## SUMMARY

This paper addresses the question, do the rotational forces in the hovering
fruit fly *Drosophila melanogaster* reflect something different (the
Magnus effect) or more of the same (circulatory-and-attached-vortex force)?
The results of an unsteady blade-element model using empirically derived force
coefficients from translating (root-oscillating) wings are compared with
recent results derived from both the measured forces on a dynamically scaled
*Drosophila* wing and the computational fluid dynamic (CFD)-modeled
forces on a virtual *Drosophila* wing. The behavior of the forces in
all three models during wing rotation supports the hypothesis that rotational
lift is not a novel aerodynamic mechanism but is caused by the same
fluid-dynamic mechanism that occurs during wing translation. A comparison of
the unsteady model with a quasi-steady model that employs empirically derived
rotational coefficients further supports the hypothesis that rotational forces
are more of the same. Finally, the overall similarity of the results between
the unsteady model, the physical wing model and the CFD model suggests that
the unsteady model can be used to explore the performance consequences of
kinematic variation and to investigate locomotor control in freely moving
animals.

- unsteady aerodynamics
- rotational circulation
- delayed stall
- Magnus effect
- attached vortex lift
- hovering
*Drosophila*flight

## Introduction

Motor-driven, scaled-up model wings of a hovering hawkmoth, *Manduca
sexta* L., and fruit fly, *Drosophila melanogaster* (Meigen 1830),
have allowed tremendous progress in our understanding of the flight forces on
hovering insect wings. Visualization of the flow past the *Manduca*
model wings confirmed the expectation
(Ellington, 1984;
Dickinson and Götz, 1993)
that an attached leading-edge vortex augments flight forces during the
translation phase of wing movement and showed that a spanwise flow stabilized
the vortex and prevented its shedding into the wake (Van den Berg and
Ellington, 1997a,
b).

Measured forces on the model wings of *Drosophila* support previous
work showing that a large portion of the lift impulse is generated during
stroke reversals (Dickinson et al.,
1999). These model *Drosophila* experiments showed two
force peaks during stroke reversal. One peak occurred while the wing was
rapidly pitching about a spanwise axis, and the timing and sign of this peak
was a function of the timing of wing rotation. Because of this behavior,
Dickinson et al. (1999) argued
that the rotation of the wing adds a rotational circulation component to the
total circulation and that the associated force component is similar to the
Magnus force occurring on translating and rotating cylinders and spheres. The
second peak occurs immediately after stroke reversal and is independent of the
timing of wing rotation. This behavior suggested to Dickinson et al.
(1999) that the aerodynamic
force is augmented because of an interaction with the wake shed by the
previous stroke. These interpretations of the rotational forces have been
challenged by recent computational fluid dynamic (CFD) results suggesting that
the rotation-dependent peak can be explained by the rapid generation of strong
vorticity due to wing rotation, while the rotation-independent peak can be
explained by the acceleration reaction (the reaction to accelerating an added
mass of fluid; Sun and Tang,
2002).

The present study explores the nature of the rotation-dependent peak and
how it can be modeled. The theoretical force (*F*), per unit span, on a
thin-airfoil in translation and rotation at low angles of attack is a function
of the superposition of four circulatory components:
1
where ρ is fluid density, *U* is fluid velocity,Γ
_{t} is the translational circulation due to the wing being
attached to a translating body, Γ_{h} is the heaving circulation
due to the wing actively or passively oscillating, Γ_{r} is the
rotational circulation due to the wing rotating around a spanwise axis, andΓ
_{M} is the Magnus circulation (*sensu stricto*), which
is also due to wing rotation (Fung,
1993). Γ_{t}, Γ_{h} andΓ
_{r} arise because the corresponding kinematic components
(translation, heaving and rotation, respectively) create an incident flow at a
finite angle (α′) with the wing section. The presence of a finite
angle between the wing section and incident flow distorts the boundary layer,
creating an asymmetry in the speed of flow over the upper and lower surface of
the wing section. Γ_{M} differs from the other circulatory
components because it is independent of α′.

