The influence of wingwake interactions on the production of aerodynamic forces in flapping flight
Department of Integrative Biology, University of California, Berkeley, CA 94720, USA
* Author for correspondence (e-mail: birchj{at}socrates.berkeley.edu)
Accepted 21 March 2003
![]() |
Summary |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Key words: insect flight, flapping flight, wake structure, insect aerodynamics, DPIV, digital particle image velocimetry, flow visualization
![]() |
Introduction |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Currently, two approaches attempt to circumvent these difficulties in
measuring force production in living insects. The first is through the use of
dynamically scaled robots programmed with kinematics derived from studies of
flying or tethered animals (Bennett,
1970; Dickinson et al.,
1999
; Ellington et al.,
1996
). A second approach is to computationally simulate a solution
to the NavierStokes equation for the given pattern of motion and
geometry (Hamdani and Sun,
2000
; Liu et al.,
1996
,
1998
). Such methods offer a
complete solution for forces and flows in space and time but require extensive
computing power and may be sensitive to model parameters.
Using dynamically scaled robots, recent studies suggest that the
aerodynamic forces generated by the back-and-forth wing motion of hovering
insects may be conveniently separated into four components due to: added-mass
acceleration, translational circulation, rotational circulation and wake
capture (Dickinson et al.,
1999; Sane and Dickinson,
2002
). The forces due to translational and rotational circulation
are well-approximated by quasi-steady models
(Sane and Dickinson, 2002
).
Thus, the time history of these force components is explained in large part by
the temporal changes in wing kinematics and not the intrinsic time
dependencies in the underlying flows. Furthermore, regardless of whether the
kinematic motion generating vorticity is a steady propeller-like revolving
motion around the wing hinge (translation) or the combination of this motion
with a constant change in angle of attack (translation plus rotation),
aerodynamic forces are generated by the prolonged attachment of a leading edge
vortex. The physical bases of the forces generated during stroke reversal are
less clear due to complications caused by the requisite change of wing motion.
First, as the wing starts and stops at the beginning and end of each stroke it
is subject to acceleration-reaction forces. These forces represent the
impulsive change in momentum within the fluid imparted by the accelerating
wing (Daniel, 1984
) and are
typically modeled in quasi-steady terms using a time-invariant added-mass
coefficient (Sarpkaya, 1996
).
Because in such models the wing's influence on the surrounding fluid is
mathematically equivalent to a time-invariant increase in wing mass, the
`added-mass' force tracks the time history of wing acceleration. However,
results from experiments with impulsively and non-impulsively started bluff
bodies show that peak transient forces are delayed with respect to wing
acceleration, and thus quasi-steady models of acceleration-reaction forces are
overly simplistic (Hamdani and Sun,
2000
; Odar and Hamilton,
1964
; Sarpkaya,
1982
,
1991
,
1992
).
The second way in which the reciprocating stroke pattern complicates force
generation is through the influence of the pre-existing wake. During stroke
reversal, the wing sheds the vorticity generated during the prior stroke, and,
as it reverses direction, the wing passes through this shed vorticity field.
Under certain circumstances, this flow field can influence force at the start
of the stroke, a mechanism previously termed `wake capture'
(Dickinson et al., 1999;
Sane and Dickinson, 2001
). Due
to the complexity of the flows at stroke reversal, it is unlikely that wake
capture will be amenable to quasi-steady approximations. Nevertheless, there
are several means by which this wingwake interaction might be
incorporated within the current quasi-steady framework. First, the forces due
to the influence of the wake might simply represent an augmentation of
circulatory forces generated by the altered flow field at stroke reversal, a
hypothesis analogous to the influence of gust on steady flight in aeroplanes
(McCormick, 1995
). This idea
could be tested by directly measuring the actual velocity and orientation of
the flow around the wing at the start of the stroke. Such `corrected' values
for instantaneous velocity and angle of attack could then be fed into a
quasi-steady model for translational and rotational forces, and the results
compared with measured values. In addition, if wake capture is simply an
augmentation of steady-state circulatory forces acting throughout the stroke,
the strength of the leading edge vortex should track the magnitude of the
instantaneous force.
Another explanation for wake capture is that it represents an
acceleration-reaction force caused by the rush of fluid against the wing at
the start of each stroke. This effect is responsible for large drag forces on
plants and sedentary animals in wave-swept environments
(Daniel, 1984;
Denny, 1988
). Prior studies of
wake capture demonstrated that a stationary wing, stopped after completing one
stroke, continues to generate force as it is impacted by a jet of fluid within
the wake (Dickinson et al.,
1999
). Simulations of this effect in two dimensions using
computational fluid dynamics (CFD; Hamdani
and Sun, 2000
) are consistent with experimental results using
identical kinematics (Dickinson,
1994
). Because the wing is moving through a vortex jet, the
acceleration-reaction forces should be greater than those expected if the wing
were to accelerate through still fluid. This hypothesis could best be tested
by careful quantification of flow structure combined with instantaneous force
measurement. However, such an analysis would be hindered by the lack of a
sufficiently accurate model of acceleration-reaction forces.
