Numerical investigation on the propagation and evolution of focused waves over a sloping bed

It is well recognized that wave focusing is one of main mechanisms leading to nonlinear extreme/rouge waves in the real sea. The evolution of this nonlinear focused wave group (i.e., wave generated by wave focusing), characterized by the wave shape and kinematics, is significantly affected by the bottom topography. Since the terrain of the nearshore area is generally in the form of a mild slope, it is of both academic and practical values to investigate its propagation and evolution over a sloping bed. In this paper, a two-dimensional fully nonlinear numerical model is established based on the potential flow theory and the time-domain high-order boundary element method. The hybrid Euler-Lagrangian method is used to track the instantaneous water surface, and the fourth-order Runge-Kutta method is used to update the water surface and the velocity potential at the next time step. The focused wave is generated by prescribing velocities from second-order Stokes wave theory at the immerged sources boundary, and is forced to focus at a pre-defined location and time by the principle of wave focusing. By changing the bed sloping and the input wave amplitude, it was found that the focused wave crest decreases with the increase of the bed sloping, and the actual focal location is somehow shifted behind the theoretical value. When the initial focusing position is pre-defined at the beginning of the slope, the free surface elevations and the wave energy of higher-order harmonics above the slope are both smaller than the corresponding values on the flat bottom. However, when the focusing position is pre-set at the back part of the slope, the conclusions are opposite. The presence of the slope facilitates the energy conversion from low frequencies to high frequencies.