Introduction

eVTOL aircraft, with its low-carbon, low-cost, and intelligent characteristics, is expected to become the main aircraft for future UAM. With the gradual relaxation of policies and the continuous improvement of infrastructure, the application of eVTOL in urban air rental, emergency medical rescue, cargo distribution and other fields will become more widespread. The design of ducted rotors in eVTOL does represent an important development trend in this field, especially in improving safety and optimizing aerodynamic performance1,2. Specifically manifested in 1)Eliminating safety hazards of open rotors: Open rotors pose a potential threat to ground personnel or objects when rotating at high speeds. The design of the ducted rotor wraps the rotor inside the duct, effectively avoiding this safety hazard3. 2)Protect the rotor from external impacts: The duct structure not only provides a physical barrier for the rotor, but also reduces the possibility of external objects (such as branches, wires, etc.) hitting the rotor, thereby reducing the risk of damage or accidents caused by impact4. 3)Improve overall structural strength: ducted design is usually closely integrated with the aircraft structure, which helps to improve the overall structural strength, enhance the stability and durability of the aircraft5. 4)Improve lift efficiency: By optimizing the airflow distribution, ducted rotors reduce the energy loss of blade tip vortices, thereby improving lift efficiency. This enables a smoother transition between vertical takeoff and horizontal flight for eVTOL, improving overall flight performance6. 5)Reduce noise: The duct structure helps to reduce the noise generated when the rotor rotates7. By wrapping the rotor inside the duct, the interaction between the airflow and the rotor blades is reduced, thereby lowering the noise level and making eVTOL quieter when operating in urban environments. The design of ducted rotors has significant advantages in improving the safety and optimizing aerodynamic performance of eVTOLs, and is one of the important trends for the future development of eVTOLs. With the continuous advancement of technology and the continuous development of the market, ducted lift systems are expected to play a more important role in the eVTOL field.

At present, many studies have investigated the influence of geometric parameters of ducted rotors on their aerodynamic performance8,9,10,11,12. Parameter influence studies have shown that reducing blade tip clearance, moving the rotor backwards within a certain range, or increasing the tail expansion angle can all improve the aerodynamic performance of ducted rotors13,14,15,16. However, these studies were conducted based on traditional ducted rotor structures. With the advancement of rotor blade and ducted manufacturing and material technology, other forms of ducted rotor structures have also been proposed. This paper proposes a method of increasing lift by embedding a rotor tip into the inner wall of a duct to improve the slip boundary and lip disturbance inside the duct. Through numerical simulation, the flow state and aerodynamic characteristics of the flow field under under hovering, axial flow, and oblique flow states were calculated and analyzed, and compared with the aerodynamic forces of isolated rotors of the same diameter and ducted rotors under the same operating conditions.

Calculation model and method

The rotor featured in this paper boasts a three-bladed shape and adopts a CLARK-Y airfoil profile. It exhibits a linear torsion of 20° spanning from the root to the tip, accompanied by a disc diameter of 710 mm and a hub diameter of 60 mm. From top to bottom, the rotation direction of the propeller is clockwise(Fig. 1a). In the duct rotor structure(Fig. 1b), the duct section employs the widely utilized original duct airfoil, featuring an inlet diameter of 730 mm, an outer diameter of 854 mm, a lip radius of 22 mm, and a duct length of 487 mm. The rotor’s axial position within the duct is positioned approximately one-third of the duct body’s height, measured from the duct lip. The clearance between the rotor tip and the inner wall of the duct is 2.86% of the blade radius. The geometric model of the embedded ducted rotors (Fig. 1c) is based on the previous ducted rotors, maintaining the rotor’s dimensions and its axial position relative to the duct unchanged. The size of the groove on the inner wall of the culvert and its relative position with the propeller tip are shown in Fig. 1d and e. This modification ensures that the distance from the rotor tip to the groove inner wall is consistent with the distance from the rotor tip to the duct inner wall in the ducted rotors. Compared to the ducted rotors, the aspect ratio of the duct in the embedded ducted rotors is reduced from 1.499 to 1.437. According to literature, this change is not expected to have a significant impact on the aerodynamic loads of the duct and rotor assembly and can thus be disregarded.

Fig. 1
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Definition of computational model.

