## 3 Optimization Strategy
### Optimization Model of Propeller Chord and Pitch Distributions  
Based on the optimization results of cross-sectional airfoils, the chord and pitch distributions of the propeller can be optimized. The chord and pitch distributions of the blade were parameterized by two 2 nd order CST curves and were respectively controlled by parameters \(c_{0}\)-\(c_{2}\) and \(d_{0}\)-\(d_{2}\), which can be expressed as follows:  
\[\left\{ \begin{array}{l}c(\vec{r})=\vec{r}^{N_{1}}(1-\vec{r})^{N_{2}}\sum \limits_{i=0}^{2}c_{i}B_{2}^{i}(\vec{r})+(1-\vec{r})\cdot(c_{\text{start}}-c_{ \text{end}})+c_{\text{end}}\\ \\ \theta(\vec{r})=\vec{r}^{N_{1}}(1-\vec{r})^{N_{2}}\sum\limits_{i=0}^{2}d_{i}B _{2}^{i}(\vec{r})+(1-\vec{r})\cdot(\theta_{\text{start}}-\theta_{\text{end}}) +\theta_{\text{end}}\\ \\ B_{2}^{i}(\vec{r})=\frac{2!}{i!(2-i)!}\vec{r}^{i}(1-\vec{r})^{2-i},i=0,1,2 \end{array} \right. \tag{11}\]  
where \(N_{1}\) and \(N_{2}\) use the same values as in Equation (4), \(\vec{r}\) is the dimensionless radial position on the blade, and is expressed by \(r/R\). \(c_{\text{start}}\) and \(c_{\text{end}}\) are the chord at \(\vec{r}=0\) and \(\vec{r}=1\), and \(\theta_{\text{start}}\) and \(\theta_{\text{end}}\) are the pitches at \(\vec{r}=0\) and \(\vec{r}=1\). To avoid unreasonable design and reduce the variables, \(c_{\text{start}}\) and \(c_{\text{end}}\) were given and fixed during the optimization. \(\theta_{\text{start}}\) and \(\theta_{\text{end}}\) were used as optimization design variables. Figure 11: Computation grid of the cross-sectional airfoil with a Gurney flap: (**a**) Grid around the airfoil; (**b**) Grid near the Gurney flap. Therefore, there were a total of eight design variables in the optimization of the propeller chord and pitch distributions. Besides, since the hub of the blade has a minor influence on the propeller performance, Gurney flaps were only used on the blade in the radial range of 0.2 \(R\)-1.0 \(R\). According to the given values of the parameters \(c_{0}\)-\(c_{2}\), \(d_{0}\)-\(d_{2}\), \(\theta_{\text{start}}\), and \(\theta_{\text{end}}\), the chord and pitch values at 17 radial positions could be calculated. Then, the three-dimensional shape of the propeller with a Gurney flap could be obtained, as shown in Figure 12.  
Then, similar to the airfoil optimization, the constraints and objective function for optimization of propeller were determined as follows:  
1. Constraint of propeller's structural mass. To limit the structural mass of the propeller, the structural form, in which high modulus carbon fiber cloth was wrapped around the rigid foam core, was adopted. The mass of the propeller \(m\) can be estimated as follows:  
\[m=S_{\text{prop}}e\rho_{c}+(V_{\text{prop}}-S_{\text{prop}}e)\rho_{f} \tag{12}\]  
where \(S_{\text{prop}}\) is the wetted area of the propeller, \(e\) is the thickness of the carbon fiber cloth, \(\rho_{c}\) is the volume density of the carbon fiber cloth, \(V_{\text{prop}}\) is the volume of the propeller, and \(\rho_{f}\) is the volume density of the rigid foam. 2. Constraint of propeller's required thrust. The aim of the propeller optimization was to maximize the efficiency at the required thrust (\(T_{\text{req}}\)). Polynomial interpolation method was used to calculate the propeller rotational velocity and aerodynamic efficiency at \(T_{\text{req}}\). Firstly, the thrusts \(T_{1}\), \(T_{2}\), and \(T_{3}\) corresponding to three different rotational velocities \(n_{\text{s1}}\), \(n_{\text{s2}}\), and \(n_{\text{s3}}\) could be calculated. Then, according to these three sets of data, a quadratic polynomial function could be fitted as \(T=T(n_{\text{s}})\). Finally, the rotational velocity \(n_{\text{s-req}}\) corresponding to the required thrust \(T_{\text{req}}\) could be solved with the equation \(T_{\text{req}}=T(n_{\text{s-req}})\). 3. Objective function for the optimization of chord and pitch distributions.