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SURFS-UP: Freak Waves and Breaking Wave Impact on Offshore Structures

Final Report Summary - SURFSUP (SURFS-UP: Freak Waves and Breaking Wave Impact on Offshore Structures)

Chapter 1

The Surfs-UP European Industry Doctorate project concerns the mathematical and engineering modelling of freak waves at sea via "Variational water wave modelling" and the modelling of "Wave slamming" against maritime structures such as offshore wind-turbine masts. These two subprojects were undertaken by the ESRs Floriane Gidel and Tomasz Salwa, and supervised by Profs Onno Bokhove and Mark Kelmanson at Leeds University as well as Dr Tim Bunnik and Ir Geert Kapsenberg from MARIN. The subprojects were integrated in that the wave modelling of the project was of relevance to the wave slamming project. In addition, the freak waves targetted in the wave modelling subproject will ultimately be aimed at the wind turbine masts subjected to wave slamming as investigated in the second subproject. Moreover, we used an integrated numerical finite element modelling environment www.firedrakeproject.org in both subprojects, which facilitated knowledge sharing and dissemination on computational developments such as parallel, i.e. fast computing.

Variational water wave modelling, by Floriane Gidel:
Motivation and achievements: A little-known fact of international commercial importance is that, every week, two large (i.e. more than 100 tonnes) ships 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 "freak waves", i.e. suddenly appearing huge waves that have traumatised mariners for centuries. Despite increased research in the last 20 years on understanding freak-wave origins, scientists are even now unable to predict their formation. Therefore, freak waves currently remain an unavoidable threat to ships, and to their crew and passengers. Thus motivated, an EU-funded collaboration between Applied Mathematics (Leeds University) and the Maritime Research Institute Netherlands supported my project, in which the ultimate goal, of importance to the international maritime sector, was to develop reliable damage-prediction tools, leading to beneficial impact in terms of both safety and costs. In order to understand the behaviour of freak waves, I have developed and coded robust computational methods (based on advanced mathematical theory and validated with laboratory experiments in large-scale wave tanks) for the low-cost simulation of freak-wave dynamics in both shallow and deep water. To examine the damage of freak waves on realistic maritime structures, I have also developed a "numerical wave tank": a computational basin with wave-makers and seabed topography. Specifically, this has enabled reliable simulation of freak waves in a target area, to measure their impact upon a vessel or a wind turbine. Finally, in order to assess our numerical wave tank results from the wave-maker, where waves are generated, to the breaking waves at the beach, we acquired a series of novel experimental wave-tank measurements in collaboration with MARIN.

Wave slamming, by Tomasz Salwa:
The aim of the project has been to develop a mathematical model and simulation tool describing a physical system consisting of water waves interacting with an offshore wind-turbine mast, using a variational formulation and Galerkin method, possibly extended with wave breaking. Coupling potential-flow water waves to a 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. I have developed a linearized model of the fluid-structure or wave-mast coupling, based on the variational principle for the fully coupled nonlinear model. Our 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 energy exchange between the subsystems is in balance, yielding a total energy that shows only small and bounded oscillations with second-order convergence in time. Similarly, (second-order) convergence is observed for spatial mesh refinement. The linearized model so far developed should be extended into a nonlinear regime and the part of the model involving the nonlinear beam structure has been implemented hitherto.

Pseudo-compressible and breaking water wave models:
The drawback of the incompressible potential flow model is that it inevitably does not allow for wave breaking. Moreover, coupling of nonlinear waves with the beam proves challenging due to the finite motion of the domains which are described at the nonlinear level in different formalisms: water -Eulerian and the beam -Lagrangian. As a result, meshes of the two subsystems no longer match, which makes nonlinear coupling prone to numerical instability. Instead of pursuing this direction, we propose another model loosely based on a Van-der-Waals fluid. 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 it is again described with a potential, but now we allow compressibility. The functional thus yields a rotation less momentum equation. 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 in density. Due to the compressibility, in addition to surface gravity waves the model enables internal gravity and acoustic waves. There is a risk involved that the latter will dominate the result, but due to their higher frequency it should be possible to filter and dampen these, using implicit time integrators. I have examined hydrostatic and linearized models as verification steps. With a proper choice of equation of state that models also a finite-width interface, the dispersion relation confirms that there are multiple wave modes present, including gravity water waves. The model also matches incompressible linear potential flow water waves, which is an important verification step. However, at the nonlinear level, the acoustic noise proves significant. When we have dealt with the control and damping of acoustic noise in the ideal gas model, the model can be readily coupled to a hyperelastic beam, similarly to the incompressible potential flow case of the first approach, but with a nonlinear fluid and (possibly) breaking waves. Ongoing efforts are made to enable this exciting development. Finally, a related pseudo-compressible mixture theory for water-air dynamics has been derived. It differs from the above approach in that for a diffusive or breaking air-water interface rotational flow components will emerge.