The role of drag in insect hovering
Theoretical and Applied Mechanics, Cornell University, Ithaca, NY 14853, USA
e-mail: jane.wang{at}cornell.edu
Accepted 11 August 2004
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Summary |
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To investigate force generation and energy cost of hovering flight using different combination of lift and drag, I study a family of wing motion parameterized by the inclined angle of the stroke plane. The lift-to-drag ratio is no longer a measure of efficiency, except in the case of horizontal stroke plane. In addition, because the flow is highly stalled, lift and drag are of comparable magnitude, and the aerodynamic efficiency is roughly the same up to an inclined angle about 60°, which curiously agrees with the angle observed in dragonfly flight.
Finally, the lessons from this special family of wing motion suggests a strategy for improving efficiency of normal hovering, and a unifying view of different wing motions employed by insects.
Key words: lift, drag, efficiency, dragonfly flight, normal hovering, unsteady aerodynamics
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Introduction |
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Differentiating lift and drag may seem to be a matter of semantics. After all, living organisms presumably only care about the net forces. However, because theories of simple systems, such as an airfoil or a paddle, have influenced our approaches to understanding more complex locomotion in nature and our choices of model systems, in order to go beyond the confines of these theories it is necessary to first borrow the conventional terminology. Towards the end of this paper, we will see that this differentiation helps us to resolve one of the puzzles in quasi-steady estimates of dragonfly flight as well as to construct more efficient hovering strokes.
Viscous drag is often studied in the context of locomotion of
microorganisms (bacteria, sperm and protozoa), which live in Stokes flow
[Reynolds number (Re)=0; Purcell,
1977; Childress,
1981
; Wu et al.,
1975
; Taylor,
1985
]. The focus here is on the non-Stokesian regime. It was
suggested that small insects might employ a drag mechanism at Re
below
100 (Horridge,
1956
); however, use of drag is often found in large insects, such
as butterflies, which use near vertical stroke plane at relatively higher
Re (
103)
(Ellington, 1984
). The drag in
these cases is dominated by pressure force. At even higher Re, some
birds and fish also use pressure drag to fly and swim
(Blake, 1981
;
Vogel, 1996
). Thus, the
Re, as long as it is sufficiently high to be outside the Stokesian
regime, does not seem to determine whether an organism uses mainly drag or
lift. Vogel (1996
) reviewed
the drag-based and lift-based thrust in aquatic motion. Using the example of a
pedal motion parallel to the forward motion, i.e. rowing, he suggested that
the `drag-based system is better when the craft is stationary but
lift-based system is superior at any decent forward speed'. The motion
considered by Vogel is appropriate for forward swimming and rowing but is
different from typical hovering motions employed by insects. Recognizing that
at high angle of attack both lift and drag resulted from the same pressure
force that acts normal to the wing, Dickinson
(1996
) suggested that `the
dichotomy between lift- and drag-based mechanisms of locomotion
(Vogel,
1996
) was blurred'. Still, in subsequent studies,
drag and lift have not been treated on equal footing. For example, most models
approximated the stroke plane to be horizontal. While this is a reasonable
simplification, it is also a special case where drag in two half-strokes is
almost equal and in opposite direction, thus making negligible net
contribution to the net force.
Some of the best hoverers dragonflies and true hoverflies
employ asymmetric strokes along an inclined stroke plane, similar to rowing.
This is in contrast to `normal hovering' used by most insects including flies,
bees and wasps, who flap their wings about a horizontal plane
(Weis-Fogh, 1973). In normal
hovering, a wing generates a vertical force in both half-strokes, while in
asymmetric strokes it generates a vertical force primarily during the
downstroke. This difference, together with the fact that normal hovering
resembles a helicopter wing motion, prompted Weis-Fogh to hypothesize that
normal hovering might be more efficient. This turns out not to be the case in
the wing motions studied here, as I will show later.
A puzzle about hovering along an inclined stroke plane is that quick
estimates of required lift coefficients based on bladeelement theory, assuming
constant lift and drag coefficient, range from 3.5 to 6
(Weis-Fogh, 1973;
Norberg, 1975
), which are
substantially higher than those estimated for normal hovering insects:
typically around 1 (see table 5 in
Weis-Fogh, 1973
). Later
inclusion of corrections due to induced downward flow predicted similarly high
coefficients (Ellington, 1984
).
