A computational study of the aerodynamic forces and power requirements of dragonfly (Aeschna juncea) hovering
Institute of Fluid Mechanics, Beijing University of Aeronautics & Astronautics, Beijing 100083, People's Republic of China
* Author for correspondence (e-mail: sunmao{at}public.fhnet.cn.net)
Accepted 8 March 2004
![]() |
Summary |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
When the midstroke angles of attack in the downstroke and the upstroke are set to 52° and 8°, respectively (these values are close to those observed), the mean vertical force equals the insect weight, and the mean thrust is approximately zero. There are two large vertical force peaks in one flapping cycle. One is in the first half of the cycle, which is mainly due to the hindwings in their downstroke; the other is in the second half of the cycle, which is mainly due to the forewings in their downstroke. Hovering with a large stroke plane angle (52°), the dragonfly uses drag as a major source for its weight-supporting force (approximately 65% of the total vertical force is contributed by the drag and 35% by the lift of the wings).
The vertical force coefficient of a wing is twice as large as the quasi-steady value. The interaction between the fore- and hindwings is not very strong and is detrimental to the vertical force generation. Compared with the case of a single wing in the same motion, the interaction effect reduces the vertical forces on the fore- and hindwings by 14% and 16%, respectively, of that of the corresponding single wing. The large vertical force is due to the unsteady flow effects. The mechanism of the unsteady force is that in each downstroke of the hindwing or the forewing, a new vortex ring containing downward momentum is generated, giving an upward force.
The body-mass-specific power is 37 W kg-1, which is mainly contributed by the aerodynamic power.
Key words: dragonfly, Aeschna juncea, hovering flight, unsteady aerodynamics, power requirements, NavierStokes 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, lift force was many times larger than
that measured from dragonfly wings under steady-state conditions. This clearly
showed that the effect of unsteady flow and/or wing interaction was important.
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 separated works.
In order to further understand the dragonfly aerodynamics, it was desirable
to determine the aerodynamic force and flow structure simultaneously and also
to know the force on the individual forewing and hindwing during their
flapping motions. Freymuth
(1990) conducted force
measurement and flow visualization on an airfoil in hover modes. One of the
hover modes was for hovering dragonflies. Only mean vertical force was
measured. It was shown that large mean vertical force coefficient could be
obtained and the force was related to a wake of vortex pairs that produced a
downward jet of stream. Wang
(2000
) used a computational
fluid dynamics (CFD) method to study the aerodynamic force and vortex wake
structure of an airfoil in dragonfly hovering mode. Time variation of the
aerodynamic force in each flapping cycle and the vortex shedding process was
obtained. It was shown that large vertical force was produced during each
downstroke and 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.
In the works of Freymuth
(1990) and Wang
(2000
), only a single airfoil
was considered. Lan and Sun
(2001c
) studied two airfoils in
dragonfly hovering mode using the CFD method. For comparison, they also
computed the flow of a single airfoil. For the case of the single airfoil,
their results of aerodynamic force and flow structure were similar to that of
Freymuth's (1990
) experiment
and Wang's (2000
) computation.
For the fore and aft airfoils flapping with 180° phase difference (counter
stroking), the time variation of the aerodynamic force on each airfoil was
broadly similar to that of the single airfoil; the major effect of interaction
between the fore and aft airfoils was that the vertical forces on both the
airfoils were decreased by approximately 20% in comparison with that of the
single airfoil.
The above works (Freymuth,
1990; Wang, 2000
;
Lan and Sun, 2001c
), which
obtained aerodynamic force and flow structure simultaneously, were done for
airfoils. It is well known that the lift on an airplane wing of large aspect
ratio can be explained by a two-dimensional wing theory. But for a dragonfly
wing, although its aspect ratio is relatively large, its motion is much more
complex than that of an airplane wing. Three-dimensional effect should be
investigated. Moreover, the effect of aerodynamic interaction between the
fore- and hindwings in three-dimensional cases is unknown. The work of Lan and
Sun (2001c
) on two airfoils
flapping with 180° phase difference showed that interaction between the
two airfoils was detrimental to their aerodynamic performance. This result is
opposite to the common expectation that wing interaction of a dragonfly would
enhance its aerodynamic performance. It is of interest to investigate the
interaction effect in the three-dimensional case.
