Numerical study on aerodynamic characteristics of two-dimensional propulsive wing in cruise and hover

Abstract

The propulsive wing vehicle is a new concept vehicle, which is driven by a cross-flow fan (CFF) embedded in the trailing edge of the wing. The propulsive wing vehicle is capable of cruising and hovering at high angles of attack with very high aerodynamic force coefficients, and has the potential to become a new type of vertical take-off and landing (VTOL) vehicle. The cruise and hover states of the propulsive wing vehicle are defined, and a numerical model of the two-dimensional propulsive wing is established. Based on the computational fluid dynamics (CFD) method, the rotation of the CFF is simulated by using the sliding mesh technique. The effects of cruise speed, angle of attack and CFF rotation speed on the aerodynamics of the two-dimensional propulsive wing are evaluated, and the mechanism of the propulsive wing flow field changes is revealed. The results show that the propulsive wing has a very high lift coefficient of up to 60 at low speed and high angle of attack cruise, and a high thrust coefficient of up to 40 at low speed and small angle of attack cruise. The aerodynamic force of the propulsive wing fluctuates periodically with the rotation of the CFF, and the amplitude of the fluctuation is related to the vorticity of the CFF blade shedding vortex. In hover, the flow field of the propulsive wing is affected by the geometry and wake deflection into an asymmetric distribution, forming a vortex on each side of the airfoil, and the vortex diameter varies with the CFF rotation speed, which in turn has an impact on the hovering performance of the propulsive wing.

Keywords

propulsive wing cross-flow fan (CFF)cruise hover aerodynamic characteristics computational fluid dynamics

Introduction

In response to the demand for high-speed cruise and vertical takeoff and landing (VTOL) capabilities, many new configurations aircraft^1–3 have been developed and the related aerodynamic characterization studies^4,5 have been conducted. The propulsive wing^6,7 vehicle is a new concept vehicle that has emerged over the last decade. The propulsive wing consists of a cross-flow fan (CFF), a main airfoil and a deflector (Figure 1). The CFF was invented in 1893,⁸ and its structure consists of a housing and a fan (Figure 2). The air flows radially into the fan from the inlet at the top of the housing, and ejected radially from the airflow channel at the bottom by the compression of the fan. The airflow is essentially in the same plane, therefore the CFF possesses good two-dimensional characteristics. Where radial space is limited, a certain airflow requirement can be met by increasing the spreading length of the CFF. When the CFF is embedded into the wing along the spreading direction, they form a propulsive wing.⁶ During flight, the CFF sucks the airflow on the upper surface of the wing, compresses the airflow and ejects it from the flow channel at the trailing edge. While forming thrust, it can also delay the separation of the airflow on the upper surface of the wing, increases the stall angle of attack and improves the aerodynamic coefficient greatly. When the CFF rotates, an eccentric vortex will be formed inside it, where the static pressure is very low, and a part of vortex induced lift^9,10 will be given to the propulsive wing under some conditions. However, as the rotation speed of the CFF increases, the strength of the eccentric vortex will also increase, resulting in poor airflow of CFF, increased vibration and noise. Therefore, the study of aerodynamic characteristics of propulsive wing vehicle is indispensable in flight vehicle design.

Figure 1.

Propulsive airfoil in reference [6].

Figure 2.

Two dimensional structure of cross-flow fan (CFF).

In fact, before the concept of propulsive wing was put forward, some researchers have applied CFF to aircrafts^11–13 to delay stall through air flow control. However, the complexity of the wing structure is greatly increased. In 2006, Kummer and Dang^6,7 formally proposed the concept of propulsive wing from the perspective of airfoil design (Figure 1). It not only optimizes the wing structure, but also makes full use of the function of airflow control and propulsion of CFF. The relevant CFD simulation⁶ and wind tunnel test results¹⁴ show that the propulsive wing vehicle can fly at an angle of attack of up to 40°, and the lift coefficient is as high as 6–7 to realize short takeoff and landing (STOL). However, the thick airfoil and large deflector increase the drag of the propulsive wing vehicle during cruise and limit its cruise speed. With the increase of cruise speed and angle of attack, the air flow is easier to separate. However, during the wind tunnel test in,¹⁴ the velocity of the airflow is only 3.92 m/s, the aerodynamic characteristics of propulsive wing in high-speed cruise state are not considered.

