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
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
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Summary |
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Key words: thrush nightingale, Luscinia luscinia, flight, aerodynamics, wake, wind tunnel, digital particle image velocimetry (DPIV)
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Introduction |
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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 ,
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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,
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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.
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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.
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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.
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Materials and methods |
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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.
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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 100500 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 1.171.25 kg m-3 and the
temperature 1620°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.
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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:
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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
1/50th pixel in the best case. In practice, one can expect
1/20th 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 34 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 2024 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.)
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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 xc=Utc, where
tc is the evolution time of this wake segment. So, if that
time is a wingbeat period, T, then a wake wavelength, is
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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 foreaft 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,
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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.
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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.
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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.).
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Results |
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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 34, 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.
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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.
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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 leftright (±x) or updown (±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.
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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.
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Medium speed (U=7 m s-1)
Characteristic patterns of y(x,z) for the
centreplane, midwing and wingtip sections are shown in
Fig. 14AC.
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.
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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).
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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
downstrokeupstroke 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.
|
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.
|
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) |
|
|
Fig. 19B shows the
streamwise distribution of vertically averaged, vertical velocities,
![]() | (10) |
Fig. 20 is a single wake
image from a vertical plane, just off the centre flight line, for flight at 9
m s-1, when the bird was briefly gliding and adjusting its position
in the test section. As in Fig.
18, the largest|y| values lie on a nearly
horizontal line with mean flow from right to left, representing a net drag.
The downwash is much stronger in the upper half-plane, quite different from
the steady wakes data and from the gliding example of
Fig. 18. The wake defect
(Fig. 21) is also much larger
than the straight gliding case (Fig.
19A). The high-drag wake of
Fig. 20 comes either from the
body and tail, or from trailing-edge shedding, close to the wing root, of a
partially stalled wing during manoeuvre. The two cases cannot be distinguished
unambiguously here, but the roughly equal amplitudes and number of structures
with positive and negative sign argue for the former.
|
|
Figs 18 and 20 isolate the contributions of body and tail to the wake structure since the wings are beating with small amplitude and acceleration, if at all. These wakes are qualitatively different from the flapping wing wakes in the standard reconstructions. The body wake disturbance itself (Fig. 18) is of very low amplitude (Fig. 19) and cannot usually be detected amongst the much stronger disturbances generated by the wings. It need not be considered further in the following momentum balance calculations.
Wake impulse and momentum balance
Calculation of wake circulation, impulse and reference
quantities
Having established a qualitative picture of the wake structure, the peak
spanwise vorticity|y|max and the circulation,
of a structure in cross-section were measured for all fields appearing
to be part of a steady level flight segment. There were 1261 of these.
In the following plots, the measured circulations are normalised in two
ways. Most obviously, one might divide by a reference 0,
which is the circulation that would be required for a wing of equal span in
steady flight to support the weight. This can be readily calculated since the
KuttaJoukowski theorem gives a simple relation between the lift per
unit span L' and the circulation,
![]() | (11) |
![]() | (12) |
![]() | (13) |
Alternatively, one might consider the case where all of the weight support
derives from the impulse of elliptical vortex loops, one shed per downstroke
(Fig. 1D). The vertical
impulse, Iz, is given by the product of the projection
onto the horizontal plane of the planar area of the vortex loop with its
circulation,
![]() | (14) |
![]() | (15) |
![]() | (16) |
![]() | (17) |
0 and
1 both decrease with flight
speed, U, as the problem of providing sufficient impulse can be
spread over an area that increases with U. In fact, there is a simple
relationship between
0 and
1 determined by
the geometry of the flat wake and the projected ellipse. To see this, one may
rewrite Equation 13 slightly:
![]() | (18) |
Combining with Equation 17, the ratio of the two circulations is:
![]() | (19) |
![]() | (20) |
Individual measurements of vortex patches in low-, medium- and
high-speed wakes
Fig. 22 shows the
distribution of measured circulation versus peak spanwise vorticity
for all measurable vortex cross-sections in steady flight at U=4 m
s-1. A measurable cross-section is defined as a contiguous
above-threshold region surrounding a local peak value in
y(x,z). Characteristically there will be one of
these in any single frame, either of positive or negative sign, from
structures created at the beginning and end of the downstroke, respectively.
Each sum in Equation 6 is made around the peak value in the whole frame, and
the thresholding and correction procedure described in Equations 7 and 8 is
then applied. Example areas and peak values are shown superimposed on the raw
data in Fig. 6.
|
In Fig. 22, the closed
symbols show positive-signed elements (from the starting vortex) and the open
symbols are for the negative-signed ones (stopping vortex). At the centre/body
sections (Fig. 22A), the range
of + (peak vorticity of positive sign) is larger than the
equivalent range of
+ (the circulation associated with the
peak maximum positive vorticity), whose values lie close to
0, but (or, given Equation 20, consequently) not close to
1, the reference most pertinent to this wake geometry. The
open symbols for the stopping vortex are significantly lower in magnitude, in
both
- and
-. They are quite closely
clustered around the mean in
-, as compared with the
+ values. Throughout
Fig. 22, the
- distributions are more compact than their
+ counterparts, which is opposite to the degree of
compactness, or coherence, of the spatial distributions of
y(x,z). Proceeding through midwing
(Fig. 22B) to wingtip
(Fig. 22C), the distribution
of starting and stopping vortices begins to overlap slightly, as the centroids
of each cloud (horizontal and vertical solid lines) approach each other,
mostly due to a small decrease in the
+. By contrast, recall
that in the individual sequences of Fig.
12, the procession from wing root to wingtip was accompanied by an
increase in
-. Each individual result is rather
sensitive to details of the cross-sectional geometry of the vortices and the
resulting fraction that contributes to
-. In
Fig. 12, based on Figs
8 and
11, all of the stopping vortex
was included in the calculation of
-. The problem of
correctly accounting for
that does not occupy a compact domain will be
revisited shortly.
The three main results from Fig.
22 are that: (i) a closed vortex loop wake with even the highest
measured circulation in the vertical centreplane would be unable to support
the weight of the bird. The average fraction of weight support provided in
this model is 45%. (ii) Significantly higher circulations
(+,
-) are not measured at more distal
vertical planes, providing no evidence for any other simple candidate vortex
topology. (iii)
- and
- are significantly
beneath
+ and
+ and so a significant
asymmetry is not accounted for by a simple closed-loop model.
A similar survey for the medium-speed (U=7 m s-1) case
is given in Fig. 23 (note the
rescaled abscissa). While there are differences in
+,- and
+,- at the
centre/body plane, they are much smaller than at the slow flight speed, and
+, in particular is clustered much more closely around the
mean value. At the midwing (Fig.
