PhD Thesis Mahdi Zakyani

7/21/2019 PhD Thesis Mahdi Zakyani

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Abstract

In modern society many applications and processes rely on combustion.

It is used in many energy conversion systems needed for transport or for

electricity production. We also use it at home to heat up or houses, and, inmany cases, to cook.

With the increasing concern for the environment more stringent regula-tions are imposed restricting emissions of pollutants such as NOx or green

house gases. In addition, because of the limited supplies of fossil fuel, thereis also a need of making the combustion more efficient.

All this requires a better understanding of the combustion process which

can only be achieved through scientific research.

Combustion is a very multidisciplinary subject which involves fluid me-

chanics, thermodynamics, heat and mass transfer, chemical kinetics andthermochemistry. In most of the combustion applications the flow regime is

turbulent. Modeling and simulation of turbulent flows is already a subject

of study on its own. Turbulence happens at a large spectrum of time andlength scales. In turbulent combustion the chemical time and length scales

interfere with those of the turbulence. The modeling of the turbulence-

chemistry interaction is an essential aspect of combustion research.

In recent decades large eddy simulation (LES) is proposed as an approach

to model turbulent flows. In LES a filter function is applied to the gov-erning equations of fluid motion, Navier-Stokes equations, to obtain the

LES equations. LES has shown excellent results in simulation of differ-

ent kinds of flow. In LES the large scale motions of the flow are resolved,

whereas the small scales, which are thought to be more universal thanthe large ones and which are essentially at the sub-grid level, are mod-

eled. This feature leads to a better prediction of the mixing field whichis important in the combustion simulation. However, applying the filter

function to the governing equations of combustion, e.g. species mass frac-

tion equations, leads to filtered chemical source term. This term cannot

be closed using simple assumptions because combustion happens at verysmall scales in comparison to turbulence smallest length scale. This re-

quires specific modeling for turbulent combustion.


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In recent years, the Conditional Moment Closure (CMC) model which hasbeen previously developed using the Reynolds averaged Navier-Stokes (RANS)

approach, is further extended to be used with LES. CMC is based on theexperimental observation that the fluctuations of temperature and speciesmass fraction is small around their conditional mean, if they are condi-

tioned on mixture fraction. CMC has shown good results in simulation

of turbulent combustive flows when used with RANS. The combination ofCMC and LES is more recent and so far the research on the topic is lim-

ited. In the present thesis, the focus is on the development and studying

the feasibility of the LES-CMC approach.

After an introduction in chapter 1, chapter 2 presents the basic equations

governing fluid flow. Next LES and the formulation of LES for reactiveflows is presented in chapter 3. This is followed by an introduction of theCMC formulation in chapter 4. Chapter 5 deals with the numerical method

used for the simulations. Chapter 6 and chapter 7 discuss the results for 2

test cases. The first test case is Sandia D, a non-premixed turbulent pilotedflame. the second test case is Delft III, a non-premixed piloted turbulent

flame. Four different simulations were performed to study the effect of

SGS model, number of CMC cells and the conditional scalar dissipationmodel.

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Jury Members

President Prof. Gert DESMETVrije Universiteit Brussel

Vice-President Prof. Rik PINTELON

Vrije Universiteit Brussel

Secretary Prof. Steve VANLANDUIT


Internal Member Prof. Jacques DE RUYCK


External Members Prof. Andrzej BOGUSAWSKITechnical University of Czestochowa

Prof. Bart MERCIUniversiteit Gent

Prof. Gerard DEGREZUniversiteLibre de Bruxelles

Promoter Prof. Chris LACOR


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3.3.2 Dynamic Smagorinsky model . . . . . . . . . . . . . . 28

3.4 Favre filtered mixture fraction equation . . . . . . . . . . . . 30

3.5 Favre filtered species mass fraction equation . . . . . . . . . 313.6 Scalar mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.6.1 Variance of mixture fraction . . . . . . . . . . . . . . . 32

3.6.2 Scalar dissipation rate . . . . . . . . . . . . . . . . . . 33

4 Conditional Moment Closure 37

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Conditional filter . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.3 Conditional moment closure formulations . . . . . . . . . . . 38

4.3.1 Conditional moment closure sub-models . . . . . . . . 41

4.4 LES-CMC coupling . . . . . . . . . . . . . . . . . . . . . . . . 464.4.1 LES-CMC data transfer . . . . . . . . . . . . . . . . . 46

4.4.2 LES-CMC solution procedure . . . . . . . . . . . . . . 49

5 Numerical Methods 53

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.2 Finite volume method . . . . . . . . . . . . . . . . . . . . . . . 54

5.3 Predictor-corrector method . . . . . . . . . . . . . . . . . . . . 56

5.4 Solving the pressure Poisson equation . . . . . . . . . . . . . 58

5.4.1 Multigrid method . . . . . . . . . . . . . . . . . . . . . 59

5.5 Solving the CMC equations . . . . . . . . . . . . . . . . . . . 61

5.6 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 62

5.6.1 Buffer zone . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.6.2 Velocity lateral boundary condition . . . . . . . . . . . 63

6 Sandia D Non-Premixed Piloted Turbulent Flame 65

6.1 Test case description . . . . . . . . . . . . . . . . . . . . . . . 65

6.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.2.1 Flow simulation . . . . . . . . . . . . . . . . . . . . . . 67

6.2.2 Combustion simulation . . . . . . . . . . . . . . . . . . 67

6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.3.1 Comparison along the centerline . . . . . . . . . . . . 696.3.2 Comparison along the radial line . . . . . . . . . . . . 70

6.3.3 Comparison in mixture fraction space . . . . . . . . . 79

6.3.4 Budget of terms of CMC equations . . . . . . . . . . . 79

7 Delft III Non-Premixed Piloted Turbulent Flame 83

7.1 Test case description . . . . . . . . . . . . . . . . . . . . . . . 83

7.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

7.2.1 Flow simulation . . . . . . . . . . . . . . . . . . . . . . 85

7.2.2 Combustion simulation . . . . . . . . . . . . . . . . . . 86

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7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 867.3.1 Effect of SGS model . . . . . . . . . . . . . . . . . . . . 86

7.3.2 Effect of number of CMC cells . . . . . . . . . . . . . . 1017.3.3 Effect of conditional scalar dissipation model . . . . . 1147.3.4 Budget of terms of CMC equations . . . . . . . . . . . 121

8 Conclusions and Perspectives 129

8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

8.2 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

List of Publications 133

Bibliography 134

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Nomenclature

Upper-case Roman

N number of speciesT temperatureR universal gas constantW atomic weight of mixtureW atomic weight of speciesT standard temperatureCp specific heat at constant pressure of mixtureCp, specific heat at constant pressure of speciesSij strain rate tensor

Uj, diffusion velocity of the speciesD mass diffusivity, diffusivityD binary mass diffusivity of speciesP r Prandtl numberSc Schmidt number of speciesLe Lewis number of speciesSc Schmidt numberM number of reaction

Q heat sourceQj progress rate

Kfj forward rate constant of reactionjKrj reverse rate constant of reactionjAfj preexponential constantTaj activation temperatureL characteristic lengthU characteristic velocityRe Reynolds numberG filter function, function for AMC modelCS Smagorinsky model constantTij sub test stress tensor

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Lij Leonard stress tensorJsgsj sub-grid scale turbulent scalar flux

C sub-grid scale variance of the mixture fraction model constantCI turbulent kinetic energy model constantP probability density functionA area

Lower-case Roman

m total massm mass of speciesm mass fraction of species

p pressurep partial pressure of speciesh enthalpy of mixtureh enthalpy of specieshf, enthalpy of formation of species

at standard temperature and standard pressure

t timexi Cartesian coordinate vector componentui Cartesian velocity vector componentqj heat flux

u smallest turbulent velocity scalek turbulent kinetic energyn unit normal vector of control surfaceu Cartesian velocity vector

Upper-case Greek

viscous dissipation, scalar quantity

symbol for species

filter width

test filter width Gamma function,Lower-case Greek

density

density of species dynamic viscosityij stress tensor

ij Kronecker delta

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Dirac delta function

rate of production or destruction of species

thermal conductivityT heat release due to chemical reaction

T heat release due to chemical reaction

mixture fraction

kinematic viscosity

j ,

j molecular stoichiometric coefficients of species in reactionjj temperature exponent

dissipation rate

smallest turbulent length scale, mixture fraction sample space

smallest turbulent time scale

characteristic time2 sub-grid scale variance of the mixture fraction scalar dissipation rateSubscripts

for species

j reactionjt turbulent quantity

Superscripts

sgs subgrid scale.| conditional Fvare filtered quantity.| conditional filtered quantity. Fvare filtered quantity. filtered quantity. Fvare test filtered quantity

