剪切流中的稳定性和转换 英文版PDF电子书下载

1 Introduction and General Results 1

1.1 Introduction 1

1.2 Nonlinear Disturbance Equations 2

1.3 Definition of Stability and Critical Reynolds Numbers 3

1.3.1 Definition of Stability 3

1.3.2 Critical Reynolds Numbers 5

1.3.3 Spatial Evolution of Disturbances 6

1.4 The Reynolds-Orr Equation 7

1.4.1 Derivation of the Reynolds-Orr Equation 7

1.4.2 The Need for Linear Growth Mechanisms 9

Ⅰ Temporal Stability of Parallel Shear Flows 15

2 Linear Inviscid Analysis 15

2.1 Inviscid Linear Stability Equations 15

2.2 Modal Solutions 17

2.2.1 General Results 17

2.2.2 Dispersive Effects and Wave Packets 33

2.3 Initial Value Problem 38

2.3.1 The Inviscid Initial Value Problem 38

2.3.2 Laplace Transform Solution 42

2.3.3 Solutions to the Normal Vorticity Equation 46

2.3.4 Example:Couette Flow 48

2.3.5 Localized Disturbances 50

3 Eigensolutions to the Viscous Problem 55

3.1 Viscous Linear Stability Equations 55

3.1.1 The Velocity-Vorticity Formulation 55

3.1.2 The Orr-Sommerfeld and Squire Equations 56

3.1.3 Squire's Transformation and Squire's Theorem 58

3.1.4 Vector Modes 59

3.1.5 Pipe Flow 61

3.2 Spectra and Eigenfunctions 64

3.2.1 Discrete Spectrum 64

3.2.2 Neutral Curves 71

3.2.3 Continuous Spectrum 74

3.2.4 Asymptotic Results 78

3.3 Further Results on Spectra and Eigenfunctions 85

3.3.1 Adjoint Problem and Bi-Orthogonality Condition 85

3.3.2 Sensitivity of Eigenvalues 89

3.3.3 Pseudo-Eigenvalues 93

3.3.4 Bounds on Eigenvalues 94

3.3.5 Dispersive Effects and Wave Packets 96

4 The Viscous Initial Value Problem 99

4.1 The Viscons Initial Value Problem 99

4.1.1 Motivation 99

4.1.2 Derivation of the Disturbance Equations 102

4.1.3 Disturbance Measure 102

4.2 The Forced Squire Equation and Transient Growth 103

4.2.1 Eigenfunction Expansion 103

4.2.2 Blains Boundary Layer Flow 105

4.3 The Complete Solution to the Initial Value Problem 106

4.3.1 Continuous Formulation 106

4.3.2 Discrete Formulation 108

4.4 Optimal Growth 111

4.4.1 The Matrix Exponential 111

4.4.2 Maximum Amplification 112

4.4.3 Optimal Disturbances 119

4.4.4 Reynolds Number Dependence of Optimal Growth 120

4.5 Optimal Response and Optimal Growth Rate 126

4.5.1 The Forced Problem and the Resolvent 126

4.5.2 Maximum Growth Rate 131

4.5.3 Response to Stochastic Excitation 133

4.6 Estimates of Growth 139

4.6.1 Bounds on Matrix Exponential 139

4.6.2 Conditions for No Growth 141

4.7 Localized Disturbances 144

4.7.1 Choice of Initial Disturbances 144

4.7.2 Examples 147

4.7.3 Asymptotic Behavior 149

5 Nonlinear Stability 153

5.1 Motivation 153

5.1.1 Introduction 153

5.1.2 A Model Problem 154

5.2 Nonlinear Initial Value Problem 155

5.2.1 The Velocity-Vorticity Equations 155

5.3 Weakly Nonlinear Expansion 160

5.3.1 Multiple-Scale Analysis 160

5.3.2 The Landau Equation 164

5.4 Three-Wave Interactions 167

5.4.1 Resonance Conditions 167

5.4.2 Derivation of a Dynamical System 168

5.4.3 Triad Interactions 172

5.5 Solutions to the Nonlinear Initial Value Problem 177

5.5.1 Formal Solutions to the Nonlinear Initial Value Problem 177

5.5.2 Weakly Nonlinear Solutions and the Center Manifold 179

5.5.3 Nonlinear Equilibrium States 180

5.5.4 Numerical Solutions for Localized Disturbances 185

5.6 Energy Theory 188

5.6.1 The Energy Stability Problem 188

5.6.2 Additional Constraints 191

Ⅱ Stability of Complex Flows and Transition 197

6 Temporal Stability of Complex Flows 197

6.1 Effect of Pressure Gradient and Crossflow 198

6.1.1 Falkner-Skan(FS)Boundary Layers 198

6.1.2 Falkner-Skan-Cooke(FSC) Boundary layers 203

6.2 Effect of Rotation and Curvature 207

6.2.1 Curved Channel Flow 207

6.2.2 Rotating Channel Flow 211

6.2.3 Combined Effect of Curvature and Rotation 213

6.3 Effect of Surface Tension 216

6.3.1 Water Table Flow 216

6.3.2 Energy and the Choice of Norm 218

6.3.3 Results 221

6.4 Stability of Unsteady Flow 223

