Unsteady aerodynamic forces of a flapping wing
Institute of Fluid Mechanics, Beijing University of Aeronautics & Astronautics, Beijing 100083, People's Republic of China
* Author for correspondence (e-mail: sunmao{at}public.fhnet.cn.net)
Accepted 6 January 2004
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Summary |
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In the Re range considered (201800), when Re was
above 100, the lift
(
L) and drag
(
D) coefficients were
large and varied only slightly with Re (in agreement with results
previously published for revolving wings); the large force coefficients were
mainly due to the delayed stall mechanism. However, when Re was below
100,
L decreased and
D increased greatly. At
such low Re, similar to the case of higher Re, the leading
edge vortex existed and attached to the wing in the translatory phase of a
half-stroke; but it was very weak and its vorticity rather diffused, resulting
in the small
L and large
D. Comparison of the
calculated results with available hovering flight data in eight species
(Re ranging from 13 to 1500) showed that when Re was above
100, lift equal to insect weight could be produced but when Re
was lower than
100, additional high-lift mechanisms were needed.
In the range of Re above 100,
from 90° to 180°
and
r from 17% to 32% of the stroke period (symmetrical
rotation), the force coefficients varied only slightly with Re,
and
r. This meant that the forces were approximately
proportional to the square of
n (n is the wingbeat
frequency); thus, changing
and/or n could effectively control
the magnitude of the total aerodynamic force.
The time course of L
(or
D) in a half-stroke
for ut varying according to the SHF resembled a half
sine-wave. It was considerably different from that published previously for
ut, varying according to a trapezoidal function (TF) with
large accelerations at stroke reversal, which was characterized by large peaks
at the beginning and near the end of the half-stroke. However, the mean force
coefficients and the mechanical power were not so different between these two
cases (e.g. the mean force coefficients for ut varying as
the TF were approximately 10% smaller than those for ut
varying as the SHF except when wing rotation is delayed).
Key words: flapping wing, insect, computational fluid dynamics, unsteady aerodynamics, delayed stall, force coefficients
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Introduction |
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Dickinson and Götz
(1993) measured the aerodynamic
forces of an airfoil started rapidly at high angles of attack in the Reynolds
number (Re) range of the fruit fly wing (Re=75225;
for a flapping wing, Re is based on the mean chord length and the
mean translation velocity at radius of the second moment of wing area). They
showed that lift was enhanced by the presence of a dynamic stall vortex, or
leading edge vortex (LEV). After the initial start, lift coefficient
(CL) of approximately 2 was maintained within 23
chord lengths of travel. Afterwards, CL started to
decrease due to the shedding of the LEV. But the decrease was not rapid,
possibly because the shedding of the LEV was slow at such low Re; and
from 3 to 5 chord lengths of travel, CL was still as high
as approximately 1.7. The authors considered that because the fly wing
typically moved only 24 chord lengths each half-stroke, the
stall-delaying behavior was more appropriate for models of insect flight than
were the steady-state approximations.
Ellington et al. (1996) and
van den Berg and Ellington
(1997a
,b
)
performed flow-visualization studies on the large hawkmoth Manduca
sexta, in tethered forward flight (speed range, 0.45.7 m
s1), and on a mechanical model of the hawkmoth wings
(Re
3500). They found that the LEV on the wings did not shed in
the translational phases of the half-strokes and that there was a spanwise
flow directed from the wing base to the wing tip. Analysis of the momentum
imparted to fluid by the vortex wake showed that the LEV could produce enough
lift for the weight support. For the hovering case, the hawkmoth wing traveled
approximately three chord lengths each half-stroke, whereas for the case of
forward flight at high speeds the wing traveled twice as far. The authors
suggested that the spanwise flow had prevented the LEV from detaching.
The above studies identified delayed stall as the high-lift mechanism of
some small and large insects. Recently, Dickinson et al.
(1999) measured the aerodynamic
forces on a revolving model fruit fly wing (Re
75) and showed that
stall did not occur and large lift and drag were maintained. Usherwood and
Ellington
(2002a
,b
)
measured the aerodynamic forces on revolving real and model wings of various
insects and a bird (quail) and, for some cases, flow visualization was also
conducted. They found that large aerodynamic forces were maintained by the
attachment of the LEV for Re
600 (mayfly) to 15 000 (quail) and
for different wing planforms. These results further showed that the delayed
stall mechanism was valid for most insects [wing length (R) 2 mm
(fruit fly) to 50 mm (hawkmoth)]. The delayed stall mechanism was confirmed by
computational fluid dynamics (CFD) analyses
(Liu et al., 1998
;
Wang, 2000
;
Lan and Sun, 2001
).