23B), the mean values of
and
are experimentally
indistinguishable, a situation that continues on into the wingtip section
(Fig. 23C). In all cases,
however,
/
0
1 and so
/
1
0.5.
|
The insufficiency of plausible vortex loops with the measured is
less worrisome than in the low-speed case because the qualitative wake
reconstructions (Fig. 15) have
already shown the presence of a significant upstroke component. Nevertheless,
while
/
1 might be expected to be less than one,
values of 0.5 or less might significantly complicate later attempts at
calculating force balances.
Fig. 24 shows a different
picture again at high-speed (U=10 s-1). In the qualitative
reconstructions, the difficulty in finding identifiable concentrations of
spanwise vorticity in the centreplane has already been noted
(Fig. 16). Consequently,
Fig. 24A has only two data
points of either sign at the centreplane. It is quite likely that these
represent slices toward the outer boundary of the region considered to be
centre/body, but there is no manual editing of the data and so they must stand
as given. +,- and
+,- at the centreplane have equal magnitude, and
both rise at midwing (Fig.
24B). Astraight-line wake composed of such vortices would more
than balance the weight. The values of
and
both fall slightly
towards the wingtip (Fig.
24C), but here the measurement is mainly dominated by flow
instabilities of the trailing vortices themselves
(Fig. 16E,F) whose magnitude
is difficult to relate to the strength of the main vortex structure, where the
primary component is now streamwise (
x), and most readily
measured at the midwing section (Fig.
16C,D).
|
Continuous variations in peak and integrated vorticity magnitudes
over the range of flight speeds
The choice of the three flight speeds covered in Figs
22,
23,
24 does not indicate anything
particular about those speeds, and the same measurements have been made at all
flight speeds in 1 m s-1 increments between 4 and 11 m
s-1. The mean and standard deviations of and
for
all measured flight speeds are shown in
Fig. 25. At the lowest flight
speeds,
is
strongly asymmetric, as is
, though less
noticeably (mirroring the result in Fig.
22). As U increases, the absolute value and asymmetry in
both measures decrease. They do so gradually, without any discontinuities or
abrupt changes.
and
fall with increasing U for the
same reason they do in fixed-wing airplanes, as reflected in Equation 12,
because the wake area per unit time available for weight support
increases.
|
The asymmetry in strengths and shapes of vortex patches complicates
estimates of the rate of wake momentum generation, but the main points can be
illustrated with quite conservative assumptions. We again make use of the
convenient reference values 0 and
1, the
approximate circulations required for weight support if the wake were composed
of straight line trailing vortices (
0) or isolated ellipses
(
1).
Fig. 26 summarises the
variation in the strength of the starting and stopping vortices at the three
different span locations. Beginning at the centreline data
(Fig. 26A), one can
immediately observe that, at slow speeds, despite the fact that the starting
vortex is significantly stronger than the stopping vortex, if the wake were
composed of elliptical vortex loops of this (highest) measured value, it would
still only provide about half the impulse required for weight support over one
wingbeat. While /
1 is only approximately 0.5,
/
0 is therefore (Equation 20) approximately 1.0, but
the best approximation of the wake structure at slow speed was isolated loops
and certainly not continuous trailing vortices.
|
As U increases, the loop model becomes less and less appropriate,
and one may then refer to values of /
0. At the
centreline, these always fall short of 1.0, but this is to be expected as the
vertical centreplane is the worst place, in principle, to measure a wake
composed primarily of streamwise (not spanwise) vorticity. Thus, at higher
speeds, one looks to more distal planes, such as the midwing
(Fig. 26B) where reasonable
cross-sections through curved trailing vortices are found. Indeed,
/
0 values here are generally equal to, or above 1.0.
The stopping vortex strengths are, if anything, higher than those of the
starting vortices. They are both stronger here than at the centreplane,
requiring/implying the existence of a significant streamwise component. This
is completely consistent with the qualitative reconstructions of the previous
section, which also argued for the existence of streamwise vortices continuing
on into an aerodynamically active upstroke.
Of course, the constant circulation wake provides a smaller vertical
projected area than the idealised rectangular wake, and so the circulation
requirement will be higher than 0, but not by huge amounts.
The measured circulations for the gliding and flapping wake of the kestrel in
Spedding
(1987a
,b
)
were 0.5 and 0.55 m2 s-1 respectively. The rather small
(10%) difference is partly due to the slightly larger lateral spreading of the
wake on the downstroke of the flapping flight wake. Here too, it is likely
that the wake width (which cannot be directly measured here with any useful
accuracy) is (i) slightly increased and (ii) has increased in the time elapsed
between generation and measurement. This time is always greater than
T, the stroke period (recall Equation 4 and the related
discussion).
The situation is quite similar at the wingtip
(Fig. 26C). At higher speeds,
/
0 is close to, or slightly above 1.0, and accounting
for sufficient vertical momentum generation in the wake seems unlikely to be
problematic. (A slightly more refined calculation will follow.)
For U≤7 m s-1, the absolute value and relative magnitudes of starting and stopping vortices are similar at all spanwise locations. Unlike the case at higher speeds, there is no implication of other strong concentrations of vorticity, for example, in different orientations. The diffuse and relatively weak stopping vortex is just as diffuse and relatively weak at the wingtip as it is at the centreline.
A complete accounting of the measured circulation distributions
In trying to account for the shortfall in momentum of the slow speed wakes,
the existence of some other topology than the ones emerging from the current
reconstructions is unlikely. However, when the patches of vorticity are as
incoherent and diffuse as they are in the stopping vortices, it is very
possible that only some fraction of the total circulation is being correctly
accounted for (recall Equations 68).
Fig. 27 shows the results of a
much less selective approach where all values of
y(x,z) of either sign above the usual threshold of
20% of the local maximum value are added toward the total circulation of that
sign
tot. (This amounts to adding up all the vorticity of
the same sign as the peak vortex in each of the panels AD of
Fig. 6.) Integrating small
values of the vorticity over large areas must be done with caution, as it
would be quite easy to accumulate an area-dependent sum that is mostly
measurement error. The conservative 20% threshold criterion avoids this
problem since there are no errors of such magnitude. Moreover, in each case a
quiescent patch of background is chosen and summed using the same criteria
over a small area. This is then rescaled and subtracted from the grand total
so that if any background noise did contribute to the total, then its average
value would be subtracted out again. The magnitude of this correction was
usually exactly zero, and always less than 1% of the total, when it could be
traced back to difficulties in finding a truly quiescent patch in the
vorticity field.
