Vortex wakes of bird flight: old theory, new data and
future prospects
A. Hedenström
Department of Theoretical Ecology, Lund University, Sweden.
Abstract
Flying birds leave a vortex wake. Fluid dynamic theory in the form of Helmholtz’ theorems dictate
the allowable topologies, and Kelvin’s circulation theorem requires that changes in wake circulation are directly proportional to force changes on the wing/aerofoil that generated the wake.
Much bird flight research has therefore been focused on the properties of trailing wake vortices
behind birds, since an accurate quantitative description of these will reveal also the aerodynamics
of bird wings. The first vortex theory of bird flight assumed the periodic shedding of discrete
vortex loops, each one generated during a downstroke, while the upstroke was considered aerodynamically functionless. This view received some support from early visualization experiments
of take-off flight or very low speeds, while experiments at a higher speed (U = 7 m/s) in one
species showed undulating wing-tip vortices of similar circulation on both down and upstroke.
The necessary force asymmetry between downstroke and upstroke was obtained by wing flexing during the upstroke. Then followed an almost 20-year drought, with no further quantitative
experiments, until recently when digital particle imaging velocimetry (DPIV) was successfully
deployed in a low-turbulence wind tunnel, and where the same small (30 g) bird could be studied
across a large range of flight speeds (4–11 m/s). These new experiments revealed a much more
complicated wake pattern than previous data suggested, mainly due to the improved experimental
resolution. The bird generated structures most closely resembling vortex loops at slow speeds,
which gradually transformed into something similar to a constant circulation wake at the highest
speeds. However, the wakes were never as clean as the idealized cartoon models of the vortex
theory of bird flight, and previous paradoxical results were shown to be attributable to the resulting
difficulty in accounting for all wake components. New DPIV data on other species indicate that
these findings are quite general.
1 Introduction
In a classic experiment Magnan et al. [1] used tobacco smoke to visualize the vortex wake of a
pigeon Columba livia, which was found to consist of vortex loops (‘tourbillons’) in the slowly
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Vortex Wakes of Bird Flight
707
flying bird. Even though the published photographs are quite indistinct and difficult to interpret,
this was the first demonstration that the vortex sheet rolls up in discrete structures associated with
the wing beat cycle. However, the first generation of quantitative aerodynamic models of bird
flight used the actuator disc approach to calculate the induced drag [2, 3]. Bird flight typically
encompasses Reynolds numbers (Re, for definition see below) in the range 8000–200,000 [4],
which from an aerodynamic point of view is a problematic range because of the transition from
laminar to turbulent boundary layers, or even laminar flow separation and laminar reattachment
[5]. In this region of Re there is an abrupt increase of drag due to this transition in the boundary
layer followed by a decrease from this higher drag with further increase of Re, which makes any
attempts of quantitative analysis complicated. Research on bird flight has a long tradition, not least
because the aerodynamic models are of great potential practical benefit to ecologists who want to
understand the strategies and constraints on migration performance in wild birds. Since the 1960s,
when the first comprehensive flight mechanical model was developed, the field has seen a steady
flow of theoretical and empirical advances. This paper summarizes some key developments with
special emphasis on vortex wake models and experimental data from real wakes in birds.
2 Some definitions
For comparative but also purely aerodynamic purposes we will have reason to refer to some often
used indices of the flow regime. First, the Reynolds number is a dynamic similarity measure and
defined as
Uc
Re =
,
ν
where U is the flight speed in relation to the fluid at rest, c is a characteristic length in the direction
of flow and usually taken as the mean chord, and ν is the kinematic viscosity. The Reynolds number
can be interpreted as the ratio between inertial and viscous forces.
In oscillatory flows the ratio of two time scales, the time required for a fluid particle to pass
over the mean chord, tc = c/U , and the time taken for one kinematic cycle, T = 1/f , is commonly
expressed as the reduced frequency
ωc
k=
.
2U
The value of k expresses the relative importance of unsteady terms with k ≈ 0.1 implying that
unsteady effects most often can be ignored, while k of order 1 indicates that unsteady phenomena
are likely to occur. When comparing two situations such as different-sized animals moving in
different fluids, the similarity of Re and reduced frequency guarantees that the flow regime and
hence aerodynamic properties will be the same. A closely related reduced frequency based on the
wing semi-span, b, is
ωb
=
.
U
The two measures of reduced frequency are related through the aspect ratio (AR = 2b/c) as
= k · AR. Alternatively, a measure of the flapping velocity of the wing tip to the forward
velocity is given by
2φbf
K=
,
U
where φ is the angular stroke amplitude and f is the flapping frequency. The advance ratio,
J = K −1 , is the ratio of forward flight velocity to the wing-tip velocity. K is closely related to the
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Strouhal number defined as
fA
,
U
where A is the double amplitude of the wake vortices in a Kármán (drag wake) or a reverse Kármán
wake (thrust wake), but in the absence of wake information the double amplitude of the wing (or
fin) tip is usually taken.
St =
3 Vortex theories of bird flight
Kelvin’s circulation theorem states that, in a homogeneous, incompressible and inviscid fluid, the
circulation around a closed circuit will have the same value when measured over the same fluid
elements and circuit at any time as the circuit is followed in the flow. This theorem will prove very
useful to our applications on bird flight, because the aerodynamic properties of the wake vortices
can be directly linked to the time-averaged aerodynamic forces on the wings having generated
the vortices. The circuit of Kelvin’s circulation theorem just has to enclose the aerofoil and the
space behind it where wake vortices will appear. In steady flight at speed U , the Kutta–Joukowski
theorem gives a relation between the lift, L, and the circulation, , as
L = ρ2bU ,
(1)
where ρ is air density and b is wing semi-span. It follows that if there is a change in lift developed
by the aerofoil also the bound circulation changes, which must be offset by an equal and opposite
circulation in the wake. This is all there is in terms of background to develop vortex based
theories of flight. The modeller only has to specify a more or less realistic geometry of the wake
vortices and enforcing a force balance between weight, drag and the force associated with the
wake momentum.
3.1 The actuator disc and momentum jet
In its simplest form the bird is replaced by an actuator disc of radius b that magically deflects
the oncoming airflow downwards (Fig. 1). At an extreme end of the spectrum of vortex wake
based flight models, the momentum jet qualifies as belonging to this family, where the vorticity
is confined to an infinitely thin, cylindrical sheet enclosing the uniform jet [6]. Notice however
that, in the wake description, nothing remains of the time varying forces developed by cyclically
beating wings. The mass flow through the actuator disc induces a downward velocity, wi , when it
passes the disc, reaching a final speed of 2wi in the far wake with a jet diameter of b. The rate of
vertical momentum flux required to support the weight at some forward speed U determines the
induced velocity wi = mg/(2Sd U ρ), where m is body mass, g is acceleration due to gravity, and
Sd (=πb2 ) is the wing disk area. This in turn gives the induced power as Pi = mgwi . For a complete
P(U )-relationship, commonly denoted the ‘power curve’ the terms of parasite and profile power
are added to Pi , resulting in the most popular and widely used flight mechanical model of bird
flight [3, 5, 7, 8].
3.2 Vortex ring theory
A step towards increased reality was taken by Rayner [9–11] who developed an aerodynamic
model of flapping flight in which the wake vortices were represented as circular or elliptic loops,
each one shed as the result of a downstroke. It was assumed that the upstroke was aerodynamically
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709
unloaded and therefore did not leave any traces in the wake. The size, geometry and orientation of
the vortex loops are determined by the wingspan, wingbeat frequency, stroke amplitude, forward
speed and by the circulation distribution along the wing, ( y). The circulation of the vortex
rings is determined by imposing the force balance condition, i.e. that the rate of wake momentum
must balance the vector sum of weight and aerodynamic drag (Fig. 2). Notice that the angle, ψ,
by which the rings are tilted with respect to the horizontal determines the lift to drag ratio as
L/D = cot ψ. The mean rate of increase of kinetic energy deposited in the rings is the induced
power. The periodicity in the wake now corresponds to the wing beat periodicity, with a close
connection between the aerodynamic force time history and the wake trace. In the first generation
of this flight model it was assumed that discrete vortex loops are shed at all speeds, which later
had to be modified as experimental data refuted this assumption (see below).
