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.

The simulation of the Navier-Stokes equations requires accurate and robust numerical methods that
take into account diffusion operators, gradient and convection terms. Operational approaches have
shown their effectiveness on simplexes. However, in some models or codes
(TrioCF, Flica5), it may be useful to improve the accuracy of solutions locally using an
error estimator or to take into account general meshes. We are here interested in staggered schemes.
This means that the pressure is calculated at the centre of the mesh and the velocities on the edges
(or faces) of the mesh. This results in methods that are naturally accurate at low Mach numbers .
New schemes have recently been presented in this context and have shown their
robustness and accuracy. However, these discretisations can be very costly in terms of memory and
computation time compared with MAC schemes on regular meshes
We are interested in the "gradient" type methods. Some of them are based on a
variational formulation with pressure unknowns at the mesh centres and velocity vector unknowns on
the edges (or faces) of the cells. This approach has been shown to be effective, particularly in terms of
robustness. It should also be noted that an algorithm with the same degrees of freedom as the
MAC methods has been proposed and gives promising results.
The idea would therefore be to combine these two approaches, namely the "gradient" method with the same degrees of freedom as MAC methods. Initially, the focus will be on recovering MAC schemes on regular meshes. Fundamental
questions need to be examined in the case of general meshes: stability, consistency, conditioning of
the system to be inverted, numerical locking. An attempt may also be made to recover the gains in
accuracy using the methods presented in for discretising pressure gradients.
During the course of the thesis, time will be taken to settle the basic problems of this method (first and
second years), both on the theoretical aspects and on the computer implementation. It may be carried
out in the Castem, TrioCFD, Trust or POLYMAC development environments. The focus will be on
application cases that are representative of the community.