Turbulence Models for Incompressible Fluids Derived from Kinetic Theory

Turbulence models for incompressible fluids are derived from kinetic theory. The kinetic model involves a relaxation time type collision operator which describes the relaxation of the probability distribution function (pdf) towards an isotropic pdf on a time scale tau. The dependence of tau upon the kinetic turbulent energy can be tuned in such a way that both the so called "viscous subrange" (dominated by molecular viscosity) and the "inertial range" (obeying the Kolmogorov law) can be described. Starting from this kinetic model and focusing on the "inertial range" regime, we derive turbulence models of k - epsilon type and compare the form an the properties of this model with the standard k - epsilon models for incompressible flows. Finally, macroscopic system of equations including the mean velocity, the turbulent energy, and an arbitrary higher order velocity-moment of the 'pdf', is derived by means of a minimisation entropy principle.