Aerodynamic modeling of ducted fans under environmental disturbances

Ducted fans are widely used in unmanned aerial vehicles (UAVs) to perform various tasks due to high efficiency, high safety and low noise. The external environment affects the stability of the ducted fan, and predicting the aerodynamic characteristics under environmental disturbances is of great significance for stable operations of ducted-fan aircraft. In this paper, we considered the influence of different environmental disturbances on the axial velocity through the rotor disk, and established an aerodynamic mechanism model based on the momentum theory and blade element theory. In order to verify the model validity, three-dimensional CFD (Computational Fluid Dynamics) numerical simulation and bench tests were carried out. The results indicate that the mechanism model can well predict the thrusts and torques of the ducted fan under different flow conditions and wall constraints, and the errors between the model and the simulation results are generally within 12%. Overall, the established model is universal and extendable for different types of disturbances, and has the potential to be applied to evaluate the impact of unknown disturbances.


Introduction
Due to high flexibility and adaptability, unmanned aerial vehicles (UAVs) are widely used in military and civilian fields to perform air operations [1][2][3].Ducted fan propulsion has therefore received extensive attention [4][5][6].The ducted fan is composed of a duct and a fan, and can produce greater thrust than the propeller in the same size, or reach a more compact structure under the same thrust because the duct can generate additional thrust.In addition, the ducted fan has the advantages of low noise and high safety, which makes it strongly competitive in UAV propulsion.However, the ducted fan is easily affected by the surrounding meteorological environment in actual flight, resulting in changes in aerodynamic performance, which poses serious challenges to UAV flight.Environmental disturbances can be divided into two categories as shown in Figure 1, namely, incoming flow disturbances and wall disturbances.Predicting the aerodynamic characteristics of ducted fan under environmental disturbance is of great significance for the stability control of the whole aircraft.Prediction models can be divided into physical-based models and data-based models.Data-based models require a large amount of real data, have poor scalability, and are difficult to reveal the mechanism of performance changes from the perspective of fluid dynamics.The physics-based model starts from the basic and underlying mechanism and has good scalability, so it's still a research hotspot.Traditional physical-based methods for propeller modeling include momentum theory, blade element theory, blade element momentum theory, lifting-line method, lifting surface method, panel method, etc [7].These methods can accurately predict the aerodynamic performance of the propeller, and some of them have been applied to the ducted fan for geometry design [8].
Currently there are few researches discussing about the modeling of ducted fan in non-hovering conditions.Ai [9] proposed a lift mechanism model of a coaxial ducted aircraft based on the momentum theory and the blade element theory.His model considered the relationship between the vertical climbing rate and the induced speed of the rotors and the induced velocity interaction, and exhibits high adaptability and accuracy.Nevertheless, Ai's model is only applicable to the ceiling effect, which is difficult to be extended to other environmental disturbances.
In this paper, we proposed a mechanism model of ducted fan under environmental disturbances, which can be used to predict the influence of different environmental disturbances on the thrust and torque.We selected three typical cases to verify the model to illustrate that the model has acceptable accuracy and scalability, and can be extended to all conditions with environmental disturbances.
The rest part is presented in the following manner.Section 2 introduces the aerodynamic modeling of the ducted fan under environmental disturbances based on momentum theory and blade element theory.Section 3 introduces the numerical setup and carries out the model validation using three different cases.Section 4 draws the conclusions.

