The most crucial part of a simulation method for two-phase flow is the way in which the fluids (e.g. LNG and its vapor) are coupled at the liquid-gas interface. Viscous effects play a role at this interface, but result in a shear layer which is very thin and expensive to resolve. We propose to model this shear layer with a tangential velocity discontinuity, resulting in a novel and truly sharp two-velocity model. In this model, each fluid gets its own velocity and corresponding momentum conservation law. The resulting two-phase Navier--Stokes equations are solved using a finite-volume method on an adaptive 3D mesh. A dimensionally unsplit geometric volume of fluid (VOF) method advects the interface. This results in a sharp representation of the interface. The transport of momentum is done using the same volume fluxes as used for mass transport. An algebraic interpolation of the fluxes is proposed, resulting in exact mass and momentum conservation, while obtaining semi-discrete conservation of kinetic energy, and an almost perfect exchange with potential and (capillary) surface energy. Whereas the traditional one-velocity model results in an artificial thickening of the shear layer, the two-velocity model instead sharply and accurately approximates the unresolved shear layer with a velocity discontinuity. We do not neglect viscous stresses altogether, and in fact mesh refinement of the two-velocity model yields the same solution as obtained with the one-velocity model. Our approach greatly reduces the required number of grid points (i.e. computational cost), making simulations of breaking wave impacts better affordable. The proposed numerical method has been validated using several academic flow problems of surface instabilities (like Kelvin--Helmholtz), and was applied to several wave impact problems (e.g. sloshing in liquid storage tanks).