Dickinson et al. (1999) and
Sane and Dickinson (2001
), by
measuring the aerodynamic forces on a mechanical model of fruit fly wing in
flapping motion, showed that when the translation velocity varied according to
a trapezoidal function (TF) with large accelerations at stroke reversal and
the wing rotation was advanced, in addition to the large forces during the
translational phase of a half-stroke, very large force peaks occurred at the
beginning and near the end of the half-stroke. Sun and Tang
(2002a
) and Ramamurti and
Sandberg (2002
) simulated the
flows of model fruit fly wings using the CFD method, based on wing kinematics
nearly identical to those used in the experiment of Dickinson et al.
(1999
). They obtained results
qualitatively similar to those of the experiment. The large forces during the
translational phase were explained by the delayed stall mechanism
(Dickinson et al., 1999
). It
was suggested that the large force peaks at the beginning of the half-stroke
were due to the rapid translational acceleration of the wing and the
interaction between the wing and the wake left by the previous strokes
(Dickinson et al., 1999
;
Sun and Tang, 2002a
;
Birch and Dickinson, 2003
), and
those near the end of the stroke were due to the effects of wing rotation
(Dickinson et al., 1999
;
Sun and Tang, 2002a
).
The experiments on revolving wings (Usherwood and Ellington,
2002a,b
;
Dickinson et al., 1999
) have
showed that similar high force coefficients (due to the delayed stall
mechanism) are obtained in the Re range of approximately 140 (model
fruit fly wing) to 15 000 (quail wing). It is of interest to investigate the
aerodynamic force behavior and the delayed stall mechanism at lower Reynolds
numbers, because Re for some very small insects is as low as 20
(Weis-Fogh, 1973
;
Ellington, 1984a
). In the
experiments (Dickinson et al.,
1999
; Sane and Dickinson,
2001
) and CFD simulations (Sun
and Tang, 2002a
; Ramamurti and
Sandberg, 2002
) on the flapping model fruit fly wings, the
translation velocity (ut) of the wing varied as a TF with
rapid accelerations at stroke reversal (the stroke positional angle followed a
smoothed triangular wave), which was an idealization on the basis of the
kinematic data of tethered fruit flies
(Dickinson et al., 1999
).
However, data of free flight in many insects
(Ellington, 1984c
;
Ennos, 1989
) showed that
ut was close to the simple harmonic function (SHF). Recent
data of free-flying fruit fly (Fry et al.,
2003
) also showed that ut was close to the
SHF. When ut varies as the SHF, the time courses of the
forces and their sensitivity to some of the kinematic parameters, such as
wing-rotation rate and stroke timing, might be significantly different from
those when ut varies as a TF with large acceleration at
stroke reversal. Therefore, it is of interest to study the aerodynamic forces
for the case of ut varying as the SHF.
In the present study, we use the CFD method to simulate the flows of a
model insect wing in flapping motion. The flapping motion consists of
translation and rotation (Fig.
1). The translation follows the SHF; the rotation is confined to
stroke reversal. As will be shown below, for a given wing with
ut varying the SHF (in the absence of wing deformation),
its aerodynamic force coefficients depend only on the following five
non-dimensional parameters: Re, stroke amplitude (), mid-stroke
angle of attack (
m), wing-rotation duration
(
r) and rotation timing (
r). We
consider the Re range of 20 to 1800; in addition, we investigate the
effects of varying
,
m,
r and
r.
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Materials and methods |
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Stroke kinematics
The velocity at the span location r2 due to wing
translation is called the translational velocity (ut).
ut is assumed to vary as the SHF:
![]() | (1) |
![]() | (2) |
In the flapping motion described above, the period of wingbeat cycle
c, the geometric angle of attack at mid-stroke
m, the rotation duration
r or the
mean angular velocity rotation
and the rotation timing
r
need to be specified. Note that since U=2
nr2
(where n is the wingbeat frequency and
is the stroke
amplitude),
c (=U/cn) is related to
by
c=2
(r2/R)(R/c).
