Hybrid CFD-CAA simulation of a heated and unheated Cylinder in turbulent cross ows

Master Thesis:

Hybrid CFD-CAA simulation of a heated and

unheated Cylinder in turbulent cross �ows

Awais Ali

Supervised by:

Dipl.-Ing. Manuel Münsch

Dr.-Ing. Julien Grilliat

Prof. Dr.-Ing. habil. Antonio Delgado

April 11, 2013

Lehrstuhl für Strömungsmechanik, Cauerstraÿe 4, 91058 Erlangen, Germany

"Ich versichere, dass ich die Arbeit ohne fremde Hilfe und ohne Benutzung anderer

als der angegeben Quellen angefertigt habe und dass die Arbeit in gleicher oder änlicher

Form noch keiner anderen prüfungsbehörde vorgelegen hat und von dieser als Teil einer

Prüfungsleistung angenommen wurde. Alle ausführungen, die wörtlich oder sinngemäss

übernommen wurden, sind also solche gekennzeichnet."

"I con�rm that I have written this thesis without any external help and not using

sources other than those I have listed in the thesis. I con�rm also that this thesis or a

similar version of it has not been submitted to any other examination board and has not

been previously accepted as part of an exam for a quali�cation. Each direct quotation

or paraphrase of an author is clearly referenced."

Awais Ali

Acknowledgement

Acknowledgements are not easy to write. I believe that a single change in past could

alter present and future. And this belief makes me thankful for many situations that

arose in my life and many people with whom I interacted. This space is not enough to

acknowledge everybody. But for everything that I have achieved in my life, I have to

be thankful to my family specially my parents. They supported me in every decision I

made without any doubt and hesitation. And thanking them is never enough for me.

People who helped me the most during this study are my advisors Dipl.-Ing. Manuel

Münsch and Dr.-Ing. Julien Grilliat, without their help and encouragement I would

have been totally lost. So Manuel and Julien a special thanks goes to you. I was lucky

enough to have very nice colleagues at LSTM Erlangen with whom I had very interesting

discussions regarding my thesis and some more general discussions. I had a lot of nice

co�ee breaks with many people and got an opportunity to learn a lot from every body. So

everybody from LSTM Erlangen who reads this, accept my special thanks. I would like to

mention some names, Naveed Iqbal always helpful, polite and a very pleasant personality

to talk to, Navaladi Arichunan for small co�ee breaks and helpful suggestions.

I would also like to take this opportunity to thank those friends who were always ready to

help me in every kind of situation. I can not leave out my friend Usman Sadiq, he is not

in Germany at the moment, but without his help in many situations I would have been

in a di�erent situation. So Usman a thanks is not enough, but at-least its a recognition

of help and advice that I always received from you.

Master thesis iii Awais Ali

Abstract

The �ow past a circular cylinder is characterized with the sequential shedding of vortices

in its wake. Each vortex being shed generates a pressure �uctuation on the cylinder

surface, which eventually turns into acoustic pressure emission. The vortex shedding

takes place at a frequency f which is related to the in�ow velocity U0 and the cylinder

Diameter D by means of Strouhal number:

St =fD

U0

= 0.2

As a result the acoustic emission takes place at a constant frequency and a whistle is

heard. This phenomenon causes lots of disturbances and its reduction is an on-going

research object.

In this study hybrid CFD-CAA technique is used to simulate the generated noise em-

ploying noise reduction techniques developed at LSTM Erlangen. The �ow is modeled

with compressible Navier-Stokes (NS) equations and acoustic analogies are used to es-

timate the sound production. To simulate turbulence Large Eddy Simulation (LES)

modeling technique is used, which has a requirement of careful application of numerical

methods. Therefore a sequential description of modeling and numerical issues observed

and employed during this study are mentioned. A comparison has been made between

two di�erent Sub-grid Scale (SGS) modeling techniques and observation regarding better

performance of one equation (kinetic energy) based SGS model are described. The issue

of spanwise extents of the �ow domain are discussed and a study of spanwise correla-

tions of streamwise velocity is also shown. The importance of estimating length scales

Master thesis iv Awais Ali

is a crucial issue in acoustics, since the cut-down of large structures (due to spanwise

dimension) could introduce a low frequency error in the pressure �uctuations. Experi-

ments conducted at LSTM resulted in a reduction in PSD for heated cylinder cases at

Re = 10,000 and 30,000. Flow over a heated cylinder is governed by two major phe-

nomena in �uid mechanics i.e. natural convection and forced convection. The shedding

of vortices is a major source of pressure �uctuation in �ow and hence is connected to

sound generation. Finally, a comparison has been made for acoustic noise emission from

cylinder with and without wall heating at two Reynolds numbers.

Master thesis v Awais Ali

Contents

Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

List of �gures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

1 Introduction 2

1.1 Cylinder Flow and Noise Phenomena . . . . . . . . . . . . . . . . . . . . 2

1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Objective of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Physical Modeling 5

2.1 Governing Equations of Fluid Dynamics . . . . . . . . . . . . . . . . . . 5

2.1.1 Thermo-physical Properties . . . . . . . . . . . . . . . . . . . . . 6

2.2 Turbulence Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 LES Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.2 Subgrid Scale Modeling . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Computational Methodology 14

3.1 Finite Volume Discretization . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1.1 Spatial Discretization . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.2 Temporal Discretization . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Navier Stokes Equations Discretization . . . . . . . . . . . . . . . . . . . 21

3.2.1 Pressure-Velocity-Density Coupling . . . . . . . . . . . . . . . . . 21

4 Cylinder Flow Simulations 25

4.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.1.1 Grid For Re 10,000 and 30,000 . . . . . . . . . . . . . . . . . . . . 26

4.1.2 Boundary and Initial Conditions . . . . . . . . . . . . . . . . . . . 26

Master thesis vii Awais Ali

CONTENTS CONTENTS

4.1.3 Numerical Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 Grid Convergence Study . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3 Spanwise Length Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.4 Comparison between Two SGS Models . . . . . . . . . . . . . . . . . . . 31

4.4.1 Instantaneous Flow Field . . . . . . . . . . . . . . . . . . . . . . . 32

4.4.2 Mean Flow Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.5 Re 10,000 Heated Cylinder Simulations . . . . . . . . . . . . . . . . . . . 34


4.5.2 Mean Flow Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.6 Re 30,000 Heated Cylinder Simulations . . . . . . . . . . . . . . . . . . . 39


4.6.2 Mean Flow Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5 Aero-Acoustic Simulations 42

5.1 Re 10,000 Aero-acoustic Simulations . . . . . . . . . . . . . . . . . . . . . 42

5.2 Re 30,000 Aero-acoustic Simulations . . . . . . . . . . . . . . . . . . . . . 44

5.3 Lighthill and Möhring Comparison . . . . . . . . . . . . . . . . . . . . . 45

6 Conclusion and Outlook 49

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Master thesis viii Awais Ali

List of Figures

3.1 Control Volume representation for FVM . . . . . . . . . . . . . . . . . . 16

4.1 O-type grid used in present study (a) Cross-sectional view of the grid in

x-y plane (b) Zoomed view of the grid around cylinder wall . . . . . . . . 27

4.2 Experimental Data for wall temperature and �tted polynomial to specify

wall temperature in simulation Re 10,000 . . . . . . . . . . . . . . . . . 28

4.3 Figure shows the two points A and B which are two locations in (x, y)

at (0.75D, 0) and (6.5D, 0) respectively, where (0, 0) is the center of the

cylinder. On right a line is shown in z-direction with equally spaced points

that represent the probe locations . . . . . . . . . . . . . . . . . . . . . . 30

4.4 Two point correlations Ruu of streamwise velocity u in spanwise direction

plotted against normalized separation distance ∆z/D for Re 10,000 (a)

Correlations for simulation with πD spanwise length (b) Correlations for

simulation with 8D spanwise length . . . . . . . . . . . . . . . . . . . . . 31

4.5 Time history of lift and drag coe�cients (≈50 vortex cycles) for Smagorin-sky and One Equation model for Re 10,000 . . . . . . . . . . . . . . . . . 32

4.6 Iso-surface of instantaneous vorticity magnitudes for Re 10,000 at ωz =

−0.0318 for two SGS models (a) oneEq model (b) Smag model . . . . . . 33

4.7 Mean velocity ( UMean =√u2x + u2

y + u2z ) �eld for simulations with two

di�erent models for Re 10,000 (a) Smag model (b) oneEq model . . . . . 33

4.8 Mean Reynold Stress component τxx = u′xu′x for Re 10,000 (a) Smag model

(b) oneEq model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.9 Relationship between St and Re from di�erent experiments [1]. Two hor-

izontal lines in the graph mark the two Re under study 10,000 and 30,000

respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.10 Time history of lift and drag coe�cients (≈50 vortex cycles) for 25oC

(25T) and 277oC (277T) . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Master thesis ix Awais Ali

LIST OF FIGURES LIST OF FIGURES

4.11 Comparison between normalized Instantaneous vorticity contours ωD/U∞

(=0.5-10) Re 10,000 (a) No heating (25T) (b) Severe heating (277T) . . . 37

4.12 Mean velocity ( UMean =√u2x + u2

y + u2z ) �eld for two simulations at

Re 10,000 (a) No Heating 25oC (b) Strong Heating 277oC . . . . . . . . . 38

4.13 Mean Reynold Stress component τxx = u′xu′x at Re 10,000 (a) No Heating

25oC (b) Strong Heating 277oC . . . . . . . . . . . . . . . . . . . . . . . 38

4.14 Comparison between mean x and y velocity with and without heating

along y direction at di�erent x locations (a) x/D = −0.75 (b) x/D = −1

(c) x/D = −2 (d) x/D = −4 . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.15 Comparison between normalized Instantaneous vorticity contours ωD/U∞

(=0.5-18) (a) No heating (25T) (b) Severe heating (277T) . . . . . . . . 40

4.16 Mean velocity �eld for two simulations at Re 30,000 (a) No Heating 25oC

(b) Strong Heating (UMean =√u2x + u2

y + u2z) 277

oC . . . . . . . . . . . 41

4.17 Mean Reynold Stress component τxx = u′xu′x at Re 30,000 (a) No Heating

25oC (b) Strong Heating 277oC . . . . . . . . . . . . . . . . . . . . . . . 41

5.1 Distribution of Sound Pressure over frequency (Re 10,000) (a) 90 Degree

point (b) 180 Degree point . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.2 Sound pressure plotted around the cylinder at 1m away probe locations

for heated and unheated cases at Strouhal Frequencies (Re 10,000) . . . . 44

5.3 Sound pressure contours (Re 10,000) (left: Heated, right: UnHeated) . . 44

5.4 Distribution of Sound Pressure over frequency (Re 30,000) (a) 90 Degree

point (b) 180 Degree point . . . . . . . . . . . . . . . . . . . . . . . . . 45


for heated and unheated cases at Strouhal Frequencies (Re 30,000) . . . . 46

5.6 Sound pressure contours (Re 30,000) (left: Heated, right: UnHeated) . . 46

5.7 Comparison between Lighthill and Möhring Analogies for Re 10,000 (a)

90 degree location (b) 180 degree location (c) 90 degree location (d) 180

degree location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47


for unheated case (Re 10,000), Lighthill and Möhring Comparison . . . . 48

Master thesis x Awais Ali

Abbreviations

CAA Computational Aero-Acoustics

CFD Computational Fluid Dynamics

PISO Pressure Implicit with Split Operator

SIMPLE Semi-Implicit Method for Pressure-Linked Equations

ν Kinematic Viscosity

Re Reynolds Number

D Cylinder Diameter

U0 Free Stream Velocity

ρ Density

T Temperature

NS Navier Stokes

FV Finite Volume

u Velocity Vector

h Speci�c enthalpy

Cp Speci�c heat at const. pressure

ui Velocity Component in ith direction

κ Heat conductivity co�.

