The Mathematical Theory of Turbulence

The Mathematical Theory of Turbulence

by M.M. Stanisic

Paperback(2nd ed. 1988)

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Product Details

ISBN-13: 9780387966854
Publisher: Springer New York
Publication date: 12/18/1987
Series: Universitext
Edition description: 2nd ed. 1988
Pages: 501
Product dimensions: 6.10(w) x 9.25(h) x 0.04(d)

Table of Contents

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 Kármán’s Similarity Hypothesis.- 4. Universal Velocity Distribution Law.- 4.1 Prandtl’s Approach.- 4.2 von Kármán’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 Weiszäcker’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.

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