MSc Thesis - Jaguar Land Rover

CRANFIELD UNIVERSITY

School of Aerospace, Transport and Manufacturing

M.Sc. Thesis

Academic Year: 2015-2016

Akshat Srivastava

The Impact of Unsteadiness on Uncertainty in AutomotiveAerodynamics Simulation using OpenFOAM

Supervisor:

Dr. Panagiotis Tsoutsanis

© Cranfield University, 2016.All rights reserved. No part of this publication may be reproduced

without the written permission of the copyright holder.

Except where acknowledged in the customary manner, the material presen-ted in this thesis is, to the best of my knowledge, original and has not beensubmitted in whole or part for a degree in any university.

________________________Akshat Srivastava

Abstract

The Detached Eddy Simulation (DES) is performed for flow over sphere. The simula-tions are carried out based on free-stream Mach number M∞ = 0.2 and sphere diame-ter D = 1m in sub-critical regime at Re = 10,000. A cartesian grid is generated usingCfMesh in OpenFOAM and computations are performed using PIMPLE Foam solver.For this Reynolds number at near to the equator of the sphere, flow separates laminarlyand in the separated shear layer the transition to turbulence occur at certain distance.The frequency spectrum using probes at different locations are described and discussedin details. The three main instabilities of different frequencies shed from sphere surfacenamely, the large-scale vortex shedding at St = fvs D/U = 0.203, the Kelvin Helmholtzand a frequency lower than the vortex shedding frequency known a low-frequency whichattributes to the shrinkage and enlargement of recirculation bubble. Additionally, turbu-lence statistics are compared with previous experimental and numerical results availablein literature for sub-critical Reynolds number. Specific consideration is dedicated to com-puting the mean flow statistics and parameters such as mean angular pressure and skinfriction coefficient, mean lift and drag coefficient, among others, to validate the solverand turbulence model used.

Keywords: turbulence, sphere flow, OpenFOAM, vortex-sheding, low-frequency, wake

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Acknowledgements

The work described in this report is the result of my 3 months thesis performed at Cran-field University, UK. Many people contributed their guidance for completion of this thesiswork. I would like to thank everyone who helped me in one way or another and few peoplein particular.

First and foremost, I would like to thank Jaguar and Land rover (JLR) for providing mean opportunity to work in this industrial thesis. My supervisor Dr. Panagiotis Tsoutsanishas been a great help through his valuable guidance, support and direction. It is my firstexperience working in turbulence subject, hence, his knowledge and expertise guided meto understand the project better.

Finally, I would like to thank all the other people for their help at various stagesthrough the project. I heartily appreciate all your sincere efforts.

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Contents

Abstract v

Acknowledgements vi

List of Figures xii

List of Tables xiii

1 Introduction 11.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Experimental Background . . . . . . . . . . . . . . . . . . . . . 21.2.2 Previous Numerical Investigation . . . . . . . . . . . . . . . . . 4

1.3 Thesis Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Physics and Modelling 82.1 Governing Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Turbulence models and Numerical methods . . . . . . . . . . . . . . . . 9

2.2.1 RANS principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 LES principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.3 Filtered Navier-Stokes equation . . . . . . . . . . . . . . . . . . 15

2.3 Turbulence closure model . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4 RANS-LES Hybrid approach . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4.1 Detached Eddy Simulation (DES) . . . . . . . . . . . . . . . . . 182.4.2 Delayed Detached Eddy Simulation (DDES) . . . . . . . . . . . 19

2.5 Improved Delayed Detached Eddy Simulation (IDDES) . . . . . . . . . . 202.5.1 Modification of the Sub-grid length-scale . . . . . . . . . . . . . 212.5.2 DDES branch of IDDES . . . . . . . . . . . . . . . . . . . . . . 232.5.3 WMLES branch of IDDES . . . . . . . . . . . . . . . . . . . . . 23

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Contents

2.5.4 Hybrid branch of DDES and WMLES . . . . . . . . . . . . . . . 25

3 Software and Methodology 273.1 OpenFOAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1.1 Utilities and Solvers . . . . . . . . . . . . . . . . . . . . . . . . 283.1.2 Case structure in OpenFOAM . . . . . . . . . . . . . . . . . . . 29

3.2 OpenFOAM Discretization . . . . . . . . . . . . . . . . . . . . . . . . . 303.2.1 Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . . 323.2.2 Time discretization . . . . . . . . . . . . . . . . . . . . . . . . . 343.2.3 Momentum-Pressure Coupling . . . . . . . . . . . . . . . . . . . 343.2.4 Implementation of Turbulence Model . . . . . . . . . . . . . . . 35

3.3 Simulation Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.4 Solvers, Smoothers and Preconditioners . . . . . . . . . . . . . . . . . . 37

4 Computational Methodology 394.1 Computational domain . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Initial and Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 404.3 Time step selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.4 Mesh Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.4.1 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.4.2 Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.5 Mesh dependent study . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5 Results and Discussions 555.1 Frequency spectrum analysis . . . . . . . . . . . . . . . . . . . . . . . . 555.2 Instantaneous flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.3 Mean flow parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.4 Mean flow statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6 Conclusion and Future work 786.1 Conclusion on thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.2 Recommendations for future work . . . . . . . . . . . . . . . . . . . . . 80

Bibliography 82

Appendices 88

A controlDict 88

B fvSolution 93

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Contents

C fvSchemes 97

D meshDict 100

E Mesh comparative study 103

ix

List of Figures

2.1 Energy spectrum of length scales. a) High energy region, b) transfer ofenergy region c) dissipation region [36] . . . . . . . . . . . . . . . . . . 13

2.2 Filtering operation for a flow variable [38] . . . . . . . . . . . . . . . . . 152.3 Examples of three mesh design during grid refinments [38] . . . . . . . . 192.4 Sub-Grid length scale [38] . . . . . . . . . . . . . . . . . . . . . . . . . 222.5 Blending function profiles [38] . . . . . . . . . . . . . . . . . . . . . . . 24

3.1 Case Structure of incompressible IDDES simulation . . . . . . . . . . . . 303.2 Finite Volume Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3 General overview of log file . . . . . . . . . . . . . . . . . . . . . . . . 373.4 Schematic overview of OpenFOAM simulation . . . . . . . . . . . . . . 38

4.1 Schematic of computational domain for simulation . . . . . . . . . . . . 404.2 ICEM CFD mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.3 Velocity magnitude contour for structured mesh generated using cfMesh . 454.4 Computational domain geometry . . . . . . . . . . . . . . . . . . . . . . 454.5 Patch file example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.6 File format conversion from STL to FMS format using command surfa-

ceToFMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.7 Volume refinement using sphere and cone objects . . . . . . . . . . . . . 484.8 Patch refinement for sphere . . . . . . . . . . . . . . . . . . . . . . . . . 484.9 Prismatic layers using boundaryLayers dictionary . . . . . . . . . . . . . 494.10 Velocity contour for mesh validation . . . . . . . . . . . . . . . . . . . . 504.11 Pressure contour for mesh validation . . . . . . . . . . . . . . . . . . . . 504.12 Y+ contours; (a) Course Mesh, M1 (b) Medium Mesh, M2 (c) Fine Mesh,

M3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.13 Cut plane and Zoom view; (a) Course Mesh, M1 (b) Medium Mesh, M2

(c) Fine Mesh, M3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.14 Time history of Lift and Drag Coefficient plots for entire simulation time

t=9.048, showing transition stage passes after 75D/U time units or 1.1sec. 53

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List of Figures

4.15 FFT analysis at Probe location-9 showing non-dimensional vortex shed-ding frequency for (a)M1=0.181 (b)M2=0.211 and (c)M3=0.203 . . . . . 54

5.1 Location of computational probes and lines . . . . . . . . . . . . . . . . 555.2 FFT analysis of the streamwise velocity fluctuation at probe P9 (x/D =

2.0, r/D = 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.3 Time history and FFT analysis at different locations: (a,b) radial velocity

and FFT of it at probe P1, (c,d) radial velocity and FFT of it at probe P2,(e,f) radial velocity and FFT of it at probe P9, (g,h) radial velocity andFFT of it at probe P4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.4 Energy dissipation in downstream of sphere wake (a) Probe location P9(x/D = 2.0, r/D = 0), (b) Probe location P4 (x/D = 3.0, r/D = 0.6) . . 60

5.5 (a)Time history of streamwise velocity at probe P9, and (b) time historyof pressure coefficient at P2 . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.6 Cross-correlation between streamwise velocity fluctuation at probe P9and pressure coefficient at probe P2 . . . . . . . . . . . . . . . . . . . . 62

5.7 Vortex shedding at every quarter time period using Q-iso-surfaces (advan-cing from (a) to (d)), in X-Y plane . . . . . . . . . . . . . . . . . . . . . 65

5.8 Vortex shedding at every quarter time period using Q-iso-surfaces (advan-cing from (a) to (d)), in X-Z plane . . . . . . . . . . . . . . . . . . . . . 66

5.9 Instantaneous contours of pressure coefficient, Cp; (a) Coarse mesh, M1;(b) Medium mesh, M2; and (c) Fine mesh, M3 . . . . . . . . . . . . . . . 68

5.10 Instantaneous contours of non-dimensional skin-friction coefficient,(τ/(ρ U2 Re0.5));(a) Coarse mesh, M1; (b) Medium mesh, M2; and (c) Fine mesh, M3 . . . 69

5.11 Angular distribution of mean pressure coefficient and skin friction coeffi-cient around sphere; compared with experimental results of Kim & Durbin[14]at Re = 4200, Bakic[16] at Re = 50000 and DNS results of Seidle etal.[24] at Re = 5000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.12 Streamwise velocity profile along the wake centre line . . . . . . . . . . 715.13 Streamwise velocity profile for M1, M2 and M3 at three locations in the

wake, compared with the experimental data of Kim & Durbin at Re=3700 725.14 Mean velocity profiles along the wake centre line . . . . . . . . . . . . . 735.15 Mean streamwise and radial (cross-stream) velocity profile at different

locations in the wake of sphere . . . . . . . . . . . . . . . . . . . . . . . 755.16 Fluctuating mean streamwise and radial (cross-stream) velocity profile at

different locations in the wake of sphere . . . . . . . . . . . . . . . . . . 755.17 Contours of normalised mean Reynolds stresses for (a) Coarse mesh, M1;

(b) Medium mesh, M2; and (c) Fine mesh, M3 . . . . . . . . . . . . . . . 76

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List of Figures

5.18 Contours of normalised mean shear stress and Turbulent kinetic energyfor (a) Coarse mesh, M1; (b) Medium mesh, M2; and (c) Fine mesh, M3 . 77

E.1 Vortex shedding at same time period for all three mesh using Q-iso-surfaces,in X-Y plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

E.2 Instantaneous streamwise velocity contours for; (a) Coarse mesh, M1 (b)Medium mesh, M2 and (c) Fine mesh, M3 . . . . . . . . . . . . . . . . . 104

E.3 Instantaneous cross-stream velocity contours for; (a) Coarse mesh, M1(b) Medium mesh, M2 and (c) Fine mesh, M3 . . . . . . . . . . . . . . . 105

E.4 Instantaneous Mach number contours for; (a) Coarse mesh, M1 (b) Me-dium mesh, M2 and (c) Fine mesh, M3 . . . . . . . . . . . . . . . . . . . 106

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List of Tables

2.1 Values of constants in Spalart-Allmaras model . . . . . . . . . . . . . . . 17

3.1 Utilities in OpenFOAM . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2 Incompressible solvers in OpenFOAM . . . . . . . . . . . . . . . . . . . 29

4.1 Initial values for simulation . . . . . . . . . . . . . . . . . . . . . . . . . 424.2 Summary of simulation time and DESModelRegions for all three mesh . 434.3 M1, M2, M3 representa the coarse, medium and fine mesh while exper-

imental data for vortex shedding Strouhal number and separation anglefrom Achenbach [1, 2] and drag coefficient from Schlichting [55] . . . . . 51

5.1 Probes locations used initially for finding the correct positions to capturesthe main frequencies associated with the fluctuations . . . . . . . . . . . 56

5.2 Mean flow statistical data, DES (present simulation) results comparedwith DNS and LES results at Re=10000 . . . . . . . . . . . . . . . . . . 67

5.3 Mean flow statistics compared with DNS results of Rodriguez et al.[7] atRe = 3700 and LES results of Constantinescu & Squires[29] at Re = 104 . 74

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Nomenclature

Abbrevations

Cd Drag Coefficient

Cf Skin-friction Coefficient

CFD Computational Fluid Dynamics

CFL Courant Number

Cl Lift Coefficient

Cp Pressure coefficient

CV Control Volume

DDES Delayed Detached Eddy Simulation

DES Detached Eddy Simulation

DNS Direct Numerical Simulation

FFT Fast Fourier Transform

FV Finite Volume

GAMG Geometric-algebraic multi-grid solver

IDDES Improved Delayed Detached Eddy Simulation

LES Large Eddy Simulation

PBiCG Preconditioned Bi-Conjugate Gradient

PCG Preconditioned Conjugate Gradient

PISO Pressure Implicit with Splitting of Operators

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Nomenclature

RANS Reynolds Averaged Navier-Stokes

S-A Spalart-Allmaras turbulence model

SGS Sub-Grid Scale

SIMPLE Semi-Implicit Method for Pressure-Linked Equations

St Strouhal number

URANS Unsteady Reynolds Averaged Navier-Stokes

WMLES Wall Modeled Large Eddy Simulation

Greek Symbols

∆ Filter width m

δ Boundary layer thickness m

ε Energy dissipation rate m2/s3

κ Von Karmann constant −

µ Dynamic viscosity m2/s

ν Kinematic viscosity m2/s

νt Turbulent viscosity m2/s

τw Wall shear stress N/m2

ν Modified turbulent kinematic viscosity m2/s

Roman Symbols

ρ∞ free-stream density kg/m3

P Cell centroid −

R Neighbouring Cell centroid −

d DES length m

c Speed of sound m/s

Ck Kolmogorov constant −

Nomenclature

dw Wall distance m

f Frequency 1/s

fb Step function - IDDES −

fe Elevating function - IDDES −

k Wave number 1/s

lhyb Hybrid RANS-LES length scale m

M∞ free-stream Mach Number −

Q Second invariant tensor −

U∞ free-stream velocity m/s

uτ Friction velocity m/s

viv j Velocity components m/s

y+ Distance in wall units −

Mathematical Symbols

u filtered velocity m

∇ Nabla operator −

lDES DES model length scale m

q Energy source term −

Re Reynolds number −

t Time s

u′ Velocity fluctuations m/s

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Chapter 1

Introduction

1.1 Background and Motivation

The flow around bluff bodies such as vehicle aerodynamics, flow around wings at hightangle of attack, interaction of gust with buildings, heat transfer improvements, amongothers are some of the large number of examples which are of great interest for vari-ous engineering applications. Prediction of flow around such bluff bodies which showsmassive separation, are still remain one of the greatest challenges to the ComputationalFluid Dynamics (CFD). Truth be told, the investigation of turbulent flow past canonicalgeometries can be valuable to explore these complex flow structures and additionally togive valuable data for validation of CFD models (e.g. LES, DES models). In this sense,the fundamental approach of the present work is to investigate the turbulent flow past asphere at sub-critical Reynolds number (laminar boundary layer separation; transition toturbulence occurs in the separated shear layer). Prediction of flow at supercritical Reyn-olds number (turbulent boundary layer separation), increments the burden on the model,essentially through the need to predict the growth of boundary layer and separation, whichis under the control of RANS model in characteristic DES applications. The cost of en-tire domain LES at supercritical Reynolds number is not a long way from that of DirectNumerical Simulation due to resolution needed to capture the turbulence structure, insidethe slender attached boundary layer.

The unsteady flow past a sphere at sub-critical Reynolds number has an intricate naturecharacterized by the transition form laminar to turbulent flow in the detached shear layer,the presence of a turbulent wake behind the sphere and unsteady shedding of vortices.This turbulent flow has been object of numerous experimental and numerical studies[1, 2, 3, 4], most of studies provided the data of flow visualization, angular distribu-tion of skin-friction and pressure coefficients over the sphere surface, vortex sheddingfrequency and drag coefficient, among others. In the most recent decades Reynolds-

1

1. Introduction

Averaged Navier-Stokes equations (RANS), Large Eddy Simulations (LES) and DirectNumerical Simulation (DNS) have turned out to be effective tool for giving time-accuratehelpful data about the flow behaviour. However, one of the important requirements of thesimulation of complex turbulent flow is the expensive measure of computational assetsexpected to convey them out. That is why, majority of numerical simulations of the flowover sphere have been carried out in the laminar regime [5, 6]. However, for turbulent re-gime there are still very few time-accurate calculations carried out [4, 7]. Besides, a largenumber of the numerical works reported since now have been performed utilizing differ-ent turbulent models, including Large Eddy Simulations (LES) [8, 9, 10] and DetachedEddy Simulations (DES) [11].

While the geometry is straightforward, the flow around the sphere is very complex toanalyse, having a significant number of difficulties that are hard to precisely capture in nu-merical models. In this work, flow field behaviour are obtained using Detached Eddy Sim-ulation (DES), a hybrid method which basically reduces to Reynolds-Averaged Navier-Stokes (RANS) treatment close to the wall and turns into Large Eddy Simulation (LES)inthe region away from solid surfaces, conditionally grid density should be sufficient[12].DES is a nonzonal method which is computationally feasible for high Reynolds numberflows, yet likewise determines time-dependent, three-dimensional turbulent motions as inLES. Past simulation results of this strategy have been good, yielding sufficient expecta-tions over a wide range of flows and likewise demonstrating that the computational costhas a weak reliance on Reynolds number, like RANS method, yet at same time give morereasonable description of unsteady effects.

Despite the fact that extensive research available, analysis of mechanism for transitionin shear-layer, behaviour and quantitative estimation of wake structure are still rare. Thereis a serious lack of detailed experimental and numerical data for sphere case, such as lowfrequency fluctuations, separation angle, recirculation length, force coefficients, amongothers, at higher Reynolds numbers. However to the best of our knowledge, there is nocomplete study for sphere case which consider the effects of low frequency fluctuationsin wake. While low-frequency fluctuations in the wake of some other bluff bodies havebeen examined by few numerical studies.

