INS seminars 52

Abstract:

We deal with Baer-Nunziato like systems for modeling mixtures of two compressible flows. The main emphasis is put on the vanishing phase regime : namely the statistical volume fraction of a phase, an unknown of the PDE model, gets asymptotically close to zero and ultimately reaches the boundary of the phase space. In this limit, general mathematical existence results are known to fail. Then and provided that the solution can be continued boundedly, the concern is to numerically determine pure phasic quantities from the statistically averaged unknowns that govern the vanishing phase.

The key observation is that entropy dissipation may play a role in handling the reported singular limit both mathematically and numerically speaking. Indeed in such a limit, the first order PDEs exhibit resonance and it can be proved that underlying additional entropy laws may not be preserved already in the case of a fraction wave with linear degeneracy (in the strong hyperbolic zone). This striking property leads us to adopt the so-called framework of kinetic relations known in other fluid flow contexts : we propose to define relevant (relative) mass flux closure laws across the fraction wave that in the one hand are entropy dissipative while in the second hand, they define Riemann solutions that keep bounded values within/along the phase space as one the volume fraction goes to zero in the initial data. To make the derivation tractable in the large, the proposed strategy is investigated when approximating solutions of the original PDEs by those of a suitable Suliciu's relaxation procedure. For prescribed relevant kinetic relations, existence and uniqueness results of self-similar solutions are obtained in this framework. The relaxation procedure provides in turn a natural approximate Riemann solver (of HLLC type with a full number of waves) for the original Baer-Nunziato equations. Its robustness and accuracy are fairly grounded on several 1D and 2D test cases involving phase vanishing.

This is a series of works with Jean-Marc Herard (EDF), Khaled Saleh (Univ. Paris 6) and Nicolas Seguin (Univ. Paris 6).