Lattice Boltzmann Computations of Grid

Transcription

Lattice Boltzmann Computations of
Grid-Generated Turbulence
Lattice-Boltzmann-Berechnungen von
gittergenerierter Turbulenz
Der Technischen Fakultät der
Friedrich-Alexander-Universität Erlangen-Nürnberg
zur Erlangung des Grades
DOKTOR-INGENIEUR
vorgelegt von
Nagihan Özyılmaz
Erlangen 2010
Als Dissertation genehmigt von der Technischen Fakultät der Universität Erlangen-Nürnberg
Tag der Einreichung: 14.05.2009
Tag der Promotion: 12.11.2009
Dekan:
Prof. Dr.-Ing. Reinhard German
Berichterstatter:
Prof. i.R. Dr. Dr. h. c. Franz Durst
Prof. Dr.-Ing. Ulrich Rüde
Prof. Dr.-Ing. Gunther Brenner
to Atom Ant (!)
Acknowledgments
The present work was realized during my research stay at the Institute of Fluid Mechanics at
the Friedrich Alexander University of Erlangen. I would like to express my appreciation to
the major contributors.
First, I would like to thank my supervisor, Prof. Dr. Dr. hc. Franz Durst, for supporting and guiding my research work. His commitment to the research will continue to inspire
me in the future. I have many other reasons to be thankful to him, but I would like to thank
him, especially for leading me to the area of the physics of turbulence. I am sure the more
time I spend doing research and the more areas of fluid mechanics this research work will
cover, the more I will appreciate the time I have spent intensively on the understanding of
fundamental flows.
I would like to thank Prof. Dr. Rüde, Prof. Dr. Brenner, Prof. Dr. Brunn and Prof.
Dr. Pflaum for accepting the examination of the thesis and also being present in the oral
exam.
An important part of this work was accomplished through discussions with Dr. Özgür Ertunç.
I am also grateful for his close cooperation during the different stages of this study.
The idea of applying the lattice Boltzmann method to the investigation of grid-generated
turbulence was suggested first by Dr. Kamen N. Beronov. Thanks are due also to him.
This work could not have been realized at all without the initial help of Dr. Peter Lammers, who provided me with the first introduction to the code employed. His early support
on SX systems in the High Performance Computing Center in Stuttgart is also acknowledged.
Special thanks are due to Dr. Thomas Zeiser, who made invaluable contributions to the
second and third chapters. I am grateful for the time he took to provide detailed suggestions.
I took my first intensive turbulence lectures from PD Dr. Jovan Jovanovic, and the knowledge
gained through these lectures contributed substantially to the presented work. His suggestions
for chapter 7 are also acknowledged.
I have spent many good years in the labyrinthine city Erlangen, enjoying different aspects,
and many people contributed to this enjoyment. In this respect, I would like to thank the
members of Ertak, Kais Haddad, Miki Aoki, Yhi Chun and especially Balkan Genc, Anuhar
Osorio Nesme, Selma Duman and Zübeyde Özsiseci.
I cannot close this section without thanking the members of my family who have been providing constant support in every aspect and all the time. And finally, very special thanks to
the new member of the family: to Nil Deniz Coban.
i
Abstract
Grid-generated turbulence was examined numerically via direct numerical simulations employing the lattice Boltzmann BGK method. A 3D flow solver developed at LSTM Erlangen,
written in Fortran and optimized for vector-parallel supercomputers and commodity clusters,
was employed for the predictions. The computations presented in this work were carried out
partly at the High Performance Computing Center in Stuttgart (HLRS) and partly at the
Leibniz Rechenzentrum in Munich (HLRB).
The main scientific motivations for this work were to address the lateral inhomogeneity of
turbulent quantities in grid turbulence, the decay laws of the turbulent kinetic energy, the
decay laws of the turbulent dissipation rate and the Reynolds stress and dissipation tensor
anisotropies. These issues were studied in terms of their Reynolds number dependence and also
their grid porosity dependence in an intermeadiate Reynolds number range. For this purpose,
direct numerical simulations were conducted at mesh-based Reynolds numbers ReM = 1400
and ReM = 2100 and for grid porosities β = 53%, 64%, 72%, 82%.
Investigations on the lateral inhomogeneity showed that in grid-generated turbulence, the
streamwise mean velocity field was homogeneous regardless of the value of the porosity, even
for a grid porosity as low as 53%. Hence spatially homogeneous mean velocity fields resulted
from the present computations, in contrast to some experimental findings published in the
literature. The simulations showed that the experimentally observed inhomogeneity in the
mean velocity field might be caused by the imperfect manufacture of the grids employed in
the measurements. On the other hand, the computed fields of the Reynolds stress tensor
and of its anisotropy were found to be strongly inhomogeneous, even for geometrically perfect
grids. The observed inhomogeneities remained even far downstream of the grid. No improvement was observed in the homogeneity of the Reynolds stress tensor with either increasing
Reynolds number or increasing porosity. Through an analysis of the terms of the turbulent
kinetic energy equation, it was shown that the early homogenization of the mean velocity field
was the main reason for the persistence of the inhomogeneity of the Reynolds stress field.
The results were analyzed so as to characterize the effects of the initial conditions (through
which the turbulence is generated) on the Reynolds stress tensor field and dissipation tensor
field. The results confirmed the expected axisymmetry of both tensors, i.e. the off-diagonal
components of both tensors were negligible compared with the diagonal components and the
vertical and spanwise diagonal components were almost equal to each other. The magnitude
of the components of the Reynolds stress tensor was strongly dependent on the value of the
grid porosity. Hence, to reach a universal coefficient of the turbulent kinetic energy, another
scaling was applied rather than using only bulk variables. Using this new scaling to normalize
the data, the coefficient for the power law decay of turbulent kinetic energy was found to be
0.15. In the examined Reynolds number range, the magnitudes of the Reynolds stress components changed slightly, hence it was suggested that the same decay coefficient be used for
the overall parameter range. The exponent, on the other hand, depended on the grid porosity
and, based on the current data, it was found to lie in the range 1.62 − 1.66, which agreed with
the data suggested in the literature.
ii
In contrast to the Reynolds stress tensor, the magnitude of the dissipation tensor was strongly
Reynolds number dependent but depended only slightly on the grid porosity. Hence the power
law coefficient for dissipation rate decay was Reynolds number dependent. On the other hand,
in the examined parameter range, the decay exponent for the dissipation tensor was universal
for x/M > 15 and was approximately 2.8.
The results were further analyzed in order to quantify the anisotropies of the Reynolds stress
and dissipation tensors. This analysis showed that the second and third invariants of the
Reynolds stress anisotropy lie on the right lower edge of the anisotropy map, corresponding
to the nearly isotropic and axisymmetric turbulence state. As expected, the Reynolds stress
anisotropy decreased with increasing streamwise distance from the grid. The anisotropy near
the grid decreased with increasing Reynolds number. Also near the grid, the flow was shown
to be more anisotropic for lower grid porosity.
The results were also examined in detail to study the axisymmetry of the Reynolds stress
and dissipation tensor anisotropies. It was shown that with increasing Reynolds number and
increasing grid porosity, axisymmetry of the anisotropy components was improved. The ratio, A, between the two kinds of anisotropies was examined in detail. It was concluded that
A showed the same trend for all initial conditions: immediately after the grid, its value decreased until X/M ≈ 8, then it started to increase. Near the grid, the value of A decreased
with increasing Reynolds number and with increasing grid porosity. The vaue of A was further compared with (IIe /IIa )0.5 . Here also the axisymmetry condition was improved for higher
Reynolds number and for higher grid porosities.
iii
Zusammenfassung
Gittergenerierte Turbulenz wurde mittels der Lattice Boltzmann BGK Methode numerisch untersucht. Für die Berechnungen wurde ein 3D-Strömungslöser eingesetzt, der am LSTM Erlangen entwickelt, in Fortran geschrieben und für Vector-Parallel-Supercomputer und CommodityCluster optimiert wurde. Die in der vorliegenden Arbeit präsentierten Simulationen wurden
teils am Höchstleistungsrechenzentrum Stuttgart (HLSR) und teils am Leibnitz Rechenzentrum München (HLRB) durchgeführt.
Die wesentliche wissenschaftliche Motivationen dieser Arbeit waren auf die laterale Inhomogenität von turbulenten Grössen, die Potenzgesetze für das Abklingen von turbulenter
kinetischer Energie und turbulenter Dissipationsrate sowie die Anisotropien des Reynoldsspannungstensors und Dissipationstensors einzugehen. Diese wurden im Bereich moderater
Reynoldszahlen bezüglich ihren Reynoldszahl- und Gitterporositatsabhängigkeiten untersucht.
Zu diesem Zweck wurden direkte numerische Simulationen aufgebaut bei Reynoldszahlen
ReM = 1400 und ReM = 2100 und für Gitterporositäten von β = 53%, 64%, 72% und
82%.
Die Untersuchungen zur lateralen Inhomogenität zeigten, dass bei der gittergenerierten Turbulenz die longitudinale mittlere Geschwindigkeit homogen ist, ungeachtet dem Wert der Gitterporosität. Sogar für eine Gitterporosität von 53% ist die mittlere Geschwindigkeit homogen.
Es wurde also ein räumlich homogenes Geschwindigkeitsfeld erreicht, was zu manchen experimentellen Untersuchungen der Literatur im Widerspruch steht. Die Simulationen zeigten weiterhin, dass das experimentell beobachtete inhomogene Geschwindigkeitsfeld die Folge der Anwendung von asymmetrischen Gittern bei den Experimenten ist. Dennoch sind das Reynoldsspannungsfeld und dessen Anisotropie stark inhomogen, sogar für geometrisch perfekte Gitter.
Diese Inhomogenitäten verbleiben sogar bei weiten Abständen vom Gitter. Die Reynoldsspannungstensorinhomogenität wurde weder mit zunehmender Re Zahl noch mit zunehmender
Porosität verbessert. Durch eine Analyse der Grössen der kinetischen Energiegleichung wurde
gezeigt, dass die frühe Homogenisierung der mittleren Geschwindigkeit der Grund für die Persistenz der Inhomogenität des Reynoldsspannungsfeldes bei weiten Abständen vom Gitter ist.
Die Ergebnisse wurden analysiert um die Einflüsse von Anfangsbedingungen (unter denen
die Turbulenz erzeugt wurde) auf den Reynoldsspannungstensor und auf den Dissipationstensor zu charakterisieren. Die Ergebnisse bestätigten die erwartete Axialsymmetrie der beiden
Tensoren, d.h. die nicht diagonalen Komponenten des Reynoldsspannungstensors sind vernachlässigbar im Vergleich zu den diagonalen Komponenten. Die lateralen und spannweitigen
Komponenten sind nahezu gleich. Die Beträge der Komponenten des Reynoldsspannungstensors sind stark von der Gitterporosität abhängig. Infolgedessen wurde eine neue Normalisierung eingeführt, um einen universalen Beiwert für kinetische Energie zu erreichen. Nach
dieser Normalisierung wurde der Beiwert für das Potenzgesetz von turbulenter kinetischer Energie als 0.15 bestimmt. Im untersuchten Parametergebiet waren die Änderungen der Beträge
der Komponenten des Reynoldsspannungstensors gering. Aus diesem Grund wurde weiter
vorgeschlagen, den gleichen Beiwert für gesamte Parametergebiet zu benutzen. Der Exponent
ist jedoch von der Gitterporosität abhängig, basierend auf den gegenwärtigen Untersuchungen
liegt er im Bereich 1.62 − 1.66, was mit Literaturdaten im Übereinstimmung ist.
iv
Im Gegensatz zum Reynoldsspannungstensor hängt der Betrag des Dissipationstensors stark
von der Re Zahl ab, jedoch ist er von der Gitterporosität nur gering abhängig. Folglich ist
der Beiwert für das Potenzgesetz des Dissipationstensors abhängig von der Re Zahl. Es wurde
aber für den gesamten Parameterbereich ein universaler Exponent für das Abklingen des Dissipationstensors vorgeschlagen, für x/M > 15 ist er ≈ 2.8.
Die Ergebnisse wurden weiterhin analysiert, um die Anisotropien des Reynoldsspannungstensors und des Dissipationstensors zu quantifizieren. Diese Analyse zeigte, dass die zweite
und die dritte Invarianten der Reynoldsspannungsanisotropie auf der rechten unteren Seite
der Anisotropie Invariantenmappe liegen, entsprechend dem beinahe isotropen und axialsymmetrischen Zustand. Wie erwartet nimmt die Anisotropie des Reynoldsspannungstensors mit
zunehmender Re Zahl ab. Die Anisotropie in der Nähe des Gitters nimmt ebenfalls mit
zunehmender Re Zahl ab. Die Strömung ist mehr anisotrop für niedrigen Gitterporositäten.
Die Ergebnisse wurden ebenfalls genutzt, um die Axialsymmetrie der Reynoldsspannungsund Dissipationstensoren im Detail zu untersuchen. Es wurde gezeigt, dass mit zunehmender
Re Zahl und mit zunehmender Gitterporosität die Anisotropiekomponenten stärker axialsymmetrisch sind. Das Verhältnis A, zwischen den beiden Anisotropien wurde im Detail
untersucht. Diese Untersuchung zeigte, dass nach dem Gitter A die gleiche Tendenz für alle
Anfangsbedingungen hat. Der Wert nimmt bis x/M ≈ 8 ab und danach nimmt er zu. In der
Nähe des Gitters nimmt der Wert von A mit zunehmender Re Zahl und mit zunehmender
Porosität ab. Der Wert von A wurde ferner verglichen mit dem Wert von (IIe /IIa )0.5 . Hier
auch wurde eine Verbesserung der Axialsymmetrie für höhere Re Zahlen und Gitterporositäten
beobachtet.
v
Contents
Acknowledgments
i
Abstract
ii
Zusammenfassung
iv
Contents
vi
Nomenclature
viii
1 Introduction and aim of the work
1.1 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Short literature survey on grid turbulence . . . . . . . . . . . . . . . . . . . .
1.3 Computations of grid-generated turbulence . . . . . . . . . . . . . . . . . . . .
1.4 Remaining problems and questions . . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 Inhomogeneity of grid turbulence . . . . . . . . . . . . . . . . . . . . .
1.4.2 Von Karman and Howarth equation and its application to grid turbulence
1.5 Motivation and the structure of the thesis . . . . . . . . . . . . . . . . . . . .
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2 Lattice Boltzmann technique and its application
2.1 Historical background . . . . . . . . . . . . . . . . . . . . . . .
2.2 Description of the method . . . . . . . . . . . . . . . . . . . .
2.2.1 The Boltzmann equation . . . . . . . . . . . . . . . . .
2.2.2 BGK approximation . . . . . . . . . . . . . . . . . . .
2.2.3 Derivation of the lattice Boltzmann equation with BGK
(LBGK) . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.4 Equilibrium distributions and the lattice . . . . . . . .
2.2.5 Basic algorithm . . . . . . . . . . . . . . . . . . . . . .
2.2.6 Boundary treatment . . . . . . . . . . . . . . . . . . .
2.3 From lattice Boltzmann equation to Navier-Stokes equations .
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3 Verification of the code
3.1 The BEST Code . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 The description of the code and the general approach
a simulation . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Optimization of the code . . . . . . . . . . . . . . . .
3.1.3 Parallelization of the code . . . . . . . . . . . . . . .
3.2 Calculations of fully developed channel flows . . . . . . . . .
vi
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approximation
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for
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the set-up
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CONTENTS
3.2.1
3.2.2
3.2.3
Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Set-up of the computations for channel flow . . . . . . . . . . . . . . .
Comparison of the results with the data available in the literature . . .
4 Set-up of the computations for grid turbulence
4.1 Parameters of the simulations . . . . . . . . . .
4.2 Boundary conditions . . . . . . . . . . . . . . .
4.3 Post-processing . . . . . . . . . . . . . . . . . .
4.4 Spatial resolution . . . . . . . . . . . . . . . . .
4.5 Reynolds number distributions . . . . . . . . . .
4.6 Mach number . . . . . . . . . . . . . . . . . . .
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5 Von Karman and Howarth analysis
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6 Inhomogeneity of grid-generated turbulence
6.1 Mean velocity field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Reynolds stress fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Reasoning of the inhomogeneity of the Reynolds stress components . . . . . .
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7 Effects of initial conditions
7.1 Decay of turbulent kinetic energy and dissipation rate
7.1.1 Effects of porosity . . . . . . . . . . . . . . . .
7.1.2 Effects of Reynolds number . . . . . . . . . .
7.2 Anisotropy of grid-generated turbulence . . . . . . .
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8 Conclusions and outlook
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Bibliography
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vii
CONTENTS
Nomenclature
Roman
B
CD
CK
D
ExpD
ExpK
I(x, y)
L
Lf
Lg
Kn
M
Ui
Um
P
RN
ReM
Reλ
Reτ
ReL
T
k
kB
Q
W
ui uj
aij
cs
d
eij
f
f (r, t)
g(r, t)
lk
lν
u
Letters
constant of the law of the wall
coefficient of the power law decay of turbulent dissipation rate
coefficient of the power law decay of turbulent kinetic energy
dimension of the flow problem
exponent of the power law decay of turbulent dissipation rate
exponent of the power law decay of turbulent kinetic energy
inhomogeneity of a variable
length of the fringe region
streamwise integral length scale
lateral integral length scale
Knudsen number
mesh size
instantaneous velocity
time and spatially averaged mean streamwise velocity
pressure
normalization factor used for the Reynolds stress tensor
mesh-based Reynolds number
Taylor microscale Reynolds number
friction Reynolds number
turbulent Reynolds number
temperature
turbulent kinetic energy
Boltzmann constant
total number of microscopic velocity directions
weighting factor
ij th component of the Reynolds stress tensor
anisotropy of Reynolds stress tensor
speed of sound
thickness of the rods
anisotropy of dissipation tensor
single particle distribution function
second-order correlation function for streamwise velocity fluctuations
second-order correlation function for lateral velocity fluctuations
Kolmogorov length scale
viscous length scale
streamwise velocity fluctuations
viii
CONTENTS
uτ
v
xb
xe
w
IIa
IIIa
wall friction velocity
transverse velocity fluctuations
starting streamwise position of the fringe region
last streamwise position of the fringe region
spanwise velocity fluctuations
second invariant of Reynolds stress anisotropy tensor
third invariant of Reynolds stress anisotropy tensor
Greek Letters
β
grid porosity
ǫ
turbulent dissipation rate
ǫN
normalization factor used for the dissipation tensor
γ
fringe function
κ
Von Karman constant for the law of the wall
λ
single time relaxation parameter
λg
Taylor’s microscale of turbulence
ρ
density of the fluid
τ
non-dimensional single time relaxation parameter
ω
collision frequency
ξ~
microscopic lattice velocity vector
Subscripts
i
flow direction
w
variable value at the wall
α
direction of the microscopic velocity vector
Superscripts
eq
equilibrium state
+
non-dimensional variable in wall units
ix
Chapter 1
Introduction and aim of the work
1.1
General remarks
Fluid mechanics plays a very crucial role in many fields of engineering, science and medicine
and also in the daily lives of human beings. It has developed into a subject that serves to
understand the phenomena which are caused by fluid flows found in many areas in both nature
and technology. Such flows exist, for instance, in blood flows, in the atmosphere, in rivers,
oceans and seas, in many technical applications of fluid flows such as combustion, various
chemical processes, multiphase flows, etc. The importance of fluid mechanics can hardly be
overestimated. Because of its relevance to the daily lives of people, the early investigations
of fluid mechanics date back to the origin of human beings. However, its development as
a separate science came about first after the important work of Leonardo da Vinci. Many
different researchers have contributed to its further development since then. For a summary
of the historical development of fluid mechanics, see for example Durst (2006).
The mathematical description of fluid flows was already known, as it is used today, at the
beginning of the nineteenth century. Louis Marie Henri Navier (1822) and Georg Gabriel
Stokes (1843) derived the basic equations for fluid mechanics, independently of each other, at
that time. These equations show that a fluid flow must obey mass and momentum conservation and the differential forms of the corresponding equations are called the Navier-Stokes
equations after the names of these two scientists. They are nowadays employed in the following form for an incompressible fluid:
Conservation of mass (ρ =constant):
∂Ui
=0
∂xi
Conservation of momentum:
∂Uj
1 ∂P
∂ 2 Uj
∂Uj
+ Ui
=−
+ν
+ ρgj
∂t
∂xi
ρ ∂xj
∂xi ∂xi
1
(1.1)
(1.2)
CHAPTER 1. INTRODUCTION AND AIM OF THE WORK
At the time the Navier-Stokes equations were derived, adequate experimental and numerical
methods were lacking. Even years after, one was bound to work either in an analytical way
using simplified forms of the equations or to use similarity laws to extraxt some knowledge
from the equation. Further, although the Navier-Stokes equations formed a closed system,
it included a non-linear term, and therefore analytical solutions were only possible for some
special cases and only for low Reynolds number flows. Thus, until the first part of the twentieth century, for the more complex fluid problems, similarity laws were used together with
experimental studies. It was in the second part of the twentieth century that experimental and
numerical methods of fluid mechanics started to be established, and with these methods, the
development of fluid mechanics was accelerated. The most important experimental methods
developed in this period were hot wire anemometry and laser Doppler anemometry. At the
same time, great progress in the development of numerical methods was also accomplished.
The numerical methods were proven to be reliable for laminar flows, but they helped to improve the understanding of turbulent flows also.
In technology, flows are almost always turbulent. For most technical flows, laminar flows
are exceptions. The importance of an adequate understanding of turbulent flows is obvious.
However, the cause and maintenance of turbulent flows remain among the unsolved problems
of our times. Nevertheless, it is known that if the flow exceeds a certain velocity limit, inertial
forces become more important than the viscous forces and the flow undergoes some kind of
instabilities, which results in a state called “turbulence”. This chaotic state of the flow is
characterized by three dimensionality and unsteadiness. The state of the flow is non-linear,
non-local, highly diffusive and dissipative and the distributions of fluid flow properties are
quasi-normal, which means the triple correlations and higher order moments are not zero.
Because deterministic approaches are not applicable to the study of turbulent flows, one has
to treat the problem statistically. Reynolds (1895) suggested decomposing the instantaneous
velocity field into a time-averaged mean flow velocity and a time-dependent fluctuating part
as follows:
Uj (xi , t) = U j (xi ) + uj (xi , t)
(1.3)
Using this definition, equations 1.1 and 1.2 can be averaged in time to obtain the continuity
and momentum equations for the mean flow:
∂U i
=0
∂xi
1 ∂P
∂
∂U j
∂U j
+ Ui
=−
+
∂t
∂xi
ρ ∂xj
∂xi
2
(1.4)
∂U j
ν
− ui uj
∂xi
(1.5)
1.1. GENERAL REMARKS
The momentum equation for the mean flow (equation 1.5) now includes a new tensor, ui uj ,
which has nine components:


u21 u1u2 u1 u3


 u2 u1 u22 u2 u3 
u3 u1 u3u2 u23
However, since it is a symmetric tensor, the number of unknowns reduces to six. It is possible
to obtain exact equations for ui uj . For this, first the mean flow equations are subtracted from
the total flow counterparts to obtain equations for instantaneous fluctuations, ui and uj , then
the resulting ui equation is multiplied by uj and the uj equation is multiplied by ui . The two
resulting equations are added and averaged in time to give
∂ui uj uk 1
∂ui uj
∂ui uj
∂Uk
∂Uj
∂p
∂p
= −uj uk
− ui uk
−
−
+ ui
+ Uk
uj
∂t
∂xk
∂xk
∂xk
∂xk
ρ
∂xj
∂xj
−2ν
∂ui ∂uj
∂ 2 ui uj
+ν
∂xk ∂xk
∂xk ∂xk
(1.6)
Equation 1.6 is called the “Reynolds stress equation” and it includes triple correlations. Following a similar procedure, it is possible also to derive equations for triple correlations. However,
the resulting equation this time includes fourth-order correlations. If one continues to derive
even higher order moments, each equation will include a term which is always one order of
magnitude higher. Hence it is never possible to reach a closed system of moments. This is
the so-called closure problem of turbulence. As in the other areas of fluid mechanics, one
can approach turbulent flows theoretically, numerically and experimentally. Since the present
thesis is concerned with a numerical study, computational methods of fluid mechanics will be
mentioned in more detail below.
With the rapid growth of computational power and the development of numerical methods
in the second half of the twentieth century, computational fluid dynamics (CFD) has become
an indispensable tool for the investigation of turbulent flows. One can differentiate three different approaches here: direct numerical simulation (DNS), large eddy simulation (LES), and
pure turbulence modeling together with the application of Reynolds-averaged Navier-Stokes
equations (RANS). In order to state the differences between these methods, an important
feature of turbulent flows must be pointed out. A turbulent flow consists of many elements
called eddies of different scales. The largest eddy has a size of the characteristic length of the
system. The sizes of the smaller eddies are determined by the energy transfer process. It is
assumed that the energy is transferred from the largest eddies to the next smaller eddy in a
cascade process. This process ends when the smallest eddies dissipate the mechanical energy
into heat. In real turbulent flows, this process is correct only for the net energy balance. The
ratio between the largest and the smallest eddies is proportional to Re3/4 . That means that,
when Reynolds number increases, the range of the scales increases correspondingly.
