Graduate School ETD Form 9 PURDUE UNIVERSITY …lyrintzi/MS_thesis_Martha.pdfChandra Sekhar Martha...

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CHANDRA SEKHAR MARTHA

LARGE EDDY SIMULATIONS OF 2-D AND 3-D SPATIALLY DEVELOPINGMIXING LAYERS

MASTER OF SCIENCE IN AERONAUTICS AND ASTRONAUTICS

Dr. Anastasios S. Lyrintzis Aerodynamics

Dr. Gregory A. Blaisdell Aerodynamics

Dr. Charles L. Merkle Aerodynamics

Dr. Anastasios S. Lyrintzis

Dr. Gregory A. Blaisdell

Dr. Anastasios S. Lyrintzis 03/05/2010

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Title of Thesis/Dissertation:

For the degree of ________________________________________________________________

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*Located at http://www.purdue.edu/policies/pages/teach_res_outreach/viii_3_1.html

LARGE EDDY SIMULATIONS OF 2-D AND 3-D SPATIALLY DEVELOPING MIXING

LAYERS

MASTER OF SCIENCE IN AERONAUTICS AND ASTRONAUTICS

CHANDRA SEKHAR MARTHA

04/30/2010

LARGE EDDY SIMULATIONS OF

2-D AND 3-D SPATIALLY DEVELOPING MIXING LAYERS

A Thesis

Submitted to the Faculty

of

Purdue University

by

Chandra Sekhar Martha

In Partial Fulfillment of the

Requirements for the Degree

of

Master of Science in Engineering

May 2010

Purdue University

West Lafayette, Indiana

ii

I dedicate this thesis to the people that care about me the most.

iii

ACKNOWLEDGMENTS

First and foremost, I offer my sincerest gratitude to my advisors Dr. Anastasios

Lyrintzis and Dr. Gregory Blaisdell, who have supported me throughout this work

with their invaluable guidance, patience and knowledge. I truly enjoyed working with

my advisors, who are very friendly and exceptionally knowledgeable. I would like to

thank Dr. Charles Merkle for serving as a third member on my thesis committee. I

would like to acknowledge the efforts of Mr. Joe Kline, our site specialist, in main-

taining the computer cluster used for the simulations of the current work. I would

also like to acknowledge my colleagues, friends for their encouragement. I would like

to thank my professors and friends at Indian Institute of Technology-Madras for mo-

tivating me to pursue graduate studies. I cannot thank my family enough for their

love and moral support. My deepest gratitude also goes to my uncle and aunt, Mr.

& Mrs. Sandanala, for their everlasting love and constant support. Without their

support, my ambition to study abroad can hardly be realized.

This work was part of Task 8, nozzle acoustic analysis, of the supersonic business

jet program, funded by Rolls-Royce and Gulfstream Aerospace Corporations.

iv

TABLE OF CONTENTS

Page

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii

CHAPTER 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Theoretical Investigation . . . . . . . . . . . . . . . . . . . . 31.1.2 Experimental Investigation . . . . . . . . . . . . . . . . . . . 41.1.3 Computational Investigation . . . . . . . . . . . . . . . . . . 5

1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

CHAPTER 2. COMPUTATIONAL PROCEDURE . . . . . . . . . . . . . . 72.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Sub-grid Scale Modeling . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Numerical Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 Pressure-based Vs. Density-based Solvers . . . . . . . . . . . 102.3.2 Spatial Discretization . . . . . . . . . . . . . . . . . . . . . . 112.3.3 Temporal Discretization . . . . . . . . . . . . . . . . . . . . 122.3.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 12

CHAPTER 3. 2-D MIXING LAYER SIMULATIONS . . . . . . . . . . . . . 133.1 Mixing Layer Parameters . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3 Inflow Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 Buffer Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.5 Simulation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 173.6 Post-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.7 Baseline Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.7.1 Baseline Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . 203.7.2 Incompressible Vs. Compressible Simulations . . . . . . . . 203.7.3 Single Precision Vs. Double Precision Computations . . . . 243.7.4 Incompressible Mixing Layer - Effect of forcing . . . . . . . . 30

v

Page3.8 Effect of Statistical Sample Size . . . . . . . . . . . . . . . . . . . . 423.9 Temporal Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . 473.10 Effect of Buffer Zone and Stream-wise Domain Size . . . . . . . . . 523.11 Optimization of 2-D Baseline Grid Size . . . . . . . . . . . . . . . . 66

3.11.1 Cross-stream Domain Size . . . . . . . . . . . . . . . . . . . 663.11.2 Grid Stretching . . . . . . . . . . . . . . . . . . . . . . . . . 733.11.3 Grid Coarsening . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.12 Grid Resolution Study . . . . . . . . . . . . . . . . . . . . . . . . . 843.13 Comparison of Computational Cost . . . . . . . . . . . . . . . . . . 90

CHAPTER 4. 3-D MIXING LAYER SIMULATIONS . . . . . . . . . . . . . 914.1 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.2.1 Inflow Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . 914.2.2 Obtaining k and ǫ for the Inlet BC . . . . . . . . . . . . . . 93

4.3 Simulation Parameters and Procedure . . . . . . . . . . . . . . . . . 954.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 964.5 3-D LES with Improved Inflow Forcing . . . . . . . . . . . . . . . . 112

4.5.1 Deficiencies of Fluent’s Vortex Method . . . . . . . . . . . . 1124.5.2 Improved Vortex Method . . . . . . . . . . . . . . . . . . . . 1164.5.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . 117

4.6 Estimate of Computational Cost for a Realistic Jet LES . . . . . . 137

CHAPTER 5. CONCLUSIONS AND FUTURE WORK . . . . . . . . . . . 1395.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395.2 Recommendations for Future Work . . . . . . . . . . . . . . . . . . 142

LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

APPENDICES

APPENDIX A. A NOTE ON DENSITY-BASED LES . . . . . . . . . . . . 147

APPENDIX B. ISSUES WITH LOWER TIME-STEPS . . . . . . . . . . . . 149

vi

LIST OF TABLES

Table Page

2.1 Comparison of the features of Pressure-based and Density-based solversfor LES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1 Simulation parameters for the 2-D incompressible and compressible runs. 22

3.2 Vorticity thickness growth rates for the forced incompressible and com-pressible mixing layers obtained using single precision computations. . 23

3.3 Comparison of the normalized peak Reynolds stresses and the growth rateof the 2-D baseline simulation with the data in the literature. . . . . . 34

3.4 Physical time of each sample in the baseline, LowT SF 1 and LowT SF 2simulations. The value of n is 28000 in all the simulations. . . . . . . . 48

3.5 Mixing layer growth rates for the baseline, LowT SF 1 and LowT SF 2simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.6 Comparison of the peaks of Reynolds stresses for the baseline, DNS ofUzun [39] and the LowT SF 1 simulations. . . . . . . . . . . . . . . . . 51

3.7 Length of the physical and buffer zones for the simulations described in3.10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.8 Number of nodes spanning the y-direction for the meshes considered insection 3.11. The baseline mesh is indicated by a (*). . . . . . . . . . . 68

3.9 Number of nodes spanning the y-direction for the grid stretching study insection 3.11.2. The revised baseline simulation is indicated by a (*). . . 74

3.10 Mixing layer growth rates obtained from the grid resolution study. . . . 86

3.11 Comparison of the Reynolds stresses obtained from the grid refinementstudy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.12 Computational resources required to perform the current 2-D single pre-cision simulations. The density-based LES is indicated by (*). . . . . . 90

4.1 Comparison of the normalized peak Reynolds stresses and vorticity growthrates of 3-D mixing layer with Fluent’s VM forcing with the data in theliterature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.2 Comparison of the features of the Fluent’s and improved VM algorithms. 117

vii

Table Page

4.3 Comparison of the normalized peak Reynolds stresses and vorticity growthrates of 3-D mixing layer using the improved VM with the data in theliterature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

4.4 Projection of the computational resources required to perform a realisticLES of a jet using the current methodology in Fluent. The actual numbersfor the 3-D baseline simulation are indicated by (*). . . . . . . . . . . . 138

viii

LIST OF FIGURES

Figure Page

3.1 Schematic of the 2-D computational domain with boundary conditions. 15

3.2 Baseline 2-D computational grid. Note: Every 6th node is shown. . . . 21

3.3 Comparison of the instantaneous vorticity contours after 12 FTC’s forincompressible, compressible mixing layers with and without inflow forcingobtained using single precision computations. . . . . . . . . . . . . . . 26

3.4 Comparison of mixing layer growth for incompressible, compressible mix-ing layers obtained using single precision computations. . . . . . . . . . 27

3.5 Center of the mixing layer for incompressible, compressible mixing layersobtained using single precision computations. . . . . . . . . . . . . . . 27

3.6 Comparison of the instantaneous vorticity contours after 12 FTC’s forincompressible, compressible mixing layers with and without inflow forcingobtained using double precision computations. . . . . . . . . . . . . . . 28

3.7 Comparison of mixing layer growth for the single and double precisioncomputations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.8 Center of the mixing layer for the single and double precision computa-tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.9 Comparison of the instantaneous vorticity contours for with and withoutforcing for incompressible mixing layer. . . . . . . . . . . . . . . . . . . 31

3.10 Incompressible mixing layer growth with and without forcing. . . . . . 32

3.11 Center of the incompressible mixing layer with and without forcing. . . 32

3.12 Comparison of the profiles of scaled velocity with and without forcing forincompressible mixing layer. . . . . . . . . . . . . . . . . . . . . . . . . 35

3.13 Comparison of the contours of σxx/∆U2 with and without forcing for in-compressible mixing layer. . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.14 Comparison of the profiles of σxx/∆U2 with and without forcing for in-compressible mixing layer. . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.15 Comparison of the contours of σyy/∆U2 with and without forcing for in-compressible mixing layer. . . . . . . . . . . . . . . . . . . . . . . . . . 38

ix

Figure Page

3.16 Comparison of the profiles of σyy/∆U2 with and without forcing for in-compressible mixing layer. . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.17 Comparison of the contours of σxy/∆U2 with and without forcing for in-compressible mixing layer. . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.18 Comparison of the profiles of σxy/∆U2 with and without forcing for in-compressible mixing layer. . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.19 Effect of the statistical sample size on the profiles of Reynolds stresses . 44

3.20 Comparison of mixing layer growth after 8 and 12 FTC’s . . . . . . . . 45

3.21 Comparison of center of the mixing layer after 8 and 12 FTC’s . . . . . 45

3.22 Comparison of the instantaneous vorticity contours at the end of the simu-lations with 2 and 4 FTC’s for transients followed by 8 FTC’s for sampling. 46

3.23 Effect of time-step on the incompressible 2-D LES . . . . . . . . . . . . 49

3.24 The profiles of Reynolds stresses for the incompressible mixing layer sim-ulation with ∆t = 5× 10−8 s. . . . . . . . . . . . . . . . . . . . . . . . 50

3.25 Comparison of the instantaneous vorticity contours after t = 4.2× 10−3 swith and without the buffer zone. . . . . . . . . . . . . . . . . . . . . . 55

3.26 Comparison of the mixing layer growth with and without the buffer zone. 56

3.27 Comparison of the mixing layer center with and without the buffer zone. 56

3.28 Comparison of the profiles of scaled velocity with and without the bufferzone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.29 Velocity profiles for the simulations with no buffer zones. . . . . . . . . 58

3.30 Comparison of the mean stream lines with and without the buffer zone. 59

3.31 Comparison of scaled σxy contours with and without the buffer zone. . 60

3.32 Comparison of the profiles of scaled σxy with and without the buffer zone. 61

3.33 Comparison of scaled σxx contours for with and without the buffer zone. 62

3.34 Comparison of the profiles of scaled σxx with and without the buffer zone. 63

3.35 Comparison of the scaled σyy contours for with and without the bufferzone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.36 Comparison of the profiles of scaled σyy with and without the buffer zone. 65

3.37 Effect of cross-stream domain size on the contours of vorticity magnitude. 69

3.38 Comparison of the mixing layer growth as the cross-domain size is reduced. 70

x

Figure Page

3.39 Effect of y domain size on the profiles of scaled velocity . . . . . . . . . 71

3.40 Velocity profiles for the mesh extending from y = −25δω(0) to +25δω(0)with symmetry and velocity boundary conditions on the side boundaries. 72

3.41 Effect of grid stretching - Comparison of the instantaneous vorticity con-tours. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.42 Effect of grid stretching on the mixing layer growth. . . . . . . . . . . . 76

3.43 Effect of grid stretching on the mixing layer center. . . . . . . . . . . . 76

3.44 Effect of grid stretching - comparison of the profiles of σxx/∆U2. . . . 77

3.45 Effect of grid stretching - comparison of the profiles of σyy/∆U2. . . . 78

3.46 Effect of grid stretching - comparison of the profiles of σxy/∆U2. . . . 79

3.47 Effect of increasing the stream-wise spacing - comparison of the instanta-neous vorticity contours. . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.48 Effect of increasing the stream-wise spacing on the profiles of Reynoldsstresses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.49 Effect of increasing the stream-wise spacing on the profiles of Reynoldsstresses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.50 Effect of grid refinement on the evolution of the mixing layer. The contoursare shown at the same physical time. . . . . . . . . . . . . . . . . . . 88

3.51 Effect of grid refinement on the mixing layer growth. . . . . . . . . . . 89

3.52 Effect of grid refinement on the mixing layer center. . . . . . . . . . . . 89

4.1 Comparison of k, ǫ for RANS simulation and the exponential function fits. 95

4.2 Mixing layer growth predicted by 3-D incompressible LES using Fluent’svortex method for forcing. . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.3 Span-wise averaged scaled velocity profiles as predicted by 3-D incom-pressible LES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.4 Span-wise averaged scaled Reynolds stresses as predicted by 3-D incom-pressible LES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.5 Iso-surface of ωz = 0.5× (∆U/δω(0)). . . . . . . . . . . . . . . . . . . . 102

4.6 Iso-surface of |ωx| + |ωy| + |ωz| = 0.5 × ∆U/δω(0) colored by span- andstream-wise vorticities. . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.7 Full isometric view of an iso-surface of |ωx|+ |ωy|+ |ωz| = 0.5×∆U/δω(0). 104

xi

Figure Page

4.8 Close-up isometric view of an iso-surface of |ωx| + |ωy| + |ωz| = 0.5 ×∆U/δω(0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.9 Iso-surface of Q = 2.2× 10−2× (∆U/δω(0))2 colored by span- and stream-wise vorticities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.10 Full isometric view of an iso-surface of Q = 2.2× 10−2 × (∆U/δω(0))2. . 107

4.11 Close-up isometric view of an iso-surface of Q = 2.2×10−2×(∆U/δω(0))2. 108

4.12 Iso-surface of smaller Q = 2.2×10−5×(∆U/δω(0))2 colored by stream-wisevorticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.13 Contours of span-wise vorticity in the plane z = 10δω(0) (mid-plane). . 110

4.14 1-D Energy spectrum of the stream-wise velocity at x = 200δω(0). . . . 111

4.15 Stream-wise velocity correlation (Q11) at x = 200δω(0). . . . . . . . . . 111

4.16 Span-wise perturbation velocity introduced by the vortices at the inletplane. The maximum and minimum contour levels are 0.08, -0.08 m/s,respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.17 Schematic of the stream-wise vortex pairs (1 and 2) generated by theimplemented VM forcing algorithm. The vortex pairs emulate hair-pinvortices developed within the boundary layer of a splitter plate wall. . . 118

4.18 1-D energy spectrum for the 3-D incompressible LES using the improvedVM algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.19 The stream-wise velocity correlation (Q11) obtained with the 3-D incom-pressible LES using the improved VM algorithm. . . . . . . . . . . . . 120

4.20 Mixing layer growth predicted by 3-D incompressible LES using the im-proved VM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.21 Span-wise averaged scaled velocity profiles for 3-D mixing layer obtainedusing the improved VM. . . . . . . . . . . . . . . . . . . . . . . . . . . 122

4.22 Span-wise averaged scaled Reynolds stresses as predicted by 3-D incom-pressible LES using the improved VM. . . . . . . . . . . . . . . . . . . 123

4.23 Full isometric view of an iso-surface of |ωx|+ |ωy|+ |ωz| = 0.5×∆U/δω(0)with one period in the span-wise direction obtained using the improvedVM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

4.24 Full isometric view of an iso-surface of Q = 2.2×10−2× (∆U/δω(0))2 withone period in the span-wise direction obtained using the improved VM. 127

xii

Figure Page

4.25 Close-up isometric view of an iso-surface of |ωx| + |ωy| + |ωz| = 0.5 ×∆U/δω(0) obtained using the improved VM. . . . . . . . . . . . . . . . 128

4.26 Close-up isometric view of an iso-surface of Q = 2.2× 10−2× (∆U/δω(0))2

obtained using the improved VM. . . . . . . . . . . . . . . . . . . . . . 129

4.27 Iso-surface of Q = 2.2 × 10−5 × (∆U/δω(0))2 colored by stream-wise vor-ticity obtained using the improved VM. . . . . . . . . . . . . . . . . . . 130

4.28 Iso-surface of Q = 2.2×10−2× (∆U/δω(0))2 colored by span-wise vorticityobtained using the improved VM. . . . . . . . . . . . . . . . . . . . . . 131

4.29 Iso-surface of |ωx|+ |ωy|+ |ωz| = 0.5×∆U/δω(0) colored by stream-wisevorticity obtained using the improved VM. . . . . . . . . . . . . . . . . 132

4.30 Iso-surface of Q = 2.2×10−2× (∆U/δω(0))2 colored by span-wise vorticityobtained using the improved VM. . . . . . . . . . . . . . . . . . . . . . 133

4.31 Iso-surface of Q = 2.2 × 10−2 × (∆U/δω(0))2 colored by stream-wise vor-ticity obtained using the improved VM. . . . . . . . . . . . . . . . . . . 134

4.32 Iso-surface of Q = 2.2×10−2× (∆U/δω(0))2 colored by span-wise vorticityafter 3.5 FTC’s using the improved VM. . . . . . . . . . . . . . . . . . 135

4.33 Iso-surface of Q = 2.2 × 10−2 × (∆U/δω(0))2 colored by stream-wise vor-ticity after 3.5 FTC’s using the improved VM. . . . . . . . . . . . . . . 136

xiii

Appendix Figure Page

A.1 Comparison of the instantaneous vorticity contours with pressure- anddensity-based solvers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

B.1 Spurious oscillations with the pressure-based compressible LES solver forsmaller ∆t’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

xiv

SYMBOLS

Roman Symbols

~Af Area vector of a cell face

c◦ Computational cell center

c∞ Speed of sound in free-stream

D Computational domain

E1 1-D turbulent energy spectrum

f(~x) Flow-field variable

f(~x) Resolved-scale component

fsg(~x) Subgrid-scale component

f(ξ) Scaled mean stream-wise velocity component

G(~x, ~x′

, ∆) Filter function

k turbulent kinetic energy

L Stream-wise size of the computational domain

Mc Convective Mach number

p Mean pressure

Q Second invariant of velocity gradient tensor

Q11 Two-point stream-wise velocity correlation

Re Reynolds number

Sij Component of resolved strain-rate tensor

t Time

U Mean stream-wise velocity component

U1 Stream-wise velocity component of high-speed flow

U2 Stream-wise velocity component of low-speed flow

Uc Convective velocity of the large-scale eddies

~u = (u, v, w) Mean velocity vector

xv

(u, v, w) Velocity vector in Cartesian system

(u′

, v′

, w′

) Perturbation velocity vector in Cartesian system

V Volume of a computational cell

V Volume space defined by a computational cell

x, y, z Cartesian coordinates

~x, ~x′

Position vector in Cartesian system

xi Alternate notation for position vector

yc(x) Center of the mixing layer

Greek Symbols

α Parameter to control the amplitude of perturbations

δij Kronecker delta

δω Vorticity thickness

δω(0) Initial vorticity thickness

∆ Grid filter width

∆U Velocity difference across the mixing layer

∆t Time-step

∆x, ∆y, ∆z Grid spacing in the x, y and z directions

∆y0 Grid spacing around the mixing layer center

ǫ Turbulent dissipation rate

η Velocity ratio

θ Flow angle with respect to the inlet face

κ Wavenumber

µ Molecular viscosity

ν Molecular kinematic viscosity

νt Sub-grid scale eddy-viscosity

ξ Scaled y-coordinate

ρ Density

σij Component of mean molecular stress tensor

σxx Reynolds normal stress in the x-direction

xvi

σyy Reynolds normal stress in the y-direction

σzz Reynolds normal stress in the z-direction

σxy, σyz, σzx Reynolds shear stresses in Cartesian system

τc Characteristic time-scale

τij Component of sub-grid scale stress tensor

φ Scalar flow variable

φf Scalar flow variable at the center of a cell face

ωx, ωy, ωz Vorticity vector components in x, y and z directions

Other Symbols

∇ Gradient operator

| | Absolute value operator

( )∞ Free-stream flow value

〈〉 Time averaging operator

∂∂t

Partial time derivative operator

∂∂x1

, ∂∂x2

, ∂∂x3

Partial spatial derivative operators in Cartesian coordinates

xvii

ABBREVIATIONS

BC Boundary Condition

BCD Bounded Central Differencing

CFD Computational Fluid Dynamics

CFL Courant - Friedrichs - Lewy

DNS Direct Numerical Simulation

FSM Fractional Step Method

FTC Flow-Through Cycle

LES Large Eddy Simulation

LST Linear Stability Theory

MUSCL Monotone Upstream-Centered Schemes for Conservation Laws

NITA Non-Iterative Time Advancement

NRBC Non-Reflecting Boundary Condition

PISO Pressure-Implicit with Splitting of Operators

PRESTO! PREssure STaggering Option

RANS Reynolds Averaged Navier Stokes

RMS Root Mean Square

SGS Sub-Grid Scale

TUI Text User Interface

UDF User Defined Function

VM Vortex Method

xviii

ABSTRACT

Martha, Chandra Sekhar. M.S.A.A.E, Purdue University, May 2010. Large EddySimulations of 2-D and 3-D Spatially Developing Mixing Layers. Major Professors:Anastasios S. Lyrintzis and Gregory A. Blaisdell.

