## SUMMARY

High force coefficients, similar to those observed for revolving model
hawkmoth wings in the accompanying paper (for which steady leading-edge
vortices are directly observed), are apparent for revolving model (mayfly,
bumblebee and quail) and real (quail) animal wings ranging in Reynolds number
(*Re*) from 1100 to 26000. Results for bumblebee and hawkmoth wings
agree with those published previously for *Drosophila*
(*Re*≈200). The effect of aspect ratio is also tested with planforms
based on hawkmoth wings adjusted to aspect ratios ranging from 4.53 to 15.84
and is shown to be relatively minor, especially at angles of incidence below
50°.

The normal force relationship introduced in the accompanying paper is supported for wings over a large range of aspect ratios in both `early' and `steady' conditions; local induced velocities appear not to affect the relationship.

## Introduction

High force coefficients are required to account for hovering flight in animals ranging from small insects (e.g. Ellington, 1984a,b,c,d,e,f) to medium-sized birds (Norberg, 1975) and bats (Norberg, 1976). Ellington et al. (1996) showed leading-edge vortices to be present over flapping real and model hawkmoth wings. These leading-edge vortices, created by dynamic stall and maintained by spanwise flow, contribute significantly to lift production in slow-flying hawkmoths. The accompanying paper (Usherwood and Ellington, 2002) shows this phenomenon, and high force coefficients, to be a stable aerodynamic characteristic of revolving model moth wings. The present paper aims to determine how robust this characteristic is to variations in wing design and Reynolds number. Model hawkmoth wings with a range of aspect ratio and real and model wings from a number of `key species' are tested.

### Aspect ratio

The basic planiform shape of many animal wings may be characterized in
simple terms (Weis-Fogh, 1973;
Ellington, 1984b). One key
variable is the aspect ratio *AR* of the wing:
1
where *S* is the total wing area and *R* is the single wing
length. Most flying animals are functionally two-winged; many four-winged
insects link fore- and hindwings, and, for these morphological parameters, the
linked wings are treated as one. Standard hawkmoth planforms
(Usherwood and Ellington,
2002), with their chords scaled by ×0.4, ×0.6,×
0.8, ×1 and ×1.4, are tested in this study, resulting in
five wing designs with constant wing length and an aspect ratio range of
4.53-15.84. Scaling the chord produces reasonably insect-wing-like planforms
with the variation of a single parameter.

Insects have wings of *AR* ranging from 2.8 (butterflies,
Dudley and DeVries, 1990) to
10.9 (craneflies, Ellington,
1984b). Vertebrates capable of hovering have wings ranging in
*AR* from 4.4 (pied flycatcher,
Norberg, 1975) to 8.2
(hummingbirds, Wells, 1993).
The aspect ratios of the wings in this study range from 4.53 to 15.84, and
angles of incidence greater than 90° are tested, so our results are
relevant to studies of animals that hover using a vertical stroke plane or
swim using drag-based propulsion.

Conventional propellers and wind turbines revolve, but delayed stall and
high force coefficients typically exist only at the wing (rotor) bases
[Himmelskamp in Schlichting,
1968 (propellers); Graham,
1992 (wind turbines)]. Otherwise, flow over high-*AR*
propellers and turbines at high angles of incidence stalls conventionally, and
blade-element analyses using coefficients derived from steady, two-dimensional
flow conditions are effective. So, it is reasonable to expect that the
high-lift mechanisms described by Usherwood and Ellington
(2002) for wings of
*AR*=6.34 might gradually or suddenly decline with increasing aspect
ratio.

