## SUMMARY

Bird flight occurs over a range of Reynolds numbers (*Re*;
10^{4}⩽*Re*⩽10^{5}, where *Re* is a
measure of the relative importance of inertia and viscosity) that includes
regimes where standard aerofoil performance is difficult to predict, compute
or measure, with large performance jumps in response to small changes in
geometry or environmental conditions. A comparison of measurements of fixed
wing performance as a function of *Re*, combined with quantitative flow
visualisation techniques, shows that, surprisingly, wakes of flapping bird
wings at moderate flight speeds admit to certain simplifications where their
basic properties can be understood through quasi-steady analysis. Indeed, a
commonly cited measure of the relative flapping frequency, or wake
unsteadiness, the Strouhal number, is seen to be approximately constant in
accordance with a simple requirement for maintaining a moderate local angle of
attack on the wing. Together, the measurements imply a fine control of
boundary layer separation on the wings, with implications for control
strategies and wing shape selection by natural and artificial fliers.

## Introduction

### Bird flight performance envelope

The equations of motion for a homogeneous, incompressible fluid with
constant density, ρ, can be expressed so that, in the absence of any
special boundary conditions, their solution for a given geometry depends only
on the magnitude of one dimensionless number, the Reynolds number *Re*,
which may be written as:
(1)
where *u* is a flow speed, *l* is a characteristic length scale
and μ is the fluid viscosity. The numerator depends on the mass, size and
speed at which fluid is moving, and one may think of *Re* as a measure
of the relative importance of inertial and viscous forces in a flow. Inertial
forces tend to destabilise a flow, while viscosity tends to smooth it out, and
so high Reynolds numbers are frequently associated with complex, and possibly
turbulent, flows. *Re* is a large number in many human-engineering
applications. For example, for a large passenger plane (such as the Boeing
747-400) with cruising flight speed *U*=250 m s^{–1} and
mean chord length *c*=8 m, a Reynolds number that uses *c* as
the length scale is approximately 10^{8}. This is a large number, and
one might expect the air motions to be characterised by enormously complex,
turbulent flows. Indeed they are in the wake of the wings and body, but
paradoxically enough, in the design and analysis of wing performance, the huge
values of *Re* make it possible to completely ignore viscous terms, and
a large and successful body of engineering literature makes accurate
calculations of the aerodynamic performance based on inviscid (no viscosity)
theories.

By contrast, birds are significantly smaller, and move more slowly, so that
the typical *Re* based on mean chord shown in
Fig. 1 is much lower, by three
orders of magnitude or more, ranging from 10^{4} to 10^{6} as
the body mass, *m*, varies from less than 10 g to over 10 kg. This can
give rise to certain difficulties in applying formulae straight from
aeronautics texts to bird flight, even while ignoring the fact that bird wings
flap and deform in ways that are quite outside the usual engineering
experience.

### Wing performance at low Reynolds number

Because aeronautics is usually practised at much higher *Re* than
appears in Fig. 1, relevant
wing performance data are more scarce, and the few reliable sources (such as
Althaus, 1980;
Schmitz, 1945;
Hoerner, 1965;
Laitone, 1997) are limited in
scope. Fig. 2 is taken from a
comprehensive collection of aerofoil data
(Lyon et al., 1997), and shows
an example of the measured performance of a two-dimensional airfoil section at
*Re* from 6×10^{4} to 3×10^{5}. The
aerofoil section is the Eppler 387, designed for sailplanes in what is termed
low-speed flight in engineering literature. When *Re* falls
significantly below the design space for this aerofoil (i.e. when
*Re*<10^{5}), the lift:drag polars are characterised by a
`tongue', where at moderate angles of attack, α, the drag increases
abruptly with little increase in lift. As α increases further, the drag
decreases again, just as abruptly. The magnitude of the effect increases as
*Re* decreases, and reasonable agreement (say, to within a factor of
two for the drag) between different wind tunnel facilities is hard to find.
The reason for the abrupt increase, and subsequent decrease of the drag withα
, is due to the dynamics of a separation region on the upper (suction)
surface of the aerofoil. The separation region may be associated with complete
detachment of smooth streamlines from the aerofoil, but for a small range ofα
, the flow may reattach again, when the affected region is called a
separation bubble. This process of separation and possible reattachment is
very sensitive to details in the aerofoil geometry, ambient turbulence and
possibly α(*t*). Some kind of large amplitude variation in
*C*_{d}(α) (*C*_{d} is a section lift
coefficient of normalised lift per unit span, used for two-dimensional
aerofoils; the coefficient of lift for a finite span wing is denoted
*C*_{D}) is not uncommon at *Re*⩽10^{5} for
smooth aerofoils with significant thickness. Because the physical process
depends on small details of the viscous boundary layer, these flows are also
very difficult to compute, and standard inverse methods of aerofoil design
that either completely ignore viscous effects, or model them in an *ad
hoc* fashion, fail completely.

