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

Brossier, Françoise; Lewandowski, Roger

doi:10.1051/m2an:2002016

Brossier, Françoise ; Lewandowski, Roger

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

Résumé

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 $ν_{t}$ . The mixing length $ℓ$ acts as a parameter which controls the turbulent part in $ν_{t}$ . 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 $ℓ \to \infty$ 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: $k_{ℓ} \to 0$ as $ℓ \to \infty,$ but for velocity $u_{ℓ}$ we obtain only weaker results. Numerical results allow to conjecture that $k_{ℓ} \to 0,$ $ν_{t} \to \infty$ and $u_{ℓ} \to 0$ as $ℓ \to \infty .$ So we can conjecture that this classical turbulent model obtained with one degree of closure regularizes the solution.

MR Zbl | 2 citations dans Numdam

DOI : 10.1051/m2an:2002016

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

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     title = {Impact of the variations of the mixing length in a first order turbulent closure system},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {345--372},
     publisher = {EDP-Sciences},
     volume = {36},
     number = {2},
     year = {2002},
     doi = {10.1051/m2an:2002016},
     mrnumber = {1906822},
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     url = {https://www.numdam.org/articles/10.1051/m2an:2002016/}
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Brossier, Françoise; Lewandowski, Roger. Impact of the variations of the mixing length in a first order turbulent closure system. ESAIM: Modélisation mathématique et analyse numérique, Tome 36 (2002) no. 2, pp. 345-372. doi : 10.1051/m2an:2002016. https://www.numdam.org/articles/10.1051/m2an:2002016/

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