## 4 Case Study and Analysis
### Wind Tunnel Test
#### 4.2.1 Test Conditions and Method  
The wind tunnel was a single return-flow and closed wind tunnel. In addition, its test section was 12 m long and its cross section was a 3 \(\times\) 3 m rounded rectangle. The  
Figure 20: Aerodynamic performances of Prop A and Prop B near the design point: (**a**) Thrust vs. rotational velocity; (**b**) Torque vs. rotational velocity; (**c**) Efficiency vs. advance ratio; (**d**) Efficiency vs. thrust.  
turbulence intensity of this wind tunnel was 0.1-0.13%. In addition, the wind speed could be continuously controlled from 3-100 m/s. A sketch of wind tunnel is shown in Figure 23.  
In the wind tunnel test of propeller, the criteria of geometric similarity, kinematic similarity, and dynamic similarity were required for a model to have similitude with the prototype [35]. The thrust coefficient (\(C_{T}\)) and power coefficient (\(C_{P}\)) of the propellers can be expressed as follows:  
\[\begin{array}{l}C_{T}=\frac{T}{\rho n_{s}^{2}D^{4}}=f_{T}(\lambda,Re,Ma)\\ C_{P}=\frac{P}{\rho n_{s}^{3}D^{5}}=f_{P}(\lambda,Re,Ma)\end{array} \tag{15}\]  
Figure 21: Aerodynamic performances of two optimized propellers under an average Reynolds number of about 1.8 \(\times\) 10\({}^{5}\): (**a**) Thrust vs. rotational velocity; (**b**) Torque vs. rotational velocity; (**c**) Efficiency vs. advance ratio; (**d**) Efficiency vs. thrust.  
Figure 22: Aerodynamic performances of Prop A and Prop B at their design advance ratios.  
Thus, the advance ratio, Reynolds number and Mach number of the test model needed to be the same as the prototype. In general, it is often impossible to strictly meet all requirements during a model test, so some unimportant aspects of similitude may be neglected [36]. In this study, the Mach number of the propeller tip was lower than 0.4 and could be ignored, and only the advance ratio (\(\lambda\)) and Reynolds number (\(Re\)) of the test models were guaranteed to be the same as that of the prototypes. Then the scaling laws of the free-stream velocity (\(V_{\infty}\)) and rotational velocity (\(n_{\rm s}\)) between the prototype and the test model could be formulated as follows:  
\[\begin{array}{l}\frac{V_{\infty\rm{1}}}{V_{\infty\rm{2}}}=\frac{\rho_{\rm{2}} D_{\rm{2}}}{\rho_{\rm{1}}D_{\rm{1}}}\cdot\frac{n_{\rm{1}}}{\mu_{\rm{2}}}\\ \frac{n_{\rm{1}}}{n_{\rm{2}}}=\frac{V_{\infty\rm{1}}}{V_{\infty\rm{2}}}\cdot \frac{D_{\rm{2}}}{D_{\rm{1}}}\end{array} \tag{16}\]  
where parameters with subscript \(\rm{1}\) correspond to prototype, and parameters with subscript \(\rm{2}\) correspond to test model.  
The test models were made of carbon fiber and resin material using a high-precision steel mold, and the diameters of the models were 1.40 m, which was 0.7 times the diameter of Prop A and Prop B, as shown in Figures 24 and 25. The mass of the test models was controlled within 200 g, and their dynamic balance performances were qualified after inspection. The test model was installed in front of the force measurement device, and the measurement device was connected with an extended strut. The extended strut was mounted on the ground of the wind tunnel, as shown in Figure 26.  
In the wind tunnel test of the propeller, the maximum blockage was 2.1%, which met the requirement of wind tunnel test. Considering the blockage effect, the test results were corrected by the blockage correction method [37]. In order to check the accuracy of the test result under low Reynolds numbers, a repeated test was first carried out. The force measurement test of the scaled model of Prop A was performed three times under the same condition, where the Re was about 1.6 \(\times\) 10\({}^{5}\) and the freestream velocity was 16 m/s. The data of each test state were obtained by averaging the results of 800 consecutive samples in steady state. As shown in Figure 27, the relative system error of three tests was within 1%, which satisfies the requirement of this study.