6.4.1 Oscillatory Flow 223

6.4.2 Arbitrary Time Dependence 229

6.5 Effect of Compressibility 237

6.5.1 The Compressible Initial Value Problem 237

6.5.2 Inviscid Instabilities and Rayleigh's Criterion 240

6.5.3 Viscous Instability 246

6.5.4 Nonmodal Growth 249

7 Growth of Disturbances in Space 253

7.1 Spatial Eigenvalue Analysis 253

7.1.1 Introduction 253

7.1.2 Spatial Spectra 255

7.1.3 Gaster's Transformation 264

7.1.4 Harmonic Point Source 266

7.2 Absolute Instability 270

7.2.1 The Concept of Absolute Instability 270

7.2.2 Briggs'Method 273

7.2.3 The Cusp Map 278

7.2.4 Stability of a Two-Dimensional Wake 281

7.2.5 Stability of Rotating Disk Flow 284

7.3 Spatial Initial Value Problem 290

7.3.1 Primitive Variable Formulation 290

7.3.2 Solution of the Spatial Initial Value Problem 291

7.3.3 The Vibrating Ribbon Problem 294

7.4 Nonparallel Effects 300

7.4.1 Asymptotic Methods 301

7.4.2 Parabolic Equations for Steady Disturbances 309

7.4.3 Parabolized Stability Equations(PSE) 318

7.4.4 Spatial Optimal Disturbances 329

7.4.5 Global Instability 337

7.5 Nonlinear Effects 344

7.5.1 Nonlinear Wave Interactions 344

7.5.2 Nonlinear Parabolized Stability Equations 346

7.5.3 Examples 349

7.6 Disturbance Environment and Receptivity 351

7.6.1 Introduction 351

7.6.2 Nonlocalized and Localized Receptivity 353

7.6.3 An Adjoint Approach to Receptivity 363

7.6.4 Receptivity Using Parabolic Evolution Equations 367

8 Secondary Instability 373

8.1 Introduction 373

8.2 Secondary Instability of Two-Dimensional Waves 374

8.2.1 Derivation of the Equations 374

8.2.2 Numerical Results 378

8.2.3 Elliptical Instability 381

8.3 Secondary Instability of Vortices and Streaks 383

8.3.1 Governing Equations 383

8.3.2 Examples of Secondary Instability of Streaks and Vortices 389

8.4 Eckhaus Instability 394

8.4.1 Secondary Instability of Parallel Flows 394

8.4.2 Parabolic Equations for Spatial Eckhaus Instability 397

9 Transition to Turbulence 401

9.1 Transition Scenarios and Thresholds 401

9.1.1 Introduction 401

9.1.2 Three Transition Scenarios 403

9.1.3 The Most Likely Transition Scenario 411

9.1.4 Conclusions 413

9.2 Breakdown of Two-Dimensional Waves 414

9.2.1 The Zero Pressure Gradient Boundary Layer 414

9.2.2 Breakdown of Mixing Layers 420

9.3 Streak Breakdown 425

9.3.1 Streaks Forced by Blowing or Suction 425

9.3.2 Freestream Turbulence 429

9.4 Oblique Transition 436

9.4.1 Experiments and Simulations in Blasius Flow 436

9.4.2 Transition in a Separation Bubble 441

9.4.3 Compressible Oblique Transition 445

9.5 Transition of Vortex-Dominated Flows 446

9.5.1 Transition in Flows with Curvature 446

9.5.2 Direct Numerical Simulations of Secondary Instability of Crossflow Vortices 450

9.5.3 Experimental Investigations of Breakdown of Crossflow Vortices 455

9.6 Breakdown of Localized Disturbances 456

9.6.1 Experimental Results for Boundary Layers 459

9.6.2 Direct Numerical Simulations in Boundary Layers 460

9.7 Transition Modeling 465

9.7.1 Low-Dimensional Models of Subcritical Transition 465

9.7.2 Traditional Transition Prediction Models 469

9.7.3 Transition Prediction Models Based on Nonmodal Growth 471

9.7.4 Nonlinear Transition Modeling 474

Ⅲ Appendix 479

A Numerical Issues and Computer Programs 479

A.1 Global versus Local Methods 479

A.2 Runge-Kutta Methods 480

A.3 Chebyshev Expansions 483

A.4 Infinite Domain and Continuous Spectrum 486

A.5 Chebyshev Discretization of the Orr-Sommerfeld Equation 487

A.6 MATLAB Codes for Hydrodynamic Stability Calculations 489

A.7 Eigenvalues of Parallel Shear Flows 503

B Resonances and Degeneracies 509

B.1 Resonances and Degeneracies 509

B.2 Orr-Sommerfeld-Squire Resonance 511

C Adjoint of the Linearized Boundary Layer Equation 515

C.1 Adjoint of the Linearized Boundary Layer Equation 515

D Selected Problems on Part Ⅰ 519

Bibliography 529

Index 551