Vortex Wakes of Bird Flight: Old Theory, New Data and Future Prospects

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

A. Hedenström Department of Theoretical Ecology, Lund University, Sweden.

Abstract

Flying birds leave a vortex wake. Fluid dynamic theory in the form of Helmholtz’theorems dictate the allowable topologies, and Kelvin’s circulation theorem requires that changes in wake circulation 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 vortices behind birds, since an accurate quantitative description of these will reveal also the aerodynamics of bird wings. The first vortex theory of bird flight assumed the periodic shedding of discrete vortex loops, each one generated during a downstroke, while the upstroke was considered aerodynamically functionless. This view received some support from early visualization experiments of take-off flight or very low speeds, while experiments at a higher speed (U = 7 m/s) in one species 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 flexing during the upstroke. Then followed an almost 20-year drought, with no further quantitative experiments, until recently when digital particle imaging velocimetry (DPIV) was successfully deployed in a low-turbulence wind tunnel, and where the same small (30 g) bird could be studied across a large range of flight speeds (4–11 m/s). These new experiments revealed a much more complicated wake pattern than previous data suggested, mainly due to the improved experimental resolution. 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 highest speeds. However, the wakes were never as clean as the idealized cartoon models of the vortex theory of bird flight, and previous paradoxical results were shown to be attributable to the resulting difficulty in accounting for all wake components. New DPIV data on other species indicate that these findings are quite general.

1 Introduction

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

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 with the wing beat cycle. However, the first generation of quantitative aerodynamic models of bird flight used the actuator disc approach to calculate the induced drag [2, 3]. Bird flight typically encompasses 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 from laminar 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 boundary layer followed by a decrease from this higher drag with further increase of Re, which makes any attempts of quantitative analysis complicated. Research on bird flight has a long tradition, not least because the aerodynamic models are of great potential practical benefit to ecologists who want to understand 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 steady flow of theoretical and empirical advances. This paper summarizes some key developments with special 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 often used indices of the flow regime. First, the Reynolds number is a dynamic similarity measure and defined as Uc Re = , ν where U is the flight speed in relation to the fluid at rest, c is a characteristic length in the direction of flow and usually taken as the mean chord, and ν is the kinematic viscosity. The Reynolds number can 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 pass over the mean chord, tc = c/U, and the time taken for one kinematic cycle, T = 1/f , is commonly expressed as the reduced frequency ωc k = . 2U The value of k expresses the relative importance of unsteady terms with k ≈ 0.1 implying that unsteady effects most often can be ignored, while k of order 1 indicates that unsteady phenomena are likely to occur. When comparing two situations such as different-sized animals moving in different fluids, the similarity of Re and reduced frequency guarantees that the flow regime and hence aerodynamic properties will be the same. A closely related reduced frequency based on the wing 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 forward velocity is given by 2φbf K = , 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

Strouhal number deﬁned as fA St = , U where A is the double amplitude of the wake vortices in a Kármán (drag wake) or a reverse Kármán wake (thrust wake), but in the absence of wake information the double amplitude of the wing (or ﬁn) tip is usually taken.

3 Vortex theories of bird ﬂight

Kelvin’s circulation theorem states that, in a homogeneous, incompressible and inviscid fluid, the circulation around a closed circuit will have the same value when measured over the same fluid elements and circuit at any time as the circuit is followed in the flow. This theorem will prove very useful to our applications on bird flight, because the aerodynamic properties of the wake vortices can be directly linked to the time-averaged aerodynamic forces on the wings having generated the vortices. The circuit of Kelvin’s circulation theorem just has to enclose the aerofoil and the space behind it where wake vortices will appear. In steady flight at speed U, the Kutta–Joukowski theorem 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 developed by the aerofoil also the bound circulation changes, which must be offset by an equal and opposite circulation in the wake. This is all there is in terms of background to develop vortex based theories of ﬂight. The modeller only has to specify a more or less realistic geometry of the wake vortices and enforcing a force balance between weight, drag and the force associated with the wake 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 deflects the oncoming airflow downwards (Fig. 1). At an extreme end of the spectrum of vortex wake based flight models, the momentum jet qualifies as belonging to this family, where the vorticity is confined to an infinitely thin, cylindrical sheet enclosing the uniform jet [6]. Notice however that, in the wake description, nothing remains of the time varying forces developed by cyclically beating wings. The mass flow through the actuator disc induces a downward velocity, wi, when it passes the disc, reaching a final speed of 2wi in the far wake with a jet diameter of b. The rate of vertical momentum flux required to support the weight at some forward speed U determines the induced velocity wi = mg/(2SdUρ), where m is body mass, g is acceleration due to gravity, and 2 Sd(=πb ) is the wing disk area. This in turn gives the induced power as Pi = mgwi. For a complete P(U)-relationship, commonly denoted the ‘power curve’ the terms of parasite and profile power are added to Pi, resulting in the most popular and widely used flight mechanical model of bird flight [3, 5, 7, 8].

