## 4 Analyses and Discussions
### Aerodynamic Loads of Airfoil under Coupled Condition
#### 4.2.1 Case 1  
In this case, the dynamic stall characteristics of the OA209 airfoil are simulated at \(\alpha=12.0^{{}^{\circ}}+8.0^{{}^{\circ}}\sin(\omega t+\pi)\) with reduced frequency of 0.075, the basic Mach number is \(M_{B}\) = 0.4, and \(\lambda\) = 0.0, 0.25, and 0.5, respectively. The variation of Mach number with AoA is shown in Figure 9, where \(\lambda\) = 0.0 represents the case of steady velocity. Figure 8: Comparisons of dynamic stall characteristics of airfoil between the 2D and 3D conditions. (**a**) Section of 0.6 R. (**b**) Section of 0.7 R.  
Figure 9: Comparison of Mach number variation with AoA. The streamlines and pressure contour around the OA209 airfoil under the coupled conditions are shown in Figure 10. It can be seen that the LEV at the AoA of \(12.84^{\circ}\) during the upstroke process is not obvious under the SFV condition. However, the LEV is observed near the leading edge of airfoil for the state of \(\lambda=0.5\). It is indicated that the LEV would form earlier under the condition of OFV. Due to the induction of LEV and the influence of trailing edge of airfoil, the trailing edge vortex (TEV) forms near the trailing edge of airfoil. The AoAs of LEV and TEV formation are shown in Table 2. It is illustrated that the AoAs of LEV and TEV formation are both decreased with the increase of normalized oscillating velocity. Figure 11 shows the pressure coefficient (\(Cp_{2}\), normalized by \(\rho_{\infty}d_{\infty}^{2}/2\)) of the OA209 airfoil on the upper surface through a pitching cycle. It is illustrated that the \(Cp_{2}\) near the leading edge of airfoil under the OFV condition is smaller than that under the SFV condition at the same AoA between the azimuth of \(60^{\circ}\) to \(180^{\circ}\), because the increase of freestream velocity under the OFV condition would enlarge the pressure. What is more, another feature can be noticed in that the negative pressure peak is restricted at \(Cp_{2}=-1.08\) due to the effect of the LEV. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(\lambda\) = 0.0 & \(\lambda\) = 0.25 & \(\lambda\) = 0.5 \\ \hline
**LEV** & \(14.47^{\circ}\) & \(14.07^{\circ}\) & \(12.84^{\circ}\) \\
**TEV** & \(15.25^{\circ}\) & \(14.87^{\circ}\) & \(13.66^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: AoAs of leading edge vortex (LEV) and trailing edge vortex (TEV) formation. Figure 10: Comparisons of pressure contour and streamlines at different oscillating velocities (k = 0.075, \(\alpha=12.0^{\circ}+8.0^{\circ}\sin(\omega t+\pi)\)). The aerodynamic loads of the OA209 airfoil are shown in Figure 12. It can be noticed that the \(C_{l}\) is increased by enlarging the normalized oscillating velocity during the upstroke process, and the lift curve slopes at a large AoA, for the different oscillating velocities are 0.371 (line a for \(\lambda=0.5\)), 0.273 (line b for \(\lambda=0.25\)) and 0.228 (line c for \(\lambda=0.0\)), respectively. As a result, the peaks of \(C_{l}\) are also enlarged under the OFV condition. This characteristic indicates that lift induced by LEV would be enlarged as the oscillating velocity increases. From Table 3, it is illustrated that the peak of \(C_{l}\) with \(\lambda=0.5\) is 14.5% higher than that of \(\lambda=0.0\). Meanwhile, the divergence angles of the \(C_{m}\) and \(C_{d}\) are decreased when the oscillating velocity increases. Additionally, the peaks of \(C_{m}\) and \(C_{d}\) loops under the OFV condition are also larger than that of the SFV condition, and the peak values are shown in Table 3. Therefore, it is illustrated that the unsteady aerodynamic characteristics of airfoil under the OFV condition can be aggravated due to the effects of dynamic stall vortex. Figure 11: Comparisons of pressure coefficient at different oscillating velocities (k = 0.075, \(\alpha=12.0^{+}+8.0^{+}\sin(\omega t+\pi)\)). (a) \(\lambda=0.0\). (b) \(\lambda=0.25\). (c) \(\lambda=0.5\). Figure 12: Comparisons of aerodynamic loads at different oscillating velocities (k = 0.075, \(\alpha=12.0^{+}+8.0^{+}\sin(\omega t+\pi)\)).