## 4 Case Study and Analysis
### Aerodynamic Analysis
#### 4.3.1 Aerodynamic Analysis of Cross-Sectional Airfoils  
Firstly, the aerodynamic characteristics of the cross-sectional airfoils of Prop A and Prop B at a radial location of 0.75 R were analyzed. The cross-sectional airfoil of Prop A can be seen as an airfoil with 0% height Gurney flap, which was referred to as Airfoil A for convenience. The cross-sectional airfoil of Prop B is an airfoil with a 1% height Gurney flap, which was denoted by Airfoil B. In addition, their geometries are shown in Figure 30. It can be seen that the cross-sectional airfoils of A and B have similar upper and lower surface shapes except for the Gurney flap. The aerodynamic performances of Airfoil A and Airfoil B at different Reynolds numbers (from \(Re\) = 4.0 \(\times\) 10\({}^{4}\) to \(Re\) = 2.0 \(\times\) 10\({}^{5}\)) were calculated. As shown in Figure 31, Airfoil B has higher maximum lift-to-drag ratio than Airfoil A when the \(Re\) is lower than \(7.0\times 10^{4}\), while the situation is opposite when the \(Re\) is bigger than 1.4 \(\times\) 10\({}^{5}\). As the Reynolds number increases from \(7.0\times 10^{4}\) to \(1.4\times 10^{5}\), the improvement of the maximum lift-to-drag ratio of Airfoil B relative to Airfoil A gradually decreases. When the Reynolds number reaches \(1.4\times 10^{5}\), the maximum lift-to-drag ratio of Airfoil B gradually changes from larger than Airfoil A to smaller than Airfoil A. To understand the reason for this change, the pressure coefficient (C\({}_{p}\)) and streamline distributions of the two airfoils under several angles of attack near the maximum lift-to-drag ratio at four Reynolds numbers (from 4.0 \(\times\) 10\({}^{4}\) to 2.0 \(\times\) 10\({}^{5}\)) are calculated, respectively (see Figures 32 and 33). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(V\infty\) (m/s) & \(n\) (rpm) & \(\lambda\) & \multicolumn{2}{c}{\(Re_{\text{average}}\)} \\  & & & Prop A & Prop B \\ \hline \multirow{3}{*}{4.2} & 150 & 1.20 & \(3.7\times 10^{4}\) & \(3.5\times 10^{4}\) \\  & 178 & 1.01 & \(4.2\times 10^{4}\) & \(4.0\times 10^{4}\) \\  & 206 & 0.87 & \(4.6\times 10^{4}\) & \(4.4\times 10^{4}\) \\  & 577 & 1.20 & \(1.4\times 10^{5}\) & \(1.4\times 10^{5}\) \\
16.0 & 685 & 1.01 & \(1.6\times 10^{5}\) & \(1.5\times 10^{5}\) \\  & 700 & 0.98 & \(1.8\times 10^{5}\) & \(1.7\times 10^{5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison between two optimized propellers at the design point. Figure 27: Results of the repeated test: (**a**) Thrust coefficient vs. advance ratio; (**b**) Power coefficient vs. advance ratio. When the Reynolds number is 4.0 \(\times\) 10\({}^{4}\), the laminar separation bubbles are large and play a major role in the airfoil performance. As shown in Figure 32, since a huge laminar separation bubble markedly reduces the equivalent camber of the airfoil and increases the equivalent thickness, the lift coefficient of the airfoils is notably decreased and the drag coefficient is increased, leading to a considerable decrease in the lift-to-drag ratio. At this point, the Gurney flap helps improve this situation by increasing the equivalent camber of the airfoil. The pressure coefficient distributions in Figure 32 show that the pressure coefficient near the lower surface of Airfoil B is obviously higher than that of Airfoil A. In addition, compared with Airfoil A, the laminar separation bubble on Airfoil B can be reduced by the vortex induced by the Gurney flap. That indicates the Gurney flap could bring a significant increase in lift coefficient to Airfoil B and improve flow behavior near the upper surface. Although the Gurney flap will also bring a bigger drag coefficient to Airfoil B, the increase in lift coefficient by the Gurney flap plays a more important role to the performance of Airfoil B. This is because the lift coefficient is relatively small and the drag coefficient is relatively large at this very low Reynolds number.