined by a mechanisticapproach based on the description of an ebullition cycle in nucleate boiling, which isformulated as:

wg ttf

+=

1(44)

The waiting period (tw) and the growth period of vapour bubbles (tg) is derived from thetransient conduction and force balance model respectively.

When transient conduction occurs, the boundary layer gets disrupted and cold liquidcomes in contact with the heated wall. Assuming that the heat capacity of the heater wallρsCpsδs is very small, the conduction process can be modelled by considering one-dimensionaltransient heat conduction into a semi-infinite medium with the liquid at a temperature Tl andthe heater surface at a temperature Ts. The wall heat flux can be approximated by:

l

lslw

TTkQδ

)( −= (45)

where δl(= tπη ) is the thickness of the thermal boundary layer. If the temperature profileinside this layer is taken to be linear (Hsu and Graham, 1976), it can be expressed as:

l

lswb

xTTTTδ

)( −−= (46)

where x is the normal distance from the wall. Based on the criterion of the incipience ofboiling from a bubble site inside the thermal boundary layer, the bubble internal temperaturefor a nucleus site (cavity) with radius rc is


gfgc

satsatb hrC

TTT

ρσ

2

2−= at crCx 1= (47)

where θθ sin/)cos1(1 +=C and θsin/12 =C . The angle θ represents the bubblecontact angle. By substituting Eq. (32) into Eq. (31), the waiting time tw can be obtained as

2

2

1

/2)()(1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−−

==cfggsatsatw

clsw rhCTTT

rCTTtt

ρσπη(48)

The cavity radius rc can be determined by applying Hsu’s criteria and tangency conditionof equations (32) and (33), viz.,

πηπησρ 222

2221 )()(2

ls

w

lls

sat

cfgg TTQkTT

TrhCC

t −⎥⎦

⎤⎢⎣

⎡=

−

⎥⎥⎦

⎤

⎢⎢⎣

⎡= (49)

From the above equation,

2/12

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

wfgg

lsatc Qh

kTFrρσ

(50)

where,2/122/1

21 cos1sin1

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=⎟⎟⎠

⎞⎜⎜⎝

⎛=

θθ

CCF (51)

According to Basu et al. (2005), the factor F indicates the degree of flooding of theavailable cavity size and the wettability of the surface. If the contact angle θ → 0, all thecavities will be flooded. Alternatively, as θ → 90o, F → 1, all the cavities will not be flooded(i.e. they contain traces of gas or vapour).

4.3.4. Force Balance Model and the Bubble Growth Time

The bubble growth time is correlated to its lift-off diameter which depends on variousforces acting on the bubble in the directions parallel and normal to a vertical heating surface.Figure 7 illustrates the forces acting on the bubble in the x-direction and y-direction whichcan be formulated according to the studies performed by Klausner et al. (1993) and Zeng etal. (1993):.

ΣFx =Fsx + Fdux + FsL + Fh + Fcp (52)


andΣFy =Fsy + Fduy + Fqs + Fb (53)

where Fs is the surface tension force, Fdu is the unsteady drag due to asymmetrical growth ofthe bubble and the dynamic effect of the unsteady liquid such as the history force and theadded mass force, FsL is the shear lift force, Fh is the force due to the hydrodynamic pressure,Fcp is the contact pressure force accounting for the bubble being in contact with a solid ratherthan being surrounded by liquid, Fqs is the quasi steady-drag in the flow direction, and Fb isthe buoyancy force. In addition, g indicates the gravitational acceleration; α, β and θi are theadvancing, receding and inclination angles respectively; dw is the surface/bubble contactdiameter; and d is the vapour bubble diameter at the wall.

Figure 7. Schematic drawings illustrating the forces balance of a growing vapour bubble attached to theheated surface.

The forces acting in the x-direction can be estimated from:

]cos[cos αββα

πσ −−

−= wsx dF ; θcosdudux FF −=

22

21 rUCF lLsL πρ Δ= ;

449 2

2 wlh

dUF

πρ Δ=


r

wcp r

dF σπ 2

4

2

= (54)

The forces acting in the x-direction can be estimated from:

]sin[sin)(

)(22 βα

βαπβαπσ +−−−

−= wsy dF ; θsinduduy FF −=

rUCF lDqs πμ Δ= 6 ; ( )grF glb ρρπ −= 3

34

(55)

From the various forces described along the x-direction and y-direction, r is the bubbleradius, ΔU is the relative velocity between the bubble centre of mass and liquid, CD and CL

are the respective drag and shear lift coefficients and rr is the curvature radius of the bubble atthe reference point on the surface x = 0, which is rr ∼ 5r (Klausner et al., 1993).

The growth force Fdu is modelled by considering a hemispherical bubble expanding in aninviscid liquid, which is given by Zeng et al. (1993) as:

⎟⎠⎞

⎜⎝⎛ += rrrCrF sldu

22

23πρ (56)

where ( ˙ ) indicates differentiation with respect to time. The constant Cs is taken to be 20/3(Zeng et al., 1993). In estimating the growth force, additional information on the bubblegrowth rate is required. As in Zeng et al. (1993), a diffusion controlled bubble growthsolution by Zuber (1961) is adopted:

tJabtr ηπ

2)( = ; fgg

satpll

hTC

Jaρ

ρ Δ= ;

pll

l

Ck

ρη = (57)

where Ja is the Jakob number, η is the liquid thermal diffusivity and b is an empiricalconstant that is intended to account for the asphericity of the bubble. For the range of heatfluxes investigated in this investigation, b is taken to be 0.21 based on a similar subcooledboiling study performed by Steiner et al. (2006), which has been experimentally verifiedthrough their in-house measurements with water as the working fluid.

While a vapour bubble remains attached to the heated wall, the sum of the parallel andnormal forces must satisfy the following conditions: ΣFx = 0 and ΣFy = 0. For a slidingbubble case, the former establishes the bubble departure diameter (Dd) while the latter yieldsthe bubble lift-off diameter (Dl). The growth period tg appearing in Eq. (29) can be readilyevaluated based on the availability of the bubble size at departure from its nucleation sitethrough Eq. (42). Details of the present wall partition model can also be found in literature(Yeoh et al., 2008) and the references therein.


4.4. Experimental Details

Brief discussions of the experimental setup for the isothermal and subcooled bubblyturbulent pipe flows are provided below.

Isothermal bubbly turbulent pipe flow experiments have been performed at the Thermal-Hydraulics and Reactor Safety Laboratory in Purdue University (2001). The test sectioncomprised of an acrylic round pipe with an inner diameter D = 50.8 mm and a length of 3061mm. Temperature of the apparatus was kept at a constant temperature (i.e. 20oC) within adeviation of ±0.2oC controlled by a heat exchanger installed in a water reservoir. Local flowmeasurements using the double sensor and hotfilm anemometer probes were performed atthree axial (height) locations of z/D = 6.0, 30.3 and 53.5 and 15 radial locations of r/R = 0 to0.95. Experiments at a range of superficial liquid velocities jf and superficial gas velocities jg

were performed covering most bubbly flow regions including finely dispersed bubbly flowand bubbly-to-slug transition flow regions.

A series of subcooled boiling experiments have been performed by Yun et al. (1997) andLee et al. (2002). The experimental setup consisted of a vertical concentric annulus with aninner rod of 19 mm outer diameter uniformly heated by a 54 kW DC power supply. Thisheated section comprised of a 1.67 m long Inconel 625 tube with a 1.5 mm wall thicknessfilled with magnesium oxide powder insulation. The outer wall comprised of two stainlesssteel tubes with 37.5 mm inner diameter. Demineralised water was used as the working fluid.Local gas phase parameters such as radial distribution of the void fraction, bubble frequencyand bubble velocity were measured by a two-conductivity probe method located 1.61 mdownstream of the beginning of the heated section. The bubble Sauter diameters (assumingspherical bubbles) were determined through the IAC, calculated using the measured bubblevelocity spectrum and bubble frequency.

4.5. Numerical Details

For isothermal gas-liquid bubbly turbulent pipe flow, the generic CFD code ANSYS-CFX 11 (2006) was utilised to handle the two sets of equations governing conservation ofmass and momentum. Numerical simulations were performed on a 60o radial sector of thepipe with symmetry boundary conditions imposed at the end vertical sides. At the test sectioninlet, uniformly distributed superficial liquid and gas velocities, void fraction and bubble sizewere specified. Details of the boundary conditions for different flow conditions aresummarized in Table 1.

At the pipe outlet, a relative averaged static pressure of zero was specified. A three-dimensional mesh containing hexagonal elements was generated resulting in a total of108,000 elements over the entire pipe domain. For all flow conditions, bubble size in therange of 0-10 mm was discretised into 10 bubble classes as tabulated in Table 2. Theprescribed 10 size groups were further divided into two velocity fields for the inhomogeneousMUSIG model to predict the transition of bubbly-slug flow. Reliable convergence wereachieved within 2500 iterations when the RMS (root mean square) pressure residual droppedbelow 1.0×10-7. A fixed physical time scale of 0.002s was adopted for all steady statesimulations.


Table 1. Bubbly flow conditions and its inlet boundary conditions employed in thepresent study

Superficial liquid velocity, fj (m/s)

Superficial gas velocity,

gj (m/s)

Hibiki et al. (2001) experiment Bubbly flow Regime Transition Regime

0.491

[00.D/zgα =(%)]

[00.D/zSD

=(mm)]

- 0.0556

[10.0] [2.5]

-

0.986

[00.D/zgα =(%)]

[00.D/zSD

=(mm)]

0.0473

[5.0] [2.5]

0.1130

[10.0] [2.5]

0.242

[20.0] [2.5]

Table 2. Diameter of each discrete bubble class for MUSIG model

Class No. Central Class Diameter di (mm)Isothermal Subcooled Boiling

1 0.5 0.452 1.5 0.943 2.5 1.474 3.5 2.025 4.5 2.586 5.5 3.147 6.5 3.718 7.5 4.279 8.5 4.83

10 9.5 5.4011 - 5.9612 - 6.5313 - 7.1014 - 7.6615 - 8.23

For subcooled boiling flow, numerical solutions were obtained from the two sets oftransport equations governing not only mass and momentum but also energy using the genericCFD code CFX 4.4. A total number 15 bubble classes were specified for the dispersed phasesin the homogeneous MUSIG model (see Table 2). Similar to the isothermal flow simulations,only one quarter of the annulus geometry was modelled due to the uniform prescription of theheat flux at the inner wall. A body-fitted conformal system was employed to generate thethree-dimensional mesh within the annular channel resulting in a total of 13 (radial) × 30


(axial) × 3 (circumference) control volumes. A standard k-ε model was applied for bothphases while additional turbulent viscosity induced by bubbles was included using Sato’smodel95. Since wall function was used in the present study to bridge the wall and the fullyturbulent region away from heater surface, the normal distance between the wall and the firstnode in the bulk liquid should be such that the corresponding x+ was greater than 30. Gridindependence was examined. In the mean parameters considered, further grid refinement didnot reveal significant changes to the two-phase flow parameters. Convergence was achievedwithin 1500 iterations when the mass residual dropped below 1.0×10-7. Experimentalconditions used for comparison with the simulated results are tabulated in Table 3.

Table 3. Subcooled boiling flow conditions measured by Yun et al. (1997) adopted in thepresent numerical study

Case Pinlet [Mpa] Tinlet [°C] Tsub (inlet)[°C] Qw [kW/m2] G [kg/m2s]

L1 0.143 96.9 13.4 152.9 474.0L2 0.137 94.9 13.8 197.2 714.4L3 0.143 92.1 17.9 251.5 1059.2

5. Results and Discussion

5.1. Isothermal Bubbly Turbulent Pipe Flow Results

Preliminary numerical simulations performed in this study over a range of superficial gasand liquid velocities within the bubbly flow regime have consistently resulted in the predictionof larger than expected bubble sizes. However, these results were found to clearly contradict themeasurements performed by Hibiki et al. (2001) as well as some experimental observations(Bukur et al., 1996; George et al., 2000). One plausible explanation for this discrepancy couldpossibly be the error embedded in the turbulent dissipation rate prediction (Bertola et al., 2003)as a consequence of the turbulence model being applied contributing in turn to excessively highcoalescence rates in the MUSIG model. As reported in Chen et al. (2005), similar observationsalso confirmed the high coalescence rates that were experienced in their bubble column study.They argued that the local coalescence rate should be reduced by an order of magnitude lowerthan the local breakage rate by a factor of about 10. Olmos et al. (2001) further demonstratedthe need to prescribe suitable calibration factor in order to aptly predict the bubble sizedistributions in their bubble column flow configuration. A value of 0.075 was assumed for thecoalescence calibration factor. According to similar arguments stipulated above, the coalescenceand breakage calibration factors (i.e. FC and FB), were set as 0.05 and 1.0 throughout allintended simulations of isothermal flow conditions (Cheung et al., 2007b).

5.1.1. Local Distributions of Void Fraction snd Interfacial Gas Velocity

Depending on the range of superficial velocities of gas and liquid experienced in thebubbly flow regime, the phase patterns of the vapour void fraction can be broadly categorized


into four types of distributions: “wall peak”, “intermediate peak”, “core peak” and“transition” (Serizawa an Kataoka, 1988). Figure 8 compares the gas void fraction profilesobtained from the homogeneous MUSIG model with the experimental data measured at thelocation of z/D = 53.5 (i.e. close to the exit of channel) for three different bubbly turbulentpipe flow conditions. The high void fractions close to wall proximity typically characterisedthe “wall peak” behaviour. Its phenomenological establishment can be best described by thebalance between the positive lift force that acted to impel the bubbles away from the centralcore of the flow channel and the opposite effect being imposed by the lubrication forcepreventing the bubbles from being obliterated at the channel walls. In spite of similar trendspredicted, the predicted void fraction peaks, on closer examination, appeared to be leaningmore towards the channel wall in contrast to the actual bubble distributions observed duringexperiments. Against the assessment on other wall lubrication models by Frank et al. (2004)and Tomiyama (1998), a much lower than expected wall force determined via Antal et al.(1991) model purported to be the most probable cause of the discrepancy. As aforementioned,the two-phase turbulence modelling that could affect the magnitude of the turbulence induceddispersion force could also contribute to the additional modelling uncertainty.

Figure 8. Predicted radial void fraction distribution and experimental data of Hibiki et al.(2001) atmeasuring station of z/D =53.5


Figure 9. Predicted interfacial gas velocity distribution and experimental data of Hibiki et al. (2001) atmeasuring station of z/D =53.5.

Figure 9 illustrates the local interfacial gas velocity distributions at the measuring stationof z/D = 53.5, close to the outlet of the pipe. Unlike in single phase flows, the presence ofbubbles has the tendency to enhance the liquid flow turbulence intensity (Serizawa et al.,1990; Hibiki et al., 2001). Additional turbulence being experienced at the core flattened theliquid velocity profile as expected. Through the interfacial momentum transfer, such effectsare brought down to the gas phase yielding similar interfacial gas velocity profiles. In general,the interfacial gas velocity profiles for all flow conditions were found to be in goodagreement with measurements, especially at the channel core. In essence, the prediction of thelocal bubble sizes evaluated by the coalescence and breakage kernels of the MUSIG modelhas a coupling effect on the phases velocities. In better determining the Sauter mean bubblediameter, a more accurate description of the interfacial forces between the two phases shouldsurface in more enhanced liquid/gas velocity predictions.


5.1.2. Local Distributions of Sauter Mean Bubble Diameter, Interfacial AreaConcentration and the Evolution of Bubble Size Distribution

Figure 10 illustrates the predicted and measured Sauter mean bubble diameter radialprofiles at the location of z/D = 53.5. As measured by Hibiki et al. (2001), the Sauter meanbubble diameter profiles remained roughly unchanged throughout the whole channel. Overall,bubble size changes were found mainly due to the bubble expansion caused by the staticpressure variation along the axial direction. As demonstrated by the good agreement betweenthe predicted and the measured bubble diameters at the channel core, the ensemble bubbleexpansion effect was adequately captured by the two-fluid approach via the ideal gasassumption. Near the wall region, slightly larger bubbles were formed due to the tendency ofsmall bubbles migrating towards the wall thereby creating higher concentration of bubbles,increasing the likelihood of possible bubble coalescence. Remarkable agreement with themeasurement at the wall region clearly illustrated the enlargement of bubble size due tobubble coalescence and bubble breakage successfully represented by the kernels in theMUSIG model. As the Sauter mean bubble diameter is closely related to the interfacialmomentum forces (i.e. drag and lift forces), appropriate evaluation of the bubble diameter isthus crucial for the prediction of interfacial gas velocities.

Figure 10. Predicted Sauter mean bubble diameter distribution and experimental data of Hibiki et al.(2001) at measuring station of z/D =53.5.


Figure 11. Predicted size fraction of each bubble classes and its evolution along radial direction at themeasuring station of z/D =53.5.

The population balance of bubbles within the bubbly turbulent pipe flow can be furtherexemplified by tracking the evolution of bubble size distribution at the measuring station.Figure 11 shows the development of the size fraction of each bubble classes along the radialdirection at the location of z/D = 53.5. Since the turbulence intensity is relatively low at thechannel core (i.e. r/R = 0.05), bubble sizes remained unaffected owning to the insignificantbubble coalescence and bubble breakage rates. With increasing number of bubbles driven bythe lift force and the rising turbulence intensity within the boundary layer towards the channelwalls (i.e. r/R=0.8 and 0.95), bubble coalescence and bubble breakage became increasinglynoticeable forming larger bubbles and re-distributed the BSD to higher bubble classes. Suchbehaviour was amplified especially for cases with relatively higher void fraction (e.g. Figure8a,c).

Based on the assumption where the bubbles are spherical in shape, the localInterfacial Area Concentration (IAC) profiles may be determined based on the relationbetween the local void fraction and Sauter mean bubble diameter according to

sgif Da /6α= . The measured and predicted local interfacial area concentration profiles

for the respective flow condition are shown in Figure 12. Similar to the void fractionprofiles, the IAC predictions at the measuring station yielded good agreement against themeasured profiles. The discrepancy between predicted results and measured data could beattributed to the error introduced by the void fraction prediction owing to theuncertainties embedded within the turbulence and wall lubrication models. Anotherpossible cause could be the invalid assumption of spherical bubbles to aptly resolve the


gas-liquid flow where large distorted bubbles prominently featured at high superficialvelocities (i.e. <jf>=0.986m/s).

Figure 12. Predicted Interfacial Area Concentration (IAC) distribution and experimental data of Hibikiet al. (2001) at measuring station of z/D =53.5.

5.2. Subcooled Boiling Flow Results

Increasing complexity of numerical simulations is now further elaborated by consideringthe bubble dynamics in conjunction with the heat and mass transfer processes typified by asubcooled boiling flow. For the local cases of L1, L2 and L3, the measured and predictedradial profiles of the Sauter mean bubble diameter, vapour void fraction and interfacial areaconcentration located at the measuring plane 1.61 m downstream of the beginning of theheated section are discussed below. In all the figures, the dimensionless parameter (r-Ri)/(Ro-Ri) = 1 indicates the inner surface of the unheated flow channel wall while (r-Ri)/(Ro-Ri) = 0indicates the surface of the heating rod in the channel.


Figure 13. Prediction Sauter mean bubble diameter distribution and experimental data at the measuringstation.

5.2.1. Local Distributions of Void Fraction, Sauter Bubble Diameter and InterfacialArea Concentration

Figure 13 illustrates the predicted Sauter mean bubble diameter profiles at the measuringplane of the heated annular channel. Experimental data and observations (Bonjour andLallemand et al., 2001; Lee et al., 2002) suggested that vapour bubbles, relatively small whendetached from the heated surface, have the tendency of significantly colliding with otherdetached bubbles at the downstream and subsequently forming bigger bubbles viacoalescence. For all three cases, the bubble size changes were found to be adequatelypredicted by the modified MUSIG model. Observed consistent trends between the predictedand measured Sauter mean bubble diameter reflected the measure of the modified MUSIGmodel in aptly capturing the bubble coalescence especially in the vicinity of the heated wall.The development of bubbles in this region stemmed from the evaporation process occurringat the heated wall and forces acting on the vapour bubbles determining the bubble size atdeparture or lift-off.


Table 4. Predicted heat partitions, bubble departure and lift-off diameter of subcooledboiling flow conditions

Case L1 Case L2 Case L3Qc (W/m2) 0% 0% 0%

Qtc (W/m2) 2.51% 4.56% 6.42%

Qtcsl (W/m2) 55.07 61.25% 65.58%

Qe (W/m2) 42.42% 34.19% 28.00%

Dd (mm) 0.56 0.58 0.57

Mea

suri

ng lo

catio

n

Dl (mm) 1.45 1.31 1.20

Table 4 illustrates the various contributing heat flux components and the associatedbubble departure and lift-off diameters evaluated by the improved heat partition model. Onthe basis of the force balance model, the bubble departure diameter were predicted with a sizeof approximately 0.56~0.58 mm while the lift-off diameters were found to range from1.2~1.45 mm. The ratio between the bubble lift-off diameter and bubble departure diameterwas thus ascertained to be between 2 and 3, which incidentally closely corresponded toexperimental observations of Basu et al. (2005). Surface quenching due to sliding bubbles andevaporation were found to be the dominant modes of heat transfer governing the heat partitionmodel. The former highlighted the prevalence of bubble sliding motions on the surfacesignificantly altering the rate of heat transfer and subsequently the resultant vapour generationrate. Away from the heated wall, bubbles entering the bulk subcooled liquid were condenseddue to the subcooling effect. Predicted trends of the Sauter mean diameter profiles clearlyshowed the gradual collapse of the bubbles from the channel centre to the outer unheatedwall. The over-prediction of the bubble diameters for cases L1 and L2 could be attributed bythe under-estimation of the subcooling condensation, which was determined by an empiricalcorrelation based on the Nusselt number description.

Although the measured bubble sizes near the heated wall were found to agree ratherwell with the measured data confirming to certain extend the appropriate estimation of thebubble lift-off diameters, a closer examination of the local void fraction profiles at themeasuring station in Figure 14 indicated a less than satisfactory prediction of the voidfraction near the heated surface where they were either over- or under-predicted asexemplified in cases L1 and L3. The void fraction distribution in case L2 comparednonetheless reasonably well with measurement. This discrepancy could be attributed to theuncertainties within the heat partition model in specifically evaluating the vapourgeneration rate. In the quest of reducing the application of empirical correlations, theconsideration of the active nucleation site density in the present study still depended on theuse of an appropriate relationship, which could be sensitive to the flow conditions. Thesignificance of active wall nucleation site density linking to the prediction of the IAC hasalso been reported in Hibiki and Ishii43. Nevertheless, as the population of cavities mayvary significantly between materials and cannot be measured directly, an adequateexpression of the active nucleation site density covering a wide range of flow conditionsremains outstanding and more concerted research is required. The IAC profiles as shown in


Figure 15 also exhibited similar trends with the void fraction distributions plotted in Figure14.

Figure 14. Predicted radial void fraction distribution and experimental data at the measuring station.

5.2.2. Evolution of Bubble Size Distribution and Bubble Generation Rate Due toCoalescence, Breakage and Condensation

Figure 16 shows the bubble size distribution expressed in terms of interfacial areaconcentration of individual bubbles classes along radial direction for the case L3 at themeasuring station describes the bubble dynamics caused by coalescence, breakage andcondensation in subcooled boiling flows. Significant vapour bubbles represented from bubbleclass 3 in the vicinity of the heated wall essentially indicated the size of the bubble lift-offdiameter which coalesced with downstream/neighbouring bubbles forming larger voidfraction peaks as indicated by bubble classes of 7 and 9. Owing to the high shear stress withinthe boundary layer, some bubbles are affected by turbulent impact due to breakage resultingin the formation of smaller bubbles as evidenced by the significant distributions indicated


within bubble classes 1 and 2. Away from the heated wall, the condensation processdominated in reducing the void fraction of each bubble classes and eventually collapsingmajority of the bubbles beyond the position (r-Ri)/(Ro-Ri)=0.6. The net generation rate due tocoalescence and breakage and condensation rate of selected bubble classes are depicted inFigure 17. Close to the wall region, the highest generation rate corresponding to the peakvalue observed in Figure 16 is represented by bubble class 7; substantial generation rate wasalso found for bubble class 3 at the same region. While the coalescence of bubbles was seento be governed mainly by bubble classes 3 and 7, bubble classes 3 and 12 also contributed tothe condensation process due to their considerably high number density and interfacial area.These two figures aptly demonstrated the mechanisms of coalescence, breakage andcondensation in the modified MUSIG model affecting the thermo-mechanical andhydrodynamics processes within the subcooled boiling flow.

Figure 15. Predicted interfacial gas velocity distribution and experimental data at the measuring station.


Figure 16. Predicted IAC of each bubbles class along radial direction for the case L3 at the measuringstation.

Figure 17. Predicted net bubble generation rate due to coalescence and breakage and condensation rateof selected bubble classes of modified MUSIG model.


5.3. Limitations and Shortcomings of Existing Models

Encouraging predictions by the homogeneous MUSIG and modified MUSIG modelsclearly demonstrated their viable applications in resolving isothermal bubbly turbulent pipeflow and subcooled boiling conditions. Nevertheless, the flow cases that have beeninvestigated from above generally possessed only weak bubble-bubble interactions andnarrow bubble size distributions. Specifically, the limitation involved solving a singlevelocity field for all bubble classes. To circumvent the problem, the inhomogeneous MUSIGmodel (Krepper et al., 2005), which divides the gaseous dispersed phase into N number ofvelocity groups, was further assessed to evaluate its feasibility in handling bubbly-to-slugtransition flow regime.

Transitional flow condition with liquid superficial velocity <jf>=0.986m/s and gassuperficial velocity <jg>=0.242m/s as measured by Hibiki et al. (2001) was investigated forthe application of both the homogeneous and inhomogeneous MUSIG models. Similar to thebubbly turbulent pipe flow simulations presented in previous section, the same boundaryconditions were specified for both models where details can be referred in Table 1. Bubblesize in the range of 0-10 mm discretized into 10 sub-size groups as tabulated in Table 2 wassimilarly considered. For the inhomogeneous model, two velocity fields were solvedrepresenting the travelling speed of small and big bubbles. The first five bubble classes (rangeof 0 – 5 mm) were assigned to the first velocity field while the remaining bubble classes(range of 5 – 10 mm) were assigned to the second velocity field. Sensitivity studies on theincreasing resolution greater than two velocity fields were also performed. With regards to themean parameters investigated, negligible differences were nonetheless found.

The measured and predicted local radical void fraction, Sauter mean bubble diameter,IAC and gas velocity distribution at the measuring station of z/D=53.5 are illustrated inFigure 18. Comparing the predicted Sauter mean diameters, the inhomogeneous MUSIGmodel was found to yield comparatively better prediction when compared against themeasured data. This could be attributed to the merit of splitting the bubble velocity with twoindependent fields which facilitated the model to re-capture the separation of small and bigbubbles caused by different lift force actuation. Nevertheless, notable discrepancies werefound when comparing against other variables (i.e. void fraction, gas velocity and IAC)against the experimental measurements. As depicted in Figure 18b, void fractions of bothmodels were obviously over-predicted at the channel core but under-predicted at the wallregion, which resulted in unsatisfactory IAC predictions (see Figure 18c). Interestinglyenough, the consideration of multiple velocity fields in the inhomogeneous MUSIG modeldid not contribute to the desired expected improvements when comparing the gas velocitypredictions against those of the homogeneous MUSIG model (see Figure 18d). This could bepossibly due to the interfacial force models which have been developed principally forisolated bubbles rather than on a swarm or cluster of bubbles. Direct applications of thesemodels for high void fraction conditions, where bubbles are closely packed, becomequestionable and introduce uncertainties in the model calculations. Such findings have alsobeen reported lately in the experimental work by Simonnet et al. (2007). Based on theirmeasurements, they concluded that the aspiration of bubbles in the wake of the leading onesbecame dominant if the void fraction exceeded the critical value 15%. This caused a sharpincrease of the relative velocity of bubbles and significantly altered the associated dragcoefficient. Interfacial forces models based on empirical correlations of isolated spherical


bubbles have also been found to be not appropriate (Jakobsen, 2001; Behzadi et al., 2004) forhigh void fraction conditions, As high void fraction (i.e. 20%) condition has been simulatedhere, the ambiguity of the interfacial drag forces could plausibly be the main source of error.Furthermore, the wall lubrication force could be under-predicted by the Antal’s model (1991)which could represent another source of error in calculating lateral interfacial forces (Lucas etal., 2007).

Figure 18. Predicted local radical void fraction, Sauter mean bubble diameter, IAC and gas velocitydistribution of transition bubbly-to-slug flow condition by homogenous and inhomogeneous MUSIGmodel.

On the other hand, coalescence due to wake entrainment and breakage of large bubblescaused by surface instability may prevail beyond the critical void fraction limit. The existingkernels which only featured coalescence due to random collision and breakage due toturbulent impact for spherical bubbles have to be extended to account for additional bubblemechanistic behaviours for cap/slug bubbles. Several papers in the literature have attemptedto deal with this problem by the development of a two-group interfacial area transportequation (Fu et al., 2002a,b; Sun et al., 2004a,b). Encouraging results attained thus far not


only suggested the feasibility of the proposed approach but more importantly the prevalenceof large bubble mechanisms that are substantially different from spherical bubble interactionsin turbulent gas-liquid flows.

6. Conclusions and Further Developments

In previous section, a complete three-dimensional two-fluid model coupled with the classmethod of population balance approach was presented to handle the complex hydrodynamicsand thermo-mechanical processes of various bubbly turbulent pipe flow conditions.Numerical study of isothermal bubbly turbulent pipe flows in a vertical pipe was first studiedin order to confine the complexity of solely modelling the bubble coalescence and bubblebreakage mechanisms. The homogenous MUSIG model which assumed all bubbles travellingwith the same velocity was applied. Comparison of the predicted results was made against themeasurements of Hibiki et al. (2001). Overall, the homogeneous MUSIG model yielded goodagreement for the local radial distributions of void fraction, interfacial area concentration,Sauter mean bubble diameter and gas and liquid velocities against measurements. Numericalresults also clearly showed that the range of bubbles sizes existed in the gas-liquid flowsrequired substantial resolution. Numerical results obtained through this study clearlydemonstrated the competence of the MUSIG model and the robustness of the bubblecoalescence and bubble breakage kernels in accommodating the interactions of finelydispersed bubbles within isothermal bubbly turbulent pipe flows.

The potential of the population balance approach was further exploited in modellingsubcooled boiling flows. Such flows by nature are inherently complex since theysimultaneously embrace all complex flow hydrodynamics, bubble coalescence and bubblebreakage accompanied by heat and mass transfer processes in the bulk flow, and varioussurface heat flux characterisations. An improved heat partition model which mechanisticallydetermined the bubble departure and lift-off diameter was also presented as a closure for theproblem in order to determine the appropriate evaporation rate of the nucleation process.Numerical results validated against the experiment data of Yun et al. (1997) and Lee et al.(2002) for low-pressure subcooled boiling annular channel flows showed good agreement forthe local Sauter mean bubble diameter, void fraction, IAC profiles. Detailed vapour sizedistribution and its corresponding generation rates were probed via the size fractions of themodified MUSIG model in order to better envisage the condensation effect in conjunctionwith the occurrence of bubble coalescence and bubble breakage within the gas-liquid flows.

In addition to the homogeneous MUSIG model, inhomogeneous MUSIG model was alsoapplied to investigate the feasibility of application in handling transition bubbly-to-slug flow.It was observed that the inhomogeneous MUSIG gave better prediction of the Sauter meanbubble diameter distributions when compared to the homogeneous model. The morecomplicated inhomogeneous MUSIG model provided the premise of better capturing thedifferent effects of the lift force acting on the small and large bubbles. Less encouragingresults were however ascertained between the predicted and measured void fraction,interfacial gas velocity and IAC profiles.

From the above numerical study, as an example of population balance modelling, someimportant numerical issues have been demonstrated. Firstly, it exposed the limitation of theexisting interfacial forces models. Most of the existing interfacial forces models were


developed and calibrated from isolated single particle which may not be strictly applicable ifparticles are closely packed. This is due to the fact that particle motion may be influenced byneighbouring particles resulting different momentum transfer. According to the recentexperimental study by Simonnet et al. (2007), significant decrement of gas bubble dragcoefficient was found if the gas void fraction exceeded 15%. In fact, very recently, someresearch studies have been performed attempting to improve the existing interfacial drag andlift force models (Hibiki and Ishii, 2007; Marbrouk et al., 2007; Liu et al, 2008)

Secondly, it unveiled the constraint of the current coalescence and breakage kernels. Forbubbly simulations presented in this review, coalescence and breakage kernels were derivedbased on the spherical bubble assumption. This assumption limits the kernels to be onlyapplicable to bubbly flow regime where bubbles are in spherical shape. This also explainswhy less encouraging results were obtained when bubbly-to-slug flow was considered. Inbubbly-to-slug flows, subject to the balance of the surface tension and surrounding fluidmotion, large bubbles may be deformed into cap or taylor bubbles. Thus, coalescence due towake entrainment may become significant which unfortunately was not modelled in currentadopted kernels. Unfortunately, fundamental knowledge of bubble coalescence and breakagemechanisms still remain elusive forming a bottleneck for the development of more robustpopulation balance kernels. In essence, not only for gas-liquid systems, similar problems andchallenges are also prevalent in other PBE applications; for example: soot formationprediction in combustion system (Frenklach and Wang, 1994). It is certainly that substantialresearch works would be centred in kernels development in near future.

Finally, although encouraging results had been obtained from the MUSIG model basedon CM, one should be also reminded that QMOM or other moment methods which representa rather sound mathematical approach and an elegant tool of solving the PBE with limitedcomputational burden could be alternatively considered for modelling practical multiphaseflow systems for future investigative studies.

Notation

aif = interfacial area concentrationAc = cross sectional area of the boiling channelAq = fraction of heater area occupied by bubblesC1, C2 = constants defined in equation (32)CD = drag coefficientCL = shear lift coefficientCp = specific heatCs = constant defined in equation (41)Cv = Acceleration coefficientd = vapor bubble diameter at heated surfacedw = surface/bubble contact diameterD = average bubble diameterDc,DB = bubble death rate due to coalescence and breakageDb = departing bubble diameterDd = bubble departure diameterDl = bubble lift-off diameter


Ds = Sauter mean bubble diameterf = bubble generation frequency

fi = scalar size fraction of the ith discrete bubble classes giif αα /=

F = degree of surface cavity floodingFc,FB = calibration factors for coalescence and breakageFb = buoyancy forceFcp = contact pressure forceFdu = unsteady drag force due to asymmetrical growth of the bubbleFh = force due to the hydrodynamic pressureFqs = Quasi-steady-drag forceFs = surface tension forceFsL = shear lift forceFlg = action of interfacial forces from vapor on liquidFgl = action of interfacial forces from liquid on vapor

dragFlg= drag force

liftFlg= lift force

ricationF lublg

= wall lubrication forcedispersionFlg

= turbulent dispersion force

g = gravitational constantg = gravitational vectorG = mass fluxGs = dimensionless shear rateh = interfacial heat transfer coefficienth0, hf = initial and critical film thicknesshfg = latent heatH = enthalpyJa = Jakob numberk = thermal or turbulent kinetic energyK = projected area of bubblehfg = latent heat of vaporizationls = sliding distanceni = number density of the discrete bubble ith classNa = active nucleation site densityP = pressurePk = turbulent kinetic energy production termPc,PB = bubble production rate due to coalescence and breakageQw = wall heat fluxQc = heat transfer due to forced convectionQe = heat transfer due to evaporationQtc = heat transfer (transient conduction) due to stationary bubbleQtcsl = heat transfer (transient conduction) due to sliding bubbler = bubble radius at heated wall or flow spacing within annular


channelrc = cavity radius at heated surfacerr = curvature radius of the bubble at heated surfaceR = source/sink term due to coalescence and breakageRe = bubble Reynolds numberRf = ratio of the actual number of bubbles lifting off to the number of

active nucleation sitesRph = source/sink term due to phase changeRi = radius of inner heated wallRo = radius of outer unheated walls = spacing between nucleation sitesSi = additional source terms due to coalescence and breakageSij = tensor of shear stressSt = Stanton numbert = timetg = bubble growth periodtsl = bubble sliding periodtw = bubble waiting periodT = temperatureΔT = difference in temperatureu = velocityu = velocity vectoruτ = friction velocityvi = specific volume of discrete bubble ith classx = cartesian coordinate along xx+ = non-dimensional normal distance from heated wally = cartesian coordinate along y

Greek Lettersα = advancing angleαg = vapor void fractionαl = liquid void fractionβ = receding angleχ = coalescence rateδ l = Thermal boundary layer thicknessε = turbulent dissipation rate

WNφ = bubble nucleation rate

CONDφ = bubble condensation rate

η = thermal diffusivity or coalescence volume matrixλ = size of an eddy

eλ = effective viscosityμ = viscosityθ = bubble contact angle


θi = inclination angleρ = densityσ = surface tension/Prandtl number

ijτ = bubble contact time

ω = turbulent frequencyΩ = breakage rateξ = size ratio between an eddy and a particle in the inertial subrange

Hξ = heated perimeter of boiling channel

Γlg = interfacial mass transfer from vapor to liquidΓgl = interfacial mass transfer from liquid to vapor

Subsriptsaxial = axial distributiong = vaporinlet = channel entrancel = liquidlocal = local distributions = surface heatert = turbulentsat = saturationsub = subcooledsup = superheatedw = heated surface wall

Acknowledgment

The financial support provided by the Australian Research Council (ARC project IDDP0877734) is gratefully acknowledged.

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Chapter 5

NUMERICAL ANALYSIS OF HEAT TRANSFERAND FLUID FLOW FOR THREE-DIMENSIONAL

HORIZONTAL ANNULI WITH OPEN ENDS

Chun-Lang Yeh*

Department of Aeronautical Engineering, National Formosa University,Huwei, Yunlin 632, Taiwan, R.O.C.

Abstract

Study of the heat transfer and fluid flow inside concentric or eccentric annuli can be applied inmany engineering fields, e.g. solar energy collection, fire protection, underground conduit,heat dissipation for electrical equipment, etc. In the past few decades, these studies wereconcentrated in two-dimensional research and were mostly devoted to the investigation of theeffects of convective heat transfer. However, in practical situation, this problem should bethree-dimensional, except for the vertical concentric annuli which could be modeled as two-dimensional (axisymmetric). In addition, the effects of heat conduction and radiation shouldnot be neglected unless the outer cylinder is adiabatic and the temperature of the flow field issufficiently low. As the author knows, none of the open literature is devoted to theinvestigation of the conjugated heat transfer of convection, conduction and radiation for thisproblem. The author has worked in industrial piping design area and is experienced in thisfield. The author has also employed three-dimensional body-fitted coordinate systemassociated with zonal grid method to analyze the natural convective heat transfer and fluidflow inside three-dimensional horizontal concentric or eccentric annuli with open ends. Owingto its broad application in practical engineering problems, this chapter is devoted to a detaileddiscussion of the simulation method for the heat transfer and fluid flow inside three-dimensional horizontal concentric or eccentric annuli with open ends. Two illustrativeproblems are exhibited to demonstrate its practical applications.

Keywords：three-dimensional horizontal concentric or eccentric annuli with open ends,conjugated heat transfer of convection, conduction and radiation, body-fitted coordinatesystem, zonal grid method.

* E-mail address: [email protected]. Tel. No.：886-5-6315527, Fax No.：886-5-6312415.

Chun-Lang Yeh172

Nomenclature

pC specific heat capacity

hD hydraulic diameter

Ec Eckert number , = ****

2*

/ refhwp kDqCV

ref

Fr Froude number , = *** / hDgVg gravitational acceleration

jkg contravariant metric tensorh convective heat transfer coefficientJ Jacobiank thermal conductivityPr Prandtl number , = ** /αυ

p hydraulic pressure , = 2**

****

V

ygp

ref

ref

ρ

ρ+

q heat fluxjq curvilinear coordinateΦR source term

Ra Rayleigh number , = ***3*** / αυβ TDg h ΔoRa modified Rayleigh number , = ****4*** / kqDg wh αυβ

T temperature(u,v,w) physical velocity , = ( *** ,, wvu ) / *V

jV contravariant velocity*V characteristic velocity, = ** / hDα

(x,y,z) Cartesian coordinates , = ( *** ,, zyx ) / *hD

α thermal diffusivity； also radiation absorptivityβ thermal expansion coefficientΦΓ diffusion coefficient

ε radiation emissivity

θ non-dimensionalized temperature , = ***

**

/ kDqTT

hw

ref−

μ viscosityυ kinematic viscosityρ density , = ** / refρρ

Numerical Analysis of Heat Transfer and Fluid Flow… 173

σ Stefan-Boltzmann constantΦ energy dissipation term； also dependent variableφ azimuthal angle

Subscripts

i inner cylindernb neighboring grid pointso outer cylinderP main grid pointref reference state（）at atmospheric pressure and room temperaturew wall

Superscripts

averaged quantity* dimensional quantity

Introduction

Heat transfer and fluid flow inside concentric or eccentric annuli can be found innumerous engineering applications, e.g. solar energy collection, fire protection, undergroundconduit, heat dissipation for electrical equipment, etc. In the past few decades, these studieswere concentrated in two-dimensional research and were mostly devoted to the investigationof the effects of convective heat transfer. However, in practical situation, this problem shouldbe three-dimensional, except for the vertical concentric annuli which could be modeled astwo-dimensional (axisymmetric); in addition, the effects of heat conduction and radiationshould not be neglected unless the outer cylinder is adiabatic and the temperature of the flowfield is very low. As the author knows, none of the open literature is devoted to theinvestigation of the conjugated heat transfer of convection, conduction and radiation for thisproblem. Numerous theoretical and experimental studies on natural convection in horizontalconcentric or eccentric annuli have been conducted. In most of these studies, a two-dimensional model was used in which the annuli were assumed to be infinitely long andcoupled with thermal boundary conditions on the cylinder surfaces specified as either withtwo constant wall temperatures or one with constant wall temperature while the other withconstant wall heat flux (including adiabatic surface) [1-56]. Some representative studies arelisted below.

1. Gju et al. [2,3] conducted experiments for the natural convection in horizontalconcentric or eccentric annuli. The inner and outer cylinders were both kept atconstant temperature. The instability and transition of the heat transfer and fluid flowwere investigated. It is found that for a concentric annulus of inner/outer diameterratio 2.36, chaos occur near Rayleigh number between 0.9×105 and 3.37×105.

Chun-Lang Yeh174

2. Kuehn and Goldstein [7-9] also performed experimental investigations for the naturalconvection in horizontal concentric or eccentric annuli. The inner and outer cylinderswere both kept at constant temperature. The instability and transition of heat transferand fluid flow were investigated. It is found that for a concentric annulus ofinner/outer diameter ratio 2.6, transition from laminar to turbulent regimes occursnear Rayleigh number of 4×106.

3. Vafai et al. [11-20] studied numerically the two- and three-dimensional naturalconvection in horizontal concentric annuli. The heat transfer and fluid flow underdifferent inner/outer diameter ratios and Rayleigh numbers were investigated. Thescope of discussion covers laminar and turbulent regimes as well as steady andtransient flows. In respect of the fluid flow field, the interaction of primary andsecond flows was examined. With respect to the thermal field, the distribution ofNusselt number was analyzed. In addition, the influence of geometry on the heattransfer and fluid flow was also discussed.

4. Yoo et al. [21-27] performed two-dimensional simulation of natural or mixedconvection in horizontal concentric annuli. The outer cylinder was kept at constanttemperature while the inner cylinder was kept at constant temperature or heat flux.The heat transfer and fluid flow were analyzed for different Rayleigh numbers,Reynolds numbers, Prandtl numbers and inner/outer diameter ratios. The scope ofdiscussion covers the transition of laminar to turbulent regimes as well as thephenomenon of chaos.

5. Shu et al. [28-33] studied numerically the two-dimensional natural convection inhorizontal concentric or eccentric annuli. The outer cylinder is a square duct kept at alower temperature while the inner cylinder is a circular one kept at a highertemperature. The heat transfer and fluid flow were investigated for different Rayleighnumbers, Reynolds numbers, Prandtl numbers, inner/outer diameter ratios andeccentricity of inner/outer cylinders. In respect of the thermal field, the distributionof Nusselt number was analyzed.

6. Mujumdar et al. [34-36] investigated numerically the two-dimensional naturalconvection and phase change in horizontal concentric annuli. The outer cylinder is asquare or circular duct while the inner cylinder is a circular or square one. The heattransfer and fluid flow were examined for different Rayleigh numbers and heatingrates on the inner or outer cylinders.

7. El-Shaarawi et al.[37-39] studied numerically the laminar mixed convection inhorizontal concentric annuli with non-uniform circumferential heating. Secondaryflow and Nusselt number were analyzed in their study. They also investigated thetransient conjugated natural convection and conduction in open-ended verticalconcentric annuli, which could be modeled as two-dimensional flow because of itsaxisymmetry. The authors also analyzed the free convection in vertical eccentricannuli with a uniformly heated boundary, which has to be simulated by three-dimensional model due to the eccentricity.

8. Mota et al. [40-42] simulated the two-dimensional natural convection in horizontalconcentric or eccentric annuli of elliptic or circular cross sections. The heat transferand fluid flow were analyzed for different inner/outer diameter ratios and eccentricityof inner/outer pipes. They found that the eccentric elliptic annulus has the lowest heatloss.


9. Char et al. [43-45] investigated numerically the two-dimensional natural or mixedconvection in horizontal concentric or eccentric annuli. The outer cylinder was keptat a lower temperature while the inner cylinder was rotating and kept at a highertemperature. Low Reynolds number turbulence model was adopted to account for theturbulence effects. The heat transfer and fluid flow were analyzed for differentRayleigh numbers, Reynolds numbers and inner/outer diameter ratios.

10. Ho et al. [46-48] performed experimental and numerical investigations for the two-dimensional natural convection in horizontal concentric or eccentric annuli. Theouter cylinder was kept at constant temperature while the inner cylinder was kept atconstant heat flux. The heat transfer and fluid flow were investigated for differentRayleigh numbers, Prandtl numbers and eccentricity of inner/outer cylinders. Theyfound that the heat and fluid flow were affected mainly by the Rayleigh number andeccentricity of inner/outer cylinders, while the Prandtl number had only minor effecton this flow.

11. Mizushima et al. [49,50] studied the instability and transition of two-dimensionalnatural convection in horizontal concentric annuli. The outer cylinder was kept at alower temperature while the inner cylinder was at a higher temperature. The heattransfer and fluid flow were analyzed for different Prandtl numbers. The authors alsoinvestigated the linear stability of the flow.

12. Hamad et al. [51,52] conducted experiments for the natural convection in inclinedannuli with closed ends. The inner cylinder was kept at constant heat flux. The heattransfer and fluid flow were investigated for different Rayleigh numbers, Nusseltnumbers, inclined angles and inner/outer diameter ratios. They found that the heatand fluid flow were affected mainly by the Rayleigh number and inner/outerdiameter ratios, while the inclined angle had only minor effect on this flow.

13. Raghavarao and Sanyasiraju [53,54] studied numerically the two-dimensional naturalconvection in horizontal concentric or eccentric annuli. The inner and outer cylinderswere both kept at constant temperature. The heat transfer and fluid flow wereinvestigated for different inner/outer diameter ratios and eccentricity of inner/outercylinders.

14. Choi and Kim [55,56] investigated numerically the three-dimensional linear stabilityof mixed-convective flow between rotating horizontal concentric cylinders. The innercylinder was rotating and kept at constant heat flux. The authors found that theheating of the inner cylinder delays the formation of Taylor vortices when therotating effect is more pronounced than the buoyancy effect. On the other hand,when rotation and buoyancy are comparable, the flow becomes unstable. Thisinstability is caused mainly by the buoyancy effect.

From the above literature review, it can be seen that there have been very few three-dimensional investigations of the heat and fluid flow in concentric or eccentric annulibetween two horizontal cylinders with open ends, except with the configuration of cavity type[1,15,16]. Owing to the broad application in practical engineering problems, this chapter isdevoted to a detailed discussion of the simulation method for the heat transfer and fluid flowinside three-dimensional horizontal concentric or eccentric annuli with open ends, with theaim to get a more thorough understanding of this problem.

Chun-Lang Yeh176

Mathematical Model

For convenience of illustration, take the underground conduit of electrical power cable asan example. In this section, two situations are discussed. One is a purely natural convectiveflow and the other is a conjugated heat and fluid flow which involves conduction, convectionand radiation.

2.1. Natural Convective Flow

Heat is generated from the electrical resistance of the power cable and the heatdissipation process in the annulus relies on the natural convection heat transfer from bothopen ends of the conduit, which penetrate onto the manhole surfaces. The heat dissipation rateis determined from the ventilation stemmed from natural convection and will affect thelifetime of the power cable. Owing to the symmetric nature of the flow field with respect tothe two open ends and to a vertical plane crossing the center of the cylinders, thecomputational domain is schematically shown in Figure1. Selected transverse andlongitudinal sections are illustrated in Figure 2.

(a)



(b)

Figure 1. Illustration of the computational domain and zonal grid distribution - (a) overall view (b)zoom-in view near the interface of the two zones.

Unlike the vertical annuli in which the fully developed thermal boundary conditions maybe achievable[39], the boundary conditions at the open ends of the present problem are muchmore troublesome. Although this plane is named as the outlet (see Figure 1) in considerationof the heat dissipation route in the conduit, it consists of inflow (fresh flowing fluid) andoutflow (heated flowing fluid) at the same plane. An approach used in the simulations of thenon-cavity type, buoyancy-induced flows is the “zonal grid” approach[57-60] which extendsthe computational domain outside the outlet plane so that the boundary conditions can bereasonably specified with the ambient flow properties. Typical examples can be found inRefs.59 and 60. Although this approach requires enormous computations for three-dimensional problems, it provides more reliable results among existing approaches. In thiswork, the “zonal grid” approach is adopted to resolve the problem of the outlet boundaryconditions. In the following discussion, two different cases are discussed. The first is anumerical investigation of the three-dimensional natural convection inside a horizontalconcentric annulus with specified wall temperature or heat flux[59] and the second is a three-dimensional natural convection inside horizontal concentric or eccentric annuli with specifiedwall heat flux[60].

The boundary conditions for the first problem as illustrated schematically in Figure 1 arestated as follows. Either adiabatic or isothermal condition is given on the outer cylindersurface, while on the inner cylinder a constant heat flux is given. For the isothermal condition,the outer cylinder surface is maintained at Tref（）i.e. 300 K . No slip condition is given to allthe three components of the velocity on the outer and inner cylinder surfaces. Since no flowcrosses the circumferentially (or longitudinally) symmetric plane, the angular (or axial)

Chun-Lang Yeh178

velocity vanishes on that plane. The angular (or axial) derivatives of the remaining velocitycomponents and temperature also vanish on the circumferentially (or longitudinally)symmetric plane.

Figure 2. Illustration of the selected - (a) transverse sections for eccentric annulus, (b) transversesections for concentric annulus, and (c) longitudinal sections.

The boundary conditions for the second problem are stated as follows. The adiabaticcondition is given on the outer cylinder surface, while on the inner cylinder a constant heatflux, which in practical situation is resulted from the heat generation of the power cable dueto electrical resistance, is given. No slip condition is given to all the three components of thevelocity on the outer and inner cylinder surfaces. Since no flow crosses the circumferentially(or longitudinally) symmetric plane, the angular (or axial) velocity vanishes on that plane.The angular (or axial) derivatives of the remaining velocity components and temperature alsovanish on the circumferentially (or longitudinally) symmetric plane.


Because the thermal boundary conditions at the inner cylinder surfaces are specified interms of heat fluxes instead of temperatures in the above two example problems, a modifiedRayleigh number is defined as follows.

)/( *****

3***

refhwrefref

hrefo kDqDg

Raαυ

β= (1)

As pointed out by Kuehn and Goldstein [8], an onset of transition from laminar toturbulent regimes starts near 6104×=Ra for gases in a concentric annulus of

6.2/ =io DD . Moreover, a study by Labonia and Gju [3] indicated that chaotic flows were

observed in the range of 55 1037.3109.0 ×≤≤× Ra for a concentric annulus of36.2/ =io DD . However, their conclusions were drawn from the experimental observations

and based upon the cases associated with the constant temperature differences between theouter and inner cylinders. Kumar [6] made a numerical investigation using a two-dimensionalmodel for an infinitely long, horizontal, concentric annulus where the inner cylinder wasspecified by a constant heat flux and the outer cylinder was isothermally cooled. He foundthat the critical oRa above which the numerical results failed to converge were 5101.3 × and

6103× at 5.1/ =io DD and 2.6, respectively. Kumar also pointed out that it was hard tojudge whether the flow would become oscillatory or three-dimensional beyond the critical

oRa for a given ratio of io DD / . The present example problems, which are essentially

three-dimensional with the medium of air（ 7.0Pr = ） , encounter the same convergence

difficulty beyond )10( 7ORao = , which may imply an onset of transition from steadylaminar to chaotic or even turbulent flows. Here, two cases of Rayleigh numbers,

510=oRa and 610=oRa , are calculated and demonstrated.The flow pattern of interest here necessitates the solution of three-dimensional fully

elliptic type of partial differential equations, which describe the natural convective flow field.Considering the steady state flow situation, the governing equations in Cartesian coordinatesread：

( ) ( ) ( ) 0ρwz

ρvy

ρux

=∂∂

+∂∂

+∂∂

(2)

( ) ( ) ( )

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

∂∂

+∂∂

+∂∂

+∂∂

⋅

+∂∂

−=∂∂

+∂∂

+∂∂

zw

yv

xu

xzu

yu

xu

xpwu

zvu

yuu

x

31Pr 2

2

2

2

2

2

ρρρ

(3)

Chun-Lang Yeh180

( ) ( ) ( )

22

2

2

2

2

2 131Pr

Frzw

yv

xu

yzv

yv

xv

ypwv

zvv

yuv

x

ρ

ρρρ

−+⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

∂∂

+∂∂

+∂∂

+∂∂

⋅

+∂∂

−=∂∂

+∂∂

+∂∂

(4)

( ) ( ) ( )

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

∂∂

+∂∂

+∂∂

+∂∂

⋅

+∂∂

−=∂∂

+∂∂

+∂∂

zw

yv

xu

zzw

yw

xw

zpww

zvw

yuw

x

31Pr 2

2

2

2

2

2

ρρρ

(5)

( ) ( ) ( )

Π−Φ⋅⋅+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

⋅

+∂∂

+∂∂

+∂∂

=∂∂

+∂∂

+∂∂

Eczpw

ypv

xpuEc

zyxw

zv

yu

x

Pr

2

2

2

2

2

2 θθθθρθρθρ(6)

where

2

222222

32

2

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

−

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎠⎞

⎜⎝⎛∂∂

=Φ

zw

yv

xu

yw

zv

xw

zu

xv

yu

zw

yv

xu

(7)

and**** / wqgVμ=Π (8)

Note that the Boussinesq approximation usually made in the formulation of the naturalconvection is not adopted here and the density is determined using the ideal gas law. Thereason of using the ideal gas law rather than Boussinesq approximation for densitydetermination is that the Boussinesq approximation is not valid in case of high temperaturedifference, which may arise when the Rayleigh number exceeds a critical value.

2.2. Conjugated Heat and Fluid Flow

In section 2.1, the effects of heat conduction and radiation are neglected. This assumptionis valid only if the outer cylinder（concrete conduit） is adiabatic and the temperature of theflow field is low enough. A practical underground power cable placed in a concrete conduitcan be schematically shown as Figure 3. To give a preliminary estimate for the influence ofthe conduction and radiation heat transfer on this problem, the two-dimensional simplified


model shown in Figure 4 is analyzed. Consider conjugated conduction and convection heattransfer first.

solar beam

concreteconduitsoil

power cable

Figure 3. Illustration of a practical underground power cable placed in a concrete conduit.

concrete conduit2cm

power cable

air

r2

r1= do /2di /2

soil

Figure 4. Illustration of a two-dimensional simplified model for the underground power cable placed ina concrete conduit.

Chun-Lang Yeh182

Assume that the concrete conduit is in thermal equilibrium with the surrounding soil,diameter of the power cable is di=0.1m, the inner radius of the concrete conduit isr1(=do/2)=0.1m，and its thickness is r2-r1=2cm, then the heat transfer（which includes bothconduction and convection）from the soil to the air is[61]

xxmsoil

xmsoil

xxmsoilxmsoil

TTTT

TTTTxPUq

Δ+

Δ+

−−

−−−Δ=

||ln

)|()|()( ,

where the conjugated heat transfer coefficient, U , is

1

21 ln11

rr

kr

h

U

concreteair

+= (9)

kconcrete=0.7W/m．K is the thermal conductivity of the concrete conduit.

From Figure 4 of Ref.7, the Nusselt number Nu= 10≅kDh ho

. Then, the convection

heat transfer coefficient for the air inside the annulus is

ho Dkh /10= = 7.21.0/107.210 2 =×× − KmW ⋅2/ (10)

where

1.01.02.0 =−=−= ioh ddD m;

thermal conductivity for air 2107.2 −×=airk KmW ⋅/ . The conjugated heat transfer

coefficient, U , can then be found by substituting oh from Eq.(10) into hair in Eq.(9).

52.2

1.012.0ln

7.01.0

7.21

1=

+=U KmW ⋅2/

The error caused by the neglect of conduction heat transfer then is

%1.752.2

52.27.2=

−=

−U

Uho

The above analysis can give us a preliminary estimate of the influence of the conductionheat transfer on this problem. In fact, it can be observed from Eq.(9) that the conjugated heat


transfer coefficient, U , is closely connected with the outer/inner radius ratio（r2/r1）of theconcrete conduit, which is related to the thickness of the concrete conduit. For a thin concreteconduit, r2/r1 approaches unity and ln(r2/r1) is very small. Under such situation, theconduction heat transfer can be neglected. However, in practical engineering problems, theconcrete conduit can not be too thin because of the high pressure it sustained（due to the soilfor a land cable or the sea water for a submarine cable）. Therefore, the conduction heattransfer should be taken into account in practical engineering applications.

To illustrate the influence of the radiation heat transfer on this problem, the two-dimensional simplified model（Figure 4）is analyzed again. Assume the surface temperatureof the power cable to be Ti , radiation emissivity 88.0≅iε （for rubber）, and the inner

surface temperature of the concrete conduit to be To , radiation emissivity 91.0≅oε （for

concrete）. Also assume that the surface of the power cable and the inner surface of theconcrete conduit are gray surfaces, i.e. radiation emissivity and absorptivity areequal（ αε = ）. Then, the net radiation heat transfer on the surface of the power cable canbe estimated as

pioiiipoiooopicabler AFTAFTAq →→ −=× 44'' σεσε

where the view factors

po

piio A

AF =→ ， 1=→oiF ， 81067.5 −×=σ 42/ KmW ⋅

Then

])([ 44''

o

iioocabler T

TTq εεσ −=

whereo

i

TT

can be written as oo

o

o

i

TT

TTT

TT Δ

+=Δ+

= 1 .

From the author’s previous studies[59,60], KT 1≈Δ , KTo 301≈ for 510=oRa ,

and KT 12≈Δ , KTo 316≈ for 610=oRa , where the modified Rayleigh

number（ oRa ）is defined in Eq.(1).

This yieldsoTTΔ

<< 1 ⇒ ooo

i

TT

TT

TT Δ

+≈⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ+=⎟⎟

⎠

⎞⎜⎜⎝

⎛411

44

.

When 510=oRa , the net radiation heat transfer from the surface of the power cablethen is

Chun-Lang Yeh184

5.8)]301141(88.091.0[3011067.5 48'' ≈×+−×××= −

cablerq W/m2 (11)

and when 610=oRa , the net radiation heat transfer from the surface of the power cable is

6.58)]3161241(88.091.0[3161067.5 48'' −≈×+−×××= −

cablerq W/m2 (12)

On the other hand, from the definition of the modified Rayleigh number（Eq.(1)）, theheat transfer from the surface of the power cable is

3*

2*

****

3***

*****

Pr1)/()/(

h

ref

ref

o

hrefo

href

refrefhrefw

DgRaDkRa

DgDkq

⋅==

υ

ββαυ

where 1.0* =hD m, 2* 107.2 −×=refk KmW ⋅/ for air , Pr = 0.711, and

72*

**

102.13 ×=ref

refg

υ

β ( Km ⋅3/1 ) .

Therefore, when510=oRa , the heat transfer from the surface of the power cable is

29.0* =wq 2/ mW (13)

and when610=oRa , the heat transfer from the surface of the power cable is

9.2* =wq 2/ mW (14)

Comparisons of Eq.(11) with (13) and Eq.(12) with (14) reveal that the radiation heattransfer plays an important role in this problem.

Although the above analysis is only a two-dimensional simplified analysis, it can give usa preliminary view of the influence of the conduction and radiation heat transfer on thisproblem. The analysis shows that conduction and radiation heat transfer should not beneglected unless the outer cylinder is adiabatic and the temperature of the flow field is verylow.

In respect of the conjugated heat transfer of convection and conduction, Ha and Jung[62]investigated numerically the three-dimensional conjugated heat transfer of conduction andnatural convection in a differentially heated cubic enclosure with a heat-generating cubicconducting body. Méndez and Treviño[63] made a numerical study for the conjugated


conduction and natural convection heat transfer along a thin vertical plate with non-uniforminternal heat generation. Liu et al.[64] performed a two-dimensional numerical analysis of thecoupled conduction-convection problem for an underground rectangular duct containing threeinsulated cables. Du and Bilgen[65] studied numerically the coupling effect of the wallconduction with natural convection in a rectangular enclosure. Kim and Viskanta[66]analyzed numerically the effects of wall conductance on the natural convection in differentlyoriented square cavities. A common feature of the above studies for the conjugatedconduction-convection heat transfer is to consider the individual heat transfer mode in eachmedium and to keep continuities of the temperature and heat flux at the interfaces of any twodifferent media. Take the underground power cable as an example, the heat transfer modesinclude conduction of the power cable (solid), convection of the air inside the concreteconduit (gas), conduction of the concrete conduit (solid) and conduction of the soil (solid).Heat transfer modes in each medium have to be solved separately. Continuities of thetemperature and heat flux at the interfaces of any two different media provide the necessaryboundary conditions.

With respect to the influence of the radiation heat transfer [67-73], the zone model andMonte Carlo method are generally recognized as more accurate ones. However, these twomethods require enormous computing time and storage. In addition, they are not ofdifferential type and therefore are more cumbersome to incorporate with the differentialequations of fluid motion and energy. Perhaps the most commonly used radiation model forengineering application is the flux model. Recently, the finite volume method （FVM）andthe discrete ordinates method（DOM）have become more and more popular because of theirsuccess in simulating irregular geometries [69-72]. In using the FVM and DOM, thesimulation domain is divided into small blocks. This is quite similar to the concept of thefinite volume method in computational fluid dynamics. The DOM needs a quadrature setwhich has a crucial influence on the accuracy of the method. On the other hand, the FVM isless restricted and is better for the conservation of radiation heat transfer.

As pointed out earlier, an onset of transition from laminar to turbulent regimes starts near710=Ra . Among the existing turbulence simulation methods, Direct Numerical

Simulation（）（）DNS and Large Eddy Simulation LES can simulate the actual physics ofturbulence more accurately. However, using them for routinely designing and analyzingengineering problems can not be achieved nowadays. DNS is limited to low Reynoldsnumber flows because of the large computer resources needed. Although LES reduces therequirement to some extent; however, much more sophisticated sub-grid scale models andnear-wall treatments are required for many flows of interest. A recent research of the LESwas conducted by Worthy and Rubini [74] who studied LES stress and scalar flux sub-gridscale models in the context of buoyant jets. Reynolds stress model (RSM) is essentially morerealistic than the Eddy viscosity model (EVM) because it involves modeled transportequations for all of the Reynolds stresses. Yilbas et al. [75] employed the RSM to examine jetimpingement onto a hole with a constant wall temperature and analyzed the influence of thehole wall temperatures and jet velocities. In their study, the Nusselt number ratio (ratio of theNusselt number predicted to the Nusselt number obtained for a fully developed turbulent flowbased on the hole entrance Reynolds number) was computed and the mass flow ratio (ratio ofmass flow rate through the hole to mass flow emanating from the nozzle) was determined. Onthe other hand, even though the EVM assumes a crude relation between the turbulence

Chun-Lang Yeh186

quantity（）（）Reynolds stress and the mean flow quantity strain rate , turbulence modelsbased on this concept, such as the standard k-ε model, have been widely used for industrialCFD calculations due to numerical stability and low cost. However, it has long beenrecognized that the standard k-ε model has defects in predicting turbulent flows with largestreamline curvature and rotational effects. Consequently, the more sophisticated two-equation models employ the Low Reynolds Number（）LRN Model, the Non-Linear EddyViscosity Model（）（）NEVM or the Algebraic Reynolds Stress Model ARSM to improve thesuitability of the standard k-ε model. Brescianini and Delichatsios [76] examined severalversions of the k-ε turbulence model as to their suitability for computational fluid dynamicsmodeling of free turbulent jets and buoyant plumes. Yang and Ma [77] investigated thepredictive performance of linear and nonlinear eddy-viscosity turbulence models for aconfined swirling coaxial jet. Abdon and Sundén [78] carried out an investigation of a singleround unconfined impinging air jet under different flow and geometrical conditions to assessthe performance of linear and nonlinear two-equation turbulence models. Wei et al. [79]proposed and formulated an algebraic turbulent mass flux model (AFM), which properlyaccounts for swirl–turbulence interactions, and incorporated this model with an algebraicReynolds stress model proposed previously to simulate the swirling turbulent flow andmixing in a combustor with helium/air jet and swirling air stream. In the author’s previousstudy[80], the performances of three LEVM, one ARSM and one DRSM（ DifferentialReynolds Stress Model） turbulence models are evaluated for the simulation of the plain-orifice atomizer and the pressure-swirl atomizer flows. These models include the standard k-εmodel[81], Launder-Sharma’s LRN k-ε model[82], Nagano-Hishida’s LRN k-ε model[83],Gatski-Speziale’s ARSM model[84], and Randriamampianina-Schiestel-Wilson’s DRSMmodel[85]. Adequate evaluations can help users to select a suitable turbulence model for theirapplications.

Numerical Method

The governing equations stated in last section can be cast into the following general form,which permits a single algorithm to be applied.

( ) ΦΦ +⎟⎟⎠

⎞⎜⎜⎝

⎛

∂Φ∂

Γ∂∂

=Φ∂∂ R

xxv

x jjj

j

ρ (15)

To facilitate the handling of complex geometry of the present problem, the body-fittedcoordinate system is used to transform the physical domain into a computational domain,which is in a rectangular coordinate system with uniform control volumes. Transformation ofEq.(15) to the body-fitted coordinates leads to

( ) ΦΦ +⎟⎟⎠

⎞⎜⎜⎝

⎛∂Φ∂

Γ∂∂

=Φ∂∂ JR

qJg

qVJ

q kjk

jj

j ρ (16)


where jq ：（curvilinear coordinates ζηξ ,, ）

J：；Jacobian ≡ ζξηηζξξηζηξζξζηζηξ zyxzyxzyxzyxzyxzyx −−−++jV ： contravariant velocity（ U,V,W）

[ ])()()(1ηζζηζηηζηζζη yxyxwzxzxvzyzyu

JU −+−+−=

[ ])()()(1ζξξζξζζξζξξζ yxyxwzxzxvzyzyu

JV −+−+−=

[ ])()()(1ξηηξηξξηξηηξ yxyxwzxzxvzyzyu

JW −+−+−=

jkg ： metric tensor

[ ]2222

11 )()()(1ηζζηζηηζηζζη yxyxzxzxzyzy

Jg −+−+−=

[ ]2222

22 )()()(1ζξξζξζζξζξξζ yxyxzxzxzyzy

Jg −+−+−=

[ ]2222

33 )()()(1ξηηξηξξηξηηξ yxyxzxzxzyzy

Jg −+−+−=

[]))((

))(())((12

2112

ζξξζηζζη

ξζζξζηηζζξξζηζζη

yxyxyxyx

zxzxzxzxzyzyzyzyJ

gg

−−+

−−+−−==

[]))((

))(())((12

3113

ξηηξηζζη

ηξξηζηηζξηηξηζζη

yxyxyxyx

zxzxzxzxzyzyzyzyJ

gg

−−+

−−+−−==

[]))((

))(())((12

3223

ξηηξζξξζ

ηξξηξζζξξηηξζξξζ

yxyxyxyx

zxzxzxzxzyzyzyzyJ

gg

−−+

−−+−−==

Chun-Lang Yeh188

The grid layout is constructed by connecting the grid points in each transverse plane,which are generated by solving the two-dimensional elliptic type of partial differentialequations governing the distribution of the grid points [86]. Numerical calculation of Eq.(16)is performed using the control-volume based finite difference procedure. The discretizedgoverning equations are solved on a non-staggered grid system in association with theSIMPLEC algorithm [87] and QUICK scheme [88].

In the use of the “zonal grid” approach[57-60], the computational domain is extendedoutside the outlet plane and is divided into two sub-domains of Zones I and II, asschematically illustrated in Figure 1. Zone I and Zone II are overlapped by three layers of gridpoints. The outer cylinder is flush with the adiabatic solid wall while the inner cylinderextends to the free boundary. The boundary conditions for Zone I have been stated in“MATHEMATICAL MODEL” and will not be repeated here. The boundary conditions ofZone II are as follows. On the surface of the inner cylinder, the same boundary conditions asspecified for Zone I are used. On the adiabatic solid wall, the no-slip condition and adiabaticwall are specified. The condition of zero normal gradients is met on the symmetric planeexcept for the normal velocity component, which vanishes naturally. On the free boundaries,the normal gradients of all the dependent variables except for the temperature are set to bezero. The temperature conditions at the free boundaries are specified as follows： When theflow at any such boundary is leaving the domain, the normal temperature gradient is taken aszero. However, when the flow comes into the domain, its temperature is assigned to that ofthe ambient.