Based on the researches of the CFF^15–18 and the CFF vehicles with hovering capability,^19,20 Fulton designed a new propulsive airfoil²¹ (Figure 3). Through conducting the CFD simulations and hovering tests (Figure 4), the cruise and hover functions of the propulsive wing vehicle were verified, and the propulsive wing vehicle has the potential to become a new type of vertical takeoff and landing vehicle. However, the CFD analysis of the propulsive wing vehicle was carried out in a steady-state, which can reduce the calculation time, but cannot simulate the changes of the transient flow fields of the propulsive wing and the CFF. This method is insufficient to support an in-depth analysis of the variation of propulsive wing aerodynamics and the characteristics of the flow field. In conclusion, for the propulsive wing vehicle, which is a new concept vehicle, it is necessary to conduct all-round aerodynamic characteristics research on its key flight states, and have a clearer understanding of its flow field structure and variation laws, so as to provide a basis for vehicle design in the future.

Figure 3.

Propulsive airfoil in reference [21].

Figure 4.

Hovering test of propulsive wing vehicle.²¹

With the continuous improvement of computer performance, the application of CFD methods in the study of aircraft aerodynamic characteristics is becoming more and more mature, and much related research work^22–24 has been carried out. For the propulsive wing, its structure is different from the conventional wing, and there is a high-speed rotating CFF inside the wing, which requires a lot of computational costs if the 3D model is directly simulated. In the past decade, many scholars had used the two-dimensional simulation method to study the aerodynamic characteristics of fan-wing,^25–28 and the fan-wing has much in common in geometric structure and working principle with the propulsive wing. Considering that the CFF has very stable two-dimensional characteristics,^8,29 this study adopts CFD method to study the aerodynamic characteristics of the two-dimensional propulsive wing under cruise and hover states, to explore the variation law of the aerodynamics of the airfoil, and to make a more in-depth analysis of the aerodynamic changes by combining the structure and variation mechanism of the airfoil flow field.

The other parts of this study is organized as follows: In Section “Problem definition and numerical method”, the cruise and hover states of propulsive wing vehicle are defined, the calculation model of two-dimensional airfoil is given, and the numerical calculation method of airfoil flow field of two-dimensional propulsive wing is established and verified. In Section “Analysis of aerodynamic characteristics of two-dimensional propulsive wing”, the effects of cruise speed and angle of attack on the aerodynamic performance of propulsive wing in cruise are studied, the reason for the high aerodynamic coefficient of propulsive wing is explained, and the periodic variation characteristics of aerodynamic force are analyzed; then, the influence of CFF rotation speed on the hovering attitude angle of the propulsive wing is studied, the aerodynamic performance and flow field structure characteristics of the propulsive wing in hover are analyzed. In Section “Conclusions”, the main conclusions of this study are given.

Problem definition and numerical method

In this section, the problem studied is defined, and the calculation model of two-dimensional propulsive wing is given. Then the numerical method for the calculation of airfoil aerodynamic forces is established, and the feasibility of the numerical method is proved through relevant verification.

Definition of research problem

Firstly, the cruise and hover states of propulsive wing vehicle are defined, and the relevant dimensionless formulas and two-dimensional airfoil parameters are given.

The three-dimensional model of the wing of a practical propulsive wing vehicle is shown in Figure 5, which is composed of main airfoil, CFF, deflector and side plates. The cruise state of the propulsive wing vehicle is similar to that of a conventional vehicle. The included angle between the airfoil chord and the incoming flow is angle of attack $α$ , the lift $F_{L}$ is perpendicular to the incoming flow $U_{\infty}$ , and the drag $F_{D}$ is parallel to the incoming flow direction. When the value of $F_{D}$ is negative, it represents thrust $F_{T}$ , and the direction is opposite to the incoming flow (Figure 6(a)).

Figure 5.

Three dimensional wing model of propulsive wing vehicle.

Figure 6.

Flight states of propulsive wing vehicle. (a) Cruise state, (b) Relative static state, (c) Hover state.

When the propulsive wing is in a relative static state with the incoming flow velocity $U_{\infty} = 0$ (Figure 6(b)), due to the action of the CFF, there are still three forces on the propulsive wing, namely, force $F_{X}$ in the horizontal direction, force $F_{Y}$ in the vertical direction and gravity G. At this time, tilt the propulsive wing clockwise until the included angle between the airfoil chord and the horizontal line is $α_{h} = \arctan (F_{X} / F_{Y})$ , so that the resultant force $F_{x}$ of the wing in the horizontal direction approaches 0, the resultant force $F_{y}$ in the vertical direction is equal to the gravity G, and the direction is opposite, then hovering of the vehicle can be realized (Figure 6(c))

In cruise state, the lift and drag coefficients are as follows:

C_{L} = \frac{F_{L}}{1 / 2 ρ U_{\infty}^{2} c}, C_{D} = \frac{F_{D}}{1 / 2 ρ U_{\infty}^{2} c}

(1)

Where,

ρ

is the air density; c is the chord length of the airfoil.

In,²¹ the hovering of propulsive wing vehicle has been realized. Therefore, the model used in this study is based on the model in,²¹ and the parameters of CFF are changed. The definition of geometric dimensions of propulsive airfoil and CFF are shown in Figure 7, and the main parameters of propulsive airfoil and CFF are listed in Table 1.

