A Comprehensive Review of Propeller Design and Propulsion Systems for High-Altitude Pseudo-Satellites

Abstract

In both scientific and industrial fields, there has been a notable increase in attention toward High-Altitude Pseudo-Satellites (HAPSs) in recent years. This surge is driven by their distinct advantages over traditional satellites and Remotely Piloted Aircraft Systems (RPASs). These benefits are particularly evident in critical areas such as intelligent transportation systems, surveillance, remote sensing, traffic and environmental monitoring, emergency communications, disaster relief efforts, and the facilitation of large-scale temporary events. This review provides an overview of key aspects related to the propellers and propulsion systems of HAPSs. To date, propellers remain the most efficient means of propulsion for high-altitude applications. However, due to the unique operational conditions at stratospheric altitudes, propeller design necessitates specific approaches that differ from those applied in conventional applications. After a brief overview of the propulsion systems proposed in the literature or employed by HAPSs, focusing on both the technical challenges and advancements in this emerging field, this review integrates theoretical foundations, historical design approaches, and the latest multi-fidelity optimization techniques to provide a comprehensive comparison of propeller design methods for HAPSs. It identifies key trends, including the growing use of CFD-based simulations methodologies, which contribute to notable performance improvements. Additionally, the review includes a critical assessment of experimental methods for performance evaluation. These developments have enabled the design of propellers with efficiencies exceeding 85%, offering valuable insights for the next generation of high-endurance, high-altitude platforms.

1. Introduction

In recent years, there has been a growing interest in unmanned stratospheric platforms for long-duration Earth observation and telecommunications, commonly referred to as HAPSs. These platforms represent a valuable complement to satellites and RPAS. Most HAPSs operate at altitudes of approximately 18–20 km, which are well above commercial air traffic. At these altitudes, the atmospheric conditions are relatively stable, with minimal convective activity and nearly constant temperatures. Wind speeds are also generally lower compared to those observed in both lower and higher atmospheric layers, making this region of the stratosphere particularly attractive for persistent operations.

This segment of the atmosphere is part of what is often referred to as Near Space, qualitatively defined as the region extending from around 20 km altitude up to the Kármán line at 100 km. Earth observation from the stratosphere benefits from the proximity to the Earth’s surface, enabling higher spatial resolution than satellite imagery, assuming equivalent optical payload characteristics. Furthermore, temporal resolution is significantly enhanced, as HAPSs can maintain a fixed position (station-keeping), thereby eliminating or substantially reducing revisit times. Until recently, HAPS technologies had received limited attention. However, several configurations have now been proposed and can generally be classified into two main categories, based on their method of lift generation: fixed-wing platforms and lighter-than-air systems. Among the fixed-wing solutions, the Airbus Zephyr platform [1] and the AE Prismatic PHASA-35 system [2] show considerable promise. In the category of lighter-than-air systems, Thales is developing the Stratobus airship [3], while the SCEYE airship [4] has already completed multiple missions in the stratosphere. These platforms differ in configuration and payload capacity, which can range from a few kilograms (5–20 kg) to several hundred kilograms, depending on the specific design.

Hybrid configurations have also been explored in the literature. For example, the Italian Aerospace Research Centre (CIRA) is actively developing a hybrid solution that combines both aerostatic and aerodynamic lift, as detailed in [5,6,7].

One of the most critical subsystems in HAPSs is the propulsion system, due to its significant impact on both power consumption and overall system weight. Achieving the required endurance and payload capacity at stratospheric altitudes and relatively low flight speeds presents a substantial challenge. The most feasible approach to address this issue is the adoption of propulsion systems characterized by high efficiency and low specific energy consumption.

The design of propulsion systems for HAPSs has undergone significant evolution in recent decades, primarily driven by technological advancements in renewable energy sources (e.g., photovoltaic systems) and energy storage technologies (e.g., batteries). In [8], a study is presented on a propulsion system for a subsonic, high-altitude unmanned aircraft capable of operating at an altitude of approximately 30 km, with a loiter time of about three hours. The propulsion system in that case consisted of a turboprop engine fueled by onboard fuel and oxidizer. At that time, the endurance record at 20 km was held by the Boeing Condor, which achieved 58 h of continuous flight using a liquid-cooled, two-stage turbocharged spark-ignition engine. A key challenge associated with air-breathing turbocharged piston engines at high altitudes is heat rejection, due to the low ambient air density. This necessitates the use of larger radiators. In addition, achieving effective combustion requires very high pressure ratios to compensate for the low atmospheric pressure. In [9], a comparative study is provided, analyzing various propulsion system types for a High-Altitude Long-Endurance (HALE) airship across two mission profiles at altitudes between 18 km and 21 km, each with a target endurance of one month. The primary difference between the two scenarios lies in the payload mass. The authors evaluate four propulsion technologies: solar photovoltaic arrays, hydrogen fuel cells, rechargeable batteries, turboshaft engines powered by hydrogen, and regenerative fuel cells. At that time, prototype regenerative fuel cells could deliver approximately 400 Wh/kg, with projected capabilities reaching 700 Wh/kg. According to the analysis in [9], turboshaft or turboprop engines—potentially hydrogen-fueled—remained the most favorable solution for high-altitude propulsion, as they offered a lower propulsion-system-to-total-mass ratio compared to other alternatives. Two critical design requirements are emphasized in [10]: the propulsion system must be extremely lightweight and highly reliable, and must be able to operate at temperatures as low as 217 K. An additional challenge is the requirement for operation across a wide altitude range, as the platform must be capable of reaching and maintaining very high altitudes. Although propellers remain the preferred choice for many near-space aircraft, the authors of [10] also investigate unconventional propulsion concepts, including those based on electrohydrodynamic forces, radiometric forces, and thermal transpiration. However, these emerging technologies are not yet viable, mainly due to the limited understanding of the underlying physical mechanisms and the presence of multiple unresolved engineering challenges. Historically, one of the main obstacles in the practical deployment of HAPSs has been the limited efficiency of solar cells and the relatively low energy density of battery systems. At present, HAPSs are typically powered by solar energy during daylight hours and rely on battery storage during nighttime operations. This architecture is motivated by findings reported in [11], where the authors compare various unmanned aircraft vehicle (UAV) platforms powered by different energy sources, including combustion engines, regenerative fuel cells, solar energy, and batteries. While most of the UAVs analyzed are tropospheric in nature, the Airbus Zephyr HAPS (Airbus Defence and Space, Farnborough, United Kingdom) is also included as a case study due to its reliance on solar energy for propulsion. The study concludes that, compared to combustion systems and regenerative fuel cells, solar energy combined with battery storage offers the most favorable performance in terms of flight time-to-total weight ratio, payload weight-to-flight time ratio, and flight time-to-cost ratio. These advantages have significantly contributed to the growing viability and development of HAPS technology [12]. In early studies on HAPS platforms, battery energy densities were approximately 100 Wh/kg. Subsequent technological advancements led to an increase to 250 Wh/kg [13]. In 2022, Amprius Technologies (Fremont, CA, USA) announced the commercial availability of lithium–ion battery cells with an energy density of 450 Wh/kg and a volumetric density of 1150 Wh/L [14]. Similarly, MicroLink Devices (Niles, IL, USA), a manufacturer of solar arrays for both satellite and HAPS applications, has developed high-efficiency solar sheets capable of delivering over 1.5 kW/kg, with a solar energy conversion efficiency of up to 37.75% [15]. Fuel cell technology has also progressed substantially. Since 2009, when state-of-the-art systems delivered power densities of around 400 W/kg, the technology has advanced considerably, with current systems reaching up to 1.6 kW/kg and targets as high as 2 kW/kg set by key stakeholders in the aviation sector [16]. Regardless of the energy source, most contemporary HAPS concepts—across various stages of development—employ electric propulsion systems, wherein electric motors drive lightweight, high-efficiency propellers. For low-speed applications typical of stratospheric flight, propellers continue to represent the most efficient propulsion method [17]. When integrated with renewable energy sources such as solar panels, these systems offer not only high precision and reliability, but also enable the long-duration, continuous operations that are essential for sustained HAPS missions in the stratosphere. Some of the most representative HAPS configurations developed over the past 40 years, each of which has successfully completed at least one flight, are presented in Figure 1 and Figure 2, along with their corresponding altitude and endurance performance data. The year indicated for each platform refers to the date when the performance was actually achieved or to the year in which such performance is expected to be reached on the basis of the design specifications. The data illustrated in these figures clearly demonstrate that, on average, HAPSs equipped with electric propulsion systems tend to achieve superior altitude and endurance capabilities compared to those powered by combustion-based propulsion.

As a result, two main categories of HAPS can currently be identified: solar-powered airplanes and solar-powered airships [28]. Among the most notable solar airplane projects, both past and present, are Helios [29], AEV-3 [30], Aquila [31], and Zephyr [32]. Regarding solar airships, key projects include HiSentinel [33], the Stratospheric Platform (SPF) [34], the Korean Stratospheric Airship Program [28], the High-Altitude Airship (HAA) [35], and Stratobus [28].

To achieve high endurance capabilities at high altitudes, it is essential not only to improve engine efficiency but also to properly design propellers to enhance their performance. A key factor in propeller design and operation is the environment in which it functions. High-altitude conditions (ranging from 17 km up to 50 km) present unique challenges for propeller operation. At these altitudes, airfoils operate under low Reynolds number conditions, typically between ${10}^4$ and ${10}^5$ [36]. Within this range, the transition from laminar to turbulent flow plays a crucial role, with viscous forces becoming a significant factor influencing flow behavior. Compared to high Reynolds number operations, some of the most notable effects include a sharp decrease in the lift-to-drag ratio after its maximum, caused by the development of thicker boundary layers; possible nonlinear behavior of airfoils at low angles of attack; laminar flow separation along the blades leading to short or long laminar separation bubbles; increased sensitivity of the flow to free-stream turbulence; and a consequent rise in drag forces resulting in reduced efficiency [36,37,38].

The laminar separation bubble is a primary factor contributing to the decrease in propeller efficiency at very low Reynolds numbers [39]. The key to designing an effective propeller lies in mitigating the detrimental effects of laminar separation bubbles on the blades [40]. Furthermore, the interactions between three-dimensional effects, such as centrifugal [41,42] and Coriolis forces, with the transition process and the development of laminar separation bubbles along the blade, are not yet fully understood. These three-dimensional effects, amplified by blade rotation, can induce cross-flow within the boundary layer and alter the transition location, thereby influencing both the extent and stability of laminar separation [43]. Traditional low-fidelity methods, such as blade element momentum theory (BEMT) and lifting-line theory, are unable to accurately capture these mechanisms, particularly under low Reynolds number conditions. Although these methods remain popular for preliminary design due to their computational efficiency, their predictive accuracy deteriorates rapidly in flow regimes dominated by viscous and unsteady effects. Consequently, high-fidelity Computational Fluid Dynamics (CFD) approaches—especially Reynolds-Averaged Navier–Stokes (RANS) solvers coupled with transition-sensitive models such as the $\gamma$–$R{e}_{\theta }$ model—have been increasingly employed to model these flows with improved accuracy [42,44]. Recent research has also integrated such solvers with shape optimization algorithms to develop propellers that actively control or delay laminar separation bubble formation, thereby enhancing aerodynamic efficiency over a wider range of operating conditions [45]. Several propeller designs tailored for various high-altitude systems have been reported in the literature. Each design is optimized to operate efficiently within specific altitude ranges and with different propulsion systems. Colozza provides an overview of existing propeller systems used in various HAPS configurations [27].

Some of the main HAPSs exploiting stratospheric propellers are the Condor aircraft [18]; Pathfinder aircraft [22,46] shown in Figure 3; Grob Strato 2C aircraft [46,47] shown in Figure 4; Grob EGRETT aircraft [48]; Perseus A and B [21,27] aircraft shown in Figure 5, Centurion and Helios aircraft [29,46,49] shown in Figure 6; and Theseus [50].

