Dragonfly flight: free-flight and tethered flow visualizations reveal a diverse array of unsteady lift-generating mechanisms, controlled primarily via angle of attack
Department of Zoology, Oxford University, South Parks Road, Oxford, OX1 3PS, UK
* Author for correspondence (e-mail: Adrian.thomas{at}zoo.ox.ac.uk)
Accepted 24 August 2004
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Summary |
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Key words: dragonfly, flight, leading edge vortex, micro-air vehicles, unsteady aerodynamics, critical point theory, spanwise flow
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Introduction |
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Such high performance is remarkable, but perhaps not surprising. Recent
theoretical analyses (Anderson et al.,
1998; Triantafyllou et al.,
1993
,
1991
;
Wang, 2000
), computational
analyses (Jones and Platzer,
1996
; Tuncer and Platzer,
1996
), and experimental analyses
(Anderson et al., 1998
;
Huang et al., 2001
) indicate
that isolated flapping foils can produce high thrust coefficients together
with very high efficiency if the kinematics are appropriately configured.
Specifically, wingbeat frequency (f), stroke double amplitude
(a) and flight speed (U) should combine to give a
dimensionless Strouhal number (St=fa/U) at which
wake formation is energetically efficient and a leading-edge vortex (LEV) is
formed on each downstroke (Taylor et al.,
2003
). The LEV should remain over the foil until at least the end
of the downstroke. Theoretical (Bosch,
1978
; Jones and Platzer,
1996
) and computational (Lan and Sun,
2001a
,b
;
Tuncer and Platzer, 1996
)
analyses indicate that adding a second, trailing foil can further increase
efficiency. Corresponding reductions in shaft torque and power have also been
measured in flight tests using helicopters modified to allow appropriate
interactions between the rotor blades
(Wood et al., 1985
). The
direct flight musculature and four-winged morphology of dragonflies make them
ideal candidates for exploiting such aerodynamic effects.
LEVs were first proposed to be a likely source of high lift forces in
flying insects by Maxworthy, who demonstrated their presence experimentally on
mechanical flapping models, or flappers (Maxworthy,
1979,
1981
). A series of analyses
using flappers with dragonfly-like wings and kinematics (Saharon and Luttges,
1987
,
1988
;
Somps and Luttges, 1985
)
showed indirectly that LEVs could be important in forward flight in
dragonflies, and this was confirmed directly by tethered dragonfly flow
visualizations (Reavis and Luttges,
1988
). Earlier analyses of tethered dragonflies `hovering' in
still air had already emphasized the role of unsteady aerodynamics
(Somps and Luttges, 1985
), but
found stalled flows completely separated at both the leading and trailing
edge, instead of a bound LEV (Kliss et
al., 1989
). Not surprisingly, studies using plunging flat plates
in zero mean flow conditions recorded similar stalled flow structures
(Kliss et al., 1989
), which
were found to be built up over several wingbeats.
Although the care taken in this sizeable body of work is impressive,
tethered flight especially in conditions of zero flow
(Somps and Luttges, 1985)
cannot be assumed to produce flows representative of those used by
free-flying insects. Previous work with tethered dragonflies refers to the use
of an `escape mode' (Reavis and Luttges,
1988
), which suggests that the insects may have been trying to
escape from the tether (Somps and Luttges,
1986
; Yates,
1986
). This interpretation is borne out by the large unbalanced
side forces registered in this mode and by the extraordinarily high transient
lift peaks measured for tethered dragonflies `hovering' in still air
(1520 times body weight; Reavis and
Luttges, 1988
). However, since the resonant frequency of the force
balance used in the latter study was only twice the 28 Hz wingbeat frequency,
these extraordinarily high lift values should be treated with caution. Whilst
previous tethered studies are at least indicative of the extreme capabilities
of dragonfly aerodynamics (Somps and
Luttges, 1986
), they are almost certainly not representative of
the aerodynamics of normal flight (Yates,
1986
).
The accompanying studies of mechanical flappers (Saharon and Luttges,
1987,
1988
,
1989
) remain among the most
comprehensive parametric analyses of the effect of individual wing kinematics
for any insect, and are the first studies to deal with the effects of phase
relationships between the wings. They also include some analysis of the
effects of wing morphology on the aerodynamics notably the effect of a
corrugated wing section (Saharon and
Luttges, 1987
). However, one important caveat to this work is that
the wings were modelled on one side of the body only, with the wing flapping
from the wall of the wind tunnel. This is problematic for two reasons.
Firstly, tunnel wall effects (Barlow et
al., 1999
) will come into play near the base of the wings.
Secondly and critically flow visualizations with other insects
(Srygley and Thomas, 2002
),
including dragonflies (Bomphrey et al.,
2002
), indicate that the LEV can extend continuously across the
body from one wing to the other. This flow topology cannot be produced without
a realistic interaction across the body between contralateral wings, and it is
therefore qualitatively different from that produced by any of the one-sided
flappers or whirling arms used to date (e.g.
Birch and Dickinson, 2001
;
Dickinson et al., 1999
;
Usherwood and Ellington, 2002
;
Van den Berg and Ellington,
1997a
,b
).
Even the construction of existing two-sided flappers appears to preclude such
interactions because either the wings are not placed in anatomically realistic
positions relative to each other, or the body is missing or anatomically
unrealistic (Dickinson et al.,
1999
; Ellington et al.,
1996
; Maxworthy,
1979
,
1981
; Van den Berg and
Ellington,
1997a
,b
).
Neither one-sided flappers, nor tethered flow visualizations alone, are
sufficient to identify with confidence the details of the flow topology and
unsteady aerodynamics associated with normal dragonfly flight. Free-flight
flow visualizations with real dragonflies are required to show whether the
same aerodynamics are used in normal flight as have been found in tethered
flight and on one-sided flappers. This lack of reliable flow visualizations
has left considerable room for speculation on the aerodynamics, with a number
of workers (Azuma et al., 1985;
Azuma and Watanabe, 1988
;
Wakeling and Ellington, 1997c
)
suggesting that dragonfly lift generation could be explained by conventional
aerodynamics with attached flows, assuming lift coefficients in the range
measured on detached dragonfly wings in steady flows
(Kesel, 2000
;
Newman et al., 1977
;
Okamoto et al., 1996
;
Wakeling and Ellington,
1997a
). Even if conventional aerodynamics could explain
dragonfly flight this does not mean that dragonflies use conventional
attached-flow aerodynamics. The qualitative nature of the flow field generated
by dragonflies has to be determined by experiment by flow
visualization.
Distinguishing the nature of flow separation in insect flight
Here we present the first flow visualizations of dragonflies flying freely
in a windtunnel. We use the smoke-wire visualization technique in a very
specific way: one that is common in the aerodynamic literature (e.g. for
studies of jets and wakes, see Perry and
Chong, 1987), but has not previously been used in studies of
animal flight. Rather than describing the flow by interpreting the observed
smoke patterns without using any other external information, we instead use
the smoke visualizations as a tool to distinguish among the simplest set of
known local analytical solutions to the NavierStokes equations. Rather
trivially, this allows us to determine the topology of attached flows (when
they are used), but much more importantly, allows us to distinguish the type
of flow separation that results in the LEV (which is usually present).
Formally, the local analytical solutions to the NavierStokes equations
yielding separated flows are the hypotheses being tested in this research;
sketches of the solutions we consider the three simplest solutions
yielding separated flow are presented in
Fig. 1. The NavierStokes
equations have no known general analytical solution, but local solutions can
be derived in the vicinity of critical points in the flow. The formal
procedure is quite classical in aerodynamic analyses of complex wakes and
jets, or of separated flows (Chong et al.,
1990
; Lim, 2000
;
Perry and Chong, 1987
,
2000
;
Perry and Fairlie, 1974
;
Tobak and Peake, 1982
).