### The implications of Reynolds number for flight

Reynolds number *Re* has a large impact on the behaviour of fluids
flowing past an object; Vogel
(1981) presents the concepts
clearly in a biological context. It is therefore reasonable to expect
*Re* to have a similar bearing on the flow (and so lift and drag)
acting on wings. Indeed, it is frequently supposed that many of the unexpected
phenomena associated with insect flight may be accounted for by the low values
of *Re* at which they operate. However, predictions based on
*Re* arguments are not always founded; while it is true that viscous
drag forces are higher for smaller animals, it is not true that the very small
and `fringe-winged' insects (*Re*<28) `row' through the air using
drag-based mechanisms (Ellington,
1984a). Indeed, the vertical stroke plane associated with
drag-based weight support is surprisingly seen in larger insects (butterflies)
at *Re* values of approximately 2800
(Maxworthy, 1981;
Ellington, 1984a;
Sunada et al., 1993).

So, at this stage, it is unclear whether insects, small, and even large, vertebrates operate in the same flow regime; it is not known whether there are significant qualitative differences in flow analogous to the transition between laminar and turbulent conditions. Is there a subtle gradient from one regime to another (this does not appear likely given the properties of normal laminar/turbulent transitions)? Or is there a biologically significant threshold above (or below) which certain aerodynamic mechanisms are unable to operate? If so, where are these boundaries?

### Key species

To gain more information of biological interest, this study investigates several key species for which the appropriate parameters are known. Bees are of particular interest as they show a considerable size range both within a species (between different castes of bumblebee) and among related genera (e.g. the Euglossini or orchid bees). A bumblebee wing was therefore tested to provide information on the aerodynamic properties of wings in revolution for an insect for which there is a great deal of morphological, kinematic and energetic data (Dudley and Ellington, 1990a,b; Cooper, 1993) and which should also be applicable to studies of euglossine bees (Casey and Ellington, 1989; Dudley, 1995; Dudley and Chai, 1996).

To determine the steady aerodynamic performance of wings in revolution at
low and high *Re*, a `mini-spinner' was built covering the range of
*Re* from 1100 to 26000. The model animals chosen for these extremes
were the mayfly *Ephemera vulgata* and blue-breasted quail *Coturnix
chinensis*, for which fresh wings were available. The `mini-spinner', a
smaller and simplified version of the more elaborate propeller described in
Usherwood and Ellington
(2002), proved a robust and
effective tool. It also allowed the use of real bird wings over a limited size
range, so both real and model quail wings were tested.

### Inferring the presence of a leading-edge vortex

Smoke observations for simple model hawkmoth wings by Usherwood and Ellington (2002) supported the finding (Ellington et al., 1996) that the mechanism for high lift is a leading-edge vortex. However, wing speeds and designs in the present study precluded such observations. As shown in Usherwood and Ellington (2002), flow separation can nevertheless be inferred if the resultant force is approximately normal to the wing surface.

## Materials and methods

Force measurements were made using two experimental propellers. The larger design, described by Usherwood and Ellington (2002), allows `early' (from the first half-revolution) and `steady' vertical and horizontal forces to be measured using foil strain gauges. The smaller, much simpler, design could only measure `steady' forces, but could do so over a much larger speed range.

Vertical and horizontal force coefficients were derived from measured vertical forces and torques as described by Usherwood and Ellington (2002). Following Usherwood and Ellington (2002), the term `propeller coefficient' is used to distinguish force coefficients derived from propeller experiments.

### Large propeller experiments

Unless otherwise stated, all aspects of the experimental method for the
large propeller experiments were identical to those described by Usherwood and
Ellington (2002). Methods of
wing construction, force measurement and data processing were suitable for a
limited *Re* range, appropriate for hawkmoths and queen bumblebees.

#### Aspect ratio

The standard hawkmoth planform was adapted to produce wing pairs with a
range of five aspect ratios (Fig.
1A): all wings were thin and flat. The wing length in every case,
including the offset due to the method of attachment to the propeller head
(see Usherwood and Ellington,
2002), was 556 mm, and the relevant second,
*r̂*_{2}(*S*), and third,
*r̂*_{3}(*S*),
non-dimensional wing moments of area remained constant:
*r̂*_{2}(*S*)=0.547 and
*r̂*_{3}(*S*)=0.588. Wing
thickness was constrained by the material used, and the angular velocity was
kept constant. The mean wing thickness (relative to mean chord) and
*Re* (defined using the conventions of
Ellington, 1984f) were
therefore confounding variables (Table
1). The constant angular velocity also resulted in smaller
signal-to-noise ratios for higher-*AR* (narrower) wings, because they
experienced smaller forces.

