Numerical simulation and analysis of a ducted-fan drone hovering in confined environments

Abstract

Ducted-fan drones are expected to become the main drone configuration in the future due to their high efficiency and minimal noise. When drones operate in confined spaces, significant proximity effects may interfere with the aerodynamic performance and pose challenges to flight safety. This study utilizes computational fluid dynamics simulation with the Unsteady Reynolds-averaged Navier–Stokes (URANS) method to estimate the proximity effects. Through experimental validation, our computational results show that the influence range of proximity effects lies within four rotor radii. The ground effect and the ceiling effect mainly affect thrust properties, while the wall effect mainly affects the lateral force and the pitching moment. In ground effect, the rotor thrust increases exponentially by up to 26% with ground distance compared with that in open space. Minimum duct thrust and total thrust are observed at one rotor radius above the ground. In ceiling effect, all the thrusts rise as the drone approaches the ceiling, and total thrust increases by up to 19%. In wall effect, all the thrusts stay constant. The pitching moment and lateral force rise exponentially with the wall distance. Changes in blade angle of attack and duct pressure distributions can account for the performance change. The results are of great importance to the path planning and flight controller design of ducted-fan drones for safe and efficient operations in confined environments.

1 Introduction

Due to the remarkable maneuverability and high flexibility, unmanned aerial vehicles (UAVs) have been widely deployed in civilian and military scenarios [1,2,3,4]. The ducted-fan drone is expected to become the main configuration of UAVs in the future owing to high efficiency and minimal noise [5]. Compared with traditional rotorcraft, ducted-fan drones have propeller or fan blades shrouded by the duct. The duct brings additional thrust and contributes to higher efficiency or a more compact structure. It also enhances flight reliability during actual task operations, especially those in confined environments [6].

When drones operate in narrow zones, such as a disaster-stricken area, the intricate surroundings may arouse aerodynamic interferences, which is a huge challenge during the flight [7, 8]. The strong and rapidly developing disturbance not only results in performance loss of the ducted fan, but also potentially leads to aircraft out of control [9,10,11]. It is crucial to study the aerodynamic behavior of ducted-fan drones for better adaptation to confined environments.

Proximity effects caused by confined environments typically include ground effect (GE), ceiling effect (CE) [12], and wall effect (WE) [13, 14]. Abundant research work has been done to investigate the impact of these effects on the quadrotor aerodynamics [15, 16]. In terms of the ground effect, Sanchez-Cuevas et al. [17] studied the ground effect and partial ground effect for quadrotors with experimental tests. They introduced corrections to traditional potential flow models to consider the influence of central body and built an analytical model for aerodynamic performance prediction. The results showed that the proposed model was useful in control strategies to compensate for ground effect. Conyers et al. [18] compared the Cheeseman-Bennett model to experiments. It was found that this model could not accurately predict the quadrotor performance in ground effect, and further study was required. Paz et al. [19, 20] conducted CFD simulations to investigate the ground effect. They verified the reliability and efficiency of the multiple reference frame (MRF) method in static quadrotor performance prediction and adopted dynamic meshes to assess the dynamic performance. Results showed that the presence of the ground aroused a decrease in the drag force, and an increase in the lift and pitching moment, which were magnified by the translational velocity.

Some researchers paid attention to the aerodynamic performance of quadrotors in ceiling effect. Jimenez-Cano et al. [21] focused on a quadrotor used for inspection of bridges. They built up the dynamic model of a skewed quadrotor in ceiling effect through CFD simulation and presented a nonlinear Lyapunov-based attitude, altitude, and position controller. The outdoor experiments on a real bridge validated their proposed design. Vong [22] in his PhD thesis reported the aerodynamic disturbances associated with quadrotor flights inside a tunnel. He mentioned that the reduction of vertical component of velocity due to the ceiling plane had a dominant effect on altering the rotor’s upstream flow, which drove an increase in the effective blade angle of attack and lift production. Nishio et al. [23] derived an aerodynamics-based thrust model under ceilings based on the momentum theory. With the proposed model, they succeeded in stabilizing the quadrotor flight using only onboard sensors.

