J. exp. Biol. (1981), 90, 143-162 TJ.3With 8 figures

Printed in Great Britain

AN AERODYNAMIC ANALYSIS OF BIRD WINGSAS FIXED AEROFOILS

BY PHILIP C. WITHERS

Department of Biology, Portland State University, P.O. Box 751,Portland, Oregon 97207

(Received 5 March 1980)

SUMMARY

The aerodynamic properties of bird wings were examined at Reynoldsnumbers of 1-5 x io 4 and were correlated with morphological parameterssuch as apsect ratio, camber, nose radius and position of maximum thickness.The many qualitative differences between the aerodynamic properties of bird,insect and aeroplane wings are attributable mainly to their differing Reynoldsnumbers. Bird wings, which operate at lower Reynolds numbers than aero-foils, have high minimum drag coefficients (o>o3-o-i3), low maximum liftcoefficients (o-8-i -2) and low maximum lift/drag ratios (3-17). Bird and insectwings have low aerofoil efficiency factors (0-2-0-8) compared to conventionalaerofoils (0-9-0-95) because of their low Reynolds numbers and high profiledrag, rather than because of a reduced mechanical efficiency of animal wings.

For bird wings there is clearly a trade-off between lift and drag perform-ance. Bird wings with low drag generally had low maximum lift coefficientswhereas wings with high maximum lift coefficients had high drag coefficients.The pattern of air flow over bird wings, as indicated by pressure-distributiondata, is consistent with aerodynamic theory for aeroplane wings at lowReynolds numbers, and with the observed lift and drag coefficients.

INTRODUCTION

The theory of animal flight can be better understood than the theory of many otheraspects of animal energetics and locomotion, by making use of classical fluid dynamicand modern aerofoil theory. The fluid-dynamic basis for lift and drag applies equallyto man-made and animal aerofoils, and our present understanding of bird flight ismostly derived from aerofoil theory with only some recourse to empirical data forbirds (e.g. Pennycuick, 1969,1975; Tucker, 1968,1972, 1973; Weis-Fogh, 1973,1976;Greenewalt, 1975; Rayner, 1979). However, aerofoil data cannot be extrapolateddirectly to avian and insect flight, but must be scaled accordingto the Reynolds number(Re). Bird wings generally operate at Re's below a critical transition range (io4-!©6),where lift and drag coefficients alter markedly (Von Mises, 1959; Hoerner, 1965;Hoerner & Borst, 1975)- Empirical studies of bird wings (Nachtigall & Kempf, 1971;Reddig, 1978; Nachtigall, 1979) and insect wings dramatically illustrate the aero-dynamic consequences of a low Re (see also Nachtigall, 1977; Jensen, 1956; Vogel,1966).

144 P- C. WITHERS

The primary objectives of the present study are to investigate the basic aero-dynamic properties of isolated, fixed bird wings at their appropriate Me, to determinehow the aerodynamic properties are affected by wing morphology, and to comparetheir performance with that of conventional aerofoils (working at high Re) and insectwings (low Re). The basic aerodynamic measurements are: lift and drag coefficients atvarying angles of attack, minimum drag coefficient, maximum lift coefficient, maximumlift/drag ratio, and aerofoil efficiency factor. The role of wing-tip slots, a complicationdue to bird wings being composed of many laminated feathers, is considered elsewhere(P. C. Withers, in preparation) but basic aerodynamic data for a single primary featherfrom a black vulture are presented here as a comparison for the data concerningcomplete wings.

METHODS

Bird icings. Bird wings, dried in an appropriate gliding position, were used. Wingswere obtained from frozen birds or were borrowed from the ornithology collection ofthe Duke University Zoology Department. A brass rod was glued inside the humeruswith epoxy resin, and the rod was mounted directly on a force transducer assembly.Separation of the primary feathers did not occur for any species at positive angles ofattack. Only the hawk wing had wing-tip slots.

