## 4 Propeller Theory for HAPSs
### Development on Theoretical Background
This method collects samples through CFD simulations, constructs a Gaussian process model, and applies a genetic algorithm to optimize the chord length and twist angle distribution of the propeller. [PERSON] [45] performed high-fidelity CFD aerodynamic optimization of a propeller incorporating a Gurney flap. While high accuracy is critical, the direct use of CFD for propeller optimization is computationally expensive due to complex flow dynamics. Gradient-based optimization methods can reduce CFD computational costs [86], but they are susceptible to convergence to local optima. To improve both accuracy and efficiency, CFD can be used to enhance BEMT. [PERSON] et al. [92] used CFD to optimize pitch angle and aerodynamic forces before applying BEMT for inverse design. A multi-stage method was proposed in [93], employing BEMT for initial design and CFD for airfoil optimization. Another approach involves decoupling the design process, as demonstrated by [94], who used discrete adjoint methods and flow field reconstruction to optimize pitch angle and chord distribution. However, although decoupling may improve efficiency, it often fails to achieve global optimization. Most studies focus on optimizing propellers using BEMT supplemented by CFD to improve design accuracy, yet CFD-based methods remain less accurate than high-precision CFD simulations. In [95], a multi-fidelity approach combining high-fidelity CFD simulations with BEMT to accelerate high-precision design is proposed. This approach uses a multi-fidelity data fusion model, based on a small number of high-fidelity and a larger number of low-fidelity samples, to improve computational efficiency and optimize propeller blade designs. Models such as random forests [96] and multi-fidelity Kriging [97] have been applied in aerospace, achieving better results than single-fidelity models. This strategy leverages the strengths of BEMT for low-fidelity data and CFD for high-fidelity data, aligning with practical requirements of multi-fidelity fusion models. Regarding the evaluation of propeller performance, the aerodynamic characteristics of blade elements are generally computed using coupled viscous-inviscid solvers such as XFOIL [98]. While these methods offer significant computational efficiency, they exhibit limitations in modeling complex aerodynamic phenomena, including the prediction of laminar separation bubbles, transition from laminar to turbulent flow on three-dimensional surfaces, and the occurrence of radial crossflow [99]. Numerous studies employ numerical simulations to calculate propeller performance, frequently utilizing the \(k-\omega\) shear-stress transport turbulence model or the \(k-\epsilon\) turbulence model [51; 85; 100; 101; 102]. In [PERSON]'s work [103], Vortex Theory-based methods often yield accurate performance predictions. However, performance evaluations of propellers operating at high altitudes are subject to increased uncertainty due to the scarcity of experimental or flight data and the challenges associated with modeling low Reynolds number flows [104]. Consequently, [PERSON] et al. [104] employed the Polynomial Chaos Expansion (PCE) method to quantify performance uncertainty for propellers in high-altitude conditions. For completeness, the following sections present the BEMT and Vortex Theory.