3.2 Vortex ring theory

A step towards increased reality was taken by Rayner [9–11] who developed an aerodynamic model 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

WIT Transactions on State of the Art in Science and Engineering, Vol 4, © 2006 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Vortex Wakes of Bird Flight 709 unloaded and therefore did not leave any traces in the wake. The size, geometry and orientation of the vortex loops are determined by the wingspan, wingbeat frequency, stroke amplitude, forward speed and by the circulation distribution along the wing, ( y). The circulation of the vortex rings is determined by imposing the force balance condition, i.e. that the rate of wake momentum must 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 as L/D = cot ψ. The mean rate of increase of kinetic energy deposited in the rings is the induced power. The periodicity in the wake now corresponds to the wing beat periodicity, with a close connection between the aerodynamic force time history and the wake trace. In the first generation of this flight model it was assumed that discrete vortex loops are shed at all speeds, which later had 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 a circular disc cross section (Sd) with wingspan as diameter, where the oncoming flow of speed U is deflected downwards by the induced speed ui so that the downward imparted momentum balances the weight. Also shown is the coordinate system (x, y, z) used throughout this paper.

Figure 2: Vortex ring wake of ﬂapping bird ﬂight. When the upstroke is aerodynamically inactive the downstroke generates a vortex loop. The associated impulse (I = ρSe, where ρ is air 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 in relation 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 at a moderate cruising speed (U = 7ms−1) in a kestrel (Falco tinnunculus). Instead, the wake was characterized 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 to achieve non-zero thrust) between down- and upstroke is achieved in many birds by the flexing of the wrist-joint, causing a reduced span during the upstroke and thereby a reduced projected wake area 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 vortices of constant circulation (hence we shall denote this wake as the cc-wake, indicating constant circulation). The generation of vortices is associated with an energy cost since the energy content in the wake is lost. Therefore, with no or minimal transverse vortices, the cc-wake could be argued to minimize the mechanical cost of cruising flight and should be a favourable configuration in bird flight where minimizing energy cost is advantageous, such as during long-distance migration or commuting between nest and foraging sites. The cc-wake can be understood as a deformed version of the trailing vortices left behind by 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 undulatory track, 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 both straightforward and elegant [6]. The impulses associated with down- and upstroke wake elements, respectively, are

Id = ρccAd and Iu = ρccAu, (2) where cc is the circulation of the trailing wing tip vortices, and Ad and Au are the wake areas circumscribed by the tip-vortices during downstroke and upstroke, respectively. The associated forces 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 vortices of constant circulation. The overall wingbeat wavelength (λ) depends on the wing beat frequency, while the wake angles may differ between downstroke and upstroke (ψd, ψu).

1 L¯ = ρ (A cos ψ + A cos ψ ) , (3) T 0 d d u u 1 D¯ = ρ (A sin ψ − A sin ψ ) , (4) T 0 d d u u 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 1 + ζ L/D = cot ψ. (5) 1 − ζ

Pennycuick [13] simpliﬁed this analysis further by assuming constant spans during down- and upstroke, 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-/downstroke transition. Then the L/D depends on the span ratio (b = bu/bd)as 1 + b L/D = cot ψ. (6) 1 − b

It follows that L/D is maximized by a high ζ or b, i.e. from a small upstroke span reduction and a shallow wing beat amplitude. In a closely related model, Rayner [14] incorporated the cc-wake in a quasi-steady lifting line analysis of flapping flight at cruising speeds. The usual elliptic wing loading was replaced by an alternative due to Jones [15] that gives slightly improved aerodynamic efficiency. A problem now presented itself by the fact that two models with quite different presumed wake geometries, the vortex 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 rings to the cc-wake at some intermediate flight speed, and the notion of gaits was introduced to the flight literature. The gait analogue to terrestrial locomotion, where the transition between different patterns 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 unimaginable topology of intermediates. In order to account for the unsteadiness of flapping flight and the influence of the wake on the induced flow near the airfoil, Phlips et al. [16] modelled the wake as a lifting line representing the current half-stroke, but previous vortex lines were collected as streamwise wing-tip vortices and with transverse vortices shed at the turn-points of each half-stroke to account for changes in the bound circulation. The validity of their analysis was restricted to reduced frequencies k < 1 ◦ and 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 ‘ladder wake’ postulated by Pennycuick [17] to apply when birds have rigid wings that cannot be flexed at the wrist joint during the upstroke. Examples of birds where a ladder wake could exist are hummingbirds and swifts, where the force asymmetry between down- and upstroke has to be achieved by variation in bound circulation, and hence the shedding of transverse vortices, rather than by reduced wingspan and maintained circulation as in the cc-wake. Yet other more elaborate theoretical wake configurations have been treated in the literature [18, 19], but it is beyond the 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 the excellent review by Spedding [6].

4 Bird wakes in reality

The development of aerodynamic models has converged with that of experimental results on the geometry and properties of real wakes. However, the vortex ring theory of Rayner [9] was developed independently from experiments carried out simultaneously by Kokshaysky [20]. Thereafter the aerodynamic modelling has been tightly connected to empirical wake data. We now proceed by 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 tracer particles 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 = 0.87 for the chaffinch and k = 1.29 for the brambling. Both species generated discrete vortex rings associated with downstroke. The results came as a timely support for the assumptions of the vortex theory of bird flight by Rayner [11], although no quantitative information regarding vorticity and circulation was available.

