The aerodynamic effects of wingwing interaction in flapping insect wings
1 Biofuture Research Group, Department of Neurobiology, University of Ulm,
89069 Ulm, Germany
2 Department of Biology, University of Washington, Seattle, WA 98195-1800,
USA
3 California Institute of Technology, MC 138-78, Pasadena, CA 91125,
USA
* Author for correspondence (e-mail: fritz.lehmann{at}uni-ulm.de)
Accepted 8 June 2005
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Summary |
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Key words: clap-and-fling, wake capture, wingwake interaction, leading edge vortex LEV, robotic wing, digital particle image velocimetry, Drosophila
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Introduction |
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Obligate use of the clap-and-fling has only been observed in small insects
(Ellington, 1984a;
Ennos, 1989
), suggesting that
the behavior might be essential for adequate lift production in small insects
operating at low Reynolds numbers. A noteworthy exception is the fruit fly
Drosophila melanogaster, which rarely employs a clap-and-fling motion
during free flight (Ennos,
1989
; Fry et al.,
2003
) but frequently exhibits a complete clap-and-fling when flown
under tethered conditions (Vogel,
1966
; Götz,
1987
; Zanker,
1990a
; Lehmann,
1994
). Larger insects may employ the behaviour while carrying
loads (Marden, 1987
), or
performing power demanding flight turns
(Cooter and Baker, 1977
).
Ellington (1984a
) suggested
that the lacewing Chrysopa carnea uses clap-and-fling, not only for
lift augmentation, but also for steering and flight control. Marden's
experiments (Marden, 1987
) on
various insects showed that insects generate approximately 25% more
aerodynamic lift per unit flight muscle (mean, 79.2 N kg1)
when they clap-and-fling than insects using conventional wing kinematics
(mean, 59.4 N kg1), although these values were based on
solely on an estimate of induced power requirements for flight. Tethered
flying Drosophila exhibit a bilateral asymmetry in wing motion during
the clap-and-fling during presentation of optomotor stimuli
(Götz, 1987
;
Zanker, 1990b
;
Lehmann, 1994
). Specifically,
the fly may delay the pronation of the wing on the inner side of a visually
induced turn by as much as 0.2 ms, possibly modifying the direction of flight
force (Lehmann, 1994
).
Moreover, electrophysiogical stimulation of the second basalare muscle (M.b2)
during flight has indicated that the delay in wing rotation during open-loop
optomotor flight condition is due to a change in angle of attack rather than
to a change in rotational speed (Lehmann,
1994
).
While the research cited above documented the use of the clap-and-fling,
other studies explored the underlying fluid mechanics of the behavior.
Weis-Fogh suggested that the function of the fling is to strengthen the
development of circulation at the beginning of the downstroke
(Weis-Fogh, 1973; see also
Lighthill, 1973
). Such
enhancement could come about by two, non-exclusive, mechanisms. First, a
low-pressure region between the two wings might initially draw fluid over the
dorsal surface of each wing, thus enhancing circulation at the onset of the
downstroke. Second, as the wings approach each other closely, the net
circulation around them drops to zero due to mutual annihilation of equal and
opposite circulation around each wing. During the following fling, the close
proximity of the trailing edges inhibits the creation of the starting
vortices, because the conservation of zero circulation for the entire system
(as required by Kelvin's Law) is fulfilled by the equal and opposite
circulation of the two wings as they move apart. The diminished strength of
starting vortices would minimize the retarded development of circulation
following an impulsive start due to the Wagner effect
(Wagner, 1925
). Lighthill
(1973
) pointed out that this
phenomenon could operate within an inviscid fluid.
Many researchers have studied the fling using analytical methods
(Lighthill, 1973;
Ellington, 1975
;
Edwards and Cheng, 1982
) and
physical models (Bennett, 1977
;
Maxworthy, 1979
;
Spedding and Maxworthy, 1986
;
Sunada et al., 1993
). However,
the effect of clap has received less attention. Based on tethered flight
kinematics in Drosophila, Götz
(1987
) argued that the fluid
ejected from the closing gap at the end of the clap produces a momentum jet
that augments force production (see also
Ellington, 1984b
). More
recently, numerical simulations on the entire clap-and-fling sequence in both
three dimensions (Sun and Yu,
2003
) and in two dimensions across a wide range of Reynolds
numbers (Miller and Peskin,
2004
,
2005
) show that clap is likely
to enhance lift. However, these conclusions have not been verified using
physical measurements.
In this paper, we explore the fluid dynamic mechanisms underlying clap-and-fling in greater detail using a dynamically scaled two-winged flapper, which enables us to measure the time course of forces throughout the entire stroke, while simultaneously visualizing the resulting flow fields using digital particle image velocimetry. By systemically varying the angular distance between the two wings at the dorsal stroke reversal, we evaluate the aerodynamic effect of wingwing interaction during clap-and-fling. Further, by varying the kinematics of the stroke, we examine how these effects were influenced by subtle changes in wing motion.
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Materials and methods |
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Kinematics
The prescribed kinematic patterns were constructed using custom MATLAB
routines in which various aspects of wing motion could be modified. All
experiments were conducted using stroke kinematics of approximately 160°
in stroke amplitude. Depending on the experiment, we varied: (1) the distance
between the wings, (2) the Reynolds number (by changing the flapping
frequency), or (3) rotational timing and angular velocity at the ventral and
dorsal stroke reversal (for a description of kinematic angles, see
Sane and Dickinson, 2001). In
experiments investigating the effect of angular separation between the wings,
we used a stroke pattern such that the mean geometric angle of attack at mid
half stroke was 45°. In these patterns, wing rotation was symmetric about
stroke reversal, with 4% of the wing rotation occurring before and 4% after
stroke reversal. The wing kinematics of a tethered flying Drosophila
were derived by slightly smoothing kinematic data published elsewhere
(Zanker, 1990a
). Throughout
the paper we use the term `angular distance' as the angle
between the
wing's spanwise or longitudinal axis and the mid plane between both robotic
wings during clap-and-fling (defined in
Lighthill, 1973
; horizontal
plane in Fig. 1). In contrast,
the term `total angular separation' between the wings is the total angle
between the spanwise axes of both wings and thus twice the angular distance
(2
). In our robotic model, angular distance depends on two factors: the
angular excursion of the wing in the stroke plane during dorsal wing rotation,
and the distance between the two wing hinges. This is important because in one
set of experiments, we added spacers between each wing and the corresponding
wing hinge in order to minimize the gap between both wings during clap
conditions. To simplify the comparison between the two sets of experiments
(with and without spacers), we ignored the changes in
due to the
spacers and plotted all results against
derived from experiments in
which the wings were mounted without spacer in line with the rotational axis
of the mechanical wing hinge. In general, there is no `natural' or `standard'
clap-and-fling kinematic behavior for experimental modeling. Instead, the term
clap-and-fling should be understood to be a whole set of kinematic patterns
within a broad parameter space. This also includes modifications of the
clap-and-fling due to wing elasticity (`peel') and fling motion following
near-clap conditions.
