Direct Numerical Simulation of Turbulent Planar Jets with Polymer Additives Nuno Filipe Cabral Pimentel Thesis to obtain the Master of Science Degree in Aerospace Engineering Supervisor: Prof. Carlos Frederico Neves Bettencourt da Silva Examination Committee Chairperson: Prof. Filipe Szolnoky Ramos Pinto Cunha Supervisor: Prof. Carlos Frederico Neves Bettencourt da Silva Member of the Committee: Prof. José Manuel da Silva Chaves Ribeiro Pereira November 2018
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Direct Numerical Simulation of Turbulent Planar Jets with
Polymer Additives
Nuno Filipe Cabral Pimentel
Thesis to obtain the Master of Science Degree in
Aerospace Engineering
Supervisor: Prof. Carlos Frederico Neves Bettencourt da Silva
Examination Committee
Chairperson: Prof. Filipe Szolnoky Ramos Pinto CunhaSupervisor: Prof. Carlos Frederico Neves Bettencourt da Silva
Member of the Committee: Prof. José Manuel da Silva Chaves Ribeiro Pereira
November 2018
ii
”Nothing in this world can take the place of persistence. Talent will not; nothing is more common than
unsuccessful people with talent. Genius will not; unrewarded genius is almost a proverb. Education will
not; the world is full of educated failures. Persistence and determination alone are omnipotent.”
- Calvin Coolidge
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Acknowledgments
Firstly, I would like to start thanking Professor Carlos Silva for giving me the opportunity to work in this
area. Also, for his availability to explain and teach many interesting things in the area, along with his
advices during the thesis.
I would like to thank Mateus Guimaraes, for the data post-processing and obtaining the first results
with the the algorithm developed within this work.
I would like to give a special thanks to my IST colleagues and friends Hugo Abreu and Afonso Ghira
for introducing me to Professor Carlos Silva and for the meaningful discussions throughout the thesis
development, which were crucial to clarify many things.
I would also like to give a special thanks to Millennium BCP, namely Dr. Antonio Carreira, for providing
a crucial time off work required to develop this work and supporting me throughout this thesis.
I would like to thank my friends and colleagues Bruno Conceicao, Diogo Antunes, Ricardo Verıssimo
and Pedro Abracos for their continuous support and motivation.
Finally, I would like to thank my family for supporting me throughout all my IST course, for their
understanding and motivation. This work is devoted to them.
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Resumo
Neste trabalho foram efetuadas simulacoes numericas diretas (DNS) do jato espacial plano turbulento
com moleculas polimericas para estudar o mecanismo de interacao do polımero no avanco do es-
coamento do jato. Estas simulacoes foram conseguidas com a implemtancao do modelo numerico
de fluido visco-elastico, representado pelo modelo constitutivo reologico FENE-P. O modelo numerico
para a equacao de transporte do tensor de conformacao foi adaptado as condicoes de fronteira nao
periodicas do escoamento, nao estando presente na literatura referencias ao caso em estudo. Na
concretizacao deste, foi tido em conta o desempenho computacional da simulacao, para o qual se
implementou um mecanismo de celulas fantasma por forma a reduzir o tempo de computacao. O
algoritmo desenvolvido foi alvo de uma extensa verificacao numerica, de forma a garantir a resolucao
correta das equacoes governantes para o escoamento com moleculas de polımero diluıdas. O algoritmo
e testado em simulacoes numericas diretas com variacao das caracterısticas fısicas das moleculas do
polımero, nomeadamente a concentracao polimerica e tempo de relaxacao. Os resultados permitiram
observer um decrescimo da espessura do jato com a presenca de fluido visco-elastico, assim como
uma reducao da dissipacao da energia viscosa. De notar que o algoritmo implementado representa um
enorme progresso nas simulacoes numericas de escoamento viscoelastico turbulento de jatos espaci-
ais, fornecendo os primeiros resultados numericos em estudos do genero.
Palavras-chave: Turbulencia visco-elastica, DNS, FENE-P, Jato plano turbulento espacial
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Abstract
In this study it was performed Direct Numerical Simulations (DNS) of a spatial turbulent planar jet with
polymer additives to further understand the mechanism of polymer interaction on jet flows. These sim-
ulations were achieved with the implementation of the visco-elastic fluid numerical model, represented
by the rheological constitutive FENE-P model. The numerical model for the conformation tensor trans-
port equation was adapted for the non-periodic boundary conditions of the flow, for which no references
are present in the literature. The development of the numerical algorithm took into consideration the
computational performance of the simulations, in which it was implement a ghost cell mechanism to
decrease the computation time. The developed algorithm has been extensively verified to ensure the
correct resolution of the governing equations for turbulent jet flow with polymer additives. The numerical
model was tested in direct numerical simulation for different polymer molecules physical characteristics,
namely the polymeric concentration and the relaxation time. The observed results verified a decrease in
the jet width on the presence of a visco-elastic flow, together with a decrease of the viscous energy dis-
sipation rate. It should be noted that the implemented numerical model represents a major progress on
the numerical simulations of turbulent spatial jet flow with polymer additives, providing the first numerical
LASEF Laboratory of Fluid Simulation in Energy and Fluids.
MDR Maximum Drag Reduction.
MHR Magneto-hydrodynamics.
MPI Message Passing Interface.
NUMA Non-uniform memory access.
PEO Polyethylene Oxide.
ppm parts per million.
SPD Symmetric Positive Definite.
T/NT Turbulent/Non-turbulent interface.
Greek symbols
αp, βp Runge-Kutta Coefficients.
β Polymer Concentration.
xix
∆t Time step.
δ Flow width.
δij Kronecker’s delta.
ε Turbulent kinetic energy dissipation.
η Kolmogorov length scale.
λ Taylor microscale.
λi Eigenvalues of Cij .
µ Dynamic Viscosity.
ν Kinematic Viscosity.
ρ Fluid density.
τη Kolmogorov time scale.
τp Zimm relaxation time of the polymer.
ξi Gauss-Legendre integration nodes.
Roman symbols
C Conformation tensor.
H Convective flux tensor.
u Velocity vector.
Ai Gauss-Legendre integration weights.
Cij Components i, j of the Conformation tensor.
E(k) Kinetic energy density function.
f(Ckk) Peterlin function.
h Width of the inlet slot.
k Wavenumber.
L Maximum polymer extension.
l Characteristic length scale.
Lx Computational domain length in the x direction.
Ly Computational domain length in the y direction.
Lz Computational domain length in the z direction.
xx
nx Number of grid points in the x direction.
ny Number of grid points in the y direction.
nz Number of grid points in the z direction.
p Pressure.
ri Instantaneous orientation of a polymer dumbbell.
Re Reynolds Number.
Reλ Reynolds Number based on the Taylor scale.
Sij Strain Rate tensor.
t Time.
Tij Stress tensor.
u First component (streamwise) of the velocity vector.
ui Velocity vector.
U1L Mean local co-flow velocity.
U1 Co-flow velocity.
U2 Peak velocity at the inlet.
Uc Mean local centreline velocity.
v Second component (normal) of the velocity vector.
w Third component (spanwise) of the velocity vector.
x First spatial (streamwise) coordinate.
y Second spatial (normal) coordinate.
z Third spatial (spanwise) coordinate.
Subscripts
i, j, k Computational indexes.
x, y, z Cartesian components.
Superscripts
[p] Polymer.
[s] Solvent.
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Chapter 1
Introduction
The objective of the current chapter is to describe the motivation underlying the present work and to
provide a brief description of the outline of the document.
1.1 Motivation
Since Toms (1948) [1] reported turbulent drag reduction by the addition of long-chain polymers by up to
80 percent, this effect has been studied for both practical and theoretical purposes. Currently, industrial
applications regarding polymer drag reduction are mostly related to long-distance liquid transportation
pipeline systems, marine vehicles and heating/cooling system water-circulating devices.
This work will focus on the study of this phenomenon on turbulent jet flows, to further understand the
mechanism of polymer interaction on jet flows and advance our understanding of fluid turbulence and
visco-elastic flows. It is important to note that the framework of this study has potential applications on
the aerospace industry, namely on heat transfer reduction by injection of micro-jets into turbine blades
[2]. There is a potential application on combining visco-elastic fluids with the micro-jet structures, as
the heat transfer behaviour of visco-elastic fluids has been studied recently, namely focused on cooling
applications for turbine disks [3, 4].
Moreover, it is possible to make a formal analogy between the Finite Extensible Nonlinear Elastic -
Peterlin (FENE-P) set of equations for visco-elastic flows and magneto-hydrodynamics (MHD) equations,
as mentioned in [5, 6]. Thus, the numerical model considered here to simulate polymer solutions (FENE-
P) will have a strong resemblance and application to the methodology used in numerical simulations of
MHD flows. As for aerospace industry applications, in recent years plasma flows have rapidly become
an important set of new technologies that find application to hypersonic propulsion, being evident a
growing interest in plasma-based flow manipulation through MHD forces [7–9]. This step, together with
the combining technological developments in electromagnetics, aerodynamics, and chemical kinetics,
might lead to a breakthrough in propulsion systems for improving aerospace vehicle performance [10].