A,A Surface are magnitude and vector

I Unit Tensor

N Neighboring Cell Centroid position

p Cell Centroid position

x Spatial Coordinate

Smag SGS model Smagorinsky

oneEq SGS model One Equation type

∇. Divergence of

∇ Gradient of

PSD Power Spectral Density

SPL Sound Pressure Level

Ma Mach Number

RANS Reynolds Averaged Navie-Stokes Equations

DNS Direct Numeric Simulation

γ Ratio of Gas constants Cp/Cv

Master thesis xi Awais Ali

Chapter 1

Introduction

1.1 Cylinder Flow and Noise Phenomena

Cylinder �ow is one of the frequently studied �ows in experimental and numerical �uid

dynamics [2]. The �ow phenomena is characterized by the non-dimensional Reynolds

number:

Re =DU

ν(1.1)

Drastic changes occur in cylinder wake phenomena as the Re increases above zero. Flow

irregularities starts appearing when Re > 40, and a vortex street starts to form in laminar

wake. Flow starts becoming three-dimensional at this point. At Re > 200 wake starts

becoming turbulent. For 300 < Re < 3e5 the wake is completely turbulent but boundary

layer is laminar and this �ow regime is referred to as sub-critical. For 3e5 < Re < 3.5e5

the boundary layer is still laminar and starts becoming turbulent at only one separation

point, this regime is called critical. This asymmetry in �ow also causes a non-zero mean

lift in this critical Re range. For 3.5e5 < Re < 1.5e6 boundary layer is partly turbulent

and laminar both at the same time and this regime is called super-critical. With 1.5e6

< Re < 4e6 boundary layer is completely turbulent just on one side and this �ow regime

is called Upper Transition. At Re > 4e6 boundary layer is completely turbulent, at

this point �ow regime is called Trans-critical [2].

Noise generation in �ow over a circular cylinder is a by product of turbulent process. Even

though the estimation of periodic structures could give an idea about the characteristic of

noise, still the simulation of turbulence is necessary to get accurate measures of the noise

generated. Measuring acoustics is not simple, as many parameters have to be taken into

account. For instance, experiments designed to measure noise levels in anechoic wind-

tunnel must be carefully designed, because wind-tunnel also has some noise emission

of its own. In numerical methods accurate prediction of acoustics requires a very well

Master thesis 2 Awais Ali

1.2. Motivation Chapter 1. Introduction

resolved simulation of �ow. The numerical methods must be set in such a manner that

the �ow properties are not in�uenced. The alteration of �ow properties from periodic

boundary conditions might not be that visible in integral �ow properties like drag or lift

but in frequency domain the errors could make big di�erences in the results. A further

study of energy spectrum could help validate the numerical techniques used.

1.2 Motivation

Noise is considered a major environmental pollutant. For instance in big industrial

areas, around busy airports and highways. Aircraft and automobile manufacturers are

trying to reduce the noise emission to a minimum possible level. For the same reason

understanding the generation of sound and control of noise is a major research interest

for academia.

Cylinder �ow and noise phenomena are representative of many blu� body �ow and noise

generation phenomena in engineering applications (e.g., landing gear, automobile radio

antenna etc.). Many experimental and numerical studies have been conducted to study

cylinder �ows in di�erent con�gurations. But the turbulent nature of �ow, boundary

layer and �ow separation has always been a challenge to simulate accurately. With

technological advancements in the �eld of computing, researchers and engineers are able

to work with models that were previously limited because of low computing power.

1.3 Objective of Thesis

This study aims to use the Hybrid CFD-CAA approach for estimation of cylinder noise

phenomena with and without wall heating. Cylinder �ow with wall heating is governed

to two main �uid mechanics phenomenon i.e. forced convection (momentum) and nat-

ural convection (buoyancy). The way these two phenomenon couple and e�ect the �ow

depends upon the �ow regime and wall heating. Previous studies carried out with cylin-

der wall heating at low Re suggested a suppression of vortices under dominant natural

convection. In this study due to high Re (higher than previous studies) natural convec-

tion doesn't a�ect the �ow regime. Still heating changes the thermodynamic properties

of �uid near the wall. The objective of present work is to simulate cylinder �ow phenom-

ena with wall heating and investigate its e�ects on the noise production. An attempt

has been made to highlight the issues concerning accurate �ow modeling of heated and

unheated case for further use with acoustic analogies. Re chosen for this study lies in

the sub-critical Re regime of cylinder �ow. Experimental data on this case has already

been generated [3]. In experiments performed at LSTM a decrease of Power Spectral


1.4. Thesis Outline Chapter 1. Introduction

Density (PSD) was observed with heated cylinder cases, at Strouhal Frequency. This

study aims to reproduce the observations made in experiments. For experimental setup

[3] can be referred.

1.4 Thesis Outline

Thesis is structured into �ve Chapters. Chapter 1 gives the motivation and objective

of the study and key ideas to move further. Chapter 2 lays out the constituent equa-

tions of Fluid Mechanics. LES simulation model and �ltering approaches speci�cally

concerning the LES of compressible �ow, are also explained. SGS modeling technique

is described and models used in current study are explained. Chapter 3 outlines the

Computational Methodology used to simulate the NS equations. An Finite Volume (FV)

discretization for general transport equation is described. Issues related to discretization

of NS with the helps of FV and mesh non-orthogonality are also explained. Discretization

schemes are specially described keeping in view their application to LES. And in the last,

solution algorithm for coupling of compressible NS equations is explained. Chapter 4

contains the result of the simulation performed. A detailed description of case setup i.e.

discretization schemes and solution algorithm is given. Results of grid convergence study

are also included. Calculation of spanwise correlations for estimation of �ow structures

in spanwise direction, are also shown. A comparison between two SGS models is done

and reasons for choosing the model are explained. Finally a comparison between heated

and non-heated cylinder for two Reynolds numbers is presented. Chapter 5 contains

the results from aero-acoustic simulations carried out during the present study. Acoustic

simulations are performed for both Re and a comparison between heated and unheated

cases for Re is presented. All simulations are performed with Möhrings analogy. For Re

10,000, apart from Möhring analogy, Lighthill analogy is also used to perform the same

simulations, to draw a �rst hand comparison between two simulated analogies. Chapter

6 �nally, a summary of the whole study is presented. Important results and conclusions

from the current work are written. Possible problems and extensions of the current study

are suggested.


Chapter 2

Physical Modeling

2.1 Governing Equations of Fluid Dynamics

In general, laws in continuum mechanics are expressed in terms of conservation or balance

principles. Fluid motion is governed by conservation principles of continuum mechanics

for mass (ρ), momentum (u) and energy (h). But these conservation principles are

insu�cient themselves to formulate a determinate problem. Nevertheless these important

constitutive equations for compressible Newtonian �uids are given as below [4, 5, 6].

∂ρ

∂t+∇.(ρu) = 0 (2.1)

∂ρu

∂t+∇.(ρuu) = −∇p+∇.S + ρf (2.2)

∂ρh

∂t+∇.(ρhu) = p+ S.D +∇.h + ρσ (2.3)

where

S = µ(∇.u)I + 2µD is the viscous stress tensor

D = ∇.u is the rate of deformation or strain tensor

h =

∫ T

T0

Cp(T )dT is the speci�c enthalpy

p =∂p

∂t+ u.(∇p) is the material time derivative of p

h = κ∇T is the heat conduction vector

f is the speci�c body force

σ is the speci�c net radiation


2.2. Turbulence Modeling Chapter 2. Physical Modeling

2.1.1 Thermo-physical Properties

Density ρ, pressure p and static temperature T are not independent from each other.

For a gas one can use the equation of state:

p = ρRT (2.4)

The gas constant R is given as R = Cp − Cv, where Cp and Cv are the speci�c heats atconstant pressure and volume, respectively.

Variations in dynamic viscosity µ is accounted for by the Sutherland's law, given as:

µ

µ0

=

(T

T0

) 32 T0 + S1

T + S1

where µ0 is the reference viscosity at reference temperature T0, S1 is the Sutherland's

constant for gaseous material and for air its value is 110.4K.

2.2 Turbulence Modeling

Turbulence undoubtedly is present in most applied �elds of engineering from earth sci-

ences to life sciences. The complexity of �ow (range of scales due to turbulence) increases

with Re (O(R11/4e )). This complexity caused by the turbulent motion of �uid renders

Direct Numeric Simulation (DNS) inapplicable to most �ows of interest. Scientists and

engineers try to tackle the problem in di�erent ways. The oldest approach to make the

NS equations solvable involve statistical average of equations over time and to model

the arising Reynolds stresses using di�erent techniques. The averaging process is called

Reynolds averaging and the averaged equations are called Reynolds averaged Navier-

Stokes (RANS). Another approach becoming popular with advancements in computing

is Large Eddy Simulation (LES) technique. LES constitutes a spatial �ltering approach

where large scale motions in �ow are solved and small scales are modeled. Following

topics layout the LES modeling approach based on the NS equations 2.1 2.2 and 2.3.

2.2.1 LES Model

The idea behind LES is based on the notion that large scale �uid motions carry the most

of �uid energy and their e�ects on transport and mixing is much more prominent than

the small scales. A spatial �ltering approach can be used to separate the large scale

motions (also called Grid Scales GS) from the small scale motions (also called Sub-Grid

Scales SGS). Since GS �uid motions are much important in transport of conserved prop-



erties, so the modeling approach which emphasizes on solving these GS motions more

precisely makes sense. Also SGS motions are a better candidate for modeling because of

their homogeneous and isotropic turbulence characteristics.

Before writing the �ltered NS equations its important to workout through some impor-

tant concepts that lead towards the SGS modeling and �ltering of the compressible NS

equations.

Favre Filtering

In LES it is assumed that as a result of the �ltering, �ow can be decomposed into two

parts. For the �ow of compressible �uids the density or mass weighted decomposition is

used.

ρ = ρ+ ρ′ (2.5)

φ = φ+ φ′′ (2.6)

and,

φ =ρφ

ρ(2.7)

where,

ρφ = ρφ (2.8)

is normally utilized.

In above equations φ′′ represent the high frequency SGS motions, while φ represent the

low frequency GS scales. Also equation 2.7 resulted from some basic assumptions about

the �lter i.e. ρφ′′ = 0, also ρ′φ = 0. Variables �ltered in this sense (.) are called Favre

Filtered variables.

LES Filtering Approach

As discussed earlier LES constitutes a �ltering approach. These �lters perform a neces-

sary action of separating GS from SGS scales in physical space. In a Fourier space the

�lter separates the low frequency and high frequency �uid motions. A �lter kernel G

may be de�ned as a convolution integral:

φ = φ(x, t) =

∫D

G(x− z,∆)φ(z, t)d3z = G ∗ φ(x, t) (2.9)



where the kernel G is the �lter function and integration is over the computational domain

D. Applying �lter over φ means that amplitude of high wave number (high frequency

SGS scales) spatial Fourier components are reduced. This �ltering of variables results in

a decreased demand for spatial resolution.

The non resolved part of φ(x, t), denoted φ′(x, t) is de�ned as:

φ′(x, t) = φ(x, t)− φ(x, t) (2.10)

Properties of Filter

For further manipulation of NS equations after �lter application, following three prop-

erties are required:

1. Consistency ∫D

G(x− z,∆)d3z = 1

2. Linearity

φ+ ψ = φ+ ψ

3. Commutation with di�erentiation

∂φ

∂x=∂φ

∂x

Three Classical Filters

Three particular convolution operators are commonly used for spatial scale separation,

the Box or top hat �lter, the Gaussian �lter and the spectral or sharp Cuto� �lter.

Their kernel functions are given in table below, both in physical and Fourier space for

one speci�c direction.

where ∆ in table 2.1 represent the �lter width. Filter described above are used in

explicit �ltering for LES simulations. Code used in present study (OpenFOAM) uses

implicit �ltering top hat �ltering.



Table 2.1: Kernels of three classical �lters

Filter Kernel in Physical Space Kernel in Fourier Space

Box Filter G(x− z) =

{1

∆if |x− z| ≤ ∆

20 otherwise

G(k) =sin(k∆/2)

k∆/2

Gaussian G(x− z) =6

π∆2exp(−6(x− z)2

∆2) G(k) = exp(

−∆2k2

24)

Sharp Cuto� G(x− z) =sin(kc(x− z))

kc(x− z)with kc = π/∆ G(k) =

{1 if |k| < kc0 otherwise

LES Filtered NS equations

Apply �ltering to NS equations (2.1,2.2,2.3) and using properties of the �lter results in

Filtered NS equations.