4.2 Slow forward ﬂight

The next major experimental development was the application of the particle image velocimetry (PIV) method in which a cloud of buoyant helium filled soap-bubbles was generated and a bird was trained to fly through this cloud. The 3D movements of the bubbles were recorded by stereophotogrammetry. This methodological breakthrough allowed, for the first time, quantitative measurements of bird wakes to be made. The first species to be evaluated was the pigeon (Columba livia) in slow flight, U = 2.4ms−1 [22], in which the wake consisted of discrete vortex loops. However, the loops appeared asymmetric in the sense that the start end was rather concentrated while the stop vortex core had a larger diameter and was spatially less well defined. Quite surprisingly, and somewhat disconcertingly, the vortex loops contained approximately 1/2 of the momentum required to support the weight of the bird, which signalled that not all vorticity was confined to the vortex core of the observed rings. In a second experiment of a jackdaw (Corvus monedula), the results from the pigeon were repeated, i.e. a significant wake momentum deficit was obtained with only 1/3 of that being required confined to the main vortex core [23]. This result was obtained by integrating centreline velocities through the wake, thus including, supposedly, flow induced by vorticity not restricted to the compact vortex cores. The shortfall nevertheless greatly exceeded the calculated experimental uncertainty. Some selected wake properties from these 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). Discrete vortex loops were also found in two species of bat in slow flight [26]. Taken together birds and bats in slow flight (U ≤ 3ms−1) shed discrete vortex loops generated by the downstroke with the upstroke being more or less aerodynamically unloaded, although the detailed geometry differ

WIT Transactions on State of the Art in Science and Engineering, Vol 4, © 2006 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Vortex Wakes of Bird Flight 713 g 1 20 20 22 23 12 49 30 30 28 12 25 24 330 HRS HRS HRS Falco Source , mean c Luscinia f c c c c c d d d d c 0.45 / Tmg , vorticity; ω Uc I / / U c 1.800.55 0.94 1.13 1.04 0.99 0.93 1.04 1.6 0.31 0.40 2.2 0.45 0.5 0.8 0.15 0.72 1.0 0.19 1.16 Corvus monedula , kestrel max ω b e b b R , vortex ring radius; / R studies of bird flight. 0 0.39 0.097 R Policephalus meyeri , thrush nightingale loop loop 0.17 18.8 7.63 0.52 loop 0.14 4.54 3.47 0.35 loop 0.16 3.6 1.16 0.54 F. montifringilla , jackdaw ) Topology , vortex core radius; − 1 0 7 loop-cc 7 loop-cc 9 cc downstroke. R 10 cc slow loop slow loop slow loop 1. ≥ , acceleration due to gravity. g 14.4 4 loop 0.11 8.7 1.28 30 1 loop 0.13 / Tmg I Strix aluco , Meyer’s conure ) (Hz) (ms 2 , flight speed; NFU U 39.6 7.7 7 cc 40.0 0 7 cc-glide 0.2 39.6 7.7 7 cc m , body mass; Fringilla coelebs , brambling (see text for symbol definitions). (see text for symbol definitions). e S 2 b /ρ U /ρ mgT , time period; T = mg 1 , wingbeat frequency; = f 0 Lonchura striata , tawny owl , impulse; I Columba livia , chaffinch (kg) (m) (m) AR (N/m Table 1: Morphology and wake properties compiled from wake-visualization Mass Semi-span Chord 0.0210.022 0.130.35 0.132 0.33 0.046 0.044 5.7 6.1 0.094 17.2 19.0 7.0 18 17 55.9 3 6.7 1.8 loop 2.4 0.017 0.11 0.047 4.7 16.1 14.8 4 0.48 0.216 0.296 0.095 6.0 38.2 5.6 2.5 , wing loading; N Erithacus rubecula. , circulation; a In relation reference circulation Hedenström, A. Rosén, M. & Spedding, G.R., unpublished data. Core radius in relation to transverse distance between wing tip vortices. Scientific names: pigeon Refers to an aggregated vortex in a confined space, not the result of a single In relation to reference circulation This quantity denotes sufficiency of supporting the weight when tinnunculus , white-rumped munia luscinia , Robin wing chord; f g c d e b Chaffinch Brambling Pigeon Pigeon a Species Kestrel, upstroke 0.21Meyer’s conure 0.338 0.12 Robin 0.076 8.8 Kestrel, gliding 0.21 0.338 0.076 8.8 Jackdaw Kestrel, downstroke 0.21 0.338 0.076 8.8 Tawny owl Thrush nightingale 0.030 0.131 0.048 5.4 23.6 White-rumped munia 0.014 0.08 AR, aspect ratio;

WIT Transactions on State of the Art in Science and Engineering, Vol 4, © 2006 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) 714 Flow Phenomena in Nature from an idealized ring/ellipse of equal core radius and vorticity at all stations. The two quantitative measurements 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] obtained wake images of a kestrel flying at a moderate cruising speed, U = 7ms−1. The wake appeared dramatically 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, showing an 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 ‘constant circulation’ wake. The lift is given by eqn (3), which was 2.15 N for the kestrel to be compared with the weight of the bird of 2.06 N (Table 1). Hence, at cruising speed the inferred wake topology and measured properties (Au, Ad, ) satisfied the force balance criterion.As explained above these findings prompted the amendment of the vortex theory of forward flapping flight to account for the appropriate wake geometry in cruising flight [12, 14].

