Numerical simulation of two and three-dimensional viscous free surface flows in partially-filled containers using a surface capturing approach

A new surface capturing method is developed for numerically simulating viscous free surface flows in partially-filled containers. The method is based on the idea that the flow of two immiscible fluids (specifically, a liquid and a gas) within a closed container is governed by the equations of motion for a laminar, incompressible, viscous, nonhomogeneous (variable density) fluid. By computing the flowfields in both the liquid and gas regions in a consistent manner, the free surface can be captured as a discontinuity in the density field, thereby eliminating the need for special free surface tracking procedures;The numerical algorithm is developed using a conservative, implicit, finite volume discretization of the equations of motion. The algorithm incorporates the artificial compressibility method in conjunction with a dual time stepping strategy to maintain a divergence-free velocity field. A slope-limited, high order MUSCL scheme is adopted for approximating the inviscid flux terms, while the viscous fluxes are centrally differenced. Two different methods are considered for solving the resulting block-banded system of equations;The capabilities of the surface capturing method are demonstrated by calculating solutions to several challenging two and three-dimensional problems. The first test case, the two-dimensional broken dam problem, is considered in detail. Results are presented for several grid sizes, upwind schemes, and limiters, and are compared to experimental data from the literature. The solutions employing high order upwind interpolants and a compressive minmod limiter on the density are found to yield the best results. The two-dimensional, viscous Rayleigh-Taylor instability is examined next. Solutions for a density ratio of two are computed for various Reynolds numbers. Computed perturbation growth rates are shown to be in good agreement with theoretical predictions. Results for the three-dimensional broken dam problem are then presented. The computed free surface motions are found to be quite similar to the two-dimensional case. Finally, two cases involving axisymmetric spin-up in a spherical container are studied. The computed free surface shapes are found to exhibit the characteristic parabolic profiles as steady state conditions are approached