3
In DNS, all turbulent quantities are calculated using conservation equations directly. That
means that, a computational domain must be chosen in such a way that the different scales of
the flow are well resolved. Since turbulent flows are always three dimensional, the number of
grid points must increase with (Re3/4 )3 . If one also takes into account the knowledge that the
characteristic time scale decreases with increasing Reynolds number, the actual proportionality should be around Re11/4 (Breuer (2001)). From this observation, it is clear that a DNS is
very demanding, even with the efficiency of today’s computers. Hence, the studies of DNS are
still restricted to low Reynolds number turbulence investigations and are not able to predict
practically relevant high Reynolds number flows. However, it is the most accurate method
among them; the only error is caused by the approximations of the numerical scheme. Therefore, it is used commonly for basic investigations in academia. At first sight, this might seem
to be unattractive. However, even in some fundamental flows at low-to-moderate Reynolds
numbers such as channel and pipe flows, there are many open questions for which further
investigations with DNS are worth carrying out.
The RANS method uses equations 1.4 and 1.5 rather than solving equations for the instantaneous values, because the computations would be too demanding in the latter case. However,
as stated above, these equations contain six independent correlations of fluctuating parts of
the velocities. In order to obtain a closed system, one has to use turbulence models to predict
Reynolds stress components. Based on the number of additional equations to be solved, they
are divided into subgroups as zero-, one- and two-equation models. Numerous models have
been offered, with k-ǫ and k-ω being two of the most popular. The somewhat more complex
Reynolds stress modelling is also an example of such models, where the unknown terms of the
Reynolds stress equation 1.6 are modelled. Although the effort in this area has been enormous,
it is still not clear which model should be used for which flow, not to mention that there is no
general model which delivers reasonable answers for all kinds of flows.
LES is a method which combines the advantages of two methods mentioned above. In this
method, only the largest eddies are resolved, as in DNS and the smaller eddies are modelled.
The idea is not to require computer power as high as in DNS, but to deliver more accurate
results than those which would be possible with RANS schemes. It has indeed a great advantage over RANS: since small structures are easier to model and the large structures are
affected by the geometry of the system, it is more probable that one can find a universal
model with LES than with RANS. Hence the decomposition of the turbulent fields into coarse
and fine structures is a crucial step in LES. Here, many filtering methods exist. After the
decomposition, large structures are calculated with the help of conservation laws and the fine
structure models are applied for small eddies. The most widely used fine structure model is
the Smagorinsky model.
4
1.2. SHORT LITERATURE SURVEY ON GRID TURBULENCE
For the numerical discretizations of the governing equations of fluid flows, different approaches
are available. The traditional CFD is based on the discretization of the Navier-Stokes equations, which are partial differential equations containing first and second derivatives in the
spatial coordinates and first derivatives in time. The discretization of the basic equations
can be accomplished by the application of various methods such as finite difference, finite
volume, finite element and spectral methods. Generally, time derivatives are discretized by
finite differencing and spatial derivatives are discretized by either of the other three methods.
Regardless of the discretization scheme chosen, the resulting system of algebraic equations is
then solved in order to obtain approximate solutions for the velocity components and pressure,
by incorporating appropriate initial and boundary conditions.
More recently, a different approach called the lattice Boltzmann method has become very
popular for the solution of flow phenomena. In this method, instead of starting from the
Navier-Stokes equations directly, the Boltzmann equation is discretized; through a multi-scale
expansion, the macroscopic fluid behavior can be recovered. Because of its two overwhelming
advantages, i.e. efficient treatment of the solid boundaries and the ease of the parallelism it
became a serious alternative also for the solution of turbulent flows. It is also the preferd
numerical approach applied in the present thesis, where flows generated through grids are examined in the low Reynolds number regime via DNS. A short literature survey on such flows
is given below, and the structure of the thesis is outlined.
1.2
Short literature survey on grid turbulence
Grid-generated turbulence (GGT) has been studied extensively in the literature. The majority
of the work carried out was either experimental or analytical. Such flows are axisymmetric
and nearly isotropic. Ertunc (2007) made an in-depth study on axisymmetric flows for his
PhD thesis. A detailed literature survey on the early theoretical and experimental studies on
isotropic, homogeneous and axisymmetric turbulent flows can be found in that study. Here,
only the aspects closely related to the present thesis are addressed.
Von Karman and Howarth (1938) worked on homogeneous, isotropic and reflection-symmetric
turbulence and derived the equation for the second-order two-point velocity correlation function for the final decay of turbulence. The derived equation is called the Von Karman-Howarth
equation. Important knowledge has been drawn from their work for decaying homogeneous
and isotropic turbulence. Sedov (1944) carried out a detailed research on the solutions of
Karman-Howarth analysis and derived an equation for the decay of uniform and isotropic
turbulence. Monin and Yaglom (1975) discussed the Karman-Howarth theorem and derived
the dynamic equation for locally isotropic turbulence. In this way, the dissipation rate could
5
be introduced from the time rate of change of velocity variance; however, the proofs were
lacking. Frisch (1995) and Lindbog (1996) gave the necessary proofs and pointed out that
the equation of Monin and Yaglom could also be applied to freely decaying turbulence. Hill
(1997) reviewed the work by Frisch and Lindborg and addressed the steps required to eliminate the pressure-velocity correlations in the Karman-Howarth theorem without assuming
isotropy. Recent studies have concentrated more on the application of the Karman-Howarth
equation for the anti-symmetric (or helical) flows for the third-order statistics, considering
velocity-vorticity correlations, such as the work by Chkhetiani (1996), L’vov et al. (1997),
Gomez et al. (2002), Kurien (2003).
Townsend and Batchelor (1947) addressed decaying isotropic turbulence. They concluded
that the rate of change of the mean-square vorticity in isotropic turbulence is proportional to
the mean cube of vorticity. In further studies on decaying nearly isotropic GGT, they studied
low Reynolds numbers, as small as ReM = 650. They identified three regions in the decay
process as the initial, transition and final periods and realized that nonlinear effects became
negligible in the final period of decay, where eddies of all scales decay through direct viscous
dissipation. For the final period of decay, they proposed an asymptotic decay with an exponent of 5/2 and a coefficient of 0.13, based on experiments on an energy spectrum at small
wavenumbers. The discussion of the value of the decay exponent has been the centerpiece
of studies by many other authors since then. Chandrasekhar (1950) extended their study for
anisotropic turbulence. He studied axisymmetric turbulence with negligible inertial effects.
He also arrived the value 5/2. Birkhoff (1954) showed that exponents of 3/2 and 5/2 are theoretically compatible with the Navier-Stokes equations. Tan and Ling (1963) made low-speed
water-tunnel measurements using a grid. They concluded that the turbulent energy decays
with an exponent of 2. Their study attracted much criticism from other researchers, especially
the fact that their data showing u2 /v 2 ≈ 1 were contrary to any other studies in that area.
Saffmann (1967) showed that if the net linear momentum of the fluid is not equal to zero, the
exponent is 3/2.
Bennet and Corrsin (1978) studied low Reynolds number GGT experimentally. The results
of their measurements were in agreement with the theoretical studies of Von Karman and
Howarth (1938) and Batchelor and Townsend (1948a, b), but they also showed that at their
smallest Reynolds number, the triple correlations were still important, in contrast to those
studies. Skrbek and Stalp (2000) made further studies on homogeneous and isotropic turbulence with a focus on the decay process. They offered a model based on the three-dimensional
energy spectra. Their analysis supported the value of Saffman rather than that offered by
Batchelor and Townsend.
Corrsin (1963) addressed the inhomogeneity of GGT and pointed out that in order to as6
1.2. SHORT LITERATURE SURVEY ON GRID TURBULENCE
sure the homogeneity of such turbulence, the grid has to have a porosity of at least β = 0.57
and the diameter of the tube must be much higher than the mesh size, M. Based on the results
of the experiments, he suggested to place the contracting part at x/M > 40, so that the homogeneity could be assured. Batchelor and Townsend (1957) and Batchelor and Steward (1950)
pointed out the inhomogeneity when very fine grids were used. Grand and Nisbet (1957) set
up experiments using grids with a porosity of 0.70. Although β was much higher than the
value suggested by Corrsin (1963), they observed inhomogeneities as high as 15% even far
downstream of the grid at x/M ≈ 80. They concluded that the inhomogeneity decreases with
increasing M. More recently, Liu et al. (2004) studied the inhomogeneity of the r.m.s. of
velocity fluctuations and concluded that inhomogeneity can reach up to 30% at x/M > 40
for a porosity of 0.65. They observed that if the porosity increases, the inhomogeneity also
increases. Overall, the inhomogeneity of GGT has not been addressed in detail until recently,
when Ertunc (2007) set up experiments to investigate axisymmetric turbulence. He observed
a highly inhomogeneous Reynolds stress field and a highly inhomogeneous anisotropy tensor.
Since grid-generated turbulence is only nearly isotropic, attempts to improve its isotropy
played another central role in related studies. Investigations on the effects of wind-tunnel
straining on the free stream turbulence started with the work of Prandtl (1932, 1933) and
Taylor (1935). They showed that the turbulent flow passing through a gross strain with an
axisymmetric area change experienced directionally selective vortex-line distortions, resulting
in changes in its axial and transverse energy niveaus. Their work was followed by many others.
Fage (1934), Hall (1938), MacPhail (1944), Dryden and Schubauer (1947), and Ribner and
Tucker (1953) were among the first researchers who studied strain effects. Uberoi and Wallis
(1966) let an initially anisotropic turbulence pass through a nozzle of small contraction and
investigated three different grids with different mesh sizes. They observed that in the contraction part there was a trend towards isotropy; however, after passing the contraction, the
turbulent quantities approached their pre-contraction values. In contrast, Comte-Bellot and
Corrsin (1966) pointed out that when the turbulent quantities were equilibriated by symmetric
contraction of the wind tunnel, isotropy could be obtained. They studied both rod and disk
grids and reached the same conclusion.
The effects of arbitrary irrotational distortions were adressed by Bathelor and Proudman
(1954). Their study led to a theoretical approach known as rapid distortion theory today.
Townsend (1954) and Tucker and Reynolds (1968) made further studies on irrotational distortion of homogeneous turbulence. Tucker and Reynolds (1968) extended Townsend’s study
to a ratio of 6 : 1, compared with 4 : 1. They ended up with some contradictory results with
respect to the Townsend’s study; they observed higher anisotropy levels, and the strained turbulence rapidly became less anisotropic if the strain was stopped. Gence and Mathieu (1979,
1980) applied two succesive plane strains without any streamwise acceleration. They observed
7
an energy transfer from the fluctuating motion of turbulence to the mean flow.
1.3
Computations of grid-generated turbulence
In spite of the great interest in grid turbulence in the literature, its numerical investigation is
new. In fact, there is only one published computational study on decaying turbulence realized
by applying grid elements in the computational domain (Djenidi (2006)). The reason why
GGT was not explored computationally until now was mostly related to the general difficulties of simulating three dimensional turbulent flows. For example, high gradients can only
be captured if the computational grid is well resolved. Additionaly, the reliable study of the
decay process in the streamwise direction also requires high resolution. This means that the
application of supercomputers is essential for such studies. Also, the general difficulties in
incorporating solid boundaries in pseudo-spectral methods might have played an important
role here. As discussed in detail in the review work on the DNS of turbulent flows by Moin and
Mahesh (1998), early numerical studies on DNS of turbulent flows just involved some novel
numerical experiments, where the objective was to study the influence of isolated physical parameters rather than implementing physical boundary conditions. Most of these simulations
was carried out in cubic boxes with the application of periodic boundary conditions in all
directions in order to study homogeneous, isotropic, decaying turbulence.
The first study of this kind dates back to 1972 with the work of Orszag and Patterson (1972).
They carried out three sets of simulations for Reλ ≤ 35 using a spectral Galerkin method based
on a Fourier representation of the velocity field. The computational domain had 64 × 64 × 64
points. This resolution was too low for the inertial range dynamics; however it was a very
important study which showed for the first time that spectral methods could be applied accurately to three-dimensional moderate Reynolds number flows to obtain large-scale dynamics.
An important outcome of this study was that the skewness factor was independent of the initial Reynolds number, which meant that the small-scale turbulence dominating the skewness
was independent of Reynolds number, even for the moderate Reynolds number range. The
numerical method offered in this work was applied and extended by many other studies.
By applying Orszag and Patterson’s spectral method, Schumannn and Herring (1976) addressed the homogeneous and axisymetric turbulence at Reλ ≈ 28 in terms of Rotta’s return
to isotropy rate and compared the results with those obtained from the direct interaction
approximation with respect to energy, dissipation, skewness and anisotropy. Schumann and
Patterson (1978a, b) extended the method of Orzsag and Patterson to study velocity and
pressure fluctuations for both isotropic and anisotropic homogeneous, incompressible and decaying turbulence. In the first study, they described the extended method and its application
to isotropic turbulence for Reλ ≤ 35. In the second, they applied this method to examine
8
1.3. COMPUTATIONS OF GRID-GENERATED TURBULENCE
the return of axisymmetric turbulence to isotropy for Reλ ≈ 40 with 323 points by using
periodic boundary conditions. They studied the pressure and velocity fluctuations and examined the return to isotropy issue with respect to pressure-strain correlations. They tried
different anisotropic Gaussian initial boundary conditions and confirmed that the return to
isotropy was caused by the pressure strain correlations and pointed out that the rate of return
to isotropy was larger at higher wavenumbers.
Rogallo (1981) supplied the most extensive numerical database ever obtained for homogeneous turbulence. He applied the method by Orszag and Patterson (1972) to homogeneous
turbulence where the fluid was subjected to either uniform deformation or rotation. He examined four cases. In each case, the flow evolved from the same initially isotropic state according
to a mean velocity gradient uniform in space and time. These four cases were irrotational plane
strain, irrotational axisymmetric strain, uniform shear and uniform rotation. He compared
his results with the linear theory, with the well-known Reynolds stress models and also with
the available experimental data. Since his data contained all Reynolds stress budget, it was a
very important contribution which allowed other researchers to investigate available turbulent
models. The DNS studies on turbulence following his study used basically his algorithm.
Domis (1981) performed LES of homogeneous, isotropic turbulence. He examined the decay
rate of kinetic energy and the pressure statistics and their dependence on the initial velocity
spectra. He observed an asymptotic decay of energy with an exponent of 1.63. The final period
of decay with an exponent of 2.5, as suggested by Townsend and Batchelor (1947) was not
observed in that study. The energy decay showed a dependence on the low-wavenumber part
of the initial spectra. On the other hand, pressure statistics did not show any dependence on
the initial velocity spectra. Lee et al. (1986) studied homogeneous turbulence subject to rapid
mean field strain. They examined plane strain, axisymmetric contraction and isotropic compression. Their database included Reynolds stresses, turbulent kinetic energy and anisotropy.
Lesieur et al. (1987) studied the decay of kinetic energy and temperature variance in threedimensional isotropic turbulence. Lesieur and Rogallo (1988) performed LES of passive scalar
diffusion in isotropic turbulence. Mansour and Wray (1994) simulated decaying isotropic
turbulence at low Reynolds numbers (Reλ < 70). They pointed out that the shape of the
Kolmogorov spectrum was independent for Reλ < 50 and that the value of the power law
exponent showed a dependence on the Reynolds number and varied in the range 1.1 − 2.5.
Chasnov (1991) performed LES on decaying three-dimensional isotropic turbulence. He developed a new eddy viscosity subgrid model and, together with an aditional force, he applied this
model to an LES in order to investigate the inertial subrange. He showed that using this new
model results in an energy spectrum of k −5/3 . He obtained the value 2.1 for the Kolmogorov
constant. Chasnov (1993a) worked on decaying isotropic turbulence on a domain with 643
9
points in order to obtain the time evolution of the Loitsianski integral for high Reynolds numbers. He pointed out that it had a power-law character at long times and concluded that
in this power-law range, the exponent was approximately 2.5. Chasnov (1993b) carried out
LES for three-dimensional decaying isotropic turbulence for the study of the time evolution of
the mean-square velocity and passive scalar variance with or without a uniform mean scalar
gradient. His LES on 2563 points were tested to confirm several theoretical studies for asymptotic scalings. Chasnov (1996) carried out DNS of decaying two-dimensional homogeneous
turbulence of a fluid with infinite extent. He discussed the influence of the initial Reynolds
number on the time evolutions of energy and enstropy. He could not obtain a universal asymptotic state but identified different ranges with respect to the initial Reynolds numbers, where
different scalings were plausible.
Using Rogallo’s algorithm, Briggs et al. (1996) studied unforced and forced turbulent mixing layers. By introducing an initially homogeneous isotropic turbulence velocity field with
a model energy spectrum for decaying turbulence, they made comparisons of the temporal
decay of kinetic energy with other experiments and simulations. They found a faster initial
decay when E(k) ≈ k 4 was used rather than a dependences of E(k) ≈ k 2 . On the other hand,
after two eddy turnover times, both simulations exhibited similar exponents. In their forced
mixing layer computations, the spatial dependence of the kinetic energy and integral length
scale were similar to those observed in oscillating grid experiments.
Benzi et al. (1996) carried out computations for anisotropic turbulent shear flow using the
lattice Boltzmann method. They investigated the flow in terms of extended self-similarity
and concluded that it is not valid for anisotropic flows, where strong shear effects are present.
Coleman et al. (2000) were the first to simulate any strain effect on three dimensional wallbounded flows. They examined two strain fields. Chen et al. (1992) carried out lattice
Boltzmann simulations for the study of isotropic decaying turbulence in a periodic geometry
on 128 × 128 × 128 grid points. They calculated the time evolution of kinetic energy, enstropy
decay, vortex evolution in space and time evolution of spectra. They compared these results
with those obtained from their spectral code. Both methods delivered very similar results.
Their study showed clearly that the lattice Boltzmann method was a serious alternative to
the traditional Navier-Stokes solver. It was 2.5 times more efficient than the spectral method
for the investigated moderate Reynolds number (ReM = 1000).
Huang and Leonard (1994) studied isotropic, homogeneous, incompressible turbulence by direct numerical simulations. They studied self-similar solutions of the Karman-Howarth equation and concluded that the self-similar solutions depend on the decay exponent and on the
initial conditions. Similarly to the study of Mansour and Wray (1993), their decay exponent
showed a Reynolds number dependence and varied in the range 1.5 − 1.25 for the Reynolds
10
1.4. REMAINING PROBLEMS AND QUESTIONS
number range Reλ = 10 − 40.
Samtaney et al. (2001) studied decaying compressible turbulence via DNS. They applied
a high-order compact finite different scheme on 2563 . Their kinetic energy data showed a
power law decay with the exponents of 1.3 − 1.7, not too far from those obtained for incompressible fluids.
Yu et al. (2005) worked on the decaying homogeneous, isotropic turbulence in inertial and rotating frames with the lattice Boltzmann method. They have carried out both DNS and LES.
They calculated the decay exponents of kinetic energy, dissipation rate and low-wavenumber
scaling of the energy spectrum from these databases and made a comparison between DNS
and LES results. The Reynolds number range was 53 < Reλ < 119. They reached the somewhat surprising conclusion that an LES preserves the instantaneous flow fields more accurately
than DNS. Also, from their LES study, they reached a value for the Smagorinsky constant
(Cs = 0.1) lower than the value used traditionally in Navier-Stokes LES. They made detailed
comparisons with experiments and DNS on homogeneous and isotropic turbulence based on
traditional CFD methods. Overall, this study showed once again that the lattice Boltzmann
method is a reliable tool for the investigation of three-dimensional isotropic turbulence.
Djenidi (2006) reported the first DNS of GGT. He applied the lattice Boltzmann method
and simulated a grid-generated turbulence by using four-by-four flat square elements with a
solidity of 0.25. He made his computations on 640 × 73 × 73 points. In this work, he found out
that the resolution he chose for the computations was not fine enough for the integral length
scale, resulting in an unreliable decay behavior of kinetic energy. Hence he did not draw any
solid conclusions regarding the kinetic energy decay.
1.4
1.4.1
Remaining problems and questions
Inhomogeneity of grid turbulence
The inhomogeneity of GGT did not attract as much attention as the other issues related to
these kinds of flows, such as the effect of straining and the decay rate laws for turbulent kinetic energy. However, it is also of very crucial importance. For instance, the appropriate
streamwise positions of the contraction elements depend on the a priori knowledge on the
homogeneity of the underlying flow so that one can examine the effect of the straining without
the influence of inhomogeneity. Hence precautions must be taken to assure homogeneity.
The inhomogeneity of a turbulent quantity is defined as the the percentage deviation of that
quantity from the absolute value of its average value calculated along a constant line. Let us
11
take the time-averaged streamwise mean velocity, U(x, y), as an example. Its inhomogeneity
is given by
P y
U(i∆x, j∆y) − N1y N
j=1 U (i∆x, j∆y)
IU (x, y) = 100.
P
N
y
| N1y j=1
U(i∆x, j∆y)|
(1.7)
Some discrepancies exist among the few available studies on the inhomogeneity of GGT.
Corrsin (1963) concluded that GGT must be homogeneous after a 40 mesh size downstream
distance as far as the mean velocity fluctuations are concerned, provided that the grid has a
minimum porosity of 57% and the diameter of the duct must be much larger than the mesh size
of the grid. On the other hand, the studies of Grant & Nisbet (1957), Batchelor & Townsend
(1948a, b) and Batchelor & Steward (1950) indicated possible inhomogeneities if very thin
grids were applied. These authors noticed also some dependencies of the inhomogeneity on
the mesh size and the porosity of the grid. Loehrke & Nagib (1972) also noted inhomogeneties
of the flow field. In a more recent study by Liu et al. (2004) on perforated grids, even 30% inhomogeneity was observed for the Reynolds stress components. Hence no solid conclusion can be
drawn on whether the mean velocity and Reynolds stress tensor fields are homogeneous or not.
Motivated by the observation of the above-mentioned discrepancies between different studies, Ertunc (2007) conducted detailed hot wire experiments in an axisymmetric wind tunnel
in order to study strained and unstrained turbulence. Among other things, he analyzed his
results with respect to inhomogeneity. The mesh Reynolds number range in these experiments
was ReM = 3200 − 8000 and the porosity was fixed at 64%, higher than the value suggested by
Corrsin (1963). He calculated the inhomogeneities of the streamwise mean velocity, Reynolds
stresses and anisotropy of Reynolds stresses and also made comparisons between the strained
and unstrained cases in terms of the observed inhomogeneity levels. He recognized high levels
of inhomogeneity far downstream of the grid, even after a 120 mesh size. Especially the components of the Reynolds stress tensor showed strongly inhomogeneous regions. For x/M > 15,
the inhomogeneities of the normal stresses were still in the range ±5%. The anisotropy of
Reynolds stresses was even more inhomogeneous: he observed inhomogeneties of the streamwise diagonal component of the anisotropy tensor (a11 ) as high as 20%. Although not as
high as Reynolds stresses, he also pointed out some inhomogeneous regions of mean velocity
component: ±10% inhomogeneity for x/M < 15. However, for x/M > 20, the inhomogeneity
of the mean velocity decreased to ±2%.
Another important result of that study was that the observed inhomogeneties did not show
any dependence on the value of ReM . In a further study on strained turbulence, where a
contraction ratio of 3.69 was applied, it was shown that whereas the turbulence became more
isotropic, the inhomogeneity of a11 decreased to ±5. In other words, application of contraction
12
helped not only to improve isotropy but also helped to decrease the inhomogeneity of a11 . On
the other hand, the Reynolds stress fields were not perfectly homogeneous. Figure 1.1 summarizes some of his results obtained at ReM = 3200 for the inhomogeneities of mean streamwise
velocity, streamwise Reynolds stress component and a11 . The differences in the level of the
inhomogeneties of the three variables can be easily observed.
That experimental study showed that the preconditions which found general acceptance for
GGT in the literature may not be enough to guarantee perfectly homogeneous turbulence. In
particular, it pointed out the high inhomogeneity of the Reynolds stress tensor field, which
was not dicussed in Corrsin’s study (1963). Hence futher studies are required, which should
give clear answers on the level of inhomogeneties of the mean velocity and Reynolds stress
tensor: are the observed inhomogeneities indeed caused by the flow instabilities or by the
imperfections in the manufacture of the rods used in the experiments? Is it possible to obtain
perfectly homogeneous fields using perfectly symmetric rods? Do the inhomogeneties of the
Reynolds stress tensor depend on the streamwise distance from the grid? Does the value of
the porosity have any kind of effect on the level of inhomogeneity?
13
(a)
(b)
(c)
Figure 1.1: Inhomogeneities observed in the Ertunc’s experiments at ReM = 3200, for: a)
mean velocity, U m , b) u2 , c) a11 (Ertunc (2007)).