4.4 Gliding ﬂight

At equilibrium gliding, flight potential energy is converted into work against the aerodynamic drag. The bird itself does not perform any work since the wings are not flapped. Gliding flight is however not effortless since by holding the wings in an outstretched position the flight muscles produce static muscle work that consumes chemical energy at a rate approximately two times the basal metabolic rate [27]. The wake in gliding flight observed in a kestrel at U = 7ms−1, consists of two straight streamwise wing-tip vortices [28]. The measured circulation matched the force balance criterion, indicating that all vorticity is accounted for in the main vortex structures observed. A Harris’ hawk (Parabuteo unicinctus) gliding in a wind tunnel also showed wing-tip vortices, 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 in this section (Table 1). In flapping flight quantitative wake data were available from two speeds, U = 2.5ms−1 (pigeon, jackdaw) and U = 7ms−1 (kestrel). Additional data, albeit qualitative, from a few other species suggested the presence of vortex loops at slow speeds (U ≤ 3ms−1). First, the paradoxical wake momentum deficit could not be satisfactorily explained, although Rayner [24] suggested that the wake deficit could be due to that the birds actually decelerated during the experiments. However, this explanation appeared to be false [30]. Second, the apparent existence of topologically two distinct wake types, the discrete loop and the cc-wake, has led some authors to introduce the notion of ‘gaits’ [14, 24, 25], in analogy to the discrete gaits as found in quadrupeds. In terrestrial locomotion the transition between gaits, such as canter, trot, gallop, is very abrupt and occur quickly at some predictable Froude number. Rayner [24, 25, 31] speculated that the transition between the wake types in bird flight could happen from one wing beat to another 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 little success when looking for indirect signs of a transition between wake types by observing various wing beat kinematic parameters across flight speeds [35–37]. Most kinematic measures actually

WIT Transactions on State of the Art in Science and Engineering, Vol 4, © 2006 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Vortex Wakes of Bird Flight 715 showed smooth curves in relation speed without any discontinuities as expected at a sudden gait transition.

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 image velocimetry (CIV) routines, and not least the introduction of a new low-turbulence wind tunnel dedicated 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 background flow is non-turbulent as this is the reference against which to reduce any effects caused by the object [38]. At Lund University a wind tunnel dedicated for bird flight was designed and has been operational since 1994 [39]. The tunnel is a Göttingen type with recirculating flow and a contraction ratio of 12.25:1 between the settling chamber and test section cross-areas. The octagonal test section is 1.2 m in diameter with a 1.5 m long closed part followed by a 0.5 m gap between the end of the closed test section and the bell mouth of the first diffuser. This opening is 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 around a pivot. A survey using a hot wire anemometer gave a turbulence level of ≤ 0.05%, measured as RMS at U = 10ms−1. The low background turbulence of the wind tunnel is a prerequisite for repeatable DPIV. In the tunnel the spatial directions (x, y, z) refer to the streamwise, spanwise and vertical direction and their associated speeds are defined as (u, v, w). A more detailed account of tunnel 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 custom designed DPIV set-up has been deployed and applied to bird flight in the Lund wind tunnel. The incentive was the imbalance in the literature between quantitative data and speculation, with data from only two points on the speed axis representing different species and showing fundamentally different wake geometries. Hence, there was a clear gap to fill in order to address the long standing momentum deficit paradox at slow speed and the possible change of the wake across a wider range of speeds. The DPIV technique relies on pairs of digital images from which the displacement of identifiable particle patterns are determined. DPIV can be applied to numerous biofluid problems and it has been used quite extensively to study swimming [41]. Previous applications to animal flight have been restricted to low Re using mechanical flapping wings [42, 43] and recently also to 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 µm fog 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 to fly 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 was arranged 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) is flying 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. Laser pulses and cameras (tm1, 2) are synchronized by two delay generators (dg1, 2), and camera 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 by two pulse delay generators (Stanford Instruments DG535). In the first set-up the light sheet was vertically aligned with the flow, and hence the images are streamwise slices of the wake. A 3D reconstruction of the wake topology could be obtained by sampling the wake at different stations along the wingspan, as recorded by an independent video camera positioned far downstream from the test section (inside the first diffuser). The system was tuned as to give minimum measurement error 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% in disturbance flow fields (u and w), and ±10% for gradient quantities such as ωy (for definition see eqn [7]). A full account of the experimental set-up, procedure and detailed error analysis is given by 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 representative of the species-rich passerine family, being a rather typical long-distance migrant of small

WIT Transactions on State of the Art in Science and Engineering, Vol 4, © 2006 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Vortex Wakes of Bird Flight 717 size with low aspect ratio wings (lean mass 20 g, AR = 5.4). Results from such a species should therefore 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 repeatable manner approximately 0.9 m upstream from the image centre of its wake. Sequences of 10 image pairs at 10 Hz (determined by the repetition rate of the laser) were taken at integer speeds in the range 4–11 m s−1 and at different locations along the wingspan. A total of 4000 wake images were sorted 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 be remembered that the entire wake consists of the inviscid induced (lift-dependent) drag as well as the pressure and viscous drag components from the body and wings. The possibility to observe them in the wake flow depends on whether they remain physically separated, and hence can be distinguished from each other, or not. Otherwise, if they are intermixed the wake will represent the sum of thrust and drag components, which will be exactly zero in steady flight at constant speed [30].