. test filtered quantity

.|

integrated conditionally filtered variable

D deviatoric part

Symbols

gradient operator

Abbreviations

CFD Computational Fluid Dynamics

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DNS Direct Numerical SimulationLDV Laser Doppler Velocimetry

LES Large Eddy SimulationMPI Message Passing InterfaceRANS Reynolds Averaged Navier-Stokes

C.S. control surface

C.V. control volume

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Chapter 1

Introduction

Contents

1.1 Combustion and energy . . . . . . . . . . . . . . . . . . 1

1.2 Turbulence modeling . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Reynolds averaged Navier-Stokes (RANS) simula-

tion of turbulent flows . . . . . . . . . . . . . . . . . 3

1.2.2 Large eddy simulation (LES) of turbulent flows . . 4

1.3 Turbulent combustion modeling . . . . . . . . . . . . . 5

1.3.1 Equilibrium chemistry model . . . . . . . . . . . . . 6

1.3.2 Steady and unsteady flamelet models . . . . . . . . 6

1.3.3 Flamelet-progress variable model . . . . . . . . . . 7

1.3.4 Conditional moment closure (CMC) model . . . . . 7

1.3.5 Probability density function (PDF) model . . . . . . 10

1.3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . 10

1.1 Combustion and energy

The very basic needs of human for having a warm place to live and food to

eat motivates mankind to control fire. It is estimated that control of fire byman was accomplished almost 400000 years ago. Controlling fire helped

human to establish civilization. Controlling energy coming from chemical

matter was and still is one of the great concerns of society. Many of the

power plants use gas turbine units, which burn fossil fuel to produce elec-tricity. Moreover, traveling around the earth and going into space is done

with the power coming from burning chemical matters inside internal com-

bustion engines, aircraft engines or rocket engines. Fossil fuels have been

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CHAPTER 1. INTRODUCTION

extensively used during the last century, and they are still one of the bigenergy resources today.

One of the great concerns of engineers, for design and development of de-vices that receive their energy from burning fossil fuel, is burning the fuel

efficiently. This concern comes from two natural limitations. One is the

shortage of the fossil fuel due to vast consumption and limited reservoirs.The other is creating tons of polluting chemical matters, which are pro-

duced during the combustion process and are poisonous and unhealthy for

the human body. Therefore, burning fossil fuel efficiently to manage lim-

ited resources and to reduce air pollution is the center of many researchesin combustion engineering.

There should be a fuel and an oxidizer for a combustion phenomenon. Thefuel can be a solid matter, e.g. charcoal, a liquid substance, e.g. oil or gas

like methane. The oxidizer can also be any chemical matter, but it de-

pends on the fuel type. As most of the fuels in the nature are hydrocarbon.The oxygen present in the air is used as an oxidizer in many combustion

processes. If the fuel and oxidizer are mixed before reaction happens, the

process is called premixed combustion. If the fuel and oxidizer are not

mixed before the reaction happens, but they are mixed while reacting, it iscalled non-premixed combustion.

When fuel and oxidizer come into contact combustion can occur dependingon the system condition, e.g. pressure and temperature. If the combustion

happens with the help of an external source of heat, usually a spark, then

it is called spark ignited combustion like in gas turbines and spark ignited(SI) internal combustion engines. If the combustion occurs without the

help of a spark, it is called auto-ignited combustion like in compression

ignited (CI) internal combustion engines. The results of combustion are

usually chemical matter other than fuel and oxidizer. The heat obtainedfrom the combustion process can be used for many purposes like heating

the water to its boiling point to produce water vapor in boilers of power

plants or increasing heat of product gases and expanding them in gas tur-bine units to produce energy.

1.2 Turbulence modeling

In practical combustion systems many phenomena happen, which are un-

known or little known to scientists and engineers. Turbulence is the most

important physical phenomenon which almost happens in all engineering

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fluid dynamical systems, including combustion systems. Roughly speak-ing, turbulence can be defined as chaotic behavior of the flowing fluid. The

drawing of Leonardo da Vinci suggests that the distorted fluid motion thatwe know as turbulence has long been known to exist. Nevertheless, thefirst quantitative investigation of this state is generally attributed to Os-

borne Reynolds. Reynolds used a glass tube of water and injected ink at

the entrance of the tube to distinguish laminar and turbulent flows. TheReynolds number is named after him to distinguish these two different

regimes of the fluid motion. Since then turbulence has been the subject of

many theoretical and experimental investigations. There is a considerableamount of literature devoted to studying turbulence in fluid flows. Only

in recent decades, with rapid development of digital computers, scientists

and engineers could simulate turbulent flows of engineering interest.

The Navier-Stokes equations are a set of non-linear partial differential

equations, which describe the fluid flow motion [118]. However, findinga closed mathematical solution for the Navier-Stokes equations is a very

difficult task. There have been valuable researches on studying proper-

ties of the Navier-Stokes equations and finding solutions for simple cases.The literature on the subject is vast and interested readers are referred

to [97, 118]. Unfortunately, most of the solutions are limited to laminar

flows and simple geometries. Needless to mention, none of the available

mathematical solutions solved the complete Navier-Stokes equations.

1.2.1 Reynolds averaged Navier-Stokes (RANS) simu-lation of turbulent flows

In recent decades with rapid development of digital processors, fluid dy-

namists started using computers to solve the Navier-Stokes equations.Nevertheless, turbulence was a big challenge like in theoretical methods.

The first method to model turbulent flows was based on Reynolds decom-

position. In Reynolds decomposition, the flow field is divided into twoparts: a time independent part and a time fluctuating part [91]. Using the

Reynolds decomposition method and applying a time averaging operator

to the Navier-Stokes equations results in the Reynolds-averaged Navier-

Stokes equations. This method is widely used in academic research. It isalso a reliable tool for turbulence modeling in the industry as well.

There has been much research done using the RANS equations to model

turbulent flows for fundamental and practical fluid flow problems. Inter-

ested readers are referred to the review papers by Bradshaw [11] for tur-

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bomachinery applications and Tulapurkara [112] for flow past airplane ap-plications.

1.2.2 Large eddy simulation (LES) of turbulent flows

In large eddy simulation (LES), the flow field is divided into the large-scale resolved field and the small-scale unresolved contributions. Applying

a spatial filter to the turbulent field removes motions of the length scale

smaller than the filter size. These smaller scales which are also calledsub-grid scales (SGS) should be modeled. Large scale motion of the flow is

affected by the geometry. The small scales of the motion tend to be more

universal and are therefore modeled. There are two significant advantages

of LES approach over RANS approach. In LES, although computationallyexpensive, the large scale motions are resolved. Therefore, most of the

energy containing information of the fluid motion which controls the dy-

namics of the turbulence is captured. as a result only sub-grid turbulentfluctuations have to be modeled with SGS models which can be much sim-

pler than the RANS turbulence model. in addition the dynamic procedure

[30] allows to avoid tuning up the constant needed in the SGS model, as inRANS case often several turbulence models constants have to be tuned.

LES has shown promising results in the simulation and modeling of non-reactive flows. The most widely used LES model was proposed by Smagorin-

sky [100]. He assumed that the eddy viscosity is proportional to the sub-

grid scale characteristic length scalex and to the characteristic turbulentvelocity based on the second invariant of the filtered field deformation ten-

sor. There exists extensive literature devoted to the LES. However, here

we mention a few of them, which are important in the field of combustion.For a complete review on the large eddy simulation, readers are referred

to [55] and [82]. Yan and Su [121] used two different SGS models to sim-

ulate the free turbulent round jet. They used the standard Smagorinsky

eddy viscosity model, and the non-eddy viscosity stimulated small scalemodel in their calculations. Recently, Zhang and Stanescu [124] applied

LES using spectral element method to study round jets. Jones and Wille[39] also used LES to simulate a plane jet in a cross-flow. Majander and

Siikonen [58] adopted LES to simulate a round jet penetrating into a cross

flow. They could capture the counter-rotating vortex pair which is traveling

at the stream-wise direction. The jet into cross flow has important appli-cation in turbine blade cooling, fuel nozzle discharge and VTOL aircraft.

Prediction of the jet noise emanating from the aircraft nozzle is another

important discipline in studying jet flows using LES. Bogey and Bailly [9]

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adopted explicit filtering with and without dynamic Smagorinsky modelto simulate a round free jet at Mach number M = 0.9. Wang et al. [116]

applied dynamic Smagorinsky model to study variable density turbulentaxisymmetric jets. A complete review of the researches done in the studyof LES of jet noise is given in DeBonis [22] and Bodony and Lele [8].