∂ρ

∂t+∇.(ρu) = 0 (2.11)

∂ρu

∂t+∇.(ρuu) = −∇p+∇.(S−B) + ρf (2.12)

∂ρh

∂t+∇.(ρhu) = ˜p+ S.D +∇.(h− b) + ρσ (2.13)

where density weighted (or Favre Filtered) SGS stress tensor (B) and �ux vector (b) has

been introduced in the equations above, which are de�ned as:

B = (ρuu− ρuu)

= (ρuu− ρuu) + (ρuu′′ + ρu′′u) + (ρu′′u′′)

= L + C + R

(2.14)

b = (ρuh− ρuh)

= (ρuh− ρuh) + (ρuh′′ + ρu′′h) + (ρh′′u′′)

= I + c + r

(2.15)

where {L, I} are Leonard Terms, {C, c} are cross terms and {R, r} are Reynold Terms.

The terms appearing in the �ltered set of equations can be divided into four categories:

1. ρ, u, h

These terms represent the GS scale of dependent variables and their evaluation is



the primary goal of the model.

2. Reynold Terms

{R, r} represent out-scatter or dissipation and require modeling.

3. Leonard Terms & Cross Terms

{L, I ; C, c} are speci�c to LES and do not appear in RANS modeling. Leonard

terms can be calculated directly since their calculation only involves dependent

variables. (ρ, u, h). Cross terms need explicit modeling. The main e�ect of

Leonard terms is distribution among grid scales while cross terms are responsi-

ble for backscatter i.e. transfer from SGS to GS scales.

4. S, f , p,S.D,h & σ

A description of non-�ltered form of these terms has been described before in this

chapter. This group is non-linear and the terms cannot be evaluated without

further assumptions. Details on evaluation of these terms can be found in [5], [6]

2.2.2 Subgrid Scale Modeling

As a result of �ltering the NS equations additional terms arise which need to be modeled.

The idea behind SGS modeling is to represent the SGS stress tensor B and �ux vector

b in terms of the resolved �elds ρ, u and h. This representation should be as close

as possible to the residual stress and �ux vector being modeled. Physically SGS model

represents the eddies or �ow motions that cannot be represented by the spatial resolution

of the domain. Model should also be able to accurately predict the energy transfer from

GS to SGS i.e. −B.D.

This transfer of energy is a two way process i.e. energy transfers from GS to SGS as

well as SGS to GS. But Leslie and Quarini[7] predicted that the �ow of energy from

GS to SGS scales is about 1.5 times the net transfer. SGS stress tensor and �ux vector

represent e�ect of this energy transfer (from GS to SGS) on resolved �ow �elds. Instead

of modeling the individual terms that need to be modeled (i.e. C, c,R, r), usually the

�ux vector and stress tensor are modeled as a whole.

Smagorinsky Model

The most popular SGS models are the eddy viscosity and di�usivity models and the

most common of them all was proposed by Smagorinsky (1963)[8]. The Smagorinsky

model assumes that the e�ects of SGS stresses and �uxes are an increase in transport

and dissipation. These phenomena are connected to viscosity in laminar �ows and hence



it is reasonable to assume that these terms can be modeled as an increase in viscosity

and di�usivity, hence these kind of models are called eddy viscosity and di�usivity type

models.

The model is based on a local equilibrium assumption which means that production and

dissipation balance at SGS scales, further it is assumed that SGS stress tensor and �ux

vector can be represented as (Fureby [5]):

B =2

3ρkI− 2µSGSDD (2.16)

b = −2κSGS∇h (2.17)

where k is the kinetic energy of SGS, I is the unit tensor, DD is the deviatoric part of

rate of strain tensor i.e. DD = D − 1

3tr(D)I , µSGS is the SGS viscosity and κSGS is

the SGS di�usivity. Using dimensional analysis SGS eddy viscosity and di�usivity can

be modelled as:

µSGS = C2sρ∆

2||D|| (2.18)

κSGS = CHρ∆2||D|| (2.19)

in above equations, equation 2.18 can be recognized as classical Smagorinsky model.

The SGS eddy di�usivity model equation 2.19 was suggested by Edison [9], who as-

sumed CH = C2s/PrT where PrT ≈ 0.9 is the turbulent Prandlt number. The classical

Smagorinsky coe�cient Cs assumes values between 0.2 � 0.065. Where a values of 0.2

has been found to be suitable for �ow without walls and a value of 0.065 has been found

to be suitable for channel �ows where turbulence has to be modeled near walls [4].

Another method to arrive at this model is by assuming a local equilibrium for transport

equation of kinetic energy.

∂ρk

∂t+∇.(ρuk) = −B.D +∇.((µSGS + µ)∇k)− CEρk

32∆−1 (2.20)

Assuming local equilibrium in the equation above:

−B.D = CEρk32∆−1

Using equation 2.16 and solving for k a model with two coe�cients can be reached.

Further details can be found in reference [5].



One Equation Model

Eddy viscosity and di�usivity models are the simplest models to account for the in�uence

for SGS scales on GS scales. Yoshizawa 1993 [10] suggested an one-equation model based

on the kinetic energy k = 1/2(u2− u2). This model again assume that stress tensor and

�ux vector can take the forms given in equations 2.16 and 2.17 respectively. Following

Yoshizawa [10], an approximate balance equation for kinetic energy can be written as

equation 2.20, where the �rst term on the right hand side represent production, the

second and third terms represent transport and dissipation/di�usion processes which

needs to be modeled.

The SGS viscosity needs to be modeled before this equation can be solved. It is assumed

that SGS viscosity and di�usivity can be expressed as µSGS = Ckρ√k∆ and κSGS =

µSGS/PrT . The dimensionless constants are assigned values Ck = 0.09 & CE = 1.00 [5].

van Driest Damping

Models discussed above have their own advantages and disadvantages. But a general

problem in eddy viscosity and di�usivity type models is that the model coe�cients are

not constants. As discussed earlier about Smagorinsky model, the model coe�cient

should assume di�erent values in di�erent �ow regions. Specially near walls, the eddy

viscosity needs to be reduced. This problem of reducing the eddy viscosity near walls is

solved using van Driest Damping, which suggests [4]:

Cs = Cs0

(1− e

−n+

A+

)2

(2.21)

where n+ is the normal distance from the wall in viscous wall units i.e. n+ = nuτ/ν,

where uτ =√τw/ρ is the wall shear velocity and τw is the wall shear stress. A+ is a

constant which is usually taken to be 25. Although this modi�cation works but this

modi�cation is di�cult to justify in context of LES as an SGS model should depend

solely on the local properties of the �ow.

LES Delta

Filter width ∆ is another parameter which complicates the simulation of �ow near solid

walls. Flow structures near wall are anisotropic. Regions of low and high speed �ow are

created that are 1000 viscous units and 30-50 viscous units wide in both spanwise and

normal directions respectively [4]. Resolving this requires highly anisotropic gird and

choice of length scales are not obvious.

Filter width ∆ represent SGS length scale. A most commonly used de�nition of LES



delta is calledCube Root Delta and is de�ned as∆ = (∆1∆2∆3)1/3, where∆i represent

the grid size in ith grid direction. Other de�nitions and choices are also possible [4].


Chapter 3

Computational Methodology

With a complete description of physical models, we have a set of equations which needs

to be discretized �rst and then a solution algorithm can be used to solve the non-linear

set of equations. Many discretization techniques have been used to solve NS equations,

these include Finite Element FE, Finite Di�erence FD, Finite Volume FV and

more exotic approaches like Lattice Boltzman Methods LBM. FD techniques have

generally been used more in research oriented investigations where higher order (4th or

higher) is of interest. There application to engineering oriented problems is limited, be-

cause of requirement of body �tted coordinates. FEM doesn't have the restriction of

body �tted coordinates but the FEM formulation doesn't guarantee the local conserva-

tion of variables [11].

Finite Volume methods have the nice property of conservation based on their CV formu-

lation, but going for higher orders for accuracy is not easy. Nevertheless FVM remains

to be the most popular method for engineering problems, as a trade o� for conservation

of �elds at the cost of accuracy is inevitable.

The chapter follows a general description of FVM discretization technique for NS equa-

tions. Before describing a solution algorithm for non-linear discretized equations, some

numerical scheme issues related to LES are also discussed.

3.1 Finite Volume Discretization

The LES �ltered equations (2.11, 2.12, 2.13) presented in Chapter 2 represent conser-

vation of basic properties of �uid �ow. In �nite volume approach these equations are

integrated over a control volume dV and time step ∆t to produce produce the integral

form of the governing equations.


3.1. Finite Volume Discretization Chapter 3. Computational Methodology

∫ t+∆t

t

[∂

∂t

∫V

ρdV +

∫V

∇.(ρu)dV

]dt = 0 (3.1)

∫ t+∆t

t

[∂

∂t

∫V

ρudV +

∫V

∇.(ρuu)dV −∫V

∇.(S−B)dV

]dt =∫ t+∆t

t

[∫V

−∇pdV]dt

(3.2)

∫ t+∆t

t

[∂

∂t

∫V

ρhdV +

∫V

∇.(ρhu)dV −∫V

S.DdV −∫V

∇.(h− b)dV

]dt =∫ t+∆t

t

[∫V

˜pdV

] (3.3)

In above equations body forces f and radiation term σ have been neglected. These

terms can always be included as source terms in to the system of equations.

Above equations can be recognized as general transport equations and each of the terms

have to be discretized individually. From here onwards discretization for general trans-

port equation will be described which will be applicable to individual terms (convection,

di�usion etc.). FVM discretization here is divided into 2 parts i.e. temporal and spatial.

In spatial domain equations are discretized over control volumes that divide the whole

solution domain into �nite number of CVs. Since the problem is transient (LES), tempo-

ral discretization involved the division of solution time into small time steps ∆t. As time

is a parabolic (solution at ti depends only on ti−k, where k is +ve integer) coordinate

only ∆t is required to specify the temporal discretization of solution domain. ∆t can be

uniform or non-uniform depending upon algorithm and solution requirements.

3.1.1 Spatial Discretization

As described earlier instead of following the discretization of each partial di�erential

equation, a generic transport equation for φ will be discretized. Each term in transport

equation for φ will be treated separately. Special cases are dealt separately. Figure 3.1

shows a representation of a generic CV. In �gure below hexahedral CVs are shown but the

methods are equally applicable to polyhedral cells resulting from unstructured meshing

techniques. In �gure P is the center of control volume, N represents a neighboring CV,

Af represents the face normal of face f and d represent the vector joining the cell centers

of P and N. Generic transport equation for a variable φ can be written as:

∂

∂t

∫V

φdV︸︷︷︸TemporalDerivative

+

∫V

∇.(uφ)dV︸︷︷︸Convection

−∫V

∇.(Γφ∇φ)dV︸︷︷︸Diffusion

=

∫V

Sφ(φ)dV︸︷︷︸Sources

(3.4)



Aff

Pd

N

Figure 3.1: Control Volume representation for FVM

In equation 3.4 φ is the transported variable i.e. velocity, mass enthalpy, turbulent

energy, Γ is the di�usivity coe�cient. This is a second order equation since the di�usion

term includes a second derivative [11]. To represent the di�usion term with acceptable

accuracy the order of the discretization must be equal to or higher than the order of the

equation. To conform to this level temporal discretization must also be second order.

And as a consequence of this requirement all dependent variables are assumed to vary

linearly around point P and time t so that:

φ(x) = φP + (x− xP ).(∇φ)P (3.5)

φ(t+ ∆t) = φt + ∆t.

(∂φ

∂t

)t

(3.6)

Gauss' Theorem

Before going further and discretizing the individual terms of the generic transport equa-

tion an important tool needs to de�ned. Gauss theorem is used to express volume

integrals to surface integrals throughout this section as:∫V

∇.φdV =

∫∂V

dA.φ (3.7)

where ∂V is the surface bounding V and dA is the in�nitesimal surface element with

outward pointing normal on ∂V .

A series of volume and surface integrals now need to be evaluated over the CV to second

order. Taking into account the variation of φ and x around P (from eqs 3.5):∫V

φ(x)dV = φPVP (3.8)



where VP is the cell volume. Now converting volume integrals into surface integrals using

Gauss' theorem and integration of divergence operator over the cell surface:∫V

∇.φdV =

∫∂V

dA.φ =∑f

(∫f

dA.φ

)=∑f

A.φf (3.9)

where f represent the face numbering and A represent the outward normal of the face.

