Geometric volume-of-fluid advection with parabolic interface reconstruction

When simulating interfacial two-phase flows with the volume-of-fluid (VOF) method, the geometric advection of the volume fraction field requires the reconstruction of cellwise volume-preserving approximations of the interface. So far, the discretely conservative VOF advection schemes that have been proposed in the literature are limited to planar interface reconstructions, since these already present the challenge of intersecting non-trivial, non-convex polyhedra with a half-space. In this work, we present the first discretely conservative VOF advection scheme that uses parabolic interface reconstructions instead of planar ones. This is achieved by deriving the exact moments of the intersection between an arbitrary polyhedron and the space bounded by a paraboloid surface. In turn, this enables the solution of local constrained optimization problems for finding the optimal paraboloid surfaces matching the volume fractions around interfacial computational cells. We introduce and test several variants of this optimization problem. The resulting semi-Lagrangian advection scheme is validated with several classical three-dimensional test-cases, from which its order of accuracy and computational cost are assessed.