Stochastic Tools in Turbulence

by Lumley, John L.

• ISBN13: 9780486462707
• ISBN10: 0486462706
• Edition: Reprint
• Format: Paperback
• Copyright: 2007-11-02
• Publisher: Dover Publications

Summary

This accessible treatment offers the mathematical tools for describing and solving problems related to stochastic vector fields. Advanced undergraduates and graduate students will find its use of generalized functions a relatively simple method of resolving mathematical questions, and extensive appendixes make it mathematically self-contained. 1970 edition.

Table of Contents

Preface	p. ix
Probability Distributions and Densities
Definition of Probability	p. 1
Formalizing the Definition	p. 3
Measure and Lebesgue Integration	p. 5
Distribution Function	p. 7
The Probability Density Function	p. 8
A Simple Example of the Distribution and Density Functions	p. 10
Expected Values	p. 11
Joint Distribution Functions	p. 12
The Joint Density Function	p. 13
Expected Values	p. 14
Conditional Probabilities	p. 15
Conditional Expectations	p. 16
Statistical Independence and Lack of Correlation	p. 16
Change of Variable	p. 17
Moments, Characteristic Functions, and the Gaussian Distribution
Moments Defined	p. 19
Significance of the Moments	p. 21
The Correlation Matrix and Principal Axes	p. 24
The Characteristic Function	p. 25
Properties of the Characteristic Function	p. 25
Two Simple Examples	p. 27
The Central-Limit Theorem for Independent Variables	p. 30
The Gaussian Distribution from a More Physical Point of View	p. 33
Moments of a Gaussian Distribution	p. 37
The Jointly Normal Distribution	p. 38
Cumulants	p. 39
The Gram-Charlier Expansion	p. 39
Random Functions
Generalities: Multipoint Characteristic Functions	p. 43
Statistics for Derivatives and Integrals	p. 44
Processes and Characteristic Functionals: The Gaussian Process	p. 47
Limit Processes of Random Functions	p. 50
The Representation Problem	p. 54
Finite Total Energy and Characteristic Eddies	p. 57
Calculation of the Characteristic Eddies	p. 59
Rate of Convergence of the Series of Eigenfunctions	p. 60
Stationarity and the Ergodic Problem	p. 62
Autocorrelations of Stationary Processes and Their Properties	p. 68
Estimation by Time Averages	p. 71
The Representation Problem for Stationary Processes: Spectra	p. 76
Estimation by Time Averages with Zero Integral Scale	p. 79
Another Type of Representation Theorem for Stationary Processes: Characteristic Eddies	p. 80
Alternate Approaches to Harmonic Decomposition for Stationary Processes
A Central-Limit Theorem for Random Functions	p. 84
Random Processes in More Dimensions
Multidimensional Vector Fields of Finite Energy	p. 95
Homogeneity, Averaging, and Ergodicity in Several Dimensions	p. 96
The Homogeneous Scalar Field: One-Dimensional Spectra	p. 98
The Homogeneous Scalar Field: The Three-Dimensional Spectrum	p. 100
The Homogeneous Scalar Field: Consequences of Isotropy	p. 100
The Homogeneous Scalar Field: General Form of the Spectra	p. 102
The Solenoidal Homogeneous Vector Field: Implications of Incompressibility	p. 104
The Solenoidal Homogeneous Vector Field: One-Dimensional Spectra	p. 107
The Solenoidal Homogeneous Vector Field: The Three-Dimensional Spectrum	p. 107
The Solenoidal Homogenous Vector Field: Consequences of Isotropy	p. 108
The Solenoidal Homogeneous Vector Field: General Form of the Spectra	p. 111
Characteristic Eddies for a Homogeneous Vector Field	p. 113
Incompletely Homogeneous Fields: Co-and Quadrature Spectra and Coherence	p. 114
Characteristic Eddies for an Incompletely Homogeneous Field	p. 117
Multiple-Valued Functions	p. 121
Distribution of Solutions for an Algebraic Equation	p. 130
Fourier Transforms
Fourier Transforms of Well-Behaved Functions	p. 137
The Inverse Transform	p. 138
The Convolution	p. 139
Symmetry Properties	p. 140
Parseval's Relation	p. 141
Relations among Derivatives	p. 142
Shift of Variables	p. 142
Multiple Variables	p. 143
Tensors
Transformation Properties: Co- and Contra variant Indices	p. 145
The Metric Tensor: Changing Indices	p. 147
Cartesian Systems, Numerical Tensors, and Tensor Densities	p. 149
Differentiation	p. 152
Eigenvalues and Eigenvectors: Representations	p. 153
Principal Invariants, The Cayley-Hamilton Theorem, and Inverses	p. 156
Theory of Generalized Functions
Generalities	p. 159
Linear Continuous Functionals	p. 160
Addition and Multiplication by a Constant and by a Function	p. 161
Convergence of Sequences of Generalized Functions	p. 162
Differentiation and Integration of Generalized Functions	p. 162
Support of a Generalized Function	p. 164
Direct Product of Generalized Functions	p. 164
Convolutions of Generalized Functions	p. 164
Fourier Transforms of Generalized Functions	p. 166
Several Variables	p. 167
Effect of a Shift of Variables	p. 167
Asymptotic Behavior of Generalized Functions	p. 168
Fourier Transforms of Generalized Functions Defined on the Space of Bounded, Infinitely Differentiable Functions	p. 172
The Fourier Transform of the Convolution	p. 174
Behavior of the Fourier Transform	p. 175
The Kernel Theorem	p. 176
Representations of Generalized Functions of Finite Total Energy	p. 176
Invariant Theory, Isotropy, and Axisymmetry
Invariance under Transformation Groups	p. 179
Independent Invariants of Tensors of Various Orders	p. 180
Representations of Tensor Functions-A General Method	p. 182
Tensor Constants and Functions of a Scalar	p. 183
Tensor Functions of a Vector	p. 183
Tensor Functions of a Tensor of Second Rank	p. 184
References	p. 187
Index	p. 191
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