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Digital particle image velocimetry
To visualize wake structure during clap-and-fling, the oil was seeded with
bubbles by pumping room air through a ceramic water purification filter. The
seeding consisted of approximately evenly sized bubbles at high concentration.
After the bubbles had been generated, we waited for about 1 min to allow the
larger bubbles to rise, thus minimizing the chances of getting spurious
vertical flow due to rising bubbles. The smaller bubbles that remained in the
fluid took approximately 1520 min to rise to the top of the tank
(Maybury and Lehmann, 2004).
This small upward velocity was not detectable using the time delay settings of
our measurements. To perform digital particle image velocimetry (DPIV) we used
a TSI dual mini-Nd:YAG laser to create two identically positioned light sheets
through the wing at five equally spaced distances from the wing base (wing
base, 33%, 50% and 75% distance between wing base and tip, and a layer close
to the wing tip) and an additional layer at the center of wing area at
approximately 65% wing length (distance at which mean force vector acts on the
Drosophila shaped wing; Birch and
Dickinson, 2003
). The paired images of approximately 185x185
mm2 were captured at 12 different phases of the stroke cycle.
Non-dimensional time within the stroke cycle,
, is normalized as
=t/T, where
t is the time from the start of the downstroke and T is the
stroke period. For convenience, when discussing the clap-and-fling, we will
express non-dimensional time after the start of the downstroke as
+1 (e.g.
=0.01=1.01). Twelve sets of chordwise
images were taken from the start of the clap to the end of the fling
(
=0.94, 0.97, 0.99, 1.00, 1.01, 1.02,
1.03, 1.05, 1.08. 1.10, 1.13 and 1.15). DPIV analysis on the velocity fields,
including calculation of vorticity, was performed using TSI Insight v5.1 and
TSI macros in Tecplot v9.0.
Translational force coefficients
We derived mean lift and drag coefficients,
L and
D, respectively, for wing
motion from mean lift and drag averaged throughout the entire stroke cycle
using:
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Fling coefficient
Weis-Fogh's kinematic two-dimensional (2-D) simplification of the
clap-and-fling, based on his observations of Encarsia Formosa
(Weis-Fogh, 1974), modeled the
wings as rigid plates rotating around their trailing edges up to a separation
angle of approximately 60° prior to the start of downstroke translation.
In the present study, the Plexiglas wings are also essentially rigid and thus
a blade-element version of Weis-Fogh's analytical model also applies to our
experimental evaluation. According to 2-D inviscid flow theory, the fling
induces circulation around each chordwise wing segment that depends on chord
width, c, the angular rate of change of incidence of a single wing,
, and a fling coefficient g(
) that describes the
angle of the v-shaped wedge to which the wings open before they begin
translating apart. If rotational motion is identical in both wings, the fling
coefficient can be estimated experimentally from measured circulation as
(Lighthill, 1973
):
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Results |
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Due to the alignment of the wing hinges and the rigidity of the robotic
wings, the generic kinematic pattern that we used did not allow a full clap,
in which both wings physically touch along their entire surface
(Fig. 1AD), so the wings
are not parallel during the clap and the wing bases are farther apart than the
wing tips (left insert, Fig. 2,
inter-wing base distance 4.3 cm). To investigate the potential
significance of a wing clap with wings oriented parallel to each other, we
added plastic spacers between the wing hinge and wing base to minimize the gap
between the wings during the clap (right inset,
Fig. 2, inter-wing base
distance
0.5 cm). To highlight the consequences of alterations in wing
gap for lift production in the robotic model,
Fig. 2 shows mean lift
coefficient augmentation for wings with and without the spacers in place,
plotted as functions of angular wing distance. In both cases, the lift
coefficient steadily decreases with increasing wing separation (exponential
decay fit; blue: y=1.0+0.038ex/5.5,
r2=0.98,
2/DF=1.4x105, N=35;
red: y=0.99+0.12ex/6.29,
r2=0.93,
2/DF=7.7x105, N=25).
The presence of spacers permits a parallel alignment during the clap, and
further enhances lift by 3.9% (from 9.4% to 13.3%) relative to the one-wing
case.
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To evaluate the relative contribution of each lift peak in Fig. 4 with respect to total lift enhancement, we measured the area under each peak. The required width of each peak was defined as the time between two successive zero crossings of the lift enhancement trace that enclose the peak. The combined effect of the fling (positive peaks III and V and negative peak IV) is an increase in mean lift and total force relative to the one-wing case of 9.6% and 9.2%, respectively, whereas the effect of the clap (negative peak I) was to decrease lift and total force by 4.3% and 2.5%, respectively. Thus, the net effect of the clap-and-fling is only 5.3% for lift and 6.7% for total force. In addition to these effects at the dorsal stroke reversal, however, we also found, quite unexpectedly, a change in force during the early upstroke (peak VI, Fig. 4). As also evident from the orientation of the force vectors in Fig. 1B,D, the total force production decreases whereas lift increases. Although the alteration in lift due to peak VI appears small, it amounts to an enhancement of 4.1%, and almost completely counterbalances the loss in lift production at the end of the upstroke preceding the clap (negative peak I, Fig. 4).
Velocity fields and leading edge vorticity
To investigate the aerodynamic mechanisms underlying the force and lift
enhancements during the fling and the force attenuation during the clap phase,
we conducted 2-D DPIV. We performed these measurements in five different
sections perpendicular to the wing axes at 12 different times during dorsal
stroke reversal (Figs 4B and
5) and in a section at the
center of wing area. The chordwise flow at the center of wing area at
approximately 65% wing length is shown in
Fig. 5. In agreement with
previous studies (Birch and Dickinson,
2001), we chose to quantify the flow velocity by averaging the
values in a small region of the DPIV images close at the trailing edge of each
wing, at a section of the wing at 65% wing length (grey areas,
Fig. 6A) from
=0.97 to 1.08
(Fig. 5CJ,OV).
