Vortex wakes of bird ﬂight: old theory, new data and future ......Much bird ﬂight research has...

Vortex wakes of bird flight: old theory, new data andfuture prospects

A. HedenströmDepartment of Theoretical Ecology, Lund University, Sweden.

Abstract

Flying birds leave a vortex wake. Fluid dynamic theory in the form of Helmholtz’ theorems dictatethe allowable topologies, and Kelvin’s circulation theorem requires that changes in wake circu-lation are directly proportional to force changes on the wing/aerofoil that generated the wake.Much bird flight research has therefore been focused on the properties of trailing wake vorticesbehind birds, since an accurate quantitative description of these will reveal also the aerodynamicsof bird wings. The first vortex theory of bird flight assumed the periodic shedding of discretevortex loops, each one generated during a downstroke, while the upstroke was considered aero-dynamically functionless. This view received some support from early visualization experimentsof take-off flight or very low speeds, while experiments at a higher speed (U = 7 m/s) in onespecies showed undulating wing-tip vortices of similar circulation on both down and upstroke.The necessary force asymmetry between downstroke and upstroke was obtained by wing flex-ing during the upstroke. Then followed an almost 20-year drought, with no further quantitativeexperiments, until recently when digital particle imaging velocimetry (DPIV) was successfullydeployed in a low-turbulence wind tunnel, and where the same small (30 g) bird could be studiedacross a large range of flight speeds (4–11 m/s). These new experiments revealed a much morecomplicated wake pattern than previous data suggested, mainly due to the improved experimentalresolution. The bird generated structures most closely resembling vortex loops at slow speeds,which gradually transformed into something similar to a constant circulation wake at the highestspeeds. However, the wakes were never as clean as the idealized cartoon models of the vortextheory of bird flight, and previous paradoxical results were shown to be attributable to the resultingdifficulty in accounting for all wake components. New DPIV data on other species indicate thatthese findings are quite general.

1 Introduction

In a classic experiment Magnan et al. [1] used tobacco smoke to visualize the vortex wake of apigeon Columba livia, which was found to consist of vortex loops (‘tourbillons’) in the slowly

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Vortex Wakes of Bird Flight 707

flying bird. Even though the published photographs are quite indistinct and difficult to interpret,this was the first demonstration that the vortex sheet rolls up in discrete structures associated withthe wing beat cycle. However, the first generation of quantitative aerodynamic models of birdflight used the actuator disc approach to calculate the induced drag [2, 3]. Bird flight typicallyencompasses Reynolds numbers (Re, for definition see below) in the range 8000–200,000 [4],which from an aerodynamic point of view is a problematic range because of the transition fromlaminar to turbulent boundary layers, or even laminar flow separation and laminar reattachment[5]. In this region of Re there is an abrupt increase of drag due to this transition in the boundarylayer followed by a decrease from this higher drag with further increase of Re, which makes anyattempts of quantitative analysis complicated. Research on bird flight has a long tradition, not leastbecause the aerodynamic models are of great potential practical benefit to ecologists who want tounderstand the strategies and constraints on migration performance in wild birds. Since the 1960s,when the first comprehensive flight mechanical model was developed, the field has seen a steadyflow of theoretical and empirical advances. This paper summarizes some key developments withspecial emphasis on vortex wake models and experimental data from real wakes in birds.

2 Some definitions

For comparative but also purely aerodynamic purposes we will have reason to refer to some oftenused indices of the flow regime. First, the Reynolds number is a dynamic similarity measure anddefined as

Re = Uc

ν,

where U is the flight speed in relation to the fluid at rest, c is a characteristic length in the directionof flow and usually taken as the mean chord, and ν is the kinematic viscosity. The Reynolds numbercan be interpreted as the ratio between inertial and viscous forces.

In oscillatory flows the ratio of two time scales, the time required for a fluid particle to passover the mean chord, tc = c/U , and the time taken for one kinematic cycle, T = 1/f , is commonlyexpressed as the reduced frequency

k = ωc

2U.

The value of k expresses the relative importance of unsteady terms with k ≈ 0.1 implying thatunsteady effects most often can be ignored, while k of order 1 indicates that unsteady phenomenaare likely to occur. When comparing two situations such as different-sized animals moving indifferent fluids, the similarity of Re and reduced frequency guarantees that the flow regime andhence aerodynamic properties will be the same. A closely related reduced frequency based on thewing semi-span, b, is

�= ωb

U.

The two measures of reduced frequency are related through the aspect ratio (AR = 2b/c) as�= k · AR. Alternatively, a measure of the flapping velocity of the wing tip to the forwardvelocity is given by

K = 2φbf

U,

where φ is the angular stroke amplitude and f is the flapping frequency. The advance ratio,J = K−1, is the ratio of forward flight velocity to the wing-tip velocity. K is closely related to the



708 Flow Phenomena in Nature

Strouhal number defined as

St = fA

U,

where A is the double amplitude of the wake vortices in a Kármán (drag wake) or a reverse Kármánwake (thrust wake), but in the absence of wake information the double amplitude of the wing (orfin) tip is usually taken.

3 Vortex theories of bird flight

Kelvin’s circulation theorem states that, in a homogeneous, incompressible and inviscid fluid, thecirculation around a closed circuit will have the same value when measured over the same fluidelements and circuit at any time as the circuit is followed in the flow. This theorem will prove veryuseful to our applications on bird flight, because the aerodynamic properties of the wake vorticescan be directly linked to the time-averaged aerodynamic forces on the wings having generatedthe vortices. The circuit of Kelvin’s circulation theorem just has to enclose the aerofoil and thespace behind it where wake vortices will appear. In steady flight at speed U , the Kutta–Joukowskitheorem gives a relation between the lift, L, and the circulation, �, as

L = ρ2bU�, (1)

where ρ is air density and b is wing semi-span. It follows that if there is a change in lift developedby the aerofoil also the bound circulation changes, which must be offset by an equal and oppositecirculation in the wake. This is all there is in terms of background to develop vortex basedtheories of flight. The modeller only has to specify a more or less realistic geometry of the wakevortices and enforcing a force balance between weight, drag and the force associated with thewake momentum.

3.1 The actuator disc and momentum jet

In its simplest form the bird is replaced by an actuator disc of radius b that magically deflectsthe oncoming airflow downwards (Fig. 1). At an extreme end of the spectrum of vortex wakebased flight models, the momentum jet qualifies as belonging to this family, where the vorticityis confined to an infinitely thin, cylindrical sheet enclosing the uniform jet [6]. Notice howeverthat, in the wake description, nothing remains of the time varying forces developed by cyclicallybeating wings. The mass flow through the actuator disc induces a downward velocity, wi, when itpasses the disc, reaching a final speed of 2wi in the far wake with a jet diameter of b. The rate ofvertical momentum flux required to support the weight at some forward speed U determines theinduced velocity wi = mg/(2SdUρ), where m is body mass, g is acceleration due to gravity, andSd(=πb2) is the wing disk area. This in turn gives the induced power as Pi = mgwi. For a completeP(U )-relationship, commonly denoted the ‘power curve’ the terms of parasite and profile powerare added to Pi, resulting in the most popular and widely used flight mechanical model of birdflight [3, 5, 7, 8].

