Discrete Exterior Calculus Discretization of Two-phase Incompressible Navier-Stokes Equations with a Conservative Phase Field Method

We extend the discrete exterior calculus (DEC)  method  for simulating single phase flow (Jagad et al. Phys. Fluid 2021) to two-phase flows of immiscible fluids with surface tension and discontinuous jumps in density and viscosity. The two-phase incompressible Navier-Stokes equations and conservative phase field equation for interface capturing are first transformed into the exterior calculus framework. The differential forms, exterior derivatives and Hodge star operators are then discretized using DEC.  With appropriate choice of phase-field parameters, boundedness of the phase field is proved for the DEC method for both a first order forward Euler and a second order time integration scheme without employing a mass redistribution technique. Several standard advection tests on planar domain and a curved surface are conducted for investigating the applicability of interface capturing in a surface flow. Our results show excellent boundedness, mass conservation and convergence properties. Various physical simulations of incompressible two-phase flows (e.g. falling drop, rising bubble, drop impact on a liquid pool) under large density and viscosity jumps as well as surface tension demonstrate the positive features and advantages of the proposed DEC scheme.