## 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
(γ_{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 γ_{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; atγ
_{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 γ_{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.

## 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 *r*_{2}, and
*r*_{2}=0.61*R*, where *R* is the wing length
(the mean flapping velocity at *r*_{2} is used as the reference
velocity in the present study).

### 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 29×77×45 in the normal direction, around the
wing and in the spanwise direction, respectively, and the background grid had
dimensions 46×94×72 in the *Y*-direction and directions
parallel and normal to the stroke-planes, respectively. The time step value
used (Δτ=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). *l*_{f} and
*d*_{f} denote the lift and drag of the forewing, respectively;
*l*_{h} and *d*_{h} 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.
*V*_{f} and *T*_{f} denote the vertical force
and thrust of the forewing, respectively; *V*_{h} and
*T*_{h} 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
*V*_{f}, *T*_{f}, *V*_{h},
*T*_{h}, *l*_{f}, *d*_{f},
*l*_{h} and *d*_{h} are denoted as
*C*_{V,f}, *C*_{T,f}, *C*_{V,h},
*C*_{T,h}, *C*_{l,f}, *C*_{d,f},
*C*_{l,h} and *C*_{d,h}, respectively. They are
defined as:
3
where ρ is the fluid density, *S*_{f} and
*S*_{h} 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=V*_{f}+*V*_{h} and
*T=T*_{f}+*T*_{h}, respectively. The
coefficients of *V* and *T* are denoted as
*C*_{V} and *C*_{T}, 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 β
(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 *r*_{2}
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 *u*_{t} and is given by:
6
where the non-dimensional translational velocity
*u*_{t}^{+}=*u*_{t}/*U*
(*U* is the reference velocity); the non-dimensional timeτ
=*tU/c* (*t̂* is the time;
*c* is the mean chord length of the forewing, used as reference length
in the present study); τ_{c} is the non-dimensional period of the
flapping cycle; and γ is the phase angle of the translation of the wing.
The reference velocity is *U*=2Φ*nr*_{2}, whereΦ
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 α_{d} for the downstroke and by α_{u}
for the upstroke (Fig. 1).
Around the stroke reversal, the angle of attack changes with time, and the
angular velocity (α) is given by:
7
where the non-dimensional form
;
is a
constant; τ_{r} is the time at which the rotation starts; andΔτ
_{r} is the time interval over which the rotation lasts.
In the time interval of Δτ_{r}, the wing rotates fromα
_{u} to α_{d}. Therefore, whenα
_{d}, α_{u} and Δτ_{r} are
specified,
can be
determined (around the next stroke reversal, the wing would rotate fromα
_{u} to α_{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*/ν=2Φ*nr*_{2}*c*/ν (ν is the
kinematic viscosity of the air), and the advance ratio (*J*) is defined
as
*J=V*_{∞}/2Φ*nR*)=*V*_{∞}/(*UR/r*_{2}).

### Non-dimensional parameters of wing motion

In the flapping motion described above, we need to specify the flapping
period (τ_{c}), the reference velocity (*U*), the
geometrical angles of attack (α_{d} and α_{u}),
the wing rotation duration (Δτ_{r}), the phase difference
(γ_{d}) between hindwing and forewing, the mean flapping angle
() and the stroke plane angle
(β). 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): β≈60°, *n*=36 Hz and Φ=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 Φ do not vary
with flight speed [data in Azuma and Watanabe
(1988) show that *n*
hardly varies with flight speed and Φ 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
(*S*_{f} and *S*_{h} are 3.81 and 5.26
cm^{2}, respectively); the reference length (*c*) is 0.81 cm;
*U*=2Φ*nr*_{2}=2.5 m s^{-1};
*Re=Uc*/ν≈1350; τ_{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 α was ∼10 000-30 000
deg. s^{-1}. Here, α is set as 20 000 deg. s^{-1},
giving Δτ_{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 β at forward flight can be
assumed to be the same as that at hovering [in Sun and Lan's
(2004) study of hovering
flight, β=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, γ_{d} and *J*
are varied systematically to study their effects, therefore they are
known.

