|
| |
|
|
|
|
||||
| Home Help Feedback Subscriptions Archive Search Table of Contents | ||||
First published online December 28, 2007
Journal of Experimental Biology 211, 234-238 (2008)
Published by The Company of Biologists 2008
doi: 10.1242/jeb.013797
Research Article, Biomechanics of Flight |
Aerodynamic efficiency of flapping flight: analysis of a two-stroke model
Theoretical and Applied Mechanics, Cornell University, Ithaca, NY 14853, USA
e-mail: jane.wang{at}cornell.edu
Accepted 22 October 2007
Summary
To seek the simplest efficient flapping wing motions and understand their relation to steady flight, a two-stroke model in the quasi-steady limit was analyzed. It was found that a family of two-stroke flapping motions have aerodynamic efficiency close to, but slightly lower than, the optimal steady flight. These two-stroke motions share two common features: the downstroke is a gliding motion and the upstroke has an angle of attack close to the optimal of the steady flight of the same wing. With the reduced number of parameters, the aerodynamic cost function in the parameter space can be visualized. This was examined for wings of different lift and drag characteristics at Reynolds numbers between 102 and 106. The iso-surfaces of the cost function have a tube-like structure, implying that the solution is insensitive to a specific direction in the parameter space. Related questions in insect flight that motivated this work are discussed.
Key words: efficiency, aerodynamics, flapping wing, flight
Introduction
Quantitative studies have made much progress in revealing various
aerodynamic mechanisms for force generation in flapping flight, as extensively
reviewed (Weis-Fogh and Jensen,
1956
; Ellingon, 1984;
Dickinson, 1996;
Sane, 2003;
Lehmann, 2004;
Wang, 2005). Less clear is the
relative cost of various wing motions that are capable of generating the same
averaged force. For example, for a given wing, how does the aerodynamic cost
of flapping flight compare to that of a fixed wing flight? A necessary first
step in addressing the question of relative efficiency is to define a fair and
relevant measure for cross-comparing different wing motions. With the model of
aerodynamic forces and actuators, we can, in principle, determine efficient
wing motions either in real animals or model systems. Suppose we succeed in
doing so, the solutions may still be non-intuitive due to the fact that a
typical wing motion is described by a large number of parameters. Moreover,
some features in the predicted motions are specific to the model rather than
the original system. The main purpose of this paper is to seek some common
features of efficient flapping motions in a minimal model, as a step toward
understanding the more complex ones.
Hovering, as opposed to forward flight, is a natural candidate for
comparing the efficiency of two flight strategies: flapping vs steady
flight. In hovering, a moving wing alone generates the required thrust without
the need for additional propulsions. Here, the word efficiency is used to mean
the inverse of the dimensionless cost. First, an energetic criterion is
defined, with which the relative aerodynamic cost of employing different wing
motions can be compared. Finding efficient wing motions requires an effective
method for reducing the parameter space without excluding all of the efficient
motions. One approach is to use the observed insect wing motions as a guide to
construct families of wing motions described by a set of physical parameters.
The idea is that the odds of finding efficient solutions among insects' or
birds' wing motions are likely to be higher than our random guesses. In the
reduced parameter space, we can identify with energy-minimizing flapping wing
kinematics for various insects, compare with observed motions, and recognize
common features (Berman and Wang,
2007). In this paper, we seek the simplest efficient flapping wing
motions that can be analyzed in detail. To this end, we consider a family of
up and down motions described by six parameters and calculate the aerodynamic
power in quasisteady limit. The parameter space was further reduced to
four-dimensions, based on the reasoning described below. An advantage of
working with the remaining four-parameter space is that it is possible to
visualize the parameter space. The sensitivity of the cost function can be
viewed with respect to the wing motion parameters, and in relation to
classical steady wing motion.
A criterion for comparing the aerodynamic cost of a wing undergoing different hovering wing motions
For animals and airplanes, the total energetic cost is measured by the consumption of their respective fuels. At the limit where the conversion rate from chemical to mechanical energy is independent of the wing motion, the total cost is directly proportional to the mechanical work. The mechanical work done by a flapping wing includes aerodynamic and inertial components. The former is the work done to overcome fluid drag and the latter, work done to accelerate and decelerate the wing in a vacuum. The inertial cost can be calculated directly from the wing kinematics, and unless the elastic storage of the muscles is perfect, the net inertial cost is non-zero. The aerodynamic cost can be measured experimentally, calculated using direct numerical simulations, or estimated by quasi-steady force models.
