High-order time integrators for front-tracking finite-element analysis of viscous free-surface flows

摘要

This paper proposes implicit Runge-Kutta (IRK) time integrators to improve the accuracy of a front-tracking finite-element method for viscous free-surface flow predictions. In the front-tracking approach, the modeling equations must be solved on a moving domain, which is usually performed using an arbitrary Lagrangian-Eulerian (ALE) frame of reference. One of the main difficulties associated with the ALE formulation is related to the accuracy of the time integration procedure. Indeed, most formulations reported in the literature are limited to second-order accurate time integrators at best. In this paper, we present a finite-element ALE formulation in which a consistent evaluation of the mesh velocity and its divergence guarantees satisfaction of the discrete geometrical conservation law. More importantly, it also ensures that the high-order fixed mesh temporal accuracy of time integrators is preserved on deforming grids. It is combined with the use of a family of L-stable IRK time integrators for the incompressible Navier-Stokes equations to yield high-order time-accurate free-surface simulations. This is demonstrated in the paper using the method of manufactured solution in space and time as recommended in Verification and Validation. In particular, we report up to fifth-order accuracy in time. The proposed free-surface front-tracking approach is then validated against cases of practical interest such as sloshing in a tank, solitary waves propagation, and coupled interaction between a wave and a submerged cylinder. Copyright (C) 2015 John Wiley & Sons, Ltd.