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Aerodynamic Characteristics, Power Requirements and Camber Effects of the Pitching-Down Flapping Hovering
Journal of Bionic Engineering 6 (2009) 120–134
Aerodynamic Characteristics, Power Requirements and Camber Effects of the Pitching-Down Flapping Hovering Peng Bai, Er-jie Cui, Hui-ling Zhan China Academy of Aerospace Aerodynamics, Beijing 100074, P. R. China
Abstract The pitching-down flapping is a new type of bionic flapping, which was invented by the author based on previous studies on the aerodynamic mechanisms of fruit fly (pitching-up) flapping. The motivation of this invention is to improve the aerodynamic characteristics of flapping Micro Air Vehicles (MAVs). In this paper the pitching-down flapping is briefly introduced. The major works include: (1) Computing the power requirements of pitching-down flapping in three modes (advanced, symmetrical, delayed), which were compared with those of pitching-up flapping; (2) Investigating the effects of translational acceleration time, 'Wt, and rotational time, 'Wr, at the end of a stroke, and the angle of attack, D, in the middle of a stroke on the aerodynamic characteristics in symmetrical mode; (3) Investigating the effect of camber on pitching-down flapping. From the above works, conclusions can be drawn that: (1) Compared with the pitching-up flapping, the pitching-down flapping can greatly reduce the time-averaged power requirements; (2) The increase in 'Wt and the decrease in 'Wr can increase both the lift and drag coefficients, but the time-averaged ratio of lift to drag changes a little. And D has significant effect on the aerodynamic characteristics of the pitching-down flapping; (3) The positive camber can effectively increase the lift coefficient and the ratio of lift to drag. Keywords: aerodynamic characteristics, flapping wing, power requirements, camber effect, pitching-down-flapping, MAVs Copyright © 2009, Jilin University. Published by Elsevier Limited and Science Press. All rights reserved. doi: 10.1016/S1672-6529(08)60109-2
Nomenclatures (X, Y, Z): The inertial coordinates fixed on the body of MAVs or fruit fly; (x, y, z): The coordinates fixed on the flapping wing; (xc, yc, zc): The coordinates, in which yc axis is the same as Y axis, zc axis is fixed along the spanwise and is the same as z axis; i, j, k: The unit vectors in the (x, y, z) coordinates; I: The azimuth angle of translation; ): The amplitude of translation; D: The angle of attack in the middle of a stroke; Dup and Ddown: The angles of attack in the middle of upstroke and downstroke, respectively;
It : The translational angular velocity; I t and D t : The non-dimensional angular velocities of translation and rotation, respectively;
r0: The inertial radius of the flapping wing; ut: The translational velocity at r0, which is called the flapping velocity; ut : The non-dimensional translational velocity; U: The reference velocity, which equals the time-averaged flapping velocity u t in one cycle; U t : The flapping velocity in the middle of a stroke,
which is a constant; R: The span of the flapping wing; c : The average chord of the flapping wing; S: The area of the wing; f: The frequency of flapping/The camber of the flapping wing; U: The density of air; Re: The Reynolds number; W: The non-dimensional time; W r : The beginning of the rotation at the end of a stroke;
D t0 : The constant to determine D t ;
'W r : The rotational time at the end of a stroke;
D and M : The angular accelerations of rotation and
'Wt: The translational acceleration and deceleration time at the end of stoke;
translation, respectively; Corresponding author: Peng Bai E-mail: [email protected]
Bai et al.: Aerodynamic Characteristics, Power Requirements and Camber Effects of the Pitching-Down Flapping Hovering
121
W 1 : The beginning of acceleration at the beginning of a
CP ,t : The coefficient of power requirement to realize
stroke; W 2 : The beginning of the deceleration at the end of
translation; CP , r : The coefficient of power requirement to realize
stoke; Wc: The non-dimensional flapping period; 'W: The non-dimensional physical time step; 'Wsub: The pseudo-time step in the subiteration; CL and CD: The coefficients of lift and drag, respectively;
rotation; CP : The coefficient of power requirement of the flap-
Qi , r : the rotational inertia moment;
ping; Pa: The power required to overcome the aerodynamic force; Ma: The aerodynamic moment around the origin; Pi: The power required to overcome the inertia force of the wing; :: The angular velocity around the origin. ds: The area element; dm: The mass element of the ds; G r : The vector from the origin to the ds; G F ( f x , f y , f z ): The aerodynamic force acting on ds;
CQ , a ,t : The coefficient of translational aerodynamic
H: The angular momentum of the wing;
moment; CQ , a , r : The coefficient of rotational aerodynamic mo-
I xx , I yy , I zz , I xy , I xz , I yz : The products of inertia of the
ment; CQ ,i ,t : The coefficient of translational inertia moment;
CP : The time-averaged coefficient of power require-
CL and CD : The time-averaged coefficients of lift and
drag, respectively; Qa ,t : the translational aerodynamic moment; Qa , r : the rotational aerodynamic moment; Qi ,t : the translational inertia moment;
CQ ,i , r : The coefficient of rotational inertia moment; CP , a : The coefficient of power requirement to overcome
the aerodynamic force; CP ,i : The coefficient of power requirement to overcome the inertia force;
1 Introduction The development of Micro Air Vehicles (MAVs) makes the flapping flight at Low Reynolds Number (LRE), which widely exists in the nature, become an attractive topic for scientists. Previous studies indicated that for MAVs shorter than 15cm flapping wing might be the only method to obtain enough lift, stability and manipulability to fly. Leonardo da Vinci studied the flapping wing in 1500[1]. In the middle of the 19th century, Etienne-Jules Marey recorded and studied the details of bird flapping flight using a camera with 11 frames per second designed by himself[1]. Later studies on flapping wing appeared in the early papers[2,3]. Systematic studies on the flapping wing were carried out through experiments and numerical simulations by Dickinson et al.[4,5], Ellington et al.[6], Sun and Tang[7], Ramamurti and
flapping wing; ment;
x : The non-dimensional x coordinate; x f : The location of the maximum camber; y : The non-dimensional y coordinate; f : The non-dimensional camber;
Sandberg[8], Bai et al.[9], et al., in the past two decades. Previous works on the fruit fly flapping[4–9] indicate that there are four mechanisms to generate high unsteady lift: (1) Delayed stall, the most important mechanism to generate high unsteady lift; (2) Decelerating translation at the end of a stroke; (3) Accelerating translation at the beginning of a stroke; (4) Pitching-up rotation at the end of a stroke, which generates the peaks of lift and drag, and the peak of drag is much higher than that of lift. Therefore the fruit fly flapping is also called “the pitching-up flapping” in this paper. In order to keep the beneficial delayed stall, accelerating and decelerating mechanisms, and get rid of the pitching-up rotation mechanism, which generates very high drag peaks, the pitching-down flapping was invented by the authors[10]. Compared with the pitching-up flapping, the pitching-down flapping can effectively decrease the drag and improve the time-averaged ratio of
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lift to drag, which means the wing rotates pitching-down at the end of a stroke. In this paper, the pitching-down flapping is briefly introduced. The power requirements, the flapping parameters effects and the camber effect of the pitching-down flapping in hovering condition were investigated with a numerical simulation method[11]. The investigation results suggest that: (1) Compared with the pitching-up flapping, the pitching-down flapping can effectively decrease the time-averaged power requirement, while maintaining the high lift; (2) The lift and drag coefficients increase with the increase in 'Wt and the decrease in 'Wr; the time-averaged ratio of lift to drag changes a little; and D significantly affects the aerodynamic characteristics; (3) For the pitching-down flapping, the camber can effectively increase the lift coefficient and the ratio of lift to drag. However, this beneficial camber effect does not exist for the pitching-up flapping.
2 Kinematics of pitching-down flapping hovering Fig. 1 sketches the fruit fly flapping, in which (X, Y, Z) is the inertial coordinates fixed on fruit fly’s body; (x, y, z) is the coordinates fixed on the flapping wing; and (xc, yc, zc) is the coordinates, in which yc axis is the same as Y axis, zc axis is fixed along the spanwise and is the same as z axis. The rotation around the Y axis is called “translation”, and the azimuth angle is I. The rotation around the
z axis is called “rotation”, and the angle of attack is D. The inertial radius of the flapping wing around the Y axis is defined by r0
³
S
r 2 dS / S .
The translational velocity at r0 is called the flapping velocity, defined by ut. And the translational angular velocity It = ut/r0. Details of the fruit fly flapping were described in Dickinson and Sun’s papers[5,7]. Then only the characteristics are presented here: (1) The order of the leading edge and trailing edge does not change; (2) The windward surface and the leeward surface alternate each other; (3) The rotation at the end of a stroke is pitching-up. Compared with the fruit fly (pitching-up) flapping, the pitching-down flapping maintains the translation, but changes the rotation. And its obvious characteristics are: (1) The leading edge and the trailing edge alternate in each stroke; (2) The windward surface and leeward surface do not change; (3) Pitching-up rotation is replaced by pitching-down rotation at the end of a strokes. The azimuth angle I, the angle of attack D, the translational and rotational velocities I t , D t of the pitching-up flapping in hovering in three modes (advanced, symmetrical, delayed) are shown in Fig. 2. The end with solid circle denotes the leading edge. The parameters of pitching-down flapping in hovering are shown in Fig. 3. The solid circles appear at both ends, indicating that the leading and trailing edges alternate each other and either end can be the leading edge. In Fig. 2 and Fig. 3, both translations are the same and the translational plane is horizontal. In the middle of a stroke, D and ut do not change. ut is given as: ut
U t sin(ʌ(W W 1 ) 'W t ),
W 1 d W d W 1 'W t / 2.
ut
U t ,
W 1 'W t / 2 d W d W 2 .
ut
U t sin(ʌ(W W 2 'W t / 2) 'W t ), W 2 d W d W 2 'W t / 2. t
Where, u
(1) ut / U , U is the reference velocity, which
equals the time-averaged flapping velocity ut in one cycle; U t is the flapping velocity in the middle of a
Fig. 1 Sketch of the fruit fly flapping (pitching-up flapping).
stroke, which is a constant; IJ = tU/c ; IJ1 and IJ1+ǻIJt /2 are the beginning and the end of acceleration at the beginning of a stroke; IJ2 and IJ2+ǻIJt /2 are the beginning and
Bai et al.: Aerodynamic Characteristics, Power Requirements and Camber Effects of the Pitching-Down Flapping Hovering
the end of deceleration at the end of stoke. The rotations of the pitching-up flapping and the pitching-down flapping are different, but the formulas of Dt are identical.
220 200 180 160 140
0.5D t0 [1 cos(2ʌ(W W r ) / 'W r )] ,
W r d W d W r 'W r .
120 100 80
(2)
60
Advanced Symmetrical Delayed
40 20 0
2
4 t (a)
6
8
Advanced Symmetrical Delayed
1.0 0.5 0.0
t
+
where, Dt = Dtc/U, D t0 is a constant; IJr and IJr +ǻIJr are the beginning and the end of the rotation at the end of a stroke. When ǻIJr and the angles of attack, Dup and Ddown, in the middle of upstroke and downstroke are determined, D t0 can be calculated. Here the difference between the two flappings is, for the pitching-up flapping, Dup = 40Û, Ddown = 140Û, and for the pitching-down flapping, Dup = 220Û, Ddown = 140Û. Other details of pitching-down flapping kinematics are given in Ref. [10].