3.2 Vortex ring theory

A step towards increased reality was taken by Rayner [9–11] who developed an aerodynamicmodel of flapping flight in which the wake vortices were represented as circular or elliptic loops,each one shed as the result of a downstroke. It was assumed that the upstroke was aerodynamically




unloaded and therefore did not leave any traces in the wake. The size, geometry and orientation ofthe vortex loops are determined by the wingspan, wingbeat frequency, stroke amplitude, forwardspeed and by the circulation distribution along the wing, �( y). The circulation of the vortexrings is determined by imposing the force balance condition, i.e. that the rate of wake momentummust balance the vector sum of weight and aerodynamic drag (Fig. 2). Notice that the angle, ψ,by which the rings are tilted with respect to the horizontal determines the lift to drag ratio asL/D = cot ψ. The mean rate of increase of kinetic energy deposited in the rings is the inducedpower. The periodicity in the wake now corresponds to the wing beat periodicity, with a closeconnection between the aerodynamic force time history and the wake trace. In the first generationof this flight model it was assumed that discrete vortex loops are shed at all speeds, which laterhad to be modified as experimental data refuted this assumption (see below).

Figure 1: Actuator-disc model for induced power of a flying bird. The bird is represented by acircular disc cross section (Sd) with wingspan as diameter, where the oncoming flow ofspeed U is deflected downwards by the induced speed ui so that the downward impartedmomentum balances the weight. Also shown is the coordinate system (x, y, z) usedthroughout this paper.

Figure 2: Vortex ring wake of flapping bird flight. When the upstroke is aerodynamically inactivethe downstroke generates a vortex loop. The associated impulse (I = ρSe�, where ρ isair density, Se is the planar area of the vortex loop and � is the circulation of the loop)and hence aerodynamic force are normal to the surface area of the loop, itself tilted inrelation to the horizontal by an angle ψ.




3.3 The constant circulation wake and other relatives

An experiment by Spedding [12] showed that the wake did not consist of discrete vortex loops ata moderate cruising speed (U = 7 m s−1) in a kestrel (Falco tinnunculus). Instead, the wake wascharacterized by a pair of wing-tip vortices of near constant circulation throughout the wing beat,without any noticeable shedding of transverse vortices. The necessary asymmetry (in order toachieve non-zero thrust) between down- and upstroke is achieved in many birds by the flexing ofthe wrist-joint, causing a reduced span during the upstroke and thereby a reduced projected wakearea in relation to that from the downstroke [12]. Because the bound circulation is constant, �0,there is no shedding of transverse vortices and so the main wake structures are the wing-tip vorticesof constant circulation (hence we shall denote this wake as the cc-wake, indicating constantcirculation). The generation of vortices is associated with an energy cost since the energy contentin the wake is lost. Therefore, with no or minimal transverse vortices, the cc-wake could be arguedto minimize the mechanical cost of cruising flight and should be a favourable configuration inbird flight where minimizing energy cost is advantageous, such as during long-distance migrationor commuting between nest and foraging sites.

The cc-wake can be understood as a deformed version of the trailing vortices left behindby a fixed wing aircraft. The wake consists of two straight wing tip vortices in gliding flight,and in cruising flapping flight, a shallow wing beat makes these vortices follow an undulatorytrack, both vertically and horizontally as they trace the 3D movement of the wing tips (Fig. 3).The constant circulation and the simple geometry of this wake make an analytical treatment bothstraightforward and elegant [6]. The impulses associated with down- and upstroke wake elements,respectively, are

Id = ρ�ccAd and Iu = ρ�ccAu, (2)

where �cc is the circulation of the trailing wing tip vortices, and Ad and Au are the wake areascircumscribed by the tip-vortices during downstroke and upstroke, respectively. The associatedforces are equal to the time rate of generation of wake momentum (=impulse), F = d (mv)/dt,which, integrated over one wing beat period, gives the time averaged lift and drag as

Figure 3: Trailing vortices at cruising flight. The wake consists of trailing wing-tip vorticesof constant circulation. The overall wingbeat wavelength (λ) depends on the wingbeat frequency, while the wake angles may differ between downstroke and upstroke(ψd, ψu).




L̄ = 1

Tρ�0 (Ad cosψd + Au cosψu) , (3)

D̄ = 1

Tρ�0 (Ad sinψd − Au sinψu) , (4)

where ψd and ψu are the titling angles that the down- and upstroke make with the horizontal,respectively (Fig. 3; [6, 12]). With a symmetric wing beat, ψd =ψu, and with Au/Ad = ζ(ζ≤ 1),L/D can be written as

L/D =(

1 + ζ1 − ζ

)cotψ. (5)

Pennycuick [13] simplified this analysis further by assuming constant spans during down- andupstroke, i.e. that the flexing of the wings before the upstroke takes place momentarily at the down-/upstroke transition and likewise that the extension of the wing takes place at the up-/downstroketransition. Then the L/D depends on the span ratio (b′ = bu/bd) as

L/D =(

1 + b′

1 − b′

)cotψ. (6)

It follows that L/D is maximized by a high ζ or b′, i.e. from a small upstroke span reduction anda shallow wing beat amplitude.

In a closely related model, Rayner [14] incorporated the cc-wake in a quasi-steady lifting lineanalysis of flapping flight at cruising speeds. The usual elliptic wing loading was replaced by analternative due to Jones [15] that gives slightly improved aerodynamic efficiency. A problem nowpresented itself by the fact that two models with quite different presumed wake geometries, thevortex ring model and the model based on the cc-wake, were used to represent flapping flight.Rayner’s [14] prescription was the postulate that there was a sudden transition from vortex ringsto the cc-wake at some intermediate flight speed, and the notion of gaits was introduced to theflight literature. The gait analogue to terrestrial locomotion, where the transition between differentpatterns of limb movement and ground contact was very sudden at predictable Froude numbers,was based on the existence of the two fundamentally different wake forms and the unimaginabletopology of intermediates.

In order to account for the unsteadiness of flapping flight and the influence of the wake on theinduced flow near the airfoil, Phlips et al. [16] modelled the wake as a lifting line representingthe current half-stroke, but previous vortex lines were collected as streamwise wing-tip vorticesand with transverse vortices shed at the turn-points of each half-stroke to account for changes inthe bound circulation. The validity of their analysis was restricted to reduced frequencies k < 1and wing beat amplitude φ≤ 60◦, where significant departures from the linear lift slope c1(α)occur due to unsteady phenomena. The far wake of Phlips et al. [16] is similar to the ‘ladderwake’ postulated by Pennycuick [17] to apply when birds have rigid wings that cannot be flexedat the wrist joint during the upstroke. Examples of birds where a ladder wake could exist arehummingbirds and swifts, where the force asymmetry between down- and upstroke has to beachieved by variation in bound circulation, and hence the shedding of transverse vortices, ratherthan by reduced wingspan and maintained circulation as in the cc-wake. Yet other more elaboratetheoretical wake configurations have been treated in the literature [18, 19], but it is beyondthe scope of this paper to go much beyond this point of model complexity for flapping flight.




For further accounts of aerodynamic models of flapping flight the reader is recommended theexcellent review by Spedding [6].

4 Bird wakes in reality

The development of aerodynamic models has converged with that of experimental results on thegeometry and properties of real wakes. However, the vortex ring theory of Rayner [9] was devel-oped independently from experiments carried out simultaneously by Kokshaysky [20]. Thereafterthe aerodynamic modelling has been tightly connected to empirical wake data. We now proceedby reviewing some classic experimental work on bird wakes.