Due to the arbitrary choice of the area under investigation and the vorticity
distribution involved, the results and data shown in
Fig. 6 must be interpreted with
some caution (see next paragraph for the limits of DPIV analysis). Despite
these potential errors, it is possible to compare the velocity fields at the
trailing edge of one wing vs two wings because the investigation area
is identical in both cases. The quantitative analysis shows that the average
magnitude of the fluid velocity field in an investigation area ventral to the
trailing edge in the one-wing case is slightly higher (18.8±3.9 cm
s1, mean ± S.D., N=8
different times from
=0.97 to
=1.08) than the fluid velocities in
the same region in the two-wing case (18.0±3.6 cm s1,
Fig. 6B). Neither the mean
velocity estimates nor mean temporal change in fluid velocity (mean
acceleration for one wing=0.11±0.51 m s2, two
wings=0.14±0.46 m s2, N=7 different times
from
=0.97 to
=1.08) in the investigation area
differ significantly between the one-wing vs two-wing flapping
condition (t-test on means, P>0.05).
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|
Previous analyses of the clap-and-fling maneuver predicted that the leading
edge vortices are stronger, due to an influx of fluid as the wings fling apart
at the onset of the downstroke. To quantify the circulation of these vortices,
we calculated the spatial distribution of vorticity in all five layers. The
vorticity distributions of the fluid layer at the wing's center of area (layer
at approximately 65% wing length) are shown as a time sequence during dorsal
stroke reversal (Fig. 7). The
LEVs formed in the two-wing case appear stronger than those generated by an
isolated wing (Fig.
7FJ,RV). To derive a quantitative description of
leading edge vorticity during clap-and-fling we estimated the magnitude of
leading edge vorticity in regions enclosing the LEV. Because of the changing
flow pattern and growth of the LEV, we had to vary the size and shape of these
regions, as indicated by the white areas in
Fig. 7. For example, because of
the stroke reversal, the region of interest is on the ventral side of the wing
during the upstroke and on the dorsal side during the downstroke. Moreover,
the proximity of the two wings during stroke reversal forced us to define much
smaller regions of interest during the initial phase of the fling compared to
regions at the end of the downstroke (clap phase) and late fling wing motion.
The changing region size implies large uncertainties for estimating leading
edge vorticity because the total circulation of a region will depend
critically on region size, and most DPIV studies address this by increasing
the region size until the circulation estimates asymptotes. Alternatively, we
also estimated LEV circulation by a line integral using the automatic
streamtrace tracking tool in Insight (TSI) to get the vortex contour and thus
to separate the vortex from the surrounding fluid. However, because of the
complex vortex shape and proximity of vortices with opposite spins, in many
cases the streamlines did not enclose the vortex core. For this reason we
decided to simplify our analysis and use the investigation areas as shown in
Fig. 7. A detailed and
excellent discussion about the limits of DPIV analysis for deriving forces
from fluid velocities and their derivatives is given by Noca et al.
(1999). More recently,
Spedding (2003
) suggested a
model in which drag coefficients of rectangular wings can be calculated from
wake momentum measured in birds flying in a wind tunnel
(Rosen et al., 2004
; Spedding
et al.,
2003a
,b
).
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Average circulation of the LEV for each sequence of the clap-and-fling
maneuver is plotted for the fluid layer close to the wing base
(Fig. 8A), at one-third
distance to the wing base (33%R,
Fig. 8B), at 50%R
(Fig. 8C), at three-quarter
distance (75%R, Fig.
8D) and close to the wing tip
(Fig. 8E). Theaveraged values
for all five layers are shown in Fig.
8F. The results show that circulation due to leading edge
vorticity is slightly higher, by up to 57 cm2 s1,
at the end of the upstroke in the two-wing case
(Fig. 8F). However, this result
is not intuitive with respect to the force and lift measurements during wing
motion preceding the clap and thus cannot explain why total force is
attenuated at this time in the stroke cycle (peak I,
Fig. 4A). During the clap
phase, LEV circulation collapses to zero. Subsequently, LEV circulation builds
up strongly during the fling phase in the two-wing case, reaching a magnitude
of approximately 196 cm2 s1 at
=1.08 fractions after the clap. The
magnitude of LEV circulation at this stroke phase is similar to the maximum
value of 207 cm2 s1 measured at the end of the
upstroke (
=0.97,
Fig. 8F). In contrast, in the
single-wing case, circulation of the LEV after clap reaches a maximum of 133
cm2 s1 at
=1.08, which is 28% less than the
value prior to the clap, corresponding to a difference of approximately 52
cm2 s1 (from 185 cm2
s1 to 133 cm2 s1;
Fig. 8F). The maximum
difference in LEV circulation during the fling phase between the one-wing and
two-wing cases was 63.0 cm2 s1, or 32% of the
two-wing case, measured at
=1.08.
Quite similar to the differences in LEV circulation between one- and
two-wing flapping, absolute peak vorticity of LEV increases during the fling
phase when flapping the image wing (Fig.
9). We found that LEV peak vorticity, averaged over all five
spanwise DPIV layers, increases up to approximately 12 s1 or
50% when flapping both wings, compared to its value for a single wing
(Fig. 9B). The maximum increase
in LEV peak vorticity was obtained at
=1.08 fractions after the clap and at
approximately two-thirds wing length: at 65%R (center of wing area;
peak vorticity of one wing=25.6 s1; two wings=50.7
s1; difference=25.1 s1) and 75%R
(peak vorticity of one wing=23.2 s1; two wings=50.2
s1; difference= 27.0 s1). These results
are consistent with the delayed increase in LEV circulation, as outlined in
the previous paragraph. Although neither the circulation estimates nor the
peak vorticity values of LEV are sufficient to derive aerodynamic forces from
the flow measurements using analytical models, the results show qualitatively
the enormous influence of the image wing on LEV strength and circulation
during fling motion.
Fling coefficients
The flow fields in Fig. 7
were also used to test the theoretical predictions of the strength of the LEV
during the fling maneuver. We calculated the rotational coefficient
g() from DPIV using the data shown in
Fig. 8, which depends on the
angle of wedge between the wings (total angular separation) prior to
translation. There is a chance that the high force values during
clap-and-fling might have slightly changed the programmed kinematics of the
wings because of some play between the gears of the robotic model. For this
reason, we directly measured the geometrical angle of attack of the rotating
wings from their location in the DPIV images. The mean angular velocity during
the fling, based on these measurements, was 74.1 deg. s1
(Fig. 10A).
