Nonlinear, nonlocal and fractional turbulence : alternative recipes for the modeling of turbulence
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書誌事項
Nonlinear, nonlocal and fractional turbulence : alternative recipes for the modeling of turbulence
Springer, c2020
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Includes bibliographical references and indexes
内容説明・目次
内容説明
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 (Levy flights), fractal geometry (fractal- , 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.
目次
I. IntroductionA. Aims and scopes of this book B. A brief tour d'horizon through today's turbulence field and modeling
II. Reynolds Averaging of the Navier-Stokes Equations (RANS)
III. The closure problem IV. Boussinesq's 'constitutive law'
V. First turbulence models for shear flows A. Shear flows and the works of Prandtl, Taylor and contemporairesB. Momentum and vorticity transfer models 1. Prandtl's mixing length model 2. von Karman's local model 3. Reichardt's inductive model 4. Prandtl's mean gradient model 5. Prandtl's shear layer model 6. Taylor's vorticity transfer modelC. Overview of deficiencies of local modelsD. More general deficiencies and fallacies
E. Questioning the logarithmic lawF. Logarithmic versus (deficit) power law
VI. Review of nonlinear and nonlocal modelsA. Nonlocality in phase spaceB. Atomic and continuum theoriesC. Stress as an objective polynomial function of the mean rate of strain tensor D. Modified diffusivity models E. Truly history-dependent and nonlocal models
VII. The Difference-Quotient Turbulence Model (DQTM)A. The discovery and Prandtl's models
B. Momentum transfer approach 1. Molecular transport 2. Transport by eddies 3. Comparison of laminar and turbu-lent flows 4. Levy-flight turbulence model and K41 4.1 Introduction 4.2 Levy walks on a one-dimensional lattice 4.3 Levy walks, Levy flights , Levy pairs and eddies in turbulence
4.4 Eddy class statistics 4.5 The life time of eddies 4.6 The eddy diameters 4.7 The fractal eddy cascade model 4.8 The occupation number 4.9 The occupation probability 4.10 The momenta of eddies 4.11 The number of eddy classes 4.12 Levy flight statistics, beta-fractal model and the DQTM
C. New nonlocal turbulence models 1. Introduction 2. Liouville fractional derivative 3. Overview of the derivation of im- portant nonlocal turbulence models 4. Liouville-Prandtl Mixing-Length Model 5. The Heaviside-Liouville -Prandtl Shear-Layer Model 6. The Liouville-Heaviside Turbu- lence Model 7. The Difference Quotient Turbu-lence Model 8. Summary
VIII. Self-similar RANS
IX. Elementary turbulent shear flow solutionsA. Plane wake flowB. Axi-symmetric jets 1. Jet in a quiescent surrounding 2. Jet in a parallel co-flow C. Plane Couette flows D. Plane Poiseuille flows E. "Wall-turbulent" flows
X. Thermodynamics of turbulence A. Introduction 1. Micro- and macroscopic theories 2. Langevin and Fokker-Planck equations 3. Reduction of degree of freedom by scaling 4. Different thermodynamic conceptsB. A brief review of some essentials of Boltzmann-Gibbs thermodynamicsC. Kraichnan's BG-equilibrium thermody-namics of 2-d and 3-d turbulent fieldsD. An introduction to extensive thermo-dynamics of TsallisE. Relation between Levy statistics and Tsallis extensive thermodynamicsF. Escort probability distribution and ex-pectation valuesG. Generalized thermodynamic potentialsH. Fractional calculus: A promising future-oriented method to describe turbulenceI. Jackson's fractional derivative and the DQTMJ. Beck-Tsallis thermodynamics of turbulence
K. Fractional generalization of Kraichnan's spectra and their validation by numerical experiments L. Velocity structure functionsM. Justification of the quadratic form of the energy as a function of space coordinatesN. A generalized temperature of turbulenceO. Final discussion on the extensive thermodynamics of turbulence
XI. Turbulence - a cooperative phenomenonA. IntroductionB. Cooperative phenomena 1. What is a critical or cooperative phenomenon? 2. Stress and order parameter 3. Symmetry breaking 4. Response functions and critical exponents 5. Pair correlation function and correlation length 6. Universality: yes, or no? 7. Turbulent phase transition with its two phasesC. Mean field theory of a paramagnetic to ferromagnetic phase transitionD. Mean field theory of turbulenceE. First experiments for a qualitative comparisionF. Discussion of results
XII. Conclusions and outlook AppendicesA. Normalization of probability distributionB. The variance of a Levy flight processC. The structure and the Weierstrass functionD. Circular mean velocity profile of plane Turbulent Poiseuille flowsE. Fourier transformation for q-generalized energy spectrum of turbulent flowsF Extremum principles for Boltzmann-Gibbs entropyG. Laplace multipliers, Tsallis factor and generalized temperature of turbulence
References
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