LAUSR

A ( $$2N+1$$ 2 N + 1 )-equation model to simulate the flow of N ( $$\textit{N}\ge 3$$ N ≥ 3 ) immiscible compressible fluids separated with interfaces is proposed. The model is based on the single velocity diffuse-interface method and includes $$N-1$$ N - 1 advection equations for fluid volume fractions. Solving the advection equations with a non-linear high-order scheme commonly results in the violation of the non-negativity constraint that any arbitrary partial sum of volume fractions should be in the interval [0, 1]. First, it is shown that this constraint can be met if the $$N-1$$ N - 1 advection equations are solved for some rational functions of volume fractions rather than for volume fractions themselves. The non-linear sub-cell slope reconstruction (MUSCL-type and THINC) with the proposed rational advection functions is proved to be non-oscillatory and provide the distribution of volume fractions satisfying the non-negativity constraint. Second, it is proved that the PV property (preservation of constant-pressure and constant-velocity equilibrium) is maintained providing that linear functions of volume fractions are used in the advection equations. We suggest two ways for resolving the contradiction in choosing the advection functions (functions of volume fractions) in accordance with the non-negativity constraint and the PV property. We also adopt two numerical methods—the Roe-type scheme and the HLLC scheme to solve the governing equations. Finally, the proposed numerical model is tested with several benchmark problems. The results obtained demonstrate robustness and effectiveness of the proposed numerical approach in solving multi-fluid flows with large interface deformations.