Unsteady Aerodynamic Forces and Power ... - Springer Link

J Bionic Eng 15 (2018) 298–312 DOI: https://doi.org/10.1007/s42235-018-0023-y

Journal of Bionic Engineering http://www.springer.com/journal/42235

Unsteady Aerodynamic Forces and Power Consumption of a Micro Flapping Rotary Wing in Hovering Flight Chao Zhou, Yanlai Zhang, Jianghao Wu* School of Transportation Science and Engineering, Beihang University, Beijing 100191, China

Abstract The micro Flapping Rotary Wing (FRW) concept inspired by insects was proposed recently. Its aerodynamic performance is highly related to wing pitching and rotational motions. Therefore, the effect of wing pitching kinematics and rotational speed on unsteady aerodynamic forces and power consumption of a FRW in hovering flight is further studied in this paper using computational fluid dynamics method. Considering a fixed pitching amplitude (i.e., 80˚), the vertical force of FRW increases with the downstroke angle of attack and is enhanced by high wing rotational speed. However, a high downstroke angle of attack is not beneficial for acquiring high rotational speed, in which peak vertical force at balance status (i.e., average rotational moment equals zero.) is only acquired at a comparatively small negative downstroke angle of attack. The releasing constraint of pitching amplitude, high rotational speed and enhanced balanced vertical force can be acquired by selecting small pitching amplitude despite high power consumption. To confirm which wing layout is more power efficient for a certain vertical force requirement, the power consumed by FRW is compared with the Rotary Wing (RW) and the Flapping Wing (FW) while considering two angle of attack strategies without the Reynolds number (Re) constraint. FRW and RW are the most power efficient layouts when the target vertical force is produced at an angle of attack that corresponds to the maximum vertical force coefficient and power efficiency, respectively. However, RW is the most power efficient layout overall despite its insufficient vertical force production capability under a certain Re. Keywords: micro air vehicle, flapping rotary wing, aerodynamic forces, power consumption, computational fluid dynamics Copyright © 2018, Jilin University.

1 Introduction Micro Air Vehicles (MAVs) have been receiving increasing research attention because of its various applications in both military and civil areas[1], such as environmental monitoring and homeland security. On the basis of wing layouts, most MAVs can be simply classified into fixed wing, Rotary Wing (RW), and Flapping Wing (FW). To adopt the advantages of various wing layouts, several compound concepts that utilize the flying principles of FW and RW have also been proposed. One of these compound layouts is the micro Flapping Rotary Wing (FRW)[2,3]. The micro FRW concept is inspired by the flight of dragonflies (Fig. 1a). When a dragonfly flies forward (Fig. 1b), it simultaneously generates vertical forces which are required to balance its body weight by vertically flapping its wings and thrusts in flight direction to overcome the drag in forward flight. Similar with dragonfly wings, FRWs also flap vertically but are ar*Corresponding author: Jianghao Wu E-mail: [email protected]

ranged in an anti-symmetric layout (Fig. 1c). Horizontal thrust is simultaneously produced with aerodynamic vertical force as FRW flaps, driving the FRW rotate passively around the central axis (Fig. 1c). An experimental FRW model is shown in Fig. 1d. Although FRW can rotate like a rotary wing, its rotating motion is passive and driven by the aerodynamic thrust-donating rotational moment produced by wing flapping and pitching. Therefore, less anti-torque is required, which is beneficial for aircraft minimization, than in traditional rotary wing. Various experimental[2–6] and numerical[4,7–9] studies have been conducted to explore the FRW aerodynamic performance and the effect of wing kinematic parameters on wing aerodynamic forces. Based on the piezoelectrical[3,4] and mechanic models[2,5,6], relations between aerodynamic forces of FRW and wing kinematic parameters (e.g., flapping frequency, geometricalbased Angle of Attack (AoA) as well as wing rotating speed) have been clarified. The mechanism of how

Zhou et al.: Unsteady Aerodynamic Forces and Power Consumption of a Micro Flapping Rotary Wing in Hovering Flight L

(b)

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T L T Vertically flapping

Pitching

L

(c) T

Vertically flapping

(d)

Rotating Vertically flapping

Vertically flapping

L

T

Pitching

Fig. 1 Dragonfly flight and FRW biomimetic aerodynamic mechanism. (a) A dragonfly; (b) a dragonfly-like flapping wing in forward flight; (c) a dragonfly-like flapping wing with rotating motion; (d) FRW model.

aerodynamic forces are produced has also been addressed by analyzing the vortex field around the wing[7,8]. Wing pitching motion has great influence on wing aerodynamic. However, different AoAs not only lead to different wing vertical force, but also result in different passive rotational speeds[5], which affect the unsteady wing aerodynamic forces as well. It is suggested that effects of wing pitching and rotating motion couple with each other. Although aerodynamic performance of FRW under different wing kinematic has been addressed, we could not tell whether the difference in aerodynamic forces comes from the variation of wing AoA or wing rotating speed. Therefore, independent effect of AoA and the rotational speed on wing aerodynamic forces is still unclear, thereby requiring further investigation. Besides, subjected to the mechanical FRW models, their wing AoAs are generally set as constant and wing pitching amplitude is achieved by wing flexible deformation. The wing pitching amplitude of FRW model studied, accordingly, is generally given a small value in the range of 20˚ – 40˚. Natural insects that use vertical flapping wings (e.g., dragonflies) generally adopt large pitching amplitudes (i.e., approximately 80˚ for dragonflies[10]). This raises an interesting question that which pitching amplitude could be better for FRW to achieve a

