Momentum and Energy in the Wake of a Pigeon (Columba Livia) in Slow Flight

Theoretical models purporting to predict the power output of flying birds have become increasingly elaborate over the years (e.g. Pennycuick, 1975; Rayner, 1979a,b), but confidence in their predictive value remains limited, owing to the difficulty of testing the predictions. Tucker (1973) found a promising degree of agreement between predictions based on mechanical analyses and measurements of oxygen consumption. However, the comparison cannot be carried to a high degree of detail because the physiological measurements do not allow different components of the power requirements to be discriminated. The measurement of oxygen consumption yields a single value, which, according to mechanical theory, is made up from several components, including induced, parasite and profile power components. Where the measured and predicted results differ, it is difficult to tell whether the discrepancy is due to an error in prediction of one or several components of power or to a flaw in the underlying concept of additive power components. Furthermore, chemical power, as reflected in oxygen consumption, is converted into mechanical power with an unknown efficiency, which may or may not itself be independent of speed or power.

This paper describes an attempt to single out one component of power, the induced power, and to measure it independently of the rest. The induced power is that which is expended in supporting the bird’s weight. In level flight, the weight must be balanced by a force equal to the rate at which momentum is imparted to the air. The induced power is the rate at which the bird imparts kinetic energy to the air in order to create the necessary momentum. It should be possible to examine the wake after the bird has passed, and measure the kinetic energy left behind.

In principle, this could be done without making any prior assumptions about the structure of the wake. However, it is much easier to estimate momentum and energy in relation to a specific model of the structure expected in the wake. The vortex ring model of the wake, developed by Rayner (1979a,b) was used as a basis for the analysis. If the wake does indeed consist of vortex rings as predicted, it should be a simple matter to calculate the energy of each ring, and hence the induced power.

The first objective was to ascertain whether vortex rings with the predicted characteristics occur. Having confirmed this, the next objective was to make a semiindependent estimate of the wake energy and hence the induced power requirement. These can then be compared with the predictions of the vortex-ring model.

Three pigeons (Columba livia Gm) were trained to fly in response to the alternate switching of two red lights. The naïve bird was introduced at one end of a flight cage (dimensions 4×2·7× l·2m, Fig. 1) with a red light illuminated at that end. When the bird’s attention appeared to be directed down the length of the cage, the lights were switched and about 0·5 s later a small disturbance created behind the bird would induce it to fly to the illuminated bulb at the other end of the flight cage. After a calming period of 10–20 s the procedure was repeated. Training sessions typically lasted 30–60 min and were repeated daily for 2–4 weeks after which the pigeons would fly in response to the light switching without any other cues. Training continued in reduced light levels and in conditions gradually approaching those of the experiment. After 10 weeks, one pigeon had been trained to the point where level and repeatable flights could be evoked at the flick of a switch. Flights were repeated under 4kW of light and filmed at 200 frames s⁻¹ with a Photosonics IPL high speed ciné-camera. Analysis of these films revealed no significant accelerations of the body and no measurable departure from horizontal motion in the test section of the flight cage. The morphology and kinematics measured directly and by film analysis respectively are given in Table 1.

A cloud of neutrally-buoyant helium bubbles (approx. 3 mm diameter) was generated by an Armfield SAI Bubble Generator with a low-speed head (No. 2). Helium, air and bubble film solution were passed at carefully metered flow rates to a mixing head where bubbles were produced at the tip of three concentric tubes. The design, construction and development of a helium bubble flow visualization system have been described by Lee (1976). The bubble generator head was positioned 0·35 m from the floor and tilted at 30 °from the vertical back wall of the flight cage to produce a cloud of bubbles with low initial velocity in the field of view of the camera system.

Two motor-driven Nikon cameras (F2 and FM) with micro-Nikkor 55 mm (f3·5) rises were mounted on wooden blocks on a rigid frame of slotted steel angle. The lens centres were separated by a horizontal distance of 0 · 25 m and the wooden mounting blocks were filed down so that the lenses were at the same height, as measured with a steel rule (±0 · 2mm). A plumb line was used to fix a planar, vertical, background grid (2 cm squares in a 2 · 45 × 1 · 22 m area) along the back wall of the flight cage, and camera orientation normal to the grid was checked by sighting down the lenses at a mirror mounted flush with the grid surface.

