First published online February 20, 2004
Journal of Experimental Biology 207, 1137-1150 (2004)
Published by The Company of Biologists 2004
doi: 10.1242/jeb.00868
Unsteady aerodynamic forces of a flapping wing
Jiang Hao Wu and
Mao Sun*
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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The unsteady aerodynamic forces of a model fruit fly wing in flapping
motion were investigated by numerically solving the Navier–Stokes
equations. The flapping motion consisted of translation and rotation [the
translation velocity (ut) varied according to the simple
harmonic function (SHF), and the rotation was confined to a short period
around stroke reversal]. First, it was shown that for a wing of given geometry
with ut varying as the SHF, the aerodynamic force
coefficients depended only on five non-dimensional parameters, i.e. Reynolds
number (Re), stroke amplitude (
), mid-stroke angle of attack
(
m), non-dimensional duration of wing rotation
(
r) and rotation timing [the mean translation velocity
at radius of the second moment of wing area (U), the mean chord
length (c) and c/U were used as reference velocity, length
and time, respectively]. Next, the force coefficients were investigated for a
case in which typical values of these parameters were used (Re=200;
=150°; m=40°; r was 20%
of wingbeat period; rotation was symmetrical). Finally, the effects of varying
these parameters on the force coefficients were investigated.
In the Re range considered (20–1800), 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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It has been shown that conventional aerodynamic theory, which was based on
steady flow conditions, cannot explain the generation of large lift by the
wings of small insects (for reviews, see
Ellington, 1984a
;
Spedding, 1992). In the past
few years, much progress has been made in revealing the unsteady high-lift
mechanisms of flapping insect wings.
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=75–225;
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 2–3
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 2–4 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, 0.4–5.7 m
s–1), 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 (Re75) 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 Re600 (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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Most of the procedures used in these CFD simulations have been described
elsewhere (Sun and Tang,
2002a,b).
The planform of the model wing used (Fig.
2) is the same as that of the robotic fruit fly wing used by
Dickinson et al. (1999). The
wing section is a flat plate of 3% thickness with round leading and trailing
edges. The ratio of the wing length (R) to the mean chord length
(c) is 3. The radius of the second moment of wing area
(r2) is 0.6R (the mean translational velocity at
r2 is used as reference velocity in this study). Two
coordinate systems are used. One is the inertial coordinate system,
OXYZ, and the other is the body-fixed coordinate system,
oxyz (Fig. 1).
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) |
where the non-dimensional translational velocity
ut+=ut/U
(U is the reference velocity); non-dimensional time
=tU/c (t is the time); and c
is the non-dimensional period of a wingbeat cycle. The azimuth-rotational
speed of the wing is related to ut. Denoting the
azimuth-rotational speed as 
, we
have
.
The geometric angle of attack of the wing is denoted by . It assumes a
constant value except at the start or near the end of a stroke. The constant
value is denoted by m, the mid-stroke angle of attack.
Around the stroke reversal, changes with time and the angular
velocity, 
, is given by:
 | (2) |
where the non-dimensional form
;
is the mean non-dimensional angular velocity of rotation [note that
here is different by a factor of R/r2 from that
defined in Ellington (1984c),
where velocity at wing tip was used as reference velocity]; r
is the non-dimensional time at which the rotation starts;
r is the non-dimensional time interval over which the
rotation lasts, which is termed as wing-rotation duration. In the time
interval of r, the wing rotates from
=m to =180–m.
Therefore, when m and r are
specified, can be determined.
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=2nr2
(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 Navier–Stokes equations. Written in the inertial
coordinate system OXYZ and non-dimensionalized, they are as follows:
 | (3) |
 | (4) |
 | (5) |
 | (6) |
where u, v and w are three components of the non-dimensional
fluid velocity and p is the non-dimensional fluid pressure. In the
non-dimensionalization, U, c and c/U are taken as reference
velocity, length and time, respectively. Re in equations 4–6 is
defined as Re=cU/
(where is the kinematic viscosity
of the fluid). The numerical method used to solve equations 3–6 is the
same as that in Sun and Tang
(2002a,b).
Once the Navier–Stokes 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) |
where 
is the fluid density and S is the wing area.