As the current work will comprise the resolution of Navier-Stokes equations coupled with polymer
models using Direct Numerical Simulations (DNS), it is of the uttermost importance the optimization of
1
the computational performance of the numerical model. Such is entailed by considering parallelization
techniques for high performance computing, with transversal applications to numerical simulations in the
industry.
1.2 Objectives
The main objectives for the work presented here are:
1. To implement and verify a computational method to simulate turbulent jets with polymer additives
(FENE-P model) through Direct Numerical Simulations (DNS) on an existing turbulent jet DNS
code.
2. To optimize the computational performance of the resulting computational model.
3. To analyse the effects of polymer additives on the development of turbulent jets.
1.3 Thesis Outline
The present document is organized into six chapters.
In the first chapter, it is described the underlying usefulness of the current work, highlighting potential
future applications, along with a statement of the main objectives to be accomplished.
In the second chapter, it is introduced the main subjects fundamental to the present work. The topics
of turbulence and turbulent planar jet are presented, along with a review on flows with diluted polymer
solutions. It is also presented a review on parallelization techniques for high performance computing.
In the third chapter, it is described the constitutive equations concerning the velocity field and the
polymer transport equation, along with the detailed definition of the numerical methods that were used
to implement the equations for the spatial discretization, temporal advancement and the boundary con-
ditions. It is provided detail regarding the parallelization strategy, namely on the enhancements that
were developed during this work.
In the fourth chapter, it is presented the code verification with analytical solutions, for example, veri-
fication was performed with the Couette flow. The verification exercise was extended to all components
from the conformation tensor transport equation and, also, the coupling term on the velocity field equa-
tions.
In the fifth chapter, it is presented the results obtained from DNS of turbulent planar jet with polymer
additives. Several quantities are analysed for different polymer physical characteristics. It is worth
mentioning that these results are the first numerical results of DNS visco-elastic turbulent planar jet.
The conclusions are presented in the sixth chapter, as well as proposals for future work.
2
Chapter 2
Background
This chapter defines the concept of turbulence and planar jet flows, along with some of its characteristics.
Furthermore, it is introduced an overview of numerical methods for the phenomenon of polymer-induced
drag reduction (FENE-P) and High Performance Computing, with emphasis on parallelization strategies.
2.1 What is turbulence?
As highlighted by [11], turbulent flows are prevalent in engineering applications and nature. Indeed,
flows such as waterfalls, rivers or flows around vehicles (e.g. airplanes, vehicles) are of turbulent nature.
Figure 2.1 illustrates examples of this phenomena.
(a) Space shuttle Atlantis launch in 2006. Fromwww.nasa.gov/centers/marshall/history.
(b) View of the Mayon volcano at January 24, 2018.From Romeo Ranoco/Reuters.
Figure 2.1: Examples of turbulent flows in engineering applications and nature.
3
Turbulence is a characteristic of the flow, being generally associated with flows that are highly un-
steady with chaotic variations of velocity and pressure in space and time. The transition from a laminar
to a turbulent flow is defined by Reynolds number [12], which is given by:
Re =ρUl
µ=Ul
ν(2.1)
where µ is the molecular viscosity, ρ is the density and ν is the kinematic viscosity. U and l are the
characteristic velocity and length scale of the flow, respectively. In most cases, when Re exceeds a
certain critical value, the flow transits from laminar to turbulent flow. The Reynolds number expresses
the ratio of inertial/inviscid forces and viscous forces and indicates the flow’s reaction to disturbances,
because perturbations can be damped by viscosity or amplified by inviscid forces.
The turbulence chaotic nature is noted by watching the velocity variation in time measured on the
centreline of a turbulent jet, as illustrated on Figure 2.2.
Figure 2.2: Time history of the axial component of velocity on the centreline of a turbulent jet. From [11].
Despite the random behaviour observed on velocity U1(t), observations show that the average ve-
locity < U1 > and its fluctuations present a stable statistically behaviour, which justifies the study of
turbulent flows by means of statistics.
The first concept of energy transfer through scales in turbulence was introduced by Richardson [13],
which stated that turbulence is produced at a large scale and is progressively broken down into smaller
and smaller scales by inviscid forces, until reaching a sufficiently small scale where viscosity becomes
dominant and the mechanical energy is dissipated into heat.
This conceptual picture was translated into a quantitative theory by Kolmogorov [14], who summa-
rized the following hypothesis to define the energy transfer process in turbulence:
• The local isotropy hypothesis: At sufficiently high Reynolds number, the small-scale turbulent
motions are statistically isotropic.
• The first similarity hypothesis: At sufficiently high Reynolds number, the statistics of small-scale
4
motions have a universal form dependent only on viscosity and dissipation rate. This region is
named universal equilibrium range.
• The second similarity hypothesis: At sufficiently high Reynolds number, the statistics of motions
at scales that are much smaller than the integral scale and much larger than the smallest scale
have a universal form dependent only by the dissipation rate. This region is named inertial sub-
range. The zone in which the viscous effects start being dominant is named dissipation sub-range.
The smallest scales of turbulence are called the Kolmogorov scales. By using the first similarity
hypothesis, and performing a dimensional analysis, length (η), time (τη) and velocity (uη) are determined
by the viscosity ν and dissipation rate ε and are given by:
η ≡(ν3
ε
)1/4
, τη ≡√ν
ε, uη ≡ (νε)
1/4 (2.2)
Obukhov [15] formulated Kolmogorov’s hypothesis in the spectral space, in which the turbulence
eddies characteristics scale l is described as a function of a wave number k = 2π/l.
Figure 2.3: Energy spectrum at high Re under the Kolmogorov hypothesis for a Newtonian fluid. Flowscales are expressed as the log of the wavenumber, k = 1/l. L is the flow characteristic length, l0 is thecharacteristics size of the largest eddies and η is the Kolmogorov scale. lEI is the start of the inertialsub-range and lDI is the start of the dissipation sub-range. From [16].
The kinetic turbulence energy distribution along the spectrum of wave numbers is expressed in Figure
2.3. Ek is the turbulence kinetic energy, L is the characteristics length of the flow, l0 is the characteristics
size of the largest eddies and η is the Kolmogorov scale. lEI is the start of the inertial sub-range and
lDI is the start of the dissipation sub-range.
Using Kolmogorov’s second similarity hypothesis, that at the inertial sub-range viscous effects are
negligible, becoming only function of the dissipation rate and the eddies length scale l, it follows that the
5
energy spectrum is proportional to ε2/3κ−5/3 for a Newtonian fluid and expressed by:
E(k) = Ckε2/3κ−5/3 (2.3)
in which k is such that lEI > l > lDI , and Ck is the Kolmogorov’s constant.
For the case of a visco-elastic fluid, the polymer interaction with the flow leads to a steepening of
the kinetic energy spectrum beyond a wavenumber lp, which is the Lumley length-scale. In [17], it is
presented a model spectrum for the visco-elastic fluid consisting of 3 major regions: (I) Inertial cascade,
(E) Elastic subrange and (V) Viscous dissipation. This model spectrum is illustrated on Figure 2.4.
It is important to notice that the model spectrum for the visco-elastic fluid on Figure 2.4 has been
verified with experimental results by [17].
Figure 2.4: Model spectrum for turbulence in polymer solutions indicating the three regions: (I) κ < 1/lp,Kolmogorov’s inertial cascade; (E) 1/lp < κ < 1/ηp, Elastic subrange where fractions of turbulencekinetic energy are transferred to elastic energy, which is then dissipated by viscous drag of relaxingpolymers and internal friction between the monomers of a single polymer; (V) κ > 1/ηp, Viscous dis-sipation range where turbulence kinetic energy is dissipated by viscous forces, just as the Newtonianspectrum. From [17].
The (I) Inertial cascade on Figure 2.4 consists on a Newtonian inertial cascade at low wavenumbers,
whose governing equations have been above-mentioned.
According to [17], the (E) Elastic subrange is a spectral region separated from the inertial cascade
by the Lumley scale lp, which is determined by the elastic properties of the fluid and the turbulence
dissipation rate. In this region, fractions of the turbulence kinetic energy arriving from the inertial sub-
range are converted into elastic energy by polymer stretching. Subsequent relaxation of the stretched
polymers dissipates a part of this elastic energy due to the viscous drag of the polymer molecules in the
solvent and interactions between monomers of a single polymer. The remaining elastic energy is trans-
formed back to turbulence kinetic energy. This phenomenon is called back reaction and, consequently,
the energy flux on this region is continuously reduced from higher to lower wavenumbers.
