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The Journal of Experimental Biology 206, 2313-2344 (2003)
Copyright © 2003 The Company of Biologists Limited
doi: 10.1242/jeb.00423

A family of vortex wakes generated by a thrush nightingale in free flight in a wind tunnel over its entire natural range of flight speeds

G. R. Spedding^1,*, M. Rosén² and A. Hedenström²

¹ Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA 90089-1191, USA
² Department of Animal Ecology, Lund University, Ecology Building, SE-223 62 Lund, Sweden

^* Author for correspondence (e-mail: geoff{at}usc.edu)

Accepted 2 April 2003

	Summary

TOP Summary Introduction Materials and methods Results Discussion References

 
In view of the complexity of the wing-beat kinematics and geometry, an important class of theoretical models for analysis and prediction of bird flight performance entirely, or almost entirely, ignores the action of the wing itself and considers only the resulting motions in the air behind the bird. These motions can also be complicated, but some success has previously been recorded in detecting and measuring relatively simple wake structures that can sometimes account for required quantities used to estimate aerodynamic power consumption. To date, all bird wakes, measured or presumed, seem to fall into one of two classes: the closed-loop, discrete vortex model at low flight speeds, and the constant-circulation, continuous vortex model at moderate to high speeds. Here, novel and accurate quantitative measurements of velocity fields in vertical planes aligned with the freestream are used to investigate the wake structure of a thrush nightingale over its entire range of natural flight speeds. At most flight speeds, the wake cannot be categorised as one of the two standard types, but has an intermediate structure, with approximations to the closed-loop and constant-circulation models at the extremes. A careful accounting for all vortical structures revealed with the high-resolution technique permits resolution of the previously unexplained wake momentum paradox. All the measured wake structures have sufficient momentum to provide weight support over the wingbeat. A simple model is formulated and explained that mimics the correct, measured balance of forces in the downstroke- and upstroke-generated wake over the entire range of flight speeds. Pending further work on different bird species, this might form the basis for a generalisable flight model.

Key words: thrush nightingale, Luscinia luscinia, flight, aerodynamics, wake, wind tunnel, digital particle image velocimetry (DPIV)

	Introduction

TOP Summary Introduction Materials and methods Results Discussion References

 
The problems in understanding bird flight aerodynamics
A complete, correct and/or detailed understanding of the aerodynamic mechanisms of importance in bird flight is complicated immensely by a number of factors. The most basic problem is that flight speeds are sufficiently slow (typical values for the mean forward speed, U, may range from 1–20 m s^-1) and the length scales are sufficiently small (mean chord, c, ranging from 1–10 cm), that the effects of viscosity are not ordinarily negligible. This fact can be written more formally by calculating a characteristic value for the dimensionless Reynolds number,

where is the kinematic viscosity. For U=10 m s^-1 and c=5 cm, Re3x10⁴. This is an extremely inconvenient number. It lies well below typical values of 10⁶ for small planes where viscous effects can safely be presumed to be restricted to thin, attached boundary layers, and it lies well above characteristic values of 10² where the flow over the body and in any wake is laminar and well-organised. On the contrary, even at moderate angles of attack, the flow over well-designed aerofoils veers notoriously between separated and non-separated states, with dramatic differences in mean and instantaneous force coefficients as a result. Over and above treatments found in standard aerodynamics texts, one must also account for the fact that in animal flight the wings themselves are moving relative to the body, and furthermore that they are not rotating steadily like a propeller, but are beating up and down, accelerating and decelerating with each cycle. To this one adds the effects of flexible wing surfaces that not only have complex geometric descriptions, but also significantly change their shape during the wing beat cycle. Although it is straightforward to compile long lists of complicating factors, it is not clear which of them are important, and why and when. Mechanical or numerical models that slavishly mimic each property lack generality while elegantly simplified analysis might simply be irrelevant.

Describing fluid motions by the vorticity field
An attractive alternative to measuring or predicting aerodynamic forces on odd-shaped bodies with high-amplitude, unsteady motions is to investigate instead the air motions in the wake that are caused by the body (the term `body' here is used in the general sense to mean solid body, and it includes all wings and appendages). It is frequently convenient to describe fluid motion by its vorticity ,

(1)

where u is the velocity vector. Both and u are vector fields, and is a measure of the direction and magnitude of the local rotation in a fluid; it is exactly twice the local angular velocity. Textbooks such as Batchelor (1967, p. 92) and Lighthill (1986, p. 43) speak clearly and elegantly about the analysis and description of fluid motions in terms of the vorticity. Here, we note that a non-zero component of vorticity accompanies any shearing motion in a fluid, and so a description of u in terms of is not only convenient mathematically, but is also directly connected to the mechanical strain deformations associated with work being done on the fluid particles.

A further mathematical convenience is to speak of vortex lines, which are three-dimensional curves along which|| is constant. In a homogeneous fluid without viscosity, there are restrictions on how vortex lines can be arranged and, if and when the vortex lines are collected in simple groups or clusters, then a description of the fluid motion in terms of its vortex lines can be quite economical. The strength of a vortex is measured by its circulation,

(2)

which is the vorticity integrated over a material surface, S. When the vortex geometry is simple, then identification of a suitable surface is simple also, and in aerodynamics applications, can be related quite readily to integrated or localised forces on the wing.

Applying these methods to the aerodynamic analysis of bird flight holds out the promise of replacing a very large and intricate computation, involving highly unsteady motion of very complex geometries, with a much simpler description of the distribution of wake vorticity. The unsteady forces on the wings themselves are either inferred or ignored as mechanical and energetic quantities are calculated directly from the wake footprint which, by Newton's laws, must contain the integrated history of the forces exerted by the body on the fluid. In particular, the kinematics of the wings themselves are important only in so far as they produce a certain disturbance in the wake.

