Statistical Fluid Mechanics: Mechanics of Turbulence, Volume II

Authors' Preface to the English Edition     iii
Editor's Preface to the English Edition     v
Mathematical Description of Turbulence. Spectral Functions     1
Spectral Representations of Stationary Processes and Homogeneous Fields     1
Spectral Representation of Stationary Processes     3
Spectral Representation of Homogeneous Fields     16
Partial Derivatives of Homogeneous Fields. Divergence and Curl of a Vector Field     23
Isotropic Random Fields     29
Correlation Functions and Spectra of Scalar Isotropic Fields     29
Correlation Functions and Spectra of Isotropic Fields     35
Solenoidal and Potential Isotropic Vector Fields     49
One-Point and Two-Point Higher-Order Moments of Isotropic Fields     58
Three-Point Moments of Isotropic Fields     75
Locally Homogeneous and Locally Isotropic Random Fields     80
Processes with Stationary Increments     80
Locally Homogeneous Fields     93
Locally Isotropic Fields     98
Isotropic Turbulence     113
Equations for the Correlation and Spectral Functions of Isotropic Turbulence     113
Definition of Isotropic Turbulence and the Possibilities of its Experimental Realization     113
Equations for the Velocity Correlations     117
Equations for the Velocity Spectra     123
Correlations and Spectra Containing Pressure     130
Correlations and Spectra Containing the Temperature     136
The Simplest Consequences of the Correlation and Spectral Equations     141
Balance Equations for Energy, Vorticity, and Temperature-Fluctuation Intensity     141
The Loitsyanskii and Corrsin Integrals     146
Final Period of Decay of Isotropic Turbulence     152
Experimental Data on the Final Period of Decay. The Decay of Homogeneous Turbulence     162
Asymptotic Behavior of the Correlations and Spectra of Homogeneous Turbulence in the Range of Large Length Scales (or Small Wave Numbers)     169
The Influence of the Spectrum Singularity on the Final Period Decay     174
Self-Preservation Hypotheses     177
The von Karman Hypothesis on the Self-Preservation of the Velocity Correlation Functions     177
Less Stringent Forms of the von Karman Hypothesis     181
Spectral Formulation of the Self-Preservation Hypotheses     185
Experimental Verification of the Self-Preservation Hypotheses     189
The Kolmogorov Hypotheses on Small-Scale Self-Preservation at High Enough Reynolds Numbers     197
Conditions for the Existence of Kolmogorov Self-Preservation in Grid Turbulence     204
The Meso-Scale Quasi-Equilibrium Hypothesis. Self-Preservation of Temperature Fluctuations     210
Spectral Energy-Transfer Hypotheses     212
Approximate Formulas for the Spectral Energy Transfer     212
Application of the Energy Transfer Hypotheses to the Study of the Shape of the Spectrum in the Quasi-Equilibrium Range     225
Application of the Energy-Transfer Hypotheses to Decaying Turbulence behind a Grid     235
Self-Preserving Solutions of the Approximate Equations for the Energy Spectrum     237
The Miliionshchikov Zero-Fourth-Cumulant Hypothesis and its Application to the Investigation of Pressure and Acceleration Fluctuations     241
The Zero-Fourth-Cumulant Hypothesis and the Data on Velocity Probability Distributions     241
Calculation of the Pressure Correlation and Spectra     250
Estimation of the Turbulent Acceleration Fluctuations     256
Dynamic Equations for the Higher-Order Moments and the Closure Problem     260
Equations for the Third-Order Moments of Flow Variables     260
Closure of the Moment Equations by the Moment Discard Assumption     267
Closure of the Second- and Third-Order Moment Equations Using the Millionshchikov Zero-Fourth-Cumulant Hypothesis     271
Zero-Fourth-Cumulant Approximation for Temperature Fluctuations in Isotropic Turbulence     286
Space-Time Correlation Functions. The Case of Stationary Isotropic Turbulence     290
Application of Perturbation Theory and the Diagram Technique     295
Equations for the Finite-Dimensional Probability Distributions of Velocities     310
Turbulence in Compressible Fluids     317
Invariants of Isotropic Compressible Turbulence     317
Linear Theory; Final Period of Decay of Compressible Turbulence     321
