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 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 → ∞ 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 → 0 as ℓ → ∞ , but for velocity u ℓ we obtain only weaker results. Numerical results allow to conjecture that k ℓ → 0 , ν t → ∞ and u ℓ → 0 as ℓ → ∞ . 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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