The basis for theoretical models of bird wakes
Are bird wakes actually composed of simple collections of vortex lines? The first and most well-known of the mechanical models of bird flight is the actuator disc model, expounded by Pennycuick (1968a, 1975) and others. Here the bird is entirely replaced by an idealised circular disc, which acts to accelerate air across it, and deflect it downwards. Implicitly the wake is indeed composed of collections of vortex lines, as the uniformly accelerated flow is separated from the unaffected ambient by a tube of circular cross-section, composed of all of the vortex lines in the otherwise undisturbed flow. The simplicity is extreme, but has made it the most widely used and robust of calculation methods in use today (e.g. Pennycuick, 1989). Some context and consequences of the actuator disc modelling strategy are considered in Spedding (2003). Note that since the kinematics of the beating wings have been disposed of entirely, the model can have little to say about the consequences of variation in wingbeat amplitude, frequency or cyclic variations in planform geometry — all topics of potential interest. Moreover, the infinite tube is unlikely to be a very close approximation of the actual wake.

The first serious attempt to construct an aerodynamic model of bird flight based on a realistic wake structure was by Rayner (1979a, 1979b, 1979c), who proposed that each wingbeat was only aerodynamically active on the downstroke. The starting and stopping vortices produced at the beginning and end of this downstroke were connected by a pair of trailing vortices shed from the wingtips, and so the wake was composed of a series of vortex rings, or more accurately, elliptical loops. This sounds deceptively simple, and the process cartooned in Fig. 1 gives some indication of the assumptions required and likely complexities.

View larger version (17K): [in this window] [in a new window]   Fig. 1. The generation of a single, closed-vortex loop during a downstroke can, in principle, lead to a simple wake model geometry. The bird body (which has no aerodynamic significance) is represented by a stick supporting the wings. The assembly moves at constant speed, U. As the wings accelerate at the beginning of the downstroke (A), they shed vorticity into the near wake, which rolls up as a concentrated starting vortex. During the downstroke (B), the starting vortex remains connected to the two wingtip vortices, which elongate as the downstroke progresses. At the end of the downstroke, the wings decelerate, shedding vorticity into the wake along the trailing edge, and then vanish (C), taking no further part in the aerodynamics until they reappear at the beginning of the next wingbeat. The hypothetical deformed loop left at C then relaxes into, or can be modelled by, a planar ellipse, and the idealised model wake (D) is composed of a sequence of these, separated by spaces left by the inactive upstroke. Although this wake-generation mechanism is ostensibly simple, the details are not, and numerous assumptions about the formation, shedding and subsequent roll-up of vortex lines or tubes with complex curvature are built in. I, wake impulse; circular arrows indicate the local sense of rotation of the induced flow.

The vortex ring model was entirely theoretical, having no experimental support, although it clearly represented an improved picture from the old vortex tube, and was argued from reasonable grounds. It received independent support in experimental work published that year by Kokshaysky (1979), who showed that cross-sections through clouds of sawdust in the wakes of small passerines revealed ring-like structures, with one shed per wingbeat. The technique was not a quantitative one, however, and so certain critical quantities such as wake momentum and energy could not be verified. The vortex ring model received further support in quantitative reconstruction of three-dimensional tracks traced by clouds of neutrally buoyant, helium-filled soap bubbles for pigeons in slow (U=2.4 m s^-1) flight (Spedding et al., 1984), and for a jackdaw in similar conditions (U=2.5 m s^-1; Spedding, 1986). In both cases, however, the measured wake momentum was insufficient to provide weight support, and it was tentatively concluded that some as yet unidentified complexities in the wake structure or its measurement were responsible for this seeming paradox, which has remained unresolved.

Unexpectedly, experiments with the same apparatus on kestrel flight at moderate (U=7 m s^-1) speeds (Spedding, 1987b) showed no wake momentum deficit and no vortex rings either. Instead of discrete loops separated by aerodynamically inactive upstrokes, two continuous undulating vortex tubes were found, one trailing behind each wingtip, and without strong concentrations of starting or stopping vortices cross-linking the two. The measured circulation of the shed vortices was the same on down- and up-strokes, supporting this interpretation, and a net thrust was achieved by the variation in wake width due to flexion of the primary feathers during the upstroke. A cartoon of a constant-circulation wake model is shown in Fig. 2. This was a qualitatively new kind of wake structure, and while vortex rings might seem like a favourable configuration because they convey the maximum momentum per unit kinetic energy, the constant-circulation wake would also appear to be advantageous in minimizing the shedding of cross-stream vorticity. (The cross-stream vorticity is not absent, but occurs in the curvature of the downstroke trailing vortices.) In order to generate net thrust some variation in impulse must occur, but it is through varying the wing geometry, and not through varying the circulation on the remainder of the wing that continues to take part in the aerodynamics. These results were originally described in a thesis (Spedding, 1981), and the same experimental apparatus was subsequently used to visualise wakes of noctule bats, but without quantitative measurements (Rayner et al., 1986). At slow speeds (1.5, 3 m s^-1), the bubble tracks were interpreted to be tracing discrete rings or loops, while at higher speeds (7.5 m s^-1) the patterns seemed closer to the constant-circulation geometry seen in the kestrel.

View larger version (11K): [in this window] [in a new window]   Fig. 2. Constant-circulation wake. In (A), the effect of the (invisible) bird moving at speed U is to leave behind a pair of undulating vortices with constant circulation (1=2=constant), in which case potential cross-stream vortices denoted by broken lines have zero strength. Here the geometry is simplified for convenience so the wake appears as if the downstroke and upstroke portions were approximate ellipses and rectangles, respectively, as drawn in (B). Although the actual geometry assumed in most models (e.g. Rayner, 1986; Spedding, 1987b) is slightly more complicated, the fundamental principle remains that the wake impulse (I) from both down- and upstrokes points upward, contributing to lift, and hence weight support. Because the wingspan is reduced on the upstroke, the projection of area S1 onto a vertical plane will be larger than that of S2, and so the net impulse of the whole wake is forward, generating thrust.

Other than isolated photographs in review-type articles or books (e.g. in Norberg, 1990; Rayner, 1991a,b; Spedding, 1992), this remains the sum total of experimental evidence on the structure of vertebrate wakes in flapping flight. There are some obvious gaps to fill; for example, on how it is that wake patterns transition from one form to another. Spedding (1981, 1987b) cautioned against interpolating between only two data points, but speculated that intermediate wake forms between constant-circulation and closed-loop wakes might involve the gradual increase in strength of cross-stream vortices, as shown in Fig. 2. Rayner (1986, 1991a,b, 2001), on the other hand, has proposed that all bird wakes must be either one of the two forms (closed-loop or constant-circulation) and that these constitute two separate gaits, analogous in some respects to terrestrial gaits, of horses, for example.