5.3.1 Wake topology At low speeds (Re = 13,000 at U = 4ms−1) the wake showed characteristic start vortices generated at the beginning of downstrokes and corresponding stop vortices that were spatially more spread out and with lower peak vorticity. The core-to-loop radius was 0.11, which should be considered as a small-cored vortex loop [46]. Some vorticity is shed during the upstroke also at slow speeds, but for the most part, the upstroke may be inferred to be more or less aerodynamically inactive. A notable feature of the slow speed wake (U ≤ 5ms−1) was that the region dominated by 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. This mixture 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 consecutive frames to generate a composite covering a whole wing beat period (Fig. 5). The 2D flow/vorticity maps are roughly consistent with a wake consisting of discrete vortex loops, each shed as the result of a downstroke but with different radii between the start and stop ends. In particular, one must take note that if the circulation is not balanced between positive (start) and negative (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 the loop (Fig. 5). When increasing the flight speed the wakes exhibit increasing vorticity originating from the upstroke, suggesting an increasing aerodynamic significance. At the maximum speed, U = 11ms−1, there was a more or less continuous vortex trail throughout the wing beat of similar strength. Cartoon reconstructions of wake topology at three different speeds are shown in Fig. 6. The reconstruction in Fig. 6 is based on wake images at different positions along the wingspan, which together could be used to deduce the 3D geometry. Even though the high-speed wake appears quite dissimilar from the other two, it represents one end of a continuous spectrum where the discrete loops are at the other extreme. As suggested by quantitative properties (see below), there is a smooth transition of wake topology mainly caused by the increasing aerodynamic function of the upstroke [30]. At cruising speeds (U = 9−11ms−1;Re= 35,000) the main features are undulating wing tip vortices that are interconnected by cross-stream vortices of alternate sign. In spite of its superficial similarities, this wake should not be considered identical to the ladder wake postulated by Pennycuick [17]. In the ladder-wake the ‘rungs’ of the ladder are cross-stream vortices shed 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 slightly more than one wing-beat vortex wake at slow speed, U = 4ms−1. The composite is constructed from a sequence of consecutive wake images obtained in an image plane aligned with the (x, z)-directions at the mid-span position (centre plane). The wake wavelength 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 about ωy = 0 with numbers at the ends showing values in units of s−1 (from [30]).

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

(and associated lift). The cross-stream vortices observed here represent a continuous shedding of vorticity during each half-stroke, but the origin of this vortex shedding remains unclear. This wake therefore mostly resembles the constant circulation wake found in the kestrel at U = 7ms−1 [12]. It may be that also the kestrel wake has these low-amplitude cross-stream vortices but that they were not detected by the helium bubble method.

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

∂w ∂u ω = − , (7) y ∂x ∂z where u (in x-direction) and w (in z-direction) are velocity components. The circulation measures the strength of a vortex and calculated as the vorticity integrated over a surface, S, in the imaging plane as = ω · dS. (8)

In practice the circulation of eqn (8) was calculated by making a discrete approximation by the 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 assuming a Gaussian distribution of vorticity around its peak value. Measures of vorticity and circulation were carried out for the different wing locations and plotted against flight speed, for the 4000 wake velocity fields [30]. Peak vorticity and circulation of main start and stop structures are shown in Fig. 7, where both quantities have been normalized as |ω|maxc/U and /Uc. The normalized vorticity expresses the ratio of timescales for convection over the wing and rotation around a vortex, while the normalized circulation expresses the integrated magnitude of shed circulation compared with a measure of the momentum flux over the wing chord. Both quantities show maximum values at the lowest speed (U = 4ms−1), and they decline to minimum values at U = 11ms−1. There is a notable difference between start and stop vorticity and circulation at slow speeds, but with increasing speed the difference decreases and from U = 6−7ms−1 it has disappeared (Fig. 7).

5.3.3 Force balance Whether the measured circulation is sufficient for balancing the forces required to fly depends critically on the appropriate interpretation of the wake geometry that determines the correct area appearing in eqn (2). It is convenient to define two reference values of circulation required to support the weight [30]. First, if the wake were to consist of two straight wing-tip vortices the Kutta–Joukowski theorem [eqn (1)] gives the lift that must balance the weight, W , if we neglect 3D and wing tip effects. Hence, rearranging eqn (1) yields the circulation required to support the weight 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 the horizontal plane Se = πb(λd/2), where λd(=UTτ and T is the wingbeat period and τ is the downstroke ratio) is the downstroke wave length, then the associated circulation required to balance the weight is 1 = mgT/ρSe. (10)

Figure 7: Quantitative wake properties in relation to speed for a thrush nightingale. Variation in standardized 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 (ﬁlled circles) and stopping vortices (open circles) are shown in relation to ﬂight speed. Error bars represent standard deviations (from [30]).

Although neither of the two idealized geometries apply directly to the nightingale wake, they may still 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 and apparently weaker than the start vortices. However, when comparing the total integrated circulation 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 vortices would contain circulation enough to support the weight, while the well defined start vortices exhibit an approximate 50% momentum deficit. This finding therefore repeats the original wake momentum deficit of the pigeon and jackdaw [22, 23]. As previously remarked there were patches of positive vorticity in the region of down/upstroke transition otherwise dominated by the negative stopping vorticity. Spedding et al. [30] made a further calculation, supposing that this additional positive vorticity has the same source as that of the main start vortex and hence that the total circulation contributing to the total aerodynamic force is given by the sum of all vorticity of the same sign (+, in the case of the start vortex with additional 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 tot from all positive (filled circles) and negative (open circles) vorticity from the wake in relation to flight speed (U). (a) The fraction of the total circulation that is not contained in the strongest vortex cross-section is higher in the stopping vortices than the starting vortices. (b) The total integrated circulation in relation to the reference circulation 1 (see eqn (10) for definition). At speeds ≤8ms−1 the total negative vorticity would be sufficient for weight support, but not the positive component. Error bars represent standard deviations (from ref. [30]). for all + signed vorticity in this way it appeared that weight balance was achieved (Fig. 9). In that case the revised geometry of the vortex loops could account for weight support since force balance 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 European robins (Erithacus rubecula) using the same procedure and camera set-up shown in Fig. 4 [47]. The robin is a close relative to the thrush nightingale but migrates a shorter distance within Europe and has a lower body mass (during experiments m = 0.017 g), and a shorter wingspan and lower aspect ratio than the nightingale (2b = 0.22 m, AR = 4.7). At U = 4ms−1 Re = 13,000 and k = 0.55 and at U = 9ms−1 Re = 29,000 and k = 0.24.

