# Numerical Study on Dynamic-Stall Characteristics of Finite Wing and Rotor
## 3 Analyses and Discussion
### Finite-Wing Dynamic Stall with Nonspanwise Flow  
Aiming at investigating the interaction between the dynamic-stall and wing-tip vortices, the dynamic-stall characteristics of the finite wing were simulated at Ma = 0.3, k = 0.075, and \(\alpha=10.0^{\circ}-8.0^{\circ}\sin(\omega t)\). Equal-vorticity (8.0) surfaces, normalized by \(\rho_{\infty}V_{\infty}^{2}\) at different AoAs, are shown in Figure 4. It is illustrated that vorticity distribution is basically symmetric, and wing-tip vortex variation at different AoAs is not obvious. With the increase of AoA (\(\uparrow\)), the dynamic-stall vortex first forms near the leading edge of the finite wing at r/R = 0.5. At an AoA of 16.93\({}^{\circ}\) in the downstroke process (\(\downarrow\)), it is illustrated that the dynamic-stall vortex separates from the upper surface of the wing at r/R = 0.5, and reattaches itself on the wing surface when it extends to the wing tip and wing root. The vorticity of the leading-edge vortex (LEV) with \(\alpha=16.93^{\circ}\) (\(\downarrow\)) at different wing sections (r/R = 0.5, 0.7, 0.9) is shown in Figure 5. It is also illustrated that maximum LEV vorticity is at r/R = 0.5, and vorticity decreases when the blade sections get close to the wing tip.  
Figure 4: Equal vorticity surfaces at different angles of attack (AoAs).  
Figure 3: Comparison of numerical \(C_{n}\) with test data of a SA349/2 rotor in forward flight.  
In order to display the vortex structure, the spatial streamlines of the finite wing are shown in Figure 6. It is illustrated that the LEV is symmetrically attached to the upper surface. In addition, airflow is rolled up from the lower surface near the wing tip due to the pressure difference of the wing surface, and then the clockwise wing-tip vortex forms on the upper surface and convects to the wing wake.  
Aiming at illustrating the variations of aerodynamic loads induced by the wing-tip vortex, pressure coefficient (Cp) wing distribution at an AoA of 16.93\({}^{\circ}\) and 18.00\({}^{\circ}\) is respectively shown in Figure 7. It can be seen from Figure 7(a) that Cp peaks would decrease when the wing section gets close to the wing tip due to the effect of the wing-tip vortex. As a result, the adverse pressure gradient is also decreased. It is illustrated in Figure 7(b) that Cp distribution on the upper surface is obviously influenced by LEV at r/R = 0.5 and 0.7. However, the LEV effect on Cp distribution is not obvious near the wing tip. It is indicated that the wing-tip vortex could effectively restrict the LEV.  
Figure 5: Finite-wing vorticity at different wing sections (\(\alpha=16.93^{\circ}\)).  
Figure 6: Spatial streamlines of wing flowfield.  
Figure 7: Cp distribution at different wing sections.  
The comparison of dynamic-stall load at different wing sections (r/R = 0.5, 0.7, 0.9, and 0.95) are shown in Figure 8. It is illustrated that the lift curve slopes of \(C_{l}\) decrease in the upstroke process, and lift-stall characteristics are alleviated when the wing sections get close to the wing tip. This is maybe due to the pressure difference between the upper and lower wing surface being decreased by the unloading effect of the wing-tip vortex. As a result, the peaks of pitching moment coefficient \(C_{m}\) and drag coefficient \(C_{d}\) are also decreased. Pressure (normalized by \(\rho_{\infty}V_{\infty}^{2}\)) and streamline distributions at \(\alpha=18.00^{\circ}\) are also shown in Figure 8 (only half the wing is shown due to flowfield symmetry). It is illustrated that airflow is separated from the upper surface at a span of r/R = 0.5-0.8 due to the effect of LEV. As a result, pressure on the upper surface is obviously reduced, and the lift would keep increasing with the AoA until the vortex is shed from the wing surface.