The relationship between wingbeat kinematics and vortex wake of a thrush nightingale
1 Department of Animal Ecology, Lund University, Ecology Building, SE-223 62
Lund, Sweden
2 Department of Aerospace and Mechanical Engineering, University of Southern
California, Los Angeles, CA 90089-1191, USA
* Author for correspondence (e-mail: m.rosen{at}usc.edu)
Accepted 14 September 2004
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Summary |
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The relationship between measured wake geometry and wingbeat kinematics can be qualitatively explained by presumed self-induced convection and deformation of the wake between its initial formation and later measurement, and varies in a predictable way with flight speed. Although coarse details of the wake geometry accord well with the kinematic measurements, there is no simple explanation based on these observed kinematics alone that accounts for the measured asymmetries of circulation magnitude in starting and stopping vortex structures. More complex interactions between the wake and wings and/or body are implied.
Key words: wingbeat kinematics, vortex wake, thrush nightingale, Luscinia luscinia, DPVI, aerodynamics, bird flight, bird, wind tunnel
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Introduction |
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A notable characteristic of aerodynamic studies using wake analysis is that
the results do not depend on, or reveal anything directly about, the wing
kinematics that create the disturbance. While observations of wake structure
were new and extensive, they were related to the kinematics only by rather
loose inference. The kinematic basis for flight in birds and bats has a long
history of careful measurement (see, for example,
Brown 1953;
Norberg, 1976
) for classic
treatments of birds and bats, respectively). In turn, inferring aerodynamic
quantities from kinematics alone is also difficult. In the absence of any
better alternative, all such studies have been obliged to make strong
assumptions about the quasi-steady (and 2D) aerodynamic properties of the wing
sections as they accelerate and deform during the wingbeat. A more recent
study by Hedrick et al. (2002
)
carried these calculations through to the point of estimating circulations of
wing sections, but this process invoked exactly the same set of quasi-steady
assumptions about how the wing motion and air flow are linked. Undoubtedly,
much remains to be done to make this connection clearer. This paper is a small
step in this direction, where a simple kinematic analysis is related to the
measured wake flow. The flow on the wing itself is inaccessible to the flow
experiments, but correlates of kinematic variation with wing speed and wake
structure will be sought.
The wingbeat kinematics were analysed for the same bird under similar
conditions as for the wake study. As in the wake measurements, it is critical
that the flight training be sufficient to ensure consistent, repeatable,
steady flight over the whole range of studied flight speeds. The kinematic
analysis allows this requirement to be checked, and more importantly, for the
changing wake structure to be correlated with observed changes in kinematics.
This study becomes more compelling in light of the rather complex wake
structures revealed in the wake analysis of Spedding, Rosén and
Hedenström (2003a).
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Materials and methods |
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Initially four juvenile thrush nightingales were caught at Ottenby Bird Observatory, Sweden, on their autumn migration in August 2001. After 2 weeks of flight training, it became clear that one bird was far more capable of prolonged and stable flights. This bird was further trained to perch on a stick mounted on the wind tunnel sidewall when not flying. During flight tests, the stick was manually tilted downwards to initiate flight. About 1 m upstream from the perch location a luminous marker was placed in the airflow to serve as a reference point for where the perch would reappear at the end of a flight episode. The marker helped to direct the flight position of the bird to the centre of the test section and repeated trials over 3 months converged towards stable and repeatable flights.
When running flow visualisation experiments the bird would fly in reduced
light conditions as required for the digital particle image velocimetry (DPIV)
measurements, but the high speed camera recordings required about 2 kW of
continuous light. The thrush nightingale is a typical nocturnal migrant and
preferred the dark settings, as reflected in the broader range of flight
speeds recorded (411 m s1 in low-light conditions
compared with 510 m s1 by the high-speed cameras).
Previous experiments using thrush nightingales in the same wind tunnel have
produced the same range of observed flight speeds as for this set-up,
510 m s1 (e.g.
