The Nonlinear Interaction and Resonance of Steep Long-Crested Bichromatic Surface Waves in a Numerical Wave Tank

Abstract

We are studying numerically the problem of generation and propagation of long-crested gravity waves in a tank containing an incompressible inviscid homogeneous fluid initially at rest with a horizontal free surface of finite extent and of infinite depth. A non-orthogonal curvilinear coordinate system, which follows the free surface is constructed which gives a realistic “continuity condition”, since it tracks the entire fluid domain at all times. A depth profile is assumed and employed to perform a waveform relaxation algorithm to decouple the discrete Laplacian along dimensional lines, thereby reducing its computation over this total fluid domain. In addition, the full nonlinear kinematic and dynamic free surface conditions are utilized in the algorithm. A bichromatic deterministic wave maker using a Dirichlet type boundary condition and a suitably tuned numerical beach is utilized. This paper pays special attention to satisfying the full nonlinear free surface conditions and presents the nonlinear interaction of the higher order components, especially near resonance.