## 4 Analyses and Discussions
### Vortex Characters of Dynamic Stall under the OFV Condition  
#### 4.3.1 Circulation of Flowfield  
Aimed at investigating the characteristics of a separated vortex, the circulation of flowfield is estimated from the local vorticity field via numerical integration based on [PERSON] theorem [39]:  
\[\Gamma=\iint\limits_{A}\omega_{z}dA \tag{8}\]  
where \(A\) represents the integral window encompassing the vortex, \(\omega_{z}\) represents the vorticity. In order to research the vorticity variation of LEV and TEV, the circulations are integrated in the simulated flowfield (\(\omega_{z}<0.0\) for the LEV, and \(\omega_{z}>0.0\) for the TEV). Meanwhile, the vorticity in the boundary layer is neglected in the integration. Since the second vortex induced by the LEV usually has the same rotation as the TEV, the TEV integral window is started behind 0.6c position of airfoil to eliminate the interference of the second vortex, just as shown in Figure 15.  
The variations of circulation at different oscillating velocities on upstroke process are shown in Figure 16, and the simulated state is the same as case \(1\) (i.e., k = 0.075, \(\alpha=12.0^{\circ}\) + 8.0\({}^{\circ}\) sin(\(\omega t+\pi\))). It can be seen from Figure 16a that the circulation of LEV at \(\lambda=0.0\) is less than that under the OFV condition when the AoA is smaller than \(15.5^{\circ}\), because the flowfield under the SFV condition is more  
Figure 14: Comparisons of pressure coefficient with different oscillating velocities (k = 0.05, \(\alpha=10.0^{\circ}+6.0^{\circ}\) sin(\(\omega t+\pi\))). (a) \(\lambda=0.0\). (b)\(\lambda=0.25\). (c) \(\lambda=0.5\).  
Figure 15: Integral windows of vortex circulation.  
stable. Meanwhile, it is indicated that the AoAs of LEV formation are decreased by increasing the oscillating velocity, since the variation of freestream velocity could increase the instability of the boundary layer of airfoil, which induces the airflow separation easily. Meanwhile, it can be seen from Figure 16b that the LEV generates earlier under OFV conditions compared with SFV conditions. However, the accumulating rate of LEV circulation at \(\lambda=0.25\) and \(\lambda=0.5\) is smaller than the circulation at \(\lambda=0.0\), because the freestream velocity is decreased when the AoA increases under the OFV condition, which could reduce the energy supplement from the airflow into the LEV. As a result, the circulation would be decreased with the increase of oscillating freestream velocity. In addition, it also can be seen that the circulation of TEV has the same variational characters due to the circulation conservation of flowfield. Therefore, it is indicated that the circulation variations under the OFV condition are milder compared with the SFV condition.  
In order to analyze the dynamic stall vortex characteristics deeply, Figure 17 shows the circulation variations of case 2 (i.e., k = 0.05, \(\alpha=10.0^{\circ}+6.0^{\circ}\sin(\omega t+\pi)\)), at different oscillating velocities. It is illustrated in this figure that the circulation of LEV at \(\lambda=0.0\) obviously increases when the AoA is larger than \(12.0^{\circ}\). However, the circulation of LEV at \(\lambda=0.5\) almost keeps stable at different AoAs (Figure 17a) or times (Figure 17b). It is indicated that the LEV is basically eliminated in the state of \(\lambda=0.5\). In addition, it also can be seen that the circulation of TEV is also stable for this case due to the circulation conservation of the flowfield. As a result, the dynamic stall characteristics of airfoil would be alleviated under the OFV condition with light dynamic stall, and this variational characteristic is similar to the aerodynamic loads.  
Figure 16: Comparisons of vortex circulation between different oscillating velocities (case 1). (a) Variation with AoA. (b) Variation with time.  
Figure 17: Comparisons of vortex circulation between different oscillating velocities (case 2). (a) Variation with AoA. (b) Variation with time.