Simulation numérique d'écoulements gaz-particules sur maillage non structuré

Accurate predictions of gas-solid ows in a wide range of industrial applications require defining suitable algorithms to cope with time-dependent evolutions in complex geometries. This motivates investigating the feasability of finite-volume schemes on unstructured meshes, following recent trends and using upwinding techniques in order to prevent spurious oscillations. It is well known that troubles quickly occur when investigating two-phase flows, using the two-fluid approach, essentially due to the fact that such sets of partial differential equations involve non-conservative convective subsets, but also to the fact that the latter may become non-hyperbolic in some time-space regions. Quite recently, some proposals based on Roe's ideas arose to overcome these diffculties. This work is devoted to the presentation of a finite-volume scheme which allows the computation of compressible two-phase flows on unstructured meshes. A four-equation two-fluid model is chosen in order to describe evolutions of gas-solid flows. The convective part of the resulting set of partial differential equations is a non-conservative conditionnaly hyperbolic set of equations. One can't solve the associated one-dimensional Riemann problem and can't prove that solutions of the latter system fulfill the maximum principle for the volumetric fraction. We proposeto solve the system with a fractional step method such that the maximum principle for the volumetric fraction is fulfilled on the whole algorithm. The finite-volume scheme used is based on Godunov's and Roe's schemes and accounts for non-conservative contributions in a suitable way. Some two-dimensional test cases including shocks and rarefaction waves within both phases are described and show that the scheme enables to handle strong shock simulations. Computational results for the stationary state of a two-phase flow in a simple nozzle and simulations of dense fluidized beds are displayed. Afterwards, a three-equation model is chosen in order to describe evolutions of gas-solid flows, assuming that there is no sliding velocity between the gas phase and the particle phase. As a consequence, the set of governing partial differential equations contains a conservative convective subset which is always hyperbolic. This is an interesting feature since it allows solving theassociated one dimensional Riemann problem for almost arbitrary initial data. It is proved that solutions of the latter system fulfill the maximum principle for the volumetric fraction, provided that the granular pressure is suitable. As a result, several numerical techniques may be used, among which the vfRoe scheme which is a rough Godunov scheme recently proposed to deal with complex hyperbolic systems. Some basic two dimensional computational results are presented, which enable the comparison with results issuing from the two-fluid approach. Somecomputations of impinging jets confirm that the maximum principle holds. But computations of a two-phase flow in a nozzle show that the hypothesis of the relative velocity equal to zero is not suitable in such cases.