### Objectives

It is interesting that, as shown in Figs 1 and 2, a substantial fraction of birds operate within a Reynolds number regime where significant aerodynamic performance variation due to the presence or absence of boundary layer separation and/or reattachment can be predicted for fixed wings. The purpose of this paper is to evaluate recent results from wind tunnel studies of flapping bird flight in the light of companion studies on fixed wing performance. We investigate the degree to which the bird flight results can be understood in the light of the new fixed wing data, with a view to understanding how birds manage their aerodynamics in a potentially unstable or unpredictable flow regime.

## Materials and methods

### Wind tunnel facilities

The wind tunnel at Lund has been described in detail
(Pennycuick et al., 1997), the
set-up for recent DPIV-based measurements has been detailed previously
(Spedding et al., 2003a;
Spedding et al., 2003b). The
facilities at the Dryden tunnel at University of Southern California (USC)
have also been described (Spedding et al.,
2006). Both tunnels have closed-loop designs, and large
contraction ratios (12.25:1 at Lund; 7:1 at USC) that follow a series of
smoothing screens (5 in Lund, 11 at USC). Consequently, the turbulence levels
in both tunnels are quite low. For all experiments reported here, the tunnels
operated at speeds from 5–10 m s^{–1}, when mean
turbulence levels, *u*′/*U*, where *u*′ is an
averaged root mean squared fluctuating velocity and *U* is the mean
speed, were approximately 0.035% (Lund) and 0.025% (USC). The comparatively
low turbulence levels are essential for obtaining reliable force balance and
wake measurements in the *Re* regime
10^{4}–10^{5}.

The birds in the Lund facility are trained to fly centred on a luminescent
marker in reduced light conditions. Their wakes are sampled far downstream,
17–22 chord lengths aft of the bird, because of safety concerns with the
high intensity laser light. In the USC tunnel, a wing is mounted vertically on
a single sting connected to a custom force balance capable of resolving lift
and drag forces of 0.1 mN (about 0.01 g force). The suspension system is
damped so only time-averaged forces can be measured. Flow measurements are
made on the suction surface, and at *x*=1*c* and 10*c*,
where the *x* (streamwise) coordinate begins at the leading edge of the
wing, with mean chord *c*. Measurements were made at various spanwise
(*y*) locations in both sets of experiments. In the Lund tunnel this is
done by monitoring the naturally occurring drift of the bird with respect to
the light slice with a synchronised CCD camera placed in the downstream
diffuser section, while in the USC tunnel, the light slice is simply moved
along the fixed wing. The results here are given for rectangular planform
wings with aspect ratio *AR*=2*b*/*c*=6 (where *b*
is the wing semispan), and are for data taken at midspan.

### Digital particle image velocimetry (DPIV) methods and data analysis

In both tunnels, a custom correlation imaging velocimetry (CIV) method
(Fincham and Spedding, 1997;
Fincham and Delerce, 2000) was
used to estimate velocity fields in vertical light slices, aligned in the
streamwise direction behind the wings. The recirculating tunnels were
gradually filled with a fine smoke composed of 1 μm diameter particles. The
particle image sizes and brightness were significantly improved after an hour
or more of continuous operation, and this was systematically included in the
experimental protocol. For a given lighting and particle set-up, the one
remaining parameter is the CIV exposure time, δ*t*, determined by
the time difference between consecutive flashes of the dual-head Nd:Yag
lasers. In the USC experiments, this varied between 100–300 μs; in
the Lund wind tunnel it varied from 200–500 μs. For a given optical
geometry, the correct choice depends on the flow complexity, both within and
across the plane of the light slice, and significant differences in background
noise can be realised with only a 20 μs change in δ*t*.