The distortion and resulting normal (lift) force on a pitched wing in a
uniform flow (in which case, only Γ_{t} applies) is similar to
the Magnus effect — the distortion of the boundary layer and resulting
lift that occur on a cylinder or sphere that is both translating and rotating
around an axis normal to the translation. The force component due toΓ
_{t} is, therefore, a Magnus-like force but is not the
Magnus-like force discussed by Dickinson et al.
(1999). Flapping and rotating
wings in a uniform flow will distort the boundary layer similarly (again,
because of their influence on α′). The force components due toΓ
_{h} and Γ_{r} are, therefore, Magnus-like
forces, but the component due to Γ_{h} is also not the
Magnus-like force discussed by Dickinson et al.
(1999). The force component due
to Γ_{r} arises from the rotational component of the wing's
motion to the incident flow and the resulting α′ (see below) and,
in this sense, it is really no different from the force components due toΓ
_{t} and Γ_{h}. Because the Γ_{r}
component is due to wing rotation, and the dynamics of the associated force
partially resemble the Magnus effect on rotating cylinders, it is this force
component that Dickinson et al.
(1999) refer to as Magnus-like.
The force component due to Γ_{M} is similar to that due toΓ
_{r} in that it is dependent on wing rotation but differs from
the Γ_{r} component in that it is independent of the angle of
the incident flow. Because Γ_{M} is also independent of the
chordwise center of the incident flow (in contrast to Γ_{r}),
the force component due to Γ_{M} exactly resembles the Magnus
effect on a rotating cylinder and it is this force that Sun and Tang
(2002) refer to as the Magnus
force. I refer to the combined force due to Γ_{t},Γ
_{h} and Γ_{r} as the
circulatory-and-attached-vortex force (see below), and the force due toΓ
_{M} as the Magnus force.

In this paper, I use a previously developed
(Walker and Westneat, 2000),
semi-empirical, unsteady blade-element (USBE) model to address the question,
do the rotational forces in the hovering fruit fly reflect something different
(Magnus circulation and corresponding Magnus force) or more of the same
(circulatory-and-attached-vortex force)? The unsteady results are quite
similar to the measured forces on the physical wing models and to CFD
estimates of the flight forces, and the rotational forces are well modeled by
unsteady coefficients measured on translating wings, which supports the
hypothesis that forces occurring during wing rotation arise from the same
fluid-dynamic mechanisms as forces occurring during wing translation
(*viz* the circulatory-and-attached-vortex force and acceleration
reaction).

## Materials and methods

### The unsteady blade-element model

A very general, semi-empirical, unsteady blade-element (USBE) model of the
dynamics of a root-oscillating limb (wing or fin) was developed previously
(Walker and Westneat, 2000).
The model has two components; one due to added mass (acceleration reaction)
and the other due to unsteady circulation and the influence of an attached
vortex. The geometry of the incident flow and incident angle of attack in the
USBE model is a large-angle generalization of Theodorsen's model of a wing in
flutter (Theodorsen, 1935),
which is described in detail by Fung
(1993). The history of the
Theodorsen and similar models and applications to insect flight are outlined
by Zbikowski (2002).
Applications of the Theodorsen model to the flapping flight of vertebrates are
found in DeLaurier (1993) and
Kamakoti et al. (2000). The
unique feature of the USBE model is that empirical force coefficients measured
from oscillating wings are used instead of theoretical coefficients based on
the Theodorsen function or its modifications. Unsteady coefficients based on
the Theodorsen function are not a function of time but are unsteady in that
they are a function of the reduced frequency parameter. The raw coefficients
used in this study are steady in both senses (although they are modified by
the unsteady Wagner function). The Theodorsen coefficients are an
approximation for low angles of incidence and attached flow. Consequently, the
Theodorsen coefficients are not particularly useful for much of the wing
stroke of a hovering *Drosophila* because the *Drosophila*
stroke is characterized by incident angles that are much larger than the stall
angle throughout much of the stroke cycle
(Dickinson et al., 1999). One
advantage of the empirically derived coefficients is that they explicitly
account for the augmenting effect of an attached vortex when the wing is above
the stall angle.