Įt+
D t
123
í0.5
160
Advanced Symmetrical Delayed
140
í1.0
120 0
100 80
2
4 t (b)
6
8
60 Downstroke
40 20 0
2
4 t (a)
6
8
+ t
Upstroke
Advanced Advanced Symmetrical Symmetrical Delayed Delayed
1.0
(c)
Fig. 3 pitching-down flapping hovering; (a) The translational angle I and the angle of attack D; (b) The non-dimensional angular velocities of translation and rotation, I t and D t ; (c) Sketch of the downstroke and upstroke in the symmetrical mode.
0.0
t
+
Įt+
0.5
í0.5 í1.0 0
2
4 t (b)
6
8
(c)
Fig. 2 Pitching-up flapping hovering; (a) The translational angle I and the angle of attack D; (b) The non-dimensional angular velocities of translation and rotation, I t and D t ; (c) Sketch of the downstroke and upstroke in the symmetrical mode.
From Torkel’s experiment[12] and Vogel’s bundling fruit fly flapping experiment[13], the density of air is 1.226 kg·mí3, the weight of the insect is 1.96 × 10í5 N, the mass of the wing is 2.4 × 10í6 g, the span of the wing R = 0.3 cm, the average chord c = 0.108 cm, the inertial radius r0 = 0.58R, the amplitude of translation ) = 2.62 rad ˙ 150Û, the frequency of flapping f = 240 Hz. Then the reference velocity U, Reynolds Number Re, the non-dimensional flapping period Wc and the hovering lift coefficient CLw can be obtained: U = 2)fr0 = 218.7 cm·sí1, Re = cU/Q = 147, Wc˙(1/f)/( c /U) | 8.42, CLw = 1.15.
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Journal of Bionic Engineering (2009) Vol.6 No.2
From Dickinson’s experiment[5], the density of the mineral oil is 0.88 × 103 kg·mí3, the span of the wing R = 0.25 m, the area of the wing S = 0.0167 m2, the average chord c = 8.79 cm, the inertial radius r0 = 0.58R, the frequency of flapping f = 0.145 Hz, the amplitude of translation ) ˙ 160Û, the reference velocity U = 2)fr0 = 0.117 m·sí1, then 0.5UU2S | 0.101 kg·m·sí2. Then the non-dimensional parameters used to study the pitching-down flapping in this paper are obtained: Wc = 8.42, 'Wt = 0.1Wc, 'Wr = 0.32Wc, ) = 160Û.
3 Computational method and grid 3.1 Computational method In this paper, the controlling equations are unsteady incompressible Navier-Stokes (INS) equations with pseudo-compressibility. The numerical calculation scheme is the artificial compressibility method developed by Rogers[11]. The numerical flux is calculated through the 3rd order Roe scheme. The viscosity is important at low Re, the viscous flux is considered and computed by the 2nd-order center difference scheme. In order to study this unsteady problem and accelerate the convergence speed, the 2nd-order two-time step method is used in the temporal direction. The LGS implicit scheme is used in the subiteration. The non-dimensional physical time step 'W = 0.02, which is determined by numerical experiment, and the pseudotime step in the subiteration 'Wsub = 1014. 3.2 Boundary conditions (1) Far-field boundary condition The velocity is given by the free flow and the pressure is interpolated at the far entry boundary. And the pressure is given by the free flow and the velocity is interpolated at the far exit boundary. (2) Wall boundary condition G G G G U U B , wp wn U a B n , (3) G G where, U B and a B , the velocity and the acceleration of G the wall boundary. n , the vector normal to the wall. The correctness of the numerical method and the program has been proven in Refs. [9,10]. 3.3 Model wing and computational grid The model wing of fruit fly of Dickinson’s experiment is shown in Fig. 4a. The average chord c =
8.79 cm, the ratio of wing span to chord O = 2.84, the thickness of the wing is 0.05 c . The leading and trailing edges are semicircular. The model wing used to study the pitching-down flapping is shown in Fig. 4b, which is taken from Fig. 4a by averaging the leading and trailing edges. And the rotational axis is located at 50% chord. Figs. 4c and d show the grid along the chordwise with camber of 0% and 10%, respectively. The numbers of the points along the three directions (circlewise × normal × spanwise) are 81 × 65 × 70. The distance between the far-field boundary and the wall is 15 c . The minimal normal distance near the wall boundary is 0.002 c .
(a) The fruit fly model wing of Dickinson’s experiment
(b) The model wing and computational grid of the pitching-down flapping
(c) Computational grid of the chordwise section with camber equals 0
Fig. 4 The model wing and the computational grid of the pitching-down flapping.
Bai et al.: Aerodynamic Characteristics, Power Requirements and Camber Effects of the Pitching-Down Flapping Hovering
(d) Computational grid of the chordwise section with camber equals 10%
Fig. 4 Continued.
4 Results and discussion 4.1 CL and CD of the pitching-down flapping hovering The lift and drag coefficients, CL and CD, and the ratios, CL/CD, of pitching-up and pitching-down flapping hovering in three modes (advanced, symmetrical and delayed) were calculated and compared in Ref. [10]. Compared with the pitching-up flapping, the high lift coefficient CL generated by the delayed stall mechanism is maintained and becomes more smooth. The second lift peak caused by pitching-up mechanism disappears. The maximum peak of the drag coefficient CD of the pitching-down flapping drops to about half of the pitching-up flapping. The second drag peak caused by pitching-up mechanism also disappears. The most important benefit of the pitching-down flapping is that the time-averaged drag coefficient CD
flapping, the time-averaged CL of the pitching-down flapping decreases by 20.8% in advanced mode, increases by 5.3% in symmetrical mode, and increases by 45.3% in delayed mode. The time-averaged CD decreases by 41.4% in advanced mode, decreases by 36.6% in symmetrical mode, and decreases by 41.9% in delayed mode. The time-averaged ratio, CL / CD , of the pitching-down flapping increases effectively, 35% in advanced mode, 66.1% in symmetrical mode, and 150% in delayed mode. The above primary works on pitching-down flapping are interesting and the results are exciting, but there still remain lots of questions which need further study. The following works are carried out in this paper: (1) The moment and the power requirement coefficients of the pitching-down flapping in three modes are calculated and compared with those of the pitching-up flapping; (2) The effects of three flapping parameters, 'Wt, 'Wr, D, on the aerodynamic characteristics of the pitching-down flapping in the symmetrical mode are studied; (3) In order to improve the aerodynamic characteristics of the pitching-down flapping further, the camber effects are studied. 4.2 Coefficients of moment and power requirement of the pitching-down flapping The coefficients of moment and power requirement are defined as ref [14] and are listed as follows: The coefficient of translational aerodynamic moment:
CQ , a ,t
[10]
decreases greatly
. Here only the time-averaged coef-
ficients, CL , CD and CL / CD are listed in Table 1. Table 1 shows that, compared with the pitching-up Table 1 Time-averaged CL , CD and CL / CD of the pitching-up and the pitching-down flapping hovering (D = 40Û, 'Wt = 0.1Wc, 'Wr = 0.32Wc) Flapping modes Pitching-up flapping
Pitching-down flapping
Advanced mode
Symmetrical mode
Delayed mode
CL
1.960
1.787
0.774
CD
2.997
2.745
2.923
CL / CD
0.6538
0.6511
0.2648
CL
1.551
1.882
1.125
CD
1.757
1.740
1.697
1.0816
0.6628
CL / CD
0.8828
125
Qa ,t 0.5 UU 2 Sc
.
(4)
The coefficient of rotational aerodynamic moment:
CQ , a , r
Qa , r 0.5 UU 2 Sc
.
(5)
The coefficient of translational inertia moment: CQ ,i ,t
Qi ,t 0.5UU 2 Sc
.
(6)
The coefficient of rotational inertia moment: CQ ,i , r
Qi , r 0.5 UU 2 Sc
.
(7)
The coefficients of power requirements can be
Journal of Bionic Engineering (2009) Vol.6 No.2
126
derived from the above coefficients of moment and the non-dimensional angular velocity. The coefficient of power requirement to overcome the aerodynamic force: CP , a
Pa 0.5 UU 3 S
CQ , a ,tIt CQ , a , rD t .
Pi 0.5UU 3 S
CQ ,i ,tIt CQ ,i , rD t .
Ma
(CQ , a ,t CQ ,i ,t )It .
(CQ , a , r CQ ,i , r )D t .
(15)
I sin ʌ D ,I cos ʌ D ,D I sin D , I cosD ,D ,
(16)
:
Ma :
Pa
³ zf y sin DI xf z zf x cos DI xf yD ds
³ ª¬ zf y sin D xf z zf x cos D º¼ I xf yD ds
(10)
S
The coefficient of power requirement to realize rotation: CP , r
(14)
Then:
S
CP ,t
G
M a | ³ ( zf y i ( zf x xf z ) j xf y k )ds ,
(9)
The coefficient of power requirement to realize translation:
G
³ r u Fd s
Assuming the wing is thin, then y | 0.
(8)
The coefficient of power requirement to overcome the inertia force: CP ,i
where,
(17)
Qa ,tI Qa , rD ,
Qa ,t
(11)
³ zf y sin D xf z zf x cos D ds,
(18)
S
The coefficient of power requirement for the flap-
Qa , r
ping:
³ xf y ds,
(19)
S
CP
CP , a CP ,i
CP ,t CP , r .
(12)
The coefficients of aerodynamic force and moment are calculated through the unsteady Navier-Stokes equations, the coefficients of inertia moment are calculated below.
The moments and products of inertia of the mass of the wing are written as
I xy
I xx
³ y
2
z 2 dm | ³ z 2 dm,
(20)
I zz
³ x
2
y 2 dm | ³ x 2 dm,
(21)
I yy
³ x
z 2 dm |I xx I zz ,
(22)
2
³ xydm, I
³ xzdm, I
xz
yz
³ yzdm.
(23)
According to Refs. [12] and [13], the moments and products of inertia are taken as follows: I xx | 0.721 × 10í7 g·cm2, I yy | 0.79 × 10í7 g·cm2, I zz | 0.069 × 10í7 g·cm2, I xy | 0.0 g·cm2, I xz | 0.148 × 10í7 Fig. 5 Sketch of the coordinates (x, y, z) fixed on the flapping wing.
g·cm2, I yz | 0.0 g·cm2.
The coordinates (x, y, z) fixed on the flapping wing G are shown in Fig. 5. Where ds is the area element, r is the vector from the origin (o: the root of the wing) to the ds. The power required to overcome the aerodynamic force, Pa, is calculated as:
as
Pa
Ma : ,
(13)
The angular momentum of the wing, H, is written
H
ª I xx « « I yx « I zx ¬
I xz º » I yz » : i I xxI sin D I xzD I zz »¼ jI yyI cos D k I zxI sin D I zzD . (24) I xy I yy I zy
Bai et al.: Aerodynamic Characteristics, Power Requirements and Camber Effects of the Pitching-Down Flapping Hovering
The inertia moment of the wing, Mi, is written as Mi
ª dH º « dt » : u H ¬ ¼ xyz
i ª¬ I xxIsin D I xz D I2 cos D sin D º¼
sin D I I2 sin 2 D D 2 º j ª I yyI cos D 2 I zzID xz ¬ ¼
k ª¬ I zzD I xzIsin D I zzI2 sin D cos D º¼ .
(25)
The power required to overcome the inertia force of the wing, Pi, is written as Mi :
Pi
Qi ,tI Qi , rD ,
(26)
I I xx sin 2 D I yy cos 2 D
Qi ,t
sin D cos D , I xz D sin D D 2 cos D 2 I zzID
I zzD I xzIsin D I zzI2 sin D cos D ,
Qi , r CQ ,i ,t
I yy 0.5 U Sc
3
[(cos 2 D
(27) (28)
I xx sin 2 D )I I yy
2 I xz D sin D D cos D I yy
2
CQ ,i , r
I zz I D sin D cos D ]. I yy
I zz 0.5 U Sc3
(29)
2 ª I xz º I sin D I sin D cos D » . «D I zz ¬ ¼ (30)
Because that the flapping wing is symmetrical to the z axis, then I xz
³ xzdm
0.0 g·cm2.
