The Mathematical Theory of Turbulence by M.M. Stanisic
"I do not think at all that I am able to present here any procedure of investiga tion that was not perceived long ago by all men of talent; and I do not promise at all that you can find here anything_ quite new of this kind. But I shall take pains to state in clear words the pules and ways of investigation which are followed by ahle men, who in most cases are not even conscious of foZlow ing them. Although I am free from illusion that I shall fully succeed even in doing this, I still hope that the little that is present here may please some people and have some application afterwards. " Bernard Bolzano (Wissenschaftslehre, 1929) The following book results from aseries of lectures on the mathematical theory of turbulence delivered by the author at the Purdue University School of Aeronautics and Astronautics during the past several years, and represents, in fact, a comprehensive account of the author's work with his graduate students in this field. It was my aim in writing this book to give to engineers and scientists a mathematical feeling for a subject, which because of its nonlinear character has resisted mathematical analysis for many years. On account vii i of its refractory nature this subject was categorized as one of seven "elementary catastrophes". The material presented here is designed for a first graduate course in turbulence. The complete course has been taught in one semester.
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Onset of Turbulence.- One -- Classical Concepts in Turbulence Modeling.- 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 Poiseui11e and Couette Flow.- 3. Length Theory.- 3.1 Prandtl's Mixing Length Theory.- 3.2 Mixing Length in Taylor's Sens.- 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.- Two -- Statistical Theories in Turbulence.- 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 Ve1ocity 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.- 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 wi(xi).- 8.3 Hypotheses of Slmilarity.- 8.4 Propagation of Correlations in Locally Isotropic Flow.- 8.5 Remarks Concerning Kolmogoroff's 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 ?-Equation.- 12.4 Elimination of Pressure Functional ? from the ?-Equation.- 12.5 Forms of the Correlation for n=l and n=2.- IV. Magnetohydrodynamic Turbulence.- 13. Magnetohydrodynamic Turbulence by Means of a Characteristic Functional.- 13.1 Formulation of the Problem in Phase Space.- 13.2 ?-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 Conservati on Laws.- 14.3 Special Case of Isotropic Magnetohydrodynamic Turbulence.- 15. Stationary Solution for ?-Equations.- 15.1 Stationary Solution for the Case ?=?=O.- 15.2 Solution to the ?-Equations for Final Stages of Decay.- 16. Energy Spectrum.- 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 ?=?=O.- 20.3 Numeri ca1 Experiments.- 21. Some Further Improvements of Dispersion Theory in Magnetohydrodynamic Turbulence.- 21.1 Remarks on the Turbulent Dispersion of Temperature for Rm?R?1.- 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 ?1, ?2, ?3, ? and Asymptotic Solution of ?-1ntegra1.- 22.5 Evolution of ?-Integral Eq. (22.29).- 23. Results for the ?2 Spectrum with Joule Heating.- 23.1 The Asymptotic Behavior of the Solutions.- 23.2 The Most Probable Form of the ?2-Spectrum.- 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).- Author Index.
The Mathematical Theory of Turbulence by M.M. Stanisic
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