The face values φf has to be calculated by interpolation, which will be covered later.

Convective Term

The discretization of the convective term can be obtained using equation 3.9 as:∫VP

∇.(uφ)dV =∑f

A.(uφ)f =∑f

(A.uf )φf =∑f

Fφf (3.10)

where F represent the volume �ux through face F = A.uf . The �ux F depends on

the face values of the Favre �ltered velocity u. The velocity �eld from which �uxes are

calculated must satisfy the continuity equation.

Cell face values can be interpolated linearly from neighboring cell center values uP & uN

as:

φf = fxφP + (1− fx)φN (3.11)

where fx is de�ned as ratio of distance fN and PN as fx = fN/PN . This linear

interpolation technique is commonly known Central Di�erencing (CD) and has been

shown to be second order accurate [4]. CD scheme however has some drawbacks as well,

chief amongst them is the unboundedness of the scheme. Here, boundedness means that

solution at a particular point is bounded by the neighboring points which in�uence it.

There are many schemes available for discretization, some of which are very exotic [4].

But another simple scheme used is the Upwind Di�erencing (FD) scheme, based upon

the Flux Direction F :

φf =

{φP if F ≥ 0

φN if F < 0(3.12)

Even though the UD scheme is bounded, it is known to be dissipative. A scheme which

produces numerical dissipation is problematic particularly in LES, because the contribu-

tion of turbulent di�usivity is very small and even smaller dissipation can produce large

inaccuracies in the �nal solution.

The best choice for LES simulations is CD scheme with local mesh re�nement to keep

the cell local Peclet number Pe = ρu∆x/Γ ≤ 2 [4]. So a local mesh re�nement in regions

of interest is the solution for CDS as its the best possible choice for LES simulations.



Di�usion Term

Using a similar approach as above the di�usion term in eq 3.4 can be discretized as

follows: ∫V

∇.(Γφ∇φ)dV =∑f

A.(Γφ∇φ)f =∑f

(Γφ)fA.(∇φ)f (3.13)

where (Γφ)f can be interpolated using CDS (eq 3.11). If the mesh is orthogonal the

vectors d and Af in �gure 3.1 are parallel. In that case face gradient of φ term can be

expressed as [11]:

A.(∇φ)f = |A|φN − φP|d|

(3.14)

this method is more accurate than simply interpolating (linear, CDS) the cell centered

gradients as:

(∇φ)f = fx(∇φ)P + (1− fx)(∇φ)N (3.15)

where

(∇φ)P =1

V

∑f

φf (3.16)

and the hat signi�es the interpolated gradient. Although both methods are second

order but the interpolation needs a larger computational molecule and produces larger

truncation errors.

If the mesh is non-orthogonal eq 3.14 is no longer second order accurate and needs to be

supplemented as follows:

A.(∇φ)f = |Ad|φN − φP|d|︸︷︷︸

orthogonal

+ A∆.(∇φ)f︸︷︷︸

non−orthogonal

(3.17)

where vector Ad represent component parallel to d and A∆ represent the remaining part

of A.

Di�usion term is bounded for orthogonal meshes but the addition of non-orthogonality

doesn't ensure boundedness, particularly for high non-orthogonality. In such a case the

contribution of non-orthogonal part needs to be limited which will e�ect the order of

accuracy. In LES where order of accuracy is of prime interest a care needs to be taken

to keep the mesh non-orthogonality to a minimum level[11].

Source Terms

Terms in the transport equation that cannot be written as convection, di�usion of tempo-

ral contributions are classi�ed as source terms. Source terms generally include a function



of φ or other variables which needs to be linearized before being added to the system

matrix.

Sφ(φ) = SC + SPφ (3.18)

where SC and SP may also depend upon φ. Following the midpoint rule for volume

integrals [4]: ∫VP

Sφ(φ)dV = SCVP + SPVPφP (3.19)

The above term is also second order accurate.

3.1.2 Temporal Discretization

As done in the previous section, a generic transport equation 3.4 is discretized in space.

In this section the same generic transport equation will be discretized in temporal co-

ordinate. As done for equations 3.1 3.2 and 3.3, the generic transport equation can be

written as:

∫ t+∆t

t

[∂

∂t

∫V

φdV +

∫V

∇.(uφ)dV −∫V

∇.(Γφ∇φ)dV

]dt =

∫ t+∆t

t

[∫V

Sφ(φ)dV

]dt

(3.20)

Replacing the terms identi�ed before as convection, di�usion and source, with their

discretized forms in the section above:∫ t+∆t

t

[(∂φ

∂t

)P

VP +∑f

Fφf −∑f

(Γφ)fA.(∇φ)f

]dt =

∫ t+∆t

t

[SCVP + SPVPφP ]

(3.21)

Terms in the equation above need not to have same discretization technique for temporal

discretization. As long as all individual terms have at-least second order accuracy, the

overall result of the system will be second order accurate.

Again as with spatial discretization many di�erent schemes have been used for time

discretization as well. Among them most famous are Crank-Nicholson Scheme, Euler

Implicit, Euler Explicit and Second Order Backward Di�erencing [4]. Here Backward

Di�erencing Scheme of second order accuracy is described in detail.



Second Order Backward Di�erencing

A linear variation of φ is assumed here in temporal coordinate. With this linearity

assumption the temporal derivative and time integral can be calculated as follows:(∂φ

∂t

)P

=φnP − φn−1

P

∆t(3.22)∫ t+∆t

t

φ(t)dt =φn−1 + φn

2∆t (3.23)

Equations stated above provide a second order approximation of the time derivative at

time t + 0.5∆t. For a second order accurate representation at t + ∆t three time levels

will be required. And a scheme produced in this way is called Backward Di�erencing

scheme:

φn−2 = φt−∆t

φn−1 = φt

φn = φt+∆t

Taylor series expansions of time level n − 2 and n − 1 around n up to third order of

truncation can be written and combined to give a second order accurate expression for

time level n, which would be written as:(∂φ

∂t

)t=

1.5φn − 2φn−1 + 0.5φn−2

∆t(3.24)

By neglecting the temporal variation in face �uxes and derivatives equation 3.24 yields

a fully implicit second order accurate discretization of general transport equation:

1.5φn − 2φn−1 + 0.5φn−2

∆tVP +

∑f

Fφnf −∑f

(Γφ)fA.(∇φ)nf = SCVP + SPVPφnP (3.25)

Backward di�erencing (BD) has been found to be considerably cheaper and easier to

implement than other second order schemes like Crank-Nicholson, but backward di�er-

encing is known to produce higher truncation errors [11]. This error manifests itself as an

added di�usion. And as discussed earlier any kind of numerical di�usion e�ects the LES

results signi�cantly, so this could be a potential problem. Fortunately a requirement of

CFL number < 1 for stability results in very small time steps, which in turn minimizes

the problem of added numerical di�usion due to BD scheme. Also a small time scale is

required to resolve the GS dynamics accurately, which again also helps to decrease the


3.2. Navier Stokes Equations Discretization Chapter 3. Computational Methodology

numerical di�usion because of temporal discretization scheme.

3.2 Navier Stokes Equations Discretization

Previous section on FVM covers the discretization techniques applied to generic transport

equation for φ. But in NS equations all the terms are not convection or di�usion terms.

As it has been stated earlier, all the terms other than Convection and Di�usion are

treated as source terms, still some terms may require special treatment. Having said that

we can have a look at the convective term in momentum equation 3.2. The term ∇.(ρuu)

introduces non-linearity and hence linearization is achieved by taking one velocity from

the previous iteration and hence generating a face �ux from previous iteration velocity.

A mathematical representation is:∫VP

∇.(ρuu)dV =

∫∂VP

dA.(ρuu) =∑f

A.(ρu.u)f

≈∑f

A.(ρu)n−1f .unf =

∑f

F n−1unf

(3.26)

3.2.1 Pressure-Velocity-Density Coupling

Solution algorithms for compressible NS equations can be classi�ed into two categories as

segregated and coupled algorithms. Coupled solvers are those which take into account

the coupling between pressure,velocity and energy and the whole system of equations

is solved simultaneously. However in segregated approach pressure-velocity coupling is

introduced through a separate equation for pressure to impose the divergence free velocity

�eld condition set by continuity (for incompressible �ows, SIMPLE). Algorithms based

on the same ideas have been extended for the compressible �ow solvers as well. But

instead of just taking into account pressure-velocity coupling, one has to think about

compressibility as well and hence the Pressure-Velocity-Density coupling.

In compressible case the algorithms generally used compute a velocity �eld based on a

guessed pressure �eld. The velocity �eld based on a guessed pressure �eld doesn't satisfy

continuity and and a calculated pressure gradient from the mass imbalance is used to

correct the velocity and pressure �eld. This procedure is continued until some speci�c

level of convergence is achieved. However in compressible �ows the mass imbalance

would not just depend upon the face �ux normal to cell face but also on density. In this

case to correct the mass �ux imbalance, both density and velocity needs to be corrected.

First step towards the design of an algorithm to solve compressible NS is to construct a

pressure equation for pressure-velocity-density coupling [4] [12].



Compressible Pressure Equation

In this section a systematic derivation of compressible pressure equation is done. In next

section the coupling algorithm using the pressure equation derived in this section will be

presented.

Discretization of momentum equation has already been described previously in this chap-

ter. Here the momentum equation in its discretized form can be stated as:

auP uP = H−∇p (3.27)

auP consists of the sum of the coe�cients of uP i.e. auP = acP +adP +atP , where ac,ad and

at are coe�cients from convection, di�usion and temporal terms. The term H vector

consists of the contribution from the neighboring matrix coe�cients multiplied by their

velocities and all the source terms except the pressure gradient term but including the

current time contributions.

H =∑N

(acN + adN)uN + atC + adC (3.28)

where atC is the portion of temporal derivative which doesn't depend upon the current

time step and adC is the part of di�usion term and doesn't depend on current time step.

Also all the coe�cients are function of velocities at previous time step. Equation 3.27

can be written as:

uP = (auP )−1H−∇p (3.29)

an equation for pressure can be derived by substituting expression for uP in continuity

equation. But before doing that density pressure relation has to be handled. Using the

change rule for ∂ρ/∂t:∂ρ

∂t=∂ρ

∂p

∂p

∂t(3.30)

For ideal gases∂ρ

∂p= ψ = 1/RT . Using this expression ρ can be expressed in terms

of p. Substituting the the velocity equation 3.29 in divergence term of continuity an

expression can be written as:

∇.(ρu) = ∇.[ρ(auP )−1H]−∇.[ρ(auP )−1∇p] (3.31)

The second term in above equation is the Laplace operator and is similar to the in-

compressible counterpart and is not modi�ed. The �rst term represents the divergence.

Using the pressure density relationship from ideal gas law, this term is converted to the



following convective form:

∇.[ρ(auP )−1H] = ∇.[ψp(auP )−1H] = ∇.(Fpp) (3.32)

where FP = ψ(auP )−1H is the �ux featuring the convective e�ects in pressure. Substitut-

ing equation 3.32 in 3.31 and using the pressure density relationship, continuity equation

is reformulated to form pressure equation (Jasak 2007[12]):

∂ψp

∂t= ∇.(Fpp)−∇.[ρ(auP )−1∇p] (3.33)

Transient Algorithm for NS

Most commonly used algorithm for solution of transient NS equations is Pressure Implicit

with Splitting of Operators (PISO) [4] [12] [13]. In PISO algorithm no under-relaxation

of variables is introduced. PISO algorithm works on a predictor-corrector approach.

Momentum equations are solved to give an intermediate velocity �eld. On the basis of

the predicted velocity �eld pressure equation is assembled and solved. A correction step

is performed using the new pressure �eld. This process is continued a speci�ed number

of times or till a speci�ed convergence criteria is satis�ed. Then the algorithm continues

to the next time step. This time values from the previous time step can be used as an

initial guess for a faster convergence towards the solution.

To make the algorithm more robust and e�cient an outer Semi-Implicit Method for

Pressure-Linked Equations (SIMPLE) [12] [14] like iteration can be added. This kind

of modi�cation of SIMPLE with PISO is also referred to as PIMPLE [15]. In actuality

PIMPLE algorithm is just a transient SIMPLE algorithm, where the pressure equation

is the same as PISO with and added option of relaxing the variables.

