Toward efficient and accurate interface capturing on arbitrary hybrid unstructured grids: The THINC method with quadratic surface representation and Gaussian quadrature

A novel interface capturing scheme is proposed to compute moving interface on arbitrary hybrid unstructured grids. Different from conventional volume of fluid (VOF) schemes that require complicated geometric manipulations for interface reconstructions, the present method can be viewed as a hybrid geometric/algebraic type VOF approach, i.e. the interface is implictly retrieved from an algebraic function that effectively makes use of the geometrical information, such as normal direction and curvature of the interface. Unlike previous versions of the THINC (tangent of hyperbola interface capturing) method, the interface is represented by a quadratic surface for grid cells of arbitrary shapes in this scheme. The Gaussian quadrature is used to estimate the integration of the cell-wise multi-dimensional hyperbolic tangent reconstruction function which is then used to retrieve the interface from the volume fraction value of the target cell. The Gaussian quadrature is also used to compute the numerical fluxes from the reconstruction function. Numerical accuracy for reconstruction and flux computation can be effectively improved by increasing quadrature points. The whole solution procedure follows the finite volume method for advection transport, and is thus simple and easy to use for cells of arbitrary shapes. As verified in the benchmark tests in this paper, the presented scheme, so-called THINC/QQ (THINC method with quadratic surface representation and Gaussian quadrature) scheme, shows significantly improved geometrical fidelity of interface representation particularly for curved surface. Despite algorithmic simplicity, the solution quality of THINC/QQ is comparable to other existing VOF methods with PLIC geometrical interface reconstructions, and thus an accurate and efficient VOF scheme of great practical significance for unstructured grids.