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The three separation patterns in Fig.
1 are the simplest local analytical solutions for flow topology
that yield separated flows (Hornung and
Perry, 1984), and they are also the commonest forms of separation
seen in experimental studies (Hornung and
Perry, 1984
; Perry and
Fairlie, 1974
; Tobak and
Peake, 1982
). A pair of negative open bifurcations (where the
surface streamlines converge asymptotically upon a bifurcation line and
separate from the surface; Fig.
1A; Hornung and Perry,
1984
; Perry and Hornung,
1984
) sharing the same origin and attachment line forms the
separated flow over a delta wing at moderate angles of attack
(Délery, 2001
). The
open bifurcation is also characteristic of the footprint where a vortex
touches down on a surface unsteady flow
(Perry and Chong 2000
). The
WerléLegendre separation is perhaps the most well-studied
separation (Délery,
2001
; Hornung and Perry,
1984
; Legendre,
1956
; Perry and Chong,
2000
; Werlé,
1962
): it occurs in the unsteady region where a dust-devil touches
down, near the apex of a delta wing at high angle of attack, and as the origin
of the LEV on the wing top surface or fuselage of many delta-winged aircraft
(Délery, 2001
). Simple
U-shaped separations form the LEV in dynamic stall
(Hornung and Perry, 1984
;
Peake and Tobak, 1980
;
Tobak and Peake, 1982
), the
unsteady post-stall flow over a wing at moderate angle of attack (for example,
sections 15.4.1 and 15.4.2 in Katz and
Plotkin, 2001
) and the horseshoe vortex flow in front of a
cylinder, or adverse pressure gradient on a surface
(Délery, 2001
;
Peake and Tobak, 1980
). In the
flow over a blunt-nosed ellipsoid, separation switches discontinuously
(stepwise) between the three topologies in turn as the angle of attack
increases (Su et al., 1990
).
More complex separations exist (the various `Owl face' separation patterns for
example; Hornung and Perry
1984
) but these involve far more complex patterns of critical
points (i.e. flow singularities) and vortex skeletons. It is possible that
these complex separations occur over insect bodies (at least of the larger
insects), but there are good energetic (evolutionary) reasons why insects
should be adapted to use the simplest forms of separation, if possible. More
complex separations involve larger sets of attached vortices, and complex
multiple separations and cross flows, so they would be energetically
unattractive as a fundamental topology for the LEV. We predict that insects
will therefore avoid them, if they can, and hypothesize that the structure of
the leading edge vortex in insect flight will involve one of the three
separation topologies shown in Fig.
1. The aim of this research is to distinguish which of these
topologies actually applies to the separation forming the LEV in dragonfly
flight.
Flow topology is defined solely by the qualitative pattern of the
streamlines, and is independent of quantitative variations in flow speed along
them. Flows are topologically identical if they share the same arrangement of
critical points (points where the streamline direction is undefined such as
stagnation points or the centres of vortices). The exact pattern of the
limiting streamlines near a surface or in the vicinity of a 3D critical point
in the fluid can be solved analytically from the NavierStokes and
continuity equations. The arrangement of their critical points constitutes the
phase portrait of the flow: `two phase portraits have the same topological
structure if a mapping from one phase portrait to the other preserves the
paths of the phase portrait' (Tobak
and Peake, 1982). That is, two flows are topologically identical
if a deformation of the streamlines exists that can transform one pattern to
the other without causing any streamline to cross itself or another. In terms
of the familiar rubber sheet analogy, a surface streamline pattern, or the
instantaneous streamline pattern in a 2D section of a (possibly unsteady) 3D
flow, drawn on the sheet remains topologically the same, no matter how the
sheet is pulled or stretched (provided there is no tearing). Qualitative flow
visualizations contain the same topological information as quantitative flow
visualizations; indeed, all of the fundamental work on the topology of 3D
unsteady separated flows is based upon qualitative visualization techniques
(for reviews, see Délery,
2001
; Perry and Chong,
1987
). With the many practical advantages that qualitative
visualization techniques carry, it should come as no surprise that they remain
an essential part of experimental aerodynamic analyses of complex separated
flows (e.g. Smits and Lim,
2000
).
Guided by our free-flight visualizations we are able to restrict our
analysis of tethered flight sequences to those where the topology of the flow
field matches what we see in free flight. Earlier studies could not reject
unrealistic tethered flight visualizations because until now, there have been
almost no free-flight flow visualizations with dragonflies to provide baseline
data for comparison; we have obtained that fundamental data and present it
here. Our free-flight visualizations are the first extensive flow
visualizations of free-flying dragonflies, and of any functionally four-winged
insect. The only other published flow visualizations of free-flying insects
are for the butterfly Vanessa atalanta
(Srygley and Thomas, 2002),
and four images of a moth and a dragonfly
(Bomphrey et al., 2002
). All
other previously published flow visualizations are either of tethered insects
or mechanical models, and while it is assumed that these produce flows similar
to those generated by free-flying insects, it remains to be demonstrated that
they do. The tethered flight visualizations, by fixing the field of view,
allow us to visualize flow structure with unprecedented resolution
sufficient to allow us to identify critical points in the flow around the
wings, on the body, and within the LEV. The free and tethered flight flow
visualizations we provide here show that whilst attached flows are typical for
the hindwings in normal counterstroking flight, the forewings almost
exclusively use separated flows when they generate lift (even though angle of
attack can be varied to maintain attached flows on both sets of wings).
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Materials and methods |
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Mechanical flapper
A 150 mmx25 mmx0.75 mm brass plate was plunged sinusoidally by
a drive box consisting of an input shaft driven by an electric motor (SD13 AC,
Parvalux Electric Motors Ltd., Bournemouth, UK), with an inverter for speed
control (Mitsubishi U120, Tokyo, Japan). Gears on the input shaft drive gears
on the output shaft, which in turn drive the vertical movements of nylon
pistons in brass cylinders via con-rods. The internal movements of
the drive box are similar to an internal combustion engine, and because the
motion is a unidirectional rotation, there is no backlash from the gears, and
the sinusoidal input through pin-joints to the piston provides negligible
backlash in the output drive. The plate was connected to the piston by a 3 mm
diameter steel rod, passing through a small aperture in the bottom of the wind
tunnel. Only the plate and supporting rod were in the flow. For the flapper
experiments the plunging plate was replaced by two 75 mm x25 mm
x0.75 mm brass plates hinged at the centreline and arranged so that mean
angle of attack could be varied repeatably. The flapper was driven by the same
drive-system, but the drive shaft was split at a Y-junction to drive each wing
of the flapper and the hinge base was attached rigidly to the force
balance.
Flow visualization experiments
Smoke visualizations were performed in the Oxford University Zoology
Department low-noise, low-speed, low-turbulence, open-return wind tunnel,
which has a contraction ratio of 32:1, working section of 0.5 m x0.5 m
x1 m, and turbulence level (measured by hot-wire survey) of less than
0.3% Root Mean Square (RMS) at the 1 m s1 and 2.75 m
s1 airspeeds used in this study. Smokelines were generated
by the smoke-wire technique using model steam engine oil or Johnson's®
Baby Oil on an electrically heated 0.1 mm nichrome wire. The flow velocity was
1.0 m s1 for free flight (sufficient for good smokelines)
and 2.75 m s1 for tethered flight (sufficient to induce
sustained tethered flight). An array of DC spotlights was used to give 650 W
even overhead illumination.
