The fluid dynamics of flight control by kinematic phase lag variation between two robotic insect wings
Department of Neurobiology, University of Ulm, Albert-Einstein-Allee 11, 89081 Ulm, Germany
* Address for correspondence (e-mail: fritz.lehmann{at}biologie.uni-ulm.de)
Accepted 30 September 2004
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Summary |
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Key words: insect flight, aerodynamics, DPIV, leading edge vortex, wake, dragonfly
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Introduction |
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In contrast, dragonflies vary the phase relationship between ipsilateral
fore- and hindwings with different behaviors
(Norberg, 1975;
Reavis and Luttges, 1988
;
Wakeling and Ellington, 1997
;
Wang et al., 2003
). Three
categories of phase relationship between fore- and hindwing have been
established: phase-shifted stroking, counterstroking and parallel stroking. A
highly consistent characteristic for the conventional flight modes is a
54-100° phase shift (the hindwing leads forewing motion) common for
dragonflies during (i) straight forward and upward flight, (ii) the escape
mode, in which a tethered animal produces peak lift in each stroke cycle of up
to approximately 20 times their body weight, and (iii) during maneuvering
flight (Somps and Luttges,
1985
; Wakeling and Ellington,
1997
; Wang et al.,
2003
). The kinematic phase shift persists even when the animals
are changing forward flight speed
(Wakeling and Ellington,
1997
). In contrast, hovering flight seems to be supported by
larger phase differences of up to 180°, in which the wings beat out of
phase (counterstroking; Alexander,
1986
; Norberg,
1975
; Wakeling and Ellington,
1997
). Counterstroking was also found in a study on maneuvering
flight in dragonflies flying freely in a wind tunnel
(Alexander, 1986
). Detailed
analysis of wing kinematics during various flight behaviours suggests that
in-phase, or parallel stroking, might produce higher aerodynamic forces and
should be favored during the energetically most demanding flight such as
hovering, take-off or load-lifting flight (Alexander,
1984
,
1986
;
Rüppell, 1989
).
Direct force measurements on tethered dragonflies flying in a wind tunnel
show that peak lift increases from approximately 2.3 to 6.3 times body weight
when the animal decreases the phase angle between both flapping wings
(Reavis and Luttges, 1988).
Although this finding supports the assumption that parallel stroking might
maximize lift production, it has been questioned by analytical modeling in
which flight efficiency and mean thrust coefficient was estimated as a
function of kinematic phase relationship
(Lan, 1979
). This study
predicts that the hindwing extracts maximum energy from the forewing downwash
when the hindwing leads by a quarter stroke cyle (90°), while the thrust
coefficient is largest when the phase relationship is 45°. As a
consequence, dragonflies exhibiting parallel stroking will increase thrust,
but at the expense of a relative increase in energetic costs.
According to biplane theory, total lift production in tandem wings depends
on the proximity and the strength of forewing downwash that interferes with
the hindwing (Milne-Thomson,
1966). In dragonflies, the hindwing flaps in close proximity to
the forewing and thus must cope with a potential reduction in the effective
angle of attack (the angle between the wing chord line and the oncoming fluid)
due to forewing downwash. The attenuation in aerodynamic performance of the
hindwing in turn critically depends on forewing wake structure and the timing
with which the hindwing interacts with the forewing downwash
(Azuma et al., 1985
). Assuming
two-dimensional (2D) flapping conditions, two long and narrow wings working
independently should have higher lift-to-drag ratios than a combined wing with
the same wing area, due to the differences in aspect ratio
(Bertin and Smith, 1979
;
Mises, 1959
). Thus tandem
wings flapping in phase should produce less total lift because the two wings
are always closer throughout the entire stroke cycle than wings flapping out
of phase (Alexander, 1984
).
It is difficult to assess the significance of phase relationship to
modulate lift production in a flying insect because kinematic phase shifts are
mostly accompanied by other changes in wing kinematics, such as stroke
amplitude or angle of attack (Reavis and
Luttges, 1988; Wakeling and
Ellington, 1997
). For this reason various investigations on the
aerodynamics of static and flapping dragonfly and damselfly wings have been
conducted under various conditions, either experimentally
(Kesel, 2000
;
Kliss et al., 1989
;
Newman et al., 1977
;
Okamoto et al., 1996
;
Saharon and Luttges, 1987
;
Somps and Luttges, 1985
) or
analytically (e.g. Azuma et al.,
1985
; Wang et al.,
2003
). Savage et al.
(1979
) modeled dragonfly
aerodynamics experimentally under 2D conditions by pulling a single model wing
on a carriage through the air, and derived forces from the resulting wake
using inviscid flow theory. Kliss et al.
(1989
) used an oscillating
flat plate with 90° angle of attack to study vortex shedding, and found
that stroke length is critical to minimize complete flow separation during
wing translation. In several elaborate studies, Saharon and Luttges
(1987
,
1988
,
1989
) demonstrated vortex
generation in a mechanical-driven dragonfly under three-dimensional (3D)
flapping conditions and described eight major vortices that are generated
throughout each wing beat cycle. They found that in most of the tested cases,
much of the interference between hindwing and forewing wake was detrimental to
maximized wing-wake interaction. Different stroke-phase relationships (90, 180
and 270°) produced different flow wing-wake patterns, and vortices
appeared to fuse under certain flapping conditions. A quantitative analysis of
vortex displacement in the wake revealed that the travelling velocity of some
vortices shed in the wake varied when phase relationship was altered
(Saharon and Luttges, 1989
).
However, none of the studies mentioned above have directly measured
aerodynamic forces produced by the flapping fore- and hindwing, nor quantified
alterations in leading edge vorticity and local flow conditions in response to
changing kinematic phase angles.
To investigate experimentally the complex wing-wake interaction in four-winged insects and to evaluate in more detail the functional significance of stroke-phase modulation on wake structure, aerodynamic force production and lift-to-drag ratio, we employed a 3D robotic dragonfly model mimicking hovering conditions at intermediate Reynolds number, in which stroke-phase relationships between fore- and hindwing could be altered systematically. While varying kinematic phase shift we measured aerodynamic forces using a miniaturized force transducer, and mapped the velocity field around the flapping wings using Digital Particle Imaging Velocimetry (DPIV) in order to quantify vorticity and vortical flow structures at the wings, including the structures shed into the wake.
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Materials and methods |
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Development of a generic kinematics and wing control
Owing to the range of stroke patterns used by dragonflies to balance their
weight and allow maneuvering, it appears to be difficult to describe a typical
dragonfly kinematics (Norberg,
1975; Rüppell,
1989
). Kinematic studies on different species of dragonfly
demonstrate that during forward and climbing flight some animals beat their
wings with a near horizontal stroke plane
(Wakeling and Ellington, 1997
;
Fig. 1A) and others with highly
inclined stroke planes (Azuma and Watanabe,
1988
; Wang et al.,
2003
). Moreover, dragonflies produce flight forces using various
combinations of stroke amplitude and stroke frequency that range from 50 to
150° and from 27 to 73 Hz, respectively
(Azuma and Watanabe, 1988
;
Rüppell, 1989
).