While the augmented circulatory force due to Γ_{r} was
included in the model, Magnus-type forces due to Γ_{M} were not.
Magnus forces have not been previously modeled in biological applications of
flapping wings, but Fung (1993)
suggests that these forces should contribute to the lift balance. The extended
model below adds this Magnus force component. The simple kinematics of a
*Drosophila* wing oscillating along a horizontal stroke plane allow the
model to be greatly simplified. The full model is detailed by Walker and
Westneat (2000).

The USBE model begins with a geometric description of wing kinematics. The
wing has length (span), *R*, and is arbitrarily divided along its span
into *p* elements with equal width: d*R=R/p*. The
length-specific radial position is
*r̂=r/R*, where *r* is the
distance from the wing base. In the following, the bracketed subscripts
indicate that a variable is a function of time (*t*) and/or radial
position along the wing span (*r*).

The azimuth position of the wing, γ_{(t)}, is the angle
between the rotational axis of the wing and a horizontal vector directed
posterior to the fly. In this coordinate system, γ_{(t)} is
0° when the rotational axis is back against the body and 90° when it
is perpendicular to the body axis. Following previous work, the wing does not
oscillate with simple harmonic motion but instead rapidly accelerates to a
constant angular velocity. The non-dimensional period of linear acceleration
at each stroke reversal is
and
the timing of the acceleration is symmetric about the point of stroke
reversal.

The pitch of a wing chord, α_{(t)}, is the angle between the
wing chord and the flapping axis (which is normal to the stroke plane and,
therefore, vertical for the hovering *Drosophila*)
(Fig. 1). α_{(t)}
is negative when the wing is pronated and is positive when the wing is
supinated. The geometric angle of attack, α_{g(t)}, is the angle
of the wing chord relative to the stroke plane and is equal toπ
/2+α_{(t)} for the case of a horizontal stroke plane
(Fig. 1). Again, the wing does
not rotate with simple harmonic motion. The non-dimensional period of each
rotational phase (there are two phases per stroke cycle) is
, and
the non-dimensional time difference between the mid-point of rotation and
stroke reversal is
(Sane and Dickinson,
2001).

The normal, ν_{n(r,t)}, and chordwise, ν_{x(r,t)},
flow due to wing translation and rotation are:
2
3
These equations assume that the velocity component due to body translation is
trivial, an assumption that is relaxed in the full model
(Walker and Westneat, 2000).
Equation 3 and the first component of Equation 2 are due to the wing element
translating (due to flapping) with a speed,
.
The second component of Equation 2 is due to the wing rotating around a
spanwise axis located
x̂_{r(r)}*c*_{(r)} from
the leading edge and the chordwise center of incident flow located
x̂_{i}*c*_{(r)} from the
leading edge, where x̂_{i} and
x̂_{r(r)} are percentage distances along
the chord, and *c*_{(r)} is chord length. It is important to
note that x̂_{r(r)} is a function of
wing shape and cannot be constant along the span. This is easily seen using,
as an example, the distal wing chord on the wing illustrated by Sane and
Dickinson (2002). If the
rotational axis, x̂_{o}, is 0.25, the
entire distal element is posterior to the rotational axis, and the
element-specific rotation axis, Δx_{r(r)}, must be less than
zero. The values of x̂_{o} used for each
analysis are found in Table 1.
The value of x̂_{i} must satisfy the
boundary condition of tangent flow. For attached flow,
x̂_{i} occurs at the three-quarter chord
location (DeLaurier, 1993;
Fung, 1993). While initial work
on the root-oscillating *Drosophila* wing model indicated thatΔ
x_{i}=0.75 for separated flow
(Dickinson et al., 1999), more
detailed work has shown that x̂_{i} is a
function of rotational velocity and can be as low as 0.57
(Sane and Dickinson, 2002). In
the present study, a constant value of 0.75 was used, although future work
could easily incorporate a x̂_{i} that
is dependent on rotational velocity.

