## SUMMARY

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 (*u*_{t}) 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 *u*_{t} varying as the SHF, the aerodynamic force
coefficients depended only on five non-dimensional parameters, i.e. Reynolds
number (*Re*), stroke amplitude (Φ), mid-stroke angle of attack
(α_{m}), non-dimensional duration of wing rotation
(Δτ_{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;Φ
=150°; α_{m}=40°; Δτ_{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
(*C̄*_{L}) and drag
(*C̄*_{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, *C̄*_{L} decreased and
*C̄*_{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 *C̄*_{L} and large
*C̄*_{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, Φ from 90° to 180°
and Δτ_{r} from 17% to 32% of the stroke period (symmetrical
rotation), the force coefficients varied only slightly with *Re*, Φ
and Δτ_{r}. This meant that the forces were approximately
proportional to the square of Φ*n* (*n* is the wingbeat
frequency); thus, changing Φ and/or *n* could effectively control
the magnitude of the total aerodynamic force.

The time course of *C̄*_{L}
(or *C̄*_{D}) in a half-stroke
for *u*_{t} varying according to the SHF resembled a half
sine-wave. It was considerably different from that published previously for
*u*_{t}, 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 *u*_{t} varying as
the TF were approximately 10% smaller than those for *u*_{t}
varying as the SHF except when wing rotation is delayed).

## Introduction

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
(*C*_{L}) of approximately 2 was maintained within 2–3
chord lengths of travel. Afterwards, *C*_{L} 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, *C*_{L} 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 (*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 (*u*_{t}) 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
*u*_{t} was close to the simple harmonic function (SHF). Recent
data of free-flying fruit fly (Fry et al.,
2003) also showed that *u*_{t} was close to the
SHF. When *u*_{t} 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 *u*_{t} varies as a TF with large acceleration at
stroke reversal. Therefore, it is of interest to study the aerodynamic forces
for the case of *u*_{t} 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
*u*_{t} 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 (Φ), mid-stroke
angle of attack (α_{m}), wing-rotation duration
(Δτ_{r}) and rotation timing (τ_{r}). We
consider the *Re* range of 20 to 1800; in addition, we investigate the
effects of varying Φ, α_{m}, Δτ_{r} andτ
_{r}.

## Materials and methods

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
(*r*_{2}) is 0.6*R* (the mean translational velocity at
*r*_{2} 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 *r*_{2} due to wing
translation is called the translational velocity (*u*_{t}).
*u*_{t} is assumed to vary as the SHF:
(1)
where the non-dimensional translational velocity
*u*_{t}^{+}=*u*_{t}/*U*
(*U* is the reference velocity); non-dimensional timeτ
=*tU*/*c* (*t* is the time); and τ_{c}
is the non-dimensional period of a wingbeat cycle. The azimuth-rotational
speed of the wing is related to *u*_{t}. Denoting the
azimuth-rotational speed as , we
have
.
The geometric angle of attack of the wing is denoted by α. It assumes a
constant value except at the start or near the end of a stroke. The constant
value is denoted by α_{m}, the mid-stroke angle of attack.
Around the stroke reversal, α changes with time and the angular
velocity, , is given by:
(2)
where the non-dimensional form
;ϖ
is the mean non-dimensional angular velocity of rotation [note thatϖ
here is different by a factor of *R/r _{2}* from that
defined in Ellington (1984c),
where velocity at wing tip was used as reference velocity]; τ

_{r}is the non-dimensional time at which the rotation starts;Δτ

_{r}is the non-dimensional time interval over which the rotation lasts, which is termed as wing-rotation duration. In the time interval of Δτ

_{r}, the wing rotates fromα =α

_{m}to α=180–α

_{m}. Therefore, when α

_{m}and Δτ

_{r}are specified, ϖ can be determined.