Other researchers made some efforts in studying the wall effect. Chui et al. [24] investigated the behavior of a quadrotor operating near a wall, which eventually collided with the wall. Based on CFD simulation, they came to the conclusion that delicate motor control was required for the investigated quadrotor to fully recover from a non-destructive collision. Vong et al. [25] managed to control a quadrotor near the sidewall in a simulated tunnel-like environment, where the drone was over one third as tall and wide as the section. Carter et al. [26] measured the thrust and wakes of a quadrotor near a ground, ceiling, and sidewall to test the applicability of well-established near-boundary models. It was revealed that the sidewall effects were small in comparison to those seen near the ground and ceiling, and the quadrotor would experience a slight drop in lift near the sidewall.

Existing research mainly focuses on the quadrotor performance in proximity effects. Few studies have discussed ducted-fan drone configurations. The existence of the duct, on the one hand, affects the flow distributions around the drones and the flow structures near the propeller, which will lead to changes in propeller performance. On the other hand, the duct provides significant additional thrust. Changes in duct aerodynamic characteristics have a great impact on the overall drone performance, especially for operations in confined environments. Prior research has paid no attention to these aspects.

This paper aims to thoroughly examine different proximity effects on the aerodynamic behavior of a ducted-fan drone by using the unsteady CFD method. The flow mechanism behind the performance change is analyzed and discussed. We believe that this work could promote the understanding of flow physics for ducted-fan drones hovering in restricted areas, and we hope it could be helpful in the flight controller design for safe and stable operations. The remainder of the paper is arranged as follows: Section 2 describes the research methodology. Section 3 focuses on CFD validation. Section 4 discusses and analyzes the results concerning the proximity effects. Section 5 draws the conclusion.

2 Methodology

To start with, this study gathers and establishes key physical parameters in the following equations.

Thrust coefficient:

$$C_{T} = - \frac{{F_{{\text{Z}}} }}{{\rho A\Omega^{2} R^{2} }}.$$

(1)

Torque coefficient:

$$C_{Q} = \frac{Q}{{\rho A\Omega^{2} R^{3} }}.$$

(2)

Power coefficient:

$$C_{P} = \frac{P}{{\rho A\Omega^{3} R^{3} }}.$$

(3)

Figure of merit (FM):

$${\text{FM}} = \frac{{C_{{T_{{{\text{total}}}} }}^{\frac{3}{2}} }}{{\sqrt 2 C_{{P_{{{\text{total}}}} }} }}.$$

(4)

Lateral force coefficient:

$$C_{{\text{X}}} = \frac{{F_{{\text{X}}} }}{{\rho A\Omega^{2} R^{2} }}.$$

(5)

Pitching moment coefficient:

$$C_{M} = \frac{{M_{{\text{Z}}} }}{{\rho A\Omega^{2} R^{3} }}.$$

(6)

Normalized velocity:

$$C_{{{\text{vel}}}} = \frac{v}{\Omega R}.$$

(7)

Pressure coefficient:

$$C_p=\frac p{0.5\rho\Omega^2R^2}.$$

(8)

Where $\rho$ represents the air density ($\rho = 1.225\ {{\text {kg/m}}^3}$ at 15 °C), $A$ represents the rotor disk area ($A = 0.0408\ {\text{m}^2}$), $\Omega$ represents the fan rotating speed ($\Omega = 4000\ {\text{rev/min}}$), and $R$ represents the rotor radius ($R = 0.114\ {\text m}$).