Figure 7.

Definition of geometric parameters of propulsive airfoil and CFF. (a) Propulsive airfoil parameters, (b) CFF parameters.

Table 1.

Parameters of propulsive airfoil and CFF

Parameters	Value
Airfoil	GOE 570
Airfoil chord length $c$ /mm	500
CFF radius $R$ /mm	51
Deflector angle of incidence $ψ$ /(°)	30
Number of CFF blades	25
Blade width $c_{b l a d e}$ /mm	16.7
Maximum blade thickness $b$ /mm	1.36
Blade inner arc radius $r_{i n}$ /mm	11.73
Blade outer arc radius $r_{o u t}$ /mm	14.54
Blade tip arc radius $r_{t i p}$ /mm	0.212
Blade angle of incidence $Φ$ /(°)	28.3

Numerical method

In this study, CFD method^30,31 is used to calculate the flow field and aerodynamic force of two-dimensional propulsive wing. In this section, the numerical method and application process of CFD software ANSYS Fluent in numerical simulation are briefly given.

The unsteady flow field of two-dimensional airfoil is calculated in this study, therefore 2-D Navier-Stokes equation is selected as the main governing equation of unsteady incompressible flow field of propulsive wing:

ρ \frac{\partial (u_{i})}{\partial x_{i}} = 0

(2)

ρ \frac{\partial u_{i}}{\partial t} + \frac{\partial (u_{i} u_{j})}{\partial x_{j}} = - \frac{\partial p}{\partial x_{i}} + \frac{\partial}{\partial x_{j}} [μ (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}})] + F_{i}

(3)

Where,

ρ

is the density;

i, j = 1, 2

represent the coordinate axis

x, y

, respectively; u is the velocity; p is the static pressure;

F_{i}

is the volume force.

The finite volume method is used to discretize the flow field in the study, the convection term is discretized by the second-order upwind scheme, and the Semi-Implicit Method for Pressure Linked Equation (SIMPLE) algorithm³² is used to solve the control equation. The turbulence is modeled by the realizable $k - ε$ model.^33,34

The setting of calculation zone, boundary conditions and coordinate axis of numerical simulation in this study are shown in Figure 8. The fixed two-dimensional Cartesian coordinate system is adopted. The size of computational domain in cruise state is $30 c \times 18 c$ and that in hover state is $20 c \times 28 c$ . In cruise state (Figure 8(a)), the left side is a uniform speed inlet ( $U_{\infty}$ ), the right side is a pressure outlet, and the upper and lower sides are symmetrical boundaries. In hover state (Figure 8(b)), the upper side is a uniform velocity inlet, the lower side is a pressure outlet, and the left and right are symmetrical boundaries. The no slip boundary conditions are adopted for the main airfoil and deflector, the circular zone inside the CFF and the external flow field are static zones, and the annular area at the CFF blades is the rotating zone (Figure 8(c)). The grid at the CFF is shown in Figure 8(d) and (e). The grid size of the first layer of the airfoil and deflector is $0.005 % c$ , y + is about 1.5, and the grid growth rate along the wall is 1.2. Considering the small size of CFF blade, the first layer grid of the blades is set to $0.002 % c$ . To accurately capture the details of the airfoil wake, the grids near the airfoil and the wake region are refined.

Figure 8.

Computational zone and sliding mesh. (a) Calculation zone division of cruise state, (b) Calculation zone division of hover state, (c) Rotating and static zone, (d) Rotating zone mesh, (e) Boundary layer mesh.

Validation

Grid and time step validation

To ensure the reliability of the calculation results and reduce the error caused by the numerical method,^35,36 the grid and time step used in the numerical calculation are verified. By changing the mesh density in the rotating zone, meshes of different sizes are generated to calculate the aerodynamic force of the propulsive wing during cruise. The boundary layer mesh settings of these grids are the same. $U_{\infty} = 10 m / s$ , $R e = 342293$ , $α = 10^{\circ}$ and $Ω = 6000 r / min$ . Firstly, case 2 (the total number of grids is 1.05 million) is taken as the baseline, case 3 (the total number of grids is 1.4 million) is obtained by increasing the mesh density in the rotating zone. The errors of lift and drag coefficient between case 3 and case 2 are 0.38% and 0.79%, respectively. Then, the mesh density in the rotating zone is further increased to obtain case 4 (the total number of grids is 1.77 million). The errors of the lift and drag coefficient between case 4 and case 2 are 0.49% and 0.89%, respectively. Considering the calculation efficiency, case 1 (the total number of grids is 0.72 million) is obtained by reducing the mesh density in the rotating zone. The errors of the lift and drag coefficient between case 1 and case 2 are 2.8% and 4.1%, respectively. The calculation results of different meshing cases are listed in Table 2. Considering the calculation efficiency and accuracy, the grid division scale of case 2 is used for simulation, and the total number of grids is about 1.05 million.

