## 5 Propeller Design for HAPS Operation
### Recent Development on Propeller Design Methodology
The results, presented as efficiency curves of the blade, are shown in Figure 14.  
The comparison between Vortex Theory and three-dimensional RANS simulations, shown in Figure 14, highlights the performance limitations of low-fidelity aerodynamic models, particularly beyond the design point. The results, displayed over a wider range of advance ratios, demonstrates good agreement between the two methods up to approximately \(J=1.25\), corresponding to the design point (indicated by the dashed vertical line). Within the range of \(J=1.0\) to 1.25, both methods follow a similar trend, although Vortex Theory slightly overestimates efficiency compared to the three-dimensional RANS model with GT option. Beyond this point, the predicted efficiency diverges significantly: Vortex Theory predicts a decreasing trend, whereas three-dimensional RANS reveals a  
Figure 13: Comparison between the optimized propeller obtained considering a spanwise-varying Reynolds number and propeller designed under constant Reynolds numbers (\(10^{5}\), \(2.5\times 10^{5}\), and \(5\times 10^{5}\)). \(N_{b}=4\). Maximum efficiency is shown as a function of the advance ratio \(\lambda\). new efficiency peak around \(J=1.4\), reaching approximately 86%. This behavior is consistent with the discrepancies observed in [141] between Vortex Theory and high-fidelity simulations, particularly at high advance ratios. In [95], the limitations of low-fidelity models such as BEMT and Vortex Theory are also highlighted, and a multi-fidelity surrogate modeling framework is proposed to better capture CFD trends under high-altitude and low-Reynolds-number conditions. These discrepancies primarily arise from the Vortex Theory inability to capture complex three-dimensional flow phenomena accurately, such as flow separation, spanwise effects, and transition. Although correction factors, including tip and hub loss models, enhance predictions, they remain insufficient under low Reynolds number, laminar separated, or transonic flow conditions. Consequently, the low-fidelity method lacks robustness and may both overpredict and underpredict performance depending on operating conditions, as confirmed by the presented results. In contrast, three-dimensional RANS simulations (e.g., Gamma Transition), provide more reliable and accurate efficiency predictions over a broader range of operating conditions. To address these issues, recent studies have proposed multi-fidelity optimization frameworks that combine Vortex Theory with high-fidelity RANS simulations. A notable example of such an approach is the work by [PERSON] et al. [128], who developed a comprehensive propeller design methodology for the high-altitude solar UAV EAV-3, intended to operate at 15-20 km. The approach combines a minimum-induced-loss BEMT framework with transition-model CFD and multi-objective optimization based on a Kriging metamodel. The design space was parametrized through variations in blade lift coefficient distributions along the span, defined using a second-order polynomial, and the aerodynamic characteristics of the selected airfoil (DAE51) were computed using transition SST CFD to capture low-Reynolds effects more accurately than XFOIL. A Design of Experiments (DOE) matrix was used to train the surrogate model, and a desirability function guided the optimization, balancing efficiency at 15 km, torque limits at sea level, and rpm constraints at 18 km. The resulting 1.2 m two-bladed propeller achieved a design-point efficiency of 66%, with performance validated through wind tunnel tests, CFD simulations, and flight measurements. The close agreement (within 3%) between predicted and experimental results confirmed the robustness of this hybrid methodology, which bridges the gap between classical BEMT and full inverse-design CFD workflows. Other studies have adopted similar multi-fidelity strategies with a focus on reducing the number of expensive high-fidelity simulations.