Towards a sharp, structure preserving two-velocity model for two-phase flow: transport of mass and momentum

The numerical modelling of convection dominated high density ratio two-phase flow poses several challenges, amongst which is resolving the relatively thin shear layer at the interface. To this end we propose a sharp discretisation of the two-velocity model of the two-phase Navier-Stokes (NS) equations. This results in the ability to model the shear layer, rather than resolving it, by allowing for a velocity discontinuity in the direction(s) tangential to the interface. In this paper we focus our attention on the transport of mass and momentum in the presence of such a velocity discontinuity. We propose a generalisation of the dimensionally unsplit geometric volume of fluid (VOF) method for the advection of the interface in the two-velocity formulation. Sufficient conditions on the construction of donating regions are derived that ensure boundedness of the volume fraction for dimensionally unsplit advection methods. We propose to interpolate the mass fluxes resulting from the dimensionally unsplit geometric VOF method for the advection of the staggered momentum field, resulting in semi-discrete energy conservation. Division of the momentum by the respective mass, to obtain the velocity, is not always well-defined for nearly empty control volumes and therefore care is taken in the construction of the momentum flux interpolant: our proposed flux interpolant guarantees that this division is always well-defined without being unnecessarily dissipative. Besides the newly proposed two-velocity model we also detail our exactly conservative (mass per phase and total linear momentum) implementation of the one-velocity formulation of the two-phase NS equations, which will be used for comparison. The discretisation methods are validated using classical time-reversible flow fields, where in this paper the advection is uncoupled from the NS solver, which will be developed in a later paper.