In Fig. 27A,
/
tot for the starting vortices starts at values very
close to one at slow flight speeds (4 and 5 m s-1), and then drops
continuously to just above 0.2 at the higher speeds. At lower U,
there are no significant concentrations of positive vorticity other than the
single, highest amplitude structure. That is not the case for the stopping
vortices in the same figure. Here, they never account for much more than one
third of the total circulation of that sign. Significant amounts of the total
circulation of negative sign are unaccounted for by concentrating only on one
structure. That being the case, then we might conduct a hypothetical case
where at slow speeds the vortex loops are imagined to have started with a
uniform circulation equal to the strength of the total circulation in the
wake. The ratio of
tot/
1 will therefore be
equal to 1.0 if such an accounting procedure would support the weight. It is
shown as a function of flight speed in
Fig. 27B. The total
negative-signed circulation would have been sufficient for weight support, if
at one time it were a good measure of the uniform circulation in an elliptical
loop with the expected geometry, one produced every downstroke.
Now the problem is that the relation,
![]() |
Where has the circulation gone? The answer is contained in the
reconstructions of y(x,z) shown in Figs
6 and
8. Careful inspection of that
part of the vorticity field attributed to the upstroke shows that this domain
contains regions of
+ as well as
-. (Look
carefully at Fig. 6A,C where
white arrows show low amplitude, positive vorticity peaks, and also the middle
of Fig. 8.) Now recall the
accounting procedure for counting the sums of each signed patch of vorticity
towards the total circulation (Equation 6). If the data includes a complete
starting or stopping vortex structure, then the peak value is found and all
circulation of that sign is eventually accounted for, but in that partial view
of the wake only. The assumption is that the sections containing either
starting or stopping vortices will have exclusively and solely vorticity of
that sign. Any positive-signed patches that appear in predominantly
upstroke-generated motions will be omitted. Similarly, any negative patches
around starting vortices will be left out. In practice there are few of the
latter, but the same is not true of the former, and while rather careful
attention has been paid to correctly including all of the diffuse patterns of
negative vorticity, the similarly scattered positive patches occurring in the
wake regions categorised as upstroke have effectively been ignored. The
pertinent parts of Figs 6 and
8 show that this contribution
should not be assumed to be negligible, and the calculation procedure must be
adjusted accordingly, one more time.
Fig. 28 shows a revised
calculation of tot/
1.
-
is calculated as before, but the estimate of
+ now includes
the contributions from sections centred at both starting vortices and stopping
vortices.
tot/
1is never experimentally
less than 1.0, for calculations based on total circulations of either sign,
which, in turn, do not differ from each other. Sufficient circulation has been
detected in the wake, so that if it is assumed to have come from some
initially coherent closed-loop structure with approximately uniform
circulation of that magnitude, then this would be sufficient for steady level
flight. This balance is only achieved if we include positive vorticity
apparently shed towards the end of the downstroke and/or the beginning of the
upstroke. The interaction of this opposite-signed vorticity with the mostly
negative-signed patches shed at the end of the downstroke might account for
the diffuse and incoherent distribution of the stopping vortices at low
speeds.
Spanwise variation
One of the implications of this result is that the measured changes in
circulation in the wake towards the end of the downstroke and beginning of the
upstroke entail changes in circulation on the wing, which in turn imply
translational or rotational acceleration. These changes may be uniform along
the span, or local. The variation in
measured|y|max and
+,- with
spanwise location might provide evidence for local shedding, but no strong
variation of the average values was shown in
Fig. 12 for the 4 m
s-1 case, other than the noted increase in
towards the wingtip.
The data for spanwise variation in
+,- and
+,- for all flight speeds are summarised in
Fig. 29.
|
Although the effect does not fall outside the error bars, at low flight speeds the peak positive vorticity decreases systematically from the centreline outward to the wingtip (Fig. 29A), while the trend is reversed for the negative vorticity peak. Both trends gradually disappear at higher flight speeds (e.g. U≥7 m s-1). At the same time, the circulations do not differ measurably or systematically from centreplane to wingtip (Fig. 29B) over any range of U. The simple interpretation of the flight model of Fig. 15 is fully consistent with both observations, as the same total circulation is spread out amongst an increasingly diffuse collection of vortex lines as the wingbeat progresses from beginning to end of downstroke.
There is some tendency discernable in
Fig. 29 for a reduction in
relative magnitude of the centreline circulation values at higher flight
speeds, which is better shown in Fig.
30, where is normalised according to the reference value
for weight support in the rectangular (gliding) wake. At higher flight speeds,
the centreline value is less than 1.0, and less than the values at more distal
sections. Again, this is exactly what one would expect for a simplified wake
model (such as Fig. 17) of
primarily streamwise vorticity, which curves to intersect the data plane much
more reliably and prominently towards the wingtips, where the streamwise
vortices are presumed to originate.
|
The low-, medium- and high-speed wake topologies that are consistent with the qualitative and quantitative arguments thus far are summarised in Fig. 31. While these simple models based on the thrush nightingale data are incomplete in some details, the successful approximate accounting for the wake momentum flux at all flight speeds supports the idea that the basic patterns are correct, and might be used as a basis for constructing, admittedly simplified, analytical and predictive models.
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![]() |
Discussion |
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![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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As U increases, the distribution of vorticity in vertical streamwise planes becomes more complicated (Fig. 14). The strength of cross-stream starting vortices gradually decreases, and the relative contribution of the upstroke gradually increases. Finally, at the higher speeds, the centreline vorticity distributions, rescaled locally on the usual colour bar, show a broad spectrum of variations throughout the wingbeat (Fig. 16). The amplitudes are significantly diminished, however (Fig. 25). It is not clear whether previous experiments were unable to distinguish these relatively low-amplitude spatial variations in the velocity field, or whether the thrush nightingale is less proficient at constant-circulation wake generation than the kestrel, which is the only other point of quantitative comparison.
Perhaps the nearest points of theoretical comparison for these gradually
varying wakes with complex cross-stream vorticity distribution are the wake
circulation distributions predicted by the optimisation models of Hall and
Hall (1996) and Hall et al.
(1998
), who numerically solved
a variational problem to find the optimum (in the sense of minimum induced
drag) spanwise, time-varying circulation distribution for rigid wings in both
low- and high-amplitude flapping flight. A series of vertical cuts through
their {x,y}-wake distributions of iso-circulation contours
(fig. 13 in
Hall and Hall, 1996
) might be
difficult in practice to distinguish from the result of the same operation on
the empirical models of Fig.
