A fast vector penalty-projection method for incompressible non-homogeneous or multiphase Navier-Stokes problems

We present a new {\em fast vector penalty-projection method (VPP$_{\eps}$)} to efficiently compute the solution of unsteady Navier-Stokes problems governing incompressible multiphase viscous flows with variable density and/or viscosity. The key idea of the method is to compute at each time step an accurate and curl-free approximation of the pressure gradient increment in time. This method performs a {\em two-step approximate divergence-free vector projection} yielding a velocity divergence vanishing as $\cO(\eps\,\dt)$, $\dt$ being the time step, with a penalty parameter $\eps$ as small as desired until the machine precision, {\em e.g.} $\eps=10^{-14}$, whereas the solution algorithm can be extremely fast and cheap. Indeed, the proposed {\em vector correction step} typically requires only a few iterations of a suitable preconditioned Krylov solver whatever the spatial mesh step. The method is numerically validated on three benchmark problems for non-homogeneous or multiphase flows where we compare it to the Uzawa augmented Lagrangian (UAL) and scalar incremental projection (SIP) methods. Moreover, a new test case for fluid-structure interaction problems is also investigated. That results in a very robust method running faster than usual methods and being able to efficiently and accurately compute sharp test cases whatever the density, viscosity or anisotropic permeability jumps, whereas other methods crash.