1. The equilibrium gliding performance of a bird is described by the relationship between sinking speed (V8) and air speed (V). When V3 is plotted against V, the points fall in a ‘performance area’ because the wing span is changed during gliding.

  2. The lowest Vs for each V in the performance area defines a ‘maximum performance curve’. This curve can be predicted by a mathematical model that changes the wing span, area and profile drag coefficient (CD pr) of a hypothetical bird to minimize drag. The model can be evaluated for a particular species given (a) a linear function relating wing area to wing span, and (b) a ‘polar curve’ that relates Cp.pr and the lift coefficient (CL) of the wings.

  3. For rigid wings, a single polar curve relates CD,pr to CL values at a given Reynolds number. The position and shape of the polar curve depend on the aerofoil section of the wing and the Reynolds number. In contrast, the adjustable wings of a laggar falcon (Falco jugger) and a black vulture (Coragyps atratus) gliding in a wind tunnel have CL, and CD,pr values that fall in a ‘polar area’ rather than on a curve. The minimum values of CD,pr at each CL bound the polar area and define a polar curve that is suitable for evaluating the model.

  4. Although the falcon and the vulture have wings that are markedly different in appearance, the data for either bird are enclosed by the same polar area, and fitted by the same polar curve for minimum CD,pr at each CL value. This curve is a composite of the polar curves for rigid wings with aerofoils similar to those found in avian wings. These observations suggest that the polar curves of other gliding birds may be similar to that of the falcon and the vulture.

  5. Other polar curves are defined by CL and CD,pr values for the falcon and the vulture gliding at a constant speed but at different glide angles. Each speed has a different polar curve; but for a given speed, the same polar curve fits the data for either bird.

  6. The falcon and the vulture gliding in the wind tunnel at a given speed were found to increase their drag by decreasing their wing span. This change increases induced drag and probably increases CD,pr for the inner parts of the wing because of an unusual property of bird-like aerofoil sections: wings with such sections have minimum values of CD,pr at CL values near 1, while conventional wings have minimum values of CD,pr at CL values near 0.

During equilibrium gliding, all the forces on a glider are balanced; it neither accelerates nor decelerates. The relationship between a glider’s air speed and the speed at which it sinks through the air completely describes the glider’s aerodynamic performance, and can be predicted from a relatively simple theory for rigid wings (Welch, Welch & Irving, 1955; Pennycuick, 1975).

Gliding birds, however, change their wing span during flight. Anyone can observe these birds gliding slowly on fully spread wings, then progressively flexing their wings as they glide faster and faster. Hankin (1913) called this behaviour ‘flex gliding’. Flex gliding influences stability (Lighthill, 1975) and gliding performance (Cone, 1964; Newman, 1958; Pennycuick, 1968; Raspet, 1960; Tucker & Parrott, 1970). In this paper, I shall analyse the relationship between lift and drag, for different wing spans, in birds gliding at equilibrium and use that relationship to predict maximum gliding performance.

Consider a bird gliding at equilibrium at speed V along a flight path inclined at angle θ (the glide angle) to the horizontal (Fig. 1). Only two forces have significant effects on the bird : gravitational force (the bird’s weight, W) and aerodynamic force. Since the sum of these two forces is zero, the aerodynamic force on the bird equals the weight in magnitude but has the opposite direction.

Fig. 1.

Forces and velocities during equilibrium gliding. The glide path is inclined at angle θ to the horizontal. Lift and drag are perpendicular and parallel, respectively, to the glide path. Air speed (V) is parallel to the glide path, and sinking speed (Vs) is vertical and downwards.

Fig. 1.

Forces and velocities during equilibrium gliding. The glide path is inclined at angle θ to the horizontal. Lift and drag are perpendicular and parallel, respectively, to the glide path. Air speed (V) is parallel to the glide path, and sinking speed (Vs) is vertical and downwards.

The aerodynamic force has two components (Fig. 1): total drag (D), the component Wsinθ that is parallel to the flight path, and lift (L), the component

Wcosθ that is perpendicular to the flight path. Sinθ and cosθ can be replaced with expressions involving the sinking speed through the air (Vs) and the air speed (V):
formula
formula
The total drag coefficient, CD, and the lift coefficient, CL, are computed by dividing the force components by 0·5ρSV2:
formula
formula
Air density (ρ) is 1·23 kg m−3 in this study — the value at sea level in the US standard atmosphere (von Mises, 1959), and S is the projected wing area, including the area of the wings intercepted by the body.
The performance of a glider, described by the relationship between V and Vs, depends on total drag, as can be seen by rearranging equation 1 :
formula
I shall analyse performance by dividing total drag into additive components: induced drag, profile drag and parasite drag. Many textbooks on aircraft aerodynamics describe this method in detail (see, for example, Pope, 1951).

Induced drag

Induced drag, D1, arises whenever the wing produces lift. Lift results from a higher pressure below the wing than above, and this pressure difference induces an upward flow of air at the tips of the wing and a downward flow along the wing span. The induced downward flow causes the induced drag on the wing (see Prandtl & Tietjens, 1957, for a detailed discussion). Induced drag depends on wing span, b (measured across the widest span of the wings), and is given by:
formula
The induced drag factor, k, has a value of 1 for a wing with a straight leading edge when viewed from the front, constant downwash and an elliptical distribution of lift along its span. Actual wings typically have k values in the range of 1–1·1 (Reid, 1932), and some wings theoretically may have k values less than 1 (Cone, 1962; van Dam, 1987). k also corrects the parasite drag of a complete aircraft at low speeds (see section on Parasite drag below). I shall follow the common practice of using a value of 1 for k when dealing with isolated wings, and a value of 1·1 when dealing with a complete aircraft. Spedding (1987) found a value of k of 1·04 in a gliding sparrow hawk, Falco tinnunculus; which is consistent with a value of 1·1 after correcting for parasite drag at low speeds. (Some authors place the induced drag factor in the denominator of equation 6, in which case it typically has values of 1 or less.)

The induced drag factor of bird wings may change during flight if the wings change their shape and span so that the lift distribution is not elliptical. There are insufficient data to account for these changes in the present theory, which assumes that k remains constant during flight. However, the effect of increases in k will be discussed qualitatively.

The induced drag is related to the induced drag coefficient, given by:
formula
If the right side of equation 6 is substituted for Di in the equation above, and L is expressed in terms of CL, one obtains a simple expression for CD,i:
formula
where A is the aspect ratio of the wing, defined as the ratio of wing span to wing chord. Wing chord is the mean width of the wing, defined as S/b. Therefore, A = b2/S.