The other inertia products are obtained from the above data. Then: CQ ,i ,t
4.6[ cos 2 D 0.91sin 2 D I 0.17ID sin D cos D ],
CQ ,i , r
2 0.4 ªD I sin D cos D º . ¬ ¼
(31) (32)
The curves of CQ,a,t, CQ,a,r and CQ,i,t, CQ,i,r during one cycle of the pitching-up flapping and pitching-down flapping are shown in Fig. 6. For the pitching-up flapping: (1) The coefficient of rotational aerodynamic moment CQ,a,r is much smaller than that of translational aerodynamic moment CQ,a,t; (2) In the middle of a stroke,
127
the angle of attack does not change, the coefficients of the inertia moment, CQ,i,t and CQ,i,r, both equal approximately 0. But at the end of a stroke, CQ,i,t is much greater than CQ,i,r. Compared with the pitching-up flapping, the coefficients of moment of the pitching-down flapping have the following characteristics: (1) The coefficient of translational aerodynamic moment CQ,a,t decreases very much, because that the rotational direction changes and the drag decreases very much; (2) The effect on the coefficient of translational inertia moment CQ,i,t is small. Meanwhile, because that the coefficient of rotational inertia moment CQ,i,r has small value, the effect can also be ignored. The curves of coefficients of power requirement in one cycle, CP,a,, CP,i, CP,t, CP,r, of the pitching-up flapping and the pitching-down flapping are shown in Fig. 7 and Fig. 8 respectively. Analyzing these figures, the conclusions can be drawn that: (1) The CP,a of pitching-down flapping is much smaller than that of pitching-up flapping. The maximum value of CP,a of pitching-up flapping is about 11.5, and that of pitching- down flapping is only about 4.8; (2) Compared with pitching-up flapping, the CP,a of pitching-down flapping removes the second peaks; (3) The CP,i of pitching-down is approximately equal to CP,i of pitching-up flapping; (4) The coefficient of power requirement to realize translation, CP,t, is much greater than the coefficient to realize rotation, CP,r; (5) The CP,t of pitching-down flapping is much smaller than that of pitching-up flapping. The maximum value of CP,t of pitching-up flapping is about 14, and that of pitching-down flapping is only about 8.4; (6) Just like the CP,a, the CP,t of pitching-down flapping also removes the second peaks. From the above analyses, the coefficient of power requirement of pitching-down flapping is much smaller than that of pitching-up flapping. The time-averaged CP and the ratio of time-averaged CL / CP are listed in Table 2. Compared with that of pitching-up flapping, the power requirement of pitching-down flapping decreases greatly, while still maintaining the high unsteady lift. The ratio of lift coefficient to power requirement coefficient, CL / CP , significantly improves. These mean that MAVs with pitching-down flapping may be more effective than that with pitching-up flapping.
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15
10
10
CQ,a,t
5
Advanced Symmetrical Delayed
CQ,i,t
Advanced Symmetrical Delayed
5 0
0
CQ,a,r
í5
CQ,i,r í5
í10 í10
í15 0.00
0.25
0.50
0.75
1.00
0.00
0.25
IJ / IJc
0.50
0.75
1.00
IJ / IJc
CQ,a,t CQ,a,r
CQ,i,t CQ,i,r
(a) Moment coefficients of the pitching-up flapping in three modes
(b) Moment coefficients of the pitching-down flapping in three modes
CP,a CP,i
CP,t CP,r
Fig. 6 The moment coefficients of the pitching-up flapping and pitching-down flapping during one cycle in three modes (Dup = Ddown =40Û, 'Wt = 0.1Wc, 'Wr = 0.32Wc).
Fig. 7 Coefficients of power requirement, CP,a, CP,i CP,t, CP,r, of the pitching-up flapping in three modes.
4
Advanced Symmetrical Delayed
CP,a 5
2
CP,t
0
CP,i
0
í2
Advanced Symmetrical Delayed
í4 0.00
0.25
0.50 IJ / IJc
0.75
1.00
í5 0.00
CP,r
0.25
0.50 IJ / IJc
0.75
Fig. 8 Coefficients of power requirement, CP,a, CP,i CP,t, CP,r, of the pitching-down flapping in three modes.
1.00
Bai et al.: Aerodynamic Characteristics, Power Requirements and Camber Effects of the Pitching-Down Flapping Hovering
129
Table 2 Time-averaged coefficient of power requirement CP and CL / CP of the two flapping in three modes (D = 40Û, 'W t = 0.1Wcˈ 'Wr = 0.32Wc)
ping, CPup
CP of the pitching-down flapping, CPdown ( CPdown ˉ CPup )/ CPup
CL / CP of the pitching-up flapping CL / CP of the pitching-down flapping
Advanced
Symmetrical
Delayed
3.594
2.979
3.401
2.106
2.269
1.939
ˉ41.4%
ˉ23.8%
ˉ43.0%
0.5452
0.5999
0.2276
CL
Flapping modes
CP of the pitching-up flap-
(a) Lift coefficient in one cycle 0.7365
0.8294
0.58
4.3 Effects of flapping parameters on pitching-down flapping The symmetrical mode is chosen to study the effects of flapping parameters on pitching-down flapping.
+ t
Įt +
4.3.1 Effect of translational acceleration time 'Wt Here, 'Wt = 0.1Wc, 0.18Wc, 0.24Wc, while 'Wr = 0.32Wc and D = 40Û, are simulated to study the effect of 'Wt on pitching-down flapping. The translational and rotational angular velocities are shown in Fig. 9. As 'Wt decreases, the translational acceleration at the end of a stroke increases, and the translational angular velocity in the middle of a stroke decreases. The unsteady lift and drag coefficients in one cycle are shown in Fig. 10. It can be found that: (1) The lift and drag coefficients in the middle of a stroke are determined by the flapping angular velocity. As 'Wt decreases, both lift and drag coefficients decrease; (2) As 'Wt decreases, the peaks of the lift and drag coefficients moves forward.
Fig. 9 Dimensionless translational and rotational angular velocities in one cycle with different 'Wt in symmetrical mode.
3
ǻIJt = 0.10, ǻIJr = 0.32 ǻIJt = 0.18, ǻIJr = 0.32 ǻIJt = 0.24, ǻIJr = 0.32
2 1 0 í1 í2 í3 0.00
0.25
0.50 IJ / IJc
0.75
1.00
(b) Drag coefficient in one cycle
Fig. 10 Lift and drag coefficients with different 'Wt in one cycle in symmetrical mode.
The time-averaged lift and drag coefficients in one cycle are listed in Table 3. It is shown that as 'Wt increases from 0.1 to 0.24, the time-averaged lift coefficient, CL , increases by 12.5%. Meanwhile the timeaveraged drag coefficient, CD , also increases. The ratio of lift to drag, CL / CD , almost does not change. Table 3 Time-averaged lift and drag coefficients and the ratio of lift to drag of the pitching-down flapping with different 'Wt in the symmetrical mode. ('Wr = 0.32Wc, D = 40Û) 'Wt
0.1Wc
0.18Wc
CL
1.882
2.028
0.24Wc 2.118
CD
1.740
1.872
1.959
CL / CD
1.082
1.083
1.081
4.3.2 Effect of rotational time 'Wr Here, 'Wr = 0.16Wcˈ0.24Wcˈ0.32Wc, while 'Wt = 0.1Wc, D = 40Û, are calculated to study the effect of 'Wr on pitching-down flapping. The translational and rotational angular velocities are shown in Fig. 11. As 'Wr increases, the rotational angular velocity at the end of a stroke decreases. The unsteady lift and drag coefficients in one cycle are shown in Fig. 12. It can be found that: (1) Because 'Wr does not affect the translational angular
Journal of Bionic Engineering (2009) Vol.6 No.2
130
velocity and the angle of attack in the middle of a stroke, the lift and drag coefficients in the middle of a stroke almost do not change. (2) 'Wr significantly affects the peaks of the lift and drag coefficients. As 'Wr decreases, both peaks increase at the beginning of a stroke. The time-averaged lift and drag coefficients are shown in Table 4. As 'Wr decreases from 0.32Wc to 0.16Wc, CL and CD both increase, and the time-averaged ratio of lift to drag, CL / CD , decreases slightly. 2
+ t
ǻIJr = 0.16 ǻIJr = 0.24 ǻIJr = 0.32
1
0
í1
í2 0
2
4 t
6
8
Fig. 11 Translational and rotational angular velocities in one cycle with different 'Wr in the symmetrical mode.
5
Table 4 Time-averaged lift and drag coefficients and the ratio of lift to drag of the pitching-down flapping with different 'Wr in the symmetrical mode. ('Wt = 0.1Wc, D = 40Û) 'Wr
CL CD CL / CD
0.16Wc 2.138 2.013
0.24Wc 2.019 1.880
0.32Wc 1.882 1.740
1.062
1.074
1.082
4.3.3 Effect of angle of attack D The effects of the angle of attack in the middle of a stroke, D, on pitching-down flapping are studied. Here D = 30Û, 40Û, 50Û, while 'Wt = 0.1Wc, and 'Wr = 0.32Wc. The rotational and translational anglular velocities at different values of D are shown in Fig. 13. As D increases, the rotational angular velocity increases. This tendency is contrary to that of pitching-up flapping. The lift and drag coefficients are shown in Fig. 14. It can be found that: (1) As D increases, the unsteady lift and drag coefficients both increase. But this tendency is nonlinear, especially for the lift coefficient; (2) As D increases, the peaks of the lift and drag coefficients at the beginning of a stroke increase. The time-averaged coefficients of lift and drag are listed in Table 5.
ǻIJt = 0.1, ǻIJr = 0.32 ǻIJt = 0.1, ǻIJr = 0.24 ǻIJt = 0.1, ǻIJr = 0.16
4
Įt+
3
+ t
2 1 0 í1 í2 0.00
0.25
0.50 IJ / IJc
0.75
1.00
(a) Lift coefficient in one cycle
Fig. 13 Translational and rotational angle velocities in one cycle with different D in the symmetrical mode.
4 3 2
ǻIJt = 0.1, ǻIJr = 0.32 ǻIJt = 0.1, ǻIJr = 0.24 ǻIJt = 0.1, ǻIJr = 0.16
1
4 3
0
2
í1
1
í2
0.00
Į = 30Û Į = 40Û Į = 50Û
0
í3 í4
í1 0.25
0.50 IJ / IJc
0.75
1.00
(b) Drag coefficient in one cycle
Fig. 12 Lift and drag coefficients with different 'Wr in one cycle in the symmetrical mode.
í2 0.00
0.25
0.50 IJ / IJc
0.75
1.00
(a) Lift coefficient in one cycle
Fig. 14 Lift and drag coefficients with different D in one cycle in the symmetrical mode.
where, the non-dimensional coordinates x x / c and y = y/c; the location of the maximum camber x f 50% ; the non-dimensional camber f f / c . The thickness of the wing is still 0.05c. Here, f = 0, r10%, r20%, are chosen, while 'Wt = 0.1Wc, 'Wr = 0.32Wc, D = 40Û. The flapping model wing with camber and computational grid are shown in Fig. 4.
