Gliding Flight: Drag and Torque of a Hawk and a Falcon With Straight and Turned Heads, and a Lower Value For The Parasite Drag Coefficient

This paper reports the aerodynamic drag and torque on wingless model raptor bodies with their heads in two positions: straight, or turned 40 ° to the right. The aerodynamics of turned heads in raptors – falcons, hawks and eagles in this study – is interesting, because these birds have three properties that seem to be in conflict. (i) Some raptors, particularly large falcons such as peregrines (Falco peregrinus), approach prey by diving at it from great distances at high speeds. (ii) The aerodynamic drag on raptor bodies is presumably minimal for given conditions when their body is symmetrical, i.e. when the head is straight rather than turned to one side. (iii) Raptors appear to have their most acute vision along a line of sight directed approximately 40 ° to either side of the head axis. So how can a raptor approach prey at high speed if it must turn its head sideways to see distant prey straight ahead, thereby increasing drag?

The present paper is the first of a series of three that investigate this conflict. It quantifies the dependence of drag on head angle in body models of a peregrine falcon (PF) and a red-tailed hawk (RT, Buteo jamaicensis). The second paper (Tucker, 2000) shows that raptors look almost exclusively sideways rather than straight at objects more than 40 m away. It also shows that raptors could approach prey by flying along a spiral path that keeps their line of sight for maximum visual acuity pointing sideways at the prey while their heads are straight for maximum speed. If they flew along a straight path, they would have to turn their heads to one side and fly slower to see the prey with maximum acuity. A higher speed could theoretically compensate for a longer spiral path and allow the raptor to reach the prey in the same time along either path. The relationship between drag and head angle established in the present paper is a prerequisite to this conclusion. The third paper (Tucker et al., 2000) shows that wild peregrines follow curved paths towards distant prey more often than straight ones. The curved paths resemble the spiral path that would allow the falcon to keep its head straight while viewing the prey to one side with maximum visual acuity.

The present paper also suggests that raptors with straight heads may have a lower parasite drag coefficient than the previously measured value (Tucker, 1990a). The model bodies in this study had the lowest drag coefficients yet measured for raptors, and a low parasite drag coefficient is consistent with speed measurements on a diving gyrfalcon (Falco rusticolus) (Tucker et al., 1998). Recent measurements on flapping birds also suggest lower values for parasite drag coefficients (Pennycuick et al., 1996).

The PF and RT body models were mounted in a wind tunnel with two head positions: ‘straight’ models were bilaterally symmetrical with heads that pointed straight ahead, and ‘bent’ models had turned heads that pointed 40 ° to the right. The models had two characteristics essential for this study: (i) their shapes accurately replicated those of the real bird bodies, and (ii) except for head position, the straight and bent models had identical shapes. The following sections describe in detail the processes for measuring frozen bird bodies, making the models from the measurements and measuring drag and torque. Table 1 shows the accuracy of the measurement and model-making procedures.

Both the measurement process and the model-making process used a large number of three-dimensional ‘surface points’ on a body or model to define shape. Each point has three values (z, r and the angle Θ) that describe it on a system of cylindrical coordinates. This system can be visualized relative to a cylinder with an axis along the z-axis and radii that extend perpendicularly in all directions from any point on the z-axis. Any point on the surface of the cylinder lies on a radius and is a distance r from the z-axis along that radius. The angle Θ identifies the radius, and a value for Θ alone identifies one of an infinite number of planes that contain the z-axis.

For the bird bodies and models described here, the body exclusive of the neck and head is bilaterally symmetrical; and the body axis, which is also the z-axis, runs from anterior to posterior in the midsagittal plane. Θ has a value of 0 ° for a ‘ventral radius’ that runs ventrally from the z-axis in the midsagittal plane and a value of 180 ° for a ‘dorsal radius’ that runs dorsally. ‘Intermediate radii’ run between the ventral and dorsal radii.

