ik - Turbulence Mechanics/CFD Group - University of Manchester

ALGERIAN REPUBLIC DEMOCRATIC AND POPULAR
Ministry of teaching and scientific research
UNIVERSITY OF SCIENCE AND TECHNOLOGY
MOHAMED BOUDHIAF ORAN
U.S.T.O
FACULTY OF MECHANICAL ENGINEERING
DEPARTEMENT OF MARINE ENGINEERING
Thesis for the degree of master in energetic
SIMULATION OF TURBULENT FLOW ACROSS IN-LINE
TUBE BUNDLE USING DIFFERENT URANS MODELS
Presented by
Miss AMMOUR Dalila
Supervisors
Pr. ADJLOUT Lahouari
Dr. ADDAD Yacine
2006-2007

Contents
List of Figures.................................................................................................................4
List of Tables..................................................................................................................7
Abstract...........................................................................................................................8
Acknowledgements......................................................................................................10
Nomenclature...............................................................................................................12
1 Introduction
1.1 Introduction...........................................................................................................15
1.2 Study objectives....................................................................................................19
1.3 Outline of the thesis..............................................................................................20
2 Literature Review
2.1 Introduction...........................................................................................................21
2.2 Literature review of tube bundles.........................................................................21
2.2.1 LES of tube bundles....................................................................................21
2.2.2 Heat transfer in tube bundles......................................................................22
2.2.3 Pressure Fluctuations..................................................................................23
2.2.4 Vortex shedding..........................................................................................24
2.2.5 Vibrations………………………………………………………….……...26
3 Governing Equations
3.1 Introduction..........:..............................................................................................28
3.2 Navier-Stokes equations......................................................................................28
3.2.1 Reynolds Averaging.....................................................................................28
3.3 Classes of turbulence models...............................................................................29
3.3.1 Algebraic turbulence models…………………………………...…………29
3.3.1.1 Baldwin-Lomax model…………………………………...……….29
3.3.1.2 Cebeci-Smith model……………………………………………….30
3.3.2 One-equation turbulence models………………………...………………...30
3.3.2.1 Prandtl’s one-equation model……………...………………………30
3.3.3 Two equation turbulence models………...…………………………………31
3.3.3.1 Boussinesq eddy viscosity assumption……………………...……..31
2

3.3.3.2 K-epsilon models……………………...…………………………...32
3.3.3.3 K-omega models…………………………….……………………..33
3.3.4 V2-f models……………………………………………………………...35
3.2.5 Reynolds stress model (RSM)…………………………………………...36
3.3.6 Large Eddy simulation (LES)…………………………………………....40
3.3.7 Detached Eddy simulation (DES)…………………….………….……...42
3.3.8 Direct numerical simulation (DNS)…………………….……………….43
3.3.9 The SST- Cas
model……………………………….……………………..43
3.4 Turbulence modelling of unsteady flows (URANS)………….……….………44
3.4.1 Introduction………………………………………….…………….……..44
3.4.2 Unsteady Reynolds Navier-Stokes equations………….……….……….44
3.4.2 Turbulence Modelling of Unsteady Cross Flow In-line Tube Bundle…..45
4 Numerical Simulations
4.1 Introduction.........................................................................................................46
4.1.1 Pre-processor………………………………….……………………….....46
4.1.2 Solver (Code-Saturne)……………………….…………………………..46
4.1.3 Post-Processor………………………………….………………………...47
4.2 The Finite Volume method.................................................................................49
4.3 Time Discritisation..............................................................................................51
4.4 Boundary Conditions..........................................................................................53
4.4.1 Inlet.............................................................................................................53
4.4.2 Outlet...........................................................................................................54
4.4.3 Walls and symmetries.................................................................................54
5 Results and Discussion of the simulation
5.1 Introduction……………………………………………...……………………...57
5.2 Case description………………………………………...………….…………...58
5.3 Grid generation…………………………………………………………………59
5.4 Discussion of the results………………………………..………………………60
6 Conclusions and Recommendations for future work
5.1 Final remarks.......................................................................................................92
5.2 Recommendations for future work......................................................................93
7 Bibliography.…...……………………………………………...………………….. 94
3

List of Figures
1.1
4.1
4.2
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
Turbulent flow around circular cylinder (Catallano et al.2003)……………………...
Steps of Numerical simulation of across flow in-line tube bundles………………….
Notations for the spatial discritisation……………………………………………….
Tube arrangements…………………………………………………………………...
Geometry of in-line tube bundles…………………………………………………….
Boundary conditions of tube bundles………………………………………………..
Cross sectional view of 2D grid (2X2 arrangement) N=5400 cells, y+= [13-70]……
Cross sectional view of 2D grid (3X3 arrangement) N=21600 cells, y+= [13-70]…..
Cross sectional view of 3D grid (3X3 arrangement) in XY, YZ and XZ sections:
N=604800 cells, y+= [13-70]………………………………………………………...
Evolution of pressure and velocity, Comparison between URANS models.
(a) Pressure, (b) Velocity…..………………………………………………………...
2D Instantaneous Pressure Contour field in a XY cross sectional view for gap ratio
1.44. (a) k-ε model, (b) RSM, (c) k-ω SST, (d) SST-C as ……………………………
2D Instantaneous velocity contour field in a XY cross Sectional view for gap ratio
1.44. (a) k-ε model, (b) RSM, (c) k-ω SST, (d) SST-C as ……………………………
2D Velocity vectors field in a XY cross Sectional view for gap ratio 1.44. (a) k-ε,
(b) RSM, (c) k-ω SST, (d) SST-C as ………………………………………………….
2D Vorticity field in a XY cross sectional view for gap ratio 1.44. (a) k-ε, (b) RSM,
(c) k-ω SST, (d) SST-C as …………………………………………………………….
3D mean pressure distribution in a XY cross view for P/D=1.44, Re=70000. (a) k-
ω SST, (b) RSM, (c) SST- Cas
, (d) DES, (e) LES of Imran for P/D=1.5, Re=15000..
3D averaged velocity field in a XY cross view for P/D=1.44, Re=70000. (a) k-ω
SST, (b) RSM, (c) SST- Cas
, (d) DES, (e) LES of Imran for P/D=1.5, Re=15000…..
Mean pressure distribution around centre tube, comparison between 2D Unsteady
RANS for P/D=1.44, Re=70000 and LES of Imran (Star-V4) for P/D=1.5
Re=15000 and Experiment of Yahiaoui et al. (2007)………………………………..
Mean pressure distribution around centre tube, comparison between 2DUnsteady
RANS for P/D=1.44, Re=70000 and LES of Imran (Star-V4) for P/D=1.5
Re=15000 and Experiment of Yahiaoui et al. (2007)………………………………..
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15
48
53
58
66
66
67
67
68
69
70
71
72
73
74
75
76
76

5.16
5.17
5.18
5.19
5.20
5.21
5.22
5.23
5.24
5.25
5.26
5.27
5.28
5.29
5.30
5.31
Mean pressure distribution around centre tube, comparison between 3D
UnsteadyRANS for P/D=1.44, Re=70000 and LES of Imran (Star-V4) for P/D=1.5
Re=15000 and Experiment of Yahiaoui et al. (2007)………………………………..
Mean velocity profile, Comparison between 2D Unsteady RANS, Re=70000 and
LES of Imran Re=15000 (Star-V4) and experiment of Aiba et al. (1982) in the
wake of centre tubes at x=4.33cm……………………………………………………
Mean velocity profile, Comparison between 2D Unsteady RSM, Re=70000 and
LES of Imran at Re=15000 (Star-V4) and experiment of Aiba et al. (1982) in the
wake of centre tubes at x=4.33cm……………………………………………………
Mean velocity profile, Comparison between 2D Unsteady RSM, SST- C ,
Re=70000 and LES of Imran (Star-V4), Re=15000 in the wake of centre tubes at
x=4.33cm……………………………………………………………………………..
Mean velocity profile, Comparison between 3D URANS, Re=70000 and LES of
Imran (Star-V4) at Re=45000 and experiment of Aiba et al. (1982) in the wake of
centre tubes at x=4.33cm……………………………………………………………..
Mean velocity profile, Comparison between SST- C ,Re=70000 and LES of Imran
(Star-V4), Re=45000 in the wake of centre tubes at x=4.33cm…………………….
Fluctuating Pressure DPS at location of probe 1……………………………………..
Fluctuating Pressure and DPS at location of probe 3………………………………...
(a) Fluctuating Pressure and DPS at location of probe 6. (b) LES
(Benhamadouche)…………………………………………………………………….
Fluctuating Velocity and DPS at location of probe 1………………………………..
Fluctuating Velocity and DPS at location of probe 3………………………………..
Fluctuating Velocity and DPS at location of probe 6………………………………..
Reynolds stresses in the wake of the centre tubes. (a) , (b) , (c) ,
(d) ……………………………………………………………………………..
Mean velocity profiles of RSM in the wake of the centre tubes. (a) , (b) ,
(c) , (d) u/uo…………………………………………………………………….
Iso-surface of parameter Q for the instantaneous flow across in-line tube bundles.
(a) RSM, (b) k-ω SST, (c) SST- Cas
, DES…………………………………………...
Comparison between Code-Saturne (in right) and Star-CD (in left). k-ω SST,
(a)Pressure, (b) Velocity, (c) Turbulent kinetic energy……………………………...
5
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as
77
77
78
78
79
79
80
81
82
83
84
85
86
87
89
90

5.32
3D mean velocity vectors, (a) k-ω SST, (b) SST- C , (c) RSM, (d) DES, (e) LES
(Benhamadouche)…………………………………………………………………….
6
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91

List of Tables
3.1 Coefficients of the standard k-ε model……………………………………………..…….31
3.2 Coefficients of the k-ω SST model………………………………………………..….......33
3.3 Coefficients of the LRR model…………………………………………………..……….37
3.4 Coefficients of the SSG model………………………………...………………..………..38
5.1 Parameters of 2D and 3D grids of the present case……………………………………....60
7

Abstract
The flow in tube bundles is of great interest to the power generation industry, not only for the
study of performance of great exchangers. Safety studies require predictions of vibrations
caused by fluid-structure interaction or large temperature fluctuations that eventually lead the
thermal stripping. The cross flow in a 2D and 3D square in-line tube bundle is computed for
pitch ratio of P/D=T/D=1.44 and Reynolds number of 70000. The grid generated is structured.
Unsteady Reynolds Navier-Stokes models are widely used for the complex unsteady flows. In
the present case URANS models are used to examine the flow predictions in in-line tube
bundle. URANS models tested are standard κ – ε, Menter`s shear stress transport (MSST) [37]
and the Reynolds Stress Models (RSM). Other models are used, the new SST- C [18] model
for 2D and 3D calculations moreover DES approach for 3D simulation. This case is computed
by Code-Saturne based on the finite volume method. Quantitative and qualitative results are
analyzed then compared with LES and experimental data. The 2D simulations fail to capture
the complete flow physics hense 3D calculations on the other hand seem to produce better
results of pressure and velocity profiles and agree better with LES and experiment. Good
predictions are retained with the new SST- C model [18]. The three models k-ω SST, SST-
Cas
as
and RSM seems to give similar predictions of the flow. Code Star-CD is used for
comparison. It gives similar results and confirms the asymmetry of the flow. When
frequencies of oscillations are given. This is done by using Density Power Spectrum (DPS)
and localizing the peak values (the most energetic frequency). By applying DPS to the
velocity's and pressure's signals, one clear peak is obtained around the frequency 45Hz
(St=0.84) similar than the LES [13]. It means that a big recirculation coexists in the bottom of
the tube then the shear stress is higher in the bottom.
Résumé
L'écoulement dans les faisceaux de tubes est d'un grand intérêt au sein de l'industrie de
production d'électricité, non seulement pour l'étude de l'exécution de grands échangeurs. Les
études de sécurité exigent des prévisions de vibrations provoquées par l'interaction fluidestructure
ou des grandes fluctuations de la température qui mènent par la suite au
dépouillement thermique. L'écoulement dans un 2D et 3D faisceau de tubes intégré carré est
simulé pour un rapport de P/D=T/D=1.44 et un nombre de Reynolds de 70000. Le maillage
8
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généré est structuré. Les modèles URANS sont largement répandus pour les écoulements
instables complexes. Dans le cas present les modèles URANS testés sont: κ – ε standard,
(MSST) [37] de Menter et (RSM). Autres modèles sont utilisés, le nouveau model SST-
C as [18] pour les calculs 2D et 3D en plus de l`approche DES pour la simulation 3D. Les
conditions aux limites sont périodiques. Le cas present est simulé par Code-Saturne basé sur
la méthode des volumes finis. Des résultats qualitatifs et quantitatifs sont alors analysés et
comparés à la LES et aux données expérimentales. Les simulations 2D ne capturent pas
complètement l'écoulement mais d`autre part les résultats des calculs 3D semblent produire de
meilleurs résultats des profils de pression et de vitesse et mieux conformes à la LES et à
l'expérience. De bonnes prévisions sont captées avec le nouveau modèle SST- C [18]. Les
trois modèles k-ω SST, SST- C et RSM semblent donner de mêmes prévisions de
as
l'écoulement. Le Code Star-CD est employé pour la comparaison. Il donne des résultats
semblables et confirme l'asymétrie de l'écoulement. Quand les fréquences des oscillations sont
indiquées. Ceci est fait en employant le spectre (DPS) et en localisant les valeurs de crête (la
fréquence la plus énergique). En appliquant le DPS aux signaux de la vitesse et de la pression,
une crête claire est obtenue autour de la fréquence 45Hz (St=0.84) vérifié avec la LES [13]. Il
signifie qu'un grand recyclage coexiste au dessous du tube alors l'effort de cisaillement est
plus grand au dessous.
9
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Acknowledgements
I would like begin my sincere gratitude to God for his help and special thanks to
Professor Dominique Laurence, my supervisors Professor Adjlout Lahouari and Dr.
Yacine Addad and also Dr. Sofiane Benhamadouche for their invaluable guidance and
continuous advice throughout my present work. I wish to offer my thanks to Dr.
Alistair Revell and Dr. Juan Uribe. They have been an endless source of
encouragement and inspiration.
I also offer my thanks to all the CFD team in University of Manchester, School of
Mace and University of Oran USTO IGCMO in particular Dr. Aounallah Mohamed.
There have many other people who have contributed to this work and to the fun
environment in which it has be carried out. I'm grateful to Dr. Charles Moulinec,
Nicolas Jarrin, and Dr. Imran Afgan. I can't forget to mention the help of Pat Shepherd
for its support during the training course.
My friends have been very important to me during this time, in particular Amel and
Zahid, Hajira and there are many others, I offer a general thanks to every one else.
Most importantly, for their love and inspiration, I would like to thank my family, in
particular my parents, my brothers, my grandmother, Mounia, Sara and Fatima. They
have constantly supported me taught me and encouraged me, and it is always a huge
motivation to show them how grateful Iam.
10

This work is dedicated to my parents
11

Nomenclature
Greek letters
α Coefficients of the κ - ω SST model [-]
1 4 ,...,α
β Thermal expansion coefficient [-]
Δ
Filter width [m]
δ ij
Kronecker delta [-]
Γ
k
Diffusion coefficient [-]
Von Karman constant [-]
λ Integral length scale [m]
μ Molecular viscosity [N.s/m²]
μ t
Function of the local boundary layer velocity profile [-]
ν Dynamic viscosity [m²/s]
ν τ
Sub grid eddy viscosity [m²/s]
ν t
Turbulent Viscosity [m²/s]
Ωij
Rotation rate tensor [1/s]
ρ Density [kg/m³]
τ ij
Viscous stress [N/m²]
ε Isotropic dissipation [N/m²]
ε ij
Turbulent dissipation rate tensor [N/m²]
Latins letters
n Unit vector representing the wall-normal direction [-]
A Lumley’s flatness parameter [-]
A+ Van Driest damping coefficient [-]
a ij
bij
Cij
C p
Anisotropy tensor [m²/s²]
Normalised anisotropy tensor [-]
Cross stress tensor [m²/s²]
Pressure Coefficient [-]
12

