Rotational lift: something different or more of the same?
Department of Biology, University of Southern Maine, 96 Falmouth St, Portland, ME 04103, USA
(e-mail: walker{at}usm.maine.edu)
Accepted 16 September 2002
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Summary |
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Key words: unsteady aerodynamics, rotational circulation, delayed stall, Magnus effect, attached vortex lift, hovering, Drosophila flight
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Introduction |
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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) |
The distortion and resulting normal (lift) force on a pitched wing in a
uniform flow (in which case, only 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 to
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
'). The force components due to
h and
r are, therefore, Magnus-like
forces, but the component due to
h is also not the
Magnus-like force discussed by Dickinson et al.
(1999
). The force component due
to
r arises from the rotational component of the wing's
motion to the incident flow and the resulting
' (see below) and,
in this sense, it is really no different from the force components due to
t and
h. Because the
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
M is similar to that due to
r in that it is dependent on wing rotation but differs from
the
r component in that it is independent of the angle of
the incident flow. Because
M is also independent of the
chordwise center of the incident flow (in contrast to
r),
the force component due to
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
t,
h and
r as the
circulatory-and-attached-vortex force (see below), and the force due to
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).
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Materials and methods |
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While the augmented circulatory force due to r was
included in the model, Magnus-type forces due to
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: dR=R/p. The
length-specific radial position is
=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, (t), is the angle
between the rotational axis of the wing and a horizontal vector directed
posterior to the fly. In this coordinate system,
(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, (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).
(t)
is negative when the wing is pronated and is positive when the wing is
supinated. The geometric angle of attack,
g(t), is the angle
of the wing chord relative to the stroke plane and is equal to
/2+
(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, n(r,t), and chordwise,
x(r,t),
flow due to wing translation and rotation are:
![]() | (2) |
![]() | (3) |
|
The angle of incidence, '(r,t), between the wing
chord and the incident stream is ±tan-1
(
n(r,t)/
x(r,t)), where the ± takes the
sign of
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
(dFn(r,t)) and parallel with
(dFx(r,t)) the wing chord:
![]() | (4) |
![]() | (5) |
![]() | (6) |
![]() | (7) |
![]() | (8) |
![]() | (9) |
![]() | (10) |
The added mass force, or acceleration reaction, normal to the wing element
is:
![]() | (11) |
![]() | (12) |
![]() | (13) |
Finally, the Magnus force due to M can be modeled by:
![]() | (14) |
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.
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Results |
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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 downstrokeupstroke 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.
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Discussion |
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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
L across the whole stroke
cycle ranges from 6% to 25% greater than the CFD estimate (similar trends
occur with the drag coefficient CD 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
'. Two-dimensional force coefficients on
root-oscillating wings across a range of attack angles have not been
published, but the two-dimensional coefficients at
'=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
CD and, especially, CL 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
CL at all ', but the CD
of lower aspect ratio wings increases more rapidly than that of higher aspect
ratio wings as
' 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,
CL=2', on a wing section at a low angle
of incidence is derived from a simple algebraic rearrangement of:
![]() | (15) |
|
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
' 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
r(r)c(r).
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Acknowledgments |
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