  This paper presents a new monolithic free-surface formulation that exhibits
correct kinetic and potential energy behavior. We focus in particular on the
temporal energy behavior of two-fluids flow with varying densities. Correct
energy behavior here means that the actual energy evolution of the numerical
solution matches the evolution as predicted by the discrete two-fluid
equations. We adopt the level-set method to describe the two-fluid surface. To
ensure the correct energy behavior we augment the interface convection equation
with kinetic and potential energy constraints. We solve the resulting
formulation consisting of the fluid and interface equations in a monolithic
fashion using a recently proposed level-set method [I. Akkerman. Monotone
level-sets on arbitrary meshes without redistancing. Computers & Fluids,
146:74-85, 2017.]. For the spatial discretization divergence-conforming NURBS
are adopted. The resulting discrete equations are solved with a quasi-newton
method which partially decouples the constraints from the rest of the problem.
The simulation of a dambreak problem numerically supports the correct energy
behavior of the proposed methodology. This improves the solution quality
significantly upon a more traditional approach. Due to the excellent accuracy
per degree of freedom one can suffice with a much lower resolution.