2.1 Ducted-fan drone

The model of the ducted-fan drone is shown in Fig. 1a. Similar to a quadrotor, the drone has four propellers with a diameter of 228 mm, evenly distributed on the 600 mm × 544 mm × 50 mm rectangular fuselage. The flat and wrapped structure makes it have better trafficability, higher safety, and greater load capacity than normal quadrotors. It is therefore more suitable for performing complicated tasks. The total weight is about 2.4 kg, and the design rotating speed for hovering is 4000 rev/min. More detailed blade parameters are displayed in Fig. 1b. The cross section of the blade is DAVIS-N airfoil. The maximum chord length is used at high blade heights to increase blade thrust. The chord length decreases at the blade tip to reduce the drag induced by the tip leakage flow. The pitch angle $\beta$ varies from $50^\circ$ near the hub to $20^\circ$ at the blade tip.

Fig. 1

Details of the ducted-fan drone used in the study: a overall configurations of the ducted-fan drone; b blade chord and pitch angle distributions of the propeller

2.2 Computational grids

Figure 2 displays the arrangement of the calculation domain in various cases of proximity effects. The overall domain is divided into two subdomains. The static subdomain is a cylinder set for simulating the external flow field. The sizes for simulating ground effect, ceiling effect, and wall effect are 88R × 88R × 48R, 88R × 88R × 48R, and 88R × 48R × 88R, respectively. The rotating subdomain refers to the zone inside the ducted fan, which rotates in absolute coordinates at the fan’s rotating speed. Interfaces are generated between the static subdomain and the rotating subdomain. Note that the distance between the obstacle plane and the ducted-fan drone is highlighted and will change accordingly in each case.

Fig. 2

Domain settings and boundary conditions for three cases in proximity effects

To encompass all essential flow areas, such as blade tips and drone fuselage, unstructured grids were carefully built using ANSYS ICEM CFD. Figure 3 displays the mesh grids for the numerical simulation. As shown in Fig. 3a, the surface mesh size of the drone fuselage was 3 mm and that of the propeller was 0.8 mm, which guaranteed the flow field calculation accuracy near the ducted fan. Figure 3b shows the cross section of the volume grid. The number of boundary layers was 11, and the height of the first layer was calculated to be 0.008 mm to ensure that the wall y⁺ was below 1. The density box was additionally generated for encryption to ensure precise capture of the flow characteristics around the drone and accurate calculation of pressure distributions on the drone surface.

Fig. 3

Generated mesh adopted for the numerical simulation

2.3 Numerical setup

The flow can be classified as incompressible due to the Mach number at the blade tip being approximately 0.14 at 4000 rev/min. Based on this, the study chose the proficient pressure-based solver available in ANSYS Fluent. The shear stress transport (SST) k-ω turbulence model was used. The coupled algorithm was employed along with a second-order upwind scheme.

Boundary conditions were set up in accordance with Fig. 2. In the static subdomain, non-slip walls were assigned to the obstacle plane, while the rest were defined as velocity inlets with zero velocity amplitude. In the rotating subdomain, the inner duct wall was specified as an absolutely stationary wall.

3 Validation and verification

Bench tests were conducted in the School of Vehicle and Mobility, Tsinghua University for validation. Figure 4 demonstrates the configuration of the test bench, including acrylic plates simulating the rigid wall, an experimental ducted-fan drone fabricated with carbon fiber, and a set of data collection systems as well as associated sensors. Multiple types of sensors were mounted to capture thrust, torque, speed, current, and voltage data in real time.

Fig. 4

Schematic diagrams of the test bench: a overall test bench; b test ducted-fan drone

In order to validate the accuracy of the CFD method, an open space hovering case was carried out and quantitatively examined. The comparisons between the simulation results and the experimental data are illustrated in Fig. 5. The thrust coefficients remained relatively constant at various rotating speeds, with the rotor thrust making up approximately 57% of the total thrust. It was clear that the simulation results agreed well with the experiments with a maximum error of 7%, indicating that the numerical simulation technique utilized could predict the drone’s performance with acceptable accuracy. Possible factors accounting for the discrepancies may be the aerodynamic interference from sensors and support columns.

