Simulating Free Boundaries in 2D FEM Flow Simulation with Direct Time Integration of the Surface Velocities

Abstract

We have derived a novel approach to simulate free surfaces. For two-dimensional finite-element flow simulations of the incompressible Navier-Stokes equations we integrate out the velocities on the surface obtained from the FEM-simulations with integrators for ordinary differential equations. The advance of the fluid front does not need additional data structures or interpolated mesh points as in the front tracking for finite difference methods, the adaptive mesh of the FEM-simulation is sufficient. As we perform the time-integration of the FEM-code, the second-order Adams-Bashforth turns out to be the most suitable integrator for the surface motion. For the speed of the wavefronts, we get excellent agreement for large viscosity with the lubrication approximation by Huppert, and for small viscosity, we get very good agreement with the experimental data for water by Martin and Moyce.