The aerodynamic efficiency of transport of a classical airfoil in steady forward flight is determined by the lift:drag ratio, which is the inverse of the aerodynamic work required to transport a unit weight over a unit distance. A similar ratio can be defined for the efficiency of endurance, which is proportional to the inverse of the aerodynamic power required to support a unit weight. These ratios are often used to compare the relative efficiency of different airfoils: the higher the ratio, the more efficient the airfoil. For flapping motions, in addition to cross-comparing different wing shapes, it is of interest to investigate the relative efficiency of the same airfoil undergoing different wing motions, which is the focus of this work.
In the case of a hovering insect, the wing and the weight are given, and we
seek the wing motion that minimizes the mechanic power subject to the
constraint of the weight balance. Specifically, the aerodynamic cost of
endurance (P*) is defined as the dimensionless aerodynamic
power to support a unit weight:
 | (1) |
 | (2) |
and
are instantaneous aerodynamic force
and wing translational velocity,
and
the instantaneous aerodynamic
torque and wing angular velocity, Mg the weight, and
T the period. P* is dimensionless, and the
reference velocity
Uref=
is
constant for a specific wing of area A and weight Mg.
The inertial cost can be added if we have a model of elastic storage of the
muscles (Berman and Wang,
2007). For the piece-wise constant motions studied here, the
inertial cost is zero except near the transition.
Note that in the case of steady wing motion
(Fig. 1B),
P* has the familiar form:
P*=CD(
)/
(),
where CL() and CD() are
lift and drag coefficients, respectively. For general wing motions, this
simple ratio no longer holds, and thus maximizing the averaged lift:drag ratio
is not equivalent to minimizing the aerodynamic power. We further note that
P* is proportional to the specific power, power per mass
supported (Ellington, 1984).
The main difference is that here the specific power is only compared among the
motions that generate the same force. This difference matters when the
aerodynamics force coefficients depend indirectly on the net force due to, for
example, the change of the Reynolds number of the flow required to generate
the specified force.
|
The steady forward flight motion is defined by the wing velocity,
U, and the angle of attack, (stroke-I,
Fig. 1A). Similarly, the rotary
wing motion is defined by the angular velocity, 
and . The
simplest flapping motion is a back-and-forth motion, which can be viewed as a
rotary motion projected onto the diameter keeping the same angle of attack and
velocity (Fig. 1B). In the
quasi-steady limit considered here, these symmetrical back and forth motions
are equivalent to the rotary motion. Among them, the one that minimizes
P* has m that minimizes
and
Um that balances the weight at
.
The next simplest flapping motion (Stroke-II,
Fig. 1E) consists of two
constant motions arranged in a V-shape, and is defined by seven parameters:
the velocity (Ud,u), the angle of attack
(d,u), the angle of stroke path (βd,u)
during down (d) and up (u) strokes, and the fraction of a period spent on the
upstroke (
). Six of them are independent if we further require that two
ends of V maintain the same altitude. A pair of mirror images of each V-shape
forms a figure-eight (Fig. 1F),
which is a hovering motion.
These two-stroke motions are piece-wise constant, thus the main unsteady
aerodynamic effect is the dynamic stall, during which the leading edge vortex
forms and remains attached while producing a force that is roughly constant at
a given angle (Francis and Cohen,
1933; Ellington,
1984; Dickinson and Götz,
1993; Ellington et al.,
1996; Wang, 2000;
Usherwood and Ellington,
2002). At this limit, the empirical formula to fit the lift-drag
at relative low Reynolds numbers (
102) can be expressed as
(Wang, 2005):
 | (3) |
 | (4) |
the angle of attack, 0l
the angle at which lift is zero and 0d the angle at which
the drag is minimal. 0l and 0d are
non-zero for asymmetrical airfoils. Similar form was also obtained to fit the
experimental measurements of lift–drag characteristics of helicopter
blades during dynamic stall at much higher Reynolds numbers of about
106 (Leishman,
2000). The fit of this formula to airfoil data is shown in
Fig. 2. Note that
Eqn 3 differs from the classical
Joukowski's lift in its angle dependence, sin2 rather than sin,
and it is to be applied to all angles of attack during dynamic stall.
|
Efficient two-stroke wing motions
In this minimal model, the parameter space is six-dimensional, which is
still difficult to visualize directly. To further reduce the number of
parameters, we make two observations. The first is that the classical airfoil
motion of a fixed or rotating wing uses only aerodynamic lift, but not drag,
to support a weight (Wang,
2004). If the airfoil is reoriented such that the net force is
vertical (Fig. 1C), as in
gliding, the wing can support an additional weight by a factor of
.
Consider the ideal case in which the downstroke is a gliding motion and the
upstroke returns instantaneously, consuming no energy, to support a specified
weight,
and
.
