Impact of the variations of the mixing length in a first order turbulent closure system

This paper is devoted to the study of a turbulent circulation model. Equations are derived from the "Navier-Stokes turbulent kinetic energy" system. Some simplifications are performed but attention is focused on non linearities linked to turbulent eddy viscosity nut. The mixing length l acts as a parameter which controls the turbulent part in nu(t). The main theoretical results that we have obtained concern the uniqueness of the solution for bounded eddy viscosities and small values of l and its asymptotic decreasing as l --> infinity in more general cases: Numerical experiments illustrate but also allow to extend these theoretical results: uniqueness is proved only for l small enough while regular solutions are numerically obtained for any values of l. A convergence theorem is proved for turbulent kinetic energy: k(l) --> 0 as l --> infinity but for velocity u(l) we obtain only weaker results. Numerical results allow to conjecture that k(l) --> 0; nu(t) -->infinity and u(l) --> 0 as l --> infinity. So we can conjecture that this classical turbulent model obtained with one degree of closure regularizes the solution.