The application of CIV techniques to these data have been described
(Spedding et al., 2003a;
Spedding et al., 2006) and
will not be repeated in detail here. The important general operational
consideration is that the correct tuning of the δ*t* parameter
can be matched by independent selection of correlation and search box sizes in
the CIV algorithms so as to maximise the bandwidth of the velocity estimates
about the likely range of disturbance values (with any mean flow subtracted)
of the computed displacement field. Finally, the displacement field is
reinterpolated onto a regular rectangular grid with patched smoothing spline
functions, which can be differentiated analytically to yield the first order
spatial derivatives. This operation also corrects for the finite displacement
of velocity vectors during the exposure time δ*t*. All the data
described here come from estimates of the *u* and *w* velocity
components in the streamwise (*x*) and vertical (*z*)
directions, respectively. The rotational part of the velocity field is then
given by the spanwise vorticity, denoted:
(2)
This quantity is displayed on discrete colour bars whose resolution reflects
the uncertainty of the measurement. Very approximately, velocity estimates are
likely to be correct within 1% and gradient quantities such asω
_{y} have a likely uncertainty of 5–10%.

## Results

### Wakes of birds and fixed wings

A number of recent studies have investigated the wakes of birds in the Lund
wind tunnel (Spedding et al.,
2003b; Rosén et al.,
2004; Hedenström et al.,
2006a; Hedenström et al.,
2006b; Rosén et al.,
2007). The birds range in mass from the thrush nightingale
*Luscinia luscinia* (*m*=30.5 g) to the robin *Erithacus
rubecula* (*m*=16.5 g). Their small size makes them good subjects
for wind tunnel study as the corrections required for tunnel blockage and
wake–wall interference are negligible. As measurements are made from
17–22*c* downstream, interpretation of the wake vorticity
patterns is complicated by the fact that they have been evolving and deforming
over this distance. Fig. 3
shows a vertical slice, aligned with the mean flow, in a plane at about the
mid-semispan position. It is a composite from four consecutive frames, where
slightly different phases of the wing beat (the wing-beat frequency,
*f*=14 Hz, while the laser repetition rate is 10 Hz) are sampled at a
fixed position in space. The data are shown in a reference frame moving with
the mean flow, with vectors of the disturbance velocity shown at half
resolution. The flight speed *U*, determined by the independently
controlled tunnel speed, is 7 m s^{–1}, which can be regarded as
close to a cruising speed. The spanwise vorticity,ω
_{y}(*x,z*), is shown on a discrete colour bar, where
light blue is zero and *extrema* are mapped asymmetrically about this
level. As has been noted before, the patches of spanwise vorticity that can be
traced to the wing acceleration at the start of a downstroke are more compact
and higher in amplitude than those appearing at the end of the downstroke,
where a more diffuse pattern of vorticity trails into the upstroke-generated
wake. This mid-wing data slice cuts obliquely through a structure shed from
the partially retracted wing on the upstroke. Although the structures
attributable to the downstroke are much stronger, as confirmed by the stronger
induced airflow between them, the upstroke is not completely inactive.

Vertical slices aligned with the mean flow show the spanwise component of
vorticity ω_{y}(*x,z*) only, and reconstructing the
three-dimensional wake geometry from large numbers of such slices at different
spanwise locations is quite lengthy
(Spedding et al., 2003b). In
particular, streamwise cuts do not show the streamwise vorticityω
_{x}(*y,z*) that trails from the wingtips, except by
implication from the changing circulation of coherent patches ofω
_{y}(*x,z*). In this paper we focus specifically on
cross-comparisons of the ω_{y}(*x,z*) component. For a
simple conceptual three-dimensional model, it is a reasonable approximation to
imagine that the spanwise vorticity shed at the beginning of the downstroke is
continuous with, and approximately the same strength as, the trailing vortices
left behind the wingtips during the first half of the downstroke.