This gliding motion is more efficient and is used as the downstroke of the
flapping motions studied below.
An upstroke must return in a finite time and it costs energy. At first
sight, the least costly upstroke seems to be the vertical upstroke at zero
angle of attack (Fig. 1D),
because it moves along the shortest path and the wing experiences the least
drag. However, this is not the case (Fig.
3). To find a more efficient upstroke, we note the angle
dependence of the aerodynamic force at small angles. As deviates from
zero, to the leading order, the lift increases as , as given by the
Kutta–Joukowski theory, while the drag increases as 2.
At small ,
2
,
thus an upstroke at small angle of attack generates a lift at a relatively
small cost, which can be advantageous compared to that with =0, which
generates no lift. These lift-generating strokes replace the vertical
upstroke. The two-stroke motion is now described by four parameters,
(u, βu, , d), and
the corresponding P* is given by:
 | (5) |
 | (6) |
 | (7) |
 | (8) |
,
βu, u) for each downstroke parameterized by
d in Matlab.
|
; Berman and Wang,
2007), which is discussed later.
|
d) two-stroke,
composed of a gliding downstroke with the same angle of attack
(d) followed by a vertical upstroke at zero angle of attack,
and two-stroke composed of the same gliding stroke followed by an optimized
upstroke. Two representative Reynolds numbers (Re) are shown:
103, close to that of an insect, and 3x105, close
to that of a low-speed airplane. The maximum lift:drag ratios are 4.7 and 60,
respectively. In both cases, there are multiple asymmetrical down-and-up
flapping motions (Fig. 3B),
whose P* are close to minimal, but slightly higher than
the minimum of
of
the steady wing motion.
By definition, the first derivative of a function at minimum (inside the
domain) is zero, and the sensitivity to parameters is given by higher order
derivatives. In the case of back-and-forth motions, the efficiency drops
relatively quickly as the angle of attack deviates from the optimal angle,
whereas in the case of up-and-down motions, the minimum is much more flat.
Fig. 4C shows the iso-surfaces
of 1/P* as a function of parameters of the upstroke,
(, u, βu) for a given
d. Their shapes near the maximum have a tube-like structure.
The longitudinal direction of the tube corresponds to the multiple solutions
found here.
Among the parameters in the quasi-steady force model, the one that is most
sensitive to Re is CD(0);
CD(0)1/
. To
investigate the effect of Re, CD(0) is varied from 0.3 to
0.003, which corresponds to Re from 102 to
106, estimated using the Blasius theory of flow past a plate
(Glauert, 1947). To see the
effect of wing shape, the calculation was repeated using ten randomly chosen
published lift–drag characteristics for airfoils in NACA-4digit, DH and
Xfoil-series (Selig, 2002).
For low Reynolds number plates, CL(
/4) and
CD(/2) are varied between 1 and 2.5, to simulate the
effect of the sharpness of the wing tip
(Wang, 2000). The extended
tube structure is found for all tested Reynolds numbers and wing shapes.
|
The above analysis was partly motivated by the question concerning the relative efficiency of flapping and steady wing motions that support the same weight. The two-stroke model suggests that at the limit where the lift and drag are described by the translational quasi-steady forces, the most efficient motion to support a given weight is the steady wing motion at the optimal angle of attack. There are multiple flapping motions that are very close to the optimum. The efficient two-stroke motions have in common that the downstroke is a gliding motion at an arbitrary angle of attack and the upstroke operates near the optimal angle of attack. We are currently investigating if flapping flight can be more efficient than the steady wing motion when the unsteady aerodynamics effects are included.
Another motivation came from our interest in understanding whether insects
are aerodynamically efficient; specifically, whether hovering insects have
found some energy-minimizing wing kinematics, given that hovering is an
energy-demanding mode of flight. The idea of optimization in biological
systems is open to debate. Without testable predictions, however, it is
difficult to make progress. Using the same criterion as discussed here,
various published wing strokes were examined
(Berman and Wang, 2007), for
fruit flies (Ennos, 1989;
Fry et al., 2003), a bumblebee
(Dudley and Ellington, 1990)
and a hawkmoth (Willmott and Ellington,
1997), and it was found that some of the specific features of the
predicted energy-minimizing hovering kinematics, e.g. the frequency and the
wing stroke pattern, are qualitatively and quantitatively similar to the
previously observed data. This, however, does not imply that all hovering
insects fly using a single pattern of wing motion. Optimal wing motion depends
on the wing morphology and lift–drag characteristics. Even for a
specific wing, there can be many solutions that are very close to optimal, as
indicated here with a four-parameter model. This multiplicity of solutions
does not contradict the idea that energy-minimizing shapes the wing
kinematics. Instead, we should expect variations in the observed wing
kinematics that have comparable efficiency near the optimal.