14
1.4.2
Von Karman and Howarth equation and its application to
grid turbulence
The Von Karman and Howarth equation is a differential equation for the dynamic behavior of
isotropic turbulence. It describes second-order correlation functions s(r, t), f (r, t) and n(r, t),
where r is the distance between two points and t is the time.
Let vi (x1 ) and vj (x1 ) denote the fluctuating velocity components at the position x1 in the
i and j directions respectively. One can define other variables, which also define the fluctuating velocity components at different positions, say for x2 , so that construction of the
correlations between fluctuations between two points (vi (x1 )vj (x2 )) is possible. Similarly, one
can consider two-point correlations between velocity and pressure fluctuations (p(x1 )vi (x2 ));
the same holds for the triple correlations (vi (x1 )vj (x1 )vk (x2 )).
In order to obtain the non-dimensionalized correlations for Reynolds stresses, one first defines the non-vanishing components of the Reynolds stresses by taking the isotropic conditions
into account, first for the case when both x1 and x2 lie on the same axis. The equation obtained is then extended to the general case and, by application of the continuity equation,
the relations between the non-zero components are established. By the application of NavierStokes equations, the equations for the second-order correlations are reached. In the case of
Reynolds stresses, this procedure gives
f (r, t) =
g(r, t) =
v1 (x1 )v1 (x2 )
v1 (x1 )v1 (x1 )
v2 (x1 )v2 (x2 )
v2 (x1 )v2 (x1 )
(1.8)
Typical profiles of f and g functions are sketched in figure 1.2. By application of the same
procedure, the non-dimensionalized second-order correlations for pressure-velocity fluctuations
and triple correlations can be derived:
p(x1 )v1 (x2 )
s(r, t) = p
p(x1 )2 v1 (x1 )2
n(r, t) =
v1 (x1 )2 v1 (x2 )
(v1 (x1 )3/2 )
(1.9)
It is possible to obtain an equation for the non-dimensional Reynolds stress correlations for
homogeneous turbulence (v = 0) from the Navier-Stokes equation at point x1 for component
i (vi equation) and also for another component, say vk , by multiplying the vi equation by vk
and the vk equation by vi and averaging both equations in time and adding the resulting two
equations. This equation contains triple correlations. By applying this equation by setting
15
Figure 1.2: Sketch of the functions f , g and h.
(i = k and summing on i) and integrating, the following equation is obtained:
q


2
v
(x
)
1 1
∂ 2
∂
4  ∂f

(v1 (x1 )f ) = 2νv1 (x1 )2
+
−
∂t
∂r r
∂r
ν
(1.10)
Equation 1.10 was first derived by Von Karman and Howarth (1938), and therefore it is called
the Von Karman and Howarth equation. It is used to study the dynamic behavior of isotropic
turbulence. Each equation of this kind has two variables; therefore, equation 1.10 contains an
additional term for the triple correlations, which is unknown. The way to treat the Karman
and Howarth equation is to make assumptions for the correlations, using similarity, statistical and physical considerations. With this approach, only solutions for the decay of the
turbulent intensity have been obtained so far. Completely self-similar solutions always lead
to a decay exponent of 1, if the Reynolds number is not zero. This state can be achieved
only if time goes to infinity. The correlation functions which are self-similar only for some
ranges of r can lead to decay exponents which are different from one. By setting r = 0, and using Taylor’s microscale for turbulence λ, the decay of the turbulence intensity can be obtained:
5
V
q
=
x + constant
MA2
2
v(x1 )
(1.11)
q
where, V is the time-averaged mean velocity, M is the mesh size and v(x1 )2 is the turbulence
intensity. The coefficient A is given in the range 1.95 − 2.2 in the literature.
16
1.5. MOTIVATION AND THE STRUCTURE OF THE THESIS
In the present thesis, Von Karman and Howarth analysis is used in order to validate the
computations (see Chapter 5 for details).
1.5
Motivation and the structure of the thesis
The present thesis was initially motivated because there were insufficient numerical data available on grid-generated turbulence. As shown by the literature survey above, the early numerical work concentrated essentially on the homogeneous and isotropic turbulence predictions.
In those studies, decaying turbulence was generated by an isotropic and homogeneous initial
spectrum and the DNS was used just to solve the subsequent decay. Therefore, such studies
are never free from the effects of the initially imposed homogeneous and isotropic turbulence.
This way of turbulence generation does not correspond exactly the generation mechanism
through a grid, where, as discussed in section 1.4.1, in fact the “homogeneity” is rather questionable. Therefore it is not appropriate to make direct comparisons between experiments on
grid-generated turbulence and results of DNS of homogeneous and isotropic turbulence. Hence
it was decided to study the potential of the application of the lattice Boltzmann method to
the detailed study of grid-generated turbulence.
As stated in section 1.4.1, some questions are still open in terms of the homogeneity of the
grid turbulence. One of the scientific motivations of the present work was to give conclusive
answers regarding the homogeneity of the mean velocity and Reynolds stress fields obtained
through grid-generated turbulence.
The discussion of the decay laws of kinetic energy played a central role in the early studies on grid-generated turbulence. It is well known that the turbulence decay can be expressed
in terms of power laws. However, there is no agreement on the value of the decay exponents
and coefficients. In this respect, it is very important to be able to show some kind of universality. The same holds for the decay of dissipation rate. It is therefore interesting to see,
regardless of the initial conditions, such as Reynolds number or geometry of the grid, if and
where a self-similar region is entered. In order to see the extent to which the initial conditions
affect the time-averaged turbulent quantities, two mesh Reynolds numbers and a wide range
of grid porosities were studied.
The remaining work is divided into seven chapters. The description of the numerical model
employed for the simulations is given in chapter 2, together with a short historical background
of the method. The details of the computer code implementing this method are outlined in
chapter 3. The verification of this code was carried out by simulating different flows on several
platforms by several authors. A summary of these validation studies is given also in chapter 3,
17
focusing especially on turbulent plane channel flows. The parameters of the computations,
and also the boundary conditions applied are introduced in chapter 4 and the procedure for
the data processing is also outlined there. Chapters 5, 6 and 7 are devoted to the analysis of
the results of the numerical simulations. A verification study was carried out by application
of Von Karman and Howarth analysis and is described in chapter 5. The inhomogeneity of
turbulent quantities is discussed in chapter 6. The investigation of the effects of the initial
conditions on the decay of the turbulent kinetic energy and dissipation rate is reported in
chapter 7. Also discussed in the same chapter are the effects of the initial conditions on the
anisotropy of the Reynolds stress and dissipation tensors and on the axisymmetry of grid turbulence. Lastly, in chapter 8, the main conclusions drawn during the study are summarized
and an outlook is provided.
18
Chapter 2
Lattice Boltzmann technique and its
application
Many different models have been suggested for the numerical solutions of fluid flows. These
computational methods depend on the discretizations of the basic equations which describe the
behavior of the underlying physics. These descriptions can be done at different levels. In one
kind of approach, motivated by the physics at the molecular level, a large number of “particles”
are followed, as in the case of the Newton-Hamilton approach; in another kind of approach,
one describes the fluid flow directly at the macroscopic limit, as in the case of the NavierStokes equations. One can distinguish basically four different levels of description, which are
summarized on the left-hand side of figure 2.1. The most detailed description, accounting for
individual particle dynamics, is given by the Hamilton equations, followed by the Liouville
equation, the Boltzmann model of mesoscopic interactions of the particles distribution functions, and finally the macroscopic Navier-Stokes equations, dealing with continuum mechanics.
Based on these equations, different numerical approaches are applicable. The corresponding
numerical methods are given on the right-hand side of figure 2.1. The first three numerical
methods include more details on the physics of the fluid problem than finally needed for a
prediction at the continuum mechanics level appropriate in most (engineering) applications.
They are usually referred to as “bottom-up” methods. They are based on the discretization
of microscopic and mesoscopic kinetic equations. In order to recover the macroscopic fluid behavior, a multi-scale expansion is applied in each of these approaches. On the other hand, in
traditional CFD, e.g., finite volume, finite difference and spectral methods, the Navier-Stokes
equations are directly discretized; therefore, such methods are called “top-down” methods.
Obviously, the more details on the fluid flow that one equation contains, the more complex
and demanding are the corresponding methods for its numerical solution. Therefore, when
small scales are not crucial for the solution of the problem, numerical methods based on the
discretization of the Navier-Stokes equations are by far the simplest methods, thus providing
an optimal solution approach. Not surprisingly, they have found widespread use in many areas.
19
CHAPTER 2. LATTICE BOLTZMANN TECHNIQUE AND ITS APPLICATION
For the detailed simulation of very complex flows with fine-scale structure, such as combustion and turbulence, it can be advantageous to use mesoscopic methods. In the present work,
which deals with DNS of turbulence, such an approach, e.g. the lattice Boltzmann method
(LBM), was applied. The reasons why this particular method was chosen, instead of a “direct”
Navier-Stokes solver, more specifically a (pseudo)spectral method, are mostly related to the
aspects of ease and efficiency of the computational method; they will be outlined first.
Hamilton Equation
Lattice Gas
Liouville Equation
Lattice Liouville
Boltzmann Equation
Lattice Boltzmann
Navier−Stokes
Equations
Traditional CFD (Finite
Difference, Finite Volume,
Spectral Methods)
Figure 2.1: The equations which describe a fluid flow at different levels (left) and the corresponding numerical
methods based on the discretization of the equations on the left-hand side (right).
First, there are two major differences between the lattice Boltzmann method (LBM) and
the traditional CFD:
• The convection operator in LBM (propagation step) is linear in velocity space, whereas
the convection term in Navier-Stokes solvers are quadratically non-linear. Nevertheless, using a multi-scale expansion, it is possible to recover from LBM this non-linear
macroscopic convection behavior.
• The calculation of pressure is different: in LBM, it is obtained by using an equation
of state, i.e. implicitly from the local density, whereas in traditional CFD, pressure
is calculated from the Poisson equation. The latter approach encounters numerical
difficulties in complex geometries.
These two features give the LBM its first advantage: it has a straightforward calculation procedure.
Another advantageous aspect of LBM is its numerical efficiency. Various authors worked on
20
this aspect and made comparisons with traditional methods. Succi et al. (1991) studied fully
developed two-dimensional forced turbulence and made a comparison between a code based
on a pseudo-spectral method and an LBM code on a 128 × 128 grid. They concluded that the
number of floating-point operations would require 150N D for LBM and (25 log2 N)N D for the
spectral code, where N is the number of lattice points and D is the dimension of the problem.
This study was of great importance, because, first, it provided a direct comparison between
the two methods for an application where the pseudo-spectral methods were established as the
best. It showed also that the factor with which the number of floating calculations increases
was constant for LBM ( N D ), whereas the spectral method had a logarithmic dependence on
the number of grid points. Their study showed practically that the LBM was 2.5 times faster
than for the two-dimensional flow they studied. A similar study was performed by Chen et
al.(1992) for isotropic turbulence in three dimensions and resulted in a similar conclusion:
LBM was 2.5 times faster than the spectral method. It is clear that when more grid points
are used, the difference in efficiency between the two methods increases in favor of LBM. This
is the case for the present calculations.
Some further advantageous features of LBM emerge because it originates from the kinetic
theory. It has the advantages of molecular dynamics, for example, a clearer physical understanding can be obtained and complex forces are readily implemented. Implementation of
different boundary conditions is also straightforward. This is a very attractive property of
LBM because the application of the no-slip boundary condition (bounce-back) requires very
little computational time. It is of crutial importance when dealing with solid boundaries,
where efficient wall-fluid interaction approaches are required, as in the present work (the grid
is given as an obstacle in the computational domain). The method is simple to parallelize due
to its locality (no spectral decomposition or pressure solver is necessary).
Lastly, the applicability and reliability of LBM in a variety of complicated applications are
already well established. A detailed survey of the different applications of LBM is given in
the next section.
The development of the LBM was originally motivated in order to circumvent the drawbacks
of the lattice gas cellular automata (LGCA). It is possible to discuss the LBM independently
of the LGCA; however, both methods have many things in common. Hence an introduction
to the LGCA concept and its drawbacks is essential for a good understanding of the lattice
Boltzmann method. Section 2.1 gives a background to the historical development of LBM
from LGCA towards the modern LB methods.
21
2.1
Historical background
Cellular automata (CA) usually employ regular arrangements of cells for computations - they
are some kind of grids. The earliest studies on CA are due to Stanislas Ulam and Von Neumann (1966). Other terms such as cellular spaces, tesselation automata and cellular structures
are also used in the literature instead of cellular automata. The idea of a cellular automaton
is to produce very complex systems by the application of simple “update rules”, which have a
purely local character. That means that the state of a cell at the next time step depends only
on the states of the neighboring cells at that time. Because of this local character, it is very
suitable for parallel computing. CAs are discrete, which means that each cell investigates its
neighboring cells and, according to its own and their current states and the update rules, it
determines its future state. After the next state of every cell has been determined, the update
is carried out simultaneously. By construction CA are unconditionally stable. They have been
used in many application areas, including the investigation of dense gases.
The idea that the large-scale behavior of a suitable cellular automaton can be used for the
approximation of the fundamental equations of fluid dynamics was first suggested in the 1970s.
However, not all CAs are suitable for the study of fluid flows. First, many types of CAs do
not obey all conservation laws relevant macroscopically, which is absolutely essential for CFD.
In addition, most of them are unable to compute the molecular transport of certain variables,
that is, to reflect the material propagation effect; this is, however, another prerequisite for
CFD. An LGCA must obey at least the two above-mentioned preconditions. In order to allow
a propagation and at the same time to use simple update rules, the propagation and update
(also called collision) are separated in LGCA. The update rules are very similar to CA in that
the rules are also local, i.e. they depend only on the states of the neighboring cells. Lattice
gas models are discrete in phase space and time, and also in geometric space; they are made
of “Boolean molecules”.
The very first model of LGCA was suggested by Hardy et al. (1973, 1975). This model
was called the HPP model after the initials of the authors. They proposed to use a 2D regular
square lattice. This lattice arrangement allowed four link directions: at a vertex, at most
four particles could exist. Each of these particles had to have a velocity in one of the four
link directions. That means that, two or more identical particles could not exist in the same
location simultaneously. The update or the collision rule of this model was as follows: at
one time step, every particle was moved to one unit in space in the direction of its velocity.
If two particles with opposite velocity directions met at a vertex, then these particles were
replaced with each other, otherwise they remained unchanged. These collision rules, like the
other LGCA models following HPP, were chosen so that the total particle number and the
momentum at one vertex were conserved. However, with this method it was not possible
22
2.1. HISTORICAL BACKGROUND
to obtain the Navier-Stokes equations in the macroscopic limit. Frisch et al. (1986) showed
that the problem in the HPP model was the insufficient symmetry of the underlying lattice.
They pointed out that, in order to have enough symmetry, the second and fourth ranks of
lattice moment tensors must be isotropic; if either fails, as in the case of the HPP model, the
Navier-Stokes equations can not be obtained. Frisch et al. (1986) suggested a new 2D scheme
called the FHP model, which had hexagonal symmetry. This study was a breakthrough which
allowed the development of LGCA for a much wider range of applications. This study was
followed by studies of three-dimensional lattices. In 1986 d’humieres et al. suggested a facecentered hypercube (FCHC) model with sufficient symmetry for three- and four-dimensional
simulations.
Motivated by these studies, a rapid development started towards today’s modern lattice Boltzmann models. The development was accelerated in order to cope with the drawbacks of the
LGCA:
• the lack of Galilean invariance
• statistical noise
• spurious invariants
• high viscosity (limitation to low Reynolds number)
• exponential complexity
• explicit dependence of the pressure on velocity
The early lattice Boltzmann models were proposed mainly in order to get rid of the statistical
noise. Shorty after this problem was treated, it was noticed that other drawbacks related to
LGCA could also be cured.
The lattice Boltzmann equation as a separate numerical method was first applied by McNamara and Zanetti (1988). The idea was to replace the “Boolean fields” by ensemble-averaged
distributions (continuous Fermi-Dirac distributions as equilibrium distributions). By applying this change, the statistical noise problem was cured, since the particle distribution was
an averaged and smooth quantity. It should be noted, however, that this model led to other
drawbacks of LGCA, including the complicated nature of the collision operator. Therefore, it
is referred to as non-linear LBM. Shortly thereafter, Higuera and Jimenez (1989) linearized
the collision operator. This progress erased the difficulty of the applicability of LBM in three
dimensions. The models of McNamara and Zanetti (1988) and Higuera and Jimenez (1989) circumvented only the statistical noise and the exponential complexity of the problem of LGCA,
but, they also motivated the further improvement of lattice Boltzmann models. The issue of
23
the limitation of LGCA to low Reynolds numbers was alleviated by Higuera et al. (1989) by
applying so-called enchanced collisions. They used the proposed model for the calculations
of two-dimensional turbulence and it was compared with spectral models by Benzi and Succi
(1990). The same model was applied also for the study of three-dimensional multiphase flows
in porous media by Cancelliere et al. (1990). The most important progress apart from these
studies was accomplished by Koelman (1991) and Qian et al. (1992), who replaced the collision operator by a single time relaxation parameter. These and further studies at that time
led together to the establishment of the modern lattice Boltzmann method (lattice Boltzmann
model with BGK appoximation) in the 1990s. They are free from all drawbacks of LGCA
and since then they have been applied in many different areas, including the investigation
of multiphase flows and multicomponent flows by Gunstensen et. al. (1991), Grunau et al.
(1993), Shan and Chen (1993) and Shan and Doolen (1995), simulation of particles suspended
in fluids by Ladd (1993, 1997), Behrend (1995) and Aidun and Lu (1995), simulation of heat
transfer and reaction by Alexander et al. (1993), Vahala et. al. (1995), Qian (1993) and
Chen et. al. (1994), granular flows by Flekkoy and Herrman (1993) and Tan et al. (1995),
viscoelastic flows by Aharonov and Rothman (1993) and magnetohydrodynamics by Martinez
et al.(1994a).
LBM was used reliably also for low-Reynolds number turbulence, as mentioned previously.
Especially its use in isotropic, decaying turbulence is well established (Benzi and Succi (1990),
Chen (1992), Martinez et al. (1994b), Yu et al. (2005)). There are a few examples where it
was applied to nonhomogenous turbulence and/or anisotropic flows, by Succi et al. (1991),
Eggels (1996), Benzi (1996), Djenidi (2006). Clearly, the lattice Boltzmann method is sufficiently mature to be applied in further DNS studies of low-to-moderate Reynolds number
flows.
2.2
2.2.1
Description of the method
The Boltzmann equation
There are various ways to describe the motion of a fluid:
• The fundamental Newton-Hamilton equations in classical mechanics describe many-body
particle dynamics and they contain huge number of particles. Therefore it is not practical
to have a microstate of such a system.
• The Liouville equations describe the next level of system descrition, the probability
distribution over the phase space of many-particle systems:
3N
∂fN X
−
∂t
j=1
∂HN ∂fN
∂HN ∂fN
−
∂qj ∂pj
∂pj ∂qj
24
=0
(2.1)
2.2. DESCRIPTION OF THE METHOD
where N is the number of particles in the system, HN is the Hamiltonian of the system, qj is the general coordinate and pj is the general momentum of particle j and
fN (q1 , p1 , ..qN , pN , t, dq1 , dp1 ...dqN , dpN ) is the probability of finding particle 1 in the interval ([q1 , q1 + dq1 ], [p1 , p1 + dp1 ]), particle 2 in ([q2 , q2 + dq2 ], [p2 , p2 + dp2 ]) and so on.
That means that, equation 2.1 describes multiple correlations between particles. Bogoljubov, Born, Green, Kirkwood and Yvon showed that a coupled system of differential
equations for reduced density distributions (Fs ) should be equivalent to equation 2.1
(Bogoljubov(1946)). The Fs s are obtained by integrating part of the phase space:
Z
s
fN (q1 , p1 , ...qN , pN , t)dqs+1dps+1 ...dqN , dpN
(2.2)
Fs (q1 , p1 , ...qs , ps , t) := V
where V s is a normalization factor. This system is called as BBGKY after the names of
the authors mentioned above.
• The Boltzmann equation is an integro-differential equation for the single particle distribution function. It is derived from the BBGKY system by assuming that only two
particle collisions are considered and that the velocities of two particles are uncorrelated
before the collision (molecular chaos hypothesis). It is also assumed that external forces
do not have any effect on the local collision dynamics. Under these assumptions and in
the absence of body forces, we have
∂f
+ ~v · ∇f = Ω(f, f )
∂t
(2.3)
where f (~x, ~v, t) is the single particle distribution function and Ω is the so-called collision
operator, which models two-particle correlations through one-particle PDF only, and
which assumes that multiple particle interactions are negligible.
• The Navier-Stokes equations treat the fluid studied as a continuum. They have a wide
range of application areas but when small scales are important, the solution of the
real fluid will be different from those obtained from the solutions of the Navier-Stokes
equations. Therefore, depending on the Knudsen number, a proper choice between
molecular dynamics model and macroscopic fluid behavior models should be made. The
Navier-Stokes equations are given as
∂Uj
1 ∂P
∂ 2 Uj
∂Uj
=−
+ν
+ ρgj
+ Ui
∂t
∂xi
ρ ∂xj
∂xi ∂xi
(2.4)
where U is the macroscopic velocity and P is the pressure.
2.2.2
BGK approximation
The form of the collision integral, Ω in equation 2.3 is very complicated in traditional kinetic
theory. This is one of the main factors when treating the Boltzmann equation numerically or
25
analytically. Therefore, different simplifying approaches have been proposed. Bhatnagar et
al. (1954) showed that a meaningful model at the macroscopic level can be obtained using
a linearized Ω having only one relaxation parameter: each collision changes the distribution
function f by an amount proportional to the departure from the equilibrium distribution
function f eq . Hence the Boltzmann equation with a BGK approximation for Ω can be written
as
1
∂f
+ ~v · ∇f = − (f − f eq )
∂t
λ
(2.5)
where λ is the single time relaxation parameter.
2.2.3
Derivation of the lattice Boltzmann equation with BGK approximation (LBGK)
In this section, various ways of obtaining the form of the equations implemented for the present
study will be introduced. Historically, as stated in section 2.1, the lattice Boltzmann equation
was developed empirically, in order to eliminate the statistical noise in LGCA, which has the
form
ni (~x + ~vi δt, t + δt) = ni (~x, t) + ∆i
(2.6)
where ni is a Boolean array, with entries assuming a value of either 1 or 0 based on whether
the cell is occupied or not, i is the direction of the velocity and ∆i is the collision function.
Taking ensemble averages over large regions of the lattice gas and over long times, the lattice Boltzmann equation was formulated so that well-resolved macroscopic quantities could be
reached.
Other approaches to reach LBGK have been suggested by Sterling and Chen (1996) and
He and Luo (1997).
Sterling and Chen
The starting point of this approach is the discrete Boltzmann equation with BGK approximation:
1 ˜
∂ f˜α
eq
˜
˜
(2.7)
fα − fα
+ ~vα ∇fα = −
λ
∂ t̃
where α is the direction of the microscopic velocity vector. This equation can be rewritten
in dimensionless form by introducing a characteristic length scale (L), a reference speed (U)
and reference density (nr ) and a reference time (tc ), which shows the time between particle
collisions:
1
∂fα ~
+ ξα ∇fα = −
(fα − fαeq )
∂t
Knτ
26
(2.8)
where ξ~ = v~α /U and the parameter Kn denotes the Knudsen number:
Kn = tc
U
L
(2.9)
Note that the relaxation parameter τ is non-dimensionalized as τ = λ/tc .
Equation 2.8 is then discretized by using an explicit Euler finite difference (FD) approach
in time and by using an upwind FD scheme in space and then setting the velocity equal to
δx/δt, to yield
fα (~x + ξ~α δt, t + δt) − fα (~x, t + δt)
fα (~x, t + δt) − fα (~x, t)
+
δt
δx
(fα (~x, t) − fαeq (~x, t))
= −
Knτ
(2.10)
The ratio between lattice spacing δx and δt, i.e. the magnitude of the velocity, is set equal to
the magnitude of the ξ~α s. In this way, the magnitude of the smallest velocity becomes unity.
