# A Numerical Study of the Performance of Point Absorber Wave Energy Converters
## 3 Results and Discussion
### Effect of WEC Shape  
The influence of the WEC design on its movement and forces was analyzed for \(H\) = 0.04 m, \(k_{2}\), and \(d_{s0}\). Figure 6a,b display the displacement of the WECs over time in the \(x\)- and \(z\)-directions. The maximum displacement appears larger for W1 than for W2 and W3 in both the horizontal and vertical directions. The horizontal displacements for W1 and W2 are not very different, though, but the vertical movement is lower for W2. The horizontal displacement of W3 seems about twice as low as those of W1 and W2. Significant horizontal velocity variations for W1 and W2 can be noticed; see Figure 6c. Figure 6d shows that the velocity differences of W1 and W3 in the vertical direction are marginal. Due to the small vertical displacement of W2, the vertical velocity is also small. The velocity variations of the WECs can be explained through the vorticity contour plots presented in Figure 7. Vorticity is present at the corners of W1 and W3 but is sparse near W2, as there is not much perturbation inherent to the cylindrical device. It can also be noticed that the pressure is low where high vorticity is present; see Figure 8.  
The pressure force exerted by the wave on the WECs in the \(x\)-direction is higher for W1 and W2; see Figure 6e. \(F_{x}\) and \(F_{z}\) are two times higher for W2 than for W3. On the other hand, the vertical force \(F_{z}\) exerted on W1 is lower than on W3 due to the higher lift force; see Figure 6f. W3 is at a greater depth, inducing higher hydrostatic pressure. Any smaller or identical displacement requires a large upward vertical force. As a result, the pressure forces in the horizontal and vertical directions are dependent on the hydrostatic force (\(F=\rho gH\)) and higher for W2. Figure 6: Effect of different WEC shapes of effective density 1000 kg/m\({}^{3}\), wave height \(H\) = 0.04 m, submergence depth \(d_{s0}\) = 0 m, and \(k_{2}\) = 50 N/m. (**a,b**) Horizontal (\(Dx\)) and vertical (\(Dz\)) displacements of the WECs in function of time. (**c,d**) Horizontal and vertical velocities of all WECs. (**e,f**) Forces on all WECs in \(x\)- and \(z\)-directions. A general relation between the force \(F\), the velocity \(v\), and the displacement/position \(D\) of the buoy in the vertical direction can be written. [PERSON]'s second law is used to investigate the motion of the WEC. The pressure and shear forces of the fluid are applied on the solid surface and generate the total vertical force. \[F^{n+1}=Ma^{n+1} \tag{1}\]  
where \(M\) and \(a^{n+1}\) are the mass and vertical acceleration of the solid WEC, respectively. The vertical movement of the WEC is calculated as follows:  
\[v^{n+1}=v^{n}+\frac{1}{2}\Big{(}a^{n}+a^{n+1}\Big{)}\Delta T \tag{2}\]  
\[D^{n+1}=D^{n}+\frac{1}{2}\Big{(}v^{n}+v^{n+1}\Big{)}\Delta T \tag{3}\]  
Figure 8: Pressure contours. (**a**) W1. (**b**) W3. Wave height \(H\) = 0.04 m, submergence depth \(d_{s}\) = 0 m, and \(k_{2}\) = 50 N/m. Figure 7: Vorticity contours. (**a**) W1. (**b**) W2. (**c**) W3. Wave height \(H\) = 0.04 m, submergence depth \(d_{s0}\) = 0 m, and \(k_{2}\) = 50 N/m. where \(n\) and \(n+1\) are the previous and current times, and \(\Delta T\) is the time step. Once the acceleration \(a^{n+1}\) is known, the vertical velocity \(v^{n+1}\) and the vertical position \(D^{n+1}\) can be calculated at the same time \(n+1\). Similar fluid-motion solver equations were used in [17]. The submergence depth of all WECs is now increased. At \(d_{s1}\), the displacement of W3 is more significant than that of W1 and W2 in both the \(x\)- and \(z\)-directions, as shown in Figure 9a,b. Correspondingly, the trend of velocity \(V_{x}\) in the horizontal direction displays similarities for W1 and W2. The \(V_{x}\) of W3 is lower, while the velocities \(V_{z}\) in the vertical direction for W3 and W1 overlap and are higher than for W2. Figure 9e,f show that the force \(F_{z}\) on W1 and W2 increases with the increase in submergence depth \(d_{s1}\). The force \(F_{x}\) on W1 also increases as the depth \(d_{s1}\) increases.