Figure 1: Actuator-disc model for induced power of a flying bird. The bird is represented by a
circular disc cross section (Sd ) with wingspan as diameter, where the oncoming flow of
speed U is deflected downwards by the induced speed ui so that the downward imparted
momentum balances the weight. Also shown is the coordinate system (x, y, z) used
throughout this paper.
Figure 2: Vortex ring wake of flapping bird flight. When the upstroke is aerodynamically inactive
the downstroke generates a vortex loop. The associated impulse (I = ρSe , where ρ is
air density, Se is the planar area of the vortex loop and is the circulation of the loop)
and hence aerodynamic force are normal to the surface area of the loop, itself tilted in
relation to the horizontal by an angle ψ.
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3.3 The constant circulation wake and other relatives
An experiment by Spedding [12] showed that the wake did not consist of discrete vortex loops at
a moderate cruising speed (U = 7 m s−1 ) in a kestrel (Falco tinnunculus). Instead, the wake was
characterized by a pair of wing-tip vortices of near constant circulation throughout the wing beat,
without any noticeable shedding of transverse vortices. The necessary asymmetry (in order to
achieve non-zero thrust) between down- and upstroke is achieved in many birds by the flexing of
the wrist-joint, causing a reduced span during the upstroke and thereby a reduced projected wake
area in relation to that from the downstroke [12]. Because the bound circulation is constant, 0 ,
there is no shedding of transverse vortices and so the main wake structures are the wing-tip vortices
of constant circulation (hence we shall denote this wake as the cc-wake, indicating constant
circulation). The generation of vortices is associated with an energy cost since the energy content
in the wake is lost. Therefore, with no or minimal transverse vortices, the cc-wake could be argued
to minimize the mechanical cost of cruising flight and should be a favourable configuration in
bird flight where minimizing energy cost is advantageous, such as during long-distance migration
or commuting between nest and foraging sites.
The cc-wake can be understood as a deformed version of the trailing vortices left behind
by a fixed wing aircraft. The wake consists of two straight wing tip vortices in gliding flight,
and in cruising flapping flight, a shallow wing beat makes these vortices follow an undulatory
track, both vertically and horizontally as they trace the 3D movement of the wing tips (Fig. 3).
The constant circulation and the simple geometry of this wake make an analytical treatment both
straightforward and elegant [6]. The impulses associated with down- and upstroke wake elements,
respectively, are
Id = ρcc Ad
and
Iu = ρcc Au ,
(2)
where cc is the circulation of the trailing wing tip vortices, and Ad and Au are the wake areas
circumscribed by the tip-vortices during downstroke and upstroke, respectively. The associated
forces are equal to the time rate of generation of wake momentum (=impulse), F = d (mv)/dt,
which, integrated over one wing beat period, gives the time averaged lift and drag as
Figure 3: Trailing vortices at cruising flight. The wake consists of trailing wing-tip vortices
of constant circulation. The overall wingbeat wavelength (λ) depends on the wing
beat frequency, while the wake angles may differ between downstroke and upstroke
(ψd , ψu ).
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711
L̄ =
1
ρ0 (Ad cos ψd + Au cos ψu ) ,
T
(3)
D̄ =
1
ρ0 (Ad sin ψd − Au sin ψu ) ,
T
(4)
where ψd and ψu are the titling angles that the down- and upstroke make with the horizontal,
respectively (Fig. 3; [6, 12]). With a symmetric wing beat, ψd = ψu , and with Au /Ad = ζ(ζ ≤ 1),
L/D can be written as
1+ζ
L/D =
cot ψ.
(5)
1−ζ
Pennycuick [13] simplified this analysis further by assuming constant spans during down- and
upstroke, i.e. that the flexing of the wings before the upstroke takes place momentarily at the down-/
upstroke transition and likewise that the extension of the wing takes place at the up-/downstroke
transition. Then the L/D depends on the span ratio (b = bu /bd ) as
L/D =
1 + b
1 − b
cot ψ.
(6)
It follows that L/D is maximized by a high ζ or b , i.e. from a small upstroke span reduction and
a shallow wing beat amplitude.
In a closely related model, Rayner [14] incorporated the cc-wake in a quasi-steady lifting line
analysis of flapping flight at cruising speeds. The usual elliptic wing loading was replaced by an
alternative due to Jones [15] that gives slightly improved aerodynamic efficiency. A problem now
presented itself by the fact that two models with quite different presumed wake geometries, the
vortex ring model and the model based on the cc-wake, were used to represent flapping flight.
Rayner’s [14] prescription was the postulate that there was a sudden transition from vortex rings
to the cc-wake at some intermediate flight speed, and the notion of gaits was introduced to the
flight literature. The gait analogue to terrestrial locomotion, where the transition between different
patterns of limb movement and ground contact was very sudden at predictable Froude numbers,
was based on the existence of the two fundamentally different wake forms and the unimaginable
topology of intermediates.
In order to account for the unsteadiness of flapping flight and the influence of the wake on the
induced flow near the airfoil, Phlips et al. [16] modelled the wake as a lifting line representing
the current half-stroke, but previous vortex lines were collected as streamwise wing-tip vortices
and with transverse vortices shed at the turn-points of each half-stroke to account for changes in
the bound circulation. The validity of their analysis was restricted to reduced frequencies k < 1
and wing beat amplitude φ ≤ 60◦ , where significant departures from the linear lift slope c1 (α)
occur due to unsteady phenomena. The far wake of Phlips et al. [16] is similar to the ‘ladder
wake’ postulated by Pennycuick [17] to apply when birds have rigid wings that cannot be flexed
at the wrist joint during the upstroke. Examples of birds where a ladder wake could exist are
hummingbirds and swifts, where the force asymmetry between down- and upstroke has to be
achieved by variation in bound circulation, and hence the shedding of transverse vortices, rather
than by reduced wingspan and maintained circulation as in the cc-wake. Yet other more elaborate
theoretical wake configurations have been treated in the literature [18, 19], but it is beyond
the scope of this paper to go much beyond this point of model complexity for flapping flight.
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712 Flow Phenomena in Nature
For further accounts of aerodynamic models of flapping flight the reader is recommended the
excellent review by Spedding [6].
4 Bird wakes in reality
The development of aerodynamic models has converged with that of experimental results on the
geometry and properties of real wakes. However, the vortex ring theory of Rayner [9] was developed independently from experiments carried out simultaneously by Kokshaysky [20]. Thereafter
the aerodynamic modelling has been tightly connected to empirical wake data. We now proceed
by reviewing some classic experimental work on bird wakes.
4.1 Take-off flight
Kokshaysky [20, 21] recorded the wakes during short take-off flights in two finches, the chaffinch
(Fringilla coelebs) and the brambling (F. montifringilla), by using paper and wood dust as tracer
particles combined with multiple flashes photography. The two closely related species are morphologically very similar (Table 1) with reduced frequency during the experiments being k = 0.87
for the chaffinch and k = 1.29 for the brambling. Both species generated discrete vortex rings associated with downstroke. The results came as a timely support for the assumptions of the vortex
theory of bird flight by Rayner [11], although no quantitative information regarding vorticity and
circulation was available.
