Lift and power requirements of hovering flight in Drosophila virilis
Institute of Fluid Mechanics, Beijing University of Aeronautics and Astronautics, Beijing 100083, People's Republic of China
* e-mail: sunmao{at}public.fhnet.cn.net
Accepted 22 May 2002
![]() |
Summary |
---|
![]() ![]() ![]() ![]() ![]() ![]() |
---|
For the fruit fly Drosophila virilis in hovering flight (with symmetrical rotation), a midstroke angle of attack of approximately 37° was needed for the mean lift to balance the insect weight, which agreed with observations. The mean drag on the wings over an up- or downstroke was approximately 1.27 times the mean lift or insect weight (i.e. the wings of this tiny insect must overcome a drag that is approximately 27 % larger than its weight to produce a lift equal to its weight). The body-mass-specific power was 28.7 W kg-1, the muscle-mass-specific power was 95.7 W kg-1 and the muscle efficiency was 17 %.
With advanced rotation, larger lift was produced than with symmetrical rotation, but it was more energy-demanding, i.e. the power required per unit lift was much larger. With delayed rotation, much less lift was produced than with symmetrical rotation at almost the same power expenditure; again, the power required per unit lift was much larger. On the basis of the calculated results for power expenditure, symmetrical rotation should be used for balanced, long-duration flight and advanced rotation and delayed rotation should be used for flight control and manoeuvring. This agrees with observations.
Key words: fruit fly, Drosophila virilis, lift, power, unsteady aerodynamics, computational fluid dynamics
![]() |
Introduction |
---|
![]() ![]() ![]() ![]() ![]() ![]() |
---|
In the past few years, much progress has been made in revealing the
unsteady high-lift mechanisms of insect flight. Dickinson and Götz
(1993) measured the aerodynamic
forces on an aerofoil started impulsively at a high angle of attack [in the
Reynolds number (Re) range of a fruit fly wing, Re=75-225]
and showed that lift was enhanced by the presence of a dynamic stall vortex,
or leading-edge vortex (LEV). Lift enhancement was limited to only 2-3 chord
lengths of travel because of the shedding of the LEV. For most insects, a wing
section at a distance of 0.5R (where R is wing length) from
the wing base travels approximately 3.5 chord lengths during an up- or
downstroke in hovering flight (Ellington,
1984b
). A section in the outer part of the same wing travels a
larger distance, e.g. a section at 0.75R from the wing base travels
approximately 5.25 chord lengths, which is much greater than 2-3 chord lengths
(in forward flight, the section would travel an even larger distance during a
downstroke).
Ellington et al. (1996)
performed flow-visualization studies on a hawkmoth Manduca sexta
during tethered forward flight (forward speed in the range 0.4-5.7 m
s-1) and on a mechanical model of the hawkmoth wings that mimicked
the wing movements of a hovering Manduca sexta [Re
3500;
in the present paper, Re for an insect wing is based on the mean
velocity at r2 (the radius of the second moment of wing
area) and the mean chord length of the wing]. They discovered that the LEV on
the wing did not shed during the translational motion of the wing in either
the up- or downstroke and that there was a strong spanwise flow in the LEV.
(They attributed the stabilization of the LEV to the effect of the spanwise
flow.) The authors suggested that this was a new mechanism of lift
enhancement, prolonging the benefit of the delayed stall for the entire
stroke. Recently, Birch and Dickinson
(2001
) measured the flow field
of a model fruit fly wing in flapping motion, which had a much smaller
Reynolds number (Re
70). They also showed that the LEV did not
shed during the translatory phase of an up- or downstroke.
Dickinson et al. (1999)
conducted force measurements on flapping robotic fruit fly wings and showed
that, in the case of advanced rotation, in addition to the large lift force
during the translatory phase of a stroke, large lift peaks occurred at the
beginning and near the end of the stroke. (In the case of symmetrical
rotation, the lift peak at the beginning became smaller and was followed by a
dip, and the lift peak at the end of the stroke also became smaller; in the
case of delayed rotation, no lift peak appeared and a large dip occurred at
the beginning of the stroke.) Recently, Sun and Tang
(2002
) simulated the flow
around a model fruit fly wing using the method of computational fluid dynamics
and confirmed the results of Dickinson et al.
