ALBERT

Note: CONTENTS Preface Preface to the Second Edition Introduction 1 Onset of Turbulence Part One - Classical Concepts in Turbulence Modeling Chapter I. Turbulent Flow 1. Equations of Fluid Dynamics and Their Consequences 1.1 Reynolds' Averaging Technique 1.2 Equations of Fluid Dynamics 1.3 Equation of Kinetic Energy 1.4 Equation of Heat Conduction 2. Reynolds' Stresses 2.1 Physical and Geometrical Interpretation of Reynolds' Stresses 2.2 Eddies and Eddy Viscosity 2.3 Poiseuille and Couette Flow 3. Length Theory 3.1 Prandtl's Mixing Length Theory 3.2 Mixing Length in Taylor's Sense 3.3 Betz's Interpretation of von Karman's Similarity Hypothesis 4. Universal Velocity Distribution Law 4.1 Prandtl's Approach 4.2 von Karman's Approach 4.3 Turbulent Pipe Flow with Porous Wall 5. The Turbulent Boundary Layer 5.1 Turbulent Flow Over a Solid Surface 5.2 Law of the Wall in Turbulent Channel Flow 5.3 Velocity Distribution in Transient Region of a Moving Viscous Turbulent Flow 5.4 A New Approach to the Turbulent Boundary Layer Theory Using Lumley's Extremum Principle Part Two - Statistical Theories in Turbulence Chapter II. Fundamental Concepts 6. Stochastic Processes 6.1 General Remarks 6.2 Fundamental Concepts in Probability 6.3 Random Variables and Stochastic Processes 6.4 Weakly Stationary Processes 6.5 A Simple Formulation of the Covariance and Variance for Incompressible Flow 6.6 The Correlation and Spectral Tensors in Turbulence 6.7 Theory of Invariants 6.8 The Correlation of Derivatives of the Velocity Components 7. Propagation of Correlations in Isotropic Incompressible Turbulent Flow 7.1 Equations of Motion 7.2 Vorticity Correlation and Vorticity Spectrum 7.3 Energy Spectrum Function 7.4 Three-Dimensional Spectrum Function Chapter III. Basic Theories 8. Kolmogoroff's Theories of Locally Isotropic Turbulence 8.1 Local Homogeneity and Local Isotropy 8.2 The First and the Second Moments of Quantities w-j(x-j) 8.3 Hypotheses of Similarity 8.4 Propagation of Correlations in Locally Isotropic Flow 8.5 Remarks Concerning Kolmogoroff1s Theory 9. Heisenberg's Theory of Turbulence 9.1 The Dynamical Equation for the Energy Spectrum 9.2 Heisenberg's Mechanism of Energy Transfer 9.3 von Weiszacker's Form of the Spectrum 9.4 Objections to Heisenberg's Theory 10. Kraichnan's Theory of Turbulence 10.1 Burgers' Equation in Frequency Space 10.2 The Impulse Response Function 10.3 The Direct Interaction Approximation 10.4 Third Order Moments 10.5 Determination of Green's Function 10.6 Summary of Results of Burgers' Equation in Kraichnan's Sense 11. Application of Kraichnan's Method to Turbulent Flow 11.1 Derivation of Navier-Stokes Equation in Fourier Space 11.2 Impulse Response, Function for Full Turbulent Representation 11.3 Formal Statement by Direct-Interaction Procedure 11.4 Application of the Direct-Interaction Approximation 11.5 Averaged Green's Function for the Navier-Stokes Equations 12. Hopf's Theory of Turbulence 12.1 Formulation of the Problem in Phase Space and the Characteristic Functional 12.2 The Functional Differential Equation for Phase Motion 12.3 Derivation of the ϕ-Equation 12.4 Elimination of Pressure Functional π from the ϕ-Equation 12.5 Forms of the Correlation for n=l and n=2 Chapter IV. Magnetohydrodynamic Turbulence 13. Magnetohydrodynamic Turbulence by Means of a Characteristic Functional 13.1 Formulation of the Problem in Phase Space 13.2 ϕ-Equations in Magnetohydrodynamic Turbulence 13.3 Correlation Equations 14. Wave-Number Space 14.1 Transformation to Wave-Number Space 14.2 The Spectrum Equations and Additional Conservation Laws 14.3 Special Case of Isotropic Magnetohydrodynamic Turbulence 15. Stationary Solution for ϕ-Equations 15.1 Stationary Solution for the Case λ=ν=0 15.2 Solution to the ϕ-Equations for Final Stages of Decay 16. Energy Spectrin 16.1 Energy Spectrum in the Equilibrium Range 16.2 Extension of Heisenberg's Theory in Magnetohydrodynamic Turbulence 17. Temperature Dispersion in Magnetohydrodynamic Turbulence 17.1 Turbulent Dispersion 17.2 Formulation of the Problem 17.3 Universal Equilibrium 18. Temperature Spectrum for Small and Large Joule Heat Eddies 18.1 Small Joule Heat Eddies 18.2 Large Joule Heat Eddies 19. The Temperature Spectrum for the Joule Heat Eddies of Various Sizes 19.1 The Viscous Dissipation Process 19.2 The Joule Heat Model 19.3 The Calculation of the Temperature Spectrum 19.4 Effect of Viscous Dissipation on the Temperature Distribution 20. Thomas' Numerical Experiments 20.1 Turbulent Dynamo Competing Processes 20.2 Nondissipative Model System λ=ν=0 20.3 Numerical Experiments 21. Some Further Improvements of Dispersion Theory in Magnetohydrodynamic Turbulence 21.1 Remarks on the Turbulent Dispersion of Temperature for Rm〉〉R〉〉l 21.2 Heat Equation for Conductive Cut-Off Wave Number for H(k) 21.3 Solution of the Heat Equation 22. A Solution for the Joule-Heat Source Term 22.1 Physical Introduciton 22.2 Form of the Source Function and Particular Solution 22.3 The Joule Heating Spectrum 22.4 The Range of Values α1, α2, α3, σ and Asymptotic Solution of τ-integral 22.5 Evolution of τ-Integral Eq. (22.29) 23. Results for the θ2 Spectrum with Joule Heating 23.1 The Asymptotic Behavior of the Solutions 23.2 The Most Probable Form of the θ2-Spectrum Chapter V. Contemporary Turbulence 24. Recent Developments in Turbulence Through Use of Experimental Mathematics - Attractor Theory 24.1 Things That Change Suddenly 24.2 Order in the Chaos 24.3 Attractor Theory in Turbulent Channel Flows 25. Recent Developments in Experimental Turbulence 25.1 Coherent Structure of Turbulent Shear Flows Appendices Appendix A -- Derivation of Correlation Equations (13.51-13.62) Appendix B -- Derivation of Spectrum Equations (14.45-14.46) Appendix C -- Fourier Transforms (18.10) Appendix D -- The Time Variation of Eq. (18.3) Appendix E -- The Time Variation of Eq. (18.19) Bibliography Author Index Subject Index