Values of z vary in steps from the base of the beak to the base of the tail, and each value of z identifies a cross section (‘section’, for short) of the bird or model body. For example, a typical model body might have 40 sections, each with 20 radii, for a total of 800 surface points. These points define the ‘armature’ for the model, and the armature becomes the finished model after turning its head in bent models and smoothing its surface with a thin layer of plaster.

The armatures are bilaterally symmetrical, because the surface points on only one side of the armature come from measurements on frozen bodies. In the example above, only 11 of the 20 points in each section would be measured: the two points on the ventral and dorsal radii and the nine points on the intermediate radii for angles between 0 ° and 180 °. The remaining nine points fall on the intermediate radii between 180 ° and 360 ° and are reflections of the measured points.

I measured surface points on the frozen bodies of a PF (intact mass 0.713 kg) and an RT (intact mass 1.017 kg) after cutting off the wings at mid-humerus to leave the proximal humeral feathers attached to the body. These feathers smoothly covered the stump of the humerus (which was folded back parallel to the body surface) and the region where the wing had joined the body. I preened and smoothed the feathers on the slightly thawed bodies, as described in Tucker (1990a).

A digitizer connected to a computer recorded the coordinates of surface points on the feathers on one side of a frozen body, as described in Tucker (1990a). The PF had 46 sections spaced at 7.93 mm intervals and 6–31 measured surface points per section, depending on the size and shape of the section. The RT had 32 sections spaced at 12.70 mm intervals and 11 measured surface points per section. For the RT, these points fell on radii with steps of 18 ° between them, i.e. the ventral and dorsal radii and the nine intermediate radii. For both the PF and the RT, the first section was at the base of the beak, and the last section was at the base of the tail.

The surface points for the PF were the same as those used by Tucker (1990a), but here they were used in each section to interpolate a new set of surface points that fell on the same radii described for the RT.

The AC machine cut model armatures from expanded polystyrene plastic (known as styrofoam) by using a hot wire to melt, or ‘cut’, an open channel (the kerf) through a styrofoam cylinder with a density of 16 or 32 kg m⁻³. This cylinder was 0.13 m in diameter and up to 0.2 m long, depending on the size of the finished model, and its axis was the z-axis of the armature. The armature could be up to 0.4 m long, because the AC machine made the armature in separate anterior and posterior parts.

The AC machine cut an armature by a process somewhat like sharpening a pencil with a knife. The AC machine could rotate a plate, and I used hot wax to stick one end of the styrofoam cylinder to the plate so that the z-axis, the axis of the cylinder and the axis of rotation were all the same. The AC machine could then rotate the plate and the cylinder as one rotates a pencil while sharpening it. With the plate held at some value of Θ, the AC machine moved the a hot wire (the knife) from one end of the cylinder to the other along a curve in the plane for that value of Θ and sliced off a thin piece of styrofoam (the pencil shaving). The surface points for that value of Θ determined the shape of the curve. The AC machine then rotated the cylinder to the next value of Θ and moved the hot wire along another curve to slice off another thin piece of styrofoam. After several rotations through the set of 20 values of Θ, the hot wire had sliced off enough styrofoam to make the armature, just as a knife makes the pencil point by removing shavings.

The AC machine used computer-controlled stepping motors to rotate the cylinder and move the hot wire along the curve at a rate of approximately 5 mm s⁻¹. The wire, made from nichrome 0.13 mm in diameter and 95 m long, was perpendicular to the plane for Θ and carried a current of 0.6 A at 5.6 V. The wire moved through the styrofoam with virtually no mechanical resistance as it melted a kerf 0.4 mm wide. When the wire reached one end of the cylinder, the slice of styrofoam fell free, and the stepping motor rotated the cylinder to the next value of Θ. The AC machine then cut off the next slice of styrofoam as it moved the hot wire back to the other end of the cylinder, again along a specified curve. After 20 rotations, Θ was back where it started, and the AC machine repeated the process until it had cut the cylinder down to the armature. The AC machine typically made four cuts for each value of Θ, and took 100 min to finish both parts of the armature.

