The Wake of a Kestrel (Falco Tinnunculus) in Flapping Flight

Previous investigations have demonstrated the existence of vortex rings in the wake of birds flying slowly down the length of an 8 m flight cage (Spedding, Rayner & Pennycuick, 1984; Spedding, 1986) although some doubts as to the adequacy of this description were raised by the apparent shortfall in the momentum measured in this wake compared with that required for weight support. While these conditions provided a stern test of the flight model under investigation (Rayner, 1979a,b), where a high degree of unsteadiness in the flow necessitates an accurate aerodynamic model in order to estimate the induced power requirement, it is perhaps of more interest to a practical ecologist to obtain information about flight at the higher speeds commonly used by most birds in migrating or commuting flights. The significance of two characteristic flight speeds, the minimum power speed, U_mp, and the maximum range speed, U_mr, was identified by Pennycuick (1968), derived from the U-shaped P(U) curves predicted by his actuator-disk flight model (P is the power required to fly, U is the flight speed). These characteristic speeds persist in the vortex ring model calculations (Rayner, 1979b) and, while the parasite and profile power requirements consume a greater fraction of the total power, the induced power contribution remains significant at these flight speeds.

Although there are well-documented differences in the wing beat kinematics of birds flying at low and high speeds (e.g. Brown, 1953, 1963) no flight model explicitly accounts for the different flight styles which may occur. In the flow visualization experiments which have been performed thus far, vortex rings have been described in the wake of small passerines (Kokshaysky, 1979) and for a pigeon and a jackdaw (Spedding et al. 1984; Spedding, 1986) in slow flight, all in flight cages or tanks. It might reasonably be expected that the lower wing beat amplitudes and accelerations more typical of cruising flight at higher flight speeds would significantly alter the distribution of wake vorticity. At the same time, at these lower reduced frequencies (the reduced frequency, Ω = cw/2U, where c is the wing chord, (O is the radian frequency and U the mean forward velocity), it may be possible to considerably simplify some of the assumptions required in the aerodynamic analysis.

In the previous paper (Spedding, 1987), the steady gliding flight of a kestrel was found to accord well with classical steady-state aerodynamic theory. Here the wake of the same kestrel under identical conditions, but in flapping flight, will be investigated to determine the extent to which these properties remain in the domain of small but finite reduced frequency.

Generally, the apparatus and methods are identical to those described in Spedding (1987), the only difference being the flight behaviour of the kestrel as it grew more accustomed to the experimental conditions and operator 2 was positioned slightly further away from the cameras to encourage flapping flight through the bubbles. The flight path is shown schematically in Fig. 1. After an initial drop from take-off at point 1, the bird would fly level at 400–450 mm from the floor down the length of the corridor before pulling up and alighting on point 2. The take-off and landing points were about 1 m in height. Six of these flights were timed with a stopwatch, recording a mean ground speed of 6ms⁻¹ from point 1 to point 2. Accounting for acceleration and deceleration at either end of the flight, the ground speed in mid-flight was estimated to be around 7ms⁻¹, in agreement with that calculated from the horizontal distance between successive tail images on the stereophotographs.

Two stereopairs were digitized and reconstructed as three-dimensional velocity fields, following previously established procedures (Spedding et al. 1984). Series of two-dimensional slices were taken through these data, and velocity profiles were recorded in the plane of the sections. The coordinate system is a right-handed one, with the origin at the intersection of the left camera lens principal axis with the back wall. The Z axis is coincident with the lens principal axis, X runs from left to right, parallel to the line of flight, and vertical distances are measured in Y (see fig. 2 of Spedding, 1987).

At these flight speeds, the wavelength of the wake (the horizontal distance between wake elements formed by two consecutive instances of a particular point in the wing beat cycle) is longer than could usually be covered by the cloud of helium bubbles, and the structure of the wake must be deduced from more than one stereopair.

