Symbol Description ............................................. ix
List of Figures ................................................ xv
List of Tables .............................................. xxvii
Preface ...................................................... xxix
1 Introduction to Instability and Transition ................... 1
1.1 Introduction ............................................ 1
1.2 What Is Instability? .................................... 4
1.3 Temporal and Spatial Instability ........................ 4
1.4 Some Instability Mechanisms ............................. 5
1.4.1 Dynamic Stability of Still Atmosphere ............ 5
1.4.2 Kelvin-Helmholtz Instability ..................... 7
2 Computing Transitional and Turbulent Flows .................. 11
2.1 Fluid Dynamical Equations .............................. 11
2.1.1 Equation of Continuity .......................... 12
2.1.2 Momentum Conservation Equation .................. 12
2.1.3 Energy Conservation Equation .................... 14
2.1.4 Alternate Forms of the Energy Equation .......... 15
2.1.5 Equations of Motion in Terms of Derived
Variables ....................................... 15
2.2 Some Equilibrium Solutions of the Basic Equation ....... 16
2.2.1 Couette Flow between Parallel Plates ............ 18
2.2.2 Flow between Concentric Rotating Cylinders ...... 19
2.2.3 Couette Flow between Parallel Plates, Driven
by Pressure ..................................... 21
2.2.4 Steady Stagnation Point Flow .................... 24
2.2.5 Flow Past a Rotating Disc ....................... 26
2.3 Boundary Layer Theory .................................. 30
2.4 Control Volume Analysis of Boundary Layers ............. 35
2.4.1 Displacement Thickness .......................... 36
2.4.2 Momentum Thickness .............................. 36
2.4.3 Separation of a Steady Boundary Layer ........... 37
2.4.3.1 Accelerated Flows (dp/dx < 0) .......... 38
2.4.3.2 Retarded Flow (dp/dx > 0) .............. 38
2.5 Numerical Solution of the Thin Shear Layer (TSL)
Equation ............................................... 40
2.5.1 Falkner-Skan Similarity Profile ................. 43
2.5.2 Separation Criterion for Wedge Flow ............. 44
2.5.3 Blasius Profile ................................. 44
2.5.4 Hiemenz or Stagnation Point Flow ................ 44
2.6 Laminar Mixing Layer ................................... 44
2.7 Plane Laminar Jet ...................................... 46
2.8 Issues of Computing Space-Time Dependent Flows ......... 48
2.8.1 Waves — Building Blocks of a Disturbance Field .. 49
2.8.2 Plane Waves ..................................... 50
2.9 Wave Interaction: Group Velocity and Energy Flux ....... 51
2.9.1 Physical and Computational Implications of
Group Velocity .................................. 52
2.9.2 Wave Packets and Their Propagation .............. 53
2.10 Issues of Space—Time Scale Resolution of Flows ......... 54
2.10.1 Spatial Scales in Turbulent Flows ............... 54
2.10.2 Two- and Three-Dimensional DNS .................. 55
2.11 Temporal Scales in Turbulent Flows ..................... 57
2.12 Computing Time-Averaged and Unsteady Flows ............. 59
2.13 Computing Methods for Unsteady Flows: Dispersion
Relation Preserving (DRP) Methods ...................... 61
2.13.1 Spectral or Numerical Amplification Factor ...... 62
2.13.2 Quantification of Dispersion Error .............. 65
2.14 DRP Schemes: Parameter Ranges for Creating q-Waves ..... 68
3 Instability and Transition in Flows ......................... 75
3.1 Introduction ........................................... 75
3.2 Parallel Flow Approximation and Inviscid Instability
Theorems ............................................... 76
3.2.1 Inviscid Instability Mechanism .................. 78
3.3 Viscous Instability of Parallel Flows .................. 79
3.3.1 Eigenvalue Formulation for Instability of
Parallel Flows .................................. 80
3.3.2 Temporal and Spatial Amplification of
Disturbances .................................... 83
3.3.2.1 Temporal Amplification Theory .......... 83
3.3.2.2 Spatial Amplification Theory ........... 83
3.3.2.3 Relationship between Temporal and
