A multiphase model with internal degrees of freedom: application to shock–bubble interaction

Abstract

The aim of this paper is the derivation of a multiphase model of compressible fluids. Each fluid has a different average translational velocity, density, pressure, internal energy as well as the energies related to rotation and vibration. The main difficulty is the description of these various translational, rotational and vibrational motions in the context of a one-dimensional model. The second difficulty is the determination of closure relations for such a system: the ‘drag’ force between inviscid fluids, pressure relaxation rate, vibration and rotation creation rates, etc. The rotation creation rate is particularly important for turbulent flows with shock waves. In order to derive the one-dimensional multiphase model, two different approaches are used. The first one is based on the Hamilton principle. This method gives thermodynamically consistent equations with a clear mathematical structure, coupling the various motions: translation, rotation and vibration. However the relaxation effects have to be added phenomenologically. In order to achieve the closure of the system and its numerical resolution we use the second approach, in which the pure fluid equations are discretized at the microscopic level and then averaged. In this context, the flow is considered to be the annular flow of two turbulent fluids. We also derive the continuous limit of this model which provides explicit formulae for the closure laws. The structure of this system of partial differential equations is the same as the one obtained by the Hamilton principle. The final issue is to determine the rate of energy (or entropy) rotation. We assume that all the entropy creation related to the various relaxation effects after the passage of the shock wave is converted to rotational motion. The one-dimensional model is validated by comparing its predictions with averaged two-dimensional direct numerical results. The problem on which this model is tested is the interaction of a shock wave propagating in a heavy gas with a light gas bubble. The results obtained by the one-dimensional multiphase model are in a very good agreement with the two-dimensional averaged results.