The angle of incidence, α′_{(r,t)}, between the wing
chord and the incident stream is ±tan^{-1}
(ν_{n(r,t)}/ν_{x(r,t)}), where the ± takes the
sign of ν_{x(r,t)} (Fig.
1). This angle is used to estimate the lift and drag coefficients
(see below) and the components of the circulatory force normal to
(d*F*_{n(r,t)}) and parallel with
(d*F*_{x(r,t)}) the wing chord:
4
5
d*L*′_{(r,t)} and d*D*′_{(r,t)}, the
components of the circulatory force normal to and parallel with the local
stream, are:
6
7
where
8
*C*_{L(r,t)}, and *C*_{D(r,t)} are empirically
derived lift and drag coefficients measured from oscillating wings at a
constant α′ (Dickinson et al.,
1999). These coefficients reflect the augmenting effects of a
stable, attached, leading-edge vortex on the downstream surface of the wing
when α′ is above a threshold (approximately 13-15°). The
additional component due to the attached vortex is known as the
attached-vortex force (Kuethe and Chow,
1986). Despite the presence of the attached vortex, the measured
coefficients on a root-oscillating wing at a constant angle of attack remain
relatively stable for several chord lengths of travel
(Dickinson et al., 1999;
Usherwood and Ellington,
2002a) and can, therefore, be considered quasi-steady
(Dickinson et al., 1999;
Sane and Dickinson, 2002). It
is important to note that the coefficients were measured following the
influence of the initial, inertial peak and Wagner-effect trough
(Dickinson et al., 1999). In
Equations 6 and 7, Φ_{(r,t)} is the Wagner function due to Garrick
(Fung, 1993) and accounts for
the interaction between the wing and the starting vortex at the beginning of a
stroke. It is this function that effectively makes the coefficients in this
model unsteady. The time-course of this wing-wake interaction effect is much
longer than that due to wake capture (which is not included in the model). The
vertical (lift) and horizontal (drag) force components on a blade element due
to the circulatory-and-attached-vortex force are:
9
10
Note that if the rotational component of the resultant stream were ignored,
lift and drag could be estimated directly from Equations 6 and 7.

The added mass force, or acceleration reaction, normal to the wing element
is:
11
where β_{n} is the added mass coefficient of the wing section (a
value of 1.0 was used). x̂_{i}=0.5 was
used for the computation of
(see Equation 2 above; DeLaurier,
1993; Fung, 1993).
The vertical (lift) and horizontal (drag) components of
d*F*_{a(r,t)} are:
12
13
The acceleration reaction is an unsteady, non-circulatory force
(Fung, 1993). Note also that
(1) the acceleration reaction generates lift even when the wing is linearly
accelerating along the horizontal plane (see the force traces in
Dickinson and Götz, 1993)
and that (2) the acceleration reaction is non-zero even when the wing is
translating without linear acceleration but is rotating (due to the derivative
of the rotational component of ν_{n(r,t)} in Equation 2).

Finally, the Magnus force due to Γ_{M} can be modeled by:
14
which is the product of the added mass
,
the translational velocity
and the angular rotation
() of the
element (Fung, 1993; section
6.7, equation 10). Note that d*F*_{r(r,t)} is a force arising
from a circulatory component due to wing rotation but, unlike the unsteady
circulatory force due to Γ_{r}, the Magnus force (due toΓ
_{M}) is independent of both the angle of incident,α′
, and the rotational moment,
(x̂_{i}-x̂_{r(r)})*c*_{(r)}.
Except where noted, the Magnus force component is not included in the unsteady
model.

### Comparisons

Results from the USBE model are compared with both the measured forces on
the dynamically scaled *Drosophila* wing
(Dickinson et al., 1999) and
the CFD-modeled forces on the virtual *Drosophila* wing
(Sun and Tang, 2002). Wing
chords for the physical wing were computed from an outline of the wing
provided by M. H. Dickinson. Wing chords for the virtual wing were computed
from the digitized outline of the illustrated wing in Sun and Tang
(2002). For the comparisons
with the virtual wing, the output forces were standardized by
.
For the comparisons with the physical wing, the output forces were left
unstandardized. Morphological and kinematic parameters for the three physical
wing (PM1-PM3) and four virtual wing (VM1-VM4) comparisons are given in
Table 1.

