The aerodynamics of revolving wings I. Model hawkmoth wings
Department of Zoology, University of Cambridge, Downing Street,
Cambridge CB2 3EJ, UK
* Present address: Concord Field Station, MCZ, Harvard University, Old Causeway
Road, Bedford, MA 01730, USA
(e-mail: jimusherwood{at}lycos.co.uk )
Accepted 21 March 2002
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Summary |
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Key words: aerodynamics, Manduca sexta, propeller, hawkmoth, model, leading-edge vortex, flight, insect, lift, drag
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Introduction |
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Experiments based on flapping models are the best way at present to investigate the unsteady and three-dimensional aspects of flapping flight. The effects of wingwing interaction, wing rotation about the supination/pronation axis, wing acceleration and interactions between the wing and the induced flow field can all be studied with such models. However, experiments with flapping models inevitably confound some or all of these variables. To investigate the properties of the leading-edge vortex over `revolving' wings, while avoiding confounding effects from wing rotation (pronation and supination) and wingwing interaction, this study is based on a propeller model. `Revolving' in this study refers to the rotation of the wings about the body, as in a propeller. The conventional use of the term `rotation' in studies of insect flight, which refers to pronation and supination, is maintained. A revolving propeller mimics, in effect, the phase of a down-(or up-) stroke between periods of wing rotation.
The unusually complete kinematic and morphological data available for the
hovering hawkmoth Manduca sexta
(Willmott and Ellington,
1997b), together with its relatively large size, have made this an
appropriate model insect for previous aerodynamic studies. This, and the
potential for comparisons with computational
(Liu et al., 1998
) and
mechanical flapping models, both published and current, make Willmott and
Ellington's (1997b
) hovering
hawkmoth an appropriate starting point for propeller experiments.
This study assesses the influences of leading-edge detail, twist and camber
on the aerodynamics of revolving wings. The similarities between the
leading-edge vortex over flapping wings and those found over swept and delta
wings operating at high angles of incidence
(Van den Berg and Ellington,
1997b) suggest that the detail of the leading edge may be of
interest (Lowson and Riley,
1995
): the sharpness of the leading edge of delta wings is
critical in determining the relationship between force coefficients and angle
of attack. Protuberances from the leading edge are used on swept-wing aircraft
to delay or control the formation of leading-edge vortices (see
Ashill et al., 1995
;
Barnard and Philpott, 1995
).
Similar protuberances at a variety of scales exist on biological wings, from
the fine sawtooth leading-edge of dragonfly wings
(Hertel, 1966
) to the adapted
digits of birds (the alula), bats (thumbs) and some, but not all, sea-turtles
and pterosaurs. The effect of a highly disrupted leading edge is tested using
a `sawtooth' variation on the basic hawkmoth planform.
Willmott and Ellington
(1997b) observed wing twists
of 24.5° (downstroke) to 19° (upstroke) in the hovering hawkmoth F1,
creating higher angles of attack at the base than at the tip for both up- and
downstroke. Such twists are typical for a variety of flapping insects (e.g.
Jensen, 1956
;
Norberg, 1972
;
Weis-Fogh, 1973
;
Wootton, 1981
;
Ellington, 1984c
), but this is
not always the case (Vogel,
1967a
; Nachtigall,
1979
). The hawkmoth wings were also seen to be mildly cambered,
agreeing with observations for a variety of insects; see, for instance,
photographs by Dalton (1977
) or
Brackenbury (1995
). Both these
features of insect wings have been assumed to provide aerodynamic benefits
(e.g. Ellington, 1984c
) and
have been shown to be created by largely passive, but intricate, mechanical
deflections (Wootton, 1981
,
1991
,
1992
,
1993
,
1995
;
Ennos, 1988
).
Previous studies of the effects of camber have had mixed results. Camber on
conventional aircraft wings increases the maximum lift coefficients and
normally improves the lift-to-drag ratio. This is also found to be true for
locust (Jensen, 1956),
Drosophila (Vogel,
1967b
) and bumblebee (Dudley and Ellington, 1990b) wings. However,
the effects of camber on unsteady wing performance appear to be negligible
(Dickinson and Götz,
1993
).
The propeller rig described here enables the aerodynamic consequences of leading-edge vortices to be studied. It also allows the importance of various wing features, previously described by analogy with conventional aerofoil or propeller theory, to be investigated.
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Materials and methods |
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The shaft of the propeller was attached via a 64:1 spur gearbox to a 12V Escap direct-current motor/tachometer driven by a servo with tachometer feedback. The input voltage was ramped up over 0.8 s; this was a compromise between applying excessive initial forces (which may damage the torque strain gauges and which set off unwanted mechanical vibrations) and achieving a steady angular velocity as quickly as possible (over an angle of 28°). The voltage across the tachometer was sampled together with the force signals (see below) at 50 Hz. Angular velocity during the experiments was determined from the tachometer signal, so any small deviations in motor speed (e.g. due to higher torques at higher angles of attack) were accounted for.
The mean Reynolds number (Re) for a flapping wing is a somewhat
arbitrary definition (e.g. Ellington,
1984f; Van den Berg and
Ellington, 1997a
), but it appears unlikely that the hovering
hawkmoths of Willmott and Ellington
(1997a
,b
,c
)
were operating anywhere near a critical value: both larger and smaller insects
can hover in a fundamentally similar way; wing stroke amplitude, angle of
attack and stroke plane are consistent for the wide range of insects that
undertake `normal hovering' (Weis-Fogh,
1973
; Ellington,
1984c
). Because of this, and the benefits in accuracy when using
larger forces, a fairly high rotational frequency (0.192 Hz) was chosen.
