Transition to Turbulence

Chapter 1. Receptivity, Instability and Transition: A Perspective; 1.1 Historical Introduction; 1.1.1 Introduction to Flow Instability; 1.1.2 Inviscid Instability Theory; 1.1.3 Role of Dissipation on Instability; 1.1.4 Viscous Instability: Linear Equations; 1.1.5 Temporal, Spatial, and Spatio-temporal Instability Studies; 1.1.6 Similarity Profile for Equilibrium Flow: Tollmien-Schlichting Waves from the Orr-Sommerfeld Equation; 1.1.7 Instability Theory, Experiments and Some Unanswered Questions; 1.2 Introduction to Receptivity Analysis; 1.3 Simple Concepts in Instability Studies; 1.3.1 Kelvin-Helmholtz Instability; 1.3.2 Taylor-Green Vortex Instability; 1.3.3 Equilibrium Solution of the Two-dimensional Taylor-Green Vortex Problem; 1.3.4 Results of the Taylor-Green Vortex Problem Simulation; 1.4 Closing Remarks; Chapter 2. Dynamical System Theory and Role of Equilibrium Flows; 2.1 Conservation Equations; 2.1.1 Conservation of Mass or Continuity; 2.1.2 Conservation of Translational Momentum; 2.1.3 Navier-Stokes Equations in Derived Variables; 2.1.4 Governing Equations for Rotational form of the (~V; ~!)-formulation; 2.1.5 Evolution Equation for Solenoidality Error in (~V; ~!)-formulation; 2.2 Boundary Layer Theory for Equilibrium Flow; 2.2.1 Thin Shear Layer Equation for Two-dimensional Steady Flow; 2.2.2 Boundary Layers Integral Properties; 2.2.3 Displacement Thickness; 2.2.4 Momentum Thickness; 2.3 Limitations of Boundary Layer Equation and Steady Separation; 2.4 Solving Boundary Layer Equation and Similarity Transformation; 2.4.1 Similarity Transform and Analysis; 2.4.2 Zero Pressure Gradient Boundary Layer; 2.4.3 Stagnation Point or the Hiemenz Flow; 2.5 Closing Remarks; Chapter 3. Fundamentals of Scientific Computing; 3.1 Computing Space-Time Dependent Flows; 3.2 The Bi-DirectionalWave Equation; 3.2.1 Three-dimensional Plane Waves; 3.2.2 Requirements for Spatial Discretization; 3.3 Upwind Schemes for Higher Reynolds Number Flows; 3.3.1 General Compact Schemes; 3.3.2 First Derivatives Obtained by Compact Scheme; 3.3.3 Selection of Compact Schemes; 3.4 Global Spectral Analysis: Resolution; 3.4.1 Spectral Accuracy of Some Compact Schemes; 3.5 Time Integration Schemes; 3.6 Analysis of Convection-Diffusion Equation; 3.6.1 Spectral Analysis of Numerical Schemes; 3.6.2 RK4-OUCS3-CD2 Scheme; 3.6.3 RK4-NCCD Scheme; 3.7 Role of Diffusion and Relation to Rotationality; 3.7.1 Enstrophy Transport Equation; 3.8 Spatial and Temporal Scales in Turbulent Flows; 3.8.1 Spatial Scales in Turbulent Flows; 3.8.2 Temporal Scales in Turbulent Flows; 3.9 Two- and Three-Dimensional Turbulent Flows; 3.10 Time-Averaged and Unsteady Flows; 3.11 Closing Remarks; Chapter 4. Instability and Transition; 4.1 Introduction; 4.2 Inviscid Instability of Parallel Flows; 4.2.1 Inviscid Instability Mechanism; 4.2.2 Is there Spatial Inviscid Instability?; 4.2.3 Role of Viscous Terms: Early Developments; 4.3 Linear Viscous Stability Theory; 4.4 Properties of the Orr-Sommerfeld Equation: Developing Solution Method; 4.4.1 Compound Matrix Method (CMM); 4.5 Instability Analysis with the Orr-Sommerfeld Equation; 4.5.1 Grid Search Method: Eigen-Spectrum; 4.6 Other Linear Instability Theories; 4.6.1 Role of Fourier-Laplace Transform: Abel and Tauber Theorems; 4.6.2 Temporal, Spatial and Spatio-temporal Growth Routes; 4.6.3 Signal Problem Assumption: Progress or Impediment?; 4.6.4 Temporal Instability Theory: A Case-Study with CMM; 4.7 Instability Properties using the Orr-Sommerfeld Equation; 4.7.1 Effects of Pressure Gradient on Instability of Boundary Layers; 4.8 Closing Remarks; Chapter 5. Receptivity Analysis: Relation with Instability Experiments; 5.1 Introduction; 5.2 Linear Receptivity of Boundary Layer: Bromwich Contour Integral Method; 5.2.1 Receptivity to Wall Excitation: Frequency Response for Signal Problem; 5.2.2 Near-field Response of Localized Excitat