For the simulations performed for nuclear safety, the flows to describe are most of the time turbulent. In this context, the aim of this work is to develop and analyse effective numerical schemes for LES in complex geometry domains (unstructured grids) for incompressible or low Mach number flows. Two requirements seem essential to build such schemes, namely to control kinetic energy and to be accurate for convection dominated flows.
The schemes under study are of fractional-step type, relying on a pressure correction method applied to the Navier-Stokes equations. The discretization is based on a low degree nonconforming finite element approximation in space (Rannacher-Turek).
Concerning the time marching algorithm, we propose a Crank-Nicolson like scheme for which we prove a kinetic energy control. This scheme has the advantage to be numerically low dissipative (numerical dissipation residual is second order in time). Concerning the Rannacher-Turek space discretization, it seems not very accurate for the simulation of convection dominatedflows, particularly with respect to the MAC scheme. To this purpose, two approaches are investigated in this work. The first approach consists in building a penalized scheme constraining the velocity degrees of freedom tangent to the faces to be written as a linear combination of some normal ones, thus giving a MAC-like scheme by making the penalization parameter to infinity. The second approach relies on the enrichment of the pressure approximation discrete space. For the discretization of the steady Stokes problem with this element, first order estimates in space for the velocity (H1 norm) and for the pressure (L2 norm) are proven in the case of uniform meshes constituted of rectangles or parallelograms.
Finally, various numerical tests are presented in both two and three dimensions and for general meshes, to illustrate the capacity of the schemes and compare theoretical and experimental results.