Unsteady forces and flows in low Reynolds number hovering flight: two-dimensional computations vs robotic wing experiments
1 Theoretical and Applied Mechanics, Cornell University, Ithaca, NY 14853,
USA
2 Integrative Biology, University of California, Berkeley, CA 94720,
USA
3 Bioengineering, California Institute of Technology, Pasadena, CA 91125,
USA
* Author for correspondence (e-mail: jane.wang{at}cornell.edu)
Accepted 3 October 2003
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Summary |
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In the range of amplitudes studied here, 35 chords, the force coefficients have a weak dependence on stroke amplitude. As expected, the forces are sensitive to the phase between stroke angle and angle of attack, a result that can be explained by the orientation of the wing at reversal. This dependence on amplitude and phase suggests a simple maneuver strategy that could be used by a flapping wing device.
In all cases the unsteady forces quickly reach an almost periodic state with continuous flapping. The fluid forces are dominated by the pressure contribution. The force component directly proportional to the linear acceleration is smaller by a factor proportional to the ratio of wing thickness and stroke amplitude; its net contribution is zero in hovering. The ratio of wing inertia and fluid force is proportional to the product of the ratio of wing and fluid density and the ratio of wing thickness and stroke amplitude; it is negligible in the robotic wing experiment, but need not be in insect flight.
To identify unsteady effects associated with wing acceleration, and coupling between rotation and translation, as well as wake capture, we examine the difference between the unsteady forces and the estimates based on translational velocities, and compare them against the estimate of the coupling between rotation and translation, which have simple analytic forms for sinusoidal motions. The agreement and disagreement between the computed forces and experiments offer further insight into when the 3D effects are important.
A main difference between a 3D revolving wing and a 2D translating wing is the absence of vortex shedding by a revolving wing over a distance much longer than the typical stroke length of insects. No doubt such a difference in shedding dynamics is responsible in part for the differences in steady state force coefficients measured in 2D and 3D. On the other hand, it is unclear whether such differences would have a significant effect on transient force coefficients before the onset of shedding. While the 2D steady state force coefficients underpredict 3D forces, the transient 2D forces measured prior to shedding are much closer to the 3D forces. In the cases studied here, the chord is moving between 3 to 5 chords, typical of hovering insect stroke length, and the flow does not appear to separate during each stroke in the cases of advanced and symmetrical rotation. In these cases, the wing reverses before the leading edge vortex would have time to separate even in 2D. This suggests that the time scale for flow separation in these strokes is dictated by the flapping frequency, which is dimensionally independent. In such cases, the 2D unsteady forces turn out to be good approximations of 3D experiments.
Key words: insect flight, computational fluid dynamics, biofluid dynamics, vorticity field, two-dimensional force, three-dimensional force
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Introduction |
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Given the complexity of modeling fluid flows in three dimensions
(Liu et al., 1998;
Sun and Tang, 2002
;
Ramamurti and Sandberg, 2002
),
it would be desirable to determine if simpler models provide results that are
consistent with those generated in experiments. Here, we compare
two-dimensional (2D) computations of hovering flight against robotic wing
experiments. 2D computations are appealing partly because of their relative
simplicity and efficiency. Obviously, 2D computations cannot predict
three-dimensional (3D) effects; on the other hand, it is almost impossible to
attach the significance of 3D effects without knowing what happens in 2D flow.
Therefore, in addition to being relevant to cases where the flow is
approximately 2D, as with large wing aspect ratio, when compared with 3D
experiments or computations, 2D computations can offer useful insight into the
relative significance of 3D effects, as we will discuss at the end of the
paper.
Comparing computations and experiments is delicate, partly because it is almost impossible to match any two setups exactly, and partly because it is tempting to present results that compare well, thereby biasing the interpretations. Therefore, it is essential to test the methods in qualitatively different flows generated by different wing kinematics.