Fig. 5

Comparisons between computation and experiments for the drone hovering in an open environment

To expedite the grid convergence test, the hovering case with a rotating speed of 4000 rev/min was chosen. Three grids of varying qualities, labeled as coarse, medium, and fine, were created and evaluated. Figure 6 displays the results. In terms of both the thrust and torque coefficients, the values obtained from the coarse mesh were notably greater than those from other meshes. The medium mesh and the fine mesh reached almost the same results. As a consequence, the medium mesh was picked for the following study to balance precision and computing time.

Fig. 6

Grid convergence test for the drone hovering in an open environment ($\Omega = 4000\ {\text{rev/min}}$)

The time step size is an important factor in transient analysis, needing to be small enough to capture all relevant transient aspects while maintaining acceptable computation cost and efficiency. We conducted time step independence verification with the medium mesh. The in-ground-effect hovering case with h = 2R and $\Omega = 4000\ {\text{rev/min}}$ was chosen for analysis, and four different sizes with corresponding rotation per time step being $0.6^\circ$, $1.2^\circ$, $2.4^\circ$, and $6^\circ$ were picked. The results, as shown in Fig. 7, indicated that a time step of t = 0.00005 s could effectively capture the transient thrust features, and was therefore selected for further investigations.

Fig. 7

Time step independence test for the drone hovering in ground effect (h = 2R, $\Omega = 4000\ {\text{rev/min}}$)

4 Results

This section assesses the aerodynamic performance of the ducted-fan drone in proximity effects, namely ground effect, ceiling effect, and wall effect, as they happen when the drone flies in restricted areas. The impact of distance from obstacle planes on the aerodynamic behavior is analyzed based on flow field details. The fan’s rotating speed remains constant at 4000 rev/min in all scenarios.

4.1 Ground effect

Results concerning ground effect are shown in Fig. 8. The small error between the experimental value and the simulation value indicates that the simulation has succeeded in capturing the key trends of thrust in ground effect with tolerable deflections. The simulation results are therefore valid for further analysis. Figure 8a shows that the ground effect has limited influence for altitudes at 4R and above. As the drone approaches the ground from 4R to 0.5R, the rotor thrust increases exponentially with ground distance by 26%. The duct thrust first decreases and then increases, with a minimum point observed at around h = 1R. The total thrust has similar trends affected by the duct thrust.

Fig. 8

Aerodynamic performance of the ducted-fan drone operating in ground effect: a thrust; b rotor torque; c FM

There is high consistency between the rotor thrust and the rotor torque. As shown in Fig. 8b, the torque coefficient increases almost exponentially with altitude, indicating that the propulsion system is subject to a heavier workload near the ground. Moreover, Fig. 8c shows that the figure of merit first decreases and then increases. It is not efficient for the ducted-fan drone to operate in ground effect since the decrease of the duct thrust reduces the increment of the total thrust. Specifically, h = 1R to 1.5R is the least favorable height for actual operations of the ducted-fan drone due to minimum hovering efficiency.

Figure 9 displays the velocity field at different altitudes in ground effect. Note that the velocity magnitude in the subfigures has been normalized with the blade tip velocity. At high altitudes such as 2R or 4R, the airflow is sucked in by the fan rotation and accelerated by the rotor disk, forming a high-speed downward moving wake. The outer wake impacts the ground and deflects along the plane. The inner wake forms a dead water zone represented by two counter-rotating ground vortices below the ducted-fan drone due to the exhaust obstruction. Axial symmetry of the velocity field is generally well preserved in this case.

Fig. 9

Velocity contours for various cases in ground effect: a h = 0.5R, b h = 1R, c h = 2R, d h = 4R

With the decrease of the altitude, the enhanced ground effect reduces the mass flow rate through the rotor disk, especially in the middle of the drone. The ground vortex gradually disappears, and the flow recirculation begins to appear at h = 1R in Fig. 9b. This can be harmful in practical applications as particles on the ground might be sucked and taken into the duct, interfering with normal operations. When the drone hovers close to the ground at h = 0.5R in Fig. 9a, the recirculation disappears because the ground severely restricts the airflow through the ducted fan. The airflow mainly passes through the ducted fan at the blade tip. In this case, the low-speed and high-vorticity flow results in significant unsteadiness inside the flow field and fluctuation in the drone performance.

