The Kolmogorov-Obukhov Theory of Turbulence: A Mathematical Theory of Turbulence (SpringerBriefs in Mathematics)

Turbulence is a major problem facing modern societies. It makes airline passengers return to their seats and fasten their seatbelts but it also creates drag on the aircraft that causes it to use more fuel and create more pollution. The same applies to cars, ships and the space shuttle. The mathematical theory of turbulence has been an unsolved problems for 500 years and the development of the statistical theory of the Navier-Stokes equations describes turbulent flow has been an open problem. The Kolmogorov-Obukhov Theory of Turbulence develops a statistical theory of turbulence from the stochastic Navier-Stokes equation and the physical theory, that was proposed by Kolmogorov and Obukhov in 1941. The statistical theory of turbulence shows that the noise in developed turbulence is a general form which can be used to present a mathematical model for the stochastic Navier-Stokes equation. The statistical theory of the stochastic Navier-Stokes equation is developed in a pedagogical manner and shown to imply the Kolmogorov-Obukhov statistical theory. This book looks at a new mathematical theory in turbulence which may lead to many new developments in vorticity and Lagrangian turbulence. But even more importantly it may produce a systematic way of improving direct Navier-Stokes simulations and lead to a major jump in the technology both preventing and utilizing turbulence. Table of Contents Cover The Kolmogorov-Obukhov Theory of Turbulence - A Mathematical Theory of Turbulence ISBN 9781461462613 ISBN 9781461462620 Preface Contents The Mathematical Formulation of Fully Developed Turbulence 1.1 Introduction to Turbulence 1.2 The Navier-Stokes Equation for Fluid Flow 1.2.1 Energy and Dissipation 1.3 Laminar Versus Turbulent Flow 1.4 Two Examples of Fluid Instability Creating Large Noise 1.4.1 Stability 1.5 The Central Limit Theorem and the Large Deviation Principle, in Probability Theory 1.5.1 Cramer'� s Theorem 1.5.2 Stochastic Processes and Time Change 1.6 Poisson Processes and Brownian Motion 1.6.1 Finite-Dimensional Brownian Motion 1.6.2 The Wiener Process 1.7 The Noise in Fully Developed Turbulence 1.7.1 The Generic Noise 1.8 The Stochastic Navier-Stokes Equation for Fully Developed Turbulence Probability and the Statistical Theory of Turbulence 2.1 Ito Processes and Ito's Calculus 2.2 The Generator of an Ito Diffusion and Kolmogorov's Equation 2.2.1 The Feynman-Kac Formula 2.2.2 Girsanov's Theorem and Cameron-Martin 2.3 Jumps and Levy� Processes 2.4 Spectral Theory for the Operator K 2.5 The Feynman-Kac Formula and the Log-Poissonian Processes 2.6 The Kolmogorov-Obukhov-She-Leveque Theory 2.7 Estimates of the Structure Functions 2.8 The Solution of the Stochastic Linearized Navier-Stokes Equation The Invariant Measure and the Probability Density Function 3.1 The Invariant Measure of the Stochastic Navier-Stokes Equation 3.1.1 The Invariant Measure of Turbulence 3.2 The Invariant Measure for the Velocity Differences 3.3 The Differential Equation for the Probability Density Function 3.4 The PDF for the Turbulent Velocity Differences 3.5 Comparison with Simulations and Experiments 3.6 Description of Simulations and Experiments 3.7 The Invariant Measure of the Stochastic Vorticity Equation 3.7.1 The Invariant Measure of Turbulent Vorticity Existence Theory of Swirling Flow 4.1 Leray's Theory 4.2 The A Priori Estimate of the Turbulent Solutions 4.3 Existence Theory of the Stochastic Navier-Stokes Equation The Bound for a Swirling Flow Detailed Estimates of S2 and S3 The Generalized Hyperbolic Distributions References Index Cover......Page 1 The Kolmogorov-Obukhov Theory of Turbulence......Page 4 Preface......Page 8 Contents......Page 10 1.1 Introduction to Turbulence......Page 12 1.2 The Navier–Stokes Equation for Fluid Flow......Page 15 1.2.1 Energy and Dissipation......Page 16 1.3 Laminar Versus Turbulent Flow......Page 17 1.4 Two Examples of Fluid Instability Creating Large Noise......Page 19 1.4.1 Stability......Page 21 1.5 The Central Limit Theorem and the Large Deviation Principle, in Probability Theory......Page 26 1.5.1 Cramér's Theorem......Page 28 1.5.2 Stochastic Processes and Time Change......Page 29 1.6 Poisson Processes and Brownian Motion......Page 31 1.6.1 Finite-Dimensional Brownian Motion......Page 35 1.6.2 The Wiener Process......Page 37 1.7 The Noise in Fully Developed Turbulence......Page 39 1.7.1 The Generic Noise......Page 42 1.8 The Stochastic Navier–Stokes Equation for Fully Developed Turbulence......Page 43 2.1 Ito Processes and Ito's Calculus......Page 46 2.2 The Generator of an Ito Diffusion and Kolmogorov's Equation......Page 48 2.2.2 Girsanov's Theorem and Cameron–Martin......Page 50 2.3 Jumps and Lévy Processes......Page 51 2.4 Spectral Theory for the Operator K......Page 53 2.5 The Feynman–Kac Formula and the Log-Poissonian Processes......Page 57 2.6 The Kolmogorov–Obukhov–She–Leveque Theory......Page 59 2.7 Estimates of the Structure Functions......Page 60 2.8 The Solution of the Stochastic Linearized Navier–Stokes Equation......Page 64 3.1 The Invariant Measure of the Stochastic Navier–Stokes Equation......Page 66 3.1.1 The Invariant Measure of Turbulence......Page 68 3.2 The Invariant Measure for the Velocity Differences......Page 70 3.3 The Differential Equation for the Probability Density Function......Page 72 3.4 The PDF for the Turbulent Velocity Differences......Page 73 3.5 Comparison with Simulations and Experiments......Page 77 3.6 Description of Simulations and Experiments......Page 80 3.7 The Invariant Measure of the Stochastic Vorticity Equation......Page 81 3.7.1 The Invariant Measure of Turbulent Vorticity......Page 84 4.1 Leray's Theory......Page 85 4.2 The A Priori Estimate of the Turbulent Solutions......Page 88 4.3 Existence Theory of the Stochastic Navier–Stokes Equation......Page 95 Appendix A The Bound for a Swirling Flow......Page 99 Appendix B Detailed Estimates of S2 and S3......Page 107 Appendix C The Generalized Hyperbolic Distributions......Page 110 Reference......Page 112 Index......Page 116

1.The Mathematical Formulation of Fully-Developed Turbulence 2.Probability and the Statistical Theory of Turbulence 3.The Invariant Measure and the Probability Density Function 4. Existence Theory of Swirling Flow The Bound for a Swirling Flow Detailed estimates of S2 and S3 The Generalized Hyperbolic Distributions References.

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