Table Des Matières 4 Finite Volume Scheme for Convection–diffusion Equation with L 1.1 Introduction 1.2 Équations Des Modèles Physiques

In this paper which is an extended version of an article submitted to the Finite Volumes for Complex Applications V conference, we design a finite-volume based numerical scheme for the solution of the nonlinear balance equations of RNG variants of the well-known k − ε model and the v − f system encountered in the k − ε− v − f model which can be seen as an extension of first order turbulence models. In this class of models, the description of the turbulence relies on two variables, the turbulent kinetic energy k and its dissipation rate ε, which, for physical reasons, must remain positive. When standard upwinding techniques for the convection terms are used, the presented scheme is proved to preserve the positivity of these two unknowns, and, through a topological degree argument, to admit at least a solution. Moreover, the computation of the values of k and ε in the near-wall regions requires that the mesh is highly refined if no treatment if enforced, since the characteristic scales of the solution decay towards the order of the length of molecular dissipation as the distance to the wall tends to zero. We described here for the sake of completness of the dissertation, the underlying ideas to design of wall-laws in the case of the k − ε and k − ε − v − f models, which amounts to enforcing Dirichlet boundary conditions for the turbulent scales at walls, which are solution to an asymptotic model. The study led to the reevaluation of some model constants, comparing to the literature, in the case of the k − ε − v − f model. Finally, a numerical convergence study has been performed to assess the properties of the scheme: a first order convergence rate in both time and space is verified in the case of upwind fluxes and when using a MUSCL discretization for the approximation of the convection terms, the scheme becomes of second order in space. 2.