1.2 Literature Review

1.2.1 Experimental Background

Turbulent flow past the sphere has been the subject of various experimental investigations[1, 2, 3, 13, 14, 15]. The essential interest for these investigations included visualization ofprimary vortical structure in the wake, understanding the mechanism of vortex shedding,

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1. Introduction

estimation of frequencies present in the wake, mean pressure coefficient around sphereand streamwise drag. These studies also attempted to understand and explain the mech-anism through which wake become unstable. As the Reynolds number increases beyondthe Re= 280, experiments demonstrate that the wake starts to shed vortices in a consistentmanner. With the further increment in Reynolds number the shedding process truns outto be more unpredictable and complex, and, in the long run, the wake structure becomeschaotic. Since vortex loops diffuse very quickly, the examination of the wake configura-tion by the method of classical visualization techniques becomes troublesome. As far asanyone is concerned, except for the recent investigation by Bakic [16] at Re = 5× 104,none of the above experimental investigation had provided the data for velocity compon-ents and Reynolds stresses in the wake which would be necessary for fully validating theexpectations of a time-accurate numerical simulations.

Chomaz et al. [13] recognized the two primary instability modes present in the wake,when vortex shedding is there. For Re > 280 large-scale vortices shed from the surface ofsphere. The vortex shedding or the first mode is identified to the large-scale shedding inthe wake. At the limit between the recirculation zone and the exterior fluid, this instabilityshows itself as a progressive wave movement with alternate fluctuations produced by theshear. These fluctuations decide the periodic shedding of the vortices that structure behindthe sphere. The recirculation zone is definitely not axisymmetric. Beginning at Re = 800there is a second high frequency mode (or spiral mode) connected with the small-scaleshear-layer Kelvin-Helmholtz (K-H) instability on the fringe of the recirculation zone.This unsteadiness is capable for the distortion of large vortex structure and produce thevortex rings (subsequently vortex tubes), which shed in a quasi-coherent form inside ofthe detached shear layers, hence results in a production of small scale vortices, and, inthe long run, transition to turbulence in the wake. The high frequency mode or the spiralmode instability is present just in an area limited to the wake immediately downstreamof the sphere and in the detached shear layers, where it is more dominating then thefirst or vortex shedding mode. These two instability modes can exists together all thewhile upto a threshold Reynolds number, however, experiment results shows some aboutits value. Although, most of the experiments discussed so far, caught both modes atRe = 104, results of Kim et al. [14] and Bakic [16] capture both modes up to Re = 105

and Chomaz et al were able to capture both modes at Re = 3× 104. While Achenbach[2] failed to detect both modes beyond Re = 6×103 and Sakamoto and Haniu [3] beyondRe = 1.5×104.

The relationship between frequencies and the structure of wake is the another issueof great interest. Sakamoto et al.[3] researched this in their experiment using hot wire aswell as flow visualization techniques. Their observe that, laminar hairpin-shaped vorticesbegins to shed at Re = 280 to form a completely laminar wake in periodic and regular

3

1. Introduction

fashion. These hairpin vortices are shed regularly from Re = 280 to around 420 withfrequency of same strength and in the same plane, so the the lateral forces coefficient per-pendicular to shedding plane is zero all times. While from Re = 420to480,they observedthat shedding direction starts oscillating which is conformed by DNS study of Mittal [17].At the point when Reynolds number surpasses 800, periodically shedding vortex tube nowcovers whole near-wake region, and hairpin like vortices which were laminar earlier nowbecomes turbulent, however entire vortex sheet is still laminar which is separating fromthe sphere. Even now, as compared to the ones at low Reynolds number, the large struc-ture vortices still appear to keep up a hairpin-shaped form. Correspondingly, some scalevortex loops formed as small-scale vortex tubes shed into it, and as the move far fromthe sphere, interface with the large vortices. Kelvin-Helmholtz instability is subjected tothese smaller vortex tubes which are laminar initially. Baric in his experiment was ableto capture transition of these vortical structures into turbulence as accompany by roll-upand pairing processes. The vortex sheet begins to undergoes transition from laminar toturbulence at Re = 3× 103 and ends around Re = 6× 103 when it becomes fully turbu-lent. The experiment of Baric conforms that, change in the wake structure and integralparameters is very little with the increase in Reynolds number until very close to the crit-ical Reynolds number or until the drag crisis. That is, at separation the boundary layerover the sphere is laminar up to Recrit . Due to the complete transition to turbulence inthe detached shear layer, stabilizing effect is generated which happens from Re = 7×103

to Recrit and results in more regular shedding pattern of the large-scale vortices. Con-versely, with the Reynolds number, the Strouhal number associated with the shear layerincreases strongly since due to the smaller wavelengths the shear layer becomes unstable.At Re = 104 the value of Strouhal number is in the range of 1.8− 2.5 and is detectablegenerally in the region of detached shear layer. Taneda [15] observe for Reynolds num-ber in between 104−105 that oscillating wake in the azimuthal plane, rotates slowly andirregularly around the axis through center of the flow, oriented parallel to the main flowdirection. Again, even Taneda in his experiment, did not observe any change in wakestructure upto critical Reynolds number

1.2.2 Previous Numerical Investigation

Several time-accurate simulations of laminar flow over sphere using finite-element meth-ods were accounted among others by Mansoorzadeh et al. [18], Shen and Loc [19],Kalro and Tezduyar [20] and Aliabadi and Tezduyar [21], while Johnson and Patel [6]and Shirayama [22] used finite-difference method approaches. Depending on the authors,generally the onset of vortex shedding was seen in the range of Re = 280− 400 and theconsidered Reynolds number were upto 103. As compared with the experimental data forRe 275−285, the onset of vortex shedding in these simulations are relatively spread very

4

1. Introduction

large which in turns implies the different level of accuracy of these codes. These codescontributes in better understanding of vortex structure in the wake ad vortex-sheddingmechanisms.

Tomboulides and collaborators [4, 8] study the transition of near wake to turbulenceby performing laminar, DNS and LES simulations. They correctly captured the onset ofvortex shedding at around Re = 250−285 in their laminar simulation. For LES and DNSsimulations, they limited the maximum Reynolds number of 2×104 and 103 respectively.To solve the incompressible Navier-Stokes equations the numerical method employed isa spectral element-Fourier algorithm, while for LES simulation the SGS model utilisedwas based on renormalization group theory.The reported value of Strouhal numbers re-lated to the shedding and spiral modes by Tomboulides and Orszag [4] is very close tothe experimental value measured by Sakamoto and Haniu [3]; the Strouhal number forshedding was St = 0.2 which is within 10% of the experimental value. Another group,Kim and Choi [23] used LES for flow over sphere from Re = 3.7× 103 to 104 to studythe change in the wake structure. These investigators used hybrid discretization (in lam-inar acceleration region, upwind and central difference elsewhere) and used the immersedboundary method in cylindrical coordinates to calculate the turbulent flow past sphere.At low Reynolds number, the quantitative (velocity profile in wake) agreements with theExperimental data of Kim and Durbin was goodwhile for both Reynolds number the meanpressure and drag coefficient and shedding frequency were also in range of experimentalresults. DNS simulation is performed by another group, Seidl et al. [24] at Reynoldsnumber of 5,000. They were able to capture the formation of initially laminar vortex tubesuccessfully in the detached shear layers, and in addition the mechanism of roll-up andpairing and transition of these vortices. They also able to get correctly the values of dragcoefficient, Strouhal number, etc with their simulation. Schmid [25] performed severalLES simulation using different SGS models (Smagorinsky, dynamic and no model) atRe = 5× 104. To precisely capture the initial formation of vortex tube and its transitionto turbulence, they utilizes the local grid refinement in the separated shear layer usingthe finite volume method. They observed that the influence of SGS model is somewhatminor on mean flow quantities. They had compared their data with the experimentalobservations of Bakic and overall agreement of mean flow velocity profile and its fluctu-ations in the near wake region at same Reynolds number was in agreement. While thiswas conversely with the RANS simulation of Poon et al. [26], which was done on sameReynolds number flow and in their agreement was poor for integral quantities as well aswake characteristics. The main observation for unmatched results is that they predictedthe transition downstream then the experimental observed location which would effectthe prediction of mean drag coefficient value. Since the value of turbulent kinetic en-ergy is very high in the free stream, it can be possible that these problems caused due

5

1. Introduction

to the set up of RANS or the mesh density in the separated shear layer. Drikakis [27]investigated the steady RANS simulation at subcritical and supercritical Reynolds num-ber using artificial compressibility solver using κ − ε model in conjunction at near wall,while another researcher group Koschel et al. [28] used no turbulence model at all for un-structured finite-element scheme. The achievement of these simulation results for gettingmean quantities like pressure of drag coefficient around sphere was restricted.

Some of the recent simulation like, Constantinescu and Squires [29] simulate the flowover sphere at Re = 104 using Large Eddy Simulation (LES) and Detached Eddy Sim-ulation (DES). They utilises the 0-type grid which is generated by revolving 2D grid inazimuthal direction and observed that both methods reproduce the main flow characterist-ics and vortex shedding phenomena successfully. This group then again performed DirectEddy Simulation (DES) for flow over sphere at subcritical and supercritical regimes attwo different Reynolds number Re = 104 and Re = 105 [30], able to capture the meanflow parameters this time which were in good agreement with the experimental results.As far as DNS results are concerned there is not much data available for time-accurateinstantaneous and statistical flow data because for three-dimensional (3D) simulation andtime-accurate results demands running simulation for much longer time, adding to thecomputational resources there are also fine grid for DNS simulation which further putload on resources. Then recently Lehmkuhl et al. [31] carried out simulation for flowdynamics of wake behind sphere at Re = 3700 and 10,000, they performed some throughanalysis on unstructured grid by rotating it in azimuthal direction, further more they alsoconsider the low-frequency fluctuation which effect the shrinkage and enlargement of re-circulation zone. They have concluded that the vortex formation region is related to thebase-suction coefficient Cpb. They are also able to successfully capture all three dominantinstabilities, i.e.large-scale vortex shedding, small-scale Kelvin-Helmholtz instability andmodulation of the recirculation which occur at very low frequency fm, further to theiranalysis, they also pointed out that with increase in Reynolds number the length of recir-culation zone decreases. Their results are in good agreement with experimental results.

1.3 Thesis Objectives

The main task of the thesis is to simulate flow over automotive body taking sphere asa test case here. Its is known that flow over bluff bodies causes turbulence in the wakewhich effects the aerodynamic properties. There are some low-frequency fluctuations inthe wake which is hard to capture and require longer simulation time in order to predict itaccurately.

In previous section we have discussed some of the instabilities associated with theflow over sphere and effect of low-frequency among them which causes shrinkage and

6

1. Introduction

enlargement of recirculation bubble. Next two chapters described that hybrid RANS-LESmodel could be an appropriate approach for such kind of flows. Therefore, following arethe important consideration which discussed in this study throughout:

1. An open source numerical tool which can able to simulate the incompressible, un-steady turbulent flow.

2. A turbulence model which can able to capture turbulent flow features such as vortexshedding, among others, and even then it should be computationally less expensive.

3. A good mesh in for open source tool since it is known that tools like OpenFOAMare highly sensitive to mesh quality.

4. Selection of suitable initial and boundary conditions, accurate discretization schemesand solver settings.

5. Time period selection in order to capture the the clear footprints of low-frequencyfluctuations.

6. Validation of results for instantaneous and mean flow parameters with previous ex-perimental and numerical results at comparable Reynolds number.

1.4 Outline

The general outline of thesis is as follow, the first chapter-1 highlights the previous experi-mental and numerical background of flow over sphere case and provides the overall intro-duction to the shedding mechanism. Chapter-2 discuss the hybrid numerical method util-ised in thesis, a detail discussion of DES, DDES and IDDES approach is given. Chapter-3provides the detail procedure of numerical tool used and set up for the case with somebasic focus on governing equations used in solver. Chapter-4 discuss the computationaldomain, mesh procedure with CfMesh and refinements used for generating all the meshes.Chapter-5 discuss the results and validate the numerical results of thesis with experimentaland numerical results present in literature. The last chapter-6 conclude the research withsome of the recommendations for future work.

7

Chapter 2

Physics and Modelling

2.1 Governing Equation

The objective of this thesis is to simulate the flow around sphere at subcritical Reynoldsnumber of 10,000 which represent the external flow aerodynamics as a test case. Sine theflow is at low Mach number M = 0.2 (Incompressible flow) and subjected to Newtonianfluid properties, hence to describe the fluid dynamics of the flow here, Navier-Stokesequation can be assumed as a governing equation. There are basically five equationsconsists in a Navier-Stokes equation, first one is the continuity ((2.1), represents mass isconserved), three momentum equations for each direction ((2.2), represents momentumis conserved) and last one is the energy equation (2.3, represents energy is conserved).

Dρ

Dt=

∂ρ

∂ t+∇.ρu = 0 (2.1)

ρuDt

=∂ρu∂ t

+∇.(ρuu) =−∇p+∇.(µ∇u)+ f (2.2)

DρeDt

=∂ρe∂ t

+∇.(ρue) =−∇pu+∇.(µu∇u)−∇q (2.3)

where, D/Dt = ∂/∂ t + u∇ is know as Substantial derivative. Whereas ρ, p,e are thedensity, pressure and total internal energy. While u represents the velocity magnitude forall three directions and the symbol ∇ refers to as Nabla operator which is defined as:

8

2. Physics and Modelling

∇ =

(∂

∂x,

∂

∂x,

∂

∂x

)(2.4)

Since the flow is consider as incompressible flow because of low Mach regime, henceit results in a homogeneous and constant density across whole domain. The assumptionsmade here, results a simpler form of Navier-Stokes equation due to the absence of anyexternal forces such as body forces or gravity. Also, the temperature is considered con-stant and assumed that it doesn’t have any influence on the flow field dynamics. All of theassumptions made here reduces the number of unknowns to just four (all three velocitycomponents and pressure), while the energy equation can be omitted, equation 2.3. Hencethe resulting equations would be:

∇.u = 0 (2.5)

∂u∂ t

+∇.(uu) =−∇pρ

+∇.(ν∇u) (2.6)

2.2 Turbulence models and Numerical methods

The external flow in Automotive engineering is inherently connected to the turbulence. Itis a phenomena which exists in various engineering and industrial applications. Due to itswide existence, it is one of the most researched topic of CFD so far and hence there arevarious ways available by which different turbulent scale which exists in a turbulent flowcan be computed. Among all the techniques available, Direct Eddy Simulation is the onewhich provides the most ’exact’ solution of Navier-Stokes equation since it solves all thescales in turbulence and hence doesn’t require modelling at all. However DNS has someserious drawbacks:

1. It has a very very high requirement of computational power (e.g. cost of DNS scaleis ∝ Re3), hence it make DNS very costly for daily use or for initial test simulations.

2. Since the number of cell require to carry out DNS simulation is ∝ toRe9/4, hencelarge domain size would result in millions of cells for large Reynolds number andreducing the domain would results some non-physical changes in flow dynamics.For Automotive applications we need a large far field to damp out any possibilityof wall effect behind the body of interest.

9


Then there is another method which models the smallest scale (kolmogorov scale)and computes the most important large scales, knows as Large Eddy Simulation (LES).Whereas the third one and the most used one in industries is the Reynolds AveragedNavier-Stokes equation (RANS) which model a very wide range of turbulent length scales.Now days, a hybrid approach is emerged which combines the advantages of RANS andLES together, this method is used in thesis. Firstly some of the basic principles of RANSand LES are discussed.

2.2.1 RANS principle

The Navier-Stokes equations 2.5 and 2.6 can be represents in cartesian coordinate system,xi(i = 1,2,3). Hence the incompressible equations can be written as:

∂u j

∂x j= 0 (2.7)

Dui

Dt≡ ∂ui

∂ t+u j

∂ui

∂x j=− 1

ρ

∂ρ

∂xi+ν

∂ui

∂x j∂x j(2.8)

where ui is the cartesian components of velocityIn RANS (Reynolds-Averaged Navier-Stokes equation) method, we averaged out all

the unsteadiness in the flow and regarded as a part of turbulence. Hence the flow velocityis represented as the sum of two terms:

ui(xi, t) = ui(xi)+u′i(xi, t), (2.9)

where,

ui(xi) = limT→∞

1T

∫ T

0ui(xi, t)dt (2.10)

Where T represents the averaging interval, it should be large as compared to the typ-ical time scale of turbulent fluctuations. While u′i represents the time averaged value offluctuation.

For unsteady flow problems, ensemble averaging is used in place of time averaging.The ensemble averaging can be explained as variable that can be controlled (boundaryconditions, energy, etc.) for a set of flows who are identical but initial conditions aregenerated randomly. This will give flows that differ considerably from one another. Hence

10


an ensemble average is defined as an average over large set of such flows. Can be writtenas:

ui(xi) =1N

N

∑n=1

uni(xi, t) (2.11)

Where N represents the number of members of the ensemble. For Reynolds averaging(unsteady flow) we apply the ensemble average approach to the incompressible continuityequation 2.7, gives

∂ u j

∂x j= 0 (2.12)

We take the mean of the left hand side of the momentum equation 2.8, since mean ofconvective term is not a easy task because of nonlinearity. Hence equation can be writtenas:

Dui

Dt=

∂ ui

∂ t+

∂ (uiu j)

∂x j(2.13)

Using equation 2.9 for non linear term gives:

uiu j = (ui +u′i)(u j +u′j) (2.14)

= uiu j +u′jui +u′iu j +u′iu′j

= uiu j +u′jui +u′iu j +u′iu′j

= uiu j +u′iu′j (2.15)

Since,

u′jui = u′jui = 0 (2.16)

Using equation (2.15) with equation (2.13), we get

Dui

Dt=

∂ ui

∂ t+ u j

∂ ui

∂x j+ ui

∂ u j

∂x j+

∂ (u′iu′j)

∂x j(2.17)

Using incompressible mean velocity, equation (2.17) simplifies to

11


Dui

Dt=

∂ ui

∂ t+ u j

∂ ui

∂x j+

∂ (u′iu′j)

∂x j(2.18)

Now taking the mean of the other terms in momentum equation results in Reynolds(RANS) equation.

