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First published online November 5, 2004
Journal of Experimental Biology 207, 4299-4323 (2004)
Published by The Company of Biologists 2004
doi: 10.1242/jeb.01262

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Dragonfly flight: free-flight and tethered flow visualizations reveal a diverse array of unsteady lift-generating mechanisms, controlled primarily via angle of attack

Adrian L. R. Thomas^*, Graham K. Taylor, Robert B. Srygley, Robert L. Nudds and Richard J. Bomphrey

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

	Summary

TOP Summary Introduction Materials and methods Results Discussion References

 
Here we show, by qualitative free- and tethered-flight flow visualization, that dragonflies fly by using unsteady aerodynamic mechanisms to generate high-lift, leading-edge vortices. In normal free flight, dragonflies use counterstroking kinematics, with a leading-edge vortex (LEV) on the forewing downstroke, attached flow on the forewing upstroke, and attached flow on the hindwing throughout. Accelerating dragonflies switch to in-phase wing-beats with highly separated downstroke flows, with a single LEV attached across both the fore- and hindwings. We use smoke visualizations to distinguish between the three simplest local analytical solutions of the Navier–Stokes equations yielding flow separation resulting in a LEV. The LEV is an open U-shaped separation, continuous across the thorax, running parallel to the wing leading edge and inflecting at the tips to form wingtip vortices. Air spirals in to a free-slip critical point over the centreline as the LEV grows. Spanwise flow is not a dominant feature of the flow field – spanwise flows sometimes run from wingtip to centreline, or vice versa – depending on the degree of sideslip. LEV formation always coincides with rapid increases in angle of attack, and the smoke visualizations clearly show the formation of LEVs whenever a rapid increase in angle of attack occurs. There is no discrete starting vortex. Instead, a shear layer forms behind the trailing edge whenever the wing is at a non-zero angle of attack, and rolls up, under Kelvin–Helmholtz instability, into a series of transverse vortices with circulation of opposite sign to the circulation around the wing and LEV. The flow fields produced by dragonflies differ qualitatively from those published for mechanical models of dragonflies, fruitflies and hawkmoths, which preclude natural wing interactions. However, controlled parametric experiments show that, provided the Strouhal number is appropriate and the natural interaction between left and right wings can occur, even a simple plunging plate can reproduce the detailed features of the flow seen in dragonflies. In our models, and in dragonflies, it appears that stability of the LEV is achieved by a general mechanism whereby flapping kinematics are configured so that a LEV would be expected to form naturally over the wing and remain attached for the duration of the stroke. However, the actual formation and shedding of the LEV is controlled by wing angle of attack, which dragonflies can vary through both extremes, from zero up to a range that leads to immediate flow separation at any time during a wing stroke.

Key words: dragonfly, flight, leading edge vortex, micro-air vehicles, unsteady aerodynamics, critical point theory, spanwise flow

	Introduction

TOP Summary Introduction Materials and methods Results Discussion References

 
Aerodynamics of dragonfly flight
Dragonflies are accomplished aerial pursuit hunters: they can hover, accelerate in almost any direction and manoeuvre precisely at high speed (Alexander, 1984, 1986; Azuma and Watanabe, 1988; May, 1991; Rüppell, 1989; Wakeling and Ellington, 1997b) to intercept other insects, with measured success rates as high as 97% (Olberg et al., 2000). The direct flight musculature of dragonflies means that stroke frequency, amplitude, phase and angle of attack can be varied independently on each of the four wings. Dragonflies put these abilities to excellent effect, using them to enlist a variety of wing kinematics in free flight (Alexander, 1984, 1986; Azuma and Watanabe, 1988; Rüppell, 1989; Wakeling and Ellington, 1997b). The wings counterstroke when cruising, but stroke in-phase during manoeuvres and when higher accelerations are required (Alexander, 1984, 1986; Rüppell, 1989). Switching to in-phase stroking allows dragonflies to attain speeds of 10 m s^–1, sustainable accelerations of 2 g, and instantaneous accelerations of almost 4 g (Alexander, 1984; May, 1991).

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 (15–20 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 Navier–Stokes 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 Navier–Stokes 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 Navier–Stokes 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).