Cs
Smagorinsky coefficient [-]
D Diameter of cylinder [m]
ν
Dij
T
Dij
F1
F2
fij
Viscous diffusion of Reynolds Stresses [-]
Turbulent diffusion of Reynolds Stresses [-]
First blending function for the SST model [-]
Second blending function for the SST model [-]
Normalised redistribution tensor [-]
G Kernel of spatial filter [-]
g Gravity [m/s²]
g ij
Velocity gradient tensor [1/s]
k Turbulent kinetic energy [kg.m2/s2]
Lz
Spanwise extrusion length (homogeneous direction) [m]
l Mixing length scale [m]
N Number of grid cells [-]
p Pressure [N/m²]
P Average pressure. ith component [N/m²]
P' Pressure fluctuation [N/m²]
Pk
Pref
Production of turbulent kinetic energy [m²/s³]
Reference pressure [N/m²]
R Radius of cylinder [m]
Re Reynolds number [ρU D/μ]
Rij
Reynolds stress tensor [m²/s²]
S Filtered strain rate magnitude [-]
Sij
Strain tensor [1/s]
St Strouhal number [-]
T Turbulent time scale [s]
U b
Bulk velocity [m/s]
Uo Inlet Velocity [m/s]
u`
Fluctuating velocity, ith component [m/s]
13

u" i
ui
uk
xi
Modelled turbulent fluctuation in RANS [m/s]
Instantaneous velocity, ith component [m/s]
Friction velocity based on k [m/s]
Coordinate, ith component [-]
y+ Nondimensional wall distance. [-]
U i
Average velocity, ith component [m/s]
U j
Filtered velocity. ith component [m/s]
Acronyms
CDS Central Differencing Scheme
CFD Computational Fluid Dynamics
DES Detached Eddy Simulation
DNS Direct Numerical Simulation
EVM Eddy Viscosity Model
GGDH Generalised Gradient Diffusion Hypothesis
LES Large Eddy Simulation
LRR Launder, Reece and Rodi model
RANS Reynolds Averaged Navier-Stokes
RSM Reynolds Stress Model
SA Spalart Allmaras
SCWF Scalable Wall Function
SMC Second Moment Closure
SSG Speziale, Sarkar and Gatski model
SST Shear Stress Transport
UDS Upwind Differencing Scheme
14

Chapter I Introduction
Chapter 1
Introduction
1.1 Introduction
Computational fluid dynamics or CFD is the analysis of systems involving fluid flow, heat
transfer and associated phenomena such as chemical reactions by means of computer-based
simulation. The technique is very powerful and spans a wide range of industrial and non-
industrial application areas. Some examples are:
• Aerodynamics of aircraft and vehicles: lift and drag.
• Hydrodynamics of ships.
• Power plant: combustion in IC engines and gas turbines.
• Turbomachinery: flows inside rotating passages, diffusers…etc.
• Electrical and electronic engineering: cooling of equipment including micro-circuits.
• Chemical process engineering: mixing and separation.
• External and internal environments of building: wind loading and heating ventilation.
• Marine engineering.
• Environment engineering: distribution of pollutants and effluents.
• Hydrology and oceanography: flows in rivers, oceans.
• Meteorology: weather prediction.
• Biomedical engineering: blood flows through arteries and veins.
Numerical Simulations are used for two types of purposes.
The first is to accompany research of a fundamental kind. By describing the physical
mechanisms governing fluid dynamics better, Numerical Simulation help to understand model
and later control these mechanisms. This kind of study requires that the Numerical Simulation
produce data of very high accuracy, which implies that the physical model chosen to represent
the behavior of the fluid must be pertinent and that the algorithm used by the computer
system, must introduce no more that a low level of error. The quality of the data generated by
the numerical simulation also depends on the level of resolution chosen. For the best possible
precision, the simulation has to take into account all the space-time scales affecting the flow
dynamics. When the range of scales is very large, as it is in turbulent flows.
15

Chapter I Introduction
Numerical Simulation is also used for another purpose: engineering analysis. Where flow
characteristics need to be predicted in equipment design phase. Here, the goal is no longer to
produce data for analyzing the flow dynamic itself, but rather to predict certain of the flow
characteristics or, more precisely, the values of physical parameters that depend on the flow,
such us the stresses exerted on an immersed body, the production and propagation of acoustic
waves. The purpose is to reduce the cost and time needed to develop a prototype. The desired
predictions may be either of the mean values of these parameters of their extremes.
In CFD, numerical algorithms are used to reach an approximate solution to the flow field,
which is commonly represented by a discrete set of nodes, defined by a mesh specifically
tailored to the geometry of the problem. The variation of physical values across the flow field
can be expressed exactly by the differential Navier-Stokes equations which, in CFD, are
replaced with sets of algebraic expressions known as discretised equations.
Thus, instead of a closed-form analytical solution, the end product of CFD is a collection of
numbers at discrete space and time locations.
Turbulence [56] is an irregular, chaotic state of fluid motion that occurs when the
instabilities present in the initial or boundary conditions are amplified, and a self sustained
cycle is established in which turbulent eddies (coherent region of vorticity) are generated and
destroyed. Turbulence is best described through its characteristics (example of turbulent flow
in figure1). The most distinguishing features of turbulent flows are:
• Randomness: Turbulent flows are extremely sensitive to initial and boundary conditions:
slight changes in term will make the development of flows that are otherwise identical
diverge, as the differences are exponentially amplified in time. The statistical properties of
the flows will, however, remain unchanged. This randomness is the reason why much of
turbulence research has relied on statistical methods of investigation and prediction.
• Vorticity: We cannot call a random flow turbulent if the curl of the velocity vector is
negligibly small, even if it has some of the other characteristics of turbulence. The random
motions of waves on the ocean surface as well as the irrotational fluctuations in the potential
flow above the boundary layer are examples of random non-turbulent flows. All turbulent
flows are rotational and exhibit high levels of fluctuating vorticity, which is usually
concentrated in regions with strong coherence (coherent structure of eddies). Vortex
stretching is an essential component of turbulence dynamics.
16

Chapter I Introduction
• Mixing: Turbulent motions greatly enhance the transport of mass, momentum and energy.
The enhanced mixing of momentum in a turbulent flow results is higher skin-friction
coefficient, while the more effective mixing of different species results in more rapid
dispersion of contaminants. The dominance of convective effects over diffusive ones is one
of the key characteristics of turbulence.
• Irregularity: Turbulent flow is irregular, random and chaotic. The flow consists of a
spectrum of different scales (eddy sizes) where largest eddies are of the order of the flow
geometry (i.e. boundary layer thickness, jet width, etc). At the other end of the spectra we
have the smallest eddies which are by viscous forces (stresses) dissipated into internal
energy. Even though turbulence is chaotic it is deterministic and is described by the Navier-
Stokes equations.
• Diffusivity: In turbulent flow the diffusivity increases. This means that the spreading rate of
boundary layers, jets, etc. increases as the flow becomes turbulent. The turbulence increases
the exchange of momentum in e.g. boundary layers and reduces or delays there by
separation at bluff bodies such as cylinders, airfoils and cars. The increased diffusivity also
increases the resistance (wall friction) in internal flows such as in channels and pipes.
• Large Reynolds numbers: Turbulent flow occurs at high Reynolds number. For example,
the transition to turbulent flow in pipes occurs that Re=2300, and in boundary layers at
Re=10000.
• Three-dimensional: Turbulent flow is always three-dimensional. However, when the
equations are time averaged we can treat the flow as two-dimensional
• Dissipation: Turbulent flow is dissipative, which means that kinetic energy in the small
(dissipative) eddies are transformed into internal energy. The small eddies receive the
kinetic energy from slightly larger eddies. The slightly larger eddies receive their energy
from even larger eddies and so on. The largest eddies extract their energy from the mean
flow. This process of transferred energy from the largest turbulent scales (eddies) to the
smallest is called cascade process.
• Continuum: Even though it has small turbulent scales in the flow they are much larger than
the molecular scale and the flow can be treated as a continuum.
17

Chapter I Introduction
Figure1: Turbulent flow around circular cylinder (Catallano et al.2003)
The difficulty in predicting turbulence arises from the nonlinearity of the Navier-Stokes
equations, which generate a broad range of length and time scales, with several orders of
magnitude between the smallest and the largest eddy. In light of this observation various
different approaches have been developed and applied, with a hierarchy of complexity, which
can be broadly categorized into three groups:
Direct Numerical Simulation (DNS), Large Eddy Simulation (LES) and Reynolds Averaged
Navier Stokes models (RANS):
• Direct Numerical Simulation: The most accurate approach to turbulence simulation is to
solve the navier-Stokes equations without averaging or approximation other than numerical
discretisation whose errors can be estimated and controlled. It is also the simplest approach
from the conceptual point of view. In such simulations, all of the motions contained in the
flow are resolved. The computed flow field obtained is equivalent to a single visualization of
a flow or a short duration laboratory experiment.
• Large Eddy Simulation: Turbulent flows contains a wide range of length and time scale;
The large scale motions are generally much more energetic than the small scale ones, their
size and strength make them by far the most effective transporters of the conserved
properties. The small scale is usually much weaker and provides little transport of these
properties. A simulation which treats the large eddies more exactly than small ones may
make sense. Large eddy simulations are three dimensional, time dependant and expensive
but much less costly than DNS of the same flow, because it is more accurate, DNS is the
preferred method whenever it is feasible. LES is the preferred method for flows in which the
Reynolds number is too high or the geometry is too complex to allow application of DNS.
18

Chapter I Introduction
• RANS models: Engineers are normally interested in knowing just a few quantitative
properties of a turbulent flow. In Reynolds Averaged approaches to turbulence, all of the
unsteadiness is averaged out. All unsteadiness is regarded as part of the turbulence. On
averaging, the non-linearity of the Navier-Stokes gives rise to terms that must be modeled,
just as they did earlier. The complexity of turbulence makes it unlikely that any single
Reynolds-Averaged model will be able to represent all turbulent flows so turbulent models
should be regarded as engineering approximations rather than scientific laws.
• Very Large Eddy Simulation: It appears that we have to either use RANS, which is
affordable, or LES, which is more accurate but rather expensive. It is natural to ask whether
there is a method that provides the advantages of both RANS and LES while avoiding the
disadvantages. The method that accomplished this is called Very Large Eddy Simulation or
Unsteady Reynolds-Averages Numerical Simulation, in this method, one uses a RANS
model but computes an unsteady flow. The results often contain periodic vortex shedding.
When the results of such a simulation are time-averaged, they often agree better with
experiment than steady RANS computations.
• Detached Eddy Simulation: DES is suggested for separated flows (Travin et al., 2000). In
this approach, RANS is used for the attached boundary layer and LES is applied to the free
shear flow resulting from separation. This requires some means of producing the initial
conditions for the LES in the separation region and this are a difficulty.
1.2 Study Objectives:
The flow in tube bundles is of great interest to the power generation industry, not only for the
study of performance of heat exchangers. Safety studies require predictions of vibrations
caused by fluid-structure interaction or large temperature fluctuations that eventually lead to
thermal stripping. The flow within the bundles experiences complex unsteady behaviour,
making it an attractive case to be studied using different numerical model. The unsteadiness in
this flow can be a result of imposed fluctuating boundary conditions. The presence of such
unsteadiness in a flow can significantly after the evolution of different parameters such as
Reynolds stresses u iu , turbulent kinetic energy κ, and dissipation rate ε. Despite the
j
existence and the utility of progressively more complex modeling.
URANS models are widely used for the complex unsteady flows. In the present case URANS
models are used to examine the flow predictions in the in-line tube bundles. The URANS
19

Chapter I Introduction
models tested are standard κ – ε, Menter`s shear stress transport (MSST) and the Reynolds
Stress Models (RSM). Other models are used, the new SST-Cas model (the standard SST
model is used alone for the first few time steps in order to initialize the calculation for the
SST-Cas model).
After obtaining results, they must be compared with LES or available experimental data and
then assess which of those models is able to reproduce an unsteady flow behaviour across the
tubes.
1.3 Outline of the thesis:
The work presented here is organized as follows. After the introduction, Chapter 2 presented
a literature review of in-line tube bundle. Chapter 3 gives a review of the existing turbulence
models and those which are used during the course of this project. The numerical aspects of
the code used and steps of the numerical study are presented in Chapter 4. In Chapter 5 a
turbulent flow across in-line tube bundle studied using different URANS models, SST-Cas
model and DES is presented together with results and discussion of the simulation in this
Chapter.
Finally, Chapter 6 includes conclusions together with suggestions for future work.
20

Chapter II Literature Review
Chapter 2
Literature review
2.1 Introduction
The flow within the bundles experiences complex unsteady behaviour, Random excitation
forces can cause low-amplitude tube motion that will result in-long-term-fretting-wear or
fatigue. All these problems attract the attention of researchers in the whole world.
2.2 Literature review of tube bundles
2.2.1 LES of tube bundles
Rollet-Miet et al. [58] presented the first LES calculations using the finite-element method
of the turbulent flow across a staggered tube bundle. Since then, few publications have
appeared although tube bundles are widely employed in cross-flow heat exchangers, as they
combine case of construction with good thermal and mechanical efficiency. Rollet-Miet et al
[58] pointed out the superiority of the LES technique because it is better suited the flows
where the size of eddies (integral length case of the turbulence) is comparable to the size of
the obstacles of the flow.
Benhamadouche and Laurence [13] performed similar LES calculations for the turbulent flow
across the staggered tube bundle using the finite-volume method on a collocated unstructured
grid. They found that the type of the subgrid scale models (whether the standard or the
dynamic Smagorinsky) is not critical for this type of application. In this same year, Charles
Moulinec, J.C.R Hunt and F.T.M Niuwstadt studied flow through a staggered array of parallel
rigid cylinders computed with the help of a three dimensional (DNS) at Re=500 and 6000,
when Re

Chapter II Literature Review
development of rms level along the flow lane. A steady recirculation region consisting of a
pair of counter-rotating vortices exists in the gap as found by Ziada and Oengoren [59].
This regime is called reattachment regime. Turbulence intensities at this regime are small
compared to the alternate vortex shedding regime. However, Liang and Papadakis [4]
predicted a clear vortex shedding frequency behind the first cylinder and this is one reason
that the LES calculations for the in-line tube bundle over-predicted the rms level behind that
cylinder.
Konstantinidis et al. [79] examined experimentally the effect of inlet flow pulsation in across
flow over a tube array with an external frequency around twice that the flow pulsation
activates the flow field behind the first cylinder and increases the turbulence intensities for the
first three cylinders.
In a later Study, Konstantinidis et al. [80] also observed a symmetrical vortex formation mode
when the external frequency is around triple that of the natural alternate vortex shedding. The
effect of pulsation on the flow field and heat transfer in a six row in-line tube array is
investigated using 3D LES technique by Chunlei Liang and George Papadakis [4].
2.2.2 Heat transfer in tube bundles
Akhilech Gupta [66], an experiment of nucleate boiling heat transfer in an electrically
heated 5X3 in-line horizontal tube bundle under pool and low cross flow. It is observed that
the heat transfer is minimum on bottom row tubes and increases in the upward direction with
maximum values on top row tubes. Also, heat transfer coefficient on central column tubes was
found to be slightly higher than those on the corresponding side tubes.
Chunlei Liang and George Papadakis [10] the aim is to study the effect of pulsation on the
field and convective heat transfer over an in-line cylinder array at a sub critical Reynolds
number Re=3400 (based on cylinder and the gap velocity across the minimum section) using
LES technique. The heat transfer rate in the front part of the second cylinder is greatly
enhanced due to the vortex shedding lock on behind the first cylinder. LES computations of
six-row cylinders demonstrated that heat transfer around the third row and downstream
cylinders are not influenced much by the external pulsation.
Haitham M.S. Bahaidrah and M. Ijaz and N.K. Anauds [2], two dimensional study developing
fluid flow and heat transfer across five in-line tube bundle with a Prandlt number of 0.7. The
tube cross-sectional shapes studied were circular, flat, oval and diamond tubes were compared
22

Chapter II Literature Review
with each other and with those for circular tubes. Flat and oval tubes offered greater flow
resistance and heat transfer rate when compared with circular cylinders for all values of
Reynolds number. Diamond tubes offered less resistance and heat transfer rate. For Re>50,
flat and oval tubes performed better.
2.2.3 Pressure Fluctuations
Ishigai et al. [60] investigates the flow pattern for a wide range of gap ratios. It is reported
that for in-line tube bundle five distinct regions are formed. However, in case of square tube
bundles only three distinct flow patters are observed; for very narrow gap ratios the free shear
layer of the front of the cylinder attaches to the downstream cylinder thus stopping the
Karman vortices to develop, for moderate gap ratio the Karman vortices are shed but are
distorted and deflected due to downstream suppression, for very wide gap ratios regular
Karman vortices are shed much like in the case of a single cylinder.
Aiba et al. [61] perform experimental study on square in-line tube banks for gap ratio of 1.2
and 1.6. It is observed that the tube response of the downstream cylinders is quite different
from the upstream ones. The pressure distribution around the cylinder surface shows highly
deflected flow with a stagnation point of 45 degrees. The flow behaviour is asymmetric both
these configurations which is owed to the narrow gap ratios.
Traub [62] conducts open wind tunnel experiments to study the influence of turbulence
intensity on pressure drop in in-line and staggered tube bundles at various Reynolds numbers.
It is observed that the drag coefficient remains more or less the same for a wide range of
Reynolds numbers and only changes slightly for very high Reynolds numbers. It is also
concluded from this study that as the Reynolds number becomes very high the circulation
region shrinks due to shifting of the point of flow separation. Due to this shifting, the pressure
drop decreases and hence the drag coefficient decreases. The paper provides experimental data
of drag coefficient over a wide range of Reynolds number for various gap ratios.
Lam and Fang [63] perform an experimental study on the effect of gap ratio on the flow over a
square four cylinder in-line configuration. The paper discusses flow pattern, pressure
distribution and lift and drag forces on cylinders at a Reynolds number of 12,800 based on
free stream velocity. It is seen that at small gap ratios due to suppression of wake region
vortex shedding is hardly formed. Moreover for these gap ratios the stagnation point is not at
zero degrees rather the shift is 20-50 degrees from flow direction.
23