We first visualised free-flying hawkers and darters during take-off and manoeuvring flight in a wind tunnel. High-speed digital video recordings were obtained using one or two synchronised cameras (NAC500; 250 frames s1; 496x358 pixels). This yielded approx. 525 informative frames, from the 38 wingbeats for which the dragonflies were flying in smoke. We then tethered the hawkers to allow us to frame the image more tightly, therefore maximising frame rate and resolution (NAC500; 500 frames s1; 496x166 pixels). The hawkers were rigidly tethered to a 6-component force balance (I-666, FFA Aeronautical Research Institute of Sweden, Stockholm, Sweden SE-17290) during the high-speed flow visualizations. The tether was a 0.5 mm sheet aluminium platform cemented with cyanoacrylate adhesive to the sternum. This yielded just over 5800 informative frames of high-speed flow-visualization. High-resolution images were obtained simultaneously, using a Canon XL1 camcorder (25 frames s1; domestic compact PAL digital video, 300 000 effective pixels), and up to three Canon MV30 camcorders (25 frames s1) were used to assist reconstruction of the 3D unsteady flows from other angles. This yielded just over 2250 informative frames, giving over 8500 informative frames in total. The images we present here are unmodified, except for adjustments in overall image brightness/contrast.
Interpretation of the flow visualizations
In steady flows, for example in the laminar flow between the smoke wire and
the leading edge of the insect wing, the streaklines formed by smoke are
coincident with streamlines of the flow. In the unsteady flow generated by the
rotating and accelerating wings (for example during pronation or supination),
however, the streaklines of the smoke will deviate from the streamlines over
time, because the shape of the smokelines represents not the current movement
of the fluid, but the current movement plus the spatially integrated time
history of recent motions. Thus care must be taken in the interpretation of
individual smoke visualizations, but the problem is greatly eased by
considering the movement of the flow field indicated by the smokelines in a
series of images making up an animation
(Perry and Chong, 2000). In
this case, the instantaneous flow can be determined from the movement of the
smokelines without the observer becoming overly distracted by discrete
features (kinks, loops), which may represent historical, rather than actual,
flow features. Animations of the high-speed video sequences referred to here
are available online as Supplementary Information for this purpose.
Smoke visualizations are also problematic where vortex stretching is a
major feature of the flow. Over long time scales, smoke particles may, in
effect, be left behind by vortex stretching so that vortices forming important
features of the flow may not be marked by smoke particles. However, problems
with vortex stretching typically involve a timescale of many seconds
(Kida et al., 1991), far
longer than is relevant to insect flight. Dense smokelines are present in our
flow visualizations within the regions of most intense vortex stretching (in
the core of the leading edge and wingtip trailing vortices), which clearly
demonstrates that smoke visualizations are appropriate for the analysis of the
flows over insect wings.
Smoke introduced into the flow in a region where vorticity is generated,
moves with the fluid. Like vorticity, the smoke pattern is Galilean invariant,
so the smoke pattern does not depend on the frame of reference of the
observer, making it uniquely suitable for studying insect flight where the
frames of reference are extremely complex. Quantitative velocimetry data, such
as instantaneous velocity vectors or streamlines, depend very much on the
observer velocity and great care must be taken in selecting the frame of
reference (R. J. Bomphrey, N. J. Lawson, G. K. Taylor and A. L. R. Thomas,
manuscript 1 submitted; Perry and Chong,
2000). In the flow visualizations presented here, smoke is
released into the flow far upstream and then passively transported by the
laminar flow through the tunnel to the insect. Inevitably a particular
smokeline enters the flow around the insect at the point where vorticity is
being generated, passing close to, or even bifurcating at, an attachment point
on the body or wings, a separation point at the leading edge, or at a
free-slip critical point in the fluid above or behind the insect. The topology
of the flow can be simply reconstructed by following those particular
smokelines and identifying the bifurcations in them that mark the position of
critical points in the flow field.
Critical point theory
A problem with previous analyses of the unsteady separated flow over the
wings of flying insects is that no formal system has been used to describe the
different types of flow field that insects generate. In contrast, the
aerodynamic literature, since the early 1980s, has relied on the formal system
provided by critical point theory to describe unsteady flow fields, especially
where complex vorticity fields, 3D unsteady flows and vortex shedding
processes are involved (Délery,
2001; Tobak and Peake,
1982
). Critical point theory was first used by Legendre
(1956
) to describe steady
separated flows, and it can be readily applied to skin friction lines on a
body surface, to the pattern revealed by the projections of the instantaneous
streamlines in any plane in a steady or unsteady 3D flow, or to the
instantaneous streamlines in a steady or unsteady 3D vector field. According
to Chong et al. (1989), `Critical points and bifurcation lines are the
salient features of a flow pattern. In fact, they are probably the
only identifiable features of flow patterns'. Critical point
theory was first introduced to insect flight research by Srygley and Thomas
(2002
) to allow unambiguous
description of the complex 3D separated flow topologies butterflies
produce.
The direction of a streamline at any point is the instantaneous direction
of motion of an infinitesimal fluid particle at that point. Legendre
(1956) noted that only one
streamline could pass through any non-singular point in the vector field
describing the flow, but this is not true at singular or critical points.
Since the flow is modelled by a vector field, it can be described by a system
of differential equations, which can lead mathematically to the existence of
singular, or critical, points points where either the direction or
magnitude of the flow velocity vector is indeterminate
(Poincaré, 1882
). If a
body is present in the flow, then there always exist at least two critical
points on the surface of the body where the direction of the streamline(s)
cannot be defined. This is a specific consequence of a more general
topological rule due to Lighthill
(1963
) that we discuss
shortly: the important point is that there are certain topological rules that
constrain the patterns and types of critical points that can exist in real
flows.
Critical points are classified into three main types: nodes, foci and
saddles. Nodal points are common to an infinite number of streamlines; an
example is the attachment point at the front of a wing or body. Foci are also
common to an infinite number of streamlines, but differ from nodes in that
none of the streamlines entering or exiting them share a common tangent line.
At a focus, an infinite number of streamlines spiral around the critical
point. The streamlines may spiral in to the critical point, spiral out from
the critical point, or form closed paths around it (in which case the critical
point is termed a centre). Centres are inherently unstable, and are transient
features that tend to degenerate into foci. An example of a focus is the
attachment point where a vortex touches down on a surface (like the point
where a dust devil or tornado touches the ground). Saddle points are common to
only two streamlines (termed separators) that pass through the critical point:
the flow along one of these separators converges upon the critical point,
whereas the flow along the other diverges from it. Adjacent streamlines curve
between the separators, so the separators at saddle points divide the flow
into distinct regions. Examples of saddle points occur wherever flows
converge, and saddle points are characteristic of separated flows in general.
Some authors even suggest that separated flows may be defined by the
occurrence of saddle points
(Délery, 2001;
Lighthill, 1963
;
Perry and Chong, 1987
;
Tobak and Peake, 1982
)).
Critical points define the topology of the flow, and they obey topological
rules in just the same way as do the classical 3D regular solid bodies, where
the number of faces plus corners must equal the number of edges plus two; for
example, a cube has six faces plus eight corners and 12 edges. Similarly,
Lighthill (1963) noted that
the skin friction lines (limiting streamlines) on the surface of a 3D body
obey the topological rule that the number of nodes plus foci must equal the
number of saddles plus two, and Tobak and Peake
(1982
) have defined
topological rules for 3D flows in general. A clear recent review of the use of
critical point theory is provided by Perry and Chong
(2000
). Importantly, there are
only a very limited number of ways of joining a set of critical points and,
for simple systems of critical points, flows with the same set of critical
points have the same topology. In other words, topological rules constrain the
patterns of the streamlines joining the critical points (i.e. the phase
portrait), so that for simple systems of critical points, knowing the nature
and number of critical points can be sufficient to specify the phase portrait
and streamlines of the flow.