4.1 Take-off flight

Kokshaysky [20, 21] recorded the wakes during short take-off flights in two finches, the chaffinch(Fringilla coelebs) and the brambling (F. montifringilla), by using paper and wood dust as tracerparticles combined with multiple flashes photography. The two closely related species are mor-phologically very similar (Table 1) with reduced frequency during the experiments being k = 0.87for the chaffinch and k = 1.29 for the brambling. Both species generated discrete vortex rings asso-ciated with downstroke. The results came as a timely support for the assumptions of the vortextheory of bird flight by Rayner [11], although no quantitative information regarding vorticity andcirculation was available.

4.2 Slow forward flight

The next major experimental development was the application of the particle image veloci-metry (PIV) method in which a cloud of buoyant helium filled soap-bubbles was generated anda bird was trained to fly through this cloud. The 3D movements of the bubbles were recordedby stereophotogrammetry. This methodological breakthrough allowed, for the first time, quanti-tative measurements of bird wakes to be made. The first species to be evaluated was the pigeon(Columba livia) in slow flight, U = 2.4 m s−1 [22], in which the wake consisted of discrete vortexloops. However, the loops appeared asymmetric in the sense that the start end was rather concen-trated while the stop vortex core had a larger diameter and was spatially less well defined. Quitesurprisingly, and somewhat disconcertingly, the vortex loops contained approximately 1/2 of themomentum required to support the weight of the bird, which signalled that not all vorticity wasconfined to the vortex core of the observed rings. In a second experiment of a jackdaw (Corvusmonedula), the results from the pigeon were repeated, i.e. a significant wake momentum deficitwas obtained with only 1/3 of that being required confined to the main vortex core [23]. This resultwas obtained by integrating centreline velocities through the wake, thus including, supposedly,flow induced by vorticity not restricted to the compact vortex cores. The shortfall neverthelessgreatly exceeded the calculated experimental uncertainty. Some selected wake properties fromthese experiments are shown in Table 1.

Discrete vortex loops have been reported also in some additional species in slow forward flight[24, 25], but no quantitative data except approximate flight speed are available (Table 1). Discretevortex loops were also found in two species of bat in slow flight [26]. Taken together birds andbats in slow flight (U ≤ 3 m s−1) shed discrete vortex loops generated by the downstroke withthe upstroke being more or less aerodynamically unloaded, although the detailed geometry differ



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from an idealized ring/ellipse of equal core radius and vorticity at all stations. The two quantitativemeasurements of ring momentum suggested a significant deficit of forces to achieve force balance.

4.3 Cruising flight

Using the same set-up as for the pigeon and jackdaw in slow flight, Spedding [12] obtainedwake images of a kestrel flying at a moderate cruising speed, U = 7 m s−1. The wake appeareddramatically different from that of slow speed by the lack of any detectable transverse vortices.The main wake features were a pair of streamwise vortices shed from near the wing tips, showingan undulatory trace tracking the path of the wing tips. The wake elements associated with down-and upstroke measured similar circulation, hence this wake is usually referred to as the ‘constantcirculation’ wake. The lift is given by eqn (3), which was 2.15 N for the kestrel to be comparedwith the weight of the bird of 2.06 N (Table 1). Hence, at cruising speed the inferred wake topologyand measured properties (Au, Ad,�) satisfied the force balance criterion. As explained above thesefindings prompted the amendment of the vortex theory of forward flapping flight to account forthe appropriate wake geometry in cruising flight [12, 14].

4.4 Gliding flight

At equilibrium gliding, flight potential energy is converted into work against the aerodynamicdrag. The bird itself does not perform any work since the wings are not flapped. Gliding flight ishowever not effortless since by holding the wings in an outstretched position the flight musclesproduce static muscle work that consumes chemical energy at a rate approximately two timesthe basal metabolic rate [27]. The wake in gliding flight observed in a kestrel at U = 7 m s−1,consists of two straight streamwise wing-tip vortices [28]. The measured circulation matched theforce balance criterion, indicating that all vorticity is accounted for in the main vortex structuresobserved. A Harris’ hawk (Parabuteo unicinctus) gliding in a wind tunnel also showed wing-tipvortices, which were shown to be influenced by the primary feather configuration [29].

4.5 Conclusion and speculation

The combined basis of bird wakes available until year 2003 are those studies referred to inthis section (Table 1). In flapping flight quantitative wake data were available from two speeds,U = 2.5 m s−1 (pigeon, jackdaw) and U = 7 m s−1 (kestrel). Additional data, albeit qualitative,from a few other species suggested the presence of vortex loops at slow speeds (U ≤ 3 m s−1).First, the paradoxical wake momentum deficit could not be satisfactorily explained, althoughRayner [24] suggested that the wake deficit could be due to that the birds actually deceleratedduring the experiments. However, this explanation appeared to be false [30]. Second, the apparentexistence of topologically two distinct wake types, the discrete loop and the cc-wake, has led someauthors to introduce the notion of ‘gaits’ [14, 24, 25], in analogy to the discrete gaits as found inquadrupeds. In terrestrial locomotion the transition between gaits, such as canter, trot, gallop, isvery abrupt and occur quickly at some predictable Froude number. Rayner [24, 25, 31] speculatedthat the transition between the wake types in bird flight could happen from one wing beat toanother and that there could not be any intermediate forms, hence the gait-transition analogy.It should be remembered that there were no wake data from any bird at more than one speed.Over time the notion of gaits gained tacit acceptance [32–34], even though researchers had littlesuccess when looking for indirect signs of a transition between wake types by observing variouswing beat kinematic parameters across flight speeds [35–37]. Most kinematic measures actually




showed smooth curves in relation speed without any discontinuities as expected at a sudden gaittransition.

5 Bird wakes in reality: digital particle image velocimetry (DPIV)

Towards the end of the previous millennium it seemed that what was lacking was new data.Thanks to the technological developments in flow visualization techniques by accurate, high-resolution CCD-array cameras, the development of efficient analysis, such as correlational imagevelocimetry (CIV) routines, and not least the introduction of a new low-turbulence wind tunneldedicated for bird flight research, the time was right to attempt new experiments.

5.1 The wind tunnel

A basic requirement for repeatable wind tunnel experiments at low Re is that the backgroundflow is non-turbulent as this is the reference against which to reduce any effects caused by theobject [38]. At Lund University a wind tunnel dedicated for bird flight was designed and hasbeen operational since 1994 [39]. The tunnel is a Göttingen type with recirculating flow anda contraction ratio of 12.25:1 between the settling chamber and test section cross-areas. Theoctagonal test section is 1.2 m in diameter with a 1.5 m long closed part followed by a 0.5 m gapbetween the end of the closed test section and the bell mouth of the first diffuser. This openingis a very important feature of the design as it readily allows access to the experimental subjects,typically live birds. To enable climbing and descending flight the entire tunnel is tiltable arounda pivot. A survey using a hot wire anemometer gave a turbulence level of ≤ 0.05%, measured asRMS at U = 10 m s−1. The low background turbulence of the wind tunnel is a prerequisite forrepeatable DPIV. In the tunnel the spatial directions (x, y, z) refer to the streamwise, spanwise andvertical direction and their associated speeds are defined as (u, v, w). A more detailed account oftunnel design and technical data can be found in Pennycuick et al. [39].

