BGK source terms for out-of-equilibrium two-phase flow models

Abstract

A modified form of BGK source terms is proposed for modeling two-phase flows with thermodynamical disequilibrium. The novelty is that three independent time-scales allow to manage the return to the thermodynamical equilibrium while remaining in agreement with the second law of thermodynamics. This is achieved thanks to the definition of a "local-in-time" equilibrium state which tends towards the asymptotic equilibrium state when time increases. The thermodynamical paths of the system are then modified with respect to the classical BGK source terms used for two-phase flow modeling, and the relaxation process of the system towards the asymptotic equilibrium state can be defined with additional degrees of freedom. In a numerical point of view, both the classical and the modified BGK source terms have advantages. The choice between these two forms strongly depends on the numerical strategy used to perform simulations.

Study of relaxation processes in a two-phase flow model

“…Note that invoking the concavity argument requires to consider a unique relaxation parameter λ. Recently a modification of BGK-like source terms which allows distinct time scales has been introduced in [12].…”

“…Finally, the entropy gradient source term allows to easily consider with different relaxation times for the mechanical, thermal exchange or the mass transfer. This is not the case for the BGK-like source term which requires the maximization of the mixture entropy with a common relaxation time, even if some recent improvements have been done on this topic [12]. function (ρ k , ϵ k ) → sk (ρ k , ϵ k ).…”

“…The aim of the present paper is to compare two possible forms of source terms. The first one, recently referred as BGK source term [12], corresponds to a linearization around the equilibrium state Y eq . Its apparition goes back to [2,7,14] and is now used in multiphase configurations with complex phasic equation of state (EOS) [11,19].…”

Simulation of a homogeneous relaxation model for a three-phase mixture with miscible phases

“…The core of this section is to analyze two possible choices of source terms. The first one, referred in [20] as BGK source term, corresponds to a linearization around the equilibrium vector Y eq . Introduced in [9,15,3], it is used in multiphase configurations with complex phasic equations of state [23,22].…”

“…Note that invoking the concavity argument requires to consider a unique relaxation parameter λ. Recently a modification of BGK-like source terms which allows distinct time scales has been introduced in [20]. Proposition 3.…”