rice university

RICE UNIVERSITY ADJOINT ANALYSIS FOR RECEPTIVITY PREDICTION by Alexander Dobrinsky A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy

Houston, Texas October 2002

RICE UNIVERSITY

Adjoint Analysis for Receptivity Prediction by Alexander Dobrinsky A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Approved, Thesis Committee:

S. Scott Collis, Chair Assistant Professor of Mechanical Engineering

Marek Behr Assistant Professor of Mechanical Engineering

Matthias Heinkenschloss Associate Professor of Computational and Applied Mathematics

Houston, Texas October 2002

Abstract Adjoint Analysis for Receptivity Prediction by

Alexander Dobrinsky

Physical knowledge of the laminar-turbulent transition process, prediction of the transition location, as well as the ability to control transition are essential in many engineering applications. However, control of the laminar-turbulent transition depends critically on various environmental sources and their ability to excite the instability waves in the flow, which are responsible for the laminar-turbulent transition. The process by which external disturbances are converted into instability waves is called receptivity. The research described in this thesis focuses on the receptivity of two– and three– dimensional boundary layers. The main objective of this research is to formulate, validate and apply adjoint analysis in order to predict receptivity. Adjoint analysis is a powerful approach for investigating the receptivity of different flows for arbitrary environmental sources. In this work, Adjoint Navier–Stokes (ANS) equations are formulated based on the sensitivity approach, and adjoint predictions are validated against Linearized Navier–Stokes (LNS) calculations. Further, Adjoint Parabolized Stability Equations (APSE) are derived as an approximation of ANS equations and compared against the ANS results. Our studies indicate that the APSE method should be constructed as an approximation to the ANS equations, not as the formal adjoint of the PSE. When implemented in this manner, we show that APSE is a viable method for receptivity prediction, even in highly nonparallel flows. The APSE is first applied to predict receptivity of weakly nonparallel two-dimensional boundary layer flows for a variety of parameters. We find that these flows are

iii

generally more receptivity to oblique disturbances although two-dimensional disturbances are less stable. We also find that favorable pressure gradient boundary layers are more receptive then adverse pressure gradient boundary layers, although adverse pressure gradients are destabilizing. The APSE are then applied to highly nonparallel three-dimensional boundary layers where we find that for the inviscidly unstable crossflow instability, stability effects typically dominate receptivity effects. Comparison of receptivity for stationary and unsteady crossflow instabilities shows that receptivity to both localized momentum sources and streamwise wall-velocity excitations is larger for unsteady modes, but that receptivity to wall-normal excitations is larger for stationary modes. Finally, we consider receptivity for the swept parabolic cylinder and observe that convex surface curvature tends to enhance receptivity to wall roughness, in agreement with prior studies. Utilizing the efficiency of adjoint methods, we also consider other excitations for the swept parabolic cylinder and show that convex surface curvature also enhances receptivity to streamwise momentum sources, but that receptivity to both normal and tangential velocity disturbances is slightly reduced.

iv

Acknowledgments I want to express my sincere gratitude and appreciation to my advisor Dr. Scott Collis for guidance and continued encouragement and support. I would like to thank him for teaching me not to be afraid of challenges and be professional in my work. With his enthusiasm, his inspiration, and his great efforts to explain things clearly and simply, he helped to make the research interesting for me. Throughout my thesiswriting period, he provided encouragement, sound advice, good teaching, and lots of good ideas. I would have been lost without him. I am also grateful to the members of my committee for taking the time to guide me through my dissertation. I would like to express my appreciation to Professor Matthias Heinkenschloss for his insightful remarks and ideas and to Professor Marek Behr for his valuable comments. Further, I want to thank my colleagues for their valuable discussions and suggestions. Special thanks to Dr. Yong Chang for his sincere willingness to help me answer my questions and for his programming expertise. I am also thankful to all the members of our research team: Steven Kellogg, Dr. R.D. Prabhu, Guoquan Chen, Srinivas Ramakrishnan, Zach Smith, and Dr. Kaveh Ghayour for helping me get through the difficult times, and for the caring they provided. Also, I wish to thank my entire extended family for providing a loving environment for me. Lastly, and most importantly, I wish to thank my parents, Tataina Dobrinskaya and Yakov Dobrinsky for their full support and encouragement in my endeavor. To them I dedicate this thesis.

This research work is based in part upon work supported by the NASA Langley Research Center under Grant numbers NG-1-2134 and NAG-1-1976.

vi

Nomenclature

Abbreviations FSC

Falkner–Skan–Cooke

OSE

Orr-Sommerfeld Equation

PSE

Parabolized Stability Equations

TS

Tollmien–Schlichting

Greek Symbols

α

Streamwise disturbance wavenumber

β

Spanwise disturbance wavenumber

δij

Kronecker delta

η

Parabolic coordinate perpendicular to the attachment line, normal to the surface of the cylinder

Γ

Denotes the boundary of the physical domain, Ω

Λu

Efficiency function

ν

Kinematic viscosity

Ω

Physical domain of the problem

ω

Nondimensional frequency vii

ξ

Parabolic coordinate perpendicular to the attachment line along the surface of the cylinder

αT S

Streamwise wavenumber for Tollmien-Schlichting wave

Roman Symbols

A

Matrix operator in LNS, PSE equations

Ai

Inviscid flux Jacobians (F i )U

B


g¯

The base value of control

C


H(U (g))

Objective function dependent on argument U

D


Di

Viscous flux Jacobians (F vi )U

E1


E2


F vi

Viscous flux vectors

Fi

Inviscid flux vectors

G


g

Wall boundary controls, momentum and mass sources

g0

The variation of control about its base value viii

U

State field

U0

The variation of state about its base value

U0

State field at time t0

¯ U

The base value of state field

v

Velocity vector {v1 , v2 , v3 } or {u, v, w}

x

Position vector {x, y, z}

B

Generalized boundary conditions

C(U , g)

Navier–Stokes equations in compact form

J (g)

Objective function dependent on argument g

N

Navier–Stokes operator

S(g)

Momentum source in Navier–Stokes operator

IR3

Three dimensional space of real numbers

L0

Global length scale

p

Pressure field

s

Coordinate on the parabolic cylinder, perpendicular to the attachment line along the surface of the cylinder.

t

Nondimensional time

t0

Initial point in time

tf

Final point in time

ix

V0

Reference velocity

x

Generalized nondimensional streamwise coordinate

xo

The observation station

y

Generalized nondimensional wall-normal coordinate

Re

Local Reynolds number

n

Coordinate on the parabolic cylinder, perpendicular to the attachment normal the surface of the cylinder.

X

Global coordinate

Y

Global coordinate

Z

Global coordinate

z

Generalized nondimensional wall normal coordinate

Superscripts and Subscripts ( )0

Perturbation or disturbance quantity

( )∗

Adjoint variables

( )max

Maximum value

(ˆ)

State shape functions

(˜)

Adjoint shape functions

(¯)

Base flow quantity

x

Table of Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iii

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Receptivity and Sensitivity Analysis . . . . . . . . . . . . . . . . . . .

3

1.3

Prior Research on the Receptivity of Shear Flows . . . . . . . . . . .

4

1.3.1

Asymptotic Theory . . . . . . . . . . . . . . . . . . . . . . . .

5

1.3.2

Finite Reynolds Number Theory (FRNT) . . . . . . . . . . . .

6

1.3.3

Direct Numerical Simulation (DNS)

. . . . . . . . . . . . . .

9

1.3.4

Experimental Studies . . . . . . . . . . . . . . . . . . . . . . .

10

1.3.5

Adjoint Methods . . . . . . . . . . . . . . . . . . . . . . . . .

11

1.4

Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

1.5

Accomplishments . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2 Formulation of Adjoint Methods for Receptivity Prediction . . . .

15

2.1

Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

2.2

Receptivity Formulation . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.3

The Sensitivity Problem . . . . . . . . . . . . . . . . . . . . . . . . .

24

2.3.1

Adjoint Formulation for the Sensitivity Problem . . . . . . . .

26

2.3.2

Receptivity as a Sensitivity Problem . . . . . . . . . . . . . .

28

Natural Receptivity . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

2.4

xi

3 Implementation of Numerical Methods . . . . . . . . . . . . . . . . .

36

3.1

LNS-ANS Formulation . . . . . . . . . . . . . . . . . . . . . . . . . .

38

3.2

PSE-APSE Formulation . . . . . . . . . . . . . . . . . . . . . . . . . .

41

3.2.1

PSE Formulation . . . . . . . . . . . . . . . . . . . . . . . . .

41

3.2.2

APSE Formulation . . . . . . . . . . . . . . . . . . . . . . . .

44

3.3

The Orr–Sommerfeld Equation and its Adjoint . . . . . . . . . . . . .

46

3.4

Implementation of PSE/APSE . . . . . . . . . . . . . . . . . . . . . .

47

3.4.1

Spatial Discretization . . . . . . . . . . . . . . . . . . . . . . .

47

3.4.2

PSE/APSE Stabilization Techniques

. . . . . . . . . . . . . .

51

Validation of Numerical Methods . . . . . . . . . . . . . . . . . . . .

57

3.5.1

Validation of LNS/ANS . . . . . . . . . . . . . . . . . . . . . .

57

3.5.2

Validation of PSE/APSE . . . . . . . . . . . . . . . . . . . . .

60

3.5

4 Receptivity of Blasius and Falkner–Skan Boundary Layers 4.1

4.2

. . . .

87

Forced Receptivity Results . . . . . . . . . . . . . . . . . . . . . . . .

87

4.1.1

Blasius Boundary Layer . . . . . . . . . . . . . . . . . . . . .

88

4.1.2

Falkner–Skan Boundary Layers . . . . . . . . . . . . . . . . .

92

Natural Receptivity Results . . . . . . . . . . . . . . . . . . . . . . .

94

4.2.1

Blasius Boundary Layer . . . . . . . . . . . . . . . . . . . . .

94

4.2.2

Falkner–Skan Boundary Layers . . . . . . . . . . . . . . . . .

96

5 Receptivity of Three-Dimensional Boundary Layers . . . . . . . . . 110 5.1

Swept Hiemenz flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.2

Falkner–Skan–Cooke Boundary Layer . . . . . . . . . . . . . . . . . . 115 5.2.1

Steady Three-dimensional Crossflow Instabilities . . . . . . . . 117

5.2.2

Unsteady Three-dimensional Crossflow Instabilities . . . . . . 122

5.2.3

Pressure Gradient Effects . . . . . . . . . . . . . . . . . . . . 122

xii

5.2.4

Importance of Nonparallel Effects . . . . . . . . . . . . . . . . 123

5.3

The Swept Parabolic Cylinder . . . . . . . . . . . . . . . . . . . . . . 124

5.4

Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6 Transient Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.1

PSE Implementation for Wall Boundary Control Sources. . . . . . . . . . . . . . . . . . . . . . 158

6.2 7

Comparison of PSE and LNS Transients . . . . . . . . . . . . . . . . . 159

Conclusions and Future Directions . . . . . . . . . . . . . . . . . . . 172 7.1

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

7.2

Future Directions for Research . . . . . . . . . . . . . . . . . . . . . . 173 7.2.1

Sensitivity of Adjoint to the Flow Parameters . . . . . . . . . 174

7.2.2

Nonmodal Algebraic Growth . . . . . . . . . . . . . . . . . . . 174

7.2.3

Nonlinear Receptivity

7.2.4

Secondary Receptivity . . . . . . . . . . . . . . . . . . . . . . 175

. . . . . . . . . . . . . . . . . . . . . . 175

A Nondimensionalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 A.1 Local and Global Nondimensionalizations . . . . . . . . . . . . . . . . 177 A.2 Forced Adjoint Nondimensionalization . . . . . . . . . . . . . . . . . 180 B Mean Flow Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 B.1 The Falkner–Skan–Cooke Boundary Layer . . . . . . . . . . . . . . . 182 B.1.1 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 B.2 The Swept Parabolic Cylinder . . . . . . . . . . . . . . . . . . . . . . 188 C Tensor Concepts of Differential Geometry . . . . . . . . . . . . . . . 197 C.1 Tensor Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 199 C.2 The Continuity Equation in a Body-fitted Coordinate System . . . . 201 xiii

D Governing Equations in Curvilinear Coordinates . . . . . . . . . . . 203 D.1 The Harmonic Linearized Navier–Stokes Operator . . . . . . . . . . . 203 D.2 The Harmonic Adjoint Navier–Stokes Operator . . . . . . . . . . . . 205 D.3 Amplitude Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 E Bi-orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

xiv

List of Tables 2.1

Adjoint Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

B.1 Local Nondimensionalization Based on δr and U∞ . . . . . . . . . . . . . 185 B.2 Local Nondimensionalization Based on δs and U∞ . . . . . . . . . . . . . 186 0 B.3 Global Nondimensionalization Based on L and U∞ . . . . . . . . . . . . 186

xv

List of Figures 2.1

Control Volume for Receptivity Prediction. . . . . . . . . . . . . . . . . .

35

3.1

The Pentadiagonal System Used in Solving LNS-ANS Equations. . . . . .

69

3.2

Schematic of Computational Domain. . . . . . . . . . . . . . . . . . . . .

70

3.3

Convergence Using Newton’s Method. . . . . . . . . . . . . . . . . . . .

71

3.4

Convergence of the LNS/ANS Method with Streamwise Resolution. . . .

72

3.5

Transient Effects in LNS. . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

3.6

Validation of the Adjoint Method. . . . . . . . . . . . . . . . . . . . . . .

74

3.7

Convergence of the PSE with Streamwise and Wall-normal Resolution. .

75

3.8

Transient Effects in PSE. . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

3.9

The Error in the Kinetic Energy of Instability. . . . . . . . . . . . . . . .

77

3.10 Comparison of the PSE Solutions for Various Stabilization Parameters, τ .

78

3.11 Convergence of the j-product with the Number of Pressure Iterations. . .

79

3.12 Convergence of APSE with the Streamwise and Wall-normal Resolutions.

80

3.13 Comparison of State and Adjoint Streamwise Velocity Profiles. . . . . . .

81

3.14 Convergence of the j-product with the Streamwise and Wall-normal Resolutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

3.15 Different Normalizations in Adjoint Using τ -stabilization. . . . . . . . . .

83

3.16 The Error in the Adjoint Kinetic Energy Using j-normalization. . . . . .

84

3.17 Comparisons of the LNS, PSE, ANS and APSE Growth Rates. . . . . . .

85

3.18 Comparison of the Two Different Normalizations in APSE. . . . . . . . .

86

4.1

Receptivity Predictions for TS waves in the Blasius Boundary Layer. . .

98

4.2

Comparison of APSE and AOSE. . . . . . . . . . . . . . . . . . . . . . .

99

4.3

Receptivity Predictions for Various Values of β. . . . . . . . . . . . . . . 100

xvi

4.4

Falkner-Skan Boundary Layer With an Adverse Pressure Gradient. . . . 101

4.5

Falkner-Skan Boundary Layer With a Favorable Pressure Gradient. . . . 102

4.6

Receptivity to Wall-normal and Streamwise Velocity Disturbances. . . . 103

4.7

Three–dimensional Disturbances in an Adverse Pressure Gradient. . . . . 104

4.8

Three–dimensional Disturbances in a Favorable Pressure Gradient. . . . 105

4.9

Adjoint Predictions for Acoustic Receptivity.

. . . . . . . . . . . . . . . 106

4.10 Acoustic Receptivity for the Blasius Boundary Layer. . . . . . . . . . . . 107 4.11 Acoustic Receptivity in an Adverse Pressure Gradient. . . . . . . . . . . 108 4.12 Acoustic Receptivity in a Favorable Pressure Gradient. . . . . . . . . . . 109 5.1

Parallel Theory Receptivity Prediction for Swept Hiemenz Flow. . . . . . 127

5.2

Efficiency to Wall Roughness for Swept Hiemenz Flow. . . . . . . . . . . 127

5.3

ANS Predictions Compared With LNS Calculations for Wall Roughness Sources in the Swept Hiemenz Flow. . . . . . . . . . . . . . . . . . . . . 128

5.4

Phase Effects on the Roughness Efficiency Function for Swept Hiemenz Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.5

Adjoint Kinetic Energy Growth Rate, ANS and APSE Comparison for FSC. 130

5.6

Schematics of the Nonparallel Region in FSC. . . . . . . . . . . . . . . . 131

5.7

Nonparallel Effects in FSC. . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.8

Comparison of APSE and LNS Predictions for FSC Boundary Layer. . . . 133

5.9

Growth Rates of Crossflow Instabilities and Corresponding Adjoint Modes in FSC Boundary Layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.10 Adjoint Kinetic Energy Using Different Norms for FSC Flow. . . . . . . . 134 5.11 Kinetic Energy Growth Rates for Crossflow Instabilities in FSC Flow. . . 135 5.12 Adjoint Spanwise Wavenumber and Kinetic Energy Growth Rate. . . . . 135 5.13 Absolute Values of Crossflow Instability Maximum Streamwise Velocity and Adjoint Maximum Streamwise Velocity. . . . . . . . . . . . . . . . . 136 xvii

5.14 Falkner–Skan–Cooke boundary layer for Various Values of β. . . . . . . . 137 5.15 Difference in the Absolute Values of Adjoint Streamwise Velocities. . . . 138 5.16 Difference in the Absolute Values of Adjoint Streamwise Wall Shear Stress. 138 5.17 Difference in the Absolute Values of Adjoint Wall Normal Velocities. . . 139 5.18 Difference in the Absolute Values of Adjoint Spanwise Velocities.

. . . . 139

5.19 Difference in the Absolute Values of Adjoint Pressure. . . . . . . . . . . . 140 5.20 The Imaginary Part of the Streamwise Wave Number for Various Values of β. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.21 The Wave Angle θ for Various Values of β . . . . . . . . . . . . . . . . . 141 5.22 Receptivity of a FSC Boundary Layer for Disturbances with Various Values of β in Global Normalization. . . . . . . . . . . . . . . . . . . . . . . . . 142 5.23 Receptivity of a FSC Boundary Layer for Disturbances with Various Values of β in Local Normalization. . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.24 Streamwise Wave Number of Adjoint Crossflow Instabilities for Various Values of ω. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 5.25 Receptivity of FSC Boundary Layer for Disturbances with Various Frequency Values F in Local Normalization. . . . . . . . . . . . . . . . . . . 145 5.26 Receptivity of FSC Boundary Layer for Disturbances with β = 0.2 for F = 150 and F = 0 in Local Normalization. . . . . . . . . . . . . . . . . 146 5.27 Effects of Pressure Gradient on Receptivity of Crossflow Instabilities in Local Normalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 5.28 Effects of Pressure Gradient on Receptivity of Crossflow Instabilities in Global Normalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.29 The Imaginary Part of the Adjoint Streamwise Wave Number for Various Values of βH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 5.30 Evaluation of Nonparallel Effects for the Crossflow Instability. . . . . . . 150

xviii

5.31 Evaluation of Nonparallel Effects for the Adjoint Mode. . . . . . . . . . . 151 5.32 Comparison of ANS and LNS Predictions for the Swept Parabolic Cylinder.152 5.33 The Imaginary Part of the Streamwise Wavenumber With and Without Curvature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.34 Curvature Effects on the Wave Angle θ. . . . . . . . . . . . . . . . . . . 153 5.35 Receptivity of Flow Over a Swept Parabolic Cylinder With and Without Curvature for Crossflow Instabilities in Local Normalization. . . . . . . . 154 5.36 Amplitude of the Receptivity Efficiency Function for β = 100. . . . . . . 155 5.37 Comparison Between Amplitudes of Incompressible and Compressible Receptivity Efficiency Functions. . . . . . . . . . . . . . . . . . . . . . . . . 156 5.38 Comparison of Kinetic Energy Growth Rates For Crossflow Mode of β = 100 in Compressible and Incompressible Flow Over a Swept Parabolic Cylinder.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

6.1 PSE Implementation Using ke-normalization Versus PSE With a Fixed Growth Rate, α. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 6.2

The Effects of a Wall-normal Boundary Suction and Blowing Source. . . 163

6.3

LNS and PSE Transient for a Source at Re = 282. . . . . . . . . . . . . . 164

6.4

The Typical Wall-normal Suction and Blowing Source. . . . . . . . . . . 165

6.5 LNS and PSE Transient for a Source at Re = 346. . . . . . . . . . . . . . 165 6.6 LNS and PSE Transient in Different Measures Using p-stabilization in the PSE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 6.7 LNS and PSE Transient in umax . . . . . . . . . . . . . . . . . . . . . . . 167 6.8

Transient Response for Different Streamwise Resolutions. . . . . . . . . . 168

6.9

Kinetic Energy for a Wide Gaussian Bump. . . . . . . . . . . . . . . . . 169

6.10 Kinetic Energy of the Instability Wave for a Distributed Source in the Form of a Sine Wave Multiplied by a Wide Gaussian bump. . . . . . . . 170 xix

6.11 Wide Gaussian Wall-normal Suction and Blowing Source. . . . . . . . . . 170 6.12 Wall-normal Suction and Blowing Source in the Form of Sine Wave Multiplied by a Wide Gaussian bump.

. . . . . . . . . . . . . . . . . . . . . 171

B.1 Schematic of the Falkner-Skan-Cooke Flow. . . . . . . . . . . . . . . . . 191 B.2 FSC Boundary Layer Profiles. . . . . . . . . . . . . . . . . . . . . . . . . 192 B.3 Schematic of the Global Normalization. . . . . . . . . . . . . . . . . . . . 192 B.4 Profiles of Nondimensional Velocities for βh = 1. . . . . . . . . . . . . . . 193 B.5 Stability of the Falkner-Skan-Cooke flow. . . . . . . . . . . . . . . . . . . 193 B.6 Profiles of Nondimensional Velocities for βh = 0.63. . . . . . . . . . . . . 194 B.7 The Geometry of the Swept Parabolic Cylinder. . . . . . . . . . . . . . . 195 B.8 Schematic of a Body-Fitted Coordinate System. . . . . . . . . . . . . . . 196

xx

Chapter 1 Introduction 1.1

Background

Turbulent flows are so prevalent in everyday life that we hardly even notice them. We observe turbulent flows when we pour cream into a coffee cup or when we watch the smoke from a smokestack or the exhaust from a car’s tailpipe. Despite the fact that turbulent flows are widespread, understanding of the physics of turbulence and the process of transition from laminar to turbulent flows is still incomplete. Physical knowledge of the transition process, prediction of the transition location, and the ability to control transition are important to many engineering applications. Accounting for the transition process is crucial in designing natural laminar flow airfoils, combustion chambers, automobiles and submarines. For instance, in order to reduce the viscous drag associated with turbulent motion over an airplane wing, one would like to maintain laminar flow over a large portion of the wing surface. One can achieve this by optimizing the shape of the airfoil or, possibly, by introducing small actuators on the airplane wing for active laminar flow control. Maintaining laminar flow is also crucial in order to reduce the noise associated with turbulent flow, which could be an important factor in submarine design. There are also situations when turbulent flow is desirable in order to promote better mixing of fuel and air, such as the flow inside combustion chambers. In all of these cases, knowledge of transition is essential in order to produce viable designs. The transition process occurs when environmental disturbances are converted into instability waves that are amplified by the flow. For small environmental excitations

1

Chapter 1. Introduction

2

in boundary layer flows, transition is typically divided into three stages: receptivity, linear stability, and non-linear breakdown. Receptivity describes the conversion of environmental excitations into boundary layer instability waves, linear stability describes an exponential amplification of disturbances and nonlinear breakdown accounts for nonlinear amplification of instability waves and their subsequent breakdown into turbulence. For environmental disturbances of high amplitude, the laminar-to-turbulent transition may not be separated into these three distinct stages. Any transition process that cannot be described by the above three-stage model is called bypass transition. Bypass transition can be affected by a variety of factors such as the high transient growth associated with the non-normality of the LNS operators [5, 12, 75, 88], and large initial excitations that lead to nonlinear interaction of excited modes in flow, or a combination of both. This dissertation focuses on the receptivity stage of transition. In order to study the receptivity of various flows within a wide range of parameters, we first derive adjoint methods in a framework of sensitivity analysis and show that receptivity is a particular case of a general flow sensitivity analysis. Next, we apply the adjoint method in order to document the receptivity of two-dimensional Falkner–Skan boundary layers to various environmental disturbances that excite two-dimensional and oblique Tollmien–Schlichting (TS) instabilities. After documenting the receptivity of weakly nonparallel boundary layers, we consider the receptivity of crossflow instabilities in the three–dimensional swept Hiemenz and Falkner–Skan–Cooke boundary layers, as well as in the swept parabolic wing boundary layer. These results document the receptivity of highly nonparallel flows while demonstrating the advantages of global adjoint based methods for this important class of flows. This Chapter begins by defining the concepts of receptivity and sensitivity of a


3

flow and by briefly reviewing the various methods used to predict receptivity. The Chapter concludes with a summary of the accomplishments of this research and an overview of the material discussed herein.

1.2

Receptivity and Sensitivity Analysis

The receptivity process was first defined by Morkovin [59] and since then has been understood as an important factor in laminar-turbulent transition. In this work we adopt the more general definition of receptivity as described by Kerschen [49]. Kerschen defines receptivity as the conversion of disturbances into the instability waves of a shear flow where he distinguishes between natural and forced receptivity. Natural receptivity considers disturbances that originate outside the shear layer such as freestream acoustic waves and freestream turbulence. Conversely, forced receptivity considers disturbances that originate inside the shear layer, such as vibrating ribbons, suction/blowing slots, wall motions, etc. In either case, one can think of receptivity as the process that provides an initial condition for a stability analysis such as quasi-parallel stability theory, Parabolized Stability Equations (PSE), or even full Navier–Stokes simulations. Receptivity can be best formulated in the context of a sensitivity analysis. In sensitivity analysis, one typically is interested in changes of a particular objective function that depends on the solution of the governing equations. The solution, in turn, depends on external disturbances that are expressed as appropriate boundary conditions and source terms. For instance, the objective function could correspond to the disturbance kinetic energy evaluated at a particular streamwise location in a boundary layer and averaged in the wall-normal direction.


1.3

4

Prior Research on the Receptivity of Shear Flows

Since the pioneering study of Lord Rayleigh [68] who developed a general linear stability theory for inviscid plane-parallel shear flows, a large amount of research has attempted to understand the transition process in two- and three-dimensional boundary layers. While linear stability analysis of fluid flows has a long history, dating back to the early twentieth century theoretical works of Orr [62], Sommerfeld [78], Tollmien [86] and Schlichting [74], and supported by experimental works of Schubauer and Skramstad [76]; receptivity was only more recently been understood to be an important factor in the transition process. Currently, receptivity is an active field of research characterized by many different approaches including asymptotic theory, Finite Reynolds Number Theory (FRNT), local adjoint based methods, Direct Numerical Simulations (DNS), and experimental studies. While many approaches have proven effective for predicting receptivity, asymptotic theory, Finite Reynolds Number theory, and local adjoint based methods have utilized either a high Reynolds number asymptotic limit or a local parallel-flow approximation and neither approach accounts for the effect of the growing mean flow on receptivity. Recently, a number of investigators have used the full linearized Navier–Stokes equations in order to predict receptivity for nonparallel flows [10, 21]. While these studies have played an important role both in validating theoretical predictions and in identifying weaknesses in the theories, their computational expense makes it difficult to cover the large receptivity parameter space. Since some of the most significant challenges in receptivity studies are those of covering the nearly infinite parameter space of disturbances types, locations, and strengths occurring in a practical transition environment, progress is needed to develop efficient tools to help alleviate this problem.


5

With the pioneering work of Hill [41–43], Tumin [89,91], Luchini and Bottaro [55], and the subsequent work of Airiau [1,2], Pralits [67] and Collis and Dobrinsky [17,18]; adjoint methods emerged as an efficient method of receptivity prediction for arbitrary sources over a wide range of flow parameters. In the next several sections we shall review available methods for receptivity prediction in greater detail and highlight their advantages and disadvantages.

1.3.1

Asymptotic Theory

Before 1983, it was well known that environmental factors can influence the transition location of boundary layer flows. However, the process by which environmental disturbances are converted into the instability waves of the boundary layer was not well understood. The mechanism for converting large wavelength acoustic waves into the small scale boundary layer instability waves was first explained by Goldstein [32, 33], and Ruban [71], and later reviewed by Goldstein and Hulgreen [34] and Heinrich and Kerschen [37]. The conversion process becomes possible when long-wavelength acoustic-waves scatter from regions of rapid changes in the pressure gradient, which are on the scale of the Tollmien–Schlichting (TS) waves. These changes in pressure gradient can occur at the leading edge of an airfoil, at a roughness element on the surface, or at places of rapid changes in surface curvature. The asymptotic theory allows one to predict receptivity of TS waves in the limit of large Reynolds numbers. In this limit, Stewartson obtained the steady flow over the variation in the wall surface using triple-deck theory [81, 82]. This theory divides the boundary layer into three distinct regions in the normal direction. The boundary layer (deck) is subdivided into a viscous layer with a thickness on the order Re −5/8 , a middle deck of inviscid but rotational flow with a thickness of Re −1/2 , and an upper inviscid deck where disturbances are assumed to decay exponentially over a length of Re −3/8 . In each


6

region or deck, the solution to the approximated steady-state Navier–Stokes (NS) equations is obtained and then matched asymptotically across the boundary layer. The unsteady disturbances are obtained by linearizing the NS equations about the steady flow, solving in each deck, and asymptotically matching across the boundary layer. The resulting equations are solved by applying the Fourier transform in time and in the streamwise and spanwise directions. The solution is then obtained in physical space by evaluating an inverse Fourier integral with an integration path in the complex wavenumber space chosen to satisfy causality requirements. The receptivity amplitude is found by evaluating the residue corresponding to the unstable TS wave. Perhaps the main result of Goldstein and Ruban’s work is the derivation of an explicit formula for the coupling coefficient, defined as the ratio of the resultant amplitude of the TS wave to the amplitude of the freestream disturbance, which for a given steady flow and acoustic disturbance, depends only on the type and size of the variations in the wall surface. Using asymptotic theory, Goldstein [32, 33] demonstrated that the results are consistent with the experimental observations of Shapiro [77]. While the asymptotic method allows one to obtain analytical receptivity predictions in the form of the coupling coefficient, the theory is only applicable for high Reynolds numbers and parallel flows. The restriction to high Reynolds numbers is relaxed by the so-called Finite Reynolds Number Theory.

1.3.2

Finite Reynolds Number Theory (FRNT)

Crouch, Choudhari, and Streett have used receptivity studies employing the finite Reynolds number approach [15, 24]. The method is widely applicable for parallel shear flows and accounts for viscous effects occuring at finite Reynolds numbers. Using this approach, Choudhari and Streett [15] calculated coupling coefficients for a


7

variety of external disturbances including wall-normal suction/blowing and surface roughness. In FRNT, one solves the inhomogeneous Linearized Navier-Stokes (LNS) equations using a parallel-flow approximation. The LNS equations are Fourier transformed in time and in the streamwise and spanwise directions, which reduces them to a fourth-order ordinary differential equation called the Orr–Sommerfeld Equation (OSE). The solution to this differential equation is sought for prescribed inhomogeneous boundary conditions. In physical space, the solution is obtained by evaluating an inverse Fourier integral, where the integration path in the complex wavenumber space is chosen to satisfy causality requirements. Since one is usually interested in convective instabilities, one chooses the integration contour in the upper half plane. The integration can be written in terms of a sum of residues over the poles in the complex wavenumber plane that are located within the contour of integration. Discrete poles as well as the continuous spectrum contribute to the Fourier integral. Note that if one is interested in receptivity for a particular TS wave corresponding to a pole at a particular wavenumber (αT S ), one only needs to calculate the residue at that pole. Using FRNT, Choudhari and Street [15] obtained receptivity predictions in a concise way by introducing efficiency functions Λ(j) u for two-dimensional TS waves in the Blasius boundary layer. Here the superscript (j) is used to indicate the type of external excitation considered. For the receptivity process characterized by scattering of acoustic waves from a suction/blowing source or from a roughness element on the wall boundary, Choudhari and Streett define the efficiency function as (j)

uT S (x, y, t) = Cu(j) uac Eu (y, ω, Re)eαT S x−ωt ,

(1.1a)

¯ (j) (αT S )Λ(j) Cu(j) = ε(j) w F u (ω, Re).

(1.1b)

where


8

In these expressions, x and y are streamwise and wall-normal coordinates, t is time, ω is nondimensional frequency, Re is the local Reynolds number based on the displacement thickness, Cu(j) is the coupling coefficient between the perturbation’s streamwise (j)

velocity uT S and the acoustic field uac , Eu is the eigenfunction profile as a function of wall-normal coordinate y, F¯ (j) is the transform of the spatial distribution of the external source evaluated at the complex unstable TS wavenumber αT S , and ε(j) w is the size of the external excitation. The above definition of the efficiency function is independent of the effects of the local geometry of the source which are entirely included in F¯ (j) . Using (1.1), Choudhari and Street report the dependence of Λ(j) u on the frequency of external disturbances for a fixed location of the wall inhomogeneity. They find that Λ(j) u for suction and blowing sources tends to be larger for low frequency values while Λ(j) u for wall roughness tends to be larger for high frequency values. They further consider the dependence of Λ(j) u on the Reynolds number, Re, based on the local displacement thickness for a fixed frequency of external excitation. The authors observe that receptivity to wall suction is two orders of magnitude larger than receptivity to wall geometry variations. The largest receptivity for wall suction sources occurs for Reynolds numbers located approximately two-thirds of the distance from the leading edge to the first neutral point; where the neutral point (or branch) is defined as a particular streamwise location in the flow where the instability wave has a zero growth-rate. The receptivity due to wall roughness, however, is larger near the second branch. Benefits of FRNT include the fact that it is relatively straightforward, it requires modest numerical computations, and it is applicable to moderate Reynolds number flows. However, it does not account for the nonparallel growth of boundary


9

layers as formulated. While some boundary layers, such as the two-dimensional Blasius boundary layer, can be well approximated by parallel theory, three-dimensional boundary layers are often highly nonparallel and their receptivity characteristics may not be well represented by parallel flow approximations [16]. In order to remedy this, Bertollotti [10] proposed to use a Taylor-series expansion of the laminar meanflow at the location of the roughness. He obtained weakly nonparallel receptivity predictions through application of the residue theorem to a series of inhomogeneous Orr–Sommerfeld type problems. His approach is computationally efficient and explicitly evaluates nonparallel contributions to the eigenmode response, but it is limited to surface disturbances that have a streamwise extent much smaller than the lengthscale of mean-flow nonparallelism. In order to investigate nonparallel effects under more general conditions, a number of researchers have conducted Direct Numerical Simulations (DNS) by solving numerically the Linearized Navier-Stokes equations.

