First published online September 16, 2005
Journal of Experimental Biology 208, 3785-3804 (2005)
Published by The Company of Biologists 2005
doi: 10.1242/jeb.01852
A computational study of the aerodynamics and forewing-hindwing interaction of a model dragonfly in forward flight
Ji Kang Wang and
Mao Sun*
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
|
|---|
The aerodynamics and forewing-hindwing interaction of a model dragonfly in
forward flight are studied, using the method of numerically solving the
Navier-Stokes equations. Available morphological and stroke-kinematic
parameters of dragonfly (Aeshna juncea) are used for the model
dragonfly. Six advance ratios (J; ranging from 0 to 0.75) and, at
each J, four forewing-hindwing phase angle differences
(
d; 180°, 90°, 60° and 0°) are considered.
The mean vertical force and thrust are made to balance the weight and
body-drag, respectively, by adjusting the angles of attack of the wings, so
that the flight could better approximate the real flight.
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
|
|---|
Scientists have always been fascinated by the flight of dragonflies.
Analysis based on quasi-steady aerodynamic theory has shown that the vertical
force required for weight support is much greater than the steady-state values
measured from dragonfly wings, suggesting that unsteady aerodynamics must play
important roles in the flight of dragonflies
(Norberg, 1975
; Wakeling and
Ellington,
1997a,b,c).
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
|
|---|
The model wings
The model fore- and hindwings (Fig.
1) are the same as those used in Sun and Lan
(2004). The thickness of the
wings is 1% of c (where c is the mean chord length of the
forewing). The planforms of the wings are similar to those of the wings of
Aeshna juncea (Norberg,
1972). The fore- and hindwings are the same length, but the chord
length of the hindwing is larger than that of the forewing. The radius of the
second moment of the forewing area is denoted by r2, and
r2=0.61R, where R is the wing length
(the mean flapping velocity at r2 is used as the reference
velocity in the present study).
View larger version (29K):
[in this window]
[in a new window]
|
Fig. 1. Sketches of the model wings, the flapping motion and the reference frames.
FW and HW denote fore- and hindwings, respectively. O,X,Y,Z is an
inertial frame, with the X and Y axes in the horizontal
plane. ß, stroke plane angle; V , incoming flow
velocity.
|
|
The flow computation method and evaluation of the aerodynamic forces
The flow equations and computational method used in the present study are
the same as those used in Sun and Lan
(2004). Only an outline of the
method is given here. The Navier-Stokes equations are numerically solved using
moving overset grids. The algorithm was first developed by Rogers and Kwak
(1990) and Rogers et al.
(1991) for single-grid, which
is based on the method of artificial compressibility, and it was extended by
Rogers and Pulliam (1994) to
overset grids. The time derivatives of the momentum equations are differenced
using a second-order, three-point backward difference formula. The derivatives
of the viscous fluxes in the momentum equation are approximated using
second-order central differences. For the derivatives of convective fluxes,
upwind differencing based on the flux-difference splitting technique is used.
A third-order upwind differencing is used at the interior points, and a
second-order upwind differencing is used at points next to boundaries. With
overset grids (Fig. 2), for
each wing there is a body-fitted curvilinear grid, which extends a relatively
short distance from the body surface, and in addition, there is a background
Cartesian grid, which extends to the far-field boundary of the domain. The
solution method for single-grid is applied to each of these grids; data are
interpolated from one grid to another at the inter-grid boundary points.
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) |
These two formulae also apply to the hindwing. The coefficients of
Vf, Tf, Vh,
Th, lf, df,
lh and dh are denoted as
CV,f, CT,f, CV,h,
CT,h, Cl,f, Cd,f,
Cl,h and Cd,h, respectively. They are
defined as:
 | (3) |
where 
is the fluid density, Sf and
Sh are the areas of the fore- and hindwings, respectively.
The total vertical force (V) and total thrust (T) of the
fore- and hindwings are V=Vf+Vh and
T=Tf+Th, respectively. The
coefficients of V and T are denoted as
CV and CT, respectively, and defined
as:
 | (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) |
where the non-dimensional translational velocity
ut+=ut/U
(U is the reference velocity); the non-dimensional time
=tU/c (
is the time;
c is the mean chord length of the forewing, used as reference length
in the present study); c is the non-dimensional period of the
flapping cycle; and is the phase angle of the translation of the wing.
The reference velocity is U=2
nr2, where
and n are the stroke amplitude and stroke frequency of the
forewing, respectively. Denoting the azimuth-rotational velocity as
, we have
.
The geometric angle attack of the wing is defined as the acute angle between
the stroke plane and the wing-surface plane, which assumes a constant value
during the translational portion of a half-stroke; the constant value is
denoted by 
d for the downstroke and by u
for the upstroke (Fig. 1).
