Numerical investigation on unsteady characteristics of ducted fans in ground effect

Yiwei LUO, Tinfu AI, Yuhng HE, Bin XU, Yuping QIAN,*,Yngjun ZHANG

a State Key Laboratory of Automotive Safety and Energy, School of Vehicle and Mobility, Tsinghua University, Beijing 100084, China

b Vehicle Research Center, School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China

c Beijing Institute of Technology Chongqing Innovation Center, Chongqing 401120, China

KEYWORDS

Abstract Ducted fans are widely used in various applications of Unmanned Aerial Vehicles(UAVs)due to the high efficiency,low noise and high safety.The unsteady characteristics of ducted fans flying near the ground are significant, which may bring stability problems.In this paper, the sliding mesh technology is applied and the Unsteady Reynolds Averaged Navier-Stokes(URANS)method is adopted to evaluate the influence of ground on the aerodynamic performance of ducted fans.The time-averaged results show that the ground leads to the decrease of duct thrust, the increase of rotor thrust and the decrease of total thrust.The transient results show that there exist small-scale stall cells with circumferential movements in ground effect.The stall cells start to appear at the blade root when the height is 0.8 rotor radius distance, and arise at both the blade root and tip when the height drops to 0.2.It is found that the unsteady cells rotate between blade passages with an approximate relative speed of 30%-80%of the fan speed,and lead to thrust fluctuations up to 37% of the total thrust.The results are essential to the flight control design of the ducted fan flying vehicle, to ensure its stability in ground effect.

1.Introduction

Due to high mobility and excellent trafficability, UAVs with vertical take-off and landing functions are widely used in military and civil fields to perform various complicated tasks.1,2Ducted fans as their propulsion units have therefore attracted more and more attention of researchers.3,4The ducted fan is composed of a duct and fan blades or propeller blades.Its high propulsion efficiency, low noise and high safety make it competitive compared to the propeller of the same size, especially in narrow terrain operations.2,5,6When the ducted fan operates in a narrow environment, the aerodynamic interference of the restrictive environment on the ducted fan is remarkable, and the main challenge lies in the Ground Effect(GE).The ground will affect the flow near the outlet and alter the natural development of its wake,resulting in changes in the duct thrust and rotor thrust.7,8Severe ground effects may even lead to the instability of the ducted fan.9

Although ducted fans have been developed for decades since 1930s,10research on ducted fans in GE began in recent years.The researchers mainly applied steady CFD numerical simulations combined with experimental verification to explore the aerodynamic characteristics of ducted fans in ground effect.A numerical study by Mi11found that the ground,water surface and dynamic waves have a blocking effect on the aerodynamic characteristics of the ducted fan.The jet flow at the outlet is bounced and re-inhaled in his case,and the soft water surface has a weaker effect compared to the rigid ground,resulting in a smaller increase in thrust.Deng et al.12adopted PIV measurements to assess the aerodynamic performance of the ducted fan under three cases including hovering, forward flight and ground effect.It was found that when the ducted fan flies near the ground, the back pressure at the exit would be increased, thereby reducing the thrust generated by the duct.Ai et al.1studied the dynamics modeling and control of a novel coaxial ducted fan aerial robot in ground effect.They found that the ground effect would result in additional thrust gain of the ducted fan,which could be compensated by robust controllers such as a nonlinear disturbance observer combined with a double-loop PID controller.Han et al.7compared the aerodynamic performances of the open rotor and the ducted fan hovering in ground effect through experimental measurement and CFD simulation.They pointed out that the aerodynamic performance of the ducted fan is more sensitive than that of an open rotor,and the ducted fan has higher efficiency than propellers of the same size in both In-Ground-Effect(IGE) and Out-of-Ground-Effect (OGE).

Some other studies discussed the aerodynamic characteristics of a contra-rotating ducted fan in ground effect.Han et al.13investigated the performance of the contra-rotating ducted fan in ground effect using CFD simulation.The influence of key parameters such as the rotating speed, the mounting angle, the rotor spacing, and the tip clearance on its performance is discussed.The results are verified by experimental data.Wang et al.14studied the thrust characteristics and the jet spectrum characteristics of the ducted fan with the distance from the ground.It was mentioned that the dynamic characteristics of the fan jet are significantly correlated with the unsteady interference effect between the upper and lower rotors when the height is decreased to 2.2R (R is the radius of the rotor).

It has been reached a consensus that the height affected by the ground effect is usually within 2R,and in some cases could extend to 4R, which is slightly wider compared with helicopters.15Moreover, with the decrease of the height above the ground, the ground effect would cause the duct thrust to decrease and the rotor thrust to increase.