∂ ui

∂ t+ u j

∂ ui

∂x j=− 1

ρ

∂ p∂xi

+ν∂ ui

∂x j∂x j−

∂u′iu′j

∂x j(2.19)

Equation (2.19) can be written in a simplified form as:

ρ

(∂ ui

∂ t+ u j

∂ ui

∂x j

)=

∂

∂x j

[− pδi j +µ

(∂ ui

∂x j+

∂ u j

∂xi

)−ρu′iu

′j

](2.20)

On the left hand side, the term in square brackets represents the sum of three stresses;namely,−pδi j represents the mean pressure, the second term represents the viscous stressfrom the momentum transfer and the last term −ρu′iu

′j, is the fluctuating velocity. This

term is called Reynolds stresses.The Reynolds stresses are components of symmetric second order tensor, where the

diagonal components represents normal stresses while non-diagonal components repres-ents shear stresses. Half the trace of the Reynolds stresses give the turbulent kineticenergy, k, given by:

k =12

ρu′iu′i (2.21)

Since, six more unknowns are introduced because of the six independent elements dueto symmetry of the Reynolds stress tensor hence, in order to close the system, i.e. numberof unknowns equal to the number of equations, we need to model the Reynolds stressesin one of the ways given in literatures [32, 33]

Turbulence consists of different size eddies, the largest eddies which are highly un-stable in a flow, break up and hence transfer their energy to smaller eddies which are alsounstable and break up again to transfer their energy to yet smaller eddies. This is know asenergy cascading and continues until the Reynolds number Re(l)≡ u(l)l/ν is sufficientlysmall so that eddy motion is stable and molecular viscosity is effective in dissipating thekinetic energy [34].

12


2.2.2 LES principle

Smagorinsky [35], gives the idea and basic theory of LES in 1963. The largest scaleeddies, according to theory of Kolmogorov are the eddies which contain most of theenergy and do most of the transportation hence these eddies are one of the most importantone in turbulence and are calculated directly. While the smallest scale eddies can beeasily modelled since they are assumed to behave uniformly. This is the concise basicprinciple of Large Eddy Simulation (LES). Generally expressed, this implies the smallerscales contributes a small amount of the total energy while the larger scale contain thedominant part of the energy. This can be outline by the turbulent energy cascade or energyspectrum, Figure 2.1. The straight dotted line is also defined in figure which representsthe Kolmogorov’s law and is defined as:

E(k) =Ckε2/3k−5/3 (2.22)

Where Ck = 1.5, ε is the energy dissipation rate, and k is the wavenumber which is in-versely proportional to the length scale.

Figure 2.1: Energy spectrum of length scales. a) High energy region, b) transfer of energyregion c) dissipation region [36]

In above Figure 2.1, energy spectrum is divided into three sub-regions:

13


1. Integral length scale: This is the first region and is characterized by largest eddieswhich contain the dominate part of energy, denoted by ki.

2. Inertial subrange: The second region in which eddies follows the Kolmogrov’slaw. In this region mostly the transfer of energy from large to small scale is happinghence it is dominated by transitive scale.

3. Dissipative range: The last and third region contains the smallest scale eddies who’sbehaviour is dominated by the viscosity and energy transfer from the larger scaleeddies.

Like in Reynolds-Averaged Navier-Stokes equations (RANS), 2.2.1; we do some aver-aging to model the large scale, in Large Eddy Simulation (LES) we apply filtering. Thescale separation is performed using this filtering, which is the locally derived weightedaverage of the flow properties over a volume of a fluid. The filter width ∆ is one of theimportant feature in filtering operation. ∆ is selected in such a way that, turbulent lengthscale larger then it are held in the flow while the Sub-Grid Scales (SGS) or the smallerscales then ∆ should be modelled. In this way we can write any turbulent flow variable,like flow velocity, as a sum of large and small scale.

u = u−u′ (2.23)

The resolved larger scale is represented by overbar while smaller scale are represented byprime. The filter process for large scale is obtain by:

u =∮

u(x′)G(x,x′;∆)dx′ (2.24)

Where (x,x′;∆) is know as filter function and it should satisfy the condition:∮G(x,x′;∆)dx′ = 1 (2.25)

The schematic representation of one-dimensional filtering operation for one of the flowvariable is shown in Figure 2.2. The implicit top-hat filter is a standard filter applied inOpenFOAM (standard filter for Finite Volume methods), which takes an average over arectangular region (Some other filters are also exist such as, Gaussian filter or sharp hatfilter [37]). The local and averaged value of u will be equal if we choose fiter width equalto the grid spacing. It is given by:

G(x,∆) =

1∆, if | x |≤ ∆

2

0, otherwise(2.26)

14


Figure 2.2: Filtering operation for a flow variable [38]

2.2.3 Filtered Navier-Stokes equation

Equation of motion is obtained for resolved large scales by applying filter to the incom-pressible Navier-Stokes equation (2.8), the filtered equations are denoted by overbar:

∂ ui

∂xi= 0 (2.27)

∂ ui

∂ t+

∂

∂x j(uiu j) =−

1ρ

∂ p∂xi

+1ρ

∂τRi j

∂x j+ν∇

2ui (2.28)

A dependency is caused between unresolved and resolved scales due to the non-linearconvective term of Navier-Stokes equation. The impact of the unresolved scales are con-solidated in the subgrid-stress tensor, which includes the residual stresses and it is char-acterized by:

τRi j = ρ(uiu j− uiu j) (2.29)

15


To define the unresolved scales, an Eddy viscosity model is utilized in LES. Hence stresstensor becomes:

τRi j = 2ρνt Si j +

13

δi jτRkk (2.30)

Where νt represents turbulent or eddy viscosity. Which gives us:

∂ ui

∂ t+

∂

∂x j(uiu j) =−

1ρ

∂ p∂xi

+2∂

∂x j[(ν +νt)Si j] (2.31)

Above equation (2.31) represents the final Filtered Navier-Stokes equation, now the laststep is to give the definition of turbulent-viscosity (νt).

2.3 Turbulence closure model

In the thesis we used Spalart-Allmaras (S-A) turbulence model to determine the turbulentviscosity (νt). Since S-A model use only one additional equation hence it is relativelysimple. The modified turbulent kinematic viscosity (ν) is introduced as the only addi-tional unknown in the equation. Modified turbulent viscosity is defined by [39]:

νt = ν fv1 (2.32)

where,

fv1 =χ3

χ3 + c3v1

χ =ν

ν

Here, ν is the molecular viscosity, cv1 is a contant and ν represents the modified turbulentviscosity or the working variable, giving the transport equation:

Dν

Dt= cb1Sν +

1cσ

[∇.((ν + ν)∇ν)+ cb2(∇ν)2]− cw1 fw

[ν

d

]2

(2.33)

S = ω +ν

κ2dfv2 (2.34)

16


fv2 = 1− χ

1+χ fv1(2.35)

Where ω represents magnitude of vorticity while function fw is given by:

fw = g

[1+ c6

w3

g6 + c6w3

]1/6

(2.36)

g = r+ cw2(r6− r) (2.37)

r =ν

Sκ2d2(2.38)

The values of constants defined above is tabulated in Table 2.1

Constant Valuecb1 0.135cb2 0.622cw2 0.3cv1 7.1cσ 2/3κ 0.41

cw3 2cw1 cb1/κ2 +(1+ cb2)/cσ

Table 2.1: Values of constants in Spalart-Allmaras model

2.4 RANS-LES Hybrid approach

In Hybrid methods, for region near the wall they typically utilises the solution of anotherset of model equations. The region where turbulent boundary layer is solved in a zonalhybrid methods is defined for a region in the vicinity of the wall. While explicit boundarycondition is prescribed for communication to the outer LES region. Where as a smoothtransition between different regions is made in a blended hybrid methods.

Spalart and Allmaras [40] was the first to propose the most widely recognized typeof a hybrid RANS-LES method in 1992, name as, Detached Eddy Simulation (DES). Itcombines the advantages of both Reynolds-Averaged Navier-Stokes (RANS) and LargeEddy Simulation (LES) together, which is the basic thought behind this approach. In

17


a more explanatory way, this hybrid RANS-LES method acts as only RANS mode inattached boundary layer and transform into LES mode only for detached flow regions.From the region of unsteady RANS equations to the region where standard LES is solved,a smooth transition is produced for these blended hybrid methods, while this kind ofswitching between RANS and LES relies on the local-grid resolution. Piomelli et al. [41]had shown that due to the interface treatment resulting form transition layers in DES,results in decrease of skin friction for these blended approaches. This section give theoverview of the kind of errors in Detached Eddy Simulation (DES) and Delayed DetachedEddy Simulation (DDES) while dealing with these models and next section 2.5 providethe solution to overcome these kinds of errors.

2.4.1 Detached Eddy Simulation (DES)

In a classic Detached Eddy Simulation (DES), a limiter combines the standard Spalart-Allmaras RANS model with its Sub-Grid Scale (SGS), defined by:

lDES = mindw,CDES∆ (2.39)

where lDES represents the model length scale, dw is the distance to the wall(given bydestructive term of Spalart-Allmaras model), CDES is the derived constant whose value is0.65 and ∆ is the largest local-grid spacing:

∆ = max∆x,∆y,∆z (2.40)

In Detached Eddy Simulation (DES), near the wall (dw <CDES∆) in a attached boundarylayer, a classic S-A RANS is acting, while away from the wall (dw >CDES∆) in a separa-tion region, a SGS model is acting with a filter CDES∆. Despite the fact that this turbulencemodel is most common and utilized for several years, regardless it experiences a few dis-advantages. Issues emerges when separation region is smaller then the thick boundarylayer in a wall bounded flows. For this situation, often the boundary layer thickness islarger then the grid spacing parallel to the wall ∆|| or in other words, it grid become fineenough in for DES length-scales, parallel to the wall such that the LES branch followthrough it in accordance to equation 2.39. Due to this a phenomena is developed which iscalled Grid Induced Separation (GIS) [42, 40] according to it, as a consequence of finergrid, the eddy viscosity reduces below the RANS level but the velocity fluctuations whichare driving the LES content (or resolved Reynolds stresses) have not replaced the modeledReynolds Stresses. Hence, these ’missing stresses’ causes the reduction in skin friction.Figure 2.3 represents the basic grid examples to give the overview of grid importance.

The Figure 2.3a shows the grid in which wall-parallel spacing ∆|| is larger then the

18


(a) Grid spacing larger then the boundary layer

(b) Grid spacing smaller then the boundarylayer,too coarse for LES

(c) Grid spacing to support LES content

Figure 2.3: Examples of three mesh design during grid refinments [38]

boundary layer thickness δ , due to it, in entire the boundary, the DES length-scale is equalto the RANS type (lDES = dw). Figure 2.3c shows the grid wall-parallel grid spacing lessthan to boundary layer thought the domain, traditionally this represents the pure LES typegrid, therefore in most of the boundary layer the SGS model is activated (lDES =CDES∆)

while only in vicinity of wall a RANS model is activated (lDES = dw). In Figure 2.3b, thewall-parallel grid spacing is not as small as for pure LES grid therefore deep in the bound-ary layer, a SGS model of DES is originated. It can not able to capture all the velocityfluctuations since the grid is not fine enough at this point. Besides, without the acquaint-ance of resolved stresses to re-established the balance the modeled Reynolds stresses andeddy viscosity will be reduced. In literature this phenomena is called Modeled StressDepletion (MSD).

2.4.2 Delayed Detached Eddy Simulation (DDES)

The equivocal grid like Figure 2.3b give rise to the problem like Modeled Stress Depletion(MSD), hence the method is formulated to avoid these error, called Delayed DetachedEddy Simulation (DDES) which is just a simple modification of classic Detached EddySimulation (DES) and similar to the shear-stress transport model proposed by Menter et

19


al. [43]. The noticeable feature of DDES is that, to define length-scales it utilises someblending functions. Even if due to the grid spacing the DES limiter is activated eventhough DDES maintains the full RANS mode by detecting the boundary layer whichis dependent on the eddy viscosity and therefore on solution as well. As explained bythe Haase et al. [44], even if blending function showing that point of interest is insidethe boundary layer, it declines to change into LES mode. As an outcome, the transitionbetween LES-RANS is more abrupt. Hence, the DDES is degigned in such a way that itwipe out the errors caused by DES to a grid refinement like MSD or GIS.

Menter et al. [43] has given the blending functions F1 and F2 which utilises theRANS model internal length scale and the wall distance. At the boundary layer, thesefunction are 1 and at the edge of the boundary layer they reduces rapidly to 0. A para-meter ”r” is utilized in one equation models (S-A model) since internal length scale is notpresent, this parameter is defined as the ratio of model length-scale to the wall distanceand is given for S-A model as:

rd =νt +ν

max[√

Ui, jUi, j,10−10].κ2d2w

(2.41)

Where Ui, j represents velocity, κ is Von Karman constant and dw represents the walldistance. ” fd” in log layer is equal to 1 while it reduces to 0 at the edges and is given by:

fd = 1− tanh(8rd)3 (2.42)

It is 0 in whole domain except in LES region (rd << 1) where it reduces to 0. In contrastto the old definition of DES length-scale given by equation 2.39, a new definition given byequation 2.43 also consider the modified length scale which depends on turbulent or eddyviscosity in comparison to to old definition where only grid dependency is considered.

lDES = dw− fdmax(0,dw−CDES∆) (2.43)

Now with the new definition of lDES, based on value of rd even if fd shows point is wellinside the boundary layer, it is possible to reject the LES mode.

2.5 Improved Delayed Detached Eddy Simulation (IDDES)

Another Improved turbulence method is the Improved Delayed Detached Eddy Simula-tion (IDDES) which overcomes the errors of previous two Classic Detached Eddy Simu-lation (DES) and Delayed Detached Eddy Simulation (DDES). This method consolidatethe advantages of Delayed Detached Eddy Simulation (DDES) and Wall Modeled Large

20


Eddy Simulation (WMLES), which is the main objective of IDDES model. An alternateapproach is applied to overcome the larger grid resolution requirement which is the basicdemand of classic LES, is known as Wall Modeled LES. Taking example, Schumann [45]has given a wall-stress model in 1975, has utilized the empirical derived wall functionsalong with velocities by considering the first off-wall point in log-layer to calculate anapproximation for wall stresses at the boundary.

Then again, it is likewise conceivable to utilize the DES for these WMLES as wassuccessfully attempted by Nikitin et al. [46]. The log-layer mismatch (LLM) error isencountered mostly with WMLES, between the LES and RANS regime. Actually sim-ulation gives two log layers: outer most layer when distance to the wall is greater thenthe local grid size while the RANS model give the inner layer. Due to the mismatch errorLLM, an error of under-prediction of 15 to 20% was noticed in inner and outer layer. Eventhough, in comparison to LES, WMLES still save lot of computing time. The IDDES isdeveloped in such a way that it gives one formula set for both WMLES and DES applic-ations and also avoid the LLM so that it can be used for complex geometry for differentflows inside a single simulation. The IDDES method can be sub-divided into four partsto demonstrate how it works [40, 44, 47, 48].

2.5.1 Modification of the Sub-grid length-scale

Common definition of sub-grid scale for classic LES in most of the literature is give asthe cube root of a cell volume, defined as:

∆ =√(∆x)2 +(∆y)2 +(∆z)2 (2.44)

Moreover, for the classic DES (section 2.4.1) the decision of the sub-grid length scale isdependent on maximum of three cell dimensions 2.40. Both definitions give rises to aproblem more precisely with the constants of SGS, which ought to have different con-stant values for various flow regimes such as free/pure turbulent flow (Decaying IsotropicHomogeneous Turbulence) or for wall-bounded flows. Hence another definition was setup to avoid the requirement of different values for different flow regimes. In this newdefinition the main idea is to include some wall-distance dependency which gives the anew definition of sub-grid length scale:

∆ = f (∆x,∆y,∆z,dw) (2.45)

Where dw represents the wall distance hence new formula depends on both, the local cellsize and the wall distance. Therefore three equations can be given by dividing compu-tational domain into three sub-domains. First one is given by the maximum local-grid

21


spacing just like for classic DES since grid is mostly isotropic away from the wall andhence it is set as a classic DES case, given by:

∆ f ree = ∆max ≡ max(∆x,∆y,∆z) (2.46)

Second one is given by equation 2.47, the sub-grid length scale in the region close to thewall should not follow the drop of the wall-normal step. The sub-grid length scale in thisregion is defined by wall parallel grid only:

∆wall = const(dw) = f (∆x,∆z) (2.47)

Third and the last one is defined as, region between the away from the wall and the regionclose to the wall is assumed to follow as a linear function of dw for the sub-grid lengthscale. Furthermore an assumption is made for ∆ that it varies in the range ∆min≤∆≤∆max.Combining all the above statements yields one single equation:

∆ = minmax[Cwdw,Cw∆max,∆wn],∆max (2.48)

Where Cw = 0.15 is a constant based on developed channel flow for LES and ∆wn is thegrid spacing in wall-normal direction. Figure 2.4 is a typical representation of sub-grid

Figure 2.4: Sub-Grid length scale [38]

length scale for a channel flow, where solid line is valid when ∆≤Cwdw. Furthermore aslong as dw≤∆max is valid the value of ∆ remains constant, at this point ∆=Cw∆max. Whenthe maximum cell size dimension becomes less then the distance to the wall, dw > ∆max

22


the SGS grows explicitly with ∆=Cwdw. As the maximum value is reached ∆max, the SGSremains constant afterwards. The dashed line represents basically a strong wall-normalstretching. The value of SGS remain constant near to the wall, Cw∆max. Once ∆wn >

Cw∆max, the value of SGS grows explicitly until maximum cell size is reached. It is wellunderstood that rate is smaller comparatively in the second case since it is unacceptablefor simulations. In contrast to the different SGS models, the IDDES approach utilizes avery complex method of assessing the grid filter. Other then the wall normal distance,cell dimensions and height of the cell in wall normal direction have their impact in theformulation of the grid filter.