## Results

Lift and drag for case PM1, the wing with unequal downstroke and upstrokeα
_{g(t)}, are illustrated in
Fig. 2. The shape of the
unsteady drag curve is nearly identical to the measured drag curve, but the
modeled peaks near the end of each stroke are approximately 20% greater than
the measured peaks. The modeled drag during the translational phase of the
upstroke is steady while the corresponding measured drag gradually rises. The
modeled lift curve closely matches the measured lift curve, with the major
differences occurring during the translation-only phase of the stroke cycle.
Relative to the modeled rate, the measured rate of increased lift during the
translation phase is slightly greater during the downstroke but much greater
during the upstroke.

Both modeled and measured lift for case PM2 show distinct lift peaks at the beginning of each stroke, but the measured lift peaks are approximately 50% greater than the corresponding modeled lift peaks (Fig. 3). Similarly, the measured lift occurring during the translation phase of each stroke is distinctly greater than the modeled lift. By contrast, the magnitude of the modeled and measured lift peaks occurring at the ends of each stroke and the negative peaks occurring at stroke reversals are nearly identical. The general shape of the modeled and measured lift curves for case PM3 is very similar, but the broad lift peaks of the unsteady model are distinctly truncated, and the negative, measured peak at the downstroke—upstroke reversal is approximately 65% greater than the modeled peak.

VM1 and VM2 were used to demonstrate the effect of the duration of the linear acceleration phase, , on the presence of large force peaks at the beginning of each stroke (Sun and Tang, 2002). The similarity between the CFD-modeled and USBE-modeled curves is quite striking (Fig. 4). The magnitude of the rotation-dependent drag peak (the peak occurring at the beginning of the stroke) is approximately the same for the CFD and USBE curves, but the rotation-dependent lift peak is approximately 30% higher in the USBE curve. The USBE lift decomposition (Fig. 5) shows two important features. First, the end-stroke lift peak is dominated by the circulatory component, although a small acceleration reaction component does occur despite the fact that the wing is translating with zero acceleration during this part of the cycle. Second, the beginning-stroke lift peak is dominated by the acceleration reaction, as suggested previously (Sun and Tang, 2002).

VM1, VM3 and VM4 were used to demonstrate the effect of the timing of wing rotation relative to stroke reversal (Sun and Tang, 2002), and the general features of the lift curve are similar to the corresponding physical wing experiments (Dickinson et al., 1999). Again, the similarity of the CFD-modeled and USBE-modeled curves is remarkable (Fig. 6). The drag curves differ in one important respect: the CFD-delayed curve presents a distinct peak at the beginning of each stroke that is nearly twice the magnitude of the corresponding peak in the USBE model. The CFD and USBE lift curves for VM3 and VM4 show differences that occur in VM1 but are greatly magnified. In other words, while the USBE estimates of lift during the translation phase are approximately 0.1 units higher than the CFD estimates for VM1, this difference is 0.25 units higher for VM3 and VM4. Importantly, however, the slope of the lift curve during the translation phase is nearly identical for the CFD and USBE models.

The addition of the Magnus force to the unsteady model has no effect on
drag because of the horizontal stroke plane. Except for perhaps PM3, the
addition of the Magnus force to the unsteady model results in a generally
worse fit to the measured results because of the more positive peaks during
wing rotation (Figs 1,
2). These results suggest that
the influence of the Magnus force is trivial, at best, during the hovering
flight of *Drosophila*. Because of the generally worse fit when the
Magnus force is included in the unsteady model, its effect was removed for all
comparisons with the virtual wing model.

## Discussion

### Accuracy of the unsteady model

The subtle kinematic changes exploited by animals to control swimming and flying forces are not well known (Taylor, 2001). The map of variation in measured forces on a pair of motor-driven wings as a function of wing kinematics provides detailed predictions on how animals might control locomotor forces (Sane and Dickinson, 2001). Comparisons of limb kinematics and body forces on tethered animals allow detailed investigation of locomotor control, but the tethering may introduce artifacts that are not found in freely flying animals. Ultimately, we want to know how freely moving animals control locomotor forces, a goal that will require a comparison of detailed limb kinematics and body dynamics.

