Legendre Spectral Element Method For Incompressible Unsteady Viscous Free-surface Flows

Unsteady incompressible viscous free-surface flows are largely encountered in engineering and natural systems. The difficulty of this kind of problems relies on how to accurately track the position of the free surface via the time evolution. In the past the people used many numerical methods to investigate the free-surface flows including FDM, FEM, BEM and so on. In this paper the Legendre spectral element method is proposed to solve the viscous incompressible free-surface flows. The main ingredients of our method consist: use of the full viscous stress tensor for natural impositions of the traction (surface tension) boundary conditions; use of ALE methods for accurate representation of moving boundaries; use of semi-implicit time-stepping procedures and improved Uzawa method to partially decouple the free-surface evolution and interior Navier-Stokes equations; and use of Stokes operator to control the gridding motion.For purposes of analysis and clarity of presentation, the attention is focused on the linear stability of the thin film flow down an inclined plane that is geometrically the simplest free surface problem. In the numerical tests we develop an efficient method based on semi-implicit scheme in time and Legendre spectral element method in space; Stokes equation is applied to solve the gridding velocity. The numerical results obtained are not only in close agreement with the results of the linearized stability of Orr-Sommerfeld equation, but also show that Stokes operator better ensures the conservation of the volume of the spectral elements than the usual elastic operator. This proves that the ALE SEM is a good method to the free-surface flow problems.