4.2 Slow forward flight
The next major experimental development was the application of the particle image velocimetry (PIV) method in which a cloud of buoyant helium filled soap-bubbles was generated and
a bird was trained to fly through this cloud. The 3D movements of the bubbles were recorded
by stereophotogrammetry. This methodological breakthrough allowed, for the first time, quantitative measurements of bird wakes to be made. The first species to be evaluated was the pigeon
(Columba livia) in slow flight, U = 2.4 m s−1 [22], in which the wake consisted of discrete vortex
loops. However, the loops appeared asymmetric in the sense that the start end was rather concentrated while the stop vortex core had a larger diameter and was spatially less well defined. Quite
surprisingly, and somewhat disconcertingly, the vortex loops contained approximately 1/2 of the
momentum required to support the weight of the bird, which signalled that not all vorticity was
confined to the vortex core of the observed rings. In a second experiment of a jackdaw (Corvus
monedula), the results from the pigeon were repeated, i.e. a significant wake momentum deficit
was obtained with only 1/3 of that being required confined to the main vortex core [23]. This result
was obtained by integrating centreline velocities through the wake, thus including, supposedly,
flow induced by vorticity not restricted to the compact vortex cores. The shortfall nevertheless
greatly exceeded the calculated experimental uncertainty. Some selected wake properties from
these experiments are shown in Table 1.
Discrete vortex loops have been reported also in some additional species in slow forward flight
[24, 25], but no quantitative data except approximate flight speed are available (Table 1). Discrete
vortex loops were also found in two species of bat in slow flight [26]. Taken together birds and
bats in slow flight (U ≤ 3 m s−1 ) shed discrete vortex loops generated by the downstroke with
the upstroke being more or less aerodynamically unloaded, although the detailed geometry differ
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0.021
0.022
0.35
0.216
0.21
0.21
0.014
0.48
0.12
0.030
0.017
0.21
Pigeon
Chaffinch
Brambling
Pigeon
Jackdaw
Kestrel, downstroke
Kestrel, upstroke
White-rumped munia
Tawny owl
Meyer’s conure
Thrush nightingale
Robin
Kestrel, gliding
0.338
0.076
0.047
0.048
0.131
0.11
0.046
0.044
0.094
0.095
0.076
0.076
Chord
(m)
0.13
0.132
0.33
0.296
0.338
0.338
0.08
Semi-span
(m)
8.8
4.7
5.4
5.7
6.1
7.0
6.0
8.8
8.8
AR
40.0
16.1
23.6
17.2
19.0
55.9
38.2
39.6
39.6
N
(N/m2 )
0
14.8
14.4
18
17
6.7
5.6
7.7
7.7
30
F
(Hz)
Topology
loop
loop
loop
loop
loop
cc
cc
loop
loop
loop
loop
loop-cc
cc
loop
loop-cc
cc
cc-glide
U
(ms−1 )
slow
3
1.8
2.4
2.5
7
7
1
slow
slow
4
7
10
4
7
9
7
0.2b
0.16
0.11
0.17
0.14
0.097b
0.39b
0.13e
R0 /R
8.7
2.2
0.8
3.6
1.6
1.0
0.99
18.8
4.54
1.80
0.55
ωmax c/U
1.28
0.45
0.15
1.16
0.31
0.19
0.93
7.63
3.47
0.94
1.13
/Uc
0.45c
0.5c
0.72d
0.54c
0.40c
1.16d
1.04d
0.52c
0.35c
1.04d
I /Tmg f
1
20
20
22
23
12
12
49
25
24
30
30
330
HRSg
HRS
HRS
28
Source
AR, aspect ratio; N , wing loading; f , wingbeat frequency; U , flight speed; R0 , vortex core radius; R, vortex ring radius; ω, vorticity; c, mean
wing chord; , circulation; I , impulse; T , time period; m, body mass; g, acceleration due to gravity.
a Scientific names: pigeon Columba livia, chaffinch Fringilla coelebs, brambling F. montifringilla, jackdaw Corvus monedula, kestrel Falco
tinnunculus, white-rumped munia Lonchura striata, tawny owl Strix aluco, Meyer’s conure Policephalus meyeri, thrush nightingale Luscinia
luscinia, Robin Erithacus rubecula.
b Core radius in relation to transverse distance between wing tip vortices.
c In relation to reference circulation = mgT /ρS (see text for symbol definitions).
1
e
d In relation reference circulation = mg/ρU 2b (see text for symbol definitions).
0
e Refers to an aggregated vortex in a confined space, not the result of a single downstroke.
f This quantity denotes sufficiency of supporting the weight when I /Tmg ≥ 1.
g Hedenström, A. Rosén, M. & Spedding, G.R., unpublished data.
Mass
(kg)
Speciesa
Table 1: Morphology and wake properties compiled from wake-visualization studies of bird flight.
Vortex Wakes of Bird Flight
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713
714 Flow Phenomena in Nature
from an idealized ring/ellipse of equal core radius and vorticity at all stations. The two quantitative
measurements of ring momentum suggested a significant deficit of forces to achieve force balance.
4.3 Cruising flight
Using the same set-up as for the pigeon and jackdaw in slow flight, Spedding [12] obtained
wake images of a kestrel flying at a moderate cruising speed, U = 7 m s−1 . The wake appeared
dramatically different from that of slow speed by the lack of any detectable transverse vortices.
The main wake features were a pair of streamwise vortices shed from near the wing tips, showing
an undulatory trace tracking the path of the wing tips. The wake elements associated with downand upstroke measured similar circulation, hence this wake is usually referred to as the ‘constant
circulation’ wake. The lift is given by eqn (3), which was 2.15 N for the kestrel to be compared
with the weight of the bird of 2.06 N (Table 1). Hence, at cruising speed the inferred wake topology
and measured properties (Au , Ad , ) satisfied the force balance criterion. As explained above these
findings prompted the amendment of the vortex theory of forward flapping flight to account for
the appropriate wake geometry in cruising flight [12, 14].
4.4 Gliding flight
At equilibrium gliding, flight potential energy is converted into work against the aerodynamic
drag. The bird itself does not perform any work since the wings are not flapped. Gliding flight is
however not effortless since by holding the wings in an outstretched position the flight muscles
produce static muscle work that consumes chemical energy at a rate approximately two times
the basal metabolic rate [27]. The wake in gliding flight observed in a kestrel at U = 7 m s−1 ,
consists of two straight streamwise wing-tip vortices [28]. The measured circulation matched the
force balance criterion, indicating that all vorticity is accounted for in the main vortex structures
observed. A Harris’ hawk (Parabuteo unicinctus) gliding in a wind tunnel also showed wing-tip
vortices, which were shown to be influenced by the primary feather configuration [29].
4.5 Conclusion and speculation
The combined basis of bird wakes available until year 2003 are those studies referred to in
this section (Table 1). In flapping flight quantitative wake data were available from two speeds,
U = 2.5 m s−1 (pigeon, jackdaw) and U = 7 m s−1 (kestrel). Additional data, albeit qualitative,
from a few other species suggested the presence of vortex loops at slow speeds (U ≤ 3 m s−1 ).
First, the paradoxical wake momentum deficit could not be satisfactorily explained, although
Rayner [24] suggested that the wake deficit could be due to that the birds actually decelerated
during the experiments. However, this explanation appeared to be false [30]. Second, the apparent
existence of topologically two distinct wake types, the discrete loop and the cc-wake, has led some
authors to introduce the notion of ‘gaits’ [14, 24, 25], in analogy to the discrete gaits as found in
quadrupeds. In terrestrial locomotion the transition between gaits, such as canter, trot, gallop, is
very abrupt and occur quickly at some predictable Froude number. Rayner [24, 25, 31] speculated
that the transition between the wake types in bird flight could happen from one wing beat to
another and that there could not be any intermediate forms, hence the gait-transition analogy.
It should be remembered that there were no wake data from any bird at more than one speed.