Profile drag

Profile drag, Dpr, is the drag on the wing in addition to induced drag. It arises from skin friction and pressure differences and is related to the profile drag coefficient, given by:
formula
The profile drag coefficient varies with the lift coefficient of the wing, and hence with V. The relationship between CD,pr and CL depends on the shape of the wing’s cross-section (the aerofoil section, since the wing is an aerofoil) and the Reynolds number (Re) of the wing, defined as
formula
c′ is the wing chord calculated from maximum wing area (Smax) and maximum wing span (bmax):
formula
μis the viscosity of the air. The ratio p/μ has the value 68 436 s m−2 for air at sea level in the US standard atmosphere.

At a given Re, CL plotted against CD,pr for a rigid wing yields a polar curve. If the wing is not twisted along its span and has the same aerofoil section everywhere, the polar curve is nearly independent of aspect ratio (Prandtl & Tietjens, 1957) and is characteristic of the aerofoil section.

CD,pr at a given CL decreases as Re increases, with a drop of several-fold over a narrow range of Re values (Hoerner, 1965; Schmitz, 1960). The midpoint of this range is the critical Re, which decreases in turbulent air or if the aerofoil surface is rough. The critical Re for a smooth, bird-like aerofoil section in non-turbulent air is 75 000 (the Gottingen 801 section; Schmitz, 1960; similar to the USATS 4 section in Fig. 4). Most soaring birds have Re values between 75 000 and 106. CD,pr changes gradually over this range according to the relationship:

formula
where F is the ratio CD,pr /( CD,pr at Re ×10−5 = 1). I derived this relationship from data in Hoerner (1965, fig. 17, pp. 6–13) and Schmitz (1960, fig. 6, p. 171) for a variety of aerofoil sections at CL = 0’7.

As Re changes, the polar curve shifts right or left without markedly changing shape (Schmitz, 1960). Thus, theCD,prvs CL curve can be shifted to any Re between the critical Re and 106 by calculating a correction term for CD,pr at a CL of 0·7 and adding that term algebraically to the CD,pr axis. Equation 12 was used to correct all polar curves and CD,pr values in this paper to Re ×10−5 = 1. An example of the correction is shown in the Appendix.

Parasite drag

Parasite drag, Dpar, is the drag on the aircraft exclusive of the drag on the wings and is related to the parasite drag coefficient, given by:
formula
In flying birds, Dpar is the drag of the head, body, tail and feet. Spar is the crosssectional area of the drag-producing structures.
The parasite drag on an aircraft is commonly expressed in terms of the area of a flat plate held perpendicular to the airflow. If the drag coefficient of the plate is assigned a value of 1, a plate of area SparCD,par has the same drag as the parasite drag of an aircraft at speed V. The area of the plate is known as the equivalent flat plate area, Sfp:
formula
If Sfp for an aircraft is known, parasite drag is given by:
formula
Sfp tends to increase at lower speeds as the tail of an aircraft drops, thereby increasing the drag of the fuselage and tail. The change in Sfp is small, and it is common practice to compensate for it by increasing the induced drag factor slightly (Pope, 1951). Sfp also tends to increase at lower speeds as Re decreases (Goldstein, 1965).

Speed and sinking speed

The total drag on an aircraft is the sum of the three components:
formula
so, from equation 5,
formula
After expanding the drag terms, this equation becomes:
formula

Models

Equation 18 becomes a model of a glider if one allows b, L, CD,pr, Sfp and V to vary in the same way that they do in the real glider. The simplest model (I shall call it the ‘constant span’ model) describes a glider with rigid wings. Speed varies, but the wing span, profile drag coefficient and the other variables on the right side of equation 18 remain constant. If sinking speed is plotted against speed for the constant span model, a curve (1 shall call it the ‘performance curve’) results.

The performance curve is often called the ‘glide polar’, but I shall use the former terminology to avoid confusion with the polar curves that relate lift and profile drag coefficients for wings. Performance curves are usually drawn on a V vs V9 diagram with V3.increasing downwards and V increasing to the right (Fig. 2).

Fig. 2.

The performance area bounded bv performance cun es for a falcon, Falco jugger. Speed for maximum performance is lowest at A, where CL is maximum at 1 ·87. Vs = V along line BC - i.e. descent is vertical at each speed. Vertical descent is slowest at B, where total drag = profile drag of the fullv extended wings with CD,pr = 1 - i.e. the wings act as flat plates held perpendicular to the airflow. Vertical descent is fastest at C, where total drag = parasite drag — i.e. the wings are fullv folded. Solid lines calculated from the theory in this paper, dashed lines interpolated.

Fig. 2.

The performance area bounded bv performance cun es for a falcon, Falco jugger. Speed for maximum performance is lowest at A, where CL is maximum at 1 ·87. Vs = V along line BC - i.e. descent is vertical at each speed. Vertical descent is slowest at B, where total drag = profile drag of the fullv extended wings with CD,pr = 1 - i.e. the wings act as flat plates held perpendicular to the airflow. Vertical descent is fastest at C, where total drag = parasite drag — i.e. the wings are fullv folded. Solid lines calculated from the theory in this paper, dashed lines interpolated.

A more realistic model for birds, which I shall call the ‘maximum performance’ model, describes gliders with adjustable wings. Birds typically extend their wings to maximum span when gliding slowly and flex glide at higher speeds. As wing span changes, so do wing area, induced drag, and the lift and profile drag coefficients. These changes make the product SCD,pr pr in equation 18 a function of wing span.

The relationship between sinking speed and speed calculated from equation 18 as wing span varies need not be a performance curve as is the case with the constant span model. If, at a given speed, b varies, then sinking speed varies as well. Birds can also increase sinking speed by lowering their feet to increase Sfp (Pennycuick, 1960, 1968, 1971b). As a result, sinking speed can have a range of values at each speed, and the points relating sinking speed and speed fall in an area (Cone, 1964). I shall call this the ‘performance area’ (Fig. 2).

The ‘maximum performance’ model describes the upper boundary of the performance area. I shall call this boundary the ‘maximum performance curve’. Maximum performance in this sense means that a bird can glide at a particular speed for the maximum time and distance through the air when it minimizes sinking speed. If one considers the motion of the bird relative to the earth, the maximum performance curve shows the minimum vertical wind velocity component required to keep the bird from losing altitude and, for a given horizontal wind, indicates the maximum distance over the ground that the bird can cover for a given loss in altitude.

The lower boundary7 of the performance area is the ‘minimum performance curve’, which describes the maximum sinking speed at each speed. At all but the lowest speeds, maximum sinking speed occurs in vertical descent when Vs = V and drag = weight. Vertical descent is slowest when the bird ‘parachutes’ on extended wings (Fig. 2, point B), and fastest when the bird folds its wings and dives (Fig. 2, point C).

