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