Fig. 10B shows the function
g(
) according to Eq. 2, using the values of LEV
circulation derived from the measured flow data shown in
Fig. 8 for each of four
different chordwise sections (33%, 50%, 75% and 100%R). The chord
length for these sections was 0.094, 0.094, 0.11 and 0.11 m, respectively. The
data show that in the 50% section, g(
) increases with
total gap angle up to a value of 3.5 at an opening angle of approximately
73°, then decreases slightly as the gap approaches 90°. This result is
consistent with prior flow measurements from a single wing moving at similar
Reynolds number, which indicates that the chordwise circulation is maximum
close to a spanwise location of 65%R
(Birch and Dickinson, 2001
;
Ramamurti and Sandberg, 2001
).
At gap openings below 30°, g(
) remains less than
0.3 in all sections. The increase of g(
) with
increasing gap angle highlights the difference between the 3-D experimental
results obtained in a real fluid and Lighthill
(1973
)'s 2-D inviscid-flow
model, which reaches maximum g(
) at the beginning of
the fling phase (blue line, Fig.
10C). The 2-D inviscid model provides a reasonable match with the
3-D experimental data at gap angles above 73°
(Fig. 10C).
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Dependencies of clap-and-fling force augmentation
To investigate the dependence of clap-and-fling force augmentation on
subtle or gross changes in stroke kinematics and local flow conditions, we
systemically varied wing motion and Reynolds number using a generic stroke
pattern (cf. Materials and methods). Fig.
11A shows that augmentation of the mean force coefficient
increases approximately threefold, from about 3% to 9%, with increasing
duration of dorsal wing rotation (linear regression,
y=1.04+8.78x104x,
r2=0.49, N=28, P<0.001). However,
although the linear regression is significant, the data show a decrease in
lift augmentation for flip durations greater than 40% of the stroke durations
suggesting that the contribution of clap-and-fling circulation to total force
coefficient peaks at intermediate rotational velocities. By varying stroke
frequency from 48.8 to 191 mHz, we could change the Reynolds number from
approximately 47 to 186. The results indicate that even small changes in
Reynolds number produce a significant effect on the mean force coefficient
(Fig. 11B). With increasing
Reynolds number, augmentation of the mean force coefficient decreases
significantly from approximately 112 to 106% of the one wing value (linear
regression, y=1.122.87x104x,
r2=0.58, N=15, P<0.001). This increase in
the effectiveness of the clap-and-fling with decreasing Reynolds number is
consistent with a recent 2-D simulation based on the immersed-boundary layer
method of Miller and Peskin
(2004). Most of this effect,
however, appears to be due to drag because
Fig. 11C shows that the
augmentation of the lift coefficient tends to decrease with decreasing
Reynolds number (linear regression,
y=1.02+3.02x104x,
r2=0.32, N=15, P=0.25).
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Discussion |
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Previous models
Most analyses of Weis-Fogh's clap-and-fling mechanism have largely focused
on the fling phase and several studies clearly show that wingwing
interaction produces higher lift at the onset of downstroke. However, given
the different experimental conditions and various assumptions of the
analytical and experimental models (viscous vs. inviscid conditions
and two- vs three-dimensional models), any comparison of our
conceptual model with the idealized initial models by Weis-Fogh
(1973), Lighthill
(1973
) and others should be
done with care. In particular, axial flow from wing base to tip during
three-dimensional wing translation may alter LEV shape and stability
(Ellington et al., 1996
;
Usherwood and Ellington,
2002
). Weis-Fogh's and Lighthill's original framework of the
clap-and-fling was primarily two-dimensional with trailing wing edges
completely connected. Because of this assumption neither author considered
trailing edge vorticity in their studies on the clap-and-fling (see
peel, below). The initial description of Weis-Fogh's and Lighthill's
leading-edge separation bubble was later replaced by Edwards and Chen (1982),
who introduced the concept of leading edge vorticity. In his two-dimensional
analytical model of the clap-and-fling, Lighthill
(1973
) assumed an inviscid
fluid. For this reason, circulation around a single wing is infinite at the
initial fling motion because a calculation without vortex separation does not
permit the accumulation of vorticity in the fluid. At small gap angles, the
function g(
) in
Fig. 10C (blue) thus starts at
infinity. Moreover, Lighthill pointed out the modifications of clap-and-fling
due to viscous effects including vortex shedding, but greatly underestimated
the magnitude of the accumulation of circulation in the separation vortex.
Maxworthy (1979
) and Spedding
and Maxworthy (1986
)
demonstrated this later for a two-dimensional system. Moreover, Maxworthy also
conducted three-dimensional experiments at two vastly different Reynolds
numbers of 32 and 13 000, and experimentally compared the dynamics of
circulation produced in the opening wing cleft at different relative fluid
viscosities. Bennett's experiments
(Bennett, 1977
) on the
clap-and-fling at Reynolds number of 83 000 are slightly different because he
used only one model wing with a vertical wall that mirrors the effect of the
second moving airfoil. This assumption works well for the inviscid case at
high Reynolds numbers because the wall gives an exact plane of symmetry. For
the viscous case at moderate and low Reynolds numbers, however, the
development of unsteady viscous boundary layers over the vertical plane most
likely destroys three-dimensional vortex structures that potentially alter
force production. Bennett reported that the model wing starting impulsively
with fling motion generates 15% more lift in the presence of the vertical wall
as compared to the control single wing. Using spacers to minimize the distance
between our two model wings, we obtained a similar enhancement of about 13% at
a moderate Reynolds number of 134 and three-dimensional flapping conditions
(Fig. 2), although Bennett
(1977
) explicitly predicted
that the effect of clap-and-fling would be negligible for small insects. Even
more important, in contrast to our fruit fly model, which performs a full
cycle motion involving both clap-and-fling and a ventral reversal, the model
wing employed in all previous experiments
(Bennett, 1977
;
Maxworthy, 1979
;
Spedding and Maxworthy, 1986
)
did not simulate the fling-preceding clap and thus ignored its deleterious
effects.
Wake history and force attenuation during clap phase
The results reported here measured the time course of force variation
between the two-wing vs one-wing case, and also examined the
influence of the wake history of previous strokes on the time course of force
production by clap-and-fling. These experiments also revealed that the force
generated at the end of the upstroke is slightly attenuated in the presence of
an image wing (negative peak I, Fig.
4). It is worth noting that the absence of vertical body motion in
our experiments matches recent findings on vertical body oscillations during
free flight hovering in Drosophila using high speed video analysis
(Fry et al., 2003). As the
wings start the clap, the presence of an image wing significantly diminishes
the ability of each wing to accelerate fluid and thus generate aerodynamic
forces. Alternatively stated, the presence of an image wing creates a
circulation of opposite sense very close to the wing, thus diminishing its
ability to build up circulation. This situation is analogous to the
interaction between the starting and bound vortex of an impulsively started
wing predicted by Wagner
(1925
). If correct, this
hypothesis predicts that due to decreased circulation, less trailing edge
vorticity is shed by the wing in the two-wing case than in the one-wing case.