Following the conventions of Ellington
(1984f
), this produces an
Re of 8071. While this is a little higher than that derived from the
data of Willmott and Ellington
(1997b
) for F1
(Re=7300), the hawkmoth selected below for a `standard' wing design,
it is certainly within the range of hovering hawkmoths.
Wing design
The wings were constructed from 500 mmx500 mmx 2.75 mm sheets
of black plastic `Fly-weight' envelope stiffener. This material consists of
two parallel, square, flat sheets sandwiching thin perpendicular lamellae that
run between the sheets for the entire length of the square. The orientation of
these lamellae results in hollow tubes of square cross section running between
the upper and lower sheets from leading to trailing edge. Together, this
structure and material produces relatively stiff, light, thin, strong wing
models.
The standard hawkmoth wing planform was derived from a female hawkmoth `F1'
described by Willmott and Ellington
(1997a,b
)
(Fig. 2A). F1 was selected as
the most representative because its aspect ratio and radii for moments of area
were closest to the average values found from previous studies
(Ellington, 1984b
;
Willmott and Ellington,
1997b
). The wing was connected to the sting on the propeller head
by a 2.4 mm diameter steel rod running down a 20 mm groove cut in the ventral
surface of the wing. The groove was covered in tape, resulting in an almost
flat surface barely protruding from the wing material. The rod also defined
the angle of attack of the wing as it was gripped by grub-screws at the sting
and bent at right angles within the wing to run internally down one of the
`tubes' formed by the lamellae. A representative zero geometric angle of
attack
was set by ensuring that the base chord of each wing was
horizontal. The rotation of each sting (about the pronation/supination axis)
could be set independently in increments of 5° using a 72-tooth
cog-and-pallet arrangement. The leading and trailing edges of the wings were
taped, producing bluff edges less than 3 mm thick. The wing thickness was less
than 1.6% of the mean chord.
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Leading-edge range
Three variations on the standard, flat, hawkmoth wing model were
constructed. `Sharp' leading edges were produced by sticking a 10 mm border of
0.13 mm brass shim to the upper surface of the leading edge of standard
hawkmoth wing models which had had 10 mm taken off the leading edges. The
converse of this, wings with `thick' leading edges, was achieved by using two
layers of the plastic wing material, resulting in wings of double thickness.
While this confounds leading-edge thickness and wing thickness, it allowed
wings to be produced that had thick leading edges without also distinct steps
in the upper or lower surface. The third design was of standard thickness and
had a `sawtooth' leading edge of 45° pitch
(Fig. 2B), with sawteeth 10 mm
deep and 10 mm long.
Twist range
Twisted wing designs were produced by introducing a second 2.4 mm diameter
steel rod, which ran down the central groove, with bends at each end running
perpendicularly down internal tubes at the wing base and near the tip. The two
ends of the rod were out of plane, thus twisting the wing, creating a lower
angle of attack at the wing tip than at the base. One wing pair had a twist of
15° between base and tip, while the second pair had a twist of 32°. No
measurable camber was given to the twisted wings.
The wing material was weakened about the longitudinal axis of the wing by alternately slicing dorsal and ventral surfaces, which destroyed the torsion box construction of the internal `tubes'. This slicing was necessary to accommodate the considerable shear experienced at the trailing and leading edges, far from the twist axis.
Camber range
Standard hawkmoth wing models were heat-moulded to apply a camber. The
wings were strapped to evenly curved steel sheet templates and placed in an
oven at 100°C for approximately 1h. The wings were then allowed to cool
overnight. The wings `uncambered' to a certain extent on removal from the
templates, but the radius of curvature remained fairly constant along the
span, and the reported cambers for the wings were measured in situ on
the propeller. For thin wings, camber can be described as the ratio of wing
depth to chord. One wing pair had a 7% camber over the basal half of the wing:
cambers were smaller at the tip because of the narrower chord. The second wing
pair had a 10% camber over the same region. The application of camber also
gave a small twist of less that 6° to the four wing models.
Wing moments
The standard wing shape used was a direct copy of the hawkmoth F1 planform
except in the case of the sawtooth leading-edge design. However, the model
wings do not revolve exactly about their bases: the attachment `sting' and
propeller head displace each base by 53.5 mm from the propeller axis. Since
the aerodynamic forces are influenced by both the wing area and its
distribution along the span, this offset must be taken into account.
Table 1 shows the relevant
wing parameters for aerodynamic analyses, following Ellington's
(1984b) conventions. The total
wing area S (for two wings) can be related to the single wing length
R and the aspect ratio
:
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Aerodynamic forces and torques are proportional to the second and third
moments of wing area, S2 and S3
respectively (Weis-Fogh,
1973). Non-dimensional radii,
2(S) and
3(S),
corresponding to these moments are given by:
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Non-dimensional values are useful as they allow differences in wing shape to be identified while controlling for wing size.
The accuracy of the wing-making and derivation of moments was checked after the experiments by photographing and analysing the standard `flat' hawkmoth wing. Differences between the expected values of S2 and S3 for the model wings and those observed after production were less than 1%.
Smoke observations
Smoke visualisation was performed independently from force measurements.