Below a compact overview of the PIMPLE algorithm is laid out in form of steps:

Start SIMPLE

1. Advance in time and calculate Density using ideal gas law

2. Construct and solve the discretized Energy Equation

3. Construct and solve the discretized Momentum Equation

Start PISO

(a) Construct and solve the Pressure Equation, also perform speci�ed number of

non-orthogonal correctors

(b) Correct �uxes with new pressure �eld



(c) Correct and Relax all variables (pressure, velocity, density)

(d) Go to 3a till speci�ed number of iteration (number of inner PISO Iterations)

4. Solve and correct for turbulence

5. Go to 2 till till speci�ed number of iteration (number of outer SIMPLE iterations)

6. Go to 1

The algorithm above describes the general solution procedure used by a PIMPLE or

Transient SIMPLE algorithm. Equations used has already been written and discretized

in previous sections. The discretized form of equations and algorithms are designed in

such a manner to remove non-linearities. The resulting set of equations can be solved

as linear algebraic equations at any step of above algorithm. E�cient choice of solver

for linear system of equations is another important matter. This choice depends upon

size, parallelism and properties of the linear equations to be solved. This work doesn't

demand a discussion of the solvers for linear system of equations. Details about the

solvers used for present study is described in the next chapter.


Chapter 4

Cylinder Flow Simulations

In this chapter the results of the cylinder �ow simulations for Re 10,000 and 30,000 will

be summarized. Up-till now in previous chapters governing equations of �uid �ow, mod-

eling techniques for turbulence and solution algorithm have been discussed. This chapter

presents the result of the �ow simulations carried out. Cylinder �ow is perhaps one of the

most interesting �ows, simulated to study many e�ects of numerical techniques, solution

algorithm and to validate models. The favoritism towards cylinder �ow geometry as a

test case is owing to the amount of data that has been generated over the years. Flow

over a cylinder is considered a simple case but not yet simple enough to ignore many

phenomena that might occur in industrial engineering problems.

Before going further and presenting the results of the present study, it would be worth

mentioning some of the important studies done in the past. Many LES simulations

have been conducted for �ow over a circular cylinder in sub-critical range (i.e. 300<

Re<3e5). An Re that has been studied the most in simulations as well as in experiments

for cylinder is Re= 3900. The main interest in all these studies have been grid type,

grid resolution and numerical method [16]. Even though the Re studied in this case is

di�erent than the one mostly studied, its still of interest to see what traditionally has

been the setup, grid type and numerical method used for similar simulations. These

details have been mentioned and discussed in [16].

Here are some observations from the references. The most popular grid type has been

O-type or C-type [17] [18] [19] [20]. Spanwise dimension of the domain in most stud-

ies is found to be πD. Breuer[17] and Kravchenko & Moin [18] also concluded that

doubling the spanwise dimension (from πD to 2πD) didn't have any e�ect on the �nal

results. Numerical Method used for discretization has mostly with FVM with exception

of Higher Order FDM and FEM used in some studies. For time integration the most

popular method has been found to be Runge-Kutta methods, however Euler-backward

has also been used by many anthers. In FVM and FEM for spatial discretization, Cen-


4.1. Simulation Setup Chapter 4. Cylinder Flow Simulations

tral Di�erencing (linear) scheme has been used in di�erent orders of accuracy. Finally all

the authors have used Eddy Viscosity type models for SGS modeling, with and without

dynamic procedure of Germano [21] and Lilly [22].

4.1 Simulation Setup

For both Re (10,000 & 30,000) studied similar simulation setup has been used.

4.1.1 Grid For Re 10,000 and 30,000

A curvilinear O-type grid has been used for this study with a cylinder diameter of 16mm.

The grid extension in radial direction is 15D and in spanwise dimension πD is used. Fig-

ure 4.1 shows two views of the grid. The grid is designed to ful�ll the dimensionless wall

distance requirements in radial and spanwise direction, as suggested by Piomelli and

Chasnov [23]. According to Piomelli and Chasnov grid for LES must satisfy y+ < 2 for

�rst grid point in wall normal direction and in spanwise direction ∆z ≈ 15− 40 must be

satis�ed. The present grid for both Re is designed to satisfy constrains imposed by the

wall dimensionless distance requirements.

For Re 10,000 the number of grid points in radial and circumferential directions are 130

and 242 respectively. In spanwise direction, after a grid convergence study 64 grid points

are selected. Grid expansion ratio in radial direction is less than 1.05.

For Re 30,000 number of grid points are only changed to meet wall normal dimension-

less distance requirement. It is assumed that previous grid (Re=10,000) is su�ciently

�ne enough in circumferential and spanwise direction, hence number of grid points in

radial direction are changed only. Number of grid points the meet y+ requirements are

increased from 130 to 170 and grid growth factor is kept less than 1.05.

4.1.2 Boundary and Initial Conditions

The inlet boundary conditions (on the left of the domain) are not considered to be

turbulent. This is in keeping with the previous observations and studies. Breuer [17]

argues that in this kind of grid design control volumes near the inlet are large in size

because of grid expansion. Thus any kind of perturbations added to the in�ow will

be highly damped and the probability of perturbed �ow reaching the cylinder is very

low. In this study the inlet is speci�ed as a �xed constant velocity inlet. At the outlet

wave-transmissive boundary conditions are applied [24]. On the cylinder wall isothermal

and no-slip boundary conditions are applied. While in the spanwise direction periodic

boundary conditions are used.


4.1. Simulation Setup Chapter 4. Cylinder Flow Simulations

(a) (b)

Figure 4.1: O-type grid used in present study (a) Cross-sectional view of the grid in x-yplane (b) Zoomed view of the grid around cylinder wall

Fluid properties considered are that of air with ideal gas assumption to account for

compressibility. A ratio of gas constants γ is taken to be 1.4. Variations in viscosity are

accounted for by using Sutherland's law. In�ow boundary conditions are used as initial

conditions.

For heated cylinder cases wall temperature data measured in experiments is used to

create a polynomial to �t the data. In all the cases a polynomial of degree 5 is used to �t

the data. A representation of experimental data and �tted polynomial for temperature

are shown in �gure 4.2 for Re 10,000 and heating of 177 oC. Polynomials used to �t data

can be found in table 4.1

Table 4.1: Expressions used to �t experimental temperature pro�le

T oC , Re Expression� x5 x4 x3 x2 x1 x0

177 , 10,000 4.53390e10 5.75248e8 −9.85540e6 −9.56760e4 1498.8 479.5277 , 10,000 2.54472e11 4.40549e8 −3.31080e7 −1.25900e5 3074.8 597.5277 , 30,000 2.33420e11 8.83579e8 −5.12782e7 −3.92287e5 5025.7 631.1

4.1.3 Numerical Setup

Large Eddy simulations in this study are carried out using OpenFOAM code (version

2.1.1) [25]. Compressible �ow solver used for the solution is called rhoPimpleFoam

and is based on SIMPLE-PISO algorithm used for pressure-velocity coupling of FVM

discretized equations as discussed in 3.2.1. All convective, viscous and pressure gradient

terms are approximated with a second order central di�erencing scheme as discussed in

3.1.1. A second order backward di�erencing is used to approximate all the terms as

discussed in 3.1.2. A constant time step is used to keep the local cell Courant number


4.2. Grid Convergence Study Chapter 4. Cylinder Flow Simulations

192

194

196

198

200

202

204

206

208

210

-0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008

Te

mp

era

ture

[d

eg

C]

X position on Cylinder Surface [mm]

5th Degree Poly FitTemp Profile Experimental

Figure 4.2: Experimental Data for wall temperature and �tted polynomial to specifywall temperature in simulation Re 10,000

less than 0.5.

In addition the average statistics are computed for approximately 50 vortex shedding

cycles. Many di�erent studies have used di�erent averaging times for statistical averaging

of LES simulations. Franke and Frank [20] studied this point and they concluded that

more than 40 vortex shedding periods are required to achieve a statistically converged

simulation for LES.

To solve the pressure and density a preconditioned Conjugate Gradient (PCG) solver

with Diagonal Incomplete Cholesky (DIC) as a preconditioner is used. While for all

other variables (velocity, enthalpy, SGS stress Tensor) a preconditioned bi-Conjugate

Gradient (PBiCG) with Diagonal Incomplete LU (DILU) as preconditioner is used. At

each time step all the dependent variables are solved with a local accuracy of 10−7.

4.2 Grid Convergence Study

A grid convergence study is carried out for the current case setup for Re 10,000 and

employing one-equation SGS model for LES (2.2.2). It is assumed that in radial and cir-

cumferential direction grid is �ne enough while keeping up with the wall y+ requirements

and the grid expansion factor. Grid density is changed only in spanwise direction to see

how this e�ects the results. Three di�erent grids are studied for grid convergence study

with 32, 64 and 96 grid points in spanwise direction. Results from the 3 grids studied

suggested almost no variation in results from 64 to 96 grid points in spanwise dimension.


4.3. Spanwise Length Scales Chapter 4. Cylinder Flow Simulations

Table 4.2: Grid convergence study results for Re 10,000

Nz St Cd Clrms⟨−Cpb

⟩ ⟨θsep⟩ ⟨

lr⟩/D

G3 96 0.2 1.252 0.5049 1.24 88.2 0.784G2 64 0.2 1.255 0.4877 1.22 88.4 0.784G1 32 0.207 1.175 0.3265 1.06 88.4 1.030

DNS [26] 0.19-0.21 1.11-1.28 0.45-0.57 1.06-1.02 - 0.82Exp[27],[26],[1] 0.19-0.20 1.19 0.38-0.53 1.11 - 0.78

Based on this analysis the gird with 64 grid points (G2 in table 4.2) in spanwise direction

was chosen. Results of the grid convergence study can be seen in table 4.2.

In table 4.2 St is the Strouhal frequency (calculated from Cl), Cd is time averaged

drag. Instantaneous coe�cient of drag is de�ned as Cd = Fd/(0.5ρ∞U2∞), Clrms is the

root mean square of �uctuating lift coe�cient de�ned as:

Clrms =

√√√√1/nn∑i=1

(Cl − Cli)2.

Cl is the instantaneous lift coe�cient de�ned in the same way as Cd.⟨−Cpb

⟩is the

negative of the coe�cient of averaged back pressure (where over-line represents averag-

ing in time and angled brackets represent averaging in spanwise direction),⟨θsep⟩is the

separation angle of the �ow and⟨lr⟩/D is the recirculation length non-dimensionalised

by the cylinder diameter D. Recirculation length is normally measured as the perpendic-

ular distance from the cylinder surface (in the wake) to the point, where the x-velocity

changes sign.

4.3 Spanwise Length Scales

Periodicity in �ow for acoustics needs to be modeled accurately. This means that the

length of �ow structures in periodic direction needs to be taken into account while consid-

ering the spatial length of domain in spanwise direction. To estimate the grid dimension

two di�erent grids were studied for two point correlations of streamwise velocity u, in

spanwise direction. Two point correlations give an indication of spanwise extent of the

domain. The size of structures is roughly twice the point of origin and point where

correlations go to zero. Hence the calculation domain is usually considered large enough

if the correlations go to zero in half of the spanwise grid extent [28, 29]. The de�nition


4.3. Spanwise Length Scales Chapter 4. Cylinder Flow Simulations

Figure 4.3: Figure shows the two points A and B which are two locations in (x, y) at(0.75D, 0) and (6.5D, 0) respectively, where (0, 0) is the center of the cylinder. On right aline is shown in z-direction with equally spaced points that represent the probe locations

of correlation coe�cient used is:

R(u(z), u(z + ∆z)) = Ruu =〈u(z)u(z + ∆z)〉√

〈u(z)u(z)〉〈u(z + ∆z)u(z + ∆z)〉(4.1)

Where u(z) and u(z + ∆z) are �uctuating velocity in x-direction at two di�erent loca-

tions in spanwise (z-direction) direction. The two point correlations are a function of

separation ∆z between two points . These correlations are calculated for two di�erent

(x, y) locations in streamwise plane with di�erent separation distances ∆z. For each ∆z

the averaging is done over as many points as possible (at-least 7 to 2, for smaller ∆z

its possible to have more points than for bigger separation distances, see �gure 4.3).

