The kinematics of broadbanded waves

Ocean waves are random and their crests evolve non-linearly. Yet commonly used wave models correctly incorporate only one of these aspects: the models either use non-linear regular wave theory whilst ignoring the randomness or use linear random theory with empirical corrections close to the free surface. For the more realistic design and re-assessment of off-shore structures, it is important to have an accurate model of the flow within large waves. In this report a numerical wave tank is used to study the water-wave problem. The numerical wave tank has been developed by Taylor and Vijfvinkel [16] and incorporates the non-linearity and the irregularity of waves. Wave-wave interactions have been simulated numerically for the full hydrodynamic equations using the scheme developed by Craig and Sulem [1]. This is based on Fourier expansion for both the free surface and the velocity potential. One advantage of this scheme is that it is possible to start with a linear group at the focus point, run the code backwards linearly in time and then restart the code running forward in time until the wave group focuses. After the restart, all the hydrodynamic non-linearities are included within the calculation. Typically, the non-linear focus point is shifted compared to the linear focus point and the peak surface displacement is higher than the linear value. In this report the focusing and dispersion of fully non-linear wave groups on different water depths is studied. A procedure for linearising the focused surface profile is introduced. This allows us to study the effects of wave non-linearity on the phase speed of individual Fourier components within the wave group. The wave properties on the surface given by the numerical wave tank are used to calculate the internal kinematics. From these profiles non-linear Morison drag forces can be calculated and compared with drag forces estimated using a empirical modification to linear wave theory known as Delta stretching [6]. For uni-directional waves, we propose a simple variant on Delta stretching which produces global force predictions much closer to the fully non-linear solutions.