The difficulty encountered in constructing a reliable estimate of
acceleration-reaction forces underscores a general problem with the
multi-component quasi-steady approach. Although the net aerodynamic force may,
for utility and convenience, be divided into components resulting from
translational circulation, rotational circulation, added mass, etc., such
divisions are to some extent arbitrary. All fluid forces acting on a submerged
body result from physical interactions succinctly expressed in the
NavierStokes equation. Although the utility of simpler time-invariant
models is obvious for applications in both biology and engineering, such
methods may not provide sufficiently accurate descriptions. Furthermore, as
illustrated by the concept of `added mass', such methods may obscure the
underlying physical basis of observed forces. As discussed by Wu
(1981), an equation that
conveniently captures all aerodynamic force (F) acting on a solid body
within a fluid is:
![]() | (1) |
![]() | (2) |
In this study, we use DPIV (Raffel et
al., 1998) to quantify chord-wise flow dynamics during the
flapping motion of a robotic wing. The DPIV is paired with simultaneous force
measurement, permitting a direct comparison of flow dynamics and force
production. Our basic approach is to examine the pattern of fluid flow and
force generation during a continuous sequence of strokes starting from rest.
Of particular interest is the comparison of the initial stroke, when the wing
begins moving through still fluid, with later strokes, when the wing must move
through the shed vorticity of prior strokes. The results show that large force
peaks, previously attributed to added mass, are best explained via
the vortex moment equation as a rapid growth in vorticity and not by any
quasi-steady formulation in which forces are proportional to the instantaneous
magnitude of vorticity. Furthermore, they show that the influence of the wake
on force generation may be divided into two phases during each stroke: an
early augmentation (wake capture), followed by a subsequent attenuation. Like
the initial force peak generated by a wing starting from rest, wake capture
cannot be explained by the instantaneous magnitude of vorticity. Rather, the
wake capture forces are best explained by the altered growth rate of vorticity
as the wing passes through the shed vorticity of the prior stroke. The later
decrease in force results from an attenuation of translational circulation
caused by downwash induced by the wake. Collectively, these findings provide a
direct view of wake dynamics during flapping flight and quantify the potential
influence of the shed vorticity of previous strokes on force production. They
also provide empirical results with which to test recent numerical simulations
based on nearly identical stroke kinematics
(Ramamurti and Sandberg, 2002
;
Sun and Tang, 2002
).
![]() |
Materials and methods |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
|
These experiments focused on simultaneous measurement of instantaneous
forces and flows using a simple back-and-forth wingbeat pattern. This pattern
was chosen because all four mechanisms of force production possible in
flapping flight are present (Sane and
Dickinson, 2001). The wing flapped through 160° of amplitude
with a 45° angle of attack at midstroke. Following the convention of Sane
and Dickinson (2001
), we
defined rotation parameters (
) as percentages of a complete wing stroke.
Thus,
0 represents the time when wing rotation begins,
f represents flip timing (when the midpoint of the flip
occurs) and
t is flip duration. In this experiment,
0=0.12, tf=0.06 and
=0.12. Thus, wing rotation was advanced relative to stroke
reversal by 12% of the stroke period and was completed at stroke reversal. The
wings did not deviate from the stroke plane, and the upstroke and downstroke
were identical by mirror symmetry. However, slippage between the teeth of the
gear box introduced inaccuracies of approximately 1-5° in the angle of
attack, which could result in small differences between up- and downstrokes.
Flapping the wing at 168 mHz generated maximum tip velocities at midstroke of
0.31 m s-1, with a mean wingtip velocity over the entire stroke of
0.26 m s-1. These kinematics corresponded to a Reynolds number of
approximately 160 based on the velocity of the chord section in which we
visualized flow. In order to correlate force and flow information, we express
time during the stroke as a non-dimensional parameter,
, such that
=0 at the start of the downstroke,
and
=1 at the end of the subsequent
upstroke (Fig. 1B).