In addition to comparing the experimental and computational forces, we also
evaluate the relative importance of unsteady effects. These include wing
acceleration, both in translation and rotation, and interactions between the
wing the existing flow. Most recent work using a robotic fruit fly focused on
kinematics based on tethered flight measurements. These kinematics have
relatively constant translational velocity in the mid-stroke and large
accelerations and sharp rotations at the end of strokes. In these strokes,
pronounced peaks appear near the end of each stroke. These peaks were
attributed to either wing rotation and wake capture
(Dickinson et al., 1999;
Birch and Dickinson, 2003
), or
rotational and translational acceleration
(Sun and Tang, 2002
;
Sane and Dickinson, 2002
). The
sinusoidally varying strokes studied here offer a set of kinematics where the
relative contribution of some of the dynamic effects can be theoretically
estimated. For example, we estimate the relative contribution that wing
rotation and acceleration make to the quasi-steady forces. We also estimate
the wing inertia relative to the fluid force, as well as the non-inertial
forces due to wing acceleration relative to the pressure forces associated
with vorticity flux. Given that the free flight kinematics of fruit flies
appear to be more sinusoidal than those derived from tethered flight
(Fry et al., 2003
), our
results, though using an idealized kinematics, may nevertheless relate to the
forces generated by the real flies.
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Materials and methods |
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We used digital particle image velocimetry (DPIV) to measure the flow
structure in a 841 cm2 area centered on the wing. The oil was
seeded with air forced through a ceramic water filter stone, creating a dense
bubble field. After the larger bubbles rose to the surface, the remaining
bubbles, although slightly positively buoyant, were effectively stationary for
the duration of each exposure pair. 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 2000 µs. These light sheets were parallel to the wing chord and
positioned at 0.65R, where R is the wing span, and timed to
fire when the wing chord was directly in front of the high-speed video camera
placed perpendicular to the laser sheet. We chose 0.65R as our point
of measurement because in a prior DPIV study in which the wake was viewed from
the rear, 0.65R was the position in which the circulation was the
greatest (Birch and Dickinson,
2003). We captured one image per stroke from a 29 cmx29 cm
area centered on the wing during each of the four strokes. After saving the
captured images, the trigger for the laser was advanced and the starting
position of the wing was adjusted to line up with the camera at the
appropriate time before starting the next trial
(Birch and Dickinson, 2003
). We
repeated this procedure until we had divided each stroke into 10 equally
spaced intervals. In this way, we quantified the fluid flow from the
perspective of the wing through four downstroke/upstroke cycles, although this
paper will only report on the wake dynamics during the fourth stroke. For each
image-pair captured, a 2-frame cross-correlation of pixel intensity peaks with
50% overlap of 64 pixelx64 pixel interrogation areas resulted in 900
velocity vectors/image. Vector validation resulted in the removal of only 2 of
9000 vector values; these were filled by interpolation of a mean value from a
3 x 3 nearest neighbor matrix. A custom program written in Matlab
(v.5.0) calculated vorticity,
=
xu, from velocity
fields smoothed using using a least-squares finite difference scheme.
Computational method
The computation models a thin wing element of elliptic cross section under
the same kinematics as performed in the experiments. The computation of flows
around this hovering wing employs a fourth-order finite difference scheme of
NavierStokes equation in vorticity-stream function formulation
(E and Liu, 1996). The scheme
is implemented in the elliptic coordinates with appropriate boundary
conditions to account for the wing motion (Wang,
2000a
,b
).
See also Russell and Wang
(2003
) for an alternative
method employing Cartitian grids appropriate for multiple wings.
The two-dimensional Navier-Stokes equation governing the vorticity in the
elliptic coordinates has the following form:
![]() | (1) |
![]() | (2) |
The NavierStokes equation is solved in a frame fixed to the wing. In
the 2D vorticity stream function formulation, the non-inertial frame
introduces only one extra term, the rotational acceleration of the wing. Other
non-inertial terms can be expressed as a gradient of a potential function.