In the flapping motion described above, the period of wingbeat cycleτ
_{c}, the geometric angle of attack at mid-strokeα
_{m}, the rotation duration Δτ_{r} or the
mean angular velocity rotation ϖ and the rotation timing τ_{r}
need to be specified. Note that since *U*=2Φ*nr*_{2}
(where *n* is the wingbeat frequency and Φ is the stroke
amplitude), τ_{c} (=*U/cn*) is related to Φ byτ
_{c}=2Φ^{•}(*r*_{2}/*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*/ν (where ν 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 ρ 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 *C*_{L} and *C*_{D}; 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, Φ, α_{m}, Δτ_{r} andτ
_{r} need to be specified. That is, the aerodynamic force
coefficients on the wing depend on five non-dimensional parameters:
*Re*, Φ, α_{m}, Δτ_{r} andτ
_{r}. When the wing rotation is symmetrical, τ_{r}
may be determined from Δτ_{r}; thus, *C*_{L}
and *C*_{D} depend only on four parameters: *Re*, Φ,α
_{m} and Δτ_{r}.

## Results

### 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, *t̂*, such
that *t̂*=0 at the start of the
downstroke and *t̂*=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
*t̂*=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
*t̂*=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 *C*_{L} and *C*_{D} are shown in
Fig. 3 [measured data are taken
from fig. 7 of Usherwood and Ellington
(2002b)]. In the whole α
range (from –20° to 100°), the computed *C*_{L}
agrees well with the measured values; both have approximately sinusoidal
dependence on α. The computed *C*_{D} also agrees well
with the measured values except when α 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 53×48×41 (around the wing
section, in the normal direction and in the spanwise direction respectively),
grids 2 and 3 had dimensions 77×70×61 and 109×93×78,
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 20*c* away from the wing surface in the normal direction and
8*c* 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, Δτ=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 Δτ=0.02 were appropriate for the present study.

### Forces and flows of a typical case

We first considered a case in which typical values of wing kinematic
parameters were used (*Re*=200, Φ=150°,α
_{m}=40°, Δτ_{r}=1.87 and wing rotation
was symmetrical; with the above values of Φ, α_{m} andΔτ
_{r}, we had τ_{c}=9.37, ϖ andΔτ
_{r}=0.2τ_{c}).

Fig. 5 shows the time
courses of *C*_{L} and *C*_{D} in one cycle.
*C*_{L} in the middle portion of a half-stroke is large and
dominates over *C*_{L} at the beginning and near the end of the
half-stroke (*C*_{D} behaves similarly). In previous studies by
Dickinson et al. (1999) and Sun
and Tang (2002a), in which
*u*_{t} 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
*C*_{L} and *C*_{D} also exist
(Fig. 5B,C) but they are very
small. This is because near the end of the half-stroke the
*u*_{t} has become very low and wing rotation cannot produce a
large force at low *u*_{t}.

The mean lift (*C̄*_{L}) and
drag (*C̄*_{D}) coefficients are
1.66 and 1.67, respectively, which are much larger than the steady-state
values [measured steady-state *C*_{L} and
*C*_{D} on a fruit fly wing in uniform free-stream in a wind
tunnel at the same *Re* (200) and same α_{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 (α 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 *C*_{L} and
*C*_{D} 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 *C*_{L}
and *C*_{D} 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 *C*_{L} and *C*_{D} in one cycle for
various *Re* (*Re* ranging from 20 to 1800; other conditions
being the same as in the typical case). In general, *C*_{L}
increases and *C*_{D} decreases as *Re* increases.
However, when *Re* is higher than ∼100, *C*_{L} and
*C*_{D} do not vary greatly, whereas when *Re* is lower
than ∼100, *C*_{L} is much smaller and
*C*_{D} much larger than at higher *Re*.

For the case of *Re*=200, as discussed above, the large
*C*_{L} and *C*_{D} during a stroke are due to
the delayed stall mechanism. For the cases of other *Re*, as seen in
Fig. 7, *C*_{L}
and *C*_{D} do not have a sudden drop during a half-stroke
(between τ=0 and τ=0.5τ_{c}, the downstroke; betweenτ
=0.5τ_{c} and τ=τ_{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
*C*_{L} and large *C*_{D}. For reference,
vorticity contour plots at various times in one cycle for the case of
*Re*=20 are shown in Fig.
9.