Vaporised Ondina EL oil (Shell, UK) from a laboratory-built smoke generator
was fed into a chamber of the propeller body and from there into the hollow
shaft. This provided a supply of smoke at the propeller head, even during
continuous revolution. Smoke was then delivered from the propeller head to the
groove in the ventral surface of the wing by 4.25 mm diameter Portex tubing. A
slight pressure from the smoke generator forced smoke to disperse down the
groove, down the internal wing `tubes' and out of the leading and trailing
edges of the wing wherever the tape had been removed. Observations were made
directly or via a video camera mounted directly above the propeller.
Photographs were taken using a Nikon DS-560 digital camera with 50 mm lens.
Lighting was provided by 1 kW Arri and 2.5 kW Castor spotlights. A range of
rotational speeds was used: the basic flow properties were the same for all
speeds, but a compromise speed was necessary. At high speeds, the smoke spread
too thinly to photograph, while at low speeds the smoke jetted clear of the
boundary layer and so failed to label any vortices near the wing. A wing
rotation frequency of 0.1 Hz was used for the photographs presented here.
Force measurements
Measurement of vertical force
The propeller body was clamped to a steel beam by a brass sleeve. The beam
projected horizontally, perpendicular to the propeller axis, over a steel
base-plate (Fig. 1B). The beam
(1.4 m long, 105 mm deep and 5 mm wide) rested on a knifeblade fulcrum, which
sat in a grooved steel block mounted on the base-plate. Fine adjustment of the
balance using a counterweight allowed the beam to rest gently on a steel shim
cantilever with foil strain gauges mounted on the upper and lower surfaces.
The shim was taped firmly to the beam and deflected in response to vertical
forces acting on the propeller on the other side of the fulcrum because of the
`see-saw' configuration. The strain gauges were protected from excessive
deflection by a mechanical stop at the end of the beam. Signals from these
`vertical force' strain gauges were amplified and fed into a Macintosh Quadra
650 using LabVIEW to sample at 50 Hz. The signal was calibrated using a 5 g
mass placed at the base of the propeller, directly in line with the propeller
axis. No hysteresis between application and removal of the mass was observed,
and five calibration measurements were made before and after each experiment.
The mean coefficient of variation for each group of five measurements was less
than 2 %, and there was never a significant change between calibrations before
and after each experiment.
The upper edge of the steel beam was sharpened underneath the area swept by the propeller wings to minimise aerodynamic interference. The beam was also stiffened by a diamond structure of cables, separated by a 10 mm diameter aluminium tube sited directly over the fulcrum.
Measurement of torque
The torque Q required to drive the wings was measured via
a pair of strain gauges mounted on a shim connected to the axle of the
propeller (Fig. 1, iv). The
signal from these strain gauges was pre-amplified with revolving electronics,
also attached to the shaft, before passing through electrical slip-rings
(through which the power supply also passed) machined from circuit board. The
signal was then amplified again before being passed to the computer, as with
the vertical force signal.
The torque signal was calibrated by applying a known torque: a 5 g mass hung freely from a fine cotton thread, which passed over a pulley and wrapped around the propeller head. This produced a 49.1 mN force at a distance of 44 mm from the centre of the axle and resulted in a calibration torque of 2.16 mN m. This procedure was extremely repeatable and showed no significant differences throughout the experiments. Five calibration readings were recorded before and after each experiment. The mean coefficient of variation for each group of five measurements was less than 6 %.
Torques due to friction in the bearings above the strain gauges and to aerodynamic drag other than that caused by the wings were measured by running the propeller without wings. This torque was subtracted from the measurements with wings, giving the torque due to the wing drag only. It is likely, however, that this assessment of non-aerodynamic torque is near the limit of the force transducers, and is somewhat inaccurate, because the aerodynamic drag measured for wings at zero angle of incidence was apparently slightly less than zero.
Experimental protocol
Each wing type was tested twice for a full range of angles of attack from
-20 to +95° with 5° increments and three times using an abbreviated
test, covering from -20° to +100° in 20° increments. Four runs
were recorded at each angle of attack, consisting of approximately 10 s before
the motor was turned on followed by 20 s after the propeller had started. The
starting head positions for these four runs were incremented by 90°, and
pairs of runs started at opposite positions were averaged to cancel any
imbalance in the wings. Overall, -20, 0, 20, 40, 60 and 80° had 10
independent samples each, 100° had six, and all the other angles of attack
had four.
Data processing
Once collected, the data were transferred to a 400 MHz Pentium II PC and
analysed in LabVIEW. Fig. 3A
shows a typical trace for a single run. The top (green) trace shows the
tachometer signal, with the wing stationary for the first 10 s. The middle
(blue) trace shows the torque signal: a very large transient is produced as
the torque overcomes the inertia of the wings, and the signal then settles
down. The bottom (red) trace shows the vertical force signal.
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The rise in the tachometer signal was used to identify the start of wing movement. Zero values for the force signals were defined as the means before the wings started moving; from then on, signal values were taken relative to their zero values.
Filtering
Force and torque signals were low-pass-filtered at 6 Hz using a finite
impulse response filter. Large-amplitude oscillations persisted in the
vertical force signal. These are due to the massive propeller and beam resting
on the vertical force strain gauge shim, thus producing a lightly damped
mass-spring system. A simple physical argument allows this oscillation to be
removed effectively. A moving average, taken over the period of oscillation,
consists only of the aerodynamic force and the damping force: mean inertial
and spring forces are zero over a cycle. When the damping force is negligible,
this method will yield the mean aerodynamic force with a temporal resolution
of the order of the oscillation period. This simple `boxcar' filtering
technique was tested on a signal created by the addition and removal of a
range of masses to the propeller head (Fig.
