Variational water-wave models and pyramidal freak waves

A little-known fact is that, every week, two ships weighing over 100 tonnes sink in oceans, sometimes with tragic consequences. This alarming observation suggests that maritime structures may be struck by stronger waves than those they were designed to withstand. These are the legendary rogue (or freak) waves, i.e., suddenly appearing huge waves that have traumatised mariners for centuries and currently remain an unavoidable threat to ships, and to their crews and passengers. Thus motivated, an EU-funded collaboration between the Department of Applied Mathematics (Leeds University) and the Maritime Research Institute Netherlands (MARIN) supported this project, in which the ultimate goal, of importance to the international maritime sector, is to develop reliable damage-prediction tools, leading to beneficial impact in terms of both safety and costs. To understand the behaviour of rogue waves, cost-effective water-wave models are derived in both deep and shallow water. Novel mathematical and numerical strategies are introduced to capture the dynamic air-water interface and to ensure conservation of important properties. Specifically, advanced variational Galerkin finite-element methods are used to provide stable simulations of potential-flow water waves in a basin with wavemakers and seabed topography, which allows reliable simulations of rogue waves in a target area. For optimised computational speed, wave absorption is considered with a beach on which waves break and dissipate energy. Robust integrators are therefore introduced to couple the potential-flow model to shallow-water wave dynamics at the beach. Experimental validation of the numerical tank is conducted at Delft University of Technology to ensure accuracy of the simulations from the wavemaker to the beach. The numerical tank is designed for subsequent use by MARIN to investigate the damage caused by rogue waves on structures in order to update maritime design practice and to ensure safety of ships, therefore leading to a competitive commercial advantage across Europe.