Now, the only kinematic parameters left to be specified areα
_{d} and α_{u}. In the present study,α
_{d} and α_{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
(*C*_{V,W}) is defined as
*C*_{V,W}=*m g*/0.5ρ

*U*

^{2}(

*S*

_{f}+

*S*

_{h}); the body-drag coefficient (

*C*

_{D,b}) is defined as

*C*

_{D,b}=body-drag/0.5ρ

*U*

^{2}(

*S*

_{f}+

*S*

_{h}). Using the above data,

*C*

_{V,W}is computed as

*C*

_{V,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

*C*

_{D,b}). Values of

*C*

_{D,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 (γ_{d}=180°, 90°,
60° and 0°; hindwing leads the forewing motion), are considered. At a
given set of *J* and γ_{d}, α_{d} andα
_{u} are chosen such that the *C*_{V}
approximately equals *C*_{V,W}, and *C*_{T}
approximately equals *C*_{D,b}. The calculation procedure is as
follows. At a given *J* and γ_{d}, a set of values ofα
_{d} and α_{u} is estimated (how the starting
values are estimated is described below). The flow equations are solved and
the corresponding *C*_{V} and *C*_{T} are
calculated. *C*_{V} is compared with *C*_{V,W}
(1.35) and *C*_{T} is compared with *C*_{D,b}
(Table 1). If
*C*_{V} is different from *C*_{V,W}, or
*C*_{T} is different from *C*_{D,b},α
_{d} and α_{u} are adjusted. The calculations are
repeated until the difference between *C*_{V} and
*C*_{V,W} is less than 0.05 and the difference between
*C*_{T} and *C*_{D,b} is less than ∼0.01 (as
will be seen below, in most cases, a difference between *C*_{T}
and *C*_{D,b} of less than 0.005 is achieved).

The case of *J*=0 (γ_{d}=180°) is computed first.
For this case, values of α_{d} and α_{u} close to
the real ones are available from Norberg
(1975). For dragonfly
*Aeshna juncea* hovering with γ_{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 α_{d} and α_{u} should be around 50°
and 20°, respectively (note that β=52°).α
_{d}=50° and α_{u}=15° are used as the
starting values, and the converged values of α_{d} andα
_{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 (γ_{d}=180°) is computed, using the
converged values of α_{d} and α_{u} of
*J*=0 (γ_{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
(γ_{d}=180°). Next, the case of *J*=0
(γ_{d}=90°) is computed, using the converged values ofα
_{d} and α_{u} at *J*=0
(γ_{d}=180°) as the starting values; then the cases of
*J*=0.15-0.75 (γ_{d}=90°) are computed in the same
way as in the corresponding cases of γ_{d}=180°. Finally,
the cases of *J*=0-0.75, γ_{d}=60° and 0° are
treated in a similar way.

The calculated results of α_{d} and α_{u} are
shown in Table 2. Since, in
each of the cases, the starting values of α_{d} andα
_{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 α_{d} and α_{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 γ_{d}, when *J* is increased,α
_{d} decreases and α_{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), α_{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), α_{u} should increase. As also seen in
Table 2, α_{u}
increases with *J* at a relatively higher rate (α_{u}
increases approximately from 8° to 65° when *J* changes from 0
to 0.75). This is reasonable because, if α_{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 α_{d} and α_{u}
with *J* also show that it is reasonable to expect that the solutions
are realistic.

In Table 2, the mean total
force coefficients (*C̄*_{V},
*C̄*_{T}), and the mean force
coefficients of the forewing
(*C̄*_{V,f},
*C̄*_{T,f}) and hindwing
(*C̄*_{V,h},
*C̄*_{T,h}) are also given
(*C̄*_{V,f},
*C̄*_{T,f}, etc. could show how
much aerodynamic force is produced by the forewing or by the hindwing).
*C̄*_{V} is close to
*C*_{V,W} and
*C̄*_{T} is closed to
*C*_{D,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,
*C̄*_{T} is only 0, 1.4 and 6.6%
of *C̄*_{V}, respectively. At a
given *J*, α_{d} and α_{u} do not change
greatly when γ_{d} is varied. For example, at *J=*0.15,α
_{d} and α_{u} are 44° and 14°,
respectively, at γ_{d}=180°; 42° and 13.2° atγ
_{d}=90°; 40° and 12.5° atγ
_{d}=60°; 38° and 9.7° atγ
_{d}=0°.