The wake geometry in Fig. 3
appears at first sight to be complex, much more so than the simple and elegant
models composed of small numbers of vortex lines that one usually sees in
flight models (cf. Rayner,
1979; Phlips et al.,
1981; Hall and Hall,
1996), and even those of Spedding et al.
(Spedding et al., 2003b),
where the models are based on such measurements, but in greatly simplified
form. There are two reasons for this: first, the wake is imaged quite far away
from its origin, and so initial order can be lost in the self-induced
deformation of the wake, which includes pairing and merging interactions
between same-signed vortices. Second, these wakes at moderate *Re* do
not appear like wake models that ignore viscosity. The wake structures
interact, deform and dissipate because they live in a real fluid, with viscous
forces generated by relative shearing motions. This is as true for fixed wing
aerofoils as it is for bird wings.

In Fig. 4, the flow behind a
cambered plate and an Eppler 387 wing are compared in vertical planes across
the midspan. The spanwise vorticity is shown at a distance
*x*=1*c* from the leading edge (i.e. immediately behind the
trailing edge) and at *x*=10*c*. At α=4° (the reasons
for this choice will become clear later), the near wakes of both aerofoils are
very compact chains of alternate-signed vortex patches. For small α, the
vortex structures have a passage frequency past an observer fixed in the wind
tunnel reference frame of approximately 400 Hz (for the cambered plate), which
is consistent with the laminar free wake instability mechanism modelled and
measured by Sato and Kuriki (Sato and
Kuriki, 1961). At higher α, the near wake regularity is
disrupted by unsteady motion of the trailing edge separation point and from
boundary layer instabilities on the pressure side of the aerofoil. In the far
wakes (bottom row of Fig. 4),
the initial order of the low-α near wake has evolved to a more complex
pattern of diffuse vorticity (note the fourfold difference in colourbar
scaling). The complexity of fixed wing wakes at moderate downstream distance
is not notably less than observed for the flapping bird wake. This is true
even when the angle of attack is small so that the early wake at
*x*=1*c* is very compact and structured. The wing wakes do not
look very much like textbook, inviscid descriptions either, and so one would
expect the bird wakes to be similarly varied. It might be reasonable to ask
why it is that bird wakes do not look more disorganised so far downstream, and
this point will be taken up later.

Comparative experiments on the various passerine species flown in the Lund
wind tunnel (Hedenström et al.,
2006a; Hedenström et al.,
2006b; Rosén et al.,
2007) have shown that the apparent complexity of the bird wakes
does contain structures with readily predictable properties. Surprisingly,
some of these properties can be predicted by very simple fixed wing
aerodynamic theory. A classical result (see
Anderson, 1984) states that the
lift per unit span, *L*′, on an aerofoil can be written as:
(3)
where ρ is the air density, *U* is the flight speed, and Γ is
the strength of the circulation on the wing. Γ is not determined from
Eqn 3, but in practice takes a
value that is required to avoid physically implausible conditions at the
trailing edge. In steady level flight, the total lift, *L*, must
balance the total weight *W*, and so an expression for Γ can be
derived:
(4)
Eqn 4 gives a prediction for the
circulation on a wing of span 2*b*, flying at steady speed *U*
and supporting a weight, *W*. As a first approximation, on a wing of
finite span, then one will also expect wake vortices of constant strengthΓ
to be shed from the wingtips as the wing travels forward.
Eqn 4 thus can also be used to
predict the strength of the most evident wake structures behind a lifting
surface, and can be used as the simplest available such model for bird wakes,
even though they are not fixed rigid wings.

In comparing flying devices of different sizes and at different flight
speeds, it is convenient to non-dimensionalise Γ and dividing
Eqn 4 by *Uc*, we have:
(5)
where *S*=2*bc* is the wing planform area.
Fig. 5 plots
Eqn 5 as a solid line together
with data collected from wake experiments on four bird species. They are
plotted as a function of *U*/*U*_{mp}, where
*U*_{mp} is the estimated minimum power speed for each species,
as calculated from a simple actuator disk flight model
(Pennycuick, 1989). Its
particular value is not important, only that it represents some approximately
equivalent speed for each bird. The collapse of data and agreement with the
solid line is good. Despite the apparent complexity of the wakes such as shown
in Fig. 3, their most simple
quantitative measurement is in good agreement with fixed wing theory, as if
the wings operate quite like a standard wing, which happens to flap.