6
Experimental data [17] suggested that this region follows a power-law with a slope of−3, as illustrated
on Figure 2.4, where the polymer stresses overcome the viscous stresses. The energy spectrum in a
turbulent structure of wavenumber k decreases according to:
E(k) = Ckε2/30 l5/3p (lpκ)
−3 (2.4)
where ε0 is equal to the energy flux from the Newtonian inertial cascade.
The (V) Viscous subrange consists on a region dominated by viscous stresses which transform
turbulence kinetic energy into heat, similarly to the dissipation range in the Newtonian spectrum, in
which the new viscous dissipation scale is given by:
ηp ≡(ν3
εv
)1/4
(2.5)
where εv is the energy flux contributing to the viscous dissipation at the end of the elastic subrange.
2.2 Turbulent Planar Jets
The turbulent plane jet flow is a type of flow belonging to the free shear flow type, which are charac-
terised by developing far away from the interaction with boundaries, and that advances along a preferred
direction. An example of a two-dimensional plane jet flow is presented in Figure 2.5.
Figure 2.5: Turbulent planar jet. From [18].
This type of flow is characterised by being statistically two-dimensional, with a dominant direction of
mean flow across the streamwise x direction and a mean velocity in the spanwise z direction of zero.
It is important to note some features of this kind of flow, namely that the velocity in the streamwise x
direction is much greater than the velocity in the normal y direction (u >> v), whilst the normal gradient
∂/∂y is much larger than the streamwise gradient ∂/∂x (∂/∂y >> ∂/∂x), as stated in [16]. Moreover, it
is worth mentioning the relation of flow scales for this type of flow, namely the shear layer thickness (δ)
and the flow streamwise length (L), for which L >> δ.
As the remaining shear flows, the turbulent plane jet is characterised by presenting a sudden change
from a turbulent rotational region, to an irrotational one. This transition is observed at the turbulent/non-
turbulent interface (T/NT), in which occurs exchanges of mass, momentum and other scalars while the
jet develops in the streamwise direction, spreading out. This dispersion of the flow makes it swell,
7
transmitting momentum to the adjoining fluid in the form of momentum and vorticity, dragging mass into
it in a phenomenon known as turbulent entrainment [16].
The plane jet can be split across different regions, as illustrated on Figure 2.6:
• The potential core region: It corresponds to the flow between the shear layers near the jet inlet,
originated by the high velocity gradient between the flow incoming from the jet inlet and that of the
remaining environment.
• The transition region: It comprises all the area in which the flow experiences transition into a fully
developed turbulent state.
• The self-similarity region: Region in which the flow characteristics, such as velocity and other
scalar properties, can be collapsed seamlessly after being properly scaled following the Kol-
mogorov’s hypothesis.
Figure 2.6: Turbulent planar jet regions. From [16].
2.3 Turbulence in dilute polymer solutions
2.3.1 Experimental discoveries
Since the discovery of the polymer drag reduction by Toms [1], there have been numerous experiments
done and theories proposed to explain the mechanisms underlying the drag reduction by dilute polymers
on turbulent flows. Indeed, Nadolink and Haigh [19] included over 2500 entries on the bibliography
review over this phenomenon. Nonetheless, Jin [20] refers that the most cited literature on polymer drag
reduction in boundary layers are Lumley [21, 22], Virk [23] and de Gennes [24], which also provide the
most successful attempts to explain the mechanism of polymer drag reduction.
8
Lumley [21] defined drag reduction as the reduction of skin friction in turbulent flow below that of
the solvent alone, by the addition of trace percentages of polymers. This phenomenon was observed
to occur in thermodynamically dilute solutions of long, flexible, expanded high-molecular-weight linear
polymers, considering typical polymer-solvent systems, such as the polyethylene oxide (PEO or Polyox)
in water. This one has become the most common solution due to its inexpensiveness, easiness to handle
and effectiveness (33% reduction in skin friction over that of water alone by the addition of 18 parts per
million (ppm) of PEO of molecular weight of 0.76× 106), as stated by [21]. An illustrative example of the
effect of PEO solution on turbulent flows is presented on Figure 2.7.
Figure 2.7: Turbulent mixing of jets of water (left) and PEO solution (right). From [25].
Further to that, Lumley [22] said that in regions where polymer molecules are elongated, viscosity is
enhanced. Moreover, Lumley stated the at sufficiently high wall shear stresses, the fluctuating strain rate
that in turbulent flows elongated polymer molecules are found in the wall layer, but not in the viscous
sub-layer, as vorticity and strain flow rate are not correlated (molecules do not expand, since there is
only simple shear on the viscous sub-layer).
Therefore, increased viscosity will only occur in the wall layer, leading to dissipation of turbulence
9
Figure 2.8: Turbulent boundary layer velocity profile. From [26].
fluctuations and mitigation of turbulence stresses. Small eddies on the wall layer will be damped by
the increased viscosity, and the resulting lower Reynolds stress at the buffer layer will thicken the vis-
cous sub-layer. The large eddies will expand on the viscous sub-layer and lead to higher streamwise
fluctuating velocities in this region. On the maximum drag reduction (MDR) regime large eddies will be
dominant.
Virk [27] compiled many experimental data available at that time on pipe flows with polymer additives
and smooth surfaces. Virk summarized three regimes of pipe flows with polymer additives:
• Laminar regime: Under a laminar flow regime, no drag reduction is observed, and the skin friction
obeys Poiseuille’s law, because the overall viscosity is very similar to the solvent viscosity in the
dilute solution.
• Polymeric regime: The flow is turbulent and the drag reduction effect is observed. Furthermore,
Virk states that the onset of drag reduction occurs at a universal characteristic wall shear stress,
and that the amount of drag reduction is dependent on the polymer characteristics, such as poly-
mer molecular weight and polymer concentration.
• Maximum drag reduction (MDR): It was determined the existence of a universal drag reduction
saturation asymptote, i.e. independent of system and polymer properties. The maximum drag
reduction is observed in the velocity profile and in the friction flow rate domain.
The drag reduction phenomenon appears to lie between two universal asymptotes: the Newtonian
turbulent flow (the so-called Prandtl-Karman law) and the MDR (see Figure 2.9). The experimental data
compiled in [23] confirmed the agreement with the MDR asymptote and drag reduction onset shear
stress relation. Indeed, it was concluded that MDR occurs when the effects of polymers are felt over all
scales, causing the buffer layer thickness to extend across the entire boundary layer.
10
Figure 2.9: Schematic illustration of the onset of drag reduction and the maximum drag reduction asymp-tote. The dotted line represents the case with a fixed polymer concentration C and increasing Re. Thedashed line is the case for a fixed Re, where the onset of drag reduction is first observed, and thepolymer concentration C is increased. f refers to the friction drag for pipe flows. From [28].
Moreover, Virk [23] observed and stated that the mean velocity profile can be divided into three zones
(see Figure 2.10):
• Standard viscous sub-layer: Usual region from the wall outward.
• Elastic sub-layer: This layer exists between the ”Newtonian plug” region and the usual viscous
sub-layer. It is characteristic of drag reduction and its extent increases with increasing drag reduc-
tion, up to the maximum drag reduction. On the latter situation, this layer is spread through the
entire cross section.
• Newtonian log layer: Also named ”Newtonian plug”, this layer has the universal slope logarithmic
profile. On this layer the velocity profile is shifted upwards, but parallel to the Newtonian law of the
wall.
The turbulence structure information can be analogously categorized into the same three zones.
Years later, experiments were performed in which it was observed that drag reduction initiates as
polymer enters lower inertial wall layer from above or below, suggesting that the source of the drag
reduction lies within the lower inertial range [24, 29]. Therefore, De Gennes [24] proposed a model based
on the elastic properties of the polymer particles, instead of the viscosity effects stated by Lumley. The
elastic theory postulates that the elastic energy stored by the partially stretched polymers is an important
variable for drag reduction and suppression of turbulence, and the increase in the effective viscosity is
small and inconsequential.
Moreover, the onset of drag reduction occurs when the cumulative elastic energy stored by the par-
tially stretched polymers becomes comparable with the kinetic energy in the buffer layer. The small
scales will be damped, leading to a buffer layer increase and reduced drag. The polymer concentration
11
Figure 2.10: Mean velocity profiles during drag reduction. From [23].
is included in the onset criterion because the cumulative elastic energy of the polymers is a function of
the concentration [24].
Recently, experiments have been made with fully developed three-dimensional turbulence that do
not support convincingly elastic theory. Instead, an approach based on an ”energy flux theory” is pro-
posed, that states that the turbulence energy flux in the cascade process is gradually reduced by the
energy transfer into the elastic motion of polymers, which becomes dominant in the small scales. An
experimental work that supports this approach is presented by [30].