Four Sunpak AZ5000 flashguns (GN48 at 100 ASA) were mounted at a small angle to the floor of the flight cage to maximize contrast between the bubbles and the background. The flashguns were triggered sequentially at intervals determined by a chain of four 555 timers. Opto-isolators before the SCRs in each timing circuit prevented interference from the high voltage switching of the flashguns. Time interval settings were calibrated with a Marconi Instruments Universal Counter Timer 2438 applied at the opto-isolator output pins. With the flashguns set at full power, the photographic depth of field spanned the entire width of the flight cage, and the gradual tail-off in light intensity (total flash duration approx. 3 · 5 ms) of each flash allowed the direction of travel of each bubble to be deduced.

In the experiments, the camera shutters were opened for 1/8 s, during which time the four flashguns were triggered; camera apertures were set at f32. Ilford HP5 film was processed in Acuspeed developer at an equivalent speed of 800 ASA and the negatives were then printed with a Leitz Focomat V35 enlarger and 40 mm Zeiss lens onto 18 × 24cm sheets of Kodalith 4556 or Ilfolith IH7 lithographic film. These positive enlarged transparencies formed the raw material for all subsequent analysis.

In this section we briefly describe how the photocoordinates (x,y) of bubble images on pairs of stereophotographs (stereopairs) were used to derive their threedimensional real space coordinates (X,Y,Z). The object space is that part of real space containing information to be used in the analysis. An exposure station is the position of the camera at the time of exposure, or more precisely in this case, the location of the point half-way between the incident nodal point and the emergent nodal point along the optical axis of the lens. This ‘lens centre’ was estimated from technical drawings supplied by Nikon. The principal point on a photograph is that point at the intersection of the focal plane with a line from the rear nodal point of the camera lens perpendicular to the focal plane.

The two enlarged lith positives of a stereopair were fixed on the carriages of a Karl Zeiss Jena Stecometer (stereocomparator) where corresponding points on the stereopair could be measured. The photographs were viewed in stereo through binoculars and their positions, absolute and relative, were adjusted until a reference floating mark appeared to rest on the desired point. At the push of a button, the stereocomparator coordinates (x̄,⁠) of the left and right photographs were recorded. For each stereopair, up to 2500 bubble images were digitized in this fashion and recorded on paper tape.

As can be seen in Fig. 3, the parallax equations assume that the cameras are aligned with their lens axes parallel to each other and normal to the ground base plane XY. Errors in the determination of H and B enter directly into equations (4). H could be fixed to ±0·5 mm (1/6 × mean bubble diameter) and B was determined to within 0·2mm. The focal length is also assumed to be fixed and affects the scaling in Z, although it is in fact known only nominally from lens specifications. As measurements have been taken from enlarged positives, scaling corrections must be applied later from analysis of a control object of known dimensions, and these will also take care of any uncertainty in the focal length. The accuracy of the measured photocoordinates (x_a,y_a) and the parallax between them, p_a, is affected by any deviation from the ideal geometry of Fig. 3. Errors in the determination of the principal point, lens aberrations, film emulsion shrinkage, film curvature in the camera or enlarger and imprecise orientation of the photographic system will all affect the fidelity of the final result.

Scaling in X, Y and Z is calculated by reference to stereopairs of a control cube analysed in the same way. A cube with sides of 0·2 m was suspended in object space and photographed and scaling factors were determined by the apparent lengths of X, Y and Z in the reconstructed image. The cameras were not moved after this calibration procedure. Subsequent reconstruction of the same cube from another stereopair produced mean standard deviations of 0·35 mm in X, 0·33 mm in Y and 0·52mm in Z. The mean difference between measured and calculated lengths on the cube was ± 0·50 mm. Given an estimate for the standard deviation of each time interval from timer calibration, the standard error of velocity measurements may be calculated (G. M. Jarman, personal communication), and measured velocities should be accurate to within 10%.

Paper tape issuing from the stereocomparator was fed to the Honeywell Multics Computer System at the University of Bristol where all photogrammetric analysis and velocity field calculations were performed. All programmes were written in FORTRAN IV ; that used for relative orientation was adapted from code written by P. R. Wolf. All model calculations which are discussed later in this paper were performed on a Research Machines 380Z microcomputer in Microsoft BASIC and ciné-film analysis was performed on a NAC Film Motion Analyser connected to the same machine.