The constant span model

In the constant span model, L is assumed to equal W (a constant); the only variable on the right side of equation 18 is V. The equation for the performance curve becomes:
formula
where k1 and k2 are constants, defined as follows:
formula
formula
This model is useful to glider pilots (Welch et al. 1955) because it approximates the characteristics of man-made gliders with simple wings. Such gliders can fly only at the combinations of V and Vs that fall on a single performance curve because their rigid wings have lift and profile drag coefficients that fall on a single polar curve. The glider cannot change its drag, and hence its sinking speed, without changing speed. [For manoeuvres such as landing, gliders are usually equipped with panels on the wings that may be raised to increase profile drag. Advanced gliders also have wing flaps that can change the aerofoil section of the wing during flight - see, for example, Jacobs (1986). The constant span model does not describe gliders that use these devices.]

The constant span model has been used for gliding birds (Newman, 1958; Pennycuick, 1971a,b, 1975, 1982; Blake, 1983). The simplicity of the model is attractive, but a gliding bird need not hold its wing span and the other variables on the right side of equation 18 constant at different speeds as the model requires. The constant span model describes a performance curve that may be wholly within the performance area, partly within it or wholly outside it, depending on the values assigned to k1 and k2 in equation 19.

The maximum performance model

In the maximum performance model, wing span, wing area and the profile drag coefficient may vary at each speed; their values are chosen to minimize sinking speed at each speed. Under these conditions, equation 18 describes the maximum performance curve.

Sinking speed may have a minimum value at a given speed when the wings are flexed to less than their maximum span. Suppose, for example, that Co.pr were constant. Then changes in wing span would have opposite effects on two of the drag terms in equation 17 : a reduction in span would increase induced drag but decrease profile drag by reducing wing area (Newman, 1958; Pennycuick, 1968; Pennycuick & Webbe, 1959; Tucker & Parrott, 1970). Indeed, a falcon gliding in a wind tunnel at different speeds has been observed to adjust its wing span to near the theoretical values for minimum sinking speeds (Tucker & Parrott, 1970).

CD,pr, however, varies with wing span because (1) CD,pr and CL are related by a polar curve in bird wings as span varies, and (2) CL is related to S (equation 4), which varies with wing span. To develop the maximum performance model, I shall represent these two relationships by polynomials:
formula
and
formula
in which C0, …, C4 are constants.

The wing span for minimum sinking speed at each speed can be found by making appropriate substitutions into equation 16, setting the derivative dD/db equal to 0 and solving for b. The minimum sinking speed can then be calculated.

First the terms in equation 16 are expanded:
formula
For economy of notation, some of the constants in equation 24 may be combined into new constants:
formula
Before differentiation, CD,pr and S in equation 25 must be replaced with functions of b. CD,pr is a function of CL (equation 22), so
formula
CL is a function of S, so from equation 4
formula
and
formula
Replacing S with the right side of equation 23,
formula
Differentiating,
formula
and, replacing L with (equation 2),
formula
The value of b that makes the derivative equal to 0 is the wing span (b′) for minimum total drag. Replacing b with b ′, L with and D with WV9/V in equation 25 yields
formula
where is the value of V3 used to calculate b ′.

The first chosen value of will probably not be a solution of equation 32 (i.e. Vs will probably not equal ), and I used Newton’s method (Sokolnikoff & Sokolnikoff, 1941) to estimate a new value of V’. This value is used to recalculate b’, and the whole process is repeated until values of and b′ are obtained that do satisfy equation 32. The procedure is repeated with different values for V to obtain the maximum performance curve.

The quantity C0 in the above equations varies with Re, and hence with V. As Re changes, the curve (equation 22) relating CD,pr to CL shifts to the right or left by an amount related to F from equation 12. This shift is accomplished mathematically by adding C(F–1) to C0, where C is the value of CD,pr calculated from equation 22 for CL = 0·7. (A computer program that calculates a maximum performance curve, including Re effects, is available from the author.)

This section describes the data that will be used to evaluate the maximum performance model and summarizes how they have been obtained. All the data have been published, and details may be found in the original publications. Curvilinear relationships between two variables are described with second-degree polynomial equations (fitted by least squares) of the form:
formula
where a0, a1 and a2 are constants.

Lift and drag

Lift and drag can be determined by measuring W and the glide angle θ, or V, and V, and then solving equations 1 and 2. In some cases, the measurements have been made on birds gliding in natural conditions by using tracking devices, mounted either on the ground or in a sailplane (Raspet, 1960; Pennycuick, 1960, 1971a), to record the position of the bird relative to the observer.

It is difficult to make accurate measurements in nature for two reasons: (1) tracking devices of sufficient accuracy are complicated and difficult to operate, (2) not only the bird must be tracked, but the air as well. Equilibrium gliding requires an inertial frame of reference relative to which the air does not move. The frame may move relative to the observer (for example, if there is a wind), but its movements must be measurable. Indeed, there may not be such a frame, for winds often change in both speed and direction as a result of gusts and topographical features (see McGahan, 1973, and Pennycuick, 1971a, for a discussion of these problems).

The glide angle θ can be measured more accurately in a wind tunnel than it can in nature. If a bird is trained to glide freely in a wind tunnel tilted at the glide angle, the bird remains stationary relative to the observer, and the motion of the air in the wind tunnel is known. This technique has been used with a pigeon (Columba livia; Pennycuick, 1968), a laggar falcon (Falco jugger, similar in size to the more familiar peregrine falcon; Tucker & Parrott, 1970) and a black vulture (Coragyps alratus; Parrott, 1970). These studies will be identified in this paper by references to ‘the pigeon’, ‘the falcon’ and ‘the vulture’.

Wing span and area

Wing span and area are changed in living birds during flight. Maximum wing span (bmax) and area (Smax) can be measured on dead birds by stretching out the wings. At lesser spans, the relationship between wing span and area was linear in the pigeon, falcon and vulture gliding in a wind tunnel. This relationship can be estimated for gliding birds from measurements of bmax and Snlax (see Appendix).

Equivalent flat plate area

Equivalent flat plate area of a bird can be measured by placing the body on a flight balance in a wind tunnel (Pennycuick, 1968; Tucker, 1973). Sfp is a function of a bird’s linear dimensions and varies with body mass (Tucker, 1973):
formula
This equation was obtained from drag measurements made at nearly constant speeds (11–12 ms −1) on birds with masses between 0·025 and 6·9 kg. It therefore includes Re effects due to body size but not those due to speed. Sfp did not change with speed in the pigeon. Variations in Sfp with speed are ignored in the present study since the drag due to Sfp is a small proportion of total drag at the speeds at which birds usually glide.

