Statistical Fluid Mechanics: Mechanics of Turbulence, Volume II
"If ever a book on turbulence could be called definitive," declared Science, "it is this book by two of Russia's most eminent and productive scientists in turbulence, oceanography, and atmospheric physics." Noted for its clarity as well as its comprehensive treatment, this two-volume set serves as text or reference. 1975 edition.
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"If ever a book on turbulence could be called definitive," declared Science, "it is this book by two of Russia's most eminent and productive scientists in turbulence, oceanography, and atmospheric physics." Noted for its clarity as well as its comprehensive treatment, this two-volume set serves as text or reference. 1975 edition.
Authors' Preface to the English Edition iiiEditor's Preface to the English Edition vMathematical Description of Turbulence. Spectral Functions 1Spectral Representations of Stationary Processes and Homogeneous Fields 1Spectral Representation of Stationary Processes 3Spectral Representation of Homogeneous Fields 16Partial Derivatives of Homogeneous Fields. Divergence and Curl of a Vector Field 23Isotropic Random Fields 29Correlation Functions and Spectra of Scalar Isotropic Fields 29Correlation Functions and Spectra of Isotropic Fields 35Solenoidal and Potential Isotropic Vector Fields 49One-Point and Two-Point Higher-Order Moments of Isotropic Fields 58Three-Point Moments of Isotropic Fields 75Locally Homogeneous and Locally Isotropic Random Fields 80Processes with Stationary Increments 80Locally Homogeneous Fields 93Locally Isotropic Fields 98Isotropic Turbulence 113Equations for the Correlation and Spectral Functions of Isotropic Turbulence 113Definition of Isotropic Turbulence and the Possibilities of its Experimental Realization 113Equations for the Velocity Correlations 117Equations for the Velocity Spectra 123Correlations and Spectra Containing Pressure 130Correlations and Spectra Containing the Temperature 136The Simplest Consequences of the Correlation and Spectral Equations 141Balance Equations for Energy, Vorticity, and Temperature-Fluctuation Intensity 141The Loitsyanskii and Corrsin Integrals 146Final Period of Decay of Isotropic Turbulence 152Experimental Data on the Final Period of Decay. The Decay of Homogeneous Turbulence 162Asymptotic Behavior of the Correlations and Spectra of Homogeneous Turbulence in the Range of Large Length Scales (or Small Wave Numbers) 169The Influence of the Spectrum Singularity on the Final Period Decay 174Self-Preservation Hypotheses 177The von Karman Hypothesis on the Self-Preservation of the Velocity Correlation Functions 177Less Stringent Forms of the von Karman Hypothesis 181Spectral Formulation of the Self-Preservation Hypotheses 185Experimental Verification of the Self-Preservation Hypotheses 189The Kolmogorov Hypotheses on Small-Scale Self-Preservation at High Enough Reynolds Numbers 197Conditions for the Existence of Kolmogorov Self-Preservation in Grid Turbulence 204The Meso-Scale Quasi-Equilibrium Hypothesis. Self-Preservation of Temperature Fluctuations 210Spectral Energy-Transfer Hypotheses 212Approximate Formulas for the Spectral Energy Transfer 212Application of the Energy Transfer Hypotheses to the Study of the Shape of the Spectrum in the Quasi-Equilibrium Range 225Application of the Energy-Transfer Hypotheses to Decaying Turbulence behind a Grid 235Self-Preserving Solutions of the Approximate Equations for the Energy Spectrum 237The Miliionshchikov Zero-Fourth-Cumulant Hypothesis and its Application to the Investigation of Pressure and Acceleration Fluctuations 241The Zero-Fourth-Cumulant Hypothesis and the Data on Velocity Probability Distributions 241Calculation of the Pressure Correlation and Spectra 250Estimation of the Turbulent Acceleration Fluctuations 256Dynamic Equations for the Higher-Order Moments and the Closure Problem 260Equations for the Third-Order Moments of Flow Variables 260Closure of the Moment Equations by the Moment Discard Assumption 267Closure of the Second- and Third-Order Moment Equations Using the Millionshchikov Zero-Fourth-Cumulant Hypothesis 271Zero-Fourth-Cumulant Approximation for Temperature Fluctuations in Isotropic Turbulence 