A complete understanding of the noise generation mechanisms is prerequisite to

reducing aircraft jet noise. A computationally intensive large eddy simulation (LES)

can help directly predict the noise. For this purpose, the Ansys-Fluent LES tool is

evaluated for its modeling accuracy and computational cost. A satisfactory computa-

tional methodology is developed first by studying a canonical problem. A mixing layer

at a Reynolds number of 720 is simulated using LES in two- and three-dimensions

without the splitter plate walls and is studied extensively to gain insights of the

flow physics. The effects of inflow-forcing and the buffer-zone at the domain exit

incorporated in 2-D LES are investigated. The flow is damped in the buffer zone

without explicitly introducing the damping terms and is found to be satisfactory for

the current work. The instabilities of the 2-D mixing layer are captured well us-

ing random inflow-forcing. The buffer-zone is found to help the prediction of the

Reynolds stresses near the exit boundary. The sensitivity of the 2-D results to time-

step, sampling time and grid is also studied. The 2-D grid is, then, optimized to

construct a mesh for 3-D LES. The deficiencies of Fluent’s in-built vortex method

(VM) algorithm to force perturbations in 3-D are discussed. An adapted VM algo-

rithm is implemented and is found to predict the three-dimensional instabilities of the

mixing layer well. The energy spectrum of the LES computations indicate that the

second-order-accurate bounded central differencing scheme, used in the present study,

is adequate for modeling turbulent flows. The computational resources to perform a

realistic jet simulation are also estimated.

1

1. INTRODUCTION

Jet noise has been a major problem for aircraft for nearly 50 years. The rapid growth

of the civil aviation industry aggravated the problem of jet noise and resulted in

increased noise levels around airport neighborhoods. As the noise is detrimental

to the communities around the airports, the aviation industry is under tremendous

pressure to reduce the aircraft noise. In order to encourage the aviation industry to

build quieter aircraft, the FAA has imposed stringent regulations on the aircraft noise

levels. The jet engine exhaust noise (or jet noise) is a major component of the total

jet aircraft noise at take off and landing and hence must be reduced further to meet

these strict noise restrictions. Therefore, there is a significant need to understand the

noise generation mechanisms in detail to reduce the jet noise.

Computational techniques such as direct numerical simulation(DNS), large eddy

simulation(LES) and detached eddy simulation(DES) can give better insight into

the noise generation mechanisms when compared to the experiments. The recent

improvements in the computing power of the processor chips have transformed these

computationally intensive techniques into feasible tools to directly predict the noise.

Therefore, there is a need to develop LES/DES methodologies for flows involving

complicated geometries to successfully predict the jet noise. The current work deals

with developing a methodology to perform an LES using Fluent. The methodology

is evaluated by modelling a canonical problem, such as a mixing layer.

A mixing layer is formed at the interface of two fluid streams moving one over the

other with different velocities. It represents an important class of free shear flows.

Mixing layers are a common occurrence in many engineering applications involving

chemically reacting flows, scalar mixing and jets, etc. A complete understanding of

the characteristic features of the mixing layers is prerequisite to control the process

of mixing in such applications.

2

The velocity profile in a mixing layer has an inflection point. The flows with

such an inflection point are unstable for certain wavy disturbances [1]. The corre-

sponding instability is essentially an inviscid one and molecular viscosity has only a

damping influence on it [2]. The flow in a mixing layer is dominated by large, pre-

dominantly two-dimensional vortex structures developed from the Kelvin-Helmholtz

instability (also referred to as fundamental instability). These vortices shed in time

with a characteristic frequency known as the fundamental frequency. These large-

scale structures are observed within a wide range of Reynolds numbers [3, 4] and,

therefore, often considered as an intrinsic feature of a turbulent mixing layer. The

pairing (or amalgamation) of these coherent vortical structures (or rollers) is the

primary mechanism contributing to the growth of a mixing layer [5].

Experimental evidence shows that the growth rate of a mixing layer can be greatly

manipulated by forcing the mixing layer near a subharmonic of the most-amplified

frequency (or fundamental frequency) at a very low forcing level [6]. The development

of a mixing layer is highly sensitive to the boundary conditions. The splitter plate

geometry [7], the free-stream turbulence intensity [8] and the velocity ratio [9, 10]

affect the evolution of mixing layer.

Experiments [11–13] show that the two-dimensional large vortex structures are

subject to three-dimensional instabilities. The three-dimensional instabilities associ-

ated with a mixing layer predominantly involve the interaction of rib or stream-wise

vortices with the span-wise rollers. The rib vortices develop in the braids between

the spanwise vortices [14]. These ribs extend from the bottom of one roller to the top

of its neighbor.

The planar mixing layer exhibits ‘self-similarity’ for sufficiently large Reynolds

numbers [15]. The linear mixing layer growth rate and collapse of the profiles of

appropriately scaled mean velocity and turbulent quantities are indicators of the self-

similarity.

3

1.1 Review

The mixing layer has been a subject of research for the last 70 years. It is in-

vestigated by two different approaches: a) temporally evolving mixing layer, and b)

spatially evolving mixing layer. In the first approach, the evolution of flow structures

is investigated as they convect downstream with respect to an observer sitting on top

of these structures. Therefore, the changes in flow structures are seen in time in this

approach. The second approach deals with the study of the flow structures as they

convect downstream with respect to a fixed frame of reference. In this approach, the

changes in flow structures are seen in space.

In what follows, a brief review of the theoretical, experimental, and computational

findings related to the mixing layer are presented.

1.1.1 Theoretical Investigation

The theoretical study mainly focused on the analytic/numerical solutions of the

perturbed flow properties associated with the instabilities of a mixing layer.

The inviscid linearized stability theory (LST) for temporally growing disturbances

[16] and for spatially growing disturbances [17,18] formed the backbone for the quan-

titative analysis of the instabilities. It was suggested in Reference [19] that some of

the essential features of the instability properties cannot be described by the inviscid

LST of temporally growing disturbances for small distances downstream. The LST

of spatially growing disturbances [18] is found to agree well with the experimentally

observed instability properties.

Theoretical studies [20] also led to the prediction of a secondary instability mech-

anism which is a consequence of the fundamental instability in a mixing layer. The

secondary instability, which leads to the paring of vortices, is most unstable in the

two-dimensional limit; i.e., when there are no span-wise disturbances [21]. The pres-

ence of span-wise disturbances results in helical pairing (or local pairing) [22] of the

span-wise vortices. The ‘translative instability’ [21] of a mixing layer bends the cores

4

of the span-wise vortices and is found to be most unstable for span-wise wavelengths

approximately 23

of the spacing between the 2-D vortex rollers. This instability is

attributed to the development of counter-rotating stream-wise vortices, which even-

tually leads to the three-dimensionality of mixing layer [21]. A sketch of the topology

of stream-wise vortices is shown in figure 15 in reference [23].

1.1.2 Experimental Investigation

The turbulent structure of an incompressible mixing region dominated by 2-D

rollers is investigated by Liepmann & Laufer [24]. Wyganski & Fiedler [25] have also

explored and documented mixing layer’s turbulence properties in detail. Browand &

Weidman [26] have used conditional sampling [27] to isolate various stages of the large-

scale interactions and concluded that the pairing process is mainly responsible for the

production of Reynolds shear stress (or transverse momentum transport). Spencer

& Jones [28] have experimentally studied the turbulent mixing in an incompressible

planar mixing layer for velocity ratios 0.6 and 0.3. The effect of inlet boundary

conditions on the development of a mixing layer is studied experimentally and can

be found in reference [29].

The coherent structures of a mixing layer are isolated by employing forcing at

the fundamental frequency and its subharmonics by various researchers. This forcing

technique was found to have a significant impact on the mixing layer in terms of

enhanced/reduced growth and turbulence suppression in certain regions [6, 30,31].

Experiments [11, 12, 32] have shown the existence of three-dimensional instabil-

ities in a mixing layer. These instabilities generate small-scale three-dimensional

eddies and leave the large-scale coherent structures relatively intact [13]. The counter-

rotating stream-wise vortices are documented in detail through the experiments and

given in reference [33]. Small-scale eddies are produced by the interactions between

the merging span-wise structures and the stream-wise vortices [34].

5

1.1.3 Computational Investigation

Earlier computational investigation is primarily focused on the temporally devel-

oping mixing layers because of its computational simplicity and low cost. The focus

then shifted to the LES/DNS of spatial mixing layers with the availability of increased

computational power over the following years.

The mixing layers observed in experimental settings are simulated through spa-

tially developing mixing layers. The numerical simulations of temporally evolving

mixing layers serve as approximations to the spatial counterparts. This approxima-

tion becomes exact as the velocity ratio of the mixing layer approaches one [14].

Moser and Rogers [14,35] have computationally studied temporally evolving mix-

ing layers using DNS. They studied the Kelvin-Helmholtz roll-up, the pairing of the

span-wise vorticity rollers in great detail and observed the stream-wise rib vortices by

employing controlled boundary conditions. They concluded that the “pairing inhibits

the growth of infinitesimal three-dimensional disturbances and triggers the transition

to turbulence in highly three-dimensional flows”.

Riley and Metcalfe [36] have performed DNS of a mixing layer to gain insights

into the effect of forcing on the mixing layer evolution. They addressed anomalies

discovered with the experimentally forced mixing layers, such as suppression of mixing

layer growth and negative turbulent energy production . They suggested that the

absence of out-of-phase subharmonic in the forcing leads to these anomalies.

Ling et al. [37] have also observed stream-wise vortices in their DNS of a 3-D

temporally evolving mixing layer and verified that the time and length scales double

through the vortex pairing process. The stream-wise large-scale structures (ribs) are

found to be most unstable for a span-wise wavelength of about two thirds of that in

the stream-wise direction.

Stanley and Sarkar [38] have performed DNS of a two-dimensional spatially evolv-

ing mixing layer by employing different inlet forcing methods. The discreet forcing

employed in their simulations fixed the locations at which the vortices pair. The

6

two-dimensional spatially developing mixing layer is also studied by Uzun [39] as a

test case for his LES code.

1.2 Objectives

The main objective of this work is to develop a computational methodology to

perform an LES using the commercial CFD tool, ANSYS-Fluent. The 2-D and 3-

D spatially developing mixing layers are investigated using LES. It is intended to

estimate the computational resources required to perform a realistic LES of jet flow

using Fluent based on the experience gained through this work.

1.3 Thesis Organization

A spatially developing 2-D mixing layer with Re = 720 is chosen as a test case to

implement the LES methodology using Fluent. The results of the baseline simulation

are validated by comparing them with the 2-D DNS results of Uzun [39]. The baseline

2-D mesh is optimized by investigating the sensitivity of the simulation results to the

grid stretching ratio, domain size in the cross-stream direction. The effect of the buffer

zone employed in the LES is also investigated. A grid-refinement study is performed

to ensure that the baseline mesh is adequate enough to capture the large-scales as

well as the small-scales in the mixing layer. The numerical schemes employed in the

present work are described in chapter 2. The details of the 2-D simulations performed

in this study are presented in chapter 3.

The optimized 2-D mesh is used to construct a preliminary mesh for 3-D LES

of mixing layer at Re = 720. Fluent’s in-built vortex method (VM) and a self-

implemented VM algorithm are used to generate unsteady perturbations for the LES.

The details and the results of the 3-D simulations are presented in chapter 4. The

conclusions and recommendations for future work are given in chapter 5.

7

2. COMPUTATIONAL PROCEDURE

The commercial CFD tool, ANSYS-Fluent (version 6.3.26) is used in the present

study. The tool is based on cell-centered finite volume discretization. The computa-

tional procedure employed in this work is presented briefly in the following sections.

The detailed description of the LES solver and the numerical schemes can be found

in reference [40].

2.1 Governing Equations

Most of the analysis performed in the present study deals with incompressible mix-

ing layers. Therefore, the governing equations for incompressible LES are discussed

in this section.

The governing equations for LES are obtained by filtering the small-scale eddies

in the time-dependent Navier-Stokes equations. The eddies that are smaller than the

filter width are filtered out in this process. The filter width is, typically, related to the

grid spacing. So, the filtering process essentially filters out all the eddies that cannot

be resolved with the mesh employed in the simulation. The small eddies dissipate

the turbulent kinetic energy and are modeled using a Sub-grid Scale (SGS) model.

Therefore, the LES governing equations deal with the dynamics of the large eddies

involved in the simulation.

The rationale behind LES is that the small-scales are isotropic and universal, and

hence, they can be modeled accurately. However, the large-scales are dependent on

the geometries and boundary conditions involved in the simulation. In addition, the

large eddies are mainly responsible for the transport of mass, momentum and energy.

Therefore, the large-scales are explicitly computed by the LES technique.

8

In contrast, the direct numerical simulation (DNS) technique resolves the whole

spectrum of turbulent scales by explicitly computing them. As resolving the whole

turbulent spectrum demands a very fine mesh, the computational cost associated

with the DNS is very high and prohibitive for engineering applications. In RANS,

however, all the turbulent scales are modeled. Therefore, LES falls in between the

RANS and DNS in terms of modeling the turbulent scales involved and the associated

computational cost of simulation.

In LES, a flow-field variable (f(~x)) is decomposed into a large-scale or resolved-

scale component (f(~x)) and a small-scale or subgrid-scale (fsg(~x)) component. This

is represented in mathematical terms as

f(~x) = f(~x) + fsg(~x). (2.1)

The resolved-scale component is obtained by filtering the flow field in the entire

computational domain (D) using a filter function (G) as follows

f(~x) =

∫

Df(~x

′

)G(~x, ~x′

, ∆)d~x′

. (2.2)

The filtering is achieved implicitly by the finite-volume discretization incorporated

within in Fluent. The corresponding implicit grid filter function, G is

G(~x, ~x′

, ∆) =

1V

~x′ ∈ V

0 otherwise, (2.3)

where V is the vector space within a cell of volume, V . The filtered Navier-Stokes

equations describing incompressible flows are

Continuity:∂

∂xi

(ui) = 0, (2.4)

Momentum: ρ

[

∂

∂t(ui) +

∂

∂xj

(uiuj)

]

=∂σij

∂xj

− ∂p

∂xi

− ∂τij

∂xj

. (2.5)

The molecular stress tensor is given by

σij =

[

µ

(

∂ui

∂xj

+∂uj

∂xi

)]

− 2

3µ

∂ul

∂xl

δij. (2.6)

And sub-grid scale stress defined by

τij = ρuiuj − ρuiuj. (2.7)

9

2.2 Sub-grid Scale Modeling

The effect of small-scale eddies that cannot be captured by the grid is modeled by

the subgrid-scale stress defined in equation 2.7. This forms the closure for the filtered

Navier-Stokes equations.

The subgrid-scale stress is modeled similar to the Boussinesq hypothesis [41] em-

ployed in the RANS models. The SGS stress is expressed as

τij −1

3τkkδij = −2νtSij, (2.8)

where νt is the SGS eddy-viscosity and Sij is the resolved strain-rate tensor defined

by

Sij ≡1

2

(

∂ui

∂xj

+∂uj

∂xi

)

. (2.9)

The SGS viscosity (νt) is determined from dynamic Smagorinsky model. This

model is a modification of the original Smagorinsky SGS model [42] proposed by

Germano et al. [43] and Lilly [44]. The details of the implemented model in Fluent

and its validation can be found in [45]. The model constant of the Smagorinsky-Lilly

model [42] is dynamically computed in this model. The model constant thus varies

in time and space over a fairly wide range. In the Fluent code, however, the range is

clipped to zero and 0.23 to avoid numerical instability.

2.3 Numerical Schemes

The numerical schemes are given for LES of incompressible mixing layer simula-

tions. The highest Mach number in all the simulations performed in the present work

is about 0.5, which makes the flow-field mildly compressible justifying the use of the

incompressible flow assumption in the present analysis. Fluent offers two algorithms

to solve the governing equations. The following section describes the choice of the

algorithm used in the present study.

10

2.3.1 Pressure-based Vs. Density-based Solvers

Table 2.1 Comparison of the features of Pressure-based and Density-based solvers for LES.

Feature Pressure-based Density-based

Central differencing schemes X ×Non-reflecting boundary conditions × X

The availability of a few features in Fluent that are important for an LES sim-

ulation is shown in Table. 2.1. The low-dissipative central differencing schemes are

available in the pressure-based solver and the highest-order for the central differ-

encing scheme is only second-order-accurate. In contrast, the density-based solver

does not offer any central differencing schemes. The third-order MUSCL [40] scheme

(which is a blend of second-order-accurate central and upwind differencing schemes)

is the highest-order scheme available in the density-based solver. The non-reflecting

boundary conditions are implemented only in the density-based solver.

As LES requires low-dissipative schemes, the pressure-based solver is used hence-

forth with no use of non-reflecting boundary conditions (NRBC’s) at the outlet. It is

believed that the buffer zone employed in the simulation would obviate the need for

NRBC. Despite the lack of central discretization schemes, a simulation is performed

with the density-based LES and the results are compared with the pressure-based

solver in Appendix. A.

The pressure-based segregated algorithm is used to solve the filtered Navier-Stokes

equations. The governing equations are solved sequentially unlike the density-based

algorithm which solves all the equations simultaneously. As the governing equations of

the flow-field are segregated from one another, a pressure-velocity coupling method

is required to couple the momentum and continuity. The Pressure-Implicit with

11

Splitting of Operators (PISO) pressure-velocity coupling scheme is used with the

default settings for the neighbor and skewness corrections.

The PRESTO! (PREssure STaggering Option) scheme is employed for pressure

interpolation scheme used in the momentum equation. This scheme interpolates the

pressure stored at the cell centers to the centroid of the faces in a manner similar

to the staggered-grid schemes used with the structured meshes [46] to avoid even-

odd grid point decoupling of the flow. The interpolated pressure is then used in the

finite-volume integrated momentum equation.

The gradients are evaluated using the ‘Green-Gauss Cell-Based’ method. Accord-

ing to this method, the expression for the gradient of a scalar φ at the cell center c◦

is discretized using the Green-Gauss theorem as

(∇φ)c◦=

1

V

∑

f

φf~Af , (2.10)

where the face value φf , is the arithmetic average of φ’s at the neighboring cell centers.

2.3.2 Spatial Discretization

The central differencing schemes are ideal for discretizing the spatial terms in the

governing equations as they offer low numerical diffusion. However, pure central dif-

ferencing schemes often produce non-physical wiggles in the flow-field. This problem

is further aggravated by low sub-grid scale diffusivity of the LES. The Bounded Cen-

tral Differencing (BCD) is implemented in Fluent to address this issue. The BCD is

a blend of pure central differencing and the second-order, first-order upwind schemes.

The convective terms in the momentum equation are discretized using second-order

BCD. It is reiterated that the central discretization schemes are only available in the

pressure-based solver. It should be noted that the highest order central discretization

scheme available in the 6.3.26 version of Fluent is only a second-order.

12

2.3.3 Temporal Discretization

In order to maintain high temporal accuracy, a second-order implicit time-stepping

scheme is used for time advancement. The implicit scheme allows one to use larger

time-steps for the simulation unlike an explicit time-stepping scheme.

