This article presents an all–Mach method for two–phase inviscid flow in the presence of surface tension. A modified version of the Hartens, Lax, Leer and Contact (HLLC) approximate Riemann solver based on the work by Garrick et al., (2017) is developed and combined with the popular Volume of Fluid (VoF) method: Compressive Interface Capturing Scheme for Arbitrary Meshes (Ubbink & Issa, 1999) (CICSAM). This novel combination yields a scheme with both HLLC shock capturing as well as accurate liquid–gas interface tracking characteristics. The sonic and two–phase properties of the scheme are assessed using the popular advecting bubble, the Gas–Liquid Riemann problem, and the under–water explosion. The implementation of surface tension in the scheme is verified via the classic static bubble configuration, and the popular Rayleigh–Plesset collapse problem. For the static bubble, the spurious currents are of the order 10−8 demonstrating that the scheme is well–balanced. For the Rayleigh—Plesset problem, the maximum relative error on the minimum radius of collapse is found to be within 5℅ on the coarsest mesh.


High–speed multi–phase compressible flow induced by shock waves is of interest to both basic science and engineering. Over the last three decades, different approaches have been proposed to model multi-phase compressible flow (Baer & Nunziato, 1986; Kapila et al., 2001; Shyue, 1998). In this work, we shall employ a homogeneous one–fluid formulation (reduced four–equation) for the inviscid compressible flow modelling of a liquid–gas system in the presence of surface tension effects.

Here, we implement the popular Hartens, Lax, and Leer Contact (HLLC) approximate Riemann solver to capture the sonic characteristics of the flow. However, this solver, on its own, suffers from one major drawback. That is, the smearing of the liquid–gas interface. This has a significant impact on the ability to compute interface curvature accurately and hence surface tension effects were often neglected.

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