1.3 Turbulent combustion modeling

Modeling and simulation of the combustion phenomenon is a challenging

topic. One should adopt different fields of physics and chemistry to be able

to simulate processes, which happen in a combustion system. The basicknowledge in thermodynamics and chemistry is essential in any combus-

tion modeling and simulation. Fluid mechanics also play an important role

in combustion simulation. Thermochemistry, chemical kinetics, heat and

mass transfer and fluid mechanics go hand in hand in the understandingof a combustion system regardless whether it is a candle or a sophisticated

aircraft engine. In other words, combustion is a multidisciplinary subject.

Another important topic in studying combustion is the modeling of the

chemical reactions. Chemical reactions are central to combustion phenom-

ena. After contacting the fuel and oxidizer, first radicals are created, and

then these radicals will interact and produce the reaction products andheat. Modeling of the chemical reactions is studied in chemical kinetics.

In chemical kinetics, the Arrhenius law relates the rate at which chemicalreactions happen to the temperature.

In laminar flames, reaction terms, which appear in the equations, will

cause no problem as no averaging or filtering procedure is used. Evenso, in turbulent flames applying the time averaging operator or filter func-

tion to the chemical source terms will give unclosed terms that need to be

modeled. Modeling the chemical source term in turbulent combustion is

a big challenge. Many models have been proposed to model the unclosedchemical reaction terms in the equations. As non-premixed combustion

is studied in this thesis, here the turbulent combustion models which aredeveloped for diffusion flames will be shortly explained. Needless to men-

tion a few of the turbulent combustion models can be used for both non-

premixed as well as premixed combustion.

First, the turbulent combustion was studied using RANS turbulence mod-

els. Interested readers are referred to the review work of Veynante and

Vervisch [115] and Bilger et al. [6]. Applying LES to turbulent combus-

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tion started in early 90s and is hence a relatively new field. Turbulentcombustion models which have been successfully adopted in RANS simu-

lations are adapted and widely used with LES. Many target flames weresimulated with LES, and they showed good agreement with the experimen-tal data. A complete review of adoption of LES for simulating combustion

systems can be found in [85, 86].

1.3.1 Equilibrium chemistry model

In the equilibrium model, the reaction happens at an infinitely fast rate,

and is reversible for all mixture fraction values. Therefore, all species are

in equilibrium at each mixture fraction value. The equilibrium chemistrymodel is the simplest method proposed for simulating non-premixed tur-

bulent flames [89]. It should be considered that all reactions happen at aninfinitely short period. The time scale of the reactions in this model should

be much smaller than the smallest time scale of the turbulent motion. This

will limit the applicability of the method to cases without extinction and

re-ignition or without flame lift-off. However, it is a good model for simpleapplications. E.g. Branley and Jones [13] used equilibrium chemistry and

dynamic Smagorinsky procedure to simulate a hydrogen diffusion flame.

1.3.2 Steady and unsteady flamelet models

The laminar flamelet method which is proposed by Peters [79] considers

that the turbulent flame is composed of thin laminar flamelets. Thatmeans the reaction zone remains laminar and diffusion of species will

occur in the direction normal to the stoichiometric mixture fraction iso-

surface. The model is widely used to simulate turbulent flames usingRANS and LES approaches. E.g. Pitsch [84] adopted unsteady flamelet

and RANS approach to study differential diffusion in the DLR A turbulent

non-premixed jet flame. Chen et al. [17] used flamelet model together withthe RANS approach to study lifted turbulent methane/air and propane/air

jet diffusion flames.

The feasibility of the laminar flamelet approach together with LES is stud-

ied in [21, 20]. Stein and Kempf [106] used LES and the steady laminar

flamelet model to simulate Sydney swirling bluff-body flames. Pitsch andSteiner [87] used LES and the unsteady flamelet model to simulate a non-

premixed methane flame, Sandia flame D.

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1.3.3 Flamelet-progress variable model

The flamelet-progress variable model was proposed by Pierce and Moin[81]. They used the model together with LES to study a coaxial jet com-

bustor. The model is similar to the flamelet model but instead of scalar

dissipation, a reactive scalar is used to model the condition of reaction.The model is applied to study a few cases and showed satisfactory results.

Ihme et al. [34] used the model with a LES approach to study a non-

premixed flame with local extinction and re-ignition effects. In anotherstudy, Ihme and Pitsch [35] used a LES and flamelet/progress variable

approach to simulate extinction and re-ignition of the laboratory flames:

Sandia D and E. In a recent study Ihme and See [36] extend the modelto account for unsteady effects to model auto-ignition of lifted methane-air

flame using a LES approach.

1.3.4 Conditional moment closure (CMC) model

The conditional moment closure model was originally proposed as an ad-

vanced turbulent combustion model by Klimenko and Bilger [49]. The

CMC model promises to be valid for a vast variety of combustion prob-lems unlike other turbulent combustion models, e.g. the laminar flamelet

model. The model can simulate non-premixed, premixed and partially pre-mixed flames which is impossible for other turbulent combustion modelsat the same time. It is also capable of predicting local or global extinc-

tion and re-ignition of the flames. It also has the ability of simulating

lifted flames. These characteristics put CMC as one of the promising tur-

bulent combustion models. The main idea behind the CMC model comesfrom experimental observations. It is observed in the experiments of tur-

bulent flames that the variation of temperature and species mass fraction

is small around their conditional mean if the conditional mean is takenover the mixture fraction. This observation leads to the proposal of the

CMC method for dealing with turbulent combustion flames. In the follow-ing paragraphs, we will review the research done on the CMC model anddiscuss the ability of this model.

RANS-CMC simulation of turbulent flames

Primarily, the CMC model was used together with RANS turbulence mod-

els. E.g. Kim et al. [43] applied CMC using thek gturbulence modelfor modeling and simulation of a complex methanol bluff body stabilized

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flame. Roomina and Bilger [94] used first-order CMC with the k tur-bulence model to simulate the Sandia D flame. In another study, Kim and

Huh [44] used also first-order CMC for simulation of a C H4/H2 bluff bodystabilized flame. Other studies with first-order CMC in the RANS con-text can be found in e.g. [24], [25]. Fairweather and Woolley [24] adopted

first-order CMC usingk and Reynolds stress/scalar flux turbulence clo-sures for simulation of a non-premixed hydrogen flame. Furthermore, inanother study [25], they used the same strategy for the simulation of a

non-premixed methane flame. Kim et al. [46] applied the CMC model

to predict flame structures and NO formation in the moderately and in-tensely low oxygen dilution combustion mode. Kim and Mastorakos [41]

used first-order CMC to study non-premixed counterflow flames. They

used the Reynolds stress turbulence model and solved an additional trans-port equation for the turbulent scalar flux. Fairweather and Woolley [26]used the CMC model to study a swirl stabilized bluff body non-premixed

methane flame. They used Reynolds stress and scalar flux second-moment

closure for turbulence modeling.

As mentioned earlier, CMC is capable of simulating lifted flames. Devaud

and Bray [23] showed the ability of the first-order CMC in predicting the

flame lift-off using thek turbulence model. They could predict the lift-off height of the hydrogen flame accurately. Kim and Mastorakos [40] used

first-order CMC to simulate hydrogen lifted flame. They have also adoptedthe kturbulence model. Yang et al. [122] applied first-order CMC modeland thekgturbulence model to study a hydrogen lifted flame as well.Patwardhan et al. [77] applied first-order CMC to study the Cabra flame,

which is a lifted turbulent jet flame in a vitiated co-flow. The CMC method

could capture the lift-off height of the flame with good accuracy.

Another characteristic of the CMC model is the ability to predict extinction

and ignition phenomena. Mastorakos and Bilger [61] used second-order

CMC to study auto-ignition in turbulent flames. The extension to second-

order is necessary to capture the auto-ignition phenomenon. Kim and Huh[45] used also second-order CMC to study the local extinction of the San-

dia flame D, E and F. It is shown that using second-order closure for somereaction mechanisms will improve the results in comparison to first-order

closure. Kim et al [42] applied first order and second order CMC to simu-

late a turbulentCH4/H2/N2jet diffusion flame. They also usedkmodelfor turbulence modeling. The flame under study has some local extinctionand re-ignition. Sreedhara and Huh [103] used the first and second order

closure together with the k turbulence model to study aC H4/H2 bluffbody flame. Three flames were studied with significant extinction involved

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as jet velocity increased. Fairweather and Woolley [27] used first order andsecond order CMC to study local extinction and re-ignition of Sandia flame

D, E and F.