5.2 DPIV for birds (BPIV)

The use of DPIV has become a widely used measurement technique of fluid flow [40], and a customdesigned DPIV set-up has been deployed and applied to bird flight in the Lund wind tunnel. Theincentive was the imbalance in the literature between quantitative data and speculation, with datafrom only two points on the speed axis representing different species and showing fundamentallydifferent wake geometries. Hence, there was a clear gap to fill in order to address the long standingmomentum deficit paradox at slow speed and the possible change of the wake across a wider rangeof speeds. The DPIV technique relies on pairs of digital images from which the displacement ofidentifiable particle patterns are determined. DPIV can be applied to numerous biofluid problemsand it has been used quite extensively to study swimming [41]. Previous applications to animalflight have been restricted to low Re using mechanical flapping wings [42, 43] and recently alsoto tethered insects [44]. In our wind tunnel a 200 mJ dual-head pulsed Nd:YAG laser was used,but it appeared that about 100 mJ/pulse was sufficient to yield reflection enough from the 1 µmfog particles used as seeding. The recirculation design of the wind tunnel made it ideal for DPIV,since it allows the entire tunnel to be filled with a homogenous thin fog. The bird is trained tofly steadily in the front half of the test section, while the laser light sheet was approximately 18c(c is wing chord) downstream from the bird (Fig. 4). An array of infrared LED-photodiodes wasarranged so that when interrupted by the bird, the laser pulses were suspended to prevent the bird




Figure 4: Experimental set-up in the Lund wind tunnel to record wake vortices. The bird (tn) isflying in the front of the test section at speed U , separated from the laser light sheet(coming from the laser, pl) gaited (gb) by the summed output from an array of LED-photodiode pairs that would suspend the laser light if interrupted by the bird. Laserpulses and cameras (tm1, 2) are synchronized by two delay generators (dg1, 2), andcamera output is read into imaging cards in a PC (ic1, 2) (based on [30, 45]).

from direct contact with the laser light. The laser pulses are synchronized with the cameras bytwo pulse delay generators (Stanford Instruments DG535). In the first set-up the light sheet wasvertically aligned with the flow, and hence the images are streamwise slices of the wake. A 3Dreconstruction of the wake topology could be obtained by sampling the wake at different stationsalong the wingspan, as recorded by an independent video camera positioned far downstream fromthe test section (inside the first diffuser). The system was tuned as to give minimum measurementerror on the wake disturbance quantities (such as u and w, rather than the mean free stream)and the final ‘add bird’ experiments were conducted with an estimated uncertainty of <0.5% indisturbance flow fields (u and w), and ±10% for gradient quantities such as ωy (for definition seeeqn [7]). A full account of the experimental set-up, procedure and detailed error analysis is givenby Spedding et al. [45].

5.3 DPIV of a thrush nightingale

In the first series of experiments a thrush nightingale (Luscinia luscinia) was used as a represen-tative of the species-rich passerine family, being a rather typical long-distance migrant of small




size with low aspect ratio wings (lean mass 20 g, AR = 5.4). Results from such a species shouldtherefore be taken as sufficiently general to apply to many species of similar size and morphology.The bird was trained, by using a movable perch in the test section, to fly steadily in a repeatablemanner approximately 0.9 m upstream from the image centre of its wake. Sequences of 10 imagepairs at 10 Hz (determined by the repetition rate of the laser) were taken at integer speeds in therange 4–11 m s−1 and at different locations along the wingspan. A total of 4000 wake images weresorted according to speed and wing position (body, arm, hand wing, and wing tip) and analysed.The following account is based on original data published in Spedding et al. [30]. It should beremembered that the entire wake consists of the inviscid induced (lift-dependent) drag as well asthe pressure and viscous drag components from the body and wings. The possibility to observethem in the wake flow depends on whether they remain physically separated, and hence can bedistinguished from each other, or not. Otherwise, if they are intermixed the wake will representthe sum of thrust and drag components, which will be exactly zero in steady flight at constantspeed [30].

5.3.1 Wake topologyAt low speeds (Re = 13,000 at U = 4 m s−1) the wake showed characteristic start vorticesgenerated at the beginning of downstrokes and corresponding stop vortices that were spatiallymore spread out and with lower peak vorticity. The core-to-loop radius was 0.11, which should beconsidered as a small-cored vortex loop [46]. Some vorticity is shed during the upstroke also atslow speeds, but for the most part, the upstroke may be inferred to be more or less aerodynamicallyinactive. A notable feature of the slow speed wake (U ≤ 5 m s−1) was that the region dominatedby the stop vortex (clockwise or negative vorticity) also contained embedded patches of positive(counter clockwise) vorticity so that there appeared to be a mosaic of alternate vorticity. Thismixture of opposing vortices may be the reason for the larger extension of the stop vortex, i.e.larger core radius, caused by vortex interaction. The wake was reconstructed by combining con-secutive frames to generate a composite covering a whole wing beat period (Fig. 5). The 2Dflow/vorticity maps are roughly consistent with a wake consisting of discrete vortex loops, eachshed as the result of a downstroke but with different radii between the start and stop ends. Inparticular, one must take note that if the circulation is not balanced between positive (start) andnegative (stop) vortices, then the true wake structure (when measured at this downstream location)must be more intricate than this. The loops induce a downwash normal to the surface plane of theloop (Fig. 5).

When increasing the flight speed the wakes exhibit increasing vorticity originating fromthe upstroke, suggesting an increasing aerodynamic significance. At the maximum speed,U = 11 m s−1, there was a more or less continuous vortex trail throughout the wing beat ofsimilar strength. Cartoon reconstructions of wake topology at three different speeds are shownin Fig. 6. The reconstruction in Fig. 6 is based on wake images at different positions along thewingspan, which together could be used to deduce the 3D geometry. Even though the high-speedwake appears quite dissimilar from the other two, it represents one end of a continuous spectrumwhere the discrete loops are at the other extreme. As suggested by quantitative properties (seebelow), there is a smooth transition of wake topology mainly caused by the increasing aerody-namic function of the upstroke [30].

At cruising speeds (U = 9−11 m s−1; Re = 35,000) the main features are undulating wingtip vortices that are interconnected by cross-stream vortices of alternate sign. In spite of itssuperficial similarities, this wake should not be considered identical to the ladder wake postu-lated by Pennycuick [17]. In the ladder-wake the ‘rungs’ of the ladder are cross-stream vorticesshed at the transitions between down/up and up/down strokes due to changed bound circulation




Figure 5: Cross-section of wake vortices from a thrush nightingale. Reconstruction of slightlymore than one wing-beat vortex wake at slow speed, U = 4 m s−1. The composite isconstructed from a sequence of consecutive wake images obtained in an image planealigned with the (x, z)-directions at the mid-span position (centre plane). The wakewavelength is λ= UT , where T is the wingbeat period, and the downstroke length (λd)and upstroke length (λu) are marked with wingspan (2b = 26 cm) as reference length.The colour bar is scaled asymmetrically aboutωy = 0 with numbers at the ends showingvalues in units of s−1 (from [30]).

Figure 6: Cartoon interpretation of wake geometry in a thrush nightingale at different speeds. Thepanels represent slow (top), medium (middle) and cruising (bottom) speeds (from [30]).