Due to this diversity of dragonfly kinematics, various authors have modeled
physically and analytically different types of dragonfly kinematics. In the
oscillating flat plate case (Kliss et al.,
1989), the authors varied stroke amplitude, frequency and aspect
ratio, but did not model other characteristic features of dragonfly wing
motion, including wing-wake interaction. The study, moreover, compared
aerodynamic flow patterns produced over a vast range of Reynolds numbers
(Re) ranging from 10 to 4300. Savage's physical model
(Savage et al., 1979
) of a
hovering dragonfly used Norberg's kinematic data of freely flying dragonfly
Aeschna (Norberg,
1975
). This model wing did not flap its wing around a root,
however, but rather translated during up- and downstroke. The more elaborate
`pitching-plunging' dragonfly model employed by Saharon and Luttges
(1988
,
1989
) flapped two ipsilateral
wings in a tilted stroke plane and at 90° stroke amplitude. In this model
the authors varied reduced frequency, which was accompanied by changes in
Re, and three distinct phase angles between fore- and hindwing, but
no other kinematic parameters (Saharon and
Luttges, 1989
). Aerodynamic characteristics of static dragonfly
wings and body were conducted under 2D conditions in a wind tunnel
(Kesel, 2000
;
Okamoto et al., 1996
). In the
latter study the authors mounted wings of a dragonfly and flat plates on a
glider and evaluated the effect of angle of attack (dragonfly wing), camber,
thickness sharpness of the leading wing edge and surface roughness (model
wing) on force production at Re=1000-10 000.
To avoid too many kinematic parameters confounding the results in the
present study, we developed a generic kinematic pattern that allowed us to
model kinematic phase shifts similar to those reported for dragonflies
(Fig. 1B). The horizontal wing
trajectories were derived from a simple sinusoidal function, which was chosen
because of a finding in previous studies that the first harmonic of a Fourier
series gives a good representation of the stroke cycle of freely flying
dragonflies (Azuma and Watanabe,
1988; Wakeling and Ellington,
1997
; Wang et al.,
2003
). We used a constant angle of attack during wing translation
with a feathering angle of 45° at mid stroke, which is similar to values
reported previously (Azuma and Watanabe,
1988
; Fig. 2B).
This angle is the optimum lift angle of a translating wing free from wake
interference and is within the range of data published for dragonflies
(Dickinson et al., 1993
;
Rüppell, 1989
). The
stroke amplitude of 100° that we used is near the average measured for
both the fore- and hindwing motion in dragonflies flying at various flight
speeds (Wakeling and Ellington,
1997
). The flapping frequency of the robotic wings was 533
mHz.
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We chose to stack the wings vertically, which seems to be sufficiently
close to the orientation of wing hinges presented by a freely flying dragonfly
with a near vertical mean thrust vector
(Fig. 1A,B). In this respect
our tandem model with vertical aligned wings differs from other dragonfly
models in which the wing hinges are aligned horizontally, yielding a `front'
and a `rear' wing rather than an `upper' and a `lower' wing (Saharon and
Luttges, 1987,
1988
). For this reason, our
model only covers a limited aspect of four-winged insect flight. It is not
intended to explain per se the various types of wing-wake interaction
assumed during the various forward and hovering flight conditions found in
freely flying dragonflies. If not stated otherwise, fore- and hindwing hinges
in our robotic model were separated vertically by 1.3 mean forewing chord
lengths, i.e. the closest distance between the wings at which the wings did
not touch physically during flapping at the various kinematic phase
relationships (Fig. 1C).
In accordance with the stroke kinematics used for an analytical dragonfly
model, we chose a symmetrical wing rotation during the ventral and dorsal
stroke reversal, in which the midpoint of rotational duration occurs when the
wing reverses its translational direction
(Wang, 2000a). A wing rotating
symmetrically starts rotating before and finishes after it has reversed its
flapping direction, which may minimize rotational lift because at that time
translational wing velocity is smallest. To minimize inertial load produced by
rotational moments in our generic kinematic pattern, wing rotation followed a
sinusoidal velocity profile. The onset of wing rotation relative to stroke
reversal, expressed as a fraction of the total wing cycle time,
0, was -0.1, indicating that wing rotation begins 10% of the
stroke period prior a stroke reversal. Flip duration,
, was 0.2,
indicating that wing rotation ends 10% after the stroke reversal (for
nomenclature, see Sane and Dickinson,
2001a
). The kinematic pattern we used in this study produces lift
due to wing rotation equivalent to 3.2% of total lift production by the
hindwing free from forewing wake interference. We estimated rotational lift
contribution from total lift by subtracting the `quasi-steady' lift estimate
during wing translation that was calculated using a conventional
`quasi-steady' analytical model, as suggested by Dickinson et al.
(1999
). In sum, considering
the small amount of rotational lift, it seems unlikely that the pronounced
modulation in measured hindwing lift production as shown in the present study
results from alterations in rotational circulation during the stroke
reversals, but rather reflects aerodynamic mechanisms during wing
translation.
The motion of the two model wings was driven by six servo motors that are controlled by self-written software developed under Visual C++.NET (Microsoft) for a conventional computer. To record force data and to control wing motion simultaneously, the computer was equipped with a 16-channel analog-to-digital data acquisition board (6036E, National Instruments, Austin, TX, USA) and a 24-bit digital interface card (6503, National Instruments) for controlling the motion of the servo motors via a micro-controller (Fig. 1C). We updated the angular position of the entire motor assembly with a maximum rate of 67 Hz (0.015 ms period), whereas the force sampling frequency was approximately 12-times higher, yielding 800 Hz. A potential problem during wing motion of robotic wings is that the actual wing kinematics may differ from the programmed kinematic pattern, whenever the actual power requirements for wing motion exceed the power supplied by the driving motors. This happens especially when the wings accelerate from rest under high inertial load. To avoid a confounding effect on our force measurements due to power constraints of the motor assembly, we modified the servo motors in order to monitor electrically their internal angular position, which is mechanically determined by the angle of the motor main shaft driving the wing. In a control procedure preceding each experimental series, we compared the actual angular position of each servo motor with the programmed wing angles and adjusted either the motor's power supply or the wing's flapping frequency until actual and programmed kinematics were indistinguishable. Besides other constraints, the high power requirement for wing flapping was a major factor that limited maximum flapping speed, and thus Re, of our model wings.