Į = 30Û Į = 40Û Į = 50Û
0.25
0.50 IJ / IJc
0.75
1.00
4.4.1 The advanced mode
(b) Drag coefficient in one cycle
Fig. 14 Continued. Table 5 Time-averaged coefficients of lift and drag and the ratio of lift to drag of pitching-down flapping with different D in the symmetrical mode ('Wt = 0.1Wc, 'Wr = 0.32Wc) D
30Û
40Û
50Û
CL
1.326
1.882
2.252
CD
1.143
1.740
2.490
CL / CD
1.161
1.082
0.904
Within the scope of D in this paper, as D increases the coefficients of lift and drag increase nonlinearly, and the ratio of lift to drag decreases. This trend is similar to that of general rigid wing. 4.4 Effect of camber on the pitching-down flapping It is well known that the flapping wing of fruit fly has no camber. Because the windward and leeward surfaces alternate in each stroke, the fixed camber will change sign in upstroke and downstroke, and the camber is unuseful to the pitching-up flapping. But for pitching-down flapping, the order of the windward and leeward surfaces does not change and the camber may be useful to improve the aerodynamic characteristics. In order to prove this idea, the effect of camber on pitching-down flapping in three modes (advanced, symmetrical, delayed) are numerically studied. The 4-figure NACA airfoil is adopted to define the camber of a flapping wing, which is determined by two parabolas. The two parabolas are tangential at the point of the maximum camber locating in the middle of the chord[15]. The formulas are listed as follows: x
x
0 ~ xf :
x f ~ 1.0 :
y
y
f
f
1 (2 x f x x 2 ) , x f2
The coefficients of lift and drag during one cycle with different camber values in the advanced mode are shown in Fig. 15. The time-averaged lift and drag coefficients and the ratio of lift to drag with different camber values are shown in Fig. 16. It can be easily found that the camber affects the lift and drag coefficients effectively. As f decreases from 0 to í20%, the lift coefficient CL decreases, the drag coefficient CD increases and the ratio of lift to drag CL / CD decreases. On the contrary, as the camber increases from 0 to 10%, the lift coefficient CL increases, the drag coefficient CD decreases a little and the ratio of lift to drag CL / CD increases. As the camber further increases from 10% to 20%, CL and CD both increase further, but CL / CD is almost unchanged. As f increases from 0 to 10%, CL increases by 9.8%, and as f increases from 0 to 20%, CL increases by 16.6% and CL / CD increases by 13.8%. 4 2 CL
4 3 2 1 0 í1 í2 í3 í4 0.00
131
Flex = í20 % Flex = í10 % Flex = 0 % Flex = 10 % Flex = 20 %
0 Advanced í2 0.00
0.25
0.50 IJ / IJc
0.75
1.00
(a) Lift coefficient in one cycle
CD
CD
Bai et al.: Aerodynamic Characteristics, Power Requirements and Camber Effects of the Pitching-Down Flapping Hovering
(33)
1 [(1 2 x f ) 2 x f x x 2 ] , (1 x f ) 2
(34)
(b) Drag coefficient in one cycle
Fig. 15 Lift and drag coefficients with different cambers in one cycle in the advanced mode.
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Journal of Bionic Engineering (2009) Vol.6 No.2
3
CL
í í í í CL/CD, CD, CL
2 1
Flex = í20 % Flex = í10 % Flex = 0 % Flex = 10 % Flex = 20 %
Symmetrical
0 í1 0.00
0.25
0.50 IJ / IJc
0.75
1.00
(a) Lift coefficient in one cycle
The above results indicate that, in the advanced mode, negative camber has some disadvantages on the aerodynamic characteristics of pitching-down flapping, and positive camber has advantages, which can effectively increase CL and CL / CD .
CD
Fig. 16 Time-averaged lift and drag coefficients and the ratio of lift to drag with different cambers in the advanced mode.
4.4.2 The symmetrical mode
4.4.3 The delayed mode Finally, the effects of the camber on the pitching-down flapping in the delayed mode are analyzed. The coefficients of lift and drag with different cambers in one cycle are shown in Fig. 19. The time-averaged lift and drag coefficients and the ratio of the lift to drag are shown in Fig. 20.
(b) Drag coefficient in one cycle
Fig. 17 Lift and drag coefficients with different cambers in one cycle in the symmetrical mode.
Symmetrical
í í í í CL/CD, CD, CL
2.0
1.5
í CL í CD CíL/CíD
1.0
0.5 í20
í10
0 í f (%)
10
20
Fig. 18 Time-averaged lift and drag coefficients and the ratio of lift to drag at different cambers, f , in the symmetrical mode.
2
CL
The coefficients of lift and drag during one cycle with different camber values in the symmetrical mode are shown in Fig. 17. The time-averaged CL , CD and CL / CD with different camber values are shown in Fig. 18. As the camber f decreases from 0 to í20%, the drag coefficient increases, the lift coefficient decreases and the ratio of the lift to drag decreases effectively. As f increases from 0 to 10%, CD increases a little by 0.3%, CL increases by 9.4%, and CL / CD increase by 9.1%. As f increases to 20% further, CL increases by 14.7%, CD increases by 7.9%, but CL / CD increases by only 6.3%. CL / CD reaches the maximum value at f =10%. The results indicate that, in the symmetrical mode, the negative camber also has some disadvantages on the pitching-down flapping, and the positive camber can improve CL and CL / CD effectively. However as f > 10% CL / CD starts to decrease. This trend in the symmetrical mode is similar to that in the advanced mode.
0
í2
í4 0.00
Flex = í20 % Flex = í10 % Flex = 0 % Flex = 10 % Flex = 20 %
Delayed
0.25
0.50 IJ / IJc
0.75
1.00
(a) Lift coefficient in one cycle
Fig. 19 Lift and drag coefficients with different cambers in one cycle in the delayed mode.
Bai et al.: Aerodynamic Characteristics, Power Requirements and Camber Effects of the Pitching-Down Flapping Hovering
Flex = í20 % Flex = í10 % Flex = 0 % Flex = 10 % Flex = 20 %
2
CD
0 Delayed í2
0.00
0.25
0.50 IJ / IJc
0.75
1.00
(b) Drag coefficient in one cycle
Fig. 19 Continued. Delayed
í í í í CL/CD, CD, CL
2.0
í CL í CD CíL/CíD
1.5
1.0
0.5 í20
í10
0 f (%)
10
20
Fig. 20 Time-averaged lift and drag coefficients and the ratio of lift to drag with different cambers in the delayed mode.
The tendencies of the effects of the camber in the delayed mode are similar to those in the advanced and symmetrical modes. Compared with the coefficients as f =0, when f =10% and 20%, the time-averaged lift coefficient, CL , increases by 14.8% and 23.7% respectively, and the time-averaged ratio of lift to drag, CL / CD , increases by 8.4% and 3.6% respectively. From the above numerical simulations and analyses of the effect of camber on the pitching-down flapping in the advanced, the symmetrical, the delayed modes, the conclusions can be drawn that: (1) The negative camber has disadvantages on the aerodynamic characteristics of the pitching-down flapping; (2) On the contrary, the positive camber can effectively improve the time-averaged lift coefficient and the ratio of lift to drag; (3) It is as the same as the general rigid wing, that the relationships between CL , CD , CL / CD and f are nonlinear; (4) Approximately, CL / CD reaches the maximum value at f =10%.
5 Conclusions A numerical method was used to simulate the
133
hovering of pitching-down flapping, which was invented by the author based on the previous studies on pitching-up flapping (fruit fly flapping). The most important and the only difference between pitching-down flapping and pitching-up flapping is the direction of rotation at the end of a stroke. Compared with pitching-up flapping, pitching-down flapping maintains the delayed stall, acceleration and deceleration mechanisms to keep the high unsteady lift. Meanwhile, the unsteady drag is greatly reduced with the removal of the pitching-up mechanism, which effectively improves the ratio of lift to drag. The power requirements of pitching-down flapping, the effects of the flapping parameters ('Wt, 'Wr, D) and the camber on the pitching-down flapping are systemically analyzed, from which the following conclusions can be drawn: (1) Compared with pitching-up flapping, the pitching-down flapping greatly reduces the power requirement to overcome the aerodynamic force and realize the translation. The time-averaged power requirement coefficient CP significantly decreases. These mean that while maintaining the high unsteady lift, the pitching-down flapping greatly reduces the power requirement and improves the mechanical efficiency; (2) For pitching-down flapping in the symmetrical mode, as the time of translational acceleration, i.e. 'Wt, increases or the time of rotation, i.e. 'Wr, decreases, the time-averaged lift and drag coefficients increase, but the ratio of lift to drag changes a little. The angle of attack in the middle of a stoke, i.e. D, significantly affects the aerodynamic characteristics. As D increases from 30Û to 50Û, CL and CD increases, CL / CD decreases nonlinearly. (3) It is well known that the camber is unuseful for pitching-up flapping. But for pitching-down flapping, the positive camber can effectively increase CL and CL / CD . In this paper, as the camber increases from 0 to 20%, CL and CD both increase nonlinearly, and CL / CD reaches the maximum value while the camber is approximately 10%. The negative camber makes CD increase, CL and CL / CD decrease, which is undesirable for pitching-down flapping. Further studies will be focused on the forward flight and the stability of pitching-down flapping.
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References [1] Burton R. Bird Flight, Facts On File, New York, USA, 1990. [2] Theodorsen T. General Theory of Aerodynamic Instability and the Mechanism of Flutter, NACA Technical Report 496, National Advisory Committee for Aeronautics, 1935. [3] Osborne M F M. Aerodynamics of flapping flight with application to insects. Journal of Experimental Biology, 1951, 28, 221–245. [4] Dickinson M H, Gotz K G. Unsteady aerodynamic performance of model wings at low Reynolds numbers. Journal of Experimental Biology, 1993, 174, 45–64. [5] Dickinson M H, Lehmann F O, Sane S P. Wing rotation and the aerodynamic basis of insect flight. Science, 1999, 284, 1954–1960. [6] Ellington C P, Van den Berg C, Willmott A P, Thomas A L R. Leading-edge vortices in insect flight. Nature, 1996, 384, 626–630. [7] Sun M, Tang J. Unsteady, aerodynamic force generation by a model fruit fly wing in flapping motion. Journal of Experimental Biology, 2002, 205, 55–70. [8] Ramamurti R, Sandberg W C. A three-dimensional computational study of the aerodynamic mechanisms of insect flight. Journal of Experimental Biology, 2002, 205,
1507–1518. [9] Bai P, Cui E J, Li F, Zhou W J. Study of high aerodynamic lift mechanics of hovering insect flapping wing at low Reynolds number. Acta Aerodynamic Sinica, 2007, 25, 175–182. (in Chinese) [10] Bai P, Cui E J, Li F, Zhou W J, Chen B Y. A new bionic MAV’s flapping motion based on fruit fly hovering at low Reynolds number. Acta Mechanica Sinica, 2007, 23, 485–493. [11] Rogers S E, Kwak D. An upwind differencing scheme for the incompressible navier-stokes equations. Applied Numerieal Mathematics, 1991, 8, 43–64. [12] Torkel W F. Quick estimates of flight fitness in hovering animals, including novel mechanism for lift production. Journal of Experimental Biology, 1973, 59, 169–230. [13] Vogel S. Flight in Drosophila: II. Variations in stroke parameters and wing contour. Journal of Experimental Biology, 1967, 46, 383–392. [14] Sun M, Tang J. Lift and power requirements of hovering flight in Drosophila virilis. Journal of Experimental Biology, 2002, 205, 2413–2427. [15] Abbott I H, von Doenhoff A E. Theory of Wing Sections, Dover Publications, New York, USA, 1958.