It has also shown that to model the extinction and ignition phenomena, themodel should be based on at least two variables: a first variable, e.g. mix-

ture fraction which describes the mixing of fuel and oxidizer and a second

variable which allows to distinguish between burning and non-burning re-

gions. Cha et al. [16] used the doubly-conditional moment closure ap-proach in order to simulate extinction and re-ignition in non-premixed

flames. They used the approach in simulating a numerically investigated

flame. Kronenburg [50] also applied the doubly-conditional moment clo-

sure approach to simulate a simple non-premixed flame. First, the flamewas studied numerically by DNS method, and then the doubly-conditional

moment closure approach is used to demonstrate the ability of the ap-proach in predicting local extinction and re-ignition. In another investi-

gation, Kronenburg and Papoutsakis [53] applied the doubly-conditional

moment closure approach to study extinction and re-ignition phenomena.Kronenburg and Kostka [52] used the proposed doubly-conditional mo-

ment closure to simulate the Sandia flame D, E and F. They could predict

the temperature, major species andN Ofor the Sandia flame D and E withexcellent accuracy. It is pointed out that using another variable apart from

mixture fraction for conditioning the species mass fraction will give far bet-ter results than the first-order closure. Patwardhan and Lakshmisha [78]

applied CMC model to study auto-ignition of hydrogen jet in a co-flow ofheated air.

Another area of application of the CMC model is in spray combustion. Kim

and Huh [48] applied the first-order CMC model to study the n-heptanespray combustion. Wright et al. [120] used the CMC model to study flame

establishment of the spray auto-ignition. There have also been a few re-

searches on applying CMC model for sooting flames. E.g. Kronenburg et

al. [51] studied soot formation of methane-air jet diffusion flames usingthe CMC model. In another study, Yunardi et al. [123] used the CMC

model to study soot formation in turbulent, non-premixed ethylene flames.Furthermore, Woolley et al. [119] applied CMC to study soot formation

in turbulent, non-premixed methane and propane flames. Using the CMC

model will help to study the effects of differential diffusion in soot forma-

tion. There are other areas in the turbulent combustion field that are stud-ied by researchers using the CMC model. Thornber et al. [108] combined

the CMC model with a compressible LES method to study a lean premixed

methane slot burner.

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LES-CMC simulation of turbulent flames

The first LES-CMC simulation of a flame is achieved by Navarro-Martinezet al. [71]. They used the conditional moment closure approach using

the dynamic Smagorinsky procedure to simulate Sandia flame D. Results

for the Sydney bluff body burner are presented in [69] using CMC to-gether with the dynamic Smagorinsky model. In another study, Navarro-

Martinez and Kronenburg [70] used first-order CMC with LES to simulate

the Cabra lifted hydrogen flame. They used the dynamic Smagorinskymodel in their calculation. Triantafyllidis et al. [110] adopted CMC and

LES to simulate the forced ignition of a bluff-body stabilized non-premixedmethane flame. Stankovic et al. [105, 104] studied the hydrogen auto-ignition in a turbulent co-flow of heated air using LES-CMC. Recently,

Garmory and Mastorakos [28] used LES-CMC to model Sandia flame D

and F. The results showed good agreement with the experiment for Sandia

flame D. Also the transient nature of local extinction and re-ignition wascaptured using LES-CMC. Lately, Ayache and Mastorakos [2] performed

LES-CMC simulation of Delft III non-premixed flame, which gave satis-

factory results for the investigated turbulent flame.

1.3.5 Probability density function (PDF) model

The probability density function (PDF) model proposed by Pope [90] solves

transport equations for joint scalar and joint scalar-velocity PDF. The main

advantage of the model is that the chemical source terms appear in a closed

form. However, the molecular mixing term is unclosed and therefore, needsmodeling. The method is vastly used together with the RANS turbulence

model for the different kinds of flames. E.g. Masri and Pope [60] used

the PDF method to simulate a piloted turbulent non-premixed methaneflame. The method later adapted using the idea of a filtered density func-

tion (FDF) to be used together with the LES model. E.g. Sheikhi et al. [99]used a scalar filtered mass density function methodology and large eddysimulation to model Sandia flame D. Complete review of the PDF method

can be found in [31].

1.3.6 Summary

In the following, in chapter 2, the governing equations of reactive flows

including gas law and thermochemistry and Navier-Stokes equations will

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be presented. In chapter 3, a short introduction of turbulent flows willbe given and the fundamentals of large eddy simulation will be explained.

Chapter 4 is devoted to introducing the conditional moment closure andthe LES-CMC formulation. Also the coupling between the LES code andthe CMC code and the details of the data transfer between the fine LES

grid and the coarse CMC grid are explained in chapter 4. In chapter 5,

the numerical methods used to solve the governing equations of fluid flowand conditional moment closure model will be discussed. The predictor-

corrector method which was implemented in the code to enhance the ef-

ficiency of the LES code, is also explained. In chapter 6 the LES-CMCresults of the Sandia D turbulent flame will be presented and discussed.

The results of the simulations for the Delft III turbulent flame will be pre-

sented in chapter 7 which is one of the first LES-CMC simulations of theflame. In this chapter the effect of sub-grid scale model, number of CMCcells and conditional scalar dissipation model on the Delft III flame will be

explained.

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Chapter 2

Governing Equations of

Reactive Flows

Contents

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Gas law and thermochemistry . . . . . . . . . . . . . . 14

2.3 The Navier-Stokes equations . . . . . . . . . . . . . . . 152.4 Species mass fraction equation . . . . . . . . . . . . . 16

2.5 Energy equation . . . . . . . . . . . . . . . . . . . . . . . 17

2.6 Mixture fraction equation . . . . . . . . . . . . . . . . . 20

2.7 Chemical kinetics . . . . . . . . . . . . . . . . . . . . . . 20

2.1 Introduction

The Navier-Stokes equations are the basic equations for modeling and sim-ulating reactive flows. For modeling and simulating reactive flows, addi-

tional species mass fraction equations should also be solved [113, 54]. This

makes the computation of reactive flows very expensive in comparison to

non-reactive flows. In the following, the governing equations of the reac-tive flows, Navier-Stokes equations and species mass fraction equations,

will be introduced.

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CHAPTER 2. GOVERNING EQUATIONS OF REACTIVE FLOWS

2.2 Gas law and thermochemistry

In the simulation of reactive flows, species mass fraction, Y, is an impor-tant parameter. The species mass fraction is defined by,

Y =m

m (2.1)

where m is the mass of species and m is the total mass in a givenvolume,V. For a mixture ofNspecies the pressure is given by,

p=

N=1

p (2.2)

wherep is the partial pressure and is given by,

p =R

WT (2.3)

whereTis the temperature,R = 8.314J/(moleK)is the universal gas con-stant andW is the atomic weight of the species. is given by,

=Y (2.4)

since the density of the multi-component gas is,

=

N=1

(2.5)

The equation of state reads,

p= R

WT (2.6)

whereWis the mean molecular weight of the mixture and is given by,

1

W =

N

=1YW

(2.7)

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In the computation of reactive flows, the enthalpy of the mixture, h, isrequired. The enthalpy of the mixture is computed from the knowledge of

species mass fraction as,

h=N

=1

hY (2.8)

where species enthalpy are computed from,

h =h

f,+ T

T

Cp,(T)dT (2.9)

wherehf, is the enthalpy of formation of species at standard temper-ature, T, and standard pressure and Cp, is the specific heat at constantpressure for species . The specific heat at constant pressure for the mix-

ture is computed as,

Cp =N

=1Cp,Y (2.10)

2.3 The Navier-Stokes equations

The Navier-Stokes equations are the fundamental equations for modeling

and simulating a vast majority of fluid flow problems. The Navier-Stokes

equations are a set of non-linear partial differential equations, which arestrongly coupled. The solution of the Navier-Stokes equations is the ve-

locity, pressure, temperature and the density field of the flow under study.

The compressible Navier-Stokes equation for a Newtonian fluid reads,

t +

uixi

= 0 (2.11)

uit

+uiujxj

= p

xi+ij

xj(2.12)

whereui is the velocity component, p is the pressure, is the density andij is stress tensor which is given by,

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ij = 2Sij 13 ukxk ij (2.13)where is the dynamic viscosity and the strain rate,Sij , defined as,

Sij =1

2

uixj

+ujxi

(2.14)

The energy equation can take many forms and is very important in thesimulation of combustion. Therefore, it will be explained in the following

in a separate section.

