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 Kármán'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. Lévy-flight turbulence model and K41 4.1 Introduction 4.2 Lévy walks on a one-dimensional lattice 4.3 Lévy walks, Lévy flights, Lévy 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 Lévy 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 Lévy statistics and Tsallis extensive thermodynamicsF. Escort probability distribution and ex-pectation valuesG. Generalized thermodynamic potentialsH. Fractional calculus: A promising future-oriented method to d
About the Author: Peter W. Egolf made an apprenticeship as a heating designer and studied at the University of Applied Sciences of Central Switzerland heating and air conditioning. After working in an industrial R&D laboratory, he studied physics at the Swiss Federal Institute of Technology (ETH Zurich). In 1984 he obtained his diploma in Dynamical Meteorology. Then he entered a R&D division at Gebrüder Sulzer AG in Winterthur, where he invented and investigated industrial air conditioning systems. At an advanced age he had the opportunity to make a PhD in a Noble Laureate 'family' in low-temperature physics (superfluidity). In 1989 Egolf introduced the Lagrange-Hamilton description of the free surface of quantum fluid He II. In 1990 he obtained his PhD at ETHZ with an innovation award. In the late 1980's he invented the Difference-Quotient Turbulence Model (DQTM). From 1990 on a ten years' employment at the Swiss Federal Institute for Materials Testing and Research followed in the field of energy and buildings. In 1994 he entered the field of research on ice slurries. In 1998 Egolf created the International Working Party on Ice Slurries of the International Institute of Refrigeration (IIR) and was its first President. From 2000-2018 he was the head of the Theory and Numerics Division (SIT) of the Thermal Sciences Institute at the University of Applied Sciences of Western Switzerland. He initiated a second International Working Party of the IIR (on Magnetic Refrigeration), and for ten years he was serving as its President. Furthermore, he was awarded the first prices of the Swiss Technology Awards 1996 and 2006. Today, Peter W. Egolf is retired, however, he is still involved in the study of fundamental research on fractional turbulence and nonextensive thermodynamics of turbulence. He was a main organizer of eight international conferences. Egolf invented or co-invented a new melting/freezing model, an innovative storage device for ice slurries, a translucent solar thermal wall, an innovative decentralized air-conditioning system, a high-frequency magnetic refrigerator, a new hyper-thermia cancer treatment method with rotating nanowires, etc. You will find more information on his biography in numerous editions of Who'sWho in the World, Who's Who in Science and Engineering and Who'sWho in America.

Kolumban Hutter received a diploma in civil engineering in 1964 from ETH Zurich and a M. Sc. & Ph. Degree in Theoretical and Applied Mechanics from Cornell University, Ithaca, New York in 1973. He has held the position of Professor of Mechanics at Darmstadt University of Technology, Germany from 1987-2006. Presently he is a guest scientist at the Laboratory of Hydraulics, Hydrology and Glaciology (VAW), ETH Zurich and was from 2006-2009 a part time researcher at Academia Sinica, Taiwan. His research interests are in geophysical mechanics with applications in the dynamics of glaciers and ice sheets, the mechanics of granular materials, avalanching flows of snow, debris and mud, physical limnology and the foundations of continuum mechanics and thermodynamics. Kolumban Hutter is author and co-author of more than 400 papers (more than 300 peer reviewed) and has written or edited more than 20 books, among these areTheoretical Glaciology, Continuum Methods of Physical Modeling (with K. Jöhnk) Avalanche Dynamics (with S. P. Pudasaini) Solid-Fluid Mixtures of Frictional Materials in Geophysical and Geotechnical Context (with L. Schneider), Physics of Lakes (with Profs Y. Wang and I.Chubarenko, 3 volumes), Fluid & Thermodynamics (with Prof Y. Wang, 2 volumes) all published by Springer-Verlag.He has also served as scientific editor of the Journal of Glaciology for 14 years and was the founder and Editor-in-Chief of Continuum Mechanics and Thermodynamics for 17 years. He is also the Founder and (now past) Editor-in-Chief of the Series "Advances in Geophysical and Environmental Mechanics" published by Springer Verlag. Kolumban Hutter was awarded the Max-Planck Prize of the Max-Planck Society and the Alexander von Humboldt Foundation (Germany) in 1994, the Alexander von Humboldt Prize of the Foundation of Polish Science in 1998, and the Seligman Crystal of the International Glaciological Society in 2003. He lives retired in Zurich, Switzerland