1.3.3

Direct Numerical Simulation (DNS)

Some of the most valuable results have been obtained using DNS. Readers interested in DNS for receptivity prediction of crossflow vortices in high speed flows near a swept leading edge of an airfoil may consult Collis and Lele [19, 21]. Details regarding DNS implementation for receptivity prediction may also be found in Streett [83]. Further information can be found in the works of Malik [56] for hypersonic boundary layers and Bertolotti [10] for three-dimensional boundary layers. DNS is used to numerically solve the LNS equations without retreating to the approximations introduced by asymptotic theory or finite Reynolds number theory. DNS is particularly beneficial in calculating the transient response of the boundary layer due to external perturbations. It is also one of the few means in attesting applicability of FRNT for receptivity prediction. For instance, using DNS, Collis


10

and Lele [21], found that FRNT overpredicts receptivity near the leading edge of the swept parabolic cylinder by as much as 29 percent. Furthermore, DNS can be easily applied to accommodate nonlinear effects that can be important in bypass transition studies. While these studies have played an important role in validating theoretical predictions as well as in identifying weaknesses in the theories, their computational expense makes it difficult to cover the large receptivity parameter space. Even though DNS studies can serve as benchmarks for evaluating various receptivity approaches, a single and most important test for validity of theoretical and computational methods is experimental investigation.

1.3.4

Experimental Studies

Since the early experiment of Schubauer and Skramstad [76] extensive experimental research has been conducted to aid in understanding the receptivity of boundary layers. Nishioka and Morkovin, [61], and more recently Saric [70] have thoroughly reviewed experimental research. Readers interested in the Soviet experimental research are advised to consult Kachanov, Kozlov and Levchenko [47] and Kozlov and Ryzhov [52]. Some of the earlier studies in acoustic receptivity of two-dimensional boundary layers with two-dimensional surface roughness include the work of Aizin and Polyakov [3], and later, Zhou et al. [95], and Wlezien [93]. Zhou et al. [95] considered acoustic receptivity due to a single strip of tape for various oblique angles. They concluded from their measurements that, at a fixed streamwise location, acoustic receptivity decreases as the oblique angle increases. However, they did not account for the reduced linear growth of the oblique TS waves. In order to carefully investigate the acoustic receptivity of the Blasius boundary layer subject to oblique disturbances, King and Breuer [50] conducted experiments to investigate the receptivity and the subsequent evolution of TS waves formed by the interaction of acoustic waves with


11

two-dimensional and oblique surface roughness. They used a thin aluminum flat plate with elliptical leading edge as their model. To avoid potential receptivity to surface roughness, the flat plate was polished to a 0.2µm (rms) surface finish. They induced receptivity by using surface waviness of given height and freestream acoustic excitations. Contrary to the conclusions of Zhou et al. [95], King and Breuer [50] found that receptivity significantly increases with an increased oblique angle. While TS instability waves are typically responsible for transition in the Blasius boundary layer flow when subject to small environmental perturbations, crossflow instability is a major contributor to transition in three-dimensional boundary layers on swept wings. A review of the stability and transition of three-dimensional boundary layers is given by Saric and Reed [72]. Detailed experiments of crossflow instability in three-dimensional boundary layers were conducted by Saric and Yeates [73], Poll [65], Kachanov and Tararykin [48] and recently by Deyhle and Bippes [27] who found that freestream turbulence intensity determines, whether stationary or travelling waves, dominate the transition process. Similar to DNS studies, experimental investigations, while precise, are expensive and challenging and do not easily cover the large parameter space of various disturbance types and locations.

1.3.5

Adjoint Methods

Adjoint methods have been developed as a means to efficiently cover the large receptivity parameter space. While, adjoint methods have long been employed in optimal control problems, they had not been used for receptivity prediction until Hill [41] and Tumin [90] suggested the use of adjoint sensitivity equations to evaluate localized receptivity in Blasius boundary layers. They demonstrated that the adjoint method, while being simple and efficient to implement, gives results identical to the FRNT


12

predictions. Adjoint methods promise to cover the large receptivity parameter space more economically and have recently been used to predict the receptivity characteristics of a wide range of flows, including pipe Poiseuille flow [89], the Blasius boundary layer [41, 43, 60, 94], laminar wall jets [91] and G¨ ortler vortices in boundary layers on concave surfaces [55]. With the exception of the work of Luchini and Bottaro [55], and the later work of Hill [43], all of these studies rely on expansion of the homogeneous solution to the locally parallel flow into a bi-orthogonal set of eigenfunctions as described by Grosch and Salwen [35]. Luchini and Bottaro [55] solved the receptivity problem for Görtler vortices using the adjoint of the linearized boundary-layer equations. This approach has the advantage of naturally including nonparallel effects within the receptivity predictions which are known to be important for streamwise oriented disturbances, such as the Görtler instability. Recently, Hill [43] extended his parallel adjoint theory to utilize Parabolized Stability Equations (PSE). He presents a limited number of receptivity predictions based on adjoint PSE (APSE) along with comparisons to forced PSE simulations and the DNS results of Crouch and Spalart [26]. Even though Hill [43] uses the APSE approach to study receptivity for nonparallel flows, his paper does not describe the implementation of the method in detail, nor does it compare APSE with LNS or ANS predictions. Furthermore, it is not clear from his study whether nonparallel effects are important for the types of flows he considers. The research presented here extends upon Hill’s work by providing a new formulation of the adjoint approach as a particular case of more general sensitivity analysis. We employ the adjoint approach as a powerful tool for exploring receptivity for various types of flows. While Hill largely restricted his receptivity analysis to parallel boundary layers using a local adjoint method, we explore receptivity of highly nonparallel boundary layers using global adjoint methods.


1.4

13

Objectives

In this research, we chiefly concern ourselves with investigating the receptivity processes in various types of flows using global adjoint methods. A global adjoint analysis has the advantage of naturally including nonparallel effects within the receptivity prediction. One of the most significant challenges of the receptivity problem has been to cover the nearly infinite parameter space of disturbance types, locations strengths, that occur in a practical transition environment. Adjoint based methods provide an efficient method for investigating receptivity over a wide set of parameters.

1.5

Accomplishments

• We have developed consistent global adjoint methods for receptivity prediction in nonparallel flows. These include an Adjoint Navier–Stokes (ANS) solver, and methods based on the Adjoint Parabolized Stability Equations (APSE). We have analyzed the applicability of these methods for receptivity studies and compared the predictions with direct simulations using a Linearized Navier–Stokes (LNS) solver. Finally, we have assembled this into a powerful set of tools that gives us the ability to predict the receptivity of flows in arbitrary coordinate systems for arbitrary locations and arbitrary excitations. • We have documented the receptivity characteristics of canonical boundary layer flows such as the Blasius and Falkner–Skan boundary layers. We have analyzed pressure gradient effects for two-dimensional and three-dimensional disturbances. Receptivity characteristics are obtained for various sources which include forced disturbances, such as suction and blowing sources on the wall boundary, tangential excitations at the wall, and momentum sources in the domain of the flow. We have compared results obtained using APSE with the local


14

adjoint results based on a parallel flow approximation obtained by Hill [41]. • We have documented the receptivity of swept Hiemenz, Falkner–Skan–Cooke and swept parabolic cylinder boundary layer flows, and have evaluated nonparallel effects on the receptivity of crossflow vortices. The Falkner–Skan–Cook family of flows serves as an excellent test case to help understand pressure gradient and nonparallel effects in three–dimensional boundary layers. On the other hand, the boundary layer on a swept parabolic cylinder is similar to the swept wing boundary layer flows encountered in aeronautical applications including the important effect of surface curvature. • We have compared PSE and LNS based stability predictions in the regions close to disturbance sources and conclude that, in those regions, PSE cannot be used to provide accurate solutions. This is particularly true near the leading edge of swept wing boundary layers where nonparallel effects are significant. In these regions, the PSE solutions are often corrupted by inaccurate transients. Another implication of this conclusion is that PSE can only be applied to investigate the evolution of one particular mode and cannot be used together with general disturbance sources that generate a wide range of modes. • We found that when APSE are formulated as an approximation to the ANS equations they are viable for receptivity prediction in nonparallel flows close to the leading edge. This is true even in regions where the PSE are not useful.

Chapter 2 Formulation of Adjoint Methods for Receptivity Prediction This chapter presents the formulation of receptivity and sensitivity analysis using adjoint methods. We begin by defining the general problem and then initiate our discussion of receptivity analysis by presenting a concrete example based on adjoint methods, which are an extension of Hill’s [17,18,41,43] approach. Utilizing sensitivity analysis, we then illustrate how adjoint methods can be used to find the sensitivity of a particular objective function due to changes in control variables. We consider two objective functions: an objective function that measures the kinetic energy of a flow, and an objective function that measures the amplitude of a particular instability wave.

2.1

Problem Formulation

Consider a time domain [t0 , tf ] and a fixed spatial domain Ω, that is an open, connected, bounded subset of IR3 with a boundary Γ. The non-dimensional fluid state is denoted by U (x, t) = {v, p/ρ}T = {v1 , v2 , v3 , p/ρ}T where v is the velocity vector, p is the pressure, ρ is the density, x is the position vector, and t is time. The state U satisfies the incompressible Navier–Stokes (NS) equations, C(U , g) = 0, in a nondimensional form

C(U , g) =

def

  N (U ) − S(g)   

B(U , g)

    U (x, t ) − U (x) 0 0

15

in [t0 , tf ] × Ω on [t0 , tf ] × Γ in Ω .

(2.1)

Chapter 2. Formulation of Adjoint Methods for Receptivity Prediction

16

The first line in the system (2.1) is the symbolic representation of the three components of the momentum equations and the continuity equation subject to the volumetric source term S(g), while the second line is the representation of the boundary conditions. The last line is the initial condition for the flow. In these equations we explicitly display the control variables g that appear in both the boundary conditions B(U , g) = 0, and in the source term S(g). We would like to emphasize that by the control variables g we assume all the forced and natural excitations presented in the flow. These can include suction and blowing sources, tangential excitations on the wall boundary, wall inhomogeneity, momentum and mass sources, and acoustics waves. In order to simplify the presentation all operators are written in a Cartesian coordinate system. The incompressible Navier–Stokes operator in non-dimensional form is N (U ) = GU ,t + F i,i (U ) − F vi,i (U ) , def

(2.2)

where the inviscid, F i , and viscous, F vi , flux vectors are     v  1          v  

  δi1      p  δi2 2 def F i (U ) = vi +   ρ    δi3    v3            

       

1

      

0

,

F vi (U ) = Re 0 def

  v1,i       v 2,i

       

  v3,i     

      

0

,

(2.3)

where G = diag(1, 1, 1, 0), δij is the Kronecker delta, and Re 0 is the reference Reynolds number. The nondimensionalization of the Navier–Stokes equations is typical (see White [92] for an example). Spatial directions are normalized by the global length scale L0 , velocities by the reference velocity V0 , time by V0 /L0 , and pressure by ρ0 V02 .


17

The global Reynolds number is given by Re 0 = V0 L0 /ν where ν is a kinematic viscosity of the fluid. Throughout the paper, we assume incompressible flow and set nondimensionaliz density ρ = 1. Our goal in sensitivity and receptivity analysis is: To determine how changes in g change the state U .

2.2

Receptivity Formulation

Consider a decomposition of the state field U (x, t) into a steady base flow component ¯ (x) and a, possibly, unsteady perturbation component εU 0 (x, t) U ¯ + εU 0 , U =U

(2.4)

¯ and ε is assumed to be much smaller than unity. For details where U 0 is of order U regarding the mean-flow implementation and solution for the flows considered in this thesis, please refer to Appendix B. The setting of a typical problem is shown in Figure 2.1 which shows the computational domain with inflow boundary Γi , outflow boundary Γo , wall boundary Γb and the freestream boundary Γt . Also, we define Γc to be a portion of the wall boundary over which control is applied. The control might take the form of a suction/blowing actuator on the wall boundary, for example, that excites a TS wave as shown in Figure 2.1 with some nonzero amplitude am . For the class of receptivity problems considered here, we are interested in the time asymptotic response of a steady, undisturbed base-flow subjected to a time-harmonic ˆ (x)e−ιωt + c.c, where disturbance at angular frequency ω such that U 0 (x, t) = U c.c denotes the complex conjugate. In the following, we do not explicitly write the complex conjugate term. The evolution of the TS wave is governed by the Linearized Navier–Stokes (LNS) equations which are obtained by substituting expression (2.4)


18

for U into (2.1). Assuming no momentum and mass sources, S(g) = 0, and dropping the initial condition leads to the time-harmonic LNS of order ε ¯ )U ˆ = 0, Û L(

(2.5)

¯ )U ˆ = −ιωGU ˆ + (Ai U ˆ ),i − (Di U ˆ ),i , Û L(

(2.6)

¯ )U ˆ is Û where L(

ι=

√

−1, and the inviscid and viscous flux Jacobians are ¯

∂Fi ¯¯ Ai = ¯ ∂U ¯U¯

¯

∂Fiv ¯¯ and Di = ¯ . ∂U ¯U¯

(2.7)

Explicitly, in Cartesian coordinates, 

2¯ u 0 0 1

   v ¯ A1 =    w ¯ 

1





 

 

v¯

u¯

0 0





 

 

w¯ 0

u¯

0

  

 0 2¯  0 w u¯ 0 0  v 0 1  ¯ v¯ 0   , A2 =   , A3 =   , (2.8)       0 w   0 0 2w  0 u¯ 0  ¯ v ¯ 0 ¯ 1     

0 0 0

0

1

0 0

0

0

1

0

and Di =

1 ∂ . Re ∂xi

(2.9)

Note that the Ai are linear algebraic operators while the Di are linear differential operators. Also, notice that the matrices, Ai , can be written more compactly in the diadic notation, Ai = v¯i ej eTj + ej eTi v¯j + ei eT4 + e4 eTi , where ei eTj denotes a fourth ¯ i ∈ IR4×4 and rank tensor δin δjm , and i, j = 1, 2, 3. Similarly, we define the matrices A ¯ i = v¯i ej eT + ej eT v¯j that will be used in specifying boundary conditions boundary A j i conditions. The boundary conditions for the LNS equations depend on the types of control sources presented in the flow. Depending on the mean flow under consideration, various types of instabilities can be excited such as TS, crossflow, or Görtler instabilities.


19

The instability waves, also referred to as modes, either grow or decay downstream and typical environmental sources excite both stable and unstable modes. In order to isolate one particular unstable mode, we begin by first computing its parallel theory approximation and then use this approximation as the inflow condition for a LNS calculation. The precise mathematical definition of a mode is only possible within the parallel theory approximation where modes are the eigenfunctions of the Orr–Sommerfeld operator. In the LNS equations, this precise definition of a mode is not available; therefore, we take the liberty of defining an unstable LNS mode as a streamwise asymptotic solution of the LNS equations obtained by placing an unstable parallel theory eigenfunction at the inflow. According to the above definition, an ˆ m satisfies equations (2.5) with the boundary conditions: unstable LNS mode U   ˆ =0 v    m

ˆ ) =0 (σ

n m     (A − D ) U ˆ m = (An − Dn ) U î n n m

on Γb on Γt

(2.10)

on Γi ,

ˆ i is where subscript ‘m’ indicates the particular mode under consideration and U m the parallel theory eigenfunction for that mode on the inflow boundary. In (2.10), ˆn = σ îj nj is the projection of the stress tensor in the outward boundary normal σ direction, and σ îj = pˆδij − vî,j /Re is the total stress tensor. On the wall boundary we set zero velocity boundary conditions, zero traction is set on the top boundary, and at the inflow we specify the flux of mass and momentum as well as viscous fluxes. Note that no condition is formally set on the outflow boundary Γo . Similar to Hill [41], we assume that the flow response is purely convective, which allows the use of a nonreflective buffer treatment at the outflow. The implementation of the buffer is explained in Section 3.1.


20

ˆ = {ˆ ˜ = {˜ For sufficiently smooth fields, U v , pˆ} and U v , p˜}, the following Euler– Lagrange identity is easily constructed ˜ T L( ¯ )U ˆ =U ˆ T L( ¯ )U ˜ + ∇ · j(U ˆ,U ˜), Û ˜U U

(2.11)

where the adjoint linear operator is given by ˜ − AT U ˜ ,i − (DT U ˜ ),i ¯ )U ˜ = −ιωGT U ˜U L( i i

(2.12)

and j is the bilinear concomitant ³

´

³

ˆ − Di U ˆ + Di U ˜ ˜ T Ai U ji = U

´T

ˆ. U

(2.13)

˜ is called the adjoint field. The field U Similar to the convective assumption on the LNS solution, we also assume that the adjoint field is convective in the streamwise direction, propagating upstream. The upstream propagation can be inferred from the fact that only convective terms in the adjoint equation have changed the sign. We define an adjoint instability mode as the asymptotic solution of the Adjoint Navier–Stokes (ANS) equations ¯ )U ˜k = 0, ˜U L(

(2.14)

with boundary conditions   ˜ =0 v    k

¯TU ˜ + (σ ˜ ) =0 A

k n k n     (AT + DT )U ˜ k = (AT + DT )U õ k n n n n

on Γb on Γt

(2.15)

on Γo ,

˜ , subscript ‘k’ corresponds to the kth adjoint mode and U õ ˜ n = (en eT4 +Dn )U where σ k is the parallel theory approximation of the adjoint mode on the outflow boundary Γo . For receptivity prediction, these equations are solved subject to an adjoint parallel


21

theory eigenfunction at the outflow boundary (Γo ). The definition of the adjoint mode corresponding to an unstable LNS mode is crucial for receptivity prediction. If, at this point, we assume that the base flow is locally parallel, we would obtain the same receptivity results as Hill [41], in which he considered homogeneous solutions of the Orr-Sommerfeld and its adjoint for receptivity prediction. However, since we are interested in receptivity prediction for nonparallel flows, we now generalize Hill’s results using the full LNS/ANS system. Consider a control volume CVI (shown in Figure 2.1) where we have adopted the convention that x = x1 denotes the streamwise direction, y = x2 the wall-normal direction, z = x3 the spanwise direction and y = 0 corresponds to the wall location. ˆ m and U ˜ k satisfy the homogeneous LNS and ANS equations respectively with If U homogeneous boundary conditions at y = 0 and y = ∞, then (2.11) shows that ˆ m, U ˜ k) = 0 . ∇ · j(U

(2.16)

ˆ m, U ˜ k ) · n dΓ = 0 , j(U

(2.17)

Using the divergence theorem, I Γ

where n is the unit outward normal vector to the boundary Γ. For the rectangular domain CVI shown in Figure 2.1, the expression (2.17) can be rewritten as R∞ 0

R xb xa

¯

ˆ m, U ˜ k ) dy ¯¯ j1 (U

¯x=xb ˆ ˜ j2 (U m , U k ) dx¯¯

y=∞

− −

R∞ 0

R xb xa

¯

ˆ m, U ˜ k ) dy ¯¯ j1 (U

¯x=xa ˆ ˜ j2 (U m , U k ) dx¯¯ y=0

+ = 0,

(2.18)

where j1 and j2 are the streamwise and wall-normal components of j respectively. After applying the homogeneous wall boundary conditions, and stress-free freestream conditions for both the state and adjoint solutions given in (2.10) and (2.15), equation


22

(2.18) reduces to Z 0

∞

¯ ¯

ˆ m, U ˜ k ) dy ¯ j1 (U ¯

x=xb

−

Z 0

∞

¯ ¯

ˆ m, U ˜ k ) dy ¯ j1 (U ¯

= 0.

(2.19)

x=xa

Since locations xa and xb are arbitrary, it follows from (2.19) that ˜ k ; x) def ˆ m, U = J(U

Z 0

∞

ˆ m, U ˜ k ) dy = constant j1 (U

(2.20)

ˆ m, U ˜ k ; x) is conserved. This is a general result for the for any value of x. Thus, J(U LNS/ANS system where the only requirement is that the flow satisfy homogeneous boundary conditions at y = 0 and y = ∞. Moreover it can be shown that ˆ m, U ˜ k ; x) = δmk , J(U

(2.21)

ˆ m, U ˜ m ; x) = 1 (see where the adjoint is assumed to be normalized such that J(U Appendix E). Note that the above orthogonality relation does not hold for arbitrary ˆ m , and U ˜ k , but only for those that satisfy (2.5) and (2.14) with boundary fields U conditions (2.10) and (2.15), respectively. We note that in general, the functional J( · , · ; · ) can accept various arguments. If the arguments are time dependent, the output of the functional is time dependent as well. To illustrate how the adjoint modes can be used to obtain receptivity predictions, consider forced receptivity with compact sources of mass and momentum, qˆ e−iωt and ˆ b e−iωt . Under these conditions, the LNS (2.5) become boundary velocities, v ¯ )U ˆ = qˆ , Û L(

(2.22)

with inhomogeneous wall boundary conditions ˆ (x, 0, z, ω) = v ˆ b (x, z, ω) . v

(2.23)


23

In general, at each streamwise location x, the full solution of (2.22) can be reconˆ m (x, y), with amplitude am , structed as a linear combination of M discrete modes, U ˆ (x, y) such that plus a field W ˆ (x, y, z)e−iωt + W ˆ (x, y, z)e−iωt , U 0 (x, t) = U ˆ (x, y, z) = U

M X

ˆ m (x, y, z) , am U

(2.24a) (2.24b)

m=1

ˆ (x, y, z) satisfies LNS equations (2.5) with respective boundary conditions where U ˆ may contain a particular solution plus contributions from continuum (2.10), W modes, and amplitude am is given by am = aim + ∆am H(x − xs ) .

(2.25)

Here aim is the initial amplitude, ∆am is the change in the amplitude due to the presence of the source, and H(x − xs ) is the step function which is zero upstream of the source and equal to one downstream of the source for x ≥ xs . In the region downstream of the source, am is changed by ∆am and remains constant unless other ˆm sources are present. Our goal is to find the amplitude, am of the natural mode U downstream of all sources. To obtain an expression for am , the following modified Euler–Lagrange identity (2.11) is used ³

´

˜ T L( ˆ T L( ˆ,U ˜ m) − U ˜ T qˆ , ¯ )U ˆ − qˆ = U ¯ )U ˜ m + ∇ · j(U Û ˜U U m m

(2.26)

which simplifies to ˜ T qˆ = 0 . ˆ,U ˜ m) − U ∇ · j(U m

(2.27)

Integrating in the wall-normal direction y and in the streamwise direction from some xa to xb , where these locations are far upstream and downstream of any sources,


24

yields ˆ,U ˜ m ; xa ) = ˆ,U ˜ m ; xb ) − J(U J(U

Z

xb

Z

xa

∞

0

˜ m dy dx − qˆ T U

Z

xb

xa

h

iy=∞

ˆ,U ˜ m) j2 (U

y=0

dx . (2.28)

Substituting (2.24b) and assuming purely convective instabilities, we obtain K X

Z

k=1 xb Z ∞

xa

K X

ˆ k, U ˜ m ; xb ) − ak J(U 0

˜ m dy dx − qˆ T U

Z

ˆ k, U ˜ m ; xa ) = aik J(U

k=1 xb h

xa

iy=∞

ˆ,U ˜ m) j2 (U

y=0

(2.29)

dx ,

ˆ m, U ˜ k ; x) = δmk , and boundary condiwhich, with the orthogonality condition J(U tions (2.23), yields am −

aim

Z

= ∆am =

xb

xa

Z 0

∞

˜ m dy dx + qˆ U T

Z

xb

xa

˜ T2 σ

¯ ¯ ˆ dx¯¯ v b

.

(2.30)

y=0

ˆ m , far downstream of all Thus, the change in amplitude of a particular mode, U sources, is determined given only the form of the sources and the eigenmodes of the regular and adjoint homogeneous problems. Nonparallel and surface curvature effects are naturally included in the receptivity prediction through the global adjoint ˜ m , while these effects are included in the stability characteristics through U ˆ m. field U Thus, a general feature of solutions to the homogeneous adjoint equations is that they can be used to “filter” a particular solution to determine the amplitude of the corresponding natural mode.

2.3

The Sensitivity Problem

Consider an objective functional J (g) = H(U (g)) that implicitly depends on the control, g, through the state U (g). As was mentioned before, control is a generic term here that can include forced excitations on the wall boundary as well as various


25

sources of mass and momentum within the flow. An example of an objective functional is the integral of the kinetic energy of the flow over an observation region Ωo and over time interval T . Just as for the receptivity problem, in sensitivity analysis we are interested in the time-asymptotic response of a steady, undisturbed base-flow subjected to a time-harmonic disturbance at angular frequency ω. For this purpose we use eιωt to filter a particular adjoint mode. The objective function is then given by 1 2T

J (g) =

Z

t0 +T

Z Ωo

t0

U T GU eιωt dΩo dt,

where U = U (g) ,

(2.31)

where we recall that G = diag(1, 1, 1, 0). In order to determine the sensitivity, we take the Fréchet derivative of the objective functional about control g¯ due to changes in the control g 0 , ´ 1 Z t0 +T Z ³ ¯ T ¯ eιωt dΩo dt , J (¯ U GU 0 + U 0T GU g )g = 2T t0 Ωo 0

0

(2.32)

which can also be written as Ã !T DJ 1 Z t0 +T Z J (¯ g )g = g 0 dΩg dt . 2T t0 Dg Ωg 0

0

(2.33)

¯ = U (¯ Here U g ), U 0 is the change of the state due to changes in the control g 0 , and DJ /Dg is the gradient of the cost function, defined over the domain of the control, Ωg . Considering only harmonic disturbances in time g 0 = gˆ e−ιωt such that ˆ e−ιωt and U0 = U ³

´

DJ /Dg = DJˆ/Dg eιωt .

(2.34)

Integrating over time yields µ

1Z ˆ +U ˆ T GU ¯ ¯ T GU J (g)g = U 2 Ωo 0

0

¶

1Z dΩo = 2 Ωg

Ã

DJˆ Dg

!T

gˆ dΩg .

(2.35)


26

Since (2.35) is valid for all gˆ , to compute the gradient of J (i.e. DJ /Dg) we need ˆ in terms of gˆ . This is conveniently achieved by introducing adjoint to express U variables.

2.3.1

Adjoint Formulation for the Sensitivity Problem

To obtain the gradient of the objective functional, we begin by defining the inner product D

E

ˆ 1, U ˆ2 = U

Z Ω

ˆ TU ˆ 2 dΩ . U 1

(2.36)

With this definition, the Lagrange identity (2.11) can be rewritten as ¿

¯ )U ˆ ˜ T , L( Û U

À

¿

À

¯ )U ˜ + ˆ T , L( ˜U = U

Z Ω

ˆ,U ˜ ) dΩ , ∇ · j(U

(2.37)

ˆ L, ˜ and j are defined in Section 2.2. where the operators L, Consider the LNS equations (2.5) with no right-hand-side sources but with nonzero ˆ b , located between streamwise stations xa and wall boundary sources, denoted by v xb . Assuming that the generated instability waves are purely convective in nature, we set homogeneous inflow conditions for the LNS equations. On the top boundary, we assume zero traction for the state, and on the wall boundary, nonslip velocity conditions are enforced in the region where there are no sources. At the outflow, a nonreflective outflow boundary treatment is used (see Section 3.1). To summarize, the state boundary conditions are given by   ˆ=0 v      b 

ˆ=v ˆ v

  ˆn = 0 σ     

ˆ =0 (An − Dn ) U

on Γb \Γc on Γc on Γt

(2.38)

on Γi ,

where Γc denotes the region on the wall boundary where sources (i.e. controls) are


27

applied. To get the sensitivity of J , we note that since G = GT then from (2.32) J 0 (g)g 0 = Setting

Z Ωo

ˆ T GU ¯ dΩo . U

  ¯ ¯ )U ˜ = GU ˜U L(  0

(2.39)

on Ωo ,

(2.40)

on Ω\Ωo ,

we get from the Lagrange identity (2.37), with the condition (2.5) ¿

¯ )U ˜ ˆ T , L( ˜U J (g)g = U 0

0

À

=−

Z Ω

ˆ,U ˜ ) dΩ , ∇ · j(U

(2.41)

which, after using Green’s theorem, reduces to 0

J (g)g

0

Z

= +

∞

0

¯

ˆ,U ˜ ) dy ¯¯ j1 (U ¯

x=xa

¯ Z xb ¯ ˆ ˜ j2 (U , U ) dx¯¯ xa

−

− y=0

Z

∞

0 xb

Z

xa

¯

ˆ,U ˜ ) dy ¯¯ j1 (U ¯

¯ ¯ ˆ ˜ j2 (U , U ) dx¯¯

(2.42) x=xb

. y=∞

By judicious selection of the adjoint boundary conditions, the sensitivity of the objective functional can be explicitly written in terms of the control variables and appropriate adjoint quantities. For this problem, we set homogeneous outflow conditions ¯TU ˜ +σ ˜ n = 0, and nonslip for the adjoint. At the freestream boundary we choose A n boundary conditions are imposed on the adjoint velocity at the wall boundary. To summarize,

  ˜=0 v   

on Γb

˜ +σ ¯TU ˜ =0 A

(2.43)

on Γt

n n     (AT + DT )U ˜ =0 n n

on Γo .

Using these adjoint boundary conditions along with the state boundary conditions (2.38), the Fréchet derivative of the objective function is simply 0

0

J (g)g =

Z

xb

xa

¯

ˆ,U ˜ ) dx¯¯ j2 (U ¯

Z

= y=0

xb

xa

˜ T2 σ

¯ ¯ ˆ dx¯¯ v b

, y=0

(2.44)


28

where the gradient of the objective functional is obtained from (2.34) with DJˆ/Dg = ˜. ˜ 2 = (e2 eT4 + D2 )U σ

2.3.2

Receptivity as a Sensitivity Problem

For receptivity analysis we use the objective functional J (φ; g) =

1 Z tf J(U , φ; xb )dt , T t0

(2.45)

where J(U , φ; xb ) is defined by (2.20), t0 À 0, tf = t0 + T , T = 2π/ω, station xb ιωt ˜ † (x)e−ιωt + φ(x)e ˜ is the station where we observe the flow, and φ(x, t) = φ is a

prescribed filtering function (here superscript † denotes the complex conjugate). The Fréchet derivative of (2.45) is 1 Z tf J (φ; g)g = J(U 0 , φ; xb )dt , T t0

(2.46)

ˆ e−ιωt + U ˆ † eιωt , U0 = U

(2.47)

0

0

where

ˆ satisfying the LNS equations (2.5) with the inhomogeneous boundary condition with U   ˆ=0 v       v ˆ=v ˆb   ˆn = 0 σ     

ˆm = 0 (An − Dn ) U


(2.48)

on Γi .

Substituting (2.47) together with the expression for φ into (2.46) and integrating over one period yields

˜ xb ) + c.c . ˆ , φ; J 0 (g)g 0 = J(U

(2.49)

To obtain the sensitivity of the objective function (2.45), we define the adjoint


29

Sources Adjoint Quantities ³ ´ 1 b u u ˜ ,y y=0 ³ Re ´ 1 b v + p ˜ v ˜ Re ,y wb q

³

1

Re w˜,y ˜ U

´

y=0

y=0

Table 2.1: Adjoint Quantities Corresponding to Various Environmental Sources, v b = {ub , v b , wb }. ˜ m that satisfies (2.14) with the following boundary conditions field U   ˜ =0 v    m

˜m ¯TU A n

on Γb ˜ ) =0 + (σ

n m     J(U ˜ xb ) ˆ,U ˜ m ; xb ) = J(U ˆ , φ;

(2.50)

on Γt on Γo .

Just as in the case of the receptivity problem, through the use of the Lagrange identity (2.26), we obtain (2.28). With the assumption of convective instabilities, ˆ,U ˜ m ; xa ) = 0, it follows from (2.28) with qˆ = 0 J(U 0

0

˜ xb ) = J(U ˆ , φ; ˆ,U ˜ m ; xb ) = − J (g)g = J(U

Z

xb

xa

h

iy=∞

ˆ (ˆ ˜ m) j2 (U g ), U

y=0

dx .

(2.51)

o õ ˜ b , y, z) to ι˜ ˜ b , y, z) = U ˜ o and the streamwise derivative of φ(x Um αm By setting φ(x m o

o ˜ is the eigenfunction and α where U ˜m is the streamwise wavenumber (i.e. eigenvalue) m

from parallel theory, we obtain the expression for the receptivity amplitude given in (2.30) with qˆ = 0. J 0 (g)g 0 = ∆am =

Z

xb

xa

h

Z

i

ˆ (ˆ ˜ m) j2 (U g ), U

y=0

dx =

xb

xa

¯ ¯

˜ T2 v ˆ b dx¯¯ σ

.

(2.52)

y=0

Table 2.1 summarizes which adjoint quantity corresponds to which source type.


2.4

30

Natural Receptivity

Consider receptivity due to planar acoustic waves propagating in the +x direction scattering from a localized roughness element or steady suction/blowing at the wall. If the unsteady sound has amplitude εs , and the steady surface disturbance has amplitude εh , then the velocity and pressure fields U = {v, p} as well as control gˆ can be expanded as ˆ s (x)e−ιωt + εh εs U ˆ (x)e−ιωt + . . . , ¯ (x) + εh U h (x) + εs U U (x, t) = U

(2.53a)

g(x, t) = g¯ (x) + εh g h (x) + εh εs gˆ hs (x)e−ιωt + . . . ,

(2.53b)

ˆ s is a harmonic disturbance where U h is perturbation to the flow due to roughness, U ˆ is the instability mode excited due to in the form of a planar acoustic wave, and U the interaction of an acoustic wave with the roughness source. Substituting expansion (2.53) into the NS equations (2.1), we obtain equations at orders O(1), O(εh ), O(εs ), O(εs εh ), and other orders. At order εh we obtain the steady LNS equations for U h ¯ )U h = 0 , Lh (U

(2.54)

with boundary conditions   vh = 0       v h (x, 0, z) = G (x, z) h h   σn = 0     h 

(An − Dn ) U = 0


(2.55)

on Γi .

The operator Lh in (2.54) is the same as the operator Lˆ in (2.6) but with the term −ιωG removed. Note also that Gh in (2.55) represents the particular wall excitation on the scale of εh which could be surface roughness or steady suction/blowing. In the


31

case of surface roughness, we have ¯

¯ ¯¯ ∂v , Gh (x, z) = −h(x, z) ¯ ∂y ¯y=0

(2.56)

where h(x, z) represents the shape of the roughness source. In the case of steady suction or blowing Gh (x, z) = v b (x, z)

(2.57)

(see for example, Collis [16] for explanation of these boundary conditions). Consider ˆ s e−iωt . Then to O(εh εs ) we obtain a sound perturbation given by v s = v ¯ )U ˆ = qˆ (U h , U ˆ s) , Û L(   ˆ=0 v       v ˆ (x, 0, z, ω) = Ghs (x, z, ω)  σ  ˆn = 0      ˆ

(An − Dn ) U = 0

where

(2.58)


(2.59)

on Γi ,

 h  v s − (ˆ v s · ∇)v h ˆ ) = −(v · ∇)ˆ qˆ (U h , U  0 s

(2.60)

ˆ hs (x, z, ω) is the wall excitation at O(εh εs ). For surface roughness and G ¯

ˆ s ¯¯ ˆ hs (x, z, ω) = −h(x, z) ∂ v G , ¯ ∂y ¯y=0

(2.61)

and for steady suction/blowing Ghs (x, z, ω) = 0, since there is no excitation at O(εh εs ). ˜ To simplify the notation in this section, we remove the subscript m and use U ˜ m , with the understanding that we always refer to a particular adjoint instead of U mode. Also in this section, ∆a represents ∆am .