*C̄*_{L} and
*C̄*_{D} at various *Re*
are plotted in Fig. 10. For
*Re* above ∼100, change in
*C̄*_{L} and
*C̄*_{D} with *Re* are
small, whereas for *Re* below ∼100,
*C̄*_{L} decreases and
*C̄*_{D} increases rapidly as
*Re* decreases. Similar to the typical case, approximately 85–90%
of *C̄*_{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
*C̄*_{L} and
*C̄*_{D} in the range ofα
_{m} from 25° to 60° (for all *Re* considered in
the above section). The slope of the
*C̄*_{L} (α_{m})
curve is approximately constant between α_{m}=25° and
35°; beyond α_{m}=35°, it decreases gradually to zero atα
_{m}≈50°.

The rate of change of *C̄*_{L}
with α_{m}
(d*C̄*_{L}/dα_{m})
from α_{m}=25° to 35° is given in
Table 1. For *Re* above∼
100,
d*C̄*_{L}/dα_{m}
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,
d*C̄*_{L}/dα_{m}
decreases greatly.

### The effects of rotation duration

In the calculations above, Δτ_{r}=1.87
(=0.2τ_{c}; ϖ=0.93). Observation of many insects in free
flight (Ellington, 1984c;
Ennos, 1989) showed that ϖ
ranged approximately from 0.8 to 1.4. Here, we investigate the effects of
varying Δτ_{r} (i.e. varying ϖ) on the aerodynamic force
coefficients.

Fig. 12 gives the time
courses of *C*_{L} and *C*_{D} in one cycle for
four values of Δτ_{r};
Table 2 gives the mean force
coefficients. Varying Δτ_{r} does not change the mean force
coefficients greatly (see Table
2); when Δτ_{r} is almost doubled (varied from
1.27 to 2.40), *C̄*_{L} and
*C̄*_{D} change only
approximately 3%. *C*_{L} and *C*_{D} in the
mid-portion of a half-stroke vary little with Δτ_{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). WhenΔτ
_{r} is relatively short (ϖ is relatively large), the
force peaks are relatively large but they occupy a short period; whenΔτ
_{r} is longer (ϖ 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 Δτ_{r} give
more or less the same contribution to the corresponding mean force
coefficient. This explains why
*C̄*_{L} and
*C̄*_{D} do not change greatly
with Δτ_{r} (or ϖ).

### The effects of rotation timing

Fig. 13 shows the time
courses of *C*_{L} and *C*_{D} in one cycle for
different rotation timing (τ_{r} can be read from
Fig. 13A; *Re*,α
_{m}, Φ and Δτ_{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
*C*_{L} and *C*_{D} 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, *C*_{L} and
*C*_{D} 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 *C*_{L} and
*C*_{D} 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,
*C*_{L} is negative and *C*_{D} 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.
*C̄*_{L} and
*C̄*_{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, Φ, ranged approximately from 90° to 180°.
Moreover, an insect might change Φ to control its aerodynamic force (e.g.
Ellington, 1984c;
Lehmann and Dickinson, 1998).
Here, we investigate the effects of Φ on the force coefficients.
Calculations were made for various Φ (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 *C*_{L} and *C*_{D} in
one cycle; Table 4 gives the
mean force coefficients. Note that when Φ is varied, the non-dimensional
period of wingbeat cycle will change
(τ_{c}=2Φ*r*_{2}/*c*); sinceΔτ
_{r} (i.e. ϖ) is fixed,Δτ
_{r}/τ_{c} is different for different Φ.
In the range of Φ from 90° to 180°, the effects of varying Φ
on the force coefficients are not large; when Φ increases or decreases by
30°, *C̄*_{L} and
*C̄*_{D} change less than 3% and
6%, respectively. When Φ is below approximately 90°, the effects of
varying Φ become larger (see the results for Φ=65°;
Fig. 14;
Table 4). It is of interest to
point out the fact that *C̄*_{L}
and *C̄*_{D} hardly vary withΦ
(in the range of Φ from 90° to 180°) means that the mean
lift and mean drag vary as Φ^{2}, because the forces are
non-dimensionalized by *U*^{2}, and *U* equals
2Φ*nr*_{2} (*n* is the wingbeat frequency).