Flow equations and evaluation of the aerodynamic forces
The governing equations of the flow are the three-dimensional
incompressible unsteady NavierStokes equations. Written in the inertial
coordinate system OXYZ and non-dimensionalized, they are as follows:
![]() | (3) |
![]() | (4) |
![]() | (5) |
![]() | (6) |
Once the NavierStokes equations are numerically solved, the fluid
velocity components and pressure at discretized grid points for each time step
are available. The aerodynamic forces (lift, L, and drag, D)
acting on the wing are calculated from the pressure and the viscous stress on
the wing surface. The lift and drag coefficients are defined as follows:
![]() | (7) |
![]() | (8) |
Non-dimensional parameters that affect the aerodynamic force coefficients
For a wing of given geometry (in the absence of deformation), when its
flapping motion is prescribed, solution of the non-dimensional
NavierStokes equations (equations 46) gives the aerodynamic
force coefficients CL and CD; the only
non-dimensional parameter in the NavierStokes equations that needs to
be specified is Re. To prescribe the flapping motion, as mentioned
above, ,
m,
r and
r need to be specified. That is, the aerodynamic force
coefficients on the wing depend on five non-dimensional parameters:
Re,
,
m,
r and
r. When the wing rotation is symmetrical,
r
may be determined from
r; thus, CL
and CD depend only on four parameters: Re,
,
m and
r.
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Results |
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Upon the suggestion of a referee of the present paper, we made a further
test of the code using the recent experimental data of Usherwood and Ellington
(2002a,b
)
on revolving model wings. In the computation, the wing rotated 120° after
the initial start, and Re was set as 1800 (this Re value was
similar to that of the bumble bee wing). In order to make comparisons with the
experimental data, lift and drag coefficients were averaged between 60°
and 120° from the end of the initial start of rotation. The computed and
measured CL and CD are shown in
Fig. 3 [measured data are taken
from fig. 7 of Usherwood and Ellington
(2002b
)]. In the whole
range (from 20° to 100°), the computed CL
agrees well with the measured values; both have approximately sinusoidal
dependence on
. The computed CD also agrees well
with the measured values except when
is larger than
60°.
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The above comparisons show that there still exist some discrepancies between the CFD simulations and the experiments but that, in general, the agreement between the computational and experimental aerodynamic forces is good. We think that the present CFD method can calculate the unsteady aerodynamic forces and flows of the model insect wing with reasonable accuracy.
Grid resolution test
Before proceeding to study the physical aspects of the flow, the effects of
the grid density, the time step and computational-domain size on the computed
solutions were considered. The sensitivity of the computed flow to spatial and
time resolution and to the far-field boundary location was evaluated for the
case of Re=1800 (this Reynolds number is the highest among the cases
considered in this study). Calculations were performed using three different
grid systems. Grid 1 had dimensions 53x48x41 (around the wing
section, in the normal direction and in the spanwise direction respectively),
grids 2 and 3 had dimensions 77x70x61 and 109x93x78,
respectively. The spacings at the wall were 0.003, 0.002 and 0.0015 for grids
1, 2 and 3, respectively. The far-field boundary for these three grids was set
at 20c away from the wing surface in the normal direction and
8c away from the wing-tips in the spanwise direction. The grid points
were clustered densely toward the wing surface and toward the wake.
Fig. 4 shows the time course
of the lift coefficient in one cycle and the contours of the non-dimensional
spanwise component of vorticity at mid-span location near the end of a
half-stroke (just before the wing starting the pitching-up rotation),
calculated using the above three grids and a time-step value of 0.02. It is
observed that the first grid refinement produced some change in the vorticity
plot; however, after the second grid refinement, the discrepancies are
considerably reduced. The differences between the computed lift coefficients
using the three grids are small; there is almost no difference between the
lift coefficients computed using grids 2 and 3. Computations using grid 3 and
two time-step values, =0.02 and 0.01, were conducted.
Discrepancies between the computed aerodynamic forces and vorticity fields
using the two time steps were very small. Finally, the sensitivity of the
solution to the far-field boundary location was considered by calculating the
flow in a large computational domain. In order to isolate the effect of the
far-field boundary location, the boundary was made further away from the wing
by adding more grid points to the normal direction of grid 3. The calculated
results showed that there was no need to put the far-field boundary further
than that of grid 3. From the above analysis, it was concluded that grid 3 and
a time step value of
=0.02 were appropriate for the present
study.
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Forces and flows of a typical case
We first considered a case in which typical values of wing kinematic
parameters were used (Re=200, =150°,
m=40°,
r=1.87 and wing rotation
was symmetrical; with the above values of
,
m and
r, we had
c=9.37,
and
r=0.2
c).
