Impact of the variations of the mixing length in a first order turbulentclosure system

Abstract

This paper is devoted to the study of a turbulentcirculation model. Equations are derived from the “Navier-Stokes turbulentkinetic energy” system. Some simplifications are performed but attentionis focused on non linearities linked to turbulent eddy viscosity  $\nu _{t}$ . The mixing length $\ell $ acts as a parameter which controls theturbulent part in $\nu _{t}$ . The main theoretical results that we haveobtained concern the uniqueness of the solution for bounded eddy viscositiesand small values of $\ell $ and its asymptotic decreasing as $\ell\rightarrow \infty $ in more general cases. Numerical experimentsillustrate but also allow to extend these theoretical results: uniqueness isproved only for $\ell $ small enough while regular solutions are numericallyobtained for any values of $\ell $ . A convergence theorem is proved forturbulent kinetic energy: $k_{\ell }\rightarrow 0$ as $\ell \rightarrow\infty ,$ but for velocity $u_{\ell }$ we obtain only weaker results.Numerical results allow to conjecture that $k_{\ell }\rightarrow 0,$ $\nu_{t}\rightarrow \infty $ and $u_{\ell }\rightarrow 0$ as $\ell \rightarrow\infty .$ So we can conjecture that this classical turbulent model obtainedwith one degree of closure regularizes the solution.