2.5.2 DDES branch of IDDES

When inflow conditions are not turbulent then the first branch corresponds to DDES isactivated. The DDES length scale is given by:

lDDES = lRANS− fdmax(0, lRANS− lLES) (2.49)

with lLES = CDESΨ∆ and lRANS = dw The delaying function is given by equation 2.42.Compared to the classic DES length scale,there is one more factor ψ . The purpose behindthe addition of ψ is on the ground that flow Reynolds number decreases due to decreasein the sub-grid eddy viscosity with grid refinement. Sooner or later, the DES will miss-translate and behave like in the vicinity of a wall. As a result, the turbulent viscositywill drop with respect to the surrounding velocity and length scales through fv and ftfunctions.

The functions for Spalart-Allamars model relies on the ratio of the turbulent viscosityto the molecular viscosity, can be defines according to S-A model: νt/ν or χ ≡ ν/ν .Moreover, CDES value increased effectively due to the Ψ factor. For Spalart-Allamarsmodel this shield function Ψ(νt/ν) is defined as:

Ψ2 = min

[102,

1− cb1cw1κ2 f ∗w

[ ft2 +(1− ft2) fv2]

fv1max(10−10,1− ft2

](2.50)

Where constant κ is equal to 0.424. If the sub-grid turbulent viscosity is larger than 10ν ,then the correction become inactive (ψ = 1) and for lower values it becomes stronger.

2.5.3 WMLES branch of IDDES

In WMLES branch, as opposed to the DDES branch, it will be activated only when flow isturbulent and unsteady and have an adequate grid fineness to resolve eddies in boundary

23


layer. The coupled RANS-LES length scale is utilised to accomplished coupling amongstthem, it is given by:

lWMLES = fB(1+ fe)lRANS +(1− fB)lLES (2.51)

Where fB, blending function is given by:

fB = min2exp(−9α20,1.0 (2.52)

Where α is equal to 0.25− dwhmax

The blending function fB is determine by the fast switch-ing mechanism profound inside the boundary layer, between pure RANS and LES modes,this transition is found in the range of 0.5hmax < dw < hmax wall distance as shown in Fig-ure 2.5. The main idea behind this function is to provide rapid transition between modes,for pure RANS mode it is equal to 1 while 0 for LES mode. Another function is defined as

Figure 2.5: Blending function profiles [38]

in equation 2.53 is called an "elevating" function. It is formulated to balance the excessivedecrease in modeled Reynolds stress (RANS), which is encountered in the region of inter-action between RANS and LES interface and thus treats the logarithmic-layer mismatch.

fe = max( fe1−1,0Ψ fe2 (2.53)

24


While the function fe1 is defined as:

fe1

(dw

hmax

)=

2exp(−11.09α2), if α ≥ 0

2exp(−9.0α2), if α < 0(2.54)

Since α = 0.25, hence clearly fe1 does not depends the solution but only depends on grid.Therefore this function works for RANS component as an elevating for blended length-scale given by equation 2.51 for RANS-LES. In transition zone when fB < 1, the functionfe = fB. While function fe2 is given by

fe2 = 1.0−max ft , fl (2.55)

In equation 2.51, for RANS component the intensity of elevating is control by the functionfe2 by utilizing function ft and fl , defined as:

ft = tanh[(c2

t rdt)3]

(2.56)

fl = tanh[(c2

l rdl)10]

(2.57)

Here turbulent νt and laminar ν viscosities are represented by subscript t and l respect-ively, while rdt and rdl are functions which are analogue to function rd and is expressedby equation 2.41. The parameters rdl and rdt will be around one laminar layer and in logregion of turbulent boundary layer, respectively. Where as ct and cl are constant para-meters defined for different models. The function fe2 depends on the solution since bothfunctions ft and fl are analogue to function rd and hence also depends on the solutionwhile in boundary layer both functions are around one and enforce functions fe1 and fe2

to become 0.

2.5.4 Hybrid branch of DDES and WMLES

For different kind of grid and simulations, the idea is to develop a method which willautomatically switch and select WMLES or DDES mode. A reformulation of lengthscale of DDES is necessary since with the current definitions of length scales for DDESor of WMLES this was not possible. The new definition is given by:

lDDES = fdlRANS +(1− fd)lLES (2.58)

25


While modified blending function is given by:

fd = max(1− fdt , fb (2.59)

Where fdt is given as:

fdt = 1− tanh[(8rdt)3] (2.60)

The coupled hybrid length scale for both branches DDES and WMLES is given now as:

lhyb = fd(1+ fe)lRANS +(1− fd)lLES (2.61)

It is pointed out that when flow is turbulent, the function fdt is around 1 since rdt << 1.Furthermore, fd = fB in order to lhyb approaches to lWMLES. While if the flow is laminar,then the function fe becomes 0 and lhyb approaches to lDDES.

26

Chapter 3

Software and Methodology

This section gives the overview of softer utilized in this thesis and the computational meth-odology used to achieve the objective. Since there are already much literature available touse such methodologies hence this chapter provide a more general overview to performLES simulations. Moreover, general structure and solver used are also addressed.

3.1 OpenFOAM

OpenFOAM is an object-oriented free open source software which refers to Open FieldOperation and Manipulation. It has basically a collection of libraries, OpenFOAM hasfunctionality to connect these libraries to solve CFD problems. OpenFOAM is written infinite volume code, hence equation are discretised in finite volume approach and the solu-tion is re-written to conform the approach[49]. The main philosophy behind OpenFOAMis to make it available for every user to modify and re-write any functionality in it (underGeneral Public License). Under GNU license, OpenFOAM Foundation ensures that itremain free for all and anyone can contribute to its development, this philosophy leads toits rapid development and hence it become most widely utilized open source CFD solver.Even though every year OpenFOAM foundation aims to provide new and improved ver-sion of OpenFOAM (till now version 3.0.1), still version 2.2 and 2.4 are widely used inacademic and industrial applications. This thesis address OpenFOAM version 2.4, it hassolvers for compressible and incompressible flows to simulate laminar or turbulent flows(RANS, LES, DES) and heat transfer problems. Another attraction with OpenFOAM isthat it the development team provides range of webinars, consultancy and training supportfor industrial users, however for academic enthusiast users it is no cost widely availableopen source solver which has a functionality to perform as the user desire.

The key feature of OpenFOAM is that its finite volume code is written in C++ lan-guage which is widely used among industry and academics. It also has a precompiled

27

3. Software and Methodology

utilities and solvers and also equipped with pre- and post-processing utilities which en-able users to perform complete simulation free of cost. Hence, the benefits mentionedabove has a potential to find users since it provides both technical and economical bene-fits.

3.1.1 Utilities and Solvers

OpenFOAM provides many utilities which helps in mesh generation, pre-precessing andpost-processing applications which are available to users in entire work flow. This sec-tion provides some of the utilities which aids in thesis, since the meshing is done withanother software hence it will be address in later part of the thesis. It is essential to bringout the fact for new users that OpenFOAM doesn’t provide GUI even though there arevarious GUI available for OpenFOAM but mostly are not endorsed by OpenFOAM de-velopers, the convenient way interact and communicate with it is via written commandsin a terminal window. In terminal the prospective user give the name of solver or utility toperform tasks and information for a particular utility can be found via entering a commant’-help’ in a terminal window. Some of the utilities are given in table 3.1.

Utility Task

blockMesh Official block mesher in OpenFOAMsnappyHexMesh Automatic mesh refiner for complex geometriescheckMesh Check mesh quality and report statisticsDESModelRegions Gives the volume percentage over which RANS

and LES is workingforceCoeffs Lift, Drag and Moment coefficientsprobes Specify the probe for samplingprobeLocations Show the probes locationsyPlus Report and calculate the y plus for wall patcheswallShearStress Report and calculate shear stress for wall patchesdecomposePar Automatically decompose the domain for parallel

processingreconstructPar Automatically reconstruct the decomposed domain

data

Table 3.1: Utilities in OpenFOAM

The solver utilised in thesis is the incompressible transient solver "pimpleFoam",which combines the PISO and SIMPLE algorithm together while maintaining the nu-merical stability even with higher time steps, this solver is used in cojuction with both

28


RANS and LES turbulence models hence for IDDES simulation it would be a perfectchoice. Besides the pimpleFoam solver there are many incompressible as well as com-pressible solver present in OpenFOAM 2.4, some of the common incompressible solverbased on case structure and set up are explained in table 3.2. For specific problem thesesolvers (computer executable) provide find the solution using the algorithms of a particu-lar solver.

Utility Task

icoFoam A incompressible solver for transient flows thatcan be applicable to low Reynolds number applic-ations like; small MAVs or UAVs, internal fluidsystem.

simpleFoam A incompressible solver for steady state flows util-ises SIMPLE algorithm and can be used for bothlaminar and turbulent flow by using RANS inbackground.

pisoFoam A incompressible transient solver utilising PISOalgorithm and can be used in conjunction with bothturbulence models RANS and LES.

pimpleFoam A incompressible transient solver utilising bothPISO and SIMPLE algorithms in conjunction.This solver can be used for larger time steps whileprovide the numerical stability during simulation.It can be used in conjunction with both turbulencemodels RANS and LES.

Table 3.2: Incompressible solvers in OpenFOAM

3.1.2 Case structure in OpenFOAM

Domain region (liquid or solid) or the system of equations are specifically connected tothe set up, case structure and design of an OpenFOAM simulation. This sub-section il-lustrate the case structure and configuration of incompressible IDDES simulation (shownin Figure 3.1). Inside case folder, there are three sub-directories which require to set upsimulation, in particular 0, constant and system. The user defines the boundary conditionsin 0 directory for simulation which contains one file each for IDDES variables, namely,pressure, velocity, Sub-grid scale viscosity and S-A turbulent variable. While the simu-lations requirements which remain same through out the simulation are defined in con-stant directory, it has sub-directories like polyMesh which contains the mesh information,

29


transportProperties file contains the fluid type, RASProperties contains the S-A closuremodel while LESProperties contains the LES model constants and turbulencePropertiescontains the information of type turbulence model (LES). The last and most importantfolder which decide stable or diverging simulation is the system folder, it contains atleastthree important files, i.e., controlDict file which contains the information of start and endtime as well as the time step size, fvSchemes file which contains the equation discret-ization schemes and last one is fvSolution which contains the information of numericalalgorithms to be used for solving system of equations.

Figure 3.1: Case Structure of incompressible IDDES simulation

3.2 OpenFOAM Discretization

As mentioned in previous section 3.1, OpenFOAM utilises Finite volume method forLES simulations. This section provides the overview of how equations are discretizedin OpenFOAM and which numerical schemes are used in thesis. Even though there are

30


some schemes which provide much better and accurate results but this is beyond the topicof research right now in this thesis. Some of the default simulation schemes are also usedin this research work.

The basic idea behind the Finite Volume Method (FVM) is to sub-divide or discretizethe domain in space. Since due to the need of solving the equation in time for unsteadysimulation, time-marching method is needed which makes the equations semi-dicretize tosolve the problem.

Figure 3.2: Finite Volume Method

A set of arbitrary shaped Control Volumes (CV’s) are needed for spatial discretization,and for each control volume a computational point P is defined (example, Figure 3.2).From the adjacent cell (N) to the current cell (P), a vector d connect the cell centres ofthe two, while A represents the normal area vector between the cells for common face.OpenFOAM defines it variables in this way which results in collocated grid.

As mentioned earlier, for unsteady simulation a temporal discretization is necessary,or in other words the time interval or time steps (∆t) are required time marching from theinitial conditions.

The task is to discretized the filtered Navier-Stokes equation (2.28) of incompressible,Newtonian fluid for LES simulation. After integration of control volume and time usingFinite Volume Method (FVM, yields:∫

V∇.udV =

∫∂V

dA.u = 0 (3.1)

31


∫ t+∆t

t

[ddt

∫V

udv+∫

V∇.(uu)dV −

∫V

∇.νe f f (∇u+∇uT )dv

]dt =

−∫ t+∆t

t

[∫V

∆ pρ

dV

]dt

(3.2)

The above equation is in second-order due to the presence of the diffusion term whichintroduced the second derivative. Since we now that in order to get accurate results, theorder of discretization of equations need to be higher than or equal to the second-order.Moreover, due to the second order spatial discretization, higher or second-order temporaldiscretization would results in much better and time accurate solution. Now by applyingTylor series to the transport quantities we wold get:

φ(x) = φP +(X−XP).(∇φ)P +ϕ

(|X−XP|2

)(3.3)

∂φ(t)∂ t

=φ(t +∆t)−φ(t)

∆t+ϕ(∆t) (3.4)

Sub-sections below provides the discretization for each term in governing equation, firstlyspatial discretization is discussed followed by temporal.

3.2.1 Spatial discretization

The general discretization integrals utilized for evaluation on control volumes are dis-cussed here, for more through knowledge one can refer to Jasak [50].

Volume Integral :∫

Vp

φ(x)dV ≈ φPVP (3.5)

Sur f ace Integral :∫

fφdA = φ f A f (3.6)

Divergence Integral :∫

VP

∇.φdV ≈∑f

A f .φ f (3.7)

32


Gradient Integral :∫

VP

∇φdV ≈∑f

A f φ f (3.8)

Convective term discretization

The Divergence integral (equation 3.7) is applied to the convective term for discretization.It gives:∫

VP

∇.(uφ)dV = ∑f

A.(uφ) f = ∑f(A.(u)φ f = ∑

fFφ f (3.9)

Where the volume flux through face is given by F = A.U f . To get values at faces, secondorder interpolation is required between the two neighbouring cell values P.

Diffusive term discretization

Discretization for diffusive term is given by:∫VP

∇.(ν∇φ)dv = ∑f

A.(ν∇φ) f = ∑f

ν f A.(∇φ) f (3.10)

By using an interpolation, the scalar term ν f can be easily found out. While the termA.(∇φ) f is highly depended on the mesh, like in equation 3.10, the face gradient φ fororthogonal mesh can be defined while the vectors A and d are parallel to each other forsuch kind of meshes.

A.(∇φ) f = |A|φN−φP

|d|(3.11)

Here P and N are neighbour cells. For non-orthogonal meshes, the equation 3.11 is notvalid any more for second order accurate equations. An additional term is introduceswhich represents the non-orthogonality.

A.(∇φ) f = |Ad|φN−φP

|d|︸︷︷︸orthogonal

+ A∆.(

∇φ

)︸︷︷︸

non−orthogonal

(3.12)

The instability can be arise if mesh non-orthogonality is very high since it will createnegative coefficients, which results in reduced accuracy due to limited correction. That iswhy a user should always aim for limited non-orthogonality while generating a mesh.

33


3.2.2 Time discretization

Temporal discretization also have various ways for discretizing time just like the spatialdiscretization. The transport equations need to be second-order accurate which is the mostcritical thing to remember here. One example of the temporal discretization is the Cranck-Nickolson scheme which is the expensive and undesired in this research but provide mostaccurate result, another option is the second order backward differencing scheme.

Backward Differencing

The temporal discretized equation is given by 3.13, which provides second order accuracyby utilising three time steps to achieve it.

∂φ

∂ t=

32φ n+1−φ n + 1

2φ n−1

∆t(3.13)

While perfoming the simulation, one try to minimize the turncation errors, still there aretime’s when some errors are introduced due to small variations in face fluxes. Thesekind or errors has some serious effects in LES simulation since they cause an additionaldiffusion. The main problem arises when these errors go beyound the sub-grid diffusion.

Due to this reason, the cell face Courant number is always try to set below 1 in orderto maintain stability.

CFL =u f .n|d|

(3.14)

This is the reason behind using very small time step in this research work since it resultsin very small temporal diffusion error.

3.2.3 Momentum-Pressure Coupling

Since the thesis consider the PIMPLE algorithm which utilises the merger PISO-SIMPLEalgorithm and hence it uses the idea of SIMPLE mode (relaxation), since we run sev-eral simulations to obtain the best inner-outer corrector loops (explained in section 3.3)therefore current simulation doesn’t utilises the relaxation factor idea instead we set thetolerance of 10−6 by observation of test simulation results which should be enough fortransient simulations.

In PIMPLE algorithm, the momentum equation is calculated first after that the pres-sure equation is calculated and this new pressure is used to re-calculate the momentumequation which in turns give new pressure by using this new momentum (explained inschematic overview of OpenFOAM in Figure 3.4).

34


Thi final Navier-Stokes equation is given below: (full derivation is not presented here)

apup = H−∑f

Ap f (3.15)

∑f

A

(1ap

)f

(∇ p) f = ∑f

A.

(Hap

)f

(3.16)

One can refer to literatures [36] and [50] for full derivation of above equations. Here ap

represents the set of coefficients depending on up. While H vector consist all the termslike source and convection part except the pressure term. Pressure Implicit with Splittingof Operators (PISO) algorithm coupled with SIMPLE advantage is preferred for transientsimulations, like in this research work.

3.2.4 Implementation of Turbulence Model

Since we are dealing with Spalart-Allmaras turbulence model in this research work whichwas already discussed in section 2.3, it equations follow the classic disctretization methodsimilar to the one which are discussed so far, hence recalling Spalart-Allmaras equationonce again:

∂ ν

∂ t+∇.(ν u)− 1

σ∇.((ν + ν)+∇ν) = cb1Sν︸︷︷︸

production

+1σ

cb2(∇ν)2︸︷︷︸transport

−cw1 fw

(ν

d

)2

︸︷︷︸dissipation

(3.17)

where table 2.1 already represents the constant values and expression for functions. Inorder to increase stability of solution, we drive the production term as an explicit functionof νn−1. Given by:

νproduction = cb1

[|∇× u|+ νn−1

κ2d2 fv2

]ν

n−1 (3.18)

Discretizing transport term as:

νtransport =cb2

σ∇nun−1.∑

fAν f (3.19)

35


Discretizing destructive term as:

νdissipation =−

[cw1 fw

νn−1

d2

](3.20)

Where n−1 refers to previous time step.

3.3 Simulation Overview

This section emphasis the procedure of solving equation for LES simulation using discret-ized Navier-Stokes equations which are already discussed. Figure 3.3 gives the generaloverview of simulation, where Final Residual is the tolerance. Steps involved in PIMPLEalgorithm are briefly discussed below:

1. For initializing the flow field and to start LES simulation, a RANS solution havingvelocity, pressure, eddy viscosity and face fluxes is utilized.