With this definition, the multiplication of equation 2.10 by δt gives the following equation:
fα (~x, t) − fαeq (~x, t)
fα (~x + ξ~α δt, t + δt) − fα (~x, t) = −
Knτ
(2.11)
Cancellation of the Knudsen number in equation 2.11 is accomplished by choosing the time
step δt equal to the reference collision time tc which was used to obtain the non-dimensional
equation 2.8, so that the LBGK equation is obtained in the following way:
1
fα (~x + ξ~α δt , t + δt ) = fα (~x, t) − (fα (~x, t) − fαeq (~x, t))
τ
(2.12)
Equation 2.12 contains only one term which has to be calculated at t + δt. This makes the
lattice Boltzmann an explicit scheme. Introducing the definition for the collision frequency
w = τ1 , and choosing new reference values, the “lattice units” for time δt and for spacing
δx = cδt = ctc , equation 2.12 can be rewritten as
fα (~x + ξ~α δt , t + δt ) = (1 − ω)fα(~x, t) + ωfα (~x, t)eq (~x, t)
(2.13)
He and Luo
The starting point of this approach is the continuous Boltzmann equation with BGK approximation (equation 2.5). There are two steps in the discretization: the first is the discretization
in time. By fixing the direction of microscopic momentum, and then using the definition
∂
d
~
=
+ ξ.∇
dt
∂t
27
(2.14)
equation 2.5 can be rewritten as an ordinary differential equation:
df
1
1
+ f = f eq
dt λ
λ
(2.15)
Integrating equation 2.15 over a time step of δt,
~ t + δt) = 1 e− δtλ
f (~x + ξδt,
λ
Z
δt
t′
δt
~ t)
e λ f eq dt′ + e− λ f (~x, ξ,
(2.16)
0
By assuming that δt is small and f eq is smooth, the equilibrium distribution can be approximated as
′
t
eq
′
′
~ , ξ,
~ t+t)= 1−
f (~x + ξt
δt
′
~ ξ,
~ t + δt) + Ø(δt2 ), 0 ≤ t′ ≤ δt
~ t) + t f eq (~x + ξδt,
(2.17)
f eq (~x + ξ,
δt
Using this approximation, equation 2.16 is rewritten as
~ ξ,
~ t + δt) − f (~x, ξ,
~ t) = (e− δtλ − 1)[f (~x, ξ,
~ t) − f eq (~x, ξ,
~ t)] +
f (~x + ξδt,
δt
λ
~ ξ,
~ t + δt) − f eq (~x, ξ,
~ t)]
1 + (e− λ − 1) × [f eq (~x + ξδt,
δt
(2.18)
δt
by expanding e− λ in a Taylor series expansion, and once again neglecting terms of order
Ø(δt2 ), one finally obtains
where τ =
λ
δt
~ t) − f eq (~x, ξ,
~ t)
~ t , ξ,
~ t + δt ) = f (~x, ξ,
~ t) − 1 f (~x, ξ,
f (~x + ξδ
τ
(2.19)
was used for the dimensionless relaxation time.
Equation 2.19 is formally first order accurate in δt. It describes the evolution of the particle
distribution function f with discrete time. Equation 2.19 resembles the lattice Boltzmann
equation strongly provided that we have a proper equilibrium distribution. Therefore, the
second step is to assure the discretization in phase space. The connection between the microscopic properties defined by the kinetic theory (Boltzmann equation) and the macroscopic
properties can be achieved by calculating the moments of the particle distribution functions
~ where ρ is
(Boltzmann-Maxwellian) from the following integrals in the momentum space, ξ,
the density and u is the macroscopic velocity. The calculation of ρ and ~u is very important,
because they are used to construct the equilibrium distributions initially. As in any kinetic
theory,
Z
Z
~
ρ = f dξ = f eq dξ~
Z
Z
~ dξ~ = ξf
~ eq dξ~
ρ~u = ξf
(2.20)
28
~ so that calculation of the
Therefore, we need a proper discretization in momentum space, ξ,
moments above could be possible. Using such an approximation, the moments can be calculated via equation 2.29. It is suggested to use the following quadrature for this approximation:
Z
X
~ eq (~x, ξ,
~ t)dξ~ ≈
ψ(ξ)f
Wα ψ(ξ~α f eq (~x, ξ~α , t)
(2.21)
α
with fα (~x, t) = Wα f (~x, ξ~α, t) and fαeq (~x, t) = Wα f eq (~x, ξ~α , t). That means, using suitable
~ α and weight Wα ) for the quadrature, that the correct
lattices (sets of discrete velocities, xi
discretization in phase space can be accomplished. And if one calculates the moments according to equation 2.29 exactly, then it is possible to construct the lattice Boltzmann equation as
given by equation 2.12. Under these constraints, it is possible to connect the lattice Boltzmann
equation with the Boltzmann equation explicitly.
2.2.4
Equilibrium distributions and the lattice
The equilibrium distributions in the lattice Boltzmann equation are of the Boltzmann-Maxwellian
type. They are derived by applying the “maximum entropy” principle under the constraints
of mass and momentum conservations up to second-order accuracy and have the following
general form independent of the chosen lattice:
!
~α~u (ξ~α~u)2 ~u.~u
ξ
fαeq = Wα ρ 1 + 2 +
− 2
(2.22)
cs
2c4s
2cs
The values of the weighting factor Wα and the choice of lattice (discrete) velocity vectors ξ~α
depend on the lattice type. Here, they will be discussed only for the commonly used lattice
type for three dimensions, D3Q19, since this lattice was also used for the present simulations.
The other popular lattice in three dimensions, D3Q15, was shown to be less stable (especially
for turbulence) and D3Q27 does not deliver better results despite its higher computational
costs. In the notation of the models, D denotes the dimension of the problem and Q denotes
the total number of the microscopic velocity directions ξ~α including the rest particles with
zero velocity. It is a multi-speed lattice type. There are three lattice speeds in this model:
√
0, 1 and 2. The total number of ξ~α components are 1 for speed 0, 6 for speed 1 and 12 for
√
speed 2 and they are equal to
ξ~0 = (0, 0, 0), ξ~1 = (+1, 0, 0),
ξ~4 = (0, −1, 0),
ξ~2 = (−1, 0, 0), ξ~3 = (0, +1, 0),
ξ~5 = (0, 0, +1), ξ~6 = (0, 0, −1),
ξ~7 = (+1, +1, 0), ξ~8 = (−1, −1, 0), ξ~9 = (+1, −1, 0),
ξ~10 = (−1, +1, 0), ξ~11 = (+1, 0, +1), ξ~12 = (−1, 0, −1),
ξ~13 = (+1, 0, −1), ξ~14 = (−1, 0, +1), ξ~15 = (0, +1, +1),
ξ~16 = (0, −1, −1), ξ~17 = (0, +1, −1), ξ~18 = (0, −1, +1)
29
(2.23)
The weighting functions for this lattice are derived by writing down the non-vanishing moments up to fourth order (odd-order moments vanish). The resulting equation system is then
solved to obtain the Wα s.
For the D3Q19 model:
0th moment:
X
Wα = W0 + 6W1 + 12W2 = ρ
(2.24)
kB T
2
ξ~ix
Wα = 2W1 + 8W2 = ρ
m
(2.25)
α
2nd moment:
X
α
4th moment:
X
α
2
kB T
= 2W1 + 8W2 = 3ρ
m
2
X
kB T
2 ~2
~
ξix ξiy Wα = 4W2 = ρ
m
α
4
ξ~ix
Wα
(2.26)
Thus the Wα for particles at rest W0 = 1/3, for speed 1 W1−6 = 1/18 and for speed
W7−18 = 1/36.
√
2
Choosing the corresponding values of Wα and ξ~α , equilibrium functions are calculated for
three speeds according to equation 2.22 for this model. Applying similar procedures, one can
also obtain the parameters for other types of lattices. Figure 2.2 shows the lattice velocities
used in some popular lattice types for two and three dimensions.
2.2.5
Basic algorithm
A typical lattice Boltzmann simulation is started from given inital values of density and
velocity. Initial distributions, fα , are usually calculated as equilibrium distributions, according
to equation 2.22. After this initialization step, each time step consists of the following three
steps, the same operation being repeated for all cells in one loop:
• calculation of the local macroscopic quantities (density and velocity)
• collision
• propagation
30
(a)
(b)
(c)
Figure 2.2: The lattice velocities of some popular lattices (Wolf-Gladrow(2000)): a) D2Q9, b)
D3Q15, c) D3Q19.
31
The first two steps are basically the implementation of equation 2.12. In the collision step,
the following equation is implemented, where f˜α (~x, t) shows the post-collision distributions:
1
f˜α (~x, t) = fα (~x, t) − (fα (~x, t) − fαeq (~x, t))
τ
(2.27)
That means, at each time step, that for all fluid points in the computational domain, the
values of the distribution functions are updated due to the collisions between the particles,
using the calculated equilibrium distributions at that time step.
In the streaming step, the following equation is satisfied:
fα (~x + ξ~α , t + 1) = f˜α (~x, t)
(2.28)
That means, at each time step, for all fluid points, that the new distributions calculated in
the collision step are propagated to the nearest neighbor in the direction of the lattice velocity
for the values at the next time step.
For the solid boundaries, the bounce-back rule is applied, as explained further below.
The macroscopic quantities (density and velocity) are then given by
X
fα
ρ=
α
ρ~u =
X
ξ~α fα
(2.29)
α
2.2.6
Boundary treatment
Bounce-back:
In order to satisfy the no-slip boundary condition at solid interfaces, the so-called bounceback rule is incorporated during the propagation step within the lattice Boltzmann simulation.
Bounce-back means that if a particle distribution streams to a solid node, it scatters back to
the node from which it came. Generally, there are two ways to define a node as boundary:
one can locate the boundary either on grid nodes, or on links. The second version, known
as “bounce-back on the link”, was shown to be second-order accurate for flat boundaries by
He et al. (1997). Hence it was also applied in the present thesis. In this method, the wall is
placed half-way between a flow node and a boundary node (bounce-back node), as shown by
figure 2.3. At each time step, the distribution function towards a solid boundary is inverted
into a post-collision fα in the opposite direction, during the stream without a collision step as
follows:
f−α (~x, t + δt ) = f˜α (~x, t)
(2.30)
fα (~x, t) = f˜α (~x − ξ~α , t − δt)
(2.31)
Here, pre-collision is given by
32
2.3. FROM LATTICE BOLTZMANN EQUATION TO NAVIER-STOKES EQUATIONS
and post-collision is given by
f˜α (~x, t) = fα (~x, t) − ω(fα − fαeq )
(2.32)
Periodic boundary conditions:
At the sides of the computational domain, spatial periodicity is introduced. This is accomplished by introducing an additional “ghost” layer beyond the domain boundary. The particle
distributions are copied in this layer. In the next step, they are streamed to the adjacent layer
of points at the domain from the corresponding last layer of points at the opposite end of the
domain. As an example, if one uses a D2Q9 model (see figure 2.2(a)) in two dimensions, the
periodic boundary condition is implemented as follows:
fα (1, y) = fα (lx + 1, y), α = 1, 5, 8; 1 < y < ly
fα (lx , y) = fα (0, y), α = 3, 6, 7; 1 < y < ly
(2.33)
and for the diagonal distributions on the corners of the computational domain:
f5 (1, 1) = f5 (lx + 1, ly + 1)
f6 (lx , 1) = f6 (0, ly + 1)
f7 (lx , ly ) = f7 (0, 0)
f8 (1, ly ) = f8 (lx + 1, 0)
2.3
(2.34)
From lattice Boltzmann equation to Navier-Stokes
equations
The macroscopic flow behavior governed by the Navier-Stokes equations can be recovered
from the LBM dynamics by the application of the Chapman-Enskog procedure (multi-scale
expansion). It contains the following steps(Wolf-Gladrow(2000)):
1. The distributions of fα (~x, t) are expanded around the equilibrium distributions as follows:
fα (~x, t) = fαeq (~x, t) + Knfα1 (~x, t) + Kn2 fα2 (~x, t) + O(Kn3 )
(2.35)
where
fα0 = fαeq ,
X
fαi (~x, t) = 0,
α
X
α
33
cα fαi (~x, t) = 0, f or i ≥ 1
(2.36)
(a)
(b)
(c)
(d)
Figure 2.3: Illustration of bounce-back on the link.
34
2.3. FROM LATTICE BOLTZMANN EQUATION TO NAVIER-STOKES EQUATIONS
Here, Kn represents as previously, the Knudsen number. The expansion is done for small
Knudsen numbers. Therefore, the contributions fα1 (~x, t) and fα2 (~x, t) are negligible for mass
and momentum conservations.
2. The left-hand side of equation 2.12 is expanded into a Taylor series up to terms of second
order:
∂fα
∂fα
~
fα (~x + ξδt , t + δt ) = fα (~x, t) + δt
+
+ cαi
∂t
∂xi
∂
∂fα
∂ ∂fα
(δt)2 ∂ 2 fα
+ cαi cαk
+ O (δt)3
(2.37)
+ 2cαi
2
∂t2
∂t ∂xi
∂xi ∂xk
3. A multiple scale expansion is introduced for time and spatial derivatives, as follows
∂
∂ (1)
∂
∂ (1)
∂ (2)
→ Kn
+ Kn2
+ O (Kn)3 ,
→ Kn
+ O (Kn)2
∂t
∂t
∂t
∂xi
∂xi
These expansions are then substituted into equation 2.12, giving
∂fα
∂fα
+
+ cαi
δt
∂t
∂xi
∂ ∂fα
∂ ∂fα
(δt)2 ∂ 2 fα
+ 2cαi (
) + cαi cαk
(
) +
2
∂t2
∂t ∂xi
∂xi ∂xk
1
fα (~x, t) − fα(0) (~x, t) = O[(δt)3 ]
τ
(2.38)
Now, using the expansions in Kn above:
∂ (1) (0)
∂ (1) (0)
Knδt
f + cαi
f
∂t α
∂xi α
(1)
∂
∂ (2) (0)
∂ (1) (1)
2
(1)
+Kn δt
f +
f + cαi
f
∂t α
∂t α
∂xi α
(1) (1)
2
∂ ∂
∂ (1) ∂ (1) (0)
∂ (1) ∂ (1) (0)
2 (δt)
f (0) + 2cαi
f + cαk cαi
f
+Kn
2
∂t ∂t α
∂t ∂xα α
∂xk ∂xi α
+Knωfα(1) + Kn2 ωfα(2) = O[Kn3 ] + O[(δt)3 ]
(2.39)
Dividing by (Knδt) and arranging in orders of Kn,
KnTα(0) + Kn2 Tα(1) = O[Kn3 ] + O[δ 3 ]
(2.40)
(2.41)
with
Tα(0)
∂ (1) (0)
∂ (1) (0)
fα + cαi
f + ωfα(1)
=
∂t
∂xi α
35
Tα(1) =
∂ (1) (1) δt ∂ (1) ∂ (1) (0)
∂ (1) (1) ∂ (2) (0)
fα +
fα + cαi
f +
f
∂t
∂t
∂xi α
2 ∂t ∂t α
∂ (1) ∂ (1) (0)
f +
+δtcαi
∂t ∂xα α
∂ (1) ∂ (1) (0) ω (2)
δt
f + fα
cαk cαi
2
∂xk ∂xi α
δt
(2.42)
(0)
The zeroth and first-order moments of the term Tα are calculated to obtain the mass and
momentum conservation in first order:
X
∂ (1)
∂ (1)
Tα(0) =
(ρui )
(2.43)
ρ+
∂t
∂x
i
α
and
X
α
∂ (1) (0)
∂ (1)
(ρui ) +
P
ξ~αi Tα(0) =
∂t
∂xk ik
(2.44)
so that the continuity equation and the Euler equation for incompressible fluids to first order
in Kn are given as follows:
∂
∂
ρ+
(ρui ) = 0
(2.45)
∂t
∂xi
∂u
1
+ u∇u = − ∇p
∂t
ρ
(2.46)
where pressure p must be given by
ξ~2
kB T
=ρ
m
3
√
~ 3.
and the speed of sound must be given by cs = ξ/
ρ
(2.47)
(1)
Similarly, taking the zeroth- and the first-order moments of (Ti ) to obtain the mass and
momentum conservation up to second order in Knudsen number, as follows:
1 ~2
∂ 2 (ρu)
= δt τ −
xi /3 ∇2 (ρu) + ∇∇(ρu)
(2.48)
∂t
2
Summation of the first- and second-order terms gives the continuity and momentum equations
with the kinematic viscosity:
δt
2
ν = cs τ −
(2.49)
2
Since δt is set to 1 in lattice Boltzmann simulations, i.e. the lattice Boltzmann time unit is
taken to be δt, τ > 1/2 or 0 < ω < 2 must hold to assure positive viscosity and therefore the
stability of the calculations. In the above expansion, the terms up to second order were kept,
which makes the LBM a second-order accurate method in terms of both the Knudsen number
and the time step (i.e. the ratio of δt to some hydrodynamically relevant time scale).
36
Chapter 3
Verification of the code
As stated in chapter 1, the scientific motivation for carrying out this research work was the
lack of numerical studies of decaying isotropic turbulence. In spite of the importance of this
kind of flow for turbulence research nad partially also to develop turbulence models, only very
few numerical results are available that provide a sound basis for the developments. To obtain
the relevant information in a reliable form from experimental studies, where turbulence is
generated through grids, is also not possible. It is even difficult to compare the experimental
and numerical results. In numerical studies, usually periodic boundary conditions are applied
for the study of homogeneous turbulence and these cannot be introduced in experiments.
A more proper boundary treatment is required in order to study grid-generated turbulence
numerically. Application of the boundary condition, which is known as the “fringe region”
condition in the literature, has been found to be a suitable approach for studying decaying
isotropic turbulence numerically. Implementation of this fringe region method will be one of
the subjects of the next chapter. However, before attempting to carry out detailed simulations in a relatively new computational area, such as grid-generated turbulence, the computer
code employed should be shown to be accurate and efficient in an application where a reasonable understanding of the flow and its properties has already been established in the literature.
Simulation of turbulent channel flows offers a perfect opportunity for testing the computer
code employed, since on the one hand, the computations of such flows have been the subject of
numerical studies employing (pseudo)spectral codes (Moser et al. (1998)), and on the other,
there are numerous experimental studies, covering a large Reynolds number range (e.g. Fischer (2000), Zanoun (2003)). These literature data include basic turbulent quantities of plane
channel flows, such as mean velocity distributions and Reynolds stress profiles. Thus, one can
validate one’s own code by making comparisons between different numerical and experimental
studies.
The lattice Boltzmann solver used by the author for predictions, developed at LSTM Erlangen, was applied to the verification of the computations for channel flows. This solver
37
CHAPTER 3. VERIFICATION OF THE CODE
implements the LBGK equations introduced in chapter 2. The code is known as the BEST
code for lattice Boltzmann computations of fluid flows. Its verification for turbulent channel
flows has been performed by several researchers, including the present author. The computational results to be discussed were obtained by Özyılmaz (2003) and by Lammers (2004).
BEST has also been applied in other flows, such as flows through porous media and packedbed reactors (Zeiser et al. (2001, 2008), Freund et al. (2003, 2005), Bernsdorf et al. (2000)),
flow past a square cylinder (Breuer et al. (2000), Zeiser (1998), Bernsdorf et al. (1998)) and
channel flows with rough walls (Lammers (2004)).
The outcome of these computational studies and a the detailed description of the BEST
code are the central points of the present chapter. Section 3.1 outlines the steps required
for realization of a typical simulation with BEST. Both optimization and parallelization are
required in order to speed up the computations. In modern computer systems, the optimization can be done either for vector-based platforms or for cache-based platforms. Section 3.1.2
describes how the optimization on each platform is realized in BEST. Section 3.1.3 outlines
the parallelization of the code. The above-mentioned comparison of the experimental and
numerical channel flow studies is presented in section 3.2.
38
3.1. THE BEST CODE
3.1
3.1.1
The BEST Code
The description of the code and the general approach for the
set-up of a simulation
BEST is a parallel lattice Boltzmann BGK solver for 2D/3D flow problems, written in Fortran
language and optimized for vector-parallel supercomputers and commodity clusters. It was
developed at LSTM Erlangen over the period from 1992 until 2007. The early work was undertaken by Jörg Bernsdorf. In this early stage, he concentrated on the development of a first
lattice Boltzmann solver at the Iinstitute, producing good agreement with the results obtained
using finite volume methods for flow through porous media (Bernsdorf (1996), Bernsdorf and
Schaefer (1997), Bernsdorf et al. (1999a)). The first simulations for 2D channel flows around
a square obstacle with a lattice BGK solver were also carried out at an early stage (Bernsdorf
et al. (1998)). These simulations were compared with finite volume methods (Breuer et al.
(2000)). Detailed comparisons between the two methods with respect to velocity profiles, drag
coefficient, recirculation length and Strouhal number showed excellent agreement between finite volume results and lattice Boltzmann simulations.
Thereafter, a very detailed study of the transport phenomena in packed-bed reactors and
porous media was performed by Bernsdorf (1999b), using computer tomography to extract
3D geometries. Bernsdorf (1999c-e) worked on the pressure drop investigations. (Zeiser et al.
(2001)). Brenner et al. (2002) studied the dispersion and reaction in a catalyst-filled tubes.
To obtain the geometric 3D structure, they applied a Monte Carlo method (Zeiser (2008)).
They investigated the influence of different tube-to-particle ratios, among others. They were
able to show that the code was capable of representing the complex physical phenomena correctly. In addition to just hydrodynamics, simple reactions and adsorption processes were
also investigated (Freund et al. (2003)). A recent PhD thesis by Zeiser (2008) gives the most
detailed outcome of work related to flows in complex geometries.
The application of BEST to the investigation of turbulent channel flows was initialized by
Lammers. He made a detailed statistical analysis at Reτ = 150 and 180 (Lammers et al.
(2001, 2003)). In his PhD thesis (Lammers (2004)), he compared different lattice models
for 3D simulations with respect to divergence, stability and accuracy and concluded that the
D3Q19 model was the most suitable model for the predictions of turbulent channel flows.
Lammers (2004) also studied channels with 2D roughness elements at Reλ = 60 and 100. He
showed that uw is not exactly zero for rough walls. Turbulent channel flows at moderate
Reynolds numbers were studied in detail by the present author in her Master thesis (Özyılmaz
(2003)) and also during the early stage of this PhD study. The emphasis of these studies was
on the near-wall statistics of channel flows.
39
Hence, by the time the present author obtained the BEST code for further applications to turbulent flows, it had already been tested for different applications, including turbulent channel
flows, as stated above. It was optimized and parallelized for simulations on high-performance
computers (optimization and parallelization of the code are explained in the following sections). The first thing to do was therefore to become familiar with the proper set-up of a
computation. A typical set-up of the code may be explained as follows.
A typical simulation with BEST is begun by reading the configuration parameters from an
external file. This file contains the information on the lattice type (e.g. D2Q9, D3Q19,D3Q27
etc.), the resolution of the computational grid and a the total number of iterations, the value
of the relaxation parameter, the frequency of the data sampling and initial and boundary
conditions. After reading the control parameters of the simulation, the memory is allocated.
If the flow domain includes obstacles, corresponding parts of the computational domain are
marked as solid points. Initial values for the macroscopic fields, e.g. density and velocity,
have to be defined. One can choose in this step one of the various velocity profiles which
are available in the code, such as a parabolic velocity profile or a log-law velocity profile to
name just a few. One can introduce additionally some streamwise or spanwise vortices to the
flow field during this initialization step. In this way, a turbulent flow field is generated at the
beginning of the computation, therefore the time to obtain converged solutions is significantly
reduced. Using the initial values of velocity and density, equilibrium distributions are calculated according to equation 2.22. If the computations are to be carried out from an old status,
the last distribution functions are read from restart files. Then, the time looping is started.
At each time step, for every fluid cell in the computational domain, hydrodynamic variables
and equilibrium distributions are calculated. Collision and propagation steps are realized.
Then, the physical boundary conditions are applied. After predefined time steps have been
reached, accumulation of the statistics is started. When the maximum number of time steps
or the maximum allowed elapsed time is reached, the main loop is left. Further statistics are
accumulated and results are written to disc.
One can set up any kind of flow simulation using the same procedure as explained above,
by choosing initial and boundary conditions suitable for the specific problem.
3.1.2
Optimization of the code
Since direct numerical simulations of turbulent flows are computationally very demanding,
one needs to consider possible optimization techniques in order to speed up the computations.
In this way, a more efficient use of the computer system is assured. BEST was optimized
for different platforms and by different authors. An MPI-based version, which was optimized
40
3.1. THE BEST CODE
for vector computers, was provided by Peter Lammers (Lammers (2004)). There is another
version, which was parallelized with OpenMP standard for cache-based platforms supplied by
Thomas Zeiser (Zeiser (2008)). In the following, details of the optimization are given.
There are mainly two different platforms in modern high-performance computer architectures: tailored vector-based systems and commodity cache-based microprocessors. Since there
are some differences between these two systems, the optimization approach for each will be
in some ways different also. An important difference between the vector-based systems and
the cache-based microprocessors must be recalled at this stage (Hager and Wellein (2008)):
vector-based architectures are designed for high-performance computing and offer not only
a high peak performance of the arithmetic units but also a high memory bandwidth. However, to exploit their computational power fully, the operations must be vectorized and long
loops are required to allow efficient use of vector pipelines. Cache-based microprocessors are
designed for the consumer (“gamer”) marked and not primarily for HPC purposes, hence a
low cost of the complete system design is important. As a consequence, clock frequencies are
rather high, memory capacity is moderate but the main memory bandwidth is very limited. To
bridge this “memory gap” at least to some extent, “caches” have been introduced as some sort
of “buffers”, with the consequence that complete “cache lines” always have to be exchanged
with the next higher/lower memory hierarchy. Reasonable performance for memory-intensive
codes such as typical engineering applications and LBM in particular can only be obtained if
all data of loaded cache lines are actually used because otherwise much of the already limited
memory bandwidth will be wasted.