Also the averaging is performed over 13 vortex shedding cycles, which means that per-

fect statistical convergence is not achieved in both the simulations (8D and πD). A

representation of two points for which correlations are calculated is shown in �gure 4.3.

The distributions of Ruu for πD and 8D length in spanwise direction are shown in

�gure 4.4. It can be seen for both the spanwise lengths of domain, that the correlations

doesn't e�ectively go to zero. The same behavior is reported by Wissink and Rodi [29]

for a spanwise length of 8D, for Re 3900. For this low Reynolds number they reported

that a domain dimension in spanwise direction of 8D is not enough to e�ectively resolve

the length of structures in spanwise direction. Norberg [1] has given an estimation of

these correlation lengths over a range of Re. His study shows a decrease in length of


4.4. Comparison between Two SGS Models Chapter 4. Cylinder Flow Simulations

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Ru

u

Deltaz / D

Point A

Point B

(a)

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Ru

u

Deltaz / D

Point A

Point B

(b)

Figure 4.4: Two point correlations Ruu of streamwise velocity u in spanwise directionplotted against normalized separation distance ∆z/D for Re 10,000 (a) Correlations forsimulation with πD spanwise length (b) Correlations for simulation with 8D spanwiselength

Table 4.3: SGS model study results for Re 10,000

SGS Model St Cd Clrms⟨−Cpb

⟩ ⟨θsep⟩ ⟨

lr⟩/D

oneEq 0.2 1.252 0.5049 1.24 88.2 0.784Smag 0.2 1.410 0.7884 1.52 89.0 0.508

Exp[27],[26],[1] 0.19-0.20 1.19 0.38-0.53 1.11 - 0.78

scales with increase in Re. In �gure 4.4 it is clear that the correlations are not going to

zero even for a domain length of 8D. Fluctuations in Ruu could probably be due to the

lack of statistical convergence for two simulations.

Based on the results from correlations study, the domain size of πD is chosen as going

for a 8D simulation doesn't seem to e�ect the results very drastically. Also a simulation

of 8D spanwise length with same grid density, is limited by the time required for the

computation (not by the size of computational facility at RRZE Erlangen).

4.4 Comparison between Two SGS Models

Two SGS models are studied for the cylinder case simulation. E�ects of models are

studied for no wall heating on grid G3 (in table 4.2). Models studied are Smagorinsky

and One Equation Model and has already been described in section 2.2.2. Results of

comparison between two models can be found in table 4.3.



-1.5

-1

-0.5

0

0.5

1

1.5

2

0 50 100 150 200

Cl,C

d

T Uinf / D

Cd SmagCl Smag

Cd oneEqCl oneEq

Figure 4.5: Time history of lift and drag coe�cients (≈50 vortex cycles) for Smagorinskyand One Equation model for Re 10,000

4.4.1 Instantaneous Flow Field

Figure 4.5 shows the lift and drag coe�cient history for two SGS models under study.

It is worth noting that the prediction of lift and drag is higher for Smag model than for

oneEq model. This behavior of higher �uctuations around mean values for lift is depicted

in Clrms where Smag models predicts a higher value of the coe�cient than oneEq model.

This behavior of higher values in lift and drag can be explained by visualizing the z-

component (in spanwise direction) of instantaneous vorticity as shown in �gure 4.6.

As the boundary layer separates from the cylinder surface, shear layers are formed

which are visible in iso-surface of vorticity in �gure 4.6. It can be observed in �gures

that a transition from 2D to 3D �ow occurs earlier in case of Smag model and to close to

the cylinder, which e�ects the lift and drag variation of the cylinder. While the length of

shear layers is longer in case of oneEq model, which is also observed in integral properties

of the simulation like a longer recirculation length and a lower back pressure (refer table

4.3).

4.4.2 Mean Flow Field

Di�erences in instantaneous �ow features due to SGS model clearly show up in the mean

�ow properties of the simulation. An evidence of the di�erences is visible in table 4.3,

where a lower recirculation length, higher back pressure and separation angle seems to

be related. A lower separation angle results from delayed separation of the �ow which in-



(a) (b)

Figure 4.6: Iso-surface of instantaneous vorticity magnitudes for Re 10,000 at ωz =−0.0318 for two SGS models (a) oneEq model (b) Smag model

Figure 4.7: Mean velocity ( UMean =√u2x + u2

y + u2z ) �eld for simulations with

two di�erent models for Re 10,000 (a) Smagmodel (b) oneEq model

turn results in lower back pressure and lower recirculation length for Smag model. Figure

4.7 show the mean velocity �elds for the two simulations with two di�erent models. A

di�erence in recirculation length is evident from mean velocity values.

Delayed separation of boundary layer can be explained by higher turbulence inten-

sity in produced by the Smag model. A higher turbulent intensity in the boundary layer

helps the �ow remain attached for a longer distance over cylinder surface. This phe-

nomena can be observed by comparing the averaged Reynold Stresses produced in two

simulations. Figure 4.8 shows Reynolds stress component �eld τxx for two simulations.

It can be observed from the �gure 4.8 that shear layers for Smag model ((a)) exhibit

higher turbulence levels. Also a a shorter roll up distance for shear layer produced by

Smag model can also be observed. This again is another reason for short recirculation


4.5. Re 10,000 Heated Cylinder Simulations Chapter 4. Cylinder Flow Simulations

Figure 4.8: Mean Reynold Stress componentτxx = u′xu

′x for Re 10,000 (a) Smag model (b)

oneEq model

length and higher back pressure produced with Smagorinsky model.

Based on the results from both the SGS models and a comparison with the experimental

data lead to the use of One Equation SGS model for further simulations in this study.

4.5 Re 10,000 Heated Cylinder Simulations

Previous sections in this chapter have described about the numerical setup, choice of

grid (grid convergence study), spanwise length of the domain and the choice of SGS

turbulence model used. This section presents the results from simulations with wall

heating for Re 10,000. Cylinder with wall heating is coupled phenomena where viscous,

buoyancy and inertial e�ects are coupled together. Many studies have been done over

the years with heated wall cylinders mostly in the range of lower Re, where buoyancy

e�ects can dominate with moderate wall heating. Sabanca et al. [30] have studied the

wall heating e�ects for lower Re less than 100 with wall heating ratios of 1.1 - 1.8, and

they showed a decrease in St number with temperature increase. They reasoned that

heating the cylinder produces compressibility e�ects and stabilizes the wake. Shi et al.

[31] also studied these e�ects through numerical simulations and observed a decrease

in St number with increasing heating temperature ratio of the wall. They attributed

this change in Strouhal number with and e�ective Reynolds number calculated using

the viscosity at an e�ective temperature rather than ambient temperature. Still the Re

numbers studied are less than 170 in a laminar incompressible �ow range. While these

studies have shed important light on the dynamics of variation in Strouhal frequency

with wall heating its still di�cult to quantify the role of individual forces on the vortex

dynamics [30]. Another important experimental study conducted on the issue was done

by Leblond and Belorgey in 1997 [32]. They studied the coupled role of viscosity and



Table 4.4: Re 10,000 heated cylinder results

Temp St Cd Clrms⟨−Cpb

⟩ ⟨θsep⟩ ⟨

lr⟩/D

25oC 0.200 1.255 0.4877 1.22 88.4 0.784177oC 0.207 1.231 0.4143 1.16 88.4 0.866277oC 0.214 1.199 0.3526 1.10 88.4 0.963

buoyancy forces on heated cylinders for low Re. They concluded that ratio of G1/2r /Re

( Gr is Grashof number given as Gr = gβ(Ts − T∞)D3/ν2, where g is acceleration due

to gravity, β is volumetric thermal expansion coe�cient, Ts surface temperature and T∞

bulk temperature ) plays an important role to determine the e�ects of heating. According

to their observation, for G1/2r /Re < 0.7 wake is ruled by forced convection and heat then

e�ects essentially the thermodynamical characteristics of �uid. For high Re this means

that inertial e�ects will dominate the �ow. For the ratio > 0.7 they predicted that

buoyancy forces dominate and suppression of vortex can also achieved.

For higher Re (here we consider 10,000 and 30,000 as higher Re previously not studied)

this means that the e�ects of heating would be that of changing the viscosity and other

thermodynamic properties of the �ow, if the ratio G1/2r /Re < 0.7. This increase in

viscosity then again can be connected to e�ective Re as suggested by Shi et al. Their

formula for e�ective temperature and Reynolds number is given below as Teff = T∞ +

C(Tw−T∞), Reff = U∞D/ν(Teff ) where Tw is the wall temperature, T∞ is the ambient

temperature, Teff is called as e�ective temperature, Reff is the e�ective Reynolds number

and C is a constant with di�erent values reported in literature. And they used a value

of C = 0.28 for their study.

Table 4.4 presents the results of the simulations conducted during this study. Among

many interesting notable factors, an increase in shedding frequency is evident from the

increase in Strouhal number. The change is integral properties of the �ow can be con-

nected with the e�ective Re phenomena. As the e�ective temperature increases, the

e�ective Re decreases due to increase in viscosity. But previous studied showed a de-

crease in shedding frequency with heating. This phenomena again is dependent upon the

Re number and can be explained by looking at relationship of Strouhal number and Re

as shown in �gure 4.9. For Re 10,000 as the viscosity increases the e�ective Re decreases

and the Strouhal number increases. Previous studies conducted for low Re observed a

decrease in Strouhal number, which again can be explained by the �gure. As the e�ective

Re decreases below 1000 decrease in Strouhal frequency is evident from the graph.



Figure 4.9: Relationship between St and Re from di�erent experiments [1]. Two hori-zontal lines in the graph mark the two Re under study 10,000 and 30,000 respectively


The di�erences in instantaneous �ow �elds for two heated cases would be very minute

and hard to identify, as the �ow is dominated by convection and heating doesn't change

the �ow regime signi�cantly. As seen in the table 4.4 rms lift coe�cient and drag

coe�cient decrease. The same can be observed in the �gure 4.5 for lift and drag plotted

for maximum and minimum temperatures studied. The reason for decrease in lift and

drag coe�cients could be delayed separation of boundary layer under the in�uence of

increased viscosity due to the heating of the �uid at the cylinder surface. The di�erence

in the instantaneous vorticity magnitudes are small to detect di�erences in shear layers,

due to heating. The delay in separation also adds stability to formed shear layers. The

stabilization of shear layer means that the length of the shear layer will be more before

it rolls up directly into the wake. Figure 4.11 shows instantaneous normalized vorticity.

Shear layers exhibit hight vorticity magnitudes. A slight di�erence in shear layer length

due to heating can be observed. The vortex shedding phase for the two �gures is not

similar. This di�erence of length in shear layers can be veri�ed by looking at the averaged

Reynold stresses and mean vorticity in �ow normal directions as well.


As discussed earlier the small �ow �uctuations are di�cult to observe in the instantaneous

�ow �eld but the accumulated results of small di�erences in instantaneous �ow are visible

in mean �ow �elds. First and foremost example of these di�erences can be observed by



-1.5

-1

-0.5

0

0.5

1

1.5

2

0 50 100 150 200

Cl,C

d

T Uinf / D

Cd 25TCl 25T

Cd 277TCl 277T

Figure 4.10: Time history of lift and drag coe�cients (≈50 vortex cycles) for 25oC (25T)and 277oC (277T)

(a) (b)

Figure 4.11: Comparison between normalized Instantaneous vorticity contours ωD/U∞(=0.5-10) Re 10,000 (a) No heating (25T) (b) Severe heating (277T)



Figure 4.12: Mean velocity ( UMean =√u2x + u2

y + u2z ) �eld for two simulations at

Re 10,000 (a) No Heating 25oC (b) StrongHeating 277oC

Figure 4.13: Mean Reynold Stress compo-nent τxx = u′xu

′x at Re 10,000 (a) No Heating

25oC (b) Strong Heating 277oC

the change in the recirculation length in table 4.4 with change in temperature. This

change in recirculation zone can be seen in mean velocity �eld as shown in �gure 4.12.

Another important aspect of di�erence between non-heated and strongly heated case

is the Reynold stresses. Shear layers in the �ow exhibit highest velocity �uctuations

and consequently higher stresses are produced. An increase in viscosity due to heating

stabilizes the shear layer. This in turn results in lower Reynolds Stresses in shear layers

for the heated case. This phenomena can be seen in mean Reynold stress component

�eld τxx of two cases as shown in �gure 4.13. In �gure 4.13 case (a) with no heating

clearly shows the higher Reynold stresses produced in the shear layer and a roll up of

shear layer slightly before the roll up of heated case (b).