Aerodynamic Characteristics, Power Requirements and Camber Effects of the Pitching-Down Flapping Hovering Peng Bai, Er-jie Cui, Hui-ling Zhan China Academy of Aerospace Aerodynamics, Beijing 100074, P. R. China
Abstract The pitching-down flapping is a new type of bionic flapping, which was invented by the author based on previous studies on the aerodynamic mechanisms of fruit fly (pitching-up) flapping. The motivation of this invention is to improve the aerodynamic characteristics of flapping Micro Air Vehicles (MAVs). In this paper the pitching-down flapping is briefly introduced. The major works include: (1) Computing the power requirements of pitching-down flapping in three modes (advanced, symmetrical, delayed), which were compared with those of pitching-up flapping; (2) Investigating the effects of translational acceleration time, 'Wt, and rotational time, 'Wr, at the end of a stroke, and the angle of attack, D, in the middle of a stroke on the aerodynamic characteristics in symmetrical mode; (3) Investigating the effect of camber on pitching-down flapping. From the above works, conclusions can be drawn that: (1) Compared with the pitching-up flapping, the pitching-down flapping can greatly reduce the time-averaged power requirements; (2) The increase in 'Wt and the decrease in 'Wr can increase both the lift and drag coefficients, but the time-averaged ratio of lift to drag changes a little. And D has significant effect on the aerodynamic characteristics of the pitching-down flapping; (3) The positive camber can effectively increase the lift coefficient and the ratio of lift to drag. Keywords: aerodynamic characteristics, flapping wing, power requirements, camber effect, pitching-down-flapping, MAVs Copyright © 2009, Jilin University. Published by Elsevier Limited and Science Press. All rights reserved. doi: 10.1016/S1672-6529(08)60109-2
Nomenclatures (X, Y, Z): The inertial coordinates fixed on the body of MAVs or fruit fly; (x, y, z): The coordinates fixed on the flapping wing; (xc, yc, zc): The coordinates, in which yc axis is the same as Y axis, zc axis is fixed along the spanwise and is the same as z axis; i, j, k: The unit vectors in the (x, y, z) coordinates; I: The azimuth angle of translation; ): The amplitude of translation; D: The angle of attack in the middle of a stroke; Dup and Ddown: The angles of attack in the middle of upstroke and downstroke, respectively;
It : The translational angular velocity; I t and D t : The non-dimensional angular velocities of translation and rotation, respectively;
r0: The inertial radius of the flapping wing; ut: The translational velocity at r0, which is called the flapping velocity; ut : The non-dimensional translational velocity; U: The reference velocity, which equals the time-averaged flapping velocity u t in one cycle; U t : The flapping velocity in the middle of a stroke,
which is a constant; R: The span of the flapping wing; c : The average chord of the flapping wing; S: The area of the wing; f: The frequency of flapping/The camber of the flapping wing; U: The density of air; Re: The Reynolds number; W: The non-dimensional time; W r : The beginning of the rotation at the end of a stroke;
D t0 : The constant to determine D t ;
'W r : The rotational time at the end of a stroke;
D and M : The angular accelerations of rotation and
'Wt: The translational acceleration and deceleration time at the end of stoke;
translation, respectively; Corresponding author: Peng Bai E-mail: [email protected]
Bai et al.: Aerodynamic Characteristics, Power Requirements and Camber Effects of the Pitching-Down Flapping Hovering
121
W 1 : The beginning of acceleration at the beginning of a
CP ,t : The coefficient of power requirement to realize
stroke; W 2 : The beginning of the deceleration at the end of
translation; CP , r : The coefficient of power requirement to realize
stoke; Wc: The non-dimensional flapping period; 'W: The non-dimensional physical time step; 'Wsub: The pseudo-time step in the subiteration; CL and CD: The coefficients of lift and drag, respectively;
rotation; CP : The coefficient of power requirement of the flap-
Qi , r : the rotational inertia moment;
ping; Pa: The power required to overcome the aerodynamic force; Ma: The aerodynamic moment around the origin; Pi: The power required to overcome the inertia force of the wing; :: The angular velocity around the origin. ds: The area element; dm: The mass element of the ds; G r : The vector from the origin to the ds; G F ( f x , f y , f z ): The aerodynamic force acting on ds;
CQ , a ,t : The coefficient of translational aerodynamic
H: The angular momentum of the wing;
moment; CQ , a , r : The coefficient of rotational aerodynamic mo-
I xx , I yy , I zz , I xy , I xz , I yz : The products of inertia of the
ment; CQ ,i ,t : The coefficient of translational inertia moment;
CP : The time-averaged coefficient of power require-
CL and CD : The time-averaged coefficients of lift and
drag, respectively; Qa ,t : the translational aerodynamic moment; Qa , r : the rotational aerodynamic moment; Qi ,t : the translational inertia moment;
CQ ,i , r : The coefficient of rotational inertia moment; CP , a : The coefficient of power requirement to overcome
the aerodynamic force; CP ,i : The coefficient of power requirement to overcome the inertia force;
1 Introduction The development of Micro Air Vehicles (MAVs) makes the flapping flight at Low Reynolds Number (LRE), which widely exists in the nature, become an attractive topic for scientists. Previous studies indicated that for MAVs shorter than 15cm flapping wing might be the only method to obtain enough lift, stability and manipulability to fly. Leonardo da Vinci studied the flapping wing in 1500[1]. In the middle of the 19th century, Etienne-Jules Marey recorded and studied the details of bird flapping flight using a camera with 11 frames per second designed by himself[1]. Later studies on flapping wing appeared in the early papers[2,3]. Systematic studies on the flapping wing were carried out through experiments and numerical simulations by Dickinson et al.[4,5], Ellington et al.[6], Sun and Tang[7], Ramamurti and
flapping wing; ment;
x : The non-dimensional x coordinate; x f : The location of the maximum camber; y : The non-dimensional y coordinate; f : The non-dimensional camber;
Sandberg[8], Bai et al.[9], et al., in the past two decades. Previous works on the fruit fly flapping[4–9] indicate that there are four mechanisms to generate high unsteady lift: (1) Delayed stall, the most important mechanism to generate high unsteady lift; (2) Decelerating translation at the end of a stroke; (3) Accelerating translation at the beginning of a stroke; (4) Pitching-up rotation at the end of a stroke, which generates the peaks of lift and drag, and the peak of drag is much higher than that of lift. Therefore the fruit fly flapping is also called “the pitching-up flapping” in this paper. In order to keep the beneficial delayed stall, accelerating and decelerating mechanisms, and get rid of the pitching-up rotation mechanism, which generates very high drag peaks, the pitching-down flapping was invented by the authors[10]. Compared with the pitching-up flapping, the pitching-down flapping can effectively decrease the drag and improve the time-averaged ratio of
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Journal of Bionic Engineering (2009) Vol.6 No.2
lift to drag, which means the wing rotates pitching-down at the end of a stroke. In this paper, the pitching-down flapping is briefly introduced. The power requirements, the flapping parameters effects and the camber effect of the pitching-down flapping in hovering condition were investigated with a numerical simulation method[11]. The investigation results suggest that: (1) Compared with the pitching-up flapping, the pitching-down flapping can effectively decrease the time-averaged power requirement, while maintaining the high lift; (2) The lift and drag coefficients increase with the increase in 'Wt and the decrease in 'Wr; the time-averaged ratio of lift to drag changes a little; and D significantly affects the aerodynamic characteristics; (3) For the pitching-down flapping, the camber can effectively increase the lift coefficient and the ratio of lift to drag. However, this beneficial camber effect does not exist for the pitching-up flapping.
2 Kinematics of pitching-down flapping hovering Fig. 1 sketches the fruit fly flapping, in which (X, Y, Z) is the inertial coordinates fixed on fruit fly’s body; (x, y, z) is the coordinates fixed on the flapping wing; and (xc, yc, zc) is the coordinates, in which yc axis is the same as Y axis, zc axis is fixed along the spanwise and is the same as z axis. The rotation around the Y axis is called “translation”, and the azimuth angle is I. The rotation around the
z axis is called “rotation”, and the angle of attack is D. The inertial radius of the flapping wing around the Y axis is defined by r0
³
S
r 2 dS / S .
The translational velocity at r0 is called the flapping velocity, defined by ut. And the translational angular velocity It = ut/r0. Details of the fruit fly flapping were described in Dickinson and Sun’s papers[5,7]. Then only the characteristics are presented here: (1) The order of the leading edge and trailing edge does not change; (2) The windward surface and the leeward surface alternate each other; (3) The rotation at the end of a stroke is pitching-up. Compared with the fruit fly (pitching-up) flapping, the pitching-down flapping maintains the translation, but changes the rotation. And its obvious characteristics are: (1) The leading edge and the trailing edge alternate in each stroke; (2) The windward surface and leeward surface do not change; (3) Pitching-up rotation is replaced by pitching-down rotation at the end of a strokes. The azimuth angle I, the angle of attack D, the translational and rotational velocities I t , D t of the pitching-up flapping in hovering in three modes (advanced, symmetrical, delayed) are shown in Fig. 2. The end with solid circle denotes the leading edge. The parameters of pitching-down flapping in hovering are shown in Fig. 3. The solid circles appear at both ends, indicating that the leading and trailing edges alternate each other and either end can be the leading edge. In Fig. 2 and Fig. 3, both translations are the same and the translational plane is horizontal. In the middle of a stroke, D and ut do not change. ut is given as: ut
U t sin(ʌ(W W 1 ) 'W t ),
W 1 d W d W 1 'W t / 2.
ut
U t ,
W 1 'W t / 2 d W d W 2 .
ut
U t sin(ʌ(W W 2 'W t / 2) 'W t ), W 2 d W d W 2 'W t / 2. t
Where, u
(1) ut / U , U is the reference velocity, which
equals the time-averaged flapping velocity ut in one cycle; U t is the flapping velocity in the middle of a
Fig. 1 Sketch of the fruit fly flapping (pitching-up flapping).
stroke, which is a constant; IJ = tU/c ; IJ1 and IJ1+ǻIJt /2 are the beginning and the end of acceleration at the beginning of a stroke; IJ2 and IJ2+ǻIJt /2 are the beginning and
Bai et al.: Aerodynamic Characteristics, Power Requirements and Camber Effects of the Pitching-Down Flapping Hovering
the end of deceleration at the end of stoke. The rotations of the pitching-up flapping and the pitching-down flapping are different, but the formulas of Dt are identical.
220 200 180 160 140
0.5D t0 [1 cos(2ʌ(W W r ) / 'W r )] ,
W r d W d W r 'W r .
120 100 80
(2)
60
Advanced Symmetrical Delayed
40 20 0
2
4 t (a)
6
8
Advanced Symmetrical Delayed
1.0 0.5 0.0
t
+
where, Dt = Dtc/U, D t0 is a constant; IJr and IJr +ǻIJr are the beginning and the end of the rotation at the end of a stroke. When ǻIJr and the angles of attack, Dup and Ddown, in the middle of upstroke and downstroke are determined, D t0 can be calculated. Here the difference between the two flappings is, for the pitching-up flapping, Dup = 40Û, Ddown = 140Û, and for the pitching-down flapping, Dup = 220Û, Ddown = 140Û. Other details of pitching-down flapping kinematics are given in Ref. [10].
Įt+
D t
123
í0.5
160
Advanced Symmetrical Delayed
140
í1.0
120 0
100 80
2
4 t (b)
6
8
60 Downstroke
40 20 0
2
4 t (a)
6
8
+ t
Upstroke
Advanced Advanced Symmetrical Symmetrical Delayed Delayed
1.0
(c)
Fig. 3 pitching-down flapping hovering; (a) The translational angle I and the angle of attack D; (b) The non-dimensional angular velocities of translation and rotation, I t and D t ; (c) Sketch of the downstroke and upstroke in the symmetrical mode.