Table 2.

Aerodynamic coefficients of propulsive wing with different mesh density

Cases	Grids	$C_{L}$	Error	$C_{D}$	Error
Case 1	0.72 million	9.134	2.8%	−2.795	4.1%
Case 2	1.05 million	9.523	baseline	−2.915	baseline
Case 3	1.4 million	9.487	0.38%	−2.938	0.79%
Case 6	1.77 million	9.476	0.49%	−2.941	0.89%

With the grid division scale of case 2, the process of one circle of the CFF rotation is divided into three time steps: 72, 360 and 720, each time step is 1.389e⁻⁵, 2.778e⁻⁵ and 1.1112e⁻⁴ s, respectively. The calculation results of different time steps are shown in Table 3. Taking the values of case 6 as baseline, the errors of lift and drag coefficient of case 7 and case 6 are 0.36% and 0.69%, respectively, and the errors of case 5 and case 6 are 8.3% and 7.9%, respectively. Considering the calculation efficiency and accuracy, the time step of case 6 is selected for simulation, that is, one time step is 2.778e⁻⁵ s.

Table 3.

Aerodynamic coefficients of propulsive wing with different time steps

Cases	Time steps/one circle	$C_{L}$	Error	$C_{D}$	Error
Case 5	72	8.735	8.3%	−2.686	7.9%
Case 6	360	9.523	baseline	−2.915	baseline
Case 7	720	9.489	0.36%	−2.935	0.69%

Wind tunnel test validation

Previously, the research group conducted wind tunnel tests on the propulsive wing model in reference⁷ and measured the aerodynamic forces of the propulsive wing under different working conditions. In this study, some experimental values ( $U_{\infty} = 6 m / s$ , $R e = 246451$ , $α = 10^{\circ}$ and $Ω = 0 \sim 2500 r / min$ ) are compared with the calculated values to verify the reliability of the numerical method. The test system includes propulsive wing model, balance, test bench and wind tunnel, as shown in Figure 9. The two-dimensional structure of the validation model is shown in Figure 10. The main airfoil is GOE 570, the chord length is 600 mm, and the CFF diameter is 80 mm. The comparison results between the calculated values and the experimental values of the aerodynamic forces of the propulsive wing are shown in Figure 11. With the increase of CFF rotation speed, the lift and thrust of the wing gradually increase, and their change trends are well fitted with the experimental data, which verifies the correctness and effectiveness of the two-dimensional propulsive wing numerical method established in this study.

Figure 9.

Test system of validation model.

Figure 10.

Two dimensional structure of validation model.

Figure 11.

Calculated and experimental values of the validation model. (a) $C_{L}$ , (b) $C_{D}$ .

Analysis of aerodynamic characteristics of two-dimensional propulsive wing

As mentioned above, the propulsive wing vehicle does not need additional power devices, it can generate horizontal thrust and vertical lift on the wing only by the rotation of CFF. The propulsive wing vehicle can realize cruise and hover at high angle of attack, and has a very wide application prospect. In this section, the numerical methods established above are used to simulate and analyze the aerodynamic characteristics of two-dimensional airfoil in cruise and hover, respectively. The variation laws of airfoil aerodynamic forces with angle of attack, cruise speed and CFF rotation speed are studied, and the reasons for the variation of airfoil aerodynamic force are explained in combination with the variation mechanism of airfoil flow field in different states.

Cruise state

In this section, CFD method is used to study the aerodynamic characteristics of two-dimensional propulsive wing in cruise. Firstly, the variation law of airfoil aerodynamic force with angle of attack and cruise speed is analyzed; Then, through the comparative analysis of the flow field of the propulsive wing at different cruise speeds, the reason for the high aerodynamic force coefficients of the propulsive wing at low speed is explained; Finally, the aerodynamic characteristics of propulsive wing are deeply analyzed from the complex periodic variation of aerodynamic force. At the same time, the change mechanism of airfoil flow field in cruise is explained. The working speed of CFF in cruise is 6000 r/min, the cruise speeds are 3, 5, 7, 10, 15, 20 and 30 m/s, respectively. And the Reynolds number varies from 102,688 to 1,026,880 correspondingly. The angle of attack varies from 0° to 40° with an increment of 15°.

Change of aerodynamic force

In this study, the aerodynamic parameter formula (1) of conventional wing is used to represent the aerodynamic performance of propulsive wing, which can intuitively show the unique performance of propulsive wing.