32

From (2.30) we obtain ∆a = −

Z

xb xa

Z

∞

h

0

Z

i

˜ dy dx − (v h · ∇)ˆ v s + (ˆ v s · ∇)v h · v

xb

xa

h

ˆ,U ˜) j2 (U

iy=∞ y=0

dx , (2.62)

˜ satisfies the adjoint equation (2.14) with adjoint boundary conditions (2.15). where, U Consider the first integral in (2.62), written in index notation Z

xb

Z

∞

0

xa

vih

Z xb Z ∞ ∂ˆ vjs ∂v h v˜j dy dx + vîs j v˜j dy dx . ∂xi ∂xi xa 0

(2.63)

By the product rule Z

Z

h s ∂vj vî v˜j ∂xi xa 0 Z xb Z ∞ ∂(ˆ vis v˜j ) vjh − ∂xi xa 0 xb

∞

dy dx =

(2.64) Z

dy dx +

xb

xa

Z

∞

0

∂(ˆ vis vjh v˜j ) dy dx , ∂xi

which leads to the following expression ∆a = −

Z

xb

xa

Z 0

∞

˜ ) dy dx − Υ(ˆ ˜, v ) − U · Q(ˆ v ,v v ,v s

h

s

h

Z

xb

xa

¯

ˆ,U ˜ )¯¯y=∞ dx , (2.65) j2 (U y=0

where s ˜ )j = vî,j Q(ˆ vs, v v˜i − (ˆ vis v˜j ),i ,

j = 1, 2, 3 ,

(2.66)

˜ )4 = 0, and Υ(ˆ ˜ , v h ) is Q(ˆ vs, v vs, v ˜ , vh) = Υ(ˆ vs, v

Z

xb

xa

Z 0

∞

(∇ · γ) dy dx , with γ i = (ˆ vis vjh v˜j ) .

(2.67)

Following Hill [41], we define a second adjoint problem with the right-hand side forcing ˜ ), Q(ˆ vs, v ¯ )U ˜ h = Q(ˆ ˜) , L˜h (U vs, v

(2.68)

¯ ) is the same as the adjoint operator L( ¯ ) but with −ιωGT removed. ˜U where L˜h (U


33

˜ ) in (2.65) leads to Substituting the left-hand-side of (2.68) for Q(ˆ vs, v ∆a = −

Z

xb

Z

xa

∞

0

µ

¶

¯ )U ˜ h dy dx − Υ − U h · L˜h (U

Z

xb

xa

¯y=∞

ˆ,U ˜ )¯¯ j2 (U

y=0

dx ,

(2.69)

where, for simplicity, we dropped the explicit arguments for Υ. Integrating by parts the first term in (2.69) and using the Lagrange identity to move L˜h onto U h yields Z

∆a =

xb

Z

∞

0

xa

˜ h ) dy dx − Υ − ∇ · j(U h , U

Z

xb

xa

¯

ˆ,U ˜ )¯¯y=∞ dx . j2 (U

(2.70)

y=0

ˆ n = 0 at Γt , setting the adjoint Since the state boundary conditions σ hn = 0 and σ ¯TU ˜h+σ ˜ hn = 0 at the top boundary yields boundary condition A n ∆a = −

Z

xb

xa

¯

˜ h )¯¯ j2 (U , U ¯

Z

0

h

dx + Υ +

xa

y=0

where 0

Z

Υ =

0

∞

xb

¯xb

˜ h )¯¯ j1 (U , U ¯ h

xa

¯

ˆ,U ˜ )¯¯ j2 (U

y=0

dx ,

(2.71)

dy − Υ .

(2.72)

Using (2.67) for Υ leads to 0

Υ =−

Z

∞ 0

Z ¯xb ¯ ˜ )¯ dy − γ1 (ˆ v ,v ,v s

h

xa

xb

xa

Z ¯∞ ¯ ˜ )¯ dx + γ2 (ˆ v ,v ,v s

h

0

0

∞

¯xb

˜ h )¯¯ j1 (U , U ¯ h

dy . xa

(2.73) Assuming convective disturbances and applying homogeneous wall and freestream ˜ we obtain boundary conditions for U h and U Υ0 =

Z 0

∞

h

¯ ¯

˜ )¯ ˜ ) + j1 (U h , U −γ1 (ˆ vs, vh, v ¯

dy .

(2.74)

xb

˜ h such that at In order to set Υ0 = 0, we choose outflow boundary conditions for U x = xb ˜ h) = 0 ˜ ) + j1 (U h , U −γ1 (ˆ vs, vh, v

(2.75)


34

This is achieved by setting ˜ hn = 0 on Γo . ˜ h = 0 and σ v

(2.76)

˜ h (see Section 3.1), At the inflow, Γi , we use nonreflective buffer conditions for U ¯TU ˜h+σ ˜ hn = 0 non-slip velocity conditions are enforced at the wall boundary, and A n is enforced on the top boundary. With the adjoint boundary conditions defined above, we get from (2.71) for the roughness element at the wall Z

∆a =

xb

xa

Λ(x)h(x, z)|y=0 dx ,

(2.77)

where Λ(x) is the receptivity efficiency factor due to a point roughness element at x and is defined as

"

#

ˆs ∂v ∂v h ˜2 + ˜2 . Λ(x) = − ·σ ·σ ∂y ∂y

(2.78)

For roughness in a two-dimensional boundary layer, (2.78) simplifies to "

1 ∂ u¯ ∂ u˜h ∂ uˆs ∂ u˜ Λ(x) = − + Re ∂y ∂y ∂y ∂y

#

.

(2.79)

y=0

Similarly, for steady suction or blowing Z

∆a =

xb

xa

¯

˜ h2 · v b ¯¯ σ

y=0

dx ,

(2.80)

˜ h2 gives the receptivity to a point suction/blowing source at x. Recall that where σ ¯

˜ h2 σ

¯ ˜h 1 ∂v ˆ ¯¯ + p˜h y = . ¯ Re ∂y y=0

(2.81)

Note, that for steady suction or blowing in the wall-normal direction the adjoint ¯

˜ h /∂y ¯¯ pressure p˜h represents the receptivity since ∂ v

y=0

= 0.


^ ~ J(U,Um ; x) = 0

Γt

y =¥

CVI

CVII

^ ~ J(U,Um ; xa)

^ ~ J(U,Um ; xb)

am = a mi

y

35

^ ~ J(U,Um ; xb)

am= ami + Dam

Γi

Γo

xa Actuator ~ J(U,Um ; x)

Γb

xb

^

Figure 2.1: Control Volume for Receptivity Prediction.

x

Chapter 3 Implementation of Numerical Methods In this chapter we describe the numerical methods used to obtain stability and receptivity characteristics of boundary layer flows. Over the past decade, there has been a substantial amount of work in developing accurate and efficient methods for predicting laminar flow transition. Numerical simulations have advanced far beyond the use of the linear stability theory with the understanding that the main disagreement between theoretical and experimental results is attributed to nonparallel effects. Prior to the early nineties, most Direct Numerical Simulations (DNS) were temporal. In the temporal formulation, a spatially periodic computational domain is assumed and the temporal evolution of the flow is computed. This method, however, cannot correctly account for the mean growth of a shear layer and therefore is only capable of giving qualitative information. A number of spatially evolving Direct Numerical Simulations (DNS) have been available since the early nineties. An overview of transition studies through numerical simulation in the early literature can be found in Kleiser and Zang [51]. Early progress in the development of spatial DNS was made by Fasel and Konzelmann [29, 30] and by Spalart [79, 80]. Even though DNS provides satisfactory results in predicting the stability of nonparallel spatially growing boundary layers, it is too expensive to investigate flow stability effectively. To remedy this, Herbert and Bertolotti developed the Parabolized Stability Equations (PSE) [40], which dramatically reduced the computational expense and are capable of capturing the effects of the growing boundary layer. Using the PSE approach, Bertolotti, Herbert and Spalart [11] investigated the non-parallel

36

Chapter 3. Implementation of Numerical Methods

37

effects and computed the neutral stability curve for a developing Blasius boundary layer. At about the same time, Joslin [45, 46] implemented a spatial DNS and simulated laminar/turbulent transition due to the interaction of two oblique TS waves. Similar to Bertolotti et al., Joslin compared his DNS results with results given by parallel theory and PSE, and found a good agreement between PSE and DNS. Recently, Berlin [6] also used DNS to simulate transition due to oblique wave interaction. Not only has DNS become more available to predict the stability of flows, receptivity research has also benefitted from DNS. The details of the implementation of DNS for receptivity prediction can be found in the work of Collis and Lele [16,21,22]. Even though advances in computer performance have enabled accurate direct numerical simulations of receptivity, the computational cost is still too high to adequately cover the large receptivity parameter space. Thus the adjoint approach becomes a method of choice for numerical analysis of receptivity. In order to develop adjoint codes, we utilized a similar approach that has been used before to implement DNS for stability analysis of nonparallel flows. Building upon previous work we have developed adjoint methods based on LNS and PSE to predict receptivity of shear flows. In this chapter, we present the detailed numerical method for implementing direct and adjoint methods for receptivity prediction. We begin by deriving the LNS and ANS equations in a body-fitted coordinate system and discuss spatial discretization and the implementation of the solver for both methods. We validate our LNS numerical method by demonstrating the convergence of the solution with mesh refinement. Likewise, ANS is validated by comparing ANS predictions with receptivity predictions obtained using direct LNS calculations. In Section 3.2, we describe how the PSE and APSE can be constructed as an approximation of the LNS and ANS equations, respectively. We then discuss the implementation of the PSE and APSE methods and evaluate their numerical stability and accuracy for various flows.


3.1

38

LNS-ANS Formulation

Given a mean flow solution, we consider the LNS equations for the perturbation ˆ (x)e−ιωt+ιβz , where U ˆ = {ˆ ˆ = {ˆ ˆ = field, U 0 (x, t) = U u, vˆ, w, ˆ pˆ}T (or U v , pˆ}T and v {ˆ v1 , vˆ2 , vˆ3 }). Here, ω is the angular frequency and β is the spanwise wave number. Similarly, we consider the ANS equations for the adjoint field in the form U ∗ (x, t) = ˜ ˜ (x)e−ι˜ωt+ιβz U with ω ˜ = −ω, and β˜ = −β.

In a body-fitted coordinate system, the LNS equations (2.22) are given in the following general form1 Û ˆ ,y + ιβ C Û ˆ + ¯ )U ˆ = −ιω G Û ˆ +A Û ˆ ,x + B Û L(

(3.1)

Û ˆ −E ˆ 1U ˆ ,xx − E ˆ ,yy + β 2 E ˆ 2U ˆ, ˆ 2U +D where all the matrix operators are given in Appendix D. The state boundary conditions are

  ˆ=v ˆb v   

ˆt ˆ =σ σ

n n    i ˆ ¯ ˆ  v ˆ=v ˆ , [L(U )U ]1 = 0

on Γb on Γt ,

(3.2)

on Γi

ˆ i is the inflow velocity. Note that inflow ˆ tn is defined in Chapter 2, and v where σ boundary conditions given in (3.2) are not the same as the inflow boundary conditions given in (2.10). In the context of a finite difference method, however, they are an approximation to (2.10). The ANS are written as ¯ )U ˜ = −ι˜ ˜U ˜ + (A ˜U ˜ ),x + (B ˜U ˜U ˜ ),y + ιβ˜C ˜U ˜ + L( ωG

(3.3)

˜U ˜ − (E ˜ 1U ˜ 2U ˜ ),xx − E ˜ ,yy + β˜2 E ˜ 2U ˜, +D 1 Unlike the LNS/ANS equations given in Chapter 2, the LNS/ANS and PSE/APSE equations are presented here in primitive form.


39

with boundary conditions   ˜=0 v   

on Γb

¯TU ˜ +σ ñ = 0 A

n     v ˜i = φ˜i ,

i = 1, 2, 3;

on Γt

,

(3.4)

¯ )U ˜ ] = 0 on Γo ˜U [L( 1

˜ are defined in Chapter 2. ¯ T , and φ ˜ n, A where σ n The matrix operators in (3.1) and (3.3) are given in Appendix D, while Appendix C briefly explains how the matrices are obtained in a body fitted coordinate system using concepts from differential geometry. To solve the LNS and ANS equations we utilize the efficient LNS solver originally developed by Streett [83]. In the streamwise direction fourth-order-accurate central differences are used for the first- and second-derivatives. Near the boundaries, finite difference stencils that are biased towards the interior are used. The wall-normal direction is discretized using a Chebyschev collocation scheme (see for example, Trefethen [87] for a discussion of spectral methods). After discretization, the resulting linear system is arranged as a block pentadiagonal system as shown in Figure 3.1 that is composed of the block matrices A, B, C, D, E, and F . The block matrices are of size 4Ny × 4Ny , where Ny is the number of collocation points in the wall normal direction. Consider for example, block matrix C, illustrated in Figure 3.1. There are four major rows with labels ‘U -momentum’, ‘V -momentum’, ‘W -momentum’, and ’Continuity’, that correspond to each of the LNS equations. The first column, labeled ’U-terms’, are all the terms that multiply the streamwise velocity components, whereas other columns contain terms, that multiply corresponding velocity components or pressure. The solution field is in vector form as shown in Figure 3.1. The linear system is reduced to upper tridiagonal form through Gaussian elimination on the block matrices and then is solved using backward substitution to find the required solution vector ˆ . The inflow, outflow, wall, and freestream boundary conditions are implemented U


40

by modifying appropriate entries in the block matrices as well as modifying the righthand side of the linear system. The right-hand side may also contain momentum forcing. Exactly the same approach is used to find the adjoint solution. The solution procedure is unchanged, only the block matrices A, B, C, D, E, and the right-hand side are modified to correspond to the adjoint equations with adjoint boundary conditions. In order to correctly account for outflow boundary conditions in the LNS, as well as the inflow boundary conditions in the ANS equations, a nonreflective boundary treatment is used. These boundary treatments are also known as buffers or sponges and are necessitated by the finite computational domain. The most effective nonreflective boundary treatments we have found are those that reduce the LNS/ANS equations ˆ ,xx is substituted by to PSE/APSE in the buffer domain. To implement the buffer, U the expression ˆ ,xx + (1 − µ(x)) (2ια(x)U ˆ ,x + α2 (x)U ˆ), µ(x)U

(3.5)

where µ(x) is a function that is equal to unity in the domain of interest, but smoothly changes to zero in the buffer region. The coefficient α(x) is an approximation of the disturbance kinetic energy logarithmic growth rate, α(x) = where

1 ∂Ke(x) , Ke(x) ∂x Z

Ke(x) =

0

∞

vî (x)† vî (x) dy .

(3.6)

(3.7)

The coefficient α(x) is typically obtained from a preliminary PSE calculation. From (3.5) we can see that in the buffer region, the effect of the second streamwise derivative ˆ ,x + α2 (x)U ˆ . A similar buffer is substituted by its parabolic approximation 2ια(x)U technique is employed for ANS calculations, with the buffer region reversed, and


41

placed upstream of the domain of interest. Also, the coefficient α(x) is replaced by α ˜ (x), which is the approximation of the adjoint kinetic energy logarithmic growth rate (frequently we use α ˜ (x) = −α(x)). Figure 3.2 illustrates the computational domain for the LNS/ANS pair including inflow and outflow boundary conditions as well as the buffer conditions. The reader interested in other methods for nonreflective boundary treatments is advised to consult the works of Streett and Macaraeg [84], and Joslin et al. [44], among others.

3.2

PSE-APSE Formulation

While solutions of the LNS and ANS equations provide an accurate account of receptivity and stability for boundary layer flows, they are often quite expensive and may require a large amount of storage. Recently, a technique based on PSE has been developed by Herbert and Bertolotti [7, 39, 40], which has been successfully used to predict the evolution of instability waves in nonparallel boundary layers. In the next section, we review the PSE formulation and derive an APSE method in Section 3.2.2.

3.2.1

PSE Formulation

¯ , is assumed to have slow growth in the For boundary layer flows, the mean flow, U streamwise direction. The growth rate is assumed to be of order O(Re −1 ). Consider disturbances U 0 that satisfy the LNS equations. Using the appropriate shape function, ˆ (x), and phase function, θ(x, z, t), one can always decompose the disturbance, U 0 , U in the form2 ˆ (x)eιθ(x,z,t) . U 0 (x, t) = U 2

(3.8)

Note that starting from this point until the end of the chapter (unless otherwise specified), the ˆ will be used as a shape function defined by equation (3.8) and not as the shape function symbol U defined by the LNS equations at the beginning of Section 3.1.


42

ˆ (x) in the streamwise direction should not be At this point, a slow variation in U ˆ by requiring that the kinetic assumed; however, we may constrain variations in U energy of the shape function is constant, which leads to Z

∞

0

ˆ† v

ˆ ∂v dy + c.c. = 0 ∂x

(3.9)

In general we do not explicitly carry the complex conjugate part in this equation. In (3.9), a superscript † denotes the complex conjugate. Further, we refer to the constraint equation (3.9) or similar constraints as the normalization condition. With this constraint, the kinetic energy growth is absorbed in the phase function and ˆ changes slowly in the streamwise direction. The fundamental the shape function U assumption in PSE is that under the constraint (3.9), the wave vector, k = ∇θ, is a slowly varying function in the streamwise and spanwise directions. Under this assumption, the wave vector, k, can be written as k = k(x, z) = {α(x, z), β(x, z), ω}, where α and β are the streamwise and spanwise wave numbers. Using condition (3.9), we rewrite (3.8) as ˆ (x, y)eι(θ(x,z,t)−θ(x0 ,z0 ,t0 )) /U ˆ (x0 , y0 ) , U 0 (x, t) = U

(3.10a)

ˆ (x0 , y0 )eιθ(x0 ,z0 ,t0 ) at location {x0 , z0 , y0 } has been used to normalize the where U state.3 θ(x, z, t) − θ(x0 , z0 , t0 ) =

Z

x

x0

Z

z

z0

ι(α(x0 , z 0 ) + β(x0 , z 0 )) dx0 dz 0 − ιω(t − t0 ) . (3.10b)

We further introduce the convenient notation Θ = eι(θ(x,z,t)−θ(x0 ,z0 ,t0 )) . 3

(3.11)

The symbol θ is later used to denote various angles. Here it is used briefly and should not cause any confusion.


43

The integration in (3.10b) is taken from the location (x0 , z0 ), where the instability amplitude is known, until the desired location (x, z). Throughout this work, we assume that the flow is homogeneous in the spanwise direction such that disturbances are periodic in z. Also, for simplicity, we choose coordinate z0 = 0, and time t0 = 0. With this assumption, (3.10b) simplifies to Rx

ˆ (x, y)e U 0 (x, t) = U

x0

ια(x0 ) d x0 +ιβz−ιωt

,

(3.12)

ˆ (x, y) is now assumed to be normalized by U ˆ (x0 , y0 ). By substituting (3.12) where U into the LNS and collecting similar terms, we obtain ¯ )U ˆ +A ¯U ˆ ,x − E ˆ 1U ˆ ,xx − ια,x E ˆ 1U ˆ = 0, Los (α, β, ω; U

(3.13a)

where ¯ )U ˆ = Los (α, β, ω; U

³

´

ˆ 1 + β 2E ˆ2 U ˆ ˆ + iαA ˆ + iβ C ˆ +D ˆ + α2 E −ιω G

Û ˆ ,y − E ˆ 2U ˆ ,yy , + B

(3.13b)

and ¯ =A ˆ − 2ιαE ˆ1 . A

(3.13c)

At this point, no approximation has been made. Equation (3.13a) is the exact analogue of the LNS in the splitting (3.12). The PSE approximation is to neglect terms of order O(Re −2 ) and dropping these terms leads to ¯ )U ˆ +A ¯U ˆ ,x = 0 , Los (α, β, ω; U

(3.14)

¯ = A. ˆ We see that (3.14) appears to be a first-order hyperbolic system in where A the streamwise direction. It is efficiently solved by a marching method as described


44

in Section 3.4.1. The system (3.14) has to be solved together with the nonlinear constraint (3.9). The solution of the system is typically achieved through an iteration on α until the constraint condition (3.9) is fulfilled. Notice that ellipticity is still present through the pressure gradient terms, often refered to as residual ellipticity, which makes PSE ill-conditioned and various stabilization strategies have to be employed to obtain stable solutions as discussed in Section 3.4.2.

3.2.2

APSE Formulation

Applying the PSE approximation to the ANS equations leads to the APSE. Starting from the ANS, we assume solutions that have both wave and shape components and proceed in a manner similar to that done with standard PSE. For example, the APSE solution can be written as4 Rx

˜ (x, y)e U ∗ (x, t) = U

xe

˜ ια(x ˜ 0 )dx0 +ιβz−ι˜ ωt

,

(3.15)

where α ˜ is the adjoint streamwise wavenumber and xe is the streamwise location of the outflow boundary Γo . As in the PSE implementation, we have assumed that the adjoint solutions are periodic in the spanwise direction. Substituting (3.15) into the ANS and collecting similar terms, one obtains ˜¯ U ˜ ω ¯ )U ˜ +A ˜ ,x + L ˜U ˜ −E ˜ 1,xx U ˜ 1,x U ˜ −E ˜ 1U ˜ ,xx − 2ι˜ ˜ − ι˜ ˜ 1U ˜ = 0, Lõs (˜ α, β, ˜; U αE α,x E (3.16) with ˜ = (B ˜ ,y + A ˜ ,x ) , L 4

(3.17)

Note that starting from this point until the end of the chapter (unless otherwise specified) the ˜ will be used as a shape function defined by equation (3.15) and not as the shape function symbol U defined at the beginning of Section 3.1.


45

where the Adjoint Orr–Sommerfeld (+ Squire) operator is given by ˜ ω ¯ )U ˜ = α, β, ˜; U Lõs (˜ ³

(3.18)

´

˜ 1 + β˜2 E ˜2 U ˜ +B ˜U ˜ ,y − E ˜ ,yy , ˜ + ι˜ ˜ + ιβ˜C ˜ +D ˜ +˜ ˜ 2U −ι˜ ωG αA α2 E

¯˜ = A ˜ − 2E ˜ 1,x − 2ι˜ ˜ 1 . Dropping the terms of order O(Re −2 ) results in and A αE ˜¯ U ˜ ω ¯ )U ˜ +A ˜ ,x + L ˜U ˜ = 0, α, β, ˜; U Lõs (˜

(3.19a)

˜¯ = A ˜. A

(3.19b)

with

Using the above definition of the adjoint, the Euler–Lagrange identity (2.11) in the PSE/APSE formulation becomes ³

´

³

´

˜¯ U ˜ T Los U ˆ T Lõs U ˜U ˜ + ∇ · j (P SE) (U ˆ,U ˜), ˆ +A ¯U ˆ ,x = U ˜ +A ˜ ,x + L U where



j (P SE)

(3.20)



˜ TA ¯U ˆ U . = T ˜ B ˜TE Û ˆ −U ˜ TE ˆ 2U ˆ ,y + U ˆ 2U ˆ U ,y

(3.21)

Similar to J, which is defined by (2.20), we define J (P SE) in (3.21) and J (P SE) is also a conserved quantity. We refer to both J and J (P SE) as the j-product, depending on the context. The constancy of the j-product is an indication of the orthogonality of the LNS/ANS or PSE/APSE pairs. Similar to the PSE normalization condition (3.9), one has to introduce a constraint condition in the adjoint to define the splitting of the adjoint shape and wave functions. Several possible normalizations have been explored. One possible choice is to assume that α ˜ = −α. For parallel boundary layer flows this is the case, which indicates that it may be a reasonable normalization requirement for an adjoint solution, especially


46

for weakly nonparallel flows. Throughout the text we refer to this normalization as the α-normalization. One drawback of this normalization is that the PSE solution has to be obtained first. Also, if the PSE solution is not a good approximation of the LNS solution, errors in the streamwise wavenumber, α, may adversely affect the accuracy or stability of the APSE solutions. Note, however, that this normalization requires no iteration over α ˜ and adjoint solutions are obtained at a small cost once the streamwise wavenumber, α, is known. Alternatively, ke-normalization can be imposed on adjoint solutions in a similar manner as imposed on the state, Z 0

∞

˜† v

˜ ∂v dy = 0 . ∂x

(3.22)

We frequently refer to this normalization as the ke-normalization. Finally, one can require that the j-product of the state and adjoint solutions be unity. Iteration on α ˜ is then employed to fulfill this requirement. For accurate PSE solutions this may be a method of choice, but as we will see later, it is not advisable if the state solution exhibits strong transients or is inaccurate. The later condition for splitting is denoted as j-normalization.

3.3

The Orr–Sommerfeld Equation (OSE) and its Adjoint

ˆ ,x which The OSE is obtained from PSE (3.14) by ignoring the terms proportional to U results in an eigenvalue problem for α given the values of β and ω ¯ )U ˆ = 0. L0 os (α, β, ω; U

(3.23)


47

The operator Los is modified (therefore it is denoted by L0 os ) such that all the streamwise derivatives of the mean flow are zero and the wall-normal component of the mean-flow is negligible. Similarly, one obtains the Adjoint Orr–Sommerfeld Equation ˜ ,x as well as the L ˜U ˜ term. (AOSE) from (3.19a) by dropping terms proportional to U This approximation is an adjoint eigenvalue problem for α ˜ given the values of β˜ and ω ˜, ˜ ω ¯ )U ˜ = 0, L˜0 os (˜ α, β, ˜; U

(3.24)

where the operator L˜0 os is the same as Lõs but with mean-flow streamwise derivatives, as well as the wall-normal component of the mean-flow velocity, ignored. Before development of PSE/APSE methods, OSE was the method of choice to study the stability of flows, whereas both OSE and AOSE have been used to predict receptivity of flows. Typically, OSE and AOSE are nondimensionalized using local length scales such as boundary layer thickness δr . However, PSE and APSE are are both global methods and require nondimensionalization based on global scales. In order to compare solutions obtained using the PSE/APSE approach with those from OSE/AOSE, we describe the conversion between global and local nondimensionalizations in Appendix A.

3.4 3.4.1

Implementation of PSE/APSE Spatial Discretization

To discretize the PSE and APSE, we use a Chebyschev collocation scheme in y with a backward Euler integration in x. ˆ ˆ ˆ ,x = U i − U i−1 , U ∆xi

˜ ˜ ˜ ,x = U i − U i+1 , U ∆xi+1

∆xi = xi − xi−1 .

(3.25)


48

Here xi is the discretizatized streamwise coordinate x. Using (3.25), the fully discrete PSE system can be written as ˆ i = Ni U ˆ i−1 , Mi U

(3.26)

¯ i +∆xi Los , Los is the discretized Orr–Sommerfeld operator evaluated where Mi = A i i ¯ evaluated at xi , and U ˆ i is a at xi , Ni is the discretized operator corresponding to A ˆ for all y at each xi (here Ni = A ¯ i ). Assembling this into a single, large vector of U system of linear equations yields   M2   −N3 M3   ...      

... −NNx −1 MNx −1 −NNx

and

M Nx                        



uˆ(i,1)

   v ˆ  (i,1)   w ˆ  (i,1)

î = U                        

  ˆ2   U    ˆ3   U    ..  .     ˆ Nx −1   U      U ˆ

pˆ(i,1) .. . uˆ

 (i,Ny )   v ˆ  (i,Ny )   w ˆ  (i,Ny )

pˆ(i,Ny )

Nx

                      

=

  ¯ 2U ˆ1  N       0   

           

          

          

.. .

0 0

,

(3.27)

                                                              

,

(3.28)

which is written more concisely as ˆ =r. MU

(3.29)


49

Similarly discretizing the continuous adjoint equations leads to a marching method ˜ iU ˜i = N ˜ i+1 . ˜ iU M

(3.30)

˜i = A ¯˜ i + ∆xi Lõs , Lõs is a discretized Adjoint Orr–Sommerfeld operator Here M i i ˜ i is a discretized operator corresponding to evaluated at streamwise location xi , N ˜¯ evaluated at x , and U ˜ i is a vector of U ˜ for all y at each xi . Similar to PSE, the A i discretized adjoint equations can be written as a global system   ˜1   U    ˜2   U    ..  .     ˜ Nx −2  U       ˜



˜ ˜  M1 −N1  ˜ 2 −N ˜2  M   ...      

... ˜ Nx −2 ˜ Nx −2 −N M ˜ Nx −1 M

U Nx −1

           

           

          

=

0 0 .. .

    0       N ˜ Nx −1 U ˜ Nx

                      

.

(3.31)

Various normalizations could be used to define the relation between the shape and the wave function component of the instability wave. For the APSE, we have defined ke-normalization, α-normalization and j-normalization as possible normalization strategies. To understand how a normalization condition can be used to iteratively solve for α, consider the kinetic energy normalization for PSE (3.9) and write the iterative method as (k+1) αke

(k)

=α

(k)

Ke1 − . Ke(k)

(3.32a)

The symbol Ke was defined in (3.7) and Z

∞

vî† vî,x dy .

(3.32b)

Ke,x = Ke1 + Ke†1 ,

(3.32c)

Ke1 =

0

Note that


50

where summation over repeated indices is implied. Upon convergence of the iteration, (k)

we have Ke1

= 0, which is exactly the requirement of (3.9). Unfortunately, the

iterative scheme (3.32a) often exhibits slow convergence. We typically accelerate convergence using a formal Newton iteration. Say one’s goal is to make Ke1 = 0. Let’s rewrite this symbolically as F (α) = 0, which makes the dependence on α explicit. To perform a Newton iteration, a linear approximation of F (α(k+1) ) is constructed about some known value, α(k) : F (α(k+1) ) ≈ F (α(k) ) +

∂F ¯¯ (α(k+1) − α(k) ) . ¯ ∂α α(k)

(3.33)

Requiring that F (α(k+1) ) = 0, leads to the following iterative method Ã

α

(k+1)

=α

(k)

∂F − ∂α

!−1

F (α(k) ) .

(3.34)

α(k)

The only complication that arises from the use of a Newton iteration is the evaluation ˆ ,α which can be easily derived by of ∂F/∂α. This derivative involves the quantities U formally taking the derivative of PSE with respect to α and evaluating it at α(k) . ¯ )U ˆ ,α + AU ˆ ,αx = Los,α (α(k) , β, ω; U ¯ )U ˆ, Los (α(k) , β, ω; U

(3.35)

¯ )U ˆ = (iA + 2αE1 ) U ˆ. Los,α (α, β, ω; U

(3.36)

where

Similarly, for the adjoint equations ˜ ,α˜ + A ˜, ˜U ˜ ,αx = Lõs,α˜ U Lõs U

(3.37)

where ³

´

˜ = ιA ˜. ˜ + 2˜ ˜1 U Los,α U αE

(3.38)


51

ˆ ,α is identical to the PSE except for the inhomogeneous right-hand The equation for U ˆ . Thus, this equation can be solved using side which depends on the PSE solution, U the same numerical methods as used for the PSE. In practice, we actually compute ˆ ,α by taking the derivative of the discretized PSE at each location xi . This allows U ˆ ,α . Figure 3.3 us to reuse the LU factorization of the PSE operator to compute U compares the convergence of the Newton iteration described here with the standard iteration given by (3.32a). Typical convergence tolerances are 10−8 which means that only three Newton iterations are required, while over nine standard iterations are needed. Since each iteration requires an LU factorization of a dense Ny × Ny matrix, Newton iteration is between two and three times faster than the conventional iteration.

3.4.2

PSE/APSE Stabilization Techniques

PSE, as well as APSE, are not fully parabolic equations. The remaining ellipticity comes from the gradient of the disturbance pressure. In order to remedy this inherent ill-posedness of PSE/APSE which manifests as a numerical instability on fine meshes, one typically uses an implicit marching scheme and a large marching step in the streamwise direction as described by Li and Malik [53]. They also found that the PSE approximation applied to the two-dimensional linearized Navier–Stokes equations with constant coefficients leads to an ill-posed Cauchy problem. To remedy this, Li and Malik proposed dropping the streamwise pressure gradient term pˆ,x . They argued that the accuracy of the solution does not depend on the presence of that term for Blasius boundary layer flow, but that this approximation may produce larger errors for three-dimensional boundary layers. Even with this fix, they noticed that some residual ellipticity remains, which is undesirable if high accuracy solutions are needed. Andersson et al. [4], introduced a stabilization procedure for PSE which we have


52

extended for three-dimensional PSE using a generalized coordinate system. Similar to the PSE stabilization procedure, we also provided a scheme for stabilizing APSE. For completeness, we present the derivation by Andersson et al. [4], since a similar derivation applies for the stabilization of APSE. ˆ ,x is approximated by To stabilize PSE, we recall that the streamwise derivative, U the backward Euler method with a truncation error

∆xi ˆ U ,xx , 2

that is O(Re −2 ). From

¯U ˆ ,xx as (3.14) we can express A ¯U ˆ ,xx = −Los x U ˆ + Los U ˆ ,x + A ˆ ,x . ¯ ,x U A

(3.39)

Andersson et al. suggest that to stabilize the PSE, one can add a term proportional ˆ ,x is O(Re −2 ) and its to the truncation error. From (3.39) we conclude that Los U addition to the PSE does not change the order of the PSE method. Thus, to stabilize ˆ ,x multiplied by a parameter ‘τ ’ and the resulting stabilized PSE PSE we add Los U are ˆ ,x = LU ˆ, (I − τ L)U

(3.40a)

¯ −1 Los , L = −A

(3.40b)

where

and ‘τ ’ is a parameter that must be determined to ensure numerical stability. To analyze the numerical stability of PSE, we follow Andersson’s approach and consider the asymptotic limit far away from the wall boundary where both streamwise and wall-normal derivatives of the mean-flow may be neglected. In this limit, and with the assumption of constant curvature, the PSE has constant coefficients and the solution ˆˆ ιηy ˆ = U can be decomposed into normal modes in y such that U e . Substituting a


53

ˆˆ solution of this form into the PSE results in a simplified operator L acting on U      L=     

−ια

−ιη

0

c/u

0

0

c + ιαu ιuη

0

0



   0 −ιη/u   ,   c/u 0  

0

(3.41)

−ια

where c = ιω − ια u − α2 /Re − η 2 /Re − ιη v. The eigenvalues of (3.41) are k1,2 = −ια ± η

and k3 = ιω/u − ια − α2 /(uRe) − η 2 /(uRe) − ιηv/u ,

(3.42)

and the eigenmodes of operator L propagate with the group velocity Cg = ω,α = −ki,α /ki,ω . Since the first two eigenvalues are independent of ω, the group velocity, Cg = ±∞, for these eigenvalues. This corresponds to information propagating instantly in the upstream and downstream direction (a more rigorous derivation is possible based on artificial compressibility). In order to stabilize the PSE, we need to eliminate the upstream propagating mode. Since the eigenvalues of the operator L/(I − τ L) are kis = ki /(1 − τ ki ), (3.40) can be written in terms of the eigenfunctions ˆ (e) as U ˆ (e) = U ,x

−ki ˆ (e) U . (1 − τ ki )

(3.43)

Applying backward Euler to the streamwise derivative in (3.43) yields (e)

(e)

ˆˆ ˆˆ U i = γm U i−1

and γm =

1 − τ km . 1 − (τ + ∆xi )km

(3.44)

For numerical stability, one requires that |γm | < 1 for all values of η. Note that |γm |2 =

(1 − τ km )2 + (τ αr )2 ≤ 1, [1 − (τ + ∆xi )km ]2 + [(τ + ∆xi )αr ]2

where αr is the real part of the streamwise wave number α.