31. There are two factors that complicate direct comparisons.
First, the wake in Hall and Hall's formulation is assumed to be left at the
trace of the wingtip trailing edge, without roll-up. While corrections to the
computed optimal circulation distribution due to wake roll-up might be small,
the experimentally measured wakes have rolled up and, particularly at low
flight speeds, have had plenty of time to do so. As Hall and Hall point out,
there is then no obvious way to infer the original circulation distribution on
the wing from the rolled up late wake. For the same reason, the late wake
would not be expected to have the same distribution of vorticity as the
theoretical one, and the significance of observed differences is not clear.
The second point is that rigid wings do not have the characteristic upstroke
flexion that leads to the wake asymmetry required for positive thrust.
Instead, the variations in spanwise circulation distribution move inboard. The
net result is equivalent, with a net positive thrust, but one is achieved by
varying the span of a wing with relatively constant circulation on the
remaining reduced span (or so the constant circulation model holds), while the
other involves a spanwise variation of the circulation itself. Future
quantitative tests against appropriately modified formulations of these
analytical/numerical models would be very interesting.
In summary, while the low and high-speed wakes are not inconsistent with previous closed-loop and constant-circulation models respectively, most wakes (at most speeds) are not exclusively of either kind, but have some intermediate structure, with amplitudes of cross-stream vorticity at the centreline that decay gradually as U increases. The three examples of U=4, 7 and 10 m s-1 in Figs 8, 14 and 16 (and in idealised summaries in Fig. 31) are simply examples on a continuum whose gradually varying quantitative properties are summarised in Fig. 25. Since most bird flight models assume either closed-loop or constant-circulation wakes, then most models are inapplicable to most flight speeds encountered here.
No evidence for gaits
Until now, the only two previous structures discovered in bird wakes indeed
appeared to be either some kind of closed loop or a pair of continuous
trailing vortices. This has led to speculation about the possible existence of
two (and only two) distinct gaits in bird flight (e.g.
Rayner et al., 1986;
Rayner and Gordon, 1998
;
Rayner, 2001
), analogous,
presumably, to the gaits encountered in terrestrial locomotion, and this
notion has even spread to the more general literature (e.g.
Alexander, 2002
).
In terrestrial locomotion the changing balance of gravitational and
inertial forces and spring forces in the muscles and tendons leads to distinct
gaits such as the well-known walktrotgallop transitions in many
quadrupeds (e.g. Hildebrand,
1965; Alexander,
1982
), when the gaits can be distinguished in the differing phase
relations and duty factors amongst the limbs involved. The notion of distinct
gaits in terrestrial locomotion then involves quite abrupt transitions between
them and also frequently involves the existence of forbidden speeds close to
their margins.
In various studies of wingbeat kinematics
(Tobalske and Dial, 1996;
Tobalske, 2000
;
Hedrick et al., 2002
), the
absence of any detectable sharp transition in any measurable kinematic
parameter has nevertheless been interpreted as marking a `gradual' transition
between `gaits'. In Hedrick et al.
(2002
), local changes in angle
of incidence,
, and relative wind velocity, urel,
were inferred from wing traces, and then converted through a presumed
two-dimensional analytic relation of CL(
) to lift
L, and thence by Equation 11 to estimates of
. Aside from the
question as to whether local two-dimensional, steady, inviscid
CL(
) relations can apply to the high-amplitude,
three-dimensional unsteady flapping motions of the wings (particularly at low
U), it is most likely that gradual changes in
urel and
would lead to correspondingly gradual
changes in
on the wing and in the wake. That is actually inconsistent
with any gait selection mechanism, but fully consistent with the gradual
variation in strengths of cross-stream vortex structures observed here.
No other study of bird or bat wakes, quantitative or qualitative, has involved more than two flight speeds. There is no indication, in any of the results in Figs 22, 23, 24, 25, 26, 27, 28, 29, 30, where numerous quantities are plotted as continuous functions of flight speed, that any sharp or discontinuous transition in wake topology occurs, at any U. Moreover, most of the wake topologies, at most flight speeds, are not closed loops or continuous vortices, but are some kind of intermediate form, where the strength of the cross-stream vorticity gradually decreases with increasing flight speed. In the case of the thrush nightingale, the notion of distinct gaits is not only non-useful, but it is qualitatively and quantitatively incorrect.
Since there are no other wake data covering a range of flight speeds, then
the reasonable working hypothesis is that gradual transitions occur from
low-speed to high-speed wakes of flying birds through gradual increases in
cross-stream vorticity, much as originally hypothesised and discussed in
Spedding
(1981,1987b
),
Pennycuick (1988
) and Spedding
and DeLaurier (1996
). We
should point out, however, that the common intermediate wake structure is not
quite as predicted by these authors. Instead of single connecting strands of
vorticity representing decrements and increments of the circulation on the
wing, the upstroke here sheds its own starting and stopping vortices, to form
a double-ringed wake. The upstroke wake is more distinct at medium to low
speeds. It forms the bridge between wakes with discrete elements at low speeds
and continuous variations at high speed. The purportedly distinct closed-loop
and constant-circulation wakes are otherwise simply single points on the
continuous curve.
The wake momentum paradox
The problem
The wake momentum paradox arose because quantitative measurements of the
wake in slow-flying pigeons (Spedding et
al., 1984) and jackdaws
(Spedding, 1986
) produced at
most 50% of the momentum required for weight support. The magnitude of the
deficit greatly exceeded any reasonable uncertainty estimate of the wake
measurements used in its calculation (primarily the diameter and circulation
of the closed-loop structures). Using the same measurement and analysis
techniques, there was no corresponding deficit for the medium-speed flight of
either gliding or flapping kestrels (Spedding,
1987a
,b
).
In the absence of further data in the intervening years, the conundrum has
remained unsolved.
Although it has been suggested (Rayner,
1991b,
2001
) that birds in very slow
flight do not or cannot fly straight and level but are in fact decelerating at
about 1/3 g, there is solid evidence to the contrary. This
includes experimental evidence from analysis of high-speed cine film
(Spedding, 1981
) of both
pigeon and jackdaw flight under very similar circumstances to the original
experiments, and from tracings of trajectories of the feet or eyeballs in the
same multiple-flash photographs used for the original wake analysis
(Spedding, 1986
). The evidence
also includes the practical difficulties that, for the particular experimental
geometry, a ballistic trajectory of constant 1/3 g
deceleration would require initial vertical speeds greater than the horizontal
flight speed, and that the birds did in fact arrive in front of the cameras
with a very small net increase in height. The original statement of the
problem should thus be taken at face value, and the low speed results continue
to provide a puzzling context for this work.