#### Bumblebee

The planform for a bumblebee (Fig.
1B) *Bombus terrestris* wing design was taken from a
previous study and used to produce a wing pair as described for the hawkmoth
(Usherwood and Ellington,
2002). Bumblebee B27 was selected because its aspect ratio and
radii for moments of area were the closest to the population means. Again, the
wing shape was kept constant so that the offset due to the attachment of the
wings to the propeller head changed the wing moments, as shown in
Table 2.

The propeller was driven slightly more slowly than for the hawkmoth tests, at 0.147 Hz, thus reducing the Reynolds number to 5496, a value appropriate for the largest queen bumblebees and large euglossines. Further reduction in speed produced very noisy results because of dominating mechanical oscillations, while reducing the wing length would have confounded the effects of the offset, which otherwise was kept constant for experiments on the large propeller.

### Small propeller (`mini-spinner') experiments

Fig. 2 shows the basic
construction of the `mini-spinner'. It uses the same principle for the
measurement of vertical forces (moments about a knife-blade fulcrum, forming a
`see-saw') as used by Usherwood and Ellington
(2002) but different
principles for torques. Unsteady force measurements and flow visualisations
are impossible with the mini-spinner, but the smaller size requires higher
frequencies of revolution for *Re* similarity, with the advantage that
low-*Re* models can be used while minimising the effects of random air
movements; random air movements will be negligible compared with the flow
generated by the wings. The size and relative stiffness of the mini-spinner
also allows the use of real bird wings. The extremes in *Re* are
represented by model mayfly forewings and both model and real quail wings.

#### Wing design

Model mayfly forewings were based on those from a 26.4 mg male mayfly
*Ephemera vulgata* (Fig.
3A). The hindwings were not included in the model because they
were small and their orientation during flapping flight was unknown. The
planform was maintained, so the small shift due to the diameter of the rotor
head (of diameter 9 mm, causing an offset of 4.5 mm) influences the wing
moments. Table 2 shows the
resulting wing parameters.

The model mayfly wings were constructed from stiff, thin (0.15 mm) card glued to 0.57 mm diameter wire running half-way down the ventral surface of the wing. This resulted in a wing thickness at the position of mean chord of 5% of wing chord.

Geometric angles of attack were set by rotating the wire wing-stems within the propeller head and measured using a inclinometer, which achieved an estimated accuracy of ±2°. Angles from 0 to 90° were used, with 10° increments. The angles of incidence were calculated as in Usherwood and Ellington (2002).

A 61.6 g blue-breasted quail *Coturnix chinensis* was killed by
decapitation as part of another study
(Askew et al., 2001). The right
wing (fresh mass 2.29 g) was removed at the base of the humerus and pinned to
dry using hypodermic needles. The pinned position mimicked a typical
mid-downstroke position determined from the video recordings of ascending
flight used by Askew et al.
(2001). Once stiff, the wing
was connected using four sutures to a rod bent to follow the humerus and
radius/ulna. It was only possible to use a single wing because a second right
wing accurately matching the first was not available and the dorsal/ventral
asymmetry of bird wings makes use of the left wing inappropriate. To balance
the propeller, the stem of the rod attached to the wing was allowed to
protrude through the propeller head. The wing was only slightly twisted
(maximally 3°) but was strongly cambered, particularly at the base. At the
`elbow' joint between the humerus and ulna/radius, the wing depth (including
camber and thickness) was 28.8% of the chord; at the `wrist', over the alula
base, this value was 24.1%; half-way between the alula and wing tip, it was
10%.