(1999
). Using simultaneously
obtained forces and flow structures, they showed that, in the case of advanced
rotation, the large lift peak at the beginning of the stroke was due to the
fast acceleration of the wing and that the large lift peak near the end of the
stroke was due to the fast pitching-up rotation of the wing. They also
explained the behaviour of forces in the cases of symmetrical and delayed
rotation. As a result of these studies
(Dickinson and Götz, 1993
;
Ellington et al., 1996
;
Dickinson et al., 1999
;
Birch and Dickinson, 2001
;
Sun and Tang, 2002
), we are
better able to understand how the fruit fly produces large lift forces.
Although the above results were mainly derived from studies on fruit flies, it
is quite possible that they are applicable to other insects that employ
similar kinematics.
The power requirements for generating lift through the unsteady mechanisms
described above cannot be calculated using methods based on steady-state
theory. In the rapid acceleration and fast pitching-up rotation mechanisms,
virtual-mass force and force due to the generation of the `starting' vortex
exist, and they cannot be included in the power calculation using steady-state
theory. In the delayed stall mechanism, the dynamic-stall vortex is carried by
the wing in its translation, and the drag of the wing, and hence the power
required, must be different from that estimated using steady-state theory. It
is of great interest to determine the power required for generating lift
through the unsteady mechanisms described above. Moreover, when the wing
generates a large lift force through these unsteady mechanisms, a large drag
force is also generated. For the fruit fly wing, the drag is significnatly
larger than the lift, as can be seen from the experimental data
(Dickinson et al., 1999;
Sane and Dickinson, 2001
) and
the computational results (Sun and Tang,
2002
). From the computational results of Sun and Tang
(2002
), it is estimated that
the mean drag coefficient over an up- or downstroke is more than 35% greater
than the mean lift coefficient. It is, therefore, of interest to determine
whether these unsteady lift mechanisms are realistic from the energetics point
of view.
Here, we investigate these problems for hovering flight in Drosophila virilis using computational fluid dynamics. In this method, the pressure and velocity fields around the flapping wing are obtained by solving the Navier-Stokes equations numerically, and the lift and torques due to the aerodynamic forces are calculated on the basis of the flow pressure and velocities. The inertial torques due to the acceleration and rotation of the wing mass can be calculated analytically. The mechanical power required for the flight may be calculated from the aerodynamic and inertial torques. The motion of the flapping wing and the reference frames are illustrated in Fig. 1.
|
![]() |
Materials and methods |
---|
![]() ![]() ![]() ![]() ![]() ![]() |
---|
|
The flapping motion considered here is an idealized one, which is similar
to that considered by Dickinson et al.
(1999) and Sun and Tang
(2002
) in their studies on
unsteady lift mechanisms. An up- or downstroke consists of the following three
parts, as shown in Fig. 1B:
pitching-down rotation and translational acceleration at the beginning of the
stroke, translation at constant speed and constant angle of attack during the
middle of the stroke, and pitching-up rotation and translational deceleration
at the end of the stroke. The translational velocity is denoted by
ut, which takes a constant value of Um
except at the beginning and near the end of a stroke. During the acceleration
at the beginning of a stroke, ut is given by:
![]() | (1) |
![]() | (2) |
The angle of attack of the wing is denoted by . It also takes a
constant value except at the beginning or near the end of a stroke. The
constant value is denoted by
m. Around the stroke reversal,
changes with time and the angular velocity
is given by:
![]() | (3) |
In the flapping motion described above, the mean flapping velocity
U, velocity at midstroke Um, angle of attack at
midstroke m, deceleration/acceleration duration
t, wing rotation duration
r,
period of the flapping cycle
c and flip timing
r must be given. These parameters will be determined using
available flight data, together with the force balance condition of the
flight.
The NavierStokes equations and the computational method
The flow equations and computational method used in the present study are
the same as in a recent paper (Sun and
Tang, 2002). Therefore, only an outline of the method is given
here. The governing equations of the flow are the three-dimensional
incompressible unsteady NavierStokes equations. Written in the inertial
coordinate system OXYZ and non-dimensionalized, they are as follows:
![]() | (4) |
![]() | (5) |
![]() | (6) |
![]() | (7) |
In the flapping motion considered in the present paper, the wing conducts translational motion (azimuthal rotation) and pitching rotation. To calculate the flow around a body performing unsteady motion (such as the present flapping wing), one approach is to write and solve the governing equations in a body-fixed, non-inertial reference frame with inertial force terms added to the equations. Another approach is to write and solve the governing equations in an inertial reference frame. By using a time-dependent coordinate transformation and the relationship between the inertial and non-inertial reference frames, a body-conforming computational grid in the inertial reference frame (which varies with time) can be obtained from a body-conforming grid in the body-fixed, non-inertial frame, which needs to be generated only once. This approach has some advantages. It does not need special treatment on the far-field boundary conditions and, moreover, since no extra terms are introduced into the equations, existing numerical methods can be applied directly to the solutions of the equations. This approach is employed here.