Although limited information is available regarding the design of propellers used in existing HAPSs, a review of their key characteristics is provided in

Table 1. Comparison of aircraft nominal thrust and propeller diameter from various high-altitude UAV platforms. Adapted from ref. [51].

Year	Aircraft Name	Nominal Thrust [N]	Propeller Diameter [m]
1987	Egrett [48]	2773	3.04
1988	Condor [27]	1129	4.90
1993	Pathfinder [27,46]	23	2.01
1994	Perseus [21]	388	4.40
1995	Strato2C [46,47]	2500	6.00
1996	Theseus [50]	409	2.74

[51].

This review aims to provide a comprehensive and critical overview of the state of the art in propulsion systems and propeller technologies for HAPSs. Special attention is given to the influence of stratospheric environmental conditions on propeller aerodynamic behavior and to the unique design challenges posed by low Reynolds number regimes. This paper systematically reviews the evolution of propulsion architectures for HAPSs, with emphasis on the modeling methodologies, ranging from low- to high-fidelity approaches, adopted in propeller design, as well as on recent advances in optimization strategies tailored to high-altitude platforms. Finally, it examines experimental methods used to validate propeller performance under the low-Reynolds-number conditions typical of HAPS operations.

2. Methodology

To ensure a comprehensive and methodologically rigorous review, relevant literature was collected from major scientific databases, including Scopus, Web of Science, IEEE Xplore, ScienceDirect, and the NASA Technical Reports Server (NTRS). Google Scholar was also used for supplementary searches and cross-referencing. Searches were conducted using combinations of the following keywords: “HAPS”, “High-Altitude Pseudo-Satellites”, “Stratospheric Aircraft”, “Propeller Design”, “Low Reynolds Number Propellers”, “Electric Propulsion”, “Hybrid Propulsion”, “High-Altitude UAVs”, “HALE”, “Propulsion systems”, “Experimental testing”. The time span of the literature spans from early foundational contributions to propeller theory (e.g., Rankine and Froude, mid-19th century) to the most recent advancements published up to 2025. This wide temporal coverage allows for both a historical perspective and a focus on emerging high-fidelity and data-driven design methodologies. Inclusion criteria required that documents (i) be peer-reviewed or published by reputable institutions; (ii) address HAPS propulsion or propellers working in low-Reynolds-number environments; and (iii) present theoretical, computational, or experimental contributions. Sources that lacked methodological clarity, quantitative rigor, or were only tangentially related to the scope of the review were excluded. This ensured both relevance and reliability across the selected literature corpus. This approach ensured the inclusion of both historical developments and cutting-edge contributions to the field, providing a comprehensive and balanced overview of the state-of-the-art in HAPS propulsion and propeller technology.

3. Propulsion Systems for HAPSs

The capability of aircraft and airships to operate at high altitudes for extended durations is strongly dependent on the performance of the propulsion system. For many years, propulsion has been the primary limiting factor for stratospheric applications, mainly due to its power consumption—it is the largest power-consuming system onboard—and its weight. Significant research on propulsion systems for HAPSs has been conducted over the past fifty years to overcome these power and weight constraints and enhance aircraft performance in terms of altitude, flight endurance, and payload capacity. This section provides an estimation of the propulsion system power requirements and presents an overview of the main propulsion concepts proposed in the literature. A propulsion system for HAPSs must satisfy multiple requirements, primarily related to power, mass, and operating conditions. Regarding operational requirements, the propulsion system must function efficiently across a wide range of environmental conditions. A HAPS may need to autonomously take off from the ground and reach the stratosphere; therefore, the propulsion system should perform efficiently throughout the entire altitude range, characterized by varying temperatures and pressures. The propulsion power requirement is mainly driven by the thrust necessary to overcome drag and by the efficiencies of the system components. The drag forces for aircraft and airships are defined, respectively, by Equations (1) and (2).

$$D={\displaystyle \frac{1}{2}}\rho V_{\infty }^{2}S{C}_D$$

(1)

$$D={\displaystyle \frac{1}{2}}\rho V_{\infty }^2{V}_a^{{\displaystyle \frac{2}{3}}}S{C}_{D,v}$$

(2)

In Equation (2), the drag D is expressed as a function of the volumetric drag coefficient, $C_{D,v}$, and the volume of the airship, $V_a$. The volumetric drag coefficient, $C_{D,v}$, can be related to the fineness ratio f (defined as the ratio of the length to the diameter or width of the airship). An equation proposed in the literature [52], valid for a maximum fineness ratio of 10 and for a cylindrical body with hemispherical ends, is presented in Equation (3).

$$C_{D,v}=0.23175-0.15757f+0.04744f^2-7.0412·{10}^{-3}{f}^3+5.1534·{10}^{-4}{f}^4-1.4835·{10}^{-5}{f}^5$$

(3)

The velocity $V_{\infty }$ in the preceding equations depends on the specific application. For an airship aiming to maintain a stationary position, $V_{\infty }$ corresponds to the ambient wind speed. Wind speeds represent a major design consideration for HAPSs, as they vary with time, location, and altitude. Many stratospheric platform concepts proposed in the literature operate at approximately 20 km altitude, where wind speeds generally reach a minimum. For high-altitude aircraft and airships, the typical thrust requirement ranges from 100 N to 100 kN. Once the drag is determined, the power required by propulsion systems employing propellers to generate thrust can be readily calculated. Numerous types of propulsion systems have been proposed and many have undergone testing. The following paragraphs provide an overview of the main propulsion systems discussed in the literature. These systems are categorized into four groups in this section: combustion propulsion, electrical propulsion, hybrid propulsion, and future propulsion concepts for high-altitude applications.

3.1. Combustion Propulsion

Combustion propulsion was the first type of propulsion system investigated for achieving stratospheric flight. Early stratospheric aircraft employed piston engines equipped with large radiators to cool the system, due to the low-density environment where heat convection efficiency is poor. The Boeing Condor, with an endurance of 56 h at approximately 20 km altitude, utilized a liquid-cooled, two-stage turbocharged spark-ignition engine. The Aurora Flight Sciences Perseus A UAV, designed to operate for several hours at an altitude of around 25 km, used a spark-ignition engine and carried liquid oxygen to support combustion. The Grob Strato 2C, designed for operations at 24 km altitude with an eight-hour endurance, employed the same propulsion system as the Boeing Condor but featured an additional turbocharging stage. Subsequently, turboprop and turboshaft engines have been investigated. In [8], a propulsion system capable of powering an aircraft to an altitude of approximately 30 km for several hours was analyzed. The described system is a turboprop engine using fuel and oxidizer carried onboard. Specifically, a stoichiometric mixture of methanol and hydrogen peroxide, diluted with water to maintain acceptable turbine inlet temperatures, was considered as the propellant. Compared to piston engines, this system offers the advantage of eliminating the need for a radiator, as all waste heat is expelled through the exhaust gases without requiring convective heat transfer to the atmosphere. Preliminary analyses demonstrated the capability of this system to operate effectively at high stratospheric altitudes, achieving flight endurance of several hours.

Recently, propulsion systems utilizing hydrogen have gained attention. Historically, hydrogen has received limited consideration in the literature for propulsion systems due to safety concerns arising from its high flammability and storage complexity. Additionally, hydrogen’s low energy density per unit volume necessitates large storage volumes. However, hydrogen offers several advantages in the HAPS domain, particularly for hybrid buoyant aircraft [53]. These aircraft, characterized by large hulls filled with low-density gases, can accommodate the storage of large hydrogen volumes without issue. Furthermore, although hydrogen’s volumetric energy density is lower than that of fuels such as kerosene, its gravimetric energy density is substantially higher. This implies that less fuel mass is required to deliver the same energy for a mission, making hydrogen especially suitable for applications where weight reduction is critical. Moreover, combustion of hydrogen with air produces water vapor, leading to reduced atmospheric pollution compared to conventional fuels. An example of a HAPS employing a liquid hydrogen-fueled propulsion system is AeroVironment’s Global Observer, designed to operate for several days at an altitude of 20 km. Although the project was discontinued years ago, a prototype successfully completed test flights. Currently, due to significant advances in batteries, solar cells, and fuel cells, hydrogen-based propulsion systems receive comparatively less attention.

3.2. Electric Propulsion

Electric propulsion is one of the most commonly investigated alternatives for HAPSs in the literature due to its advantageous properties. In general, a HAPS equipped with a properly designed pure electric propulsion system can operate for extended periods in the stratosphere (on the order of months), compared to vehicles employing combustion or hybrid propulsion systems, albeit with a significant reduction in payload mass. An electric propulsion system (EPS) can be divided into three main components: the energy device, the drive device, and the propulsion device. The energy device typically consists of photovoltaic arrays (solar cells) or fuel cells. Photovoltaic arrays convert the incoming solar energy into electrical energy with an efficiency of approximately 10–20%, supplying power to the engines, payload, and, when in excess during daylight hours, recharging batteries or regenerative fuel cells for use during night flights. An innovative storage method related to excess daytime solar energy collected by solar cells is presented in [54]. This method involves converting the surplus power into gravitational potential energy; specifically, the excess power during daylight hours is used to climb to a higher altitude. During the night, the HAPS descends to its initial altitude without engine use. The analysis in [54] indicates that this ascent and descent maneuver can result in mass savings compared to storing excess power in batteries and regenerative fuel cells or simply rejecting the surplus energy. Therefore, this approach may be preferable if maintaining a constant altitude is not a mission requirement. Fuel cells electrochemically combine two elements, commonly hydrogen and oxygen, to produce electricity with high efficiency (up to approximately 50%), outperforming combustion processes limited by Carnot efficiency. Regenerative fuel cells (RFCs), which can operate both as fuel cells and as electrolysers to store energy, are of particular interest for high-altitude long-endurance applications. If other types of hydrogen–oxygen fuel cells are used, an electrolyser may be required to replenish hydrogen and oxygen in storage tanks by splitting the water produced during fuel cell operation; this electrolyser can be powered by energy from solar arrays. In many HAPS configurations, fuel cells are commonly used in conjunction with solar cells to recharge fuel tanks.

Previous studies have assessed the suitability of using fuel cells alone as energy devices for HALE platforms [55]. In particular, phosphoric acid and polymer electrolyte fuel cells, which utilize hydrogen and external air compressed by a turbo-compressor, have been considered as potential energy sources. To drive the turbine, the working fluid exiting the fuel cell is heated further through a burner and a burner preheater, thereby increasing its energy content. The results of the analysis presented in [55] indicate lower specific fuel and air consumption compared to conventional aircraft propulsion systems. However, the high specific heat rejection, which necessitates large heat exchangers, combined with the substantial volume required for hydrogen storage, render this approach non-viable as an energy source for HALE aircraft. The drive device comprises the electric motor, inverter, and controller. Electric motors are generally classified into two categories: AC and DC motors, each with its own advantages and disadvantages. DC motors operate on direct current and, for these applications, can be further divided into brushed and brushless types based on their operating principles. A primary advantage of DC motors is compatibility with power sources such as solar arrays and fuel cells, which produce DC power, thereby eliminating the need for a DC-to-AC converter (inverter). Additional benefits include ease of torque-speed control, precision, power density, versatility, and quiet operation. However, drawbacks include increased complexity—due to the presence of brushes, commutators, and controllers that wear out relatively quickly, especially in low-pressure and low-density environments—higher maintenance requirements, susceptibility to malfunctions compared to AC motors, greater cost, and lower energy efficiency.