In the present study, we extend our previous two-dimensional study
(Lan and Sun, 2001c) to a
three-dimensional case. As a first step, we study the case of hovering flight.
For the dragonfly Aeschna juncea in free hovering flight, detailed
kinematic data were obtained by Norberg
(1975
). Morphological data of
the dragonfly (wing shape, wing size, wing mass distribution, weight of the
insect, etc.) are also available (Norberg,
1972
). On the basis of these data, the flows and aerodynamic
forces and the power required for producing the forces are computed and
analyzed. Because of the unique feature of the motion, i.e. the forewing and
the hindwing move relative to each other, the approach of solving the flow
equations over moving overset grids is employed.
![]() |
Materials and methods |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
|
The flapping motion of a wing is simplified as follows. The wing translates
downward and upward along the stroke plane and rotates during stroke reversal
(Fig. 1B). The translational
velocity is denoted by ut and is given by:
![]() | (1) |
The angle of attack of the wing is denoted by . It assumes a
constant value in the middle portion of a half-stroke. The constant value is
denoted by
d for the downstroke and by
u
for the upstroke. Around the stroke reversal,
changes with time and
the angular velocity (
) is given
by:
![]() | (2) |
The NavierStokes equations and solution method
The NavierStokes equations are numerically solved using moving
overset grids. For flow past a body in arbitrary motion, the governing
equations can be cast in an inertial frame of reference using a general
time-dependent coordinate transformation to account for the motion of the
body. The non-dimensionalized three-dimensional incompressible unsteady
NavierStokes equations, written in the inertial coordinate system
oxyz (Fig. 1C), are as
follows:
![]() | (3) |
![]() | (4) |
![]() | (5) |
![]() | (6) |
The equations are first transformed from the Cartesian coordinate system
(x,y,z,) to the curvilinear coordinate system
(
,
,
,
) using a general time-dependent coordinate
transformation. For a flapping wing, in order to make the transformation
simple, a body-fixed coordinate system
(o'x'y'z') is also
employed (Fig. 1C). In terms of
the Euler angles
and
(defined in
Fig. 1C), the inertial
coordinates (o,x,y,z) are related to the body-fixed coordinates
(o',x',y',z')
through the following relationship:
![]() | (7) |
The time derivatives of the momentum equations are differenced using a
second-order, three-point backward difference formula. To solve the time
discretized momentum equations for a divergence free velocity at a new time
level, a pseudo-time level is introduced into the equations, and a pseudo-time
derivative of pressure divided by an artificial compressibility constant is
introduced into the continuity equation. The resulting system of equations is
iterated in pseudo-time until the pseudo-time derivative of pressure
approaches zero; thus, the divergence of the velocity at the new time level
approaches zero. 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. Details of this algorithm can be found in Rogers
and Kwak (1990) and Rogers et
al. (1991
). For the
computation in the present work, the artificial compressibility constant is
set to 100 (it has been shown that when the artificial compressibility
constant varied between 10 and 300, the number of sub-iterations changes a
little but the final result does not change).
With overset grids, as shown in Fig.
2, for each wing there is a body-fitted curvilinear grid, which
extends a relatively short distance from the body surface; in addition, there
is a background Cartesian grid, which extends to the far-field boundary of the
domain (i.e. there are three grids). The solution method for a single grid is
applied to each of the three grids. The wing grids capture features such as
boundary layers, separated vortices and vortex/wing interactions. The
background grid carries the solution to the far field. The two wing grids are
overset onto the background Cartesian grid and parts of the two wing grids
overlap when the two wings move close to each other. As a result of the
oversetting of the grids, there are regions of holes in the wing grids and in
the background grid. As the wing grids move, the holes and hole boundaries
change with time. To determine the hole-fringe points, the method known as
domain connectivity functions by Meakin
(1993) is employed. Intergrid
boundary points are the outer-boundary points of the wing grids and the
hole-fringe points. Data are interpolated from one grid to another at the
hole-fringe points and, similarly, at the outer-boundary points of the wing
grids. In the present study, the background grid does not move and the two
wing-grids move in the background grid. The wing grids are generated by using
a Poisson solver that is based on the work of Hilgenstock
(1988
). They are of O-H type
grids. The background Cartesian grid is generated algebraically. Some portions
of the grids are shown in Fig.