2.4 Species mass fraction equation

The equation of continuity for species reads,

Yt

+ujY

xj+Uj,Y

xj= (2.15)

where is the rate of production or destruction of speciesand Uj,is the

diffusion velocity of the species. Determination of the diffusion velocityis a difficult task, and normally it is found using Ficks law of diffusion,

Uj,Y= DYxj

(2.16)

whereD is the binary mass diffusivity of species . If the oxidizer is air,then nitrogen is abundant in the mixture. Therefore, all other species canbe treated as trace species and the binary diffusion coefficient can be calcu-

lated based on the binary diffusion coefficient of the species into nitrogen.

Since N=1 = 0summation of equation 2.15 over all species gives,

t +

ujxj

=

xj

N=1

Uj,Y

= 0 (2.17)

and therefore,

N

=1Uj,Y= 0 (2.18)

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From equation 2.17 it is clear that if Ficks law of diffusivity is used, themass conservation is not always guaranteed unless all diffusion coeffi-

cients are equal D = D. Therefore, the equation for the mass fractionof species will be,

Yt

+ujY

xj=

xj

D

Yxj

+ (2.19)

2.5 Energy equation

The Energy equation is written using the first law of the thermodynamics.

The energy equation should be formulated carefully because it can takemany forms. The energy equation can be written in terms of enthalpyor internal energy. However, working with enthalpy equation is easier in

many fluid flow problems. The energy equation for enthalpy of the mixture

hfor a multi-component reactive flow is,

h

t +

ujh

xj=p

t + uj

p

xj qjxj

+ Q + (2.20)

where Q is the heat source due to radiation or electric spark for ignition,

qj is the heat flux and is the viscous dissipation term which is,

=ijuixj

(2.21)

In a multi-component system the heat flux, qj , is not only due to the heatconduction but two other effects are also present that cause the heat flux.

One is the heat flux due to the energy of a component if the velocity of that

component is different from the mass averaged velocity of the mixture andis called the Soret effect. The other effect is called the Dufour effect and

is due to the increase of diffusion velocity and is a reciprocal effect. TheDufour effect is normally neglected as it has a small contribution to theheat flux. Therefore, the heat flux is given by,

qj = T

xj+

N=1

hYUj, (2.22)

where is the thermal conductivity. In most of the cases, combustion hap-

pens at very low speeds and the variation of pressure is small. Therefore,

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the variation of pressure in time and space is negligible, and the pressureterms can be safely removed from the equation of enthalpy. The viscous

dissipation term, , is also small and can be safely removed. The heatsource, Q, can also be removed if there is no electric spark for ignition orthe radiation effect is neglected like what is done in the present thesis.

Accordingly, the enthalpy equation using equation 2.22 and mentioned as-

sumptions can be rewritten as,

h

t +

ujh

xj=

xj

T

xj

N=1

hYUj,

(2.23)

Using Ficks law of diffusivity for diffusion velocity the enthalpy equationwill be,

h

t +

ujh

xj=

xj

T

xj+

N=1

DhYxj

(2.24)

The following identity holds if we use the relations 2.8 and 2.9.

dh= CpdT+

N=1

hdY (2.25)

Using the relation 2.25 in heat flux terms of equation 2.24, the energy

equation can be written as,

h

t +

ujh

xj=

xj

P r

h

xj+

N=1

P r

Sc 1

h

Yxj

(2.26)

In most of the cases in combustion simulations, the Lewis number whichis defined as follows is assumed to be unity.

Le= Sc

P r =

CpD(2.27)

whereSc is the Schmidt number of species and is given by,

Sc=

D(2.28)

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andP ris the Prandtl number and is given by,

P r=Cp

(2.29)

Assumption of unity Lewis number is also used in this thesis. Using this

assumption the equation 2.26 will be simplified to,

h

t +

ujh

xj=

xj

Cp

h

xj

(2.30)

It is useful sometimes to write energy equation in terms of temperature.

Using relations 2.8, 2.9 and 2.25 in equation 2.23, an equation for thetemperature can be found,

CpT

t +Cpuj

T

xj=

xj

T

xj

N=1

Cp,YUj,

T

xj+ T (2.31)

where Tis the heat release due to chemical reactions and is given by,

T = N

=1

h (2.32)

Using Ficks law of diffusivity for the diffusion velocity equation 2.31 can

be written as,

CpT

t +Cpuj

T

xj=

xj

T

xj

+ N

=1

Cp,D Yxj

Txj

+ T (2.33)

Assuming unity Lewis number equation 2.33 can be written,

CpT

t +Cpuj

T

xj=

xj

T

xj

+

D

N

=1Cp,

Yxj

T

xj+ T (2.34)

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Sometimes we can assume that the specific heat at the constant pressureis constant for all species in the mixture, using this assumption equation

2.31 can be simplified to,

CpT

t +Cpuj

T

xj=

xj

T

xj

+ T (2.35)

where T is given by,

T = N

=1hf, (2.36)

2.6 Mixture fraction equation

The mixture fraction, , is a conserved scalar which describes in any point

in the flow field the mixing condition of the fuel and oxidizer. When it isequal to 1 then it is pure fuel and when it is 0, it is pure oxidizer. Mixture

fraction definition is useful in studying non-premixed flames. The mixture

fraction appears in the form of a transport equation without any sourceterm as,

t +

uj

xj=

xj

D

xj

(2.37)

where the diffusivity,D, is defined as,

D=

Sc (2.38)

In equation 3.31is the kinematic viscosity and Sc is the Schmidt number,which is adjusted accordingly.

2.7 Chemical kinetics

Considering a system ofNspecies andM reactions,

N

=1j

N

=1j f or j= 1, M (2.39)

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where is a symbol for species , j and

j are the molecular stoichio-metric coefficients of species in reactionj. From the mass conservation

we have,

N=1

jW=N

=1

jW f or j= 1, M (2.40)

The mass reaction rate for species which appears in equation 2.19 is thesum of rates j produced by allM reactions,

=

M

j=1

j =W

M

j=1

jQj (2.41)

where

j =

j

j (2.42)

It is obvious that,

N=1

= 0 (2.43)

which shows that the total mass is conserved. The progress rate Qj isgiven by,

Qj =Kfj

N=1

YW

j

Krj

N=1

YW

j

(2.44)

whereKfj andKrj are the forward and reverse rate constant of reaction

j. AlsoY/W is the molar concentration of species. The rate constantis usually given by the Arrhenius law as,

Kfj =AfjTj exp

EjRT

= AfjT

j exp

TajT

(2.45)

Therefore to determine the progress rates Qj for every reaction, the datafor the preexponential constantAfj , the temperature exponent j and theactivation temperatureTaj should be provided.

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Chapter 3

Large Eddy Simulation

Contents

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 Foundation of large eddy simulation . . . . . . . . . . 24

3.2.1 Filter function . . . . . . . . . . . . . . . . . . . . . 26

3.3 Favre filtered Navier-Stokes equations . . . . . . . . 27

3.3.1 Constant Smagorinsky model . . . . . . . . . . . . 28

3.3.2 Dynamic Smagorinsky model . . . . . . . . . . . . . 28

3.4 Favre filtered mixture fraction equation . . . . . . . 30

3.5 Favre filtered species mass fraction equation . . . . 31

3.6 Scalar mixing . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.6.1 Variance of mixture fraction . . . . . . . . . . . . . 32

3.6.2 Scalar dissipation rate . . . . . . . . . . . . . . . . 33

3.1 Introduction

LES is a promising tool in practical and fundamental turbulent combus-tion research. The idea of LES comes from the fact that energy of a tur-

bulent motion that comes from large scales of the system is transferred to

smaller scales until it is dissipated at the smallest scale. This process is

called the energy cascade, and it is a very important characteristic of tur-bulent motion of fluids. In the current chapter, we first explain the time

and length scales which are present in a turbulent flow and then LES for-

mulation will be introduced. Next the sub-grid scale models are introduced

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CHAPTER 3. LARGE EDDY SIMULATION

and finally, the mixture fraction transport equation, mixture fraction vari-ance and scalar dissipation rate, which are important in combustion sim-

ulation will be presented.

3.2 Foundation of large eddy simulation

The fundamental problem associated with modeling and simulation of tur-

bulent reactive flows as well as turbulent non-reactive flows is the closureproblem. For modeling the turbulent flows, we have to deal with unknown

terms that come from applying a time averaging operator, e.g. in RANS,

or applying a filter, e.g. in LES, on the governing equations. The Navier-Stokes equations are the fundamental equations that describe the fluid

flow motion [118]. The Navier-Stokes equations are a set of non-linear

partial differential equations, which are strongly coupled. Before the eraof digital computers, many attempts have been made to find a closed form

mathematical solution for the Navier-Stokes equations. However, most of

the mathematical solutions are only available for simple cases and geome-tries. There is a valuable resource of the mathematical solution of the

Navier-Stokes equations in the literature [97, 4].