2. The next step is to use previous time step for updating turbulence properties.

3. The momentum and pressure coupling is solved thrice using PIMPLE algorithmsince nOuterCorrectors or nCorrPIMPLE is set to three (external loop correctors,which are set to 3).

4. The pressure within the PIMPLE loop is corrected twice using PISO algorithmsince nCorrectors or nCorrPISO is set to two (internal loop corrector, which is setto 2).

5. The velocities are solved using previous flow field equation. A solver that re-quire a smoother smoothSolver applicable to symmetric matrices is applied. WhilesymGaussSeidel "Symmetric Gauss-Seidel" is used as a choice of smoother.

6. The pressure equation are solved using PCG "Preconditioned Conjugate Gradient"solver which is applicable to symmetric matrices. While for a symmetric precon-ditioner "Diagonal Incomplete-Cholesky" DIC is used to improve computationalefforts of simulation.

7. To sustain the convergence for pressure equation, the non-orthogonal correctornNonOrthogonalCorrector which means the pressure field is solved more oftenwith new calculated value is set to 1. Due to it we can see the pressure equation

36


is solved one more time in PISO loop. It is generally set to either 1 or 0 for LESsimulations.

∇2 p= f (U,∇p)→ pnew→∇

2 pnew = f (U,∇pnew)→ till nNonOrthogonalCorr is reached

8. The number of outer and inner correctors are chosen in such a way that a certaintolerance (in this case it is 10−6) on the quantities is achieved. For this researchcase the outer corrector 3 and inner corrector 2 with non-orthogonal correctors 1 isfound sufficient to achieve certain tolerance in each iteration.

Figure 3.3: General overview of log file

3.4 Solvers, Smoothers and Preconditioners

Solvers, Smoothers and Preconditioners can save a lot of computational efforts when usedproperly. Figure 3.1 represent sub-directory fvSolution under system directory whichcontains these settings and conditioners. In Appendix B, applied setting for PIMPLEalgorithm is provided. Generally considered most computational demanding equation isthe pressure equation among others. The Diagonal Incomplete-Cholesky (DIC) was selec-ted as a preconditioner for pressure equation. According to CFDdirect [51], Geometric-algebric multi-grid solver (GAMC) would provide much accurate solution and increasethe speed by corsen/refine mesh in stages but during this research work it is found outthat DIC is give more accurate results while GAMC is diverging (reason behind this is

37


Figure 3.4: Schematic overview of OpenFOAM simulation

beyond the scope of this thesis). Moreover it is realised that DIC speed up the solutionand the tolerance limit is achieved in much earlier than GAMC. The Preconditioner Con-jugate Gradient (PCG) solver is utilized for solving pressure equation. The other termsare solved using smoothSolver in conjunction with Symmetric Gauss-Seidel (symGauss-Seidel) as a smoother for symmetric matrices.

38

Chapter 4

Computational Methodology

In previous section we have presented the physics and modelling approach (chapter 2)used in this thesis which is very small in comparison to the functionality of DES solutionmethod; fortunately this part is widely explained in various literatures, text books andpublications, hence only the important portion of DES simulation needed (e.g., filteredNavier-Stokes equations, momentum-pressure coupling and solver used for resultant matrices)in thesis is presented in previous sections. However this section focus entirely on simula-tion set-up by firstly introduction of domain selection (including boundary and initial con-ditions) then time-step selection followed by mesh generation strategy and finally meshdependent study.

4.1 Computational domain

The unsteady simulations were performed in OpenFOAM with a sphere diameter D = 1,whose centre is located at coordinate (0, 0, 0). The computational domain (Figure 4.1)extends 4.5 diameters in upstream direction as well as in radial outward direction from thecentre of sphere which corresponds to a blockage (area) ratio of approximately 1.2%. Theblockage ratio here is defined as the ratio of sphere frontal (Asphere = 0.785398163) area tothe test section area (Adomain = 63.62). This is proportional to the sphere is hanging insidethe circular pipe with a diameter of D = 9D (here D always represent the diameter ofsphere). Here, the rate of decay of perturbation away from the body determines the extentof the upstream and radial computational domain. Since it is well known [52] that rate ofdecay of perturbation due to the presence of sphere is 1/r3, here ′r′ represents the distancefrom centre of the sphere. For two-dimensional bluff bodies (i.e. cylinder) the rate ofdecay of perturbations is 1/r2, for three-dimensional sphere case this factor is ′r′ timessmaller than the corresponding one. Accordingly, the velocity at x = −4.5 is just 0.1%different form the initial velocity or the free-stream velocity, as per potential hypothesis.

39

4. Computational Methodology

The same contention additionally holds in the radial direction. Moreover, the blockagefactor of the order of 1% are viewed as insignificant when performing simulations orexperiments as already mentioned in few literatures [2, 14]. While computational domainin downstream direction extends 25 diameter form the centre of sphere. As reported byTomboulides and Orszag [4], simulation by performing domain length of 20 diameter indownstream direction (with same outflow boundary condition) didn’t have any effect inupstream direction. Still 25D is chosen due to the reason to have atleast 3 to 4 vorticalstructures shedding behind the sphere within the domain, it will increase the grid elementsdrastically which indeed increase the computational efforts.

Figure 4.1: Schematic of computational domain for simulation

Some mesh were generated with ICEM CFD during initial investigation representedin Figure 4.2 with more then 1 million cells for both structured and unstructured mesh butthe simulation diverges every time after converting mesh in OpenFOAM format by usingfluent3DMeshToFoam. It is observe that after conversion the domain size increased in allthree direction which can lead to the divergence due to no element present in the increasedlength. Further investigation is beyound the scope of current thesis work (remember thatOpenFOAM is an open source code which can have bugs moreover the version used forthis research is 2.4 which is older then the 2016 version 3.0.1). Further details about meshis presented in section 4.4.

4.2 Initial and Boundary conditions

The boundary conditions in OpenFOAM are provided in files for initial velocity, pres-sure and turbulent fields in 0 directory (Figure 3.1). Four patches are generated duringmeshing procedure namely; sphere patch, inlet patch, outlet patch and sides (representsthe radial side of the computational domain). Numerous boundary conditions are imple-ment on final set-up mesh according to simulations and experiments performed earlier,since OpenFOAM uses different names for defining boundary conditions and due to lack

40


(a) Structure mesh with 1.4 million elements

(b) Unstructured mesh with 1.8 million elements

Figure 4.2: ICEM CFD mesh

of detailed documentation on it, the best and the most suitable boundary conditions areimplemented. They are described in detailed below:

1. Sphere patch: The velocity field is set as fixedValue with value of (0,0,0) whichas the name implies, the velocity over sphere patch is kept fixed at 0 m/s in alldirections. The pressure field is set as zerogradient, which represents the normalgradient of pressure is zero or Neumann boundary condition. while fixedValue of′0′ for turbulent fields or Dirichlet boundary condition.

2. Inlet patch: The Dirichlet boundary condition or fixedValue with a uniform value offree-stream in x-direction only U∞ =(68.058,0,0)m/s is specified. The free-streamvelocity is calculated by using the Mach number of M∞ = 0.2, while dynamic andkinematic viscosity is calculated by using the Reynolds number formula (equation4.1), taking free-stream conditions for density (ρ∞ = 1.225kg/m3), pressure p∞ =

101325 pascal, Reynolds number Re∞ = 10,000 and diameter of sphere D = 1m.

Re∞ =ρ∞U∞D

µ∞

(4.1)

While for initializing pressure field the Neumann type boundary condition is pre-

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scribed hence zeroGradient is used. Furthermore, turbulent field is calculated usingturbulent intensity of 0.1% which gives the value 0.071444734, a fixedValue is setusing this.

3. Outlet patch: Neumann and Dirichlet boundary condition for velocity and pressurefield respectively, is prescribed in literature [53]. While in OpenFOAM is found thatNeumann type boundary condition for both fields (pressure and velocity) generatesnumerically induced oscillations which can easily be avoided by using inletOutletboundary condition for velocity and turbulent field (using same values as for inletpatch). Hence for a daring experiment this boundary condition is used and foundsuitable in current research work, while the pressure field is set as fixedValue witha value of ′0′.

4. Sides patch: It represents the cylindrical domain around the sphere at it is set assymmetry boundary condition for every field which is according to OpenFOAMguide represents the slip wall for non-planer patches.

Initial Condition ValueRe∞ 10,000M∞ 0.2ρ∞ 1.225kg/m3p∞ 101325 pascalµ∞ 8.336×10−3 kg/msν∞ 6.805×10−3 m2/s

nuTilda 0.071444734D 1m

Table 4.1: Initial values for simulation

4.3 Time step selection

Since this research work used PIMPLE algorithm whose main attraction is stability ofsimulation for higher time steps, which can be used to advance in time much faster forunsteady simulation without compromising the stability of solution but some importantphysics can be skip if we use higher time step. Considering the total time required forrunning the simulation it would be expected to use higher time steps but in this thesis weare using much smaller time step to capture the effect of low-frequency fluctuation whichis the main task of thesis. Hence a time step is selected using the convection time which

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50 times less then it. The convection time is given by time required by the flow to passthe sphere which is equal to tconvection = D/U∞, calculated as 0.01469. Therefore the timestep selected to carry out simulation is 0.00029, ∆t = 0.00029. Moreover the maximumCourant Number, CFL is set to 0.5 for whole simulation, if the mean Courant numbervalue exceed 0.5 any time in simulation, the OpenFOAM reports divergence.

Considering the low frequency fluctuation, the time required to run entire simula-tion should be long enough to capture such low frequencies (more explanation is re-fchap:results), hence the current simulations are run for 9.048 seconds (see A) whichcorresponds to 120 vortex shedding cycles. According to the knowledge of literatures,this is the longest simulation for flow over sphere at this Reynolds number). For calcu-lating the vortex shedding, the experimental value of vortex shedding Strouhal numberfvs = 0.195 is used, which gives the frequency of one vortex shedding equals to 13.2713therefore, corresponding time for one vortex shedding is equal to 0.07535. Using thesecalculations it is found out that for one vortex shedding almost 260 time steps are requiredwhich should allow enough time to capture the low-frequency content accurately.

Mesh Number of CPUs Clock time RANS volume% LES volume%Coarse mesh, M1 32 34 : 47 : 30 ∼ 1.4Days 37.2 62.8

Medium mesh, M2 32 121 : 04 : 00 ∼ 5Days 30.7 69.3Fine mesh, M3 32 195 : 13 : 01 ∼ 8Days 35.1 64.9

Table 4.2: Summary of simulation time and DESModelRegions for all three mesh

4.4 Mesh Generation

The most challenging part faced during this process is the post-processing, since it notonly require a complete understanding of how OpenFOAM work but also a requires asolid understanding of flow physics (which is already discussed in 1) work for flow oversphere case. Therefore the most crucial part of pre-processing is to generate a suitablemesh which represents the flow physics as close as to the nature. Since a mesh generationis a time consuming process for complex geometries even with commercial software’slike ICEM CFD. Some initial meshes were generated using this software package whichgenerate very good results in Ansys FLUENT but diverges in OpenFOAM after convert-ing mesh in it’s format. The potential reason behind it is already mentioned in section 4.1.Hence ICEM meshes are neglected for current research work.

OpenFOAM has its own mesher snappyHexMesh which is considered very powerfulmeshing tool among OpenFOAM users. Even though it generates high quality meshautomatically, still it is not very popular among community due to two reasons, firstly

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it requires significant amount of RAM memory for implementation and secondly, fromusability viewpoint it is consider very cumbersome for generating inflation layers for highquality boundary layer grids. Therefore another open source software is consider in thisthesis work which is explained in next section.

4.4.1 Procedure

Due to the problem faced with commercial meshing package, ICEM CFD and due to thecumbersome nature of OpenFOAM own mesher snappyHexMesh, another open sourcesoftware cfMesh is used to generate grid for simulations,it is a mesher for OpenFOAMprovided by Creative Field company. It is a cross-platform library for automatic meshgeneration that is built on top of OpenFOAM and it is compatible with all versions ofOpenFOAM [54].

The Cartesian Mesh is generated using cfMesh, which offers structured mesh as wellas Unstructured Tetrahedral mesh, since Tetrahedral meshes are well-known to not beoptimal for OpenFOAM simulations among users (this statement wasn’t investigated dueto time constrain). Hence a premilary test with Structured as well as Cartesian mesh wasinvestigated to make the decision for final type of mesh to be run. Since purely Structuredmesh are better, at the condition that cells have a low skewness and/or are aligned with theflow. While, Cartesian meshes ensures an optimized mesh quality (zero skewness, aspectratio equalling one) in the quasi-totality of the domain.

Structured mesh are known to converge better than tetrahedral meshes. It was shownthat OpenFOAM could give bad results when cells are skewed or not well aligned with themesh. The best illustration in our case is that the potentialFoam solver, used to initializedthe velocity field, failed to give physical results on a Structured mesh, when it worked fineon the Cartesian mesh. The main problem to create a high quality structured mesh in ourcase is the presence of the sphere, which prevent cells to be aligned with the flow closeto it. Here, an illustration of this after few iterations on the structured mesh Figure 4.3a.For all these reasons, Cartesian mesh type is better than structured in our case. TheCartesian mesh generated using cfMesh generated a 3D mesh which containd mostlyhexahedral elements while in transition region, polyhedral elements are present betweenthe elements of different sizes. The procedure is detailed below, it worth mentioning herethat Carterian mesh generated, introduced automatically one boundary layer which canbe refined further.

1. Geometry: The tool used to prepare the domain’s geometry (Figure 4.1) was Sa-lome, which provides a CAD module.

2. Patch Creation: Using Salome all patches were created separately namely; sphere,inlet, outlet and sides. Figure 4.4 shows the patch names.

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(a) Global view

(b) Zoomed view

Figure 4.3: Velocity magnitude contour for structured mesh generated using cfMesh

Figure 4.4: Computational domain geometry

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3. STL file preparation: Each patch was then exported in STL format. The name ofthe patch was manually added into each STL file (Figure 4.5 shows the exampleof sphere patch file). Finally, all STL files were concatenated (copy-pasted) into asingle one.

Figure 4.5: Patch file example

4. File conversion: The STL input file was converted into FMS format using commandsurfaceToFMS, which is recommended in the cfMesh documentation and allowsthe user to define OpenFOAM patch types (like patch, wall, symmetry, empty, etc.)before meshing instead of modifying them in the constant/polyMesh/boundary fileafter each mesh generation. Figure 4.6 shows the heading of the FMS file whichwas used:

Figure 4.6: File format conversion from STL to FMS format using command surfa-ceToFMS

5. meshDict file: Finally two mandatory field are edited in meshDict file (see, Ap-pendix D) to start the meshing process using cfMesh:

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surfaceFile: It points to a geometry file. The patch in geometry file shouldmatch to the case patches. The geometry file name used is input.fms

maxCellSize: It represent the default cell size used for the meshing job. It is themaximum cell size generated in the domain.

A max cell size (maxCellSize) was defined (here 0.25m, leading to around onemillion cells in the mesh). Then, different volume refinements made of spheresand cones and associated with given refinement levels (additionalRefinementLevels)were created, which is explained in next sub section.

6. Mesh generation: Lastly, generation of the mesh using the cartesianMesh commandfrom cfMesh, which generates Cartesian mesh for whole domain and also generatesone boundary layer over sphere patch.

4.4.2 Refinement

Refinement is another crucial step in mesh generation process to capture the flow wherethe data has to be collected (or area of interest) while maintain the limit of number ofelements to save computational efforts. Since the mesh generated using cartesianMeshcommand is very coarse to carry out simulation hence three kind of refinements are usedto get final mesh. Moreover using refinements three other meshes were generated to carryout mesh study. The detailed process for mesh refinement is described below (Table ??shows values for all three mesh):

1. Volume refinement: The volume refinement is made using the dictionary objectRe-finment which have few objects like sphere, boxes, cone, etc. for specifying refine-ment region. The sphere and two cone objects are chosen here for the purpose. Thesphere object with center at (0, 0, 0) and with radius of 1.25 while cone1 and cone2with starting point at 0 and end point at 3 and 12 respectively. The additionalRe-finementLevels dictionary within the object dictionary used for further refinement tostudy the mesh independence. This dictionary is used to specifies the number of ad-ditional refinement levels compared to maxCellSize. Figure 4.7 shows the volumerefinements made.

2. Patch refinement: Using the localRefinment dictionary which allow local refine-ment regions at the boundary, the sphere (patch) surface was refined with up to agiven distance from it (refinementThickness).In the following view (Figure 4.8), thesurface refinement on the sphere’s wall is well visible:

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Figure 4.7: Volume refinement using sphere and cone objects

Figure 4.8: Patch refinement for sphere

3. Prismatic Layers: There was finally defined prismatic layer parameters on thesphere’s patch using boundaryLayers dictionary. The first layer thickness was cal-culated using the formula:

y =y+µ

ρu∗(4.2)

where u∗ is defined as friction velocity and is given by:

u∗ =√

τw

ρ(4.3)

While the shear stress τw is calculated by:

τw =C f .12

ρ∞U2∞ (4.4)

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and skin friction coefficient C f is calculated using formula

C f = 2[log10(Re)−0.65]−2.3 (4.5)

Using above equation, the first layer thickness (maxFirstLayerThickness) was foundto be y = 1.085e− 3m for y+ = 1, so we used y = 1e− 3m in the meshDict file.The value of 1.2 for the ratio between adjacent prismatic layer thicknesses (thick-nessRatio) is an usual one known to give good results. Then, the number of layers(nLayers) of five was chosen so that the total thickness never exceed the local cellsize. Indeed, cfMesh doesn’t allow prismatic layers thickness to exceed the localcell size and stretches them when there are too many layers. Limiting the numberof layers ensures the first layer thickness is conserved even when the cell size isscaled down, for example during a mesh sensitivity study.In Figure 4.9, one can seethe prismatic layers on the sphere’s surface:

Figure 4.9: Prismatic layers using boundaryLayers dictionary

Figure 4.10 shows, a non-interpolated plot of the velocity field at t = 9s shows that themesh refinement allowed to well catch the recirculation area and the wake of the sphereup the to end of the area of interest while limiting the mesh fineness in the rest of thedomain.A contour of the pressure field (Figure 4.11) confirms also that the refinementwas made in the areas of highest gradients.As expected, the y+ value (Figure 4.12) wasbelow one everywhere on the sphere, with a max value of around 0.75.

