《The Kolmogorov-Obukhov Theory of Turbulence: A Mathematical Theory of Turb...
本帖最后由 请叫我雷锋 于 2017-10-15 11:50 编辑《The Kolmogorov-Obukhov Theory of Turbulence: A Mathematical Theory of Turbulence》
湍流的柯尔莫哥洛夫-奥布霍夫理论：湍流的数学理论
作者：
Bj¨orn Birnir
University of California
Department of Mathematics
出版社：Springer
出版时间：2013年
目录
1 The Mathematical Formulation of Fully Developed Turbulence . . . . . . 1
1.1 Introduction to Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Navier–Stokes Equation for Fluid Flow . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Energy and Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Laminar Versus Turbulent Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Two Examples of Fluid Instability Creating Large Noise . . . . . . . . . . 8
1.4.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 The Central Limit Theorem and the Large Deviation Principle,
in Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5.1 Cram´er’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5.2 Stochastic Processes and Time Change . . . . . . . . . . . . . . . . . . 18
1.6 Poisson Processes and Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . 20
1.6.1 Finite-Dimensional Brownian Motion . . . . . . . . . . . . . . . . . . . 24
1.6.2 The Wiener Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.7 The Noise in Fully Developed Turbulence . . . . . . . . . . . . . . . . . . . . . . 28
1.7.1 The Generic Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.8 The Stochastic Navier–Stokes Equation for Fully Developed
Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2 Probability and the Statistical Theory of Turbulence . . . . . . . . . . . . . . . 35
2.1 Ito Processes and Ito’s Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2 The Generator of an Ito Diffusion and Kolmogorov’s Equation . . . . 37
2.2.1 The Feynman–Kac Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2.2 Girsanov’s Theorem and Cameron–Martin . . . . . . . . . . . . . . . 39
2.3 Jumps and L´evy Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4 Spectral Theory for the Operator K . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.5 The Feynman–Kac Formula and the Log-Poissonian Processes . . . . 46
2.6 The Kolmogorov–Obukhov–She–Leveque Theory . . . . . . . . . . . . . . . 48
2.7 Estimates of the Structure Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.8 The Solution of the Stochastic Linearized Navier–Stokes
Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
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x Contents
3 The InvariantMeasure and the Probability Density Function . . . . . . . 55
3.1 The InvariantMeasure of the Stochastic Navier–Stokes Equation . . . 55
3.1.1 The InvariantMeasure of Turbulence . . . . . . . . . . . . . . . . . . . . 57
3.2 The InvariantMeasure for the Velocity Differences . . . . . . . . . . . . . . 59
3.3 The Differential Equation for the Probability Density Function . . . . 61
3.4 The PDF for the Turbulent Velocity Differences . . . . . . . . . . . . . . . . . 62
3.5 Comparison with Simulations and Experiments . . . . . . . . . . . . . . . . . 66
3.6 Description of Simulations and Experiments . . . . . . . . . . . . . . . . . . . . 69
3.7 The InvariantMeasure of the Stochastic Vorticity Equation . . . . . . . . 70
3.7.1 The InvariantMeasure of Turbulent Vorticity . . . . . . . . . . . . . 73
4 Existence Theory of Swirling Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.1 Leray’s Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2 The A Priori Estimate of the Turbulent Solutions . . . . . . . . . . . . . . . . 78
4.3 Existence Theory of the Stochastic Navier–Stokes Equation . . . . . . . 85
Appendix A The Bound for a Swirling Flow . . . . . . . . . . . . . . . . . . . . . . . . . 89
Appendix B Detailed Estimates of S2 and S3 . . . . . . . . . . . . . . . . . . . . . . . . . 97
Appendix C The Generalized Hyperbolic Distributions . . . . . . . . . . . . . . . 101
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
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