The wing, once attached to the rod, was scanned (Fig. 3B), and the appropriate moments were calculated. A print-out of the scanned image was used as a template for a wing model. The model wing was constructed from stiff, thin (0.3 mm) card glued to 1.4 mm diameter wire running half-way down the ventral surface of the wing. This resulted in a wing thickness at the position of mean chord of 4% of the chord. The single model wing was counterbalanced in the same way as the real wing. Again, the propeller head was considered when calculating wing moments (Table 2).

Angles of attack α were set by rotating the wire wing stems within the propeller head and measured using an inclinometer. The arbitrary `representative' α was taken across the wing chord from the base of the alula to the tip of the innermost primary. The angle of incidenceα′ was calculated as in Usherwood and Ellington (2002).

*Frequency and* Re

A variable power supply was used to drive the propeller head, using a 22 mm
diameter, 12 V motor (RS) connected to a 24 mm diameter 7.2:1 gearhead. The
rotational frequency was varied using the power supply until it reached 3.3 Hz
for the model mayfly wing pairs, as judged with the use of a Drelloscop Strob
2009S07 stroboscope. Rotational frequency was set before and checked after
each test. A rotational frequency of 3.3 Hz resulted in an *Re* based
on the mean chord (Ellington,
1984f) of 1100, close to values estimated from video recordings of
mayflies in ascending flight taken in the field and reasonable for the
parameters described by Brodsky
(1973) for the same
species.

The rotational frequency for the quail wing and model was 12.5 Hz,
resulting in an *Re* of 26000 based on the mean chord. Askew et al.
(2001) have observed a maximum
downstroke angular flapping velocity ω of 190° s^{-1} for a
quail with a wing length of 95 mm. This corresponds to a maximum *Re*
of 48000, so 26000 for the propeller implies that it is operating in a similar
flow regime to the flapping wing for most of the downstroke. The upstroke has
little aerodynamic effect.

### The mini-spinner for low Re: model mayfly wings

#### Vertical forces

The mini-spinner as shown in Fig. 2A has the motor, gearbox and propeller head oriented vertically. During steady revolution, a moment is created about the fulcrum due to the vertical force and the arm length to the right of the fulcrum. This is equal and opposite to the moment created by the tension force applied from a wire connected to the under-hook of a Mettler BasBal BB240 balance situated directly above, and the appropriate arm length to the left, of the fulcrum. This arrangement was calibrated with the repeated application of a 1 g mass to the centre of the propeller head, which resulted in an imperceptible deflection and produced values consistent with the geometry of the arrangement and the accuracy of the balance. The inherent linearity of the `see-saw' arrangement was confirmed during set-up and testing. Thereafter, a single point calibration was sufficient. Five (or 10 at values of α of particular interest) vertical force measurements were made at each angle of attack.

#### Torques

Aerodynamic torques were measured by rotating the motor, gearbox and propeller head unit to a horizontal orientation as shown in Fig. 2B. During steady revolution, the moment about the fulcrum is equal to the aerodynamic torque from the revolving propeller head and wings. This torque can thus be calculated given the distance from the fulcrum to the wire attachment (140 mm) directly below the balance. The same number of measurements was made as for the vertical forces, and the aerodynamic effects of the motor head and stings were determined from tests without wings and removed.

Each vertical force and torque value was the mean of a pair of runs, starting with the wings in opposite positions. The measurements taken for each run consisted of a `zero' and a 9s average after steady revolution had been achieved. This takes into account any error due to an imbalance between the wings.

### The mini-spinner for high Re: real and model quail wings

#### Vertical forces

Vertical forces were measured exactly as for the mayfly wings except that the moments were opposed by a stiff steel shim on which was glued a pair of strain gauges instead of the vertical wire leading to the balance: forces were too large and variable for the balance to provide accurate results. Signals from the strain gauges were amplified electronically before being sampled at 50 Hz using a Macintosh Quadra 650. Vertical force signals were averaged over 50s. Five values from 10 paired runs, taking imbalance into account as above, were found for each angle of attack.

