Accurate interface normal and curvature estimates on three-dimensional unstructured non-convex polyhedral meshes

This paper presents a framework for extending the height-function technique for the calculation of interface normals and curvatures to unstructured non-convex polyhedral meshes with application to the piecewise-linear interface calculation volume-of-fluid method. The methodology is developed with reference to a collocated node-based finite-volume two-phase flow solver that utilizes the median-dual mesh, requiring a set of data structures and algorithms for non-convex polyhedral operations: truncation of a polyhedron by a plane, intersection of two polyhedra, joining of two convex polyhedra, volume enforcement of a polyhedron by a plane, and volume fraction initialization by a signed-distance function. By leveraging these geometric tools, a geometric interpolation strategy for embedding structured height-function stencils in unstructured meshes is developed. The embedded height-function technique is tested on surfaces with known interface normals and curvatures, namely cylinder, sphere, and ellipsoid. Tests are performed on the median duals of a uniform cartesian mesh, a wedge mesh, and a tetrahedral mesh, and comparisons are made with conventional methods. Across the tests, the embedded height-function technique outperforms contemporary methods and its accuracy approaches the accuracy that the traditional height-function technique exemplifies on uniform cartesian meshes.