Fig. 4 is typical of the approximately 900 stereopairs obtained which indicate that a ring-like structure does exist in the wake. We may note that the helium bubbles appear to be following the air flow faithfully.

In 210 of these stereopairs, conditions were determined accurately enough for quantitative work. Owing to the large number of computer and man hours required fully five stereopairs were fully analysed. The selection criteria were that there should be an extensive but well distributed coverage of bubbles in the wake and that the wake structure produced during one wingbeat cycle (downstroke + upstroke) should be visible in stereo. After inspecting 900 stereopairs, the development and general structure of the wake are quite familiar and readily identified. The following results are outlined with reference to one representative stereopair. The consistency of the wake parameters measured in all five stereopairs and summarized in Table 2 is thought to justify such an approach.

Figs 5 and 6 are simple reconstructions of the wake. The pigeon flew from right to left across the field of view and the multiple exposure of the tail has been outlined in Fig. 5A. There are 1828 bubble images represented here, most of them in the foreground, as stereo views of bubbles far from the cameras were often obscured or confused by the intervening bubble cloud. The wingbeat is assumed to be symmetrical about the mid-plane and the wake is also taken to be symmetrical about the same plane. In the plan view of Fig. 5B the position of the outspread tail feathers is shown together with a scale bar showing the extent of the wingspan. The same format is shown in Fig. 6 which shows four vertical slices, 100 mm thick, in sections 1–4 moving from the background towards the cameras. Sections 1 and 2 straddle the mid-plane and sections 3 and 4 cut through the outer regions of a ring-like structure. The general wake structure revealed is similar to that which might be expected for sections through a small-cored vortex ring. The hypothesis that such a ring is present may be further tested by constructing velocity profiles for various sections through the wake; these may be compared with those reported in experimental studies of vortex rings by Sullivan, Widnall & Ezekiel (1973), Sallet & Widmayer (1974), Didden (1977), Maxworthy (1977) and Gühler & Sallet (1979), among others. Fig. 7 shows the coordinate systems and notation used in this study. Note that the vortex structures in the wake are generally inclined from the horizontal, the angle ψ being the angle between the horizontal and a convenient longitudinal axis in the wake. This corresponds to the ring momentum angle of Rayner (1979a) for a wake composed of vortex rings. Note also that experiments were conducted with the bird flying both from left to right (Fig. 4) and from right to left across the field of view (Figs 5, 6, 7). This causes few problems in the analysis, the major change being the change in sign of ψ. There is some asymmetry in the wake however (cf. Fig. 8) which will be discussed later and all following analysis and discussion assumes the bird to be flying from right to left.

The v(X′) velocity profile of Fig. 8 shows the variation in fluid velocity normal to the ring plane as we move along the ring plane at the centre line Z₀. v(X′) profiles were obtained for a series of sections over the whole range of Z and conformed to expected flow patterns for sections across a vortex ring. At the midline, Z₀, dv/dX′ is greatest as v changes sign across the vortex core. Remembering that Fig. 8 is a superposition of the circulatory flow of the vortex together with its own translational motion (the ring is convecting downwards away from the bird), the vortex core location is given by the intersection of the line of v(X′) with a horizontal line at U_s, the ring convection velocity. U_s is a mean value taken by measuring the distance between successive rings on photographs where this could be discerned; there were 32 of these. The distance travelled by one ring during one stroke period (T) could be calculated, T is known from ciné-film analysis, and U_s was thus estimated at 1·25 ms⁻¹, S.D. 0·3. This value is marked on the v axis of Fig. 8. We note that the slope of v(X′) is not symmetrical about X′₀. The steeper, better defined slope on the right of the figure corresponds to the starting vortex shed by the wings while the left-hand side of the curve intersects the stopping vortex produced at the end of the downstroke. In an ideal vortex rise these two cross-sections would be identical. The departure from this condition, which can be seen on close inspection of many stereopairs, is small but consistent. It is not obvious whether this might be due to distortion by the next ring in the chain, incomplete roll-up of the more recently shed wake vorticity or if it reflects a time history of wing circulation and hence vortex shedding into the wake which precludes the formation of a truly axisymmetric vortex ring. During the course of this investigation we found that each ring was affected by the development of the next ring in the chain, indicated by an increase in the ring momentum angle with time which was pronounced on the downstroke producing the newest vortex ring. Of the choices outlined above, the first thus seems to be implied, but in the absence of any further evidence we note that the effect is small, ignore it, and concentrate on the velocity profiles across the righthand vortex core which is less distorted by interaction with other wake elements.