Thus they can be absorbed into the pressure term. The curl of the gradient of
pressure is zero. The body motion is reflected in the far field boundary
conditions, and the no-slip boundary condition at the wing is enforced
explicitly through the vorticity and stream function boundary conditions. More
specifically, on the wing, we set =c, where c is a constant, to satisfy
the no-penetration boundary condition, and
/
n=0 to
satisfy the no-slip boundary condition. At far field, we set
x
=(U0+rx
0),
where U0 and
0 are the translational and
rotational velocity of the wing, respectively, r is the position
relative to the wing center, and
=0. The exact boundary condition on
can be recovered by solving the Poisson equation twice
(Wang, 1999
). For this
computation, the far field boundary condition is correct to the dipole
order.
A fourth-order RungeKutta scheme is used for the time iterations,
which exhibits a stability domain for this explicit scheme. The stability
condition includes two CourantFriedrichLevy (CFL)-like
conditions related to the convection and diffusion time scales over a mesh
size:
![]() | (3) |
![]() | (4) |
The forces on the ellipse can be computed from integrating the stress
tensor along the body. Writing the NavierStokes equation in the
coordinates fixed to the wing, we have:
![]() | (5) |
![]() | (6) |
![]() | (7) |
The velocity and vorticity are solved in the non-inertial coordinates,
which are then transformed into the inertial frame. The forces are calculated
in the inertial frame by integrating the viscous stress:
![]() | (8) |
![]() | (9) |
Wing motion, choice of parameters and normalization
The wing follows a sinusoidal flapping and pitching motion. Specifically,
the wing sweeps in the horizontal plane and pitches about its spanwise axis
with a single frequency f:
![]() | (10) |
![]() | (11) |
The translational motion of the wing is completely specified by two
dimensionless parameters, the Reynolds number,
Re=Umaxc/=
fA0c/
,
and A0/c, where Umax is the maximum
wing velocity, and c the chord. In the subsequent studies, we fix
f but vary A0/c and study its effect on
the flow. For clarity, we will report the value of
A0/c directly instead of Re.
A0/c varies from 2.8 to 4.8, with resulting
Re from 75 to 115, appropriate for fruitflies. Other parameters
0, ß and f are fixed to be
/2,
/4 and
0.25 Hz, respectively.
A main variable of interest in this study is the phase delay between
rotation and translation, , which was shown to be a sensitive parameter
in force generation (Dickinson et al.,
1999
; Wang,
2000b
). Three cases,
=
/4, 0 and
/4,
corresponding to the advanced, symmetrical and delayed rotation
(Dickinson et al., 1999
), will
be studied for each A0/c.
We normalize the computational and experimental forces by the maxima of
their corresponding quasi-steady forces, as described in the next section. Our
choice for the normalization is dimensionally the same as the conventional
choice, Gu2rmsc, but has the
advantage of normalizing away features specific to the wing, such as its
thickness and geometry. This is because that force dependence of the wing
geometry is sometimes relatively simple. For example, the force coefficients
of ellipses of different thickness were shown to have almost the same
functional dependence on the angle of attack but different magnitude (see
fig. 6 in
Wang, 2000a
). The experimental
force coefficients of the robotic fly wing also show little dependence on the
wing planform. If the dependence on the geometry in the steady and unsteady
forces is similar, then their ratio does not depend sensitively on the
geometry of the wing. This would allow us to compare wings of different cross
sections and planforms. Although comparing the force coefficients appears to
be a natural thing to do, one must be cautious when comparing 2D and 3D force
coefficients. The lift coefficient of 1 has different meanings in experiments
and computations, unless the sectional lift coefficient in a 3D wing is a
constant. Strictly speaking, the numbers should only be compared within the
computations or the experiments.
|
Quasi-steady forces
Before discussing the unsteady forces, we first describe the calculation of
the quasi-steady forces based on both the translational and rotational
velocity. Because the wing operates at a large range of angles of attack, from
0° to 135°, the KuttaJoukowski lift, which works for attached
flow associated with small angles of attack, is clearly inapplicable. Instead,
we determine the quasi-steady coefficients empirically, using both the robofly
experiments and computation of a steady translating wing at a fixed angle of
attack. The lift and drag coefficients, defined with respect to the far field
flow, are measured at a time when the forces reach a temporary plateau after
the initial transients (see for example,
fig. 2 in
Dickinson et al., 1999;
fig. 5 in
Wang, 2000a
). Forces at all
angles are measured at a fixed time, t=2 in dimensionless time
scale.
|
|
From the 3D experiments, the lift and drag coefficients are well
approximated by:
![]() | (12) |
![]() | (13) |
![]() | (14) |
![]() | (15) |
![]() | (16) |
![]() | (17) |
|
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In the case of a translating and rotating wing, the instantaneous velocity
of the wing varies along the chord, as
u(x,t)=u0(t)+0(t)xx,
where x is the position on the chord measured from the pitching axis.