Over time the notion of gaits gained tacit acceptance [32–34], even though researchers had little
success when looking for indirect signs of a transition between wake types by observing various
wing beat kinematic parameters across flight speeds [35–37]. Most kinematic measures actually
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showed smooth curves in relation speed without any discontinuities as expected at a sudden gait
transition.
5 Bird wakes in reality: digital particle image velocimetry (DPIV)
Towards the end of the previous millennium it seemed that what was lacking was new data.
Thanks to the technological developments in flow visualization techniques by accurate, highresolution CCD-array cameras, the development of efficient analysis, such as correlational image
velocimetry (CIV) routines, and not least the introduction of a new low-turbulence wind tunnel
dedicated for bird flight research, the time was right to attempt new experiments.
5.1 The wind tunnel
A basic requirement for repeatable wind tunnel experiments at low Re is that the background
flow is non-turbulent as this is the reference against which to reduce any effects caused by the
object [38]. At Lund University a wind tunnel dedicated for bird flight was designed and has
been operational since 1994 [39]. The tunnel is a Göttingen type with recirculating flow and
a contraction ratio of 12.25:1 between the settling chamber and test section cross-areas. The
octagonal test section is 1.2 m in diameter with a 1.5 m long closed part followed by a 0.5 m gap
between the end of the closed test section and the bell mouth of the first diffuser. This opening
is a very important feature of the design as it readily allows access to the experimental subjects,
typically live birds. To enable climbing and descending flight the entire tunnel is tiltable around
a pivot. A survey using a hot wire anemometer gave a turbulence level of ≤ 0.05%, measured as
RMS at U = 10 m s−1 . The low background turbulence of the wind tunnel is a prerequisite for
repeatable DPIV. In the tunnel the spatial directions (x, y, z) refer to the streamwise, spanwise and
vertical direction and their associated speeds are defined as (u, v, w). A more detailed account of
tunnel design and technical data can be found in Pennycuick et al. [39].
5.2 DPIV for birds (BPIV)
The use of DPIV has become a widely used measurement technique of fluid flow [40], and a custom
designed DPIV set-up has been deployed and applied to bird flight in the Lund wind tunnel. The
incentive was the imbalance in the literature between quantitative data and speculation, with data
from only two points on the speed axis representing different species and showing fundamentally
different wake geometries. Hence, there was a clear gap to fill in order to address the long standing
momentum deficit paradox at slow speed and the possible change of the wake across a wider range
of speeds. The DPIV technique relies on pairs of digital images from which the displacement of
identifiable particle patterns are determined. DPIV can be applied to numerous biofluid problems
and it has been used quite extensively to study swimming [41]. Previous applications to animal
flight have been restricted to low Re using mechanical flapping wings [42, 43] and recently also
to tethered insects [44]. In our wind tunnel a 200 mJ dual-head pulsed Nd:YAG laser was used,
but it appeared that about 100 mJ/pulse was sufficient to yield reflection enough from the 1 µm
fog particles used as seeding. The recirculation design of the wind tunnel made it ideal for DPIV,
since it allows the entire tunnel to be filled with a homogenous thin fog. The bird is trained to
fly steadily in the front half of the test section, while the laser light sheet was approximately 18c
(c is wing chord) downstream from the bird (Fig. 4). An array of infrared LED-photodiodes was
arranged so that when interrupted by the bird, the laser pulses were suspended to prevent the bird
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Figure 4: Experimental set-up in the Lund wind tunnel to record wake vortices. The bird (tn) is
flying in the front of the test section at speed U , separated from the laser light sheet
(coming from the laser, pl) gaited (gb) by the summed output from an array of LEDphotodiode pairs that would suspend the laser light if interrupted by the bird. Laser
pulses and cameras (tm1, 2) are synchronized by two delay generators (dg1, 2), and
camera output is read into imaging cards in a PC (ic1, 2) (based on [30, 45]).
from direct contact with the laser light. The laser pulses are synchronized with the cameras by
two pulse delay generators (Stanford Instruments DG535). In the first set-up the light sheet was
vertically aligned with the flow, and hence the images are streamwise slices of the wake. A 3D
reconstruction of the wake topology could be obtained by sampling the wake at different stations
along the wingspan, as recorded by an independent video camera positioned far downstream from
the test section (inside the first diffuser). The system was tuned as to give minimum measurement
error on the wake disturbance quantities (such as u and w, rather than the mean free stream)
and the final ‘add bird’ experiments were conducted with an estimated uncertainty of <0.5% in
disturbance flow fields (u and w), and ±10% for gradient quantities such as ωy (for definition see
eqn [7]). A full account of the experimental set-up, procedure and detailed error analysis is given
by Spedding et al. [45].
5.3 DPIV of a thrush nightingale
In the first series of experiments a thrush nightingale (Luscinia luscinia) was used as a representative of the species-rich passerine family, being a rather typical long-distance migrant of small
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size with low aspect ratio wings (lean mass 20 g, AR = 5.4). Results from such a species should
therefore be taken as sufficiently general to apply to many species of similar size and morphology.
The bird was trained, by using a movable perch in the test section, to fly steadily in a repeatable
manner approximately 0.9 m upstream from the image centre of its wake. Sequences of 10 image
pairs at 10 Hz (determined by the repetition rate of the laser) were taken at integer speeds in the
range 4–11 m s−1 and at different locations along the wingspan. A total of 4000 wake images were
sorted according to speed and wing position (body, arm, hand wing, and wing tip) and analysed.
The following account is based on original data published in Spedding et al. [30]. It should be
remembered that the entire wake consists of the inviscid induced (lift-dependent) drag as well as
the pressure and viscous drag components from the body and wings. The possibility to observe
them in the wake flow depends on whether they remain physically separated, and hence can be
distinguished from each other, or not. Otherwise, if they are intermixed the wake will represent
the sum of thrust and drag components, which will be exactly zero in steady flight at constant
speed [30].
5.3.1 Wake topology
At low speeds (Re = 13,000 at U = 4 m s−1 ) the wake showed characteristic start vortices
generated at the beginning of downstrokes and corresponding stop vortices that were spatially
more spread out and with lower peak vorticity. The core-to-loop radius was 0.11, which should be
considered as a small-cored vortex loop [46]. Some vorticity is shed during the upstroke also at
slow speeds, but for the most part, the upstroke may be inferred to be more or less aerodynamically
inactive. A notable feature of the slow speed wake (U ≤ 5 m s−1 ) was that the region dominated
by the stop vortex (clockwise or negative vorticity) also contained embedded patches of positive
(counter clockwise) vorticity so that there appeared to be a mosaic of alternate vorticity. This
mixture of opposing vortices may be the reason for the larger extension of the stop vortex, i.e.
larger core radius, caused by vortex interaction. The wake was reconstructed by combining consecutive frames to generate a composite covering a whole wing beat period (Fig. 5). The 2D
flow/vorticity maps are roughly consistent with a wake consisting of discrete vortex loops, each
shed as the result of a downstroke but with different radii between the start and stop ends. In
particular, one must take note that if the circulation is not balanced between positive (start) and
negative (stop) vortices, then the true wake structure (when measured at this downstream location)
must be more intricate than this. The loops induce a downwash normal to the surface plane of the
loop (Fig. 5).
When increasing the flight speed the wakes exhibit increasing vorticity originating from
the upstroke, suggesting an increasing aerodynamic significance. At the maximum speed,
U = 11 m s−1 , there was a more or less continuous vortex trail throughout the wing beat of
similar strength. Cartoon reconstructions of wake topology at three different speeds are shown
in Fig. 6. The reconstruction in Fig. 6 is based on wake images at different positions along the
wingspan, which together could be used to deduce the 3D geometry. Even though the high-speed
wake appears quite dissimilar from the other two, it represents one end of a continuous spectrum
where the discrete loops are at the other extreme. As suggested by quantitative properties (see
below), there is a smooth transition of wake topology mainly caused by the increasing aerodynamic function of the upstroke [30].
