## 1 Introduction  
Ultra-high-altitude UAVs have significant advantages in cruising altitude, endurance, and long-term fixed-point parking [1, 2]. Therefore, they have garnered considerable research attention in both military and civilian applications. Due to the difficulty in energy supply, ultra-high-altitude UAVs commonly use solar panels and batteries to supply power, and brushless motors are used to drive the propeller to provide thrust. Since the balance of energy is quite weak, the efficiency of the propeller is one of the key factors for ultra-high-altitude UAVs to survive day and night [3]. Complex environmental characteristics are the main challenge for the propeller design of ultra-high-altitude UAVs [4]. Because of the low air density (0.0889 kg/m\({}^{3}\) at 20 km altitude) and low cruising velocity (0.1 Ma) at high altitudes, the working Reynolds number of the propeller is lower than \(1\times 10^{5}\). Because of the influence of the low Reynolds number, the aerodynamic efficiency of such a propeller may be far lower than a conventional propeller, resulting in greater power consumption [5]. Therefore, improving the aerodynamic performance of low Reynolds number propellers is of great significance to enhance the long-endurance capability of ultra-high-altitude UAVs. Generally, the performance of a propeller can be improved by reducing the induced loss of the blade and increasing the lift-to-drag ratio of the cross-sectional airfoil [6]. Based on the minimum energy loss principle, which is also known as the Betz optimum condition, and the blade element momentum theory (BEMT), [PERSON] et al. [7] proposed a fast propeller design method. Once the required thrust, rotational velocity, and diameter of the propeller were known, a design with minimum induced loss could be quickly achieved. Coupled with the free form deformation method, dynamic mesh interpolation technology, optimization algorithm, and surrogate model, [PERSON] et al. [8] developed an integrated aerodynamic and aeroacoustic design method of a propeller. [PERSON] et al. [9] proposed an efficient optimization method of a propeller with a minimal number of high-fidelity model evaluations. In his method, the classical blade-element momentum theory and a high-fidelity computational fluid dynamics tool were linked by a space mapping algorithm to improve computing efficiency, but the blade cross-sectional airfoil was not optimized. Based on the hierarchical Kriging model, [PERSON] et al. [10] proposed an aerodynamic/aeroacoustic optimization method in the design of a helicopter rotor and achieved considerable reduction in noise without compromising aerodynamic performance[PERSON] Considering the influence of a low Reynolds number and its variation along the radial direction in experimental data, [PERSON] et al. [11] designed a low Reynolds number propeller for a stratospheric airship adopting NACA4412 as the cross-sectional airfoil. Compared with the traditional method, the efficiency of the propeller was improved by 5%[PERSON] The existing propeller design methods provide a rich reference for the construction of optimization methods for low Reynolds number propellers. When a propeller is working at a very low Reynolds numbers, laminar separation bubbles can be easily generated on the blade and result in a significant drop in the propeller's efficiency. In order to accurately capture the effects of laminar bubbles on the performances of cross-sectional airfoils and blades, a high-precision CFD simulation method that can consider laminar transition is necessary in the optimization of low Reynolds number propellers [12]. The flow behavior on the blade is closely related to the aerodynamic characteristics of the propeller's cross-sectional airfoil at very low Reynolds numbers [13,14]. Thus, in the design of a low Reynolds number propeller, the aerodynamic performance of cross-sectional airfoils is more important [15,16] than that in a conventional propeller.