The computer program that controlled the stepping motors used the surface points in several calculations to determine the path of the hot wire in the plane for a given value of Θ. For example, the program decided how deep each cut could be, and checked the sequence of cuts for both previous and subsequent values of Θ to avoid interference between cuts (described in the next section). It also computed the combination of z and r steps that gave the shortest path between surface points on adjacent sections, and it scaled the surface points to make the armature a particular size.

The stepping motors moved the hot wire in 0.25 mm steps along the z axis, in 0.18 mm steps along the r axis and in 0.2 ° steps around the z-axis for Θ. The computer program used the length of the maximum radius for a surface point as a measure of armature size and scaled all the armatures so that this radius had a length of 55.9 mm. This value made the PF armature life size and the RT armature 0.87 times life size.

I glued the anterior and posterior parts of an armature together with water-soluble glue, drilled a hole through the armature and glued a hardwood dowel into the hole. The hole was 9.5 mm in diameter and passed through the armature in the midsagittal plane near the middle of the sternum and parallel to the dorsal and ventral radii. The dowel served as a mount for the model in the wind tunnel, and a hole drilled along its axis received a threaded rod from the strut of the drag and torque balances. The strut ended flush with the ventral surface of the model, and a flush cap plugged the 9.5 mm hole on the dorsal side of the model. The distances from the base of the beak to the mounting strut were 0.152 m and 0.143 m for the PF and RT models, respectively.

The hot wire of the AC machine left the armature with a facetted surface (Fig. 1), and the ridges between the facets established the approximate surface of the finished model. The wire also fluffed out the armature, because the wire could not move into concavities on the perimeter of a section without cutting short the radii adjacent to the concavity. i.e. interfering with them (Fig. 1). The program that ran the AC machine recognized concavities and prevented such interference.

I coated the armature with a thin layer of plaster to round out the facets and sanded the model to its final shape. Next, I cut a beak and tail from styrofoam with a hot wire, glued them to the model, filled any crevices with plaster and sanded these appendages to their final shape. The result was a smooth model with a symmetrical body and surface roughness approaching that of a smooth plaster wall. The mass of the finished models was between 0.034 and 0.051 kg.

To turn the head on an armature, I cut two 20 ° wedges from the right side of the neck so that the sharp edge of each wedge was parallel to the dorsal and ventral radii and in the midsagittal plane (Fig. 2). One sharp edge was at the point where the neck joined the head, and the other was where the neck joined the shoulder girdle. I sliced the head from the armature by cutting from the line where the sharp edge of the anterior wedge had been, to the left side of the neck, and sliced off a section of the neck with a similar cut for the posterior wedge. I then glued the head, neck section and wedges back onto the armature with the wedges on the left side of the neck, and used a hot wire to smooth the neck to natural contours.

The result was a life-like armature with the head turned 40 ° to the right.

In addition to models cut from styrofoam, this study used the PF model described by Tucker (1990a). The armature of this model was made from plywood laminations shaped to fit the same surface points used to interpolate the surface points for the styrofoam PF armature cut by the AC machine. The model with the laminated armature – the laminated model – had a higher drag than the model with the styrofoam armature, but I was able to eliminate the difference by modifying the shape of the laminated model with plaster. This study discusses both the laminated model and the modified laminated model.

The wind tunnel (described by Tucker and Parrott, 1970) ran at a constant air speed (V) of 11.7 m s⁻¹ in the region of the test section occupied by the body models. This value was measured with a pitot tube connected to a micromanometer and monitored by a digital counter that displayed the period between rotations of the wind tunnel fan shaft. The air density (ρ), temperature and atmospheric pressure during the measurements did not deviate significantly from 1.23 kg m⁻³, 22 °C and 101.33 kPa (1 atmosphere), respectively, at sea level. The drag balance (described in detail by Tucker, 1990b) was of the parallelogram type, with four strain gauges connected as a Wheatstone bridge cemented to the parallelogram arms. The voltage across the bridge, displayed by a digital voltmeter, was linearly related to the force component along the sensitive axis of the balance.