2.
|
For far-field boundary conditions, at the inflow boundary, the velocity components are specified as freestream conditions while pressure is extrapolated from the interior; at the outflow boundary, pressure is set equal to the free-stream static pressure, and the velocity is extrapolated from the interior. On the wing surfaces, impermeable wall and no-slip boundary conditions are applied, and the pressure on the boundary is obtained through the normal component of the momentum equation written in the moving coordinate system. On the plane of symmetry of the dragonfly (the XZ plane; see Fig. 1A), flow-symmetry conditions are applied (i.e. w and the derivatives of u, v and p with respect to y are set to zero).
Evaluation of the aerodynamic forces
The lift of a wing is 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); the drag of a wing is the component that
is parallel to the translational velocity. 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 directions gives the vertical force and thrust of a wing.
Lf and Tf denote the vertical force
and thrust of the forewing, respectively; Lh and
Th denote the vertical force and thrust of the hindwing,
respectively. For the forewing:
![]() | (8) |
![]() | (9) |
![]() | (10) |
![]() | (11) |
![]() | (12) |
Data of hovering flight in Aeschna juncea
High-speed pictures of the dragonfly Aeschna juncea in hovering
flight were taken by Norberg
(1975), and the following
kinematic data were obtained. For both the fore- and hingwings, the chord is
almost horizontal during the downstroke (i.e.
d
ß)
and is close to the vertical during the upstroke; the stroke plane angle
(ß) is approximately 60°; the stroke frequency (n) is 36 Hz,
the stroke amplitude (
) is 69°; the hindwing leads the forewing in
phase by 180°. 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
fore- and hindwings are 0.81 cm and 1.12 cm, respectively; the moment of
inertial of wing-mass with respect to the fulcrum (I) is 4.54 g
cm-2 for the forewing and 3.77 g cm-2 for the hindwing
(Norberg, 1972
).
On the basis of the above data, the parameters of the model wings and the
wing kinematics are determined as follows. The lengths of both wings
(R) are assumed to be 4.7 cm; the reference length (the mean chord
length of the forewing) c=0.81 cm; the reference velocity
U=2nr2=2.5 m s-1; the Reynolds
number Re=Uc/
1350; the stroke period
c=U/nc=8.58.
is set as 180° and
0° for the fore- and hindwings, respectively. Norberg
(1975
) did not provide the
rate of wing rotation during stroke reversal. Reavis and Luttges
(1988
) made measurements on
similar dragonflies and it was found that maximum
was 10 00030 000 deg.
s-1. Here,
is set as
20 000 deg. s-1, giving
and
r=3.36.
![]() |
Results and analysis |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
As a first test, it is noted that in the initial stage of the starting
motion of a sphere, because the boundary layer is still very thin, the flow
around the sphere can be adequately treated by potential flow theory, and the
flow velocity around the sphere and the drag (added-mass force) on the sphere
can be obtained analytically. The acceleration of the sphere during the
initial start is a cosine function of time; after the initial start, the
sphere moves at constant speed (Us). In the numerical
calculation, the Reynolds number [based on Us and the
radius (a) of the sphere] is set as 103.
Fig. 3A shows the numerical and
analytical drag coefficients (Cd) vs
non-dimensional time (s)
(Cd=drag/0.5
Us2
a2;
s=tUs/2a). Between
s=0 and
s
0.2, the numerical result is in
very good agreement with the analytical solution.
Fig. 3B shows the azimuthal
velocity (u
) at
s=0.1 as a function
of r/2a (r is radial distance) with fixed azimuthal
angle
/2. The numerical result agrees well with the analytical solution
outside the boundary layer.
|
In the second test, the flow around the starting sphere is computed by the
single-grid code, and the results computed using the single grid and moving
overset grid are compared (also in Fig.