Chapter II Literature Review
Ishigai et al. [60] study fluid flow over in-line tube bundles for very low Reynolds numbers
using finite element technique. The main idea behind the paper is to study the effect of
pressure drop on heat transfer. The research concludes that recirculation between the cylinders
increases with an increase in Reynolds number. This causes the separation point to move
further away from the rear stagnation point. In other words the angle of separation when
measured from the front stagnation point decreases with an increase in Reynolds number.
At a low Reynolds number of 200 Lam et al. [64] performed a particle image velocitymetry on
a four cylinder square array where the gap ratio was 4.
2.2.4 Vortex Shedding
Van Atta et al. [65] studied chaotic and organized vortex shedding behind self excited
cylinder wakes at fairly low Reynolds number using hot wire measurements and smoke flow
visualization techniques. Their research revolves around two cases, first one being the ordered
lock-in case in which only a single high order harmonic vibration frequency is excited, results
indicated that the wake structure is span-wise periodic. The second case is the fully chaotic
one where several high order vibration modes are simultaneously excited, results from this
case show that the vortex street is disorganized and is definitely not span-wise periodic,
however the statistical properties such as velocity signals are independent of the span-wise
position.
For a three cylinder arrangement a similar flow behavior is seen but with a switching in
direction. Thus the flow pattern of the three by three arrangements is termed to be meta-stable
by Zdravkovich and Stonebanks [67]. Kim and Durbin [68] suggest that this biased flow
behavior is due to the turbulent perturbations in the incoming flow and is an intrinsic property
of flow. Sayers [69] presents yet another experimental study for a four cylinder arrangement
showing either a total suppression of vortex shedding or an asymmetrical pattern for very
narrow gap ratios.
Lam and Lo [70] have done extensive water tunnel experimentations on the wake formation
and vortex shedding frequency of a square cylinder bundle at low Reynolds number of 2100
with different angles of attack. Our interest is only the zero angle of attack that is the in-line
arrangement. The bundle gap ratio varies from 1.28 to 5.96 along with the angle of attack.
Interesting thing to note from this study is that a bitable state of wide and narrow wake exists
for aspect ratios below 1.54. The asymmetric mode is initiated by an outward deflection of the
outer shear layer of upstream cylinders which rolls up besides the downstream cylinders.
24

Chapter II Literature Review
Based on these observations it is thus concluded that a distinct oscillation of wake exists in
the downstream flow. Finally the Reynolds number has little effect on the size and shape of
the wake since low spacing prohibits lengthening of the shear layers in the down stream
direction. Ogengoren and Zaida [59] have done an experimental study on the vortex shedding
and resonance in an in-line tube bundle. According to the authors when resonance occurs,
pressure pulsations at discrete frequencies are produced. The high amplitude of these pressure
pulsations causes vibrations and noise problems. The modes consisting of standing waves in
normal direction to the flow is most likely to cause resonance. The flow pattern of a non
resonant mode and a resonant mode is entirely different. When resonance occurs vortices
forming behind tubes have the same sense of rotation and are shed simultaneously from the
same sides of the tubes. This means that the vortices behind all the tubes have the same sense
and phase. Surface wave resonance is stated to be the reason behind this synchronization.
Another interesting thing to note from this study is that when the stream-wise gaps between
tube arrays is less than the tube diameters wake velocity profile does not develop. Under these
circumstances the gap scan is regarded as cavities bounded by shear layers which separate
from the tube edges. The instability of these shears layers which is triggered and synchronized
by the resonance mode causes the asymmetric flow pattern.
Finally the study concludes that this asymmetric behavior is only seen when the stream-wise
gap distances fairly narrow. For wider gap ratios the resonance occurs when the frequency of
vorticity shedding approaches the resonance frequency, the causes of this deviation from
standard or classical reasoning is still under investigation and is an unsolved dilemma.
Summer et al. [71] have done an extensive experimental study on two and three cylinders
placed side by side for a Reynolds number range of 500-3000. It has been observed that both
the two cylinder and three cylinder configurations show various flow patterns for different gap
spaces. In this study the transverse ratio (T/D) is varied from 1.0 to 6.0. The three regions
classified for this variation in T/D ratios are, small (T/D < 1.2), intermediate (1.2 < T/D 2.2). In the intermediate region which is also the interest region of our study
an asymmetrical flow pattern is reported. This is also reported by Sumner et al. [71] and Kim
&Durbin [68]. It is also observed by Kim & Durbin [68] that the biased flow pattern switches
intermittently from being directed towards one cylinder to the other. Thus such a flow pattern
is termed to be bistable. They concluded that this flow behavior is independent of Reynolds
number and purely dependent upon the T/D ratio. Summer et al. [71] further correlates the two
showing that the flow deflection decreases as the T/D ratio is increased.
25

Chapter II Literature Review
Wolfe and Ziada [78] used a feedback control on vortex shedding from two tandem cylinders.
It was concluded that when a cylinder is placed in the wake of another cylinder then its
unsteady loading is not only dependent upon the flow behavior in its own wake but also on the
flow pattern in the wake of the upstream cylinder. On the basis of this, a feedback control was
applied to reduce the response of the downstream cylinder to both turbulence excitations and
vortex shedding. The study was based on two cases, the resonant case (lower flow speed, Re =
41100) where the cylinder frequency of the vortex shedding coincides with the resonance
frequency of the downstream cylinder and the non resonant case (slightly higher speed, Re =
57900). The feedback control did not reduce the velocity fluctuations at the vortex shedding
frequency instead it shifted the vortex shedding frequency to a higher level.
Samir Ziada [77] describes the vorticity shedding excitation in tube bundles and its relation to
the acoustic resonance mechanism. These phenomena are investigated by means of velocity
and pressure measurements, as well as with the aid of extensive visualization of the unsteady
flow structure at the presence and absence of acoustic resonance. Vorticity shedding excitation
is shown to be generated by either jet, wake, or shear layer instabilities. The tube layout
pattern (in-line or staggered), the spacing ratio, and Reynolds number determine which
instability mechanism will prevail, and thereby the relevant Strouhal number for design
against vorticity shedding and acoustic resonance excitations. Strouhal number design charts
for vortex shedding in tube bundles are presented for a wide range of tube patterns and
spacing ratios. Regarding the acoustic resonance mechanism, it is shown that the natural
vorticity shedding, which prevails before the on set of resonance, is not always the source
exciting acoustic resonance. This is especially the case for in-line tube bundles. Therefore,
separate "acoustic" Strouhal number charts must be used when appropriate to design against
acoustic resonances. To this end, the most recently developed charts of acoustic Strouhal
numbers are provided.
2.2.5 Vibrations:
Weaver et al. [72] show another interesting study in which flexible cantilever in-line
cylinder arrays have been experimentally tested using wind tunnel for a P/D ratio of 2.01 and
3.56. The paper has two different configurations; smooth cylinders and finned cylinders. For
the smooth cylinder configuration the study is based on three different models; (i) a single
flexible tube amongst a set of fixed tubes, (ii) a whole array of flexible tubes and (iii) a bundle
26

Chapter II Literature Review
of flexible tubes. The paper presents data relating to root mean square tip amplitude at various
pitch flow velocities, where the pitch flow velocity is defined in the same way as the gap
velocity for a square tube bundle array. The study revolves around the fluid elastic instability
which is defined as the excitation mechanism which causes the most violent vibrations leading
to rapid tube failure. The flow velocity at which this failure occurs is termed as critical or
threshold velocity. Price and Paidoussis [73] investigate a single flexible cylinder placed
inside a rigid tube bundle configuration with a gap ratio of 1.5. Data for this case corresponds
gap velocities and turbulence intensities.
Feenstra et al. [74] experimentally study flow induced vibrations in a cantilevered tube bundle
array with single and two phase cross-flow. For a single phase flow the study addresses two
cases; a single flexible tube in an otherwise rigid tube bundle and a fully flexible tube bundle
configuration. It was observed that for the single flexible tube configuration fluid elastic
instability was achieved at 25% higher flow velocity and symmetric vortex shedding occurred
at 50% higher flow velocities. The paper presents an excellent comparison with previous
experimental studies conducted by Paidoussis [75] and Weaver & Fitzpatrick [76] at a gap
ratio of 1.5 using water tunnel.
27

Chapter III Turbulence modelling
Chapitre 3
Turbulence modelling
3.1 Introduction
Turbulence modeling is a key issue in most Computational Fluid Dynamics simulations.
Virtually all engineering applications are turbulent and hence require a turbulence model.
3.2 Navier-Stokes equations
The Navier-Stokes equations are the basic governing equations for a viscous, heat
conducting fluid. It is a vector equation obtained by applying Newton's Law of Motion to a
fluid element and is also called the momentum equation. It is supplemented by the mass
conservation equation, also called continuity equation and the energy equation. Usually, the
term Navier-Stokes equations are used to refer to all of these equations.
The Navier-Stokes equation is:
3.2.1 Reynolds Averaging
2
∂ui
∂ui
1 ∂P
∂ ui
+ u j = − + v
∂t
∂u
ρ ∂x
∂x
∂x
j
i
j
j
(3.1)
In Reynolds ensemble averaging the solution variables of the Navier-Stokes equations are
decomposed into mean and fluctuating parts
u = u + u'
(3.2)
i
Where ui , ui and u'i
are the instantaneous, mean and fluctuating components respectively. The
scale quantities such as pressure are also decomposed the same principle.
i
i
P = P + P'
(3.3)
Inserting the decomposition form equation (3.2) and (3.3) into equation (3.1) gives
2
( u + u'
) ∂(
u + u'
) 1 ∂(
P + P)
∂ ( u + u )
∂ i i
i i
i '
+ u j = − + v
∂t
∂x
ρ ∂x
∂x
∂x
j
i
j
j
(3.4)
Expanding equation (3.4), taking time averaging, ignoring mean of the fluctuating quantities
and keeping mean of mean quantities gives
2
∂ui
∂u
u'
i j ∂u'i
1 ∂P
∂ ui
+ u j + = − +
∂t
∂x
∂x
ρ ∂x
∂x
∂x
j
j
28
i
j
j
(3.5)

Chapter III Turbulence modelling
The third term on the left hand side of equation (3.5) can now be further modified as
u'
j ∂u'i ∂u'
j u'i
u'i
∂u'
= −
∂x
∂x
∂x
j
j
j
j
(3.6)
Where the 2nd term on the right hand side of equation (3.6) can be dropped out for an
incompressible case. Thus the final Reynolds Averaged Navier-Stokes equations become
2
∂ui
∂ui
1 ∂P
∂ u ∂u'
i i u'
j
+ u j = − + v −
∂t
∂x
ρ ∂x
∂x
∂x
∂x
j
i
j
j
j
(3.7)
Where the quantities with over-bar are mean ensemble averaged quantities. The last term in
equation (3.7) is the Reynolds stresses which need to be modelled for closure of RANS
equations.
3.3 Classes of turbulence models
3.3.1 Algebraic turbulence models
Algebraic turbulence models or zero-equation turbulence models are models that do not
require the solution of any additional equations, and are calculated directly from the flow
variables. As a consequence, zero equation models may not be able to properly account for
history effects on the turbulence, such as convection and diffusion of turbulence energy. These
models are often too simple for use in general situations, but can be quite useful for simple
flow geometries or in start-up situations. The two most well known zero equation models are:
3.3.1.1 Baldwin-Lomax model
Baldwin and Lomax [20] is a two-layer algebraic 0-equation model which gives the eddy
viscosity, μ t as a function of the local boundary layer velocity profile. The model is suitable
for high speed flows with thin attached boundary-layers, typically present in aerospace and
turbomachinery applications. It is commoly used in quick design iretation where robustness is
more important than capturing all details of the flow physics. The Baldwin-Lomax model is
not suitable for cases with large separated regions and significant curvature/rotation effects.
For more information see also [21], [22], [23], [24].
29

Chapter III Turbulence modelling
3.3.1.2 Cebeci-Smith model:
Smith and Cebeci [25] is a two-layer algebraic 0-equation model which gives the eddy
viscosity, μ t as a function of the local boundary layer velocity profile. The model is suitable
for high-speed flows with thin attached boundary-layers, typically present in aerospace
applications. Like the Baldwin-Lomax model, this model requires the determination of a
boundary layer edge.
Other even simpler models, such a models written as ( ) , are sometimes used in
+
μ = f y
particular situations (e.g boundary layers or jets). See also [26]
3.3.2 One-equation turbulence models
One equation turbulence models solve one turbulent transport equation, usually the turbulent
kinetic energy. The original one-equation model is Prandtl’s one-equation model [27], [28],
[29].
3.3.2.1 Prandtl’s one-equation model
Kinematic eddy viscosity
Equation of the model
∂k
+ U
∂t
j
∂k
− C
∂x
Closure coefficients and auxilary relations
ε =
CD
σ k
C D
3
k
l
2
= 0.
08
= 1
where
2
τ ij = 2vT Sij
− kδij
3
j
1
vt D
D
t
2
k
2 = k l = C
(3.8)
ε
3
k
l
2
+
30
∂
∂x
j
⎡⎛
v ⎞ ⎤
T ∂k
⎢ ⎜
⎜v
+
⎟ ⎥
⎢⎣
⎝ σ k ⎠ ∂x
j ⎥⎦
(3.9)

Chapter III Turbulence modelling
Another popular one-equation model is the Spallart-Allmaras model [32], [33], [34], that uses
a transport equation for the viscosity including eight closure coefficients and three damping
functions, in a similar way to the Baldwin-Barth models [30], [31].
3.3.3 Two equation turbulence models
Two equation turbulence models are not of the most common type of turbulence models.
Models like the k-epsilon and the k-omega model have become industry standard models and
are commonly used for most types of engineering problems. Two equation turbulence models
are also very much still an active area of research and new refined two-equation models are
still being developed.
By definition, two equation models include two extra transport equations to represent the
turbulent properties of the flow. This allows a two equation model to account for history
effects like convection and diffusion of turbulent energy.
Most often one of the transported variables is the turbulent kinetic energy, k. The second
transported variable varies depending on what type of two-equation model it is. Common
choices are the turbulent dissipation, ε, or the specific dissipation, ω. The second variable can
be thought of as the variable that determines the scale of the turbulent (length-scale of timescale).
Whereas the first variable, k, determines the energy in the turbulence.
3.3.3.1 Boussinesq eddy viscosity assumption
The basis for all two equation models is the Boussinesq eddy viscosity assumption, which
postulates that the Reynolds stress tensor, τ ij is proportional to the mean strain rate tensor,
and can be written in the following way:
τ
2
= 2 +
(3.10)
3
ij μtS
ij ρκδij
Where μ t is a scalar property and is called the eddy viscosity which is normally computed
from the two transported variables. The last term is included for modelling incompressible
flow to ensure that the definition of turbulence kinetic energy is obeyed:
u'i
u'i
κ =
(3.11)
2
31
Sij