In addition to kinematic modifications of wing motion due to power constraints, wing kinematics may also change due to wing flexing and bending. Moreover, wing flexing can potentially produce inertial peaks that might complicate the measured force traces. Although inertial forces produced during the acceleration and deceleration periods of the wing are relatively small (see below and Fig. 2A), strong flexing of the wing might add brief periods of acceleration/deceleration components to the overall wing acceleration/deceleration profile that is produced by the translational and rotational wing motion. Fig. 2B shows that the hindwing flapping free of the forewing downwash experiences a maximum combined aerodynamic and inertial load between 0.4 and 0.6 N for each mid-halfstroke. To derive a rough estimate of the magnitude of wing flexing during the various times of the stroke cycle, we statically loaded the wing in air with small weights that we placed either at a distance of two-thirds wing length on the upper wing surface or at the wing tip, and measured the deflection of the wing at both locations (Fig. 1D). Loading the wing at the wing tip is thought to produce a rather conservative estimate because the main force vector during wing translation is thought to act close to the two-thirds wing position. The results show that with an average load of 0.3 N, which is equal to the average force measured throughout the entire stroke cycle, fore- and hindwing solely flex approximately 1.0 mm at two-thirds distance from the wing holder, and up to 1.7 mm under the maximum load of 0.5 N that occurs approximately at mid-halfstroke. Due to the sinusoidal velocity profile during wing translation, however, we assume that the wing builds up and releases its deflection more gradually at the beginning and the end of each halfstroke, respectively, which should in turn minimize sudden accelerations and thus high inertial peaks. In sum, we feel rather confident that the measured alteration in force development due to the various kinematic phase shifts between fore- and hindwing are not primarily caused by extensive wing flexion but are likely to represent aerodynamic alterations due to wing-wake interaction.
Force measurements
Fore- and hindwing planforms were based on the wings of the dragonfly
Polycanthagyna melanictera Selys and were made from 2 mm Plexiglas.
Since wing velocity of each wing blade element depends on its distance to the
rotational axis of the robotic hinge, we calculated total wing length as the
distance between the vertical rotational axis of the gear box and wing tip.
Thus total forewing length was 190 mm with an aspect ratio of 6.8, and total
hindwing length was 185 mm with an aspect ratio of 7.4, assuming that gear
box, force sensor and wing holder add to wing length but not to wing area
(Fig. 1C). However, the length
of the wings sensu strictu (without wing holder, force sensor and
gear box) was only 135 mm for the upper forewing (aspect ratio=3.6) and 140 mm
for the lower hindwing (aspect ratio=4.2;
Fig. 1C). Each wing was mounted
on a robotic hinge with three-degrees of freedom, with all axes crossing a
single origin.
In this study we modeled hovering flight conditions of a four-winged insect, which are thought to differ from flow conditions produced during steady forward flight. Advanced ratio as well as reduced frequency are ratios of the `steady motion' caused by the body of an insect flying through the air at constant speed, whereas `unsteady motion' is due to motion of a wing oscillating back and forth about its root. Advanced ratio and reduced frequency are thus measures that indicate which velocity component (free stream due to body motion or wing flapping) dominates the incident flow on the wing. Both quantities are important for `quasi-steady' analytical modeling and the development of dynamically scaled robotic hinges. Since we modeled hovering flight conditions, however, all flow components acting on the two wings are generated by the wing's own motion and thus advance ratio is zero and reduced frequency is infinity.
The two scaled wings were immersed in a 0.6 mx0.6 mx1.2 m glass
tank filled with pharmaceutical white oil (density, 0.88x103
kg m-3; kinematic viscosity, 120 cSt). The size of the tank was
chosen to minimize wall and ground effects that were calculated based on a set
of equations derived from a robotic wing flapping in oil at similar speed
(Dickinson et al., 1999). A
modified force/torque sensor (Nano17, ATI, Apex, NC, USA) was alternately
attached to the base of each wing, and the experiments were repeated to obtain
measurement on both wings. The sensor recorded shear forces and moments along
and around all three wing axes. We converted forces measured normal and
parallel to the wing surface into lift and drag using commercial Active-X
controls (ATI) and software written in Visual C++.NET. We typically recorded
six successive stroke cycles. It has been shown that the first stroke cycle
produces slightly higher forces because the downwash velocity is minimal under
these conditions (Birch and Dickinson,
2001
). For further data analysis we thus averaged only four stroke
cycles (cycles 2-5) in order to avoid confounding effects from the initial
downwash acceleration or any transient forces when the wing was started or
halted at the end of the experiment. Mean total force, lift and drag were
averaged subsequently throughout the entire stroke cycle. The force traces
showing the maximum modulation in lift performance were filtered using a 5 Hz
FFT smoothing filter in Origin 7.0 (Microcal, Northampton, MA, USA).
To derive mean lift coefficient
L, for wing motion from
mean lift and drag averaged throughout the entire stroke cycle, we used the
equation:
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Wing inertia and added mass effects
In real and model wings the forces at the wing base consist of at least
three different components: (i) aerodynamic forces due to both the pressure
distribution around the wing and viscous forces in the fluid, (ii) inertial
forces due to wing and added mass acceleration, and (iii) gravitational
forces. The gravitational component on the force sensor is due to the mass of
the wing and was subtracted from the measured forces by recording the lift
component acting on the resting wing at each point of the stroke cycle. We
estimated the contribution of inertial forces due to wing mass analytically,
assuming that all mass of the wing, mw, including the mass
of the wing holder, is concentrated in the center of wing mass. The center of
wing mass we have indicated by a red dot for each wing shown in
Fig. 1C. Although the mass of
the wing holder is in close distance to the mounting surface of the force
sensor, to which all forces and moments refer, its total mass of approximately
7.0 g is about 54% of the mass of the larger forewing and 62% of the smaller
hindwing, and should thus be considered for inertial effects. According to
Ellington (1984a), inertial
forces during flapping flight in the horizontal plane are proportional to the
first moment of wing mass m1 that is equal to:
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In this equation ' is
normalized wing mass per unit wing length and
is the normalized radial position
along the wing. Since the wing holder is a complex 3D piece that was difficult
to model analytically, we derived the first moment of total wing mass
experimentally by balancing the model wing, including the wing holder, on a
small pin with an approximately 1 mm2 support area. The results of
these measurements show that the moment arm, lx, between
the force sensor and the wing's center of mass is 48.3 mm for the forewing and
50.8 mm for the hindwing (Fig.
1C). Inertial forces
associated to accelerations in the horizontal stroke plane are then:
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![]() | (4) |
![]() | (5) |
When a wing accelerates within the fluid it sets the surrounding air in
motion, resulting in inertial forces by the fluid (added mass effect).
