A graded-adaptive grid projection method to solve the Navier-Stokes equations for incompressible interfacial flows characterized by large density ratios is presented. The numerical model is similar to the one we proposed in  and extended to 3D problems in .
The free surface is described using a level set method. A Godunov-type method and a Crank-Nicholson temporal discretization scheme are used to solve the advection equation of the level set function and to update of the momentum equation. The reinitialization procedure of the distance function is based on solving a hyperbolic equation to steady state using third-order Runge-Kutta and fifth-order WENO schemes.
The conservation equations are discretized on a rectangular adaptive grid with an octree data structure and the pressure stored at the grid cell vertices. In order to avoid spurious pressure oscillations, the velocity components are stored at the cell edges. This new storage scheme combines the advantages of vertex-based schemes, in which the nodes where the pressure is stored are aligned, and cell center-based schemes, which avoid pressure-velocity coupling problems.
The numerical model incorporates a continuous surface tension model based on the balanced-force algorithm proposed in . A special treatment of T-nodes (nodes located at vertices, edges or faces of cells belonging to two different refinement levels) is proposed that considerably improves the efficiency of the method.
Several tests in two and three dimensions have been carried out to assess the accuracy and efficiency of the proposed method. In this work we present some numerical results for a 3D kinematic test, which are compared with those obtained by other authors. We also present results for the impact of a drop of water onto a liquid surface, which are compared with experimental visualization results.