Chapter III Turbulence modelling
The same equation can be written more explicitly as:
⎛ U U ⎞
i j 2
ρu'
i u'
j μ ⎜
∂ ∂
− = t + ⎟ + ρκδ
⎜
ij
(3.12)
x j x ⎟
⎝ ∂ ∂ i ⎠ 3
The boussinesq assumption is both the strength and the weakness of two equation models.
This assumption is a huge simplification which allows one to think of the effect of turbulence
on the mean flow in the same way as molecular viscosity effects a laminar flow. The
assumption also makes it possible to introduce intiutive scalar turbulence variables like the
turbulent energy and dissipation and to relate these variables to even more intuitive variables
like turbulence intensity and turbulence length scale.
The weakness of the Boussinesq assumption is that it is not in general valid. There is nothing
which says that the Reynolds stress tensor must be proportional to the strain rate tensor, it is
true in simple flows like straight boundary layers and wakes, but in complex flows, like flows
with strong curvature, or strongly accelerated or decellerated flows the Boussinesq assumption
is simply not valid. This gives two equation models inherent problems to predict strongly
rotating flows and other flows where curvature effects are significant. Two equation models
also often have problems to predict strongly decellerated flows like stagnation flows.
3.3.3.2 K-epsilon models
The k-epsilon model is one of the most common turbulence models. It is a two equation
model that means, it includes two extra transport equations to represent the turbulent
properties of the flow. This alows a two equation model to account for history effects like
convection and diffusion of turbulent energy. The first transported variable is turbulent kinetic
energy k. It is the variable that determines the energy in the turbulence. The second
transported variable in this case is the turbulent dissipation. ε. It is the variable that determines
the escale of the turbulence. The usual k-epsilons models are:
• Standard k-epsilon model
• Realisable k-epsilon model
• RNG k-epsilon model
The model used in the present work is standard k-epsilon model.
32

Chapter III Turbulence modelling
Transport equations for standard k-epsilon model
For turbulent kinetic energy k
∂
∂t
For dissipation ε
∂
∂t
∂
∂x
∂ ∂ ⎡⎛
μ ⎞ ∂k
⎤
ρ
xi
x ⎜
+
⎟
(3.13)
∂ ∂ j ⎢⎣
⎝ σ k ⎠ ∂x
j ⎥⎦
t
( k)
+ ( ρkui
) = ⎢⎜
μ ⎟ ⎥ + Pk
+ Pb
− ρε − YM
+ Sk
∂
∂x
⎡⎛
μ ⎞ ∂ε
⎤
⎜
+
⎢⎣
σ ⎟
⎝ ε ⎠ ∂x
j ⎥⎦
t
( ρε ) + ( ρεui
) = ⎢⎜
μ ⎟ ⎥ + C ε ( Pk
+ C3ε
Pb
) − C2ε
ρ + Sε
Modelling turbulent viscosity
Production of k
i
j
P
k
ε
k
2
ε
k
1 (3.14)
2
k
= ρC
(3.15)
ε
μt μ
= − '
∂U
j
ρ u'i
u j
(3.16)
∂xi
Pk t
= μ S
Where S is the modulus of the mean rate-of-strain tensor, defined as:
The constants have the values shown in Table (3.1)
C 1ε
2ε
C μ
2
ij ij S
(3.17)
S S = 2
(3.18)
C σ k
σ ε
1.44 1.92 0.09 1.0 1.3
3.3.3.3 K-omega models
Table 3.1: Coefficients of the standard k-ε model.
The k-omega model is one of the most common turbulence models. It is a two equation
model that means, it includes two extra transport equations to represent the turbulent
properties of the flow. This alows a two equation model to account for history effects like
convection and diffusion of turbulent energy. The first transported variable is turbulent kinetic
energy k. It is the variable that determines the energy in the turbulence. The second
33

Chapter III Turbulence modelling
transported variable in this case is the turbulent dissipation, ω. It is the variable that
determines th escale of the turbulence. The common used k-omega models are :
• Wilcox’s k-omega model [35]
• Wilcox’s modified k-omega model [36]
• SST k-omega model [37], [38]
The model used in the present work is SST k-omega model.
SST k-omega model
The SST k-ω turbulence model, Menter [37], is a two-equation eddy-viscosity model which
has become very popular. The use of a k-ω formulation in the inner parts of the boundary
layer makes the model directly usable all the way down to the wall through the viscous sub-
layer, hense the SST k-ω model can be used as Low-Re turbulence model without any extra
damping functions. The SST formulation also switches to a k-ε behaviour in the free-stream
an avoid the common k-ω problem that the model is too sensitive to the inlet free-stream
turbulence properties. Authors who use the SST k-ω model often merit it for its good
behaviour in adverse pressure gradients and separating flow. The SST k-ω model does
produce a bit too large turbulence levels in regions with large normal strain, like stagnation
regions with strong acceleration. This tendency is much less pronounced than with a normal kε
model though.
Kinematic eddy viscosity
Turbulence kinetic energy
∂k
+ U
∂t
Specific dissipation rate
j
∂k
∂x
j
v T
= P
k
−
a1k
=
max F
β *
( a ω,
Ω )
1
∂
kω
+
∂x
j
⎡
⎢
⎢⎣
2
( v + σ v )
k
T
∂k
⎤
⎥
∂x
j ⎥⎦
∂ω ∂ω
⎡
⎤
2 2 ∂
∂ω
1 ∂k
∂ω
+ U j = αS
− βω + ⎢(
v + σ ωvT
) ⎥ + 2( 1−
F1
) σ ω 2
∂t
∂x
j
∂x
j ⎢⎣
∂x
j ⎥⎦
ω ∂xi
∂xi
34
(3.19)
(3.20)
(3.21)

Chapter III Turbulence modelling
Closure coefficients and auxillay relation
4
⎧
⎫
⎪⎪⎧
⎡ ⎛ k 500v
⎞ 4σ
⎤⎪⎫
⎪
⎨⎨
⎢ ⎜ ⎟ ω 2k
F1
= tanh min max
⎜
,
⎟
,
* 2
2 ⎥⎬
⎬
⎪⎪⎩
⎢
⎥⎪⎭
⎩ ⎣ ⎝ β ωy
y ω ⎠ CDkω
y
⎦ ⎪
⎭
(3.22)
⎛ 1 ∂k
∂ω
−10 ⎞
CD =
⎜
⎟
kω
max 2ρσω
2 , 10
(3.23)
⎝ ω ∂xi
∂xi
⎠
2
⎡⎡
⎛
⎤
⎢
2 k 500v
⎞⎤
F ⎢ ⎜ ⎟⎥
⎥
2 = tanh max
⎢ ⎜
, * 2 ⎟
(3.24)
⎥
⎣
⎢⎣
⎝ β ωy
y ω ⎠⎥⎦
⎦
⎛
⎞
⎜
∂Ui
*
P
⎟
k = min τ β kω
⎜ ij , 20
(3.25)
⎟
⎝ ∂x
j ⎠
1F1 + φ2(
1− F1
φ = φ
)
(3.26)
The constants have the values shown in Table 3.2
α 1 α 2 β 1 β 2
*
β σ k1
σ k 2 σ ω1
σ ω 2
5/9 0.44 3/40 0.0828 9/100 0.85 1 0.5 0.856
3.3.4 V2-f models
Table 3.2: Coefficients of the k-ω SST model
2
The υ − f model [39] is similar to the standard k-ε model. Additionally, it incorporates also
some near-wall turbulence anisotropy as well as non-local pressure-strain. It is a general
turbulence model for low Reynolds numbers. That does not need to make use of wall
2
functions because it is valid upto solid walls. Instead of turbulent kinetic energy k. the υ − f
model uses a velocity scale
2
2
υ for evaluation of the eddy viscosity. υ can be thougth of as
the velocity fluctuation normal to the streamlines. It can provide the right scaling for the
presentation of the damping of turbulent transport close to the wall. The anisotropic wall
effects are modelled through the elliptic relaxation function ƒ, by solving a separate elliptic
equation of the Helmholtz type. In order to improve the computational performances of the
35

Chapter III Turbulence modelling
2
υ − f model, a variant of this eddy-viscosity model is derivied when the change of variables
is introduced. Instead of using the wall-normal velocity fluctuation
2
υ as the velocity scale,
2
the normalised wall-normal velocity scale ζ = υ / k is used. This turbulence variable can be
regarded as the ratio of the two time scales: scalar k/ε (isotropic), and lateral
2
υ /
ε (anisotropic). Following the definition of ζ, the new transport equation is derivied from
2
2
the equation for υ and k, and solved instead of the transport equation forυ
. See also [40].
3.2.5 Reynolds stress model (RSM)
The Reynolds stress model (RSM) [41] is a higher level, elaborate turbulence model. It is
usually called a second order closure. This modelling approach originates from the work by
Launder. [41]. In RSM, the eddy viscosity approach has been discarded and the Reynolds
stresses are directly computed. The exact Reynolds stress transport equation accounts for the
directional effects of the Reynolds Stress fields.
The Reynolds stress model involves calculation of the individual Reynolds stresses, ρ u'i u'
j
using differential transport equations. The individual Reynolds stresses are then used to obtain
closure of the Reynolds-averaged momentum equation.
The exact transport equations for the transport of the Reynolds stresses, u' i u'
j may written as
follows:
Or
∂
∂t
∂
∂
( ρu'
u'
) + ( ρu
u'
u'
) = − [ ρu'
u'
u'
+ P(
δ u'
+ δ u'
) ]
∂ ⎡ ∂
+ ⎢μ
∂xk
⎣ ∂xk
i
j
∂x
⎤ ⎛ ∂u
j ∂u
⎞ i
( u'
u'
) − ρ⎜u
' u'
+ u'
u'
⎟ − ρβ ( g u'
θ + g u'
θ )
i
k
j
⎥
⎦
⎛ u u ⎞
i j u ∂u
i j
P⎜
∂ ' ∂ '
⎟
∂ ' '
+ + − 2μ
− 2ρΩ
⎜
k
x j x ⎟
⎝ ∂ ∂ i ⎠ ∂xk
∂xk
k
i
⎜
⎝
j
i
k
∂x
∂x
Local time derivative+ =
ij
k
k
j
i
k
j
k
∂x
k
⎟
⎠
kj
( u'
j u'mε
ikm + u'i
u'mε
jkm ) + Suser
i
i
ik
j
j
j
i
+
(3.27)
C D T , ij + DL,
ij + Pij
+ Gij
+ φij
− ε ij + Fij
(3.28)
+ User defined source term.
36

Chapter III Turbulence modelling
Where is the convection term, equals the turbulency diffusion, stands for the
Cij D T , ij
D L,
ij
molecular diffusion, is the term for stress production, equals buoyancy production, ε
Pij Gij ij
stands for the dissipation and Fij
is the production by system rotation.
Dissipation
The dissipation term can be written as the sum of the isotropic and deviatoric parts:
2
ε ij = εδij
+ Dε ij
3
(3.29)
In most models, the isotropic part is calculated via a transport equation and the deviatoric part
is lumped into the pressure-strain correlation:
φ −
ij mod elled = φij
Dε ij
(3.30)
The transport equation for the isotropic part of the dissipation rate can be derived from the
fluctuating momentum equation [42]. The result is a transport equation with higher order
terms that require further modelling. Although the inaccuracy of the modelling involved on
the dissipation equation is an obvious deficiency, the resulting equation is widely used and is
almost standard for all the models, including many two-equation models. The standard
transport equation for the dissipation rate proposed by Hanjali´c and Launder [43] is:
∂ε
+ U
∂t
Where Pk is defined as:
k
∂ε
= C
∂x
k
Pkε
− C
k
ε ∂ ⎛
+ ⎜C
∂x
⎜
⎝
k
∂ε
⎞
⎟
∂x
⎟
ij ⎠
ε1
ε 2
2
k j
ε u'i
u'
j
ε
(3.31)
∂U
P = −u
'
k
i ' i u j
(3.32)
∂x
j
The coefficients , and C vary according to the pressure-strain closure but in general,
Cε Cε1 ε 2
Cε 2 is set to 1.9 to match the decay rate of isotropic turbulence; Cε
is set between the values
0.15 and 0.18 and C usually takes the value of 1.44 [42].
Diffusion
ε1
The most popular way of representing the diffusive terms is the generalized gradient
diffusion hypothesis, GGDH [54] which can be written as:
37

Chapter III Turbulence modelling
D
T
ij
= −
∂
∂x
k
⎛
⎞
⎜ k ∂u'i
u'
j
C
⎟
s u'k
u'l
⎜
⎟
⎝
ε ∂xl
⎠
(3.33)
Where Cs = 0.22 is a constant determined from model optimization [42]. Other models have
been proposed and can be found in [43], [46] and [42]. It is important to note that some of the
diffusion models do not take into account the pressure-diffusion terms. This might be seen as a
source of inaccuracy and although it does not seem to be critical in the modelling; it has been
called into question [45].
The inclusion of the velocity-pressure correlation can be seen in the value of the empirical
constant Cs. The model constant obtained without the inclusion of the pressure diffusion is
about 20% greater than the usual 0.22.
Pressure strain
The pressure-strain correlation term is the term where most of the modelling effort has been
made over the last thirty years. This term is important because it has the same order as the
production term and it tends to redistribute the energy between the Reynolds stress
components, diminishing the difference between them.
Derived from the Navier-Stokes equations and the continuity equation, the pressure field
satisfies the Poisson equation:
and the fluctuating pressure satisfies:
u ∂u
2 ∂ i j
∇ p = −ρ
∂x
∂x
j
i
( u'
u'
−u'
)
i
j
j
−
∂x
j
2
∂xi∂x
j
i j i u j
(3.34)
U u'
2 ∂ ∂ ∂
∇ p'= −2ρ
ρ
'
(3.35)
∂x
rapid
The first part is directly dependent on the mean velocity which makes it respond rapidly to the
velocity changes. The second part is nonlinear and involves interaction between fluctuating
velocities. This decomposition is also carried out into most of the commonly used Second
Moment Closures, the pressure-strain term is divided into a slow and a rapid part and then a
correction term is added such as the wall echo term:
φ +
slow
ij = φij1
+ φij
2 φω
(3.36)
38

Chapter III Turbulence modelling
One of the most popular models is the linear Launder, Reece and Rodi (LRR) [41] which
models the pressure strain term as:
( b Ω + b )
⎛
2 ⎞
φ ij = −C1εbij
+ C2kSij
+ C3k⎜
bikS
jk + bjkS
ik − bmnSmnδij
⎟ + C4k
ik jk jkΩik
(3.37)
⎝
3 ⎠
Where bij is the normalised anisotropy tensor, Sij is the strain tensor and Ω ij is the rotation
rate tensor defined as:
aij
u'i
u'
j 1
bij = = − δij
(3.38)
2k
2k
3
S
ij
Ω
ij
1 ⎛ ⎞
⎜ ∂U
∂U
i j
= + ⎟
2 ⎜ ⎟
⎝
∂x
j ∂xi
⎠
1 ⎛ ⎞
⎜ ∂U
∂U
i j
= − ⎟
2 ⎜ ⎟
⎝
∂x
j ∂xi
⎠
The constants have the values shown in Table 3.3
C 1
2 C C 3
4 C C ε1
C ε 2
3.0 0.8 1.75 1.31 1.44 1.90
Table 3.3: Coefficients of the LRR model.
(3.39)
(3.40)
Another popular model for the pressure-strain correlation term is the Speziale, Sarkar and
Gatski model (SSG) [47] which has a quadratic behaviour included originally as a higher
order correction to the slow part of the pressure-strain correlation [42]. The model for the
pressure-strain term is:
⎛ 1 ⎞
φij
= −C1εbij
+ C'1
ε⎜
bikbkj
− bmnbnm
⎟ + C2kS
⎝ 3 ⎠
⎛
2 ⎞
+ C3k⎜
bikS
jk + bjkSik
− bmnSmnδij
⎟
⎝
3 ⎠
+ C
4)
( b Ω + b Ω )
ik
jk
jk
ik
39
ij
(3.41)

Chapter III Turbulence modelling
The model uses the coefficients showed in Table 3.4
C 1
1 ' C 2 C C 3
4 C C ε1
C ε 2
3.4+1.8 P/ε 4.2 ( ) 5 . 0
0. 8 − 1.
3 bijb
1.25 0.4 1.44 1.83
ij
Table 3.4: Coefficients of the SSG model.
1
In the coefficient C1 , dependence on P/ ε (with P = Pii
) is introduced to achieve the correct
2
asymptotic behaviour of the Taylor series expansion ofφ ij .
3.3.6 Large Eddy simulation (LES)
Large eddy simulation (LES) [48] is a popular technique for simulating turbulent flows. An
implication of Kolmogorov’s (1941) theory of self similarity is that the large eddies of the
flow are dependant on the geometry while the smaller scales more universal. This feature
allows one to explicitly solve for the large eddies in a calculation and implicitly account for
the small eddies by using a subgrid-scale model (SGS model).
Mathematically, one may think of separating the velocity field into a resolved and sub-grid
part.
The resolved part of the field represent the “large” eddies, while the sub grid part of the
velocity represent the “small scales” whose effect on the resolved field is included through the
subgrid-scale model. Formally, one may think of filtering as the convolution of a function
with a filtering kernel G:
resulting in
( x)
G(
x ξ) u(
x)
dξ,
u i ∫ −
= (3.42)
u = u + u '
(3.43)
i
Where ui the resolvable scale is part and u' is the sub-grid scale part. However, most
practical (and commercial) implementations of LES use the grid itself as the filter and perform
no explicit filtering.
The filtered equations are developed from the incompressible Navier-Stokes equations of
motion:
i
40
i
i