better aerodynamic performance. Therefore, the effect of pitching motion on aerodynamic performance should be studied. Before a specific layout is used in the MAV design, the advantages and disadvantages of using such wing layout should be initially considered. Aerodynamic vertical force and aerodynamic power efficiency are two important criterions to which the aerodynamic performances of various wing layouts are usually compared. Lentink et al.[11] and Kruyt et al.[12] experimentally studied the aerodynamic power efficiency in the vertical force generation of HFW and RW. They found that RW is superior to (Horizontally Flapping Wing) HFW. Kruyt et al.[12] further compared the aerodynamic power efficiency of a series of hummingbird wings with that of a commercial micro-helicopter rotor and suggested that their efficiencies are remarkably similar. Similar comparison conclusions have also been drawn from several numerical studies. Wu et al.[8] conducted a systematic comparison of the aerodynamic power efficiency of various wing layouts under certain vertical force production conditions. They found that RW is the most efficient when a small vertical force is required. However, if a larger vertical force is required, FRW is the only wing layout that can satisfy the vertical force and

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power efficiency requirements. Relevant works include Refs. [9,13–15]. The comparison conclusions above are generally drawn based on the baseline condition with a constant Reynolds number (Re). Although the baseline condition is reasonable for scientific research, engineers would not consider Re as a primary designing parameter and a constant. Therefore, the aerodynamic performances of various wing layouts should be further compared based on the new baseline condition that must be built in terms of practical application. In this study, the unsteady aerodynamic performance of a micro FRW, including the wing vertical force and power consumption, is further investigated using Computational Fluid Dynamics (CFD) method based on our previous study. The independent effect of wing pitching and rotating kinematics on the vertical forces of the wings is initially studied to explain how different vertical forces are produced given different pitching and rotating kinematics. Then, the coupling influences of the two typical parameters are further investigated based on the balancing status of the rotational moment. The influence of pitching amplitude on balanced aerodynamic forces and power consumption is also explored. Finally, the aerodynamic power consumption of various wing layouts that could achieve hovering flight is compared with constant vertical force constraints. The baseline conditions with and without the constant Re constraint are considered to reveal which wing layout is more power efficient.

vestigate the aerodynamic performance of the four wing motions. The wing kinematics and coordinate systems are presented in Fig. 2. In this study, three Cartesian coordinate systems are introduced to prescribe the wing motions, namely, inertia system O-XYZ, co-rotating system O-xyz, and body-fixed system O-x′y′z′, as shown in Fig. 2a. The three coordinate systems share origin O, which is a fixed point on the wing root. The OXZ plane is in the horizontal plane. The co-rotating system O-xyz initially coincides with the O-XYZ while it co-rotates as the wing rotates around the OY axis. The O-x′ and O-z′ axes in the body-fixed system O-x′y′z′ are along the chordwise and spanwise directions, respectively. The wing kinematics are prescribed by three angular parameters, namely, flapping angle ϕ (the angle between axes Ox and Ox′),

2 Model and method In this investigation, the unsteady motion of a FRW is modeled as a pitching motion around one-quarter of the chord line, combined with vertical flapping and horizontal rotating motion (Fig. 1c). The wing’s geometric parameters and the FRW’s flapping kinematic parameters are based on a dragonfly wing at Re = 2325[10]. The aerodynamic vertical force and the power consumption of FRW are computed by solving the 3D incompressible Navier–Stokes equations using the CFD method. 2.1 Wing model and kinematics For simplicity, a rigid rectangular plate wing with thickness of 1% of wing chord length (c) is used to in-

Fig. 2 Coordinate systems and wing kinematics in (a) overall and (b) airfoil section views in the middle of downstroke and upstroke, respectively.

Zhou et al.: Unsteady Aerodynamic Forces and Power Consumption of a Micro Flapping Rotary Wing in Hovering Flight

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wing AoA α (the angle between the wing surface and Ox′-Oz′ plane), and rotating angle ψ (the angle between axes OX and Ox′). The sinusoidal function[8,10] of ϕ is adopted and the non-dimensional angular velocity ϕ is given as:

ϕ = π/360 Φ sin(2πτ),

(1)

where Φ and τ are the flapping amplitude and nondimensional time, respectively. Kinematic parameters, including wing geometric and wing flapping parameters, are nondimensionalized. The wing chord length and the average flapping velocity at the wing tip in a flapping stroke are selected as the reference length and velocity (U), respectively. U is expressed as πcλfΦ/180, where λ and f are the wing aspect ratio and flapping frequency, respectively. Reference time Tref is expressed as c/U. The non-dimensional time τ is t/Tref, where t is real time. α is defined using a trapezoidal function with symmetric rotation[8,10]. The constant AoAs between the wing and the plane x′Oz′ in the upstroke and the downstroke are defined as αu and αd, respectively. In the reversal phase at the end of the stroke, the non-dimensional angular velocity ( α ) is given as:

α = ωr{1 − cos[2π(τ-τr)/Δτr]} τr≤ τ ≤ τr + Δτr, (2) where ωr, τr, and Δτr are the mean non-dimensional angular velocity, non-dimensional time when reverse starts, and non-dimensional reversal time interval, respectively. During the time of Δτr, the wing AoA changes by Δα, that is, from αu to αd. Δα and the mean AoA (α0) are computed as αu − αd and (αu + αd)/2, respectively. For ψ, the constant rotating speed of ψ is considered in this study, and the ratio of flapping period to rotation motion (n) is also introduced. 2.2 Governing equations and solution method The 3D incompressible unsteady Navier–Stokes equations govern the flow around the wing, which are written in the inertia coordinate systems in the following dimensionless form:

∇⋅u = 0 ⎧ ⎪ , 1 2 ⎨ ⎪⎩ut + (u ⋅ ∇)u + ∇p − Re ∇ u = 0

(3)

where u is the velocity vector and p is the static pressure. Re is defined as Uc/ν, where ν is the kinematic viscosity

Fig. 3 O-H grid used for computation. (a) Complete grid; (b) surface mesh.

of fluid. The artificial compressibility method developed by Rogers et al.[16,17] is employed to derive the velocity and the pressure in the flow field around the wing using an O-H grid (shown in Fig. 3). Details of the flow solver can be found in a previous study[7,8]. Once the Navier–Stokes equations are solved, the flow pressure and the velocity components in any time in physics are obtained. The force (F) and moment (M) vectors acting on the wings in the inertia coordinate systems are computed by integrating the pressure and viscous stress along the wing surface. The same reference nondimensionalized wing kinematic parameters are also used to nondimensionalize the force and moment. The force and moment coefficients are computed as CF = F/(0.5ρU2S) and CM = M/(0.5ρU2Sc), respectively, where ρ is fluid density, and S is wing area given as λc2. The components of CF and CM in the vertical direction (i.e., OY direction) are defined as vertical force coefficient (CV) and torque coefficient (CQ), respectively. Wing power consumption is also studied in this paper and it is mainly used to overcome the work conducted by wing drag, pitching moment, and wing inertia force, regardless of the wing layout used for the hovering flight. Given that inertia closely depends on various factors such as wing structure and material, the present study only focuses on aerodynamic work. Power consumption here is studied based on the non-dimensional aerodynamic power coefficient. Based on the CM and non-dimensionalized angular velocity vectors (ω) in the inertia coordinate systems, nondimensionalized aerodynamic power coefficient (CP) is given as CP = CM · ω. Aerodynamic power consumption of a FRW contributes to wing flapping, pitching and rotating motions at an arbitrary rotating speed. Here note that power consumptions of FRW only at torque equilibrium state are

considered in this paper. At these states, FRW rotates at a constant speed and its average torque is zero, so the average power consumption coming from rotating motion becomes zero and power consumed is mainly from wing flapping and pitching. The average vertical force coefficient ( CV ) and aerodynamic power coefficient ( CP ) are then computed by averaging CV and CP in a flapping period, respectively. 2.3 Validation Validations on the computational code and the grid model (i.e., an O-H grid[8]) are conducted to ensure the rationality of the simulation. Grid density, computational domain size, and time step are verified first. In this paper, three wing layouts, that are RW, HFW and FRW, are considered. Given that the motion of FRW is most complicated among this studied wing layouts, a typical case of FRW is selected and tested with the following kinematic parameters: Re = 2325, λ = 5.8, αu = 60˚, αd = −20˚, Φ = 70˚,τ = 14.17, Δτ = 0.25τ, and ψ = 0.074.

The time courses of CV and CQ computed by three grids with different densities in a flapping cycle are plotted in Fig. 4. The node qualities in normal, chordwise, and spanwise directions are 81 × 81 × 91 (grid 1), 51 × 53 × 57 (grid 2), and 31 × 33 × 37 (grid 3), respectively. The outer boundary is located 30 c away from the wing surface and 15 c away from the wing tip. The first grid spacing near the wing is 0.0005, 0.001, and 0.002, respectively. The computation used a time step of 400 in a flapping cycle. Non-dimensional time tˆ is defined to clearly describe the time courses of wing motion during one flapping cycle. tˆ = 0 and tˆ = 1 indicate the start of the downstroke and the end of the upstroke, respectively. As shown in Fig. 4, CV and CQ are converged when the grid is increased from grids 2 to 1. Therefore, the grid density of grid 2 is selected and computational domain size and time step are further validated based on grid 2. The computational domain size and the time step are suggested to be 30 c and 400, respectively, in one flapping cycle for this study. In summary, the preceding analysis indicates that grid 2 with 400 time steps in one flapping cycle is appropriate and therefore selected for all cases in this study. The code validation for HFW and RW has been

CV


CQ

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Fig. 4 Computed results of (a) CV and (b) CQ in one flapping period by different grids.

verified in several previous studies[10,18]. The computational code should also be tested to conduct the unsteady aerodynamic analysis of FRW because the flow of FRW is more complex than those of the HFW and RW. However, rare cases of FRW, both experimental and numerical, are found and compared. Thus, a case presented by Gopalakrishnan and Tafti[19] is taken as the validation case. In this case, a forward-flight rigid plate with vertical flapping and pitching motions is studied. Most of the FRW features, except for the rotational effects, are included. However, the uniform inflow represents the rotational effect to a certain extent. In the work of Gopalakrishnan and Tafti[19], a rectangular plate with an aspect ratio of 3.5 was analyzed. The wing root was 0.5c away from the flapping axis, and the pitching axis was located on a quarter-chord line. The advance ratio (defined by the ratio of forward-flight velocity to flapping velocity at the wing tip) was 0.5 and Re was 10,000. The wing motion parameters were Φ = 30˚ and Δα = 30˚. Gopalakrishnan and Tafti[19] examined the unsteady aerodynamic force of the flapping wing using Large-eddy simulation on a deforming body-fitted fluid grid.