The torque balance was made by adding to the drag balance a vertical shaft that could rotate on low-friction ball bearings. The mounting strut for the models attached to the shaft and was approximately co-linear with it. The strut transmitted any torque around its axis to the shaft, and an arm on the shaft kept the shaft from rotating by pressing on the sensitive axis of the drag balance. With this arrangement, the voltmeter responded linearly to any torque produced by aerodynamic forces acting on the model.

The measurement procedure produced paired values of drag or torque, followed by calibration values. After mounting a model on the strut, I turned the wind tunnel on and recorded the voltmeter reading. Then I turned the tunnel off, waited for the wind speed to drop, turned the tunnel on again and recorded another voltmeter reading. With the tunnel off, I calibrated the balance by hanging weights on a thread attached to the strut of the drag balance or to the moment arm of the torque balance and running over a pulley on a low-friction ball bearing. The weights produced voltmeter readings that bracketed those for the model, and drag or torque were then calculated by interpolation.

The mounting strut was 90 mm long from the top of the shroud of the drag balance to the surface of the model, and its cross-sectional shape was that of a Navy Number 1 strut (Tucker, 1990b). Its chord, from leading to trailing edges, was 6.2 mm, and its maximum thickness perpendicular to the chord was 2.5 mm. The strut was always aligned so that airflow in the wind tunnel was parallel to a line between its leading and trailing edges.

I calculated the drag of a model by subtracting strut drag and interference drag from the drag measured by the drag balance. Strut drag is the measured drag of the strut alone on the drag balance, and interference drag is drag that arises because of the mutual interference of the strut and the model with the airflow in the region where the strut attaches to the model. Interference drag in this study equals the strut drag multiplied by the ratio interference length/strut length, where interference length is 33.1 mm (Tucker, 1990b). The total of strut drag and interference drag was always less than 12 % of the drag measured by the drag balance.

The yaw angle is the angle between the direction of the z-axis and the direction of air flow in the wind tunnel, when both directions are projected onto a horizontal plane. The yaw angle is zero when the head of the model points directly upwind, and the angle increases when the z-axis rotates counterclockwise, as seen from above. A model could be set to any yaw angle by rotating it on the mounting strut.

The pitch angle is the angle between the direction of the z-axis and the direction of air flow in the wind tunnel, when both directions are projected onto a vertical plane. The pitch angle is zero when the yaw angle is zero and the model has minimum drag. The pitch angle increases when the z-axis rotates clockwise, as seen from the model’s left side. A model could be set to pitch angles between −20 ° and 20 ° by rotating the mounting strut around the point where it attached to the drag or torque balance.

This study discusses measurements on six models of bird bodies, and it also discusses mathematical models and torque models. Mathematical models are sets of equations, and torque models are mathematical models that calculate the drag required to generate torque. To distinguish these types of model, I shall henceforth refer to body models, mathematical models and torque models.

The body models include four made from styrofoam armatures cut by the AC machine: the straight PF, bent PF, straight RT and bent PF. The other two – the laminated model and the modified laminated model – are similar to the straight PF model.

The drag coefficients of body models depended on the species, on the pitch and yaw angles and on whether the model was straight, bent or laminated (Figs 3, 4). Table 2 gives equations for parabolas fitted by the least-squares method to the data. The following sections frequently describe the drag coefficient when the yaw and pitch angles are zero, and I shall use the symbol C_D for the drag coefficients measured under these conditions. In other cases, the text describes the measurement conditions for C_D.

The drag coefficients had three notable features. (i) The straight PF and RT models had C_D values of 0.11 and 0.12, respectively; both less than the previous lowest published value for a raptor (0.14, Tucker, 1990a). However, Nachtigall (1998) reports a lower value of 0.08 for a starling (Sturnus vulgaris). (ii) Turning the head 40 ° to the right increases the C_D values of both the PF and RT models by more than 50 % compared with their straight counterparts. (iii) C_D values are minimal for the bent PF and RT models when the models are yawed 7–10 ° in the direction opposite to that of the turned head.