Pennycuick et al., 1996).
Here, in the digital video recordings, a typical flight episode would last for
1030 s (limited by the bird's accurate flight position in the test
section), followed by a few minutes of rest on the perch. The cycle could be
repeated for about 90 min before the bird was put back in the aviary for a
longer rest. Typically one morning and one afternoon flight session were
performed each day. In total the training was scheduled daily for 3 months
prior to the start of experiments. The experiments using flow visualisation
and high-speed cameras lasted for another 50 days (end 11 January, 2002). The
bird was released into the wild the following spring, seemingly unaffected by
the experience. For bird morphological characteristics, see
Table 1.
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Sampling and analysis of wingbeat kinematics
The number of steady flight episodes analysed at each flight speed
U were 5, 5, 5, 4, 5 and 4 at 1 m s1 intervals from
5 to 10 m s1.
A RedLake digital video camera system (MotionScope PCI500, USA, operating
at frame rate 125 s1, shutter speed 1/1850 s) was mounted 4
m downstream from the bird's flight position. The location of the camera far
back in the first diffuser causes a negligible effect on the airflow around
the bird (see Pennycuick et al.,
1997). From the camera AVI-output, strings of JPEGs were extracted
and used for the analysis.
Time sequences of y, z coordinates (Fig. 1A) of the wingtip and shoulder joint for both wings (mean presented) were digitised using a custom program written in PV-WAVE (Visual Numerics, USA) command language, the same language used for accessing and calculating quantities in the wakes database. The body width was used as a reference length scale to calculate physical distances.
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The projected wingspan b'(t) (m), was measured from
tip to tip, throughout the wingbeat cycle. b'd is
the projected span at mid downstroke ±10° from horizontal
(Fig. 1A). The horizontal
position is defined as a line connecting the left and right shoulder joints.
b'u is the equivalent measure for the upstroke
(Fig. 1A). In the upstroke the
wing is moving rapidly, almost vertically, so registrations of the wingtip in
the horizontal position are scarce and the data were interpolated graphically
from a diagram of b'(y,z) constructed from
b'(t) sequences. The span ratio, R, is
calculated from
![]() | (1) |
The wingtip position in the vertical ztip was
calculated as the vertical distance between the wingtip and the horizontal
line, hence ztip=0 is a wing held in the horizontal
position (Fig. 1A). The
beginning and end of a wing stroke were defined as the points where
ztip reaches maximum values above
(ztip,max) and below (ztip,min) the
horizontal, respectively (Fig.
1A). The downstroke ratio is the ratio of the downstroke
duration to the total stroke period where start and stop points of the
downstroke and upstroke were determined by the maximum and minimum values of
ztip.
The wingbeat frequency f and wingbeat amplitude,
A1, were derived from fitting a single frequency sine
function to the wingtip trace, so the vertical direction of the wingtip is
described by
![]() | (2) |
1/f, the stroke period, is denoted T. The horizontal
stroke wavelength (see Fig.
1B) of a wingbeat cycle was calculated from wingbeat frequency and
speed U as
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The reduced frequency k is proportional to the ratio of two time
scales, the average time required for the mean flow to pass over the mean
chord, tc=c/U, and the time taken for
one wingbeat, T=1/f. It is conventionally expressed in terms
of the mean half chord, c/2, and radian frequency,
=2
f, as
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The magnitude of k is frequently used as an indicator of the
relative importance of unsteady terms in the aerodynamics, and in classical,
small amplitude theories, it is exactly this (e.g.
Theodorsen, 1934). Some
caution is due in a simple-minded application of this number to complex
geometries and kinematics of animal flight, but very generally when k
is on the order of 0.1 then unsteady effects can usually be ignored, while
k of order 1 signals a probable strong influence of unsteady
phenomena (see also remarks by Spedding,
1993
).