Evidence for this hypothesis is provided by a comparison of the peak trailing
edge vorticity as it leaves the wing in
Fig. 7BD,NP (absolute peak vorticity in grey area of
Fig. 6A, time slice
=0.97: one wing=1.13
s1; two wings= 0.90 s1). The DPIV images
suggest that trailing edge vorticity, and thus the bound circulation at
<0.99, are considerably stronger
for the single-wing case than for the two-wing case at an identical time
interval. As the wing rotates in preparation for clap, and the stopping vortex
is shed, the leading edge vorticity builds up and total force begins to rise
due an increase in bound circulation (Fig.
7BE,NQ). Although in large part this build-up
results from rotational circulation, the presence of an image wing enhances
the leading edge vorticity generated per unit time, suggesting some
modification of circulation during clap.
At the instant of clap, there is a slightly higher force in the two-wing
case than the single-wing case (Fig.
4B, time slice =1.0).
Ellington (1984b
) and also
Götz (1987
) suggested
that a modest increase is likely due to a jet of fluid squeezed out of the
closing gap between the trailing edge during the later portion of the clap
phase. Brodsky (1991
), using
smoke streams, also suggested a jet mechanism during wing pronation in the
peacock butterfly Inachis. As evident from
Fig. 5O,P, there is indeed a
small downward jet of fluid, which may account for this modest increase in
force in the final phase of clap. In Fig.
12 we estimated the velocity of this fluid jet during two wing
flapping conditions. The data suggest that at the end of clap motion (time
slice
=1.0), the velocity of the
fluid leaving the gap reaches up to 15 cm s1, corresponding
to approximately 50% of the maximum velocity we measured during the entire
sequence of clap-and-fling wing motion. For comparison, a recently published
2-D simulation of the clap-and-fling at Reynolds number of 17 by Sun and Yu
(2003
) suggests that the wings
generate consistently higher forces for the two-wing case than the one-wing
case, rather than an initial decrement (clap attenuation, peak I) followed by
a modest rise as seen in the experiments performed here. At present, it is not
clear if this discrepancy results from the differences in Reynolds number or
the two-dimensionality of the simulations in Sun and Yu
(2003
).
|
Force enhancement during fling
The time traces following the fling reveal a complex pattern of force
augmentation and attenuation. In particular, lift decreases below the single
wing case approximately 0.1 fractions of the stroke cycle after the fling
(negative peak IV, Fig. 4)
followed by a sharp increment (positive peak V,
Fig. 4). This fluctuation
arises from a delayed rise in peak forces in the two-wing case. In the initial
part of the fling, the one-wing case reaches a peak slightly earlier than in
the two-wing case. For a short duration after the fling, the total force for
both the two-wing and one-wing cases rises at roughly the same rate, but peaks
earlier in the single-wing case than the two-wing case. For the fling, this
peak continues to rise to a higher value even as the single wing force begins
to fall, resulting in the sharp fling-related peak centered around
=1.08
(Fig. 4B). Following these
early events, the single wing force falls to a slightly higher value than the
fling case, resulting in a modest negative peak IV, between peaks III and V.
Whereas the net forces slowly reach steady values in both cases, in the case
of fling they attain a steady value earlier, leading to the second peak (peak
V).
The corresponding events are visualized using DPIV in Figs 5 and 7. Although the flows were imaged at five different sections along the wing, in addition, we chose a section through the center of the wing area most representative of the total force on the wing (65%R). As in case of the clap, these images reveal substantial differences between the present and previous studies on this topic. As the wings fling apart, there is a rapid build-up of leading edge vorticity (Fig. 7EL,QX). In this initial stage (Fig. 7FH,RT), two prominent effects appear to influence the circulation around each wing. First, the presence of counter-circulation on the image wing appears to inhibit the initial rise in circulation (Fig. 7FG,RT). Second, as the wings fling apart further, the influx of fluid from above into the low pressure region between the wings induces a strong leading edge vorticity, causing a sharp rise in force. The possible reasons for the inhibition of the early growth of LEV might be the following: first, unlike in Weis-Fogh's original model that was derived from the wing kinematics of the small wasp Encarsia formosa, wing separation in our robotic model allowed trailing edge vortices to form at the early fling phase (Fig. 7T). The presence of trailing edge vorticity may inhibit the development of leading edge vorticity because fluid is accelerated in the opening cleft from below, potentially interacting with the development of LEV and creating an upward momentum.
Effects of wingwing proximity
Some past studies on insect flight have described variations of
clap-and-fling such as `near clap-and-fling'
(Ellington, 1984a;
Ennos, 1989
) and the
clap-and-`peel' (Götz,
1987
). The relative contributions of these various mechanisms to
aerodynamic forces are functions of the proximity of the two wings as well as
their independent patterns of motion. Our work addressed these effects in two
ways. First, to estimate the effect of near-clap conditions, we varied the
angular distance
between the wings at the start of the dorsal reversal
from 4.95° to 10° (total angular separation= 9.9 to
20°, Figs 1 and
2). At a fixed distance between
the wing hinges, the tips of the two wings nearly touch each other at
=4.95°, whereas at 0° the wings are exactly parallel at
stroke reversal, but their tips are at a greater distance from each other.
Under these conditions, the effects of wingwing interactions are quite
evident (Fig. 1). At a starting
angle
of approximately 10°, the difference in mean force between
one and two wings is zero, suggesting that at these large spatial separations,
the wings do not influence the development of circulation on each other. The
total angular separation required for significant wingwing interaction
must vary with Reynolds number, which determines the degree to which the wings
will develop their own starting vortex. However, from the results shown here,
we can conclude that for insects flying in a Reynolds number range comparable
to our Drosophila-based model, wing kinematics with total angular
separation (2
) of less than 20° at dorsal stroke reversal is
sufficient to result in a significant wingwing interaction.
Second, to examine the effects of the clap involving a close juxtaposition
of the entire area of the wings, we attached spacers to the wings to allow
them to interact more closely (Fig.