Few relevant studies investigated the unsteady characteristics of ducted fans in ground effect, although they are highly significant.Jin et al.16built a Moore-Greitzer model for ducted fans in ground effect and validated it using three-dimensional unsteady simulations.Three cases were investigated in the research: h = 1.2R, h = 0.5R and h = 0.2R (h is the height from the ground).The ducted fan is in stable operating state when h = 1.2R, and enters into the rotating stall state when h reaches 0.5R.The Fast Fourier Transformation (FFT)results of pressure signals illustrate that the stall cells(vortices that transfer across passages in the opposite direction of fan rotation)appear in the tip area,of which circumferential speed is 21.4% of the rotating speed.When the height is further reduced to 0.2R,the ground effect causes large radial flow distortion at the outlet.Jin’s research provides a new perspective for the unsteady characteristics of ducted fans in ground effect but lacks evidence to make it a general theory.The evolution process of stall cells is not systematically revealed in the flow field, and the correlation between thrust characteristics of ducted fans and stall cells is not explored, either.

In this paper,the unsteady CFD method is applied to investigate the time-averaged and transient aerodynamic performances of ducted fan in ground effect, and the physical principles behind the phenomena are analyzed.We hope that this study provides a theoretical basis for the flight control design of the ducted fan flying vehicle.The remaining parts are organized as follows: Section 2 introduces the geometric model and numerical simulation methods of ducted fan.Section 3 presents the validation of the CFD method.Section 4 compares and analyzes the time-averaged and transient results under different heights above the ground.Section 5 draws the conclusions.

2.Methodology

2.1.Geometrical models

The investigated ducted fan shown in Fig.1(a) has three equally spaced blades.The propeller and duct diameters are d = 206 mm and D≈292 mm, respectively, while the duct length is l = 50 mm.The gap distance between the blade tip and the duct is δ = 0.91 mm.The entire duct is flat and can be integrated with the fuselage, which is suitable for UAV and similar applications.Its specific parameters are shown in Fig.1(b).The pitch angle ranges from 40.63° at the hub to 20.35°at the tip.The maximum chord length appears at nearly half the blade height.

2.2.Grid generation

The entire computational domains are divided into stationary zones without the ducted fan (b1 and b2) and a rotating zone(c) including the ducted fan, as shown in Fig.2.The overall domains are of cylindrical shape,and the ducted fan is located above the bottom of the cylinder,which is used to simulate the ground.The vertical position of ground and ducted fans will vary in different numerical simulation cases.

Fig.1 Configurations of investigated ducted fan.

Fig.2 Scheme of overall zone settings and mesh grids.

The structural and hexahedron grids were carefully constructed to address all important flow regions: surface boundary regions, blade tip regions and the ground regions.The structural mesh in static zone was generated based on ANSYS ICEM CFD.Due to the complexity of the flow field near the ground, the Geometric1 and Geometric2 methods were used to set the number of nodes in grid division to ensure that the grid near the ducted fan and the ground was fine, while the grid far away was relatively coarse.The meshing strategy was effective in enhancement of the calculation accuracy.The overall mesh quality of the static zone was above 0.7.The structural mesh in rotating zone was generated using NUMECA AutoGrid5, which is a powerful grid generation software in turbomachinery.The tip refining region and boundary layer mesh are shown in Fig.3.The number of boundary layers at the blade was set to 17.The minimum skewness of grids was beyond 19, the maximum aspect ratio was about 600, and the maximum expansion ratio was 3.85.The mesh quality has far exceeded the requirements of 3D mesh.The entire numerical domain had approximately 12 million cells, of which the rotating domain and the static domain each account for half.

2.3.CFD method

In this paper, the numerical simulation of the ducted fan in ground effect was conducted using the URANS approach in the commercially available ANSYS FLUENT solver.The blade tip speed is below Mach number 0.15 with the rotating speed Ω = 4000 r/min.Hence flow can be regarded as incompressible flow and the pressure-based solver was selected for simulation.The Shear Stress Transport (SST) k-ω turbulence model was adopted for better capture of the flow characteristics of the interactions between the blade and the ground17,18.The coupled algorithm was used to solve continuity, momentum and energy equations simultaneously owing to its high robustness, and the second-order upwind scheme was applied for better accuracy.The Multiple Reference Frame (MRF)model was chosen for steady-state calculation, the results of which served as the input for transient simulation.The sliding mesh technique was developed to simulate the unsteady flow phenomenon brought by the ground.

Fig.3 Detailed mesh of surface of ducted fan.

For the boundary conditions,Fig.2 exhibits the settings in the static zone and the rotating zone.There is a sum of four pairs of interfaces adopted for connection between zones,which are all set as matching interfaces to ensure accurate data exchanges.The static zone contains the outer duct,the ground,and ‘‘Far”.‘‘Far” was set to the Velocity inlet with a magnitude of zero, while the others were kept Wall.The dynamic zone is the zone between two interfaces and the inner duct.Since the inner duct does not rotate with the propeller, it was separately set as the non-rotating Wall in the global reference frame.