At cruising speeds (U = 9−11 m s−1 ; Re = 35,000) the main features are undulating wing
tip vortices that are interconnected by cross-stream vortices of alternate sign. In spite of its
superficial similarities, this wake should not be considered identical to the ladder wake postulated by Pennycuick [17]. In the ladder-wake the ‘rungs’ of the ladder are cross-stream vortices
shed at the transitions between down/up and up/down strokes due to changed bound circulation
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Figure 5: Cross-section of wake vortices from a thrush nightingale. Reconstruction of slightly
more than one wing-beat vortex wake at slow speed, U = 4 m s−1 . The composite is
constructed from a sequence of consecutive wake images obtained in an image plane
aligned with the (x, z)-directions at the mid-span position (centre plane). The wake
wavelength is λ = UT , where T is the wingbeat period, and the downstroke length (λd )
and upstroke length (λu ) are marked with wingspan (2b = 26 cm) as reference length.
The colour bar is scaled asymmetrically about ωy = 0 with numbers at the ends showing
values in units of s−1 (from [30]).
Figure 6: Cartoon interpretation of wake geometry in a thrush nightingale at different speeds. The
panels represent slow (top), medium (middle) and cruising (bottom) speeds (from [30]).
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(and associated lift). The cross-stream vortices observed here represent a continuous shedding
of vorticity during each half-stroke, but the origin of this vortex shedding remains unclear. This
wake therefore mostly resembles the constant circulation wake found in the kestrel at U = 7 m s−1
[12]. It may be that also the kestrel wake has these low-amplitude cross-stream vortices but that
they were not detected by the helium bubble method.
5.3.2 Quantitative properties of the wake
The spanwise (or cross-stream) component of vorticity is defined and calculated as
ωy =
∂w ∂u
− ,
∂x
∂z
(7)
where u (in x-direction) and w (in z-direction) are velocity components. The circulation measures
the strength of a vortex and calculated as the vorticity integrated over a surface, S, in the imaging
plane as
=
ω · dS.
(8)
In practice the circulation of eqn (8) was calculated by making a discrete approximation by
the sum of all contiguous cells in a local neighbourhood around a vortex cross-section, where
|ω|y exceeds a threshold value. The below-threshold vorticity was then estimated by assuming
a Gaussian distribution of vorticity around its peak value. Measures of vorticity and circulation
were carried out for the different wing locations and plotted against flight speed, for the 4000 wake
velocity fields [30]. Peak vorticity and circulation of main start and stop structures are shown in
Fig. 7, where both quantities have been normalized as |ω|max c/U and /Uc. The normalized
vorticity expresses the ratio of timescales for convection over the wing and rotation around a
vortex, while the normalized circulation expresses the integrated magnitude of shed circulation
compared with a measure of the momentum flux over the wing chord. Both quantities show
maximum values at the lowest speed (U = 4 m s−1 ), and they decline to minimum values at
U = 11 m s−1 . There is a notable difference between start and stop vorticity and circulation at
slow speeds, but with increasing speed the difference decreases and from U = 6−7 m s−1 it has
disappeared (Fig. 7).
5.3.3 Force balance
Whether the measured circulation is sufficient for balancing the forces required to fly depends
critically on the appropriate interpretation of the wake geometry that determines the correct area
appearing in eqn (2). It is convenient to define two reference values of circulation required to
support the weight [30]. First, if the wake were to consist of two straight wing-tip vortices the
Kutta–Joukowski theorem [eqn (1)] gives the lift that must balance the weight, W , if we neglect
3D and wing tip effects. Hence, rearranging eqn (1) yields the circulation required to support the
weight for a gliding wake as
0 = mg/(ρ2bU ).
(9)
If instead the wake appeared in the form of discrete vortex loops with projected area onto the
horizontal plane Se = πb(λd /2), where λd (=UT τ and T is the wingbeat period and τ is the
downstroke ratio) is the downstroke wave length, then the associated circulation required to
balance the weight is
1 = mgT /ρSe .
(10)
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Figure 7: Quantitative wake properties in relation to speed for a thrush nightingale. Variation in
standardized peak vorticity |ω|max (c/U ), where C is wing chord and U is forward speed
(a) and standardized total measured circulation /Uc (b). Measures from starting (filled
circles) and stopping vortices (open circles) are shown in relation to flight speed. Error
bars represent standard deviations (from [30]).
Although neither of the two idealized geometries apply directly to the nightingale wake, they may
still be useful as reference quantities against which measured circulation can be compared.
The start vortices contain more concentrated circulation than the stop vortices (Fig. 8), quantitatively illustrating the feature of Fig. 5 where the stop vortex is spatially more distributed and
apparently weaker than the start vortices. However, when comparing the total integrated circulation associated with positive (start) and negative (stop) vortices with the reference values 0 and
1 it appears that only structures with uniform strength equal to that measured in the stop vortices
would contain circulation enough to support the weight, while the well defined start vortices
exhibit an approximate 50% momentum deficit. This finding therefore repeats the original wake
momentum deficit of the pigeon and jackdaw [22, 23].
As previously remarked there were patches of positive vorticity in the region of down/upstroke
transition otherwise dominated by the negative stopping vorticity. Spedding et al. [30] made a
further calculation, supposing that this additional positive vorticity has the same source as that
of the main start vortex and hence that the total circulation contributing to the total aerodynamic
force is given by the sum of all vorticity of the same sign (+, in the case of the start vortex with
additional low-amplitude vorticity from the area dominated by the stop vortex). After accounting
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Figure 8: Circulation in relation to speed in a thrush nightingale. Total integrated circulation tot
from all positive (filled circles) and negative (open circles) vorticity from the wake in
relation to flight speed (U ). (a) The fraction of the total circulation that is not contained
in the strongest vortex cross-section is higher in the stopping vortices than the starting
vortices. (b) The total integrated circulation in relation to the reference circulation 1
(see eqn (10) for definition). At speeds ≤8 m s−1 the total negative vorticity would
be sufficient for weight support, but not the positive component. Error bars represent
standard deviations (from ref. [30]).
for all + signed vorticity in this way it appeared that weight balance was achieved (Fig. 9). In
that case the revised geometry of the vortex loops could account for weight support since force
balance was achieved, and the wake momentum paradox was resolved [30].
5.4 DPIV of a robin
Another similar set of data as for the thrush nightingale has since been collected from two European
robins (Erithacus rubecula) using the same procedure and camera set-up shown in Fig. 4 [47]. The
robin is a close relative to the thrush nightingale but migrates a shorter distance within Europe and
has a lower body mass (during experiments m = 0.017 g), and a shorter wingspan and lower aspect
ratio than the nightingale (2b = 0.22 m, AR = 4.7). At U = 4 m s−1 Re = 13,000 and k = 0.55 and
at U = 9 m s−1 Re = 29,000 and k = 0.24.
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Figure 9: Total circulation in relation to speed. The total integrated circulation with the positive
vorticity (+ , filled squares) also including all above threshold traces of positive vorticity found in the neighbourhoods of the predominantly negative vorticity associated with
the transition between down and upstrokes. Both positive and negative vorticity measured in this way are sufficient for weight support at slow speeds. Error bars represent
standard deviations (from ref. [30]).
The wake topology showed striking similarities with that of the thrush nightingale, with characteristic discrete loops at slow speeds. An example of a cross-section of vortex loop associated
with a downstroke at U = 6 m s−1 is shown in Fig. 10. Also the robin exhibited the asymmetry
between start and stop vortices, typified by a more diffuse and spatially spread out stop vortex
than the start vortex, even if the illustrated example shows quite a clean vortex cores of both
ends of the loop. When increasing the speed the wake undergoes similar gradual changes as in
the nightingale, characterized by increasing vorticity shed from the upstroke. By U = 9 m s−1
the same magnitude vorticity is shed throughout the down and upstrokes, in agreement with the
cartoon reconstruction of Fig. 6 (the streamwise thick vortices in the lower panel, Fig. 6). The
pattern was apparently the same in the two robins investigated.