3B). The removal of the oscillation from the signal was highly
effective (Fig. 3C), and the
full change in signal was observed after a single oscillation period (0.32s)
had passed. This `step' change corresponded to the static calibration of the
vertical strain gauge, confirming that the damping force was indeed
negligible. The longer-period oscillation visible in the vertical force signal
trace (Fig. 3A) is due to a
slight difference in mass between the wings. The effect of this imbalance is
cancelled by averaging runs started in opposite positions.
A similar filtering technique was used on the torque signal. Unlike the vertical force signal, however, several modes of vibration were observed. A large filter window size (1.28s) was needed to remove the dominant mode, but resulted in a poorer temporal resolution (equivalent to approximately a quarter-revolution).
Pooling the data into `early' and `steady' classes
The filtered data for each angle of attack was pooled into `early' or
`steady' classes. `Early' results were averaged force coefficients relating to
the first half-revolution of the propeller, between 60 and 120° from the
start of revolution, 1.5-3.1 chord-lengths of travel of the middle of the
wing. This excluded the initial transients and ensured that the large filter
window for the torque signal did not include any data beyond 180°. A
priori assumptions were not made about the time course for development of
the propeller wake, so force results from between 180 and 450° from the
start of revolution were averaged and form the `steady' class. The large angle
over which `steady' results were averaged and the relative constancy of the
signal for many revolutions (Fig.
4) suggest that the `steady' results are close to those that would
be found for propellers that have achieved steady-state revolution, with a
fully developed wake. However, it should be noted that brief high (or low),
dynamic and biologically significant forces, particularly during very early
stages of revolution, are not identifiable with the `early' pooling
technique.
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Coefficients
Conversion into `propeller coefficients'
Calibrations before and after each experiment were pooled and used to
convert the respective voltages to vertical forces (N) and torques (N m).
`Propeller coefficients' analogous to the familiar lift and drag coefficients
will be used for a dimensionless expression of vertical and horizontal forces
Fv and Fh, respectively: lift and drag
coefficients are not used directly because they must be related to the
direction of the oncoming air (see below).
The vertical force on an object, equivalent to lift if the incident air is
stationary is given by:
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The `mean coefficients' method of blade-element analysis (first applied to
flapping flight by Osborne,
1951) supposes that a single mean coefficient can represent the
forces on revolving and flapping wings. So, the form of equation 4 appropriate
for revolving wings is:
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![]() | (8) |
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In this case the term in parentheses is the third moment of wing area
S3 for both wings. The mean horizontal force coefficient
Ch is given by:
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Conversion into conventional profile drag and lift coefficients
If the motion of air about the propeller wings can be calculated, then the
steady propeller coefficients can be converted into conventional coefficients
for profile drag CD,pro and lift CL.
The propeller coefficients for `early' conditions provide a useful comparison
for the results of these conversions; the induced downwash of the propeller
wake has hardly begun, so CD,pro and
CL approximate Ch,early and
Cv,early. However, wings in `early' revolution do not
experience completely still air; some downwash is produced even without the
vorticity of the fully developed wake. Despite this,
Ch,early and Cv,early provide the best
direct (though under-) estimates of CD,pro and lift
CL for wings in revolution.
Consider the wing-element shown in Fig.
5, which shows the forces (where the prime denotes forces per unit
span) acting on a wing element in the two frames of reference. A downwash air
velocity results in a rotation of the `lift/profile drag' from the
`vertical/horizontal' frame of reference by the downwash angle . In the
`lift/profile drag' frame of reference, a component of profile drag acts
downwards. Also, a component of lift acts against the direction of motion;
this is conventionally termed `induced drag'. A second aspect of the downwash
is that it alters the appropriate velocities for determining coefficients;
Ch and Cv relate to the wing speed
U, whereas CD,pro and CL
relate to the local air speed Ur.
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If a `triangular' downwash distribution is assumed, with local vertical
downwash velocity w0 proportional to spanwise position
along the wing r (which is reasonable, and the analysis is not very
sensitive to the exact distribution of downwash velocity; see
Stepniewski and Keys, 1984),
then there is a constant downwash angle
for every wing chord. Analysis
of induced downwash velocities by conservation of momentum, following the
`RankineFroude' approach, results in a mean vertical downwash velocity
given by:
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![]() | (14) |
![]() | (15) |
![]() | (16) |
![]() | (17) |
![]() | (18) |
![]() | (19) |
The appropriate air velocities for profile drag and lift coefficients may
be described conveniently as proportions of the wing velocity. In simple
propeller theories, a vertical downwash is assumed, and the local air velocity
Ur at each element, as a proportion of the velocity of the
wing element U, is given by:
![]() | (20) |
![]() | (21) |
Given the rotation of the frames of reference described in equations 18 and
19, and the change in relevant velocities discussed for equations 20 and 21,
profile drag and lift coefficients can be derived from `steady' propeller
coefficients:
![]() | (22) |
![]() | (23) |
Display of results
Angle of incidence
The definition of a single geometric angle of attack is clearly
arbitrary for cambered and twisted wings, so angles were determined with
respect to a zero-lift angle of attack
0. This was found
from the x-intercept of a regression of `early'
Cv data (Cv,early) against a range of
from -20° to +20°. The resulting angles of incidence,
'=
-
0, were thus not predetermined; the
experimental values were not the same for each wing type, although the
increment between each
' within a wing type is still 5°. The
use of angle of incidence allows comparison between different wing shapes
without any bias introduced by an arbitrary definition of geometric angle of
attack.
