Christopher Ivey

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.

Conservative and bounded volume-of-fluid advection on unstructured grids

Abstract This paper presents a novel Eulerian–Lagrangian piecewise-linear interface calculation (PLIC) volume-of-fluid (VOF) advection method, which is three-dimensional, unsplit, and discretely conservative and bounded. The approach is developed with reference to a collocated node-based finite-volume two-phase flow solver that utilizes the median-dual mesh constructed from non-convex polyhedra. The proposed advection algorithm satisfies conservation and boundedness of the liquid volume fraction irrespective of the underlying flux polyhedron geometry, which differs from contemporary unsplit VOF schemes that prescribe topologically complicated flux polyhedron geometries in efforts to satisfy conservation. Instead of prescribing complicated flux-polyhedron geometries, which are prone to topological failures, our VOF advection scheme, the non-intersecting flux polyhedron advection (NIFPA) method, builds the flux polyhedron iteratively such that its intersection with neighboring flux polyhedra, and any other unavailable volume, is empty and its total volume matches the calculated flux volume. During each iteration, a candidate nominal flux polyhedron is extruded using an iteration dependent scalar. The candidate is subsequently intersected with the volume guaranteed available to it at the time of the flux calculation to generate the candidate flux polyhedron. The difference in the volume of the candidate flux polyhedron and the actual flux volume is used to calculate extrusion during the next iteration. The choice in nominal flux polyhedron impacts the cost and accuracy of the scheme; however, it does not impact the methods underlying conservation and boundedness. As such, various robust nominal flux polyhedron are proposed and tested using canonical periodic kinematic test cases: Zalesaks disk and two- and three-dimensional deformation. The tests are conducted on the median duals of a quadrilateral and triangular primal mesh, in two-dimensions, and on the median duals of a hexahedral, wedge and tetrahedral primal mesh, in three-dimensions. Comparisons are made with the adaptation of a conventional unsplit VOF advection scheme to our collocated node-based flow solver. Depending on the choice in the nominal flux polyhedron, the NIFPA scheme presented accuracies ranging from zeroth to second order and calculation times that differed by orders of magnitude. For the nominal flux polyhedra which demonstrate second-order accuracy on all tests and meshes, the NIFPA methods cost was comparable to the traditional topologically complex second-order accurate VOF advection scheme.