A computational study of the aerodynamics and forewing-hindwing interaction of a model dragonfly in forward flight
Ministry-of-Education Key Laboratory of Fluid Mechanics, Institute of Fluid Mechanics, Beijing University of Aeronautics and Astronautics, Beijing 100083, People's Republic of China
* Author for correspondence (e-mail: m.sun{at}263.net)
Accepted 12 August 2005
![]() |
Summary |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
At hovering and low J (J=0, 0.15), the model dragonfly uses separated flows or leading-edge vortices (LEV) on both the fore- and hindwing downstrokes; at medium J (J=0.30, 0.45), it uses the LEV on the forewing downstroke and attached flow on the hindwing downstroke; at high J (J=0.6, 0.75), it uses attached flows on both fore- and hindwing downstrokes. (The upstrokes are very lightly loaded and, in general, the flows are attached.)
At a given J, at d=180°, there are two
vertical force peaks in a cycle, one in the first half of the cycle, produced
mainly by the hindwing downstroke, and the other in the second half of the
cycle, produced mainly by the forewing downstroke; at
d=90°, 60° and 0°, the two force peaks merge
into one peak. The vertical force is close to the resultant aerodynamic force
[because the thrust (or body-drag) is much smaller than vertical force (or the
weight)]. 55-65% of the vertical force is contributed by the drag of the
wings.
The forewing-hindwing interaction is detrimental to the vertical force (and
resultant force) generation. At hovering, the interaction reduces the mean
vertical force (and resultant force) by 8-15%, compared with that without
interaction; as J increases, the reduction generally decreases (e.g.
at J=0.6 and d=90°, it becomes 1.6%). A
possible reason for the detrimental interaction is as follows: each of the
wings produces a mean vertical force coefficient close to half that needed for
weight support, and a downward flow is generated in producing the vertical
force; thus, in general, a wing moves in the downwash-velocity field induced
by the other wing, reducing its aerodynamic forces.
Key words: dragonfly, forward flight, unsteady aerodynamics, forewing-hindwing interaction, Navier-Stokes simulation
![]() |
Introduction |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Force measurement on a tethered dragonfly was conducted by Somps and
Luttges (1985). It was shown
that over some part of a stroke cycle, vertical force was many times larger
than the dragonfly weight. They considered that the large force might be due
to the effect of forewing-hindwing interaction. Flow visualization studies on
flapping model dragonfly wings were conducted by Saharon and Luttges
(1988
,
1989
), and it was shown that
constructive or destructive wing/flow interactions might occur, depending on
the kinematic parameters of the flapping motion. In these studies, only the
total force of the fore- and hindwings was measured and, moreover, force
measurements and flow visualizations were conducted in separate works.
Experimental (Freymuth, 1990
)
and computational (Wang, 2000
)
studies on an airfoil (two-dimensional wing) in dragonfly hovering mode showed
that large vertical force was produced during each downstroke and that the
mean vertical force was enough to support the weight of a typical dragonfly.
During each downstroke, a vortex pair was created; the large vertical force
was explained by the downward two-dimensional jet induced by the vortex pair
(Wang, 2000
). In these works
(Freymuth, 1990
;
Wang, 2000
), because only a
single airfoil was used, the effects of interaction between the fore- and
hindwings and the three-dimensional flow effects could not be considered. Flow
visualization studies on free-flying and tethered dragonflies were recently
conducted by Thomas et al.
(2004
). It was shown that
dragonflies fly by using unsteady aerodynamic mechanisms to generate
leading-edge vortices (LEVs) or high lift when needed and that the dragonflies
controlled the flow mainly by changing the angle of attack of the wings. Their
results represent the only existing data on the flow around the wings of
free-flying dragonflies.