(3.45)


54

Multiplying both sides of (3.45) by the denominator in (3.45) and expanding squares yields 2 2τ ∆xi αr2 + ∆x2i αr2 + km ∆x2i − 2(1 − τ km )(km ∆xi ) ≥ 0 .

(3.46)

Note, that if the minimum of the left-hand-side of (3.46) is greater than or equal to zero the whole expression is greater than zero and the method is stable. The minimum of (3.46) corresponds to km = 1/(2τ + ∆xi ), which after substitution into (3.46), yields ∆xi ≥

1 − 2τ . |αr |

(3.47)

According to (3.47), the allowable mesh spacing ∆xi is determined by the value of ‘τ ’. Similar to the PSE stabilization, the APSE stabilization is given by ˜ ,x = L˜U ˜, ˜U (I + τ L)

(3.48a)

where −1

˜¯ L˜ . L˜ = −A os

(3.48b)

˜˜ e−ιηy , and ignoring ˜ =U Assuming a normal mode decomposition in y, such that U curvature terms, as well as derivatives of the mean flow, results in the simplified operator L˜ = −L. Performing a similar stability analysis on APSE when using backward Euler we obtain ˜˜ (e) = γ U ˜˜ (e) , U m i i+1

γm =

1 − τ km , 1 − (τ + ∆xi )km

(3.49)

which under condition (3.47) results in a stable APSE method. The general form of the PSE/APSE given by (3.27) and (3.31) do not change after


55

˜ i , Ni , and N ˜ i become stabilization is applied; however, matrices Mi , M ¯ i + f Los , Mi = A i

¯ i + (f − 1)Los , Ni = A i

f = 1 + τ ∆xi ,

(3.50a)

and ˜¯ + f L˜ , ˜i=A M i osi

˜¯ + (f − 1)L˜ . ˜i = A N i osi

(3.50b)

Similarly the source terms equations (3.35) and (3.37) used for Newton iteration on α are must also be modified to include stabilization terms. There are several things one should note. With the stabilization technique described above, the APSE are no longer strictly adjoint to the PSE. However, the error is comparable with the truncation error of the method. One may argue that adding ˆ to the PSE will make APSE exactly adjoint to the PSE method, however, Los,x U ˆ leads to an unstable method. Besides, stability of the method with adding Los,x U ˆ cannot be analyzed based on the eigenvalues km of the operator L the term Los,x U ¯ −1 Los,x has different eigenvalues. In Section 3.5.2 we evaluonly, since the operator A ate the accuracy of the solution with and without stabilization. For future reference, the stabilization strategy defined in this subsection will be referred as τ -stabilization. Pressure iteration technique Another stabilization technique is the removal of the pressure gradient term in PSE as proposed by Li and Malik [53], which we refer to as p-stabilization. However, with this stabilization technique, the PSE solutions are potentailly inaccurate. To correct this, we introduce an iterative method that incorporates the influence of the streamwise pressure gradient to improve solution accuracy and this method (called pstabilization with pressure-iteration ) can be easily employed for both PSE and APSE ¯ is modified by zeroing the solutions. Starting from equation (3.14), the matrix A


56

term A14 . This pressure term is then moved to the right-hand side and lagged in the iteration. The PSE method becomes:

¯ )U ˆ (k+1) + AU ˆ (k+1) Los (α(k+1) , β, ω; U ,x

 (k)  −c1 pˆ,x       0 =   0      0

              

,

(3.51)

where c1 is a curvature term defined in Appendix D and pˆ(0) ,x = 0. Thus, the first iteration gives the standard PSE solution with no streamwise derivative of the pressure shapefunction. Subsequent iterations solve a forced PSE equation, where the source term is computed based on the prior iteration. Typically, only two iterations are required to adequately correct the PSE solution for pressure gradient effects. If the number of iterations becomes larger, say greater than four, the instability reappears, since the solution converges to the one that would be obtained if the pressure term had ¯ matrix. Each k iteration involves solving PSE and recomputing been retained in the A α at each marching station, xi . However, after the first iteration, the changes in α are slight such that the α iteration converges quickly, especially when using the Newton iteration described in Section 3.4.1. Similar iteration strategies can be used to obtain accurate APSE solutions. Start¯ is modified by zeroing the term A41 . The ing from equation (3.19a), the matrix A adjoint streamwise velocity term is then moved to the right-hand side and lagged in the iteration. The APSE become:

¯ )U ˜ (k+1) + A ¯˜ U ˜ (k+1) + L ˜U ˜ (k+1) = Lõs (˜ α(k+1) , β, ω; U ,x

       

0

      

0

0 (k) −c1 u˜,x

              

,

(3.52)


57

where u˜(0) ,x = 0. The first iteration gives the standard APSE solution with no streamwise derivative of the adjoint streamwise velocity shape function, and subsequent iterations solve a forced APSE equation, where the source term is computed based on the prior iteration. Only two iterations are usually required to adequately correct the APSE solution for the u˜ gradient term. In Section 3.5.2 we evaluate the accuracy of PSE and APSE solutions with and without p-stabilization +pressure-iteration along with a comparison to results using τ -stabilization.

3.5

Validation of Numerical Methods

Validation of the LNS/ANS and PSE/APSE methods is presented in this section both for a Tollmien–Schlichting disturbance of frequency F = ω/Re 0 × 106 = 150 in a Blasius boundary layer with global Reynolds number Re 0 = 200, and for a crossflow instability with spanwise wave number β = 0.2 in the Falkner–Skan–Cooke boundary layer with sweep angle Λ = 45o , Hartree parameter βh = 0.65, and global Reynolds number Re 0 = 100. We begin by first investigating the convergence of the LNS/ANS methods for different streamwise and spanwise resolutions. We then illustrate the use of the the global adjoint approach by comparing ANS predictions with direct LNS calculations. We then validate the PSE and APSE methods by comparing PSE and APSE results for various normalizations and for various stabilization techniques with LNS and ANS results.

3.5.1

Validation of LNS/ANS

We begin by stating the main findings of our LNS/ANS validation studies • LNS and ANS solutions converge with streamwise and wall normal mesh refinement for both two–dimensional and three–dimensional boundary layers.


58

• LNS solutions large transients, especially for three–dimensional boundary layers, due to approximate inflow data. • The ANS method gives accurate receptivity predictions in two- and threedimensional boundary layer flows as compared to direct LNS simulations. (While this is the expected result, it serves as a mutual validation of our LNS and ANS implementations.) The convergence of a numerical solution with mesh refinement is an important verifications of the validity of a numerical scheme. To demonstrate convergence of the LNS solution with streamwise resolution, Figure 3.4(a) shows the relative error in kinetic energy for two different resolutions Nx = 500 and 800. The mesh with Nx = 500 corresponds to 17 points per instability wavelength, while Nx = 800 corresponds to 28 points per wavelength. We have also calculated an LNS solution on a very fine mesh (Nx = 1106 corresponding to 41 points per wavelength) as a reference solution. In each case, we use Ny = 64 with is sufficient to make the solution converged in y. The relative error E is calculated using E=

Ke low resolution (x) − Ke high resolution (x) , Ke high resolution (x)

(3.53)

where Ke is the kinetic energy of an instability wave defined in (3.7) and the LNS calculation with Nx = 1106 is used to normalize the error. Notice that even for the coarse resolution, the relative error in kinetic energy does not exceed 8%, whereas at the higher resolution the error is smaller than 2% in the transient free region (Re = 300 − −650). The oscillations in the error are caused by the imperfect outflow boundary treatment where the buffer is used. Figure 3.4(b) shows the convergence of the adjoint solution with the streamwise mesh refinement indicated by the constancy of the j-product (the j-product is defined in Section 3.2.2). For high resolutions Figure 3.4(b) shows that the ANS solution is highly orthogonal to


59

the LNS unstable mode. Similar convergence studies have been conducted for various wall-normal resolutions for a constant high streamwise resolution. We find that LNS and ANS methods also converge with wall normal mesh refinement and these results are not shown here. Even though the LNS solutions converge with mesh refinement, they may exhibit strong sensitivity to inflow data producing a large transients near the inflow boundary. These transients represent the evolution of a collection of both stable and unstable modes that are initiated due to the imposition of an approximate inflow profile computed using parallel theory. These transients should be ignored if the evolution of a single mode is desired. For two–dimensional boundary layers, the transient due to approximate inflow data is mild, but for some three–dimensional boundary layers, such as the Falkner–Skan–Cooke boundary layer, the transients can be severe making it difficult to obtain an accurate prediction of the first neutral point location for the dominant unstable mode (see Bertolotti [9] for discussion). To investigate the effect of approximate inflow data on the overall quality of the LNS solution, Figure 3.5 shows the the disturbance kinetic energy growth rate for LNS solutions started at several different inflow locations with parallel-theory eigenfunctions applied on the inflow boundary. Near the inflow boundary, all the solutions exhibit a strong transient. Notice that the LNS growth rates from all three simulation are in good agreement for x ≥ 400. Given the large receptivity of stable modes near the leading edge, it is difficult to obtain a transient-free LNS solution for x < 400 when using a parallel-theory eigenfunction on the inflow. However, as shown in Chapter 5, this will not prevent us from making accurate receptivity predictions using adjoint methods even in regions where accurate LNS solution cannot be obtained.


60

To validate the ANS method, we consider the forced receptivity of the Blasius and Falkner–Skan–Cooke boundary layers to suction/blowing sources at the wall boundary. Figure 3.6 illustrates how adjoint methods can be effectively used to predict the receptivity of these flows. For example, Figure 3.6(a) compares three forced LNS runs for the Blasius boundary layer with corresponding ANS predictions. We see that the ANS predictions are in perfect agreement with the LNS results. While the adjoint theory requires this, it is a good indicator of the validity of our numerical solutions. Similar agreement is obtained for the receptivity of the Falkner–Skan–Cooke boundary layer for a stationary crossflow instability as shown in Figure 3.6(b).

3.5.2

Validation of PSE/APSE

Since the unmodified PSE/APSE method is not stable on fine meshes, one needs to investigate the accuracy and convergence of the PSE/APSE methods when using different stabilization strategies. Likewise, different normalizations may also affect convergence and accuracy. Starting with the PSE method, we find that • PSE solutions converge with both wall-normal and streamwise mesh refinement for two-dimensional boundary layers. • PSE solutions can exhibit large transients, especially for three–dimensional boundary layers, due to approximate inflow data. • Different stabilization strategies have an insignificant impact on the accuracy of the PSE solution for two–dimensional boundary layers, but can affect accuracy for highly nonparallel three–dimensional boundary layers. First, to validate of the PSE method, Figure 3.7(a) shows convergence of the PSE solution with streamwise mesh refinement for the two–dimensional Blasius boundary layer. We consider three different streamwise resolutions: Nx = 130, 300 and 800,


61

which corresponds to 4.8, 11, and 30 mesh points per wavelength. Figure 3.7(a) shows the error in the PSE solutions relative to a well resolved LNS run (Nx = 1106, Ny = 64) where the error is computed using E=

Ke low resolution (PSE) − Ke high resolution (LNS) Ke high resolution (LNS)

.

(3.54)

The oscillations seen in Figure 3.7 are due to slight oscillations in the LNS solution, which are caused by the imperfect outflow boundary conditions where the buffer is used. We see that in the transient-free region (Re = 300 − −600), the PSE solutions are all within 2% of the LNS solution. Similar convergence results, but in the wallnormal direction, are shown in Figure 3.7(b). Again, the PSE results within 2%, apart from the transient region, where the error is slightly larger. In general, we conclude that our PSE results are in excellent agreement with LNS for the Blasius boundary layer. We now investigate the effect of approximate inflow boundary data on the overall quality of PSE solutions similar to the studies done above for the LNS equations. Similar to LNS, for the Blasius flow, the inflow transients due to approximate inflow boundary data are mild, but for the Falkner–Skan–Cooke boundary layer, it is again difficult to obtain a transient free solution that accurately predicts the location of the first neutral point as can be seen from Figure 3.8. We have tested several different inflow locations for which parallel eigenfunctions could be found, and similar to LNS, we have confidence in the PSE solution only for x > 400. As was pointed out before, the discretized PSE are numerically unstable for fine streamwise meshes. Various stabilization methods, in general, can affect the accuracy of the PSE solution as compared with LNS. To show the effect of stabilization on the PSE solution, consider Figure 3.9, which shows the relative error of the PSE solution compared with LNS using either p-stabilization or τ -stabilization on a coarse mesh


62

(Nx = 130, which corresponds approximately to four mesh points per wavelength) and on a finer mesh (Nx = 300). We find that stabilization produces no significant change in the relative error either on the coarse or fine mesh. Overall, we conclude that for weakly nonparallel boundary layers, such as the Blasius boundary layer, the effects of stabilization in the PSE solutions are negligible. This is not the case, however, for three–dimensional boundary layers where stabilization can adversely affect the growth of crossflow instability waves. To illustrate this, Figure 3.10 compares the growth rates obtained using pressure-iteration (denoted in the figure as τ = 0) with τ -stabilization for τ = 1 and τ = 2. We can see from Figure 3.10(a) that the major difference between the three cases occurs at the location where the kinetic energy growth rate exhibits the largest change in second derivative. Note that such change is typical for three-dimensional nonparallel boundary layers and this behaviour is not exhibited by weakly nonparallel boundary layers such as the Blasius boundary layer. The τ -stabilization tends to give higher values for the growth rate than the pressureiteration approach. The difference in the kinetic energy amplitudes is shown in Figure 3.10(b), where it is seen that τ -stabilization underpredicts the kinetic energy. It is important to note, however, that the differences seen with different stabilization strategies are largely confined to the transient region (x < 400). Downstream of the transient region the PSE solutions using different stabilization techniques are in good agreement as can be seen from the similar growth rates in Figure 3.10(a). However, we show below that τ -stabilization can adversely affect the accuracy of APSE solutions. After verifying accuracy and stability of our PSE implementation, we now consider the APSE method. Note that in the APSE approach, not only do streamwise and wall-normal resolutions affect solution accuracy, but also the stabilization strategy combined with normalization can also play a important role. Our principal findings for APSE are:


63

• Accurate APSE solutions require the inclusion of the streamwise pressure-gradient term for both two- and three-dimensional boundary layers. • APSE solutions converge to the ANS solutions with streamwise and wall normal mesh refinement in the α-normalization with p-stabilization if the PSE solution is well resolved. • τ -stabilization, in general, gives adjoint solutions that are worse than the solutions obtained using p-stabilization with pressure-iteration . Also, the detrimental effects of τ -stabilization are most pronounced with the ke-normalization. • For three–dimensional boundary layers in the regions where the PSE solutions are influenced by inflow transients, the adjoint solution should be obtained using ke-normalization, since in this normalization, it is completely independent of the PSE solution. In order to assess the accuracy of adjoint solutions, we frequently plot j-product, where a constant j-product is a good indication that the adjoint solution is correct since it demonstrates the orthogonality of the PSE and APSE solutions. With this in mind, we begin by considering the effect of p-stabilization with and without pressureiteration on the APSE method for two streamwise meshes (Nx = 300 and Nx = 800) both computed with a high resolution in y (Ny = 64). Figure 3.11 shows the j-products computed using the cooresponding PSE solutions. We see that with no iteration on pressure (i.e. p-stabilization alone) the j-product is not constant and the loss in orthogonality is 16%. Thus, p-stabilization alone is not a viable method of stabilization. However, the solution is considerablly improved with only one pressure iteration and subsequent iterations continue to improve the accuracy of the solution, which is demonstrated by the virtually constant value of the j-product in the large portion of the domain for (Nx = 800). Note that the rapid oscillations


64

near the inflow and outflow boundaries are an indication of the onset of numerically unstable in those regions and subsequent pressure iterations may lead to the spread of instability throughout the computational domain. However, by the third iteration, the orthogonality is within 2% for Nx = 300 and less then 1% for Nx = 800. Thus, when using p-stabilization, pressure-iteration is necessary to obtain viable PSE/APSE pairs and we use pressure-iteration whenever required. However, self consistency between PSE and APSE in the form of constant j-product does not necessarily ensure that an APSE solution gives accurate receptivity predictions. The true test of accuracy is to compare APSE solutions to accurate ANS solutions. Such a comparison is shown in Figure 3.12 which plots the relative error in APSE solutions for various streamwise and wall-normal resolutions measured against a high accuracy ANS solution computed on a Nx = 1106 and Ny = 64 mesh. Clearly, the APSE solution converges to the ANS solution. We note that the APSE results in Figure 3.12 are computed using the α-normalization. If other adjoint normalizations are employed, the errors in the APSE solutions differ. For example, using the ke-normalization we find that the APSE solution underpredicts the ANS solution on the coarse streamwise mesh (Nx = 130) but has a much smaller error for the higher streamwise resolutions. We return in the effect of normalization below. Focusing on Figure 3.12(b), we notice that errors in the APSE are relatively large for Ny = 32, which is quite different from the behavior of the PSE solution (see Figure 3.7) where the solution is well predicted at this wall normal resolution. This difference can be understood by considering Figure 3.13 where we plot the state and adjoint streamwise velocity profiles. The adjoint velocity profile is more localized near the wall boundary as compared with the state velocity profile; therefore, greater resolution is required to obtain an accurate adjoint solution and a similar result has been reported for the AOSE [41].


65

In Figure 3.14 we plot the j-products for the same resolutions used in Figure 3.12. Not surprisingly, the j-product is poor for Ny = 32. However, in general the quality of the j-product tracks well with the relative error with respect to ANS as shown in Figure 3.12. From this, we surmise that the j-product is a good indicator of accuracy as well as consistency between the PSE and APSE pair. As in Figure 3.11, we note that the oscillations near the inflow and outflow are due to the onset of numerical instability as additional pressure-iterations are taken. Generally, the j-products are nearly constant for all streamwise resolutions considered and for Ny = 48 and 64, indicating that the adjoint solution is accurate. After verifying the convergence of the APSE solution for different streamwise and wall-normal resolutions, we now investigate the effects of different normalizations and stabilizations. We begin by considering stabilizations on a coarse streamwise grid (Nx = 130, Ny = 64). While no stabilization is required for this grid when using Backward Euler; nevertheless, it is useful to consider the impact of stabilization on accuracy. Results are shown in Figure 3.15 for both ke-normalization and α-normalization with and without τ -stabilization. Frame (a) shows the j-product for the ke-normalization while frame (c) shows the j-product for the α-normalization. Note, that whether or not stabilization is employed, the ke-normalization does not perform as well as the α-normalization. Note in general, τ -stabilization produces detrimental effects on the quality of the adjoint solutions especially in the ke-normalization. Furthermore, from Figure 3.15(c), we see that for the α-normalization, the error in the j-product is less than 1% without stabilization, while with stabilization it is approximately 3%. To explore the effects of stabilization on a finer mesh (Nx = 300, ≈ 9 mesh points per wave-length), we plot in Figure 3.15(b) and (d), the j-products again for ke-normalization and α-normalization. The best orthogonality is exhibited when


66

pressure iteration is employed (τ = 0 curve) using the α-normalization as shown in in frame (d). With pressure-iteration the error in the j-product is less than 0.5% while with τ -stabilization the error is about 3%. In Figure 3.15(b), we have used the ke-normalization to get and we see that with this normalization, the adverse effects of τ -stabilization are significant. Besides the α-normalization and ke-normalization, the normalization based on iteration in adjoint variable α ˜ can be employed to enforce the constancy of the jproduct, which is referred to as j-normalization. The j-product in j-normalization is not plotted, since by construction, the j-product is identically one for that normalization. However, to attest to the accuracy of the resulting APSE solutions for various τ -stabilization parameters, Figure 3.16 shows the relative error in the APSE solution as compared to a well resolved ANS run (the ANS solution is computed with Nx = 1106, Ny = 64). The error with large values of τ may be a few percent, which is still acceptable, and is smaller than the error in adjoint kinetic energy with the ke-normalization for the same values of τ . As we have seen for two–dimensional boundary layers, the adjoint solutions obtained with α-normalization using pressure-iteration are the best approximation to the ANS. This is not true in the case of three–dimensional boundary layers, such as the Falkner–Skan–Cooke boundary layer. The reason for this disagreement is the fact that neither the LNS nor PSE solutions are accurate for small local Reynolds numbers due to the approximate inflow boundary data. To emphasize this point, consider in Figure 3.17 the growth rate of the PSE solution obtained by imposing a parallel theory eigenfunction at the inflow. The eigenfunction is only an approximation to the PSE solution, and therefore, one observes transients at the inflow. Moreover, the transients obtained using PSE are nonphysical, since they do not correspond to the physical LNS transients at the inflow (see Chapter 6). If the adjoint solution is


67

constructed as the adjoint to the PSE, then it can inherit the nonphysical behavior in the region where PSE exhibits the transient. This can happen, for instance, if the α-normalization or j-normalization is employed to obtain the adjoint solution. To see this, consider Figure 3.18, which shows the APSE solution obtained using two different methods: first using the ke-normalization, and second using the α-normalization both using pressure-iteration with p-stabilization. With the α-normalization, the adjoint solution depends on the PSE solution, which may not be correct in the region near the inflow, while the adjoint solution based on ke-normalization does not depend on the PSE solution and is a better approximation of the ANS method. In Figure 3.18, the APSE solution with α-normalization inherits the incorrect transient behavior near the leading edge, resulting in incorrect adjoint predictions. We have also tried the τ -stabilization, however, adjoint solutions obtained with τ -stabilization lead to incorrect overall growth rates in the adjoint when α-normalization is used. Using τ -stabilization with ke-normalization, however, does gives accurate adjoint solutions as can be seen by examining the adjoint growth rates from Figure 3.18. We conclude this section by reiterating the main points of this validation study. Comparisons of ANS preditions with direct LNS simulations serve to both mutally validate our implementations as well as demonstrate the utility of global adjoint methods for studying receptivity in nonparallel flows. Careful comparisons to both LNS and ANS have guided the implementation and validation of our PSE and APSE approaches. In particular, we find that to obtain accurate PSE/APSE solutions both the stabilization and normalization conditions have to be carefully selected. While stabilization strategies do not affect the accuracy of PSE solutions for weakly nonparallel boundary layers, they do impact accuracy for nonparallel three–dimensional boundary layers. Furthermore, we find that using p-stabilization with pressure iteration produces accurate APSE solutions, especially in the α-normalization, provided


68

that an accurate PSE solution exists. Otherwise, the ke-normalization should be employed in the adjoint, preferably using pressure-iteration with p-stabilization, to obtain accurate adjoint predictions. With this validation of the LNS/ANS and PSE/APSE methods, the following Chapter utilizes these methods to document receptivity for two-dimensional boundary layers while three-dimensional boundary layers are considered in Chapter 5.


A

B

D

C

A

B

E

D

C

A

B

E

D

C

A

B

E

D

C

A

B

E

D

C

A

1....Ny

V terms

W terms

x1

r

= B

C

A

B

E

D

C

Matrix C

U V W P

{

{

D

{

{

C

U terms

69

U V W P

x Nx

P terms

Ny+1....2Ny

{} u1

4Ny

U momentum U=

V momentum

uN

y

W momentum

4Ny

Continuity

Figure 3.1: The pentadiagonal system used in solving LNS-ANS equations.


70

(a) Bu r

^

ffe

t σ

^

vb

w flo

S LN

in

ow

S AN

(b)

^ T

λ+

infl

~ n=0 σ

A n

r feerr uuffffe

BBBu

r Ze

nce rba istu d o

Figure 3.2: Computational domain and Boundary conditions for LNS and ANS solutions.


71

Tolerance Criteria

1

1e-10

1e-20

0

2

4

6 8 10 12 Number of Iterations

Figure 3.3: Convergence using the newton method iteration strategy .

14

16

18

compared with regular


72

0.1 Nx=800 Nx=500

Relative error in Ke

(a) 0.06 0.02 -0.02 -0.06 -0.1

250

300

350

400

450 Re

500

550

600

650

700

1.01 Nx=500 Nx=800 Nx=1100

(b)

J product

1.005

1

0.995

0.99

300

350

400

450 Re

500

550

600

Figure 3.4: Convergence of the LNS/ANS method with streamwise resolution for TS wave in the Blasius boundary layer; (a) error in kinetic energy for Nx = 500 and Nx = 800. The average wavelength of the instability wave is ≈ 78.5, therefore, mesh with Nx = 500 corresponds to 17 pts. per wavelength, Nx = 800 corresponds to 28 mesh pts. per wavelength, and Nx = 1100 corresponds to 41 pts.; (b) j-product for three different resolutions. For all cases, Ny = 64.


73

Disturbance Kinetic Energy Growth Rate

0.01 xi =70.52 148.87 227.21

0.006

0.002 0 -0.002

-0.006

-0.01

200

400

600

800

x

1000 1200 1400 1600 1800

Figure 3.5: LNS kinetic energy growth near the first neutral point for crossflow instability in the Falkner–Skan–Cooke boundary layer for several inflow conditions at xi = 70.52, 148.87 and 227.21. For these runs Ny = 64 and Nx = 882.


74

180 160

(a)

Disturbance Ke

140 120 100 80 60 40 20

0 250

300

350

400

450

500

550

600

Re

2000

Disturbance Ke

(b) 1600 1200 800 400 0 100

200

300

400

500

Re

Figure 3.6: Validation of the adjoint method; (a) adjoint predictions for the Blasius boundary layer compared with the LNS calculations for suction and blowing sources at three different locations, Re = 316, 346 and 400: adjoint predictions LNS simulations; (b) adjoint predictions compared with the LNS calculations for crossflow instability in the Falkner–Skan–Cooke boundary layer. The suction and blowing source is located at Re = 235: LNS simulation APSE prediction.


75

0.03

Error in Ke

0.02

PSE, Nx=130 PSE, Nx=300 PSE, Nx=800

(a)

0.01 0 0.01 0.02 0.03

250 300 350 400 450 500 550 600 Re

Error in Ke

0.04

PSE, Ny=32 PSE, Ny=48 PSE, Ny=64

(b)

0.02 0 -0.02 -0.04 200

300

400

Re

500

600

700

Figure 3.7: Convergence of the PSE with streamwise and wall-normal resolution for the Blasius boundary layer; (a) relative error in the kinetic energy of the PSE solution for various streamwise resolutions Nx = 130, 300 and 800; (b) the error in the kinetic energy of the state field between PSE and LNS solutions for different wallnormal resolutions with Nx = 300. For all meshes the p-stabilization is used where applicable. The reference LNS run is with Nx = 1106 and Ny = 64. For all the PSE runs Ny = 64.

Disturbance Kinetic Energy Growth Rate


76

0.02 xi =70.52 109.70 148.87 188.04 227.21 266.39

0.015 0.01 0.005 0 -0.005 0

500

1000

x

1500

2000

2500

Figure 3.8: PSE kinetic energy growth rates near the first neutral point for inflow conditions at several different streamwise stations xi = 70.52, 109.70, 148.87, 188.04, 227.21, 266.39. The streamwise resolution Nx = 300 (about 6 mesh pts. per instability wavelength) and Ny = 64.


Error in Disturbance Ke

0.04

77

pressure iteration τ=0.5 τ=1 τ=2

(a)

0.02 0

-0.02 -0.04 200

300

400

500

600

700

Error in Disturbance Ke

Re

0.04

pressure iteration τ=0.5 τ=1 τ=2

(b)

0.02 0

-0.02 -0.04 200

300

400 Re

500

600

Figure 3.9: The error in the disturbance kinetic energy for various stabilization strategies for TS instability with frequency F = 150 in the Blasius boundary layer; (a) the relative error for the PSE solution on coarse mesh Nx = 130; (b) the relative error for the PSE solution on a fine mesh Nx = 300 and Ny = 64.


78

Kinetic Energy Growth Rate

0.02 τ=0 τ=1 τ=2

(a) 0.015 0.01 0.005 0 -0.005 0

500

1000

1500

2000

2500

2000

2500

x

Disturbance Kinetic Energy

16 τ=0 τ=1 τ=2

14 12 10

(b)

8 6 4 2 0 0

500

1000

1500 x

Figure 3.10: Comparison of the PSE solutions for various stabilization parameters, τ , for crossflow instability in Falkner–Skan–Cooke boundary layer; (a) comparison of the growth rates; (b) comparison of the amplitudes. Nx = 300,Ny = 64.


79

1.02 1

j-product

0.98 0.96 0.94 0.92 0.9 0.88

no iteration 1st iteration 2nd iteration 3rd iteration

0.86 0.84 0

(a) 500

1000

1500

2000

2500

x 1.05

j-product

1

0.95

0.9

0.85 0

no iteration 1st iteration 2nd iteration 3rd iteration (b) 500

1000 x

1500

2000

2500

Figure 3.11: The convergence of the j-product with the number of pressure iterations for the TS wave in Blasius boundary layer; (a) Nx = 300; (b) Nx = 800. These cases were chosen to illustrate convergence of adjoint solution with streamwise resolution. For all the cases considered Ny = 64.


80

Relative Error in Adjoint Kinetic Energy

0.01 APSE, Nx=130 Nx=300 Nx=800

0.006

0.002

(a)

0 -0.002

-0.006

-0.01

300

350

400

450 Re

500

550

600

Relative Error in Adjoint Kinetic Energy

1 APSE, Ny=32 Ny=48 Ny=64 0.6

0.2

(b)

0 -0.2

-0.6

1

300

350

400

Re

450

500

550

600

Figure 3.12: Convergence of APSE with the streamwise and wall-normal resolutions for the TS wave in the Blasius boundary layer; (a) error in the adjoint kinetic energy of the APSE solution for various streamwise resolutions Nx = 130, 300 and 800; (b) error in the adjoint kinetic energy of the APSE solution for various wall-normal resolutions Ny = 32, 48 and 64 (for all meshes the pressure-stabilization is used when applicable. The adjoint runs were done using the α-normalization).


81

Streamwise state velocity profiles

1 (a)

0.8

PSE(Nx=300, Ny=64) Re{u} PSE(Nx=300, Ny=64) Im{u}

0.6 0.4 0.2 0 -0.2 -0.4 0

20

40

y

60

80

100

Streamwise adjoint velocity profiles

1.5 ~ APSE(Nx=300, Ny=64) Re{u} ~ APSE(Nx=300, Ny=64) Im{u}

(b) 1 0.5 0 -0.5 -1 -1.5 -2

0

20

40

y

60

80

100

Figure 3.13: (a) instability streamwise velocity instability profiles; (b) adjoint streamwise velocity profiles. Both the state and adjoint profiles are extracted at Re = 371, and all the profiles are obtained from the run with Ny = 64 and Nx = 300.


82

1.6 Ny=64 Ny=48 Ny=32

j-product

1.4

1.2

1 (a) 0.8 0

500

1000

1500

2000

2500

x 1.01 Nx=800 Nx=300 Nx=130

j-product

1.005

1

0.995 (b) 0.99 0

500

1000

1500

2000

2500

x

Figure 3.14: Convergence of the j-product with mesh refinement for the TS wave in the Blasius boundary layer; (a) convergence of the j-product with the wall-normal mesh refinement (Nx = 300); (b) convergence of the j-product with the streamwise mesh refinement (Ny = 64). All the adjoint runs are done using α-normalization, and pressure-stabilization is used when needed.


j-product (Nx=130, ke-normalization)

1

1

j-product (Nx=300, ke-normalization)

1.02

(a)

83

(b)

0.98 0.96

0.95

0.94 τ= 0 τ = 0.5 τ =1 τ =2

0.9

0

1.04

500

1000

x

1500

2000

2500

j-product (Nx=130, a-normalization) τ= 0 τ = 0.5 τ =1 τ =2

(c) 1.03 1.02

τ= 0 τ = 0.5 τ =1 τ=2

0.92 0.9 0.88 0

1.05 1.04

500

1000

1500

x

2000

2500

j-product (Nx=300, a-normalization) τ= 0 τ = 0.5 τ =1 τ =2

(d)

1.03 1.02

1.01

1.01

1 0.99 0

1 500

1000 1500 x

2000

2500

0.99 0

500

1000

x

1500

2000

2500

Figure 3.15: Comparison of different normalizations for the TS wave in the Blasius boundary layer; (a) the j-products obtained using ke-normalization for different τ s; (c) the j-products obtained using α-normalization for different τ s. The APSE resolution is Nx = 130, Ny = 64; the cases (b) and (d) are similar to (a) and (c), respectively, but the mesh has Nx = 300, and Ny = 64. For the τ = 0 cases, pressure iteration is used for Nx = 300, but no pressure iteration is employed for Nx = 130. For nonzero values of τ pressure iteration is not employed. Unless otherwise specified, the j-products are always computed using PSE and the corresponding APSE solutions.


84

Relative Error in Adjoint Ke

0.05

τ =0.0 τ =0.5 τ =1.0 τ =2.0

0.04 0.03 0.02 0.01 0 -0.01

350

400

450

Re

500

550

600

650

Figure 3.16: The error in the adjoint kinetic energy using j-normalization for various values of τ for the adjoint TS wave in the Blasius boundary layer. The APSE resolution is Nx = 130, Ny = 64. For the τ = 0 case, pressure iteration is used.

(1/Ke) dKe/dx (Kinetic Energy Growth Rate)


x 10

85

-3

2 LNS 1 PSE 0 -1

APSE

-2

ANS

-3

-3

x 10 2

100

200

300

400 Re

500

600

700

1 0 -1 -2 40 60 80 100

140

Figure 3.17: Comparisons of the LNS, PSE, ANS and APSE growth rates for crossflow disturbances in the Falkner–Skan–Cooke boundary layer.


Adjoint kinetic energy growth rate

8

x 10

86

-3

ke-normalization with τ=1 ke-normalization with pressure iteration a -normalization with τ=1 a -normalization with pressure iteration

6 4 2 0 -2 -4 -6 -8

0

50

100

150

Re

200

250

300

350

Figure 3.18: Comparison of the two different normalizations in APSE for crossflow disturbance in the Falkner–Skan–Cooke boundary layer. with and without τ stabilization. When τ -stabilization is not employed pressure iteration is used to stabilize adjoint solution.