3). They are in good agreement. For the case of the single grid,
the grid is of O-O type, where the numerical coordinates (,
,
)
lie along the standard spherical coordinates. It has dimensions
100x65x129. The outer boundary is set at 30a from the
sphere. The non-dimensional time step is 0.01. Grid sizes of
68x41x81 and 46x27x51 were also used. By comparing the
results from these three grids, it was shown that the grid size of
100x65x129 was appropriate for the computation. For the case of
moving overset grids, the grid system consists of two grids: one is the
curvilinear grid of the sphere and the other is the background Cartesian grid.
The outer boundary of the sphere grid is at 1.4a from the sphere
surface and the out-boundary of the background grid is 30a from the
sphere. The grid density is made similar to that of the single grid.
In the third test, the set-up of Sane and Dickinson
(2001) is followed and the
aerodynamic forces are computed for the flapping model fruit fly wing. The
computed lift and drag are compared with the measured values in
Fig. 4. In the computation, the
wing grid has dimensions of 109x50x52 around the wing section, in
normal direction and in spanwise direction, respectively; the outer boundary
of the wing grid is approximately 2.0c from the wing. The background
Cartesian grid has dimensions of 90x85x80 and the outer boundary
is 20c from the wing. The non-dimensional time step is 0.02. The grid
density test was conducted and it was shown that the above overset grids were
appropriate for the computation. In Fig.
4A,B, the flapping amplitude is 60° and the midstroke angle of
attack is 50°; in Fig.
4C,D, these quantities are 180° and 50°, respectively. The
magnitude and trends with variation over time of the computed lift and drag
forces are in reasonably good agreement with the measured results.
|
The total vertical force and thrust; comparison with insect weight
In the calculation, the wings start the flapping motion in still air and
the calculation is ended when periodicity in aerodynamic forces and flow
structure is approximately reached (periodicity is reached approximately
23 periods after the calculation is started).
Fig. 5 shows the total
vertical force and thrust coefficients in one cycle, computed by two grid
systems, grid system 1 and grid system 2. In both grid systems, the outer
boundary of the wing-grid was set at about 2c from the wing surface
and that of the background grid at about 40c from the wings. For grid
system 1, the wing grid had dimensions 29x77x45 in the normal
direction, around the wing and in the spanwise direction, respectively, and
the background grid had dimensions 90x72x46 in the X
(horizontal), Z (vertical) and Y directions, respectively
(Fig. 2 shows some portions of
grid system 1). For grid system 2, the corresponding grid dimensions were
41x105x63 and 123x89x64. For both grid systems, grid
points of the background grid concentrated in the near field of the wings
where its grid density was approximately the same as that of the outer part of
the wing grid. As seen in Fig.
5, there is almost no difference between the force coefficients
calculated by the two grid systems. Calculations were also conducted using a
larger computational domain. The domain was enlarged by adding more grid
points to the outside of the background grid of grid system 2. The calculated
results showed that there was no need to put the outer boundary further than
that of grid system 2. It was concluded that grid system 1 was appropriate for
the present study. The effect of time step value was considered and it was
found that a numerical solution effectively independent of the time step was
achieved if 0.02. Therefore,
=0.02 was used in the
present calculations.
|
From Fig. 5, it is seen that
there are two large CL peaks in one cycle, one in the
first half of the cycle (while the hindwing is in its downstroke) and the
other in the second half of the cycle (while the forewing is in its
downstroke). It should be noted that by having two large
CL peaks alternatively in the first and second halves of a
cycle, the flight would be smoother. Averaging CL (and
CT) over one cycle gives the mean vertical force
coefficient () [and the mean thrust coefficient
(
)]:
=1.35 and
=0.02. The
value of
1.35 gives a vertical force of 756 mg, approximately equal to the insect mass
(754 mg). The computed mean thrust (11 mg) is close to zero. That is, the
force balance condition is approximately satisfied. In the calculation, the
stroke plane angle, the midstroke angles of attack for the downstroke and the
upstroke have been set as ß=52°,
d=52° and
u=8°, respectively. These values of ß,
d and
u give an approximately balanced
flight and they are close to the observed values [ß
60°; during
the downstroke the chord is almost horizontal (i.e.
d
ß), and during the upstroke the chord is close to
vertical].