The implicit time-stepping schemes, however, need a few inner iterations between

two successive time-steps to converge to the next. The number of inner iterations

needed to converge depends on the time-step. Typically, a smaller time-step results in

faster convergence to the next time-step. An LES needs to be advanced to sufficiently

large time to account for transients and to gather flow statistics. Therefore, a smaller

time-step would result in a higher number of time-steps leading to an increase in

computational cost. The simulation time, thus, depends on the time-step as well as

the number of inner iterations between time-steps. A general rule of thumb is that

the time-step be chosen so that the solution converges within 20 inner iterations to

keep the computational cost low. In the present study, the number of inner iterations

is 20 for all the simulations. It should be noted that the time-step needs to be small

enough to simulate unsteady simulations accurately as discussed in section 3.5.

2.3.4 Boundary Conditions

The boundary conditions and inflow forcing are described for the current 2-D and

3-D mixing layer simulations separately in the corresponding chapters.

13

3. 2-D MIXING LAYER SIMULATIONS

The eventual goal of this work is to study spatially developing 3-D mixing layers using

LES. As the LES of 3-D mixing layer is computationally intensive, it is validated with

a spatially developing 2-D mixing layer first. This strategy facilitates the testing of

various LES solver parameters of Fluent and thus helps us build a satisfactory LES

methodology for the future 3-D runs. It also helps us investigate the 2-D mixing layer

further to optimize the number of grid points in the 2-D grid without compromising

the accuracy of the computational results. The optimized grid would, then, be used

as a baseline to construct a mesh for the future 3-D runs.

The planar mixing layers have been investigated experimentally by Wygnanski

and Fiedler [25], Spencer and Jones [28] as well as Bell and Mehta [29]. They have

been studied computationally by Rogers and Moser [14,35], Stanley and Sarkar [38],

and Uzun [39], among others.

A 2-D mixing layer is picked as a test case to establish and validate LES solver

methodology using Fluent. The intention is to successfully reproduce the 2-D DNS

results of Uzun [39] with the developed methodology. The LES is performed on a

baseline 2-D mesh, which is similar to the mesh used by Uzun [39], to make a fair

comparison with his results. This baseline simulation is presented in section 3.7.

Additional 2-D simulations are performed to investigate the effect of time-step, grid

refinement and to optimize the grid points in the baseline mesh. The procedure

employed in all these simulations is described in the subsequent sections followed by

the discussion of results of each simulation.

14

3.1 Mixing Layer Parameters

The planar mixing layer is simulated by specifying the following hyperbolic tangent

velocity profile for the stream-wise mean velocity at the inlet

u(y) =U1 + U2

2+

U1 − U2

2tanh

(

2y

δω(0)

)

; (3.1)

and the mean cross-stream velocity is considered to be,

v(y) = 0. (3.2)

The terms U1, U2 represent the velocities of high-speed and low-speed streams, re-

spectively. The symbol y, is the coordinate of a point in the cross-stream direction

with respect to a coordinate system with its origin at the center of the mixing layer.

And δω(0) is the initial vorticity thickness, which is defined as

δω(0) =U1 − U2

∣

∣

∣

∂u∂y

∣

∣

∣

max

. (3.3)

The convective velocity of the large-scale eddies of the mixing layer is

Uc =U1 + U2

2= 0.375c∞; (3.4)

and the relative convective Mach number of the mixing layer is

Mc =U1 − U2

2c∞= 0.125, (3.5)

where c∞ is the speed of sound at ambient temperature. The Reynolds number based

on the initial vorticity thickness and the velocity difference across the mixing layer is

Re =(U1 − U2)δω(0)

ν= 720. (3.6)

The velocity ratio of the mixing layer is given by

η =U1 − U2

U1 + U2

=1

3. (3.7)

15

3.2 Boundary conditions

As mentioned in section 3.1, the mixing layer is simulated artificially by specifying

a hyperbolic tangent velocity profile. The velocity profile and the inflow forcing is

implemented using user defined functions(UDF’s). The pressure outflow boundary

condition is used at the exit of the buffer zone. Uniform back pressure is specified

at the pressure outflow boundary condition(BC). A schematic of the physical, buffer

zones and the boundary conditions enforced in the incompressible LES simulations is

shown in figure 3.1. Perturbations on the cross-stream component of velocity are spec-

ified at the velocity inlet. The inflow forcing method used to generate perturbations

is described in section 3.3.

Symmetry BC

Hyperbolictangentinletvelocity

Symmetry BC

High speed stream

Low speed stream

Pressu

reoutletBC

Buffer zonePhysical zone

Figure 3.1. Schematic of the 2-D computational domain with boundary conditions.

It should be noted that the velocity inlet BC is, typically, compatible with incom-

pressible flow simulations in Fluent. This BC might pose some stability problems

when it is used for compressible flows. Hence, a mass-flow BC is used for the com-

pressible LES simulations performed in section 3.7.2.

16

3.3 Inflow Forcing

The mixing layer is simulated with a hyperbolic tangent velocity profile. This

profile can be considered a good approximation to the flow over a splitter plate after

the splitter plate wake effects have vanished. The splitter plate walls generate turbu-

lence in the boundary layers, which is then convected downstream. Therefore, it is

necessary to force perturbations within the mixing layer at the inlet to emulate the

turbulence generated off the walls of the splitter plate.

The forcing based on random perturbations [39,47] is used in the 2-D simulations

to supply realistic inflow boundary conditions to the LES solver. The perturbations

are forced only on the cross-stream velocity at the inflow boundary using

v(y) = ǫαUcexp

(

− y2

∆y20

)

, (3.8)

where ǫ is a random number between −1 and 1, α = 0.0045, and ∆y0 is the grid

spacing around the center line of the mixing layer. A random number is assigned to

ǫ after each time-step. In other words, the amplitude of the Gaussian profile given

in equation 3.8 varies randomly for each time-step. The inflow forcing algorithm is

implemented so that the amplitude of the Gaussian profile remains the same for all

the 2-D simulations at a given time-step. The implemented inflow forcing algorithm

ensures that the amplitude of the Gaussian profile is the same for all the 2-D runs

at a given time-step. In the present study ∆y0 is set to 0.16δω(0) in all the 2-D

simulations to better compare the flow-field.

The inflow forcing for compressible flows is discussed in section 3.7.2. The effect

of random inflow forcing is discussed in section 3.7.4. In grid refinement simulations

discussed in section 3.12, the value of ∆y0 from the baseline mesh is used to retain

the amplitude of the perturbations of the baseline simulation.

17

3.4 Buffer Zone

The pressure-outflow BC at the outflow boundaries might influence the solution

within the computational domain. Therefore, a buffer zone is attached to the physical

computational domain to suppress the vorticity before it reaches the outflow bound-

aries. In the current work, the damping is achieved implicitly by the stream-wise grid

stretching within buffer zone, where as explicit damping is employed by Uzun [39]

by introducing damping terms in the governing equations. The effectiveness of the

buffer zone is tested in section 3.10.

3.5 Simulation Procedure

The best practices for an LES as described by Georgiadis [48] have shaped the

simulation procedure employed in the present study. The velocity profile and the

perturbations are incorporated into the solver by implementing appropriate UDF’s.

The time-step is chosen to ensure that the large scales, i.e., the Kelvin-Helmholtz

vortices in the mixing layer are temporally well resolved. A characteristic time-scale

is defined for this purpose. It is constructed from the initial vorticity thickness and

the convective velocity of the vortices and is given by

τc =δω(0)

Uc

. (3.9)

A reasonable temporal resolution is achieved in the current LES simulations by choos-

ing a time-step which is a fraction of the characteristic time-scale (τc). It is also

ensured that this chosen time-step satisfies the CFL criterion to make sure that the

flow field does not advance more than one grid point downstream during a time-step.

The time-step (∆t) is chosen to be 10−7 s, which corresponds to ∆t/τc = 0.1.

This time-step corresponds to a CFL number of 0.25 in the 2-D baseline simulation.

This time-step is twice the time-step used in the DNS simulations of Uzun [39]. The

sensitivity of the LES to the time-step is investigated in section 3.9 by considering

18

a time-step of 5 × 10−8 s and the results are compared with his simulation. The

time-step is 10−7 s for all other 2-D simulations.

A second time-scale is defined to quantify the time a flow particle resides within the

computational domain. This time-scale is referred to as Flow-Through Cycle (FTC)

throughout this work. It is defined as the ratio of the length of the computational

domain (L) to the convective velocity of the vortices (Uc) or in mathematical terms

FTC =L

Uc

. (3.10)

This time-scale would be useful to estimate the physical time to which the simulation

needs to be advanced in time for the transients to leave the computational domain

and to collect flow statistics.

A steady RANS solution is obtained prior to the start of simulation using the

standard k−ǫ turbulence model [40]. A spectral synthesizer [40] is then used to super-

impose turbulent velocity fluctuations over the mean velocity field obtained from the

RANS solution. This is done using the Fluent’s TUI command, ’/solve/initialize/init-

instantaneous-vel’. The perturbed velocity field is used as an initial condition to

advance the LES solver in time. This practice can help substantially reduce the

simulation time to get to the statistically stationary state [40].

Unless otherwise stated explicitly, all the 2-D simulations in the present study are

single precision and advanced in time for 4 FTC’s to account for the transients to

exit the domain. After that the random number generator for the inflow forcing is

reinitialized and the flow statistics are gathered for the next 8 FTC’s at a sampling

frequency of 1 time-step. In section 3.8, the sensitivity of the results to the sampling

time as well as the sampling frequency is presented.

3.6 Post-processing

The results of each 2-D simulation are post-processed after the flow statistics are

gathered. The flow-field variables are scaled as described below to examine the self-

19

similarity of mixing layer. The mean stream-wise velocity is scaled with the velocity

difference across the mixing layer and is denoted by

f(ξ) =U − Uc

U1 − U2

, (3.11)

where U is the mean stream velocity, and

ξ =y − yc(x)

δω(0), (3.12)

yc(x) =1

2[y0.1(x) + y0.9(x)], (3.13)

δω(x) = y0.9(x)− y0.1(x). (3.14)

The center of the mixing layer is denoted by yc(x); y0.1(x), y0.9(x) are the coordinates

in the cross-stream direction where

U − Uc

U1 − U2

= 0.1, 0.9 (3.15)

respectively.

The Reynolds stresses are scaled by the square of the velocity difference (∆U =

U1 − U2) across the mixing layer. The normalized Reynolds stresses are defined as

σxx =〈u′

u′〉

∆U2, (3.16)

σyy =〈v′

v′〉

∆U2, (3.17)

σxy =〈u′

v′〉

∆U2, (3.18)

where 〈〉 denotes time averaging and u′

, v′

are the unsteady stream-wise and cross-

stream components of flow velocity respectively.

The scaled mean stream-wise velocity is plotted against the scaled cross-stream

direction (ξ) at different stations downstream. These profiles are also compared with

Pope’s [49] error-function

f(ξ) =1

2erf

(

ξ

0.5518

)

. (3.19)

The profiles of scaled Reynolds stresses are also plotted at various downstream loca-

tions for each simulation.

20

3.7 Baseline Simulation

The developed LES methodology is tested on a baseline mesh and the results of

the simulation are compared with Uzun’s [39] results.

3.7.1 Baseline Mesh

The computational grid used in this validation study is shown in figure 3.2. As

shown in the figure, the computational domain extends from 0 to 350δω(0) in the

stream-wise (x) direction and from −300δω(0) to 300δω(0) in the cross-stream (y)

direction. The number of grid points in the x, y directions are 576 and 575 respec-

tively. The nodes are stretched in the y-direction from the origin and mirrored about

the centerline of the mixing layer. The grid stretching ratio in the y-direction is ∼1.01 (or ∼1%). The minimum grid spacing in the y-direction is 0.16δω(0) around the

centerline.

The computational domain has a physical domain, which extends from 0 to

200δω(0) in the x-direction and a buffer zone (or sponge region) that fills the rest

of the domain downstream. The physical domain has 500 nodes that are uniformly

spaced with a spacing of ∆x = 0.4δω(0) in the x-direction. The buffer zone has 77

nodes that are stretched in the x-direction with a stretching ratio of approximately

1.04.

This mesh is similar to the one used by Uzun [39] except that the grid points in

the buffer zone and the cross-stream direction of his mesh are stretched using the

stretching functions utilized by Colonius [50]. In the current mesh, the nodes are

stretched using a geometric progression.

3.7.2 Incompressible Vs. Compressible Simulations

The highest Mach number in the flow-field is 0.5 corresponding to the high-speed

stream. The flow is simulated modeling the flow as incompressible and compressible

21

x/δω(0)

y/δ

ω(0

)

0 100 200 300 400-300

-200

-100

0

100

200

300

Figure 3.2. Baseline 2-D computational grid. Note: Every 6th node is shown.

and the results are compared for the two cases. For each case, the flow is simulated

using sigle precision computations with and without inflow forcing resulting in 4 sim-

ulations. The objective is to test the behavior of the LES to these flow assumptions.

The ideal gas law is used in the compressible case to couple the energy equation

to the momentum. The velocity boundary condition at the inlet is found to produce

stability problems for the compressible flow. Hence, the hyperbolic tangent velocity

at the inlet is imposed indirectly by specifying mass flow. The mass flux profile is

computed by simply multiplying the velocity profile with density (= 1.225 kg

m3 ). The

stagnation temperature profile is computed from the adiabatic relation assuming that

the static temperature is constant (= 293.15K).

For the incompressible case, the density is assumed to be equal to 1.225 kg

m3 . The

hyperbolic tangent velocity profile is specified at the inlet. The perturbations are

applied in the cross-stream velocity through random forcing described in section 3.3.

22

The forcing for the compressible case is applied by specifying a profile of flow angle

with respect to the inlet boundary. The profile is computed using

θ(y) = tan−1

[

v′

(y)

u

]

, (3.20)

where θ is the flow-angle with respect to the inlet face, v′

is cross-stream perturbation

velocity as computed from equation 3.8 and u is the stream-wise mean velocity.

The time-step is chosen to be 10−7 s, which corresponds to ∆t/τc = 0.1. This

time-step is twice the time-step used in the 2-D DNS simulations of Uzun [39]. The

time-step is kept constant for all the four simulations. Each simulation takes about

3.5 days on 4 cores of 4 processors each. This simulation time includes 4 FTC’s for

transients and 8 FTC’s for data sampling. The data is sampled after each time-step

during the sampling period. The number of time-steps required to complete one FTC

is 3500. The following table shows the list of simulations performed to compare the

results of the incompressible and compressible simulations.

Table 3.1 Simulation parameters for the 2-D incompressible and compressible runs.

Case ∆t (sec) Transients Sampling Frequency

Incompressible - unforced 10−7 4 FTC 8 FTC 1

Incompressible - forced 10−7 4 FTC 8 FTC 1

Compressible - unforced 10−7 4 FTC 8 FTC 1

Compressible - forced 10−7 4 FTC 8 FTC 1

The instantaneous vorticity contours at the end of each simulation, corresponding

to the same physical time, are shown in figure 3.3. As shown in the figure 3.3(a),

the unforced incompressible mixing layer stays laminar for a longer distance than the

unforced compressible mixing layer in figure 3.3(c). In the presence of forcing, the

instability begins to grow upstream around x = 60δω(0) for both the incompressible

and compressible cases as shown in figures 3.3(b), 3.3(d).

23

The growth and center (yc, as defined in Eq. 3.12) of the mixing layer are plotted

in figures 3.4 and 3.5, respectively. The figure 3.4 shows that the mixing layer has a

low growth rate when it is laminar followed by a near linear growth rate further down-

stream. The unforced incompressible mixing layer grows linearly for x > 130δω(0).

Whereas, the unforced compressible mixing layer achieves approximate linear growth

rate around x = 90δω(0). Both the incompressible, compressible forced mixing layers

start to grow linearly around x = 70δω(0) and have similar vorticity growth rates.

Table 3.2 shows the slope of the vorticity thickness growth in the linear growth

region for the incompressible and compressible forced mixing layer. It is interesting

to see that the unforced compressible mixing layer growth rate is approximately the

same as its forced counterpart.

Table 3.2 Vorticity thickness growth rates for the forced incompress-ible and compressible mixing layers obtained using single precisioncomputations.

Case Growth rate, dδω(x)dx

Incompressible - unforced 0.0330

Incompressible - forced 0.0563

Compressible - unforced 0.0602

Compressible - forced 0.0597

An ideal mixing layer should remain laminar when it is unforced. However, the

numerical errors incurred in a simulation get amplified downstream and cause the

vortex sheet to roll-up, which eventually leads to an increased mixing layer growth

rate through vortex diffusion and possibly through vortex pairing. The numerical

errors involved in a simulation are of two types: the truncation errors due to the

approximation of derivatives and the round-off errors caused by the limited precision

of the floating point numbers. The round-off errors of the single precision solver may

have caused the compressible unforced mixing layer to roll up quickly in compari-

24

son to its incompressible counterpart. Therefore, double precision computations are

performed in the following section to investigate this issue in detail.

3.7.3 Single Precision Vs. Double Precision Computations

Both single and double precision computations are performed without the inflow

forcing to investigate how the round-off and truncation errors affect the stability

of the unforced mixing layer. In addition, the forced, incompressible mixing layer

is simulated using the double precision solver to compare the results to its single

precision counterpart. As expected, the run times for these simulations are 60%

higher than those of single precision runs on 16 processors.

The instantaneous vorticity contours are shown in figure 3.6 for the three cases. It

can be seen from figures 3.3(a) and 3.6(a) that the incompressible, unforced mixing

layer stays laminar for a longer distance with the double precision solver due to smaller

round-off errors. The compressible, unforced mixing layer rolls up around same loca-

tion for both single and double precision computations as shown in figures 3.3(c) and

3.6(c). This suggests that truncation errors of the compressible solver are dictating

the evolution of the mixing layer and therefore, the round-off errors are not signifi-

cant. It should also be noted that the energy equation, required in the compressible

simulations, may have led to the dominance of truncation errors over the round-off

errors. Figures 3.3(b) and 3.6(b) indicate that the instantaneous vorticity contours

for the single and double precision computations are almost identical for the forced,

incompressible mixing layer. Therefore, it can be concluded that the main driving

force behind the evolution of the incompressible, forced mixing layer is the inflow

forcing and not the round-off errors.

The mixing layer growth and the center, obtained using the single and double

precision solvers, are plotted in figures 3.7 and 3.8, respectively. It should be noted

that the incompressible, forced mixing layer is insensitive to the precision of the solver

and correspondingly the growth rate, center, and the Reynolds stresses are the same

25

for the single and double precision cases. Hence, its growth and center are not shown

in figures 3.7 and 3.8 for the double precision case.

A good numerical simulation needs to keep an unforced mixing layer laminar as

long as possible within the computational domain. As shown earlier, the unforced

incompressible mixing layer stays laminar for a longer distance downstream than

the compressible one. Thus the mixing layer is modeled as incompressible for the

rest of the present study. The precision (single or double) of the computations is

not important for the incompressible solver as the evolution of the mixing layer is

dominated by the inflow forcing and not the round-off errors. Therefore, the single

precision solver is used henceforth.

26

x/δω(0)

y/δ

ω(0

)50 100 150 200

-25

0

25

(a) Single precision, incompressible LES - No forcing.

x/δω(0)

y/δ

ω(0

)

50 100 150 200-25

0

25

(b) Single precision, incompressible LES - Random forcing.

x/δω(0)

y/δ

ω(0

)

50 100 150 200-25

0

25

(c) Single precision, compressible LES - No forcing.

x/δω(0)

y/δ

ω(0

)

50 100 150 200-25

0

25

(d) Single precision, compressible - Random forcing.

Figure 3.3. Comparison of the instantaneous vorticity contours after12 FTC’s for incompressible, compressible mixing layers with andwithout inflow forcing obtained using single precision computations.

27

x/δω(0)

δω(x)/

δω(0

)

0 50 100 150 2000

1

2

3

4

5

6

7

8

9

10

Incompressible - unforced

Incompressible - forced

Compressible - unforced

Compressible - forced

Figure 3.4. Comparison of mixing layer growth for incompressible,compressible mixing layers obtained using single precision computa-tions.

x/δω(0)

yc/

δ ω(0

)

0 50 100 150 200-1.5

-1

-0.5

0





Figure 3.5. Center of the mixing layer for incompressible, compressiblemixing layers obtained using single precision computations.

28

x/δω(0)

y/δ

ω(0

)

50 100 150 200-25

0

25

(a) Double precision, incompressible LES - No forcing.

x/δω(0)

y/δ

ω(0

)

50 100 150 200-25

0

25

(b) Double precision, incompressible LES - Random forcing.

x/δω(0)

y/δ

ω(0

)

50 100 150 200-25

0

25

(c) Double precision, compressible LES - No forcing.

