The spectral element method applied to the viscous free surface flows

Abstract

In this paper, viscous free surface flows are analyzed in the context of the Navier-Stokes (N-S) equations and a numerical model for their simulation is presented. The governing equations are solved through the Spectral Element Method (SEM), which possesses the property of high-order spatial accuracy as proposed by Karniadakis & Sherwin (2005). In order to deal with the free surface motion in time, the Arbitrary Lagrangian-Eulerian (ALE) method is utilized. It is shown that the proposed SEM and its present implementation produce very accurate and convergent solutions to the Poisson equation, the Helmholtz equation, and a manufactured problem of the N-S equations. Numerical results of the lid-driven cavity flow are compared with other numerical results. As for the viscous free surface flows, viscous sloshing with a prescribed initial free surface profile is simulated and it is shown that the present methodology gives very reasonable results for the small-amplitude problem.