Chapter 4 Receptivity Analysis for the Blasius and Falkner–Skan Boundary Layers This chapter presents receptivity results for the Blasius and Falkner–Skan boundary layers. We analyzed both the effects of pressure gradients and nonparallel effects on receptivity of these boundary layers to both two– and three–dimensional disturbances. The receptivity characteristics are obtained for various sources, which include forced disturbances, such as suction and blowing sources on the wall boundary, tangential excitations at the wall, and momentum sources in the domain of the flow. Further, we documented the receptivity of these boundary layers to acoustic disturbances for both two– and three–dimensional instabilities. At the end, we summarize the findings of this chapter. We begin by considering forced receptivity.

4.1

Forced Receptivity Results

Forced receptivity results are computed based on solutions to either the ANS or APSE. In order to validate our methods, we performed a detailed comparison of direct LNS, ANS and APSE solutions showing that all methods give consistent predictions (refer to Figure 3.6 for a typical comparison). For brevity, only the APSE results are reported in the form of the adjoint quantities for a range of unstable frequencies and Reynolds numbers. These adjoint quantities can be used to directly predict the initial amplitude of disturbances due to impulse sources of mass, momentum, and boundary velocities. ˆ m (x, y) is normalized at For all receptivity results presented here, the state U each location x by the streamwise component of disturbance velocity at y = y˜max ˆ m (x, y˜max )| = maxy |U ˆ m (x, y)|. The adjoint mode U ˜ m is normalized by where |U 87

Chapter 4. Receptivity of Blasius and Falkner–Skan Boundary Layers

88

ˆ m ; x) = 1 at a particular x location. Recall that subscript ‘m’ ˜ m, U requiring J(U refers to a particular mode. All receptivity results using APSE are compared to parallel theory predictions using the AOSE as described by Hill [41] for the Blasius boundary layer. For accurate comparisons, we recomputed all the parallel theory results for comparison to APSE, and these recomputed results are in agreement with those of Hill. Further, note that unless otherwise specified, the shape functions for ˆ and U ˜ are respectively used as defined in Chapter 3. When, state and adjoint U it is important to show results with the phase function included, we use U 0 and U ∗ notation for state and adjoint respectively. To simplify comparisons to parallel theory, all receptivity results for both the Blasius and the Falkner–Skan boundary layers are presented based on local scaling where the reference length-scale is δs , the similarity length scale for either the Blasius or Falkner–Skan boundary layers, and the reference velocity is Ue the local edge velocity. Thus, the local Reynolds number is Re = Ue δs /ν. Since the APSE is the global method we define a global Reynolds number, Re 0 = Ue L/ν, where L is the appropriate length scale (typically displacement thickness at the inflow), and the frequency parameter is defined as F = ω × 106 /Re 0 .

4.1.1

Blasius Boundary Layer

We begin by considering the receptivity characteristics of the Blasius boundary layer (for details on the Blasius mean flow, see White [92]) to two-dimensional disturbances. Figure 4.1(a) shows the maximum amplitude, in y, of the streamwise component of the adjoint velocity computed at every x location in the flow, which measures the receptivity to a point source of streamwise momentum at this location, which can be used to predict the receptivity due to a vibrating ribbon (see Hill [41]). The y locations where the streamwise component of adjoint velocity achieves its maximum


89

amplitude are plotted in Figure 4.1(b). Placing a streamwise impulse forcing at this location will induce the largest excitation of the flow. Whereas receptivity to momentum sources is given by the adjoint streamwise velocity, receptivity to wall normal and tangential excitations is given by adjoint wall pressure and adjoint wall shear stress as shown in Figures 4.1(c) and 4.1(d) respectively. To obtain the receptivity amplitude of the flow for an impulse source at the source location, one simply reads the values of am from the figure. For example, from Figure 4.1(c), one can see that for the case of F = 150, a point wall normal excitation at Branch II, Re = 534 induces an instability wave of amplitude am = 0.2019. Note, that as the frequency of the unsteady source is decreased, the receptivity to both types of excitations generally increases. Overall, we observe that the boundary layer is more receptive to lower frequency excitations, although these disturbances are more stable. From Figure 4.1, one can immediately conclude that the receptivity to wall normal excitation is an order of magnitude higher than the receptivity to tangential excitation, while the receptivity to the momentum sources is the highest at the distance from the wall as indicated by Figure 4.1(b). From Figures 4.1(a), (c) and (d) one also observes that the receptivity to a point force in the streamwise direction generally increases downstream, whereas receptivity to normal and tangential boundary velocities tends to be larger near the first neutral point. For all disturbance types considered here, the parallel receptivity theory is in excellent agreement with APSE, indicating that nonparallel effects for localized twodimensional receptivity are nearly negligible under these conditions. Although parallel theory does a good job of predicting localized receptivity for this flow, for nonlocal sources, the errors due to the incorrect growth rate predictions from parallel theory will influence the receptivity results. The discrepancy between purely parallel and nonparallel approaches can be clearly seen in Figure 4.2. Here we compare the


90

adjoint wall pressure predictions from parallel theory with the results obtained using APSE where we plot the amplitude for F = 150 of the spatial adjoint solution |U ∗m4 (x, 0, z, t)| = |˜ pm (x, 0, z)| exp(−

Rx xe

2Im(˜ αm (x0 ))dx0 ), which includes both the

shape function and wave function contributions. For convenience, the state U 0m and adjoint U ∗m are normalized at the first neutral point so that the curves can be used to directly predict the amplitude of the instability wave at that location. For example, an impulse source located at Re ≈ 300 gives the maximum amplitude at Branch I due to the combined effects of receptivity and instability. Notice that while parallel theory and PSE agree well at the normalization location, due to the difference in the growth rates there are errors in the parallel theory for other streamwise locations. This indicates that for an impulse source placed at the normalization location, parallel theory would give accurate predictions to the disturbance amplitude, which are exactly the conditions documented in Figure 4.1. However, for sources placed upstream or downstream of the normalization location, the amplitude given by the parallel theory will be incorrect due to the incorrect spatial growth rate from parallel theory. If instead of an impulse source, one has a distributed source, say a Gaussian distribution, then the amplitude is the integral in x of the source weighted by the adjoint pressure |˜ p|y=0 plotted in Figure 4.2. Note, while we have dropped subscripts, we still understand that adjoint quantities refer to a particular adjoint mode ˜ m . Even for a source centered at the normalization point, parallel theory will be U incorrect with the error increasing for more distributed sources. However, it is worth noting that parallel theory underpredicts upstream, and overpredicts downstream of the normalization location so that these errors will tend to cancel. The receptivity of the Blasius boundary layer to three-dimensional disturbances is summarized in Figure 4.3 for a single frequency F = 20, and various spanwise wave


91

numbers β. We can immediately conclude that the Blasius boundary layer is generally more receptive to three–dimensional disturbances by comparing the receptivity of oblique disturbances with different spanwise wave numbers, β. The increase in receptivity is related to the disturbance wave-angle, θ, defined as an angle between the streamwise direction x and the direction of wave propagation, θ = tan−1 (β/αr ). The larger θ, the greater the receptivity. In general, parallel theory slightly underpredicts receptivity near the first neutral point and overpredicts near the downstream neutral point. Compared to the two–dimensional results, there is a pronounced increase in receptivity near Branch II for wall velocity disturbances. Notice also, that in Figure 4.3(d) the discrepancy between local parallel theory and APSE is more pronounced downstream of the second neutral point compared to the two–dimensional results. While this may seem largely academic since the disturbances decay downstream of Branch II, this could have important implications for bypass transition. In bypass transition, highly receptive stable modes may be excited by a disturbance, and through nonlinear interactions, excite an unstable mode. Alternatively, bypass transition can also be due to the large transient growth of the linear superposition of various non-orthogonal modes [5, 28, 69]. Thus, the receptivity of stable disturbances is relevant for bypass transition, and the current results demonstrate that nonparallel effects become more pronounced for stable modes downstream of their Branch II locations. To predict the actual amplitude of an instability wave at a desired location, both stability and receptivity characteristics must be considered. If the state and adjoint are normalized at some particular x location, say at xo slightly downstream of Branch I for two–dimensional disturbances, we would directly obtain the disturbance amplitude at that location due to an impulse source from the adjoint solution. Consider first an impulse source located near Branch I. Since the receptivity of three–dimensional


92

disturbances is higher, an oblique wave will have a higher amplitude than a two– dimensional disturbance at the source location. In Figure 4.3(f ) we also see that the growth rates of oblique instabilities are increasing near the first neutral point of the two–dimensional instability wave (Re ≈ 900), with the increasing spanwise wavenumber β. Notice, for example, that for an oblique wave with β = 0.1, the growth rate −αi is larger then the growth rate of the two–dimensional disturbance for Reynolds numbers smaller then Re ≈ 1200. Near Branch I of the two–dimensional disturbance the oblique disturbances may be significantly more unstable then their two–dimensional counterpart. For instance, at Re ≈ 900 the three–dimensional TS wave with β = 0.1 is about twice as unstable as the two–dimensional TS wave. This indicates that three–dimensional disturbances, due to their high receptivity and high initial instability, may lead to transition at the location near the first neutral point. If the impulse source is moved upstream, there is some location where the amplitudes of the two–dimensional waves at Branch II become larger because of their higher growth rates at higher Reynolds numbers. This implies that there is a location in the flow, where introducing an impulse source produces equal response for two– and three–dimensional waves at Branch II.

4.1.2

Falkner–Skan Boundary Layers

To assess the influence of mean pressure gradient on receptivity, we have used APSE to document the receptivity of Falkner–Skan wedge flows (see White [92]) which have edge velocity distributions given by U ∼ xm . Since we are interested primarily in the general trend in receptivity as pressure gradients are varied, we have explored only two values for m = ±0.021 . The two–dimensional receptivity results for an adverse pressure gradient are shown in Figure 4.4, and the favorable pressure gradient results 1

It should be clear from the context that it is not the same m as used in the subscripts for modes.


93

are shown in Figure 4.5. In general, an adverse pressure gradient decreases receptivity for all source types, while a favorable pressure gradient increases receptivity. This observation is more vividly shown in the Fig. 4.6 which shows the receptivity to wall velocities disturbances at F = 100 for different pressure gradients. Returning to Figs. 4.4 and 4.5, one sees that the agreement between parallel theory and APSE is excellent over the entire parameter range considered here. Similar to the Blasius results, however, nonparallel effects will become more important for distributed sources and 3-d disturbances. We note that our results for local suction/blowing in adverse gradient flows, are consistent with the parallel theory of Choudhari et al. [14], who showed that adverse pressure gradients reduce natural receptivity to wall suction and wall admittance variations. Again, there is a trade-off between receptivity and instability. In an adverse pressure gradient flow, receptivity is reduced while instability is enhanced and the opposite occurs for favorable pressure gradients. This leads to an interesting possibility. In natural laminar flow airfoils, favorable pressure gradients are utilized to help stabilize disturbances, and therefore delay transition. By doing so, however, the receptivity of disturbance may be markedly increased. Thus, while linear instability is reduced, these designs may be more susceptible to bypass transition caused by high linear receptivity of stable modes. This is an area for future research. To investigate the combined effects of pressure gradients and three–dimensionality, consider the receptivity of three–dimensional disturbances in Falkner–Skan boundary layers shown in Figures 4.7 and 4.8. Here, as in the Blasius boundary layer, receptivity to oblique disturbances is higher for greater values of θ. The greatest receptivity is achieved when the effects of favorable pressure gradients and oblique disturbances are combined, although the instability is quite weak under these conditions. Similar to the Blasius boundary layer flows, receptivity to wall normal excitations is


94

significantly larger than receptivity to tangential disturbances, and receptivity to a momentum impulse source can be three orders of magnitude higher than tangential disturbances at the wall. For three–dimensional disturbances, local parallel theory tends to overpredict receptivity near their second neutral point, which again may be important for bypass mechanisms where stable modes play an important role.

4.2

Natural Receptivity Results

We now consider the receptivity of the Blasius and Falkner–Skan boundary layers to free-stream acoustic waves. In the limit of incompressible flow, acoustic waves take the form of an imposed time-harmonic oscillation of the freestream. Here, we consider the receptivity due to scattering of freestream acoustic disturbances by a small roughness element or a steady suction or blowing source on the wall boundary. Details regarding the formulation of nonparallel adjoint methods for natural receptivity can be found in Chapter 2, Section 2.4.

4.2.1

Blasius Boundary Layer

In order to verify the accuracy of the adjoint method we first compare receptivity predictions based on the adjoint formulation with results from a direct LNS simulation. The LNS calculations are obtained using a two-step procedure. First, the LNS equations are solved with a linearized wall boundary conditions given by (2.55) to find a steady perturbation U h . The unsteady sound perturbation U s , (i.e., Stokes boundary layer), is found by solving the linearized unsteady boundary layer equations. Then v h and v s are used to calculate the bilinear momentum forcing (2.60), ˆ subject to this forcing on the right hand-side, and and the LNS are solved for v the boundary condition (2.55). Figure 4.9 compares the kinetic energy, Ek , of the instability wave obtained using LNS with the APSE predictions demonstrating that


95

APSE predictions, are in excellent agreement with direct LNS. With this validation, we proceed to document the natural receptivity characteristics of the Blasius and Falkner–Skan boundary layers. Figure 4.10(a) shows the efficiency function Λ = Λr defined in (2.79), which gives the receptivity amplitude, am , for the sound source of frequency F scattered from a point roughness element. For example, if a point roughness element is located at Re = 600, then the excited instability wave at F = 100, (F Re 3/2 ≈ 1.5), has amplitude am ≈ 0.07. We observe that higher frequency disturbances tend to be more receptive for acoustic/roughness receptivity. We note that receptivity amplitudes in this case are on the same order as forced receptivity for tangential excitations shown in Figure 4.1(d). Figure 4.10(b) shows the receptivity due to sound waves scattered from a steady blowing source as given by the adjoint pressure |˜ phm |y=0 . Note that acoustic/blowing receptivity is two orders of magnitude larger than acoustic/roughness receptivity, and an order of magnitude larger than forced receptivity due to unsteady, wallnormal suction and blowing. Receptivity tends to increase downstream of Branch I for acoustic/roughness disturbances, while receptivity decreases downstream for acoustic/blowing excitations. Also, for acoustic/blowing disturbances, receptivity decreases with increased frequency, similar to forced receptivity for unsteady suction/blowing. Figures 4.10(a) and (b) show an acceptable agreement between parallel theory and APSE, where the maximum difference of about 4% occurs for the high frequency instabilities near the second neutral point. As mentioned before, this only pertains to localized excitations. For distributed sources the incorrect spatial growth rates from parallel theory will produce errors in receptivity prediction. Acoustic receptivity of the Blasius boundary layer to three-dimensional disturbances is summarized in Figures 4.10(c) and (d). Note that receptivity for the oblique


96

disturbances considered here is strongly dependent on wave angle θ. Similar to forced receptivity, oblique disturbances are highly receptive for large wave angles. Note also that parallel theory is in good agreement with the results obtained using APSE. Note, however that parallel theory tends to overpredict at Branch II for the highly oblique disturbances. The nonparallel effects tend to increase for oblique TS waves, since the streamwise wavelength of these disturbances becomes larger than the wavelength of two–dimensional disturbances.

4.2.2

Falkner–Skan Boundary Layers

Figures 4.11 and 4.12 give natural receptivity results for two– and three–dimensional disturbances in favorable and adverse pressure gradient Falkner–Skan boundary layers. Similar to the forced receptivity results, favorable pressure gradients tend to increase receptivity, while adverse pressure gradients decrease receptivity. Likewise, oblique disturbances again are more receptive than two–dimensional disturbances and the combination of three–dimensional disturbances with favorable pressure gradients leads to the greatest receptivity. In Figure 4.11 one can observe a slight kink in the absolute value of the efficiency function |Λb | for the frequency F = 10. This is related to the fact that we normalize ˆ m (x, y)| = 1 at every x station in the flow. the PSE solution by requiring that maxy |U At the x location corresponding to the local Reynolds number Re where F Re 3/2 ≈ 1.2, the local maxima is found at a new location in y, and the streamwise derivative of ˆ m (x, y)| acquires a discontinuity. maxy |U The high receptivity of three-dimensional disturbances can be important in transition prediction since, in some cases, the increased receptivity may outweigh the diminished growth rate of these disturbances compared to their two–dimensional


97

counterparts. For instance, the receptivity of the β = 0.05 disturbance at F = 10 produces an instability wave with transitional amplitudes sooner than the corresponding two–dimensional disturbance would. Finally, parallel theory is still in a tolerable agreement with the nonparallel predictions for oblique disturbances even though there are larger differences observed for adverse Falkner–Skan boundary layer flow. The difference can be explained by examining the instability wavelength. In adverse pressure gradient flows, the instability wavelength is typically larger compared with the instability wavelength Blasius and favorable Falkner–Skan flows. The increase is more pronounced near the second neutral point, leading to the greater influence of nonparallel effects in the neighborhood of the second branch.


35

1.4

10 o

(a)

150 100

10

0 0

oo o

50

15

5

1.2

20

o

o o o o ooo

o 100 o 50

1

o20

o o o

1000

2000

3000

0.6 0

4000

1000

2000

Re 1.4

10 o

0.8

o

o

150 o

o

ymax

|˜ umax |

o

200

20

(b)

200

30 25

98

3000

4000

Re

(c)

0.035

(d)

1.2 0.03

0.4 0.2 0 0

0.02

o

0.015

200

y=0

o o o o o o o 50 o 200 o o150100

1000

ooo

Re

0.6

0.025

¯ ¯ ¯ ∂ u˜ ¯ ¯ ∂y ¯

0.8

10 o o 20

1

|˜ p|y=0

1

o o o

o o 100 o 150

50

0.01

2000

Re

3000

4000

0.005 0

1000

o

2000

10 o

20

3000

4000

Re

Figure 4.1: Receptivity predictions based on APSE for two–dimensional disturbances in the Blasius boundary layer; (a) the maximum receptivity to a streamwise point force; (b) the distance from the wall for maximum receptivity to a streamwise point force; (c) the receptivity due to wall-normal suction/blowing at the wall; (d) the receptivity to streamwise velocity disturbances at the wall: APSE results, local parallel theory, ◦ denotes first and second neutral points. Here we have chosen to use the following global Reynolds numbers: Re 0 = 200, 200, 200, 400, 1000, 1000 corresponding to the frequencies F = 200, 150, 100, 50, 20, 10. We have used this choice for all studied cases.


99

0.9

|˜ p|y=0

0.7 0.5 0.3 0.1

_ 0.1 200 250 300 350 400 450 500 550 600

Re Figure 4.2: Comparison of the amplitude of the adjoint wall pressure values using APSE with the predictions of adjoint parallel theory .

Chapter 4. Receptivity of Blasius and Falkner–Skan Boundary Layers 1.4

(a)

100

β = 0.1

o

0.05 o

ymax

|˜ umax |

β = 0.1

1.2

60 o o

40

1.0

1000

1400

1800

o

0.8

0

o oo

1000

2200

Re

1400

Re

(c)

o 0

0.02 oo

0.4 1000

1400

1800

0.03

o

o o

0.02

0.05 o

0.02 0 oo

x 10 0.5 0

1400

1800

-3

(f)

o

40

0.05

o

o

1

0.07

o

2200

Re

β = 0.1

50

0.07

o

1000

β = 0.1

o

o

(d)

o

2200

Re

2200

0.01

αi

θ = tan−1 (β/αr )

(e)

Re

0.05

o

o o

60

¯ ¯ ¯ ∂ u˜ ¯ ¯ ∂y ¯

o

o

0.8

y=0

0.07

o o

1

|˜ p|y=0

o o

β = 0.1

1.2

1800

β = 0.1

0.04

1.6

0.05 0.02 oo

o

o

0

oo oo

0.07 o

0.02 oo

20

2

(b)

0.07

80

0

100

0.05 0.07

0.02 0

2 3

30

4 20 o

o

500

1000

1500

2000

Re

0.02

2500

3000

5

1000

1400

1800

Re

2200

2600

Figure 4.3: Receptivity predictions based on APSE for three–dimensional disturbances for various values of the spanwise wavenumber β, in the Blasius boundary layer for frequency F = 20; (a) the maximum receptivity to a streamwise point force; (b) the distance from the wall for maximum receptivity to a streamwise point force; (c) the receptivity due to wall-normal suction/blowing at the wall; (d) the receptivity to streamwise velocity disturbances at the wall; (e) the oblique angle θ; (f ) the growth rates αi . For Figures (a, b, c, d): APSE results, local parallel theory, ◦ denotes first and second neutral points.


20

1.6 50

(a)

100

150

15

10

20

1.4

o

200 o

10

1.2

ymax

|˜ umax |

101

o o

o o

ooo

o 50 20 10

o

o

5

(b)

o

1

o

200 o 150 o 100 o

o

0.8

o o

0 0

1000

2000

3000

0.6 0

4000

1000

2000

Re 1

y=0

0.015

Re

o

0 0

0.02

¯ ¯ ¯ ∂ u˜ ¯ ¯ ∂y ¯

o

0.01

o

0.6 o o o

o 100 o o150 200

o 50

1000

(d)

0.025 ooo

o

10

20

1

|˜ p|y=0

0.8

0.2

4000

0.03

(c)

0.4

3000

Re

Re

3000

4000

o

o o

0.005

2000

o

0 0

o 200 150 100

o

1000

50

10

20

2000

3000

4000

Re

Figure 4.4: Receptivity of the Falkner-Skan boundary layer with an adverse pressure gradient, m = −0.02; (a) maximum receptivity to a streamwise point force; (b) the distance from the wall for maximum receptivity to a streamwise point force; (c) receptivity due to wall-normal suction/blowing at the wall; (d) receptivity to streamwise velocity disturbances at the wall: APSE, local parallel theory.


40

(a)

o

30 o

20

200

50

100 150

10

o ooo o

0 0

20

1

200

(b)

oo 150 o o 100 o 50

o

ymax

|˜ umax |

1.2

10

o20

0.8

10 o

o

o o

1000

o

0.6

o

2000

3000

0.4 0

4000

1000

2000

Re 0.035

(c) y=0 o o

0.6

150

10 o

o o20

o 200 o o o 100 o 50

0.2 0

1000

2000

Re

3000

4000

Re

0.8

¯ ¯ ¯ ∂ u˜ ¯ ¯ ∂y ¯

1

1

|˜ p|y=0

4000

(d)

0.03

1.2

0.4

3000

Re

1.6 1.4

102

o

0.025 200 oo o 150

0.02

o o

o

100

o 50

0.015 0.01 0

o 20

1000

2000

10 o

3000

4000

Re

Figure 4.5: Receptivity of the Falkner-Skan boundary layer with a favorable pressure gradient, m = 0.02; (a) maximum receptivity to a streamwise point force; (b) distance from the wall for maximum receptivity to a streamwise point force; (c) receptivity due to wall-normal suction/blowing at the wall, (d) receptivity to streamwise velocity disturbances at the wall: APSE, local parallel theory.


103

0.8

(a)

0.7 0.6

|˜ p|y=0

0.5 0.4 0.3 0.2 0.1 200

300

400

500

600

700

800

900

Re 0.04

(b)

Re

1

¯ ¯ ¯ ∂ u˜ ¯ ¯ ∂y ¯

y=0

0.035 0.03 0.025 0.02 0.015 0.01 0.005 200

300

400 500

600

Re

700

800 900

Figure 4.6: Receptivity to wall-normal (a) and streamwise (b) velocity disturbances at the wall for F = 100 based on APSE: favorable pressure gradient (m = 0.02), Blasius, adverse pressure gradient (m = −0.02).


160

1.4

(a) β = 0.1

120

(b)

β = 0.1 o

0.07

0.07

1.2

0.05 0.02

o o

80

0.05 o

o

oo

ymax

|˜ umax |

104

o

0

1 o

40

0.02 oo

o o oo

0.8

0

oooo

0

800

1200

1600

2000

Re

2400

600

1000

1400

Re

1800

2200

2600

0.04

(c)

1.4

β = 0.1

0.03

o o o

β = 0.1 o

o o

0.02

Re

0.05 o

o

0.6

o o

y=0

¯ ¯ ¯ ∂ u˜ ¯ ¯ ∂y ¯

o

o

1

|˜ p|y=0

o

1

(d)

o

0.07

o

0.07

0.02 o

0.01

0.05

0.02 oo

0.2

600

1000

1400

1800

0

2200

Re

0 oo

0

2600

x 10

(e)

o

0.07

40

1800

2200

Re

2600

β = 0.1

0.05 0.07

0.02

(f) 0

2

0.05

o

o

1400

3

1 o

o

1000

0

β = 0.1

o

αi


60

600

3 4 5

20

o

o

0.02

6 7

0

500

1000

1500

Re

2000

2500

3000

600

1000

1400

1800

Re

2200

2600

Figure 4.7: Receptivity of three–dimensional disturbances in the Falkner-Skan boundary layer with adverse pressure gradient, m = −0.02, for frequency F = 20, and various values of spanwise wave number β; (a) maximum receptivity to a streamwise point force; (b) distance from the wall for maximum receptivity to a streamwise point force; (c) receptivity due to wall-normal suction/blowing at the wall; (d) receptivity to streamwise velocity disturbances at the wall; (e) the oblique angle θ; (f ) the growth rates αi . For Figures (a, b, c, d) : APSE, local parallel theory.

Chapter 4. Receptivity of Blasius and Falkner–Skan Boundary Layers 60

(a)

1

β = 0.1 0.07 o

0.05

o

0.9

o

0.02 oo

20

0.07

o o

ymax

|˜ umax |

40

(b)

β = 0.1

o

o

105

0.8

0.05

o

0.02 o o 0

o

0 o o o

oo

oo

0.7

0 1200

2400

1200

β = 0.1 o

0.04 y=0

o

0.07 o

Re

2000

2400

0.05

o

1

o o 0

(d) β = 0.1 o

o o

o

0.07 o

0.05

0.03

Re

o

1

1600

0.05

(c) o

|˜ p|y=0

2000

Re

¯ ¯ ¯ ∂ u˜ ¯ ¯ ∂y ¯

1.5

1600

0.02

o

o o 0 0.02

0.5

oo

0.02 o o

1200

1600

Re

2000

0.01

2400

1200

1600

Re

2000

2400

-3

x 10

50 40

o

o

0

β = 0.1

β = 0.1

(e) 0.07

o

o

o

o

0.07

0.05

0.02

0

1

αi


60

0.05

30

(f)

2 3

20

0.02 o

o

1200

1600

2000

Re

2400

4 2800

1200

1600

2000

2400

Re

Figure 4.8: Receptivity of three–dimensional disturbances in the Falkner-Skan boundary layer with a favorable pressure gradient, m = 0.02, for frequency F = 20, and various values of spanwise wave number β; (a) maximum receptivity to a streamwise point force; (b) distance from the wall for maximum receptivity to a streamwise point force; (c) receptivity due to wall-normal suction/blowing at the wall; (d) receptivity to streamwise velocity disturbances at the wall; (e) the oblique angle θ; (f ) the growth rates αi . For Figures (a, b, c, d) : APSE, local parallel theory.


106

Kinetic Energy

4

3

2

1

0 300 350

400

450

500

550

600

650

700

Re Figure 4.9: Adjoint predictions compared with the LNS calculations for acoustic receptivity subject to a Gaussian roughness element located at Re=330, F = 150 and Re 0 = 200. Roughness element’s width is comparable to the instability wavelength: APSE predictions, LNS simulations.


107

11 10

(a)

200 o o o

0.07

|Λ|

o

150 100

8

o

o o

0.06

50

o

o

o

20

o

7

o

6

o

50

5

o

o

4

100

10

3

150 200

o

o o

o

o

o

o

2

0.04 0.4

0.6

0.8

1

1.2

1.4

F Re

1.6

3/2

1.8

2

2.2

0.4

2.4

0.6

0.8

1

1.2

1.4

F Re

1.6

3/2

1.8

2

2.2

2.4

20

0.2

β = 0.1

β = 0.1

0.16

o

0.04 0.5

o

0.07

o

0.12 oo o

(d)

0.07

16

o

|˜ ph |y=0

(c)

|Λ|

20

o

0.05

0.08

(b)

10 o

9

|˜ ph |y=0

0.08

0.05

0.02

o

12

0.05 o o o

8

o o 0

oo

o o

o

0.02 0

1

1.5

F Re 3/2

2

2.5

4 0.5

1

oo

1.5

F Re 3/2

2

2.5

Figure 4.10: Acoustic receptivity for the Blasius boundary layer; (a) receptivity due to the point surface roughness for two–dimensional instabilities; (b) receptivity due to the point steady wall-normal blowing source for two–dimensional instabilities; (c) receptivity due to the point surface roughness for oblique instabilities with various values of β for frequency F = 20; (d) receptivity due to the point steady wall-normal blowing source for oblique instabilities with various values of β for frequency F = 20: APSE, local parallel theory, ◦ denotes first and second neutral points.


108

10 0.06

o

o

o

20

7

200 150

o

0.045

|Λ|

o

o o

o 100

0.04

10

0.03

50

4

o

100 o 150 o 200 o

oo o

o

2

2.2

0 0.6

0.8

1

1.2

1.4

1.6

1.8

F Re 3/2

2

2.2

0.4

2.4

0.6

0.8

1

1.2

1.4

1.6

F Re 3/2

1.8

14

0.12

β = 0.1 β = 0.1

(d)

o

o

10 o

0.07

o o o

|˜ ph |y=0

(c)

0.08

|Λ|

5

1 0.4

0.04

o

2

o

0.025

0.06

20

6

3

50

0.035

(b)

o

8

|˜ ph |y=0

o

0.05

0.1

10

9

(a)

0.055

0.05

o

o o

0.02

0

oo o

o

o o

6

0.07

o

0.05

2

oo 0

oo

0.02 0.5

1

1.5

F Re 3/2

2

2.5

0

1

2

F Re 3/2

0.02

3

Figure 4.11: Acoustic receptivity for the Falkner-Skan boundary layer with an adverse pressure gradient, m = −0.02; (a) receptivity due to the point surface roughness for two–dimensional instabilities; (b) receptivity due to the point steady wall-normal blowing source for two–dimensional instabilities; (c) receptivity due to the point surface roughness for oblique instabilities with various values of β for frequency F = 20; (d) receptivity due to the point steady wall-normal blowing source for oblique instabilities with various values of β for frequency F = 20: APSE results, local parallel theory, (o) denotes first and second neutral points, respectively.


12

0.12

11 0.11

(a)

o 0.08

50

o

o 20 o 10

o

|˜ ph |y=0

|Λ|

0.09

o o

20

8

o 6

50 o

4

0.05 0.8

1

1.2

1.4

1.6

1.8

F Re 3/2

2

2.2

2

2.4

o

150

0.4

0.6

0.8

o o

1

1.2

o

1.4

1.6

F Re 3/2

200 1.8

2

2.2

2.4

25

0.25

(c)

0.07

β = 0.1

(d)

o

0.2

20 0.05

0.15 o o

0.02

o o

oo

|˜ ph |y=0

o o

0.1

o

100

3 0.6

o o

7

5

o

0.06

0.4

(b)

o

9

200 150 100 o o

0.07

10

10

0.1

|Λ|

109

β = 0.1 o

15

0.05 o

o o

10

o

0 o

0

0.05 0.8

1.2

1.6

F Re

3/2

2

2.4

0.07 o

5

o o

0.8

0.02 oo

1.2

1.6

F Re

3/2

2

2.4

Figure 4.12: Acoustic receptivity for the Falkner-Skan boundary layer with a favorable pressure gradient, m = 0.02; (a) receptivity due to the point surface roughness for two–dimensional instabilities; (b) receptivity due to the point steady wall-normal blowing source for two–dimensional instabilities; (c) receptivity due to the point surface roughness for oblique instabilities with various values of β for frequency F = 20; (d) receptivity due to the point steady wall-normal blowing source for oblique instabilities with various values of β for frequency F = 20: APSE, local parallel theory, ◦ denotes first and second neutral points.

Chapter 5 Receptivity of Three-Dimensional Boundary Layers In recent years, there have been a variety of approaches used to analyze receptivity of the three-dimensional boundary layers based on Finite Reynolds Number Theory (FRNT) (see References 13, 25), DNS results by Collis [21], and adjoint based methods (see Reference 43). The methods based on FRNT utilized a local parallel-flow approximation and therefore the effect of the growing mean flow on receptivity is not accounted for. The main advantage of the FRNT method is that it is much more efficient than DNS. However, due to recent advances in computing technology, DNS has become less expensive and the the full linearized Navier–Stokes equations can be discretized and solved in order to predict receptivity for nonparallel flows (see References 8, 21, 54, 56, 83). While DNS studies have played an important role in both validating theoretical predictions, as well as identifying weaknesses in the FRNT, they are still computationally expensive and cannot efficiently cover the large receptivity parameter space. Recently, Bertolotti [10] introduced an alternative approach based on FRNT that represents the nonparallel boundary layer growth using a Taylor-series expansion of the mean flow. Weakly nonparallel receptivity predictions are obtained by application of the residue theorem to a series of inhomogeneous Orr–Sommerfeld-type problems. While this method is computationally efficient and explicitly evaluates nonparallel contributions to the eigenmode response, it is limited to surface disturbances that

110

Chapter 5. Receptivity of Three-Dimensional Boundary Layers

111

have a streamwise extent much smaller than the length scale of mean-flow nonparallelism. Fortunately, these conditions are well satisfied for the local receptivity problems in three-dimensional boundary layers as presented in Reference 10, which shows excellent agreement with both DNS and the DLR Prinzip experiment [27]. While Bertolotti’s [10] method provides valuable insight, as formulated, it requires the solution of different forced Orr–Sommerfeld problems for each disturbance type and location, thereby limiting its ability to easily cover the large receptivity parameter space. An alternative approach to studying receptivity in nonparallel boundary layers is through adjoint analysis. Adjoint methods have the advantage of naturally incorporating nonparallel effects within the receptivity predictions. As opposed to the methods based on the Orr-Sommerfeld equation, they are easier to implement, more economical and can give receptivity information for a wide range of parameters. Similar to the work of Hill [43], we apply adjoint analysis to study receptivity characteristics of nonparallel boundary layers. While the work of Hill [43] has laid the foundation for efficient prediction of the receptivity of nonparallel boundary layers, it has neither explored the receptivity characteristics of nonparallel boundary layers for various sets of parameters, nor analyzed receptivity in the highly nonparallel boundary layers. While Hill has done some preliminary calculations for receptivity prediction of a swept wing flow, it is not clear in that work whether nonparallel effects are important in that particular boundary layer. Further, Hill’s adjoint predictions deviate from direct PSE predictions by about 7% close to the leading edge. Not only it is not clear from Hill’s work whether adjoint methods can be used to predict the receptivity of nonparallel flows, the PSE calculations near the leading edge are not compared with DNS or experimental results for these types of flows. Our work is distinguished from work of Hill [43] in several ways:


112

• We confirm the validity of the APSE methods for receptivity prediction in threedimensional nonparallel boundary layers by comparing them with LNS and ANS methods. • We analyze the validity of PSE calculations for strongly nonparallel flows. • We resolve some implementational issues that must be addressed when constructing adjoint solutions for these flows. • We explore the effects of oblique disturbances. • We evaluate frequency effects. • We evaluate pressure gradient effects. • We provide a comparison against parallel theory predictions to highlight the influence of nonparallel effects on receptivity • We analyze a variety of different three-dimensional boundary layers. We begin this chapter by first considering receptivity of the swept Hiemenz flow discussed in Bertolotti [10]. To validate our adjoint method, we compare parallel and nonparallel receptivity predictions based on the ANS method with the receptivity results of Bertolotti [10]. We further validate the ANS approach by comparing ANS predictions with direct LNS simulations for a roughness element in the form of a Gaussian bump at several locations in the flow. Finally, we compare the ANS and APSE methods to show that APSE methods can be successfully used to predict receptivity in nonparallel flows. After validating the adjoint approach for the swept Hiemenz flow, we consider receptivity of the Falkner–Skan–Cooke family of flows. For these flows we study the effects of pressure gradients, disturbance wave angle and disturbance frequency on receptivity; therefore, fulfilling one of the main goals of this


113

research: Efficient receptivity predictions of highly nonparallel boundary layers for a variety of sources and flow parameters. Finally, to show that ANS and APSE methods can be applied successfully to more physically realistic flows, we compare the adjoint approach with direct LNS calculations for the receptivity of the boundary layer over a swept parabolic cylinder.

