Stochastic Tools in Turbulence
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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
Product details
- No. of pages: 208
- Language: English
- Copyright: © Academic Press 1970
- Published: January 1, 1970
- Imprint: Academic Press
- eBook ISBN: 9780323162258
About the Author
John L. Lumey
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