## 5 Propeller Design for HAPS Operation
### Recent Development on Propeller Design Methodology
Other models for aerodynamic corrections in three-dimensional flow are presented in [135; 136; 137]. One of the key strengths of two-dimensional methods lies in their computational efficiency. These approaches allow for rapid estimation of aerodynamic coefficients using simplified tools such as panel methods or viscous/inviscid interaction solvers, which are significantly less computationally intensive than 2D RANS or 3D CFD simulations. These rapid methods, often combined with viscous/inviscid boundary layer interaction models, provide quick estimations. However, while solvers like XFOIL are well established for conventional airfoil shapes, their accuracy may decrease when applied to unconventional geometries or complex flow conditions, which are common in gradient-free optimization and low Reynolds number flows. As a result, the design space is often limited to regions where classical propeller analysis methods remain valid and computationally inexpensive, restricting exploration of more innovative or complex propeller configurations. A study presented in [38] compares the results of a design methodology based on a two-dimensional method (Larrabee method) with those from CFD analysis of the resulting propeller. The propeller was specifically designed for flight at 18 km altitude, with a target thrust of 22 N, a flight speed of 22 m/s, and a diameter slightly below 1 m. The findings reveal that the thrust predicted by the CFD simulation is consistently lower than the thrust used in the design process, with an average discrepancy of 20% across the cases examined. These results demonstrate that two-dimensional methods may introduce errors in the calculation of thrust and power coefficients, particularly at low Reynolds numbers. To improve the design methodology, these approaches should be complemented by high-fidelity methods such as three-dimensional CFD. This issue arises because separation bubbles form more easily at low Reynolds numbers [138], a phenomenon that is challenging to capture even with CFD simulations and significantly impacts lift, drag, and overall efficiency [139]. In general, two-dimensional methods can accurately predict the drag coefficient \(C_{D}\) and thus the effect of separation bubbles, while pressure gradient effects along the blade and their influence on separation bubbles and transition can only be observed through CFD analysis. Furthermore, as the Reynolds number decreases, the accuracy of these methods diminishes due to increasing interaction between inviscid and viscous regions [140]. In [68], a comparison was made between low-fidelity and high-fidelity methods. In particular, the work focuses on optimizing the efficiency of a propeller for HAPSs using a process that includes iterative geometry optimization cycles based on low-fidelity methods, namely BEMT coupled with Vortex Theory, with airfoil performance data obtained from XFOIL (for viscous analysis) and two-dimensional RANS simulations (via Ansys Fluent with the Spalart-Allmaras turbulence model). Optimization was performed using the Particle Swarm Optimization (PSO) algorithm, selected for its ability to find the global maximum with fewer function evaluations. Operational conditions were set at an altitude of 16 km and an airspeed of 77.64 m/s, corresponding to a flight Mach number of 0.26, and a fixed shaft power of 30 kW. Subsequently, the BEMT-based predictions were compared with those obtained from a high-fidelity three-dimensional RANS simulation with the \(k-\omega\) SST turbulence model (including Gamma Transition (GT) options) to assess the loss of accuracy when using low-fidelity methods compared to more precise simulations. The aim was to determine whether simpler methods could still provide reliable results for preliminary design, saving time and computational resources.