The treatment of the interface of the two zones follows the overlapping grid method [57].Starting with guessed values, solutions in Zone I together with the overlapped region areupdated by one sweep of iteration. The updated solutions at the outlet plane of Zone I are theninterpolated by bilinear interpolation as the boundary conditions of Zone II. The solutions inZone II are then updated by one sweep of iteration and are used, in turn, to interpolate thevalues in the overlapped region, which provide the new boundary conditions for theresolutions of Zone I together with the overlapped region. This completes a full solutioncycle. In each solution cycle, continuity of the dependent variables and conservation of fluxesare preserved across the interface. In fact, this is the key to the success of the approach. Thesolution cycle is repeated until the convergence criterion is satisfied. The convergencecriterion is described below.

The general form of the discretized governing equations can be written as

Φ+Φ=Φ ∑ bAAnb

nbnbPP (17)

Define

ΦΦ −Φ−Φ= ∑ bAABnb

nbnbPPi (18)

ΦΦ +Φ+Φ= ∑ bAAHnb

nbnbPPi (19)


( )∑ Φ

ΦΦ =

ii

i

HN

B1max

λ (20)

where i is an arbitrary grid point in the computational domain and N is the total number ofgrid points. When 4100.1 −Φ ×≤λ for each dependent variable Φ in both the two zones,the iteration process is convergent.

Illustrative Problems

In this section, two illustrative problems are discussed. The first is a numericalinvestigation of the three-dimensional natural convection inside a horizontal concentricannulus with specified wall temperature or heat flux[59] and the second is a three-dimensional natural convection inside horizontal concentric or eccentric annuli with specifiedwall heat flux[60]. Illustration of the computational domain and zonal grid distribution areschematically shown in Figure 1. The inner and outer diameters of Zone I are 0.1m and 0.2m,respectively, and its length is 3m (equivalent to 30Dh

* ’s). Zone II is constructed by a cubewith side length of 20 Dh

* ’s to assure its boundary conditions being reasonably specified bythe ambient properties. Numerical tests reveal that the maximum difference in mass inflowrate at the outlet plane between 51×51×101（）radial by angular by axial and 61×61×121grid meshes for Zone I, while the corresponding grid meshes for zone II are 21×41×21（ x byy by z） and 31×61×31, respectively, is less than 0.5﹪. Therefore, the former set of gridmesh is adopted in the present work. Figure 2 illustrates the section positions of theconfiguration being studied.

4.1. Natural Convection inside a Horizontal Concentric Annulus withSpecified Wall Temperature or Heat Flux

For the first illustrative problem, both the adiabatic and isothermal conditions for theouter cylinder surface are examined for the conditions of ro / ri = 2 and Rao=106 to investigatethe natural convection heat dissipation inside the conduit.

Figure 5 shows the velocity vector plots at selected transverse and longitudinal sectionsfor the examined cases. Note that the results for sections A-A, B-B and D-D are the verticalprojections of the flow field and therefore the ordinate is y, while the result for section C-C isthe horizontal projection of the flow field and therefore the ordinate is x. The flow patternsfor the two cases are rather different. As observed from the velocity vector plots on thelongitudinal sections, there appear secondary flows in all the six longitudinal sections for bothcases. The secondary flow of the adiabatic case is stronger than that of the isothermal case.Further downstream, the secondary flows evolve into counter-rotating recirculation zones forthe adiabatic case. Another interesting phenomenon can be observed from the velocity vectorplots on the transverse sections. For both cases, the inflow paths are clearly observed in theportion below the inner cylinder, whereas the outflow paths are in the portion above. Suchflow patterns result obviously from the buoyancy effect, which can be observed in thermal

Chun-Lang Yeh190

plume phenomena. This can also be seen from the velocity vector plots at a distance slightlyoutside the open end (i.e. in Zone II) shown in Figure 6, in which the entrainment effectcaused by the upward motion of the buoyant flows can also be seen.

0 10 20 30z

0.9

1

1.1

1.2

y

Section B-B

0 10 20 30z

0.6

0.8

1

x

Section C-C

0 10 20 30z-0.2

-0.1

0

0.1

y

Section D-D

0 10 20 30z-0.5

0

0.5

1

1.5y

Section A-Aadiabatic

1000V*

0 0.5 1-0.5

0

0.5

1

1.5

x

y

500V*z=0

0 0.5 1-0.5

0

0.5

1

1.5

x

y

z=6

0 0.5 1-0.5

0

0.5

1

1.5

y

x

z=12

0 0.5 1-0.5

0

0.5

1

1.5

y

x

z=18

0 0.5 1-0.5

0

0.5

1

1.5

y

x

z=24

0 0.5 1-0.5

0

0.5

1

1.5

y

x

z=30



0 10 20 30z

0.6

0.8

1

x

Section C-C0 10 20 30z

0.9

1

1.1

1.2

y

Section B-B0 10 20 30z

-0.5

0

0.5

1

1.5

y

Section A-Aisothermal

1000V*

0 10 20 30z-0.2

-0.1

0

0.1

y

Section D-D

0 0.5 1-0.5

0

0.5

1

1.5

x

y

1000V*z=0

0 0.5 1-0.5

0

0.5

1

1.5

x

y

z=6

0 0.5 1-0.5

0

0.5

1

1.5

y

x

z=12

0 0.5 1-0.5

0

0.5

1

1.5

y

x

z=18

0 0.5 1-0.5

0

0.5

1

1.5

y

x

z=24

0 0.5 1-0.5

0

0.5

1

1.5

y

x

z=30

Figure 5. Velocity vector plots at selected transverse and longitudinal sections.

Chun-Lang Yeh192

0 1 2x

-2

0

2

4

y

adiabatic 1000V*

0 1 2x

-2

0

2

4

isothermal

y

Figure 6. Velocity vector plots near the end of the annulus (z=30.6).

0 0.5 1-0.5

0

0.5

1

1.5

3

5

4

32

1

5Level T(K)6 3155 3134 3113 3092 3071 305

y

x

z=12

0 0.5 1-0.5

0

0.5

1

1.5

23

45

6

78Level T(K)9 3158 3147 3136 3125 3114 3103 3092 3081 307

x

y

z=0

0 0.5 1-0.5

0

0.5

1

1.5

2

3

4

5

1

6Level T(K)6 3145 3124 3103 3082 3061 304

y

x

z=24

0 0.5 1-0.5

0

0.5

1

1.5

5

43

2

6

1

Level T(K)6 3145 3124 3103 3082 3061 304

y

x

z=30

adiabatic



0 0.5 1-0.5

0

0.5

1

1.5

1

2

354

Level T(K)6 302.55 3024 301.53 3012 300.51 300

y

x

z=12

0 0.5 1-0.5

0

0.5

1

1.5

1

2

3

5

4

Level T(K)6 302.55 3024 301.53 3012 300.51 300

x

y

z=0

0 0.5 1-0.5

0

0.5

1

1.5

1

2

35

4

Level T(K)6 302.55 3024 301.53 3012 300.51 300

y

x

z=24

0 0.5 1-0.5

0

0.5

1

1.5

1

2

354

Level T(K)6 302.55 3024 301.53 3012 300.51 300

y

x

z=30

isothermal

Figure 7. Temperature contours at selected longitudinal sections.

0 1 2 3 4x

-2

0

2

4

6

1257

1

23

4 3

Level T(K)8 3087 3076 3065 3054 3043 3032 3021 301

y

300 K

adiabatic

0 1 2 3 4x

-2

0

2

4

6

1

1

239

Level T(K)12 30311 302.7510 302.59 302.258 3027 301.756 301.55 301.254 3013 300.752 300.51 300.25

isothermal

300 K

y

Figure 8. Temperature contours near the end of the annulus (z=30.6).

Chun-Lang Yeh194

0 60 120 180306

308

310

312

314

316

φ(o)

T(K)z=0z=12z=24z=30

adiabatic

0 60 120 180301

301.5

302

302.5

φ(o)

T(K)z=0z=12z=24z=30

isothermal

Figure 9. Azimuthal distributions of the inner cylinder surface temperatures at four longitudinalsections.

Figure 7 displays the temperature contours at four selected longitudinal sections along theannulus in Zone I (refer to Figure 2 for the section positions), while Figure 8 shows thetemperature contours at a distance slightly outside the open end for the examined cases. It isobserved that higher temperature regions around the inner cylinder surface locate in the upperportion. This can also be seen from the azimuthal temperature distributions along the innercylinder surface shown in Figure 9. As pointed out in the above discussion of the flow patternon the transverse sections, the (hot) outflow is observed in the portions above the inner cylinder,whereas the (cold) inflow in the portions below. The temperature contours at a position slightlyoutside the annulus, as shown in Figure 8, consistently reflect the above observation.


Figure 9 shows the azimuthal temperature distributions along the inner cylinder surface atfour selected longitudinal sections. It is clearly seen that the highest temperature occurs rightat the top of the inner cylinder (i.e. 00=φ ). Also from the figure it is observed that the innercylinder surface temperatures decrease toward the outlet plane for the adiabatic case, whileremains relatively constant for the isothermal case. In addition, the variation of the surfacetemperatures is smaller for the isothermal case. The maximum inner cylinder surfacetemperature of the isothermal case is lower than that of the adiabatic case by about 11 K.

4.2. Natural Convection inside Horizontal Concentric or Eccentric Annuliwith Specified Wall Heat Flux

For the second illustrative problem, two cases of modified Rayleighnumber, 510=oRa and 610=oRa , are examined. However, the temperature variation in

the entire flow field for the case of 510=oRa is not obvious. Therefore, only the case of610=oRa , which may lead to more distinct temperature variations on the inner cylinder

surface, is demonstrated here. Figure 10 shows the velocity vector plots at selected transverseand longitudinal sections for the concentric and eccentric annuli. The flow patterns for thetwo cases are rather different. As observed from the velocity vector plots on the longitudinalsections, there appear secondary flows in all the six longitudinal sections for both cases.

0 10 20 3000.20.40.60.8

1 Section A-Aeccentric case

1000V*

0 10 20 300

0.25

0.5

0.75

Section B-B

0 10 20 30-0.5

0

0.5

Section C-C

0 10 20 30-0.75

-0.5-0.25

00.25

Section D-D


Chun-Lang Yeh196

0 0.5 1-1

-0.5

0

0.5

1

x

y

z=6

0 0.5 1-1

-0.5

0

0.5

1

x

y

1000V*z=0

0 0.5 1-1

-0.5

0

0.5

1

y

x

z=12

0 0.5 1-1

-0.5

0

0.5

1

y

x

z=18

0 0.5 1-1

-0.5

0

0.5

1

y

x

z=24

0 0.5 1-1

-0.5

0

0.5

1

y

x

z=30

0 10 20 300.9

1

1.1

1.2Section B-B

0 10 20 30

0.6

0.8

1 Section C-C

0 10 20 30-0.2

-0.1

0

0.1Section D-D

0 10 20 30-0.5

0

0.5

1

1.5 Section A-Aconcentric case

1000V*



0 0.5 1-0.5

0

0.5

1

1.5

y

x

z=30

0 0.5 1-0.5

0

0.5

1

1.5

y

x

z=24

0 0.5 1-0.5

0

0.5

1

1.5

y

x

z=18

0 0.5 1-0.5

0

0.5

1

1.5

y

x

z=12

0 0.5 1-0.5

0

0.5

1

1.5

x

y

z=6

0 0.5 1-0.5

0

0.5

1

1.5

x

y

1000V*z=0

Figure 10. Velocity vector plots at selected transverse and longitudinal sections.

Further downstream, the secondary flows evolve into counter-rotating recirculationzones. Another interesting phenomenon can be observed from the velocity vector plots on thetransverse sections. For the concentric case, the inflow paths are clearly observed in theportion below the inner cylinder, whereas the outflow paths are in the portion above. For theeccentric case, since the inner cylinder sits on the bottom of the outer cylinder, the inflow andoutflow paths coexist in each transverse plane. Such flow patterns result obviously from thebuoyancy effect that can be observed in thermal plume phenomena. This can also be seenfrom the velocity vector plots at a distance slightly outside the open end (i.e. in Zone II)shown in Figure 11, in which the entrainment effect caused by the upward motion of thebuoyant flows can also be seen. Another point to be drawn from Figure 10 is that the axialflow motion is significant and cannot be neglected. Therefore the adoption of a three-dimensional formulation is necessary.

Figure 12 displays the temperature contours at four selected longitudinal sections alongthe annulus in Zone I (see Figure 2 for the section positions), while Figure 13 shows thetemperature contours at a distance slightly outside the open end for the examinedconfigurations. It is observed that higher temperature regions around the inner cylindersurface locate in the lower portions of the annulus for the eccentric annulus, whereas in theupper portion for the concentric annulus. This can also be seen from the azimuthaltemperature distributions along the inner cylinder surface shown in Figure 14. Thetemperature contours at a position slightly outside the annulus, as shown in Figure 13,consistently reflect the corresponding flow pattern at a distance slightly outside the open endsshown in Figure 11.

Chun-Lang Yeh198

Figure 11. Velocity vector plots near the end of the annulus (z=30.6).



Figure 12. Temperature contours at selected longitudinal sections.

Figure 13. Temperature contours near the end of the annulus (z=30.6).

Chun-Lang Yeh200

Figure 14. Azimuthal distributions of the inner cylinder surface temperatures at four transverse sections.

Figure 14 shows the azimuthal temperature distributions along the inner cylinder surfaceat four selected longitudinal sections. It is clearly seen that the highest temperatures occurright at the bottom of the inner cylinder (i.e. 0180=φ ) for the eccentric annulus, whereas


occur right at the top of the inner cylinder (i.e. 00=φ ) for the concentric annulus. Also fromthe figure it is observed that the inner cylinder surface temperatures decrease toward theoutlet plane. In addition, the variation of the surface temperatures is smaller for the case ofconcentric annulus. A more evenly distributed inner cylinder surface temperature is desirableas far as the lifetime of the inner cylinder is concerned. With the concentric annulus, about 30K decrement of the maximum inner cylinder surface temperature can be achieved ascompared to the eccentric annulus.

Conclusion

In this chapter, the simulation method for the heat and fluid flow inside three-dimensionalhorizontal concentric or eccentric annuli with open ends is discussed. The simulationmethod discussed includes conjugated heat transfer model, turbulence model and zonal gridapproach, which extends the outlet boundary from the open end of the conduit to a far enoughoutside position that can be reasonably specified with the ambient flow properties. Twoillustrative problems are discussed. The first is a three-dimensional natural convection insidea horizontal concentric annulus with specified wall temperature or heat flux and the second isa three-dimensional natural convection inside horizontal concentric or eccentric annuli withspecified wall heat flux. For the first illustrative problem, it is found that higher temperaturesaround the inner cylinder occur in the region near its top. The maximum inner cylindersurface temperature occurs right at the top of the inner cylinder. The inner cylinder surfacetemperatures decrease toward the outlet plane for the adiabatic case, while remains relativelyconstant for the isothermal case. The variation of the inner cylinder surface temperatures issmaller for the isothermal case, as compared to the adiabatic case. For the second illustrativeproblem, it is found that the eccentric annulus has a poorer natural convection heat dissipationrate, as compared to the concentric annulus. The inner cylinder surface temperatures decreasetoward the outlet plane. It is also found that higher temperatures around the inner cylinderoccur in the region near its bottom（）the contacting point of the inner and outer cylinders forthe eccentric annulus, whereas in the region near its top for the concentric annulus. Thevariation of the inner cylinder surface temperatures is smaller for the case of concentricannulus, as compared to the eccentric annulus. The maximum inner cylinder surfacetemperatures occur right at the bottom of the inner cylinder for the eccentric annulus, whereasright at the top of the inner cylinder for the concentric annulus.

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vol.98, 108-118.


Chapter 6

CONVECTIVE HEAT TRANSFER IN THE THERMALENTRANCE REGION OF PARALLEL FLOW

DOUBLE-PIPE HEAT EXCHANGERSFOR NON-NEWTONIAN FLUIDS

Ryoichi Chiba1*, Masaaki Izumi2 and Yoshihiro Sugano3

1Yamagata University, Jonan 4-3-16, Yonezawa 992-8510, Japan2Ishinomaki Senshu University, Shinmito 1, Minamisakai, Ishinomaki 986-8580, Japan

3Iwate University, Ueda 4-3-5, Morioka 020-8551, Japan

Abstract

In this chapter, the conjugated Graetz problem in parallel flow double-pipe heat exchangers isanalytically solved by an integral transform method—Vodicka’s method—and an analyticalsolution to the fluid temperatures varying along the radial and axial directions is obtained in acompletely explicit form. Since the present study focuses on the range of a sufficiently largePéclet number, heat conduction along the axial direction is considered to be negligible. Animportant feature of the analytical method presented is that it permits arbitrary velocitydistributions of the fluids as long as they are hydrodynamically fully developed. Numericalcalculations are performed for the case in which a Newtonian fluid flows in the annulus of thedouble pipe, whereas a non-Newtonian fluid obeying a simple power law flows through theinner pipe. The numerical results demonstrate the effects of the thermal conductivity ratio ofthe fluids, Péclet number ratio and power-law index on the temperature distributions in thefluids and the amount of exchanged heat between the two fluids.

Key words: heat transfer, forced convection, heat exchanger, non-Newtonian fluid, analyticalsolution

* E-mail address: [email protected]. Telephone: +81-238-26-3219. Fax: +81-238-26-3205

Ryoichi Chiba, Masaaki Izumi and Yoshihiro Sugano206

Nomenclature

a: radius of outer pipe, mb: radius of inner pipe, mB: radius ratio (= b/a)c: heat capacity, J/[kg·K]f: dimensionless total diffusivityg: expansion coefficienth: heat transfer coefficient, W/[m2·K]J0()/J1(): the first kind of Bessel functions of order zero/onek: overall heat transfer coefficient, W/[m2·K]K: constant in the convective term, Eq. (9)n: the number of partitionsNu: Nusselt number defined by Eqs. (25) and (26)

Pe : Péclet number ratio (= I II II I/[ ( )]u b u a bα α − )r: radial coordinate, mR: heat capacity flow rate ratio of fluids defined by Eq. (C4)T: temperature, Ku: fully developed velocity profile, m/su : average velocity, m/sU: dimensionless velocity (= /u u )w: mass flow rate, kg/sx: axial coordinate, mX: eigenfunctionY0()/Y1(): the second kind of Bessel functions of order zero/oneα: thermal diffusivity, m2/sε: heat exchanger effectiveness defined by Eq. (29)εH: eddy diffusivity of heat, m2/sγ: eigenvalueη: dimensionless radial coordinate (= r/a)λ: thermal conductivity, W/[m·K]λ : thermal conductivity ratio (= λII/λI)ν: power-law indexθ: dimensionless temperature (= II I II

0b 0b 0b( ) /( )T T T T− − )

ξ: dimensionless axial coordinate (= I 2 I/( )x a uα )Ψ: the number of heat transfer units defined by Eq. (C3)

Subscripts

0: entranceb: bulki: region number

Convective Heat Transfer in the Thermal Entrance Region... 207

l: eigenvalue number∞: asymptotic

Superscripts

I: fluid III: fluid II

1. Introduction

The usual double-pipe parallel flow heat exchanger consists of two concentric circularpipes, and fluids with different temperatures enter the annulus and inner pipe at the same sideof the heat exchanger. As the fluids flow through their respective channels, heat is transferredfrom the hot fluid to the cold fluid. The traditional methods for predicting heat transfer insuch situations are based on the assignment of heat transfer coefficients for each streamindependently of the actual coupling of the boundary conditions [1]. The methods areemployed under the two essential assumptions that are customarily made [2]: (i) the heattransfer coefficients are considered to be insensitive to the longitudinal distributions of boththe heat flux and surface temperature, and (ii) they are taken to be uniform irrespective of theheat exchanger length.

There is substantial evidence [3] that under turbulent flow conditions the above-mentioned methods are acceptable for nonmetallic fluids because the local heat transfercoefficients are scarcely influenced by thermal boundary conditions in those cases, and thethermal entrance region usually covers a small part of the heat transfer length. On the otherhand, under laminar flow conditions, which are encountered in low Reynolds number flowheat exchangers1, local heat transfer coefficients become very sensitive to the thermalboundary condition. In addition, the thermal entrance length can often be of the same order ofmagnitude as the heat exchanger length [2]. This leads to spatial variations in heat transfercoefficients in much of the heat transfer surfaces. Since the two streams are thermally coupledvia the wall separating them, the varying heat transfer coefficients on both surfaces of thewall, i.e., overall heat transfer coefficient, cannot be defined a priori.

In the case of spatially varying overall heat transfer coefficient, one has to solve thecoupled forced convection heat transfer problem between heating and heated fluids—that is,the conjugated Graetz problem—for an accurate evaluation of the performance of the heatexchanger. Since the influence of coupling of the boundary conditions can be important in thethermal entrance regions, especially with laminar flow, the conjugated Graetz problem forparallel flow laminar heat exchangers with a relatively short heat transfer length has beenanalytically investigated.

The earliest investigations on the heat transfer problems of parallel flow laminar heatexchangers were conducted by Stein [4, 5]. Subsequently, the same problem was treated byMikhailov et al. [6] using a specialized version of the method for conjugated Graetzproblems. Pagliarini et al. [2] theoretically investigated thermal interaction between the 1 Laminar or low Reynolds number turbulent flows are the consequences of either high viscosity fluids, compact

flow passages (i.e., small hydraulic diameter), or low fluid velocities.


streams of laminar flow double-pipe heat exchangers, while considering axial heat conductionin the wall separating the fluids. Neto et al. [7] employed the generalized integral transformtechnique, deriving an analytical solution to a mixed lumped-differential formulation ofdouble-pipe heat exchangers. Plaschko [8] used matched asymptotic expansions to obtain anapproximate solution to the heat transfer in parallel flow heat exchangers with high Pécletnumbers. Huang et al. [9] developed a control algorithm of heat flux imposed on the externalsurface of the outer pipe in order to obtain the desired thermal entrance length and fluidtemperatures in parallel flow double-pipe heat exchangers.

The problems of heat transfer in laminar co-current flow of two immiscible fluids, suchas those studied in [10-14], are also governed by the same type of differential equations asthose for parallel flow heat exchangers, and are therefore included in the conjugated Graetzproblem. Bentwich et al. [10] and Nogueira et al. [13] analyzed the temperature distributionand heat transfer in core-annular two-phase liquid-liquid flow subject to constant walltemperature boundary condition. The heat transfer problem of core-annular laminar flow in apipe with constant wall heat flux boundary condition was theoretically investigated by Leib etal. [11]. The same type of problem in a pipe with the third-kind boundary condition wasstudied by Su [14]. Davis et al. [12] examined three types of conjugated boundary valueproblems related to conjugated multiphase heat and mass transfer problems, and developedsystematic procedures for their solutions. Simultaneous heat and mass transfer in internal gasflow in a duct whose walls are coated with a sublimable material [15] and the membrane gasabsorption process [16] are also known to come under the category of conjugated Graetzproblem.

The above-cited papers considered fluids to be Newtonian. In view of the fact that heatexchange between Newtonian and non-Newtonian fluids is of importance in engineeringapplications as well as that between Newtonian fluids, it is desirable to develop a fullyanalytical method that is not restricted by the rheology characteristics of fluids. In thischapter, the conjugated Graetz problem in parallel flow double-pipe heat exchangers isanalytically solved by an integral transform method—Vodicka’s method, and an analyticalsolution to the fluid temperatures varying along the radial and axial directions is obtained in acompletely explicit form. Since the present study focuses on the range of sufficiently largePéclet number, heat conduction along the axial direction is considered to be negligible. Animportant feature of the analytical method presented is that it permits arbitrary velocitydistributions, i.e., arbitrary rheology characteristics, of the fluids as long as they arehydrodynamically fully developed. Numerical calculations are performed for the case inwhich a Newtonian fluid flows in the annulus of the double pipe whereas a non-Newtonianfluid obeying a simple power law flows through the inner pipe. The numerical resultsdemonstrate the effects of the thermal conductivity ratio of the fluids, Péclet number ratio andpower-law index on the temperature distributions in the fluids and the amount of exchangedheat between the two fluids.


2. Theoretical Analysis

2.1. Analytical Model and Mathematical Formulation

Figure 1 shows the physical model and coordinate system. Two fluids, fluid I and fluid II,flow concurrently with fully developed velocity distributions uI(r) and uII(r) in a double-pipeheat exchanger. The heat exchanger consists of two concentric thin pipes: outer pipe of radiusa and inner pipe of radius b. While the fluid II flows through the inner pipe, the fluid I flowsin the annulus between the pipe walls. The fluid temperatures at the entrance are I

0T (r) andII

0T (r). The surface at r = a is insulated and heat is exchanged between both fluids through theinner pipe wall.

u=uI(r)T= I

0T (r)

x

r

II

I

u=uII(r)T= II

0T (r)

Thin concentric pipes

λI, αI

λII, αII

a

b

O

Figure 1. Physical model and related cylindrical coordinate system for a parallel flow double-pipe heatexchanger.

The following assumptions are introduced:

(i) physical properties are independent of temperature and are therefore constant,(ii) the heat resistance of the pipes is negligible,(iii) the axial heat conduction and turbulent axial diffusion are negligible,(iv) molecular and turbulent transport of momentum and heat are additive,(v) viscous dissipation and compression work are negligibly small,(vi) turbulent diffusion of heat can be described by available thermal eddy diffusivity

expressions.

The item (i) indicates that no natural convection occurs in the fluids. Since the effects ofpipe wall conductivity on the heat exchange effectiveness are minor in parallel flow heatexchangers [2], the item (ii) is a reasonable assumption. The item (iii) is valid only for highPéclet number flow.

In this case, the steady-state heat balance is expressed as follows [3]:

H( , ) 1 ( , )( ) [ ( )]

m mm m mT x r T x ru r r r

x r r rα ε

⎧ ⎫∂ ∂ ∂= +⎨ ⎬∂ ∂ ∂⎩ ⎭

, m = I, II, (1)


where α and εH denote the thermal diffusivity and eddy diffusivity of heat, respectively, andthe superscript m indicates the fluid number. The boundary conditions and continuitycondition at r = b are then

0(0, ) ( )m mT r T r= , m = I, II, (2)

II I( ,0) ( , ) 0T x T x ar r

∂ ∂= =

∂ ∂, (3)

I III II( , ) ( , )T x b T x b

r rλ λ∂ ∂

=∂ ∂

, I II( , ) ( , )T x b T x b= . (4)

To simplify manipulations, we obtain the dimensionless forms of Eqs. (1)-(4):

( ) I1 ( )

(1 )( ) II

mm

mm

mm

U mf

BU Pe mB

θηξθη η

η η η θηξ

⎧ ∂=⎪ ∂⎡ ⎤∂ ∂ ⎪= ⎨⎢ ⎥∂ ∂ − ∂⎣ ⎦ ⎪ =⎪ ∂⎩

, (5)

0(0, ) ( )m mθ η θ η= , m = I, II, (6)

II I( ,0) ( ,1) 0θ ξ θ ξη η

∂ ∂= =

∂ ∂, (7)

I II( , ) ( , )B Bθ ξ θ ξλη η

∂ ∂=

∂ ∂, I II( , ) ( , )B Bθ ξ θ ξ= , (8)

where we introduce the following dimensionless quantities: /r aη = , /B b a= ,

( ) ( ) /m m mU u r uη = ; m = I, II, I II II I/ [ ( )]Pe u b u a bα α= − , II I/λ λ λ= ,II I II

0b 0b 0b( ) / ( )m mT T T Tθ = − − ; m = I, II, I 2 I/ ( )x a uξ α= , H( ) 1 ( ) /m m mf rη ε α= + ; m = I, II.Equation (5) is a partial differential equation with variable coefficients; therefore, it is

very difficult to obtain the exact solution for arbitrary velocity profiles and total diffusivitydistribution. In order to solve Eq. (5), we divide the annular and circular channels into nI andnII regions in the radial direction (η-axis direction), respectively, and approximate fm(η) andUm(η) as constants fi and Ui in each region [17]. In this case, the dimensionless energyequation in the ith region (i = 1, 2,..., nI+nII) is:


2

2

1i i iiK θ θ θ

ξ η η η∂ ∂ ∂

= +∂ ∂ ∂

, (9)

whereII

II I II

(1 ) /( ) 1/ 1

i ii

i i

U Pe B B f i nKU f n i n n

⎧ − ⋅ ≤ ≤⎪= ⎨+ ≤ ≤ +⎪⎩

.

Note that the subscripts of θ, K, λ and η denote the region number, not the fluid number.The continuity conditions at the imaginary interfaces (including the real interface at η = B)and boundary conditions are expressed as:

1

1

( , ) ( , )i i i i i

i

λ θ ξ η θ ξ ηλ η η

+

+

∂ ∂=

∂ ∂, 1( , ) ( , )i i i iθ ξ η θ ξ η+= , (10)

0(0, ) ( )miθ η θ η= , if 1 ≤ i ≤ nII, then m = II,

if nII+1 ≤ i ≤ nI+nII, then m = I, (11)

I II( )1( ,1)( ,0) 0n n

θ ξθ ξη η

+∂∂

= =∂ ∂

, (12)

where ηi is the outer radius of the ith region, that is, IInBη = and I II( )

1n n

η+

= . The initial-

boundary value problem expressed by Eqs. (9)-(12) is identical to that for transient heatconduction case in a composite medium consisting of nI+nII layers.

2.2. Derivation of Analytical Solution

We employ Vodicka’s method [18] to derive an analytical solution of temperatures in thefluids and their related Nusselt numbers. Since it allows us to easily analyze the transient heatconduction in a composite medium with a large number of sub-regions, this method is appliedto the analysis of temperature fields in a wide range of objects [17, 19]. Although it ispossible, in principle, to analyze the problem under consideration using Laplace transform[20], the inverse transform is mathematically quite complicated for a large number of sub-regions. Therefore, it is the authors opinion that Vodicka’s method is to be preferred due to itsadvantage in facility. Note that the analytical treatment proposed by Mikhailov et al. [21] maybe used as an alternative.

Using Vodicka’s method, the solution to Eqs. (9)-(12) is assumed to be

1( , ) ( ) ( )i l il

lXθ ξ η φ ξ η θ

∞

∞=

= +∑ , (13)


where θ∞ is the temperature of the fluids for ξ → ∞. An overall heat balance shows that withan infinite heat transfer area, the outlet temperature of each stream in parallel flow is

11

BPe B B

θλ∞

+=

+ +. (14)

The eigenfunction ( )ilX η is obtained as

0 0( ) ( ) ( )il il i l il i lX A J K B Y Kη γ η γ η= + , (15)

where J0() and Y0() are the first and second kind of Bessel functions of order 0, respectively.The constants Ail and Bil in Eq. (15) are determined from the conditions that are obtained

by substituting Eqs. (13) and (15) into the continuity and boundary conditions, Eqs. (10) and(12). Consequently, the continuity conditions of the temperature field at the interfacesbetween neighboring regions can be fulfilled. The eigenvalues γl (l = 1, 2,...) are obtainedfrom the condition under which all the Ail and Bil values are nonzero and are thereforepositive roots of the following transcendental equation (for details, see the appendix A):

I II I II1 2 ( 1) ( )

const.0n n n n+ − +

⎡ ⎤⋅ ⋅ = ⎢ ⎥

⎣ ⎦E E E b , (16)

where

1i i i+= ⋅E C D , I II

I II

I II

1

( )1

( )

( )

ln n

n nln n

Y K

J K

γ

γ+

+

+

⎡ ⎤⎢ ⎥=⎢ ⎥−⎣ ⎦

b ,

1 1 0

1 1 0

/ ( ) ( )

/ ( ) ( )i i i l i l i i l i

ii i i l i l i i l i

K Y K Y K

K J K J K

λ λ γ γ η γ η

λ λ γ γ η γ η+

+

⎡ ⎤− −= ⎢ ⎥

⎢ ⎥⎣ ⎦C ,

0 1 0 11

1 1 1 1 1 1

( ) ( )

( ) ( )i l i i l i

ii l i l i i l i l i

J K Y K

K J K K Y K

γ η γ η

γ γ η γ γ η+ +

+

+ + + +

⎡ ⎤= ⎢ ⎥

− −⎢ ⎥⎣ ⎦D , (17)

and J1() and Y1() are the first and second kind of Bessel functions of order 1, respectively.By substituting Eq. (13) into Eq. (11), the following equation is obtained:

01

( ) (0) ( ) ( )ml il

lG Xη φ η θ η θ

∞

∞=

= = −∑ , m = I or II. (18)


The eigenfunction Xil(η) given by Eq. (15) has an orthognal relationship withdiscontinuous weight functions; it is expressed as follows (see the appendix B):

I II

11

const. ( )( ) ( )d

0 ( )

i

i

n n

i i il iki

l kK X X

l k

η

η

λ η η η η−

+

=

=⎧= ⎨ ≠⎩

∑ ∫ . (19)

G(η) can be expanded into an infinite series of Xil(η) as follows:

1( ) ( )l il

lG g Xη η

∞

=

= ∑ , (20)

where the expansion coefficient gl is given by

[ ]

I II

1I II

1

1

2

1

( ) ( )d

( ) d

i

i

i

i

n n

i i ili

l n n

i i ili

K G Xg

K X

η

ηη

η

λ η η η η

λ η η η

−

−

+

=

+

=

=∑ ∫

∑ ∫. (21)

Taking Eq. (15) into account, we substitute Eq. (13) into Eq. (9). This yields a first-orderlinear ordinary differential equation for φl(ξ) as follows:

2d 0d

ll l

φ γ φξ

+ = . (22)

By solving Eq. (22) with the condition φl(0) = gl, which is obtained from the comparisonbetween Eqs. (18) and (20), we obtain φl(ξ) as

2

( ) ll lg e γ ξφ ξ −= . (23)

Finaly, the dimensionless temperature in the ith region η ∈ [ηi−1, ηi] inside the double-pipe heat exchanger, θi(ξ, η), is derived as:

2

0 01

1( , ) ( ) ( )1

li l il i l il i l

l

Bg e A J K B Y KPe B B

γ ξθ ξ η γ η γ ηλ

∞−

=

+⎡ ⎤= + +⎣ ⎦ + +∑ . (24)

Local Nusselt numbers for outer and inner streams are defined in the usual manner as

I II

I Ib

2 ( ) 2( 1)

BB

h a b BNuηη

θλ θ θ η

==

− − ∂= =

− ∂ for the outer stream, (25)


II IIII

II IIb

2 2

BB

h b BNuηη

θλ θ θ η

==

∂= =

− ∂ for the inner stream. (26)

3. Numerical Calculation

In this study, there is no analytical restriction on the form of the velocity profiles of theflowing fluids. Thus, as a numerical example, a double-pipe heat exchanger in which aNewtonian fluid flows in the annulus whereas a non-Newtonian fluid flows through the innerpipe is considered here. The non-Newtonian fluid is assumed to obey a power law, which canapproximate the non-Newtonian viscosity of many types of fluids with good accuracy over awide range of shear rates. The fluids have mutually different uniform temperatures at theentrance, that is, I

0θ (η) = 1 and II0θ (η) = 0. Under laminar flow conditions, I II

H H 0ε ε= = andthe fully developed velocity profile for each fluid is [1, 22]:

2 2I I

2 2

(1 ) ln ( 1) ln( ) 2( 1) ( 1) ln

B Bu uB B B

η ηη − + −=

− − +, (27)

1

II II 3 1( ) 11

u uB

ννν ηη

ν

+⎡ ⎤+ ⎛ ⎞⎢ ⎥= − ⎜ ⎟⎢ ⎥+ ⎝ ⎠⎣ ⎦

, (28)

where ν is the power-law index. For values of ν less than unity, the behavior ispseudoplastic, whereas forν greater than unity the behavior is dilatant. For ν = 1, it reduces toNewton’s law of viscosity.

In the numerical calculations, we radially divide the each channel into equal intervalswith the number of partitions nI = nII = 20. Our ealier work [17] showed that the number ofpartitions 20 per a channel provides sufficiently accurate results. The number of terms in theinfinite series in Eq. (24) is taken as 200. It should be noted that this value is used for theverification of sufficient convergence of the numerical results.


To show the accuracy of the present analytical solution, we first compare local Nusseltnumbers, bulk temperatures and wall temperatures to existing results [23, 24]. The valuesshown in Table 1 are in good agreement between the present analytical solution and theexisting solutions: the similarity approach solution [23] and the eigenvalue solution given in[24].

Table 1. A comparison of the entrance region results obtained by the present analytical method and existing methods [23, 24] for a

parallel flow double-pipe heat exchanger with ν = 1, Pe = 2/3, λ = 2 and B = 0.5

x [23, 24] ξ IIBθ II

Bθ * θ|η = B θ|η = B* NuI NuI* NuII NuII*

0.001 0.00016667 0.010589 0.0103 0.43057 0.426 24.123 23.96 16.224 16.42

0.002 0.00033333 0.016574 0.0163 0.43345 0.431 19.215 19.15 12.892 12.98

0.01 0.0016667 0.047178 0.0470 0.44917 0.449 11.560 11.56 7.6210 7.64

0.02 0.0033333 0.073899 0.0737 0.46062 0.460 9.4119 9.42 6.1655 6.17

0.04 0.0066667 0.11529 0.115 0.47611 0.476 7.7722 7.78 5.0914 5.09

0.1 0.016667 0.20501 0.205 0.50591 0.506 6.2719 6.28 4.1998 4.20

0.2 0.033333 0.31095 0.311 0.54061 0.541 5.6245 5.63 3.9329 3.93

0.4 0.066667 0.45267 0.453 0.59330 0.594 5.3742 5.38 3.9267 3.92

0.6 0.10000 0.54102 0.541 0.62927 0.629 5.3390 5.35 3.9403 3.94* Existing solutions, the data cited from [23] for 0.01x ≤ and [24] for 0.02x ≥ .


0

0.2

0.4

0.6

0.8

1

10-4 10-3 10-2 10-1 100

ν = 0.3 3

θ

ξ

Fluid I

Fluid II

Inner pipe wall

η = 0, 0.2, 0.4, 0.5, 0.6, 0.8, 1

(a)

0

0.2

0.4

0.6

0.8

1

10-4 10-3 10-2 10-1 100

ν = 0.3 3

θ

ξ

Inner pipe wall

Fluid II

Fluid I η = 0, 0.2, 0.4, 0.5, 0.6, 0.8, 1

(b)



0

0.2

0.4

0.6

0.8

1

10-4 10-3 10-2 10-1 100

ν = 0.3 3

θ

ξ

Inner pipe wall

Fluid Iη = 0, 0.2, 0.4, 0.5, 0.6, 0.8, 1

(c)

Figure 2. Temperature variation along the streamwise direction at different radial locations with B = 0.5and λ =10, for (a) Pe = 0.5, (b) Pe = 2 and (c) Pe = 10.

0

0.5

1

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1

ν = 3 1 0.3

UII

η/B

Figure 3. Velocity profiles for different power-law index values.

Figure 2 shows the axial distributions of dimensionless temperature θ at some radiallocations. In this figure, the solid lines and dashed lines represent the cases for ν = 0.3 and ν= 3, respectively, for which fluid II exhibits the velocity profiles shown in Figure 3. It isobserved that for a constant thermal conductivity ratio λ , a greater Péclet number ratio Peproduces a larger temperature variation in fluid I. Moreover, although we omit the graphicalrepresentation, for a constant Pe a greater λ causes a larger temperature variation in fluid I.These results can be predicted easily from Eq. (14). With regard to the effect of ν on the


temperature distributions, an increase in ν increases the values of θ in both fluids throughoutthe heat exchanger length except the vicinity of the inner pipe axis. Especially, the effect ispronounced at η = B, or the inner pipe wall, and it decreases with distance from the inner pipewall. Not surprisingly, the temperatures of fluid II, a power-law fluid, are directly affected byν. However, we can also observe a slight influence of ν on fluid I in the temperaturedistributions.

0

0.2

0.4

0.6

0.8

1

10-4 10-3 10-2 10-1 100

ν = 0.3 3

θ|η=

B

ξ

Pe = 0.1, 0.2, 0.5, 1, 2, 5, 10

(a)

0

0.2

0.4

0.6

0.8

1

10-4 10-3 10-2 10-1 100

ν = 0.3 3

θ|η=

B

ξ

Pe = 0.1, 0.2, 0.5, 1, 2, 5, 10

(b)



0

0.2

0.4

0.6

0.8

1

10-4 10-3 10-2 10-1 100

ν = 0.3 3

θ|η=

B

ξ

Pe = 0.1, 0.2, 0.5, 1, 2, 5, 10

(c)

Figure 4. Temperature variation along the streamwise direction at the inner pipe wall with B = 0.5, for(a) λ = 0.1, (b) λ = 1 and (c) λ = 10.

The temperatures of the inner pipe wall, which are the most susceptible to the effect of ν,are plotted against ξ in Figure 4. As λ becomes small, θ of the inner pipe wall increases. Inaddition, the larger the value of ν is, the higher θ always becomes. It is interesting that thevariation behavior of θ along the ξ-axis direction depends on the value of Pe ; while θ of theinner pipe wall increases monotonically for a small Pe , it may have a maximal value for alarge Pe , decreasing from a certain axial location.

This is due to the following reasons: in the immediate vicinity of the entrance of the heatexchanger, large hot/cold heat flows from fluid I to fluid II because of great radialtemperature gradients near the inner pipe wall. This leads to an increase in θ of the inner pipewall. Additionally, since the temperature variation in fluid II becomes smaller than that influid I for a large Pe (see Figure 2-c), fluid I greatly varies its temperature over the fullsection of the annulus along the axial direction. In contrast, fluid II varies its temperature onlyin the immedeate vicinity of the inner pipe wall. Up to a downstream location, θ of the innerpipe wall remains elevated due to high heat flux passing through the wall. When the fluidsreach downstream to some extent, the heat flux passing through the wall, i.e., radialtemperature gradients decrease and meanwhile heat conduction in the negative η directiongradually increases in fluid II. As a result, the temperature of the inner pipe wall begins todecrease and then it converges at the asymptotic temperature given by Eq. (14).

It is known that in parallel flow heat exchangers, the bulk temperatures of heating andheated fluids vary monotonically from each entrance temperature to the asymptotic one.However, local temperatures, especially those near the inner pipe wall, do not necessarily


vary in a monotone along the streamwise direction and may have a maximal value at a certainaxial location. This phenomenon tends to occur in the case of large Péclet number ratio.

0

5

10

15

20

25

30

10-4 10-3 10-2 10-1 100

ν = 3 1 0.3Const. heat flux B.C.Const. temperature B.C.

NuI

ξ(a)

0

5

10

15

20

25

30

10-4 10-3 10-2 10-1 100

ν = 3 1 0.3Const. heat flux B.C.Const. temperature B.C.

NuII

ξ(b)

Figure 5. Local Nusselt numbers of (a) outer stream and (b) inner stream for B = 0.5, λ = 1 and Pe = 1.


The local Nusselt numbers defined by Eqs. (25) and (26) are shown in Figure 5-(a) and(b), respectively. The discrete plots in Figure 5 indicate the local Nusselt numbers of singleflows for a Newtonian fluid with constant temperature/constant heat flux boundary condition.The data are cited from [25] for flow in the pipe and [26, 27] for flow in the annulus. Figure 5demonstrates that the local Nusselt number of the outer stream (fluid I) is little affected by ν.On the other hand, the Nusselt number of the inner stream (fluid II) depends on the value ofν, increasing with a decrease in ν. This trend can be also found in the case of single flow [28].In summary, the local Nusselt number of each stream in parallel flow heat exchangers isdetermined by own rheology characteristics, irrespective of the rheology characteristics ofcounterpart.

Next, we compare the curve for ν = 1 with the discrete plots in both streams. In the innerstream, the local Nusselt number obtained from the present numerical calculations fallssomewhere in between the two local Nusselt numbers for single flow, being, on the whole,closer to the one for constant temperature boundary condition. On the other hand, in the outerstream, the Nusselt number obtained from the present numerical calculations is smaller thanboth Nusselt numbers for single flow; it is closer to the one for constant temperatureboundary condition again. However, it is found difficult to accurately approximate localNusselt numbers in parallel flow heat exchangers with those for constant temperature/heatflux boundary condition.

0

0.2

0.4

0.6

0.8

1

10-4 10-3 10-2 10-1 100

ν = 0.3 3

θI b, θII

b

ξ

θIb

θIIb

(a)



0

0.2

0.4

0.6

0.8

1

10-5 10-4 10-3 10-2 10-1 100

ν = 0.3 3

θI b, θII

b

ξ

θIb

θIIb

(b)

Figure 6. Bulk temperatures of the heating and heated fluids with different power-law index values forB = 0.5, λ = 1 and (a) Pe = 1 and (b) Pe = 10.

Figure 6 shows the axial variations in the bulk temperatures of heating and heated fluids.Since a smaller ν leads to a larger Nusselt number in the inner stream, as shown in Figure 5-(b), heat exchange between both fluids is found to be enhanced for ν = 0.3. From Figure 6-(a),maximum difference between the bulk temperatures for ν = 0.3 and 3 is calculated to beapproximately 1% for I

bθ and 10% for IIbθ , and from Figure 6-(b) approximately 7% for both

Ibθ and II

bθ .

Figure 7 makes comparisons of the heat exchanger effectiveness1 ε defiined by

Ib1 ( )( )

1θ ξε ξ

θ∞

−=

−, (29)

between the value by the present analytical solution and that calculated by means of the bulktemperatures of the fluids and a constant overall heat transfer coefficient (measured by overallNusselt number in the figure). θ∞ in Eq. (29) is already given by Eq. (14). For the derivationdetails of the latter heat exchanger effectiveness, q.v. the appendix C. Figure 7 demonstratesthat it is impossible to approximate well the heat exchanger effectiveness throughout all

2 The heat exchanger effectiveness is defined as the ratio of the actual over-all rate of heat transfer to the maximum

possible as computed for an exchanger with the same operating conditions but with infinite heat transfer area[29].


values of ξ with any constant overall heat transfer coefficients. Hence, in designing heatexchangers with laminar flow, i.e., low Reynolds number flow heat exchangers, it is crucial toconsider streamwise variations in the overall heat transfer coefficient.

0

0.2

0.4

0.6

0.8

1

10-4 10-3 10-2 10-1 100 101

Present analytical solutionBulk temp. & const. overall Nu

ε

ξ

Overall Nu = 1

2

3

4

5

(a)

0

0.2

0.4

0.6

0.8

1

10-4 10-3 10-2 10-1 100 101

Present analytical solutionBulk temp. & const. overall Nu

ε

ξ

Overall Nu = 2

4

6

8

10

(b)

Figure 7. Axial variation of heat exchanger effectiveness with B = 0.5, Pe = 1 and ν = 1, for (a) λ = 1and (b) λ = 10.


5. Conclusion

In this chapter, the conjugated Graetz problem in parallel flow double-pipe heatexchangers has been analytically solved by an integral transform method—Vodicka’smethod—and an analytical solution to the fluid temperatures varying along the radial andaxial directions has been obtained in a completely explicit form. An important feature of theanalytical method presented is that it permits arbitrary velocity distributions of the fluids aslong as they are hydrodynamically fully developed. Numerical calculations have beenperformed for the case in which a Newtonian fluid flows in the annulus of the double pipe,whereas a non-Newtonian fluid obeying a simple power law flows through the inner pipe.The numerical results have demonstrated the effects of the thermal conductivity ratio of thefluids, Péclet number ratio and power-law index, on the temperature distributions in the fluidsand the amount of exchanged heat between the two fluids.

Acknowledgements

The authors would like to thank Dr. S. Jian, Universidade Federal do Rio de Janeiro, forhis useful comments. Thanks are also given to Prof. A. Haji-Sheikh, the University of Texasat Arlington, for helpful suggestions.

Reviewed by

This manuscript was peer-reviewed by Dr. Su Jian, Universidade Federal do Rio deJaneiro, Brazil.

Appendix A: The Eigenvalue Equation

The eigenfunctions Xil (i = 1, 2,..., nI+nII) must satisfy

21( ) ( ) ( ) 0il il i l ilX X K Xη η γ ηη

′′ ′+ + = , (A1)

with the continuity and boundary conditions

( 1)( ) ( )il i i l iX Xη η+= , (A2)

1 ( 1)( ) ( )i il i i i l iX Xλ η λ η+ +′ ′= , (A3)

1 (0) 0lX ′ = , (A4)

I II( )(1) 0

n n lX

+′ = , (A5)


where primes denote differentiation with respect to η. Substituting Eq. (15) into Eqs. (A2)-(A5) yields

0 1 0 10 0

0 1 1 0 1 1 0 2 1 0 2 1

1 0 1 1 1 0 1 1 2 0 2 1 2 0 2 1

0 2 2 0 2 2 0 3 2 0 3 2

2 0 2 2 2 0 2

( ) ( ) 0 0

( ) ( ) ( ) ( ) 0 0

( ) ( ) ( ) ( ) 0 0

0 0 ( ) ( ) ( ) ( )

( ) (

l l

l l l l

l l l l

l l l l

l

J K Y K

J K Y K J K Y K

J K Y K J K Y K

J K Y K J K Y K

J K Y K

η ηγ η γ η

γ η γ η γ η γ η

λ γ η λ γ η λ γ η λ γ η

γ η γ η γ η γ η

λ γ η λ γ

= =′ ′

− −

′ ′ ′ ′− −

− −

′ ′ 2 3 0 3 2 3 0 3 2) ( ) ( ) 0 0

0

l l lJ K Y Kη λ γ η λ γ η

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

′ ′− −⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

I II

I II

1

1

2

2

( )

( )

0000

00

l

l

l

l

n n l

n n l

ABAB

A

B+

+

⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⋅ =⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎩ ⎭⎩ ⎭

(A6)

Since Eq. (A6) is a homogeneous system of equations, the determinant of the matrix mustbe zero to have a nontrivial solution for all Ail and Bil. However, the procedure for obtainingthe determinant is complicated. Therefore, we perform the manipulation shown below.

The boundary condition Eq. (A5) is expressed as:

I II( )0

n n+⋅ =G a , (A7)

where I II I II1 1( ) ( )l ln n n nJ K Y Kγ γ

+ +⎡ ⎤= ⎣ ⎦G , I II I II I II

T( ) ( ) ( )n n n n l n n l

A B+ + +

⎡ ⎤= ⎣ ⎦a and the

superscript T is the transpose operator. The continuity conditions Eqs. (A2) and (A3) areexpressed as:

1 1i i i i+ +⋅ = ⋅F a D a , (A8)

where

0 0

1 1 1 1

( ) ( )

/ ( ) / ( )i l i i l i

ii i i l i l i i i i l i l i

J K Y K

K J K K Y K

γ η γ η

λ λ γ γ η λ λ γ γ η+ +

⎡ ⎤= ⎢ ⎥

− −⎢ ⎥⎣ ⎦F , (A9)

and Di+1 is given in Eq. (17). With 11i i i

−+= ⋅H F D , Eq. (A8) can be written as

I II I II1 1 ( )i i i n n n n+ + − += ⋅ ⋅a H H H a . (A10)


From Eq. (A7), we obtain

I II

I II I II

I II

1T( ) ( )

1

( )1

( )ln n

n n n n lln n

J KA

Y K

γ

γ+

+ ++

⎡ ⎤⎢ ⎥= −⎢ ⎥⎣ ⎦

a . (A11)

Since the constant B1l must be zero due to the nature of Bessel functions2,

[ ]T1 1 0lA=a . This leads to the following equation, considering Eqs. (A10) and (A11):

I III II I II

I II

111 2 1 ( )

1

1

( )0

( )

lln nn n n n l

ln n

A J K AY K

γ

γ++ − +

+

⎡ ⎤⎢ ⎥⎡ ⎤

= ⋅ ⋅ ⎢ ⎥⎢ ⎥ −⎣ ⎦ ⎢ ⎥⎢ ⎥⎣ ⎦

H H H . (A12)

In order that all the Ail and Bil (i = 1, 2,..., nI+nII) are nonzero, the eigenvalues γl mustsatisfy Eq. (A12) with I II( )

0n n l

A+

≠ . Here one can write as Hi = Ci·Di+1/det Fi = Ei/det Fi (Ci

and Ei are given in Eq. (17)), and therefore obtain Eq. (16).

Appendix B: Orthogonality of the Eigenfunctions

The orthogonality of the eigenfunctions Xil (i = 1, 2,..., nI+nII) will now be established.Equation (A1) is written for l = j and then for l = k with j ≠ k. The equation for l = j ismultiplied by ηXik, and the equation for l = k is multiplied by ηXij. The two resultingequations are subtracted, multiplied by λi, and then integrated between η = ηi−1 and η = ηi.The following results:

11

2 2( ) d ( ) ( ) ( ) ( )i

i

ii

i i k j ij ik i ik ij i ij ikK X X X X X Xη

η

ηη

λ γ γ η η λη η η λη η η−

−

′ ′⎡ ⎤− = −⎣ ⎦∫ , (B1)

where primes denote differentiation with respect to η. Taking the summation of both sides ofEq. (B1) for i = 1, 2,..., nI+nII,

I II I II

1

2 2

1 1( ) d ( ) ( ) ( ) ( )

i

i

n n n n

i i k j ij ik i i ik i ij i ij i ik ii i

K X X X X X Xη

η

λ γ γ η η λ η η η η η−

+ +

= =

′ ′⎡ ⎤− = −⎣ ⎦∑ ∑∫

1 1 1 1 1( ) ( ) ( ) ( )i ik i ij i ij i ik iX X X Xη η η η η− − − − −′ ′⎡ ⎤− −⎣ ⎦ . (B2)

2 In other words, X1l (η) must have a finite value at η = 0.


Equations (A2) and (A3) are written for l = j and then for l = k, respectively. Theequation (A2) for l = j is multiplied by the equation (A3) for l = k at each side, and similarlythe equation (A2) for l = k is multiplied by the equation (A3) for l = j. The two resultingequations make Eq. (B2) a simpler form:

I II

1

2 2

1( ) d

i

i

n n

i i k j ij iki

K X Xη

η

λ γ γ η η−

+

=

− =∑ ∫

I II I II I II I II I II( ) ( ) ( ) ( ) ( )(1) (1) (1) (1)

n n n n k n n j n n j n n kX X X Xλ

+ + + + +⎡ ⎤′ ′−⎣ ⎦ . (B3)

Considering Eq. (A5), Eq. (B3) reduces to

I II

11

d 0i

i

n n

i i ij iki

K X Xη

η

λ η η−

+

=

=∑ ∫ . (B4)

For the case of j = k = l, Eq. (A1) leads to a normalizing factor defined as thedenominator of Eq. (21).

Appendix C: Heat Exchanger Effectiveness for a ConstantOverall Heat Transfer Coefficient

Ignoring the radial distributions of velocity and temperature of the fluids, we derive theheat exchanger effectiveness by means of their cross-sectional average values and a constantoverall heat transfer coefficient. The bulk temperatures of the fluids at the axial location x inthe parallel flow heat exchanger shown in Figure 1 can be written in dimensionless forms as[30]

( 1) ( )Ib

1( )1

RR eR

ξ

θ ξ− + ⋅Ψ+ ⋅

=+

, (C1)

III bb

1 ( )( )R

θ ξθ ξ −= , (C2)

where Ψ(ξ) and R are the number of heat transfer units and heat capacity flow rate ratiowhich occur frequently in heat exchanger analysis, respectively; they are given by

II II

2 2( )(1 )

k b x Nuw c Pe Bπξ ξ

λ⋅ ⋅ ⋅

Ψ = =⋅ −

, (C3)


II II

I I 1w c Pe BRw c B

λ⋅ ⋅= =

+. (C4)

Nu in Eq. (C3) is the overall Nusselt number based on the outer stream properties beingdefined as Nu = ka/λI and k denotes the usual overall heat transfer coefficient of the inner pipewall. Consequently, the heat exchanger effectiveness ε expressed by Eq. (29) is obtained as afunction of the heat exchanger length ξ as follows:

( 1) ( )( ) 1 Re ξε ξ − + ⋅Ψ= − . (C5)

References

[1] Mikhailov, M. D., and Ozisik, M. N.: Unified Analysis and Solutions of Heat and MassDiffusion, Dover Publications, New York (1994)

[2] Pagliarini, G., and Barozzi, G. S.: Thermal Coupling in Laminar Flow Double-Pipe HeatExchangers. Trans ASME Journal of Heat Transfer, 113, 526-534 (1991)

[3] Stein, R. P.: Liquid Metal Heat Transfer. In: Irvine, T. F., Hartnett, J. P. (eds.) Advancesin Heat Transfer, Vol. 3, Academic Press, New York, London (1966)

[4] Stein, R. P.: Heat Transfer Coefficients in Liquid Metal Concurrent Flow Double PipeHeat Exchangers. Chemical Engineering Progress Symposium Series, 59, 64-75 (1965)

[5] Stein, R. P.: The Graetz Problem in Concurrent Flow Double Pipe Heat Exchangers.Chemical Engineering Progress Symposium Series, 59, 76-87 (1965)

[6] Mikhailov, M. D., and Shishedjiev, B. K.: Coupled at boundary mass or heat transfer inentrance concurrent flow. International Journal of Heat and Mass Transfer, 19, 553-557(1976)

[7] Neto, F. S., and Cotta, R. M.: Lumped-differential analysis of concurrent flow double-pipe heat exchanger. Canadian Journal of Chemical Engineering, 70, 592-595 (1992)

[8] Plaschko, P.: High Péclet number heat exchange between cocurrent streams. Archive ofApplied Mechanics (Ingenieur Archiv), 70, 597-611 (2000)

[9] Huang, C. H., and Yeh, C. Y.: An optimal control algorithm for entrance concurrentflow problems. International Journal of Heat and Mass Transfer, 46, 1013-1027 (2003)

[10] Bentwich, M., and Sideman, S.: Temperature distribution and heat transfer in annulartwo-phase (liquid-liquid) flow. Canadian Journal of Chemical Engineering, 42, 9-13(1964)

[11] Leib, T. M., Fink, M., and Hasson, D.: Heat transfer in vertical annular laminar flow oftwo immiscible liquids. International Journal of Multiphase Flow, 3, 533-549 (1977)

[12] Davis, E. J., Venkatesh, S.: The solution of conjugated multiphase heat and masstransfer problems. Chemical Engineering Science, 34, 775-787 (1979)

[13] Nogueira, E., Cotta, R. M.: Heat transfer solutions in laminar co-current flow ofimmiscible liquids. Heat and Mass Transfer, 25, 361-367 (1990)

[14] Su, J.: Exact Solution of Thermal Entry Problem in Laminar Core-annular Flow of TwoImmiscible Liquids. Chemical Engineering Research and Design, 84, 1051-1058 (2006)

[15] Sparrow, E. M., and Spalding, E. C.: Coupled laminar heat transfer and sublimationmass transfer in a duct. Trans ASME Journal of Heat Transfer, 90, 115-124 (1968)


[16] Wang, W. P., Lin, H. T., and Ho, C. D.: An analytical study of laminar co-current flowgas absorption through a parallel-plate gas-liquid membrane contactor. Journal ofMembrane Science, 278, 181-189 (2006)

[17] Chiba, R., Izumi, M., and Sugano, Y.: An analytical solution to non-axisymmetric heattransfer with viscous dissipation for non-Newtonian fluids in laminar forced flow.Archive of Applied Mechanics (Ingenieur Archiv), 78, 61-74 (2008)

[18] Vodicka, V.: Linear heat conduction in laminated bodies (in German). MathematischeNachrichten, 14, 47-55 (1955)

[19] Chiba, R.: Stochastic heat conduction analysis of a functionally graded annular disc withspatially random heat transfer coefficients. Applied Mathematical Modelling, 33, 507-523 (2009)

[20] Carslaw, H. S., and Jaeger, J. C.: Conduction of Heat in Solids, 2nd Edition, ClarendonPress, Oxford (1986)

[21] Mikhailov, M. D., Ozisik, M. N., and Vulchanov, N. L.: Diffusion in composite layerswith automatic solution of the eigenvalue problem. International Journal of Heat andMass Transfer, 26, 1131-1141 (1983)

[22] Bird, R. B., Stewart, W. E., and Lightfoot, E. N.: Transport Phenomena, John Wiley &Sons Inc, New York (1960)

[23] Gill, W. N., Porta, E. W., and Nunge, R. J.: Heat transfer in thermal entrance region ofcocurrent flow heat exchangers with fully developed laminar flow. International Journalof Heat and Mass Transfer, 11, 1408-1412 (1968)

[24] Nunge, R. J., and Gill, W. N.: An analytical study of laminar counterflow double-pipeheat exchangers. AIChE Journal, 12, 279-289 (1966)

[25] Shah, R. K.: Thermal entry length solutions for the circular tube and parallel plates. 3rdNational Heat Mass Transfer Conference, 1, HMT-11-75 (1975)

[26] Kays, W. M., Lundberg, R. E., and Reynolds, W. C.: Heat transfer with laminar flow inconcentric annuli with constant and variable wall temperature and heat flux. NASATechnical reports, AHT-2 (1961)

[27] Worsoe-Schmidt, P. M.: Heat transfer in the thermal entrance region of circular tubesand annular passages with fully developed laminar flow. International Journal of Heatand Mass Transfer, 10, 541-551 (1967)

[28] Cotta, R. M., and Ozisik, M. N.: Laminar forced convection of power-law non-Newtonian fluids inside ducts. Heat and Mass Transfer, 20, 211-218 (1986)

[29] Stein, R. P., and Sastri, V. M. K.: A heat transfer analysis of heat exchangers withlaminar tube-side and turbulent shell-side flows–A new heat exchanger Graetz problem.AIChE Symposium Series, 118, 81-89 (1972)

[30] Quick, K.: Direct calculation of exchanger exit temperatures in cocurrent flow.Chemaical Engineering, 93, 92 (1986).


Chapter 7

NUMERICAL SIMULATION OF TURBULENTPIPE FLOW

M. Ould-Rouis and A.A. FeizUniversité Paris-Est, Laboratoire Modélisation et Simulation Multi Echelle,

MSME, FRE3160 CNRS, 77454 Marne-la-Vallée, France

Abstract

Many experimental and numerical studies have been devoted to turbulent pipe flows due tothe number of applications in which theses flows govern heat or mass transfer processes: heatexchangers, agricultural spraying machines, gasoline engines, and gas turbines for examples.The simplest case of non-rotating pipe has been extensively studied experimentally andnumerically. Most of pipe flow numerical simulations have studied stability and transition.Some Direct Numerical Simulations (DNS) have been performed, with a 3-D spectral code, orusing mixed finite difference and spectral methods. There is few DNS of the turbulent rotatingpipe flow in the literature. Investigations devoted to Large Eddy Simulations (LES) ofturbulence pipe flow are very limited. With DNS and LES, one can derive more turbulencestatistics and determine a well-resolved flow field which is a prerequisite for correctpredictions of heat transfer. However, the turbulent pipe flows have not been so deeplystudied through DNS and LES as the plane-channel flows, due to the peculiar numericaldifficulties associated with the cylindrical coordinate system used for the numericalsimulation of the pipe flows.

This chapter presents Direct Numerical Simulations and Large Eddy Simulations of fullydeveloped turbulent pipe flow in non-rotating and rotating cases. The governing equations arediscretized on a staggered mesh in cylindrical coordinates. The numerical integration isperformed by a finite difference scheme, second-order accurate in space and time. The timeadvancement employs a fractional step method. The aim of this study is to investigate theeffects of the Reynolds number and of the rotation number on the turbulent flowcharacteristics. The mean velocity profiles and many turbulence statistics are compared tonumerical and experimental data available in the literature, and reasonably good agreement isobtained. In particular, the results show that the axial velocity profile gradually approaches alaminar shape when increasing the rotation rate, due to the stability effect caused by thecentrifugal force. Consequently, the friction factor decreases. The rotation of the wall haslarge effects on the root mean square (rms), these effects being more pronounced for thestreamwise rms velocity. The influence of rotation is to reduce the Reynolds stress component⟨Vr'Vz'⟩ and to increase the two other stresses ⟨Vr'Vθ'⟩ and ⟨Vθ'Vz'⟩. The effect of the Reynolds

M. Ould-Rouis and A.A. Feiz232

number on the rms of the axial velocity (⟨Vz'2⟩1/2) and the distributions of ⟨Vr'Vz'⟩ is evident,and it increases with an increase in the Reynolds number. On the other hand, the ⟨Vr'Vθ'⟩-profiles appear to be nearly independent of the Reynolds number. The present DNS and LESpredictions will be helpful for developing more accurate turbulence models for heat transferand fluid flow in pipe flows.

Nomenclature

Cf=2τw/ρUb2 friction coefficient

Lz length of the computational domainN rotation numberNθ, Nr, Nz grids in θ, r and z directionsr dimensionless coordinate in radial direction scaled by the pipe

radiusR pipe radius (m)Reb=UbD/ν Reynolds number based on bulk velocityReτ=uτD/ν Reynolds number based on friction velocityRep=UpD/ν Reynolds number based on Up.Sij rate of strain tensoruτ shear stress velocityUb=Up/2 bulk velocityUp centreline streamwise velocity of the laminar Poiseuille flowv’r,v’θ, v’z fluctuating velocity componentsy dimensionless distance from the wall, y=1-ry+=(1-r) uτ/ν distance from the wall in viscous wall unitsz coordinate in axial direction

Greek letters

δ* defined by δ*(2R-δ*) = 2 0

R

∫ r(1-Vz(r)/Up)dr

Δ=(r Δr Δθ Δz)1/3 characteristic gridspacingΔr gridspacing in radial directionΔθ gridspacing in circumferential directionΔz gridspacing in axial directionν kinematic viscosityθ coordinate in circumferential direction

θ∗ defined by θ*(2R-θ*) = 2 0

R

∫ r Vz(r)/Up (1-Vz(r)/Up)dr

Numerical Simulation of Turbulent Pipe Flow 233

1. Introduction

The turbulent circular pipe flow has attracted the interest of many investigators. Thesimplest case of non-rotating pipe has been extensively studied experimentally [Laufer(1954), Lawn (1971)] and numerically. Most of pipe flow numerical simulations have studiedstability and transition [Itoh (1977), Patera and Orszag (1981)]. Some Direct NumericalSimulations (DNS) have been performed. Using mixed finite difference and spectral methods,Nikitin (1993) was able to obtain satisfactory agreement with experimental data, inside theReynolds number range of 2250-5900. Unger et al. (1993) obtained excellent agreement withexperiments, using a second order accurate finite difference method. They confirmed thatpipe flow at low Reynolds number deviates from the universal logarithmic law. Zhang et al.(1994) reported simulation of low to moderate Reynolds number turbulent pipe flow obtainedwith a 3-D spectral code. Their initial results agree satisfactorily with both experiments andprevious numerical simulations. Eggels et al. (1994a) have carried out DNS and experimentsin order to investigate the differences between fully developed turbulent flow in anaxisymmetric pipe and a plane channel geometry. Most of the statistics on fluctuatingvelocities appear to be less affected by the axisymmetric pipe geometry.

When a flow is introduced to an axially rotating pipe, fluids are given a tangentialcomponent of velocity by the moving wall and the flow in the pipe exhibits a complicatedthree-dimensional nature. The high levels of turbulence and large shearing rates associatedwith swirling flows enhance the mixing process and provide a more homogeneous flow offluids. The role of the swirl flow is of great importance for the overall performance of the gasturbine. Recently, the numerical simulation of turbulent rotating pipe flow has received someinterest. Eggels et al. (1994b) used a DNS of the turbulent rotating pipe flow for moderatevalues of the rotation number. They confirmed numerically the drag reduction observed inexperiments. Orlandi and Fatica (1997) have also performed DNS of the turbulent rotatingpipe flow. Their investigation was devoted to the study of the range of the rotation number,N, not considered by Eggels et al. (1994b), that is the investigation of the flow field at highvalues of N (N ≤ 2) but not enough high to include re-laminarization, and to analyze themodifications of the near-wall vortical structures, for a more satisfactory explanation of thedrag reduction. They showed that a degree of drag reduction is achieved in the numericalsimulations just as in the experiments, and that the changes in turbulence statistics are due tothe tilting of the near-wall streamwise vortical structures in the direction of rotation. Themore recent study by Orlandi and Ebstein (2000) is an extension of the previous one. N hasbeen increased up to 10. These authors have evaluated the budgets for the Reynolds stressesat high rotation rates. These budgets are useful to those interested in developing new one-closure turbulence models for rotating flows.

Using a modified mixing length theory, Kikuyama et al. (1983) conducted calculations ofthe flow in an axially rotating pipe in a region far downstream from its inlet section. Theyobserved that when a turbulent level is introduced into the rotating pipe, a flow laminarizationis set up through an increase in the rotational speed of the pipe while destabilizing effectsoccur when the flow is initially laminar. Malin and Younis (1997) used two Reynolds stresstransport closures for modeling flow and heat transfer in fully developed axially rotating pipeflow. They showed that both models reproduce the observed influences of rotation. Theseinclude reduction in skin friction and wall heat transfer, suppression of radial turbulent


transport of heat and momentum, laminarization of the flow. An encouraging level ofagreement between calculations and available findings of literature was generally found.

Speziale et al. (2000) presented both the analysis and modeling of turbulent flow in anaxially rotating pipe flow. A particular attention was paid to tracing the origin of each of thetwo central physical features: the rotationally dependent axial mean velocity and therotationally dependent mean azimuthal or swirl velocity relative to the rotating pipe, in orderto gain a better physical insight into this turbulent flow. It was shown that second-orderclosure models provide good description of this flow and can describe both these featuresfairly well. In all the previous numerical studies, only one Reynolds number has beenconsidered.