Hedenström et al. (Hedenström
et al., 2006b) and Rosén et al.
(Rosén et al., 2007)
also noted that Eqn 5 is very
like a lift coefficient, as can be seen by substituting the dynamic pressure,
*q*=½ρ*U*^{2} into
Eqn 5, so since
*W*=*L*,
(6)
When *U*/*U*_{mp}=1, close to some kind of cruising
operation, Γ/*Uc*=0.2, and so all birds in
Fig. 5 appear to operate with a
time-averaged lift coefficient of approximately 0.4 at
*U*=*U*_{mp}.

This entire analysis presumes a steady fixed wing, and it is not obvious why the data in Fig. 5 agree so closely with predictions, particularly when the measured wake geometry [see stick figures in Spedding et al. (Spedding et al., 2003b)] clearly differs (as it must) from that of a powered fixed-wing glider. It is possible that the basic wake shape and its strength can be imagined as that of a powered glider, and then that modifications to that basic shape occur due to flapping, so that, on average, steady fixed wing predictions still work. Although no insight can be claimed into the magnitude and importance of the unsteady forces, it does encourage a re-examination of local wing kinematics in a quasi-steady framework.

### Wing kinematics in flapping flight

The wingtip trace of the house martin *Delichon urbica* can be well
represented by a reconstruction from only two Fourier modes, whose relative
amplitude and phase varies with flight speed
(Rosén et al., 2007),
as shown in Fig. 6. This
includes a pause phase, visible as a secondary dip in the vertical component
of the wingtip speed, *w*_{tip}, at the two higher flight
speeds of *U*=8 and 10 m s^{–1}.
Fig. 6 shows that the
normalised tip speed varies considerably during the course of the wing beat,
particularly at the slower flight speeds. The gradients of
*w*_{tip} show the wingtip acceleration, and the peak
amplitudes are larger than, or comparable to the flight speed, *U*. It
is clear that an analysis of the local wing section properties must therefore
take into account this variation, and a local Reynolds number,
*Re*_{loc}, can be calculated from:
(7)
where *c*(*r*) is the wing chord length at position *r*
along the semispan, *b, u*_{loc}(*r*) is the local
relative air speed (which depends on *w*_{tip} and *U*)
incident on the wing section at position *r*, and ν is the kinematic
viscosity. *u*_{loc}(*r*) is not measured, but
estimated, from a wing model that assumes a rigid wing flapping at a single
hinge at the root and with kinematics given by the Fourier coefficients
responsible for Fig. 6. This
simple wing model allows a quick estimate of the approximate conditions on the
wing at different spans, shown in Fig.
7 for *r*=0.2*b*, 0.5*b* and
0.8*b*.

*Re*_{loc} fluctuates much as *w*_{tip}
fluctuates, but the amplitude of the fluctuations is much smaller at the wing
root, where, because of the higher mean chord, *Re*_{loc} is
actually highest. Here, *Re*_{loc} is still less than
2×10^{4}, however, and it falls to even lower values towards the
tip, and *Re*_{loc} at *r*=0.8*b* is usually less
than 1.4×10^{4}. This example is for *U*=6 m
s^{–1}, which is slightly below an estimated cruising speed,
*U*_{mp}=8.5 m s^{–1}. However, it shows that for
the small-sized passerines whose aerodynamic performance has been measured
thus far, local Reynolds numbers are on the lower end of the range considered
in Fig. 1. The difference is
important because the propensity for laminar boundary layer separation and the
possibility for its reattachment on the wing is very strongly affected by
*Re*.

### Aerodynamic performance of wing sections at moderate *Re*

Here we summarise properties of time-averaged lift:drag polars for fixed
wings, measured for values of *Re* and *AR* that are similar to
those for small bird wings. The implied positions of time-averaged performance
of the birds will be noted on these steady-state polars. Although bird wing
aerodynamics are not always likely to be well described on a time-averaged
basis (particularly at low flight speeds), comparing their average performance
on average polars is at least a consistent operation, and the results might be
instructive. It should also be noted carefully that even though time-averaged
performance calculations might appear to be consistent, it still does not mean
that they are actually correct, and still further does not mean that
instantaneous forces and/or unsteady effects are not important. The purpose is
restricted solely to examining the degree of agreement that can be explained
using the most simple and parsimonious model.