In a lattice Boltzmann simulation, the highest demand on memory bandwidth is due to the
calculations during the relaxation and propagation steps. It is possible to reduce the computational time noticably if these two steps are combined (Wellein et al (2006)). Whereas
the relaxation step only accesses the values of the local cells, the propagation step exchanges
information with the non-local next-neighbor cells. This data dependence must be eliminated
in order to be able to carry out the collision and propagation steps at the same time. This
can be achieved by introducing separate variable fields for the current time step and for the
next time step. The resulting data structure is a 5-D array, which includes the information on
the spatial coordinates and the values of the microscopic velocities. The order of the indices
influences the memory access characteristics.
In the vector-based (optimized) version of the BEST code, the microscopic velocities are
addressed by the first position of these arrays. The spatial coordinates are addressed by the
next three positions. The last position addresses the current time step. In the cache-based
(optimized) version of the BEST code, the microscopic velocities are addressed by the fourth
position. The first three positions contain the information on the spatial coordinates. That
41
means that, the distribution functions are not addressed one after the other but are ordered
in terms of their velocity directions. The consequence is that the density functions of a cell
are consecutive in the memory, which is ideal for the collision step (Wellein et al. (2006)).
3.1.3
Parallelization of the code
In order to cope with the huge datasets of turbulence flow calculations in a reasonable time,
one should consider the parallelization so that the code instructions are processed simultaneously. Thanks to this simultaneous treatment, it is possible to increase the usable memory
and thus to reduce the computing time. From the programming point of view, there are two
possibilities of parallelization: 1) Open MP and 2) MPI.
Open MP is based on the concept of the shared memory. In this method, parallelization
is carried out inside one node. In a multi-processor system, one node consists of many CPUs,
which can access the same memory at the same time. One can therefore, divide the job among
different processors inside one node, without having to take the data exchange between different processors into consideration. The greatest advantage of this method is that it is easy to
accelerate an existing serial code by parallelizing it in a stepwise approach.
The message-passing interface (MPI) exhanges data between the processes explicitly. Therefore, the parallelization requires much more effort than Open MP, but MPI can be applied
both in shared memory and distributed memory systems. MPI is used together with domain
partitioning to assign parts of the total grid to individual processes. The computational domain is decomposed in such a way that each process uses only one local memory space. The
processes communicate with each other by sending and receiving messages. Cooperative operations are needed for the data transfer. That means that each “send” operation must have
a corresponding “receive” operation (Körner et al (2005)). The data exchange between the
processors is managed by calling standard libraries. These kinds of message-passing libraries
are available for many common programming languages, including Fortran.
In this work, the vector-optimized and MPI version of BEST was applied, both for plane
channel flows calculations which are treated in the next section and for the calculations of
grid-generated turbulence which are the subject of the remaining chapters.
3.2
Calculations of fully developed channel flows
Two-dimensional turbulent channel flows were investigated with lattice Boltzmann DNS for
a moderate Reynolds number range of 106 < Reτ < 180. In the following, basics of such
flows are recalled, computational details are given and the results are compared with different
42
3.2. CALCULATIONS OF FULLY DEVELOPED CHANNEL FLOWS
experimental and numerical studies available in the literature.
3.2.1
Fundamentals
A fully developed two-dimensional channel flow configuration can be realized by letting the
fluid flow through a rectangular duct which has a length L and a channel half-height δ. The
length of the channel (L) is large compared with the channel height (2δ). The flow is twodimensional: the mean cross-stream velocity is zero (W ). The mean flow is predominantly in
streamwise direction x. For large x values, the flow is fully developed: that means that the
time-averaged statistics do not vary with the main flow direction. The extent in the spanwise
direction (z) of the channel is large in comparison with the channel half-height (δ), so there
is not a significant change of the statistics in this direction. Under these circumtances, timeaveraged statistics vary only with wall normal direction (y) and the mean continuity equation
1.4 can be written as
dV
=0
dy
(3.1)
since W = 0 and U does not depend on x. Since at the walls V = 0, equation 3.1 shows that
mean vertical velocity must be zero for all y values.
The mean momentum equation 1.5 for the lateral direction reduces to
0=−
1 ∂P
d 2
(v ) −
dy
ρ ∂y
(3.2)
Since at y = 0, v 2 is also zero, from equation 3.2 we can write
v2 +
P
Pw (x)
=
ρ
ρ
(3.3)
where Pw (x) shows the pressure distribution at the wall. From equation 3.3, it is clearly seen
that the mean axial pressure is uniform across the channel:
∂P
dPw
=
∂x
dx
(3.4)
The mean momentum equation for the axial direction can be written as
0=ν
1 ∂P
d2 U
d
uv
−
−
dy 2
dy
ρ ∂x
(3.5)
− ρuv,
Defining τ = ρν dU
dy
dPw
dτ
=
dy
dx
43
(3.6)
Since τ is a function only of y and Pw is a function only of x, it can be seen from equation 3.6
w
that both dτ
and dP
are constant. To sum up, the flow is driven by a constant pressure drop
dy
dx
in the streamwise direction which is balanced by the mean shear stress gradient.
In wall-bounded flows, it is customary to introduce wall variables because of the universality aspect. By non-dimensionalization, it is possible to compare different wall-bounded
flows. From the wall shear stress and viscosity, two new viscous scales can be defined: the velocity scale relevant close to the wall is the wall friction velocity, uτ , which is defined as follows:
uτ =
r
τw
ρ
(3.7)
In the region close to the wall, the dynamics of turbulence are substantially influenced by the
fluid kinematic viscosity, ν, so that the relevant length scale for flow phenomena is the viscous
length scale, lν , defined as
r
ρ ν
(3.8)
δν = ν
τw uτ
The wall units lν and uτ lead to the definition of a friction velocity Reynolds number:
Reτ =
uτ δ
δ
=
ν
δν
(3.9)
and non-dimensionalized length scale:
y+ =
uτ y
y
=
,
δν
ν
u+ =
U
uτ
(3.10)
Since the statistics are homogeneous in the streamwise and spanwise directions, one is allowed to average the time-averaged statistics spatially in these two directions. Therefore, in
turbulent plane channel flows, it is customary to examine the turbulent statistics in terms of
the non-dimensionalized distance from the wall (y + ).
44
3.2.2
Set-up of the computations for channel flow
As shown in the previous section, the investigation of a two-dimensional fully developed channel flow is, in principle, the same as the investigation of a one-dimensional flow problem whose
statistics vary only with the direction normal to the wall (y).
Numerically, it is possible to realize this kind of flow by the application of periodic boundary
conditions in the statistically homogeneous directions (i.e. in the streamwise (x) and spanwise
directions (z)) and by satisfying the no-slip velocities at the walls. Hence, as soon as we
have a means to drive the flow, we can examine a plane channel flow numerically without any
difficulty. Realization of periodic boundaries and the no-slip conditions are done as explained
in section 2.2.6.
In order to drive the flow, two methods may be considered: one can introduce either a mass
flow or a pressure gradient. The latter was applied in the calculations below. In this method,
the pressure drop was added as a body force term during the calculations of equilibrium distributions. The magnitude of the velocity was modified before using its value for the equilibrium
distributions. This was accomplished by applying the following form of the lattice Boltzmann
equation:
2−ω~ ~
eq
~
ξα f (3.11)
fα (~x + ξα , t + 1) − fα (~x, t) = −ω fα (~x, t) − fα (ρ, ~u + f/2ρ, ~x, t)) + 3Wα
2
where f~ is the imposed pressure gradient (f~ = ∆p).
While setting up the simulations, a value for the pressure drop in the streamwise direction
was given. Using the definition of Reτ and uτ , the value of the channel half-height δ was
dP
. The value of Reτ was then determined by
determined for a fixed dx
1
Reτ =
(dp/dx)δ
ν
(3.12)
The calculations discussed in the following section were carried out for Reynolds numbers
100 < Reτ < 150 by Özyılmaz (2003), and for Reτ = 180 by Lammers (2004). The computational domain had 512 points in the streamwise and 128 points in the other two directions.
The turbulent calculations were initialized by imposing a logarithmic velocity profile on the
flow field in all these simulations. The calculations were performed using the D3Q19 lattice
model, since Lammers (2004) showed that employing the D3Q27 model did not bring any
advantages as far as the accuracy was concerned, although it was computationally much more
demanding.
45
3.2.3
Comparison of the results with the data available in the literature
Information on fully developed, turbulent plane channel flow is available in the literature over
a wide range of Reynolds numbers and is continuously updated with data from measurements
and numerical predictions. The channel flow database used for the verification of our code is
summarized in table 3.1. Overall, the data cover a large range of Reynolds numbers, starting
at very low values where the onset of channel flow turbulence can be investigated and also
covering those where the flow can be considered to provide all relevant high Reynolds number
effects (88 < Reτ < 4800).
Source
Zanoun (2003)
Moser et al. (1999)
Fischer (1999)
Lammers (2004)
Özyılmaz (2003)
Technique
Experimental:
Hot-wire
DNS: Chebyshev–Pseudo-spectral
Experimental:
Laser–Doppler
DNS:
Lattice–Boltzmann
DNS:
Lattice–Boltzmann
Reτ
1163 — 4783
178 — 587
88 — 350
180
106 — 150
Table 3.1: The database on fully developed, turbulent, plane channel flow used for the verification of the
code.
As discussed in section 3.2.1, in fully developed, turbulent plane channel flows, the meanflow velocity has only one non-zero component, and that is along the axis of the channel.
It was also shown in that section that the relevant length and velocity scales in near-wall
region are lν and uτ . By using these inner scales, it is possible to show a universality in the
region very close to wall. This is done in figure 3.1, where the mean velocity distributions
obtained from the database given in table 3.1 are summarized. A good match between the
time-averaged mean velocity data obtained by different numerical and experimental methods
is observable from this figure. The vertical axis shows the streamwise mean velocity component which is normalized by uτ . The horizontal axis shows the distance from the wall scaled by
lν . The velocities are averaged spatially in the streamwise and spanwise coordinate directions.
When scaled by inner variables, the velocity profile indicates a universality with respect to
the distance from the wall in the region, which is referred to as the viscous sublayer in the
literature (y + > 10). The velocity profile can be described by U + ≈ y + in the viscous sublayer.
Far from the wall (y + > 10), mean velocity profiles indicate a strong Reynolds number dependence at low Reynolds numbers. Both experimental values (Fischer (1999)) and numerical
studies carried out by different computational methods (Moser et al. (1999), Özyılmaz (2003))
show the same trend. The discussion of the difference in the Reynolds number dependence
behavior of low and high Reynolds number studies is beyond the scope of this section. What
counts is the observation that for a similar Reynolds number range, mean velocities obtained
46
from lattice Boltzmann simulations were confirmed by LDA measurements and numerical
studies based on spectral methods.
The high Reynolds number experimental data (Zanoun (2003)) do not show a similar dependence on Reynolds number for the same y + range. Starting from distances as low as
y + ≈ 150, a universal behavior is observable with respect to Reynolds number. In the region
far from the wall, the velocity profile can be described by a logarithmic law, which has the
form
U + = (1/κ) ln y + + B
(3.13)
The study of Zanoun (2003) showed that the region y + > 150 can be approximated with
κ = 1/e and B = 10/e for the Reynolds number range 1100 < Reτ < 4800.
Figure 3.2 compares the root mean square of streamwise turbulent fluctuations. The peak
of turbulent intensity is observed at y + ≈ 11 − 12. This is confirmed by all datasets. The peak
value obtained at low Reynolds numbers in experiments and LB simulations lie in the range
2.5 − 2.6. The spectral simulations show a higher values for the peak of the turbulent intensity. This difference may be attributed to the calculation of uτ , which is very prone to small
deviations. However, overall, it can be seen that the expected profile of turbulent intensity is
obtained by LB simulations. Same is true for the distribution of the only non-zero off-diagonal
element of Reynolds stress tensor, which is demonstrated in figure 3.3: lattice Boltzmann simulations confirm the trends observed by the other studies. The peak value of −uv depends
strongly on the Reynolds number. It shows a clear trend of growth with increasing Reynolds
number.
Overall, it is seen that the lattice Boltzmann code used for the present thesis was efficient and
accurate enough for moderate Reynolds number turbulent flows, especially for plane channel
flows.
47
25
20
+
15
u
X
+$
.
$*∇
.
X
+ ∇
$ *∇
.
*
*$.
X
+
∇
$
*.
∇
$∇
*
*X
+ .
$∇
* .
X
+$∇
* ∇.
$
∇
.
$
*∇
X
+
*∇
*$.∇
∇
$
*∇.
*$ X
*$
.∇
++
**.$
X
+X
∇
*.$
*+
*$*.$X
*∇
+
$*$.X
$
**∇
$*∇
*∇
.X
+.
$
*
.
$∇
*+
.
*$∇
.X
$
*∇
$
∇
* .
$∇
*.X
+.
*∇
*$
X
+.
∇
*$.∇
$
*X
.
*. ∇
+
$
∇
∇
*$
.*X
+
* $*.∇X
+
*$.∇
X
+
*∇X
+
*∇$.X
+
+
*$
∇X
* X
.
+
*∇
X
+
*∇$
+.
$* X
*∇
X
+
.
$*∇
.
+
* X
*∇.
*$ X
+
∇
$*X
+.
*∇
$ .
*∇
X
+
88
106
118
130
150
160
180
211
250
300
350
395
595
1167
1543
1850
2155
2573
2888
3046
3903
4040
4605
4783
$
10
+
X
*
∇
$
.
5
0
10
1
+
10
2
10
3
y
Figure 3.1: The full set of mean-flow velocity profiles obtained in the studies listed in Table 3.1 are shown,
scaled on inner variables. These data correspond to the range 88 < Reτ < 4800. Laboratory measurements
are indicated by symbols and numerical simulations by lines.
3
2.5
u’+
2
1.5
1
395
1167 1850
4783
0.5
10
0
10
1
10
+
2
10
3
y
Figure 3.2: Intensity of streamwise velocity fluctuations, normalized by uτ .
48
1
2155 (Zanoun)
1167 (Zanoun)
590 (Moser,Kim,Mansour)
395 (Moser,Kim,Mansour)
350 (Fischer)
300 (Fischer)
250 (Fischer)
+ +
0.9
0.8
0.7
-u+v+
0.6
0.5
0.4
0.3
0.2
0.1
0
1
∗∗∗∗∗∗∗∗∗∗∗
∗
211 (Fischer)
∗
∗∗
∗
∗
180 (Kim,Moin,Moser)
∗
∗
∗
∗+∗+
160 (Fischer)
+∗
∗+
∗+∗+
∗
150 (Lammers)
+
+∗
y+/Reτ
+
∗ +
∗
+
130 (Beronov,Ozyilmaz)
∗
∗+
114 (Fischer)
+
+
∗+
∗
98 (Fischer)
+
∗+
∗
+
+
106 (Beronov,Ozyilmaz)
∗
∗
∗+
∗+
+
∗
∗+
+
+
+
∗+
+
∗+
∗
88 (Fischer)
+
+
∗+
∗
∗
Mean momentum equation with 4 terms
∗
∗
∗+
+∗
∗
dU+/dy+=[ 1-(y+/Reτ) ]+u+v+
∗
∗
∗+
∗
∗+
+ + +++
++ +
++
+
++
+
+
+
+
+
+
+
+
+
-u+v+
0
0.2
0.4
0.6
+
0.8
y /Reτ
Figure 3.3: Reynolds shear stress profiles
49
1
Chapter 4
Set-up of the computations for grid
turbulence
The numerical treatment of any turbulent flow requires one to employ, above all, appropriate boundary conditions. In some cases, the boundary conditions correspond to the physical
problem perfectly, as in the case of plane channel flows, as discussed in the previous section. In these flows, the statistics obtained from the experiments are indeed independent of
the streamwise and spanwise directions in the fully developed region of the channel and the
flow is bounded by two parallel walls. Therefore, the application of periodic boundary conditions in the streamwise and spanwise directions and the non-slip velocity condition at the
walls correspond exactly to the underlying physical problem. In unbounded flows, i.e. free
turbulent flows, however, the numerical treatment is not that straightforward. The flow is
not confined by a certain boundary, and therefore, in order to simulate the whole flow field,
the computational domain must be over-dimensioned. One needs to consider some artificial
boundary conditions, which permit the use of a finite computational domain as an acceptable
approximation. Grid-generated turbulence belongs to this kind of turbulent flow. As far as
the set-up of the realistic simulations of this kind of turbulence is concerned, the question of
which boundary condition type should be used is of the upmost importance.
Only a few numerical studies dealing with the boundary condition aspects on free turbulent flows have been reported in the literature. In the majority of these studies, an inflow
velocity profile was imposed at the inlet of the computational domain and at the outlet either a zero gradient condition was applied or a convective boundary condition was imposed.
Boersma et al. (1997) carried out direct numerical simulations of spatially evolving turbulent
jet flows using inflow velocity conditions and convective outflow boundary conditions. The
authors commented that the application of the convective outflow boundary condition was
stable but not really very realistic, because at every point in the computational domain, the
same outflow velocity was implemented. Therefore, they did not take into consideration the
results obtained in the region close to the outflow.
50
One may consider applying a zero gradient condition as an outflow boundary instead of convective boundaries, expecting more realistic flow to emerge there. This was done by Djenidi
(2006) who simulated grid-generated turbulence. He compared convective and zero gradient
conditions, but he did not observe any significant change in the results.
There is yet another way to handle far-field boundary conditions when studying unbounded
turbulent flows. It is not as common as applying the above-mentioned types of inflow/outflow
boundary conditions, but is becoming more popular now, especially in the spectral method
community, where boundary conditions are traditionally an uneasy issue. The idea of this
method is the following. Since in practice it is never possible to avoid non-physical results
close to artificial boundaries, some neighborhood of the computational domain boundaries is
consciously sacrificed with regard to physical results. So far, there is no qualitative difference
from what should be properly done when the other types of far-field conditions are used.
However, this alternative approach then calls for the use of periodic boundary conditions in
a natural way, but they are physically correct only along so-called homogeneous spatial directions. The numerical advantages are used even along the inhomogeneous directions, at the
price of neglecting part of the computed results. This approach is known as the “fringe region”
method. By introducing a “sponge region” adjacent to the outflow domain, in which the turbulent quantities are damped, the governing equations are modified in such a way over that
region that the full numerical solution is forced to become spatially periodic. This idea was
first suggested by Spalart (1988), who examined the turbulent boundary layer using spectral
methods and also gave a justification for the new approach. It was shown to work well enough
in a number of subsequent studies, especially in turbulent boundary layers (Guo et al. (1994),
Colonius and Ran (2002), Schlatter et al. (2005)).
For the computations in the present thesis, it was decided to apply the latter method, because,
first, the “fringe region method” allows one to apply periodic boundary conditions. This was a
major advantage for the current study. Since, as outlined before, the code employed was tested
mainly for flows with periodic boundaries, such as channel flows, the new approach allowed
the code to be kept essentially unchanged and this was a great advantage. Also, convective
outflow boundary condition and zero-gradient boundary condition do not have any apparent
advantage over the “fringe region method”, as explained above.
The details of the implementation of the fringe region method employed along with the other
important parameters of the computations are described in the rest of this chapter. Some preliminary results obtained from the analysis of these computations were published in Özyılmaz
et al. (2008). The results shown in those studies are only partly related to the discussions
given in the thesis.
51
CHAPTER 4. SET-UP OF THE COMPUTATIONS FOR GRID TURBULENCE
4.1
Parameters of the simulations
As already pointed out in chapter 1, in this work some remaining issues regarding flows behind
grids were readdressed. These issues were investigated in terms of the effects of the porosity
(β) and mesh-based Reynolds number (ReM ). Mainly two areas were covered: (1) questions
regarding the inhomogeneity of GGT (chapter 6) and (2) effects of initial conditions on the
distribution of the time-averaged statistics. (chapter 7), totally seven direct numerical simulations were performed. The parameters used for these simulations are summarized in table
4.1. A porosity range of 53% − 82% was examined. Most of the computations were conducted
at the same mesh-based Reynolds number, ReM = 1400.
The porosity (β) and ReM are given as
β=
d
1−
M
2
;
ReM =
UM M
ν
(4.1)
where M is the mesh size of the grid and d is the thickness of the rods. The lower limit of
the β range was chosen in order to allow comparisons to be made with the available experimental data in the literature (Corrsin (1963), Ertunc (2007)). The value of β was modified by
changing only the thickness of the rods (d) while the same mesh size (M) was applied, which
was equal to 40 in lattice units. For all computations, square grid elements were used; their
shape can be inferred from figure 4.1, where instantaneous streamwise velocity isosurfaces
were visualized at the entrance of the computational mesh. Results shown in this figure were
obtained from case C in table 4.1.
Apart from one simulation, the same computational grid resolution was applied in all simulations, which had 2400 × 160 × 160 points in the streamwise, lateral and spanwise directions,
respectively. Thus, the downstream distance extended to x/M = 60. As will be discussed
later in detail, as far as the questions regarding the inhomogeneity were concerned, it was
necessary to see the effect of the spatial resolution in the lateral and spanwise directions. To
that end, one more computation was set up (case F in table 4.1). In this respect, it was also
of paramount importance to see the assess the effect of the asymmetry of the grid elements
employed on the inhomogeneity of the time-averaged mean velocity. It was therefore decided
to carry out another direct numerical simulation (case C*).
In order to elucidate the effect of the Reynolds number on the downstream distribution of
the time-averaged statistics, ReM was increased to 2100, using the grid with 72% porosity.
This run was represented by case E in table 4.1.
52
4.1. PARAMETERS OF THE SIMULATIONS
Figure 4.1: Instantaneous streamwise velocity component, showing the square rods used in the simulations.
In this the computation, a β = 72% porosity grid was introduced (case C).
Case
A
B
C,C*
D
E
F
Porosity % ReM
53
1400
64
1400
72
1400
82
1400
72
2100
72
1400
Computational domain
2400 × 160 × 160
2400 × 160 × 160
2400 × 160 × 160
2400 × 160 × 160
2400 × 160 × 160
2400 × 400 × 400
Table 4.1: Parameters of the simulations conducted in this work.
53
4.2
Boundary conditions
For the set-up of the computations, the way in which periodic boundary conditions were implemented was kept the same in all three directions. In simulations of grid turbulence (without
channel wall effects), applying such conditions is physically justified along the lateral directions, but not in the streamwise direction of the mean flow, along which all statistics are
spatially inhomogeneous. Along that direction, periodic boundaries are combined with the
fringe region method introduced above. According to that method, an additional region is
appended at the downstream end of the computational domain. The Navier-Stokes equations
are modified in this region in such a way that their solution is forced to be spatially periodic:
an artificial forcing term is added to the physical flow driving. This term acts to suppress flow
disturbances with respect to a prescribed in/outflow condition U see Schlatter et al. (2005).
It is designed to vanish outside the fringe region, allowing a natural flow. The force term
is formally denoted G(x, y, z, t) and it is specified below. The right-hand side of the Navier
Stokes equation is denoted N[U].
∂U
(x, y, z, t) = NS[U] + G(x, y, z, t)
∂t
G(x, y, z, t) = γ(x)(U(x, y, z, t) − um (x, t))
(4.2)
In short, velocity fluctuations are damped using a prescribed linear damping function, γ(x).
This fringe region is effective , γ 6= 0 only for xb < x < xb + L. The fringe region is thus a slab
orthogonal to the mean flow direction x. In the above equation, no other forcing is applied:
the flow can be driven by prescribing U(x, t) alone, which is effectively giving the mass flow
rate. If the mean flow is known, e.g., to be spatially evolving in the downstream direction
x, as in turbulent boundary layer, then U(x, t) would have a non-trivial x dependence with
appropriate inflow/outflow behavior. In the present work, however, this is not the case.
Instead of computing the deviation to this prescribed ideal mean flow U using the local velocity, in equation 4.2 the instantaneous streamwise velocity is represented by um (xi , t), an
instantaneous spatial average along the lateral directions. In this work, the following form of
the damping function was applied, and its spatial distribution is plotted in 4.2.
γ(x) =
1
(1 − cos(2π(x − xb )/L))
2
(4.3)
In all computations, the streamwise extent of the computational mesh had the same number
of points. Thus, the fringe always had the same length. It started from xb = 2000 and had a
length of L = 400 in lattice units. This range corresponds to 50 < x/M < 60. Therefore, in
54
4.2. BOUNDARY CONDITIONS
the remaining sections of the thesis, only the results up to x/M = 50 will be discussed.