A similar measure, K, can be constructed from the ratio of mean
tip speed in the vertical (wtip) to forward flight speed:
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Since the arc length s travelled by the wingtip is
s=b, and this is accomplished twice per wingbeat
cycle, then
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![]() | (6) |
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The measured inclination angles of the wingtip trace are
kin,d and
kin,u (deg.) for the down- and
upstrokes (Fig. 1B),
respectively. Values for
kin are calculated from
2A1 and
as
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Measuring the wake topology
The method used for estimating the velocity field behind a bird in the wind
tunnel is described by Spedding et al.
(2003b) and this particular
analysis of the thrush nightingale is accounted for in detail by Spedding,
Rosén and Hedenström
(2003a
).
Inclination angles of the wake
The inclination angle of the wake was estimated using two different
methods:
Method 1: wake trace
The flapping wings generate sections of wake on the down and upstroke that
can be reasonably well approximated as plane sections
(Spedding, Rosén and
Hedenström, 2003a) (
trace,
Fig. 2). The inclination angle
of these sections was estimated by drawing a line parallel to the section
plane. At low speeds this section is defined by a line from the centre of the
start vortex
+ to the centre of the dominating (stop) vortex
at the end of downstroke.
trace is the angle
between this line and the horizontal. At higher speeds, where a full stroke
wavelength could not be captured in a single image, a line was fit to the wake
trace originating from a mid up- or downstroke.
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Method 2: induced flow
Alternatively, the mean inclination angle could be taken from a line normal
to the direction of induced downwash at the centre of the vortex structure
(ind, Fig. 2).
So derived,
ind may be supposed to be a reasonable measure of
the average direction of the mean resultant force from that part of the
wingbeat. When the induced flow is normal to the wake plane, then
ind=
trace.
ind and
trace were measured from upstroke
and downstroke segments separately using wake data sampled behind the mid-wing
position and central-body position only
(Spedding, Rosén and
Hedenström, 2003a
). The analysis included only segments of
the wake that were typical for steady flight.
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Results |
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Time traces describing some of these parameters (vertical position and projected wingspan) can be seen in Fig. 4 for U=10 m s1. No extra smoothing is imposed on the raw data here, and within the measurement resolution the vertical position is a regular periodic function, which can be quite well fit with a single frequency component sine wave. The projected wingspan is more complex, reflecting the folding of the wing itself. The time traces are quite repeatable over the eight wingbeats shown. The extensive training makes it simple to find useable traces of this kind.
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Wingbeat kinematics from high-speed filming
Timing of the wingbeat cycle
The amplitude of the wingbeat does not change significantly with U
(Fig. 5,
Table 2). Similarly, the
asymmetry of the angular excursion, as measured by A0,
remains almost constant (Table
2). The wingbeat frequency, f, is a very weak U-shape,
with a mean of 14.4 Hz and a minimum of 14.0 Hz at 6 m s1
(Fig. 6,
Table 2). The overall variation
in f is only 7% between the extreme values at 6 and 10 m
s1, respectively.
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The successful single frequency sine fit is a consequence of a highly
repeatable wing stroke time traces. The 10 point/cycle sampling rate is
sufficient to show that smaller scale fluctuations are unlikely to appear in
the wingbeat trace. This was tested by approximating the time traces in a
Fourier series. The function ztip(t) digitised
over N discrete points can be expressed as a discrete sum of Fourier
coefficients:
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![]() | (10) |
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Since f changes only a little, the reduced frequency k
(equation 4) declines
systematically with U, ranging from approximately 0.4 to 0.2 over
U=510 m s1
(Fig. 8). Moreover, since
A1, hence , does not change significantly with
U, then J(U) is also a simple function, not
significantly different from a straight line, whose constant slope can be
predicted from
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In Fig. 9 the vertically and
horizontally projected wingtip traces for three different speeds have been
plotted side by side to show how the wingbeat patterns vary qualitatively over
the speed range. The plots are on a grid with aspect ratio of 1, and show
wingtip trace ztip (open circles) and the projected
wingspan b' (filled circles) over the course of several
wingbeats. Since the wingbeat frequency f is approximately constant
(Fig. 6), the wavelength
of the wingbeat cycle increases with increasing U. In fact,
this is the only notable difference between the projected wing traces, which
otherwise appear similar in form. The exception to this otherwise geometric
similarity is the slightly increased time spent on the upstroke, visible in
Fig. 9 as an increasing
percentage of non-hatched area, and measured by decreasing downstroke
fraction,
in Fig.
