## 5 Numerical Analysis  
### Stall Performance  
Because the stall performance of the wing cannot be obtained directly by experimental means, the relevant research can bce performed with the help of the numerical simulation method established in Chapter 4. As portrayed in Figure 22, under the three incoming velocities, when \(\alpha\) is in the range of \(-4^{\circ}\)-\(8^{\circ}\), the lift coefficient curves of the two wings have a high degree of coincidence, indicating that the membrane deformation does not affect the lift characteristics at medium and small angles of attack, which is consistent with the wind tunnel experiment results. When \(\alpha>8^{\circ}\), the two lift curves begin to indicate slight differences, and the lift characteristics of polyimide-skin wings are better than those of rigid-skin wings. When the angle of attack continues to increase, the stall characteristics of the two wings are significantly different. The stall angle of the rigid-skin wing is about \(13^{\circ}\), while the membrane deformation delays the stall angle of the polyimide-skin wing to \(14^{\circ}\). When \(v=6.4\) m/s and \(\alpha=14^{\circ}\), the lift coefficient of the polyimide-skin wing is 0.089 higher than that of the rigid-skin wing. Figure 23 can visually portray the increment of aerodynamic coefficient caused by membrane deformation under different inflow conditions. In the figure, \(\Delta C_{L}\), \(\Delta C_{D}\), and \(\Delta C_{M}\), represent the changes of lift coefficient, drag coefficient and pitching moment coefficient when the skin changes from rigid to flexible, respectively. For the lift coefficient, except for the three data points of low wind speed and small angle of attack, the membrane deformation brings a small increment of lift. Below a \(12^{\circ}\) angle of attack, the effect of membrane deformation on the increment becomes more obvious with the increase of wind speed. Above a \(12^{\circ}\) angle of attack, with the increase of wind speed, the effect of lift increasing becomes weaker. However, the increment of lift coefficient is significantly larger than that at small angle of attack, which is related to the improvement of stall performance caused by membrane deformation. When \(\alpha=12^{\circ}\), the effect of membrane on lift is hardly affected by wind speed. Figure 21: Membrane deformation calculated by finite element method: (**a**) Upper surface; (**b**) Lower surface. Figure 20: Finite element model of membrane wing. For the pitch moment coefficient, in most cases, the skin deformation will lead to the increase of the pitch moment, which is consistent with the conclusion of the wind tunnel experiment. At each angle of attack, the increment of pitch moment becomes larger with the increase of wing load. Although the wing loads are different, the curves in Figure 23b indicate a similar law, that is, the greater the angle of attack, the greater the increment. When \(\alpha>12^{\circ}\), the slope of the curves increases sharply, which is caused by the improvement of stall performance. When \(W/S=7.8\) kg/m\({}^{2}\) and \(\alpha=16^{\circ}\), the increment of pitch moment exceeds 10%, while when \(W/S=2.6\) kg/m\({}^{2}\) and \(\alpha=-4^{\circ}\), the pitch moment decreases. Figure 24 portrays the streamlines of the polyimide-skin wing at a 50% span section when \(\alpha=4^{\circ}\) and \(v=6.4\) m/s. It can be observed that the membrane deformation on the upper surface makes the rear of the airfoil more flat and the turbulence separation position is delayed from 70.5% to 73.6% chord length so that the airfoil can generate more lift. The three-dimensional streamline demonstrated in Figure 25 can be used to visually explain the stall delay phenomenon. The turbulent separation points of each section along spanwise form a separation line. Under aerodynamic load, the membrane in the middle area of the upper surface is convex, so that the middle of the separation line bends toward the trailing edge of the wing, and then brings differential aerodynamic characteristics along spanwise.