AC motors are electric motors that operate with alternating current. They can be classified into two main categories: synchronous and asynchronous. The primary advantages of AC motors include their simplicity, reliability (requiring less maintenance), efficiency (consuming less power), and wide availability. The main disadvantage is the necessity of an inverter in the system to supply power to the motor. Permanent magnet synchronous motors (PMSMs) and brushless DC motors (BLDCs) are, respectively, the most common AC and DC electric motors used in industrial applications. Due to their advantages, AC motors are generally preferred for HAPS propulsion systems. However, some studies in the literature have analyzed propulsion systems for HAPSs employing DC electric motors, demonstrating their feasibility for high-altitude applications [56].

In the literature, several curve fits have been developed using data from electric motors reported in open sources, which can be used to estimate the mass and efficiency of an electric motor based on the required power, P [57]. The mathematical expression for the mass curve is presented in Equation (4).

$$m_{em}=0.2669+9.845·{10}^{-4}{P}_{max}-1.120·{10}^{-10}{P}_{max}^2$$

(4)

The motor efficiency, in general, is a function of angular speed and power. A motor has a design rpm in which the efficiency is higher and working at a different rpm means working inefficiently. Often, a gearbox is necessary, and its use comes from a compromise between efficiency and weight. In [57], there is a simplified relationship to make a preliminary estimate of the efficiency, knowing the power P, of a four-pole AC motor; this formula is shown in Equation (5).

$${\eta }_{em}=75.844+8.6059log\left({\displaystyle \frac{P}{1000}}\right)$$

(5)

Extensive research has been conducted to improve electric motor efficiency, such as the work by Yu [58] on PMSMs. Yu proposes an integrated optimization control method that combines loss-model-based efficiency optimization control methods (which rely on loss models of PMSMs [59]) with search-based optimization control methods (which continuously regulate the stator current to drive system efficiency toward the optimum). This approach involves first determining the optimal value of the d-axis current using the loss-model-based efficiency optimization method and then, utilizing this value, applying the search-based optimization control method with a narrowed search interval. This strategy improves convergence time and efficiency, overcoming the limitations of applying the two methods separately.

Motor efficiency affects not only power consumption but also heat generation. A less efficient motor dissipates more energy as heat, resulting in higher temperatures and potential performance issues, particularly in low-density environments. Electric motors designed for high-altitude applications must therefore achieve both high efficiency and effective cooling to dissipate heat. Some passive cooling solutions proposed by manufacturers include introducing cuts along the motor casing or open-frame designs (e.g., holes or fissures designed to promote airflow through the motor body) to enhance cooling performance. Several studies in the literature address the thermal response of motors operating in low-density environments. For instance, ref. [60] reports testing of a commercial off-the-shelf motor inside a large vacuum chamber capable of reaching high vacuum levels (${10}^{-10}$ atm). Results indicate that motor temperature increases with decreasing pressure and increasing angular velocity (power output), as expected. However, the motor remains adequately cooled throughout the simulations as long as the propeller continues spinning. Following shutdown, without the cooling effect of propwash, the motor temperature rises, leading to overheating. Depending on the mission profile, the most demanding condition for the motor may not be cruise but take-off. According to [61,62], power losses during take-off can exceed those during cruise, making heat dissipation critical in this phase and representing the greatest challenge for the thermal management system. An alternative approach proposed in the literature involves integrating heat exchangers within the motor design. Designing such heat exchangers for these applications is challenging due to numerous constraints, including requirements for lightweight construction, compact volume, high mechanical strength to support active motor components, low thermal resistance between heat source and sink, and effective integration of thermal management within the propulsion system.

In [61,62], two innovative heat exchangers with different surface areas, designed to be placed inside the stator inner bore of electric motors for HAPSs, were developed. These motors are intended to operate in environmental conditions ranging from 45 °C to −85 °C and pressures from 101 kPa to 5.5 kPa, corresponding to sea level and 20 km altitude, respectively. To meet the stringent requirements described above, the authors employed additive manufacturing techniques using powder bed selective laser melting, with an aluminum alloy (AlSi10Mg) as the powder material. Although this manufacturing method is unconventional for heat exchangers, it proved capable of fulfilling the design criteria. This technique enabled the authors to create heat exchangers featuring a very high active surface area-to-volume ratio and internal structures that would be unachievable with traditional manufacturing methods.

Propulsion device refers to elements like propellers and fans that, driven by the engine shaft, generate thrust. A detailed analysis of propellers for HAPSs is presented in the following chapters. An initial feasibility assessment of a HAPS, equipped with an electric propulsion system that exploits both solar and fuel cells to power an electric motor, has been conducted in [57]. The feasibility assessment would provide the viability for operating a HAPS, with also the goal of pointing out any limitations and constraints. The analysis takes in consideration the environmental model of the US cost and the state of the art of power and propulsion technologies (exploiting conservatives “off-the-shelf” values for most of the components exploited in the system) to determine the capabilities and limitations of a HAPS. The results of the analysis state that a HAPS equipped with this type of propulsion system could operate on West US cost continuously, using the technology that was available on market at the time in [57]. On East US coast, where the winds are stronger, the operation was possible during the whole year except in winter months. Therefore, at the time of [57], further technological advancements of the propulsion systems were necessary to support operations during the winter months. Nowadays this kind of propulsion is the most considered for HAPSs due to recent remarkable developments in the field of renewable energy sources [13,14].

3.3. Hybrid Propulsion

Hybrid propulsion for HAPSs refers to propulsion systems that combine more propulsion technology in the same aircraft to reduce the drawbacks related to each propulsion system type. A total electric aircraft could remain in flight for months, but it necessitates high aspect ratios and huge wingspans; this is due to the high specific weight of electric storage devices for night flight. An aircraft that exploits combustion engines, instead, could carry more payload mass, but it is unable to remain in flight for long time.

In [9], an investigation was conducted on the combination of various power sources and propulsion systems currently employed in HAPSs. Specifically, the authors first analyzed the performance of electric motors powered by solar photovoltaic arrays, hydrogen fuel cells, regenerative fuel cells, and rechargeable batteries, as well as turboshaft and turboprop engines fueled by hydrogen. Subsequently, they evaluated the performance of hybrid propulsion systems. Among the hybrid propulsion systems studied in [9] are configurations that use photovoltaic cells to power the electric motor during daytime flight, while relying on rechargeable batteries, fuel cells, regenerative fuel cells (which generate hydrogen from water during the day by utilizing excess solar power, and consume water and power during the night), or hydrogen-fueled turboshafts or turboprops for nighttime operation. At that time, prototype regenerative fuel cells were capable of delivering 400 Wh/kg, with expectations to reach 700 Wh/kg.

The analysis conducted by the authors in [9], which considered the requirements for a flight at 20 km altitude with a target endurance of one month, showed that the propulsion system mass represented approximately 10% of the total airship mass when using photovoltaic cells that provide energy only during daylight. This ratio increased to 14–24% for regenerative fuel cells, 21–36% for batteries, and 30–50% for polymer electrolyte membrane (PEM) fuel cells. The mass of the engine itself was not included in these estimates, but its impact was approximately 2.1%. At that time, the turboshaft or turboprop engine—potentially fueled by hydrogen—remained the preferred solution for high-altitude propulsion, offering a propulsion system mass to total mass ratio of about 28.5%.

Instead, for longer missions, combining an electric motor powered by solar cells during hours of daylight and a turboprop/turboshaft propulsion system exploited during hours of darkness could be better. This hybrid solution leads to important mass savings compared to an electric propulsion system and a longer endurance compared to a turboprop/turboshaft propulsion system. It is evident that, for long endurance (weeks or months), solar power during the day should be exploited; a long endurance flight will be impossible using consumable sources of energy. However, when going further away from the equator and particularly for arctic region flights, exploiting hydrogen-fueled turboshafts (or turboprops) both for day- and night-flight is mandatory.

Other feasibility studies on hybrid propulsion systems combining electric motors and combustion engines have been conducted in [54] with the aim of leveraging the advantages of both propulsion types, resulting in a more reasonably sized vehicle at the expense of some flight endurance. Additionally, as interface methods between the electric and combustion engines, the use of a clutch—which allows each engine to independently power the propeller—and the subsequent use of the combustion engine as an electricity generator to supply power to the electric motor alongside solar cells have been investigated. The feasibility assessment results indicate that a reasonably sized vehicle suitable for long-endurance (several days) stratospheric flight can be achieved by employing a hybrid propulsion system combining electric and combustion engines with a clutch. However, due to power losses, using the combustion engine to drive the electric motor during nighttime was found to be less effective. Moreover, hybrid propulsion demonstrated longer endurance compared to internal combustion engines alone, especially during summer, and a more practical HAPS size compared to a fully electric aircraft. Another type of hybrid propulsion system, a turbo-electric distributed propulsion system powered by hydrogen combustion, has been investigated in [63] for use on HAPSs carrying payloads with high electric power demands. Specifically, this system utilizes a lightweight gas turbine engine that burns hydrogen to generate electricity, which powers multiple electric motors and the payload. A small amount of thrust is also produced from the gas expelled through the turbo-generator’s nozzle. Applied to a suitably designed stratospheric aircraft, this system demonstrated good endurance capabilities (on the order of days), with slight endurance improvements as the number of fans increases, while simultaneously generating sufficient power for high-demand payloads.

3.4. Future Propulsion Concepts

Observing all the HAPS projects present in the literature, it can be seen that they are all designed to operate in the lowest part of the stratosphere. In order to operate at higher altitudes (more than 25–30 km), new types of vehicles with entirely new propulsion systems are required. Probably, propulsion systems to operate at these altitudes will not exploit propellers to generate thrust, unless more innovative materials are developed; propellers, in fact, are already characterized by large diameters for many proposed HAPSs and will scale larger with the increase in altitude. A promising propulsion technology, based on electric power, seems to be atmospheric ion propulsion. In [64], an initial feasibility assessment on this kind of propulsion was successfully carried, demonstrating its feasibility. Additionally, in [10], an overview of advanced concepts for high altitude propulsion is provided. According to the authors, the main propulsion systems candidate for these applications could be the electro-hydrodynamics propulsion systems, the radiometric force propulsion systems, and a thermal transpiration-based propulsion system. Electro-hydrodynamics propulsion systems generate thrust by ionizing and accelerating air flow through a high potential difference between two electrodes. They are possible candidates for stratospheric loitering applications for their light weight and simplicity. However, this kind of propulsion is characterized by a very low thrust efficiency. Despite studies being conducted to enhance its efficiency, there are still several concerns regarding the devices that must be addressed. Concerning the radiometric force, it is a force that arises when there is a temperature difference between a hot side and a cold side of a plate or vane that is immersed in a rarefied gas. Numerous theories suggest that relatively high forces per unit area could be produced, enough to support a high-altitude craft. A possible application is shown in [65], in which the temperature gradient, exploited to drive this force, generated by continuous beaming of microwave energy from a ground station impacting a lightweight carbon fiber sail, produces a sufficient lift force to maintain the sail in flight. The main drawbacks of this technology are that this force became very small at higher or lower altitudes than the designed altitude and the extremely high cost of maintaining a constant beam power to generate lift. In addition, nowadays, this mechanism is still not fully understood; a better comprehension of the physical mechanism of radiometric forces generation is necessary to assess the true potential of these forces for high altitude flight applications. The latter mechanism, thermal transpiration, is a physical phenomenon where a temperature gradient across a tube containing a rarefied gas results in gas motion. The gas flow moves from the colder side of the tube to the hotter part. The main advantage is that, instead of converting solar power into electrical energy and then to propulsive force, solar beams can directly power the system. The main limitations of this kind of propulsion are the low energy efficiencies, despite numerous efforts being made to improve it, and the too low thrust level for these applications. An additional propulsion system, based on breakthrough physics, described in [10], relies on antimatter. The antimatter annihilation mechanism has the highest energy density, and less than one milligram of this material could be sufficient to operate for months. However, nowadays, the low production rates and the energy efficiency of the current production process make the use of such technology impracticable. All the advanced propulsion concepts for HAPSs, just described, face substantial technological hurdles, necessitating significant advancements for their feasibility.