Figure 4.14 also shows a comparison of mean x and y velocity at di�erent locations along

the wake. Distance shown in the wake is measured from the cylinder center line i.e.

origin (0,0) is placed at center of the circle. Distance in the graphs is normalized with

diameter D and velocity are normalized by U∞. A decrease in velocity with heating is

expected in the wake. This is also con�rmed by a change in recirculation length with

heating.



-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

-4 -3 -2 -1 0 1 2 3 4

Ux,U

y

Y Distance

Ux 277 TUx 24 T

Uy 277 TUy 24 T

(a)

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

-4 -3 -2 -1 0 1 2 3 4

Ux,U

y

Y Distance

Ux 277 TUx 24 T

Uy 277 TUy 24 T

(b)

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

-4 -3 -2 -1 0 1 2 3 4

Ux,U

y

Y Distance

Ux 277 TUx 24 T

Uy 277 TUy 24 T

(c)

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

-4 -3 -2 -1 0 1 2 3 4

Ux,U

y

Y Distance

Ux 277 TUx 24 T

Uy 277 TUy 24 T

(d)

Figure 4.14: Comparison between mean x and y velocity with and without heating alongy direction at di�erent x locations (a) x/D = −0.75 (b) x/D = −1 (c) x/D = −2 (d)x/D = −4

4.6 Re 30,000 Heated Cylinder Simulations

In previous section a detailed study of the results from Re=10,000 simulations has been

presented. Integral properties of the �ow show the same behavior as for Re 10,000. Table

4.5 shows the results for two cases studied at Re 30,000.

As observed in previous section heating e�ects the �ow as an increase in �uid viscosity

around the cylinder. This phenomena is again observed in the results. Integral �ow

properties vary with temperature as an e�ective Re decreases with increase is viscosity.

Table 4.5: Re 30,000 heated cylinder results

Temp St Cd Clrms⟨−Cpb

⟩ ⟨θsep⟩ ⟨

lr⟩/D

25oC 0.204 1.347 0.6389 1.37 87.9 0.646277oC 0.213 1.281 0.4785 1.22 86.1 0.811



(a) (b)

Figure 4.15: Comparison between normalized Instantaneous vorticity contours ωD/U∞(=0.5-18) (a) No heating (25T) (b) Severe heating (277T)

An increase in Strouhal frequency can again be explained by looking at the �gure 4.9.

Second line in the �gure marks Re 30,000 it can be seen that a decrease in Re from this

point will increase the Strouhal number and hence Strouhal Frequency.


Instantaneous properties of �ow vary as the thermodynamics of �uid change around

the heated cylinder. Again a visualization of normalized instantaneous magnitude of

vorticity is presented in �gure 4.15. Figure (a) and (b) in �gure 4.15 are plotted with

the same scale. Reduction in vorticity is observed in (b) which can be expected with an

increase in viscosity. As observed in the previous section, estimating the length of shear

layers from instantaneous �ow is not easy. But these �uctuations add up to show clear

di�erences in the mean �ow �eld of the same property.


Here some �ow visualizations of the mean �ow �eld are presented. The �ow phenomena

and di�erences occurring in �ow due to heating are very similar as observed with Re

10,000. Figure 4.16 shows the mean velocity �eld for two simulations. Di�erences in

mean velocity �eld can be observed and change in recirculation length under the e�ect

of heating can be seen. Di�erences in viscosity also e�ect the turbulent characteristics of

�ow. As noted in the previous section an increase in viscosity stabilizes the shear layer

and the length of shear layers increase. Figure 4.17 shows the mean The Reynold stress



Figure 4.16: Mean velocity �eld for twosimulations at Re 30,000 (a) No Heat-ing 25oC (b) Strong Heating (UMean =√u2x + u2

y + u2z) 277

oC

Figure 4.17: Mean Reynold Stress compo-nent τxx = u′xu

′x at Re 30,000 (a) No Heating

25oC (b) Strong Heating 277oC

component �eld τxx of two cases i.e. without heating and with severe heating.


Chapter 5

Aero-Acoustic Simulations

Aero-acoustics deals with the sound generated aerodynamically. Application of aero-

acoustics can be found in wide areas of industry like, aircrafts and automobiles, high

speed trains, noise in valves and nozzles, noise in vacuum cleaner etc. The perception of

sound for human ears is by varying pressure in �uid, which could vary in amplitude and

frequency. A study of the phenomena is important to control it, for it can be harmful

and a cause of environmental pollution. With an increasing number of aircrafts and

automobiles the study of aero-acoustics to decrease noise is becoming important. This

chapter presents the results of aero-acoustic simulations. Acoustic simulations are carried

out for two di�erent Re (i.e. 10,000 and 30,000) and two di�erent wall heating scenarios

(i.e. no heating and severe heating). Results between two di�erent heating cases are

compared and a reduction in sound level with heating is demonstrated. A comparison

between two di�erent aero-acoustic analogies is also performed for heated and non heated

cases at Re 10,000. Analogies compared are Lighthill [33] and Möhring [34].

Highest frequency studied in all the simulations is 2000 Hz. All the simulations are

performed using Möhring's acoustic analogy. CFD results are interpolated to acoustic

mesh to calculate Möhring volume source on the acoustic mesh. A propagation region of

10D is considered outside of the 15D radius of the CFD simulation domain. Probes are

put around 1m radius circle and in�nite elements are used to propagate sound to these

locations. Following these details results will demonstrate a di�erence between heated

and unheated case for two Re. A comparison between Möhring and Lighhill analogies is

also presented in the last section of this chapter.

5.1 Re 10,000 Aero-acoustic Simulations

To analyze the results in this section Sound Pressure [dB] from the resulting simulation

are studied at two di�erent locations i.e. 90 degree ( (x,y) = (0,1) ) and 180 degree ((x,y)


5.1. Re 10,000 Aero-acoustic Simulations Chapter 5. Aero-Acoustic Simulations

0

10

20

30

40

50

60

70

10 100 1000

So

un

e P

ressu

re [

dB

]

f [Hz]

25T277T

(a)

-20

-10

0

10

20

30

40

50

60

70

10 100 1000

So

un

e P

ressu

re [

dB

]

f [Hz]

25T277T

(b)

Figure 5.1: Distribution of Sound Pressure over frequency (Re 10,000) (a) 90 Degreepoint (b) 180 Degree point

= (-1,0)). The SPL is de�ned as:

SPL = 10log10

(p2rms

p2ref

), pref = 2 ∗ 10−5Pa (5.1)

In cylinder aero-acoustics Strouhal frequency has the largest amplitude. By studying

the frequency spectrum at the 90 degree point one can clearly observe the high amplitude

peaks at Strouhal Frequency. With heating a frequency shift is also visible in acoustic

simulations as cab be observed by comparing two curves in �gure 5.1.

An overall reduction of magnitude in sound pressure can be observed from �gure

5.1. In �gure 5.1 (a) shows the [dB] comparison between heated and unheated case at

90 degree probe. Here Strouhal Frequency has the greatest magnitude and a sharp rise

is also observed at the 1st harmonic (2 * Strouhal Frequency). In (b) at the Strouhal

Frequency there is not sharp rise in magnitude. as at 180 degree (or 360/0 degree) of the

cylinder, a zone of silence is produced due to the shape of the sound source at Strouhal

Frequency. Overall a reduction in sound is evident from the �gure. Its also interesting

to see the magnitudes of the sound pressure at Strouhal Frequency, around the whole

cylinder. It can be achieved by plotting the sound pressure on polar coordinates for all

the points at same frequency.

Figure 5.2 shows the polar plot of sound pressure for heated and unheated cases

at Strouhal Frequencies. An overall reduction of maximum sound pressure around the

whole cylinder can be observed from the �gure. Its also interesting to see the contours

of sound pressure at shedding frequency, where one can observe a dipole and a di�erence

in magnitude of sound pressure for heated and unheated cases. Figure 5.3 shows the

contours of sound pressure for heated and unheated cases. The contours are plotted on


5.2. Re 30,000 Aero-acoustic Simulations Chapter 5. Aero-Acoustic Simulations

0

10

20

30

40

50

60

70

80

0 10 20 30 40 50 60 70 80

Soune Pressure in dB

25T 122 Hz

277T 124 Hz

Figure 5.2: Sound pressure plotted around the cylinder at 1m away probe locations forheated and unheated cases at Strouhal Frequencies (Re 10,000)

Figure 5.3: Sound pressure contours (Re 10,000) (left: Heated, right: UnHeated)

a log-scale to make the di�erences prominent. As observed from previous section an

overall decrease of 5 dB in sound pressure is observed at shedding frequency.

5.2 Re 30,000 Aero-acoustic Simulations

Experiments conducted showed an even higher decrease in magnitude of sound pressure

for higher Re. The same simulation setup and frequency range has been used to study

the e�ects of heating on sound reduction for Re 30,000. Sound pressure is studied at

same two points as discussed in previous section i.e. 90 Degree and 180 Degree. Figure

5.4 demonstrates sound pressure plotted against frequency in Hz. A reduction in sound

level can be observed for both Strouhal and 1st harmonic in the frequency range. A


5.3. Lighthill and Möhring Comparison Chapter 5. Aero-Acoustic Simulations

20

30

40

50

60

70

80

90

100

10 100 1000

So

un

e P

ressu

re [

dB

]

f [Hz]

25T277T

(a)

20

30

40

50

60

70

80

90

10 100 1000

So

un

e P

ressu

re [

dB

]

f [Hz]

25T277T

(b)

Figure 5.4: Distribution of Sound Pressure over frequency (Re 30,000) (a) 90 Degreepoint (b) 180 Degree point

reduction of 5 dB at the shedding frequency can be observed. This reduction in sound

pressure is even higher at the 1st harmonic.

Figure 5.5 shows the polar plot of sound pressure for heated and unheated cases at

Strouhal Frequencies. A a very smooth dipole like shape can be observed in the �gure

for two cases at two frequencies. A reduction in sound pressure can also be observed in

the �gure.

It is interesting to visualize the contours of sound pressure at Strouhal Frequency.

Figure 5.6 shows the contour of sound pressure for two cases. To make the di�erences

in sound pressure prominent, a log scaling of the contours has been used. A stronger

dipole for the unheated case is very clear from the �gure (right), while the dipole on the

left appears to be weaker under the e�ect of heating. An overall reduction in the sound

level all-around the cylinder can be observed from the contours.

5.3 Lighthill and Möhring Comparison

One major di�erence that arise in numerical solution of the two analogies is that Möhring

analogy takes into account the mean �ow while Lighthill analogy models the convective

e�ects as sources (e.g. shear layers etc.). In Lighthill analogy sources are calculated

considering the �ow to be stationary. For present study as the �ow speeds are still very

low (Ma<0.1), the two analogies are expected to produce similar results.

To study di�erences between two analogies heated and unheated cases at Re 10,000

are compared for Lighthill and Möhring analogy. Figure 5.7 shows the sound pressure

distribution over frequency range, for two di�erent locations. It can be observed from

the �gures that even though changes in magnitude occur between two analogies the



0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80 90 100

Soune Pressure in dB

25T 360 Hz

277T 373 Hz

Figure 5.5: Sound pressure plotted around the cylinder at 1m away probe locations forheated and unheated cases at Strouhal Frequencies (Re 30,000)

Figure 5.6: Sound pressure contours (Re 30,000) (left: Heated, right: UnHeated)



variation of sound pressure over frequency range is very similar for two analogies. These

di�erences in magnitude can arise from the way both analogies treat mean �ow in sources

computation.

10

20

30

40

50

60

70

10 100 1000

So

un

e P

ressu

re [

dB

]

f [Hz]

25T Lighthill25T Mohring

(a)

-20

-10

0

10

20

30

40

50

60

70

10 100 1000

So

un

e P

ressu

re [

dB

]

f [Hz]


(b)

0

10

20

30

40

50

60

70

10 100 1000

So

un

e P

ressu

re [

dB

]

f [Hz]


(c)

0

10

20

30

40

50

60

70

10 100 1000

So

un

e P

ressu

re [

dB

]

f [Hz]


(d)

Figure 5.7: Comparison between Lighthill and Möhring Analogies for Re 10,000 (a) 90degree location (b) 180 degree location (c) 90 degree location (d) 180 degree location

It is also interesting to see how the two analogies compare all around the cylinder at

maximum amplitudes (shedding frequency). Figure 5.8 demonstrates polar plot of sound

pressure around cylinder, for unheated case at Re 10,000. Di�erences in magnitude can

be observed. These di�erence in magnitude are higher at 0 Degree/180 Degree locations.