Chapter III Turbulence modelling
∂u
⎛ ⎞
i ∂ui
1 ∂P
∂
⎜
∂ui
+ u = − + ⎟
j
v
∂t
∂x
∂ ∂ ⎜ ⎟
j ρ xi
x j ⎝ ∂x
j ⎠
(3.44)
Substituting in the decomposition ui = ui
+ u'i
and Pi = Pi
+ P'i
then filtering the resulting
gives the equation of motion for the resolved field:
∂ui
∂ui
1 ∂P
∂ ⎛ u ⎞ ∂
i
ij
u
⎜
∂
j
v ⎟
1 τ
+ = − + +
∂t
∂x
j ρ ∂xi
∂x
⎜
j x ⎟
⎝ ∂ j ⎠ ρ ∂x
j
(3.45)
We have assumed that the filtering operation commute, which is not generally the case. It is
thought that the errors associated with this assumption are usually small, though that commute
with differentiation has been developed. The extra term
advection terms, due to the fact that
And hence
j
∂τ ij
arises from the non-linear
∂x
∂ui
∂ui
ui
≠ u j
∂x
∂x
Similar equation can be derived for the subgrid-scale field.
ij
i
j
i
j
j
j
(3.46)
τ = u u − u u
(3.47)
Subgrid-scale turbulence models usually employ the Boussinesq hypothesis, and seek to
calculate the SGS stress using:
where
S ij is the rate of strain tensor for the resolved scale defined by:
1
τ ij − τ kkδ
ij = −2μt
Sij
(3.48)
3
S
ij
1 ⎛ ⎞
⎜ ∂u
∂u
i j
= + ⎟
2 ⎜ ⎟
⎝
∂x
j ∂xi
⎠
(3.49)
And v is the subgrid-scale turbulent viscosity. Substituting into the filtered Navier-Stokes
t
equations, we then have:
∂u
1 ⎛
i ∂ui
∂P
∂
+ u = − + ⎜
j
∂t
∂x
∂ ∂ ⎜
j ρ xi
x j ⎝
41
∂u
⎞
∂x
⎟
j ⎠
i [ v + v ] ⎟,
t
(3.50)

Chapter III Turbulence modelling
Where we have used the incompressibility constraint to simplify the equation and the pressure
is now modified to include the trace term τ / 3
Subgrid scale models
kkδ
ij
• Smagorinsky model, Smagorinsky. [51]
• Algebraic Dynamic model, Germano et al. [48]
• Localized Dynamic model, Kim & Menon. [49]
• WALE (Wall-Adapting Local Eddy-viscosity) model, Nicoud and Ducros. [50]
• RNG-LES model
3.3.7 Detached Eddy simulation (DES)
The difficulties associated with the use of the standard LES models, particularly in near-wall
regions, has lead to the development of hybrid models that attempt to combine the best aspects
of RANS and LES methodologies in a single solution strategy. An example of a hybrid
technique is the detached-eddy simulation (DES), Spalart et al. [52] approach. This model
attempts to treat near-wall regions in a RANS-like manner, and treat the rest of the flow in an
LES-like manner. The model was originally formulated by replacing the distance function d in
the Spallart-Allmaras (S-A) model with a modified distance function d min[ d,
C Δ],
= DES
where C is a constant and Δ is the largest dimension of the grid cell in equation. This
DES
modification of the S-A model, while very simple in nature, changes the interpretation of the
model substantially. This modified distance function causes the model to behave as a RANS
model in regions close to walls, and in a Smagorinsky-like manner away from walls the
model. This is usually justified with arguments that the scale-dependence of the model is
made local rather than global, and that dimensional analysis backs up this claim.
The DES approach may be used with any turbulence model that has an appropriately defined
turbulence length scale (distance in S-A model) and is a sufficiently localized model. The
Baldwin-Barth model. While very similar to the S-A model is probably not a candidate for use
with DES. The standard version of this model contains several van Driest-types damping
functions that make the distance function more global in nature. Menter’s SST model [37] is a
good candidate, and has been used by a number of researchers. Menter’s SST model uses a
turbulence length scale obtained from the model’s equations and compares it with the grid
length scale to switch between LES and RANS, Streets. [53]
42

Chapter III Turbulence modelling
In practical, more programming is needed than simply changing the calculation of the length
scale. Many implementations of the DES approach allow for regions to be explicitly
designated as RANS or LES regions, overruling the distance function calculation. Also, many
implementations use differencing in RANS regions (e.g. upwind differences) and LES regions
(e.g. central differences).
3.3.8 Direct numerical simulation
A direct numerical simulation (DNS) is a simulation in Computational Fluid Dynamic in
which the Navier-Stokes equations are numerically solved without any turbulence model. This
means that the whole range of spatial and temporal scales of the turbulence must be resolved.
All the spatial scales of the turbulence must be resolved in the computational mesh, from the
smallest dissipatives scales (kolmogorov scales); up to the integral scale L, associated with the
motions containing most of the kinetic energy.
3.3.9 The SST- Cas
model
The SST- C [18] model is used in the present work. The SST model described in section
as
(3.2.3.3) requires only small modifications to incorporate the Cas model [18].
The Cas model
In contrast to eddy viscosity models, the Reynolds stress model calculates an exact
production which is explicitly linear in the mean strain rate. Indeed, one can write exactly
P as
k = C k S where C a non-dimensional parameter is representing the degree of alignment
between stresses and strains:
as
C
as
aijSij
= −
(3.51)
S
The aim of the present work is to test the performance of the SST-Cas model to reproduce an
unsteady flow across in-line tube bundle.
Model derivation
From the definition of C in equation (3.51) its total derivative can be obtained using the
as
product rule ( Dφ / Dt = ∂φ
/ ∂t
+ U ∂φ
/ ∂x
). After derivation:
k
k
43

Chapter III Turbulence modelling
DC
Dt
+
as
* ( α + α A S )
3
∂
+
∂x
k
ε
= α1
C
k
SijaikΩ
+ α5
S
⎡
⎢
⎣
3
jk
2
as
−
*
+ α S C
1
S
∂C
DS
Dt
⎤
as
( v + σ casvt
) ⎥
∂xk
⎦
1
ij
2
as
Sijaika
+ α4
S
⎛
⎜a
⎜
⎝
Sijaika
−α
2
η
ij
kj
2SijC
+
S
as
kj
⎞
⎟
⎠
(3.52)
The constants α1… α5 are related to those in the original underlying pressure-strain model via:
*
*
α = ( + ) = ( − )
1 1 C
*
3
1
*
C3
=
2
C = α
α 1 1 C1
2 2
⎛ 4 ⎞
⎜ − C3
⎟
⎝ 3
α
⎠
3 =
2
α α = 2( 1−
C ) α = 2( 1−
C )
4
Tables 3.3 and 3.4 lists the values of the model constants from the LRR and SSG pressure-
strain models (C1… C5)
The SST model requires small modifications to incorporate theC model. Initially, the
modification was intended to be applied to the production rate of turbulence kinetic energy
term only, but it can be applied in a more coherent manner by means of a simple modification
to the turbulent eddy viscosity in equation 3.12, as follows:
⎛
⎞
⎜
1 a1
Cas
v =
⎟
t k min ; ;
⎜
⎟
⎝ ω S F2
S ⎠
4
5
5
as
(3.53)
The value of C in equation (3.53) is limited to ±0.31 for the calculation of the production
as
terms in equations 3.20 and 3.21, while when evaluating diffusion terms, the absolute value,
|Cas|, is used to avoid negative values which could lead to numerical difficulties.
3.4 Turbulence modelling of unsteady flows (URANS)
3.4.1 Introduction
An alternative to LES for industrial flows can then be unsteady RANS (Reynolds-Averaged-
Navier-Stokes), often denoted URANS (unsteady RANS) or TRANS (Transient RANS).
44

Chapter III Turbulence modelling
3.4.2 Unsteady Reynolds Navier-Stokes equations
In URANS the usual decomposition is employed,
1
U =
2T
T
∫
−T
U
() t
dt , U = U + u"
(3.54)
The URANS equations are the usual RANS equations, but with the transient (unsteady) term
retained (on incompressible flow)
∂ U
∂t
i
∂
+
∂x
j
( U U )
i
j
2
1 ∂P
∂ Ui
= − + v
ρ ∂x
∂x
∂x
∂U
∂x
i
i
i
= 0
j
j
∂u"
i u"
−
∂x
j
j
(3.55)
Note that the dependant variables are now not only a function of the space coordinates, but
also a function of time,
= U ( x,
y,
z,
t)
, P = P(
x,
y,
z,
t)
, " u"
= u"
u"
( x,
y,
z,
t)
Ui i
u i j i j
The results from URANS [55] can be decomposed as a time averaged part U , a resolved
fluctuation u', and the modelled turbulent fluctuation u".
U = U + u"
= U + u'+
u"
(3.56)
3.4.2 Turbulence Modelling of Unsteady Cross flow In-line Tube Bundle
In URANS, part of turbulence is modelled (u") and part of the turbulence is resolved (u'). If
we want to compare computed turbulence with experimental turbulence, we must add these
two parts together.
All RANS models (High Reynolds number k-ε model, k-ω SST model and the new SST-Cas
model, Reynolds Stress model (RSM)) are used in this case to compute an unsteady flow. The
aim of the present work is to assess which of these models is able to reproduce the cross flow
in-line tube bundle which is an unsteady flow.
45

Chapter IV Numerical simulation
Chapter 4
Numerical Simulation
4.1 Introduction
At Electricite De France (EDF) development of in-house codes has been a resolute strategic
choice for more than fifteen years. In order to solve the Navier-Stokes equations a Finite-
Volume code is used. Code -Saturne, general- purpose Computational Fluid Dynamic Code
for laminar and turbulence flows in complex two and three dimensional geometries. The code
is used for industrial applications and research activities in several fields related to energy
production (nuclear thermal-hydraulics, gas and coal combustion, turbomachinery, heating,
ventilation and air conditioning...).
CFD codes are structured around the numerical algorithms that can tackle fluid flow problems.
Hence all codes contain three main elements: the first is a pre-processor, the second is a solver
and the third is a post-processor.
4.1.1 Pre-processor
Pre-processing consists of the input of a flow problem to a CFD program by means of an
operator-friendly interface and the subsequent transformation of this input into a form suitable
for use by the solver. In the present work the pre-processor is GAMBIT and the activities at
the pre-processing stage involve:
� Definition of the geometry of the region of interest: the computational domain.
� Grid generation the sub division of the domain into a number of smaller, nonoverlapping
sub-domain: a grid (or mesh) of cells for control volumes or elements).
� Selection of the physical and chemical phenomena that need to be modelled.
� Definition of fluid properties
� Specification of appropriate boundary conditions at cells which coincide with or
touch the domain boundary.
4.1.2 Solver (Code- Saturne)
The numerical methods that form the basis of the solver Code-Saturne perform the following
steps:
46

Chapter IV Numerical simulation
� Approximation of the unknown flow variables by means of simple functions.
� Discritisation by substitution of the approximations into the governing flow equations
and subsequent mathematical manipulations.
� Solution of the algebraic equations.
Code-Saturne is well suited for two- and three-dimensional calculations of steady or transient
single-phase, incompressible, laminar or turbulent flows. It supports two Reynolds Averaged
Navier Stokes (RANS) models: the standard k-ε model, the launder Sharma model and a
Second Moment Closure (LRR) model. It also contains the LES Smagorinsky and dynamic
models. The new models which are also implemented in Code -Saturne are: The SST model,
elliptic relaxation models υ²-f, the SSG model and the scalable wall function. The flow solver
is based on a finite volume approach, with a fully collocated arrangement for all variables.
The time discretisation is similar to the method used in other commercial codes. It is based on
a predictor-corrector scheme for the Navier-Stokes equations. An important asset of Code-
Saturne is relies on its ability to deal with any kind of mesh (hybrid, containing arbitrary
interfaces and any type of cell).
4.1.3 Post-processor
As in pre-processing a huge amount of development work has recently taken place in the
post-processing field. Owing to the increased popularity of engineering workstations, many of
which have outstanding graphics capability, the leading CFD packages are now equipped with
versatile data visualisation tools. There include:
� Domain geometry and grid display
� Vector plots
� Line and shaded contour plots
� Particle traching
� View manipulation (translation, rotation, scaling, etc.)
� Color postscript output
47

Chapter IV Numerical simulation
usclim.F
(Boundary conditions
Symmetry and walls)
usini1.F
(Initialization,
time step,
averaging…etc)
Pre-processing
Grid Generation
(Structured mesh)
Processing
Solver
Code-Saturne
Principal subroutines of
the present test case
Post-processing
48
usproj.F
(Present case)
Pressure and
Velocity profiles
(Contour of pressure,
velocity…, stream-lines,
iso-surfaces…)
ustsns.F
Correction of
the flow
(periodicity)
Figure 4.1: Steps of Numerical Simulation of cross flow in-line tube bundles

Chapter IV Numerical simulation
4.2 The Finite Volume method
The method relies on:
� Dividing the domain into control volumes.
� Formal integrations of the governing equations of fluid flow over all the control
volumes of the solution domain.
� Discretisation involves the substitution of a variety of finite-differences-type
approximations for the terms in the integrated equation representing flow processes
such us convection, diffusion and sources. The converts the integral equations into a
system of algebraic equations.
� Solution of the algebric equations by an iterative method.
The conservation of a general flow variable Φ. For example a velocity component, within a
finite control volume can be expressed as a balance between the various processes tending to
increase or decrease it:
Rate of change of
Φ in the control
volume with respect to
time
=
Net flux of Φ due
to convection into
the control volume
49
+
+
Net rate of creation
of Φ inside the
control volume
Net flux of Φ due
to diffusion into the
control volume
In this chapter, (I, J) refer to the cell on either side of a face (see Figure 4.2) while the lower
case letters reference the usual tensor quantities
∫
V
∂ρφ
dV
∂t
+
∫
V
∂
∂x
j
∂ ⎛ ⎞
⎜
∂φ
( ρu
⎟
jφ
) dV = ∫ Γ dV +
∂ ⎜ ⎟ ∫ SdV (4.1)
x j ⎝ ∂x
V
j ⎠ V
Where “S” represents the source term, “V” the volume of the cell and “Г” the diffusion
coefficient. Using Gauss theorem the volume integrals of the divergence can be transformed
into surface integrals, which can be written as:

Chapter IV Numerical simulation
∫
V
∂ρφ ∂φ
+ ∫ ρφu
j n j dA = ∫ Γ n j dA +
∂
∂x
t A
A j
V
n
50
∫
SdV
(4.2)
Where “A” is the area of the face and “ ” is the face normal vector. The volume integrals
j
are approximated by the product of the value at the cell centre and the cell volume V.
Face integrals
∫
V
SdV I
≅ S V
S
I
(4.3)
The surface integrals can be approximated by the mid point rule, the product of the face
centre value and the area of the face. In the collocated arrangement, an interpolation is
required in order to obtain the values at the face centres, since all the variables are stored at
the cell centres. In Code-Saturne the convection can be calculated either by using an upwind
differencing scheme (UDS) or a central differencing scheme (CDS). The code has also a slope
test based on the product of the gradients at the cell centres to dynamically switch from CDS
to UDS. Using the upwind scheme, the value at the face can be obtained as:
φ = φ if = 0 U
F
F
I
j
i i F n
φ = φ if = 0 U
i i F n
(4.4)
This scheme is robust and stable but introduces additional numerical diffusion which can
become large if the grid is coarse. For a centred scheme, the value at the face can be computed
as:
With
φ
F
αφ
⎛
⎞
⎜ 1 ⎡ ∂φ
∂φ
⎤
+ ( 1 − α ) φ
⎟
j +
⎜
⎢ I + j ⎥OF
⎟
(4.5)
⎝
2 ⎢⎣
∂x
j ∂x
j ⎥⎦
⎠
= I
j
FJ '
a = . The last term in equation (4.5) is added for non-orthogonal grids, where the
I'
J '
centre of the face does not lie in the mid point between the cell centres. The diffusion integral
can be computed as:
∫
A
∂φ
n
∂x
j
j
dA
=
∑
Neigh
∂φ
Γ n
∂x
j
j
A
(4.6)