Although added mass effects appear to be small during wing motion of a
slightly larger Drosophila model wing that moves at similar speed and
Reynolds number (Sane and Dickinson,
2001a), we calculated the potential contribution of added mass
inertia using an analytical model for an infinitesimally thin 2D plate moving
in an inviscid fluid modified towards 3D conditions using a blade-element
approach (Sane and Dickinson,
2001a
; Sedov,
1965
). Similar to the wings of a fruit fly, the model wings used
in this study rotate approximately at one quarter chord length from the
leading edge. Total force normal to the wing surface due to the added mass
acceleration of the fluid may be then expressed as (cf. erratum on
equation 1 in
Sane and Dickinson, 2001b
, but
additional corrections also apply; S. Sane, personal communication):
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Reynolds number
In aerodynamics the fluid flow around a wing depends on Re, which
is the ratio of inertial forces to viscous forces within the fluid. In
flapping flight of insects this measure is conventionally defined by the
product of mean wing chord and time-averaged wing tip velocity divided by the
fluid's kinematic viscosity (Ellington,
1984c). Reynolds number for wing motion in our experiments was 105
for the forewing and 125 for the hindwing, which is thought to be at the lower
end of Re observed for dragonflies
(Rüppell, 1989
). For
example, the smallest dragonfly (Nannophya pygmaea), with a hindwing
length of 10.5 mm, will fly at Re=250-500, assuming a stroke
amplitude of 50-100° and a wingbeat frequency of 80 Hz. Reynolds number
for wing motion of Polycanthagyna melanictera as shown in
Fig. 1A is higher and amounts
to approximately 2000 (Wakeling and
Ellington, 1997
). However, Kliss et al.
(1989
) modeled hovering
dragonfly aerodynamic using a flat plate at Re=10-4300, as mentioned
above. The difference in Re between our model wing of
100 and an
averaged sized dragonfly flying at Re>1000 appears to be important
in this study and might be troublesome for interpreting the data. However,
there are several reasons that the flow conditions at Re=100 are
sufficiently similar to the flow conditions we expect at Re>1000.
First, empirical data on static plates in uniform flow show that the force
coefficients vary only little between Re=100 and 1000. Although this
variation is slightly higher than the stable coefficients above
Re=1000, it is much less than between Re=10 and 100
(Hoerner, 1965
). Most of the
transition from attached flow conditions to flow separation at which flow is
shed at reasonable intervals seems to happen within the range
Re=10-100.
Second, the shedding frequency in static plates is a function of
Re and changes in Re domain between 100 and 1000 that would
be relevant for our experimental approach. The force coefficients of our model
fore- and hindwing depend critically on the time of vortex shedding relative
to the stroke reversals. However, the strong dependency of vortex shedding on
Re is questionable in root-flapping wings at which the spanwise wing
blade elements face different flow velocities and thus different Re
during translatory motion. Previous studies using mechanical flappers have
shown that root-flapping wings may stabilize a leading edge vortex (LEV), and
vortex shedding at the stroke reversals at which the wing changes the sign of
the angle of attack may occur before the vortex grows too large to be shed
during wing translation (Birch and
Dickinson, 2001). Moreover, a recent paper on LEV stability
reported that even in a continuous rotating propeller mimicking wing
translation of the hawkmoth Manduca sexta, the LEV remains stable and
no vortex shedding occurs, similar to those expected in a flat plate
translating through the fluid at similar Re
(Usherwood and Ellington,
2002a
). Usherwood and Ellington
(2002b
) concluded that the
shifts from early to steady flow conditions are relatively constant throughout
a large range of Re. Thus it appears possible that the wings of a
fruit fly Drosophila (Re=100-200) exhibit a similar force
coefficient to the flapping wings of a quail (Re=26,000) because the
high force coefficients in both animals are supposedly due to leading edge
vorticity (Usherwood and Ellington,
2002b
). This view is supported by an analytical model on flow
separation (Miller and Peskin,
2004
) suggesting that shedding behaviour in wings is only affected
at Re<50, which is consistent with the experimental data on flat
plate in uniform flow obtained by Hoerner
(1965
). In sum, all the above
results suggest that investigating dragonfly wing-wake interaction at
Re=100 seems to be less troublesome than would be expected from flat
plate data. Moreover, the low Re used in this study was helpful for
conducting DPIV because the high viscosity of the fluid (mineral oil, see
below) minimized the buoyancy of the seeding particles (air bubbles) that we
tested experimentally using different mineral oils. Thus, the Re
selected in this experiment was the best compromise between matching the flow
conditions to dragonfly hover flight between 1000 and 2000 and the
experimental constraints on visualizing reliably the flow around the wings
using DPIV.
Particle image velocimetry
To visualize wake structure, the oil was seeded with bubbles by pumping air
through a ceramic water-purifier filter. The seeding consisted of evenly sized
small bubbles with low upward velocity (<0.5 mm s-1) and high
concentration. We used a 50 mJ per pulse dual mini-Nd:YAG laser (Insight v.
5.1, TSI, Shoreview, MN, USA) to create two identical positioned light sheets
approximately 5 mm thick separated in time by 2500 µs. Paired images of a
250 mm2 flow field were captured using a PowerView 2M (TSI) camera.
A two-frame cross-correlation of pixel intensity using the Hart Correlator
engine (TSI) for a final interrogation area of 32x32 pixels, resulted in
more than 10 000 vectors. Each DPIV experiment consisted of a seven-stroke
wingbeat cycle, and the flow fields from the last five strokes were recorded,
averaged and analyzed. No further smoothing was applied to the flow field
vectors. The light sheet intersected the hindwing at 50% wing length,
perpendicular to the long axis of the wing. DPIV analysis, including
calculation of vorticity, was done using Insight v 5.1 and TSI macros in
Tecplot v 9.0.
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Results |
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Time traces of lift production
To uncover the aerodynamic mechanisms behind the phase modulation effects,
we mapped the wing lift forces throughout a stroke cycle and identified
positions where the interaction between the hindwing and the forewing wake has
the largest effect on hindwing lift (Fig.
4). Due to stroke symmetry in both halfstrokes, the time course of
force production is similar during the up- and downstroke and thus differs
from the aerodynamic forces produced by a tethered dragonfly, flying with a
steeply inclined stroke plane (Reavis and
Luttges, 1988). Fig.
4A shows time traces of lift production for three different
flapping conditions of the hindwing: hindwing lift free from forewing wake
interference (black line), hindwing lift when the forewing leads by 25% stroke
cycle (blue line) and hindwing lift when the hindwing leads by 25% stroke
cycle (red line). The data show that positive lift is produced throughout the
stroke with a small negative lift peak (lift force for all three cases =
-0.013±0.003 N, mean ± S.D.) during wing rotation.