Figs 2 and 3 are two stereopairs showing partial coverage of the vortex wake behind a kestrel in flapping flight. In Fig. 2, the bird, flying from right to left, is near the end of an upstroke during which the primary feathers have been flexed and inclined downstream of the mean flow. In the last of the four kestrel images, the primaries are being re-extended out into the flow. The bubbles mark the flow of the previous downstroke and the beginning of the upstroke, and are situated mainly in the background so as to visualize the wake behind the starboard wing. Behind a central induced downward flow, a curved vortex core is visible, marked by an axial flow along the core and a circulatory flow around it. The axial flow is towards the wingtip and is therefore a viscous drag wake. The vortex core can be clearly seen continuing along a straight cylindrical component of the wake, produced during the upstroke. The continuity between these two sections is unambiguously shown by the axial core flow, and Fig. 3 shows the upstroke portion of the wake as a straight cylindrical vortex of larger diameter than that produced by the downstroke. Here,bubble tracks show the foreground wake shed by the port wing. The principal features of the wake are visible in Fig. 4 which shows, from right to left, a linear vortex core of large diameter shed during an upstroke, a curved, smaller diameter core produced by the following downstroke with the flow induced by this vortex clearly visible towards the centre of the wake, and finally, a continuation of this vortex core into the next upstroke as demonstrated by the axial flow along the core. At their lowest extent the wake trailing vortices are about 250 mm from the ground. The propensity for bubbles to become trapped inside a vortex and move axially along its length was found to be quite convenient for tracing the core location and Fig. 5 shows the wingtip trailing vortex behind the starboard wing during the first half of a downstroke marked in this way. In this photograph, the continuity between upstroke and downstroke wake vortices is clearly visible.

In all stereopairs examined, the observed wake structure was consistent with a continuous undulating vortex pair without detectable concentrations of transverse, or cross-stream vorticity joining the two cores. If transverse lines of vorticity are present, then they are not aggregated into recognizable vortex structures. Fig. 6 is a diagram of the wake as judged by qualitative inspection of all stereopairs of the wake. The fitness of such a description of the wake may be tested with some simple quantitative wake measurements.

The three-dimensional velocity field was calculated from a full stereophotogram-metric analysis of two of the kestrel flights. Examples of the reconstructed bubble field in one of these are shown in Figs 7 and 8, which are elevation and plan views of the wake produced by half a downstroke followed by half an upstroke. Bubbles are confined to the foreground in this case and the wake is assumed to be symmetrical about the vertical plane along the line of flight. The line of flight is an average estimate from all stereopairs and the standard deviation in Z is ±50 mm. This corresponds to less than 0·15 of the wing semispan, b. Fig. 8 also shows the location of numbered and lettered sections taken vertically across (spanwise) and along (streamwise) the length of the wake, respectively.

If, as postulated in the previous section, the trailing vortices in the wake are continuous and not connected by vortex lines running transversely across the wake, then according to laws of vorticity established for inviscid incompressible fluids by Helmholtz and Kelvin (as outlined, for example, by Batchelor, 1967), the circulation around each vortex must be invariant along its length. Accordingly, the circulations measured by integration around the edge of the vortex core should be equal at all downstream core cross-sections. Figs 9 and 10 show the w(Y) velocity distributions in two cross-sections taken across the mid-sections of the downstroke and upstroke wake vortices (sections 8 and 3 of Fig. 8). The curves have been drawn by eye to account for the finite thickness of the transects in Z and the width of the core may be estimated from the distance between w_max and w_min-If the vortex core is assumed to be circular and to contain all the vorticity then the vortex has circulation г,

The circulation measurements reported above support the notion that the wake of a flapping kestrel is composed of a continuous vortex pair of constant circulation without significant components of cross-stream vorticity connecting the two. The wake structure is thus quite different from that reported for small passerines (Kokshaysky, 1979) or for the slow flight of the pigeon or jackdaw (Spedding et al. 1984; Spedding, 1986). In those cases where vortex rings are produced in the wake, the rapid acceleration and deceleration of the wings at the beginning and end of the downstroke alters the circulation on the wing with the concomitant shedding of vorticity into the wake, and the trailing vortex lines are closed up into a loop by these starting and stopping vortices. In the medium-speed flight of the kestrel, these rapid accelerations of the wings are absent, as are the cross-stream vortices. There is transverse vorticity (a Z-component) in the wake of course, manifested in the outward curvature of the downstroke trailing vortex (in the Z direction) when it is not parallel to the X axis.