The flow equations are solved using the algorithm developed by Rogers and
Kwak (1990) and Rogers et al.
(1991
). The algorithm is based
on the method of artificial compressibility, which introduces a pseudotime
derivative of pressure into the continuity equation. Time accuracy in the
numerical solutions is achieved by subiterating in pseudotime for each
physical time step. The algorithm uses a third-order flux-difference splitting
technique for the convective terms and a second-order central difference for
the viscous terms. The time derivatives in the momentum equation are
differenced using a second-order, three-point, backward-difference formula.
The algorithm is implicit and has second-order spatial and time accuracy. For
details of the algorithm, see Rogers and Kwak
(1990
) and Rogers et al.
(1991
). A body-conforming grid
was generated using a Poisson solver based on the work of Hilgenstock
(1988
). The grid topology used
in this work was an OH grid topology. A portion of the grid used for
the wing is shown in Fig.
2.
Description of the coordinate systems
In both the flow calculation method outlined above and the force and moment
calculations below, two coordinate systems are needed. They are described as
follows. One is the inertial coordinate system OXYZ. The origin
O is at the root of the wing. The X and Y axes are
in the horizontal plane with the X axis positive aft, the Y
axis positive starboard and the Z axis positive vertically up (see
Fig. 1A). The second is the
body-fixed coordinate system oxyz. It has the same origin as the
inertial coordinate system, but it rotates with the wing. The x axis
is parallel to the wing chord and positive aft, and the y axis is on
the pitching-rotation axis of the wing and positive starboard (see
Fig. 1A). In terms of the Euler
angles and
(defined in Fig.
1A), the relationship between these two coordinate systems is
given by:
![]() | (8) |
![]() | (9) |
Evaluation of the aerodynamic forces
Once the NavierStokes equations have been numerically solved, the
fluid velocity components and pressure at discretized grid points for each
time step are available. The aerodynamic force acting on the wing is
contributed by the pressure and the viscous stress on the wing surface.
Integrating the pressure and viscous stress over the wing surface at a time
step gives the total aerodynamic force acting on the wing at the corresponding
instant in time. The lift of the wing, L, is the component of the
total aerodynamic force perpendicular to the translational velocity and is
positive when directed upwards. The drag, D, is the component of the
total aerodynamic force parallel to the translational velocity and is positive
when directed opposite to the direction of the translational velocity of the
downstroke. The lift and drag coefficients, denoted by CL
and CD, respectively, are defined as follows:
![]() | (10) |
![]() | (11) |
Evaluation of the aerodynamic torque and power
The moment around the root of a wing (point o) due to the
aerodynamic forces, denoted by -Ma, can be written as
follows (assuming that the thickness of the wing is very small):
![]() | (12) |
![]() | (13) |
|
The power required to overcome the aerodynamic moments, called aerodynamic
power Pa, can be written as:
![]() | (14) |
![]() | (15) |
![]() | (16) |
![]() | (17) |
![]() | (18) |
![]() | (19) |
![]() | (20) |
Evaluations of the inertial torques and power
The moments and products of inertia of the mass of a wing, with respect to
the oxyz coordinate system (see
Fig. 3), can be written as
follows, assuming that the thickness of the wing is very small:
![]() | (21) |
![]() | (22) |
![]() | (23) |
![]() | (24) |
![]() | (25) |
![]() | (26) |
![]() | (27) |
![]() | (28) |
![]() | (29) |
![]() | (30) |
![]() | (31) |
![]() | (32) |
![]() | (33) |
![]() | (34) |
![]() | (35) |
![]() | (36) |
![]() | (37) |
![]() | (38) |
![]() | (39) |
Evaluation of the total mechanical power
The total mechanical power of the wing, P, is the power required
to overcome the combination of the aerodynamic and inertial torques and can be
written as:
![]() | (40) |
![]() | (41) |
![]() | (42) |
![]() | (43) |
Data for hovering flight in Drosophila virilis
Data for free hovering flight of the fruit fly Drosophila virilis
Sturtevant were taken from Weis-Fogh
(1973), which were derived
from Vogel's studies of tethered flight
(Vogel, 1966
). Insect weight
was 1.96x10-5 N, wing mass was 2.4x10-6 g
(for one wing), wing length R was 0.3 cm, the area of both wings
St was 0.058 cm2, mean chord length c
was 0.097 cm, stroke amplitude
was 2.62rad and stroke frequency
n was 240 s-1.