Momentum Theory
Based on reasonable assumptions, the momentum theory has been widely used to predict the aerodynamic performance of turbomachines such as wind turbines and rotors.The momentum theory assumes that the flow is a one-dimensional steady flow of ideal gas, and ignores the influence of the duct lip shape.The momentum theory simplifies the propeller into an infinitely thin rotor disk, and the airflow is accelerated through the rotor disk, forming a pressure difference above and below the disk plane.The rotor thrust is generated consequently.As is shown in Figure 2, Plane 0 represents the plane infinitely far upstream of the rotor disk.Plane 1 and 2 represent the upper and lower surfaces of the rotor disk, and Plane 3 represents the plane infinitely far downstream.For the hover condition, the inflow velocity magnitude at infinity upstream is 0, the induced velocity at the rotor disk is i v , and the slip velocity at the outlet of the duct diffuser is w .According to the area of each plane, the axial velocity of each region satisfies where d  refers to the expansion ratio, defined as the ratio of the equivalent wake area at the outlet to the rotor disk area.Therefore, the mass flow rate along the axial direction is given by According to the momentum theory, the total thrust of the single-propeller ducted fan is equal to the momentum change throughout the control body, namely rotor T is generated by the pressure difference between the upper and lower surfaces of the rotor disk.
Using the Bernoulli equation for Plane 0 and Plane 1, it can be obtained that Using the Bernoulli equation for Plane 2 and Plane 3, it can be obtained that Based on equation ( 4) and (5), we have Therefore, the rotor thrust is derived as The ratio of rotor thrust to total thrust of the ducted fan is given by

Blade Element Theory
Blade element theory is a theoretical model commonly used in the industrial field for blade design and performance evaluation.It contains several key blade geometric parameters such as the blade height, blade airfoil, chord length, pitch angle and incidence angle, and can calculate the three-dimensional aerodynamic characteristics of the blade quickly and accurately in a wide range of operating conditions.The blade element model divides the blade into infinite microelement along the blade height.Each microelement is a quasi-two-dimensional airfoil and the interaction between adjacent airfoils is neglected.The blade aerodynamic force and torque can be then obtained by integrating along the blade height.
The schematic diagram of the blade element theory is shown in Figure 3, where T U is the tangential velocity of the blade element, and P U is the axial velocity.They are expressed using the following equations.In the above formula, i v is the induced velocity through the rotor disk, 0 V is the incoming velocity, and  is the rotating speed of the ducted fan.Regardless of the radial velocity of the blade element, the total velocity and the angle between the total velocity and the blade element rotating plane are expressed as ( ) The relationship between the blade incidence angle  and the blade angles of attack  satisfies    =− (12) Based on the relationship between the airfoil angle of attack and the aerodynamic characteristics, the lift and drag of the blade element are The lift coefficient changes almost linearly with the angle of attack in a certain range, so the lift coefficient of the blade element can be expressed as 00 ( ) ( ) where L C  represents the slope of lift coefficient versus the angle of attack, and 0  represents zero-lift angle of attack.Considering that the pitch angle changes linearly with the blade height, that is, (15) According to the force analysis in Figure 3, the forces acting on the blade element in the plane parallel to and perpendicular to the rotor disk are respectively shown as The thrust, torque and power of the blade element are Since the rotor aerodynamic force is mainly generated at the blade tip region where the tangential velocity is much larger than the axial velocity, it can be assumed that the angle between the total velocity and the rotating plane is small.We therefore have sin  and cos 1   .The thrust and torque can be simplified as Combined with equation ( 5)-( 7), the thrust and torque of the blade element can be deduced as The chord length could be regarded as a constant value due to the small changes in the radial direction.

The chord length
c at the radius of ( ) is taken as the average chord length.The axial velocity P U can be neglected compared to the tangential velocity T U , especially for the blade tip region, so we have

Effect of Environmental Disturbances
The rotor thrust is obtained by both the momentum theory and the blade element theory.Combining equations ( 7) and ( 21), it can be obtained that ( ) ( ) We assume that the axial velocity where ( ) ( ) However, the above momentum equations are no longer applicable under environmental disturbances due to the changes in the inlet and outlet flow.Our previous work shows that environmental disturbances arouse changes in the mass flow rate through the fan, resulting in the changes in the rotor aerodynamic characteristics.It occurs to us that it's reasonable to use the mass flow rate to evaluate the impact of environmental disturbances.The mass flow rate under disturbances is given by ( ) where x is a dimensionless operating parameter.Equation (27) needs to satisfy ( ) kx→ when x → or 0 x → depending on the cases.In the meanwhile, the environmental disturbances have an impact on the pressure distributions around the duct, leading to the changes in the ratio of rotor thrust to total thrust.We use a correction factor ( ) gx to take the effect into account, which writes ( ) where ( ) gx needs to satisfy ( ) gx→ when x → or 0 x → depending on the cases.Eventually, the rotor thrust, rotor torque and total thrust of the ducted fan under environmental disturbances are given by combining equations ( 21)-( 22), ( 24)-(28).

