Two-Phase Outflows: Boundary Conditions and Algorithm

<p>We present an effective outflow boundary condition,&nbsp;and an associated numerical algorithm, within the phase-field&nbsp;framework for dealing with two-phase outflows or&nbsp;open boundaries.&nbsp;The set of two-phase outflow boundary conditions for the phase-field&nbsp;and flow variables are designed to prevent the un-controlled growth in the total&nbsp;energy of the two-phase system, even in situations&nbsp;where strong backflows or vortices may be&nbsp;present&nbsp;at the outflow&nbsp; boundaries.&nbsp;We also present an additional boundary condition&nbsp;for the phase field function, which together&nbsp;with the usual Dirichlet condition can&nbsp;work effectively as the phase-field inflow&nbsp;conditions.&nbsp;The numerical algorithm for dealing with these boundary&nbsp;conditions is developed on top of a strategy for de-coupling the computations&nbsp;of all flow variables and for&nbsp;overcoming the performance bottleneck&nbsp;caused by variable&nbsp;coefficient matrices associated with variable&nbsp;density/viscosity.&nbsp;The algorithm contains special constructions,&nbsp;for treating the variable dynamic viscosity in the outflow boundary condition, and for&nbsp;preventing a numerical locking at the outflow boundaries&nbsp;for time-dependent problems.&nbsp;Extensive numerical tests &nbsp;with incompressible two-phase flows involving&nbsp;inflow and outflow boundaries&nbsp;demonstrate that, the&nbsp;two-phase outflow&nbsp;boundary conditions and the numerical algorithm&nbsp;developed herein allow for the fluid interface and the &nbsp;two-phase flow to pass through the outflow&nbsp;or open boundaries in a smooth and seamless&nbsp;fashion,&nbsp;and that our method produces stable&nbsp; simulations&nbsp;when large density ratios and&nbsp;large viscosity ratios are involved&nbsp;and when strong backflows are&nbsp;present at the outflow boundaries.</p>