Current status
To date, there have been no quantitative data on bird wakes at more than one particular flight speed for the same individual or species. Furthermore, all existing quantitative studies are based on the three-dimensional bubble-cloud seeding technique, where large parts of the overall wake volume can be simultaneously observed, but at the expense of rather low spatial resolution. Typically, 2500 bubble traces were recorded over a volume of approximately 600 mmx600 mmx400 mm (numbers from Spedding et al., 1984; Spedding, 1986), which is equivalent to a mean inter-bubble spacing of 27 mm in each direction, comparable to mean core radii of 35 mm and 30 mm for the vortex rings observed in the pigeon and jackdaw experiments, respectively. The inconsistent quantities at slow flight speeds could have been caused by structural details whose presence would only be discernible at higher resolution, and the assumptions forced upon the experimental analysis by the limited spatial resolution might closely reflect assumptions in the model under test, thereby rendering the test non-independent. Recalling the first part of this introduction, one might be especially wary when Reynolds numbers are in the range where quite disorganised and turbulent motions might be anticipated at small scales (of the order of a core radius), and in some cases at large scales (of the order of a mean chord, c) too.

Objectives
This paper reports on the results of an extensive series of experiments in measuring bird wakes over a continuous range of flight speeds in a low-turbulence wind tunnel. The measurement technique has been customised extensively for this particular application and offers improvement in spatial resolution by a factor of 10 and a similar improvement in accuracy of estimation of velocity fields and their spatial gradients. A companion paper (M. Rosén, G. R. Spedding and A. Hedenström, in preparation) describes the detailed wing kinematics of the same bird flying under the same conditions, allowing connections between the wingbeat and wake structure to be deduced. Here the motion of the wings themselves is ignored almost entirely, and we focus on a correct reconstruction of the most likely three-dimensional wake structure. Qualitative and quantitative changes in wake structure with flight speed will be presented. At most flight speeds, the wake is dissimilar to those previously reported and the consequences will be discussed.

	Materials and methods

TOP Summary Introduction Materials and methods Results Discussion References

 
The experiment
The experiment and its data analysis methods are quite new, and their design, implementation, validation and performance analysis are given in some detail in Spedding et al. (2003). A brief summary only will be given here.

Apparatus and bird training
Experiments were carried out using a closed-loop, low-turbulence wind tunnel designed for bird flight experiments (Pennycuick et al., 1997), and the general setup is shown in Fig. 3. Four juvenile thrush nightingales Luscinia luscinia L. were caught in southern Sweden on migration in August 2001, and brought to the wind tunnel aviary in Lund. After a period of acclimatization, daily flight training began, and soon revealed that two, and eventually one, bird would fly for prolonged periods in the test section. The bird was trained to sit on a perch that could be lowered for take-off and flight, and raised before landing. The bird was trained to fly at a position near the centre of the test section in low light conditions with an upstream luminescent marker as the sole reference point. The training was prolonged and rigorous, beginning more than 2 months prior to experiments, and progressing in conditions that gradually resembled the experiment, beginning with low ambient light conditions, and eventually to the introduction and maintenance of fog particles and occasional bursts of high intensity laser light.

View larger version (8K): [in this window] [in a new window]   Fig. 3. The bird (tn) is trained to fly at constant speed, U, the independently controlled speed of the wind tunnel. Two Stanford DG 535 delay generators (dg1,2), configured to run off a single crystal base timing clock, generate synchronised timing pulses to control the dual-head Nd:YAG (pl) laser output flash timing (ta) and the asynchronous reset (tb) for the two CCD array cameras (tm1,2). The timing of the reset pulses is determined by the mean speed, U, and by the downstream displacement of tm2 from tm1, and is designed to remove the mean flow from the measured displacement field. Digital images are acquired at independent interface cards (ic1,2) and transferred directly to PC RAM. The laser timing pulses are gated (gb) with the summed output from an array of LED-photodiode pairs so that if any one or more beams are interrupted by the bird, laser output stops. (Modified from Spedding et al., 2003.)

The laser was a Quanta Ray PIV II, dual head Nd:YAG from Spectra Physics, with a maximum flash intensity of 200 mJ pulse^-1, although it was used mostly at about 120 mJ. The timing between pulses in a pulse-pair can be as little as 1 ns. Settings of 100–500 µs were used in all experiments reported here. The double-pulsed laser beam was spread into a planar sheet by a sequence of converging and then two cylindrical lenses, before reflecting off a 45° inclined front surface mirror into the test section through a clear Plexiglass panel (Fig. 3).

A vertical grid of infrared LED-photodiode pairs was arranged so that if any beam was interrupted by the bird, the laser pulses would be automatically suspended. The flight speed, U, varied between 4 and 11 m s^-1, air density was 1.17–1.25 kg m^-3 and the temperature 16–20°C. From previous wind tunnel calibration data, turbulence intensities were calculated to be <0.06% of U in the speed range used (Pennycuick et al., 1997). These levels are too low to be measured directly by digital particle image velocimetry (DPIV) methods themselves (see Spedding et al., 2003).

Tables 1 and 2 give some basic morphological data together with some common aerodynamic performance measures for this experiment.

Wind tunnel corrections
The bird is small compared with the wind tunnel test section, and interactions with the side walls can be ignored. This can be demonstrated with a simple lumped vortex model of a thin airfoil (Katz and Plotkin, 2001, p. 119), from which one can write an expression for the modified lift, L', due to presence of solid boundaries in a confined duct of height h:

(3)

The magnitude of the correction is negligible (<10^-2) for all values of c/h ≤0.2, which is true even when the span, 2b, is taken as a length scale. This criterion perhaps should be taken as a lower limit, because possible proximity of the wake to the tunnel walls at the measuring station is of equal importance. In fact, it will be seen that the wake growth rates in both the y (spanwise) and z (vertical) directions were interestingly low, and corrections based on subsequently measured wake widths at the measuring station did not exceed 2x10^-2 (2%).