Chapter IV Numerical simulation
With a linear approximation for the gradient the face centre, the diffusion is computed as:
The values of φJ ' and I '
Gradient Reconstruction
∂φ
φ J ' − φ I '
∑ Γ n j A = ∑ Γ n j A
(4.7)
∂x
IJ
Nei
j
51
Neigh
φ can be computed by using the gradient at the cell centre:
j
j
∂φ
φ I ' = φ I + I I ' I j
(4.8)
∂x
The calculation of the gradients is achieved by an iterative solver in which the gradient is
expressed as:
∂φ
1
I = ∑ n jdA
∂x
j V ∫ φ
Neigh A
The surface integral can be approximated using the midpoint rule so that is becomes:
∂φ
1
I =
∂x
V
j
∑
Neigh
φ A
to obtain the value of φF a Taylor series expansion can be applied to obtain:
∂φ
∂x
j
O
F
F
n
j
(4.9)
(4.10)
∂φ
φ F = φ 0 + OF j 0 + O
(4.11)
∂x
The value of φ0 can be obtained by a linear interpolation and the gradient at the same
∂φ
point, from an averaged between the values of I
∂x
solve can be written as:
∂φ
∂x
j
I
1
=
V
4.3 Time discretisation
∑
Neigh
1
j
j
⎛ ∂φ
∂φ
and J
∂x
{ αφ + ( 1−
α)
φ j + OF ⎜
j I + J } AF
nj
j
. Finally the system to
1 ⎜
(4.12)
2 ∂x
j ∂x
j
The time discretisation in Code-Saturne is achieved through a fractional step scheme (Euler
implicit) that can be associated the SIMPLEC method. The solution algorithm consists a
prediction -correction method. In the first step the momentum equation is solved using an
⎝
∂φ
⎟ ⎞
⎠

Chapter IV Numerical simulation
n n
explicit pressure gradient from the previous time step. With u = ρu
being the momentum
at time step n, the system to solve at the first step of the method is:
Q
−
Q
∂
⎛
* n
*
n
i i
* n
i
*
+ ⎜ u i Q j − μ ⎟ = − + S
t x ⎜
j
x ⎟
(4.13)
Δ ∂
∂ j ∂x
i
⎝
Where S includes all the source terms that can be made implicit or explicit,
n n
S = A + B u and Δ t is the time step. After this prediction step a new velocity field is
i
i
i
obtained (denoted by (*)) which is usually not divergence free. The second step consists of
calculating the pressure gradient in order to satisfy the continuity equation. By taking the
divergence of the momentum equation, the Poisson equation for the pressure can be written
as:
52
∂u
* * * ( P − P )
∂ ⎛ ∂ ⎞
⎜
∂
Δt
⎟ =
∂x
⎜
j x ⎟
⎝ ∂ j ⎠ ∂x
⎞
⎠
i
∂P
j
i
*
φ i
(4.14)
Finally, once the updated pressure (P**) has been obtained, the velocity field is corrected.
This is done by neglecting convection and diffusion variations:
Q
* *
i
* * * ( P − P )
* ∂
− Qi
= −Δt
(4.15)
∂x
n 1 * *
The velocities and the pressure are updated, that is Q = Q and
i
+ n + 1 * *
P = P
. When a
turbulence model is used, the resolution of the turbulent variables takes place after the
velocities are computed. The dependence of other variables is fully explicit so that each
equation is solved separately.

Chapter IV Numerical simulation
4.4 Boundary conditions
Figure 4.2: Notations for the spatial discritisation.
For the resolution of discritised equations described previously, the boundary conditions have
to be prescribed for any variable Φ.
4.4.1 Inlet
At the inlet a Dirichlet condition is prescribed for all transport variables (velocity, scalars,
n+
1
turbulent variables…) so the values for and Q are prescribed by the user. A
homogenuous Neumann condition (zero flux) is imposed on the pressure. The convection term
Q
n
j n j
ϕ is calculated directly from the prescribed values. For the diffusion terms, the boundary
value is calculated as:
φ
n + 1
inlet
53
inlet

Chapter IV Numerical simulation
⎛ φ ⎞
⎜
∂
φ −
Γ n ⎟
j = Γ
⎜ x ⎟
⎝ ∂ j ⎠ I ' Fjn
inlet
For the source terms, the prescribed value φ n+1
inlet
54
n+
1 *
inlet φI
'
j
(4.16)
is used as a boundary value. The pressure
gradient normal to the face is prescribed as zero although as extrapolation from the previously
obtained results is possible. The term
⎛
⎜
⎝
∂ P ⎞
t n ⎟
∂x
⎟
j ⎠
δ
is set to zero in the Poisson equation and
Δ j
the term on the right hand side is calculated as:
4.4.2 Outlet
Q
* n+
1 n+
1
jn j = ρinletu
j(
inlet)
n j
(4.17)
For the outlet, a homogeneous Neumann condition is imposed on the velocity, scalars and
n+
1
turbulent variables. For the pressure, a Dirichlet condition is used on P . The boundary
values for the diffusion terms set to zero, and for the source terms a first order approximation
is used to setφ = φ I ' . The Dirichlet condition for pressure provides the boundary value used
for the computation of the pressure gradient. The Dirichlet condition is also used in δP in the
Poisson equation. In the momentum correction equation, the boundary value forδ P is set to
zero.
4.4.3 Walls and symmetries
For the walls and symmetry faces, a zero mass flux is imposed and both Dirichlet and
Neumann conditions can be applied to scalars. For the tangential velocity, a homogeneous
Dirichlet boundary condition is used at the wall whereas at the symmetry faces a
homogeneous Neumann condition is applied. The pressure gradient normal to the face is set to
zero although it can also be computed via an explicit extrapolation of the value at the
boundary cell.
For convections terms, the boundary value of the flux is set to zero. For diffusion terms when
a Neumann condition is applied to the variable Φ; the value prescribed is used directly. If a
Dirichlet condition is applied, the boundary value is calculated in the same way as in the inlet.
outlet

Chapter IV Numerical simulation
If the variable has a Dirichlet boundary condition and a source term requiring the gradient of
the variable, the prescribed value is used as boundary valueφ = φ . If a Neumann condition is
used, an extrapolated value from the boundary cell is used with a first order approximation.
For the velocity component normal to the wall, the boundary value is set to zero to ensure zero
mass flow.
For the pressure gradient calculation, if the flux of the variable at the face is prescribed, the
boundary value is:
∂φ
φ = φ + I'
F n
∂δ
P
To solve the Poisson equation, the values of Δt
n j
∂x
b
55
b
I ' j b
(4.18)
∂x
j
j
and Q jn j for the momentum
*
correction equation, the boundary value for δP is obtained from the cell value from a first
∂ P
order approximation using the fact that Δt n j = 0
∂x
δ
j
The value for the boundary conditions at the wall is prescribed for all velocities and
turbulent variables. The way this boundary condition is treated depends on the turbulence
model used. Many turbulence models have been designed under a local equilibrium
assumption and use the universal logarithmic law to bridge the viscous sublayer, hence
solving for the flow outside the buffer layer. To prescribe the velocity at the wall, the total
shear stress is required:
u uk
*
τω = ρ
(4.19)
1 / 2
with u* is the friction velocity andu
k = k / Cμ
. In the code, the shear stress is calculated
by using the wall function approach. Defining the tangential velocity at the wall as
u = u − u n n
tg
j
j
i
with κ =0.42 and C = 5.2.
i
j
. The shear stress can be calculated from:
u
*
uk
=
−1
/ 4 1/
2
= Cμ
k
u
1
ln
k
tg
j
u
tg
j
+ ( y ) + C
(4.20)
(4.21)

Chapter IV Numerical simulation
For the turbulent variables, the boundary conditions are obtained from:
56
∂k
= 0
∂y
∂ε u
= −
∂y
ky
In the case of k − ε model and for the second moment closure the conditions are:
∂ u'
u'
i
∂x
j
j
n
j
= 0
3
k
2
(4.22)
(4.23)
if i = j (4.24)
u u u u
*
' = (4.25)
1 '2
k
u u = u u'
= 0
(4.26)
1
3
'2 3
∂ε u
= −
∂y
ky
This in a local frame where x2
is normal to the wall.
3
k
2
(4.27)

Chapter V Results and Discussion of the Simulation
Chapter 5
Results and discussion of the simulation
5.1 Introduction
In the present we are looking upon clusters of density packed cylinders called tube bundles.
In group of cylinders can be arranged in in-line (straight), staggered (rotate square), normal
triangle or parallel (rotate normal triangular) configurations (see figure 5.1). Our particular
interest is in-line configuration as this is widely used in heat exchangers of chemical and
nuclear/coal power plants. The frequently parameters characterizing the tube bundle's
geometry are:
The gap ratio
In a group of circular cylinders, the arrangement of the cylinders is important. For example,
different cylinder arrays are specified by the pitch to the diameter:
P
D
Pitch
=
Diameter
Square array (90°)
P
Tube Row
57
P
D
Rotated Square array (45°)
P

Chapter V Results and Discussion of the Simulation
Triangular array (30°)
Reynolds number (Re)
P
Figure 5.1: Tube arrangements
Square array (60°)
The Reynolds number is a dimensionless number that is significant in the design of a model
of any system in which the effect of viscosity is important in controlling the velocities or the
flow pattern of a fluid. For the case of circular cylinder, the characteristic length is typically
taken to be the tube diameter; it can be shown that the Reynolds number is representative of
the ratio of inertia force to viscous force in the fluid. It is given by:
Inertia force
Re= (5.1)
Viscous force
The Reynolds numbers gives a measure of transition from laminar to turbulent flow, boundary
layer thickness, and fluid field across cylinders.
Strouhal number (St)
The inverse of the reduced flow velocity is called the Strouhal number, provided that the
frequency is the frequency associated with flow field, such as the vortex shedding, the
Strouhal number is related to the oscillation frequency of periodic motion of a flow.
Where ƒ is the vortex shedding frequency.
D
St=ƒ (5.2)
U
5.2 Case Description
In the present work an in-line (array) tube bundle configuration is tested to understand the
complex flow behaviour across tube bundles with a gap ratio P/D (aspect ratio) which is
defined in section 5.1 of 1.44 and transverse length to diameter ratio T/D of 1.44 and
58
P

Chapter V Results and Discussion of the Simulation
Reynolds number of 70000. The current configuration is square (P/D=T/D) (see figure 5.2).
The present test case requires only periodic and wall boundary conditions (see Figure 5.3).
The simulation configuration is reduced to a few set of cylinders as seen in Benhamadouche et
al. [13]. For 3D calculation the bundle is assumed infinite in all directions but it has been
observed that the extrusion length plays an important role in the development of the flow
physics. Results with one diameter extrusion length and also with 2 by 2 tubes case shows that
the flow did not develop an anticipated resulting. Hence a complete two points correlation
study was carried out and it was concluded that one needs to take at least two diameter depths
and 3 by 3 configuration for numerical simulations.
5.3. Grid generation
Numerical accuracy and stability of any simulation is strongly dependent on the quality of
the grid used hence one of the most tedious jobs in any numerical simulation is the grid
generation. Numerical accuracy issue relates to mesh density and parameters such as the first
+
cell's width y (next to the solid wall) and moreover the adimensional distance y which is
proportional to y (see equation 5.3)
+
y ν
y = (5.3)
u
Where u*
is the friction velocity (It was read from listing Code-Saturne)
+
For URANS models and especially standard k-ε model, the y must lie between 30 and 70. In
the present case the grid generated at the first time is 2D mesh of the geometry 2 by 2 tubes.
The number of cells in 2D indicates the number of 2D cells obtained by cutting the
computational domain with a plane orthogonal to the spanwise direction. Seven blocks were
first generated with different faces and were then copied 24 times to get the smaller domain of
2 by 2 tubes (see figure 5.4) and after were copied 96 times to get the larger domain of 3 by 3
tubes (see figure 5.5). 2D grid is structured. The larger domain is used in the present
simulation because if the domain size in the streamwise or spanwise directions is smaller than
the size of the largest structures then errors might be occurred. It was decided to use 3 by 3
configuration and to use the spanwise depth of two diameters L =2D to obtain 3D mesh (see
figure 5.6). All parameters of 2D and 3D grids generated are shown in table 5.1:
*
59
z

Chapter V Results and Discussion of the Simulation
2D
Grid
3D
Grid
Number
of
cells
21600
604800
Number
of
faces
44400
65600
Size of
Grid
(Mo)
2.617
34.445
Number
of
elements
Quad4
44400
Hexa8
21600
Quad4
65600
Hexa8
604800
User's
Time
CPU
(sec)
1.15
18.21
System's
Time
CPU
(sec)
0.12
2.56
Total
Time
(sec)
1.33
24.45
TTCPU
T Time
0.96
0.85
Table 5.1: Parameters of 2D and 3D grids in the present case.
+
y
[13,70]
[13,70]
2D and 3D Grids are generated in a machine of a CPU processor Pentium4, processor's speed
2.4 GHz and RAM memory 512 MB.
In first time, 2D calculations with 2D grid were run in a machine's CPU processor is Pentum4;
processor's speed is 2.4 GHz and RAM memory is 512 MB. About 3D grid which is so large,
2 CPU processors were needed in minimum for running 3D calculations. Consequently, a
cluster was used. The cluster's CPU processor is INTEL XEON, processor's speed is 3.2 GHz
and Total RAM memory is 4GB (total number of CPU's is:8.) For the present work, it was
used just 2 CPU processors from the 8.
5.4 Discussion of the results
The asymmetric flow inside tube bundle can best be understood by concentrating on the
wake of the centre cylinder. In this section, the pressure distribution around centre cylinder,
velocity profiles in the wake of tubes and turbulence intensities are discussed.
Figure (5.7) shows the convergence curves. It means evolution of the pressure (see figure 5.7
(a)) and velocity (see figure 5.7 (b)) according to number of iterations and comparison
between various URANS models. For the pressure, URANS models show that the pressure
and velocity fluctuate according to time except k-ε model which is stable and does not show
any instability of pressure and velocity along time. When averaging the two quantities of
pressure and velocity. They become stable after 15000 iterations for a pressure and after
30000 iterations for a velocity.
60
y
(m)
0.6E-03
0.6E-03

Chapter V Results and Discussion of the Simulation
Figure 5.8 shows the instantaneous pressure contour lines distribution in a XY cross
sectional view together with a comparison between different 2D URANS models used in the
present case. For accurate comparison all the plots are drawn on a same scale of -1 to1. All
models show that the flow is unstable and asymmetric except standard k-ε model; it fails to
capture the instability of the flow. The pressure has the same behavior around all tubes. It
appears a high pressure region on the top of the cylinder with a low pressure just downstream
which believed to be the correct solution according to LES. The slight of stagnation region on
the top is a direct result in deflection of mean flow due to suppression and distortion of vortex
shedding from downstream cylinder.
Figure (5.9) shows the instantaneous velocity contour lines distribution in a XY cross section
and a comparison between various 2D URANS models. For accurate comparison all the plots
are drawn on a same scale of 0 to 2.5. The low velocity is seen in the stagnation point and
close to the wall caused by the high viscosity turbulent. The flow seems to be going down.
Shear layer separation is seen to originate from the bottom between 180 degree and 270
degree. Due to small gap spacing the recirculation bubbles are not of the same size and shape,
where the bottom recirculation region is substantially larger in size and bigger in intensity than
the top one (see figure 5.32). Because of high viscosity at the wall, velocity is minimum and
maximum in free stream flow far from cylinders and in zones of recirculation. K-ω SST, RSM
and SST- C show unsteadiness of the flow but standard k-ε model does not show any
as
instability.
Figure (5.10) shows velocity vectors distribution in a XY cross section for various 2D
URANS models. The same scale than the velocity is used. This figure illustrate better zones of
recirculation, RSM and SST- C models show recirculation in bottom of cylinders better than
as
the others URANS models (see figure 5.10 (b), (d)).
Figure (5.11) shows 2D Vorticity in XY sectional view for different URANS models. The kε
model fails to capture the shear layer separation for the same scale of 0 to 6. Hence the other
URANS models show two recirculation bubbles. The shear layer separates from the bottom of
the cylinder surface as well resulting in two recirculation regions behind every tube. However,
the flow is still asymmetrical with one recirculation bubble larger than the other.
61