Model Validation
In this section, the proposed aerodynamic mechanism model of the ducted fan under environmental disturbances is verified by three-dimensional numerical simulation.We first introduce the simulation methods and the corresponding experimental verification, and then select three typical operating conditions for simulation to fully demonstrate the effectiveness and universality of our model.
For illustration purposes, the following physical parameters used are defined as follows: Thrust coefficient: Torque coefficient: Velocity coefficient: where  is the air density, A is the rotor disk area,  is the rotor rotating speed, and R is the rotor radius.

CFD Methods
The geometry of the investigated ducted fan is shown in    .446-In terms of numerical setup, we divided the overall computational domain into a static domain and a rotating domain, where the static domain covers the external flow field and the rotating domain covers the internal flow field around the ducted fan. Figure 5 shows the mesh details in each domain.We generated hexahedral structured grids in the static domain using ANSYS ICEM CFD.We encrypted the grid near the ducted fan by setting the node distribution law in order to improve the calculation accuracy.The mesh quality in the static domain reached 0.7.We used NUMECA AutoGrid5 to create the mesh in the rotating domain.The number of boundary layers at the blade was 17, the minimum skewness was over 19, the maximum aspect ratio was around 600, and the maximum expansion ratio was 3.85.On the whole, the number of grids in the whole computational domain was about 12 million, with half of them in the static domain and half in the rotating domain.The mesh quality could well meet the requirements for ducted fan simulation.Based on ANSYS FLUENT, we carried out the numerical investigation using the Unsteady Reynolds Averaged Navier-Stokes (URANS) method.The pressure-based solver and the shear stress transport (SST) k  − turbulence model were adopted in the simulation.The coupled algorithm and second-order upwind scheme were used to improve the calculation accuracy.The sliding mesh technique was developed to capture flow structures around the ducted fan under environmental disturbances.
For verification purposes, we conducted experimental tests of the ducted fan hovering in open space in Chongqing Innovation Center of Beijing Institute of Technology.Results are shown in Figure 6.It can be seen that the simulation results well match the experimental results, with the errors in total thrust and rotor torque predictions lower than 10%.In view of the aerodynamic interference brought by the test support and measurement error, the agreement between simulation and experiments is acceptable.Current numerical setup could be used for validation of our proposed aerodynamic mechanism model.

Case 1: Hovering in Winds
Case 1 refers to the incoming flow disturbance.For illustration purposes, we consider the effect of crosswinds on the aerodynamic performance of ducted fan.The dimensionless wind speed is defined as where V is the actual wind speed.Cases with dimensionless wind speeds of 0, 0.0232, 0.1159 and 0.1854 (corresponding to 0, 1, 5 and 8 m/s crosswinds) are drawn for comparison.Figure 7 shows the contours of velocity magnitude in the y-z plane.It can be seen that the air is sucked in from open environment, accelerated by the fan, and forms symmetric wake downwards in windless hover.Little flow separation is seen at the duct diffuser section.With the increase of the wind speed, the flow field of the duct fan gradually shifts to the downwind side.The flow separation of the duct inner wall at the upwind side and the duct outer wall at the downwind side is greatly enhanced.Excessive flow separation makes the net flow on the upwind side decrease significantly, while less impact is observed on the downwind side.22), ( 24)-( 28) and ( 30)-(31), and the comparisons between model predictions and CFD predictions is illustrated in Figure 8.It can be seen that as the wind speed increases, the rotor thrust increases, the rotor torque increases, and the total thrust decreases.The proposed model could well predict the ducted fan performance in wind conditions because the maximum error in thrust predictions and torque predictions is 9%.