There is very limited literature on investigations devoted to Large Eddy Simulations(LES) of turbulence pipe flow. The first LES work on fully developed turbulent pipe flow isgiven by Unger and Friedrich (1991). LES have been applied to flows in complex geometriesto a very limited extent. The major reasons for this are due to the need for describing the non-trivial geometry accurately whilst limiting the number of computational grid points. LESpredictions on turbulent pipe flow with rotation are extremely rare. There are only threeworks which deal with the turbulent rotating pipe flow. In 1993, Eggels and Nieuwstadtperformed Large Eddy Simulations (LES) for the fully developed turbulent flow inside anaxially rotating pipe. The main objective of this study was to investigate the influences ofpipe rotation on mean flow properties and mean velocity profiles as well as on the profiles ofthe Reynolds stress components. They showed that the mean flow properties are fairly wellpredicted by LES, especially about the reduction of wall friction, deformation of the meanaxial velocity profile and parabolic distribution of the mean circumferential velocity. Yangand McGuirk (1999) reported LES of turbulent pipe flow for the rotating and non rotatingcases. Their numerical results compare reasonably well with the experimental data. Theyconfirmed the experimental observations that turbulence decreases with an increase in piperotation due to the stability effect of the centrifugal force.

In the present study, DNS and LES of fully developed turbulent pipe flow are performedto report the effects of the rotation and Reynolds numbers on the flow characteristics. Thedirect simulations have been carried out at two Reynolds numbers, Re=4900 and Re=7400,for different rotation rates ranging from N = 0 to N= 18. To elucidate the impact of higherReynolds and rotation numbers, large eddy simulations with dynamic model have beenconducted for a Reynolds number up to 20600. The present paper is organized as follows: themathematical formulation and the numerical methods (DNS and LES) are described in section2. Section (3.1) presents DNS predictions of the turbulent pipe flow, and compares ourcomputations to the results reported in the archival literature. Section (3.2) deals with the LESpredictions of the turbulent pipe flow. The validation of DNS and LES approaches areachieved by comparing the predicted profiles and statistics to the numerical and experimentaldata available in the literature. The effects of the Reynolds and rotation numbers on differentthermal statistics (mean velocity profiles, root mean square of fluctuating velocitycomponents, Reynolds shear stresses, skewness and flatness factors, velocity and vorticityfields) are investigated and discussed in section 3. The conclusion in section 4 ends this work.


2. Governing Equations and Numerical Method

2.1. Direct Numerical Simulation

The continuity and momentum equations governing 3D-incompressible turbulent floware written in a cylindrical coordinate system, Figure 1, in terms of the variables qr=r.Vr,qθ=Vθ and qz=Vz, in order to avoid the singularity at the axis r=0. The dimensionlessequations are obtained using Up, the centerline streamwise velocity of the laminar Poiseuilleflow, and the pipe radius R as velocity and length scales respectively:

(1)

(2)

(3)

(4)

where the stresses, τij=2Sij, are given by:

(5)


Figure 1. Sketch of control volumes.

A mean pressure gradient in the qz equation maintains a constant bulk velocity, Ub. TheReynolds number and the rotation rate are defined as Reb=UbD/ν and N=ΩR/Ub, respectively.

2.2. Large Eddy Simulation

The governing equations for Large Eddy Simulation (LES) are given below. The overbarindicates the filtering of the instantaneous fields which leads to the resolved scale fields (thesmaller scales of turbulent motion being removed by the spatial filtering approach).

0=∂

∂+

∂

∂+

∂

∂

zzq

rq

rrq

θθ

,ˆ ˆ 1ˆr 11 1

21 2

2 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

∂

∂+

∂

∂+

∂

∂+

∂∂

−=+∂

∂+

∂

∂+

∂

∂+

∂

∂

zz

rrr

rReP

rrqNz

zqqqq

rrqqr

rt

q θτ

θθθτθτ

θθ

θθθθθ

,ˆˆ ˆ ˆr 1

-

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−

∂

∂+

∂

∂+

∂

∂+

∂∂

=−∂

∂+

∂

∂+

∂

∂+

∂

∂θθτ

τ

θθττ

θθθ

θ θ zrzrr

rrr

RerP

rNrqqz

zqrqr

rqq

rrqrq

rtrq q

.ˆ ˆ 1ˆr 11

1 1

⎥⎥⎦

⎤

⎢⎢⎣

⎡∂

∂+

∂

∂+

∂

∂+

∂∂

−=∂

∂+

∂

∂+

∂

∂+

∂

∂

zzzz

rrrz

rRezP

zzqzqzqq

rrzqrq

rtzq τ

θθττ

θθ


The total stresses ijijij 'ˆ τττ += are ijSReTij )1(ˆ ντ += where the strain tensor

expressed by the variables iq is:

The eddy viscosity νT has different expressions according to the subgrid model used.

Smagorinsky Model

In this model, the subgrid scale eddy viscosity is related to the deformation of theresolved velocity field as:

[ ] 2/1 22(2( ijSijS Δ)

sCSΔ)

sCTν ==

In the present study, the Smagorinsky coefficient CS is set equal to 0.15. For a discussionon the value and the interpretation of this constant, we refer to Mason and Callen (1986). Thissubgrid model largely used in LES of isotropic turbulence produced good results. When itapplied to inhomogeneous, and in particular to wall bounded flows, the constant wasmodified.

Dynamic Eddy Viscosity Model

The dynamic model provides a methodology for determining an appropriate local valueof the Smagorinsky coefficient. The model was proposed by Germano et al. (1991), withimportant modifications and extensions provided by Lilly (1992). In this model, the constantCd is not given a priori, but is computed during the simulation from the flow variables. Theturbulent viscosity is expressed using an eddy viscosity assumption as:

[ ] 2/1 22

ijSijS) (Δd

CTν =

but Cd is dynamically determined as follows.


Two different filter widths are introduced; the test filter Δ~ is larger than thecomputational filter Δ and it is applied to the momentum equations. Germano et al. (1991)derived an exact relationship between the subgrid scale stress tensors at the two different filter

widths (Germano identity). Substitution of a Smagorinsky form ijSijS|S| 2= for the subgrid

scale stress into Germano identity, along with some additional assumptions [Lilly (1992)],leads to the expression for the constants

ijM

ijM

ijM

ijL

ΔdC

22

1−= (1)

where the second order tensors Lij and Mij are given as follows:

,22

~~~

ij

MΔd

C-jqiqjqiqij

L =−= ~~~~2

2

ijS S ijS S Δ

ΔijM −=

The constant could be positive or negative. The positive values are linked to energyflowing from large to small scales and the negative to energy going from small to large scales(backward energy transfer). The angled brackets in equation (1) denotes averages in thehomogeneous direction.

The governing equations are discretized on a staggered mesh in cylindrical coordinates.The numerical integration is performed by a finite difference scheme, second-order accuratein space and in time. The time-advancement employs a fractional-step method. A third-orderRunge Kutta explicit scheme and a Crank-Nicolson implicit scheme are used to evaluate thenon-linear and viscous terms respectively. Uniform computational grid and periodic boundaryconditions are applied to the circumferential and axial directions. In the radial direction, non-uniform meshes specified by a hyperbolic tangent function are employed. On the pipe wall,the usual no-slip boundary condition is applied. For Reb = 4900 and Reb = 7400, weperformed simulations on a pipe of length Lz=20R using a 65x39x65 grid in the θ-, r- and z-directions. We have investigated the influence of different grids on the accuracy of thesolution. The finest grid (129x49x129) leads to well resolved simulations. However, the grid65x39x65 seems to predict and capture all features of the flow although small differencesoccur between some of the statistics obtained with these two grids. Since the fine gridrequires much larger CPU-time and storage requirements, we performed calculations on the65x39x65 grid which gives a good compromise between the required CPU-time andaccuracy. Tables 1 and 2 list the details of the grid resolutions used in the simulations for theother Reynolds numbers (Reb = 5300, Reb =10300 and Reb = 20600). The final statistics areaccumulated by spatial averaging in the homogeneous streamwise and circumferentialdirections, and by time-averaging.


Table 1. Grids resolutions for LES (Dynamic model)

Model Reb N Reτ (Lθ, Lr, Lz) Grid0 628 128x95x128

103006 6070 1242

DynamicModel (D. M.)

206006 1218

(2π, 1, 15) 128x129x256

Table 2. Grids resolutions for DNS

Reb Reτ N (Lθ, Lr, Lz) Grid360 0 256x257x256360 2 256x257x256334 6 256x257x2565300

231 18

(2π, 1, 15)

128x257x256


3.1. Direct Numerical Simulations

Mean Velocity Profiles

The axial mean velocity profiles (mean axial velocity normalized by the bulk meanvelocity Ub) are compared in Figure 2a with the DNS and experimental data by Eggels et al.(1994a) for a stationary pipe flow. The calculated profiles are in satisfactory agreement withthese data. For both Reynolds numbers, the streamwise velocity increases near the centre anddecreases near the wall when the pipe is rotating. An examination of the velocity profilesshows a gradual approach towards a parabolic shape (Poiseuille profile) when increasing N,and correspondingly the effect of turbulence suppression due to the pipe rotation becomesmore and more noticeable. Rotation has thus a very marked influence on the damping of theturbulent motion and drag reduction. This is in agreement with the experiments conducted byNishibori et al. (1987), Reich and Beer (1989) and Imao and Itoh (1996). At Re=7400, thecomputed velocities have lower values than those for Re=4900 in the central region of thepipe, whatever the rotation rate is. A similar observation has been reported in the experimentsby Reich and Beer (1989). It appears thus that the Reynolds number dependence of the meanvelocity profile decreases when the rotation rate increases. To better investigate the effect ofrotation on the turbulent pipe flow, we conducted four DNS with a rotation rate changinggradually from zero to a limit value N = 18. For this value (N = 18), a tendency to therelaminarization of the flow is observed. Table 2 summarizes the grid resolutions. As shownin Figure 2b, the mean velocity profile is significantly affected by the Coriolis force. It isworth pointing that the extend of the near-wall region is reduced when N increases, inaccordance with the experimental observation of Kikuyama et al. (1983).


Figure 2a. Axial mean velocity normalized by the bulk velocity Ub as function of the wall distance for 0≤ N ≤ 2.

Figure 2b. Axial mean velocity normalized by the bulk velocity Ub as function of the wall distance forN=0 (), N=2 (---), N=6 (···) and N=18 (··).


Mean Axial Velocity Normalized on the Friction Velocity

In Table 3, the mean flow parameters are compared to those of literature, for Reb ≈ 5300.The DNS calculations are first compared to the findings of Eggels et al. (1994a) for which theReτ is close to our Reτ. There is a good agreement between the present predictions and theresults of Eggels et al. (1994a). The slight difference between the data sets can be attributed tothe difference in the grid resolutions between the two simulations. The axial mean velocitynormalized by the friction velocity uτ versus the distance from the wall is shown in Figure 3ain wall units y+. This computed profile is consistent with the DNS profile of Eggels et al.(1994a). The validation of the present predictions has also been achieved by comparing ourcalculated kinetic energy, Figure 3b, and vorticity fluctuations, Figure 3c, to those of Eggelset al. (1994a).

Table 3. Mean-flow parameters for Direct Numerical Simulation (DNS) compared toliterature, at Reb = 5300 and N=0

DNS Eggels et al.(1994b) Loulou (1996)

Westerweel etal. (1996)

[PIV]

Westerweel etal. (1996)

[LDA]Nθ 256 128 160 - -

Nr 257 96 72 - -

Nz 256 256 192 - -

RΔθ+max 8.89 8.84 7.50 - -

Δr+min 0.11 0.94 0.39 - -

Δr+max 4.03 1.88 5.70 - -

Δz+ 14.10 7.00 9.90 - -

Rep 6954 6950 7248 7100 7200

Reb 5299 5300 5600 5450 5450

Re 362 360 380 366 371

Uc/u 19.19 19.31 19.12 19.38 19.39

Ub/u 14.63 14.73 14.77 14.88 14.68

Uc/Ub 1.31 1.31 1.29 1.30 1.32

Cf (x10-3) 9.35 9.22 9.16 9.03 9.28

*/R 0.127 0.127 0.121 0.124 0.130

θ*/R 0.069 0.068 0.066 0.068 0.071

H= */θ* 1.85 1.86 1.84 1.83 1.83


Eggels et al, 1994a.

Figure 3a. Axial mean velocity normalized by the friction velocity uτ as function of the distance forN=0 and Reb=5300.: lignes (present DNS), symbols.


Figure 3b. Kinetic energy profile for N=0: lignes (present DNS), symbols.



Figure 3c. Vorticity fluctuations versus the wall distance for N=0 and Reb=5300.: lignes (present DNS),symbols.

Figure 3d. Axial mean velocity normalized by uτ: a comparison between the present DNS and the dataof literature.


Figure 3e. Axial mean velocity normalized by uτ as function of the distance (in wall units) from the wallwith Reb and N as parameters.

To ascertain the reliability and accuracy of the present numerical simulation, the presentpredictions are compared to the available results of literature at Re=7442 in the case of astationary pipe: the experimental data of Eckelmann (1974) and Kim et al. (1987) for achannel flow. They are also compared to the LDA-mean flow distribution by Durst et al.(1995) for a turbulent stationary pipe flow. Figure 3d shows that all the velocity profiles meetfairly well with our numerical predictions. The satisfactory agreements confirm that the meanflow field is well predicted by the present numerical simulations. In Figure 3e, the solid linerepresents the universal velocity distributions in the viscous sublayer, in the buffer layer andin the inertial sublayer. The viscous sublayer is well resolved in the numerical simulations,yielding the linear velocity distribution Vz

+=y+ for y+<5. The buffer region is also wellpredicted in accordance with the log-law Vz

+=-3.05+5*ln y+. For the two Reynolds numbersconsidered, the agreement with the log-law at larger distances from the wall (y+>30) is lessfor the present DNS results, and also for the experimental data by Eggels et al. (1994a). Whenthe pipe rotates, the differences between the computed mean velocity and the log-laws are dueto the relaminarization of the flow when the rotation rate increases. Similar observations havebeen reported by Orlandi and Fatica (1997) and by Zhang et al. (1994). The reason is that thelog-laws are not observed in the pipe flow for Reb ≤ 9600 (Reb=UbD/ν), in contrast to planechannel flows. Theoretically, the log-laws are only justified at large Reynolds numbers(Tennekes and Lumley, 1972). The variation of the velocity friction with the rotation number

N is sketched in Table 4, for Reb = 5300. Here, each velocity friction uτ is normalized by that

of the non-rotating case, 0uτ . Note that the friction velocity diminishes with an augmentationin the rotation number, inducing a marqued reduction in the friction coefficient at the wall.


The friction factor decreases with an increase in the rotation number and this tendencybecomes more remarkable for larger values of the Reynolds number. For the highest rotationnumber, N = 18, the reduction in the friction coefficient f is about 60%. For stationary pipeflows, the simulation predicts a friction factor close to the value evaluated by using theBlasius relation for Re=4900 (f = 0.3164 Re-1/4 = 0.0378).

Table 4. Variation of the velocity friction with the rotation number at Reb = 5300.

Reb N u / uτ0

0 1.002 1.006 0.935300

18 0.64

Root Mean Square (rms)

The radial distribution of the root mean square (rms) of the fluctuating velocities in anon-rotating pipe are plotted in Figure 4a, for Reb = 5300, along with DNS calculations ofEggels et al. (1994a). There is a satisfactory agreement between the present predictions andthe measurements and DNS by Eggels et al. (1994a). The rms values of all velocitycomponents are also depicted in Figures 4b-d and compared to the numerical computation ofKim et al. (1987), to the measured rms of Kreplin and Eckelmann (1979) obtained with a X-hot film probe and to the LDA measurements by Durst et al. (1995). The comparison betweenthe present DNS results and the available experimental and numerical data of literature seems

Figure 4a. Root-mean-square profiles of azimuthal, radial and axial velocity components for N=0.Lines: Present DNS; Symbols (Eggels et al., 1994b): (Δ) Vr , rms , (O) Vθ , rms , ( ) Vz , rms.


to be better satisfactory. However, there are some slight differences. These differences aremost likely caused by the coarse numerical resolution of our computations. For Reb = 4900,Figure 4b shows that the peak in the distribution of the streamwise rms is located at the inneredge of the buffer region while that of the normal and tangential components are located atthe outer edge, as it can be seen in Figures 4c,d.

Figure 4b. Root-mean-square of the velocity fluctuations: axial velocity component, N=0.

Figure 4c. Root-mean-square of the velocity fluctuations: radial velocity component, N=0.


Figure 4d. Root-mean-square of the velocity fluctuations: tangential velocity component, N=0.

Figure 5a. Root-mean-square profiles of axial velocity with Reb and N as parameters (0≤N≤2).


Figure 5b. Root-mean-square profiles of velocity fluctuations for 0≤N≤18, Re=5300: <(Vz’)2>1/2 () ,<(Vθ’)2>1/2 (---), <(Vr’)2>1/2 (···).

The rotation of the wall has large effects on the rms, these effects being morepronounced for the streamwise rms velocity. Similar observations have been reported by


Eggels et al. (1994a) and by Orlandi and Fatica (1997). For N=2 and N=1, the rmsdistributions are almost the same (Figure 5a). Figure 5b depicts the turbulence intensitiesnormalized by the friction velocity for Reb = 5300 and higher rotation numbers. At amoderate rotation number (N = 1), the fluctuation levels of the three velocity componentsare increased in comparison to the non-rotating case, Figure 5b. From N ≥ 2, this tendencyremains for the radial and azimuthal components. On the contrary, a significant reductionof the axial turbulence intensity is observed. The intensification of the radial and azimuthalturbulence intensities in the core region of the rotation pipe denotes that the turbulencethere tends to be isotropic, especially at high rotation number. Near the wall (0 ≤ y ≤ 2), theazimuthal turbulence intensity exhibits remarkable peak values with increasing N, while themaximum of the axial fluctuations is considerably reduced (-10% for N = 2 and -30% for N= 18).

From the three rms velocities, the effect of the Reynolds number on the rms of the axialvelocity (⟨Vz'2⟩1/2) is evident, and it increases with an increase in the Reynolds number, Figure5a. Similar observations are reported by Zhang et al. (1994). For the other rms velocities,⟨Vθ'2⟩1/2 and ⟨Vr'2⟩1/2, this trend is observed in the core region of the flow.

Reynolds Shear Stresses

Figure 6a shows the distributions of the Reynolds stress components ⟨Vr'Vz'⟩. TheReynolds stresses in a stationary pipe allow to check the accuracy of the simulations. Thecomputed values of ⟨Vr'Vz'⟩ are compared with the measurements of Eggels et al.(1994a). The agreement between the present predictions and these results is satisfactory.The total shear stress (the sum of the turbulent and viscous stresses) is also plotted in thisfigure (solid line). As it can be seen the Reynolds shear stress dominates in the coreregion of the flow since the viscous shear stress is small. When the rotation rateincreases, the decrease of the viscous stress in the core region leads to a decrease of⟨Vr'Vz'⟩. On the other hand, the stress ⟨Vr'Vz'⟩ increases near the wall in accordance withthe increase of the viscous stresses. For Re=7400, the values of ⟨Vr'Vz'⟩ are slightly largerthan the corresponding values obtained for Re=4900, especially near the wall. The⟨Vr'Vθ'⟩ and ⟨Vθ'Vz'⟩ stresses, which are zero in a stationary pipe flow, become non-zerowhen the pipe rotates. The influence of rotation is to reduce the ⟨Vr'Vz'⟩ shear stress andto increase the two other stresses ⟨Vr'Vθ'⟩ and ⟨Vθ'Vz'⟩ as it is shown in Figure 6b. Nearthe rotating wall, the high values of ⟨Vθ'Vz'⟩ are related to the tilting of the near wallvortical structures (Orlandi and Fatica, 1997). In the core region of the flow, the radialoscillations of ⟨Vθ'Vz'⟩ are due to the large-scale structures within this region. Thebehavior of ⟨Vr'Vθ'⟩ is almost linear in the central part of the pipe. The Reynolds numberhas larger effects on the component ⟨Vθ'Vz'⟩, these effects being more pronounced whenthe rotation rate increases. The largest variations occur in the central region of the pipe.At N=1, the distributions of ⟨Vr'Vz'⟩ for both Reynolds numbers are well separated, andthe ⟨Vr'Vz'⟩-values are larger at Re=7400. On the other hand, the ⟨Vr'Vθ'⟩-profiles appearto be nearly independent of the Reynolds number.


Figure 6a. Distribution of the Reynolds shear stress component <Vr’Vz’> in wall units with Reb and Nas parameters.

Figure 6b. Reynolds shear stresses distributions in wall units for N=1.


Velocity and Vorticity Fields : Structures and Statistics

The instantaneous velocity and instantaneous vorticity fields are of great help in revealingthe influence of the rotation effect on turbulence coherent structures. Figure 7a depicts theisosurface of the vorticity modulus (ω = 3) for various rotation numbers N. It clearly showsthe impact of the rotation rate on the turbulence structures:

• the more and more flat shape of the isosurface near the wall, with increasing N,denotes a gradually reduction of the turbulence activity,

• As moving away from the wall, the augmentation in the rotation number induces aset of turbulence structures with strong vorticity developing up to the pipe centre,

• These vortical structures are inclined and better organized with increasing N.

Figure 7a. Isosurface of the vorticity modulus (ω=3) for 0≤N≤18.

The rotation seems to have a tendency to organize the flow near the pipe wall and in thecore region, leading to a relaminarization of flow.

A plot of a cross-sectional view of the streamwise vorticity field on an axial-azimuthalplane, for many rotation numbers, also exhibits the vertical structures in the pipe flow, Figure7b. At moderate rotation rate (N = 2), the vertical structures concentrate in some regions ofthe flow, while the other regions appear rather quiet. The strong longitudinal expansion of


these regions denotes presence of elongated vorticity streaks, similar to those observed inexperiments. The impact of rotation on the axial velocity fluctuation can be seen in Figure 7c.The patterns clearly illustrate a better organization of the pipe flow when N increases, and amore pronounced inclination of the axial velocity isosurface.

Figure 7b. Cross-sectional view of the streamwise vorticity field ans axial-azimuthal plane, for0≤N≤18: N=0 (a), N=2 (b), N=6 (c), N=18 (d).

The rms values of all three vorticity components, for the four direct numericalsimulations, are plotted in Figure 7d. Once again, one can clearly see the impact of theCoriolis force which gradually reduces the vorticity fluctuations. It is interesting to point outthe increasing anisotropy of the vorticity fluctuations. Indeed, with increasing N, the axialvorticity fluctuations predominate those of the two other components.


Figure 7c. Isosurface of the axial velocity for 0≤N≤18.

Figure 7d. Continued on next page.


Figure 7d. Root mean square of the three vorticity components for 0≤N≤18: <(ωz’)2>1/2 () , <(ωθ’)2>1/2

(---), <(ωr’)2>1/2 (···).

High-Order Statistics

Many experimental investigations of turbulent pipe flow at moderate Reynolds numbershave been published (Durst et al., 1995; Den Toonder, 1995; Westerweel et al., 1996). Theyused essentially Laser techniques to measure data: PIV (Particle Image Velocimetry) andLDA (Laser Doppler Anemometers). Figure 8 compares our DNS results for higher orderstatistics (skewness and flatness factors for axial velocity component) to the experimentaldata of Westerweel et al. (1996). There is a satisfactory agreement between the numerical andexperimental profiles. However, the present DNS slightly overpredicts the peak of the axialcomponent Vz’. Durst et al. (1995) reported similar observation when comparing their resultswith many DNS of literature.

The skewness factors of the axial, radial and tangential velocity components (S1, S2 andS3, respectively) are compared in Figures 8c,d,e to the DNS of Kim et al. (1987), to the laser-Doppler measurements of Niederschulte et al. (1990) for channel flow, and also to the LDAmeasurements of Durst et al. (1995) in the case of a stationary pipe. S3 is also compared to theLDV measurements of Karlsson and Johansson (1986) for a boundary layer flow.

About S1 (Figure 8c) the trends observed in the present simulations, in the experimentaldata and DNS of literature are similar, with a positive skewness near the wall and a negativeskewness factor approaching a value of –0,5 at larger y+ in the stationary pipe. The agreementwith the experimental or numerical findings of the referenced literature is not so good, butstill reasonable. Our predictions of S1 are somewhat overpredicted, especially near the wall.Figure 8d shows the variations of the skewness factor of S2 in wall region: the differencesbetween the present predictions and the results of literature are larger than for S1. Thesediscrepancies could be attributed to the too coarse grids used, but it should be also taken into


account that the experimental data are obscured by noises close to the wall, which lead to asmaller skewness factor. The predicted values of S3, are shown in Figure 8e: for a stationarypipe, almost zero-values are predicted over most of the pipe diameter because of symmetry.This is in good agreement with the data from DNS or measurements reported in the literature.


Figure 8. Skewness (a) and Flatness (b) factors for axial velocity component, N=0 and Reb=5300:lignes (present DNS), symbols.

Figure 8c. Skewness factors of the velocity fluctuations in the near-wall region: the streamwise velocitycomponent with Reb and N as parameters.


Figure 8d. Skewness factors of the velocity fluctuations in the near-wall region: the normal-to-the-wallvelocity component with Reb and N as parameters.

Figure 8e. Skewness factors of the velocity fluctuations in the near-wall region: the circumferentialvelocity component with Reb and N as parameters.


For a rotating pipe, the skewness factors S1 and S2 show large variations, especially in thenear wall region. However S1 approaches the negative value of –0,5 at larger distances fromthe wall (y+ > 70), while S2 tends towards zero. These large variations of the skewnesses withthe rotation rate are a further indication that changes of the orientation in the vorticalstructures occur near the wall (Orlandi and Fatica, 1997). When the pipe rotates, the skewnessfactor of the tangential velocity component, S3, is larger for N=1 near the wall. Its non zero-values across the flow field suggest that the reflection symmetry is broken by rotation.

In contrast to the skewness factor, the flatness factor of the axial velocity component, F1,is in better agreement with the results of literature as it can be seen in Figure 9a. However,higher flatness values are predicted at the wall region while the agreement is very good in thecore region. The behavior of the radial velocity component, F2, shows similar trend (Figure9b). The agreement is good with the data reported in previous studies for y+ > 25 (Durst et al.,1995; Niederschulte et al., 1990; Kim et al., 1987; Karlsson and Johansson, 1986). Theflatness factor is slightly larger than 3, which is the value of flatness factor for a Gaussiandistribution. However, near the wall, there is a strong increase of the flatness factor whereasthe experimental data show a decreasing flatness factor as the wall is approached (Durst et al.,1995; Niederschulte et al., 1990). Clearly, noticeable discrepancies between the differentresults for the flatness factor of the radial velocity component are observed, but the origin ofthese discrepancies is not clarified. The distribution of the flatness factor of the tangentialvelocity component, F3, is plotted in Figure 9c. The agreement between the present simulationfor stationary pipe flow and the results of literature is fairly good.

Figure 9a. Flatness factors of the velocity fluctuations in the near-wall region: the streamwise velocitycomponent with Reb and N as parameters.


Figure 9b. Flatness factors of the velocity fluctuations in the near-wall region: the normal-to-the-wallvelocity component with Reb and N as parameters.

Figure 9c. Flatness factors of the velocity fluctuations in the near-wall region: the circumferentialvelocity component.


In the case of a rotating pipe, Figures 9a,c show that large variations of the flatnessfactors F1, F2 and F3 occur near the rotating wall while they are almost unchanged in theremaining part of the pipe. This is an indication of the intermittent nature of the wall region.

Durst et al. (1995) showed that the higher order-moments and turbulence intensity arehighly interconnected, especially close to the wall. Therefore, we have examined the inter-relations between these quantities over the cross-section of the pipe. We have plotted inFigure 10 the distribution of the turbulence intensity together with the distributions of theskewness and flatness factors of the streamwise velocity component. It can be seen that theincrease of the turbulence intensity results from a decrease of the non-Gaussian behavior ofthe skewness and flatness factors. The points for the maximum intensity, zero skewness andminimum flatness coincide. This result agrees with the measurements of Durst et al. (1995).

Figure 10. Distributions of turbulent intensity, skewness and flatness factors of the axial velocitycomponent.

3.2. Large Eddy Simulation

This section is devoted to Large Eddy simulation (LES) of turbulent pipe flow. Theobjective of these LESs is to improve upon the description of such flow and to elucidate theeffect of higher Reynolds numbers and higher rotation numbers on the field flow. Theaccuracy of the LES technique depends significantly on the ability of the subgrid-scale (SGS)model. Feiz (2005) examined the performance of two turbulence models: the classicalalgebraic eddy-viscosity model of Smagorinsky (S.M.) and the dynamic model (D.M.) for therange 556 ≤ Reτ ≤ 697. Details of the corresponding grid resolutions are listed in Table 5.


The results show that the flow field and its statistics are better predicted with the dynamicmodel. For example, Figure 11 compares our LES computations with the DNS data ofWagner et al. (2001). The reader can find more details in Feiz (2005). In the this section, wepresent some LES results obtained with the dynamic model. The motivation of these LESstudies is to deal with the behaviour of the turbulent pipe flow in the rotating and non-rotatingpipes for higher Reynolds numbers (Reτ > 1000), and to examine the effectiveness of the LESmethod for predicting such turbulent flows. Table 6 gives informations on the grid resolutionfor Reτ = 1242, using LES with the dynamic model. Table 7 lists the mean flow parameters ofthe present LES, along with the DNS data of Wagner et al. (2001), for Reb = 10300. Theagreement between them is reasonable good.

Table 5. Grids resolutions for LES (Smagorinsky and Dynamic models) at Reb = 10300and N=0

Model Reb Reτ N (Lθ, Lr, Lz) GridS. M. 556D. M. 10300 628 0 (2π, 1, 15) 128x95x128

Table 6. Grids resolutions for LES (Dynamic model) at Reb = 20600 and N=0

Model Reb Reτ N (Lθ, Lr, Lz) GridD. M. 20600 1242 0 (2π, 1, 15) 128x129x256

Table 7. Mean-flow parameters for Large Eddy Simulation (LES) and literature at Reb

= 10300 and N=0

LES Wagner et al. (2001)Nθ 128 240Nr 95 70Nz 128 486RΔθ+

max 8.89 8.36Δr+

min 0.11 0.64Δr+

max 4.03 7.68Δz+ 14.10 6.58Rep 13181 13210Reb 10300 10300Reτ 628 640Uc/u 20.38 20.64Ub/u 15.94 16.09Uc/Ub 1.28 1.28Cf (x10-3) 7.87 7.70

*/R 0.115 0.115θ*/R 0.070 0.071H= */θ* 1.65 1.63

Wagner et al., 2001.


Figure 11. Axial mean velocity normalized by the friction velocity as function of the wall distance forReb=10500 and N=0. Lignes (present LES): (a) Smagorinsky model, (b) dynamic model. Symboles:Wagner et al (2001).

The rms velocity fluctuations nondimensionalized by the friction velocity are given onFigures 12a,b,c for three Reynolds numbers Reb = 5300, Reb = 10300 and Reb = 20600. Thepredicted fluctuations in the wall-normal, Figure 12a, azimuthal, Figure 12b, and streamwise,Figure 12c, directions are similar to the experimental data of Westerweel et al. (1996). Thefluctuations in the streamwise direction are more intense than those in the two otherdirections. The locations of the maxima for each velocity fluctuation are different: themaxima of azimuthal and radial turbulence intensities are located in the logarithmic region,while the maxima of the axial turbulence intensities occur in the buffer region. Furthermore,

the peak in the axial velocity fluctuation is located at a distance in wall unit y+ ≈ 13,


Figure 12. Root mean square velocity fluctuations in wall units, for N=0 and Reb=5300 (), Reb=10300(---), Reb=20600 (···): (a) tangential component<(Vθ’)2>1/2/uτ; (b) radial component<(Vr’)2>1/2/uτ; (c)axial component <(Vz’)2>1/2/uτ

Figure 13. Reynolds shear stresses distributions in wall units for N=0 and Reb=5300 (), Reb=10300 (---), Reb=20600 (···).


irrespective of the Reynolds number, i.e. the Reynolds number has no effect on the peak valueof the axial velocity fluctuation. For the two other velocity components, the positions of thecorresponding peaks shift farther off from the wall with increasing Reynolds number,suggesting that the maxima of <Vr’Vr’> and <Vθ’Vθ’> are not always located in the boundarylayers. Similar to the behaviour of the rms of velocity fluctuations in the radial and azimuthaldirections, the Reynolds shear stresses move away from the wall when Reb increases,Figure 13.

Figures 14a,b depicts the distribution of the skewness and flatness factors of the axialvelocity fluctuation near the wall region of a non-rotating pipe, for many Reynolds numbers.These distributions point out two interesting observations:

• first, the skewness and flatness coefficients of the velocity fluctuations seem to bevery sensitive to the variation of the Reynolds number, near the wall,

• in the core region, the skewness and flatness values for fully developed turbulentflow are not equal to those of a Gaussian distribution (S = 0 and F = 3). This implies,for the skewness of the radial velocity fluctuation (not represented here), that theturbulent energy is convected from the wall to towards the pipe axis.

Close to the pipe axis, the probability distribution of the axial fluctuations is asymmetric.The negative values of S1 indicate that the negative fluctuation of Vz’ are dominant inprobability. In the vicinity of the wall, S1 becomes positive and is enhanced. This trend ismore apparent with increasing Reb. Similarly, the profiles of the flatness coefficient of theaxial velocity component increases near the wall with increasing Reb, suggesting that the axialturbulence fluctuation in the wall region become more intermittent with increasing Reynoldsnumber.

Figure 14. Skewness (a) and Flatness (b) factors of axial velocity component for N=0 and Reb=5300(), Reb=10300 (---), Reb=20600 (···).


Figure 15. Effect of Re on the skewness and flatness factors of the axial velocity fluctuation for high N(N=6): Reb=5300 (), Reb=10300 (---), Reb=20600 (···).

Figure 16. Effect of N on the skewness and flatness factors of the axial velocity fluctuation for high Reb

(Reb=20600): N=0(), N=6 (---).

Figures 15 a,b shows the effect of Reynolds number on the skewness and flatness factorsof the axial velocity fluctuation, for high rotation rate (N = 6). It seems that the impact of theReynolds number on the flatness factor is more important than that on the skewness factor.For the highest Reynolds number (Reb = 20600), the influence of the rotation number on theskewness factor is more pronounced than that on the flatness factor, Figures 16 a,b. In both


cases, the effect of Reb or N on the skewness and flatness coefficients are mainly restricted inthe near wall region. The present LES predictions compare reasonably well with the results ofliterature. We can conclude that all phenomena in rotating and non-rotating turbulent pipeflow can be captured directly by LES with dynamic model.

4. Conclusion

This chapter was devoted to DNS and LES of the fully developed turbulent flow in anaxially rotating pipe for various Reynolds numbers and different rotation rates. Differentstatistical turbulence quantities including the mean and fluctuating velocity components,friction coefficient, the Reynolds shear stresses and higher order statistics are obtained andanalyzed. An effort to reveal the effects of the Reynolds number and the rotation number onthe turbulent pipe flow is sketched. The validation of the present DNS and LES has beenachieved by comparing our predictions with some available results of literature. It is shownthat the computed deformation of the mean axial velocity profile agrees with the deformationobserved experimentally and that rotation has a very marked influence on the suppression ofthe turbulent motion and on the drag reduction. A tendency to the relaminarization of the flowis observed for the highest rotation number. The Reynolds number dependence of the meanvelocity profile decreases when the rotation rate increases. The friction factor decreases withan increase in the rotation number and this tendency becomes more remarkable for largervalues of the Reynolds number.

The rms and Reynolds stresses profiles showed an encouraging level of agreement withthe measured data and DNS results of literature. From N ≥ 2, a significant reduction of theaxial turbulence intensity is observed, while the fluctuation levels of the radial and azimuthalcomponents are increased denoting an isotropization of the in the core region of the rotation.The rotation also reduces the ⟨Vr'Vz'⟩ shear stress and increases the two other stresses ⟨Vr'Vθ'⟩and ⟨Vθ'Vz'⟩. The Reynolds number has larger effects on the component ⟨Vθ'Vz'⟩, these effectsbeing more pronounced when the rotation rate increases. On the other hand, the ⟨Vr'Vθ'⟩-profiles appear to be nearly independent of the Reynolds number.

The overall agreement between the predicted skewness and flatness factors and theresults reported in the literature is satisfactory. For rotating pipe, the large variations of theskewness and flatness factors of the velocity components, in the near wall region denote theintermittent nature of the wall region and indicate that changes of the orientation in thevortical structures occur near the wall. For the highest rotation rate, the impact of theReynolds number on the flatness factor is more important than that on the skewness factor.For the highest Reynolds number, the influence of the rotation number on the skewness factoris more pronounced than that on the flatness factor. In both cases, the effect of the Reynoldsor rotation numbers on the skewness and flatness coefficients are mainly restricted in the nearwall region. Visualizations of the instantaneous velocity and vorticity fields exhibitturbulence structures with strong vorticity developing up to the pipe centre. These vorticalstructures are inclined and better organized with increasing N.

An interesting outcome of the present investigation is to establish databases of variousturbulence statistics of the turbulent pipe flow at different Reynolds and rotation numbers.These databases will undoubtedly helpful for evaluating and developing turbulence models.


References

[1] Durst F. , Jovanovic J. and Sender J., 1995, J. Fluid Mech., 295, p. 305.[2] Eckelmann H., 1974, J. Fluid Mech., 65, p. 439.[3] Eggels, J. G. M. and Nieuwstadt, F. T. M., 1993, In Proc. 9th Symp. on Turbulent Shear

Flows, Kyoto, Japan, p.310.[4] Eggels, J. G. M., Unger, F., Weiss, F., M. H., Westerweel, J., Adrian, R. J., Friedrich, R.

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(Ed. R. J. Adrian), Lisbon, Portugal, p. 273.[11] Kikuyama , K., Murakami, M. and Nishibori, K., 1983, Bull. J.S.M.E., 26, p. 506.[12] Kim J., Moin P. and Moser R., 1987, J. Fluid Mech., 177, p. 133.[13] Kreplin H. and Eckelmann H., 1979, Phys. Fluids, 22, p. 1233.[14] Laufer, J., 1954, NACA Report 1174.[15] Lawn, C. J., 1971, J. Fluid Mech., 48, p. 477.[16] Lilly, D. K., 1992, Phys. Fluids A, 4, p. 633.[17] Loulou P., 1996, Thèse de doctorat, Stanford University, Department of Aeronautics and

Astronautics.[18] Malin M. R. and Younis B. A., 1997, Int. Comm. Heat Mass Transfer, 24, p. 89.[19] Mason, P. J. and Callen, N. S., 1986, J. Fluid Mech., 162, p. 439.[20] Niederschulte N. A., Adrian R. J. and Hanratty T. J., 1990, Exps. Fluids, 9, p. 222.[21] Nikitin, N. V., 1993, In Bulletin of APS, Vol. 38, No. 12, p. 2311.[22] Nishibori, K., Kikuyama, K. and Murakami, M., 1987, JSME Intl J., 30, p. 255.[23] Orlandi, P. and Fatica, M., 1997, J. Fluid Mech., 143, p. 43.[24] Orlandi, P. and Ebstein, D., 2000, International Journal of Heat and Fluid Flow, 21, p.

499.[25] Patera, A. T. and Orszag, S. A., 1981, J. Fluid Mech., 112, p. 467.[26] Reich, G. and Beer, H., 1989, Intl J. Heat Mass Transfer, 32, p. 551.[27] Tennekes, H. and Lumley, J. L., 1972, “A First Course in Turbulence”, MIT Press.[28] Toonder Den J. M. J., 1995, Thèse de doctorat, Delft, University of Technology,

Department of Aero- and Hydrodynamics.[29] Unger, F., Eggels, J. G. M., Friedrich, R. and Nieuwstadt, F. T. M., 1993, In Proc. 9th

Symp. on Turbulent Shear Flows, Kyoto, Japan, pages 2/1/1-2/1/6.[30] Unger, F. and Friedrich, R., 1991, In Proc. 8th Symp. on Turbulent Shear Flows,

Munich, Germany, pp. 19-3-1 – 19-3-6.[31] Wagner C., Huttl T. J. and Friedrich R., 2001, Computers & Fluids, 30, p. 581.[32] Westerweel J., Draad A. A., Hoeven van der J. G. Th. and Oord van J., 1996,

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[33] Yang, Z. and McGuirk, J.J., 1999, Proc. of Turbulence and Shear Flow Phenomena,USA, p. 863.

[34] Zhang, Y., Gandhi, A., Tomboulides, A. G. and Orszag, S. A., 1994, AGARD Conf.Proc., 551, pp. 17.1-17.9.


Chapter 8

PIPE FLOW ANALYSIS OF URANIUM NUCLEARHEATING WITH CONJUGATE HEAT TRANSFER

G.H. Yeoh* and M.K.M. HoAustralian Nuclear Science and Technology Organisation (ANSTO),

PMB 1, Menai,NSW 2234, Australia

Abstract

The field of computational fluid dynamics (CFD) has evolved from an academic curiosity to atool of practical importance. Applications of CFD have become increasingly important innuclear engineering and science, where exacting standards of safety and reliability areparamount. The newly-commissioned Open Pool Australian Light-water (OPAL) researchreactor at the Australian Nuclear Science and Technology Organisation (ANSTO) has beendesigned to irradiate uranium targets to produce molybdenum medical isotopes for diagnosisand radiotherapy. During the irradiation process, a vast amount of power is generated whichrequires efficient heat removal. The preferred method is by light-water forced convectioncooling—essentially a study of complex pipe flows with coupled conjugate heat transfer.Feasibility investigation on the use of computational fluid dynamics methodologies intovarious pipe flow configurations for a variety of molybdenum targets and pipe geometries aredetailed in this chapter. Such an undertaking has been met with a number of significantmodeling challenges: firstly, the complexity of the geometry that needed to be modeled.Herein, challenges in grid generation are addressed by the creation of purpose-built body-fitted and/or unstructured meshes to map the intricacies within the geometry in order to ensurenumerical accuracy as well as computational efficiency in the solution of the predicted result.Secondly, various parts of the irradiation rig that are required to be specified as compositesolid materials are defined to attain the correct heat transfer characteristics. Thirdly, the use ofan appropriate turbulence model is deemed to be necessary for the correct description of thefluid and heat flow through the irradiation targets, since the heat removal is forced convectionand the flow regime is fully turbulent, which further adds to the complexity of the solution. Ascomplicated as the computational fluid dynamics modeling is, numerical modeling hassignificantly reduced the cost and lead time in the molybdenum-target design process, andsuch an approach would not have been possible without the continual improvement ofcomputational power and hardware. This chapter also addresses the importance of

* E-mail address: [email protected], Phone no:+61-2-9717 3817, Fax no.:+61-2-9717 9263. Corresponding

Author:Dr Guan Heng YEOH, B40, ANSTO, Private Mail Bag 1, Menai, NSW 2234, Australia.

G.H. Yeoh and M.K.M. Ho270

experimental modeling to evaluate the design and numerical results of the velocity and flowpaths generated by the numerical models. Predicted results have been found to agree well withexperimental observations of pipe flows through transparent models and experimentalmeasurements via the Laser Doppler Velocimetry instrument.

Introduction

Australian Nuclear Science and Technology Organisation (ANSTO) is Australia’snational nuclear research centre, and it possesses two reactors: the long-serving and recently-decommissioned High-Flux-Australia-Research (HIFAR) Reactor and the newly-commissioned Open Pool Australian Light-water (OPAL) research reactor. The mainfunctions of both reactors are to produce radiopharmaceutical products for medicine, togenerate neutron beams for scientific research and to irradiate silicon ingots forsemiconductor applications.

This paper aims to examine the state-of-the-art application of Computational FluidDynamics to obtain sensible solutions to three separate pipe flow cooling problems of themolybdenum-99 irradiation facility, of which target cans containing uranium are irradiatedand subsequently processed for the production of medical isotopes. The diversity andreliability of the CFD methodology will be demonstrated by the degree of detail modeled ineach study as well as by the consistency of agreement among simulation results andexperimental validation data and other numerical forms of verification comparisons. Threecase designs are examined. They are:

1. The ‘rocket-can’ design as used in the HIFAR reactor2. The proposed ‘annular can’ replacement design for HIFAR3. The current ‘Molybdenum-plate’ design used in the new OPAL reactor

In simple terms, the study of reactor thermo-hydraulic characteristics is the evaluation ofa reactor coolant system’s heat removal capacity. To this regard, it is an evaluation of flowthrough what is often a complex system of pipe-work containing various reactor componentsacting as blockages. Such complex systems are difficult to evaluate for pressure drop andflow velocity characteristics using standard correlation-based formula describing pipe flow.To complicate matters, the heat transfer characteristics of components are closely dependenton the unique flow characteristics of each specific pipe-flow system. In the past, such uniqueheat transfer characteristics had been evaluated by the use of scaled and prototype heat-transfer rigs, but this has become increasingly and prohibitively expensive. With the advent ofcomputers, many factors have made the direct simulation of pipe flows, by means of CFD,feasible. The exponential increase in computational power, and ever-increasing improvementsin numerical methods and graphical post processing, met with a similar decrease incomputational cost has made CFD a viable method to directly solve for such distinctive pipeflow systems.

Pipe Flow Analysis of Uranium Nuclear Heating with Conjugate Heat Transfer 271

Background

Fluid dynamics deals with both heat transfer and fluid flow where the fluid paths can becomplicated and governing equations are non-linear. For most real-life problems associatedwith fluid dynamics, obtaining an analytical solution is difficult, if not impossible.Experiments that simulate fluid flow using scaled models in air tunnels, water tunnels ortowing tanks are used to obtain accurate information, but these experiments can be difficultand expensive to set up.

CFD takes the advantage of modern computing power in the application of numericalmethods to solve complex fluid dynamic problems. Retrospectively, CFD emphasizes theresolution of the physical processes through the use of digital computers which proceeds byfirst negotiating the sub-division of the domain into a number of finite, non-overlapping sub-domains. This leads to the construction of an overlay mesh of cells covering the wholedomain. In general, the set of fundamental mathematical equations are required to beconverted into suitable algebraic forms, which are then solved via suitable numericaltechniques.

The mathematical equations governing the heat transfer and fluid flow of the pipe flowsystems within HIFAR and OPAL are those of the conservation of mass, momentum andenergy. Nevertheless, the majority of pipe flows in the reactor are turbulent in nature. DirectNumerical Simulation (DNS) is the most accurate approach to turbulence simulation, whichdirectly solves the governing transport equations without undertaking any averaging orapproximation other than the numerical approximations performed on them. Through suchsimulations, all of the fluid motions contained in the flow are considered to be resolved; allsignificant turbulent structures are required to be adequately captured (i.e., the domain ofwhich the computation is carried out needs to accommodate for the smallest and largestturbulent eddy). Alternatively, Large Eddy Simulation (LES), which essentially resolves thelarge eddies exactly through the availability of mesh requirement but approximates the smalleddies, is still expensive but much less costly than DNS. The results of a DNS or LESsimulation contain very detailed information about the flow, producing an accurate realizationof the flow while encapsulating the broad range of length and time scales. DNS and LESapproaches usually require high usage of computational resources and often cannot be used asa viable design tool in reactor design and analysis because of the enormity of the numericalcalculations and the large number of grid nodal points.

Meanwhile, a more pragmatic approach is to adopt computational procedures that canstill supply adequate information about the turbulent processes but avoid the need to predictall of the effects associated with each and every eddy in the flow. For most engineeringpurposes, especially in reactor design and analysis, information about the time-averagedproperties of the flow (e.g., mean velocities, mean pressures, mean stresses, etc.) aresufficient to satisfy regulatory requirements. In this sense, all details concerning the state ofthe flow contained in the instantaneous fluctuations can be discarded. This process of onlyobtaining mean quantities can be achieved by adopting a suitable time-averaging operation onthe equations governing the conservation of mass, momentum and energy; these time-averaged equations are generally known as the Reynolds-Averaged Navier-Stokes (RANS)equations.


Turbulence Modeling

The Reynolds-averaged approach to turbulence results in the formulation of the two-equation turbulence model—the standard k-ε model proposed by Launder and Spalding(1974). This model is well established, widely validated and gives rather sensible solutions tomost industrially relevant flows. As an alternative to the standard k-ε model, other eddyviscosity models such as RNG k-ε model and reliazable k-ε model proposed by Yakhot et al.(1992) and Shih et al. (1995) are possible recommendations. The improved features of thesemodels have been shown to be aptly applicable to predict important flow cases having flowseparation, flow re-attachment and flow recovery.

For wall attached boundary layers, turbulent fluctuations are suppressed adjacent to thewall and the viscous effects become prominent in this region known as the viscous sub-layer.The modified turbulent structure of near-wall flow generally precludes the application of thetwo-equation models such as standard k-ε model, RNG k-ε model and reliazable k-ε model atthe near-wall region. One common approach is to adopt the so-called wall-function method;the near-wall region is bridged with logarithmic wall functions to avoid resolving the viscoussub-layer. Nevertheless, it is possible to totally resolve the viscous sub-layer by theapplication of the model standard k-ω model developed by Wilcox (1998) where ω is afrequency of the large eddies of which the model has also shown to perform splendidly closeto walls in boundary layer flows. The standard k-ω model is nevertheless very sensitive to thefree-stream conditions and unless great care is exercised, spurious results are obtained in flowregions away from the solid walls. To overcome such problems, the SST (Shear StressTransport) variation of Menter’s model (1993, 1996) was developed with the aim ofcombining the favorable features of the standard k-ε model with the standard k-ω model inorder that the inner region of the boundary layer is adequately resolved by the latter while theformer is employed to obtain numerical solutions in the outer part of the boundary layer. Thismodel works exceptionally well in handling non-equilibrium boundary layer regions such asflow separation.

Grid Generation

The arrangement of discrete number of points throughout the flow field, normallycalled a mesh, is a significant consideration in CFD. For the Molybdenum-99irradiation facility, grid generation poses the most challenging task because of itsinherent intricate geometrical details. In the three case designs considered, theconstruction of suitable meshes accounts for almost the entire reactor design andanalysis. Owing to the complexity of geometry, application of structured and/orunstructured meshes is required.

By definition, a structured mesh is a mesh containing cells having either a regular-shapeelement with four-nodal corner points in two dimensions or a hexahedral-shape element witheight-nodal corner points in three dimensions. Commonly applied in numerous CFDinvestigations, it basically deals with the straightforward prescription of either an orthogonalmesh or a body-fitted mesh. An unstructured mesh can however be described as a mesh


overlaying with cells in the form of either a triangle-shape element in two dimensions or atetrahedron-shape in three dimensions.

For a body-fitted mesh, the mesh construction of the internal region of the physicaldomain can normally be achieved via two approaches. On one hand, the Cartesian coordinatesmay be algebraically determined through interpolation from the boundary values. Thismethodology requires no iterative procedure and it is computationally inexpensive. On theother hand, a system of partial differential equations of the respective Cartesian coordinatesmay be solved numerically with the set of boundary values as boundary conditions in order toyield a highly smooth mesh in the physical domain. The former is commonly known as thetransfinite interpolation method and the latter is typically the elliptic grid generation method(Smith, 1982 and Thompson, 1982).

For an unstructured mesh, triangle and tetrahedral meshing are by far the most commonforms of unstructured grid generation. In Delaunay meshing, this most commonly adoptedgrid generation procedure entails the initial set of boundary nodes of the geometry to betriangulated according to the Delaunay triangulation criterion. Here, the most importantproperty of a Delaunay triangulation is that it has the empty circumcircle (circumscribingcircle) property (Shewchuk, 2002). All algorithms for computing Delaunay triangulations relyon the fast operations for detecting when a grid point is within a triangle's circumcircle and anefficient data structure for storing triangles and edges. The most straightforward way ofcomputing the Delaunay triangulation is to repeatedly add one vertex at a time, thenretriangulating the affected parts thereafter. When a vertex is added, a search is done for alltriangles’ circumcircles containing the vertex. Then, those triangles are removed whosecircumcircles contain the newly inserted point. All new triangulation is then formed byjoining the new point to all boundary vertices of the cavity created by the previous removal ofintersected triangles. Delaunay triangulation techniques based on point insertion extendnaturally to three dimensions by considering the circumsphere (circumscribing sphere)associated with a tetrahedron. More details on Delaunay triangulation and meshing can bereferred in Mavriplis (1997). Besides Delaunay method, other meshing algorithms inunstructured grid generation include the advancing front method (Lo, 1985; Gumbert et al.,1989; Marcum and Weatherill, 1995) and quadtree/octree method (Yerry and Shepard, 1984;Shepard and Georges, 1991).

It should be noted that the use of hybrid grids that combine different element types suchas triangular and quadrilateral in two dimensions or tetrahedral, hexahedral, prisms andpyramids in three dimensions can provide the maximum flexibility in matching mesh cellswith the boundary surfaces or internal solid regions where heat transfer due to conductionneeds to be resolved, and allocating cells of various element types in other parts of thecomplex flow regions. Grid quality can be enhanced through the placement of quadrilateral orhexahedral elements in resolving boundary layers near solid walls or composite materials insolid regions within the pipe flow systems whilst triangular or tetrahedral elements aregenerated for the rest of the flow domain. This generally leads to both accurate solutions andbetter convergence for the numerical solution methods.


Outline of CFD Model

Governing Equations

The unsteady RANS equations for the CFD model that reflect the conservation of mass,momentum and energy can be written as:

Mass

( ) ( ) ( ) 0u w

t x y zρ ρ ρρ ∂ ∂ ∂∂

+ + + =∂ ∂ ∂ ∂

v(1)

x-Momentum

( ) ( ) ( ) ( )

xyxx xzu

u uu u wut x y z

u u u Sx x y y z z x y z

ρ ρ ρ ρ

ττ τμ μ μ

∂ ∂ ∂ ∂+ + + =

∂ ∂ ∂ ∂′∂′ ′⎡ ⎤ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂⎡ ⎤ ⎡ ⎤+ + + + + +⎢ ⎥⎢ ⎥ ⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦⎣ ⎦

v

(2)

y-Momentum

( ) ( ) ( ) ( )

xy yy yz

u u wt x y z

Sx x y y z z x y z

ρ ρ ρ ρ

τ τ τμ μ μ

∂ ∂ ∂ ∂+ + + =

∂ ∂ ∂ ∂′ ′ ′∂ ∂ ∂⎡ ⎤∂ ∂ ∂ ∂ ∂ ∂⎡ ⎤ ⎡ ⎤+ + + + + +⎢ ⎥⎢ ⎥ ⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦⎣ ⎦

v

v v v v

v v v(3)

z-Momentum

( ) ( ) ( ) ( )

yzxz zzw

w u w w wwt x y z

w w w Sx x y y z z x y z

ρ ρ ρ ρ

ττ τμ μ μ

∂ ∂ ∂ ∂+ + + =

∂ ∂ ∂ ∂′∂′ ′⎡ ⎤ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂⎡ ⎤ ⎡ ⎤+ + + + + +⎢ ⎥⎢ ⎥ ⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦⎣ ⎦

v

(4)

Energy

( ) ( ) ( ) ( )

x y z

yx z

q q q

H u H H wH pt x y z t

qq qT T Tx x y z z z x y z

ρ ρ ρ ρ

λ λ λ

∂ ∂ ∂ ∂ ∂+ + + = +Φ +

∂ ∂ ∂ ∂ ∂′∂′ ′⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂

+ + + + +⎢ ⎥ ⎢ ⎥ ⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦ ⎣ ⎦

v

(5)


From the above equations, ( ) indicates time-averaged quantities, t is the time, ρ is thedensity, u, v and w are the respective velocity components along the Cartesian coordinatedirections x, y and z, H is the enthalpy, T is the temperature, μ is the fluid dynamic viscosityand λ is the fluid thermal conductivity. The time-averaged source or sink terms uS , Sv and

wS are given by

up u wSx x x y x z x

μ μ μ′∂ ∂ ∂ ∂ ∂ ∂ ∂⎡ ⎤ ⎡ ⎤ ⎡ ⎤= − + + +⎢ ⎥ ⎢ ⎥ ⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦ ⎣ ⎦

v(6)

( )refp u wS gy x y y y z y

μ μ μ ρ ρ′ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤∂ ∂ ∂ ∂ ∂ ∂ ∂

= − + + + − −⎢ ⎥ ⎢ ⎥ ⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦ ⎣ ⎦v

v(7)

wp u wSz x z y z z z

μ μ μ′∂ ∂ ∂ ∂ ∂ ∂ ∂⎡ ⎤ ⎡ ⎤ ⎡ ⎤= − + + +⎢ ⎥ ⎢ ⎥ ⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦ ⎣ ⎦

v(8)

From above, p′ is the modified averaged pressure defined by 23p p kρ′ = + +

23 ref i ig xμ ρ∇⋅ −V where p is the dynamics pressure, k is the turbulent kinetic energy,

( ), ,u w≡V v is the velocity vector, refρ is the reference density, gi are the gravitational

acceleration components ( )0, ,0g− and xi are the coordinates relative the Cartesian datum

( ), ,x y z and the effects due to the viscous stresses in the energy equation (5) are described

by the averaged dissipation function Φ . For weakly compressible flows, it is commonpractice to transform the energy equation by replacing the heat flux according to the localenthalpy gradient instead of the temperature gradient, viz.,

xHq

Pr xμ ∂

=∂

yHq

Pr yμ ∂

=∂

zHq

Pr zμ

= −∂

(9)

where Pr is the Prandtl number. Also, the pressure work term p t∂ ∂ and the averaged

dissipation function Φ that represents the source of energy due to work done deforming thefluid element are usually ignored in most practical applications.

On the basis of the eddy viscosity model, the Reynolds stresses defined by the stressvector ijτ′ ′≡τ where i, j = x, y, z in the momentum equations (2)–(4) is given by

( )( ) 2 23 3

Tij t t kτ μ μ δ ρ δ′ ′− ≡ − = ∇ + ∇ − ∇⋅ −V V Vτ (10)


where tμ is the turbulent eddy viscosity. The Reynolds flux in the energy equation (5)

defined by iq′ may also be modelled analogously to the eddy viscosity hypothesis as

ti

t i

HqPr xμ ∂

=∂

(11)

where tPr is the turbulent Prandtl number. To satisfy dimensional requirements, at least twoscaling parameters are required to relate the Reynolds stress to the rate of deformation. Inmost engineering flow problems, the complexity of turbulence precludes the use of anysimple formulae. A feasible choice is the turbulent kinetic energy k and another turbulentquantity which is the rate of dissipation of turbulent energy ε . The local turbulent viscosity

tμ can be obtained either from dimensional analysis or from analogy to the laminar viscosity

as t tv lμ ρ∝ . On the latter definition, based on the characteristic velocity tv defined as k

and the characteristic length l as 3/ 2k ε , the turbulent viscosity tμ can thus be ascertainedaccording to

2

t μkCμ ρε

= (12)

where Cμ is an empirical constant. In order to evaluate the turbulent viscosity in equation(12), the values of k and ε must be known which are generally obtained through solution oftheir respective transport equations.

After a fair amount of algebra, the final forms of equations for the standard k-ε modeldeveloped by Launder and Spalding (1974) for the turbulent kinetic energy k and dissipationof turbulent energy ε can be written as:

( ) ( ) ( ) ( )

t t t

k k k

k u k k wkt x y z

k k w P Gx x y y z z

ρ ρ ρ ρ

μ μ μ ρεσ σ σ

∂ ∂ ∂ ∂+ + + =

∂ ∂ ∂ ∂

⎡ ⎤ ⎡ ⎤ ⎡ ⎤∂ ∂ ∂ ∂ ∂ ∂+ + + + −⎢ ⎥ ⎢ ⎥ ⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦ ⎣ ⎦

v

(13)

( ) ( ) ( ) ( )

( )1 3 2t t t

u wt x y z

C P C G Cx x y y z z k ε ε

ε ε ε

ρε ρ ε ρ ε ρ ε

μ μ με ε ε ε ρεσ σ σ

∂ ∂ ∂ ∂+ + + =

∂ ∂ ∂ ∂

⎡ ⎤ ⎡ ⎤ ⎡ ⎤∂ ∂ ∂ ∂ ∂ ∂+ + + + −⎢ ⎥ ⎢ ⎥ ⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦ ⎣ ⎦

v

(14)

where P is the shear production defined by


( )( ) ( )23

Tt tP kμ ρ μ= ∇ ⋅ ∇ + ∇ − ∇⋅ + ∇⋅V V V V V (15)

and G is the production due to the gravity, which is valid for weakly compressible flows canbe written as

tGρ

μ ρρσ

= − ⋅∇g (16)

where ( )0, ,0g≡ −g is the gravity vector and 3C and ρσ are normally assigned values of

unity and G in equation (16) is the imposed condition whereby it always remains positive,

i.e. ( )max ,0G The constants for the standard k-ε model have been arrived through

comprehensive data fitting for a wide range of turbulent flows (see Launder and Spalding,1974): Cμ = 0.09, σk = 1.0, σε = 1.3, 1Cε = 1.44 and 2Cε = 1.92. Note that the effect ofbuoyancy is included in equations (14) and (15) to account for regions in the fluid with verylow flow velocities.

For the solid regions comprising of various materials, the heat-conduction equation canbe expressed as

s s s ss ps s s s

T T T TC St x x y y z z

ρ λ λ λ⎡ ⎤∂ ∂ ∂ ∂∂ ∂ ∂⎡ ⎤ ⎡ ⎤= + + +⎢ ⎥⎢ ⎥ ⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦⎣ ⎦

(17)

where Ts is the solid phase temperature and S is the internal volumetric heat sourceaccounting for the total heat generated due to the irradiation process. In equation (17),

sρ , psC and λs denote the thermophysical properties of the solid materials.

Boundary Conditions

Based on the mass flow rate, the normal velocity can be calculated and prescribed at theinlet boundary. The temperature of the fluid is also prescribed at this boundary. At the outlet,the static pressure is normally imposed according to fully-developed flow condition. For allvariables, the normal derivative at the outlet boundary is equivalent to zero.

One possible approach of overcoming the difficulty of modeling the near-wall region of asolid wall is through the prescription of logarithmic wall functions for the standard k-ε model.In order to construct these functions, the region close to the wall can usually be characterizedby considering the dimensionless velocity U + and wall distance y+ with respect to the local

conditions at the wall. The dimensionless wall distance y+ is defined as ( )u d yτρ μ−

where very near the wall, y = d, while the dimensionless velocity U + can be expressed in the


form as U uτ where U is taken to represent some averaged velocity representing either of

the mixture velocity parallel to the wall, uτ is the wall friction phase velocity which is

defined with respect to the wall shear stress wτ as wτ ρ .

For wall distance of y+ < 5, the boundary layer is predominantly governed by viscousforces that produce the no-slip condition; this region is subsequently referred to as the viscoussub-layer. By assuming that the shear stress is approximately constant and equivalent to thewall shear stress wτ , a linear relationship between the averaged velocity and the distancefrom the wall can be obtained yielding

U y+ += for 0y y+ +< (18)

With increasing wall distance y+ , turbulent diffusion effects dominate outside the viscoussub-layer. A logarithmic relationship is employed:

( )1 lnU E yκ

+ += for 0y y+ +> (19)

The above relationship is often called the log-law and the layer where the wall distancey+ lies between the range of 30 < y+ < 500 is known as the log-law layer. Values of κ

(~0.4) and E (~9.8) in equation (19) are universal constants valid for all turbulent flows pastsmooth walls at high Reynolds numbers. The cross-over point 0y+ can be ascertained bycomputing the intersection between the viscous sub-layer and the logarithmic region based onthe upper root of

( )0 01 lny E yκ

+ += (20)

A similar universal, non-dimensional function can also be constructed to the heat transfer.The enthalpy in the wall layer is assumed to be:

H Pry+ += for Hy y+ +< 21)

( )lnTH

PrH F yκ

+ += for Hy y+ +> (22)

where HF is determined by using the empirical formula of Jayatilleke (1969):


0.75

exp 9.0 1 1 0.28exp 0.007Ht t

Pr PrF EPr Pr

κ⎧ ⎫⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞⎪ ⎪⎢ ⎥= − + −⎨ ⎬⎢ ⎥⎜ ⎟ ⎜ ⎟

⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦⎪ ⎪⎣ ⎦⎩ ⎭(23)

By definition, the dimensionless enthalpy H + is given by:

( ) 0.25 0.5w

H

H H C kH

Jμρ

+−

= (24)

where wH is the value of enthalpy at the wall and the diffusion flux JH is equivalent to the

normal gradient of the enthalpy ( )wallH n∂ ∂ perpendicular to the wall. The thickness of the

thermal conduction layer is usually different from the thickness of the viscous sub-layer, andchanges from fluid to fluid. As demonstrated in equation (20), the cross-over point Hy+ canalso be similarly computed through the intersection between the thermal conduction layer andthe logarithmic region based on the upper root of

( )1 lnH T H HPr y Pr F yκ

+ += (25)

For the rest of the boundary conditions at a solid wall, all wall temperatures aredetermined using the energy balance. Boundary conditions for the turbulent kinetic energyand the dissipation have its normal derivative at the wall equal to zero and obtained throughthe relation

3 4 3 2C kd

μεκ

= (26)

where d is the distance of the nearest grid point from the wall boundary.

Computational Procedure

The algebraic forms of the governing equations to be solved by matrix solvers can beformed by integrating the system of equations using the finite volume method over smallcontrol volumes. By employing the general variableφ , the generic form of the governingequations can be written initially in the form as

( ) ( ) St φ φ

ρφρ φ φ

∂⎡ ⎤+∇ ⋅ = ∇ ⋅ Γ ∇ +⎣ ⎦∂

V (27)


In order to bring forth the common features, terms that are not shared between theequations are placed into the source term Sφ . By setting the transport variable φ equal to 1,

u , v , w , H , k and ε and selecting appropriate values for the diffusion coefficient φΓ

and source term Sφ , special forms of each of the partial differential equations for the

continuity, momentum and energy as well as for the turbulent scalars can thus be obtained.The cornerstone of the finite volume method is the control volume integration. In order to

numerically solve the approximate forms of equation (27), it is convenient to consider itsintegral form of this generic transport equation over a finite control volume. Integration of theequation over a three-dimensional control volume ΔV yields:

( ) ( )V V V V

dV dV dV S dVt φ φ

ρφρ φ φ

Δ Δ Δ Δ

∂⎡ ⎤+ ∇ ⋅ = ∇ ⋅ Γ ∇ +⎣ ⎦∂∫ ∫ ∫ ∫V (28)

By applying the Gauss’ divergence theorem to the volume integral of the advection anddiffusion terms, equation (28) can now be expressed in terms of the elemental dA as

( ) ( )V A A V

dV dA dA S dVt φ φ

ρφρ φ φ

Δ Δ Δ Δ

∂⎡ ⎤+ ⋅ = Γ ∇ ⋅ +⎣ ⎦∂∫ ∫ ∫ ∫V n n (29)

Equation (29) needs also to be further augmented with an integration over a finite time stepΔt. By changing the order of integration in the time derivative terms,

( ) ( )t t t t

V t t A

t t t t

t A t V

dt dV dA dtt

dA dt S dVdtφ φ

ρφρ φ

φ

+Δ +Δ

Δ Δ

+Δ +Δ

Δ Δ

⎛ ⎞ ⎛ ⎞∂+ ⋅ =⎜ ⎟ ⎜ ⎟∂ ⎝ ⎠⎝ ⎠

⎛ ⎞⎡ ⎤Γ ∇ ⋅ +⎜ ⎟⎣ ⎦

⎝ ⎠

∫ ∫ ∫ ∫

∫ ∫ ∫ ∫

V n

n

(30)

In essence, the finite volume method discretises the integral forms of the transportequations directly in the physical space. If the physical domain is considered to be subdividedinto a number of finite contiguous control volumes, the resulting statements express the exactconservation of property kφ from equation (21) for each of the control volumes. In a controlvolume, the bounding surface areas of the element are, in general, directly linked to thediscretisation of the advection and diffusion terms. The discretised forms of these terms are:

( ) ( )k fffA

dA Aρ φ ρ φΔ

⋅ ≈ ⋅ Δ∑∫ V n V n (31)


fffA

dA Aφ φφ φΔ

⎡ ⎤ ⎡ ⎤Γ ∇ ⋅ ≈ Γ ∇ ⋅ Δ⎣ ⎦ ⎣ ⎦∑∫ n n (32)

where the summation in equations (31) and (32) is over the number of faces of the elementand fAΔ is the area of the face of the control volume. The source term can be subsequently

approximated by:

V

S dV S Vφ φΔ

≈ Δ∫ (33)

For the time derivative term, the commonly adopted first order accurate approximationentails:

( ) ( ) ( )1n nt t

t

dtt tρφ ρφ ρφ++Δ ∂ −

≈∂ Δ∫ (34)

where Δt is the incremental time step and the superscripts n and n + 1 denote the previous andcurrent time levels respectively. Equation (21) can then be iteratively solved accordingly tothe fully implicit procedure by

( ) ( ) ( )11

1

1

nn n

fff

n

nff

f

At

A S Vφ φ

ρφ ρφρ φ

φ

++

+

+

⎛ ⎞−+ ⋅ Δ =⎜ ⎟Δ ⎝ ⎠

⎛ ⎞⎡ ⎤Γ ∇ ⋅ Δ + Δ⎜ ⎟⎣ ⎦

⎝ ⎠

∑

∑

V n

n

(35)

Consider the particular control volume element in question of which point P is taken torepresent the centriod of the control volume, which is connected with the respective centroidsof other surrounding control volumes. Equation (35) can thus be expressed in terms of thetransport quantities at point P and surrounding nodal points with a suitable prescription ofnormal vectors at each control volume face and dropping the superscript n +1 which bydefault denotes the current time level as:

( )1 1 1 1 1n

Pn n n n n PP P nb nb off non u P

nb

Va a S S S V

tρφ

φ φ+ + + + + Δ= + + + Δ +

Δ∑ (36)

where

( ) 11

nPn P

P nb P P fnb

Va a S V F

tρφ +

+ Δ= + Δ + +

Δ∑ ∑ (37)


and the added contribution due to non-orthogonality of the mesh is given by 1nnonS + , which is

required to be ascertained especially for body-fitted and unstructured meshes. Note that the

convective flux is given by ( ) 11 nnf ff

F Aρ φ++ = ⋅ ΔV n For the sake of numerical treatment, the

source term for the control volume in equation (35) has been treated by

( )1 1 1n n nu P P PS V S S Vφ φ+ + +Δ = − Δ (38)

In equation (36), Pa is the diagonal matrix coefficient of 1nPφ+ , 1n

fF +∑ are the mass

imbalances over all faces of the control volume and PS is the coefficient that is extractedfrom the treatment of the source term in order to further increase the diagonal dominance. Thecoefficients of any neighboring nodes for any surrounding control volumes k

nba in equation(36) can be expressed by

( )1 1max ,0n nnb f fa D F+ += + − (39)

where 1nfD + is the diffusive flux containing the diffusion coefficient φΓ along with the

geometrical quantities of the particular element within the mesh system. The treatment of theadvection term which results in the form presented in equation (39) is known as upwinddifferencing to guarantee diagonal dominance. In order to reduce the effect of false diffusioncaused by upwind differencing, the well-known deferred correction approach is adopted totreat the off-diagonal contributions 1n

offS + in equation (36) due to higher resolution

differencing schemes.In this study, the coupled solution approach, a more robust alternative to the segregated

approach, is adopted to solve the velocity and pressure equations simultaneously. AlgebraicMultigrid accelerated Incomplete Lower Upper (ILU) factorization technique is employed toresolve each of the discrete system of linearized algebraic equations in the form of equation(36). The advantages of a coupled treatment over a segregated approach are: robustness,efficiency, generality and simplicity. Nevertheless, the principal drawback is the high storagerequirements for all the non-zero matrix entries.

Results and Discussion

For the analysis of reactor thermo-hydraulic safety, the central focus often befalls on a‘bounding case study’ which demarks what can be loosely regarded as the operationalenvelop of the reactor cooling system. This bounding case study is a simulation of all extremeoperational conditions that may occur simultaneously during the life of the reactor. Themeaningful operands of these bounding operational conditions are represented by inputparameters (such as coolant temperatures, mass flow, etc.) of the system which become theparametric specifications of the computer model.


However, before these inputs are specified, a computer model of the irradiation rig mustfirst be generated; complete with flow passages, internal structures and irradiation target(s).Components of the model rig assembly must be built in accordance to engineering blueprintsand where possible, minor geometric simplifications are introduced into the model so thatsimulations remained physically representative without having to waste computationalresources or prolong solution times over unnecessarily detailed control volumes. After themodel is generated, it is geometrically discretised by the generation of a ‘mesh’ which furthersegregates each control volumes into sub-volumes. The physical properties of each geometricfeature – such as the aluminium in the rig structure are then specified in the correspondingvolumes of the model.

It is standard practice to conduct a grid-convergence test before the modelled geometry isdeemed representative of the physical prototype. To perform this, the amount of controlvolumes is increased in all three axes until the increase in mesh resolution does not result inany appreciable difference in final key results, such as in the model’s maximum attainabletemperature. Finally, the results are analysed and compared to the safety limit.

This procedure for this safety analysis can be summarised in five steps:

1. Determining the bounding case scenario2. Modelling the geometry, the physics and physical properties3. Solving the CFD model by numerical approximation techniques4. Checking the validity of our solution by mesh sensitivity analysis as well as by

comparisons with other simpler numerical models such as from one-dimensionalsimulations reactor nuclear thermo-hydraulic code

5. Analysing and comparing results with safety requirement

‘Rocket-Can’ Design as Used in the HIFAR Reactor

The first pipe-cooling irradiation system to be examined is the ‘rocket-can’ design usedin HIFAR. This study investigated the maximum temperature of 2.2% 235U enriched UO2

pellets during irradiation.The recently decommissioned HIFAR reactor, a 10 MW nuclear research reactor at the

Australian Nuclear Science and Technology Organisation (ANSTO), produced a steadysupply of technicium-99 (99Tc) radiopharmaceutical for domestic and international use.Technicium-99 is formed by the radioactive-decay of Molybdenum-99 (99Mo), which itself isa fission product of Uranium-235 (235U).

The process of generating technicium-99 started with the loading of seventeen UO2

pellets into a thick aluminium tube shaped like a ‘rocket-can’ (Figure 2). Granulatedmagnesium oxide (MgO) was packed in with the pellets to assist heat conduction and tocontrol pellet spacing. The rocket-cans were then sealed by welding on a cap and inserted intoslots inside a long vertical ‘stringer’ assembly which looked much like a ‘cake-stand’. Inturn, the stringer was inserted into a ‘liner’, a hollow tube with flow by-pass inlets at itsconical tip. Finally the liner was inserted into the centre of four concentric annular fuel plateswhich formed the fuel assembly (Figure 1). The purpose of the liner was to separate highspeed flow passing through the four outer concentric fuel plates from the slower flow goingthrough the liner cooling the cans. All components: cans, stringer and liner were fabricated


using aluminium on account of its unique qualities of neutron transparency, anti-corrosionand good thermal conductivity.

Figure 1. Schematic drawing of the HIFAR fuel element.

When the HIFAR reactor was operating, 235U in fuel elements underwent fission whichabsorbed and generated neutrons in a self-sustaining process. The nuclear reaction producedlarge amounts of heat that was removed by upwardly flowing heavy water (deuterium oxide,D2O) through the fuel element. One and a half percent of coolant flow bypassed into thecentre liner for rocket-can cooling. Also, the neutrons produced by the nuclear process in thefuel were absorbed by the UO2 pellets to produce molybdenum that had a short half-life of 2.7days before beta-decaying into technicium-99. After irradiation, the molybdenum-99 waschemically separated and quickly packaged so that it arrived at clinics in its usable beta-decayed form as technicium.

IDENTIFICATIONNUMBER

SALVAGE COUPLING PHALANGEFOR ASSEMBLY TO SHIELD PLUG

EMERGENCY COOLINGWEIR AND SPRAY RING

INTERMEDIATE SECTION

COOLANT FLOWTHERMOCOUPLE TUBE

PERFORATEDEXTENSION

THERMOCOUPLECOMB

EMERGENCY COOLINGWATER TUBE (NOTUSED IN HIFAR)

THERMOCOUPLETUBE

VIEW OF ARROW A

DOWEL

FUEL TUBES

OUTLINE OFLINER

LOWER COMB

GUIDE NOSE

SPHERICALSEAT

SKIRT

COOLANTFLOW


Figure 2. Schematic drawing of the inadiation rig and placement of target cans in the liner.

During irradiation, a vast amount of heat was generated in the uranium pellets whichmust be evacuated. For safety and licensing purposes, it was necessary to demonstrate that thepellet would not melt during irradiation. Thus, neutronic and thermal-hydraulic analysesusing numerical methods were used to determine the pellet maximum temperature. Thismethod directly solved the three conservation equations of mass, momentum and energy inthe coolant flow domain which was coupled with the conduction physics of the solid domainsof the can, magnesium oxide and uranium oxide pellets. The result of this work was critical tothe licensing requirements of HIFAR as bounded by operation licence conditions (OLCs) andevaluated safety limits.

Grid Generation of Rocket-Can

The complete geometry of can, stringer, liner, fuel plates and flow passages wasgenerated simultaneously and their respective volumes patched as separate logical entities asshown in Figure 3. The simultaneous volume generation meant that separate componentsurfaces were logically connected and no surface patching was required between components.


On the outset, minor geometric details were simplified in regions of little flow-dynamicconsequence in-order to accelerate the time to solution convergence. Some work was neededto organise the arrangement of each entity with respects to other parts but the effort expendedwas outweighed by the benefit of not needing to specify intra-component surface patching.Note in Figure 3 that hexahedral body-fitted volumes were used in the geometry to properlymodel the thin conduction regions of the liner and fuel plates. The thin solid region of theliner and stringer could only be practically constructed by using a structured mesh because ofthe high thermal flux across the thin stretch of aluminium. Numerical diffusion over such athin area of high thermal flux given if we were using unstructured mesh would have beenexceedingly pronounced. The additional benefit of using hexahedral control volumes was intheir superiority over tetrahedral elements for the modelling of near-wall flows where thecorrelation-model of wall flow-profiles work best with body-fitted rectangular mesh. Thiswas also represented an economy in the number of rectangular mesh required as compared totetrahedral mesh to attain the same level of solution accuracy.

The pellet-stack geometry consisted of a 119.5 mm vertical cylinder of MgO with aradius of 5 mm. The pellets modelled inside the MgO stack was 3.5mm high with a radius4.5mm as shown in Figure 4a. In Figure 5a, the rocket-can and stringer are shownsimultaneously. Notice the relief windows at the top of the stringer which allowed the coolantflow to pass through as was identical to the real physical prototype. Finally, five volumes ofthe same model as shown in Figure 3b were stacked on each other to produce the completestringer assembly such as shown in Figure 9.

(a)

(b)

Figure 3. The modelled ‘block’ of multiple components is shown with the (a) rocket-can and (b) slottedstringer highlighted.


(a)

(b)

Figure 4. The modelled stack of (a) UO2 pellets in MgO was built to fit inside (b) the rocket-can’sinternal cavity.

(a)

(b)

Figure 5. The model of (a) rocket-can inserted inside stringer and (b) multiple stringers were assembledto produce a complete stringer assembly.


Specification of Material Properties

The total D2O mass flow through the rig was 16 kg/s with 98.5% passing through the fueland 1.5% passing through the liner. The material properties of aluminium and heavy water aresummarised in Table 1 and the variable conductivities of magnesium oxide and uraniumdioxide are shown in Figure 6.

Table 1. Material Properties of Aluminium and Heavy Water

MATERIAL PROPERTIES Al D2O Density kg.m-3 2702 1094.92 Specific Heat Capacity J.kg-1.K-1 903 4.12849 Thermal Conductivity W.m-1.K-1 273 0.614259 Dynamic Viscosity μ Pa.s 0.000712

Magnesium Oxide conductivity vs. Temp.

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

300 3000Temp. [K]

Ther

mal

Con

duct

ivity

[W/(m

.K)]

(a)

Uranium Oxide conductivity vs. Temp.

2

3

4

5

6

7

8

200 418 1800 3000

Temp. [K]

Ther

mal

Con

duct

ivity

[W/(m

.K)]

(b)

Touloukian 1970.

Figure 6. Temperature dependent thermal conductivities of (a) magnesium oxide and (b) uraniumdioxide.

Computational Geometry

The domains of the simulation were specified as follows (the default solid-domainmaterial was aluminium):

• ‘Fuel1’, ‘Fuel2’, ‘Fuel3’, ‘Fuel4’– Solid Domain• ‘Liner’ – Solid Domain• ‘Stringer’ – Solid Domain• ‘Can’ – Solid Domain• ‘MgO’ – Solid Domain, material: Magnesium Oxide• ‘UO2’ – Solid Domain, material: Uranium Oxide• ‘Fluid’ – Fluid Domain, fluid: heavy water


In summary the CFD problem can be simplified as a list of input and outputs:

Inputs• Geometry• Material Properties• Heat Source• Boundary Conditions

Outputs• Temperature Profile T (x,y,z)• Wall Heat Flux q (x,y,z)

UO2 pellets producde heat at a steady rate. The heat was conducted through the MgOinto the aluminium rocket can then into the flowing heavy water D2O. The solutions ofinterest given by CFD included: the temperature profile T (x,y,z) of the whole region and wallheat flux q (x,y,z) over the surface of the rocket can.

The subdomain ‘Pellets’ was defined for the entire UO2 domain as an energy source witha volumetric power of 8.4556 x 108 W.m-3. The fuel plates, the structural material(aluminium) and the coolant (heavy water) would also produce heat but the heating from fuelplates was not modeled because it was known that they were cooled efficiently by the fasterflowing coolant outside the liner and heating from other aluminium components wasnegligible.

For the boundary conditions, there were two inlets, one at ‘INLET FUEL’ with NormalSpeed 2.58816 m.s-1, another at ‘INLET LINER’ with Normal Speed 0.0717642m.s-1 (thesewere calculated from coolant mass flow value, inlet areas and bypass ratios), both with astatic temperature of 318K (45°C) in the 16-pellet simulation. In these simulations, there weretwo separate flow regions.

Power Density Calculation

The power emitted from 2.2% enriched UO2 pellets were calculated by using the MonteCarlo N-Particle (MCNP) transport code for simulating neutronic reactions. For thissafety analysis a conservative reactor power of 11MW was assumed. The 16 pellets of a canwere modeled discretely (Dia. 9mm × 3.5mm high). Total power per m-3 for 16 pellets =3204.7 W.m-3 / 3.79 × 10-6 m3 (Volume of 16 Pellets) = 8.4556 × 108 W.m-3. This powerdensity signified the source term for the conduction equation.

Turbulence Solver and Convergence Criteria

To account for the turbulent pipe flow, the standard k-ε model was used. Computationalpredictions were deemed to be converged when the normalized residual mass was less than 1x 10-4.

Computational Predictions

Simulation for the heating of sixteen pellets in a rocket can had been previously modeledby Yeoh and Storr (2000) which was interested in the rocket-can’s maximum surface heat


flux to determine the margin for Onset of Nucleate Boiling (ONB). The CFD model of Yeoh& Storr’s simulation had a lesser degree of resolution and modeled the stack of pellets as asingle entity instead of discrete pellets. This was partly attributed to the more limitedcomputational resources available at the time. Yeoh & Storr’s CFD result’s were validated toa degree by simple irradiation experiments where metal samples were introduced inside thepellet stack and subsequently examined for evidence of melting in order to estimatemaximum temperatures. In this investigation calculated by CFX-10, we were primarilyinterested in the maximum temperature of the UO2 pellets. Temperature contours in Figure 7aindicate a maximum pellet temperature of 2656°C which remained below the pellet meltingtemperature of 2847°C. For clarity, the four concentric annular fuel plates, liner, stringer andcan in Figure 7 are shown in silhouette. This result was consistent with the maximumtemperature of Yeoh & Storr’s large-volume pellet stack at 2277°C. The temperature wasunderstandably lower in Yeoh & Storr’s simulation because the same power was distributedover a larger volumetric space.

(a)

(b)

Figure 7. Section view of temperature contours showing maximum temperatures of (a) UO2 pellets:2656°C and (b) MgO: 2439°C.

Other maximum temperatures of interest include: the magnesium oxide at 2439°C (mp.2800°C) in Figure 7b; the aluminium can at 307°C (mp. 660°C) in Figure 8a and the heavywater D2O at 79°C (bp. 120°C at 2 atm) in Figure 8b. The temperatures were highest in themiddle of the pellets, decreasing steeply across the width of the pellet and still more steeplyacross the MgO powder, reflecting its low thermal conductivity. This was in contrast with thelow thermal gradient across the aluminium rocket can, due to the high thermal conductivity ofaluminium.

Figures 9 and 10 show the wall heat flux at the surface of rocket cans, the higher fluxeswere colored blue, due to outward flux being defined as negative. There were four high heatflux regions corresponding to the slot opening. Opposite the slot opening in Figure 10, forcedconvection cooling was restricted by the stringer enclosure and as a consequence caused alower heat-flux as indicated by the green region of the can surface. The sudden expansion ofthe open slot at the side of the stringer created a highly turbulent region which assists in


forced convection cooling and this resulted in the highest thermal heat flux achieved asindicated in Figure 10 by the dark blue region of the can.

(a)

(b)