Analysis
Properties of Correlation Imaging Velocimetry (CIV)
The two laser pulses were imaged onto two Pulnix TM9701N full-frame transfer CCD array cameras, in upstream-downstream sequence. The digital image pairs (768x484x8 bits) were analysed using a custom variant of standard DPIV methods, known as Correlation Imaging Velocimetry (CIV). The collection of CIV techniques is described in detail in Fincham and Spedding (1997). CIV was developed to maximise the accuracy of estimation of very small particle displacements, regardless of computational cost. Arbitrary sized and shaped cross-correlation boxes can be defined and are completely decoupled from the similarly arbitrarily defined search domain. No FFTs (Fast Fourier Transforms) are used in the computation, and sub-pixel displacements can be estimated to 1/50^th pixel in the best case. In practice, one can expect 1/20^th pixel uncertainty. When mean pixel displacements are 5 pixels, the uncertainty is approximately 1%, and the velocity bandwidth is 1:100. In order to profit from the advanced numerical techniques it is essential to properly control/select the value of the timing interval, t, between exposures of the two images in a pair. In this two-camera variant, t is partly determined by the mean flow and camera separation so that the mean displacement field is zero. t is then tuned, on top of this value, to ensure that disturbance quantities (i.e. displacements due to the bird wake) fill out the range of displacements up to 5 pixels. Constraints on this calculation are the three-dimensional, cross-plane motion in the wake, and the light sheet (or slab) thickness, which is set to between 3–4 mm.

Customisations for bird flight measurements
Because the two successive images come from two separate cameras, there are extra distortions introduced by having two different lenses and two slightly different (unavoidably, within the manufacturing tolerances of the cameras) camera geometries, effective focal lengths and optical axis orientations. An extensive series of tests (described in Spedding et al., 2003) with test backgrounds of pseudo-particles and pseudo-displacement fields, allowed the CIV calculations themselves to be used to compute a mean distortion field at the same, or higher, resolution as the experimental data. The test or residual fields can be stepwise ramped up in complexity, from stationary object to fixed displacement, to moving object, to wind tunnel background flow. Finally, in the last stage, the flying bird is added to the set-up, and only differences between this case and the background flow are computed. Thus, the effects of optical misalignments and distortion are automatically compensated for, and the CIV calculation bandwidth is focused entirely on the wake displacement field due to the presence of the bird and its beating wings.

The disturbance displacement fields are calculated with 20–24 pixel correlation boxes, overlapping by 50% to yield pixel displacements on a nominally rectangular 58x54 grid, with aspect ratio one, and resolution of approximately =3.5 mm Note that is comparable to the light sheet thickness, which governs the averaging distance normal to the plane. The sampling volume is thus roughly cubical. This field is corrected for the finite displacement of the source correlation box and the flow is reinterpolated onto a grid with the same dimensions, using a two-dimensional, thin-shell smoothing spline. Adjustment of the smoothing parameter allows certain nonphysical displacement errors (if present) to be removed. The smoothing parameter is equivalent in the spline formulation of specifying a non-zero viscosity for the fluid (for details, see Spedding and Rignot, 1993), and does not involve any neighbourhood-averaging, which would be guaranteed to underestimate peak gradient quantities. Spatial gradients are calculated directly from the spline coefficients without recourse to further smoothing or averaging.

The analysis is performed in a frame of reference moving with the mean speed, U, and u, v and w are velocity components in the streamwise (x), spanwise (y) and vertical (z) directions in this reference frame. (This choice of coordinate system reflects the most common one for aerofoil or aircraft analysis, where y is almost always a spanwise location.) Data were taken in vertical planes aligned with the freestream. The bird would sometimes take up slightly different positions in y, or would drift slowly. With the position of the light slice fixed, its location relative to the bird could be checked from standard video images taken by a camera downstream of the test section. Silhouettes of the bird were visible against the bright vertical stripes of the over-exposed laser sheet image. The slice positions were categorised as centre/body, left/right midwing, left/right wingtip and left/right outer field, as illustrated in Fig. 4. All data described in this paper come from vertical slices at centre/body, left midwing and left wingtip. (Data from left and right wings did not differ, and there were many fewer right wing data runs as they represent unusually large departures from the standard position for the bird.)

View larger version (10K): [in this window] [in a new window]   Fig. 4. Classification of all spanwise locations by character code (bottom) and the three named categories appearing in this paper, centre/body (lr), midwing (lx) and wingtip (ly).

Safety considerations required the leftmost point of the data (governed by the left margin of the right camera image, which was determined by the light sheet fan-out and x-location) to be approximately 84 cm downstream of the bird. The wake left behind during the course of a wingbeat extends downstream by a distance x_c=Ut_c, where t_c is the evolution time of this wake segment. So, if that time is a wingbeat period, T, then a wake wavelength, is

(4)

The wingbeat frequency changes rather little as U ranges from 4–11 m s^-1 (M. Rosén, G. R. Spedding and A. Hedenström, in preparation), so increases steadily with increasing U. The downstream measuring location is 2–3 at U=4 m s^-1, and only 1 at U=11 m s^-1.

Quantitative analysis and wake-specific measurements
In each vertical slice, we have velocity components u and w as functions of x and z. These can be argued to be the most interesting components: since w is parallel to the gravitational vector, g, it describes the momentum changes and forces that counteract g. Similarly, variations in u are directly related to the fore—aft forces on the bird, which are the drag and thrust, opposed to, and aligned with the direction of motion. From the data the only measurable component of vorticity (Equation 1) is the spanwise vorticity _y, normal to the plane of the light slice,

(5)

The y subscript will occasionally be dropped for clarity. Since it is primarily maps of _y(x,z) that will be used to describe the wakes, it is very important to estimate this quantity as accurately as possible and to know the likely uncertainty. A usual rule of thumb for reasonable (and credible) estimates of uncertainty in gradient quantities in fluid flows of moderate complexity is ±10%. For rather more rigorous and quantitative statements of the likely uncertainties in application of the CIV method, see Fincham and Spedding (1997). Here, the extra care taken in isolating the disturbance field (which contains all the vorticity), and in saturating the measurement bandwidth through appropriate choice of t, gives a likely uncertainty in _y of ≤5%. This is discussed in detail, with evidence from extensive control experiments, in Spedding et al. (2003).

