On variational modelling of wave slamming by water waves

This thesis is concerned with the development of both mathematical (variational formulation) models and simulation (finite-element Galerkin) tools for describing a physical system consisting of water waves interacting with an offshore wind-turbine mast. In the first approach, the starting point is an action functional describing a dual system comprising a potential-flow fluid, a solid structure modelled with nonlinear elasticity, and the coupling between them. Novel numerical results for the linear case indicate that our variational approach yields a stable numerical discretization of a fully coupled model of water waves and an elastic beam. The drawback of the incompressible potential flow model is that it inevitably does not allow for wave-breaking. Therefore another approach loosely based on a van-der-Waals gas is proposed. The starting point is again an action functional, but with an extra term representing internal energy. The flow can be assumed to have no rotation, so although it is again described with a potential, compressibility is now introduced. The free surface is embedded within the compressible fluid for an appropriate van-der-Waals-inspired equation of state, which enables a pseudo-phase transition between the water and air phases separated by a sharp or steep transition variation in density. Due to the compressibility, in addition to gravity waves the model enables acoustic ones, which is confirmed by a dispersion relation. Higher-frequency acoustic waves can be dampened by the appropriate choice of time integrators. Hydrostatic and linearized models have been examined as verification steps. The model also matches incompressible linear potential flow. However, at the nonlinear level, the acoustic noise remains significant.