Profile drag coefficient

Rigid wings: polar curves for bird-like and conventional aerofoil sections

The profile drag coefficient, CD,pr, may be determined by measuring the total drag of a rigid wing and subtracting the induced drag from it. The total drag can be measured by mounting the wing on a flight balance in a wind tunnel, and induced drag can be calculated using equation 6. Alternatively, induced drag may be. eliminated so that profile drag equals total drag. This is accomplished by mounting, an untwisted, rectangular wing with a constant aerofoil section in a wind tunnel so that it spans the tunnel from wall to wall (Pope & Harper, 1966). The profile drag coefficient is obtained by dividing profile drag by 0·5 ρ SVz.

The shapes of bird-like aerofoil sections are quite different from the shapes of conventional ones. Bird wings (for example see Nachtigall, 1985) have highly cambered upper and lower surfaces at the base of the wing and less cambered surfaces towards the tip (Fig. 3). Wings of conventional aircraft have lower surfaces that are nearly flat to convex, as in the Clark Y section (Fig. 4). This section was used in many aircraft built between 1920 and 1940, and it often appears as an example in textbooks on low-speed aerodynamics. Its aerodynamic characteristics are well-known for a range of Re values, and it will be used in this paper as a reference for comparison with the less conventional, bird-like sections.

Fig. 3.

Bird-like aerofoil sections and their polar curves. The sections progress (top to bottom) from a highly cambered section typical of the base of a bird wing to a less-cambered section typical of the tip of a bird wing. Shown for comparison are the polar curve for minimum drag (equation 35) for the falcon and the vulture, and polar curves for three different pigeon wing sections (Nachtigall, 1979). Polar curves have been corrected to a Reynolds number of 105 and smoothed by fitting points to a second-degree polynomial equation. Data for the bird-like sections and their polar curves mav be found in the National Advisory Committee on Aeronautics Technical Reports listed in References. The number of the Technical Report is given in parentheses: Gottingen 462 (286), RAF 19 (93), Gottingen 400 (124), Eiffel 35 (93).

Fig. 3.

Bird-like aerofoil sections and their polar curves. The sections progress (top to bottom) from a highly cambered section typical of the base of a bird wing to a less-cambered section typical of the tip of a bird wing. Shown for comparison are the polar curve for minimum drag (equation 35) for the falcon and the vulture, and polar curves for three different pigeon wing sections (Nachtigall, 1979). Polar curves have been corrected to a Reynolds number of 105 and smoothed by fitting points to a second-degree polynomial equation. Data for the bird-like sections and their polar curves mav be found in the National Advisory Committee on Aeronautics Technical Reports listed in References. The number of the Technical Report is given in parentheses: Gottingen 462 (286), RAF 19 (93), Gottingen 400 (124), Eiffel 35 (93).

There are also differences between the polar curves of bird-like aerofoil sections and those of conventional aerofoils. Conventional sections have a minimum CD,pr value at a CL value near 0, and CD,pr is nearly constant for CL values between 0 and 1· 0 (Clark Y, Fig. 4). Highly cambered bird-like sections such as the RAF 19 (Fig. 3) have a minimum value for CD,pr near a CL value of 1, and CD,pr increases two-or three-fold as CL drops towards 0. In addition, the minimum CD,pr values for bird-like sections are higher than that for the Clark Y section.

Fig. 4.

The top three aerofoil sections have the same thickness and progressively less-cambered lower surfaces. The lift coefficient for the minimum profile drag coefficient decreases as the camber of the lower surface decreases. The bottom aerofoil section (Clark Y) is typical of those used in low-speed man-made aircraft flying at Reynolds numbers between 1·5 × 105 and 4 ×106. Polar curves have been corrected to a Reynolds number of 105 and smoothed by fitting points to second-degree polynomial equations (except Clark Y data, which were fitted by free-hand curve). Data for these sections and their polar curves may be found in the National Advisory Committee on Aeronautics Technical Reports listed in References. The number of the Technical Report is given in parentheses: RAF 19 (93), USATS 4 (93), Durand propeller 13 (93), Clark Y (244).

Fig. 4.

The top three aerofoil sections have the same thickness and progressively less-cambered lower surfaces. The lift coefficient for the minimum profile drag coefficient decreases as the camber of the lower surface decreases. The bottom aerofoil section (Clark Y) is typical of those used in low-speed man-made aircraft flying at Reynolds numbers between 1·5 × 105 and 4 ×106. Polar curves have been corrected to a Reynolds number of 105 and smoothed by fitting points to second-degree polynomial equations (except Clark Y data, which were fitted by free-hand curve). Data for these sections and their polar curves may be found in the National Advisory Committee on Aeronautics Technical Reports listed in References. The number of the Technical Report is given in parentheses: RAF 19 (93), USATS 4 (93), Durand propeller 13 (93), Clark Y (244).

These differences are related to the amount of camber of the lower surface of the wing. The more the camber of the lower surface, the higher the CL at which the minimum CD,pr occurs (Fig. 4).

Not all studies of avian wing sections agree with the above description. Nachtigall (1979) investigated model wings with the same aerofoil sections as those of the pigeon wing. The polar curves for the model sections were similar to those for conventional sections in that CD,pr was at a minimum near a CL of 0 (Fig. 3). Also, the maximum Co.pr values for the model sections were several times higher than those for the birdlike or conventional sections in Figs 3, 4.

Adjustable wings: the polar area for bird wings

The profile drag of a wing attached to a living bird can be determined by measuring the total drag of the bird and subtracting induced drag and parasite drag. In contrast to rigid wings, CD,pr and CL values for the adjustable wings of living birds do not fall on a single polar curve. Birds may change wing span, aerofoil sections and degree of wing twist from base to tip at a given speed. Consequently, their wings have a range of CD,pr values for each CL - The points representing this range fall within an area (I shall call it the polar area) of the CD,prvs CL diagram rather than on a curve. (Fig. 7 shows the polar area for the falcon and the vulture.)

Bird wings can have a polar curve if the bird glides under conditions that select one curve from the infinite number that comprise the polar area. For example, the pigeon, falcon and vulture wings had polar curves when the birds were gliding in wind tunnels at their shallowest glide angles (i.e. with minimum sinking speed and drag) for a given speed. These curves are the left-hand bounds of the polar areas, and I shall call them ‘polar curves for minimum drag’ (e.g. Fig. 5).

Fig. 5.

Polar curve for minimum drag for the falcon (open circles) and the vulture (filled circles), corrected to a Reynolds number of 105. CL increases as speed decreases. Data from Tucker & Parrott (1970, table 2) and Parrott (1970, table 1).