In the case of constant small pitching amplitude and constant translating
velocity, the potential theory (Munk,
1925
) predicts the associated lift to be:
![]() | (18) |
Note that Equation 18
includes both the pressure lift of a translating and rotating wing in the
absence of circulation (Magnus force) and the lift due to circulation given by
the Kutta's condition. There is no a priori reason as to which of the
quasi-steady forms should fit in the unsteady case better, since both use
assumptions that are invalid in unsteady cases. Recently, revised quasi-steady
models have been proposed to fit these forces
(Sane and Dickinson, 2002).
For the purpose of this paper, we simply compare LR,
LT and DT with the unsteady forces. It
turns out that for the prescribed motions here, LR
deviates substantially from the unsteady forces, while LT
approximates the unsteady forces reasonably well. Therefore, in the subsequent
discussions, we will use LT as an estimate for the
quasi-steady forces.
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Results |
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Figs 2,
3,
4 summarize the results for
variation in . Each figure shows the comparison of experimental and
computational force coefficients over the first four cycles. In addition,
instantaneous force vectors are shown in superposition with the traveling wing
during the second stroke.
The wing motion in these cases differs in the angle of attack at the end of
stroke. The angles of attack are /4,
/2 and 3
/4, respectively, in
the advanced (
=
/4), symmetrical (
=
/2) and delayed
(
=
/4) rotation cases, as shown in the force vector plots. The
delayed rotation case is unusual from the point of view of operating an
airfoil. After each reversal, the wing has angles of attack greater than
/2, which leads to a downward lift (see
Fig. 4). However, insects or
bio-mimetic devices may use such a stroke to reduce the force on one wing, and
thus generate a torque to turn. In addition, when the wing is moving at an
angle greater than
/2, the flow separates quickly, which is qualitatively
different from the other two cases. Thus it also provides a good case for
testing computations and experiments in different scenarios.
In all cases, the forces quickly settle into an almost periodic state after two strokes. The computational drag follows the experimental drag closely in all three cases. Lift agrees well in the first two cases (Figs 2, 3), but shows a clear phase delay in the case of delayed rotation (Fig. 4). Notice that the shift occurs only after the first stroke. The averaged experimental lift and drag coefficients are (0.93, 1.28), (0.86, 1.34) and (0.38, 1.10), for the advanced, symmetrical and delayed rotation, respectively. The averaged computational lift and drag coefficients are (1.10, 1.36), (0.82, 1.44) and (0.19, 1.21), respectively, for the corresponding cases. We will return to the presence and absence of the phase shift in lift in these three different cases when we discuss the 3D effects.
The averaged force coefficients depend weakly on stroke amplitude, as shown
in Fig. 5. In the case of
=
/4, the average experimental lift coefficients are 0.93, 0.99, 0.95
and 0.93 at A0/c=2.8, 3.6, 4.2 and 4.8,
respectively. The corresponding computational lift coefficients are 1.07, 1.0,
0.9 and 0.9. The drag coefficients are 1.28, 1.19, 1.12 and 0.93 in
experiments, and 1.36, 1.34, 1.24 and 1.16 in computation. This indicates that
the total force scales roughly with
, as
expected. Within this range of amplitude variation, the flows are
qualitatively similar for a given choice of
. Although we only show the
=
/4 case, the results are similar for
=
/4 and
=0.
