Turbulence plays an important role in many industrial applications (flow, heat transfer, chemical reactions). Since Direct Simulation (DNS) is often an excessive cost in computing time, Reynolds Models (RANS) are then used in CFD (computational fluid dynamics) codes. The best known, which was published in the 70s, is the k - epsilon model.
It results in two additional non-linear equations coupled to the Navier-Stokes equations, describing the transport, for one, of turbulent kinetic energy (k) and, for the other, of its dissipation rate (epsilon). ). A very important property to check is the positivity of the parameters k and epsilon which is necessary for the system of equations modeling the turbulence to remain stable. It is therefore crucial that the discretization of these models preserves the monotony. The equations being of convection-diffusion type, it is well known that with classical linear schemes (finite elements, finite volumes, etc ...), the numerical solutions are likely to oscillate on distorted meshes. The negative values of the parameters k and epsilon are then at the origin of the stop of the simulation.
We are interested in nonlinear methods allowing to obtain compact stencils. For diffusion operators, they rely on nonlinear combinations of fluxes on either side of each edge. These approaches have proved their efficiency, especially for the suppression of oscillations on very distorted meshes. We can also take the ideas proposed in the literature where it is for example described nonlinear corrections applying on classical linear schemes. The idea would be to apply this type of method on the diffusive operators appearing in the k-epsilon models. In this context it will also be interesting to transform classical schemes of literature approaching gradients into nonlinear two-point fluxes. Fundamental questions need to be considered in the case of general meshes about the consistency and coercivity of the schemes studied.
During this thesis, we will take the time to solve the basic problems of these methods (first and second year), both on the theoretical aspects and on the computer implementation. This can be done in Castem, TrioCFD or Trust development environments. We will then focus on regular analytical solutions and application cases representative of the community.