VOF evaluation of the surface tension by using variational representation and Galerkin interpolation projection

In this work we propose a variational approach with cell-to-point Galerkin projections for studying two-phase interface advection problems dominated by surface tension. A Volume Of Fluid (VOF) algorithm is used for tracking and locating the evolution of the two-phase interface on a Cartesian grid and a finite element numerical scheme for solving the velocity-pressure state. The velocity field that drives the evolution of this interface is computed from the weak form of the Navier-Stokes equation where the surface tension force is represented in variational form by the continuous surface force (CSF) and continuous surface stress (CSS) methods. Standard numerical approaches solve the strong form of the Navier-Stokes equations and define the CSS term by taking the divergence of the surface tension tensor. This computation of the divergence term results in a singular force which is difficult to compute when the grid is refined since the tensor is computed in a discontinuous cell-by-cell way. In this work we use the variational formulation of the Navier-Stokes equation and avoid differentiation. The tensor, which is a function of the unit normal, is evaluated over regular Sobolev spaces by using a cell-to-point Galerkin projection. This allows a regular piece-wise continuous representation of the surface tensor and the unit normal based on the VOF reconstruction. In standard approaches the CSF surface force is computed by using the curvature, which is the divergence of the unit normal. In this paper we recover the curvature with point-wise Galerkin projection avoiding direct differentiation. Tests on convergence for two and three-dimension in the static and dynamical cases are reported to show the correct representation in the desired spaces. This method is also natural for coupling non uniform grid computation of the fluid with Cartesian grid of the VOF algorithm.