The Green–Naghdi (GN) wave models are categorized into different levels based on the assumptions made for the velocity field. The low-level GN model (Level I GN model or called the GN-1 model) is a weakly dispersive, strongly nonlinear wave model. As the level goes up, the high-level GN model becomes a strongly dispersive, strongly nonlinear wave model. This paper introduces the algorithm to solve the Green–Naghdi wave models of different levels in three dimensions. The high-level GN (GN-3 and GN-4) models are applied to three-dimensional wave problems for the first time. Three test cases are considered here. First one is on the wave evolution in a closed basin. The symmetry, in the xx and yy directions in this case, verifies that the algorithm introduced here works well. The GN-3 results are also compared with the linear analytical results for a small wave elevation in a closed basin, and the agreement is good. The last two cases involve wave diffraction problems caused by an uneven seabed. In both of the last two cases, the GN-3 model is proved to be the converged GN model. The agreement between the GN-3 model and the experimental data and numerical predictions of the fully nonlinear Boussinesq model of others is also very good.
- High-level Green-Naghdi model
- Wave evolution
- Wave shoaling
- Wave diffraction
- Three-dimensional shallow-water wave model