Numerical Simulation of Droplet-deformation by a Level Set Approach with Surface Tension

In this paper we present a three-dimensional Navier–Stokes solver for incompressible two-phase flow problems with surface tension and apply the proposed scheme to the simulation of droplet-deformation. Our approach employs a standard finite difference/finite volume discretization on uniform Cartesian staggered grids and uses Chorin’s projection approach. The free surface between the two fluid phases is tracked with a level set technique. Here, the interface conditions are implicitly incorporated into the momentum equations by the continuum surface force method. Surface tension is evaluated using a smoothed delta function and third order interpolation. The problem of mass conservation for the two phases is treated by a reinitialization of the level set function employing a regularized signum function and a global fixed point iteration. All convective terms are discretized by a WENO scheme of fifth order. Altogether, our approach exhibits a second order convergence away from the free surface. The discretization of surface tension requires a smoothing scheme near the free surface which leads to a first order convergence in the smoothing region. We discuss the details of the proposed numerical scheme and present the results of several numerical experiments concerning mass conservation, convergence of curvature and the application of our solver to the simulation of droplet-deformation due to a shear flow in three space dimensions.