Such comparisons are uncommon. The pectoral fin kinematics during the
flapping flight of the bird wrasse *Gomphosus varius* were compared
with measures of instantaneous dorsoventral and anteroposterior accelerations
of the body in order to infer control of swimming forces
(Walker and Westneat, 1997).
Measures of body dynamics in freely moving animals require the numerical
differentiation of the measured displacement of the body with respect to time.
The errors involved in this type of analysis have been discussed previously
(Harper and Blake, 1989;
Walker, 1998).

A complimentary approach to estimating body dynamics with a numerical
differentiation method is the application of a fluid-dynamic model to the limb
kinematics. The USBE model explored in the present study was initially
developed to compare the performance of oscillating limbs undergoing a variety
of motions at Reynolds numbers (*Re*) above 1000
(Walker and Westneat, 2000)
and below 100 (Walker, 2002).
The results of the present study suggest that predictions from the simulations
are probably robust against small errors in the unsteady model. The
application of the unsteady model to the investigation of locomotor control in
freely moving animals should also require reasonable model accuracy. While the
ability of the unsteady model to estimate measured forces on motor-driven
plates was briefly discussed previously
(Walker and Westneat, 2000),
it is worth checking the accuracy of the model more formally. The comparison
of the unsteady results to the measured forces on the physical model
(Dickinson et al., 1999) with
the CFD results on the virtual model allows this comparison
(Sun and Tang, 2002).

In general, the shape and elevation of the force curves from the USBE model
are quite similar to the corresponding physical model and CFD curves. The
slopes of the USBE drag and, especially, lift curves during the translation
phases differ from those of the physical model curves but are strikingly
similar to the slopes seen in the CFD model curves. Although the slopes during
the translation phase are similar for the USBE and CFD models, the USBE lift
estimates are consistently 12-18% greater than the CFD estimates. Indeed, the
USBE estimate of the mean lift coefficient
*C̄*_{L} across the whole stroke
cycle ranges from 6% to 25% greater than the CFD estimate (similar trends
occur with the drag coefficient *C*_{D} but, because of the
larger scale, these trends are only conspicuous during the rotational peaks).
The relationship between the USBE and physical model during the translation
phase is more variable, largely because of the variability in the measured
forces. The source of this variability is unknown but is clearly not being
accounted by either the unsteady or CFD models.

While the USBE model captures the rotation-dependent force peaks effectively (e.g. at the end of the strokes in Figs 2, 3), it fails to capture the rotation-independent peaks when these are large (e.g. at the beginning of each stroke in Fig. 3). While the modeled peaks reflect an inertial contribution (acceleration reaction) to the force balance (Fig. 5), this mechanism is clearly insufficient to account for the large measured peaks. These results, then, support the original interpretation that these peaks reflect wake capture (Dickinson et al., 1999) and not the acceleration reaction (Sun and Tang, 2002).

The similarity between the USBE model results and the physical and,
especially, CFD model results suggests that, with few caveats, the USBE model
is sufficient to investigate both simulated kinematic parameter spaces
(Walker and Westneat, 2000;
Walker, 2002) and locomotor
control from measured kinematic variables in freely moving animals. The
principal caveats of the broad application of the model are accounting for the
effects of reduced frequency, *Re* and model wing shape on the measured
force coefficients. Unfortunately, there is no work measuring the force
coefficients on root-oscillating wings with finite reduced frequencies (i.e.
in translation). The lift and drag coefficients employed in this study reflect
the influence of the induced downwash, which influence should decrease as the
reduced frequency approaches zero. To account for this decreased influence,
one could use two-dimensional coefficients and apply a model of the induced
downwash (DeLaurier, 1993;
Kamakoti et al., 2000;
Hedrick et al., 2002) to the
estimate of α′. Two-dimensional force coefficients on
root-oscillating wings across a range of attack angles have not been
published, but the two-dimensional coefficients at α′=45° are
8% greater than the three-dimensional coefficients
(Birch and Dickinson,
2001).