Sane and Dickinson (2001)
studied the effects of varying Φ 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 Φ fell below approximately 120°,
*C̄*_{L} decreased and
*C̄*_{D} increased with
decreasing Φ (*C̄*_{D}
increased rapidly as Φ became small). In the present simulation
(Fig. 14;
Table 4), when Φ is below∼
90°, we also found
*C̄*_{D} increasing and
*C̄*_{L} decreasing with a
decrease in Φ, but the rates of change in
*C̄*_{L} and
*C̄*_{D} are smaller than those
reported by Sane and Dickinson
(2001). In their experiment,
when Φ changed, *Re* and Δτ_{r} also changed, but
the ratio of Δτ_{r}/τ_{c} did not change; in the
present simulation, *Re* and Δτ_{r} did not change
when Φ changed. To make further comparison with their results, we made
some calculations in which *Re* and Δτ_{r} changed
with Φ but Δτ_{r}/τ_{c} was kept unchanged
(=0.2). The results are given in Fig.
15 and Table 5. The
trends of variation in *C̄*_{L}
and *C̄*_{D} with Φ are
similar to those in Sane and Dickinson
(2001): when Φ is below
approximately 120°, *C̄*_{L}
decreases and *C̄*_{D} increases
with Φ decreasing, and when Φ is below 90°,
*C̄*_{D} increases rapidly.

## Discussion

### 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, *C̄*_{L} and
*C̄*_{D} vary only slightly with
*Re*, in agreement with the previous results. However, when *Re*
is below ∼100, *C̄*_{L}
decreases and *C̄*_{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
*C̄*_{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*=2Φ*nr*_{2}, *Re*=*Uc*/ν and
*C̄*_{L,W}=*m g*/0.5ρ

*U*

^{2}

*S*

_{t}, where

*and*

**g***S*

_{t}were the gravitational acceleration and the area of both wings, respectively).

*Re*and

*C̄*

_{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
*C̄*_{L,W} is 2.87; when its wing
area is extended to include the brim hairs
(Ellington, 1975), its
*C̄*_{L,W} is still as high as
1.62. As seen in Fig. 11, at
such a low *Re*, the maximum
*C̄*_{L} is ∼1.15 (atα
_{m}≈45°), which is much smaller than its
*C̄*_{L,W}. In the computations
that gave the results in Fig.
11, symmetrical rotation was used and Φ was 150°. For
reference, we made another calculation in which advanced rotation was used,Φ
=180° and α_{m}=45° (this combination of parameters
was expected to maximize
*C̄*_{L}). The computation gave
*C̄*_{L}=1.25, which was also
much smaller than the *C̄*_{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 anα
_{m} between 30° and 50°, a
*C̄*_{L} equal to
*C̄*_{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 Φn

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
*u*_{t} varying as the SHF depend only on *Re*,α
_{m}, Φ and Δτ_{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 Δτ_{r} and also vary
only slightly with Φ in the range of Φ approximately from 90° to
180°.

When Φ and/or *n* is varied, *Re* will change (note that
*Re*=2Φ*nr*_{2}*c*/ν). Since the force
coefficients hardly vary with Φ and *Re*, the mean lift
(*L̄*) and drag
(*D̄*) vary approximately with
(Φ*n*)^{2}. That is, changing Φ and/or *n* can
effectively control the aerodynamic forces. For instance, increasing Φ by
15%, *L̄* can be increased by
approximately 32% [note that by increasing α_{m} by 15% (e.g.
from 40° to 46°), *L̄* increases
only by ∼10% (see Fig.
11)].

In the above discussion, we have assumed that the force coefficients hardly
vary with *Re*, Φ and Δτ_{r} (or ϖ). 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 α_{m} from 25° to 60°, *Re*, Φ andϖ
were simultaneously increased by 20% from those of the typical case.
Fig. 16 shows the results. At
all α_{m} considered, the force coefficients vary only
slightly.