For numerical calculations, this paper employed the incompressible viscous Navier-Stokes equation, utilizing a turbulence model. The rotor adopted a multiple reference frame model, while the duct and guide cone adhered to solid wall boundary conditions to meet the no-slip condition. To enhance calculation accuracy, second-order upwind schemes were applied to momentum, turbulent flow energy, and turbulent dissipation rate. Convergence studies were conducted for both ducted rotors and embedded ducted rotors using three successive mesh densities: coarse, medium, and fine. Specifically, the ducted rotors configuration featured a coarse mesh with 3.99 million elements, a medium mesh with 5.30 million elements, and a fine mesh with 11.71 million elements. For the embedded ducted rotors configuration, the respective mesh densities comprised 4.34 million, 6.11 million, and 13.42 million elements. The study revealed significant differences in results between the coarse and medium meshes, whereas differences between the medium and fine meshes were negligible. Consequently, the medium mesh was selected as the baseline for subsequent simulations. To account for air viscosity’s impact on rotor torque, a triangular prismatic boundary layer grid was applied to the geometric shape’s surface, ultimately generating an unstructured nested grid. Figure 2 illustrates grid space slices. This approach ensures a consistent overlap area between subdomain grids during dynamic nesting in numerical calculations, eliminating the need for repetitive “hole digging” or background grids. The rapid and efficient updating of interpolation contribution elements renders this method ideal for numerical simulations involving rotating boundaries. Theoretically, the computational domain for ducted and embedded ducted rotors should be infinite. However, practical considerations of computational resources and time necessitate the use of a finite domain. Provided the chosen domain size does not significantly affect the flow field near the model, it suffices for numerical calculations. The domain’s shape is typically determined by the computational state. Given this article’s focus on aerodynamic characteristics in axial and oblique flow states, a cubic computational domain was selected. The cube’s side length was set at twenty times the duct body’s outer diameter, serving as the reference length, with the computational model centered within the domain. To simulate rotor rotation within the duct, a sliding grid divided the computational domain into rotating and stationary parts. The rotating domain and rotor spun at the same angular velocity around the axis, with flow field information exchanged between them through an interface. To accurately capture the flow field around the model and expedite calculations, the mesh density was higher near the model surface and vicinity, and sparser in distant areas.

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Grid space slice diagram.

NASA’s research center conducted a series of wind tunnel tests in the 1960s to study the power systems of vertical takeoff and landing aircraft(VTOL). Kalman and Kenneth conducted wind tunnel tests on a ducted rotor’s aerodynamic performance at the Langley Research Center17. The tests mainly focused on measuring the aerodynamic loads at different angles of attack within the typical transitional speed range of ducted rotors for VTOL aircraft, with an advance ratio ranging from 0 to 0.595. Due to the detailed experimental data provided by these tests, this study selected some of the experimental results as examples to validate the numerical calculations of ducted rotors. Figure 3 present the comparison between the calculated and experimental values of the lift and thrust coefficients of a ducted rotors operating at 8000 rpm with an incoming flow velocity of 30 m/s, showing the variation with angle of attack. The comparison revealed that the maximum error between the numerical calculations and experimental values did not exceed 10%, thus confirming the effectiveness of the proposed method.

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Comparison chart illustrating the correlation between numerical calculation results and experimental data.

Calculation results and analysis

Hover state

This section conducts numerical calculations on the thrust, power, and power load of embedded ducted rotors in hover state, while also conducting a comparative analysis with isolated rotors and ducted rotors under the same operating conditions. The thrust and power requirement curves of isolated rotors, ducted rotors, and embedded ducted rotors with respect to rotation speed are shown in Fig. 4a and b. When the rotor speeds are equal, Fig. 4a indicates that the ducted rotors has increased thrust relative to the isolated rotors by 7.8%, the embedded ducted rotors increased by 14.2%, while the power consumed by the rotor at this point is essentially the same. As shown in Fig. 4c, under the same power consumption conditions, the embedded ducted rotors generates more thrust than the ducted rotors and isolated rotors. The above results indicate that embedding part of the rotor blade tip into the duct wall can effectively improve the overall aerodynamic efficiency, achieving a lift-enhancing effect.

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Comparison of aerodynamic characteristics of three states.