4. Propeller Theory for HAPSs

4.1. Development on Theoretical Background

Nowadays, as discussed in previous chapters, the propulsion system of HAPS platforms typically employs propellers. The low atmospheric density in the stratosphere, which at 20 km altitude is approximately 1/14 of the air density at sea level, coupled with low flight speeds (10–30 m/s), results in HAPS platforms operating at low Reynolds numbers and advance ratios (depending on rpm and propeller diameter) [66,67,68,69]. Additionally, the lower temperature increases the Mach number, especially at the blade tip. Consequently, numerous studies have focused on optimizing propellers for HAPS platforms. The development of propeller theory dates back to the 19th century with contributions from Rankine and Froude [70,71]. Drzewiecki introduced Blade Element Theory (BET) in [72], although his approach did not consider the induced velocity of the propeller. The method aimed at minimizing induced losses was initially proposed by Betz and Prandtl [73] and further developed by Goldstein [74]. Betz and Prandtl established criteria for minimizing energy loss, primarily focusing on propellers operating under light loads, while Theodorsen in [75] demonstrated that Betz’s condition could also be applied to propellers with heavy disk loadings. Glauert formulated an estimation of the airflow around helicoidal vortex sheets, which provides more accurate predictions for propellers with lower advance ratios and becomes increasingly effective as the number of blades increases [76].

Integrating momentum theory, BET and Vortex Theory, Larrabee in [49,77] introduced a practical design theory for propellers with minimal induced loss. The aerodynamic design of propellers is still primarily conducted using two-dimensional methods, such as BEMT or Vortex Theory, even for propellers operating under high-altitude conditions [51,67,69,78,79,80,81,82,83,84,85]. BEMT enables rapid evaluation of a propeller when planform parameters and airfoil cross-section characteristics are known. Furthermore, given specific flow conditions, airfoil characteristics, and thrust requirements, BEMT facilitates rapid inverse design of a propeller’s planform, including chord length and pitch angle distribution, based on Minimum Induced Loss (MIL) criteria. However, designs based on BEMT often lack high precision, leaving room for performance improvement. To enhance propeller aerodynamic efficiency, optimization techniques have been applied to blade shape design. Liu et al. [67] refined chord length and pitch angle distribution using NSGA-II and a multi-island genetic algorithm (MIGA) based on BEMT, achieving a lightweight and efficient propeller. Ning [80] improved BEMT accuracy by correcting its residual sequence and optimizing chord length and pitch angle through gradient-based methods. Marinus [68] performed multi-stage optimization for high-altitude pseudo-satellites, considering factors such as airfoil type, diameter, planform shape, and thickness, revealing important trends in propeller design.

However, BEMT struggles to capture three-dimensional rotational effects like radial flow and root and tip influences, particularly at low Reynolds numbers. Its accuracy depends heavily on blade element aerodynamic coefficients, limiting computational precision. Thus, BEMT is primarily suitable for initial optimization and low-fidelity evaluations. More precise assessment methods are necessary to further enhance propeller efficiency. With the advancement of numerical simulation tools, the CFD method has become an essential aid in optimizing propeller design [86,87,88,89,90]. Compared to BEMT, CFD-based numerical simulations provide more accurate evaluations of propeller aerodynamic forces, thereby enhancing design precision. Kou et al. [91] proposed a propeller design optimization approach using Bayesian optimization. This method collects samples through CFD simulations, constructs a Gaussian process model, and applies a genetic algorithm to optimize the chord length and twist angle distribution of the propeller. Yao [45] performed high-fidelity CFD aerodynamic optimization of a propeller incorporating a Gurney flap. While high accuracy is critical, the direct use of CFD for propeller optimization is computationally expensive due to complex flow dynamics. Gradient-based optimization methods can reduce CFD computational costs [86], but they are susceptible to convergence to local optima. To improve both accuracy and efficiency, CFD can be used to enhance BEMT. Guo et al. [92] used CFD to optimize pitch angle and aerodynamic forces before applying BEMT for inverse design. A multi-stage method was proposed in [93], employing BEMT for initial design and CFD for airfoil optimization. Another approach involves decoupling the design process, as demonstrated by [94], who used discrete adjoint methods and flow field reconstruction to optimize pitch angle and chord distribution. However, although decoupling may improve efficiency, it often fails to achieve global optimization.

Most studies focus on optimizing propellers using BEMT supplemented by CFD to improve design accuracy, yet CFD-based methods remain less accurate than high-precision CFD simulations. In [95], a multi-fidelity approach combining high-fidelity CFD simulations with BEMT to accelerate high-precision design is proposed. This approach uses a multi-fidelity data fusion model, based on a small number of high-fidelity and a larger number of low-fidelity samples, to improve computational efficiency and optimize propeller blade designs. Models such as random forests [96] and multi-fidelity Kriging [97] have been applied in aerospace, achieving better results than single-fidelity models. This strategy leverages the strengths of BEMT for low-fidelity data and CFD for high-fidelity data, aligning with practical requirements of multi-fidelity fusion models.

Regarding the evaluation of propeller performance, the aerodynamic characteristics of blade elements are generally computed using coupled viscous-inviscid solvers such as XFOIL [98]. While these methods offer significant computational efficiency, they exhibit limitations in modeling complex aerodynamic phenomena, including the prediction of laminar separation bubbles, transition from laminar to turbulent flow on three-dimensional surfaces, and the occurrence of radial crossflow [99]. Numerous studies employ numerical simulations to calculate propeller performance, frequently utilizing the $k-\omega$ shear-stress transport turbulence model or the $k-ϵ$ turbulence model [51,85,100,101,102].

In Nigam’s work [103], Vortex Theory-based methods often yield accurate performance predictions. However, performance evaluations of propellers operating at high altitudes are subject to increased uncertainty due to the scarcity of experimental or flight data and the challenges associated with modeling low Reynolds number flows [104]. Consequently, Nikolaos et al. [104] employed the Polynomial Chaos Expansion (PCE) method to quantify performance uncertainty for propellers in high-altitude conditions. For completeness, the following sections present the BEMT and Vortex Theory.

4.2. Blade Element Momentum Theory

The Blade Element Momentum Theory, well described in [105], operates under the same assumptions as Blade Element Theory, where the blade is divided into finite segments called blade elements, and the aerodynamic characteristics of each element are calculated using principles derived from airfoil theory [66], while also incorporating insights from Momentum Theory. Generally, BEMT accounts for the effect of propeller rotation on the induced angle of attack and provides an efficient approach for propeller design and analysis [106,107], with minimal computational cost during the early design stages. However, it cannot accurately capture the three-dimensional aerodynamic effects of propeller rotation, which leads to relatively lower accuracy [68,95,108].

In [109], a method for designing an optimum propeller is presented, where the search for the optimal propeller involves solving a constrained minimization problem aimed at minimizing power consumption while maximizing thrust by applying BEMT. This optimization problem has been addressed by various authors in the literature [109,110]. By combining the Momentum Theory equations with those from the BET, it is possible to derive the BEMT formulation, as thoroughly explained in [111]. From the BET, the thrust and torque distributed over each annular section $dr$ are expressed in Equations (6) and (7). The fundamental forces acting on the airfoil are illustrated in Figure 7. Here, $V_{\infty }$ represents the forward velocity of the propeller, and $N_b$ is the number of blades. $\Omega$ denotes the angular velocity of the propeller. The pitch angle $\theta$ is defined relative to the zero-lift line of the airfoil section and varies with the radial position r. The advance angle is $\varphi$, and $V_e$ is the total relative velocity of the airfoil section, calculated from the velocity triangle in Figure 7 as $[V_e = \sqrt{{\left(V_{\infty }(1+a)\right)}^2+{\left(\Omega r(1-a^{\prime })\right)}^2}]$ where a is the axial induction factor and $a^{\prime }$ is the tangential induction factor. The chord length of the propeller blade section is denoted by c, and $\Omega$ is the angular velocity.

$$dT=N_b{\displaystyle \frac{1}{2}}\rho V_e^2(C_{l}cos\varphi -C_{d}sin\varphi )cdr$$

(6)

$$dQ=N_b{\displaystyle \frac{1}{2}}\rho V_e^2(C_{l}sin\varphi +C_{d}cos\varphi )crdr$$

(7)

By imposing that $C_n = C_{l}cos\varphi -C_{d}sin\varphi$ (normal force coefficient) and $C_a = C_{l}sin\varphi +C_{d}cos\varphi$ (axial force coefficient) and introducing solidity defined as $\sigma = {\displaystyle \frac{N_{b}c}{2\pi r}}$, the final expressions utilized in the BEMT for thrust and torque are obtained in Equations (8) and (9).

$${\displaystyle \frac{dT}{dr}} = \sigma \pi C_{n}r\rho V_e^2$$

(8)

$${\displaystyle \frac{dQ}{dr}} = \sigma \pi C_a{r}^2\rho V_e^2$$

(9)

By equating the thrust and torque relations provided by the BET theory (Equations (6) and (7)) with those obtained from the Momentum Theory, $dT = 4\pi r\rho V_{\infty }^2(1+a)a dr$, $dQ = 4\pi r^3\rho V_{\infty }(1+a)a^{\prime }\Omega dr$, $dP = 4\pi \rho \left[V_{\infty }^3{(1+a)}^{2}ar dr+{\Omega }^2{V}_{\infty }(1+a)a^{\prime 2}{r}^3 dr\right]$ using the conservation of axial momentum to the stream tube swept out by each radial blade element [83,112], and taking into account the relationships derived from the velocity triangle, $V_e^2 = {\displaystyle \frac{V_{\infty }^2{(1+a)}^2}{{sin}^2\varphi }}+{\displaystyle \frac{{\Omega }^2{r}^2{(1+a^{\prime })}^2}{{cos}^2\varphi }},$ it is possible to derive the fundamental expressions for the axial and rotational induction factors, denoted as a and $a^{\prime }$ (Equations (10) and (11)).

$$a = {\left[{\displaystyle \frac{4{sin}^2\varphi }{\sigma C_n}}-1\right]}^{-1}$$

(10)

$$a^{\prime } = {\left[{\displaystyle \frac{4sin\varphi cos\varphi }{\sigma C_a}}+1\right]}^{-1}$$

(11)

The impact of induced velocity within the propeller plane is most significant near the blade tips. However, classical BEMT does not account for the influence of vortices shed from the blade tips into the slipstream, which affect the induced velocity distribution. To address this limitation, a correction factor known as the momentum loss factor F, originally introduced by Prandtl [73], is commonly applied (Equation (12)), where f is defined by Equation (13). ${\varphi }_t$ represent the flow angle of the blade tip and R denotes the radius of the propeller.

$$F = {\displaystyle \frac{2}{\pi }}arccos\left(e^{-f}\right)$$

(12)

$$f = {\displaystyle \frac{N_b}{2}}\left({\displaystyle \frac{R-r}{Rsin\left({\varphi }_t\right)}}\right)$$

(13)

In practical applications, the performance of a propeller is commonly assessed using nondimensional coefficients such as thrust coefficients $C_T$, torque coefficients $C_Q$, power coefficients $C_P$, as well as propeller efficiency $\eta$ (Equations (14)–(17)). These parameters are well established in the literature and facilitate meaningful comparisons across different designs and operating conditions [46,76].

$$d{C}_T = {\displaystyle \frac{{\pi }^3}{4}}\sigma C_n{\left({\displaystyle \frac{r}{R}}\right)}^3{\displaystyle \frac{{(1-a^{\prime })}^2}{co{s}^2\varphi }}d\left({\displaystyle \frac{r}{R}}\right)$$