5.1

Swept Hiemenz flow

We begin the discussion of adjoint methods for three-dimensional boundary layers by considering the well known swept Hiemenz flow. Bertolotti [10], using his modified FRNT method, found that nonparallel effects are significant for this flow not only in the close proximity to the attachment line, but in the region near and downstream of the first neutral point. To validate our adjoint analysis we begin by first comparing parallel theory receptivity predictions based on the AOSE with the similar results of Bertolotti [10] shown in Figure 5.1. Here, receptivity is presented for an impulse suction and blowing source. We see from Figure 5.1, that adjoint analysis based on parallel theory agrees well with Bertolotti’s data obtained using the FRNT method. To validate the receptivity predictions of the nonparallel adjoint analysis, we compare the amplitudes of the efficiency function |Λr | defined by (2.79)1 for the impulse roughness sources in Figure 5.2. The receptivity results are obtained using APSE. The efficiency function Λr indicates the amplitude and phase of the instability wave measured at the first neutral point due to an impulse roughness source. As Figure 5.2 indicates, the agreement of our adjoint method with Bertolotti’s results is quite good overall, but differs more near the first branch where Bertolotti’s approximate method underpredicts the correct response. 1

In Chapter 2 the efficiency function is denoted by Λ. Here we use subscript r to emphasize that this efficiency function is for wall roughness.


114

To further validate our adjoint methods, we conducted a series of LNS runs for Gaussian sources given by exp[(x − xl )2 /σ 2 ) log(0.01)], where xl is the location of the source and σ is the source width. Figure 5.3 shows the LNS simulations for sources with σ = 4.0 for xl = 120, 140, 160, 170 and 180 along with ANS predictions. We see that for all the sources considered, the ANS predictions give perfect agreement with the direct LNS simulations. Note also that the receptivity to a source located at xl = 160 is smaller than the receptivity to a source located at xl = 180, even though the amplitude of the efficiency function is larger for xl = 160 then for xl = 180, as can be seen from Figure 5.2. This can be understood by looking at the phase of the efficiency function plotted in Figure 5.4(a). The complex amplitude of the instability wave is obtained by integrating the product of the complex efficiency function and the roughness source, which in Figure 5.4(a) is indicated by the thicker line. As can be seen from Figure 5.4(a), the roughness source at xl = 160 is “more out of phase” with the efficiency function than the source at xl = 180. This is more clearly seen in Figures 5.4(b, c) where, for roughness sources located at xl = 160 and 180, the amplitude of the instability wave is given by the area under the curve plotted in Figures 5.4(b) and (c). Here, the area under the solid curve corresponds to the real component of the receptivity amplitude and the area under the dashed curve corresponds to the imaginary component. We see that the area under the solid curve (Figure 5.4(b)) corresponding to the amplitude due to a source at xl = 160 is smaller than that for a source located at xl = 180. Finally, to validate the APSE approach, Figure 5.5 compares the adjoint kinetic energy growth rate from ANS and APSE. We see that both methods gives identical adjoint predictions, even in the highly nonparallel regions (for x = 50, for example). This indicates that APSE is a viable method even in highly nonparallel regions of the flow.


5.2

115

Falkner–Skan–Cooke Boundary Layer

We now apply our global adjoint methods to study the receptivity characteristics of Falkner–Skan–Cooke boundary layers. We start by evaluating nonparallel effects. To indicate the portion of the wedge that we consider for receptivity analysis, consider Figures 5.6(a) where we plot a schematic of the problem and Figure 5.6(b) which shows the growth rate for a crossflow instability in a Falkner–Skan–Cooke boundary layer. Here we consider the evolution of the crossflow instability wave with spanwise wavenumber β = 0.2 in the flow with Hartree parameter βH = 0.65. The neutral points are indicated by the zero crossing of the growth rate. We primarily consider receptivity of crossflow instabilities to environmental excitations (controls) located prior to and in the vicinity of the first neutral point since this is the region where nonparallel effects are important and where excitations are most effective in generating large amplitude instability waves. To see that in these regions the boundary layer is highly nonparallel, Figure 5.7 compares boundary layer profiles for the Falkner– Skan–Cooke boundary layer near the first neutral point of the crossflow instability with a Blasius boundary layer profile at the TS wave first neutral point (here we have used a TS wave with F = 150). In Figure 5.7 we took the difference between boundary layer profiles located one instability wavelength apart from each other. For both the Falkner–Skan–Cooke and Blasius boundary layer, the first profile was chosen at there respective first neutral points. The Falkner–Skan–Cooke flow profiles are taken in the coordinate system aligned with the local streamline. From Figures 5.7(a) and (b) one can conclude that the boundary layer growth over one instability wavelength is about twice as large in the Falkner–Skan–Cooke boundary layer as in the Blasius boundary layer. In order to access the accurate of the adjoint predictions, Figure 5.8 compares


116

APSE predictions with the direct LNS results obtained using steady wall-normal suction and blowing excitation with spanwise wavenumber β = 0.2 for the Falkner–Skan– Cooke flow with βH = 0.65 and global sweep angle θs = 45o . We see that the kinetic energy of the unstable mode predicted using APSE is in excellent agreement with the direct LNS method. Just as in the case for the swept Hiemenz flow, APSE when constructed as an approximation to the ANS, gives accurate receptivity predictions for crossflow instabilities in Falkner–Skan–Cooke boundary layers. Note, however, that as mentioned in Section 3.5.2, the APSE solution may be less accurate if it is constructed as the adjoint to the PSE since the PSE solutions typically have large transients at inflow. To emphasize this point, consider the kinetic energy growth rates for a crossflow instability and its adjoint in Figure 5.9. Clearly, there is a large transient in the PSE solution due to approximate inflow data obtained using parallel theory. Moreover, since the flow is highly nonparallel near the inflow region, the parallel theory eigenfunction is a poor approximation to the inflow conditions for the PSE. Further, from the high amplitudes of the adjoint solution near the inflow, we see that the PSE solution is highly sensitive to small changes in the inflow conditions. Thus, approximation errors can have a significant effect on the quality of the PSE solution near the inflow. In contrast, the adjoint solution does not suffer as severely from the approximate adjoint eigenfunction imposed on the outflow, since near the outflow, the boundary layer is approximately parallel so that the parallel theory eigenfunction is a good approximation to the APSE solution. From Figure 5.9 one can see that the adjoint kinetic energy growth rate does not exhibit a transient, the adjoint solution has a well defined neutral point, and the adjoint field can be effectively used to predict receptivity in highly nonparallel regions. To obtain useful adjoint solutions, one has to choose a normalization that does not require prior knowledge of the PSE solution. This ensures that transients present


117

in PSE do not corrupt the APSE solution. For example, choosing ke-normalization, yields accurate adjoint solutions, whereas choosing α-normalization as the adjoint normalization criteria yields incorrect solutions at the inflow (see Figure 5.10 for comparison of two normalizations). It is possible that using the incorrect normalization condition prevented Hill [43] from obtaining accurate adjoint results near the leading edge of the swept wing. Now that we have established the validity of the APSE approach for highly nonparallel boundary layers, we next consider how receptivity varies with changes in parameters.

5.2.1

Steady Three-dimensional Crossflow Instabilities

We begin by exploring the dependence of receptivity on the spanwise wavenumber. Unless otherwise specified, all the receptivity results in the remainder of this section are computing using APSE for the flow with βH = 0.65, global sweep angle θs = 45o , and Re 0 = 100. We start by plotting kinetic energy growth rates for disturbances with β = 0.101, β = 0.1 and β = 0.099 in Figure 5.11. We deliberately choose spanwise wavenumbers to vary within 1% to identify the marginal effects of changes in spanwise wavenumber on receptivity. As expected, we observe transients at the inflow. The transients propagate downstream until about x = 600. Downstream of that location, the instability wave with β = 0.101 has the largest growth rate as compared to the β = 0.1 and β = 0.099 modes. No transients are present, however, in the adjoint growth rates shown in Figure 5.12(b). Since the β = 0.101 mode has the largest growth rate we expect it to have the largest amplitude at some downstream location (say at Re = 260), unless other crossflow modes have much higher receptivity then the β = 0.101 mode. Figure 5.12(a) shows the adjoint streamwise wavenumber αr associated with each spanwise wavenumber β as a function of nondimensionalized


118

streamwise coordinate x. Note that the instability with β = 0.101 has the highest streamwise wavenumber αr . This indicates that the total wavenumber k =

q

αr2 + β 2

increases with increasing spanwise wavenumbers β. On the other hand, the wave angle θ = tan−1 (β/αr ) decreases as β increases (not shown). In order to study the receptivity of these modes, we choose a particular normalization location, Re = 260, where the receptivity amplitude of all three instability modes is evaluated (see Figure 5.13(a) for a plot of umax and the normalization location). Figure 5.13(b) shows the receptivity for the three different spanwise wavenumbers due to a momentum point source. For example, a point source positioned at Re = 100 excites a disturbance of amplitude am ≈ 1.4 at Re = 260 for β = 0.0099. The same source will excite a disturbance of am ≈ 1.45 for the β = 0.101 mode. From Figure 5.13(b) we can immediately conclude that stability effects outweigh receptivity effects and instability waves with larger growth rates (such as β = 0.101 mode), reach larger amplitudes at Re = 260. Receptivity to wall-normal steady blowing is shown in Figure 5.14(a). Figure 5.14(b) gives receptivity to wall tangential excitations in the streamwise direction, whereas Figure 5.14 (c) gives receptivity to tangential excitations in the spanwise direction. Finally, Figure 5.14(d) gives receptivity to a wall roughness source. The efficiency function Λr is defined in Chapter 2 by equation (2.78). In all the cases, stability effects outweigh receptivity effects for the crossflow instability. Higher amplitudes are achieved by more unstable modes. The amplitudes of the modes at Re = 260 are largely unaffected by the relatively minor differences in receptivity, unless environmental sources are located in close proximity to Re = 260. To further study the receptivity of crossflow modes with slightly different spanwise wave numbers, we consider the difference in absolute values of adjoint profiles for two local Reynolds numbers Re = 71 and Re = 260. From Figures 5.15, 5.16,


119

and 5.18 we observe that for |˜ u|, the streamwise adjoint wall shear stress, and |w| ˜ receptivity is higher for instability waves with β = 0.101 at both Reynolds numbers. Note, however, that the situation is slightly different for the |˜ v | component of the adjoint field as shown in Figure 5.17. At Re = 260, the β = 0.101 mode is more receptive to wall-normal momentum sources located near the wall than the β = 0.099 mode; however, wall normal momentum sources located further from the wall, excite the β = 0.099 mode with higher amplitudes. Also, examining differences in the amplitudes of adjoint pressure in Figure 5.19, we find that for wall-normal steady blowing sources, the β = 0.0099 mode is more receptive than the β = 0.101 mode at Re = 260. However, due to its slower growth, the β = 0.0099 mode has a smaller amplitude then the β = 0.101 mode when excited at Re = 71. Note, the small oscillations seen in Figure 5.19(a). They are caused by minute numerical errors which are on a much smaller scale then the absolute value of the adjoint pressure shown in Figure 5.19(b). In order to better understand the effects of spanwise wavenumber on the crossflow instability, we document receptivity for β = 0.1, 0.2, 0.3 and 0.4. However, before we begin discussing receptivity we first consider their stability characteristics as they relate to the receptivity characteristics of these modes. To illustrate the stability characteristics of the crossflow modes, we plot in Figure 5.20 the imaginary component of the streamwise wavenumber αi which corresponds to the negative of the kinetic energy growth rate. We notice that the crossflow mode with β = 0.2 tends to be more unstable than the β = 0.1 and β = 0.3 modes. Meanwhile, the modes β = 0.1 and β = 0.3 tend to have similar growth rates for higher Reynolds numbers. Further, we find from Figure 5.20 that β = 0.4 mode is stable. Also, in Figure 5.20 we plot the adjoint kinetic energy growth rates. The adjoint crossflow mode with β = 0.2 has the largest growth rate upstream, whereas the adjoint modes β = 0.1 and β = 0.3 have


120

similar growth rates for higher Reynolds numbers. It is instructive also to consider the wave angles θ = (180/π) tan−1 (β/αr ) for a variety of spanwise wavenumbers shown in Figure 5.21. We recall that both the Blasius and Falkner–Skan boundary layers are more receptive to excitations with high wave angles then two–dimensional TS waves. Figure 5.21 shows that the larger the spanwise wavenumber, for a given instability wave, the smaller its corresponding wave angle θ. Therefore, for two instabilities with streamwise and spanwise wavenumbers (αr1 , β1 ) and (αr2 , β2 ) we observe the following αr2 β2 < . β1 αr1

(5.1)

when β2 is taken to be larger than β1 . We now consider the receptivity characteristics of this Falkner–Skan–Cooke boundary layer for a variety of spanwise wavenumbers in Figure 5.22. Figure 5.22 (a) shows the maximum amplitude, in y, of the streamwise component of the adjoint velocity computed at every x location in the flow. This adjoint quantity gives receptivity to a point source of streamwise momentum. The y locations where the streamwise component of adjoint velocity achieves its corresponding maximum amplitude are shown in Figure 5.22(b). Placing a streamwise momentum forcing at this location will induce the largest excitation in the flow. Note the small jumps in the curves plotted in Figure 5.22 (b). These jumps are due to the fact that that absolute value of the adjoint streamwise velocity profile is flat near its maximum value and the numerical procedure we used did not have enough precision to locate an exact maximum. By interpolating to a higher resolution mesh in the wall-normal direction the curves could be made smoother. The larger jumps in y for β = 0.2 at about Re = 155 are associated with a new global maximum value of the absolute adjoint streamwise velocity.


121

The receptivity to wall-normal excitations is given by the adjoint wall-pressure shown in Figure 5.22(c), whereas receptivity to streamwise and the spanwise tangential excitations is given by the streamwise and spanwise components of the adjoint wall shear stress shown in Figures 5.22(d) and (e). Finally, receptivity to wall roughness is given by the amplitude of efficiency function |Λr | shown in Figure 5.22(e). All of the ˜ m, U ˆ m ; x) = 1 adjoint quantities are normalized at Re = 260 by requiring that J(U ˆ m (x, y)| = 1. Therefore, these adjoint quantities indicate the amand that maxy |U plitude of an instability wave at Re = 260. For example, from Figure 5.22(c), one can see that for the case of β = 0.1, a localized wall-normal excitation at Re = 100 produces an instability wave of amplitude am = 0.356 at Re = 260. Due to different adjoint growth rates for the crossflow instabilities with different spanwise wavenumbers, we observe competing receptivity and stability effects. For example, since the adjoint growth rate of the β = 0.2 mode is higher than all other modes considered (see Figure 5.20), we expect it to have larger adjoint values in the inflow region. This can be easily observed in Figures 5.22(a)-(f ). To obtain a more insightful perspective on the receptivity of Falkner–Skan–Cooke boundary layers for crossflow instabilities with different spanwise numbers β, consider the adjoint solution normalized at each location x. By normalizing the adjoint in this way, we can explore the receptivity amplitudes of the instability waves at the location of the source. The drawback of this normalization is that since the adjoint is nor˜ m, U ˆ m ; x) = 1 at each location, transients presented in U ˆ m near the malized by J(U inflow introduce artificial transients in the adjoint solution. The receptivity characteristics in this normalization are shown in Figure 5.23. We see in Figure 5.23 (a) that the receptivity to streamwise momentum sources tends to be larger for larger values of spanwise wavenumber, whereas the receptivity for wall-normal excitations is larger for crossflow instabilities with smaller spanwise wavenumbers. Note that similar to


122

the receptivity characteristics of the two–dimensional boundary layers, receptivity is an order of magnitude higher for wall-normal excitations than for tangential excitations. Also, note that receptivity of stationary crossflow instabilities is comparable in amplitude with the receptivity of TS waves in the Blasius boundary layer.

5.2.2

Unsteady Three-dimensional Crossflow Instabilities

To assess the receptivity dependence on the frequency, F = ω × 106 /Re 0 , of the crossflow instability waves, consider first Figure 5.24 which shows the real and imaginary parts of the streamwise wavenumber for the mode with β = 0.2. From Figure 5.24, we find that for larger values of F the instability growth rate decreases and the instability waves become more oblique. This is in agreement with prior results (see e.g., [13]) which show that stationary crossflow modes are more unstable than traveling modes. In order to explore the receptivity of Falkner–Skan–Cooke boundary layer for nonstationary crossflow instabilities, consider Figures 5.25(a)-(d) which are similar to Figures 5.22(a)-(d), with the only difference being that receptivity here is given for unsteady momentum and wall sources. From Figures 5.22(a)-(d), we find that for streamwise momentum excitations, receptivity increases with frequency and this is also true for wall normal and streamwise tangential excitations. The receptivity of stationary and unsteady (F = 150) crossflow instabilities for spanwise wavenumber β = 0.2 is shown in Figures 5.26(a)-(d), where we find that for forced receptivity, unsteady crossflow modes are more receptive to wall boundary and momentum sources.

5.2.3

Pressure Gradient Effects

To illustrate the effects of pressure gradient on the receptivity of crossflow instabilities Figure 5.27(a), (c), (d) shows receptivity to streamwise momentum, wall-normal and streamwise wall tangential point sources respectively at the location of the source.


123

Meanwhile, Figure 5.27(b) shows the maximum amplitude, in y, of the streamwise component of the adjoint velocity computed at every x location in the flow. While it is not as clear as the Falkner–Skan boundary layer, the Falkner–Skan–Cooke boundary layer flow with βH = 0.55 tends to be more receptive to streamwise momentum point sources than the flow with βH = 0.65. Also, from Figure 5.27(c) we find that for more favorable pressure gradients (βH = 0.65), the flow tends to be more receptive to wall-normal excitations downstream of local Reynolds number Re = 110. Figure 5.28, documents the same results, but using the global nondimensionalization with the adjoint solutions normalized at Re = 260. Notice, that due to the high receptivity of the βH = 0.55 mode (see Figure 5.29 for comparison of adjoint kinetic energy growth rates), streamwise momentum sources located downstream of Re ≈ 70 excite larger instability waves for the flow with βH = 0.55 at the location Re = 260. In general, however, due to the higher growth rate the crossflow mode, βH = 0.65 flow tends to have larger amplitudes at Re = 260 for wall excitations of various types.

5.2.4

Importance of Nonparallel Effects

As we have mentioned before, the major benefit of a global adjoint method, either ANS or APSE, is that they efficiently incorporate nonparallel effects. In Figure 5.30 we compare PSE profiles and in Figure 5.31 APSE profiles with corresponding profiles obtained using parallel theory at few different streamwise locations. The profiles are normalized such that maxima of streamwise velocity is equal to one. Note, the disagreement of the state and adjoint y-profiles obtained with OSE and AOSE with the corresponding PSE and APSE predictions. This is a clear indication that the inclusion of nonparallel effects is essential for obtaining accurate receptivity results just as inclusion of these effects is essential for stability analysis.


5.3

124

The Swept Parabolic Cylinder

In this Section we apply our global adjoint approach (i.e. APSE) to investigate receptivity of the three-dimensional boundary layer on a swept parabolic cylinder. This flow gives us the opportunity to explore both nonparallel and surface curvature effects. For all results presented in this section we consider the evolution of the crossflow instability with spanwise wavenumber β = 100 in the boundary-layer on a swept parabolic cylinder with sweep angle θs = 35o and Reynolds number Re = 105 . This test case models one of the cases studied by Collis and Lele [21], although their computations were for compressible flow at a Mach number of 0.8. Figure 5.32 compares APSE predictions with direct PSE results for a steady blowing source on the wall boundary. Figures 5.32(a) and (b) show receptivity results without curvature for sources located at s = 1.4 and s = 0.3, while Figure 5.32 shows receptivity with surface curvature effect included for a blowing source at s = 0.3. We find that, in all cases, APSE predictions are in excellent agreement with direct LNS results. To see how the presence of convex surface curvature affects the state and adjoint solutions, Figure 5.33 shows the imaginary parts of streamwise wavenumber with and without curvature for both state and adjoint solutions. Note that the presence of convex surface curvature decreases the growth rate of the crossflow instability and similar results have been reported by Malik and Balakumar [57], Masad and Malik [58] and Collis and Lele [21]. Meanwhile, the wave angle θ is slightly smaller when curvature is accounted for, although the difference is small (Figure 5.34). The receptivity to forced momentum, wall normal, and wall tangential excitations are documented in Figure 5.35 with and without curvature. Figure 5.35(a) shows that receptivity to a momentum point source at the source location is larger when curvature is included; however, Figures 5.35(c) and (d) show that receptivity to wall


125

normal and streamwise tangential excitations is smaller with curvature. Similar to the Falkner–Skan–Cooke boundary layers, receptivity to wall normal disturbances is higher in general than the receptivity to tangential excitations. To study the effects of convex surface curvature on receptivity to localized wall roughness, consider Figure 5.36 which shows the receptivity efficiency function obtained with and without curvature. Similar to the results of Collis and Lele [16], we find that curvature increases receptivity to roughness. The difference however is small, no larger then 2%. By comparing our results with the compressible results of Collis and Lele [16] at M = 0.8 in Figure 5.37, we find that receptivity is larger for the incompressible flows by a factor of 5 ∼ 10. At the same time we observe that in the incompressible flow the growth rate of the crossflow mode is smaller than in the M = 0.8 compressible flow [16]2 . Recall that similar trends have been observed before that flows tend to be more receptivity to more stable modes.

5.4

Discussion of Results

We conclude this Chapter by reiterating some of the most important points. First, we showed that the ANS and APSE can be used to give accurate receptivity predictions for a variety of nonparallel three–dimensional boundary-layer flows, such as swept Hiemenz, Falkner–Skan–Cooke and swept parabolic cylinder flows. We also pointed out the issues that must be accounted for when constructing APSE solutions. We used APSE to explore the dependence of receptivity on spanwise wavenumber, and frequency of crossflow modes. In particular we found that for steady crossflow instabilities, stability effects typically dominate receptivity. For unsteady crossflow instabilities, we observed that receptivity to a point momentum source as well as 2

Please note that in order to compare the amplitudes of the efficiency function with the one given √ by Collis [16], we have multiplied our results by 2π


126

streamwise tangential excitations is larger for higher frequency modes, but receptivity to wall normal excitations is larger for the lower frequency modes. Finally, we investigated pressure gradient effects, evaluated nonparallel effects, and demonstrated how adjoint methods can be applied in the context of more complex flows, such as the swept parabolic cylinder. For the parabolic cylinder, we find that convex surface curvature enhances receptivity to momentum sources as well as to wall roughness, in agreement with prior results. On the other hand, receptivity to wall normal and wall streamwise tangential excitations is reduced by convex surface curvature. Comparing amplitudes of receptivity efficiency functions with results of Collis [16] we find that our incompressible flow is more receptive to wall roughness, however, the resulting crossflow modes are more stable then the corresponding modes in the compressible flow.


127

5 4

|p~wall |

3 2 1 neutral point

0 -1

20

40

60

80

100

120

140

160

180

200

x

Figure 5.1: Receptivity to the impulse wall suction source based on the parallel theory prediction for Hiemenz flow. Bertolotti results, ◦ adjoint predictions

-3 14 x 10 neutral point

12 10

Efficiency | Λ r |

8 6 4 2 0 -2 0

50

100

150

200

250

300

350

X

Figure 5.2: Efficiency function to the wall roughness. ◦ Bertolotti results, APSE predictions


128


0.12 0.1 0.08 0.06 0.04 0.02 0

100 120 140 160 180 200 220 240 260 280 300 x

Figure 5.3: ANS predictions compared with the LNS calculations for wall roughness sources in the form of Gaussian bumps located at xl = 120, 140, 160, 170 and 180.


129

Efficiency function Λr

(a) 0.01 0

-0.01 140 150 160 170 180 190 x

ΛrΗ(x)

15 x 10

-3

(b)

-3 4 x 10

10

0

5

-4

0

-8

(c)

-12 -5 170 174 178 182 186 190 150 154 158 162 166 170 x x

Figure 5.4: The effect of the phase of efficiency function on receptivity prediction. (a) Re{Λ}, Im{Λ}, thick solid lines are the roughness sources scaled by 0.01 at xl = 160 and 180; (b) efficiency function Λ multiplied by the roughness source H(x) at location xl = 160, Re{ΛH(x)}, Im{ΛH(x)}. The area under the solid and dashed curves corresponds to the real and imaginary parts of the receptivity amplitude respectively; (c) similar to (b), efficiency function Λ multiplied by the roughness source H(x) at location xl = 180

Adjoint Kinetic Energy Growth Rate


130

0.4 APSE ANS

0.3

0.2

0.1

0

50

60

70

80

90

100

110

120

X

Figure 5.5: Comparison of the adjoint kinetic energy growth rate obtained using ANS and APSE methods.


131

(a)


x 10

-3

(b)

6 4 2 0 -2 50

100 150 200 250 300 Re

Figure 5.6: The FSC flow: (a) Schematic of the excitation near the leading edge; (b) the growth rate of kinetic energy for the crossflow instability. The grey region corresponds to the region where boundary layers are highly nonparallel.

∆u

(a)

∆u

|

0.04

-0.01

132

(b)

∆u, ∆w

|

-0.02

|

|

0.06

0 |

∆u = u(xn) - u(xn+λ)


|

0.02 0

|

∆w

-0.03 0

10

20 30 y

40

50

-0.02

0

5

10 15 20 25 30 y

|

Figure 5.7: The nonparallel effects of three–dimensional boundary layers. (a) the growth of the Blasius boundary layer over one instability wavelength near the first neutral point. The considered instability is the TS wave with F = 150; (b) the growth of the Falkner–Skan–Cooke boundary layer over one instability wavelength near the first neutral point. The mean flow profiles are computed near the first neutral point of the crossflow instability. The Hartree parameter βH = 0.65, global sweep angle θs = 45o , Re 0 = 100. and the instability spanwise wavenumber is β = 0.2.


133

3500 Suction and blowing

(a)

APSE


3000 2500 2000 1500 1000 500 0

0

100

200

300

Re

400

500

600

700

Kinetic Energy

(b)

20

10

0 Re = 65

Figure 5.8: Comparison of APSE and LNS predictions for Falkner–Skan–Cooke boundary layer. (a) APSE predictions compared with the LNS calculations for instability with spanwise wave number β = 0.2, with Reynolds number R0 = 100, for a suction and blowing source at Re = 65 in the Falkner–Skan–Cooke boundary layer with βH = 0.65, global sweep angle θs = 45o , and Re 0 = 100. (b) the zoomed view near the source location.

Instability and Adjoint Ke Growth Rates


134

0.01 Instability Ke growth rate Adjoint Ke growth rate

0.005

0

-0.005

-0.01 0

100

200

300

400

x

500

600

700

800

900

Figure 5.9: Growth rates of crossflow instabilities and corresponding adjoint modes in Falkner–Skan–Cooke boundary layer with βH = 0.65, Re 0 = 100 and global sweep angle θs = 45o , for crossflow instability with spanwise wavenumber β = 0.2.

30 Adjoint Ke with ke-normalization Adjoint Ke with a-normalization

Adjoint Ke

25 20 15 10 5 0

0

100 200 300 400 500 600 700 800 900 1000 x

Figure 5.10: Comparison of adjoint kinetic energy obtained using ke-normalization with the one obtained using α-normalization. The results are for crossflow instability with spanwise wavenumber β = 0.2 in in Falkner–Skan–Cooke boundary layer with βH = 0.65, global sweep angle θs = 45o , and Re 0 = 100.


Kinetic Energy Growth Rates

0.002

135

β=0.101 β=0.1

0.001

β=0.099

0

-0.001

-0.002

0

100

200

300

400

x

500

600

700

800

900

Figure 5.11: Kinetic energy growth rates for crossflow instabilities in Falkner–Skan– Cooke boundary layer with βH = 0.65, global sweep angle θs = 45o , and Re 0 = 100.

0.09

0.001

β=0.101 β=0.1 β=0.099

(a)

Adjoint Ke Growth Rates

Adjoint Wavenumber αr

0.1

0.08 0.07 0.06 0.05

0

-0.0005 -0.001 -0.0015

0.04 0.03

β=0.101 β=0.1 β=0.099

(b)

0.0005

-0.002 200

400 x

600

800

0

200

400 x

600

800

Figure 5.12: (a) adjoint spanwise wavenumber; (b) kinetic energy growth rates for crossflow instabilities in Falkner–Skan–Cooke boundary layer with βH = 0.65, global sweep angle θs = 45o , and Re 0 = 100.


136

2

β=0.101 β=0.1 β=0.099

(a)

^ |u max |

1.5

1

|

Q

|

normalization location

0.5 0

50

100

150

200 Re

250

300

350

400

2

~ |u max | |

1.6

β=0.1

1.4

β=0.099

1.2 1

|

~

Q

β=0.101

(b)

1.8

0.8 0.6 0.4 0.2

50

100

150

200 Re

250

300

350

Figure 5.13: (a) absolute value of crossflow instability maximum streamwise velocity for different spanwise wave numbers β; (b) absolute value of adjoint maximum streamwise velocity for different spanwise wave numbers β. Here |Θ| is the absolute value of the wave function Θ defined by (3.11). The Falkner–Skan–Cooke boundary layer is with βH = 0.65, global sweep angle θs = 45o , and Re 0 = 100.


x

(a)

0.4

4

β=0.101 β=0.1 β=0.099

-2

10

(b)

3 2

~

0.2

|

|

~

Q du Re dy

y=0

y=0

0.3

p

137

Q

~

|

|

~

1

0.1 0 50

x 10

-2

3

100 150 200 250 300 350 Re 10

-3

(d)

2

Lr

|

3 |

~

|

x

(c)

|

~

Q dw Re dy y=0

4

0 50

100 150 200 250 300 350 Re

2

1

1 0 50

100 150 200 250 300 350 Re

50

100 150 200 250 300 350 Re

Figure 5.14: Receptivity of Falkner–Skan–Cooke boundary layer with βH = 0.65, global sweep angle θs = 45o , and Re 0 = 100. (a) the receptivity due to wall-normal suction/blowing at the wall; (b) the receptivity to streamwise velocity disturbances at the wall; (c) the receptivity to spanwise velocity disturbances at the wall; (d) Receptivity due to the roughness source.


138

0.04 ~

~ |u(β=0.101)| - |u(β=0.1)|

0.03

~

Difference in |U|

~ ~ |u(β=0.099)| - |u(β=0.1)|

Re = 71

0.02 0.01

Re = 260

0 Re = 260

-0.01 -0.02

Re = 71

-0.03 -0.04 0

5

10

15

20 y

25

30

35

40

Figure 5.15: Difference in the absolute values of adjoint streamwise velocities at two different streamwise locations Re = 71 and Re = 260 for Falkner–Skan–Cooke boundary layer with βH = 0.65, global sweep angle θs = 45o , and Re 0 = 100.

~

0.08

~

~

du dy

0.04

β=0.101

- du dy

β=0.1

~

du dy

- du dy β=0.099

β=0.1

Re = 71

du Difference in dy

~

260

0 260

-0.04 Re = 71

-0.08 0

5

10

15

20 y

25

30

35

40

Figure 5.16: Difference in the absolute values of adjoint streamwise wall shear stress at two different streamwise locations Re = 71 and Re = 260 for Falkner–Skan–Cooke boundary layer with βH = 0.65, global sweep angle θs = 45o , and Re 0 = 100.


~

~

|v(β=0.101)| - |v(β=0.1)| ~

0.02

~

|v(β=0.099)| - |v(β=0.1)|

Re = 71

0.01

~

Difference in |V|

139

Re = 260

0

Re = 260

-0.01 Re = 71

-0.02 0

5

10

15

20 y

25

30

35

40

Figure 5.17: Difference in the absolute values of adjoint wall normal velocities at two different streamwise locations Re = 71 and Re = 260 for Falkner–Skan–Cooke boundary layer with βH = 0.65, global sweep angle θs = 45o , and Re 0 = 100.

0.06 ~

~

|w(β=0.101)| - |w(β=0.1)| 0.04 ~

0.02 Re = 260

~

Difference in |W|

~

|w(β=0.099)| - |w(β=0.1)|

Re = 71

0 Re = 260

-0.02 Re = 71

-0.04

-0.06

0

5

10

15

20 y

25

30

35

40

Figure 5.18: Difference in the absolute values of adjoint spanwise velocities at two different streamwise locations Re = 71 and Re = 260 for Falkner–Skan–Cooke boundary layer with βH = 0.65, global sweep angle θs = 45o , and Re 0 = 100.


x 10

8

140

-3

~ ~ |p(β=0.101)| - |p(β=0.1)|

(a) 6

~ ~ |p(β=0.099)| - |p(β=0.1)| Re = 71

~

Difference in |P|

4 2

Re = 260

0 Re = 260

-2 Re = 71

-4 -6 -8

0

5

10

15

20 y

25

30

35

40

0.4 β=0.101

(b)

0.35

β=0.1 β=0.099

|P|

~

0.3 0.25

0.1

0.2

0.09

β=0.101

Re = 71

β=0.1

0.08 0.15

Re = 71

β=0.099

~

|P|

0.07

0.1

0.06 0.05

0.05

0.04 0.03

0 0

10 y

20

30

40

50 0.02 4

60

70 6

80 8

90 10 y

100 12

14

16

Figure 5.19: (a) Difference in the absolute values of adjoint pressure at two different streamwise locations Re = 71 and Re = 260; (b) Profiles of absolute values of adjoint pressure for three different spanwise wavenumbers for Falkner–Skan–Cooke boundary layer with βH = 0.65, global sweep angle θs = 45o , and Re 0 = 100.