Figure 11 displays representative velocity profiles through the start and stop vortices of Fig. 10,
as the u(z) and w(x) components in relation to the peak vorticity of each vortex at coordinates
(x0 , y0 ). The vortex core diameter measured by the distance between velocity peaks is about 2.3 and
2.8 cm for the start-and-stop vortices, respectively, while the overall streamwise loop diameter
is about 20 cm at this speed. This latter estimate comes from a hypothetical loop with simple
geometry generated during the downstroke at U = 6 m s−1 , and wing beat period T = 0.068 s and
a downstroke fraction of the wingbeat period τ ≈ 0.5 (as calculated from companion kinematic
measurements). For a cross-stream diameter close to the wingspan (22 cm) this would imply a
span efficiency of 0.91 [25]. The ratio R0 /R = 0.11 for the start vortex and R0 /R = 0.14 for the
stopping vortex are similar to previously measured values for slow flight in the thrush nightingale
[0.1; 30], while the values measured in a pigeon (0.17; 22] and a jackdaw (0.14; 23] are somewhat
larger (Table 1). However, at U = 5 m s−1 R0 /R = 0.16 for a start vortex in the robin (Table 1),
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Figure 10: Cross-section of a vortex loop from a robin. The starting (rightmost) vortex is slightly
more concentrated than the stopping (leftmost) vortex. The bird is flying to the left
at U = 6 m s−1 . The 14-step symmetrical colour bar shows negative and positive
vorticity (s−1 ).
Figure 11: Profiles of the velocity components of wake vortices in a robin. u(z) (a, c) and w(x)
(b, d) for the staring vortex (a, b) and stopping vortex (c, d) shown in Fig. 10. (x0 , z0 )
is the location of the peak in vorticity ωy . The slightly more diffuse stopping vortex is
seen as a larger diameter (2.3 and 2.8 cm, respectively).
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showing that this parameter may vary rather little and that all hitherto measured birds actually
show similar relative dimensions of vortex loops.
The quantitative wake properties /Uc and ωmax c/U also showed the same pattern with speed
as the thrush nightingale, i.e. they both declined monotonically from maximum values at the
slowest speed (Table 1). The normalized circulation showed similar values between the two
species, while the vorticity was more than twice as large in the nightingale compared with the
robin at the slowest speed but the values converged towards the maximum speed (Table 1).
The comparison of the measured circulation associated with concentrated vortices against the
reference values 0 and 1 revealed a nominal wake momentum deficit also in the robin, of
the same magnitude around 50% as in previously investigated species at slow speeds (Table 1).
However, a similar accounting for positive vorticity embedded with the opposite-signed stop
vortex as for the thrush nightingale resulted in enough total positive circulation associated with
the downstroke to claim weight support at slow speeds for the robin experiment as well [47].
5.5 Wakes and kinematics
The wake vortices described and analysed in the previous sections were recorded approximately 0.9 m downstream from the position of the bird, which represents about 3 wing beats at
U = 4 m s−1 and 1.3 wing beats at U = 10 m s−1 , which allow the wake to evolve and change since
the time of shedding off the wings. At slow speeds in particular, when vorticity is at maximum,
the start end of vortex loops will be particularly prone to move due to self-induced convection,
because the oldest portion of the vortex loop (the start end) is convected downwards due to its
own vorticity and the bound vorticity on the wing [10]. Therefore, when we measure the spatial
coordinates of wake elements and their relative position, for instance as the wake inclination
angles (ψd , ψu ), the measured value of ψd would be lower than when the vortex was created. Any
such effects due to self-induced convection will likely be reduced at higher flight speeds where
peak vorticity and time since shedding are reduced (Table 1), but the magnitude of the effects and
whether they can be ignored remains to be investigated. Some basic wing beat kinematic parameters that should correlate with wake geometry were analysed for the same thrush nightingale as
used in the wake analysis [48].
The kinematic parameters investigated were wingbeat amplitude (A0 ), wing beat frequency ( f ),
downstroke fraction (τ) and span ratio (b = bu /bd ). Notice that if b = 1 the span of the downstroke
and upstroke is the same and a net thrust must be obtained from differential circulation between
downstroke and upstroke. However, in the investigated species b was always <1. Notably, both
amplitude and wing-beat frequency changed very little with speed. The wing beat frequency
showed a very weak U-shaped function of speed, which was also found in a previous study of
the same species in the same wind tunnel [49], but the variation in f was only 7% between the
extreme values [48]. Therefore, changes in measures of reduced frequency (k, , K), depend
mainly on U and hence display essentially linear functions of U .
The wake geometric properties were estimated in two ways: (i) from cross-stream vorticity
maps such as Fig. 5, the inclination angles were measured between centres of identifiable vortex
blobs (ψwake ); (ii) the induced downwash should be normal to the plane of vortex elements, and
hence measuring the direction of the induced flow should give an estimate of the wake orientation
(ψind ). These measures of wake orientation were then compared with the expected geometry if
assuming that the wake remains stationary along the path of the wing tip, inferred from amplitude,
flapping frequency and forward speed. During downstroke it appeared that all three measures
differed significantly at slow speeds, but that they converged at the highest speeds (Fig. 12). The
angles of the wake trace increased with increasing speed, while the angles of the kinematic trace
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Figure 12: Wake inclination angles. Angles are measured directly from vorticity maps (ψtrace )
and the induced flow (ψind ) and deduced from wing tip kinematics (ψkin ) during
(a) downstroke and (b) upstroke in a thrush nightingale in relation to speed U (from
ref. [47]).
declined. This may seem contradictory, but reasoning based on self-induced convection of the
wake vortices combined with apparent angular rotation of the wing, affecting the local angle
of attack, could plausibly explain the pattern observed [48]. The agreement between kinematics
and wake geometry agreed better for the upstroke (Fig. 12), but also here there are systematic
discrepancies.
These data demonstrate that the wake vortices do not remain stationary where left by the
wing tip, but evolve due to self-induced convection especially at slow flight speed. Therefore,
a simplistic 1 : 1 correspondence between wing beat kinematics and wake geometry is not a
valid basis for a vortex based flight model. Even if Rosen et al. [48] offers only a limited set
of kinematics data in their first analysis, they nevertheless suggest interesting links between
kinematics and wake properties that certainly require further attention to be fully understood. Of
particular interest should be to measure the wing rotation and so the local angle of attack, which
has so far not been done with any useful detail in bird flight.
5.6 Comparing wake properties
Most wake visualization experiments have concerned slow flight, where one expects a greater contribution from the wake-generated vorticity towards the total drag/energy/power budget, because
the self-generated downwash need to be larger than at cruising speeds. In the wake, the result has
invariably been interpreted as some variant on a vortex ring/loop generated by the aerodynamic
action of the downstroke. Until recently, the only other point of comparison was the kestrel study of
Spedding [10], where the wake was instead composed of undulating wing tip vortices of constant
circulation. The successful development of the DPIV method in a variable speed, low-turbulence,
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wind tunnel allowed, for the first time, the study of changes in wake geometry, vorticity and circulation across the natural speed range in a bird [30]. It appeared that the wake gradually transforms
from discrete loops at slow speeds into a cc-like wake at typical cruising speeds. This transformation from a discrete loop to the cc-wake is probably achieved by the addition of an upstroke loop,
end to end with the main downstroke loop, and as speed increases the upstroke wake structure
is increased in strength to elongation until the downstroke and upstroke wake components form
a continuous trailing wake of streamwise vortices with low amplitude cross-stream vortices (cf.
