Stability and robustness of a consistent and mass-conserving dual-grid Volume-of-Fluid method.

We consider Volume-of-Fluid methods for the Direct Numerical Simulation of the Navier-Stokes equations for multiphase incompressible flow with capillary forces and a large contrast in material properties. These methods are notoriously unstable for high density ratios, further compounded by the presence of large surface tension and small viscosity (as for example with air and water fluid properties). In order to alleviate these difficulties we use as discrete variables the Volume-of-Fluid and the momentum density. We explore the performance of the method on a staggered ``MAC'' grid. The staggered character of the grid makes it difficult to compute the fluxes of mass and momentum in a consistent manner. Thus we use a dual grid with a twice finer resolution for Volume-of-Fluid than for momentum and pressure, in a manner similar to the method of Rudman (1998). We perform numerous tests to quantify the stability and robustness of the method, including a dynamic spurious current tests in which a spherical droplet or bubble is advected at uniform velocity. We explore a large number of regimes for this flow characterized by the Reynolds, Weber and CFL numbers, the density ratio and the grid resolution. The stability limit is determined in this five-parameter space.