Formal derivation of a stratified compressible two-phase flow 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. Furthermore, the source terms governing the relaxation to thermodynamical equilibrium between the phases (pressure, velocity, temperature) are closed by empirical relations, see [DP99, IH11] for instance. It is a key issue to provide a mathematical theory of derivation of inviscid compressible multiphase models with no empirical closures. Starting from multiphase Navier-Stokes models, mathematical methodologies may enable to derive original average multiphase models, or to validate existing models. In this talk, I will present a formal derivation of an averaged model for a stratified compressible two-phase flow in a thin domain [LVSS25]. The fluid layers being thin, we use a dimension reduction technique to obtain our final model [GP01, SW84]. More precisely, we average the two sets of compressible Navier-Stokes equations — one for each fluid — in the stratification direction. While standard procedures seem to be profoundly related to one-velocity models, our approach suppresses this constraint. However, it is important to note that this modelling process necessarily leads to one-pressure models, as in [DP99, IH11]. We are also able to recover one-velocity, one-pressure models when we study the standard framework — with continuity of the velocity field at the interface. Our derivation process also adapts to the energy equation in the Navier-Stokes-Fourier case, enabling the obtention of either one or two-temperature models.

HYP26 conference