Recently, Sun and Lan
(2004) studied the
aerodynamics and the forewing-hindwing interaction of the dragonfly Aeshna
juncea in hover flight, using the method of computational fluid dynamics
(CFD). Three-dimensional wings and wing kinematics data of free-flight were
employed in the study. They showed that the vertical force coefficient of the
forewing or the hindwing was twice as large as the quasi-steady value and that
the mean vertical force could balance the dragonfly weight. They also showed
that the large vertical force coefficient was due to the LEV associated with
the delayed stall mechanism and that the interaction between the fore- and
hindwings was not very strong and was detrimental to the vertical force
generation. The result of detrimental interaction is interesting. But Sun and
Lan (2004
) investigated only a
specific case of flight in Aeshna juncea, i.e. hovering with 180°
phase difference between the fore- and hindwings. Whether the result that
forewing-hindwing interaction is detrimental is a local result due to the
specific kinematics used or is a more general result is not known. It is
desirable to make further studies on dragonfly aerodynamics at various flight
conditions and on the problem of forewing-hindwing interaction.
In the present study, we address the above questions by numerical
simulation of the flows of a model dragonfly in forward flight. The vertical
force and thrust are made to balance the insect weight and body-drag,
respectively, by adjusting the angles of attack of the wings, so that the
simulated flight could better approximate the real flight. The phasing and the
incoming flow speed (flight speed) of the model dragonfly are systematically
varied. At each flight speed, four phase differences -0°, 60°, 90°
and 180° (the hindwing leads the forewing motion) - are considered.
Dragonflies vary the phase difference between the fore- and hindwings with
different behaviours (Norberg,
1975; Azuma and Watanabe,
1988
; Reavis and Luttges,
1988
; Wakeling and Ellington,
1997b
; Wang et al.,
2003
; Thomas et al.,
2004
). It has been shown that a 55-100° phase difference (the
hindwing leads forewing motion) is commonly used in straight forward flight
(e.g. Azuma and Watanabe, 1988
;
Wang et al., 2004) and a 180° phase difference is used in hovering (e.g.
Norberg, 1975
). Recent
observation by Thomas et al.
(2004
) has shown that 180°
phase difference is also used in forward flight. We chose 60°, 90° and
180° to represent the above range of phase difference. Although 0°
phase difference (parallel stroking) has been mainly found in accelerating or
manoeuvring flight (e.g. Alexander,
1986
; Thomas et al.,
2004
), this phase difference is also included for reference. As in
Sun and Lan (2004
), the
approach of solving the flow equations over moving overset grids is employed
because of the unique feature of the motion, i.e. the fore- and hindwings move
relative to each other.
![]() |
Materials and methods |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
|
|
Only the flow on the right of the plane of symmetry
(Fig. 1A) is computed; the
effects of left wings are taken into consideration by the central mirroring
condition. The overset-grid system used here is the same as that in Sun and
Lan (2004). Each of the wing
grids had dimensions 29x77x45 in the normal direction, around the
wing and in the spanwise direction, respectively, and the background grid had
dimensions 46x94x72 in the Y-direction and directions
parallel and normal to the stroke-planes, respectively. The time step value
used (
=0.02) is also the same as that in Sun and Lan
(2004
).
In the present study, the lift of a wing is defined as the component of the
aerodynamic force on the wing that is perpendicular to the translational
velocity of the wing (i.e. perpendicular to the stroke plane), and the drag of
a wing is defined as the component that is parallel to the translational
velocity (note that these are not the conventional definitions of lift and
drag; the conventional ones are the components of force perpendicular and
parallel to the relative airflows, respectively). lf and
df denote the lift and drag of the forewing, respectively;
lh and dh denote the lift and drag of
the hindwing, respectively. Resolving the lift and drag into the Z
and X axes gives the vertical force and thrust of a wing.
Vf and Tf denote the vertical force
and thrust of the forewing, respectively; Vh and
Th denote the vertical force and thrust of the hindwing,
respectively. For the forewing:
![]() | (1) |
![]() | (2) |
![]() | (3) |
![]() | (4) |
![]() | (5) |
Conventionally, reference velocity used in the definition of force coefficients of a wing is the relative velocity of the wing. In the above definition of force coefficients, U is used as the reference velocity. At hovering, U is the mean relative velocity of the wings. It should be noted that at forward flight, U is not the mean relative velocity of the wings and the above definition of force coefficients is different from the conventional one.