0.0
t
+
Įt+
0.5
í0.5 í1.0 0
2
4 t (b)
6
8
(c)
Fig. 2 Pitching-up flapping hovering; (a) The translational angle I and the angle of attack D; (b) The non-dimensional angular velocities of translation and rotation, I t and D t ; (c) Sketch of the downstroke and upstroke in the symmetrical mode.
From Torkel’s experiment[12] and Vogel’s bundling fruit fly flapping experiment[13], the density of air is 1.226 kg·mí3, the weight of the insect is 1.96 × 10í5 N, the mass of the wing is 2.4 × 10í6 g, the span of the wing R = 0.3 cm, the average chord c = 0.108 cm, the inertial radius r0 = 0.58R, the amplitude of translation ) = 2.62 rad ˙ 150Û, the frequency of flapping f = 240 Hz. Then the reference velocity U, Reynolds Number Re, the non-dimensional flapping period Wc and the hovering lift coefficient CLw can be obtained: U = 2)fr0 = 218.7 cm·sí1, Re = cU/Q = 147, Wc˙(1/f)/( c /U) | 8.42, CLw = 1.15.
124
Journal of Bionic Engineering (2009) Vol.6 No.2
From Dickinson’s experiment[5], the density of the mineral oil is 0.88 × 103 kg·mí3, the span of the wing R = 0.25 m, the area of the wing S = 0.0167 m2, the average chord c = 8.79 cm, the inertial radius r0 = 0.58R, the frequency of flapping f = 0.145 Hz, the amplitude of translation ) ˙ 160Û, the reference velocity U = 2)fr0 = 0.117 m·sí1, then 0.5UU2S | 0.101 kg·m·sí2. Then the non-dimensional parameters used to study the pitching-down flapping in this paper are obtained: Wc = 8.42, 'Wt = 0.1Wc, 'Wr = 0.32Wc, ) = 160Û.
3 Computational method and grid 3.1 Computational method In this paper, the controlling equations are unsteady incompressible Navier-Stokes (INS) equations with pseudo-compressibility. The numerical calculation scheme is the artificial compressibility method developed by Rogers[11]. The numerical flux is calculated through the 3rd order Roe scheme. The viscosity is important at low Re, the viscous flux is considered and computed by the 2nd-order center difference scheme. In order to study this unsteady problem and accelerate the convergence speed, the 2nd-order two-time step method is used in the temporal direction. The LGS implicit scheme is used in the subiteration. The non-dimensional physical time step 'W = 0.02, which is determined by numerical experiment, and the pseudotime step in the subiteration 'Wsub = 1014. 3.2 Boundary conditions (1) Far-field boundary condition The velocity is given by the free flow and the pressure is interpolated at the far entry boundary. And the pressure is given by the free flow and the velocity is interpolated at the far exit boundary. (2) Wall boundary condition G G G G U U B , wp wn U a B n , (3) G G where, U B and a B , the velocity and the acceleration of G the wall boundary. n , the vector normal to the wall. The correctness of the numerical method and the program has been proven in Refs. [9,10]. 3.3 Model wing and computational grid The model wing of fruit fly of Dickinson’s experiment is shown in Fig. 4a. The average chord c =
8.79 cm, the ratio of wing span to chord O = 2.84, the thickness of the wing is 0.05 c . The leading and trailing edges are semicircular. The model wing used to study the pitching-down flapping is shown in Fig. 4b, which is taken from Fig. 4a by averaging the leading and trailing edges. And the rotational axis is located at 50% chord. Figs. 4c and d show the grid along the chordwise with camber of 0% and 10%, respectively. The numbers of the points along the three directions (circlewise × normal × spanwise) are 81 × 65 × 70. The distance between the far-field boundary and the wall is 15 c . The minimal normal distance near the wall boundary is 0.002 c .
(a) The fruit fly model wing of Dickinson’s experiment
(b) The model wing and computational grid of the pitching-down flapping
(c) Computational grid of the chordwise section with camber equals 0
Fig. 4 The model wing and the computational grid of the pitching-down flapping.
Bai et al.: Aerodynamic Characteristics, Power Requirements and Camber Effects of the Pitching-Down Flapping Hovering
(d) Computational grid of the chordwise section with camber equals 10%
Fig. 4 Continued.
4 Results and discussion 4.1 CL and CD of the pitching-down flapping hovering The lift and drag coefficients, CL and CD, and the ratios, CL/CD, of pitching-up and pitching-down flapping hovering in three modes (advanced, symmetrical and delayed) were calculated and compared in Ref. [10]. Compared with the pitching-up flapping, the high lift coefficient CL generated by the delayed stall mechanism is maintained and becomes more smooth. The second lift peak caused by pitching-up mechanism disappears. The maximum peak of the drag coefficient CD of the pitching-down flapping drops to about half of the pitching-up flapping. The second drag peak caused by pitching-up mechanism also disappears. The most important benefit of the pitching-down flapping is that the time-averaged drag coefficient CD
flapping, the time-averaged CL of the pitching-down flapping decreases by 20.8% in advanced mode, increases by 5.3% in symmetrical mode, and increases by 45.3% in delayed mode. The time-averaged CD decreases by 41.4% in advanced mode, decreases by 36.6% in symmetrical mode, and decreases by 41.9% in delayed mode. The time-averaged ratio, CL / CD , of the pitching-down flapping increases effectively, 35% in advanced mode, 66.1% in symmetrical mode, and 150% in delayed mode. The above primary works on pitching-down flapping are interesting and the results are exciting, but there still remain lots of questions which need further study. The following works are carried out in this paper: (1) The moment and the power requirement coefficients of the pitching-down flapping in three modes are calculated and compared with those of the pitching-up flapping; (2) The effects of three flapping parameters, 'Wt, 'Wr, D, on the aerodynamic characteristics of the pitching-down flapping in the symmetrical mode are studied; (3) In order to improve the aerodynamic characteristics of the pitching-down flapping further, the camber effects are studied. 4.2 Coefficients of moment and power requirement of the pitching-down flapping The coefficients of moment and power requirement are defined as ref [14] and are listed as follows: The coefficient of translational aerodynamic moment:
CQ , a ,t
[10]
decreases greatly
. Here only the time-averaged coef-
ficients, CL , CD and CL / CD are listed in Table 1. Table 1 shows that, compared with the pitching-up Table 1 Time-averaged CL , CD and CL / CD of the pitching-up and the pitching-down flapping hovering (D = 40Û, 'Wt = 0.1Wc, 'Wr = 0.32Wc) Flapping modes Pitching-up flapping
Pitching-down flapping
Advanced mode
Symmetrical mode
Delayed mode
CL
1.960
1.787
0.774
CD
2.997
2.745
2.923
CL / CD
0.6538
0.6511
0.2648
CL
1.551
1.882
1.125
CD
1.757
1.740
1.697
1.0816
0.6628
CL / CD
0.8828
125
Qa ,t 0.5 UU 2 Sc
.
(4)
The coefficient of rotational aerodynamic moment:
CQ , a , r
Qa , r 0.5 UU 2 Sc
.
(5)
The coefficient of translational inertia moment: CQ ,i ,t
Qi ,t 0.5UU 2 Sc
.
(6)
The coefficient of rotational inertia moment: CQ ,i , r
Qi , r 0.5 UU 2 Sc
.
(7)
The coefficients of power requirements can be
Journal of Bionic Engineering (2009) Vol.6 No.2
126
derived from the above coefficients of moment and the non-dimensional angular velocity. The coefficient of power requirement to overcome the aerodynamic force: CP , a
Pa 0.5 UU 3 S
CQ , a ,tIt CQ , a , rD t .
Pi 0.5UU 3 S
CQ ,i ,tIt CQ ,i , rD t .
Ma
(CQ , a ,t CQ ,i ,t )It .
(CQ , a , r CQ ,i , r )D t .
(15)
I sin ʌ D ,I cos ʌ D ,D I sin D , I cosD ,D ,
(16)
:
Ma :
Pa
³ zf y sin DI xf z zf x cos DI xf yD ds
³ ª¬ zf y sin D xf z zf x cos D º¼ I xf yD ds
(10)
S
The coefficient of power requirement to realize rotation: CP , r
(14)
Then:
S
CP ,t
G
M a | ³ ( zf y i ( zf x xf z ) j xf y k )ds ,
(9)
The coefficient of power requirement to realize translation:
G
³ r u Fd s
Assuming the wing is thin, then y | 0.
(8)
The coefficient of power requirement to overcome the inertia force: CP ,i
where,
(17)
Qa ,tI Qa , rD ,
Qa ,t
(11)
³ zf y sin D xf z zf x cos D ds,
(18)
S
The coefficient of power requirement for the flap-
Qa , r
ping:
³ xf y ds,
(19)
S
CP
CP , a CP ,i
CP ,t CP , r .
(12)
The coefficients of aerodynamic force and moment are calculated through the unsteady Navier-Stokes equations, the coefficients of inertia moment are calculated below.
The moments and products of inertia of the mass of the wing are written as
I xy
I xx
³ y
2
z 2 dm | ³ z 2 dm,
(20)
I zz
³ x
2
y 2 dm | ³ x 2 dm,
(21)
I yy
³ x
z 2 dm |I xx I zz ,
(22)
2
³ xydm, I
³ xzdm, I
xz
yz
³ yzdm.
(23)
According to Refs. [12] and [13], the moments and products of inertia are taken as follows: I xx | 0.721 × 10í7 g·cm2, I yy | 0.79 × 10í7 g·cm2, I zz | 0.069 × 10í7 g·cm2, I xy | 0.0 g·cm2, I xz | 0.148 × 10í7 Fig. 5 Sketch of the coordinates (x, y, z) fixed on the flapping wing.
g·cm2, I yz | 0.0 g·cm2.
The coordinates (x, y, z) fixed on the flapping wing G are shown in Fig. 5. Where ds is the area element, r is the vector from the origin (o: the root of the wing) to the ds. The power required to overcome the aerodynamic force, Pa, is calculated as:
as
Pa
Ma : ,
(13)
The angular momentum of the wing, H, is written
H
ª I xx « « I yx « I zx ¬
I xz º » I yz » : i I xxI sin D I xzD I zz »¼ jI yyI cos D k I zxI sin D I zzD . (24) I xy I yy I zy
Bai et al.: Aerodynamic Characteristics, Power Requirements and Camber Effects of the Pitching-Down Flapping Hovering
The inertia moment of the wing, Mi, is written as Mi
ª dH º « dt » : u H ¬ ¼ xyz
i ª¬ I xxIsin D I xz D I2 cos D sin D º¼
sin D I I2 sin 2 D D 2 º j ª I yyI cos D 2 I zzID xz ¬ ¼
k ª¬ I zzD I xzIsin D I zzI2 sin D cos D º¼ .
(25)
The power required to overcome the inertia force of the wing, Pi, is written as Mi :
Pi
Qi ,tI Qi , rD ,
(26)
I I xx sin 2 D I yy cos 2 D
Qi ,t
sin D cos D , I xz D sin D D 2 cos D 2 I zzID
I zzD I xzIsin D I zzI2 sin D cos D ,
Qi , r CQ ,i ,t
I yy 0.5 U Sc
3
[(cos 2 D
(27) (28)
I xx sin 2 D )I I yy
2 I xz D sin D D cos D I yy
2
CQ ,i , r
I zz I D sin D cos D ]. I yy
I zz 0.5 U Sc3
(29)
2 ª I xz º I sin D I sin D cos D » . «D I zz ¬ ¼ (30)
Because that the flapping wing is symmetrical to the z axis, then I xz
³ xzdm
0.0 g·cm2.