10.
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The maximum wingspan, measured at mid-downstroke, did not change significantly with flight speed, but the ratio of the span during the upstroke to that of the downstroke increased with increasing speed (Fig. 11).
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Kinematics and wakes
Of the primitive kinematics parameters measured in the previous section,
the only systematic variations with flight speed were in the downstroke ratio
, the span ratio R and, to some degree, the wingbeat frequency
f. As far as can be measured from these data, there are no
significant quantitative or qualitative changes in wingbeat amplitude and it
is only the relative contribution of the upstroke within the wingbeat cycle
that can be associated with variation in U. This restricted range of
variation in kinematics can be linked with measured wake structure.
Fig. 12 shows vertical
centre plane cross sections of the wake at flight speeds of 5, 8 and 11 m
s1, complementing those shown in Spedding, Rosén and
Hedenström (2003a) given
for 4, 7 and 10 m s1). At U=5 m
s1, the downstroke wake is the most notable feature,
beginning with a strong cross-stream starting vortex (+ arrow in
Fig. 12). By contrast, the
stopping vortex ( arrow) is more diffuse and weak. While the upstroke
appears to be mostly aerodynamically inactive, as indicated by the
low-magnitude induced flow vectors in the upstroke part of
Fig. 12, and it was only by
including contributions to the total circulation from elements that appear
during the upstroke traces that a satisfactory crude balance between weight
and vertical impulse could be achieved in Spedding, Rosén and
Hedenström (2003a
). The
trace upstroke vorticity in Fig.
12A is thus important.
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At 8 m s1 the downstroke again begins with a strong starting vortex, while the stopping vortex and upstroke leave behind a complex pattern of vorticity, which can now be seen throughout the entire upstroke region. The induced flow and its downward component are more noticeable, indicating upstroke generation of lift (at the expense of additional drag).
At 11 m s1 the wake pattern trails through the entire
wingbeat, with neither the beginning nor end of the downstroke being
particularly distinct. This implies a mode of flight that most closely
resembles a constant circulation model, with an aerodynamically loaded wing on
both down- and upstroke (Rayner,
1986; Spedding,
1987
), although the constant shedding of cross-stream vorticity is
not the same as the proposed absence of such shedding in the constant
circulation model.
Just as in the kinematics analysis, the most evident qualitative variations in wake structure come from the relative magnitude of the contribution from the upstroke.
Wake geometry
The angle of the induced downwash ind, the actual wake
trace
trace defined by the core vortex structures, and the
path of the wingtip
kin as it progresses through the air
(equation 8) are presented in
Fig. 13A,B (downstroke) and
Fig. 13C,D(upstroke). There
are significant differences between the kinematics data and the data from the
actual wake measurements. Also, the two different wake measurements differ in
both upstroke and downstroke. Most of the differences simply show that the
wake does not remain frozen in place on the path left by the wing trace.
Others indicate interactions between upstroke and downstroke components. These
will be discussed in context in the next section.
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Discussion |
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![]() | (12) |
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The demands of weight support thus become easier to satisfy as flight speed increases. The drag on the wings and body rises steeply, however (as U2), and the wings/wake must also generate a net horizontal force to balance the sum of all drag components. In the absence of a separate thrust generator, the wake geometry itself must be arranged to produce the thrust. The differing functions of the downstroke and upstroke account for this.