#### Torques

The forces due to the faster, heavier quail wings were such that the above
method of measuring aerodynamic torques was impossible without adding large
masses to stabilise the beam, which resulted in excessive loading on the
strain-gauge shim. The torques were high enough, however, to be determined
with sufficient accuracy from the power consumption of the motor. The current
*I* passing through, and the voltage *V* across, the motor were
measured five times for each angle of attack. The electrical power input
(*IV*) is converted into aerodynamic power by the motor, with certain
losses. These motor losses can be categorised
(Electro-Craft Corporation,
1980) as being either speed-sensitive (which covers losses due to
eddy currents, hysteresis, windage, friction, short circuits and brush
contact) or torque-sensitive (winding resistance). The speed-sensitive
components of electrical losses will be a constant *C* because a
constant rotational frequency was used. *C* was determined by measuring
the electrical power required to drive the motor with no wings attached. The
torque-sensitive power loss due to the winding *P*_{winding} is
given by:
2
where *r*_{e} is the resistance of the motor. Tests showed that
*r*_{e} varied only very slightly with the time spent at the
maximum torque, so the internal resistance of the motor did not change as a
result of internal heating. Thus, the value of *r*_{e} taken
for the stationary motor can also be used during revolution. Subtracting the
two power losses from the power input yields the aerodynamic power
*P*_{aero} required to overcome the aerodynamic torque on the
wings:
3
and torque *Q* is given by:
4
where Ω is the angular velocity of propeller revolution.

## Results

### Aspect ratio series

Fig. 4 shows `early' and
`steady' results for the hawkmoth wings over a range of aspect ratios. In each
case, the `pooled' data for the flat hawkmoth wings shown in Usherwood and
Ellington (2002) are presented
(both `early' and `steady' values) for comparison. The shift between `early'
and `steady' values seen at intermediate angles of incidence for the standard
hawkmoth wings is visible for all aspect ratios. The relationship between both
*C*_{h,early} and *C*_{h,steady}
(Fig. 4A,C) and α′
at low angles is very consistent for wings of every *AR* tested.
However, under both conditions, *AR* has a progressively greater effect
at higher α′. Low-*AR* wings achieve considerably higher
maximum horizontal force coefficients, peaking at
*C*_{h,early}=3.4 and *C*_{h,steady}=3.5 nearα′
=90°, while the highest-*AR* wings achieve maximum
horizontal force coefficients of only 2.5
(Fig. 5).

The relationship between both *C*_{v,early} and
*C*_{v,steady} (Fig.
4B,D) and α′ is dependent on *AR* While the
maximum values reached, approximately 1.7 for *C*_{v,early} and
1.3 for *C*_{v,steady}, are very similar for the entire range
of aspect ratios and occur at similar values of α′, between 40 and
60°, the initial gradients differ significantly. The relationships are
approximately linear between α′=-20 and +20°. The gradients
d*C*_{v}/dα′, with their 95 % confidence intervals
over this range, are given in Fig.
6. Lower-*AR* wings, and wings in `steady' revolution, have
lower gradients.

### Bumblebee results

Fig. 7 shows the results for
the *Bombus* wings. *C*_{h} and *C*_{v},
both `early' and `steady', show remarkably few differences compared with the
`pooled' hawkmoth results from Usherwood and Ellington
(2002).

### Steady results for range of species

Fig. 8 shows the `steady'
force coefficients for the model mayfly and model and real quail wings derived
from force measurements using the `mini-spinner'. These are plotted with the
`steady' coefficients for *Bombus* and pooled hawkmoth wings. Slight
differences are visible in the horizontal force coefficients, with the mayfly
showing lower coefficients (although with high standard errors) and the quail
higher coefficients. The relationship between *C*_{v,steady}
and α′ was remarkably consistent over the whole range of wings
tested. All wings achieved maximum vertical force coefficients well above 1 at
values of α′ between 40 and 60°.