The forces of the forewing and the hindwing
The total vertical force (or thrust) coefficient is the sum of the vertical
force (or thrust) coefficient of the fore- and hindwings.
Fig. 6 gives the vertical force
and thrust coefficients of the fore- and hindwings. The hindwing produces a
large CL,h peak during its downstroke (the first half of
the cycle) and very small CL,h in its upstroke (the second
half of the cycle); this is true for the forewing, but its downstroke is in
the second half of the cycle. Comparing
Fig. 6 with
Fig. 5 shows that the hindwing
in its downstroke is responsible for the large CL peak in
the first half of the cycle, and the forewing in its downstroke is responsible
for the large CL peak in the second half of the cycle. The
contributions to the mean total vertical force by the forewing and hindwing
are 42% and 58%, respectively. The vertical force on the hindwing is 1.38
times that on the forewing. Note that the area of the hindwing is 1.32 times
that of the forewing. That is, the relatively large vertical force on the
hindwing is mainly due to its relatively large size.
|
The vertical force and thrust coefficients of a wing are the results of the lift and drag coefficients of the wing. The corresponding lift and drag coefficients Cl,f, Cd,f, Cl,h and Cd,h are shown in Fig. 7. For the hindwing, Cd,h is larger than Cl,h during the downstroke of the wing, and ß is large (52°). As a result, a large part of CL,h is from Cd,h (approximately 65% of CL,h is from Cd,h and 35% is from Cl,h). This is also true for the forewing. That is, the dragonfly uses drag as a major source for its weight-supporting force when hovering with a large stroke plane angle.
|
The mechanism of the large vertical force
As shown in Fig. 6, the peak
value of CL,h is approximately 3.0 (that of
CL,f is approximately 2.6). Note that in the definition of
the force coefficient, the total area of the fore- and hindwings
(Sf+Sh) and the mean flapping velocity
U are used as reference area and reference velocity, respectively.
For the hindwing, if its own area and the instantaneous velocity are used as
reference area and reference velocity, respectively, the peak value of the
vertical force coefficient would be
3.0x[(Sf+Sh)/Sh]xU2/(U/2)2=2.1.
Similarly, for the forewing, the peak value would be
2.6x[(Sf+Sh)/Sf]xU2/(
U/2)2=2.4.
Since the thrust coefficients CT,f and
CT,h are small, CL,f and
CL,h can be taken as the coefficients of the resultant
aerodynamic force on the fore- and hindwings, respectively. The above shows
that the peak value of resultant aerodynamic force coefficient for the
forewing or hindwing is 2.12.4 (when using the area of the
corresponding wing and the instantaneous velocity as reference area and
reference velocity, respectively). This value is approximately twice as large
as the steady-state value measured on a dragonfly wing at
Re=7301890 [steady-state aerodynamic forces on the fore- and
hindwings of the dragonfly Sympetrum sanguineum were measured in a
wind tunnel by Wakeling and Ellington
(1997a
); the maximum resultant
force coefficient, obtained at an angle of attack of
60°, was
approximately 1.3].
There are two possible reasons for the large vertical force coefficients of the flapping wings: one is the unsteady flow effect; the other is the effect of interaction between the fore- and hindwings (in the steady-state wind-tunnel test, interaction between fore- and hindwings was not considered).
The effect of interaction between the fore- and hindwings
In order to investigate the interference effect between the fore- and
hindwings, we computed the flow around a single forewing (and also a single
hindwing) performing the same flapping motion as above.
Fig. 8A,B gives vertical force
(CL,sf) and thrust (CT,sf)
coefficients of the single forewing, compared with CL,f
and CT,f, respectively. The differences between
CL,sf and CL,f and between
CT,sf and CT,f show the interaction
effect. A similar comparison for the hindwing is given in
Fig. 8C,D. For both the fore-
and hindwings, the vertical force coefficient on a single wing (i.e. without
interaction) is a little larger than that with interaction. For the forewing,
the interaction effect reduces the mean vertical force coefficient by 14% of
that of the single wing; for the hindwing, the reduction is 16% of that of the
single wing. The interaction effect is not very large and is detrimental to
the vertical force generation.
|
The unsteady flow effect
The above results show that the interaction effect between the fore- and
hindwings is small and, moreover, is detrimental to the vertical force
generation. Therefore, the large vertical force coefficients produced by the
wings must be due to the unsteady flow effect. Here, the flow information is
used to explain the unsteady aerodynamic force.