When grids with different porosities were applied, it was necessary to apply different values of the pressure force in order to obtain the same bulk properties, i.e. to reach the same
ReM = 1400. The applied pressure forces for the computations carried out with different
porosities are shown in figure 4.2. As expected, for higher porosity the resistance is lower and
the necessary force to drive the flow decreases, if the mesh-based Reynolds number is fixed.
55
1
0.8
γ
0.6
0.4
0.2
0
0
0.2
0.4
∆x
0.6
0.8
1
Figure 4.2: The profile of the fringe region function used in the computations: γ = 12 (1 − cos(2πx/L)),
=
π
L sin(x);
for 0 < x < 2π. L = 400. ∆x was normalized by L.
0.005
(-dp/dx)/(ρUm 2 /2)
dγ
dx
0.004
0.003
0.002
0.001
0
0.5
0.55
0.6
0.65
0.7
porosity
0.75
0.8
Figure 4.3: Distributions of −dp/dx values as driving force for different porosities for ReM = 1400.
56
4.3. POST-PROCESSING
4.3
Post-processing
The results presented in the remaining parts of the thesis were all time-averaged values, which
were obtained as follows. First, a statistically steady state of the flow was achieved. Then,
mean velocity components were averaged in time. To that end, in every time interval of
“t = d/U m ”, one sample was taken, where d is the thickness of the rods and U m is the
streamwise velocity component averaged in time and in the lateral and spanwise directions.
Using this characteristic time, 10000 samples were collected. That means, for instance,that
for the simulation with β = 72% (d = 6 in lattice units), the time averaging was continued
for around 2 × 106 iterations. At first glance this number might seem to be unnecessarily
high. Indeed, the time and space-averaged statistics were converged much earlier, when they
were investigated with respect to the streamwise distance from the grid. However, for the
clarification of the inhomogeneity issue, longer time-averaging was required so that one could
rely on the somewhat surprising outcome of the study, as will be discussed in chapter 6. After
the time-averaging of the mean velocities, other turbulence statistics were averaged in order
to obtain yet another 10000 samples.
In GGT, it is customary to investigate the turbulent quantities with respect to the streamwise
distance, which is normalized by the mesh size, here x/M, averaging the time-averaged statistics also in the lateral and spanwise directions. Most of the results shown in the following
sections, especially in chapter 7, were also processed in this way. To that end, 95 different
streamwise positions were chosen, which define slices (squares in y, z planes, with periodic
boundaries) over which the spatial averaging takes places. The distribution of these positions
was not perfectly regular: closer to the grid, more samples were collected.
4.4
Spatial resolution
The present simulations were conducted initially based on the knowledge that the total num11/4
bers of the points required for a direct numerical simulation is proportional to ReL (Breuer
(2001), see chapter 1). Considering this relation, around 2.88 × 107 points were required to
√
be able carry out simulations at Reλ = 25 (Reλ ≈ ReL ). To be on the safe side, around
6.14 × 107 points were used. On the other hand, as discussed in the previous section, the
present simulations were set up so that the same mesh-based Reynolds number could be simulated, which resulted in higher Reλ values than 25 for some parameters (see section 4.5).
The grid resolution can be also checked based on the time-averaged quantities obtained from
the simulations. In a direct numerical simulation, one needs to account for energy at all length
scales. The largest are comparable to the domain size, the smallest is the Kolmogorov length
57
scale denoted here by lk , defined as
lk =
ν3
ǫ
1/4
(4.4)
The numerical mesh step size ∆ has to be around lk if all relevant scales are to be captured.
Since the lattice Boltzmann simulations carried out in this work were conducted in a uniform
grid, the grid size, ∆x = ∆y = ∆z = ∆, was always the same and was equal to unity. Therefore, a value of lk ≈ 1 would show that the results are perfectly resolved in that specific region.
From the outcome of the computations of the current work, reasonably well-resolved results
were observed, as shown by figure 4.4, which illustrates the value of the Kolmogorov length
scale for the computations with different porosities and different Reynolds numbers. The values
of lk were calculated using the time- and space-averaged dissipation rates. While interpreting
these results, one should keep in mind that the Kolmogorov length scale was derived based on
an order-of-magnitude consideration, hence to describe the spatial resolution of the smallest
scale, a prefactor might be needed (Fröhlich(2006)). That means, the condition ∆/lk = 1 does
not have to be held strictly. For all parameters, ∆/lk < 5, even for the nearest grid points.
Hence, the initial computation resolution was adequate enough.
2
lk
1.5
1
0.5
0
10
20
30
40
50
x/ M
Figure 4.4: Distributions of lk different porosities and ReM . Squares, case A; stars, case B; diamonds (open),
case C; triangle, case D; diamonds (closed), case E. The vertical lines point out the downstream distance where
lk reaches unity.
58
4.5. REYNOLDS NUMBER DISTRIBUTIONS
4.5
Reynolds number distributions
Figure 4.5 demonstrates the streamwise distributions of the Taylor microscale-based Reynolds
number, Reλ , for different runs. This number is defined as
20
Reλ =
3
k2
νǫ
1/2
(4.5)
In figure 4.5(a), the distributions for different porosities are compared and in figure 4.5(b),
the distributions for different ReM are compared.
The values shown in the figure were calculated using time- and space-averaged kinetic energy and dissipation rate values. Immediately after the grid, variation of Reλ in the range
20 − 50 was found. The Reynolds number distributions obtained showed a clear effect of the
value of β: although the flow had the same properties in the bulk (the same mean velocity was
obtained since the same ReM = 1400 was always simulated), for the same streamwise position,
the level of Reλ was lower if the porosity was higher. Also, if there was a small increase in the
mean property, this resulted in higher values of Reλ as expected.
4.6
Mach number
Another important parameter of the lattice Boltzmann simulations must be mentioned here:
the incompressibility condition is never exactly satisfied and a small, numerical compressibility
remains in the computational flow fields. Its level depends on parameters such as spatial and
temporal resolution of the LBM scheme. A corresponding Mach number “U/c” is defined,
which should be below 0.1 everywhere in the domain, in order to keep the compressibility
effects below the level of the physical flow fluctuations. Here, U is a local velocity, taken as
the pointwise value of the streamwise velocity fluctuations and c is the speed of sound and is
√
equal to 1/ 3 in lattice units. In the present study, this Mach number was about 0.06 for the
runs conducted at ReM = 1400 and 0.09 for those at ReM = 2100. Thus, incompressibility
was satisfied to a good approximation.
59
50
Reλ
30
20
15
10
5
10
x/ M
20
50
20
50
(a)
50
Reλ
30
20
15
10
5
10
x/ M
(b)
Figure 4.5: Distributions of Taylor microscale-based Reynolds numbers, Reλ , for a) different
porosities and b) different ReM values. Squares, case A; stars, case B; diamonds (open), case
C; triangle, case D; diamonds (closed), case E.
60
Chapter 5
Von Karman and Howarth analysis
In the previous chapter, the parameters of the direct numerical simulations conducted for the
present thesis were introduced. Before considering the outcome of these simulations in detail,
a preliminary analysis was performed to validate the results. As outlined in chapter 1, this
analysis was carried out by the application of the Von Karman and Howarth theory, which was
shown to be valid for isotropic turbulence. As will be discussed in more detail in chapter 7, grid
turbulence is slightly anisotropic in the vicinity of the grid but far donwnstream it becomes
isotropic. Hence the Von Karman and Howarth approach is applicable for the far field in grid
turbulence. In section 1.4.2, the principles of this approach were explained and its application
to the decay of turbulence intensity was described: the experiments in the literature showed
that the decay of turbulence behind the grid obeyed equation 1.11 with varying values of the
coefficient A (1.95 < A < 2.2), with lower values for higher Reynolds numbers. The results of
the simulations were used to calculate the distribution of the turbulent intensity at different
porosities at the same mesh Reynolds number (cases A, B, C and D) in order to see if the
numerically found A values agree with those observed experimentally. The outcome of this
study is given in figure 5.1 for x/M ≥ 30. As can be seen, the value of A depends strongly on
the value of the porosity: with decreasing porosity, A increases. On the other hand, the values
obtained are well within the range given in the literature, considering the Reynolds number
range, which is lower than that usually applied in the literature (for details, see section 1.2).
Furthermore, the results of the simulations were processed in order to obtain the distributions of the correlation functions, f (x) and g(x), which were normalized by the streamwise
(Lf ) and lateral integral length scales (Lg ), respectively. The results obtained through this
analysis are shown in figures 5.2(a) and 5.3(a) for f (x) and g(x) calculated at x/M = 30 and
in figure 5.4 for f (x) and g(x) calculated at different streamwise positions (x/M = 20, 30, 40).
First, the results comfirmed the expected distributions which could be obtained through the
Von Karman and Howarth theory (chapter 1); having the strongest correlation at the smallest downstream distance, the correlation decreases with increasing streamwise distance and
61
CHAPTER 5. VON KARMAN AND HOWARTH ANALYSIS
it then reaches almost zero. The results showed that increasing the grid porosity increased
the correlations between both streamwise and lateral velocity fluctuations (figures 5.2(a) and
5.3(a)). According to the present calculations, the distributions of the correlation functions
do not depend strongly on the streamwise positions where they are calculated (figure 5.4).
It should be recalled that computational studies carried out on grid turbulence are bounded
by the computational resources. That means, due to the computational power limitations,
that it is not possible to have a long streamwise extent as applied in the experimental studies.
As explained in the previous chapter, the streamwise extent of the present calculations is up
to x/M = 50. In addition to this, due to the same restrictions, there is a Reynolds number
limit. Hence there exists no experimental data obtained under exactly the same numerical
conditions applied in the current study.
On the other hand, the experimental dataset of Ertunc (2007) includes low Reynolds number
data; therefore, his results are used for the comparison in the following. The lowest Reynolds
number investigated in the experimental study of Ertunc (2007) was ReM = 1584 using a
porosity of β = 64% and his experimental set-up extended up to x/M = 160. The distribution functions obtained from that study at ReM = 1584 and x/M = 80 are compared in
figure 5.2(a). As mentioned, the experimental study was carried out over a wider distance,
hence it was possible to calculate the correlations over a wider distance (figure 5.2(b)). The
experimental study shows a stronger dependence on the streamwise position where the correlations were calculated. Especially between the correlations obtained at x/M = 30 (red
solid line) and the correlations calculated at far downstream (blue solid line), there is a clear
difference in the correlation magnitude. In figure 5.3(b), the same procedure was applied for
the lateral correlation function and exactly the same conlusions can be drawn: there exists a
difference in the magnitudes of the correlation functions at the same streamwise position calculated from different kinds of studies. On the other hand, taking into account the differences
in the parameters used in both studies, the overall agreement is acceptable.
62
A=2.10
U/u’
60
*
*
*
*
*
*
*
*
*
*
*
A=2.5
*
40
*
*
30
35
40
45
50
x/M
Figure 5.1: Distribution of the turbulence intensity at different porosities. The values of the coefficient A
used in equation 1.11 are shown. Squares, case A (53%); stars, case B (64%); diamonds, case C (72%); and
triangles, case D (82%).
63
1
increasing porosity
0.8
f(x)
0.6
0.4
0.2
0
0
2
4
6
8
10
6
8
10
x/Lf
(a)
1
0.8
x/M=100
f(x)
0.6
0.4
0.2
x/M=30
0
0
2
4
x/Lf
(b)
Figure 5.2: The distribution of the correlation function, f (x), with respect to the streamwise
distance normalized with the integral length scale Lf : a) correlation function, f (x), obtained
from the simulations (symbols as in figure 5.1, calculated at x/M = 30, ReM = 1400) compared
with those of the experiments (solid red line, calculated at x/M = 80, ReM = 1584; b) correlation function, f (x), obtained from the experiments of Ertunc (2007) calculated at different
streamwise distances compared with the distribution obtained from case A at x/M = 30.
64
1
0.8
increasing porosity
g(x)
0.6
0.4
0.2
0
0
2
4
6
8
10
6
8
10
X/Lg
(a)
1
0.8
x/M=100
g(x)
0.6
0.4
x/M=30
0.2
0
0
2
4
X/Lg
(b)
Figure 5.3: The distribution of the correlation function, g(x), with respect to the streamwise
distance normalized with the integral length scale Lg : a) correlation function, g(x), obtained
from the simulations (symbols as in figure 5.1, calculated at x/M = 30, ReM = 1400) compared
with those of the experiments (solid red line, calculated at x/M = 80, ReM = 1584); b) correlation function, g(x), obtained from the experiments of Ertunc (2007) calculated at different
streamwise distances compared with the distribution obtained from case A at x/M = 30.
65
1
0.8
f(x)
0.6
0.4
0.2
0
0
2
4
6
8
10
6
8
10
x/Lf
(a)
1
0.8
g(x)
0.6
0.4
0.2
0
0
2
4
X/Lg
(b)
Figure 5.4: The distributions of the correlation functions, f (x) and g(x), calculated for case
A at different streamwise positions, x/M = 20, 30, 40: a) distribution of f (x); b) distribution
of g(x).
66
Chapter 6
Inhomogeneity of grid-generated
turbulence
As outlined in section 1.4.1, Ertunc (2007) carried out detailed experimental work on gridgenerated turbulence and focused on an aspect of the subject, that is rarely studied: the
inhomogeneity of GGT. The main outcome of this study was that the turbulent quantities
showed highly inhomogeneous fields for a porosity of 64%, which was above the recommended
value for the assurance of the homogeneity. The Reynolds number range of his experiments
was 3200 < ReM < 8000 and the results did not change substantially with increasing Reynolds
number. Although not as high as Reynolds stresses, inhomogeneities were also observed for
the streamwise mean velocity component.
The main objectives of the study in the present chapter were, above all, to see whether the
experimentally observed inhomogeneties could be confirmed and to see whether the porosity
of the grids has any influence on the level of the inhomogeneity. In order to establish if there
is somehow a dependence on the porosity in terms of inhomogeneity, a wide range of porosity
was applied: β = 53%, 64%, 72% and 82%. These simulations were introduced as cases A, B,
C and D respectively, in chapter 4. As seen from table 4.1, the computational domain had
the same number of grids in these computations. For the purpose of the current discussion,
the effect of the transverse and spanwise extents of the computational domain was important;
therefore, a further simulation was conducted with an approximately six times larger grid (case
F). For further discussions, another simulation with a modified grid geometry was carried out
(case C*).
The details and the outcome of these simulations are discussed in the remaining sections
of this chapter. Results were processed as follows. Once the turbulence field was developed,
the mean velocity statistics were averaged in time. After obtaining approximately 15000 (tend )
independent time-averaged samples for the mean velocity components, samples for Reynolds
stress components were collected, using the mean velocity fields corresponding to tend . Sam67
CHAPTER 6. INHOMOGENEITY OF GRID-GENERATED TURBULENCE
pling of the time-averaged data also continued until 15000 samples for the Reynolds stresses
had been collected. An intermeadite post-processing was carried out after 10000 (tmid ) samples. As far as the discussion related to this work is concerned, no important differences were
observed between the statistics of tmid and tend . On the other hand, the results discussed in
the following sections correspond to tend .
The figures showing the contours of the inhomogeneties were prepared as follows: a fixed
spanwise position was selected and the value of the inhomogeneity was calculated according
to equation 1.7 to give the distribution of the inhomogeneity with respect to distances in the
streamwise and transverse directions. The procedure was repeated, changing the spanwise
positions. However, the qualitative conclusion was the same for every slice in the spanwise
direction. Hence, in the following sections, only one dataset is shown for the whole discussion,
whose spanwise position lies in the middle of the computational domain.
6.1
Mean velocity field
Figure 6.1 shows the inhomogeneties of time-averaged streamwise mean velocity component
for cases A, B, C and D. The fields shown in red correspond to values higher than the average value of the velocity component and the blue fields correspond to values lower than the
value of the average velocity. The selected contour levels correspond exactly to those of the
experiments discussed in section 1.4.1: a level of ±2% was selected. The results given here
show clearly that only very close to the grid (x/M ≤ 10), when the fluid flows through the
obstacles, are high inhomogenities observable. On the other hand, with increasing streamwise
distance, a decrease in the level of inhomogeneity is observable. For x/M > 30, no trace of
inhomogeneity is left. It is also remarkable that the inhomogeneity fields indicated in figure
6.1 do not show any clear trend in terms of the porosity, contrary to the expectation based
on the literature survey. The data in the far field show a perfectly homogeneous mean velocity distribution, even for the lowest porosity (β = 53%). On the other hand, these results
are in agreement with Corrsin’s study (1963) in the sense that the streamwise mean velocity
component is homogeneous in grid turbulence. This expectation is confirmed here, even for a
Reynolds number as low as ReM = 1400.
However, as discussed in the previous section, although not too high, the measurements of
Ertunc (2007) revealed some regions of inhomogeneous mean velocity fields (figure 1.1), also
in the far field, especially for 20 < x/M <60. This important difference between the experimental and computational results has yet to be understood.
A probable failure source which might come from the numerical side is the insufficient number of the rods applied in the computational domain. In order to elucidate this influence,
68
6.1. MEAN VELOCITY FIELD
another simulation was carried out, which had 2.5 times more points, in both the transverse
and spanwise directions, resulting in an approximately six-fold increase in the overall computational size. In the computations discussed so far, four rods were employed. In order to see
if the number of rods had any effect on the homogenization, the thickness of the rods and
the mesh size were kept the same. Instead of using four rods, 10 rods were used and one of
the previous calculations (case C with β = 72%) was repeated with a computational resolution of 2400 × 400 × 400. The inhomogeneity field resulting from this simulation (case C*) is
demonstrated in figure 6.2. This result shows very clearly that the previously applied number
of rods was sufficient to examine the level of inhomogeneity of the mean velocity. Obviously,
high inhomogeneity is observable only in the vicinity of the rods, exactly as in the case of
the simulations conducted with a smaller mesh resolution. There is no difference between the
four- and ten-rod arrangements.
Thus, in contrast to the experimental study by Ertunc (2007), the numerical study discussed
so far shows that the streamwise mean velocity component in grid turbulence is homogeneous.
There is, however, another factor apart from the mesh resolution which might cause a difference between an experimental and a numerical analysis in terms of the inhomogeneity of the
mean velocity: a possible difference in the geometry of the grids applied in each study. Since
the porosity was defined in the same way and both studies employed square rods, the only
difference in this sense might be a possible imperfection of the grids used in the experiments.
In order to check this, another simulation was carried out, details of which are given below.
The rods employed in the numerical simulations were perfectly symmetrical in all directions.
In order to check the above reasoning, slightly modified rods were applied. Since the extension
of the transverse and spanwise directions of the computational domain was sufficient, the new
computation was carried out with the smaller mesh resolution which had a four-rod arrangement and for β = 72% (case F). In this new simulation, the two rods in the upper half of
the computational domain were kept exactly the same as the rods employed in the previous
computations. In order to introduce a kind of asymmetry to the geometry, the dimensions
of the lower rods were modified. The number of points in both the transverse and spanwise
directions was increased by 1 in lattice units for the lowest rod. Similarly, by decreasing the
number of points by 1 in lattice units in both the vertical and spanwise directions, the other
rod became narrower in these two directions. The grid was positioned in the streamwise direction in the same location as before.
The symmetrical rods applied in the previous simulations (cases A, B, C, D and C*) had
a thickness of 6 in lattice units. Hence modification of its width by 1 makes a difference of
around 17% in each direction. At first glance this modification might seem to be unrealistically
high. However, the main aim of this simulation was not to be able to reproduce the results
69
observed by the experiments but to be able to show the trend in a very clear way which could
be caused by possibly asymmetric grids.
A very clear observation is made through this computation, the results of which are given
by figure 6.3. The lower part of the computational domain, where the asymmetric rods were
employed now demonstrates a highly inhomogeneous mean velocity field. Figure 6.3 provides
convincing proof of the cause of the inhomogeneous mean velocity distribution which can be
observed in an experimental study, when asymmetric rods are employed.
The discussion up to now allows one to draw the fair conclusion that the streamwise mean
velocity component is homogeneous in grid-generated turbulence, even for a Reynolds number
as low as ReM ≈ 1400, provided that one is sure about using perfectly symmetric geometries.
Regardless of the value of the porosity, the streamwise mean velocity is homogeneous; it is
homogeneous even for a porosity as low as 53%.
The factor of paramount importance in the experiments mentioned above was, however, not
the inhomogeneity of the mean velocity but the high inhomogeneity level of Reynolds stresses
and their anisotropy. The next section deals with this point.
70
6.1. MEAN VELOCITY FIELD
(a)
(b)
(c)
(d)
Figure 6.1: Inhomogeneity of streamwise mean velocity component, U for: a) case A, b) case
B, c) case C, d) case D.
71
Figure 6.2: Inhomogeneity of streamwise mean velocity component, U : case F
Figure 6.3: Inhomogeneity of streamwise mean velocity component, U : case C*
72
6.2. REYNOLDS STRESS FIELDS
6.2
Reynolds stress fields
The inhomogeneities of the diagonal Reynolds stress component, u1 u1 , for different porosities
are shown in figure 6.4. As before, red regions correspond to higher values than the mean of
the Reynolds stress component and blue regions correspond to values lower than the mean of
the Reynolds stress component. Clearly seen in figure 6.4 are regions of large deviations from
the mean value, which extend in the whole domain. Note that the number of time averaged
data is the same as the mean velocity component discussed in the previous section. Therefore, it is astonishing to observe that, after such a long time averaging, the Reynolds stress
component remains inhomogeneous, even in the far-field region. There is no decrease in the
level of the inhomogeneity with increasing downstream distance. It is also worth emphasizing
that the same observations can be made for all porosities investigated: increasing the porosity
of the grid does not make the Reynolds stress component more homogeneous, if the Reynolds
number remains constant.
Inhomogeneities of the other diagonal Reynolds stress tensor component, u2u2 , and the offdiagonal component, u1 u2 , are shown in figures 6.5 and in 6.6 respectively. Here also, high
positive and negative deviations from the mean values are observable throughout the whole
domain. A comparison between figures 6.4, 6.5 and 6.6 indicates that the degree of inhomogeneity is of comparable order for u1u1 and u2 u2 . On the other hand, the level of the
off-diagonal component of the Reynolds stress tensor is much higher than the inhomogeneities
of the diagonal components. These qualitative observations are in agreement with the experiments of Ertunc (2007) .
The inhomogeneity of the a11 anisotropy component of the Reynolds stress tensor is shown in
figure 6.7. This component is also strongly inhomogeneous.
Although, there may be quantitative deviations from the experimental study presented above,
the overall trend is also confirmed by the numerics: Reynolds stress components are strongly
inhomogeneous and the level of u1 u2 inhomogeneity is stronger than the other two components.
DNS for different porosities showed that there is no trend towards increasing homogeneity either with increasing x1 /M or with increasing porosity at fixed mesh size, M.
The results of the numerical analysis so far furnish a strongly inhomogeneous Reynolds stress
tensor and homogeneous streamwise mean velocity. On the other hand, no dependence on the
porosity of the grid was observed in terms of the inhomogeneity.
73
(a)
(b)
(c)
(d)
Figure 6.4: Inhomogeneity of Reynolds stress component, u1 u1 : a) case A, b) case B, c) case
C, d) case D.
74
(a)
(b)
(c)
(d)
C, d) case D.
75
(a)
(b)
(c)
(d)
C, d) case D.
76
(a)
(b)
(c)
(d)
Figure 6.7: Inhomogeneity of anisotropy component, a11 : a) case A, b) case B, c) case C, d)
case D.
77
6.3
Reasoning of the inhomogeneity of the Reynolds
stress components
The previous analysis of the numerical simulations showed that the mean streamwise velocity
component can be made inhomogeneous by applying asymmetric grid geometries. In addition,
experiments by Ertunc (2007) showed that the inhomogeneity of the mean velocity component
has a tendency to decrease with increasing downstream distance from the grid. The same does
not hold, however, for the Reynolds stress component: both the mentioned experiments and
the current numerical simulations indicated that the strong inhomogeneity of the stresses extends in the whole streamwise domain. That means that, despite the fact that a mean velocity
field which is inhomogeneous in the vicinity of the grid becomes homogeneous in the far-field
region, the Reynolds stress components persist with their inhomogeneity even in the region
far downstream.
The reason why the Reynolds stress components remain inhomogeneous once the mean velocity component becomes homogeneous can be explained by an analysis of the terms of the
kinetic energy equation. The turbulence kinetic energy is described by the following equation:
∂k
∂k
∂Ui
∂
+ Uk
= − ui uk
−
∂t
∂xk
∂x
∂x
|
| {z k} | k
{z
}
(I)
(II)
∂ui ∂ui
p
∂2k
+ k uk − ν
+ν
ρ
∂xk ∂xk
∂xk ∂xk
} | {z
}
{z
} | {z
(III)
(IV )
(6.1)
(V )
The physical meaning of this equation is that the sum of the change in the kinetic energy of
turbulence per unit mass and time and the convective transport by the mean motion (I) must
be equal to the sum of the production of turbulence by the mean velocity gradients (II), the
transport of the total turbulence mechanical energy by turbulent fluctuations (III), the major
part of the total viscous dissipation of turbulent energy (IV) and the viscous diffusion (V).