Explaining these unusually large lift coefficients motivated a shift of focus
from quasi-steady analysis to the investigation of unsteady mechanisms in
hovering flight. Recently investigated mechanisms, such as dynamic stall
(Dickinson and Götz, 1993
;
Ellington et al., 1996
;
Wang, 2000b
), wing rotation
and wingwake interaction (Dickinson
et al., 1999
), can explain an increase of averaged lift up to a
factor of two, but not a factor of four. This raises the question of whether
the high coefficients seen in hovering with inclined stroke plane result from
unsteady mechanisms alone or other assumptions made in the theoretical
analyses.
Without getting into the details of the unsteady mechanisms, an obvious
feature of a downward stroke along an inclined stroke plane is that the
associated pressure drag has an upward vertical component, which can have a
non-negligible contribution to weight balance. In the previous quasi-steady
analyses, drag was assumed to be much smaller than lift. For example, the lift
to drag ratio was assumed to be 7 by Weis-Fogh
(1973
) and 6 by Norberg
(1975
). These were estimates
based on the maximal value of lift to drag ratio from experiments on a locust
wing (Jensen, 1956
). Ellington
(1984
) used the relation
CD,pro
7/
Re based on flow past a
cylinder and deduced a value of 0.150.2 at an Re of
103 for the profile drag coefficient. While these values might
be reasonable at a small angle of attack, they are considerable underestimates
of drag at stalled angles during the downstroke.
Given that our ability to quantify unsteady forces, at least of model wings, is much improved, it seems worthwhile to re-examine the force generation and energy cost of hovering flight using different strategies. The first goal of this paper is to analyze these quantities in a family of hovering motions, which are parameterized by the inclined angle of the stroke plane and, correspondingly, different combinations of lift and drag in supporting the weight of an insect. The results offer an explanation of the discrepancy by a factor of four in the quasi-steady analysis. They also suggest a strategy for improving hovering efficiency and a unifying view of hovering motions used by different insects.
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Models and methods |
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In particular, I consider a two-dimensional cross-section of a wing
executing the following motion:
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![]() | (2) |
|
Special cases of these wing motions have been studied theoretically
(von Holst and Kuchemann,
1941), experimentally (Freymuth
et al., 1991
) and computationally
(Gustafson et al., 1992
;
Wang, 2000a
). Here I compute
and extend the forces and flows for
0 and ß.
Computational methods
The flow around the wing is governed by the NavierStokes equation,
which is solved with a fourth-order compact finite-difference scheme
(E and Liu, 1996) in elliptic
coordinates (Wang,
2000a
,b
;
Wang et al., 2004
).
The NavierStokes equation in the coordinates fixed to the wing has
the form:
![]() | (3) |
![]() | (4) |
![]() | (5) |
![]() | (6) |
![]() | (7) |
The velocity and vorticity are obtained in the non-inertial coordinates,
which are then transformed into the inertial frame. The forces are calculated
in the inertial frame by integrating the viscous stress along the wing:
![]() | (8) |
![]() | (9) |
The translational motion of the wing is specified by two dimensionless
parameters: the Reynolds number (Re
Umaxc/
=
fA0c/
)
and A0/c. The typical Re of a dragonfly
is
103, and that of a fruit fly is
102. The
Re dependence of the force was previously studied for similar wing
motions from Re=15.7 to Re=1256 and it was shown that the
averaged force does not have a strong dependence when
150<Re<1256 (Wang,
2000a
), where the force is dominated by pressure
(Wang et al., 2004
). In the
following computations, Re=150,
A0/c=2.5, f=1, B=
/4 and
, which are in the range of observed values in insect hovering.
These parameters are also where two-dimensional computations and
three-dimensional experiments agree well
(Wang et al., 2004
).
