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

Experts of fluid dynamics agree that turbulence is nonlinear and nonlocal. Because of a direct correspondence, nonlocality also implies fractionality. Fractional dynamics is the physics related to fractal (geometrical) systems and is described by fractional calculus. Up-to-present, numerous criticisms of linear and local theories of turbulence have been published. Nonlinearity has established itself quite well, but so far only a very small number of general nonlocal concepts and no concrete nonlocal turbulent flow solutions were available. This book presents the first analytical and numerical solutions of elementary turbulent flow problems, mainly based on a nonlocal closure. Considerations involve anomalous diffusion (Lévy flights), fractal geometry (fractal-β, bi-fractal and multi-fractal model) and fractional dynamics. Examples include a new ‘law of the wall’ and a generalization of Kraichnan’s energy-enstrophy spectrum that is in harmony with non-extensive and non-equilibrium thermodynamics (Tsallis thermodynamics) and experiments. Furthermore, the presented theories of turbulence reveal critical and cooperative phenomena in analogy with phase transitions in other physical systems, e.g., binary fluids, para-ferromagnetic materials, etc.; the two phases of turbulence identifying the laminar streaks and coherent vorticity-rich structures. This book is intended, apart from fluids specialists, for researchers in physics, as well as applied and numerical mathematics, who would like to acquire knowledge about alternative approaches involved in the analytical and numerical treatment of turbulence. Preface......Page 5 References......Page 15 Contents......Page 16 Roman Symbols......Page 20 Greek Symbols......Page 31 Special Symbols......Page 36 1.1 Aims and Scopes of This Book......Page 37 1.2 A Brief Tour d ́Horizon Through Today ́s Turbulence Field and Modeling......Page 39 References......Page 43 Chapter 2: Reynold ́s Averaging of the Navier-Stokes Equations (RANS)......Page 49 References......Page 54 Chapter 3: The Closure Problem......Page 55 References......Page 58 Chapter 4: Boussinesq ́s ``Constitutive Law ́ ́......Page 60 References......Page 62 5.1 Shear Flows and the Works of Prandtl, Taylor, and Contemporaries......Page 63 5.2.1 Prandtl ́s Mixing Length Model......Page 65 5.2.2 von Krmn ́s Local Model......Page 68 5.2.3 Reichardt ́s Inductive Model......Page 69 5.2.4 Prandtl ́s Mean Gradient Model......Page 71 5.2.5 Prandtl ́s Shear Layer Model......Page 72 5.2.6 Taylor ́s Vorticity Transfer Model......Page 73 5.3 Overview of Deficiencies of Local Models......Page 76 5.4 More General Deficiencies and Fallacies......Page 78 5.5 Questioning the Logarithmic Law......Page 93 5.6 Logarithmic Versus (Deficit) Power Law......Page 98 References......Page 104 6.1 Nonlocality in Phase Space......Page 109 6.2 Atomic and Continuum Theories......Page 112 6.3 Stress as an Objective Polynomial Function of the Mean Rate of Strain Tensor......Page 114 6.4 Modified Diffusivity Models......Page 116 6.5 Truly History Dependent and Nonlocal Models......Page 122 References......Page 136 7.1 The Discovery and Prandtl ́s Models......Page 140 7.2.1 Molecular Transport......Page 144 7.2.2 Transport by Eddies......Page 150 7.2.4.1 Introduction......Page 157 7.2.4.2 Lévy Walks on a One-Dimensional Lattice......Page 159 7.2.4.3 Lévy Walks, Lévy Flights, Lévy Pairs, and Eddies in Turbulence......Page 162 7.2.4.4 Eddy Class Statistics......Page 165 7.2.4.5 The Lifetime of Eddies......Page 167 7.2.4.6 The Eddy Diameters......Page 168 7.2.4.7 A Fractal Eddy Cascade Model......Page 170 7.2.4.8 The Occupation Number......Page 173 7.2.4.9 The Occupation Probability......Page 175 7.2.4.10 The Momenta of Eddies......Page 177 7.2.4.11 The Number of Eddy Classes......Page 178 7.2.4.12 Lévy Flight Statistics, β-Fractal Model, and the DQTM......Page 181 7.3.1 Introduction......Page 190 7.3.2 Liouville Fractional Derivative......Page 193 7.3.3 Overview of the Derivation of Important Nonlocal Turbulence Models......Page 194 7.3.4 Liouville-Prandtl Mixing Length Model......Page 195 7.3.6 The Liouville-Heaviside Turbulence Model......Page 197 7.3.7 The Difference-Quotient Turbulence Model......Page 198 7.3.8 Summary......Page 201 References......Page 203 Chapter 8: Self-Similar RANS......Page 206 Reference......Page 211 9.1 Plane Wake Flows......Page 212 9.2.1 Jet in a Quiescent Surrounding......Page 221 9.2.2 Jet in a Parallel Co-flow......Page 253 9.3 Plane Couette Flows......Page 258 9.4 Plane Poiseuille Flows......Page 280 9.5 ``Wall Turbulent ́ ́ Flows......Page 304 References......Page 326 10.1.1 Microscopic and Macroscopic Theories......Page 330 10.1.3 Reduction of the Degrees of Freedom by Scaling......Page 331 10.1.4 Different Thermodynamic Concepts......Page 332 10.2 A Brief Review of Some Essentials of Boltzmann-Gibbs Thermodynamics......Page 334 10.3 Kraichnan ́s BG Equilibrium Thermodynamics of 2-d and 3-d Turbulent Fields......Page 336 10.4 An Introduction to the Nonextensive Thermodynamics of Tsallis......Page 347 10.5 Relation Between Lévy Statistics and Tsallis Nonextensive Thermodynamics......Page 353 10.6 Escort Probability Distribution