Maximum lift forces of 0.37, 0.41 and 0.23 N were obtained just after the mid
halfstroke (28% stroke cycle) for the three conditions: (i) single wing
flapping, (ii) hindwing leads and (iii) forewing leads wing motion by 25%
stroke cycle, respectively. The difference between hindwing lift produced
during the two different phase-shift relationships and the single wing
performance is shown in Fig.
4B. In the best phase, when the hindwing leads by a quarter
stroke, hindwing lift force is attenuated at the start of the stroke by
approximately 0.10 N, but then develops a larger peak force (0.10 N) at a
later position in the stroke cycle than a single hindwing free from forewing
downwash (Fig. 4B, red trace).
For the worst phase, when the forewing leads by a quarter stroke, lift
throughout the stroke is considerably reduced, producing 0.14 N less lift at
peak attenuation (Fig. 4B, blue
trace). The worst phase peak attenuation, and the best phase peak enhancement
both occur at approximately 35% and 85% of stroke cycle. The high magnitude of
lift alteration and its dynamic change within the stroke cycle is thought to
reflect major changes in the complex wake structure formed by the flapping
wings.
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Particle image velocimetry
To gain more insights into the relationship between lift modulation and
wake structure during wing-wake interaction, we characterized the flow
conditions around the wings during both lift attenuation at the beginning of
the stroke cycle (Figs 4B,
7, *1) and for the
peak effects on lift later in the stroke (*2). As a first step, we
mapped the potential alterations in the strength of the leading edge vortex on
the hindwing in a defined region around the wing's leading edge using DPIV
(Fig. 8, white box), because it
has been shown previously that leading edge vorticity may contribute
significantly to total lift production
(Ellington et al., 1996;
Polhamus, 1971
;
van den Berg and Ellington,
1997
). As a second step, we derived the local flow conditions,
including the effective angle of attack and the velocities of the wing
relative to the surrounding oil, from DPIV analysis in a region between the
free stream and the lower surface of the wing
(Fig. 7, white box). The local
flow conditions are of great importance since they determine the magnitude of
lift production due to circulation bound to the wing during wing translation
and circulation produced by the LEV, because total lift is proportional to the
product of local flow velocity and circulation
(Ellington, 1984b
).
When the forewing leads by a quarter stroke cycle, the strength of the hindwing LEV is attenuated by 23% compared to a single wing flapping free of forewing wake interference, as measured at the beginning of the stroke cycle (Figs 4B, 7A,B, *1). The difference in vortex strength is even higher (31%) later in the stroke cycle (Figs 4B, 7D,E, 8, *2) after the wing segment has travelled approximately 1.2 chord widths after stroke reversal and the LEV has gained size. The smaller leading edge vorticity in the hindwing, when the forewing leads, coincides with the attenuation of lift in the stroke cycle, as shown in Fig. 4B. In contrast, the hindwing's LEV develops differently when the hindwing leads by a quarter stroke cycle. At both the early (15% of stroke cycle) and the late time (35% of stroke cycle) within each half stroke the hindwing's LEV achieves a strength similar to that of a flapping wing free of wake interference (Fig. 7A,C,D,F). This result suggests that, at least when the hindwing leads the stroke, the local flow conditions must have changed in order to explain both the lift attenuation at the beginning of the stroke (15% of stroke cycle) and the increases in instantaneous lift forces above the lift that can be achieved by single wing flapping at 35% of the stroke cycle (Fig. 4B).
To assess the effect of local flow conditions in order to explain the
changes in lift production of the hindwing, we calculated the mean orientation
of the flow towards the wing (effective angle of attack) and its mean
velocities from the combined orientation and velocities of the downwash, and
the motion and geometric angle of the wing, similar to a procedure suggested
previously (Birch and Dickinson,
2001). At 15% of hindwing stroke cycle, the fluid vector
reconstruction reveals that the effective angle of attack
eff and flow velocities for the hindwing flapping in the
forewing downwash, are favorable for the forewing leading phase
(
eff=12.6°; Fig.
9). In contrast, when the hindwing leads, the effective angle of
attack decreases close to zero (
eff=1.6°;
Fig. 9). Local flow velocities
remain approximately constant in all three cases (0.25-0.28 m s-1;
Fig. 9). Later in the hindwing
stroke cycle (35% of stroke cycle) the local fluid vector is only favorable
for the hindwing leading case and not for the forewing leading phase, which
matches the respective enhancement and attenuation of lift production in our
direct force measurements.
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Discussion |
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Wake structure in physical dragonfly models
The force measurements in our dragonfly model show that in a horizontal
stroke, both half strokes contribute to aerodynamic lift production
(Fig. 4). The major flow
structures we visualized in the wake are thus similar to the two major
vortical structures found in other physical insect models, mimicking a 3D
complete stroke cycle in the horizontal: a large starting vortex shed at the
beginning of each half stroke and a leading edge vortex during wing
translation (Figs 5 and
6;
Birch and Dickinson, 2001;
Dickinson et al., 1999
;
Ellington et al., 1996
). Due
to the complex flow pattern, we could not clearly identify stop vortices at
the end of each half stroke. In contrast, Saharon and Luttges
(1988
) described eight
vortices that are shed into the wake of flapping dragonfly model wings: four
vortices by each wing throughout the stroke cycle. The authors found that each
simple element of wing motion, such as the transition from pitching to
plunging motion, initiated its own vortex structure. Similar patterns are
described for vortex shedding patterns in a 2D model wing
(Savage et al., 1979
). Savage
et al. found that a LEV (first vortex) is initiated during wing translation,
which is common in most insect model wings moving at high angle of attack and
similar to the present study (Birch and
Dickinson, 2001
; Ellington et
al., 1996
). During wing rotation (supination) for the subsequent
half stroke, a second vortex is shed from the trailing wing edge in
conjunction with trailing edge vorticity (third vortex) left in the wake in
order to satisfy the Kutta condition when the wing starts to translate
(Savage et al., 1979
). In most
cases, these vortex structures are displaced in the 3D model in the horizontal
direction or move downstream when reduced frequency (based on wing
beat/plunging cycle) is increasing from 0.18 to 5.0
(Saharon and Luttges, 1988
).
In many instances, however, the changes in vortex travel velocity were small,
suggesting that there might be only minor alteration in overall wake pattern
when the animal is changing forward speed (or reduced frequency;
Saharon and Luttges, 1988
).
Moreover, the smoke traces used to visualize the wake in the 3D dragonfly
model suggested constructive vortex fusion that might amplify downwash
patterns and enhance vortex persistence of the wings. In contrast, in the
present robotic model we did not observe that vortices with the same spin
fused in the wake, but found instead that hindwing LEV stability and
persistence appears to be influenced by trailing edge vorticity shed from the
forewing.