Figure 8. Section view of temperature contours showing maximum temperatures of (a) rocket can:307°C and (b) D2O: 79°C.

Figure 9. Heat-flux contour at the surface of the rocket cans. The view is through the opening in thestringer. The silhouette of the stringer is made out in transparent purple.

Inside the liner, the mean flow accelerated when squeezed into tighter paths between thecan and the stringer, as indicated by yellow and orange vectors in Figure 11 and deceleratedwhen entering an expansion, as shown by blue vectors near the tip of the rocket can. Thedeceleration was basically due to reasons of mass conservation as explained by the Bernoulliequation. The length of an arrow in Figure 11 is proportional to the flow velocity at that point.The long red arrows belonged to the faster flow outside the liner, with velocities of up to4m.s-1.


Figure 10. Heat-flux contour at the surface of the rocket cans. The view is through the enclosed back ofthe stringer.

Figure 11. Flow velocity vector plot of coolant channels inside outside the liner.

Proposed ‘Annular Can’ Replacement Design for HIFAR

Flow visualization and LDV measurements were performed to better understand the fluidflow around the narrow spaces within the X216 irradiation rig, prototypes of annular targetcans and liner. A three-dimensional computational fluid dynamics model was used to


investigate the hydraulics behavior within a HIFAR fuel element liner model. An interestingfeature of the computational model was the use of unstructured meshing, which consisted oftriangular elements and tetrahedrons within the flow space (Figure 13), to model the “scaled-up” experimental model. This present investigation focused on the evaluation of the CFDmodel in its capability to predict the complex flow structures inside the liner containing themock-up X216 rig with two targets. The reliability of the model was validated againstexperimental observations and measurements.

Description of the Water Tunnel Experimental Apparatusand Methods

Figure 12 describes the transparent model of the prototype design of the rig and annulartarget cans that were placed in the water tunnel facility. In this test section, the liner wassealed so that the flow path was only through the liner inlet holes. An orifice plate was usedto measure the flow rates within the facility. Flow visualization of the fluid flow wasperformed using an Argon Ion laser light sheet, high-resolution digital camera and standardvideo equipment. The flow was seeded with Iriodin powder concentrations of 0.5 to 1 gramper 3000 liters of water. Particles in the flow were illuminated while they were in the field ofthe laser light sheet. With the digital images and video footage the flow field was clearlyvisible.

Figure 12. Points of velocity measurements in X216 rig.


Figure 13. Unstructured mesh of the X-216 annular target.

Velocities were measured using a two-dimensional (2D) Dantec LDV system, operated in1D mode measuring axial velocity components. Data were stored on the computer attached tothe LDV hardware. Uncertainty in measurement using the LDV equipment was determined tobe a maximum of ± 3.5% for a particular optics configuration but generally less than ±1.8%.Points of velocity measurement are shown in Figure 12. The points measured in the planeshown had been taken with the laser entering at the open side of the mock-up rig and theforward scatter detector viewing through the closed side of the rig. Measurements were alsotaken using the LDV probe in backscatter mode, and were verified at a number of points byusing the forward scatter mode. Backscatter mode was used only for measurements in the gridpattern between the heights of 185 mm and 360 mm, since at the other locations the beamswere sufficiently attenuated due to the additional influences of the acrylic interfaces, givingvery low data rates.

Computational Details

A three-dimensional CFD program ANSYS-CFX5.6 has been employed to simulate thecomplex thermal-hydraulics behavior in the space within a HIFAR fuel element liner modelin the water tunnel. The CFD code solved the conservation equations of mass, momentum andenergy. Turbulence of the fluid flow was accounted via a standard k-ε model. Buoyancy


effect was included for regions where low velocities were present. This effect was includedwithin the source terms of the momentum equations as well as in the turbulence models.

The unstructured meshing was adopted for the construction of the computational modelof the mock-up because of the inherent intricate details of the irradiation rig and theplacements of the annular targets within the rig. The mesh was prepared in two stages. Asurface mesh of triangular mesh elements was initially generated on all the surfaces coveringthe model components. The volume mesh consisted of tetrahedral elements was thensubsequently generated within the fluid flow domain from the surface mesh elements. Figure13 shows the mesh layout of triangular surface elements around the pin and support tube nosecone of the irradiation rig. For the entire geometrical structure that included the assembly oftwo annular target cans placed within the prototype rig in the liner and mock-up fuel element,a volume mesh of 605158 tetrahedrons and a surface mesh of 42030 triangular elements wereallocated.

The governing equations were solved by matrix solution techniques formulated byintegrating the system of equations using the finite volume method over small elementalvolumes. For each elemental volume, relevant quantity (mass, momentum and turbulence)was conserved in a discrete sense for each control volume. Here, a coupled solver, whichsolved the hydrodynamic equations (for velocities and pressure) as a single system, wasemployed. It has been found in the segregated approach that the strategy to first solve themomentum equations using a guessed pressure and an equation for a pressure correctionresulted in a large number of iterations to achieve reasonable convergence. By adopting thecoupled solver, it has been established that such a coupled treatment significantly outweighedthe segregated approach in terms of robustness, efficiency, generality and simplicity. Toaccelerate convergence for each of the discretised algebraic equations, the AlgebraicMultigrid solver was adopted.

Validation Against Water Tunnel Observationsand Measurements

CFD simulation of the fluid flow through the various components of the fuel elementmodel that included the mock-up rig and annular target cans was performed. Figure 14illustrates the computed flow distribution inside and outside of the liner nose cone.

Based on the experimental flow rate of 1.6965 kgs-1 and a base diameter of 0.3 m of themock-up fuel element model, there observed a very low flow velocity outside of the linernose cone in Figure 14(b). Nevertheless, the fluid after being squeezed through the small sizebottom and side holes of the liner nose cone caused these interacting merging flows to yield avery highly complicated flow structure consisting of multiple vortices of recirculating flows(see Figure 14(a)). It was also evident that due to the significant acceleration of the flowfound near the liner holes, the velocities increased dramatically to a magnitude of 5.0 ms-1

and resulted in large pressure drops. Near the bottom hole of the liner nose cone, the CFDmodel predicted a normal velocity of approximately of 3 ms-1. This predicted value has beenfound to be in good agreement with experimental LDV measurement of 2.6 ms-1, whichprovided confidence to the reliability of the models in the CFD computer program. As thefluid moved vertically upwards, the flow gradually became more uniform.


(a)

(b)

Figure 14. (a) Velocity vectors and (b) velocity contours around perforated liner.

(a) (b)

Figure 15. Flow separation near the pin as examined by (a) CFD and (b) PIV capture.

Another important consideration for the rig and target specification was the incorporationof a pin situated at some distance below the placement of the rig as can be seen in Figure 15.This design feature was implemented in the liner because of safety concerns in the event of apossible accident scenario of the rig falling through to the bottom and impacting on the linernose cone. The selection of pin size was an important requirement for the rig and targetspecification. From the flow predictions in Figure 15(a), it could be ascertained that the pinsize chosen contributed to only minor flow disturbances in the area between the pin and thebottom surface of the rig nose cone. It was also observed that the majority of the bulk fluidflow was unperturbed and diverged smoothly as it approached the rig. Flow visualization


performed during experiments (see Figure 15(b)) projected a similar flow pattern, whichfurther confirmed the reliable predictions of the CFD model.

(a) (b)

Figure 16. CFD study of flow distribution around annular can.

Figure 16 presents the flow distribution of the fluid traveling between the inner annularcan wall and rig outer surface designated for the purpose of illustration as region 1 and thearea between the outer annular can wall and the liner inner surface designated as region 2.The fluid flowing within these spaces was found to be rather uniform. These favorable flowstructures indicated that axial cooling in the reactor rig along the length of the target couldremove the heat effectively for the design where uranium foils are embedded in the sealedannular targets during the irradiation process. An interesting aspect of the model predictionsthrough the velocity contours in Figure 16(b) showed succinctly more fluid moving verticallyupwards in region 2 than in region 1. Based on the LDV flow measurements at the discretelocations in regions 1 and 2 in Figure 12, the experiments confirmed the CFD predictions ofthe different velocities in the two regions. Velocity values of 0.44 ms-1 and 0.502 ms-1 weremeasured during experiments for regions 1 and 2 respectively. The predicted velocities asdepicted by the velocity contours in Figure 16(b) demonstrated the similar trend predictedthrough the CFD model where region 1 yielded lower velocities compared to the highervelocities in region 2.

Figure 17 illustrates the LDV measurements for the unmodified and modified designs ofthe can flutes affecting the axial velocity distributions for two volumetric flow rates. Changesintroduced to the flute designs through the removal of any sharp edges significantly alteredthe axial velocity distributions. The profiles were flatter thereby resulting in more uniformand lesser wake flow structures. In the same figure, comparison between predicted andmeasured vertical velocities is also presented. The predicted velocity profiles through theCFD model showed similar encouraging distributions with the measured profiles in the inner


and outer channels. Good qualitative agreement was achieved between the predicted andmeasured velocities.

X216 Rig Channel Axial Velocities0 degree plane

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 50 100 150 200 250 300 350 400 450 500

Axial Distance From Stem Tip (mm)

Flui

d Ve

loci

ty m

/s

1.61 l/s Inner Channel1.61 l/s Outer Channel1.61 l/s IC;RMS1.61 l/s OC;RMS1.68 l/s IC1.68 l/s OC1.68 l/s IC;RMS1.68 l/s OC;RMS1.61 l/s mod IC1.61 l/s mod OC1.61 l/s mod IC;RMS1.61 l/s mod OC;RMS1.61 l/s IC CFD prediction1.61 l/s OC CFD prediction

NOTES:1) Run @ 1.61 l/s was performed before fin smoothing2) Run @ 1.61 l/s mod was performed after fin smoothing

Figure 17. Inner and outer channel LDV velocity measurements along X216 Rig vertical axis.

Current ‘Molybdenum-Plate’ Design Used in the New OPALReactor

The third pipe-cooling irradiation system to be examined is the ‘molybdenum-plate’design used in the new Open Pool Australian Light-water (OPAL) research reactor. Thisstudy investigated the maximum temperature of 20% 235U enriched U-Al compound sealed inaluminium cladding plates. It will also demonstrate the incremental development of the modeland the affect this has on the maximum temperature, as more physics and boundaryconditions were introduced. The OPAL research reactor is an open-pool design constructedby INVAP, Argentina. The core rests thirteen meters under an open pool of light-water whichprovides both cooling and radiation protection. Surrounding the core is a Heavy WaterReflector Vessel that is physically isolated from the bulk light water coolant. Its primarypurpose is to reflect neutrons back into the core to maintain the critical (nuclear) conditionsnecessary for steady fission reaction rates during normal reactor operation. The secondarypurpose of the reflector vessel is to provide a relatively large volume with high flux forirradiation and neutron beam facilities.

A compact reactor core measuring only 0.35 m x 0.35 m x 0.62 m (width, breadth,height) is located 13m below the reactor pool surface. The Reflector Vessel surrounding theCore contains heavy water and accommodates facilities such as irradiation rigs and the coldNeutron Source (see Figure 18). A number of irradiation positions penetrating the ReflectorVessel provide facilitates for the production of radioisotopes. These positions are sealed to


prevent the H2O and the D2O from mixing. Moreover, the irradiation positions contain built-in mechanisms for cooling which draws cold H2O from the main pool downwards into theirradiation targets which is then conveyed away through pipes.

Figure 18. OPAL reactor facilities (top view).

Molybdenum-Plate Life Cycle

A primary aspect of the OPAL reactor is the production of Technetium-99 (99mTc) for themedical supply of local and international markets. To manufacture Molybdenum-99, aproduct made of the compound U-Al contained in an aluminium matrix (called meat) is rolledin between two parallel plates of aluminium alloy A96061. The result is a simple, self-contained plate of aluminium measuring 230 mm x 28 mm x 1.64 mm that contains asandwich of U-Al ‘meat’ in the centre (see Figure 19). It is of note that the isotopic content offission uranium is high, at an enrichment of 20% 235U but low enough to abide byinternational guidelines for LEU (Low Enriched Uranium) fuels and irradiation targets.

Assembly of the irradiation rigs are performed in a hot-cell situated at pool level. Fresh,un-irradiated molybdenum plates are loaded—four at a time—into a holder known as the‘target’ (see Figures 20 and 21). A rig stem is then inserted through the target’s hollow centreto form an assembled irradiation ‘rig’. The rig incorporates two to three irradiation targetsalong with other flow control devices like nozzles and flow restrictors (see Figure 22). Onceassembled, the rigs are transported from the hot-cells into the service pool (underwater) via ahermetically sealed elevator. These rigs could then be remotely lowered to the reflectorvessel’s bulk irradiation facilities (see Figure 18).


Figure 19. One irradiation target with four molybdenum plates.

Figure 20. Isometric view of inadiation target model.

Figure 21. Cross sectional view of the irradiation target with four molybdenum plates inside.

Before the reactor is brought to critical, all primary cooling and secondary coolingsystems are switched on. The rigs cooling flow draws light water from the main pool down


into the irradiation rigs before being passed onto radiation decay tanks and heat exchangersafter. The fast neutrons radiating from the fuel plates are slowed by the H2O core coolant andare completely thermalised by the D2O between the core and the facility. Having beenslowed, these neutrons could then be captured by the Uranium-235 meat resulting in fission.

A maximum number of three targets could be loaded in one irradiation rig. Thisconstituted the bounding case to be modeled.

Figure 22. Molybdenum plate target and rig (shown on its side but loaded vertically).

Design Characteristics

Compared to the previous method of molybdenum manufacture, the philosophy of thispurpose-built system was for high molybdenum yield and low radio-active waste output. Theprevious design of molybdenum irradiation in the recently decommissioned HIFAR reactor(an old British design of the DIDO class), utilized 2.2% enriched UO2 reactor power fuelpellets packed in bulky aluminium cans. These cans were centrally inserted intoconcentrically arranged annular fuel plates and differed from the new irradiation rigs that lieoutside the core itself.

Consequently, the two main changes in molybdenum targets as compared to the previousHIFAR (High Flux Australian Research) reactor are: (1) the geometry, which has changedfrom a can to a plate, (2) the enrichment, increasing from 2.2% to 20% and (3) the location ofthe irradiation rig outside the core. This new configuration allows the irradiation of more 235Ubut also uses a substantially lower mass of aluminium and 238U, the result of which is anincrease in yield and a decrease in aluminium waste.

Despite these gains in efficiency, the increase in power generated by such a highconcentration of enriched uranium poses challenging conditions for heat removal. The thinlayer of aluminium gives little impedance to the flow of heat away from the U-Al centre butat the same time, any interruption to the forced convection flow of coolant could cause thecladding to blister or at the very worst melt, as there is very little aluminium mass to whichthe very large amounts of heat produced can conduct away. Thus, the prediction of flow onand over the molybdenum plates is crucial to ensure no isolated heat spots could develop thatwould result in the blistering of the clad or in the most extreme case, the melting of cladduring irradiation (burn out).

Thus, the aim of this numerical study was to show the thermo-hydraulic design of themolybdenum holder was sufficient to remove the heat generated by fission with a sizablemargin between bounding operational conditions and extreme (i.e. almost impossible)conditions.

After a maximum of 14 days of irradiation, the rig assembly is removed from thereflector vessel and placed in the service pool storage racks to ‘cool’. Loading and unloading


of irradiation rigs can be performed whilst the reactor is at full power. The removal of the rigfrom its irradiation position halts the neutron-irradiation heating, leaving only the nuclear-decay heat to be removed by means of natural convection. The decay-heat produced by weak-nuclear reactions is much lower than the fission heat—initially at approximately 8% of thepower produced during irradiation and decreases exponentially with time.

Methods of Calculations and Model Set-Up

The maximum steady state temperatures of the molybdenum-99 rigs as a result of thecoupled action of nuclear heating and light-water forced convection cooling were determined.These values could then be used to determine two important margins: (1) the differencebetween the maximum attainable temperature in the plate and the aluminium cladding blistertemperature and (2) the difference between the operational maximum heat flux and the heatflux at the onset of nuclear boiling.

For the simulation of heat transfer during irradiation, different parts of the irradiationassembly were separately modeled, meshed and combined by what is termed as ‘patching’.Afterwards, material properties were explicitly specified in each domain, along with therelevant boundary conditions. The built model incorporated nothing more than the necessaryareas under examination to optimize the use of computational resources. Other fundamentalfeatures such as the setting of appropriate turbulence model, the solution’s mass-residualconvergence criteria and the time & space discretisation schemes for the iterative solutionprocess were then set (Tu et al., 2008). Some features of the numerical model, such as themodeling of buoyancy were deemed to have a negligible effect to the overall flowcharacteristics and were thus not modeled.

In order to validate the CFD model, a comparison with a set of independent results wasnecessary. Available INVAP results using a more simplified one-dimensional code were usedas comparative data against the CFD results. In this way, we were able to evaluate theconsistency of the numerical models. For those cases which have no corresponding INVAPstudy, the only method for validation was by means of grid convergence testing and by thecomparison of results using different initial values or different convergence criteria. Theapproach with this numerical study was to start with the most simplified assumptions in theCFD model and to introduce increasing complexities to examine the relative importance ofeach effect. A series of CFD simulations, utilizing a three-dimensional CFD programANSYS-CFX10.0, were thus conducted to attain a final validation-case which included all theidentified bounding conditions.

Power Distribution of Molybdenum-Plate Targets

To undergo irradiation, assembled rigs loaded with two or three targets are insertedvertically inside the reflector vessel. When the reactor is at power, nuclear reactions occur inthe molybdenum-plates as thermal neutrons are absorbed by uranium with the effect ofproducing heat. Since the horizontal neutron flux from the core radiates uniformly in the nearvicinity of the circular reflector pool, there is little flux variation in the horizontal plane of theirradiation rig. However, the irradiation rig is very long, so the neutron flux distribution varies


appreciably in the vertical direction, resulting in a cosine power distribution along the verticalaxis of the molybdenum-rig. Using the Monte Carlo N-Particle (MCNP) transport codefor simulating neutronic reactions, it was found that for a three target configuration, themiddle position of three targets generated the most power. A separate and independentneutronic calculation conducted by INVAP (MOLY-0100-3OEIN-004, 2003) found that themaximum total power loading was 40.2 kW for one target and 79.6 kW for two targets. Sincethese results indicate a trend for proportionately lower powers for each additional target, itwas deemed that each target could generate no more than 40.2kW by itself during normalreactor operations. However, to increase the safety margin for this analysis, it was specifiedthat each of the three targets would generate a power of 60 kW.

The power distribution is defined in terms of the ratio between the maximum and theaverage power along the vertical position of the plates in the irradiation rig. This can bedefined either in terms of the plate surface heat flux or in terms of the volumetric powerdensity, with a ratio known as the Power Peaking Factor (PPF). In ref. Moly-0100, the PPFhas been shown to be equal to 1.1 for a single target and 1.3 for a two-target loading. It wasthus assumed that a PPF of 1.3 could be used for three targets. This assumption has beenvalidated later by an MCNP model of the three targets loaded for irradiation with a core offresh fuel.

Thus, the bounding case was defined as an irradiation rig loaded with three targets, eachcontaining four plates. The power density was defined as follows:

( )( ) ( )( )plate0 -9

Q 15000cos 1.3 cos 3.134453 0.413V 5188.8 10

q PPF B z z z= − = − (40)

where

q : Power density, [W.m-3]PPF : Power Peaking Factor, [-]Qplate : Total power per plate, [W]V : Meat volume, [m3]B : Adjustable constant, [m-1]z : Coordinate on the z axis, [m]z0 : offset of the centre, [m]

Under nominal conditions, the average flow velocity through most of the rig’s crosssection along the plates is 3.8 m.s-1. It was decided to set a more conservative limit with areduced average flow velocity of 3 m.s-1 for this study.

Boundary Conditions and Domain Restriction

Since coolant for the molybdenum-rigs is drawn from the pool, the temperature isessentially the pool water temperature which is regulated at a steady 37°C. To establish asafety margin of 3°C, a value of 40°C was conservatively applied for the inlet coolantboundary condition. It is difficult to ascertain the coolant velocity profile entering the rig


because the initial pool-water flow conditions could not be reliably predicted. But seeing asthe purpose of this investigation was to examine the heat transfer characteristics of the rig, theexact flow pattern entering the rig was not of primary importance. Furthermore, the reducer-expansion arrangement of the nozzle preceding the three targets had the intended effect ofstraightening the flow before it passed over the molybdenum-plates. Therefore the flow canbe assumed to have a developed turbulent velocity profile before entering the targets’ coolantchannels.

By these considerations, the total control volume modeled in these simulations onlyextended from the rig inlet nozzle to the outlet restrictor (see Figure 22) and a fully developedturbulent flow profile was assumed as the inlet condition. To allow the average velocity todevelop into a turbulent velocity flow profile, an artificial pipe extension of fifteen hydraulicdiameters was modeled before the inlet. Also, lest the flow after the restriction nozzle affectthe preceding flow domain, an artificial pipe of 15 hydraulic diameters was also attached tothe end of the restrictor. Finally, the outer surface of the rig was conservatively assumed to beadiabatic to attain the highest possible temperature in the coolant, now set to be solelyresponsible for the removal of heat from the rig.

Figure 23. Cross-sectional view of control volumes representing the irradiation rig.

Figure 24. The control volumes grouped in their respective materials shows each separate entity moreclearly.


Since it is obvious that the rig geometry is symmetrical in two planes (see Figures 23 and24), only one-quarter of the irradiation rig needed to be modeled with symmetrical boundariesapplied to the ‘cut planes’ to limit the number of mesh elements. In truth, a one-eighth modelwould have sufficed but it was decided a complete coolant channel would be modeled so thatno assumptions need be made of what was subsequently demonstrated as a symmetricalturbulent velocity profile through the pipe.

The inlet condition was defined at mass flow rate of 0.6259 kg.s-1, corresponding to the 3m/s average velocity in the major channel (see Figure 25). The inlet coolant temperature wasalso conservatively set at a higher temperature of 40°C, providing a 3°C margin above thenominal 37°C pool temperature. At the outlet, a 0 Pa gauge pressure was specified and thecylindrical outer wall was defined as adiabatic. The standard k-ε model was chosen for thishighly forced convective problems with automatic adjustments for kinetic and dissipationvalues as calculated by local velocities.

(a)

(b)

(c)

Figure 25. Quarter-sections of the: (a) nozzle (b) target (c) restrictor; blue indicates flow areas.

Grid Generation

For the sake of accuracy and computational efficiency, a body fitted orthogonal mesh wasselected to model the irradiation rig. Building quadrilateral surfaces for the body fitted meshrequired more time than automatically generated grids but this resulted in the use of lessnodes and reduced the amount of mesh-induced solution diffusivity when compared totetrahedral mesh. The occurrence of this artificial diffusion is a result of interpolation errorsarising from highly skewed mesh contact angles of tetrahedral-meshes.

In areas where large gradients were expected, the volumes were meshed with relativelyhigh density to capture sharp temperature changes. Similarly, grid optimization wasperformed in those areas where the temperature and velocity gradients were small by areduction in mesh density. Within the flow domain, the change of mesh sizes was gradual toprevent numerical instability in the solution. Whilst mesh size transitions in the solid domainwere not required to be as stringent because the harmonic-mean interpolation for thermalconduction between solid nodes is not as geometrically sensitive as the solution forconvection and diffusion in fluid nodes. As mentioned, highly skewed elements were avoidedwhere possible to minimize numerical diffusion. In Figure 26, the circled area in (a) indicatesthe mesh is too skewed and has been re-meshed in (b) so that all cell face angles stoodbetween 38° and 142°. This optimization improved the solution maximum temperature resultby a difference of 6°C.


(a) (b)

Figure 26. Circled area re-meshed from (a) to (b) to reduce the presence of skewed mesh.

Mesh refinement was checked using three different meshes. Due to limits incomputational resources, it was not possible to double the mesh seeding in every direction atthe same time. Thus, two new meshes were created, one with a doubling of mesh in thestreamwise directions and one with a doubling in the crosswise. In each case, a simulationwas run and the maximum temperatures did not vary by more than 1°C. Thus it was shownthat grid convergence had been established, with a final node element count of 432,667.

Validation Case

This simulation compared the temperatures attained in INVAP’s correlation-based one-dimensional code with our CFD model under the same flow and power conditions. Thenominal condition study conducted by INVAP evaluated a two target loading scenario with atotal power output of 79.4 kW, a cooling flow of 3.6 m.s-1 and an oxide layer thickness of 7μm as calculated by the Argonne National Lab (ANL) model for oxide layer growth.

Since INVAP’s model calculated for two targets whilst the CFD model was built withthree, one four-plate target was completely ‘switched off’, so that only two targets producedpower as simulated in the INVAP scenario. The inlet mass flow of the CFD model wasincreased from the safety scenario of 0.6258 kg.s-1 to 0.7510 kg.s-1 to correspond with theaverage flow velocity increase from 3 m.s-1 (safety study) to 3.6m.s-1 (validation study).INVAP’s input parameters of solid physical properties and oxide layer thickness were alsoadopted in this validation study. A summary of settings is displayed in Tables 2 and 3.

Details of the model can have effects on the criterion maximum temperature. Thesedetails include:

1. modeling of the oxide layer2. modeling of contact conditions between the Plate and the Target Holder3. power density distribution in the U-Al meat

To appreciate differences between model details, a simple CFD model was first built withthe assumptions:


1. without oxide layer.2. perfect contact between the plate and the holder3. a homogeneous power profile distribution across the U – Al meat

Table 2. Summary of the settings for the validation case

Oxide thickness 7 μm Number of targets 2 Total Power 79.4 kW Inlet flow rate 0.7510 kg.s-1

Inlet temperature 37°C Outlet gauge pressure 0 Pa

Table 3. Summary of the material properties for the validation case

MATERIAL PROPERTIES Al U2Al and Al matrix Oxide Water Molar Mass g.mol-1 26.98 270 18.02 Density kg.m-3 2702 4625.34 997 Specific Heat Capacity J.kg-1.K-1 903 900 4181.7 Thermal Conductivity W.m-1.K-1 165 148 2.25 0.6069 Dynamic Viscosity μ kg.m-1.s-1 0.0008899

Effect of Oxide Layer Modelling

During irradiation, the layer of aluminium oxide that forms on the molybdenum-plate isrelatively thin at 7μm. Physically, this thin layer could not be directly represented in thecomputer model because the mesh was not fine enough to virtually interpret what wasphysically there. The oxide layer could not be ignored either because though small, itsreduced conductivity had a significant effect on the molybdenum-plate’s maximumtemperature.

Figure 27. Step-function thermal conductivity profile.


Figure 28. Graphical representation of distance vs. temperature for realistic and modeled thermalconductivity for aluminum and aluminum oxide.

A first attempt was made to simply prescribe a thermal conductivity profile onto thecladding elements as shown in Figure 27. The thermal conductivity of the aluminium varieswith respects to cladding depth, with the circled regions of aluminium-oxide possessing amuch lower thermal conductivity. However, the result of this simulation was proven a failureas the maximum temperature proved no different from a separate identical simulation runwith no oxide-layer.

To circumvent this problem without resorting to the use of more computationallyexpensive mesh, the effect of the oxide layer was mathematically modeled as an integral partof the cladding. This was done by calculating an equivalent conductivity via the formulabelow which was derived from flux conservation principles (Patankar, 1980):

1

Al ox

1 f fkk k

−⎛ ⎞−

= +⎜ ⎟⎝ ⎠

(41)

where ox

Al ox

=f δδ δ+

. The oxide layer thickness was assumed to be 7 μm corresponding with

INVAP’s assumption. The equivalent conductivity of the cladding result was thus:

11 1

1 1

1 1

7350 7 1 0.02 0.020.02 67.44 . .

165 . . 350 165 2.252.25 . .

ox

Al ox

Al

ox

mm

f k W m Kk W m K

k W m K

δ μδ δ μ −

− −− −

− −

=⎧⎪ + = −⎪ ⎛ ⎞⇒ = = ⇒ = + =⎨ ⎜ ⎟= ⎝ ⎠⎪⎪ =⎩

This formula modified the aluminium conductivity to attain the correct temperature at thealuminium boundary but at the expense of correct temperatures inside the aluminium itself, asgraphically explained in Figure 28.

Maximum temperatures from the step-conductivity profile and the average conductivityprofile technique are displayed in detail in Table 4. The largest difference in temperature


proved to be between the no-oxide and average-conductivity oxide simulations, with adifference of 3.9°C in the U-Al meat.

However, the stepwise-conductivity profile technique resulted in only a 0.5°C gain in U-Al meat maximum temperature. It is obvious from these results that the step-wiseconductivity profile technique was unsuitable. The average conductivity profile techniquewas adopted as the method for simulating the effect of aluminium oxide on claddingconductivity.

Table 4. Comparison of models

Max. Temperatures (°C)

Reference Oxide model Oxide layer absent step average Contact type perfect perfect perfect Power distribution uniform uniform uniform Surface (water side) 59.2 59.2 59.0 Surface (cladding side) 99.1 99.3 100.5 Cladding – Meat interface 100.9 101.3 104.7 Meat – centre line 102.0 102.5 105.9

Effect of Contact Surface

Since the thermal conductivity of water was much less than aluminium, a break in thesolid conduction path between the molybdenum-plate and the target holder would bedetrimental for the conduction of heat away from the plate. Also, as there was a largetolerance between the plate and the recess in which the plate is held, a water gap existedbetween one side of the plate and the holder. This study examined the maximum temperaturedifference caused by the presence of this water gap (see Figure 29).

During the creation of this geometry, a space between the plate and its holder was madeand patched separately to allow different contact conditions to be simulated. Since the innerchannel was slightly larger than the outer channel, slight difference in local velocities oneither side of the plate would produce a minute pressure difference that forces the plate to oneside. To model the reduced thermal-conduction effects of the water gap, solids with thethermal conductivity of water were patched in between the plate and the holder. Effectively,the water gap acted as a low conductivity solid with no coolant advection. This was areasonable and conservative assumption because the very restricted flow path of the gapwould result in very small flow rates that would provide negligible amounts of forcedconvection cooling. The alternative was to mesh the gap at a higher resolution to adequatelysolve for the fluid advection which was in practical terms, unnecessary.


Figure 29. Detailed view of contact realistic conditions between the plate and its holder.

Table 5. Effect of the contact surface

Temperatures (°C) Thermal conduction profile type: average average Contact type perfect partial Power distribution uniform uniform Surface (water side) 59.0 62.1 Surface (cladding side) 100.5 103.6 Cladding - Meat interface 104.7 107.6 Meat - centre line 105.9 108.8

The difference in contact conditions only proved to be marginal as can be seen in Table5. The reduction from ‘full contact’ to ‘half contact’ between the plate and the holder onlyincreased the overall maximum temperature by 3°C. This proved to be positive from anengineer’s standpoint because it was undesirable to have heat transfer characteristics that aretoo sensitive to unpredictable contact conditions. The half-contact condition between plateand holder was thus adopted for future simulations for its more conservative assumption.

Effect of Power Distribution

In this study, the effect of a uniform and cosine power distribution on the maximumtemperature was examined. A simpler study by INVAP using a one-dimensional heat transfercode provided a simple comparative assessment. According to MCNP calculations conductedby ANSTO (Geoff, 2006), a power density profile distributing a total power of 79.4 kW overeight plates with a power peaking factor (PPF) of 1.3 was calculated as:

( )( ) ( )( )plate0 -9

79400Q 8cos 1.3 cos 4.97424 0.288

V 5188.8 10q PPF B z z z= − = − (41)


where

q : Power density, [W.m-3]PPF : Power Peaking Factor, [-]Qplate : Total power per plate, [W]V : Meat volume, [m3]B : Adjustable constant, [m-1]z : Coordinate on the z axis, [m]z0 : Offset of the centre, [m]

Power Density vs. Vertical Position2 Targets

8.00E+08

1.00E+09

1.20E+09

1.40E+09

1.60E+09

1.80E+09

2.00E+09

2.20E+09

2.40E+09

2.60E+09

0 0.2 0.4 0.6

z [m]

q [W

.m^-

3]

Figure 30. Power distribution along the plates (two positions).

Table 6. Power distribution effect on temperatures

Temperatures (°C)

Validation case INVAP study Oxide layer average average — Contact type partial partial — Power distribution uniform cosine — Surface (water side) 62.1 61.9 Surface (cladding side) 103.6 114.9

105

Oxide - Cladding interface — — 109 Cladding - Meat interface 107.6 120.2 111 Meat - centre line 108.8 121.6 113

This definition allowed the use of the same maximum power density (i.e., 2.5 109 W.m-3)as the one used by INVAP to calculate the maximum meat temperature, and also the same


total power in two plates. It also respected the PPF of 1.3 for the two position case. Applyingthe cosine power distribution to the model gave the simulation more realism than if a constantpower was applied throughout the plate. Two simulations were run to observe the effect of thedifference between a constant power and a cosine distribution power.

Using a cosine-shape distribution for the power increased the maximum meattemperature by 12.8°C as shown in Table 5. Since a cosine power distribution (see Figure 30)is more conservatively representative, this model feature was retained for the “bounding case”scenarios.

Conclusive Remarks for the Validation Case

Beginning from a simple model and making successive improvements with increasingdetailed features, a final validation case was attained which has been proven to be consistentand conservative against INVAP’s one-dimensional heat-transfer study. This course of actionhas also allowed an appreciation of the sensitivity of temperature changes with respect tovarying engineering assumptions.

The temperatures on the interface between the cladding and the oxide layer could not beattained in the CFD model, and therefore has not been compared. The CFX commercialsoftware appears to quote the temperature value held at the centre of the control volume oneither side of the plate-water interface, so that these results could not be directly comparedagainst INVAP’s study. However, the cladding side surface temperature yields valid andcomparable results to INVAP’s plate surface temperature results.

Table 7. Summary of the simulations for the validation case.

Temperatures (°C)

Validation Case INVAP study

Oxide layer absent step average average average — Contact type perfect perfect perfect partial partial —

Power distribution uniform uniform uniform uniform cosine —

Surface (water side) 59.2 59.2 59.0 62.1 61.9 Surface (cladding side) 99.1 99.3 100.5 103.6 114.9

105

Oxide - Cladding interface — — — — — 109 Cladding - Meat interface 100.9 101.3 104.7 107.6 120.2 111

Meat - centre line 102 102.5 105.9 108.8 121.6 113

Various hypotheses were tested on the validation case. Several simulations were runwhich revealed:

• The application of an average thermal conductivity over the whole cladding wasmore realistic than the definition of a step function to lower the conductivity in theoxide layer.


• The assumption of one-side contact between the plate and its holder was morerealistic than assuming full aluminium contact on both sides of the plate.

• The application of a variable power density was more realistic than a simpledefinition of the average power density.

In conclusion, these results were in good agreement with INVAP’s calculations and assuch, this model could now be used to examine even more conservative flow conditions forthe purpose of safety analysis. Summary of results are presented in Table 7.

Bounding Case

Having demonstrated the consistency of the CFD model against INVAP’s verificationdata, the CFD model could now be used to investigate the flow conditions of the boundingcase scenario. A summary of the material properties are detailed in Table 8.

Similar to the validation simulation, a cosine power profile (see Figure 31) for threetargets was set for this bounding simulation and a ‘high resolution’ advection scheme withautomatic timescale was adopted for the solution process in-order to accelerate solutionconvergence. A summary of temperature results for oxide thicknesses of 20 μm and 10 μmare displayed in Table 9 below:

Table 8. Summary of physical properties

MATERIAL PROPERTIES Aluminium U2Al and Al matrix oxide Water Molar Mass g.mol-1 26.98 270 18.02 Density kg.m-3 2700 4625.34 996.2 Specific Heat Capacity J.kg-1.K-1 903 900 4178.8 Thermal Conductivity W.m-1.K-1 165 148 2.25 0.609 Dynamic Viscosity μ kg.m-1.s-1 0.0008327

Pow er Density vs. Vertical Position 3 Targets

1.50E+09

2.00E+09

2.50E+09

3.00E+09

3.50E+09

4.00E+09

0 0.2 0.4 0.6 0.8

z [m]

q [W

.m^-

3]

Figure 31. Power distribution over 3 plates.


Table 9. Calculated temperatures for the bounding case

Temperatures (°C) Oxide layer 20 μm 10 μm Surface (water side) 92.2 92.5 Surface (cladding side) 183.1 181.1 Cladding - Meat interface 199.5 190.9 Meat - centre line 201.6 193

Thus, assuming an average flow velocity of 3 m.s-1 at 40°C, an oxide thickness of 20 μm,and a PPF of 1.3 for a total power of 45 kW over three plates, the main results were obtained:

• Maximum water temperature: 93°C• Maximum meat temperature: 202°C

These results showed a sufficient margin from plate melting and water boiling.

Conclusion

The cooling systems for three separate molybdenum irradiation facilities have beensolved for maximum mean temperatures using the computational fluid dynamicsmethodology.

In order to accomplish this, the geometry of all three systems were first modeled andgeometrically discretised in a variety of structured and unstructured mesh. Secondly, accuratematerial properties were attributed to corresponding areas of the numeric model. Uraniumvolumes were then attributed Power densities as calculated by MCNP to simulate the powerproduced within irradiated targets. The high velocities and thus high Reynolds numbers of allthree case studies placed the flow regime within a turbulent setting. As the study wasconcerned with time-averaged results, the RANS approach for turbulence modeling was mostsuited and the simple standard k-ε turbulence model was selected over other models for itsapplicability to fully turbulent flows. Other types of flows, such as large swirling flows orflows with large amounts of laminar-turbulent blending, would have required moresophisticated RANS modeling like the RNG k-ε model, reliazable k-ε model and SST Menter’smodel. However, this level of sophistication was unnecessary for the solution of these pipeflow systems and was thus not employed.

Computational results of all three case studies have been demonstrated to agree well withindependent experimental and numerical studies. The success of these studies furtherconfirms the robustness and versatility of CFD methods in the field of nuclear engineeringand will remain a continual feature in the field of thermo-hydraulics.


Acknowledgements

The authors would like to thank Dr. George Braoudakis (Head of Nuclear AnalysisSection) for providing the MCNP power input parameters in these simulations and for proof-reading this document; past internship students Mr. Tony Chung (UNSW, Sydney) and Mr.Guillaume Bois (INSA, Lyon) for their assistance in attaining the computational simulations;Mr. David Wassink (Water Tunnel Manager) for his experimental validation data; and,finally, to ANSTO for allowing this work to be published.

References

[1] Durance, G. (2006), Private communication, ANSTO.[2] Gumbert, C., Lohner, R., Parikh, P. & Pirzadeh, S. (1989), A package for unstructured

grid generation and finite element flow solvers, AIAA Paper 89-2175.[3] Jayatilleke, C. L. V. (1969), The influence of Prandtl number and surface roughness on

the resistance of the laminar sublayer to momentum and heat transfer, Prog. Heat MassTransfer, 1, 193 -321.

[4] Launder, B. E. & Spalding, D. B. (1974), The numerical computation of turbulent flows,Comp. Meth. Appl. Mech. Eng., 3, 269-289.

[5] Lo, S. H. (1985), A new mesh generation scheme for arbitrary planar domains, Int. J.Numer. Methods Eng., 21, 1403-1426.

[6] Marcum, D. L. & Weatherill, N. P. (1995), Unstructured grid generation using iterativepoint insertion and local reconnection, AIAA Paper 94-1926.

[7] Mavriplis, D. J. (1997), Unstructured grid techniques, Ann. Rev. Fluid Mech., 29, 473-514.

[8] Menter, F. R. (1993), Zonal two equation k-ω turbulence models for aerodynamicsflows, AIAA paper 93-2906.

[9] Menter, F. R. (1996), A comparison of some recent eddy-viscosity turbulence models, J.Fluids Eng., 118, 514-519.

[10] MOLY-0100-3OEIN-004 (2003), Heating calculation of Molybdenum targets, INVAPReport.

[11] Patankar, S. V. (1980), Numerical Heat Transfer and Fluid Flow, HemispherePublishing Corporation, New York.

[12] Shepard, M. S. & Georges, M. K. (1991), Three-dimensional mesh generation by finiteoctree technique, Int. J. Numer. Methods Eng., 32, 709-749.

[13] Shewchuk, J. S. (2002), Delaunay refinement algorithms for triangular mesh generation,Computational Geometry: Theory and Applications, 22, 21–74.

[14] Shih, T.-H., Liou, W. W., Shabbir, A., Yang, Z. & Zhu, J. (1995). A new k-ε eddyviscosity model for high Reynolds number turbulent flows, Comp. Fluids, 24, 227-238.

[15] Smith, R. E. (1982), Algebraic grid generation, Numerical Grid Generation, Thompson,J. F. (Ed.), North-Holland, Amsterdam, 137.

[16] Thompson, J. F. (1982), General curvilinear coordinate systems, Numerical GridGeneration, Thompson, J. F. (Ed.), North-Holland, Amsterdam, 1-30.

[17] Touloukian, Y.S., Powell, R.W., Ho, C.Y., Klemens, P.G. (1970). ThermophysicalProperties of Matter, 2.


[18] Tu, J. Y., Yeoh G. H. & Liu C. Q. (2008), Computational Fluid Dynamics – A Practicalapproach. Butterworth-Heinemann, Oxford.

[19] Wilcox, D. C. (1998), Turbulence Modelling for CFD, DCW Industries, Inc.[20] Yakhot, V., Prszag, S. A., Tangham, S., Gatski, T. B. & Speciale, C. G. (1992),

Development of turbulence models for shear flows by a double expansion technique,Phys Fluids A: Fluid Dynamics, 4, 1510-1520.

[21] Yeoh, G. H. and Storr, G. J. (2000), A three-dimensional study of heat and mass transferwithin the irradiation space of the HIFAR fuel element, Advance ComputationalMethods in Heat Transfer VI, Sunden, B. and Brebbia, C. A. (Eds.), WIT Press,Southampton, 343-351.

[22] Yerry, M. A. & Shepard, M. S. (1984), Automatic three-dimensional mesh generationby the modified-octree technique, Int. J. Numer. Methods Eng., 20, 1965-1990.


Chapter 9

FIRST AND SECOND LAW THERMODYNAMICSANALYSIS OF PIPE FLOW

Ahmet Z. Sahin*

Mechanical Engineering DepartmentKing Fahd University of Petroleum and Minerals

Dhahran 31261, Saudi Arabia

Introduction

In a fully developed laminar flow through a pipe, the velocity profile at any cross sectionis parabolic when there is no heat transfer. But when a considerable heat transfer occurs andthe thermo-physical properties of the fluid vary with temperature, the velocity profile isdistorted. If the thermo-physical properties of the fluids in a heat exchanger vary substantiallywith temperature, the velocity and the temperature profiles become interrelated and, thus, theheat transfer is affected. Viscosity of a fluid is one of the properties which are most sensitiveto temperature. In the majority of cases, viscosity becomes the only property which may haveconsiderable effect on the heat transfer and temperature variation and dependence of otherthermo-physical properties to temperature is often negligible. The viscosity of the liquidsdecreases with increasing temperature, while the reverse trend is observed in gases (Kreithand Bohn, 1993). Heat transfer and pressure drop characteristics are affected significantlywith variations in the fluid viscosity. When the temperature is increased from 20 to 80 oC, theviscosity of engine oil decreases 24 times, the viscosity of water decreases 2.7 times and theviscosity of air increases 1.4 times. Therefore, selection of the type of fluid and the range ofoperating temperatures are very important in the design and performance calculations of aheat exchanger.

In the process of designing a heat exchanger, there are two main considerations. Theseare the heat transfer rates between the fluids and the pumping power requirement to overcomethe fluid friction and move the fluids through the heat exchanger. Although the effect ofviscous dissipation is negligible for low-velocity gas flows, it is important for high-velocity

* E-mail address: [email protected]

Ahmet Z. Sahin318

gas flows and liquids even at very moderate velocities (Shah, 1981; Kays and Crawford,1993). Heat transfer rates and pumping power requirements can become comparableespecially for gas-to-gas heat exchangers in which considerably large surface areas arerequired when compared with liquid to liquid heat exchangers such as condensers andevaporators. In addition, the mechanical energy spent as pumping power to overcome thefluid friction is worth 4 to 10 times as high as its equivalent heat (Kays and London, 1984). Therefore, viscous friction which is the primary responsible cause for the pressure drop andpumping power requirements is an important consideration in heat exchanger design.

On the other hand, efficient utilization of energy is a primary objective in designing athermodynamic system. This can be achieved by the minimization of entropy generation inprocesses of the thermodynamic system. The irreversibilities associated with fluid flowthrough a pipe are usually related to heat transfer and viscous friction. Various mechanismsand design features contribute to the irreversibility terms (Bejan, 1988). There may exist anoptimal thermodynamic design which minimizes the amount of entropy generation. For agiven system, a set of thermodynamic parameters which optimizes the operating conditionsmay be obtained.

The irreversibility associated with viscous friction is directly proportional to the viscosityof the fluid in laminar flow. Therefore, it is necessary to investigate the effect of a change ofviscosity during a heating process for an accurate determination of entropy generation and ofthe required pumping power.

Heat transfer and fluid pumping power in a piping system are strongly dependent uponthe type of fluid flowing through the system. It is important to know the fluid properties andtheir dependence to temperature for a heat exchanger analysis. As a result of heat transfer, thetemperature changes in the direction of flow and the fluid properties are affected. Thetemperature variation across the individual flow passages influences the velocity andtemperature profiles, and thereby influences the friction factor and the convective heattransfer coefficient. If the thermo-physical properties of the fluids in a piping system varysubstantially with temperature, the velocity and the temperature profiles become interrelatedand, thus, the heat transfer is affected. Viscosity of a fluid is one of the properties which aremost sensitive to temperature. Therefore, selection of the type of fluid and the range ofoperating temperatures are very important in the design and performance calculations of apiping system.

On the other hand, the exergy losses associated with fluid flow through a pipe are usuallyrelated to heat transfer and viscous friction. Dependence of the thermo-physical properties onthe temperature affects not only the viscous friction and pressure drop, but also the heattransfer. This implies that the exergy losses associated with the heat transfer and the viscousfriction are also affected. The contributions of various mechanisms and design features on thedifferent irreversibility terms often compete with one another. Therefore, an optimalthermodynamic design which minimizes the amount of entropy generation may exist. In otherwords, a set of thermodynamic parameters which optimize operating conditions could beobtained for a given thermodynamic system.

In this chapter, the entropy generation for during fluid flow in a pipe is investigated. Thetemperature dependence of the viscosity is taken into consideration in the analysis. Laminarand turbulent flow cases are treated separately. Two types of thermal boundary conditions areconsidered; uniform heat flux and constant wall temperature. In addition, various cross-sectional pipe geometries were compared from the point of view of entropy generation and

First and Second Law Thermodynamics Analysis of Pipe Flow 319

pumping power requirement in order to determine the possible optimum pipe geometry whichminimizes the exergy losses.

Pipe Subjected to Uniform Wall Heat Flux

We consider the smooth pipe with constant cross section shown in Figure 1. A constantheat flux q′′ is imposed on its surface. An viscous fluid with mass flow rate m and inlet

temperature 0T enters the pipe of length L . Density ρ , thermal conductivity k , and

specific heat pC of the fluid are assumed to be constant. Heat transfer to the bulk of the fluid

occurs at the inner surface with an average heat-transfer coefficient h , which is a function oftemperature dependent viscosity.

The effect of viscosity on the average heat transfer coefficient is given by Kays andCrawford (1993)

. .. .

n

b

c pc p w

h Nuh Nu

μμ

⎛ ⎞= = ⎜ ⎟

⎝ ⎠(1)

where the exponent n is equal to 0.14 for laminar flow. In the case of turbulent flow theexponent n is equal to 0.11 for heating and 0.25 for cooling.

For fully developed laminar flow, constant property heat transfer coefficient . .c ph is

given by Incropera and DeWitt (1996).

DkNu

Dkh pcpc 11

48.... == (2)

Figure 1. Sketch of constant cross sectional area pipe subjected to uniform wall heat flux.

Ahmet Z. Sahin320

And for fully developed turbulent flow, constant property heat transfer coefficient . .c ph is

given by

( ) ( )( )

. .. . 2 / 3

/ 8 Re 1000 Pr

1 12.7 Pr 1 /8c pc p

fk kh NuD D f

−= =

+ −(3)

where the Reynolds number is

Reb

UDρμ

= . (4)

The average Darcy friction factor, f , for a smooth pipe is also considered to be afunction of temperature dependent viscosity and is given by Kays and Crawford (1993)

m

w

b

pcff

⎟⎟⎠

⎞⎜⎜⎝

⎛=

μμ

..

for liquids (5a)

m

w

b

pc TT

ff

⎟⎟⎠

⎞⎜⎜⎝

⎛=

..

for gases (5b)

where the exponent 14.0−=m for laminar flow and 25.0−=m for turbulent flow case.The friction factor for constant properties for laminar flow is given by Incropera and

DeWitt (1996) as

Re64

.. =pcf (6)

and for turbulent flow as

[ ] 2. . 0.79 ln(Re) 1.64c pf −= − (7)

To account for the variation of the bulk temperature along the pipe length, bμ andtherefore Re and Pr, in Eqs. (3) – (7), are related to the bulk fluid temperature halfwaybetween the inlet and outlet of the pipe, as suggested by Kreith and Bohn (1993). Since thetemperature variation along the pipe is initially unknown and depends on h , a trial and error

procedure is followed to determine both h and f .


Total Heat Transfer Rate

The rate of heat transfer to the fluid inside the control volume shown in Figure 1 is

pQ mC dT q Ddxδ π′′= = (8)

where 2 / 4m U Dρ π= . In writing Eq. (8), the pipe is assumed to have a circular crosssection. However, the analysis is not affected by assuming cross-sectional areas other thancircular. Integrating Eq. (8), the bulk-temperature variation of the fluid along the pipe isobtained as

04

p

qT T xUDCρ

′′− = . (9)

The temperature variation along the pipe is linear when the viscous dissipation and axialconduction effects are neglected. The temperature gradient for the fluid in this case dependsmainly on the magnitude of the heat flux.

For a constant heat flux q′′ and average heat transfer coefficient h evaluated at the bulktemperature halfway between inlet and outlet of pipe, the temperature difference between thewall surface and bulk of the fluid is given as

wqT Th′′

− = . (10)

Eq. (9) may be rewritten as

4 St xD

θ = , (11)

where θ is the dimensionless temperature defined as

0

/T Tq h

θ −=

′′(12)

and the Stanton number is

p

hStUCρ

= . (13)

Ahmet Z. Sahin322

Total Entropy Generation

The entropy generation within the control volume of Figure 1 can be written as (Bejan,1996)

genw

QdS mdsTδ

= − , (14)

where the entropy change is

pdT dPds CT Tρ

= − . (15)

Substituting Eq. (8) into Eq. (14),

1wgen p

w p

T TdS mC dT dPTT C Tρ

⎡ ⎤−= −⎢ ⎥

⎣ ⎦. (16)

The pressure drop is (Kreith and Bohn, 1993)

2

2f UdP dx

Dρ

= − , (17)

where f is the Darcy friction factor.Integrating Eq. (16) along the pipe length L , using Eqs. (9) and (17), the total entropy

generation becomes (Sahin, 1999 and Sahin, 2002)

( ) ( ) ( )31 4 1 1ln ln 1 4

1 4 8gen pSt f US mC St

St qτλ τ ρ τλ

τ τλ⎧ ⎫+ +⎡ ⎤⎪ ⎪= + +⎨ ⎬⎢ ⎥ ′′+ +⎪ ⎪⎣ ⎦⎩ ⎭

, (18)

where the dimensionless wall heat flux is

0

/q hT

τ′′

=(19)

and the dimensionless length of the pipe is

LD

λ = . (20)


Eq. (18) can be written in a dimensionless form as

0/TQSgen=ψ (21)

where the total heat transfer is

( )0 4pQ mC T Stτλ= (22)

Thus, the entropy generation per unit heat transfer rate becomes

⎭⎬⎫

⎩⎨⎧

++⎥⎦⎤

⎢⎣⎡

++++

= )41ln(81

41)1)(41(ln

41 τλ

τλτττλ

τλψ St

StEcf

StSt

St(23)

where the Eckert number is defined as

2 2

( ) ( / )p w p

U UEcC T T C q h

= =′′−

. (24)

Two dimensionless groups arise naturally in Eq. (23), namely,

1 StλΠ = (25)and

2EcfSt

Π = (26)

Thus, Eq. (23) becomes in compact form

⎭⎬⎫

⎩⎨⎧

Π+Π+⎥⎦

⎤⎢⎣

⎡Π+++Π+

Π= )41ln(

81

41)1)(41(ln

41

121

1

1

τττ

τττ

ψ (27)

or

( ) ⎥⎦

⎤⎢⎣

⎡Π+

Π+++

Π= Π+ 28

111

11

4141

1ln4

1 ττττ

τψ . (28)

The first and second terms in Eq. (27) are related to entropy generation due to heattransfer and due to viscous friction respectively. Eq. (27) contains three non-dimensionalparameters, namely, τ , 1Π and 2Π . Among these parameters, τ represents the heat flux

imposed on the wall of the pipe q′′ and 1Π represents the pipe length L . Once the type of

the fluid and the mass flow rate are fixed, the parameter 2Π can be calculated on the basis of

Ahmet Z. Sahin324

temperature analysis. Thus, τ and 1Π are the two design parameters that can be varied fordetermining the effects of pipe length and/or wall heat flux on the entropy generation.

For small values of the wall heat flux ( 1τ << ), Eq. (27) is reduced to

)41ln(321

11

2 Π+ΠΠ

= ττ

ψ . (29)

For low viscosity, 2 / 8 1Π << and Eq. (28) shows that entropy generation becomes

( )⎥⎦

⎤⎢⎣

⎡Π+++Π+

Π=

1

1

1 411)41(ln

41

ττττ

τψ (30)

Amount of heat transfer is the primary concern in a heat exchanger design. However, aconsiderable amount of exergy is destroyed during the heat exchange process. For efficientutilization of energy the entropy generation needs to be minimized. Thus, exergy destructionper unit amount of heat transfer is considered to be a suitable quantity in dealing with thesecond law analysis of a pipe flow. In the present analysis this ratio is represented by thedimensionless entropy generation, ψ , defined as the entropy generation per unit heat transferrate for specified pipe inlet temperature.

Since 1Π represents the length of the pipe, the dimensionless entropy generation per unitheat transfer rate to the pipe ψ decreases with the pipe length. The dimensionless entropygeneration ψ approaches a common value for short pipes with a limit

20 81

1lim

1

Π++

=→Π τ

τψ . (31)

For the case of constant viscosity, ψ decreases with a slope of

⎥⎦

⎤⎢⎣

⎡Π+

+−−=

Π∂∂

→Π 221

0 81

)1(112lim

1 ττψ

. (32)

ψ decreases and approaches zero asymptotically (and therefore the total heat transfer) aspipe length increases.

Since τ represents the amount of heat flux, the dimensionless entropy generation ψ ,defined on the basis of total heat transfer rate to the pipe, goes to infinity when 0τ = . ψdecreases sharply as τ increases and then starts increasing.

A second dimensionless entropy generation may be defined, on the basis of unit heatcapacity rate of fluid in the pipe, as


)/( TQS

CmS gen

p

gen

Δ==φ , (33)

where TΔ is the increase 0( )T L T− of the bulk temperature of the fluid in the pipe. Since

/ pQ T mCΔ = is constant for a fixed mass flow rate, the total entropy generation (φ ),

which increases with an increase in pipe length, can be given as

ψτφ 14 Π= . (34)

For the case of constant viscosity, φ starts to increase with a rate of

⎥⎦⎤

⎢⎣⎡ Π++

=Π∂∂

→Π 21

0 81

14lim

1 τττφ

(35)

and approaches to infinity as pipe length increases.It should be noted that 2Π is inversely proportional to τ and therefore the propipe 2τΠ

is constant. Thus the value of φ is finite for 0τ =

)(21lim 210

ΠΠ=→

τφτ

(36)

φ decreases with a slope of

)(lim 2210

ΠΠ−=∂∂

→τ

τφ

τ. (37)

Pumping Power to Heat Transfer Rate Ratio

The ratio of pumping power to heat transfer rate is

2

4rD PUP

Qπ Δ

= . (38)

Using Eqs. (8) and (17), the pumping power to heat transfer ratio rP is

218rP = Π . (39)

Ahmet Z. Sahin326

Recalling that the product 2τΠ is invariable with varying τ , it can be concluded fromEq. (39) that the pumping power ratio is inversely proportional to τ .

Effect of Pipe Cross Section

The dimensionless entropy generation given in Eq. (28) can be expressed in terms of Renumber as (Sahin, 1998)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛ +

++

+=

+ 22 Re1

1

11 Re1

Re1

1lnRe CCCC

τττ

ττ

ψ (40)

where 11 Re4 Π=C and 22

2 Re8Π

=C .

For any given pipe geometry and stream of constant thermophysical property fluid,dimensionless numbers C1 and C2 are constants. Therefore, the dimensionless total entropygeneration as given in Eq. (40) is a function of Re and τ only.

Values of Nu and Re)( f for fully developed laminar flow are given in Shah andLondon (1978) for a variety of pipe geometries. The hydraulic diameter DH for any given pipegeometry can be calculated using the relation:

pAD c

H 4= . (41)

Table 1 gives the hydraulic diameters for some common pipe geometries.Similarly, the modified dimensionless entropy generation becomes

ψτφRe

1C= (42)

and the pumping power to heat transfer ratio is obtained to be

22 ReCPr = (43)

As the temperature difference between the fluid inlet and the surface of the pipe, τ , isincreased, the entropy generation increases, however, the pumping power requirement perunit heat transfer decreases. In the limit when τ approaches zero, the contribution of heattransfer to the entropy generation disappears and


Relim 210τψ

τCC=

→(44)

which indicates a linear increase in entropy generation that is solely due to the viscousfriction as expected.

Table 1. Hydraulic diameters of some common pipe geometries.

Pipe Geometry Hydraulic diameter (DH)

CircularcH AD

π2

=

SquarecH AD =

TrianglecH AD 4 3

32

=

Rectangle*

cH A

XYX

YD⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+=

1

12

Sinusoidal* **

cH A

XY

XY

EXY

XYD

1

2

2

2

12

,1

1224

−

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎠⎞

⎜⎝⎛+

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛+=

π

π

ππ

ππ

* Y/X is the aspect ratio and

** ∫ −=σ

σσσ0

22 sin1),( dkkE is the elliptic integral of second kind (Tuma, 1979)

Figure 2 shows the variation of ψ with Re for five selected pipe geometries. The cross-

section cA and 0T were constant. The sinusoidal pipe geometry gives the lowestdimensionless entropy generation. As Re is increased, the circular geometry becomes the bestchoice. The total entropy generation tends to decrease initially and then increases with Re. ψincludes terms form heat transfer and viscous friction. As Re is increased, the heat transfercontribution decreases and that of viscous friction increases. Thus, for low Re number flow,those geometries with higher surface areas are favorable. Circular geometry is the best choicefor high Re numbers. Square geometry appears to be good choice after the sinusoidal andcircular ones. ψ is found to be considerably high for triangular and rectangular pipes.Therefore, they are inferior choices. Rectangular pipes would yield the lowest ψ when the

Ahmet Z. Sahin328

aspect ratio becomes 1 (square geometry). For rectangular pipes with lower aspect ratio, theentropy generation is higher and there may not even exist a minimum.

As 0Re → , the effect of geometry disappears and the dimensionless entropy generationbecomes a function of )(τ , i.e.,

( )τψ +=→

1lnlim0Re

(45)

Comparison of pipe geometries using φ may be appropriate when the total heat-transferrate is important as shown in Figure 3. Favorable pipe geometry appears to be circular butsinusoidal and square geometries yield comparable results. Triangular and rectangular pipegeometries are inferior choices because there is no optimum Re with minimum entropygeneration.

The pumping power required to overcome viscous friction is shown in Figure 4. Thecircular geometry is superior to all other geometries. The results are similar to those for themodified dimensionless entropy generation.

Figure 2. Dimensionless entropy generation ψ vs Re for various pipe geometries; 26 m104 −×=cA

and 2 W/m500=′′q .


Figure 3. Modified dimensionless entropy generation φ vs Re for various pipe geometries;26 m104 −×=cA and 2 W/m500=′′q .

Figure 4. Pumping power to heat-transfer ratio rP ve Re for various pipe geometries;26 m104 −×=cA and 2 W/m500=′′q .

Ahmet Z. Sahin330

Pipe Subjected to Constant Wall Temperature

We consider the smooth pipe with constant cross section shown in Figure 5. The surfacetemperature Tw is kept constant. An viscous fluid with mass flow rate m and inlettemperature 0T enters the pipe of length L . Density ρ , thermal conductivity k , and

specific heat pC of the fluid are assumed to be constant. Heat transfer to the bulk of the fluid

occurs at the inner surface with an average heat-transfer coefficient h , which is a function oftemperature dependent viscosity.

The effect of viscosity on the average heat transfer coefficient is given by Shah andLondon (1978)

. .. .

n

b

c pc p w

h Nuh Nu

μμ

⎛ ⎞= = ⎜ ⎟

⎝ ⎠(46)

where the exponent n is equal to 0.14 for laminar flow. In the case of turbulent flow theexponent n is equal to 0.11 for heating and 0.25 for cooling.

For fully developed laminar flow, constant property heat transfer coefficient . .c ph is

given by Incropera and DeWitt (1996)

DkNu

Dkh pcpc 66.3.... == . (47)

Constant property heat transfer coefficient . .c ph for fully developed turbulent flow is

given by Eqn. (3). Correlations for the average Darcy friction factor, f , for a smooth pipe aregiven in Eqns. (5) – (7) for laminar and turbulent flow cases.

Temperature Distribution

The rate of heat transfer to the fluid inside the control volume shown in Figure 1 is

dxTTDhdTCmQ wp )( −== πδ (48)

where2 / 4m U Dρ π=

Integrating Eq. (48), the bulk temperature variation of the fluid along the pipe can beobtained as:

⎥⎥⎦

⎤

⎢⎢⎣

⎡−−−= x

DCUhTTTT

poww ρ

4exp)( (49)


Figure 5. Sketch of constant cross sectional pipe subjected to constant wall temperature.

The temperature variation along the pipe approaches the pipe wall temperatureexponentially assuming a uniform heat transfer coefficient evaluated at the bulk temperaturehalfway between the inlet and outlet of the pipe.

Eq. (49) can be re-written as:

⎟⎠⎞

⎜⎝⎛−= x

DSt4expθ (50)

where θ is the dimensionless temperature defined as,

wo

w

TTTT

−−

=θ

and St is the Stanton number defined as

p

hStUCρ

= .

Total Entropy Generation

The total entropy generation within the control volume in Figure 1 can be written as:

genw

QdS mdsTδ

= −(51)

where the entropy change is,

Ahmet Z. Sahin332

pdT dPds CT Tρ

= −

Substituting Eq. (48) in Eq. (51), the total entropy generation becomes

1wgen p

w p

T TdS mC dT dPTT C Tρ

⎡ ⎤−= −⎢ ⎥

⎣ ⎦ (52)

The pressure drop in Eq. (52) is given by Bejan (1988)

2

2f UdP dx

Dρ

= − (53)

Integrating Eq. (52) along the pipe length, L, using Eqs. (49) and (53), the total entropygeneration is obtained as (Sahin, 1998 and Sahin 2000)

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−−

+−−⎥⎦

⎤⎢⎣

⎡−

−= −

−

τττ

ττ λ

λλ

1ln

81)1(

11ln

44

4 StSt

St

pgene

StEcfeeCmS (54)

where τ is the dimensionless temperature difference

w

ow

TTT −

=τ ,

λ is the dimensionless length of pipe

DL

=λ ,

and Ec is the Eckert number defined as

wpTCUEc

2

= .

The total rate of heat transfer to the fluid is obtained by integrating Eq. (48) along thepipe length and can be written as:

).1)(( 4 λStowp eTTCmQ −−−= (55)


Now defining a dimensionless entropy generation as:

)/( ow

gen

TTQS−

=ψ

Eq. (54) can be written as

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−−

+−−⎥⎦

⎤⎢⎣

⎡−

−−

= −−

− τττ

ττψ

λλ

λ

λ 1ln

81)1(

11ln

11 4

44

4

StSt

St

Ste

StEcfee

e(56)

Therefore, two dimensionless groups naturally arise in Eq. (56) as:

1 StλΠ = (57)

and

2EcfSt

Π = (58)

Thus Eq. (56) can be written in a compact form for the constant viscosity assumption as:

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−−

Π+−−⎥⎦

⎤⎢⎣

⎡−

−−

=Π

Π−Π−

Π− τττ

ττψ

1ln

81)1(

11ln

11 1

11

1

4

24

4

4eee

e (59)

which is a function of three non-dimensional parameters, namely τ , 1Π and 2Π . Among

these parameters, τ represents the fluid inlet temperature To and 1Π represents the pipe

length L. Once the type of the fluid and the mass flow rate are fixed, the parameter 2Π can

be calculated based on temperature analysis. Thus, τ and 1Π are the two design parametersthat can be varied for determining the effects of pipe length and/or inlet fluid temperature onthe entropy generation.

Since 1Π represents the length of the pipe, the dimensionless entropy generation definedon the basis of total heat transfer rate to the pipe, ψ , decreases initially and then startsincreasing along the pipe length. The rate of increase in entropy generation approaches aconstant value as the total heat transfer rate to the fluid approaches its maximum value of

)(max owp TTCmQ −= (60)

For long pipes where 114 <<Π−e and τ>>Π14e , it can be shown from Eq. (59) that theentropy generation increases linearly with the slope

Ahmet Z. Sahin334

22

1

Π=

Πddψ

for the case of constant viscosity.Since τ represents the difference between the temperature of the pipe surface and that of

the inlet fluid, the dimensionless entropy generation defined on the basis of total heat transferrate to the pipe, ψ , increases as τ increases due to the increase in the gap between the bulkand wall temperatures. For small values of τ , the total entropy generation is due to theviscous friction; and in the limit when 0=τ , the total dimensionless entropy change usingEq. (59) becomes:

1421

0 121lim Π−→ −

ΠΠ=

eψ

τ. (61)

It should be noted that the dimensionless entropy generation, ψ , in the above analysis is

a function of the total heat transfer rate, Q , which in turn depends on the length of the pipeand inlet fluid temperature. However, a modified dimensionless entropy generation can bedefined on the basis of unit heat capacity rate of fluid in the pipe as:

)/( TQS

CmS gen

p

gen

Δ==φ (62)

where TΔ is the increase of the bulk temperature of the fluid in the pipe, oTLTT −=Δ )( .

Noting that pCmTQ =Δ/ is constant for fixed mass flow rate and

ψφ )1( 14Π−−= e . (63)

φ , indicating the total entropy generation along the pipe, is expected to increase with theincrease in pipe length.

φ and ψ differ only for small values of 1Π . For large values of 1Π , ψφ = as can beseen from Eq. (63).

The modified dimensionless entropy generation defined, based on the unit heat capacityrate through the pipe, φ , shows similar behavior to that of ψ . This was expected, since φand ψ are related through a constant factor of ( )141 Π−− e as given in Eq. (63).


Pumping Power to Heat Transfer Rate Ratio

The pumping power to heat transfer rate ratio is

2

4rD PUP

Qπ Δ

= . (64)

Using Eqs. (50) and (53), the pumping power to heat transfer rate ratio, rP is obtained as

)1(21

1421Π−−

ΠΠ=

ePr τ

. (65)

Due to an increase in the bulk temperature and a decrease in viscosity, the pumpingpower to heat transfer rate ratio may decrease initially and then increase, as the total heattransfer rate to the fluid decreases as the bulk temperature approaches the wall temperature asψ is increased. For large values of ψ and constant viscosity, it can be shown from Eq. (65)that the pumping ratio, Pr, increases linearly with the slope

τ22

1

Π=

ΠddPr .

For small values of 1Π the effect of the viscosity variation is small and in the limit,

τ8lim 2

01

Π=

→Π rP (66)

which is clearly a function of the fluid viscosity.

Effect of Pipe Cross Section

The dimensionless entropy generation as given in Eqn. (59) can be re-written as functionof Re as (Sahin, 1998)

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−−

+−−⎥⎦

⎤⎢⎣

⎡−

−−

= −−

− τττ

ττψ

1lnRe)1(

11ln

11 Re/

22

Re/Re/

Re/

11

1

1

CC

C

CeCee

e(67)

where 11 Re4 Π=C and 22

2 Re8Π

=C .

Ahmet Z. Sahin336

For any given pipe geometry and stream of constant thermophysical property fluid,dimensionless numbers C1 and C2 are constants. Therefore, the dimensionless total entropygeneration as given in Eq. (67) is a function of Re and τ only.

Values of Nu and Re)( f for fully developed laminar flow are given in Shah andLondon (1978) for a variety of pipe geometries. The hydraulic diameter DH for any given pipegeometry can be calculated using the relation:

pAD c

H 4= .

Similarly, ψφ )1( Re/1Ce−−= (68)

and

)1(Re

Re/21

1Cr eCCP −−

=τ

(69)

For very low Re number flow, it is interesting to note that the importance of geometrydisappears. In the limit as

ττ

ψ +⎟⎠⎞

⎜⎝⎛+

=→ 1

1lnlim0Re

(70)

As the temperature difference between the fluid inlet and the surface of the pipe, τ , isincreased, the entropy generation increases, however, the pumping power requirement perunit heat transfer decreases. In the limit when τ approaches zero, the contribution of heattransfer to the entropy generation disappears and

Relim 210CC=

→ψ

τ(71)

which indicates a linear increase in entropy generation that is solely due to the viscousfriction as expected.

Figure 6 shows the variation of dimensionless entropy generation, ψ , with Re numberfor five selected pipe geometries; namely circular, square, equilateral triangle, rectangular

(aspect ratio of 1/2) and sinusoidal (aspect ratio 2/3 ). The cross sectional area of eachpipe, Ac, and the inlet fluid temperature, To, are kept constant. For low Re numbers,sinusoidal pipe geometry gives lowest entropy generation of all the pipe geometriesconsidered. However, as the Re number is increased, circular pipe geometry becomes the bestchoice. In general, the total entropy generation tends to decrease initially and then increaseswhile the Re number is increased.


The dimensionless total entropy generation, ψ , includes contributions due to heattransfer and viscous friction. As the Re number is increased, the contribution due to heattransfer decreases and that of viscous friction increases. Therefore, for low Re number flowfor which the viscous frictional contribution can be neglected, those pipe geometries withhigher surface areas are favorable. This is because of the fact that the total heat transfer is afunction of the surface area for fixed cross sectional area and length of pipe. On the otherhand, circular pipe geometry is the best choice for high Re numbers where viscousfrictional contribution to the entropy generation becomes dominant. Sinusoidal and squarepipe geometries appear to be good choices after the circular pipe for the high Re numberregion. The dimensionless entropy generation is found to be high for triangular andrectangular pipes almost for the entire Re number region. Therefore, they are the worstchoices from the point of view of entropy generation. This conclusion is obtained only ifthe above comparison of entropy generation is made by keeping the flow cross sections ofthe pipes constant. Comparison of pipe geometries with same equivalent diameter, on theother hand, leads to the results in which triangular and square pipes seem to be betterchoices than the circular pipe geometry. This may be misleading. Rectangular pipes wouldyield the best results when the aspect ratio becomes 1, which corresponds to squaregeometry, and the square geometry is almost nowhere better than the circular one. Forlower aspect ratio rectangular pipes there may not even exist a minimum entropygeneration.

For very low Re number flow, it is interesting to note that the importance of geometrydisappears and all the curves approach to the same value as Re number approaches zero. Thislimit of dimensionless entropy generation is a function of the temperature difference betweenthe inlet fluid stream and surface of the pipe, τ , as

ττ

ψ +⎟⎠⎞

⎜⎝⎛+

=→ 1

1lnlim0Re

(72)

The modified dimensionless entropy generation, φ , versus Re number is given in Figure7. The most favorable pipe geometry appears to be the sinusoidal type in this case, for thewhole range of Re numbers. The circular geometry is the next favorable geometry as far asthe modified dimensionless entropy generation is concerned; however, the differences inentropy generation for the geometries selected are not significant. It should be noted that,there is no optimum Re number to provide a minimum modified entropy generation, since thetotal heat transfer increases as Re number is increased.

Figure 8 shows the pumping power requirement to overcome the viscous friction. Clearlythe circular geometry is superior to all other geometries. Sinusoidal geometry appears to be abad choice from the point of view of pumping power to heat transfer ratio. Triangular pipe isthe worst choice in this case.

Ahmet Z. Sahin338

Figure 6. Dimensionless entropy generation ψ versus Re number for various pipe geometries,27 m 104 −×=cA and 01.0=τ .

Figure 7. Modified dimensionless entropy generation φ versus Re number for various pipegeometries m 104×=cA and 01.0=τ .


Figure 8. Pumping power to heat transfer ratio rP versus Re number for various pipe geometries27 m 104 −×=cA and 01.0=τ .

Appendix: Viscosity Dependence on Temperature

Experimentally it is known that the viscosity of liquids vary considerably withtemperature. Around room temperature (293 K), for instance, a 1% change in temperatureproduces a 7% change in viscosity of water and approximately a 26% change in viscosity ofglycerol (Sherman, 1990).

As a first approximation, the variation of viscosity with temperature can assumed to belinear,

( ) ( )ref refT b T Tμ μ= − −

where b is a fluid dependent dimensional constant and refT is a reference temperature (=

293 K). This would be a reasonable approximation if the bulk temperature variation is small.However, for highly viscous liquids the variation of viscosity with temperature is exponentialand a more accurate empirical correlation of liquid viscosity with the temperature is given bySherman (1990) as

1 1( ) expn

refref ref

TT BT T T

μ μ⎡ ⎤⎛ ⎞ ⎛ ⎞

= −⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

Ahmet Z. Sahin340

where n and B are fluid dependent constant parameters. In the present work, both the linearand the exponential viscosity models were used. In addition, a constant viscosity model (inwhich the numerical value of viscosity at a reference bulk temperature is taken as constant) isalso included in order to see the significance of the variation of viscosity on the results.

Nomenclature

pC Specific heat (J/kg K)

D Diameter (m)

Ec Eckert number, ( )2 / /pU C q h⎡ ⎤′′⎣ ⎦f Average Darcy friction factor

h Average heat transfer coefficient (W/m2 K)k Thermal conductivity (W/m K)L Length of the pipe (m)m Mass flow rate (kg/s)

Nu Average Nusselt number, /hD kP Pressure (N/m2)Pr Prandtl number

rP Pumping power to heat transfer rate ratioq′′ Heat flux (W/m2)

Q Total heat transfer rate (W)

Re Reynolds number, / bUDρ μs Entropy (J/kg K)

genS Entropy generation (W/K)

St Stanton number, ( )/ ph UCρT Temperature (K)

0T Inlet fluid temperature (K)

refT Reference temperature (=293 K)

wT Wall temperature of the pipe (K)

U Fluid bulk velocity (m/s)x Axial distance (m)

Greek Symbols

TΔ Increase of fluid temperature (K)μ Viscosity (N s/m2)


bμ Viscosity of fluid at bulk temperature (N s/m2)

refμ Viscosity of fluid at reference temperature (N s/m2)

wμ Viscosity of fluid at wall temperature (N s/m2)

λ Dimensionless axial distance, /L D1Π Modified Stanton number, Stλ

2Π Dimensionless group, /fEc St

ψ Dimensionless entropy generation, ( )0/ /genS Q T

φ Modified dimensionless entropy generation, ( )/ /genS Q TΔρ Density (kg/m3)

τ Dimensionless wall heat flux, ( ) 0/ /q h T′′

θ Dimensionless temperature, ( ) ( )0 / /T T q h′′−

. .c p Constant properties

References

[1] Bejan, A., Advanced Engineering Thermodynamics, John Wiley and Sons, New York(1988).

[2] Bejan, A., Entropy Generation Minimization, CRC Press, New York (1996).[3] Incropera, F.P. and DeWitt, D.P., Introduction to Heat Transfer, Third Ed., John Wiley,

New York (1996).[4] Kays, W. M. and Crawford, M. E., Convective Heat and Mass Transfer, Third Ed.

McGraw Hill, New York (1993).[5] Kays, W. M. and London, A. L., Compact Heat Exchangers, 3rd Ed., McGraw Hill,

New York (1984).[6] Kreith, F. and Bohn, M.S., Principles of Heat Transfer, fifth ed., West Publ. Co., New

York (1993).[7] Shah, R. K., Compact Heat Exchanger Design Procedures, in: Heat Exchangers,

Thermal-Hydraulic Fundamentals and Design, Ed: Kakac, S., Bergles, A.E., andMayinger, F., Mc-Graw Hill, New York (1981).

[8] Shah, R.K., and London, A.L., Laminar Flow Forced Convection in Ducts, AcademicPress, New York (1978).

[9] Sahin, A.Z., Irreversibilities in Various Duct Geometries with Constant Wall Heat Fluxand Laminar Flow, Energy - The International Journal, vol. 23, no. 6, pp. 465- 473,(1998).