Determination of significance of differences
Because the zero-lift angle differs slightly for each wing type, the types
cannot be compared directly at a constant angle of incidence. Instead, it is
useful to plot the relationships between force coefficients and angles with a
line width of ± one mean standard error (S.E.M.): this allows plots to
be distinguished and, at these sample sizes (and assuming parametric
conditions are approached), the lines may be considered significantly
different if (approximately) a double line thickness would not cause overlap
between lines. The problems of sampling in statistics should be remembered, so
occasional deviations greater than this would be expected without any
underlying aerodynamic cause.
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Results |
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Standard hawkmoth
Fig. 6 shows
Ch and Cv plotted against
' for the standard flat hawkmoth model wing pair. The minimum
Ch is not significantly different from zero and is, in
fact, slightly negative. This illustrates limits to the accuracy of the
measurements. Significant differences are clear between `early' and `steady'
values for both vertical and horizontal coefficients over the mid-range of
angles. Maximal values of Ch occur at
'
around 90°, and Cv peaks between 40 and 50°. The
error bars shown (± 1 S.E.M.) are representative for all wing
types.
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In Figs 7,8,9, standard hawkmoth results are included as an underlying grey line and represent 0° twist and 0% camber.
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Leading-edge range
Fig. 7 shows
Ch and Cv plotted against
' for hawkmoth wing models with a range of leading-edge forms.
The relationships between force coefficients and
' are strikingly
similar, especially for the `steady' values (as might be expected from the
greater averaging period). The scatter visible in the polar diagram
(Fig. 7C) incorporates errors
in both Ch and Cv, making the scatter
more apparent than in Fig.
7A,B.
Twist range
Fig. 8 shows
Ch and Cv plotted against
' for twisted hawkmoth wing models. Results for the 15° twist
are not significantly different from those for the standard flat model. For
the 32° twist, however, Ch and Cv
plotted against
' both decrease under `early' and `steady'
conditions at moderate to large angles of incidence. This is emphasised in the
polar diagram (Fig. 8C), which
shows that the maximum force coefficients for the 32° twist are lower than
for the less twisted wings. The degree of shift between `early' and `steady'
force coefficients is not influenced by twist.
Camber range
Fig. 9 shows
Ch and Cv plotted against
' for cambered hawkmoth wing models, and the corresponding polar
diagrams are presented in Fig.
9C. Consistent differences, if present, are very slight.
Conversion into profile drag and lift coefficients
Fig. 10 shows the results
of the three methods for estimating CD,pro and
CL derived above, based on the mean values for all wings
in the `leading-edge' range. The `small-angle' model uses equations 14 and 15;
the `no-swirl' model uses the large-angle equations 18 and 19 and the
assumption that the downwash is vertical (equation 20); the `with-swirl' model
uses the large-angle expressions and the assumption that the induced velocity
is inclined to the vertical (equation 21).
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The `small-angle' model is inadequate; calculated profile drag and lift
coefficients are very close to `steady' propeller coefficients and do not
account for the shift in forces between `early' and `steady' conditions. Both
models using the large-angle expressions provide reasonable values of
CD,pro and CL for ' up to
50°; agreement with the `early' propeller coefficient polar is very good.
Above 50°, both models, especially the `no-swirl' model, appear to
underestimate CL.
Air-flow observations
Smoke emitted from the leading and trailing edges and from holes drilled in
the upper wing surface labels the boundary layer over the wing
(Fig. 11). At very low angles
of incidence (Fig. 11A), the
smoke describes an approximately circular path about the centre of revolution,
with no evidence of separation or spanwise flow. Occasionally at small angles
of incidence (e.g. 10°), and consistently at all higher angles of
incidence, smoke separates from the leading edge and travels rapidly towards
the tip (`spanwise' or `radially'). The wrapping up of this radially flowing
smoke into a well-defined spiral `leading-edge vortex' is visible under steady
revolution (Fig. 11B) and
starts as soon as the wings start revolving.
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Near the wing tip, the smoke labels a large, fairly dispersed tip- and trailing-vortex structure. At extreme angles of incidence (including 90°), flow separates at the trailing edge in a similar manner to separation at the leading edge (the Kutta condition is not maintained): stable leading- and trailing-edge vortices are maintained behind the revolving wing, and both exhibit a strong spanwise flow.
The smoke flow over the `sawtooth' design gave very similar results.
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Discussion |
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Vertical force coefficients are large
If `early' values for Cv provide minimum estimates for
`propeller' lift coefficients (since the propeller wake, and thus also the
downwash, is not fully developed), then the maximum lift coefficient
CL,max for the `pooled' data is 1.75, found at
'=41°. Willmott and Ellington
(1997c
) provide steady-state
force coefficients for real hawkmoth wings in steady, translational flow over
a range of Re. Their results for Re=5560 are shown with the
`pooled' data for flat wings in Fig.