Kinematics of flapping wings
The flapping motions of the wings are shown in
Fig. 1. The free-stream
velocity, which has the same magnitude as the flight velocity, is denoted by
V, and the stroke plane angle is denoted by ß
(Fig. 1B). The azimuthal
rotation of a wing is called `translation', and the pitching (or flip)
rotation of the wing near the end of a half-stroke and at the beginning of the
following half-stroke is called rotation. The speed at r2
is called the translational speed. The wing translates downwards and upwards
along the stroke plane and rotates during stroke reversal
(Fig. 1B). The translational
velocity is denoted by ut and is given by:
![]() | (6) |
![]() | (7) |
Non-dimensional parameters of wing motion
In the flapping motion described above, we need to specify the flapping
period (c), the reference velocity (U), the
geometrical angles of attack (
d and
u),
the wing rotation duration (
r), the phase difference
(
d) between hindwing and forewing, the mean flapping angle
(
) and the stroke plane angle
(ß). For the flow computation, we also need to specify Re and
J.
For the dragonfly Aeshna juncea in hovering flight, the following
kinematic data are available (Norberg,
1975): ß
60°, n=36 Hz and
=69° for
both wings;
; and
17.5° for the forewing and hindwing, respectively; geometrical angles of
attack are approximately the same for fore- and hindwings. Morphological data
for the insect have been given in Norberg
(1972
): the mass of the insect
(m) is 754 mg; forewing length is 4.74 cm; hindwing length is 4.60
cm; the mean chord lengths of the forewing and the hindwing are 0.81 cm and
1.12 cm, respectively. In the present study, we assume that for the dragonfly,
, n and
do not vary
with flight speed [data in Azuma and Watanabe
(1988
) show that n
hardly varies with flight speed and
is increased only at very high
speed]. On the basis of the above data, we use the following parameters for
the model dragonfly: the length of both wings (R) is 4.7 cm
(Sf and Sh are 3.81 and 5.26
cm2, respectively); the reference length (c) is 0.81 cm;
U=2
nr2=2.5 m s-1;
Re=Uc/
1350;
c=U/nc=8.58. Norberg
(1975
) did not provide the rate
of wing rotation during stroke reversal. Reavis and Luttges
(1988
) made measurements on
some dragonflies and it was found that maximum
was
10 000-30 000
deg. s-1. Here,
is set as 20 000 deg. s-1,
giving
r=3.36. In hovering, the body of dragonfly
Aeshna juncea is horizontal
(Norberg, 1975
). We assume it
is also horizontal at forward flight. The angle between the body axis and the
stroke plane hardly changes (Azuma and
Watanabe, 1988
; Wakeling and
Ellington, 1997b
), therefore ß at forward flight can be
assumed to be the same as that at hovering [in Sun and Lan's
(2004
) study of hovering
flight, ß=52° was used; the same value is used here]. We also assume
that at all speeds considered, geometrical angles of attack are the same for
fore- and hindwings. In the present study,
d and J
are varied systematically to study their effects, therefore they are
known.
Now, the only kinematic parameters left to be specified are
d and
u. In the present study,
d and
u are not treated as known input
parameters but are determined in the calculation process; they are chosen such
that the computed mean vertical force of the wings approximately equals the
insect weight and the computed mean thrust approximately equals the body drag.
The mean vertical force coefficient required for balancing the weight
(CV,W) is defined as
CV,W=mg/0.5
U2(Sf+Sh);
the body-drag coefficient (CD,b) is defined as
CD,b=body-drag/0.5
U2(Sf+Sh).
Using the above data, CV,W is computed as
CV,W=1.35. The body-drag of Aeshna juncea is not
available. Here, the body-drag coefficients for dragonfly Sympetrum
sanguineum (Wakeling and Ellington,
1997a
) are used (converted to the current definition of
CD,b). Values of CD,b at various
J are shown in Table
1.
|
![]() |
Results |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
The case of J=0 (d=180°) is computed first.