Fig. 5 shows the time
courses of CL and CD in one cycle.
CL in the middle portion of a half-stroke is large and
dominates over CL at the beginning and near the end of the
half-stroke (CD behaves similarly). In previous studies by
Dickinson et al. (1999) and Sun
and Tang (2002a
), in which
ut varied as a TF with large accelerations at stroke
reversal, large force peaks occurred near the end of the half-stroke. They
were caused by the pitching-up rotation of the wing while it was still
translating at relatively large velocity. In the present case, peaks in
CL and CD also exist
(Fig. 5B,C) but they are very
small. This is because near the end of the half-stroke the
ut has become very low and wing rotation cannot produce a
large force at low ut.
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The mean lift (L) and
drag (
D) coefficients are
1.66 and 1.67, respectively, which are much larger than the steady-state
values [measured steady-state CL and
CD on a fruit fly wing in uniform free-stream in a wind
tunnel at the same Re (200) and same
m (40°)
are 0.6 and 0.75, respectively (Vogel,
1967
)]. As seen in Fig.
5, the major part of the mean lift (or drag) comes from the
mid-portions of the half-strokes. During these periods, the wing is in pure
translational motion (
is constant). From the results in
Fig. 5, it is estimated that
88% of the mean lift is contributed by the pure translational motion. As was
shown previously (Ellington et al.,
1996
; Liu et al.,
1998
; Dickinson et al.,
1999
; Sun and Tang,
2002a
), the large CL and
CD during the translatory phase of a half-stroke were due
to the delayed stall mechanism. That is, the delayed stall mechanism is mainly
responsible for the large aerodynamic forces produced. The flow-field data
provide further evidence for the above statement. The contours of the
non-dimensional spanwise component of vorticity at mid-span location are given
in Fig. 6. The LEV does not
shed in an entire half-stroke, showing that the large CL
and CD in the mid-portion of the half-stroke are due to
the delayed stall mechanism.
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The effects of Re
Fig. 7 shows the time
courses of CL and CD in one cycle for
various Re (Re ranging from 20 to 1800; other conditions
being the same as in the typical case). In general, CL
increases and CD decreases as Re increases.
However, when Re is higher than 100, CL and
CD do not vary greatly, whereas when Re is lower
than
100, CL is much smaller and
CD much larger than at higher Re.
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For the case of Re=200, as discussed above, the large
CL and CD during a stroke are due to
the delayed stall mechanism. For the cases of other Re, as seen in
Fig. 7, CL
and CD do not have a sudden drop during a half-stroke
(between =0 and
=0.5
c, the downstroke; between
=0.5
c and
=
c, the upstroke), i.e.
stall is also delayed for an entire half-stroke.
Fig. 8 shows the vorticity
contour plots at half-wing length near the end of a half-stroke. It is seen
that, at all Re considered, the LEV does not shed and the delayed
stall mechanism exists. However, for Re lower than
100, the LEV
is very diffused and weak compared with that for higher Re (comparing
Fig. 8D,E with
Fig. 8AC; the strength
of the LEV can be estimated from the values of vorticity represented by the
contours and the spacing between the contours), resulting in small
CL and large CD. For reference,
vorticity contour plots at various times in one cycle for the case of
Re=20 are shown in Fig.
9.
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|
L and
D at various Re
are plotted in Fig. 10. For
Re above
100, change in
L and
D with Re are
small, whereas for Re below
100,
L decreases and
D increases rapidly as
Re decreases. Similar to the typical case, approximately 8590%
of
L is contributed by the
pure translational motion for all the values of Re considered.
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Dependence of the force coefficients on mid-stroke angle of attack
Fig. 11 gives
L and
D in the range of
m from 25° to 60° (for all Re considered in
the above section). The slope of the
L (
m)
curve is approximately constant between
m=25° and
35°; beyond
m=35°, it decreases gradually to zero at
m
50°.
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The rate of change of L
with
m
(d
L/d
m)
from
m=25° to 35° is given in
Table 1. For Re above
100,
d
L/d
m
hardly varies with Re and its value is approximately 3.0, which is
almost the same as the measured value (2.93.1) for the revolving wings
[see fig. 6 of Usherwood and Ellington
(2002b
); the cited value is
for the case of aspect ratio equal to 6 (R/c=3)]. For Re
below
100,
d
L/d
m
decreases greatly.