(14)

$$d{C}_Q = {\displaystyle \frac{{\pi }^3}{8}}\sigma C_a{\left({\displaystyle \frac{r}{R}}\right)}^4{\displaystyle \frac{{(1-a^{\prime })}^2}{co{s}^2\varphi }}d\left({\displaystyle \frac{r}{R}}\right)$$

(15)

$$C_P = 2\pi C_Q$$

(16)

$$\eta = {\displaystyle \frac{J}{2\pi }}{\displaystyle \frac{C_T}{C_Q}}$$

(17)

However, classical BEMT exhibits several fundamental limitations when applied to propellers operating in the HAPS regime. A first and major limitation arises from the strong dependence of airfoil aerodynamic characteristics on Reynolds number in this regime. At low Reynolds numbers (${10}^4$–${10}^5$), phenomena such as laminar separation bubbles, delayed transition to turbulence, and early flow separation significantly alter the lift and drag behavior of airfoils. Consequently, the aerodynamic polars exhibit sharp variations with Reynolds number, unlike in higher-Reynolds conditions where they tend to be relatively insensitive. BEMT does not generally account for the substantial spanwise variation in the Reynolds number along the blade. Even if the same airfoil geometry is used throughout the span, the local Reynolds number, governed by the local relative velocity and chord length, can vary by more than an order of magnitude from root to tip. Because low-Reynolds airfoil characteristics are so sensitive to the Reynolds number, each section effectively operates with a different aerodynamic polar. Classical BEMT formulations typically ignore this, applying a single representative polar to the entire blade, which introduces significant errors in the estimation of aerodynamic forces. Consequently, applying BEMT without incorporating spanwise-resolved aerodynamic data may result in sub-optimal performance estimates and hinder the identification of a truly optimized blade geometry. As shown in [108], neglecting the influence of Reynolds number variation can lead to an overestimation of thrust and efficiency by up to 25% under typical stratospheric conditions. Moreover, BEMT inherently neglects viscous flow effects, relying on an inviscid formulation that assumes the forces on each blade section depend solely on local flow angles and circulation. At low Reynolds numbers, viscous phenomena dominate the flow behavior, affecting boundary layer development, delaying or preventing reattachment after separation, and increasing overall drag. Such effects fall entirely outside the scope of classical BEMT, compromising its ability to accurately predict thrust and power requirements in these regimes. The theory also employs simplified models to account for tip and root losses, most commonly through Prandtl’s tip loss factor. These corrections were derived under assumptions valid primarily for high-Reynolds, inviscid, attached flow and do not reliably capture the diffused, viscous-dominated tip vortices or altered induction seen at low Reynolds numbers. Taken together, these limitations highlight that, while BEMT can still provide useful first-order estimates for low Reynolds number propeller performance, it must be applied with caution. To mitigate these issues, an enhanced BEMT approach was proposed in [112], validated for Reynolds numbers between $4\times {10}^4$ and $8\times {10}^4$. Their method integrates locally computed airfoil polars using XFOIL for each blade section and operating point, ensuring that aerodynamic input reflects the actual operating conditions. Additionally, it incorporates a rotational stall delay correction by scaling the lift coefficient using a function of local blade radius and inflow angle: $C_L = C_l·f_{sd}(r,\varphi )$ where $f_{sd}(r,\varphi )$ represents a stall-delay factor that captures the increased lift observed near the root due to centrifugal and Coriolis effects in rotating systems. To extend the applicability beyond stall, the method also blends XFOIL-derived pre-stall curves with empirically constructed post-stall extensions. The transition is implemented linearly over a ${10}^{\circ }$ interval centered at the stall angle, ensuring both numerical robustness and aerodynamic continuity. This treatment is particularly important for propellers with high pitch angles, which may operate at low advance ratios or in static thrust conditions. In such cases, local blade sections can experience angles of attack well beyond the linear regime, even reaching deep post-stall behavior. Accurate modeling in this region is essential for reliable thrust and torque prediction. When direct XFOIL computations fail to converge at high angles of attack—often beyond ${15}^{\circ }$—extrapolation techniques become necessary to complete the polar curves. One established method is that of Montgomerie [113], which analytically extends lift and drag coefficients based on geometric and asymptotic assumptions. Another widely used approach is the Viterna–Corrigan model [114], originally developed for wind turbines, which constructs smooth and physically consistent polar extensions using empirical formulations. These models are commonly integrated into BEMT codes to ensure stability and accuracy when simulating propellers under extreme inflow conditions.

4.3. Solidity

The solidity, also known as Activity Factor (AF), is a critical parameter in the design of a propeller, particularly for applications involving high-altitude platforms. The AF is a dimensionless number representing the ratio of the blade area to the disk area of the propeller. Selecting an appropriate AF ensures that the propeller operates at optimal efficiency. If the AF is too low, the propeller blades might be too narrow, resulting in insufficient thrust. Conversely, a very high AF might cause excessive drag and power loss. For evaluating the performance of high-altitude pseudo-satellites, it is important to note that high-altitude platforms, unlike conventional aircraft, should work in a wide range of altitudes, and so in a wide range of densities. A propeller designed with an appropriate AF can maintain better performance under varying air densities, ensuring consistent thrust and efficiency at different altitudes. In [115], wind tunnel tests were conducted on five propellers with different blade tip widths to explore the effects of blade planform and advance ratio on both power coefficient and propulsive efficiency. The aim was to identify the optimal blade planform for achieving high thrust and a low advance ratio. Tests were conducted on five distinct propellers with varied blade planforms, assessing pitch angles ranging from ${20.5}^{\circ }$ to ${40}^{\circ }$ while maintaining specific geometric parameters and AFs as outlined in [115,116]. In that study, a Reynolds number of $1.5\times {10}^5$ for the propeller, based on the chord length at $r/R=0.7$, has been imposed. The AF increases with the widening of the propeller tip as shown in Figure 8. Analysis of the characteristic curve of this propeller with respect to the power coefficient ($C_P$) shows that the propeller with the highest AF yields a greater power coefficient, as illustrated in Figure 9.

A comparison of the max efficiency value and power coefficients of blades with different AF has been illustrated in Figure 10 and Figure 11. The main characteristics of the described propellers are shown in

Table 2. Characteristics of different propellers designed for various HAPS operations [115,116,117,118,119,120,121].

Propeller	AF	D (m)	${\mathit{N}}_{\mathit{b}}$	${\mathit{\eta }}_{\mathit{max}}$
Propeller 1	43.4	0.85	2	0.87
Propeller 2	70.0	0.85	2	0.85
Propeller 3	96.6	0.85	2	0.83
Propeller 4	123.2	0.85	2	0.83
Propeller 5	149.8	0.85	2	0.85

. One important result from this study, as shown the figures mentioned before, is that, for propellers operating at a low advance ratio ($J<0.8$), a blade with a narrower tip is advantageous for propulsive efficiency, although it limits power performance. Conversely, for propellers with an advance ratio exceeding 0.8 ($J>0.8$), those with wider tips exhibit superior power absorption capabilities while maintaining acceptable propulsive efficiency. The variation in the efficiency with blade planform and advance ratio is determined by the radial distributions of airflow angle and interference angle, which directly affects the efficiency of the blade elements. At low advance ratio, the low airflow angle and high interference angle of the blade element can result in reduced blade element efficiency, thereby decreasing the overall propulsive efficiency of the propeller. As the advance ratio increases, although the higher interference angle still causes the blade element efficiency to decrease, the increase in airflow angle could enhance the efficiency of the blade elements, leading to an overall improvement in propulsive efficiency.

4.4. Vortex Theory

Vortex Theory offers a more comprehensive framework for analyzing the aerodynamic performance of rotating propellers by explicitly modeling the circulation bound to the blades and the helical wake shed downstream. This approach accounts for the three-dimensional induced velocity field created by the rotation of the blades and the finite span effects, which play a critical role near the blade tips. Unlike classical BEMT, which combines simplified one-dimensional momentum balances with local two-dimensional blade element analysis, Vortex Theory directly relates the distribution of circulation along the blade to the induced velocities in the surrounding flow. This leads the determination of the induced angle of attack and, consequently, more accurate predictions of local aerodynamic forces. For its ability to capture the effects of bound circulation and trailing vortices, Vortex Theory remains a widely cited framework for preliminary propeller sizing and first-order performance estimates. However, at the low Reynolds numbers typical of high-altitude applications, it is usually complemented by CFD analyses during the design process. With reference to Figure 12, the induced angle of attack $a_i$, results from the induced velocity w, which is the vector sum of the tangential and axial components of induced velocity, $w_t$ and $w_a$ (Equation (18)).

$$a_i = arctan\left({\displaystyle \frac{w_t}{w_a}}\right)-\varphi$$

(18)

The Vortex Theory yields the expression for the total circulation around the propeller as given by Equation (19).

$$N_b\Gamma = 4\pi r{w}_t$$

(19)

By equating the expression derived from the Kutta–Jukowsky theorem for a blade element (Equation (20)) with the aforementioned expression (Equation (19)) and incorporating the solidity term $\sigma$, the equation for the tangential velocity, denoted as $w_t$, is derived (Equation (21)).

$$l = \rho V_e\Gamma$$

(20)

$$w_t = {\displaystyle \frac{\sigma }{4F}}{V}_e{C}_l\left(\theta -\varphi -a_i\right) = {\displaystyle \frac{\sigma }{4F}}{V}_e{C}_l\left(\theta -arctan\left({\displaystyle \frac{w_t}{w_a}}\right)\right)$$

(21)

To determine the axial velocity as a function of the tangential velocity, it is necessary to equate the thrust derived from the BET with that from the general impulsive theory. This process involves obtaining the relationships between the axial and rotational interference factors (Equation (22)).

$${\displaystyle \frac{1+a}{{\chi }^2}} a = a^{\prime }(1-a^{\prime })$$

(22)

where $\chi$, a, and $a^{\prime }$ are defined as $\chi = {\displaystyle \frac{\Omega r}{V_{\infty }}}$, $a = {\displaystyle \frac{w_a}{V_{\infty }}}$, and $a^{\prime } = {\displaystyle \frac{w_t}{\Omega r}}$. By substituting these relations into Equation (22), a relation between the two components of the induced velocity is obtained in Equation (23), which can be solved for $w_a$ as a function of $w_t$, as defined in Equation (24).

$$(V_{\infty }+w_a)w_a = w_t(\Omega r-w_t)$$

(23)

$$w_a = {\displaystyle \frac{1}{2}}\left[-V_{\infty }+\sqrt{V_{\infty }^2+4w_t(\Omega r-w_t)}\right]$$

(24)

The calculation process is discussed in [85]. After determining $w_a$ and $w_t$ through an iterative process (for example, solving the implicit equation given in Equation (21) by applying Newton’s method to find the zeros of the function), it is possible to obtain the induced angle $a_i$ using Equation (18). The estimation of $a_i$ allows for a more accurate determination of the dimensionless force coefficients compared to those obtained with classical BEMT. This is achieved by properly modifying the BEMT equations to include $a_i$.

5. Propeller Design for HAPS Operation

5.1. Design Issues for High Altitude Propellers

Several challenges are associated with the design of a propeller for high-altitude platforms. The primary concern is the selection of the propeller rotational speed (rpm), which is limited by considerations related to the Mach number at the propeller tip. Typically, the Mach number at the propeller tip is maintained between 0.65 and 0.75 to prevent the formation of shock waves on the blade.

The secondary and most critical challenge, as previously mentioned, concerns the operating environment. Due to the high altitude, airfoils operate under low Reynolds number conditions (from ${10}^4$ to ${10}^5$). This flow regime is characterized by the onset of laminar-to-turbulent transition, where viscous effects dominate the aerodynamic behavior. Compared to high Reynolds number conditions, several phenomena become more pronounced: the lift-to-drag ratio decreases significantly after reaching its peak due to the development of thicker boundary layers; symmetric airfoils may exhibit nonlinear aerodynamic responses even at small angles of attack; and flow separation along the blades can generate either short or extended laminar separation bubbles. Moreover, the flow becomes increasingly sensitive to external disturbances such as free-stream turbulence, which increases drag and degrades overall propeller performance [36,37,38].