Having thus located the core position in X′, X′_core, a profile of u(Y′) may be taken where the velocity parallel to the ring plane is plotted along a line running normal to the ring plane, through the vortex core, centred at Z₀. The resulting profile of Fig. 9 has the affected shape as u reaches peak values ±u_max at the edge of the vortex core, decaying thereafter with | Y′ – Y′₀ |. The diameter of the vortex core is given by the distance in Y′ between u_min and u_max. The curve has been fitted by eye and this, together with the scatter in the data, lends some uncertainty to the value of R₀. However, this did take into account information from neigh bouring profiles at different Z locations and the fact that each section has a finite thickness when a least squares best fit would always underestimate u_min, u_max and du/dY′ and similarly overestimate R₀, to an extent depending on the thickness of the section. A conservative error estimate would be ±20 % which is close to the standard deviation for all measurements (Table 2).

In all cases, velocity profiles obtained from sections through the wake were consistent with the hypothesis that it is composed of vortex rings. The results are summarized in Table 2.

The vortex ring model (Rayner, 1979a,b) used here for comparison with experiment is based on the logic that in steady level flight the mean forces acting on the bird are in equilibrium. The vortex-based forces of lift and induced drag must balance weight and the frictional drag forces. If it is assumed that useful aerodynamic force is generated by the downstroke alone, then momentum to support and propel the bird is transferred to the wake in the form of discrete bursts of vorticity, each representing a single downstroke. By the laws of vortex conservation these bursts must be composed of closed vortex loops, and it is natural to choose to model these structures as small-cored vortex rings. As pointed out by Lighthill (1973), a vortex ring has the property of conveying a given quantity of momentum with the least energy (or induced drag), and is therefore likely to be selected by birds -if feasible -since power economy in flight is critical. It is well known that subsonic aircraft wakes roll up into small-cored line vortices, and there is limited previous experimental evidence (Magnan, Perrilliat-Botonet & Girerd, 1938; Kokshaysky, 1979) that, in some birds at least, the wake of a flapping wing consists of discrete vortex rings. The experiments described here provide further strong evidence for this hypothesis, and give the first opportunity to measure the properties of the vortices in comparison with theoretical descriptions of flight performance.

Model predictions of wake parameters and induced power for the morphological and kinematic data of Table 1 are shown in Table 3, together with measured values. As the bird is flying very slowly, parasite and profile drags are small compared with weight: equivalent flat plate area of the body is estimated as l · 42 × l0 ^{− 3}m² so that parasite drag is 4 · 9 × 10^{− 3}N, while the horizontal component of profile drag is estimated as l · 3 × 10^{− 2}N if profile drag coefficient is 0 · 02 (cf. Rayner, 1979a). These drag forces are negligible compared with weight Mg = 3· 43 N, even allowing for substantially higher drag coefficients than quoted, and therefore the wake momentum angle ψ(to the vertical Z axis) should be small (0 · 30 °). Measured angles range between 8 · 0 ° and 13 · 5 °, and the increase in angle is largely the result of ‘solid-bodY′ rotation of the vortex ring induced by vorticity generated by previous wing strokes ; such rotation can be readily distinguished on many of the photographs obtained.

The form of the vortex-ring model used to predict the wake configuration is essentially that described in detail by Rayner (1979a), but with certain simplifications incorporated, partly as a response to these experiments. Profile drag and power are now calculated by blade-element methods for the upstroke as well as for the downstroke, but since at this flight speed profile drag is insignificant, the change is not critical. The more important change is possible because wing-beat kinematics have been determined from film in parallel with flow visualizations. In its original form the model included a two-dimensional equilibrium condition expressing the balance of mean forces in horizontal and vertical planes : this condition fixed the vortex-ring momentum angle and vortex momentum, and imposed a constraint on wing-beat kinematics so that the vortex sheet shed by the wings during the downstroke could deform under its self-induced downwash into a vortex ring of the appropriate configuration. It is the second constraint that has been relaxed, since with full kinematic data available it would be redundant. Moreover, these experiments indicate that while vortex roll-up does occur, at these speeds at least it is a more complex process than originally envisaged, and, pending further experimental evidence, it is unjustified to include a description of vortex development in the model. The condition now takes the less strict form of a constraint on wing-beat geometry, in that the mean angle of wing-tip travel to the horizontal (ψ ₁) must be greater than the momentum angle ψ. Since here ψ₁ is approximately 45 ° there is no difficulty : the constraint only becomes important in fast flight, where it imposes an upper limit on flight speed. As speed increases ψ₁ falls, whereas ψ rises as profile and parasite drags become bigger; beyond a certain speed the wing beat cannot generate sufficient thrust. It is not known whether this constraint is more critical than that of obtaining sufficient power output.