3 Results and discussion 3.1 Aerodynamic forces with fixed Δα The effect of wing rotation and pitching kinematics on wing aerodynamic performance is studied independently based on typical cases. The parameters of these typical cases are the same with those of our previous study[8] and set as: Re = 2325, λ = 5.8, Φ = 70˚, Δα = 80˚,τ = 14.17, and Δτ = 0.25τ. The rotational speed here is provided actively from n = 1/12 to n = 1/3 and αd is set from –30˚ to 10˚ on the basis of Δα = 80˚. The mean vertical force and rotating moment coefficients computed from the cases with different downstroke AoA (ad) and rotational speeds are illustrated in Fig. 6.

As shown in Fig. 6, given the same pitching kinematics, CV monotonically increases with n for the wing, which suggests that larger rotational speed is conducive to CV augment. Both positive and negative CQ values are produced in the parameter spaces studied, and CQ decreases with the increment of n. The wing with positive CQ could acquire a higher rotational speed than the current one, while the wing with a negative CQ could not maintain the current wing rotational speed. As n increases, the αd corresponding to the equilibrium status of rotational moment, that is, CQ = 0, also decreases. Therefore, small ad is designed to achieve a higher rotational speed. This finding corresponds to the previous study that investigated the thrust produced by flapping

CV , CT

The comparison of the vertical force and thrust coefficients in one flapping cycle in the present study with those in the work of Gopalakrishnan and Tafti[19] is shown in Fig. 5. The two methods show a reasonable agreement in vertical force and thrust trends. The differences between the present study and the work of Gopalakrishnan and Tafti[19] are less than 5% and 10% at the peak value of CV and CT (thrust coefficient, CT = T/0.5ρU2S, where T is the thrust of the flapping wing), respectively.. The average aerodynamic coefficients in one flapping cycle are CV = 0.574 and CT = 0.319 in the present study, and CV = 0.571 and CT = 0.376 in the work of Gopalakrishnan and Tafti[19]. The slight variations are attributed to the difference in wing thickness, which was not clearly introduced in the reference. However, the code used in the previous study is suitable for the present study.

303

Fig. 5 Comparison of Cv and CT from numerical results with those from Gopalakrishnan and Tafti[19]. CQ CV

η = 1/ 6 η = 1/ 6

η = 1/ 3

η = 1/ 2

η = 1/ 3

η = 1/ 2

2

1.0

0.8

0

0.6 −2 0.4 −4

0.2 −30

−20

−10 αd (˚)

0

10

Fig. 6 CQ and CV at different αd at fixed rotating speed.

wing or airfoil. At the same rotational speed plotted in Fig. 6, CV varies with αd in parabolic rule, and the peak value of CV is acquired at high αd (i.e., 0˚ – 10˚). Although high αd is beneficial for higher CV , it also decrements CQ (as shown in Fig. 6a) and leads to a decelerated rotational motion, which may be harmful to the increment of CV . To further explain the reasons for the large CV given a high rotational speed or αd, the instantaneous aerodynamic forces and rotating moment are analyzed. Fig. 7 plots the time courses of FRW’s CV and CQ given αd = 10˚ with different n, that is, n = 1/12, 1/6, and 1/3. As shown in Fig. 7, CV and CQ in the downstroke rarely change as n increases, and the enhanced CV and decreased CM are mainly attributed to the upstroke. This may be explained by the variation of the effective AoA[7,8] and the real incoming velocity[8] (Fig. 8). The


304 n

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2 1 0 −1 0.0

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(b)

0 −2 0.0

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Downstroke

Upstroke 0.5

1.0

Upstroke 1.5 ^t

Upstroke

Downstroke 2.0

2.5

3.0

Fig. 7 Time courses of CV (a) and CQ (b) produced by cases given αd = 10˚ with different n.

effective AoA is described as αe = α − β, where β is the angle between the rotational speed and the flapping speed as shown in Fig. 8. The wing’s effective incoming flow velocity is enhanced but αe decreases due to the increment of n. The influence of the two aspects contradicts in improving the vertical force and rotational moment. Hence, downstroke CV and CQ are approximately the same in the range of n from 1/12 to 1/3. Different from the downstroke, both αe and the incoming velocity in the upstroke increase, thus CV and CQ both increase with n. The time courses of CV and CQ given n = 1/6 with different αd, that is, −30˚, −10˚, and 10˚, are studied to discuss the effect of wing pitching kinematics on aerodynamic performance (shown in Fig. 9). As shown in Fig. 9, downstroke mainly contributes to the production of CV regardless of the αd selected; upstroke accounts for negligible positive vertical force and even provides a large negative component. This phenomenon is also observed in a dragonfly wing during its forward flight. As αd increases from –30˚ and 10˚, instantaneous CV is always enhanced, thereby accounting for the increment of the mean CV in a full flapping period. Although the time courses of CV at different αd resemble each other, instantaneous CQ varies severely with αd. At a small αd, positive rotational moment is always produced in a flapping period. However, as αd increases, the rotational moments in both downstroke and upstroke gradually become negative. The difference of instantaneous aerodynamic forces and moments varies with αd and reflects in the flow structure (Fig. 10). In the middle of