For this case, values of
d and
u close to
the real ones are available from Norberg
(1975
). For dragonfly
Aeshna juncea hovering with
d=180°, Norberg
(1975
) observed that in the
mid-portion of the downstroke, the wing chord was almost horizontal, and in
the mid-portion of the upstroke it was close to the vertical; that is the real
values of
d and
u should be around 50°
and 20°, respectively (note that ß=52°).
d=50° and
u=15° are used as the
starting values, and the converged values of
d and
u are 52° and 8°, respectively. Using starting
values that are not far from the real values can reduce the number of
iterations. More importantly, there could be more than one solution due to the
nonlinearity in aerodynamic force production, and by so doing, the calculation
can generally converge to the realistic solution. Second, the case of
J=0.15 (
d=180°) is computed, using the
converged values of
d and
u of
J=0 (
d=180°) as the starting values. Since
J is not changed greatly, it is expected that these starting values
are not very different from the realistic solution. The same is done,
sequentially, for the cases of J=0.3, 0.45, 0.6 and 0.75
(
d=180°). Next, the case of J=0
(
d=90°) is computed, using the converged values of
d and
u at J=0
(
d=180°) as the starting values; then the cases of
J=0.15-0.75 (
d=90°) are computed in the same
way as in the corresponding cases of
d=180°. Finally,
the cases of J=0-0.75,
d=60° and 0° are
treated in a similar way.
The calculated results of d and
u are
shown in Table 2. Since, in
each of the cases, the starting values of
d and
u are expected to be not far from the real values, it is
reasonable to expect that these solutions are realistic. Let's examine how the
calculated
d and
u vary with advance
ratio, which can give some information on whether or not the solutions are
realistic. As seen in Table 2, at a given
d, when J is increased,
d decreases and
u increases. This should
be the correct trend of variation for the following reasons. When J
is increased, in the downstroke the relative velocity of the wing increases
and, to keep the total vertical force from increasing (vertical force is
mainly produced during the downstroke and it needs to be equal to the weight
of the dragonfly),
d should decrease; in the upstroke, the
relative velocity decreases and, to produce enough thrust (thrust is mainly
produced during the upstroke and a larger thrust is needed as J is
increased),
u should increase. As also seen in
Table 2,
u
increases with J at a relatively higher rate (
u
increases approximately from 8° to 65° when J changes from 0
to 0.75). This is reasonable because, if
u does not increase
with J fast enough, the effective angle of attack of the wing would
become negative (generally, operating at negative effective angle of attack is
not realistic). The variations of
d and
u
with J also show that it is reasonable to expect that the solutions
are realistic.
|
In Table 2, the mean total
force coefficients (V,
T), and the mean force
coefficients of the forewing
(
V,f,
T,f) and hindwing
(
V,h,
T,h) are also given
(
V,f,
T,f, etc. could show how
much aerodynamic force is produced by the forewing or by the hindwing).
V is close to
CV,W and
T is closed to
CD,b, as they should be. The mean thrust (the body-drag)
is much smaller than the mean vertical force (the weight); e.g. at
J=0, 0.3 and 0.6,
T is only 0, 1.4 and 6.6%
of
V, respectively. At a
given J,
d and
u do not change
greatly when
d is varied. For example, at J=0.15,
d and
u are 44° and 14°,
respectively, at
d=180°; 42° and 13.2° at
d=90°; 40° and 12.5° at
d=60°; 38° and 9.7° at
d=0°.
The fact that changing d from 180° to 0° does not
influence
d and
u values greatly indicates
that the forewing-hindwing interaction might not be very strong. This is
because the interaction between the wings is expected to be sensitive to the
relative motion, or to the phase difference, between the wings, and if strong
interaction exits, the values of
d and
u
would be greatly influenced by varying
d from 180° to
0°.
The time courses of the aerodynamic forces
The effects of phasing
Fig. 3 gives the time
courses of CV and CT in one cycle for
various forewing-hindwing phase differences for hovering flight
(J=0). For a clear description of the time courses of the forces and
flows, we express time during a cycle as a non-dimensional parameter,
, such that
=0 at the start of the downstroke of
the hindwing and
=1 at the end of the
following upstroke. At
d=180°, there are two large
CV peaks in one cycle, one in the first half-cycle
(
=0-0.5) and the other in the second
half-cycle (
=0.5-1.0) [this case has
been investigated in Sun and Lan
(2004
) and is included here
for comparison]. When the phase difference is changed to
d=90°, these two peaks merge into a large
CV peak between
=0 and
=0.75. The result at
d=60° is similar to that at
d=90°,
except that the CV peak is between
=0 and
=0.62 and is higher. For the case of
d=0°, the CV peak is between
=0 and
=0.5 and is even higher.
CV is the sum of CV,f and
CV,h. Fig.