From the above data, the mean translational velocity of the wing U
(the reference velocity), the Reynolds number Re, the non-dimensional
period of the flapping cycle c and mean lift coefficient
required for supporting the insect's weight
L,w were calculated as
follows: U=2
nr2=218.7 cm s-1;
Re=cU/v=147 (v=0.144 cm2 s-1);
c=(1/n)/(U/c)=8.42;
L,w=1.96x10-5/0.5
U2St=1.15
(
=1.23x10-3 g cm-3). Note that, in our
previous work (Sun and Tang,
2002
), a smaller
L,w (approximately 0.8)
was obtained for the same insect under the same flight conditions. This is
because a larger reference velocity, the velocity at midstroke, was used
there. The moments and product of inertia were calculated by assuming that the
wing mass was uniformly distributed over the wing planform, and the results
were as follows: Ixx=0.721x10-7 g
cm2; Iyy=0.069x10-7 g
cm2; Izz=0.790x10-7 g
cm2; Ixy=0.148x10-7 g
cm2.
![]() |
Results and discussion |
---|
![]() ![]() ![]() ![]() ![]() ![]() |
---|
Measured unsteady lift in Dickinson et al.
(1999) was presented in
dimensional form. For comparison, we need to convert it into the lift
coefficient. In their experiment, the fluid density
was
0.88x103 kg m-3, the model wing length R
was 0.25 m and the wing area S was 0.0167 m2. The speed at
the wing tip during the constant-speed translational phase of a stroke, given
in Fig. 3D of Dickinson et al.
(1999
), was 0.235 m
s-1 and, therefore, the reference speed (mean speed at
r2=0.58R) was calculated to be U=0.118 m
s-1. From the above data,
0.5
U2S=0.102 N. Using the definition of
CL (equation 10), the lift in
Fig. 3A of Dickinson et al.
(1999
) can be converted to
CL (Fig. 4)
and compared with the calculated CL for the cases of
advanced rotation (Fig. 4A),
symmetrical rotation (Fig. 4B)
and delayed rotation (Fig. 4C).
The aspect ratio of the wing in the experiment was calculated as
R2/S=3.74, and a wing of the same aspect ratio
was used in the calculation. The magnitude and trends with variation over time
of the calculated CL are in reasonably good agreement with
the measured values.
|
In the above calculation, the computational grid had dimensions of
93x109x71 in the normal direction, around the wing section and in
the spanwise direction, respectively. The normal grid spacing at the wall was
0.002. The outer boundary was set at 10 chord lengths from the wing. The time
step was 0.02. Detailed study of the numerical variables such as grid size,
domain size, time step, etc., was conducted in our previous work on the
unsteady lift mechanism of a flapping fruit fly wing
(Sun and Tang, 2002), where it
was shown that the above values for the numerical variables were appropriate
for the flow calculation. Therefore, in the following calculation, the same
set of numerical variables was used.
Force balance in hovering flight
Since we wanted to study the power requirements for balanced flight, we
first investigated the force balance. For the flapping motion considered in
the present study, the mean drag on the wing over each flapping cycle was
zero, and the horizontal force was balanced. Therefore, we needed only to
consider under what conditions the weight of the insect was balanced by the
mean lift.
As noted above, the kinematic parameters required to describe the wing
motion are U, Um+, c,
t,
r,
r and
m. Of these, U and
c were
determined above using the flight data given by Weis-Fogh
(1973
). Ennos
(1989
) made observations of the
free forward flight of Drosophila melanogaster (the flight was
approximately balanced). His data (his Figs
6D,
7A) showed that symmetrical
rotation was employed by the insect. Data on the hovering flight of
craneflies, hoverflies and droneflies also showed that symmetrical rotation
was employed by these insects (see Figs
8,
9 and
12, respectively, of
Ellington, 1984a
). Therefore,
it was assumed here that symmetrical rotation was employed in hovering flight
in Drosophila virilis; as a result,
r was determined.