Fig. 2 showed that as
*Re* drops from 10^{5} to 6×10^{4}, the
performance characteristics of an aerofoil such as the Eppler 387 change
dramatically. Fig. 8, from
measurements in the USC Dryden wind tunnel, show that the performance
characteristics change again as *Re* continues to drop. Recall that
Fig. 7 suggests that
*Re*_{loc}, the local Reynolds number on any wing section,
falls mostly between 1 and 2×10^{4}.
Fig. 8 shows that at such low
*Re*, the curve of *C*_{L}(*C*_{D}) for a
wing with *AR*=6 is quite smooth, in strong contrast to the abrupt
jumps (in both *C*_{D} and *C*_{L}) seen at
higher *Re*. As *Re* falls, *C*_{L,max} also
falls considerably, but this is most likely irrelevant to the cruising bird.
Recall further that the wake measurements of
Fig. 5 suggest a performance
that is commensurate with a time-averaged lift coefficient of approximately
0.4. The horizontal line drawn at *C*_{L}=0.4 in
Fig. 8 shows how this positions
the wing comfortably below regions where *C*_{D} rises steeply.
Fig. 9 shows that the angles of
attack required for an *AR*=6, Eppler 387 wing to generate these
moderate lift coefficients are 4, 3, 2.5 and 1° for *Re*=1, 2, 3
and 6×10^{4}, respectively. The higher the Reynolds number, the
smaller the required angle of attack for a given lifting performance, and the
more conservative a regime that can be occupied.

## Discussion

### Inferring wing section properties of bird wings

This paper is centred around the medium-speed, or cruising performance, of
bird wings. The first example result in
Fig. 3 showed that the far wake
is moderately complex in appearance. However, given the complexity of similar
far wakes of simple wing shapes in Fig.
4, the bird wake begins to look comparatively simple, and the
strengths of the largest coherent vortex patches are quite simple to predict,
based only on classical wing theory arguments
(Fig. 5). The normalised wake
circulation, Γ/*Uc*, is very well matched by predictions for
fixed wings of the same size, carrying the same load, and flying at the same
speed. Indeed, Γ/*Uc* can be expressed as one half of the
time-averaged lift coefficient (Eqn
6), and its quite moderate value, approximately 0.4 at
*U*_{mp}, suggests, in turn, a wing that is at commensurately
moderate angles of attack. The fixed wing wake results of the cambered plate
and Eppler 387 aerofoil in Fig.
4 are therefore quite likely to be representative of the types of
flow that can be expected in the bird wake.

In the Introduction, it was noted that wing performance at moderate
*Re* is notoriously sensitive to both *Re* and to small changes
in geometry and environmental conditions. However, in deducing likely local
sectional Reynolds numbers *Re*_{loc} in
Fig. 7, it appears that for
these small birds, *Re*_{loc} is just below values where the
aerodynamic performance becomes strongly affected by the stability and
transitional flows in and around laminar separation bubbles
(Fig. 8). The lowest
*Re*_{loc}, where fixed wing properties are the most stable,
are found at the wingtips, and the higher *Re*_{loc} are found
towards the root, where the oscillation amplitude is at its lowest.

The detailed wake measurements discussed here are available only for
small-sized birds, occupying the lower *Re* regime of all birds in
Fig. 1, and it is reasonable to
wonder how these results scale as the size and mean-chord Reynolds number
increase. If the fixed wing flight model remains correct, at least to a first
order of approximation, then Eqn
6 implies that the dimensionless lift coefficient,
*C*_{L}, will also be constant for a given
*U*/*U*_{mp}. The angle of attack required to achieve
*C*_{L}≈0.4 decreases as *Re* increases
(Fig. 9), and so as *Re*
increases from values of 1×10^{4} where the steady-state flow is
always stable, to 6×10^{4} where abrupt performance jumps appear
due to flow separation and subsequent reattachment, so the required angle of
attack falls to move the cruising performance point further away from the
unstable region. It is also notable that when *Re* does rise
sufficiently to bring the wing into a regime that has performance jumps due to
separation, the cruising angle of attack is never more than a couple of
degrees beneath the point at which these effects come into play. Thus, small
changes in angle of attack can produce large changes in the force direction
and magnitude on the wing. This would be a useful condition for control
purposes.