Quasi-steady estimate
In addition to solving the NavierStokes equation, it was instructive
to apply a quick quasi-steady estimate with a liftdrag polar obtained
for a translating wing at Re=200 at an angle of attack
(A) from [0,
] (Wang
et al., 2004
):
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![]() | (11) |
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Results |
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Fig. 2 shows a side-by-side comparison of the wing motion, forces, vorticity field and mean flow in the two cases. In the case of a symmetric stroke (Fig. 2A), each half-stroke generates almost equal lift in the vertical direction and almost equal drag in the opposite horizontal direction. The averaged vertical and horizontal force coefficients are 1.07 and 1.61, respectively, resulting in a ratio of 0.66. By contrast, the asymmetric stroke (Fig. 2B) generates most of its vertical force during the downstroke, in which the lift and drag coefficients are 0.45 and 2.4, respectively; they are 0.50 and 0.68 during the upstroke. The vertical and horizontal force coefficients averaged over one period are 0.98 and 0.75, resulting in a ratio of 1.31, which is twice the value of the symmetric stroke. In this case, 76% of the vertical force is contributed by aerodynamic drag.
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Comparing the vorticity field in the two cases shows a faster downward jet
produced by the asymmetric stroke. Fig.
2Biv shows the time-averaged velocity below the wing. The velocity
is plotted in physical space, which is interpreted from the computed velocity
in the body coordinates. The symmetric stroke generates a jet whose width is
comparable to the flapping amplitude, and it penetrates down for
45 chords. By contrast, the asymmetric stroke generates a jet
whose width is comparable to the chord, and it penetrates downward for
7
chords. This difference may be significant when the wing is hovering above a
surface, where the ground effect is non-negligible.
Ten cases from 0=0 to
0=90°
Next, I investigate how the flows, forces and specific power vary with the
angle of the stroke plane (ß). Fig.
3 shows the vorticity field of four representative cases in the
fourth period. Ten snapshots are taken, equally spaced in time. The downward
dipole jets are in the approximately opposite direction to the net force. The
jet speed can be estimated by the travel distance over one period. It
increases with ß. At ß=4°, 30°, 48°, 63° and
75°, the dipole pair travels over 2.4, 3, 3.5, 3.9 and 4 chords,
respectively.
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Fig. 4 shows the time-dependent vertical and horizontal forces for five cases. The averaged vertical forces are similar in all cases, as shown in Fig. 5. The fluctuation of the vertical force increases with ß, while the fluctuation of the horizontal force decreases with ß. For example, the maximum vertical force is approximately a factor of two higher at ß=75° compared with ß=4°, but the maximal horizontal force is approximately a factor of two lower. These variations are consistent with the fact that at larger ß the downward jet is faster and narrower. The narrower jet at larger ß makes sense since the wing sweeps less horizontal distance at a given A0.
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The variation of the force fluctuation may correlate with the body
orientation during hovering. For an elongated body, it is preferable to hover
with a horizontal body when employing a highly inclined stroke plane and with
a vertical body when using a horizontal stroke plane
(Weis-Fogh, 1973).
Fig. 4 suggests that the body
aligns in the direction where the force fluctuation is small, which would
reduce body oscillations.
How do these different hovering styles affect the net forces and the
specific power? Fig. 5 compares
them as a function of ß. It illustrates two interesting points. First, as
the stroke plane tilts up, the average vertical force coefficient,
V, remains almost constant
up to ß
60°. The horizontal force averages zero, but its average
magnitude,
H, decreases
with ß. Thus, the ratio
V/
H
increases by a factor of two as ß increases from 0° to 60°.
Second, the averaged power exerted by the wing to the fluid is given by
= FD(t)u(t), where
FD(t) is the drag. Comparing this power with the
ideal power based on the actuator disk theory
(Leishman, 2000
) gives a
non-dimensional measure:
![]() | (12) |
The mechanism for this cut-off requires further investigation, but it is
worth noting that ß 60° agrees with one of the largest angles
observed in free hovering flight of Aeschna juncea
(Norberg, 1975
); studies of
tethered flight reported smaller inclined angles
(Ellington, 1984
;
Wakeling and Ellington,
1997
).
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Discussion |
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An explanation of anomalously high lift coefficients obtained in previous estimates
Now I return to the question discussed in the Introduction about the high
lift coefficients obtained in quasi-steady analysis of dragonflies
(Weis-Fogh, 1973;
Norberg, 1975
;
Ellington, 1984
). The lift to
drag ratio during downstroke was assumed to be
6.5
(Weis-Fogh, 1973
;
Norberg, 1975
), which is based
on the value at small angle of attack in locust flight
(Jensen, 1956
). According to
the current computation in the corresponding case of ß=63°, the drag
contributes about 76% of the net vertical force. Therefore, the assumption of
a lift to drag ratio of 6.5 is equivalent to assuming a drag contribution of
about (24/6.5)% of the vertical force. Consequently, it ignores about 72%
[76(24/6.5)] of the net vertical force. This would result in
approximately a factor of four increase in the estimate of the lift
coefficient. If we include the computed drag, Norberg's estimate
(CL
3.56.1;
Norberg, 1975
) would yield a
CL of
0.91.5, which is much more
reasonable.
