## 3 Results and Discussions  
### Fluctuating Displacements from Aeroelastic Tests  
Figure 5 shows fluctuating displacements in the along- and across-wind directions for tower-only cases. For tower-only cases, one wind direction that is normal to one of the surfaces was considered. The abscissa indicated reduced velocity defined by a reference wind speed at hub height \(U_{\textit{ref}}\), a reference length \(D_{\textit{ref}}\), and a natural tower frequency \(f_{\textit{lower}}\). The ordinate indicates reduced displacements in the along- (\(D_{\textit{std}}\)) and across-wind (\(L_{\textit{std}}\)) directions normalized by a reference length \(D_{\textit{ref}}\). Fluctuating displacements in the along-wind direction (Figure 5a) increased proportionally to the square of mean wind speed and decreased with increasing number of tower sides, showing a maximum value for the square tower. However, the slope for the square tower was greater than those of the other towers. For the across-wind direction, as shown in Figure 5b, fluctuating displacements for the square tower were much greater than those for the other towers, showing instability vibrations. For the other polygonal towers, limited vibrations such as vortex-induced vibration were found near the reduced velocity of 6.3, showing large vibrations for the octagonal tower. Fluctuating displacements near limited vibrations decreased with increasing number of sides, and the difference seemed to be small for the polygonal towers with number of sides larger than 10. Generally, as limited vibrations by vortex shedding were strongly related to vortex formation and shedding, it can be understood that vortex shedding weakened as the number of sides increased. For reduced velocities larger than 15, the fluctuating displacements for the decagonal tower became large. The aspect ratio (\(H_{\textit{ref}}/D_{\textit{ref}}\)) and tapering ratio (((\(D_{\textit{bottom}}-D_{\textit{top}}\))/\(H_{\textit{ref}}\)) \(\times\) 100) of \begin{table}
\begin{tabular}{c c c} \hline \hline
**Items** & **Aeroelastic Test** & **Pressure Measurement** \\ \hline Top diameter of circular tower \(D_{\textit{top}}\) & 0.04 m \\ \hline Bottom diameter of circular tower \(D_{\textit{bottom}}\) & 0.06 m \\ \hline Representative length \(D_{\textit{ref}}\) & 0.05 m \\ (diameter of circular tower at 0.5 \(H_{\textit{ref}}\)) & 0.9 m \\ \hline Sampling frequency & 600 Hz \\ \hline Number of 10-min samples & 10 \\ \hline \multirow{2}{*}{Reference wind speed} & Tower-only & up to 6.3 m/s \\ \cline{2-3}  & Wind turbine & up to 5.6 m/s \\ \hline Reduced velocity \(U^{*}=U_{\textit{ref}}(D_{\textit{ref}}\times f_{0})\) & Tower-only & up to 23 & - \\ \hline Wind direction \(\theta_{\textit{ref}}\) & Wind turbine & up to 32 & - \\ \hline \multirow{2}{*}{Azimuth angle \(\theta_{\textit{ref}}\)} & Tower-only & 0\({}^{\circ}\) & 0\({}^{\circ}\) \\ \cline{2-3}  & Wind turbine & 0\({}^{\circ}\) and 90\({}^{\circ}\) & 0\({}^{\circ}\):10\({}^{\circ}\):90\({}^{\circ}\) \\ \hline \multirow{2}{*}{Azimuth angle \(\theta_{\textit{ref}}\)} & Tower-only & - \\ \cline{2-3}  & Wind turbine & 0\({}^{\circ}\), 30\({}^{\circ}\), 60\({}^{\circ}\), 90\({}^{\circ}\) & 0\({}^{\circ}\), 30\({}^{\circ}\), 60\({}^{\circ}\), 90\({}^{\circ}\) \\ \hline Natural frequency \((f_{\textit{lower}}\) or \(f_{\textit{turbine}}\)) & Tower-only & 5.6 Hz & - \\ \cline{2-3}  & Wind turbine & 3.5 Hz & - \\ \hline \multirow{2}{*}{Damping ratio} & Tower-only & 0.8\% & - \\ \cline{2-3}  & Wind turbine & 1.1\% & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Details of wind tunnel test. the circular tower were 18 and 2%, respectively; thus, it was assumed that aeroelastic characteristics depending on cross-sectional shapes were clearly found. Figure 6 shows the fluctuating displacements in the along- and across-wind directions for a square wind turbine. In the present study, a wind turbine with a square-cross-section tower was simply referred to as a square wind turbine, which is true for all other polygonal cross-section towers.