It should be noted clearly that the arguments here are that one can understand something of the aerodynamics of the complex, deformable, feathered, flapping wings by comparing predictions from simple fixed, rigid wing models. Even when flapping motions are considered, they are for a rigid wing of the same size, flapping about a simple hinge at the root. The idea is not that such simplifications can provide a complete or even sufficient explanation of the true wing geometry and motions, but that since the quantitative data and observations do agree well with model predictions based on fixed wings, this approach shows the simplest tenable baseline approximation, upon which more complex and realistic theories might be constructed.

### What fixes the Strouhal number?

The inverse of the ratio of wingtip speed to forward flight speed is
commonly termed an advance ratio, which indicates the forward distance
travelled relative to the tip motion of an oscillating or rotating propulsor.
While Fig. 6 shows that
*w*_{tip}/*U* varies greatly during the course of a wing
beat, the time-averaged (root mean square) value can be a convenient measure
of the relative importance of flows induced by the unsteady wing motion
compared with the steady (approximately constant) flight speed. It is very
simply related to another common measure of relative timescales in unsteady
wakes, the Strouhal number *St*:
(8)
where *A* is either the maximum lateral distance between shed vortices
in an unsteady wake, or the tip-to-tip amplitude of an oscillating (flapping,
or heaving/pitching) wing. The average
*w̄*_{tip}/*U* can be
expressed as:
(9)
for a wingtip that travels a vertical distance of *A* twice every
wing-beat period, *T*. The right hand side of
Eqn 9 re-expresses the ratio of
speeds on the left hand side as a ratio of distances, where 2*A* is the
wingtip travel distance and *UT* is the horizontal distance covered at
speed *U* during one wing beat. From Eqn
8 and
9,
(10)
It has been noted previously (Rosén
et al., 2004) that since both *f* and *A* tend to
vary little with *U*, then *St* (or any equivalent measure) just
decreases as 1/*U*, and is not constant for any given bird over its
range of natural flight speeds.

The Strouhal number tends to fall within a restricted range of possible
values for flying birds, bats and insects alike, taking values between
0.2–0.4, with a mean of 0.29 in a literature sample for animals flying
at reported preferred flight speed (Taylor
et al., 2003); the median value for `direct' fliers (birds that do
not practice intermittent flight) was 0.2. Interestingly, the root mean square
of the varying *w*_{tip}/*U*(*t*) in
Fig. 6 shows that
*w̄*_{tip}/*U* is 0.71 for
*U*=6 m s^{–1} and 0.54 for *U*=8 m
s^{–1} for the house martin. This is equivalent to
*St*=0.35 and 0.27, respectively, and since a `preferred' flight speed
most likely lies between these two values
(Rosén et al., 2007),
then the equivalent *St*=0.31, has a value not inconsistent with Taylor
et al.'s analysis.

Explanations for the observed 0.2⩽*St*⩽0.4 range in animal
swimming and flying propulsion have concentrated on unsteady mechanisms (see
also Wang, 2000), which can
most generally be described as requiring a balance between time scales of
growth and then separation of leading edge vortices and time scales of the
oscillating propulsor motion itself. Wang also describes the constraint on
*St* (finding preferred values between 0.16 and 0.27) in terms of the
maximum angle of attack without stall, but these maximum angles are from
45°–60° and are for an unsteady stall in a viscous flow at
*Re*=10^{3} dominated by large-scale separation.

From wing kinematic constraints alone (i.e. ignoring the contributions due
to induced flow by the wing itself) the local aerodynamic angle of attack is
determined by a combination of the stroke plane angle, the local twist and the
section speed relative to the mean flow
(Fig. 10). For the purposes of
argument, let us suppose that the local aerodynamic angle of attack is fixed
during each wingstroke. We may then describe the stroke plane angle and local
twist together as a summed quantity, α_{0}, which is constrained
mechanically to operate within a certain range. Then the aerodynamic angle of
attack, α, is:
(11)
where α_{w} is simply related to
*w̄*_{tip}/*U* by:
(12)
as can be seen from Fig. 10.
So, operating a flapping wing at a preferred α depends on the tip speed
to forward speed ratio, which is proportional to *St*. The tendency to
maintain a constant *St* close to *U*_{mp} (or other
measure of preferred flight speed) can be seen as simply the maintenance of a
low positive angle of attack at which the wing section performance is
efficient (in terms of *L*/*D*) and safe (in avoiding abrupt
separation). The fact that *St* is not actually constant over the range
of *U* for any given bird shows that α_{0} is not
constant either, but is tailored to adapt to the varying advance ratio. For
the foregoing argument to apply, we need not require that α_{0}
be constant, only that it has a fixed range, which is similar amongst the
different species, and that the preferred flight speed condition occurs at the
centre of that range.