2). These modifications enhance lift modestly by an additional
3.9% of the performance of a single wing, suggesting that a closer
juxtaposition of the entire area might have potential benefits towards force
generation. However, since the spacers also displaced the wing surface from
the rotational axis of the robot's gear box, some of the enhancement might be
due to the changes in the overall performance of the two wings. Because of the
distance between the left and right wing hinge in both the robot and in
insects, a close juxtaposition of the entire area of the wings can only be
achieved when the wings bend at the wing root, which was shown by high-speed
photographs in tethered flying Drosophila
(Götz, 1987
;
Lehmann, 1994
). Because the
model wings used in these experiments are rigid, they only approximate a full
clap by allowing a greater interaction between the area of the two wings and
do not exactly replicate it. Thus it is likely that the elasticity of the
wings allows the animal to increase the size of wing contact area during a
full clap (time of stroke reversal) in order to enhance the efficacy of LEV
induction during fling motion.
Upward momentum during fling and a new explanation for the peel
In contrast to the rectangular shape of model wings used in previous
studies on the clap-and-fling (Bennett,
1977; Maxworthy,
1979
; Spedding and Maxworthy,
1986
; Sunada et al.,
1993
), most insect wings such as Drosophila are more or
less oval in shape. As a result, the clap-and-fling motion in rectangular
wings occurs with its trailing edges closely aligned, and the trailing edge
region remains impervious to fluid throughout the entire motion of the fling.
In contrast, the oval wings in our experiments were only close in a small
region at the tip of the wing during fling, and fluid is admitted into the
region between the wings from all directions. In other words: in oval wings,
it is impossible to have full trailing wing contact (as assumed in Weis-Fogh's
and Lighthill's original models) during fling unless the elasticity of the
wings allows the wings to bend along their chords. This kinematic maneuver is
termed peel, and has been described in many larger insects such as
various species of butterflies, bush cricket, mantis and locust
(Dalton, 1975
;
Cooter and Baker, 1977
;
Ellington, 1984a
; Brackenbury,
1990
,
1991a
,b
;
Brodsky, 1991
). The peel
kinematics was originally suggested as a mechanism to accelerate the fluid
into the cleft more smoothly (Weis-Fogh,
1973
; Ellington,
1984b
). In rigid oval wings, the opening between the trailing wing
edges allows fluid to enter the cleft from the rear while lowering lift
production due to the fluid's momentum (see direction of fluid vectors in
Fig. 5U, theincrease in
vorticity in the rear cleft in Fig.
7U, and Fig. 12).
The opening might also allow trailing edge vorticity to shed into the wake,
similar to the shedding of the starting vortex in the single-flapping wing
case (Fig.
7IL,UX). Thus the possible explanation of the sudden
decrease in force/lift augmentation (peak IV,
Fig. 4) is that the generation
of strong trailing edge vorticity close to the wing's surface lowers the
benefit of leading edge circulation. The peel offers a solution to
this problem because it may function as a mechanism to prevent fluid being
sucked into the cleft from the rear. This would increase lift production
associated with the clap-and-fling maneuver by preventing the generation of an
upward-directed momentum jet.
Wake capture force
In addition to far-field effects, clap-and-fling also influences near-field
phenomena such as wingwake interactions generated by flapping wings
immediately following stroke reversal
(Dickinson et al., 1999). The
extraction of kinetic energy from the wake behind a freely flying insect has
been demonstrated in a study of butterflies during take-off, in which smoke
trails were used to visualize the flow around the moving wings
(Srygley and Thomas, 2002
).
Because wingwake interactions may depend on the distribution of
vorticity shed at the start of each stroke, they may be weaker when less
vorticity is shed at the end of the prior half stroke. This is similar to
increasing the strength of the inter-vortex stream produced by trailing and
leading edge vorticities (Fig.
13; Dickinson,
1994
). As seen in Fig.
7BD,NP, it seems that there is a small decrease in
the vorticity shed at the trailing edge as the wing approaches clap. Because
this change potentially lessens the velocity of the inter-vortex stream
(Fig. 5BD,NP), as
indicated by the decrease in the difference of flow velocity between both
experimental conditions (Fig.
6B), we expect a decrease in forces immediately following stroke
reversal for the two-wing case as compared with the one-wing case. This
hypothesis is borne out by the presence of negative peak II
(Fig. 4A).
|
Even so, if we assume that modification in wake capture force is
responsible for the primary peak of force enhancement during clap-and-fling
(peak III), then peak IV in the force trace should not be regarded as a
decrease in LEV induced force/lift enhancement
(Fig. 4), but rather a delay
between the wake effect and a subsequent enhanced growth of the LEV. In this
scenario, peak V would simply result from strong leading edge vorticity at
large separation angle when the wing has reached its maximum translational
velocity during the downstroke (Fig.
10B,C). It is worth noting, however, that peak V appears
relatively late in the downstroke at
=1.2, whereas fling circulation, as
indicated by g(
), has fully developed by
=1.08
(Fig. 10C). Since at
=1.1 the wing's leading edge has
moved out of the DPIV field of view, we were not able to quantify the entire
time course of LEV development during the downstroke, and thus cannot offer a
detailed explanation for the transient peak of force/lift enhancement (peak V,
Fig. 4).
Vortex symmetry and Reynolds number effect
In the conventional view, the inhibitory effect of viscosity on fluid
acceleration commonly predicts that lift forces produced during wing motion
should decrease with decreasing Reynolds number. For example, the performance
of a single Drosophila model wing flapping with a generic kinematic
pattern decreases linearly with decreasing Reynolds number (Re=
30200, 2-D in Dickinson and
Götz, 1993; 3-D in
Lehmann, 2002
), which is
consistent with recent CFD (computational fluid dynamics) predictions on the
aerodynamics of wing flapping in small insects at low Reynolds numbers
(Miller and Peskin, 2004
;
Wu and Sun, 2004
). Miller and
Peskin argue that mean lift, but not drag, produced by a single, isolated wing
decreases with decreasing Reynolds number, due to the prolonged attachment of
the trailing edge vortex, which they termed `vortex symmetry'. This
effect may be explained by noting that if the leading and trailing edge
vortices move together there is no change in the moment of vorticity and thus
no lift generated in the direction orthogonal to motion
(Wu at al., 1991
). This
symmetry deteriorates at Reynolds numbers above approximately 32. At this
point the trailing edge vortex sheds at the start of translation and the
moment of vortices changes steadily throughout motion, resulting in lift. The
same 2-D CFD model also suggests an increase in relative contribution of the
clap-and-fling to lift as Reynolds number decreases
(Miller and Peskin, 2005
).