2.3.2 Numerical simulations
Over the past 20 years, Direct Numerical Simulations (DNS) have played an increasingly important
role in the investigation of turbulent drag reduction mechanisms by polymer additives, namely for wall-
bounded turbulent flows. Despite the limitedness of DNS accuracy, due to the model for polymer stress,
inability to resolve all polymer scales and potential numerical instability, DNS provides the advantage of
describing the orientation of the polymer micro-structure in addition to the velocity field and Reynolds
stresses. This is especially relevant over laboratory experiments, allowing additional meaningful and
comprehensive insight over drag reduction mechanisms.
Typically, numerical simulations consider polymer molecules as two beads connected by an elastic
spring. The polymer dynamics are then described by the evolution of the end-to-end vector connecting
the two beads (see Figure 2.11), using constitutive equations such as Oldroyd-B and FENE-P [31].
Recently, the FENE-P polymer model has been widely used in DNS to study turbulence polymer drag
reduction, in such works as [32], [33] and [34].
The polymer orientation is represented as a continuous second-order tensor field, the so-called con-
formation tensor. This conformation tensor is defined as the normalised second moment of the end-to-
12
Figure 2.11: FENE-P dumbbell model with a spring between two beads. From [35].
end vector between the two beads [36], described as:
Cij =〈rirj〉
13 〈r2〉eq
(2.6)
where ri is the instantaneous orientation of a polymer dumbbell, r2eq is the square of the equilibrium
separation distance, and the angle brackets imply an ensemble average over the configuration space of
the dumbbell.
The conformation tensor Cij is a symmetric and positive definite (SPD) matrix. This property chal-
lenges the DNS of turbulence coupled FENE-P, as it is required that Cij remains semi-positive. While
mathematically both constraints are satisfied by the governing equations, these properties can be lost
due to cumulative numerical errors. Early attempts at numerical simulation of visco-elastic turbulence
were plagued by Hadamard instabilities that resulted from the numerical loss of the SPD property [37].
Sureshkumar and Beris [38] overcame these instabilities by introducing a stress diffusion term into
the equation for the conformation tensor. Variations of this approach were used in several investigations.
In 1997, the first DNS of channel flow [32] was able to show the Drag Reduction phenomenon, although
with lower Reynolds numbers in the numerical simulations in comparison to the conditions under which
Drag Reduction is experimentally observed with dilute polymers.
Vaithianathan [39] exploited the SPD property of Cij to derive independent equations for the eigen-
vectors and eigenvalues of the conformation tensor. In this formulation, Cij must remain greater than
zero and the eigenvalues of Cij should comply with the finite extensibility of the polymer:
λ1 + λ2 + λ3 ≤ L2 (2.7)
where λi are the eigenvalues of Cij and L is the maximum polymer extension (non-dimensional). Even
though this implicit formulation guarantees that Equation 2.7 is satisfied, the compact finite-difference
method that was used by [40] did not guarantee the eigenvalues remain positive. Instead, realizability
was enforced by setting the negative eigenvalues to zero before constructing the conformation tensor,
ensuring numerical stability. However, the uncontrolled, spatially distributed adjustments of the eigenval-
ues destroyed overall conservation of the conformation tensor, and spatial averages of the conformation
tensor contained spurious contributions from the convective term.
13
Thus, the decomposition applied by [39] guaranteed stability (by providing easy access to the eigen-
values) but did not guarantee conservation. This issue can be traced back to early numerical approaches
to compressible flows, which often suffered from loss of conservation [41]. It is related with the hyper-
bolic nature of the equation for Cij in the Oldroyd-B, FENE-P and Giesekus models, which admits shocks
(discontinuities) in the polymer stress tensor [42]. Discontinuities in the polymer stress cannot be fully
resolved by a grid, and so the main responsibility of the numerical scheme is to correctly predict the
jump magnitude. Jumps in the conformation tensor should satisfy the overall conservation balance to
guarantee correct elastic wave propagation. This behaviour is illustrated on Figure 2.12.
Figure 2.12: One-dimensional schematic of a shock (thick, solid line). Black dots represent the gridpoints. Thick dashed line is an ideal representation of the shock on the grid. A spectral representation,without an artificial stress diffusivity, would look like the thin dashed line, with overshoots and under-shoots (Gibbs phenomenon). The dotted line indicates the effect of adding the stress diffusivity to thespectral representation. From [36].
The solid line indicates a jump in the polymer stress tensor across a discontinuity. The other curves
illustrate the equivalent numerical representation based on a finite-difference and spectral scheme. The
Gibbs phenomenon observed in the spectral representation can be attenuated by introducing an artificial
diffusivity. However, artificial diffusion can also reduce the magnitude of the jump. There are more
sophisticated approaches to filtering the spectral modes to (just) eliminate the ringing, but they are still
at a relatively early stage of development [43]. In contrast, specific finite-difference schemes have been
designed to maintain the magnitude of the jump and avoid excessive spreading of the discontinuity.
Early hyperbolic solvers were first-order in space, robust and reliable, yet often overly dissipative
[41]. More recent approaches have overcome these shortcomings while still preserving the simplicity
of implementation and robustness. The approach taken by [36, 44] consists on an algorithm based
on the method of Kurganov and Tadmor (KT). This second-order scheme guarantees that a positive
scalar will remain so at all points and it was generalized to guarantee that an SPD tensor also remains
SPD. Furthermore, the method dissipates less elastic energy than methods based on artificial diffusion,
resulting in stronger polymer–flow interactions. This approach has been successfully used in DNS of
Shear Flow, Decaying and Forced HIT [45–47], and, as such, will be used in this dissertation.
14
2.4 High performance computing and parallelization strategies
As stated in [48], there is an increasing need of computational resources to perform increasingly complex
DNS simulations. In the case of large computational simulations of fluid flow, and to take advantage of
these resources, it is necessary to develop codes that can run in parallel and are suited to the cluster
configurations that exist nowadays [49].
2.4.1 Parallelization Efficiency
In this subsection, it is presented the metrics used for the performance of the code, namely the speed-up
and the efficiency of the parallel code. While the speed-up provides a good measure of the performance,
it would be irrelevant if the number of processors was not known, while the efficiency provides a way of
comparison between the different configurations.
The speed-up is defined as the ratio between the sequential time ts and the parallel equivalent tp,
given by equation (2.8):
Speed− up =tstp
=σ(n) + ψ(n)
σ(n) + ψ(n)p + κ(n, p)
(2.8)
in which σ is the time in inherently serial tasks, ψ is the time spent in parallelizable tasks, p is the number
of processing units, n is a measure of the problem size (such as mesh points), and κ is the measure of
communication introduced by parallelization, which is a function of both the problem size and the number
of processors. The σ(n) + ψ(n) reflects the total sequential time of the code. It is convenient to define
this metric in terms of fractions of the code with reference to the total time. Thus, it is introduced the
parallel Fp and sequential Fp fractions of the code. Ideally the parallel fraction would be 100%, but it is
not possible since there are always some initialization procedures that cannot be performed in parallel.
The parallel and sequential fractions are related according to equation (2.9):
Fp =ψ(n)
σ(n) + ψ(n)= 1− Fs (2.9)
If one considers that κ is neglectable, it is obtained the Amdahl’s Law from (2.10). Amdahl’s Law provides
an upper limit to the performance of a parallel code and it shows that unless the sequential fraction is
very small, there will be no gain in increasing the number of processors.
Speed− up =1
1 + Fp
(1p − 1
) (2.10)
The efficiency of a parallel code (equation (2.11)) is given by dividing the speed-up by the number
of processes. It provides a comparable measure of performance between configurations that differ in
problem size and/or number of processors.
Efficiency =1
p+ Fp (1− p)(2.11)
In Figure 2.13 it is shown the evolution of efficiency with the number of processors. It becomes clear why
15
scalability is critical, as larger numbers of processors will yield less and less performance improvements.
Figure 2.13: Evolution of efficiency with the number of processors. From [18].
2.4.2 Load Balancing
Load balancing is an important concept to assess performance of parallelization code. It consists on
dividing a task among different processing units in a synchronized way, that is each unit takes the same
amount of time to complete its part, without having processes finishing earlier than the others and
waiting, resulting in idling resources. The slowest process will be the limiting factor when assessing load
balancing.
Another form of imbalance is related to memory imbalance between processors. On machines with
little memory per node, load imbalance can become burdensome.
2.4.3 Granularity
Code granularity refers to measure between computation and communication, once a process in parallel
programming is split across multiple processing units that work simultaneously. The following types of
granularity are defined in [49]:
• Coarser grain: The task is divided in large components, with greater computation work between
communication stages, which implies a better opportunity for performance increase. The load
balancing is more difficult to achieve, with possibly longer synchronization waiting periods.
• Finer grain: The task is divided into small components and less computation is required between
communication events. Despite being an advantage for load balancing, it will lead to higher com-
munication overheads, thus leading to potential worse performance.