Describing the set of critical points in the flow around an insect therefore provides a rigorous description of the topology of the flow field. The use of critical points is relatively straightforward where a complete instantaneous 3D vector field of the flow is available. Where time-dependent techniques, such as smoke visualization, are involved, the position and nature of the critical points must be inferred indirectly, but this can still be done without ambiguity. The process we use to identify critical points from the smokelines is explained in the second section of the results (see Figs 12, 14), and while we cannot identify the streamlines of the flow directly, we can identify and objectively define its topology by identifying the critical points of the flow field.
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Results |
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The distribution of dragonfly free-flight flow patterns, number of
wingbeats, and flight sequences in which they occur are presented in
Table 1. Almost three quarters
of all wingbeats (28/38) were counterstroking with a LEV on the forewing. 5/38
of wingbeats involved attached flows, usually during manoeuvres, and 4/38
involved simultaneous in-phase wingbeats associated with
accelerations. A free-slip critical point on the midline was observed during
five wingbeats but those were all the wingbeats where the smoke was on the
midline (dragonflies rarely crossed through the smoke). Spanwise flows were
observed during ten wingbeats, and in seven of those cases the spanwise flow
was from wing tip towards the wing root. Each of those cases involved sideslip
either due to yaw or roll, and the inwards flow was from the leading wingtip
towards the thorax. A LEV over the hindwing, stalled flow on the forewing, and
zero-lift aerodynamics were each observed on two wingbeats in free flight in
the windtunnel during control manoeuvres. However, our dragonflies flew gently
in the windtunnel never even approaching their maximum aerodynamic
performance; speeds of 10 m s1, sustainable accelerations of
2 g, and instantaneous accelerations of almost 4
g (Alexander,
1984; May,
1991
).
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Unstructured wake in free-flying dragonflies
The interaction between the wings and the shed LEV leads to an unstructured
wake devoid of the vortex loops that have been assumed to connect vortices
shed at the top and bottom of each stroke in most theoretical models of insect
flight.
The wake of free-flying dragonflies is illustrated in
Fig. 2, where smoke is on the
centreline of the animal, and Fig.
3, where the smoke plane is close to the wingtip (also see video
S1 in supplementary material). In both cases the wake is characterised by a
lack of any coherent structure. The wingtip vortices are clear in
Fig. 3, but can be seen in the
wake visualizations for less than 1/50th of a second. Whether they dissipate
on this timescale or not is uncertain. Comparing Figs
2 and
3, it is clear that although
the wingtip vortices form discrete wake elements in
Fig. 3, these have no
counterpart there is no starting vortex at the centreline in
Fig. 2. Therefore the shape of
the wake shed from each wingbeat is complex, lacking a starting vortex, but
with the curved paths of the wingtip trailing vortices following the curved
path taken by the wingtips, and then being closed off by the vortex shed into
the wake at the end of the downstroke. This flow pattern is strikingly
reminiscent of the flow generated by a jet in a cross-stream
(Smits and Lim, 2000).
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Use, formation and structure of the leading edge vortex in free flight
Counterstroking is the normal flight mode used by dragonflies. The LEV in
counterstroking is visualised in Fig.
4, where the dragonfly has just taken off and is flying sideways,
holding station next to its perch, against the 1 m s1 flow
through the windtunnel. The LEV in counterstroking flight is bounded by a
separation near the leading edge, with the separatrix touching down at a
stagnation point on the top surface of the forewing near the trailing edge.
Thus the separation containing the LEV is similar in size to the wing chord.
In the image-sequence of Fig. 4
the intersection of the smoke plane with the wing moves from the wingtip
towards the centreline, and the LEV is similar in size and consistent in
structure at each station along the length of the wing. The structure of the
LEV at the midline, over the wing hinge and thorax is clear in Figs
8,
9, the LEV is continuous across
the wing span, and is unchanged as it crosses the wing hinge onto the thorax.
More detail of the structure at the centre of the LEV above the thorax is
provided by the tethered flight flow visualizations below.
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Free-flying dragonflies switch from counterstroking to in-phase stroking to
generate elevated forces (Alexander,
1984; Rüppell,
1989
). The qualitative aerodynamic consequences of in-phase
stroking are shown in video S1 in supplementary information and in
Fig. 7. A LEV forms on the
forewing during pronation. The forewing then undergoes a curtailed downstroke,
at the same time as the hindwing undergoes an extended downstroke with
particularly extreme supination, such that the forewing catches up with the
hindwing as it begins its upstroke. The LEV remains attached to the forewing
throughout supination, but the point of reattachment shifts back off the
forewing and onto the hindwing. The single LEV so formed remains over both
pairs of wings throughout the first half of their combined upstroke. This
results in an even higher degree of flow separation, with a single LEV
extending over the combined chord of both wings, as if over a single
continuous surface: the flow separates at or near the leading edge of the
forewing and reattaches on the upper surface of the hindwing. As in the
counterstroking flight mode, the qualitative results are clear: the LEV is
continuous across the thorax, with a free-slip focus over the midline. The
flow topology becomes complex and variable towards the wingtips: the LEV
inflects to form a single tip vortex when the wings are held close together,
but the structure of the tip vortex is complex, and as wing spacing increases
it separates into two distinct tip vortices. Although their wings are
completely unlinked, in-phase stroking in dragonflies resembles in-phase
stroking in functionally two-winged insects, with qualitatively the same flow
topology as visualised on free-flying butterflies, where the LEV is very much
smaller relative to the wing chord
(Srygley and Thomas,
2002
).
|
Unloaded upstrokes in free-flying dragonflies
The smokelines were usually scarcely deflected by the forewing during the
upstroke (e.g. Fig. 6),
indicating that it was only weakly loaded if at all. Negative loading (force
directed towards the morphological ventral surface of the wings indicated by a
dorsally directed deflection of the smokestreams) was never observed in free
flight, and in tethered flight was only seen for brief periods at the end of
the forewing upstroke. On the rare occasions when negative loading was
observed it caused torsion and marked ventral spanwise bending of the wings,
which could have important implications for the structural mechanics and
aerofoil design of dragonfly wings (Kesel,
2000; Sunada et al.,
1998
; Wootton et al.,
1998
).
Attached flows on loaded and unloaded downstrokes in free-flying dragonflies
In some decelerating or sinking flight manoeuvres requiring low aerodynamic
force coefficients, the flow over both pairs of wings remains attached on the
downstroke as well (Fig. 6).
Attached flows cannot provide the very high lift coefficients that LEVs can,
but since they are also expected to produce much less drag, and higher
lift-to-drag ratios, they might be used for efficient cruising flight (a
behaviour we observed in only two free flight sequences in the constricted
space of the windtunnel). Attached flows are only achieved at very low angles
of attack, which reinforces our conclusion that angle of attack is the most
important kinematic variable governing aerodynamic mechanism in dragonflies.
In some attached flow sequences, the wings slice the smokelines like a knife
(Fig. 6, the same flow pattern
is seen in more detail in tethered flight in
Fig. 11), indicating that
dragonflies can accurately select zero angle of attack for zero lift
production. This mechanism is adopted during decelerating manoeuvres involving
loss of altitude. The same mechanism was used in pitch or roll manoeuvres as a
means of generating large force imbalances between ipsilateral or
contralateral wing pairs, without the need to take a negative load.
|
On the insignificance of spanwise flow in free-flying dragonflies
Previous work has implicated tipward spanwise flow through the vortex core
in stabilising the LEV (Willmott et al.,
1997). Spanwise flow is visualised in some of our images by
smokelines drawn out of plane. For example, Figs
8 and
9 show dragonflies flying with
a degree of sideslip, with the LEV visualised over the leading wing in
Fig. 8 and over the trailing
wing in Fig. 9. The plane of
the smoke streams is distorted in opposite directions in the two images
bulging towards the centreline in
Fig. 8 and towards the wingtip
in Fig. 9. The bulge in the
smoke plane indicates spanwise flow in opposite directions in the two images.