From the data of Ennos (1989
)
and Ellington (1984a
), the
deceleration/ acceleration duration
t was estimated to
be approximately 0.2
c. We assumed that
t=0.18
c (this value was used in
previous studies on unsteady force mechanism of the fruit fly wing;
Dickinson et al., 1999
;
Sun and Tang, 2002
). Using
t and U, Um+ could be
determined.
|
|
|
|
|
Dickinson et al. (1999) and
Sun and Tang (2002
) used
r
0.36
c. In the present study, we
first assumed
r
0.36
c and then
investigated the effects of varying
r. At this point,
all the kinematic parameters had been determined except
m,
which was determined using the force balance condition.
The calculated lift coefficients versus time for three values of
m are shown in Fig.
5. The mean lift coefficient
plotted
against
m is shown in
Fig. 6. In the range of
m considered (
m=25-50°),
increases with
m; at
m=35°,
, which is the
value needed to balance the weight of the insect. At
m=35°, the mean drag coefficient during an up- or
downstroke (represented by
) was calculated to be
1.55, which is significantly larger than
(=1.15). The lift-to-drag ratio is 1.15/1.55=0.74.
|
Power requirements
As shown above, at m=35°, the insect produced enough
lift to support its weight. In the following, we calculated the power required
to produce this lift and investigated the mechanical power output of insect
flight muscle and its mechanochemical efficiency.
Aerodynamic torque
As expressed in equation 20, the aerodynamic power consists of two
components, one due to the aerodynamic torque for translation and the other
due to the aerodynamic torque for rotation. The coefficients of these two
torques, CQ,a,t and CQ,a,r, are shown
in Fig. 7B.
CQ,a,t is much larger than CQ,a,r. The
CQ,a,t curve is similar in shape to the
CD curve shown in Fig.
5C for obvious reasons.
One might expect that, during the deceleration of the wing near the end of a stroke, CQ,a,t would change sign because of the wing being `pushed' by the flow behind it. But as seen from Fig. 7B (e.g. during the downstroke), CQ,a,t becomes negative only when the deceleration is almost finished because, while decelerating, the wing rotates around an axis that is near its leading edge. Therefore, a large part of the wing is effectively not in deceleration and does not `brake' the pushing flow.
Inertial torque
The coefficients of the inertial torques for translation
(CQ,i,t) and for rotation (CQ,i,r) are
shown in Fig. 7C. The inertial
torques are approximately zero in the middle of a stroke, when the
translational and rotational accelerations are zero. At the beginning and near
the end of the stroke, the inertial torque for translation has almost the same
magnitude as its aerodynamic counterpart. Similar to the case of the
aerodynamic torques, the inertial torque for translation is much larger than
the inertial torque for rotation.
At the beginning of a stroke, the sign of CQ,i,r is
opposite to that of
(Fig. 7C). Near the end of the
stroke, the sign of CQ,i,r is also opposite to that of
. In this part of the
stroke, although
has
the same sign as
has the opposite sign to
. In equation 37,
is multiplied by
Ixy, which is much larger than Iyy;
thus, its effect dominates over that of other terms in the equation, leading
to the above result. This shows that, for the flapping motion considered, the
inertial torque of rotation will always contribute to `negative' work.
Power and work
From the above results for the aerodynamic and inertial torque
coefficients, the power coefficients can be computed using equations 41-43.
The coefficients of power for translation (CP,t) and
rotation (CP,r) are plotted against time in
Fig. 8.
CP,t is positive for the majority of a stroke and becomes
negative only for a very short period close to the end of the stroke.
CP,r is negative at the beginning and near the end of a
stroke and is approximately zero in the middle of the stroke. Throughout a
stroke, the magnitude of CP,t is much larger than that of
CP,r. Two large positive peaks in CP,t
appear during a stroke. One occurs during the rapid acceleration phase of the
stroke as a result of the larger aerodynamic and inertial torques occurring
there. The other is in the fast pitching-up rotation phase of the stroke and
is due to the large aerodynamic torque there.
Integrating CP,t over the part of a cycle where it is positive gives the coefficient of positive work for translation, which is represented by CW,t+. Integrating CP,t over the part of the cycle where it is negative gives the coefficient of `negative' work for translation; this is represented by CW,t-. Similar integration of CP,r gives the coefficients of the positive and negative work for rotation; they are denoted by CW,r+ and CW,r-, respectively. The results of the integration are: CW,t+=15.96, CW,t-=-1.00, CW,r+=0.56 and CW,r-=-2.30.