Flow visualization
We used digital particle image velocimetry (DPIV) to quantify the flow
structure in a slice of fluid centered on the wing. Prior to each experiment,
we seeded the oil with air forced through a ceramic water filter stone to
create a dense bubble field. After larger bubbles rose to the surface, the
remaining bubbles, although slightly positively buoyant, did not rise
perceptively during capture of the paired DPIV images. Forces measured with
bubbles in the tank were identical to those measured in the absence of
bubbles, indicating that their introduction did not alter the basic properties
of the medium. A commercial software package controlling a dual Nd-YAG laser
system (Insight v. 3.2, TSI Inc., St Paul, MN, USA) created two identically
positioned light sheets approximately 2.5 mm thick separated by 2 ms
(Fig. 1C). These light sheets
were positioned at 0.65R (R=length of one wing) and timed to
fire when the wing chord was directly in front of a high-speed video camera
placed perpendicular to the laser sheet
(Fig. 1D). We chose
0.65R as our point of measurement because, in a prior DPIV study,
this was the position at which the leading edge vortex was still attached and
exhibited near-maximal spanwise vorticity
(Birch and Dickinson, 2001).
After saving the captured images from the two laser flashes, the trigger for
the laser was advanced 100 ms, and the starting position of the wing moved
backwards so that at the start of the next sequence it would pass in front of
the camera at a slightly later point in the sequence of stroke cycles. In this
way, we captured the fluid flow around the wing through four
downstroke/upstroke cycles at 100 ms intervals. In order to facilitate a more
intuitive interpretation of fluid motion, we subtracted wing speed from fluid
velocity, so that the fluid is visualized from a frame of reference that
follows the span of the wing (Fig.
2).
|
For each image-pair captured, a cross-correlation of pixel intensity peaks with 50% overlap of 64 pixelx64 pixel interrogation areas yielded a 30x30 array of vectors. The effectiveness of our bubble-seeding density was evident during vector validation. After creating velocity vectors for 236 images (59 images stroke-1 x four strokes), the magnitude of only two out of a possible 212 400 vectors was greater than three standard deviations of the mean length in their respective images. These two deleted values were filled by interpolation of a mean value from a 3x3 nearest-neighbor matrix. Aside from these two corrections, we performed no additional filtering or modification of the flow data.
A program written in MATLAB was used to calculate vorticity from the velocity fields and perform all subsequent flow measurements. By gathering a complete time history of fluid velocity through four strokes, we could observe the growth of vorticity in select regions around the wing throughout each stroke and compare velocity and vorticity between strokes. In some calculations, vorticity panels were subject to a threshold mask that recognized only the top 10% of vorticity values. This criterion was applied to all panels of all strokes and resulted in the description and quantification of four major regions of vorticity: the leading edge vortex (LEV), the under wing shear layer (USL), the translational starting vortex (TSV) and the rotational starting vortex (RSV). To isolate and visualize the wake created by prior strokes from the flows generated during a current stroke, we subtracted the instantaneous velocity fields measured during the first stroke from those measured during the fourth stroke. These resulting differences in both flow and forces represented the influence of vorticity shed during prior strokes on subsequent aerodynamic performance.
In order to estimate sectional forces from the DPIV images, we calculated
the first moment of the vorticity field, , using the center of the wing
as the origin for the position vector. A custom program in MATLAB then
calculated the time derivative of this term. Multiplying by -1 and fluid
density provides the instantaneous sectional force predictions to compare with
measured forces.
Estimating the aerodynamic angle of attack
To test whether a corrected quasi-steady model can explain the influence of
the wake, it is necessary to estimate the distortion of the aerodynamic angle
of attack, aero, caused by shed vorticity. Although the
concept of
aero is straightforward, in practice its
measurement is problematic. Because the wing functions as an impermeable
boundary, fluid upstream is deflected gradually downward from the free stream
orientation until it flows parallel to the wing's surface. This deflection is
not due to the presence of chord-wise vorticity and is present even in
two-dimensional flow. In three-dimensional flow, the presence of chord-wise
vorticity from the wing tip or other sources deflects the flow downward to an
even greater degree than in the two-dimensional case. Thus, any experimental
measurement of downwash should be made relative to the deflection required by
the boundary condition of tangential flow at the surface of the wing. In the
case of the model fly wing, the downwash caused by the wake of past strokes
may be conveniently measured by comparing flows of a starting stroke, when the
wake is just developing, with later strokes, when the wake is fully entrained.
Another complication in measuring
aero is that the effect of
fluid incidence on circulatory forces is not restricted to any specific region
ahead of the wing. Thus, it may be misleading to measure the effect of
downwash within a defined interrogation region. We chose a region upstream of
the wing that was large enough so that both the mean
aero
and fluid velocity during a starting stroke were similar to values dictated by
wing kinematics.
![]() |
Results |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
|
|
|
|
|
Plots of vorticity and velocity through the downstroke of the first and
fourth strokes provide a quantitative comparison of flow dynamics. In the
first stroke (Fig. 4), the
initial force peak, which we attribute to acceleration-reaction force, occurs
as the LEV grows (compare the red arrow and blue region,
=0.030.07). The growth of the
LEV occurs as a sheet of counter vorticity under the wing rolls up into a
translational starting vortex (TSV). The LEV reaches its final size after
approximately 1.5 chord lengths of travel. During subsequent translation in
which the wing moves at constant velocity away from the TSV, the LEV reaches a
stable size and force production remains relatively constant
(
=0.170.38). At
=0.39, the wing begins to rotate,
which increases both the size of the LEV and the magnitude of the net force.