Fig. 8 Schematic of FRW airfoil and its effective AoA and incoming velocity in (a) downstroke and (b) upstroke.

the downstroke (Figs. 10a and 10b), the Leading Edge Vortex (LEV) is attached to the Leading Edge (LE) at αd = −30˚, which causes a LE suction that contributes to the vertical force and the thrust-donating rotational

Zhou et al.: Unsteady Aerodynamic Forces and Power Consumption of a Micro Flapping Rotary Wing in Hovering Flight αd

(a)

−30˚

−10˚

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10˚

2 1 0 −1 0.0 (b)

0.5

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3.0

6 3 0 −3 Upstroke −6 Downstroke 0.0 0.5

Downstroke 1.0

Upstroke 1.5 ^t

Downstroke 2.0

Upstroke 2.5

3.0

Fig. 9 Time courses of (a) CV and (b) CQ produced by cases given n = 1/6 with different αd.

moment. However, when the wing flaps at a high αd (e.g., αd = −10˚), the LEV nearby separates from the LE and the Trailing Edge Vortex (TEV), which is weak and attached to the trailing edge at small αd, is greatly enhanced and sheds from the trailing edge. Hence, vertical force and the negative rotational moment are mainly produced by the pressure on the entire wing (Fig. 10d, αd = 10˚). When the wing moves to the middle of the upstroke with a small αd (−30˚), despite a decreased αe due to the wing rotational speed, αe is still negative and a LEV is formed on the lower surface of the wing (Fig. 10c), thereby resulting in a negative vertical force but a positive rotational moment. As αd increases, αe becomes positive and LEV occurs in the wing’s upper surface. However, the wing flaps in an almost vertical gesture because αd is very large, thus the varied LEV mainly results in the decrement of the negative rotational moment and only causes a negligible vertical force in the upstroke. High αd and n are both beneficial for vertical force production. However, a combination of high αd and n could not be achieved because high αd is against high n. Consequently, the aerodynamic forces of FRW are produced at a balanced status and the real operating conditions of FRW are considered. Similar to our previous study[8], the case with CQ = 0 is taken as the balanced status. The computed wing balanced rotational velocity and the computed CV with different αd values are plotted in Fig. 11. As shown in Fig. 11, as αd increases from −20˚ to 0˚, the balanced rotational speed approximately decreases with αd. Rotational speed is

closely related to average rotational moment, which can be simply separated into thrust-causing and drag-causing rotational moments. Thrust-causing rotational moment is mainly attributed to wing pitching and flapping, while drag-causing moment is produced by wing rotation. Wu et al.[7] computed the rotational moment with different α0 values but the same rotational speed and Δα (seen in Ref. [7], Table 20). CQ decreases with the increment of α0 that is, αd given the constant Δα, and is nearly proportional to −α02. With constant Δα, the rotational moment produced by the thrust, which is the projection of total force on the horizontal plane, can be reasonably thought to be constant at small α0. Therefore, the change in CQ that varies with αd at same rotational speed is mainly attributed to the drag-causing rotational moment. The averaged drag-causing rotational moment coefficient ( CQD , nondimensionalized by wing rotational speed) can be assumed to change as CQD ∝ (−α02). At the balanced status, the thrust- causing rotational moment remains approximately the same with constant Δα. Therefore, the drag-causing moment, which is described as 0.5ρUr2 CMD , is also constant with different α0. Given CQD ∝ (−α02), Ur or n can be deduced as almost proportional to α0. Under the coupling effect of both αd and rotational speed, CV sharply increases as αd increases to −20˚, while it slightly decreases as αd continues to increase. Fig. 12 shows that the instantaneous CV at αd = −20˚ and αd = 0˚. As plotted in Fig. 12, both the upstroke and downstroke contribute to the enhanced CV produced at αd = −20˚. Compared with Fig. 9a, the upstroke vertical


306

αd = 30˚

αd = 10˚

αd = 0˚

(a)

(b) −1.0 Pressure coefficient

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force at αd = −20˚ at a balanced status approaches αd = 0˚ as a fast wing rotational speed counteracts the opposite effect of small αd on the wing vertical force. A fast rotational speed leads to a positive αe at αd = −20˚ and significantly enhances the positive vertical force production. Therefore, by increasing the rotational speed, FRW could obtain an enhanced vertical force by selecting a moderate downstroke AoA with a constant Δα.

η

Fig. 10 Flow structure around FRW given n = 1/6 with different αd in the middle of (a, b) downstroke and (c, d) upstroke.

3.2 Aerodynamic forces and power varied with Δα

FRW can acquire the benefit of high vertical force by increasing its rotational speed, which is related with wing thrust production. Numerous studies on thrust performance of the flapping wing or plunging airfoil

Fig. 11 Aerodynamic vertical force coefficient and rotational speed at different αd at balanced condition.