Specific power
The body-mass-specific power, denoted by P*, is defined
as the mean mechanical power over a flapping cycle (or a stroke in the case of
normal hovering) divided by the mass of the insect. P* can
be written as follows:
![]() | (44) |
When calculating CW, one needs to consider how the
`negative' work fits into the power budget
(Ellington, 1984c). There are
three possibilities (Ellington,
1984c
; Weis-Fogh,
1972
,
1973
). One is that the
negative power is simply dissipated as heat and sound by some form of an end
stop; it can then be ignored in the power budget. The second is that, during
the period of negative work, the excess energy can be stored by an elastic
element, and this energy can then be released when the wing does positive
work. The third is that the flight muscles do negative work (i.e. they are
stretched while developing tension, instead of contracting as in `positive'
work), but the negative work uses much less metabolic energy than an
equivalent amount of positive work. Of these three possibilities, we
calculated CW (or P*) on the basis of
the assumption that the muscles act as an end stop. CW is
written as:
![]() | (45) |
Using CW,t+ and CW,r+ calculated above, CW was calculated using equation 45 to be 16.52. The specific power P* was then calculated using equation 44: P*=36.7 W kg-1.
Effects of the timing of wing rotation on lift and power
In the flight considered above, symmetrical rotation was employed.
Dickinson et al. (1999) showed
that the timing of the wing rotation had significant effects on the lift and
drag of the wing. It is of interest to see how the lift and the power required
change when the timing of wing rotation is varied.
Fig. 9 shows the calculated
lift and drag coefficients for the cases of advanced rotation and delayed
rotation (results for the case of symmetrical rotation are included for
comparison). The value of r used can be read from
Fig. 9A. The case of advanced
rotation has a larger CL and CD than
the case of symmetrical rotation, and the case of delayed rotation has a much
smaller CL and a slightly larger CD
than the case of symmetrical rotation. An explanation for the above force
behaviours was given by Dickinson et al.
(1999
) and Sun and Tang
(2002
). The mean lift
coefficient
for the advanced rotation case is
1.47, 28 % larger than that for the symmetrical rotation case (1.15);
for the delayed rotation case is only 0.39,
which is 66 % smaller than that for the symmetrical rotation case.
The aerodynamic and inertial torque coefficients for the advanced rotation and delayed rotation cases are shown in Figs 10 and 11, respectively. Similar to the symmetrical rotation case, the CQ,a,t curve for the case of advanced rotation (or delayed rotation) looks like the corresponding CD curve. For the advanced rotation case, CQ,a,t is much larger than that of the symmetrical rotation case, especially from the middle to the end of the stroke (compare Fig. 10B with Fig. 7B). CQ,i,t is the same as that of the symmetrical rotation case, because the translational acceleration is the same for the two cases (compare Fig. 10C with Fig. 7C). For the delayed rotation case, CQ,a,t is larger than that of the symmetrical rotation case in the early part of a stroke (compare Fig. 11B with Fig. 7B). CQ,i,t is the same as that of the symmetrical rotation case for the same reason as above. Similar to the symmetrical rotation case, for both the advanced and delayed rotation cases, CQ,a,r and CQ,i,r are much smaller than CQ,a,t and CQ,i,t, respectively.
|
|
The non-dimensional power coefficients for the cases of advanced rotation and delayed rotation are shown in Fig. 12 (the results of symmetrical rotation, taken from Fig. 8, are included for comparison). CP,r is much smaller than CP,t because, as shown in Figs 10 and 11, CQ,a,r and CQ,i,r were much smaller than CQ,a,t and CQ,i,t, respectively. CP,t behaves approximately the same as CQ,a,t. The most striking feature of Fig. 12 is that CP,t for advanced rotation is much larger than that for symmetrical rotation from the middle to near the end of a stroke.
Integrating the power for the cases of advanced rotation and delayed
rotation in the same way as above for the case of symmetrical rotation, the
corresponding values of CW,t+,
CW,t-, CW,r+ and
CW,r- were obtained, from which the work
coefficient per cycle, CW, was calculated. The results are
given in Table 1 (results for
symmetrical rotation are included for comparison). For the advanced rotation
case, CW is approximately 80 % larger than for symmetrical
rotation case. As noted above, is 28 % larger
than that of the symmetrical rotation case. This shows that advanced rotation
can produce more lift but is very energy-demanding. For the delayed rotation
case, CW is approximately 10 % larger than for the
symmetrical rotation case but, as noted above, its
is 66 % smaller; therefore, the energy spent per
unit
is much larger than for the symmetrical
rotation case. The above results show that advanced rotation and delayed
rotation would be much more costly if used in balanced, long-duration
flight.
|
For reference, we calculated another case in which advanced rotation timing
was employed in balanced flight but m was decreased to
22° so that the mean lift was equal to the weight
(Table 1). In this case,
CW was 30.10, which is approximately 82 % larger than for
the case employing symmetrical rotation, showing clearly that advanced
rotation is very energy-demanding.