This increase in force is due to the contribution of rotational circulation
and the influence of the increased angle of attack on translational
circulation (Sane and Dickinson,
2002
). As the wing rotates, the under wing shear layer (USL) rolls
up into a vortex under the trailing edge. This vortex counterbalances the
additional vorticity within the LEV that results from rotation and thus
represents a rotational starting vortex (RSV). Force drops precipitously at
stroke reversal when translational velocity falls to zero. Rotation results in
the shedding of the LEV and RSV, which form a counter-rotating pair upstream
of the wing at the onset of the next stroke.
At the beginning of the fourth stroke, fluid flow displays a more
complicated pattern than in the first stroke due to the presence of shed
vorticity within the wake (Fig.
5). As the wing travels into the vortex pair composed of the shed
LEV and RSV, these two vortices direct a jet of high velocity fluid towards
the underside of the wing
(=0.000.09), resulting in an
instantaneous force peak of 1.1 N at
=0.05. As the wing continues
translating, the vorticity from the remnants of the last LEV, combined with a
new USL, eventually roll up into a new starting vortex, which is substantially
larger than the starting vortex created during the first stroke (compare
Fig. 4,
=0.10 with
Fig. 5,
=0.10). The RSV lies directly beneath
this new combined translational starting vortex, directing another jet of
fluid rearward under the trailing edge. With continued motion, the wing passes
through the wake (Fig. 5;
=0.15) and forces drop to nearly half
the peak level (Fig. 5; 0.56 N
at
=0.15).
An additional vortex structure seen from
=0.00 to
=0.19 in the fourth stroke (but
absent during the same period in stroke one) results from the two-dimensional
view of the complicated three-dimensional structure of the wake at stroke
reversal. Two concentrations of clockwise vorticity (blue) are visible at the
start of stroke four: the starting vortex from rotation described above and a
slice through the arc-shaped tip vortex of the prior stroke
(Fig. 6). Note that as the wing
moves through the stroke, it eventually reaches fluid where the tip vortex has
moved downward. By
=0.19, the only
remaining influence of the prior stroke's tip vorticity is the induced
downward flow, which may be seen by comparing the flow pattern at
=0.19 in Figs
4 and
5.
|
Fig. 7 provides a schematic
summary representation of the growth and movement of the major areas of
vorticity as the wing goes through one complete stroke cycle. At the start of
the downstroke, the arrangement of vortices is relatively simple, consisting
of a developing LEV, an attached TSV and a USL. As the stroke proceeds, the
TSV sheds (=0.20). Near the end of
the downstroke, the wing slows and rotates and an RSV develops at the trailing
edge. After reversing direction, the wing slices through both the LEV and the
RSV at the start of the upstroke. This movement through the previous vorticity
accelerates slightly the build-up of the new LEV (LEV2 at
=0.52). Just after translation
begins, a new USL forms (USL2 at
=0.55) that is connected to the shed
TSV (TSV2). This new starting vortex (TSV2) is part of a
doublet with the shed RSV from the prior stroke (RSV1). This
doublet is shed as the wing progresses through translation and, upon stroke
reversal, the process begins again.
|
First moment of vorticity during an impulsive start
In order to measure flow near the wing with sufficient spatial resolution,
we deliberately chose a field of view that did not capture all vorticity
throughout repeating strokes. In addition, an enlarged planar view would be
insufficient to measure the salient features of the three-dimensional flow
structures generated by the flapping wing. For these reasons, it was not
possible to estimate forces from the time-derivative of the first moment of
vorticity (equation 1) throughout the entire stroke. However, the flow
structure at the beginning of the first stroke was sufficiently compact to
allow an estimate of forces based on equation 1. This calculation should
provide insight into the physical basis of the force transient at the start of
the first stroke. While the magnitude of the acceleration term in equation 1
contributed little to the overall force, the time course of the calculated
sectional force based on the vortex moment matches well the measured lift and
drag over the first 16% of the stroke (Fig.
8). After that, vorticity leaving the field of view renders the
estimate inaccurate. If the predicted sectional lift is multiplied by wing
length, the resulting peak is roughly 50% greater than that of the measured
lift. Such a discrepancy is expected because we deliberately measured flow
within the section where chord-wise vorticity has been shown to be greatest
(Birch and Dickinson, 2001).