The LEV is stable throughout such manoeuvres (Figs
8 and
9), so in a qualitative sense
spanwise flow is not necessary to stabilise the LEV. This clear qualitative
result is consistent with recent experiments showing that blocking any
spanwise flow on a flapping wing does not destabilise the LEV
(Birch and Dickinson, 2001
).
More recent experimental analyses suggest that spanwise flows are only
present, even on mechanical flappers, at the higher Reynolds numbers relevant
for Manduca sexta (Birch et al.,
2004
), and which are in the range used by our dragonflies.
However, recent theoretical analyses point out that spanwise flows may never
be necessary for LEV stabilisation, given the kinematics and aerodynamic
timescales used by real insects (Wang et
al., 2004
).
Variations in the aerodynamics of free-flying dragonflies
Further variations in the aerodynamics are apparent during free-flight
manoeuvres, and confirm that changing angle of attack is important in LEV
formation. Fig. 5 (Video S2 in
supplementary material) includes a LEV formed over the hindwing by supination
during a pitching manoeuvre in free flight (a LEV was also frequently formed
on the hindwing during tethered flight performances by A. mixta, with
identical flow topology to that on the forewing downstroke). We also observed
(on one wingbeat) a dragonfly with stalled flow on the forewing during a climb
(penultimate frame of video S2). Stalled flows are a common feature of the
tethered flight performances, where they may be artefacts of tethering.
Tethered flight flow visualizations
Tethered flight is not free flight, and tethered flight flow visualizations
should be treated with caution, because tethered insects can produce flow
patterns that are never seen in free flight. However, by constraining the
insects to one position we are able to zoom in and focus on the smoke plane,
increasing the resolution of the flow-visualization images. Uniquely, here, we
have the free-flight data to guide us in identifying flow-patterns in tethered
flight that correspond to flow patterns observed in free flight, and more
importantly (and in contrast to all previous work on tethered flying insects)
we are able conservatively to treat with caution those visualization images
showing flow patterns that do not match those seen in free flight.
Baseline data flow over static dragonflies, or dragonflies flapping but generating no lift
To highlight the components of flow that are due to active flapping and
lift generation by the dragonflies, Fig.
10 shows the flow around tethered dragonflies that are static and
in Fig. 11 the dragonflies are
flapping but feathering their wings to zero aerodynamic angle of attack (as
told by the shearing flows with negligible smokeline deflection).
|
The flow around stationary insects (Fig. 10) consists of a bluff-body wake behind the body of the insect, and a set of Karman streets behind the wings. The flow over the head and forward thorax is attached, but separates near the hindwing root to form an unstructured wake, with no obvious periodicity or concentrations of vorticity. As expected with a bluff-body wake, the disturbance due to the thorax is limited to the region downstream, and does not extend above the body to any appreciable degree. The wings shed vortices periodically in a Karman street, indicating that they maintain some small angle of attack even when the dragonfly is quiescent.
In contrast, in both free flight (e.g.
Fig. 6) and in tethered flight
the dragonflies would occasionally choose to flap with their wings held at an
angle of attack so close to zero that no Karman vortex street was generated
(Fig. 11). The absence of a
Karman street behind the wings shows that the angle of attack is very close to
zero it is a well-known result for sharp-edged flat plates that flow
separates at less than 2° positive or negative angle of attack, and that
once the flow separates the flow field becomes time dependent, with wake
oscillations generated by the unstable shear layer behind the trailing edge
forming a Karman street in the wake (see, for example,
Werlé, 1974;
Van Dyke, 1988
, plates 35 and
36; Katz and Plotkin, 2001
, p.
508). Tethered dragonflies and dragonflies in free flight regularly achieved
this flight condition during active flapping, but not during inactive flight
(as can be seen in Fig. 10),
suggesting that the wing is not feathering to the flow passively, and
therefore suggesting that active control is involved. We were unable to
replicate this flow pattern with isolated dragonfly wings even with 0.1°
precision control of angle of attack at the base. This is presumably because
the wings acquire a twist once they are removed from the insect, and further
supports the suggestion that precise control of angle of attack is necessary
to generate this flow.
The shape of the displacement of the smoke streams where they are cut by
the wings in Fig. 11 reflects
the nature of the boundary layer. Distortions of the smoke streams (and the
complex frame of reference) make detailed interpretation difficult. However,
the wake is roughly 1 mm thick immediately behind the trailing edges of both
the fore- and hindwings in Fig.
11A and behind the forewing in
Fig. 11B. The boundary layer
is expected to remain laminar over the whole chord at the Reynolds numbers at
which the wing operates (Re4300). For comparison the Blasius
solution for the boundary layer thickness
99 of a flat plate
at a point a distance x downstream of the leading edge is
99=5x/Re1/2
(Katz and Plotkin, 2001
, p.
461), which takes a value of 0.75mm at x=10 mm (i.e. immediately
behind the trailing edge), assuming a local wing velocity of 5 m
s1; this suggests that the boundary layer over the
dragonflies wing is not dissimilar to the laminar boundary layer on a thin
flat plate, despite the corrugations of the profile.
Identifying critical points in the dragonfly flow visualizations
Fig. 12 presents a
collection of tethered flight flow visualizations, with the dragonflies
flapping actively, where critical points in the flow are particularly clearly
marked. In these visualizations, by chance, individual smokelines hit
precisely at a critical point, or on a line of critical points such as an
attachment line. Smoke particles can only be passively transported with the
fluid, so that bifurcation of the smokeline at a discrete point implies a
splitting of the streamline at this point (historical if not instantaneous).
At the point where the smokeline bifurcates, the direction and velocity of the
flow is obviously undefined, which is diagnostic of a critical point (because
a critical point is the only place where streamlines cross, where velocity and
direction are undefined it is a singularity in the flow field):
smokeline bifurcation unambiguously identifies the position and nature of a
critical point.
The simplest critical points to understand are at attachment points and attachment lines. These are indicated in Fig. 12 by the blue arrows. Attachment points on the head are clearly marked by smokeline bifurcations in Fig. 12E,G. Attachment lines on the undersurface of the wing are unambiguously marked by smoke bifurcations in Fig. 12AC, and on the top surface of the hindwing in Fig. 12B.
Two forms of free-slip critical point occur. The free-slip critical point (focus) above the thorax is indicated by a yellow arrow throughout Fig. 12 whenever it is visualized, and although the structure of this critical point is complex it is unambiguously marked by smoke bifurcation in the diagonal close-up views of Fig. 12H,I, where the smokelines match remarkably well the solution trajectories (streamlines) of the open U-shaped separation predicted from local analytical solution of the NavierStokes equations (Fig. 1). There is also a free-slip critical point in the form of a saddle indicated by the red arrow and unambiguously marked by smoke bifurcations in Fig. 12DF,I. This saddle point marks a pressure maximum in the shear flow between the LEV over the wing, and the shed vortex in the wake. Its presence is diagnostic of the fact that the wake is one-sided consisting of a series of vortices each of the same sign (starting vortices would have opposite sign; they are not found in the flow generated by dragonflies).