The change in mass flow rate induced by ground effect influences the blade angle of attack, leading to changes in its aerodynamic characteristics. To better examine the rotor thrust, chordwise pressure distributions at 10%, 50%, and 90% blade heights are calculated and depicted in Fig. 10, where airfoils at each blade height are shown in Fig. 10a. According to Fig. 10c-d, the pressure difference between the blade pressure side and the suction side shows that there is high local blade loading observed at the leading edge, which increases significantly with the decrease of the altitude. On the contrary, low blade loading is seen at the trailing edge, and it remains almost unchanged in ground effect. The expansion of the pressure peak at the leading edge indicates that the ground effect increases the angle of attack in the mid-span and tip regions, leading to the increased rotor thrust. Of all the blade heights, the main factor leading to increased thrust is the blade tip. The thrust at the blade tip increases by 28% as the ducted-fan drone descends from 4R to 0.5R. In addition, the irregular variation of the pressure distributions in Fig. 10b indicates that the ground effect enhances the flow complexity near the fan hub. Flow instability may occur in the blade root regions.

Fig. 10

Chordwise pressure distributions along the blade in ground effect: a diagram of airfoils; b at 10% blade height; c at 50% blade height; d at 90% blade height

In order to study flow features near the duct, Fig. 11 demonstrates the surface pressure distributions and the vorticity distributions characterized by the iso-surface of Q-criterion = 300,000. When the ducted-fan drone is hovering out of ground effect, as shown in Fig. 11d, the pressure on the duct surface is basically the same as the surrounding ambient pressure with noticeable low-pressure areas located at duct lips that create suction on the upper surface. Duct thrust is generated in this way. As the altitude decreases, the duct thrust is affected by both the high-pressure zones at the bottom of the drone and the low-pressure zones at the lip. The shrinkage of the low-pressure zones dominates as the altitude decreases from h = 4R to h = 1R, so the duct thrust decreases. As the altitude decreases below h = 1R, however, the expansion of the high-pressure zones becomes dominant, and the duct thrust consequently increases. This is consistent with the results shown in Fig. 8.

Fig. 11

Pressure distributions on the duct surface and iso-surface of Q-criterion = 300,000 for different cases in ground effect: a h = 0.5R; b h = 1R; c h = 2R; d h = 4R

In terms of the vorticity distributions, there are typical hub vortex and tip leakage vortex shown in the flow field at h = 4R. As the altitude decreases to h = 2R in Fig. 11c, the flow remains relatively stable despite the enhanced hub vortex and tip leakage vortex. For altitudes below 2R in Fig. 11a and Fig. 11b, the decrease in mass flow rate induced by intense ground effect results in the reverse flow near the rotor disk. The mixing of the reverse flow and the mainstream enhances the flow complexity, and a large number of vortex structures begin to appear between blade passages. These unsteady vortex structures are mainly concentrated between the fan and the ground, accounting for the unsteady behavior of the drone in ground effect.

4.2 Ceiling effect

Figure 12 shows the aerodynamic features of the ducted-fan drone hovering in ceiling effect. Consistent results are demonstrated between the experimental and simulation data with a maximum discrepancy of 7%. The aerodynamic interference of the wires and support columns and the limited area of the acrylic plates may cause differences in the results. According to Fig. 12a, the ceiling effect has a limited influence on the drone performance for distances larger than 4R. The rotor thrust accounts for 55% of the total thrust, which is the main contributor in this case. As the drone approaches the ceiling, the total thrust increases exponentially with the distance. Thrust gains are particularly significant below 2R, with a maximum value of 33% at z = 0.5R. The duct thrust gradually replaces the rotor thrust, becoming the primary source of the total thrust.

