## 2 Numerical Method and Validation  
In this study, the CFD method was used to realize high-fidelity calculations of the airfoil and propeller [25]. Reynolds-averaged Navier Stokes (RANS) equations were the governing equations, and finite volume method and second-order upwind scheme were used. Since the propeller in this study worked at a low Reynolds number (\(10^{4}\)-\(10^{5}\) level), laminar transition had to be taken into account in the numerical calculation. The \(\gamma\)-\(Re_{\theta}\) model proposed by [PERSON] and [PERSON] [26; 27] was adopted as the transition model, which combines the advantages of the \(\gamma\)-\(Re_{\theta}\) transition formula and the shear stress transport (SST) \(k\)-\(\omega\) turbulence model. This model incorporates two additional transport equations of the local-transition Reynolds number \(\widetilde{Re}_{\theta t}\) and the intermittent factor \(\gamma\). The \(\gamma\)-\(Re_{\theta}\) model is widely used in the calculation of low Reynolds number flow problems. To precisely capture the transition boundary layers, \(y+\) of the calculation grid was considered to be close to 1 [28]. The expression of \(y+\) is as follows:  
\[y^{+}=\frac{\sqrt{\tau_{\omega}/\rho}\cdot\Delta y}{v} \tag{1}\]  
where \(\tau_{\omega}\) is the shear stress of the wall, \(\Delta y\) is the height of the first grid cell off the wall, and \(v\) is kinematic viscous coefficient [29]. To verify the accuracy of this numerical calculation method for airfoil, the aerodynamic data of an Eppler 387 airfoil at different angles of attack under Re = 1.0 \(\times\)\(10^{5}\) were calculated. The results were then compared with the experimental results from a low turbulence pressure tunnel (LTPT) at the Langley research center [30]. The test conditions and parameters of the test model are displayed in Table 1, where _Span_ and \(c\) are the chord and span length of the test wing segment, _Ma_ is the Mach number, \(P\) is the air pressure, and \(Re\) is the Reynolds number. More detailed information about the test model and the test method of Eppler 387 are introduced in [30] The computation grid of Eppler 387 is displayed in Figure 1. As shown in Figure 2, both the force coefficients and the laminar separation location calculated by CFD are in good agreement with the experimental results, which proves that this CFD method has high calculation accuracy for the aerodynamic characteristics of airfoil at low Reynolds numbers. To verify the accuracy of this numerical calculation method for low a Reynolds number propeller, a propeller designed for a near-space solar-powered drone was selected for testing [31]. Figure 3 shows the chord and pitch distributions of the test propeller, and Figure 4 presents the three-dimensional shape of the test propeller. According to similarity criterion of equal Reynolds number and equal advance ratio (\(\lambda\)), ground experiments for the designed propeller, including thrust and torque measurement at a Reynolds number of about \(1.7\times 10^{5}\), were carried out [31]. The relation between test states on the ground and flight states at different altitudes are displayed in Table 2, where \(\lambda\) is the advance ratio, \(V_{\infty}\) is the free-stream velocity, \(n\) is the propeller rotational speed (revolutions per minute), and \(Re_{\text{average}}\) is the average Reynolds number of the blade sections of the propeller. In the test, the advance velocity of the propeller is controlled by a high accuracy railcar, as shown in Figure 5. More detailed information about the test propeller and the test method are introduced in [31].