The predicted maps of _y(x,z) for ideal vortex loop and constant-circulation models (Figs 1 and 2) are shown in Fig. 5. Vertical cuts through a vortex loop should show two vortex cross-sections of equal strength for all vertical cuts except those at the wingtip, where disturbances on top of the streamwise vortices might be visible. By contrast, centreplane cuts through the constant-circulation wake should show almost nothing at all. Moving away from the centreline, cuts through the upper and lower curved branch of the trailing downstroke vortex should be seen. The wingtip pattern will look much the same as for the closed-loop.

View larger version (17K): [in this window] [in a new window]   Fig. 5. Idealised predicted spanwise vorticity y(x,z) in vertical cross-sections through (A) the vortex loop and (B) the constant-circulation wake models of Figs 1D and 2, respectively. Although sections further towards the wingtip cut more obliquely through the presumed vortex lines, the effect on peak|y| measurements would be small, and in A the sections through the closed loop are shown with unchanged amplitude. (The circulation will be unchanged.) If the wake has continuous trailing vortices (as in the constant-circulation model), then at the centreplane|y|=0 (B). Midwing cuts may have more complicated cross-sectional geometries if, as anticipated, they cut through transition regions between down and upstroke-generated vortices.

When and if the data do not conform to simple predicted geometries, the main challenge is in performing the inverse of Fig. 6, deducing the most likely three-dimensional structure based on stacks of two-dimensional slices. It is not impossible to do this, partly because of a classical result in mathematics due to Helmholtz, showing that in a homogenous field/fluid, objects such as vortex lines must either terminate at a boundary or form closed loops on themselves. When combined with the symmetry of the wing and body geometry and of the normal wingbeat kinematics, this quite strongly constrains (i) the number of possible vortex wake topologies that could plausibly be produced, and (ii) the number of self-consistent interpretations of limited data, such as vertical centreplane slices, or stacks of slices from centreline to wingtip. If, for example, in Fig. 5A the peak value, or integrated magnitude, of the cross-section contours of spanwise vorticity changes from slice to slice, then some component of streamwise vorticity must exist to account for the difference. If, on the other hand, the diagnostic values do not change within measurement uncertainty, then the most parsimonious explanation is that the cross-sections are passing through a single structure that intersects both measurement planes. It is the application of this reasoning that allows iterative testing and re-evaluation of postulated three-dimensional structures in the wake, so not only can existing theories be tested, as illustrated in Fig. 5, but new wake geometries can also, in principle, be proposed.

View larger version (103K): [in this window] [in a new window]   Fig. 6. (A—D) Four consecutive fields of y(x,z) with velocity vectors superimposed at half their true spatial resolution. The reference frame is moving with the mean flow, and so it is as if the bird had passed from right to left, leaving behind these traces in still air. The colour bar intervals correspond roughly to the measurement uncertainty. The colour bar is scaled asymmetrically about y=0, and the numbers at the ends show values in units of s-1. The circle-ended line shows the scale of the wingspan, 2b. The window size x,z is approximately 20 cmx18 cm. The circles drawn around locally maximum positive values of y(x,z) show the regions within which normalised circulation + is calculated. tot is calculated by including all above-threshold values in the same frame, regardless of whether they are within the local neighbourhood, or connected. Similarly, the negative peak is identified by the broken circle. The trailing vorticity attributable to the upstroke contains both negative and positive local peaks (large white arrows). In A these low-amplitude, positive peaks will be included in the sum for tot (because they have the same sign as the peak value), but in C, they will not. The development of an accounting procedure that correctly accounts for the real (as opposed to idealised) measured vorticity distributions is given in Figs 27 and 28 and their associated text.

View larger version (16K): [in this window] [in a new window]   Fig. 27. Total integrated circulation tot from all positive (filled circles) and negative (open circles) vorticity in the observation window, plotted as a function of flight speed U. Although no single window contains the entire wake structure, each selected window, centered on peak values of either sign, contains all of the vorticity shed either at the beginning of the downstroke, or at the end of the downstroke and beginning of the upstroke. (A) The fraction of the total circulation that is not contained in the strongest vortex cross-section is very much higher in the stopping (negative) vortices than in the starting (positive) vortices. (B) The total negative vorticity would be sufficient for weight support, but not the positive component. The sum of the two, which ought to be zero (recall the convention of plotting the absolute value of the negative components), is not.

View larger version (15K): [in this window] [in a new window]   Fig. 28. As Fig. 27B, but + (filled squares) now includes all traces of above-threshold positive vorticity found in the neighbourhoods of the predominantly negative signed vorticity. Neither component is significantly different from the other (the sums balance), and both are within experimental uncertainty of sufficiency for weight support (tot=1).

(6)

where the strength (circulation) of vortex A is calculated from the sum of all contiguous cells where _y exceeds some threshold value, such as 20% of its maximum. The calculation is robust and simple, but difficulties can arise when the area occupied is very diffuse, and the result must additionally be constrained to be inside some local spatial area. Moreover, using a fixed-fraction threshold ensures that some low-amplitude contributions will be omitted, and so controlling the unruly spread of vortex A by imposing a high threshold increases the severity of this underestimate. Here, we assume that the true distribution of below-threshold vorticity is something like a similarly thresholded Gaussian function, G. For this, or any other known or presumed functional form, one can calculate the fraction omitted for any arbitrary fixed threshold, and add that to the sum of Equation 6. For example, for the normalised Gaussian function with amplitude, A, and half-width, ,

(7)

where r is the radial distance from the centre, then the fraction of G above threshold T_G (where T_G varies from 0 to 1) is

(8)

When T_G=0.2, the above-threshold fraction of G is 0.8 of its total. This is the procedure followed for all estimates of circulation reported herein. Threshold values of 20% of the local maximum ensure that directly summed values remain above any likely noise level and the correction represents a reasonable compromise in presuming and/or estimating the contribution of the low-amplitude tails of the distribution.

For economy of presentation, normalised measures of|_y|_maxc/U for positive and negative-signed vortices will be named ⁺ and ^-, and their corresponding normalised circulations, /Uc, will be denoted ⁺ and ^-. Means for a particular flight speed and span location will be denoted by overbars in the text if the context is otherwise ambiguous. All error bars in the figures show standard deviations (S.D.).