Figure 3.6. Comparison of the instantaneous vorticity contours after12 FTC’s for incompressible, compressible mixing layers with andwithout inflow forcing obtained using double precision computations.

29

x/δω(0)

δω(x)/

δω(0

)

0 50 100 150 2000

1

2

3

4

5

6

7

8

9

10

11

12Incompressible - unforced




Incompressible - unforced, double precision

Compressible - unforced, double precision

Figure 3.7. Comparison of mixing layer growth for the single anddouble precision computations.

x/δω(0)

yc/

δ ω(0

)

0 50 100 150 200-1.5

-1

-0.5

0





Incompressible - unforced, double precision

Compressible - unforced, double precision

Figure 3.8. Center of the mixing layer for the single and double pre-cision computations.

30

3.7.4 Incompressible Mixing Layer - Effect of forcing

The incompressible mixing layer LES results obtained in the previous section

are discussed in detail in this section. The effect of forcing is also presented. The

instantaneous vorticity contours after the end of the simulation (corresponding to

12 FTC’s or 42000 time-steps) are shown in figure 3.9 for the unforced and forced

simulations. The growth of the mixing layer is plotted for the two cases in figure 3.10.

It is evident from figure 3.9 that the perturbations cause the mixing layer to roll-up

quickly. The unforced mixing layer stays laminar until x = 120δω(0). The vorticity

growth for x < 120δω(0) is due to the molecular viscosity and the corresponding

vorticy growth rate in this region is dδω(x)dx

= 0.0046. The unforced mixing layer

begins to roll-up downstream due to the amplification of numerical errors involved in

the simulation. In the absence of forcing, there is no pairing of vortices as shown in

figure 3.9(a).

The forced mixing layer rolls up quickly around x = 60δω(0). The vortices are

found to undergo pairing at random locations due to random inflow forcing. Fig-

ure 3.9(b) shows the vortices undergoing pairing process around x = 140δω(0) and

170δω(0) when the mixing layer is forced. This is in contrast to the observations

reported by Stanley and Sarkar [38]. They were able to obtain pairing at fixed loca-

tions using discreet forcing in phase at both the fundamental and first subhormonic

frequencies. The mixing layer grows almost linearly for x > 60δω(0) as shown in

figure 3.10. The linear fit to the growth of the mixing layer for x > 60δω(0) is also

plotted in figure 3.10. The slope of the mixing layer growth is approximately 0.0563,

which is about 12% higher than the value reported by Uzun [39]. This could be due

to the second-order-accurate BCD scheme used in the present simulation, as opposed

to sixth-order compact-differencing scheme used by Uzun [39]. The dissipation asso-

ciated with the lower order spatial discretization may have caused the mixing layer

to grow at a higher rate. The higher time-step used in the simulation may have also

caused the mixing layer to grow faster.

31

The center of the mixing layer, as plotted in figure 3.11, indicates the tendency of

high-speed flow to penetrate into the low-speed flow. This is more pronounced when

the mixing layer is forced as seen in the figure.

x/δω(0)

y/δ

ω(0

)

50 100 150 200-25

0

25

(a) No forcing.

x/δω(0)

y/δ

ω(0

)

50 100 150 200-25

0

25

(b) Random forcing.

Figure 3.9. Comparison of the instantaneous vorticity contours forwith and without forcing for incompressible mixing layer.

32

x/δω(0)

δω(x)/

δω(0

)

0 50 100 150 2000

1

2

3

4

5

6

7

8

9

10

no forcing

forcing

linear fit, slope = 0.0563

Figure 3.10. Incompressible mixing layer growth with and without forcing.

x/δω(0)

yc/

δ ω(0

)

0 50 100 150 200-1.5

-1

-0.5

0

no forcing

forcing

Figure 3.11. Center of the incompressible mixing layer with and without forcing.

33

The simulation data is also examined for self-similarity. The profiles of scaled

mean velocity and Reynolds stresses are plotted at five stream-wise locations. The

scaled mean velocity is plotted against the scaled cross-stream coordinate and is

shown in figure 3.12. The Pope’s error function [49] is also plotted in figure 3.12. The

profiles collapse well for both the unforced and forced cases at all locations.

The profiles of scaled Reynolds stresses are often good gauges to assess self-

similarity. The contours of scaled Reynolds stresses are plotted in figures 3.13, 3.15

and 3.17. A quick comparison of these figures with the instantaneous vorticity con-

tours in figure 3.9 reveals that the vortex roll-up is the main mechanism for the

generation of the Reynolds stresses. The profiles of the Reynolds stresses are plotted

in figures 3.14, 3.16, and 3.18. The Reynolds stresses attain maxima around the

center of the mixing layer. The normal Reynolds stresses (i.e. σxx, σyy) vary from

ξ = −2 to 2, where as the Reynolds shear stress (σxy) varies between ξ = −1 and 1.

Figure 3.13 shows that the centerline value of σxx increases downstream until

x = 160δω(0) and remains constant further downstream. In addition, the centerline

value of σyy increases monotonically all along the centerline as shown in figure 3.15.

However, the corresponding σxy increases until x = 130δω(0) and remains constant for

130δω(0) < x < 160δω(0) and then decreases downstream as shown in figure 3.17. This

is attributed to the localized pairing of the vortices between x = 130δω(0), 160δω(0)

caused by the inflow forcing. The σxx profiles, as plotted in figure 3.14, collapse in the

far downstream suggesting that mixing layer is self-similar in that region. However,

the profiles of the other two stresses do not collapse in this region as shown in figures

3.15 and 3.17. It is believed that increasing the sampling size might result in the

collapse of all the Reynolds stresses downstream. The sampling size is increased to

12 FTC’s from 8 in section 3.8 and the results are discussed in that section.

The peaks of the normalized Reynolds stresses in the entire computational domain

and the mixing layer growth rate are compared with the available experimental and

computational data and is shown in table 3.3. It can be seen from the table that

2-D simulations over-predict the normal Reynolds stresses. It is reiterated that the

34

discrepancies between 2-D DNS of Uzun [39] and the present simulation might be a

combined effect of using a lower-order spatial discretization and a higher time-step.

The time-step is decreased in section 3.9 and the results are compared with Uzun’s

DNS.

The 2-D incompressible forced mixing layer discussed in this section is referred to

as 2-D baseline simulation in the subsequent discussion.

Table 3.3 Comparison of the normalized peak Reynolds stresses andthe growth rate of the 2-D baseline simulation with the data in theliterature.

Reωσxx

∆U2

σyy

∆U2

σxy

∆U2

1η

dδω(x)dx

Reference

- 0.031 0.019 0.009 0.19 Experiment [25]

- 0.036 0.014 0.013 0.16 Experiment [28]

1,800 0.032 0.020 0.010 0.163 Experiment [29]

720 0.040 0.084 0.023 0.15 2-D DNS [38]

720 0.048 0.078 0.012 0.15 Uzun’s 2-D DNS [39]

720 0.044 0.080 0.015 0.17 Present 2-D LES

35

ξ

f(ξ)

-3 -2 -1 0 1 2 3-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δω(0)x = 190 δ

ω(0)

error function

(a) No forcing.

ξ

f(ξ)

-3 -2 -1 0 1 2 3-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δω(0)x = 190 δ

ω(0)

error function

(b) Random forcing.

Figure 3.12. Comparison of the profiles of scaled velocity with andwithout forcing for incompressible mixing layer.

36

(a) No forcing.

(b) Random forcing.

Figure 3.13. Comparison of the contours of σxx/∆U2 with and with-out forcing for incompressible mixing layer.

37

ξ

σxx/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)

(a) No forcing.

ξ

σxx/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)

(b) Random forcing.

Figure 3.14. Comparison of the profiles of σxx/∆U2 with and withoutforcing for incompressible mixing layer.

38

(a) No forcing.

(b) Random forcing.

Figure 3.15. Comparison of the contours of σyy/∆U2 with and withoutforcing for incompressible mixing layer.

39

ξ

σyy/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)

(a) No forcing.

ξ

σyy/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)

(b) Random forcing.

Figure 3.16. Comparison of the profiles of σyy/∆U2 with and withoutforcing for incompressible mixing layer.

40

(a) No forcing.

(b) Random forcing.

Figure 3.17. Comparison of the contours of σxy/∆U2 with and withoutforcing for incompressible mixing layer.

41

ξ

σxy/∆U2

-3 -2 -1 0 1 2 3-0.02

-0.018

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)

(a) No forcing.

ξ

σxy/∆U2

-3 -2 -1 0 1 2 3-0.02

-0.018

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)

(b) Random forcing.

Figure 3.18. Comparison of the profiles of σxy/∆U2 with and withoutforcing for incompressible mixing layer.

42

3.8 Effect of Statistical Sample Size

The 2-D baseline simulation is ran for 4 FTC’s to account for transients followed

by 8 additional FTC’s to obtain the mean and RMS values of relevant quantities.

In an LES, typically the first-order quantities such as mean values reach statistically

stationary states quicker than the second-order statistics (i.e. Reynolds stresses).

Therefore, it is important to verify that all the statistics are converged at end of an

LES.

The baseline simulation is advanced in time for an additional sampling period

of 4 FTC’s and the results are post-processed after 12 FTC’s. Figure 3.19 shows

the comparison of the profiles of scaled Reynolds stresses with those of the baseline

simulation. As seen in the figure, the profiles remain mostly unaffected suggesting

that the original samping size of 8 FTC’s ensures that the second-order statistics

converge. As expected, the growth and center of the mixing layer that are obtained

from the mean velocity, as shown in figures 3.20 and 3.21, also remain the same.

Computations involving LES, typically, require significant amount of resources in

terms of the number of processors and wall clock time. Thus, it is also worthwhile

to investigate whether the transient sample size can be reduced in order to decrease

the simulation turn-around time. For this purpose, the size of the transient sample

is reduced to 2 FTC’s from 4 and the statistics are gathered for the next 8 FTC’s.

It should be noted that the random number generator of the current inflow forcing

is reinitialized just before the sampling begins in all the present 2-D simulations.

The instantaneous vorticity contours obtained at the end of the simulation (or 10

FTC’s) are compared with the baseline simulation in figure 3.22. It can be seen

from figure 3.22 that the contours are virtually identical. This indicates that the

initial state of the flow field does not play a role in the long term evolution of the

mixing layer. Consequently, the mixing layer growth rate and the center, as shown

in figures 3.20 and 3.21, are nearly the same for the simulations allocating 2 and 4

43

FTC’s for the transients. This is observed at the final stages of the current work, and

hence, a transient sample size of 4 FTC’s is used for the subsequent simulations.

44

ξ

σxx/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)

(a) σxx/∆U2, 8 FTC’s.

ξ

σxx/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)

(b) σxx/∆U2, 12 FTC’s.

ξ

σyy/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)

(c) σyy/∆U2, 8 FTC’s.

ξ

σyy/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)

(d) σyy/∆U2, 12 FTC’s.

ξ

σxy/∆U2

-3 -2 -1 0 1 2 3-0.02

-0.018

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)

(e) σxy/∆U2, 8 FTC’s.

ξ

σxy/∆U2

-3 -2 -1 0 1 2 3-0.02

-0.018

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)

(f) σxy/∆U2, 12 FTC’s.

Figure 3.19. Effect of the statistical sample size on the profiles of Reynolds stresses

45

x/δω(0)

δω(x)/

δω(0

)

0 50 100 150 2000

1

2

3

4

5

6

7

8

9

10 unforced

forced, sampling = 8 FTC


forced, sampling = 8 FTC with 2 FTC for transients

Figure 3.20. Comparison of mixing layer growth after 8 and 12 FTC’s

x/δω(0)

yc/

δ ω(0

)

0 50 100 150 200-1.5

-1

-0.5

0

unforced



forced, sampling = 8 FTC with 2 FTC for transients

Figure 3.21. Comparison of center of the mixing layer after 8 and 12 FTC’s

46

x/δω(0)

y/δ

ω(0

)

50 100 150 200-25

0

25

(a) 2 FTC’s for transients and 8 for sampling.

x/δω(0)

y/δ

ω(0

)

50 100 150 200-25

0

25

(b) 4 FTC’s for transients and 8 for sampling.

Figure 3.22. Comparison of the instantaneous vorticity contours atthe end of the simulations with 2 and 4 FTC’s for transients followedby 8 FTC’s for sampling.

47

3.9 Temporal Sensitivity

The time-step is reduced by half to ∆t = 5 × 10−8 s to match the time-step

used in Uzun’s 2-D DNS [39]. He used a time-step of 6 × 10−8 s in his simulation.

The decreased time-step increases temporal resolution of the large-scale structures

involved in the flow field and captures the fine-scale structures that are otherwise

unresolved with higher time-steps. However, it increases the resources required for

the simulation. Two Sampling Frequencies (SF) are considered in this section to

evaluate the temporal sensitivity of the results.

1. LowT SF 2

In this simulation, the data is sampled after every 2 time steps so the

physical time of each sample is same as that of the baseline simulation. It is to

be noted that the sample size is the same for both the baseline and this case.

2. LowT SF 1

In this simulation, the data is sampled after each time step, doubling the

sample size with respect to the baseline. Therefore, the statistical data in this

case contains the contribution from the time steps that are not present in the

baseline case.

The tags of physical time of each sample in the LowT SF 1, LowT SF 2 and the

baseline simulations are given schematically in table 3.4. It is important to note

that all the simulations have been time-advanced to the same physical time. The

simulations are ran for 4 FTC’s for transients and the flow statistics are sampled

for 8 FTC’s. Each FTC requires 7000 time steps, where as this number is 3500 in

the baseline simulation. The LowT SF 1 simulation closely matches the Uzun’s 2-D

DNS. In his simulation, the flow has been sampled each time step for 35000 time-steps

corresponding to 6 FTC’s. In the LowT SF 1 simulation, the flow has been sampled

for 56000 time-steps corresponding to 8 FTC’s.

The growth and center of the mixing layer for each case are compared with the

baseline simulation in figure 3.23. The figure suggests that the growth and center for

48

Table 3.4 Physical time of each sample in the baseline, LowT SF 1 andLowT SF 2 simulations. The value of n is 28000 in all the simulations.

Baseline: t1 t2 t3 . . . . tn

LowT SF 2: t1 t2 t3 . . . . tn

LowT SF 1: t1 t1.5 t2 t2.5 t3 . . . . tn

the two simulations match closely with the results of baseline case for x < 150δω(0).

The vorticity growth of the LowT SF 1 simulation matches better with the baseline

for x > 150δω(0) as shown in figure 3.23(b). The deviation in the growth rate in

this region is higher for the LowT SF 2 simulation as shown in figure 3.23(a). This

is probably due to the reduced sample size with respect to the LowT SF 1 simula-

tion. The vorticity growth rates for the LowT SF 1, LowT SF 2 and the baseline

simulations are listed in table 3.5.

Table 3.5 Mixing layer growth rates for the baseline, LowT SF 1 andLowT SF 2 simulations.

Case Growth rate, Deviation from ∆t, s Sampling

dδω(x)dx

Uzun’s DNS [39] time

Baseline 0.056 12% 10−7 8 FTC’s

LowT SF 2 0.055 10% 5× 10−8 8 FTC’s

LowT SF 1 0.052 4% 5× 10−8 8 FTC’s

Uzun’s 2-D DNS 0.050 - 6× 10−8 6 FTC’s

The LowT SF 1 simulation closely matches the growth rate observed by Uzun [39].

The Reynolds stress profiles for this simulation are shown in figure 3.24. The peak

values of the Reynolds stress profiles at various downstream locations are compared for

the baseline, LowT SF 1 and Uzun’s 2-D DNS [39] and listed in table 3.6. The table

49

x/δω(0)

δω(x)/

δω(0

)

0 50 100 150 2000

1

2

3

4

5

6

7

8

9

10

∆t = 10-7 s∆t = 5*10

-8s

(a) LowT SF 2

x/δω(0)

δω(x)/

δω(0

)

0 50 100 150 2000

1

2

3

4

5

6

7

8

9

10

∆t = 10-7 s∆t = 5*10

-8s

(b) LowT SF 1

x/δω(0)

yc/

δ ω(0

)

0 50 100 150 200-1.5

-1

-0.5

0

∆t = 10-7 s∆t = 5*10

-8s

(c) LowT SF 2

x/δω(0)

yc/

δ ω(0

)

0 50 100 150 200-1.5

-1

-0.5

0

∆t = 10-7 s∆t = 5*10

-8s

(d) LowT SF 1

Figure 3.23. Effect of time-step on the incompressible 2-D LES

shows that both the baseline and LowT SF 1 simulations match the peak Reynolds

stresses predicted by Uzun reasonably well for x > 130δω(0).

In summary, it can be concluded that the flow sampling at every time-step im-

proves the prediction of the mixing layer growth rate with lower time-step.

During the earlier stages of this work, it is found that the pressure-based LES

solver is unstable at time-steps lower than 10−7 seconds for compressible mixing

layers. This is described in detail in Appendix B.

50

ξ

σxx/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)

(a) σxx/∆U2, 8 FTC’s.

ξσyy/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)

(b) σyy/∆U2, 8 FTC’s.

ξ

σxy/∆U2

-3 -2 -1 0 1 2 3-0.02

-0.018

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)

(c) σxy/∆U2, 8 FTC’s.

Figure 3.24. The profiles of Reynolds stresses for the incompressiblemixing layer simulation with ∆t = 5× 10−8 s.

51

Table 3.6 Comparison of the peaks of Reynolds stresses for the base-line, DNS of Uzun [39] and the LowT SF 1 simulations.

Baseline LowT SF 1 Uzun’s DNS [39]

dδω(x)/dx 0.056 0.052 0.050

70δω(0) 0.004 0.002 0.013

100δω(0) 0.015 0.013 0.034

Peak σxx/∆U2 at x = 130δω(0) 0.036 0.035 0.042

160δω(0) 0.044 0.040 0.049

190δω(0) 0.043 0.052 0.046

70δω(0) 0.007 0.004 0.027

100δω(0) 0.039 0.038 0.051

Peak σyy/∆U2 at x = 130δω(0) 0.054 0.052 0.071

160δω(0) 0.069 0.070 0.078

190δω(0) 0.080 0.070 0.080

70δω(0) 0.004 0.002 0.008

100δω(0) 0.007 0.008 0.015

Peak |σxy/∆U2| at x = 130δω(0) 0.015 0.016 0.014

160δω(0) 0.015 0.012 0.009

190δω(0) 0.012 0.014 0.011

52

3.10 Effect of Buffer Zone and Stream-wise Domain Size

The baseline computational domain consists of a physical region and a buffer

zone attached to it. In the baseline mesh, the length of the physical and buffer zones

are 200δω(0) and 150δω(0), respectively. The pressure outflow boundary condition is

imposed on the exit of the buffer zone.

The buffer zone (or sponge zone) suppresses the vorticity before it reaches the

exit boundary. The grid is stretched in the stream-wise direction in the buffer zone.

The coarser grid in the buffer zone increases the numerical dissipation, which damps

the vorticity. Thus, the damping is achieved implicitly through grid stretching as

opposed to explicit damping used by Uzun [39]. The effectiveness of such an implicit

damping method is investigated in this section. The stream-wise domain length is

also increased in this study to evaluate the presence of the buffer zone on a longer

physical domain. Three simulations are performed for this purpose.

1. Simulation-I

The first simulation is performed with the buffer zone removed from the

baseline grid. The results from this simulation are compared with the baseline

simulation to assess the use of buffer zone. The pressure outflow BC is imposed

at the exit boundary of the physical region ,which is located at x = 200δω(0).

2. Simulation-II

The second simulation uses a mesh with a longer physical domain and a

buffer zone. The stream-wise length of the physical domain is increased from

200δω(0) to 250δω(0). The nodes in the stream-wise direction are increased to

612 from 501 to maintain the baseline stream-wise resolution. The length of

the buffer zone is kept constant at 150δω(0). The pressure outflow BC is used

at the exit of the buffer zone by specifying a uniform back pressure. The effect

of the stream-wise domain size in the presence of buffer zone is evaluated with

this simulation.

53

3. Simulation-III

The last simulation is performed using only the physical domain of the

mesh used in second simulation. The length of the computational domain is

250δω(0) and the outflow boundary condition is applied at x = 250δω(0).

The stream-wise lengths of the physical and buffer zones for the meshes considered

in these simulations are listed in table 3.7. All the three simulations are time-advanced

to the same physical time as that of the baseline simulation. The solver settings are

identical for all the simulations. In addition, the same perturbation profile is used at

the inlet for all the simulations at a given time step.

Table 3.7 Length of the physical and buffer zones for the simulationsdescribed in 3.10.