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

The wake topology showed striking similarities with that of the thrush nightingale, with characteristic discrete loops at slow speeds. An example of a cross-section of vortex loop associated with a downstroke at U = 6ms−1 is shown in Fig. 10. Also the robin exhibited the asymmetry between start and stop vortices, typified by a more diffuse and spatially spread out stop vortex than the start vortex, even if the illustrated example shows quite a clean vortex cores of both ends of the loop. When increasing the speed the wake undergoes similar gradual changes as in the nightingale, characterized by increasing vorticity shed from the upstroke. By U = 9ms−1 the same magnitude vorticity is shed throughout the down and upstrokes, in agreement with the cartoon reconstruction of Fig. 6 (the streamwise thick vortices in the lower panel, Fig. 6). The pattern 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 and 2.8 cm for the start-and-stop vortices, respectively, while the overall streamwise loop diameter is about 20 cm at this speed. This latter estimate comes from a hypothetical loop with simple geometry generated during the downstroke at U = 6ms−1, and wing beat period T = 0.068 s and a downstroke fraction of the wingbeat period τ ≈ 0.5 (as calculated from companion kinematic measurements). For a cross-stream diameter close to the wingspan (22 cm) this would imply a span efficiency of 0.91 [25]. The ratio R0/R = 0.11 for the start vortex and R0/R = 0.14 for the stopping 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 somewhat −1 larger (Table 1). However, at U = 5ms 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 slightly more concentrated than the stopping (leftmost) vortex. The bird is ﬂying to the left at U = 6ms−1. The 14-step symmetrical colour bar shows negative and positive vorticity (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 is seen as a larger diameter (2.3 and 2.8 cm, respectively).

WIT Transactions on State of the Art in Science and Engineering, Vol 4, © 2006 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) 724 Flow Phenomena in Nature showing that this parameter may vary rather little and that all hitherto measured birds actually show similar relative dimensions of vortex loops. The quantitative wake properties /Uc and ωmaxc/U also showed the same pattern with speed as the thrush nightingale, i.e. they both declined monotonically from maximum values at the slowest speed (Table 1). The normalized circulation showed similar values between the two species, while the vorticity was more than twice as large in the nightingale compared with the robin 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 the reference values 0 and 1 revealed a nominal wake momentum deﬁcit also in the robin, of the 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 stop vortex as for the thrush nightingale resulted in enough total positive circulation associated with the 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 approximately 0.9 m downstream from the position of the bird, which represents about 3 wing beats at U = 4ms−1 and 1.3 wing beats at U = 10ms−1, which allow the wake to evolve and change since the 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 its own vorticity and the bound vorticity on the wing [10]. Therefore, when we measure the spatial coordinates of wake elements and their relative position, for instance as the wake inclination angles (ψd, ψu), the measured value of ψd would be lower than when the vortex was created. Any such effects due to self-induced convection will likely be reduced at higher flight speeds where peak vorticity and time since shedding are reduced (Table 1), but the magnitude of the effects and whether they can be ignored remains to be investigated. Some basic wing beat kinematic parameters that should correlate with wake geometry were analysed for the same thrush nightingale as used 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 downstroke and upstroke is the same and a net thrust must be obtained from differential circulation between downstroke and upstroke. However, in the investigated species b was always <1. Notably, both amplitude and wing-beat frequency changed very little with speed. The wing beat frequency showed a very weak U-shaped function of speed, which was also found in a previous study of the same species in the same wind tunnel [49], but the variation in f was only 7% between the extreme values [48]. Therefore, changes in measures of reduced frequency (k, , K), depend mainly on U and hence display essentially linear functions of U. The wake geometric properties were estimated in two ways: (i) from cross-stream vorticity maps such as Fig. 5, the inclination angles were measured between centres of identifiable vortex blobs (ψwake); (ii) the induced downwash should be normal to the plane of vortex elements, and hence 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 if assuming 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 measures differed significantly at slow speeds, but that they converged at the highest speeds (Fig. 12). The angles 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 (from ref. [47]). declined. This may seem contradictory, but reasoning based on self-induced convection of the wake vortices combined with apparent angular rotation of the wing, affecting the local angle of attack, could plausibly explain the pattern observed [48]. The agreement between kinematics and wake geometry agreed better for the upstroke (Fig. 12), but also here there are systematic discrepancies. These data demonstrate that the wake vortices do not remain stationary where left by the wing 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 a valid basis for a vortex based flight model. Even if Rosen et al. [48] offers only a limited set of kinematics data in their first analysis, they nevertheless suggest interesting links between kinematics and wake properties that certainly require further attention to be fully understood. Of particular interest should be to measure the wing rotation and so the local angle of attack, which has 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 contribution from the wake-generated vorticity towards the total drag/energy/power budget, because the self-generated downwash need to be larger than at cruising speeds. In the wake, the result has invariably been interpreted as some variant on a vortex ring/loop generated by the aerodynamic action of the downstroke. Until recently, the only other point of comparison was the kestrel study of Spedding [10], where the wake was instead composed of undulating wing tip vortices of constant circulation. The successful development of the DPIV method in a variable speed, low-turbulence,