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The effects of rotation duration
In the calculations above, r=1.87
(=0.2
c;
=0.93). Observation of many insects in free
flight (Ellington, 1984c
;
Ennos, 1989
) showed that
ranged approximately from 0.8 to 1.4. Here, we investigate the effects of
varying
r (i.e. varying
) on the aerodynamic force
coefficients.
Fig. 12 gives the time
courses of CL and CD in one cycle for
four values of r;
Table 2 gives the mean force
coefficients. Varying
r does not change the mean force
coefficients greatly (see Table
2); when
r is almost doubled (varied from
1.27 to 2.40),
L and
D change only
approximately 3%. CL and CD in the
mid-portion of a half-stroke vary little with
r (see
Fig. 12). The force peaks
around the stroke reversal are due to the effects of wing rotation
(Dickinson et al., 1999
;
Sun and Tang, 2002a
); at a
given translation velocity, the peaks increase with rotation rate
(Sane and Dickinson, 2002
;
Hamdani and Sun, 2000
). When
r is relatively short (
is relatively large), the
force peaks are relatively large but they occupy a short period; when
r is longer (
is smaller), the force peaks become
smaller but they occupy a longer period. As a result, the force peaks around
the stroke reversal for the cases of different
r give
more or less the same contribution to the corresponding mean force
coefficient. This explains why
L and
D do not change greatly
with
r (or
).
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|
The effects of rotation timing
Fig. 13 shows the time
courses of CL and CD in one cycle for
different rotation timing (r can be read from
Fig. 13A; Re,
m,
and
r are the same as those
in the typical case). In the case of advanced rotation (the major part of
rotation is conducted before stroke reversal), the peaks in
CL and CD near the end of a
half-stroke are larger than those in the case of symmetrical rotation; this is
because the wing conducts pitching-up rotation at a higher translational
velocity (see Fig. 13A). At
the beginning of the next half-stroke, CL and
CD are also larger than their counterparts in the case of
symmetrical rotation; this is because the wing does not conduct pitching-down
rotation in this period (the wing rotation is almost finished before this
period). In the case of delayed rotation (the major part of rotation is
conducted after stroke reversal), no CL and
CD peaks occur near the end of the half-stroke because the
wing does not rotate in this period; in the beginning of the next half-stroke,
CL is negative and CD is large
compared with that in the case of symmetrical rotation because all of the wing
rotation is conducted in this period and the rotation is pitching-down
rotation.
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The mean force coefficients are given in
Table 3.
L and
D for the case of advanced
rotation are approximately 40% and 30% larger than those for the case of
delayed rotation, respectively.
|
The effects of stroke amplitude
Free-flight data collected from many insects
(Ellington, 1984c;
Ennos, 1989
;
Fry et al., 2003
) showed that
the stroke amplitude,
, ranged approximately from 90° to 180°.
Moreover, an insect might change
to control its aerodynamic force (e.g.
Ellington, 1984c
;
Lehmann and Dickinson, 1998
).
Here, we investigate the effects of
on the force coefficients.
Calculations were made for various
(65°, 90°, 120°, 150°
and 180°) while other parameters were fixed (they are the same as those in
the typical case). Fig. 14
shows the time courses of CL and CD in
one cycle; Table 4 gives the
mean force coefficients. Note that when
is varied, the non-dimensional
period of wingbeat cycle will change
(
c=2
r2/c); since
r (i.e.
) is fixed,
r/
c is different for different
.
In the range of
from 90° to 180°, the effects of varying
on the force coefficients are not large; when
increases or decreases by
30°,
L and
D change less than 3% and
6%, respectively. When
is below approximately 90°, the effects of
varying
become larger (see the results for
=65°;
Fig. 14;
Table 4). It is of interest to
point out the fact that
L
and
D hardly vary with
(in the range of
from 90° to 180°) means that the mean
lift and mean drag vary as
2, because the forces are
non-dimensionalized by U2, and U equals
2
nr2 (n is the wingbeat frequency).
