## 3 Optimization Strategy
### Iterative Optimization Method for Low Reynolds Number Propeller with Gurney Flap  
To obtain the optimal performance of the cross-sectional airfoils, each airfoil with a Gurney flap needs to be optimized under a specific Reynolds number and the lift coefficient determined by the actual working conditions of the propeller. However, in the traditional design method of conventional propellers, the Reynolds number and lift coefficient of the cross-sectional airfoil are unavailable before the design is completed. This problem can be solved by the iterative optimization of the cross-sectional airfoils and the chord and pitch distributions. The procedure for the iterative optimization strategy for low Reynolds number propeller is described as follows:  
Step 1. Input the initial design parameters  
To start the optimization, the following parameters must be initialized:  
1. Free-stream velocity (\(V_{\infty}\));
2. Propeller radius (\(R\));
3. Blade number (\(N_{\text{B}}\));
4. Required thrust (\(T_{\text{req}}\));
5. Propeller rotation velocity (\(n_{\text{s}}\));
6. Aerodynamic data of the initial cross-sectional airfoils. where the rotational velocity is only empirically given as \(n_{\text{s}}\) in this step, and will change into \(n_{\text{s-req}}\) (the rotational velocity corresponding to the required thrust) after the optimization of the propeller in Step 4. Step 2. Obtain the initial chord and pitch distributions by Betz method. According to the given parameters in step 1, an initial propeller can be designed by the Betz method [7]. In addition, the initial chord and pitch distributions can be obtained. Step 3. Optimize the cross-sectional airfoils with a Gurney flap. Seventeen cross-sectional airfoils with a Gurney flap with an interval of 0.05 R in the radial range of 0.2 \(R\)-1.0 \(R\) on the blade were optimized according to the following steps:  
1. Firstly, the airfoil was parameterized by two 4 th CST curves. In addition, the relative height of the Gurney flap for the Gurney flap airfoil was fixed as \(\overline{h}\). The design space of the airfoil was composed of 10 variables (\(a_{0}\)-\(a_{4}\) and \(b_{0}\)-\(b_{4}\)), from which the samples were generated. The constraints of the lift coefficient (\(C_{l}\)-target) and working Reynolds number (\(Re_{\text{target}}\)) of the airfoil were calculated according to the initial propeller in Step 2 in the first iteration. In addition, in the following iterations, they were obtained from Step 6. 2. For each sample, the lift coefficients \(C_{l1}\) and \(C_{l2}\) at the angles of attack \(\alpha_{1}\) and \(\alpha_{2}\) were calculated by the CFD method. Then, the required angle of attack \(\alpha_{\text{target}}\) corresponding to \(C_{l}\)-target was obtained by linear interpolation. So, the lift-to-drag ratio \(K\) corresponding to \(\alpha_{\text{target}}\) for each sample could be calculated. 3. Multi-island genetic algorithm (MIGA) [29] was used to evaluate existed samples and generate new samples from the design space that potentially have bigger lift-to-drag ratios. 4. Once the relative difference in the lift-to-drag ratio (\(K\)) between two adjacent samples was less than \(\varepsilon_{1}\) for 20 consecutive times, the optimization was finished and the sample  
Figure 13: Computation grid of the propeller with a Gurney flap: (**a**) Adjustable blocks around the propeller; (**b**) Grid around cross-sectional airfoil with a Gurney flap. with the maximum \(K\) was considered as the optimal solution. Otherwise, (2) and (3) needed to be repeated. Step 4. Optimize the chord and pitch distributions of propeller. 1. Firstly, the chord and pitch distributions were parameterized by two 2 nd CST curves, respectively. In addition, the design space of the airfoil was composed of eight variables (\(c_{0}\)-\(c_{2}\), \(d_{0}\)-\(d_{2}\), \(\theta_{\text{start}}\) and \(\theta_{\text{end}}\)), from which the samples were generated. 2.