Deflections were visible in the revolving quail wings, with the tips of both real and model wings bending backwards, especially at higher values ofα . The values of α′ shown for the quail wings in Fig. 8 must therefore be considered approximate and lower than the true values.

## Discussion

### Steady high-life mechanisms exist for a wide range of revolving wings

#### Force coefficients for a range of AR

Aspect ratio appears to have remarkably little effect on the force
coefficients that can be achieved by revolving wings. Wings with values of
*AR* from 4.53 to 15.84 produce indistinguishable maximum vertical
force coefficients between α′=40° and 60° of 1.70
(`early') and 1.30 (`steady'). There is no distinct reduction in force
coefficient that would be associated with `stall', at least belowα′
=65° (and so of any relevance to insects hovering with a
horizontal stroke plane), even for wings of very high *AR* Above this
angle, however, low-*AR* wings achieve higher force coefficients, which
are dominated by *C*_{h}. At α=90°, there is a
considerable range in *C*_{h}
(Fig. 5): for
*AR*=15.84, *C*_{h,early}=2.53 and
*C*_{h,steady}=2.29; for *AR*=4.53,
*C*_{h,early}=3.42 and *C*_{h,steady}=3.52. For
the lower-*AR* wings, these values are well above those predicted for
flat plates in steady translational flow. Ellington
(1991) gives an approximate
relationship for the drag coefficient of an infinite flat plate
*C*_{D,FP} appropriate for *Re* in the range
10^{2} to 10^{3}:
5
where *C*_{D,FP} should be equivalent to
*C*_{h,steady} at α′=90°. *Re* is at
least several thousand for the wings described here, so the predicted
horizontal force coefficient is very close to 2 and varies only slightly over
the range of *Re* covered by the wings. Furthermore, the
three-dimensional effect of air `sneaking' around the ends of the wing instead
of flowing around its width would lead to even lower values of
*C*_{h} (Hoerner,
1958) and incorrectly predict the direction of the relationship
between maximum *C*_{h} and *AR*. The cause of the
observed relationship is uncertain, but analogy with the vortices
characteristically found over delta wings at high angles of incidence suggests
that interference between leading- and trailing-edge vortices at highα′
may be more significant for wings with higher *AR*.

These results suggest that blade-element analyses of revolving, perpendicular `wings' may be in serious error if conventional, steady, two-dimensional force coefficients are used. In particular, older analyses of pectoral-fin swimming in fish (Blake, 1978) may have to be re-assessed.

#### Bumblebee force coefficients

The measurements made on the bumblebee wings are near the lower limits of
the large propeller rig. However, all propeller coefficients
(*C*_{h,early}, *C*_{h,steady},
*C*_{v,early} and *C*_{v,steady}) agree so well
with the values found for hawkmoth wings that little comment is possible,
other than to observe that similar aerodynamic mechanisms are almost certainly
available to bumblebees and hawkmoths.

#### Steady force coefficients from mayfly to quail

Remarkably consistent, high force coefficients are achieved for simple,
thin, flat model wings in steady revolution at *Re* from 1100 to 26000;
the real quail wing, with thickness and camber, not to mention feathers,
produces very similar force coefficients. Drovetski
(1996) gives polar diagrams
from 0 to 25° for simple model galliform (game bird) wings. The video
recordings of Askew et al.
(2001) (and, consequently, the
wing and wing model used in this study) do not show the trailing-edge notch
described by Drovetski (1996);
it appears that such a notch is present only in gliding flight or is an
artefact of pinning the wings in a fully extended position. The maximum lift
coefficients cited by Drovetski
(1996) for wing models ranging
from California quail *Callipepla californica* to turkey *Meleagris
gallopavo* were between 0.61 and 0.80; it seems that some aspect of
revolution may as much as double the vertical force coefficients. Values for
blackbird *Turdus merula*, house sparrow *Passer domesticus* and
mallard *Anas platyrhynchus*
(Nachtigall and Kempf, 1971)
range from 0.9 to 1.1, higher than for the galliforms of Drovetski
(1996) but still considerably
lower than those for revolving quail wings.