The other inertia products are obtained from the above data. Then: CQ ,i ,t
4.6[ cos 2 D 0.91sin 2 D I 0.17ID sin D cos D ],
CQ ,i , r
2 0.4 ªD I sin D cos D º . ¬ ¼
(31) (32)
The curves of CQ,a,t, CQ,a,r and CQ,i,t, CQ,i,r during one cycle of the pitching-up flapping and pitching-down flapping are shown in Fig. 6. For the pitching-up flapping: (1) The coefficient of rotational aerodynamic moment CQ,a,r is much smaller than that of translational aerodynamic moment CQ,a,t; (2) In the middle of a stroke,
127
the angle of attack does not change, the coefficients of the inertia moment, CQ,i,t and CQ,i,r, both equal approximately 0. But at the end of a stroke, CQ,i,t is much greater than CQ,i,r. Compared with the pitching-up flapping, the coefficients of moment of the pitching-down flapping have the following characteristics: (1) The coefficient of translational aerodynamic moment CQ,a,t decreases very much, because that the rotational direction changes and the drag decreases very much; (2) The effect on the coefficient of translational inertia moment CQ,i,t is small. Meanwhile, because that the coefficient of rotational inertia moment CQ,i,r has small value, the effect can also be ignored. The curves of coefficients of power requirement in one cycle, CP,a,, CP,i, CP,t, CP,r, of the pitching-up flapping and the pitching-down flapping are shown in Fig. 7 and Fig. 8 respectively. Analyzing these figures, the conclusions can be drawn that: (1) The CP,a of pitching-down flapping is much smaller than that of pitching-up flapping. The maximum value of CP,a of pitching-up flapping is about 11.5, and that of pitching- down flapping is only about 4.8; (2) Compared with pitching-up flapping, the CP,a of pitching-down flapping removes the second peaks; (3) The CP,i of pitching-down is approximately equal to CP,i of pitching-up flapping; (4) The coefficient of power requirement to realize translation, CP,t, is much greater than the coefficient to realize rotation, CP,r; (5) The CP,t of pitching-down flapping is much smaller than that of pitching-up flapping. The maximum value of CP,t of pitching-up flapping is about 14, and that of pitching-down flapping is only about 8.4; (6) Just like the CP,a, the CP,t of pitching-down flapping also removes the second peaks. From the above analyses, the coefficient of power requirement of pitching-down flapping is much smaller than that of pitching-up flapping. The time-averaged CP and the ratio of time-averaged CL / CP are listed in Table 2. Compared with that of pitching-up flapping, the power requirement of pitching-down flapping decreases greatly, while still maintaining the high unsteady lift. The ratio of lift coefficient to power requirement coefficient, CL / CP , significantly improves. These mean that MAVs with pitching-down flapping may be more effective than that with pitching-up flapping.
Journal of Bionic Engineering (2009) Vol.6 No.2
128
15
10
10
CQ,a,t
5
Advanced Symmetrical Delayed
CQ,i,t
Advanced Symmetrical Delayed
5 0
0
CQ,a,r
í5
CQ,i,r í5
í10 í10
í15 0.00
0.25
0.50
0.75
1.00
0.00
0.25
IJ / IJc
0.50
0.75
1.00
IJ / IJc
CQ,a,t CQ,a,r
CQ,i,t CQ,i,r
(a) Moment coefficients of the pitching-up flapping in three modes
(b) Moment coefficients of the pitching-down flapping in three modes
CP,a CP,i
CP,t CP,r
Fig. 6 The moment coefficients of the pitching-up flapping and pitching-down flapping during one cycle in three modes (Dup = Ddown =40Û, 'Wt = 0.1Wc, 'Wr = 0.32Wc).
Fig. 7 Coefficients of power requirement, CP,a, CP,i CP,t, CP,r, of the pitching-up flapping in three modes.
4
Advanced Symmetrical Delayed
CP,a 5
2
CP,t
0
CP,i
0
í2
Advanced Symmetrical Delayed
í4 0.00
0.25
0.50 IJ / IJc
0.75
1.00
í5 0.00
CP,r
0.25
0.50 IJ / IJc
0.75
Fig. 8 Coefficients of power requirement, CP,a, CP,i CP,t, CP,r, of the pitching-down flapping in three modes.
1.00
Bai et al.: Aerodynamic Characteristics, Power Requirements and Camber Effects of the Pitching-Down Flapping Hovering
129
Table 2 Time-averaged coefficient of power requirement CP and CL / CP of the two flapping in three modes (D = 40Û, 'W t = 0.1Wcˈ 'Wr = 0.32Wc)
ping, CPup
CP of the pitching-down flapping, CPdown ( CPdown ˉ CPup )/ CPup
CL / CP of the pitching-up flapping CL / CP of the pitching-down flapping
Advanced
Symmetrical
Delayed
3.594
2.979
3.401
2.106
2.269
1.939
ˉ41.4%
ˉ23.8%
ˉ43.0%
0.5452
0.5999
0.2276
CL
Flapping modes
CP of the pitching-up flap-
(a) Lift coefficient in one cycle 0.7365
0.8294
0.58
4.3 Effects of flapping parameters on pitching-down flapping The symmetrical mode is chosen to study the effects of flapping parameters on pitching-down flapping.
+ t
Įt +
4.3.1 Effect of translational acceleration time 'Wt Here, 'Wt = 0.1Wc, 0.18Wc, 0.24Wc, while 'Wr = 0.32Wc and D = 40Û, are simulated to study the effect of 'Wt on pitching-down flapping. The translational and rotational angular velocities are shown in Fig. 9. As 'Wt decreases, the translational acceleration at the end of a stroke increases, and the translational angular velocity in the middle of a stroke decreases. The unsteady lift and drag coefficients in one cycle are shown in Fig. 10. It can be found that: (1) The lift and drag coefficients in the middle of a stroke are determined by the flapping angular velocity. As 'Wt decreases, both lift and drag coefficients decrease; (2) As 'Wt decreases, the peaks of the lift and drag coefficients moves forward.
Fig. 9 Dimensionless translational and rotational angular velocities in one cycle with different 'Wt in symmetrical mode.
3
ǻIJt = 0.10, ǻIJr = 0.32 ǻIJt = 0.18, ǻIJr = 0.32 ǻIJt = 0.24, ǻIJr = 0.32
2 1 0 í1 í2 í3 0.00
0.25
0.50 IJ / IJc
0.75
1.00
(b) Drag coefficient in one cycle
Fig. 10 Lift and drag coefficients with different 'Wt in one cycle in symmetrical mode.
The time-averaged lift and drag coefficients in one cycle are listed in Table 3. It is shown that as 'Wt increases from 0.1 to 0.24, the time-averaged lift coefficient, CL , increases by 12.5%. Meanwhile the timeaveraged drag coefficient, CD , also increases. The ratio of lift to drag, CL / CD , almost does not change. Table 3 Time-averaged lift and drag coefficients and the ratio of lift to drag of the pitching-down flapping with different 'Wt in the symmetrical mode. ('Wr = 0.32Wc, D = 40Û) 'Wt
0.1Wc
0.18Wc
CL
1.882
2.028
0.24Wc 2.118
CD
1.740
1.872
1.959
CL / CD
1.082
1.083
1.081
4.3.2 Effect of rotational time 'Wr Here, 'Wr = 0.16Wcˈ0.24Wcˈ0.32Wc, while 'Wt = 0.1Wc, D = 40Û, are calculated to study the effect of 'Wr on pitching-down flapping. The translational and rotational angular velocities are shown in Fig. 11. As 'Wr increases, the rotational angular velocity at the end of a stroke decreases. The unsteady lift and drag coefficients in one cycle are shown in Fig. 12. It can be found that: (1) Because 'Wr does not affect the translational angular
Journal of Bionic Engineering (2009) Vol.6 No.2
130
velocity and the angle of attack in the middle of a stroke, the lift and drag coefficients in the middle of a stroke almost do not change. (2) 'Wr significantly affects the peaks of the lift and drag coefficients. As 'Wr decreases, both peaks increase at the beginning of a stroke. The time-averaged lift and drag coefficients are shown in Table 4. As 'Wr decreases from 0.32Wc to 0.16Wc, CL and CD both increase, and the time-averaged ratio of lift to drag, CL / CD , decreases slightly. 2
+ t
ǻIJr = 0.16 ǻIJr = 0.24 ǻIJr = 0.32
1
0
í1
í2 0
2
4 t
6
8
Fig. 11 Translational and rotational angular velocities in one cycle with different 'Wr in the symmetrical mode.
5
Table 4 Time-averaged lift and drag coefficients and the ratio of lift to drag of the pitching-down flapping with different 'Wr in the symmetrical mode. ('Wt = 0.1Wc, D = 40Û) 'Wr
CL CD CL / CD
0.16Wc 2.138 2.013
0.24Wc 2.019 1.880
0.32Wc 1.882 1.740
1.062
1.074
1.082
4.3.3 Effect of angle of attack D The effects of the angle of attack in the middle of a stroke, D, on pitching-down flapping are studied. Here D = 30Û, 40Û, 50Û, while 'Wt = 0.1Wc, and 'Wr = 0.32Wc. The rotational and translational anglular velocities at different values of D are shown in Fig. 13. As D increases, the rotational angular velocity increases. This tendency is contrary to that of pitching-up flapping. The lift and drag coefficients are shown in Fig. 14. It can be found that: (1) As D increases, the unsteady lift and drag coefficients both increase. But this tendency is nonlinear, especially for the lift coefficient; (2) As D increases, the peaks of the lift and drag coefficients at the beginning of a stroke increase. The time-averaged coefficients of lift and drag are listed in Table 5.
ǻIJt = 0.1, ǻIJr = 0.32 ǻIJt = 0.1, ǻIJr = 0.24 ǻIJt = 0.1, ǻIJr = 0.16
4
Įt+
3
+ t
2 1 0 í1 í2 0.00
0.25
0.50 IJ / IJc
0.75
1.00
(a) Lift coefficient in one cycle
Fig. 13 Translational and rotational angle velocities in one cycle with different D in the symmetrical mode.
4 3 2
ǻIJt = 0.1, ǻIJr = 0.32 ǻIJt = 0.1, ǻIJr = 0.24 ǻIJt = 0.1, ǻIJr = 0.16
1
4 3
0
2
í1
1
í2
0.00
Į = 30Û Į = 40Û Į = 50Û
0
í3 í4
í1 0.25
0.50 IJ / IJc
0.75
1.00
(b) Drag coefficient in one cycle
Fig. 12 Lift and drag coefficients with different 'Wr in one cycle in the symmetrical mode.
í2 0.00
0.25
0.50 IJ / IJc
0.75
1.00
(a) Lift coefficient in one cycle
Fig. 14 Lift and drag coefficients with different D in one cycle in the symmetrical mode.
where, the non-dimensional coordinates x x / c and y = y/c; the location of the maximum camber x f 50% ; the non-dimensional camber f f / c . The thickness of the wing is still 0.05c. Here, f = 0, r10%, r20%, are chosen, while 'Wt = 0.1Wc, 'Wr = 0.32Wc, D = 40Û. The flapping model wing with camber and computational grid are shown in Fig. 4.
