# Stochastic Tools in Turbulence

## 1st Edition

**Authors:**John L. Lumey

**eBook ISBN:**9780323162258

**Imprint:**Academic Press

**Published Date:**1st January 1970

**Page Count:**208

## Description

Stochastic Tools in Turbulence discusses the available mathematical tools to describe stochastic vector fields to solve problems related to these fields.

The book deals with the needs of turbulence in relation to stochastic vector fields, particularly, on three-dimensional aspects, linear problems, and stochastic model building. The text describes probability distributions and densities, including Lebesgue integration, conditional probabilities, conditional expectations, statistical independence, lack of correlation. The book also explains the significance of the moments, the properties of the characteristic function, and the Gaussian distribution from a more physical point of view. In considering fields, one must account for single-valued functions of one or more parameters, or collections of single-valued functions of one or more parameters such as time or space coordinates. The text also discusses multidimensional vector fields of finite energy, the characteristic eddies for a homogenous vector field, as well as, the distribution of solutions of an algebraic equation.

Engineers, algebra students, and professors of statistics and advanced mathematics will find the book highly useful.

## Table of Contents

Preface

1. Probability Distributions and Densities

1.1 Definition of Probability

1.2 Formalizing the Definition

1.3 Measure and Lebesgue Integration

1.4 Distribution Function

1.5 The Probability Density Function

1.6 A Simple Example of the Distribution and Density Functions

1.7 Expected Values

1.8 Joint Distribution Functions

1.9 The Joint Density Function

1.10 Expected Values

1.11 Conditional Probabilities

1.12 Conditional Expectations

1.13 Statistical Independence and Lack of Correlation

1.14 Change of Variable

2. Moments, Characteristic Functions, and the Gaussian Distribution

2.1 Moments Defined

2.2 Significance of the Moments

2.3 The Correlation Matrix and Principal Axes

2.4 The Characteristic Function

2.5 Properties of the Characteristic Function

2.6 Two Simple Examples

2.7 The Central-Limit Theorem for Independent Variables

2.8 The Gaussian Distribution from a More Physical Point of View

2.9 Moments of a Gaussian Distribution

2.10 The Jointly Normal Distribution

2.11 Cumulants

2.12 The Gram-Charlier Expansion

3. Random Functions

3.1 Generalities: Multipoint Characteristic Functions

3.2 Statistics for Derivatives and Integrals

3.3 Processes and Characteristic Functionals: The Gaussian Process

3.4 Limit Processes of Random Functions

3.5 The Representation Problem

3.6 Finite Total Energy and Characteristic Eddies

3.7 Calculation of the Characteristic Eddies

3.8 Rate of Convergence of the Series of Eigenfunctions

3.9 Stationarity and the Ergodic Problem

3.10 Autocorrelations of Stationary Processes and Their Properties

3.11 Estimation by Time Averages

3.12 The Representation Problem for Stationary Processes: Spectra

3.13 Estimation by Time Averages with Zero Integral Scale

3.14 Another Type of Representation Theorem for Stationary Processes: Characteristic Eddies

3.15 Alternate Approaches to Harmonic Decomposition for Stationary Processes

3.16 A Central-Limit Theorem for Random Functions

4. Random Processes in More Dimensions

4.1 Multidimensional Vector Fields of Finite Energy

4.2 Homogeneity, Averaging, and Ergodicity in Several Dimensions

4.3 The Homogeneous Scalar Field: One-Dimensional Spectra

4.4 The Homogeneous Scalar Field: The Three-Dimensional Spectrum

4.5 The Homogeneous Scalar Field: Consequences of Isotropy

4.6 The Homogeneous Scalar Field: General Form of the Spectra

4.7 The Solenoidal Homogeneous Vector Field: Implications of Incompressibility

4.8 The Solenoidal Homogeneous Vector Field: One-Dimensional Spectra

4.9 The Solenoidal Homogeneous Vector Field: The Three-Dimensional Spectrum

4.10 The Solenoidal Homogenous Vector Field: Consequences of Isotropy

4.11 The Solenoidal Homogeneous Vector Field: General Form of the Spectra

4.12 Characteristic Eddies for a Homogeneous Vector Field

4.13 Incompletely Homogeneous Fields: Co- and Quadrature Spectra and Coherence

4.14 Characteristic Eddies for an Incompletely Homogeneous Field

4.15 Multiple-Valued Functions

4.16 Distribution of Solutions for an Algebraic Equation

Appendix 1. Fourier Transforms

A1.1 Fourier Transforms of Well-Behaved Functions

A1.2 The Inverse Transform

A1.3 The Convolution

A1.4 Symmetry Properties

A1.5 Parseval's Relation

A1.6 Relations Among Derivatives

A1.7 Shift of Variables

A1.8 Multiple Variables

Appendix 2. Tensors

A2.1 Transformation Properties: Co- and Contravariant Indices

A2.2 The Metric Tensor: Changing Indices

A2.3 Cartesian Systems, Numerical Tensors, and Tensor Densities

A2.4 Differentiation

A2.5 Eigenvalues and Eigenvectors: Representations

A2.6 Principal Invariants, The Cayley-Hamilton Theorem, and Inverses

Appendix 3. Theory of Generalized Functions

A3.1 Generalities

A3.2 Linear Continuous Functionals

A3.3 Addition and Multiplication by a Constant and by a Function

A3.4 Convergence of Sequences of Generalized Functions

A3.5 Differentiation and Integration of Generalized Functions

A3.6 Support of a Generalized Function

A3.7 Direct Product of Generalized Functions

A3.8 Convolutions of Generalized Functions

A3.9 Fourier Transforms of Generalized Functions

A3.10 Several Variables

A3.11 Effect of a Shift of Variables

A3.12 Asymptotic Behavior of Generalized Functions

A3.13 Fourier Transforms of Generalized Functions Defined on the Space of Bounded, Infinitely Differentiable Functions

A3.14 The Fourier Transform of the Convolution

A3.15 Behavior of the Fourier Transform

A3.16 The Kernel Theorem

A3.17 Representations of Generalized Functions of Finite Total Energy

Appendix 4. Invariant Theory, Isotropy, and Axisymmetry

A4.1 Invariance under Transformation Groups

A4.2 Independent Invariants of Tensors of Various Orders

A4.3 Representations of Tensor Functions—A General Method

A4.4 Tensor Constants and Functions of a Scalar

A4.5 Tensor Functions of a Vector

A4.6 Tensor Functions of a Tensor of Second Rank

References

Index

## Details

- No. of pages:
- 208

- Language:
- English

- Copyright:
- © Academic Press 1970

- Published:
- 1st January 1970

- Imprint:
- Academic Press

- eBook ISBN:
- 9780323162258