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|
Sane and Dickinson (2001)
studied the effects of varying
and other parameters using a dynamically
scaled mechanical model of the fruit fly. Their results [see fig. 5A,C of Sane
and Dickinson (2001
)] showed
that when
fell below approximately 120°,
L decreased and
D increased with
decreasing
(
D
increased rapidly as
became small). In the present simulation
(Fig. 14;
Table 4), when
is below
90°, we also found
D increasing and
L decreasing with a
decrease in
, but the rates of change in
L and
D are smaller than those
reported by Sane and Dickinson
(2001
). In their experiment,
when
changed, Re and
r also changed, but
the ratio of
r/
c did not change; in the
present simulation, Re and
r did not change
when
changed. To make further comparison with their results, we made
some calculations in which Re and
r changed
with
but
r/
c was kept unchanged
(=0.2). The results are given in Fig.
15 and Table 5. The
trends of variation in
L
and
D with
are
similar to those in Sane and Dickinson
(2001
): when
is below
approximately 120°,
L
decreases and
D increases
with
decreasing, and when
is below 90°,
D increases rapidly.
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![]() |
Discussion |
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From the flight data of an insect, the mean lift coefficient needed for
supporting its weight (denoted by
L,W) can be determined.
Data of free hovering (or very low-speed) flight in eight species were
obtained. Six species were from Ellington
(1984b
,c
)
[the wing length of these species ranges from 9.3 mm (in Episyrphus
balteatus) to 14.1 mm (in Bombus hortorum)]; two smaller ones,
Drosophila virilis and Encarsia formosa, were from
Weis-Fogh (1973
). These data
include: insect mass (M), wing length, mean chord length, radius of
second moment of wing area, stroke amplitude and wingbeat frequency (see
Table 6). On the basis of these
data, the reference velocity, Re and mean lift coefficient needed for
supporting insect weight were computed
(U=2
nr2, Re=Uc/
and
L,W=mg/0.5
U2St,
where g and St were the gravitational
acceleration and the area of both wings, respectively). Re and
L,W are given in
Table 6.
|
Now, we compare the data in Table
6 with the results of model-wing simulation in
Fig. 11 [here, we assume that
the wing planform does not have a significant effect on lift coefficient; this
is true for revolving wings (Usherwood and
Ellington, 2002b)]. Of the insects considered, Encarsia
formosa has the lowest Re (13) and its
L,W is 2.87; when its wing
area is extended to include the brim hairs
(Ellington, 1975
), its
L,W is still as high as
1.62. As seen in Fig. 11, at
such a low Re, the maximum
L is
1.15 (at
m
45°), which is much smaller than its
L,W. In the computations
that gave the results in Fig.
11, symmetrical rotation was used and
was 150°. For
reference, we made another calculation in which advanced rotation was used,
=180° and
m=45° (this combination of parameters
was expected to maximize
L). The computation gave
L=1.25, which was also
much smaller than the
L,W
of Encarsia formosa. These results show that using the flapping
motion described above, the insect could not produce enough lift to support
its weight; i.e. at such low Re, high-lift mechanisms, in addition to
the delayed stall mechanism, are needed [Weis-Fogh
(1973
) suggested the `clap and
fling' mechanism]. For other insects, Re is above 100 and, as seen in
Fig. 11, at an
m between 30° and 50°, a
L equal to
L,W can be produced.
The above comparison shows that when Re is higher than 100,
the delayed stall mechanism can produce enough lift for supporting the
insect's weight and when Re is lower than
100, additional
high-lift mechanisms are needed.
Lift and drag vary approximately with the square of n
The non-dimensional NavierStokes equations (equations 36),
the equations prescribing the flapping motion (equations 1, 2) and the
equations defining the aerodynamic force coefficients (equations 7, 8) show
that the mean force coefficients of a wing of given geometry with
ut varying as the SHF depend only on Re,
m,
and
r (assuming symmetrical
rotation). As already discussed above, when Re is above
100, the
force coefficients vary only slightly with Re; results in Tables
2,
4 show that the force
coefficients vary only slightly with
r and also vary
only slightly with
in the range of
approximately from 90° to
180°.
When and/or n is varied, Re will change (note that
Re=2
nr2c/
). Since the force
coefficients hardly vary with
and Re, the mean lift
(
) and drag
(
) vary approximately with
(
n)2. That is, changing
and/or n can
effectively control the aerodynamic forces. For instance, increasing
by
15%,
can be increased by
approximately 32% [note that by increasing
m by 15% (e.g.
from 40° to 46°),
increases
only by
10% (see Fig.
11)].
In the above discussion, we have assumed that the force coefficients hardly
vary with Re, and
r (or
). However,
in testing the effects of a particular parameter on the force coefficients, we
varied one parameter while keeping all others the same as in the typical case.