During the upstroke of the kestrel, the wings are not folded close to the body, but the primary feathers are flexed and closed up, parallel to the incident air flow, while the secondary feathers remain extended into the flow. The secondary feathers remain loaded and perform aerodynamic work during the upstroke, as confirmed by the presence of trailing vortices in the wake during this period. Moreover, the circulation around the secondary feathers appears to be maintained at a constant level and the wing beat can be conceived as a cycle of loading and unloading of the primary feathers as they are extended on the downstroke and flexed during the upstroke.

During the upstroke, the folding of the primary feathers appears to be such that the trailing vortex is shed from the end of the secondary feathers, or around the flexed primaries. This shift in location of the shedding point is consistent with the experimental results of McCormick & Padakannaya (1971) showing that the effect of a drooped wingtip is to increase the strength of the inner trailing vorticity at the hinge, causing the wingtip trailing vortex to move inboard towards the hinge.

These results also support the careful observations of Brown (1953, 1963) who deduced from high-speed cinematography of birds in medium-speed flight that the secondary feathers provide aerodynamic lift during the upstroke. The flight speed and wing beat kinematics of the kestrel in this study appear to be similar to those of free-flying kestrels as recorded by high-speed ciné film (Scholey, 1983) and are quite characteristic of the low-amplitude wing beat cruising flight of many species (Brown, 1953, 1963). This being the case, there is no reason.to suspect that the resulting wake structures would be substantially different either, and it is proposed that the continuous undulating vortex pair wake structure, described here for the kestrel, is the usual pattern in this type of flight.

The fact that this wake pattern has a simple geometric description with a constant value of the circulation around the trailing vortex lines allows an aerodynamic analysis to proceed with several simplifications. The way in which such an analysis might be conducted is outlined in the following section.

Consider a bird which flies in steady horizontal flight at speed U, flapping its wings with a frequency f, through an angle, ϕ, the wing beat amplitude (Fig. 11). The wingspan is 2b and the wing beat is such that the wing is fully extended on the downstroke but is gradually flexed and then re-extended during the upstroke so that, at mid-stroke, the effective wingspan is the distance between the ends of the secondary feathers in addition to the small spanwise extent of the flexed primary feathers. Let this distance be denoted 2b₂. The result of this upstroke is assumed to leave a straight section of wake behind, as measured in the kestrel wake, of spanwise extent R’2b₂-R’ is a dimensionless wake spacing, which was measured to be 0·76 in Spedding (1987) for steady-state gliding, close to the value of 0·785 for an elliptically loaded wingpair. In the absence of a direct experimental measurement, either value could be used, although somewhat higher R’ values of 0·85–0·95 were measured in the wake in flapping flight. A compromise choice to reflect this higher possibility might be 0·85, pending further experimental evidence. Similarly, the greatest spanwise extent of the wake produced on the downstroke is R’2b. For convenience, I define:

After a straightforward consideration of the wake geometry and enforcing the equilibrium condition of force balance on the bird, equations 9 and 10 may be solved for T and one other unknown wing beat kinematic parameter. If ϕ and T are given, together with the wing shape as expressed by 2b and 2b₂, then the remaining unknown is the stroke period τ, which must take on a value such that the momenta represented by areas A₁ and A₂ satisfy equations 9 and 10. This is not the sole possible choice of governing parameters however, and equations 9 and 10 could equally well be solved for the wing beat amplitude ϕ, given a reasonable value of r. Note also that the formulation of a solvable model does not depend on a particularly rigid adherence to the exact geometry of Fig. 11. It is obvious from even the most casual of observations of birds in cruising flight, that the degree of flexure, or sweep, of the primary feathers, during both upstroke and downstroke, varies widely according to species and according to flight speed, and this can be readily incorporated via a judicious selection of 2b and 2b₂. Similarly, a curved upstroke wake geometry could be written into equation 6 for A₂.