The results, combined with an inspection of
Fig. 4, suggest that the early
force transient is due to the rapid growth of vorticity at the start of the
stroke.
|
Influence of wake on force production
Unfortunately, it was not possible to calculate force from the first moment
of vorticity during wake capture because the flow distribution extended well
beyond our field of view (Fig.
5). Although methods exist to compensate for vorticity flux across
the boundary of a control volume in two dimensions
(Noca et al., 1997), these
methods did not prove robust when applied to our data. However, because both
strokes follow identical kinematics, subtracting flow fields of stroke one
from those of stroke four provides an explicit picture of the wake from prior
strokes, independent of the fluid motion created by wing motion within the
stroke itself. Such reconstructions should provide insight into the physical
basis of forces caused by the presence of the wake.
Fig. 9 shows the reconstructed
fluid velocity of the wake, representing the point-by-point difference in flow
between the fourth and first stroke. From
=0.00 to
=0.05, the primary feature of the
difference in flows between the fourth and first strokes is an area of
elevated fluid velocity upstream of the wing. This barbell-shaped jet of fluid
has two regions of peak velocity: an upper area corresponding to the flow
between the counter-clockwise LEV and the clockwise remnant of the tip vortex
from the prior stroke and a lower area representing the jet produced from the
prior stroke between the counter-clockwise LEV and the clockwise RSV. This jet
moves rearward, just under the trailing edge from
=0.02 to
=0.12. The panels also show an
induced downward flow, which is strongest from
=0.09 to
=0.15 and then dissipates from
=0.22 to
=0.33 as the stroke proceeds. By
stroke reversal (
=0.430.48),
the influence of the wake is barely measurable.
The vorticity fields corresponding to the reconstructed velocity fields of
Fig. 9 are shown in
Fig. 10. From
=0.02 to
=0.07, there is a small region of
clockwise vorticity (blue) at the leading edge, signifying that the LEV is
slightly stronger at the start of stroke four compared with stroke one. The
situation reverses, however, and by
=0.09 a counter-clockwise vorticity
(red) appears at the leading edge, indicating that the LEV of stroke four is
weaker than in stroke one. By
=0.12,
a clockwise layer of vorticity (blue) forms just above the surface of the
wing, bounded above by a counter-clockwise layer (red). This indicates that
from
=0.12 to
=0.26, the LEV in stroke four is more
closely attached to the surface of the wing than in stroke one. After
=0.30, the difference in the
structure of the LEVs of the two strokes is quite small, consistent with the
near identical force records at this phase in the cycle.
To test whether quasi-steady equations corrected for the values of the
instantaneous velocity field might be capable of explaining either the early
augmentation or late attenuation of forces during stroke four
(Fig. 3A), we quantified the
mean velocity and orientation of fluid within a 260 cm2 region in
front of the wing (Fig. 11A).
Fluid velocity within this interrogation region, measured from
=0.00 to
=0.20, was approximately that of the
robotic flapping speed during stroke one
(Fig. 11B). In addition, the
wing intercepts fluid during the first downstroke at an angle very close to
the 45° kinematic angle of attack
(Fig. 11C). Although the mean
fluid velocity within this same region was 52% higher during the fourth stroke
(Fig. 11B), the aerodynamic
angle of attack,
aero, estimated from the mean orientation
of flow within the rectangular region was between 10° and 20°.
Substituting these values into the quasi-steady model of translational force
(Sane and Dickinson, 2002
)
explains less than 65% of the peak force during wake capture
(Fig. 11D). This suggests that
the additional forces created as the wing passes through the wake of the
previous stroke result from spatial and temporal changes in flow that are not
accounted for in a quasi-steady model of a circulatory force. By contrast, the
corrected quasi-steady model does predict with reasonable accuracy the later
drop in force due to downwash.
|
![]() |
Discussion |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Quasi-steady models and the first moment of vorticity
Forces generated once the LEV has reached a stable size are
well-approximated by a quasi-steady semi-empirical model
(Sane and Dickinson, 2002).
This is of course expected, because this model uses an empirical force
coefficient that is itself measured under conditions of constant velocity.
When a stable distribution of vorticity is attached to the wing, changes in
the first moment of vorticity (
; equation 1) result from the motion of
the vorticity distribution through the fluid, conditions that satisfy the
quasi-steady assumptions. This convergence was described by Wu
(1981
), who noted that one may
derive the KuttaJoukowski theorem from equation 1 for a wing moving at
constant velocity with bound circulation.
