## 5 Propeller Design for HAPS Operation
### Recent Development on Propeller Design Methodology
However, this approach is not applicable in low Reynolds number conditions due to significant variation in airfoil curves along the blade. To address this, especially for HAPS propellers, viscous-potential numerical tools (e.g., XFOIL, Javafoil) have been employed in [84] to compute \(C_{l}\) and \(C_{d}\) curves over a wide range of Reynolds and Mach numbers. These curves are then interpolated to find the required values of \(C_{l}\) and \(C_{d}\) for any combination of angle of attack, Reynolds number, and Mach number during the design process. This method enables optimal determination of geometric characteristics of the propeller, such as chord and pitch distribution, significantly improving overall efficiency. Indeed, this approach has demonstrated an increase in propeller efficiency by more than 5% compared to conventional methods, as shown in Figure 13, in which a comparison between the efficiencies of the design of an optimized propeller, considering a variable Reynolds number along the span, and designs assuming constant Reynolds numbers equal to \(10^{5}\), \(2.5\times 10^{5}\), and \(5\times 10^{5}\) is presented. The methodology has been applied for a small HAPS operating at an altitude of 20 km, a wind speed of 10 m/s, and a required thrust of 100 N (each propeller). According to [84], this improvement corresponds to a potential payload increase of up to 25%. Other recent studies using two-dimensional methods, such as BEMT and Vortex Theory, to evaluate the performance of various propeller designs, are reported in [51; 67; 69; 83; 85; 128; 129]. Although these methods have been extensively validated across a range of flow conditions, direct evaluation in stratospheric environments remains limited due to the scarcity of experimental data in this regime. In [130], verification of the applicability of Vortex Theory to high-rotation, low Reynolds number propellers was performed through experimental tests. For a 0.33 m propeller operating at 10,000 rpm, the maximum error in predicted efficiency was approximately 9%, highlighting the limitations of classical theories for designing propellers operating in this regime. Instead, top-performing designs from optimization processes are typically verified using three-dimensional RANS simulations. In general, two-dimensional methods provide reasonable predictions of efficiency trends and serve as valuable tools for rapid assessment of new designs. However, their accuracy decreases when three-dimensional effects become significant or when lift and - \(\alpha\) and \(C_{d}\)
- \(\alpha\)) for blade sections are less reliable. These limitations are especially relevant for complex blade geometries, with large variations in thickness and chord distributions along the blade span, or for swept blade designs. Furthermore, applying two-dimensional theory under the assumption of laminar flow (including laminar separation bubbles) does not guarantee similar behavior in three-dimensional flow, where centrifugal and Coriolis forces are present. These forces can significantly alter flow patterns and influence the separation point [131]. Centrifugal forces drive the boundary layer airflow toward the blade tip, while the Coriolis force produces a favorable pressure gradient along the chordwise direction, accelerating airflow toward the leading edge [132; 133]. This leads to a stall delay phenomenon, necessitating aerodynamic corrections. To account for these three-dimensional effects, several models have been developed in recent years, such as the Corrigan and Schillings method [134]. This method models these effects as a shift in the angle of attack, \(\Delta\alpha\), which depends on the local chord, blade station, and the strength of centrifugal terms. Subsequently, lift is corrected using Equation (25). \[C_{L}(\alpha+\Delta\alpha)\ =\ C_{l}(\alpha)+\left(\frac{\partial C_{l}}{ \partial\alpha}\right)\Delta\alpha \tag{25}\]  
In this equation, \(\frac{\partial C_{l}}{\partial\alpha}\) represents the lift curve slope in the linear region of the lift.