The robotic dragonfly model suggested by Saharon and Luttges
(1988) differs from the
present hovering model in several respects. First, Saharon's and Luttges'
model was placed in a wind tunnel with a freestream velocity of 76 cm
s-1. From the data provided, we calculated a mean wing tip velocity
of 540 cm s-1 that results in an advance ratio of approximately
0.14, whereas advance ratio in the present model is zero. Second, in addition
to that, the robotic model of Saharon and Luttges mimicked the dragonfly
kinematics during escape mode found by Norberg
(1975
), which is characterized
by a highly inclined stroke plane while the dragonfly body is held horizontal.
The tilted stroke plane, in turn, requires that a large proportion of total
lift is produced during the downstroke at which the angle of attack of the
hindwing is close to 90°, whereas during the upstroke the wing flapped at
0° angle of attack (fig. 3 in Reavis
and Luttges, 1988
). Third, the kinematic pattern shown by Saharon
and Luttges suggests that the robotic model rotated its wings rapidly at the
stroke reversals, when translational wing velocity was approaching zero. This
kinematic pattern exhibited rather discrete translational and rotational
phases, and this might be the reason why these authors found that each simple
element of wing motion, such as the transition from pitching to plunging
motion, initiated its own vortex structure. In contrast, the onset of wing
rotation in our model wing began 10% of the stroke period prior a stroke
reversal and ended 10% after the stroke reversal, which resulted apparently in
a combined shedding of vortices produced during wing rotation and
translation.
Changes in aerodynamic forces due to phase modulation
Phase modulation effects on the forewing were small and only occurred in
phase-shift cases where the fore- and hindwing were moving close to each other
throughout the stroke cycle (Fig.
3D). Thus it seems likely that some of the modulation of forewing
lift is caused by wall effects due to physical distortion of forewing downwash
by the hindwing (Dickinson et al.,
1999; Rayner,
1991
). We measured the maximum increase in forewing lift compared
to the performance of a forewing flapping separately from the hindwing, when
the forewing leads by 2.5-5% of stroke cycle. In this case the forewing
downwash is directed completely onto the dorsal surface of the hindwing
throughout the stroke cycle (Fig.
5). However, at most kinematic phase shifts we measured a small
decrease in forewing lift, although hindwing downwash effects on forewing lift
should be considerably less than forewing downwash effects on hindwing lift
(Fig. 6). Two effects might be
responsible for this difference. First, downwash flow velocities are thought
to be considerably larger below a wing than above it
(Demoll, 1918
;
Hoff, 1919
). Because the wing
accelerates flow downwards, the resultant flow below the wing will have a
smaller cross-sectional area than the flow above it, according to Venturi's
principle, and consequently the flow velocities in the region below the wing
will be higher than above. Thus, the high flow velocities in the forewing wing
downwash potentially influence hindwing lift to a greater extent than the low
flow velocities produced by the hindwing influence forewing lift. Second, the
vortical structures in the wake travel in the direction of the fluid jet
acceleration and thus it is likely that vortices shed by the hindwing have
less interaction with the forewing than vice versa. Nevertheless, the
small but significant modulation in forewing lift disappears when the two wing
hinges are separated by more than 5 wing chords, supporting our hypothesis
that forewing lift modulation might be due to wall effects caused by the
hindwing (Fig. 3B, open red
circles).
In contrast to the forewing, the stroke-phase relationship between both wings alters hindwing lift production by a factor of approximately 2 (Fig. 3C). Quite similar to the finding on forewing lift, the modulation ceases when we increase the distance in vertical separation between the two wing hinges, resulting in an approximately constant loss of hindwing lift production (Fig. 10). This result suggests that the phase modulation of hindwing lift production is likely to be due to transient forewing wake structures, because at 5-chord-width depth the forewing wake velocities are rather homogenized within the fluid. One potent transient vortex structure likely to influence hindwing lift is the forewing starting vortex that is left in wake while the traveling wing builds up aerodynamic circulation after starting from rest (Figs 5 and 6). Because of vortex interaction, we were not able to identify reliably the two vortices as single structures at all phases of the stroke cycle when flapping both wings; however, results obtained from so-called `static' wing experiments might be able explain the relative decrease in leading edge vorticity of the hindwing, as shown in Fig. 8. We studied the potential threat of starting vortical structure on hindwing lift in DPIV experiments in which the hindwing remained static at its 15% of stroke cycle position throughout the forewing stroke (using identical fore- and hindwings, aspect ratio=3.6). These experimental conditions show that the position of the forewing's starting vortex is close to the hindwing's leading edge, next to the position of the developing LEV, potentially attenuating its development and thus decreasing hindwing lift.
|
The theoretical work by Lan
(1979), who predicted that the
optimum kinematics to maximize hindwing lift is a 25% phase shift, supports
the finding in our physical dragonfly model but runs counter to lift
measurements on a tethered flying dragonfly Aeshna palmatta
(Reavis and Luttges, 1988
). On
the force balance, Aeshna (body weight 0.6 g) produces approximately
1.4 g lift when the `beta angle' is
87°. Reavis and Luttges
(1988
) defined the `beta
angle' as the angle between the freestream flow and the distance between the
fore-aft wing tips. For this reason, the `beta angle' is not identical with
the phase-shift angle used in this study, although the `beta angle' appears to
be a comparable measure for the kinematic phase difference between the two
flapping wings. The force measured in the animal increases to
approximately 3.7 g lift when the `beta angle' decreases to a value
of approximately 52°, which appears to be opposite to the finding in our
dragonfly model. Nevertheless, the tethered flight data apparently indicate
that a change in kinematic phase relationship between the fore- and hindwing
may modulate total peak lift by a factor of 2.6. This value is approximately
twice the modulation we found in the present study for the performance of the
combined wings (Fig. 3D) and is
close to the modulation we found for the hindwing
(Fig. 3C). Apossible
explanation for the discrepancy in sign between the data derived from the
dragonfly and the analytical/physical model is that while varying phase shift,
the dragonfly modulates simultaneously other kinematic parameters such as
stroke amplitude (varies in the hindwing between 60 and 75°), stroke
frequency (varies between 34 and 37 Hz) and maximum angle of attack of both
wings (forewing range is 65-90°, hindwing range is 35-55°;
Reavis and Luttges, 1988
).
Since the force data derived from the tethered dragonfly imply that maximum
lift increases linearly with an increase in all three kinematic parameters, a
phase advance of the hindwing, in conjunction with a pronounced
decrease in amplitude, frequency and/or angle of attack, would
explain the decrease in lift measured in the tethered flying animal.