Fig. 6). Only a subset of the studies presented in Table 1 present any quantitative measures of the
wake and so any comparisons among species are tentative at best. Both the normalized vorticity
and circulation are declining functions of airspeed, but at U = 4 m s−1 and very similar reduced
frequency (k = 0.54 and k = 0.55, respectively) the thrush nightingale show a more than twice as
large peak spanwise vorticity as the robin. The wing loading is 47% larger in the nightingale compared with the robin. The highest normalized vorticity was found in the pigeon (18.8, k = 0.82;
Table 1), suggesting a correlation between wing loading and vorticity. However, the correlation
is not perfect since the jackdaw, with a comparatively high wing loading, shows a lower peak
vorticity value than the thrush nightingale. A straightforward comparison is further confounded
by the fact that the start vortex core contains variable amounts of the total same-signed vorticity,
where the robin wake appears to represent the largest fraction of the total circulation. It may be
that the vorticity shed is a composite function of several variables, for example the wing loading and reduced frequency (k), and a more robust and strongly-linked measure might be that
of a suitably-defined circulation measure that accounts for the amount of vorticity that actually
gets into the wake structures of different kinds. Interestingly, the peak vorticity differed between
down- and upstrokes in the kestrel (while its integrated magnitude in the circulation did not) while
gliding flight values were intermediate between the up- and downstroke flapping flight values.
If one takes the steady gliding wake as a baseline case, then the wing accelerations on downand upstroke in flapping flight might be viewed as tuned perturbations about this baseline for the
purpose of generating net forward thrust from an asymmetric wing beat while maintaining, on
average, sufficient downward momentum for weight support.
The relative loop dimensions seem to be quite similar among species (Table 1), with the exception of the white-rumped munia (Lonchura striata) which was flying in a confined space at slow
speed and with very high wing beat frequency, with a resulting vortex ring diameter approximately
three times larger than the wingspan [50]. The authors interpreted this as an aggregated structure
generated by a number of downstrokes.
Even though Table 1 contains a complete summary of existing data from vortex wakes in birds
the quantitative information is still limited and selective. However, it does suggest a rich future
for careful comparative investigations.
6 Discussion
The vortex wake approach to aerodynamic modelling of bird flight has been available for a
quarter of a century, following in the footsteps of experimental and theoretical analyses of classical
aerodynamics ([51]). In its simple abstraction it offers an elegant way of analysing flight mechanics
[9–11, 52, 53]. Initially, the wake was thought to consist of discrete vortex rings at all speeds [10], a
supposition which gained some experimental support at slow flight speeds [20, 22, 23]. However,
the demonstration of the constant circulation wake in a kestrel at a moderate flight speed [12]
showed that the picture was more complicated than originally assumed. For a long time there was
little development in research on bird wakes, and yet there was a growing interest from ecologists
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applying flight mechanical theory for understanding variation in flight-related morphology and
behaviour [8]. Clearly, if an aerodynamic model is fraught with uncertainty and it is incorporated
as a component of another, say, ecological model, then the uncertainty due to the aerodynamic
model will be propagated also to the predictions derived from the ecological model. In this light,
the recent application of DPIV to bird flight is clearly a development with significantly positive
consequences, not only for mathematical model-builders, but also for the wider set of scientists
using flight mechanical models of bird flight. In the following sections I will briefly discuss some
issues regarding bird flight in relation to vortex wakes.
6.1 The topology of the wake and its properties
Since the demonstration of the cc-wake [12], the paradigm has been the existence of discrete
gaits associated with the wake geometry, rings at slow speed and cc-wake at cruising speed, and
that birds adjust their kinematics so that the output is one or the other of these ‘gaits’ [14, 25,
35, 36]. However, when analysing the wake in relation to speed for a thrush nightingale [30],
it appeared that the wake topology transforms continuously from discrete loops to continuous
wing tip vortices across the natural speed range. This transformation occurs by the reduction in
strength of the cross-stream vortices and an increase in vorticity shed during the upstroke. This
apparent continuous change is well supported by quantitative measures of the wake, as well as
by the change in wing beat kinematics in relation to speed [48, 54].
The momentum deficit of the early wake visualization experiments has been a disturbing condition and some solutions to the paradox have been proposed. Recently, Tytell and Ellington [55]
investigated the evolution of a vortex ring after formation. If the ring Reynolds number, defined as
Re0 =
DA
,
ρνf
where DA (=mg/πb2 ) is the disc loading, ν is the kinematic viscosity and f is the wingbeat
frequency, exceeds a certain value the ring will be turbulent and will shed off vorticity that may
cancel through interaction with opposing vortices. By the time the structure is imaged in the
far wake some vorticity could be missing which would cause an apparent momentum deficit.
The pigeon has a Re0 = 84,000, which is well above the threshold value for initially turbulent
vortex rings [55]. In the thrush nightingale Re0 = 21, 000 and in the robin Re0 = 17, 000, which
are at the low end of the range of initially turbulent vortex rings [46, 56]. The wake momentum
deficit was similar in magnitude in the nightingale and pigeon when measuring the circulation
of the strongest start vortex structure, and the recently studied robin showed a very similar wake
momentum deficit (Table 1). If there is a transition to turbulent vortex rings somewhere near
or below the robin Re0 so that the vortex rings would have shed momentum until the time of
recording, this could result in the observed momentum deficit. However, the detailed analysis
by Spedding et al. [30] showed that the vorticity is not irretrievably lost. It only occurs in an
unexpected place and is mixed in with the opposite vorticity of the stop vortex by the time it is
recorded in the far wake. Importantly, it was possible to recover all vorticity and to achieve a force
balance by a careful matching between start and associated stop vortices. The generality of this
solution to the wake deficit problem remains to be shown in further similar experiments, but it
currently looks as if the paradox is expunged. It also cautions against overly simple comparisons
of wakes generated by flapping wings with ring-like structures generated by pistons in cylindrical
tubes.
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6.2 Optimum kinematics
Inspired by work on fish locomotion [57, 58] regarding the possible optimal propulsion kinematics
as characterized by the Strouhal number (St, see Section 2 for definition), Taylor et al. [59]
proposed that also birds adjust their wing beat kinematics to an ‘optimal’ range of about St ∈ [0.2,
0.4]. The aerodynamic incentive for this is that at some St the energy input to produce optimum
bound vortices is minimum per unit energy output, i.e. the propulsion efficiency is maximum. The
presence of an optimal reduced frequency has also been found in numerical simulations about
pitching and rotating wing segments at Re = 1000 [60]. The idea of optimal St in fish propulsion
is the formation of ‘optimal’ vortices [61] and so, in principle, there is no lower bound on St
[62]. If vortices are shed to close to each other they might interact adversely with reduced overall
propulsion efficiency. Interestingly, the flapping frequency did not change very much across the
speed range in the thrush nightingale [48], and the same applied to the robin [47]. Since wing
beat amplitude also changed little with speed [48], measures of reduced frequency such as k
and St, change due to changes in flight speed. Hence, birds seem not to maintain some ‘optimal’
wing beat kinematics in order to keep variation in reduced frequency minimal. Particularly at
cruising speeds where the wake consists of continuous vortices any adverse vortex interactions
seems unlikely, as would be possible in a reverse Kármán wake where the optimum efficiency was
encountered. Rosén et al. [48] point out that even though the agreement between ‘optimal’ St and
the range of actual St in birds may look as more than a pure coincidence [59], the true underlying
reason may well be a complex of aerodynamic constraints together with structural/mechanical
ones, such as in the tendon-muscular system, mechanical resonance in relation to morphology, or
some other physiological trait being optimized.
