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, p. 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.

DOI : https://doi.org/10.1051/m2an:2002016
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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