The fact that changing γ_{d} from 180° to 0° does not
influence α_{d} and α_{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 α_{d} and α_{u}
would be greatly influenced by varying γ_{d} from 180° to
0°.

### The time courses of the aerodynamic forces

#### The effects of phasing

Fig. 3 gives the time
courses of *C*_{V} and *C*_{T} 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,
*t̂*, such that
*t̂*=0 at the start of the downstroke of
the hindwing and *t̂*=1 at the end of the
following upstroke. At γ_{d}=180°, there are two large
*C*_{V} peaks in one cycle, one in the first half-cycle
(*t̂*=0-0.5) and the other in the second
half-cycle (*t̂*=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 toγ
_{d}=90°, these two peaks merge into a large
*C*_{V} peak between
*t̂*=0 and
*t̂*=0.75. The result atγ
_{d}=60° is similar to that at γ_{d}=90°,
except that the *C*_{V} peak is between
*t̂*=0 and
*t̂*=0.62 and is higher. For the case ofγ
_{d}=0°, the *C*_{V} peak is between
*t̂*=0 and
*t̂*=0.5 and is even higher.
*C*_{V} is the sum of *C*_{V,f} and
*C*_{V,h}. Fig.
4 gives the time courses of *C*_{V,f} and
*C*_{V,h} for the above cases. In all these cases, the hindwing
produces a large *C*_{V,h} peak during its downstroke and a
very small *C*_{V,h} during its upstroke; this is also true for
the forewing. At γ_{d}=180°, the downstroke of the hindwing
is in the first half-cycle (*t̂*=0-0.5)
and the downstroke of the forewing is in the second half-cycle
(*t̂*=0.5-1.0), resulting in the two
*C*_{V} peaks (one between
*t̂*=0 and
*t̂*=0.5 and the other between
*t̂*=0.5 and
*t̂*=1.0; see the *C*_{V}
curve for γ_{d}=180° in
Fig. 3). Atγ
_{d}=90°, the downstroke of the hindwing is still in the
first half-cycle (between *t̂*=0 and
*t̂*=0.5), but the downstroke of the
forewing is between *t̂*=0.25 and
*t̂*=0.75, resulting in the
*C*_{V} peak between
*t̂*=0 and
*t̂*=0.75 (see the *C*_{V}
curve for γ_{d}=90° in
Fig. 3). The
*C*_{V} peak for the cases of γ_{d}=60° and
0° in Fig. 3 can be
explained similarly.

Fig. 5 gives the
*C*_{V} and *C*_{T} 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 γ_{d} is decreased from
180° to 90° (and below), the two *C*_{V} peaks (between
*t̂*=0 and
*t̂*=0.5 and between
*t̂*=0.5 and
*t̂*=1.0, respectively) merge into one
*C*_{V} peak. This is generally true for other advance ratios
considered.

#### The effects of flight speed

Fig. 6 gives the time
courses of *C*_{V} and *C*_{T} in one cycle for
various advance ratios. For clarity, only the *C*_{V} and
*C*_{T} curves for *J*=0, 0.3 and 0.6 are plotted [the
*C*_{V} (or *C*_{T}) curve for *J*=0.15
is between those of *J*=0 and 0.3; the *C*_{V} (or
*C*_{T}) curve for *J*=0.45 is between those of
*J*=0.3 and 0.6; and the *C*_{V} (or
*C*_{T}) curve for *J*=0.75 is close to that for
*J*=0.6].

At γ_{d}=180° (Fig.
6A), as *J* is increased, the distributions of
*C*_{V} in the first half-cycle
(*t̂*=0-0.5) change greatly:
*C*_{V} between *t̂*=0 and
*t̂*=0.3 is decreased and
*C*_{V} around *t̂*=0.4 is
increased. As discussed above, *C*_{V} in the first half-cycle
is due to the hindwing downstroke. The decrease in *C*_{V}
between *t̂*=0 and
*t̂*=0.3 is caused mainly by two factors;
(1) α_{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
*C*_{V} around *t̂*=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
*t̂*=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 *C*_{V} around
*t̂*=0.4.