Overall two analogies in question have shown similar behavior.



0

10

20

30

40

50

60

70

80

0 10 20 30 40 50 60 70 80

Soune Pressure in dBLighthillMohring

Figure 5.8: Sound pressure plotted around the cylinder at 1m away probe locations forunheated case (Re 10,000), Lighthill and Möhring Comparison


Chapter 6

Conclusion and Outlook

This chapter is the last chapter of this study. This chapter concludes this study. Apart

from conclusion an outlook and further extensions of the work are also discussed. There

are some improvements which have been suggested in this chapter and some problems

discussed which might help improve further investigations.

6.1 Conclusions

This study has presented and demonstrated CFD simulations for two Re with two dif-

ferent wall heatings. Aero-acoustic simulations are performed for two the Re 10,000 and

30,000. The main objective to observe sound reduction due to heating is very well met.

Reasons for frequency shift due to heating are also discussed. It is concluded that fre-

quency shift is dependent on Strouhal vs Reynolds number relationship. Heating cylinder

wall at higher Re a�ects thermodynamics of �uid. Change in thermodynamic viscosity

due to heating changes the e�ective Re. Depending upon the shift of e�ective Re from

ambient Re, frequency increases or decreases. The issue of spanwise length scales is also

discussed and its shown from correlations that πD domain size in spanwise dimension is

not enough to compute the spanwise structures accurately. Even though questions about

8D simulations have been raised in literature for Re 3,500 [29], but these structures are

expected to reduce in length for higher Re [1]. So a spanwise domain size of 8D would

be enough for Re 10,000 and it could be reduced for higher Re.

Regarding the SGS modeling its been noted that one equation (eddy viscosity and di�u-

sivity) kinetic energy based model outperformed the classical Smagorinsky model. Both

models are compared for integral properties with experimental and DNS data available.

It is observed that the one equation model performs better for this case. Possible reasons

for Smagorinsky model to lack accuracy could be the default implementation of the coef-

�cients in openFOAM. Similar results for the similar case has been reported in literature


6.2. Outlook Chapter 6. Conclusion and Outlook

as well [16].

The �nal goal of this study to observe the sound reduction due to heating of cylinder

has been demonstrated. Möhring analogy is used to perform all the simulations for com-

parison between heated and unheated cases. A signi�cant reduction of noise for the Re

10,000 and 30,000 is shown when the cylinder is heated to an approximate temperature

of 277 oC. An approximate 5 dB of reduction is observed in sound pressure for both Re.

Overall the aim of this study are successfully accomplished.

6.2 Outlook

There are many parts in this study which could be improved and many possible exten-

sions. As discussed in previous chapters, wall heating of a cylinder in cross �ow causes

the wake to go under the e�ect of coupled buoyancy and viscous forces. During this study

an important result is to observe a frequency shift with heating and visualizing it on St

vs Re curve obtained from experiments [1]. The frequency shift is quite understandable

after considering the e�ect of heating on viscosity and looking at St variation against

Re. But an important matter to look into is the e�ect of statistical averaging time of the

simulation. Franke and Frank [20] observed a dependence of St on statistical averaging

time. For an Re 3900, they concluded that at least 50 vortex cycles must be averaged

before considering the simulation to be statistically converged. It could be important

specially when a study aims at predicting minor changes in shedding frequency due to

heating. Present simulations are averaged for at least 50 vortex cycles. So, it could be a

good idea to see if any signi�cant di�erences arise with increasing averaging time.

In a part of this study spanwise correlations are also studied to estimate the length of

structures in spanwise direction. The reason to estimate length of these structures is

to design a domain that at-least accommodates these structures. A domain size smaller

than length of structures in spanwise direction mean that an error of low frequency will

be introduced due to cut-down or wrong estimation of these structures. It is concluded

that a simulation of 8D might be enough to estimate the spanwise scales accurately.

Simulations performed in this part lack some statistical convergence (13 vortex cycles of

averaging for correlations). Results with longer averaging times (more than 13 vortex

cycles) might not be similar to this study. A step forward would be to check these results

with longer average times.

The issue of heated cylinder still needs more attention to develop a deep understanding

into the coupled phenomena. To con�rm the frequency shift due to a viscosity change

only simple numerical experiment can also help. A simulation could be conducted to

suppress any changes in viscosity due to heating and observing the change in frequency.


6.2. Outlook Chapter 6. Conclusion and Outlook

If the frequency variation is only due to variation in viscosity, this simulation might not

show a shift in frequency. It could also help understand how would wake be e�ected if

there is only decrease in density in the wake due to heating, but no changes in shear

layer (as viscosity is constant).

The aero-acoustic simulations conducted in the present study doesn't simulate the ex-

perimental conditions. Instead this study just aimed to view the cylinder in a uniform

�ow. In experiments, wind-tunnels has a signi�cant e�ect as they have a noise of their

own and they produce shear layers starting from their exit. Also the present study has

considered propagation only in x-y plane, this is also a considerable simpli�cation at the

expense of real experimental setup. Simulating experimental setup could help increase

con�dence in the simulation setup, and achieve close results as observed in experiments.


Bibliography

[1] C. Norberg, �Fluctuating lift on circular cylinder: review and new measurements,�

Journal of Fluids and Structures, vol. 17, no. 1, pp. 57�96, 2003.

[2] B. Sumer and J. Fredsøe, Hydrodynamics around cylindrical structures, vol. 12.

World Scienti�c Publishing Company Incorporated, 1997.

[3] A. Hahnenkamm, R. Masood, M. Münsch, H. Lienhart, E. Lopez, L. Zhou, J. Gril-

liat, and A. Delgado, �On the impact of thermal sources on noise generation,�

PAMM, vol. 12, no. 1, pp. 553�554, 2012.

[4] J. Ferziger and M. Peri¢, Computational methods for �uid dynamics, vol. 2. Springer

Berlin etc, 1999.

[5] C. Fureby, �On subgrid scale modeling in large eddy simulations of compressible

�uid �ow,� Physics of Fluids, vol. 8, no. 5, pp. 1301�1311, 1996.

[6] E. Garnier, N. Adams, and P. Sagaut, Large eddy simulation for compressible �ows.

Springer, 2009.

[7] D. Leslie and G. Quarini, �The application of turbulence theory to the formulation

of subgrid modelling procedures,� Journal of �uid mechanics, vol. 91, no. 01, pp. 65�

91, 1979.

[8] J. Smagorinsky, �General circulation experiments with the primitive equations,�

Monthly weather review, vol. 91, no. 3, pp. 99�164, 1963.

[9] T. M. Eidson, �Numerical simulation of the turbulent rayleigh�bénard problem using

subgrid modelling,� Journal of Fluid Mechanics, vol. 158, no. 1, pp. 245�268, 1985.

[10] A. Yoshizawa, �Bridging between eddy-viscosity-type and second-order turbulence

models through a two-scale turbulence theory,� Physical Review E, vol. 48, no. 1,

p. 273, 1993.


BIBLIOGRAPHY BIBLIOGRAPHY

[11] E. De Villiers, The potential of Large Eddy Simulation for the modelling of wall

bounded �ows. 2006.

[12] H. Jasak, Numerical Solution Algorithms for Compressibe Flows. 2007.

[13] R. I. Issa, �Solution of the implicitly discretised �uid �ow equations by operator-

splitting,� Journal of computational physics, vol. 62, no. 1, pp. 40�65, 1986.

[14] S. V. Patankar and D. B. Spalding, �A calculation procedure for heat, mass and

momentum transfer in three-dimensional parabolic �ows,� International Journal of

Heat and Mass Transfer, vol. 15, no. 10, pp. 1787�1806, 1972.

[15] C. Open, �Openfoam user guide,� 2011.

[16] D. A. Lysenko, I. S. Ertesvåg, and K. E. Rian, �Large-eddy simulation of the �ow

over a circular cylinder at reynolds number 3900 using the openfoam toolbox,� Flow,

Turbulence and Combustion, pp. 1�28, 2012.

[17] M. Breuer, �Large eddy simulation of the subcritical �ow past a circular cylinder:

numerical and modeling aspects,� International Journal for Numerical Methods in

Fluids, vol. 28, no. 9, pp. 1281�1302, 1998.

[18] A. G. Kravchenko and P. Moin, �Numerical studies of �ow over a circular cylinder

at re= 3900,� Physics of �uids, vol. 12, p. 403, 2000.

[19] P. Moin and R. Mittal, �Suitability of upwind-biased �nite di�erence schemes for

large-eddy simulation of turbulent �ows,� AIAA journal, vol. 35, no. 8, 2012.

[20] J. Franke and W. Frank, �Large eddy simulation of the �ow past a circular cylinder

at red= 3900,� Journal of wind engineering and industrial aerodynamics, vol. 90,

no. 10, pp. 1191�1206, 2002.

[21] M. Germano, U. Piomelli, P. Moin, and W. H. Cabot, �A dynamic subgrid-scale

eddy viscosity model,� Physics of Fluids A: Fluid Dynamics, vol. 3, p. 1760, 1991.

[22] D. Lilly, �A proposed modi�cation of the germano subgrid-scale closure method,�

Physics of Fluids, vol. 4, pp. 633�635, 1992.

[23] U. Piomelli and J. R. Chasnov, �Large-eddy simulations: theory and applications,�

Turbulence and transition modelling, pp. 269�336, 1996.

[24] T. J. Poinsot and S. Lelef, �Boundary conditions for direct simulations of compress-

ible viscous �ows,� Journal of computational physics, vol. 101, no. 1, pp. 104�129,

1992.


BIBLIOGRAPHY BIBLIOGRAPHY

[25] H. G. Weller, G. Tabor, H. Jasak, and F. C, �A tensorial approach to computa-

tional continuum mechanics using object-oriented techniques,� Computer in physics,

vol. 12, p. 620, 1998.

[26] S. Dong and G. Karniadakis, �Dns of �ow past a stationary oscillating cylinder at

re = 10000,� Journal of Fluids and Structures, vol. 20, pp. 519�531, 2005.

[27] R. Gopalkrishnan, �Vortex-induced forces on oscillating blu� cylinders,� tech. rep.,

DTIC Document, 1993.

[28] J. Frohlich, C. P. Mellen, W. Rodi, L. Temmerman, and M. A. Leschziner, �Highly

resolved large-eddy simulation of separated �ow in a channel with streamwise peri-

odic constrictions,� Journal of Fluid Mechanics, vol. 526, pp. 19�66, 2005.

[29] J. Wissink and W. Rodi, �Numerical study of the near wake of a circular cylinder,�

International journal of heat and �uid �ow, vol. 29, no. 4, pp. 1060�1070, 2008.

[30] M. Sabanca and F. Durst, �Flows past a tiny circular cylinder at high temperature

ratios and slight compressible e�ects on the vortex shedding,� Physics of Fluids,

vol. 15, p. 1821, 2003.

[31] J.-M. Shi, D. Gerlach, M. Breuer, G. Biswas, and F. Durst, �Heating e�ect on

steady and unsteady horizontal laminar �ow of air past a circular cylinder,� Physics

of Fluids, vol. 16, p. 4331, 2004.

[32] N. Michaux-Leblond and M. Belorgey, �Near-wake behavior of a heated circu-

lar cylinder: viscosity-buoyancy duality,� Experimental thermal and �uid science,

vol. 15, no. 2, pp. 91�100, 1997.

[33] M. J. Lighthill, �On sound generated aerodynamically. i. general theory,� Proceed-

ings of the Royal Society of London. Series A. Mathematical and Physical Sciences,

vol. 211, no. 1107, pp. 564�587, 1952.

[34] W. Mohring, �On vortex sound at low mach number,� J. Fluid Mech, vol. 85, no. 4,

pp. 685�691, 1978.


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Hybrid CFD-CAA simulation of a heated and unheated Cylinder in turbulent cross ows

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