We present a class of kinetic models which may reproduce the phenomenology of turbulent incompressible flows. The fluid model derived from the macroscopic limit of this kinetic model reduces to the usual Navier-Stokes in the case of laminar flow. In the case of turbulent flow, it describes the evolution of the distribution of turbulent energy subject to the interactions between the fluid structures (production, diffusion and dissipation). The typical interaction time scale can be chosen so as to recover the molecular viscosity in the non-turbulent limit and to fit with the Kolmogorov law otherwise.
Joint work with M. Lemou.