	Results

TOP Summary Introduction Materials and methods Results Discussion References

 
Reconstruction of the vortex wakes
The vortex wake structure will be reconstructed from series of vertical slices for three flight speeds, U=4, 7 and 10 m s^-1. As will later become clear, there is no special significance to these speeds, and they are used as examples of low, medium and high speed flight over the range 4–11 m s^-1 achievable by the thrush nightingale. Measurements are summarised as combined velocity and vorticity fields, with velocity vectors shown amplified (the arrow length corresponds to some factor greater than one times the real spatial displacement over the exposure time t) and halved in spatial resolution. These are superimposed upon _y(x,z) mapped onto a discrete colourbar whose effect is to show colour contours, where the contour interval is commensurate with the claimed measurement uncertainty in _y. Thus, if a feature can be seen in the data, it probably does exist.

More than 4000 velocity fields have been analysed over the range of flight speeds, and there is no way to show all of the supporting evidence and measurements for all of the reconstructions. The slow-speed case will be presented in some detail, and then subsequent cases will be summaries only, even though they have been based on similar amounts of both qualitative and quantitative evidence.

Deducing the wake structure from multiple vertical slices at different spanwise stations is an iterative process. Plausible, but temporary conceptual models of the wake structure are formulated and tested through repeated inspection and measurement of large numbers of velocity/vorticity maps. Qualitative models guide quantitative tests, which in turn support or contradict the models. The presentation of the qualitative wakes data precedes the quantitative measurements in this paper, because appreciation of the former is required to understand the significance of the latter. For this reason, the qualitative reconstructions will be summarised and completed in this section, requiring a certain amount of interpretation to be mixed in with the raw data. The benefit is that the conceptual and physical models can act as an organising structure within which the significance of the extensive quantitative measurements can be understood and evaluated.

Slow speed (U=4 m s^-1)
Fig. 6 shows four consecutive frames of the vertical centreplane velocity and vorticity fields. Since the wingbeat frequency is approximately 14 Hz (at all flight speeds) while the sampling rate, determined by the maximum laser repetition rate, is 10 Hz, each frame shows a portion of the wake from a different wingbeat, slightly phase-shifted, so the wake self-samples as it is advected by the mean flow past the fixed cameras. The starting vortex at the left of Fig. 6A is succeeded in Fig. 6B by another which is shifted to the right (increasing x). In the next frame (Fig. 6C), no starting vortex is visible, the whole frame being occupied by upstroke-generated motions. Subsequently (Fig. 6D) a third starting vortex appears. Approximately 4.2 wake periods have passed by the cameras in four frames. The wingbeat frequency f calculated from this phase-shifted time series is 14 Hz. f calculated from high-speed video kinematic analysis is 14.2 Hz.

A second interesting consequence of these phase-shifted data is that, to some extent, the degree of steadiness of the wingbeat can be inferred from the repeatability of the wake pattern. Thus we note that while the starting vortex is always the most visible object in the wake, its location in z does not change very much. The wake structure is quite level, and the flight must have been also. This can now be turned into an important criterion for further selection of data, since the only other control on the bird position is months of training. If, and only if, a wake pattern is repeated along the 10 Hz sampling sequence, then the data are accepted as having come from steady level flight.

Regarding the vorticity field itself, it is immediately obvious that positive-signed, starting vortices (or those so-presumed) are significantly higher in amplitude and more coherent than their negative-signed counterparts. This is always the case, without exception, and the sequence shown here is completely typical in this regard. The two frames showing upstroke-generated vorticity (Fig. 6A,C) show very broadly distributed, low amplitude (but measurable) traces with little clear structure.

The asymmetry in peak vorticity is readily quantifiable, as in Fig. 7, where a simple time series is plotted of the strongest absolute value vorticity in each frame. Values shown as filled circles come from the remnants of starting vortices and those as open circles from the stopping vortices appearing at the end of the downstroke. Not only are the peak values different, by a factor of 3–4, but the total integrated circulations (also plotted as squares in Fig. 7) are different too, albeit by a smaller amount. It is not simply that the same amount of vorticity has been spread over a larger area; the total amounts are apparently different. We will later revisit this topic in some detail.

View larger version (11K): [in this window] [in a new window]   Fig. 7. The peak absolute value of the spanwise vorticity||max (circles), rescaled by the wing chord c and mean speed U as a function of time t in wingbeat periods T. The normalised circulation (squares) is also plotted on the second ordinate. Filled symbols, positive vortices; open symbols, negative ones. The first four time steps correspond to the data in Fig. 6. The time series represents successive sections of the wake passing through the observation window. The field is strongly asymmetric in both the peak vorticity and its integrated total strength.

A more compact and easily interpretable summary of the data of Fig. 6 is given in Fig. 8, where segments of the time series have been patched together to show the spatial structure of the wake from one complete wingbeat. Since each frame is a phase-shifted view of a repeated wake structure, neighbouring frames are overlaid with the first in time located rightmost, and passing right to left through the original time series. Although the detailed structure varies somewhat from wingbeat to wingbeat, this basic wake pattern is seen in all centreplane slices. None of the vortices are perfectly circular in cross-section, the starting vortex is significantly more compact and pronounced than the stopping vortex, and although there are trails of negative vorticity continuing on into the upstroke (again, this is always the case), qualitatively, it appears quite weak. By implication, the upstroke is mostly aerodynamically inactive. Other than the weak stopping vortex, a closed-loop wake model with most or all aerodynamically useful forces occurring on the downstroke would be a reasonable approximation of this structure.

View larger version (47K): [in this window] [in a new window]   Fig. 8. A reconstruction from three consecutive frames of Fig. 6 to show the vortex wake over slightly more than one wingbeat cycle. The wake is shown as if left in still air by the bird passing from right to left. The silhouette is drawn approximately to scale and in the correct vertical (z) position but its horizontal (x) location should in fact be displaced by about 3 to the left (upstream) because the measuring station is that far downstream of the bird in the test section. During the time required for the wake to advect past the cameras (approx. 3T, or 0.21 s), the wake has moved downwards under its self-induced convection speed. The three component frames are matched approximately but the data are not edited or reinterpolated to improve the fit, and the borders are left outlined so their location is clear. The wingspan bar (2b) is placed to begin at the start of the downstroke. The wake wavelength is determined by the flight speed and wingbeat period and is shown as a double-arrowed bar. The relative time spent on downstroke and upstroke is given by the downstroke ratio, and can be verified from the wake picture. The colour bar and its scaling are as given in Fig. 6, and are fixed for all low-speed wake images (Figs 6, 8, 9, 11).