Fig. 5.

Polar curve for minimum drag for the falcon (open circles) and the vulture (filled circles), corrected to a Reynolds number of 105. CL increases as speed decreases. Data from Tucker & Parrott (1970, table 2) and Parrott (1970, table 1).

The falcon and vulture also glided over a range of glide angles at a particular speed. Measurements under these conditions yield polar curves that I shall call ‘polar curves for constant speed’. They will be described after the next section.

Polar curves for minimum drag

The CD,pr and CL values for the falcon and the vulture fall on the same polar curve for minimum drag (Fig. 5) when corrected to an Re value of 105. The equation for the curve is:
formula
This curve is a composite of the polar curves for the bird-like aerofoil sections found in different regions of the wing (Fig. 3). As the wing span varies, these sections are exposed to the airflow to different degrees, so it is not surprising that the polar curve for minimum drag shares points with the polar curves for the aerofoil sections.

The pigeon’s polar curve for minimum drag has CD,pr values that are much greater than those for the falcon and vulture at high CL values (Fig. 6), comparable at CL values near 0·5, and lower at CL values below 0·5. Pennycuick cautions that low CD,pr values for the pigeon may be unreliable because of uncertainties in measuring parasite drag.

Fig. 6.

Polar curves for minimum drag for various birds and the wing of a red-shouldered hawk. Also shown are curves relating the lift coefficient to the total drag coefficient for the falcon and the vulture. These total drag coefficients are less than the profile drag coefficients alone of the hawk wing and the pigeon at some lift coefficients. All polar curves have been corrected to a Reynolds number of 105 and smoothed by fitting points to second-degree polynomial equations. See text for references.

Fig. 6.

Polar curves for minimum drag for various birds and the wing of a red-shouldered hawk. Also shown are curves relating the lift coefficient to the total drag coefficient for the falcon and the vulture. These total drag coefficients are less than the profile drag coefficients alone of the hawk wing and the pigeon at some lift coefficients. All polar curves have been corrected to a Reynolds number of 105 and smoothed by fitting points to second-degree polynomial equations. See text for references.

Using data from gliding vultures, obtained by tracking from sailplanes (Raspet, 1960; Pennycuick, 1971a), presumably with the vultures at minimum sinking speeds, I have constructed polar curves for minimum drag. The CD,pr values calculated from Raspet’s data for black vultures (see Appendix) are negative (Fig. 6), probably because the vulture and the sailplane were gliding in different air masses (Tucker & Parrott, 1970; Pennycuick, 1971a).

Pennycuick (1971a) assumed that CD,pr was independent of CL in African white-backed vultures and estimated its value to be 0·0073 at an Re value of 2·9×105. Correcting this value to Re = 105 yields 0 ·011, somewhat lower than the minimum Co,Pr value of 0 018 for the falcon and the vulture (Fig. 6).

The lift coefficients and total drag coefficients for dried bird wings mounted on a flight balance in a wind tunnel have been measured for several species (Withers 1981). A polar curve constructed from the data (see Appendix) for the one soaring bird he investigated, the red-shouldered hawk (Buteo lineatus), will be used here. Although the measurements were made at one speed with a constant wing span, the results are interesting when compared with those for the wings of living birds (Fig. 6).

The hawk wing had CD,pr values that are greater than the total drag coefficients of the intact falcon and the vulture. Clearly, the dried wing is not aerodynamically similar to living wings. Withers attributes the high drag of the dried wing to a low Re value (0·5 × 105, which is below the critical Re for smooth wings), surface roughness, fluttering of feathers and wing twist. In addition, the aerofoil sections of living bird wings during gliding can change markedly from those of anaesthetized or dead birds (Biesel, Butz & Nachtigall, 1985).

It is remarkable that the same polar curve for minimum drag describes the data for both the falcon and the vulture, as these birds are distinctly different in appearance and size. At maximum span, the falcon’s wings have pointed tips and an aspect ratio of 7·7 (calculated from data in Table 1). The vulture’s wings have square tips ending in separated primary feathers, and an aspect ratio of 5·6. The vulture weighs three times as much as the falcon.

The polar curve for these birds is also similar to curves for rigid wings with birdlike aerofoils, although it differs from the curve for the gliding pigeon. The pigeon is a flapping bird rather than a gliding bird. Perhaps measurements on other flapping birds will show that they too have different polar curves from gliding birds.

These observations suggest that polar curves for minimum drag may be similar to equation 35 in all gliding birds with highly cambered aerofoils - a hypothesis that will be interesting to test. In the absence of polar curves for minimum drag for gliding birds other than the falcon and the vulture, I shall use equation 35 to evaluate the maximum performance model for gliding birds in general.

Polar curves for constant speed

Both the falcon and the vulture glided in the wind tunnel over a range of glide angles at a given speed. Data for both birds fell on the same polar curve for a given speed, but each speed had a different polar curve (Fig. 7). CL increased and wing span decreased with increasing CD,pr at each speed.

Fig. 7.

Polar curves for constant speed for the falcon (open circles) and the vulture (filled circles). Points for both birds at a particular speed are connected by lines. (To correct for slight speed differences between the two birds, CL values for the falcon have been interpolated from a curve relating V and CL.) At a given speed, the wing span of each bird decreases as the profile drag coefficient increases. All polar curves are corrected to a Reynolds number of 105. Data from Tucker & Parrott (1970, fig. 5) and Parrott (1970, fig. 2).

Fig. 7.

Polar curves for constant speed for the falcon (open circles) and the vulture (filled circles). Points for both birds at a particular speed are connected by lines. (To correct for slight speed differences between the two birds, CL values for the falcon have been interpolated from a curve relating V and CL.) At a given speed, the wing span of each bird decreases as the profile drag coefficient increases. All polar curves are corrected to a Reynolds number of 105. Data from Tucker & Parrott (1970, fig. 5) and Parrott (1970, fig. 2).

Wing span and profile drag

How do birds change their wing spans to glide at various combinations of V and Vs in the performance area? Either an increase or a decrease in wing span could increase drag, depending on where the bird is flying in the performance area.

The falcon and the vulture when gliding at a given speed in the wind tunnel invariably decreased their wing spans to increase drag. This change increased induced drag, but paradoxically it also increased profile drag (as calculated in this study) in both birds (Fig. 8). One might expect that profile drag would decrease with wing span because of the reduction in wing area. The lift coefficient increased with the reduction in wing area but not enough to account for the increase in CD,pr according to the polar curve for minimum drag (Fig. 5).

Fig. 8.