4 gives the time courses of CV,f and
CV,h for the above cases. In all these cases, the hindwing
produces a large CV,h peak during its downstroke and a
very small CV,h during its upstroke; this is also true for
the forewing. At
d=180°, the downstroke of the hindwing
is in the first half-cycle (
=0-0.5)
and the downstroke of the forewing is in the second half-cycle
(
=0.5-1.0), resulting in the two
CV peaks (one between
=0 and
=0.5 and the other between
=0.5 and
=1.0; see the CV
curve for
d=180° in
Fig. 3). At
d=90°, the downstroke of the hindwing is still in the
first half-cycle (between
=0 and
=0.5), but the downstroke of the
forewing is between
=0.25 and
=0.75, resulting in the
CV peak between
=0 and
=0.75 (see the CV
curve for
d=90° in
Fig. 3). The
CV peak for the cases of
d=60° and
0° in Fig. 3 can be
explained similarly.
|
|
|
The effects of flight speed
Fig. 6 gives the time
courses of CV and CT in one cycle for
various advance ratios. For clarity, only the CV and
CT curves for J=0, 0.3 and 0.6 are plotted [the
CV (or CT) curve for J=0.15
is between those of J=0 and 0.3; the CV (or
CT) curve for J=0.45 is between those of
J=0.3 and 0.6; and the CV (or
CT) curve for J=0.75 is close to that for
J=0.6].
|
At d=90°, 60° and 0°
(Fig. 6C, E and G,
respectively), the effects of increasing J on CV
are similar to those in the case of
d=180°.
The lift and drag coefficients of the fore- and hindwings
The vertical force coefficient of a wing is related to the lift and drag
coefficients (see Eqn 1).
Fig. 7 shows the vertical
force, lift and drag coefficients of the hindwing and the forewing,
respectively, for the case of J=0.3 and d=180°.
Fig. 8 shows the corresponding
results for the case of J=0.6 and
d=180°. It is
seen that for the forewing or the hindwing, the drag coefficient is larger
than, or close to, the lift coefficient. Furthermore, ß is large
(52°). As a result (see Eqn
1), a large part of the vertical force coefficient is contributed
by the drag coefficient. This is also true for other flight conditions. Our
computations show that for all cases considered in the present study, 55-67%
of the total vertical force is contributed by the drag of the wings. The
results here are for hovering and forward flight conditions. For hovering,
similar results have been obtained previously: Sun and Lan
(2004
) showed that for the
same dragonfly as in the present study, 65% of the weight-supporting force is
contributed by the wing drag; Wang
(2004
), using two-dimensional
model, showed that a dragonfly might use drag to support about three-quarters
of its weight.
|
|
|
|
|
|
|
|
|
First, we examine the cases of d=180°. At
J=0 for the forewing (Fig.
9A), during the downstroke a LEV of large size appears (see plots
at
2 and
3 in
Fig. 9A); during the upstroke,
there is no LEV and the vorticity layers on the upper and lower surfaces of
the wing are approximately the same (see plots at
5 and
6 in Fig. 9A),
indicating that the effective angle of attack is close to zero. For the
hindwing (Fig. 9B), during the
downstroke the flows are generally similar to those of the forewing, except
that the LEV is a little smaller and a vortex layer shed from the trailing
edge (trailing-edge vortex layer) of the forewing is around the hindwing at
its mid-upstroke (see plot at
5 in
Fig. 9B). At J=0.3
(Fig. 10), the LEVs of the
wings during their downstrokes are smaller than those at J=0 (compare
Fig. 10 with
Fig. 9); in fact, the LEV of
the hindwing has the form of a thick vortex layer (see plots at
2 and
3 in
Fig. 10B), indicating that the
flow is effectively attached. Another difference is that the trailing-edge
vortex layer of the forewing is less close to the hindwing at its mid-upstroke
than in the case of J=0 (comparing the plot at
5 in
Fig. 10B with the plot at
5 in Fig. 9B).
At J=0.6 (Fig. 11),
the LEVs of both the forewing and hindwing during their downstrokes have the
form of a thick vortex layer (see plots at
2 and
3 in Fig. 11A
and Fig. 11B), indicating that
flows are effectively attached. The flow attachment during the downstrokes at
relatively large J can be clearly seen from the sectional streamline
plots shown in Fig. 15: as
J increases, flows around the forewing and hindwing become more and
more attached.