The laminar separation bubble is a primary factor contributing to the reduction in propeller efficiency at low Reynolds numbers [39]. Designing an effective propeller requires mitigating the detrimental effects of laminar separation bubbles on the blades [40]. Moreover, at low Reynolds numbers, the variation in Reynolds number along the propeller blade can become significant due to the increasing linear velocity from the hub to the tip, especially if the airfoil chord does not decrease consistently. This variation, which is negligible at high Reynolds numbers, is typically not considered in the design of conventional propellers. In fact, during the design phase of a conventional aircraft propeller operating in a high Reynolds number regime, constant airfoil characteristics such as lift and drag curves are used for each propeller section, derived from an average constant Reynolds number [110]. Consequently, applying traditional design methods to HAPSs will not yield optimal aerodynamic performance. A poorly optimized design may cause unnecessary drag, increase demands on the power system, and ultimately compromise the mission feasibility.

Additionally, to efficiently design a propeller, it is crucial to account for the variation in Reynolds number with wind speed. Although wind intensity is minimal at the altitudes where HAPSs operate [122], wind speed remains a critical factor. It affects not only the thrust required for station keeping but also the Reynolds number experienced by each blade section and the relative inflow velocity at the propeller disk, which combines the vehicle’s forward speed with any ambient wind along the flight direction. Therefore, analyzing the wind characteristics in the mission region is essential.

5.2. 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 ${\eta }_v$, 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 Theodorsen [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 [112,124,126]. 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 [112,127] 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 Theodorsen’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. 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 drag coefficient data ($C_l$ − $\alpha$ 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).

$$C_L(\alpha +\Delta \alpha ) = C_l\left(\alpha \right)+\left({\displaystyle \frac{\partial C_l}{\partial \alpha }}\right)\Delta \alpha$$

(25)

In this equation, ${\displaystyle \frac{\partial C_l}{\partial \alpha }}$ represents the lift curve slope in the linear region of the lift. 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. 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 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 Park 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. 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 $20 \mathrm{km}$, with an airspeed of $32 \mathrm{m}/\mathrm{s}$ and a thrust of $20 \mathrm{N}$ (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.

Another type of approach, in order to include CFD in the design process, is proposed in [94]; the authors present a high-fidelity multi-level optimization framework for the propeller, which is characterized by direct optimization using CFD. They propose a series of criteria to quantify the influence of shape changes on the viscosity loss and induced loss of the propeller, which can avoid the consumption of computing power at unnecessary design points and thus keep the computational cost within an acceptable range without sacrificing the computational accuracy. First of all, the optimization result, based on a variable-fidelity model based on BEMT and CFD, is taken as the baseline propeller [145,146,147]. Subsequently, the discrete adjoint method is used to optimize the airfoil cluster in a three-dimensional state. Next, parametric perturbation and flow pattern reconstruction techniques are employed to refine the chord length distribution and twist angle. To assess viscosity losses, the concepts of perturbation loss and additional efficiency are introduced, enabling a more precise evaluation of chord length optimization at each blade section. The optimization accuracy is verified through CFD simulations, which closely align with experimental data. By effectively balancing high accuracy with low computational costs, this approach optimizes the propeller shape while accounting for complex three-dimensional aerodynamic effects. Ultimately, the methodology provides valuable insights for designing efficient propellers for high-altitude, long-endurance UAVs. In this study, the operational conditions considered are an altitude of $15 \mathrm{km}$, a constant undisturbed flow velocity of $17 \mathrm{m}/\mathrm{s}$, and a thrust of approximately $13 \mathrm{N}$ per propeller. The characteristics of the propeller are as follows: $D = 1.4 \mathrm{m}$, $N_b = 2$, rpm = 1150. The optimization resulted in a total propeller efficiency of 79.29%, 2% higher than that of a baseline propeller designed, under the same conditions, using the conventional variable-fidelity methods described before.

In propeller design, both efficiency and weight are critical factors in improving the endurance or payload capacity of a HAPS. This topic is investigated in [97], where a parallel multi-fidelity, multi-objective optimization framework is developed. The framework combines advanced metamodeling techniques and acquisition functions to efficiently handle large datasets while balancing computational cost and accuracy, aiming to optimize propellers for high-altitude operations. The framework targets two main objectives: the aerodynamic performance of the propeller and the weight of the propeller blades, specifically their total volume, without specifying the material. Both objectives significantly influence the endurance of unmanned aerial vehicles. A notable innovation is the Deep Mean Subset of Data (DMSoD), a novel Kriging-based model that surpasses traditional Sparse Gaussian Process models in processing high-dimensional datasets. For acquisition, the framework employs the Variable Fidelity Expected Improvement Matrix (VFEIM), which effectively selects designs by optimizing the trade-off between cost and accuracy across various fidelity levels. This approach proves particularly beneficial when fidelity cost ratios are high, such as those between Vortex Theory and 3D RANS simulations. Additionally, a three-level Hierarchical Kriging model integrates evaluations from Vortex Theory and two types of 3D RANS simulations (first- and second-order upwind schemes), enhancing optimization robustness.

A batch-based infill strategy outperforms sequential design selection, accelerating the identification of high-performance solutions. Nevertheless, challenges occur in high-fidelity simulations, as many designs fail to converge using the second-order upwind scheme. Specifically, the authors report that 50% of the cases evaluated with second-order discretization exhibited numerical instabilities or divergence issues, highlighting the practical limitations of high-fidelity methods in large-scale design exploration. Despite these difficulties, the framework successfully identifies propeller designs with efficiencies between 80% and 85%, illustrating the benefits of multi-fidelity modeling in optimizing aerodynamic performance while managing computational costs. The optimization is conducted under design conditions representative of high-altitude cruise, namely a flight altitude of $16 \mathrm{km}$, a freestream velocity of $55 \mathrm{km}$, and shaft power availability ranging from 11 to $30 \mathrm{kW}$. The resulting designs span diameters up to $3.0 \mathrm{m}$, with rotational speeds constrained according to a maximum tip Mach number of 0.8. The integration of DMSoD, VFEIM, and Hierarchical Kriging increases the reliability and efficiency of the optimization process, providing a powerful tool for the development of high-altitude propellers.

An innovative method that integrates Bayesian optimization, Vortex Theory, and 3D RANS simulations is presented in [141] to maximize efficiency during cruise (i.e., minimize energy consumption for the required thrust) while simultaneously minimizing volume to reduce weight and aerodynamic drag. The study considers operational conditions at altitudes of $16 \mathrm{km}$ and $20 \mathrm{km}$, with a constant undisturbed flow velocity of $55 \mathrm{km}$ and shaft power availability ranging from 11 to $30 \mathrm{kW}$.

The proposed method identified blade designs achieving up to 86% cruise efficiency at $16 \mathrm{km}$ and 83% at $20 \mathrm{km}$, which are highly promising results for the design of stratospheric propellers. The efficiency curves, as a function of the advance ratio, of the optimized propeller, derived from CFD simulations using 3D RANS with the $\gamma$-$R{e}_{\theta }$ transition model, are presented in Figure 15.

With the advent of artificial intelligence and neural network-based techniques, modern design methodologies are increasingly leveraging these technologies to optimize complex aerodynamic components such as propellers. One such example, although not originally developed for HAPS, is a recently proposed methodology based on Multi-Fidelity Neural Networks (MFNNs), which integrates low-fidelity BEMT data with high-fidelity CFD aerodynamic data [95]. In particular, the methodology was applied for the design of a propeller for an electric aircraft operating at $28 \mathrm{m}/\mathrm{s}$, with a thrust of $190 \mathrm{N}$ under cruise conditions. A comparison between a baseline propeller, obtained using BEMT, and the optimized propeller is presented through the efficiency trends shown in Figure 16. These trends clearly indicate that the MFNN model effectively captures the nonlinear relationship between low-fidelity BEMT and high-fidelity CFD, improving aerodynamic predictions with minimal computational cost. As a result, the optimized propeller features increased chord length at the root and tip, along with a reduced pitch angle. This configuration helps to minimize tip vortex intensity and enhances efficiency both at cruise conditions (design point) and at higher rotational speeds. Furthermore, the MFNN model demonstrates significantly higher accuracy compared to single-fidelity surrogate models. Performance predictions were derived using CFD with a 3D RANS $k-\omega$ SST turbulence model. At the design point, the propeller thrust is $190 \mathrm{N}$. When the advance ratio falls below 0.32, corresponding to relatively high rotational speeds, the optimized propeller exhibits higher efficiency over a wide range of operating conditions, accompanied by a slight increase in thrust.

Some common findings from the optimizations methodologies for HAPS propellers illustrated before are as follows: the diameter in every optimization consistently converges toward the maximum allowable value, so once clearances are determined, it is recommended to use the largest possible diameter. Regarding chord distribution, the results show low values at the root, peak values in the 60-80% span range, and low values at the tip. Additionally, it has been found that properly increasing the camber of the sectional airfoil can improve the aerodynamic efficiency of the propeller at low Reynolds numbers [40]. An innovative design aimed at further enhancing propeller efficiency is presented in [45], involving the addition of a Gurney flap to the propeller airfoil. A Gurney flap is a small, flat plate mounted vertically at the trailing edge of the airfoil, typically with a height between 0.5% and 5% of the chord. Its primary function is to increase lift by increasing pressure on the lower surface of the airfoil. The design process described by the authors is iterative and consists of several steps. Initially, design parameters such as free-stream velocity, propeller radius, blade number, required thrust, rotation speed, and initial airfoil data are established. The initial chord and pitch distributions are then determined using the Betz method. Subsequently, the cross-sectional airfoils with a Gurney flap are optimized using MIGA, where the airfoils are parameterized and iteratively optimized to maximize the lift-to-drag ratio. The chord and pitch distributions of the propeller are also optimized using MIGA, focusing on maximizing efficiency while meeting thrust requirements. The optimization considers Reynolds number effects through iterative updates based on CFD and BEMT results. CST parameterization is used to define the airfoil shape, and each design iteration is validated against aerodynamic constraints. After each iteration, convergence is assessed by comparing efficiencies between iterations. If convergence is not achieved, the airfoil constraints are updated based on CFD data using the BEMT method, and the process repeats until a final optimal solution with maximum efficiency is obtained. The optimal Gurney flap height was found to be between 1% and 2% of the chord, depending on airfoil geometry and operating Reynolds number. The CFD model used in this study was validated against wind tunnel data, showing agreement within 4% for thrust and surface pressure distributions, confirming its accuracy for low-Reynolds-number applications.

Figure 17 and Figure 18 show the blades obtained from the optimization process and the corresponding efficiency results, comparing the optimized blade without a Gurney flap and the blade equipped with a Gurney flap, at an average Reynolds number of $4\times {10}^4$, supporting the reliability of the proposed design approach.

As shown in Figure 18, the optimized propeller equipped with a Gurney flap consistently outperforms the configuration without the flap across a range of advance ratios. The maximum efficiency increases from approximately 80.2% to 82.0%, with a simultaneous reduction in power consumption. At low Reynolds numbers (${10}^4$ or less), provided that the three-dimensional separation bubble can be accurately predicted, the Gurney flap significantly enhances the lift coefficient of the airfoil with minimal drag penalty by effectively increasing camber and limiting the growth of laminar separation bubbles. This improvement leads to a higher ratio of lift to drag for the airfoil and consequently increases the overall propeller efficiency. Furthermore, the aerodynamic performances of the two propellers under higher Reynolds numbers have been calculated in the study. However, as the Reynolds number increases, the efficiency gains from the Gurney flap diminish and become negligible at Reynolds numbers above approximately $1.4\times {10}^5$ (Figure 19). This behavior occurs because laminar separation bubbles, which the Gurney flap helps control, naturally decrease with increasing Reynolds numbers, reducing their adverse effects. Once these bubbles nearly vanish, the Gurney flap no longer provides benefits in terms of camber increase or separation prevention. Instead, the additional drag introduced by the flap slightly reduces the ratio of lift to drag for the airfoil, leading to a decrease in propeller efficiency at higher Reynolds numbers.