3.Validation

The experimental tests of the investigated ducted fan were carried out in Chongqing Innovation Center of Beijing Institute of Technology.The test bench is shown in Fig.4, with the top view of the test propeller.PWM signal was used to control the motor throttle position, and the data acquisition system with acquisition frequency of 20 Hz was applied to record the results.The tension sensor accuracy was 0.1% ± 20g(0.1% of the displayed reading plus 20g) during the test.

The calculated thrust and torque coefficients of the ducted fan in both the experimental and computational tests are shown in Fig.5, where the coefficients are defined as

where T is the thrust, Q is the torque of the rotor,ρ=1.225 kg/m is the air density,and A is the rotor disk area.According to Fig.5,the numerical results of different rotating speeds are in good agreement with the experimental data,with all errors less than 10%.It should be noted that due to the arrangement of the device in the experiment,some wiring harnesses and controllers will interfere with the aerodynamics of the ducted fan, which is ignored in the simulation.This accounts for the calculation error to some extent.

In terms of grid independence verification, we changed the clearance of the nodes in the static zone,while kept the grids in the rotating zone basically unchanged.The total cells of generating grids were 6 million,8 million and 12 million.According to Table 1,the simulation results of thrust coefficients of these grids were 0.06378, 0.06497 and 0.06493, and the corresponding errors were 3.36%, 1.56%, and 1.62%, respectively, indicating that 8 million mesh cells were enough to meet the requirements of calculation accuracy of the ducted fan.In this paper,the best mesh with 12 million cells was selected for subsequent computational research.

The unsteady characteristics of the ducted fan in ground effect are prominent, especially when it operates fairly close to the ground, and thus the independence of time step should be paid further attention to.Based on the verified computational grid,we selected different time steps based on our experience in the turbomachinery field to evaluate the most appropriate value for the aerodynamic performance prediction of the ducted fan in ground effect.Time steps were set to Δt = 0.00025 s, 0.0001 s, 0.00005 s, and 0.000025 s, corresponding to the rotation angles per step of 6°, 2.4°, 1.2°and 0.6°.We chose the case of hovering at the height of h = 0.5R as an example, the rotating speed was 4000 r/min,and the range of results processed was from the beginning to the 6th revolution.The simulation results are shown in Table 2.It should be noted that the time statistical characteristics of the thrust were considered rather than the specific value at a certain moment.The calculated data with Δt = 0.00005 s and 0.000025 s have basically the same statistical values.Therefore,the time step was set to Δt = 0.00005 s, and all unsteady simulations were performed based on this time step.

Fig.4 Experimental test configurations of ducted fan.

Fig.5 Aerodynamic thrusts and torques of ducted fan in open space.

Table 1 Thrust coefficient of ducted fan with different mesh density.

Table 2 Transient performance of ducted fan with different time steps (h = 0.5R, Ω = 4000 r/min).

4.Results and discussion

Owing to the symmetrical structure of the ducted fan, according to the existing studies,19,20the most significant changes of the ducted fan in ground effect lie in the rotor thrust and duct thrust, while the changes of force and torque in other directions are apparently ignorable.This section reveals the timeaveraged and transient thrust characteristics of the ducted fan in ground effect, analyzes the causes of the timeaveraged characteristics on the basis of three-dimensional flow field, and analyzes the causes of the transient characteristics based on three-dimensional flow field and one-dimensional pressure signal.

In order to evaluate the aerodynamic performance of the ducted fan, the following dimensionless coefficients will be used for quantitative analysis, including power coefficient CP,Figure of Merit FM,pressure coefficient Cp,velocity coefficient Cveland dimensionless vorticity Cvor.

where P denotes the mechanical power,p denotes the absolute pressure, p∞denotes the atmospheric pressure, V denotes the velocity magnitude, and ω denotes the vorticity.

4.1.Performance of ducted fan in ground effect

Based on Ref.11 and Ref.15 for single propellers,the influence range of ground effect is no more than h/R = 2.0, while for ducted fans, the range limit is h/R = 4.0.Zones above h/R=4.0 can be considered out of ground effect.In this paper,the cases with heights above the ground given by h/R = 0.2,0.3,0.5,0.8,1.0,1.5,2.0,3.0,4.0,5.0 are investigated at a constant rotating speed of Ω = 4000 r/min.The corresponding results are shown in Fig.6, where TIGErefers to the thrust in ground effect and TOGErefers to the thrust out of ground effect.Statistical features are captured from the 10th revolution to the 15th revolution.The figure demonstrates two thrust characteristics of the ducted fan in ground effect.

