A new numerical formulation for incompressible viscous free surface flow without smearing the free surface

Abstract

A new numerical formulation is presented in this paper for incompressible viscous free surface flow without smearing the free surface. Laws of conservation for the entire physical domain, including both liquid and air, are formulated with a single set of governing equations. To properly handle the discontinuities of the physical properties across the free surface, an extended weighting function scheme is developed for the numerical solution on a fixed and nonstaggered Cartesian grid system. Unlike existing numerical methods, the force balance equation is imposed on the free surface through the use of the NAPPLE algorithm without smearing the free surface. During the solution procedure, a harmonic function referred as an “extrapolated velocity” from the liquid is computed based on the velocity solution at the grid points adjacent to the free surface on the liquid side. With a migration velocity interpolated from such a harmonic function, the free surface profile for the next time step is estimated. This gives rise to a smooth free surface profile and thus circumvents high frequency noises on the curvature of the resulting free surface. Furthermore, advancement of the free surface is not restricted to one grid mesh in a single time step. Performance of the present method is examined through three well-documented dam-breaking examples. The results reveal the existence of an induced vortex in a layer of air adjacent to the free surface. Good agreement between the computed water front and the experimental data is observed. Although only two-dimensional cases are demonstrated in this paper, concept of the present numerical method is equally applicable to three-dimensional problems with moving free surfaces. It also applies to convective heat and mass transfer problems such as filling process in gravity and die casting processing.