Consistent with other experimental studies of impulsively started plates
and bluff bodies (Hamdani and Sun,
2000; Odar and Hamilton,
1964
; Sarpkaya,
1982
,
1991
,
1992
), we measured a large
force transient at the onset of motion. Although this transient has been
previously attributed to added mass (Sane
and Dickinson, 2002
), quasi-steady approximations do not
accurately capture the precise time course of this early force peak. The
results of DPIV show that this early force occurs before the LEV attains its
final size (Fig. 4) and thus
cannot be explained by a steady-state circulation. However, the time course of
force at the start of translation matches well with the time derivative of
(Fig. 8). This confirms
the computational work of Hamdani and Sun
(2000
), who used CFD to
simulate the forces created by an impulsively started flat-plate in two
dimensions. The forces in their simulated flows, calculated according to
equation 1, accurately predicted those measured in a prior experimental study
(Dickinson and Götz,
1993
). Together with this previous work, our results suggest that
the time course of force production by an impulsively started wing may be
roughly divided into two parts: an early phase in which the LEV grows rapidly
followed by a subsequent period in which a stable LEV remains attached as the
wing moves away from the starting vortex. During the first phase,
increases largely due to the growth of vorticity, whereas later it rises more
slowly due to the increasing separation between the LEV and the TSV. These two
phases are directly observed as a change in the slope of
in the
two-dimensional simulations of Hamdani and Sun
(2000
).
The physical basis of the wake capture
A corrected quasi-steady model that accounts for the altered flow field
caused by the shed vorticity of prior strokes cannot account for the elevated
forces generated during wake capture (Fig.
11D). This result is consistent with the observation that the LEV
is quite small at the time when the wake capture effect is greatest
(Fig. 5). This condition is
similar to that which occurs during the early transient following an impulsive
start, in that the instantaneous magnitude of vorticity cannot account for the
forces generated by the wing. Thus, wake capture is a truly unsteady effect,
and, in order to derive forces from the flow, it is necessary to employ a
model, such as equation 1, that accounts for time-dependent changes in the
flow. An inspection of vorticity fields indicates that their structure changes
both in space and time at the start of the stroke. Unfortunately, attempts to
calculate the first moment of vorticity proved unreliable in the current data
set due to the flux across the boundaries of the visualized region. Thus, we
could not confirm whether the derivative of measured within a
span-wise section was consistent with the time course of forces during wake
capture. However, given that the wing must move through a complex system of
several shed vortices (Figs 5,
9,
10), the influence on the
first moment of vorticity will be quite large. A thorough quantitative
analysis will require a larger map of the flow around the wing that encloses
all the salient features of the flow. Future studies must also take into
account the three-dimensional nature of the flow.
The influence of downwash
Unlike the case of wake capture, our results suggest that quasi-steady
models can account for the wake-dependent drop in force at midstroke
(Fig. 3). This result is not
surprising, given that the structure of the flow within the wake at this time
consists of a relatively constant unidirectional downward flow below the wing.
Thus, the flow pattern fulfills the basic assumptions for a classic induced
drag model, in which downwash lowers the angle of aerodynamic angle of attack,
thereby altering the steady-state circulation created by a wing or propeller.
However, the present condition differs from induced drag in two important
ways. First, the downwash effect is pulsatile; the downward flow generated by
the shed vortices grows over the first half of the stroke but then slowly
decreases (Fig. 10). This time
dependence of the downwash explains why the force traces for the first and
fourth strokes converge by =0.30.
Second, in standard models of downwash
(McCormick, 1995
), the
decrease in
aero causes an increase in drag but no change in
lift. However, this simplification is only valid for small angle
approximations and unseparated flow conditions that do no apply in the
current case. For a flat wing moving at a large angle of attack, the effect of
downwash will be a decrease in both lift and drag
(Birch and Dickinson, 2001
),
which, except for the small drag component due to skin friction, are simply
orthogonal components of a single pressure force that operates perpendicular
to the surface of the wing (Dickinson,
1996
; Usherwood and Ellington,
2002
).
Comparison of experimental results with CFD simulations
Sun and Tang (2002) and
Ramamurti and Sandberg (2002
)
have recently presented CFD models based on wing kinematics nearly identical
to those used in this study, providing an opportunity to compare
experimentally measured forces and flows with state-of-the-art computational
techniques. The present study corroborates many aspects of these simulations.
First, mean forces generated in the CFD models are within 15% of those
generated here [mean lift coefficient for a down/upstroke sequence: 1.2
(Sun and Tang, 2002
); 1.29
(Ramamurti and Sandberg,
2002
); 1.4 (present study)]. The source of this slight discrepancy
is difficult to identify and might be due to variations in the precise
kinematic patterns and wing morphologies used, computational inaccuracies or
experimental error. Second, the general shape of the force traces in the
simulations resembles the measured forces, particularly the translation phase
during the middle of the stroke and the rotational lift phase at the end of
each stroke. Sun and Tang
(2002
) attribute the increased
lift and drag at the end of each stroke to the instantaneous increase in
translational forces due to the `pitching-up rotation' of the wing, a
hypothesis that they claim contradicts that of Dickinson et al.