0.02 0.015

PSE

αi

APSE

αi

β=0.1 0.2 0.3 0.4

0.01

~ αi, α i

141

0.005 0

-0.005 -0.01

~

-0.015

β=0.1 0.2 0.3 0.4

-0.02 50

100

150

Re

200

250

300

350

Figure 5.20: The imaginary part of the wave number αi as the function of the spanwise wavenumber β in the Falkner–Skan–Cooke boundary layer with βH = 0.65, global sweep angle θs = 45o , and Re 0 = 100.

75

o

70

Wave angle θ

65

β=0.1 0.2 0.3 0.4

0.1

60 55

0.2

50

0.4

0.3

45 40 35 50

100

150

200 Re

250

300

350

Figure 5.21: The wave angle θ as a function of spanwise wavenumber β for the Falkner–Skan–Cooke boundary layer with βH = 0.65, global sweep angle θs = 45o , and Re 0 = 100.


6

β=0.1 β=0.2 β=0.3 β=0.4

4

(b) Ymax

(a)

4

|

~ Q |u max |

6

142

|

~

2

2

0 50

100

150

200 Re

250

300

0 50

350

1

15

x

(c)

150

200 Re

250

300

350

150

200 Re

250

300

350

150

200 Re

250

300

350

10-2

y=0

5

Q du

y=0

10

Re dy

(d)

~

0.5

|

|

~

Q p

100

|

|

~

~

0 50

100

150

-2

10

200 Re

250

300

0 50

350

x 10

100

0.01 (e) |

Lr |

5

0.005

|

~

Re dy

|

~

Q dw

y=0

(f)

0 50

100

150

200 Re

250

300

350

0 50

100

Figure 5.22: Receptivity predictions based on APSE for crossflow instabilities for various values of the spanwise wavenumber β, in the Falkner–Skan–Cooke boundary layer for frequency F = 0. Receptivity results are reported in the global normalization, where adjoint results are normalized at Re = 260. (a) the maximum receptivity to a streamwise point force; (b) the distance from the wall for maximum receptivity to a streamwise point force; (c) the receptivity due to wall-normal suction/blowing at the wall; (d) the receptivity to streamwise velocity disturbances at the wall; (e) the receptivity to spanwise velocity disturbances at the wall; (f ) the receptivity to roughness on the wall boundary.


6

5 Ymax

4 3

(b)

4 3

β=0.1 β=0.2 β=0.3 β=0.4

2

2

1

1

0 50 100 150 200 250 300 350 Re

0 50 100 150 200 250 300 350 Re 0.05

0.4 (c)

0.04

~

p

0.3 0.2

1 du Re dy y=0

0.5

~

y=0

~ | u max |

5

6 (a)

143

0.03 0.02

0.1

0.01

0

0

50 100 150 200 250 300 350 Re

(d)

50 100 150 200 250 300 350 Re

Figure 5.23: Receptivity predictions based on APSE for crossflow instabilities for various values of the spanwise wavenumber β, in the Falkner–Skan–Cooke boundary layer for frequency F = 0. Receptivity results are reported in the local normalization; (a) the maximum receptivity to a streamwise point force; (b) the distance from the wall for maximum receptivity to a streamwise point force; (c) the receptivity due to wall-normal suction/blowing at the wall; (d) the receptivity to streamwise velocity disturbances at the wall.


b = 0.2

0.3

(a)

4 x 10 0

~

~

ai

ar

0.2

b = 0.2

(b)

F=200 150 100 50

0.25

-3

144

-4

0.15 -8

0.1 0.05

0

100

200 Re

300

400

-12 0

100

200 Re

300

400

Figure 5.24: (a) real part of the adjoint streamwise wave number, α ˜ r , as the function of the frequency F ; (b) imaginary part of the adjoint streamwise wave number, α ˜ i , as the function of the frequency F .


2.5

145

3

(a)

(b)

Ymax

~ |u max |

2.5

2 F=200 150 100 50 1.5 50

100

150

200 Re

250

300

2 1.5 1 0.5 50

350

0.5

250

300

350

150

200 Re

250

300

350

1 du Re dy y=0

0.035

~

y=0

200 Re

(d)

0.4

~

150

0.04

(c)

p

100

0.3

0.2

0.1 50

0.03

0.025

100

150

200 Re

250

300

350

0.02 50

100

Figure 5.25: Receptivity predictions based on APSE for crossflow instabilities for various values of the frequency F , in the Falkner–Skan–Cooke boundary layer for spanwise wavenumber β = 0.2. Receptivity results are reported in the local normalization; (a) the maximum receptivity to a streamwise point force; (b) the distance from the wall for maximum receptivity to a streamwise point force; (c) the receptivity due to wall-normal suction/blowing at the wall; (d) the receptivity to streamwise velocity disturbances at the wall.


3.5

0.4 p

y=0

2.5 ~

~ |u max |

0.5 F=150, β=0.2 F=0

3

146

0.3

2 0.2

1.5 1 50

100 150 200 250 300 350 Re

0.1 50

0.08

5

0

a~ i

~

1 du Re dy y=0

0.06 0.04

-5

0.02 0 50

100 150 200 250 300 350 Re -3 x 10

100 150 200 250 300 350 Re

-10 50

100 150 200 250 300 350 Re

Figure 5.26: Receptivity predictions based on APSE for crossflow instabilities with β = 0.2 in the Falkner–Skan–Cooke boundary layer for frequency F = 0 and 150. Receptivity results are reported in the local normalization; (a) the maximum receptivity to a streamwise point force; (b) the receptivity due to wall-normal suction/blowing at the wall; (c) the receptivity to streamwise velocity disturbances at the wall; (d) imaginary part of the adjoint streamwise wave number, α ˜ i , as the function of the frequency F .


8 6

4

βH =0.65 0.55

(b) 3 Ymax

~ |u max |

(a)

147

4

0.55 2 0.65

2 0 50

1 0 50

100 150 200 250 300 350 Re

0.6

0.08 (d)

0.5

0.2 0.1 50

~

0.4

1 du Re dy y=0

~

p

y=0

(c)

0.3

100 150 200 250 300 350 Re

0.65 0.55 100 150 200 250 300 350 Re

0.06 0.04 0.02 0

50

0.55 0.65

100 150 200 250 300 350 Re

Figure 5.27: Receptivity predictions based on APSE for crossflow instabilities in the Falkner–Skan–Cooke boundary layer with two Hartree parameters βH = 0.65 and βH = 0.55. Receptivity results are reported in the local normalization; (a) maximum receptivity to a streamwise point source; (b) the distance from the wall for maximum receptivity to a streamwise point force; (c) receptivity to the streamwise velocity disturbances; (d) receptivity to the wall normal velocity disturbances.


~ Q |u max |

6

10

βH=0.65

(a)

(b)

0.55

|

Ymax

4

~

0 50

100 150 200 250 300 350 Re

|

1 y=0 |

|

0.5

~

~

5

|

|

100 150 200 250 300 350 Re

-2 15 x 10 (d) 10

Re dy

Q du ~

~

Q p

y=0

(c)

0 50 10

x

0 50

100 150 200 250 300 350 Re 10-2

0.01 (f)

(e)

Lr

|

y=0

100 150 200 250 300 350 Re

|

5

0.005

|

~

Re dy

~

|

5

2 0 50

Q dw

148

0

50

100 150 200 250 300 350 Re

0

50

100 150 200 250 300 350 Re

Figure 5.28: Receptivity predictions based on APSE for crossflow instabilities in the Falkner–Skan–Cooke boundary layer with two Hartree parameters βH = 0.65 and βH = 0.55. Receptivity results are reported in the global normalization; (a) maximum receptivity to a streamwise point source; (b) the distance from the wall for maximum receptivity to a streamwise point force; (c) receptivity to the streamwise velocity disturbances; (d) receptivity to the wall normal velocity disturbances.


4

x 10

149

-3

βH=0.65

2

0.55

~ α i

0

-2

-4

-6

-8 0

50

100

150

Re

200

250

300

350

Figure 5.29: The imaginary part of the adjoint wave number α ˜ i as a function of the Hartree parameter βH in the Falkner–Skan–Cooke boundary layer.


150

0.14

1 (a)

(b)

0.12

0.8 0.1 0.08

u

v

0.6

0.06

0.4

0.04 0.2

0.02 0

0 0

2

4

6

8

y

10 12 14 16 18 20

0

5

10 15 20 25 30 35 40 45 50 y

1.2 0.018

(c)

1

(d)

0.014

w

0.8 p

0.6 0.4

0.01 0.006

0.2 0 0

2

4

6

8

10 12 14 16 18 20 y

0.002 0 0

5

10

15

20 y

25

30

35

40

Figure 5.30: Evaluation of nonparallel effects for the crossflow instability with spanwise wavenumber β = 0.2 in Falkner–Skan–Cooke boundary layer with βH = 0.65. (a) The streamwise velocity profiles, (b) the wall normal velocity, (c) the spanwise velocity profiles, (d) pressure profiles in the state field obtained using PSE and OSE at neutral point x ≈ 100. PSE results, OSE.


1

0.8

(a)

151

(b)

0.8 v

~

~u

0.6 0.6

0.4 0.4 0.2

0.2 0 0

0 5

10 15 20 25 30 35 40

0

5

10 15 20 25 30 35 40

y

y 1.4 1.2

(c)

~ w

1 0.8 0.6 0.4 0.2 0 0

5

10 15 20 25 30 35 40

y

Figure 5.31: Evaluation of nonparallel effects for the adjoint mode with spanwise wavenumber β = 0.2 in Falkner–Skan–Cooke boundary layer with βH = 0.65. (a) The streamwise adjoint velocity profiles, (b) the wall normal adjoint velocity, (c) the spanwise adjoint velocity profiles, (d) adjoint pressure profiles in the adjoint field obtained using APSE and AOSE at neutral point x ≈ 100. APSE results, AOSE.

Crossflow Kinetic Energy

1 0.8

(a)

0.6 0.4 0.2 0

1.4

1.8

s

2.2

2.6

3




152

0.1 (b)

0.08 0.06 0.04 0.02 0

0.2

0.4

0.6 s

0.8

1

0.05 0.04

(c)

PSE Simulation

0.03

Adjoint predictions

0.02

(a), (b) No curvature (c) Curvature

0.01 0 0.2

0.4

0.6 s

0.8

1

Figure 5.32: Adjoint predictions compared with the LNS calculations for instability with spanwise wave number β = 100, in boundary layer over a swept parabolic cylinder with Reynolds number Re 0 = 105 and sweep angle θs = 35o , for a suction and blowing sources with the width corresponding to the wavelength. (a) Prediction of the crossflow kinetic energy for the source located at s = 1.4, without the curvature; (b) Prediction of the crossflow kinetic energy for the source located at s = 0.3, without the curvature; (c) Prediction of the crossflow kinetic energy for the source located at s = 0.3, with the curvature.


25

153

β=100, with curvature No curvature

20 15 ~ α i

~ α α i i,

10 5 0

-5 -10 -15 -20

0.3

0.4

0.5

0.6 s

0.7

0.8

0.9

1

Figure 5.33: The imaginary part of the wavenumber αi as the function of the spanwise wavenumber β in the boundary layer over a swept parabolic cylinder. Both the crossflow and the adjoint wavenumbers are shown.

50

o

45

β=100, with curvature No curvature

40

θ

35 30 25 20 15 0.2

0.3

0.4

0.5

0.6 s

0.7

0.8

0.9

1

Figure 5.34: The oblique angle θ for the boundary layer over a swept parabolic cylinder with and without curvature.


20

4.5 (a)

(b)

With Curvature No Curvature

15

4 Ymax

~ |u max |

154

10

3.5 3

5

2.5

0

0.4

0.6 s

0.8

2

1

0.25

0.8

1

0.4

0.6 s

0.8

1

(d) 0.05

~

y=0

1 du Re dy y=0

0.2 p

0.6 s

0.06 (c)

~

0.4

0.15 0.1 0.05

0.04 0.03

0.4

0.6 s

0.8

1

0.02

Figure 5.35: Receptivity predictions based on APSE for the crossflow instabilities in the swept parabolic cylinder boundary layer with and without curvature. Receptivity results are reported in the local normalization; (a) the maximum receptivity to a streamwise point force; (b) the distance from the wall for maximum receptivity to a streamwise point force; (c) the receptivity due to wall-normal suction/blowing at the wall; (d) the receptivity to streamwise velocity disturbances at the wall.


155

1000 curvature no curvature

900 800

|Λ|

700 600

500

400

0.3

0.4

s

0.5

0.6

0.7

0.8

Figure 5.36: Amplitude of the receptivity efficiency function for β = 100 with and without curvature for the flow conditions Re = 1 × 105 and θs = 35o .


10

156

4 Incompressible flow with curvature Incompressible flow, no curvature

2p |

Lr|

3 10 Compressible flow (M=0.8) with curvature

2 10

10

1 10 -1

s

1 10

10 0

Figure 5.37: Comparison between amplitudes of incompressible and compressible receptivity efficiency functions for β = 100 and flow conditions Re = 1 × 105 and θs = 35o . Compressible efficiency functions are taken from Collis [20].


4 3 2 1 0 -1

0

1

s

2

3

Figure 5.38: Comparison of kinetic energy growth rates for crossflow mode of β = 100 in incompressible flow and compressible flow for β = 100, M = 0.8, 5 o Re = 1 × 10 , and θ = 35 . Compressible kinetic energy growth rate is taken from Collis [20].

Chapter 6 Transient Effects In this Chapter, we compare spatial transients obtained using the LNS and PSE methods. This is an important study, since the recent use of the PSE/APSE methods for control of laminar-turbulent boundary layer transition has been utilized by Pralits et al. [66]. Their approach assumes that PSE can be used as an efficient and accurate method for simulating transition in nonparallel boundary layers subject to various environmental controls. Based upon this assumption, they construct APSE as formally adjoint equations to the PSE and use them to find an optimal control strategy for drag reduction. Unfortunately, they do not compare their results with experimental predictions, nor do they verify their results with DNS calculations. As valuable as the PSE method can be, it cannot be used as a substitute for the DNS calculations, and, as shown in this Chapter, does not provide the correct transient response as compared with LNS solution. The PSE are able to predict the evolution of a particular unstable mode, but they fail to give correct results when multiple modes are excited. We begin this Chapter by first describing how the PSE method is modified to account for control disturbances. We comment on two approaches, ours and the one used by Pralits et al. [66], and find that their approach is unstable for large controls. We then compare PSE and LNS results for wall-normal controls and consider how different stabilization strategies, as well as streamwise resolution affect the transient part of the PSE solution. We conclude the Chapter by comparing LNS and PSE solutions for several distributed control strategies, which resemble the control strategies employed by Pralits et al. [66].

157

Chapter 6. Transient Effects

6.1

158

PSE Implementation for Wall Boundary Control Sources.

In order to obtain PSE solutions with boundary controls, Pralits et al. [66] first solve the PSE subject to a parallel theory eigenfunction on the inflow where they employ the PSE normalization condition, given in (3.9). Note that iteration in α as described in Section 3.4.1 is used to enforce the normalization condition, which causes the PSE to be a nonlinear set of equations. Using this approach, the boundary controls ˆ and wave function (i.e. α) of the PSE solution. affect both the shape function U To obtain the effect of a boundary control on the disturbance, they re-run the PSE solver subject to the same inflow conditions, but with homogeneous wall boundary conditions to obtain a second solution and subtract the two solutions to find the effect of the control. An alternative procedure that we propose also requires two runs but is more efficient. First we obtain the PSE solution subject to a parallel theory eigenfunction on the inflow and with homogeneous wall boundary conditions. The growth rate, α, is then saved and used for a second solution that is obtained with homogeneous inflow boundary conditions but with nonzero wall-boundary controls. This procedure is more efficient, since no iteration in α is required for the second solution. Using this approach, controls affect only the shape functions of the PSE solution and the system is linear. While we have confirmed that both approaches give identical results for small amplitude control disturbances (see Figure 6.1), the first approach becomes unstable to the wall-normal excitations of amplitudes v b equal or larger than 0.1. In Figure 6.2 we illustrate the changes in the growth rate αi due to the application of wall-normal suction and blowing at the location Re = 346 for two different amplitudes v b = 0.001 and 0.01 obtained using the approach of Pralits et al. [66]. We notice that both the


159

real and imaginary parts of the growth rate exhibit large variations when the control amplitudes are increased. Due to the artificial nonlinearity of the PSE, the numerical method becomes unstable for large wall-normal control excitations, which produces discontinuities in the PSE growth rate. On the other hand, with our approach, the PSE are linear and stable and the results simply scale with control amplitudes.

6.2

Comparison of PSE and LNS Transients

Using our PSE approach we now compare the transient solutions from PSE with those from LNS. Figure 6.3 plots umax for both methods for a wall-normal suction and blowing source located at Re = 282. The source is in the form of a Gaussian bump and is shown in Figure 6.4 ( The values of umax are calculated at each streamwise location xi , and the maximum is computed in the wall normal direction. This is the typical experimental measure). The wall-normal control suction and blowing source can be expressed as

R x0 b

b

v = vˆ e

x0 0

ια(x0 ) d x0 +ιβz+ιωt

.

(6.1)

Figure 6.4(a) shows a typical wall-normal control v b , whereas Figure 6.4(b) shows vˆb , which is used as the wall-normal suction and blowing source in PSE. The solution in Figure 6.3(a) has been obtained on a fine mesh (about 30 mesh points per instability wavelength), and p-stabilization with pressure-iteration was employed to find the PSE solution. In Figure 6.3(b) we illustrate the convergence of the state field for several pressure iterations, and we see that the first iteration produces incorrect amplitudes for the state field, while the second and third iterations converge to the LNS solution downstream of the transient region. The transient region obtained using PSE with several pressure iterations, however, exhibits large oscillations in solution as compared to the LNS solution. The oscillations are probably due to the increasing instability of the PSE method when the number of pressure


160

iterations is increased. Similar to Figure 6.3(a), Figure 6.5(a) shows the transient in LNS and PSE solutions for a wall-normal suction and blowing source in the form of a Gaussian bump located further downstream, at Re = 346. Independent of the source location, we find that the PSE solution tends to overpredict the response of the field in the transient region. Figure 6.6 shows the transient solution in the PSE and LNS solutions for other quantities of interest. For this case, the amplitude of the boundary control is normalized to one. Figure 6.6(a) illustrates the transient in the disturbance kinetic energy, Figure 6.6(b) illustrates the transient in the maximum of wall-normal velocity (vmax ) while Figure 6.6(c) shows the transient in the maximum value of pressure (pmax ). Here the PSE solution is obtained again using p-stabilization with pressure iteration. Note that the transient is quite different from the LNS solution for all of the above quantities. To evaluate the effects of stabilization, we compare the transient solutions of the PSE and LNS using τ -stabilization for τ = 1 and τ = 2 in Figure 6.7. Note that the transient solution is smoothed out by the stabilization strategy, and the amplitude of the resulted disturbances is slightly overpredicted with τ -stabilization. We emphasize that the transient in the LNS solution results from the sum of stable and unstable modes, which are excited with various amplitudes due to the control at the wall boundary. The LNS transient is closely related to the receptivity of stable modes and evolution of the transient is dependent on the stability of these modes. The transient in PSE, however, is accounted for by modifying the shape function of a single unstable mode. Due to the different nature of the PSE and LNS transients it is not surprising that they disagree. To illustrate how a transient PSE solution is converges with streamwise resolution,


161

we consider three streamwise resolutions Nx = 130, Nx = 300, and Nx = 800. In Figure 6.8(a) we plot the transient for three different streamwise resolutions using pressure-iteration with p-stabilization. Note that in the transient region, the PSE solutions do not converge with mesh refinement, and exhibit higher oscillations on the finer meshes. Note, however, convergence of the PSE results obtained using τ stabilization in Figure 6.8(b). In either case, the PSE solution does not converge to the LNS solution in the transient region, and with τ -stabilization the asymptotic amplitude is slightly overpredicted. Finally, to further illustrate the inconsistency of the PSE method for the prediction of the transient region of a flow, we consider in Figures 6.9 and 6.10, the PSE and LNS solutions for distributed suction and blowing sources as shown in Figure 6.11 and 6.12, respectively. Note that the suction and blowing source in Figure 6.12 is obtained by multiplying the wide Gaussian bump located at Re = 400 with the base extending from about Re = 300 to Re = 500 by the sine wave with the wavenumber approximately equal that of the instability wave. It is clear from the figures that the PSE transient is inconsistent with the LNS transient solution. To summarize, we find that the PSE method cannot be used as a substitute for the LNS calculations, since it does not provide the correct transient response as compared with LNS methods. PSE, however, can be used if one is only interested in the asymptotic response of the flow to a localized wall boundary source.


162

0.007 0.006

using iteration in α no iteration in α

umax

0.005 0.004 0.003 0.002 0.001 0

200 250 300 350 400 450 500 550 600 650 700 Re

Figure 6.1: Comparison of the PSE solution using ke-normalization vs the PSE solution obtained with a fixed growth rate, α. The suction and blowing source has the amplitude v b = 0.001. The source is located at Re = 346.


-3 0.088 (c) Im{α}

Re{α}

15 x 10 (a) 10

163

5

0.084 0.08

0 0.076 -5 200

300

400

500

600

700

200

Re

300

400

500

600

700

500

600

700

Re

0.02 (b)

0.1

0

Im{α}

Re{α}

0.01

-0.01

(d)

0.08

-0.02 -0.03 200

300

400

500 Re

600

700

0.06 200

300

400 Re

Figure 6.2: The effects of wall-normal boundary suction and blowing source at Re = 346 on the growth rates of the PSE solution with ke-normalization; (a) and (c) are the real and imaginary parts of α for the suction source of amplitude v b = 0.001; (b) and (d) are for v b = 0.01.


164

0.01

0.01

(a)

0.008 0.006

0.006

umax

umax

0.008

0.004

0.004 0.002

0.002 0 200

No iteration First iteration Second iteration

(b)

LNS PSE

300

400

500 Re

600

700

0 200

300

400

500

600

700

Re

Figure 6.3: Comparison of the LNS and PSE transient; (a) Comparison of the LNS and PSE transient for a wall-normal suction and blowing source at Re = 282; (b) the convergence of the PSE solution with pressure iteration.


1

x10

165

-3

-3

4

(a)

x 10

(b)

0.8

2

v

v^ b

b

0.6 0

0.4 -2

0.2 0

240

260

280

300

320

-4 240

340

260

280

300

320

340

Re

Re

Figure 6.4: The typical wall-normal suction and blowing source in the form of a Gaussian bump; (a) the source for LNS (v b ); (b) the source for PSE (ˆ v b ).

0.008

LNS PSE

(a)

0.008

0.004 0.002 0 200

No iteration First iteration Second iteration

0.006

umax

umax

0.006

(b)

0.004 0.002

300

400

500 Re

600

700

0

200

300

400

500

600

700

Re

Figure 6.5: Comparison of LNS and PSE transient; (a) comparison of the LNS and PSE transient for a wall-normal suction and blowing source at Re = 346; (b) the convergence of the PSE solution with the pressure iteration with a wall-normal suction and blowing source at the same location.


1400

166

3

(a)

PSE LNS

1200 1000

2

800

vmax

Ke

PSE LNS

(b) 2.5

600

1.5 1

400

0.5

200 0 200

300

400

500

600

700

Re

0 200

300

400

500

600

700

Re

4 PSE LNS

(c)

pmax

3

2

1

0 200

300

400

500

600

700

Re

Figure 6.6: Comparison of the LNS and PSE transient in different measures using pstabilization in the PSE; (a) comparison of the LNS and PSE transient in disturbance kinetic energy; (b) comparison of the LNS and PSE transient in vmax ; (c) comparison of the LNS and PSE transient in pmax . All comparisons are made for a wall-normal suction and blowing source at Re = 346 using p-stabilization.


167

-3

x 10

τ=1 τ=2 LNS

umax

6

4

2

0 200

300

400

Re

500

600

700

Figure 6.7: Comparison of the LNS and PSE transient in umax for various values of τ . The suction and blowing source is located at Re = 346.


168

0.01 Nx=800 Nx=300 Nx=130

(a)

umax

0.008

0.006

0.004

0.002

0 200

300

400

500

600

700

Re

0.008

Nx=130 (coarse mesh) Nx=300 (τ=1) Nx=800 (τ=1)

(b)

umax

0.006

0.004

0.002

0

200

300

400

500

600

700

Re

Figure 6.8: Comparison of the transient response for several different streamwise resolutions (Nx = 130, Nx = 300, and Nx = 800) corresponding to approximately 4.8, 14, and 30 mesh points per instability wave: (a) the resolution study for the PSE solution using p-stabilization; (b) the resolution study for the PSE solution using τ -stabilization. For all of the cases, the suction and blowing source is located at Re = 282.


169

0.0002 LNS PSE

0.00016

Ke

0.00012

8e-05

4e-05

0

200

300

400

Re

500

600

700

Figure 6.9: Comparisons of the kinetic energy of the instability wave based on the LNS and PSE predictions for a wall-normal suction and blowing source at Re = 400 in the form of a wide Gaussian bump.


170

0.006 LNS PSE

Ke

0.004

0.002

0 350

400

450

500 Re

550

600

650

Figure 6.10: Comparisons of the kinetic energy of the instability wave based on the LNS and PSE predictions for a wall-normal suction and blowing source at Re = 400 in the form a Gaussian bump multiplied by a sine wave (see Figure 6.12 for details about the source).

x 10 10

-4

x 10

(a)

3

-3

(b)

9 2 8 7

1

vb

v^b

6 0

5 4

-1

3 -2 2 -3

1 0 300

400

500 Re

300

400 Re

500

Figure 6.11: The distributed wall-normal suction and blowing source at Re = 400 in the form of a wide Gaussian bump; (a) the LNS suction and blowing source; (b) the PSE source.


x 10 10

171

-4

x 10

-4

(b)

(a)

25

8 20

6 4

15

v^b

vb

2 0 -2

10 5

-4 0 -6 -5

-8 300

400 Re

500

300

350

400 Re

450

Figure 6.12: The distributed wall-normal suction and blowing source ar Re = 400 in the form of sine wave multiplied by a wide Gaussian bump. The spatial wavenumber of the sine wave corresponds to the wavenumber of the instability wave; (a) the LNS suction and blowing source; (b) the PSE source.

Chapter 7 Conclusions and Future Directions 7.1

Conclusions

Adjoint analysis is a powerful approach for investigating the receptivity and sensitivity of flows. In this work, the receptivity of two– and three–dimensional boundary layers has been investigated using the Linearized Navier–Stokes (LNS) and Adjoint Navier– Stokes (ANS) equations together with Parabolized Stability Equations (PSE) and Adjoint Parabolized Stability Equations (APSE). First, the global adjoint equations have been derived based on a general sensitivity analysis that enables receptivity prediction in nonparallel flows. The major advantage of our derivation is that the adjoint equation and boundary conditions follow directly from the formulation once an objective function is specified. Given recent progress in the development of adjoint methods for sensitivity and receptivity prediction, it becomes essential to organize, classify and attest to the validity of the various adjoint formulations especially with regard to the APSE. To address this, a careful investigation of different adjoint approaches has been conducted. APSE solutions have been compared with ANS solutions as well as direct numerical simulation based on the LNS. Our studies indicate that derivation of the APSE should be done as an approximation of ANS which, depending on the quality of the PSE, may or may not be fully adjoint to the PSE. With the APSE validated, the forced and natural receptivity characteristics of the Blasius and Falkner–Skan boundary layers to two– and three–dimensional disturbances are documented and compared with predictions based on local parallel theory. In weakly nonparallel boundary layers, parallel theory is found to be in 172

Chapter 7.

Conclusions and Future Directions

173

good agreement with APSE predictions for two– and three–dimensional disturbances. Further, we find that oblique disturbances generally lead to higher receptivity than two-dimensional disturbances for both Blasius and Falkner–Skan flows. Based on the receptivity analysis of the Falkner–Skan boundary layers, favorable pressure gradient boundary layers tend to be more receptive while adverse pressure gradient boundary layers are less receptive. There appears to be a trade-off between receptivity and instability – factors that decrease instability tend to increase receptivity. One of the main motivations for constructing global adjoint methods such as APSE and ANS, is to study the receptivity of highly nonparallel boundary layers which cannot be accurately explored with parallel theory approximations such as Finite Reynolds Number Theory (FRNT). Our observations show that both the ANS and APSE methods can be used effectively for receptivity prediction in highly nonparallel boundary layers, such as swept Hiemenz flow, Falkner-Skan-Cooke boundary-layers, as well as the boundary layer over a swept parabolic cylinder. Finally, comparisons of the transient PSE and LNS solutions occurring in close proximity to a wall boundary inhomogeneity indicate that PSE transients disagree with LNS. Thus, PSE may not be an appropriate model for boundary layer control studies especially if the transient response is important.

7.2

Future Directions for Research

There are multiple avenues for future research in analyzing the receptivity of shear flows. Perhaps the most straightforward continuation of this research is the analysis of receptivity to the variation of flow parameters.

Chapter 7.

7.2.1


174

Sensitivity of Adjoint to the Flow Parameters

Analysis of receptivity to various flow parameters is most naturally achieved through analysis of the sensitivity of the adjoint field to changes in flow parameters. The interesting parameters include instability frequencies, spanwise and streamwise wave numbers, as well as variations in meanflow parameters such as the Hartree parameter, βH . The changes in adjoint quantities due to changes in parameters can be computed by differentiating the adjoint equations with respect to an appropriate parameter. For example, the change of the adjoint field to the changes in spanwise wave number, ˜ k,β , is obtained by solving U ¯ , β)U ˜ k,β = −L( ¯ , β),β U ˜k. ˜U ˜U L(

(7.1)

Note that equations governing the evolution of the derivative of the adjoint field with respect to a parameter are the same ANS/APSE equations but with an inhomogeneous right hand side. They can be easily solved with existing methods.

7.2.2

Nonmodal Algebraic Growth

As we have mentioned before, receptivity is a part of a more general sensitivity analysis. Knowledge of the overall changes in the kinetic energy of a flow due to changes in a control is important for various engineering applications. The sensitivity analysis of Section 2.3.1 allows one to explore this problem. Adjoint solutions obtained for this particular sensitivity problem allow one, for example, to predict the locations and types of control sources that create the largest changes in the kinetic energy of the flow. Preliminary results indicate that sources placed close to Ωo , where the change in kinetic energy of the flow is observed, produce large kinetic energy changes similar to the sources located very far away from the observation location, Ωo . Using knowledge of this adjoint field, one can construct control excitations that produce the

Chapter 7.


175

largest changes in the kinetic energy of the flow at a particular streamwise location xi . This can lead to identifying so-called “optimal excitations” for the flow.

7.2.3

Nonlinear Receptivity

Besides studying nonmodal growth with adjoint methods, one can attempt to study the nonlinear receptivity of the flow resulting from control sources, which are relatively large, so that the nonlinear term in the LNS equations cannot be entirely ignored. One way to attack this nonlinear problem is through a perturbation expansion of the Navier–Stokes solution based on a small parameter, ε, which is proportional to the amplitude of the environmental excitations. In this approach, the state field is decomposed in the form ¯ + εU 1 + ε2 U 2 . . . U =U

(7.2)

substituted into the Navier–Stokes equations and solved at different orders of ε.

7.2.4

Secondary Receptivity

In the nonlinear phase, instability waves reach high amplitudes and modify the mean flow, giving rise to streamwise and spanwise variations in the mean flow. In a coordinate system moving with the local phase velocity of the instability wave, the base flow appears to be locally steady and periodic in the streamwise and spanwise directions. Due to the nonlinear interaction of excited instabilities, the disturbances usually saturate; however, the modified mean flow often becomes highly unstable to secondary instability modes. Secondary instability modes can be analyzed using Flóquet analysis of the LNS equations with periodic coefficients (see Herbert [38] for a discussion of Flóquet analysis, and Fasel et al. [30], Orszaq and Patera [63, 64] for more information on secondary instabilities). For the saturated unstable mean flow, one can define secondary receptivity as the process by which environmental disturbances are

Chapter 7.


176

converted into highly unstable secondary instability modes. While there has been some research in secondary instability analysis, there has been virtually no attempt to address the secondary receptivity process. Analysis of secondary receptivity could be a very interesting and rewarding area of research.

Appendix A Nondimensionalization A.1

Local and Global Nondimensionalizations

Frequently, results from global methods such as LNS/ANS or PSE/APSE have to be compared with local methods such as OSE/AOSE; therefore, we present the guidelines for comparing results in different nondimensionalizations. In this Chapter, the physical variables/coordinates are denoted by the superscripts 0 , and nondimensional quantities have no superscripts. Some nondimensional quantities have the subscript (l), which indicates local nondimensionalization. In general, we would like to compare the nondimensionalization of equations based on the global Reynolds number, Re 0 = U∞ L/ν, and the local Reynolds number, Re = U∞ δ/ν, where δ and L are the local and global scales, respectively. Here, L denotes a global length scale, which can be, for instance, the cord length of the object in consideration and δ = δ(x) is a local length scale, such as the boundary layer thickness. In order to nondimensionalize spatial coordinates x0 = {x0 , y 0 , z 0 }, we require that x0 , xl = δ x0 x= . L

(A.1) (A.2)

Since L/δ = Re 0 /Re = 1/r therefore, xl =

Re 0 x = (1/r)x . Re

177

(A.3)

Appendix A. Nondimensionalization

178

The time is nondimensionalized by the δ/U∞ , or by L/U∞ t0 U∞ , δ t0 U∞ t= , L

tl =

(A.4) (A.5)

and tl =

Re 0 t = (1/r)t . Re

(A.6)

Further consider nondimensionalization of the wave numbers α0 , β 0 and ω 0 , which appear in PSE/APSE, as well as in OSE/AOSE. Since we require that αx = α0 x0 , βz = β 0 z 0 and ωt = ω 0 t0 therefore αl = δα0 ,

(A.7)

βl = δβ 0 ,

(A.8)

ωl = δω 0 ,

(A.9)

α = Lα0 ,

(A.10)

β = Lβ 0 ,

(A.11)

ω = Lω 0 .