It is interesting to note how the shape of the wing is accounted for directly in the differing geometries of the downstroke and upstroke wakes, due to this folding of the wing back to the secondary feathers during the upstroke. This is reflected in the relative magnitudes of the thrust and lift components of the wake which must satisfy equation 8 overall. A parametric analysis along these lines might use b₂/b as a controlling parameter.

In Spedding (1987), equation 10 was used to estimate the magnitude of ground effect in gliding flight and it was concluded that ground effect was of limited importance at flying height, h, of the order of the semispan, b, but that its effect in reduction of D_i would increase rapidly as h approaches the mean chord c. The calculation is not so straightforward in the unsteady case and one can only speculate as to the possible effects on this wake. The lowest extent of the trailing vortex core in the wakes which provided material for the quantitative analyses was about 280 mm, or about six mean core radii from ground level. This is about 3·7 mean chord lengths, when a 25 % reduction in D_i would be expected in steady horizontal flight at this altitude. The effect on wake kinetic energy and induced drag has been neglected here as the appropriate unsteady corrections are not known. The fluid dynamics resulting from the interaction of linear wingtip trailing vortices with the ground were described by Harvery & Perry (1971), who observed a rebound of the vortex once it reached a minimum distance from the ground. This lifting of the vortex was attributed to the formation of a secondary vortex due to boundary layer separation between the two. Barker & Crow (1977) examined the two-dimensional vortex pair in ground effect, and Boldes & Ferreri (1973), among others, have investigated the behaviour of vortex rings near a wall boundary. The unsteady separation of a wall boundary layer due to its interaction with vortex rings embedded in an impinging jet has been studied by Didden & Ho (1985). These studies give some pointers as to the likely fluid dynamics in this case, and in general, it appears that the vortices do not approach closely enough to the ground for the wake structure to be qualitatively affected by its presence. No substantial secondary vortices or boundary layer separation were apparent near the floor in the wake photographs. The qualitative wake structure reported here is probably not significantly influenced by ground effect although, at some stages of the wing beat cycle, particularly towards the end of the downstroke, the induced velocities behind the wing trailing edge may be.

Since the completion of these experiments, preliminary results reported by Rayner (1986a), using this flow visualization technique, indicate that the wake of the bat Nyctalus noctula may be composed of closed vortex loops at flight speeds of 4ms⁻¹, while the wake structure at higher speeds of 9ms⁻¹ was interpreted as a continuous undulating vortex pair, following the pattern observed in the kestrel wake. The apparent occurrence of this type of wake in a different class of animal lends support to the foregoing arguments that such a wake structure should be observable in the wake of many, if not most, flying animals with broadly similar wing beat kinematics. Rayner (1986a) then goes on to discuss gait changes with flight speed. It is difficult to infer a distinct gait change (sharp discontinuity in wing beat kinematics in the case of flapping flight) from observations at just two flight speeds, and consequently this distinction has been avoided in this discussion. The observed wake structure of the kestrel in cruising flight is indeed quite different from any previously described but one can only speculate as to how small changes in flight speed might affect the qualitative structure. With decreasing flight speed, the strength of cross-stream vortex elements might gradually increase as the circulation around the wing decreases on the upstroke and a correspondingly greater amount of vorticity is shed into the wake at each end of the downstroke, consistent with the greater angular acceleration of the wing at these stages. With further decrease in flight speed, the limit of the closed vortex loop is approached. If the preceding scenario is both plausible and consistent with the gradual kinematic changes with flight speed measured by ciné film analysis (e.g. Scholey, 1983), it still lacks solid experimental evidence. We therefore concur with the arguments of Brown (1953, 1963) who observed different flight kinematics at different flight speeds but was careful to remain equivocal concerning the possible existence of distinct gait changes with speed.