The average lift depends sensitively on , as emphasized before
(Dickinson et al., 1999
;
Wang, 2000b
). For example, in
the case of A0/c=2.8, the averaged values for
experimental lift, CL are 0.93, 0.86 and 0.38, for
values of 0.25
, 0 and 0.25
, respectively. The comparable
quasi-steady lift coefficients are 0.75, 0.95 and 0.75. This dependence on
can be understood intuitively, based on two facts. First, the deviation
between the unsteady forces and quasi-steady forces occurs mostly after the
flip of each stroke. Second, the instantaneous forces in all these cases are
typically normal to the wing, as indicated in the force vector plots in Figs
2,
3,
4, and as discussed in
Materials and methods. Therefore, the contribution to lift and drag can be
correlated with the instantaneous orientation of the wing at the end of each
stroke:
/4,
/2 and 3
/2, for
=
/4,
=0 and
=
/4, respectively. One expects an increase in both lift and
drag when
=
/4, a decrease of lift and increase of drag when
=
/4, and relatively small change in lift, but a large
increase in drag for
=0. These indeed are consistent with both the
experimental and computational forces.
Comparison of experimental and computed vorticity fields
As a further comparison between computation and experiments, we show side
by side the snapshots of the vorticity field near the wing from experimental
DPIV measurements and computational results
(Fig. 6). Ten frames are shown
for each period. The colors indicate the strength of the vorticity field. In
Fig. 6, columns A and C are
computed vorticity, and B and D are experimental vorticity in a 2D slice. The
simulations appear to capture the major features of vortex dynamics through a
complete stroke cycle. Notice that the fluid momentum is directed downward by
pairs of vortices, similar to those shown in asymmetric strokes that model
dragonfly wing kinematics (Wang, 2002b). In 3D, the pairs of vortices can be
cross-sections of a donut-shaped vortex ring. The structure of the downward
momentum jet, characterized by the averaged velocity field over a cycle, is
examined elsewhere for both the symmetric and asymmetric strokes strokes (Z.
J. Wang, manuscript submitted for publication). Also notice that even the
kinematics of left and right strokes are identical, but the flow field differs
slightly. This can be seen by comparing columns A and C in
Fig. 6. The wing positions are
mirror images, but the flows deviate slightly from the mirror symmetry. Such a
deviation may be inconsequential in terms of average lift, but worth keeping
in mind when interpreting the precise time course of forces.
Unsteady forces vs quasi-steady forces
The differences between the unsteady and quasi-steady forces have been
analyzed extensively for fruitfly kinematics, based on results from tethered
animals, with relatively constant translational velocity in the mid-stroke and
large acceleration and sharp rotations at the end of strokes. The unsteady
effects were dominant near the wing reversal, where they contribute to the
rotational effect and the wake capture
(Dickinson et al., 1999). The
discrepancy between experimental measures of forces and flows
(Birch and Dickinson, 2003
) and
a CFD model of nearly identical conditions
(Sun and Tang, 2002
) raises a
debate about the physical basis of these unsteady effects. Here we do not
attempt to resolve these discrepancies, but probe the presence of unsteady
effects in a different set of kinematics. In
Fig. 7, we compare the unsteady
forces with the steady state forces based on the translational velocity. In
the case of advanced rotation, the unsteady effects can contribute an
additional 50% to the total lift, and in the case of delayed rotation, they
can reduce the total lift by a factor of 23. It is also clear that the
quasi-steady forces based on translational velocity alone do not predict the
time-dependent forces during a sinusoidal flapping.
|
To gauge the relative importance of different unsteady effects, we examine the difference between unsteady forces and estimates based on the translational velocity, as shown in Fig. 7DF. Ideally we wish to decompose, if possible, the unsteady force approximately into a sum of separate terms, which can be related to wing acceleration, the coupling between rotation and translation, wingwake interaction, etc. However, in the absence of quantitative prediction of these effects, we can only offer a plausible decomposition by correlating the force peaks with the time course of translational and rotational velocity.