The drag coefficients of the laminated and modified laminated models also have three notable features. (i) The laminated model in the present study has the same C_D as that found in a previous study of this model (Tucker, 1990a). (ii) The C_D for the laminated model is higher than that for the straight PF model, although both models were constructed using the same surface points (see above). The models looked similar, but careful examination showed that the AC machine introduced a difference into the armatures that it made: the sections of the armatures had no concavities in their perimeters, while the laminated model had shallow concavities on the perimeters of some of its sections (Fig. 5). In fact, the laminated model had concavities that ran lengthwise for several sections where the concavities of adjacent sections were in register. (iii) Filling in the concavities of the laminated model with plaster reduced its C_D to that of the straight PF model. I added plaster in two steps: the first step filled in the concavities posterior to the shoulder region, and the second step filled in those anterior to the shoulder region. The first filling reduced the difference between the C_D values of the straight PF and laminated models by two-thirds, and the second filling eliminated the remaining difference.

The largest and deepest concavities were around the thighs and in the vicinity of the vent, where the added plaster was up to 3 mm thick (Fig. 5). Evidently, lengthwise concavities on the body of a bird can markedly increase drag.

The bent models of both the PF and the RT developed substantial torque to the right at pitch and yaw angles of 0 °, and this torque increased rapidly when the model yawed towards its right (decreasing yaw angle) (Fig. 6). The torque decreased when the model yawed to its left, and became 0 ° at approximately the same angles where C_D was minimal for bent models (Figs 3, 4).

The body models in this study have lower drag coefficients than those measured previously for raptors, and turning the head on a model markedly increases drag. What are the consequences of these findings for a gliding raptor? The drag on the body is only one component of the aerodynamic force on a flying bird, and the answer to this question requires the analysis of several other properties of the bird. To introduce these properties, I shall use the concept of an ‘ideal falcon’ (Tucker, 1998), which is an abstract falcon with mathematically defined morphological and aerodynamic properties. The name comes from the similarity between an ideal falcon and the ‘ideal gas’ of physical chemistry. Both are simplified, mathematically defined abstractions of their real counterparts, yet they represent reality closely enough to provide an understanding of how their counterparts behave.

For birds, parasite drag is usually defined as the measured drag of wingless bodies in wind tunnels (Pennycuick et al., 1988; Tucker, 1990a). This definition tacitly assumes that there is no interference drag between the body and the wings when, in fact, the drag of the body with wings attached may differ from the total drag of the isolated body and isolated wings (Gesser et al., 1998; Hoerner, 1965; Nachtigall, 1998; Von Mises, 1959). The difference is the interference drag, which may be accounted for by adding it algebraically to the parasite drag.

The profile drag of ideal falcons varies with the lift produced by the wings, and the relationship between these variables can be calculated from a ‘polar curve for the wings’, which shows the minimum profile drag coefficient (CD,pr) plotted against the lift coefficient (C_L) (Tucker, 1987). The profile drag used for the polar curve depends on parasite drag, since profile drag is left-over drag, i.e. the total drag minus the sum of induced drag and parasite drag. Thus, in the context of the polar curve, any change in measured parasite drag causes an equal and opposite change in profile drag, since there is no change in total and induced drag.

The drag measurements on the straight PF model in the present study suggest that C_D,par should be reduced again. Fig. 7 shows the effect of a range of C_D,par values on the polar curves for the same total and induced drag values used for equations 3 and 4. I shall refer to the curve for C_D,par=0.1 again; the equation for this curve is:

A changed value for C_D,par affects not only profile drag but also the gliding performance of an ideal falcon, as I shall illustrate with examples. The performance curves in the examples describe an ideal falcon with a body mass of 1 kg (equivalent to that of a large peregrine or a small gyrfalcon), a range of C_D,par values between 0.05 and 0.20 and polar curves for the wings from Fig. 7.