For the kind of flows under consideration here, the time derivative on the left-hand side can
be neglected (ergodic stationary turbulence field).
Since dissipation can be accepted to be a local phenomenon, according to equation 6.1, the
only remaining terms responsible for the homogenization are the turbulent transport terms
and the viscous diffusion terms. That means that, if mean velocity gradients exist, i.e. term
II is not equal to zero then, terms (III) and (V) become active and the flow field becomes
homogeneous. The process of homogenization can be understood in a clearer way if one considers the non-dimensional form of equation 6.1, which is defined as
78
6.3. REASONING OF THE INHOMOGENEITY OF THE REYNOLDS STRESS
COMPONENTS
∗
∗ ∂k
Uk ∗
∂xk
∂ p∗
q
+ k ∗ u∗k
=
−
U 1 ∂x∗k ρ∗
2 ∗
∗
∗
Lg
q
1
1
∗ ∂ui ∂ui
∗ ∂ k
ν
ν
−
+
Reλ
λg
∂x∗k ∂x∗k
ReL
∂x∗k ∂x∗k
U1
∗
∗ ∂U i
−ui uk
∂x∗k
(6.2)
where
q 2 = 2k
(6.3a)
ReL = U 1 Lg /ν
(6.3b)
Reλ = qλg /ν
(6.3c)
and Lg and λg are integral length scale and Taylor’s micro-scale of turbulence, respectively.
Equation 6.2 shows that the contribution of the turbulent transport term to the homogenization is directly proportional to the turbulent intensity. Since the magnitude of the turbulent
intensity decreases very rapidly downstream from the grid (see chapter 5), the contribution of
the turbulent transport term to the homogenization decreases with increasing distance from
the grid.
According to equation 6.2, the contribution of the dissipation term is directly proportional
to the turbulent intensity and to the ratio Lg /λg . It is inversely proportional to Reλ . In
grid turbulence, Reλ decays with increasing streamwise distance. On the other hand, Ertunc
(2007) showed that the ratio Lg /λg remains constant with increasing streamwise distance.
Therefore, the dissipation term is not expected to decay as fast as the turbulent transport
term.
Equation 6.2 shows also that the contribution of diffusion term is inversely proportional to
the Reynolds number (ReL ). This term can have positive or negative values based on the
distribution of the spatial gradient of kinetic energy. However, its amplitude should decrease
with increasing streamwise distance, because Lg increases dowstream. On the other hand,
the decay of this term is expected to be the slowest among the three terms on the right-hand
side of equation (6.2), because it is not scaled with the turbulence intensity. This expectation
leads one to conclude that far downstream of the grid, the only remaining term which can be
responsible to homogenize the flow must be the diffusion term.
In order to check if the above expectations are reasonable, one of the simulations is chosen (case A) and the terms of the kinetic energy equation are examined based on the results
of this simulation. Figure 6.8(a) shows the kinetic energy field for 0 < x/M < 50, which
79
is calculated in the middle of the spanwise direction as before. The lines shown on the figures are selected as follows: line 1 (Y /M = 3.05) crosses the middle of the open area, line 3
(Y /M = 3.55) crosses the middle of the solid rod, and line 2 (Y /M = 3.3) is centered between
lines 1 and 3. That means that, line 1 is representative of a jet-like flow, line 3 is representative
of a wake-like flow and similarly line 2 is representative of flow in a shear layer. In figure 6.9,
the development of the longitudinal and transverse mean velocities and the turbulent kinetic
energy are plotted. Comparison of the mean velocity curves and the turbulent kinetic energy
curves in the vicinity of the grid shows clearly the generation of turbulence by a static grid: the
mean velocity gradient between the accelerated jet-like flow and back-flow at the wake of the
obstacle increases the production term (II). As a result, the turbulent kinetic energy reaches
its maximum at the wake of the rod (line 3) and has the lowest value always downstream of the
open area (line 1). Nevertheless, the gradients of streamwise and transverse mean velocities
decay in a very short distance. Thus for x/M > 4, turbulence is not produced any longer but
it is only transported, diffused and dissipated.
The terms on the right-hand side of the kinetic energy equation are shown in figure 6.10
using the three lines introduced above. For all lines, the production term is only active for
x/M < 4. The turbulent transport term is negligible compared with other terms over the complete domain. The dissipation term shows a gradual increase up to x/M ≈ 3 and a gradual
decrease in the whole donwstream region. The viscous diffusion fluctuates around zero up to
x/M ≈ 3. The amplitude of viscous diffusion increases up to x/M ≈ 2 and decays gradually
in its negative amplitude. In general, the peak locations of the dissipation peak are the last.
For 5 < x/M < 10, dissipation and viscous diffusion are almost at the same level. At the far
downstream region (x/M > 20), viscous diffusion becomes much higher than the dissipation,
except for line1, but with a very low amplitude.
As can be seen, the above considerations are satisfied by the numerical analysis: the diffusion term decays the slowest. On the other hand, the magnitude of the diffusion term in the
far field is very low. Further, because of the low levels of turbulent kinetic energy and high
Reynolds number of the bulk flow, the diffusion time-scale becomes much smaller than the
convective flow. In other words, for complete homogenization via diffusion, the time required
is longer than what one can observe in the laboratory. Hence the reason for inhomogeneous
turbulent field in the far-downstream region can be expressed as follows: fast homogenization
of the mean velocity field causes a rapid decrease in production and, consequently, the dissipation and viscous diffusion processes, so that the generated inhomogeneity of turbulent field
does not have the means for homogenization rather than complete dissipation.
80
COMPONENTS
(a)
(b)
(c)
Figure 6.8: Kinetic energy field obtained from simulation case A: a) kinetic energy field for
0 < x/M < 50, b) kinetic energy field for x/M < 10, c) inhomogeneity of kinetic energy.
81
2.5
line1
line2
line3
U/U m
2
1.5
1
0.5
0
10
0
10
1
x/M
(a)
0.15
line1
line2
line3
V/U m
0.1
0.05
0
10
0
10
1
x/M
(b)
0.6
line1
line2
line3
2
k/U m
0.4
0.2
0
5
10
15
20 25 3035
x/M
(c)
Figure 6.9: Distributions along lines 1, 2 and 3: a) streamwise mean velocity component, b)
transverse mean velocity component, c) turbulent kinetic energy.
82
COMPONENTS
0
production
transport
diffusion
dissipation
0.001
-5E-07
0.0005
-1E-06
0
-1.5E-06
production
transport
diffusion
dissipation
-0.0005
10
0
10
-2E-06
20
1
25
30
x/M
35
40
45
50
x/M
(a)
(b)
0
production
transport
diffusion
dissipation
0.002
-2E-06
0.0015
0.001
-4E-06
0.0005
0
-6E-06
production
transport
diffusion
dissipation
-0.0005
-0.001
10
0
10
-8E-06
20
1
25
30
x/M
35
40
45
50
x/M
(c)
(d)
0
production
transport
diffusion
dissipation
-5E-06
0.004
-1E-05
0.002
-1.5E-05
0
-2E-05
-0.002
10
0
10
-2.5E-05
20
1
x/M
production
transport
diffusion
dissipation
25
30
35
40
45
50
x/M
(e)
(f)
Figure 6.10: Distributions of the terms of the kinetic energy equation along a) line 1, for
0.5 < x/M < 50, b) line 1, for 20 < x/M < 50, c) line 2, for 0.5 < x/M < 50, d) line 2, for
20 < x/M < 50, e) line 3, for 0.5 < x/M < 50, e) line 3, for 20 < x/M < 50. The quantities
4
were normalized by Um
/ν.
83
Chapter 7
Effects of initial conditions
The self-preservation and similarity considerations play a central role both in the study of
the physics of turbulence and, as discussed in text books on turbulence, in the development
of turbulence models. The assumption behind the self-similarity idea of turbulence is that
at high Reynolds numbers, small-scale turbulence reaches a local equilibrium, where the turbulent statistics such as kinetic energy are independent of the way in which turbulence is
produced, which means that they are independent of the initial conditions. However, in real
life situations, Reynolds numbers are always finite, therefore the fluid flow is far from the local
equilibrium. Hence, especially in the case of flows at low-to-moderate Reynolds numbers, a
strong influence of the initial conditions in time and in space may be observed.
In most theoretical studies devoted to similarity and self-preservation of turbulence, isotropy
and homogeneity assumptions are made. On the other hand, grid turbulence is only nearly
isotropic and, as discussed in detail in the previous chapter, the Reynolds stress tensor is
strongly inhomogeneous. It is generally expected that the turbulent kinetic energy obeys a
power law decay, but it has long been known that the exponents and coefficients of the power
law decay can only be given in a range and also the power law decay of kinetic energy spectrum is not self-preserving (Gence (1983)). Taking into account the Reynolds number range
of the present numerical simulations (ReM = 1400 − 2100), it is expected that a considerable
dependence of turbulence quantities on the initial conditions will be observed. In this chapter,
the time-averaged turbulent quantities are examined with respect to their distributions in the
streamwise distance from the grid to analyze this point. Two kinds of initial conditions are
used in this analysis and their parameters are given in chapter 4: Reynolds number and grid
porosity.
In section 7.1, components of the Reynolds stress tensor and dissipation tensor are discussed,
i.e. constants and exponents of the power law decay of turbulent kinetic energy and dissipation
rate are given as obtained from the present numerical predictions. How the initial conditions
affect the Reynolds stress and dissipation tensors’ anisotropies is treated in section 7.2.
84
7.1. DECAY OF TURBULENT KINETIC ENERGY AND DISSIPATION RATE
7.1
Decay of turbulent kinetic energy and dissipation
rate
7.1.1
Effects of porosity
Streamwise distributions of the diagonal components of the Reynolds stress tensor obtained
from the numerical simulations of turbulence generated through different grid porosities at
ReM = 1400 are shown in figure 7.1. The profiles are normalized by the square of the bulk
2
velocity (Um ). It is recalled that these simulations correspond to cases A to D introduced in
table 4.1, whose porosities are equal to β = 53%, 64%, 72% and 82% respectively.
The diffferent grid porosities have important effects on the magnitude of the Reynolds stress
components, especially in the near-grid region. The influence of increasing porosity is to decrease the magnitude of the Reynolds stress components, when they are normalized by the
bulk velocity. Although different in magnitude, Reynolds stresses follow similar distributions
at different porosities. On the other hand, three distict decay behaviors are observed in figure 7.1: one in the vicinity of the grid (x/M < 7 − 8), one in the far field, where the decay of
Reynolds stresses shows a power law dependence (8 < x/M < 12), and a region where again
a power law decay is observed but with a different component (x/M > 12).
Before discussing the power law decay of turbulent kinetic energy, a pecularity of grid turbulence must be recalled at this position, that is, the axisymmetry of the time-averaged moments.
This condition requires equality of the Reynolds stress components v 2 and w 2 . In addition
to this restriction, the off-diagonal components must be negligible with respect to diagonal
components of the Reynolds stress tensor. To check if the current simulations satisfy this condition, v 2 and w 2 profiles for case A (β = 53%) are compared in figure 7.2(a). As can be clearly
seen, the difference between two components is negligible. The off-diagonal Reynolds stress
component uv is compared with the u2 component again for the same case in figure 7.2(b). The
Reynolds stress component uv is around 100 times smaller than u2 . The other two off-diagonal
components of Reynolds stress tensor (vw, uw) are even smaller, down to 10−6 − 10−7 . They
are not shown in order not to disturb the clearity of figure 7.2(b). Furthermore, figure 7.2
shows only one case, but results from the remaining simulations keep the same trend. Because
of the axisymmetry, the discussion is limited to u2 and v 2 from here on.
85
CHAPTER 7. EFFECTS OF INITIAL CONDITIONS
In order to exclude the effects of the geometry of the grid, it is suggested that another
scaling should be applied besides bulk velocity, which includes the influence of the porosity.
It is of the form
RN (n) =
2
Um
(1 − Md )n
(7.1)
Figure 7.3 makes a comparison between the distributions of the Reynolds stress component
2
u2 normalized by Um
and normalized by RN (n), where n has the value of 2.60, for the region
x/M > 7. Much better collapse is obtained for the distributions at different porosities when
RN (n) scaling is applied. Similarly, figure 7.4 compares the Reynolds stress component v 2 ,
normalized in different ways for x/M > 7. These figures show that by using a scaling of the
form of RN (n), it is possible to obtain better agreement of the Reynolds stress distributions
for turbulence produced by different grid porosities. Thus, RN (n) scaling is preferred also for
the discussion of the coefficient and exponent of the decay law for turbulent kinetic energy.
Turbulent kinetic energy decay is shown in figure 7.5 for x/M > 12. The profiles in fig2
ure 7.5(a) are normalized by Um
and those in figure 7.5(b) are normalized by RN (n). The
solid lines in figure 7.5(b) are of the form
k = CK ∗ (x/M)ExpK
(7.2)
Two power law equations are fitted to the data in Figure 7.5, which show that the value of
the power law coefficient CK is ∼ 0.15 at ReM = 1400 for different porosities. The power
law exponent ExpK is more sensitive to the value of the porosity and in the investigated
parameter range it is between 1.62 and ExpK = 1.66.
86
10
-1
u2/U2m
10-2
10-3
10
-4
10
20
30
40
50
20
30
40
50
20
30
40
50
x/M
(a)
10
-1
v2/U2m
10-2
10-3
10
-4
10
x/M
(b)
10
-1
w2/U2m
10-2
10-3
10
-4
10
x/M
(c)
2
Figure 7.1: The diagonal components of the Reynolds stress tensor normalized by Um
for
different values of β. Squares, case A (β = 53%); circles, case B (β = 64%); diamonds, case
C (β = 72%); triangles, case D (β = 82%): a) distribution of u2 , b) distribution of v 2 , c)
distribution of w 2 .
87
-1
(v2/,w2)/U 2m
10
10-2
10-3
10
-4
10
20
30
40 50
x/M
(a)
10-3
10
2
2
u /U m
-4
2
(uv)/U m
10-5
20
30
40
50
x/M
(b)
Figure 7.2: a) Comparison between v 2 and w 2 (case A). b) Comparison between u2 and uv
(case A).
88
u2/U 2m
10-2
10
-3
10
-4
10
20
30
40
50
30
40
50
x/M
(a)
(1-d/M)2.6u2/U 2m
10-2
10-3
10-4
10
20
x/M
(b)
Figure 7.3: Distribution of u2 for x/M > 7. Squares, case A (β = 53%); circles, case B
(β = 64%); diamonds, case C (β = 72%); triangles, case D (β = 82%). Quantities are
2
normalized in two ways: a) by UM
, b) by RN (2.60).
89
v2/U 2m
10-2
10
-3
10
-4
10
20
30
40
50
30
40
50
x/M
(a)
(1-d/M)2.6v2/U 2m
10-2
10-3
10-4
10
20
x/M
(b)
Figure 7.4: Distribution of v 2 for x/M > 7. Squares, case A (β = 53%); circles, case B
2
, b) by RN (2.60).
90
k/U 2m
10-2
10
-3
10-4
20
30
40
50
40
50
x/M
(a)
0.005
0.004
0.003
(1-d/M)2.6kU 2m
0.002
0.001
k=0.15 (x/M)1.62
k=(0.148 x/M)1.66
20
30
x/M
(b)
Figure 7.5: Distribution of turbulent kinetic energy for x/M > 12. Squares, case A (β = 53%);
circles, case B (β = 64%); diamonds, case C (β = 72%); triangles, case D (β = 82%).
2
2
Quantities are normalized in two ways: a) by UM
, b) by UM
/(1 − Md )2.60 .
91
In the following, a similar procedure is followed for the decay of the turbulent dissipation
4
tensor, whose diagonal components ǫ11 , ǫ22 and ǫ33 normalized by the bulk properties (Um
/ν)
are shown in Figure 7.6. Here also, there is a strong dependence of the profiles of the dissipation components on the porosity, especially in the near-grid region. This dependence reduces
for x/M > 7, resulting in a power law decay for all porosities; however, the magnitude of the
dissipation tensor still depends on the porosity. Hence, similarly to Reynolds stress normalization for the far field, the following scaling is introduced for the dissipation tensor:
ǫN (n) =
4
Um
ν(1 − Md )n
(7.3)
The dissipation tensor components normalized by ǫN (n) are given in figures 7.7 and 7.8.
The results show that using the same exponent n = 2.6 as Reynolds stresses, it is possible
to reach a very good data collapse for x/M > 12 at different porosities. Therefore, the same
normalization is applied to obtain the power law equation describing the decay of turbulent
dissipation rate (figure 7.9). The results show that once the scaling ǫN (2.60) is used, a universal behavior of the dissipation rate decay can be obtained for a broad range of porosities
at the same mesh-based Reynolds number:
ǫ = CD ∗ (x/M)ExpD
,with the power law coefficient CD ≈ 0.0016 and power law exponent of ExpD ≈ 2.8.
92
(7.4)
-4
10
-5
νε11/U4m
10
10-6
10-7
10
-8
10
20
30
40
50
20
30
40
50
20
30
40
50
x/M
(a)
-4
10
-5
νε22/U4m
10
10-6
10
-7
10
-8
10
x/M
(b)
-4
10
-5
νε33/U4m
10
10-6
10-7
10
-8
10
x/M
(c)
4
Figure 7.6: The diagonal components of dissipation tensor normalized by Um
/ν for different
values of β, squares, case A (β = 53%); circles, case B (β = 64%); diamonds, case C (β = 72%)
triangles, case D (β = 82%): a) distribution of ǫ11 , b) distribution of ǫ22 , c) distribution of ǫ33 .
93
10
-5
νε11/U 4m
10-6
10-7
10
-8
10
20
30
40
50
30
40
50
x/M
(a)
(1-d/M)2.6νε11/U 4m
10-6
10
-7
10-8
10
20
x/M
(b)
Figure 7.7: Distribution of ǫ11 for x/M > 7. Squares, case A (β = 53%); circles, case B
4
/ν, b) by ǫN (2.60).
94
10
-5
νε22/U 4m
10-6
10
-7
10
-8
10
20
30
40
50
30
40
50
x/M
(a)
(1-d/M)2.6νε22/U 4m
10-6
10
-7
10-8
10
20
x/M
(b)
Figure 7.8: Distribution of ǫ22 for x/M > 7. Squares, case A (β = 53%); circles, case B
4
/ν, b) by ǫN (2.60).
95
-5
10
-6
νε/U 4m
10
10-7
10
-8
20
30
40
50
40
50
x/M
(a)
10
-6
(1-d/M)2.6νε/U 4m
ε=0.0016*(x/M)
10
-7
10
-8
20
-2.8
30
x/M
(b)
Figure 7.9: Distribution of turbulent dissipation rate for x/M > 12. Squares, case A (β =
53%); circles, case B (β = 64%); diamonds, case C (β = 72%); triangles, case D (β = 82%).
4
Quantities are normalized in two ways: a) by UM
/ν, b) by ǫN (2.60).
96
7.1.2
Effects of Reynolds number
Keeping the porosity value the same, the Reynolds number of one of the previous simulations
is increased by 50% for the analysis in the present section. It is recalled that this simulation carried out at ReM = 2100 corresponds to case E in table 4.1. Hence, in this section,
the Reynolds stress and dissipation tensor components are compared at ReM = 1400 and
ReM = 2100 at β = 72%. The Reynolds stress components at different Reynolds numbers
2
are compared in figure 7.10. The quantities are normalized by Um
. The influence of Reynolds
number in the near-grid region is to decrease the magnitude of the Reynolds stress components, if they are normalized by bulk quantities. Although there is a considerable difference in
Reynolds number between two simulations, the components of Reynolds stress depend weakly
on the value of ReM in the investigated parameter range: for x/M > 8, Reynolds number
effects already disappear. This observation suggests that the Reynolds stress tensor is sensitive to the change in the geometry of the grid rather than the Reynolds number: as stated
in the last section, when there is an increase of 35% in the porosity (from β = 53% to 72%),
magnitudes of the Reynolds stress components change considerably (compare figure 7.1).
The decay of turbulent kinetic energy at different Reynolds numbers is shown in figure 7.12(a).
The new scaling including the porosity effects introduced in the previous section (RN (2.60))
is used for normalization of the distributions. The power law fit shown on the figure has the
coefficient CK = 0.153 and the exponent of ExpK = 1.65. Hence for the Reynolds number
range examined, the decay of turbulent kinetic energy can be taken as universal for x/M > 18.
The dissipation tensor is more sensitive to the changes in Reynolds number than the Reynolds
stress tensor (figure 7.11). Here also, the effect of increasing Reynolds number is to decrease
the magnitude of the dissipation tensor components, if they are normalized using the bulk
properties; however, dependence on the Reynolds number is observable also for x/M > 8.
Therefore, the decay coefficient for turbulent dissipation rate is Reynolds number dependent
in the investigated parameter range (figure 7.12(b)). The decay exponent for dissipation rate
is Reynolds number independent and is equal to 2.8.
97
10-1
u2/U2m
10-2
10-3
10
20
30
40
50
20
30
40
50
20
30
40
50
x/M
(a)
10-1
v2/U2m
10-2
10-3
10
x/M
(b)
10-1
w2/U2m
10-2
10-3
10
x/M
(c)
2
Figure 7.10: The diagonal components of the Reynolds stress tensor normalized by Um
for
different values of ReM : open diamonds, case C (β = 72%, ReM = 1400); closed diamonds,
case E (β = 72%, ReM = 2100): a) distribution of u2 , b) distribution of v 2 , c) distribution of
w2.
98
10
-4
νε112/U4m
10-5
10-6
10-7
10
-8
5
10
15
20
25 30 354045
50
15
20
25 30 354045
50
15
20
25 30 354045
50
x/M
(a)
10
-4
νε222/U4m
10-5
10-6
10-7
10
-8
5
10
x/M
(b)
10
-4
νε332/U4m
10-5
10-6
10-7
10
-8
5
10
x/M
(c)
Figure 7.11: The diagonal components of dissipation tensor for different values of ReM : open
diamonds, case C (β = 72%, ReM = 1400); closed diamonds, case E (β = 72%, ReM = 2100):
a) distribution of ǫ11 , b) distribution of ǫ22 , c) distribution of ǫ33 .
99
0.0025
0.002
0.0015
(1-d/M)2.6k/U 2m
0.001
k=0.153(x/M)-1.65
0.0005
15
20
25
30
35
40
45
50
40
45
50
x/M
(a)
(1-d/M)2.6νε/U 4m
10-6
ε=0.00118(x/M)-2.8
10-7
ε=0.0008(x/M)-2.8
10
-8
15
20
25
30
35
x/M
(b)
Figure 7.12: Decay of turbulence at different values of ReM : open diamonds, case C (β = 72%,
ReM = 1400); closed diamonds, case E (β = 72%, ReM = 2100): a) kinetic energy, b)
dissipation rate.
100
To summarize, a parameter range of β = 53% − 82% and ReM = 1400 − 2100 is examined
for turbulence obstructed by square grid elements and the decay of turbulent kinetic energy
and dissipation rate are discussed. For the normalization of the turbulent quantities in the
far-field region, a new scaling is proposed. The exponent n of this new scaling is the same
for the Reynolds stress and dissipation tensor and is equal to 2.60. If this scaling is used for
the normalization, the difference in the magnitudes of Reynolds stress components becomes
smaller, and one can then speak of a universal power law coefficient for kinetic energy decay
for a broad range of grid porosities, which, based on the present calculations is approximately
0.15. The profiles of the Reynolds stress components depend strongly on the porosity of the
grid, not only near the grid but also far downstream. Therefore, based on the current results,
only a range of power law exponents can be suggested, which is 1.62 < ExpK < 1.66. In
the Reynolds number range examined, the profiles of the Reynolds stress components show
negligible dependencies on the value of ReM . Hence it is suggested to take the same coefficient
of the power law for the kinetic energy decay in the investigated parameter range (CK ≈ 0.15).
It is worth emphasizing that the study at different ReM values shows that the kinetic energy
distribution is independent of Reynolds number for x/M > 18 in the parameter range examined (figure 7.12).
A detailed literature survey of the exponents for the power law decay of the turbulent kinetic energy is provided in sections 1.2 and 1.3. As can be seen from that survey, the overall
range given for the power law exponent is 1.1 − 2.5. The value of ExpK tends to have higher
values for lower mesh Reynolds numbers than those applied here; for example, the study of
Townsend and Batchelor (1947) showed that at ReM = 650, the value of ExpK was 2.5. Considering the theoretical study of Birkhoff (1954) which stated that the value of ExpK must be
expected to lie in the range 1.5 − 2.5 depending on the Reynolds number, one can easily see
that the present calculations are perfectly in agreement with the data given in the literature.
