Continuous method for calculating the transport equations for a multicomponent heterogeneous system on fixed Euler grids

A new numerical method for solving the transport equations of a multicomponent heterogeneous system on fixed Eulerian grids is considered. The system consists of an arbitrary number of components. Any two components are separated by a boundary (interface). Each component is characterized by a characteristic function — the volume fraction, which is transported in a given velocity field and determines the instantaneous distribution of the component in space. The feature of this system is that it requires two conditions to be satisfied. First, the volume fraction of each component should be in the interval [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 to solve the transport equations with respect to them. We prove that this approach ensures the fulfillment of the above conditions. The method is compatible with various TVD schemes (MINMOD, Van Leer, Van Albada, Superbee) and interface-sharpening methods (Limited downwind, THINC, Anti-diffusion, Artifical compression). The method is verified in the calculation of a number of test problems, using all the above schemes. Numerical results show the accuracy and reliability of the proposed method.