(and associated lift). The cross-stream vortices observed here represent a continuous sheddingof vorticity during each half-stroke, but the origin of this vortex shedding remains unclear. Thiswake therefore mostly resembles the constant circulation wake found in the kestrel at U = 7 m s−1

[12]. It may be that also the kestrel wake has these low-amplitude cross-stream vortices but thatthey were not detected by the helium bubble method.

5.3.2 Quantitative properties of the wakeThe spanwise (or cross-stream) component of vorticity is defined and calculated as

ωy = ∂w∂x

− ∂u

∂z, (7)

where u (in x-direction) and w (in z-direction) are velocity components. The circulation measuresthe strength of a vortex and calculated as the vorticity integrated over a surface, S, in the imagingplane as

�=∫ω · dS. (8)

In practice the circulation of eqn (8) was calculated by making a discrete approximation bythe sum of all contiguous cells in a local neighbourhood around a vortex cross-section, where|ω|y exceeds a threshold value. The below-threshold vorticity was then estimated by assuminga Gaussian distribution of vorticity around its peak value. Measures of vorticity and circulationwere carried out for the different wing locations and plotted against flight speed, for the 4000 wakevelocity fields [30]. Peak vorticity and circulation of main start and stop structures are shown inFig. 7, where both quantities have been normalized as |ω|maxc/U and �/Uc. The normalizedvorticity expresses the ratio of timescales for convection over the wing and rotation around avortex, while the normalized circulation expresses the integrated magnitude of shed circulationcompared with a measure of the momentum flux over the wing chord. Both quantities showmaximum values at the lowest speed (U = 4 m s−1), and they decline to minimum values atU = 11 m s−1. There is a notable difference between start and stop vorticity and circulation atslow speeds, but with increasing speed the difference decreases and from U = 6−7 m s−1 it hasdisappeared (Fig. 7).

5.3.3 Force balanceWhether the measured circulation is sufficient for balancing the forces required to fly dependscritically on the appropriate interpretation of the wake geometry that determines the correct areaappearing in eqn (2). It is convenient to define two reference values of circulation required tosupport the weight [30]. First, if the wake were to consist of two straight wing-tip vortices theKutta–Joukowski theorem [eqn (1)] gives the lift that must balance the weight, W , if we neglect3D and wing tip effects. Hence, rearranging eqn (1) yields the circulation required to support theweight for a gliding wake as

�0 = mg/(ρ2bU ). (9)

If instead the wake appeared in the form of discrete vortex loops with projected area onto thehorizontal plane Se = πb(λd/2), where λd(=UTτ and T is the wingbeat period and τ is thedownstroke ratio) is the downstroke wave length, then the associated circulation required tobalance the weight is

�1 = mgT/ρSe. (10)




Figure 7: Quantitative wake properties in relation to speed for a thrush nightingale. Variation instandardized peak vorticity |ω|max(c/U ), where C is wing chord and U is forward speed(a) and standardized total measured circulation �/Uc (b). Measures from starting (filledcircles) and stopping vortices (open circles) are shown in relation to flight speed. Errorbars represent standard deviations (from [30]).

Although neither of the two idealized geometries apply directly to the nightingale wake, they maystill be useful as reference quantities against which measured circulation can be compared.

The start vortices contain more concentrated circulation than the stop vortices (Fig. 8), quan-titatively illustrating the feature of Fig. 5 where the stop vortex is spatially more distributed andapparently weaker than the start vortices. However, when comparing the total integrated circula-tion associated with positive (start) and negative (stop) vortices with the reference values �0 and�1 it appears that only structures with uniform strength equal to that measured in the stop vorticeswould contain circulation enough to support the weight, while the well defined start vorticesexhibit an approximate 50% momentum deficit. This finding therefore repeats the original wakemomentum deficit of the pigeon and jackdaw [22, 23].

As previously remarked there were patches of positive vorticity in the region of down/upstroketransition otherwise dominated by the negative stopping vorticity. Spedding et al. [30] made afurther calculation, supposing that this additional positive vorticity has the same source as thatof the main start vortex and hence that the total circulation contributing to the total aerodynamicforce is given by the sum of all vorticity of the same sign (+, in the case of the start vortex withadditional low-amplitude vorticity from the area dominated by the stop vortex). After accounting




Figure 8: Circulation in relation to speed in a thrush nightingale. Total integrated circulation �totfrom all positive (filled circles) and negative (open circles) vorticity from the wake inrelation to flight speed (U ). (a) The fraction of the total circulation that is not containedin the strongest vortex cross-section is higher in the stopping vortices than the startingvortices. (b) The total integrated circulation in relation to the reference circulation �1(see eqn (10) for definition). At speeds ≤8 m s−1 the total negative vorticity wouldbe sufficient for weight support, but not the positive component. Error bars representstandard deviations (from ref. [30]).

for all + signed vorticity in this way it appeared that weight balance was achieved (Fig. 9). Inthat case the revised geometry of the vortex loops could account for weight support since forcebalance was achieved, and the wake momentum paradox was resolved [30].

5.4 DPIV of a robin

Another similar set of data as for the thrush nightingale has since been collected from two Europeanrobins (Erithacus rubecula) using the same procedure and camera set-up shown in Fig. 4 [47]. Therobin is a close relative to the thrush nightingale but migrates a shorter distance within Europe andhas a lower body mass (during experiments m = 0.017 g), and a shorter wingspan and lower aspectratio than the nightingale (2b = 0.22 m, AR = 4.7). At U = 4 m s−1 Re = 13,000 and k = 0.55 andat U = 9 m s−1 Re = 29,000 and k = 0.24.




Figure 9: Total circulation in relation to speed. The total integrated circulation with the positivevorticity (�+, filled squares) also including all above threshold traces of positive vortic-ity found in the neighbourhoods of the predominantly negative vorticity associated withthe transition between down and upstrokes. Both positive and negative vorticity mea-sured in this way are sufficient for weight support at slow speeds. Error bars representstandard deviations (from ref. [30]).

The wake topology showed striking similarities with that of the thrush nightingale, with char-acteristic discrete loops at slow speeds. An example of a cross-section of vortex loop associatedwith a downstroke at U = 6 m s−1 is shown in Fig. 10. Also the robin exhibited the asymmetrybetween start and stop vortices, typified by a more diffuse and spatially spread out stop vortexthan the start vortex, even if the illustrated example shows quite a clean vortex cores of bothends of the loop. When increasing the speed the wake undergoes similar gradual changes as inthe nightingale, characterized by increasing vorticity shed from the upstroke. By U = 9 m s−1

the same magnitude vorticity is shed throughout the down and upstrokes, in agreement with thecartoon reconstruction of Fig. 6 (the streamwise thick vortices in the lower panel, Fig. 6). Thepattern was apparently the same in the two robins investigated.

Figure 11 displays representative velocity profiles through the start and stop vortices of Fig. 10,as the u(z) and w(x) components in relation to the peak vorticity of each vortex at coordinates(x0, y0). The vortex core diameter measured by the distance between velocity peaks is about 2.3 and2.8 cm for the start-and-stop vortices, respectively, while the overall streamwise loop diameteris about 20 cm at this speed. This latter estimate comes from a hypothetical loop with simplegeometry generated during the downstroke at U = 6 m s−1, and wing beat period T = 0.068 s anda downstroke fraction of the wingbeat period τ≈ 0.5 (as calculated from companion kinematicmeasurements). For a cross-stream diameter close to the wingspan (22 cm) this would imply aspan efficiency of 0.91 [25]. The ratio R0/R = 0.11 for the start vortex and R0/R = 0.14 for thestopping vortex are similar to previously measured values for slow flight in the thrush nightingale[0.1; 30], while the values measured in a pigeon (0.17; 22] and a jackdaw (0.14; 23] are somewhatlarger (Table 1). However, at U = 5 m s−1 R0/R = 0.16 for a start vortex in the robin (Table 1),




Figure 10: Cross-section of a vortex loop from a robin. The starting (rightmost) vortex is slightlymore concentrated than the stopping (leftmost) vortex. The bird is flying to the leftat U = 6 m s−1. The 14-step symmetrical colour bar shows negative and positivevorticity (s−1).