Fig. 12

Aerodynamic performance of the ducted-fan drone operating in ceiling effect: a thrust; b rotor torque; c FM

According to Fig. 12b, the variation of rotor torque with the distance from the ceiling is consistent with the rotor thrust, and the increase is rather significant below 2R. Figure 12c focuses on the figure of merit in ceiling effect. The figure of merit increases exponentially as the ducted-fan drone approaches the ceiling due to the improvement of duct thrust. It reaches a maximum value at z = 0.5R, with an increase of more than 39% compared with the out-of-ceiling-effect case. The growth in figure of merit suggests that, on the one hand, operating near the ceiling is beneficial to improve hovering efficiency. On the other hand, the design of the flight controller needs to consider the significant thrust gain brought by the ceiling to maintain the drone stability during operations.

Figure 13 displays the velocity contours at various distances from the ceiling. According to Fig. 13d, the presence of a ceiling does not impact the flow features around the ducted-fan drone for distances from the ceiling above 2R. The fan rotation draws air from the surrounding atmosphere, with most of the airflow being accelerated at the blade tips and directed downwards. The combined outlet flow from the four propellers forms a symmetrical high-speed wake below the drone. However, when the drone hovers below 2R as illustrated in Fig. 13a-c, the ceiling limits the incoming flow direction. The narrowing of the flow area at the duct lip leads to significant airflow acceleration, resulting in the expansion of low-pressure zones and increased duct thrust. Flow patterns demonstrate that the ceiling effect diminishes the mass flow rate, particularly near the drone's center. At z = 0.5R in Fig. 13a, the ducted-fan drone experiences a pronounced ceiling effect, causing a notable reduction in the mass flow rate and triggering complex vortex movements. The flow complexity is greatly improved under this circumstance. In actual operation, the ducted-fan drone should maintain a distance of more than 0.5R from the ceiling to prevent the performance fluctuation caused by the movement of the vortex structures from damaging safety.

Fig. 13

Velocity contours for various cases in ceiling effect: a z = 0.5R; b z = 1R; c z = 2R; d z = 4R

Detailed chordwise pressure distributions at various blade heights in ceiling effect are depicted in Fig. 14. Figure 14a shows the schematic airfoils at each blade height. According to Fig. 14b-d, the region near the blade leading edge experiences the highest blade loading, which rises as the distance from the ceiling decreases. The widening of the low-pressure peak suggests that the ceiling effect increases the blade angle of attack, leading to increased rotor thrust. Among different blade heights, the root region and the mid-span region are greatly influenced by ceiling effect, and the local thrust increases by 48% and 38% when the distance changes from 4R to 0.5R, respectively. In contrast, the blade tip region makes little contribution to the thrust gain. The thrust at 90% blade heights at 0.5R even decreases by 5% compared with the out-of-ceiling-effect case. This can be well explained by the changes in the mass flow rate due to the ceiling effect in Fig. 13. That is, the ceiling effect mainly reduces the flow rate in regions with low blade heights and has an insignificant effect on that in regions with high blade heights.

Fig. 14

Chordwise pressure distributions along the blade in ceiling effect: a diagram of airfoils; b at 10% blade height; c at 50% blade height; d at 90% blade height

Figure 15 illustrates the pressure distributions on the duct surface as well as the vorticity distributions characterized by the iso-surface of Q-criterion of 300,000. When the ducted-fan drone operates away from the ceiling as shown in Fig. 15d, pressure distributions on the fuselage are largely uniform with noticeable low-pressure regions located at duct lips, creating suction on the upper surface. As the distance from the ceiling becomes smaller in Fig. 15a-c, low-pressure regions on the upper surface start to expand over the duct lip and extend to adjacent ducts. This leads to a higher pressure difference between the upper and lower sides, resulting in an increase in duct thrust.

