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. Introduction Reynolds Averaging of the Navier Stokes Equations (RANS) The closure problem Boussinesq’s ‘constitutive law’ First turbulence models for shear flows Review of nonlinear and nonlocal models The Difference Quotient Turbulence Model (DQTM) Self similar RANS Elementary turbulent shear flow solutions Thermodynamics of turbulence Turbulence – a cooperative phenomenon Conclusions and outlook.