[10] Sahin, A.Z., A Second Law Comparison for Optimum Shape of Duct Subjected to Constant Wall Temperature and Laminar Flow, Heat and Mass Transfer, vol. 33, pp. 425-430, 1998.

Ahmet Z. Sahin342

[11] Sahin, A.Z., Second Law Analysis of Laminar Viscous Flow Through a Duct Subjectedto Constant Wall Temperature, ASME Journal of Heat Transfer, vol. 120, no. 1, pp. 76-83, (1998).

[12] Sahin, A.Z., The Effect of Variable Viscosity on the Entropy Generation and PumpingPower in a Laminar Fluid Flow through a Duct Subjected to Constant Heat Flux, Heatand Mass Transfer, vol. 35, pp. 499-506, (1999).

[13] Sahin, A.Z., Entropy Generation in Turbulent Viscous Flow Through a Smooth Duct Subjected to Constant Wall Temperature, International Journal of Heat and Mass Transfer, vol. 43, no. 8, pp. 1469-1478, (2000).

[14] Sahin, A.Z., Entropy Generation and Pumping Power in a Turbulent Fluid Flow througha Smooth Pipe Subjected to Constant Heat Flux, Exergy, an International Journal, vol.2, no. 4, pp. 314-321, (2002).

[15] Sherman, F.S., Viscous Flow, McGraw Hill Book Co., New York (1990).[16] Tuma, J.J., Engineering Mathematics Handbook, Second Ed., McGraw Hill Book. Co.,

New York (1979).


Chapter 10

SINGLE-PHASE INCOMPRESSIBLE FLUID FLOWIN MINI- AND MICRO-CHANNELS

Lixin Cheng*

School of Engineering, University of Aberdeen, King’s College, Aberdeen,AB24 3UE, Scotland, the UK

Abstract

This chapter aims to present a state-of-the-art review on single-phase incompressible fluidflow in mini- and micro-channels. First, classification of mini- and micro-channels isdiscussed. Then, conventional theories on laminar, laminar to turbulent transition andturbulent fluid flow in macro-channels (conventional channels) are summarized. Next, a briefreview of the available studies on single-phase incompressible fluid flow in mini- and micro-channels is presented. Some experimental results on single phase laminar, laminar to turbulenttransition and turbulent flows are presented. The deviations from the conventional frictionfactor correlations for single-phase incompressible fluid flow in mini and micro-channels arediscussed. The effect factors on mini- and micro-channel single-phase fluid flow are analyzed.Especially, the surface roughness effect is focused on. According to this review, the futureresearch needs have been identified. So far, no systematic agreed knowledge of single-phasefluid flow in mini- and micro-channels has yet been achieved. Therefore, efforts should bemade to contribute to systematic theories for microscale fluid flow through very carefulexperiments.

Keywords: Single-phase flow; Mini-channel; Micro-channel; Incompressible fluid flow;Pressure drop; Friction factor; Surface roughness.

1. Introduction

Miniaturization has recently become the key word in many advanced technologies as wellas traditional industries. The miniaturized systems are being progressively applied incommercial sectors such as the electronics, chemical, pharmaceutical, and medical industries, * E-mail address: [email protected]

Lixin Cheng344

as well as in the military sector, for biological and chemical warfare defense, as an example.With the emergence of micro-scale thermal, fluidic, and chemical systems, the developmentof ultra-compact heat exchangers, miniature and micro pumps, miniature compressors, micro-turbines, micro-reactors, micro thermal systems for distributed power production has becomean important agenda of many researchers, research institutions, and funding agencies.Miniaturization is essential not simply from the standpoint of producing compact systems, butfor the challenge of fluid flow with yet smaller and smaller channel sizes, as is the case forcooling of electronic devices, microfluidic components and sensors. In addition, it has beenshown that proper miniaturization by use of mini- and micro-channels can result in highersystem efficiency. However, fluid flow phenomena in mini- and micro-channels are quitedifferent from those in conventional size channels. Therefore, it is of great significance tounderstand these fundamental aspects.

Due to the significant differences of fluid flow phenomena in mini- and micro-channelsas compared to conventional channels or macro-scale channels, one very important issueshould be clarified about the distinction between mini- and micro-scale channels and macro-scale channels. However, a universal agreement is not yet clearly established in the literature.Furthermore, although research in mini- and micro-channels has been greatly increasing dueto the rapid growth applications which require fluid flows in the tiny channels, the availablestudies reveal contradictory results in single-phase friction factors, the transition for laminarand turbulent flows and the effect factors. There are still significant discrepancies between theexperimental results obtained by different researchers.

At first, one very important issue should be the clarification of the distinction betweenmicro-scale channels and macro-scale channels when studying fluid flow in mini- and micro-channels. However, a universal agreement is not yet clearly established in the literature.Instead, there are various definitions on this issue [1-6]. Here, just to show one example,based on engineering practice and application areas such as refrigeration industry in the smalltonnage units, compact evaporators employed in automotive, aerospace, air separation andcryogenic industries, cooling elements in the field of microelectronics and micro-electro-mechanical-systems (MEMS), Kandlikar et al. [6] defined the following ranges of hydraulicdiameters dh which are attributed to different classifications:

Conventional channels: dh > 3 mm.Mini-channels: 200 μ m < dh ≤ 3 mm.Micro-channels: 10 μ m < dh ≤ 200μ m.Transitional micro-channels: 1 μ m < dh ≤ 10μ m.Transitional nano-hannels: 0.1 μ m < dh ≤ 1μ m.Nano-channels: dh ≤ 0.1 μ m.

In the case of non-circular channels, it is recommended that the minimum channeldimension, for example, the short side of a rectangular cross-section should be used in placeof the diameter d.

In the available studies on fluid flow in mini- and micro-channels, some researchers haveconcluded that the conventional theories work for mini- and micro-channels while others haveshowed that the conventional theories do not work well. Therefore, these controversies shouldbe clarified. As the channel size becomes smaller, some of the conventional theories for(bulk) fluid, energy and mass transport need to be revisited for validation. There are two

Single-Phase Incompressible Fluid Flo in Mini- and Micro-channels 345

fundamental elements responsible for departure from the conventional theories at mini- andmicro-scale. For example, differences in modeling fluid flow in mini- and micro-channelsmay arise as a result of (i) a change in the fundamental process, such as a deviation from thecontinuum assumption for fluid flow, or an increase influence of some additional forces, suchas electrokinetic forces, etc. (ii) uncertainty regarding the applicability of empirical factorsderived from experiments conducted at larger scales, such as entrance and exit losscoefficients for fluid flow in pipes, etc., (iii) uncertainty in measurements at mini- and micro-scale, including geometrical dimensions and operating parameters.

In this chapter, the available studies of single-phase incompressible fluid flow in mini-and micro-channels are reviewed to address these relevant important issues.

2. Single-Phase Frictional Pressure Drop Methodsin Macro-channels

2.1. Laminar and Turbulent Flow

The flow of a fluid in a pipe may be laminar or turbulent flow, or in between transitionalflow. Figure 1 shows the x component of the fluid velocity as a function of time t at a point A inthe flow for laminar, transitional and turbulent flows in a pipe. For laminar flow, there is onlyone component of velocity. For turbulent flow, the predominant component of the velocity isalso along the pipe, but it is unsteady (random) and accompanied by random componentsnormal to the pipe axis. For transitional flow, both laminar and turbulent features occur.

VA

AQ

t

Turbulent

Transitional

Laminar

x

Figure 1. Time dependence of fluid velocity at a point.

Lixin Cheng346

For pipe flow, the most important dimensionless parameter is the Reynolds number Re,the ratio of the inertia to viscous effects in the flow, which is defined as

Re Vdρμ

= (1)

where V is the average velocity in the pipe. The flow in a pipe is laminar, transitional orturbulent provided the Reynolds number Re is small enough, intermediate or large enough. Itshould be pointed out here that the Reynolds number ranges for which laminar, transitional orturbulent pipe flows are obtained cannot be precisely given. The actual transition fromlaminar to turbulent pipe flow may take place at various Reynolds numbers, depending onhow much the flow is distributed by vibrations of the pipe, roughness of the entrance region,and the like. For general engineering purposes (i.e. without undue precautions to eliminatesuch disturbances), the following values are appropriate: the flow in a round pipe is laminar ifthe Reynolds number Re is less than approximately 2100 and the flow in is turbulent if theReynolds number Re is greater than approximately 4000. For the Reynolds numbers betweenthese two limits, the flow may switch between laminar and turbulent conditions in anapparently random fashion (transitional flow).

2.2. Entrance Region (Developing Flow) and Fully Developed Flow

The region of flow near where the fluid enters a pipe is termed the entrance region(developing flow) and is illustrated in Figure 2. The fluid typically enters the pipe with anearly uniform velocity profile at section 1. As the fluid moves through the pipe, viscouseffects cause it to stick to the pipe wall (the no-slip boundary condition). Thus, a boundarylayer in which viscous effects are important is produced along the pipe wall such that theinitial velocity profile changes with distance along the pipe, x, until the fluid reaches the endof the entrance length, section 2, beyond which the velocity profile does not vary with x. Theboundary layer has grown in thickness to completely fill the pipe. Viscous effects are ofconsiderable importance within the boundary layer. For fluid outside the boundary layer(within the inviscid core surrounding the centerline from 1 to 2), viscous effects arenegligible.

The shape of the velocity profile in the pipe depends on whether the flow is laminar orturbulent, as does the length of the entrance region, le. As with many other properties of pipeflow, the dimensionless entrance length le/d, correlates quite well with the Reynolds numberRe. Typical entrance lengths are given by

0.06 Reeld= for laminar flow (2)

1/64.4Reeld= for turbulent flow (3)

Calculation of the velocity profile and pressure distribution within the entrance region(developing flow) is quite complex. However, once the fluid reaches the end of the entranceregion, section 2 in Figure 2, the flow is simpler to describe because the velocity is a function


of only the distance from the pipe centerline, r, and independent of x. The flow after section 2is termed fully developed flow.

Entrance region flow le Fully developed flow

Inviscid core

Boundary layer1 2

x

d

rV

Figure 2. Entrance region, developing flow and fully developed flow in a pipe.

2.3. Single-Phase Friction Pressure Drop Methods

The nature of the pipe flow is strongly dependent on weather the flow is laminar orturbulent. This is a direct consequence of the differences in the nature of the shear stress inlaminar and turbulent flows. The shear tress in laminar flow is a direct result of momentumtransfer among the randomly moving molecules (a microscopic phenomenon). The shearstress in turbulent flow is largely a result of momentum transfer among the randomly moving,finite-sized bundles of fluid particles (a macroscopic phenomenon). The net result is that thephysical properties of the shear stress are quite different for laminar than for turbulent flow.The friction pressure drop for both laminar and turbulent flow can be expressed as

2

2l Vp fdρ

Δ = (4)

For fully developed laminar flow in a circular pipe, the friction factor f is simply as

64Re

f = (5)

and the value of f is independent of the relative roughness dε .

For turbulent flow, the dependence of the friction factor on the Reynolds number Re ismuch more complex than that given by Eq. (5) for laminar flow. For fully developedturbulent flow and transition from laminar to turbulent flow. The Moody chart [7] shown inFigure 3 provides the friction factor f, which can be expressed as

Re,fdεφ ⎛ ⎞= ⎜ ⎟

⎝ ⎠(6)

Figure 3. Friction factor as a function of the Reynolds number and relative roughness for circular pipes-the Moody chart [7].


For turbulent flow, the friction factor f is a function of the Reynolds number Re and

relative roughness dε . The results are obtained from an exhaustive set of experiments and

usually presented in terms of a curve-fitting formulae or the equivalent graphical form.In commercially available pipes, the roughness is not as uniform and well defined as in

the artificially roughened pipes. However, it is possible to obtain a measure of the effectiverelative roughness of typical pipes and thus to obtain the friction factor. Typical roughnessvalues for various pipe surfaces are also given in Figure 3. It should be noted that the values

of dε do not necessarily correspond to the actual values obtained by a microscopic

determination of the average height of the roughness of the surface. They do, however,provide the correct correlation for friction factor f. The following characteristics are observedfrom Figure 3 for laminar flow, friction factor is calculated by Eq. (2), which is independentof relative roughness. For very large Reynolds number Re, the friction factor is a function of

the relative roughness dε as

fdεφ ⎛ ⎞= ⎜ ⎟

⎝ ⎠(7)

which is independent of the Reynolds number Re. For such flows, commonly termedcompletely turbulent flow (or wholly turbulent flow), the laminar sub-layer is so thin (itsthickness decreases with increasing the Reynolds number Re) that the surface roughnesscompletely dominates the character of the flow near the wall. Hence, the pressure droprequired is a result of an inertia-dominated turbulent shear stress rather than the viscosity-dominated laminar shear stress normally found in the viscous sublayer.

For flows with moderate values of Re, the friction factor is indeed dependent on both the

Reynolds number Re ad relative roughness dε . Flow in the range of 2100< Re <4000 is a

result of the fact that the flow in this transition range may be laminar or turbulent (or anunsteady mix of both) depending on the specific circumstances involved.

Note that even for smooth pipes (ε = 0), the friction factor is not zero. That is, there is ahead loss in any pipe, no mater how smooth the surface is made. This is a result of the no-slipboundary condition that requires any fluid to stick to any solid surface it flows over. There isalways some microscopic surface roughness that produces the no-slip behavior (and thus thefriction factor f does not equal 0) on the molecular level, even when the roughness isconsiderably less that the viscous sublayer thickness. Such pipes are called hydraulicallysmooth, and for turbulent flow in this flow, the Blasius [8] equation is used for friction factoras

0.25

0.3164Re

f = (8)

where 4000< R e < 100,000.

Lixin Cheng350

The Moody chart shown in Figure 3 is universally valid for all steady fully developedincompressible pipe flows. The following equation from the Colebrook and White [9] is validfor the entire non-laminar range of the Moody chart, as

1 / 2.512log3.7 Re

df f

ε⎛ ⎞= − +⎜ ⎟⎜ ⎟

⎝ ⎠(9)

In fact, the Moody chart is a graphical representation of this equation, which is anempirical fit of the pipe flow pressure drop data. A difficulty with its use is that it is implicit

in the dependence of f. that is, for given conditions (Re and dε ), it is not possible to solve f

without some sort of iterative scheme.For easy-to-use, Swamee and Jain [10] proposed an explicit Colebrook-White equation as

follows:

2

0.9

0.25

/ 5.74log3.7 Re

fdε

=⎡ ⎤⎛ ⎞+⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

(10)

which matches the Colebrook-White equation within 1% for 10-6 < ε/d < 10-2 and 5000 < Re< 108.

Churchill [11] proposed a more complicated expression for all flow regimes and all therelative roughnesses, which agrees well with the Moody diagram:

( )

1/121.51612 16

0.9

8 1 375308 2.457 lnRe Re7 / Re 2.7 /

fdε

−⎧ ⎫⎡ ⎤⎛ ⎞⎪ ⎪⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟= + +⎨ ⎬⎜ ⎟ ⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠+⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭

(11)

For noncircular ducts: the hydraulic diameter dh is used. The hydraulic diameter of a non-circular enclosure is four times the cross sectional area divided by the wetted perimeter. Thewetted perimeter is the edge of the cross sectional area that is in direct contact with the flowmedium, as

4h

wetted

AdP

= (12)

Given the complexities of viscous sublayers, turbulence shear stress and surfaceroughness, etc., the use of the hydraulic diameter method introduces some limitations. Forone, the use of the hydraulic diameter dh is only valid if the ration of the duct’s height towidth is less than about 3 or 4 [8]. Use the hydraulic diameter method to estimate head loss inlaminar flow can also introduce large errors. This is due to friction from the action ofviscosity throughout the whole body of the flow. It is independent of surface roughness and isnot associated with the region close to the boundary walls. Although the details of the flows


in non-circular conduits depend on the exact cross-sectional shape, many round pipe resultscan be carried over, with slight modification, to flow in conduits of other shapes.

Practical, easy-to-use results can be obtained as follows: regardless of the cross-sectionalshape, there are no inertia effects in fully developed laminar pipe flow. Thus, the frictionfactor can be written as

Reh

Cf = (13)

where the constant C is the Poiseuille number which depends on the particular shape of theduct, and Reh is the Reynolds number based on the hydraulic diameter dh. The hydraulicdiameter is also used in the definition of the friction factor and the relative roughness. Typicalvalues of C for concentric annulus and rectangle channels are given in Table 1 along with thehydraulic diameter. Note that the value of C is relatively insensitive to the shape of theconduit. Unless the cross section is very “thin” in some sense, the value of C is not toodifferent from its circular pipe value, C = 64 as shown in Eq. (5). Shah and London [12]provide the following correlation for a rectangular channel with short side a and long side b,and a channel aspect ratio ac = a/b.

( )2 3 4 5Re 24 1 1.3553 1.9467 1.7012 0.9564 0.2537c c c c cC f a a a a a= = − + − + − (14)

Table 1. Friction factors for laminar flow in noncircular ducts [8]

Shape Hydraulicdiameter Parameter RehC f=

d1/d2

0.0001 71.80.01 80.10.1 89.40.6 95.6

Concentric annulus

2 1hd d d= −

1.0 96

a/b0 96

0.05 89.90.10 84.70.25 72.90.50 62.20.75 57.9

Rectangle

a

b

2h

abda b

=+

1.00 56.9

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Once the friction factor is obtained, the calculations for noncircular conduits are identicalto those for circular pipes.

Calculations for fully developed turbulent flow in ducts of noncircular cross section areusually carried out by the Moody chart data for circular pipes with the diameter replaced bythe hydraulic diameter and the Reynolds number based on the hydraulic diameter. Suchcalculations are usually accurate to within about 15%.

3. Single-Phase Frictional Pressure Drop Methodsin Mini- and Micro-Channels

To explore the fundamental physical mechanisms of fluid flow in mini- and micro-channels, many effects, including the size effect, the surface roughness effect, viscous effect,electrostatic force effect in the channel wall, surface geometry and the measurement errors,etc. should be taken into account. The conventional theories for macro-channels are thefundamental to investigation of single-phase fluid flow in mini- and micro-channels. So far, anumber of studies have been performed to study the hydrodynamic characteristics ofincompressible fluid flow in mini- and micro-channels. However, divergences of conclusionsstill exist in quite a few fundamental understandings of the microscale fluid flow phenomena.In this section, some recent studies of incompressible fluid flow and in mini- and micro-channels are reviewed and the main conclusions from these studies are summarized.

There is big contradictory regarding the applicability of the conventional theories tomicroscale fluid flow phenomena. Several good reviews on fluid flow in mini- and micro-channels were presented by Kandliar [6], Celata et al. [13] Guo and Li [14], Sobhan andGarimella [15] and Morini [16]. There are large deviations among the available studies in theliterature. Little agreement among the experimental results of single-phase pressure drops inthe laminar and turbulent flow regimes by different researchers has been reached. Althoughsome experimental results agree with the conventional theories, most of the experimentalresults deviate from the conventional theories, with both under- and over-predictionsobtained. Especially, the experimental results also show different trends of variation relativeto the conventional predictions in the laminar and turbulent flow regimes. In addition, quitedifferent results on the transition from laminar to turbulent regimes have been obtained bydifferent investigators. Some reported that the transition occurred at lower Reynolds numberswhile others reported that the transition agreed with the conventional theory. In general, thefundamental aspects of fluid flow in mini- and micro-channels are not well understood so far.Here only some representative studies in the literature are reviewed below to show thesedifferences.

Mala and Li [17] conducted experiments of water flowing in micro-channels withdiameters from 50 to 254 μm. Their length-to-diameter ratio was from 1200 to 5000 and thematerials were fused silica and stainless steel. Their experimental results showed significanthigher values than those predicted by the conventional theory. The difference increased as themicrotube diameter decreased. The effect is caused by the surface roughness or an earlytransition from laminar to turbulent flow regime. They found an early transition from laminarto turbulent flow in the Reynolds number from 300 to 900, while the flow changed to fullydeveloped turbulent flow at the Reynolds number larger than 1000 to 1500.


Yu et al. [18] conducted experiments on nitrogen gas and water flowing in micro-channels with diameter 19, 52 and 102 μm. The test Reynolds number ranged from 250 to20,000. Their experimental results showed that the friction factor was lower than theconventional theory for fully developed laminar flow while the transition from laminar toturbulent flow occurred in the Reynolds number range 2000 to 6000.

Xu et al. [19] conducted experiments on water flowing in micro-channels with hydraulicdiameter from 50 to 300 um and their test Reynolds number was from 50 to 1500. Theirexperimental results showed that the transition to from laminar to turbulent flow region didnot occur in their test Reynolds number range. They found that the flow characteristicsdeviated from conventional theory when the channel dimensions were below 100 μm. Theirfriction factor was smaller than the predicted results by the conventional theory. Later on, Xuet al. [20] reported a similar study of water flow in micro-channels with hydraulic diameterfrom 30 to 344 and the Reynolds number from 20 to 4000. Their experimental results showedthat the experimental friction factors agreed with the conventional theories. Although thechannel sizes were similar using the same test fluid, their experimental results are quitedifferent.

Judy [21] conducted experiments of water, hexane and isopropanol flowing in fusedsilica capillary tubes with diameters from 20 to 150 μm. Their experimental results showedthat when the tube diameter was lower than 100 μm, the friction factor deviated significantlyfrom the conventional theory. This is in consistence with that of Xu et al. [19]. The deviationis independent of the Reynolds number and depended on the tube diameter. However, lateron, they did very accurate experiments on fluid flow in micro-channels with diameters from15 to 150 μm with the same fluids and the similar test conditions used in their previous study[22]. Their experimental results showed that the friction factors were in good agreement withthe conventional theory. The value of the Poiseuille number C was found close to 64,considering the experimental error and it was independent of the Reynolds number for Re <2000.

Gao et al. [23] conducted experiments on water flowing in micro-channels with diametersfrom 200 to 1923 μm. Their experimental results were in good agreement with theconventional theories. Figure 4 shows their results on the influence of the Reynolds numberand channel height on the Poiseuille number. Both the Shah and London [12] and the Balsius[8] correlations suitably predicted their experimental data for laminar and turbulent regimes.No scale effects were found in their experiments for flow hydrodynamics. Similarconclusions were obtained by Caney et al. [24], Grimella and Singal [25], Qi et al. [26], Leeand Lee [27], Warrier et al. [28] and Qu and Mudawar [29] in their single-phase frictionpressure drop experiments in microchannels.

Celata et al. [30] conducted experiments of R114 liquid flowing in 6 parallel stainlesssteel microtubes with a diameter of 130 μm and a tube length of 90 mm for both laminar andturbulent flows (Reynolds number from 100 to 8000). The relative roughness of the test tubeswas 2.65%. Their experimental results showed that the single-phase pressure drop in laminarflow agreed with the conventional theory. The laminar to turbulent flow transition was in theReynolds number range of 1880 to 2480. They noted that the high relative roughness playedan important role in the laminar to turbulent transition. Figure 5 shows their test results offriction factor in both laminar, transitional and turbulent flows compared to the Blasius

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equation and the Colebrook correlation. The Blasuis correlation underpredicts their data whilethe Colebrook correlation overpredicted their data.

Figure 4. Influence of Reynolds number and channel height on Poiseuille number: + e = 1 mm, × e =0.7 mm, e = 0.5 mm, e = 0.4 mm, e = 0.3 mm, ◊ e = 0.2 mm, e = 0.1 mm, – – – Blasius law[23].

Figure 5. Comparison of experimental friction factors to the Colebrook and the Blasius correlations[30].


Wibel and Ehrhard [31] conducted experiments on the laminar to turbulent transition ofdeionized water flowing in rectangular stainless steel microchannels with a hydraulicdiameter of about 133 μm. Three aspect ratios of 1:1, 1:2 and 1:5 were studied. Their resultsshowed that the laminar to turbulent transition was in the Reynolds number range of 1900-2200, which agreed with the conventional theory.

Li et al. [32] conducted experiments on single-phase friction pressure drops of deionizedwater flowing in smooth fused silica microtubes and rough stainless steel microtubes withdiameters of 50 to 100 μm and 373 to 1570 μm, respectively. Their test Reynolds number wasfrom 20 to 2400. For the stainless steel tubes, the surface relative roughnesses of 2.4%, 1.4%and 0.95% were tested. Figure 6 shows their experimental friction factors for the fused silicatubes compared to the conventional theory and Figure 7 shows those for stainless steel tubes.Their experimental friction factors were well predicted by the conventional theory for thesmooth silica tubes while for the roughness stainless steel tubes, their experimental frictionfactors were higher than the predictions by the conventional theory and increased with therelative roughness. For deionized water flowing through stainless steel micro-tubes with arelative surface roughness less than about 1.5%, the friction factors also agreed with theconventional theory. When the relative surface roughness was larger than 1.5%, theexperimental friction factor of the microtube flow showed an appreciable deviation from theconventional theory.

Figure 6. Comparison of experimental friction factors to the conventional theory for the fused silicatubes [32].

In general, the available experimental results on microscale fluid flow in the literaturehave demonstrated that there are big disagreements among the studies on the single phasefriction pressure drops in terms of the friction factors for laminar and turbulent flows and thelaminar to turbulent flow transition. It seems that conventional theory works for relative largechannels while it does not work well for relative smaller channels. So far, no systematicagreed theories on microscale single-phase fluid flows have yet been achieved. Quite big

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contradictory exists in the available studies. Some researchers reported that the frictionfactors for laminar fully developed flow were lower than the conventional theory, somereported that the friction factors for laminar fully developed flow were higher than theconventional values while some reported that the friction factor could be predicted by theconventional theory. Some reported that the Poiseuille number for laminar fully developedflow depends on the Reynolds number while some reported that it did not. In fact, the frictionfactor depends on the material of the micro-channel walls (metals, semi-conductors and soon) and the test fluid (polar fluid or not), thus evidencing the importance of electro-osmoticphenomena to microscales. Furthermore, the dependence of the friction factor on the relativeroughness of the micro-channel wall also exits in laminar flow regime. Regarding thelaminar-to-turbulent flow transition in microchannels, no agreement on the transitionalReynolds number has been reached so far.

Figure 7. Comparison of experimental friction factors to the conventional theory for the stainless steeltubes [32].

It should be pointed out that no study on the entrance length (developing flow) in mini-and micro-channels is available in the literature so far. This poses a question: whether thecorrelation for entrance length in macro-channels can be used for mini- and micro-channels?No doubt, it is very important to identify the developing and the fully developed flowcorrectly in mini- and micro-channels. Simply using the criteria for macrochannels mightresults in very big errors in the microscale fluid flow research.


4. Channel Size Effect on Single Phase Incompressible Fluid Flowin Mini- and Micro-channels

The effects of the channel size are of significance in fluid flow. These can be the effect ofexperimental uncertainties, the effect of surface roughness and the like. Furthermore, theeffect of surface electrostatic charges may also be significant but beyond the scope of thisreview. The accurate measurements become more difficult in mini- and micro-channels thanin macrochannels. For example, how can we accurately measure the very small flow rate inmini- and micro-channels? For example, Celeta et al. [30] used 6 parallel stainless steelmicrotubes in their experiments as shown by their test section in Figure 8. This test sectionprovided an easy method to measure the flow rate in the microtubes. However, it posesanother question: how could the flow rate be uniformly distributed in each sub-microtube? Infact, the accuracy of the flow rate measurements can greatly affect the pressure drop results.Furthermore, for pressure drop measurements, how could we implement more accuratemeasurements for the mini- and micro-channels? All these issues should be carefullyconsidered in the experimental facility and test section design [33].

Figure 8. Test section consisting of 6 parallel micro-tubes in the experimental study of Celeta et al.[30].

Careful experimental operation should also be performed for experiments of fluid flow inmini- and micro-channels. As already mentioned in section 3, the studies for the similar testconditions with the same fluids in [19] and [20] showed quite different results. The same case

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is for the studies in [21] and [22]. These two examples have shown the importance of verycareful experimental operations in the study of fluid flow in mini- and micro-channels. Toperform accurate measurements, very careful test facility design and experimental operationare necessary. Steinke et al. [33] developed an experimental facility for the investigation ofsingle-phase liquid flow in microchannels with a variety of geometries considering themicroscale effects. Furthermore, careful experimental uncertainties should be analyzed [13,34].

The channel geometry evaluation is also very important because the channel dimensionshave a major effect on the friction factor calculation. Figure 9 shows the SEM image of therectangular micro-channel geometry tested by Steinke and Kandlikar [34]. It can be seen howa rectangle channel looks like in microscope, which indicates that the channel shape actuallybecomes significance for microchannels. Figures 10 and 11 show the SEM images of thecircular microchannel geometries measured by Li et al. [32]. Obviously, a very little error inthe cross-section area measurement may cause very big errors in the flow rate calculation andthus in the friction factor calculation.

Figure 9. Actual cross section of the tested rectangular microchannel by Steinke and Kandlikar [34]: a =194 μm, b = 244 μm, L = 10 μm, θ= 85 degrees and dh = 227μm.

The surface roughness plays a very important role in the study of microscale fluid flow[6, 30-32, 35-38]. For fluid flow in rough microchannels, a number of the available studiesconsidering the surface relative roughness effect have shown that the friction factors inmicrochannels are higher than those in macro-channels, and the surface roughness also leadsto the earlier transition from laminar to turbulent. However, some studies have shown that the


dh = 1570 μmdh = 624.4μmdh = 373 μm

Figure 10. The SEM images of cross-section and inner surface of three stainless steel test tubes in theexperimental study of Li et al. [32].

Cross section[

Surface roughness

Figure 11. The SEM images of cross-section and inner surface of a fused silica test tube in theexperimental study of Li et al. [32].

relative roughness does not affect the friction factor. Thus, one issue is still open to discuss aswhat is the relative roughness limitation below or beyond which the channel can be regardedas smooth or rough. For example, to investigate the role of the surface relative roughnesseffect on pressure drop characteristics in microtubes, Kandlikar et al. [35] conductedexperiments of water flowing in two microtubes with diameters of 1067 and 620 μm. For the1067 μm diameter tube, the effects of the relative roughness from 0.178% to 0.3% on

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pressure drop are insignificant and the tube can be considered smooth. For the 620 μmdiameter tube with the same relative roughness values, the pressure drops are increased. Inthis case, the tube having a relative roughness 0.3% cannot be considered as smooth.Although several studies on the relative surface roughness effects are available [35-38], nogeneral agreed conclusions have been obtained so far. Therefore, it is worthwhile tounderstand under what condition the surface relative roughness effect can be ignored orcannot be ignored for the fluid flow in mini- and micro-channels.

5. Conclusion

The conventional fluid dynamic theories developed for macro-channels are not alwaysapplicable to fluid flow in mini- and micro-channels. The available studies on microscalefluid flow in the literature reveal quite a few contradictory and there are still quite bigdiscrepancies among the experimental results by different researchers. Various causes forsuch deviations are analyzed in this review. Furthermore, the effects of channel size onmicroscale fluid flow are analyzed. The surface roughness is a very important factor but stillnot well investigated. So far, no systematic agreed knowledge of microscale fluid flow has yetbeen achieved. Therefore, efforts should be made to achieve complete theories on fluid flowin mini- and micro-channels.

Nomenclature

A cross-sectional area of flow channel, m2

a length of rectangle, mac channel aspect ratiob width of rectangle, mC constant in Eq. (13)d internal tube diameter, me thickness of channel base, mf friction factorl length, mPwetted wetted perimeter, mRe Reynolds numberr radical directionV mean velocity, m/sx x-axis

Greek symbols

Δp pressure drop, Paε surface roughness, mμ dynamic viscosity, Ns/m2


ρ density, kg/m3

surface tension, N/m

Subscripts

crit criticale entranceh hydraulic

References

[1] S. G. Kandlikar, Fundamental issues related to flow boiling in minichannels andmicrochannels, Exp. Therm. Fluid Sci., 26 (2002) 389-407.

[2] L. Cheng, and D. Mewes, Review of two-phase flow and flow boiling of mixtures insmall and mini channels, Int. J. Multiphase Flow, 32 (2006) 183-207.

[3] P.A. Kew, and K. Cornwell, Correlations for the prediction of boiling heat transfer insmall-diameter channels, Appl. Therm. Eng., 17 (1997) 705-715.

[4] S.S. Mehendale, A.M. Jacobi, and R.K. Ahah, Fluid flow and heat transfer at micro- andmeso-scales with application to heat exchanger design, ASME Appl. Mech. Rev., 53(2000) 175-193.

[5] L. Cheng, G. Ribatski, and J.R. Thome, Gas-liquid two-phase flow patterns and flowpattern maps: fundamentals and applications, ASME Appl. Mech. Rev., 61 (2008)050802-1-050802-28.

[6] S.G. Kandlikar, S. Garimella, D. Li, S. Colin, and R. King Michael, Heat Transfer andFluid Flow in Mini-channels and Micro-channels, Elsevier Science & Technology, UK,2005.

[7] N.E. Moody, Friction factors for pipe flow, Trans. ASME, (1944) 671-684.[8] R.M. Olson, Essentials of Engineering Fluid Mechanics, 4th Ed., Harper & Row, New

York, 1980.[9] C.F. Colebrook, and C.M. White, Experiments with fluid friction in roughened pipes,

Proc Roy Soc. (A), 161 (1937) 367.[10] P.K. Swamee, and A.K. Jain, Explicit equations for pipe-flow problems, J. Hydraulic

Div., 102 (1976) 657-664.[11] S.W. Churchill, Friction factor equation spans all fluid flow regimes, Chemical Eng., 7

(1977) 91-92.[12] R.K. Shah, and A.L. London, Laminar Flow Forced Convection in Ducts, Academic

Press Inc., New York, 1978.[13] G.P. Celata, Heat Transfer and Fluid Flow in Microchannels, Begell house Inc, New

York, 2004.[14] Z.Y. Guo and Z.X. Li, size effect on single-phhase channel flow and heat transfer at

microscale, Int. J. Heat Fluid Flow, 24 (2003) 284-298.[15] C.B. Sobhan, and S.V.A. Garimella, Comparative analysis of studies on heat transfer

and fluid flow in microchannels, Microscale Thermophysical Eng., 5 (2001), 293-311.

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[16] G.L. Morini, Single-phase convective heat transfer in microchannels: a review ofexperimental results, Int. J. Therm. Sci. 43 (2004) 631-651.

[17] G.M. Mala, and D. Li,, Flow characteristics of water in microtubes, Int. J. Heat fluidFlow, 20 (1999) 142-148.

[18] D. Yu, R. Warrington, R. Barron, and T. Ameel, An experimental and theoreticalinvestigation of fluid flow and heat transfer in microtubes, ASME/JSME ThermalEngineering Conference, Vol. 1, ASME, 1995.

[19] B. Xu, K.T. Ooi, N.T. Wong, C.Y. Li, and W.K. Choi, Liquid flow in microchannels,Proceedings of the 5the ASME/JSME Joint Thermal Engineering Conference, SanDiego, California, March 15-19, 1999.

[20] B. Xu, K.T. Ooi, N.T. Wong, and W.K. Choi, Experimental method to calculate thefriction factor for liquid flow in microchannels, Int. Comm. Heat Mass Transfer, 27(2000) 1165-1176.

[21] J. Judy, D. Maynes, and B.W. Webb, Liquid flow pressure drop in microtubes,Proceedings of the International Conference on Heat Transfer and Transport Phenomenain Microscale, Banff, Canada, October 15-20, 2000.

[22] J. Judy, D. Maynes, and B.W. Webb, Characterization of frictional pressure drop forliquid flows through microchannels, Int. J. Heat Mass Transfer, 45 (2002) 3477-3489.

[23] P. Gao, S. L. Person, and M. Favre-Marinet, Scale effects on hydrodynamics and heattransfer in two-dimensional mini and microchannels, Int. J. Therm. Sci., 41 (2002) 1017-1027.

[24] N. Caney, P. Marty, and J. Bigot, Friction losses and heat transfer of ingle-phase flow ina mini-channel, Appl. Therm. Eng., 27 (2007) 1715-1721.

[25] S.V. Garimella, and V. Singhal, Single-phase flow and heat transport and pumpingconsiderations in micro heat sinks, Heat Transfer Eng., 25(1) (2004) 15-25.

[26] S.L. Qi, P. Zhang, R.Z. Wang, and L.X. Xu, Single-phase pressure drop and heattransfer characteristics of turbulent liquid nitrogen flow in micro-tubes, Int. J. Heat MassTransfer, 50 (2007) 1993-2001.

[27] H.J. Lee, and S.Y. Lee, Heat transfer correlation for boiling flows in small rectangularhorizontal channels with low aspect ratios, Int. J. Multiphase Flow, 27 (2001) 2043-2062.

[28] G.R. Warrier, V.K. Dhir, and L.A. Momoda, Heat transfer and pressure drop in narrowrectangular channels, Exp. Therm. Fluid Sci., 26 (2002) 53-64.

[29] W. Qu, I. Mudawar, experimental and numerical study of pressure drop and heat transferin a single-phase micro-channel heat sink, Int. J. Heat Mass Transfer, 45 (2002) 2549-2565.

[30] G.P. Celata, M. Cumo, M. Guglielmi, and G. Zummo, Experimental investigation ofhydraulics and single phase heat transfer in 0.130 mm capillary tube, MicroscaleThermophysical Eng., 6 (2002) 85-97.

[31] W. Wibel, and P. Ehrhard, Experiments on the laminar/turbulent transition of liquidflows in rectangular micochannels, Heat Transfer Eng., 30(1-2) (2009) 70-77.

[32] Z. Li, Y.L. He, G.H. Tang and W.Q Tao, Experimental and numerical studies of liquidflow and heat transfer in microtubes, Int. J. Heat Mass Transfer, 50 (2007) 3447-3460.

[33] M.E. Steinke, S.G. Kandlikar, J.H. Magerlein, E.G. Colgan, and A.D. Raisanen,Development of an experimental facility for investigating single-phase liquid flow inmicrochannels, Heat Transfer Eng., 27(4) 2006 41-52.


[34] M.E. Steinke, and S.G. Kandlikar, Single-phase liquid friction factors in microchannels,Int. J. Therm. Sci., 45 (2006) 1073-1083.

[35] S.G. Kandlikar, D. Joshi, and S. Tian, Effect of channel roughness on heat transfer andfluid flow characteristics at low Reynolds numbers in small diameter tubes, in:Proceedings of 35th National Heat Transfer Conference, Anaheim, CA, USA, 2001,Paper: 12134.

[36] D. Gloss and H. Herwig, Microchanel roughness effects: a close up view, Heat TransferEng., 30(1-2) (2009) 62-69.

[37] P. Young, T.P. Brackbill, and S.G. Kandlikar, Comparison of roughness parameters forvarious microchannels surface in single-phase flow applications, Heat Transfer Eng.,30(1-2) (2009) 78-690.

[38] Z.X. Li, D.X. Du, and Z.Y. Guo, Experimental study on flow characteristics of liquid incircular microtubes, Microscale Thermophysical Eng. 7, (2003) 253-265.


Chapter 11

EXPERIMENTAL STUDY OF PULSATING TURBULENTFLOW THROUGH A DIVERGENT TUBE

Masaru Sumida*

School of Engineering, Kinki University1 Takaya Umenobe, Higashi−Hiroshima, 739−2116 JAPAN

Abstract

An experimental investigation was conducted of pulsating turbulent flow in a conicallydivergent tube with a total divergence angle of 12°. The experiments were carried out underthe conditions of Womersley numbers of α =10∼40, mean Reynolds number of Reta =20000and oscillatory Reynolds number of Reos =10000 (the flow rate ratio of η = 0.5). Time-dependent wall static pressure and axial velocity were measured at several longitudinalstations and the distributions were illustrated for representative phases within one cycle. Therise between the pressures at the inlet and the exit of the divergent tube does not become toolarge when the flow rate increases, it being moderately high in the decelerative phase. Theprofiles of the phase-averaged velocity and the turbulence intensity in the cross section arevery different from those for steady flow. Also, they show complex changes along the tubeaxis in both the accelerative and decelerative phases.

Keywords: divergent tube, pulsating flow, turbulent flow, pressure distribution, velocitydistribution.

1. Introduction

Divergent tubes are an important pipeline component and are widely used as diffusers toconvert kinetic energy into pressure energy and as devices to connect two tubes of differentdiameters in pipework systems and fluid machinery. Therefore, there have been a number ofstudies, e.g., Singh and Azad [1, 2], Gan and Riffat [3], and Xu et al. [4], devoted to the flowin divergent tubes to date. That is, much attention has been given to the tube geometry, which

* E-mail address: [email protected]. Tel: +81−82−434−7000, Fax: +81−82−434−7011

Masaru Sumida366

affects pressure recovery efficiency and losses; the relationship between performance and theoptimum geometry has been investigated. The flow in divergent tubes is highly unstable andthen, recently, computational fluid dynamics has been applied to the prediction of mean andturbulent flow characteristics (Gan and Riffat [3], and Xu et al. [4]). However, all thesestudies concerned steady flow.

On the other hand, unsteady flow has become important in connection with problemsconcerning the starting and stopping or undesirable accidents of pumps and blowers, becausefluid machines are becoming better and fluid transport is becoming diversified and faster.Nevertheless, there have been few studies on unsteady flow in divergent tubes. Mizuno andOhashi [5], and Mochizuki et al. [6] conducted experiments for a two-dimensional diffuser.The former group oscillated one plane, whereas the latter group used periodic flows of a wakegenerated by a cylinder into the diffuser entrance. Thus their studies were aimed at graspingthe flow features, including unsteady separation, and/or establishing a method of controllingthe flow. Unfortunately, the periodically volume-cycled, unsteady flow has never beentreated, to the author’s knowledge.

The purpose of the present study is to treat the problem of volume-cycled, pulsatingturbulent flow through a conical divergent tube with a total divergence angle of 2θ =12°.First, the distribution of wall static pressure is measured for the various pulsation frequencies.Subsequently, periodical changes of profiles, such as the phase-averaged velocity andturbulent intensity, are examined under a representative flow condition. Furthermore,knowledge of their characteristics is obtained through comparison with those in the case ofsteady flow.

2. Experimental Apparatus and Measurement Method

2.1. Experimental Apparatus

A schematic diagram of the experimental apparatus is shown in Figure 1. The workingfluid is air. The system consists of a pulsating-flow generator, a test tube and measuringdevices. Moreover, the pulsating-flow generator is composed of a steady flow and anoscillating flow. The steady flow, i.e., time-mean flow, was supplied, through a surge tank, bya suction blower, which ensured that the flow rate was independent of the superimposedoscillation and the length of the test tube. On the other hand, the volume-cycled oscillatingflow was introduced by a piston, with a diameter of 300 mm and a stroke length adjustablefrom 0 to 1000 mm, controlled by a personal computer.

The test tube had a total divergence angle 2θ of 12° and an area ratio m of 6.25. Here theratio m was (d2/d1)2, d1 (=2a1=80 mm) and d2 (=200 mm) being the diameters at the inlet andthe exit, respectively, of the divergent tube. These dimensions are shown in Figure 2, togetherwith the coordinate system. The divergent tube was constructed from 5 accurately machined,transparent acrylic blocks connected via a slip ring. The ring, furthermore, had static pressureholes, 0.8 mm in diameter and spaced 90° apart, and a small hole for inserting a hot-wireprobe. Straight transparent glass tubes with lengths of 3700 mm (=46.3d1) and 4200 mm(21d2) were connected to the inlet and the exit of the divergent tube, respectively.

Experimental Study of Pulsating Turbulent Flow through a Divergent Tube 367

Figure 1. Schematic diagram of experimental apparatus – 1) upstream tube, 2) divergent tube, 3)downstream tube, 4) oscillating flow generator, 5) surge tank, 6) blower.

Figure 2. Coordinate system and dimensions of test tube.

2.2. Measurement Procedure and Data Acquisition

The wall static pressure was measured, using a diffusive-type semiconductor pressuretransducer (Toyoda MFG, DD102-0.1F), at 11 stations between z = −22.1d1 in the upstreamstraight tube and z = 21.9d1 in the downstream one, where z is the length measured along thetube axis from the inlet of the divergent tube. Velocity measurements were performed with ahot-wire anemometer, and the velocity w in the axial direction was obtained at 8 stationsincluding sections of z/d1= -2 and 9.6.

The voltage output from the DC amplifier of the pressure transducer and from theanemometer was sampled, with the personal computer, synchronously with a time-markersignal that indicates the position of the piston. The data was processed as follows. For aperiodically unsteady turbulent flow, any instantaneous quantity, e.g., the axial velocity w inthe pulsation cycle, can be written as

w = W + wt . (1)

Masaru Sumida368

Here, W is the phase-averaged velocity and wt is the fluctuating one. The data at eachmeasuring point was collected, in equal time intervals, for 200 to 500 pulsation cycles. Therecorded data were ensemble phase-averaged to obtain W at each phase. Moreover, theturbulence intensity w’ was obtained as the square root of the ensemble phase average of wt

2.This procedure was also applied to the phase-averaged value P of the wall static pressure. Itwas confirmed beforehand that averaging over 200 cycles has no influence on the values ofthe quantities. The scatter in the results was less than 4 and 6% for the phase-averaged valuesand the turbulence intensity, respectively. In the hot-wire measurement, it is difficult todetermine accurately the direction of flow at the position and time when the ratio of the radialto the axial component of the velocity is rather large. For such a case, errors of a limitedextent are not avoidable. Furthermore, it was also checked that the flow properties aresymmetric with respect to the tube axis.

Additionally, in order to gain insight into the pulsating flow features, the visualizationexperiment using water was executed for a divergent tube with d1=22 mm. In the experiment,the flow in the horizontal plane including the tube was rendered visible by a solid tracermethod using polystyrene particles of about 0.2 mm diameter.


3.1. Experimental Conditions

Pulsating flow in divergent tubes is governed by five nondimensional parameters: thetotal divergence angle 2θ, the area ratio m, the Womersley number α, the mean Reynoldsnumber Reta and the oscillatory Reynolds number Reos (or the flow rate ratio η). Here, theformer two and the remainder are related to the geometry of the divergent tubes and the fluidmotion, respectively. The Womersley number is defined as α = a1(ω/ν)1/2, ω being theangular frequency of pulsation and ν the kinematic viscosity of the fluid. The mean Reynoldsnumber is expressed as Reta =d1Wa1,ta /ν and the oscillatory Reynolds number as Reos

=d1Wa1,os /ν. Here, Wa1 is the cross-sectional average velocity in the upstream tube and thesubscripts ta and os indicate the time-mean and amplitude values, respectively. Finally, theflow rate ratio is given by η =Wa1,os /Wa1,ta (=Reos /Reta). The experiments were performedsystematically under the conditions of α =10~40, Reta=20000 and Reos = 10000 (η = 0.5).These conditions were chosen referring to works on pulsating straight-tube flows (forexample, Iguch et al. [7]).

In order to check the flow entering the divergent tube, preliminary measurements wereconducted for the axial velocity at z/d1 = -2 and the pressure gradient in the upstream tube. Inthe former, integrating the measured velocities over the cross section, a time-dependent flowrate Q was calculated. The result is shown in Figure 3. In the figure, the solid line denotes thefirst fundamental component developed by the Fourier series, in which the second and highercomponents are less than 3% compared with the first one and are considerably small. Thisconfirms that the prescribed flow is realized, and also indicates that a sinusoidal flow rate isachieved in the form

Q = Qta + Qos sinΘ , (2)


Θ (= ωt) being the phase angle and t the time.

Figure 3. Flow rate variation.

Furthermore, the results obtained for the varying pressure gradient in the upstream tubewere in good agreement with the previous ones obtained by Ohmi and Iguchi [8]. Forexample, the phase difference Φp between the variation of the pressure gradient and the flowrate is shown in Figure 4. Therefore, it is reconfirmed that a fully developed pulsatingturbulent flow enters the divergent tube.

Figure 4. Phase difference between axial pressure gradient in the upstream tube and flow rate variation.

Ohmi and Iguchi [8] showed that the pulsating turbulent flow in straight tubes isclassified roughly into three regimes depending on a characteristic number of α/Reta

3/8. Theflow conditions taken up in this study are in the α/Reta

3/8 region of 0.2 to 1, which is theintermediate regime in their classification. Thus we can say that the entry flow conditions intothe divergent tube in the present experiment are of higher interest in this research field.

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3.2. Wall Static Pressure

3.2.1. Distribution of Wall Static Pressure and Its Variation with Time

In divergent tubes, for steady flow, the kinetic energy of the flow converts into thepressure energy and then the static pressure rises in the downstream direction. On the otherhand, for pulsating flow with a periodic change of the flow rate, the drop and rise of thepressure in the longitudinal direction are needed to drive the fluid acceleratively anddeceleratively, respectively, with time. Hence the amplitude of the pressure variation ispredicted to be proportional to the pulsation frequency ω. Consequently, the pressure in thedivergent tube shows a complex distribution with time.

The wall static pressure P is shown in Figure 5 for the various Womersley numbers. Thepressure P is expressed in the form of the pressure coefficient Cp, which is defined as Cp = (P-Pref) / (ρ Wa1,ta

2 /2), Pref being the pressure at the station of z/d1 = -2 in the upstream tube andρ the density of the fluid. In the figure, the broken lines indicate the time-averaged values andthe chain line the result for steady flow at the same Reynolds number as Reta. In the upstreamtube, Cp changes in the phase leads of Φp, shown in Figure 4, with flow rate variation. Thetime-averaged Cp is slightly larger than that for the steady flow at Re=2000. On the otherhand, the pressure in the divergent tube rises with an increase of z/d1, except for the part ofthe period for moderate and high α. That is, the variation of Cp becomes larger with anincrease of α. Furthermore, the phase with the largest value of Cp changes from the Θ ≈ 90°(α=10) of the maximum flow rate to the Θ ≈ 150° (α=40) of the middle of the decelerativephase. Conversely, the phase with the minimum value of Cp shifts from the Θ ≈ 270° (α=10)of the smallest flow rate to the Θ ≈ 330° (α=40) of the first half of the accelerative phase. Atthese phases with the lowest Cp, for the low α (α=10), there is little change of Cp in thelongitudinal direction. However, for the moderate and high α (α=20, 40), Cp takes negativevalues, and the pressure in the divergent tube shows a favorable gradient.

(A)



(B)

(C)

Figure 5. Longitudinal distribution of wall static pressure. (: Θ=0°, : Θ= 90°, Δ: Θ= 180°, :Θ=270°, ⎯ - ⎯ : time-averaged, ⎯ ⎯ ⎯ : steady flow at Re=20000). (a) α=10, (b) α=20, (c) α=40

We examine the pressure rise ΔPL in the length L, i.e., between the inlet and the exit ofthe divergent tube. The representative result is shown in Figure 6, in which ΔPL isnondimensionalized by the dynamic pressure based on Wa1,ta in the upstream tube. In thefigure, the waveform of ΔPL is developed with the Fourier series denoted by the solid line.The broken line denotes the result that is theoretically obtained using Bernoulli’s theorem fora quasi-steady flow. It is expressed as

ΔPL = (1 + η sinΘ)2 (1− m-2) . (3)

Moreover, the symbol ← indicates the pressure rise ΔPL,s for steady flow, at Re=20000,with the same cross-sectional average velocity as Wa1,ta. The pressure rise ΔPL in the pulsatingflow changes almost sinusoidally. However, it lags behind the variation of the flow rate. Thephase difference ΦΔP between the fundamental waveform of ΔPL and the pulsating flow rateQ becomes large with an increase in the Womersley number. Incidentally, ΦΔP changes from -

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5° to -60° when α increases from 10 to 40. On the other hand, ΔPL is lower than thetheoretical value for quasi-steady flow. Furthermore, ΔPL takes approximately zero values forphase in the range of 230 ~ 340° with a small flow rate.

Figure 6. Difference between pressures at the inlet and exit of a divergent tube (← : steady flow atRe=20000).

As has been stated above, for pulsating flow in a divergent tube, the pressure at the exitof the divergent tube rises when the cross-sectional average velocity is large and is in adecelerative phase. That is, ΔPL becomes large from the latter half of the accelerative phase tothe middle of the decelerative phase (Θ ≈ 50 ~ 180°), as seen in Figure 6. By contrast, Cp

exhibits a small change in the axial direction from the ending of the decelerative phase to thefirst half of the accelerative phase (Θ ≈ 230 ~ 330°). This is because the kinetic energy to beconverted to pressure energy is low and because, to accelerate the fluid in the axial direction,the downstream pressure must be lowed. Therefore, it can be understood that the pressuredistribution at the beginning of the accelerative phase shows the favorable larger gradient forhigher Womersley number where the fluid is strongly accelerated in the streamwise direction.

3.3. Axial Velocity

3.3.1. Changes in Centerline Velocity with Time and along the Tube Axis

In this section, we discuss the axial velocity. To start, we will elucidate the outline of theflow features by focusing on the axial velocity on the tube axis. Figure 7 shows thewaveforms of instantaneous axial velocity w, together with those of its phase-averaged one Wand its turbulence intensity w’. For comparison with those on the tube axis, the results of themeasurement at the radial position r/Rz = 0.75 near the tube wall are also shown in Figure 7.Moreover, the changes in Wc and wc’ along the tube axis are illustrated in Figure 8. Thesubscript c indicates the values on the tube axis and the broken lines indicate the time-averaged values.


Figure 7. Velocity waveforms. (A) Instantaneous velocity w (B)Phase-averaged velocity W (C)Turbulence intensity w’.

The phase-averaged velocity W exhibits an almost sinusoidal change in time, as displayedin Figure 7(b). However, since the flow extends with increasing z/d1 in both the divergenttube and the downstream straight tube, Wc in these sections is reduced (see Figure 8(a)). Inparticular, the degree of reduction in the divergent section becomes smaller and larger for themiddle, i.e., Θ = 180 and 0°, of the decelerative and the accelerative phase, respectively.

Thus, the phase angle at which Wc shows a peak is discernibly delayed compared with theflow rate variation as z/d1 increases. This lag is equivalent to the time at which the fluidflowing into the divergent tube at the maximum flow rate (Θ = 90°) reaches each section. Asa result, the time lag at the divergence exit (z/d1 = 7.1) amounts to approximately one-tenth ofone cycle. On the other hand, W near the wall (r/Rz = 0.75) in the divergent tube changesalmost in synchronous phase with the flow rate. That is, W takes a maximum value at about Θ= 90°. In the case of steady flow, Wc/Wa1 at the Reynolds numbers of 10000 and 30000,corresponding to the minimum and maximum flow rates in pulsating flow, respectively,attenuates axially in the same manner as the flow at Re = 20000. Hence, the Reynolds numberhas little effect on Wc/Wa1 (the illustration is omitted here). It can be inferred from the abovediscussion that the differences in the change in Wc along the tube axis in phase are attributedto an unsteady effect.

The turbulence intensity w’c becomes large, at any phase, with an increase in z/d1 in thefirst half of the divergent tube. It shows the peak magnitude in the second half of thedecelerative phase, as seen in Figures 7(c) and 8(b). Furthermore, w’c near the divergence exitin the first half of the decelerative phase is about 2.6 times the value at the divergence inlet.Consequently, the variation of w’c over one cycle becomes larger.

Masaru Sumida374

(A)

(B)

Figure 8. Change in Wc and w’c along the tube axis (: Θ=0°, : Θ= 90°, Δ: Θ= 180°, : Θ=270°, ⎯⎯ ⎯ : time-averaged). (a) Phase-averaged velocity Wc (b) Turbulence intensity w’c.

On the other hand, w’ near the wall (r/Rz=0.75) in the upstream tube is twice as large asw’c, but the phase difference between w’ and Wa is small. In the section immediately behindthe divergence inlet, the turbulence intensity is high in the phase with the high flow rate.However, the change of w’ gradually becomes similar to that of waveform w’c with anincrease in z/d1.

3.3.2. Distributions of Phase-Averaged Velocity and Turbulence Intensity

Figure 9 shows distributions of W and w’ at four representative phases. The broken linesin the figure denote results for steady flow at the Reynolds number of Reta. Moreover,sketches of the stream are displayed in Figure 10, which is based on, with moderate


exaggeration, the observation of the water flow. The painted parts enclosed by the brokenlines show the main current of the flow.

Figure 9. Distributions of W and w’ in upper and lower figures, respectively (: Θ=0°, : Θ= 90°, Δ:Θ= 180°, : Θ=270°, ⎯ ⎯ ⎯ : steady flow at Re=20000).

In pulsating flow, the phase difference among fluid motions in the divergent tubebecomes larger as z/d1 increases. Consequently, the difference between the flow states of theaccelerative and decelerative phases becomes considerable. Therefore, the phase-averagedvelocity and turbulence intensity exhibit complicated distributions.

In the upstream tube (Figure 9; z/d1=−2), the periodic change of the axial velocity leadsslightly near the tube wall, whereas it is delayed in the central part of the cross section.Nevertheless, on the whole, the velocity at each phase shows a profile similar to the steadyone with a simple 1/7-th-power law. In the divergent tube, the pressure rise in the first half ofthe accelerative phase is smaller than the theoretical one derived for quasi-steady flow, asshown in Figure 5(b). Moreover, there is little pressure difference in the divergent tube. As aresult, the main current in the central part of the cross section reaches the exit plane withoutextending too much towards the tube wall (Figure 10(b); Θ ≈ 0°). Meanwhile, near the inlet, alocal pressure drop occurs, as seen in Figure 5(b). This accelerates the fluid in theneighborhood of the wall, and the phase of the velocity variation is more advanced (z/d1=1.4). When the flow rate increases and becomes maximum, eddies generated in the shearlayer surrounding the main current grow rapidly, accompanying the acceleration of the flow,and become massive vortices. These vortices strongly interact with each other near thedivergence exit. Following this, the main stream suddenly begins to meander in the centralpart of the cross section, resulting in a rapid increase in w’ (Θ = 90°; Figure 10(b)). By themiddle of the decelerative phase (Θ = 180°), the pressure at the divergence exit is, as before,about twice as high as that in the case of steady flow at the same flow rate, because the

Masaru Sumida376

pressure there decelerates the fluid flow (Figures 5(b) and 6). Thus the stream becomesunstable. As a result, the massive vortices in the section downstream of the divergence exitcollapse and start to decay. In addition, the main current is mixed. At the end of thedecelerative phase (Θ = 270°), the turbulence intensity in the section downstream of thedivergence exit is considerably attenuated and part of the main current retreats to the first halfof the divergent tube (Figure 10(b)). The fluid particle in the vicinity of the wall is carried,locally and momentarily, upstream along the wall. Nevertheless, flow separation andbackward flow are indistinct, and the fluid almost always flows downstream.

(A)

(B)

Figure 10. Sketches of flow pattern obtained by visualization experiment. (A) Steady flow at Re=20000(B) Pulsating flow (α=20, Reta=20000, η=0.5).


From the above description, in the phase at which the flow rate increases, the degree ofacceleration is high in the central part of the cross section. As a result, W near the divergenceexit (z/d1 = 5.5) at the maximum flow rate (Θ = 90°) shows a profile with convexity near thetube axis, as seen in Figure 9. In the decelerative phase, on the other hand, the velocity isreduced almost uniformly throughout the cross section because of the stronger action ofturbulent mixing in the section further downstream.

The distribution of the turbulence intensity differs appreciably from that in a steady flow,as seen in Figure 9. That is, the turbulence intensity is high in the phase from the end of theincrease to the first half of the decrease of the flow rate. This is because, during this phase,the shearing layer between the main current and the tube wall rolls up and becomes massivevortices. On the other hand, w’ is low in the first half of the accelerative phase when thesevortices are almost decayed. To put it concretely, when the flow rate is large, the turbulenceintensity immediately behind the divergence inlet (z/d1 = 0 ∼1.4) takes large values near thewall. Downstream, the region with high w’ extends radially, accompanied by the formation ofa shearing layer. Furthermore, the maximum w’/Wa1,ta distribution in this section occurs in thefirst half of the decelerative phase and becomes about 18% at z/d1=2.7 from the 14% in theupstream straight tube. Downstream, in the middle section of the divergent tube, at z/d1 = 2.7~ 5.5, w’ becomes maximum in the vicinity of the inflection point of the W distribution whena fluid flowing at high speed is passed through the section. Subsequently, the location ofmaximum w’ shifts toward the tube axis. In this phase (Θ ≈ 180°), w’ is about 20 percenthigher than that of steady flow with Reta. The maximum value corresponds to 20% of Wa1,ta.In the section downstream of the divergence exit, the distribution of the turbulence intensity isuniformalized in the cross section (z/d1 = 9.6).

5. Conclusion

Experimental investigations were performed for pulsating turbulent flow in a divergenttube and the flow field was examined. The principal findings of this study are summarized asfollows.

(1) The axial distributions of the pressure coefficient Cp in the divergent tube are high inthe phase from the latter half of acceleration to the middle of deceleration. Incontrast, they are low in other phases, namely, from the ending of the deceleration tothe first half of the acceleration.

(2) The time-mean value of the pressure rise ΔPL between the inlet and the exit of thedivergent tube is larger than that in the steady flow with the Re = Reta. The variationof ΔPL and the phase lag ΦΔP behind the flow rate depend on and increase with α2.

(3) The phase-averaged velocity W shows profile swelling in the central part of the crosssection when the flow rate increases. The phase at which the value becomesmaximum in each plane is delayed from Θ = 90° at the maximum flow rate with anincrease in z/d1. However, as the flow rate decreases, the W profile becomes flatdownstream of the divergent tube.

(4) The turbulence intensity w’ for a high flow rate is maximum at the position on theradius near the inflection point of the W profile. As the phase proceeds, the region

Masaru Sumida378

with a large w’ extends towards the wall and the tube axis, and the w’ profilebecomes level at the plane of the divergence exit.

Acknowledgements

The author would like to thank Mr. J. Morita of Tokuyama Corporation for his assistance.

References

[1] R. K. Singh and R. S. Azad, Exp. Thermal Fluid Sci. 1995, vol. 10, 397-413.[2] R. K. Singh and R. S. Azad, Exp. Thermal Fluid Sci. 1995, vol. 11, 190-203.[3] G. Gan and S. B. Riffat, Appl. Energy 1996, vol. 54-2, 181-195.[4] D. Xu, M. A. Leschziner, B. C. Khoo and C. Shu, Comput. Fluids 1997, vol. 26-4,417-

423.[5] Mizuno and H. Ohashi, Trans. Jpn. Soc. Mech. Eng. Ser. B 1984, vol. 50-453, 1223-

1230 (in Japanese).[6] O. Mochizuki, M. Kiya, Y. Shima and T. Saito, Trans. Jpn. Soc. Mech. Eng. Ser. B

1997, vol. 63-605, 54-61 (in Japanese).[7] M. Iguchi, M. Ohmi and S. Tanaka, Bull. JSME 1985, vol. 28-246, 2915-2922.[8] M. Ohmi and M. Iguchi, Bull. JSME 1980, vol. 23-186, 2029-2036.

In: Fluid Mechanics and Pipe FlowEditors: D.Matos and C. Valerio, pp. 379-397

ISBN 978-1-60741-037-9c© 2009 Nova Science Publishers, Inc.

Chapter 12

SOLUTION OF AN AIRFOIL DESIGN INVERSE

PROBLEM FOR A VISCOUS FLOW USING A

CONTRACTIVE OPERATOR

Jan Šimák and Jaroslav PelantAeronautical Research and Test Institute (VZLÚ), Prague, Czech Republic

Abstract

This chapter deals with a numerical method for a solution of an airfoil design in-verse problem. The presented method is intended for a design of an airfoil based ona prescribed pressure distribution along a mean camber line, especially for modifyingexisting airfoils. The main idea of this method is a coupling of a direct and approxi-mate inverse operator. The goal is to find a pseudo-distribution corresponding to thedesired airfoil with respect to the approximate inversion. This is done in an iterativeway. The direct operator represents a solution of a flow around an airfoil, describedby a system of the Navier-Stokes equations in the case of a laminar flow and by thek−ω model in the case of a turbulent flow. There is a relative freedom of choosing themodel describing the flow. The system of PDEs is solved by an implicit finite volumemethod. The approximate inverse operator is based on a thin airfoil theory for a poten-tial flow, equipped with some corrections according to the model used. The airfoil isconstructed using a mean camber line and a thickness function. The so far developedmethod has several restrictions. It is applicable to a subsonic pressure distribution sat-isfying a certain condition for the position of a stagnation point. Numerical results arepresented.

1. Introduction

The method described in this chapter is assumed for an airfoil design corresponding toa given pressure distribution. It is an extension of a method for a potential flow and later foran inviscid compressible flow [1], [2] and a laminar viscous flow [3]. It is useful in the caseswhere a specific distribution is desired. The method is based on the use of an approximateinversion and the solution of the flow around an airfoil. Since the flow is assumed turbulent,a model of turbulence is used to improve the quality of the solution. In the following textthe detailed description of the method is given.

380 Jan Šimák and Jaroslav Pelant

2. Description of the Method

The presentedmethod, as most of the others inverse methods, is based on the idea ofan approximate inversion. This approximate inverse operator, in the following text denotedby L, is accompanied by a so-called direct operator, denoted byP. This direct operatorrepresents a solution of the flow around and airfoil. The pressure distributionsfup andflo

on the upper and lower part of the airfoil are prescribed along the mean camber line abovethe chord line, in the direction from the leading edge to the trailing edge. The chord line isof lengthb. A new functionf on the interval〈−b, b〉 is defined by the following relations

f(x) = fup(−x), for x ∈ 〈−b, 0),

f(x) = flo(x), for x ∈ 〈0, b〉.

Then the inverse problem can be defined as the following:Find a functionu : 〈−b, b〉 → R such that

PL(u) = f. (2.1)

The functionu will be referred to as a pseudo-distribution.The equation (2.1) is transformed to a contractive operator whose fixed point is searched

for. This yields a sequence of pseudo-distributions

uk∞k=0 , uk+1 = uk + α (f − PLuk) . (2.2)

If this sequence converges, the limit is the solution of (2.1). In order to ensure that, asuitable parameterα ∈ (0, 1〉 is chosen. The numerical results indicate that the values lessthen one are usually sufficient. In many examples the value was set toα = 0.6.

3. Inverse Operator

The approximate inverse operatorL, described in this section, is a mapping between avelocity distributionf and a curveψ representing an airfoil. Since the given distribution isa pressure, it is necessary to transform it to a velocity distribution. The pressure is assumedto be constant across the boundary layer in the normal direction to the airfoil. From thatreason, the velocity on the edge of a boundary layer is computed from the pressure on theairfoil. The transformation is derived from relations for the pressure, density, Mach numberand speed of sound,

p = p0

(

1 +γ − 1

2M2

)

−γ/(γ−1)

,

ρ = ρ0

(

1 +γ − 1

2M2

)

−1/(γ−1)

,

c2 = γp/ρ.

The derived formula for the velocity related to the velocity in the free stream is(

v(x)

v∞

)2

=2/M2

∞+ γ − 1

γ − 1

(p0/p(x))(γ−1)/γ − 1

(p0/p(x))(γ−1)/γ

, (3.3)

Solution of an Airfoil Design Inverse Problem... 381

wherep0 is the pressure at zero velocity,M∞ is the Mach number in the free stream andγis the Poisson adiabatic constant.

The velocity distribution, obtained by the above mentioned transformation (or is simplygiven), is on an upper and lower side of the airfoil above its chord. This is a straight lineconnecting the leading and trailing edge of the airfoil. The derivation of the operator willbe in two steps. At first, a function describing a mean camber line is derived. This is a linein the middle of an airfoil, its end points are the same as the end points of the chord. Thesecond step is construction of a thickness function. It is the distance between points on asurface and points on a mean camber line. Putting the function describing the mean camberline and the function describing the thickness together we get the operatorL.

The prescribed distribution has to satisfy a condition, that the stagnation point on theleading edge is located at the beginning of the chord line. Otherwise the method can havetroubles near the stagnation point. The required position is ensured by the determination ofthe appropriate angle of attack.

3.1. Construction of the Mean Camber Line

The derivation of the function describing the mean camber line is based on the theoryof thin airfoils. In this theory, the airfoil thickness is neglected and the airfoil shape issimplified to a line. From this reason we assume that the mean camber line represents anairfoil in the following text.

At first, the origin of the coordinate system is set at the beginning of the chord line onthe leading edge. Thex-axis is set so that the chord line, which length isb, lies on the axis.Let us consider the direction of the flowv∞ parallel to thex-axis or with a small angle ofattackα∞. The main idea is to consider a system of vortices on the airfoil surface. Thissystem determines a circulation along the airfoil.

A vortex with an intensityΓ generates a velocity according to the relation

2vπr = Γ,

wherev is the size of a velocity at some point of the plane and its direction is perpendicularto a line connecting this point and the point of the vortex. The symbolr denotes the distanceof these two points.

According to this, if a system of vortices is assumed on the interval〈0, b〉 on thex-axis,then there will be a velocity generated by this system in the direction of they-axis at thepointx ∈ 〈0, b〉, given by the formula

vy(x) =1

2π(PV)

∫ b

0

dΓ(ξ)

ξ − x. (3.4)

The symbol(PV) means a principal value of an integral, defined as a limit

(PV)

∫ b

af(x) dx = lim

ǫ→0

(∫ c−ǫ

af(x) dx+

∫ b

c+ǫf(x) dx

)

,

wherec is a point at which the functionf(x) has a singularity.


Let us denote the coordinates of the mean camber line by[x, s(x)], x ∈ 〈0, b〉. Since theboth end points of the mean camber line lie on thex-axis, the conditions(0) = s(b) = 0must be satisfied. Thus, although the formula (3.4) is valid only for points on thex-axis, itcan be also used for points on the mean camber line causing only a small error, especiallyif the camber is small.

The vortices on the chord line causing the circulation are unknown but they are deter-mined by the given velocity distribution. The relation between them is quite easy to derive.Assume a closed curve whose image lies over the mean camber line (Figure 1). The cir-

vup

vloA

A′

B

B′

dξ

y

Figure 1. Derivation of the circulation along the mean camber line.

culation alonga closed curve is defined as a curvilinear integral of the tangent componentof the velocity along this curve. The flow on the upper and lower side has the velocitiesequal tovup, vlo, the velocities across the mean camber line are zero because it is assumedimpermeable. So it holds for the elementary circulation

dΓ(ξ) =(

vup(ξ) − vlo(ξ))

dξ,

wheredξ is the length of the segmentsAB andA′B′.Next, the velocity of the free stream is denoted byv∞, its components are

vx∞

= v∞ cosα∞, vy∞

= v∞ sinα∞,

whereα∞ is the angle of attack. Byv(x) =(

vx(x), vy(x))

is denoted the velocity on themean camber line, which is the free stream velocity disturbed by the vortices. Letα be theangle between the vectorv(x) and thex-axis. Then the following must be true:

tanα =vy(x)

vx(x)= tanα∞ +

vy(x)

vx∞

. (3.5)

Since themean camber line represents an airfoil, the vector of velocityv(x) must betangential to this line. This yields the relation

tanα = s′. (3.6)

Introducing a specific circulation

γ(ξ) =dΓ(ξ)

dξ

and usingrelations (3.4) and (3.6) in (3.5), the differential equation for the mean camberline is obtained,

s′(x) = tanα∞ +1

2πvx∞

(PV)

∫ b

0

γ(ξ)

ξ − xdξ. x ∈ 〈0, b〉 , (3.7)


The equation is completed with boundary conditionss(0) = s(b) = 0.In the following let us assume the velocityv∞ = 1 (this can be achieved using a suitable

normalization). It is also assumed thatvx∞

=v∞ = 1, that meansvy∞ is small compared to

vx∞

. The equation (3.7) can be rewritten as

s′(x) = a− c (PV)

∫ b

0

γ(ξ)

x− ξdξ,

wherec = 1/(2π) anda = tanα∞. The angleα∞ is left as a parameter in order to satisfythe boundary conditions. The solution to the differential equation is expressed as

s(x) = ax− c (PV)

∫ x

0

(

(PV)

∫ b

0

γ(ξ)

t− ξdξ

)

dt,

using theboundary condition forx = 0.Changing the order of integration and setting the parametera to ensure the boundary

condition forx = b is satisfied, the final results are obtained. The functions(x) and itsderivatives′(x) are given by

s′(x) =1

2πb(PV)

∫ b

0γ(ξ) ln

∣

∣

∣

∣

b− ξ

ξ

∣

∣

∣

∣

dξ − 1

2π(PV)

∫ b

0

γ(ξ)

x− ξdξ, (3.8)

s(x) =x

2πb(PV)

∫ b

0γ(ξ) ln

∣

∣

∣

∣

b− ξ

ξ

∣

∣

∣

∣

dξ − 1

2π(PV)

∫ b

0γ(ξ) ln

∣

∣

∣

∣

x− ξ

ξ

∣

∣

∣

∣

dξ, (3.9)

whereγ(ξ) = vh(ξ) − vd(ξ).

3.2. Construction of the Thickness Function

During the derivation of the mean camber line, an airfoil with zero thickness was as-sumed. By contrast, in this part of the text an airfoil with zero camber is assumed. In theprevious part, a system of vortices was assumed in order to derive the mean camber linefunctions(x). This time, a system of sources is assumed on the chord in order to derive thethickness function. This system generates a velocity potentialϕ(x, y) at some point of thexy-plane. The potential is given by the relation

ϕ(x, y) =1

2π

∫ b

0q(ξ) ln

√

(x− ξ)2 + y2 dξ, (3.10)

whereq(ξ) describes the intensity of the sources.As will be shown in the following text, it is possible to derive a relation betweenq(x)

and the airfoil thicknesst(x). From this reason a flow parallel to thex-axis is considered.According to the symmetry of the airfoil it is sufficient to work only with the upper halfof the airfoil. By v = (vx, vy) is denoted a velocity vector produced by the distribution ofsources on the chord. The relation between the thickness and the source intensity is basedon the flow rate along the chord (see Figure 2). The difference between velocities at pointsx andx+dx is dvx and the difference between the thickness at these points isdt. Thus it isnecessary (since the flow is considered incompressible) to add or remove some mass usingsources in order to achieve balance.


0 b

v∞ + vx v∞ + vx + dvx

tt+ dt

x

dx

x+ dx

Figure 2. Illustration for derivation of the thickness function.

Mathematically formulated,the balance is

(v∞ + vx) t+1

2q dx = (v∞ + vx + dvx) (t+ dt) .

After some rearrangement and omitting the termdvx dt, the relation is

1

2q(x) =

d

dx

(

v∞ + vx(x))

t(x).

Under the assumption the airfoil is thin enough, it is true|vx| ≪ v∞ inside the interval(0, b). Hence, the relation can be simplified by neglecting the termvx and the result can bewritten as

q(x) = 2v∞ t′(x). (3.11)

Differentiation of the equation (3.10) yields

vx(x) =∂ϕ(x, 0)

∂x=

1

2π(PV)

∫ b

0q(ξ)

1

x− ξdξ

and puttinghere forq(ξ) results in

vx(x) =v∞π

(PV)

∫ b

0t′(ξ)

dξ

x− ξ. (3.12)

The velocity on the chord is assumed as an average of velocities on the upper and lower sideof the airfoil. Thus the velocity incrementvx can be expressed asvx = (vh + vd)/2 − v∞.Assuming again the free stream velocityv∞ = 1, a new functionvp(x) can be defined,

vp(x) =vh(x) + vd(x)

2− 1. (3.13)

Substituting thisinto (3.12) yields

vp(x) =1

π(PV)

∫ b

0t′(ξ)

dξ

x− ξ. (3.14)

The unknown in this equation is the derivative of the thicknesst′(x) and it is possible tofind an analytical solution. This integral equation is a special case of a more general integralequation of the form

αϕ(x) +β

πi

∫

L

ϕ(ξ)

ξ − xdξ = f(x), x ∈ R,


where∫

L denotes acomplex-valued curvilinear integral along a smooth curveL. In ourcaseα = 0, β = −πi and the curveL is an identity on the interval〈0, b〉. According to [4],the solution of our problem can be written in the form

t′(x) = − 1

π(PV)

∫ b

0

vp(ξ)

x− ξ

√

ξ(b− ξ)

x(b− x)dξ +

C√

x(b− x).

After integration with respect tox, the functiont(x) is determined by

