Reduction of parasitic currents in level-set calculations with a consistent discretization of the surface-tension force for the CSF model

Parasitic currents may develop in grid-based interface simulations because of inaccurate representation of the surface forces in the discretized equations. This is due to two causes : firstly, inconsistent discretization of the surface tension force and the pressure gradient, such that the force balance is not fulfilled for a drop or a bubble at rest. Secondly, the problem is inaccurate approximation of the curvature. The least you should demand from a discretization is that it preserves a stationary solution. In this article, it is shown that this can be accomplished by rewriting the interfacial force term in the momentum equation. Using exact curvature, the exact solution for a drop is preserved to machine accuracy. In general, with this discretization, the calculation of the curvature is the only remaining source of spurious currents. Contrary to common practice for the level-set method, we stress that the curvature should be evaluated at the point on the interface whose normal cross the discretization point, and not at the gridpoint in the smeared-out region outside the interface. In 2D, a simple geometrical argument may be used to find the curvature at the interface, whereas in 3D we use extrapolation normal to the interface to create the correct curvature field in a small region around the interface