First-order Reynolds Averaged Navier–Stokes (RANS) turbulence models are studied in this thesis.
These latter consist of the Navier–Stokes equations, supplemented with a system of balance equations describing the evolution of characteristic scalar quantities called “turbulent scales”. In so doing, the contribution of the turbulent agitation to the momentum can be determined by adding a diffusive coefficient (called “turbulent viscosity”) in the Navier-Stokes equations, such that it is defined as a function of the turbulent scales. The numerical analysis problems, which are studied in this dissertation, are treated in the frame of a fractional step algorithm, consisting of an approximation on regular meshes of the Navier–Stokes equations by the nonconforming Crouzeix–Raviart finite elements, and a set of scalar convection–diffusion balance equations discretized by the standard finite volume method.
A monotone numerical scheme based on the standard finite volume method is proposed so as to ensure that the turbulent scales, like the turbulent kinetic energy (k) and its dissipation rate (ε), remain positive in the case of the standard k − ε model, as well as the k − ε RNG and the extended k − ε − v2 − f models.
The convergence of the proposed numerical scheme is then studied on a system composed of the incompressible Stokes equations and a steady convection–diffusion equation, which are both coupled by the viscosities and the turbulent production term. This reduced model allows to deal with the main difficulty encountered in the analysis of such problems: the definition of the turbulent production term leads to consider a class of convection–diffusion problems with an irregular right-hand side belonging to L1.
Finally, to step towards the unsteady problem, the convergence of the finite volume scheme for a model convection–diffusion equation with L1 data is proved. The a priori estimates on the solution and on is time derivative are obtained in discrete norms, for which corresponding continuous spaces are not dual. Consequently a more general compactness result than the Kolmogorov theorem is proved, which can be seen as a discrete counterpart of the Aubin–Simon lemma. This result allows to conclude to the convergence in L1 of a sequence of discrete functions to a solution of the continuous problem.