View larger version (31K): [in this window] [in a new window]   Fig. 1. Sketches of three solutions to the Navier–Stokes and continuity equations that lead to local flow separation patterns. These three types of flow separation are commonly observed in experimental situations. (A) The open negative bifurcation line consists of a negative bifurcation line from which a separatrix emerges at the front of the separation. The negative bifurcation always occurs in a pair with a positive bifurcation line. This kind of separation is often found when a vortex approaches and impacts with a surface; it is also involved in the separation over delta wings at moderate angle of attack when two symmetric negative bifurcation lines form at the leading edges and a single positive bifurcation line forms down the centreline of the delta. The negative bifurcation contains no discrete critical points, but the bifurcations – attachment and separation lines – are formed from a critical point in a cross flow. (B) The Werlé–Legendre separation has been studied since the 1960s, and occurs at the base of a dust-devil, or over a delta wing at high angles of attack. The Werlé–Legendre separation is a combination of a saddle point, from which a negative bifurcation line emerges, and a focus. The separatrix arises from the saddle point and negative bifurcation line. (C) The simple U-shaped separation occurs in dynamic stall, or in the post-stall flow over a wing. It contains a free-slip critical point (focus) above the line of symmetry, combined with a node of attachment, and the separatrix emerges from a saddle-point and the negative bifurcation (separation) lines that emerge from it at the front of the separation.

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 Werlé–Legendre 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 Navier–Stokes 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).

	Materials and methods

TOP Summary Introduction Materials and methods Results Discussion References

 
Animals
We netted brown hawkers Aeshna grandis L., migrant hawkers A. mixta Latreille and ruddy darters Sympetrum sanguineum Muller in the Oxford University Parks. All flow visualizations were made either immediately following capture, or occasionally later that day or on the following day, in which case the dragonflies were kept cool in a refrigerated room overnight to prevent them from damaging themselves.

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 s^–1 and 2.75 m s^–1 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 s^–1 for free flight (sufficient for good smokelines) and 2.75 m s^–1 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 s^–1; 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 s^–1; 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 s^–1; domestic compact PAL digital video, 300 000 effective pixels), and up to three Canon MV30 camcorders (25 frames s^–1) 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.

View larger version (127K): [in this window] [in a new window]   Fig. 12. Collection of flow visualization images selected to show the process of identification of critical points. Bifurcations in smoke streaklines are diagnostic of critical points. Blue arrows point to stagnation points either where a smoke stream hits the wings or head, or where the flow over the LEV touches down on the top surface of the wings, or on the body. Yellow arrows point to the free-slip critical point above the body. Red arrows point to smoke bifurcation at the saddle point in the wake caused by the shear layer between the downwash behind the attached LEV and the upwash of the LEV that was shed from the previous downstroke. (A–C) Flow over the wings at about half wing length. (D–F) Flow over the midline and interaction with the wake. (G–H) The free-slip critical point above the midline.

View larger version (91K): [in this window] [in a new window]   Fig. 14. Characteristic smoke patterns associated with the forewing downstroke in normal counterstroking flight. The video images show a tethered hawker Aeshna grandis; the topological interpretation is the same for all three species. The critical points in the 3D flow field are denoted by black spots (N=node; F=focus; S=saddle); dotted lines represent hypothetical surface streamlines. Visualizations are shown for 5 spanwise stations along the wing (A–E), marked by colour-coded slices in the figure. The LEV is continuous with the vortices trailing from the wingtips (A). The LEV diameter is similar across the wing, and the flow is topologically similar at all three stations inboard of the wingtip (B–D). The flow over the midline of the insect clearly shows that the LEV is continuous across the midline (E), indicating the existence of a free-slip focus above the thorax. The topology is the same throughout the downstroke: we have chosen those images that show the downstroke flow structures most clearly for each spanwise station.

	Results

TOP Summary Introduction Materials and methods Results Discussion References

 
Free-flight flow visualizations
Free-flying insects are, by definition, unconstrained. The vast majority of free-flight sequences with the dragonflies occur either before smoke was released, or after it had dissipated. In most cases where smoke and dragonflies were both present at the same time the dragonflies chose to fly out of the line of the smoke. Nevertheless we were able to obtain eight sequences in which the dragonfly flew through smoke and was in the field of view of at least two perpendicular cameras. In total we were able to clearly identify the flow pattern around the wings during 38 wingbeats. These represent the only existing data on the flow around the wings of free-flying dragonflies, and also represent the most comprehensive set of flow visualization data for any free-flying insect. These free-flight flow visualizations are presented as composite sequences extracted from the high-speed video recordings, to provide baseline data that can be used to check the validity of further tethered flight, mechanical model or Computational Flow Dynamics (CFD) analyses. The original video sequences are presented as supplementary material on the web.

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 s^–1, sustainable accelerations of 2 g, and instantaneous accelerations of almost 4 g (Alexander, 1984; May, 1991).

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).

View larger version (48K): [in this window] [in a new window]   Fig. 2. Free-flight flow visualization of the wake of the dragonfly Sympetrum sanguineum in counterstroking flight. (A–F) Composite figure of sequential images extracted from a 250 Hz high speed video recording (video S1 in supplementary material). The dragonfly is moving from left to right through the smoke plane, which is approximately at the near wing hinge in (A). Wake structure is incoherent. There is no sign of a starting vortex, but some sort of vortex structure (stopping vortex? Wingtip vortex in oblique view?) is apparent in (C–E) (green arrows) and a wake element of sorts can be seen between the green arrows in (D). This wake element rapidly loses its identity after it is shed, being hard to detect after two frames (1/125th of a second). The visualised wake is not consistent with a series of discrete vortex elements such as, for example, vortex rings.

