# Performance Prediction and Design of Stratospheric Propeller
## 1 Introduction  
The capability to fly continuously for months in the stratosphere is an elusive goal. Stratospheric airships have enormous potential for long-endurance vehicles, which do not need to stay in motion to remain in the air, since lift is generated through the buoyancy effect rather than aerodynamics. The efficient propulsion of propellers is essential for the maneuverability and sustainability of airships and cannot be achieved through turbojet and turbofan engines [1; 2; 3; 4; 5].  
However, the environment in the stratosphere, which is above 20 km, has a much lower air density. Additionally, the velocity of advance of a stratospheric airship is lower than 30 m/s, which is extremely low. The low Reynolds number and small advance ratio create a significant challenge for the design of an efficient propeller.  
The airfoil of high-altitude vehicles was researched by Prof. [PERSON] [6; 7]; the aerodynamic characteristics of six typical low-Reynolds-number, high-lift, and low-resistance airfoils were analyzed, and wind tunnel experiments were carried out. [PERSON] [8] proposed a method of designing the chord length and twist angle to maximize propeller efficiency. [PERSON] [9] improved [PERSON]'s method to align it with the requirements of engineering applications.  
[PERSON] [10] and [PERSON] [11] first developed the theory of propellers in the 19 th century, and [PERSON] [12] proposed the blade element theory, which neglected the induced velocity of the propeller. The design method based on minimum induced losses was firstly proposed by [PERSON] [13] and [PERSON] [14]. [PERSON]'s [15] research indicated that the Betz condition for minimum energy loss could also be applied in heavy disk loadings. [PERSON] [16] discovered an approximation of the flow around helicoidal vortex sheets, which is better if the advance ratio of the propeller is smaller, and the result improves as the number of blades increases.  
Numerical simulation technology has been widely used in propeller performance calculations with the development of computer technology. The \(k\)-\(\epsilon\) turbulence model was used by [PERSON] [17] to simulate the performance of a high-altitude propeller, and the result was in good agreement with the scaled wind tunnel experiment. [PERSON] [18] proposed an improved design method of propellers, which could be optimized under specific operative condition and according to the profile distribution along the blade, and the wind tunnel test showed that the method was feasible.  
Reasonable accurate performance prediction could be obtained based on vortex theory and blade element theory [19; 20]. [PERSON] [21] studied the propeller's aerodynamic characteristics utilizing the vortex theory and a propeller model [22; 23], and the results indicated that the S-A model is more suitable for calculating aerodynamic parameters. The performance predicted by the vortex theory was similar to that determined by scale wind tunnel tests. However, in the design process, the airfoil data of each section are needed for calculating the performance of the propeller.  
The S-A model was performed in Fluent to calculate the airfoil performance at different angles of attack and Reynolds numbers. The performance curves of the lift and drag coefficient underwent polynomial interpolation to obtain the database, which was used to calculate the performance of the propeller. The CFD model with different mesh numbers was applied to verify the mesh independence, and the CFD method at different altitudes was applied with the similarity theory to prove the feasibility of the method.  
In this paper, a prediction method is proposed to reduce calculations or experimental times and to predict a propeller's performance on the basis of characteristic blade element performance. The prediction parameters are discussed, and a realistic and accurate prediction model is obtained which reduces the calculation amount with respect to the vortex theory, in regard not only to airfoil performance calculation, but also to propeller performance calculation.