Downstroke ratio decreases with U
The differing roles of the down- and upstrokes were seen in the traces of
Fig. 9, and quantitatively in
Fig. 10. Thedownstroke ratio
decreases significantly from values of close to 0.5 at low speed to
about 0.45 at high speeds. At low speeds, the largest contributor to weight
support is the downstroke, which is responsible for the generation of a
large-scale structure with properties similar to a vortex loop
(Spedding, Rosén and
Hedenström, 2003a
). This structure lies at a relatively small
angle to the horizontal (Fig.
13B), and so most of the impulse, directed normal to the plane of
the structure, points upwards. The forward component is comparatively small
because at low speeds the total viscous and pressure drags are also small. The
direct contribution of the upstroke to weight support is difficult to
ascertain (Spedding, Rosén and
Hedenström, 2003a
), but it appears not to be zero.
Nevertheless, a large drag penalty from an exotic upstroke motion cannot be
balanced by the downstroke structure, and so the direct contribution of the
upstroke to weight support can be inferred to be small. As U
increases, the downstroke can more readily provide an increasing thrust
component, while the upstroke assumes a more significant role in weight
support. As the wake structures left behind during the upstroke become more
evident (Spedding, Rosén and
Hedenström, 2003a
; Fig.
12), it occupies a larger fraction of the total wingbeat.
A decrease of with increasing U has been observed also in
other species (Park et al.,
2001
; Tobalske and Dial,
1996
), and even if the absolute range of
varies between
species it commonly starts at about 0.5 at the slowest flight speeds measured
in a wind tunnel.
Geometry of the wake and its induced flow
Following the classical textbook treatments, where a velocity field can be
conveniently described as the flow induced by the presence of a certain
distribution of vortex elements, one searches for similar experimental
descriptions of the measured flows. In Spedding, Rosén and
Hedenström (2003a), it
was noted that the true wake flows behind a flapping bird were complex and the
most appropriate reduction of the velocity fields to simpler forms generated
by a small number of vortex lines in three dimensions was an intricate
exercise.
To make a correspondence between wing kinematics and wake structure, a
simpler data subset is considered here; only the vertical centreplane velocity
fields and their associated spanwise vorticity distributions, as in
Fig. 12.
Fig. 14 summarizes the three
different wake inclination angles, ind,
trace
and
kin, for the downstroke and upstroke, as a function of
flight speed U, condensing the full measurements of
Fig. 13.
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In the downstroke (Fig. 14A), all three angles have measurably different trends with increasing U, but the same measures from the upstroke (Fig. 14B) are similar in both sign and magnitude. The differences in downstroke can be explained in a qualitative manner in two steps.
(1) trace
kin. Since the relative
changes in both wingbeat frequency f
(Fig. 6) and downstroke ratio
(Fig. 10) are small
compared with
U/
=2/3 [where
is a
characteristic flight speed, e.g.
(UmaxUmin), the advance ratio,
J=1/K, increases almost in direct proportion to U,
and the inclination angle of the path of the wingtip trace,
kin, decreases similarly. The decrease of
kin
with U is a poor predictor, however, of the position of the vortex
wake structure as measured by
trace, which increases with
U. First, we note that the wake structure is measured about 17.5
chord lengths downstream from the bird and so represents the pattern as the
flow has evolved over about 2.4
at 5 m s1 and
1.3
at 10 m s1 (or 2.4T and 1.3T,
respectively). During this time, the wake has moved by self-induced
convection, downwards from its starting point; let us denote the vertical
component of this velocity wself. Since the starting
vortex is created first in the wingbeat cycle, at any later time it has had
longer to move than other parts, and
trace, measured under the
connecting line from start vortex to stop vortex, will always be lower than
the trace of the wingtip
kin. At the lowest flight speed of 5
m s1, the strength of shed vortices is higher, induced
downwash velocities are higher, and wself will be higher
compared with the forward speed. As U increases, wake element
circulations decrease, and the ratio wself/U
decreases rapidly. At Umax,
trace is
almost equal to
kin, approaching it from below.
