## 4 Case Study and Analysis
### Aerodynamic Analysis
#### 4.3.2 Aerodynamic Analysis of Three-Dimensional Blades  
Then, based on the analysis results of the cross-sectional airfoils, the aerodynamic analysis of the three-dimensional blades was performed to further illustrate this change of the performance of Prop A and Prop B with the Reynolds number. Two states of Prop A and Prop B were calculated by CFD method. As shown in Table 7, the design point with a Reynolds number of about 4.0 \(\times\) 10\({}^{4}\) is denoted by State 1, and the state with a higher Reynolds number of about 1.8 \(\times\) 10\({}^{5}\) is denoted by State 2. Figures 34 and 35 show the distributions of blade element thrust and element efficiency of Prop A and Prop B at State 1 and State 2. As shown in Figure 34, the blade element efficiency of Prop B is notably higher than that of Prop A in each station at State 1. As a result, Prop B achieved 1.8% higher efficiency than Prop A at State 1. However, Figure 35 shows that the blade element efficiencies of Prop B at most stations are slightly lower than that of Prop A at state 2. As a result, the efficiency of Prop B is 0.4% lower than that of Prop A.  
Figure 32: Pressure coefficient and streamline distributions at \(Re\) = 4.0 \(\times\) 10\({}^{4}\): (**a**) Airfoil A; (**b**) Airfoil B. To illustrate the reason for this change, the pressure coefficient distributions on the blades and the streamlines on the local sections (\(r/R=0.25\), \(r/R=0.5\), \(r/R=0.75\), \(r/R=0.95\)) of two propellers at State 1 and State 2 are calculated, respectively (see Figures 36 and 37). Besides, the comparisons of pressure distributions between the cross-sectional airfoils with and without Gurney flaps at three radial stations of Prop B and Prop A are shown in Figures 38 and 39. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Propeller**} & \multirow{2}{*}{\(V_{\infty}\) **(m/s)**} & \multicolumn{3}{c}{**State 1**} & \multicolumn{3}{c}{**State 2**} \\  & & \(Re_{\textbf{average}}\) & \(n\) **(rpm)** & \(T\) **(N)** & \(\eta\) & \(Re_{\textbf{average}}\) & \(n\) **(rpm)** & \(T\) **(N)** & \(\eta\) \\ \hline Prop A & 32 & \(4.2\times 10^{4}\) & 978 & 20 & 80.2\% & \(1.9\times 10^{5}\) & 978 & 115 & 86.3\% \\ Prop B & 32 & \(4.0\times 10^{4}\) & 948 & 20 & 82.0\% & \(1.8\times 10^{5}\) & 948 & 117 & 85.9\% \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Performances of two optimized propellers at the states with different Reynolds numbers. Figure 33: Pressure coefficient and streamline distributions at \(Re=2.0\times 10^{5}\): (**a**) Airfoil A; (**b**) Airfoil B. As shown in Figure 36, when the \(Re_{\text{average}}\) is about 4.0 \(\times\) 10\({}^{4}\) (State 1), there are obvious laminar separation bubbles near the trailing edge of the propellers. Similar to the case of airfoils, the laminar separation bubbles cause a significant decrease in the efficiency of the propellers at this Reynolds number. However, in this case, the performance of Prop B can be improved by a Gurney flap. On the one hand, as shown in Figure 36, the Gurney flap inhibits the laminar separation bubbles on Prop B by the induced vortex. On the other hand, the pressure distribution results at different stations (see Figure 38) show that the Gurney flap increases the upper surface suction and the lower surface high pressure on Prop B, especially near the trailing edge, which partially offsets the lift loss caused by laminar separation bubbles. At this time, the drag caused by the Gurney flap is relatively small compared to the huge pressure drag caused by the laminar separation bubbles. As a result,  
Figure 34: Aerodynamic characteristics along the blade of two propellers at State 1: (**a**) Distributions of blade element thrust; (**b**) Distributions of blade element efficiency. Figure 35: Aerodynamic characteristics along the blade of two propellers at State 2: (**a**) Distributions of blade element thrust; (**b**) Distributions of blade element efficiency.