Fast Numerical Computation of 2D Free Surface Jet Flow with Surface Tension

Abstract

A system of partial differential equations that approximate the governing equations for inviscid free surface flow subject to surface tension is presented. The approximation is based on linearizing the velocity together with a small scale approximation of the perturbation of the velocity. Two Dirichlet problems must be solved to form the approximate system, after which it can be evolved without solving Dirichlet problems. The accuracy of the solution is determined by how often the velocity term is linearized. This time-interval is called ΔT. We show that the error in the solution of the approximate system at a fixed timeTis of the order O(ΔT2). We demonstrate numerically that the error is closely correlated to the size of the normal velocity and that there is a stability limit of the form ΔT≤C/(|un|∞)γ, whereundenotes the normal velocity of the free surface and γ ≈ 2.6. Importantly,Cis independent of the resolution, so the time-step ΔTcan be chosen independently of the number of grid points,N. This is in contrast to the original system, where the stability limit of the time-step is Δt≤ O(N−3/2) and a fixed number of Dirichlet problems have to be solved per time-step. By numerical experiments, we demonstrate that the approximate system requires less than 10% of the CPU time used by the original system to solve the problem very accurately.