Figure 11: Profiles of the velocity components of wake vortices in a robin. u(z) (a, c) and w(x)(b, d) for the staring vortex (a, b) and stopping vortex (c, d) shown in Fig. 10. (x0, z0)is the location of the peak in vorticity ωy. The slightly more diffuse stopping vortex isseen as a larger diameter (2.3 and 2.8 cm, respectively).




showing that this parameter may vary rather little and that all hitherto measured birds actuallyshow similar relative dimensions of vortex loops.

The quantitative wake properties �/Uc and ωmaxc/U also showed the same pattern with speedas the thrush nightingale, i.e. they both declined monotonically from maximum values at theslowest speed (Table 1). The normalized circulation showed similar values between the twospecies, while the vorticity was more than twice as large in the nightingale compared with therobin at the slowest speed but the values converged towards the maximum speed (Table 1).The comparison of the measured circulation associated with concentrated vortices against thereference values �0 and �1 revealed a nominal wake momentum deficit also in the robin, ofthe same magnitude around 50% as in previously investigated species at slow speeds (Table 1).However, a similar accounting for positive vorticity embedded with the opposite-signed stopvortex as for the thrush nightingale resulted in enough total positive circulation associated withthe downstroke to claim weight support at slow speeds for the robin experiment as well [47].

5.5 Wakes and kinematics

The wake vortices described and analysed in the previous sections were recorded approxi-mately 0.9 m downstream from the position of the bird, which represents about 3 wing beats atU = 4 m s−1 and 1.3 wing beats at U = 10 m s−1, which allow the wake to evolve and change sincethe time of shedding off the wings. At slow speeds in particular, when vorticity is at maximum,the start end of vortex loops will be particularly prone to move due to self-induced convection,because the oldest portion of the vortex loop (the start end) is convected downwards due to itsown vorticity and the bound vorticity on the wing [10]. Therefore, when we measure the spatialcoordinates of wake elements and their relative position, for instance as the wake inclinationangles (ψd,ψu), the measured value ofψd would be lower than when the vortex was created. Anysuch effects due to self-induced convection will likely be reduced at higher flight speeds wherepeak vorticity and time since shedding are reduced (Table 1), but the magnitude of the effects andwhether they can be ignored remains to be investigated. Some basic wing beat kinematic param-eters that should correlate with wake geometry were analysed for the same thrush nightingale asused in the wake analysis [48].

The kinematic parameters investigated were wingbeat amplitude (A0), wing beat frequency ( f ),downstroke fraction (τ) and span ratio (b′ = bu/bd). Notice that if b′ = 1 the span of the downstrokeand upstroke is the same and a net thrust must be obtained from differential circulation betweendownstroke and upstroke. However, in the investigated species b′ was always <1. Notably, bothamplitude and wing-beat frequency changed very little with speed. The wing beat frequencyshowed a very weak U-shaped function of speed, which was also found in a previous study ofthe same species in the same wind tunnel [49], but the variation in f was only 7% between theextreme values [48]. Therefore, changes in measures of reduced frequency (k, �, K), dependmainly on U and hence display essentially linear functions of U .

The wake geometric properties were estimated in two ways: (i) from cross-stream vorticitymaps such as Fig. 5, the inclination angles were measured between centres of identifiable vortexblobs (ψwake); (ii) the induced downwash should be normal to the plane of vortex elements, andhence measuring the direction of the induced flow should give an estimate of the wake orientation(ψind). These measures of wake orientation were then compared with the expected geometry ifassuming that the wake remains stationary along the path of the wing tip, inferred from amplitude,flapping frequency and forward speed. During downstroke it appeared that all three measuresdiffered significantly at slow speeds, but that they converged at the highest speeds (Fig. 12). Theangles of the wake trace increased with increasing speed, while the angles of the kinematic trace




Figure 12: Wake inclination angles. Angles are measured directly from vorticity maps (ψtrace)and the induced flow (ψind) and deduced from wing tip kinematics (ψkin) during(a) downstroke and (b) upstroke in a thrush nightingale in relation to speed U (fromref. [47]).

declined. This may seem contradictory, but reasoning based on self-induced convection of thewake vortices combined with apparent angular rotation of the wing, affecting the local angleof attack, could plausibly explain the pattern observed [48]. The agreement between kinematicsand wake geometry agreed better for the upstroke (Fig. 12), but also here there are systematicdiscrepancies.

These data demonstrate that the wake vortices do not remain stationary where left by thewing tip, but evolve due to self-induced convection especially at slow flight speed. Therefore,a simplistic 1 : 1 correspondence between wing beat kinematics and wake geometry is not avalid basis for a vortex based flight model. Even if Rosen et al. [48] offers only a limited setof kinematics data in their first analysis, they nevertheless suggest interesting links betweenkinematics and wake properties that certainly require further attention to be fully understood. Ofparticular interest should be to measure the wing rotation and so the local angle of attack, whichhas so far not been done with any useful detail in bird flight.

5.6 Comparing wake properties

Most wake visualization experiments have concerned slow flight, where one expects a greater con-tribution from the wake-generated vorticity towards the total drag/energy/power budget, becausethe self-generated downwash need to be larger than at cruising speeds. In the wake, the result hasinvariably been interpreted as some variant on a vortex ring/loop generated by the aerodynamicaction of the downstroke. Until recently, the only other point of comparison was the kestrel study ofSpedding [10], where the wake was instead composed of undulating wing tip vortices of constantcirculation. The successful development of the DPIV method in a variable speed, low-turbulence,




wind tunnel allowed, for the first time, the study of changes in wake geometry, vorticity and circu-lation across the natural speed range in a bird [30]. It appeared that the wake gradually transformsfrom discrete loops at slow speeds into a cc-like wake at typical cruising speeds. This transforma-tion from a discrete loop to the cc-wake is probably achieved by the addition of an upstroke loop,end to end with the main downstroke loop, and as speed increases the upstroke wake structureis increased in strength to elongation until the downstroke and upstroke wake components forma continuous trailing wake of streamwise vortices with low amplitude cross-stream vortices (cf.Fig. 6). Only a subset of the studies presented in Table 1 present any quantitative measures of thewake and so any comparisons among species are tentative at best. Both the normalized vorticityand circulation are declining functions of airspeed, but at U = 4 m s−1 and very similar reducedfrequency (k = 0.54 and k = 0.55, respectively) the thrush nightingale show a more than twice aslarge peak spanwise vorticity as the robin. The wing loading is 47% larger in the nightingale com-pared with the robin. The highest normalized vorticity was found in the pigeon (18.8, k = 0.82;Table 1), suggesting a correlation between wing loading and vorticity. However, the correlationis not perfect since the jackdaw, with a comparatively high wing loading, shows a lower peakvorticity value than the thrush nightingale. A straightforward comparison is further confoundedby the fact that the start vortex core contains variable amounts of the total same-signed vorticity,where the robin wake appears to represent the largest fraction of the total circulation. It may bethat the vorticity shed is a composite function of several variables, for example the wing load-ing and reduced frequency (k), and a more robust and strongly-linked measure might be thatof a suitably-defined circulation measure that accounts for the amount of vorticity that actuallygets into the wake structures of different kinds. Interestingly, the peak vorticity differed betweendown- and upstrokes in the kestrel (while its integrated magnitude in the circulation did not) whilegliding flight values were intermediate between the up- and downstroke flapping flight values.If one takes the steady gliding wake as a baseline case, then the wing accelerations on down-and upstroke in flapping flight might be viewed as tuned perturbations about this baseline for thepurpose of generating net forward thrust from an asymmetric wing beat while maintaining, onaverage, sufficient downward momentum for weight support.