Finally, to comment on the transition between the up- and downstrokes, the
cost of which is assumed to be small. Although muscles in insects are known to
be capable of actively pitching the wing
(Ellington, 1984;
Dickinson et al., 1993), the
net power required to pitch the wing in the observed motions of a dragonfly, a
fruit fly and a hawkmoth was found to be negative
(Bergou et al., 2007). Thus in
theory, wing pitching can be aided by aerodynamic torque and does not require
additional power. This, and the fact that almost all insects maintain the same
leading edge (Fig. 5), suggest
that insects may benefit from passive wing pitch reversal during steady flight
to simplify control.
Acknowledgments
The work is supported by NSF, AFOSR, and Packard Foundation.
References
Bergou, A., Xu, S. and Wang, Z. J. (2007). Passive wing pitch reversal in insect flight. J. Fluid Mech. 591,321 -337.
Berman, G. and Wang, Z. J. (2007). Energy minimizing kinematics in hovering insect flight. J. Fluid Mech. 582,153 -168.[CrossRef]
Dickinson, M. H. (1996). Unsteady mechanisms of force generation in aquatic and aerial locomotion. Am. Zool. 36,537 -554.
Dickinson, M. H. and Götz, K. G. (1993). Unsteady aerodynamic performance of model wings at low Reynolds numbers. J. Exp. Biol. 174,45 -64.[Abstract]
Dickinson, M. H., Lehmann, F. O. and Götz, K. G. (1993). The active control of wing rotation by Drosophila. J. Exp. Biol. 182,173 -189.[Abstract]
Dudley, R. and Ellington, C. P. (1990).
Mechanics of forward flight in bumblebees. II. Quasi-steady lift and power
requirements. J. Exp. Biol.
148, 53.
Ellington, C. P. (1984). The aerodynamics of hovering insect flight. I–V. Philos. Trans. R. Soc. Lond. B Biol. Sci. 305,1 -181.[CrossRef]
Ellington, C. P., van den Berg, C., Willmott, A. P. and Thomas, A. L. R. (1996). Leading-edge vortices in insect flight. Nature 384,626 -630.[CrossRef]
Ennos, A. R. (1989). The kinematics and
aerodynamics of the free flight of some diptera. J. Exp.
Biol. 142,49
-85.
Francis, R. H. and Cohen, J. (1933). The flow near a wing which starts suddenly from rest and then stalls. Aero. Res. Counc. Rep. Memo. 1561,90 .
Fry, S. N., Sayaman, R. and Dickinson, M. H.
(2003). The aerodynamics of free-flight manuevers in Drosophila.
Science 300,495
-498.
Glauert, H. (1947). The Elements of Aerofoil and Airscrew Theory. Cambridge: Cambridge University Press.
Lehmann, F. O. (2004). The mechanisms of lift enhancement in insect flight. Naturwissenschaften 91,101 -122.[CrossRef][Medline]
Leishman, J. (2000). Principles of Helicopter Aerodynamics. Cambridge: University of Cambridge.
Sane, S. (2003). The aerodynamics of insect
flight. J. Exp. Biol.
206,4191
-4208.
Selig, M. (2002). Online database for airfoil lift-drag characteristics. http://www.aae.uiuc.edu/m-selig/pub/lsat/vol1-3.
Usherwood, J. R. and Ellington, C. P. (2002).
The aerodynamics of revolving wings. II. Propeller force coefficients from
mayfly to quail. J. Exp. Biol.
205,1565
-1576.
Wang, Z. J. (2000). Vortex shedding and frequency selection in flapping flight. J. Fluid Mech. 410,323 -341.[CrossRef]
Wang, Z. J. (2004). The role of drag in insect
hovering. J. Exp. Biol.
207,4147
-4155.
Wang, Z. J. (2005). Dissecting insect flight. Annu. Rev. Fluid Mech. 37,183 -210.[CrossRef]
Weis-Fogh, T. and Jensen, M. (1956). Biology and physics of locust flight. I. Basic principles in insect flight. A critical review. Proc. R. Soc. Lond. B Biol. Sci. 239, 239, 415-458.
Willmott, A. P. and Ellington, C. P. (1997). The mechanics of flight in the hawkmoth Manduca sexta. I. Kinematics of hovering and forward flight. J. Exp Biol. 200,2705 -2722.[Abstract]
Related articles in JEB:
| |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