The coupling between translation and rotation can be modeled by
Crot(t)u(t), a form
predicted by classical theory of a translating and pitching motion
(Munk, 1925
), and tested in
robotic fruitfly experiments (Sane and
Dickinson, 2002
), where Crot is assumed to be
a constant that depends on the center of rotation. Part of
Crot
(t)u(t) is the
Magnus force caused by the pressure difference due to velocity difference,
given by Bernoulli's law, and another part is due to additional circulation
caused by the rotational motion to satisfy the Kutta condition
(Munk, 1925
). In the three
kinematic patterns studied here, the peaks (labeled `r' in
Fig. 7) associated with
rotation, are picked by matching (with a small shift) the force curve to the
maximum of
(t)u(t). The positions of these
measured force peaks vary in the three cases, in accordance with the shift of
the peak positions of u(t)
(t). The variation
occurs at the same time scale as
u(t)
(t).
Other unsteady effects occur near the wing reversal (labeled `u' in Fig. 7). The position of these effects occur roughly at the same time in all three kinematics. These force peaks do not follow the trace of du(t)/dt, thus do not behave as the classical added mass. These force peaks are likely related to the unsteady growth of vorticity and wakewake interaction, which do not have simple analytic expressions in general. Regardless of its physical basis, the most substantial contribution of this unsteady effect is on drag (Fig. 7B,C,E,F).
We also note that the peaks alternate in size from stroke to stroke in the experimental lift, most obvious in Fig. 7A,D. One possible explanation is that this asymmetry reflects the mechanical artifact due to gear backlash. However a small degree of asymmetry is also observed in the computational data, e.g. the vorticity field as shown in Fig. 6 and the forces in Figs 2, 3, 4. At Re=0, the left and right strokes, which are mirror images about the vertical axis, would generate forces that have the same symmetry; i.e. the lift from left and right strokes are identical and the drag forces are of equal size but in the opposite direction. Here, the Reynolds number is finite and sufficiently large that the force can depend on the history of flow. One possibility for breaking the symmetry in forces is by the initial condition.
While identification of the above features is useful when dissecting the
unsteady force traces, it is also relevant to determining their net
contributions. The force directly proportional to the linear acceleration can
have sharp peaks, but it has a zero contribution in reciprocal motions. The
pressure force of a rotating and translating plate, approximated by
u(t)(t), has a non-zero net contribution in
the cases where
0, since
[sin(2
ft)cos(2
ft+
)]
sin
. The
unsteady vortex force due to wing acceleration has eluded simple analytical
expressions, except for power-law start up flow, where both the added mass
term and the vortex force are calculated analytically and numerically
(Pullin and Wang, 2003
). The
unsteady forces contribute to both lift and drag, both predicted in theory
(Pullin and Wang, 2003
) and
seen here in Fig. 7.
Fluid forces and wing inertia
Among various terms contributing to the fluid forces, the pressure force
dominates. The viscous force is smaller by roughly a factor proportional to
1/Re. The pressure force due to non-inertial effects resulting
from translational and rotational acceleration averages to zero in hovering
when the pitching axis is centered at the chord. The magnitude of the
instantaneous non-inertial translational force is also small for these
sinusoidal motions. In particular:
![]() | (19) |
Fig. 8 illustrates the
contribution of F (broken line) to the total force
(solid line). In kinematics where there is a fast acceleration at the end of
the stroke, the force will have sharp peaks at the end of the stroke, but the
net contribution is zero, as discussed before.
|
Finally, we estimate the inertial force associated with wing acceleration
with respect to the fluid force. The inertial force in the experiment turns
out to be negligible compared to the fluid forces, as shown experimentally
(Sane and Dickinson, 2001).
Here is a simple estimate to explain why this is so:
![]() | (20) |
![]() | (21) |
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Discussion |
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First, a notable difference between the experimental and computational
forces is seen in the delayed rotation, where there is a clear phase shift
between the computed and measured lift. In the canonical example of flow past
a 2D cylinder and a 3D sphere, the forces during von Karman vortex shedding
also show a phase shift, which was argued to be a generic feature between 2D
and 3D flows (Mittal and Balachandar,
1995). In view of this, the absence of the phase shift in the
advanced and symmetrical cases are particularly interesting. The difference in
flow structure in the three cases may be worth further investigation.