4.1.1.Time-averaged thrust characteristics

The impact of ground effect on thrust shows an exponential change as a whole.For heights h/R above 3.0, the ground shows little effect on the thrust.The impact gradually becomes significant with h/R lower than 3.0.When h/R drops below 1.0,the thrust changes sharply with the height above the ground.Compared with the open space without any interference, the rigid ground observably increases the rotor thrust, reduces the duct thrust, and leads to a decrease in the total thrust.An abnormal value occurs at h/R=0.2,because a large number of stall cells appear at the blade tip region, and the increased leakage results in the loss of rotor thrust.This phenomenon will be further explored in Section 4.3.1.The minimum total thrust TIGE= 0.52TOGEis achieved when h/R reaches 0.2.In order to provide a reasonable estimate of the time-averaged total thrust ratio, an empirical exponential model is applied to fit the curve,21which is shown as

Fig.7 illustrates the variation of FM with the heights above the ground.The FM basically remains unchanged for heights above h/R=2.0,and decreases significantly with the decrease of height below h/R = 2.0.When the height h/R = 0.2, the FM is decreased by 0.60 compared with that without ground effect.The change of FM is generally consistent with the change of total thrust, because the ground effect brings relatively little effect on the rotor torque.

4.1.2.Transient thrust characteristics

In addition to the time-averaged characteristics of the thrust of the ducted fan, the transient characteristics come into prominence with the decrease of the height, which is not mentioned in other literatures.With the decrease of the height,the ground effect leads to the fluctuation of the rotor thrust and the duct thrust.The left figures in Fig.6 show the range of thrust fluctuation, which is represented by error bars.

Fig.6 Thrust characteristics of ducted fan at different heights above ground.

Fig.7 Figure of merit of ducted fan in ground effect.

There is an identical trend in the variation of the rotor thrust and the duct thrust among all the computational results,that is,with the decrease of height,the degree of thrust fluctuation gradually increases.When h/R is above 2.0, the thrust fluctuation is approximately nonexistent, and the thrust can be represented by a constant.When h/R = 1.0, the unsteady characteristics caused by ground effect are significant, and the fluctuation is remarkably enhanced.When h/R = 0.2,the value of rotor thrust fluctuation accounts for 17% of the mean rotor thrust, the value of duct thrust fluctuation accounts for 89% of the mean duct thrust, and that of total thrust fluctuation accounts for 37% of the mean one.The results imply that the thrust fluctuation is non-negligible for thrust modeling and controller design.In the meanwhile, by comparing the fluctuation of the duct thrust with that of the rotor thrust, it is apparent that the fluctuation amplitude of the duct thrust is larger than that of the rotor thrust, and the closer it is to the ground,the larger the difference is.There is an interesting phenomenon that when the height above the ground is reduced to a threshold,the duct thrust may turn negative.To be specific,when the height h/R equals 0.3,there is a negative duct thrust sometimes during the process, and when the height h/R equals 0.2, the duct thrust is always in negative value.It can be deduced that the contribution of the duct to the total thrust may be negative when the exhaust at the outlet of the duct fan is severely restricted.Standard Deviation (SD)is an indicator to describe the degree of dispersion of the series in statistics.The larger the SD is,the larger the deviation of the time series data as a whole is.The figures on the right of Fig.6 shows the change of SD with height, from which it is obvious that the SDs of all three thrust forces grow larger with the decrease of height.The results of range and SD together indicate the intensification of thrust fluctuation with decreased height.

Aiming to explore the influence of rotational speed on thrust fluctuations, we selected the operating condition of h/R = 0.5 for further investigation, and set the rotational speed as 2000 r/min, 4000 r/min and 6000 r/min respectively.The range and SD of thrust variation were calculated and summarized in Fig.8.With the increase of rotational speed, the absolute values of the range and SD of the duct thrust, rotor thrust and total thrust increase,while they are in a similar proportion to their average values.The contribution of the rotor thrust is greater than the duct thrust in the fluctuation of the total thrust, which is consistent at different rotating speeds.

4.2.Analysis of time-averaged thrust characteristics

In the previous section, it is mentioned that in ground effect the duct thrust drops,the rotor thrust rises,and there is a loss existing in total thrust.The change in the flow field brought by ground is the key to the thrust characteristics of the ducted fan.In this section, the time-averaged flow fields at different heights are compared and analyzed.

As indicated from Fig.9,when there is no ground effect,as shown in Fig.9(b), the inlet flow forms a high-speed zone at the duct lip, leading to a low-pressure zone.Most of the inlet flow passes through the rotor to form a vertical downward outlet wake, while a small part produces tip leakage vortex through tip clearance.Through the rotor rotation, the kinetic energy is converted into the pressure potential energy,resulting in pressure difference between the upper and lower rotor planes.In contrast, the ground mainly affects the outlet flow of the ducted fan, as shown in Fig.9(a).The ground induces the flow in the duct diffuser section to shift to the inner wall of the duct,and reduces the pressure in the nearby region.The downwash wake flow is turned and stretched outward by the ground.A high-speed wall jet is thereby formed along the ground plane.At the center of the outlet, a dead water zone is generated due to the exhaust obstruction.Large-scale and counterrotating ground vortex appears between the rotor and the ground, which occupies half the outlet zone.The flow of low speed and high vorticity leads to the unsteadiness in the performance of the ducted fan.

