剪切流中的稳定性和转捩

书籍目录

1 Introduction and General Results

1.1 Introduction

1.2 Nonlinear Disturbance Equations

1.3 Definition of Stability and Critical Reynolds Numbers

1.3.1 Definition of Stability

1.3.2 Critical Reynolds Numbers

1.3.3 Spatial Evolution of Disturbances

1.4 The Reynolds-Orr Equation

1.4.1 Derivation of the Reynolds-Orr Equation

1.4.2 The Need for Linear Growth Mechanisms

I  Temporal Stability of Parallel Shear Flows

2 Linear Inviscid Analysis

2.1 Inviscid Linear Stability Equations

2.2 Modal Solutions

2.2.1 General Results

2.2.2 Dispersive Effects and Wave Packets

2.3 Initial Value Problem

2.3.1 The Inviscid Initial Value Problem

2.3.2 Laplace Transform Solution

2.3.3 Solutions to the Normal Vorticity Equation

2.3.4 Example: Couette Flow

2.3.5 Localized Disturbances

3 Eigensolutions to the Viscous Problem

3.1 Viscous Linear Stability Equations

3.1.1 The Velocity-Vorticity Formulation

3.1.2 The Orr-Sommerfeld and Squire Equations

3.1.3 Squire's Transformation and Squire's Theorem

3.1.4 Vector Modes

3.1.5 Pipe Flow

3.2  Spectra and Eigenfunctions

3.2.1 Discrete Spectrum

3.2.2 Neutral Curves

3.2.3 Continuous Spectrum

3.2.4 Asymptotic Results

3.3  Further Results on Spectra and Eigenfunctions

3.3.1 Adjoint Problem and Bi-Orthogonality Condition

3.3.2 Sensitivity of Eigenvalues

3.3.3 Pseudo-Eigenvalues

3.3.4 Bounds on Eigenvalues

3.3.5 Dispersive Effects and Wave Packets

4 The Viscous Initial Value Problem

4.1 The Viscous Initial Value Problem

4.1.1 Motivation

4.1.2 Derivation of the Disturbance Equations

4.1.3 Disturbance Measure

4.2 The Forced Squire Equation and Transient Growth

4.2.1 Eigenfunction Expansion

4.2.2 Blasius Boundary Layer Flow

4.3 The Complete Solution to the Initial Value Problem

4.3.1 Continuous Formulation

4.3.2 Discrete Formulation

4.4 Optimal Growth

4.4.1 The Matrix Exponential

4.4.2 Maximum Amplification

4.4.3 Optimal Disturbances

4.4.4 Reynolds Number Dependence of Optimal Growth

4.5 Optimal Response and Optimal Growth Rate

4.5.1 The Forced Problem and the Resolvent

4.5.2 Maximum Growth Rate

4.5.3 Response to Stochastic Excitation

4.6 Estimates of Growth

4.6.1 Bounds on Matrix Exponential

4.6.2 Conditions for No Growth

4.7 Localized Disturbances

4.7.1 Choice of Initial Disturbances

4.7.2 Examples

4.7.3 Asymptotic Behavior

5 Nonlinear Stabilitv

5.1 Motivation

5.1.1 Introduction

5.1.2 A Model Problem

5.2 Nonlinear Initial Value Problem

5.2.1 The Velocitv-V10rticity Equations

5.3 Wleakly Nonlinear Expansion

5.3.1 Multiple-Scale Analysis

5.3.2 The Landau Equation

5.4 Three-Wave Interactions

5.4.1 Resonance Conditions.

5.4.2 Derivation of a Dynamical System

5.4.3 niad Interactions

5.5 Solutions to the Nonlinear Initial Value Problem

5.5.1 F0rmal Solutions.to the Nonlinear Initial Value Problem

5.5.2 Wleaklv Nonlinear Solutions and the Center Manifold

5.5.3 Nonlinear Equilibrium States.

5.5.4 Numerical Solutions for Localized Disturbances

5.6 Energy Theory

5.6.1 The Energy Stability Problem

5.6.2 Additional Constraints.

II  Stability 0f Complex Flows and Transition

6 Temporal Stability of Complex Flows

6.1 Effect of Pressure Gradient and Crow flow

6.1.1 Falkner-Skan(FS)Boundary Layers

6.1.2 Falkner-Skan-Cooke(FSC) Boundary layer

6.2 Effect of Rotation and Curvature

6.2.1 Curved Channel Flow.

6.2.2 Rotating Channel Flow

6.2.3 Combined Effect of Curvature and Rotation

6.3 Effect of Surface Tension

6.3.1 Wlater Table Flow

6.3.2 Energy and the Choice of Norm

6.3.3 Results.

6.4 Stability of Unsteady Flow