Buffer zone xphysical xbuffer

Baseline X 200δω(0) 150δω(0)

Simulation-I × 200δω(0) -

Simulation-II X 250δω(0) 150δω(0)

Simulation-III × 250δω(0) -

The contours of instantaneous vorticity are shown for the three simulations in fig-

ure 3.25 after t = 4.2×10−3 s. The contours for the baseline simulation are also shown

in the figure for reference. The vorticity contours, shown in figures 3.25(c), 3.25(d) for

the simulation with longer physical domain clearly indicate that the spacing between

the vortices doubles after the pairing. Figure 3.25 suggests that the stream-wise size

of the physical zone as well as the presence of buffer zone have no impact on the time

history of the flow field. This evidence is further bolstered by the observation of nearly

identical vorticity thickness growths for all the simulations as shown in figure 3.26.

The center of the mixing layer as predicted by all the simulations is shown in

figure 3.27. The figure illustrates the discrepancy in the center of the mixing layer

near the exit of the computational domain when the buffer zone is absent. This is

54

mainly due to marginal reduction of the far-field mean stream-wise velocity close to

the exit boundary. The constant pressure specified at the outflow (or exit) boundary

slows down the flow mildly. The profiles of scaled velocity are plotted in figure 3.28.

The slight reduction of the mean stream-wise velocity f(ξ), around the exit boundary

for ξ < −1 & ξ > 1 can be seen in figures 3.28(c), 3.28(d) when the buffer zone is

absent. The value of f(ξ), as defined in equation 3.11, must be equal to -0.5 for ξ < −1

and 0.5 for ξ > 1. Therefore, f(ξ) < −0.5 (or 0.5) when the mean stream-wise velocity

U < U2 (or U1). The figure 3.29 also shows the effect of the outflow BC on the edge

of the mixing layer. It is found that the uniform pressure outflow BC does not allow

the flow to be entrained into the mixing layer resulting in slightly lower velocity at

the edges of the mixing layer. The mean stream lines are plotted in figure 3.30 with

and without the buffer zone for the baseline simulation. As seen in the figure, the

stream lines in the presence of the buffer zone curve slightly towards the center of

the mixing layer at x = 200δω(0) indicating flow entrainment. They almost intersect

the exit boundary at normal angles when the buffer zone is absent confirming the

inhibition of flow entrainment caused by the outflow BC. The difference is noticeable

for the stream line starting at y = −15δω(0) in the figures 3.30(a), 3.30(b).

The contours of scaled Reynolds stresses are shown in figures 3.31, 3.33 and 3.35.

The corresponding profiles are shown in figures 3.32, 3.34 and 3.36. The Reynolds

shear stress contours have a distinct eye-shaped patch when there is no buffer zone.

Figures 3.31(b), 3.31(d) show the patch near the end of the computational domain

due to the influence of the pressure outlet boundary condition. The straightening

of the stream lines around the exit boundary may have caused this anomaly. The

effect of the BC, however, is felt roughly 10δω(0) upstream of the exit boundary for

xphysical = 200δω(0) & 250δω(0). Figures 3.33 and 3.35 suggest that the contours of

scaled σxx, σyy are tilted slightly upwards and downwards, respectively near the exit

boundary. The levels of the Reynolds stresses in the rest of the domain agree pretty

well with the results obtained using the buffer zone. The profiles of all stresses collapse

well for x = 190δω(0), 220δω(0) and 250δω(0), as shown in figures 3.32(b), 3.34(b)

55

x/δω(0)

y/δ

ω(0

)50 100 150 200

-25

0

25

(a) Baseline.

x/δω(0)

y/δ

ω(0

)

50 100 150 200-25

0

25

(b) Baseline without buffer zone (Simulation - I).

x/δω(0)

y/δ ω

(0)

50 100 150 200 250-25

0

25

(c) xphysical = 250δω(0) + buffer (Simulation - II).

x/δω(0)

y/δ ω

(0)

50 100 150 200 250-25

0

25

(d) xphysical = 250δω(0), no buffer(Simulation - III).

Figure 3.25. Comparison of the instantaneous vorticity contours aftert = 4.2× 10−3 s with and without the buffer zone.

56

x/δω(0)

δω(x)/

δω(0

)

0 50 100 150 200 2500

1

2

3

4

5

6

7

8

9

10

11

12

xphysical

= 200δω(0) + buffer zone

xphysical

= 200δω(0), no buffer zone

xphysical


xphysical


Figure 3.26. Comparison of the mixing layer growth with and withoutthe buffer zone.

x/δω(0)

yc/δ

ω(0

)

0 50 100 150 200 250-1.5

-1

-0.5

0

xphysical


xphysical


xphysical


xphysical


Figure 3.27. Comparison of the mixing layer center with and withoutthe buffer zone.

and 3.36(b) suggesting self-similarity in this region. The Reynolds number based

on the vorticity thickness corresponding to the beginning of this region is around

57

ξ

f(ξ)

-3 -2 -1 0 1 2 3-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δω(0)x = 190 δ

ω(0)

error function

(a) Baseline

ξ

f(ξ)

-3 -2 -1 0 1 2 3-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δω(0)x = 190 δ

ω(0)

error function

(b) xphysical = 250δω(0) + buffer

ξ

f(ξ)

-3 -2 -1 0 1 2 3-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δω(0)x = 190 δ

ω(0)

error function

(c) Baseline without buffer zone

ξ

f(ξ)

-3 -2 -1 0 1 2 3-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δω(0)x = 190 δ

ω(0)

error function

(d) xphysical = 250δω(0), no buffer

Figure 3.28. Comparison of the profiles of scaled velocity with andwithout the buffer zone.

6000. This is in agreement with the experimental observations in the literature. In

experiments, the transition of mixing layer occurs at Reynolds numbers (based on

visual thickness and velocity difference) between 5600 and 17000 [51], which leads to

self-similarity.

In summary, it can be concluded that the effect of the pressure outflow boundary

is seen within 10 vorticity thicknesses from the exit boundary. Therefore, the buffer

zone is required at least from the perspective of obtaining accurate statistics near

58

y/δω(0)

f(ξ)

-25 -20 -15 -10 -5 0 5 10 15 20 25-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)

(a) Baseline without the buffer zone.

y/δω(0)

f(ξ)

-25 -20 -15 -10 -5 0 5 10 15 20 25-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)

(b) xphysical = 250δω(0), no buffer.

Figure 3.29. Velocity profiles for the simulations with no buffer zones.

the exit boundary and to prevent inhibition of the flow entrainment into the mixing

layer caused by the exit boundary condition. The results also suggest that increasing

the stream-wise size of the physical domain has no effect on the flow evolution in the

59

x/δω(0)

y/δ

ω(0

)

170 180 190 200-15

-10

-5

0

5

10

15

(a) With buffer zone.

x/δω(0)

y/δ

ω(0

)

170 180 190 200-15

-10

-5

0

5

10

15

(b) No buffer zone.

Figure 3.30. Comparison of the mean stream lines with and withoutthe buffer zone.

presence of the buffer zone. The mixing layer achieves a state of self-similarity for

x > 190δω(0) corresponding to a Reynolds number of 6000.

60

(a) Baseline

(b) Baseline without buffer zone

(c) xphysical = 250δω(0) + buffer


Figure 3.31. Comparison of scaled σxy contours with and without the buffer zone.

61

ξ

σxy/∆U2

-3 -2 -1 0 1 2 3-0.02

-0.018

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)

(a) Baseline

ξσxy/∆U2

-3 -2 -1 0 1 2 3-0.02

-0.018

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)x = 220 δω(0)x = 250 δω(0)


ξ

σxy/∆U2

-3 -2 -1 0 1 2 3-0.02

-0.018

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)


ξ

σxy/∆U2

-3 -2 -1 0 1 2 3-0.02

-0.018

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)x = 220 δω(0)


Figure 3.32. Comparison of the profiles of scaled σxy with and withoutthe buffer zone.

62

(a) Baseline




Figure 3.33. Comparison of scaled σxx contours for with and withoutthe buffer zone.

63

ξ

σxx/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)

(a) Baseline

ξσxx/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)x = 220 δω(0)x = 250 δω(0)


ξ

σxx/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)


ξ

σxx/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)x = 220 δω(0)


Figure 3.34. Comparison of the profiles of scaled σxx with and withoutthe buffer zone.

64

(a) Baseline




Figure 3.35. Comparison of the scaled σyy contours for with andwithout the buffer zone.

65

ξ

σyy/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)

(a) Baseline

ξσyy/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)x = 220 δω(0)x = 250 δω(0)


ξ

σyy/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)


ξ

σyy/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)x = 220 δω(0)


Figure 3.36. Comparison of the profiles of scaled σyy with and withoutthe buffer zone.

66

3.11 Optimization of 2-D Baseline Grid Size

Due to the limited computational resources at hand, it is desired to reduce the

node count in the baseline 2-D mesh in order to maximize the span-wise grid points

in the future 3-D mesh. The objective, thus, is to optimize the 2-D mesh without

compromising the accuracy of the simulation results.

The cross-stream domain size is varied to decrease the number of nodes. Once the

optimal size of the cross-stream is obtained, the stretching in that direction is varied

to reduce the nodes further. The sensitivity of the results to the cross-stream domain

size is presented in section 3.11.1. The grid stretching investigation is described in

section 3.11.2.

3.11.1 Cross-stream Domain Size

The baseline domain extends from −300δω(0) to 300δω(0) in the cross-stream

(y) direction. The domain size is shrinked in steps of 50δω(0) in the −y and +y

directions, keeping the node distribution in the y-direction unchanged with respect

to the baseline mesh. Thus, the cross-stream grid stretching ratio is kept constant at

1.01. The number of grid points spanning the entire cross-stream direction for the

meshes considered here is given in table 3.8. The first entry in the table corresponds

to the baseline grid. The stream-wise node distribution is identical for all the grids.

The smallest domain size considered in this study extends from y = −25δω(0) to

25δω(0).

All the simulations have identical solver settings and the same perturbation profile

is used at the inflow at a given time step. The simulations are advanced in time

corresponding to 12 FTC’s with a time step of 10−7 s.

The instantaneous vorticity contours and vorticity growths are plotted in fig-

ures 3.37 and 3.38, respectively. The figures suggest that the vorticity contours as

well as the growth rates are nearly identical for all the simulations. The profiles of

Reynolds stresses are also identical for all the cases. However, the scaled velocity

67

profiles, shown in in figure 3.39, demonstrate that the free stream velocity does not

match the inlet free stream velocity for smaller domains. The effect is more pro-

nounced for the smallest domain as shown in figure 3.39(f). This is similar to the

effect of the pressure outlet boundary observed earlier in section 3.10. It is found that

the symmetry boundary condition applied on the high- and low-speed side boundaries

does not allow the flow to be entrained into the mixing layer resulting in marginal

reduction of the stream-wise velocity.

It is believed that a velocity BC at these boundaries might allow the flow entrain-

ment. Hence, a new simulation is performed on the smallest domain enforcing the free

stream velocity on the top and bottom boundaries by imposing a velocity boundary

condition. The growth of the mixing layer obtained from this simulation is shown

in figure 3.38 with a (*) label. As shown in the figure, the mixing layer growth is

unchanged due to the velocity boundary condition. The velocity profiles for this case

are shown in figure 3.40, reiterate the free-stream velocity deficit observed earlier.

In summary, the results indicate that the mixing layer results are insensitive to the

cross-stream domain size except for smaller sizes. The boundary condition is found

to inhibit the flow entrainment at the exit for smaller domains. A non-reflecting

boundary condition is expected to work better in such simulations.

At this point, the simulation with y extending from −50δω(0) to 50δω(0) is taken

as a revised baseline simulation. The growth rate remains unchanged for this simu-

lation. The profiles of the Reynolds stresses are also similar to those of the baseline

simulation. The number of grid points is thus reduced to half without compromising

the quality of the results.

68

Table 3.8 Number of nodes spanning the y-direction for the meshesconsidered in section 3.11. The baseline mesh is indicated by a (*).

Domain extentions in y Nodes in y Total nodes

−300δω(0) to 300δω(0)(∗) 575 331,200

−200δω(0) to 200δω(0) 503 289,728

−150δω(0) to 150δω(0) 453 260,928

−100δω(0) to 100δω(0) 385 221,760

−50δω(0) to 50δω(0) 277 159,552

−25δω(0) to 25δω(0) 185 106,560

69

x/δω(0)

y/δ

ω(0

)

50 100 150 200-25

0

25

(a) Baseline: y = −300δω(0) to +300δω(0).

x/δω(0)

y/δ

ω(0

)50 100 150 200

-25

0

25

(b) y = −200δω(0) to +200δω(0).

x/δω(0)

y/δ

ω(0

)

50 100 150 200-25

0

25

(c) y = −150δω(0) to +150δω(0).

x/δω(0)

y/δ

ω(0

)

50 100 150 200-25

0

25

(d) y = −100δω(0) to +100δω(0).

x/δω(0)

y/δ

ω(0

)

50 100 150 200-25

0

25

(e) y = −50δω(0) to +50δω(0).

x/δω(0)

y/δ

ω(0

)

50 100 150 200-25

0

25

(f) y = −25δω(0) to +25δω(0).

Figure 3.37. Effect of cross-stream domain size on the contours ofvorticity magnitude.

70

x/δω(0)

δω(x)/

δω(0

)

0 50 100 150 2000

1

2

3

4

5

6

7

8

9

10

y/δω(0) = -25 to 25

y/δω(0) = -50 to 50

y/δω(0) = -100 to 100y/δ

ω(0) = -150 to 150

y/δω(0) = -200 to 200y/δω(0) = -300 to 300y/δω(0) = -25 to 25 (*)

Figure 3.38. Comparison of the mixing layer growth as the cross-domain size is reduced.

71

ξ

f(ξ)

-3 -2 -1 0 1 2 3-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δω(0)x = 190 δ

ω(0)

error function

(a) Baseline: y = −300δω(0) to +300δω(0).

ξ

f(ξ)

-3 -2 -1 0 1 2 3-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δω(0)x = 190 δ

ω(0)

error function

(b) y = −200δω(0) to +200δω(0).

ξ

f(ξ)

-3 -2 -1 0 1 2 3-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δω(0)x = 190 δ

ω(0)

error function

(c) y = −150δω(0) to +150δω(0).

ξ

f(ξ)

-3 -2 -1 0 1 2 3-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δω(0)x = 190 δ

ω(0)

error function

(d) y = −100δω(0) to +100δω(0).

ξ

f(ξ)

-3 -2 -1 0 1 2 3-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δω(0)x = 190 δ

ω(0)

error function

(e) y = −50δω(0) to +50δω(0).

ξ

f(ξ)

-3 -2 -1 0 1 2 3-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δω(0)x = 190 δ

ω(0)

error function

(f) y = −25δω(0) to +25δω(0).

Figure 3.39. Effect of y domain size on the profiles of scaled velocity

72

x/δω(0)

f(ξ)

-25 -20 -15 -10 -5 0 5 10 15 20 25-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)

(a) y = −25δω(0) to +25δω(0), symmetry

y/δω(0)

f(ξ)

-25 -20 -15 -10 -5 0 5 10 15 20 25-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)

(b) y = −25δω(0) to +25δω(0), velocity inlet

Figure 3.40. Velocity profiles for the mesh extending from y =−25δω(0) to +25δω(0) with symmetry and velocity boundary con-ditions on the side boundaries.

73

3.11.2 Grid Stretching

The revised baseline simulation obtained in section 3.11 after trimming the cross-

stream domain size is examined further to optimize the node count. The grid stretch-

ing in the cross-stream direction is varied for this purpose keeping the same stream-

wise node distribution from the revised baseline grid. The grid stretching ratio in the

revised baseline grid is 1.01. Table 3.9 lists the three grid stretching ratios and the

corresponding total node counts in the meshes considered in this investigation.

The three simulations have identical solver settings and advanced to the same

physical time. The perturbation profile at the inflow is again the same for all the three

simulations for a given time step. The instantaneous vorticity contours obtained after

42,000 time steps (or 12 FTC’s) are compared with the revised baseline simulation

in figure 3.41. The figure shows that the evolution of vorticity is unaffected by the

stretching in y-direction. The growth and center of the mixing layer, as shown in

figures 3.42 and 3.43, also indicate the insensitivity of the mixing layer with respect

to the grid stretching. Figures 3.44, 3.45, and 3.46 show the scaled Reynolds stresses

σxx, σyy and σxy, respectively. The profiles are nearly identical for all the simulations.

A close examination at the peak values of each profile at a given stream-wise location

reveals that the peaks of σxx remain the same and those of σyy drop slightly as the

grid stretching ratio is increased, The peaks of σxy, in particular at x = 160δω(0),

increase as the grid is stretched.

At this point of the investigation, the 2-D mesh with the grid stretching ratio

of 1.10 is used to construct a baseline mesh for the 3-D mixing layer simulations

discussed later.

74

Table 3.9 Number of nodes spanning the y-direction for the gridstretching study in section 3.11.2. The revised baseline simulationis indicated by a (*).

Stretching in y-direction Nodes in y Total nodes

1.01(*) 277 159,552

1.025 175 100,800

1.050 115 66,240

1.100 73 42,048

75

x/δω(0)

y/δ

ω(0

)50 100 150 200

-25

0

25

(a) Revised baseline

x/δω(0)

y/δ

ω(0

)

50 100 150 200-25

0

25

(b) Grid stretching factor = 1.025

x/δω(0)

y/δ

ω(0

)

50 100 150 200-25

0

25

(c) Grid stretching factor = 1.050

x/δω(0)

y/δ

ω(0

)

50 100 150 200-25

0

25

(d) Grid stretching factor = 1.100

Figure 3.41. Effect of grid stretching - Comparison of the instanta-neous vorticity contours.

76

x/δω(0)

δω(x)/

δω(0

)

0 50 100 150 2000

1

2

3

4

5

6

7

8

9

10

revised baseline

grid stretching factor = 1.025



Figure 3.42. Effect of grid stretching on the mixing layer growth.

x/δω(0)

yc/

δ ω(0

)

0 50 100 150 200-1.5

-1

-0.5

0

revised baseline




Figure 3.43. Effect of grid stretching on the mixing layer center.

77

ξ

σxx/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)

(a) Revised baseline.

ξσxx/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)

(b) Grid stretching factor = 1.025.

ξ

σxx/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)

(c) Grid stretching factor = 1.050.

ξ

σxx/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)

(d) Grid stretching factor = 1.100.

Figure 3.44. Effect of grid stretching - comparison of the profiles of σxx/∆U2.

78

ξ

σyy/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)


ξσyy/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)


ξ

σyy/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)


ξ

σyy/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)


Figure 3.45. Effect of grid stretching - comparison of the profiles of σyy/∆U2.

79

ξ

σxy/∆U2

-3 -2 -1 0 1 2 3-0.02

-0.018

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)


ξσxy/∆U2

-3 -2 -1 0 1 2 3-0.02

-0.018

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)


ξ

σxy/∆U2

-3 -2 -1 0 1 2 3-0.02

-0.018

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)


ξ

σxy/∆U2

-3 -2 -1 0 1 2 3-0.02

-0.018

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)


Figure 3.46. Effect of grid stretching - comparison of the profiles of σxy/∆U2.

80

3.11.3 Grid Coarsening

The stream-wise grid spacing is doubled everywhere to 0.8δω(0) in the revised

baseline mesh to decrease the mesh size further. The solver settings are unchanged.

The simulation is advanced to the same physical time as that of the revised baseline.

The inflow perturbation profile is also the same at a given time step.

The instantaneous vorticity contours are plotted at the end of the simulation

corresponding to 12 FTC’s in figure 3.47. The contours for the revised baseline

simulation are also shown in the figure. It is evident from the figure that the coarsened

grid dissipates the vortices for x > 70δω(0). The cores of the vortices in this region

are devoid of vorticity due to the increased grid dissipation. The vortex roll-up is

delayed until x = 140δω(0) due to the re-stabilization of the mixing layer around

x = 90δω(0). The slope of the vorticity thickness growth in the linear region is

increased to 0.06, corresponding to an increase of 7% with respect to the revised

baseline simulation. The contours of the Reynolds stresses are shown in figure 3.48

and the corresponding profiles are plotted in figure 3.49. As shown in these figures,

the development of the stresses is delayed until x = 120δω(0). The Reynolds shear

stress for this coarsened mesh, shown in figures 3.48(f) and 3.49(f), has unusually

high values for 130δω(0) < x < 170δω(0). The reason for this anomalous behavior is

unknown.

