## 5 Propeller Design for HAPS Operation
### Recent Development on Propeller Design Methodology
In [142], a hierarchical Kriging model was applied to the aerodynamic optimization of high-altitude propellers, combining blade element/Vortex Theory as the low-fidelity source with RANS simulations as the high-fidelity data. Compared to a standard Kriging approach relying solely on RANS evaluations, the hierarchical model reduced the number of high-fidelity simulations from 120 to 75 (\(-\)37.5%) while maintaining the same optimal thrust. Similarly, [143] used a surrogate-based multi-objective optimization framework in which Vortex Theory was employed to explore off-design points (e.g., take-off), while 3D RANS simulations were reserved for cruise and climb conditions. This surrogate-based framework not only preserved accuracy but also enabled reliable performance prediction across a wide range of operating conditions with fewer CFD calls. In Ref. [144], a 3D RANS database of high-altitude propeller designs is analyzed following the optimization process outlined in Ref. [143]. The performance of the designs was compared between Vortex Theory and CFD simulations. The comparison showed good agreement for low thrust and power consumption, but higher values resulted in increased variance and the presence of outliers. Efficiency predictions were more accurate for efficiencies above 70%. A variance-based sensitivity analysis identified key geometric and operational factors, such as sweep, twist, and rotational velocity, as the most influential on performance. Two multi-fidelity surrogate models, Co-Kriging and Hierarchical Kriging,were trained using data from both 3D RANS and Vortex Theory, and their performance was compared with a single-fidelity model (Kriging). Hierarchical Kriging outperformed the other models in most performance metrics and showed good agreement with Vortex Theory when sufficient training data were available. Quantitatively, Hierarchical Kriging reduced the mean squared error by approximately 48% and the mean absolute error by 36% relative to the single-fidelity Kriging baseline [144], confirming its superior predictive capability when combining low- and high-fidelity data. Notably, multi-fidelity models performed significantly better than the single-fidelity Kriging model, particularly when only CFD data were used. The best design obtained from this optimization process achieved an efficiency of 84.2%, with a propeller diameter of 3.4 m, four blades, and an angular speed of 480 rpm. The main conclusion is that, although Vortex Theory presents challenges due to outliers and noise in its predictions, combining it with multi-fidelity optimization approaches helps create more accurate surrogate models. This approach enhances propeller performance prediction while optimizing computational costs. This type of approach is impractical for iterative methodologies, where thousands of calculations are required, each taking several hours to complete. A solution proposed in [40] involves the use of CFD to simulate a propeller blade to account for centrifugal and Coriolis effects. Then, pseudo-2D curves for the profiles up to 40% of the span are extracted and used in BEMT methods, which are faster, evaluating the performance of the optimized propellers. The methodology in [40] suggests optimizing not only the chord distribution and pitch angle iteratively, but also the sectional airfoil, which is parameterized using two fifth-order class-shape transformation (CST) curves, under low Reynolds conditions. The optimized propeller, designed for a HAPS working at 20 km, with an airspeed of 32 m/s and a thrust of 20 N (each propeller), achieved a 3.6% higher efficiency than the Betz-based design at cruise (\(Re\approx 4.0\times 10^{4}\)) [40]. This approach results in improved efficiency compared to traditional methods. Such optimization would not be feasible with a non-iterative method, as the design parameters and operating conditions of the propeller remain unknown until the optimization process is complete.