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(a) Global view

(b) Zoomed view

Figure 4.10: Velocity contour for mesh validation

Figure 4.11: Pressure contour for mesh validation

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4.5 Mesh dependent study

As mentioned in the previous section, three meshes are generated for mesh independencestudy. The coarse mesh is generated with 1.1 MCVs, medium and fine mesh has 4 and6 MCVs respectively, for all the cases the first element height is selected as y = 1e−3maccording to y+= 1. The mesh refinement was carried out by utilizing the additionalRe-finementLevels dictionary by cfMesh in the refinment regions only while the global max-imum element size was remain unchanged. Table 4.3 represents the statistics obtained byaveraging 540 tD/U time units since initial 75 tD/U time units are used to pass the trans-ition stage which can be seen in Drag Coefficient graph,represent by large fluctuationswhich damped out after 1.1sec or 75 tD/U time units.

Mesh MCVs fvs Cd Cl φs Error%Coarse Mesh, M1 1.1 0.181 0.387 0.0035 93.6 3.3%

Medium Mesh, M2 4 0.211 0.392 0.0007 84.8 2%Fine Mesh, M3 6 0.203 0.394 0.0004 84.1 1.5%Experimental − 0.195 0.4 − 82.5 −

Table 4.3: M1, M2, M3 representa the coarse, medium and fine mesh while experimentaldata for vortex shedding Strouhal number and separation angle from Achenbach [1, 2]and drag coefficient from Schlichting [55]

All meshes run for the same time period (9.048) or for 120 vortex shedding. Thedrag coefficient Cd , separation angle φs and the vortex shedding frequency St agrees quitewell with the experimental data, while the biggest difference is observe with the coarseto fine mesh where error in between experimental and simulation value (drag coefficient)decreases from 3.3% to 2%, on the other hand difference between the medium and finemesh is just 0.5%. Figure 4.15, shows the FFT analysis for Probe location-9 (discussin next chapter 5). The vortex shedding Strouhal number for coarse, medium and finemesh is 0.181, 0.211 and 0.203 respectively, while the experimental value for Reynoldsnumber 10,000 is 0.195. Remember that all the values represented in this thesis are non-dimensionalized, this is why the frequencies are represented in non-dimensional Strouhalnumber. One interesting observation about FFT analysis noted here that energy contentof vortex shedding is quite high for coarse mesh in comparison to other two.

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Figure 4.12: Y+ contours; (a) Course Mesh, M1 (b) Medium Mesh, M2 (c) Fine Mesh,M3

Figure 4.13: Cut plane and Zoom view; (a) Course Mesh, M1 (b) Medium Mesh, M2 (c)Fine Mesh, M3

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(a) Lift Coefficient

(b) Drag Coefficient

Figure 4.14: Time history of Lift and Drag Coefficient plots for entire simulation timet=9.048, showing transition stage passes after 75D/U time units or 1.1sec.

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(a) Coarse Mesh, M1

(b) Medium Mesh, M2

(c) Fine Mesh, M3

Figure 4.15: FFT analysis at Probe location-9 showing non-dimensional vortex sheddingfrequency for (a)M1=0.181 (b)M2=0.211 and (c)M3=0.203

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Chapter 5

Results and Discussions

The simulation have been started with initially homogeneous flow field. As already dis-cussed with the Drag coefficient plot (figure 4.14b), some large fluctuations end before75 tD/U time units or 1.1 sec. Hence in order to ensure that the flow becomes three di-mensional and wake behind sphere achieve turbulence after going through transition, wehave advanced the flow field in time for initial duration of 75 tD/U time unites. It ensuresthat the initial transition is washed out so that data is collected afterwards. The data hasbeen collected for approximately 615 tUD time units which give about 105 vortex shed-ding cycles. By comparing with the vortex shedding, this is by far the longest simulationfor sphere at Re = 10,000. This long simulation time integration ensures that convergedstatistics have been obtained moreover the large time span provides data to analyse theexistence of low-frequency fluctuations in vortex formation region. However it is worthto mention here that the statistics presented in this thesis have been averaged in span-wisedirection both in time and space.

5.1 Frequency spectrum analysis

Figure 5.1: Location of computational probes and lines

Several probes are set up using Probe utility in OpenFOAM to carry out single point

55

5. Results and Discussions

measurements. Sine the exact locations of probes doesn’t mention in any previous experi-mental or numerical simulations hence total 11 probes are set up initially for investigationwhile results are presented for only those probes where results can be extracted, for thesake of brevity all probes and line from where statistics have been collected are presentedin Figure 5.1. The location of the probes are as follow P1 and P8 are at the location wheretransition to turbulence is suppose to occur or with-in the shear layer; P2, P3 and P9 arejust after recirculation bubble closer; P4, P5 and P10 are in the wake of sphere whileP7 is at the sphere base (table 5.1 shows the locations of all the probes during computa-tion). Moreover lines L1, L2, L3, L4, and L5 are located at locations 0.2, 1.6, 2, 3, and 5respectively.

Probe Number Probe LocationP1 x/D = 1.0, r/D = 0.6P2 x/D = 2.0, r/D = 0.6

FP3 x/D = 1.7, r/D = 0.59P4 x/D = 3.0, r/D = 0.6P5 x/D = 5.0, r/D = 0.5P6 x/D =−0.171, r/D = 0.465P7 x/D = 0.5, r/D = 0P8 x/D = 1, r/D = 0P9 x/D = 2.0, r/D = 0

P10 x/D = 5.0, r/D = 0P11 x/D = 10, r/D = 0

Table 5.1: Probes locations used initially for finding the correct positions to captures themain frequencies associated with the fluctuations

In order to gain insight the turbulent wake structure of the flow behind sphere, thesingle point measurements are carried out for the dominating frequencies in the flow fieldvia FFT(Fast Fourier Transform) analysis with Tecplot for the turbulent velocity com-ponents (radial velocity or cross-stream velocity components). The resulted frequenciesare presented in a non-dimensional Strouhal number, St = f D/U while the the amplitudeis also non-dimensionalised using Amplitude = Amp/U2D, FFT analysis are presentedhere for only finest mesh M3.

Figure 5.3 presents the instantaneous radial velocity and its frequency spectra at probesP1, P2, P9 and P4 over a period of 540 tU/D (which is about 105 vortex shedding cyclec)by using Fast Fourier Transform (FFT). It can be easily observe, depending on the probeslocation in shear layer or in wake that the radial velocity fluctuations contributes differentfrequencies at different locations. Indeed, the probe just after the transition region P2(figure 5.3(d)) at location X = 1.0, r/D = 0.6 records the dominant peak associated with

56


the frequency of large scale vortex shedding St = fvs D/U = 0.205, this frequency peakis present at all probe locations considered here. However difference between recordedvortex-shedding frequency and experimental (St = 0.195) result of Achenbach [1] is about5%, it is worth to mention here that there is wide range observed for this frequency peakin different experiments and numerical simulations [56, 29, 31] ranging from 0.18−0.20with the same Reynolds number. Moreover the magnitude of vortex-shedding frequencydecreases greatly with the increase in the distance form the sphere, as can be seen in fig-ure 5.4 for locations P9 (x/D = 2.0, r/D = 0) and P4 (x/D = 3.0, r/D = 0.6). However,Rodriguez et al.[31] also observe this phenomena at Reynolds number 3700 but the de-crease in energy content of this peak is comparably less. This difference is due to thevortex shedding occur at comparably closer to the sphere surface at transition region ofabout x/D = 1 − 1.2 for Reynolds number 10,000 while for Reynolds number 3700 itis x/D = 1.8 − 2.6. Due to this earlier transition to turbulence, it is expected that therecirculation bubble would be shorter than that at Reynolds number 3700.

Figure 5.2: FFT analysis of the streamwise velocity fluctuation at probe P9 (x/D =2.0, r/D = 0)

Another frequency peak known as Kelvin-Helmholtz instability is recorded in additionto the fvs or the vortex shedding frequency, this secondary frequency peak is associatedwith the separating shear layer at fKH D/U = 0.700 which is recorded only at the onelocation P1 (x/D= 1.0, r/D= 0.6) in the region where transition to turbulence is supposeto occur(figure 5.3(b)). At mentioned earlier, at this Reynolds number the shedding isweaker than 3700 moreover as also observed by the Bakic [16] in his experiment that the

57


higher frequency associated with the separating shear layer is not a clear frequency, thatis besides the dominant instability frequency there is also a presence of sub-harmonic ofthis frequency, this can be seen in the present result also where its second harmonic isrecorded at comparably lower frequency of fKH D/U = 0.641. The dominant shear layerinstability is also predicted by Rodriguez at fKH D/U = 0.72 and in some experiments[3, 14] also, our results are in fair agreement with their results. These unsteadiness canbe distinguish as fluctuation of high frequencies of velocity components (radial velocity)as can be easily observe form figure 5.3(a) and figure 5.3(c), i.e. as we move downstreamform the sphere surface the magnitude grows comparably. Such behaviour have also beenobserved in previous experiment by Prasad and Williamson [57] for flow over circularcylinder which studied the shear layer instabilities and also observed with simulations forflow over sphere at Reynolds number 650 and 1000 using DNS by Mittal and Najjar [5]and Tomboulides and Orszag [4] in 1999 and 2000 respectively, followed by Rodriguezet al. [7] in 2011 for DNS simulation at Reynolds number 3700. The observed highfrequency is more consistent to the results of Rodriguez et al. while there is a differencewith results obtained by Sakamoto and Haniu [3] who reported it between 0.97−1.22.

In addition to the small scale frequencies fKH and large scale frequencies fvs there isanother peak besides these two at much lower frequency than the large scale vortex shed-ding frequency which is captured at all probe locations. The value of this observed lowfrequency is fm D/U = 0.038 which corresponds to ∼ 5.1 vortex shedding cycles. Largesampling period for measuring such low frequencies is required which should be largerthan few time unites than 1/( fm D/U) = 26, in order to obtain clear footprint in frequencyspectrum. Rodriguez et al. suggest that at least 3 or 4 such periods should be recorderhowever with our time integration we should obtain almost 6 of such behaviour ideally.The most important feature observed with low frequency peaks is that, this phenomenonis taking place within formation region since it is recorded in the recirculation region andin the shear layer. The low frequency fluctuation is recorder in earlier simulations also forother bluff bodies like circular cylinder, flat plate and circular disc [58, 59, 60]. Howeversimilar to the large scale vortex shedding and small scale shear layer instability, the amp-litude of low frequency is much lower than other bluff bodies moreover it occurs at muchhigher frequency in comparison to circular cylinder almost 5 time, therefore it is possibleto run simulations for much lower time in comparison to other bluff bodies and still able tocapture low frequency properly. This frequency is also measured by Tomboulides and Or-szag [53] at Re= 500 and by Rodriguez et al. [31] at Re= 10,000 and 3700. Tomboulidesand Orszag recorded its value of fm = 0.045D/U but failed to solve its frequency due toinsufficient temporal data however Rodriguez et al run their simulation for almost 30vortex shedding and recorder the value of fm = 0.0178D/U and fm = 0.033D/U forRe = 3700 and 10,000 respectively. This thesis results for low frequency fluctuation is

58


(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 5.3: Time history and FFT analysis at different locations: (a,b) radial velocity andFFT of it at probe P1, (c,d) radial velocity and FFT of it at probe P2, (e,f) radial velocityand FFT of it at probe P9, (g,h) radial velocity and FFT of it at probe P4

in more consistence to the Rodriguez et al. results. Another group did the simulation forsphere at subcritical and super-critical reynolds number, Constantinescu and Squire [30]but couldn’t able to resolve low frequency fluctuation due to very short sampling period(∼ 10 vortex shedding cycles).

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(a) (b)

Figure 5.4: Energy dissipation in downstream of sphere wake (a) Probe location P9(x/D = 2.0, r/D = 0), (b) Probe location P4 (x/D = 3.0, r/D = 0.6)

While Tomboulides and Orszag attribute this frequency to the irregularly rotation ofthe separation point, however with other simulations focused on low frequency fluctu-ations had pointed out that this frequency is the attribute of periodic shrinkage and en-largement of recirculation region. Figure 5.2 show the FFT of stream-wise velocity fluc-tuation at probe P9 (x/D = 2.0, r/D = 0), it records the dominant peak of low frequencyat fm D/U = 0.038 which is the same value as recorder earlier specially on probe locationP2 (x/D = 2.0, r/D = 0.6). Probe P2 is located just downstream of recirculation bubblehence it is suppose to capture the pumping motion or recirculation region. Thus thereis more probability of shrinkage and enlargement of recirculation zone due to this lowfrequency than to the rotation of the vortex separation point.

One approach to investigate if two signals are associated with each other is by ana-lysing the cross-correlation between them. This ought to give a measure of the rate atwhich information of one signal affected by the other signal. For this research work, cor-relations between the pressure at probe P2 (which is located just after the closer of meanrecirculation bubble) and fluctuating streamwise velocity in wake centre line at probe P9have been utilised to quantify the relation. When this probe register positive value at anyparticular time period during simulation, this is demonstrative of the shrinkage of recir-culation region, however when it register near to zero value or negative value it representsenlargement of this region. Figure 5.5(a) and figure 5.5(b) represents the time series ofquantities for both probe locations, together with partially averaged signals for every 2vortex shedding cycles. If the quantities at both probe are correlated, then as per expecta-tion we would have high consistency between them with some frequency. The correlationcan be define, at τ time lag, of two time series (ϕ1(t) and ϕ2(t)) as:

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ρ(τ) =〈ϕ ′1(t)ϕ ′2(t + τ)〉〈ϕ ′1(t)2〉2〈ϕ ′2(t)2〉2

(5.1)

Where ϕ ′(t) = ϕ(t)− ϕ , is defined for the fluctuation of variables, here ϕ is used asthe mean value at the probes location.

(a) Probe P9

(b) Probe P2

Figure 5.5: (a)Time history of streamwise velocity at probe P9, and (b) time history ofpressure coefficient at P2

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Figure 5.6 plots the obtained cross-correlation between the signals at two probe loca-tions. On an average of τ = 30 tU/D period of a well defined periodic oscillation can beobserved in the figure. This time period obtained with cross-correlation matches with thelow-frequency ( fm = 0.033) recorded in frequency spectrum at various probe locationsconsidered, which is by all accounts the footprint of the streamwise velocity at probe P9.From the observation of figure 5.6, it is clearly visible that cross-correlation starts witha negative value which may be explained as the 180 phase angle between both signals.It can be explained by taking certain time period, for example at tU/D ∼ 450 (in fig-ure 5.5(b)) the pressure get more negative, at wake centreline the streamwise velocity (infigure 5.5(a)) increases, which represents the shrinkage of the recirculation bubble andhence the vortex formation zone. On the other hand, now if the pressure becomes lessnegative then streamwise velocity at wake centre line decreases, which represents the en-largement of recirculation bubble. Here H corresponds to the shorting of vortex formationzone while L related with a time period in which this zone gets longer.

Figure 5.6: Cross-correlation between streamwise velocity fluctuation at probe P9 andpressure coefficient at probe P2

5.2 Instantaneous flow

From the instantaneous statistics, the wake structure and the vortex formation can providethe good understanding of the flow dynamics but it requires the identification of the co-herent structures properly. Since there are many techniques to identify these structures(for example [61, 62]) however for accurate identification the Q-criterion given by Huntet al. [61] has been utilized. Hunt et al. given the definition of Q-iso-surfaces as an

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eddy structures as a region whose velocity gradient tensor ∆u ([63]) is the positive secondinvariant of it and defines as:

Q =12(||Ω||2−||S||2) (5.2)

Where Ω is the antisymmetric component and S is the symmetric components of thevelocity gradient tensor and are given by:

||Ω||2 = [Tr(ΩΩt ] and, (5.3)

||S||2 = [Tr(SSt ] (5.4)

While the positive value of Q implies that strength of rotation has overpower the strain ormore precisely vorticity overcomes the strain.

Figure 5.7 and figure 5.8 represents the series of Q-iso-surfaces at each quarter period.These series of figures represents the time advancement of the wake of the sphere forthe vortical structures behind it at two different planes or azimuthal position, figure 5.7represents the X-Y plane and figure 5.8 corresponds to a plane perpendicular to the X-Y plane, here X-Z plane is presented, viewed from the top. While in Appenxix E, acomparison of all the mesh are provided however to explain the vortical structures onlyfinest mesh is considered here upto the refinement length only.

A laminar boundary layer which is axisymmetric, separates at a separation angle ofφs = 84.1 from the equator of the sphere. This separated shear layer becomes unstableat certain distance from sphere until then it remains laminar. Vortex sheet behind thesphere starts to roll up due to presence of instabilities in the shear layer which show up atany azimuthal location which results in transition to turbulence and flow becomes threedimensional, this transition zone occurs closer to the separation point at this Reynoldsnumber; x/D = 1 − 1.2. Due to these instabilities in the shear layer which get ampli-fied further down in wake, large velocity fluctuations can be observed in the region oftransition to turbulence; x/D = 1.0, r/D = 0.6 (figure 5.3(b)).

It can be easily observe from figure 5.7 and figure 5.8 that, in the wake of sphere andjust behind the recirculation bubble there is a presence of extensive range of scales in theseparated zone. The vortex loops which are seen to be detached from opposite positionare not always detached with a separation of 180. The vortices which are formed in theseparated zone, breaks into small scale vortices which are drained into the formation zoneor region behind the sphere as well as also feed the turbulent wake. Even though the largescale vortices are not organized in the same plane, yet the wake shows prominent helical-like configuration. This same helical configuration was beforehand seen by Achehbach

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[2] at Re = 6000 and by Taneda [15] at Re = 104 to 3.8× 105 in their experiment andalso by Constantinescu & Squires [30] and Yun et al. [64] at Re = 104 in their numericalresults. While from the experimental results of Achehbach, he pointed out that vortexshedding happens at a particular position around sphere which rotates with the frequencyof vortex-shedding. However, Taneda experiment pointed out that the wake rotates ir-regularly about the axis of separation point. On the other hand from the recent numerialsimulation, the DNS results of Rodriguez [7] and LES results of Yun et al. shows thatthese structures travel almost straight in downstream direction, moreover they advocatethat, along the azimuthal direction the wall pressure changes in the sphere are related tothe helical like structures.