#### High force coefficients as a robust phenomenon

The aerodynamic phenomenon resulting in high force coefficients, presumably
associated with the creation and maintenance of leading-edge vortices, appears
remarkably robust. Some of the force measurements on the flapping
*Drosophila* model of Dickinson et al.
(1999) are equivalent to the
`early' measurements described here, and their simple harmonic relationships
are shown in Fig. 9 together
with the `early' results for the hawkmoth *AR* range. The
*Drosophila* model shows a higher minimum horizontal force coefficient
at low values of α because of relatively larger viscous forces. However,
at higher values of α′, there is very good agreement in both
*C*_{h} and *C*_{v} with the values shown for
hawkmoth planforms. If it is reasonable to suppose that shifts from `early' to
`steady' conditions are relatively constant throughout the *Re* range,
then it appears that similar force coefficients are possible from
*Drosophila* (*Re*≈200) to quail (*Re*≈26 000). If
the mechanism for these high force coefficients is indeed the leading-edge
vortex, then the insensitivity to *Re* is not as surprising as it may
appear. Leading-edge vortices over sharp, thin delta wings are effective
lift-producers for slow paper aeroplanes, Concorde and the space shuttle; a
vast range of *Re*.

### Further implications of aspect ratio

#### dC_{v}/dα′ and aspect ratio

Fig. 6 shows relationships
between aspect ratio and the rate of change of vertical force coefficient with
angle of incidence, d*C*_{v}/dα′. The relationships
for both `early' and `steady' conditions are very similar: the gradients for
regression lines through each plot on Fig.
6 are not significantly different. This phenomenon is well known
for translating wings and is due to the larger downwash of lower-*AR*
wings, which produce greater forces for the same wing length. This results in
a greater downwash angle ϵ, and so a smaller increase in `effective angle
of incidence' (α_{r}′=α′-ϵ) for a given
increase in α′. The non-zero slope of the
d*C*_{v}/dα′ relationship for `early' conditions
shows that the `early' induced downwash, while small, is not negligible; even
before development of the propeller wake, the tip vortex appears to produce a
downwash analogous to that for wings in translation.
*C*_{h,early} and *C*_{v,early} therefore
provide slight underestimates for *C*_{D,pro} and
*C*_{L} (see Usherwood and
Ellington, 2002). However, the significance of this effect is
minor compared with the surprisingly similar magnitudes of force coefficients
for the *AR* range discussed above.

#### Implications of aspect ratio for wing design

The similarity in aerodynamic characteristics of wings over a considerable
range of aspect ratio for α′<65° suggests that, all other
factors being equal, lower-*AR* wings should require less power to
support body weight than higher-*AR* wings. For a given wing length
*R* and wingbeat frequency *n*, the vertical force
*F*_{v} during hovering is:
6
and the aerodynamic power *P*_{aero} is:
7
because lift is related to area and the square of wing velocity, while power
is proportional to wing area and the cube of wing velocity. The power required
to support a given body weight is therefore proportional to *n*
(∝*P*_{aero}/*F*_{v}). The frequency and
hence the power can be reduced by increasing wing area *S* (equation 6)
which, for a given wing length, is equivalent to decreasing the *AR*
(equation 1):
8

Clearly, many other aspects influence wing design in insects: aspect ratios may be determined by inertial power or weight considerations or by the energetics of unsteady or forward-flight aerodynamics. Also, manoeuvrability, visibility, protection when folded and developmental cost may all push wing design towards non-energetically adaptive optima. However, the above relationships do suggest a possible pressure towards broader wings in insects for which efficient hovering with a horizontal stroke plane is of selective significance.

