# Numerical Study on Dynamic-Stall Characteristics of Finite Wing and Rotor
## 3 Analyses and Discussion
### Finite-Wing Dynamic Stall with Spanwise Flow  
Due to the forward-flight effect, spanwise flow appears on the rotor blade, and spanwise-flow velocity changes with different rotor azimuths. As a result, spanwise flow is a critical-influence factors on the unsteady aerodynamic characteristics of a rotor. Therefore, in order to research the effects of the spanwise flow on the dynamic stall, an inflow angle of \(30.0^{\circ}\) (deviated from the vertical direction) is considered in this study, as shown in Figure 9, where \(V\), \(V_{n}\), and \(V_{t}\) represent the resultant velocity, normal velocity, and tangential velocity, respectively. In this case, AoA variation was \(\alpha=10.0^{\circ}-8.0^{\circ}\sin(\omega t)\), the Mach number of \(V_{n}\) was 0.3, and reduced frequency was 0.075 (normalized by \(V_{n}\)). Figure 8: Dynamic-stall loads at different wing sections. (**a**) \(C_{l}\), (**b**) \(C_{m}\), (**c**) \(C_{d}\), (**d**) pressure and streamlines. Figure 9: Schematic of a finite wing with spanwise flow. Figure 10 shows equal vorticity surfaces (vorticity is 7.5) of the wing at different AoAs. It is illustrated that vortex structures of a finite wing with a spanwise flow are very different from the vortex structures with a nonspanwise flow. First, the wing-tip and wing-root vortices have an inclination angle of 30.0\({}^{\circ}\) along the freestream. Second, the LEV is accumulated near the wing tip and is restricted near the wing root due to the influence of the spanwise flow. In addition, the LEV bursts and sheds from the upper surface near the middle part of the wing due to nonuniform vorticity accumulation. The vorticity contours of the dynamic-stall vortex at different wing sections are shown in Figure 11. It is clearly illustrated that the LEV and trailing-edge vortex (TEV) near the wing tip (r/R = 0.9) shed earlier compared with those near the wing root (r/R = 0.1). It is indicated that the spanwise flow of the finite wing could promote dynamic stall near the wing tip, and delay it near the wing root. The pressure and streamline distributions of the finite wing on the upper and lower surfaces at \(\alpha=18.00^{\circ}\) are shown in Figure 12. It can be seen that airflow convection near the wing surface is contrary to the freestream flow due to the influence of LEV. As a result, airflow separation appears on the upper surface. In addition, it can be seen from the streamlines on the upper surface that an irregular flow appears near the wing tip (r/R = 0.9-1.0) due to the interaction of the wing-tip vortex with the spanwise flow. On the contrary, streamlines on the lower surface are more stable compared with the upper surface, and streamlines on the lower surface regularly incline to the wing tip due to the effect of the spanwise flow. It can also be seen that streamline inclination angles are increased when they get close to the wing tip due to wing-tip vortex induction. Figure 11: Vorticity of dynamic-stall vortex at different wing sections. Figure 10: Equal-vorticity surfaces at different AoAs. Comparisons of \(C_{l}\) at different wing sections are shown in Figure 13. It can be seen from the \(C_{l}\) curves that \(C_{l}\) presents light dynamic-stall characteristics near the wing root. Meanwhile, the slopes of the lift curves were also decreased due to the effect of the wing-tip vortex. On the contrary, the unsteady characteristics near the wing tip present deep dynamic-stall characteristics. It can be seen that the slopes of the lift curves were also decreased. However, the decrement is smaller than that of the wing root (symmetrical sections). This is maybe due to LEV induction being more obvious for the wing tip since LEV is stronger. From the comparison of \(C_{m}\) curves at different wing sections in Figure 13, it is illustrated that \(C_{m}\) peaks are decreased when the sections get close to both the wing tip and wing root. However, the \(C_{m}\) peak at the wing root is smaller than that of the wing tip, because the dynamic-stall vortex is restricted by the spanwise flow.