A finite-volume method is presented for the computation of compressible flows of two immiscible fluids at very different densities. A novel ingredient in the method is a linearized, two-fluid Osher scheme, allowing for flux computations in the case of different fluids (e.g., water and air) left and right of a cell face. A level-set technique is employed to distinguish between the two fluids. The level-set equation is incorporated into the system of hyperbolic conservation laws. Fixes are presented for the solution errors (pressure oscillations) that may occur near two-fluid interfaces when applying a capturing method. The fixes are analyzed and tested. For two-fluid flows with arbitrarily large density ratios, a simple variant of the ghost-fluid method appears to be a perfect remedy. Computations for compressible water–air flows yield perfectly sharp, pressure-oscillation-free interfaces. The masses of the separate fluids appear to be conserved up to first-order accuracy.
Koren, B., Lewis, M. R., Brummelen, van, E. H., & Leer, van, B. (2002). Riemann-problem and level-set approaches for homentropic two-fluid flow computations. Journal of Computational Physics, 181(2), 654-674. https://doi.org/10.1006/jcph.2002.7150