Multiphase Flows of N Immiscible Incompressible Fluids: An Outflow/Open Boundary Condition and Algorithm

摘要

We present a set of effective outflow/open boundary conditions and anassociated algorithm for simulating the dynamics of multiphase flows consistingof $N$ ($N\geqslant 2$) immiscible incompressible fluids in domains involvingoutflows or open boundaries. These boundary conditions are devised based on theproperties of energy stability and reduction consistency. The energy stabilityproperty ensures that the contributions of these boundary conditions to theenergy balance will not cause the total energy of the N-phase system toincrease over time. Therefore, these open/outflow boundary conditions are veryeffective in overcoming the backflow instability in multiphase systems. Thereduction consistency property ensures that if some fluid components are absentfrom the N-phase system then these N-phase boundary conditions will reduce tothose corresponding boundary conditions for the equivalent smaller system. Ournumerical algorithm for the proposed boundary conditions together with theN-phase governing equations involves only the solution of a set of de-coupledindividual Helmholtz-type equations within each time step, and the resultantlinear algebraic systems after discretization involve only constant andtime-independent coefficient matrices which can be pre-computed. Therefore, thealgorithm is computationally very efficient and attractive. We presentextensive numerical experiments for flow problems involving multiple fluidcomponents and inflow/outflow boundaries to test the proposed method. Inparticular, we compare in detail the simulation results of a three-phasecapillary wave problem with Prosperetti's exact physical solution anddemonstrate that the method developed herein produces physically accurateresults.