The solution
The initial results in this study (Fig.
22) replicated and confirmed the original wake momentum paradox,
with circulation values between 3060% (/
1 in
Fig. 22) of that required for
weight support from planar vortex loops with the measured size. It became
clear (Fig. 27) that a
significant fraction of the total circulation could reside in low amplitude,
diffuse patches of vorticity, and it was only when positive vorticity from the
end of the downstroke part of the wake was included in the total
(Fig. 28) that the sums
finally added up, equally, to 1.0. The previous wake momentum deficit was
caused by substantial amounts of circulation being effectively omitted from
the calculations because it was (i) beneath the measurement resolution, and
(ii) not in the expected place.
The bubble cloud method calculated the wake circulation by integrating in a
straight line down the centreline of the ring-like wake structure
(Spedding et al., 1984).
Application of Stokes's theorem to fluid flows that can be represented by
concentrated vortex lines or tubes embedded in an otherwise irrotational flow
shows that any line integral around a closed path containing the vortex lines
will converge to the same value of the circulation. However, the total
circulation cannot be measured if the closed curve does not include all the
vorticity. The straight line approximation presumes that all the vorticity of
opposite signs lies either side of the dividing streamline. We now see that
this is not the case, and so the previous technique was bound to underestimate
the total wake circulation. Furthermore, the comparatively limited resolution
of the bubble cloud method did not allow the other important but diffuse
traces of vorticity to be distinguished. In this study, the initial
measurements of the circulation in Fig.
26 effectively share the same assumptions, confining all the
measurements to particular contiguous blobs, and that is why they show the
same apparent momentum deficit. The deficit disappears when all the
above-threshold vorticity, in its complex distribution, is properly taken into
account.
The difficulty in using measured circulation values to deduce wake momentum
generation rates in a real experiment should not come as a surprise, for three
reasons. First, although making force estimates from wake surveys is a
classical wind tunnel technique, in practice it requires careful measurement
in tightly controlled conditions, and recent studies
(Spedding et al., 2003;
Spedding, 2003
) have shown the
non-negligible uncertainty in estimating drag (for example) from DPIV wake
measurements behind even a simple fixed wing model geometry at Reynolds
numbers and aspect ratios comparable to the bird flight experiments reported
here. The measurement difficulties are compounded when the downstream
measurement location, x/c, is far away, and when one is
trying to exploit a simplified conceptual model of a complex generation
mechanism. Both concerns apply here, particularly at low flight speeds.
The second point is that efficient locomotion of well-trained animals in a properly controlled experiment will probably not generate large excesses in momentum above that required for propulsion and weight support (if applicable). The more rigorous the experiment, the smaller will be the excess. The wake impulse to correctly balance the known (or presumed) body forces is therefore a maximum measurable quantity, and most measurement techniques will approach this value from below, erring on the low side. The story told by Figs 22, 23, 24, 27, 28, reveals how to do this for the family of wakes discovered for the thrush nightingale.
The third and final point concerning the likely difficulty in making
momentum-balancing calculations in turbulent flows at moderate Reynolds number
is that phenomena such as cancellation of vorticity and reconnection of vortex
lines can significantly modify the qualitative and quantitative properties of
fluid flows. Cantwell and Coles
(1983) measured circulation
deficits of up to 50% in moderate Re flows behind circular cylinders.
The abrupt change in topology of neighbouring vortex structures through mutual
interactions has been extensively investigated (e.g.
Boratav et al., 1992
;
Zabusky et al., 1995
) for
initially parallel tubes and for orthogonal orientations. The fact that the
sums eventually did balance in the bird wake could be used to argue that these
dissipative interactions did not in fact occur (or rather that they were not
significant in the overall energy budget), but the situation may not be so
simple, and the crude accounting method where all circulation of either sign
was lumped into simplified down- and upstroke structures may camouflage a more
complex and intrinsically difficult problem. Luckily, this problem currently
lies beneath the accuracy of our simple vortex-wake model reconstructions, and
an approximate balancing of the forces can be considered to have been
achieved. However, one may note that significantly improved measurement
resolution in future studies might in fact uncover significantly harder wake
measurement and force balance problems.
To the degree of accuracy that one might reasonably claim from these wind
tunnel experiments, the momentum balance puzzle can be considered to be
solved. Ultimately, the resolution of the long-standing wake momentum paradox
was only possible because the new, customised data acquisition and analysis
methods allowed estimates of the velocity field and its gradients with
superior resolution in space and amplitude. It is not likely that any method
that relies on tracking of individual bubbles, even if it is in three
dimensions, will have such resolution. Finally, it is worth inspecting once
again superimposed images of the velocity and vorticity fields such as
Fig. 8. If the accurate
calculation of y(x,z) were not available,
qualitative inspection of the velocity field alone would give no hint of the
true complexity of its gradient field. Similarly, any conclusions that rely
solely upon qualitative interpretation of even well-resolved velocity fields
or bubble tracks have no chance of providing unambiguous information about
likely vortex wake structure, which can only be deduced with assistance from
quantitative measurements.
High-lift mechanisms at low flight speeds?
The major contribution to previous shortfalls in wake momentum measurements
seems to be related to shedding of positive vorticity in the latter part of
the downstroke or early part of the upstroke. The source and significance of
this are not known. If it indicates a commensurate temporary increase in
circulation on the wing, this may be either some kind of high-lift mechanism
using control of separated flows, or it may only mark a control adjustment
whose effect is not noticeable until the beginning of the upstroke.
The continued immersion of the wings and body of slowly flying birds in the
downstroke-generated wake has been noted before (Spedding,
1981,
1986
) and is an inevitable
consequence of the high reduced frequency at low flight speeds
(Table 2), where the smallest
fraction of a wingspan of forward travel is achieved with each wingbeat cycle
(
/2b=UT/2b
1.1 for U=4 m
s-1). One can expect not only bodywake interactions but also
wakewake interactions as the spacing between successive structures is
small, or comparable to their size. Again there are two issues that may or may
not be related. First, the various interference effects of bodywake and
wakewake interactions are very likely to disrupt any preexisting
orderly wake structure, making it more difficult to account for the both the
energy and momentum when the wake is eventually measured, at least one
wingbeat later. Second, there is the possibility that these interactions might
be advantageous, either directly in increasing lift or reducing drag, or
indirectly, in assisting flow control either through the appropriate
positioning of wake structures or the maintenance of favourable pressure
gradients.