Effects of flip duration
In the above analyses, the duration of wing rotation (or flip duration) was
taken as r=0.36
c. Below, the effects of
changing the flip duration were investigated.
Fig. 13 shows the lift and
drag coefficients of the wing for two shorter flip durations
r=0.24
c and
r=0.19
c, with the above results for
r=0.36
c included for comparison. If the
flip duration is varied while other parameters are kept unchanged, the mean
lift coefficient will change. To maintain the balance between mean lift and
insect weight,
m was therefore adjusted. For
r=0.24
c,
m was
changed to 36.5° to give a
of 1.15; for
r=0.19
c,
m was
changed to 38.5°.
|
From Fig. 13C, it can be
seen that, when r is decreased, the
CD peak at the beginning of a stroke becomes much smaller
and the CD peak near the end of the stroke is delayed and
becomes slightly smaller. A smaller CD at the beginning of
the stroke means reduced aerodynamic power there. Since the wing decelerates
near the end of the stroke, delaying the CD peak at this
point means that the peak would occur when the wing has a lower velocity,
resulting in reduced aerodynamic power. The power coefficients are shown in
Fig. 14; at the beginning and
at the end of a stroke, CP,t is smaller for smaller
r.
|
By integrating the power coefficients in
Fig. 14,
CW,t+, CW,t-,
CW,r+ and CW,r-
were obtained (Table 2).
CW was computed using equation 45, and the results are
also shown in Table 2. When
r is decreased to 0.24
c,
CW is 12.96, much smaller than for
r=0.36
c. When
r
is further decreased to 0.19
c, CW was
slightly greater (13.06). This is because when
r is
decreased to 0.19
c, CW,t+ also
decreases; however, CW,r+ increases (possibly
due to the wing rotation becoming very fast).
|
For r=0.24
c (which has approximately
the same CW as
r=0.19
c), the mass-specific power
P* was computed to be 28.7 W kg-1. If the ratio
of the flight muscle mass to the body mass is known, the power per unit flight
muscle or muscle-mass-specific power (Pm*) can
be calculated from the body-mass-specific power. Lehmann and Dickinson
(1997
) obtained a value of 0.3
for the ratio for fruit fly Drosophila melanogaster. This value
gives:
![]() | (46) |
![]() | (47) |
It is very interesting to look at the drag on the wing. In the above case
(r=0.24
c,
m=36.5°), the mean drag coefficient
over an up- or downstroke is 1.46;
.
We see that, for this tiny hovering insect, its wings must overcome a drag
that is 27 % larger than its weight to produce a lift that equals its weight.
(This is very different from a large fast-flying bird, which only needs to
overcome a drag that is a small fraction of its weight, and from a hovering
helicopter, the blades of which need to overcome a drag that represents an
even smaller fraction of its weight.)
Comparison between the calculated results and previous data
The above results showed that, for a duration of wing rotation
r=0.19
c and
r=0.24
c, the power expenditure for
hovering flight in Drosophila virilis was relatively small compared
with that for larger values of
r. The corresponding
non-dimensional mean rotational velocities are approximately 1 (maximum
is approximately 2,
as seen in Fig. 13A). This
mean rotational velocity is close to the value of 0.95 measured in the free
forward flight of Drosophila melanogaster [data in Table 4 of Ennos
(1989
) multiplied by
R/r2=1/0.58 because a different reference velocity was
used]. It is also close to that measured in free hovering flight in
craneflies, hoverflies and droneflies: approximately 0.9, 1.2 and 0.9,
respectively [data in Table 2
of Ellington (1984a
) multiplied
by 1/0.58]. Therefore, both from measurements from similar insects and from
the calculated efficiency, it is reasonable to assume a
r of approximately 20 % of
c.
The calculated midstroke angle of attack m is
approximately 37° (see Table
2;
m=36.5° and
m=38.5°
for
r=0.24
c and
r=0.19
c, respectively). Vogel
(1967
) measured the angle of
attack for tethered Drosophila virilis flying in still air;
m was approximately 45°. Our calculated value is smaller
than this value, which is reasonable since the calculated value is for free
and balanced flight whereas the measured value was for tethered flight in
which the insect could use a larger angle of attack. Ellington
(1984a
) observed many small
insects in hovering flight, including the fruit fly, and found that the angle
of attack employed was approximately 35°. The predicted value thus is in
good agreement with observations.