Regain of hindwing lift
Despite vortex interaction in the wake produced by the combined fore- and
hindwing downwash it is remarkable that the hindwing, whilst flapping in the
wake of the forewing, is able to restore lift to a level close (within 2.5%)
to that of the hindwing flapping free from forewing downwash. Although this
can only be achieved at a flapping condition where the hindwing motion leads
by a quarter stroke cycle, it is quite unexpected because recent studies have
shown that for two-winged hovering insects the first wingbeat produces more
lift than subsequent wingbeats (Birch and
Dickinson, 2001). A likely explanation of this finding is that the
first stroke moves through undisturbed air and all subsequent strokes move
through the downwash of the previous stroke, which may reduce lift by more
than 10% (Birch and Dickinson,
2001
). The same phenomenon is found in helicopter aerodynamics,
where each rotor blade passes through the downwash generated by the preceding
blade (Stepniewski and Keys,
1984
). Closely related to helicopter technology (single and
coaxial rotor blades) is the counter-rotating propeller technology (tandem
propeller) in some long-range reconnaissance aircrafts such as the Shackleton.
At small forward speeds, a single propeller imparts a significant amount of
rotational flow to the air passing through the propeller disk. This rotational
flow does not contribute to thrust, and lowers the lift-to-drag ratio and thus
the efficiency of the aircraft. A second propeller close to the first
propeller and turning in the opposite direction, however, may turn the
rotational motion of the fluid into useful thrust, which appears to be widely
related to the fluid dynamic phenomena found in our root flapping dragonfly
wings. Our direct force measurements show that the regain in hindwing lift in
the dragonfly model results from a complex temporal pattern in which hindwing
lift is attenuated at the early stroke phase (15% of stroke cycle) but then
produces lift in excess of that produced by a wing flapping separately later
in the half stroke cycle (35% of stroke cycle).
The estimates of wing inertia and added mass inertia as shown in
Fig. 2 suggest that the
alterations in hindwing lift are not easily attributable to inertial
components because those components are typically less than 5% of the measured
force. For this reason, it appears more likely that the changes in hindwing
lift result from aerodynamic phenomena rather than from pronounced inertial
effects. Thus to understand the nature of hindwing lift attenuation and
enhancement for the best phase case in more detail, we estimated both leading
edge vorticity and the local flow conditions, because lift depends on fluid
velocity and circulation (Ellington,
1984b). At the early stage in the half stroke (at 15% of hindwing
stroke cycle), the small change in effective angle of attack from 2.1° to
1.6° might explain why hindwing lift
(Fig. 4B, blue trace,
*1) slightly decreases compared to a single hindwing, because LEV
circulation would be similar (Fig.
9A,E, 51.7 vs 56.0 cm2 s-1).
Despite the reasonable development of LEVs, the small effective angles of
attack raise the question of why the model hindwing produces such large lift
during wing translation. One possible explanation is that we underestimated
the effective angle of attack because of leading edge vorticity. A translating
wing that produces leading edge vorticity, causes the oncoming flow to behave
as it does around a cambered wing. A cambered wing, however, is able to
generate large lift even at low geometrical angle of attack close to zero.
Although this view might explain the elevated flight forces early in the
stroke cycle (Fig. 4A,
*1), it cannot easily explain the difference in hindwing
lift production during one- and two-wing flapping conditions because leading
edge vorticity is similar in both cases, as mentioned above
(Fig. 9A,E). Instead, it
appears likely that in the flapping tandem wings, subtle static pressure
distributions (especially the expected over pressure on the lower forewing
surface) might attenuate hindwing lift, which was not estimated in the present
study.
A similar aerodynamic mechanism to that described above (change in effective angle of attack) appears to apply later in the stroke (at 35% hindwing stroke cycle), at which lift increases above single wing performance due to an increase in angle and magnitude of the local flow of approximately 70% and 58%, respectively, compared to the single wing, while leading edge vorticity is approximately equal in both flapping conditions (123 vs 129 cm2 s-1; Fig. 9B,F). To explain the favorable gain in local flow conditions for the hindwing, we suggest the following hypothesis. Fig. 5 shows that the downwash produced by the wings is not directed exactly vertically downward because the inclined wings pull the fluid into the direction of wing motion (re-actio component of drag). As a consequence, at stroke conditions in which the hindwing faces the forewing downwash produced in a preceding or subsequent halfstroke, the vector angle of the forewing downwash is less corruptive than the angle of the oncoming fluid when both wings translate in the same direction (Fig. 9C). The hindwing in Fig. 9F thus yields a high angle of incidence towards the oncoming flow (28.5°) because the local downwash is determined partly by the forewing downwash produced in the previous forewing halfstroke (cf. inclination of green arrows in Fig. 9). In addition to that, the velocity of the forewing downwash contributes to the flow velocity that the hindwing experiences while moving through the fluid, which in turn amplifies aerodynamic force production at this moment of the stroke cycle (Fig. 4B). Assuming that this explanation is valid, then we would also expect a favorable downwash at 35% downstroke cycle when the forewing leads wing motion, because at this moment the forewing downwash is thought to be directed similarly towards the hindwing (Fig. 9D). The reconstruction of local flow conditions, however, has shown that under these flow conditions the local flow vector points into the direction of the hindwing downwash (green arrow points in the direction of hindwing motion) and thus lowers the hindwing's effective angle of attack (Fig. 9D). A possible reason for this phenomenon is that the LEV on the forewing is not fully developed at this moment of the stroke, indicated by the small decrease in total lift at 25% kinematic phase lag (Fig. 3B). Therefore, we suggest that the decrease in aerodynamic performance of the forewing at 35% stroke cycle, due to a possible reduction in leading edge vorticity, might lower the hindwing's capability to produce lift because of unfavorable local flow conditions. We further assume that this hindwing-wake interaction might be highly sensitive to subtle changes in stroke kinematics that alter leading edge vorticity at the beginning of the stroke cycle, such as timing and speed of wing rotation during the ventral and dorsal stroke reversals. The dependency of hindwing lift modulation on stroke cycle timing, as shown by our generic kinematic model, might even indicate that by adjusting more kinematic parameters in the stroke cycle, a higher gain in lift performance might be achieved than the one shown here.
Wing-wake interaction between contralateral wings
The small stroke amplitude of typically 50-100° found in flying
dragonflies limits the interaction of flow structures produced by the
ipsilateral and contralateral wings because the biofoils are well separated
during ventral and dorsal stroke reversal (Alexander,
1982,
1984
,
1986
;
Azuma and Watanabe, 1988
;
Chadwick, 1940
;
Norberg, 1975
;
Reavis and Luttges, 1988
;
Rudolph,
1976a
,b
;
Rüppell, 1985
,
1989
;
Wakeling and Ellington, 1997
;
Weis-Fogh, 1967
). High-speed
film sequences of tethered flight kinematics in dragonflies show only one
example in which the dragonflies Libellula luctosa and Celithemis
elisa performed a physical interaction between the wings during the
dorsal stroke reversal (Alexander,
1984
). However, unlike dragonflies, damselflies typically show
dorsal wing interaction and may use an unsteady lift enhancing mechanism
termed the clap-and-fling or partial fling (Rudolph,
1976a
,b
;
Wakeling and Ellington, 1997
).