At γ_{d}=90°, 60° and 0°
(Fig. 6C, E and G,
respectively), the effects of increasing *J* on *C*_{V}
are similar to those in the case of γ_{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 γ_{d}=180°.
Fig. 8 shows the corresponding
results for the case of *J*=0.6 and γ_{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, β 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.

### The flows around the forewing and the hindwing

Here, we present flows around the forewing and the hindwing for six
representative cases: γ_{d}=180° and *J*=0, 0.3 and
0.6; γ_{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 ofγ
_{d}=180°; Figs
12,
13,
14 show the corresponding
results for the cases of γ_{d}=60°. Since the variation in
*J* causes considerable changes in α_{d} andα
_{u}, to guard against possible misinterpretation of the
results, in each of Figs 9,
10,
11,
12,
13,
14, α_{d} andα
_{u} are specified at the same time as *J* (this is also
done in Fig. 15). In Figs
9,
10,
11,
12,
13,
14, τ_{1},τ
_{2} and τ_{3} represent the times at
0.1τ_{c} after the start of the downstroke, the mid-downstroke and
0.4τ_{c} after the start of the downstroke of a wing,
respectively; τ_{4}, τ_{5} and τ_{6}
represent the corresponding times of the upstroke of the wing.

First, we examine the cases of γ_{d}=180°. At
*J*=0 for the forewing (Fig.
9A), during the downstroke a LEV of large size appears (see plots
at τ_{2} and τ_{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 τ_{5} andτ
_{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 τ_{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 atτ
_{2} and τ_{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 τ_{5} in
Fig. 10B with the plot atτ
_{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 τ_{2} andτ
_{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 γ_{d}=60° (Figs
12,
13,
14). The flows vary with
*J* in the same way as in the cases of γ_{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 γ_{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 forγ
_{d}=60° and γ_{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 *V*_{sf} and
*T*_{sf}, respectively; those for the single hindwing are
denoted as *V*_{sh} and *T*_{sh}. The
coefficients of *V*_{sf}, *T*_{sf},
*V*_{sh} and *T*_{sh} are denoted as
*C*_{V,sf}, *C*_{T,sf},
*C*_{V,sh} and *C*_{T,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
*C*_{V,sf}, *C*_{V,sh},
*C*_{T,sf} and *C*_{T,sh} with those of
*C*_{V,f}, *C*_{V,h}, *C*_{T,f}
and *C*_{T,h}, respectively. The differences between
*C*_{V,sf} and *C*_{V,f}, etc., show the
interaction effects. At a given γ_{d} and *J* (e.g.γ
_{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 γ_{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* orγ
_{d} is varied.

The total vertical force without interaction (*V*_{NI}) is
the sum of *V*_{sf} and *V*_{sh}. The
coefficient of *V*_{NI} is denoted as *C*_{V,NI}
and defined as:
9
Let *C̄*_{V,NI} be the mean value
of *C*_{V,NI}. ThusΔ
*C*_{V}=(*C̄*_{V}-*C̄*_{V,NI})/*C̄*_{V,NI}
represents the percentage of increment in mean total vertical force
coefficient due to the forewing-hindwing interaction (whenΔ
*C*_{V} is negative, the interaction is detrimental to
vertical force generation). The value of Δ*C*_{V} 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 Δ*C*_{V}. 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% forγ
_{d}=180° and 90°, 8% for γ_{d}=60°,
and 3% for γ_{d}=0°. As *J* increases, forγ
_{d}=180°, 90° and 60°, the reduction decreases;
but for γ_{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 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
(γ_{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 γ_{d}≈60-180°, the total vertical force
is reduced by 7.8-15% due to the interaction (see
Table 3, *J*=0).

## Discussion

### The forewing-hindwing interaction is detrimental to the vertical force generation

Results in the present computations (24 cases of different phasing and
advance ratios) show that for the forewing or the hindwing, although its
vertical force coefficient at certain periods of the stroke cycle can be
slightly increased by the forewing-hindwing interaction effects, its mean
vertical force coefficient is decreased by the interaction effects. That is,
the forewing-hindwing interaction is detrimental to the vertical force
generation (and also to the resultant force generation; as mentioned above,
vertical force is very close to the resultant force because the thrust is much
smaller than the vertical force). This is remarkable but not totally
unexpected. For all the cases considered, each of the fore- and hindwings
produces a mean vertical force coefficient close to half that needed to
support the insect weight (see
*C̄*_{V,f} and
*C̄*_{V,h} in
Table 2). In producing an
upward force, a downward flow must be generated. Thus, in general, a wing
would move in the downwash-velocity field induced by the other wing, reducing
its vertical force.

