Nonlinear, Nonlocal and Fractional Turbulence: Alternative Recipes for the Modeling of Turbulence

Preface
	References
Contents
List of Symbols
	Roman Symbols
	Greek Symbols
	Special Symbols
Chapter 1: Introduction
	1.1 Aims and Scopes of This Book
	1.2 A Brief Tour d´Horizon Through Today´s Turbulence Field and Modeling
	References
Chapter 2: Reynold´s Averaging of the Navier-Stokes Equations (RANS)
	References
Chapter 3: The Closure Problem
	References
Chapter 4: Boussinesq´s ``Constitutive Law´´
	References
Chapter 5: First Turbulence Models for Shear Flows
	5.1 Shear Flows and the Works of Prandtl, Taylor, and Contemporaries
	5.2 Momentum and Vorticity Transfer Models
		5.2.1 Prandtl´s Mixing Length Model
		5.2.2 von Krmn´s Local Model
		5.2.3 Reichardt´s Inductive Model
		5.2.4 Prandtl´s Mean Gradient Model
		5.2.5 Prandtl´s Shear Layer Model
		5.2.6 Taylor´s Vorticity Transfer Model
	5.3 Overview of Deficiencies of Local Models
	5.4 More General Deficiencies and Fallacies
	5.5 Questioning the Logarithmic Law
	5.6 Logarithmic Versus (Deficit) Power Law
	References
Chapter 6: Review of Nonlinear and Nonlocal Models
	6.1 Nonlocality in Phase Space
	6.2 Atomic and Continuum Theories
	6.3 Stress as an Objective Polynomial Function of the Mean Rate of Strain Tensor
	6.4 Modified Diffusivity Models
	6.5 Truly History Dependent and Nonlocal Models
	References
Chapter 7: The Difference-Quotient Turbulence Model (DQTM)
	7.1 The Discovery and Prandtl´s Models
	7.2 Momentum Transfer Approach
		7.2.1 Molecular Transport
		7.2.2 Transport by Eddies
		7.2.3 Comparison of Laminar and Turbulent Flows
		7.2.4 Lévy Flight Turbulence Model and K41
			7.2.4.1 Introduction
			7.2.4.2 Lévy Walks on a One-Dimensional Lattice
			7.2.4.3 Lévy Walks, Lévy Flights, Lévy Pairs, and Eddies in Turbulence
			7.2.4.4 Eddy Class Statistics
			7.2.4.5 The Lifetime of Eddies
			7.2.4.6 The Eddy Diameters
			7.2.4.7 A Fractal Eddy Cascade Model
			7.2.4.8 The Occupation Number
			7.2.4.9 The Occupation Probability
			7.2.4.10 The Momenta of Eddies
			7.2.4.11 The Number of Eddy Classes
			7.2.4.12 Lévy Flight Statistics, β-Fractal Model, and the DQTM
	7.3 New Nonlocal Turbulence Models
		7.3.1 Introduction
		7.3.2 Liouville Fractional Derivative
		7.3.3 Overview of the Derivation of Important Nonlocal Turbulence Models
		7.3.4 Liouville-Prandtl Mixing Length Model
		7.3.5 The Heaviside-Liouville-Prandtl Shear Layer Model
		7.3.6 The Liouville-Heaviside Turbulence Model
		7.3.7 The Difference-Quotient Turbulence Model
		7.3.8 Summary
	References
Chapter 8: Self-Similar RANS
	Reference
Chapter 9: Elementary Turbulent Shear Flow Solutions
	9.1 Plane Wake Flows
	9.2 Axi-Symmetric Jets
		9.2.1 Jet in a Quiescent Surrounding
		9.2.2 Jet in a Parallel Co-flow
	9.3 Plane Couette Flows
	9.4 Plane Poiseuille Flows
	9.5 ``Wall Turbulent´´ Flows
	References
Chapter 10: Thermodynamics of Turbulence
	10.1 Introduction
		10.1.1 Microscopic and Macroscopic Theories
		10.1.2 Langevin and Fokker-Planck Equations
		10.1.3 Reduction of the Degrees of Freedom by Scaling
		10.1.4 Different Thermodynamic Concepts
	10.2 A Brief Review of Some Essentials of Boltzmann-Gibbs Thermodynamics
	10.3 Kraichnan´s BG Equilibrium Thermodynamics of 2-d and 3-d Turbulent Fields
	10.4 An Introduction to the Nonextensive Thermodynamics of Tsallis
	10.5 Relation Between Lévy Statistics and Tsallis Nonextensive Thermodynamics
	10.6 Escort Probability Distribution and Expectation Values
	10.7 Generalized Thermodynamic Potentials
	10.8 Fractional Calculus: A Promising Future-Oriented Method to Describe Turbulence
	10.9 Jackson´s Fractional Derivative and the DQTM
	10.10 Beck-Tsallis Thermodynamics of Turbulence
	10.11 Fractional Generalization of Kraichnan´s Energy-Enstrophy Spectrum and Its Validation by Numerical Experiments
	10.12 Velocity Structure Functions
	10.13 Justification of the Quadratic Form of the Energy as a Function of the Space Coordinates
	10.14 A Generalized Temperature of Turbulence
	10.15 Final Discussion on Nonextensive Thermodynamics of Turbulence
	References
Chapter 11: Turbulence: A Cooperative Phenomenon
	11.1 Introduction
	11.2 Cooperative Phenomena
		11.2.1 What Is a Critical or a Cooperative Phenomenon?
		11.2.2 Stress and Order Parameter
		11.2.3 Symmetry Breaking
		11.2.4 Response Functions and Critical Exponents
		11.2.5 Pair Correlation Function and Correlation Length
		11.2.6 Universality: Yes or No ?
		11.2.7 Turbulent Phase Transition with Its Two Phases
	11.3 Mean Field Theory of a Paramagnetic to Ferromagnetic Phase Transition
	11.4 Mean Field Theory of Turbulence
	11.5 First Experiments for a Qualitative Comparison
	11.6 Discussion of Results
	References
Chapter 12: Conclusions and Outlook
	References
Appendices
	Appendix A: Normalization of Probability Distribution
	Appendix B: The Variance of Lévy Flight Processes
	Appendix C: The Structure Function
	Appendix D: Circular Mean Velocity Profile of Plane Turbulent Poiseuille Flows
	Appendix E: Fourier Transformation for q-Generalized Energy Spectrum of Turbulent Flows
	References
Author Index
Subject Index