WIT Transactions on State of the Art in Science and Engineering, Vol 4, © 2006 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) 726 Flow Phenomena in Nature wind tunnel allowed, for the first time, the study of changes in wake geometry, vorticity and circulation across the natural speed range in a bird [30]. It appeared that the wake gradually transforms from discrete loops at slow speeds into a cc-like wake at typical cruising speeds. This transformation 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 structure is increased in strength to elongation until the downstroke and upstroke wake components form a 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 the wake and so any comparisons among species are tentative at best. Both the normalized vorticity and circulation are declining functions of airspeed, but at U = 4ms−1 and very similar reduced frequency (k = 0.54 and k = 0.55, respectively) the thrush nightingale show a more than twice as large peak spanwise vorticity as the robin. The wing loading is 47% larger in the nightingale compared 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 correlation is not perfect since the jackdaw, with a comparatively high wing loading, shows a lower peak vorticity value than the thrush nightingale. A straightforward comparison is further confounded by 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 be that the vorticity shed is a composite function of several variables, for example the wing loading and reduced frequency (k), and a more robust and strongly-linked measure might be that of a suitably-defined circulation measure that accounts for the amount of vorticity that actually gets into the wake structures of different kinds. Interestingly, the peak vorticity differed between down- and upstrokes in the kestrel (while its integrated magnitude in the circulation did not) while gliding 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 the purpose of generating net forward thrust from an asymmetric wing beat while maintaining, on average, 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 slow speed and with very high wing beat frequency, with a resulting vortex ring diameter approximately three times larger than the wingspan [50]. The authors interpreted this as an aggregated structure generated by a number of downstrokes. Even though Table 1 contains a complete summary of existing data from vortex wakes in birds the quantitative information is still limited and selective. However, it does suggest a rich future for careful comparative investigations.

6 Discussion

The vortex wake approach to aerodynamic modelling of bird flight has been available for a quarter of a century, following in the footsteps of experimental and theoretical analyses of classical aerodynamics ([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], a supposition 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 was little development in research on bird wakes, and yet there was a growing interest from ecologists

WIT Transactions on State of the Art in Science and Engineering, Vol 4, © 2006 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Vortex Wakes of Bird Flight 727 applying flight mechanical theory for understanding variation in flight-related morphology and behaviour [8]. Clearly, if an aerodynamic model is fraught with uncertainty and it is incorporated as a component of another, say, ecological model, then the uncertainty due to the aerodynamic model 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 positive consequences, not only for mathematical model-builders, but also for the wider set of scientists using flight mechanical models of bird flight. In the following sections I will briefly discuss some issues 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 discrete gaits associated with the wake geometry, rings at slow speed and cc-wake at cruising speed, and that 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 continuous wing tip vortices across the natural speed range. This transformation occurs by the reduction in strength of the cross-stream vortices and an increase in vorticity shed during the upstroke. This apparent continuous change is well supported by quantitative measures of the wake, as well as by 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 condition 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

D Re = A , 0 ρνf

2 where DA (=mg/πb ) is the disc loading, ν is the kinematic viscosity and f is the wingbeat frequency, exceeds a certain value the ring will be turbulent and will shed off vorticity that may cancel through interaction with opposing vortices. By the time the structure is imaged in the far 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 turbulent vortex rings [55]. In the thrush nightingale Re0 = 21, 000 and in the robin Re0 = 17, 000, which are at the low end of the range of initially turbulent vortex rings [46, 56]. The wake momentum deficit was similar in magnitude in the nightingale and pigeon when measuring the circulation of the strongest start vortex structure, and the recently studied robin showed a very similar wake momentum deficit (Table 1). If there is a transition to turbulent vortex rings somewhere near or below the robin Re0 so that the vortex rings would have shed momentum until the time of recording, this could result in the observed momentum deficit. However, the detailed analysis by Spedding et al. [30] showed that the vorticity is not irretrievably lost. It only occurs in an unexpected place and is mixed in with the opposite vorticity of the stop vortex by the time it is recorded in the far wake. Importantly, it was possible to recover all vorticity and to achieve a force balance by a careful matching between start and associated stop vortices. The generality of this solution to the wake deficit problem remains to be shown in further similar experiments, but it currently looks as if the paradox is expunged. It also cautions against overly simple comparisons of wakes generated by flapping wings with ring-like structures generated by pistons in cylindrical tubes.

6.2 Optimum kinematics

Inspired by work on fish locomotion [57, 58] regarding the possible optimal propulsion kinematics as 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 optimum bound vortices is minimum per unit energy output, i.e. the propulsion efficiency is maximum. The presence of an optimal reduced frequency has also been found in numerical simulations about pitching and rotating wing segments at Re = 1000 [60]. The idea of optimal St in fish propulsion is 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 overall propulsion efficiency. Interestingly, the flapping frequency did not change very much across the speed range in the thrush nightingale [48], and the same applied to the robin [47]. Since wing beat amplitude also changed little with speed [48], measures of reduced frequency such as k and 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 at cruising speeds where the wake consists of continuous vortices any adverse vortex interactions seems unlikely, as would be possible in a reverse Kármán wake where the optimum efficiency was encountered. Rosén et al. [48] point out that even though the agreement between ‘optimal’ St and the range of actual St in birds may look as more than a pure coincidence [59], the true underlying reason may well be a complex of aerodynamic constraints together with structural/mechanical ones, such as in the tendon-muscular system, mechanical resonance in relation to morphology, or some other physiological trait being optimized.