Fig.8 Thrust fluctuations under different rotating speeds (h/R = 0.5).

Fig.9 Time-averaged velocity and pressure fields for different heights above ground plane.

In an effort to further study the mechanisms for the thrust change in ground effect, the time-averaged Mach numbers of the blade section at both 50% blade height and 90% blade height with the change of height are extracted from the computational results.As indicated in Fig.10, the rolled-up ground vortex forms an intake distortion for the rotor, which pushes the stagnation point at the blade tip to move towards the pressure side.It demonstrates that the angle of attack of the blades is gradually increased when the ducted fan approaches the ground.In the meanwhile, there is no flow separation found on the suction side at both 50% and 90% blade heights,though the boundary layer at the trailing edge becomes thicker, indicating that the flow is stable for all cases.As a result, the turning angle and aerodynamic load of the blade increase,which contributes to the rise of rotor thrust in ground effect.

The change of time-averaged static pressure on the suction surface and pressure surface of the blade induced by the change of angle of attack is shown in Fig.11.The direction of the arrow indicates the rotational direction of the rotor.When the ducted fan is in hovering state, a low-pressure zone appears near the blade tip and close to the Leading Edge(LE)on the suction side of the blade,which is related to the high tip linear velocity and the flow separation over the suction surface.On the contrary, a high-pressure zone arises near the Trailing Edge (TE) on the pressure side.The pressure difference between the two sides produces the rotor thrust.The presence of the ground changes the location and size of the low- and high-pressure zones.As the height above the ground decreases,the low-pressure zone on the suction side undergoes a gradual shift to the blade tip in the chord direction,and extends along the spanwise direction from the blade tip to the blade root.The spread of low-pressure zone is gradually obvious, especially during the change from h/R=1.0 to h/R=0.5,and the average static pressure decreases.The change trend is true of the high-pressure zone on the pressure side, which experiences a major extension towards the blade tip and leads to an increase in the average pressure.As a consequence, the rotor thrust increases when the ducted fan is approaching the ground.

Fig.12 demonstrates the static pressure distribution around the duct section, which is azimuthally averaged and exhibited in two-dimensional coordinates.As the flow accelerates into the rotor disk in hovering,the pressure from A to B stays basically equal to atmospheric pressure with little change, while a dramatic drop is noticed in proximity to B.The zone from B to C corresponds to the duct lip section and a large low-pressure area is obtained.There exists a turning point of the pressure coefficient at C, and another low-pressure zone appears from C to D, which is related to the flow separation in the diffuser section.From D to A,the pressure rises as a result of changes in the velocity according to Bernoulli’s principle and forms a high-pressure zone around the duct.Considering that the duct thrust is generated from the vertical component of the pressure difference between the inner duct and the outer duct, the lowpressure zone from B to C and the high-pressure zone from D to E act as the main sources of the duct thrust.The zone from C to D contributes little to the duct thrust because its normal direction is almost perpendicular to the vertical direction.Due to ground effect,the airflow is decelerated at the duct lip and a high back pressure zone is formed at the duct outlet.As the height decreases, the low-pressure zone of section B-C shrinks dramatically, of which the peak value increases.In the meanwhile, the pressure in the high-pressure zone corresponding to section D-E increases slightly.As a result, the reduction in pressure difference between B-C section and D-E section contributes to the drop in the duct thrust.When the height is low enough, chances are that the pressure in the section B-C rises beyond that in the section D-E.Negative duct thrust is generated then.

4.3.Analysis of transient thrust characteristics

Fig.10 Time-averaged Mach number of blade section for different heights above ground plane.

In accordance with Fig.9, the time-averaged flow field diagram of y-z section indicates the existence of tip leakage vortex and ground vortex when the ducted fan hovers in ground effect.It is highly likely that the above unsteady flow phenomena contribute to the thrust fluctuation.In this section, the influence of the vortex structures on thrust fluctuation from the perspectives of three-dimensional flow field analysis and one-dimensional pressure analysis is presented and illustrated.

4.3.1.Transient thrust characteristics

According to Fig.13,the visualization of the vortex structures is achieved by applying the Q criterion of 4000000 to the instantaneous flow field.They appear mainly due to the unsteady flow in the tip clearance and blade root region.When h/R = 2.0, the vortex shown in the blade tip region is the tip leakage vortex, which is a three-dimensional vortex structure with large span formed along the tip clearance, and occurs commonly in all operating conditions.The vortex structure in the blade root-hub corner region is mainly wall corner vortex.Due to the ground effect, a conical stagnation zone is formed between the hub and the ground.The wall corner vortex is therefore unable to extend directly towards the outlet of the duct,but flows back to the hub plane and develops towards the leading edge of downstream blades.With the decrease of height,the number of vortex cores near the blade root encounters a slight increase, indicating that the conical stagnation region between the blade and the ground is expanded.There is no significant change of the vortex structure in the tip region until h/R = 0.2.At this height, the leakage flow in the tip region becomes intensified and disordered, which explains the fact that the thrust drops suddenly at h/R = 0.2 in Fig.6.