View larger version (102K): [in this window] [in a new window]   Fig. 3. Free-flight flow visualization of the wake of the dragonfly Sympetrum sanguineum in counterstroking flight with the smoke-plane close to the right (far) wingtip. (A–J) Consecutive images from a 250 Hz high-speed video recording. In contrast to the centreline flow shown in Fig. 2, here the wingtip vortices are clear (purple arrows), and form wake elements (green arrows) that persist for several frames. However even here at the wingtips, where the wake structure is at its most coherent, the wake elements lose their identity after five frames (A–E, 1/50th of a second). The difference between the apparent structure of the wake elements between the tip region and the centreline region suggests that wake elements have a complex structure, consistent with the lack of any defined starting vortex.

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 s^–1 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.

View larger version (44K): [in this window] [in a new window]   Fig. 4. Free-flight smoke visualization of the flow around the wings of Sympetrum sanguineum. There is a leading edge vortex (yellow arrows) on the fore-wing counterstroking flight. (A–H) Consecutive images from a 250 Hz high-speed video recording. Perpendicular views from the b and c cameras show that the dragonfly has taken off and cleared the perch and is holding station, flying sideways in (A) as the forewing completes the upstroke. The downstroke begins in (B), and the LEV is already present when the wing cuts the smoke in (C). The structure of the LEV is consistent as the intersection of the smoke and the wing moves towards the midline (D,E), and the internal flows within the LEV are clear in (F). The LEV is shed at the start of the upstroke in (G). There is no evidence of spanwise flow.

View larger version (110K): [in this window] [in a new window]   Fig. 8. Free-flight smoke visualization of Aeshna mixta in counterstroking flight flying with increasing left roll and yaw and consequent side-slip. There is a leading edge vortex near the midline (above the wing hinge), which exhibits spanwise flow running from the wingtip towards the centreline. (A–J) Consecutive images from a 250 Hz high-speed video recording. (A) shows the end of the upstroke, the dragonfly is aligned with the flow, with little roll or yaw, and the smoke streams form a vertical plane. In (B) the dragonfly begins the downstroke and a LEV is formed (yellow arrow), the dragonfly has also begun to roll and yaw to the left. In (C) the LEV grows, and the vertical plane of the smoke streams is distorted so that the centre of the LEV bulges towards the midline at the yellow arrow indicating a spanwise flow towards the midline. In (E–H) as the yaw increases and the LEV grows during the downstroke the bulge in the smokestreams caused by spanwise flow towards the midline also increases. The LEV is still present at the end of the downstroke in (I) and at the beginning of the upstroke in (J).

View larger version (106K): [in this window] [in a new window]   Fig. 9. Free-flight smoke visualization of the flow over the wings of Aeshna grandis flapping in-phase in level flight, but with a slight yaw to the left. The process of leading edge vortex formation is visualised, and the LEV has spanwise flow from the centreline towards the wingtip. (A–J) Consecutive images from a 250 Hz high-speed video recording. The leading edge vortex forms over the forewing in image sequence (A–C) at the start of the downstroke. Although the fore-wing moves upwards between images B and C, the wing rotates in a nose-down sense about an axis of rotation close to the mid-chord. This must cause a local increase in angle of attack at the leading edge, and the separation bubble that develops into the LEV forms during this phase of motion (yellow arrows). The smoke streams at the centre of the LEV are distorted in (D), bulging out towards the wingtip, which shows that there is a spanwise flow from centreline towards the wingtip – the opposite direction to that seen in Fig. 6. The bulge in the leading edge vortex is still present in (E), but decreases in (F) and is no longer apparent in (G–J), indicating that there is no longer a spanwise flow within the leading edge vortex as the wings approach the end of the downstroke and the LEV expands to cover both fore- and hindwings. The shear layer (secondary vortices?) within the leading edge vortex is apparent in (H–J), and the LEV has lifted off from the leading edge of the forewing in (J) as indicated by the presence of a smoke bifurcation at the point of the yellow arrow.

View larger version (45K): [in this window] [in a new window]   Fig. 6. Free-flight smoke visualization of the flow around the wings of Aeshna mixta executing a roll to the right in counterstroking flight. The flow field matches that which would be expected with conventional attached-flow aerodynamics. (A–H) Consecutive images from a 250 Hz high speed video recording. In (A–C) the wing is completing the upstroke and can be seen (blue arrows) to have sliced through the smoke streams like a knife – causing no vertical displacement. This suggests that the sections of the wing intercepting the smoke plane are generating little or no lift. The wing rotates in (C) and (D) at the beginning of the downstroke, and the flow exhibits a downwards deflection indicating lift-generation, but the smoke streams pass smoothly over the wing with no evidence of flow separation. The flow remains attached until the end of the downstroke in (H), as the dragonfly executes a roll to the right.