Fig. 8 shows a ‘velocity polar diagram’ (Tucker, 1998) for the gliding falcon. The four curves drawn with thick lines are ‘maximum performance curves’ for the range of C_D,par values. A line on this diagram between the origin and a point on a maximum performance curve represents a two-dimensional velocity vector for the highest speed that the falcon can reach for that vector direction (the glide angle). Rays from the origin show glide angles, the axes show the horizontal and vertical components of the vector, and concentric circles show the magnitude (speed) of the vector. The maximum speed of the falcon at any glide angle depends markedly on C_D,par, particularly at the steep angles and high speeds that occur during diving. Parasite drag becomes the largest component of drag at high speeds (Tucker, 1998), primarily because the falcon folds its wings as speed increases and reduces the wing area that generates profile drag.

Fig. 9 shows how the falcon’s speed increases with time as it dives at 45 ° from a speed of 15 m s⁻¹ to 95 % of the speed at maximum performance. Again, the C_D,par value has its greatest effect on a curve at the highest speeds, for the reasons mentioned above.

The extremely high speeds that the falcon could reach in a vertical dive, given enough space and time, are worth mentioning (in the 230–300 miles h⁻¹ range in English units) for C_D,par values of 0.1 or less. The body model measurements in the present study suggest that real falcons could have C_D,par values in this range if C_D,par decreases at high speeds because of Reynolds number effects, and there are reports of falcons reaching speeds above 200 miles h⁻¹ (Tucker, 1998; Franklin, 1999).

The turned head of the bent PF body model causes changes in two measured quantities relative to the straight model: an increment in drag (ΔD) and an increment in torque. The bent model did not accelerate and change its position perceptibly in response to these increments because the mounting strut provided equal and opposite compensating forces and torques, but a gliding bird lacks a strut. The bird can avoid acceleration when it turns its head only by altering its body shape or flight path to produce compensating forces and torques from gravitational components or aerodynamic forces. Thus, it incurs both the drag increment ΔD measured on the bent body model and the drag D_T that accompanies the generation of the compensating torque.

In one case, ΔD describes the bent PF model with no net torque acting on it – when the model is yawed 11.6 ° in the opposite direction to that in which the head is turned (Fig. 6). ΔD for the model at this yaw angle is 1.45 times the drag on the straight model at yaw and pitch angles of 0 °.

However, this method for torque compensation would be counter-productive for a real falcon that turns its head in an attempt to direct the line of sight for maximum visual acuity of one eye along the flight path. Yawing would swing the line of sight back towards where it started, and the falcon would have to turn its head even more. That motion would again increase torque, and the falcon would have to yaw even more to compensate. Eventually, the falcon might reach equilibrium with its head turned more than 40 °, but at an unknown value for ΔD.

In this section, the ideal falcon turns its head and generates compensating forces equal to ΔD and D_T to keep from accelerating. I shall estimate ΔD from measurements on the PF model bodies, and use three torque models to estimate D_T. These models make several assumptions for which there is little supporting evidence, but the calculations do show how compensating torque may arise from adjustments in body shape, and they also provide estimates of how large D_T might be.

The torque on the ideal falcon with its head turned is greater than that on the bent PF body model, because the ideal falcon is bigger (m=1 kg, where m is body mass) than the PF (m=0.713 kg) from which the body model was made. Since the product S_hM is proportional to body mass, the torque on the ideal falcon at 11.7 m s⁻¹ is 1/0.713 times that on the bent PF body model at pitch and yaw angles of 0 °, and is proportional to V².

For example, assuming that M is b/2 (i.e. 0.061 m), then R_par=6.5. That is, turning the head in flight multiplies the parasite drag of the straight body by this ratio. This example estimates a relatively large value for D_T, since R_par=1.6 when D_T is zero.