The leading edge vortex in dragonflies is continuous across the midline with a free-slip critical point above the thorax
Fig. 13 shows a series of
smoke visualizations stepping across the thorax of Aeshna grandis in
tethered flight. The flow pattern, shape, size and structure of the LEV is
consistent at all positions across the thorax, and from wingbeat to wingbeat.
A LEV is present in all images, and the shape and size of the LEV is
consistent across the thorax and out onto the wing. The shape and size of the
leading edge vortex is strikingly consistent, even though the wing chord and
velocity changes dramatically as we step along the wing, across the narrow
wing base onto the thorax. This is a remarkable result, suggesting that while
the wings form the LEV the local details of their shape, size and motion are
not amongst the principal parameters controlling LEV morphology.
|
Counterstroking aerodynamics the leading edge vortex in normal flight
The same smoke pattern (Fig.
14) typifies counterstroking in all three species of dragonfly,
appearing in c. 75% of wingbeats in free flight. This seems to be the normal
mode of flight in dragonflies. The forewing downstroke is characterised by
almost-circular smokelines immediately above the wing, suggesting the presence
of a large LEV over the forewings (Fig.
4). Conventional attached flows characterise the forewing upstroke
and the entire hindwing stroke.
A stagnation point is present on the undersurface of the wing near to the leading edge (blue arrows in Fig. 14, particularly well marked in Fig. 14D), and this is the simplest critical point to identify: smokelines hitting ahead of this stagnation point pass forwards to the leading edge, whereas smokelines hitting aft of the stagnation point run back to the trailing edge. Smokelines that hit exactly at the stagnation point bifurcate (Fig. 14D). In images where smoke does not impact the underside of the wing close enough to the stagnation line to bifurcate, its existence is implied by the smokelines impacting aft of the stagnation line, which run straight to the trailing edge (Fig. 14B), and by smokelines impacting just ahead of it, which run forwards around the wing leading edge the divergence of these smokelines implies that somewhere between them a streamline would hit the surface and stop at a critical point, all adjacent streamlines diverging towards either the leading or trailing edges of the wing. Each smokeline bifurcation therefore marks one of the 2D critical points that in 3D form a stagnation line (line of attachment) running parallel to the wing leading edge and emanating from a node of attachment (N1) on the head. That node of attachment is visualised directly whenever a smokeline hits between the insect's eyes and splits above and below the head (Fig. 14E). One smokeline hits the node of attachment between the insect's eyes and splits. Streamlines adjacent to that one radiate out from the node of attachment as the flow passes around the insect's head and on towards its thorax.
Smokeline bifurcation also occurs just ahead of the forewing trailing edge on the wing's upper surface, whenever a smokeline curves down to impact upon the top surface of the wing (Fig. 14B,D). Each bifurcation marks one of the 2D critical points that in 3D form a line of attachment emanating from a second node of attachment (N2). The smokeline bifurcation indicates reverse flow ahead of the line of attachment, because one of the arms of the bifurcation runs forwards from this point. Inevitably, this reverse flow running from the line of attachment near the trailing edge forwards towards the leading edge must converge with the flow running backwards from the leading edge towards the trailing edge. Separation occurs where these converging flows meet. Flows converging along the line of bilateral symmetry of the thorax will run parallel to this centreline, so symmetry requires a saddle point (S) to exist between the wing bases, on the centreline of the animal (in asymmetric flight, a saddle would still exist but would be of non-canonical form and might be displaced from the midline).
The rules of critical point theory of topology require that
there be two more nodes than saddles on a surface in a flow
(Lighthill, 1963), so a node
of detachment (N3) must exist at the back of the thorax or on the abdomen,
continuous with the rearward separation line (line of detachment) at the wing
trailing edge. By the time the smoke has reached the abdomen, it is too
disrupted by flow separation and the unsteady flow fields it has passed
through to reveal this node of detachment directly. However, the existence of
the rearward separation line along the trailing edge of the wing is indicated
by the presence of a shear layer, or vortex sheet, smoothly leaving the
trailing edge (Fig. 14D). This
shear layer is visualized by smokelines flowing back along the undersurface of
the wing from the forward stagnation line. Although the smokelines show that
the flow departs smoothly from the trailing edge, the resulting vortex sheet
quickly rolls up into a series of small transverse vortices under
KelvinHelmholtz instability
(Saffman and Baker, 1979
).
Whenever a smokeline passes close enough to the separation line on the upper surface of the wing, it lifts off the wing surface and spirals in on itself (Fig. 14C). Although smokelines can become curved in the absence of a vortical flow, spiralling smokelines can only be formed in the presence of a vortex. Spiralling of the smokelines close to the separation line therefore indicates that the separation surface becomes entrained in a vortex structure. Because the flow separates at or near the leading edge, this vortex is classified as a leading-edge vortex (LEV), and because the flow reattaches on the wing behind the vortex, the vortex is a bound LEV.
Smokelines over the thorax adopt the same pattern as over the wings (Figs
12D-I,
13,
14E,
16), so symmetry implies that
there is a free-slip focus (F) above and between the forewings. Although the
terminology for 2D critical points and 3D critical points on a surface is
clear and well defined, 3D free-slip critical points are altogether more
complex. The structure above the centreline is a free-slip critical point
(Tobak and Peake, 1982)
specifically, a free-slip 3D focus (Perry
and Chong, 1987
; Tobak and
Peake, 1982
). In the case where there is a free-slip critical
point on the line of symmetry of a simple U-shaped separation, the separatrix
from the node of separation on the head or thorax is open. A narrow band of
streamlines between the separatrix and the streamline that impacts the node of
attachment spirals in to the free-slip critical point under the influence of
vortex stretching as the arms of the U-shaped separation extend into the wake.
This complex 3D flow can indeed be seen in Figs
12H and I and
14E. The LEV extends out from
this free-slip 3D focus to the tips of the wings, where it is continuous with
the wing-tip vortices (Fig.
14A). The critical points identified above and in
Fig. 14 are the minimum number
both consistent with the topological rules of fluid flow and compatible with
the smokeline patterns we observed. The flow topology in counterstroking
flight in dragonflies is not consistent with either the open negative
bifurcation (Fig. 1A) or
WerléLegendre (Fig.
1B) solutions for separated flows, because of the existence of a
3D free-slip critical point above the midline (Figs
12,
13). The topology of the
dragonfly LEV is entirely consistent with the solution of the
NavierStokes equations that yields a simple U-shaped separation
(Fig. 1C).
|
Smokelines passing around the LEV become thinner
(Fig. 14D) and bunch together
(Figs 12AI,
14AE), indicating that
flow is accelerated around the vortex. Smokelines accelerated around the
vortex core develop undulations through KelvinHelmholtz instability
(Saffman and Baker, 1979) in
the shear layer at the boundary of the vortex core
(Fig. 14E,D). This instability
occurs where flows of differing velocity are separated by a distinct boundary
layer: its occurrence above the wing conclusively demonstrates the presence of
a shear layer (i.e. the separation surface). Instabilities just ahead of the
separation line sometimes develop into secondary vortices, which may
subsequently detach from the wing and travel around the vortex core
(Fig. 14D).
Formation of a leading edge vortex through changes in angle of attack
The LEV typically forms during pronation, as the forewing rotates nose-down
at the top of the stroke. Fig.
15 shows the sequence of LEV formation at midwing, and
Fig. 16 shows the sequence of
LEV formation and shedding above the thorax. Provided the relative timing of
pronation and stroke reversal is appropriate to the rotational axis used
(Dickinson et al., 1999;
Sane and Dickinson, 2002
), the
angle of incidence of the freestream at the leading edge increases rapidly.