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 withγ
_{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 γ_{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 *C*_{V} curve forγ
_{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 *r*_{2} from the wing
root at *J*=0.6. Using the diagram in
Fig. 20, the relative velocity
is estimated as 1.78*U* [*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
(γ_{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**

- c
- mean chord length of forewing
*C*_{D,b}- body-drag coefficient
*C*_{d,f}- drag coefficient of forewing
*C*_{d,h}- drag coefficient of hindwing
*C*_{l,f}- lift coefficient of forewing
*C*_{l,h}- lift coefficient of hindwing
*C̄*_{T}- mean total thrust coefficient
*C*_{T}- total thrust coefficient
*C*_{T,f}- thrust coefficient of forewing
*C*_{T,h}- thrust coefficient of hindwing
*C*_{T,sf}- thrust coefficient of single forewing
*C*_{T,sh}- thrust coefficient of single hindwing
*C̄*_{V}- mean total vertical force coefficient
*C*_{V}- total vertical force coefficient
*C*_{V,f}- vertical force coefficient of forewing
*C*_{V,h}- vertical force coefficient of hindwing
*C*_{V,NI}- total vertical force coefficient without interaction
*C̄*_{V,NI}- mean total vertical force coefficient without interaction
*C*_{V,sf}- vertical force coefficient of single forewing
*C*_{V,sh}- vertical force coefficient of single hindwing
- Δ
*C*_{V} - percentage of increment in mean total vertical force coefficient due to forewing-hindwing interaction
*C*_{V,W}- mean vertical force required for balancing the weight
*d*_{f}- drag, forewing
*d*_{h}- drag, hindwing
- J
- advance ratio
*l*_{f}- lift, forewing
*l*_{h}- lift, hindwing
- m
- mass of the insect
- n
- flapping frequency
- O
- origin of the inertial frame of reference
- r
- radial position along wing length
- R
- wing length
- r
_{2} - radius of the second moment of wing area of forewing
- Re
- Reynolds number
*S*_{f}- area of one wing (forewing)
*S*_{h}- area of one wing (hindwing)
- t̂
- time
- t̂
- non-dimensional parameter expressing time during a cycle
(
*t̂*=0 at the start of the downstroke of the hindwing and*t̂*=1 at the end of the following upstroke) - T
- total thrust
*T*_{f}- thrust of forewing
*T*_{h}- thrust of hindwing
*T*_{sf}- thrust of single forewing
*T*_{sf}- thrust of single hindwing
- U
- reference velocity
*u*_{t}- translational velocity of a wing
*u*_{t}^{+}- non-dimensional translational velocity of a wing
- V̄
- mean total vertical force
- V
- total vertical force
*V*_{∞}- free-stream velocity or flight velocity
*V*_{f}- vertical force of forewing
*V*_{h}- vertical force of hindwing
*V*_{NI}- vertical force without interaction
*V*_{sf}- vertical force of single forewing
*V*_{sh}- vertical force of single hindwing
- X,Y,Z
- coordinates in inertial frame of reference (
*Z*in vertical direction) - angular velocity of flip rotation
- non-dimensional angular velocity of flip rotation
- a constant
- αd
- geometrical angle of attack of downstroke
- αu
- geometrical angle of attack of upstroke
- β
- stroke plane angle
- γ
- phase angle of the translation of a wing
- γd
- difference in phase angle between the hindwing and the forewing
- angular velocity of azimuthal rotation
- π
- azimuthal or positional angle
- mean flapping angle
- Φ
- stroke amplitude
- ν
- kinematic viscosity of the air
- ρ
- density of fluid
- τ
- non-dimensional time
- τc
- period of one flapping cycle (non-dimensional)
- τr
- time when pitching rotation starts (non-dimensional)
- Δτr
- duration of wing rotation or flip duration (non-dimensional)

## 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, 10472008).

- © The Company of Biologists Limited 2005