The variation of aerodynamic forces and coefficients of propulsive wing with cruise speed $U_{\infty}$ and angle of attack $α$ are shown in Figure 12, and the shaded part in Figure 12(d) represents thrust As can be seen from Figure 12(c), the lift coefficient of the propulsive wing decreases with the increase of the cruise speed, indicating that the aerodynamic performance of the propulsive wing is better at low speed. At low speed, the lift coefficient is as high as 30–60, which is precisely the key for the propulsive wing vehicle to take off and land vertically.

Figure 12.

Variation of propulsive wing aerodynamic forces and coefficients with $α$ at $Ω = 6000 r / min$ and $Ω = 8000 r / min$ . (a) $F_{L}$ at $Ω = 6000 r / min$ , (b) $F_{D}$ at $Ω = 6000 r / min$ , (c) $C_{L}$ at $Ω = 6000 r / min$ , (d) $C_{D}$ at $Ω = 6000 r / min$ , (e) $F_{L}$ at $Ω = 8000 r / min$ , (f) $F_{D}$ at $Ω = 8000 r / min$ , (g) $C_{L}$ at $Ω = 8000 r / min$ , (h) $C_{D}$ at $Ω = 8000 r / min$ .

The high speed rotation of the CFF will form thrust on the propulsive wing. It can be seen from Figure 12(d) that the thrust coefficient( $C_{T} = - C_{D}$ ) of the propulsive wing decreases with the increase of the cruise speed and is very high in the low speed state, which further illustrates the excellent low speed performance of the propulsive wing. However, with the increase of cruise speed and angle of attack, the thrust of the propulsive wing gradually decreases until it becomes drag, and the propulsive wing enters the stall state, which is the non-shaded area in Figure 12(d). It shows that the stall of the propulsive wing is not only related to the angle of attack, but also to the cruise speed. The smaller the cruise speed, the higher the stall angle of the propulsive wing.

Figure 12(e)–(h) show the variations of aerodynamic forces and coefficients of the propulsive wing at $Ω = 8000 r / min$ . The overall change trends of aerodynamic forces are consistent with the variation laws in Figure 12(a)–(d). However, there are some noteworthy points. Firstly, there is no doubt that the higher the rotation speed of the CFF, the greater the aerodynamic forces of the propulsive wing. Secondly, as can be seen from Figure 12(a) and (e), when $U_{\infty} = 15 m / s$ and $α = 30^{\circ}$ , the propulsive wing has stalled at $Ω = 6000 r / min$ , but it still works normally at $Ω = 8000 r / min$ .The higher the rotation speed of the CFF, the greater the stall angle of attack of the propulsive wing.

The instantaneous static pressure clouds, streamlines and vorticity contours at $U_{\infty} = 3 m / s, 30 m / s$ are given in Figures 13 and 14, respectively. Comparing the streamlines in the two sets of figures, it can be found that the streamlines fit more closely to the airfoil surface in the low speed state, and the laminar flow starts to separate when the speed increases to 30 m/s. In Figure 14, due to the attraction of the CFF, the separated airflow is drawn in again by the CFF and a separation bubble^37–39 is formed at the back of the airfoil. The airfoil laminar flow turning-point is roughly at the position of $0.464 c$ . The entire separation bubble occupies the position of the junction of the main airfoil and the CFF, and the inlet area and the inlet flow quality of the CFF is greatly reduced. Therefore, the propulsion performance is reduced.

Figure 13.

Instantaneous static pressure clouds and vorticity contours of the propulsive wing when $α = 0^{\circ}$ , $U_{\infty} = 3 m / s$ , $R e = 102688$ , $Ω = 6000 r / min$ . (a) n = 12,005, (b) n = 12,013, (c) n = 12,020.

Figure 14.

Instantaneous static pressure clouds and vorticity contours of the propulsive wing when $α = 0^{\circ}$ , $U_{\infty} = 30 m / s$ , $R e = 1026880$ , $Ω = 6000 r / min$ . (a) n = 12,005, (b) n = 12,013, (c) n = 12,020.

Combined with the vorticity contours in Figures 13 and 14, it can be found that in the low speed state, the bound vortex on the upper surface of the main airfoil flows into the CFF along the junction of the airfoil and the CFF, and then is compressed and ejected. In the high speed state, the vorticity on the upper surface of the main airfoil increases, and due to the blockage of the separation bubble, this part of the vortex can only flow from the middle of the CFF inlet, occupying part of the airflow inlet, which in turn leads to the reduction of the CFF inlet flow quality. While the vortex flow occupies the air flow channel of the fan, the CFF blade shedding vortexes mixed in the vortex flow, the vortex groups that accumulate at the bottom of the deflector are not replenished, causing a decrease in propulsive wing performance.