View larger version (124K): [in this window] [in a new window]   Fig. 9. Magnified view of the rightmost starting vortex in Fig. 8. The apparent centre of rotation deduced from the arrows does not lie on the peak of the spanwise vorticity.

View larger version (112K): [in this window] [in a new window]   Fig. 11. (A) Vertical cross-section through a midwing plane in the 4 m s-1 wake. Plotting conventions are as described in Figs 6 and 8, so relative bird motion is from right to left. The colour bar scaling is fixed to that established in the centre plane (Fig. 6), so saturation of the negative part indicates a relatively stronger stopping vortex contribution. (B) Vertical cross-section through the wingtip plane in the 4 m s-1 wake. A and B, together with Fig. 8, can be compared with the three idealised patterns of Fig. 5.

View larger version (17K): [in this window] [in a new window]   Fig. 10. Profiles of the velocity components u(z) (A) and w(x) (B), where (x0,z0) is the location of the peak in y. The vertical dotted lines projected from z=z0 and x=x0 intersect the curves of u(z) and w(x) slightly offset from the u=0 and w=0 lines. Original data points are shown as diamonds (A) and triangles (B), joined by straight lines. Just noticeable are dotted line curves that join profiles either side of x0 and z0, respectively.

Figs 9 and 10 demonstrate that the spatial resolution is sufficient to estimate these subtle effects and to measure the shear gradients with low uncertainty (as previously claimed in Materials and methods). A shift by one grid point left—right (±x) or up—down (±z), as shown by the dotted lines very close to the solid line profiles in Fig. 10, makes very little difference to the profile gradients. There are approximately seven points across the core in each profile, and the core diameter defined by the distance between velocity peaks is approximately 2 cm in both x and z.

An equivalent reconstruction to Fig. 8, but for the midwing and wingtip sections, is given in Fig. 11. At the midwing (Fig. 11A), the two vortex cross-sections are now separated by a smaller distance, consistent with intersections further out through a curved structure. The stopping vortex has a higher peak value, both relative to the starting vortex, and absolutely, as shown by the black saturation of the lower end of colour bar. As in Fig. 8, there is little coherence in the upstroke regions, and no systematic shrinking of their streamwise extent in cross-section as we proceed from wing root to wingtip. The starting vortex cross-section at midwing, however, is more complex than closer to the centreline, appearing double-, or even triple-peaked. Again, this is quite characteristic of the many (250) midwing wake sections analysed at this flight speed. The outer region of the vortex loop is altogether less coherent (in this cross-section) than at the centreline.

The three-dimensional picture is completed by the wingtip reconstruction of Fig. 11B. From the starting vortex (left), which has a quite distinct second peak, the more complex cross-sectional structure noted in the previous figure is maintained. The stopping vortex is again more distinct than in the more central sections, but also has two strong peaks. There are some trace negative patches in a cloud around the main stopping vortex, but nothing at all in the upstroke part.

The evidence accumulated from the vertical sections at three spanwise locations points to a relatively simple vortex topology, where the majority of the vorticity (and circulation) is contained within a curved loop traceable to the downstroke. If this is the primary structure then the circulation of the vortices should be the same in each section. Fig. 12 shows the peak vorticity and the circulation of the strongest vortex in the data comprising the reconstructions of Figs 8 and 11. The peak vorticity ⁺ and circulation ⁺ of the positive (starting) vortices does not change significantly from wing root to wingtip. Neither does ^-. However, the magnitude of ^- increases towards the wingtip. This confirms: (i) that the starting vortex loop is continuous and unbranched, and (ii) that during the downstroke the shed vorticity becomes more diffuse, not all of it collected in a single concentrated lump. This numerically confirms what was already qualitatively readily apparent in Fig. 8, but with consistent support from the off-centre slices.

View larger version (14K): [in this window] [in a new window]   Fig. 12. Variation in peak vorticity magnitude||max (circles) and circulation (squares) rescaled by the wing chord c and mean speed U for positive (filled symbols) and negative (open symbols) vortices in the slow-speed wake as a function of spanwise distance divided by the semispan y/b.

Fig. 13 summarises the most likely three-dimensional topology of vortex lines making up the slow-speed wake. It is a simplification, but has the following essential properties: (i) the initial starting vortex is concentrated, (ii) during the downstroke, vortex elements become separated, (iii) the stopping vortex is quite diffuse, with elements trailing into the upstroke, and (iv) the upstroke nevertheless does not appear to generate significant coherent motion.

View larger version (16K): [in this window] [in a new window]   Fig. 13. A possible representation of the slow-speed wake by a small number of vortex lines, based on data such as Figs 8, 10, 11, 12. The primary wake structure is a collection of loops, drawn as solid ellipses. They intersect the centre/body plane of observation along the major axis marked ad, which makes an angle d with the horizontal (downstroke; au, u, respectively, for upstroke). a is the measured strength of the starting vortex. b is the total measured strength of the more diffuse collection of vortex lines left at the end of the downstroke. c is small compared with both a and b, and the collection of rectangular upstroke wake vortices (broken lines) is an idealised cartoon version of the observed trace patterns that are quite disorganised and weak. Their primary effect is to disrupt the structure of the measured stopping vortex, which they do because vortex lines of opposite sense lie close together. When their strengths go to zero, a standard closed-loop wake model results. The projection of the downstroke wake length in x, d, onto the centreline is denoted by the double-headed arrow. U is the mean flight speed; bold arrow indicates direction of flight.