6.4.1 Oscillatory Flow

6.4.2 Arbitrary Time Dependence

6.5 Effect of Compressibility

6.5.1 The Compressible Initial Value Problem

6.5.2 Inviscid Instabilities and Rayleigh’S Criterion

6.5.3 Viscous Instability

6.5.4 Nonmodal Growth

7  Growth of Disturbances in Space

7.1 Spatial Eigenvalue Analysis

7.1.1 Introduction

7.1.2 Spatial Spectra

7.1.3 Gaster’S Transformation

7.1.4 Harmonic Point Source

7.2 Absolute Instability

7.2.1 The Concept of Absolute Instability

7.2.2 Briggs’Method

7.2.3 The Cusp Map

7.2.4 Stability of a Two-Dimensional Wake

7.2.5 Stability of Rotating Disk Flow

7.3 Spatial Initial Value Problem

7.3.1 Primitive Variable Formulation

7.3.2 Solution of the Spatial Initial Value Problem

7.3.3 The Vibrating Ribbon Problem

7.4 Nonparallel Effects.

7.4.1 Asymptotic Methods

7.4.2 Parabolic Equations for Steady Disturbances

7.4.3 Parabolized Stability Equations(PSE)

7.4.4 Spatial Optimal Disturbances

7.4.5 Global Instability

7.5 Nonlinear Effects

7.5.1 Nonlinear Wave Interactions

7.5.2 Nonlinear Parabolized Stability Equations

7.5.3 Examples

7.6 Disturbance Environment and Receptivity

7.6.1 Introduction

7.6.2 Nonlocalized and Localized Receptivity

7.6.3 An Adjoint Approach to Receptivity

7.6.4 Receptivity Using Parabolic Evolution Eauations

8  Secondary Instability

8.1 Introduction.

8.2 Secondary Instability of Two-Dimensional Waves

8.2.1 Derivation of the Equations

8.2.2 Numerical Results.

8.2.3 Elliptical Instability

8.3 Secondary Instability of Vortices and Streaks

8.3.1 Governing Equations

8.3.2 Examples of Secondary Instability of Streaks and Vortices

8.4 Eckhaus Instability

8.4.1 Secondary Instability of Parallel Flows

8.4.2 Parabolic Equations for Spatial Eckhaus Instability

9  nansition to Turbulence

9.1 Transition Scenarios and Thresholds

9.1.1 Introduction

9.1.2 Three Transition Scenarios

9.1.3 The Most Likely Transition Scenario

9.1.4 Conclusions.

9.2 Breakdown of Two-Dimensional Waves

9.2.1 The Zero Pressure Gradient Boundary Layer

9.2.2 Breakdown of Mixing Layers

9.3 Streak Breakdown

9.3.1 Streaks Forced by Blowing or Suction

9.3.2 Freestream Turbulence

9.4 Oblique Transition

9.4.1 Experiments and Simulations in Blasius Flow

9.4.2 nansition in a Separation Bubble

9.4.3 Compressible Oblique Transition

9.5 Transition of Vortex.Dominated Flows

9.5.1 Transition in Flows with Curvature

9.5.2 Direct Numerical Simulations of Secondary Instability of Crossflow Vortices

9.5.3 Experimental Investigations of Breakdown of Cross flow Vortices

9.6 Breakdown of Localized Disturbances

9.6.1 Experimental Results for Boundary Layers

9.6.2 Direct Numerical Simulations in Boundary Layers

9.7 nansition Modeling

9.7.1 Low-Dimensional Models of Subcritical Transition

9.7.2 Trnaditional Transition Prediction Models

9.7.3 Trnansition Prediction Models Based on Nonmodal Growth

9.7.4 Nonlinear Transition Modeling

III Appendix

A Numerical Issues and Computer Programs

A.1 Global versus Local Methods

A.2 Runge-Kutta Methods

A.3 Chebyshev Expansions

A.4 Infinite Dommn and Continuous Spectrum.

A.5 Chebyshev Discretization of the Orr―Sommerfeld Equation

A.6 MATLAB Codes for nydrodyrnamic Stability CalCUlations

A.7 Eigenvalues of Parallel Shear Flows.

B Resonances and Degeneracies

B.1 Resonances and Degeneracies

B.2 Orr-Sommerfeld-Squire Resonance.

C Adjoint of the Linearized Boundary Layer Equation

C.1 Adjoint of the Linearized Boundary Layer Equation.

D Selected Problems on Part I

Bibliography

Index

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