Chapter V Results and Discussion of the Simulation
Figure (5.12) shows distribution of mean pressure in XY cross view for pitch ratio of 1.44
and Reynolds number of 70000 and comparison between various 3D URANS models, DES
and LES of Imran. URANS models seem to give similar results. Pressure has a same behavior
around all tubes. The peak of pressure is on the top of tube because of stagnation region. It
agrees with LES and minimum values locate in the bottom of tubes caused by detachment of
shear layer and recirculation region. Results of RSM and SST- C agree better with DES and
LES than k-ω SST.
Figure (5, 13) shows distribution of velocity in the wake along a line at x=4.33cm for
P/D=1.44 and comparison between 3D URANS models, present DES and LES of Imran for
P/D=1.5. It appears a same sense deflection across the set of cylinders. URANS models
capture the four peaks of velocity maxima and minima. Velocity is maximum far from
cylinder's walls and it appears clearly minimum near the wall and in the gap spacing. RSM
model is closer to LES than the other URANS models.
Figure (5.14) shows the C evolution of normalized pressure coefficient along the central
tube for gap ratio of 1.44, for better comparison the mean normalized C profile is shown
where it is defined as:
p
C p
2
( P - P ) ρ
= 2 U
(5.4)
The angle measurement which is show in figure (5.2) is similar to a clockwise and starts from
the inlet free stream direction that is from left to right. LES of Imran Afgan for aspect ratio of
1.5 and experimental data of Yahiaoui et al. (2007) for aspect ratio of 1.44 are used for
comparisons. The effect of flow deflection is observed in term of stagnation pressure region
located somewhere around 45 degrees from the flow direction. This shift of stagnation point
location is also validated from experimental study of Yahiaoui (2007) and LES. It is the result
in deflection of mean flow due to suppression and distortion of vortex shedding from
downstream cylinder. The minimum pressure is located at around 90 degrees because of a
separation of a shear layer and recirculation region. Interestingly all 2D URANS models k-ω
SST, SST- C and RSM (see figure 5.14) seems to give similar predictions and agree with
as
experimental data of Yahiaoui et al. for the stagnation point and the pressure minima except in
the gap space. RSM is closer to SST- C (see figure 5.15). There is a delay of reattachment for
as
2D URANS models. It can be caused by a high Reynolds number. When looking to 3D
URANS results (see figure 5.16) one observes that k-ω SST, RSM, DES and experimental
62
ref /
0
as
p

Chapter V Results and Discussion of the Simulation
data of Yahiaoui et al. and LES of Imran are in close agreements whereas about the LES of
Imran the peak of the pressure around 45 degrees is over. They also slightly over predict this
value to 40 degrees then they are closer to LES and experimental data (45 degrees).
For an incompressible flow such as the one under consideration the bulk gap velocity is
calculated by:
⎡ T / D ⎤
U g = U 0 ⎢
⎣T
/ D −1⎥
(5.5)
⎦
For 2D URANS cases the averaged velocity along a vertical line in XY plane at X=4.33 is
shown in Figure 5.17. It observed two high peaks at Y/D=1 and Y/D=4. It means that velocity
is maxima in mean flow free and fully developed far from the wall in a streamwise direction.
Velocity decreases while bringing closer to the wall. RSM model is closer to LES and
experiment of Aiba et al. [61] than the other 2D URANS models (maximum velocity is around
1.5 and minimum around zero), (see figure 5.18). Two models k-ω SST and SST- C are
closest, their curves have the same pace like LES and experimental data. Moreover they have
negative values of velocity and a little peak (see figure 5.17). It means that velocity change
direction, it might be caused by high Reynolds number and a recirculation region. k-ε model is
far to predict complete flow physics and its curve is very low comparing with LES and
experiment.
Figure (5.20) shows the 3D mean velocity along a vertical line in the wake of tube in XY
plane at X= 4.33 and length extrusion of L =2D. URANS models capture peaks of velocity
z
around U= 1 m/s. All curves are similar but SST-C model's prediction here is far better than
the other URANS models (see figure 5.21). Since it tends to capture all the local peaks in
velocity profile. It agrees better with LES and experiment.
Figures 5.22 to 5.27 show fluctuating velocity and pressure and their density power spectrum
(DPS) in different probes (1, 3, and 6). An analytical formula for the Strouhal number is
proposed in Chen’s book (see equation 5.6).
as
1 ⎛ P ⎞
St = ⎜ −1⎟
(5.6)
2 ⎝ D ⎠
It depends only on the tube spacing but not on the Reynolds number which is not really
realistic but gives a good order of the Strouhal number. For P/D=1.44 one finds St = 1.13. By
applying DPS to the pressure and velocity signals, one clear peak is obtained around the
frequency 45 Hz (see figure 5.23, 5.24, 5.25, 5.26). This value corresponds respectively to the
63
as

Chapter V Results and Discussion of the Simulation
Strouhal number of 0.84. It means that the vortex shedding detected in the shear regions with
URANS models is realistic. The Strouhal numbers obtained with Star CCM and Code-
Saturne, Benhamadouche [13] are 0.8 and 1.25.
A thorough breakdown of Reynolds stresses is shown in Figure 5.28 for RSM model. It
shows the time averaged Reynolds stresses profiles < u′u′ >, < v′v′ >, < w′w′ > and ,
, along the same line. One notices from this figure that the peaks in the
streamwise and normal Reynolds stresses (< u′u′ >, < v′v′> respectively) are quite decently
captured by the RSM model. Usually has the opposite sign to the boundary layer shear
strain.
Figure 5.29 shows averaged velocities field , , . . Mean streamwise
velocity represent sharp due to a minimum flow passage within the bundle geometry and a
variation from almost 0.2 to a maximum value of around 1. Normal velocity has a same
behavior but higher than mean streamwise velocity and a variation from 0 to 2. Usually mean
velocity has a same pace but an opposite sign to streamwise and normal velocity.
Normalized average velocity (see figure 5.29 (d)) is compared with experimental data of Aiba
et al. [61] and with LES of Imran [14]. The velocity profile are close to the experiment and
LES when the modelled quantity u" is added to the averaged quantity u which is equal to
+u'. It concludes that the URANS method produce good predictions of turbulent flow.
In order to draw a comparison from the numerical results, the structure parameter, Q, is
calculated, which was found by Hunt et al. (1988) to be an effective way to visualize the
regions of coherent Vorticity due to rotational motion (as opposed to those from shear), and is
defined as:
= ij -1/2
( S − Ω Ω )
S
Q (5.5)
Where Q should take a positive value. Figure 5.30 shows instantaneous iso-surfaces for the
three URANS models and DES approach. The lack of coherent structures is apparent in the
results from the SST model, for the entire wake region except very close to cylinders and the
side walls. For this model it is the over-predicted values of turbulent kinetic energy, leading to
high values of the turbulent viscosity, that are responsible for the early damping out of these
structures. As a result the vortex dislocations are not seen in the SST results (see figure 5.30
(b)), whereas those from the RSM clearly show the lack of structures in the space between
cylinders for the RSM results is seen clearly from the oblique angle (see figure 5.30 (a)),
whereas both the standard SST and the SST- Cas
model and DES approach show more
64
ij
ij
ij

Chapter V Results and Discussion of the Simulation
structures in this region. The vortex dislocations can also be seen in the results from the SST-
Cas
model and there are significantly more spanwise structures than seen with either of the
two other models. In fact, the wake predicted by DES approach (see figure 5.30 (d)) and
SST-C scheme (see figure 5.30 (c)) displays much more structures.
as
Figure 5.31 shows comparison between results of Code-Saturne and Star-CD. For accurate
comparison all the plots are drawn on a same scale. A non-symmetrical solution is observed.
The behavior has been confirmed by STAR-CD calculations with P/D=1.44 and Reynolds
number of 70000.
Figure 5.32 represent mean velocity vectors field. Two recirculations coexist. The shear layer
separates from the bottom of cylinder surface as well resulting in two recirculation regions
behind every tube. However, the flow is still asymmetrical with one recirculation bubble in the
bottom (see figures 5.32 (a), (b), (c), (d)) due to the acceleration of the fluid and a small
recirculation on the top of the first recirculation. The shear stress in the bottom is higher than
on the top. SST- C , k-ω SST, RSM and DES approach show the two recirculation bubbles;
as
RSM shows close agreements comparing with LES of Benhamadouche (see figure5.32 (e)).
65

Chapter V Results and Discussion of the Simulation
U∞ Ug
Periodicity in "y" direction
Figures
Figure5.2: Geometry of in-line tube bundles
Figure5.3: Boundary conditions of tube bundles
P
Periodicity in "x" direction
66
D
T
Wall

Chapter V Results and Discussion of the Simulation
Figure5.4: Cross sectional view of 2D grid (2X2 arrangement) N=5400 cells, y+= [13-70]
Figure5.5: Cross sectional view of 2D grid (3X3 arrangement) N=21600 cells, y+= [13-70]
67

Chapter V Results and Discussion of the Simulation
Figure5.6: Cross sectional view of 3D grid (3X3 arrangement) in XY, YZ and
XZ sections: N=604800 cells, y+= [13-70]
68

Chapter V Results and Discussion of the Simulation
Figure 5.7: Evolution of pressure and velocity, Comparison between URANS models.
(a) Pressure, (b) Velocity.
(a)
69
(b)

Chapter V Results and Discussion of the Simulation
(a) (b)
(c)
Figure 5.8: 2D Instantaneous Pressure Contour field in a XY cross sectional view
for gap ratio 1.44. (a) k-ε model, (b) RSM, (c) k-ω SST, (d) SST- C
70
as
(d)

Chapter V Results and Discussion of the Simulation
(a)
(c) (d)
Figure 5.9: 2D Instantaneous velocity contour field in a XY cross Sectional view
for gap ratio 1.44. (a) k-ε model, (b) RSM, (c) k-ω SST, (d) SST- C
71
as
(b)

Chapter V Results and Discussion of the Simulation
(a)
(c) (d)
Figure 5.10: 2D Velocity vectors field in a XY cross Sectional view for gap ratio 1.44.
(a) k-ε, (b) RSM, (c) k-ω SST, (d) SST- C
as
72
(b)

Chapter V Results and Discussion of the Simulation
(c)
(a) (b)
Figure 5.11: 2D Vorticity field in a XY cross sectional view for gap ratio 1.44.
(a) k-ε, (b) RSM, (c) k-ω SST, (d) SST- C
73
as
(d)

Chapter V Results and Discussion of the Simulation
(a) (b)
(c) (d)
(e)
Figure 5.12: 3D mean pressure distributions in a XY cross view for P/D=1.44, Re=70000.
(a) k-ω SST, (b) RSM, (c) SST- Cas
, (d) DES, (e) LES of Imran for P/D=1.5, Re=15000.
74

Chapter V Results and Discussion of the Simulation
(a) (b)
(c)
(e)
Figure 5.13: 3D averaged velocity field in a XY cross view for P/D=1.44, Re=70000.
(a) k-ω SST, (b) RSM, (c) SST- Cas
, (d) DES, (e) LES of Imran for P/D=1.5, Re=15000.
75
(d)

Chapter V Results and Discussion of the Simulation
Figure 5.14: Mean pressure distribution around centre tube, comparison between 2D
Unsteady RANS for P/D=1.44, Re=70000 and LES of Imran (Star-V4) for P/D=1.5
Re=15000 and Experiment of Yahiaoui et al. (2007)
Figure 5.15: Mean pressure distribution around centre tube, comparison between 2D
Unsteady RANS for P/D=1.44, Re=70000 and LES of Imran (Star-V4) for P/D=1.5
Re=15000 and Experiment of Yahiaoui et al. (2007)
76

Chapter V Results and Discussion of the Simulation
Figure 5.16: Mean pressure distribution around centre tube, comparison between 3D
Unsteady RANS for P/D=1.44, Re=70000 and LES of Imran (Star-V4) for P/D=1.5
Re=15000 and Experiment of Yahiaoui et al. (2007)
Figure 5.17: Mean velocity profile, Comparison between 2D Unsteady RANS, Re=70000
and LES of Imran Re=15000 (Star-V4) and experiment of Aiba et al. (1982) in the wake
of centre tubes at x=4.33cm.
77

Chapter V Results and Discussion of the Simulation
Figure 5.18: Mean velocity profile, Comparison between 2D Unsteady RSM, Re=70000
and LES of Imran at Re=15000 (Star-V4) and experiment of Aiba et al. (1982) in the wake
of centre tubes at x=4.33cm.
Figure 5.19: Mean velocity profile, Comparison between 2D Unsteady RSM, SST- Cas
,
Re=70000 and LES of Imran (Star-V4), Re=15000 in the wake of centre tubes at
x=4.33cm.
78

Chapter V Results and Discussion of the Simulation
Figure 5.20: Mean velocity profile, Comparison between 3D URANS, Re=70000
and LES of Imran (Star-V4) at Re=45000 and experiment of Aiba et al. (1982) in
the wake of centre tubes at x=4.33cm.
Figure 5.21: Mean velocity profile, Comparison between SST- Cas
, Re=70000
and LES of Imran (Star-V4), Re=45000 in the wake of centre tubes at x=4.33cm.
79

Chapter V Results and Discussion of the Simulation
Figure 5.22: Fluctuating Pressure DPS at location of probe 1
80

Chapter V Results and Discussion of the Simulation
Figure 5.23: Fluctuating Pressure and DPS at location of probe 3
81

Chapter V Results and Discussion of the Simulation
(a)
(b)
Figure 5.24: (a) Fluctuating Pressure and DPS at location of probe 6.
(b) LES (Benhamadouche)
82

Chapter V Results and Discussion of the Simulation
Figure 5.25: Fluctuating Velocity and DPS at location of probe 1
83

Chapter V Results and Discussion of the Simulation
Figure 5.26: Fluctuating Velocity and DPS at location of probe 3
84

Chapter V Results and Discussion of the Simulation
Figure 5.27: Fluctuating Velocity and DPS at location of probe 6
85

Chapter V Results and Discussion of the Simulation
Figure 5.28: Reynolds stresses in the wake of the centre tubes.
(a) , (b) , (c) , (d)
86

Chapter V Results and Discussion of the Simulation
(a)
(c)
Figure 5.29: Mean velocity profiles of RSM in the wake of the centre tubes.
(a) , (b) , (c) , (d) u/uo
87
(b)
(d)

Chapter V Results and Discussion of the Simulation
(a)
(b)
88

Chapter V Results and Discussion of the Simulation
(c)
(d)
Figure 5.30: Iso-surface of parameter Q for the instantaneous flow across in-line
tube bundles. (a) RSM, (b) k-ω SST, (c) SST-C , (d) DES.
as
89

Chapter V Results and Discussion of the Simulation
(a)
(b)
(c)
Figure 5.31: Comparison between Code-Saturne (in right) and Star-CD (in left).
k-ω SST, (a)Pressure, (b) Velocity, (c) Turbulent kinetic energy
90

Chapter V Results and Discussion of the Simulation
(a) (b)
(b)
(e)
Figure 5.32: 3D mean velocity vectors, (a) k-ω SST, (b) SST- Cas
, (c) RSM, (d) DES,
(e) LES (Benhamadouche)
91
(d)

Chapter VI Conclusions and Recommendations for future work
Chapter 6
Conclusions and recommendations for future work
6.1 Final remarks
The cross flow over 2D and 3D square in-line tube bundles of gap ratio 1.44 and Reynolds
number 70000 is investigated via different 2D and 3D URANS models and DES. The results
were compared to Large Eddy Simulation study for the same configuration. The complex flow
for this configuration has already been shown to be asymmetric in nature with a high
deflection in the mean flow direction.
It is seen that the 2D and 3D results tend to capture the basic asymmetry of the flow.
Moreover, the pressure predictions on the surface of tubes are also hugely under or over
estimated by all 2D URANS models. 3D URANS simulations on the other hand seem to
produce better results. The k-ε model failed to capture any flow physics in the spanwise
direction. k-ω SST, RSM and SST-C seems to give similar predictions of the flow physics.
The new SST-C model agrees better with LES and experimental data. For the present case,
final remarks are highlighted:
as
� The flow is asymmetric and instable.
as
� URANS models k-ω SST, RSM and SST-C moreover DES approach show the
instability of flow across tubes.
� K-ε model fail to capture any instability of the flow across tubes.
� Distribution of pressure around centre tube shows high pressure on the top (in
stagnation region). It is a result in deflection due to suppression and distortion of
vortex shedding.
� Distribution of velocity in a wake of centre tubes shows low velocity in free flow in
streamwise direction far from cylinder's walls and also in recirculation regions.
� Stagnation point is located somewhere around 45 degrees. It agrees with LES of Imran
and experiment of Yahiaoui et al.
� From DPS results applying to pressure's and velocity's signals, it concludes that there
is a peak around 45 Hz (St=0.84) which proves existence of recirculation.
� By drawing the structure parameter Q. It observes structures in the gap space between
cylinders. There are more streamwise and spanwise structures for SST- C and DES
than other URANS models.
92
as
as

