Unsplit geometric volume-of-fluid method with iterative piecewise-paraboloid interface reconstruction on arbitrary three-dimensional grids

Recent improvements in geometric volume-of-fluid methods, in which the fluid interface is implicitly represented using the volume-of-fluid fraction function, make them an excellent approach for solving complex interfacial dynamics problems. However, mainly due to the complexity involved in implementing them, these methods are typically limited to the use of linear approximations for interface reconstruction and volume-of-fluid fraction advection, especially in three dimensions. This typically produces second-order interface representations, thus making it difficult to achieve interface curvature convergence with grid refinement. This geometric complexity means that the literature contains no evidence of geometric volume-of-fluid methods capable of producing high-order accuracies for interface representation in time-dependent problems with deformed interfaces on arbitrary three-dimensional grids. The current work addresses this gap by presenting a new geometric volume-of-fluid method that uses an iterative least-squares paraboloid fitting to initial piecewise-linear approximations for interface reconstruction and an unsplit flux-based scheme for the advection of the volume-of-fluid fraction on unstructured three-dimensional grids with arbitrary cells, either convex or non-convex. To accurately determine the volume-of-fluid fraction advected out of grid cells, a new method based on local grid refinement and polyhedral approximation is proposed to efficiently compute the intersection between the piecewise reconstructed paraboloidal fluid regions and an arbitrary polyhedron. A detailed assessment of the proposed method is carried out for several commonly used canonical tests and more realistic scenarios, such as the rise of a bubble under gravity in a quiescent liquid, for which the effects of surface tension must be accurately computed. Comparisons with the few results available in the literature show generally favorable results.