Ebook: The Kolmogorov-Obukhov Theory of Turbulence: A Mathematical Theory of Turbulence

​​​​​​​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