Spatial Theories ....................... 84
3.4 Properties of the Orr-Sommerfeld Equation and
Boundary Conditions .................................... 85
3.4.1 Compound Matrix Method .......................... 87
3.5 Instability Analysis from the Solution of the
Orr-Sommerfeld Equation ................................ 91
3.5.1 Local and Total Amplification of Disturbances ... 95
3.5.2 Effects of the Mean Flow Pressure Gradient ...... 97
3.5.3 Transition Prediction Based on Stability
Calculation .................................... 102
3.5.4 Effects of Free Stream Turbulence .............. 103
3.6 Receptivity Analysis of the Shear Layer ............... 106
3.6.1 Receptivity Mechanism by a Linearized
Approach: Connection to Stability Theory ....... 107
3.6.1.1 A Brief Review of Laplace-Fourier
Transforms ............................ 107
3.6.1.2 Fourier and Laplace Transforms ........ 108
3.6.1.3 Inversion Formula for Laplace
Transforms ............................ 110
3.6.1.4 A Short Tutorial on Fourier Integral
and Transforms ........................ 112
3.6.1.5 Some Useful Laplace-Fourier
Transforms ............................ 115
3.6.2 Receptivity to Wall Excitation and Impulse
Response ....................................... 118
3.6.2.1 Near-Field Response Created by
Localized Excitation .................. 120
3.6.2.2 Outer Solution ........................ 122
3.6.2.3 Inner Solution ........................ 122
3.6.3 Vibrating Ribbon at the Wall ................... 127
3.6.4 Receptivity to Free Stream Excitation .......... 130
3.6.5 General Excitation and Upstream Propagating
Modes .......................................... 132
3.6.6 Low Frequency Free Stream Excitation and the
Klebanoff Mode ................................. 138
3.7 Direct Simulation of Receptivity to Free Stream
Excitation ............................................ 140
3.7.1 Coupling between the Wall and Free Stream
Modes .......................................... 144
3.7.2 Receptivity to a Train of Convected Vortices
in the Free Stream ............................. 147
3.7.3 Further Explanation of Free Stream Periodic
Excitation ..................................... 151
3.8 Nonparallel and Nonlinear Effects on Instability and
Receptivity ........................................... 155
3.8.1 Time Varying Receptivity Problem vis-a-vis
the Signal Problem ............................. 158
3.8.2 Evidence of Nonparallel and Nonlinear Effects .. 159
3.8.3 Limitations of Linearized Nonparallel
Theories ....................................... 166
4 Bypass Transition: Theory, Computations, and Experiments ... 171
4.1 Introduction .......................................... 171
4.2 Transition via Growing Waves and Bypass Transition .... 177
4.3 Visualization Study of Vortex-Induced Instability as
Bypass Transition ..................................... 179
4.4 Computations of Vortex-Induced Instability as
a Precursor to Bypass Transition ...................... 189
4.5 Instability Mechanism in Vortex-Induced Instability ... 192
4.6 Instability at the Attachment Line of Swept Wings ..... 195
5 Spatio-Temporal Wave Front and Transition .................. 201
5.1 Introduction .......................................... 201
5.2 Transient Energy Growth ............................... 205
5.3 Bromwich Contour Integral Method and Energy-Based
Receptivity Analysis .................................. 206
5.4 Spatio-Temporal Wave Front Obtained by the Bromwich
Contour Integral Method ............................... 207
5.5 Nonlinear Receptivity Analysis: Transition by the
Spatio-Temporal Front and Bypass Route ................ 217
5.5.1 Governing Equations and Boundary Condition ..... 219
5.5.2 Nonlinear Receptivity to Vortical Wall
Excitation ..................................... 223
5.5.3 Low Amplitude, Moderate Frequency Excitation ... 226
5.5.4 High Amplitude Cases and Spot Regeneration
Mechanism ...................................... 229
5.5.5 Low Frequency Excitation Cases: Different
Route of Transition ............................ 234
5.6 Calculation of the N Factor ........................... 240
Nonlinear Effects: Multiple Hopf Bifurcations and
Proper ................................................