First, the case of the single wing is considered.
Fig. 9 shows the iso-vorticity
surface plots at various times during one cycle. In order to correlate force
and flow information, we express time during a stroke cycle as a
non-dimensional parameter, , such
that
=0 at the start of the cycle and
=1 at the end of the cycle. After the
downstroke of the hindwing has just started
(
=0.125;
Fig. 9A), a starting vortex is
generated near the trailing edge of the wing, and a leading edge vortex (LEV)
is generated at the leading edge of the wing; the LEV and the starting vortex
are connected by the tip vortices, forming a vortex ring. Through the
downstroke (Fig. 9B,C), the
vortex ring grows in size and moves downward. At stroke reversal (between
0.36 and
0.65), the wing rotates and the
LEV is shed. During the upstroke, the wing almost does not produce any
vorticity. The vortex ring produced during the downstroke is left below the
stroke plane (Fig. 9DF)
and will convect downwards due to its self-induced velocity. The vortex ring
contains a downward jet (see below). We thus see that, in each cycle, a new
vortex ring carrying downward momentum is produced, resulting in an upward
force. This qualitatively explains the unsteady vertical force production.
Fig. 10 shows the velocity
vectors projected in a vertical plane that is parallel to and 0.6R
from the plane of symmetry of the insect. The downward jet is clearly
seen.
|
|
Fig. 11 shows the iso-vorticity surface plots for the fore- and hindwings (in the first half of the cycle the hindwing is in its downstroke; in the second half of the cycle the forewing is in its downstroke). Similar to the case of the single wing, just after the start of the first half of the cycle, a new vortex ring is produced by the hindwing (Fig. 11A); this vortex ring grows in size and convects downwards (Fig. 11AC). Similarly, just after the start of the second half of the cycle, a new vortex ring is produced by the forewing (Fig. 11D), which also grows in size and convects downwards as time increases. Fig. 12 gives the corresponding velocity vector plots. The qualitative explanation of the large unsteady forces on the fore- and hindwings is similar to that for the single wing.
|
|
On the basis of the above analysis of the aerodynamic force mechanism, we give a preliminary explanation for why the forewinghindwing interaction is not strong and is detrimental. The new vortex ring, which is responsible for the large aerodynamic force on a wing, is generated by the rapid unsteady motion of the wing at a large angle of attack. As a result, the effect of the wake of the other wing is relatively small. Moreover, the wake of the other wing produces downwash velocity, resulting in the detrimental effects.
Power requirements
As shown above, the computed vertical force is enough to support the insect
weight and the horizontal force is approximately zero; i.e. the force balance
conditions of hovering are satisfied. Here, we calculate the mechanical power
output of the dragonfly. The mechanical power includes the aerodynamic power
(work done against the aerodynamic torques) and the inertial power (work done
against the torques due to accelerating the wing-mass).
As expressed in equation 20
of Sun and Tang (2002), the
aerodynamic power consists of two parts, one due to the aerodynamic torque for
translation and the other to the aerodynamic torque for rotation. The
coefficients of these two torques (denoted by CQ,a,t and
CQ,a,r, respectively) are defined as:
![]() | (13) |
![]() | (14) |
|
The inertial power also consists of two parts (see equation 35 of
Sun and Tang, 2002): one due
to the inertial torque for translation and the other to the inertial torque
for rotation. The coefficient of inertial torque for translation
(CQ,i,t) is defined as:
![]() | (15) |
The power coefficient (Cp), i.e. power
non-dimensionalized by
0.5U3(Sf+Sh),
is:
![]() | (16) |
![]() | (17) |
![]() | (18) |
|
Integrating Cp over the part of a wingbeat cycle where
it is positive gives the coefficient of positive work
() for translation. Integrating
Cp over the part of the cycle where it is negative gives
the coefficient of `negative' work (C-W) for
`braking' the wing in this part of the cycle.