The relative loop dimensions seem to be quite similar among species (Table 1), with the excep-tion of the white-rumped munia (Lonchura striata) which was flying in a confined space at slowspeed and with very high wing beat frequency, with a resulting vortex ring diameter approximatelythree times larger than the wingspan [50]. The authors interpreted this as an aggregated structuregenerated by a number of downstrokes.

Even though Table 1 contains a complete summary of existing data from vortex wakes in birdsthe quantitative information is still limited and selective. However, it does suggest a rich futurefor careful comparative investigations.

6 Discussion

The vortex wake approach to aerodynamic modelling of bird flight has been available for aquarter of a century, following in the footsteps of experimental and theoretical analyses of classicalaerodynamics ([51]). In its simple abstraction it offers an elegant way of analysing flight mechanics[9–11, 52, 53]. Initially, the wake was thought to consist of discrete vortex rings at all speeds [10], asupposition which gained some experimental support at slow flight speeds [20, 22, 23]. However,the demonstration of the constant circulation wake in a kestrel at a moderate flight speed [12]showed that the picture was more complicated than originally assumed. For a long time there waslittle development in research on bird wakes, and yet there was a growing interest from ecologists




applying flight mechanical theory for understanding variation in flight-related morphology andbehaviour [8]. Clearly, if an aerodynamic model is fraught with uncertainty and it is incorporatedas a component of another, say, ecological model, then the uncertainty due to the aerodynamicmodel will be propagated also to the predictions derived from the ecological model. In this light,the recent application of DPIV to bird flight is clearly a development with significantly positiveconsequences, not only for mathematical model-builders, but also for the wider set of scientistsusing flight mechanical models of bird flight. In the following sections I will briefly discuss someissues regarding bird flight in relation to vortex wakes.

6.1 The topology of the wake and its properties

Since the demonstration of the cc-wake [12], the paradigm has been the existence of discretegaits associated with the wake geometry, rings at slow speed and cc-wake at cruising speed, andthat birds adjust their kinematics so that the output is one or the other of these ‘gaits’ [14, 25,35, 36]. However, when analysing the wake in relation to speed for a thrush nightingale [30],it appeared that the wake topology transforms continuously from discrete loops to continuouswing tip vortices across the natural speed range. This transformation occurs by the reduction instrength of the cross-stream vortices and an increase in vorticity shed during the upstroke. Thisapparent continuous change is well supported by quantitative measures of the wake, as well asby the change in wing beat kinematics in relation to speed [48, 54].

The momentum deficit of the early wake visualization experiments has been a disturbing con-dition and some solutions to the paradox have been proposed. Recently, Tytell and Ellington [55]investigated the evolution of a vortex ring after formation. If the ring Reynolds number, defined as

Re0 = DA

ρνf,

where DA (=mg/πb2) is the disc loading, ν is the kinematic viscosity and f is the wingbeatfrequency, exceeds a certain value the ring will be turbulent and will shed off vorticity that maycancel through interaction with opposing vortices. By the time the structure is imaged in thefar wake some vorticity could be missing which would cause an apparent momentum deficit.The pigeon has a Re0 = 84,000, which is well above the threshold value for initially turbulentvortex rings [55]. In the thrush nightingale Re0 = 21, 000 and in the robin Re0 = 17, 000, whichare at the low end of the range of initially turbulent vortex rings [46, 56]. The wake momentumdeficit was similar in magnitude in the nightingale and pigeon when measuring the circulationof the strongest start vortex structure, and the recently studied robin showed a very similar wakemomentum deficit (Table 1). If there is a transition to turbulent vortex rings somewhere nearor below the robin Re0 so that the vortex rings would have shed momentum until the time ofrecording, this could result in the observed momentum deficit. However, the detailed analysisby Spedding et al. [30] showed that the vorticity is not irretrievably lost. It only occurs in anunexpected place and is mixed in with the opposite vorticity of the stop vortex by the time it isrecorded in the far wake. Importantly, it was possible to recover all vorticity and to achieve a forcebalance by a careful matching between start and associated stop vortices. The generality of thissolution to the wake deficit problem remains to be shown in further similar experiments, but itcurrently looks as if the paradox is expunged. It also cautions against overly simple comparisonsof wakes generated by flapping wings with ring-like structures generated by pistons in cylindricaltubes.




6.2 Optimum kinematics

Inspired by work on fish locomotion [57, 58] regarding the possible optimal propulsion kinematicsas characterized by the Strouhal number (St, see Section 2 for definition), Taylor et al. [59]proposed that also birds adjust their wing beat kinematics to an ‘optimal’ range of about St ∈ [0.2,0.4]. The aerodynamic incentive for this is that at some St the energy input to produce optimumbound vortices is minimum per unit energy output, i.e. the propulsion efficiency is maximum. Thepresence of an optimal reduced frequency has also been found in numerical simulations aboutpitching and rotating wing segments at Re = 1000 [60]. The idea of optimal St in fish propulsionis the formation of ‘optimal’ vortices [61] and so, in principle, there is no lower bound on St[62]. If vortices are shed to close to each other they might interact adversely with reduced overallpropulsion efficiency. Interestingly, the flapping frequency did not change very much across thespeed range in the thrush nightingale [48], and the same applied to the robin [47]. Since wingbeat amplitude also changed little with speed [48], measures of reduced frequency such as kand St, change due to changes in flight speed. Hence, birds seem not to maintain some ‘optimal’wing beat kinematics in order to keep variation in reduced frequency minimal. Particularly atcruising speeds where the wake consists of continuous vortices any adverse vortex interactionsseems unlikely, as would be possible in a reverse Kármán wake where the optimum efficiency wasencountered. Rosén et al. [48] point out that even though the agreement between ‘optimal’ St andthe range of actual St in birds may look as more than a pure coincidence [59], the true underlyingreason may well be a complex of aerodynamic constraints together with structural/mechanicalones, such as in the tendon-muscular system, mechanical resonance in relation to morphology, orsome other physiological trait being optimized.