12. The differences are remarkable: the revolving model wings
produce much higher force coefficients. The maximum vertical force coefficient
for the real wings in translational flow, 0.71, is considerably less than the
1.5-1.8 required to support the weight during hovering. Willmott and Ellington
(1997c
) therefore concluded
that unsteady aerodynamic mechanisms must operate during hovering and slow
flight. The same conclusions have previously been reached for a variety of
animals for which the values of CL required for weight
support are well above 1.5 and sometimes greater than 2.
|
The result presented in this study, that high force coefficients can be
found in steadily revolving wings, suggests that the importance of unsteady
mechanisms, increasingly assumed since Cloupeau et al.
(1979)
(Ennos, 1989
; Dudley and
Ellington, 1990b; Dudley,
1991
; Dickinson and Götz,
1993
; Wells, 1993
;
Wakeling and Ellington, 1997b
;
Willmott and Ellington,
1997c
), particularly after the work of Ellington
(1984a
,b
,c
,d
,e
,f
),
may need some qualification. It should instead be concluded that unsteady
and/or three-dimensional aerodynamic mechanisms normally absent for
wings in steady, translational flow are needed to account for the high lift
coefficients in slow flapping flight.
Most wind-tunnel experiments on wings confound the two factors: flow is
steady, and the air velocity at the wing base is the same as that at the wing
tip. Such experiments have resulted in maximum lift coefficients of around or
below 1: dragonflies of a range of species reach 0.93-1.15
(Newman et al., 1977;
Okamoto et al., 1996
;
Wakeling and Ellington,
1997b
), the cranefly Tipula oleracea achieves 0.86
(Nachtigall, 1977
), the
fruitfly Drosophila virilis 0.87
(Vogel, 1967b
) and the
bumblebee Bombus terrestris 0.69 (Dudley and Ellington, 1990b).
Jensen (1956
), however,
created an appropriate spanwise velocity gradient by placing a smooth, flat
plate in the wind tunnel, near the wing base, so that boundary effects
resulted in slower flow over the base than the tip. He measured
CL,max close to 1.3, which is considerably higher than
values derived without such a procedure and partly accounts for his conclusion
that steady aerodynamic models may be adequate. Nachtigall
(1981
) used a propeller system
to determine the forces on revolving model locust wings, but did not convert
the results to appropriate coefficients.
The descent of samaras (such as sycamore keys) provides a case in which a
steadily revolving, thin wing operates at high . Azuma and Yasuda
(1989
) assume a
CL,max of up to 1.8 in their models, but appear to find
this value unremarkable. Norberg
(1973
) calculates high
resultant force coefficients
,
but does comment that this `stands out as a bit high'. Crimi
(1996
) has analysed the falling
of `samarawing decelerators' (devices that control the descent rate of
explosives) at much higher Reynolds numbers and found that the samara wings
developed a considerably greater `aerodynamic loading' than was predicted
using their aerodynamic coefficients.
`Propeller' versus `unsteady' force coefficients
Although the steady propeller coefficients are of sufficient magnitude to
account for the vertical force balance during hovering, this does not negate
the possibility that unsteady mechanisms may be involved
(Ellington, 1984a). Indeed, it
would be surprising if unsteady mechanisms were not operating to some extent
for flapping wings with low advance ratios. However, the results presented
here suggest that the significance of unsteady mechanisms may be more limited
to the control and manoeuvrability of flight (e.g.
Ennos, 1989
;
Dickinson et al., 1999
) than
recently thought, although unsteady phenomena may have an important bearing on
power requirements (Sane and Dickinson,
2001
). Steady-state `propeller' coefficients (derived from
revolving wings) may go much of the way towards accounting for the lift and
power requirements of hovering and, while missing unsteady aspects, present
the best opportunity for analysing power requirements in those insects, and
those flight sequences, in which fine kinematic details are unknown.
The relationship between Cv and Ch for sharp,
thin wings
The polar diagrams displayed in Fig.
12 show that horizontal force coefficients are also considerably
higher for revolving wings. The relationship between vertical and horizontal
force coefficients is of interest as it gives information on the cost (in
terms of power due to aerodynamic drag) associated with a given vertical force
(required to oppose weight in the case of hovering). Flow separation at the
thin leading edge of the wing models described here must produce a quite
different net pressure distribution from that found for conventional wings and
is likely to be the cause of the Cv/Ch
relationship described here.
Under two-dimensional, inviscid conditions, flow remains attached around
the leading edge. This results in `leading-edge suction': flow around the
leading edge is relatively fast and so creates low pressure. The net pressure
distribution results in a pure `lift' force; drag due to the component of
pressure forces acting on most of the upper wing surface is exactly
counteracted by the leading-edge suction. This is true even for a thin
flat-plate aerofoil: as the wing thickness approaches zero, the pressure due
to leading-edge suction tends towards -, so that the leading-edge
suction force remains finite. The pressure forces over the rest of the wing
act normal to the wing surface. The horizontal component of the leading-edge
suction force cancels the drag component of the pressure force over the rest
of the wing. Under realistic, viscid conditions, this state can be achieved
only by relatively thick wings with blunt leading edges operating at low
angles of incidence.
Viscid flow around relatively thin aerofoils at high angles of incidence
separates from the leading edge, and so there is no leading-edge suction. If
viscous drag is also relatively small, the pressure forces acting normal to
the wing surface dominate, so the resultant force is perpendicular to the wing
surface and not to the relative velocity. In the case of wings in revolution,
the high vertical force coefficients can be attributed to the formation of
leading-edge vortices. Leading-edge vortices are a result of leading-edge
separation and so are directly associated with a loss of leading-edge suction;
high vertical (or lift) forces due to leading-edge vortices must inevitably
result in high horizontal (or drag) forces
(Polhamus, 1971).