286Space-Time Correlation Functions. The Case of Stationary Isotropic Turbulence 290Application of Perturbation Theory and the Diagram Technique 295Equations for the Finite-Dimensional Probability Distributions of Velocities 310Turbulence in Compressible Fluids 317Invariants of Isotropic Compressible Turbulence 317Linear Theory; Final Period of Decay of Compressible Turbulence 321Quadratic Effects; Generation of Sound by Turbulence 328Locally Isotropic Turbulence 337General Description of the Small-Scale Structure of Turbulence at Large Reynolds Numbers 337A Qualitative Scheme for Developed Turbulence 337Definition of Locally Isotropic Turbulence 341The Kolmogorov Similarity Hypotheses 345Local Structure of the Velocity Fluctuations 351Statistical Characteristics of Acceleration, Vorticity, and Pressure Fields 368Local Structure of the Temperature Field for High Reynolds and Peclet Numbers 377Local Characteristics of Turbulence in the Presence of Buoyancy Forces and Chemical Reactions. Effect of Thermal Stratification 387Dynamic Theory of the Local Structure of Developed Turbulence 395Equations for the Structure and Spectral Functions of Velocity and Temperature 395Closure of the Dynamic Equations 403Behavior of the Turbulent Energy Spectrum in the Far Dissipation Range 421Behavior of the Temperature Spectrum at Very Large Wave Numbers 433Experimental Data on the Fine Scale Structure of Developed Turbulence 449Methods of Measurement; Application of Taylor's Frozen-Turbulence Hypothesis 449Verification of the Local Isotropy Assumption 453Verification of the Second Kolmogorov Similarity Hypothesis for the Velocity Fluctuations 461Verification of the First Kolmogorov Similarity Hypothesis for the Velocity Field 486Data on the Local Structure of the Temperature and other Scalar Fields Mixed by Turbulence 494Data on Turbulence Spectra in the Atmosphere beyond the Low-Frequency Limit of the Inertial Subrange 517Diffusion in an Isotropic Turbulence 527Diffusion in an Isotropic Turbulence. Statistical Characteristics of the Motion of a Fluid Particle 527Statistical Characteristics of the Motion of a Pair of Fluid Particles 536Relative Diffusion and Richardson's Four-Thirds Law 551Hypotheses on the Probability Distributions of Local Diffusion Characteristics 567Material Line and Surface Stretching in Turbulent Flows 578Refined Treatment of the Local Structure of Turbulence, Taking into Account Fluctuations in Dissipation Rate 584General Considerations and Model Examples 584Refined Similarity Hypothesis 590Statistical Characteristics of the Dissipation 594Refined Expressions for the Statistical Characteristics of Small-Scale Turbulence 640More General Form of the Refined Similarity Hypothesis 650Wave Propagation Through Turbulence 653Propagation of Electromagnetic and Sound Waves in a Turbulent Medium 653Foundations of the Theory of Electromagnetic Wave Propagation in a Turbulent Medium 653Sound Propagation in a Turbulent Atmosphere 668Turbulent Scattering of Electromagnetic and Sound Waves 674Fluctuations in the Amplitude and Phase of Electromagnetic and Sound Waves in a Turbulent Atmosphere 685Strong Fluctuations of Wave Amplitude 704Stellar Scintillation 721Fluctuations in the Amplitude and Phase of Star Light Observed on the Earth's Surface 721The Effect of Telescope Averaging and Scintillation of Stellar and Planetary Images 729Time Spectra of Fluctuations in the Intensity of Stellar Images in Telescopes 733Chromatic Stellar Scintillation 737Functional Formulation of the Turbulence Problem 743Equations for the Characteristic Functional 743Equations for the Spatial Characteristic Functional of the Velocity Field 743Spectral Form of the Equations for the Spatial Characteristic Functional 751Equations for the Space-Time Characteristic Functional 760Equations for the Characteristic Functional in the Presence of External Forces 763Methods of Solving the Equations for the Characteristic Functional 773Use of a Functional Power Series 773Zero-Order Approximation in the Reynolds Number 783Expansion in Powers of the Reynolds Number 791Other Expansion Schemes 798Use of Functional Integrals 802Bibliography 813Supplementary Remarks to Volume 1 853References 854Errata to Volume 1 855Author Index 863Subject Index 871
