Consistent pressure Poisson splitting methods for incompressible multi-phase flows: eliminating numerical boundary layers and inf-sup compatibility restrictions

For their simplicity and low computational cost, time-stepping schemes decoupling velocity and pressure are highly popular in incompressible flow simulations. When multiple fluids are present, the additional hyperbolic transport equation in the system makes it even more advantageous to compute different flow quantities separately. Most splitting methods, however, induce spurious pressure boundary layers or compatibility restrictions on how to discretise pressure and velocity. Pressure Poisson methods, on the other hand, overcome these issues by relying on a fully consistent problem to compute the pressure from the velocity field. Additionally, such pressure Poisson equations can be tailored so as to indirectly enforce incompressibility, without requiring solenoidal projections. Although these schemes have been extended to problems with variable viscosity, constant density is still a fundamental assumption in existing formulations. In this context, the main contribution of this work is to reformulate consistent splitting methods to allow for variable density, as arising in two-phase flows. We present a strong formulation and a consistent weak form allowing standard finite element spaces. For the temporal discretisation, backward differentiation formulas are used to decouple pressure, velocity and density, yielding iteration-free steps. The accuracy of our framework is showcased through a wide variety of numerical examples, considering manufactured and benchmark solutions, equal-order and mixed finite elements, first- and second-order stepping, as well as flows with one, two or three phases.