In conclusion, the stream-wise grid spacing ,which is as large as the initial vorticity

thickness, suppresses the vortices and stabilizes the mixing layer. It also verifies that

the revised baseline mesh is adequate for the resolution of the large scale structures in

the flow field. The Reynolds shear stresses are over predicted in the region 130δω(0) <

x < 170δω(0). Hence, the stream-wise spacing in the revised baseline mesh cannot be

increased to optimize the node count further.

81

x/δω(0)

y/δ

ω(0

)

50 100 150 200-25

0

25

(a) Revised baseline

x/δω(0)

y/δ

ω(0

)

50 100 150 200-25

0

25

(b) Coarsened in stream-wise direction by a factor of 2

Figure 3.47. Effect of increasing the stream-wise spacing - comparisonof the instantaneous vorticity contours.

82

(a) Baseline: σxx/∆U2. (b) Stream-wise spacing doubled.

(c) Baseline: σyy/∆U2. (d) Stream-wise spacing doubled.

(e) Baseline: σxy/∆U2. (f) Stream-wise spacing doubled.

Figure 3.48. Effect of increasing the stream-wise spacing on the pro-files of Reynolds stresses.

83

ξ

σxx/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)

(a) Baseline: σxx/∆U2.

ξ

σxx/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)

(b) Stream-wise spacing doubled.

ξ

σyy/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)

(c) Baseline: σyy/∆U2.

ξ

σyy/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)

(d) Stream-wise spacing doubled.

ξ

σxy/∆U2

-3 -2 -1 0 1 2 3-0.02

-0.018

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)

(e) Baseline: σxy/∆U2.

ξ

σxy/∆U2

-3 -2 -1 0 1 2 3-0.022

-0.02

-0.018

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

x = 70 δω(0)x = 100 δ

ω(0)

x = 130 δω(0)x = 160 δ

ω(0)

x = 190 δω(0)

(f) Stream-wise spacing doubled.

Figure 3.49. Effect of increasing the stream-wise spacing on the pro-files of Reynolds stresses.

84

3.12 Grid Resolution Study

As the simulations of this work utilize a second-order-accurate spatial discretiza-

tion scheme, more grid points are needed to achieve accuracies comparable to those

of higher-order schemes used in Uzun’s 2-D DNS [39]. The resolution of the revised

baseline grid is increased to investigate whether the second-order BCD scheme is

capturing the fine scales well with the revised baseline mesh.

As mentioned before, the revised baseline grid uses a uniform spacing of 0.4δω(0)

in the stream-wise (x) direction with stretching employed in the buffer zone. The

grid is stretched at a ratio of 1.01 in the cross-stream (y) direction with a minimum

spacing of 0.16δω(0) around the centerline. The number of nodes spanning across

the initial mixing layer thickness is 7 grid points. The following modifications to the

revised baseline grid are considered in this study.

1. The revised baseline mesh is refined only in y direction by a factor of 2. There-

fore, this mesh has a spacing of 0.08δω(0) in the y-direction. This change in-

creases the number of grid points across the mixing layer at the inlet to 13.

2. The mesh is refined in only x direction by a factor of 2. The stream-wise spacing

reduces to 0.2δω(0).

3. The mesh is refined in x and y directions by a factor of√

2.

4. At last, the mesh is refined in x and y directions by a factor of 2.

All the simulations are advanced to the same physical time. The inflow pertur-

bation profiles are obtained from equation 3.8 by using the minimum grid spacing

(∆y0) of the revised baseline mesh. These profiles are used to force the flow in all

the simulations. The simulations are advanced to 12 FTC’s keeping the time-step

constant at ∆t = 10−7 s.

The instantaneous vorticity contours, obtained after the same physical time, are

shown in figure 3.50. The number of vortices within the computational domain re-

main the same for all the simulations. The figure indicates that the revised baseline

85

mesh captures all the large scales of the mixing layer. It can be deduced from fig-

ures 3.50(a), 3.50(b) that the contours are nearly identical when the y-resolution is

doubled. However, the vortices seem to be stretched a little around x = 170δω(0)

and 200δω(0) possibly due to the change in the aspect ratio of the grid cells intro-

duced by changing the y-resolution alone. Figures 3.50(c) and 3.50(e) indicate that

the vorticity contours are almost the same except that vortices that are involved in

pairing (shown around x = 150δω(0), 190δω(0)) have co-rotated more in the clock-

wise direction when the resolution is increased in y direction. Note that the vortex at

x = 170δω(0), which is not undergoing pairing, is identical in both the figures. With

this thought in mind, a quick look back at figures 3.50(a) and 3.50(b) reveals that

the vortices around x = 170δω(0), 200δω(0) are not stretched but co-rotated more

about each other in figure 3.50(b). This effect is also noticeable for the vortices that

are undergoing pairing at x = 150δω(0). Thus, it is believed that increasing the cross-

stream resolution causes the vortices to pair sooner. It is to be noted that increasing

the cross-stream resolution also increases the resolution of the perturbation profile at

the inlet and makes it smoother. The smoother perturbation profile may have caused

the vortices to pair up quicker.

Figures 3.50(a) and 3.50(c) indicate that the time history of the vortex evolution

changes when the grid is better resolved in the stream-wise direction. The resolution

of smaller scales with the refined mesh may have caused the flow to evolve differently

in these cases. The effect of grid refinement in both the directions is shown in fig-

ures 3.50(d) and 3.50(e). The evolution of the vorticity is more or less unchanged for

x < 170δω(0). The second vortex from x = 200δω(0) undergoes pairing with the third

when the grid is refined by a factor of√

2, whereas it undergoes pairing with first

vortex from x = 200δω(0) when the grid is refined by a factor of 2 in both directions.

The growth and center of the mixing layer, obtained after 12 FTC’s, are plotted

in figures 3.51 and 3.52, respectively. As shown in these figures, the growth and the

center remain more or less the same for all the simulations. The slope of the linear fit

to the growth rate for each simulation in the region x > 70δω(0) is obtained and listed

86

in table 3.10. The mixing layer growth is highest when the grid resolution is increased

by 2 in x and y. The peak values of profiles of scaled Reynolds stresses for various

stream-wise locations are given for the simulations in table 3.11. The peak values, in

general, increase slightly as the grid resolution increases. The increase is maximum

for the finest grid as shown in the table. The finest grid would resolve the smaller

scales better, which would have contributed to an increase in the Reynolds stresses.

There is some discrepancy , however, in the peak value of Reynolds shear stresses at

x = 160δω(0) for the revised baseline simulation. It appears that shear stress is over-

predicted for the baseline simulation at x = 160δω(0). Aside from this discrepancy,

the revised baseline simulation compares well with the refined simulations. Therefore,

the refined baseline mesh is used in subsequent three-dimensional grids.

Table 3.10 Mixing layer growth rates obtained from the grid resolution study.

Case Vorticy thickness growth,dδω(x)dx

Revised baseline 0.0556

y-resolution doubled 0.0548

x-resolution doubled 0.055

x and y resolutions increased by√

2 0.0544

x and y resolutions doubled 0.0562

87

Table 3.11 Comparison of the Reynolds stresses obtained from thegrid refinement study.

Peak x ∆x, ∆y ∆x, ∆y

2∆x2

, ∆y ∆x√2, ∆y√

2∆x2

, ∆y

2

Growth rate 0.0556 0.052 0.050 0.054 0.0562

70δω(0) 0.004 0.004 0.004 0.004 0.004

100δω(0) 0.014 0.018 0.016 0.017 0.019

σxx/∆U2 130δω(0) 0.036 0.036 0.038 0.038 0.040

160δω(0) 0.044 0.042 0.045 0.044 0.046

190δω(0) 0.044 0.052 0.051 0.056 0.052

70δω(0) 0.006 0.010 0.004 0.006 0.008

100δω(0) 0.039 0.040 0.044 0.044 0.044

σyy/∆U2 130δω(0) 0.054 0.057 0.057 0.056 0.059

160δω(0) 0.070 0.072 0.076 0.074 0.079

190δω(0) 0.079 0.074 0.081 0.076 0.080

70δω(0) 0.0036 0.0052 0.0034 0.0044 0.0048

100δω(0) 0.0070 0.0078 0.0094 0.0086 0.0092

|σxy/∆U2| 130δω(0) 0.0152 0.0166 0.0156 0.0162 0.0170

160δω(0) 0.0160 0.0120 0.0136 0.0130 0.0120

190δω(0) 0.0114 0.0130 0.0120 0.0130 0.0136

88

x/δω(0)

y/δ

ω(0

)

50 100 150 200-25

0

25


x/δω(0)

y/δ

ω(0

)

50 100 150 200-25

0

25

(b) Cross-stream resolution doubled.

x/δω(0)

y/δ

ω(0

)

50 100 150 200-25

0

25

(c) Stream-wise resolution doubled.

x/δω(0)

y/δ

ω(0

)

50 100 150 200-25

0

25

(d) x and y resolutions increased by√

2.

x/δω(0)

y/δ

ω(0

)

50 100 150 200-25

0

25

(e) x and y resolutions doubled.

Figure 3.50. Effect of grid refinement on the evolution of the mixinglayer. The contours are shown at the same physical time.

89

x/δω(0)

δω(x)/

δω(0

)

0 50 100 150 2000

1

2

3

4

5

6

7

8

9

10

revised baseline

x refined by a factor of 2

y refined by a factor of 2

x,y refined by a factor of 1.414

x,y refined by a factor of 2

Figure 3.51. Effect of grid refinement on the mixing layer growth.

x/δω(0)

yc/

δ ω(0

)

0 50 100 150 200-1.5

-1

-0.5

0

revised baseline

x refined by a factor of 2

y refined by a factor of 2

x,y refined by a factor of 1.414

x,y refined by a factor of 2

Figure 3.52. Effect of grid refinement on the mixing layer center.

90

3.13 Comparison of Computational Cost

The computational cost for the 2-D single precision, LES cases performed in the

present study are listed in table 3.12 for the smallest to largest 2-D meshes. The

resourses correspond to the incompressible pressure-based LES with the second-order

implicit time-stepping method. The implicit scheme requires a few inner iterations

between successive time-steps to converge in time. The number of inner iterations

used in the present study is 20 for all the 2-D simulations. As shown in the table, the

time required for one time-step varies from 2 to 12 seconds. Therefore, the simulation

time, corresponding to 42000 time-steps (or 12 FTC’s), varies from 1 to 6 days. The

simulation performed with the compressible 2-D density-based solver is also listed in

the table with a (*). The compressible density-based solver takes about 1.5 times

more computational time than that of the incompressible pressure-based solver on

the same number of processors.

Table 3.12 Computational resources required to perform the current2-D single precision simulations. The density-based LES is indicatedby (*).

Grid points Time in seconds Simulation time Processors

in thousands per 1 time-step in days used

42 to 66 2 1.0 16

100 3 1.5 16

288 6 3.0 16

318 to 331 7 3.5 16

(*)331 10 5.0 16

359 8 4.0 16

403 to 475 9 4.5 16

637 12 6.0 16

91

4. 3-D MIXING LAYER SIMULATIONS

4.1 Mesh

An optimized 2-D grid is obtained by decreasing the cross-stream domain size to

−50δω(0) to 50δω(0) and increasing the cross-stream grid stretching ratio to 1.10 with

respect to the 2-D baseline mesh. The optimized mesh contains roughly 1/8th of the

grid points in the baseline mesh. It is shown in section 3.11.2 that the optimized

mesh does not affect the quality of the LES results. This mesh is extruded in the

span-wise direction to construct a 3-D baseline mesh. The number of grid points in

the span-wise direction is 51. The grid points in the stream-wise and cross-stream

directions are 576, 73, respectively. The total number of grid points is approximately

2.15 million. The size of the computational domain is 350δω(0)× 100δω(0)× 20δω(0).

The minimum spacings around the mixing layer centerline are (∆x, ∆y, ∆z ) =

(0.4δω(0), 0.16δω(0), 0.4δω(0)). It is important to note that the mesh has a physical

region until x = 200δω(0) followed by a buffer zone, which extends to x = 350δω(0).

4.2 Boundary Conditions

The boundary conditions imposed in the 2-D simulations are also used here. The

span-wise boundaries of the computational domain are modelled as periodic.

4.2.1 Inflow Forcing

Fluent offers two algorithms to model the velocity perturbations for an LES. These

algorithms are limited to generating perturbations at a velocity inlet BC in version

6.3.26. As velocity inlets are typically used for incompressible flow computations to

avoid stability issues, it can be implied that these methods are applicable only for

92

an incompressible LES. Both these methods need realistic profiles of k and ǫ to be

specified at the inlet to generate meaningful fluctuating velocity components. The

two methods are described briefly as follows.

1. Random Vortex Method (VM) [52]

In this approach, a perturbation is added on a specified mean inlet velocity

via a fluctuating 2-D vorticity field. The method makes use of the Lagrangian

form of the 2-D vorticity evolution equation and the Biot-Savart law. The

Biot-Savart law is solved by introducing vortex particles within the inlet plane.

The number of vortex particles is specified by the user. The method requires

meaningful profiles of turbulent kinetic energy and dissipation profiles as a user

input to the solver. The circulation of each vortex particle is computed from

the turbulent kinetic energy, whereas the turbulent dissipation determines its

size. The sign of circulation for each vortex particle is changed after each

characteristic time scale, which is the time required for a 2-D vortex to convect

100 times its average size in the stream-wise direction at average inlet velocity.

These vortices move randomly each timestep within the inlet plane at 5% of the

average inlet velocity. The vortices yield perturbations only in the inlet plane.

The stream-wise velocity perturbations are obtained from

u′

= −~v′ .~∇U

|~∇U |, (4.1)

where ~v′ is the planar fluctuating 2-D velocity field computed from the vortex

method and ~∇U is the gradient of magnitude of the mean inlet velocity.

2. Spectral Synthesizer

The VM does not generate perturbations that are divergence free. This

might be problematic for a numerical simulation. Divergence-free 3-D fluctuat-

ing velocity components can be generated using the following. The perturba-

tions are computed by obtaining a velocity field from the summation of Fourier

harmonics. The number of Fourier components is a user-defined parameter.

This method also requires realistic profiles of k and ǫ at the inlet.

93

In this work, the vortex method (VM) is used to generate time varying inlet

boundary conditions to supply realistic turbulent velocity components at the inlet.

The following section describes the method used to obtain the profiles of turbulent

kinetic energy and dissipation as required by the VM algorithm.

4.2.2 Obtaining k and ǫ for the Inlet BC

These profiles are obtained from a 2-D RANS simulation of incompressible flow

over a splitter plate with zero thickness. The length of the plate is 1 mm. The

computational domain extends to 6 lengths downstream of the splitter plate edge

and one length in the lateral direction. The velocity of the flows above and below the

splitter plate is matched to that of the low- and high-speed flows of the mixing layer.

The velocity profile at four lengths downstream of the edge is found to approxi-

mately match the initial vorticity thickness of the mixing layer. This velocity profile is

compared with the hyperbolic tangent profile in figure 4.1(a). As shown in the figure,

the velocity profile obtained from the 2-D RANS simulation matches the hyperbolic

profile for y/δω(0) < 1. The RANS simulation over-predicts the velocity around

y/δω(0) = 0.6, possibly due to the wake of the boundary layer in the high-speed flow.

The profiles of k and ǫ are extracted at this location and plotted in figures 4.1(b)

and 4.1(c), respectively. These profiles are approximated using the equations

k = k0 exp

(

−2y

δω

)2

+ k∞, (4.2)

ǫ = ǫ0 exp

(

−2y

δω

)2

+ ǫ∞, (4.3)

where k0, ǫ0 are the peak values of the turbulent kinetic energy and dissipation rate.

The subscript ∞ refers to the free-stream. The peaks of k and ǫ as predicted by

the RANS simulation are 150, 2.7× 107 in SI units respectively. The approximations

to the k and ǫ profiles of RANS simulation are also plotted in figures 4.1(b), 4.1(c)

assuming no free-stream turbulence. The figure shows that the approximated profiles

94

match the computed profiles reasonably well within the mixing layer. The peaks of

k and ǫ can be further parameterized using

k0 =3

2(uy=0I0)

2, (4.4)

ǫ0 = C3

4

µ

k0

3

2

l, (4.5)

where uy=0, I0 are the velocity and turbulent intensity at the mixing layer center; l is

the mixing length and Cµ = 0.09. The values of I0 and l, as obtained from the RANS

simulation, are 8%, 0.1δω(0) respectively.

The approximated profiles given in equations 4.2 and 4.3 are supplied to the VM

algorithm as input. The amplitudes of the profiles are determined by equations 4.4

and 4.5. The turbulent length scale l, used in equation 4.5, determines the size of

the vortices generated by the VM method. As the minimum resolution of the grid

is 0.16δω(0) in the mixing layer center, the turbulent length scale is increased from

0.1δω(0) to 0.5δω(0) to ensure that the vortices are properly resolved within the mixing

layer. The value of I0, obtained from the 2-D RANS calculation, is taken as 8% to

compute k0. Small values of k∞, ǫ∞ are specified to avoid the divergence caused by

the VM algorithm. The perturbations are limited to the region of the inlet bounded

by −2δω(0) < y < 2δω(0) to limit the perturbations within the mixing layer. The

number of vortices is chosen to be 22 after some experimentation with the VM as it

results in good initial distribution of vortices within the center of the mixing layer.

The vortices are thus allowed to move randomly after each timestep within the region

−2δω(0) < y < 2δω(0). The size of the vortex particle is maximum around the center

of the mixing layer and decreases away from it. The sign of the vortices changes about

4 times during 1 FTC. It is important to note that Fluent offers no control over how

often the vortices change their sign to the user.

95

y/δω

(U-U

2)/(U

1-U

2)

-3 -2 -1 0 1 2 3-0.2

0

0.2

0.4

0.6

0.8

1

1.2

RANS

Hyperbolic

(a) Velocity profiles.

y/δω

k/k0

-3 -2 -1 0 1 2 3-0.2

0

0.2

0.4

0.6

0.8

1

1.2

RANS

Approximation

(b) k profiles.

y/δω

ε/ε 0

-3 -2 -1 0 1 2 3-0.2

0

0.2

0.4

0.6

0.8

1

1.2

RANS

Approximation

(c) ǫ profiles.

Figure 4.1. Comparison of k, ǫ for RANS simulation and the expo-nential function fits.

4.3 Simulation Parameters and Procedure

The parameters of the mixing layer are defined in section 3.1. The Reynolds

number based on the mixing layer thickness and velocity difference is 720. The

numerical methods are the same as those of 2-D simulations and summarized as

follows.

The mixing layer is simulated by specifying a hyperbolic tangent velocity profile

at the inlet. The unsteady perturbations are generated by the VM algorithm in-

96

built in Fluent. The profiles of turbulent kinetic energy and dissipation are obtained

using a 2-D RANS and approximated by the equations 4.2, 4.3. These approximated

profiles are supplied as input to the VM algorithm to compute the turbulent velocity

perturbations at the inlet. The flow is modeled as incompressible due to the reasons

described in section 3.7.2. The pressure-based solver is used in conjunction with the

PISO coupling scheme. The second-order BCD is used for spatial discretization of

the filtered momentum equation. The sub-grid scale viscosity is modeled using the

dynamic-Smagorinksy model. The LES is advanced with a time-step of 10−7 s using

a second-order implicit time-stepping scheme.

4.4 Results and Discussion

The simulation is ran for 4 FTC’s to account for the transients and subsequently

the flow statistics are gathered for 8 FTC’s. The growth of the mixing layer is plotted

in figure 4.2. The figure shows a slow growth rate for x < 100δω(0) corresponding to

the laminar region of the mixing layer. The slope (dδω(x)dx

) of the linear growth region

is approximately 0.09. The corresponding growth rate is 0.27. The scaled velocity

profiles at various downstream locations are plotted in figure 4.3. The profiles collapse

onto the error-function profile [49] for all downstream locations.

The scaled normal Reynolds stresses and the primary Reynolds shear stress, ob-

tained through span-wise averaging, are shown in figure 4.4. As shown in figure 4.4(a),

the peaks of profiles of σxx increase until x = 160δω(0) followed by a decrease fur-

ther downstream. The peaks of σyy increase monotonically from x = 100δω(0) to

x = 200δω(0). The profiles of σzz collapse downstream. The peaks of σxy increase

monotonically downstream until x = 180δω(0) followed by a decrease at x = 200δω(0).

The mixing layer growth rate and the peaks of the Reynolds stresses are compared

with the data available in the literature and are shown in table 4.1. The growth rate

and the maximum values of the stresses are higher compared to the experiments. This

could be probably attributed to the deficiencies of the forcing employed in present

97

simulation, as discussed in section 4.5.1. The maximum value of σzz/∆U2, however,

matches with the experimental value.

x/δω(0)

δω(x)/

δω(0

)

0 50 100 150 2000

1

2

3

4

5

6

7

8

9

10

forced


Figure 4.2. Mixing layer growth predicted by 3-D incompressible LESusing Fluent’s vortex method for forcing.