In contrast to the Reynolds stress tensor, turbulent dissipation is more sensitive to Reynolds
number changes and less sensitive to differences in grid porosity. Hence the power law coefficient for the dissipation rate is strongly dependent on the Reynolds number. On the other
hand, the exponent is independent of the Reynolds number. Therefore, from the analysis of
the present simulations, it is possible to draw the following important conclusion: the power
law exponent for turbulent dissipation rate is constant (≈ 2.8).
101
7.2
Anisotropy of grid-generated turbulence
In this section, the results of the present direct numerical simulations are analyzed to examine
the influence of β and Reynolds number, ReM , on the Reynolds stress anisotropy (aij ) and
dissipation tensor anisotropy (eij ), which are described by the following expressions:
1
ui uj
− δij
2k
3
ǫij
1
=
− δij
ǫ
3
aij =
eij
(7.5)
where k and ǫ express the turbulent kinetic energy and dissipation rate, respectively. Their
distributions with respect to the streamwise distance from the grid due to the changes in the
initial conditions are discussed in detail in the previous section. The reader should recall the
important contribution made by Lumley and Newman (1977) in providing an insight into the
Reynolds stress anisotropy. They suggested expressing the anisotropy of the Reynolds stress
tensor in terms of its second (IIa ) and third (IIIa ) scalar invariants, which are given as
IIa = aij aji
IIIa = aij ajk aki
(7.6)
If the second invariant is plotted against the third invariant, a triangular area is formed, which
is given in figure 7.13. The map shown is called the anisotropy-invariant map and it represents
the anisotropy behavior of any kind of physical flow. As shown, there are three limiting states:
isotropic turbulence, one-component turbulence and isotropic two-component turbulence. The
left edge of the map represents the flow generated through all axisymmetric contractions and
the right edge corresponds to the flow generated through axisymmetric expansion.
As discussed later, the anisotropy of unstrained grid turbulence lies on the right edge of
the triangle, i.e. axisymmetric state, and it is very close to the limit of isotropic turbulence. It
should be kept in mind that work on the strained grid turbulence, such as return to isotropy,
successive contraction and expansion studies, depends on the physical understanding of the
unstrained grid turbulence. It should also be noted that it is possible that more complex
flows than those investigated in the current thesis may have anisotropy invariants which lie
on the axisymmetric states of the invariant map as well. That means that, in some parts of
the flow geometry, more complex flows may show similar features to some more basic flows
such as axisymmetric flows. For instance, it is known from experiments and direct numerical
simulations on turbulent plane channel flows that these flows start to follow the right edge of
the invariant map for y + > 50 up to the channel center. That means that, the information
gained through studies on simpler flows, in the present case on unstrained grid turbulence,
may be very valuable for understanding the physics of more complex flows which possess axisymmetric flow conditions in some regions of the flow geometry.
102
7.2. ANISOTROPY OF GRID-GENERATED TURBULENCE
1-Component
turbulence
A=1
0.6
0.5
nt
ne
p o ce
m
o le n
2 -C r b u
tu
IIa
0.4
Axisymmetric
turbulence
(axisymmetric
expansion)
0.3
2-Component
isotropic
turbulence
0.2
A=1
0.1
(axisymmetric
contraction)
A~1 if Reλ
A~0 if Reλ
0
-0.1
isotropic
turbulence
0
infinity
-0.05
0
0.05
0.1
0.15
0.2
0.25
IIIa
Figure 7.13: Anisotropy invariant map.
If the flow is perfectly axisymmetric, the following equalities must hold:
a11 = −2a22 = −2a33
e11 = −2e22 = −2e33
(7.7)
In axisymmetric turbulence, apart from the requirement in equation 7.7, the directions of aij
and eij must be aligned, which means there must be a linear relation between them of the
kind
eij = Aaij
(7.8)
Jovanovic and Otic (2000) showed that if the flow is exactly axisymmetric, the value of A can
be expressed in terms of the second invariants as follows:
A=
IIe
IIa
1/2
103
(7.9)
where the second invariant for the dissipation tensor anisotropy is described by
IIe = eij eji
(7.10)
The limiting values of the coefficient A are shown in figure 7.13 as corner points of the
anisotropy map.
In what follows, the author’s studies regarding the porosity and Reynolds number effects on
the magnitude of the anisotropies are summarized, the equalities in equation 7.7 are discussed
and the validity of equation 7.8 for the present calculations is examined.
A general impression of the effects of the initial conditions on Reynolds stress anisotropy
can be obtained if the invariants calculated at different streamwise distances from the grid are
examined on the anisotropy-invariant map (figure 7.13). The second and third invariants of
simulations carried out with different geometries are plotted in figure 7.14(a) and the invariants calculated from the simulations carried out at different Reynolds numbers are plotted in
figure 7.14(b). From there, the following conclusions can be drawn:
1. The invariants lie on the right boundary of the map at all streamwise distances after
the grid.
2. The anisotropy of the Reynolds stress tensor decays with increasing streamwise distance
from the grid for all cases. Immediately after the grid, it has its maximum value, then it starts
to decrease until around x/M = 2, where the anisotropies of all cases collapse in the vicinity
of the isoptropic turbulence limit.
3. Near the grid, the flow is more isotropic for lower values of porosity. In figure 7.14(a), it
can be seen that IIa and IIIa have much higher values for β = 82% and 72% than for lower
porosities.
4. With increasing Reynolds number, the anisotropy near the grid decreases (figure 7.14(b)).
104
β=0.82, x/M=0.35
0.15
β=0.72, x/M=0.35
0.1
β=0.64, x/M=0.35
IIa
β=0.53, x/M=0.35
x/M increases
0.05
x/M ~2
0
-0.01
0
0.01
0.02
IIIa
(a)
ReM=1400, x/M=0.35
0.15
IIa
0.1
ReM=2100, x/M=0.35
0.05
0
-0.01
0
0.01
0.02
IIIa
(b)
Figure 7.14: Anisotropy-invariant map of grid generated turbulence: a) porosity effect, b)
Reynolds number effect.
105
0.25
0.25
0.2
0.2
a11
0.3
a11
0.3
0.15
0.15
0.1
0.1
0.05
0.05
0
10
20
30
40
0
50
10
20
x/M
30
40
50
30
40
50
x/M
(a)
(b)
-0.05
-0.05
a22
0
a22
0
-0.1
-0.1
-0.15
-0.15
10
20
30
40
50
10
x/M
20
x/M
(c)
(d)
Figure 7.15: Anisotropy tensor components a11 and a22 . Squares, case A (β = 53%, ReM =
1400); circles: case B (β = 64%, ReM = 1400); open diamonds, case C (β = 72%, ReM =
1400); triangles, case D (β = 82%, ReM = 1400); closed diamonds, case D (β = 72%,
ReM = 2100). Left, influence of β; right, influence of ReM : a) a11 at different porosities, b)
a11 at different Reynolds numbers, c) a22 at different porosities, d) a22 at different Reynolds
numbers.
106
A more detailed picture can be obtained if the individual components of aij are analyzed.
Figure 7.15 compares the distributions of the anisotropy components a11 and a22 for different
porosities (figure 7.15(a) and 7.15(c)) and for different mesh Reynolds numbers (figure 7.15(b)
and 7.15(d)). Starting from ∼ 0.3 just after the grid, the level of the anisotropy component,
a11 , decreases first with increasing porosity (from 0.53 to 0.64), it then starts to increase
and it has its maximum value for β = 82%. On the other hand, in the case of a22 , there is
more clearer trend with respect to porosity value: with increasing porosity, this anisotropy
component increases. The effects of porosity on the anisotropy remain in the far-field region
(x/M > 20) for both components. This is as expected; as discussed in the previous section,
grid geometry has a strong influence on the magnitudes of the individual components of the
Reynolds stress tensor. On the other hand, the influence of the Reynolds number is emphasized only near the grid. For x/M < 5 − 6, the anisotropy of Reynolds stress decreases with
increasing Reynolds number.
The validity of equation 7.7 for aij , e.g. the axisymmetry, is checked in figures 7.16 and
7.17. Since aij is normalized by kinetic energy, small differences in individual Reynolds stress
components may result in important deviations in anisoptropy. This situation can be seen
in figure 7.16(a), where the lateral and spanwise components of the Reynolds stress tensor
are compared for β = 53%. As can be seen, there are some minor differences between them.
On the other hand, if three components of aij are examined, it is seen that this case deviates from the perfect axisymetric state. The line in figure 7.16(b) is equal to −0.5a11 . That
means that, any deviation from this line shows the deviation from axisymmetry. The different
components of aij are compared for other cases in figure 7.17. It can be seen that apart from
the case with β = 82%, some deviations from axisymmetry are observable. On the other
hand, at ReM = 2100 and with β = 72%, the anisotropy tensor is perfectly axisymmetric,
which suggets that increasing Reynolds number may improve the axisymmetry condition for
the anisotropy tensor also for lower porosities.
A similar procedure is applied to the anisotropy of dissipation tensor (eij ). The distributions
of e11 and e22 are plotted in Figure 7.18. The anisotropy of the dissipation tensor decreases
with increasing streamwise distance from the grid. Near the grid, the highest porosity case
has the highest anisotropy. Similarly to the Reynolds stress anisotropy, dissipation tensor
anisotropy decreases with increasing Reynolds number. The validity of equation 7.7 for eij is
checked in figures 7.19 and 7.20. Although the components ǫ22 and ǫ33 differ from each other
slightly, once they are normalized by dissipation rate to calculate anisotropy, they start to
deviate from an axisymmetric condition. On the other hand, these deviations disappear at
ReM = 2100 also for the anisotropy of the dissipation tensor.
107
0.006
v2,w2
0.004
0.002
10
20
30
40
50
x/M
(a)
0.2
a11,a22,a33
0.1
0
-a11 /2
-0.1
10
20
30
40
50
x/M
(b)
Figure 7.16: Axisymmetry of aij for case A (β = 53%): a) comparison between v 2 and w 2 , b)
comparison between a11 , a22 and a33 .
108
0.1
0.1
a11,a22,a33
0.2
a11,a22,a33
0.2
0
0
-a11/2
-a11/2
-0.1
-0.1
10
20
30
40
50
10
20
x/M
(a)
40
50
40
50
(b)
0.2
0.1
0.1
a11,a22,a33
0.2
a11,a22,a33
30
x/M
0
0
-a11/2
-a11/2
-0.1
-0.1
10
20
30
40
50
10
x/M
20
30
x/M
(c)
(d)
Figure 7.17: Axisymmetry of aij for a) case B (β = 64%, ReM = 1400), b) case C (β = 72%,
ReM = 1400), c) case D (β = 82%, ReM = 1400), d) case E (β = 72%, ReM = 2100).
109
0.3
0.3
0.2
0.2
e11
0.4
e11
0.4
0.1
0.1
0
0
-0.1
10
20
30
40
-0.1
50
10
20
x/M
(a)
40
50
30
40
50
(b)
-0.05
-0.05
e22
0
e22
0
-0.1
-0.1
-0.15
-0.15
-0.2
30
x/M
10
20
30
40
-0.2
50
x/M
10
20
x/M
(c)
(d)
Figure 7.18: Anisotropy tensor components e11 and e22 . Squares, case A (β = 53%, ReM =
1400); circles, case B (β = 64%, ReM = 1400); open diamonds, case C (β = 72%, ReM =
1400); triangles, case D (β = 82%, ReM = 1400); closed diamonds, case D (β = 72%,
ReM = 2100). Left, influence of β; right, influence of ReM : a) e11 at different porosities, b)
e11 at different Reynolds numbers, c) e22 at different porosities, d) e22 at different Reynolds
numbers.
110
2E-06
ε22,ε33
1.5E-06
1E-06
5E-07
0
10
20
30
40
50
x/M
(a)
0.2
e11,e22,e33
0.1
0
-e11 /2
-0.1
10
20
30
40
50
x/M
(b)
Figure 7.19: Axisymmetry of eij for case A (β = 53%): a) comparison between ǫ22 and ǫ33 , b)
comparison between e11 , e22 and e33 .
111
0.1
0.1
e11,e22,e33
0.2
e11,e22,e33
0.2
0
0
-e11/2
-e11/2
-0.1
-0.1
10
20
30
40
50
10
20
x/M
30
40
50
40
50
x/M
(a)
(b)
0.1
0.1
e11,e22,e33
0.2
e11,e22,e33
0.2
0
0
-e11/2
-e11/2
-0.1
-0.1
10
20
30
40
50
10
x/M
20
30
x/M
(c)
(d)
Figure 7.20: Axisymmetry of eij for a) case B (β = 64%, ReM = 1400), b) case C (β = 72%,
ReM = 1400), c) case D (β = 82%, ReM = 1400), d) case E (β = 72%, ReM = 2100).
112
In the remaining part of this section, the alignment bewteen aij and eij is examined.
Figure 7.21 compares the limiting value of A = 1 with the data obtained from the simulations. The data show that two types of anisotropies are aligned with each other for most of
the streamwise region. However, it is difficult to have a constant value for the coefficient A
based on this dataset. This can especially be seen in figures 7.21(a) and 7.21(b), where the
magnitude of the ratio of Reynolds stress anisotropy to dissipation tensor anisotropy changes
considerably for changing porosity. For different Reynolds numbers, the difference is not so
pronounced, with the exception that near the grid the low Reynolds number case starts with
higher anisoptropy. To demonstrate the slope of the alignment bewteen two anisotropies with
one constant slope, another equations were fitted as shown on the plots in figure 7.21. In this
case, the slope aproaches 1.3, but one needs to add an additional constant to the equation, as
shown on the plots.
The analysis above shows that the ratio eij /aij changes with streamwise distance. Hence,
the value of A is also a function of the streamwise distance. To see in detail how A changes
with increasing distance from the grid, it was calculated from equation 7.8 for the first component (e11 /a11 ) and is plotted against x/M in figures 7.22 and 7.23. The distributions of
the invariants IIa and IIe and their ratio (IIe /IIa )0.5 are also shown. This means that, the
validity of axisymmetry based on equation 7.9 is checked in figures 7.22 and 7.23.
The distribution of A has the same trend for all cases. Immeadiately after the grid, it starts
to decrease until x/M ≈ 7 − 8, then it starts to increase. The effect of increasing porosity near
the grid is to decrease the A value. Near the grid, the value of A also decreases with increasing
Reynolds number (figure 7.23). The A values calculated from equation 7.9 predict the data
fairly well for all cases. Especially when the porosity (figure 7.22(d)) and Reynolds number
(figure 7.23), the agreement is perfect are high. This observation is as expected; shown by the
previous analysis of different components of aij and eij (figures 7.17 and 7.20), exactly these
cases are perfectly axisymmetric. With increasing porosity and ReM , the distribution of A in
the far field also becomes smoother.
113
0.35
x/M increases
0
0.3
0.25
-0.05
0.2
0.15
A=1
e22
e11
e11=1.3a11-0.08
A=1
-0.1
e22=1.3a22+0.04
0.1
0.05
x/M increases
-0.15
0
0.05
0.1
0.15
0.2
0.25
0.3
-0.15
-0.1
-0.05
a11
a22
(a)
(b)
0.35
0
0
e11=1.3a11-0.07
0.3
0.25
A=1
-0.05
e22
e11
0.2
0.15
A=1
-0.1
e22=1.3a22+0.038
0.1
0.05
-0.15
0
0.05
0.1
0.15
0.2
0.25
0.3
-0.15
a11
-0.1
-0.05
0
a22
(c)
(d)
Figure 7.21: aij vs. eij . Squares, case A (β = 53%); circles, case B (β = 64%); open
diamonds, case C (β = 72%); triangles, case D (β = 82%); closed diamonds, case E (β =
72%, ReM = 2100): a) a11 -e11 , comparison for different porosities; b) a22 -e22 , comparison for
different porosities; c) a11 -e11 , comparison for different ReM values; d) a22 -e22 , comparison for
different ReM values.
114
1
1
0.8
0.8
A
0.6
0.6
0.4
0.4
(IIe/IIa)
0.5
0.2
0.2
IIa
0
0
IIe
10
0
10
1
10
0
x/M
10
1
10
1
x/M
(a)
(b)
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
10
0
10
1
10
x/M
0
x/M
(c)
(d)
Figure 7.22: Distributions of invariants IIa (closed diamonds), IIe (open diamonds) and the
ratio (IIe /IIa )1/2 and the coefficient A calculated from equation 7.8, for a) case A (β = 53%),
b) case B (β = 64%), c) case C (β = 72%), d) case D (β = 82%).
115
1
0.8
0.6
0.4
0.2
0
10
0
10
1
x/M
Figure 7.23: Distributions of invariants IIa (closed diamonds), IIe (open diamonds) and the ratio
(IIe /IIa )1/2 and the coefficient A calculated from equation 7.8, for case E (ReM = 2100, β = 72%).
116
Chapter 8
Conclusions and outlook
Decaying, nearly isotropic turbulence generated through passive grids was examined numerically. Although extensively studied experimentally in the literature, the numerical treatment
of such flows is recent. Early direct numerical simulations of turbulent flows concentrated
mainly on the decaying turbulence, where homogeneous and isotropic flow conditions were
created at the beginning of the simulations through periodic boundary conditions in all directions. On the other hand, the homogeneity of the grid-generated turbulence is only an
ad hoc assumption and in the vicinity of the grid the flow is anisotropic. Hence a proper
treatment of these flows requires one to resolve the grid elements in the computational domain to be able to describe the anisotropic and inhomogeneous parts of the flow. This point
distinguish the present work from the majority of the remaining computational studies carried out on decaying turbulence in the literature. Conducting direct numerical simulations
with the application of the lattice Boltzmann BGK method and using square grid elements in
the computational domain, some remaining issues regarding grid-generated turbulence were
addressed. A mesh-based Reynolds number range of 1400 < ReM < 2100 and a grid porosity
range of 53% < β < 82% were covered.
The results were first analyzed to study an aspect of grid-generated turbulence which has not
attracted much attention in the literature, i.e. the inhomogeneity of the turbulent quantites.
Some discrepancies existed among the few available studies on this issue regarding the effects
of grid porosity, hence the results were examined to explain whether the inhomogeneity of gridgenerated turbulence was related to the grid porosity in some way. The analysis confirmed
recent experimental findings regarding the Reynolds stress inhomogeneity (Ertunc (2007)).
The Reynolds stress field and its anisotropy field were shown to be strongly inhomogeneous
in the lateral direction, having elongated intense negative and positive regions coalescing with
each other. The lateral inhomogeneity of the streamwise mean velocity was also examined. It
was shown that the experimentally observed inhomogeneous mean velocity fields were related
to the application of asymmetric grid elements. Once it is assured that the grid elements used
in the experiments are perfectly symmetric, the mean velocity field inhomogeneity must stop
117
CHAPTER 8. CONCLUSIONS AND OUTLOOK
very rapidly after the grid. The numerical investigations showed further that in contrast to
the study by Corrsin (1963), the mean velocity field did not become more homogeneous when
more porous grid elements were used. In fact, regardless of the value of the grid porosity, the
mean velocity was always homogeneous for x/M > 5. Similarly, the Reynolds stress terms did
not become more homogeneous with increasing porosity, they remained inhomogeneous even
for x/M > 30 for all grid porosities examined. Hence the results concerning the inhomogeneity
of the grid turbulence were parallel to the findings of Grant and Nisbet (1957) and showed
that the precautions suggested by Corrsin (1963) did not necessarily assure the homogeneity
of turbulence.
The analysis mentioned above showed that despite the fact that in the far-field region (x/M >
30) the mean streamwise velocity was perfectly homogeneous, the Reynolds stress remained
inhomogeneous also far downstream. To explain the reason why the Reynolds stress tensor
kept its inhomogeneity also far downstream, the results of the simulations were used to examine the terms of the turbulent kinetic energy equation, where the turbulent transport and
viscous diffusion terms are possibly responsible for the homogenization of the flow field. The
simulation results showed that the turbulent transport term was negligible over the whole domain and the viscous diffusion terms decayed the slowest among all terms. On the other hand,
the magnitude of the viscous diffusion term in the far-field region was very low. Considering
that the diffusion time-scales were slower than the convective time-scales, the viscous diffusion
term could not be responsible for homogenizing the flow in the far field once there were no
mean velocity gradients. Hence the analysis of the kinetic energy equation showed that the
early homogenization of the mean velocity field was the main reason for the persistence of the
inhomogeneity of the Reynolds stress fields.
The results were further analyzed to elucidate an issue often studied in the literature for
grid turbulence, namely universality of turbulent quantities in terms of the initial conditions.
The results in this respect showed that the magnitude of the kinetic energy depended strongly
on the value of the grid porosity at the same Reynolds number. On the other hand, the effects
of the porosity on the magnitude of the kinetic energy could be easily excluded if another scaling was used to normalize the data, which was calculated using the grid porosity value. The
data normalized in this way resulted in a decay coefficient of 0.15 for the turbulent kinetic energy. The decay exponent changed considerably for different grid porosities at fixed Reynolds
number. Based on the data, the value of the decay exponent is in the range 1.62 − 1.66, being
in aggrement with the data given in the literature. Increasing the mesh-based Reynolds number 50% from ReM = 1400 to 2100 did not result in a considerable change in the value of the
coefficient and the exponent of the power law decay of the turbulent kinetic energy. On the
other hand, the magnitude of the dissipation tensor depended considerably on the Reynolds
number An important conclusion could be drawn regarding the decay of dissipation rate: for
118
the overall parameter range, the exponent of the dissipation tensor was 2.8.
Based on the results of the numerical simulations, another important aspect of grid turbulence was addressed: the anisotropies of the Reynolds stress and dissipation tensor with
respect to mesh Reynolds number and grid porosity. The outcome of this study may be summarized as follows. The invariants calculated from all simulations lie on the right boundary
of the anisotropy-invariant map at all streamwise distances after the grid, corresponding to
the nearly isotropic axisymmetric case. As expected, the anisotropies of the Reynolds stress
and dissipation tensors decayed with increasing streamwise distance from the grid for all
cases. Near the grid, the flow was more isotropic for lower values of porosity. With increasing
Reynolds number, the anisotropies of both tensors near the grid decreased.
The results were examined in detail with respect to the axisymmetry condition. The study
confirmed the experimentally observed axisymmetry of grid turbulence in the broad sense, i.e.
the lateral and spanwise diagonal components of the Reynolds stress tensor were very close to
each other and the off-diagonal components were negligible when compared with the diagonal
components. On the other hand, a closer look into the axisymmetry, i.e. the axisymmetry of
the anisotropy of Reynolds stress and dissipation tensors, showed that when lower porosities
and low Reynolds numbers were applied, this condition might not be exactly satisfied either.
The study of the alignment between anisotropy of the Reynolds stress tensor and anisotropy
of the dissipation tensor showed that the ratio, A, between these anisotropies was not constant. Its value was slightly higher than 1.0 after the grid for low porosities; with increasing
grid porosity after the grid, the value decreased and then it started to increase again. On
the other hand, the results suggested that if higher Reynolds numbers and more porous grids
were applied it would be expected that there would be a constant A value for x/M > 20. The
A values for different initial conditions were further compared with the (IIe /IIa )0.5 values
calculated from the database. Here also, the results showed that the turbulence became more
axisymmetric once more porous and higher Reynolds numbers were applied.
From the discussion above, one can easily conclude that the LBGK method can be succesfully
employed to make a detailed analysis of grid-generated turbulence. On the numerical side,
a step forward is to carry out direct numerical simulations on the strained grid turbulence,
which means the study of grid turbulence subject to expansion and/or contraction. In this
respect, especially studies on the axisymmetric expansion will be very valuable, because at
the moment, no direct numerical simulation data are available describing the whole right edge
of the anisotropy-invariant map up to the one-component limit, which, once obtained, will
be very valuable for the validation of second-order turbulence closure models. On the other
hand, accomplishing this will not be very straightforward, because with the application of
the straining, the velocity gradients will become higher with increasing streamwise distance
119
CHAPTER 8. CONCLUSIONS AND OUTLOOK
and this will cause an important computational domain resolution problem. Hence it is not
realistic to expect to be able to carry out a DNS study on strained turbulence by the application of the standard uniform lattice Boltzmann method which was applied in the present
calculations. Hence for future work, it is suggested that one should analyze the potential
of the unstructured formulations of the lattice Boltzmann method when examining the grid
turbulence subject to straining.
A very important conclusion drawn from the present predictions was the strong inhomogeneity of the Reynolds stress field. For the successful interpretation of the flow generated
through strained grid turbulence mentioned above, it must be assured that the underlying
flow is homogeneous. In addition, while the time-averaged three dimensional turbulent field is
normalized in the lateral and spanwise directions to obtain the distributions with streamwise
directions, such as to obtain the decay of turbulent kinetic energy, the existing lateral inhomogeneity may affect the distributions obtained considerably. Hence different means must be
investigated which can help to assure the homogeneity of grid turbulence. For example, Ertunc
(2007) showed that once isoptropy was improved through contraction, it was also possible to
improve the lateral homogeneity. Another way to help to make the flow more homogeneous
might be the application of active grids. Hence the future experimental and numerical studies should also focus on the effects of active grid arrangements on the inhomogeneity of grid
turbulence.
120
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