To analyze the reasons why the thrust generated by the embedded duct rotors is greater than that produced by the duct rotors, we constructed static pressure distribution diagrams on the mid-perpendicular planes of both, as shown in Fig. 5. Figure 5a and b are side and top views of the isolated rotors streamline in hover. According to its streamline diagram, it can be seen that: (1) the rotating rotor can gather the air around it and accelerate it to discharge to the lower part of the propeller; (2) The slipstream boundary under the rotor shrinks constantly, and the streamline under it is spiral. Figure 5c and d are top and side views of ducted rotors streamline in hover. It can be seen from the figure that the air near the culvert lip of the ducted rotors bypasses the culvert lip and flows into the interior of the culvert. Due to the effect of the inner wall of the culvert, there is no obvious necking of the air flow under the ducted rotors, which is different from the isolated rotors. But similar to the isolated rotors, the air flow under the ducted propeller is also spiral. The streamline diagram of the embedded ducted rotors is similar to that of the ducted rotors, which needs to be observed and compared from the peripheral pressure field (Fig. 5e and f). The focus here is on observing the pressure distribution at the duct inlet and outlet, as the pressure difference between these two sections is the source of the thrust generated by the duct. At approximately the same position near the duct inlet, the values of the isobar lines for the two are − 279.647 Pa and − 217.13 Pa (relative pressure), with the former’s suction value significantly greater than the latter. Similarly, employing the same method, it was observed that at the tail end of the duct, the pressure of the former is greater than that of the latter.

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Comparison of streamline and pressure nephogram in hover state.

The primary difference between the embedded ducted rotors and the ducted rotors lies in the internal wall of the duct. The former has a rough surface on the inner wall, while the latter is smooth. Hence, the increase in the suction force at the embedded duct lip is related to the rough surface. The cross-sectional streamline diagram of the groove on the internal wall of the duct in Fig. 6 reveals the presence of a local vortex structure within the groove, with its rotation direction opposite to that of the rotor tip vortex. Due to the relatively large aspect ratio of the rotor used in this study, induced drag is not the dominant part of the total drag, hence its effect on the power required for the rotor is almost negligible. Since the pressure in the laminar flow region is lower than that of the surrounding gas, especially within the duct, there will be a local recirculation at the rear of the duct. In the case of the embedded ducted rotors, the flow of vortices within the groove intensifies this phenomenon, causing an increase in the upward flow velocity. This portion of the airflow passes upwards over the rotor and is then drawn back down into the downward flow, creating a local low-pressure area near the duct lip, which is greater than that of the ducted rotors under the same conditions. It is the existence of this low-pressure area that accelerates the recirculation at the duct lip.

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XY section streamline diagram of embedded ducted rotors.

Figure 7 depict the axial velocity distributions at 0.05 m above the rotor, revealing that the downwash velocity generated by the embedded rotor exceeds that of the ducted rotors. It is understood that the downwash diminishes the effective angle of attack of each blade section, consequently reducing the thrust produced by the entire rotor blade. This phenomenon elucidates why the thrust generated by the embedded ducted rotors is further decreased compared to the ducted rotors. Another contributing factor is the increase in downwash velocity, which elevates the airflow speed over the duct lip surface, resulting in lower pressures at the lip compared to the ducted rotors, thereby enhancing the duct thrust. Furthermore, the suction effect of the low-pressure vortices generated at the cavitator is favorable for the duct, collectively manifesting that the duct thrust enhancement is the primary reason for the increased thrust of the embedded ducted rotors. Table 1 provides data on the aerodynamic performance improvements of the embedded ducted rotors and ducted rotors compared to the isolated rotors. The analysis indicates that: (1) the ratio of thrust generated by the embedded ducted rotors to the total thrust is higher than that of the ducted rotors; (2) the lift enhancement effect of the embedded ducted rotors is approximately 1.75 times that of the ducted rotors.

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Axial velocity distribution contour map.