Zhou et al.: Unsteady Aerodynamic Forces and Power Consumption of a Micro Flapping Rotary Wing in Hovering Flight αd

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0˚

−20˚

CV

2 1 0 −1

Downstroke 0.0

Upstroke 0.5

Downstroke 1.0

Upstroke

Downstroke

1.5 ^t

2.0

Upstroke 2.5

3.0

Fig. 12 Time courses of CL at balanced status given Δα = 80˚ but different αd .

As shown in Fig. 13a, the quantitative differences between the n and CV values at different Δα are computed. The phenomenon where n linearly decreases with α0 is still suitable for the wing with different values of Δα from 40˚ to 100˚. However, the slope between n and α0 is changed, and the smaller Δα, the larger the value of |dn/dα0|. |dn/dα0| at Δα = 40˚, that is, (0.0553 rad−1), is much larger than that at Δα = 60˚ (0.0234 rad−1), and it slightly decreases to 0.0141 rad−1 as Δα increases to 100˚. The wing balanced rotational speed is much sensitive to the variation of the wing AoA at small Δα than at large Δα. For wing pitches with Δα > 40˚, the wing balanced rotational speed acquired at the same α0 from 10˚ to 40˚ also decreases as Δα increases. Thus, if a fast rotational speed is desired, a small Δα may be selected. However, n with Δα = 40˚ is different from the other three conditions of Δα. When α0 is less than 20˚, the wing rotational speed at Δα = 40˚ is larger than that at Δα > 40˚, and a small Δα is still beneficial for high

(a)

1.4 1.2

Δα = 100˚

Δα = 80˚

Δα = 60˚

Δα = 40˚

1.0

η

0.8 0.6 0.4 0.2 0.0

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10

20 α0 (˚)

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40

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have revealed that thrust is strongly related to wing pitching kinematics[20–23], especially wing pitching amplitude[23]. FRW could achieve a higher rotational speed and subsequently acquire a larger vertical force by altering the wing pitching amplitude[8]. Therefore, the cases that release constant AoA pitching amplitude (i.e., Δα = 80˚) are further considered. In our previous work[8], CV and CP with different Δα values, that is, 100˚, 80˚, 60˚, and 40˚, have been computed and revealed that a small Δα is beneficial for high CV and power efficiency. However, the reason why high CV is produced by small Δα is not clearly explained. Therefore, the aerodynamic performance with different Δα is further discussed in this section. The wing rotational speed and aerodynamic forces with different Δα are computed and plotted in Fig. 13.

0.6 0.4 0.2 0.0

Fig. 13 n and CV at balanced status given Δα = 40˚ – 100˚. Note that the dots and line segments in the insets of (b) mean the airfoil schematics in the middle of upstroke and downstroke, respectively.

rotational speed. Ignoring the influence of the antirotational moment caused by frictions, n larger than 1 could also be theoretically acquired at α0 < 10˚ and Δα = 40˚, which means that the wing could rotate one cycle even when the wing does not undergo a complete flapping period. However, due to the severe change of n with α0, when α0 goes beyond 20˚, n at Δα = 40˚ becomes smaller than Δα > 40˚. Therefore, the wing AoA

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should be designed carefully if Δα = 40˚ is selected. The decrement of Δα not only results in the increment of n, but also enhances the CV production. The maximum CV acquired at α0 = 15˚ and Δα = 40˚ is 1.15, which is almost 60% larger than the maximum CV acquired at Δα = 80˚. Although wing pitching with small Δα could effectively enhance CV production, it also leads to severe fluctuation of CV that varies with α0, which brings great difficulties for the wing AoA selection and structural design. Despite Δα, the curve of CV versus α0 is still approximately parabolic. When the wing flaps with large pitching amplitude, such as Δα > 40˚, a maximum CV of FRW is acquired with α0 = 20˚ as it is produced with α0 around 15˚ as Δα decreases to 40˚. This phenomenon matches our previous experimental results[5,6] based on the mechanical model. Therefore, for the sake of maximum CV , the geometrical AoA of 15˚ – 20˚ is suggested. The AoA for the maximum CV corresponds to modest rotational speed other than the fastest rotational speed (Fig. 14a). Therefore, for FRW, a fast rotational speed is not always beneficial for CV enhancement, which is much different for RW. Wing power consumption is another important parameter in primary design, and it is closely related to the design of wing aerodynamics and power system. Hence, aerodynamic power consumption is also studied. In several previous studies, power factor[8,13,14] is investigated to measure the power efficiency of specific wing layouts in producing the same vertical force. Considering that wing power is directly computed based on the power coefficient in practical applications, the wing power coefficient is investigated here instead of the power factor. The curves of CP versus CV with different Δα are shown in Fig. 14b. The envelope of the curves of CP versus CV that are drawn based on the results computed in the range of Δα from 40˚ to 100˚ has a similar shape with that of the power factor versus CV [8]. A small Δα could enhance CV but at a high power consumption cost. The wing power factor still increases due to the faster increment of CV than CP as Δα decreases. The change of the curves of CP with CV at Δα = 40˚ is significantly different from that at Δα > 40˚. In the case of Δα > 40˚, the increments of CP and CV are synchronized. However, when Δα decreases to 40˚,