16
2.4.4 Distributed vs. Shared memory parallelization
Shared memory parallelization
Shared memory parallelization refers to code being executed by various processors accessing a shared
memory. Thread-based models deal with each thread behaving as a processing unit that has no sepa-
rate memory of its own and shares its memory with the other threads, each thread doing its part of the
processing on the same shared memory, contributing to the desired result. This implies that all threads
or processes see and have access to the same memory. An illustration of this set-up is presented in
Figure 2.14.
Figure 2.14: Illustration of OpenMP architecture. From [50].
The standard application for shared memory configuration is OpenMP. It uses the Fork-Join model, in
which the master thread is responsible for synchronizing the run and distributing the code amongst other
threads. Then, the code is executed between threads and, after the parallelization step, the remaining
threads are joined with the master thread so the code proceeds in sequential mode.
According to [50], this configuration presents the following limitations:
• All processes or processors demand access to the same memory. Even though a software layer
that deals with simulating a shared memory can be used, the performance will be severed.
• Scalability is physically limited by the size of the shared memory node on which it is running.
• Some variables and locations in memory also must be duplicated to avoid race conditions, where
multiple threads try to update the same shared variable simultaneously. Care must be taken not to
overlap multiple processors changing the same place in memory simultaneously.
• Not suitable for dynamic problems (i.e. when the workload fluctuates rapidly during the execution).
Distributed memory parallelization
Contrary to shared memory parallelization, distributed memory relies on communication between pro-
cessing units, each one with its individual memory.
17
Message Passing Interface, or MPI, is the standard API used to parallelize programs that run on
distributed memory architectures. It provides different types of communications, such as point-to-point
and collective (e.g. gathering or broadcasting of data). The communications must be explicitly handled
by the user. Moreover, it provides other functionalities, such as parallel I/O and derived datatypes. An
illustration of this configuration is presented in Figure 2.15.
Figure 2.15: Illustration of MPI architecture. From [50].
It is important to note that the current work considers a parallelized code under MPI.
According to [50], this configuration presents the following limitations:
• Final speed-up limited by the purely sequential fraction of the code.
• Scalability is limited due to the additional costs related to the MPI library and load balancing man-
agement.
• MPI is a flat model, treating each MPI process as a separate physical processor and memory,
independently of physical architecture. This means that unnecessary communications might take
place between processes that physically have access to the same memory.
• Certain types of collective communications become more and more time-consuming as the number
of processes increases, namely MPIAlltoall, which is used in this work.
• As each processor gets its own portion to process before the task is under-way, the assumption of
equal load distribution is taken, which might not be the case as the processing progresses. This
load unbalance might be accounted for by the user, but it involves great programming effort as MPI
is not developed with this taken into consideration, and overheads of more frequent communication
for load distribution appear.
2.4.5 Hybrid parallelization
Hybrid parallel programming consists of mixing several parallel programming paradigms in order to ben-
efit from the advantages of the different approaches. In general, MPI is used for communication between
18
processes, and another paradigm, such as OpenMP, is used inside each process. An illustration of this
configuration is presented in Figure 2.16.
Figure 2.16: Illustration of Hybrid MPI+OpenMP architecture. From [50].
A hybrid programming configuration presents the following advantages in comparison to the dis-
tributed and shared memory [49]:
• Improved scalability through a reduction in both the number of MPI messages and the number of
processes involved in collective communications. This gain is specially verified if the non-hybrid
code uses communications of type MPI AlltoAll.
• More adequate to the architecture of modern supercomputers (e.g. interconnected shared-memory
nodes, NUMA machines, ...), whereas MPI used alone is a flat approach.
• Optimization of the total memory consumption, due to the OpenMP shared-memory approach;
less replicated data in the MPI processes. According to [49], benchmark of pure MPI to Hybrid
provided a memory gain between 80% to 480%.
• Fewer simultaneous accesses in I/O and a larger average record size, with potential significant
time savings on a massively parallel application.
• Better load balancing, by joining OpenMP load balancing features with MPI.
However, implementation of hybrid programming comes with higher level of complexity, with gains in
performance not being guaranteed according to the final application [50].
19
20
Chapter 3
Governing equations and Numerical
Methods
This chapter describes in detail the implementation of the Navier-Stokes equations that govern viscous
fluid flows, considering a coupling mechanism to represent the polymer additive-turbulence interaction,
along with the numerical methods considered in the main code that were used for the simulation of turbu-
lent planar jet with polymer additives. It is highlighted the discretization schemes considered throughout
jet simulations, as well as the parallelization techniques.
It is important to note that the starting point of the numeric work here presented is a DNS code for
turbulent jet described in Ricardo Reis’ PhD thesis [16] and Diogo Lopes’ PhD thesis [18]. Both thesis
were concerned about DNS of Newtonian fluids, presenting extensive verification and validation of the
DNS of turbulent jets. Furthermore, the core of the parallelization strategy was developed in the above-
mentioned works. Subsequently, the main features that were added to the DNS code are as follows:
• Introduction of the Conformation tensor transport equation;
• Coupling of the Navier-Stokes momentum equation with the Polymer Stress Tensor;
• Use of ghost cells data on streamwise slab configuration.
An overview regarding the discretization schemes considered on this work is now presented:
• The Navier-Stokes momentum equation is solved considering a pseudo-spectral scheme for spa-
tial discretization, together with fully explicit third order Runge-Kutta for temporal advancement;
• The Conformation tensor transport equation is solved according to the following structure:
– The convective term is discretized according to the Kurganov-Tadmor (KT);
– The stretching term is discretized using the finite difference method;
– The method used for temporal advancement is the explicit third order Runge-Kutta method.
21
3.1 Velocity field
On the velocity field governing equations, it is assumed that the fluid is incompressible and that it satisfies
the continuity and momentum equations with an additional term related with divergence of polymer
stress. Below it is presented the continuity and momentum, respectively:
∂ui∂xi
= 0 (3.1)
∂ui∂t
+ uk∂ui∂xk
= −1
ρ
∂p
∂xi+
1
ρ
∂Tij∂xj
(3.2)
where ui is the i component of the velocity vector, ρ is the constant fluid density, p is the local pressure
and Tij is the combination of the viscous and polymer stress tensors. The stress tensor can be ex-
pressed as a linear sum of contributions from the Newtonian stress (T [s]ij ) and the polymer stress (T [p]
ij ),
as shown below:
Tij = T[s]ij + T
[p]ij (3.3)
where the Newtonian stress tensor T [s]ij is given by:
T[s]ij = 2ν[s] ∂Sij
∂xj(3.4)
being ν[s] the kinematic viscosity of the fluid. The Sij is the Strain Rate Tensor defined as:
Sij =1
2
(∂ui∂xj
+∂uj∂xi
)(3.5)
The Polymer Stress Tensor T [p]ij is described in the next sub-section.
3.2 The FENE-P constitutive model
Among the conformation models used in turbulence polymer simulations, the Finitely Extensible Non-
Linear Elastic Peterlin Model (FENE-P) is the most widely used configuration [28, 45]. FENE-P model
considers the polymer solution as a flowing suspension sufficiently dilute that the polymer molecules
do not interact with each other and that each molecule is idealized as an elastic dumbbell composed
by two beads connected by a non-linear spring with a maximum length [31]. Under this configuration,
and considering a uniform polymer concentration field, the constitutive equation of the Polymer Stress
Tensor yields the following relationship:
T[p]ij =
ρν[p]
τp[f(Ckk)Cij − δij ] (3.6)
where Cij is the conformation tensor, ν[p] the zero shear-rate polymeric viscosity, δij is the Kronecker
delta, L is the maximum possible extension of polymers and τp is the Zimm relaxation time of the
22
polymer. The polymer viscosity ν[p] is included in the model by a non-dimensional parameter β, which
represents the ratio between the solvent and the zero-shear-rate viscosity:
β =ν[s]
ν[p] + ν[s](3.7)
The function f(Ckk) is the Peterlin function given by:
f(Ckk) ≡ L2 − 3
L2 − Ckk(3.8)
where Ckk = Cxx + Cyy + Czz is the trace of the conformation tensor, which represents the extension
length. This function ensures finite extensibility, as it gives rise to a non-linear spring force that diverges
as√Ckk → L, preventing the spring from extending beyond L [36].
To complement the continuity equation and the conservation of momentum equation, a transport
equation for the conformation tensor Cij is required. Under the FENE-P model, the conformation tensor
evolution equation is given by:
∂Cij∂t
+ uk∂Cij∂xk
=∂ui∂xk
Cjk +∂uj∂xk
Cik −1
ρ
T[p]ij
ν[p](3.9)
The former equation is solved simultaneously with the velocity field flow equations, ensuring polymer-
flow interaction as the polymer molecules are deformed by the velocity field and, in turn, the resulting
conformation tensor introduces a polymeric stress on the flow structure.