Case 2: Hovering in Ground Effect
Case 2 refers to the wall disturbance.Hovering in ground effect is one of the most common operating conditions, which will cause large aerodynamic interference to the flying vehicles.The dimensionless ground height is defined as where H is the actual height above the ground.Cases with dimensionless ground height of 0.5, 1, 2 and hover are simulated to explore the ground effect on the ducted fan. Figure 9 shows the velocity contours.The ground mainly has an impact on the outlet flow of the ducted fan.The ground effect causes the outlet airflow to turn and stretch outward along the ground, forming a triangular low-speed stagnation zone in the middle.As the height from the ground decreases, the stagnation zone gradually extends to and even above the fan hub, and leads to the blockage at the blade root.As a result, the flow velocity at the blade root and the net mass flow passing through the fan decreases due to ground effect.24)-( 28) and ( 33)-(34), the aerodynamic performance of the ducted fan in ground effect can be obtained.Figure 10 shows the comparisons between model predictions and CFD predictions.It can be found that the influence range of the ground lies within 3R, and the ground effect can be ignored beyond this range.With the decrease of ground height, the rotor thrust increases, the rotor torque increases, and the total thrust decreases.The maximum error in thrust and torque of 5% indicates that the proposed model could be used to predict the ducted fan aerodynamic performance in ground effect.

Case 3: Hovering in Ceiling Effect
Case 3 refers to the wall disturbance.Hovering in ceiling effect is another dangerous condition during flying vehicle operation, which may lead to a crash.For illustration purposes, the dimensionless ceiling distance is defined as where Z is the actual distance from the ceiling.We carried out case study with dimensionless ceiling distance of 0.5, 1, 2 and hover to evaluate the ceiling effect on the ducted fan.According to Figure 11, the ceiling mainly affects the inlet flow.A "valve" is formed between the ceiling and the duct.As the ducted fan approaches the ceiling, the valve lift becomes smaller, and the air flow at the inlet section gradually accelerates and enhances the flow separation at the top side of the duct.In addition, the radial high-speed airflow at the inlet leads to the enhancement of flow separation in the duct diffuser section due to inertia.In general, the ceiling effect is prone to decrease the mass flow due to the enhanced flow separation.The relationship between the mass flow rate and ceiling distance is collected and fitted as follows  21)-( 22), ( 24)-( 28) and ( 36)-(37).As is shown in Figure 12, the influence range of ceiling on the aerodynamic performance of ducted fan is 0 to 3R.The ceiling effect will cause the rotor thrust to rise slightly, the rotor torque to rise slightly, and the total thrust to rise sharply.The maximum total thrust is achieved at a distance of 0.4R, which is 1.74 times larger than the total thrust out of ceiling effect.Our model may be utilized to predict the ducted fan aerodynamic performance in ceiling effect, as evidenced by the greatest error in thrust and torque of 12%.

Conclusions
Environmental disturbances have a significant impact on the aerodynamic performance of ducted fan vehicles, which is worth deep investigation.In this study, the axial velocity and flow rate without disturbance are derived based on blade element theory and momentum theory.By considering the influence of environmental disturbance on the flow rate, a correction of blade element theory is introduced, and the aerodynamic mechanism model of ducted fan under environmental disturbance is then established.Based on the three-dimensional CFD simulation, we used three typical working conditions to verify the mechanism model.Results show that our model could well predict the ducted fan aerodynamic characteristics in given conditions with a maximum error of 12% and could be potentially extended to other environmental disturbances.Future research may include combining deep learning methods to build up models which take dynamic environmental disturbances into consideration.