Quadratic Effects; Generation of Sound by Turbulence     328
Locally Isotropic Turbulence     337
General Description of the Small-Scale Structure of Turbulence at Large Reynolds Numbers     337
A Qualitative Scheme for Developed Turbulence     337
Definition of Locally Isotropic Turbulence     341
The Kolmogorov Similarity Hypotheses     345
Local Structure of the Velocity Fluctuations     351
Statistical Characteristics of Acceleration, Vorticity, and Pressure Fields     368
Local Structure of the Temperature Field for High Reynolds and Peclet Numbers     377
Local Characteristics of Turbulence in the Presence of Buoyancy Forces and Chemical Reactions. Effect of Thermal Stratification     387
Dynamic Theory of the Local Structure of Developed Turbulence      395
Equations for the Structure and Spectral Functions of Velocity and Temperature     395
Closure of the Dynamic Equations     403
Behavior of the Turbulent Energy Spectrum in the Far Dissipation Range     421
Behavior of the Temperature Spectrum at Very Large Wave Numbers     433
Experimental Data on the Fine Scale Structure of Developed Turbulence     449
Methods of Measurement; Application of Taylor's Frozen-Turbulence Hypothesis     449
Verification of the Local Isotropy Assumption     453
Verification of the Second Kolmogorov Similarity Hypothesis for the Velocity Fluctuations     461
Verification of the First Kolmogorov Similarity Hypothesis for the Velocity Field     486
Data on the Local Structure of the Temperature and other Scalar Fields Mixed by Turbulence     494
Data on Turbulence Spectra in the Atmosphere beyond the Low-Frequency Limit of the Inertial Subrange     517
Diffusion in an Isotropic Turbulence     527
Diffusion in an Isotropic Turbulence. Statistical Characteristics of the Motion of a Fluid Particle     527
Statistical Characteristics of the Motion of a Pair of Fluid Particles     536
Relative Diffusion and Richardson's Four-Thirds Law     551
Hypotheses on the Probability Distributions of Local Diffusion Characteristics      567
Material Line and Surface Stretching in Turbulent Flows     578
Refined Treatment of the Local Structure of Turbulence, Taking into Account Fluctuations in Dissipation Rate     584
General Considerations and Model Examples     584
Refined Similarity Hypothesis     590
Statistical Characteristics of the Dissipation     594
Refined Expressions for the Statistical Characteristics of Small-Scale Turbulence     640
More General Form of the Refined Similarity Hypothesis     650
Wave Propagation Through Turbulence     653
Propagation of Electromagnetic and Sound Waves in a Turbulent Medium     653
Foundations of the Theory of Electromagnetic Wave Propagation in a Turbulent Medium     653
Sound Propagation in a Turbulent Atmosphere     668
Turbulent Scattering of Electromagnetic and Sound Waves     674
Fluctuations in the Amplitude and Phase of Electromagnetic and Sound Waves in a Turbulent Atmosphere     685
Strong Fluctuations of Wave Amplitude     704
Stellar Scintillation     721
Fluctuations in the Amplitude and Phase of Star Light Observed on the Earth's Surface     721
The Effect of Telescope Averaging and Scintillation of Stellar and Planetary Images     729
Time Spectra of Fluctuations in the Intensity of Stellar Images in Telescopes     733
Chromatic Stellar Scintillation     737
Functional Formulation of the Turbulence Problem     743
Equations for the Characteristic Functional     743
Equations for the Spatial Characteristic Functional of the Velocity Field     743
Spectral Form of the Equations for the Spatial Characteristic Functional     751
Equations for the Space-Time Characteristic Functional     760
Equations for the Characteristic Functional in the Presence of External Forces     763
Methods of Solving the Equations for the Characteristic Functional     773
Use of a Functional Power Series     773
Zero-Order Approximation in the Reynolds Number     783
Expansion in Powers of the Reynolds Number     791
Other Expansion Schemes     798
Use of Functional Integrals     802
Bibliography     813
Supplementary Remarks to Volume 1     853
References     854
Errata to Volume 1     855
Author Index     863
Subject Index     871