3.3 Properties of the conformation tensor
The effects of polymer stretching on the flow development may be understood as either an elastic effect
or a viscous effect. In the framework of the FENE-P model, if the problem is approached from the
perspective of the elastic theory of drag reduction, the transport equation for the elastic energy can be
studied to understand the energy transfer between the polymers and the flow [51, 52]. Alternatively, if
polymer interaction is approached from the perspective of the viscous theory, Benzi [53] highlighted the
different physical roles played by the different components of the conformation tensor Cij , highlighting
the important role of Cyy which appears in the momentum and kinetic energy equations as an effective
viscosity.
Furthermore, the conformation tensor Cij is a measure of the second-order moment of the end-
to-end distance vector of the polymer dumbbell. It can be written as (2.6) where the vector r is the
separation vector between the two beads of the dumbbell. From the definition, it follows that the confor-
mation tensor is a symmetric positive definite (SPD) matrix [39]. Hulsen [54] proved that during exact
time evolution the conformation tensor must remain positive definite if it were initially. Nonetheless, cu-
mulative numerical errors that arise from virtually all initial value problem algorithms can give rise to
negative eigenvalues, which in turn cause the unbounded growth of Hadamard instabilities that quickly
overwhelm the calculation [55].
23
Another important property of the conformation tensor is that the trace, which represents the square
of the separation distance, must always be less than the square of the maximum extension, i.e., r2 <=
L2. The model guarantees this property through the force term f(Ckk), which diverges in strength as
this limit is approached, as mentioned previously. Hence for flows of arbitrary strength, the restoring
force is always sufficient to maintain this constraint. However, numerical errors in the evaluation of Ckk
can lead to violations of this constraint. Extension past L2 causes the force to change sign, resulting in
the rapid divergence of the calculation [39].
3.4 Computational domain
The computational domain for the turbulent jet spatial simulation consists of a box with uniform grid and
with Lx,Ly and Lz dimensions in each of the x (streamwise), y (normal) and z (spanwise) directions
(see Figure 3.1). The y and z directions have periodic boundary conditions, whilst the streamwise x is a
non-periodic direction with inflow and outflow boundaries.
Figure 3.1: View of the computational domain with reference frame and notation. From [16].
3.5 Spatial discretization
This section details all the spatial discretization procedures. The code employs the following structure:
Momentum equation terms highly accurate pseudo-spectral schemes to solve derivative on the normal
and spanwise directions and uses a compact scheme on the streamwise direction. As for the Con-
formation Tensor Transport equation, it is employed the 2nd order scheme Kurganov-Tadmor and finite
difference method.
3.5.1 Pseudo-spectral scheme
Pseudo-spectral schemes are used in spatial discretization in the normal and spanwise direction, as
these schemes are limited to the treatment of periodic functions. These schemes allow the representa-
24
tion of periodic functions in the Fourier space (also called spectral) in terms of frequencies for temporal
functions, and in terms of wavenumbers for spatial functions. Due to their high-level of accuracy without
numerical dispersion (to the machine precision), pseudo-spectral schemes are extensively used in CFD
[56]. Given a periodic flow variable along the normal y and spanwise directions z in the physical space,
φ(x, y, z, t), an inverse 2D discrete Fourier transform can be used to expand it through (3.10),
φ(x, y, z, t) =
ny2 −1∑
j=−ny2
nz2 −1∑
k=−nz2
φ(x, ky, kz, t)eι(kyy+kzz) (3.10)
where j, k are the indexes of the points along the normal y and spanwise z direction, respectively, and
(ky, kz) are the Fourier wave numbers defined by,
ky =2π
Lyj
kz =2π
Lzk. (3.11)
where i =√−1 is the imaginary unit, Ly and Lz are the spatial dimensions along the y and z directions,
and ny and nz are the number of discrete mesh points along those same directions, respectively.
Each Fourier coefficient φ(x, y, z, t), can then be obtained using the (direct) 2D discrete Fourier
transform,
φ(x, ky, kz, t) =1
nynz
ny−1∑j=0
nz−1∑k=0
φ(x, y, z, t)e−ι(kyy+kzz) (3.12)
It is important to note that the spectral variable φ is defined as a function of streamwise distance x,
as this direction does not present a periodic behaviour on the planar jet. Having defined the general
Discrete Fourier Transform (DFT), it is important to note that one of the main features of this method
is the simplicity of most operations in the spectral space [57]. In fact, derivatives in the physical space
result from the simple multiplication in the Fourier space, like
∂φ
∂y= ιkyφ
∂φ
∂z= ιkzφ (3.13)
resulting in a computationally light operation, with the accuracy only suffering from the rounding errors
and the discretization for the Fourier Transform. In addition to the high-level accuracy, the Fourier Trans-
form became more computationally interesting in 1965 with the publication of the Fast Fourier Transform
(FFT) algorithm. This algorithm presents a computational cost proportional to N log2N , where N is the
number of discretization points used. This cost is quite low comparing to the straightforward computation
of one-dimension Fourier transform with a proportional computational cost of N2 [58].
25
3.5.2 Compact finite differences discretization
The streamwise direction does not present a periodic behaviour, with one boundary being the jet inlet
with a predefined velocity and noise, and the other boundary being the outlet which ensures the flow is
not perturbed. Hence, spectral schemes are excluded to discretize the streamwise direction. In order
to maintain high accuracy, the code employs a 6th order compact finite difference scheme, described
in various references such as [40], [16] and [59]. It has been shown that these schemes provide easily
high order accuracy and possess some good ”spectral characteristics” [59] [18]. The basic idea of this
scheme is the definition of a derivative in terms of its neighbouring derivatives, which means the scheme
is implicit and requires solving a linear system of equations.
Considering a uniform grid with spatial coordinates xi = (i − 1)∆x, with ∆x = Const. (i ∈ [1, nx],
xi ∈ [0, Lx]), and using Taylor expansions of the function fi = f(xi) = f(x) with first derivative f ′i =
f ′(xi) = dfdx (xi) and second derivative f ′′i = f ′′(xi) = d2f
dx2 (xi), one can defined the general formula as
p∑j=−p
αjf′i+j =
q∑k=−q
akfi+k +O(∆xn) (3.14)
with p, q ∈ N. The order of the approximation n depends on the restrictions imposed upon the numbers
αj and ak. For the 6th order compact scheme, p = 2 and q = 3 in (3.14). Thus, (3.15) is obtained for the
Figure 4.14: Cij numerical (n) and analytical (a) solutions comparison considering a v velocity profile inthe streamwise x direction.
It is possible to observe that the numerical and analytical solutions present the same values for
all steps in the verification exercise of the conformation tensor transport equation. Thus, the code is
deemed as verified.
57
4.2 Velocity - Conformation Tensor Coupling Term Verification
As mentioned before, the velocity - conformation tensor coupling term was verified by imposing a known
Cij field and comparing the obtained result from the numerical scheme with an analytical solution. It
should be noted that this verification assumes a frozen in time Cij field to match the analytical solution.
The velocity - conformation tensor coupling term, expressed in (3.2) and (3.6), can be written as:
∂T[p]ij
∂xj=ν[p]
τp
∂ [f(Ckk)Cij − δij ]∂xj
(4.28)
On (4.28) the terms ν[p], τp and δij are constants. Moreover, the coupling term must be verified for each
of the velocity components. Therefore, for the purpose of obtaining an analytical solution for the coupling
term, it was assumed that the main diagonal values of the Cij tensor are set to one, as this setting would
nullify the coupling term’s derivative on the same direction of the respective velocity component. The
non-diagonal values of the Cij tensor were imposed and adapted to each velocity component coupling
term.
The analytical solution is derived with detail in this section for the u velocity component. It should be
noted that the coupling term for the remaining velocity components was derived analogously.
The coupling term for the u velocity component can be expanded to the following form:
∂T[p]ij
∂x=ν[p]
τp
(∂ [f(Ckk)C11 − δ11]
∂x+∂ [f(Ckk)C12 − δ12]
∂y+∂ [f(Ckk)C13 − δ13]
∂z
)(4.29)
From (4.29) it is noticeable that the ∂x term is zero, as the Peterlin function f(Ckk) and C11 are
equal to one. For the terms C12 and C13, it was assumed a linear profile along the y and z directions,
respectively.
Considering the previous remarks, equation (4.29) can be rewritten as follows:
∂T[p]ij
∂x=ν[p]
τp
(∂C12
∂y+∂C13
∂z
)(4.30)
Therefore, C12 and C13 profiles were computed as a function with independent variable equal to the
respective partial derivative on equation (4.30). The considered profiles are shown in 4.15.
