# Computational Study on Rogue Wave and Its Application to a Floating Body
## 3 Results and Discussion
### Bull's-Eye Waves Simulation  
The parameters of water and air are the same as the regular wave simulation. The domain extent and boundary conditions of a numerical wave tank are shown in Figure 4. The Cartesian coordinate was selected and the generated waves on the inlet boundary were propagated in the x-y plane. The half domain was considered because the domain and boundary conditions were symmetric about the x-z plane (y = 0 plane). The circular-shaped inlet boundary was considered. The generated waves from the inlet boundary were overlapped at the center of the circle and dissipated in the downstream. The radius of the inlet boundary was 7.5 m. The whole domain extent was 20.45 m in the x-direction, 7.5 m in the y-direction, and 5 m in the z-direction. The wave absorption zone was located in front of the outlet boundary. The x-directional length of the absorption zone was set to 6.475 m, twice the wavelength [18; 19]. The location of the focal point was x = 0 m and y = 0 m.  
For the velocity field, the Dirichlet boundary condition was applied on the inlet boundary while the Neumann boundary condition was applied on the outlet boundary. At the side boundary, a no-slip boundary condition was used. An atmospheric condition was applied on the top boundary and a no-slip condition was applied on the bottom boundary. Figure 5 shows a typical structured grid of 2,800,000 meshes. In a wave, 100 meshes were used in the one wavelength and 20 meshes were used in the wave height. From the inlet boundary to the center of the circular-shaped inlet boundary, the meshes generated are aligned to the wave propagating direction in order to decrease the numerical error caused by false diffusion. Figure 3: Wave elevation of regular wave. A regular wave with a wavelength \((\lambda)\) of 3.2375 m and the wave period \((T)\) of 1.44 s was generated on the inlet boundary. For the wave height \((H)\), 0.02, 0.03, and 0.04 m were considered as listed in Table 2. The regular waves propagated towards the focal point, and 3-D bull's-eye waves were generated. The grid convergence test was carried out with a wave height of 0.02 m for four types of grids. As listed in Table 3, the four types of coarse, medium fine, and extra fine grids were considered. The grid size was increased or decreased in all directions by 1.26 times. Figure 6 shows the wave height history in time at the focal point for the four grids. Figure 7 \begin{table}
\begin{tabular}{c c c c} \hline  & **Wave Height (H)** & **Wavelength (\(\lambda\))** & **Wave Period (T)** \\ \hline Wave condition 1 & 0.02 m & & \\ Wave condition 2 & 0.03 m & 3.2375 m & 1.44 s \\ Wave condition 3 & 0.04 m & & \\ \hline \end{tabular}
\end{table} 
Table 2: Simulated wave condition. Figure 4: Domain extent and boundary condition of a bull’s-eye waves simulation. Figure 5: Typical surface meshes and focal point of a bull’s-eye waves simulation. shows the relative errors of wave crest height at the focal point for the four grids. The relative error defined as follows:  
\[\epsilon_{ij}=\left|\frac{\text{RMS}(A_{\text{cresti}})-\text{RMS}(A_{\text{ cresti}})}{\text{RMS}(A_{\text{cresti}})}\right| \tag{12}\]  
where, the subscripts \(i\) and \(j\) indicate the index of the grid system. The relative errors between the four meshes were \(\epsilon_{21}=1.88\%\), \(\epsilon_{32}=0.88\%\), \(\epsilon_{43}=0.31\%\), respectively. Mesh 3 was selected from the grid convergence test results. Figure 8 shows the rogue wave heights for various incident wave amplitudes. The simulated wave heights were compared with the experimental data [22]. The rogue wave height was calculated as the vertical distance between the crest and the trough. The simulated wave heights showed the same trend as the experimental results. Figure 9 shows the wave profile at the focal point along \(x=0\). The simulated wave profile was well matched with the experimental results [22]. Figure 10 shows the wave height history in time at the focal point. The wave heights were captured by a volume fraction of 0.5.