An interaction between the wake vorticity and wing surface has been
demonstrated by Dickinson et al.
(1999) (see also
fig. 4 in
Yan et al., 2002
) for
mechanical model simulations of the hovering flight of the fruit fly.
Intermittent, high-lift forces at the beginning of up- and down-strokes were
measured and correlated with the wing section intersecting favourable induced
flows from previous parts of the wingstroke. This phenomenon was termed `wake
capture', though it can also be viewed more generally as a strategic placement
of the wing in a pre-existing, non-uniform flow field. In hovering flight, the
reduced frequency, k=2
fc/2U, is infinite, and
the hovering fruit fly and its model both sit directly on top of the
pre-existing wake, increasing the likelihood of significant interactions. The
Reynolds number is also significantly lower (Re
102)
and the flow around and behind the wings is dominated by large vortices
generated by boundary layer separation at the leading and/or trailing edge. It
is possible that an analogue of wake capture occurs in bird flight at very low
speeds, and that it both provides useful mechanical force and complicates the
wake measurement. The existence and importance of strongly separated flows on
bird wings at low flight speeds is currently a matter of conjecture, direct
measurement being very difficult. The most promising approach will likely be
adaptation of mechanical models to non-zero forward speeds and to geometries
and Reynolds numbers approaching that of bird flight.
The footprints of gliding and control: quick estimates of horizontal
and vertical momentum generation
The gliding and control wakes shown in Figs
18 and
20 help to interpret the usual
steady flapping flight case by contrast. One aspect is to clarify the role and
relative importance of the wake shed by the body alone. When a wake is
measured continuously throughout the wingbeat, as in Figs
14 and
16 for moderate and high
speeds, respectively, one might suspect that the effect of the body itself
(rather than the wings) is being measured, especially in vertical centreplane
cuts. To some extent this is necessarily true, as the body is treated as part
of the wings whenever the wings are aerodynamically active. One cannot
therefore distinguish between induced downwash due to wing loading or due to
body loading since, to a first approximation, they are the same thing. The
dominance of the wing-induced downwash can be clearly seen in the gliding wake
of Fig. 18, superimposed on
which is a relatively low-amplitude drag wake attributable to the body.
The interpretation would be considerably different if there were
significantly different patterns of vortex shedding at the wing root than at
the midwing, or if the peak vorticity (and circulation) magnitudes were
different in the two locations. Fig.
14A,B shows this is not the case. Furthermore, the primary
structures observed in Fig.
14A trace the up-and-down path of the wing and not the body, whose
vertical excursions are very much smaller, and not in phase [data in
Pennycuick et al. (2000) show
the vertical body position leads the vertical wing position by 90°].
The mean wake profiles such as Fig.
19A can in principle be integrated to find the rate of change of
momentum in x. The details can be found both in current textbooks
(e.g. White, 2003) and the
classics (a detailed account appears in
Prandtl and Tietjens, 1934
).
Far downstream of the original disturbance, the pressure fluctuations can be
neglected and the drag force can be estimated from
![]() | (21) |
Nevertheless, comparative estimates can be made between the gliding and
control cases. The mean velocity defect profiles of Figs
19A and
21 are both taken from the
vertical centreplane, behind the body, but with an uncertainty in y
position of about 50% of a body diameter. If Equation 21 is applied to these
profiles, using UX(z) as a measure of
u'(y,z) by assuming that the profile is circular in
cross-section, then FD,Control2FD,Glide. The
detectable difference shows that control manoeuvres can be detected and
measured in the wake. Here a drag force of twice the usual body drag is used,
presumably to adjust the x-position in the wind tunnel, moving
further downstream from the perch and reference point. The adjustment to move
in the other direction (upstream, back towards the reference marker) will
not be observable in a directly comparable wake (same structure but
mean positive defect), but as a flapping wake with stronger measured
circulation. The calculations are approximate only, but they show the
potential of the method for measurements of wake features from control or
unsteady manoeuvres, and also for drag measurements in general.
There is strong interest in making correct estimates of total drag derived
directly from the fluid motions, because the experimental estimation of
frictional and profile drag coefficients of wings and bodies of animals, dead
or alive, tethered or in free flight, is very difficult, even more so at
Reynolds numbers typical of bird flight
(Table 2), despite various
ingenious experimental attempts to do it (e.g.
Pennycuick, 1968b; Tucker,
1990a
,b
;
Pennycuick et al., 1992
,
1996
). At the Reynolds numbers
and aspect ratios in question, estimating drag even from the wake of a fixed
wing by integrating wake profiles and calculating forms of Equation 21 is not
as straightforward a procedure as it might first appear
(Spedding et al., 2003
;
Spedding, 2003
), because
unsteady and three-dimensional effects are always present. Nevertheless it is
possible, and further experiments are recommended in repeatable conditions
designed to assure steady gliding (tilting the wind tunnel, for example) so
that systematic profile surveys can be taken across the wake to include all
contributions to the drag.
Equation 21 does not include contributions from the induced drag, which
introduces terms with velocity components in the spanwise and vertical
directions. However, there are yet simpler expressions for the vertical
induced velocity of a finite lifting wing (see
Prandtl and Tietjens, 1934).
In the far wake, the induced vertical velocity, w1, is
twice the value on the wings themselves, and the lift force is the product of
w1 times the mass flux affected by the presence of the
wings,
![]() | (22) |
Having shown that simple fixed-wing aerodynamic models can be applied without great problems to the appropriately selected data sets, we now consider a general model for the more complex flapping flight case, based on the measured bird wake data.
A simple flight model: the ER wake
Here, we attempt to construct an empirical flight model. It is based
specifically on the thrush nightingale data, but contains features that may be
quite general. In this spirit, and given the difficulties in describing the
precise wake geometry, the model need not be complicated and some quite broad
assumptions can be allowed. A good starting point for a general model that can
approximate the geometry and transitions in
Fig. 31 might be the
EllipseRectangle (ER) geometry, whose basic principles were
originally introduced in Fig.
2B. Fig. 32 shows
how the ER model is constructed. The downstroke always sheds a wake
covering an area described by an ellipse. The aspect ratio of this ellipse
depends primarily on the forward speed, U, and the wingspan,
2b. The upstroke always sheds a rectangular wake, with circulation
that gradually increases from zero at the lowest flight speed, to equal to the
downstroke circulation at the highest flight speed. The extremes (in
U) of this model thus replicate simple analogues of the closed-loop
and constant circulation wakes. Just as in the experiment, most wakes are of
some intermediate form.
|
The vertical impulse of the two wake segments can be written:
![]() | (23) |
![]() | (24) |
d and
u are the unknown circulations,
whose values will be taken initially from experiment. The relative wake width
in y is determined by the span ratio, R, which is a number
less than 1.0 representing the projected relative span of the flexed wing in
its upstroke position. The rather sparse data that exist
(Spedding, 1987b
) suggest that
this can be approximated as constant. The current data at single vertical
slices here do not help in improving the certainty of this estimate.
