# Power Control Optimization of an Underwater Piezoelectric Energy Harvester
## 4 Computational Results  
The turbulent flow over a circular cylinder for Reynolds numbers: Re = 3000, 6000, 9000, and 12,000 is investigated in the present study using a two-dimensional finite volume method. The Re number is based on the cylinder diameter and has been calculated by Equation (1), where \(\rho\) and \(\upmu\) correspond to the density and the dynamic viscosity of the water, respectively, at 15 \({}^{\circ}\)C of fluid temperature. \[\text{Re = }\frac{\text{V}_{\text{water}}\cdot\text{D}\cdot\rho}{\upmu} \tag{1}\]  
The different Re numbers were achieved by changing the water speed \(\text{V}_{\text{water}}\) at the inlet. The solutions were computed during a time from 0 to 20 s. The vortex shedding is observed and the drag and lift forces are also calculated using CFD techniques. The computations have been validated by the experimental work of [PERSON] [46] which locates the beginning of the laminar-to-turbulent transition at Reynolds numbers between 200 and 300. Further than those Reynolds number but less  
Figure 3: Mesh distribution around the cylinder. Figure 2: Computational domain dimensions (not to scale). than around 3 \(\times\) 10\({}^{5}\) the wake of the cylinder is completely turbulent. In finite volume methods, the flow domain is divided into a number of control volumes or cells. Therefore, a discretized form of Equation (2) must be set up at a nodal point placed within each control volume in order to solve the problem. For control volumes neighboring to the domain boundaries, the general equation is modified to include boundary conditions. The resulting system of linear algebraic equations is then solved to obtain the velocity and pressure distribution n at each nodal point. The drag coefficient is computed as follows:  
\[\mathrm{C_{D}}\ =\ \frac{\mathrm{F_{D}}}{0.5\ \rho_{\mathrm{water}}\mathrm{U_{ \infty}^{2}}\ \mathrm{D}} \tag{2}\]  
where \(\mathrm{F_{D}}\) is the drag force on the cylinder, \(\rho_{\mathrm{water}}\) is the density of the water at standard conditions and \(\mathrm{U\infty}\) is the free stream velocity of the fluid depending on the Re number. A mesh independency study has been carried out to verify sufficient mesh resolution. To that end, the Richardson extrapolation method has been applied to the drag coefficient calculations. Figure 4 shows the comparison between the experimental data of [PERSON] [46] and the current CFD results of \(\mathrm{C_{D}}\) with three different meshes: 14,000 (coarse), 28,000 (medium), and 56,000 (fine) cells. Table 1 shows the results of the mesh independency study where RE represents the extrapolated solution, R the ratio of errors, and p the order of accuracy. Since the value of R is less than 1, all three integral quantities converge monotonically for all Reynolds number cases Re = 3000, 6000, 9000, and 12,000. As shown in Figure 4, the results of the fine mesh predict quite well the drag coefficients. Therefore, this higher resolution mesh has been used for the current computations. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & **Mesh** & & & **Richardson Extrapolation** \\ \hline
**Re** & **Coarse** & **Medium** & **Fine** & **RE** & **P** & **R** \\ \hline
3000 & 0.69 & 1.02 & 1.05 & 1.05 & 3.46 & 0.09 \\
6000 & 0.71 & 1.05 & 1.08 & 1.08 & 3.37 & 0.10 \\
9000 & 0.78 & 1.14 & 1.18 & 1.18 & 3.34 & 0.10 \\
12,000 & 0.79 & 1.15 & 1.19 & 1.19 & 3.35 & 0.10 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Results of the mesh independency study. Figure 4: Drag coefficients comparison between the experiments and the current Computational Fluid Dynamics (CFD) simulations. The vortex shedding is visualized by means of the contours of stream function vorticity. Figure 5 shows the vorticity contour lines for Re = 3000, 6000, 9000, and 12,000 cases at the time of 20 s. The alternating formation, convection, and diffusion of the vortices can be clearly observed.