C+W and C-W for
the forewing are 8.33 and 2.16, respectively. For the hindwing, they
are 8.93 and 1.14, respectively.
The body-mass-specific power (P*) is defined as the
mean mechanical power over a flapping cycle divided by the mass of the insect,
and it can be written as follows (Sun and
Tang, 2002):
![]() | (19) |
![]() | (20) |
![]() | (21) |
![]() | (22) |
![]() | (23) |
![]() |
Discussion |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
The value (single airfoil) computed by Wang
(2000
) is approximately 1.97
[in fig. 4 of Wang
(2000
), maximum of
ut is used as reference velocity and the
value is approximately 0.8; if the mean of
ut is used as reference velocity, the
value becomes
0.8x(0.5
)2=1.97]; approximately the same
value (single airfoil) was obtained by Lan and
Sun (2001c
). In the present
study, the
values for the single forewing and
single hindwing are 1.51 and 1.64, respectively, approximately 20% less than
the 2-D value. This shows that the 3-D effect on
is significant. The wing length-to-chord ratio is not small (approximately 5);
one might expect a small 3-D effect. But for a flapping wing (especially in
hover mode), the relative velocity varies along the wing span, from zero at
the wing base to its maximum at the wing tip, which can increase the 3-D
effect. Note that although
is reduced by the 3-D
effect significantly, the time course of CL of the
forewing or the hindwing is nearly identical to that of the airfoil (compare
Fig. 6A with
fig. 3 of
Wang, 2000
).
Lan and Sun's results for the fore and aft airfoils showed that the
interaction effect decreased the vertical forces on the airfoils by
approximately 22% compared with that of the single airfoil
(Lan and Sun, 2001c). For the
fore- and hindwings in the present study, the reduction is approximately 15%,
showing that 3-D forewinghindwing interaction is weaker than in the 2-D
case.
Aerodynamic force mechanism and forewinghindwing interaction
Recent studies (e.g. Ellington et al.,
1996; Dickinson et al.,
1999
; Wu and Sun,
2004
) have shown that the large unsteady aerodynamic forces on
flapping model insect wings are mainly due to the attachment of an LEV or the
delayed stall mechanism. This is also true for the fore- and hindwings in the
present study. The LEV dose not shed before the end of the downstroke of the
fore- or hindwing (Fig. 11).
If the LEV sheds shortly after the start of the downstroke, the LEV would be
very close to the starting vortex, and a vortex ring that carries a large
downward momentum (i.e. the large aerodynamic forces) could not be produced.
Generation of a vortex ring carrying large downward momentum is equivalent to
the delayed stall mechanism.
Data presented in Fig. 8 show that the forewinghindwing interaction is not very strong and is detrimental. In obtaining these data, the wing kinematics observed for a dragonfly in hovering flight (e.g. 180° phase difference between the forewing and the hindwing; no incoming free-stream) have been used. Although some preliminary explanation has been given for this result, we cannot currently distinguish whether or not this result will exist when the phasing, the incoming flow condition, etc., are varied. Analysis based on flow simulations in which the wing kinematics and the flight velocity are systematically varied is needed.
Power requirements compared with quasi-steady results and with Drosophila results
Wakeling and Ellington
(1997b,c
)
computed the power requirements for the dragonfly Sympetrum
sanguineum. In most cases they investigated, the dragonfly was in
accelerating and/or climbing flight. Only one case is close to hovering
(flight SSan 5.2); in this case, the flight speed is rather low (advance
ration is approximately 0.1) and the resultant aerodynamic force is close to
the insect weight (see fig. 7D
of Wakeling and Ellington,
1997b
; fig. 5 of
Wakeling and Ellington,
1997c
). Their computed body-mass-specific aerodynamic power is
17.1 W kg-1 (see table 3 of
Wakeling and Ellington, 1997c
;
note that we have converted the muscle-specific power given in the table to
the body-mass-specific power), only approximately half the value calculated in
the present study. Lehmann and Dickinson
(1997
) and Sun and Tang
(2002
), based on experimental
and CFD studies, respectively, showed that for fruit flies, calculation by
quasi-steady analysis might under-estimate the aerodynamic power by 50%. A
similar result is seen for the hovering dragonflies.