This curious result, relative to the isolated wing case, results from the
effect of Reynolds number on the interaction between the trailing edge
vortices. At low Reynolds numbers the proximity of the trailing edges at the
onset of translation inhibits the formation of trailing edge vorticity. Thus,
at lower Reynolds number, the leading edge vortices are stronger and serve as
each other's starting vortex to maintain Kelvin's Law. The data we estimated
for total normal force enhancement due to the clap-and-fling in the
Drosophila model wing show the same trend
(Fig. 11B). At a Reynolds
numbers of approximately 50, the force augmentation due to the fling is
greater than at a Reynolds numbers of approximately 200. However, in contrast
to the results of the 2-D simulations, we found no evidence that the lift
produced during the fling increases with decreasing Reynolds number
(Fig. 11C). Such a discrepancy
might arise from 3-D effects, or the fact that the separation between the
trailing edge of the wings was greater in our experiments than in the 2-D
simulations of Miller and Peskin
(2005
).
An alternative explanation for this effect is that the relative increase in viscous forces at low Reynolds number requires higher forces to pull the wings apart during fling motion and thus drag coefficient augmentation increases with decreasing Reynolds number (Fig. 11D). However, this explanation is not inconsistent with that which emphasizes the role of Reynolds number on the formation of the leading edge vortices; both phenomena are different manifestations of the influence of viscosity. It is important to emphasize that while viscosity can subtly enhance lift generated during the fling at low Reynolds number, it increases drag to an even greater degree. Thus, the lift-to-drag ratio becomes less favorable with decreasing Reynolds number (Fig. 1G), and the clap and fling cannot be viewed as a mechanism to improve this performance parameter. This is not necessarily inconsistent with the observations that tiny insects are more dependent on the flap and fling, assuming that they are limited by lift and not power.
Conclusions
The results of this study on the dorsal clap-and-fling mechanism in
flapping wing motion of a `hovering' robotic fruit fly wing has revealed an
unknown complexity of flight force modifications throughout the entire stroke.
The main differences between our findings and previous analytical and
experimental studies are: first, the clap part of wing motion attenuates total
force and lift and does not generate even larger lift force than the initial
phase of the fling, as found in another 3-D model wing
(Maxworthy, 1979). Second,
potential modifications of wake capture and viscous forces fling might explain
some of the changes in total force and lift production during clap-and-fling.
And third, the clap-and-fling wing beat seems to distort wake structure
throughout the stroke cycle, as indicated by the unexpected peak of lift
enhancement at the beginning of the upstroke. In sum, knowledge of the fluid
mechanical mechanisms and physical constraints underlying wingwake
interactions in flapping flight may broaden our understanding of how the
different forms of wing kinematic patterns, including the clap-and-fling, have
evolved through their long evolutionary history. Given the constraints upon
circulation development and endurance during clap-and-fling in real fluids, it
seems evident that although total lift enhancement is modest, the
clap-and-fling is a useful mechanism by which insects can elevate force
production. Considering the short time over which fling-induced flight forces
act, the enhanced locomotor performance might favor steering control in some
insects performing elaborate aerial maneuvers. However, it is also important
to consider the alternative hypothesis that clap-and-fling occurs as a
concomitant byproduct of insects trying to maximize their stroke amplitude.
For a given stroke frequency, maximizing stroke amplitude increases wing
velocity. Since total flight forces are dependent on the square of wing
velocity, an average 15° increase in stroke amplitude during steering
behavior produces an increase in total lift of approximately 20% (Lehmann and
Dickinson, 1997
,
1998
). Besides its role for
total lift enhancement, a concomitant 10% lift augmentation due to
clap-and-fling beneficially counterbalances pitch moments on the animal body
produced by the increase in ventral stroke amplitude. Ultimately, this view is
supported by high-speed video observations showing that in tethered flying
Drosophila clap-and-fling wingbeat vanishes at small stroke
amplitudes (low flight force production), whereas the tethered fly permanently
uses clap-and-fling wing beat while producing elevated forces at stroke
amplitudes near the mechanical limit of the thorax of 190°
(Lehmann and Dickinson,
1998
).
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Bennett, L. (1977). Clap and fling aerodynamics an experimental evaluation. J. Exp. Biol. 69,261 -272.
Birch, J. M. and Dickinson, M. H. (2001). Spanwise flow and the attachment of the leading-edge vortex on insect wings. Nature 412,729 -733.[CrossRef][Medline]
Birch, J. M. and Dickinson, M. H. (2003). The
influence of wingwake interactions on the production of aerodynamic
forces in flapping flight. J. Exp. Biol.
206,2257
-2272.
Brackenbury, J. (1990). Wing movements in the bush cricket Tettigonia viridissima and the mantis Ameles spallanziana during natural leaping. J. Zool. Lond. 220,593 -602.
Brackenbury, J. (1991a). Kinematics of take-off and climbing flight in butterflies. J. Zool. Lond. 224,251 -270.
Brackenbury, J. (1991b). Wing kinematics during natural leaping in the mantids Mantis religiosa and Iris oratoria. J. Zool. Lond. 223,341 -356.
Brodsky, A. K. (1991). Vortex formation in the tethered flight of the peacock butterfly Inachis io L. (Lepidoptera, Nymphalidae) and some aspects of insect flight evolution. J. Exp. Biol. 161,77 -95.
Cooter, R. J. and Baker, P. S. (1977). Weis-Fogh clap and fling mechanism in Locusta. Nature 269, 53-54.[CrossRef]
Dalton, S. (1975). Borne On The Wind. New York: Reader's Digest Press.
Dickinson, M. H. (1994). The effects of wing
rotation on unsteady aerodynamic performance at low Reynolds numbers.
J. Exp. Biol. 192,179
-206.
Dickinson, M. H. and Götz, K. G. (1993).
Unsteady aerodynamic performance of model wings at low Reynolds numbers.
J. Exp. Biol. 174,45
-64.
Dickinson, M. H., Lehmann, F.-O. and Sane, S.
(1999). Wing rotation and the aerodynamic basis of insect flight.
Science 284,1954
-1960.
Edwards, R. H. and Cheng, H. K. (1982). The separation vortex in the Weis-Fogh circulation-generation mechanism. J. Fluid Mech. 120,463 -473.
Ellington, C. P. (1975). Non-steady-state aerodynamics of the flight of Encarsia formosa. In Symposium on Swimming and Flying in Nature (ed. T. Y. Wu), pp. 729-762. Passadena: California.
Ellington, C. P. (1984a). The aerodynamics of insect flight. III. Kinematics. Phil. Trans. R. Soc. Lond. B 305,41 -78.
Ellington, C. P. (1984b). The aerodynamics of insect flight. IV. Aerodynamic mechanisms. Phil. Trans. R. Soc. Lond. B 305,79 -113.
Ellington, C. P. (1984c). The aerodynamics of insect flight. VI. Lift and power requirements. Phil. Trans. R. Soc. Lond. B 305,145 -181.