The dominance of the normal pressure force allows a `normal force
relationship' to be developed which relates vertical and horizontal force
coefficients to
and the geometric angle of attack
(see also
Dickinson, 1996
;
Dickinson et al, 1999
).
Fig. 5 shows the forces acting
on a wing element if the resultant force FR' per
unit span is dominated by normal pressure forces. This results, in terms of
coefficients, in the relationships:
![]() | (24) |
![]() | (25) |
These combine to produce the useful expressions:
![]() | (26) |
![]() | (27) |
Fig. 13 compares the measured vertical and horizontal coefficients with those predicted from the normal force relationship for the standard flat wing data. The success of the model for both `early' and `late' conditions suggests that pressure forces normal to the wing surface dominate the vertical and horizontal forces. At very low angles of incidence, it is likely that viscous forces largely comprise the horizontal (equivalent to drag) forces, but this cannot be determined from the data. At higher angles of incidence, however, Ch is clearly dominated by pressure forces acting perpendicular to the wing surface.
|
The trigonometry of the forces shown in
Fig. 5 is such that the same
physical arguments, this time with
,
and the effective angle of attack
r, result in:
![]() | (28) |
![]() | (29) |
From this:
![]() | (30) |
This account of the pressure distribution over thin aerofoils and the
normal force relationship should be applicable whenever the flow separates
from a sharp leading edge. Indeed, Fig.
14 shows that the division into vertical and horizontal force
components using equations 24 and 25 fits very well for the real hawkmoth
wings in translating flow, for which the leading-edge vortex is
two-dimensional and unstable (Willmott and
Ellington, 1997c). The model underestimates Ch
at small angles of attack, but that is simply because skin friction is
neglected. However, hawkmoth wings typically operate at much higher angles, at
which the model fits the data very well for both translating and revolving
wings.
|
The effects and implications of wing design
Leading-edge detail
The production of higher coefficients than would be expected in translating
flow appears remarkably robust and is relatively consistent over quite a
dramatic range of leading-edge styles. This may be surprising because the
leading-edge characteristics of swept or delta wings are known to have effects
on leading-edge vortex properties (Lowson
and Riley, 1995) and are even used to delay or control the
occurrence of leading-edge vortices at high angles of incidence. Wing features
of some animals, such as the projecting bat thumb or the bird alula, may
perform some role in leading-edge vortex delay or control analogous to wing
fences and vortilons on swept-wing aircraft (see
Barnard and Philpott, 1995
).
Such aircraft wings, and perhaps the analogous vertebrate wings, experience
both conventional (attached) and detached (with a leading-edge vortex) flow
regimes at different times and positions along the wing. However, the results
presented here suggest that it is unlikely that very small-scale detail of
leading edges, such as the serrations on the leading edges of dragonfly wings
(e.g. Hertel, 1966
), would
influence the force coefficients for rapidly revolving wings. The peculiar
microstructure of dragonfly wings may be more closely associated with their
exceptional gliding performance (Wakeling
and Ellington, 1997a
).
Twist
The `early' and `steady' polar diagrams for the hawkmoth wing design with
moderate (15°) twist are virtually identical to those for the flat wing
design (Fig. 8). The only
difference is that the zero-lift angle 0 was approximately
-10° for the twisted wing, so angles of incidence
' ranged
from -30 to 90° instead of -20 to 100° as for the flat wings. Thus,
the bottom left of the polar diagram was slightly extended and the bottom
right shortened. The effect was even more pronounced for the highly twisted
(32°) wing design. This design also showed a substantial reduction in the
magnitude of the force coefficients at high angles of incidence, but this is
readily explained: even when the wing base is set to a high angle of
incidence, the tip of a highly twisted wing will be at a much lower angle.
Twist is desirable in propeller blades and has been assumed to be desirable
for insects by analogy. The downwash angle is typically smaller towards
the faster-moving tip of a propeller, so a lower angle of incidence
' is needed to give the same effective angle of incidence
r' (=
' -
). Thus, a twisted blade
allows some optimal effective angle of incidence to be maintained at each
radial station despite the varying effects of downwash. However, what this
optimal effective angle of incidence should be is unclear for insects. These
revolving, low- Re wings show no features of conventional stall;
changes from high Cv to high Ch with
increasing angle of incidence can be related entirely to the normal pressure
force and not to the sudden development of a stalled wake. So it is not,
presumably, stall that is being avoided with the twisted wing.
The characteristic normally optimised in propeller design is the
`aerodynamic efficiency' or lift-to-drag ratio. This occurs at
r' well below 10° for conventional propellers and
at
(
r' at these small angles) around
10° for the translating hawkmoth wings
(Willmott and Ellington,
1997c
). The maximum lift-to-drag ratio could not be determined in
this study because of noise in the torque transducer at small angles of
incidence, but it is reasonable to suppose that the optimal
r' for aerodynamic efficiency is low, probably below
10°. This is certainly below the angles used by hawkmoths, in which
ranges from 21 to 74°
(Willmott and Ellington,
1997b
) or by many hovering insects: Ellington
(1984c
) gives
=35°
as a typical value. So, twist is not maintaining an
r'
along the wing that maximises the lift-to-drag ratio. The angles of attack for
hovering insects suggest that a compromise between high lift and a reasonably
small drag might be more important than maximising the lift-to-drag ratio.