When the parameters are simultaneously varied, do the force coefficients still
vary only slightly? We conducted some further calculations in which, for a
range of
m from 25° to 60°, Re,
and
were simultaneously increased by 20% from those of the typical case.
Fig. 16 shows the results. At
all
m considered, the force coefficients vary only
slightly.
|
Comparison of the present results with those of ut varying as the TF
In some recent experimental (Dickinson
et al., 1999; Sane and
Dickinson, 2001
) and computational (Sun and Tang,
2002a
,b
;
Ramamurti and Sandberg, 2002
;
Sun and Wu, 2003
) studies,
ut varying as a TF with rapid accelerations at stroke
reversal has been employed. It is of interest to discuss the differences
between the present results for ut varying as the SHF and
those for ut varying as a TF with rapid accelerations at
stroke reversal.
The force coefficients
The time courses of force coefficients are considerably different between
the two cases; for the case of ut varying as a TF with
rapid accelerations at stroke reversal, the CL (or
CD) curve is flat in the mid-portion of a half-stroke and
has large peaks at the beginning and near the end of the half-stroke [see fig.
3A,B of Dickinson et al. (1999)
and fig. 6 of Sun and Tang
(2002a
)], whereas for the case
of ut varying as the SHF, the CL (or
CD) curve grossly resembles a half sine-wave (see
Fig. 5). As a result, very
large time gradients of aerodynamic force exist in each half-stroke in the
case of ut varying as the TF but not in the case of
ut varying as the SHF.
However, in spite of the large differences in instantaneous force
coefficients, the mean force coefficients are not so different. To examine the
quantitative differences of the mean force coefficients between the two cases,
we made two sets of computations. In the first set, ut
varied as a TF with rapid accelerations at stroke reversal [the duration of
translational acceleration at stroke reversal was 0.18c,
similar to that used in Dickinson et al.
(1999
) and Sun and Tang
(2002a
,b
)];
symmetrical rotation, advanced rotation (the major part of rotation conducted
before stroke reversal) and delayed rotation (the major part of rotation
conducted after the stroke reversal) were considered; other conditions
(Re,
,
m,
r) were the
same as those in the typical case. In the second set, ut
was replaced by the SHF. The results showed that when wing rotation was
symmetrical or advanced,
L
and
D for
ut varying as the TF are approximately 12% and 8% smaller
than those for ut varying as the SHF, respectively, and
when wing rotation was delayed,
L for
ut varying as the TF was approximately 35% smaller than
that for ut varying as the SHF but
D was approximately the
same for the two cases.
Power requirements
In the studies on power requirements of fruit fly flight by Sun and Tang
(2002b) and Sun and Wu
(2003
), a TF similar to that
used in Sun and Tang (2002a
)
was used for ut (wing rotation was symmetrical). As seen
above,
L and
D for
ut varying as the TF are 12% and 8% smaller, respectively,
than those for ut varying as the SHF. It is of interest to
know the effects of replacing the TF by the SHF on the results presented in
Sun and Tang (2002a
) and Sun
and Wu (2003
).
To quantify the effects of the new kinematic model, we made calculations in
which the same model wing and kinematic parameters as those in Sun and Tang
(2002b) and Sun and Wu
(2003
) were used, except that
the TF for ut was replaced by the SHF. For hover flight,
the results in Sun and Tang
(2002b
) and Sun and Wu
(2003
) are as follows: mean
lift equal to the insect weight is produced at
m=36.5°
and the body-mass-specific power is 29 W kg1; with
ut varying as the SHF, the corresponding
m and body-mass-specific power are 30.5° and 31.5 W
kg1, respectively. That is, with the SHF, the
m needed is a few degrees smaller and the body-mass-specific
power is approximately 10% larger than that with the TF (similar results were
obtained for forward flight).
The reason for the needed m becoming smaller is obvious.
Seeing that
L and
D with the SHF are larger
than their counterparts with the TF by approximately the same percentage (i.e.
the
L to
D ratio is not very
different for the two cases), one might expect that the power results with the
SHF are approximately the same as those with the TF. However, as seen above,
the specific power becomes a little larger. This is because in the case of
ut varying as the SHF, both CD and
ut at the middle portion of a stroke are larger than their
counterparts in the case of ut varying as the TF (note
that aerodynamic power is proportional to the mean of the product of
CD and ut over a stroke cycle, not to
D).
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Acknowledgments |
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References |
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