For a clearer view of the vortex structures, Fig.14 shows the z-direction vorticity in one rotation period at different heights above the ground.The intercept time is selected for the 10th revolution to the 11th revolution,and the cut-off slice at z-planes is located at the trailing edge of the blade.The results in hovering condition demonstrate that there are two kinds of vortices in each blade passage,namely tip leakage vortices at the blade tip and hub vortices at the root.These vortices stay in a single passage and no cross-passage transfer occurs.In contrast,the periodic circumferential transfer across passages of vortices, which are defined as stall cells, can be clearly distinguished for cases in ground effect.The stall cells mainly occur at the blade root region when h/R lies at 0.2,0.3, or 0.8, and their intensity increases with the decrease of height.When h/R=2.0 or even higher,the stall cells at the root region principally disappear and there exists no obvious crosspassage transfer, which is verified by the nearly zero thrust fluctuation in Fig.6.The intensity of tip leakage vortices increases with the decrease of height as well.When h/R falls to 0.2, the intensity of stall cells at the root region remains almost unchanged, while the leakage vortex at the tip region is enhanced and transfers among three blade passages.During the process, large-scale stall cells are generated in the tip region, forming a circular closed-loop in the whole space.

Fig.11 Time-averaged pressure coefficient distribution on rotor for different heights above ground plane.

Fig.12 Pressure coefficient distribution around duct section for different heights above ground plane.

For the comparison of phenomena at different time,taking h/R = 0.3 and h/R = 0.8 as examples, we marked one of the stall cells with yellow circle in each figure to illustrate their transfer process between passages.The transfer direction of stall cells relative to the blade is opposite to the direction of blade rotation, and their transfer rate is less than the blade rotation speed.As demonstrated in Fig.14, when the blade rotates 0.8 revolutions, the stall cells in both cases rotate an angle of about 120° relative to the blade.It indicates that the decrease of height may have little effect on the relative transfer rate of the stall cells at the same region.

The relationship between z-vorticity and rotating speed is explored in Fig.15.The height above the ground is fixed to h/R = 0.5, and several different rotating speeds are studied,namely Ω = 2000 r/min, 4000 r/min, and 6000 r/min.Similar to Fig.14, stall cells mainly appear and transfer from one blade passage to another in the blade root region, while no obvious stall is observed in the blade tip region.The figure clearly indicates that the increase of the rotating speed leads to almost a linear increase in the vorticity.Moreover,it is easy to distinguish from the comparisons among three cases that the rotating speed of the blade has an effect on the relative transfer rate of stall cells between passages.When the blade rotates by 0.8 revolutions from the 10th revolution, the stall cells in Fig.15(a) travel at an angle of less than 120° relative to the blade, while the angles travelled by those in Fig.15(b)and Fig.15(c) are approximately equal to and more than 120°, respectively.It indicates that the increase of rotational speed greatly increases the vorticity, but in the meantime, it slightly inhibits the relative motion of stall cells between passages.A possible explanation is that the increase of rotational speed enhances the intensity of all stall cells, and afterwards,the interaction among different cells, resulting in a relatively slow transfer rate of stall cells.

Fig.13 Iso-surface of Q-criterion = 4000000 of ducted fan under various heights above ground.

4.3.2.Pressure signal analysis

The flow field diagram is an intuitive but not accurate approach to explain the thrust fluctuation of the ducted fan underground effect.The stall cells and their propagation in the passages would induce significant pressure drop and cause dramatic pressure fluctuations.In order to quantitatively investigate the relationship between the pressure fluctuation characteristics and the thrust fluctuation characteristics at different positions of the duct fan,multiple monitoring points are arranged at the inlet, middle section and outlet of the ducted fan blade passage,and upstream and downstream of the blade with different blade heights of 10%,20%,and 95%,as shown in Fig.16.In this way, pressure fluctuations at each location can be precisely monitored and recorded.The monitoring points along the central axis of the blade passage are Point 1,Point 2,Point 3 and Point 4.The monitoring points located upstream of the blade leading edge are Point 5 and Point 6,and the monitoring point located downstream of the blade trailing edge is Point 7.All monitoring points are set in the rotating coordinate system, which rotates synchronously with the blades in the global reference frame.The simulation cases in each condition include 15 complete cycles.After 5 rotating cycles, the pressure fluctuation information in time domain and frequency domain of each monitoring point is output and collected.