t(x) = − 1

π(PV)

∫ b

0

(

vp(ξ)√

ξ(b− ξ) (PV)

∫ x

0

dy√

y(b− y)(y − ξ)

)

dξ+

+ C arctan

(

x− b/2√

x(b− x)

)

− c,

wherec andC are two constants determined later. The second integral can be written in theform

(PV)

∫ x

0

dy√

y(b− y)(y − ξ)= − 1

√

ξ(b− ξ)ln

∣

∣

∣

∣

∣

∣

∣

1 +√

ξb−ξ

√

b−xx

1 −√

ξb−ξ

√

b−xx

∣

∣

∣

∣

∣

∣

∣

.

Finally weobtain the equation describing the airfoil thickness

t(x) =1

π(PV)

∫ b

0vp(ξ) ln

∣

∣

∣

∣

∣

∣

∣

1 +√

ξb−ξ

√

b−xx

1 −√

ξb−ξ

√

b−xx

∣

∣

∣

∣

∣

∣

∣

dξ + C arctan

(

x− b/2√

x(b− x)

)

+ c.

(3.15)This equationcontains two parametersC andc. If we setx = 0 or x = b, the above men-tioned integral equals zero. The second term containing the parameterC is a monotonousfunction which equals−Cπ/2 in the case ofx = 0 or is equal toCπ/2 in the case ofx = b. Since we are interested in an airfoil which is closed on both sides of the chord, wechooseC = c = 0. This choice is fully natural because it agrees with the situation wherethe functionvp is set to zero. In this case the velocity on the airfoil equals the free streamvelocity and from this reason the corresponding airfoil has zero thickness. Under this con-dition the mapping between the velocityvp(x) and the thickness functiont(x) is unique. Itis also obvious that we can obtain an airfoil with prescribed thickness of the trailing edgeby a suitable choice ofC andc.

The resulting equation is

t(x) =1

π(PV)

∫ b

0vp(ξ) ln

∣

∣

∣

∣

∣

∣

∣

1 +√

ξb−ξ

√

b−xx

1 −√

ξb−ξ

√

b−xx

∣

∣

∣

∣

∣

∣

∣

dξ. (3.16)

At the end, there is necessary to mention one drawback. The airfoil thickness should bepositive, of course. But the procedure mentioned above should result in a negative thick-ness, depending on the input velocity distribution. Since the input is the so-called pseudo-distribution, it is necessary to include some control mechanism into the implementation.


3.3. Construction of the Airfoil

(−s′(x), 1)

(1, s′(x))

s(x)

x

t(x)

t(x)

x

y

0 b

Figure 3. Construction of the airfoil.

Construction ofthe airfoil is based on the mean camber lines(x) and the thicknessfunction t(x). Both functions are dependent on the velocity distributionsf = vup, vlo.The airfoil is expressed asψ(x) =

(

ψ1(x), ψ2(x))

, wherex ∈ 〈0, b〉. Since the distancebetween points on the surface and on the mean camber line is given byt(x) (Figure 3), thecoordinates of the airfoil are expressed in the form

ψ1(x) = x∓ t(x)s′(x)

√

1 + s′2(x), (3.17)

ψ2(x) = s(x) ± t(x)1

√

1 + s′2(x). (3.18)

The uppersign is meant for the upper part of the airfoil and the bottom sign for the lowerpart. It is easy to see that the following is true,

ψ1(0) = 0, ψ1(b) = b,

ψ2(0) = 0, ψ2(b) = 0.

Thus we get a closed airfoil over the chord〈0, b〉. The derived approximate inverse operatoris the mappingL(f) = L(vup, vlo) = (ψ1, ψ2).

According to the formulation of the inversion, it is necessary to deal with continuousfunctions. In order to have an airfoil with an acceptable geometry, the functionss(x) andt(x) must be continuous and smooth. This is achieved by assuming a subsonic flow withcontinuous pseudo-distributionsuk for all k. Thus the distributions obtained in each iter-ation as a solution of the flow problem must be continuous. From the numerical point ofview, this restriction is not so strong. The prescribed distribution must be continuous butthe intermediate distributions can have discontinuities with small jumps which are smearedby the iterative process and by the integral formulation of the functionss(x) andt(x).


3.4. Numerical Realization

The evaluation of the functionsψ1(x), ψ2(x) includes evaluations of integrals, whichare done by a suitable numerical quadrature. The integrands are functions in two variablesxandy and have singularities forx = y. From this reason it is necessary to avoid this points.The quadrature used in this method is the Chebyschev-Gauss quadrature. The integrals arediscretized in the way of

∫ b

0f(x, y) dy =

∫ b

0

f(x, y)√

y(b− y)√

y(b− y)dy =

N/2∑

k=1

wkf(x, yk)√

yk(b− yk)+RN , (3.19)

wherex ∈ 〈0, b〉,N is an even number,wk = 2π/N are quadrature coefficients,

RN =2π

2NN !

∂N(

f(x, η)√

η(b− η))

∂yN

is anerror of the quadrature andyk = (b+ bxk)/2 are nodes. In this casexk = cos(

(2k −1)π/N

)

are roots of the Chebyschev polynomial

TN/2(x) = cos

(

N

2arccosx

)

.

Finally, a sequence of pointsxi is defined,

xi =b

2

(

1 + cosiπ

N

)

, i = 0, 1, . . . , N.

In the formula (3.19) is setx = xi, i = 0, 2, 4, . . . , N

and the desired quadrature formula

∫ b

0f(xi, y) dy ≈ 2π

N

N

2∑

k=1

f(xi, x2k−1)√

x2k−1(b− x2k−1), i = 0, 2, 4 . . . , N (3.20)

is obtained.The distribution of nodes using the Chebyschev polynomial has a favourable property.

The density of nodes is higher near the ends of the chord. From this reason the shape ofthe airfoil is expressed more precisely. The velocity is needed to be known at pointsxi,i = 1, 3, . . . , N − 1 and the resulting airfoil is evaluated at pointsxi, i = 0, 2, 4, . . . , N .The number of evaluated points along one side isN/2+1 and the total number of differentpoints of the airfoil is2(N/2 + 1) − 2 = N.

3.5. Viscous Correction

The approximate inversion described so far can deal with a pressure distribution regard-less of viscosity. However, the inverse operator is based on the fact that an airfoil can be


represented by streamlines. In the case of an inviscid flow, streamlines are attached totheupper and lower surface and so the airfoil contours can be replaced by them. But this isnot true in the case of a viscous flow. Streamlines are influenced by a boundary layer andthus the obtained shape is different from the physical shape. The airfoil is thicker and thecamber is smaller. Although the inverse operator has some capability to overcome this fact,in the cases with strong influence of the boundary layer it is not enough. From this reasonit is useful to introduce a correction to eliminate this problem. This correction is based onthe displacement thickness corresponding to the pressure and the airfoil geometry.

Using the mentioned inverse operator, the mean camber line function and the thicknessfunction are computed. Now the correction is applied. First, the thickness is reduced bythe displacement thickness. Second, the mean camber line is corrected. Since the geometryof the airfoil is closed and the effective shape is open, it is not possible to simply subtractthe displacement thickness form the thickness function. The remedy is to start from thedefinition of the displacement thickness and derive the correction.

The displacement thickness is defined as the following integral,

δ∗ =

∫ δ

0

(

1 − ρu

ρeue

)

dy, (3.21)

wherethe subscripte denotes the values on the edge of the boundary layer,u is the velocityacross the boundary layer andδ is the boundary layer thickness. Sinceu/ue ≈ 1 fory ≥ δ (in this caseue means a velocity outside the boundary layer) and the inverse operatorassumes an incompressible flow, the formula can be rewritten

δ∗ ≈ d−∫ d

0

u

uedy, for d ≥ δ.

Denotingf ′x(y) = ux(y)/ue(x) (the subscriptx denotes the dependency on the locationon the airfoil), the relation for the corrected airfoil thickness is

tvis(x) = t(x) − δ∗(x) = t(x) − d+ fx(d), for d ≥ δ.

That meanstvis = fx(t) for t ≥ δ. Assuming the velocity profileu nonnegative, we canextend this relation for allt nonnegative. Thus the corrected airfoil thickness is determinedby

tvis = fx(t) (3.22)

providing that the velocity profile on the airfoil is known.The velocity profile is obtained by the Pohlhausen’s method [5], which was derived for

laminar boundary layers. The approximation of the velocity by the polynomial of fourthorder is assumed

ux(y)

ue(x)= 2

y

δ(x)− 2

(

y

δ(x)

)3

+

(

y

δ(x)

)4

+due(x)

dx

δ(x)

6νy

(

1 − y

δ(x)

)3

. (3.23)

The velocity described by this polynomial satisfies the boundary conditions

ux(0) = 0, ux(δ) = ue(x),

∂2ux(0)

∂y2= −ue(x)

ν

due(x)

dx,∂ux(δ)

∂y= 0,

∂2ux(δ)

∂y2= 0. (3.24)


This representation makes sense for a parameterΛ = (δ2/ν)(due/dx) from an interval〈−12, 12〉. If Λ > 12, the velocity profile overshoots the value of the velocityue(x). IfΛ < −12, a separation occurs and the velocity profile is not nonnegative.

The boundary layer thicknessδ is obtained from the von Kármán’s momentum equationand relations for the boundary layer parameters:

τwall

ρ=due

dx(2θ + δ∗)ue + u2

e

dθ

dx, (3.25)

δ∗ = δ

(

3

10− 1

120

due

dx

δ2

ν

)

, (3.26)

θ = δ

(

37

315− 1

945

due

dx

δ2

ν− 1

9072

(

due

dx

δ2

ν

)2)

, (3.27)

τwall =µue

ρ

(

2 +1

6

due

dx

δ2

ν

)

. (3.28)

The new thicknesstvis is computed using the formula (3.22). After that, the meancamber line is modified to eliminate viscous effects. To do so, a correction function

∆(x) =δ∗up − δ∗lo

2

is subtractedfrom the mean camber line functions(x). The functions(x) is then trans-formed to satisfys(0) = s(b) = 0.

3.6. Finding the Mean Camber Line of a General Airfoil

As was mentioned earlier, the pressure distribution is given along a mean camber line.The inverse operator evaluates the function describing this line, so the mean camber line ofthe designed airfoil is known. But if the given pressure distribution is based on a knownairfoil, it is useful to have a way how to get its mean camber line.

Assume the airfoil coordinates are known. The process of the airfoil construction(3.17)–(3.18) can be rewritten as

ψup

(

x− t(x)s′(x)

√

1 + s′2(x)

)

= s(x) + t(x)1

√

1 + s′2(x),

ψlo

(

x+ t(x)s′(x)

√

1 + s′2(x)

)

= s(x) − t(x)1

√

1 + s′2(x), x ∈ 〈0, b〉 (3.29)

under the assumptionss(x) ∈ C1 〈0, b〉, t(x) ∈ C 〈0, b〉. The symbolsψup(x), ψlo(x)denote they-coordinates of the upper and lower part of the airfoil (corresponding tox-coordinatex). Linearization of the left hand sides and rearrangement leads to a differentialequation for an unknown functions(x),

s′(x)(

ψup(x)ψ′

lo(x) + ψ′

up(x)ψlo(x) − s(x)(

ψ′

up(x) + ψ′

lo(x))

)

= 2s(x) − ψup(x) − ψlo(x), x ∈ (0, b) . (3.30)


This equation is equipped with boundary conditions

s(0) = 0, s(b) = 0.

The equation was derived under the assumptions thats′(x)ψ′

up(x) 6= −1 ands′(x)ψ′

lo(x) 6= −1. In other words, the tangential vectors of the airfoil shape and themean camber line are not perpendicular to each other. This is true with an admissible ge-ometry. The expression enclosed by parentheses in (3.30) can be zero at some points fromthe interval〈0, b〉. Thus the derivatives′(x) at these points is undefined by (3.30). How-ever, the equation then sayss(x) = (ψup(x) + ψlo(x))/2, which corresponds to the ideaof a mean camber line. If the airfoil is closed and the curves representing upper and lowerparts are smooth, then there exists at least one pointx ∈ 〈0, b〉 such thatψ′

up(x) = ψ′

lo(x).At this point, eithers′(x)ψ′

up(x) = −1 or the term in parentheses in (3.30) is zero. Fromthat reason it is necessary to be careful when solving the differential equation and utilizeboth boundary conditions. Generally, the existence and uniqueness of the solution is notguaranteed, but in the common cases the solution is unique.

4. Direct Operator

This operator represents the solution of the flow problem. Depending on the model offlow used, its formulation can vary. In this case, where the viscous compressible flow isassumed, the model is described by the system of the Navier-Stokes equations.

4.1. Mathematical Formulation

The Navier-Stokes equations are given by

∂ρ

∂t+

2∑

j=1

∂(ρvj)

∂xj= 0, (4.31)

∂(ρvi)

∂t+

2∑

j=1

∂(ρvivj + p δij)

∂xj=

2∑

j=1

∂τij∂xj

, i = 1, 2, (4.32)

∂E

∂t+

2∑

j=1

∂(

(E + p)vj

)

∂xj=

2∑

j=1

∂

∂xj

(

τj1v1 + τj2v2 +

(

µ

Pr+

µT

PrT

)

γ∂e

∂xj

)

.

(4.33)

Since mostof the flow in a real situation is turbulent, the laminar model seems insuffi-cient. To improve the quality of the predicted flow and also the stability of the method, amodel of turbulence is included (hence the term with the subscriptT in (4.33)). In the casedescribed in this chapter thek − ω turbulence model is used [6], [7]. New variables de-scribing the turbulence properties are introduced, a turbulent kinetic energyk and a specific


turbulent dissipationω. Thesetwo variables are linked together by equations

∂ρk

∂t+

2∑

j=1

∂ρkvj

∂xj=

2∑

j=1

∂

∂xj

(

(µ+ σkµT )∂k

∂xj

)

+ Pk − β∗ρωk, (4.34)

∂ρω

∂t+

2∑

j=1

∂ρωvj

∂xj=

2∑

j=1

∂

∂xj

(

(µ+ σωµT )∂ω

∂xj

)

+ Pω − βρω2 + CD. (4.35)

For the simplicity, the system can be rewritten into the vector form

∂w

∂t+

2∑

j=1

∂Fj(w)

∂xj=

2∑

j=1

∂Gj (w,∇w)

∂xj+ S (w,∇w) . (4.36)

By µT an eddyviscosity coefficient is denoted. This coefficient is given by the formula

µT =ρk

ω.

The stresstensor in the N.-S. equations is given by relations

τ11 = (µ+ µT )

(

4

3

∂v1∂x1

− 2

3

∂v2∂x2

)

− 2ρk

3,

τ22 = (µ+ µT )

(

−2

3

∂v1∂x1

+4

3

∂v2∂x2

)

− 2ρk

3,

τ12 = τ21 = (µ+ µT )

(

∂v1∂x2

+∂v2∂x1

)

.

The productionof turbulencePk on the right hand side of the eq. (4.34) and the productionof dissipationPω in eq. (4.35) are expressed as

Pk = τ11∂v1∂x1

+ τ12

(

∂v1∂x2

+∂v2∂x1

)

+ τ22∂v2∂x2

,

Pω = αωωPk

k,

whereτ ij = τij for µ = 0. Finally, the cross-diffusion termCD is given by the relation

CD = σDρ

ωmax

(

∂k

∂x1

∂ω

∂x1+

∂k

∂x2

∂ω

∂x2, 0

)

.

The turbulence model is closed by parametersβ∗ = 0.09, β = 5β∗/6, αω = β/β∗ −σωκ

2/√β∗ (whereκ = 0.41 is the von Kármán constant),σk = 2/3, σω = 0.5 a σD =

0.5. This choice of parameters resolves the dependence of thek − ω model on the freestream values [7]. In the standard Wilcox model without cross diffusion the parameters areβ∗ = 0.09, β = 5β∗/6, αω = β/β∗ − σωκ

2/√β∗, σk = 0.5, σω = 0.5 aσD = 0.

If the turbulent kinetic energyk is set to zero, the turbulence model has no influenceupon the N.-S. equations and the laminar model can be solved.


4.2. Numerical Treatment

In the numerical method, a dimensionless system of equations is solved. The vari-ablesp, ρ, v1, v2, k andω are normalized with respect to the critical values of densityρ∗,velocity c∗ and pressurep∗ and to the characteristic lengthl∗. The critical values are thevalues corresponding to a unit Mach number. The dimensionless system has the same formas (4.31)–(4.35). The relations between the original and normalized variables are the fol-lowing:

v1 =v1c∗, v2 =

v2c∗, ρ =

ρ

ρ∗, p =

p

ρ∗c2∗,

k =k

c2∗

, ω =ωl∗c∗, µT =

µT

ρ∗c∗l∗. (4.37)

The problemis solved by an implicit finite volume method. Details about this methodand the numerical solution of a flow in general can be found in many textbooks, for ex-ample [8]. Since the coupling between the equations describing the flow and the equationsdescribing the turbulence is only by the viscous terms, it is possible to solve the problemin two parts [6]. When the flow variablesp, ρ, v1, v2 in the timetk+1 are computed using(4.31)–(4.33), the turbulent variablesk andω are assumed to be constant in time with valuescorresponding to the timetk. On the contrary, when computing the turbulent variablesk,ω in a timetk+1 using (4.34) and (4.35), the flow variablesp, ρ, v1, v2 are assumed to beconstant in the timetk. That means the solution of the problem consists of two systems,

1. (vk+11 , vk+1

2 , ρk+1, pk+1) = NS(vk+11 , vk+1

2 , ρk+1, pk+1, kk, ωk),

2. (kk+1, ωk+1) = Turb(vk1 , vk2 , ρ

k, pk, kk+1, ωk+1).

These systems of equations can be solved independently of each other.The solution of the system of equations (4.31)-(4.33) is similar to the way how a laminar

problem is solved. The equations are identical with the pure N.-S. equations except thestress tensorτij in viscous terms and the heat flux, which depend onk andω. Sincek andωare taken as parameters in the system, it is quite easy to modify the existing laminar solver.

By wh will be denoted a vector of 6-dimensional blockswi of the values of an approx-imate solution on finite volumesDi ∈ Dh. For wh ∈ R

n the vectorΦ(wh) consists of6-dimensional blocksΦi(wh) given by

Φi(wh) =1

|Di|∑

j∈S(i)

(

2∑

s=1

nsFs,h (wh; i, j) |Γij | −2∑

s=1

nsGs,h (wh; i, j) |Γij |)

− Sh (wh; i, j) , (4.38)

where|Di| denotes the cell area,|Γij | denotes the length of the edge betweenDi andDj ,nij = (n1, n2) denotes the outer normal toDi. FunctionsFs,h (wh; i, j), Gs,h (wh; i, j)andSh (wh; i, j) are approximations ofF(w), G(w,∇w) andS(w,∇w) on the gridDh.In order to have a higher order method we apply the Van Leerκ-scheme or the Van Albadalimiter inside the functionsFs,h. By S(i) is denoted a set of indices of neighbouring ele-ments and by the symbolτ will be denoted the time step. Thus the implicit finite volume


scheme in a cellDi can bewritten as

wk+1i = w

ki − τk

Φi(wk+1h ). (4.39)

The nonlinear equation above is linearized by the Newton method. The arising systemof linear algebraic equations is solved by the GMRES method (using softwareSPARSKIT2[9]). The convective termsFi are evaluated using the Osher-Solomon numerical flux in thecase of the flow part and by the Vijayasundaram numerical flux in the turbulent part. Thenumerical evaluation of a gradient on the edgeV2V3 on a boundary (Fig. 4) is done by thefollowing formulae (the index denotes the value in the corresponding vertex)

∂f

∂x1

∣

∣

∣

∣

V2V3

≈ − (V3,y − V2,y) (f1 − fwall)

|(V2,x − V1,x)(V3,y − V1,y) − (V3,x − V1,x)(V2,y − V1,y)|,

∂f

∂x2

∣

∣

∣

∣

V2V3

≈ (V3,x − V2,x) (f1 − fwall)

|(V2,x − V1,x)(V3,y − V1,y) − (V3,x − V1,x)(V2,y − V1,y)|. (4.40)

If the edge is inside the domain, another scheme according to Fig. 5 is used,

[V1,x, V1,y]

[V2,x, V2,y][V3,x, V3,y]

Figure 4. Scheme for a derivative on a wall.

[V1,x, V1,y]

[V2,x, V2,y]

[V3,x, V3,y]

[V4,x, V4,y]

Figure 5. Scheme for a derivative inside the domain.

∂f

∂x1

∣

∣

∣

∣

V1V3

=(f3 − f1)(V4,y − V2,y) − (f4 − f2)(V3,y − V1,y)

|(V3,x − V1,x)(V4,y − V2,y) − (V4,x − V2,x)(V3,y − V1,y)|,

∂f

∂x2

∣

∣

∣

∣

V1V3

= − (f3 − f1)(V4,x − V2,x) − (f4 − f2)(V3,x − V1,x)

|(V3,x − V1,x)(V4,y − V2,y) − (V4,x − V2,x)(V3,y − V1,y)|. (4.41)

The pointsV2 andV4 arecentres of corresponding cells,f2 andf4 are values in these cells.The valuesf1 andf3 are obtained as an arithmetic mean of values of the four neighboringcells.


4.3. Boundary Conditions

In our problem, three types of boundary conditions occur: a condition on a wall, acondition at an inlet boundary and a condition at an outlet boundary. Due to the viscosity,the zero velocity on the wall is prescribed, further the zero turbulent kinetic energy anda static temperature are prescribed. The value of the specific turbulent dissipationω isobtained by the formula

ωwall =120µ

ρy2c

,

whereyc is thedistance between the wall and the centre of a cell in the first row. At theinlet part of the boundary, the velocity vector(v1, v2), the densityρ, turbulent energykand dissipationω are prescribed. At the outlet part of the boundary, three variables areprescribed, the static pressurep, turbulent energyk and dissipationω. The other variablesare evaluated from values inside the domain. The values ofk andω on the boundary arevalues of the free stream and are given in the form of a turbulent intensityI and a viscosityratio ReT = µT /µ. The turbulent intensity is defined as

I =

√

2

3

k

v∞. (4.42)

Following this, the relations fork∞ andω∞ are obtained,

k∞ =2

3(v∞I)

2 , (4.43)

ω∞ =ρk

µ

(

µT

µ

)

−1

. (4.44)

4.4. Mesh Deformation

During the inverse method iterations, the direct operator is applied a number of time.This is of course true, a new airfoil shape is designed in each iteration. Moreover, thereis a need to find an appropriate angle of attack, which ensures the required position of thestagnation point on the leading edge. This is done by the rotation of the airfoil. This allresults in changes of the computational domain and of the mesh, of course. In order toremove the dependency on the mesh generator used, the actual mesh is deformed to fit thenew geometry.

The deformation is based on the linear elasticity model which is described in manytextbooks. The grid cells are stretched or shrinked and moved to fit the new domain. Nogrid points are created or deleted, the neighbouring cells are the same in the new grid as inthe original grid.

The linear elasticity model is described by the equation

divσ = f in Ω, (4.45)

whereσ is a stress tensor andf is some body force. The symbolΩ denotes the domain, itsboundary will be denoted byΓ. The stress tensor is expressed using a strain tensorǫ as

σ = λTr(ǫ)I + 2µǫ


and the components of the strain tensor can be expressed using a displacement functionu

as

ǫij =1

2

(

∂ui

∂xj+∂uj

∂xi

)

.

The symbolsλ andµ arethe Lamé constants which describe the physical properties of thesolid. They can be expressed using the Young’s modulusE and Poisson’s ratioν,

λ =νE

(1 + ν)(1 − 2ν), µ =

E

2(1 + ν).

Substituting theabove mentioned relations into (4.45), the following problem for thedisplacement is obtained:

Find an unknown functionu : Ω → R2 suchthat

(λ+ µ)∇(div u) + µ∆u = f in Ω,

u = uD onΓ. (4.46)

The functionuD is the displacement on the boundary, which is known. This problemis reformulated into the weak sense. Thus the problem for the mesh deformation can beformulated in the form:

Find a functionu ∈ H1(Ω)2 such thatu−u∗ ∈ H1

0 (Ω)2, whereu∗ represents Dirichletboundary conditions (that meansu∗ ∈ H1(Ω)2, u

∗|Γ = uD) and the functionu satisfiesthe equation

−µ∫

Ω∇u :∇ϕT dx− (λ+ µ)

∫

Ωdiv u · divϕ dx =

∫

Ωf ·ϕ dx (4.47)

for all ϕ ∈ H10 (Ω)2. The colon operator is defined by the relation

A :B =2∑

i=1

2∑

j=1

aijbji, A,B ∈ R2×2.

The weak problem described above can be solved by a finite element method. Thedomain is discretized by a triangular mesh with nodal points, which are the same as in themesh for the flow problem. This leads to the fact, that the solution, which represents themovement of the nodal points of the original mesh, is evaluated at the appropriate points.The parameterE is set proportional to the reciprocal values of the cell volumes. Thisensures that the most deformation is carried out on large cells instead of the small ones.

5. Numerical Examples

5.1. Example 1

In this example the given pressure distribution is obtained as a result of a flow around theNACA4412 airfoil. The inlet Mach number isM∞ = 0.6, angle of attackα∞ = 1.57 andthe Reynolds numberRe = 6 ·106. The flow is described by the use of the turbulencek−ωmodel. The resulting airfoil is compared to the original one and thus shown the correctnessof the method. The results are in Figure 6. The relative error of pressure distribution(measured inL2-norm) after 40 iterations is8.32 · 10−4.


X

P

0 0.2 0.4 0.6 0.8 10.4

0.6

0.8

1

1.2

1.4

Y0

0.2

0.4

0.6

0.8

X

||e||

0 0.2 0.4 0.6 0.8 10

0.0001

0.0002

0.0003

X

p-f

0 0.2 0.4 0.6 0.8 1-0.005

0

0.005

0.01

Figure 6. Example 1. Pressure distribution and resulting airfoil shape, error ofthe resultingairfoil measured as a norm‖ψresult − ψNACA4412‖e, difference between the prescribed andresulting pressure distribution on the chord (values are normalized).

5.2. Example 2

In this example a laminar flow with low Reynolds number is computed. The startingpressure distribution is computed on the NACA3210 airfoil with parametersRe = 1000,M∞ = 0.6, α∞ = 4.56. At first, the problem is computed without any viscous correctionand then the mentioned correction based on the Pohlhausen’s method is used. From theresults is evident, that for very low Reynolds numbers the correction is necessary. Thecomparisons are in Figure. 7.

X

P

0 0.2 0.4 0.6 0.8 10.6

0.8

1

1.2

1.4

Y0

0.2

0.4

0.6

0.8

X

y

0 0.2 0.4 0.6 0.8 1

0

0.2

X

p-f

0 0.2 0.4 0.6 0.8 1-0.02

-0.01

0

0.01

0.02

Figure 7. Example 2. Pressure distributions and resulting airfoil shapes (solid -correction,dashed - without correction), comparison of the airfoils with the NACA3210 (dotted), dif-ference between the prescribed and resulting pressure distribution along the chord (valuesare normalized).


6. Conclusion

A numericalmethod for a solution of an inverse problem of flow around an airfoil wasdescribed. The advantage of this method is the weak dependency on the model describingthe flow. It is easy to modify the method by assuming a different model. The correction dueto viscosity is necessary only with low Reynolds numbers. The presented method has stillsome drawbacks. These are the dependency of the angle of attack on the prescribed distri-bution and the applicability on the subsonic regimes only. The method can be improved inthe future.

Acknowledgment

This work was supported by the Grant MSM 0001066902 of the Ministry of Education,Youth and Sports of the Czech Republic.

References

[1] Pelant, J.Inverse Problem for Two-dimensional Flow around a Profile, Report No.Z–69; VZLÚ, Prague, 1998.

[2] Šimák J.; Pelant J.A contractive operator solution of an airfoil design inverse prob-lem; PAMM Vol. 7, No. 1 (ICIAM07), pp. 2100023–2100024.

[3] Šimák, J.; Pelant, J.Solution of an Airfoil Design Problem With Respect to a GivenPressure Distribution for a Viscous Laminar Flow, Report No. R–4186; VZLÚ,Prague, 2007.

[4] Michlin, S. G. Integral Equations; Pergamon Press, Oxford, 1964.

[5] Schlichting, H.Boundary-Layer Theory; McGraw-Hill, New York, 1979.

[6] Wilcox, D. C. Turbulence Modeling for CFD; DCW Industries Inc., 1998; 2nd ed.

[7] Kok, J. C.Resolving the Dependence on Freestream Values for thek − ω TurbulenceModel; AIAA Journal 2000, vol.38, No. 7, pp. 1292–1295.

[8] Feistauer, M.; Felcman, J.; Straškraba, I.Mathematical and Computational Methodsfor Compressible Flow; Clarendon Press, Oxford, 2003.

[9] Saad, Y.Iterative Methods for Sparse Linear Systems; SIAM, 2003; 2nd ed.



Chapter 13

SOME FREE BOUNDARY PROBLEMS IN POTENTIAL

FLOW REGIME USING THE L EVEL SET M ETHOD

M. Garzon1, N. Bobillo-Ares1 and J.A. Sethian2

1Dept. de Matematicas, Univ. of Oviedo, Spain2Dept. of Mathematics, University of California, Berkeley, and

Mathematics Department, Lawrence Berkeley National Laboratory.

Abstract

Recent advances in the field of fluid mechanics with moving fronts are linked to theuse of Level Set Methods, a versatile mathematical technique to follow free boundarieswhich undergo topological changes. A challenging class of problems in this contextare those related to the solution of a partial differential equation posed on a movingdomain, in which the boundary condition for the PDE solver has to be obtained froma partial differential equation defined on the front. This is the case of potential flowmodels with moving boundaries. Moreover, the fluid front may carry some materialsubstance which diffuses in the front and is advected by the front velocity, as for ex-ample the use of surfactants to lower surface tension. We present a Level Set basedmethodology to embed this partial differential equations defined on the front in a com-plete Eulerian framework, fully avoiding the tracking of fluid particles and its knownlimitations. To show the advantages of this approach in the field of Fluid Mechanicswe present in this work one particular application: the numerical approximation of apotential flow model to simulate the evolution and breaking of a solitary wave propa-gating over a slopping bottom and compare the level set based algorithm with previousfront tracking models.

1. Introduction

In this chapter we present a class of problems in the field of fluid mechanics that canbe modeled using the potential flow assumptions, that is, inviscid and incompressible fluidsmoving under an irrotational velocity field. While these are significant assumptions, in thepresence of moving boundaries, the resulting equations is a non linear partial differentialequation, which adds considerable complexity to the computational problem. In the litera-ture this model is often called the fully non linear potential flow model (FNPFM). Several

400 M. Garzon, N. Bobillo-Ares and J.A. Sethian

interesting and rather complicated phenomenon are described using the FNPFM, asfor ex-ample, Helle-Shaw flows, jet evolution and drop formation, sprays and electrosprays, wavepropagation and breaking mechanisms, etc, see [21], [22], [30], [13].

Level Set Methods (LSM) [31], [33], [34] [37] are widely used in fluid mechanics, aswell as other fields such as medical imaging, semiconductor manufacturing, ink jet print-ing, and seismology. The LSM is a powerful mathematical tool to move interfaces, once thevelocity is known. In many physical problems, the interface velocity is obtained by solvingthe partial differential equations system used to model the fluid/fluids flow. The LSM isbased on embedding the moving front as the zero level set of one higher dimensional func-tion. By doing so, the problem can be formulated in a complete Eulerian description andtopological changes of the free surface are automatically included. The equation for themotion of the level set function is an initial value hyperbolic partial differential equation,which can be easily approximated using upwind finite differences schemes.

Recently, the LSM has been extended to formulate problems involving the transportand diffusion of material quantities, see [3]. In [3] model equations and algorithms arepresented together with the corresponding test examples and convergence studies. Thisled to the realization that the nonlinear boundary conditions in potential flow problemscould also be embedded using level set based methods. As a result, the FNPFM can also beformulated with an Eulerian description with the associated computational advantages. Twodifficult problems that have been already approximated using this novel algorithm are wavebreaking over sloping beaches [16], [17] and the Rayleigh taylor instability of a water jet[20]. Moreover, related to drop formation and wave breaking, it has been recently reportedin the literature [46], [45] that the presence of surfactants on the fluid surface affects theflow patterns. The models described in this chapter are the groundwork for solving thesecomplex problems.

This chapter is organized as follows: in section 2. we have made an effort to obtain dy-namic equations valid for any spatial coordinate system. To do so, we derive the equationsusing only objects defined in an intrinsic way (i.e., independent of any coordinate system).At the same time, in accordance with the level set perspective, we have avoided as muchas possible, the “ material description” (Lagrangian coordinates). Geometric quantities aredefined using the level sets and tensor fields in the space. In section 3. a brief descriptionof the Levels Set Method is given using this intrinsic approach. Section 4. is devoted todescribe two particular potential flow models, the first one related to drop formation in thepresence of surfactants, which combines all the models derived in section 2.. The wavebreaking problem is modeled in 2D, code development in 3D is underway. In section 5., wepresent the numerical approximation and algorithm for the wave breaking problem. Numer-ical results and accuracy tests are also presented in section 6.. Precise definitions of certainneeded geometrical tools, throughout used in this chapter, are shown in Appendix III.

2. Some Physical Models

Here, we discuss the derivations of fluid problems and their corresponding reformula-tion using the Level Set Method (LSM) techniques. The brief derivation of known physicallaws is used also as a pretext to introduce some preliminary concepts and notation.

Some Free Boundary Problems in Potential Flow Regime... 401

2.1. Kinematic Relationships

Referenceconfiguration. The configuration of a continuous medium at certain timetis known when the position of each particle is specified. We nameΩt the space regionoccupied by the continuous medium at that time.

Kinematics require the movement description of each particle. To this aim, we must:

i. Label the particles.

ii. Specify the movement of each particle.

The first step is done considering the configuration at an arbitrary instantt0 (referenceconfiguration). Particles are marked by the pointP0 ∈ Ωt0 they occupy. Points inΩt0 aregood labels because they are in a 1 to 1 correspondence with the particles (“particles can notpenetrate each other”). In what follows we will abbreviate the phrase “particle with labelP0” by “particleP0”.

Once all the particles are labeled, it is now possible to undertake the second step. LetP0 ∈ Ωt0 be a particle. Its positionP at instantt is given by the function:

P = R(P0, t), P ∈ Ωt, P0 ∈ Ωt0 . (1)

According to the reference configuration definition, we have:

R(P0, t0) = P0. (2)

The mappingRt,Rt(P0) := R(P0, t) = P , must be invertible:

P0 = R−1t (P ) ∈ Ωt0 , P ∈ Ωt. (3)

Lagrangian/Eulerian descriptions. Any tensor fieldw may be described in two ways,using (1):

w = w(P, t) = w(R(P0, t), t) = w0(P0, t). (4)

Functionw(P, t) corresponds to the so called Eulerian description andw0(P0, t) corre-sponds to the Lagrangian description. As a consequence any tensorial fieldw admits twotime partial derivatives. The “spatial” derivative, corresponding to the Eulerian description:

∂tw :=d

dǫw(P, t+ ǫ)

∣

∣

∣

∣

ǫ=0

, (5)

measures the variation rate with time ofw from a fixed point in the space. The “convective”derivative, corresponding to the Lagrangian description:

Dtw :=d

dǫw0(P0, t+ ǫ)

∣

∣

∣

∣

ǫ=0

, (6)

gives the variation rate ofw following the particleP0.


Velocity field The velocityu = u(P0, t) of the particleP0 is obtained using the convec-tive derivative (“following the particle”) of the positionP = R(P0, t):

u = DtP, P = R(P0, t). (7)

Obviously,u admits both descriptions:

u = u(P, t) = u0(P0, t), P = R(P0, t). (8)

Given an arbitrary tensor fieldw, its spatial and convective derivatives are related usingthe calculus chain rule and the definition (7):

Dtw = ∂tw + ∂uw. (9)

Here,∂uw designates the directional derivative ofw alongu (see Appendix III). The ac-celeration of particleP0 is obtained by the convective derivative of the velocity field. Using(9), we have:

Dtu = ∂tu + u · ∇u. (10)

Transport of a vector due to a moving medium. A fluid particle is located at point1 Pat timet. After a time∆t, the same particle is at pointR(P, t+ ∆t). Clearly, the functionR must verify thatR(P, t+ 0) = P . A nearby particle at same timet is located atP + ǫa,and att + ∆t is at pointR(P + ǫa, t + ∆t). We have againP + ǫa = R(P + ǫa, t + 0).The vectorǫa that connects both particles varies as they move. Denote byDtǫa its rate ofchange with time:

Dtǫa = lim∆t→0

1

∆t[(R(P + ǫa, t+ ∆t) −R(P, t+ ∆t)) − (R(P + ǫa, t) −R(P, t))]

= lim∆t→0

R(P + ǫa, t+ ∆t) −R(P + ǫa, t)

∆t− lim

∆t→0

R(P, t+ ∆t) −R(P, t)

∆t.

The first term of the right hand side of previous equation is, by definition, the particlevelocity atP + ǫa, u(P + ǫa, t), and the second term the particle velocity atP , u(P, t).Thus we have:

Dtǫa = u(P + ǫa, t) − u(P, t).

Letting ǫ→ 0, we obtain the rate of change with time of an infinitesimal vector dragged bythe medium:

Dta := limǫ→0

1

ǫDtǫa = lim

ǫ→0

u(P + ǫa, t) − u(P, t)

ǫ=

d

dǫu(P + ǫa, t)

∣

∣

∣

∣

ǫ=0

= ∂au. (11)

We denote∂a the operator that performs the directional derivative along the vectora (seeAppendix III).

1For this calculation we use here the configuration att as the reference configuration.


Fluid volume change as it is transported by the velocity field. Let a,b andc be threesmallvectors with origin at pointP . The volume of the parallelepiped spanned by vectorsa,b, c is given by the trilinear alternate form

δV = [a,b, c] = a · b × c.

The rate of change of this volume, when particles located on its vertices move, is given byDtδV , and thus we have

DtδV = Dt[a,b, c] = [Dta,b, c] + [a,Dtb, c] + [a,b,Dtc].

Using now (11) we get

DtδV = [∂au,b, c] + [a, ∂bu, c] + [a,b, ∂cu],

which is also a trilinear alternate form. As in the tridimensional space all these forms areproportional, we can set

Dt[a,b, c] = (div u)[a,b, c], (12)

which gives us an intrinsic definition of the divergence of the fieldu. If the continuousmedium is incompressible, the volumeδV does not change,DtδV = 0, and we arrive atthe incompressibility condition

div u = 0. (13)

2.2. Dynamic Relationships

Conservation of mass. Denote byρ = (P, t) the volumetric mass density of the contin-uous medium at pointP and at timet. The rate of change of the mass in a small volumeδV dragged by the velocity field is, using definition (12),

Dt(ρδV ) = (Dtρ)δV + ρDtδV = (Dtρ+ ρ div u)δV.

The mass conservation law is thus

Dtρ+ ρ div u = 0. (14)

Applying general formula (9) toρ, we haveDtρ = ∂tρ + ∂uρ. In the case of an homoge-neous and incompressible medium with uniform initial densityρ0, using equations (14) and(13), we haveDtρ = 0 which gives (P, t) = ρ0.

Conservation of the momentum associated with a small piece of continuous medium.From Newton’s second law applied to a fluid volumeV we get the relation

Dt

∫

Vu ρdV =

∫

Vg ρdV +

∫

∂Vτ(ds). (15)

The term in left hand side of this equation is the rate of change with time of the momentumassociated with volumeV when dragged by the continuous medium. The first term in theright hand side corresponds to the volumetric forces insideV , generated by a vector field


per unit massg, usuallythe gravitational field. The second term represents the “contact”forces applied by the rest of the medium over the part inV . The Cauchy’s tensorτ is a linearoperator field that is obtained from specific relationships which depend on the material, theso called constitutive relations. We are interested in inviscid fluids which verify the Pascal’slaw:

τ(ds) = −pds,

wherep is the pressure scalar field. Green’s formula,∫

∂V−p ds =

∫

V−∇p dV,

shows that contact forces may be computed as a kind of volume forces with density−∇p.For a small volumeδV dragged by the fluid, equation (15) can be written:

Dt(u ρδV ) = (g ρ−∇p)δV. (16)

Due to the mass conservation law,Dt(ρδV ) = 0, equation (16) leads to the Euler equation:

Dtu = ∂tu + ∂uu = g − 1

ρ∇p. (17)

If g is auniform field it comes from the gradient of a potential function:

g = −∇U(P ), U(P ) = −g · (P −O),

whereP −O is the position vector of the pointP .

2.3. Potential Flow

Assuming an irrotational flow regime, curlu = 0, there exists an scalar fieldφ suchthat

u = ∇φ. (18)

Outside of the fluid domain, and separated by a free boundary, there is a gas at pressurepa

that is assumed to be constant. This means that, within the gas, the time needed to restorethe equilibrium is very small compared with the time evolution of the fluid. Therefore, atthe fluid free boundary, the boundary condition is just:

p = pa. (19)

Using the vectorial relationship∇u2/2 = ∂uu + u × (curl u), and relations (18) and (13)

we have

∇(

∂tφ+1

2u

2 +p

ρ+ U

)

= 0.

Performing the first integration,

∂tφ+1

2u

2 +p

ρ+ U = C(t),


whereC(t) is anarbitrary function of time, which can be chosen in such a way that theprevious relation can be written:

∂tφ+1

2u

2 +p− pa

ρ+ U = 0.

Now using the obvious relation∂tφ+ u2 = ∂tφ+ ∂uφ = Dtφ, we finally obtain

Dtφ− 1

2u

2 +p− pa

ρ+ U = 0. (20)

2.4. Advection

On the surface of a continuous medium with a known movement, a certain substance isdistributed, which will be named as “charge”. This is adhered to the fluid particles and itis transported by them. In this way a set of particles will always carry the same amount of“charge”. This phenomenon is called advection.

The continuous medium surface is implicitly described as the zero level set of a certainscalar functionΨ = ψ(P, t):

Γt = Q|ψ(Q, t) = 0. (21)

Vectorsa tangent to the surface are characterized by the condition

∂aΨ = a · ∇Ψ = 0;

thus, the tangent vectorial plane at each point of the surface is given by the normal unitvector2

n =∇Ψ

|∇Ψ| .

The functionψ by itself does not specify the particle movement on the surface, just itsshape. We need to add the information about how these particles move, e.g., specifying thevelocity field on the surface

Q ∈ Γt, u = u(Q, t).

A small vectora connecting two nearby particles on the surface and dragged by them asthey move, has a rate of change given by (11),

Dta = ∂au. (22)

Note thata is a tangent vector,n · a = 0.

Surface areas. Using the normal vector to the surface,n, a 2–form to calculate surfaceareas can be constructed:3

ω(a,b) := [n,a,b] = n · a × b,

2Ψ must increasefrom the interior to the exterior of the surface to getn outwards.3The surface area definition is not made using the Gram determinant of two tangent vectors, because this

procedure involves a particular parametrization of the surface. Instead, a 2-form is defined from the volumeform in space (“the parallelogram area is the volume of a rectangular prism of unit height”).


whereω(a,b) is thearea spanned by tangent vectorsa,b, and[n,a,b] the volume form inthe 3D space.

As the tangent vectorsa,b are transported by the surface movement, the parallelogramarea associated to them changes. The rate of change with time is easily obtained:

Dtω(a,b) = (Dtn) · a × b + n · (Dta) × b + n · a × Dtb.

First term of the right hand side of previous equation is zero sincea× b is a normal vectorandDtn is tangent: indeed, asn2 = 1, we haveDtn

2 = 2n · Dtn = 0.

Using (22) we have

Dtω(a,b) = ∂au · b × n + ∂bu · n × a.

This expression is bilinear and alternate with respect the tangent vectors. It must be, ateach point on the surface, proportional to the 2–formω. We denote byDiv u, “surfacedivergence”, the proportionality coefficient:

∂au · b × n + ∂bu · n × a := (Div u) ω(a,b) (23)

This definition ofDiv u does not depend upon the choice of tangent vectorsa andb. InAppendix I, the expression for the surface divergence of an arbitrary vector fieldw usingrectangular coordinates is shown.

Advection law. Now, let be

σ = σ(Q, t), Q ∈ Γt

the “charge” surface density. The “charge”δq carried by a small parallelogram, spanned bytwo small tangent vectors(a,b), of areaω(a,b) is

δq = σ ω(a,b).

As the “charge” is conserved, the advection law is

Dt δq = 0.

Now, by definition (23), we have

Dt(σω) = (Dtσ)ω + σDtω = (Dtσ + σDiv u)ω. (24)

Hence we arrive to the intrinsic equation for the advection phenomena:

Dtσ + σDiv u = 0.


2.5. Advection-Diffusion

Next, we are going to assume that the “charge” diffuses along particles on the surfaceaccording to the Fick’s law:4

j = −α∇σ,whereα is the diffusion coefficient,j is the “charge” flux and∇σ is the “charge” surfacedensity gradient. Asσ is only defined on the surfaceΓt, ∇σ is only defined for tangentvectors:

∇σ · a := ∂aσ, a tangent vector.

On the surfaceΓt let us consider a surface regionS, bounded by a curve∂S. Let beννν theunit vector field tangent toΓt and orthogonal to the curve∂S at each point. The “charge”that leaves the surface per unit time is the outward flux through the boundary∂S:

∫

δSj · ννν dl = −

∫

∂Sj · n × dl =

∫

∂Sn × j · dl.

Applying now Stokes’ theorem, we have∫

∂Sn × j · dl =

∫

S

A ω(d1P,d2P ). (25)

The 2-form of the surface integral is obtained using the intrinsic formula

A ω(a,b) = ∂a(n × j · b) − ∂b(n × j · a). (26)

We interpretA ω(a,b) as the “charge” per unit time that, by diffusion, leaves the smallparallelogram spanned by the tangent vectors(a,b).

Now it is straightforward to set the condition for the advection-diffusion mechanism

“charge” rate of changewithin the tangent

parallelogram(a,b)

= −

“charge” that leavesthe parallelogram

by diffusion

,

that isDt(σω(a,b)) = −A ω(a,b).

In Appendix II the following expression forA is obtained:

A = Div j − (Div n) j · n.

Hence, using (24) we arrive at the general equation for the advection-diffusion model:

Dtσ + σ Div u = −Div j + (Div n) j · nj = −α ∇σ

orDtσ + σ Div u = α Div∇σ − α(Div n) ∇σ · n (27)

4Fick’s diffusion law applies when the “charge” particles move randomly without any preferential direction(Brownian movement).


The Cartesian expressions5 for Div u, Div∇σ andDiv n are:

Div n = (δij − ninj)∂jni =1

|∇Ψ|3[

(∇Ψ)2∂i∂iΨ − ∂jΨ∂iΨ∂j∂iΨ]

, (28)

Div∇σ =1

|∇Ψ|2[

(∇Ψ)2∂i∂iσ − ∂iΨ∂jΨ∂i∂jσ]

, (29)

Div u =1

|∇Ψ|2[

(∇Ψ)2∂iui − ∂iΨ∂jΨ∂iuj

]

. (30)

Expanding theimplicit summands, we obtain the following expressions for the 3D space(i, j = 1, 2, 3):

Div n =1

|∇Ψ|3[

(∂1Ψ)2(∂22Ψ + ∂2

3Ψ) + (∂2Ψ)2(∂21Ψ + ∂2

3Ψ)+

+(∂3Ψ)2(∂21Ψ + ∂2

2Ψ) − 2∂1Ψ∂2Ψ∂1∂2Ψ −− 2∂1Ψ∂3Ψ∂1∂3Ψ − 2∂2Ψ∂3Ψ∂2∂3Ψ

]

, (31)

Div∇σ =1

|∇Ψ|2[

(∂1Ψ)2(∂22σ + ∂2

3σ) + (∂2Ψ)2(∂21σ + ∂2

3σ)+

+(∂3Ψ)2(∂21σ + ∂2

2σ) − 2∂1Ψ∂2Ψ∂1∂2σ −− 2∂1Ψ∂3Ψ∂1∂3σ − 2∂2Ψ∂3Ψ∂2∂3σ

]

, (32)

Div u =1

|∇Ψ|2[

(∂1Ψ)2(∂2u2 + ∂3u3) + (∂2Ψ)2(∂1u1 + ∂3u3)+

+(∂3Ψ)2(∂1u1 + ∂2u2) − ∂1Ψ∂2Ψ(∂1u2 + ∂2u1) −− ∂1Ψ∂3Ψ(∂1u3 + ∂3u1) − ∂2Ψ∂3Ψ(∂2u3 + ∂3u2)

]

. (33)

To obtain the formulas for the plane we assume axial symmetry in the direction3:

∂3Ψ = 0, ∂23Ψ = 0, ∂3σ = 0, u3 = 0, ∂3ui = 0,

|∇Ψ|2 = (∂1Ψ)2 + (∂2Ψ)2.

Inserting these values in (31), (32) and (33) we get:

Div n =1

|∇Ψ|3[

(∂1Ψ)2(∂22Ψ) + (∂2Ψ)2(∂2

1Ψ) − 2∂1Ψ∂2Ψ∂1∂2Ψ]

, (34)

Div∇σ =1

|∇Ψ|2[

(∂1Ψ)2(∂22σ) + (∂2Ψ)2(∂2

1σ) − 2∂1Ψ∂2Ψ∂1∂2σ]

, (35)

Div u =1

|∇Ψ|2[

(∂1Ψ)2(∂2u2) + (∂2Ψ)2(∂1u1)−

− 2∂1Ψ∂2Ψ(∂1u2 + ∂2u1)]

. (36)

5In thefollowing expressions we use only subscripts because orthonormal bases coincides with their corre-sponding reciprocal ones. Then, the position of the indices becomes irrelevant.


3. The Level Set Method

The Level Set method is a mathematical tool developed by Osher and Sethian [31] tofollow interfaces which move with a given velocity field. The key idea is to view the movingfront as the zero level set of one higher dimensional function called the level set function.One main advantage of this approach comes when the moving boundary changes topology,and thus a simple connected domain splits into separated disconnected domains.

Let beΓt the set of points lying in the surface boundary at time t. This surface is definedthrough the zero level set of the scalar fieldΨ = ψ(P, t):

Γt = Q|ψ(Q, t) = 0. (37)

To identify the fluid particles, the configuration att0 (reference configuration) is used:

Γt0 = Q0|ψ(Q0, t0) = 0. (38)

The particle movement is specified through the function

Q = R(Q0, t), (39)

which gives the positionQ ∈ Γt of the fluid particleQ0 ∈ Γt0 . The particleQ0 velocity iscalculated using the convective derivativeDt (“following the particle”):

u = DtQ =d

dǫR(Q0, t+ ǫ)

∣

∣

∣

∣

ǫ=0

. (40)

According to definition (37), we haveψ(R(Q0, t), t) = 0. Deriving with respect to timeand applying the chain rule, we obtain

∂tΨ + u · ∇Ψ = 0. (41)

which has to be completed with the value of the level set function at timet = 0, usually setto be the signed distance function to the initial front,

Ψ(P, 0) = s(P )d(P ),

beingd(P ) the distance from the pointP to the surface at the initial configurationΓ0,s(P ) = −1 if P ∈ Ω0 ands(P ) = +1 if P /∈ Ω0.

Now, if we take a fixed 3D domainΩD that contains the free surface for all times, wecan define the initial value problem for the level set functionΨ posed onΩD:

∂tΨ + u · ∇Ψ = 0 in ΩD (42)

Ψ(P, 0) = s(P )d(P ) in ΩD (43)

A graphical interpretation of the level set function evolution is depicted in figure 1 Equation(42) moves all the level set ofΨ, not just the zero level set, and in many physical applica-tions the front velocity is just defined for points lying on the free boundary. Therefore forthis equation to be valid on the whole domain we have to extend the velocityu off the front.There exist several extension procedures which will be briefly commented below.


0

(P, 0) (P, t)

t

t

Figure 1. Evolution of the level set function.

3.1. Extensionof Functions Defined on the Front

Let us consider a classical result from functional analysis: suppose a domainΩ boundedby a closed surface∂Ω. If for k ≥ 1 the surface∂Ω ∈ Ck, then for all functionsF (x) ∈Ck(∂Ω) there exists a functionFext(x) ∈ Ck(Ω) such thatFext|∂Ω = F (x).

In practice, there are several ways to extend any magnitudeF defined on the front ontoΩD. As shown in [10] for the numerical stability of the level set equation it is convenientto preserveΨ as a signed distance function, which is characterized by the property|∇Ψ| =1. One way is to perform reinitializations of the level set function at chosen times. Ifthis is done periodically, it will smooth the level set function. However, done too often,especially using poor reinitialization techniques, spurious mass loss/gain will occur. Thus,it is important to perform reinitialization both sparingly and accurately. For the potentialflow problems presented in this chapter we follow the strategy introduced in [2]. The idea isto extrapolateF given at the front along its gradient. Mathematically the extended variableFext is the solution of

∇Fext · ∇Ψ = 0. (44)

It is straightforward to show that this choice maintains the signed distance function forthe level sets ofΨ for all times. For the numerical approximation we proceed as follows:given a level set functionΨ at timen, namelyΨn, one first obtains a distance functionΨn

around the zero level set. Simultaneous with this construction, the extended quantityFextis obtained satisfying Eq. (44). For a complete explanation of this extension method see[2].

4. Examples of Potential Flow Models with Moving Boundaries

In this section the governing equations of two interesting physical problems will be for-mulated using a level set framework. First, drop formation is a complex 3D phenomenadriven mainly by capillary forces, which can be modeled using the potential flow assump-


tions. It is well-known that the presence of surface surfactants lowers the surface tensionaffecting drops shape. Secondly, propagation and wave breaking over sloping beaches canalso be modeled with the potential flow equations, which are valid until the jet of the waveimpinges against the flat water surface. In this case we can formulate the equations in 2Dtaking a vertical section of the beach, which facilitates the algorithm and code development.The wave numerical simulations will be presented in section 6..

4.1. Governing Equations for Surface Tension Driven Flows with MaterialAdvection-Diffusion

LetΩt be the3D closed fluid domain surrounded by air andΓt the free surface boundaryat timet. Suppose that initially a certain amount of surfactants, which are assumed to beinsoluble in water, are uniformly distributed on the surface (see Figure 2).

Figure 2. A fluid volume with surface surfactants.

For an incompressible and inviscid fluid, the governing equations are the Euler equa-tions (17). On the free boundary the following partial differential equations apply:

• The advection-diffusion equation for the surfactant is (27):

Dtσ + σDiv u = α(Div∇σ − κ ∇σ · n) on Γt,

whereσ is the surface density of the surfactant,u is the free boundary velocity,α isthe surface diffusion coefficient,κ = Div n = 1

R1+ 1

R2andR1, R2 the principal

radii of curvature ofΓt at each point.

• Continuity of the stress tensor between water and air leads to the balance of thesurface tension forces,p = pa+γ( 1

R1+ 1

R2), whereγ is the surface tension coefficient


that may depend on the surfactant concentrationσ. ThusEq. (20) becomes

∂tφ+1

2(∇φ · ∇φ) +

γ

ρκ+ U = 0 on Γt.

• Finally, if Q = R(Q0, t) is the position of a fluid particleQ0 on the free surface, thedefinition (40) states

DtQ = u(Q, t), Q ∈ Γt. (45)

The complete model equations in 3D are therefore,

u = ∇φ in Ωt (46)

∆φ = 0 in Ωt (47)

DtQ = u on Γt (48)

Dtφ = −U +1

2(∇φ · ∇φ) − γ

ρκ on Γt (49)

Dtσ = −σDiv u + α(Div∇σ − κ ∇σ · n) on Γt. (50)

This is the Lagrangian-Eulerian formulation of the model equations. Classical methodsto approximate this set of equations are the so-called “front tracking methods”, in which afixed number of markers are chosen initially and the trajectories of this markers are followedas time evolves. This method suffers difficulties when the free boundary changes topology:these problems are avoided by a level set formulation.

Level Set Framework

Equation (48) can be directly formulated as the level set Eq. (41). For the velocity fieldu(Q, t), the trajectory of a fluid particle at initial positionQ0 is given by the solution of

DtQ = u(R(Q0, t), t),

R(Q0, 0) = Q0 . (51)

Next, letΩD be a fixed 3D domain that contains the free surface for all times and letG(P, t)andS(P, t) be two functions defined onΩD such that for everyQ ∈ Γt

G(Q, t) = φ(Q, t) , (52)

S(Q, t) = σ(Q, t) . (53)

It is important to remark here thatG(P, t) andS(P, t) are auxiliary functions defined inΩD that can be chosen arbitrarily, the only restriction is that they equalφ(Q, t) andσ(Q, t)on Γt respectively. Figure 3 gives an interpretation of this property for a moving curve in2D. Applying the chain rule in both identities (52) and (53) we obtain

∂tG+ u · ∇G = −U +1

2(∇φ · ∇φ) − γ

ρκ, (54)

∂tS + u · ∇S = −σDiv u + α(Div∇σ − κ ∇σ · n). (55)


f(Q , 0)

Q

f(Q , t)

QG

t

G0

G(P, 0)G(P, t)

t

Figure 3. Extension of the velocity potential off the front.

which holdson Γt. Note thatu and the right hand side of Eq. (54) and Eq. (55) are onlydefined onΓt, and thus, in order to solve these equations over the fixed domainΩD, thesevariables must be extended off the front. This strategy has been discussed in Section 3..Naming

f = −U +1

2(∇φ · ∇φ) − γ

ρκ,

h = −σDiv u + α(Div∇σ − κ ∇σ · n),

the system of equations, written in a complete Eulerian framework, is

u = ∇φ in Ωt (56)

∆φ = 0 in Ωt (57)

Ψt + uext · ∇Ψ = 0 in ΩD. (58)

Gt + uext · ∇G = fext in ΩD (59)

St + uext · ∇S = hext in ΩD (60)

Here the subscript “ext” denotes the extension off , h andu ontoΩD.

4.2. Governing Equations for the Wave Breaking Problem

We now derive our coupled level set/extension potential equations for breaking waves intwo dimensions for which a numerical approximation will be also presented. LetΩt be the2D fluid domain in the vertical plane(x, z) at timet, with z the vertical upward direction(andz = 0 at the undisturbed free surface), andΓt the free boundary at timet (see Figure4).

We assume also an inviscid and incompressible fluid, and capillary forces are disre-garded on the free boundary curve. The model equations in the Lagrangian-Eulerian for-mulation are therefore:


-

6

6

?hΓ1

Γb

Γb

Γ2

ΓtQ

Ωt

x

z

Figure 4.The domain.

u = ∇φ in Ωt (61)

∆φ = 0 in Ωt (62)

DtQ = u on Γt (63)

Dtφ = −gz +1

2(∇φ · ∇φ) on Γt (64)

φn = 0 on Γb ∪ Γ1 ∪ Γ2, (65)

Let Ω1 be a fixed 2D domain that containsΓt for all times. Following the same embed-ding procedure for the potential function as in previous section, we obtain the complete 2DEulerian formulation:

u = ∇φ in Ωt (66)

∆φ = 0 in Ωt (67)

Ψt + uext · ∇Ψ = 0 in Ω1. (68)

Gt + uext · ∇G = fext in Ω1 (69)

φn = 0 on Γb ∪ Γ1 ∪ Γ2 (70)

being heref = 12(∇φ · ∇φ) − gz andfext the extension off ontoΩ1.

5. Numerical Approximations and Algorithms

In this section, we provide overviews of the numerical schemes used to approximatethe wave model equations. The integral formulation of Eq. (66) is approximated using aliner boundary element method (BEM), which will provide the velocity of the front noderepresentation. More detailed discussions of the various components may be found in thecited references.

5.1. Initialization

The initial front positionΓ0 and initial velocity potentialφ(Q, 0), Q ∈ Γ0, are neededto solve equations (68) and (69) respectively. Given an initial solitary wave amplitude (H0)


and the physical length of the domain (L), Tanaka’s method gives a way of calculating thesequantities.Here, we briefly discuss the theoretical basis of this method.

Assuming constant depth, the flow field can be reduced to steady state by using a coor-dinate system that moves horizontally with speed equal to the wave celerityc. The streamfunctionψ(x, z) is also harmonic and takes constant values at the bottom and at the freesurface of the domain.

¿From the definition of stream function and velocity potential we have

φx = ψy, φy = −ψx.

Under sensible assumptions about the smoothness ofφ andψ, these are just the Cauchy-Riemann equations which are satisfied by the real and imaginary parts of the functionW =φ+iψ, which is called the complex potential and it is a an analytical function of the complexvariableZ = x + iz in the domain occupied by the fluid. By interchanging the role of thevariablesZ andW , we can takeφ andψ as independent variables, sinceW = φ + iψprovides a one to one correspondence between the physical and complex potential planes.With this transformation, the fluid region is mapped into the strip0 < ψ < 1, −∞ < φ <∞ in theW plane withψ = 1 on the free surface,ψ = 0 on the bottom andφ = 0 at thewave crest. Denote byu, v the horizontal and vertical components of the velocityu, q = |u|andθ the angle between the velocity and thex axis. The complex velocity is defined by

dW

dZ= φx + iφy = u− iv = qeiθ

andit is also analytic in the flow domain. Therefore, the quantity

ω = ln(dW

dZ) = ln q − iθ,

is an analytic function ofW , so τ = ln q must be harmonic in the strip0 < ψ < 1,−∞ < φ < ∞. The Bernoulli condition at the free surface and the bottom condition canbe expressed in terms ofq andθ as:

dq3

dφ= − 3

F 2sin θ onψ = 1 (71)

θ = 0 onψ = 0, (72)

whereF is the Froude number defined byF = c√

gh.

The problemof finding a solitary wave solution can thus be transformed into the prob-lem of finding a complex functionω that is analytic with respect toW within the region ofthe unit strip0 < ψ < 1, decays at infinity, and satisfies the boundary conditions (71) and(72). Tanaka’s method provides a way to solve the previous outlined equations in terms ofthe new variablesτ , θ and a full description of the algorithm can be found in [40].

5.2. The Level Set and Velocity Potential Updating

We use the standard Narrow Band Level Method, introduced by Adalsteinsson andSethian [2], which limits computation to a thin band around the front of interest. Follow-ing the algorithm discussed in [31], we use second order in space upwind differences to


approximate the gradient in the level set equation, and a first order time scheme toupdatethe solution. For boundary conditions, homogeneous flux boundary conditions are usuallychosen, which are implemented by creating an extra layer of ghost cells around the domainwhose values are simply direct copies of theΨ values along the actual boundary. The levelset function is built from the initial position of the front by computing the signed distancefunction. This is done using the Fast Marching Method [36], which is a Dijkstra-like fi-nite difference method for computing the solution to the Eikonal equation inO(N logN),whereN is the total number of points in the computational domain.

The velocity and the velocity potential are both initially defined only on the interface.In order to create values throughout the narrow band, which are required to update the fixedgrid Eulerian partial differential equations, we use the extension methodology developed byAdalsteinsson and Sethian in [2] to construct appropriate extensions. The idea of buildingextension velocities was first introduced in [26]; in that approach, the extension velocityat any grid point in the domain was taken as equal to the velocity on the closest point onthe front itself. As shown in [7], this is equivalent to solving the equations∇u · ∇Ψ = 0,∇v · ∇Ψ = 0 for the velocity components, and in that paper, the equation was solvedusing a finite difference iteration. In [2], Adalsteinsson and Sethian present a techniquefor computing this extension velocity using the very efficient Fast Marching methodology.Finally, in [3], this approach was developed to build extension values for arbitrary materialquantities whose evolution affects the underlying interface dynamics.

The potential equation (69) is a convection equation with a strong non-linear sourceterm, and homogeneous Newmann boundary conditions are imposed on the boundary ofΩ1.To update in time this equation, note that it is similar to (68) except that it has a nonlinearsource term, and therefore we use similar schemes. For example a straightforward firstorder scheme is

Gn+1i,j = Gn

i,j − ∆t(max(uni,j , 0)D−xi,j + min(un

i,j , 0)D+xi,j +

max(vni,j , 0)D−zi,j + min(vni,j , 0)D+z

i,j ) + ∆tfni,j

where

D−xi,j = D−x

i,j Gni,j =

Gni,j −Gn

i−1,j

∆x

D+xi,j = D+x

i,j Gni,j =

Gni+1,j −Gn

i,j

∆x

are thebackward and forward finite approximation for the derivative in thex direction (wehave the same expressions for forD−z

i,j andD+zi,j .) Note that for simplicity we have written

u, v,G, f instead ofuext, vext, Gext, fext, and we describe a first order explicit schemewith a centered source term. Initial values ofG0

i,j are obtained by extendingφ(x, z, 0)|Γ0

as previously discussed. However, at any time stepn it is always possible to perform a newextension ofφn(x, z, n∆t) to obtain a better value ofGn

i,j .A key issue is how one obtainsfext in the grid points ofΩ1. There are several ways

of doing so. Here we calculatef = 12(∇φ · ∇φ) − gz on free surface nodes, and use

these values together with the condition∇f · ∇Ψ = 0 to obtainfext. This algorithmfor extending quantities defined on the front off the front works very well for the velocity


field in the case of equation (68), because it maintains the signed distance function forthelevel sets ofΨ. However, regarding equation (69) for this particular wave problem, anddue to the high variations off along the front together with its topological structure whenoverturning, the previous method creates strongG andf gradients inΩ1. This fact limitsthe grid spacing inΩ1 and the time step needed to maintain accuracy (see the section onnumerical experiments).

5.3. The Boundary Integral Equation and the BEM Approximation

A first order boundary element method is used to approximate equation (66). Boundaryintegral equations are well suited to moving boundary problems for two principal reasons.First, determining the surface velocity generally requires computing function derivatives onthis boundary, which are accurately evaluated within this formulation. Second, remeshingthe moving boundary is clearly simpler than remeshing the entire domain.

The Laplace equation for the velocity potential (67) is solved by approximating thecorresponding boundary integral equation. Boundary conditions are given by (70) and, onthe free boundary, at each time step, by the updated potential velocity given by equation(69). The approximation of the integral equation is done by the BEM, which calculates thepotential and the potential gradient on the free surface, that is, its velocityu.

The boundary integral equation for the potentialφ(P ), in a domainΩ(t) having bound-aryΣ = ∂Ω(t), can be written as

P(P ) = φ(P ) + limPI→P

∫

Σ

[

φ(Q)∂G∂n

(PI , Q) − G(PI , Q)∂φ

∂n(Q)

]

dQ = 0 , (73)

wheren = n(Q) denotes the unit outward normal on the boundary surface andPI areinterior points converging to the boundary pointP . The Green’s function or fundamentalsolution (in two dimensions) is

G(P,Q) = − 1

4πlog(r) . (74)

The integral equation is usually written with the∂G∂n

singular integral evaluated as a CauchyPrincipal Value (CPV), resulting in a ‘interior angle’ coefficientc(P ) multiplying the lead-ing φ(P ) term [5, 6]. The reason for employing the seemingly more complicated limitprocess will become clear in the discussion of gradient evaluation. The exterior limit equa-tion

limPE→P

∫

Σ

[

φ(Q)∂G∂n

(PE , Q) − G(PE , Q)∂φ

∂n(Q)

]

dQ = 0 . (75)

yieldsprecisely the same equation: the jump in the CPV integral as one crosses the bound-ary accounts for the ‘free term’ difference.

In this work, a Galerkin (weak form) approximation of Eq. (73) has been employed,and the boundary and boundary functions are interpolated using the simplest approximation,linear shape functions. Thus, the equations that are solved are of the form

∫

Σψk(P )P(P ) dP = 0 , (76)


where the weight functionsψk(P ) are comprisedof all shape functions which are non-zero at a particular nodePk [5]. The calculations reported herein employed the simplestapproximation, linear shape functions. These approximations reduce the integral equationto a finite system of linear equations, and invoking the boundary conditions allows thesolution of the unknown values of potential and flux on the boundary. Details concerningthe limit evaluation of the singular integrals can be found in [14].

As noted above, for the wave problem, and moving boundary problems in general,knowledge of the normal flux is not sufficient, the complete gradient ofφ is required tocompute the surface velocity. The remainder of this section will present the algorithm forcomputing this gradient.

¿From Eq. (73) a gradient component can be expressed as

∂φ(P )

∂Ek= lim

PI→P

∫

Σ

[

∂G∂Ek

(PI , Q)∂φ

∂n(Q) − φ(Q)

∂2G∂Ek∂n

(PI , Q)

]

dQ . (77)

Once the boundary value problem has been solved, all quantities on the right hand side areknown: a direct evaluation of nodal derivatives would therefore be easy were it not for well-known difficulties with the hypersingular (two derivatives of the Green’s function) integral[28, 29, 27]. As described in [15], a Galerkin approximation of this equation,

∫

Σψk(P )

∂φ(P )

∂EkdP = (78)

limPI→P

∫

Σψk(P )

∫

Σ

[

∂G∂Ek

(PI , Q)∂φ

∂n(Q) − φ(Q)

∂2G∂Ek∂n

(PI , Q)

]

dQdP

allows a treatment of the hypersingular integral using standard continuous elements.Interpolating∂φ(P )/∂Ek as a linear combination of the shape functions results in a

simple system of linear equations for nodal values of the derivative everywhere onΣ; thecoefficient matrix is obtained by simply integrating products of two shape functions. How-ever, the complete boundary integrations required to compute the right hand side are quiteexpensive.

The computational cost of this procedure can be significantly reduced by exploitingthe exterior limit equation, Eq. (75). It appears to be useless for computing tangentialderivatives, since, lacking the free term, the corresponding derivative equation takes theform

0 = limPE→P

∫

Σ

[

∂G∂Ek

(PE , Q)∂φ

∂n(Q) − φ(Q)

∂2G∂Ek∂n

(PE , Q)

]

dQ , (79)

and the derivatives obviously do not appear. However, subtracting this equation from Eq.(77) yields (with shorthand notation)

∂φ(P )

∂Ek=

limPI→P

− limPE→P

∫

Σ

[

∂G∂Ek

∂φ

∂n(Q) − φ(Q)

∂2G∂Ek∂n

]

dQ . (80)

Theadvantage of this formulation is that nowonly the terms that are discontinuous crossingboundary contribute to the integral. In particular, all non-singular integrations, by far themost time consuming, drop out. The calculation of the right hand side in Eq. (80) reduces


to a few ‘local’ singular integrations, and as these integrations are carried out partiallyanalytically, this produces an accurate algorithm. Further details about the evaluation ofEq. (80) can be found in [15].

5.4. Regridding of the Free Surface

In a level set formulation the position of the front is only known implicitly throughthe node values of the level set functionΨ. In order to extract the front, it is possible toconstruct first order and second order approximations of the interface using local data ofΨon the mesh (see [9] for example.) Here we use a first order linear approximation of the freesurface, which yields a polygonal interface formed by unevenly distributed nodes, whichwe call LS nodes. As a result of this extraction technique, occasionally one gets front nodeswhich are very close together, and this can cause difficulties and instabilities for boundaryelement calculations. To overcome this problem, and also to achieve more front resolutionwhen needed, we employed a front node regridding technique. An initialization point onthe front is selected according to a particular criterion, such as maximum value of height,velocity modulus, or front curvature. This point divides the front in two halves and newnodes are chosen so that, lying in the same polygon, they are redistributed by arclengthaccording to the formula:

si+1 − si = d0(1 + si(f0 − 1))

wheresi denotes the arclength distance from nodei to the initialization point (i = 0) andd0, f0 are user selected parameters. These regridded nodes on the front are used to createthe input file for the BEM calculations and are denoted by BEM nodes.

5.5. The Algorithm

To initialize the position of the front and the velocity potential on the front, we useTanaka’s method for computing numerical exact solitary waves.

The basic algorithm can be summarized as follows:

1. Compute initial front position and velocity potentialφ(Q, 0) onΓ0.

2. Extendφ(Q, 0) onto the grid points ofΩ1 to initializeG.

3. GenerateΩt and solve (67), using the Boundary Element Method. This yields thevelocityu and source termf on the front nodes.

4. Extendu andf off the front ontoΩ1.

5. UpdateG using (69) inΩ1.

6. Move the front with velocityu using (68) inΩ1

7. Interpolate (bi-cubic interpolation)G from grid points ofΩ1 to the front nodes toobtain new boundary conditions for(67). Go back to step 3 and repeat forward intime.


A more detailed algorithm including regridding is:Initialization: GivenΓ0 = Γ0, φ

0 = φ(Q, 0)

1. CalculateΨ0 and LS nodes.

2. Extendφ0 to obtainG0.

3. Redistribute LS nodes to obtain BEM nodes.

4. Calculateu0 at BEM nodes.

5. Findu0 andf0 at LS nodes and extend ontoΩ1.

Steps: GivenΨn, φn,un

1. CalculateΨn+1 and LS nodes.

2. CalculateGn+1 in Ω1 grid points.

3. Redistribute LS nodes to obtain BEM nodes.

4. InterpolateG on BEM nodes to findφn+1.

5. Calculateun+1 at BEM nodes.

6. Findun+1 andfn+1 at LS nodes and extend ontoΩ1. Go to step 1 and repeat.

7. If reinitialization

(a) Take LS nodes and reinitializeΨn+1.

(b) Take BEM nodes and extendφn+1.

5.6. Numerical Accuracy

The model equations imply that the wave mass and its total energy should be conservedas the wave evolves in time. One way to check the numerical accuracy of the discretizedequations is to compute these quantities each time step. The wave mass abovez = 0 isgiven by

m(t) =

∫

Ωt

dΩ =

∫

∂Ωt

znzds =

∫

Γt

znzds

and the total energy isE(t) = Ep(t)+Ek(t), whereEp(t),Ek(t) denotes the potential andkinetic wave energy respectively. They can be calculated using the expressions

Ep(t) =1

2ρg

∫

Ωt

zdΩ =1

2ρg

∫

Γt

z2nzds,

which isthe potential energy with respectz = 0, and

Ek(t) =1

2ρ

∫

Ωt

∇φ · ∇φdΩ =1

2ρ

∫

∂Ωt

φ∂φ

∂nds =

1

2ρ

∫

Γt

φ∂φ

∂nds,


where the divergence theorem has been applied to the three formulas and we have used thefact ∂φ

∂n = 0 onΓb,Γ1,Γ2 for the kinetic energy formula. These integrals are approximatedby a composite trapezoidal rule, using the values of the quantities on the free boundaryBEM nodes. Note that LS nodes could have been used form(t) andEp(t) approximationsbut we also used BEM nodes for simplicity. The components of the normal vector to the freesurface are computed using the level set embedding function to obtain surface geometricalvariables.

A common procedure to study the accuracy and convergence properties of the dis-cretized equations with respect the mesh sizes and the time step is by means of an analyticalsolution. A solitary wave propagating over a constant depth is a traveling wave that movesin the x direction with speed equal to the celerity of the wave (c). The velocity potential andthe velocity on the front as functions ofx are also translated with the same speedc. There-fore, in this case, by calculating initial wave data with Tanaka’s method and translating it,we are able to compute theL2 norms of the errors for the various magnitudes. For the caseof a solitary wave shoaling over a sloping bottom, the accuracy can only be checked lookingat the mass and energy conservation properties and comparing breaking wave characteristicobtained here with those reported elsewhere, for example in [22].

6. Numerical Results

The system of equations to be discretized is a non-linear system of strongly coupledpartial differential equations. First order in time and second order in space schemes areused for equation (68); first order in time and in space schemes are used for equation (69);and a first order BEM solver is used for the velocity updating.

To study the convergence properties of this method and its capability to predict wavebreaking characteristics, the numerical results corresponding to the following physical set-tings are presented: A solitary wave propagating over a constant depth and the shoaling andbreaking of a solitary wave propagating over various sloping bottoms.

6.1. Constant Depth Test

In order to tune the discretization parameters and see how they affect numerical accu-racy we performed a series of numerical tests with a solitary wave ofH0 = 0.5 m (waveheight at the crest) propagating over a constant depth of 1 m. The wave crest is initiallylocated atx = 6.5 m and the domain hasL = 15 m of length. In what follows, the unitsare taken as meters and seconds for length and time, respectively.

Let Ω1 = [0, 15]× [−0.3, 1] be the fictitious domain that contains the free boundary forall t ∈ [0, 0.5], ∆x = ∆z the grid size and∆t the time step. To discretize∂Ωt, in order togenerate the input BEM file, a variable mesh size is used:∆l = 0.1 for Γ1 andΓ2, ∆l = 0.2for Γb, and the regridding parameters forΓt are chosen to bed0 = 0.005, f0 = 10. Thisgives193 BEM nodes on the moving front and98 nodes on the fixed boundaries.

The mesh size∆x = ∆z for Ω1 should be chosen in order to achieve accurate interpo-lated values of front position and potential on the front. For the time step selection, a firstlimitation is the CFL condition. While this condition is enough for the stability of the nu-merical approximation of equations (68) and (69), the accuracy in the numerical solution of


equation (69) requires a smaller time step. This is due to the fact thatG and thesource termf , for this particular wave problem, develops high gradients inΩ1. Therefore we presentthe results for the following test cases:

• (a)∆x = 0.1, ∆t = 0.01.

• (b) ∆x = 0.1, ∆t = 0.001.

• (c) ∆x = 0.01, ∆t = 0.001.

• (d) ∆x = 0.01, ∆t = 0.0001.

For a given solitary wave parameters (H0 and length L in thex direction) Tanaka’smethod gives us the initial wave magnitudes, front location, velocity potential, velocitycomponents at front points and wave celerityc. At any timet, let (xex, zex), φex, uex, vex

be the values of these variables obtained by translating initial values a distancect alongthe x direction and spline interpolating in LS nodes. Denote by(xc, zc), φc, uc, vc thecomputed values at LS nodes,L2(z) =‖ zc − zex ‖L2(Γt), L2(φ) =‖ φc − φex ‖L2(Γt),L2(u) =‖ uc−uex ‖L2(Γt) andL2(v) =‖ vc−vex ‖L2(Γt) theL2 norm of the errors. Table1 shows these errors at the final timet = 0.5 for the various test cases.

Table 1. Values of theL2 error norms at t = 0.5

Test L2(z) L2(φ) L2(u) L2(v)

(a) 0.007239 0.095254 0.025147 0.025856(b) 0.009762 0.021451 0.039635 0.035685(c) 0.001476 0.011363 0.0099744 0.009356(d) 0.001699 0.00424601 0.0106674 0.010188

Figures 5 and 6 showL2(z), L2(φ), L2(u), L2(v) versus time for cases (c) and (d)respectively. As observed from these results, theL2 error norm in front location and velocitycomponents decreases with mesh size (∆x) but not with the time step. Only the velocitypotential gains accuracy when∆t is reduced accordingly to the above mentioned facts.

Regarding wave mass and energy conservation, at each time step we calculatem(t) andE(t) as explained in 5.6. Figures 7 and 8 show the values of|m(t)−m(0)| and|E(t)−E(0)|versus time and same behavior of these quantities with respect discretization parameters isobserved.

Next, to see if we gain accuracy in the velocity calculations by increasing the numberof BEM nodes, we take∆l = 0.05 on Γ1 andΓ2, ∆l = 0.1 on Γb, andd0 = 0.001,f0 = 5 onΓt. This gives1720 BEM nodes on the moving front and196 nodes for the fixedboundaries. For this discretization of the bEM boundary we run two more cases:

• (e)∆x = 0.01, ∆t = 0.001.

• (f) ∆x = 0.01, ∆t = 0.0001.


Values of theL2 error normsfor case (e) and (f) are almost identical to those obtainedfor case (c) and (d) respectively, which means that accuracy in velocity is not gained byincreasing the number of bEM nodes. However, as is shown in Figure 7,|m(t) − m(0)|has decreased by almost an order of magnitude due to the accuracy in front position and theimprovement in the integral approximation to calculatem(t). Figure 9 shows for case (e)the absolute errors inEp(t), Ek(t), E(t) versus time and, in agreement with the previousdiscussion, the kinetic energy is much less accurate than the potential energy.

From these numerical experiments we conclude that the proposed algorithm converges,but we do not achieve exactly first order convergence with respect discretization param-eters. This is due to the strong interdependence of the equations. Note thatf dependsnonlinearly onu and linearly onz and that the boundary condition imposed onΓt for thebEM solver builds up numerical and round off error as we step forward in time; we note thatthe level set approach is stable and robust with respect to these small sawtooth instabilitiesresulting from velocity calculations on very closely spaced nodes, and the use of filteringor smoothing was not required.

Case (c) discretization parameters give sufficient accuracy and we show wave profiles,velocity potential and velocity components for various times in Figures 10, 11 and 12 re-spectively.

6.2. Sloping Bottom Test

A solitary wave propagating over a sloping bed changes its shape gradually, slightlyincreasing maximum height and front steepness, till a point where a vertical front tangent isreached. This is usually called the breaking point BP=(tbp, xbp, zbp), wherexbp representsthex coordinate,zbp the height atxbp andtbp the time of occurrence. Beyond the BP, thewave tip develops, with velocities much bigger than the wave celerity, causing the waveoverturning and the subsequent falling of the jet toward the flat water surface. Denote thisendpoint as EP=(tep, xep, zep). Total wave mass and total energy should be, theoretically,conserved until EP. However beyond the BP a lost in potential energy and the correspondinggain in kinetic energy is expected, due to the large velocities on the wave jet.

Wave breaking characteristics change, mainly according to initial wave amplitude(H0)and bottom topography. To study how our numerical method predicts wave breaking werun the following test cases:

• (a)H0 = 0.6, L = 25, slope=1 : 22,xc = 6.05, xs = 6

• (b)H0 = 0.6, L = 18, slope=1 : 15,xc = 5.55, xs = 5.4

and compare the results obtained here for case (b) with those reported in [21]. Herexc

denotes thex coordinate at the crest for the initial wave andxs thex coordinate where thebottom slope starts.

A series of numerical experiments have been made, and optimal discretization param-eters found are:∆x = 0.01, ∆t = 0.0001 andd0 = 0.005, f0 = 10 (approximately 193BEM nodes) for all cases. Front regridding has been made according to maximum heightbefore the BP and according to maximum velocity modulus beyond BP. Beyond the BP, anddue to the complex topography of the wave front, reinitialization ofΨ and newφ(x, z, t)extension has been performed every 1000 time steps.


Table 2 shows the breaking characteristics for the test cases. Grilli et all reported in[21] for test (b) values oftbp = 2.41, xbp = 15.64 andzbp = 0.67. The discrepancies canbe attributed to the slightly different position of the initial wave (xc = 5.5) and the higherorder approximations used in their Lagrangian-Eulerian formulation.

Table 2. Breaking characteristics

Test tbp xbp zbp tep xep

(a) 2.76 17.39 0.674 3.36 20.2(b) 2.34 15.20 0.662 2.90 17.8

In Figure 13 we showm(t) versustime for case (a) and (b) and Figures 14 and 15show the evolution ofEp, Ek andE with time for cases (a) and (b) respectively. Maximumabsolute error in wave mass is0.01 beforeBP and0.02 beyond BP and maximum absoluteerror in total wave energy is0.02 near the BP. Although this errors could be improved byincreasing the number of BEM nodes on the free boundary (as shown in the constant depthcases), it would require considerably more CPU time per run due to the high cost of theBEM solver. Regarding the evolution of the potential and kinetic energy of the wave weobserve the expected behavior beyond the BP.

Figure 16 shows wave shape for case (a) att = 0, 1, 2, 2.76, 2.94, 2.14, 3.34 and Figure17 shows wave shape for case (b) att = 0, 1, 2, 2.34, 2, 48, 2.68, 2.90. In Figures 18 and19 we show in more detail the wave profiles from the BP to the EP for cases (a) and (b)respectively. Finally in Figure 20 the front BEM nodes for case (a) and time3.34 are shown.

¿From these numerical experiments we conclude that the numerical methodpresented here is capable of reproducing wave shoaling and breaking till the touchdown

of the wave jet. Considering that we use only first order approximations of the modelequations, a piecewise linear approximation of the free boundary, and a first order linearBEM, the results are quite accurate. The absolute errors in mass and energy seem to behigher than those reported in [21]. This is not surprising due to the fact that in [21] a higherorder BEM is used (both higher order elements to define local interpolation between nodesand spline approximation of the free boundary geometry) and time integration for the freeboundary conditions is at least second order in time.

6.3. Sinusoidal Bottom Test

To see how wave shape and breaking characteristics change with bottom topography,we consider two more tests, this time with a sinusoidal shape bottom:

• (c)H0 = 0.6, L = 25, xc = 6.05,Ab = 0.5, hmin = 0.5

• (d)H0 = 0.6, L = 25, xc = 6.05,Ab = 0.8, hmin = 0.2

whereAb denote the amplitude of the sinusoidal function that represents the bottom andhmin the minimum depth.


As can be seen in Table 3, the breaking characteristics are considerably differentforthese simulations, and, in particular, case (c) behaves like a spilling breaker rather than theplunging breaker of case (a) and (b). In Figures 21 and 22 we show wave profiles for various

Table 3. Breaking characteristics

Test tbp xbp zbp tep xep

(c) 1.6 12.5 0.71 1.96 14.1(d) 1.0 10.5 0.55 1.38 13.6

times corresponding to case (c) and (d) respectively. Measurements for themass and totalenergy conservation behave similar to previous cases. In Figure 23 we show the evolutionof wave mass for cases (c) and (d). Finally, Figures 24 and 25 show the evolution ofEp,Ek

andE corresponding to cases (c) and (d) respectively.These results show that, in response to the bottom topography, wave height follows a

sinusoidal curve, as does the potential and kinetic wave energies, with an amplitude relatedto the sinusoidal bottom amplitude.

7. Conclusion

To summarize, in this chapter we have derived some physical models related to movinginterfaces in an intrinsic way, that is, independent of any coordinate system. Based onthese models a complete Eulerian description of potential flow problems for a single fluid,with or without advection-diffusion of material quantities on the front has been stablished.For the case of two-dimensional breaking waves over sloping beaches a coupled level set-boundary integral algorithm has been developed. Numerical results and convergence testsshow that even first order level set schemes produce quantitative results in a robust andefficient fashion.