Second, these results are relevant to recent discussions about the role of
3D effects on delayed stall. Insects are known to flap their wings at angles
of attack much higher, around 35°
(Ellington, 1984), than the
stall angle of a conventional airfoil, about 12°. As suggested by
Ellington et al. (1996
), the
pressure gradient from root to tip within the vortex core might drive spanwise
flow that stabilizes the leading edge vortex by convecting away the vorticity.
The spanwise flow was indeed seen by smoke visualization in the robotic
hawkmoth experiment, where Re
5000
(Ellington et al., 1996
;
Willmott et al., 1997
). This
proposed mechanism is thought to be analogous to that occurring on delta wing
aircraft, in which spanwise flow through the vortex core maintains the
stability. But as discussed previously
(Ellington et al., 1996
), the
exact conditions for establishing spanwise flow in leading-edge vortex for
rotary wings are not completely understood. For example, a helicopter rotor
also experiences a pressure gradient, centrifugal and Coriolis forces, but no
large-scale spanwise flow is observed
(Harris, 1966
). Recent smoke
visualization of free-flying butterflies also did not observe substantial
spanwise flow, but reported high variability of 3D flow patterns
(Srygley and Thomas, 2003
).
DPIV images of flow field in a robotic fruitfly experiment, where
Re
150, showed no substantial spanwise flow inside the core of
leading edge vortex, but instead indicted substantial spanwise flow behind the
leading edge vortex, which connects to the tip vortex
(Birch and Dickinson, 2001
).
Strictly speaking, there is no contradiction among these experiments regarding
the spanwise flow. It is likely that the spanwise flow within the vortex core
occurs only at sufficiently large Reynolds number as in the case of hawkmoth,
but not at low Reynolds number, as in the case of fruitflies. The details of
the spanwise flow can also depend on the wing shape and kinematics. The high
variability of the 3D flow patterns shown by these different experiments,
however, makes it difficult to conclude that spanwise flow is crucial for
generating sufficient lift by a hovering insect.
An alternative explanation for why the conventional stall does not seem to
affect a flapping insect wing relates to the time scale governing the flow
separation that leads to stall. For example, a 2D translating wing at an angle
of attack of 40° and Re=1000, does not show a drop in
time-dependent force until the chord travels for about 4 chords, after which
the forces become oscillatory due a von Karman shedding
(fig.6 in
Wang, 2000a
). Therefore, in
theory there is no need for additional mechanisms to stabilize the leading
edge vortex if the wing travels less than about 4 chords. The early data
compiled by Weis-Fogh (1973
)
showed that ratios of stroke-arc to wing-chord of different species during
hovering, including bats (Plecotus auritus), birds (hummingbirds),
butterflies and moths (Lepidoptera), wasp and bees (Hymenoptera) and flies
(Diptera), have values less than 4. The beetles (Coleoptera) have values
between 5 and 6. A main difference between a 3D revolving wing and a 2D
translating wing, as noted in recent literature, is that a revolving wing does
not appear to shed its leading edge vortex after a distance much longer than
the stroke length of a typical insect
(Dickinson et al., 1999
;
Usherwood and Ellington,
2002
). No doubt such a difference would affect the force
coefficients observed in 2D and 3D in the steady state. On the other hand, the
difference in terms of vortex shedding may not affect the transient values. It
is worth re-examining the results of 3D experiments on a flapping wing
(fig. 2D in
Dickinson et al., 1999
), which
show that while the 2D steady state lift coefficients underpredict
substantially their the 3D counterparts, the 2D transient values follow
closely the 3D coefficients, up to an angle of attack of about 72°. The 3D
steady force is slightly lower than the unsteady 2D counterpart, due the
well-known downwash due to tip vortices. Similarly, in the cases studied here,
the chord is moving between 3 to 5 chords, and the leading edge vortex does
not appear to separate during each stroke in the cases of advanced and
symmetrical rotation, as indicated by the absence of phase shift between the
2D and 3D forces. In these cases, the 2D forces are good approximations of 3D
experiments.
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Acknowledgments |
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References |
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