In order to further improve the design methodologies described previously, it is crucial to consider wind speed and its effects on the Reynolds number during the design phase. One possible approach, proposed in [84], involves performing a statistical study of wind conditions at the mission location to obtain a probabilistic wind distribution. This distribution can be discretized, and a cost function, defined as a weighted sum based on the probability of each wind value, can be maximized. In Ref. [148], an algorithm based on generalized polynomial chaos expansion theory, which utilizes wind intensity probability density functions (PDFs), is proposed to convert the stochastic design problem into a deterministic one. This approach demonstrated, in the authors’ case study, that stochastic design improves mean net propulsion efficiency by approximately 5% compared to a classical deterministic design. Finally, a fundamental aspect to consider is the implementation of a mechanism that allows in-flight adjustment of propeller pitch. During the design phase of a propeller for a stratospheric aircraft, assessing the need for a variable-pitch propeller is important to enhance the overall propulsion system performance. When a propeller optimized for high-altitude operation is used at lower altitudes, where air density is significantly higher, the maximum rotational speed is limited by the maximum torque constraint. This leads to reduced shaft power and thrust. A widely used solution to enhance the compatibility between the propeller and electric motor across varying altitudes is the implementation of a variable-pitch propeller. A study detailed in [149] compares the aerodynamic performance of a fixed-pitch propeller—optimized through surrogate-based methods for operation at 25 km altitude, 40 m/s wind speed, and 1200 rpm rotational speed—with variable-pitch propellers analyzed via RANS solvers.

Table 3. Percentage performance differences of a variable-pitch propeller compared to a fixed-pitch propeller at various altitudes [149].

Altitude (km)	Wind Speed (m/s)	rpm	Shaft Power	Max Thrust	Efficiency
20	25	+43.54%	+42.35%	+36.47%	−4.06%
15	15	+112.17%	+112.11%	+75.06%	−17.48%
10	10	+139.89%	+140.66%	+83.14%	−23.85%
5	8	+141.37%	+141.38%	+79.46%	−29.19%
0	6	+156.81%	+159.09%	+78.85%	−30.86%

provides a comparison between fixed-pitch and variable-pitch propellers in terms of thrust and efficiency at different altitudes and wind speed, under a constrained rated torque.

The results show that, although the efficiency of a variable-pitch propeller is lower than that of a fixed-pitch propeller, the rotational speed achieved and subsequently its maximum thrust is significantly higher, especially at low altitude and low wind speed, because the shaft power of the variable-pitch propeller is much greater than that of the fixed-pitch propeller. The maximum relative increase in thrust can exceed 80%, expanding the flight envelope of the aircraft. However, while this improves performance in off-design conditions, it also increases the weight and complexity of the platform due to the need for actuation systems. A thorough analysis should therefore be carried out to assess whether the performance benefits of a variable pitch propeller outweigh the additional weight and resulting increase in energy consumption.

Finally, a summary of the methodologies, optimization techniques, and aerodynamic models adopted for the design of propellers in HAPS propulsion systems is provided in

Table 4. Overview of the techniques, methodologies, and design models developed over the years for high-altitude platforms propellers and the maximum efficiency values achieved. Adapted from [141].

References	Max Efficiency [%]	Optimization Technique and Design Method	Model Used in Optimization	Operating Condition	Diameter and ${\mathit{N}}_{\mathit{b}}$	Airfoil
Mourousias et al. [141], 2024	86.11	Multi-fidelity multi-objective Bayesian optimization	3D RANS/Vortex Theory	V = 55 m/s, h = 16 km, T = 185 N	D = 3 m, $N_b$ = 3	Optimized airfoil
Mourousias et al. [97], 2023	85.25	Multi-fidelity Bayesian optimization	3D RANS/Vortex Theory	V = 55 m/s, h = 16 km, T = 217 N	D = 2.7 m, $N_b$ = 3	Optimized airfoil
Gutiérrez et al. [84], 2020	85	Based on Wald design method [150]	BEMT	V = 10 m/s, T = 100 N, h = 20 km	D = 8 m, $N_b$ = 4	NACA4412
Mourousias et al. [144], 2022	84.2	Bayesian optimization	3D RANS	V = 55 m/s, h = 16 km, T = 220 N	D = 3.4 m, $N_b$ = 4	Optimized airfoil
Marinus et al. [68], 2020	82.3	Genetic algorithm (PSO)	Vortex Theory	V = 77.64 m/s, T = 318 N, h =16 km	D = 1.9 m, $N_b$ = 3	Clark-Y
Yao et al. [45], 2022	82/80.2 (with/without Gurney Flap)	Multi-level: Level 1 Betz method, Level 2 GA, Level 3 GA	2D RANS/3D RANS	V = 32 m/s, T = 20 N, h = 20 km	D = 2 m, $N_b$ = 2	Optimized airfoil
Xu et al. [149], 2019	81.84/78.52 (Fixed/Variable Pitch propeller)	Bayesian optimization	3D RANS	V = 25 m/s, T = 22.89–31.24 N, h = 20 km	D = 2 m, $N_b$ = 2	E387
Yang et al. [94], 2023	79.29	Multi-level: adjoint, perturbation, flow reconstruction	2D RANS/3D RANS/BEMT	V = 17 m/s, T = 13.3 N, h = 15 km	D = 1.4 m, $N_b$ = 2	Optimized airfoil
Tang et al. [129], 2019	78.07 (VLM), 65.2 (CFD)	Kerwin method with VLM for contra-rotating props	VLM	V = 20 m/s, T = 380 N, h = 20 km	D = 6 m, $N_b$ = 4	S1223
Jiao et al. [85], 2018	75 (calc), 70.5 (exp)	Genetic algorithm (NSGA-II)	Vortex Theory	V = 20 m/s, T = 900 N, h = 20 km	D = 6.8 m, $N_b$ = 2	Custom airfoil sections
Zheng et al. [69], 2017	73.48 (VLM), 66 (CFD)	Coney method with VLM	VLM	V = 20 m/s, T = 484 N, h = 20 km	D = 7.2 m, $N_b$ = 3	S1223
Morgado et al. [51], 2015	73.2	Adkins–Liebeck design method	BEMT	V = 30 m/s, T = 6790 N, h = 16 km	D = 6 m, $N_b$ = 2	Custom airfoil
Park et al. [128], 2018	65.4	Inverse design based on Adkins and Liebeck + RSM + desirability function	BEMT	V = 14.6 m/s, T = 21 N, h = 15 km	D = 1.2 m, $N_b$ = 2	DAE51

. Reported values include maximum propeller efficiency, reference diameter, and operating conditions, to allow for a more meaningful comparison across different works. A clear performance gap emerges between earlier approaches that rely on low-order models, such as classical BEMT and VLM, which generally achieve efficiencies below 75%, and more recent optimization frameworks that incorporate high-fidelity solvers (e.g., 3D RANS) and advanced techniques such as Bayesian or adjoint-based optimization. These modern methods often yield efficiencies above 80%, with some exceeding 85%, reflecting the importance of accurate aerodynamic modeling and integrated optimization workflows. Interestingly, high efficiency is not exclusive to large-diameter designs. While configurations with $D>6 \mathrm{m}$ (e.g., Gutiérrez [84], Jiao [85], Zheng [69]) benefit from reduced induced losses, several compact propellers—notably Yang ($1.4$ m), Yao and Xu [45,149] ($2.0$ m)—also achieve 78–82% efficiency, thanks to refined design methods and airfoil tailoring. This underlines the potential of surrogate-assisted and multi-level strategies in overcoming geometric limitations. Overall, the most effective optimization pipelines appear to be those combining physics-based models with data-driven tools, striking a balance between aerodynamic fidelity, computational cost, and adaptability to mission-specific constraints. Given the variability in operating conditions and performance targets, direct comparison remains non-trivial; however, general trends clearly support the use of hybrid and high-fidelity frameworks in future HAPS propeller design. The comparative analysis of optimization techniques for high-altitude propellers highlights a clear evolution toward robust and adaptive methodologies.

6. Experimental Methods for Evaluating HAPS Propeller Performance

A critical comparison of the optimization strategies discussed in Section 5.2 highlights key trade-offs between fidelity, cost, and reliability. CFD-only approaches, such as those by Morgado [51] and Loureiro [108], capture complex aerodynamic phenomena like stall delay and tip losses at low Reynolds numbers ($\mathrm{Re}\approx 5\times {10}^4$ to $2\times {10}^5$), but are sensitive to mesh quality and transition modeling. Efficiency discrepancies of up to 15% were reported when comparing to XFOIL and BEMT models. In contrast, surrogate and multi-fidelity approaches aim to balance accuracy and speed. Mourousias [104] employed a three-level Hierarchical Kriging model (BEMT, Vortex Theory, RANS) to solve a multi-objective problem with convergence in just 35 iterations. Yang [94] demonstrated a 20–25% increase in propulsive efficiency through high-fidelity optimization with reduced computational effort. However, both methods are fundamentally limited by the scarcity of validation data. Experimental approaches to propeller testing for HAPS are therefore critical for validating aerodynamic predictions and assessing performance in low-density atmospheric conditions. In [108], the results obtained from wind tunnel tests were compared with those derived from low-fidelity (BEMT) and high-fidelity (CFD) methods for a propeller operating at low Reynolds number. The tested propeller was an APC 14 × 7e model, made of fiberglass composite, with a diameter of 35.56 cm and a pitch of 17.78 cm. The wind tunnel experiments were conducted at the Experimental Aerodynamics Laboratory of the Federal University of Minas Gerais (Brazil), using a closed-circuit wind tunnel with a test section measuring 1.0 m in width and 1.2 m in height. The propeller was mounted in a tractor configuration and operated at a constant rotational speed of 7900 rpm throughout all tests. The freestream air velocities were varied from 0 to 25 m/s, resulting in advance ratio values ranging from 0 to 0.5. The air density during testing was $\rho = 1.185$ kg/m³, and the kinematic viscosity was $\nu = 1.4515\times {10}^{-5}$ m²/s. Turbulence intensity in the test section was maintained at approximately $0.8\%$, ensuring a uniform and steady flow. Thrust measurements were taken using a G-model load cell with a sensitivity of $2\pm 10\%$ mV/V and a reading range between 5 kg and 50 kg, with $\pm 10$ g accuracy. These experimental results were compared to different numerical approaches (Figure 20): BEMT models such as QProp and JBLADE showed good agreement with wind tunnel data at low advance ratios ($J\le 0.3$), where the flow is dominated by low Reynolds numbers and rotational effects. JBLADE, in particular, provided consistent results even at higher J due to its incorporation of 3D flow effects. Conversely, the CFD-AMI (Arbitrary Mesh Interface) model exhibited superior accuracy at higher advance ratios, benefiting from transient rotor–stator interaction modeling and turbulence resolution, albeit at a significantly higher computational cost. The CFD-FR (Frozen Rotor) model maintained a relatively constant error (∼20%) across conditions but predicted efficiency reliably. These comparisons underscore the importance of selecting the appropriate method based on flow regime and computational resources, while also highlighting the critical role of experimental testing in validating and benchmarking both low- and high-fidelity numerical predictions.