Profiles shown on 4.15 can be written as:
C12 =
zhπ5 + π, if yh < 0
− zhπ5 + π, if yh ≥ 0
(4.31)
C13 =z
h
π
5(4.32)
For this verification exercise, the following simulation parameters were considered. It should be noted
that these parameters are valid for the coupling term verification exercise in all velocity components.
Thus, for the verification test, given the properties from Table 4.2 and profiles from Figure 4.15, the
58
(a) C12 profile (b) C13 profile
Figure 4.15: C12 and C13 profiles prescribed for the coupling term on the u velocity component. H is theunit of length used in the code.
General properties Polymer properties
Box height h 2π Zero shear-rate viscosity ν[p] 0.002
Box dimensions 10× 10× 10 Maximum molecular extensibility L 100
Mesh 128× 128× 128 Relaxation time (s) τp 0.1
Polymer concentration β 0.8
Table 4.2: Coupling term simulation parameters.
analytical solution for tensor Cij is the following:
∂T[p]ij
∂x'
0.0398, if yh < 0
0, if yh ≥ 0
(4.33)
Having computed the analytical solution, it is performed the comparison with the numerical solution.
The result for the coupling term on the u velocity is presented in Figure 4.16.
From Figure 4.16, it is possible to see that the dashes lines regarding the numerical (red colour) and
analytical (blue colour) solutions follow the same trend and present the same values.
As mentioned before, the verification exercise was performed analogously for the remaining velocity
components. Now it is presented the verification results for the coupling term in the v velocity component
in Figures 4.17 and 4.18.
As observed on the coupling term for the u velocity, the dashes lines regarding the numerical (red
colour) and analytical (blue colour) solutions follow the same trend and present the same values on the
coupling term for the v velocity.
Similarly, to the v velocity component, it is shown the verification results for the coupling term in the
w velocity component in Figures 4.19 and 4.20.
As observed on the coupling term for the u velocity, the dashes lines regarding the numerical (red
colour) and analytical (blue colour) solutions follow the same trend and present the same values on the
coupling term for the w velocity.
59
Figure 4.16: Coupling term for u velocity for numerical (red) and analytical (blue) solutions.
(a) C12 profile (b) C23 profile
Figure 4.17: C12 and C23 profiles prescribed for the coupling term on the v velocity component. h is theunit of length used in the code.
Having obtained the results for the coupling term verification exercise, it is possible to conclude that
the code is verified given the precision of the results between analytical and numerical solutions.
60
Figure 4.18: Coupling term for u velocity for numerical (red) and analytical (blue) solutions.
(a) C13 profile (b) C23 profile
Figure 4.19: C13 and C23 profiles prescribed for the coupling term on the w velocity component. h is theunit of length used in the code.
Figure 4.20: Coupling term for u velocity for numerical (red) and analytical (blue) solutions.
61
62
Chapter 5
Results
This chapter is devoted to the analysis of new results obtained with the code developed within this thesis,
namely the study of the influence of polymer additives on the turbulent planar jet. It should be noted that
the results were produced as part of an ongoing investigation with Mateus C. Guimaraes [69].
The addition of polymer additives to a Newtonian solvent has proven to strongly influence the flow
physics. However, the flow physics of the visco-elastic scenario still lack understanding in comparison
to the Newtonian case, mainly due to its complexity and lack of data at an experimental and numerical
level. Some studies have been performed considering homogeneous isotropic turbulence [35, 47, 57],
but flows with a non-zero mean shear, such as planar jets, have not yet been investigated numerically
or experimentally.
It is important to note that the validation of the Newtonian code with other literature references is not
presented in this section. There has been an extensive validation process presented by [16, 18] for the
Newtonian turbulent planar jet.
Having the code been verified (see Chapter 4 for further detail), studies were carried out using DNS
of FENE-P turbulent plane jets considering the simulation parameters shown in Tables 5.1 and 5.2.
The physical length of the computational box in the x streamwise direction was chosen to allow for
the simulation of a fully developed turbulent jet. Concerning the normal direction y, the computational
domain must be large enough to accommodate the jet without interacting with the normal boundaries.
According to [18], the mesh size should be of the order of the Kolmogorov small scale, to be able to
capture the turbulent phenomenon. Moreover, since turbulence is isotropic at the small scales, the
mesh size is required to be approximately equal in all direction. In addition to this, the number of points
in the y normal and z spanwise directions are required to be decomposable into powers of 2 for better
performance of the FFTW library. Also, due to the parallelization strategy the number of points in the x
streamwise and y spanwise directions must be even.
63
nx ny nz Lx/h Ly/h Lz/h
512 512 128 18 18 4
Table 5.1: Computational domain parameters for the simulations. Number of grid points (nx, ny, nz);non-dimensional grid size (Lx/h, Ly/h, Lz/h).
It should be noted that parameters from Table 5.1 are equal to all cases presented on Table 5.2.
Moreover, the grid is uniform in the three spatial directions and that the total number of mesh points
amounts to a total of just over 33 million.
As for the physical simulation parameters, results in this section considered the cases presented in
Table 5.2.
Case Wi Re Reλ τp(s) β L ∆x/η14h
A 0 3500 100 0 1 0 4.0
B 0.33 3500 110 0.025 0.8 100 4.0
C 0.65 3500 110 0.05 0.8 100 4.0
D 1.19 3500 110 0.10 0.8 100 3.9
E 1.66 3500 140 0.20 0.8 100 3.4
F 0.20 3500 120 0.025 0.6 100 3.8
G 0.27 3500 160 0.20 0.6 100 3.3
H 1.35 3500 160 0.025 0.4 100 3.3
I 1.55 3500 180 0.20 0.4 100 3.1
J 2 3500 200 0.40 0.8 100 2.8
K 2.6 3500 220 0.60 0.8 100 2.5
L 3.3 3500 250 0.80 0.8 100 2.3
Table 5.2: Physical parameters for the simulations. Weissenberg number Wi; slot width, initial maximumjets velocity, solvent viscosity based Reynolds number Re; Averaged Taylor turbulence micro-scale cen-treline velocity solvent viscosity based Reynolds number on the x streamwise direction Reλ; polymerrelaxation time τp; ratio of the solvent to solvent plus polymer viscosity β; polymer normalized maximumextensibility L; mesh resolution normalized by the Kolmogorov small scale at x/h = 14.
The Wi represents the Weissenberg (non-dimensional) number, which is given by the ratio between
the polymer relaxation time and Kolmogorov time scale:
Wi =τpτs
(5.1)
where τs represents the Kolmogorov time scale (τs = (ν[s]/ε[s])(1/2)).
The Reynolds number based on the Taylor microscale is defined as:
Reλ =u′λ
ν[s](5.2)
where u′ is the reference perturbation velocity and λ is the Taylor scale. The Taylor scale λ is given by
64
the viscous dissipation rate assuming local isotropy at the jet centreline.
ε[s] = 15ν[s]
⟨(∂u′
∂x
)2⟩
= 15ν[s] u′2
λ2⇔ λ =
√15ν[s]u′
2
ε[s](5.3)
As the Taylor scale is a function of the x streamwise direction, the value of Reλ presented in Table
5.2 refers to an averaged value on z spanwise direction and time along the centreline for y = 0.
In order to simulate the smallest scales that are present in the flow, the grid size must be of the order
of the Kolmogorov scale η, defined in terms of the solvent turbulent viscous energy dissipation rate ε[s]
and the solvent viscosity ν[s], as shown in equation (5.4).
η =
((ν[s])3
ε[s]
)1/4
(5.4)
According to [18], several authors state that the ratio of the element size ∆x to the Kolmogorov scale
η should have a value of the order of 2.0 to 3.0. In [11], it is stated that ∆x/η = 2.1 is a good reference
value for DNS simulations of isotropic turbulence.
∆x/η presents an evolution across the x streamwise direction on the centreline, with the highest
value located at the beginning of the self-similar region and decreasing from then on. It was opted to
select to analyse this quantity at x/h = 14, as this region refers to a fully developed turbulent region
of the jet, subject to the current study. By having mesh resolution normalized by the Kolmogorov small
scales within the values stated in the literature, it was considered that the mesh size was adequate for
the study of turbulence in the fully developed region of the jet, namely for cases J to L which are the
focus of this work.
The simulations are started with variable time step and constant Courant number (CFL = 1/6).
Once self-similarity is attained, the time step is kept constant at a lower level than the minimum reached
during the first part of the simulation. At this stage, statistical quantities are computed by accumulating
data over 150.000 iterations (3 weeks computing run).
The half-width or mean flow thickness δ is defined as follows according to [69]:
δ =
∫ ∞0
u− U2L
Uc − U2Ldy (5.5)
where Uc is the centreline velocity and U2L is the local co-flow velocity.
The streamwise velocity is non-dimensionalized according to the following expression:
(U1 − U2
〈Uc〉 − U2
)2
(5.6)
where U1 is the inlet centreline velocity and U2 is the inlet co-flow velocity.