With the introduction of digital computers; researchers started solving

simplified and later full Navier-Stokes equations on complex geometries.Solving the Navier-Stokes equations for complex geometries was an achieve-

ment for a new field namely Computational Fluid Dynamic (CFD) [1]. Sci-entists and engineers used the new tool for understanding and designing

fluid dynamical systems. However, simulating the practical engineering

problems was a complex issue for users of the numerical methods in fluid

dynamic. Practical engineering devices are working at very high Reynoldsnumber, and therefore, the flow regime is turbulent for most of them. In

laminar flow, there exists usually one characteristic length and time scale

whereas in turbulent flow, there is a spectrum of time and length scales.

To understand the mechanisms involved in the production, maintaining

and the dissipation of turbulence in fluid flows, we need to distinguish dif-ferent time and length scales in the turbulent flows. Generally, the spec-

trum of time and length scales of turbulence can be divided into three main

parts. The large scales of the flow which are usually affected by the bound-

aries of the flow. these large scales which are also called energy containingrange of the spectrum are anisotropic. The next part of the spectrum is

the universal equilibrium range in contrast to energy containing range.

This part is divided in two ranges: the inertial subrange and the dissi-

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pation range. The eddies at the inertial subrange transfer the energy ofthe large scales to the smaller scales until they are at the smallest scales.

At the smallest scale which is also called dissipation range, the energy ofturbulence is dissipated into heat. Kolmogorov argued that at sufficientlylarge Reynolds number, unlike the large scales, small scales of the flow

are isotropic. The parameter governing the small scales of the motion are

, the kinematic viscosity, and , the dissipation rate. Relating them us-ing Kolmogorovs theory, we are able to find the length, time and velocity

scales of the smallest motion.

= (3

)1/4

(3.1)

u = ()1/4 (3.2)

= (

)1/2 (3.3)

Defining the largest length scale of the motion as L and the largest char-

acteristic velocity as U, the large scale dissipation rate can be calculatedas the ratio of kinetic energy and the time scale of the large eddies,

= U3

L (3.4)

From the Kolmogorov theory following Pope [91] one can obtain the rela-tion between the largest scales of the flow and the smallest scales of the

flow. Assuming that all the energy of the large scales of the motion is

transferred to the small scales of the motion gives us,

L Re3/4 (3.5)

uU Re1/4 (3.6)

Re1/2 (3.7)

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where in equations (3.5), (3.6) and (3.7)Re is the Reynolds number whichis given by,

Re=U L

(3.8)

Following Kolmogorovs theory, the energy supplied from the largest scale

of the motion is transferred to smaller scales until the smallest lengthscale. At the smallest length scales, it will be dissipated and transformed

to heat. This process is called the energy cascade, and it is very important

in modeling turbulent flows in LES. In LES, we need to separate the largescales and the small scales of the turbulent flow. To separate the large and

the small scales, a filter function is required.

3.2.1 Filter function

The filter operator which is used to separate the large and small scales in

LES is a low-pass filter. That means it only filters out high frequencies.

The filtering operator is the convolution of a non-filtered quantity with afilter function,

(x, t) = (x, t)G(x x)dx (3.9)

where (x, t)is the filtered quantities. Normally, the filter function, G(xx), used in LES is a homogeneous filter. That means the filter shouldsatisfy the following condition,

G(x x)dx = 1 (3.10)

In variable density flows a mass weighted filtering, namely Favre filter, is

used, which is defined as,

=

(3.11)

For a complete review on the mathematics and the physics of LES readers

are referred to [95, 29].

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3.3 Favre filtered Navier-Stokes equations

Applying the filter operator to the Navier-Stokes equations and using equa-tion 3.11 for variable density flows, leading to,

t +

uixi

= 0 (3.12)

ui

t +

ui

uj

xj=

p

xi+

ij

xj

sgsij

xj(3.13)

whereuiis the resolved velocity component,p is the resolved pressure andij is resolved stress tensor, which is defined as,ij = 2Sij 1

3

ukxk

ij

(3.14)

with the Favre filtered strain rate,

Sij , defined as,

Sij =12

uixj

+ujxi

(3.15)

and the sub-grid scale (SGS) stress, sgsij , defined as,

sgsij =( uiuj uiuj) (3.16)

and can be modeled using the gradient hypothesis as,

sgsij

1

3sgskk ij = 2t

Sij 13Smmij (3.17)

and t is the turbulent diffusion that should be computed using a SGSmodel. In the following two SGS models that are used in this thesis are in-troduced: the constant Smagorinsky model and the dynamic Smagorinsky

model.

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3.3.1 Constant Smagorinsky model

In the constant Smagorinsky approach the turbulent viscosity is obtainedby a simple approximation,

t =(CS)2|Sij | (3.18)

where

|

Sij |=

2

Sij

Sij (3.19)

andCS is the Smagorinsky constant. is the filter width which is takenhere as the cubic root of the local grid cell volume. CS= 0.1is used in thepresent work, following [55].

3.3.2 Dynamic Smagorinsky model

The main drawback of the constant Smagorinsky model is the unrealistic

prediction of the turbulent viscosity in near wall regions. The prediction

of the flow behavior near the wall region is very important for a successful

simulation [83]. The turbulent viscosity on the wall does not decay as fastas it happens in reality using the constant Smagorinsky model. Therefore,

the model has difficulties handling wall bounded flows. To overcome thislimitation a damping function can be used in the near wall region. How-

ever, the problem of properly tuningCS remains. The dynamic procedureproposed by Germano [30] uses the scale similarity [63] of the turbulent

motion of the fluid to determine the Smagorinsky constant locally.

Applying the test filter with the length to the filtered Navier-Stokesequation one gets the sub test stress tensor as,

Tij = uiuj ui uj (3.20)where the symbol .is defined as,

ui =ui (3.21)Modeling the deviatoric part of the sub test stress tensor gives,

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and using the fact that LDijMDij =LijM

Dij , the constant of the Smagorinsky

model can be computed dynamically each time step and in each grid point

following,

C2S=1

2

LijMDij

MDkl MDkl

(3.29)

3.4 Favre filtered mixture fraction equation

The Favre filtered mixture fraction equation is given by,

t

+ujxj

=

xj

D

xj

Jsgsj

(3.30)

where the diffusivity,D, is defined as,

D=

Sc (3.31)

A constant Schmidt number,Sc= 0.7

, is used in this work. In the equation(3.30) sub-grid scale turbulent scalar flux,Jsgsj , is,

Jsgsj =uj uj (3.32)

To close the Favre filtered mixture fraction equation, the above term should

be modeled. To model the sub-grid scale turbulent scalar flux a gradientmodel is used,

Jsgsj = Dt xj (3.33)whereDt is the turbulent diffusivity and is defined as,

Dt = t

Sct(3.34)

where the dynamic turbulent diffusivity, t, is given by the sub-grid scale

model and is computed as,

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equation 3.36 is not computable using simple assumptions. The last termof equation 3.36 is the rate of production or destruction of species . This

term is a non-linear function of species mass fraction and temperature,and the following relation cannot be held.

(T, Y) = (T , Y) (3.41)

The production-destruction rate term in species mass fraction equation is

the central problem of turbulent combustion modeling. This term is highly

non-linear because the temperature appears in the exponent of the expo-

nential function of chemical progress rate constant, equation 2.45. How-ever, models are proposed to overcome the major problem of turbulent com-

bustion modeling.

3.6 Scalar mixing

3.6.1 Variance of mixture fraction

The sub-grid scale variance of the mixture fraction is an important param-eter in studying turbulent combustive flows. It is used to determine the

shape of the probability density function, PDF, of the random variablesused in combustion. To find the mixture fraction variance a local equilib-rium assumption is used following [80],

2 =C

2

xk

xk

(3.42)

where is the cubic root of the local cell volume and C is a constantthat should be set prior to calculations. Branley and Jones [13] suggest avalue equal to 0.1 for the constant, but it is chosen to be 0.09 in this study.

Pierce and Moin [80] suggested to use the dynamic procedure to fine-tunethe constant. Jimenez et al. [37] proposed to solve the following transportequation for the variance of the mixture fraction.

2

t +

uj 2xj

=

xj

(D+ Dt)

2

xj

+ 2(D+ Dt)

xj

xj

(3.43)

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whereD andDt are molecular and turbulent diffusivity and are given byequations 3.31 and 3.34 respectively. The second term in the right-hand

side of the equation 3.43 is called scalar dissipation rate and is defined as,

= 2D xj

xj(3.44)

The scalar dissipation term mentioned above cannot be computed from the

resolved field and should be modeled. Models for the scalar dissipation will

be introduced in the following.

