## 3 Optimization Strategy
### Optimization Model of Cross-Sectional Airfoil with Gurney Flap  
In the iterative optimization of a low Reynolds number propeller, each cross-sectional airfoil needs to be optimized under corresponding constraints which are constantly updated with the iteration. The optimization process of a cross-sectional airfoil with a Gurney flap is introduced in this subsection. Firstly, the upper and lower curves of the airfoil were parameterized by the class-shape transformation (CST) method [32]. Two 4 th order curves controlled by the parameters \(a_{0}\)-\(a_{4}\) and \(b_{0}\)-\(b_{4}\) were used to describe the airfoil, which can be expressed as follows:  
\[\left\{ \begin{array}{l}C_{upper}(\overline{x})=\overline{x}^{N_{1}}(1- \overline{x})^{N_{2}}\sum\limits_{i=0}^{4}a_{i}B_{4}^{i}(\overline{x})\\ C_{lower}(\overline{x})=\overline{x}^{N_{1}}(1-\overline{x})^{N_{2}}\sum \limits_{i=0}^{4}b_{i}B_{4}^{i}(\overline{x})\\ B_{4}^{i}(\overline{x})=\frac{4!}{2!(4-i)!}\overline{x}^{i}(1-\overline{x})^{4 -i},i=0,1,\ldots,4\end{array} \right. \tag{2}\]  
where \(N_{1}=0.5\) and \(N_{2}=1\) represent an airfoil with a round head and a pointed tail, and the parameters \(a_{i}\) and \(b_{i}\) are the weight coefficients of Bernstein polynomials \(B_{n}^{i}\) (\(\varphi\)). In addition, \(\overline{x}\) is the dimensionless chordwise position on the airfoil chord, which is expressed by \(x/c\). Further, the relative height of the Gurney flap to the airfoil chord on the Gurney flap airfoil is expressed by \(\overline{h}=h/c\), which is given and fixed in the airfoil optimization to avoid the irregular shape of the propeller trailing edge (such as a wavy shape). When \(\overline{h}\) is 0, the airfoil degenerates into a conventional airfoil without a Gurney flap. Therefore, there are a total of ten design variables in the optimization of the cross-sectional airfoil, as shown in Figure 9.  
Then, the constraints and optimization objective of the cross-sectional airfoil are determined as follows:  
1. [label=()]
2. Constraint of cross-sectional airfoil's relative thickness  
The variation in the relative thickness distribution of the blade usually depends on the structural strength requirements. Here, the relative thickness (\(t_{\text{target}}\)) distribution of the propeller is constrained as follows:  
\[t_{\text{target}}=0.481(\frac{r}{R})^{4}-1.616(\frac{r}{R})^{3}+2.036(\frac{r} {R})^{2}-1.143\frac{r}{R}+0.305 \tag{3}\]  
where \(t_{\text{target}}\) is the ratio of the actual thickness to the chord length of the airfoil, and \(r\) is the radial position of the blade element. 3. Constraint of the cross-sectional airfoil's lift coefficient  
The lift coefficient distribution of the cross-sectional airfoils should be constrained by the working state of the designed propeller. In the iterative optimizations, the chord and pitch distributions can be obtained after the first iteration. Then the lift coefficient  
Figure 9: Parametric method for a Gurney flap airfoil. distribution (\(C_{l\text{-target}}\)) of the propeller at a working state can be calculated by blade element momentum theory [33] (BEMT) as follows [34]:  
\[C_{l\text{-target}}=\frac{4\sin(\varphi_{0}+\beta)\tan\beta}{\sigma(1-K^{-1}\tan \beta)} \tag{4}\]  
where \(\beta\) is the induced angle which can be solved by Newton iterative method, \(K\) is the ratio of lift coefficient to drag coefficient of a local blade element, and \(\sigma\) is the local blade solidity which is expressed by:  
\[\sigma=\frac{N_{B}c}{2\pi r} \tag{5}\]  
where \(N_{\text{B}}\) is the number of propeller blades. \(C_{l\text{-target}}\) is used as the aerodynamic constraint in the optimization of cross-sectional airfoils. Constraint of cross-sectional airfoil's working Reynolds number  
Each cross-sectional airfoil must be optimized under the corresponding Reynolds number (\(Re_{\text{target}}\)) determined by the working status of the propeller.