## 4 Case Study and Analysis
### Optimization Results  
The designs of the propeller with a Gurney flap and the propeller without a Gurney flap were completed after iterative optimization. Figure 16 shows the convergence histories of the chord and pitch distribution optimization for two propellers in the last iteration. For convenience, the optimized propeller without a Gurney flap is denoted by Prop A, and the optimized propeller with a Gurney flap is denoted by Prop B in this paper. Figure 17 shows the geometry of Prop A and Prop B, where the red part is a Gurney flap. Their chord and pitch distributions are displayed in Figures 18 and 19, respectively.  
As shown in Table 5, Prop B (propeller with a Gurney flap) can achieve 1.8% higher efficiency than Prop A (propeller without a Gurney flap) under the required thrust at the design point. That means, compared with the optimal propeller without a Gurney flap, adopting the optimal propeller with a Gurney flap can reduce the output power of the ultra-high-altitude unmanned aerial vehicle by 2.2%. It is of great significance to enhance the long-endurance capability of the ultra-high-altitude UAV. Therefore, the optimization of the propeller with a Gurney flap has evident effects.  
The aerodynamic performances of Prop A and Prop B near the design point were calculated, as shown in Figure 20. In addition, the average Reynolds numbers (\(Re_{\text{average}}\)) of the two propellers at different rotational velocities were in the range of 3.5 \(\times\) 10\({}^{4}\) to 4.6 \(\times\) 10\({}^{4}\). It can be seen that Prop B can maintain a significantly higher efficiency than Prop A in a wide range of thrusts near the design point. Thus, the improvement of the propeller with a Gurney flap is reliable.  
Figure 19: Blade pitch distribution of the two optimized propellers.  
Figure 17: The geometries of the optimized propellers: (**a**) Prop A; (**b**) Prop B.  
Figure 18: Blade chord distribution of the two optimized propellers.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Propeller** & \(T_{\text{req}}\)**(N)** & \(n\) **(rpm)** & \(\lambda\) & \(\eta\) & \(Re_{\text{average}}\) \\ \hline Prop A & 20 & 978 & 0.9816 & 80.2\% & 4.2 \(\times\) 10\({}^{4}\) \\ Prop B & 20 & 948 & 1.0105 & 82.0\% & 4.0 \(\times\) 10\({}^{4}\) \\ \hline \end{tabular}
\end{table}
Table 5: Comparison between two optimized propellers at the design point.  
In addition, the aerodynamic performances of Prop A and Prop B under higher Reynolds numbers (about 1.8 \(\times\) 10\({}^{5}\)) were also calculated, as shown in Figure 21. The increase in Reynolds number is achieved by changing the air density and dynamic viscosity, and the rotational velocity of the propeller and free-stream velocity remain unchanged. The contrast point in Figure 21 is working state that has the same advance ration with the design point in Figure 20. It can be seen that both propellers can achieve more than 86% aerodynamic efficiency within this Reynolds number range, but Prop B no longer has higher aerodynamic efficiencies than Prop A near the contrast point. This indicates that Reynolds number may weaken the aerodynamic improvement of the propeller with a Gurney flap.  
To explain the effect of Reynolds number, the efficiencies of Prop A and Prop B at five Reynolds numbers under were calculated, where the advance ratios of two propellers were consistent with their design cruise states. As shown in Figure 22, with the Reynolds number increasing, the efficiency improvement of Prop B relative to Prop A gradually decreases. In addition, as the Reynolds number increases to about 1.4 \(\times\) 10\({}^{5}\), Prop B no longer has an efficiency advantage relative to Prop A.