To further demonstrate how experimental testing supports the development of propellers for HAPSs, another notable study is described in [128]. In particular, a comprehensive experimental evaluation of a propeller designed for the solar-powered HALE UAV EAV-3 was presented. To validate the aerodynamic predictions of the 1.2 m diameter propeller, optimized for high efficiency under stratospheric conditions using BEMT with minimum induced loss and low-Reynolds CFD airfoil data, a wind tunnel campaign was conducted at the Korea Aerospace Research Institute, using a closed-circuit low-speed tunnel (4 m × 3 m test section, max speed 120 m/s, turbulence intensity 0.07% at 88 m/s). The test apparatus featured a custom measurement rig integrating an industrial AC motor (Mitsubishi HG-JR 153, max 6000 rpm), a torque sensor (HBM T-22), and a load cell (Dacell UMI, 20 kgf), ensuring precise force and moment measurements. Propeller tests were performed at fixed rotational speeds (700–1050 rpm) while varying freestream velocity (up to 10 m/s) to explore a range of advance ratios. Performance coefficients ($C_T$, $C_P$, and $\eta$) were computed from directly measured thrust and torque. A Glauert correction was applied to account for wall blockage effects, shifting J by approximately 0.005–0.01. The wind tunnel experiments were conducted under standard atmospheric conditions (near sea level) without simulating the low-density environment of high-altitude flight. However, by adjusting rotational speeds and freestream velocities, the tests reproduced equivalent advance ratios and Reynolds numbers consistent with stratospheric operation, enabling a meaningful aerodynamic assessment of the propeller performance in realistic flight regimes. The experimental results show good agreement with CFD predictions obtained using the transition SST model in Fluent, with efficiency differences limited to 1–1.5 percentage points across the tested conditions. The residual discrepancies, especially at low advance ratios, were primarily attributed to wake contraction and blockage effects, which were subsequently confirmed through additional CFD simulations that included the physical test apparatus. Propellers designed for high-altitude applications typically have large diameters, which often exceed the dimensions of standard wind tunnel test sections. To overcome this limitation, experiments are commonly conducted on scaled models, applying similitude principles to ensure geometric, kinematic, and dynamic similarity with the full-scale propeller. Key non-dimensional parameters, such as advance ratio, Reynolds number, and Mach number, are matched, allowing the scaled models, with appropriately adjusted diameters, rotational speeds, and inflow velocities, to accurately reproduce the aerodynamic behavior of the full-scale propeller [151]. However, since it is often impossible to fully satisfy all similarity requirements during model testing, attention is generally focused on matching the two parameters most relevant to the problem. This is the case of [45], in which, to verify the results of CFD simulations with wind tunnel tests, only the advance ratio and Reynolds number were ensured to match those of the prototypes. Since the Mach number at the propeller tip was below 0.4, compressibility effects were considered negligible and thus disregarded. The goal was to verify that a Gurney flap would enhance the performance of a propeller (with a diameter of 2 m) working in a low Reynolds number environment, as previously predicted by CFD simulations The experiments were conducted in a closed, single return-flow wind tunnel. The test section measured 12 m in length and featured a rounded rectangular cross-section of $3 \mathrm{m}\times 3 \mathrm{m}$. It featured a low turbulence intensity, between 0.1% and 0.13%, and allowed for continuous wind speed control ranging from 3 to 100 m/s. The scaled propellers tested had a diameter equal to 0.7 times that of the full-scale propellers. Two tests were performed: one at a free-stream velocity of 4.2 m/s, corresponding to a Reynolds number of approximately $4\times {10}^4$, which represents the design point of the propeller; and the other at a free-stream velocity of 16 m/s, corresponding to a Reynolds number of about $1.6\times {10}^5$. The results show good agreement with those obtained from CFD. In particular, the performance trends with respect to the advance ratio observed in the CFD, although slightly underestimating the experimental values, closely matched those from the wind tunnel experiments, with an average difference of less than 1% in propeller efficiency (Figure 21). Similar agreement was also found for the thrust and power coefficient curves.

However, in addition to the difficulty of testing large-diameter propellers, the high cost of wind tunnel experiments often limits their applicability to high-altitude propeller research. To address these limitations, recent studies have introduced alternative methodologies based on mobile testing platforms. One such approach is presented by [152], who developed an innovative experimental methodology specifically designed to evaluate the performance of propellers for HAPSs. Their work focused on characterizing the aerodynamic behavior of a 6.8 m-diameter propeller intended to operate at 20 km altitude, with a rotational speed of 500 rpm and a flight speed of 20 m/s. To this end, two mobile experimental testing systems were designed: one for testing scaled propellers under simulated high-altitude conditions, and another for testing the full-scale propeller at lower altitudes. The experimental setup consists of three core components: the test platform, the measurement and control system, and the delivery platform. For scaled tests, special attention was given to minimizing external disturbances and accurately capturing the relatively small aerodynamic forces. The test rig is mounted on a modified vehicle and elevated above the cab to maintain undisturbed flow conditions. Propellers are driven by a DC motor with a speed reducer and powered by a stabilized voltage supply. Measurement systems are adapted to the scale of the propeller. For the full-scale configuration, thrust and torque are measured using a custom dynamometer with four S-shaped load cells detecting structural deformations, and a separate axial force sensor mounted on slide rails. In contrast, the scaled system uses an inline S-shaped load cell for thrust and a rotary torque sensor for torque, both placed between the motor and the propeller to improve sensitivity. Real-time data acquisition is managed through a National Instruments cDAQ-9188 system with multiple I/O modules, while wind conditions are recorded using an ultrasonic anemometer and a barometer. A LabVIEW-based interface controls motor operation and processes measurement data through dedicated modules for control, acquisition, and logging. Testing is conducted along straight, flat roadways to generate relative wind, although maintaining a constant vehicle speed remains a limitation. Nonetheless, the method offers a practical, low-cost alternative to wind tunnel testing for high-altitude applications. Data segments corresponding to steady-state conditions are isolated and processed using statistical filtering (3$\sigma$ criterion) to ensure the robustness of the aerodynamic performance results. Validation of these mobile testing methods was carried out independently for each system. The full-scale 4.6 m propeller was tested at ground level under static (V = 0 m/s) and dynamic (V = 5 m/s) conditions over a speed range of 100–200 rpm. The experimental results showed good agreement with CFD predictions, especially for static thrust, with mean relative errors below 8% in dynamic conditions. Deviations at low advance ratios were primarily attributed to the small aerodynamic loads and the influence of vibrations and road irregularities. The scaled mobile testing system was validated through wind tunnel experiments involving a 1.0 m standard propeller in the NF-3 low-speed facility at Northwestern Polytechnical University ($12 \mathrm{m}\times 3.5 \mathrm{m}\times 2.5 \mathrm{m}$ test section), operating at 1500–3500 rpm and wind speeds up to 10 m/s. Aerodynamic loads were measured using a six-component strain gauge balance. Results from the wind tunnel closely matched those from the scaled mobile tests, with thrust and torque errors below 6%, attributed to the controlled environment and reduced vibration levels. Following validation, the full-scale 6.8 m propeller was tested with the full-scale propeller testing system at altitudes of 0.5 km and 3.6 km to obtain its performances at low altitudes. To replicate its performance at 20 km altitude, a geometrically scaled model was tested at 4 km, under similitude conditions, with the scaled propeller testing system. Specifically, the scaled test was conducted using a propeller with a diameter of 0.78 m, at an airspeed of 22 m/s and a rotational speed of approximately 4900 rpm. These values were selected to match the advance ratio, Reynolds number, and Mach number of the full-scale propeller operating at high altitude. The obtained performance has been compared with those obtained by CFD simulations. The comparison shows that, although the measured thrust, power coefficient, and efficiency are slightly lower than the predicted values, the overall trends are consistent. At the design point, the measured efficiency was approximately $3\%$ lower than that estimated by CFD (Figure 22). These findings confirm that, despite a moderate degree of accuracy, the proposed testing method is a cost-effective and sufficiently reliable alternative for the aerodynamic evaluation of high-altitude propellers.

As described in the previous work, predictions of full-scale propeller performance commonly rely on model-scale experiments. While flow around full-scale propellers is generally assumed to be fully turbulent, different flow regimes—including laminar, transitional, and turbulent—may coexist at model scale [153]. Additionally, flow separation that does not occur at full scale can arise at model scale. Complementing experimental tests, advances in high-performance computing enable increasing use of RANS solvers for performance prediction and scale-effect quantification. Use of turbulence models capable of simulating laminar-to-turbulent transition is critical, with several such models having been proposed in the literature [154,155]. Given the scarcity of experimental and flight data and the complexity of modeling low Reynolds number flows, Ref. [104] employs the PCE to assess performance uncertainty of high-altitude propellers. This study quantifies overall uncertainty considering both epistemic and aleatoric sources, applying PCE to two propeller flow modeling approaches: BEMT and three-dimensional RANS with the $\gamma$-$R{e}_{\theta }$ transition model. Multipoint uncertainty quantification (UQ) analyses were conducted on two benchmark propeller geometries across operating conditions with available experimental data. For BEMT, uncertainty bounds encompassed experimental data for two distinct propellers operating at medium to low Reynolds numbers over a broad range of advance ratios. However, at high altitudes, these uncertainty bounds widened excessively, limiting predictive accuracy particularly at the highest advance ratios. Conversely, 3D RANS-based UQ yielded well-defined uncertainty bounds consistently containing experimental values. Predictions at high altitudes showed low standard deviation. Sensitivity analysis identified freestream velocity and the structure parameter ${\alpha }_1$ as the most influential factors. Consequently, future UQ efforts using 3D RANS may focus primarily on these variables, enabling substantial computational cost reductions.

7. Conclusions

The design of propulsion systems for HAPSs presents a complex challenge, closely linked to the extreme environmental conditions characterizing the upper atmosphere. Low air density, along with strict constraints on weight and power, necessitates an innovative and highly engineering-focused approach.

Currently, for low-speed applications, propulsion systems employing propellers remain the most efficient propulsors. This review has shown that optimizing propeller performance is essential to enhancing the endurance and payload capacity of stratospheric platforms. Traditional propeller design methods often result in suboptimal solutions because they neglect Reynolds number effects under HAPS operating conditions.

After a concise review of the main propulsion systems used on stratospheric platforms and their evolution over time, this study delves into propeller theory, design methodology, and performance estimation for such applications, emphasizing how design methodology is crucial for maximizing efficiency at these altitudes. Optimization results reveal common trends, such as maximizing propeller diameter to improve efficiency and distributing chord length with maximum values between 60% and 80% of the blade span. A significant limitation in the design process is the considerable computational time required. The most effective approach to designing a high-efficiency propeller involves integrating 3D CFD simulations within each step of the iterative process, although this can be time-consuming. To reduce computational effort, employing potential 2D codes or limiting the use of 3D CFD simulations may be viable alternatives, albeit with some loss in accuracy. Among the methodologies proposed in the literature, promising results have been achieved using a multi-fidelity optimization technique which combines low-fidelity BEMT with high-fidelity CFD. Additional methods to enhance performance, such as incorporating Gurney flaps and accounting for wind speed in the design process, have demonstrated significant potential compared to traditional propeller design techniques, which were originally developed for low-altitude propulsion systems. Moreover, various methodologies have been presented for evaluating the performance of high-altitude propellers, ranging from numerical analyses to experimental testing. Among these, scaled-model experiments stand out as particularly advantageous, offering greater reliability and practical insight compared to CFD simulations alone. These experimental approaches also provide essential validation data for advanced optimization techniques, which, from a practical perspective, offer concrete benefits for future HAPS developments. Improved aerodynamic efficiency can directly enhance mission endurance and payload capacity, while simultaneously reducing energy requirements. Furthermore, coupling optimization algorithms with probabilistic wind models could lead to adaptive propeller designs tailored to specific mission profiles and geographic locations. Future research should aim to integrate these optimization methods into early-stage design workflows of HAPSs, supported by experimental validation under low-Reynolds number conditions, to enable robust and cost-effective development of next-generation stratospheric propulsion systems.