3.6.2 Scalar dissipation rate

The scalar dissipation rate is a quantity that measures the rate of mixing

of fuel and oxidizer. It is an important parameter in the study of turbulent

non-premixed flames [75]. Decomposition of the filtered scalar dissipationrate, equation 3.44, gives,

= 2D

xj+

xj

xj+

xj

(3.45)

working out further equation 3.45 gives,

= 2D xj

xj

+ 2D xj

xj

+ 4D

xj

xj

(3.46)The first term in equation 3.46 is due to interaction among the resolved

scales, the second term is due to the interaction between unresolved scales

and the last term is due to the interaction between resolved and unresolvedscales. Therefore, neglecting the last term in equation 3.46, the filtered

scalar dissipation rate is given by,

= m+ sgs (3.47)wherem can be directly computed from the resolved field as,

m = 2D

xj

xj

2D

xj

xj

(3.48)

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and sgs is the interaction between the SGS part which is modeled usingan equilibrium assumption following Pierce and Moin [80] as,

sgs = 2D

xj

xj

= 2Dt

xj

xj

(3.49)

Finally, the equation for scalar dissipation reads,

= 2(D+ Dt)

xj

xj

(3.50)

whereD andDt are molecular and turbulent diffusivity and are given byequations 3.31 and 3.34 respectively. Using the scalar dissipation modelbased on equation 3.50 for the variance of the mixture fraction transport

equation, equation 3.43, will lead to an inert scalar transport equation,

and it will only convect the value of the variance of the mixture fractionat the inlet. The reason is that the model proposed in equation 3.50 for

the scalar dissipation assumes a local equilibrium hypothesis in which the

unsteady, the convective, and the diffusive terms are negligible and esti-

mates that production and dissipation are equal.

Therefore a model for the scalar dissipation can be taken from RANS mod-eling of dissipation in terms of a characteristic mixing time assumed pro-

portional to the turbulent characteristic time. In LES, the SGS scalar mix-

ing time can be defined as,

1 =2 (3.51)A SGS turbulent characteristic time can be introduced as the ratio be-tween the SGS kinetic energy,k, and the Favre filtered kinetic energy dis-sipation rate,, which are defined respectively as,

k= 12

(uiui uiui) (3.52)

=

(uixj

uixj

) (3.53)

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Assuming a proportionality between both times, a model for

can be de-

rived as,

2

= 1 C =C k (3.54)

whereCis a constant and Jimenez et al. [37] proposed that we can setC=

1

Sc . The implementation of the proposed model has to use the approx-

imation ofkandderived from the SGS turbulence model. If a Smagorin-sky eddy viscosity model is used for the SGS stresses, following [37] we

have,

k= 2CI2 Sij Sij (3.55)= 2+ (CS)2|Sij |Sij Sij (3.56)

whereCIis a constant and Jimenez et al. [37] proposed to use CI = 0.07.Therefore the final implementation for

would be,

= + (CS)2|Sij |ScCI2

2 (3.57)

whereCSis the Smagorinsky constant.

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Chapter 4

Conditional Moment

Closure

Contents

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Conditional filter . . . . . . . . . . . . . . . . . . . . . . 38

4.3 Conditional moment closure formulations . . . . . . 38

4.3.1 Conditional moment closure sub-models . . . . . . 414.4 LES-CMC coupling . . . . . . . . . . . . . . . . . . . . . 46

4.4.1 LES-CMC data transfer . . . . . . . . . . . . . . . . 46

4.4.2 LES-CMC solution procedure . . . . . . . . . . . . 49

4.1 Introduction

In this chapter, the CMC equations are introduced as well as the sub-

models required to model the unclosed terms of the CMC equations. The

foundation of CMC relies on experimental observations that the variationof species mass fraction and temperature around their mean is small if

they are conditioned on the mixture fraction [49]. In non-premixed flames,a passive scalar namely the mixture fraction can describe the mixing be-

tween the fuel and oxidizer if the differential molecular diffusion is ne-

glected. Mixture fraction informs at any point how much of the mixture

in that point originates from the fuel stream. For more clarification on themixture fraction readers are referred to [54, 113]. More detailed informa-

tion about the research on the CMC model is presented in chapter 1.

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CHAPTER 4. CONDITIONAL MOMENT CLOSURE

4.2 Conditional filter

The conditional filter which is applied to the species mass fraction equa-tions and the energy equation is defined using the fine-grained probabilitydensity function as [107],

|(; x, t) = (x, t)(x, t)[ (x, t)]G(x x)dx(x, t) P(; x, t) (4.1)

In the equation 4.1, can be either mass fraction Y of species or en-thalpy h. G(x x) is the filter function which is used for LES filtering

and[ (x, t)]is the fine-grained probability density function.P()is adensity weighted filtered probability density function, FDF, which is givenby,

P(; x, t) = 1(x, t)

(x, t)[ (x, t)]G(x x)dx (4.2)

The conditional and unconditional values are related to each other by,

(x, t) = |(; x, t) P(; x, t)d (4.3)whereP(; x, t)is the FDF defined in equation 4.2.4.3 Conditional moment closure formulations

In the CMC model transport equations for the conditional species massfractions as well as conditional enthalpy are solved in time, physical space

and mixture fraction space. The transport equation for the density weighted

conditional filtered scalar, variable , is given by,

|t

+uj | |xj

=|2

2|2

+S|

|+ e+ eD (4.4)

wheree andeD are given by,

|

P()e =

xj |

P()

uj|

uj |

|

(4.5)

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| P()eD = xj

| P() Dxj

|

xj

| P() D

xj|

+|

xj

| P() D

xj|

(4.6)

and

| is the conditional scalar dissipation rate defined as,

|= 2 D xj

xj| (4.7)

Using equation 4.4, the equation for the conditional species mass fraction

takes the form,

Y|

t +

uj |

Y|

xj=

|

2

2

Y|

2 +|

|+ eY + eDY (4.8)

whereeY andeDYare given by,

| P()eY = xj

| P()ujY| uj |Y| (4.9)

|

P()eDY =

xj

|

P()

DYxj

|

xj

| P() DY Yxj

|+ Y|

xj

| P() D

xj|

(4.10)

The conditional turbulent flux term needs modeling, and the model forthat will be presented. The molecular transport term, eDY, is normallyneglected as high Reynolds number flows are studied.

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| P()eT = xj | P()ujT| uj |T| (4.15)The CMC sub-models for modeling the conditional velocity, the conditional

scalar dissipation rate and the conditional turbulent flux will be discussed

in the next section.

4.3.1 Conditional moment closure sub-models

Conditional scalar dissipation rate

Modeling the conditional scalar dissipation rate is very important in the

CMC model. The Conditional scalar dissipation controls the mixing be-

tween the fuel and the oxidizer in the mixture fraction space, especially

near the reaction zone. The importance of the conditional scalar dissipa-tion is pointed out in [56, 49]. Sardi et al. [96] performed the measure-

ment of the conditional scalar dissipation rate of a turbulent counterflow.

Kronenburg et al. [51] found an exact solution for the conditional scalardissipation rate of the locally self-similar turbulent reacting jet. They also

applied their method to investigate a hydrogen-air jet diffusion flame using

the CMC model. They also found that NOx prediction is affected by theconditional scalar dissipation model. An accurate prediction of the con-

ditional scalar dissipation will improve the predicted conditional species

mass fraction and conditional temperature drastically. In a recent study,Mortensen [66] and Mortensen and Andersson [67] used a transport equa-

tion of the conserved scalar probability density function to find the condi-

tional scalar dissipation rate.

In this thesis, two different methods for calculating the conditional scalar

dissipation rate are used; conditional volume averaging and amplitude

mapping closure model.

Conditional volume averaging procedure The first method is simply

conditional volume averaging of LES data in a given CMC cell to find the

conditional statistics. This method is used by Navarro-Martinez et al. [71].

The advantage of this method is the simplicity of the implementation. Themajor drawback of the conditional volume averaging is the dependency

on the number of LES cells inside a CMC cell. To have good conditional

statistics inside a CMC cell, there should be a sufficient amount of LES

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!a

"a

!c

"c

!b

"b

!d

"d

0

b,!

c,!

d

1

Figure 4.1: Schematic representation of conditional volume averaging procure, red

cell is the CMC cell and blue cells are LES cells

cells inside that CMC cell. This is a limitation in the usage of the con-

ditional volume averaging procedure. However, it is shown that in many

cases challenging combustion phenomena like flame lift-off [70] or turbu-lent bluff body flames [69] can be accurately described with only a limited

amount of CMC cells.