For example, the damselfly Calopteryx splendens performs the
clap-and-fling similar to the motion of the wings described by Weis-Fogh
(1973
) for the small wasp
Encarsia formosa. As the wing reaches the top of the upstroke, the
upper wing surfaces meet and then, as the wings rotate and separate, air is
drawn into the opening gap, enhancing wing circulation and thus wing lift
(Bennett, 1977
;
Edwards and Cheng, 1982
;
Ellington, 1975
;
Lighthill, 1973
;
Maxworthy, 1979
;
Spedding and Maxworthy, 1986
;
Sunada et al., 1993
;
Weis-Fogh, 1973
). In addition
to damselflies, the clap-and-fling was found in various other insect species
such as various Diptera (Ellington,
1984b
; Ennos,
1989
), lacewings (Antonova et
al., 1981
) and a whitefly
(Wootton and Newman, 1979
). It
has been shown that insects performing clap-and-fling wing motion produce 25%
more muscle mass-specific lift than insects flying with conventional wing beat
(Marden, 1987
). The
clap-and-fling mechanism is not modelled by our generic kinematics for
dragonfly because we employed solely two ipsilateral wings. Besides the
clap-and-fling, a contralateral wing might also influence force production and
thus phase-shift modulation of lift on an ipsilateral wing via the
extension of LEV over the midline of the animal. This has been demonstrated in
the red admiral butterfly Vanessa atlanta, flying freely in a wind
tunnel with a free stream velocity at around 1-2 m s-1
(Srygley and Thomas, 2002
). At
the moment of take-off, the body angle of the animal with respect to the
oncoming air and the wing's angle of attack approaches high values, supposedly
inducing flow separation on the dorsal side of the body. As a consequence, the
separation bubble on the dorsal body surface might facilitate the LEVs of both
wings to expand over the body midline towards the contralateral wing. It
remains open whether the qualitative description of flow pattern in the
butterfly can be necessarily carried across to hovering flight in dragonflies
at zero advance ratio, because under these conditions the wing root and the
body of the animal only face the downwash that is orientated downwards and
thus would be likely to initiate flow separation on the lower side of the
animal's body. The high body angle and the relatively high flow velocity in
the wind tunnel might, in case of the butterfly, provide an explanation for
why the expansion of a LEV across the midline was not described in physical
models that mimic hovering flight conditions in insects so far.
Concluding remarks
The present study on kinematic phase relationship in a hovering dragonfly
model suggests that under certain kinematic conditions, lift production in
tandem wings is maximized when the hindwing leads wing motion by approximately
a quarter stroke cycle. It is possible that this result only holds for a
limited range of wing kinematics and is limited to hovering flight conditions,
although systematical variations in forward speed (reduced frequency) of the
larger dragonfly model of Saharon and Luttges
(1988) did not produce
significant changes in flow structures. Additional flow components due to fast
forward flight potentially influence local flow conditions, vortex initiation
and vortex travel velocity in the wake produced by the wings (Wang,
2000a
,b
).
To evaluate the robustness of our findings to changes in forward flight speed
or reduced frequency, we simulated changes in vortex travel velocity by
varying the vertical separation of the two wing hinges. The results in
Fig. 10 show that the optimum
phase relationship between two model wings (maximum hindwing lift) decreases
with increasing distance between the two wings (peak force moves to the left).
A possible explanation for this phenomenon is that the duration between the
time at which the forewing sheds vortices and when those vortices interfere
with the hindwing is increasing with increasing distance between the wings.
This result implies that any change in stroke-phase relationship must be seen
at least in conjunction with the magnitude of wing separation, because both
kinematic parameters appear to determine the best phase for lift production in
the tandem wing. The finding that the travel velocity of some vortices also
depends on phase relationship (hindwing phase leads produces faster travel
velocity due to an increase in downwash velocity) at constant wing separation
might even complicate the aerodynamic consequences of the two kinematic
parameters (Saharon and Luttges,
1989
).
A similar picture might appear for aerodynamic effects due to more subtle
changes in wing kinematics such as wing torsion, flexing, and changes in wing
camber during flight (Song et al.,
2001; Sunada et al.,
1998
), including effects due to corrugation of dragonfly wings
(Kesel, 2000
). Wing flexing,
for example, has been discussed as a modification of the clap-and-fling termed
the `clap-and-peel', which might alter force production during the fling part
of the wing motion (Ellington,
1984b
). This modified clap-and-fling kinematics was found in fixed
flying Drosophila (Götz,
1987
) and larger insects such as butterflies
(Brackenbury, 1991a
;
Brodsky, 1991
), bush cricket,
mantis (Brackenbury, 1990
,
1991b
), and locust
(Cooter and Baker, 1977
). In
contrast, in our dragonfly model we used rigid flat plates that deformed only
slightly during wing translation or wing rotation
(Fig. 1D). Studies on the
aerodynamic characteristics of dragonfly, for example, show that corrugated
wings may have a slightly higher lift coefficient under 2D conditions than
flat plates (Kesel, 2000
;
Okamoto et al., 1996
).
In sum, we have shown that by using a generic stroke pattern derived from
dragonfly kinematics, the phase relationship between a robotic fore- and
hindwing may modulate hindwing lift force due to two separate, though not
independent, effects. One seems to be the attenuation of hindwing leading edge
vorticity (LEV destruction), and the second is the speed and angle of local
flow conditions. The hindwing leading edge vorticity seems to be dependent
upon hindwing proximity to the forewing starting vortex, the wing position
within the stroke cycle and the local flow conditions. Timing between the
fore- and hindwing can modulate the wake interference effects and can achieve
instantaneous lift force greater than that achieved by a wing free from wake
interference. The small decrease in lift-to-drag ratio does not necessarily
imply that there is a small energetic cost associated with having two pairs of
wings, because profile costs depend on the product between wing velocity and
drag (Fig. 3D). This issue of
the fluid dynamics in four-winged insects we will address in a subsequent
paper on the power requirements and aerodynamic efficiency of root-flapping
tandem wings. The major benefit from the ability to modulate forces through
fore- and hindwing phase relationships might be that it allows an insect to
control lift production without further changes in stroke kinematics, thus
offering an additional parameter for flight control. As suggested by several
previous studies, right-left asymmetry in phase shift might allow functionally
four-winged insects the ability to modulate forces asymmetrically, and this
might explain why many dragonflies have been reported to vary phase shift
during some turning maneuvers (Alexander,
1986; Norberg,
1975
; Reavis and Luttges,
1988
; Rüppell,
1989
).
List of symbols
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Acknowledgments |
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References |
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