Mathematical Modelling and Numerical Analysis ESAIM: M2AN Modelisation Mathematique et Analyse Numerique Will be set by the publisher DOI: (will be inserted later) IMPACT OF THE VARIATIONS OF THE MIXING LENGTH IN A FIRST ORDER TURBULENT CLOSURE SYSTEM

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 νt. The mixing length ` acts as a parameter which controls the turbulent part in νt. 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, νt → ∞ and u ` → 0 as `→∞. So we can conjecture that this classical turbulent model obtained with one degree of closure regularizes the solution. Mathematics Subject Classification. | Please, give AMS classication codes |. Received: October 3, 2001. 1