Now, the total vertical impulse is
![]() | (25) |
![]() | (26) |
![]() | (27) |
![]() | (28) |
![]() | (29) |
![]() | (30) |
It is not the purpose of the current exercise to devise a sophisticated or accurate model, but rather to test and demonstrate the self-consistency of the concept, using the simple criterion of sufficiency of weight support. It is quite instructive to follow the calculation of Iz, step-by-step, as illustrated in Fig. 33.
|
Fig. 33A shows the increase in projected horizontal wavelength with increase in flight speed, U. Since the wingbeat frequency changes little, the stride-length or advance ratio increases with U. It corresponds to a decrease in reduced frequency. Commensurately, if the wake height is assumed to be constant (this is not quite true, as indicated in Fig. 31, but at this level of detail the correction is unimportant), the angle to the horizontal of both upstroke and downstroke wake segments decreases with increasing U (Fig. 33B) and the horizontal projected area of each segment also increases. The projected area (Fig. 33C) is normalised by a wing disc area calculated from the wing semispan. At 6 m s-1, the ratio of projected to disk area is approximately 1.0, and the bird progresses forward by about one wingspan per wingstroke (see also Fig. 33A).
Fig. 33D shows the measured
circulations of the strongest single wake vortex (open circles) and total mean
circulation of either sign (closed circles). At slow flight speeds, the best
estimate of for the wake is
tot. With increasing
U it becomes less appropriate because there is no good reason to
suppose that the almost continuous shedding of small concentrations of
vorticity can be simply lumped together as if there were one big wake
structure. Instead a better estimate might now be
v, the
circulation of the strongest vortex. For this measure, the value taken at
midwing is used, which can include oblique sections through continuous
trailing vortices having smaller circulations at the centreplane because they
do not usually extend over the midline. As represented by the dash-dot line, a
weighted average of the two different
measures is used to model the
most likely value.
The final result is shown in Fig.
33E, where the fraction of weight support provided by the
ER model wake vertical impulse is shown. The model results based on
experimental lie close to 1. The worst case is at the highest flight
speed, and here the shortfall is likely to be the difficulty in correctly
estimating the strength of vortices from streamwise cuts when the main
structure is primarily streamwise also, and not spanwise. All data points lie
within reasonable limits of Iz/WT=1. It is
concluded that the wake model geometry and circulation estimates are
self-consistent, and that this framework could be used to model bird
wakes.
The only difficult input here was in knowing how to choose , which
was determined from experimental data in
Fig. 33D. It would be useful
to have a more general function, and so two empirical forms are given. The
first approximates
with an arbitrary second order polynomial as:
![]() | (31) |
The vector of best-fit polynomial coefficients is C=[6.1, -1.08,
0.05] for the thrush nightingale wake. Alternatively, since one expects
L and to be related through Equation 11
(L'=
U
), then one might also predict that
![]() | (32) |
The results of the ER model have been expressed mostly in dimensionless form with a view to scaling them out to other cases than the single species examined here. At the same time, we also will refrain from further complicating a model whose basis still rests on the one (albeit extensive) dataset. Nevertheless, it is hoped that the current data, ER model and limited generalisations from them will form the basis for a general model approach that can be successfully applied towards understanding and analysis of other aerodynamical problems in bird flight.
![]() |
Acknowledgments |
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Footnotes |
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References |
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Alexander, R. McN. (1982). Locomotion of Animals. Glasgow: Blackie.
Alexander, D. E. (2002). Nature's Flyers: Birds, Insects and the Biomechanics of Flight. Baltimore: Johns Hopkins University Press.
Batchelor, G. K. (1967). An Introduction to Fluid Dynamics. Cambridge: Cambridge University Press.
Boratav, O. N., Pelz, R. B. and Zabusky, N. J. (1992). Reconnection in orthogonally interacting vortex tubes: direct numerical simulations and quantifications. Phys. Fluids A 4,581 -605.[CrossRef]
Cantwell, B. J. and Coles, D. (1983). An experimental study of entrainment and transport in the turbulent near wake of a circular cylinder. J. Fluid Mech. 136,321 -374.
Dickinson, M. H., Lehmann, F.-O. and Sane, S. P.
(1999). Wing rotation and the aerodynamic basis of insect flight.
Science 284,1954
-1960.
Farge, M. (1987). Normalization of high-resolution raster display applied to turbulent fields. In Advances in Turbulence I (ed. G. Comte-Bellot), pp.111 -123. Berlin: Springer-Verlag.
Farge, M. (1990). L'imagerie scientifique: choix des palettes de couleurs pour la visualisation des champs scalaires bidimensionnels. L'Aeronautique et l'Astronautique 140, 24-33.
Fincham, A. M. and Spedding, G. R. (1997). Low-cost, high resolution DPIV for measurement of turbulent fluid flow. Exp. Fluids 23,449 -462.[CrossRef]
Hall, K. C. and Hall, S. R. (1996). Minimum induced power requirements for flapping flight. J. Fluid Mech. 323,285 -315.
Hall, K. C., Pigott, S. A. and Hall, S. R. (1998). Power requirements for large-amplitude flapping flight. J. Aircraft 35,352 -361.
Hedrick, T. L., Tobalske, B. W. and Biewener, A. A.
(2002). Estimates of circulation and gait change based on a
three-dimensional kinematic analysis of flight in cockatiels (Nymphicus
hollandicus) and ringed turtle-doves (Streptopelia risoria).
J. Exp. Biol. 205,1389
-1409.
Hildebrand, M. (1965). Symmetrical gaits of horses. Science 150,701 -708.[Medline]
Katz, J. and Plotkin, A. (2001). Low-Speed Aerodynamics. 2nd edition Cambridge: Cambridge University Press.
Kokshaysky, N. V. (1979). Tracing the wake of a flying bird. Nature 279,146 -148.
Lighthill, M. J. (1986). An Informal Introduction to Theoretical Fluid Mechanics. Oxford: Clarendon Press.
Norberg, U. M. (1990). Vertebrate Flight. Berlin: Springer-Verlag.
Pennycuick, C. J. (1968a). Power requirements for horizontal flight in the pigeon Columba livia. J. Exp. Biol. 49,527 -555.