It is important to notice that all variables presented in this section are an average on time and on the
Figure 5.1: Streamwise evolution of non-dimensional (a) jet’s half-width and (b) centreline velocity forfluids with different relaxation time for cases A, J, K and L. The Newtonian case refers to τp = 0. From[69].
From Figure 5.1, it is possible to identify the self-similar region by fitting straight lines to the linear
region of each curve. A good linear fit was found between x/h = 6 to x/h = 17 for the Newtonian case,
whilst for the visco-elastic case with τp = 0.8 a good linear fit was observed spanning from x/h = 10 to
x/h = 17. These windows identify the self-similar region. Despite the good linear fit observed on Figure
5.1 (b), results from case τp = 0.8 display the need of additional convergence.
Moreover, it is possible to observe that for higher polymer relaxation time there is a different flow
behaviour between the Newtonian case and the FENE-P solution. From 5.1 (a) it is noticeable a lower jet
width on the visco-elastic cases in comparison to the Newtonian case. Similarly, the centreline velocity
shows a larger value on the visco-elastic cases than on the Newtonian case. Both these remarks are
indicative of a lower energy dissipation rate for the FENE-P solutions due to the interaction of the polymer
particles on the flow physics. This result follows the expected trend, as polymer particles are expected
to absorb elastic energy at a higher rate from the flow, reducing the overall energy dissipation of the flow
and, consequently, inducing drag reduction.
On Figure 5.2 it is shown the normalized mean velocities by the local Uc centreline velocity for a fully
turbulent streamwise cross-section (x/h = 12) for cases A, J, K and L. It is also shown the root mean
squared turbulent velocity perturbation components and the Reynolds shear stress profiles for different
relaxation time.
It is possible to observe that the streamwise u velocity profiles collapse for different relaxation time.
This behaviour is like the one observed on a Newtonian case for different streamwise cross-sections, for
which the streamwise u velocity profiles collapse for different self-similar stations locations [18]. Con-
cerning the normal y velocity profile, it is noted that a higher polymer relaxation time leads to, typically,
a lower average normal velocity magnitude. It is important to notice that the spanwise w average ve-
locity is not presented, since it presents an average zero value for a planar jet [18]. As for the velocity
perturbation root mean squared components, it is observed an overall decrease of all components with
increasing polymer relaxation time. Moreover, it is seen that all perturbation velocities converge to zero
with higher distances from the centreline, as the flow is moving towards zones not perturbed by the jet.
66
Concerning the Reynolds shear stresses, the examination of Figure 5.2 indicates that for higher polymer
relaxation time the shear stress reduces, namely for cases K and L. This observation reinforces the
previous remarks, indicating a smaller energy dissipation rate for the polymer solution case due to the
energy-absorption effect of the polymers.
0 0.5 1 1.5 2 2.50
0.5
1
(u−U2L)/(U
c−U2L)
(a)
τp = 0.0τp = 0.4τp = 0.6τp = 0.8
x/h = 12
0 1 2-0.06
-0.04
-0.02
0
0.02
v/(Uc−U2L) (b)
τp = 0.0τp = 0.4τp = 0.6τp = 0.8
x/h = 12
0 1 2 30
0.1
0.2
√
u′2/(Uc−U2L)
(c)
τp = 0.0τp = 0.4τp = 0.6τp = 0.8
x/h = 12
0 1 2 30
0.1
0.2√
v′2/(Uc−U2L)
(d)
τp = 0.0τp = 0.4τp = 0.6τp = 0.8
x/h = 12
0 1 2 3
y/δ
0
0.1
0.2
√
w′2/(Uc−U2L)
(e)
τp = 0.0τp = 0.4τp = 0.6τp = 0.8
x/h = 12
0 1 2 3
y/δ
0
0.01
0.02
0.03
u′v′/(Uc−U2L)2 (f)
τp = 0.0τp = 0.4τp = 0.6τp = 0.8
x/h = 12
Figure 5.2: Normalized mean velocity: (a) streamwise and (b) normal direction components, (c-e) ve-locity root mean squared components and (f) Reynolds shear stress profiles for fluids with differentrelaxation time for cases A, J, K and L. From [69].
67
The conformation tensor Cij are plotted for the same streamwise cross-section (x/h = 12) in Fig-
ure 5.3. It is worth mentioning that the averaged conformation tensor components C13 and C23 are not
presented in Figure 5.3, as due to the nature of the turbulent flow is it possible to prove that the aver-
age value of these components is zero. This was observed on the simulation results. Concerning the
remaining conformation tensor components, it is seen that with increasing polymer relaxation time, the
magnitude of the averaged Cij increases. Moreover, one notices that the average values converges to
the polymer equilibrium state, that is value 1 for the diagonal tensor terms and 0 for the remaining tensor
terms, on the non-perturbed region of the flow, as expected. As the elastic energy stored by a stretched
polymer is proportional to the trace of the conformation tensor [70], the observed higher average values
of the conformation tensor components with increasing polymer relaxation time indicate that the polymer
molecules have an increasing absorbed elastic energy.
0 1 2 3
0
20
40
Cij
(a)ij = 11
ij = 22
ij = 33
ij = 12
τp = 0.4
0 1 2 3
-50
0
50
100
150
Cij
(b)ij = 11
ij = 22
ij = 33
ij = 12
τp = 0.6
0 1 2 3
y/δ
0
100
200
Cij
(c)ij = 11
ij = 22
ij = 33
ij = 12
τp = 0.8
0 1 2 30
50
100√
c′2 ij
(d)ij = 11ij = 22ij = 33ij = 12ij = 13ij = 23
τp = 0.4
0 1 2 30
50
100
150
√
c′2 ij
(e)ij = 11ij = 22ij = 33ij = 12ij = 13ij = 23
τp = 0.6
0 1 2 3
y/δ
0
100
200
√
c′2 ij
(f)ij = 11ij = 22ij = 33ij = 12ij = 13ij = 23
τp = 0.8
Figure 5.3: Mean (a-c) and root mean squared (d-f) components of the conformation tensor for fluidswith different relaxation time at x/h = 12 for cases J, K and L. From [69].
68
On Figure 5.4 it is presented the velocity perturbation components on the streamwise direction nor-
malized by the local centreline velocity, along with the root mean squared conformation tensor compo-
nents.
It is observed that with increasing polymer relaxation time, the perturbation velocity profile (typically
starting around x/h = 3) for all velocity components presents a smoother transition behaviour with a
lower perturbation velocity magnitude.
As for the root mean squared conformation tensor components, it is noticeable their values are
increasing with higher polymer relaxation time, with foremost relevant changes on terms C11 and C12.
Also, it is clear the presence of an abrupt increase on the root mean squared conformation tensor
components, which is indicative of a regime modification in the flow.
0 5 10 15 200
0.1
0.2
√
u′2/(Uc−U2L)
(a)
τp = 0.0τp = 0.4τp = 0.6τp = 0.8
0 5 10 15 200
0.1
0.2
√
v′2/(Uc−U2L)
(b)
τp = 0.0τp = 0.4τp = 0.6τp = 0.8
0 5 10 15 20
0
0.05
0.1
0.15
√
w′2/(Uc−U2L)
(c)
τp = 0.0τp = 0.4τp = 0.6τp = 0.8
-5 0 5 10 150
20
40
60
80
√
c′2 ij
(d)
ij = 11ij = 22ij = 33ij = 12ij = 13ij = 23
τp = 0.4
-5 0 5 10 15
x/h
0
50
100
√
c′2 ij
(e)ij = 11ij = 22ij = 33ij = 12ij = 13ij = 23
τp = 0.6
-5 0 5 10 15
x/h
0
50
100
150
√
c′2 ij
(f)ij = 11ij = 22ij = 33ij = 12ij = 13ij = 23
τp = 0.8
Figure 5.4: Centreline root mean squared evolution of (a) streamwise, (b) normal and (c) spanwise direc-tions velocity components and (d-f) conformation tensor components for fluids with different relaxationtime for cases J, K and L. From [69]
69
The Weissenberg number compares the elastic forces to the viscous forces and an analysis for the
visco-elastic simulations is presented in Figure 5.5. It is observed that for higher polymer relaxation
time with the same β (especially clear on cases with β = 0.8), Wi number increases. Simultaneously,
the same cases when viewed from a turbulent kinetic energy dissipation perspective (see Figure 5.6)
present a lower viscous energy dissipation, which in turn would contribute to a lower Wi number. Thus,
this indicates that the polymer elastic forces on the flow are far more significant than the solvent viscous
forces.
If one considers cases with an equal τp, it is noticeable that higher β leads to a higher Wi number.
This is observed, for example, in cases with τp = 0.2s. Nonetheless, the variation on the ratio between
elastic and viscous forces on the flow is clearly smaller with the influence of β than with the influence of