Next, we examine the cases of d=60° (Figs
12,
13,
14). The flows vary with
J in the same way as in the cases of
d=180°
discussed above; that is, as J increases, the LEVs on the forewing
and the hindwing downstrokes decease in size (becoming a vortex layer at
relatively large J), and the hindwing in its downstroke meets less
and less of the trailing-edge vortex layer of the forewing (compare Figs
12,
13 and
14). At a given J,
the flows of the fore- and hindwings are not greatly different from those in
the case of
d=180°, except that the hindwing in its
upstroke meets the trailing-edge vortex layer of the forewing at an earlier
time (compare Figs 12,
13 and
14 with Figs
9,
10 and
11, respectively). The fact
that there do not exist large differences between the flows for
d=60° and
d=180° indicates that
the forewing-hindwing interaction might not be very strong.
The forewing-hindwing interaction
In order to obtain quantitative data on the interaction between the fore-
and hindwings, we made two more sets of computations. In the first set, the
hindwing was taken away and the flows around the single forewing were
computed; in the second set, the forewing was taken away and the flows around
the single hindwing were computed. The vertical force and thrust for the
single forewing are denoted as Vsf and
Tsf, respectively; those for the single hindwing are
denoted as Vsh and Tsh. The
coefficients of Vsf, Tsf,
Vsh and Tsh are denoted as
CV,sf, CT,sf,
CV,sh and CT,sh, respectively, and are
defined as:
![]() | (8) |
Figs 16,
17,
18,
19 compare the time courses of
CV,sf, CV,sh,
CT,sf and CT,sh with those of
CV,f, CV,h, CT,f
and CT,h, respectively. The differences between
CV,sf and CV,f, etc., show the
interaction effects. At a given d and J (e.g.
d=180° and J=0.6;
Fig. 16E), the vertical force
coefficient of a wing is decreased at certain periods and increased at some
other periods of a cycle due to forewing-hindwing interaction. When J
is varied (e.g. comparing Fig.
16A,C,E) or
d is varied (e.g. comparing Figs
16A,
17A and
18A), the interaction effect
occurs at different periods of the cycle and its strength may change. This is
because, at a given time in the stroke cycle, a wing is at a different
position relative to the wake of the other wing when J or
d is varied.
|
|
|
|
![]() | (9) |
|
Recently, Maybury and Lehmann
(2004) conducted experiments on
interaction between two robotic wings. In their experiment, the two wings are
stacked vertically (forewing on the top), the stroke planes are horizontal and
the wings operate in still air. Although their experimental set-up is
different from the set-up of our simulation, there is some resemblance between
their experiment and our hovering simulation: the hindwing operates in the
wake of the forewing and the forewing is also influenced by the disturbed flow
due to the hindwing. Thus, the results on interaction effects obtained by
these two studies might be similar to some extent. Data in fig. 3D of Maybury
and Lehmann (2004
) show that
between a phase shift of 0 and 50% of the stroke cycle
(
d
0-180°), the total vertical force is reduced by
approximately 6-16% due to the interaction. The results in the present study
show that between
d
60-180°, the total vertical force
is reduced by 7.8-15% due to the interaction (see
Table 3, J=0).
![]() |
Discussion |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Somps and Luttges (1985),
based on their experiments, suggested that forewing-hindwing interaction might
enhance aerodynamic force production. Results in the present study, however,
show that the interaction is detrimental. It is of interest to discuss the
present results in relation to those of Somps and Luttges
(1985
). In their experiment
with a tethered dragonfly (in still air; wings flapping with
d
80°), Somps and Luttges
(1985
) measured the time
course of the total vertical force, which has a single large peak in each
cycle (see fig. 2c of Somps and Luttges,
1985
); the mean vertical force is more than twice the body weight.
Based on the fact that one single large vertical force peak is produced in
each cycle (rather than the double peaks they expected from the sum of the
forces produced independently by the fore- and hindwings), they considered
that the forewing-hindwing interaction must be strong and suggested that it
played an important role in generating the large vertical force. Our vertical
force time histories for
d=60° and 90° at hovering
are very similar to those in Somps and Luttges
(1985
), also having a single
large peak in each cycle [compare the CV curve for
d=60° or 90° in
Fig. 3A with the curve in fig.
