## 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

*Re*,which may be written as:

*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*.

*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:

_{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. 3shows 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. 3appears 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.

*L*′, on an aerofoil can be written as:

*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:

*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.

*Uc*, we have:

*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.

*q*=½ρ

*U*

^{2}into Eqn 5, so since

*W*=

*L*,

*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}.

{u,v,w} | L/T | disturbance velocities in {x,y,z} |

{x,y,z} | L | streamwise, spanwise and vertical directions |

A | L | tip-to-tip wingbeat amplitude |

AR | wing aspect ratio (=2b/c) | |

b | L | wing semispan |

c | L | mean chord length in flightwise direction |

C_{d} | section drag coefficient | |

C_{D} | wing drag coefficient | |

C_{L} | wing lift coefficient | |

C_{p} | section lift coefficient | |

f | 1/T | wingbeat frequency |

g | L/T^{2} | acceleration due to gravity |

l | L | characteristic length |

L | M.L.T.^{–2} | lift |

m | M | body/aircraft mass |

q | M/LT^{2} | dynamic pressure |

r | L | position along wingspan |

Re | Reynolds number (=Uc/v) | |

Re_{loc} | local Re | |

S | L^{2} | wing planform area |

St | Strouhal number (=fA/U) | |

t | T | time |

T | T | wingbeat period |

U | L/T | flight speed |

u_{loc} | L/T | local incident velocity on a wing section |

U_{mp} | L/T | estimated flight speed for minimum power |

u′ | L/T | root mean square of velocity fluctuations |

W | M.L.T^{–2} | bird/aircraft weight (=mg) |

w_{tip} | L/T | vertical component of wingtip speed |

α | deg. | aerodynamic angle of attack |

α_{0} | deg. | local twist from stroke plane |

α_{w} | deg. | local α due to vector sum of U and w_{tip} |

δt | T | small time interval (DPIV effective exposure time) |

Γ | L^{2}/T | circulation/strength of a vortex patch, or on a lifting wing |

λ | wavelength | |

μ | M/LT | viscosity |

ν | L^{2}/T | kinematic viscosity (=μ/ρ) |

ρ | M/L^{3} | air density |

ω_{i} | 1/T | vorticity component in the ith direction {i=x,y,z} |

{u,v,w} | L/T | disturbance velocities in {x,y,z} |

{x,y,z} | L | streamwise, spanwise and vertical directions |

A | L | tip-to-tip wingbeat amplitude |

AR | wing aspect ratio (=2b/c) | |

b | L | wing semispan |

c | L | mean chord length in flightwise direction |

C_{d} | section drag coefficient | |

C_{D} | wing drag coefficient | |

C_{L} | wing lift coefficient | |

C_{p} | section lift coefficient | |

f | 1/T | wingbeat frequency |

g | L/T^{2} | acceleration due to gravity |

l | L | characteristic length |

L | M.L.T.^{–2} | lift |

m | M | body/aircraft mass |

q | M/LT^{2} | dynamic pressure |

r | L | position along wingspan |

Re | Reynolds number (=Uc/v) | |

Re_{loc} | local Re | |

S | L^{2} | wing planform area |

St | Strouhal number (=fA/U) | |

t | T | time |

T | T | wingbeat period |

U | L/T | flight speed |

u_{loc} | L/T | local incident velocity on a wing section |

U_{mp} | L/T | estimated flight speed for minimum power |

u′ | L/T | root mean square of velocity fluctuations |

W | M.L.T^{–2} | bird/aircraft weight (=mg) |

w_{tip} | L/T | vertical component of wingtip speed |

α | deg. | aerodynamic angle of attack |

α_{0} | deg. | local twist from stroke plane |

α_{w} | deg. | local α due to vector sum of U and w_{tip} |

δt | T | small time interval (DPIV effective exposure time) |

Γ | L^{2}/T | circulation/strength of a vortex patch, or on a lifting wing |

λ | wavelength | |

μ | M/LT | viscosity |

ν | L^{2}/T | kinematic viscosity (=μ/ρ) |

ρ | M/L^{3} | air density |

ω_{i} | 1/T | vorticity component in the ith direction {i=x,y,z} |

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

*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:

*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?

*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*:

*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:

*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,

*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.

_{0}, which is constrained mechanically to operate within a certain range. Then the aerodynamic angle of attack, α, is:

_{w}is simply related to

*w̄*

_{tip}/

*U*by:

*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.

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