It is of interest to note that the value of P* for the
dragonfly in the present study (37 W kg-1) is not very different
from that computed for a fruit fly (30 W kg-1;
Sun and Tang, 2002), even
though their sizes are greatly different (the wing length of the fruit fly is
0.3 cm and that of the dragonfly is 4.7 cm). For the fruit fly, the mechanical
power is mainly contributed by aerodynamic power
(Sun and Tang, 2002
). It is
approximately the case with the dragonfly in the present study (see
Fig. 14). From
equation 15 of Sun and Tang
(2002
), the aerodynamic torque
of a wing can be written as:
![]() | (24) |
![]() | (25) |
List of symbols
![]() |
Acknowledgments |
---|
![]() |
References |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Alexander, D. E. (1984). Unusual phase relationships between the forewings and hindwings in flying dragonflies. J. Exp. Biol. 109,379 -383.
Dickinson, M. H., Lehman, F. O. and Sane, S. P.
(1999). Wing rotation and the aerodynamic basis of insect flight.
Science 284,1954
-1960.
Ellington, C. P. (1984). The aerodynamics of hovering insect flight. (6). Lift and power requirements. Phil. Trans. R. Soc. Lond. B 305,145 -181.
Ellington, C. P., van den Berg, C. and Willmott, A. P. (1996). Leading edge vortices in insect flight. Nature 384,626 -630.[CrossRef]
Freymuth, P. (1990). Thrust generation by an airfoil in hover modes. Exp. Fluids. 9, 17-24.
Hilgenstock, A. (1988). A fast method for the elliptic generation of three dimensional grid with full boundary control. In Numerical Grid Generation in CFM'88 (ed. S. Sengupta, J. Hauser, P. R. Eiseman and J. F. Thompson), pp.137 -146. Swansea, UK: Pineridge Press Ltd.
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.
Lan, S. L. and Sun, M. (2001b). Aerodynamic interactions of two foils in unsteady motions. Acta Mech. 150,39 -51.
Lan, S. L. and Sun, M. (2001c). Aerodynamic force and flow structures of two airfoils in flapping motions. Acta Mech. Sinica. 17,310 -331.
Lehmann, F.-O. and Dickinson, H. D. (1997). The
changes in power requirements and muscle efficiency during elevated force
production in the fruit fly Drosophila melanogaster. J. Exp.
Biol. 200,1133
-1143.
Meakin, R. (1993). Moving body overset grid methods for complete aircraft tiltrotor simulations. AIAA Paper 93-3350.
Norberg, R. A. (1972). The pterostigma of insect wings and inertial regulator of wing pitch. J. Comp. Physiol. 81,9 -22.
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 NavierStokes equations. AIAA J. 28,253 -262.
Rogers, S. E., Kwak, D. and Kiris, C. (1991). Steady and unsteady solutions of the incompressible NavierStokes 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.
Sane, S. P. and Dickinson, M. H. (2001). The
control of flight force by a flapping wing: lift and drag production.
J. Exp. Biol. 204,2607
-2626.
Somps, C. and Luttges, M. (1985). Dragonfly flight: novel uses of unsteady separation flows. Science 28,1326 -1328.
Sun, M. and Tang, J. (2002). Lift and power requirements of hovering flight in Drosophila virilis. J. Exp. Biol. 205,2413 -2427.[Medline]
Sun, M. and Wu, J. H. (2003). Aerodynamic force
generation and power requirements in forward flight in a fruit fly with
modeled wing motion. J. Exp. Biol.
206,3065
-3083.
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, Z. J. (2000). Two dimensional mechanism for insect hovering. Phys. Rev. Lett. 85,2216 -2219.[CrossRef][Medline]
Weis-Fogh, T. (1972). Energetics of hovering flight in hummingbirds and in Drosophila. J. Exp. Biol. 56,79 -104.
Wu, J. H. and Sun, M. (2004). Unsteady
aerodynamic forces of a flapping wing. J. Exp. Biol.
207,1137
-1150.