Į = 30Û Į = 40Û Į = 50Û
0.25
0.50 IJ / IJc
0.75
1.00
4.4.1 The advanced mode
(b) Drag coefficient in one cycle
Fig. 14 Continued. Table 5 Time-averaged coefficients of lift and drag and the ratio of lift to drag of pitching-down flapping with different D in the symmetrical mode ('Wt = 0.1Wc, 'Wr = 0.32Wc) D
30Û
40Û
50Û
CL
1.326
1.882
2.252
CD
1.143
1.740
2.490
CL / CD
1.161
1.082
0.904
Within the scope of D in this paper, as D increases the coefficients of lift and drag increase nonlinearly, and the ratio of lift to drag decreases. This trend is similar to that of general rigid wing. 4.4 Effect of camber on the pitching-down flapping It is well known that the flapping wing of fruit fly has no camber. Because the windward and leeward surfaces alternate in each stroke, the fixed camber will change sign in upstroke and downstroke, and the camber is unuseful to the pitching-up flapping. But for pitching-down flapping, the order of the windward and leeward surfaces does not change and the camber may be useful to improve the aerodynamic characteristics. In order to prove this idea, the effect of camber on pitching-down flapping in three modes (advanced, symmetrical, delayed) are numerically studied. The 4-figure NACA airfoil is adopted to define the camber of a flapping wing, which is determined by two parabolas. The two parabolas are tangential at the point of the maximum camber locating in the middle of the chord[15]. The formulas are listed as follows: x
x
0 ~ xf :
x f ~ 1.0 :
y
y
f
f
1 (2 x f x x 2 ) , x f2
The coefficients of lift and drag during one cycle with different camber values in the advanced mode are shown in Fig. 15. The time-averaged lift and drag coefficients and the ratio of lift to drag with different camber values are shown in Fig. 16. It can be easily found that the camber affects the lift and drag coefficients effectively. As f decreases from 0 to í20%, the lift coefficient CL decreases, the drag coefficient CD increases and the ratio of lift to drag CL / CD decreases. On the contrary, as the camber increases from 0 to 10%, the lift coefficient CL increases, the drag coefficient CD decreases a little and the ratio of lift to drag CL / CD increases. As the camber further increases from 10% to 20%, CL and CD both increase further, but CL / CD is almost unchanged. As f increases from 0 to 10%, CL increases by 9.8%, and as f increases from 0 to 20%, CL increases by 16.6% and CL / CD increases by 13.8%. 4 2 CL
4 3 2 1 0 í1 í2 í3 í4 0.00
131
Flex = í20 % Flex = í10 % Flex = 0 % Flex = 10 % Flex = 20 %
0 Advanced í2 0.00
0.25
0.50 IJ / IJc
0.75
1.00
(a) Lift coefficient in one cycle
CD
CD
Bai et al.: Aerodynamic Characteristics, Power Requirements and Camber Effects of the Pitching-Down Flapping Hovering
(33)
1 [(1 2 x f ) 2 x f x x 2 ] , (1 x f ) 2
(34)
(b) Drag coefficient in one cycle
Fig. 15 Lift and drag coefficients with different cambers in one cycle in the advanced mode.
132
Journal of Bionic Engineering (2009) Vol.6 No.2
3
CL
í í í í CL/CD, CD, CL
2 1
Flex = í20 % Flex = í10 % Flex = 0 % Flex = 10 % Flex = 20 %
Symmetrical
0 í1 0.00
0.25
0.50 IJ / IJc
0.75
1.00
(a) Lift coefficient in one cycle
The above results indicate that, in the advanced mode, negative camber has some disadvantages on the aerodynamic characteristics of pitching-down flapping, and positive camber has advantages, which can effectively increase CL and CL / CD .
CD
Fig. 16 Time-averaged lift and drag coefficients and the ratio of lift to drag with different cambers in the advanced mode.
4.4.2 The symmetrical mode
4.4.3 The delayed mode Finally, the effects of the camber on the pitching-down flapping in the delayed mode are analyzed. The coefficients of lift and drag with different cambers in one cycle are shown in Fig. 19. The time-averaged lift and drag coefficients and the ratio of the lift to drag are shown in Fig. 20.
(b) Drag coefficient in one cycle
Fig. 17 Lift and drag coefficients with different cambers in one cycle in the symmetrical mode.
Symmetrical
í í í í CL/CD, CD, CL
2.0
1.5
í CL í CD CíL/CíD
1.0
0.5 í20
í10
0 í f (%)
10
20
Fig. 18 Time-averaged lift and drag coefficients and the ratio of lift to drag at different cambers, f , in the symmetrical mode.
2
CL
The coefficients of lift and drag during one cycle with different camber values in the symmetrical mode are shown in Fig. 17. The time-averaged CL , CD and CL / CD with different camber values are shown in Fig. 18. As the camber f decreases from 0 to í20%, the drag coefficient increases, the lift coefficient decreases and the ratio of the lift to drag decreases effectively. As f increases from 0 to 10%, CD increases a little by 0.3%, CL increases by 9.4%, and CL / CD increase by 9.1%. As f increases to 20% further, CL increases by 14.7%, CD increases by 7.9%, but CL / CD increases by only 6.3%. CL / CD reaches the maximum value at f =10%. The results indicate that, in the symmetrical mode, the negative camber also has some disadvantages on the pitching-down flapping, and the positive camber can improve CL and CL / CD effectively. However as f > 10% CL / CD starts to decrease. This trend in the symmetrical mode is similar to that in the advanced mode.
0
í2
í4 0.00
Flex = í20 % Flex = í10 % Flex = 0 % Flex = 10 % Flex = 20 %
Delayed
0.25
0.50 IJ / IJc
0.75
1.00
(a) Lift coefficient in one cycle
Fig. 19 Lift and drag coefficients with different cambers in one cycle in the delayed mode.
Bai et al.: Aerodynamic Characteristics, Power Requirements and Camber Effects of the Pitching-Down Flapping Hovering
Flex = í20 % Flex = í10 % Flex = 0 % Flex = 10 % Flex = 20 %
2
CD
0 Delayed í2
0.00
0.25
0.50 IJ / IJc
0.75
1.00
(b) Drag coefficient in one cycle
Fig. 19 Continued. Delayed
í í í í CL/CD, CD, CL
2.0
í CL í CD CíL/CíD
1.5
1.0
0.5 í20
í10
0 f (%)
10
20
Fig. 20 Time-averaged lift and drag coefficients and the ratio of lift to drag with different cambers in the delayed mode.
The tendencies of the effects of the camber in the delayed mode are similar to those in the advanced and symmetrical modes. Compared with the coefficients as f =0, when f =10% and 20%, the time-averaged lift coefficient, CL , increases by 14.8% and 23.7% respectively, and the time-averaged ratio of lift to drag, CL / CD , increases by 8.4% and 3.6% respectively. From the above numerical simulations and analyses of the effect of camber on the pitching-down flapping in the advanced, the symmetrical, the delayed modes, the conclusions can be drawn that: (1) The negative camber has disadvantages on the aerodynamic characteristics of the pitching-down flapping; (2) On the contrary, the positive camber can effectively improve the time-averaged lift coefficient and the ratio of lift to drag; (3) It is as the same as the general rigid wing, that the relationships between CL , CD , CL / CD and f are nonlinear; (4) Approximately, CL / CD reaches the maximum value at f =10%.
5 Conclusions A numerical method was used to simulate the
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hovering of pitching-down flapping, which was invented by the author based on the previous studies on pitching-up flapping (fruit fly flapping). The most important and the only difference between pitching-down flapping and pitching-up flapping is the direction of rotation at the end of a stroke. Compared with pitching-up flapping, pitching-down flapping maintains the delayed stall, acceleration and deceleration mechanisms to keep the high unsteady lift. Meanwhile, the unsteady drag is greatly reduced with the removal of the pitching-up mechanism, which effectively improves the ratio of lift to drag. The power requirements of pitching-down flapping, the effects of the flapping parameters ('Wt, 'Wr, D) and the camber on the pitching-down flapping are systemically analyzed, from which the following conclusions can be drawn: (1) Compared with pitching-up flapping, the pitching-down flapping greatly reduces the power requirement to overcome the aerodynamic force and realize the translation. The time-averaged power requirement coefficient CP significantly decreases. These mean that while maintaining the high unsteady lift, the pitching-down flapping greatly reduces the power requirement and improves the mechanical efficiency; (2) For pitching-down flapping in the symmetrical mode, as the time of translational acceleration, i.e. 'Wt, increases or the time of rotation, i.e. 'Wr, decreases, the time-averaged lift and drag coefficients increase, but the ratio of lift to drag changes a little. The angle of attack in the middle of a stoke, i.e. D, significantly affects the aerodynamic characteristics. As D increases from 30Û to 50Û, CL and CD increases, CL / CD decreases nonlinearly. (3) It is well known that the camber is unuseful for pitching-up flapping. But for pitching-down flapping, the positive camber can effectively increase CL and CL / CD . In this paper, as the camber increases from 0 to 20%, CL and CD both increase nonlinearly, and CL / CD reaches the maximum value while the camber is approximately 10%. The negative camber makes CD increase, CL and CL / CD decrease, which is undesirable for pitching-down flapping. Further studies will be focused on the forward flight and the stability of pitching-down flapping.
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Journal of Bionic Engineering (2009) Vol.6 No.2
References [1] Burton R. Bird Flight, Facts On File, New York, USA, 1990. [2] Theodorsen T. General Theory of Aerodynamic Instability and the Mechanism of Flutter, NACA Technical Report 496, National Advisory Committee for Aeronautics, 1935. [3] Osborne M F M. Aerodynamics of flapping flight with application to insects. Journal of Experimental Biology, 1951, 28, 221–245. [4] Dickinson M H, Gotz K G. Unsteady aerodynamic performance of model wings at low Reynolds numbers. Journal of Experimental Biology, 1993, 174, 45–64. [5] Dickinson M H, Lehmann F O, Sane S P. Wing rotation and the aerodynamic basis of insect flight. Science, 1999, 284, 1954–1960. [6] Ellington C P, Van den Berg C, Willmott A P, Thomas A L R. Leading-edge vortices in insect flight. Nature, 1996, 384, 626–630. [7] Sun M, Tang J. Unsteady, aerodynamic force generation by a model fruit fly wing in flapping motion. Journal of Experimental Biology, 2002, 205, 55–70. [8] Ramamurti R, Sandberg W C. A three-dimensional computational study of the aerodynamic mechanisms of insect flight. Journal of Experimental Biology, 2002, 205,
1507–1518. [9] Bai P, Cui E J, Li F, Zhou W J. Study of high aerodynamic lift mechanics of hovering insect flapping wing at low Reynolds number. Acta Aerodynamic Sinica, 2007, 25, 175–182. (in Chinese) [10] Bai P, Cui E J, Li F, Zhou W J, Chen B Y. A new bionic MAV’s flapping motion based on fruit fly hovering at low Reynolds number. Acta Mechanica Sinica, 2007, 23, 485–493. [11] Rogers S E, Kwak D. An upwind differencing scheme for the incompressible navier-stokes equations. Applied Numerieal Mathematics, 1991, 8, 43–64. [12] Torkel W F. Quick estimates of flight fitness in hovering animals, including novel mechanism for lift production. Journal of Experimental Biology, 1973, 59, 169–230. [13] Vogel S. Flight in Drosophila: II. Variations in stroke parameters and wing contour. Journal of Experimental Biology, 1967, 46, 383–392. [14] Sun M, Tang J. Lift and power requirements of hovering flight in Drosophila virilis. Journal of Experimental Biology, 2002, 205, 2413–2427. [15] Abbott I H, von Doenhoff A E. Theory of Wing Sections, Dover Publications, New York, USA, 1958.