2c of Somps and Luttges
(1985
)]. However, analyses in
the present study clearly show that the large single force peak is not due to
forewing-hindwing interaction but rather to the overlap of the single force
peak produced by the hindwing with that by the forewing.
Separated and attached flows
As seen in Figs 9,
10,
11,
12,
13,
14,
15, at hovering
(J=0), flows on both the forewing and hindwing during the loaded
downstroke are separated and large LEVs exist. As J increases, the
LEVs become smaller and smaller and the flows become more and more attached.
The flows of the hindwing downstroke are effectively attached at
J=0.3 and those of the forewing downstroke are effectively attached
at J=0.6 (see e.g. Fig.
15). That is, in producing the aerodynamic forces needed for
flight, the model dragonfly uses separated flows with LEVs at hovering and low
J, uses both separated and attached flows at medium J, and
uses attached flow at high J.
At hovering and low J, the relative velocity of a wing is mainly due to the flapping motion and is relatively low. Thus, high `aerodynamic force coefficients' are needed (in the present section, aerodynamic force coefficients are coefficients defined in the conventional way; that is, the reference velocity used is the relative velocity of the wing; note that reference velocity used in the definition of the aerodynamic force coefficients in the proceeding sections is U, which is smaller than the relative velocity of the forewing or the hindwing in the case of forward flight). The dragonfly must use the separated flows with LEVs to generate the high aerodynamic force coefficients.
At high J, the relative velocity is contributed by both the flapping motion and the relatively high forward velocity and is relatively high. Thus, relatively low aerodynamic force coefficients are needed. The dragonfly does not need to use separated flows; instead, it uses attached flows. As an example, we estimate the mean relative velocity of a section of the forewing (or hindwing) at a distance r2 from the wing root at J=0.6. Using the diagram in Fig. 20, the relative velocity is estimated as 1.78U [U is the mean relative velocity of this section at hovering (J=0)]. The mean relative velocity is 1.78 times as large as that at hovering, and the vertical force coefficient needed would be about one-third of that needed for hovering. Therefore, at J=0.6, attached flows could produce the required aerodynamic force coefficients.
|
Comparison with flow visualization results of free-flying dragonflies
Recently, Thomas et al.
(2004) presented flow
visualization results for free-flying and tethered dragonflies. Some of their
visualization tests were made for the dragonfly Aeshna mixta flying
freely at V
=1.0 m s-1 (see, for example,
fig. 6 of Thomas et al.,
2004
). Their results show that the dragonfly uses counter-stroking
(
d=180°), with an LEV on the forewing downstroke and
attached flow on the hindwing down- and upstrokes. The model dragonfly in the
present study is modelled using the available morphological and kinematic data
of the dragonfly Aeshna juncea, which is of the same genus as the
dragonfly in the experiment. Moreover, in the flight of the model dragonfly,
force-balance conditions are satisfied, and the flight could be a good
approximation of the real flight. Therefore, we can make comparisons between
the computed and experimental results. At U=0.3,
V
of the model dragonfly is 1.23 m s-1,
close to that in the experiment. Our results show that at this flight velocity
there is a LEV on the forewing downstroke and the flows on the hindwing down-
and upstrokes are approximately attached (Figs
10B,
15), in agreement with the
flow visualization results of the free-flying dragonfly.
The above comparison is for an intermediate advance ratio. For high and
very low advance ratios, there are also similarities between the
visualizations of Thomas et al.
(2004) and the simulation of
the present study. Based on two available free flight sequences, Thomas et al.
(2004
) suggested (p. 4308)
that at fast flight (high advance ratio), flows on the forewing and the
hindwing were both attached; our results show that at J=0.6 (Figs
11,
14), the flows on both the
forewing and the hindwing are approximately attached. At very low speed, they
showed (video S2 in their supplementary material) that flows were separated on
the hindwing as well as on the forewing; our simulation gives similar results
(Figs 9,
12).
List of symbols
![]() |
Acknowledgments |
---|
![]() |
References |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Alexander, D. E. (1986). Wing tunnel studies of turns by flying dragonflies. J. Exp. Biol. 122, 81-98.[Abstract]
Azuma, A. and Watanabe, T. (1988). Flight performance of a dragonfly. J. Exp. Biol. 137,221 -252.
