# A Numerical Study of the Performance of Point Absorber Wave Energy Converters
## 2 Numerical Model
### Flow Settings  
Figure 2 shows the three WEC submergence depths for three float shapes. Different wave parameters are considered, such as a wave height \(H\) = 0.04 m and a time period \(T_{p}\) = 1.2 s. These parameters are used to examine the free-surface elevation of the wave through the empirical expression from [PERSON] and Dalrymple [18]. The expression of the surface elevation is then defined as \(\eta^{*}=(\frac{H}{2})cos(kx-\omega t+\Phi)\), where \(k\) is the wave number, \(\omega\) the angular frequency, \(t\) the time, and \(\Phi\) the phase shift. Figure 1: (**a**) Tank design. (**b**) Mesh of numerical wave tank. (**c**) Rectangular, circular cylindrical, and trapezoidal fin shapes of the wave energy converter. The shallow water theory was applied to the absorbed incident wave at the outlet. A wave with the same velocity and height was generated in the opposite direction to cancel out the incident wave [19]. All parameters used in the simulations are provided in Table 1. The rectangular WEC (W1) dimensions and wave conditions considered here are similar to those described in [17]. Two-dimensional transient simulations were carried out with the open-source CFD code OpenFOAM v-2006. Mass and momentum equations were solved with the flow considered incompressible, and the VOF model [20; 21; 22] was applied to track the water-air interface, similar to the research carried out in [19; 23]. The thermophysical properties were thus calculated in each cell of the computational domain using the phase fraction [24], and the interaction between the fluid and the structure was resolved through a finite volume discretization method. The densities of air and water considered are 1.161 and 1000 kg/m\({}^{3}\), respectively. A similar approach was applied in [16] to simulate a SWEC in an infinitely deep liquid for a submerged 2D circular cylinder at various wave frequencies. [PERSON]'s second law was used to investigate the motion of the WEC. The pressure and shear forces of the fluid were applied on the solid surface to generate the total vertical \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Shape**} & **Density** & **Wave Height** & **Spring Stiffness** & **Submergence Depth** & **Time Period** \\  & **(kg/m\({}^{3}\))** & **(m)** & **(k)** & **(d\({}_{s}\))** & **(s)** \\ \hline Rectangular & 1000 & 0.04, 0.06, 0.1 & 0, 50, 100 & 0, 0.1, 0.2 & 1.2 \\ Circular cylindrical & 1000 & 0.04 & 50 & 0, 0.1, 0.2 & 1.2 \\ Trapezoidal fin & 1000 & 0.04 & 50 & 0, 0.1, 0.2 & 1.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Details of parameters used in the simulations. Figure 2: Schematic representation of floating (\(d_{s0}=0\) m) and submerged (\(d_{s1}=0.1\) m, \(d_{s2}=0.2\) m). (**a**) Rectangular box (W1). (**b**) Circular cylinder (W2). (**c**) Trapezoidal fin (W3). force. The second-order Crank-Nicolson integration scheme was used for the WEC velocity calculation. Similar fluid-motion solver equations are proposed in the literature (see [17; 23], for instance). Four boundaries were defined: the inlet is set as the'velocity-inlet'; the outlet and top are defined as 'pressure outlets', which are conditions typically considered for an incompressible flow; and the last one (bottom part) is set as a 'wall'-type condition. A wave was generated at the system inlet, and, as mentioned previously, a wave absorption condition was imposed at the outlet section of the numerical domain for the absorption of the incident wave. A no-slip condition was applied, and no penetration was considered at the walls of the numerical tank (bottom) and other solid surfaces. Finally, atmospheric pressure was applied at the top surface of the tank. The OpenFOAM 'OverInterDyMFoam' solver was used in all simulations. A second-order upwind scheme with linear interpolation was applied for the divergence, gradient, and Laplacian terms in the governing equations.