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

Françoise Brossier; Roger Lewandowski

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 36, Issue: 2, page 345-372
  • ISSN: 0764-583X

Abstract

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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 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 and its asymptotic decreasing as in more general cases. Numerical experiments illustrate but also allow to extend these theoretical results: uniqueness is proved only for small enough while regular solutions are numerically obtained for any values of . A convergence theorem is proved for turbulent kinetic energy: k 0 as , but for velocity u we obtain only weaker results. Numerical results allow to conjecture that k 0 , ν t and u 0 as . So we can conjecture that this classical turbulent model obtained with one degree of closure regularizes the solution.

How to cite

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Brossier, Françoise, and Lewandowski, Roger. "Impact of the variations of the mixing length in a first order turbulent closure system." ESAIM: Mathematical Modelling and Numerical Analysis 36.2 (2010): 345-372. <http://eudml.org/doc/194108>.

@article{Brossier2010,
abstract = { 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 \rightarrow \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: $k_\{\ell \}\rightarrow 0$ 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 obtained with one degree of closure regularizes the solution. },
author = {Brossier, Françoise, Lewandowski, Roger},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Turbulence modelling; energy methods; mixing length; finite-elements approximations.; turbulence modelling; finite element approximation; turbulent eddy viscosity; uniqueness; convergence},
language = {eng},
month = {3},
number = {2},
pages = {345-372},
publisher = {EDP Sciences},
title = {Impact of the variations of the mixing length in a first order turbulent closure system},
url = {http://eudml.org/doc/194108},
volume = {36},
year = {2010},
}

TY - JOUR
AU - Brossier, Françoise
AU - Lewandowski, Roger
TI - Impact of the variations of the mixing length in a first order turbulent closure system
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 36
IS - 2
SP - 345
EP - 372
AB - 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 \rightarrow \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: $k_{\ell }\rightarrow 0$ 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 obtained with one degree of closure regularizes the solution.
LA - eng
KW - Turbulence modelling; energy methods; mixing length; finite-elements approximations.; turbulence modelling; finite element approximation; turbulent eddy viscosity; uniqueness; convergence
UR - http://eudml.org/doc/194108
ER -

References

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  12. D. Martin and Melina, .  URIhttp://www.maths.univ-rennes1.fr/dmartin
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