Acknowledgements

All work was performed at the Lawrence Berkeley National Laboratory, and the Math-ematics Dept. of the University of California at Berkeley. First author was partially sup-ported by the Spanish Project MTM2007-65088. Second author was supported by SpanishCGL2006-06401-BTE and CGL2008-03786-BTE projects, both funded by Ministerio deEducacion y Ciencia and Fondo Europeo de Desarrollo Regional (FEDER). We want tothank E. Suarez Dıaz for his help with some of the figures.

Appendix I. The Surface Divergence in Rectilinear Coordinates

Given a vector fieldw, we want to find an expression for the surface divergenceDiv w

using rectilinear coordinates. We start from the definition:

ω(a,b)Div w := ∂aw · b × n + ∂bw · n × a, (81)


beinga andb arbitrary tangentvectors to the surface,ω(a,b) the area of the correspondingparallelogram andn the unit vector normal to the surface at the same point.

To abbreviate the computations we use indices for basis vectors:6

a1 = a, a2 = b, a3 = n. (82)

The reciprocal basis, designed byai (i = 1, 2, 3),

ai · aj = δi

j , i, j = 1, 2, 3, (83)

is calculated by the formulae:

a1 =

a2 × a3

[a1,a2,a3], a2 =

a3 × a1

[a1,a2,a3], a3 =

a1 × a2

[a1,a2,a3]= n. (84)

According todefinition (81), we have forDiv w:

Div w =a2 × a3

[a1,a2,a3]· ∂a1

w +a3 × a1

[a1,a2,a3]· ∂a2

w

= a1 · ∂a1

w + a2 · ∂a2

w = aα · ∂aα

w. (85)

In thelast expression and below we have used the summation convection: when in a mono-mial expression we have two repeated indices it must be interpreted as a summation, from1 to 2 for greek indices and from 1 to 3 for latin indices.

Notice that the basisai is in general different in each surface point. We want now toexpressDiv w using the components and coordinates in a fixed basis (global)ei and thereciprocal oneej , defined7 by the nine equationsej · ei = δj

i .We set:

ai = hjiej , a

i = f ike

k, w = wjej ; (f i

khkj = δi

j). (86)

Substituting this expressions in the last term of (85), we have:

Div w = fαk e

k · ∂hiαeiwj

ej = fαj h

iα∂iw

j . (87)

Considering thatnj = a3 · ej = f3

j andni = a3 · ei = hi3, the coefficient in the previous

result becomes:fα

j hiα = fk

j hik − f3

j hi3 = δi

j − njni. (88)

Substituting this result in equation (87), the searched expression is obtained:

Div w =(

δij − njn

i)

∂iwj . (89)

Notice that, as we have anticipated, the final result does not depend on the selected tangentvectorsa andb.

6Latin indicesi, j,... go through the values1, 2 y 3 and greek indicesα, β,... go through1 y 2. The vectorsai (i = 1, 2, 3) must accomplish the condition[a1,a2,a3] > 0.

7Whenei is an orthonormal basis (Cartesian coordinates) then it coincides with the corresponding reciprocalbasis:ei = ei.


Appendix II. The Differential Operator A

Weare going to show that the differential operator

A =1

ω(a,b)(∂a(n × j · b) − ∂b(n × j · a)) , (90)

appearing in(25), may be written using surface divergences ofj andn. To do that, weperform in the previous definition the indicated derivatives,

A =1

ω(a,b)(b × n · ∂aj + n × a · ∂bj) +

+1

ω(a,b)(b × ∂an + ∂bn × a) · j.

According todefinition (81), the first term isDiv j. Let be:

A = Div j +B. (91)

In order to identify de second termB, we select the basisai, following the specified notationin (82). As∂aα

n (α = 1, 2) are tangent vectors to the surface, we can set

∂aαn = Nβ

αaβ , α = 1, 2. (92)

Also, for the two terms inB we obtain:

j · b × ∂an

ω(a,b)= j · a2 ×N1

1a1

ω(a1,a2)= −N1

1 j · n,

j · ∂bn × a

ω(a,b)= −N2

2 j · n.

Therefore:B = −Nα

α j · n. (93)

On theother hand, asNβα = a

β · ∂aαn, makingα = β, summing and using the result (85),

we have:Nα

α = aα · ∂aα

n = Div n. (94)

Finally, substituting this result in (93) we arrive to the searched expression:

A = Div j − (Div n)j · n. (95)

Appendix III. Useful Definitions

Points and vectors. In our euclidean space we can define two useful operations. Given apointP and a displacement vectora, we defineP + a as the point that results translatingpointP by vectora. Also, given two pointsA andB, we defineA − B as a vectorc, sothat:

A−B := c ⇔ B + c = A. (96)


Directional derivative. Given a tensor fieldw = w(P, t), function of the positionP (apoint of our Euclidean space) and the timet, we define the directional derivative along thevectora as

∂aw :=d

dǫw(P + ǫa, t)

∣

∣

∣

∣

ǫ=0

. (97)

The result∂aw is a tensor of the same rank thatw. Differentiability of the fieldw impliesthat∂aw is a linear function of the vectora.

Whenw is a vector field, we use, as customary, the special notation:

a · ∇w := ∂aw. (98)

In this case,∂aw is a linear operator acting on the vectora.

Gradient of a scalar field. Let us consider a scalar fieldφ = ϕ(P, t). As ∂aφ is a realvalued linear function on the argumenta, we can define the gradient vector field,∇φ,

a · ∇φ := ∂aφ. (99)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.002

0.004

0.006

0.008

0.01

0.012L2 Errors. (H0=0.5, depth=1)

time

L2 e

rror

frontpotentialuv

Figure 5.L2(z), L2(φ), L2(u), L2(v) vs time for case (c).


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.002

0.004

0.006

0.008

0.01

0.012L2 Errors. (H0=0.5, depth=1)

time

L2 e

rror

frontpotentialuv

Figure 6.L2(z), L2(φ), L2(u), L2(v) vs time for case (d).

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.5

1

1.5

2

2.5

3x 10

−3 Absolute error in wave mass. (H0=0.5, depth=1)

time

abs(

m(t

)− m

(0))

(a)(b)(c)(d)(e)

Figure 7. Absolute error in wave mass.


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

1

2

3

4

5

6

7x 10

−3 Absolute error in total wave energy. (H0=0.5, depth=1)

time

abs(

E(t

)− E

(0))

(a)(b)(c)(d)

Figure 8. Absolute error in wave total energy.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

2

4

6

8x 10

−4 Wave Energy. (H0=0.5 depth=1)

time

abs(

E(t

)−E

(0)

EpEkE

Figure 9. Absolute error in potential, kinetic and total energy. Case (e).

0 5 10 15−1

−0.5

0

0.5

1wave shape at several times. (H0=0.5, depth=1)

x

z

Figure 10. Front location att = 0, 0.1, 0.2, 0.3, 0.4, 0.5. Case (c).


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−3

−2

−1

0

1

2

3velocity potential. (H0=0.5, depth=1)

s

pote

ntia

l

Figure 11. Velocity potential att = 0, 0.25, 0.5. Case (c).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5

0

0.5

1

1.5

2u velocity. (H0=0.5, depth=1)

s

u

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6v velocity. (H0=0.5, depth=1)

s

v

Figure 12. Velocity components att = 0, 0.25, 0.5. Case (c).

0 0.5 1 1.5 2 2.5 3

1.88

1.9

1.92

1.94

1.96

1.98

2Wave mass

time

mas

s

slope 1:22slope 1:15

Figure 13. Wave mass vs time. Case (a) and (b).


0 0.5 1 1.5 2 2.5 3 3.50.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8Wave Energy. (H0=0.6 slope=1:22)

time

E

EpEkE

Figure 14. Wave energy. Case (a).

0 0.5 1 1.5 2 2.5 30.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9Wave Energy. (H0=0.6 slope=1:15)

time

E

EpEkE

Figure 15. Wave energy. Case (b).


0 5 10 15 20 25−4

−2

0

2

4H0=0.6 slope1:22

x

z

Figure 16. Wave shape at various times. Case (a)

0 2 4 6 8 10 12 14 16 18−4

−3

−2

−1

0

1

2

3

4H0=0.6 slope1:15

x

z

Figure 17. Wave shape at various times. Case (a).

16 16.5 17 17.5 18 18.5 19 19.5 20 20.5 21−2

−1.5

−1

−0.5

0

0.5

1

1.5

2H0=0.6 slope1:22

x

z

Figure 18. Wave shape at various times. Case (a).


14 14.5 15 15.5 16 16.5 17 17.5 18−2

−1.5

−1

−0.5

0

0.5

1

1.5

2H0=0.6 slope1:15

x

z

Figure 19. Wave shape at various times. Case (b).

18 18.5 19 19.5 20 20.5 21 21.5 22−2

−1.5

−1

−0.5

0

0.5

1

1.5

2H0=0.6 slope1:22

x

z

Figure 20. Front BEM nodes at t=3.34. Case (a).


2 4 6 8 10 12 14 16 18 20−4

−3

−2

−1

0

1

2

3

4H0=0.6 , sinusoidal bottom

x

z

Figure 21. Wave shape at various times. Case (c).

2 4 6 8 10 12 14 16 18 20−4

−3

−2

−1

0

1

2

3

4H0=0.6 , sinusoidal bottom

x

z

Figure 22. Wave shape at various times. Case (d).

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

1.88

1.9

1.92

1.94

1.96

1.98

2Wave mass

time

mas

s

(c)(d)

Figure 23. Wave mass vs time. Case (c) and (d).


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9Wave Energy

time

E

EpEkE

Figure 24. Wave energy. Case (c).

0 0.2 0.4 0.6 0.8 1 1.2 1.40.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75Wave Energy

time

E

EpEkE

Figure 25. Wave energy. Case (d).


References

[1] Adalsteinsson,D., and Sethian, J.A., A Fast Level Set Method for Propagating Inter-faces,J. Comp. Phys., 118, 2, pp. 269–277, 1995.

[2] Adalsteinsson, D., and Sethian, J.A., The Fast Construction of Extension Velocities inLevel Set Methods, 148,J. Comp. Phys., 1999, pp. 2-22.

[3] Adalsteinsson, D., and Sethian, J.A., Transport and Diffusion of Material Quantitieson Propagating Interfaces via Level Set Methods,J. Comp. Phys, 185, 1, pp. 271-288,2002.

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Chapter 14

A NEW APPROACH FOR POLYDISPERSED

TURBULENT TWO-PHASE FLOWS: THE CASE

OF DEPOSITION IN PIPE-FLOWS

S. Chibbaro∗

Dept. of Mechanical Engineering, University of “Tor Vergata”,via del politecnico 1 00133, Rome, Italy

Abstract

This article is basically a review of recent works that is aimed at putting forward themain ideas behind a new theoretical approach to turbulent wall-bounded flows, no-tably pipe-flows, in which complex physics is involved, such as combustion or particletransport. Pipe flows are ubiquitous in industrial applications and have been studiedintensively in the last century, both from a theoretical and experimental point of view.The result of such a strong effort is a good comprehension of the physics underlyingthe dynamics of these flows and the proposition of reliable models for simple turbu-lent pipe-flows at large Reynolds number Nevertheless, the advancing of engineeringfrontiers casts a growing demand for models suitable for the study of more complexflows. For instance, the motion and the interaction with walls of aerosol particles, thepresence of roughness on walls and the possibility of drag reduction through the in-troduction of few complex molecules in the flow constitute some interesting examplesof pipe-flows with some new complex physics involved. A good modeling approachto these flows is yet to come and, in the commentary, we support the idea that a newangle of attack is needed with respect to present methods. In this article, we analyzewhich are the fundamental features of complex two-phase flows and we point out thatthere are two key elements to be taken into account by a suitable theoretical model:1) These flows exhibit chaotic patterns; 2) The presence of instantaneous coherentstructures radically change the flow properties. From a methodological point of view,two main theoretical approaches have been considered so far: the solution of equa-tions based on first principles (for example, the Navier-Stokes equations for a singlephase fluid) or Eulerian models based on constitutive relations. In analogy with thelanguage of statistical physics, we consider the former as a microscopic approach andthe later as a macroscopic one. We discuss why we consider both approaches unsatis-fying with regard to the description of general complex turbulent flows, like two-phase

∗E-mail address:xxxx

442 S. Chibbaro

flows. Hence, we argue that a significant breakthrough can be obtained bychoosinga new approach based upon two main ideas: 1) The approach has to be mesoscopic(in the middle between the microscopic and the macroscopic) and statistical; 2) Somegeometrical features of turbulence have to be introduced in the statistical model. Wepresent the main characteristics of a stochastic model which respects the conditionsexpressed by the point 1) and a method to fulfill the point 2). These arguments arebacked up with some recent numerical results of deposition onto walls in turbulentpipe-flows. Finally, some perspectives are also given.

1. Introduction

Turbulent flows are ubiquitous in nature. The boundary layer in the earth’s atmosphere,rivers and canals, the photosphere of the sun, the interstellar medium, most combustionprocesses, the flow of natural gas and oil in pipelines are just a few examples of turbu-lent motions. Most turbulent flows are bounded (at least in part) by one or more solidsurfaces. Examples include internal flows such as the flow through pipes and ducts; exter-nal flows such as the flow around aircraft. Since Reynolds’ experiment in 1883, pipe flowhas played an important role in the development of our understanding of turbulent flows.In particular, it is quite simple to measure the drop in pressure over a length of fully de-veloped turbulent pipe flow and hence to determine the skin-friction coefficient. LaminarPoiseuille flow occurs when a fluid in a straight channel, or pipe, is driven by a constantupstream pressure gradient, yielding a symmetric parabolic stream-wise velocity profile. Inturbulent states, the mean stream-wise velocity profile remains symmetric, but is flattenedin the center because of the increase in velocity fluctuations. A lot of research has beencarried out for turbulent wall flows, [1, 3, 4, 5] and, in particular, in the case of pipe flow,experiments for measuring the mean-velocity profile have been successfully performed atmoderate to high Reynolds numbers [6, 7]. Thus, we can say that the basic physics of theseflows is well-understood, even though the fundamental understanding of how these pro-files change as functions of the Reynolds number and of the dissipative mechanisms haveyet to be assessed. However, this is not at all such cases where some complex phenom-ena are added like combustion [35], particle dispersion (two-phase flows) [42], presenceof wall-roughness or of complex molecules which cause a drag reduction [8, 9, 10]. Inthese cases physical understanding remains limited and appears to be scarce compared tothat obtained for simpler turbulent flows. The purpose of the present work is to analyse asuitable modeling approach, which has simplified rules compared to the real phenomena,and which is used to simulate the overall and collective behaviour of a complex system.The question is therefore whether the model contains the right physics (thus the need tounderstand clearly the important phenomena) and then how to reach an acceptable com-promise between the simplicity of the model versus its physical realism (thus the need ofan appropriate formalism). In this commentary, which tries to propose an overview of re-cent modeling developments [42, 44, 47], we analyze the case of two-phase flows and, inparticular, particle deposition onto walls.

A New Approach for Polydispersed Turbulent Two-Phase Flows 443

2. A Sketch of the Physics of Turbulent Two-Phase Wall Flows

In this section, we would like to outline the present knowledge about near-wall physicsand, more specifically, the physical mechanisms which can be considered as important forparticle deposition. Generally speaking, two elements constitute the most significant signa-tures of those turbulent flows: i) the flow is chaotic ii) quasi-coherent structures are present.It is important to underline that the first point gives information on the statistical nature ofthese flows, while the second concerns the geometrical one.

For the first point, Navier-Stokes equations, which describe accurately a turbulentflow [11], represent a dynamical system with a very large number of degree of free-dom [13, 14, 15, 16, 17]. Turbulence is characterized by non-Gaussian velocity fluctuationson a wide range of scales and frequencies. The number of degrees of freedom is of the orderof Re9/4 for a Reynolds number Re that is typically105 ÷ 108. The existence of such ofwide range of scales, and of the acute sensitivity of turbulent flows to small perturbations ininitial and boundary conditions (which are never known absolutely) explain the search of astatistical description of such flows.

Turbulent structures are identified by flow visualization, by conditional sampling tech-niques, or by other eduction methodologies; but they are difficult to define precisely. Theidea is that they are regions of space and time (significantly larger than the smallest flow orturbulence scales) within which the flow field has a characteristic coherent pattern. Klineand Robinson [18] and Robinson [19] provide a useful categorization of quasi-coherentstructures in channel flow and boundary layers. The eight categories identified are the fol-lowing:

1. Low-speed streaks in the region (0 < y+ < 10).

2. Ejections of low-speed fluid outward from the wall.

3. Sweeps of high-speed fluid toward the wall.

4. Vortical structures of several proposed forms.

5. Strong internal shear layers in the wall zone (y+ < 80).

6. Near-wall pockets, observed as areas clear of marked fluid in certain types of flowvisualizations.

7. Backs: surfaces (of scale S) across which the stream wise velocity changes abruptly.

8. Large-scale motions in the outer layers.

The deposition of very small particles is mainly led by diffusion process and Brownianmotion. At the same time, it is largely accepted that particle transfer in the wall regionand also deposition onto walls are processes which are dominated by near-wall turbulentcoherent structures [20, 21, 22]. As seen above, there are many different quasi-coherentstructures in wall-flows. Among all these structures, four appear to be determinant forparticle deposition: the low-speedstreaksin the region0 < y+ ≤ 10, the Ejectionsoflow-speed fluid outward from the wall, theSweepsof high-speed fluid towards the wall

444 S. Chibbaro

andVortical structuresof various size and intensity [23, 24, 25, 26, 27, 28, 29]. More-over, several experiments have investigated particle transfer in near-wall region and theyhave found that particles tend to remain trapped along the streaks when in viscous-layer.This migration phenomenon results in overall mean fluxes and it has been referred to asTurbophoresis[30]. This name may be, however, rather misleading since it suggests theexistence of some hidden force or mechanism which bring particles towards walls. Instead,a net migration happens because the transfer of particles towards the wall is more efficientthan the transfer, due to entrainment, of particles from the wall into outer flow. Further-more, this net particle flux to the wall is related to quasi-coherent phenomena, which areinstantaneous realizations of the Reynolds stress [29, 31].

Sweeps events have been found to be strongly correlated with particle trapping in low-speed streaks, while ejections events are correlated with particle entrainment in the outerflow [27, 21]. Ejections topology has been found more efficient than sweeps events and thisexplains particle accumulation in near-wall region. Furthermore, the importance of thesemechanisms for particle deposition depends on particle inertia. In particular, light particlesfollow much more closely sweeps and ejections and their motion towards the wall appearsto be very well-correlated with turbulent structures. On the contrary, heavy particles are notso well-correlated with turbulent structures and their motion is less influenced by them inthe near-wall region.

Given the physical picture, one first important conclusion can be drawn: a model whichaims to be appropriate for the description of particle deposition in turbulent two-phase flowshas to be

(i) a statistical model, to be capable to describe the solutions of the basic equationsas being random or stochastic processes. It is necessary to cope with a reduced orcontracted description of continuous fields.

(ii) To take into account the effect of the most important geometrical structures, we be-lieve that a model which does not consider this step is likely to fail a proper andgeneral description of particle deposition in turbulent flows.

3. Modeling

Given the framework put forward in the previous section, that is a statistical one whichcan include some geometrical information, it is necessary to determine which kind of ap-proach can be included in this framework.

Let us introduce briefly the basic equations for turbulent two-phase flows [33]. Forheavy particles whereρp ≫ ρf , the drag and gravity forces are the dominant forces and theparticle equation of motion is reduced to

dUp

dt=

1

τp(Us − Up) + g. (1)

Thedrag force has been written in this form to bring out the particle relaxation time scale

τp =ρp

ρf

4 dp

3 CD|Ur|. (2)


In Eq. (1),τp appears asthe only scale, and is the time necessary for a particle to adjust tofluid velocities. In the limit whenRep ≪ 1, it is seen from the expression ofCD in thatrange that

τp =ρp

ρf

d2p

18 νf, (3)

which is the Stokes value. The drag coefficient is an empirical coefficient that can be es-timated through experiments. Various expressions have been put forward,cf. Clift et al.[32], among which an often retained form is

CD =

24

Rep

[

1 + 0.15Re0.687p

]

if Rep ≤ 1000,

0.44 if Rep ≥ 1000.(4)

The particle relaxation time scale is a non-linear function of particle properties. In theStokes regime, it is quadratic in the particle diameterdp. Outside the Stokes regime, thedependence ofτp on particle properties and variables, such asUs andUp, is more compli-cated.

Broadly speaking, there are three classes of approaches to compute (two-phase) turbu-lent flows, which we classify with regard to the level of reduction :

1. Direct numerical simulation (DNS)

2. Large eddy simulations (LES)

3. Probability density function (PDF)

4. Reynolds average Navier Stokes (RANS) methods

Using an analogy with the statistical physics language, we can say that the two first ap-proaches can be considered “microscopic”, in the sense that they a have a least degree ofmodeling. DNS is model-free and can be thought to describe correctly turbulence [12]. LESapproach is based upon the idea of modeling only some degree of freedoms, with the pur-pose of describing accurately the largest part of the degree of freedoms of the problem. ThePDF approach is “mesoscopic”. The construction of a reduced state vector can be achievedby a coarse-graining procedure where the system is described on a large enough scale toeliminate some degrees of freedom. Information is therefore lost and this lack of completeknowledge will be reflected by the use of a stochastic description for the remaining degreesof freedom. In this method, the model is not directly written in terms of macroscopic vari-ables but it is introduced at a mesoscopic level: the idea is to build a modeled equation forthe pdf. Finally, the last approach is “macroscopic”, it starts by applying some averagingor filtering operator to the exact equations and, hence, we obtain exact but unclosed meanequations in which closure relations are then introduced. Closed mean equations result.Therefore, closure is attempted directly at the macroscopic level, and when it is performedavailable information is of course strictly limited to the very macroscopic variables thathave been explicitly retained in the second step of the procedure, that is a limited number ofmoments at each point (usually not more than two). It is important to underline here that thePDF approach is different. Here, the exact instantaneous equations are replaced by models

446 S. Chibbaro

but still at the instantaneous level. Thus, in the PDF approach, the introduction ofa modelis made at an upstream level, where far more information is still available (since we modela probability density function) and has not yet been eliminated. From a practical point ofview, in PDF approach to two-phase flows mean-field equations (Rij − ǫ ) are used for thefluid, whereas a particle PDF equation is solved by a Monte Carlo method using a trajectorypoint of view. The PDF model is therefore formulated as a particle stochastic Lagrangianmodel (a set of Langevin SDEs) [35, 34]. In this formulation, this approach corresponds toa Eulerian/Lagrangian method.

There are, finally, some other techniques like Proper orthogonal (POD) [36] or group(SO(3)- SO(2)) [37] decomposition, which have been conceived specifically for the de-scription of turbulent structures and which can be linked to the above methods. These areeduction techniques, which accessing directly to some actual information can rebuild thevelocity field and identify the structure components.

All these approaches are statistical and, therefore, belong to the suitable framework.Nevertheless, the computational cost is very different among them. More specifically, mi-croscopic approaches are very demanding and, in practice, are both not of great help inengineering applications. Indeed, in the case of a large number of particles and/or of tur-bulent flows at high Reynolds numbers, the number of degrees of freedom is huge and onehas to resort to a contracted probabilistic description.

At this level, we can already point out an important drawback which concerns themacroscopic approach. When particle diameters vary considerably from particle to particleor when we are confronted with a situation where particles have completely different his-tories (highly complicated butlocal laws), deriving partial differential equations for meanquantities is a thorny issue. The case of complicated source terms happens whenever wewant to have particle evaporation or combustion with complex expressions in terms of in-dividual particle properties. When dealing with a distribution of particle diameters, one isfaced with the problem of expressing, as a function of mean velocities〈Us〉, 〈Up〉 and themean particle diameter〈dp〉, quantities such as

〈Us

τp〉, 〈Up

τp〉. (5)

These arecomplicated functions, due to the complex dependence ofτp on particle diam-etersdp and also on particle and fluid velocities, Eq. (2) and Eq. (4). In theses cases, theLagrangian PDF (mesoscopic) approach is particularly attractive since it treats these phe-nomena without approximation while the derivation of closed moment equations is next toimpossible unless very crude simplifications are introduced.

Concerning the turbulent structure one consideration and two questions will help us todetermine which is the best-suited approach, in our opinion.

The consideration is that the macroscopic approach is not able to represent properly tur-bulent structures. Beyond the technical difficulties to achieve physical meaningful closureof two-phase turbulent equations, as said above, this approach is placed at the level of meanequations and the effect of many disparate scales are modeled at the same time throughsome constitutive relations. In such methods, it appears at least courageous any tentative toadd some physical-sound term which take into account of the statistical effect of instanta-neous and zero-mean phenomena like quasi-coherent structures. Ad-hoc terms are neither


general neither under complete control and, therefore, should be considered very carefullyas a possible solution of this issue. Therefore, our first main conclusion is that macroscopicRANS approaches should be discarded as appropriate models for polydispersed two-phaseturbulent flows and, notably, for particle deposition in wall-bounded flows.

Two questions arise about turbulent structures:

1. Which structures are really important?

2. How does one must include these structures?

Actually, it is very hard to answer to the first question and it is necessary to analyse itcase-by-case. In the example discussed here, DNS simulations and experimental resultsseem to indicate that sweeps and ejections are particularly relevant. However, much smallerstructures, like worms or point-vortex, might be much more effective for explaining internalintermittency in isotropic turbulence [38].

The second question is more probing. In principle, two ways can be explored. The first,the most usual, is to compute directly the turbulent structures, that is to resolve all the scaleswhich are responsible for those geometrical features. Since quasi-coherent structures arefluid-velocity structures, this approach consists in obtaining (in some way) an instantaneousvelocity field which contains such information. It is worth emphasising that this approachis based upon the idea of calculating a very accurate instantaneous fluid-velocity field and,therefore, can be pursued only within a microscopic approach.

The second possible route is a mesoscopic one. The modeling issue is transferred to theparticle phase and, notably, to the problem of building a suitable model for the velocity ofthe fluid seen. The general form of the Langevin model chosen for the velocity of the fluidseen consists in writing

dUs,i = As,i(t,Z) dt + Bs,ij(t,Z) dWj , (6)

where the drift vectorA and the diffusion matrixB have to be modelled. The completeLangevin equation model can therefore be written

dxp,i = Up,i dt, (7a)

dUp,i = Ap,i dt, (7b)

dUs,i = As,i(t,Z) dt + Bs,ij(t,Z) dWj , (7c)

where the particle acceleration isAp,i = (Us,i−Up,i)/τp+gi. This formulation is equivalentto a Fokker-Planck equation given in closed form for the corresponding pdfp(t;yp,Vp,Vs)which is, in sample space.

∂p

∂t+

∂

∂yp,i[Vp,i p ] +

∂

∂Vp,i[Ap,i p ] +

∂

∂Vs,i[As,i p ] =

1

2

∂2

∂Vs,i∂Vs,j

[

(BsBTs )ij p

]

. (8)

In this approach, the statistical effect of the most important turbulent structures should beincluded in eq. (7)c via appropriate terms.

Some comments are in order. Although the first way can appear attractive for its accu-racy and conceptual simplicity, some major drawbacks undermines its use for engineering

448 S. Chibbaro

applications. DNS is certainly a very accurate approach since it is able to reproduce allthescales for both phases and, therefore, together with the experimental studies should beviewed as the preferred method to investigate the fundamentals of turbulent flows [39, 40].Further, since it is difficult to analyse experimentally many important quantities (frequencyand strength of intermittent phenomena, geometrical features of small scales just to makefew examples), it appear as an unavoidable searching tool mainly for modeling purposes.Indeed, physical models are based upon such information to guarantee some physical soundbasis. However, DNS is a very demanding tool and its practical use in high-Reynolds num-ber and/or geometrical complex flows is at least doubtful. LES is very similar to DNSfor wall-bounded flows in terms of computational cost and thus the same remark appliesas well [41]. Moreover, LES is a model, even though the model concerns only a part ofthe energy spectrum. In such sense, it is questionable if this approach is able to reproduceaccurately quasi-coherent structures. Of course, it does reproducesomequasi-coherentstructures but not all. In particular, small scale structures, which are modeled, are probablynot present or, at least, not necessarily well treated. POD and SO(3)-SO(2) techniques arenot predictive methods, because they need some information which is provided by previousDNS computations. In particular, SO(3) needs some ”a priori” knowledge of the whole sta-tistical properties of the velocity field at all scales. After getting it, geometry-by-geometry,one may hope for a reduction of the important degrees of freedom by keeping track only ofthose statistical correlations, isotropic or anisotropic, which are more relevant in the SO(3)decomposition These approaches still aim to compute, even though indirectly, the completevelocity field and, therefore, they join the same category of microscopic approaches. More-over, they are based upon a projection on a finite (often small for practical purposes) numberof chosen eigenstates and, thus, they constitute a contracted model of the complete field.In conclusion, none of these microscopic models appear to be adequate for engineeringapplication and in particular two-phase turbulent pipe flows.

On the other hand, mesoscopic approach is radical different. We can say that it isan “active” approach. It is based on “a-priori” choice and not on “a-posteriori” analysis.Indeed, if we know in advance the problem we want to tackle, we can analyse the physicalproblem on the basis of experimental and DNS results. On this basis, we shallchoosewhichstructures appear to be particularly relevant for our own problem. In this way, we avoid theproblem which affects LES, POD and SO(3) approaches. Then, we shall try to include thestatistical effect of those (and only those) geometrical features which seem more important.Eventually, the model will be written in terms of the known typical statistical signatures(time-frequency, characteristic length scales, life-time...). For this step, DNS simulationswill result particularly interesting.

4. Numerical Results

In this section, we try to illustrate this view with few examples describing particle depo-sition in a turbulent pipe-flow. The point of depart is the Langevin PDF approach developed


in the last decade by Minier and his collaborators [42, 34, 43]:

dxp,i = Up,idt (9)

dUp,i =1

τp(Us,i − Up,i)dt (10)

dUs,i = − 1

ρf

∂〈P 〉∂xi

dt + (〈Up,j〉 − 〈Uf,j〉)∂〈Uf,i〉

∂xjdt

− 1

T ∗

L,i

(Us,i − 〈Uf,i〉) dt

+

√

〈ǫ〉(

C0bik/k +2

3(bik/k − 1)

)

dWi. (11)

10-2

10-1

100

101

102

103

τp+

10-5

10-4

10-3

10-2

10-1

100

k p/u*

exp.

Standard

Stand. + structures

Figure 1. Deposition rate velocity for the different model used. In all numerical casesthe continuous phase is solved via standardk − ǫ model. Experimental results are givenfor reference (triangle down). The standard results are indicated by the curve labeled with“Standard model” (circles). The results obtained with the new phenomenological model areshown by the curve indicated by “Stand. + structures” (crosses). The results obtained withthe phenomenological model derived from DNS are in good agreement with experimentalresults, in particular small particles deposit only rarely.

This model was developed without taking into account quasi-coherent structures and,therefore, it fails to describe correctly the particle deposition, see fig. 1a. Thus, as simpleand first step toward considering some of the signatures of coherent structures, a new phe-nomenological model, built on the basis of DNS results, has been proposed and introduced

450 S. Chibbaro

in the numerical simulations [44]. The results obtained with this model are in good agree-ment withexperiments [45] and, hence, show that to take into account some geometricalfeatures of the flow improves significantly the statistical description of particle deposition.Following this suggestion, a more systematic introduction of geometrical features in statis-tical PDF approach, where coherent structures are introduced as new stochastic terms in themodeled equations, has been recently attempted [47]. The sketch of that model is given infigure 2a, for more details refer to the papers [46, 47].

Sweep Ejection

Diffusion

Diffusion

y+ = 100

Interface

Standard Langevin Model

Possiblereentrainment

Outer zone

Core flow

Inner zone 0.0001 0.001 0.01 0.1 1 10 100 1000

τp

+

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

k+

p

Figure 2. (a) Sketch of the stochastic model of sweeps and ejections structures.(b)Deposition ratefor the standard model (down-triangle), the new stochastic model (Fullcircle) and experimental results (star). The new model is used in a large range of diameters.The deposition rate surges with particle inertia in the range of1 < τ+

p < 70 and slightlydecreases for greater inertia. (Courtesy from Physics of fluids).

The deposition velocity computed by the new stochastic model for 12 classes of parti-cles is represented in Fig. 2b; it is compared with the experimental data gathered by Pa-pavergos and Hedley [48] and with the results provided by the standard Langevin model. Itcan be observed that the deposition rate computed by the present model is in fair agreementwith the experiments.

5. Perspectives

There are many complex flows in which geometrical features play a major role andwhose modeling is still lacking. A recent stochastic description of geometrical asperitiesfor problems of particle resuspension joins to this category [49]. Nevertheless, in author’s


opinion, the most intriguing direction to pursue is to put forward a suitable model forbubblyflows. These flows have an enormous importance in a vast spectrum of applications and arevery hard to simulate within present methods (microscopic and macroscopic). In these flowsthe interactions between quasi-coherent structures and particles is particularly relevant sinceit is well known that particles tend to concentrate within the vortexes [50], at variance withheavy particles, showing a strong preferential-concentration behaviour. The fact that bubblyparticles pass most of their time within turbulent structures should make certainly difficultto propose an accurate model without taking them into account. While, for heavy particles,it appears essential to take into account geometrical features in certain situations (like inparticle deposition), for bubbly flows it might be true for almost all flows (even for isotropicsymmetry). A recent hard effort in DNS simulation of these flows will be of great help forthe modeling.

6. Conclusions

In this commentary, which is based upon recent modeling developments, we have dis-cussed what features should characterize a suitable model for complex turbulent two-phaseflows. The most of the attention has been devoted to the case of particle deposition ontowalls in turbulent pipe-flows. However, we believe that the rationale can be applied to manyother situations and in particular to all those situations in which geometrical features havea not negligible role.

We have discussed why a suitable model should be statistical and should be easilylinked to geometrical characteristics of the flows. Then, we have analyzed the differentstatistical approaches available for the description of turbulent two-phase flows and we havetried to explain why the “mesoscopic” PDF approach should be preferred to the others. Inparticular, we have discussed in some details the issue of modeling quasi-coherent structuresand we have emphasized that an active choice of the modeler permits to avoid both too muchdemanding numerical simulations and a misleading treatment of such structures. Finally, wehave shown some recent results obtained through this PDF approach for particle depositionin a turbulent pipe-flow which lend support to this point of view.

Acknowledgments

S. Chibbaro’s work is supported by a ERG EU grant. He greatly acknowledges thefinancial support given also by the consortium SCIRE. More information is available athttp://www.consorzio-cometa.it. The author desires to thank in the most particular wayJean-Pierre Minier, since the present author’s point of view is incommensurately related tohis highest scientific and pedagogical lessons. Furthermore, I would like to thank him forinteresting suggestions for the present manuscript. I thank Dott. Mathieu Guingo for hiscourtesy in giving me figure 2.

452 S. Chibbaro

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Reviewed By Prof. Luca BiferaleProfessor of Theoretical PhysicsDept. of Physics and INFN, University of Roma, Tor VergataVia della Ricerca Scientifica 1, 00133, Roma, Italyph +39 067259.4595, fax +39 062023507http://www.fisica.uniroma2.it/ biferale/skype callto://lucabiferaleemail [email protected]; [email protected]

INDEX

A

absorption, 208, 229academic, xi, 269accidents, 366accounting, 127, 141, 277accuracy, xi, 121, 185, 214, 238, 244, 249, 259, 269,

286, 305, 357, 400, 417, 420, 421, 422, 423, 447acetylene, 166actuation, 156acute, 443adiabatic, ix, 171, 173, 177, 178, 180, 184, 188, 189,

190, 192, 193, 195, 201, 304, 305, 381advection-diffusion, 407, 411, 425AEA, 114, 166aerosol, xiii, 162, 164, 167, 441aerosols, viii, 117, 163aerospace, 344AFM, 186Africa, 30, 31, 36, 37agent, 38aggregation, 36, 166agricultural, x, 231aid, 42, 44air, 43, 80, 86, 103, 105, 106, 108, 110, 114, 167,

179, 182, 184, 185, 186, 271, 317, 344, 366, 411air pollution, 167algorithm, xii, 53, 122, 163, 186, 188, 208, 228, 399,

400, 411, 415, 416, 418, 419, 420, 423, 425alkali, 38alternative, 211, 272, 282, 309alters, 131aluminium, 283, 284, 286, 288, 289, 290, 298, 299,

301, 302, 307, 308, 309, 313aluminum oxide, 308ambiguity, 157amplitude, 368, 370, 414, 423, 424, 425Amsterdam, 315, 452anisotropy, 252anode, 22appendix, 212, 213, 222

application, ix, xii, 113, 126, 127, 128, 152, 156,158, 162, 164, 171, 175, 185, 270, 271, 272, 312,313, 344, 361, 399, 448

applied mathematics, 163aqueous solution, 8, 11aqueous solutions, 8, 11Arabia, 317Argentina, 298argument, 43, 108, 428arithmetic, 393aspect ratio, 327, 328, 336, 337, 351, 355, 360, 362aspiration, 156assessment, 146, 310assignment, 207assumptions, 45, 207, 209, 238, 302, 305, 306, 312,

389, 390, 399, 415asymptotic, 207, 208, 219asymptotically, 324atmosphere, 442atmospheric pressure, 173Australia, 41, 117, 269, 270Australian Research Council, 162availability, 43, 120, 142, 271averaging, 45, 46, 99, 238, 271, 368, 445azimuthal angle, 173

B

baths, 167beaches, 400, 411, 425, 438beams, 270, 294behavior, 29, 34, 62, 63, 64, 65, 110, 111, 168, 214,

219, 249, 257, 259, 293, 294, 334, 349, 422, 424behaviours, 129, 157benchmark, 53, 75, 111Bessel, 206, 212, 226bioreactors, 118birth, ix, 117, 118, 121, 124blocks, 185, 366, 392boiling, 129, 131, 133, 136, 137, 139, 142, 143, 144,

145, 150, 152, 153, 154, 156, 158, 159, 162, 163,164, 165, 166, 168, 169, 302, 314, 361, 362

Boston, 164

Index456

bottleneck, 159boundary conditions, xi, 53, 84, 95, 96, 143, 144,

173, 177, 178, 179, 185, 188, 189, 207, 210, 211,212, 224, 273, 279, 289, 302, 318, 383, 388, 390,394, 395, 400, 415, 416, 418, 419, 424, 443

boundary surface, 273, 417boundary value problem, 211, 418bounds, 164Boussinesq, 180Brazil, 166, 224breakage rate, 121, 135, 145, 149, 162bubble, ix, 43, 44, 50, 51, 52, 104, 105, 106, 108,

109, 110, 112, 113, 114, 118, 123, 126, 127, 128,129, 130, 131, 133, 134, 135, 136, 137, 138, 139,140, 141, 142, 143, 144, 145, 146, 147, 148, 149,150, 151, 152, 153, 154, 155, 156, 157, 158, 159,160, 161, 162, 163, 164, 165, 166, 167, 168, 169

bubbles, ix, 44, 49, 50, 51, 52, 95, 97, 103, 104, 105,106, 108, 109, 110, 112, 113, 114, 118, 127, 129,130, 131, 132, 133, 135, 136, 137, 138, 139, 143,145, 146, 147, 148, 149, 150, 151, 152, 153, 154,155, 156, 157, 158, 159, 161, 163, 164, 166, 168,169, 453

buffer, 98, 101, 103, 105, 244, 246, 261bun, 406burn, 301bypass, 289

C

cables, 185calculus, 402calibration, 38, 126, 136, 145, 160Canada, 362canals, 442candidates, 28capacity, 119, 120, 139, 172, 206, 227, 270, 288,

307, 313, 324, 334capillary, 16, 353, 362, 410, 413carbonates, 38carrier, vii, viii, 27, 29, 41, 42, 43, 44, 45, 46, 53, 56,

58, 59, 62, 64, 65, 67, 68, 75, 80, 83, 84, 88, 94,95, 100, 101, 104, 110, 111

Cartesian coordinates, 172, 179, 273, 426case study, 282cast, 186categorization, 443cathode, 22cation, 36, 114cavities, 131, 140, 152, 185C-C, 189, 190, 191, 195, 196cell, 36, 164, 167, 305, 392, 393, 394, 395CFD, viii, xi, 41, 43, 95, 120, 121, 123, 125, 132,

143, 144, 162, 166, 167, 186, 269, 270, 271, 272,274, 283, 289, 290, 293, 294, 295, 296, 297, 298,302, 306, 312, 313, 314, 316, 397

channels, vii, xi, 1, 2, 3, 5, 20, 24, 25, 33, 35, 207,210, 292, 298, 304, 343, 344, 351, 352, 353, 355,361, 362

chaos, 173, 174charge density, 3, 4chemical composition, 36chemical reactions, 9chromatography, 1, 2, 11, 13, 17, 18, 20, 21, 22, 23,

24, 25circulation, 381, 382cladding, 298, 301, 302, 308, 309, 310, 311, 312,

314classes, viii, 41, 42, 44, 53, 54, 56, 58, 59, 62, 63, 64,

111, 125, 126, 143, 144, 149, 153, 154, 155, 156,160, 163, 445, 450

classical, 259, 410classification, xi, 114, 343, 369climate change, 167clinics, 284closure, 44, 113, 122, 158, 164, 165, 233, 234, 445,

446clustering, 97coagulation, 163, 166coagulation process, 163codes, 166collisions, 46colon, 395combustion, xiii, 129, 159, 164, 165, 441, 442, 446combustion processes, 129communication, 315communities, 1competence, 158complex systems, 270complexity, xi, 44, 68, 95, 119, 150, 158, 269, 272,

276, 399components, 5, 7, 10, 121, 139, 152, 177, 178, 232,

234, 245, 246, 249, 252, 254, 263, 265, 270, 275,283, 285, 286, 289, 294, 295, 344, 345, 368, 382,395, 414, 415, 416, 421, 422, 423, 426, 431, 446

composition, 28, 36, 38, 39, 118comprehension, xiii, 441computation, 95, 97, 115, 122, 245, 271, 315, 415,

437Computational Fluid Dynamics (CFD), viii, ix, xi,

41, 117, 118, 129, 185, 186, 269, 292, 314, 316,366

computational grid, 234, 238computer technology, 127computing, 118, 122, 185, 271, 273, 278, 392, 416,

417, 418, 419, 439concentration, ix, 2, 4, 5, 9, 10, 11, 29, 35, 49, 53,

118, 126, 148, 149, 150, 153, 158, 159, 164, 301,412

concentric annuli, ix, 171, 173, 174, 175, 229concrete, 180, 181, 182, 183, 185condensation, 131, 133, 136, 152, 153, 154, 155,

158, 161conductance, 5, 7, 185

Index 457

conduction, ix, x, 137, 138, 139, 160, 171, 173, 174,176, 180, 181, 182, 183, 184, 185, 205, 208, 209,211, 219, 229, 273, 279, 283, 285, 286, 289, 305,309, 310, 321

conductivity, x, 3, 7, 172, 182, 205, 206, 208, 209,217, 224, 275, 284, 288, 290, 307, 308, 309, 312,319, 330, 340

confidence, 295configuration, viii, 42, 99, 145, 175, 189, 294, 301,

303, 401, 402, 409conformity, 64conservation, 36, 127, 143, 185, 188, 271, 274, 280,

285, 291, 294, 308, 403, 404, 421, 422, 425constraints, 127construction, 271, 272, 273, 295, 381, 389, 410, 445contact time, 52, 136, 162continuity, 45, 50, 53, 188, 210, 211, 212, 224, 225,

235, 280control, 53, 136, 145, 186, 208, 228, 236, 279, 280,

281, 282, 283, 286, 295, 299, 304, 312, 321, 322,330, 331, 385, 447

convection, ix, xi, 118, 138, 160, 165, 171, 173, 174,175, 176, 177, 180, 181, 182, 184, 185, 189, 201,205, 207, 209, 229, 269, 290, 291, 301, 302, 305,309, 416, 426

convective, ix, 53, 137, 171, 172, 173, 176, 179, 206,282, 305, 318, 362, 393, 401, 402, 409

convergence, 49, 96, 143, 179, 188, 214, 273, 286,295, 302, 306, 313, 400, 421, 423, 425

convergence criteria, 302conversion, 8cooling, xi, 35, 269, 270, 282, 283, 284, 290, 291,

297, 298, 299, 300, 302, 306, 309, 314, 319, 330,344

copper, 55, 57, 59, 60, 61, 62, 63, 64, 65, 111correlation, 47, 48, 50, 152, 339, 349, 351, 354, 356,

362correlations, xi, 42, 152, 156, 343, 353, 354, 448cosine, 7, 303, 310, 311, 312, 313costs, 42couples, 53coupling, xii, 14, 16, 43, 44, 45, 49, 53, 147, 185,

207, 379, 392covering, 105, 126, 143, 152, 271, 295CPU, 424crack, 438CRC, 341critical value, 122, 156, 180, 392cross-sectional, 137, 227, 251, 321, 351, 360, 368,

371, 372cryogenic, 344crystalline, 39crystallization, vii, 27, 33, 34crystals, 31, 33curiosity, xi, 269curve-fitting, 349cycles, 368cyclone, 113Czech Republic, 379, 397

D

damping, 239Darcy, 320, 322, 330, 340data set, 241data structure, 273death, ix, 51, 117, 118, 121, 124, 128, 159death rate, 51, 121, 128, 159decay, 301, 302, 376decomposition, 446, 448decompression, 29, 34, 35, 37decoupling, 120deduction, 110defects, viii, 27, 36, 186defense, 344definition, 14, 17, 20, 68, 138, 184, 272, 276, 279,

311, 312, 313, 351, 388, 401, 402, 403, 405, 406,409, 412, 415, 425, 426, 427

deformation, 31, 34, 35, 37, 38, 43, 234, 237, 265,276, 394, 395

degrees of freedom, 443, 445, 446, 448dehydration, 34, 35delivery, 42demand, xiii, 441Denmark, 163densitometry, 164density, viii, 2, 7, 28, 35, 36, 37, 41, 42, 43, 44, 45,

46, 50, 51, 52, 62, 63, 64, 65, 66, 95, 110, 113,122, 125, 127, 129, 133, 137, 138, 152, 154, 160,162, 165, 166, 172, 180, 275, 289, 303, 305, 306,310, 311, 313, 361, 370, 380, 387, 392, 394, 403,404, 406, 407, 411, 439, 445

dependent variable, 118, 173, 188, 189deposition, xiii, 42, 442, 443, 444, 447, 448, 449,

450, 451deposition rate, 450derivatives, 178, 401, 402, 417, 418, 427destruction, 35, 43, 324detachment, 131, 169detection, 22deuterium oxide, 284deviation, 143, 345, 353, 355diesel, 80, 111differential equations, 208differentiation, 142, 225, 226diffusion, 2, 7, 9, 10, 22, 35, 36, 53, 118, 142, 172,

209, 278, 279, 280, 282, 286, 305, 391, 400, 407,411, 443, 447

diffusion process, 36, 443diffusivity, 12, 14, 20, 142, 161, 172, 206, 209, 210,

305digital images, 293dilute gas, viii, 41, 44direct measure, 5, 20discontinuity, 28discretization, 421, 422, 423dislocations, 35

Index458

dispersion, vii, 1, 2, 3, 10, 13, 14, 15, 16, 17, 18, 19,22, 23, 25, 44, 52, 105, 112, 113, 114, 118, 133,136, 146, 160, 166, 442

displacement, 388, 395, 427distribution, ix, xii, 3, 5, 10, 15, 16, 22, 23, 51, 59,

62, 63, 97, 108, 109, 110, 112, 118, 119, 120, 121,125, 126, 129, 130, 131, 135, 143, 146, 147, 148,149, 150, 151, 152, 153, 154, 156, 157, 158, 162,165, 166, 167, 174, 177, 188, 189, 208, 210, 228,234, 244, 245, 246, 257, 259, 263, 295, 297, 302,303, 306, 307, 309, 310, 311, 312, 313, 346, 365,366, 370, 371, 372, 377, 379, 380, 381, 382, 383,385, 386, 387, 389, 395, 396, 397, 446

distribution function, 109, 110divergence, xii, 102, 280, 365, 366, 368, 373, 374,

375, 376, 377, 378, 403, 406, 421, 425diversity, 44, 118, 270dominance, 282Doppler, xi, 56, 254, 270dust, 42, 164dynamic viscosity, 275, 360dynamical system, 443

E

earth, 442eddies, 56, 271, 272, 375elasticity, 394electric conductivity, 3electric current, 2, 7electric field, vii, 1, 2, 3, 7, 11, 14, 23electrical power, 176electrical resistance, 176, 178electrolyte, 2, 3, 4, 5, 7, 11, 22, 24electromigration, 11electroosmosis, 3, 9, 11electrophoresis, 1, 2, 9, 11, 12, 15, 16, 17, 20, 21, 22,

23, 24, 25electrostatic force, 352electrostatic interactions, 2, 11email, 454energy, ix, 8, 43, 44, 46, 47, 48, 88, 94, 102, 115,

132, 139, 144, 160, 171, 173, 185, 210, 238, 241,242, 263, 271, 274, 275, 276, 279, 280, 285, 289,294, 318, 324, 344, 365, 370, 372, 390, 391, 394,420, 421, 422, 423, 424, 425, 430, 432, 436, 448

energy transfer, 238engines, x, 42, 231England, 437enlargement, 148entropy, xi, 8, 318, 322, 323, 324, 325, 326, 327,

328, 329, 331, 332, 333, 334, 335, 336, 337, 338,341

environment, 28, 38, 43, 86equilibrium, 10, 35, 56, 68, 104, 164, 182, 404equipment, ix, 42, 171, 173, 293, 294erosion, 114estimating, 142

ethane, 166ethylene, 166Euclidean space, 428Euler equations, 411Eulerian, xii, xiii, 43, 45, 49, 111, 113, 164, 399,

400, 401, 413, 414, 416, 425, 437, 441, 446evaporation, 133, 137, 151, 152, 158, 160, 165, 446evolution, xii, 118, 120, 122, 124, 126, 129, 131,

149, 163, 164, 399, 400, 404, 409, 416, 424, 425exaggeration, 375exercise, 111expansions, 208experimental condition, 68, 95extraction, 113, 419

F

failure, 308fauna, 43fax, 454feedback, 44feeding, 112film, 53, 106, 136, 160, 245film thickness, 53, 105, 136, 160financial support, 162, 451finite differences, 400finite element method, 395finite volume, xii, 53, 185, 279, 280, 295, 379, 392finite volume method, xii, 185, 279, 280, 295, 392fire, ix, 171, 173first principles, xiii, 441fission, 283, 284, 298, 299, 301, 302flatness, 234, 254, 257, 259, 263, 264, 265flexibility, 120, 121, 124, 127, 273flooding, 140, 160flora, 43flora and fauna, 43flow, vii, viii, ix, x, xi, xii, xiii, 1, 2, 3, 4, 6, 7, 8, 9,

11, 12, 13, 14, 15, 16, 17, 19, 35, 41, 42, 43, 44,45, 47, 49, 54, 56, 58, 64, 65, 66, 67, 68, 69, 70,71, 72, 73, 74, 75, 80, 82, 83, 84, 85, 86, 87, 94,95, 96, 97, 99, 100, 101, 102, 103, 104, 105, 106,107, 108, 111, 112, 113, 114, 115, 118, 119, 123,124, 126, 127, 129, 130, 131, 134, 136, 137, 141,143, 144, 145, 146, 147, 149, 150, 152, 154, 156,157, 158, 159, 160, 162, 163, 164, 165, 166, 167,168, 171, 173, 174, 175, 176, 177, 178, 179, 180,184, 185, 186, 188, 189, 194, 195, 197, 201, 205,206, 207, 208, 209, 212, 214, 215, 219, 221, 223,224, 227, 228, 229, 231, 232, 233, 234, 235, 237,238, 239, 241, 244, 249, 251, 252, 254, 257, 259,260, 263, 265, 269, 270, 271, 272, 273, 276, 277,282, 283, 284, 285, 286, 288, 289, 291, 292, 293,294, 295, 296, 297, 299, 300, 301, 302, 303, 304,305, 306, 307, 309, 313, 314, 315, 317, 318, 319,320, 323, 324, 325, 326, 327, 330, 333, 334, 336,337, 340, 343, 344, 345, 346, 347, 349, 350, 351,352, 353, 355, 356, 357, 358, 360, 361, 362, 363,

Index 459

365, 366, 367, 368, 369, 370, 371, 372, 373, 374,375, 376, 377, 379, 380, 381, 382, 383, 386, 388,390, 392, 393, 395, 396, 397, 399, 400, 404, 410,411, 415, 425, 437, 439, 441, 442, 443, 444, 450

flow behaviour, 118flow field, x, 43, 44, 45, 68, 124, 171, 174, 176, 179,

180, 184, 189, 195, 231, 233, 244, 257, 260, 272,293, 377, 415, 443

flow rate, viii, xii, 42, 96, 99, 101, 103, 104, 105,106, 107, 108, 112, 185, 206, 227, 277, 293, 295,297, 305, 307, 309, 319, 323, 325, 330, 333, 334,340, 357, 358, 365, 366, 368, 369, 370, 371, 372,373, 374, 375, 377, 383

flow value, 289fluctuations, 53, 56, 59, 68, 69, 75, 82, 95, 112, 241,

243, 246, 247, 248, 249, 252, 255, 256, 257, 258,261, 262, 263, 271, 272, 442, 443

fluid, vii, viii, ix, x, xi, xii, xiii, 1, 2, 3, 4, 5, 6, 7, 8,9, 11, 12, 14, 20, 22, 24, 27, 28, 29, 32, 34, 35, 36,37, 41, 43, 54, 56, 58, 65, 68, 80, 95, 97, 102, 106,112, 115, 120, 129, 132, 134, 142, 143, 159, 162,163, 164, 171, 173, 174, 175, 176, 177, 185, 186,201, 205, 207, 208, 209, 210, 211, 214, 217, 218,219, 221, 224, 232, 269, 271, 275, 277, 279, 288,292, 293, 294, 295, 296, 297, 305, 309, 314, 317,318, 319, 320, 321, 323, 324, 325, 326, 330, 332,333, 334, 335, 336, 337, 339, 340, 341, 343, 344,345, 346, 347, 349, 352, 353, 355, 356, 357, 358,360, 361, 362, 363, 365, 366, 368, 370, 372, 373,375, 376, 377, 399, 400, 402, 403, 404, 405, 409,411, 412, 413, 415, 425, 437, 441, 442, 443, 445,446, 447

fluid mechanics, xii, 399, 400fluid transport, 366fluidized bed, 112, 118, 129, 164, 165focusing, 120, 130, 372foils, 297Fourier, 368, 371Fox, 123, 164, 166, 167fragmentation, 166France, 164, 168, 231freedom, xii, 379, 443, 445, 446, 448freedoms, 445friction, x, xi, 95, 98, 99, 101, 105, 106, 108, 109,

114, 161, 231, 232, 233, 234, 241, 242, 244, 245,249, 261, 265, 278, 317, 318, 320, 322, 323, 327,328, 330, 334, 336, 337, 340, 343, 344, 347, 349,350, 351, 352, 353, 354, 355, 356, 358, 359, 360,361, 362, 363

FTIR, vii, 27, 29, 30, 31, 37, 38fuel, 283, 284, 285, 286, 288, 289, 290, 293, 294,

295, 301, 303, 316functional analysis, 410funding, 344

G

gas, viii, x, 2, 4, 41, 42, 45, 46, 47, 48, 49, 50, 53,54, 55, 56, 57, 58, 62, 65, 68, 69, 78, 80, 94, 95,96, 97, 99, 100, 101, 102, 103, 104, 105, 106, 107,108, 111, 112, 115, 118, 126, 127, 130, 131, 133,134, 135, 140, 143, 144, 145, 146, 147, 148, 154,156, 157, 158, 159, 163, 164, 180, 185, 208, 229,231, 233, 317, 318, 353, 404, 442

gas phase, 45, 46, 47, 48, 49, 50, 53, 54, 56, 58, 62,65, 68, 69, 80, 131, 133, 135, 143, 147

gas turbine, x, 42, 231gases, vii, 43, 179, 317, 320gasoline, x, 231gauge, 305, 307Gaussian, 122, 257, 263gel, 213generation, xi, 8, 34, 69, 133, 137, 139, 152, 154,

155, 158, 160, 178, 185, 283, 285, 315, 316, 318,322, 323, 324, 325, 326, 327, 328, 329, 331, 332,333, 334, 335, 336, 337, 338, 340, 341

geochemical, 39geochemistry, 28geophysical, 28Germany, 27, 114, 165, 168, 266glass, 56, 58, 59, 60, 61, 62, 64, 65, 66, 79, 80, 111,

366glycerol, 339grain, 35grains, 29gravitational constant, 160gravitational field, 404gravity, 46, 50, 277, 444greek, 426grid generation, xi, 269, 272, 273, 315grid resolution, 238, 239, 241, 259, 260grids, 115, 232, 238, 254, 273, 305groups, 44, 51, 52, 106, 112, 125, 126, 143, 156,

323, 333growth, 118, 119, 130, 138, 139, 140, 141, 142, 160,

161, 164, 166, 306, 344growth rate, 142growth time, 140guidelines, 299

H

half-life, 284handling, 97, 119, 120, 132, 134, 156, 158, 186, 272haze, 164heat, ix, x, xi, 115, 118, 120, 129, 131, 133, 137,

138, 139, 142, 143, 144, 150, 152, 158, 160, 163,164, 165, 167, 168, 171, 172, 173, 174, 175, 176,177, 178, 179, 180, 181, 182, 183, 184, 185, 189,201, 205, 206, 207, 208, 209, 210, 211, 212, 213,214, 215, 218, 219, 220, 221, 222, 223, 224, 227,228, 229, 231, 232, 233, 234, 269, 270, 271, 273,

Index460

275, 277, 278, 283, 284, 285, 289, 290, 291, 297,301, 302, 303, 304, 309, 310, 315, 316, 317, 318,319, 320, 321, 322, 323, 324, 325, 326, 327, 330,331, 332, 333, 334, 335, 336, 337, 339, 340, 341,344, 361, 362, 363, 392

heat capacity, 139, 206, 227, 334Heat Exchangers, 228, 341heat removal, xi, 269, 270, 301heat transfer, ix, x, xi, 115, 131, 133, 137, 152, 160,

164, 165, 167, 168, 171, 172, 173, 174, 175, 176,180, 182, 183, 184, 185, 201, 205, 206, 207, 208,212, 222, 223, 227, 228, 229, 231, 232, 233, 269,270, 271, 273, 278, 302, 304, 310, 315, 317, 318,319, 320, 321, 323, 324, 325, 326, 327, 330, 331,332, 333, 334, 335, 336, 337, 339, 340, 361, 362,363

heating, 137, 140, 150, 174, 175, 207, 219, 222, 289,302, 318, 319, 330

heavy particle, 113, 444, 451heavy water, 284, 288, 289, 298height, 2, 4, 16, 20, 22, 24, 53, 54, 57, 59, 67, 68, 69,

75, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91,92, 93, 94, 95, 97, 111, 143, 298, 349, 350, 353,354, 405, 419, 421, 423, 425

helicopters, 42helium, 186heterogeneity, 31heterogeneous, 28hexane, 353high pressure, 183high resolution, 313high temperature, 180high-speed, 443histogram, 30, 33Holland, 204homogenous, 44, 69, 114, 157, 158host, 29, 31, 32, 34, 35, 36human, 42, 43hybrid, 97, 134, 273hydration, 34, 37hydrodynamic, 2, 10, 13, 15, 16, 18, 19, 22, 25, 141,

160, 295, 352hydrodynamics, 26, 154, 158, 165, 353, 362hydrogen, 28, 36, 38, 39hydroxide, 39hyperbolic, 7, 238, 400hypothesis, 276

I

IAC, 126, 143, 149, 150, 152, 155, 156, 157, 158identification, 33identity, 238, 385images, 293, 358, 359imaging, 400imbalances, 282implementation, ix, 117, 120, 125, 127, 385inclusion, 36, 37, 167

incompressible, vii, xi, 3, 343, 345, 350, 352, 383,388, 399, 403, 411, 413, 437, 439

independence, 145independent variable, 415indication, 257, 259indices, 20, 24, 392, 408, 426industrial, viii, ix, xiii, 42, 44, 114, 117, 118, 120,

121, 122, 171, 186, 441industrial application, xiii, 441industry, 344inert, 43inertia, 43, 56, 80, 95, 346, 351, 444, 450infinite, 212, 213, 214, 222infrared, vii, 38, 39injection, viii, 22, 42, 43, 95, 96, 99, 100, 101, 102,

103, 104, 106, 107, 108, 110, 112, 131, 163injections, 42insertion, 273, 315insight, 234, 368instabilities, 419, 423instability, 122, 157, 173, 174, 175, 305, 400institutions, 344instruments, 44insulation, 143integration, x, 231, 238, 280, 383, 385, 404, 424intensity, xii, 102, 147, 149, 249, 259, 265, 365, 366,

368, 372, 373, 374, 375, 376, 377, 381, 383, 394,444

interaction, viii, xiii, 28, 34, 41, 47, 48, 51, 53, 111,113, 174, 207, 441

interactions, 2, 11, 43, 110, 115, 131, 156, 158, 186,451

interdependence, 423interdisciplinary, 42interface, 133, 177, 188, 211, 309, 310, 311, 312,

314, 400, 416, 419, 437interference, 85international markets, 299internship, 315interphase, 42interpretation, 237, 409, 412interrelations, 259interval, 51, 380, 381, 384, 385, 389, 390intrinsic, vii, 27, 28, 35, 38, 400, 403, 406, 407, 425inversion, xii, 124, 379, 380, 386, 387Investigations, x, 231investigative, 159ionic, 4, 11ions, 3, 5, 7, 95, 123, 143, 194, 197, 302, 417, 418,

422, 423iron, 39irradiation, xi, 269, 270, 272, 277, 283, 284, 285,

290, 292, 295, 297, 298, 299, 300, 301, 302, 303,304, 305, 307, 314, 316

isothermal, 45, 49, 113, 129, 131, 133, 143, 144,145, 156, 158, 163, 177, 189, 191, 192, 193, 194,195, 201

isotopes, xi, 269, 270isotropic, 51, 53, 135, 136, 237, 249, 447, 448, 451

Index 461

Italy, 113, 164, 441, 454iteration, 53, 188, 189, 386, 394, 416iterative solution, 302

J

Jacobian, 122, 172, 187Japan, 203, 205, 266Japanese, 378judge, 179Jung, 184, 204

K

kernel, 168kinetic energy, 44, 46, 48, 88, 94, 102, 160, 241,

275, 276, 279, 365, 370, 372, 390, 391, 394, 421,423, 424

King, 317, 343, 361Kolmogorov, 452

L

Lagrangian, 43, 104, 115, 400, 401, 446Lagrangian approach, 104lamellar, 31lamina, vii, x, xi, xii, 45, 162, 163, 166, 174, 179,

185, 207, 208, 214, 223, 228, 229, 231, 232, 233,235, 276, 315, 317, 318, 319, 320, 326, 330, 336,343, 344, 345, 346, 347, 349, 350, 351, 352, 353,355, 356, 358, 362, 379, 388, 390, 391, 392, 396

laminar, vii, x, xi, xii, 45, 162, 163, 166, 174, 179,185, 207, 208, 214, 223, 228, 229, 231, 232, 233,235, 276, 315, 317, 318, 319, 320, 326, 330, 336,341, 342, 343, 344, 345, 346, 347, 349, 350, 351,352, 353, 355, 356, 358, 361, 362, 379, 388, 390,391, 392, 396, 397, 442

laminated, 229land, 183language, xiii, 441, 445large-scale, 249laser, 254, 293, 294law, x, 97, 98, 180, 205, 208, 214, 224, 233, 324,

354, 375, 403, 404, 406, 407laws, 400, 446lead, xi, 7, 42, 68, 69, 94, 111, 195, 255, 269liberation, viii, 27, 34, 35, 37licensing, 285life forms, 43lifetime, 176, 201likelihood, 148limitation, 156, 158, 359, 421limitations, ix, xii, 44, 118, 120, 123, 350, 399linear, 124, 139, 175, 186, 213, 244, 249, 278, 321,

327, 336, 339, 340, 393, 394, 399, 404, 417, 418,419, 424, 428

linear function, 428liquid film, 105, 106liquid nitrogen, 362liquid phase, 49, 50, 51, 68, 69, 101, 103, 104, 126,

133, 135liquids, vii, 113, 163, 228, 317, 318, 320, 339lithosphere, 39location, 3, 59, 63, 65, 102, 103, 118, 146, 148, 149,

219, 220, 227, 301, 377, 388, 422, 430London, 26, 228, 318, 326, 330, 336, 341, 351, 353,

361longevity, 42long-term, 28losses, xi, 43, 318, 319, 362, 366LSM, 400lubrication, 133, 146, 149, 157, 160lying, 409, 419

M

machinery, 365machines, x, 231, 366magma, vii, 27, 28, 29, 34, 35, 37, 39magnesium, 28, 37, 143, 283, 285, 288, 290maintenance, 42manipulation, 225mantle, vii, viii, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38,

39manufacturing, 400mapping, 380, 385, 386, 401markets, 299mass loss, 410mass transfer, x, 50, 118, 120, 128, 129, 131, 132,

133, 150, 158, 162, 168, 208, 228, 231, 316mass transfer process, x, 150, 158, 231matrix, 32, 36, 80, 111, 123, 124, 161, 225, 279, 282,

295, 299, 307, 313, 418, 447measurement, 148, 152, 163, 294, 295, 352, 358,

368, 372measures, 401meat, 299, 301, 306, 307, 309, 311, 312, 314mechanical energy, 318media, 185medications, 42medicine, 270melt, vii, 27, 28, 29, 33, 34, 35, 37, 285, 301melting, 28, 290, 301, 314melts, vii, 27, 33, 34, 37, 38, 39MEMS, 344mesoscopic, xiii, 442, 445, 446, 447, 448, 451metals, 356metric, 53, 172, 187Mexico, 39Mg2+, 36microbial, 118microelectronics, 344microscope, 358microscopy, 29

Index462

microstructures, 38microtubes, 353, 355, 357, 359, 362, 363migration, 11, 22, 23, 25, 444military, 344mimicking, 69mineralogy, 39minerals, 28, 38miniaturization, 344Ministry of Education, 397misleading, 337, 444, 451MIT, 266mixing, 56, 58, 80, 95, 113, 114, 186, 233, 299mobility, 2, 5, 6, 7, 9, 12, 13, 14, 20, 22, 25modeling, xi, xiii, 47, 115, 122, 164, 167, 186, 233,

234, 269, 270, 277, 302, 306, 314, 345, 439, 441,442, 445, 447, 448, 450, 451

models, vii, viii, ix, x, xi, xii, xiii, 42, 43, 44, 46, 95,97, 107, 108, 112, 113, 114, 118, 127, 130, 131,134, 146, 149, 156, 158, 159, 164, 166, 167, 185,186, 232, 233, 234, 259, 260, 265, 270, 271, 272,283, 295, 302, 309, 314, 315, 316, 340, 399, 400,425, 441, 445, 447, 448

modulation, 46, 58, 62, 63, 64, 65, 103, 108, 118modulus, 251, 395, 419, 423molar conduct, 3molar volume, 36molecules, xiii, 28, 119, 347, 441, 442molybdenum, xi, 269, 284, 299, 300, 301, 314MOM, ix, 117, 121, 122, 124, 126momentum, 42, 43, 44, 45, 49, 50, 53, 87, 95, 118,

132, 143, 144, 147, 148, 159, 209, 234, 235, 238,271, 274, 275, 280, 285, 294, 295, 315, 347, 389,403

monotone, 220Monte Carlo, 113, 121, 185, 303, 446Monte Carlo method, 185, 446Moscow, 27, 37, 39motion, xiii, 3, 6, 7, 9, 42, 43, 44, 68, 86, 95, 104,

112, 118, 131, 137, 159, 185, 190, 197, 236, 239,265, 368, 400, 441, 443, 444

motivation, 260movement, 395, 401, 405, 406, 407, 409multidimensional, 113, 121multiphase flow, ix, 113, 117, 118, 119, 120, 122,

125, 127, 128, 134, 164, 167, 168

N

nanofabrication, vii, 1nanometers, 3, 31, 33nanotubes, 25NASA, 229national, 270natural, vii, ix, 27, 38, 39, 42, 119, 171, 173, 174,

175, 176, 177, 179, 180, 184, 185, 189, 201, 209,302, 385, 442

natural gas, 442

Navier-Stokes, xii, xiii, 44, 271, 379, 390, 437, 441,443

Navier-Stokes equation, xii, xiii, 379, 390, 437, 441,443

neglect, 182negotiating, 271net migration, 444neutrons, 284, 298, 301, 302New York, 26, 167, 203, 228, 229, 315, 341, 342,

361, 397, 452, 453Newton, 214, 393, 403Newtonian, x, 205, 208, 214, 221, 224, 229nitrogen, 353, 362nitrogen gas, 353nodes, 273, 282, 305, 387, 416, 419, 420, 421, 422,

423, 424, 434nodules, vii, 27, 28, 29nonlinear, 186, 238, 271, 393, 400, 416, 421, 445non-Newtonian, x, 205, 208, 214, 224, 229non-Newtonian fluid, x, 205, 208, 214, 224, 229non-uniform, vii, 1, 10, 16, 51, 174, 185non-uniformity, 10, 16normal, 36, 37, 101, 102, 103, 112, 139, 140, 142,

145, 161, 188, 246, 277, 279, 281, 295, 298, 303,345, 380, 392, 405, 406, 417, 418, 421, 426

normalization, 383norms, 421, 422, 423nuclear, xi, 113, 164, 269, 270, 283, 284, 298, 302,

314nucleation, 34, 35, 118, 119, 131, 133, 137, 138,

139, 142, 152, 158, 160, 161, 164, 165, 166nuclei, 118nucleus, 139numerical analysis, 185Nusselt, 133, 152, 174, 175, 182, 185, 206, 211, 213,

214, 220, 221, 222, 228, 340

O

observations, viii, xi, 27, 33, 35, 37, 104, 108, 131,134, 145, 151, 152, 179, 234, 244, 248, 249, 263,270, 293

oceans, 43OECD, 168OH-groups, 28oil, 75, 80, 111, 317, 442operator, xii, 225, 379, 380, 381, 386, 387, 388, 389,

390, 394, 395, 397, 402, 404, 427, 428, 445optical, 29optical microscopy, 29optics, 294optimization, vii, 1, 114, 305order statistic, 265organization, 252orientation, 257, 265orthogonality, 226oscillation, 34, 35, 37, 366oscillations, 249

Index 463

oxide, 143, 283, 285, 288, 290, 306, 307, 308, 309,312, 313, 314

oxide thickness, 313, 314oxygen, 28

P

pairing, 112paper, viii, 41, 43, 49, 53, 67, 110, 113, 120, 203,

234, 270, 315, 363, 416parabolic, 6, 234, 239, 317, 442parameter, 5, 44, 54, 65, 68, 111, 150, 323, 333, 346,

380, 383, 385, 389, 395Paris, 113partial differential equations, xii, 179, 273, 280, 379,

399, 400, 411, 416, 421, 446particle nucleation, 164particles, viii, xii, xiii, 41, 42, 43, 44, 45, 53, 54, 55,

56, 57, 58, 59, 62, 63, 64, 65, 66, 67, 68, 69, 79,80, 83, 85, 86, 87, 88, 94, 95, 111, 113, 114, 117,118, 119, 121, 122, 125, 128, 159, 162, 164, 347,368, 399, 401, 402, 403, 405, 407, 409, 441, 443,444, 446, 449, 450, 451

partition, 133, 137, 139, 142, 152, 158, 168pathways, 33Peclet number, 2, 14, 24pedagogical, 451performance, 20, 25, 186, 207, 233, 259, 317, 318,

366periodic, 238, 366, 370, 375permeability, 96permittivity, 3, 4personal, 366, 367perturbations, 443petrographic, 29, 33Petroleum, 317pharmaceutical, 343philosophy, 301physical mechanisms, 352, 443physical properties, 28, 209, 283, 306, 313, 347, 395physics, xiii, 42, 110, 112, 167, 185, 283, 285, 298,

441, 442, 443, 445pipelines, 442planar, 315plants, 42plausibility, 36play, viii, 42, 450point defects, 35Poisson, 4, 5, 381, 395Poisson equation, 4Poisson-Boltzmann equation, 5pollen, 42pollution, 43, 167polymer, 42polymers, 43polynomial, 387, 388polystyrene, 368poor, 112, 410

population, vii, viii, ix, 51, 113, 114, 117, 118, 119,120, 121, 122, 124, 125, 126, 128, 129, 131, 149,152, 158, 159, 163, 164, 165, 166, 167, 168

porous, 95Portugal, 203, 266potential energy, 420, 423powder, 143, 290, 293power, viii, x, xi, 41, 42, 43, 118, 143, 176, 178, 180,

181, 182, 183, 184, 185, 205, 208, 214, 224, 269,270, 271, 289, 290, 301, 302, 303, 306, 307, 310,311, 312, 313, 314, 315, 317, 318, 319, 325, 326,328, 329, 335, 336, 337, 339, 340, 344

power plant, 42power plants, 42power-law, x, 205, 206, 208, 214, 217, 218, 222,

224, 229powers, 303pragmatic, 43, 271Prandtl, 46, 135, 162, 172, 174, 175, 275, 276, 315,

340precipitation, viii, 117prediction, 56, 62, 75, 80, 97, 112, 134, 145, 147,

148, 149, 152, 156, 158, 159, 162, 168, 169, 298,301, 361, 366

pressure, xii, 2, 4, 6, 11, 14, 16, 23, 27, 28, 33, 34,35, 37, 45, 49, 50, 53, 56, 80, 94, 95, 96, 141, 143,148, 160, 165, 172, 173, 183, 236, 270, 275, 277,282, 295, 305, 307, 309, 317, 318, 322, 332, 346,347, 349, 350, 352, 353, 355, 357, 359, 360, 362,365, 366, 367, 368, 369, 370, 371, 372, 375, 376,377, 379, 380, 381, 387, 388, 389, 392, 394, 395,396, 404, 442

printing, 400probability, 121, 263, 446probability density function, 121, 446probability distribution, 263probable cause, 146probe, 43, 143, 245, 294, 366production, 47, 48, 51, 102, 128, 160, 270, 276, 277,

298, 299, 344, 391program, 121, 294, 295, 302propagation, 400, 411property, 273, 280, 317, 319, 320, 326, 330, 336,

387, 410, 412proportionality, 406proposition, xiii, 441protection, ix, 171, 173, 298prototype, 270, 283, 286, 293, 295PSD, 120, 122, 123, 124, 125, 126, 129, 132pseudo, 385pumping, xi, 317, 318, 319, 325, 326, 328, 335, 336,

337, 362pumps, 344, 366purification, viii, 27, 36

Q

quadtree, 273

Index464

R

radial distribution, ix, 118, 143, 158, 227, 245radiation, ix, 171, 172, 173, 176, 180, 183, 184, 185,

298, 301radical, 156, 157, 360, 448radiopharmaceutical, 270, 283radiotherapy, xi, 269radius, vii, 1, 139, 140, 142, 160, 161, 182, 183, 206,

209, 211, 232, 235, 286, 377random, 52, 136, 157, 229, 345, 346, 444range, x, 8, 15, 24, 42, 43, 44, 80, 102, 103, 110,

112, 125, 126, 131, 142, 143, 145, 152, 156, 158,179, 205, 208, 211, 214, 233, 259, 271, 277, 278,317, 318, 337, 349, 350, 353, 355, 372, 443, 445,450

Rayleigh, 172, 173, 174, 175, 179, 180, 183, 184,195, 400

reactants, 36, 42reaction rate, 298realism, 312, 442recovery, 272, 366recrystallization, 37rectilinear, 425recycling, 39reduction, viii, xiii, 11, 13, 18, 42, 43, 98, 99, 101,

105, 108, 110, 112, 114, 138, 233, 234, 239, 244,245, 249, 251, 265, 305, 310, 373, 441, 442, 445,448

refining, 54, 110reflection, 257reforms, 137refrigeration, 344refrigeration industry, 344regulatory requirements, 271relationship, 152, 213, 238, 278, 366, 404relationships, 35, 404relaxation, 46, 54, 68, 444, 445relaxation time, 46, 54, 68, 444, 445relevance, 121reliability, xi, 244, 269, 270, 293, 295research, ix, xi, xii, 1, 42, 119, 131, 152, 159, 171,

173, 185, 269, 270, 283, 298, 343, 344, 356, 369,442

research and development, 1researchers, 44, 344, 352, 356, 360reservoir, 143resistance, 176, 178, 209, 315resolution, vii, 1, 2, 20, 24, 98, 126, 156, 158, 246,

260, 271, 282, 283, 290, 309, 313, 419resources, 119, 126, 185, 271, 283, 290, 302, 306response time, 80retardation, 9retention, 20Reynolds, viii, x, xii, xiii, 41, 42, 46, 50, 51, 53, 54,

64, 67, 101, 102, 105, 111, 161, 174, 175, 185,186, 207, 223, 229, 231, 232, 233, 234, 236, 238,239, 244, 245, 249, 250, 254, 259, 260, 261, 262,

263, 264, 265, 275, 276, 278, 314, 315, 320, 340,346, 347, 348, 349, 351, 352, 353, 354, 355, 356,360, 363, 365, 368, 370, 373, 374, 395, 396, 397,441, 442, 443, 444, 445, 446

Reynolds number, viii, x, xii, xiii, 41, 42, 50, 51, 53,54, 64, 67, 101, 105, 111, 161, 174, 175, 185, 207,223, 231, 232, 233, 234, 236, 238, 239, 244, 245,249, 254, 259, 260, 261, 263, 264, 265, 278, 314,315, 320, 340, 346, 347, 348, 349, 351, 352, 353,354, 355, 356, 360, 363, 365, 368, 370, 373, 374,395, 396, 397, 441, 442, 443, 446

Reynolds stress model, 185, 186rheology, 208, 221rivers, 442robustness, 158, 282, 295, 314Rome, 441room temperature, 173, 339roughness, xi, xiii, 315, 343, 346, 347, 348, 349,

350, 351, 352, 353, 355, 356, 357, 358, 359, 360,363, 441

rubber, 183Russia, 27, 37Russian, 27, 37, 39Russian Academy of Sciences, 27, 37

S

safety, viii, xi, 41, 45, 113, 164, 269, 282, 283, 285,289, 296, 303, 306, 313

sample, 30, 32, 33, 37, 447sampling, 443sand, 42saturation, 131, 162Saudi Arabia, 317scalar, 125, 126, 133, 160, 185, 404, 405, 409, 428scalar field, 404, 409, 428scaling, 5, 68, 276scatter, 62, 294, 368scientists, 43SCP, 113, 163, 168search, 273, 443searching, 448seeding, 306segregation, 31, 35selecting, 280selectivity, vii, 1, 2, 20, 21, 23, 24SEM, 358, 359semiconductor, 270, 367, 400sensitivity, 283, 312, 443sensors, 344separation, vii, 1, 2, 10, 20, 22, 23, 24, 25, 26, 34,

35, 37, 156, 272, 296, 344, 366, 376, 389series, 119, 125, 126, 143, 213, 214, 302, 368, 371,

421, 423shape, x, 14, 31, 43, 136, 149, 159, 231, 239, 251,

346, 351, 358, 381, 387, 388, 390, 394, 396, 405,411, 417, 418, 423, 424, 430, 433, 434, 435

sharing, 111

Index 465

shear, 50, 68, 69, 94, 95, 106, 111, 112, 115, 134,141, 142, 153, 159, 160, 161, 163, 164, 214, 232,234, 249, 250, 262, 263, 265, 276, 278, 316, 347,349, 350, 375, 443

shear rates, 214sign, 48, 386silica, 39, 352, 353, 355, 359silicate, 35, 38silicates, 28, 37silicon, 270similarity, 19, 64, 214simulation, ix, x, 11, 42, 43, 44, 45, 80, 99, 104, 106,

110, 112, 114, 123, 126, 163, 166, 167, 169, 171,174, 175, 185, 186, 201, 231, 233, 237, 244, 245,257, 259, 270, 271, 282, 288, 289, 290, 295, 302,306, 308, 312, 313, 445, 451

simulations, viii, ix, x, 41, 49, 95, 96, 99, 106, 108,117, 121, 123, 126, 127, 129, 130, 143, 144, 145,150, 156, 159, 177, 231, 233, 234, 238, 241, 244,249, 252, 254, 271, 283, 289, 302, 304, 309, 310,312, 315, 411, 425, 438, 445, 447, 448, 450, 451

singular, 417, 418, 419, 437, 438singularities, 387sites, 131, 137, 138, 161skewness, 234, 254, 255, 257, 259, 263, 264, 265skin, 98, 99, 101, 105, 106, 108, 109, 114, 233smoothing, 298, 423smoothness, 415software, 125, 312, 393soil, 181, 182, 183, 185solar, ix, 171, 173solar energy, ix, 171, 173solid phase, 277solubility, 33, 38solutions, ix, 2, 20, 24, 43, 117, 120, 121, 144, 188,

208, 214, 215, 228, 229, 270, 272, 273, 289, 439,444

soot, 128, 159, 164, 166South Africa, 30, 31, 36, 37Southampton, 316, 438Spain, 399spatial, 22, 118, 121, 132, 207, 236, 238, 400, 401,

402spatial location, 118species, 2, 3, 9, 10, 20, 22specific heat, 159, 172, 319, 330Specific Heat, 288, 307, 313spectrum, 118, 143, 448, 451speed, 2, 9, 10, 11, 12, 13, 14, 20, 22, 23, 24, 25,

156, 233, 283, 377, 380, 415, 421, 444stability, x, 175, 186, 231, 233, 234, 390, 410, 421,

439stages, 29, 119, 295stainless steel, 352, 355, 356, 357, 359standard deviation, 3, 23standard model, 450standards, xi, 269statistical mechanics, 164statistics, x, 231, 233, 234, 238, 254, 260, 265

steady state, 53, 96, 143, 179, 302, 415steel, 143, 352, 353, 355, 356, 357, 359stochastic, xiii, 51, 112, 135, 163, 229, 442, 444,

445, 446, 450stochastic model, xiii, 442, 450stochastic processes, 444stoichiometry, 36storage, 28, 39, 43, 185, 238, 282, 301strain, 47, 186, 232, 237, 394, 395strategies, 438streams, 207, 208, 213, 221, 228strength, 448stress, x, 45, 102, 106, 115, 153, 161, 185, 186, 231,

232, 233, 234, 238, 249, 250, 265, 275, 276, 278,347, 349, 350, 391, 392, 394, 411, 444

stroke, 366students, 315subsonic, xii, 379, 386, 397Sun, 126, 157, 166, 168, 186, 204superiority, 286supply, 143, 271, 283, 299suppression, 167, 233, 239, 265surface area, 50, 51, 118, 121, 122, 135, 280, 318,

327, 337, 405surface diffusion, 411surface region, 407surface roughness, xi, 315, 343, 349, 350, 352, 355,

357, 358, 360surface tension, xii, 141, 159, 160, 162, 361, 399,

411surfactant, 42, 411, 412surfactants, xii, 43, 399, 400, 411, 439swarm, 156, 168swelling, 377symbolic, 437symbols, 6, 242, 243, 255, 360, 389, 395symmetry, 4, 11, 143, 255, 257, 383, 408, 451synchronous, 373systems, 42, 52, 53, 118, 119, 120, 121, 122, 128,

136, 159, 165, 166, 167, 169, 270, 271, 273, 300,314, 315, 343, 344, 365, 392

T

Taiwan, 171talc, 28, 31, 33, 34, 35, 38tangible, 42tanks, 118, 271, 301targets, xi, 269, 293, 295, 297, 299, 301, 302, 303,

304, 306, 307, 313, 314, 315technology, 165TEM, vii, 27, 28, 29, 31, 33, 35, 38temperature, ix, x, xi, 2, 4, 28, 39, 131, 137, 139,

143, 161, 169, 171, 172, 173, 174, 175, 178, 179,180, 183, 184, 185, 188, 194, 195, 197, 200, 201,205, 206, 207, 208, 209, 211, 212, 213, 217, 218,219, 220, 221, 224, 227, 275, 277, 283, 285, 289,290, 291, 298, 302, 303, 304, 305, 306, 307, 308,

Index466

309, 310, 311, 312, 313, 314, 317, 318, 319, 320,321, 324, 325, 326, 330, 331, 332, 333, 334, 335,336, 337, 339, 340, 341, 394

temperature dependence, xi, 318temperature gradient, 188, 219, 275, 321tensor field, 400, 401, 402, 428Texas, 224textbooks, 119, 392, 394theory, xii, 46, 47, 115, 233, 352, 353, 355, 356, 379,

381Thermal Conductivity, 279, 288, 307, 313thermal equilibrium, 182thermal expansion, 172thermodynamic, 318thermodynamic parameters, 318thermo-mechanical, 154, 158thin film, 105three-dimensional, ix, 132, 144, 158, 171, 173, 174,

175, 177, 179, 184, 189, 201, 233, 280, 292, 294,302, 316

threshold, 22time, x, xi, 2, 3, 9, 22, 23, 34, 37, 43, 44, 46, 52, 54,

63, 68, 85, 96, 105, 114, 121, 126, 132, 136, 138,140, 142, 143, 161, 185, 231, 238, 269, 271, 273,275, 280, 281, 286, 290, 299, 301, 302, 305, 306,345, 368, 369, 370, 372, 373, 383, 392, 394, 400,401, 402, 403, 404, 405, 406, 407, 409, 410, 411,412, 413, 416, 417, 418, 419, 420, 421, 422, 423,424, 428, 429, 430, 431, 432, 435, 436, 443, 444,445, 446, 451

time consuming, 418tolerance, 309topological, xii, 399, 400, 417topology, 409, 412, 444total energy, 420, 423, 430tracers, 80tracking, xii, 43, 44, 115, 122, 126, 149, 399, 412trajectory, 123, 412, 446transducer, 367transfer, ix, x, xi, 42, 46, 50, 51, 115, 118, 126, 131,

133, 137, 147, 152, 159, 160, 164, 165, 167, 168,171, 172, 173, 174, 175, 176, 180, 181, 182, 183,184, 185, 201, 205, 206, 207, 208, 212, 222, 223,227, 228, 229, 231, 232, 233, 238, 269, 270, 271,273, 278, 302, 304, 310, 315, 317, 318, 319, 320,321, 323, 324, 325, 326, 327, 330, 331, 332, 333,334, 335, 336, 337, 339, 340, 347, 361, 362, 363,443, 444

transformation, viii, 27, 35, 38, 121, 122, 380, 381,415

transition, x, xi, 28, 143, 146, 156, 157, 158, 168,173, 174, 175, 179, 185, 231, 233, 343, 344, 346,347, 349, 352, 353, 355, 356, 358, 362

transitions, 163, 305transparency, 284transparent, xi, 270, 291, 293, 366transport, vii, xiii, 1, 2, 3, 4, 9, 10, 11, 25, 33, 42, 44,

46, 47, 48, 53, 123, 124, 126, 132, 134, 144, 157,

164, 165, 166, 168, 185, 209, 233, 234, 271, 276,280, 281, 289, 303, 344, 362, 366, 400, 441

transpose, 225transverse section, 178, 189, 194, 197, 200travel, 51, 52, 87, 123, 124, 131trend, 18, 80, 87, 94, 101, 102, 221, 249, 257, 263,

297, 303, 317trial, 320trial and error, 320triangulation, 273turbulence, viii, x, xi, xii, xiii, 41, 42, 44, 46, 47, 48,

49, 51, 52, 53, 58, 59, 63, 64, 65, 69, 94, 95, 97,102, 104, 105, 108, 111, 112, 113, 114, 115, 118,122, 134, 135, 136, 145, 146, 147, 149, 166, 167,175, 185, 186, 201, 231, 232, 233, 234, 237, 239,249, 251, 259, 261, 263, 265, 269, 271, 272, 276,295, 302, 314, 315, 316, 350, 365, 368, 372, 373,374, 375, 376, 377, 379, 390, 391, 392, 395, 442,443, 445, 447

turbulent, vii, viii, x, xi, xii, xiii, 41, 42, 43, 44, 45,46, 47, 48, 51, 52, 53, 58, 62, 66, 88, 94, 95, 97,99, 102, 103, 110, 111, 112, 113, 114, 115, 118,129, 131, 133, 134, 135, 136, 138, 143, 145, 146,149, 153, 156, 157, 158, 160, 161, 162, 164, 166,168, 174, 179, 185, 186, 207, 209, 229, 231, 233,234, 235, 236, 237, 239, 241, 243, 244, 245, 247,249, 253, 254, 255, 257, 259, 260, 261, 263, 265,266, 267, 269, 271, 272, 275, 276, 277, 278, 279,280, 289, 290, 304, 305, 314, 315, 318, 319, 320,330, 342, 343, 344, 345, 346, 347, 349, 352, 353,355, 358, 362, 365, 366, 367, 369, 371, 373, 375,377, 379, 390, 391, 392, 393, 394, 441, 442, 443,444, 445, 446, 447, 448, 449, 451, 453

turbulent flows, xi, xiii, 110, 179, 186, 207, 260,277, 278, 314, 315, 343, 344, 345, 347, 353, 355,441, 442, 443, 444, 445, 446, 447, 448

turbulent mixing, 377two-dimensional (2D), ix, 25, 126, 130, 171, 173,

174, 175, 179, 180, 181, 184, 185, 188, 294, 362,366, 425

two-way, 43, 44, 53

U

ubiquitous, viii, xiii, 27, 37, 441, 442Ukraine, 27uncertainty, 146, 345uniform, xi, 4, 10, 15, 16, 53, 59, 62, 63, 96, 144,

186, 207, 214, 295, 297, 309, 310, 311, 312, 318,319, 331, 346, 349, 403, 404

universal gas constant, 4updating, 421uranium, xi, 269, 270, 285, 288, 297, 299, 301, 302uranium oxide, 285uti, 52, 136

Index 467

V

vacancies, 36valence, 3validation, ix, 111, 118, 120, 125, 134, 163, 234,

241, 265, 270, 302, 306, 307, 312, 313, 315, 344validity, 11, 43, 283values, 6, 20, 30, 33, 59, 62, 63, 80, 81, 98, 135, 188,

212, 214, 217, 218, 222, 223, 227, 233, 238, 239,245, 249, 252, 255, 257, 263, 265, 273, 276, 277,280, 297, 302, 305, 324, 334, 335, 346, 349, 351,352, 356, 360, 368, 370, 372, 377, 380, 388, 391,392, 393, 394, 395, 396, 408, 415, 416, 418, 419,421, 422, 424, 426

vapor, 159, 160, 161, 162, 169variable, 5, 38, 125, 131, 173, 189, 210, 229, 280,

288, 313, 410, 415, 421variables, 49, 50, 53, 118, 120, 121, 123, 125, 132,

156, 235, 237, 277, 387, 390, 391, 392, 394, 413,415, 421, 422, 445

variance, 22, 451variation, 17, 18, 29, 119, 148, 195, 201, 217, 219,

223, 244, 263, 272, 302, 317, 318, 320, 321, 327,330, 331, 335, 336, 339, 340, 352, 369, 370, 371,373, 375, 377, 401

vector, 50, 121, 124, 160, 161, 189, 190, 191, 192,195, 197, 198, 275, 277, 292, 382, 383, 391, 392,394, 402, 403, 404, 405, 406, 407, 421, 425, 426,427, 428, 445, 447

velocity, ix, x, xi, xii, 4, 5, 6, 7, 10, 11, 12, 14, 15,16, 19, 22, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54,55, 56, 58, 59, 60, 67, 68, 79, 80, 84, 88, 94, 95,96, 97, 98, 100, 101, 102, 103, 104, 105, 106, 107,108, 111, 114, 118, 121, 123, 124, 127, 129, 131,136, 138, 142, 143, 144, 147, 154, 156, 157, 158,161, 166, 168, 172, 177, 178, 187, 188, 189, 190,195, 197, 205, 206, 208, 209, 210, 214, 217, 224,227, 231, 232, 233, 234, 235, 236, 237, 239, 240,241, 242, 243, 244, 245, 246, 247, 248, 249, 251,252, 253, 254, 255, 256, 257, 258, 259, 261, 262,263, 264, 265, 270, 275, 276, 277, 278, 282, 291,292, 293, 294, 295, 296, 297, 298, 303, 304, 305,306, 314, 317, 318, 340, 345, 346, 360, 365, 366,367, 368, 371, 372, 373, 374, 375, 377, 380, 381,382, 383, 384, 385, 386, 387, 388, 389, 392, 394,

399, 400, 402, 403, 405, 409, 411, 412, 413, 414,415, 416, 417, 418, 419, 421, 422, 423, 431, 442,443, 446, 447, 448, 449, 450

ventilation, 176versatility, 314Victoria, 117viscosity, xi, 3, 4, 9, 17, 44, 45, 46, 48, 49, 51, 95,

97, 104, 114, 134, 135, 145, 161, 167, 172, 185,207, 214, 232, 237, 272, 275, 276, 315, 317, 318,319, 320, 324, 325, 330, 333, 334, 335, 339, 340,349, 350, 368, 387, 391, 394, 397

visible, 35, 293, 368visualization, 108, 292, 293, 296, 368, 376, 443voids, 32, 33, 35vortex, 381vortices, 43, 63, 65, 104, 112, 175, 295, 375, 376,

377, 381, 382, 383

W

wall temperature, xi, 173, 177, 185, 189, 201, 214,229, 279, 318, 331, 334, 335, 341

warfare, 344water, vii, viii, 27, 28, 29, 33, 34, 35, 37, 38, 39, 65,

102, 103, 112, 142, 143, 166, 183, 271, 290, 293,294, 298, 300, 303, 309, 310, 311, 312, 314, 317,339, 352, 353, 355, 359, 362, 368, 375, 400, 411,423, 437, 440

weakness, 120welding, 283wettability, 140wind, 42windows, 286worms, 447writing, 321, 447

Y

yield, 7, 22, 24, 44, 126, 130, 131, 136, 156, 273,295, 301, 327, 328, 337

Z

zeta potential, 3, 4, 5, 16

117694972 Fluid Mechanics and Pipe Flow

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