6 Orthogonal Decomposition ................................... 245
6.1 Introduction .......................................... 245
6.2 Receptivity of Bluff-Body Flows to Background
Disturbances .......................................... 246
6.2.1 Numerical Simulation of Flow Past a Cylinder .. 251
6.3 Multiple Hopf Bifurcations, Landau Equation and Flow
Instability ........................................... 256
6.4 Instability of Flow Past a Cylinder ................... 258
6.5 Role of FST on Critical Reynolds Number for
a Cylinder ............................................ 260
6.6 POD Modes and Nonlinear Stability ..................... 263
6.7 Landau-Stuart-Eckhaus Equation ........................ 274
6.8 Universality of POD Modes ............................. 276
7 Stability and Transition of Mixed Convection Flows ......... 293
7.1 Introduction .......................................... 293
7.2 Governing Equations ................................... 297
7.3 Equilibrium Boundary Layer Flow Equations ............. 298
7.3.1 Schneider's Similarity Solution ................ 299
7.4 Linear Spatial Stability Analysis of the Boundary
Layer over a Heated Plate ............................. 305
7.4.1 Fundamental Solutions of the OSE ............... 310
7.4.2 Compound Matrix Method for the Sixth Order
OSE ............................................ 312
7.4.3 Initial Conditions for an Auxiliary System of
Equations ...................................... 315
7.4.4 Dispersion Relation ............................ 316
7.4.5 The Grid Search Method and the Newton-Raphson
Technique for Obtaining Eigenspectrum .......... 316
7.4.6 Neutral Curve and Wavenumber Contours .......... 319
7.4.7 Precision in Computing ......................... 321
7.4.8 Results of the Linear Spatial Stability
Theory ......................................... 322
7.5 Nonlinear Receptivity of Mixed Convection Flow over
a Heated Plate ........................................ 333
7.5.1 Boundary and Initial Conditions ................ 336
7.5.2 Eigenfunction Structure and DNS of the Mixed
Convection Problem ............................. 356
7.6 Concluding Remarks .................................... 358
8 Instabilities of Three-Dimensional Flows ................... 361
8.1 Introduction .......................................... 361
8.2 Three-Dimensional Flows ............................... 361
8.3 Infinite Swept Wing Flow .............................. 364
8.4 Attachment Line Flow .................................. 365
8.5 Boundary Layer Equations in the Transformed Plane ..... 367
8.6 Simplification of Boundary Layer Equations in the
Transformed Plane ..................................... 368
8.7 Instability of Three-Dimensional Flows ................ 369
8.7.1 Effects of Sweep Back and Cross Flow
Instability .................................... 370
8.8 Linear Stability Theory for Three-Dimensional Flows ... 372
8.8.1 Temporal Instability of Three-Dimensional
Flows .......................................... 373
8.8.2 Spatial Instability of Three-Dimensional
Flows .......................................... 374
8.9 Experimental Evidence of Instability on Swept Wings ... 376
8.10 Infinite Swept Wing Boundary Layer .................... 377
8.11 Stability of the Falkner-Skan-Cooke Profile ........... 381
8.12 Stationary Waves over Swept Geometries ................ 384
8.13 Traveling Waves over Swept Geometries ................. 386
8.14 Attachment Line Problem ............................... 387
8.15 Empirical Transition Prediction Method for Three-
Dimensional Flows ..................................... 389
8.15.1 Streamwise Transition Criterion ................ 389
8.15.2 Cross Flow Transition Criteria ................. 389
8.15.3 Leading Edge Contamination Criterion ........... 390
9 Analysis and Design of Natural Laminar Flow Airfoils ....... 393
9.1 Introduction .......................................... 393
9.2 Airfoil Nomenclature and Basic Aerodynamic
Properties ............................................ 394
9.3 Pressure Distribution and Pressure Recovery of Some
Low Drag Airfoils ..................................... 402
9.4 Flapping of Airfoils .................................. 406
9.5 Effects of Roughness and Fixing Transition ............ 407
9.6 Effects of Vortex Generator or Boundary Layer
Re-Energizer .......................................... 409
9.7 Section Characteristics of Various Profiles ........... 410
9.8 High Speed NLF Airfoils ............................... 412
9.9 Direct Simulation of Bypass Transitional Flow Past
an Airfoil ............................................ 420
10 Epilogue ................................................... 425
10.1 Introduction .......................................... 425
10.2 Relevance of Two-Dimensional Turbulence ............... 426
10.3 Role of Formulation in the Numerical Solution in
Two-Dimensional DNS ................................... 427
10.4 Dynamical System Representation of Turbulent Flows .... 429
10.5 Role of the Computational Domain ...................... 431
10.5.1 Renewal Mechanism and Intermittent Nature of
Turbulence ..................................... 433
10.6 Free and Forced Turbulence ............................ 437
Selected Problems ............................................. 441
Bibliography .................................................. 447
Index ......................................................... 473
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