Mathematical analysis and numerical simulation of a Kapila model

From the combustion chamber of space launchers to circuits of pressurized water reactors of nuclear plants, the understanding of compressible multiphase flows has become a real challenge and a crucial domain of research during the last decades. Because of their complexity and the limited capacity of numerical computations, compressible multiphase flows are often described by macroscopic models obtained by averaging the Navier-Stokes equations. To that effect, Le Vourc’h et al. [2] recently presented a formal derivation process enabling to obtain, among others, a two-dimensional Kapila model [3] with one velocity and pressure equilibrium from a three-dimensional compressible two-phase Navier-Stokes system. In this work, we study the properties of the Kapila model derived in [2]. Namely, we prove that this model is well-posed, strictly hyperbolic, symmetrizable, and we determine its characteristic fields — one linearly degenerate field and two genuinely nonlinear fields. We then solve the associated Riemann problem in order to numerically compute the solution of the system using a finite-volume scheme in the case of shock tubes problems.

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