The effects of scale have not been measured on root-oscillating wings at
low *Re* (<100), although two-dimensional results on static airfoils
predict a large *Re* effect (Thom
and Swart, 1940). Experiments with the *Drosophila* wing
model show that force coefficients are stable for *Re* between 100 and
1000 (Sane and Dickinson,
2002). More surprisingly, only small differences in
*C*_{D} and, especially, *C*_{L} were found for
a series of real and model wings tested at *Re* between 1100 and 26 000
(Usherwood and Ellington,
2002b). With a more systematic study of *Re* effects on
wing performance, even this small *Re* influence can be easily
incorporated into the USBE model.

Both wing aspect ratio and the distribution of wing area might influence
wing performance, but the effects of wing-area distribution have not been
investigated. Wing aspect ratio has only a very small effect on
*C*_{L} at all α′, but the *C*_{D}
of lower aspect ratio wings increases more rapidly than that of higher aspect
ratio wings as α′ increases
(Usherwood and Ellington,
2002b). These results suggest that wing shape effects should be
more systematically explored and incorporated into future studies with the
USBE model.

### Comparison with a rotational-coefficient model

The theoretical lift coefficient,
*C*_{L}=2πα′, on a wing section at a low angle
of incidence is derived from a simple algebraic rearrangement of:
15
Indeed, the small incident angle assumption reduces Equations 2 and 3 to the
more familiar equations given in Fung
(1993) or Zbikowski
(2002). For higher angles of
incidence, the results presented here suggest that empirically derived
coefficients as a function of α′ result in good predictions of
measured forces. An alternative to estimating a single coefficient as a
function of all three kinematic components would be to compute a separate
coefficient for each kinematic component. Indeed, a quasi-steady model of
hovering flight in *D. melanogaster* using separate heave and
rotational force coefficients was developed recently
(Sane and Dickinson, 2002).
Rotational force coefficients were computed by standardizing the estimated
rotational force component, which is the residual force remaining after
subtracting the quasi-steady translational force (the quasi-steady force
estimated using only the translational component of ν_{n}) from the
measured force at a point in the stroke cycle when inertial forces are
trivial. The lift for cases PM2 and PM3 estimated by the USBE model and the
rotational-coefficient quasi-steady (RCQS) model were compared. For the RCQS
model, rotational coefficients, which are a function of both
*x̂*_{o} and the non-dimensional
angular velocity,
(Sane and Dickinson, 2002),
were computed using a linear interpolation between the coefficients estimated
from the regression parameters for the pair of
in
table 1 of Sane and Dickinson
(2002) that bounded the
instantaneous . The close match
between the USBE and RCQS curves (Fig.
7) suggests that the empirically derived rotational force
coefficients do not reflect a contribution from the Magnus circulation,Γ
_{M}. Instead, the close match simply indicates that the USBE
and RCQS models are simply alternative methods for incorporating the effects
of Γ_{r} into a semi-empirical blade-element model.

### Rotational lift: something different or more of the same?

The rotation-dependent force that has been compared to the Magnus effect is quantitatively explained by a model of the same circulatory-and-attached-vortex force that also dominates the force balance during wing translation. Indeed, if Magnus forces are included in the model, the predicted lift and drag peaks are much larger than the corresponding peaks estimated from the physical wing or virtual wing (CFD) models. A recent CFD flow reconstruction clearly shows the attached vortex throughout the rotational phase (Sun and Tang, 2002). It is, therefore, not surprising that the unsteady coefficients (Dickinson et al., 1999), which reflect the augmenting effects of an attached vortex on a translating wing, can explain the forces occurring during wing rotation.