For example, if M is 2b (i.e. 0.244 m), then R_par>1.7. This value for R_par indicates a relatively small value for D_T compared with R_par=1.6 when D_T is zero. However, a falcon probably does not have a stiff enough tail to produce the entire compensating torque by this method, because the required lift on the tail at 70 m s^{− 1} is 56 % of the ideal falcon’s weight. When I loaded the end of a dead peregrine’s tail with a force equal to one-quarter of the falcon’s weight, the tail feathers bent abnormally from a horizontal position at the base to 40 ° below horizontal at the point of loading.

Real raptors commonly turn their heads while gliding slowly, probably to change their line of sight and to initiate or control yawing. However, I do not remember seeing a falcon turn its head in a high-speed dive, and Fig. 10 suggests why: raptors probably pay an increasing speed penalty for head-turning as speed increases. For example, the torque models for head-turning show that the parasite drag coefficient could plausibly more than double if a falcon turned its head at 70 m s⁻¹, and doubling the parasite drag coefficient has a marked effect on the velocity polar diagram and the acceleration during a dive (Figs 8, 9).

Thus, to summarize this discussion of turned heads and the conflict between the requirements for acute vision and high-speed flight: if diving raptors turn their heads in a dive to view the prey straight ahead with maximum acuity, they will incur a substantial speed penalty from increased drag. Tucker (2000) shows show how they could avoid this penalty by approaching the prey along a spiral path.

Several mathematical models predict the motions and energy requirements of flapping birds (for reviews, see Pennycuick, 1989; Rayner, 1999) and gliding birds (Tucker, 1998). For a given bird, these models use a constant value for the parasite drag coefficient (C_D,par). This value usually comes from measurements made on wingless bird bodies in wind tunnels, since a direct method is not available for measuring the parasite drag of a flying bird. However, a single value for C_D,par measured in this way is only an approximation for a real bird for two reasons: first, C_D,par may vary in flight with yaw and pitch angles and head position (present study) or with Reynolds number (Tucker, 1998); and second, how accurately the measured values describe a living bird is unknown. There are reasons to suspect that the accuracy is low (see below), yet values of C_D,par measured in wind tunnels make useful predictions of the behavior of birds in flight.

Some authors have recently suggested another method for determining C_D,par: choose a value that improves the accuracy of the mathematical model. Pennycuick et al. (1996) used a mathematical model (Pennycuick, 1989) that predicted flight speeds from C_D,par and other assumed values, and then adjusted C_D,par to match the predicted speeds with those observed in a bird flying in a wind tunnel. They suggested reducing the C_D,par values in the original model by a factor of up to 8 (to give C_D,par as low as 0.05) to make the model more accurate.

Tucker et al. (1998) used a similar approach and compared the predictions of a mathematical model for diving falcons (Tucker, 1998) with speeds measured during dives at various angles by a gyrfalcon. The predictions matched the measured values more closely with a C_D,par of 0.07, rather than the value of 0.18 estimated from wind tunnel measurements. At least part of this difference could be due to Reynolds number effects, since the wind tunnel speeds were less than one-quarter of those reached by the gyrfalcon.

Ideally, the two methods for determining C_D,par would yield the same result, but both methods depend on several uncertain assumptions. Here, I discuss the uncertainties in determining C_D,par from measurements in wind tunnels and then suggest a compromise value for C_D,par that bridges the gap between the results of the two methods.

One of the differences between a living bird body and a model body is that the feathers of the bird are a compliant surface whose shape may be controllable by the bird. For example, a perched hawk obviously changes the thickness of its feathers as it fluffs them and swells up or compresses them to become sleek and slender. In flight, such changes may influence the boundary layer of air around the body and also the distribution of air pressure on the body. Both these changes may change C_D,par (Du and Karniadakis, 2000; Tucker, 1990a). Some wind tunnel studies show how feathers could influence parasite drag. In this paper, the drag coefficient of the laminated body model was more than 25 % higher than that of the straight PF model made from the same set of surface points because the laminated model had shallow concavities on its surface. Eliminating these concavities, particularly those around the thighs, made the drag coefficients equal, and the feathers of living birds may fill in such concavities.

