Schmid P.J., Henningson D.S. Stability and Transition in Shear Flows

Springer, 2001. 572 p. — ISBN:0-387-98985-4.Introduction and General ResultsNonlinear Disturbance Equations
Definition of Stability and Critical Reynolds Numbers
Definition of Stability
Critical Reynolds Numbers
Spatial Evolution of Disturbances
The Reynolds-Orr Equation
Derivation of tin; Reynolds-Orr Equation
The Need for Linear Growth Mechanisms
Temporal Stability of Parallel Shear Flows
Linear Inviscid Analysis
Inviscid Linear Stability Equations
Modal Solutions
General Results
Dispersive Effects and Wave Packets
Initial Value Problem
The Inviscid Initial Value Problem
Laplace Transform Solution
Solutions to the Normal Vorticity Equation
Example: Couette Flow
Localized Disturbances
Eigensolutions to the Viscous Problem
Viscous Linear Stability Equations
The Velocity-Vorticity Formulation
The Orr-Sommerfeld and Squire Equations
Squire's Transformation and Squire's Theorem
Vector Modes
Pipe Flow
Spectra and Eigenfunctions
Discrete Spectrum
Neutral Curves
Continuous Spectrum
Asymptotic Results
Further Results on Spectra and Eigenfunctions
Adjoint Problem and Bi-Orthogonality Condition
Sensitivity of Eigenvalues
Pseudo-Eigenvalues
Bounds on Eigenvalues
Dispersive Effects and Wave Packets
The Viscous Initial Value Problem
The Viscous Initial Value Problem
Motivation
Derivation of the Disturbance Equations
Disturbance Measure
The Forced Squire Equation and Transient Growth
Eigenfunction Expansion
Blasius Boundary Layer Flow
The Complete Solution to the Initial Value Problem
Continuous Formulation
Discrete Formulation
Optimal Growth Ill
The Matrix Exponential Ill
Maximum Amplification
Optimal Disturbances
Reynolds Number Dependence of Optimal Growth
Optimal Response and Optimal Growth Rate
The Forced Problem and the Resolvent
Maximum Growth Rate
Response to Stochastic Excitation
Estimates of Growth
Bounds on Matrix ExponentialConditions for No Growth
Localized Disturbances
Choice of Initial Disturbances
Examples
Asymptotic Behavior
Nonlinear Stability
MotivationA Model Problem
Nonlinear Initial Value Problem
The Velocity-Vortieity Equations
Weakly Nonlinear Expansion
Multiple-Scale Analysis
The Landau Equation 1G
Three-Wave Interactions 1(J
Resonance Conditions
Derivation of a Dynamical System
Triad Interactions
Solutions to the Nonlinear Initial Value Problem
Formal Solutions to the Nonlinear Initial Value Problem
Weakly Nonlinear Solutions and the Center Manifold
Nonlinear Equilibrium States
Numerical Solutions for Localized Disturbances
Energy Theory
The Energy Stability Problem
Additional Constraints
Stability of Complex Flows and Transition
Temporal Stability of Complex Flows
Effect of Pressure Gradient and Crossfiow
Falkner-Skan (FS) Boundary Layers
Falkner-Skan-Cooke (FSC) Boundary layers
Effect of Rotation and Curvature
Curved Channel Flow
Rotating Channel Flow
Combined Effect of Curvature and Rotation
Effect of Surface Tension
Water Table Flow
Energy and the Choice of Norm
Results 22'
Stability of Unsteady Flow Oscillatory Flow
Arbitrary Time Dependence
Effect of Compressibility
The Compressible Initial Value Problem
nviscid Instabilities and Rayleigh's Criterion
scous Instability
Nonmodal Growth
Growth of Disturbances in Space
Spatial Eigenvalue AnalysisSpatial Spectra
Gaster's Transformation
Harmonic Point Source
Absolute Instability
The Concept of Absolute Instability
Briggs' Method
The Cusp Map
Stability of a Two-Dimensional Wake
Stability of Rotating Disk Flow
Spatial Initial Value Problem
Primitive Variable Formulation
Solution of the Spatial Initial Value Problem
The Vibrating Ribbon Problem
Nonparallel Effects
Asymptotic Methods
Parabolic Equations for Steady Disturbances
Parabolized Stability Equations (PSE)
Spatial Optimal Disturbances
Global Instability
Nonlinear Effects
Nonlinear Wave Interactions
Nonlinear Parabolized Stability Equations
Examples
Disturbance Environment and ReceptivityNonlocalized and Localized Receptivity
An Adjoint Approach to Receptivity
Receptivity Using Parabolic Evolution Equations
Secondary Instability
ntroduction
Secondary Instability of Two-Dimensional Waves
Derivation of the Equations
Numerical Results Elliptical Instability
Secondary Instability of Vortices and Streaks
Governing Equations
Examples of Secondary Instability of Streaks and Vortices ;i8!)
Eckhaus Instability
Secondary Instability of Parallel Flows
Parabolic Equations for Spatial Eckhaus Instability
Transition to Turbulence
Transition Scenarios and ThresholdsThree Transition Scenarios
The Most Likely Transition Scenario
Conclusions
Breakdown of Two-Dimensional Waves
The Zero Pressure Gradient Boundary Layer
Breakdown of Mixing Layers
Streak Breakdown
Streaks Forced by Blowing or Suction
Freestreain Turbulence
Oblique Transition
Experiments and Simulations in Blasius Flow Transition in a Separation Hubble
Compressible Oblique Transition
Transition of Vortex-Dominated Flows
Transition in Flows with Curvature
Direct Numerical Simulations of Secondary Instability of Crossflow Vortices
Experimental Investigations of Breakdown of Cross
How Vortices
Breakdown of Localized Disturbances
Experimental Results for Boundary Layers
Direct Numerical Simulations in Boundary Layers
Transition Modeling
Low-Dimensional Models of Subcritieal Transition
Traditional Transition Prediction Models
Transition Prediction Models Based on Nomnodal Growth
Nonlineai Transition Modeling
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 Domain and Continuous Spectrum
A.5 Chebyshev Discretization of the Orr-Sommerfeld Equation
A.6 MatLAB Codes for Hydrodynamic 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