To find the conditional average of a quantity e.g. scalar dissipation rate

in a given CMC cell by conditional volume averaging, first the mixture

fraction space which starts at 0 and ends at 1, is divided into the desirednumber of intervals[0, 1, 2,..., n1, 1]. Then starting from the first inter-val, [0, 1], all the LES cells inside the CMC cell are searched to find theLES cells with a mixture fraction value that fits into the interval. Next thescalar dissipation rate in those CFD cells is averaged to obtain |

1in the

CMC cell. This is illustrated in figure 4.1 where it is assumed that 4 CFD

cells were found with mixture fraction in the interval [0, 1]. The resultingconditional scalar dissipation rate|

1is then,

|1

=a+b+ c+ d

4 (4.16)

The above procedure will be performed for all intervals in the mixture

fraction range and conditional scalar dissipation |i, will be found for eachinterval.

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!

G

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Figure 4.2: Distribution of the functionG()of the AMC model in mixture fractionspace

Amplitude mapping closure model The second method used in this

study to calculate the conditional scalar dissipation rate is the amplitudemapping closure, AMC, model proposed by OBrien and Jiang [74]. An

evaluation of the conditional scalar dissipation models with DNS in homo-

geneous turbulence showed that the AMC model provides excellent datapredictions [62]. The AMC model was adopted in the simulations of lifted

jet flames with two-dimensional conditional moment closure using RANS

model [40]. In another study the model is used for the simulation of coun-terflow flames [41]. The model equation following [41] reads,

|

= G()

1

0 G()

P()d

(4.17)

where is the scalar dissipation rate from the LES solution, equation(3.44).P()is the Favre filtered density function and G()is,

G() =exp(2[erf1(2 1)]2) (4.18)

A plot of theG()is shown in figure 4.2. The advantage of the AMC modelover conditional volume averaging is its independency from the number of

LES cells inside a CMC cell.

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Conditional turbulent flux

To solve the CMC equations, the terms eY ,ehand eTin equations 4.8, 4.11and 4.14 should be written in terms of conditional species mass fractions,enthalpy and temperature. To do so a gradient hypothesis is used as fol-

lows,

ujY| uj |Y|= Dt| Y|

xj(4.19)

ujh| uj |h|= Dt|h|xj

(4.20)

ujT| uj |T|= Dt|T|

xj(4.21)

whereDt| is the conditional turbulent diffusivity.Dt| is obtained bytransferring the turbulent diffusivity, Dt, from the LES grid to the CMCgrid, hereby using a conditional volume averaging method.

The conditional turbulent flux for any scalar, e.g. species mass fraction,

enthalpy or temperature, therefor becomes,

e = 1

| P() xj| P()uj| uj ||

= 1

|

P()

xj

|

P()

Dt|

|xj

(4.22)

further expansion gives,

e =

xj

Dt| |xj

+

Dt|| P()

|xj

| P()xj

(4.23)

The last term of equation 4.23 is the gradient of the FDF. It is assumed to

be small in comparison with the gradient of

|and is therefore neglected.

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Conditional velocity

Another important parameter in the CMC model is the conditional velocity.There are models available for the conditional velocity in the RANS con-text. However, there is no report of using them in LES. In this study, two

methods are used to determine the conditional velocity. The first method

is to apply the volume averaging procedure, used for the determination ofthe conditional turbulent flux and the conditional scalar dissipation rate,

to obtain the conditional velocity from LES data. The second method which

is proposed in [109] and [47] is to use a constant value for the conditionallyfiltered velocity.

ui|=ui (4.24)whereui is taken from the LES solution.Filtered density function

As FDF a -function is used whose shape depends on the first and the

second moment of the mixture fraction.

P() = r1(1 )s1

B(r, s) (4.25)

where

B(r, s) = (r)(s)

(r+ s) (4.26)

and (x)is the Gamma function. The parameters r and s are given as,

r=(1

)

2 1 (4.27)s= r

1 (4.28)The first moment is the filtered mixture fraction,, for which a transportequation is solved at every iteration, and the second moment is the vari-

ance of the mixture fraction,2, which can be found from the equilibriummodel, equation (3.42).

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X

Y

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

Figure 4.3: Coarse CMC mesh (green grid) superposed on fine CFD mesh (red grid)

4.4 LES-CMC coupling

The CMC equations are five dimensional partial differential equations.The CMC equations are written in time, three spatial and mixture frac-

tion dimensions. That means that the conditional species mass fractions

and temperature equations are very expensive to solve. Moreover for ev-ery species present in the chemical mechanism used for the simulation an

equations should be solved. Taking further into account the fine resolution

of the LES grid that is required, it is obvious that LES-CMC simulationsneed important computational power.

The conditional quantities have a different spatial dependency than theirunconditional counterparts because the micro-mixing effect is resolved inmixture fraction space and not in physical space. That often allows to re-

duce the dimensionality of the CMC equations to 1-D, 2-D or to use coarser

CMC grid. The common approach in LES-CMC simulations is to use lessCMC cells in comparison to LES cells in a given domain [71, 110, 105],

making LES-CMC simulations also less expensive, figure 4.3.

In this thesis also fewer CMC cells than LES cells are used. The CMC

equations are therefore solved on a CMC grid, with typical grid spacing

CMC, which is coarser than the LES grid, characterized by grid spacing

LES. Therefore, data transfers are required between the two different

grids. This will be explained in the following.

4.4.1 LES-CMC data transfer

To obtain the conditionally filtered variable at the CMC grid level,|C, in-tegrating the conditionally filtered variable

| over the volume of a CMC

cell yields,

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|C = VCMC P()|dV P()dV (4.29)

where all the terms under the integral are calculated on the LES grid for

a given CMC cell. Also, the terms with superscriptCin are calculated onthe CMC grid and the terms without superscript Care computed on theLES grid. The conditionally filtered variable at the CMC grid level can

also be integrated in space to obtain the unconditional value,

= |C P()d (4.30)where|C is solved in the CMC code on the CMC grid level andP()andare available on the LES grid level.The CMC equation for the integrated conditionally filtered variable is writ-ten as,

Y|Ct

+uj |CY|Cxj

=|2

C2Y|C2

+|

C

|C

+ eCY (4.31)

whereeCYis given, following 4.9, as,

|CPC()eCY = xj

|C PC()Dt|CY|Cxj

(4.32)

h|C

t +uj |Ch|C

xj=

|2

C2h|C2

+ eCh (4.33)

whereeCh is given by,

|C

PC()eCh =

xj

|

C

PC()

Dt|

C

hC

xj

(4.34)

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Figure 4.4: Procedure to find the unconditional temperature. LES cells (blue) are

superposed on the CMC cell (red)

equal for all LES cells inside the CMC cell, are integrated with the FDF.

Note that the FDF is different for each LES cell as a -function is used

that depends on the mixture fraction,, and the sub-grid scale variance ofmixture fraction,2, figure 4.4.4.4.2 LES-CMC solution procedure

In order to solve the CMC equations data coming from the LES solution

are required. The conditional turbulent flux, the conditional scalar dissi-pation and the conditional velocities are all calculated based on data from

LES solution. They can either be calculated directly from the LES data,

e.g. in the conditional volume averaging procedure or computed using a

model which needs the LES data to scale the modeled quantity, e.g. theAMC model for the conditional scalar dissipation rate.

The solution procedure which is shown in figure 4.5 can be summarized as:

1. Solving for the flow field and the mixture fraction field. The variance

of the mixture fraction is determined using 3.42. The scalar dissipa-tion rate is obtained using equation 3.44.

2. Solving for the conditional moment equations and integration of the

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Figure 4.5: Solution procedure

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conditional density and the conditional temperature with a pdf toobtain the unconditional density and temperature

T = T| P(; ,2)d (4.38)

Y= Y| P(; ,2)d (4.39)

1

= 1|

P(; ,2)d (4.40)3. Solving for the flow field with the new density field

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Chapter 5

Numerical Methods

Contents

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.2 Finite volume method . . . . . . . . . . . . . . . . . . . 54

5.3 Predictor-corrector method . . . . . . . . . . . . . . . 56

5.4 Solving the pressure Poisson equation . . . . . . . . . 58

5.4.1 Multigrid method . . . . . . . . . . . . . . . . . . . 59

5.5 Solving the CMC equations . . . . . . . . . . . . . . . . 615.6 Boundary conditions . . . . . . . . . . . . . . . . . . . . 62

5.6.1 Buffer zone . . . . . . . . . . . . . . . . . . . . . . . 62

5.6.2 Velocity lateral boundary condition . . . . . . . . . 63

5.1 Introduction

In the simulation of flows with combustion, low Mach number variable

density flows are frequently encountered. The reason is that in most of theapplications, e.g

PhD Thesis Mahdi Zakyani

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