Analytical and numerical studies of Riemann problems for a multiphase mixture model

Abstract

In this thesis, we study submodels of a diffuse interface multiphase mixture model which was proposed by Dreyer, Giesselmann, and Kraus in [24]. This model is a type of phase field model which describe the chemically reacting viscous fluid mixtures consisting of N constituents that may develop a transition between a liquid and a vapor phases. The submodel has N partial mass balance equations, a balance equation of the momentum, and a transport equation of the phase field variable. The phase variable indicates the present phase. The model is supplied with a complicated equation of state. We will consider one space dimension and assume an isothermal flow. We consider the homogeneous part of this model which is a hyperbolic system of partial differential equations for two phase mixture flow with N components. The main aim of this work is to study the sub-model analytically and numerically. The analytical study reveals that this model is strictly hyperbolic. We presented the analytical structure and the mathematical properties of the sub-model such as the eigenstructure and the wave types of the solutions. We also obtain the exact solution of Riemann initial value problem for the pure phases flow, i.e. N = 1 as well as for the multicomponent flow, i.e. N > 1. Any discretization of the full model in [24] has to contain a correct and stable implementation of the homogeneous part. Therefore, it is justified to deal with the problem of numerics for the submodel separately. This is what we do. In the numerical study, we first consider a vapor-vapor flow. We solve the model using different Riemann solvers and present the results. We compare the numerical solution with the exact results obtained in the analytical study. Further, we consider vapor-liquid flows. In this case, major difficulties appear such as negative pressures, i.e. unphysical results. We overcome these difficulties using a tracking the interface approach. But actually, these methods are generally not easy to implement. So we also consider discontinuity capturing methods. For these, we also develop a new strategy to deal with this situation. The new approach is called estimating the pressure approach. We applied the new method to several test cases. This gave an undeniable improvement but still leaves some open problems for future research. Finally, in this work, we include the source term in the sub-model, and we discuss the ability of this model to deal with chemical reactions.