Around the stroke reversal, the angle of attack changes with time, and the
angular velocity () is given by:
 | (7) |
where the non-dimensional form
;
is a
constant; r is the time at which the rotation starts; and
r is the time interval over which the rotation lasts.
In the time interval of r, the wing rotates from
u to d. Therefore, when
d, u and r are
specified,
can be
determined (around the next stroke reversal, the wing would rotate from
u to d, and the sign of the right-hand
side of Eqn 7 should be
reversed). The axis of the flip rotation is located at a distance of 24% of
the mean chord length of the wing from the leading edge. With U and
c as the reference velocity and reference length, respectively, the
Reynolds number (Re) is defined as
Re=Uc/
=2nr2c/ ( is the
kinematic viscosity of the air), and the advance ratio (J) is defined
as
J=V/2nR)=V/(UR/r2).
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=2nr2=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/0.5U2(Sf+Sh);
the body-drag coefficient (CD,b) is defined as
CD,b=body-drag/0.5U2(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
|
|---|
Force balance in the flight
In the present study, six advance ratios (J=0, 0.15, 0.30, 0.45,
0.60, 0.75; V=0-3.1 m s-1) and, at each
J, four phase differences (d=180°, 90°,
60° and 0°; hindwing leads the forewing motion), are considered. At a
given set of J and d, d and
u are chosen such that the CV
approximately equals CV,W, and CT
approximately equals CD,b. The calculation procedure is as
follows. At a given J and d, a set of values of
d and u is estimated (how the starting
values are estimated is described below). The flow equations are solved and
the corresponding CV and CT are
calculated. CV is compared with CV,W
(1.35) and CT is compared with CD,b
(Table 1). If
CV is different from CV,W, or
CT is different from CD,b,
d and u are adjusted. The calculations are
repeated until the difference between CV and
CV,W is less than 0.05 and the difference between
CT and CD,b is less than 0.01 (as
will be seen below, in most cases, a difference between CT
and CD,b of less than 0.005 is achieved).
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.
Fig. 5 gives the
CV and CT results for forward flight
at J=0.3. The effects of varying the phasing are similar to those in
the cases of J=0, i.e. when d is decreased from
180° to 90° (and below), the two CV peaks (between
=0 and
=0.5 and between
=0.5 and
=1.0, respectively) merge into one
CV peak. This is generally true for other advance ratios
considered.
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=180° (Fig.
6A), as J is increased, the distributions of
CV in the first half-cycle
(=0-0.5) change greatly:
CV between =0 and
=0.3 is decreased and
CV around =0.4 is
increased. As discussed above, CV in the first half-cycle
is due to the hindwing downstroke. The decrease in CV
between =0 and
=0.3 is caused mainly by two factors;
(1) d of the hindwing is smaller at higher speeds
(Table 2) and (2) at higher
speeds, the forewing-hindwing interaction decreases the vertical force on the
hindwing in this period (see below). The large increase in
CV around =0.4 is
due to the effect of pitching-up rotation of the hindwing. It is known that
when a wing pitches up in an incoming flow, large aerodynamic forces could be
produced; the higher the incoming flow speed, the larger the forces
(Dickinson et al., 1999; Lan
and Sun, 2001; Sun and Tang,
2002). The hindwing undergoes pitching-up rotation at
=0.4. At higher J, the
relative velocity is larger and, in addition, the portion of wing area behind
the rotation-axis is relatively large for the hindwing (see
Fig. 1A), resulting in the
large CV around
=0.4.
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.
View larger version (26K):
[in this window]
[in a new window]
|
Fig. 7. Time courses of vertical force, lift and drag coefficients for the hindwing
(A) and the forewing (B) in one cycle at d=180° and
J=0.3. CV,h, Cl,h and
Cd,h, vertical force, lift and drag coefficients of the
hindwing, respectively; CV,f, Cl,f and
Cd,f, vertical force, lift and drag coefficients of the
forewing, respectively; d, difference in phase angle between
the hindwing and forewing; J, advance ratio;
, non-dimensional time.
|
|
View larger version (25K):
[in this window]
[in a new window]
|
Fig. 8. Time courses of vertical force, lift and drag coefficients for the hindwing
(A) and the forewing (B) in one cycle at d=180° and
J=0.6. CV,h, Cl,h and
Cd,h, vertical force, lift and drag coefficients of the
hindwing, respectively; CV,f, Cl,f and
Cd,f, vertical force, lift and drag coefficients of the
forewing, respectively; d, difference in phase angle between
the hindwing and forewing; J, advance ratio;
, non-dimensional time.