Fig. 15

Pressure distributions on the duct surface and iso-surface of Q-criterion = 300,000 for different cases in ceiling effect: a z = 0.5R; b z = 1R; c z = 2R; d z = 4R

With regard to vorticity distributions, there are typical hub vortex and tip leakage vortex when the drone hovers out of ceiling effect. The intensity of the vortex increases slightly with the decrease of the distance. There is no notable enhancement until about 0.5R. At a distance of z = 0.5R in Fig. 15a, the reduced mass flow rate results in a significant enhancement of the hub vortex and tip leakage vortex. The intense ceiling effect also leads to the emergence of active ceiling vortices between the ceiling and the drone, which continue to grow beneath the rotor disk. The movement of complex vortices is then responsible for complex flow fields and unsteady drone behavior. It can be seen by comparing the drone performance in ground effect and ceiling effect that both of them have a significant impact on the thrust, but the ground effect limiting the outlet flow is more likely to cause the flow instability.

4.3 Wall effect

Figure 16 displays the aerodynamic performance of the ducted fan drone hovering in wall effect. The experimental results well match the simulation data with a maximum error of 5%, indicating that the CFD results are valid for further analysis. It can be distinguished in Fig. 16a-b that the wall effect has an insignificant effect on the thrust and torque of the ducted-fan drone. The total thrust stays almost constant as the drone approaches the wall, with a maximum change of up to 2%. The change of figure of merit in Fig. 16e is consistent with the change of total thrust, which stays constant for distances greater than d = 3R and falls afterwards.

Fig. 16

Aerodynamic performance of the ducted-fan drone operating in wall effect: a thrust; b rotor torque; c lateral force; d pitching moment; e FM

Changes in lateral force and pitching coefficient, nevertheless, are particularly noticeable in wall effect. Figure 16c displays the lateral force at various distances from the wall. As the drone approaches the wall from a distance, the lateral force coefficient rises exponentially from 0 to 0.00737 when the distance changes from 10R to 0.5R. Similarly, the pitching moment coefficient in Fig. 16d follows the same trend that it rises gradually to 0.0301 from d = 10R to d = 0.5R. Note that the maximum lateral force at 0.5R only accounts for 3% of the total thrust, so the lateral force has minimal effect on the drone’s flight stability. More attention should be paid to reducing the pitching moment since it may pose great difficulties for the control of the aircraft by impairing the maneuverability and versatility of the ducted-fan drone.

Figure 17 demonstrates the normalized velocity fields at different distances from the wall. At a great distance such as d = 4R shown in Fig. 17d, the general flow patterns around the ducted-fan drone are identical to those for hovering in open space. The inlet and outlet flows maintain good symmetry because of the weak wall effect. As the drone approaches the wall in Fig. 17a-c, however, the wall effect pushes the inlet velocity contour to shift away from the wall and the outlet flow to tilt towards the wall, which leads to significant flow asymmetry. Due to the presence of the solid wall, the airflow near the wall has zero normal speed and the streamlines are redirected to vertical abruptly. The flow in the narrow gap between the wall and the drone is apparently accelerated, thereby inducing uneven pressure distributions on different sides of the fuselage. On the contrary, little influence is observed in the performance of the four propellers because the mass flow rates through the four ducted fans are basically identical. From this aspect, the duct is the key component to determining the aerodynamic performance of the drone in wall effect since the lateral force and pitching moment mainly come from it.

Fig. 17

Velocity contours for various cases in wall effect: a d = 0.5R; b d = 1R; c d = 2R; d d = 4R

Figure 18 shows the pressure distributions on the duct surface as well as the vorticity distributions characterized by the iso-surface of Q-criterion of 300,000. It can be found from the figure that the wall effect mainly affects the pressure distributions instead of the flow stability. When the drone operates away from the wall in Fig. 18d, the flow field and surface pressure distributions exhibit a relatively average distribution, so the thrust in the axial direction is worth attention while the forces and moments in other directions can be ignored. As the drone hovers near the wall, the acceleration of the near-wall airflow leads to a decrease in the static pressure on the near-wall vertical plane, while the pressure on the vertical plane on the other side remains unchanged. The drone has a tendency to move towards the wall as a result of the generated lateral force. Besides, the pressure difference between the two sides increases with the decrease of the distance from the wall, which explains the reason for the increase of lateral force in Fig. 16.