(2) ind is derived from the mean measured downwash angle at
the mid-point of a wake element. At low flight speeds, the downwash is tilted
further aft than would be inferred from the isolated wake vorticity (at the
centreplane only) left behind during the downstroke. The arguments for the
measured
trace, based mainly on self-induced wake motion, do
not apply because the trend for
ind is opposite; it starts
higher than
trace and then declines with increasing
U.
There is no simple correlate with the measured wingbeat kinematics to
explain the magnitude and trend of ind(U). Spedding,
Rosén and Hedenström
(2003a
) reported no
significant out-of-plane deformations of the inferred 3D wake structures, even
though the supporting data are sparse. There are no obvious trends with
amplitude (Fig. 5) or stroke
plane inclination angle (M.R., G.R.S. and A.H., unpublished data from
high-speed side-view sequences) to indicate a commensurate change in
large-scale kinematics, which would then in any case be inconsistent with the
previous discussion of
trace vs.
kin.
Fig. 3 shows a strong
pronation of the leading edge during mid-downstroke for all three flight
speeds (Fig. 3Ai, Bh, Ci). The
middle case (7 m s1) has the largest projected area of the
wing on to the image plane. Strong pronation will lead to an effective
aerodynamic angle of attack that gives a backward component to the induced
downwash. As U increases, this component will become comparatively
smaller, and so the net downwash angle should increase, decreasing
ind. The decrease of
ind is actually delayed
until middle-range flight speeds, due to the increased pronation up to that
point. Lacking detailed data on the 3D flow field close to the wing and on
similar details of the wing kinematics, these explanations are speculative.
Nevertheless, the trend of
ind(U) seems very likely
to be linked most closely with variations in the effective local angle of
attack of the wing, and it would be interesting to know more.
The upstroke wake angles in Fig.
14B all decrease in magnitude with decreasing U. The same
considerations apply as for the corresponding downstroke angles (with
appropriate changes in sign), but since the wake flow is weaker, then so are
departures due to self-deformation or convection. ind is
consistently smaller in magnitude at lower flight speeds (closer to
horizontal) than both
trace and
kin, and the
explanation is again likely to be related to details of effective angle of
attack during the upstroke.
While the relationships between trace,
ind
and
kin are not completely obvious a priori, they can
be understood in a consistent way. They are clearly significantly different
from the most simple ideas about wingbeat kinematics and aerodynamics
the vortex wake does not lie frozen along the wingtip trace, the downwash is
not normal to the plane of the wake and the wake structure could not readily
be predicted or inferred from the wing motions alone. Appropriate and
as-yet-unrealizable computations would be required to properly calculate the
unsteady flow field. When that time arrives, data such as these can serve as
diagnostics for comparison.
Span ratio and wing flexing
One measure that characterises the unusually flexible geometry of bird
wings is the span ratio R. It increases linearly with U
(Fig. 11), and its effect can
be seen qualitatively in Fig.
9. Flexion of the wing during the upstroke is thought to be
essential in providing positive thrust in cruising flight and in avoiding
excess drag in slow speed flight. The wake measurements in Spedding,
Rosén and Hedenström
(2003a) and
Fig. 12 suggest a lifting
upstroke at most flight speeds and this is supported by the gradual increase
of R with U in Fig.
11. This idea is consistent with the corresponding decrease of
with U in Fig.
10. The mid-downstroke effective wingspan does not change
significantly with U but the mid-upstroke does, so the change in
R is mostly related to changes in the upstroke (see
Fig. 11).
R is also reported to increase with U in the black-billed
magpie Pica pica, pigeon Columba livia, ringed turtle-dove
Streptopelia risoria and cockatiel Nymphicus hollandicus
(Tobalske and Dial, 1996;
Hedrick et al, 2002
). This
simple relationship is not universal, however. In the case of the barn swallow
Hirundo rustica, R has been observed to decrease with U. R
falls from 0.5 to 0.15 at flight speeds between 4 and 13 m
s1 (Park et al.,
2001
). Both the morphology and flight of the barn swallow are very
different from that of the thrush nightingale (the swallow is lighter, with
thinner, more pointed wings; wing loading Q=13 N
m2;
=7.4). The barn swallow is adapted for slow
manoeuvring flight and exhibits an intermittent flight style with a
characteristic wingbeat pause at high speed, which is not present in the
thrush nightingale. The morphological and behavioural differences ought to be
manifest in the wake structure and it would be useful to conduct combined wake
and kinematic studies such as the present study on the barn swallow, partly to
provide a test of the generality of the conclusions for the thrush nightingale
wake.
