A Legendre spectral element method for simulation of unsteady incompressible viscous free-surface flows

Abstract

A new Legendre spectral element method is presented for the solution of viscous incompressible free-surface flows. It is based on the following extensions of the fixed-domain spectral element method: use of the full viscous stress tensor for natural imposition of traction (surface tension) boundary conditions; use of arbitrary-Lagrangian-Eulerian methods for accurate representation of moving boundaries; and use of semi-implicit time-stepping procedures to partially decouple the free-surface evolution and interior Navier-Stokes equations. For purposes of analysis and clarity of presentation, attention is focused on the stability of falling films. Analysis of the spectrum of the linear stability problem (Orr-Sommerfeld equation) associated with film flow reveals physical effects that limit the stability of semi-implicit schemes and suggests optimal formulas for temporal discretization of the spectral element equations. Detailed results are presented for the spectral element simulation of the film flow problem.