Random Functions and Turbulence

ContentsForeword to the English Edition Introduction Part I. Elements of the Theory of Random Functions Chapter 1. Certain Data on the Theory of Probability § 1. Random Variables and Distribution Functions § 2. Numerical Characteristics of Random Variables § 3. Multidimensional Random Variables with Spherical Symmetry § 4. Functional Transformations of Random Variables § 5. Certain Generalizations § 6. The Characteristic Function § 7. The Determination of Statistical Moments by Means of Characteristic Functions Chapter 2. Random Processes § 1. The Definition of the Random Function of One Variable. The Probability Distribution of a Random Function § 2. Statistical Moments. The Autocorrelation Function § 3. The Two-dimensional Random Process. The Cross Correlation Function § 4. The Stationarity and Ergodicity of Random Processes § 5. Fundamental Characteristics of the Autocorrelation and Cross Correlation Functions with Stationary Random Processes § 6. The Differentiation of Random Functions § 7. The Integration of Random Functions § 8. The Normally Distributed Random Processes § 9. The Harmonic Analysis of Random Processes § 10. Generalized Harmonic Analysis. Spectral Expansions § 11. Random Processes with Stationary Increment. Structure Functions § 12. The Determination of the Correlation Function with Experimental Data § 13. The Influence of Finiteness of the Interval of Averaging Chapter 3. Random Fields § 1. Supplementary Information § 2. Scalar and Vector Random Fields. The Random Functions of Several Variables § 3. Statistical Moments § 4. Homogeneous and Isotropic Random Fields § 5. Normal Random Fields § 6. The General Form of Tensor Statistical Moments § 7. The Structure and Certain General Characteristics of Tensor Moments § 8. Spectral Expansions § 9. The Correlation of Random Solenoidal Vector Fields § 10. The Correlation of Random Potential Vector Fields § 11. The Joint Correlation of Solenoidal and Potential Random Vector Fields § 12. The Correlation of Certain Derived Fields § 13. Locally Homogeneous and Isotropie Random Fields. Structure Functions § 14. Some Additional Problems Concerning the Theory of Random Fields Part II hydrodynamic Turbulence Chapter 4. The Statistical Theory of Turbulence-The Method of Similarity and Dimensionality § 1. Some Data from the Theory of Dimensionality § 2. The Emergence of Turbulent Motion § 3. Turbulence with Very Large Reynolds Numbers § 4. Locally Isotropie Turbulence. The Theory of Kolmogorov § 5. The Microstructure of a Temperature Field in a Locally Isotropie Turbulent Flow. The Theory of Obukhov Chapter 5. The Statistical Theory of Turbulence-The Correlation Method § 1. Isotropie Turbulence. The Equation of Kärmän-Howarth § 2. The Invariant of Loitzianskii § 3. Fundamental Laws of Decay of Isotropie Turbulence § 4. On the Hypothesis of Millionshchikov and its Generalizations §5. Locally Isotropie Turbulence-Kolmogorov's Equation § 6. The Spatial Correlation of Pressure § 7. The Spatial Correlation of Acceleration § 8. The Spatial Correlation of Temperature § 9. Correlation of the Vorticity Chapter 6. The Statistical Theory of Turbulence- The Spectral Method § 1. The Turbulent Energy Balance Equation § 2. The Formulation of Fundamental Concepts and Laws in Terms of the Spectral Theory § 3. Obukhov's Spectral Theory § 4. Heisenberg^ Spectral Theory § 5.