Not only does the projected wingspan change between downstroke and
upstroke, the relative folding of the wing also changes. The most notable
difference is that the primary wing feathers are more exposed to the oncoming
air than the arm section of the wing in the flexed upstroke. It is usually
assumed that the inner part of the wing is responsible for most of the active
aerodynamics generated in the upstroke. Here it appears that when any part of
the wing is aerodynamically active during the upstroke in the thrush
nightingale, it is the hand section of the wing (e.g.
Fig. 3). The notion of the
importance of the arm wing for the upstroke circulation originates from an
analysis of the wake of a kestrel in flapping flight where the arm evidently
was important (Spedding,
1987), and this idea agreed well with long-standing kinematic data
(Brown, 1953
). The results
presented here suggest a more varied phenomenon.
Frequency is near constant
The wingbeat frequency is relatively constant, with only a weak U-shape
across the speed range (Fig.
6). This feature of thrush nightingale flight was also reported by
Pennycuick et al. (1996).
Consequently, reduced frequency parameters such as k and J
(equations 4,
7) cannot be constant. That is
indeed shown in Fig. 8, where
k decreases and J increases, both almost linearly with
increasing U. Apparently it is unimportant for the bird to maintain a
constant reduced frequency over its range of flight speeds, and there is no
attempt to adjust f and/or A1 to achieve even
short ranges of constant k or J. The ratio of timescales
between stroke period and a mean flow convective time does not seem to be the
determining factor in selection of flapping frequency.
A number of theoretical models predict optimal reduced frequencies for
flapping flight. In a 2D, direct numerical simulation about pitching and
rotating wing segments at Reynolds number Re=103, Wang
(2000) finds an optimal
reduced frequency equivalent to k
0.2 for values of
K=1/J
0.2, or J<5.0. k and
J are never this low for thrush nightingale flight
(Fig. 8). Hall et al.
(1997
) constructed an unsteady
inviscid model (with quasi steady viscous corrections) for large amplitude
flapping flight and showed optimal advance ratios of between
and
for minimum power loss.
This translates to 0.3<J<0.4 for our definition of J,
which is based on
rather than fixed 2
as an amplitude measure. The
optimal values of k and J produced by these different
formulations are quite close to each other, although, as noted by Wang
(2000
), it is not clear why
this should be. In Wang's simulation the optimal frequency is determined by
natural timescales of growth and shedding of both leading and trailing edge
vortices. The result of Hall et al.
(1997
) also concerns a ratio of
advective to wing cycle times, but for entirely different reasons as there are
no separation vortices in their inviscid model. In the thrush nightingale the
measured value of k never falls to Wang's optimum of 0.2 and
J does not fall to the 0.30.4 range predicted by Hall et al.
(1997
).
The perils in simply comparing apparently similar values of a reduced
frequency parameter are further illustrated by reference to the thorough and
careful analysis of Lewin and Haj-Hariri
(2003). In a numerical solution
of the 2D flow around a heaving aerofoil they define a usual Strouhal number
St by
![]() | (14) |
It would be extremely interesting if the thrush nightingale were found to be operating at a preferred reduced frequency, because it would imply a close coupling or resonance between fluid dynamical timescales of the mean and fluctuating motions. Then the notion of preferred wingbeat kinematic styles might be supportable. However, the constant frequency data do not show this. The fact that the wingbeat frequency of the thrush nightingale only varies by 7% over the range of U investigated suggests that other factors ultimately dictate the frequency selection. These might include preferred strain rates in muscles and tendons, preferred mechanical resonances dictated by the morphology, or preferred parameter ranges in the associated physiological systems.
