# Aerodynamic Characteristics of a Square Cylinder with Vertical-Axis Wind Turbines at Corners
## 3 Results and Discussions
### Force Coefficients and Vortex Shedding Characteristics
#### 3.2.2 Rotating Wind Turbines at Cylinder Corners  
\(\overline{C_{d}}\), \(C_{l}^{RMS}\), and \(S_{t}\) of the nine rotating cases and the plain cylinder are listed in Table 6. Ratios of these values to that for the plain cylinder are shown in Figure 10. All cases of the square cylinder with rotating corner wind turbines have lower mean drag coefficients (\(\overline{C_{d}}\)) than the plain cylinder. Compared with the plain cylinder, when the rotation speed is \(0.2f_{\rm s}\) (\(R1\)), \(\overline{C_{d}}\) of the cylinder with wind turbines dramatically decreases by about 34.2%, and then the drop rate gradually decreases as the rotating speed increases before reaching \(f_{\rm s}\). When the rotation speed increases to \(2f_{\rm s}\) (\(R6\)), the drop rate of \(\overline{C_{d}}\) again increases to 34.2%. Similarly, all cases of the square cylinder with rotating corner wind turbines have lower \(C_{l}^{RMS}\) than the plain cylinder. As rotation speed increases, the variation in the drop rate of \(C_{l}^{RMS}\) is similar to that of \(\overline{C_{d}}\), but the reduction in \(C_{l}^{RMS}\) is more significant than \(\overline{C_{d}}\). Compared with the plain cylinder, \(C_{l}^{RMS}\) of the cylinder with corner wind turbines dramatically decreases by about 86.0% as the wind turbine rotation speed is \(0.2f_{\rm s}\)(\(R1\)) and the drop rate becomes 71.3% when the rotating speed increases to \(2f_{\rm s}\) (\(R6\)). When the rotation speed is \(4f_{\rm s}\)(\(R8\)), both \(\overline{C_{d}}\) and \(C_{l}^{RMS}\) have the least reduction. It is worth noting that even though \(\overline{C_{d}}\) and \(C_{l}^{RMS}\) have a significant reduction when the wind turbine has a rotating speed of \(0.2f_{\rm s}\) and \(2f_{\rm s}\), it is still smaller than the reduction in the case with the stationary wind turbines (\(R0\)). Figure 8c-e present the comparison of time history of \(Cl\) of \(R1\), \(R6\), and \(R8\). It can be found that the fluctuation of \(Cl\) of \(R1\) is smallest, which corresponds to the largest drop rate of \(C_{l}^{RMS}\) of \(R1\). Figure 8d,e shows that the fluctuation of \(Cl\) of \(R6\) is also effectively suppressed, but the effect of fluctuation inhibition of \(R8\) is not obvious.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Case**} & \multicolumn{2}{c}{**Cell**} & \multirow{2}{*}{**y\({}^{\star}\)(Max)**} & \multirow{2}{*}{**s/W**} & \multirow{2}{*}{\(\overline{C_{d}}\)} & \multirow{2}{*}{\(C_{l}^{RMS}\)} & \multirow{2}{*}{\(S_{t}\)} \\  & **Numbers** & & & & & \\ \hline Plain cylinder & \(2.0\times 10^{4}\) & \(1.4\) & \(2\times 10^{-4}\) & \(1.99\) & \(1.29\) & \(0.123\) \\ \(R0\) (S2) &  
Figure 11 presents the power spectrum of the lift force for the nine rotating cases and plain cylinder. Figure 11a refers to the cases with \(V_{t}/f_{s}\) less than 1 and Figure 11b shows the cases with \(V_{t}/f_{s}\) equal to or larger than 1. For Strouhal number (\(S_{t}\)) shown in Figure 11, all of the rotating cases have larger Strouhal numbers than the plain cylinder, except for \(R6\). The vortex shedding energy represented by the peak value of the spectrum of the nine rotating cases is evidently smaller than that of the plain cylinder. Moreover, the vortex shedding energy of the cylinder with corner wind turbines has the lowest value when the wind turbine rotation speed is \(2f_{s}\) (\(R6\)).