Moreover, as shown in the streamlines of Figures 13 and 14, the streamline forms a stagnation point⁴⁰ when it flows over the airfoil surface, and this point has a velocity of 0. When $U_{\infty} = 3 m / s$ , the stagnation point is located approximately at $0.115 c$ on the lower surface of the airfoil. When $U_{\infty} = 30 m / s$ , the stagnation point is located approximately at $0.012 c$ on the lower surface of the airfoil. It is obvious that at low speed, the effective angle of attack of the propulsive wing is larger, represents a greater aerodynamic coefficient. This is also the reason why the propulsive wing has higher aerodynamic force coefficient at low speed. It can be predicted that with the further reduction of cruise speed, the lift coefficient of the propulsive wing will increase, but this does not mean that the lift of the propulsive wing is very high, just because the dynamic pressure in formula (1) is very small at low speed.

Periodic variation of transient aerodynamic force

In actual flight conditions, the aerodynamic forces of the propulsive wing fluctuate constantly with the high-speed rotation of the CFF. In order to have a deeper understanding of the aerodynamic performance of the propulsive wing, the periodic variations of the aerodynamic forces are analyzed from the perspective of transient.

The variations of the aerodynamic coefficients of the propulsive wing in the time domain at $α = 0^{\circ}$ and $U_{\infty} = 7 m / s$ are given in Figure 15. For the convenience of representation, the time step n is used as the horizontal axis, with one step representing 2.778e⁻⁵ s. Figure 15(a) and (b) show the variation of the lift and drag coefficients of the propulsive wing in periods $N = 30 \sim 33$ , respectively. Obviously, the aerodynamic force on the airfoil has shown a clear periodicity at this time, with 360 time steps for one period, i.e., one rotation of the CFF. Within a single period, the fluctuation of aerodynamic force is very frequent, especially the fluctuation amplitude of the lift coefficient is more variable. The aerodynamic forces of the propulsive wing for the period $N = 33$ are given in Figure 15(c) and (d), the dash-dotted lines represent the time average of $C_{L}$ and $C_{D}$ . In a single period, the aerodynamic forces of the propulsive wing have 25 waves of different heights, which is exactly equal to the number of CFF blades. When the blade turns through 14.4 time steps, the aerodynamic forces of the propulsive wing fluctuate once to form a wave, and different blades cause unequal amplitude of fluctuations.

Figure 15.

$C_{L}$ , $C_{D}$ of propulsive wing when $α = 0^{\circ}$ , $U_{\infty} = 7 m / s$ , $R e = 239605$ , $Ω = 6000 r / min$ . (a) $C_{L}$ , (b) $C_{D}$ , (c) $C_{L}$ of single period, (d) $C_{D}$ of single period.

In order to explore the reasons for the unequal amplitude of aerodynamic fluctuation of the propulsive wing, the change of aerodynamic force of the propulsive wing and its components around 2/25 period is shown in Figure 16, and the corresponding steps are $n = 12001 \sim 12030$ . In Figure 16(a), the lifts on the main airfoil and the deflector are positive, while the lift on the CFF is negative. The change trend of the lift of the deflector is more consistent with the change trend of the total lift of the propulsive wing. This shows that the high fluctuation of propulsive wing lift is mainly related to the deflector, but the main airfoil provides the main part of the lift of the propulsive wing. In Figure 16(b), the drag coefficient on the CFF is positive and the drag coefficients on the main airfoil and deflector are negative. However, the total thrust on the propulsive wing does not vary as much as the lift because the trends of drag on the CFF and main airfoil are opposite to the deflector.

Figure 16.

Transient $C_{L}$ , $C_{D}$ of propulsive wing components in 2/25 period. (a) $C_{L}$ , (b) $C_{D}$ .

Combined with the instantaneous static pressure clouds and vorticity contours^41,42 of the propulsive wing in Figures 17 and 18, we can analyze these phenomena more deeply. Comparing the five instantaneous static pressure clouds in Figure 17, it can be found that there is a high pressure zone at the bottom of the deflector, and the pressure difference between the upper and lower sides forms the lift on the deflector. As the CFF rotates, the pressure in this zone is constantly changing, causing fluctuations in lift. In the vorticity contours of Figure 18, we have made some annotations, red vortices are positive, denoted by Pi, and blue vortices are negative, denoted by Ni. Where, i = 1, 2… represents the order of the vortexes shed on the CFF blades. In Figure 18(a), P1 merges with the positive vortex group accumulated at the bottom of the deflector, resulting in an increase in vorticity here, a decrease in airflow velocity and an increase in static pressure, which acts as a lift enhancement to the deflector. In Figure 18(b), the counterclockwise N1 merges with the positive vortex group at the bottom of the deflector, dissipating part of the vorticity and leading to an increase in the airflow velocity, which causes a decrease in lift. In Figure 18(c), the influence of N1 on the deflector gradually disappears and the lift reaches the trough, while P2 is just flowing to the bottom of the deflector and is about to merge with the vortex group. In Figure 18(c)–(e), the lift of the deflector gradually rises to another crest as P2 merges with the vortex group. Meanwhile, the change of vorticity of blade shedding vortex Pi causes the change of lift force fluctuation amplitude.