(1999
). It is not clear
whether their `pitching-up hypothesis' implies that rotational forces are
caused by the increased angle of attack, which could thus be explained by a
quasi-steady model, or a circulatory force due to the time-derivative of the
angle of attack. If the former, this hypothesis is not consistent with
experimental data (Dickinson et al.,
1999
; Sane and Dickinson,
2001
,
2002
). Collectively, these
experiments show that wing rotation causes an augmentation of force that is
not explained by a quasi-steady translation model that takes into account the
instantaneous angle of attack. If, however, the hypothesis of Sun and Tang is
that rotational forces are dependent on the angular velocity of the wing about
its long axis (d
/dt), then their view is entirely consistent
with experimental data. Forces during rotation are proportional to
d
/dt and vary linearly with the cord-wise position of the
rotational axis (Dickinson, et al.,
1999
; Sane and Dickinson,
2002
), in close agreement with theoretical predictions
(Fung, 1969
).
Thus, the only significant discrepancy between the simulated and empirical
results is the prediction of an early force peak in the first and subsequent
strokes. Sun and Tang (2002)
claim to find no evidence of this wake capture peak, whereas the CFD
simulations of Ramamurti and Sandberg
(2002
) show a wake capture
peak that is consistent with prior experimental studies
(Dickinson et al., 1999
). Sun
and Tang suggest that the force peak following stroke reversal is due to the
rapid acceleration of the wing rather than the recapture of vorticity from the
wake. However, this hypothesis was tested in the current study by directly
comparing forces and flows in the presence and absence of prior strokes (Figs
3,
10). Although an impulsively
started wing generates an acceleration-dependent force, the forces created by
identical kinematics in the presence of a wake are unequivocally higher.
General significance for insect flight
Together with other recent experimental studies
(Ellington et al., 1996;
Dickinson et al., 1999
; Sane
and Dickinson, 2001
,
2002
;
Srygley and Thomas, 2002
),
these results help to solidify an emerging picture of the force and flow
dynamics of flapping wings. This study presents force measurements and flow
patterns for an arbitrary pattern of wing motion, chosen in part because it
creates forces by all the mechanisms currently known to function on single
wings. For this reason, it represents a convenient model system for analyzing
the underlying fluid mechanics of insect flight. This simple set of kinematics
and the resulting time history of forces and flows should not be
misinterpreted, however, as being characteristic of insects in general. The
actual patterns of wing motion used by different insects, or any individual at
different moments, are diverse (Srygley
and Thomas, 2002
). It remains to be determined how the relative
importance of different mechanisms or the interactions among them change with
evolution and behavior.
![]() |
Acknowledgments |
---|
![]() |
Footnotes |
---|
![]() |
References |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Bennett, L. (1970). Insect flight: lift and the rate of change of incidence. Science 167,177 -179.
Birch, J. M. and Dickinson, M. H. (2001). Spanwise flow and the attachment of the leading-edge vortex on insect wings. Nature 412,729 -733.[CrossRef][Medline]
Brodsky, A. K. (1994). The Evolution of Insect Flight. Oxford: Oxford University Press.
Cloupeau, M., Devillers, J. F. and Devezeaux, D. (1979). Direct measurements of instantaneous lift in desert locust; comparison with Jensen's experiments on detached wings. J. Exp. Biol. 80,1 -15.
Daniel, T. L. (1984). Unsteady aspects of aquatic locomotion. Am. Zool. 24,121 -134.
Denny, M. W. (1988). Biology and the Mechanics of the Wave-Swept Environment. Princeton: Princeton University Press.
Denny, M. W. (1993). Air and Water. The Biology and Physics of Life's Media. Princeton: Princeton University Press.
Dickinson, M. H. (1994). The effects of wing
rotation on unsteady aerodynamic performance at low Reynolds numbers.
J. Exp. Biol. 192,179
-206.
Dickinson, M. H. (1996). Unsteady mechanisms of force generation in aquatic and aerial locomotion. Am. Zool. 36,537 -554.
Dickinson, M. H. and Götz, K. G. (1993).
Unsteady aerodynamic performance on model wings at low Reynolds numbers.
J. Exp. Biol. 174,45
-64.
Dickinson, M. H. and Götz, K. G. (1996).
The wake dynamics and flight forces of the fruit fly Drosophila
melanogaster. J. Exp. Biol.
199,2085
-2104.
Dickinson, M. H., Lehmann, F. O. and Sane, S.
(1999). Wing rotation and the aerodynamic basis of insect flight.
Science 284,1954
-1960.
Ellington, C. P. (1984). The aerodynamics of hovering insect flight. V. A vortex theory. Phil. Trans. R. Soc. Lond. B 305,115 -144.