The LEV grows maximally during the translation phase of the downstroke. The
LEV remains attached throughout supination, as the forewing rotates nose-up at
the bottom of the stroke: this rotation appears to occur around an axis near
the leading edge of the wing, resulting in further dynamic increases in angle
of attack which stabilises the LEV. Formation, growth and stabilisation of the
LEV are therefore all associated with increases in angle of attack, resulting
from either rotation or translation. Rapid increases in angle of attack can
lead to LEV formation at any stage in the wingbeat even on the
hindwings (video S2 in supplementary material). Rapid decreases in angle of
attack can likewise induce vortex shedding at any stage.
|
Shedding of the leading edge vortex
Occasionally, a LEV persisted on the dorsal surface of the forewings well
into the upstroke, but usually the LEV was shed near the beginning of the
upstroke and passed back over the hindwings
(Fig. 16C,D), and the hindwing
kinematics seem specifically configured to permit this mode of wake-capture,
which has not previously been described for real insects (see also videos S1,
S2 in supplementary material). In contrast to previous analyses of tethered
hovering (Kliss et al., 1989),
vortices are not built up over consecutive strokes of the same wing in forward
flight: the wing kinematics seem configured to prevent interactions with wake
elements shed on previous strokes, at least in forward flight.
Absence of starting vortices
A striking qualitative feature of LEV formation is the absence of a
corresponding discrete starting vortex (Figs
12,
13,
14,
15,
16). Kelvin's theorem on
persistence of circulation requires that the total circulation around any
closed curve of particles in a fluid is constant, so any circulation generated
by the wing must be balanced by opposite circulation in the wake. For
impulsively started wings, the opposing circulation quickly rolls up into a
starting vortex, inducing an unfavourable downwash at the wing. This
diminishes lift, with full lift production only achieved after the wing has
moved several chord lengths a phenomenon called the Wagner effect
(Wagner, 1925;
Weihs and Katz, 1986
). Real
insect wings are not impulsively started, so aerodynamicists will not be
surprised to find that a discrete starting vortex is not formed in dragonfly
flight. Instead, the vortex sheet shed at the trailing edge rolls up into a
series of small transverse vortices, rather than a single large starting
vortex of comparable size to the LEV. These are clearly visualised in
Fig. 14BD. It is
possible that the qualitative difference between the flows generated by
free-flying dragonflies and those involved in the Wagner effect mean that the
latter does not apply to dragonflies. Indeed, the flow visualised in
Fig. 14BD is strikingly
consistent with the classical lumped-vortex solution for the circulation and
lift of an accelerated flat plate (see, for example,
Katz and Plotkin, 2001
,
section 13.7), a solution which indicates that when the wake consists of a
series of discrete vortices, rather than a single large starting vortex, there
is only a slight loss of lift due to the downwash of the wake vortices (see,
for example, Katz and Plotkin,
2001
, fig. 13.8). Confirmation of this hypothesis would require
quantitative data, but it is likely that the interactions between successive
vortices in the shear layer behind the wings and viscous decay of the
individual shear-layer vortices eliminates, or at least greatly reduces, the
magnitude of the reduction in lift due to the Wagner effect. Qualitatively,
the flow visualizations of Figs
14,
15,
16 make it unequivocally clear
that at the start of the downstroke the dragonfly's wings operate in a flow
field dominated by the upwash induced by the LEV shed from the previous
downstroke (Fig. 15A).
Irrespective of the Wagner effect, operating a wing in an upwash must increase
the total lift-vector and should allow it to be tilted forwards at the start
of the downstroke, which could lead to a substantial reduction in drag and
increase in lift. Quantitative data are urgently required to measure the gain
due to the beneficial interaction, at the start of the downstroke, between the
wings and the wake shed from the previous downstroke.
Leading edge vortex formation with simplified kinematics life-size flappers
We were able to replicate exactly both the gross flow topology and detailed
qualitative features of the flow over dragonfly forewings with a mechanical
model consisting merely of a flat plate in simple harmonic flapping, pitching
or plunging motion (Fig. 17
and video S3 in supplementary material, which is an animation of
Fig. 17; see also
Taylor et al., 2003). Provided
the flow velocity, frequency and amplitude combined to give a Strouhal number
in the range
(approximately the same as used by real
dragonflies), detailed features of the flow topology over dragonfly forewings
(Fig. 14) are accurately
reproduced by the plunging or flapping plates
(Fig. 17; see
Taylor et al., 2003
). A LEV
forms as the angle of attack increases through translation at the start of the
downstroke (Fig. 17BD).
Secondary vortices can be seen close to the separation line
(Fig. 17CH), and the
smokelines can be seen spiralling into the vortex
(Fig. 17CF). As in real
dragonflies, and in the parameter range
, there is no
discrete starting vortex (Fig.
17BF); instead the vortex sheet shed from the trailing edge
rolls up under KelvinHelmholtz instability
(Saffman and Baker, 1979
) into
a series of small transverse vortices of opposite sense to the LEV
(Fig. 17CG). Starting
vortices could only be visualised outside of the range
. The LEV grows through most of the downstroke
(Fig. 17BF), rolling
back from the leading edge towards the end of the downstroke
(Fig. 17G). The LEV is shed
earlier than in dragonflies. This is probably because the angle of attack
decreases rapidly at the end of the downstroke: in dragonflies, the angle of
attack is maintained or even increased as the wing rotates rapidly during
supination at the end of the downstroke, which apparently stabilises the LEV.
Complex wing kinematics are not necessary for LEV growth and
formation.
|
The mechanical flapper demonstrates unequivocally that the LEV structure, including fine details such as the shear layer behind the trailing edge and secondary vortices, can be replicated even by a flat plate in flapping or plunging motion at the appropriate Strouhal number. This is critical because in a plunging motion there are no velocity gradients along the span to generate the pressure gradients required to produce a spanwise flow. In the absence of a spanwise flow there is no mechanism to transport chordwise vorticity along the axis of the LEV and out into the wingtip vortices. Nevertheless the LEV dwells on the wing for the duration of the downstroke provided the Strouhal number range is appropriate: the bottom of the downstroke is reached before the vortex grows so large as to be shed because of its size. This controlled qualitative experiment therefore demonstrates that spanwise flow is not necessary for the LEV to be stable throughout the downstroke, provided the Strouhal number is appropriate.
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Discussion |
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One of the most important conclusions of this study is the finding that in
free-flying and tethered dragonflies, angle of attack controls aerodynamic
mechanism through the wingbeat. Formation, growth and stabilisation of the LEV
are all associated with increases in angle of attack, whereas vortex shedding
is associated with decreases in angle of attack. Changes in angle of attack
can result from rotational and translational movements of the wings alike: it
remains unclear whether rotational mechanisms differ qualitatively from other
aerodynamic mechanisms involving changes in angle of attack
(Walker, 2002). These
conclusions are consistent with earlier work with mechanical flappers, which
also found changes in angle of attack to play a key role in the dynamics of
LEV formation, growth and shedding (Saharon and Luttges,
1987
,
1988
,
1989
).