6.3 Aerodynamic mechanisms
Recent work on insect flight has focused on various mechanisms by which sufficient aerodynamic
lift is generated to support the weight and allow manoeuvres [42, 43]. Dynamic stall and associated
leading edge vortices and wake capture are examples of mechanisms used by insects. Bird flight
research has been less concerned with the search for esoteric aerodynamic mechanisms than
research on insect flight, although for example induced drag reduction by wing tip slots (splayed
primary feathers) in gliding flight has been quantified [29]. Close-to-wing flow visualization on
freely flying birds has not been possible for risk of injury (exposure of the birds eye to high intensity
laser radiation is carefully avoided). The use of dead birds is unlikely to produce representative results [63], while experiments with oscillating airfoils can potentially give useful insights.
Close-to-wing flow visualization is probably easiest to obtain during gliding flight, but until this
is made bird aerodynamics must proceed from wake flow visualization.
6.4 A vortex wake theory
The most widely known and used aerodynamic model for bird flight is that due to Pennycuick
[3], where the wake is a momentum jet. This is a drastic simplification compared with the current
representation of bird wakes as illustrated in Fig. 6. But is this reason enough to abandon the
momentum jet model in favour of a model that includes a more accurate wake representation?
All models are, by default, caricatures of the real-world system they are supposed to describe,
where the purpose of a model guide with simplifications can and cannot be accepted [64]. A
main prediction from the momentum jet-based flight model is a P(U ) function, usually referred
to as the power curve. The few direct measurements available do not differ enough from the
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Vortex Wakes of Bird Flight
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Figure 13: The geometry of wake structures in a vortex wake model of flapping flight. The
relative contribution from the circulation of the upstroke is weighted in relation to
the circulation of the downstroke, ranging between 0 at slow sped to 1 at cruising
speed. Symbols are downstroke wavelength (λd ), upstroke wavelength (λu ), downstroke wake inclination angle (ψd ) and upstroke wake inclination angle (ψu ) (based on
ref. [30]).
model [65–67], and with uncertainty estimates of the model prediction empirical data must differ
quite a lot in order to falsify the prediction [64, 68]. However, various tests of components of the
momentum jet model do suggest that it contains anomalies that are not easily rectified by changing
parameters only. For example, Pennycuick et al. [49] concluded that the value representing the
parasite drag coefficient should be reduced from the original default value of 0.4 in small birds
[3] to 0.1 or even less. Their basis for this conclusion was a comparison of the speed of minimum
wing beat frequency and the speed of minimum power (Ump ) as calculated from the model,
with the underlying assumption that the two curves should have the same speed of minima. The
consequence is that the power required to fly at speeds > Ump becomes very low, in fact lower
than the power required to glide at some speed [69], which seems unlikely. Also, the predicted
speed of Ump in very large birds such as swans becomes higher than the speed actually observed
in these birds. Hence, even if the simple momentum jet model has proven enormously successful,
not least by the many valid predictions derived from it [70] and its high citation frequency [8],
there remain some unsettling facts concerning quantitative predictions.
Spedding et al. [30] proposed a vortex wake model as a composite between a downstroke
elliptic and an upstroke rectangular wake structure (Fig. 13). The relative circulation between
down and upstroke is allowed to vary as a function of speed as u = Cu d , where Cu varies from
0 at hovering to 1 at a fully developed cruising flight cc-wake. The model has to satisfy the weight
balance criterion. In principle, the net thrust (related to the power output) could be determined by
the areas of the wake structures projected onto vertical planes. It was shown that this model was
self-consistent with the thrush nightingale data [30]. There are two principal difficulties facing
such a program, however. First, the drag is much smaller than the lift-supporting component in
the wake (by the ratio of L/D), so practical measurement uncertainty will be a significant problem.
Second, and much more significant, a reasonable drag measurement could only be made if the
wake structure representing the inviscid induced drag model were clearly separable from the
viscous drag wake. In steady self-propulsion (of any body), the net fore-aft momentum balance
will be zero, as thrust balances drag. As duly noted in [30], only if the viscous drag wakes can
be identified and isolated in the wake, could their magnitude be determined, even in principle.
A sober analysis of the more recent complex wake structures actually measured behind real flying
birds (Fig. 10) suggests that much more careful research work lies ahead.
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730 Flow Phenomena in Nature
6.5 The ecology and evolution of flight
Ecologists are concerned with various problems related to flight, such as optimum selection of
flight speeds during migration, foraging flight, display flight, predator evasion, load lifting with
respect to prey/food or fuel (fat) stores used for long non-stop flights, etc. [71, 72]. In all these
examples the ‘optimum’ behaviour can be understood on the basis of flight mechanical theory in
combination with some appropriate currency assumption and an optimization rule. Perhaps surprising, also decisions regarding optimal departure time and associated fuel load from a stopover
in migratory birds can be derived on the basis of aerodynamic principles [72]. During moult—the
periodic replacement (typically once per year) of flight feathers—the wings have reduced area
due to missing or growing feathers and will change in shape due to moult gaps. The consequences
on flight performance from moult gaps have recently been analysed from an aerodynamic perspective [73–75]. Aerodynamic performance is tightly linked to morphology and so aerodynamic
models are well suited for understanding the adaptive significance of flight-related morphology.
How animals once evolved an ability to fly is a popular and controversial topic [76], Two
scenarios—the trees-down and the ground-up—have long been the two competing hypotheses
about the evolutionary trajectory that lead to powered flight. From a vortex wake perspective it
is easy to see a natural (gradual) transition from gliding flight to flapping flight with an initially
low amplitude wing beat transforming a straight glide wake to a shallow cc-wake [77]. Aerodynamically, take-off from the ground, even if running to gain speed, seems more problematic than
going via gliding to powered flight [78], while recent fossil finds from China of unambiguously
feathered theropod dinosaurs suggest a ground-dwelling protobird [79]. However, even more
recently a bizarre ‘four-winged’ dromaeosaur Microraptor gui [80], also from China, suggests a
gliding animal and hence new evidence in favour of the arboreal theory for the origin of flight.
The fossils do not leave any behavioural evidence more than the overall morphology in extinct
animals, but an aerodynamic analysis of wake types and possible interaction between forelimb
and hind limb vortices might help us understand how these dinosaurs flew and to follow the
evolutionary trajectory that led to powered flight.
6.6 Future prospects
The quantitative visualization of wakes in freely flying birds has only begun. In addition to
a low turbulence wind tunnel it requires a co-operative and well-trained bird [30]. It is now
important to get more data from a range of sizes and wing morphologies in order to establish a
more generally valid vortex wake based theory of animal flight. On the technical side there are
foreseeable improvements, such as 3D PIV and increased repetition rate of pulsed lasers, which
will improve the geometric characterization of wake vortices and their dynamics. This could
yield time-series animation data on wake dynamics and the 3D wake topology would help in
the partitioning of the wake disturbances in drag and lift components. Near-wing PIV data will
elucidate the aerodynamic mechanisms of gliding and, hopefully, flapping flight in birds. Since
birds have evolved adaptations for efficient flight during about 150 million years, it is perhaps not
surprising that their flight endurance surpasses that of current man made flying vehicles of similar
size by two orders of magnitude. The study of migratory birds as a model system is therefore
likely to continue to be scientifically rewarding for some time to come.
Acknowledgements
The results presented in this paper are based on the joint effort with Geoff Spedding and Mikael
Rosén, to whom I am very thankful for a long-term collaboration. Geoff Spedding,Adrian Thomas,
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731
and Roland Liebe helped improve the manuscript significantly. A.H. is a Royal Swedish Academy
of Sciences Research Fellow supported by a grant from the Knut andAlice Wallenberg Foundation.
The wind tunnel research at Lund University has been funded by The Swedish Research Council,
The Knut and Alice Wallenberg Foundation, the Tryggers Foundation and the Swedish Foundation
for International Cooperation in Research and Higher Education (STINT).
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