# Aerodynamic Characteristics of a Square Cylinder with Vertical-Axis Wind Turbines at Corners
## 3 Results and Discussions
### Force Coefficients and Vortex Shedding Characteristics  
#### 3.2.1 Stationary Wind Turbines at Cylinder Corners  
\(\overline{C_{d}}\), \(C_{l}^{RMS}\), and \(S_{t}\) of four stationary cases and the plain cylinder are listed in Table 5. The percentage of the values relative to the plain cylinder is shown in Figure 7.  
As shown in Table 5 and Figure 7, four cases with stationary wind turbines have lower \(\overline{C_{d}}\) than the plain cylinder since the suction on the two sides and leeward face is lower than the plain cylinder, as shown in Figure 5.  
As shown in Figure 7, \(\overline{C_{d}}\) is reduced 37.7% compared to the plain cylinder when \(\theta\) is 45\({}^{\circ}\)(\(S2\)), which is the most significant for all cases. The lowest suction on the leeward face shown in Figure 5 contributes to the lowest \(\overline{C_{d}}\) of \(S2\). The variation in \(C_{l}^{RMS}\) is similar to that of \(\overline{C_{d}}\) and the \(RMS\) lift coefficients has a 90.7% reduction when \(\theta\) is 45\({}^{\circ}\) (\(S2\)). That is to say, the fluctuation of \(C_{l}\) was suppressed effectively. There is still a drop of 66.7% of \(C_{l}^{RMS}\) compared with the plain cylinder when \(\theta\) is 135\({}^{\circ}\) (\(S4\)). Figure 8a,b presents the comparison of time history of \(Cl\) for \(S2\) and \(S4\). The red curve represents the time history of \(Cl\) of the plain cylinder and the black curves represent the time history of the typical cases. It can be found that the fluctuation of \(Cl\) for \(S2\) has a greater inhibition than \(S4\).  
Figure 7: Comparisons of \(RMS\) lift coefficients, mean drag coefficients, and Strouhal number for the four stationary cases and plain cylinder.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & **Cell** & \multirow{2}{*}{\(\boldsymbol{\forall^{*}(\text{Max})}\)} & \multirow{2}{*}{\(\boldsymbol{\delta/\text{W}}\)} & \multirow{2}{*}{\(\overline{C_{d}}\)} & \multirow{2}{*}{\(\boldsymbol{C_{l}^{RMS}}\)} & \multirow{2}{*}{\(\boldsymbol{S_{t}}\)} \\  & **Numbers** & & & & & \\ \hline Plain cylinder & \(2.0\times 10^{4}\) & \(1.4\) & \(2\times 10^{-4}\) & 1.99 & 1.29 & 0.123 \\ S1 & \(2.4\times 10^{5}\) & \(0.41\) & \(4\times 10^{-5}\) & 1.26 & 0.19 & 0.144 \\ S2 & \(2.4\times 10^{5}\) & \(0.43\) & \(4\times 10^{-5}\) & 1.24 & 0.12 & 0.145 \\ S3 & \(2.4\times 10^{5}\) & \(0.44\) & \(4\times 10^{-5}\) & 1.35 & 0.21 & 0.141 \\ S4 & \(2.4\times 10^{5}\) & \(0.59\) & \(4\times 10^{-5}\) & 1.48 & 0.43 & 0.143 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Force coefficient for four stationary cases and plain cylinder.  
Figure 9 illustrates the power spectrum of the lift force for the four stationary cases and plain cylinder; the vortex shedding energy is represented by the peak value of the spectrum. The frequency corresponding to the maximum value of the power spectrum is the vortex shedding frequency, and then \(S_{t}\) can be calculated by \(f_{s}W/U\) (\(W\) is the width of the cylinder and \(U\) is the wind speed). As shown in Figure 9, the spectrum of the lift coefficient of the plain cylinder fits well with that in [31]. Four stationary cases have a much larger Strouhal number, approximately 20% larger than that of the plain cylinder. The vortex shedding energy presented by the peak value of the spectrum of four stationary cases is evidently smaller than that of the plain cylinder. Even though the stationary wind turbines installed at two leading corners increase the frequency of vortex shedding of the square cylinder, they can effectively reduce the energy of vortex shedding. Furthermore, the vortex shedding energy of the cylinder with corner wind turbines exhibits the lowest value when \(\theta\) is 45\({}^{\circ}\) (\(S2\)).  
Figure 8: Comparison of time history of \(Cl\): (**a**) \(S2\); (**b**) \(S4\); (**c**) \(R1\); (**d**) \(R6\); (**e**) \(R8\).