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
Navier–Stokes equations (equations 4–6) gives the aerodynamic
force coefficients CL and CD; the only
non-dimensional parameter in the Navier–Stokes 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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Code validation and grid resolution test
Code validation
The code used in this study is the same as that in Sun and Tang
(2002a). It was tested by
measured unsteady aerodynamic forces on a flapping model fruit fly wing
(Sun and Tang, 2002b;
Sun and Wu, 2003). The
calculated drag coefficient agreed well with the measured value [see fig. 2A,C
of Sun and Wu (2003)]. For the
lift coefficient, in the translation phase during the middle, and in the
rotation phase at the end, of each half-stroke, the computed value agreed well
with the measured value, whereas in the beginning of the stroke, the computed
peak value was much smaller than the measured value [see fig. 2B,D of Sun and
Wu (2003) and fig. 4 of Sun
and Tang (2002b)]. Recently,
Birch and Dickinson (2003)
visualized the vorticity patterns around the flapping model fruit fly wing
using digital particle image velocimetry. It is of interest to compare the
vorticity patterns calculated by Sun and Tang
(2002a) using the code with
the experimentally visualized ones. For convenience, we define a
non-dimensional parameter, 
, such
that =0 at the start of the
downstroke and =1 at the end of the
subsequent upstroke. At the beginning of the half-stroke, difference in the
positions of shed vortices exists between the computation and the experiment
[compare fig. 4A of Sun and Tang
(2002a) with the panel at
=0.02 in fig. 5 of Birch and
Dickinson (2003)]; during the
translation phase at the middle, and the rotation phase at the end, of the
half-stroke, the computed vorticity patterns agree well with the
experimentally visualized patterns [compare fig. 4B–E,G,H of Sun and
Tang (2002a) with panels at
=0.07, 0.12, 0.19, 0.26, 0.38 and
0.48 in fig. 5 of Birch and Dickinson
(2003)]. The vorticity
comparison is consistent with the force comparison described above: both show
that discrepancy exists at the beginning of the half-stroke. The discrepancy
might be because the CFD code does not resolve satisfactorily the complex flow
at stroke reversal. There is also the possibility that it is due to variations
in the precise kinematic patterns, especially at stroke reversal.
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°.
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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Fig. 4. Effects of grid density on computed lift coefficient
(CL) and vorticity field. (A) The time course of
CL in one cycle. (B) Vorticity contour plots at half-wing
length near the end of a stroke.
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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=0.2c).
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.
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.
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.
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 =0.5c, the downstroke; between
=0.5c 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. 8A–C; 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.
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 85–90%
of L is contributed by the
pure translational motion for all the values of Re considered.
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
m50°.
The rate of change of L
with m
(dL/dm)
from m=25° to 35° is given in
Table 1. For Re above
100,
dL/dm
hardly varies with Re and its value is approximately 3.0, which is
almost the same as the measured value (2.9–3.1) 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,
dL/dm
decreases greatly.
The effects of rotation duration
In the calculations above, r=1.87
(=0.2c; =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 ).
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.
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=2r2/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
2nr2 (n is the wingbeat frequency).
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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The influence of Re and comparison between the lift coefficients and insect flight data
Previous studies on revolving wings (Usherwood and Ellington,
2002a,b;
Dickinson et al., 1999) showed
that large aerodynamic force coefficients were produced due to the delayed
stall mechanism in the Re range of approximately 140 (model fruit fly
wing) to 15 000 (quail wing) and that the force coefficients were not
sensitive to Re. The present study on a flapping wing has provided
results for lower Re. As seen in
Fig. 10, when Re is
above 100, L and
D vary only slightly with
Re, in agreement with the previous results. However, when Re
is below 100, L
decreases and D increases
greatly. This is because at such low Re (20, 60), although the LEV
still exists and attaches to the wing in the translational phases of the
half-strokes, it is rather weak and its vorticity is considerably diffused
(see Figs 8D,E,
9).
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=2nr2, Re=Uc/ and
L,W=mg/0.5U2St,
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
m45°), 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 Navier–Stokes equations (equations 3–6),
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
TOP
Summary
Introduction
, positional
angle of the wing; 
) and azimuthal
rotation (
); (B) time
courses of lift coefficient (CL) and (C) drag coefficient
(CD) in one cycle. Reynolds number (Re)=200,
stroke amplitude (
