An effective implementation of surface tension using the marker and cell method for axisymmetric and planar flows

This work presents a method for simulating axisymmetric and planar free-surface flows dominated by surface tension forces. The surface tension effects are incorporated into the free surface boundary conditions through the computation of the capillary pressure. The required curvature is evaluated by fitting a least squares curve to the free surface using the tracking markers in the cell and in its close neighbors. To avoid short wavelength perturbations on the free surface, a mass-conserving local four-point stencil filter is employed. This filter is an extension of the trapezoidal subgrid undulations removal (TSUR) method. The TSUR technique consists of modifying the positions of two ``internal' markers of the four-point stencil in such a way that the surface length and the curvature are minimized, while still preserving the volume. The computation of the curvature is modified at cells adjacent to solid boundaries in order to apply contact angle boundary conditions. To identify neighboring cells efficiently, an implementation is effected through a dual representation of the cell data: in addition to a matrix representation, a list structure is also employed which permits the representation of specific groups of cells and associated data. This technique was implemented in the GENSMAC code, and it has been proved to be robust. The code is shown to produce accurate results when compared with exact solutions of selected fluid dynamical problems involving surface tension. Additionally, it is demonstrated that the method is applicable to complex free-surface flows, such as containers filling, splashing drops, and bursting bubbles.