This same result may also explain some of the most important differences
between our findings for free-flying dragonflies and the results of Luttges
and colleagues. The first difference is that the flows they observed on
tethered dragonflies (Kliss et al.,
1989; Reavis and Luttges,
1988
; Somps and Luttges,
1985
) were always completely stalled. This is almost certainly an
artefact of tethering, because we observed similar flow structures frequently
in tethered flight, but only once in free flight (during a climb, where the
flow over the forewing appears mildly stalled on a single downtroke;
penultimate frame of video S2 in supplementary material). In steady flows,
stall occurs when the angle of attack is above a certain critical value
(roughly 15° for a high aspect ratio wing with a classical aerofoil such
as the NACA0012). The same is true of unsteady flows, but rather higher angles
of attack may be maintained at least temporarily if the flow
separates from the leading edge and reattaches downstream such that the flow
is not globally separated. Stalled flows may perhaps be important in hovering
and during climbing manoeuvres, where extremely high local angles of attack
may be unavoidable on some parts of the wings as a result of the flapping
kinematics. However, our dragonflies' ability to maintain zero aerodynamic
incidence during feathered strokes shows they can adjust aerodynamic angle of
attack with a high degree of accuracy, and the fact that stalled flows do not
normally appear in slow forward flight or even in turning manoeuvres is
probably the deliberate result of controlling angle of attack. The fact that
dragonflies maintain attached flows over the hindwings during normal
counterstroking flight is further evidence that they usually control angle of
attack so as to avoid stalled flows.
A second important difference between our results with free-flying
dragonflies and those of Luttges and colleagues is that their mechanical
flappers formed and shed multiple discrete vortex structures on each stroke
(Saharon and Luttges, 1987,
1988
,
1989
). These vortices were
found to interact in sequence to form a `continuous multi-vortex structure',
in which `each vortex structure, however, remains a discrete entity'
(Saharon and Luttges, 1989
).
This is quite different from any structure we found on either our tethered or
free-flying dragonflies. Although secondary vortices were sometimes present in
front of the LEV, these appear to result from flow instabilities at the
separation line and from shear layer instabilities in the separation surface.
We occasionally observed a small vortex to be shed immediately after
pronation, but the normal pattern was for a single discrete LEV to form
through each stroke. It is possible that the mechanical construction of
Saharon and Luttges' flappers caused the formation of discrete vortex
structures via backlash and step changes in angle of attack.
A third difference between our results and those of Luttges and colleagues
cannot be explained by factors relating to adjustments in angle of attack.
Saharon and Luttges describe the LEV generated by their mechanical flapper as
forming `a cone pattern' with `gradual reductions [in vortex
size] associated with outboard span locations', and state that
`the apex of this conical flow structure focuses on site [sic] where the
apex of the wing tip vortex helix appears to originate'
(Saharon and Luttges, 1987).
This is diametrically opposite to the form of the conical LEVs observed on
other mechanical flappers and whirling arms
(Birch and Dickinson, 2001
;
Dickinson et al., 1999
;
Usherwood and Ellington, 2002
;
Van den Berg and Ellington,
1997a
,b
),
which apparently have a conical vortex originating from an apical focus near
to the base of the wing. It is also completely different to the cylindrical
form of the LEV observed in live dragonflies, which is continuous across the
midline of the body and inflects near the wingtip to continue into the wingtip
vortices with no apparent reduction in size
(Fig. 4A). It is difficult to
envisage why Saharon and Luttges
(1987
) would have found the
LEV to decrease in size outboard along the span, when flapping velocity
increases along the wing, and this may simply be an error in interpretation.
In any case, our finding that the LEV is continuous across the midline in
tethered and free-flying dragonflies cautions against drawing conclusions with
respect to the flow topology from one-sided flappers, which cannot possibly
generate this structure.
The result that the dynamics of LEV formation, growth and shedding in
dragonflies can be accurately reproduced using a flat plate with simple
kinematics configured to have a Strouhal number in the range
is consistent with results from the mechanical
flappers of Saharon and Luttges
(1987
,
1988
,
1989
). They found that the
dynamics of LEV growth, formation and shedding depended strongly upon the
reduced frequency parameter k=fc
/U. For a fixed
wing morphology and stroke amplitude, the Strouhal number
St=fA/U differs from the reduced frequency by a
constant factor. In the case of Saharon and Luttges' flapper, and assuming a
stroke angle of 45°, St may be calculated from reduced frequency
as St
0.5k, so the reduced frequencies of k=0.18
and k=0.5 that they used correspond to approximately the same range
of St as we used.
In fact, the Strouhal number is probably the fundamental aerodynamic
parameter governing LEV dynamics (Anderson
et al., 1998; Triantafyllou et al.,
1993
,
1991
;
Wang, 2000
): the reduced
frequency is also significant, but is unaffected by changes in stroke
amplitude. Stroke amplitude is important because it is the product of stroke
frequency and amplitude (which forms the numerator of the Strouhal number),
rather than the product of stroke frequency and wing chord (which forms the
numerator of the reduced frequency), that governs the speed of flapping. This
in turn defines the maximum angle of attack that is attained through the
stroke: for a given static angle of attack, the maximum angle of attack during
flapping increases with increasing Strouhal number. We have already said that
angle of attack is the most important kinematic variable governing the
aerodynamics of dragonfly flight, and this is consistent with the results from
Saharon and Luttges' flappers (Saharon and Luttges,
1987
,
1988
,
1989
). It is therefore natural
that Strouhal number governs LEV dynamics, because it determines both the
maximum angle of attack and the intrinsic timescales of flapping, which acts
as a forcing function for the aerodynamics
(Taylor et al., 2003
).
It is nevertheless remarkable that even fine details of the flow topology
on a dragonfly's wing can be replicated by the simple harmonic plunging of a
flat plate at an appropriate Strouhal number. This suggests that a LEV may not
be difficult for an animal with thin flapping wings to evolve. Fine-tuning of
tandem wing interactions may be rather harder, and it is possible that the
reason that counterstroking dragonflies use attached flows on the hindwings is
to avoid complex interactions between vortices generated separately on the
fore- and hindwing pairs. Parametric studies of the robustness of the
hindwingforewing interactions we have described single LEV
formation during in-phase stroking, and hindwing capture of the LEV shed by
the forewings during counterstroking must await future work with
four-winged (i.e. two-sided) flappers. Saharon and Luttges
(1989) also found that the
hindwing of their tandem-winged flapper could trap vortex structures created
by the forewing, but since the vortex structures then fused with those created
by the hindwing, the details of the process are quite different (the hindwing
generates no LEV in normal counterstroking flight in either tethered or
free-flying dragonflies).
The smoke visualizations we have presented clearly distinguish between the
hypotheses of Fig. 1. The LEV
produced by dragonflies has the topology of a simple open U-shaped separation.
The other hypotheses of Fig. 1
are rejected because there is a free-slip critical point over the midline, in
free and tethered flight in dragonflies. The same topology has also been found
in red-admiral butterflies (Srygley and
Thomas, 2002). Dragonflies and butterflies bracket the entire
range of wing morphology in insects dragonflies have amongst the
highest aspect ratios, butterflies amongst the lowest. The open U-shaped
separation has the simplest possible vortex skeleton of any pattern of
separation. It is also the natural separation that occurs in the unsteady
post-stall flow over a high aspect ratio wing, or in dynamic stall. We
therefore predict that formation of a LEV by means of an open U-shaped
separation will be the typical high-lift aerodynamic mechanism involving flow
separation in insects (and perhaps flying animals in general) with high aspect
ratio wings.
One major implication of this result is that the root-flapping motion characteristic of all flying animals may be a constraint, imposed by the pre-existing musculo-skeletal structure, rather than an adaptation. A root-flapping motion is not necessary for any feature of the dragonfly LEV, and the dragonfly high-lift aerodynamic mechanism could be replicated (as we have shown) with a combination of appropriate pitching and plunging motions. This may be of some comfort to the micro-air-vehicle community.
![]() |
Acknowledgments |
---|
![]() |
Footnotes |
---|
Present address: Smithsonian Tropical Research Institute, Unit 0948, APO AA
34002-0948 USA
Present address: School of Biology, Leeds University, L. C. Miall Building,
Clarendon Way, Leeds, LS2 9JT, UK
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