Toward free-surface flow simulations with correct energy evolution: An isogeometric level-set approach with monolithic time-integration

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 [26]. 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. As we focus on the energy behavior of time integration in case of varying densities, we restrict ourselves to low-Reynolds-number flow allowing simple Galerkin discretizations. High-Reynolds-number two-fluid flows that require stabilization are beyond the scope of the current paper. The simulation of a dambreak problem numerically supports the correct energy behavior of the proposed methodology. The proposed methodology 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.