Total drag and profile drag at different wing spans for the falcon gliding at constant speeds in a wind tunnel. At a given speed, the tunnel was tipped to various glide angles. The falcon responded by decreasing its wing span to increase drag and sinking speed. Data for low speeds (8·4 and 9·1 ms−1) are similar and are pooled. Data for high speeds (11·2, 12·3 and 14·1ms−1) are also similar and are pooled. Data from Tucker & Parrott (1970, fig. 5).

Fig. 8.

Total drag and profile drag at different wing spans for the falcon gliding at constant speeds in a wind tunnel. At a given speed, the tunnel was tipped to various glide angles. The falcon responded by decreasing its wing span to increase drag and sinking speed. Data for low speeds (8·4 and 9·1 ms−1) are similar and are pooled. Data for high speeds (11·2, 12·3 and 14·1ms−1) are also similar and are pooled. Data from Tucker & Parrott (1970, fig. 5).

Pennycuick (1971b) offered an explanation for the increased profile drag. African vultures appear to increase drag when landing by twisting their wings so that the lift increases at the outer parts of the wing and decreases at the inner parts of the wing. Consequently, the lift distribution along the wings becomes less elliptical, and the induced drag factor (k in equation 6) increases. The extra induced drag appears aS, profile drag, since k in this study has a constant value of 1·1.

Twisting the wings as described above may also increase the actual profile drag of the inner parts of the wings. These parts have aerofoil sections with highly cambered lower surfaces. Polar curves for such sections have minimum CD,pr values at CL values greater than 1 (Fig. 3), and CD,prr increases two-fold or more as CL drops towards 0. If the wings are twisted so that the lift, and therefore CL of the inner wing, drops towards 0, the profile drag of the inner wing will increase. In contrast, birds flying on the maximum performance curve adjust their wing spans and keep their CL values high, thereby avoiding the high-drag regions of the polar curves for their wings.

Man-made gliders with simple wings have lower and lower CL values as they fly faster and faster along their maximum performance curve. Consequently, designers of these aircraft select aerofoils that, like the Clark Y, have minimum CD,pr values when CL is near 0 (Fig. 4).

Maximum performance model

Given equation 35, very little additional information about a bird is needed to predict a curve for maximum performance - only weight, maximum wing span and a linear function relating wing area and wing span (Table 1). The predictions of the maximum performance model can be compared with measured performance curves of birds that presumably were gliding at the minimum sinking speed at each speed. The falcon and the vulture in the wind tunnel fit this criterion when the wind tunnel was tipped to the shallowest angles at which they would glide.

Table 1.

Characteristics of different species used with the maximum performance model

Characteristics of different species used with the maximum performance model
Characteristics of different species used with the maximum performance model

Sinking speeds were close to those predicted by the model, and wing spans showed less agreement with the predictions (Figs 9, 10). The observed sinking speeds would have been slightly lower if both birds had chosen different wing spans at some speeds. The falcon usually kept its wings too flexed, and the vulture sometimes kept its wings too flexed and at other times too spread out.

Fig. 9.

Predicted maximum performance curve and wing spans for maximum performance in the falcon. Points represent measurements made at minimum glide angles in the wind tunnel. Wing span is expressed as a fraction of maximum wing span.

Fig. 9.

Predicted maximum performance curve and wing spans for maximum performance in the falcon. Points represent measurements made at minimum glide angles in the wind tunnel. Wing span is expressed as a fraction of maximum wing span.

Fig. 10.

Predicted maximum performance curve and wing spans for maximum performance in the vulture. Points represent measurements made a minimum glide angles in the wind tunnel. Wing span is expressed as a fraction of maximum wing span.

Fig. 10.

Predicted maximum performance curve and wing spans for maximum performance in the vulture. Points represent measurements made a minimum glide angles in the wind tunnel. Wing span is expressed as a fraction of maximum wing span.

The predicted maximum performance curve for African white-backed vultures fits the data of Pennycuick ( 1971a), presumably obtained at minimum sinking speeds. Pennycuick used a constant span model to fit an empirical curve to the data (Fig. 11). However, the data are too variable to discriminate between the two curves.

Fig. 11.

Bottom: predicted maximum performance curve (M), empirically fitted performance curve (E) and measured data points for the African white-backed vulture. The empirical curve assumes a constant wing span and a constant profile drag coefficient. Measured data and empirical curve from Pennycuick (1971a). Top: predicted wingspans for maximum performance. Wing span is expressed as a fraction of maximum wing span.

Fig. 11.

Bottom: predicted maximum performance curve (M), empirically fitted performance curve (E) and measured data points for the African white-backed vulture. The empirical curve assumes a constant wing span and a constant profile drag coefficient. Measured data and empirical curve from Pennycuick (1971a). Top: predicted wingspans for maximum performance. Wing span is expressed as a fraction of maximum wing span.

Pennycuick (1960) estimated a performance curve for fulmars (Fulmarus glacialis) by tracking them while they soared near a cliff top. This curve is below the predicted maximum performance curve (Fig. 12), suggesting that the fulmars were not gliding at the minimum sinking speeds for their speeds.

Fig. 12.

Predicted maximum performance curves and wing spans in the fulmar and the wandering albatross. Points represent measurements on the fulmar (Pennycuick, 1960). Wing spans are expressed as fractions of maximum wing span.

Fig. 12.

Predicted maximum performance curves and wing spans in the fulmar and the wandering albatross. Points represent measurements on the fulmar (Pennycuick, 1960). Wing spans are expressed as fractions of maximum wing span.

The wandering albatross (Diomedea exulans) is one of the largest gliding birds; its pointed, high aspect ratio wings are quite different from the wings of the falcon and the vulture. The maximum performance model predicts poorer performance (Fig. 12; Table 2) for this bird than has been estimated by others. For example, Pennycuick (1982) estimated a maximum lift to drag ratio of 23·2 for the albatross, compared to 14·2 estimated by the maximum performance model. Pennycuick used a constant span model and a combined drag coefficient of 0·20 for profile and parasite drag to make his estimate.

Table 2.

Predictions of the maximum performance model

Predictions of the maximum performance model
Predictions of the maximum performance model

The combined drag coefficient is given by the sum CD,pr + Sfp/S, and a value for it of 0·20 represents much lower profile and parasite drag values than those used in the maximum performance model. Sfp in the maximum performance model is computed from equation 34, so Sfp/S for the albatross is 0-023. This value alone is higher than the combined drag coefficient of 0·20. If Sfp /S is halved to 0·012, then CD,pr becomes 0-008 for the constant span model, which is still less than the lowest value of CD,pr (0·011) used in the maximum performance model. Albatrosses certainly look streamlined in comparison with hawks and vultures and may indeed have better performance than the maximum performance model predicts. Measurements on a living albatross gliding in a wind tunnel would resolve the matter.