Nachtigall (1998) compared the drag coefficients of a starling (S. vulgaris) gliding in a wind tunnel with those of a smooth model on a drag balance. The minimum drag coefficient of the living bird was lower than that of the model, and Nachtigall (1998) attributed the difference to the presence of feathers. The model itself had an exceptionally low minimum drag coefficient (0.08 with the wings removed) compared with a minimum drag coefficient of 0.11 for the straight PF body model in this study.

However, frozen bird bodies in wind tunnels have a higher drag than smooth models, probably partly because the feathered surface of a frozen body is not as smooth that of a living bird in flight (Tucker, 1990a). A frozen, feathered body had 70 % more drag than a smooth plaster model of the same body (Tucker, 1990a), and Pennycuick et al. (1988) reduced the drag on a frozen body by 15 % by smoothing the feathers and holding them in place with hair spray.

Measurements on frozen bodies probably overestimate the drag of living birds, and this suggestion is consistent with an evolutionary argument: that the lowest C_D that can be measured on a reasonably anatomically correct smooth model could also occur in a living bird with naturally arranged feathers, given the many other refined avian characters for flight produced by the evolutionary process.

However, the drag of isolated body models, feathered or smooth, might underestimate the drag of the same body attached to the wings of a living bird, because a measurement on an isolated body does not include the interference drag between the body and the wings (Gesser et al., 1998; Hoerner, 1965; Nachtigall, 1998; Von Mises, 1959). Or, the presence of the wings might reduce drag by preventing flow separation around the body. These uncertainties cause Pope and Harper (1966, p. 269) to question the value of drag measurements on a wingless aircraft fuselage without information on interference drag.

Taking all the evidence together, the value for C_D,par of 0.1 used for the ideal falcon in this paper appears to be a reasonable compromise for gliding raptors, together with the relationship between C_D,pr and C_L given by equation 5. Historically, C_D,par has declined as new methods have been developed for measuring drag in wind tunnels and as data have accumulated on the accuracy of mathematical flight models. Now, the results from measurements and mathematical models seem remarkably close, considering the difference in the methodologies.

 
• AC

  armature-cutting

 
• b

  wing span

 
• C₁, C₂, C₃

  coefficients of parabolic equations

 
• C_D

  drag coefficient of body models, usually a minimum

 
• C_D,par

  parasite drag coefficient

 
• C_D,pr

  profile drag coefficient

 
• C_L

  lift coefficient

 
• C_T

  torque coefficient

 
• D

  total drag

 
• D′

  total drag of the ideal falcon with a turned head

 
• D_i

  induced drag

 
• D_par

  parasite drag

 
• D_par,s

  parasite drag with a straight head

 
• D_pr

  profile drag

 
• D_T

  drag to compensate for torque

 
• d

  magnitude of displacement

 
• d

  displacement vector

 
• F

  magnitude of force

 
• F

  force vector

 
• g

  magnitude of gravitational acceleration

 
• L

  lift

 
• L_L

  lateral lift

 
• m

  body mass

 
• M

  moment arm

 
• PF

  peregrine falcon

 
• RT

  red-tailed hawk

 
• R_par

  ratio of parasite drag values before and after head-turning

 
• R_tot

  ratio of total drag values with head turned and straight

 
• r

  radius for cylindrical coordinates

 
• S_b

  maximum area of a body section

 
• S_h

  cross-sectional area of the head

 
• T

  magnitude of torque

 
• T

  torque vector

 
• t

  time

 
• V

  speed through the air

 
• x

  pitch or yaw angle

 
• y

  parasite drag coefficient or head angle

 
• z

  axis for cylindrical coordinates

 
• α

  angle between force and displacement for torque

 
• Δ

  change in a quantity

 
• Θ

  angle for cylindrical coordinates

 
• π

  ratio of circumference to diameter of a circle

 
• ρ

  air density

This study was partially supported by a grant (DEB-9107222) from the National Science Foundation.