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

@article{M2AN_2002__36_2_345_0, author = {Brossier, Fran\c coise and Lewandowski, Roger}, title = {Impact of the variations of the mixing length in a first order turbulent closure system}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, publisher = {EDP-Sciences}, volume = {36}, number = {2}, year = {2002}, pages = {345-372}, doi = {10.1051/m2an:2002016}, zbl = {1040.35057}, mrnumber = {1906822}, language = {en}, url = {http://www.numdam.org/item/M2AN_2002__36_2_345_0} }

Brossier, Françoise; Lewandowski, Roger. Impact of the variations of the mixing length in a first order turbulent closure system. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 36 (2002) no. 2, pp. 345-372. doi : 10.1051/m2an:2002016. http://www.numdam.org/item/M2AN_2002__36_2_345_0/

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