Table 1 Performance Comparison.
Full size table

Axial flow state

This section conducts corresponding numerical simulations on the flow field characteristics of isolated rotors, ducted rotors, and embedded ducted rotors when the axial inflow velocity ranges from 1 to 14 m/s. The aim is to investigate the differences in aerodynamic forces, power requirements, and power load between embedded ducted rotors, isolated rotos, and ducted rotos under varying axial inflow conditions. Figure 8a illustrates the variation of thrust for isolated rotors, ducted rotors, and embedded ducted rotors with inflow velocity. It is evident from the graph that the thrust of all three decreases as the axial inflow increases. This reduction primarily stems from two aspects: (1) the increase in axial inflow leads to a decrease in the effective angle of attack of the rotor blade sections, resulting in reduced thrust generated by the rotor; (2) the axial inflow weakens the flow around the duct lip, consequently reducing the thrust produced by the duct as the axial inflow increases. Upon careful observation of figure, we found that when the axial flow is less than 5 m/s and the inflow velocity is equal, similar to the hover state, the thrust generated by the isolated rotors, ducted rotors, and embedded ducted rotors increases in sequence. However, as the axial inflow velocity increases, the difference between the three becomes smaller. At an inflow velocity of 5 m/s, the thrust generated by the ducted rotors is equal to that generated by the isolated rotors, which means that the ducted rotors lose its original lift increasing effect. At the same time, the ducted body also has a certain weight. In this case, the advantage of the ducted rotors in lift increasing no longer exists, and the thrust generated by the embedded ducted rotors is still greater than that of the isolated rotors and ducted rotors. During the inflow velocity ranges from 5 m/s-7 m/s, the magnitude relationship of the pulling force generated by the three changed to a certain extent, from large to small, namely embedded ducted rotors, isolated rotors, and ducted rotors. During this process, the difference between the isolated rotors and the ducted rotors increases continuously, while the difference between the embedded ducted rotors decreases continuously. When the incoming flow is 7 m/s, the thrust values generated by the embedded ducted rotors and the isolated rotors are equal, similar to the previous analysis. At this point, the embedded ducted rotors also lose its advantage in increasing lift. When the axial flow is greater than 7 m/s, both the embedded ducted rotors and the ducted rotors generate lower thrust values than the isolated rotors, and the difference continues to increase. As the axial flow further increases, the thrust generated by the embedded ducted rotors is almost equal to that of the ducted rotors, which means that the ducted rotors has completely lost its original function. The curve presented in Fig. 8b illustrates the variation of power required by the three rotor struchures with axial inflow velocity. Generally, the values of power increase initially with the increase in axial inflow velocity and then decrease. A comparison of the three curves reveals that the power consumed by the isolated rotors is greater than the latter two within the calculated range. In contrast, the power consumed by the embedded ducted rotors is initially greater than the ducted rotors, but later becomes less than the ducted rotors. In Fig. 8c. A comparison with Fig. 8a reveals similarities between the variations in power load with axial inflow velocity and the changes in thrust with axial inflow. Through analysis, it is determined that the lift enhancement effect of the embedded ducted rotors diminishes continuously as the axial inflow velocity increases, eventually causing the duct to function as a component that decreases thrust. Therefore, it is advisable for VTOL aircraft equipped with embedded ducted rotors or ducted rotors to not have excessive climbing speeds during vertical take-off.

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Aerodynamic comparison curves of three rotors structures.

Through the analysis of the reasons for the increase in lift of the embedded ducted rotors in hover state, we know that the main source of its increased thrust is still the ducted. For vertical climbing, as the axial inflow velocity increases, the flow around the lip of the ducted rotors continuously weakens. Figure 9 show the side view of streamline diagram of the embedded ducted rotors under axial flow condition when the axial flow is 1 m/s and 14 m/s, respectively. It can be seen that the flow around the lip of the ducted rotors in Fig. 9a is significantly stronger than that in Fig. 9b. At the same time, the slipstream boundary above the ducted rotors is reduced, and the disturbed air is also reduced.

Fig. 9
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Side view of streamline diagram of embedded ducted rotors under axial flow condition.

