Conservative two-phase flow simulation using piecewise-parabolic interface reconstructions

To simulate interfacial two-phase flows using finite-volume moment-based methods, e.g., the volume-of-fluid (VOF) or moment-of-fluid (MOF) method, cellwise moment-preserving approximations of the interface are needed in order to conservatively advect the indicator function of the phases. Conservative VOF and MOF methods have so far relied on piecewise-planar interface approximations, necessitating tools for intersecting non-trivial, non-convex polyhedra with a half-space. In this work, we present a new conservative approach that uses piecewise-parabolic interface approximations instead of planar ones. To that end, we first introduce the tools that we have developed for solving the "forward" problem, i.e., calculating the geometrical moments of any non-convex polyhedron intersected by a paraboloid, and the "backward" problem, i.e., reconstructing the paraboloid that optimally and conservatively matches a set of local moments of fluid. We then provide an example of application of these tools in a multiphase flow solver, including the corresponding treatment of surface tension. The resulting framework is validated with canonical and realistic three-dimensional test-cases, from which its accuracy and computational cost are assessed.