The role of drag in normal hovering
Dragonflies and hoverflies, which use a highly inclined stroke plane, are
examples where ignoring drag can lead to obvious contradictions. One might ask
to what degree drag is relevant in understanding normal hovering, which is
employed by most insects including flies and bees. The wing tip of different
insects typically traces out shapes of an oval, a parabola or a figure of
eight, under different experimental conditions (tethered vs free
flight; Marey, 1868;
Hollick, 1940
;
Jensen, 1956
;
Nachtigall, 1974
;
Ellington, 1984
;
Zanker and Götz, 1990
;
Fry et al., 2003
), but the
aerodynamic consequences of these variations have not been much discussed.
Here, I suggest that a figure of eight, an oval or a parabola can all be
decomposed into pairs of dragonfly-like strokes, as illustrated in
Fig. 1C,D. The deviation from a
horizontal stroke plane permits the insect to use some of the drag to support
its weight during the plunging-down motion. Recent force measurements on a
robotic wing mimicking hovering of fruitflies show that the upward force has a
substantial component in the direction of drag (see fig. 3A in
Fry et al., 2003). These new
results, together with the analysis here, suggest that normal-hovering insects
can also use part of drag to support their weight. Another implication is that
the instantaneous orientation of the stroke plane is a relevant parameter when
constructing model wing motions.
Improving efficiency by eliminating half of a stroke in normal hovering
The magnitude of drag in normal hovering considered here (see
Fig. 2A) is greater than that
of lift (CD=1.61 and CL=1.07) yet,
because of the use of strict horizontal stroke plane, the drag makes no
contribution to the net force. Large drag was also found in simulations of a
family of normal hovering (Wang et al.,
2004), in particular near the wing reversal, and in an extensive
experimental study of 191 hovering kinematics, where stroke amplitude, angle
of attack, deviation of the stroke plane, and timing and duration of wing
rotation were varied (Sane and Dickinson,
2001
).
Here is a strategy to benefit from the large drag found in these symmetric strokes. Instead of using both half-strokes, take a half-stroke and make it a downstroke by tilting the stroke plane such that the net force points vertically up (Fig. 6). The upstroke simply returns to the starting position with a zero angle of attack, which generates a negligible amount of force but also consumes a negligible amount of power.
|
If one applies this procedure to the case of symmetric stroke
(Fig. 2A), the downstroke has a
net coefficient of 1.612+1.072=1.98. The stroke
plane should be tilted by approximately tan-1(1.61/1.07)
56°, so the net force points upward. Since the upstroke contributes
almost no force, the averaged force coefficient in a complete stroke is
1.98/2=0.99. The total power is also reduced by a factor of two since the
upstroke does almost no work. Comparing this new stroke with the symmetric
one, the specific power (total power per supported weight) is reduced by a
factor of (1/2)(1.07/0.99)=0.54. Similarly, one can apply the same procedure
to the experimental case of
=50° and
=180° in Sane and
Dickinson (2001
), where
CD=3.16 and CL=1.87 and the net force
coefficient during the downstroke is 3.67. The stroke plane should be tilted
by
63°. The average force in the new stroke is almost the same as in
the original stroke, while the specific power over a period is reduced by a
factor of two. In both cases, by eliminating half of the stroke, the wing
supports about the same weight but consumes half of the power.
This conceptual example shows that a rowing-like motion can, in some cases,
be more efficient than an airfoil-like motion, which is quite the opposite to
what Weis-Fogh (1973) had
anticipated.
Concluding remarks
I hope that the collection of lessons learned here helps to bring unsteady
drag on an equal footing with unsteady lift in studies of flapping motions in
fluids. This also suggests a need for developing better theories of predicting
unsteady drag in separated flows (Pullin
and Wang, 2004) and experiments and computations to examine the
role of drag in the locomotion in fluids
(Wang, 2005
).
List of symbols
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Acknowledgments |
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