Figure 17.

Instantaneous static pressure clouds of the propulsive wing when $α = 0^{\circ}$ , $U_{\infty} = 7 m / s$ , $R e = 239605$ , $Ω = 6000 r / min$ . (a) n = 12,005, (b) n = 12,009, (c) n = 12,013, (d) n = 12,017, (e) n = 12,020.

Figure 18.

Instantaneous vorticity contours of the propulsive wing when $α = 0^{\circ}$ , $U_{\infty} = 7 m / s$ , $R e = 239605$ , $Ω = 6000 r / min$ . (a) n = 12,005, (b) n = 12,009, (c) n = 12,013, (d) n = 12,017, (e) n = 12,020.

Hover state

In hover, the airflow velocity of the flow field is small, and the wake of the CFF will have a large impact on the performance of the propulsive wing. In order to have an in-depth understanding of the hover performance of the propulsive wing, the aerodynamic characteristics of the two-dimensional propulsive wing in the hover state based on CFD method will be analyzed in this section. According to the previous definition of the hover state of the propulsive wing vehicle, the rotation of the CFF will still form a horizontal force $F_{X}$ and a vertical force $F_{Y}$ on the propulsive wing in a relative static state with an incoming flow velocity of 0 m/s. At this time, rotate the wing clockwise until the angle between the wing chord and the horizontal line is $α_{h}$ , so that the combined force $F_{x}$ in the horizontal direction of the propulsive wing tends to 0, and the combined force $F_{y}$ in the vertical direction is offset by gravity G, then hovering can be achieved.²¹ From the definition, the magnitude of the hovering angle $α_{h}$ is directly affected by the horizontal force $F_{X}$ and the vertical force $F_{Y}$ . Therefore, firstly, the numerical simulation of the two-dimensional propulsive wing in the relative static state is conducted in this section, the hovering angle $α_{h}$ is calculated from $F_{X}$ and $F_{Y}$ , and the change of $α_{h}$ is explained by combining the airfoil flow. Then the two-dimensional propulsive wing in hover state is simulated and the change law of aerodynamic force is summarized. Finally, the structural characteristics of the flow field of the hovering propulsive wing are analyzed. The rotation speed range of the CFF varies from 4000 r/min to 10,000 r/min with an increment of 1000 r/min.

Change of the hover angle

Firstly, the relative static state in Figure 6(b) is simulated to find the horizontal force $F_{X}$ and vertical force $F_{Y}$ numerically, and then the hover angle of the propulsive wing in the hover state is calculated by $α_{h} = \arctan (F_{X} / F_{Y})$ .

The variation of horizontal force $F_{X}$ and vertical force $F_{Y}$ of propulsive wing with CFF rotation speed in relative static state is shown in Figure 19. Obviously, $F_{X}$ and $F_{Y}$ increase with the increase of rotation speed, and $F_{X}$ increases faster, indicating that rotation speed has a greater impact on thrust^9,10 However, when $Ω = 8000 r / min$ , $F_{X}$ decreased significantly. The change of hover angle $α_{h}$ of propulsive wing is shown in Figure 20. It can be seen that $α_{h}$ increases rapidly when $Ω = 8000 r / min$ , indicating that the growth rate of horizontal thrust $F_{X}$ is greater than that of $F_{Y}$ , which corresponds to the change of Figure 19.

Figure 19.

Horizontal force $F_{X}$ and vertical force $F_{Y}$ in relative static state.

Figure 20.

Variation of propulsive wing hover angle $α_{h}$ with CFF rotation speed.

The streamlines of the propulsive wing at $Ω = 6000, 8000 r / min$ are given in Figure 21. In Figure 21(a), the stagnation point of the propulsive wing is at the lower surface of the airfoil. From the vorticity contours, it can be found that a small part of the shedding vortex on the CFF blade does not merge with the vortex group at the bottom of the deflector, but is entrained by the negative vortex flow out of the trailing edge channel, creating some disturbance to the wake, which in turn has an effect on the airflow at the lower surface of the airfoil. However, in Figure 21(b), the wake flow carries more blade shedding vortexes, as the disturbance increases, the vortex diameter under the airfoil increases, which causes the stagnation point of the propulsive wing to move to the upper surface of the airfoil, and the inflow angle is negative, causing a decrease of the force in the vertical direction of the airfoil (Figure 19). In general, the hover angle of the propulsive wing varies roughly in the range of 60°–65° with the CFF rotation speed.

Figure 21.