Fig. 18

Pressure distributions on the duct surface and iso-surface of Q-criterion = 300,000 for different cases in wall effect: a d = 0.5R; b d = 1R; c d = 2R; d d = 4R

The pitching moment is created due to the difference in the duct thrust in each zone. For zones near the wall, the wall effect leads to the extension of high-pressure regions on the upper surface towards the wall and the shrinkage of high-pressure regions on the lower surface, so the pressure difference between the upper and lower sides decreases. The duct thrust in these zones correspondingly decreases. For zones away from the wall, there exist no significant changes in the surface pressure distributions compared with the case in open space. As a consequence, the pitching moment created by asymmetric duct thrust makes the drone tend to overturn towards the wall. It is necessary to use a robust flight controller to resist lateral force and pitching moment because the drone has a risk of crashing on the wall.

5 Discussions and conclusions

The paper explores the unsteady aerodynamic behavior of a ducted-fan drone in typical proximity effects using the URANS method and the sliding mesh technique. The findings lead to the following conclusions.

• (1) Ground effect

The ground effect limits the outlet flow of the ducted-fan drone, causing the wake to turn radially and form obvious ground vortices. The reduced mass flow rate causes the blade angle of attack to rise, especially in the tip region. The rotor thrust experiences an exponential rise with increasing height from the ground, reaching a maximum value of 26%. The duct thrust is couplingly influenced by low-pressure regions at the duct lip and the high-pressure regions at the lower surface of the drone. It decreases first and then increases with height, and a minimum value appears at h = 1R. The change of total thrust aligns with the change of duct thrust.

• (2) Ceiling effect

The ceiling effect mainly influences the incoming air flow of the ducted-fan drone, causing the airflow to transition from radial to axial above the inlet. The reduced mass flow rate leads to an increase in the blade angle of attack, especially in the root and mid-span regions. The rotor thrust increases exponentially as the drone ascends, with a maximum value of 19%. In the meanwhile, low-pressure regions formed on the upper surface expand over the duct lip and extend to adjacent ducts, enhancing the duct thrust. Overall, the total thrust displays an upward trend. The total thrust rises by nearly 33% in ceiling effect compared with that in open space.

• (3) Wall effect

The wall effect mainly affects the flow symmetry, resulting in the generation of forces and moments along the non-z direction. The thrust properties stay largely constant regardless of the wall distance. The main influence is reflected in the lateral force and pitching moment, and the duct plays a key role in the changes of aerodynamic performance. The acceleration of the near-wall airflow leads to an increase in the static pressure between the near-wall vertical plane and the opposite plane, which contributes to the growing lateral force. Moreover, the extension of high-pressure regions on the upper surface and the shrinkage of high-pressure regions on the lower surface at the near-wall side of the duct increase the pitching moment. The generated lateral force and pitching moment are dangerous during actual flight because they make it possible for the drone to crash towards the wall.

The current work thoroughly examines the aerodynamics of a ducted-fan drone in proximity effects and reveals a feasible range for steady and safe operations. The suggested minimum ground distance and ceiling distance are 2R and 1R, respectively, since intense proximity effects may lead to potential flow instability. The suggested minimum wall distance is 0.5R, as long as the lateral force and pitching moment can be controlled by a robust flight controller. The results provide a solid and essential theoretical basis and data verification for similar-type ducted-fan drones operating in restricted spaces. Nonetheless, our research does not evaluate the influence of geometric parameters on performance or take the dynamic characteristics into consideration. The combination of ground, ceiling, and wall effects has not been assessed either. These factors have a profound impact during the actual flight, which may require future investigations.