Interface-Capturing Method for Calculating Transport Equations for a Multicomponent Heterogeneous System on Fixed Eulerian Grids

In this paper, we consider a new numerical method for solving the transport equations for a multicomponent heterogeneous system on fixed Eulerian grids. The system consists of an arbitrary number of components. Any two components are separated by a boundary (interface). Each component is determined by a characteristic function, i.e., a volume fraction that is transported in a specified velocity field and determines the spatial instantaneous component distribution. A feature of this system is that its solution requires two conditions to be met. Firstly, the volume fraction of each component should be in the range [0, 1], and, secondly, any partial sum of volume fractions should not exceed unity. To ensure these conditions, we introduce special characteristic functions instead of volume fractions and propose solving transport equations with respect to them. It is proved that the fulfillment of these conditions is ensured when using this approach. In this case, the method is compatible with various TVD schemes (MINMOD, Van Leer, Van Albada, and Superbee) and interface-sharpening methods (Limited downwind, THINC, Anti-diffusion, and Artificial compression). The method is verified by calculating a number of test problems using all of these schemes. The numerical results show the accuracy and reliability of the proposed method.