Δα = 100˚ Δα = 60˚

(a) 1.5

Δα = 80˚ Δα = 40˚

α0 1.2

0.9 η

α0 0.6

0.3 0.0 0.0

0.2

0.4

(b) 1.0

CV

0.6

0.8

1.0

1.2

CP

α0

0.5 Cp − CV envelope α0

0.0 0.0

0.2

0.4

CV

0.6

0.8

1.0

1.2

Fig. 14 CP and balanced rotational speed varied with CV at different Δα and α0.

despite a severe variation in CV , CP slightly changes and is kept in the range of 0.68 – 0.84, which suggests that CV could be enhanced with less and even without extra aerodynamic power consumption. In these conditions, FRW could obtain high power efficiency during vertical force production. 3.3

Comparison of power consumption among various wing layouts

The aerodynamic performances of various wing layouts capable of hovering flight are compared in this part. In the primary phase of MAV design, aerodynamic design initially aims to select a specific wing layout and design wing kinematics and geometrical parameters to satisfy the aircraft weight requirement, which is generally a fixed criterion. Thereafter, with a certain CV generated, the aerodynamic power consumption is usually estimated to conduct a power system design. Power


carried by MAV is finite, thus wing layout consumes less power with the same vertical force production is preferred and previous literatures usually compare aerodynamic performance of various wing layouts using power factor. However, those works are usually conducted based on the prerequisite with constant Re, which is against practical applications. Therefore, a common constraint here is given certain vertical force requirement while Re is variable. Before comparison, the aerodynamic vertical force and power consumption at different kinematic and Re conditions should be built as reference for the MAV aerodynamic design. Therefore, the aerodynamic power consumption of various wing layouts that vary with CV at a constant Re is investigated. Generally, MAV wings move in a Re at the scale of O (103). When Re > 1000, the effect of Re on wing aerodynamic forces and moments becomes weak. For simplification, the aerodynamic force coefficients and power coefficient at Re > 1000 are considered constant and equal to the computed typical cases at Re = 2325. RW, flapping wing (including Horizontally Flapping Wing (HFW) and (VFW)), and FRW are the four wing layouts capable of hovering flight. Only the CP and CV of RW, HFW, and FRW are compared because the aerodynamic power efficiency of VFW is lower than those of the other three wing layouts at Re = 2325[8]. The CV − CP curves of all wing layouts at Re = 2325 have similar CV -power factor shapes. The regime can be divided into four zones based on the CV and CP results from the three wing layouts. (1) CV < 0.4. All wings can fulfill this CV requirement, while RW is the most efficient with the least power consumption. (2) RW could not satisfy such CV requirement. FRW and HFW are still capable of producing CV in the range of 0.4 < CV < 0.688. FRW appears to be more efficient than HFW as it consumes less power given a certain CV . In addition, HFW may require high aerodynamic power to obtain a high CV that is larger than 0.68 as CP sharply increases with CV in this condition (0.68). (3) CV > 0.688. Given Re = 2325, only FRW could satisfy the high CV requirement. (4) CV > 1.097. The CV requirement is beyond the capability of vertical force production for all wing layouts if geometrical parameters and Re are unchanged. In terms of vertical force generation and aerodynamic power

FRW

1.0

RW II

I

309 HFW IV

III

0.8

0.6

0.4

0.2

0.0 0.0

0.2

0.4

CV

0.6

0.8

1.0

1.2

Fig. 15 Comparison of CV and CP among FRW, RW, and HFW at Re = 2325.

consumption, it could be concluded that FRW, to some extent, is superior to RW and HFW under certain Re in two aspects. The first is that it could produce higher vertical force than RW and HFW so that it may be a better layout when designing MAVs with heavy loads. Another advantage of FRW is that it is of a wide CP distribution at a specified CV due to a wide selection of wing pitching kinematics. It means we could further achieve a more power efficient flight of FRW by designing the pitching kinematics properly with the requirement of certain vertical force satisfied. According to the results of CV and CP at Re = 2325, we compare the aerodynamic power consumed by the three wing layouts (i.e., RW, HFW, and FRW) on the basis of certain target dimensional vertical forces. To achieve the target vertical force in the primary design phase, apart from the geometrical parameters, wing AoA and Re are two important kinematic parameters that are usually confirmed under the following procedures. Generally, wing AoA and Re are confirmed successively. The choice of wing AoA has two strategies, that is, maximum CV (Strategy 1, S1) and maximum power factor (Strategy 2, S2). To achieve the target vertical force, the Re of three wing layouts with specific wing kinematics may be adjusted by altering the wing flapping or rotational speed, which is the normal practice in MAV design to acquire the required vertical force. Then, power consumption could be computed based on Re after adjustment. To make the aerodynamic per-


310

formance comparable with that at Re = 2325 (Fig. 15), the target vertical force (FVT) is non-dimensioned using the reference parameters at Re = 2325 and the target vertical force coefficient ( C 'V,Re= 2325 ) is given as: 1 ρU '2 SC 'V, Re U ' 2 C'= = = ( ) 2 C 'V,Re , (4) 1 1 U 2 2 ρU S ρU S 2 2 FVT

where (·)′ donates variables at the Re after adjustment and (·)′Re = 2325 donates variables at Re = 2325. At the scale of Re ~ O(103), C 'V,Re is approximately regarded as unchanged and is replaced by CV,Re=2325 . Consequently, C 'V, Re=2325 can be expressed as: C 'V, Re = 2325 = (

U' 2 ) C V, Re = 2325 . U

(5)

Similarly, C 'P, Re= 2325 can be described as: C 'P, Re = 2325

1 ρU '3 SC 'P , Re U' = = 2 = ( )3 C 'P, Re , 1 1 U ρU 3 S ρU 3 S 2 2 P

(6) where P is the power consumption that produces FVT. 1/2

⎛C' ⎞ U' by ⎜ L, Re= 2325 ⎟ , then C 'P, Re= 2325 can be ⎜ C ⎟ U L, Re ⎝ ⎠ expressed as: Substitute

C 'P, Re = 2325 = (

C 'V, Re = 2325 C V, Re

)3/ 2 C 'P, Re .