Chapter VI Conclusions and Recommendations for future work
� Code Star-CD confirms asymmetry of a flow and gives similar results with Code-
Saturne.
� Two recirculations behind the centre tube are observed. One is larger than the other. It
is located in the bottom and a small one in upstream. Shear stress in the bottom is
higher than on the top of tube.
Recommendations for the future work
This work has made some significant progress towards the original objectives outlined in the
Introduction, but it represents only the start of the development and validation process that
would be necessary in order to confidently apply a generic form of these models to a broad
range of test cases. Certain elements of the modelling described in this thesis were, of
necessity, not explored as fully as they could have been, in order to implement and test a
robust scheme within the given time frame. There thus remain several areas within which
further work would appear to be beneficial:
� Compute the flow across in-line tube bundle using LES and DES.
� Simulate the tube bundle case together with heat transfer.
� Test the new SST-C as model [18] for different new cases.
93

Bibliography
[1] C. MOULINEC, J.C.R. HUNT and F.T.M. NIEUWSTADT. "Disappearing Wakes and
Dispersion in Numerically Simulated Flows through Tube Bundles". Flow, Turbulence and
Combustion 73: 95–116, 2004.
[2] Haitham M. S. Bahaidarah and M. Ijaz and N. K. Anand. "Numerical Study of Fluid
Flow and Heat Transfer over a Series of In-line Noncircular Tubes Confined in a Parallel-
Plate Channel". Numerical Heat Transfer, Part B, 50: 97–119, 2006.
[3] A. Gatto, K.P. Byrne, N.A. Ahmed and R.D. Archer. "Mean and Fluctuating Pressure
Measurements over a Circular Cylinder In-cross Flow using Plastic Turbine". Experiments in
fluids 30 (2001) 43-46.
[4] Chunlei Liang & George Papadakis. "Large Eddy Simulation of Cross Flow over In-line
and Staggered Tube Bundles". Flow Induced Vibration, de Langre & Axisa ed. Ecole
Polytechnique, Paris, 6-9th July 2004.
[5] H.R. Barsamian, Y.A. Hassan. "Large eddy simulation of turbulent cross flow in tube
bundles". Nuclear Engineering and Design 172 (2004) 103-122.
[6] Pierre Sagaut. "Large Eddy Simulation for Incompressible Flows". Title of the original
French édition: introduction a la simulation des grandes échelles pour les écoulements de
fluides incompressibles, Mathématique et applications. Springer Berlin Heidelberg 1998.
[7] W. RODI and D. LAURENCE. "Engineering Turbulence Modelling and Experiments 4".
Proceeding of the 4 th International Symposium on Engineering Turbulence Modelling and
Measurements Ajaccio, Corsica, France, 24-26 May, 1999.
[8] H.K. Versteeg and W. MALALASEKERA. "An Introduction to Computational Fluid
Dynamic, The Finite Volume Method". First published 1995.
[9] J.H. Ferzige and M. Peric. "Computational Method for Fluid Dynamics". 3 rd edition.
Springer Verlag Berlin Heidelberg.2002.
94

[10] Chunlei Liang and George Papadakis. "Study of the Effect of Flow Pulsation on the
Flow Field and Heat Transfer over In-line Cylinder Array Using LES". Engineering
Turbulence Modelling and Experiments 6. W. Rodi editor, 2005.
[11] Aiba Shinya, Ota Ternakazn and Tuchida Hajime. "Heat Transfer of tubes closely
spaced in an in-line bank". International Journal of Heat and Mass Transfer, volume 23, Issue
3 March 1980, pages 311, 319.
[12] JUAN C. URIBE and DOMINIQUE LAURENCE. Two Velocities Field Approach to
Hybrid RANS-LES. Under consideration for publication in J.Fluid Mech.
[13] Sofiane Benhamadouche, Dominique Laurence, Nicolas Jarrin, Imran Afgan,
Charles Moulinec. "Large Eddy Simulation of Flow Across In-line Tube Bundle", 2003
[14] Imran Afgan. "Numerical Simulation of Cross Flow over Square In-Line Tube Bundle
Arrays using Large Eddy Simulation". PhD. Thesis: Fluid Structure Interaction of Cylinder
Bodies using LES by I. Afgan.
[15] Imran Afgan. "Fluid Structure Interaction of Cylinder Bodies using Large Eddy
Simulation". PhD. Thesis: Fluid Structure Interaction of Cylinder Bodies using LES by I.
Afgan (Second Year Report).
[16] Imran Afgan. "Large Eddy Simulation of cylindrical Bodies incorporating unstructured
finite volume mesh". A thesis submitted to The University of Manchester for the degree of
Doctor of Philosophy in the Faculty of Engineering and Physical Sciences, March 2007.
[17] Juan C. Uribe. "An Industrial Approach to Near-Wall Turbulence Modelling for
Unstructured Finite Volume Method". A thesis submitted to The University of Manchester for
the degree of Doctor of Philosophy in the faculty of Engineering and Physical Sciences,
September 2006.
[18] Alistair Revell. "A Stress-Strain Lag Eddy Viscosity Model for Mean Unsteady
Turbulent Flows". A thesis submitted to the University of Manchester for the degree of doctor
of philosophy in The Faculty of Engineering and Physical Sciences, June 2006.
95

[19] Frederic Archambeau, Namane Michitoua and Marc Sakiz. "Code Saturne: A finite
Volume Code for the Computation of turbulent Incompressible Flows Industrial
Applications". International Journal of Finite Volumes
[20] Baldwin, B. S. and Lomax, H. (1978), "Thin Layer Approximation and Algebraic
Model for Separated Turbulent Flows", AIAA Paper 78-257.
[21] Granville, P. S. (1987), "Baldwin-Lomax Factors for Turbulent Boundary Layers in
Pressure Gradients", AIAA Journal, Vol. 25, No. 12, pp. 1624-1627.
[22] Mavriplis, D. J. (1991), "Algebraic turbulence modeling for unstructured and adaptive
meshes", AIAA Journal, Vol. 29, pp. 2086-2093.
[23] Turner, M. G. and Jennions, I. K. (1993), "An Investigation of Turbulence Modeling in
Transonic Fans Including a Novel Implementation of an Implicit Turbulence Model",
Journal of Turbomachinery, Vol. 115, April, pp. 249-260.
[24] Wilcox, D.C. (1998), Turbulence Modeling for CFD, ISBN 1-928729-10-X, 2nd Ed.,
DCW Industries, Inc.
[25] Smith, A.M.O. and Cebeci, T. (1967), "Numerical solution of the turbulent boundary
layer equations", Douglas aircraft division report DAC 33735.
[26] Wilcox, D.C. (1998), Turbulence Modeling for CFD, ISBN 1-928729-10-X, 2nd Ed.,
DCW Industries, Inc.
[27] Wilcox, D.C. (2004), Turbulence Modeling for CFD, ISBN 1-928729-10-X, 2nd Ed.,
DCW Industries, Inc.
[28] Emmons, H. W. (1954), "Shear flow turbulence", Proceedings of the 2nd U.S. Congress
of Applied Mechanics, ASME.
[29] Glushko, G. (1965), "Turbulent boundary layer on a flat plate in an incompressible
fluid", Izvestia Akademiya Nauk SSSR, Mekh, No 4, P 13.
[30] Wilcox, D.C. (2004), Turbulence Modeling for CFD, ISBN 1-928729-10-X, 2nd Ed.,
DCW Industries, Inc.
96

[31] Baldwin, B.S. and Barth, T.J. (1990), A One-Equation Turbulence Transport Model for
High Reynolds Number Wall-Bounded Flows, NASA TM 102847.
[32] Dacles-Mariani, J., Zilliac, G. G., Chow, J. S. and Bradshaw, P. (1995),
"Numerical/Experimental Study of a Wingtip Vortex in the Near Field", AIAA Journal, 33(9),
pp. 1561-1568, 1995.
[33] Spalart, P. R. and Allmaras, S. R. (1992), "A One-Equation Turbulence Model for
Aerodynamic Flows", AIAA Paper 92-0439.
[34] Spalart, P. R. and Allmaras, S. R. (1994), "A One-Equation Turbulence Model for
Aerodynamic Flows", La Recherche Aerospatiale n 1, 5-21.
[35] Wilcox, D.C. (1988), "Re-assessment of the scale-determining equation for advanced
turbulence models", AIAA Journal, vol. 26, pp. 1414-1421.
[36] Wilcox, D.C. (2004), Turbulence Modeling for CFD, ISBN 1-928729-10-X, 2nd Ed.,
DCW Industries, Inc.
[37] Menter, F. R. (1993), "Zonal Two Equation k-ω Turbulence Models for Aerodynamic
Flows", AIAA Paper 93-2906.
[38] Menter, F. R. (1994), "Two-Equation Eddy-Viscosity Turbulence Models for
Engineering Applications", AIAA Journal, vol. 32, pp. 269-289.
[39] Durbin, P. Separated flow computations with the model, AIAA Journal,
33, 659-664, 1995.
[40] Popovac, M., Hanjalic, K. Compound Wall Treatment for RANS Computation of
Complex Turbulent Flows and Heat Transfer, Flow Turbulence and Combustion, 78, 177-202,
2007.
[41] Launder, B. E., Reece, G. J. and Rodi, W. (1975), "Progress in the Development of a
Reynolds-Stress Turbulent Closure." Journal of Fluid Mechanics, Vol. 68(3), pp. 537-566.
[42] T.B. Gatski. Turbulent flows: model equations and solution methodology. In R. Peyret,
editor, Handbook of Computational Fluid Dynamics, chapter 6. Academic Press, 1996.
97

[43] K. Hanjali´c, M. Popovac, and M. Had˘ziabdi´c. A robust near-wall elliptic-relaxation
eddy-viscosity turbulence model for CFD. International Journal of Heat Fluid Flow, 24:1047–
1051, 2004.
[44] W.P. Jones and B.E. Launder. The prediction of laminarization with a two-equation
model of turbulence. International Journal of Heat Mass Transfer, pages 301–314, 1972.
[45] S. Parneix, D. Laurence, and P.A. Durbin. A procedure for using DNS databases.
Journal of Fluid Engineering, pages 40–47, 1998.
[46] S. Pope. Turbulent flows. Cambridge University Press, Cambridge, 2000.
[47] C.G. Speziale, S.Sarkar, and T.B. Gatski. Modelling the pressure-strain correlation of
turbulence: an invariant dynamical system approach. Journal of Fluid Mechanics, pages 245–
272, 1991.
[48] Germano, M., Piomelli, U., Moin, P. and Cabot, W. H. (1991), "A dynamic sub-grid
scale eddy viscosity model", Physics of Fluids, A (3): pp 1760-1765, 1991.
[49] Kim, W and Menon, S. (1995), "A new dynamic one-equation subgrid-scale model for
large eddy simulation", In 33rd Aerospace Sciences Meeting and Exhibit, Reno, NV, 1995.
[50] Nicoud, F. and Ducros, F. (1999), "Subgrid-scale modelling based on the square of the
velocity gradient tensor", Flow, Turbulence and Combustion, 62: pp- 183-200, 1999.
[51] Smagorinsky, J (1963), "General circulation experiments with the primitive equations,
in the basic experiment. Monthly Weather Review", 91: pp 99-164, 1963.
[52] Spalart, P. R., Jou, W.-H., Stretlets, M., and Allmaras, S. R. (1997), "Comments on
the Feasibility of LES for Wings and on the Hybrid RANS/LES Approach", Advances in
DNS/LES, Proceedings of the First AFOSR International Conference on DNS/LES.
[53] Strelets, M. (2001), "Detached Eddy Simulation of Massively Separated Flows", AIAA
2001-0879.
[54] B.J. Daly and F.H. Harlow. "Transport equations in turbulence". Physics of Fluids,
pages 2634–2649, 1970.
98

[55] Lars Davidson. "MTF270 Turbulence Modelling", P79-45 March 5, 2007.
[56] C.Benocci, J.P.A.J Van Beak and U.Piomell. "Large Eddy Simulation and Related
Techniques: Theory and Applications". Von Karman Institute For Fluid Dynamics. Lecture
Series 2006-04.
[57] Shoei-Sheng Chen. "Flow-Induced Vibration of Circular Cylindrical Structures".
Argonne National Laboratory. Argonne, llionois.
[58] Rollet-Miet, P., Laurence, D. R., Ferziger, J. 1999. "LES and RANS of turbulent flow
in tube bundles". International J. of Heat and Fluid Flow, 20.241-254.
[59] Ogengoren A., Zaida, S. 1992. "Vorticity shedding and acoustic resonance in an In-line
tube bundle, Part II: Acoustic resonance". J. of Fluids and Structures 6. PP. 293-309.
[60] Ishigai, S., Nishikawa, E., Yagi, E. 1973. "Structures of Gas flow and vibration in tube
banks with tube axes normal to flow". Inst. Sym. on Marine Engineering, Tokyo. PP. 1-5-23
to 1-5-33; in Flow induced vibration of circular cylinder structures by Chen, S. S.
[61] Aiba, S., Tsuchida, H., Ota, T. 1982. "Heat transfer around tubes in in-line tube banks".
Bull JSME, 25. 919-926; in Flow around circular cylinders Vol 2: Applications by
Zdravkovich, M. M.
[62] Traub, D. 1990. "Turbulent heat transfer and pressure drop in plain tube bundles".
Chem. Engineering Process, 28. 173-181.
[63] Lam, K., Fang, X. 1995. The effect of interference of four equispaced cylinders in cross
flow on pressure and force coefficients. J. of Fluids and Structures, 9. 195-214.
[64] Lam, K., Li, J. Y., Chan, K. T., So, R. M. C. 2003. Flow pattern and velocity field
distribution of cross-flow around four cylinders in a square configuration at a low Reynolds
number. J. of Fluids and Structures 17. 665-679.
99

[65] Van Atta, C. W., Gharib, M., Hammache, M. 1988. "Three dimensional structure of
ordered and chaotic vortex streets behind circular cylinders at low Reynolds numbers". Fluid
Dynamics Research 3. 127-132.
[66] Akhilesh Gupta. 2003. "Enhancement of boiling heat transfer in 5X3 tube bundles".
[67] Zdravkovich, M. M. and Stonebanks, K. L. 1990. "Intrinsically non uniform and
metastable flow in and behind tube arrays". J. of Fluids and Structures 4. 305-319.
[68] Kim, H. J., Durbin, P. A. 1988. "Investigation of the flow between a pair of circular
cylinders in the flopping regime". J. of Fluid Mechanics 196. 431-448.
[69] Sayers, A. T. 1988. "Flow interference between four equispaced cylinders when
subjected to a cross flow". J. of Wind Engineering and Industrial Applications, 31. 9-28.
[70] Lam, K., Lo, S. C. 1992. "A visualization study of cross flow around four cylinders in a
square configuration". J. of Fluids and Structures, 6. 109-131.
[71] Sumner, D., Wong, S. S. T., Price, S. J., Paidoussis, M., P. 1999. "Fluid behavior of
side by side circular cylinders in steady cross-flow". J. of Fluids and Structures 13. 309-338.
[72] Weaver, D. S., Zaida, S., Sun, Z., Feenstra, P. 2001. "The effect of platen fins on the
flow induced vibrations of an in-line tube array". ASME, PVP-Vol 420-1, pp. 91-100.
[73] Price, S. J., Paidoussis, M. P. 1989. "The flow induced response of a single flexible
cylinder in an in-line array of rigid cylinders". J. of Fluids and Structures, 3. 61-82; in Flow
around circular cylinders Vol 2: Applications by Zdravkovich, M. M.
[74] Feenstra, P. A., Weaver, D. S., Nakamura, T. 2003. "Vortex shedding and Fluid elastic
instability in a normal square tube array excited by two-phase cross-flow. J. of Fluids and
Structures, 17. 793-811".
[75] Paidoussis, M. P. 1982. "A review of flow induced vibrations in reactors and reactor
components". Nuclear Engg. and Design, Vol 74. 31-60.
100

[76] Weaver, D. S., Fitzpatrick, J. A. 1988. "A review of cross flow induced vibrations in
heat exchanger tube arrays". J. of Fluids and Structures, 2. 73-93.
[77] Samir Ziada. 2006. "Vorticity Shedding and Acoustic Resonance in tube bundles". J. of
the Braz. Soc. of Mech. Sci. & Eng.
[78] Wolfe.D and Ziada.S 2003. "Feedback control of vortex shedding from two tandem
cylinders". Journal of Fluids and Structures. Vol. 17, no. 4, pp. 579-592. Mar. 2003
[79] E. Kanstantinidis et al. 2000. "On the Flow and Vortex Shedding Characteristics of an
in-Line Tube Bundle in Steady and Pulsating Cross flow".
[80] E. Konstantinidis, S. Balabani, and M. Yianneskis. 2002. "A Study of Vortex
Shedding in a Staggered Tube Array for Steady and Pulsating Cross-Flow
101