As explained above, ring convection velocity and self-energy are calculated on the basis of constant vorticity across the vortex core, so that Ā = 0 · 25. Predicted selfenergy is calculated for the elliptic vortex ring by the formulae given by Rayner (1979a, equations 42 and 43), while measured self-energy is obtained for the equivalent circular ring using equation (9); if the observed ring is assumed to be elliptical then it has very low eccentricity, and the exact elliptical self-energy is close to the value given in Table 3.

It is immediately apparent from Table 3 that the vortex rings measured in the wake are significantly different from expected ; they are smaller, and also have lower circulation, and therefore the momentum convected into the wake with each ring is substantially lower than expected, being only 60% of that required to support the bird’s weight. (The momentum associated with a vortex ring is given by its vector area times the circulation times the fluid density). Even allowing for the rough-and-ready nature of some of the calculation, the size of the measured wake momentum deficit is disturbing. Apparently the reaction force from air accelerated into the wake by aerofoil action of the wings is only about three-fifths of that required for weight support.

Primarily because of the size and strength discrepancy, the estimated ring selfenergy E_s = 0 · 76 J is also 60% of predicted (1· 29 J). If we calculated induced power P_i as E_s/T as explained above, then P_i is only 5 · 05 W, a surprisingly low value for a bird of this size at this flight speed. This estimate should not, however, be seen as fully reliable until the momentum deficit has been satisfactorily explained.

The wake element spacings and ring convection velocities also differ from prediction, due in part to the different size and strength of the measured ring: however, the equivalent circular ring to that measured would travel at 2 · 43 m s ^{− 1} if isolated, and it is unlikely that vortex interactions would reduce this to as low as 1 · 72 ms ^{− 1}. It is feasible that this discrepancy is diagnostic of misinterpretation of the nature of the vortex field, but in view of the regularity with which near-planar, near-circular vortex loops were discerned this appears to be unlikely.

Inspection of equation (9) shows that the critical quantities affecting estimation of E_s, and hence P_i, are circulation K and ring radius R. All other quantities including the core radius R₀ and the constant Ā are the same in both cases. The predicted and measured circulations do not differ appreciably, and the root of the discrepancy in momentum and induced power calculations appears to lie in the value of R measured from the photographs.

Before taking the comparison of results any further, it is worth summarizing the assumptions made in each case and considering their effects on the outcome. As explained above, calculation for the vortex ring model is based on the premise that forces on the bird are balanced : it follows that the momentum of each vortex ring must balance the bird’s weight, together with the vector sum of profile and parasite drags, for the duration T of each wing-beat. Ring geometry is calculated from wing-beat kinematics by assuming near-elliptical loading distribution on the wing during the down-stroke, and effectively no loading during the upstroke. Momentum balance determines the circulation of the wake vortex, and wake energy and induced power are computed. Momentum balance is thus an inviolable condition, vital to the model calculations, but it is hard to see how the condition, derived directly from Newton’s second law of motion, can be relaxed if the bird is in steady level flight.

At this very low flight speed the bird must be experiencing the most demanding of all flight conditions, having to generate virtually all wing force by the wing-beat alone as there is relatively little forward air speed. It is to be expected, therefore, that this flight pattern will present the vortex-ring model with its most strenuous test. One outcome of the experiments which is not surprising is that the rings are rather smaller than expected ; ring size is determined by loading distribution on the wing, and clearly this is somewhat less favourable than the assumed near-elliptical distribution. To compensate, the circulation should be much higher (approx. 3 · 28 m²s ^{− 1}) for these small rings to support the weight, but evidently this is not the case.

