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

TL;DR

This paper introduces energy-stable, reduction-consistent outflow boundary conditions and an efficient algorithm for simulating multiphase flows with multiple immiscible fluids, effectively handling open boundary challenges.

Contribution

It proposes a novel set of boundary conditions and an associated algorithm that ensure energy stability, reduction consistency, and computational efficiency for N-phase multiphase flow simulations.

Findings

Boundary conditions prevent backflow instability.

Algorithm solves decoupled Helmholtz equations efficiently.

Numerical results match exact solutions in complex flow scenarios.

Abstract

We present a set of effective outflow/open boundary conditions and an associated algorithm for simulating the dynamics of multiphase flows consisting of $N$ ($N\geqslant 2$) immiscible incompressible fluids in domains involving outflows or open boundaries. These boundary conditions are devised based on the properties of energy stability and reduction consistency. The energy stability property ensures that the contributions of these boundary conditions to the energy balance will not cause the total energy of the N-phase system to increase over time. Therefore, these open/outflow boundary conditions are very effective in overcoming the backflow instability in multiphase systems. The reduction consistency property ensures that if some fluid components are absent from the N-phase system then these N-phase boundary conditions will reduce to those corresponding boundary conditions for the equivalent smaller system. Our numerical algorithm for the proposed boundary conditions together with the N-phase governing equations involves only the solution of a set of de-coupled individual Helmholtz-type equations within each time step, and the resultant linear algebraic systems after discretization involve only constant and time-independent coefficient matrices which can be pre-computed. Therefore, the algorithm is computationally very efficient and attractive. We present extensive numerical experiments for flow problems involving multiple fluid components and inflow/outflow boundaries to test the proposed method. In particular, we compare in detail the simulation results of a three-phase capillary wave problem with Prosperetti's exact physical solution and demonstrate that the method developed herein produces physically accurate results.