|
|
The flows around the forewing and the hindwing
Here, we present flows around the forewing and the hindwing for six
representative cases: d=180° and J=0, 0.3 and
0.6; d=60° and J=0, 0.3 and 0.6. Figs
9,
10,
11 show the contours of the
non-dimensional spanwise component of vorticity at half-wing length at various
times of the stroke cycle, for the cases J=0, 0.3 and 0.6 of
d=180°; Figs
12,
13,
14 show the corresponding
results for the cases of d=60°. Since the variation in
J causes considerable changes in d and
u, to guard against possible misinterpretation of the
results, in each of Figs 9,
10,
11,
12,
13,
14, d and
u are specified at the same time as J (this is also
done in Fig. 15). In Figs
9,
10,
11,
12,
13,
14, 1,
2 and 3 represent the times at
0.1c after the start of the downstroke, the mid-downstroke and
0.4c after the start of the downstroke of a wing,
respectively; 4, 5 and 6
represent the corresponding times of the upstroke of the wing.
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) |
Note that they are defined in the same way as in the case of two wings in
interaction (see Eqn 3).
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.
View larger version (20K):
[in this window]
[in a new window]
|
Fig. 16. Time courses of vertical force coefficients of forewing
(CV,f), single forewing (CV,sf),
hindwing (CV,h) and single hindwing
(CV,sh) and thrust coefficients of the forewing
(CT,f), single forewing (CT,sf),
hindwing (CT,h) and single hindwing
(CT,sh) in one cycle: (A,B) d=180°,
J=0; (C,D) d=180°, J=0.3; (E,F)
d=180°, J=0.6. d, difference
in phase angle between the hindwing and forewing; J, advance ratio;
, non-dimensional time.
|
|
View larger version (19K):
[in this window]
[in a new window]
|
Fig. 17. Time courses of vertical force coefficients of forewing
(CV,f), single forewing (CV,sf),
hindwing (CV,h) and single hindwing
(CV,sh) and thrust coefficients of the forewing
(CT,f), single forewing (CT,sf),
hindwing (CT,h) and single hindwing
(CT,sh) in one cycle; (A,B) d=90°,
J=0; (C,D) d=90°, J=0.3; (E,F)
d=90°, J=0.6. d, difference
in phase angle between the hindwing and forewing; J, advance ratio;
, non-dimensional time.
|
|
View larger version (19K):
[in this window]
[in a new window]
|
Fig. 18. Time courses of vertical force coefficients of forewing
(CV,f), single forewing (CV,sf),
hindwing (CV,h) and single hindwing
(CV,sh) and thrust coefficients of the forewing
(CT,f), single forewing (CT,sf),
hindwing (CT,h) and single hindwing
(CT,sh) in one cycle; (A,B) d=60°,
J=0; (C,D) d=60°, J=0.3; (E,F)
d=60°, J=0.6. d, difference
in phase angle between the hindwing and forewing; J, advance ratio;
, non-dimensional time.
|
|
View larger version (18K):
[in this window]
[in a new window]
|
Fig. 19. Time courses of vertical force coefficients of forewing
(CV,f), single forewing (CV,sf),
hindwing (CV,h) and single hindwing
(CV,sh) and thrust coefficients of the forewing
(CT,f), single forewing (CT,sf),
hindwing (CT,h) and single hindwing
(CT,sh) in one cycle; (A,B) d=0°,
J=0; (C,D) d=0°, J=0.3; (E,F)
d=0°, J=0.6. d, difference in
phase angle between the hindwing and forewing; J, advance ratio;
, non-dimensional time.
|
|
The total vertical force without interaction (VNI) is
the sum of Vsf and Vsh. The
coefficient of VNI is denoted as CV,NI
and defined as:
 | (9) |
Let V,NI be the mean value
of CV,NI. Thus
CV=(V-V,NI)/V,NI
represents the percentage of increment in mean total vertical force
coefficient due to the forewing-hindwing interaction (when
CV is negative, the interaction is detrimental to
vertical force generation). The value of CV is
given in Table 3. From the
total vertical force and the total thrust, the total resultant force can be
calculated. The increment in mean total resultant force coefficient due to the
forewing-hindwing interaction is obtained in the same way as above, which is
also given in Table 3. It is
very close to CV. This is because, under the
present flight conditions, the wings produce a much larger vertical force than
thrust. As seen in Table 3, at
all phase angles and advance ratios considered, the interaction is detrimental
to the vertical force (or resultant force) generation. At hovering, the
interaction reduces the mean total vertical force coefficient (or the mean
total resultant force coefficient) by around 15% for
d=180° and 90°, 8% for d=60°,
and 3% for d=0°. As J increases, for
d=180°, 90° and 60°, the reduction decreases;
but for d=0°, the reduction changes little from hovering
to medium advance ratios (J=0-0.3) and increases to 6-13% at higher
advance ratios (J=0.45-0.75).
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 forewi
TOP
Summary
Introduction