Fig.17 shows the time domain and frequency domain diagrams of relative pressure signal at 20% blade height at z/h = 1.0.The relative pressure is defined as

where p and pavgdenote the transient static pressure and timeaveraged static pressure at each monitoring point,respectively.According to Fig.17(a), by comparing the pressure changes with time among monitoring points 1–4 and 5–7, a general conclusion can be drawn that the pressure fluctuation of the monitoring points upstream is much smaller than that measured downstream.A reasonable explanation is given by the fact that vortex structures merge with each other, separate from large ones and propagate across passages around and below the rotor disk plane, most of which propagate downward to form unsteady swirling flow in the wake.As a consequence, the background noise upstream is small while the corresponding monitoring points can reflect pressure fluctuations transmitted from downstream area.Fig.17(b)shows that all monitoring points contain similar frequency information and verifies the validity of monitoring points upstream.By comparing monitoring points at the same height,for example,Point 1 and Point 5, we can judge that the monitoring points near the blade will suffer additional interference.Point 1 and Point 2 are then the most appropriate points for monitoring pressure fluctuations,and Point 2 is eventually adopted for following studies.

Fig.14 z-vorticity fields at the trailing edge of blade in one revolution under different heights above ground.

Based on the above monitoring settings, analysis of the pressure signal in the time and frequency domains is carried out.Fig.18 shows the pressure time-domain signals in the tip region and hub regions of different passages of the ducted fan at three different heights.It should be noted that the absolute ordinate value has no physical meaning,but the unit scale value can reflect the magnitude in the pressure fluctuation.As indicated from the figure, there exist pressure fluctuations of low amplitude in the tip and hub regions when h/R = 2.0.When the height is reduced to h/R = 0.8, the periodic fluctuations among different passages are easy to distinguish.The blue dashed line represents the transmission process of pressure fluctuations, which is the key signal to prove the movement of stall cells.By comparing the pressure signals in the tip region and the hub region, it can be found that the disturbance amplitude in the hub region is slightly larger than that in the tip region though they share the same periodicity, indicating that the stall cells are concentrated in the hub region in this case.When the height is further reduced to h/R = 0.2, the pressure fluctuations in the hub region increase slightly, and their transfer rate slows down.By contrast,the pressure fluctuations in the tip area become extremely remarkable.It indicates that there is a larger group of stall cells in the tip area than in the hub area, which is evidently confirmed in Fig.14.

Fig.19 shows the frequency spectrums of pressure and thrust by applying Fast Fourier Transform (FFT) for the 5th revolution to the 15th revolution under different heights above the ground.Among all the cases,the figures on the left refer to the frequency spectrums of the duct thrust and rotor thrust within the above time range, while the frequency spectrum of the instantaneous static pressure measured at monitoring Point 2 in different operating conditions is on the right.At the rotating speed of Ω = 4000 r/min, the fundamental frequency and half frequency of the blade rotation are 66.67 Hz and 33.33 Hz respectively.Since the frequency of rotating stall cells observed in the flow field is usually less than the fundamental frequency,we mainly focus on the frequencies corresponding to the high amplitude below 66.67 Hz.Four conclusions can be drawn as follows:

Fig.15 z-vorticity fields at the trailing edge of blade in one revolution with different rotating speeds.

Fig.16 Locations of monitoring points for transient static pressure.

(1) The pressure frequency spectrum is in good agreement with thrust frequency spectrum, especially at high altitude for the ducted fan.All the core frequencies of thrust fluctuation can be reflected by the pressure frequency spectrums.For example, when the height h/R is fixed at 0.8, the highlighted core frequency of thrust fluctuation is 19.99 Hz,which is consistent in the pressure spectrum.As the height goes down,there may be some slight differences between the two spectra, for example, when h/R = 0.2.The essential frequency of stall cells is still reflected by both of them.

(2) There are differences in the amplitudes of key frequencies of monitoring points with different blade heights.Due to the subsonic flow, all monitoring points can reflect the fluctuation of the flow field pressure signals,but are more sensitive to the adjacent stall cells.When the stall cells appear at the tip of the blade (h/R = 0.2), the monitoring points located at 95 % blade height are more effective in reflecting the fluctuating frequency of stall cells while in other cases are not.Furthermore, due to the fact that the flow in the root and tip regions is relatively complicated, large signal noise is observed at 10%and 95%monitoring points.The monitoring points closer to the middle of blade, such as the 20% monitoring point, are therefore more suitable.

Fig.17 Pressure signals at different monitoring points (h/R = 1.0).

Fig.18 Time domain data of pressure signals at different heights above the ground: (a) h/R = 0.2, (b) h/R = 0.8, (c) h/R = 2.0.

Fig.19 Frequency spectrums of pressure fluctuation and thrust fluctuation by FFT at different heights above ground.