The wingbeat frequency of a thrush nightingale studied at the same facility
in Lund by Pennycuick et al.
(1996) showed a similarly weak
dependence on U. The slightly lower absolute values of f are
consistent with the 10% smaller mass.
Concluding remarks
This paper serves both as a record of the wing kinematics that produce the
wake structures observed in Spedding, Rosén and Hedenström
(2003a) and also as
corroborating evidence for some of the points discovered there. [Note that it
is not an exhaustive attempt to quantify all aspects of thrush nightingale
kinematics but is restricted to a dataset that can be compared to the wake
data presented in Spedding, Rosén and Hedenström
(2003a
).] One notable feature
of both the wake studies and the associated wing kinematic parameters is the
absence of any sign of a discontinuous or sudden variation in any of the
measured quantities with U. Much speculation has arisen concerning
the possible existence of gaits in animal flight
(Rayner, 1993
;
Tobalske, 2000
;
Hedrick et al., 2002
), but none
of the quantitative experimental data covering a continuous range of flight
speeds (as opposed to one or two instances) show any measured quantity with
abrupt variation, commensurate with some qualitative change in flight style.
The data from this paper, and its companion wakes study
(Spedding, Rosén and
Hedenström, 2003a
) are the only source of quantitative data
for aerodynamic measurements at more than one single flight speed, and offer
no support for the notion of gaits. If it seems convenient to continue to
describe categories such as `slow flight' and `fast flight', it should be
noted that there is no particular (aerodynamic, mechanical, physiological)
reason for singling out two, or any other number, of discrete conditions that
appear simply as points on a smoothly varying continuum. If behavioural and/or
ecologically interesting points are identified, such as minimum power speed,
or maximum range speed, then these occur because of their location on equally
smooth curves that predict or measure mechanical or metabolic power
requirements, and not because they represent qualitatively different wake
structures or wing kinematics.
The variations observed in overall wake geometry can be explained by, or at least reconciled with, the overall wing kinematic data, although observed wake structures and their position would not be easy to deduce from the kinematics alone.
Technology does not yet permit time-resolved, 3D DPIV studies of the wake at these flow speeds, and safety considerations limit the proximity of the bird (and its field of vision) to the laser light sheet. As a result, some care and caution is required in making limited inferences back from some complex flow structure to conditions on the wing when it was originally created.
One feature of the wake that is not yet adequately explained through the
available kinematic data is the strong asymmetry between starting and stopping
vortices at the beginning and end of the downstroke. This asymmetry has been
noted in all quantitative wake studies of slow speed flapping flight (Spedding
et al., 1984,
2003a
,b
;
Spedding, 1986
) but cannot be
readily identified with a similar asymmetry in the wingbeat.
Fig. 3 shows that the wings
meet much closer together at the end of the downstroke than after the upstroke
and interference between the wings and the shed vorticity was suggested by
Spedding (1986
).
Fig. 7 shows that acceleration
magnitudes of the wingtip do not differ at the end of each half-stroke. The
diffuse distribution of vorticity produced at the end of the downstroke, at
the very least, is a significant nuisance in calculating wake-based force
estimates (Spedding et al.,
2003a
) and the story of its origin is likely to be
interesting.
Finally, as in Spedding et al.
(2003a), we should recall that
these conclusions have been obtained from studies in just one species, and the
generality of the results is not yet clear. The success of the present work in
associating many wake features with their generating kinematic conditions
bodes well for future studies using other wing geometries and kinematics.
List of symbols
Subscripts u and d denote upstroke and downstroke, respectively.
![]() |
Acknowledgments |
---|
![]() |
Footnotes |
---|
Present address: Department of Theoretical Ecology, Lund University,
Ecology Building, SE-223 62 Lund, Sweden
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References |
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