Medium speed (U=7 m s^-1)
Characteristic patterns of _y(x,z) for the centreplane, midwing and wingtip sections are shown in Fig. 14A–C. Fig. 14A is a composite of several frames. It shows a surprising, but quite characteristic, new wake structure that can be seen at a number of flight speeds. The upstroke is aerodynamically active, as judged by the downwash inclined normal to a complex upstroke-generated vortex structure that is distinct from the downstroke vorticity. The cross-section through the upstroke wake is complex, but has mostly positive vorticity at the beginning and mostly negative vorticity at the end. This suggests that a different circulation (it must drop towards the end of the downstroke and then increase again at the beginning of the upstroke) is established on the wings during the upstroke, so that the whole wake is a sequence of alternating structures from up- and downstrokes.

View larger version (95K): [in this window] [in a new window]   Fig. 14. Composites of the wake at moderate speed U=7 m s-1. The plotting conventions are as previously given in Fig. 8. The colour bar scaling is fixed for all centreplane (A), midwing (B) and wingtip (C) sections.

At midwing (Fig. 14B), the only trace of the upstroke structure is from the small upward induced flow. Vortex cross-sections can have complicated geometry, and there is an interesting mix of positive and negative patches at the junction between down- and upstroke. A similar composite, more towards the wingtip (Fig. 14C), shows another complex mosaic of positive and negative patches at this junction. Upward-induced flows can be detected at the beginning and end of the upstroke region where the section is closer to the main wake structure. The most likely collection of vortex lines to account for these figures (and many others like them) is shown in Fig. 15. Each repeating wake segment (one per wingbeat) contains two conjoined closed-loop structures. The way in which the slow-speed wake evolves into this one is by the increase in relative strength of the cross-stream vortices associated with the upstroke. It does so gradually as the speed increases. Note that while the relative magnitude increases, the absolute value does not, as the colour bar scaling for the negative vorticity component has decreased from -250 s^-1 to -160 s^-1 (cf. Figs 6 and 14).

View larger version (17K): [in this window] [in a new window]   Fig. 15. Most likely wake topology deduced from all data at U=7 m s-1. The basic form is quite similar to the low-speed wake in Fig. 13 (the symbols and notation are the same), but the upstroke-generated portion (broken lines) is stronger, and more distinct from the downstroke-generated loops (solid ellipses).

High speed (U=10 m s^-1)
At high speeds (Fig. 16), the mapping of _y(x,z) at the centreplane (Fig. 16A,B) onto the locally rescaled colour bar shows measurable cross-stream vorticity at almost every instant during both upstroke and downstroke. No single structure or pair dominates, and there is a quite seamless transition between the down- and upstroke-generated downwash. The wake wavelength, =UT, continues to increase (inevitably). Fig. 16B also shows a second section through the downstroke—upstroke transition that is closer to the true centreline than the main composite, and the absence of any large/strong stopping vortex is notable. Progressing further out towards the midwing (Fig. 16C,D), the strongest downwash (flow moving mostly vertically downwards) is confined to the downstroke. Already the upstroke trailing vortex is inboard of this section and very little disturbance can be seen during this wingbeat phase. The vorticity distribution can be quite complex as shown in Fig. 16D. The large black region in the section through the negative vortex shows that the fixed colourbar scaling established by the centreline section has been saturated. It is much easier to identify both starting and stopping vortices than was the case at the centre/body section. The oblique cut through the stopping vortex in both (Fig. 16C,D) then runs through the upwards-induced flow induced by the vortex that has projected through the page towards the viewer. Further out towards the wingtip (Fig. 16E,F), there is only a downward and then upward induced flow at the downstroke-generated portion.

View larger version (67K): [in this window] [in a new window]   Fig. 16. (A) Composite for high-speed (U=10 m s-1) flight, close to the vertical centreplane, but slightly offset, showing the structure over an entire wavelength. (B) Closer to the true centerline. (C,D) Similar sections through proximal and distal midwing locations; (E,F) the same for the wingtip section.

The sections of Fig. 16 are consistent with steadily moving outwards through a curved vortex structure that does not all meet at the centreline, but mostly extends on into the upstroke. The pattern in Fig. 16E also shows a shear layer developing above the obliquely cut wingtip vortex, with two locations where vectors point from right to left. This component is probably a viscous drag wake that is entrained along the vortex core. In high-speed wakes it is very common to see this close to the wingtip, and the free shear layer instabilities riding on top of the core structure are also common. It is doubtful whether the instabilities themselves have any impact on the bird, but the viscous drag wake is an important component of the force balance at high speeds.

Fig. 17 shows the most likely wake structure based on Fig. 16, completing the three samples of the family of wake structures. The tentative three-dimensional wake models of Figs 13, 15 and 17 are based on these and other data, and also on certain of the quantitative results in the following section, where quantitative data are organised primarily towards making estimates of wake impulse and momentum balance at different flight speeds. Some of these results, however, particularly involving circulation estimates at different spanwise locations, provide strong support for the reconstructions in this section (as also noted in Fig. 5 and its discussion), which were only completed following this analysis.

View larger version (14K): [in this window] [in a new window]   Fig. 17. Most likely collection of vortex lines for the high-speed wake, based on data such as shown in Fig. 16. Now the primary wake structures are primarily oriented in the streamwise direction and there is no preferred location for the comparatively weak cross-stream vortices y. The correctness of this structure can be determined by testing whether d=u. For an explanation of other symbols, see Fig. 13.

The wake reconstructions are based on assemblages of independent vertical slice data from multiple wingbeats, and this procedure only works if the flights themselves are steady and repeatable. Mostly, the predominant structures self-select because they can be seen repeatedly on hundreds of occasions, but there are exceptions whose appearance can be traced to some unusual (in this context) flight behaviour. Before proceeding with the quantitative analysis of the proposed wake structures in steady flight, two non-standard examples will be briefly given, first because they shed some light on the normal wake structures and their interpretation, and second because they point to further studies of important flight modes.

Other wakes
Fig. 18 shows the vertical centreplane wake for a brief period of gliding flight at 11 m s^-1. The patches of largest|_y| mark a wake that extends straight back behind the bird. Here and elsewhere, the velocity field is dominated by the induced downwash, which in general points downward and backward. Fig. 19A shows a vertical profile of the streamwise-averaged horizontal velocity,

(9)

where the sum at each vertical z location is taken over all the discrete streamwise data points, x_i, in the field of view. The leftwards pointing peak represents the departure from the mean profile due to the body drag.

View larger version (117K): [in this window]