The maintenance of a small local angle of attack along a flapping wing is
analogous to control of the proportional feathering parameter identified by
Lighthill (Lighthill, 1969;
Lighthill, 1970) for efficient
propulsion in oscillating fins, and extended to three-dimensional geometries
(Karpouzian et al., 1990). In
birds, *St* is allowed to vary with *U*, so that both *f*
and *A* can be maintained almost constant. The variation is possible
because variation in α_{0} can give reasonable values ofα
, despite the variation in α_{w}. The tendency to
constant *St* at some preferred flight speed is a result of operating
in the middle of the range of available α_{0}. The comparatively
simple form of the measured bird wakes, even when measured far downstream
(Fig. 3) indeed strongly
suggests a fine degree of boundary layer control through manipulation of localα
, so that large-scale separation at the trailing edge, and its
attendant shedding into the wake, are avoided.

### Limitations

The preceding analysis applies observations from fixed wings, with simple
shape, to the unsteady problem of flapping bird wings, with complex and
time-varying shape. The justification for so doing is provided partly by the
reasonable agreement between simple steady aerodynamic models and the gross
overall features in the bird wake. The paper thus advances the simplest
explanation for the observations. This is not to assert that unsteady
aerodynamics must play no role, nor that significant dynamic separation
effects cannot occur. Even when taking into account the wing kinematics
themselves, we have applied a quasi-steady approach, where at each instant
wing sections are analysed as if they had the same properties had they been
frozen in time at each instant. The insufficiency of this quasi-steady
approach in low speed and/or hovering flight has been famously demonstrated
(Norberg, 1976;
Ellington, 1984a;
Ellington, 1984b), and a more
detailed quantitative analysis from flapping and translating models in a tow
tank has been published (Sane and
Dickinson, 2002; Dickson and
Dickinson, 2004). There will likely be interesting unsteady
phenomena, possibly involving momentary flow separation, that contribute
significantly to a fuller understanding of the aerodynamics of bird wings, but
the current results, for the particular case of cruising flight at mean chord
Reynolds numbers between 5 and 10×10^{4}, suggest that fixed
wing behaviour can explain much.

Not all airfoil sections behave as the Eppler 387 does, which makes an
interesting test case in the severity of the separation bubble effects, but is
not necessarily representative of wing sections that are actually designed for
*Re*<10^{5}. Although this type of behaviour is common for
smooth aerofoils with finite thickness, many low-*Re* aerofoils, such
as the Davis 3R (Lyon et al.,
1997), are significantly thinner and have fewer problems in large
drag performance variations. Their section profiles are more similar to the
cambered plate of Fig. 4, which
has superior *L*/*D* to the Eppler 387 when
*Re*<10^{5} (Spedding et
al., 2006), and further research will be carried out on such
shapes, where now the instantaneous flow field can be measured as well as
time-averaged forces. A more detailed consideration of the combination of
variation in section shape and local angle of attack with time and along the
span in real flapping wings will be required to fully demonstrate the
aerodynamic flow control that is suggested in rather crude terms here.

## Conclusions

The flow around and behind simple fixed wings at Reynolds numbers similar to bird flight is not necessarily simple itself, and the wakes of flying birds are not significantly more complex than that. This observation suggests that simple aerodynamic models might help to understand many features of bird flight, as complex kinematics and geometry are reduced to simple principles. One of these simple principles might be that the constant Strouhal number arguments advanced for flapping wing flight can be explained as a simple consequence of maintaining a moderate angle of attack on the lifting wing (or propelling tail). The data presented are for small birds, because they are easiest to study in facilities with finite size. Since the performance characteristics of fixed wings vary significantly with Reynolds number, the design constraints suggested here may apply only to a fixed range of sizes, and we may find that larger bird wings are designed differently.

- © The Company of Biologists Limited 2008