The energetic advantage to butterflies of low-*AR* wings is clearer.
The large cabbage white *Pieris brassicae* hovers with a vertical
stroke plane (Ellington
1984a), which means that horizontal force coefficients as defined
here act in the vertical plane. While use is made of unsteady mechanisms such
as the `clap and fling', the benefits due to a low-*AR* wing can be
seen by considering steady propeller coefficients. The lowest-*AR* wing
tested had a maximum horizontal force coefficient of 3.52, 1.4 times that of
the highest-*AR* wing. Thus, lower-*AR* wings produce larger
forces because of their larger areas and because of their higher force
coefficients. This should allow the butterfly to flap disproportionately
slowly, lowering the power requirements for hovering.

#### Conversion of propeller coefficients into C_{D,pro} and
C_{L}

Fig. 10 shows the results
for the *AR* range of the three transformations described in Usherwood
and Ellington (2002) that
convert *C*_{h,steady} and *C*_{v,steady} into
*C*_{D,pro} and *C*_{L}, respectively. At values
of α′ greater than 50°, the models progressively underestimate
*C*_{L} with increasing *AR*. However, both large-angle
models give good fits to *C*_{h,early} and
*C*_{v,early} for values of α′ below 50°, which
are more realistic for hovering insects.

### The `normal force relationship' is unaffected by induced downwash

The `normal force relationship' between *C*_{h},
*C*_{v} and α described for standard hawkmoth wings in
Usherwood and Ellington (2002)
is also accurate at very low *Re* values
(Dickinson et al., 1999) at
high angles of attack. The effectiveness of the model for different
*AR*, and its insensitivity to induced velocities, is shown in
Fig. 11: the observed
resultant force coefficient *C*_{R} can be accurately divided
into *C*_{h} and *C*_{v} by:
9
and
10
respectively. The fits are very good, even at very high α′,
despite the various induced air velocities associated with the range of
*AR*, and `early' and `steady' conditions.

In conclusion, the aerodynamics of revolving wings appears quite
insensitive to variations in both wing morphology and kinematics: force
coefficients for a range of model insect wings and for the wing of one small
bird closely match those previously found for *Drosophila* wings. In
addition, aspect ratio has remarkably little influence on aerodynamic force
coefficients, at least at low-to-moderate angles of attack.

- List of symbols
- AR
- aspect ratio
- C
- sum of speed-sensitive components of electrical power loss
*C*_{D,FP}- drag coefficient for a flat plate in perpendicular flow
*C*_{D,pro}- profile drag coefficient
*C*_{h}- horizontal force coefficient
*C*_{L}- lift coefficient
*C*_{R}- resultant force coefficient
*C*_{v}- vertical force coefficient
*F*_{v}- vertical force
- I
- electrical current
- n
- wingbeat frequency
*P*_{aero}- aerodynamic power
*P*_{winding}- power due to winding in electric motor
- Q
- torque
- r̂
_{2}(S) - non-dimensional second moment of area
- r̂
_{3}(S) - non-dimensional third moment of area
*r*_{e}- electrical resistance
- R
- wing length
- Re
- Reynolds number
- S
- area of a pair of wings
- V
- voltage
- α
- geometric angle of attack
- α′
- angle of incidence
- αr′
- effective angle of incidence
- ϵ
- downwash angle
- ω
- downwash angular flapping velocity
- Ω
- angular velocity of the propeller

- Subscripts
- early
- before propeller wake has developed (e.g.
*C*_{v,early}) - steady
- after propeller wake has developed (e.g.
*C*_{v,steady})

## ACKNOWLEDGEMENTS

The help of Ian Goldstone and Steve Ellis and the support of members of the Flight Group, both past and present, are gratefully acknowledged.

- © The Company of Biologists Limited 2002