Figure 1 .
Figure 1.Schematic diagram of two kinds of environmental disturbances.Prediction models can be divided into physical-based models and data-based models.Data-based models require a large amount of real data, have poor scalability, and are difficult to reveal the mechanism of performance changes from the perspective of fluid dynamics.The physics-based model starts from the basic and underlying mechanism and has good scalability, so it's still a research hotspot.Traditional physical-based methods for propeller modeling include momentum theory, blade element theory, blade element momentum theory, lifting-line method, lifting surface method, panel method, etc[7].These methods can accurately predict the aerodynamic performance of the propeller, and some of them have been applied to the ducted fan for geometry design[8].Currently there are few researches discussing about the modeling of ducted fan in non-hovering conditions.Ai[9] proposed a lift mechanism model of a coaxial ducted aircraft based on the momentum theory and the blade element theory.His model considered the relationship between the vertical climbing rate and the induced speed of the rotors and the induced velocity interaction, and exhibits high adaptability and accuracy.Nevertheless, Ai's model is only applicable to the ceiling effect, which is difficult to be extended to other environmental disturbances.In this paper, we proposed a mechanism model of ducted fan under environmental disturbances, which can be used to predict the influence of different environmental disturbances on the thrust and torque.We selected three typical cases to verify the model to illustrate that the model has acceptable accuracy and scalability, and can be extended to all conditions with environmental disturbances.The rest part is presented in the following manner.Section 2 introduces the aerodynamic modeling of the ducted fan under environmental disturbances based on momentum theory and blade element theory.Section 3 introduces the numerical setup and carries out the model validation using three different cases.Section 4 draws the conclusions.

Figure 2 .
Figure 2. Schematic diagram of the momentum theory.According to the area of each plane, the axial velocity of each region satisfies

Figure 3 .
Figure 3. Schematic diagram of the blade element theory.In the above formula,

C
where  denotes the air density, c denotes the local chord length, are the lift coefficient and drag coefficient related to the blade profile.

PU
is proportional to the mass flow rate m through the rotor disk, namely.Combining equations (23) and (24), the mass flow rate of the ducted fan in hover is obtained to be

Figure 4 .
The ducted fan is composed of a propeller and a duct.The duct has a diameter of 292 mm and a height of 50 mm.The propeller has a diameter of 206 mm with three equally distributed blades.

Figure 4 (
b) shows the diagram of the real ducted fan placed on the test bench.The specific parameters of the ducted fan used in the study are listed in Table1.

Figure 4 .
Figure 4. Configurations of the investigated ducted fan.

Figure 5 .
Figure 5. Mesh details in the numerical setup.

Figure 6 .
Figure 6.Comparisons between thrusts and torques obtained by the sliding mesh technique and experiment.

Figure 7 .
Figure 7. Contours of velocity magnitude at different dimensionless wind speed: (a) v=0, (b) v=0.0232,(c) v=0.1159;(d) v=0.1854.The above velocity flow field diagram shows that the wind effect will lead to a decrease in the net mass flow rate through the rotor disk.We collected the mass flow rate at the duct inlet at different wind speeds and the results are fitted as ( )

Figure 8 .
Figure 8. Comparisons of thrust and torque predictions in wind conditions by CFD method and the proposed model: (a) thrust predictions, (b) torque predictions.

Figure 9 .
Figure 9. Contours of velocity magnitude at different dimensionless ground height: (a) hover, (b) h=2, (c) h=1, (d) h=0.5.Similar to the wind effect, the ground effect tends to lead to a decrease in the net mass flow rate.We collected the mass flow rate at the duct inlet at different ground height and the results are fitted as ( )

Figure 10 .
Figure 10.Comparisons of thrust and torque predictions in ground effect by CFD method and the proposed model: (a) thrust predictions, (b) torque predictions.
The correction factor of rotor thrust to total thrust ratio is given by The aerodynamic performance of the ducted fan in ceiling effect can be calculated by combining equations (

Figure 12 .
Figure 12.Comparisons of thrust and torque predictions in ceiling effect by CFD method and the proposed model: (a) thrust predictions, (b) torque predictions. )

Table 1 .
Parameters of the investigated ducted fan. b-7