Fully Nonlinear Energy Flux Calculations for Long-Crested Surface Waves

We employ a fully nonlinear numerical model to generate and propagate long-crested gravity waves in a tank containing an incompressible invicid homogeneous fluid, initially at rest, with a horizontal free surface of finite extent and of finite depth. A non-orthogonal curvilinear coordinate system is constructed which follows the free surface and is “fitted” to the bottom topography of the tank and therefore tracks the entire fluid domain at all times. A waveform relaxation algorithm provides an efficient iterative method to solve the resulting discrete Laplace equation, and the full nonlinear kinematic and dynamic free surface boundary conditions are employed to propagate the solution. In addition, a monochromatic deterministic theoretical wave-maker, employing a Dirichlet type boundary condition, and a suitably tuned numerical beach are utilized in the numerical model. We first show that the model generates the appropriate dispersion relation for both the deep water and shallow water conditions. Subsequently, we calculate the energy and energy flux for small steepness waves and show that the model produces the expected linear results. We complete the paper by considering steeper waves where the linear results are suspect.