View larger version (48K): [in this window] [in a new window]   Fig. 5. Free-flight smoke visualization of the flow around the wings of Aeshna mixta executing a pitch-down manouver. (A–F) Consecutive images from a 250 Hz high-speed video recording. Rotation of the hindwing at the end of the downstroke causes a rapid increase in angle of attack, and initially attached flow over the hindwing separates to form a large leading edge vortex. In (A) the flow is still attached over the hindwing (yellow arrow), but in (B), as the wing rotates, increasing angle of attack, the flow separates (yellow arrow) forming a small separation bubble. This increases in size in (C), and in (D) the stagnation point where the separatrix touches down on the top surface of the hindwing is visualised (blue arrow). The LEV continues to grow as angle of attack increases in (E), and still has not been shed in (F), at the beginning of the upstroke. There is no evidence of any spanwise flow.

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).

View larger version (44K): [in this window] [in a new window]   Fig. 7. Free-flight smoke visualization of the flow around the wings of Sympetrum sanguineum accelerating vertically with the wings stroking in-phase. A leading edge vortex (yellow arrows) forms and grows to extend over both sets of wings. (A–H) Consecutive images from a 250 Hz high-speed video recording. The dragonfly is moving from left to right through the smoke plane and the smoke is approximately 1/4 wing-length in (A) and coincident with the wing hinge in (H). (A) The end of the upstroke. (B) During the forewing rotation prior to the downstroke, there is some evidence of the start of LEV formation. In (C) the LEV is already clearly formed (yellow arrow). In (D–F) the LEV rapidly grows, the smoke streams within the LEV are thinned by the increased velocities in that region making it darker, and the stagnation point where the separatrix touches down moves aft from the forewing onto the hindwing. In (F–H), as the downstroke ends and the wing rotates, the LEV is shed into the wake. There is a saddle-point (red arrows) in the wake where smoke-streams bifurcate in the shear layer between the current LEV and the wake-vortex representing the LEV shed from the previous wake. There is no evidence of any spanwise flow.

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.

View larger version (101K): [in this window] [in a new window]   Fig. 11. Smoke visualization of tethered dragonflies flapping, but not generating any lift. (A) The dragonfly Aeshna grandis is flapping, but the aerodynamic angle of attack is sufficiently close to zero to generate no lift – as evidenced by the lack of any vertical displacement of the near-wake (red arrows). The wake shows that the wings have swept a straight path during the downstroke. In (B) (Aeshna mixta) the wake again shows that the wings can maintain an angle of attack at, or close to, zero, even when the forewing sweeps a curved path on the upstroke.

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).

View larger version (48K): [in this window] [in a new window]   Fig. 10. (A–E) Smoke visualization of static tethered dragonflies. The dragonflies are still, and the images represent baseline data showing what the flow around the dragonflies looks like when they are not flapping. The successive images step from the right (far) wing hinge across the thorax and out along the near wing. In (A) the smoke plane is aligned with the far wing hinge, and smoke flows smoothly past the 5 mm diameter mount below the insect, becoming incorporated in the Karman street (red arrow) behind the mount far downstream. The flow over the thorax is attached back to the hinge of the hindwings, and then separates to form an unstructured wake behind the body. In (B) the smokeplane is on the midline, and the smoke hits the dragonfly between the eyes. Below the dragonfly the smoke is entrained into the Karman street (red arrow) behind the mount support. Smoke streams flowing over the top of the thorax are attached back to a point behind the forewing hinge, but then separate as the top surface of the thorax descends towards the abdomen. Flow above the thorax is essentially linear and undisturbed. Flow behind the thorax is separated forming an unorganised bluff-body wake. In (C) the smoke intersects the wing at 1/4 wing length. The wings are stationary, but a Karman street behind the wings (red arrow), and slight downwards deflection of the smoke-streams indicates that they are held at some small positive static angle of attack. The flow below the insect is disturbed by the Karman street behind the mount support at the far downstream end of the image. In (D) the smoke intersects the wings at 3/4 wing length. As in (C) the flow over the wings themselves is attached, but a Karman street (red arrow) behind the trailing edge shows that the wings are held at some small positive static angle of attack. The flow is otherwise apparently laminar. (E) Here smoke hits the wing near the wingtip. The flow pattern remains similar to that seen further inboard in C and D, with a trailing Karman vortex street (red arrow).

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/Re^1/2 (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 s^–1; 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. 12A–C, 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 Navier–Stokes 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. 12D–F,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