For all of the birds mentioned above, the model predicts that the wing span should be maximum for maximum performance at all speeds up to a critical speed. Above the critical speed, wing span should decrease as speed increases. All the birds achieved their minimum sinking speed and maximum lift to drag ratio (Table 2) at speeds below the critical speed, i.e. with their wings at maximum span.

Constant span model

A performance curve predicted by the constant span model has a different shape from the maximum performance curve. For example, the maximum performance model predicts that the falcon will have its maximum lift to drag ratio when gliding at a speed of 10 ms −1 (Table 2), with values for wing span and CD,pr of 1·01 m and 0·02, respectively. Using these values in the constant span model, the predicted performance curve is above the maximum performance curve at speeds less than 10ms−1 (Fig. 13). Sinking speeds are too low because profile drag is too low (Fig. 13). At speeds greater than 10ms−1, the sinking speeds are greater than those for maximum performance because wing span, and hence profile drag, is too high. Induced drag is low, but not low enough to compensate for the higher profile drag (Fig. 13).

Fig. 13.

Comparison of predictions for the falcon by the maximum performance model (M) and constant span model (C). The constant span model uses a wing span of 101 m and a CD,pr of 0·02. The bottom panel shows performance curves. Other panels show the profile drag coefficient (CD,pr), profile drag (Dpr) and induced drag (D1) at different speeds.

Fig. 13.

Comparison of predictions for the falcon by the maximum performance model (M) and constant span model (C). The constant span model uses a wing span of 101 m and a CD,pr of 0·02. The bottom panel shows performance curves. Other panels show the profile drag coefficient (CD,pr), profile drag (Dpr) and induced drag (D1) at different speeds.

Although the constant span model is not accurate for the entire maximum performance curve, it can predict a part of it. In the falcon, for example, the predicted wing span and CD,pr values for maximum performance are nearly constant at speeds near 10 ms−1 (Figs 9 and 13, respectively). At these speeds, the constant span model predicts nearly the same performance curve as the maximum performance model if one chooses the right wing span and CD,pr values (Fig. 13).

Raspet’s data for the black vulture

Raspet (1960) gives the following data for the black vulture: mass = 2·3 kg, weight = 22·7 N, b= l·44m, S = 0·365 m2. He also gives sinking speeds (Vs) for a range of speeds (V) for a soaring vulture.

The profile drag coefficient (CD,pr′) before correction to Re ×10−5 = 1 is given by :
formula
The drag terms on the right-hand side of this equation are given, respectively, by equations 1, 6 and 15. Sfp comes from equation 34.

CD,pr′ is corrected to CD,pr at Re × 10−5 = 1 by the method described below for the hawk wing. Reynolds number for Raspet’s vulture is 68 426c′V, where c′ is the wing chord, given by S/b.

Withers’s data for the red-shouldered hawk wing

Withers (1981) gives the following relationship between CL and CD at Re×10−5 = 0·5 and aspect ratio (A) = 3:
formula
The profile drag coefficients (CD,pr′) at this Re are given by:
formula
To correct CD,pr′ to the equivalent profile drag coefficient (CD,pr) at Re ×10−5 = 1, a term must be subtracted from all CD,pr′ values. This term is 0 ·203(1–1/F), where 0 ·203 is the value of CD,pr′ at CL = 0·7, and F is computed from equation 12 for Re ×10 −5 = 0 ·5.

Wing span and area for the white-backed vulture

The relationship between wing span and wing area for the African white-backed vulture was estimated from that for the black vulture, since both birds have similar wing shapes. The dimensionless relationship between wing span and wing area for the black vulture (Table 1) is:
formula
The maximum wing area (Smax) of the white-backed vulture is 0 ·69 m2, and bmax is 2 ·18 (Pennycuick, 1971a). Multiplying both sides of the equation by Smax and then substituting values for Smax and bmax yields the desired relationship:
formula

Wing span and area for gliding birds

The wings of some gliding birds are very different in appearance from those of the falcon and the vulture. For example, the wandering albatross has wings with pointed tips and an aspect ratio twice that of the falcon or the vulture. This section estimates the relationship between span and area for a simplified, hypothetical wing of any aspect ratio. The equations are then adjusted to estimate the relationship between wing span and area for actual gliding birds. For the purposes of this paper, an estimation will be made for the wandering albatross. First, consider wing span.

The hypothetical wing has three rigid, rectangular elements of lengths r1 r2 and r3 (Fig. 14), connected by joints at the shoulder, elbow and wrist. Two sets of elements and the body width (bB) make up the wing span. The maximum wing span is given by:

Fig. 14.

Hypothetical bird wing showing changes in wing area with flexing. See text for explanation.

Fig. 14.

Hypothetical bird wing showing changes in wing area with flexing. See text for explanation.