(7) Fig. 16 Comparison of power consumption among FRW, RW, and HFW with different target vertical force requirement.

Re, after adjustment (Re′), is given as: Re′ = 2325 (

C 'V, Re = 2325 C V, Re = 2325

)1/ 2 .

(8)

The maximum CV of HFW (0.688) and FRW (1.097) at Re = 2325 are selected as target C 'V, Re= 2325 , which represent the conditions with moderate and high vertical force requirements, respectively. The C 'P, Re= 2325 of the three wing layouts using strategies S1 and S2 with target C 'V, Re= 2325 fulfilled are computed based on Eq. 7 and plotted in Fig. 16. Taking the first target C 'V, Re= 2325 (0.688) as an example, when S1 is selected, the target C 'V , Re= 2325 is larger than the maximum CV of RW and smaller than that of the FRW acquired at Re = 2325. Therefore, to achieve the target CV, the Re of RW is

increased to 2953 while that of FRW is decreased to 1842. The C 'P, Re= 2325 of RW at 0.641 is closer to that of HFW at 0.681, but is almost 53% larger than the C 'P, Re= 2325 of FRW. Therefore, FRW may be more efficient than RW and HFW if the target C 'V, Re= 2325 is produced at the angle corresponding to maximum CV (i.e., AoA strategy S2). Given that the CV with highest aerodynamic efficiency is smaller than the maximum CL , the Re′ at S2 is slightly higher than that using S1, which is suitable for the three wing layouts. Although the increment of Re could increase the power consumption to some extent, strategy S2 is still beneficial for the decrement of power consumption. The C 'P, Re= 2325 values


of the three wing layouts in S2 are lower than the values computed in S1. Therefore, the pitching kinematics that corresponds to the maximum power efficiency is suggested to be selected in MAV design to achieve better power performance. The comparison of the C 'P, Re= 2325 of a specific wing using the pitching kinematics strategy reveals that the C 'P, Re= 2325 of RW is only 45% after the wing pitching kinematic strategy is changed to S2 and the C 'P, Re= 2325 of HFW decreases by 24%. Comparatively, regardless of the pitching strategy selected, the C 'P, Re= 2325 of FRW slightly changes and is about 0.4 because the case of FRW that produces the maximum CV is the case with the highest power efficiency. The C 'P, Re= 2325 of RW at 0.266 is much smaller than those of FRW at 0.385 and HFW at 0.519. Among all the cases and using the two strategies, RW using S2 has the smallest power consumption followed by FRW, while HFW produces the same C 'V, Re= 2325 at the highest cost of power consumption. Therefore, under a moderate C 'V, Re= 2325 target and without the Re constraint, RW is the most efficient wing layout. Although a high target C 'V, Re= 2325 at 1.091 is considered, the comparison conclusions above still hold, and RW is always the most efficient wing layout that produces the required vertical force by rotating at the AoA that corresponds to the highest power efficiency at Re > 1000.

4 Conclusion The aerodynamic performance of FRW is greatly influenced by wing pitching and rotating motions. In this paper, the effects of wing pitching and rotating kinematics on unsteady aerodynamic forces and the power consumption of FRW are further studied by CFD analysis. Without Re constrained, the power consumption of FRW is further compared with those of RW and HFW to confirm which wing layout is more power efficient at a particular vertical force requirement. It was found that considering a fixed pitching amplitude (i.e., 80˚), a high downstroke AoA and high wing rotational speed are both beneficial to FRW acquiring high vertical force. However, high downstroke AoA conflicts with the wing’s fast rotational speed, which could not be acquired simultaneously at the balanced status of FRW. Consequently, the peak vertical force at the balance status of rotational moment is only acquired at a comparatively

311

small negative downstroke AoA. Releasing the pitching amplitude constraint, it is found that FRW could rotate in a higher balanced speed if a small pitching amplitude is selected, in which the vertical force at balanced status could also be significantly enhanced despite the high power consumption. By comparing the power consumed by FRW with Rotary Wing (RW) and flapping wing releasing Re constrained, FRW and RW are the most power efficient layouts when the target vertical force is produced at an AoA that corresponds to the maximum vertical force coefficient and power efficiency, respectively. In summary, RW is the most power efficient layout among the three studied wing layouts if it is designed to rotate at the angle of attack with higher power efficiency in vertical force production. These findings have great significance in FRW kinematics design as well as guiding us to select an appropriate wing layout based on designing requirements in primary design phase to ensure efficiency in micro-air-vehicle design.

Acknowledgment This research was primarily supported by the National Natural Science Foundation of China (Grant number: 11672022).

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