Rayner (1986b) has also outlined a flight model which contains some similarities to that introduced in this paper, based as it is on the same experimental data. There are some differences in the specification of the wake geometry, here fixed by and ⁠, whereas his is based on the semispan b, thus ignoring span efficiency. He also specifies a spanwise circulation distribution which minimizes the wing root bending moment as given by Jones (1980), but which differs little from the elliptical loading distribution assumed here in its effect on the calculation of induced drag. His parameters are fixed to allow calculation of wingbeat amplitude, 0, and circulation, г, with changing frequency, while the approach here has been to calculate the downstroke ratio, τ, and г for a given wing shape. The two different formulations exemplify two ways in which knowledge of the simple wake geometry with constant circulation of the wake vortices enables approximate but elementary calculation to be made of the effects of changing wing beat kinematics and geometry, respectively.

The implications of these results for aerodynamic or energetic models of bird flight depend on their proposed applications. Clearly, the wake geometry is quite unlike the closed vortex loop model of Rayner (1979a,b) and the changes in circulation, both across the span and with time, are different. In this respect, the wake geometry adopted by Phlips, East & Pratt (1981) is closer to that described here. Their wake model is split into a far wake consisting of streamwise vortices joined by transverse elements formed at the ends of each wing stroke, and a near wake consisting of the surface traced out by the wings on the current wingstroke with transverse vorticity collected at the wing position at the beginning of the stroke. From lifting line theory, a first-order unsteady spanwise loading distribution on the wing was deduced for a simplified wing beat. The results reported here support this line of investigation and prescription of the wake geometry, which may be readily amended to incorporate refinements such as changing effective span on the upstroke. As far as including a more precise description of the variable wing geometry itself, the relatively low reduced frequencies prevalent in this mode of flight (Ω of the order of 10⁻¹ based on the wing chord c, Ω of the order of 1 based on the semispan, b) enable an analytical solution to the unsteady lifting-line problem to be made by considering small perturbations of the linearized boundary conditions. A recent example is the asymptotic analysis by Cheng & Murillo (1984) of an oscillating lifting surface with curved centre line and local wing twist, which yields the surface lift distribution on a smooth planar geometry in small amplitude pitching and heaving motion.

However, the assumption of a wake structure of the type described for the medium-speed flight of the kestrel enables a number of simplifications in the calculation of induced drag and induced power to be made, so that the constant circulation model introduced in the previous discussion might profitably be used for quick estimates of these quantities. The unsteadiness of the wake is accounted for by the accurate description of the wake geometry but the constant circulation on the wings allows a simple quasi-steady estimate for the magnitude of the induced drag. On occasions where an estimate of the induced power requirement at moderate flight speeds, typically around U_mp or U_mr, is required as a component in the calculation of energy budget of a flying bird, the added complexity of a more rigorous and complete aerodynamic model is not justified by the small improvement in accuracy in estimating one component of the total aerodynamic power requirement, and this approximate calculation should suffice.

The structure of the vortex wake behind a kestrel in medium-speed flight shows considerable differences from the wakes previously described for slow flight. The absence of any significant concentrations of transverse vorticity in the wake implies a constant circulation on the wing and makes calculation of the induced power requirement quite a simple matter. If the wing beat kinematics of the kestrel are taken to be reasonably representative of birds in cruising flight, then their wakes should also be similar and the flight model based on such a wake can be applied to predict the flight power requirement. Ground effect has been neglected in the results reported here, but its effect could be investigated by enforcing a minimum flying height with obstacles of various heights placed before and after the measurement area.

It is a pleasure to thank Dr C. J. Pennycuick for his steady supervision during the course of this research, and for a careful reading of the first version of this paper. Thanks also to colleagues Drs J. M. V. Rayner and K. D. Scholey for helpful advice and discussion. The financial support of the Science and Engineering Research Council is gratefully acknowledged.