Why do previous quasi-steady models (with the exception of the RCQS model
of Sane and Dickinson, 2002)
fail to capture the magnitude of the rotation-dependent force? Both the
quasi-steady force of Sun and Tang
(2002) and the quasi-steady
translational forces of Dickinson et al.
(Dickinson et al., 1999; Sane
and Dickinson, 2001,
2002) are modeled as a
function of the angle of the chord relative to the stroke plane, which is
horizontal in a hovering *Drosophila*. The incident flow vector,
therefore, was modeled by both groups as if the wing was not rotating. Indeed,
the quasi-steady model of Sun and Tang
(2002) did not account for the
changing angle of the wing chord during rotation. But, as shown above, the
incident flow vector and corresponding α′ and circulatory force
are a function of the tangential velocity (due to rotation) of the chordwise
center of incident flow relative to the chordwise center of rotation in
addition to the translational velocity. By accounting for this rotational
component to the incident flow, the USBE model can explain the
rotation-dependent forces occurring during wing rotation.

Comparisons of the rotation-dependent force component with the Magnus
effect, then, are misleading. While the behavior of the rotation-dependent
force component partially resembles the Magnus effect, this behavior results
from the distortion of the boundary layer due to the changing geometry of the
incident flow and, therefore, the mechanism precisely resembles that of a
pitched wing in a uniform flow. This interpretation of the rotation-dependent
force peak occurring during wing rotation suggests that rotational lift is not
a novel aerodynamic mechanism but is a consequence of a kinematic mechanism
that augments incident-angle-dependent circulation and the resulting
circulatory-and-attached-vortex force. Finally, this kinematic mechanism has
interesting but unexplored implications for the evolution of wing shape, as
the magnitude of the vortex force occurring during wing rotation is a function
of
*x̂*_{r(r)}*c*_{(r)}.

- List of symbols
*C̄*_{L}- mean lift coefficient
*C*_{L(r,t)}- lift coefficient of wing element
*C*_{D(r,t)}- drag coefficient of wing element
*c*_{(r)}- chord length
- c̄
- mean chord
- d
*D′*_{(r,t)} - circulatory drag
- d
*D*_{a(r,t)} - aft component of added mass force
- d
*D*_{c(r,t)} - aft component of circulatory force
- d
*F*_{a(r,t)} - added mass force
- d
*F*_{n(r,t)} - normal component of circulatory force
- d
*F*_{r(r,t)} - Magnus force
- d
*F*_{x(r,t)} - chordwise component of circulatory force
- d
*L′*_{(r,t)} - circulatory lift
- d
*L*_{a(r,t)} - upward component of added mass force
- d
*L*_{c(r,t)} - upward component of circulatory force
- d
*R* - wing element span
*ḣ*_{(r,t)}- heaving velocity
- R
- wing length
- r
- radial position along wing span
- r̂
- non-dimensional radial position along wing span
- S
- wing area
- t
- time
- t̂
- standardized time
- U
- fluid velocity
*v*_{n(r,t)}- normal velocity of wing element
*v*_{(r,t)}- velocity of wing element
*v*_{x(r,t)}- chordwise velocity of wing element
*v̇*_{n}- first derivative of normal velocity of wing element
*x̂*_{o}- rotational axis
*x̂*_{i}- non-dimensional chordwise location of center of incident flow
*x̂*_{r(r)}- non-dimensional chordwise location of center of rotation
- α(t)
- wing pitch
- αg(t)
- geometric angle of attack
- α′(r,t)
- angle of incidence
- βn
- added mass coefficient
- Φ(r,t)
- Wagner function
- γ(t)
- azimuth position of wing
- ρ
- fluid density
- non-dimensional rotational timing parameter
- non-dimensional duration of wing rotation
- non-dimensional duration of wing translational acceleration
- Γ
_{t} - translational circulation
- Γ
_{h} - heaving circulation
- Γ
_{r} - rotational circulation
- Γ
_{M} - Magnus circulation
- non-dimensional angular velocity

## ACKNOWLEDGEMENTS

I would like to extend special thanks to M. H. Dickinson and S. Sane for
their time and patience with my work on modeling *Drosophila* hovering.
I also thank two anonymous reviewers for greatly improving the clarity of this
manuscript. This work was supported by ONR (N00014-01-1-0506), NSF
(IBN-0119643) and the Bioscience Research Institute of Southern Maine.

- © The Company of Biologists Limited 2002