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Numerical methods for multi-phase mixture conservation laws with phase transition
MSc. Ali Zein
Multi-phase mixtures occur very commonly in nature and technology. Several mathematical models have been developed to describe the flow of such mixtures. But both the mathematical modelling and numerical computation of multi-phase flows are associated with certain difficulties. The difficulties is modelling concern the physical transfer processes taking places across the interface such as mass, momentum and heat transfer, and phase change. By using averaging technique of the single phase equations results additional terms, which describe those transfer processes. The exact expressions for the transfer terms are usually unknown. Also there appear differential terms that are extracted from the transfer terms that prevent the system from being in divergence form. Therefore, they are referred to as the non-conservative terms. The numerical difficulties arise the resulting model cannot be written in divergence form (conservative form) due to the existence of non-conservative terms. And in this case one cannot define a weak solution for the systems of governing equations in the standard sense of distributions, as it is done for the systems of conservation laws. The primary goal of this project is to improve and validate numerical schemes for the solution of two-phase flow equations concerning non-conservative terms. There exist a large number of numerical methods for conservation laws which use an exact or approximate solution of the local Riemann problem at the cell interfaces. These algorithms belong to the family of Godunov-type methods. To apply these methods to two pase flows we need to improve an efficient and robust Riemann solver for the non-conservative systems. Also we need to improve an accurate methods for the discretization of the non-conservative terms. Another problem in the numerical solution of two-phase flows occurs when pure phases are present in the domain. Then for the other phase, the situation is analogous to the occurence of vacuum in the solution of the usual fluid dynamics equations. For the Euler equations, there are two different ways to attack the problem of vacuum occurrence. One is to track the gas-vacuum interface explicitly. However in multi-D this becomes very complicated due to topological problems, like merging, breaking, and creating of the interfaces. An alternative is to admit a negligible amount of the phase, which is supposed to disappear. It is important to use a positively conservative method for the solution of the interface problems between almost pure phases. Otherwise a smallest numerical inaccuracy would lead to negative pressure or densities. The doctorate was successfully completed in 2010.
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