From the examination of the instantaneous data for the large number of vortex shed-ding considered in this simulation, it can be pointed out that the instabilities of shear layercan occur at any arbitrary position and also periodical shedding of vortices can occur atrandom azimuthal position. This gives the well know helical like configuration to the vor-tices in downstream of the sphere wake but without any circulation in azimuthal direction.Moreover, instead or circulation it can be claim that these helical like pattern which areobserved in wake is because of the way the vortices are shedding in time. It is observedthat two antisymmetric vortices are shed at any particular shedding cycle, but due to thearbitrariness of the shear layer instability, the vortex shedding at next shedding period,does not occur at same azimuthal location as the past one. Thus due to these randomshedding of vortices, the direction of helix vary with time.

Breaking down the shedding cycles, first start with the figure 5.7a and figure 5.8a,shows the shear layer which represents the breaking off first instability marked with I1from the vortex sheet. While the instability marked with I2 in axisymmetric shear layershows the small instabilities which appearing continuously. In the azimuthal direction to-wards the end of the axisymmetric bubble, some corrugated structures are also observed.The corrugated structured are develop due to the presence of small scales inside of recir-culation bubble as well as from the leftover of the past ones broken off from the shearlayer. The extending corrugates structure is marked as C0, which is formed by the previ-ous separated roller of the shear layer from the opposite side, while other vortices are alsomarked as C1 and C2. After quarter period time, the instability I1 moves downstreamand get amplified, shown by figure 5.7b and figure 5.8b. One interesting structure can beseen here which is represented by R, it is a long-rib structure which connects at one endwith the structure CO. As the instability I2 broken off completely from the shear layerin azimuthal direction, the corrugated structures are more evident now at the end of thevortex sheet. The small stream-wise vortices which appears to be interweaved with firstvortex structure I1 is also observed in this top view.

Now after another quarter period, the first vortex structure I1 is completely breaks off

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(a)

(b)

(c)

(d)

Figure 5.7: Vortex shedding at every quarter time period using Q-iso-surfaces (advancingfrom (a) to (d)), in X-Y plane

form shear layer, which can be seen in figure 5.7c and figure 5.8c. Also, the structureC0 has grown which is fed from the rib R whose tail is now distorted and warped arounditself in azimuthal direction and also recirculation bubble fed small scale vortices into it.Another vortex structure can be seen opposite to the I1 which is the instability I2 who isnow completely separated from shear layer. The last figure 5.7d and figure 5.8d which

65


(a)

(b)

(c)

(d)

Figure 5.8: Vortex shedding at every quarter time period using Q-iso-surfaces (advancingfrom (a) to (d)), in X-Z plane

66


show another quarter period, the instability I1 has moved downwards while instability I2is in its final stage. Additionally, the vortex structure C0 has grown in size due to thepairing of vortices.

The large scale structures which are show by C0, C1 and C2 are composed of about 9azimuthal vortex rings[31], however due to insufficient grid resolution behind the sphere,the current simulation doesn’t able to predict it at all hence these results are not veri-fied. Nevertheless, the large scale vortical structures; C0, C1 and C2 travels downstreamwithout undergoes any change in azimuthal direction which is also observed in previousDNS and LES simulations. A wavy motion of the whole wake can be observed as thesestructures moves downstream and display helical like configuration.

5.3 Mean flow parameters

The mean flow statistics are recorded for entire flow time period after passing throughtransition stage (72 tD/U), due to the large number of time steps (more than 32000) aMatLab code is used to post-process some of the data presented here, while for showingthe contour plots, another free open source software is used which comes pre-installedwith OpenFOAM; known as ParaFoam which is an OpenFOAM enabled version of Para-view (for more information about it, one can refer CFD Direct [65]).

Mesh Minimum Cp ¯Cpb Maximum C f L/DCoarse Mesh, M1 71.5 −0.262 53.7 1.26

Medium Mesh, M2 70.5 −0.232 51.2 1.77Fine Mesh, M3 70.5 −0.240 49.5 1.71

Rodriguez (DNS) 71 −0.272 50 1.657Constantinescu & Squires (LES) 71 −0.25 49.5 1.364

Table 5.2: Mean flow statistical data, DES (present simulation) results compared withDNS and LES results at Re=10000

Figure 5.9 and figure 5.10, shows the instantaneous contours of pressure coefficientand skin friction coefficient respectively. The pressure coefficient presented here is calcu-lated using formula; Cp = (p− p∞)/ (1/2ρ∞U2

∞), while skin friction coefficient is madedimensionless by using C f = τ/(ρU2)Re0.5, this would be sufficient for laminar bound-ary layer over sphere. As can be seen from the pressure coefficient contours figure 5.9 andfrom angular distribution mean pressure coefficient (Cp) over sphere (figure 5.18a), thepressure coefficient is at maximum in front of the sphere which conforms the favourablepressure gradient since the flow accelerates at front of sphere. While the minimum pres-sure can be observed at back of the sphere which corresponds to adverse pressure gradient

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since now the flow de-accelerates to maintain the constant throughput. The angular meanpressure coefficient is also compared with experimental results of Kim & Durbin[14] atRe = 4200 and Bakic[16] at Re = 50000. Some experimental investigations[3, 16] showsthat at the same time period there is very little variation in wake structure hence the de-pendence on Reynolds number is not so important for pressure coefficient, therefore itis possible to compare it with some other results which are different from the simulationReynolds number.

Figure 5.9: Instantaneous contours of pressure coefficient, Cp; (a) Coarse mesh, M1; (b)Medium mesh, M2; and (c) Fine mesh, M3

While the both experimental results are in very good agreement with the simulationresults within experimental uncertainties, however the finest mesh show more accurateresults with Bakic experimental results at Re = 50000. The angular value of minimumpressure is well captured by all three meshes, being at φ = 71.5, 70.5, 70.5 for meshM1, M2, M3 respectively. This position of minimum pressure is in agreement with exper-imental values of Seidl et al. for Re = 5000 and with DNS simulation of Rodriguez [31]at Re = 10,000 who reported it at φ = 71. However, among all three meshes, some largedifference is recorded for base pressure coefficient Cpb which is define as the time-meanpressure coefficient at angle φ = 180 or at the rear point of sphere, where the coarse mesh

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Figure 5.10: Instantaneous contours of non-dimensional skin-frictioncoefficient,(τ/(ρ U2 Re0.5)); (a) Coarse mesh, M1; (b) Medium mesh, M2; and (c)Fine mesh, M3

is giving more accurate result among all three simulations (see table 5.2). Nevertheless,a good agreement with the experimental results is obtained specially near the stagnationpoint where Cp rises slightly but after the separation region it remains relatively constantthrough out. Due to this inconsistency in pressure measurements behind the sphere, leadsto a compensating effect which results in reasonably accurate measurement of drag coef-ficient. The behaviour of pressure coefficient after separation is not clear but it can bepossible that the transition region is influencing the flow after separation which we modelwith DES in RANS mode, and as we know that RANS cannot able to capture the detailsof transition accurately.

Figure 5.10 shows the instantaneous contours of dimensionless skin friction coeffi-cient C f , here one can easily observe the separation region, which occurs because asflow works its way against the adverse pressure gradient in the boundary layer, its en-ergy dissipate due to viscous drag such that the flow velocity in the layer adjacent to thewall approaches to zero and hence the velocity gradient also approaches to zero and so theshear stress and after the point of separation the flow reverses in the boundary layer, there-fore we can say that at the point of separation the shear stress approaches to zero. Thiscan also be seen in figure 5.18b where angular distribution of mean skin friction coeffi-cient is compared with the DNS results of Seidl et al. [24] at Re = 5000. The separationangle is recorded when the skin friction coefficient approaches to zero it corresponds toφs = 93.6, 84.8 and 84.1 for M1, M2 and M3 respectively, while for the coarse meshthe separation occurs very early compared to the medium and fine mesh. This value isin very good agreement with experimental results of Achenbach [1] at DNS results ofRodriguez et al. [31] at Re = 10000 who recorded it at angle 82.5 and 84.7 respect-ively. While the maximum value of skin friction coefficient occurs at angular position53.7, 51.2 and 49.5 for M1, M2 and M3 respectively, which is also compares well

69


(a) Mean pressure coefficient; Cp

(b) Non Dimensional skin-friction coefficient; (τ/(ρ U2 Re0.5))

Figure 5.11: Angular distribution of mean pressure coefficient and skin friction coefficientaround sphere; compared with experimental results of Kim & Durbin[14] at Re = 4200,Bakic[16] at Re = 50000 and DNS results of Seidle et al.[24] at Re = 5000.

with DNS results of Rodriguez et al. [7] at Re = 3700 and Seidl et al. [24] of 48 and 50

respectively.

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5.4 Mean flow statistics

(a) Time-averaged streamwise velocity (b) Mean fluctuating streamwise velocity

Figure 5.12: Streamwise velocity profile along the wake centre line

The averaged streamwise velocity vx and averaged streamwise velocity fluctuation v′xnormalised with free stream velocity U , along the wake centre line is represented in fig-ure 5.12 for all the three mesh, both the figures are plotted form x/D = 0. As can beseen, the biggest difference among the three mesh is between coarse and medium mesh,however medium and fine mesh doesn’t shown much fluctuation in statistical data. fig-ure 5.12(a), shows the recirculation bubble length which is defined as the distance behindthe sphere upto the distance where mean streamwise velocity changes sign, for M1, M2and M3 it is L/D = 1.26, 1.77 and 1.71 respectively. These are the same positions wheremean streamwise fluctuating velocity (figure 5.12(b)) reaches its maximum for corres-ponding meshes, which is define as formation zone according to Norberg[66]. Thesevalues are in agreement with LES and DNS results performed earlier for same Reynoldsnumber (see table 5.2). However Rodriguez et al.[7] observe two peaks of fluctuatingvelocity at Re = 3700, the pronounced one occur at x/D = 1.98, which shows that withincrease in Reynolds number the recirculation length decreased and the formation zoneoccur at a particular location now.

The profile of mean streamwise velocity at three locations x/D = 0.2, 1.6 and 3 in thewake of sphere is shown in figure 5.13. Due to the lack of experimental data available onlythe half profile is compared with experimental data of kim & Durbin, however figure 5.14shows the streamwise and cross-stream velocity profiles. One can observe that, there isno significant difference in velocity profiles near the sphere where the grid is fine enoughto resolve most of the flow features accurately. However significant difference is observe

71


at location x/D = 3 where according to experimental results and previous DNS resultsof Rodriguez[7], the flow is in recovery zone which is define as the region in between ofthe end of recirculation zone and the location where the flow accelerate regardless of theadverse pressure gradient. There is some disagreement in the location of this region inliterature, Yun et al. [10] observe that flow is at end of recirculation bubble.

Figure 5.13: Streamwise velocity profile for M1, M2 and M3 at three locations in thewake, compared with the experimental data of Kim & Durbin at Re=3700

Figure 5.15 represents the mean streamwise and cross-stream velocity profiles at fivedifferent locations, x/D= 1.6, 2, 3, 5 and 10, DES results are compared with DNS resultsof Rodriguez [7] at Re = 3700. As can be seen, the negative velocities at location x/D =

1.6 and 2 represents the recirculation zone, however the streamwise velocity vx behind thewake of sphere is negative until the free-stagnation point. At position x/D = 3 which veryclose to end of recirculation bubble, the streamwise velocity is positive but near to zero,due to it we can record the minimum value of cross-stream velocity on the side of freestagnation point, and its value is vr = −0.148U (table 5.3). The minimum streamwisevelocity or the largest backward velocity in the recirculation bubble is vx = −0.4. Theolny difference with respect to the DNS results is observed at location x/D = 10 wherethe grid density is low in comparison the other near wake locations, hence its values arenot considered here.

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(a) Streamwise velocity profile normalised with free-stream velocity

(b) Cross-stream velocity profile normalised withfree-stream velocity

Figure 5.14: Mean velocity profiles along the wake centre line

The mean fluctuating streamwise v′x/U and cross-streamwise v′r/U velocities areplotted in figure 5.16. In figure 5.17 and figure 5.18(a), comparative contours of averagedReynolds stresses of all three meshes (only finest mesh is considered for comparativeanalysis) are depicted for streamwise (v′x v′x), cross-streamwise (v′r v′r) and shear stress(v′x v′x) respectively. The contour plots for Reynold stresses are in quantitative and qual-itative agreement with the experimental measurement of Jang and Lee [67] using PIV(Partical Image Velocimetry) method at Re = 11000, however there is difference in min-imum and maximum values when compared with DNS simulation of Rodriguez et al. [7]at Re = 3700. Although stresses are found to be of higher magnitude with Re = 10000,still the peaks in the stresses are within the range with respect to DNS results at Re= 3700,i.e. within a distance of 1.8D − 3.1D from the sphere. According to figure 5.16 and fig-ure 5.17(a), the maximum streamwise turbulent intensity is at location x/D = 2.5, in factthe maximum value of streamwise turbulent intensity occurs in the shear layer, at radialposition upto r/D = 0.4. Moreover, the location where the maximum value of cross-stream Reynolds stress occur is along the wake centre line at x/D = 2.9 (see figure 5.16and figure 5.17(b)). All the maximum and minimum values are compared with DNS res-ults of Rodriguez et al.[] at Re = 3700 and LES results of Constantinescu & Squries[29]at Re = 10000 (see table 5.3), one observation can be notice that these values occur atnearly same radial location even at different Reynolds number however there is some dif-ference in streamwise locations due to shrinkage of recirculation bubble with increase inReynolds number.

The normalised mean turbulent kinetic energy (tke) contour plot is depicted in fig-

73


ure 5.18(b) for all three mesh, due to the contribution of streamwise fluctuations it isproduce mostly in shear layer and then convected down to the wake centre line with themaximum value of this quantity occur at location x/D = 3.

Value x/D y/D

Minimum mean streamwise velocity vx/UDES, Re = 104 -0.4 1.40 0.0DNS, Re = 3700 -0.321 2.133 0.0LES, Re = 104 -0.4 1.41 0.0

Maximum mean streamwise velocity vx/UDES, Re = 104 1.2 0.99 0.69DNS, Re = 3700 1.175 1.282 0.702LES, Re = 104 - - -

Minimum mean radial velocity vr/UDES, Re = 104 -0.148 1.75 0.52DNS, Re = 3700 -0.198 2.499 0.572LES, Re = 104 -0.15 1.82 0.56

Maximum mean radial velocity vr/UDES, Re = 104 0.3 0.0009 0.41DNS, Re = 3700 0.207 0.0014 0.565LES, Re = 104 - - -

Maximum mean streamwise turbulent intensity v′x v′x/U2

DES, Re = 104 0.061 2.5 0.5DNS, Re = 3700 0.055 2.606 0.423LES, Re = 104 0.063 1.78 0.46

Maximum mean radial turbulent intensity v′r v′r/U2

DES, Re = 104 0.081 2.8 0.0DNS, Re = 3700 0.069 3.090 0.0LES, Re = 104 - - -

Maximum mean Reynolds shear stress v′x v′r/U2

DES, Re = 104 -0.036 2.42 0.29DNS, Re = 3700 -0.029 2.565 0.392LES, Re = 104 -0.039 2.04 0.39

Table 5.3: Mean flow statistics compared with DNS results of Rodriguez et al.[7] at Re =3700 and LES results of Constantinescu & Squires[29] at Re = 104

74


Figure 5.15: Mean streamwise and radial (cross-stream) velocity profile at different loca-tions in the wake of sphere

Figure 5.16: Fluctuating mean streamwise and radial (cross-stream) velocity profile atdifferent locations in the wake of sphere

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(a) Reynolds streamwise normal stress v′x v′x/U2

(b) Reynolds cross-stream normal stress v′r v′r/U2

Figure 5.17: Contours of normalised mean Reynolds stresses for (a) Coarse mesh, M1;(b) Medium mesh, M2; and (c) Fine mesh, M3

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(a) Reynolds cross-stream normal stress v′x v′r/U2

(b) Turbulent kinetic energy tke/U2

Figure 5.18: Contours of normalised mean shear stress and Turbulent kinetic energy for(a) Coarse mesh, M1; (b) Medium mesh, M2; and (c) Fine mesh, M3

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Chapter 6

Conclusion and Future work

The main task of the thesis is to capture the unsteadiness in the wake of sphere consideringthe low-frequency fluctuations using an open source software. Various parameters needto be validated and the time duration for the simulation need to be determine as well asthe effect of low-frequency fluctuations. The use of the hybrid method in OpenFOAMdemonstrate good results with previous numerical and experimental results. This chaptergives the conclusion on results obtained with simulation and future recommendationsbased on errors and problems encountered using sphere as a test case for automotiveaerodynamics simulations.

6.1 Conclusion on thesis

To reach the aim of this thesis for flow over sphere, several methods are available inOpenFOAM, like RANS, LES and DNS. However considering the computational effortswith DNS simulation and unresolved flow features with RANS, the LES would prove tobe a better choice. In spite of the that, LES is considered computationally expensive sinceit require much coarser grid, therefore a hybrid RANS-LES method is used in this thesiswork which proved to be more faster and less expensive comparatively. The main findingswith respect to the sphere test case is summarised below:

1. The Improved Delayed Detached Eddy Simulation (IDDES) of the turbulent flowpast a sphere at Reynolds number 10,000 have been carried out using OpenFOAM.The main features of the flow past sphere like frequency spectrum using FFT atselected locations in near wake, instantaneous and mean flow characteristics havebeen recorded and compared.

2. The simulation have been performed on cartesian grids of about 1.1, 4 and 6 MCVsgenerated by CFMesh software provided by Creative Field company. The time

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6. Conclusion and Future work

averaged statistics have been collected by integrating data over 105 vortex-sheddingcycles after passing through transition stage of about 75 tU/D time units whichcorresponds to 1.1 seconds, the total simulation time is over 9 seconds.