### Comparison of the present results with those of u_{t} 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,
*u*_{t} 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 *u*_{t} varying as the SHF and
those for *u*_{t} 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 *u*_{t} varying as a TF with
rapid accelerations at stroke reversal, the *C*_{L} (or
*C*_{D}) 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 *u*_{t} varying as the SHF, the *C*_{L} (or
*C*_{D}) 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 *u*_{t} varying as the TF but not in the case of
*u*_{t} 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, *u*_{t}
varied as a TF with rapid accelerations at stroke reversal [the duration of
translational acceleration at stroke reversal was 0.18τ_{c},
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*, Φ, α_{m}, Δτ_{r}) were the
same as those in the typical case. In the second set, *u*_{t}
was replaced by the SHF. The results showed that when wing rotation was
symmetrical or advanced, *C̄*_{L}
and *C̄*_{D} for
*u*_{t} varying as the TF are approximately 12% and 8% smaller
than those for *u*_{t} varying as the SHF, respectively, and
when wing rotation was delayed,
*C̄*_{L} for
*u*_{t} varying as the TF was approximately 35% smaller than
that for *u*_{t} varying as the SHF but
*C̄*_{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 *u*_{t} (wing rotation was symmetrical). As seen
above, *C̄*_{L} and
*C̄*_{D} for
*u*_{t} varying as the TF are 12% and 8% smaller, respectively,
than those for *u*_{t} 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 *u*_{t} 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 α_{m}=36.5°
and the body-mass-specific power is 29 W kg^{–1}; with
*u*_{t} varying as the SHF, the correspondingα
_{m} and body-mass-specific power are 30.5° and 31.5 W
kg^{–1}, respectively. That is, with the SHF, theα
_{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 α_{m} becoming smaller is obvious.
Seeing that *C̄*_{L} and
*C̄*_{D} with the SHF are larger
than their counterparts with the TF by approximately the same percentage (i.e.
the *C̄*_{L} to
*C̄*_{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
*u*_{t} varying as the SHF, both *C*_{D} and
*u*_{t} at the middle portion of a stroke are larger than their
counterparts in the case of *u*_{t} varying as the TF (note
that aerodynamic power is proportional to the mean of the product of
*C*_{D} and *u*_{t} over a stroke cycle, not to
*C̄*_{D}).

## List of symbols

- c
- mean chord length
*C*_{D}- drag coefficient
*C̄*_{D}- mean drag coefficient
*C*_{L}- lift coefficient
*C̄*_{L}- mean lift coefficient
*C̄*_{L,W}- mean lift coefficient for supporting the insect's weight
- D
- drag
- D̄
- mean drag
- L
- lift
- L̄
- mean lift
- M
- mass of insect
- n
- wingbeat frequency

- O,o
- origins of the inertial frame of reference and the non-inertial frame of reference
- p
- non-dimensional fluid pressure
- R
- wing length
- r
_{2} - radius of the second moment of wing area
- Re
- Reynolds number
- S
- area of one wing
*S*_{t}- area of a wing pair
- t
- time
- t̂
- non-dimensional parameter (being zero at the start of downstroke and 1 at the end of the subsequent upstroke)
- U
- reference velocity
*u*_{t}- translational velocity of the wing
*u*_{t}^{+}- non-dimensional translational velocity of the wing
- x,y,z
- coordinates in non-inertial frame of reference
- X,Y,Z
- coordinates in inertial frame of reference (
*Z*in vertical direction) - α
- geometric angle of attack
- angular velocity of pitching rotation
- non-dimensional angular velocity of pitching rotation
- αm
- mid-stroke geometric angle of attack
- ϕ
- azimuthal or positional angle
- angular velocity of azimuthal rotation
- non-dimensional angular velocity of azimuthal rotation
- Φ
- stroke amplitude
- ν
- kinematic viscosity
- ρ
- density of fluid
- τ
- non-dimensional time
- τc
- period of one wingbeat cycle (non-dimensional)
- τr
- time when pitching rotation starts (non-dimensional)
- Δτr
- duration of wing rotation or flip duration (non-dimensional)
- ϖ
- mean non-dimensional angular velocity of wing rotation

## ACKNOWLEDGEMENTS

We thank the two referees whose helpful comments and valuable suggestions greatly improved the quality of the paper. This research was supported by the National Natural Science Foundation of China (10232010).

- © The Company of Biologists Limited 2004