Numerical modeling of two-fluid interfacial flows - Smolianski, Anton (2001)
The present work is devoted to the study on unsteady flows of two immiscible viscous fluids separated by free moving interface. The goal of the present work is to elaborate a unified strategy for numerical modeling of all kinds of two-fluid interfacial flows, having in mind possible interface topology changes (like merger or break-up) and realistically wide ranges for physical parameters of the problem. The presented computational approach essentially relies on three basic components: finite element method for spatial approximation, operator-splitting for temporal discretization and level-set method for interface representation. Finite element discretization is based on variational formulation of the problem and, thus, allows to naturally incorporate discontinuous material coefficients and singular interface-concentrated forces. The use of finite elements permits to localize the interface precisely, without introduction of any artificial parameters like interface thickness. We also show that interface normal and curvature can be recovered with the second-order accuracy after applying a gradient averaging technique; that allows us to compute accurately the surface tension force. For temporal discretization we employ an operator-splitting, thus, separating all major difficulties of the problem. This approach enables us, in particular, to implement equal-order interpolation for the velocity and pressure. In order to model the phenomena involving interface topology changes we make use of the level-set approach, the finite element implementation of which brings some additional benefits as compared to the standard finite difference level-set realizations. We introduce also a simple mass-correction procedure allowing to maintain an optimal, second order accurate mass conservation. Diverse numerical examples including simulations of bubble dynamics, bifurcating jet flow and Rayleigh-Taylor instability are presented to validate the proposed computational method.