Streamlines and vorticity contours of the propulsive wing when $Ω = 6000, 8000 r / min$ . (a) $Ω = 6000 r / min$ , (b) $Ω = 8000 r / min$ .

Change of aerodynamic force

The hover attitude of the propulsive wing is derived from the hover angle $α_{h}$ , and then the aerodynamic characteristics of the propulsive wing in the hover state are analyzed. The variation of the force $F_{y}$ in the vertical direction of the hovering propulsive wing and the torque $T_{C F F}$ of the CFF with the rotation speed are given in Figure 22. In Figure 22(a), $F_{y}$ increases with the increase of rotation speed. In Figure 22(b), when the rotation speed is greater than 8000 r/min, the CFF torque $T_{C F F}$ increases steeply, indicating that $Ω = 8000 r / min$ is the critical rotation speed for this configuration, and exceeding the critical speed will lead to a steep increase in the required power of propulsive wing and performance degradation. Therefore, the operating speed of the propulsive wing should be set within 8000 r/min.

Figure 22.

Variation of aerodynamic force of propulsive wing with CFF rotation speed in hover. (a) Vertical force $F_{y}$ in hover, (b) CFF torque $T_{C F F}$ .

Structure of hovering flow field

According to the above analysis, the flow field in the hover state of the propulsive wing differs greatly from that in the cruise. In order to have a more comprehensive understanding of the hover state, the flow field distribution of the propulsive wing at $α_{h} = 60^{\circ}$ , $U_{\infty} = 0 m / s$ , $Ω = 5000 r / min$ is given in Figure 23. It is obvious that in the hover state, the left and right flow fields of the propulsive wing are asymmetrically distributed. With the airfoil wake as the division, there is a vortex on the left and right sides of the flow field, with the left side having a smaller vortex diameter and the right side having a larger one. It is easy to understand that the airflow on the right side of the airfoil is attracted by the CFF inlet, and more airflow enters the CFF from the right side, therefore the vortex diameter on the right side is larger. In Figure 23(a), there is a stagnation point on the lower surface of the airfoil. With the stagnation point as the dividing point, the airflow on the upper side of the point is drawn into the CFF around the leading edge of the airfoil, while the airflow on the lower side of the point is induced by the wake to form a vortex of smaller diameter. Moreover, from Figure 23(b), it can be found that there is a deflection⁴³ at the end of the propulsive wing wake, and the direction of the deflection is related to the aerodynamic force. The deflection of the wake further interferes with the right side airflow, and the right side vortex diameter further increases. The influence of multiple factors forms a special flow field structure in the hover state of the propulsive wing, which needs to be taken into account in the vehicle design.

Figure 23.

Streamline and vorticity contour of the propulsive wing when $α = 60^{\circ}$ , $U_{\infty} = 0 m / s$ , $Ω = 5000 r / min$ . (a) Streamline, (b) Vorticity contour.

Conclusions

In this study, CFD method adopting the sliding mesh technology is used to simulate the flow field of the two-dimensional propulsive wing, and the feasibility of the numerical method is proved by comparing the calculated values of the aerodynamic forces of the propulsive wing with the wind tunnel test data. Based on this numerical method, the aerodynamic characteristics of the two-dimensional propulsive wing under cruise and hover states are studied. The results show that:

In cruise state:

The aerodynamic force coefficients of the propulsive wing are related to both the angle of attack and the cruise speed. The lift coefficient increases with the increase of angle of attack and decreases with the increase of cruise speed, and the lift coefficient is as high as 60 at low speed and high angle of attack. The thrust coefficient decreases with the increase of angle of attack and cruise speed, and the thrust coefficient is as high as 40 at low speed and small angle of attack.

The aerodynamic force of the propulsive wing changes periodically with the rotation of the CFF. The period of aerodynamic force is consistent with the rotation period of the CFF, the fluctuation frequency is consistent with the rotation frequency of the CFF blade, and the amplitude of the fluctuation is related to the vorticity of the shedding vortex of the CFF blade.

In hover state:

The hover angle $α_{h}$ of the propulsive wing is related to the rotation speed of the CFF and varies roughly in the range of 60°–65°.

The vertical aerodynamic force of the propulsive wing increases with the CFF rotation speed, but there is a critical speed. Exceeding the critical rotation speed will lead to a sudden increase in the torque of the CFF.

The hovering flow field of the propulsive wing becomes an asymmetric structure due to the propulsive wing geometry and the wake deflection. A large diameter vortex is formed on the side of CFF inlet, and a small diameter vortex is formed on the lower surface of the airfoil. The diameter of the vortex changes with the CFF rotation speed, causing the movement of the stationary point on the airfoil surface, which in turn causes changes in the aerodynamic force of the propulsive wing.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Jiaxin Lu

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