Table 4.1 Comparison of the normalized peak Reynolds stresses andvorticity growth rates of 3-D mixing layer with Fluent’s VM forcingwith the data in the literature.

Reωσxx

∆U2

σyy

∆U2

σzz

∆U2

σxy

∆U2

1η

dδω(x)dx

Reference

- 0.031 0.019 0.0225 0.009 0.19 Experiment [25]

- 0.036 0.014 0.0225 0.013 0.16 Experiment [28]

1,800 0.032 0.020 0.022 0.010 0.163 Experiment [29]

720 0.040 0.084 - 0.023 0.15 2-D DNS [38]

720 0.048 0.078 - 0.012 0.15 2-D DNS [39]

720 0.044 0.080 - 0.015 0.17 Present 2-D LES

720 0.048 0.052 0.021 0.025 0.27 Present 3-D LES, VM

98

ξ

f(ξ)

-3 -2 -1 0 1 2 3-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

x = 100 δω(0)

x = 120 δω(0)x = 140 δ

ω(0)

x = 160 δω(0)x = 180 δω(0)x = 200 δω(0)error function

Figure 4.3. Span-wise averaged scaled velocity profiles as predictedby 3-D incompressible LES.

99

ξ

σxx/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

x = 100 δω(0)x = 120 δ

ω(0)

x = 140 δω(0)x = 160 δ

ω(0)

x = 180 δω(0)x = 200 δ

ω(0)

(a) σxx.

ξσyy/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

x = 100 δω(0)x = 120 δ

ω(0)

x = 140 δω(0)x = 160 δ

ω(0)

x = 180 δω(0)x = 200 δ

ω(0)

(b) σyy.

ξ

σzz/∆U2

-3 -2 -1 0 1 2 30

0.005

0.01

0.015

0.02

0.025

0.03

x = 100 δω(0)x = 120 δ

ω(0)

x = 140 δω(0)x = 160 δ

ω(0)

x = 180 δω(0)x = 200 δ

ω(0)

(c) σzz.

ξ

σxy/∆U2

-3 -2 -1 0 1 2 3-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

x = 100 δω(0)x = 120 δ

ω(0)

x = 140 δω(0)x = 160 δ

ω(0)

x = 180 δω(0)x = 200 δ

ω(0)

(d) σxy.

Figure 4.4. Span-wise averaged scaled Reynolds stresses as predictedby 3-D incompressible LES.

100

The mixing layer is examined for the presence of three-dimensionality and rib

vortices after 12 FTC’s. An iso-surface of span-wise vorticity is shown in figure 4.5

to observe the span-wise vortex rollers after the vortex sheet is rolled up. The figure

shows that the vortex roll-up begins at around x = 110δω(0). There are 4 spanwise

rollers at this instant for 110δω(0) < x < 145δω(0). An iso-surface of |ωx|+ |ωy|+ |ωz|is plotted to investigate the presence of rib vortices in figures 4.6 and 4.7. The iso-

surface is colored by stream-wise vorticity in figures 4.6(a) and 4.6(b) to facilitate the

visualization of stream-wise (rib) vortex pairs. The iso-surface is colored by span-wise

vorticity in figures 4.6(c) and 4.6(d) to investigate if the rib vortices correspond to

the regions of positive span-wise vorticity in the flow-field. Figures 4.6(a) and 4.6(b)

indicate what appears to be a rib vortex pair between the first two span-wise rollers

around x = 115δω(0) with a rectangular cross section for the vortex cores. These

vortices are shown in an oblique view in figure 4.8. Figures 4.6(c) and 4.6(d) show

the regions of positive span-wise vorticity developed from the three-dimensionality of

the mixing layer further downstream. As shown in these figures, the iso-surface of ωz

or |ωx|+ |ωy|+ |ωz| does not clearly indicate the cores of the vortices especially in the

downstream region where the flow is significantly three-dimensional.

The second invariant of velocity gradient tensor, denoted by Q, is often considered

a good parameter to visualize the vortex cores present in the flow field [53]. The

regions with strong vorticity, can be defined as those with high positive values of

Q, while regions with high values of the kinetic energy dissipation can be associated

with strong negative values of Q. Figure 4.9 shows the top and bottom views of

an iso-surface of Q = 2.2 × 10−2 × (∆U/δω(0))2 colored by stream- and span-wise

vorticities. The iso-surface is shown in three-dimensions in figure 4.10. The vortex

cores are distinctly visible in these figures when compared to figures 4.6 and 4.7, as

expected. The rib vortices are seen, however not in distinct pairs, for x > 150δω(0)

as shown in figures 4.9 and 4.10. Rogers and Moser [14, 35] have observed the rib

vortices in their temporally evolving mixing layer simulations with carefully controlled

boundary conditions at the inlet. This, perhaps, explains the disorganized nature

101

of the rib vortices observed in the present simulation. A close-up view of the iso-

surface of Q is depicted in figure 4.11. The counter-rotating vortex pair apparent in

figure 4.8, between the first two rollers, is not present in figure 4.11, indicating that

the cores of the vortex pair are too weak to be captured by the iso-surface of Q =

2.2× 10−2× (∆U/δω(0))2. Therefore, an iso-surface of Q = 2.2× 10−5× (∆U/δω(0))2

is plotted in figure 4.12. It appears that the vortex pair is still too weak to be visible

in the figure. However, the cores of the vortex pair are seen on the surface of the

vortex roller at x = 120δω(0).

The contours of the span- and stream-wise vorticity in the plane z = 10δω(0),

located at the mid-span of computational domain, are plotted in figure 4.13. The

positive span-wise vorticity caused by the three-dimensional rib vortices is shown in

figure 4.13(a). The stream-wise vorticity contours are shown in the mid-plane in

figure 4.13(b). The cut-off for the contour levels in this figures does not indicate

the minimum and maximum values of the stream-wise vorticity found in the entire

computational domain. The cut-off values are set so that the development of the

stream-wise vorticity can be seen more clearly in the computational domain. It can

be seen from this figure that the stream-wise vorticity begins to develop even before

the vortex sheet roll-up begins in the regions of 70δω(0) < x < 100δω(0). It should

be noted that first vortex roll-up occurs around x = 110δω(0) as shown in figure 4.5.

The one-dimensional energy spectrum of the stream-wise velocity fluctuations and

the stream-wise velocity correlation (Q11) are plotted in figures 4.14 and 4.15 corre-

sponding to the unsteady velocity data obtained at x = 200δω(0) at the centerline.

Taylor’s hypothesis of frozen turbulence is used to construct the spectrum from the

time history of the stream-wise velocity. The grid cut-off frequency, which is cal-

culated based on 9 grid points per wave length, is also shown in figure 4.14. It is

illustrated in the figure that the -5/3 energy decay is maintained for more than half-

a-decade of frequency range until the grid cut-off. The energy, then, drops off rapidly

at a higher rate. The steam-wise velocity correlation, as shown in figure 4.15, at zero

separation distance matches well with the peak value of σxx profile at x = 200δω(0)

102

(a) High-speed flow side.

(b) Low-speed flow side.

Figure 4.5. Iso-surface of ωz = 0.5× (∆U/δω(0)).

as indicated in figure 4.4(a). The value of Q11 does not go to zero for large separation

distances probably due to the coherent large-scale structures of the low Reynolds

number mixing layer. It should be noted that the mixing layer is not fully turbulent

at x = 200δω(0). This also explains the non-zero values of Q11 for larger distances.

In summary, the 3-D baseline simulation results indicate that the developed LES

methodology captures the instabilities of the mixing layer reasonably well. The 1-D

energy spectrum indicates an energy decay rate of −53

in the inertial range of the

length scales with the second-order BCD spatial discretization. The growth of the

mixing layer as well as the Reynolds stresses are over-predicted due to the non-physical

nature of the perturbations incorporated into the simulation by the Fluent’s in-built

VM algorithm. The deficiencies of the in-built perturbation algorithm are addressed

in section 4.5.1 and a new simulation is performed with an improved version of the

VM algorithm. The results of the improved LES are described in detail in section 4.5.

103



(c) High-speed flow side.

(d) Low-speed flow side.

Figure 4.6. Iso-surface of |ωx|+ |ωy|+ |ωz| = 0.5×∆U/δω(0) coloredby span- and stream-wise vorticities.

104

(a) Colored by stream-wise vorticity.

(b) Colored by span-wise vorticity.

Figure 4.7. Full isometric view of an iso-surface of |ωx| + |ωy| + |ωz|= 0.5×∆U/δω(0).

105



Figure 4.8. Close-up isometric view of an iso-surface of |ωx|+|ωy|+|ωz|= 0.5×∆U/δω(0).

106



(c) High-speed flow side.

(d) Low-speed flow side.

Figure 4.9. Iso-surface of Q = 2.2 × 10−2 × (∆U/δω(0))2 colored byspan- and stream-wise vorticities.

107



Figure 4.10. Full isometric view of an iso-surface of Q = 2.2× 10−2 × (∆U/δω(0))2.

108



Figure 4.11. Close-up isometric view of an iso-surface of Q = 2.2 ×10−2 × (∆U/δω(0))2.

109



Figure 4.12. Iso-surface of smaller Q = 2.2 × 10−5 × (∆U/δω(0))2

colored by stream-wise vorticity.

110

x/δω(0)

y/δ

ω(0

)

50 100 150 200-25

0

25

-9.0E+05 -6.0E+05 -3.0E+05 0.0E+00 3.0E+05Spanwise vorticity

(a) Span-wise vorticity

x/δω(0)

y/δ

ω(0

)

50 100 150 200-25

0

25

-1.5E+05 -7.5E+04 0.0E+00 7.5E+04 1.5E+05Stream-wise vorticity

(b) Stream-wise vorticity

Figure 4.13. Contours of span-wise vorticity in the plane z = 10δω(0) (mid-plane).

111

κxδ

ω(0)

E1(κxδ ω

(0))/∆U2

δω(0

)

10-1

100

10110

-8

10-6

10-4

10-2

100

Grid cut-off

(κxδ

ω(0))−5/3

Figure 4.14. 1-D Energy spectrum of the stream-wise velocity at x = 200δω(0).

x/δω(0)

Q11/∆U2

0 2 4 6 8 10 12-0.01

0

0.01

0.02

0.03

Figure 4.15. Stream-wise velocity correlation (Q11) at x = 200δω(0).

112

4.5 3-D LES with Improved Inflow Forcing

4.5.1 Deficiencies of Fluent’s Vortex Method

The contours of the span-wise perturbation velocity induced by the vortices at

the inlet plane are shown in figure 4.16 corresponding to the beginning, 1 FTC and 2

FTC’s of the baseline simulation. The baseline simulation revealed some deficiencies

of the in-built VM algorithm of Fluent for LES of a spatially developing mixing layer.

These are listed as follows.

1. As the velocity of the vortices within the inlet plane is fixed at 5 percent of

the bulk inlet velocity, the vortices appear to be moving very slowly between

the time-steps. The velocity of the vortices cannot be changed in Fluent. In

the baseline simulation, the vortices move randomly to a distance of approxi-

mately 0.5% of the initial vorticity thickness within the inlet plane after each

time-step. This corresponds to only 2% of the span-wise grid spacing. The

figures 4.16(a), 4.16(b) indicate the slow movement of the vortices. Therefore,

this might localize the development of the rib vortices only to certain span-wise

locations.

2. The position of the vortices is initialized randomly at the inlet plane bounded by

−2δω(0) < y < 2δω(0). As the strength of the vortices decreases when they are

away from the centerline, a larger number of vortices is needed within the inlet

plane to have a reasonable number of high-strength vortices along the centerline.

In the baseline simulation, 22 vortices are used to generate perturbations at the

inlet. The initial position of the vortices is shown in figure 4.16(a).

3. The vortices at the inlet seem to overlap each other after a few time-steps. It

appears that Fluent does not restrict the vortices from overlapping. This is

more pronounced in figure 4.16(c). This results in non-physical perturbations

at the inlet and thus might lead to non-physical evolution of the mixing layer.

113

4. The vortices can move anywhere within the mesh region shown in figure 4.16

corresponding to −2δω(0) < y < 2δω(0). As the mixing layer secondary insta-

bility, which results in rib vortices, is sensitive only to the span-wise wavelength

of the perturbations [21], it is desirable to restrict the movement of the vor-

tices only along the span-wise direction. This would essentially change the

wavelength in the span-wise direction as the vortices move leading to a more

realistic development of the mixing layer.

5. It is found that the vortices in the inlet plane move out of the boundaries of the

inlet plane. This is problematic especially as the vortices cross the span-wise

periodic boundaries resulting in problems with the periodicity of the mixing

layer in the span-wise direction.

6. The level of perturbations generated by the in-built VM is too low, which may

have caused the mixing layer to roll up very far downstream in the baseline

simulation. In other words, a lot of computational domain is just used to

capture the laminar region of the mixing layer in the baseline simulation. As

shown in figure 4.16, the magnitude of the span-wise perturbation velocity is

about 0.08 m/s or about 0.06% of the centerline velocity. The cross-stream

perturbation velocity has also about the same magnitude. The magnitude of

the cross-stream perturbation velocity is about 0.45% of the centerline velocity

in the 2-D simulations performed earlier.

7. The vortices change sign roughly 5 times during one FTC. The length of the

vortex tubes is around 70δω(0) before it changes sign, which is pretty large

compared to the span-wise domain size of 20δω(0). This might probably require

large number of FTC’s for the flow statistics to converge.

8. The vortices change their sign abruptly after a few hundred time-steps within

Fluent. This abrupt change of sign might lead to unrealistic evolution of the

mixing layer downstream.

114

9. It is desirable to have counter-rotating vortex pairs at the inlet to ensure the

development of counter-rotating rib vortex pairs downstream in a mixing layer

simulation. The counter-rotating vortex pairs at the inlet also serve as a way

to model the hair-pin vortices developed in the boundary layers of the splitter

plate. In Fluent, it appears that the sign of each vortex is initialized randomly.

This results in disorganized counter-rotating pairs of stream-wise vortices, if

there are any at the inlet plane, as shown in figure 4.16(a) and leads to a

disorganized development of the large-scale structures downstream.

These deficiencies make it hard to correlate the number of rib vortices found in

the baseline simulation to the number of vortices at the inlet.

Therefore, the Vortex Method algorithm is implemented through a UDF to ad-

dress the deficiencies of the in-built algorithm. The method is described in detail

in reference [52]. A second simulation is performed with the improved VM and its

results are discussed in section 4.5.3.

115

(a) Time-step = 1

(b) 1 FTC

(c) 2 FTC’s

Figure 4.16. Span-wise perturbation velocity introduced by the vor-tices at the inlet plane. The maximum and minimum contour levelsare 0.08, -0.08 m/s, respectively.

116

4.5.2 Improved Vortex Method

The improvements incorporated into the modified VM algorithm are compared

against the features of Fluent’s VM in table 4.2. The parameters used in the im-

plemented VM algorithm for the second simulation are as follows. The number of

vortices used to generate the perturbations at the inlet are 4. The size of each vortex

is 1.5δω(0) and remains constant as the vortices are restricted to move only along the

span-wise direction within the center of the mixing layer. The stream-wise perturba-

tion velocity is computed using equation 4.1. The velocity of the vortices within the

inlet plane is increased to 75% of the mean inlet velocity to ensure that the vortices

move significantly between successive time-steps. The sign of each vortex changes

after approximately 250 time-steps resulting in a vortex tube length of 25δω(0). In

other words, the frequency of the sign change is tripled to 14 with respect to that of

the baseline simulation. The strength of each vortex, which remains constant until

the sign change, reduces to zero before changing the sign leading to a gradual change

in the sign unlike the Fluent’s VM. The maximum magnitude of the span-wise or

cross-stream velocity induced by the vortices at the inlet plane is around 0.1 m/s

( 0.08% of the center line velocity).

A schematic of the vortex tubes as they pass through the inlet plane is shown

in figure 4.17. As shown in the figure, the vortices are symmetric about the mid-

plane to ensure that there are two counter-rotating pairs of stream-wise vortices at

the inlet. In other words, the position of the first two vortex cores from the left in

the figure are computed randomly each time-step and the position of the other two

are simply obtained from the symmetry about the mid-span plane. The numbers

under the vortices in the figure denote which pair the vortices belong to. The two

vortices in the middle form the first counter-rotating pair and the other two vortices

form the second pair due to the span-wise periodicity imposed in the simulation.

Each pair emulates a hair-pin vortex developed in the boundary layer of a splitter

plate wall. As the simulation advances in time, the distance between the vortices in

117

each pair changes randomly, which in turn changes the span-wise wave length of the

perturbations. It should be noted that there are two independent span-wise wave

lengths that are varied in the simulation corresponding to the two hair-pin vortices.

It is believed that the artificial symmetry introduced at the mid-span plane would not

affect the development of the mixing layer downstream as the mixing layer becomes

self-similar further downstream.

Table 4.2 Comparison of the features of the Fluent’s and improved VM algorithms.

Feature of the vortices Fluent’s VM Improved VM

Strength Depends on k Depends on k but scalable

Initial position Random User-defined

Velocity Fixed User-specified

Mobility Anywhere in Span-wise along

the inlet plane the mixing layer center

Non-overlapping × X

Split vortices due to out-of-bounds X ×movement within the inlet plane

Frequency of the sign change Fixed User-defined

Type of sign change Abrupt Gradual

Span-wise periodicity × X

Organized counter-rotating pairs × X

4.5.3 Results and Discussion

The one-dimensional energy spectrum of the stream-wise velocity perturbations,

obtained from Taylor’s hypothesis of frozen turbulence, is plotted in figure 4.18 corre-

sponding to the velocity history obtained at x = 200δω(0). The grid cut-off, computed

by assuming 9 stream-wise points per wavelength, is also shown in the figure. The

118

Figure 4.17. Schematic of the stream-wise vortex pairs (1 and 2)generated by the implemented VM forcing algorithm. The vortexpairs emulate hair-pin vortices developed within the boundary layerof a splitter plate wall.

figure indicates that the rate of energy decay of −53

is maintained for more than

half-a-decade of wave numbers. This reiterates that the present LES methodology

of using second-order-accurate BCD is satisfactory. The stream-wise velocity corre-

lation (Q11) is shown in figure 4.19. The correlation becomes zero as the separation

distance is increased suggesting that the mixing layer is can be approximated as fully

turbulent.

The growth and the center of the mixing layer, obtained after 12 FTC’s are plotted

in figure 4.20. The scaled velocity profiles, shown in figure 4.21,collapse well onto the

error function profile for all the downstream locations. The scaled Reynolds stresses

are plotted in figure 4.22. The profiles match reasonably well further downstream

especially for σzz. The maximum values of Reynolds stresses and the growth rate

observed are compared with those found in the literature in table 4.3. The growth

119

rate and the Reynolds stresses are lower compared to the simulation with Fluent’s

VM algorithm and thus are closer to experimental values. However, they are still

higher than the experimental values except for σzz which closely matches with the

experiments. The over-prediction is probably due to the presence of the hair-pin

vortices, which strongly influence the development of the mixing layer downstream,

all through the simulation. In an experimental setting, the hair-pin structures are fed

into the mixing layer intermittently resulting in reduced growth as well as Reynolds

stresses.

κxδ

ω(0)

E1(κxδ ω

(0))/∆U2

δω(0

)

10-1

100

10110

-8

10-6

10-4

10-2

100

Grid cut-off

(κxδ

ω(0))−5/3

Figure 4.18. 1-D energy spectrum for the 3-D incompressible LESusing the improved VM algorithm.

120

x/δω(0)

Q11/∆U2

0 2 4 6 8 10 12-0.01

0

0.01

0.02

0.03

0.04

Figure 4.19. The stream-wise velocity correlation (Q11) obtained withthe 3-D incompressible LES using the improved VM algorithm.

Table 4.3 Comparison of the normalized peak Reynolds stresses andvorticity growth rates of 3-D mixing layer using the improved VMwith the data in the literature.