(3) With the decrease of the height above the ground, the number of peaks increases gradually, and their corresponding amplitude increases as well.The increase in amplitude indicates that the fluctuation becomes severe accordingly,which matches the result of Fig.6.When h/R=2.0 or 1.0, there is only one peak below the fundamental frequency, namely 6.66 Hz, which is related to the tip leakage flow of high probability.The stall cells arise at h/R=0.8,and a significant change occurs when the height changes from h/R=0.8 to h/R=0.5,where the peaks appear at 26.66 Hz and 46.65 Hz.It can be inferred that at h/R=0.8,the stall cells transfer between passages as a whole without obvious separation.On the contrary, when the height is lower than or equal to h/R = 0.5, large stall cells are broken into multiple small pieces, resulting in an increase in the peaks of fluctuations.A great number of high-frequency stall cells show up apart from the typical rotating stall cells in these cases.

(4) With the decrease of height, the relative frequency of stall cells is steadily increased.The typical stall cells begin to appear in the h/R = 0.8 case, with a relative rotating speed of 30% of the blade speed.When h/R equals 0.5, the relative rotating speeds of stall cells are increased to 40% (26.66 Hz) and 70% (46.65 Hz).The actual flow is the superposition of multiple stall cells with different rotating frequencies, which is confirmed in Fig.14.When h/R is further reduced to 0.2, a significant peak appears at 53.32 Hz,and accounts for 80%of the blade speed.

Unlike the frequency spectrum of time series data,the spectrum of the static pressure at the averaged-distributed special points in a circle reveals the spatial characteristics of stall cells.Generally speaking, the fundamental wave number, which is calculated by the number of blades divided by 2π,and its multiple wave numbers will dominate in the spectrum.If there are other obvious frequency characteristics different from the above, it indicates that there is a large-scale circumferential disturbance in the circular space.Fig.20 collects the spatial spectrums of pressure fluctuations at different heights.The spectrums are obtained from the transient results at the 10th revolution, and the difference of spectrums at different times can be ignorable based on our tests.The monitoring positions of pressure include the leading edge and trailing edge of the blade, with different blade heights varying from 20% to 98%.The fundamental wave number is calculated to be 3/2π≈0.477 in all cases.As demonstrated in the figure,the fundamental wave number and its multiple wave numbers are dominant, indicating that the stall cells in all cases refer to comparatively small disturbance in the circumferential direction.Despite all this,fluctuations at the leading edge are larger than those at the trailing edge for cases h/R = 0.5 and h/R = 0.2, which means that the stall cells develop closer to the leading edge.The stall cells of the largest scale appear around the leading edge when h/R = 0.2.Via comparison of results at different blade heights,the stall cells are mainly concentrated near 20%of the blade height rather than 98%of the blade height,except when h/R=0.2.It is a verification of evidence for the judgement about Fig.14.

5.Conclusions and future work

In this paper,the sliding mesh technique is used to simulate the rotation of the propeller,and SST model is used for numerical calculation.The time-averaged and transient performances of the ducted fan in ground effect at different heights are compared and analyzed in details.The conclusions are drawn as follows:

(1) Time-averaged characteristics

When the ducted fan hovers in ground effect, the ground causes the flow in the duct diffuser section to shift to the inner wall of the duct.The downwash wake flow is then turned and stretched outward.A high-speed wall jet along the ground plane and a dead water zone at the center of the outlet are thereby formed.As the height above the ground decreases,on the one hand, the intake distortion induced by the rolledup ground vortex increases the angle of attack of the blade,and pushes the low-pressure zone on the suction side and the high-pressure zone on the pressure side to extend along the spanwise direction from the blade tip to the blade root,resulting in an increase in the rotor thrust.On the other hand, the narrowing of the low-pressure zone at the duct lip section and the deviation of the minimum pressure point cause a drop in the duct thrust.The total thrust shows a downward trend.

(2) Transient characteristics

There are significant thrust fluctuations of the ducted fans in ground effect,and the amplitude of the fluctuation increases with the decrease of the ground height.When the height drops below h/R = 0.3, the duct thrust may even be negative.The results of FFT of the static pressure and thrust in time series at the monitoring points show that the thrust fluctuation is directly related to the stall cells travelling between blade passages in the flow field.The stall cells initially appear at the blade root when h/R drops to 0.8, and their relative rotating speed decreases with the decrease of the height.When the height above the ground h/R decreases to 0.2, the stall cells appear at the blade tip as well.The spatial FFT results show that the stall cell of the ducted fan in ground effect is a small-scale disturbance in the circumferential direction, which occurs near the leading edge of the blade.

The present study comprehensively analyzes the aerodynamic characteristics of the ducted fan in ground effect, and clarifies its flow mechanisms based on CFD methods.However, the unsteady effect of the ducted fan under other conditions such as forward flight in ground effect and ceiling effect and hovering in crosswinds have not been explored yet, and the flow mechanisms under dynamic conditions may be much more complicated, which require further investigations.Additional efforts should be made to develop the corresponding control methods in order to reduce the thrust fluctuation.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This study was co-supported by the National Key Research and Development Program of China (No.2020YFC1512500),The Advanced Aviation Power Innovation institution, The Aero Engine Academy of China, and Tsinghua University Initiative Scientific Research Program.