The calculated body-mass-specific power P* and muscle
efficiency were 28.7 W kg-1 and 17 %, respectively. Lehmann
and Dickinson (1997
) studied
the muscle efficiency of the fruit fly Drosophila melanogaster by
simultaneously measuring the metabolic rate and the flight kinematics. Using
the measured stroke amplitude and frequency, they estimated the mean specific
power using a quasi-steady aerodynamics method. Their estimate of
P* for hovering flight was 17.9 W kg-1, only
approximately half the value calculated using the present unsteady flow
simulation method. Their measured metabolic rate was approximately 199 W
kg-1. As a result, they obtained a muscle efficiency of
approximately 9 %, approximately half that obtained in the present study. In
their recent work on unsteady force measurements on a model fruit fly wing,
Sane and Dickinson (2001
)
showed that the drag on the wing was much larger than the quasi-steady
estimate of Lehmann and Dickinson
(1997
). On the basis of the
measured drag, they suggested that the previous value of muscle efficiency
presented by Lehmann and Dickinson
(1997
) should be adjusted to
approximately 20 %. This is similar to the value calculated in the present
study.
The calculated results show that, for the advanced rotation and delayed
rotation cases, the energy expended for a given mean lift is much larger than
that in the case of symmetrical rotation. On the basis of these results,
symmetrical rotation should be employed by the insect for balanced,
long-duration flight and advanced rotation and delayed rotation should be
employed for manoeuvring. This agrees with observations on balanced flight
(Ennos, 1989) and manoeuvring
(Dickinson et al., 1993
) in the
fruit fly Drosophila melanogaster.
![]() |
Acknowledgments |
---|
![]() |
References |
---|
![]() ![]() ![]() ![]() ![]() ![]() |
---|
Birch, J. M. and Dickinson, M. H. (2001). Spanwise flow and the attachment of the leading-edge vortex on insect wings. Nature 412,729 -733.[Medline]
Dickinson, M. H. and Götz, K. G. (1993).
Unsteady aerodynamic performance of model wings at low Reynolds numbers.
J. Exp. Biol. 174,45
-64.
Dickinson, M. H., Lehman, F. O. and Götz, K. G.
(1993). The active control of wing rotation by Drosophila.J. Exp. Biol. 182,173
-189.
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. (1984a). The aerodynamics of hovering insect flight. III. Kinematics. Phil. Trans. R. Soc. Lond. B 305,41 -78.
Ellington, C. P. (1984b). The aerodynamics of hovering insect flight. IV. Aerodynamic mechanisms. Phil. Trans. R. Soc. Lond. B 305,79 -113.
Ellington, C. P. (1984c). The aerodynamics of hovering insect flight. VI. 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.
Ennos, A. R. (1989). The kinematics and aerodynamics of the free flight of some Diptera. J. Exp. Biol. 142,49 -85.
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 of 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 interaction between two airfoils in unsteady motions. Acta Mech. 150,39 -51.
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.
Lehmann, F.-O., Dickinson, M. H. and Staunton, J.
(2000). The scaling of carbon dioxide release and respiratory
water loss in flying fruit flies (Drosophila spp.). J.
Exp. Biol. 203,1613
-1624.
Rogers, S. E. and Kwak, D. (1990). Upwind differencing scheme for the time-accurate incompressible Navier-Stokes equations. AIAA J. 28,253 -262.
Rogers, S. E., Kwak, D. and Kiris, C. (1991). Numerical solution of the incompressible Navier-Stokes equations for steady-state and dependent problems. AIAA J. 29,603 -610.
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.
Sun, M. and Tang, J. (2002). Unsteady
aerodynamic force generation by a model fruit fly wing in flapping motion.
J. Exp. Biol. 205,55
-70.
Vogel, S. (1966). Flight in Drosophila. I. Flight performance of tethered flies. J. Exp. Biol. 44,567 -578.
Vogel, S. (1967). Flight in Drosophila. II. Variations in stroke parameters and wing contour. J. Exp. Biol. 46,383 -392.[Medline]
Weis-Fogh, T. (1972). Energetics of hovering flight in hummingbirds and in Drosophila. J. Exp. Biol. 56,79 -104.
Weis-Fogh, T. (1973). Quick estimates of flight fitness in hovering animals, including novel mechanism for lift production. J. Exp. Biol. 59,169 -230.