Pennycuick, C. J. (1968b). A wind tunnel study of gliding flight in the pigeon Columba livia. J. Exp. Biol. 49,509 -526.
Pennycuick, C. J. (1975). Mechanics of Flight. In Avian Biology, vol. 5 (ed. D. S. Farmer, J. R. King and K. C. Parkes), pp. 1-75. London: Academic Press.
Pennycuick, C. J. (1988). On the reconstruction of pterosaurs and their manner of flight, with notes on vortex wakes. Biol. Rev. 63,299 -331.
Pennycuick, C. J. (1989). Bird Flight Performance: A Practical Calculation Manual. Oxford: Oxford University Press.
Pennycuick, C. J., Alerstam, T. and Hedenström, A.
(1997). A new low-turbulence wind tunnel for bird flight
experiments at Lund University, Sweden. J. Exp. Biol.
200,1441
-1449.
Pennycuick, C. J., Hedenström, A. and Rosén, M.
(2000). Horizontal flight of a swallow (Hirundo rustica)
observed in a wind tunnel, with a new method for directly measuring mechanical
power. J. Exp. Biol.
203,1755
-1765.
Pennycuick, C. J., Heine, C. E., Kirkpatrick, S. J. and Fuller, M. R. (1992). The profile drag of a hawk's wing, measured by wake sampling in a wind tunnel. J. Exp. Biol. 165, 1-19.
Pennycuick, C. J., Klaasen, M., Kvist, A. and Lindström,
A. (1996). Wing beat frequency and the body drag anomaly:
wind tunnel observations on a thrush nightingale (Luscinia luscinia)
and a teal (Anas crecca). J. Exp. Biol.
199,2757
-2765.
Prandtl, L. and Tietjens, O. G. (1934). Applied Hydro- and Aeromechanics. New York: Dover.
Rayner, J. M. V. (1979a). A vortex theory of animal flight. I. The vortex wake of a hovering animal. J. Fluid Mech. 91,697 -730.
Rayner, J. M. V. (1979b). A vortex theory of animal flight. II. The forward flight of birds. J. Fluid Mech. 91,731 -763.
Rayner, J. M. V. (1979c). A new approach to animal flight mechanics. J. Exp. Biol. 80, 17-54.
Rayner, J. M. V. (1986). Vertebrate flapping flight mechanics and aerodynamics, and the evolution of flight in bats. In Biona Report, vol. 5 (ed. W. Nachtigall), pp. 27-74. Stuttgart: Gustav Fischer Verlag.
Rayner, J. M. V. (1991a). Wake structure and force generation in avian flapping flight. In Bird Flight. Proc. 20th Int. Orn. Cong., Symp. 9, 702-715.
Rayner, J. M. V. (1991b). On aerodynamics and the energetics of vertebrate flapping flight. Contemp. Math. 141,351 -400.
Rayner, J. M. V. (2001). Mathematical modeling of the avian flight power curve. Math. Meth. Appl. Sci. 24,1485 -1514.
Rayner, J. M. V., Jones, G. and Thomas, A. (1986). Vortex flow visualizations reveal change in upstroke function with flight speed in bats. Nature 321,162 -164.[CrossRef]
Rayner, J. M. V. and Gordon, R. (1998). Visualization and modelling of the wakes of flying birds. In Biona Report, No. 13, Motion Systems (ed. R. Blickhan, A. Wisser and W. Nachtigall), pp. 165-173. Jena: Gustav Fischer Verlag.
Saffman, P. G. (1992). Vortex Dynamics. Cambridge: Cambridge University Press.
Spedding, G. R. (1981). The vortex wake of flying birds: an experimental investigation. PhD thesis, University of Bristol.
Spedding, G. R. (1986). The wake of a jackdaw (Corvus monedula) in slow flight. J. Exp. Biol. 125,287 -307.
Spedding, G. R. (1987a). The wake of a kestrel (Falco tinnunculus) in gliding flight. J. Exp. Biol. 127,45 -57.
Spedding, G. R. (1987b). The wake of a kestrel (Falco tinnunculus) in flapping flight. J. Exp. Biol. 127,59 -78.
Spedding, G. R. (1992). The aerodynamics of flight. In Adv. Comp. Physiol. The Mechanics of Animal Locomotion (ed. R. McN. Alexander), pp.51 -111. Berlin: Springer.
Spedding, G. R. (2003). Comparing fluid mechanics models with experiment. Phil. Trans. R. Soc. Lond. B (in press).
Spedding, G. R., Rayner, J. M. V. and Pennycuick, C. J. (1984). Momentum and energy in the wake of a pigeon (Columba livia) in slow flight. J. Exp. Biol. 111,81 -102.
Spedding, G. R. and Rignot, E. J. M. (1993). Performance analysis and application of grid interpolation techniques for fluid flows. Exp. Fluids 15,417 -430.
Spedding, G. R. and DeLaurier, J. D. (1996) Animal and ornithopter flight. In Handbook of Fluid Mechanics and Fluid Machinery, Vol. 3: Applications of Fluid Dynamics (ed. J. A. Schetz and A. E. Fuhs), pp. 1951-1967. New York, John Wiley and Sons.
Spedding, G. R., Rosén, M. and Hedenström, A. (2003). Quantitative studies of the wakes of freely-flying birds in a low-turbulence wind tunnel. Exp. Fluids 34,291 -303.
Tobalske, B. W. (2000). Biomechanics and physiology of gait selection in flying birds. Physiol. Biochem. Zool. 73,736 -750.[CrossRef][Medline]
Tobalske, B. W. and Dial, K. P. (1996). Flight
kinematics of black-billed magpies and pigeons over a wide range of speeds.
J. Exp. Biol. 199,263
-280.
Tucker, V. A. (1990a). Body drag, feather drag and interference drag of the mounting strut in a peregrine falcon, Falco peregrinus. J. Exp. Biol. 149,449 -468.
Tucker, V. A. (1990b). Measuring aerodynamic interference drag between a bird body and the mounting strut of a drag balance. J. Exp. Biol. 154,439 -461.
White, F. M. (2003) Fluid Mechanics (5th edition). New York: McGraw-Hill.
Yan, J., Avadhanula, S. A., Birch, J., Dickinson, M. H., Sitti, M., Su T. and Fearing, R. S. (2002). Wing transmission for a micromechanical flying insect. J. Micromechatronics 1, 221-237.
Zabusky, N. J., Fernandez, V. M. and Silver, D. (1995). Collapse, intensification and reconnection in vortex dominated flows: visiometrics and modeling. Physica D 86, 1-11.