6.3 Aerodynamic mechanisms

Recent work on insect flight has focused on various mechanisms by which sufficient aerodynamic lift is generated to support the weight and allow manoeuvres [42, 43]. Dynamic stall and associated leading edge vortices and wake capture are examples of mechanisms used by insects. Bird flight research has been less concerned with the search for esoteric aerodynamic mechanisms than research on insect flight, although for example induced drag reduction by wing tip slots (splayed primary feathers) in gliding flight has been quantified [29]. Close-to-wing flow visualization on freely flying birds has not been possible for risk of injury (exposure of the birds eye to high intensity laser radiation is carefully avoided). The use of dead birds is unlikely to produce representative 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 this is 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 current representation of bird wakes as illustrated in Fig. 6. But is this reason enough to abandon the momentum 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]. A main prediction from the momentum jet-based flight model is a P(U) function, usually referred to 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. The relative contribution from the circulation of the upstroke is weighted in relation to the circulation of the downstroke, ranging between 0 at slow sped to 1 at cruising speed. Symbols are downstroke wavelength (λd), upstroke wavelength (λu), downstroke wake inclination angle (ψd) and upstroke wake inclination angle (ψu) (based on ref. [30]).

model [65–67], and with uncertainty estimates of the model prediction empirical data must differ quite a lot in order to falsify the prediction [64, 68]. However, various tests of components of the momentum jet model do suggest that it contains anomalies that are not easily rectified by changing parameters only. For example, Pennycuick et al. [49] concluded that the value representing the parasite 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 minimum wing 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. The consequence is that the power required to fly at speeds > Ump becomes very low, in fact lower than the power required to glide at some speed [69], which seems unlikely. Also, the predicted speed of Ump in very large birds such as swans becomes higher than the speed actually observed in 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 downstroke elliptic and an upstroke rectangular wake structure (Fig. 13). The relative circulation between down and upstroke is allowed to vary as a function of speed as u = Cud, where Cu varies from 0 at hovering to 1 at a fully developed cruising flight cc-wake. The model has to satisfy the weight balance criterion. In principle, the net thrust (related to the power output) could be determined by the areas of the wake structures projected onto vertical planes. It was shown that this model was self-consistent with the thrush nightingale data [30]. There are two principal difficulties facing such a program, however. First, the drag is much smaller than the lift-supporting component in the 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 the wake structure representing the inviscid induced drag model were clearly separable from the viscous drag wake. In steady self-propulsion (of any body), the net fore-aft momentum balance will be zero, as thrust balances drag. As duly noted in [30], only if the viscous drag wakes can be 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 flying birds (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 of flight speeds during migration, foraging flight, display flight, predator evasion, load lifting with respect to prey/food or fuel (fat) stores used for long non-stop flights, etc. [71, 72]. In all these examples the ‘optimum’ behaviour can be understood on the basis of flight mechanical theory in combination with some appropriate currency assumption and an optimization rule. Perhaps surprising, also decisions regarding optimal departure time and associated fuel load from a stopover in migratory birds can be derived on the basis of aerodynamic principles [72]. During moult—the periodic replacement (typically once per year) of flight feathers—the wings have reduced area due to missing or growing feathers and will change in shape due to moult gaps. The consequences on flight performance from moult gaps have recently been analysed from an aerodynamic perspective [73–75]. Aerodynamic performance is tightly linked to morphology and so aerodynamic models 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], Two scenarios—the trees-down and the ground-up—have long been the two competing hypotheses about the evolutionary trajectory that lead to powered flight. From a vortex wake perspective it is easy to see a natural (gradual) transition from gliding flight to flapping flight with an initially low 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 than going via gliding to powered flight [78], while recent fossil finds from China of unambiguously feathered theropod dinosaurs suggest a ground-dwelling protobird [79]. However, even more recently a bizarre ‘four-winged’ dromaeosaur Microraptor gui [80], also from China, suggests a gliding 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 extinct animals, but an aerodynamic analysis of wake types and possible interaction between forelimb and hind limb vortices might help us understand how these dinosaurs flew and to follow the evolutionary trajectory that led to powered flight.

6.6 Future prospects

The quantitative visualization of wakes in freely flying birds has only begun. In addition to a low turbulence wind tunnel it requires a co-operative and well-trained bird [30]. It is now important to get more data from a range of sizes and wing morphologies in order to establish a more generally valid vortex wake based theory of animal flight. On the technical side there are foreseeable improvements, such as 3D PIV and increased repetition rate of pulsed lasers, which will improve the geometric characterization of wake vortices and their dynamics. This could yield time-series animation data on wake dynamics and the 3D wake topology would help in the partitioning of the wake disturbances in drag and lift components. Near-wing PIV data will elucidate the aerodynamic mechanisms of gliding and, hopefully, flapping flight in birds. Since birds have evolved adaptations for efficient flight during about 150 million years, it is perhaps not surprising that their flight endurance surpasses that of current man made flying vehicles of similar size by two orders of magnitude. The study of migratory birds as a model system is therefore likely 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 Mikael Rosén, to whom I am very thankful for a long-term collaboration. Geoff Spedding,Adrian Thomas, WIT Transactions on State of the Art in Science and Engineering, Vol 4, © 2006 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Vortex Wakes of Bird Flight 731 and Roland Liebe helped improve the manuscript signiﬁcantly. A.H. is a Royal Swedish Academy of 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 Foundation for International Cooperation in Research and Higher Education (STINT).

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