A momentum-conserving, consistent, Volume-of-Fluid method for incompressible flow on staggered grids

The computation of flows with large density contrasts is notoriously difficult. To alleviate the difficulty we consider a partially momentum-conserving discretization of the Navier-Stokes equation. Incompressible flow with capillary forces is considered, and the discretization is performed on a staggered grid of Marker and Cell type. The Volume-of-Fluid method is used to track the interface and a height-function method us used to computed surface tension. The advection of the volume fraction is performed using either the Lagrangian-Explicit / CIAM method or the Weymouth and Yue Eulerian-Implicit method. To improve the stability of the method momentum fluxes are advected in a manner ``consistent'' with the volume-fraction fluxes, that is a discontinuity of the momentum is advected at the same speed as a discontinuity of the density. To find the density on the staggered cells on which the velocity is centered an auxiliary reconstruction of the density is performed. The method is tested for a droplet without surface tension in uniform flow, for a droplet suddenly accelerated in a carrying gas at rest at very large density ratio without viscosity or surface tension, for the Kelvin-Helmholtz instability, for a falling raindrop and for an atomizing flow in air water conditions.