They operate near the upper left corner of the polar diagram, and the observed
moderate wing twists might sustain the appropriate
r'
along the wing. However, it must be emphasised that the polar diagrams for the
flat and moderately twisted wings were almost identical. The same point on the
polar diagram could be attained by either wing design simply by altering the
geometric angle of attack, so there are no clear benefits to the twisted
wing.
Less direct aerodynamic functions of twist should also be considered. Ennos
(1988) shows that camber may
be produced through wing twist in many wing designs, so any aerodynamic
advantages of camber might drive the evolution of twisted wings. It is also
possible that twisting may have no aerodynamic role whatever or may even be
aerodynamically disadvantageous. The null hypothesis for this discussion
should be that wing twist is just a structural inevitability for ultra-light
wings experiencing rapidly changing aerodynamic and inertial forces. Twist may
simply occur as a result of rotational inertia during pronation and supination
and be maintained because of aerodynamic loading on a slightly flimsy wing.
The lack of twist in flapping Drosophila wings has been explained by
the higher relative torsional stiffness of smaller wings
(Ellington, 1984c
). If
twisting had aerodynamic advantages, the evolution of more flexible materials
(which, if anything, should be less costly) might be expected. Of course,
these arguments are confounded in many aspects, including Re.
However, it is difficult for any description of an aerodynamic function of
twist to account for the purpose of wings twisted in the opposite sense, where
the base operates at lower
than the tip. This appears to be the case
for Phormia regina (Nachtigall,
1979
).
Camber
Fig. 9 agrees with results
on the performance of two-dimensional model Drosophila wings in
unsteady flow (Dickinson and Götz,
1993); any changes in the aerodynamic properties of model hawkmoth
wings due to camber are slight. Shifts in maximum Ch or
Cv appear to be within the experimental error, so these
trends should not be put down to aerodynamic effects. The similarities of the
polar diagrams show that camber provides little improvement in lift-to-drag
ratios at relevant angles of incidence.
Camber is beneficial in conventional wings because it increases the angle of incidence gradually across the chord. This shape deflects air downwards gradually, and the abrupt and undesirable breaking away of flow from the upper surface is avoided. So, the conventional reasoning behind the benefits of cambered wings to insects appears flawed: flapping insect wings use flow separation at the leading edge as a fundamental part of lift generation. A reasonable analogy exists with aeroplane wings. The thin wings of a landing Tornado jet use leading- and trailing-edge flaps to increase wing camber, maintaining attached flow and allowing higher lift coefficients than would otherwise be possible. Concorde, however, uses the high force coefficients associated with leading-edge vortices created by flow separation from the sharp, swept leading edges: no conventional leading-edge flaps are used because flow separation from the leading edge is intentional.
Camber still has a role in improving the aerodynamic performance of gliding wings, but any beneficial aerodynamic effects for flapping insect wings will require experimental evidence and not analogy with conventional wings designed (or adapted) for attached flow.
Accounting for differences between `early' and `steady' propeller
coefficients
Fig. 4 and Figs
6,7,8,9
show that there is a considerable change in force production between `early'
and `steady' conditions. There are two possible reasons for this change.
First, the wings cause an induced flow in steady revolution that is absent at
the start, and this decreases the effective angle of incidence. Second, there
may be a fundamental change in aerodynamics due, for instance, to the shedding
of the leading-edge vortex (and a resulting stall), as is seen for translating
wings (Dickinson and Götz,
1993). Simple accounts are taken of the induced downwash in the
calculation of CD,pro and CL from
steady coefficients (Fig. 10).
Below
'=50°, the downwash alone appears to account for the
shift between `early' and `steady' propeller coefficients; the calculated
values of CD,pro and CL fit the
observed values of Ch,early and
Cv,early well. Also, the observation
(Fig. 11) that leading-edge
vortices can be maintained during steady revolution supports the view that the
shift in propeller coefficients can be accounted for by the effects of
downwash alone, without a fundamental change in aerodynamics.
At very high ', the downwash models for determining
CD,pro and CL provide poorer results.
A change in the value of
at high
' can
improve the fit of CD,pro and CL to
Ch,early and Cv,early: both
kind and R (separation at the wing tip may reduce
the effective wing length) in equation 11 may be altered. However, varying
correction factors in the high
' range without a priori
justification (such as more accurate flow visualisation) limits the
possibility of aerodynamic inferences. Both fundamental changes in
aerodynamics and failure of the RankineFroude actuator disc model for
calculating induced downwash are also reasonable explanations for part of the
shift in propeller coefficients between `early' and `steady' conditions at
very high
'. The appearance of trailing-edge vortices at high
angles of incidence may be a relevant aerodynamic shift and may also account
for the relatively high force values for 45°<
<75°. An
aerodynamic change due to a shift in the position of the vortex core breakdown
is particularly worthy of consideration. Ellington et al.
(1996
) and Van den Berg and
Ellington (1997b
) noted that
the core of the spiral leading-edge vortex broke down at approximately
two-thirds of the wing length, resulting in a loss of lift in outer wing
regions. Liu et al. (1998
)
postulated that this breakdown is due to the adverse pressure gradient over
the upper wing surface caused by the tip vortex. The development of the full
vortex wake with its associated radial inflow over the wings might well shift
the position of vortex breakdown inwards under `steady' conditions at higher
', producing a quantitative reduction in the lift coefficient
compared with the `early' state.
![]() |
Acknowledgments |
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