Time domain simulation of the unidirectional and directional focused waves

The phase focusing is considered to be a significant mechanism for the formation of "short-lived" extreme waves, and this attracts many experimental and numerical investigations. Laboratory studies were carefully performed to provide a benchmark to investigate the focused waves (Baldock et al., 1996; Johannessen and Swan, 2001; Grue and Jensen, 2006). Among that, Baldock et al. (1996) is widely used as the benchmark of the two dimensional focused wave in deep water. And Johannessen and Swan (2001) is another benchmark for the experimental 3-D focused wave. It is still attractive to develop an accurate and efficient wave model to investigate the focused wave. In order to satisfy this demand, the High-Level Irrotational Green-Naghdi (HLIGN) equations derived by Kim et al. (2001) are utilized in this investigation. As the name indicates, the HLIGN equations are categorized into different levels. The higher level IGN equations are more accurate (Zhao et al., 2015) as shown before. The Harmonic Polynomial Cell (HPC) method is applied for further verification of the HLIGN results. The HPC method is originally derived by Shao and Faltinsen (2012, 2014). The comparison with other potential solvers shows that the HPC method has good accuracy and efficiency. The theoretical framework of the HLIGN equations and the HPC method is briefly described Sections 2 and 3, respectively. The numerical cases including the 2D and 3D focused wave as well the effects of water depth and directions are presented in Section 4. Summaries and conclusions of this study are provided in Section 5.