No momentum balance condition of course is either assumed or stipulated in the direct computation of momentum from ring size and circulation or of self-energy E_s from equation (9). Here, the approximation is valid for circular, planar, small cored-vortex rings where R₀ < 0 · 25 R (Norbury, 1973). The latter condition is satisfied but the rings are not circular, and we cannot claim to have shown that they are truly planar. The small departures from circular shape do not affect ring energy, but departures from planarity may cause a small increase. The model also includes the interactive energy of successive rings in the wake. The magnitude of this is assumed to be small, as is indicated by the wake photographs; it is estimated for the predicted wake as 0· 14 J, approximately 10 % of the estimated self-energy, and accordingly has been ignored. Induced power is likely therefore to have been slightly underestimated, but it will suffice as a first approximation at the rate of energy flow in the wake. In view of the as yet unexplained momentum deficiency, we are not in a position to argue whether this represents the true induced power demanded from the bird.

The nature and magnitude of the wake momentum deficit imply that there is a conceptual flaw in the vortex ring theory, or the measurements, or both. In both cases, it is assumed that the deformed, unstable, vortex sheet initially shed from the wingtips and trailing edges rolls up into a vortex ring without loss of momentum. This argument is subject to certain modifications and qualifications to allow for the effects of viscosity

Maxworthy (1974) has described the formation and motion of turbulent vortex rings in the Reynolds number (Re = 2RU_s/v: v is the kinematic viscosity) range 0 · 25 × 10⁴ – 0 · 75 × 10⁴; similarly defined, Re = 3 · 4 × 10⁴for our rings in the wake. Two fundamental processes were identified as entrainment and subsequent rejection of fluid into a turbulent wake, and the resulting decrease in ring impulse as vorticity is deposited into the wake appears as a momentum deficit therein. Maxworthy (1977) also described an axial flow along the core of a turbulent vortex ring; such an axial component can be detected in photographs like Fig. 4, and Widnall, Bliss & Zalay (1971) have remarked that a radial force would be necessary to support a momentum flux around a vortex ring. While noting these considerations, it is difficult to make valid comparisons when the generation mechanisms of the vortex flows are so different. We may further note that the azimuthal wave propagation described by Maxworthy took place about 10 ring diameters downstream of the generating nozzle and therefore far beyond the range we are considering, and also that there is no obvious large scale turbulent structure aside from the vortex ring visible in our photographs.

Similarly, viscous dissipation of the ring energy and observable wake momentum has been ignored. This is justified, according to Saffman (1970) for t << R²/v where t is a suitable time scale, vis the kinematic viscosity of the fluid and R is the ring radius. In our case, v = 0 · 15× 10 ^{− 4}m²s^{− 1} and R ≈ 0 · 2 m, so that viscous decay of the ring is not critical within 2 · 7 × 10³ s, rather more than the stroke period 0 · 15 s.

It seems unlikely then that these turbulent and viscous effects will modify the flow field to anything like the extent required to explain the measured wake momentum deficit. Neither can we identify any other large scale structures in the wake which could account for the missing momentum. For example, a flow normal to the tail surface can be seen towards the end of the downstroke, just after depression of the tail, but the vertical component of the reaction force is estimated to be less than 10 % of that required.

and should contribute little towards the overall lift at these low flight speeds. Furthermore, a significant contribution towards weight support from ⁠, for example, would be accompanied by a concomitant increase in the drag which would, in turn, have to be compensated by increasing the vortex wake momentum. The energy expended against and is assumed to pass into a separate viscous wake and there is no evidence from the photographs to suggest that such a wake might contain the required momentum.

The measured wake momentum deficit appears to imply a physical impossibility and it is difficult to identify possible sources of error. Indeed we are in no position to state categorically that the deficiency actually exists. Similarly, we cannot claim to have made a true measurement of P_i and we must remain equivocal concerning the accuracy of the various theoretical estimates of P_i.

As guidance for future attempts to resolve this paradox, we might bear in mind the fact that the conceptual model used in the measurements bears many similarities and shares many of the assumptions of the model which we are attempting to test. Also, the crucial importance of accurate measurements of the ring radius, R and, to a lesser extent the circulation, K, has been highlighted by the somewhat puzzling resume presented here and these quantities should be very carefully measured in the further work which seems to be necessary.

Special thanks are due to Professor R. T. Severn of the Department of Civil Engineering and Professor B. K. Follet of the Department of Zoology for their support and advice at critical stages of this research. We would also like to thank Dr K. D. Scholey for his assistance in ciné-film analysis and Mr G. Gooding at the University Computer Centre, Bristol for his noble efforts on our behalf. The financial support of the Science and Engineering Research Council is gratefully acknowledged.