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

Brossier, Françoise; Lewandowski, Roger

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

[1] A. Belmiloudi and F. Brossier, Numerical study of a control method computing a three-dimensional flow from the observed surface pressure. IRMAR publication (2000).

[2] C. Bernardi, T. Chacon, F. Murat and R. Lewandowski, A model for two coupled turbulent fluids. Part II: Numerical analysis of a spectral discretization, to appear in Siam Journ. of Num. An. Part III: Numerical approximation by finite elements, submitted in Advance in Mathematical Science and Application. | MR | Zbl

[3] S. Clain and R. Touzani, Solution of a two-dimensional stationary induction heating problem without boundedness of the coefficients. RAIRO Modél. Math. Anal. Numér. 31 (1997) 845-870. | Numdam | Zbl

[4] J. Duchon and R. Robert, Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations. Nonlinearity 13 (2000) 249-255. | Zbl

[5] T. Gallouët and R. Herbin, Existence of a solution to a coupled elliptic system. Appl. Math. Lett. 17 (1994) 49-55. | Zbl

[6] T. Gallouët, J. Lederer, R. Lewandowski, F. Murat and L. Tartar, On a turbulent system with unbounded eddy viscosity, J. Nonlinear Analysis Theory, Methods and Analysis, in press. | Zbl

[7] V. Girault and P.A. Raviart, Finite element methods for Navier-Stokes equations, theory and algorithm, Springer-Verlag (1986). | MR | Zbl

[8] R. Lewandowski, Analyse Mathématique et Océanographie. Collection RMA, Masson (1997).

[9] R. Lewandowski, The mathematical analysis of the coupling of a turbulent kinetic energy equation to the Navier-Stokes equation with an eddy viscosity. Nonlinear Anal. 28 (1997) 393-417. | Zbl

[10] R. Lewandowski (in preparation).

[11] J.L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Dunod (1968). | Zbl

[12] D. Martin and Melina, http://www.maths.univ-rennes1.fr/$\sim$dmartin.

[13] B. Mohammadi and O. Pironneau, Analysis of the $k$-epsilon model. Collection RMA, Masson (1994). | MR

[14] J. Oxtoby, Categories and measures. Springer-Verlag (1979).

[15] O. Pironneau, Méthode des éléments finis pour les fluides, Masson (1988). | Zbl