3. The resulted data have been compared at comparable Reynolds number wheneverpossible with numerical and experimental data available in literature. It has beenobserve that near to the equator of the sphere, flow separates laminarly and in theseparated shear layer the transition to turbulence occur at a distance between x/D= 1.0 and 1.2. The flow statistics, like vortex-shedding frequency, small and largescale instabilities, lift and drag coefficient and separation angle, have been observedto be reliable with past works.

4. Good agreement with pressure coefficient and skin-friction coefficient distributionover sphere has also been observed. However after the separation region it remainsrelatively constant through out. The behaviour of coefficients after the separation isnot clear but it can be possible that the transition region is influencing the flow afterseparation which we model with DES in RANS mode, and as we know that RANScannot able to capture the details of transition accurately.

5. The frequency spectrum in the wake of sphere of four selected probes at differentlocations have been performed and depending on the location of probes, the differ-ent frequency contributions recorded. The three main instabilities of different fre-quencies have been observed namely, the vortex shedding or large-scale instabilityfrequency at St = fvs D/U = 0.203 for fine mesh (see table 4.3), the Kelvin Helm-holtz or small-scale instability of the shear layer at fKH = 0.70 and a frequencylower than the vortex shedding frequency known a low-frequency and attributes tothe shrinkage and enlargement of recirculation bubble at fm = 0.038. The first twofrequency are in very good agreement with previous numerical and experimentalresults where data scattered for same Reynolds number, while the low-frequency isalso in good agreement with previous DNS result.

6. The low-frequency fluctuations is reorganised as a mechanism through which flowinfluence the fundamental turbulence and therefore, the wake characteristics. As aconsequence of wake tempering, to accomplish converged statistics in wake regionand to accurately capture the shrinkage and enlargement of recirculation bubble, thelarge integration time is required. Hence in this thesis almost 120 vortex sheddingare considered to achieve converged statistics.

7. The Visualization of vortical structures using Q-iso surface over this large timeintegration shows the presence of helical-like configuration due to the vortex shed-ding in the shear layer at random azimuthal direction. However it is observe that

79


during the period of vortex shedding, the vortex loops are not detaching at 180 ofseparation even though the coherent structures are antisymmetric.

8. From the observation, at every time period the vortex shedding occurs at somedifferent circumferential location and also its azimuthal position changes in randomdirection; i.e., the new vortex will be shed either to the right or left location ofthe previous vortex. Moreover, the large scale vortical structures moves withoutcirculation in azimuthal direction to the downstream of the flow, however due to thepresence of their relative positions, it gives the appearance of helical configurationand wavy motion.

9. The mean flow parameters like the streamwise and cross-streamwise velocity pro-files are also found in agreement with experimental results. The Reynolds stressesand Turbulent kinetic energy are also comparable to the previous numerical results.

6.2 Recommendations for future work

This section list some of the problems encountered during this thesis work which requiremore attention to improve results. Some of the recommendations are also provided tocarry out future work even with some other geometry.

1. The most important recommendation and error encountered during initial set upof simulation is the meshing software. It seems that conversion of ICEM mesh inOpenFOAM format is not working accurately since it increasing the domain sizeafter conversion, however it is worth to point out that the version of OpenFOAMused here is quite old so there are possibility of some bug with this version. There-fore a latest version is recommended for any future work on this thesis.

2. The current work is based on hybrid RANS-LES method however it is observedthat this approach is not working accurately in region after separation. Since it isvery challenging to identify the modelling error with hybrid approach, thereforequestion arises here that is the communication or interaction between the RANSans LES mode working properly?

3. With IDDES simulation, another most important factor is the meshing. Since themesh is made using another open source software which is new to work with, hencea proper mesh for IDDES isn’t able generate. However with the current mesh, a bestquality is observed where even the coarse mesh performs very well. Even though anhybrid mesh with structured mesh on sphere and cartesian mesh in domain wouldgive much better results with same number of elements.

80


4. Comparison with some other solvers or other turbulence model can provide solidverification and validation of current thesis work.

5. Even though hybrid RANS-LES simulations are considered as cost saving meth-ods, long simulation time due to presence of low-frequency fluctuation is still themain obstacle. However some methods available in literature which could reducethe calculation of such flow statistics as for example, work of Mockett et al.[68].Moreover solver settings which could reduce the computational time, especially thepressure calculation is crucial to a faster turn-over time.

As a closing remark, It is very beneficial for anyone to first learn the basics of Open-FOAM software to know how to get the desired results direct form the solver. Startingwith some simple geometries like backward facing step or flat plate where even coarsegrid can be used to set up the case which leads to faster turn-over time and hence helpsin gaining the knowledge of software. Afterwords the complex geometry can taken toexamine such flows for even longer duration of time.

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87

Appendix A

controlDict

/ *−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*− C++ −*−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*\| ========= ||| \ \ / F i e l d | OpenFOAM : The Open Source CFD Toolbox|| \ \ / O p e r a t i o n | V e r s i o n : 2 . 4|| \ \ / A nd | Web : www. OpenFOAM . org|| \ \ / M a n i p u l a t i o n ||

\*−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−* /FoamFi le

v e r s i o n 2 . 0 ;f o r m a t a s c i i ;c l a s s d i c t i o n a r y ;l o c a t i o n " sys tem " ;o b j e c t c o n t r o l D i c t ;

/ / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * / /

a p p l i c a t i o n pimpleFoam ;

s t a r t F r o m s t a r t T i m e ;/ / s t a r t F r o m l a t e s t T i m e ;

88

A. controlDict

s t a r t T i m e 0 ;

s t o p A t endTime ;

endTime 9 . 0 4 8 ; / / 120 v o r t e x s h e d d i n g

d e l t a T 0 . 0 0 0 2 9 ;

w r i t e C o n t r o l t i m e S t e p ;

w r i t e I n t e r v a l 6 5 ;

p u r g e W r i t e 8 ;

w r i t e F o r m a t b i n a r y ;

w r i t e P r e c i s i o n 6 ;

w r i t e C o m p r e s s i o n on ;

t imeForma t g e n e r a l ;

t i m e P r e c i s i o n 6 ;

r u n T i m e M o d i f i a b l e t r u e ;

a d j u s t T i m e S t e p no ;

maxCo 0 . 5 ;

f u n c t i o n s

p r o b e s

t y p e p r o b e s ;f u n c t i o n O b j e c t L i b s ( " l i b s a m p l i n g . so " ) ;o u t p u t C o n t r o l t i m e S t e p ;

89

A. controlDict

o u t p u t I n t e r v a l 1 ;p r o b e L o c a t i o n s(

( 1 . 0 0 . 6 0 ) / / i( 2 . 0 0 . 6 0 ) / / i i( 1 . 7 0 . 5 9 0 ) / / i i i( 3 . 0 0 . 6 0 ) / / i v( 5 . 0 0 . 5 0 ) / / v(−0.172917187637 0.470213404977 0 ) / / v i( 0 . 5 0 1 0 0 ) / / v i i(1 0 0 ) / / v i i i(2 0 0 ) / / i x(5 0 0 ) / / x(10 0 0 ) / / x i

) ;f i e l d s(

pU

) ;

# i n c l u d e " c e n t e r _ l i n e "# i n c l u d e " v e r t i c a l _ l i n e _ 0 p 2 "# i n c l u d e " v e r t i c a l _ l i n e _ 1 p 6 "# i n c l u d e " v e r t i c a l _ l i n e _ 2 "# i n c l u d e " v e r t i c a l _ l i n e _ 3 "# i n c l u d e " v e r t i c a l _ l i n e _ 5 "# i n c l u d e " v e r t i c a l _ l i n e _ 1 0 "

/ / f o r c e s/ / / / t y p e f o r c e s ;/ / f u n c t i o n O b j e c t L i b s ( " l i b f o r c e s . so " ) ;/ / o u t p u t C o n t r o l t i m e S t e p ;/ / o u t p u t I n t e r v a l 1 ;/ // / p a t c h e s ( " s p h e r e " ) ;

90

A. controlDict

/ / pName p ;/ / UName U;/ / rhoName r h o I n f ;/ / l o g t r u e ;/ // / CofR (0 0 0 ) ;/ // / r h o I n f 1 . 2 2 5 ;/ /

f o r c e s C o e f f s

t y p e f o r c e C o e f f s ;f u n c t i o n O b j e c t L i b s ( " l i b f o r c e s . so " ) ;o u t p u t C o n t r o l t i m e S t e p ;o u t p u t I n t e r v a l 1 ;

p a t c h e s ( " s p h e r e " ) ;pName p ;UName U;rhoName r h o I n f ;l o g t r u e ;

l i f t D i r ( 0 1 0 ) ;d r a g D i r ( 1 0 0 ) ;p i t c h A x i s ( 0 0 0 ) ;CofR ( 0 0 0 ) ;

r h o I n f 1 . 2 2 5 ;magUInf 6 8 . 0 5 8 ;

l R e f 1 ;Aref 0 . 7 8 5 3 9 8 1 6 3 ;

d e s F i e l d

t y p e DESModelRegions ;

91

A. controlDict

f u n c t i o n O b j e c t L i b s ( " l i b u t i l i t y F u n c t i o n O b j e c t s . so " ) ;l o g t r u e ;/ / r e g i o n " r e g i o n 0 " ;e n a b l e d on ;s t o r e F i l t e r on ;

/ / t i m e S t a r t 0 . 0 ;/ / t imeEnd 1 0 . 0 ;/ / o u t p u t C o n t r o l t i m e S t e p ;

o u t p u t C o n t r o l ou tpu tT imeo u t p u t I n t e r v a l 1 ;

/ / minMax/ / / / t y p e f ie ldMinMax ;/ / f u n c t i o n O b j e c t L i b s ( " l i b f i e l d F u n c t i o n O b j e c t s . so " ) ;/ / w r i t e yes ;/ / l o g yes ;/ / l o c a t i o n yes ;/ / mode magni tude ;/ / f i e l d s/ / (/ / U/ / p/ / ) ;/ /

/ / ************************************************************************* / /

92

Appendix B

fvSolution

/ *−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*− C++ −*−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*\| ========= ||| \ \ / F i e l d | OpenFOAM : The Open Source CFD Toolbox|| \ \ / O p e r a t i o n | V e r s i o n : 2 . 4|| \ \ / A nd | Web : www. OpenFOAM . org|| \ \ / M a n i p u l a t i o n ||

\*−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−* /FoamFi le

v e r s i o n 2 . 0 ;f o r m a t a s c i i ;c l a s s d i c t i o n a r y ;l o c a t i o n " sys tem " ;o b j e c t f v S o l u t i o n ;

/ / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * / /

p o t e n t i a l F l o w

n N o n O r t h o g o n a l C o r r e c t o r s 3 ;

93

B. fvSolution

s o l v e r s

Ph i

s o l v e r GAMG;smoothe r DIC ;c a c h e A g g l o m e r a t i o n on ;a g g l o m e r a t o r f a c e A r e a P a i r ;n C e l l s I n C o a r s e s t L e v e l 1 0 ;mergeLeve l s 1 ;

t o l e r a n c e 1e−06;r e l T o l 0 . 0 1 ;

p

s o l v e r PCG;p r e c o n d i t i o n e r DIC ;t o l e r a n c e 1e−6;r e l T o l 0 . 0 1 ;

p F i n a l

$p ;r e l T o l 0 ;

U

s o l v e r s m o o t h S o l v e r ;smoo the r symGaussSe ide l ;t o l e r a n c e 1e−6;r e l T o l 0 . 0 1 ;

94

B. fvSolution

UFina l

$U ;r e l T o l 0 ;

n u T i l d a

s o l v e r s m o o t h S o l v e r ;smoo the r symGaussSe ide l ;t o l e r a n c e 1e−9;r e l T o l 0 ;

n u T i l d a F i n a l

$ n u T i l d a ;r e l T o l 0 ;

SIMPLE

r e s i d u a l C o n t r o l

p 1e−6;U 1e−6;" ( k | omega | e p s i l o n | n u T i l d a ) " 1e−6;

n N o n O r t h o g o n a l C o r r e c t o r s 1 ;p R e f C e l l 0 ;pRefValue 0 ;

PIMPLE

momentumPredic tor yes ;

95

B. fvSolution

n O u t e r C o r r e c t o r s 3 ;n C o r r e c t o r s 2 ;n N o n O r t h o g o n a l C o r r e c t o r s 1 ;rhoMin 0 . 5 ;rhoMax 2 . 0 ;

r e l a x a t i o n F a c t o r s/ / f i e l d s/ / / / p 0 . 3 ;/ / / / e q u a t i o n s/ / / / U 0 . 7 ;/ / k 0 . 7 ;/ / e p s i l o n 0 . 7 ;/ / n u T i l d a 0 . 7 ;/ / omega 0 . 7 ;/ /

e q u a t i o n s

" . * " 1 ;

/ / ************************************************************************* / /

96

Appendix C

fvSchemes

/ *−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*− C++ −*−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*\| ========= ||| \ \ / F i e l d | OpenFOAM : The Open Source CFD Toolbox|| \ \ / O p e r a t i o n | V e r s i o n : 2 . 4| \ \ / A nd | Web : www. OpenFOAM . org|| \ \ / M a n i p u l a t i o n ||

\*−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−* /FoamFi le

v e r s i o n 2 . 0 ;f o r m a t a s c i i ;c l a s s d i c t i o n a r y ;l o c a t i o n " sys tem " ;o b j e c t fvSchemes ;

/ / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * / /

ddtSchemes/ / d e f a u l t s t e a d y S t a t e ;/ / / / d e f a u l t E u l e r ; / / Lowest p r e c i s i o n

d e f a u l t backward ; / / H i g h e s t p r e c i s i o n

97

C. fvSchemes

gradSchemes

d e f a u l t Gauss l i n e a r ;

g r ad ( n u T i l d a ) c e l l L i m i t e d Gauss l i n e a r 1 ;g r ad (U) c e l l L i m i t e d Gauss l i n e a r 1 ;

divSchemes

d e f a u l t none ;

d i v ( phi ,U) Gauss LUST u n l i m i t e d G r a d (U ) ;/ / d i v ( phi ,U) Gauss c u b i c ; / / H i g h e s t p r e c i s i o n −> d i v e r g e s

d i v ( phi , n u T i l d a ) Gauss l i m i t e d L i n e a r 1 ;/ / / / d i v ( phi , n u T i l d a ) Gauss c u b i c ; / / H i g h e s t p r e c i s i o n

/ / d i v ( ( nuEf f * dev2 ( T ( g rad (U ) ) ) ) ) Gauss l i n e a r ;d i v ( ( nuEf f * dev ( T ( g rad (U ) ) ) ) ) Gauss l i n e a r ;

l a p l a c i a n S c h e m e s

d e f a u l t Gauss l i n e a r l i m i t e d c o r r e c t e d 0 . 3 3 ;/ / d e f a u l t Gauss l i n e a r c o r r e c t e d ;

i n t e r p o l a t i o n S c h e m e s

d e f a u l t l i n e a r ;

snGradSchemes

d e f a u l t l i m i t e d c o r r e c t e d 0 . 3 3 ;

98

C. fvSchemes

/ / d e f a u l t c o r r e c t e d ;

w a l l D i s t

method meshWave ;

f l u x R e q u i r e d

d e f a u l t no ;p ;Ph i ;

/ / ************************************************************************* / /

99

Appendix D

meshDict

/ *−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*− C++ −*−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−*\| ========= ||| \ \ / F i e l d | OpenFOAM GUI P r o j e c t : c r e a t i v e F i e l d s|| \ \ / O p e r a t i o n | V e r s i o n : 0 . 8 . 9 . 0|| \ \ / A nd | Web : www. c− f i e l d s . com|| \ \ / M a n i p u l a t i o n ||

\*−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−* /

FoamFi lev e r s i o n 2 ;f o r m a t a s c i i ;c l a s s d i c t i o n a r y ;l o c a t i o n " sys tem " ;o b j e c t meshDict ;

/ / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * / /

100

D. meshDict

s u r f a c e F i l e " i n p u t . fms " ;

maxCe l lS i ze . 2 5 ;

o b j e c t R e f i n e m e n t s

s p h e r e 1

a d d i t i o n a l R e f i n e m e n t L e v e l s 3 ;t y p e s p h e r e ;c e n t r e (0 0 0 ) ;r a d i u s 1 . 2 5 ;

cone1

a d d i t i o n a l R e f i n e m e n t L e v e l s 3 ;t y p e cone ;p0 (0 0 0 ) ;p1 (3 0 0 ) ;r a d i u s 0 1 ;r a d i u s 1 1 ;

cone2

a d d i t i o n a l R e f i n e m e n t L e v e l s 2 ;t y p e cone ;p0 (0 0 0 ) ;p1 (12 0 0 ) ;r a d i u s 0 1 ;r a d i u s 1 2 ;

l o c a l R e f i n e m e n t

" s p h e r e "

101

D. meshDict

a d d i t i o n a l R e f i n e m e n t L e v e l s 4 ;r e f i n e m e n t T h i c k n e s s . 1 ;

b o u n d a r y L a y e r s

nLaye r s 0 ;

p a t c h B o u n d a r y L a y e r s

" s p h e r e "

nLaye r s 5 ;t h i c k n e s s R a t i o 1 . 2 ;m a x F i r s t L a y e r T h i c k n e s s 0 . 0 0 1 ;

/ / ************************************************************************* / /

102

Appendix E

Mesh comparative study

(a) Coarse Mesh, M1

(b) Medium Mesh, M2

(c) Fine Mesh, M3

Figure E.1: Vortex shedding at same time period for all three mesh using Q-iso-surfaces,in X-Y plane

103

E. Mesh comparative study

Figure E.2: Instantaneous streamwise velocity contours for; (a) Coarse mesh, M1 (b)Medium mesh, M2 and (c) Fine mesh, M3

104


Figure E.3: Instantaneous cross-stream velocity contours for; (a) Coarse mesh, M1 (b)Medium mesh, M2 and (c) Fine mesh, M3

105


Figure E.4: Instantaneous Mach number contours for; (a) Coarse mesh, M1 (b) Mediummesh, M2 and (c) Fine mesh, M3

106