formula
Expressing bB as a proportion of bmax,
formula
equation 36 becomes, after substitution and rearrangement,
formula
If the angles at the wing joints remain equal as the wing flexes at angle a,
formula
or, after substitution,
formula
Now consider the area of the wing as it flexes. The wing loses area HIJK (Fig. 14) at the wrist joint as the feathers overlap, but it gains area DEFG at the elbow joint as overlapped feathers are exposed. The areas lost and gained at the wrist and elbow are equal, so the net loss in area with flexing is due to overlap of the body by the base of the wing (area ABC). The area lost (S ′) at the base of both wings is:
formula
where c is the wing chord at the wing base, given by:
formula
In a tapered wing, the chord at the base of the wing is larger than the value given by the above equation, and the changes in area at the elbow and wrist are no longer equal when the wing flexes. In addition, αmay be constrained anatomically from reaching 90° in an actual wing. For simplicity, I shall attribute the entire reduction in area of a flexed, tapered wing to overlap of the body by a wing of chord at its base. The ‘taper factor’, e, is 1 for a rectangular wing with equal flex angles at each joint. Actual wings will have taper factors greater than 1, to be determined empirically. Equation 41 for actual wings becomes:
formula
The area of both wings when flexed at angle α is
formula
The relationship between b and S that results when a range of values for α is substituted into equations 40 and 44 is approximately linear for .
The above equations describe the measured relationships between wing span and wing area for the falcon and the vulture when taper factors are 1·6 and 1·5, respectively. Since the wing of the wandering albatross appears to be nearly rectangular near the wing joints, I chose a taper factor of 1·2 for it. Using k7 = 0·093, Smax= 0·611 and bmax = 3·03 for the albatross (data from Pennycuick, 1982) yields the following relationship between wing span and area:
formula
Biesel
,
W.
,
Butz
,
H.
&
Nachtigall
,
W.
(
1985
).
Erste Messungen der Flügelgeometrie bei frei Gleitfliegenden Haustauben (Columba livia var
.
domestica) unter Benutzung neu ausgearbeiteter Verfahren der Windkanaltechnik und der Stereophotogrammetrie. In BIOXA Report 3, Bird Flight - Vogelflug
(ed.
W.
Nachtigall
), pp.
139
160
.
Stuttgart
:
Gustav Fischer
.
Blake
,
R. W.
(
1983
).
Mechanics of gliding in birds with special reference to the influence of ground effect
.
J. Biomech
.
16
,
649
654
.
Cone
,
C. D.
(
1962
).
The theory of induced lift and minimum induced drag of non-planar lifting systems
.
NASA Tech. Rept
R
139
.
Cone
,
C. D.
(
1964
).
A mathematical analysis of the dynamic soaring flight of the albatross with ecological interpretations
.
Virginia Institute of Marine Science, Special Scientific Report
50
,
1
104
.
Goldstein
,
S.
(
1965
).
Modern Developments in Fluid Dynamics
, vol.
II
.
New York
:
Dover Publications
.
Hankin
,
E. H.
(
1913
).
Animal Flight: A Record of Observation
.
London
:
Iliffe
.
Hoerner
,
S. F.
(
1965
).
Fluid-dynamic Drag
. (Published by S. F. Hoerner).
Jacobs
,
D.
(
1986
).
The LS-6: one pilot’s views
.
Soaring
50
(
1
),
30
33
.
Lighthill
,
J.
(
1975
).
Aerodynamic aspects of animal flight
.
In Swimming and Flying in Nature
, vol.
2
(ed.
T.
Wu
,
C. J.
Brokaw
&
C.
Brennen
), pp.
423
491
.
New York
:
Plenum Press
.
Mcgahan
,
J.
(
1973
).
Gliding flight of the Andean condor in nature
.
J, exp. Biol
.
58
,
225
237
.
Nachtigall
,
W.
(
1979
).
Der Taubenflügel in Gleitflugstellung: Geometrische Kenngrössen der Flugelprofile und Luftkrafterzeugung
.
J. Om
.
120
,
30
40
.
Nachtigall
,
W.
(ed.) (
1985
).
BIOXA Report 3, Bird Flight - Vogelflug
.
Stuttgart
:
Gustav Fischer
.
National Advisory Committee On Aeronautics
(
1920
).
Tech. Rept
93
,
257
336
.
Washington
:
US Government Printing Office
.
National Advisory Committee On Aeronautics
(
1921
).
Tech. Rept
124
,
421
474
.
Washington
:
US Government Printing Office
.
National Advisory Committee On
Aeronautics
(
1926
).
Tech. Rept
244
,
329
368
.
Washington
:
US Government Printing Office
.
National Advisory Committee On
Aeronautics
(
1928
).
Tech. Rept
286
,
139
183
.
Washington
:
US Government Printing Office
.
Newman
,
B. G.
(
1958
).
Soaring and gliding flight of the black vulture
.
J, exp. Biol
.
35
,
280
285
.
Parrott
,
G. C.
(
1970
).
Aerodvnamics of gliding flight of a black vulture Coragvps atratus
.
J. exp.Biol
.
53
,
363
374
.
Pennycuick
,
C. J.
(
1960
).
Gliding flight of the fulmar petrel
.
J. exp. Biol
.
37
,
330
338
.
Pennycuick
,
C. J.
(
1968
).
A wind-tunnel study of gliding flight in the pigeon Columba livia
.
J. exp. Biol
.
49
,
509
526
.
Pennycuick
,
C. J.
(
1971a
).
Gliding flight of the white-backed vulture Gvps africanos
.
J. exp. Biol
.
55
,
13
38
.
Pennycuick
,
C. J.
(
1971b
).
Control of gliding angle in RüppelUS griffon vulture Gvps riippellii
.
J. exp. Biol
.
55
,
39
46
.
Pennycuick
,
C. J.
(
1975
).
Mechanics of flight
.
In Avian Biology
, vol.
V
(ed.
D. S.
Earner
&
J. R.
King
), pp.
1
75
.
New York
:
Academic Press
.
Pennycuick
,
C. J.
(
1982
).
The flight of petrels and albatrosses (Procellariiformes), observed in South Georgia and its vicinity
.
Phil. Trans. R. Soc. Ser. B
300
,
75
106
.
Pennycuick
,
C. J.
&
Webbe
,
D.
(
1959
).
Observations on the fulmar in Spitsbergen
.
Br. Birds
37
,
321
332
.
Pope
,
A.
(
1951
).
Basic Wing and Airfoil Theory
.
New York
:
McGraw-Hill
.
Pope
,
A.
&
Harper
,
J. J.
(
1966
).
Lowspeed Wind Tunnel Testing
.
New York
:
Wilev & Sons
.
Prandtl
,
L.
&
Tietjens
,
O. G.
(
1957
).
Applied Hydro- and Aero-mechanics
.
New York
:
Dover Publications
.
Raspet
,
A.
(
1960
).
Biophysics of bird flight
.
Science
132
,
191
200
.
Reid
,
G. R.
(
1932
).
Applied Wing Theory
.
New York
:
McGraw-Hill
.
Schmitz
,
F. W.
(
1960
).
Aerodynamik des Flugmodells
.
Duisburg
:
Carl Lange Verlag
.
Sokolnikoff
,
I. S.
&
Sokolnikoff
,
E. S.
(
1941
).
Higher Mathematics for Engineers and Physicists
.
New York
:
McGraw-Hill
.
Spedding
,
G. R.
(
1987
).
The wake of the kestrel (Falco tinnunculus) in gliding flight
.
J, exp. Biol
.
127
,
45
57
.
Tucker
,
V. A.
(
1973
).
Bird metabolism during flight: evaluation of a theory
.
J. exp. Biol
.
58
,
689
709
.
Tucker
,
V. A.
&
Parrott
,
G. C.
(
1970
).
Aerodynamics of gliding flight in a falcon and other birds
.
J. exp. Biol
.
52
,
345
367
.
Van Dam
,
C. D.
(
1987
).
Efficiency characteristics of crescent-shaped wings and caudal fins
.
Nature, Lond
.
325
,
435
437
.
Von Mises
,
R.
(
1959
).
Theory of Flight
.
New York
:
Dover Publications
.
Welch
,
A.
,
Welch
,
L.
&
Irving
,
F. G.
(
1955
).
The Soaring Pilot
.
London
:
Murray
.
Withers
,
P. C.
(
1981
).
An aerodynamic analysis of bird wings as fixed airfoils
.
J. exp. Biol
.
90
,
143
162
.