Oblique flow state

Numerical calculations were carried out on the lift and thrust of the embedded ducted rotors at a speed of 6000 rpm, an inflow of 10 m/s, and forward inclinations of 0 °, 10 °, 20 °, 30 °, and 40 °, respectively. The variation curves of the above parameters with the forward inclination angle and the flow field morphology of the embedded ducted rotors at different forward inclination were obtained. At the same time, a comparative analysis was conducted on the aerodynamic characteristics of isolated rotors and ducted rotors under oblique flow conditions, including the lift and drag of embedded ducted rotors and the structure of the ducted itself. Figure 10 shows a comparison of the lift generated by embedded ducted rotors, ducted rotors, and isolated rotors with respect to the change in forward tilt angle under oblique flow conditions. From Fig. 10a, it can be seen that under the same forward tilt angle, the lift generated by the embedded ducted rotors and the ducted rotors is greater than that of the isolated rotors. The lift generated by the embedded ducted rotors first increases and then decreases with the increase of the forward tilt angle, and the rate of decrease is smaller than that of the isolated rotors, similar to the ducted rotors. There is a critical relationship between the lift generated by the embedded ducted rotors and the ducted rotors at different forward inclinations. When the current inclination angle is less than this value, the lift of the embedded ducted rotors is greater than that of the ducted rotors. When it is greater than this value, the lift of the embedded ducted rotors is less than that of the ducted rotors. The calculated critical value is about 30 °. The reason for the emergence of a critical value is mainly because as the inclination angle increases, the projection of the incoming flow velocity in the axial direction of the culvert increases, and the flow around the culvert lip weakens. Figure 10b shows a comparison of the variation of ducted lift between an embedded ducted rotors and a ducted rotors with respect to the forward inclination angle. It can be observed from the figure that as the forward inclination angle increases, the difference between the two decreases continuously, which precisely indicates that the lift of the embedded ducted rotors is smaller than that of the ducted rotors after the forward inclination angle exceeds 30 °. Figure 10c shows a comparison of the thrust generated by the embedded ducted rotors, ducted rotors, and isolated rotors as a function of the forward inclination angle. It can be seen from the figure that the thrust of all three increases with the increase of the forward inclination angle, and the rate of increase for the embedded ducted rotors and ducted rotors is greater than that for the isolated rotors. At the same forward inclination angle, the thrust generated by the three in descending order is isolated rotors, embedded ducted rotors, and ducted rotors. This means that the performance of embedded ducted rotors in generating thrust is superior to that of ducted rotors. The reason why the thrust generated by the embedded ducted rotors is greater than that of the ducted rotors at the same forward inclination angle is mainly because the drag generated by the ducted rotors is smaller than that of the ducted rotors, as shown in Fig. 10d.

Fig. 10
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Aerodynamic characteristics in oblique flow state.

Figure 11 is a side view of the streamline of the embedded ducted rotors, ducted rotors and isolated rotors in a oblique flow state at forward inclinations of 0 ° and 40 °. From a macro perspective, the difference between embedded ducted rotors flow field and ducted rotors is not significant. From the isolated rotors streamline, it can be seen that the air flow flows to the rotating plane of the propeller at a certain angle of attack. After the propeller is accelerated, the air flow direction above it deflects a certain angle to the lower right of the propeller. With the increase of the forward inclination angle, the projection of the incoming flow velocity in the direction of the rotor axis becomes larger, so that the thrust generated by the rotor with large forward inclination angle under the same incoming flow size is smaller than that generated by the propeller with small forward inclination angle.

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Streamline diagram of three kinds of rotors structures with different forward inclination angle.

Conclusion and discussion

This paper presents the configuration of embedded ducted rotors. The aerodynamic performance of isolated rotors, ducted rotors and embedded ducted rotors in hover, axial flow and oblique flow was calculated. The aerodynamic characteristics of the three configurations in the same state were compared and analyzed in detail. The aerodynamic advantages of embedded ducted rotors compared with ducted rotors and isolated rotors were obtained, which provided a theoretical basis for the design and optimization of eVTOL aircraft using ducted as propulsion.

The conclusion obtained is as follows:

1) In the hover state, the thrust generated by isolated rotors, ducted rotors and embedded ducted rotors increases in turn, and the required power of the three rotors remains basically unchanged, that is to say, the power loads of the three rotors also increase in turn.

2) In the axial flow state, with the increase of climbing speed, the flow around the duct lip is continuously weakened, and the lift generated by the duct is continuously reduced. When the climbing speed is greater than a critical value, the thrust generated by the isolated rotors is greater than that generated by the ducted rotors, so the embedded ducted rotors has the advantage of increasing the thrust compared with the ducted rotors at low speed.

3) In the oblique flow state, the thrust generated by the embedded ducted rotors is greater than that of the ducted rotors at the same forward inclination angle, mainly because the drag generated by the embedded ducted rotors is less than that of the ducted rotors.

While this study establishes the embedded design’s performance advantages at engineering-relevant scales, future publications will dissect the underlying fluid-structure interaction mechanisms – particularly boundary layer modulation effects – using advanced diagnostics and scale-resolving simulations.

There remains a significant amount of work to be done in the aerodynamic analysis of eVTOL aircraft that utilize ducted fans as their propulsion force. Through means such as duct configuration optimization, rotor configuration and parameter optimization, multidisciplinary design optimization, the application of novel materials and manufacturing processes, research into the impacts of environmental and meteorological conditions, as well as experimental validation and flight testing, the aerodynamic performance of ducted rotors can be continuously improved, laying a solid foundation for the widespread application of eVTOL aircraft. In the future, with the continuous advancement of technology and the expansion of the market, eVTOL aircraft are poised to become the mainstream mode of urban air transportation, bringing revolutionary changes to human travel.