Dickinson, M. H., Lehman, F. O. and Sane, S. P.
(1999). Wing rotation and the aerodynamic basis of insect flight.
Science 284,1954
-1960.
Freymuth, P. (1990). Thrust generation by an airfoil in hover modes. Exp. Fluids 9, 17-24.[CrossRef]
Lan, S. L. and Sun, M. (2001a). Aerodynamic properties of a wing performing unsteady rotational motions at low Reynolds number. Acta Mech. 149,135 -147.[CrossRef]
Maybury, W. J. and Lahmann, F. (2004). The
fluid dynamics of flight control by kinematic phase lag variation between two
robotic insect wings. J. Exp. Biol.
207,4707
-4726.
Norberg, R. A. (1972). The pterostigma of insect wings and inertial regulator of wing pitch. J. Comp. Physiol. 81,9 -22.[CrossRef]
Norberg, R. A. (1975). Hovering flight of the dragonfly Aeschna juncea L., kinematics and aerodynamics. In Swimming and Flying in Nature (ed. T. Y. Wu, C. J. Brokaw and C. Brennen), pp. 763-781. New York: Plenum Press.
Reavis, M. A. and Luttges, M. W. (1988). Aerodynamic forces produced by a dragonfly. AIAA Paper 88-0330.
Rogers, S. E. and Kwak, D. (1990). Upwind Differencing scheme for the time-accurate incompressible Navier-Stokes equations. AIAA J. 28,253 -262.
Rogers, S. E., Kwak, D. and Kiris, C. (1991). Steady and unsteady solutions of the incompressible Navier-Stokes equations. AIAA J. 29,603 -610.
Rogers, S. E. and Pulliam, T. H. (1994). Accuracy enhancements for overset grids using a defect correction approach. AIAA Paper 94-0523.
Saharon, D. and Luttges, M. (1988). Visualization of unsteady separated flow produced by mechanically driven dragonfly wing kinematics model. AIAA Paper 88-0569.
Saharon, D. and Luttges, M. (1989). Dragonfly unsteady aerodynamics: the role of the wing phase relations in controlling the produced flows. AIAA Paper 89-0832.
Somps, C. and Luttges, M. (1985). Dragonfly flight: novel uses of unsteady separation flows. Science 28,1326 -1328.
Sun, M. and Lan, S. L. (2004). A computational
study of the aerodynamic forces and power requirements of dragonfly
(Aeshna juncea) hovering. J. Exp. Biol.
207,1887
-1901.
Sun, M. and Tang, J. (2002). Unsteady
aerodynamic force generation by a model fruit fly wing in flapping motion.
J. Exp. Biol. 205,55
-70.
Thomas, A. L. R., Taylor, G. K., Srygley, R. B., Nudds, R. L.
and Bomphrey, R. J. (2004). Dragonfly flight: free-flight and
tethered flow visualizations reveal a diverse array of unsteady
lift-generating mechanisms, controlled primarily via angle of attack.
J. Exp. Biol. 207,4299
-4323.
Wakeling, J. M. and Ellington, C. P. (1997a).
Dragonfly flight (1). Gliding flight and steady-state aerodynamic forces.
J. Exp. Biol. 200,543
-556.
Wakeling, J. M. and Ellington, C. P. (1997b).
Dragonfly flight (2). Velocities, accelerations and kinematics of flapping
flight. J. Exp. Biol.
200,557
-582.
Wakeling, J. M. and Ellington, C. P. (1997c).
Dragonfly flight (3). Quasi-steady lift and power requirements. J.
Exp. Biol. 200,583
-600.
Wang, H., Zeng, L. J., Liu, H. and Yin, C. Y.
(2003). Measuring wing kinematics, flight trajectory and body
attitude during forward flight and turning maneuvers in dragonflies.
J. Exp. Biol. 206,745
-757.
Wang, Z. J. (2000). Two dimensional mechanism for insect hovering. Phys. Rev. Lett. 85,2216 -2219.[CrossRef][Medline]
Wang, Z. J. (2004). The role of drag in insect
hovering. J. Exp. Biol.
207,4147
-4155.