and Expectation Values......Page 357 10.7 Generalized Thermodynamic Potentials......Page 359 10.8 Fractional Calculus: A Promising Future-Oriented Method to Describe Turbulence......Page 360 10.9 Jackson ́s Fractional Derivative and the DQTM......Page 362 10.10 Beck-Tsallis Thermodynamics of Turbulence......Page 364 10.11 Fractional Generalization of Kraichnan ́s Energy-Enstrophy Spectrum and Its Validation by Numerical Experiments......Page 365 10.12 Velocity Structure Functions......Page 369 10.13 Justification of the Quadratic Form of the Energy as a Function of the Space Coordinates......Page 371 10.14 A Generalized Temperature of Turbulence......Page 375 10.15 Final Discussion on Nonextensive Thermodynamics of Turbulence......Page 380 References......Page 382 11.1 Introduction......Page 387 11.2.1 What Is a Critical or a Cooperative Phenomenon?......Page 390 11.2.2 Stress and Order Parameter......Page 392 11.2.3 Symmetry Breaking......Page 394 11.2.4 Response Functions and Critical Exponents......Page 396 11.2.5 Pair Correlation Function and Correlation Length......Page 400 11.2.6 Universality: Yes or No ?......Page 402 11.2.7 Turbulent Phase Transition with Its Two Phases......Page 404 11.3 Mean Field Theory of a Paramagnetic to Ferromagnetic Phase Transition......Page 406 11.4 Mean Field Theory of Turbulence......Page 411 11.5 First Experiments for a Qualitative Comparison......Page 418 11.6 Discussion of Results......Page 421 References......Page 423 Chapter 12: Conclusions and Outlook......Page 426 References......Page 431 Appendix A: Normalization of Probability Distribution......Page 433 Appendix B: The Variance of Lévy Flight Processes......Page 434 Appendix C: The Structure Function......Page 435 Appendix D: Circular Mean Velocity Profile of Plane Turbulent Poiseuille Flows......Page 436 Appendix E: Fourier Transformation for q-Generalized Energy Spectrum of Turbulent Flows......Page 443 References......Page 449 Author Index......Page 450 Subject Index......Page 456 Experts of fluid dynamics agree that turbulence is nonlinear and nonlocal. Because of a direct correspondence, nonlocality also implies fractionality. Fractional dynamics is the physics related to fractal (geometrical) systems and is described by fractional calculus. Up-to-present, numerous criticisms of linear and local theories of turbulence have been published. Nonlinearity has established itself quite well, but so far only a very small number of general nonlocal concepts and no concrete nonlocal turbulent flow solutions were available. This book presents the first analytical and numerical solutions of elementary turbulent flow problems, mainly based on a nonlocal closure. Considerations involve anomalous diffusion (Lévy flights), fractal geometry (fractal-[beta], bi-fractal and multi-fractal model) and fractional dynamics. Examples include a new 'law of the wall' and a generalization of Kraichnan's energy-enstrophy spectrum that is in harmony with non-extensive and non-equilibrium thermodynamics (Tsallis thermodynamics) and experiments. Furthermore, the presented theories of turbulence reveal critical and cooperative phenomena in analogy with phase transitions in other physical systems, e.g., binary fluids, para-ferromagnetic materials, etc.; the two phases of turbulence identifying the laminar streaks and coherent vorticity-rich structures. This book is intended, apart from fluids specialists, for researchers in physics, as well as applied and numerical mathematics, who would like to acquire knowledge about alternative approaches involved in the analytical and numerical treatment of turbulence Experts of fluid dynamics agree that turbulence is nonlinear and nonlocal. Because of a direct correspondence, nonlocality also implies fractionality. Fractional dynamics is the physics related to fractal (geometrical) systems and is described by fractional calculus. Up-to-present, numerous criticisms of linear and local theories of turbulence have been published. Nonlinearity has established itself quite well, but so far only a very small number of general nonlocal concepts and no concrete nonlocal turbulent flow solutions were available. This book presents the first analytical and numerical solutions of elementary turbulent flow problems, mainly based on a nonlocal closure. Considerations involve anomalous diffusion (Lévy flights), fractal geometry (fractal- β , bi-fractal and multi-fractal model) and fractional dynamics. Examples include a new 'law of the wall' and a generalization of Kraichnan's energy-enstrophy spectrum that is in harmony with non-extensive and non-equilibrium thermodynamics (Tsallis thermodynamics) and experiments. Furthermore, the presented theories of turbulence reveal critical and cooperative phenomena in analogy with phase transitions in other physical systems, e.g., binary fluids, para-ferromagnetic materials, etc.; the two phases of turbulence identifying the laminar streaks and coherent vorticity-rich structures. This book is intended, apart from fluids specialists, for researchers in physics, as well as applied and numerical mathematics, who would like to acquire knowledge about alternative approaches involved in the analytical and numerical treatment of turbulence.