Velocity and Energy Relaxation in Two-Phase Flows

In this study, we investigate analytically the process of velocity and energy relaxation in two-phase flows. We begin our exposition by considering the so-called six equations two-phase model [1, 2]. This model assumes each phase to possess its own velocity and energy variables. Despite recent advances, the six equations model remains computationally expensive for many practical applications. Moreover, its advection operator may be nonhyperbolic, which poses additional theoretical difficulties to construct robust numerical schemes [3]. To simplify this system, we complete momentum and energy conservation equations by relaxation terms. When relaxation characteristic time tends to zero, velocities and energies are constrained to tend to common values for both phases. As a result, we obtain a simple two-phase model that was recently proposed for simulation of violent aerated flows [4]. The preservation of invariant regions and incompressible limit of the simplified model are also discussed. Finally, several numerical results are presented.