Reωσxx

∆U2

σyy

∆U2

σzz

∆U2

σxy

∆U2

1η

dδω(x)dx

Reference

- 0.031 0.019 0.0225 0.009 0.19 Experiment [25]

- 0.036 0.014 0.0225 0.013 0.16 Experiment [28]

1,800 0.032 0.020 0.022 0.010 0.163 Experiment [29]

720 0.040 0.084 - 0.023 0.15 2-D DNS [38]

720 0.048 0.078 - 0.012 0.15 2-D DNS [39]

720 0.044 0.080 - 0.015 0.17 Present 2-D LES

720 0.048 0.052 0.021 0.025 0.27 Present 3-D LES, Fluent VM

720 0.042 0.046 0.018 0.021 0.225 Present 3-D LES, Improved VM

121

x/δω(0)

δω(x)/

δω(0

)

0 50 100 150 2000

1

2

3

4

5

6

7

8

9

10

forced


x/δω(0)

yc(x)/

δ ω(0

)

0 50 100 150 200-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

Figure 4.20. Mixing layer growth predicted by 3-D incompressibleLES using the improved VM.

122

ξ

f(ξ)

-3 -2 -1 0 1 2 3-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

x = 100 δω(0)

x = 120 δω(0)x = 140 δ

ω(0)

x = 160 δω(0)x = 180 δ

ω(0)

x = 200 δω(0)error function

Figure 4.21. Span-wise averaged scaled velocity profiles for 3-D mixinglayer obtained using the improved VM.

123

ξ

σxx/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

x = 100 δω(0)x = 120 δω(0)x = 140 δ

ω(0)

x = 160 δω(0)x = 180 δ

ω(0)

x = 200 δω(0)

(a) σxx.

ξσyy/∆U2

-3 -2 -1 0 1 2 30

0.01

0.02

0.03

0.04

0.05

0.06

x = 100 δω(0)x = 120 δω(0)x = 140 δ

ω(0)

x = 160 δω(0)x = 180 δ

ω(0)

x = 200 δω(0)

(b) σyy.

ξ

σzz/∆U2

-3 -2 -1 0 1 2 30

0.005

0.01

0.015

0.02

0.025

0.03

x = 100 δω(0)x = 120 δω(0)x = 140 δ

ω(0)

x = 160 δω(0)x = 180 δ

ω(0)

x = 200 δω(0)

(c) σzz.

ξ

σxy/∆U2

-3 -2 -1 0 1 2 3-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

x = 100 δω(0)x = 120 δω(0)x = 140 δ

ω(0)

x = 160 δω(0)x = 180 δ

ω(0)

x = 200 δω(0)

(d) σxy.

Figure 4.22. Span-wise averaged scaled Reynolds stresses as predictedby 3-D incompressible LES using the improved VM.

124

The mixing layer is examined for three-dimensionality by plotting the iso-surfaces

of |ωx|+|ωy|+|ωz| and second invariant of velocity gradient tensor (Q) after 12 FTC’s.

Figures 4.23 and 4.25 show an iso-surface of |ωx| + |ωy| + |ωz| = 0.5 × ∆U/δω(0) in

two different oblique views. The figures illustrate the stream-wise counter rotating

vortex pairs between the vortex rollers at x = 120δω(0) and 130δω(0). The number

of pairs is 2, which correlates with the number of hair-pin vortices at the inlet. The

span- and stream-wise vortex cores are better illustrated in figures 4.24 and 4.26

by plotting an iso-surface of Q = 2.2 × 10−2 × (∆U/δω(0))2 in the same oblique

views. These figures illustrate the translative instability, as shown in figure 9 in

reference [21], of the span-wise rollers located at x = 125δω(0) and 130δω(0). The

translative instability is most unstable for span-wise wavelengths approximately 23

of the space between vortex centers in the stream-wise direction. In the present

simulation, the distance between the first two vortex cores is about 10δω(0), and the

average distance between the two-vortices forming a hair-pin vortex is about 7.5δω(0).

As the simulation employs emulating hair-pin vortices within the boundary layer as a

means to force the perturbations at the inlet, it can be concluded that the translative

instability seen in the present simulation is a consequence of the hair-pin vortices

within the splitter plate boundary layer. It is believed that the translative instability

seen in experiments is also due to these hair-pin vortices.

The two pairs of stream-wise vortices between the rollers for 110δω(0) < x <

130δω(0), as seen in figure 4.25, are too weak to be captured by the iso-surface of Q

corresponding to Q = 2.2×10−2×(∆U/δω(0))2 in figure 4.26. This might be probably

due to the weak cores of the rib vortices. Therefore, a close-up view of an iso-surface

of smaller Q = 2.2×10−5×(∆U/δω(0))2 is shown in figure 4.27. The figure still shows

that the stream-wise vortices between the rollers for 110δω(0) < x < 130δω(0) have

weaker cores and are not captured by even an iso-surface corresponding to a smaller Q.

It is important to note that the mixing layer is expected to be symmetric in the region

of the first vortex roll-up. However, figure 4.27 shows asymmetry for 110δω(0) < x <

120δω(0), possibly due to the numerical loops involved in the simulation. The multi-

125

block topology used by Fluent for a parallel simulation may also have contributed to

the observed asymmetry.

The top and bottom views of iso-surfaces of |ωx| + |ωy| + |ωz| and Q are shown

in figures 4.28, 4.29, 4.30 and 4.31 corresponding to 12 FTC’s. It should be noted

that the computational domain has been extended to two span-wise periods in these

figures. The figures reiterate the effectiveness of using the second invariant of veloc-

ity gradient to extract vortex cores of the flow-field. The counter-rotating rib vortex

pairs are seen clearly in figures 4.30 and 4.31, which start from the first vortex core

undergoing translative instability at x = 125δω(0) and end on the other. The layout

of the rib vortices matches well with the schematic shown in figure 15 given in ref-

erence [23]. The mixing layer is significantly three-dimensional for x > 150δω(0) as

shown in figures 4.30 and 4.31. It should also be noted that vortex core pairings are

not seen in the current simulation due to the perturbation scheme used. Therefore,

the current mixing layer simulation growth is fueled by the translative instability,

which bends the cores in a sinusoidal fashion entraining more fluid into the mixing

layer. This could perhaps explain the higher mixing layer growth and the Reynolds

stresses seen here when compared to the experiments as shown in table 4.3. The

pairings might be seen if the strength of the stream-wise vortices at the inlet is varied

randomly after each time-step, which is not the case in the present simulation.

The cores of the vortices are also shown corresponding to 3.5 FTC’s (or 12,300

time-steps) in figures 4.32, 4.33. These figures indicate that the structure of the

mixing layer significantly differs from that corresponding to 12 FTC’s as shown in

figures 4.30 and 4.31. Thus, the current forcing method ensures that the mixing layer

is dynamic and the rib vortex pairs are not fixed at a particular span-wise location.

In the current simulation, the vortex sheet roll-up first occurs around x = 100δω(0).

Therefore, significant portion of the computational domain is wasted just to capture

the laminar shear layer upstream of the roll-up location. It is found that increas-

ing the magnitude of perturbations at the inlet does not trigger the vortex sheet to

roll-up further upstream. However, it does change the structure of the mixing layer

126

significantly after the roll-up, causing a significant three-dimensionality in the form

of fine-scale three-dimensional structures.



Figure 4.23. Full isometric view of an iso-surface of |ωx|+ |ωy|+ |ωz|= 0.5×∆U/δω(0) with one period in the span-wise direction obtainedusing the improved VM.

127



Figure 4.24. Full isometric view of an iso-surface of Q = 2.2× 10−2×(∆U/δω(0))2 with one period in the span-wise direction obtained usingthe improved VM.

128



Figure 4.25. Close-up isometric view of an iso-surface of |ωx|+ |ωy|+|ωz| = 0.5×∆U/δω(0) obtained using the improved VM.

129



Figure 4.26. Close-up isometric view of an iso-surface of Q = 2.2 ×10−2 × (∆U/δω(0))2 obtained using the improved VM.

130



Figure 4.27. Iso-surface of Q = 2.2× 10−5 × (∆U/δω(0))2 colored bystream-wise vorticity obtained using the improved VM.

131



Figure 4.28. Iso-surface of Q = 2.2× 10−2 × (∆U/δω(0))2 colored byspan-wise vorticity obtained using the improved VM.

132



Figure 4.29. Iso-surface of |ωx|+ |ωy|+ |ωz| = 0.5×∆U/δω(0) coloredby stream-wise vorticity obtained using the improved VM.

133



Figure 4.30. Iso-surface of Q = 2.2× 10−2 × (∆U/δω(0))2 colored byspan-wise vorticity obtained using the improved VM.

134



Figure 4.31. Iso-surface of Q = 2.2× 10−2 × (∆U/δω(0))2 colored bystream-wise vorticity obtained using the improved VM.

135



Figure 4.32. Iso-surface of Q = 2.2× 10−2 × (∆U/δω(0))2 colored byspan-wise vorticity after 3.5 FTC’s using the improved VM.

136



Figure 4.33. Iso-surface of Q = 2.2× 10−2 × (∆U/δω(0))2 colored bystream-wise vorticity after 3.5 FTC’s using the improved VM.

137

4.6 Estimate of Computational Cost for a Realistic Jet LES

The baseline 3-D LES with 2.15 million grid points requires about 11 days to

advance to 12 FTC’s (or 42,000 time-steps) on 64 processors. This information is

used to project the computational time needed to perform LES on larger meshes

using the current simulation methodology. It is assumed that the Fluent’s LES scales

up linearly with the number of processors in obtaining the projected computational

resources. The estimated computational resources are given in table 4.4. The table

indicates that the number of processors needed is about 25 times the number of

grid points in millions for a simulation turn-around time of 2 weeks. If the parallel

efficiency of Fluent is 50%, then the processor count scales as 50 times the number of

grid points in millions for the same turn-around time. It should be noted that these

estimates are optimistic because the parallel efficiency of a typical code decreases as

the number of processors increases. The estimates are obtained by assuming that

the number of time-steps remains the same for larger mesh sizes. This is not a valid

assumption because: a) smaller time-steps are required for finer meshes to maintain

the same CFL number, b) the duration of a flow-through-cycle (FTC) is a problem

dependent parameter.

Uzun et al. [54] have used a mesh containing 100 million grid points to perform LES

of flow through a Chevron nozzle at a Reynolds number of 100,000. Their simulation

required about 12 days of computational time using 512 processors to advance to

50,000 time-steps on Xeon Linux Cluster at National Center for Supercomputing

Applications. The number of processors needed to do the same simulation for 42,000

time-steps for a turn-around time of 2 weeks would require about 2,500 processors

using Fluent.

138

Table 4.4 Projection of the computational resources required to per-form a realistic LES of a jet using the current methodology in Fluent.The actual numbers for the 3-D baseline simulation are indicated by(*).

Grid points Simulation time Minimum processors

in millions in days required

(*)2.15 (*)11 (*)64

2 14 50

10 14 250

25 14 625

50 14 1,250

100 14 2,500

1,000 14 25,000

139

5. CONCLUSIONS AND FUTURE WORK

5.1 Conclusions

A successful numerical methodology has been implemented to perform an LES

using the finite-volume based Ansys-Fluent CFD software. The methodology has

been devoloped and validated by studying a spatially developing mixing layer in two-

and three-dimensions at a Reynolds number of 720 without modeling the splitter

plate walls.

The results from 2-D DNS of a spatially developing mixing layer performed by

Uzun [39] are used to validate the methodology. The LES is performed on a baseline

2-D mesh with a buffer zone attached to it downstream, which is similar to the mesh

used by Uzun [39], to make a fair comparison with his results. The baseline simulation

results match well with the 2-D DNS results of Uzun [39]. Numerous 2-D simulations

are performed to investigate the effect of sampling time, buffer zone, time-step, grid

refinement and to optimize the grid points in the baseline mesh. Vortex pairing is

observed , albeit at random locations, due to the random inflow forcing employed in

the 2-D simulations. The forcing significantly alters the evolution of the mixing layer

resulting in higher growth rate and Reynolds stresses consistent with the observations

in the literature.

The simulation time of 12 FTC’s, 4 for the transients and 8 for gathering the

flow statistics, ensures that the mean and RMS flow quantities become statistically

stationary in the present 2-D simulations.

The effectiveness of using a buffer zone is evaluated by varying the stream-wise

domain size with and without the buffer zones. The mixing layer growth rate is

insensitive to the presence of buffer zone and so is the time history of flow evolu-

tion. However, the effect of the pressure outflow boundary is seen within 10 vorticity

140

thicknesses from the exit boundary when the buffer zone is absent. Therefore, it is

concluded that the buffer zone is required at least from the perspective of obtaining

accurate statistics near the exit boundary and to allow flow entrainment into the

mixing layer. The flow is damped within the buffer zone implicitly by the numerical

diffusion introduced by the coarser mesh in the current work. The use of such an im-

plicit damping scheme is found to be satisfactory in the present analysis. The results

also suggest that increasing the stream-wise size of the physical domain has no effect

on the flow evolution in the presence of the buffer zone. The mixing layer achieves a

state of self-similarity for x > 190δω(0) corresponding to a Reynolds number of 6000.

The sensitivity of the 2-D LES to the time-step is also investigated by reducing

the time-step to half of the baseline simulation. The LowT SF 1 simulation closely

matches the growth rate observed by Uzun [39]. The Reynolds stresses remain un-

changed with respect to the baseline simulation, suggesting that the time-step of 10−7

used in the baseline simulation, resolves the length scales well in time.

The grid resolution study indicates that the baseline grid resolution is sufficient to

resolve the length scales spatially well. The flow evolution is almost the same when

the cross-stream resolution alone is increased. The stream-wise resolution results in

a different flow evolution than that of the baseline simulation. However, the results

illustrate that the mixing layer growth rate and Reynolds stresses of the baseline case

are almost the same as those of the refined meshes. The stream-wise grid spacing

which is as large as the initial vorticity thickness, suppresses the vortices and stabilizes

the mixing layer. It also verifies that the baseline mesh is adequate for resolving the

large scale structures in the flow field.

Due to the limited computational resources, the node count in the 3-D baseline

mesh is optimized by studying the sensitivity of the 2-D results to the domain size and

the grid stretching ratio in the cross-stream direction. The cross-stream domain size

has no impact on the simulation results for the domain size of up to 100δω(0). The

symmetry boundary condition on the side boundaries inhibits the flow entrainment

for domain sizes less than 100δω(0). The grid stretching has no effect on the mixing

141

layer growth rate and evolution for the stretching factors considered in this study. The

total node count, thus, is decreased by a factor of 8 by decreasing the cross-stream

domain size and increasing the grid stretching ratio to 10% without compromising

the accuracy of the results. This optimized grid is then used to construct a baseline

mesh for the 3-D simulations with 51 span-wise grid points.

The baseline 3-D simulation is performed using Fluent’s in-built Vortex Method

(VM) to force the perturbations at the inlet. The growth rate and the Reynolds

stresses are overpredicted due to the non-physical nature of the perturbations at the

inlet. The vortex method is improved for the current work. The hair-pin vortices

found in the splitter plate boundary layer are mimicked by the improved VM to force

perturbations at the inlet. The second 3-D LES with the improved VM captures

the counter-rotating rib vortices as well as the translative instabilty [21]. The mixing

layer is significantly three-dimensional further downstream. The number of rib vortex

pairs correlates with the number of modelled hair-pin vortices at the inlet. Hence,

it is believed that the hair-pin vortices result in the rib vortex pairs through the

translative instability in an experimental settings. The pairing of vortex cores is not

observed due to the nature of the forcing employed. The mixing layer growth rate

and Reynolds stresses are overpredicted, albeit lower than those of LES with Fluent’s

VM, compared to the experiments. The presence of stream-wise vortices at the inlet

all through the simulation is believed to have resulted in the over-prediction. The

one-dimensional energy spectrum, obtained using a second-order-accurate bounded

central differencing scheme (BCD) in the 3-D computations, has a decay rate of −53

for more than half-a-decade of wave numbers until the grid cut-off. The current work

indicates the potential of a second-order-accurate BCD to reduce the computational

cost per grid point for LES based modeling of complex flows.

The second invariant of velocity gradient tensor, used to extract the vortex cores

in the current work, is found to be a good tool to visualize the regions of vorticity in

the flow-field.

142

Based on the experience gained in the present study, the computational cost for a

realistic LES simulation of a jet using Fluent is estimated. The number of processors

required to perform a simulation scales as 25 times the number of grid points in

millions for a simulation time of 2 weeks. A jet simulation with 100 million grid

points using Fluent’s LES would require about 2,500 processors to complete in two

weeks.

5.2 Recommendations for Future Work

The sensitivity of the 3-D LES results to the span-wise domain size needs to be

investigated. The vortex method algorithm, used to force perturbations at the inlet,

needs to be improved further to match the mixing layer growth rate and Reynolds

stresses with experiments. Firstly, the stream-wise vortices at the inlet need to be

modeled as intermittent unlike in the present simulations. Secondly, the strength of

the vortices at the inlet should to be varied over time to see vortex pairing, which is

absent in the current 3-D simulations. An LES on a finer grid should be performed

to assess the quality of the present results and to ensure that the baseline 3-D grid

captures the length- and time-scales involved in the simulation reasonably well.

LIST OF REFERENCES

143

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APPENDICES

147

A. A NOTE ON DENSITY-BASED LES

A simulation is performed using the density-based solver even though it lacks the

central differencing schemes that the pressure-based solver offers. The baseline mesh

is used for the simulation. It should be noted that only the density-based solver

offers non-reflecting boundary conditions (NRBC’s) for pressure outflow boundaries.

It is found that the NRBC’s at the outlet, top and bottom free-stream boundaries

have caused the solution to diverge. Therefore, the outlet boundary is modeled as

constant pressure outflow and top, bottom boundaries are modeled as inviscid walls.

The mass flux profile is specified at the inflow, while the perturbations are imposed by

specifying a profile for the flow angle. The magnitude of the flow angle changes with

time step, and is computed using equation 3.20. The third-order MUSCL scheme is

used for spatial discretization. The time step is 10−7 s, which is same as that of the

baseline simulation.

The instantaneous vorticity contours for the density-based simulation obtained af-

ter 12 FTC’s are plotted in figure A.1. The contours obtained with the pressure-based

simulation using a second-order BCD on the same mesh are also shown in figure A.1

for comparison. The figure shows that the vorticity contours are smoother in the case

of the density-based solver probably due to higher diffusivity of the MUSCL scheme.

The spiral arms of the vortex core are distinctly visible around x = 170δω(0) in fig-

ure A.1(b). The strong coupling of the flow variables in the density-based solver might

also have played a role in obtaining smoother contours. In contrast, the pressure-based

solver shows fine structures in the vorticity for 150δω(0) < x < 200δω(0), as shown in

figure A.1(a).

The pressure-based solver is about 1.5 times faster than the density-based. The

pressure-based solver also offers time-advancement schemes like non-iterative time

advancement (NITA) and fractional step method (FSM). These schemes, which are

148

recommended for mildly compressible simulations, can further decrease the computa-

tional cost associated with LES. Therefore, it is deemed that using the pressure-based

solver is necessary for computationally intensive LES using Fluent. The reduced

CPU time coupled with the availability of the central differencing schemes makes the

pressure-based solver attractive for the present study.

x/δω(0)

y/δ

ω(0

)

50 100 150 200-25

0

25

(a) Pressure-based.

x/δω(0)

y/δ

ω(0

)

50 100 150 200-25

0

25

(b) Density-based.

Figure A.1. Comparison of the instantaneous vorticity contours withpressure- and density-based solvers.

149

B. ISSUES WITH LOWER TIME-STEPS

The pressure-based solver is found to exhibit stability issues with lower time steps

for LES of compressible flows. This is observed in simulations with either velocity or

mass flow BC specified at the inlet. The simulations employ the second-order BCD

for momentum.

High-frequency spurious oscillations develop in the velocity field when a low time

step is used to advance the LES from a RANS calculation. These oscillations originate

from the inflow boundary and propagate downstream as the simulation advances. As

these oscillations convect downstream, the amplitude increases eventually leading to

flow divergence after a few hundred time steps. Figure B.1 shows the oscillations after

two instances of time for a compressible LES with the pressure-based solver. The time

step used for the simulation is 5 × 10−8 s. The figure illustrates the propagation as

well as the amplification of the oscillations as the simulation is advanced. It should

be noted that the same simulation runs smoothly when the time step is increased

to 10−7 s. And also, the simulation does not diverge when the flow is modeled as

incompressible with a time step of 5× 10−8 s.

This problem persists even when the LES is initialized differently, e.g., using a

second-order RANS solution as a starting point as opposed to a first-order one, which

is used through out this study as an initial condition to LES. However, using a highly

diffusive first-order-accurate upwind scheme for momentum suppresses these spurious

oscillations and stabilizes the flow with lower time steps. Therefore, it is believed that

the low diffusivity of the BCD causes these oscillations and makes the pressure-based

solver unstable for compressible flows at lower time steps.

150

(a) time step = 400 (b) time step = 800

Figure B.1. Spurious oscillations with the pressure-based compressibleLES solver for smaller ∆t’s

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