6.3 Aerodynamic mechanisms

Recent work on insect flight has focused on various mechanisms by which sufficient aerodynamiclift is generated to support the weight and allow manoeuvres [42, 43]. Dynamic stall and associatedleading edge vortices and wake capture are examples of mechanisms used by insects. Bird flightresearch has been less concerned with the search for esoteric aerodynamic mechanisms thanresearch on insect flight, although for example induced drag reduction by wing tip slots (splayedprimary feathers) in gliding flight has been quantified [29]. Close-to-wing flow visualization onfreely flying birds has not been possible for risk of injury (exposure of the birds eye to high intensitylaser radiation is carefully avoided). The use of dead birds is unlikely to produce representa-tive results [63], while experiments with oscillating airfoils can potentially give useful insights.Close-to-wing flow visualization is probably easiest to obtain during gliding flight, but until thisis made bird aerodynamics must proceed from wake flow visualization.

6.4 A vortex wake theory

The most widely known and used aerodynamic model for bird flight is that due to Pennycuick[3], where the wake is a momentum jet. This is a drastic simplification compared with the currentrepresentation of bird wakes as illustrated in Fig. 6. But is this reason enough to abandon themomentum jet model in favour of a model that includes a more accurate wake representation?All models are, by default, caricatures of the real-world system they are supposed to describe,where the purpose of a model guide with simplifications can and cannot be accepted [64]. Amain prediction from the momentum jet-based flight model is a P(U ) function, usually referredto as the power curve. The few direct measurements available do not differ enough from the




Figure 13: The geometry of wake structures in a vortex wake model of flapping flight. Therelative contribution from the circulation of the upstroke is weighted in relation tothe circulation of the downstroke, ranging between 0 at slow sped to 1 at cruisingspeed. Symbols are downstroke wavelength (λd), upstroke wavelength (λu), down-stroke wake inclination angle (ψd) and upstroke wake inclination angle (ψu) (based onref. [30]).

model [65–67], and with uncertainty estimates of the model prediction empirical data must differquite a lot in order to falsify the prediction [64, 68]. However, various tests of components of themomentum jet model do suggest that it contains anomalies that are not easily rectified by changingparameters only. For example, Pennycuick et al. [49] concluded that the value representing theparasite drag coefficient should be reduced from the original default value of 0.4 in small birds[3] to 0.1 or even less. Their basis for this conclusion was a comparison of the speed of minimumwing beat frequency and the speed of minimum power (Ump) as calculated from the model,with the underlying assumption that the two curves should have the same speed of minima. Theconsequence is that the power required to fly at speeds > Ump becomes very low, in fact lowerthan the power required to glide at some speed [69], which seems unlikely. Also, the predictedspeed of Ump in very large birds such as swans becomes higher than the speed actually observedin these birds. Hence, even if the simple momentum jet model has proven enormously successful,not least by the many valid predictions derived from it [70] and its high citation frequency [8],there remain some unsettling facts concerning quantitative predictions.

Spedding et al. [30] proposed a vortex wake model as a composite between a downstrokeelliptic and an upstroke rectangular wake structure (Fig. 13). The relative circulation betweendown and upstroke is allowed to vary as a function of speed as �u = Cu�d, where Cu varies from0 at hovering to 1 at a fully developed cruising flight cc-wake. The model has to satisfy the weightbalance criterion. In principle, the net thrust (related to the power output) could be determined bythe areas of the wake structures projected onto vertical planes. It was shown that this model wasself-consistent with the thrush nightingale data [30]. There are two principal difficulties facingsuch a program, however. First, the drag is much smaller than the lift-supporting component inthe wake (by the ratio of L/D), so practical measurement uncertainty will be a significant problem.Second, and much more significant, a reasonable drag measurement could only be made if thewake structure representing the inviscid induced drag model were clearly separable from theviscous drag wake. In steady self-propulsion (of any body), the net fore-aft momentum balancewill be zero, as thrust balances drag. As duly noted in [30], only if the viscous drag wakes canbe identified and isolated in the wake, could their magnitude be determined, even in principle.A sober analysis of the more recent complex wake structures actually measured behind real flyingbirds (Fig. 10) suggests that much more careful research work lies ahead.




6.5 The ecology and evolution of flight

Ecologists are concerned with various problems related to flight, such as optimum selection offlight speeds during migration, foraging flight, display flight, predator evasion, load lifting withrespect to prey/food or fuel (fat) stores used for long non-stop flights, etc. [71, 72]. In all theseexamples the ‘optimum’ behaviour can be understood on the basis of flight mechanical theory incombination with some appropriate currency assumption and an optimization rule. Perhaps sur-prising, also decisions regarding optimal departure time and associated fuel load from a stopoverin migratory birds can be derived on the basis of aerodynamic principles [72]. During moult—theperiodic replacement (typically once per year) of flight feathers—the wings have reduced areadue to missing or growing feathers and will change in shape due to moult gaps. The consequenceson flight performance from moult gaps have recently been analysed from an aerodynamic per-spective [73–75]. Aerodynamic performance is tightly linked to morphology and so aerodynamicmodels are well suited for understanding the adaptive significance of flight-related morphology.

How animals once evolved an ability to fly is a popular and controversial topic [76], Twoscenarios—the trees-down and the ground-up—have long been the two competing hypothesesabout the evolutionary trajectory that lead to powered flight. From a vortex wake perspective itis easy to see a natural (gradual) transition from gliding flight to flapping flight with an initiallylow amplitude wing beat transforming a straight glide wake to a shallow cc-wake [77]. Aerody-namically, take-off from the ground, even if running to gain speed, seems more problematic thangoing via gliding to powered flight [78], while recent fossil finds from China of unambiguouslyfeathered theropod dinosaurs suggest a ground-dwelling protobird [79]. However, even morerecently a bizarre ‘four-winged’ dromaeosaur Microraptor gui [80], also from China, suggests agliding animal and hence new evidence in favour of the arboreal theory for the origin of flight.The fossils do not leave any behavioural evidence more than the overall morphology in extinctanimals, but an aerodynamic analysis of wake types and possible interaction between forelimband hind limb vortices might help us understand how these dinosaurs flew and to follow theevolutionary trajectory that led to powered flight.

6.6 Future prospects

The quantitative visualization of wakes in freely flying birds has only begun. In addition toa low turbulence wind tunnel it requires a co-operative and well-trained bird [30]. It is nowimportant to get more data from a range of sizes and wing morphologies in order to establish amore generally valid vortex wake based theory of animal flight. On the technical side there areforeseeable improvements, such as 3D PIV and increased repetition rate of pulsed lasers, whichwill improve the geometric characterization of wake vortices and their dynamics. This couldyield time-series animation data on wake dynamics and the 3D wake topology would help inthe partitioning of the wake disturbances in drag and lift components. Near-wing PIV data willelucidate the aerodynamic mechanisms of gliding and, hopefully, flapping flight in birds. Sincebirds have evolved adaptations for efficient flight during about 150 million years, it is perhaps notsurprising that their flight endurance surpasses that of current man made flying vehicles of similarsize by two orders of magnitude. The study of migratory birds as a model system is thereforelikely to continue to be scientifically rewarding for some time to come.

Acknowledgements

The results presented in this paper are based on the joint effort with Geoff Spedding and MikaelRosén, to whom I am very thankful for a long-term collaboration. Geoff Spedding,Adrian Thomas,




and Roland Liebe helped improve the manuscript significantly. A.H. is a Royal Swedish Academyof Sciences Research Fellow supported by a grant from the Knut andAlice Wallenberg Foundation.The wind tunnel research at Lund University has been funded by The Swedish Research Council,The Knut andAlice Wallenberg Foundation, the Tryggers Foundation and the Swedish Foundationfor International Cooperation in Research and Higher Education (STINT).

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