Ellington, C. P. (1984d). The aerodynamics of insect flight. II. Morphological parameters. Phil. Trans. R. Soc. Lond. B 305,17 -40.
Ellington, C. P., Berg, C. V. D., 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.
Fry, S. N., Sayaman, R. and Dickinson, M. H.
(2003). The aerodynamics of free-flight maneuvers in
Drosophila. Science 300,495
-498.
Götz, K. G. (1987). Course-control, metabolism and wing interference during ultralong tethered flight in Drosophila melanogaster. J. Exp. Biol. 128, 35-46.
Lehmann, F.-O. (1994). Aerodynamische, kinematische und electrophysiologische Aspekte der Flugkrafterzeugung und Flugkraftsteuerung bei der Taufliege Drosophila melanogaster. PhD thesis, University of Tübingen, Germany.
Lehmann, F.-O. (2002). The constraints of body size on aerodynamics and energetics in flying fruit flies: an integrative view. Zoology 105,287 -295.
Lehmann, F.-O. (2004). The mechanisms of lift enhancement in insect flight. Naturwissenschaften 91,101 -122.[CrossRef][Medline]
Lehmann, F.-O. and Dickinson, M. H. (1997). The
changes in power requirements and muscle efficiency during elevated force
production in the fruit fly, Drosophila melanogaster. J. Exp.
Biol. 200,1133
-1143.
Lehmann, F.-O. and Dickinson, M. H. (1998). The
control of wing kinematics and flight forces in fruit flies
(Drosophila spp.). J. Exp. Biol.
201,385
-401.
Lighthill, M. J. (1973). On the Weis-Fogh mechanism of lift generation. J. Fluid Mech. 60, 1-17.
Marden, J. H. (1987). Maximum lift production during take-off in flying animals. J. Exp. Biol. 130,235 -258.
Maxworthy, T. (1979). Experiments on the Weis-Fogh mechanism of lift generation by insects in hovering flight Part 1. Dynamics of the `fling'. J. Fluid Mech. 93, 47-63.
Maybury, W. J. and Lehmann, F.-O. (2004). The
fluid dynamics of flight control by kinematic phase lag variation between two
robotic insect wings. J. Exp. Biol.
207,4707
-4726.
Miller, L. A. and Peskin, C. S. (2004). When
vortices stick: an aerodynamic transition in tiny insect flight. J.
Exp. Biol. 207,3073
-3088.
Miller, L. A. and Peskin, C. S. (2005). A
computational fluid dynamics of `clap and fling' in the smallest insects.
J. Exp. Biol. 208,195
-212.
Noca, F., Shiels, D. and Jeon, D. (1999). A comparison of methods for evaluating time-dependent fluid dynamic forces on bodies, using only velocity fields and their derivatives. J. Fluids Struct. 13,551 -578.[CrossRef]
Ramamurti, R. and Sandberg, W. C. (2001). Computational study of 3-D, flapping foil, flows. AIAA-2001, 605.
Rosen, M. W., Spedding, G. R. and Hedenström, A.
(2004). The relationship between wingbeat kinematics and vortex
wake of a thrush nightingale. J. Exp. Biol.
207,4255
-4268.
Sane, S. (2003). The aerodynamics of insect
flight. J. Exp. Biol.
206,4191
-4208.
Sane, S. and Dickinson, M. H. (2001). The
control of flight force by a flapping wing: lift and drag production.
J. Exp. Biol. 204,2607
-2626.
Spedding, G. R. (2003). Comparing fluid mechanics models with experimental data. Phil. Trans. R. Soc. Lond. B 358,1567 -1576.[CrossRef][Medline]
Spedding, G. R. and Maxworthy, T. (1986). The generation of circulation and lift in a rigid two-dimensional fling. J. Fluid Mech. 165,247 -272.
Spedding, G. R., Hedenström, A. and Rosen, M. W. (2003a). Quantitative studies of the wakes of freely flying birds in a low-turbulence wind tunnel. Exp. Fluids 34,291 -303.[CrossRef]
Spedding, G. R., Rosen, M. W. and Hedenström, A.
(2003b). A family of vortex wakes generated by a thrush
nightingale in free flight in a wind tunnel over its entire natural range of
flight speeds. J. Exp. Biol.
206,2313
-2344.
Srygley, R. B. and Thomas, A. L. R. (2002). Unconventional lift-generating mechanisms in free-flying butterflies. Nature 420,660 -6604.[CrossRef][Medline]
Sun, M. and Yu, X. (2003). Flows around two airfoils performing fling and subsequent translation and translation and subsequent clap. Acta Mech. Sinica 19,103 -117.
Sunada, S., Kawachi, K., Watanabe, I. and Azuma, A.
(1993). Fundamental analysis of three-dimensional `near fling'.
J. Exp. Biol. 183,217
-248.
Usherwood, J. R. and Ellington, C. P. (2002).
The aerodynamic of revolving wings. I. Model hawkmoth wings. J.
Exp. Biol. 205,1547
-1564.
Vogel, S. (1966). Flight in Drosophila. I. Flight performance of tethered flies. J. Exp. Biol. 44,567 -578.
Wagner, H. (1925). Über die Entstehung des dynamischen Auftriebes von Tragflügeln. Z. Angew. Math. Mech. 5,17 -35.
Wakeling, J. M. and Ellington, C. P. (1997).
Dragonfly flight. II. Velocities, accelerations, and kinematics of flapping
flight. J. Exp. Biol.
200,557
-582.
Weis-Fogh, T. (1973). Quick estimates of flight fitness in hovering animals, including novel mechanisms for lift production. J. Exp. Biol. 59,169 -230.
Weis-Fogh, T. (1974). In Swimming and Flying in Nature, vol. 2 (ed. Y. T. Wu, C. J. Brokaw and C. Brennen), pp. 729-762. New York: Plenum Press.
Wu, J. H. and Sun, M. (2004). Unsteady
aerodynamic forces of a flapping wing. J. Exp. Biol.
207,1137
-1150.
Wu, J. Z., Vakili, A. D. and Wu, J. M. (1991). Review of the physics of enhancing vortex lift by unsteady excitation. Prog. Aerosp. Sci. 28,73 -131.[CrossRef]
Zanker, J. M. (1990a). The wing beat of Drosophila melanogaster. I. Kinematics. Phil. Trans. R. Soc. Lond. B 327,1 -18.
Zanker, J. M. (1990b). The wing beat of Drosophila melanogaster. III. Control. Phil. Trans. R. Soc. Lond. B 327,45 -64.