## 5 Propeller Design for HAPS Operation
### Recent Development on Propeller Design Methodology  
The traditional approach to propeller design begins with determining the thrust required for the platform to accomplish its mission [123]. After establishing this requirement, the next step involves designing the most suitable propeller to produce the necessary thrust [51; 124]. The propeller is designed with specific dimensions to operate efficiently within a defined range of rotational speed and torque. Finally, an engine is selected that offers optimal performance within this rotational speed and torque range. The current approach to propeller design has evolved to incorporate various methods aimed at improving performance and efficiency. One of the most commonly used techniques is the inverse design method, which determines blade geometry based on a predefined operating point, with the goal of optimizing the overall propeller efficiency, \(\eta\). This efficiency is often expressed as the product of two components: viscous efficiency \(\eta_{\
u}\), related to drag losses caused by friction, and induced efficiency \(\eta_{i}\), associated with losses from vortex generation. The primary focus during the design process is typically on minimizing induced drag, thereby maximizing \(\eta_{i}\), following guidelines established by [PERSON] [75]. Most propeller designs tend to neglect the impact of Reynolds number variations across different blade sections, simplifying the process but limiting accuracy. This is the case of the simple multi-objective genetic algorithm (thrust and efficiency maximization) for designing propellers for HALE UAVs proposed in [125], where a constant Reynolds number of \(10^{6}\) is considered for estimating the aerodynamic coefficients of blade airfoils. This estimation is performed using viscous/inviscid panel codes (e.g., XFOIL), in which the flow outside the boundary layer is solved as inviscid, while the region near the surface, known as the boundary layer, is solved using boundary layer equations that account for viscous effects; the interaction is also extended to separated flow. By modeling laminar separation bubbles accurately, these methods also allow quite precise prediction of both lift and drag, even in cases where extensive separation bubbles are present. However, keeping the Reynolds number constant at every spanwise station may mislead the optimization process. Recent studies have focused on adapting those design methods to address challenges posed by low Reynolds number regimes. For example, three-dimensional flow equilibrium models, post-stall formulation, and adjustments for rotational effects have been introduced in [124; 126; 124]. Despite these advances, there remains no consensus on the optimal approach to handle viscous losses (\(\eta_{v}\)). A common practice is to experiment with different designs [127; 112] and select the one offering the best performance. However, these approaches often do not explicitly address low Reynolds number phenomena or the variation in aerodynamic performance along the propeller blade sections during the optimization process. A more efficient and low-cost iterative method for preliminary design of stratospheric propellers is presented in [84]. This method accounts for low Reynolds number effects and variations in Reynolds number along the blade sections by combining traditional design approaches with modern numerical tools. The core of the method uses [PERSON]'s analytical theory to minimize induced drag, while leaving one free parameter (since two variables are constrained by a single condition, creating a degree of freedom) that is optimized using a cost function, such as \(C_{L}/C_{D}\), which depends on the Reynolds number. In high Reynolds number regimes, the standard assumption is that aerodynamic curves remain constant across different propeller sections (assuming the same airfoil is used along the blade), and the lift coefficient \(C_{l}\) at maximum efficiency, the same for all airfoils, can be readily calculated from the airfoil polar.