(A.12)

From which follows Re α = rα , Re 0 Re βl = β = rβ , Re 0 Re ω = rω . ωl = Re 0

αl =

(A.13) (A.14) (A.15)


179

Consider now the homogeneous (LNS) equation in primitive variables. For instance, the the u-momentum equation can be written in local nondimensionalization ∂ul ∂ul ∂pl ∂ul ∂ u¯ ∂ u¯ + ul + vl + u¯ + v¯ + ιβl wu ¯ l+ − ∂tl ∂xl ∂yl ∂xl ∂yl ∂xl Ã ! 1 ∂ 2 ul ∂ 2 ul + = 0. Re ∂xl 2 ∂yl 2

(A.16)

In order to convert this local formulation into the global one, lets apply transformations for the coordinates as given by equations (A.3, A.6 and A.14) to (A.16) r

∂ul ∂ul ∂ul ∂pl ∂ u¯ ∂ u¯ + rul + rvl + r¯ u + r¯ v + rιβ wu ¯ l+r − ∂t ∂x ∂y ∂x ∂y ∂x Ã ! 1 ∂ 2 ul ∂ 2 ul r + = 0. Re 0 ∂x2 ∂y 2

(A.17)

Since equation (A.17) is homogeneous, any scaled solution is also a solution of the above equation; therefore, conveniently cancelling r, we obtain an equation in the global formulation ∂ul ∂ul ∂ul ∂pl ∂ u¯ ∂ u¯ + ul + vl + u¯ + v¯ + ιβ wu ¯ l+ − ∂t ∂x ∂y ∂x ∂y ∂x Ã ! 1 ∂ 2 ul ∂ 2 ul + = 0. Re 0 ∂x2 ∂y 2

(A.18)

Notice that the solution of the equation in the local formulation is identical to the solution of the equation in the global formulation. This is true for both the LNS and ANS equations. Since {ul , vl , wl , pl } = {u, v, w, p}, we will use only the {u, v, w, p} notation. Adjoint field is denoted by {˜ u, v˜, w, ˜ p˜}. We use this notation for adjoint field prior its normalization by j-product. The j-product can be computed using local or global nondimensionalization. We denote j-product in the local nondimensionalization by Jl and in the global nondimensionalization by J. The adjoint solution normalized by Jl is denoted by {˜ unl , v˜ln , w˜ln , pñl }. When adjoint is normalized by J


180

we denote it by {˜ un , vñ , w˜ n , pñ }. The superscript n indicates that adjoint field is normalized, and subscript l indicates local normalization. Let us compare the adjoint solutions in global and local normalizations. In the local normalization we normalize the adjoint by the Jl which, in the PSE/APSE approximation (using non-conservative formulation), is given by Z

Jl = Z

∞

0

·

∞

0

2αl (u˜ u + v˜ v + ww) ˜ dyl = Re ¸

(A.19)

2α (u˜ u + v˜ v + ww) ˜ dy , Re 0

(A.20)

u¯(u˜ u + v˜ v + ww) ˜ + u˜ p + p˜ u−

u¯(u˜ u + v˜ v + ww) ˜ + u˜ p + p˜ u−

2αl (u˜ u + v˜ v + ww) ˜ (1/r)dy . Re

In the global formulation the J is given by Z

J=

0

∞

u¯(u˜ u + v˜ v + ww) ˜ + u˜ p + p˜ u−

From (A.19) and (A.20) it follows that 1 1 =r , Jl J

(A.21)

therefore, adjoint in the local and global normalization are related through {˜ v nl , pñl } = r{˜ v n , pñ } .

A.2

(A.22)

Forced Adjoint Nondimensionalization

Consider the problem of sound receptivity described in Chapter 2, Section 2.4. In the local formulation1 ¯ , Re)U ˜ h = Ql (ˆ ˜l) L˜hl {U v sl , v l 1

(A.23)

Here, just as in the Chapter 2, Section 2.4, subscript m is suppressed from definition of adjoint mode.


181

where ˜ l )j = {ˆ Ql (ˆ v sl , v vls }i,j {˜ vl }i − ({ˆ vls }i {˜ vl }j ),i ,

j = 1, 2, 3

(A.24)

˜ l )4 = 0. From §A.1, it follows that and Ql (ˆ v sl , v ¯ ; Re} = rL˜h {U ¯ ; Re 0 } . L˜hl {U

(A.25)

s

ˆ , as well as the mean flow field U ¯ , are not Now, notice that the Stokes sound field, U ˆl = U ˆ, U ˜ n = rU ñ transformed in the global and local formulation. Recall also that U l and partial derivatives with respect to local coordinates are r times greater then the partial derivatives with respect to global coordinates. Then, transforming (A.23), we get ¯ , Re 0 )U ˜ h = r2 Q(ˆ ˜) vs, v rL˜h {U l

(A.26)

˜ ) is in global nondimensionalization. At the same time, in the global where Q(ˆ vs, v nondimensionalization, we have ¯ , Re 0 )U ˜ h = Q(ˆ ˜ m) , L˜h {U vs, v

(A.27)

˜h. ˜ h = rU U l

(A.28)

from which it follows that

Notice of course, that no specific adjoint normalization is required for this problem. Once we obtained (A.28), the efficiency function given by (2.79) in the local nondimensionalization 1 Λl = − Re

Ã

∂us ∂ uñl ∂ u¯ ∂ u˜hl + ∂yl ∂yl ∂yl ∂yl

!

(A.29)

is related to the global efficiency function Λl = r2 Λ .

(A.30)

Appendix B Mean Flow Solutions In this Appendix we describe the formulation and nondimensionalization of the mean flow solutions used herein to explore receptivity. Specifically, consider the family of Falkner–Skan–Cooke boundary layers, as well as the boundary layer over a parabolic cylinder. Contrary to the notation in Appendix A, the physical variables/coordinates have no superscripts, but the nondimensionalized quantities are marked with a (∗ ). This change of notation was applied in this chapter for convenience, since physical quantities appear more often throughout the derivation.

B.1

The Falkner–Skan–Cooke Boundary Layer

The Falkner–Skan–Cook flow is an approximation of the flow over the swept wedge as illustrated in Figure B.1. Near the leading edge of the wedge, the flow is singular and therefore, nonphysical. Nevertheless the Falkner–Skan–Cooke boundary layer flow is a useful tool for analyzing the effects of pressure gradients and three-dimensionality on receptivity of the boundary layers. Derivation of the Falkner-Skan-Cooke equation is given in Cooke’s original paper [23]. Here we review the details of derivation in the current notation. Consider the steady boundary layer equations for incompressible flow over an infinitely long swept wedge, as drawn in Figure B.1. u,x + v,y + w,z = 0 , uu,x + vu,y + wu,z = −p,x + νu,yy , p,y = 0 , 182

(B.1)

Appendix B. Mean Flow Solutions

183

uw,x + vw,y + ww,z = −p,z + νw,yy . The boundary conditions are u(x, 0, z) = v(x, 0, z) = w(x, 0, z) = 0 , u(x, ∞, z) = u∞ = Cxm ,

(B.2)

w(x, ∞, z) = w∞ = const

Note that since the wedge is infinitely long in z, the velocity field is independent of z; therefore, u,x + v,y = 0 ,

(B.3)

uu,x + vu,y = u∞ u∞ ,x + νu,yy , p,y = 0 , uw,x + vw,y = νw,yy , where we have used Bernoulli’s equation to express the streamwise pressure gradient term together with the boundary layer approximation that p,y = 0. Since in boundary layer flows the viscous term, νu,yy , is comparable in magnitude with the inertial term uu,x , it follows that u2∞ /x ≈ νu∞ /δ from which it follows that the boundary layer 1

thickness, δ ≈ δr (x) (νx/u∞ ) 2 , and δr is referred as the similarity length scale. The boundary layer thickness is the natural length scale; therefore, one defines a similarity variable η = y/δr (x). Further, the functions f = f (η) and g = g(η) are chosen such that the stream function is given by 1

ψ(x, y) = (νu∞ x) 2 f (η)

(B.4)

and u(x, y) = ψ(x, y)y = u∞ f 0 (η), w = w∞ g(η), where w∞ is a constant. The wall


184

normal component is v(x, y) = −ψ(x, y)x =

·

1 u∞ ν 2 x

¸1 2

[f 0 (η)η(1 − m) − (m + 1)f (η)] .

(B.5)

Using the above expressions for u, v and substituting them into u-momentum equation yields f 000 +

(m + 1) 00 f f − m(f 0 )2 + m = 0 , 2

(B.6)

f 0 (0) = 0 , f 0 (∞) = u∞ , which is known as the Falkner-Skan equation. Letting m = 0 results in Blasius equations. In order to obtain Cooke’s equation, substitute for u, v, w in the w momentum equation 1 g 00 + (m + 1)f g 0 = 0 , 2

(B.7)

g(0) = 0 , g(∞) = 1 .

Equations (B.6), and (B.7) correspond to the well known set of Falkner-Skan-Cooke equations. Equations can be solved for a set of various parameters m, w∞ and a typical solution for m = 1 is shown in Figure B.2. There are various nondimensionalization schemes reported in the literature. The most obvious is the nondimensionalization by the similarity length scale δr , together with U∞ or u∞ at a particular station, x, in the flow as the reference velocity scale. Alternatively, often in the literature, authors nondimensionalize using momentum thickness δs , or the global length scale L, which could be, for example, the chord


Rer x∗r yr∗ u∗∞ u∗r vr∗ wr∗ αr∗ ωr∗ θ θe t∗r

185

δr , U∞ U∞ δr /ν x/δr y/δr u∞ /U∞ = cos(θe ) cos(θe )f 0 (η) 1/(2Rer )[ηf 0 (1 − m) − (m + 1)f ] sin(θe )g(η) αδr ωδr /u∞ tan−1 (w∞ /u∞ (x = xref )) tan−1 (w∞ /u∞ ) U∞ t/δr

Table B.1: Local Nondimensionalization Based on the Similarity of the Length Scale, δr , and Freestream Edge Velocity, U∞ as indicated in Figure B.1. Alternatively, one can nondimensionalize by u∞ as shown in Figure B.1. length of an airfoil. In Table B.1 we summarize nondimensionalization using the similarity scale δr . For global methods such as LNS/ANS or PSE/APSE it is best to use the global nondimensionalization scheme. Consider the reference scales L, U∞ where ReL = U∞ L/ν. Typically we define the domain along the plate by two local Reynolds numbers Re1 and Re2 based on the boundary layer similarity scale δr , calculated at the inflow and outflow, respectively. Meanwhile, we use a global Reynolds number, Re0 , in order to calculate the nondimensional streamwise coordinate, x∗ . We further define the global sweep angle θ. Since the global Reynolds number, Re0 , is used only to nondimensionalize boundary layer equations, it is convenient to chose Re0 = Rer at the location x where the local sweep angle is equal to the global, θ = θe . This arrangement is illustrated in Figure B.3. Using these three Reynolds numbers, we define all the nondimensional variables 0 in the global normalization. We introduce a new notation, U∞ , to denote the edge

velocity of the flow at the reference location x = xmin0 .


Res x∗s ys∗ u∗∞ u∗s vs∗ ws∗ αs∗ ωs∗ θ θe t∗s

186

δs , U∞ rRer (1/r)x∗r (1/r)yr∗ u∞ /U∞ = cos(θe ) cos(θe )f 0 (η) r/(2Res )[ηf 0 (1 − m) − (m + 1)f ] sin(θe )g(η) rαr∗ rωr∗ tan−1 (w∞ /u∞ (x = xref )) tan−1 (w∞ /u∞ ) (1/r)t∗r

Table B.2: Local Nondimensionalization Based on Momentum Thickness δs and Freestream Edge Velocity U∞ .

u∞ w∞ Re x∗min0 x∗min x∗max Rer yL∗ u∗∞ ∗ w∞ u∗L vL∗ wL∗ L/δr αL∗ ωL∗ θ θe t∗L

0 L, U∞ U∞ cos(θe ) U∞ sin(θe ) = const 0 U∞ L/ν Re/ cos(θ) [Re21 /(Re cos(θ))(x∗min0 )m ]1/(m+1) = xmin /L [Re22 /(Re cos(θ))(x∗min0 )m ]1/(m+1) = xmax /L u∞ δr /ν = (x∗ /x∗min0 )m/2 (x∗ Re cos(θ))1/2 y/L 0 u∞ /U∞ = cos(θ)(x∗ /x∗min0 )m sin(θ) ∗ 0 u ∞ f (η) q (1/2) u∗∞ /(x∗ Re)[ηf 0 (1 − m) − (m + 1)f ] sin(θ)g(η) ∗ u∞ Re/Rer = Rer /x∗ αL = αr∗ u∗∞ (Re/Rer ) = αr∗ Rer /x∗ ωL/u∞ = ωr∗ u∗∞ (Rer /x∗ ) tan−1 (w∞ /u∞ (x = xref )) −1 tan (w∞ /u∞ ) = tan−1 (sin(θ)/u∗∞ ) U∞ t/L

Table B.3: Global Nondimensionalization Based on the Global Length Scale, L, and 0 Streamwise Component of Freestream Edge Velocity, U∞ .


187

In order to understand the nature of the crossflow instability, we plot the FalknerSkan-Cooke flow in the local coordinate system where the xs axis is tangent to the streamline, y is out of the page and zs is normal to the streamline. In this rotated coordinate system, we can easily see that the component in the zs direction has an inflection point that leads to the inviscid instability as shown in Figure B.4.

B.1.1

Validation

To validate the base flow solutions for the Falkner–Skan flow, Figure B.5 compares stability results using the OSE for our base flow solution with the stability results obtained by Choudhari [14]. Since stability of the flow is very sensitive to the perturbations in the mean flow, we conclude that based upon the match in growth rate of the instability wave with Choudhari, we have obtained correct Falkner-Skan mean flow, and that global normalization has been consistent. In Figure B.6 we further validate the Falkner-Skan-Cooke solutions by comparing them to the Fischer and Dallmann [31] Falkner-Skan-Cooke profiles. In order to match their results, we consider Rer = 564.38, βH = 0.630, and θ = 46.9o , where Re r is a local Reynolds number, βH is a Hartree parameter, and θ is a local sweep angle. Note, our local Reynolds number is different than the one provided in the paper by Fischer and Dallmann [31], since we use u∞ to define Rer whereas the authors use U∞ to define Reynolds number Re 0r = 826. (Here (0 ) is used to distinguish between two Reynolds numbers Re r ). Therefore, Rer = cos θRe 0r = cos(46.9o )826 = 564.38. Also, since we use global nondimensionalization, we have to convert our results to a local nondimensionalization. For instance, yL∗ = y/L = (y/δr )(δr /L) = yδ∗r (1/u∗∞ )Rer /Re, if we chose Re = Rer , then yδ∗r = cos θyL∗ .


B.2

188

The Swept Parabolic Cylinder

Consider a swept wing of infinite span as illustrated in Figure B.7. The leading edge radius of the parabolic cylinder is the reference length scale, the reference velocity is the chordwise component of the freestream velocity u∞ , the spanwise component of the velocity is denoted by w∞ and the total amplitude of the freestream velocity is given by U∞ . The parabolic cylinder is swept at the sweep angle θ, shown in Figure B.7. We introduce three coordinate systems over the cylinder. One is the body fitted coordinate system {s, n, z}, where s is the chordwise coordinate, n is the coordinate normal to the cylinder surface and z is in the spanwise coordinate (see Figure B.7 for details). The local coordinate system {xs , zs , n} is introduced as the coordinate system along the streamlines of the velocity field as shown in Figure B.7. We see that in this streamline coordinate system the spanwise component of the mean flow exhibits an inflection point. Finally the third global coordinate system {X, Y, Z} is introduced, as shown in Figure B.7. Since we consider a parabolic cylinder infinite in the streamwise direction, the problem of obtaining mean flow is reduced to the two–dimensional problem in the s, n plane, with velocity having three non-zero components due to the wing sweep. The flow over the parabolic cylinder is most conveniently obtained if we consider the following coordinate transformation η X = ξ2 − √ 2 2ξ + 1 √ √ 2ξη Y = 2ξ + √ 2 , 2ξ + 1

(B.8) (B.9)

where variables ξ and η are new coordinates as shown in Figure B.8. These coordinates


189

relate to the coordinate system {s, n} as q

ds =

4ξ 2 + 2 dξ,

q 1 q 1 √ s = ξ 4ξ 2 + 2 + ln( 2ξ + 2ξ 2 + 1) , 2 2

(B.10)

n = η, where the body contour is obtained by setting η = 0. We define r to be a position vector to an arbitrary point {X, Y }, τ a unit tangential vector and n a unit normal to the body surface as shown in Figure B.8. The vectors r, τ , and n are given in the coordinate system {ξ, η} as √ ) √ 2ξη η r = {X, Y } = ξ 2 − √ 2 , 2ξ + √ 2 , 2ξ + 1 2ξ + 1 √ √ r ,ξ 1 1 τ = = { 2, −2ξ} , n = {2ξ, − 2} , |r ,ξ | (2ξ 2 + 1)3/2 (2ξ 2 + 1)3/2 (

(B.11)

where r ,ξ denotes derivative of the vector r with the coordinate ξ. In general, we use “comma” notation to denote derivatives. We further write expression of the curvature of the parabolic cylinder κ = kτ ,s k =

1 , (2ξ 2 + 1)3/2

(B.12)

which will be used to derive the LNS equations in the generalized coordinate system. The general form of the velocity in the {s, n, z} coordinate frame can be written as v = uτ + vn + wz ,

(B.13)

where z is the unit normal in the z direction. Sometimes we also use notation u = v1 , v = v2 , w = v3 as convenient. The general state vector is defined as in Chapter 2, Section 2.1, U (x, t) = {v, p}T = {u, v, w, p}T , which can be decomposed in the mean


190

flow part and the perturbation part as ¯ + U 0, U (x, t) = U

¯ = {¯ U u, v¯, w, ¯ p},

U 0 = {u0 , v 0 , w0 , p0 } .

(B.14)

The mean flow solutions are obtained by solving nonsimilar boundary layer equations written in the Görtler variables ξg , ηg Z

ξg =

s

0

u∞ (s)n ηg = q , 2ξg

u∞ (x0 ) dx0 ,

(B.15)

where u∞ is the chordwise component of the freestream velocity, which in case of a parabolic cylinder, is given by √

u∞ = √

2ξ . 2ξ 2 + 1

(B.16)

The nonsimilar boundary layer equations are given by 1 f,η η − vg f,ηg − βH (f 2 − 1) − 2ξg f,ξg = 0 Re g g

(B.17)

vg ,ηg + f + 2ξg f,ξg = 0 , where the variables f, vg are q

u¯ f= , u∞

vg =

2ξg

u∞

v¯ +

2ξg f ηg ,s u∞

(B.18)

and the Hartree pressure gradient parameter βH = 2ξg u∞,ξg /u∞ . The boundary conditions for nonsimilar boundary layer equations are given by f = 0,

vg = 0, f = 1,

ηg = 0

(B.19)

ηg → ∞

Solutions of (B.17) are obtained using the spectral boundary layer equations code to


191

x

z

Y θ

U∞

X

y Z

x Ue

z

u∞

θ U∞

zs

Edge Streamline

w∞

wmax

θe xs zs

xs

θe

x

Figure B.1: Schematic of the Falkner-Skan-Cooke Flow. compute nonsimilar incompressible laminar boundary layer equations in body fitted coordinate system developed originally by Streett, Zang, and Hussaini [85]. The ηg direction is discretized using the Chebyschev collocation method, and a marching scheme based on the three-point second order backward difference method is used in the ξg direction, since equations (B.17) are parabolic in that direction. The initial condition for the marching schemes are obtained by solving self-similar flows at ξg = 0, where the ξg -derivatives are zero in (B.17). For more details on the numerical method, implementation and results, see [85].


192

1 1

(a)

(b)

Cooke's data g( η )

g(η)

f 0 (η)

0.8

0.6

0.4

0.6

0.4

0.2

0

0.8

0.2

0

1

2

3

η

4

5

0

6

0

1

2

3

η

4

5

6

7

Figure B.2: Typical plots of f 0 (η) (a) and g(η) (b). The Hartree parameter βh = 2m/(m + 1) = 1.

xmax Re2

x z u∞

θ

w∞

xmin Re1 θ

Re0 xs xmin0

U∞

Figure B.3: Schematic of the Global Normalization.


193

1.2

0 -0.02

(a)

0.8

-0.04

0.6

-0.06

w∗

u∗

1

0.4

-0.08

0.2

-0.1

0

(b) o

inflection point

-0.12

0

2

4

y

∗

6

8

0

10

2

4

y∗

6

8

10

Figure B.4: Profiles of Nondimensional Velocities in Falkner–Skan–Cooke boundary layer for βh = 1; (a) u∗ , (b) w∗ .

8

x 10 3

−Im{αs∗ }

7.5 7 6.5 6 5.5 5 0.15

0.155

ωs∗

0.16

Figure B.5: Stability of the Falkner-Skan-Cooke flow, circle - OSE calculations.

0.165

Choudhari’s [14] data,


6

6

5

5

4

(a)

4

3

w∗

u∗

194

3

2

2

1

1

0 0

0.1

0.2

0.3

0.4

0.5

yδ∗r

0.6

0.7

0.8

0.9

1

(b)

0 _ 0.1

_ 0.08

_ 0.06

_ 0.04

yδ∗r

_ 0.02

0

Figure B.6: Profiles of Nondimensional Velocities in Falkner–Skan–Cooke boundary layer for βh = 0.63; (a) u∗ , (b) w∗ . The Hartee parameter βh = 0.63, θ = 46.9o , Rer = 564.3. The doted line correspond to the results of Fischer and Dallmann [31].


195

s

L=r n

Y θ

U∞

X

z n

Z

x Ue z

u∞

θ U∞

zs

Edge Streamline

w∞

wmax

θe xs zs

xs

θe

x

Figure B.7: The geometry of the swept parabolic cylinder (see Collis [16]).


196

y η ξ

n τ r 0

x

Figure B.8: The schematic of the body-fitted coordinate system.

Appendix C Tensor Concepts of Differential Geometry The reader who is interested in differential geometry and tensor analysis is advised to consult Heinbockel [36]. In this Appendix we introduce several concepts from differential geometry, which will enable us to construct the LNS/ANS equations in curvilinear coordinate systems. We begin by defining a curvilinear coordinate system through the transformation ξ = ξ(x, y, z)

(C.1)

η = η(x, y, z) ζ = ζ(x, y, z) which will be assumed to have an inverse transformation x(ξ, η, ζ), y(ξ, η, ζ), z(ξ, η, ζ). We, also frequently use index notation {x, y, z} = {x1 , x2 , x3 } and {ξ, η, z} = {ξ1 , ξ2 , ξ3 } for coordinate systems. In the curvilinear coordinate system of coordinates, we define the base vectors Ei =

∂r ∂ξi

(C.2)

where ∂r(x1 , x2 , x3 )/∂ξi is the derivative of a position vector in the coordinate system {x1 , x2 , x3 } with respect to the curvilinear coordinate system {ξ1 , ξ2 , ξ3 }. The base vectors Ei given by (C.2) may not necessarily be orthogonal, nevertheless, one can introduce a biorthogonal set of bases Ej , called reciprocal bases, such that Ej · Ei = δij . Using the base vectors defined above, any vector field, v, such as the velocity field,

197

Appendix C. Tensor Concepts of Differential Geometry

198

can then be written using either of two sets of bases. v = v i E i = vi E i .

(C.3)

The components v i are called contravariant components of the vector, v, whereas vi denotes covariant components of that vector. Contravariant components, v i , of a vector, v, are obtained by multiplying v by the corresponding base vector Ei , v i = v · Ei . To obtain an expression for the length of a vector in curvilinear coordinates, consider the vector, r, written as dr = dξ i Ei . Then dr2 = (Ei · Ej )dξi dξj . Note that the product (Ei · Ej ) forms a matrix, g ij , which is called the metric of the coordinate system, given by base vectors Ei . In index notation, the metric is given by m g ij = Ei · Ej = xm ,i x,j

(C.4)

Unless otherwise specified, summation is assumed over two repeated indices. The coordinate system is said to be orthogonal if g ij = 0 for i 6= j

and g (i)(i) = h2(i)

for i = 1, 2, 3

(C.5)

(here parenthesis indicate that there is no summation over i). In an orthogonal coordinate system, the square of the length of the velocity vector is |v|2 = u2 + v 2 + w2 = (v i )2 h2i ,

(C.6)

where defined physical components of the velocity vector u, v and w are equal to the corresponding contravariant components v 1 , v 2 and v 3 multiplied by h1 , h2 and h3 . At this point, it seems rather arbitrary why we would introduce the definition of the metric. However, it can be shown that equations of motion are most easily


199

written in generalized the coordinate system using the metric notation.

C.1

Tensor Transformations

We define a scalar quantity to be any quantity that remains unchanged by coordinate transformation. A contravariant vector, on the other hand, transforms with the coordinate transformation according to the following rule, vi =

∂xi ¯j v , ∂ξ j

(C.7)

whereas contravariant tensors are transformed according to the law ∂xi ∂xk ¯ jm T = j mT . ∂ξ ∂ξ ik

(C.8)

Through the chain rule, it is easy to see dxi =

∂xi j dξ ∂ξ j

(C.9)

that contravariant vector components transform as the coordinate increments dxi , while contravariant tensor components transform as the product dxi dxj . Finally, consider the transformation of the partial derivative ∂ξj ∂ ∂ = . ∂xi ∂xi ∂ξj

(C.10)

Sometimes it may be inconvenient to work with partial derivatives of the form ∂ξj /∂xi . Noticing that ∂ξj ∂xi = δjk ∂xi ∂ξk and denoting J ik =

∂xi ∂ξk

(C.11)

we see that, ∂ξj = J −1 , ∂xi

(C.12)


200

therefore, ∂ ∂ = J −1 . ∂xi ∂ξj

(C.13)

Once we know how to transform contravariant vectors, contravariant tensors and partial derivatives from one coordinate system to another, we can transform the Navier-Stokes equations into any curvilinear coordinate system once the Jacobian, J , is specified. To illustrate this, consider the transformation of the continuity equation from the Cartesian coordinate system into the general curvilinear coordinate system. The continuity equation in a Cartesian system is given by C = u,x + v,y + w,z =

∂ui = 0. ∂xi

(C.14)

Since this is a scalar equation, it should not change in the new coordinate system, ¯ Recall, that using the coordinate transformation xi = xi (ξ j ), we therefore C = C. can write Ã

!

Ã

!

∂ ∂ξ j ∂ ∂ui ∂xi k ∂xi k = u ¯ = u¯ = ∂xi ∂xi ∂ξ k ∂xi ∂ξ j ∂ξ k ∂ξ j ∂ 2 xi k ∂ u¯k ∂ξ j ∂ 2 xi k ∂ξ j ∂xi ∂ u¯k u ¯ + = u¯ + k , ∂xi ∂ξ k ∂ξ j ∂xi ∂ξ k ∂ξ j ∂xi ∂ξ k ∂ξ j ∂ξ where, in the curvilinear coordinate system, we obtain an additional term,

(C.15)

∂ξ j ∂ 2 xi u¯k . ∂xi ∂ξ k ∂ξ j

It can be shown (see Heinbockel [36] for example) that ∂ξ j ∂ 2 xi ∂Ek = Em j δmj = Γm jk δmj , i k j ∂x ∂ξ ∂ξ ∂ξ

(C.16)

where the symbol, Γm jk , has a physical meaning, and indicates the the rate of change of the base vector, Ek , with coordinate ξ j in the Em direction. The symbol Γm jk , appears very often in differential geometry, and is called Christoffel’s symbol of the second kind. It is easy to verify that Christoffel’s symbols in the coordinate system ξ i , can


201

be calculated directly using metric tensor as Ã

Γm jk

∂gnk ∂gjn ∂gjk 1 = g mn + − 2 ∂ξ j ∂ξ k ∂ξ n

!

.

(C.17)

With this background we can obtain the LNS and ANS equations in the curvilinear coordinate system. Using the above notation, we illustrate the power of differential geometry by deriving the continuity equation in the body-fitted coordinate system for a parabolic cylinder.

C.2

The Continuity Equation in a Body-fitted Coordinate System

The metric tensor for the body fitted coordinate system for a parabolic cylinder is given by  

g=  

(1 + κη)2 0 0



0

  . 1 0  

0

0 1

(C.18)

Using above metric only two non-vanishing Christoffel’s symbols are Γ111 =

1 ∂h1 , h1 ∂ξ

Γ112 =

1 ∂h1 , h1 ∂η

(C.19)

where coordinates ξ, η have been introduced in Appendix B. Using Christoffel’s symbols, the continuity equation in this coordinate system is reduced to ∂ u¯k 1 ∂h1 1 1 ∂h1 2 ∂ u¯1 ∂ u¯2 ∂ u¯3 ∂ui j k u ¯ u¯ + + + = 0. = Γ u ¯ + = + jk ∂xi ∂ξ k h1 ∂ξ h1 ∂η ∂ξ ∂η ∂z

(C.20)

Substituting the physical components of velocity, u, v and w, defined through u¯1 = u/h1 , u¯2 = v/h2 , u¯3 = w/h3 , into (C.20), applying the chain rule and recalling that


202

h2 = h3 = 1 leads to 1 ∂h1 1 ∂u ∂v ∂w ∂ui + v + = 0, = + ∂xi h1 ∂ξ h1 ∂η ∂η ∂z

(C.21)

which is a continuity equation in the body fitted coordinate system with the metric given by (C.18). Similarly, with more algebra, we obtain expressions for momentum equations in the generalized coordinate system. All equations are summarized in Appendix D. The notation used in this Section is applicable for this Section only.

Appendix D Governing Equations in Curvilinear Coordinates D.1

The Harmonic Linearized Navier–Stokes Operator

The Harmonic Linearized Navier–Stokes operator in the nonconservative form is given in (3.1). In body-fitted coordinates (s, η, z), for a body of curvature, κ = κ(s, η), we define the following coefficients

c2 = h,y /h = c1 κ,

c3 = h,x /h,

c1 = 1/h,

h = 1 + κy

(D.1)

c4 = h,xy /h,

c5 = h,xx /h

(D.2)

Note that c1,x = −c1 c3 ,

c1,y = −c1 c2 ,

c1,xx = 2c1 c23 − c1 c5

(D.3)

ˆ Using the above definition of the coefficients, the matrices in (3.1) are: matrix G: 

1 0 0 0



     0 1 0 0       0 0 1 0   

0 0 0 0

203

(D.4)

Appendix D. Governing Equations in Curvilinear Coordinates ˆ matrix A:



c2 c

−2c1 c2

1 3 c u¯ + Re  1

     

ˆ matrix B:

Re 2 c1 c3 c1 u¯ + Re

2c1 c2

Re 0

       

c1

0

0  

c2 c

c2 v¯ − Re 0

0

0

0

0

1  

0

c2 v¯ − Re 0

0

1 

w¯ 0

   0    0 

0

0 0

(D.5)



0

c2 v¯ − Re 0

 

0  

c1

ˆ matrix C:



0 1 3 c1 u¯ + Re 0

0

204

0   

(D.6)



0   0

  

w¯ 0 0    0 w¯ 1   0

(D.7)

1 0

ˆ matrix D:        



c2

3 −c1 c4 2 c1 u¯,x + c2 v¯ + Re c2 u¯ + u¯,y + c1 c2 cRe 0 0  2  c2 c1 c2 c3 −c1 c4 c1 v¯,x − 2c2 u¯ − v¯,y + Re 0 0   Re  ¯,x w¯,y 0 0  c1 w 

0

c2

(D.8)

0 0

ˆ 1: matrix E ˆ 1 = c2 1 G ˆ E 1 Re

(D.9)

ˆ2 = 1 G ˆ E Re

(D.10)

ˆ 2: matrix E

Appendix D. Governing Equations in Curvilinear Coordinates

D.2

205

The Harmonic Adjoint Navier–Stokes Operator

The Harmonic Adjoint Navier–Stokes operator in the nonconservative form is gener˜ = −G, ˆ ically written in (3.3), where the constitutive matrix operators are: G ˜ = −A ˆT, B ˜ = −B ˆT, C ˜ = −C ˆT, D ˜ = D ˆ T, E ˜1 = E ˆ T , and E ˜2 = E ˆ T . ExpandA 1 2 ˜U ˜ ),x , (B ˜U ˜ ),y , and (E ˜ 1U ˜ ),xx , we get ing terms (A ˜ ˜U ˜ ,x + A ˜ ,x U ˜U ˜ ),x = A (A

(D.11)

˜U ˜ ),y = B ˜ ˜U ˜ ,y + B ˜ ,y U (B ˜ 1U ˜ ),xx = E ˜ + 2E ˜ 1,x U ˜ ,x + E ˜ ,xx ˜ 1,xx U ˜ 1U (E ˜ ,x = A ˆT matrix A ,x 

˜ ,x A

cu

  2c (c −2c2 c3 )  − 1 4 Re =   0 

−c1 c3

2c1 (c4 −2c2 c3 )

Re cu

0

−c1 c3

0

0

0

cu

0

0

0

0

cu = c1 u¯,x − c1 c3 u¯ +

    ,   

c21 (c5 − 3c23 ) Re

ˆT ˜ ,y = B matrix B ,y 

˜ ,y B

   =   

c2

2 v¯,y + Re 0

0 c2

0

2 v¯,y + Re 0

0

0

ˆ 1, E ˜ 1,xx = −(c5 − 3c2 )E ˆ1 ˜ 1,x = −2c3 E matrix E 3



0

0

0

0  

c2

2 v¯,y + Re 0

  

0   0

(D.12)

Appendix D. Governing Equations in Curvilinear Coordinates

D.3

206

Amplitude Flux

In the curvilinear body-fitted coordinate system, the amplitude flux is given by 



˜ TA Û ˆ −U ˜ TE ˆ 1U ˆ ,x + (U ˆ 1 ),x U ˆ ˜ TE U   J= T T T ˜ B Û ˆ −U ˜ E ˆ 2U ˆ ,y + U ˆ 2U ˆ ˜ E U ,y

(D.13)

Appendix E Bi-orthogonality ˆ m, U ˜ k ; x) = δmk . In this Appendix we briefly justify the orthogonality condition J(U ˆ m and U ˜ n , are said to be orthogonal with respect The regular and adjoint solutions, U ˆ m, U ˜ n ; x) = δmn for any two modes, m and n. Without loss of generality, to J if J(U ˆ m and U ˜ n in the PSE/APSE form as arguments assume that β = 0. Substituting U to the functional J(, ; ) yields J(U

∗

0 n , U m ; x)

˜ n, U ˆ m ; x)e = J(U Rx

Ce

x0

Rx x0

ιαm (x0 ) dx0

i∆mn (x0 )dx0

Rx

e

xe

ια ˜ n (x0 ) dx0

=

(E.1)

˜ n, U ˆ m ; x) = const J(U

where we define ∆mn (x0 ) ≡ αm (x0 ) + α ˜ n (x0 ) and C ≡ exp(−

R xe

ι˜ αn (x0 ) dx0 ) = const.

x0

For convenience, we use splitting defined by αm = −˜ αm . Consider first the case when m = n. The exponential term in (E.1) becomes one, and we get a necessary condition, ˜ n, U ˆ m ; x) = const, or after normalizing the adjoint, J(U ˜ n, U ˆ m ; x) = 1. J(U Consider now the case when n 6= m. In order for (E.1) to be satisfied at each Rx

˜ n, U ˆ m ; x) = const/e location in x, one requires that either J(U

x0

i∆mn (x0 )dx0

or that

˜ n, U ˆ m ; x) = 0. Since by construction, J(U ˜ n, U ˆ m ; x) does not vary exponentially, J(U ˜ n, U ˆ m ; x) = 0. we conclude that for n 6= m, J(U

207

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