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

Brossier, Françoise; Lewandowski, Roger

ESAIM : Modélisation mathématique et analyse numérique, Tome 36 (2002) no. 2, pp. 345-372

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 ${\nu }_t$. The mixing length $\ell$ 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 $\ell$ and its asymptotic decreasing as $\ell \to \infty$ in more general cases$.$ Numerical experiments illustrate but also allow to extend these theoretical results: uniqueness is proved only for $\ell$ small enough while regular solutions are numerically obtained for any values of $\ell$. A convergence theorem is proved for turbulent kinetic energy: $k_{\ell }\to 0$ as $\ell \to \infty ,$ but for velocity $u_{\ell }$ we obtain only weaker results. Numerical results allow to conjecture that $k_{\ell }\to 0,$ ${\nu }_t\to \infty$ and $u_{\ell }\to 0$ as $\ell \to \infty .$ So we can conjecture that this classical turbulent model obtained with one degree of closure regularizes the solution.

Classification : 35Q30, 76M10, 76DXX, 76FXX, 46TXX, 65NXX
Keywords: turbulence modelling, energy methods, mixing length, finite-elements approximations

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