Ellington, C. P., Van Den Berg, C., Willmott, A. P. and Thomas, A. L. R. (1996). Leading-edge vortices in insect flight. Nature 384,626 -630.[CrossRef]
Fung, Y. C. (1969). An Introduction to the Theory of Aeroelasticity. New York: Dover.
Grodnitsky, D. L. and Morozov, P. P. (1993).
Vortex formation during tethered flight of functionally and morphologically
two-winged insects, including evolutionary considerations on insect flight.
J. Exp. Biol. 182,11
-40.
Hamdani, H. and Sun, M. (2000). Aerodynamic forces and flow structures of an airfoil in some unsteady motions at small Reynolds number. Acta Mech. 145,173 -187.
Liu, H., Ellington, C. P., Kawachi, K., Van Den Berg, C. and
Willmott, A. (1998). A computational fluid dynamic study of
hawkmoth hovering. J. Exp. Biol.
201,461
-477.
Liu, H., Wassersug, R. J. and Kawachi, K.
(1996). A computational fluid dynamics study of tadpole swimming.
J. Exp. Biol. 199,1245
-1260.
McCormick, B. W. (1995). Aerodynamics, Aeronautics, and Flight Mechanics. Second edition. New York: John Wiley & Sons.
Noca, F., Shiels, D. and Jeon, D. (1997). Measuring instantaneous fluid dynamic forces on bodies, using only velocity fields and their derivatives. J. Fluids Struct. 11,345 -350.[CrossRef]
Noca, F., Shiels, D. and Jeon, D. (1999). A comparison of methods for evaluating time-dependent fluid dynamic forces on bodies, using only velocity fields and their derivatives. J. Fluids Struct. 13,551 -578.[CrossRef]
Odar, F. and Hamilton, W. S. (1964). Forces on a sphere accelerating in a viscous fluid. J. Fluid Mech. 18,302 -314.
Raffel, M., Willert, C. and Kompenhans, J. (1998). Particle Image Velocimetry: a Practical Guide. Berlin: Springer-Verlag.
Ramamurti, R. and Sandberg, W. C. (2002). A
three-dimensional computational study of the aerodynamic mechanisms of insect
flight. J. Exp. Biol.
205,1507
-1518.
Rayner, J. M. V. (1979). A new approach to animal flight mechanics. J. Exp. Biol. 80, 17-54.
Sane, S. P. and Dickinson, M. H. (2001). The
control of flight force by a flapping wing: lift and drag production.
J. Exp. Biol. 204,2607
-2626.
Sane, S. P. and Dickinson, M. H. (2002). The
aerodynamic effects of wing rotation and a revised quasi-steady model of
flapping flight. J. Exp. Biol.
205,1087
-1096.
Sarpkaya, T. (1982). Impulsively-started flow about four types of bluff body. J. Fluids Eng. 104,207 -213.
Sarpkaya, T. (1991). Nonimpulsively started steady flow about a circular cylinder. AIAA J. 29,1283 -1289.
Sarpkaya, T. (1992). Brief reviews of some time dependent flows. J. Fluids Eng. 114,283 -298.
Sarpkaya, T. (1996). Unsteady flows. In Handbook of Fluid Dynamics and Fluid Machinery, vol.1 (ed. J. A. Schetz and A. E. Fuhs), pp.697 -732. New York: John Wiley & Sons.
Spedding, G. R., Rayner, J. M. V. and Pennycuick, C. J. (1984). Momentum and energy in the wake of a pigeon (Columba livia) in slow flight. J. Exp. Biol. 111,81 -102.
Srygley, R. B. and Thomas, A. L. R. (2002). Unconventional lift-generating mechanisms in free-flying butterflies. Nature 420,660 -664.[CrossRef][Medline]
Sun, M. and Tang, J. (2002). Unsteady
aerodynamic force generation by a model fruit fly wing in flapping motion.
J. Exp. Biol. 205,55
-70.
Usherwood, J. R. and Ellington, C. P. (2002).
The aerodynamics of revolving wings I. Model hawkmoth wings. J.
Exp. Biol. 205,1547
-1564.
Wilkin, P. J. and Williams, M. H. (1993). Comparison of the aerodynamic forces on a flying sphingid moth with those predicted by quasi-steady theory. Phys. Zool. 66,1015 -1044.
Willmott, A. P., Ellington, C. P. and Thomas, A. L. R. (1997). Flow visualization and unsteady aerodynamics in the flight of the hawkmoth Manduca sexta. Phil. Trans. R. Soc. Lond. 352,303 -316.[CrossRef]
Wu, J. C. (1981). Theory for aerodynamic force and moment in viscous flows. AIAA J. 19,432 -441.
Zanker, J. M. and Götz, K. G. (1990). The wing beat of Drosophila melanogaster. II. Dynamics. Phil. Trans. R. Soc. Lond. B 327,19 -44.