Wings from the following species were studied: European starling (Sturnidae;Stumus vulgaris); common nighthawk (Caprimulgidae; Chordeiles minor); chimneyswift (Apodidae; Chaetura pelagica); Leach's petrel (Hydrobatidae; Oceanodromaleucorhoa); bobwhite quail (Phasianidae; Colinus virginumus); wood duck (Anatidae;Aixsponsa); American Woodcock (Scolopidae; Philohela minor); red-shouldered hawk(Accipitridae; ButeO lineatus). The tip of a primary feather from a black vulture(Coragyps atratus) was also studied. Body mass and wing loading of these species are:starling, 0-0850 kg, 4-3 kg m"2; nighthawk, 0-064 kg, i"9kgrrr2; chimney swift,0-017 kg, 1-7 kg m~2; petrel, 0-027 kg, I - I kg m~2"> quail, 0-150 kg, 9-2 kg m~2; woodduck, 0-590 kg m~2; woodcock, 0-200 kg, 5-6kgm~~2; hawk, 0-550 kg, 3-3 kg m~2

(Poole, 1938; Hartman, 1961).The length of the wing, from tip to point of attachment at the humerus (i.e. wing

semi-span), was measured to the nearest millimetre and the projected wing area wascalculated by weighing tracings of the wing outline. Average wing cord (c) and aspectratio (AR) were then calculated as:

c = wing area/wing semi-span, AR = wing area/(wing semi-span)2.

Chord-wise cross-sections of the large wings were determined by cutting and fittingpaper templates to the upper and lower wing surfaces; for the small wings and vulturefeather a three-dimensional micromanipulator was employed. These local cross-sections were used to calculate the average value for the wing of section chord tosection perimeter. The surface area of the wing ('wetted area') was then calculatedfrom the projected area and average ratio of wing chord/perimeter.

Wind tunnel. A Kenney Engineering Corporation model 1057 closed-circuit windtunnel (single return) was used. The working section was 0-26 x 0-30 m. The turbu-lence factor foi the wind tunnel was determined by measuring the drag coefficient of adsphere (o-i m diam.) at different Re (Pope & Harper, 1966). The drag coefficient of tht

Bird wings as fixed aerofoils 145

sphere declined markedly at Re = i-6 x io6, hence the turbulence factor is quite low,at about 2-4.

Wind velocity in the tunnel was measured by using a pilot tube (United sensorPAC-12-ki) and a Barocell electric manometer (sensitivity = io~~* mmHg). Thestatic and dynamic pressure heads measured with the pitot tube were converted to airvelocity as described by Pope and Harper (1966).

Lift I drag measurement. Lift and drag forces were measured by using two rectilinearstrain-gauge assemblies, with their movement axes mounted at 900. Each assemblywas composed of four strain-gauges for thermal stability. The strain-gauge assemblieswere mounted under the floor of the tunnel, and could be rotated through an angle ofabout ioo° with a motorized, rotating support. The lift and drag forces were calculatedtrigonometrically from the measured forces and the angle of rotation, since the wholestrain-gauge assembly rotated. The gauges were operated with a Beckman RSDynograph (type 462) with type 9802 strain-gauge couplers and 416 B pre-amplifiers.The output signals were time-averaged over about 5 s using a Heath UniversalDigital Instrument EU-805. The strain-gauge assembly was calibrated daily by usinga o-o-i N Pesola scale. Lift and drag measurements were repeatable to ±0-002 N.

The bird wings were mounted vertically in the tunnel because the strain-gaugeassembly was located under the floor of the tunnel. This attitude did not affect theaerodynamic properties of the wing to a significant extent, but meant that the massof the wing was acting at 900 to its normal direction relative to the wing. However,the weight of the wing is generally small compared to the aerodynamic forces and canbe ignored. The wings were not mounted with the root flush against the tunnel wall(as model aeroplane wings are) because of technical difficulties. The wing root henceshed a vortex wake, and the pattern of air flow around the wing root was atypical.This might increase CD\a, but would have little effect on minimum CD, or C c values.The most significant effect would be to underestimate aerofoil efficiency factor. How-ever, the data obtained with this mounting system do not indicate any significanteffects (see below). Standard wall interference factors for the wind tunnel werecalculated for a close-circuit, rectangular tunnel (Pope & Harper, 1966), but nocorrections were required except for the induced drag of the largest wing (red-shouldered hawk) which was adjusted for the proximity of the ground-plane.

The local angle of attack of the wing was measured at mid-span with a protractormounted on the top of the wind tunnel. Angle of attack varies along the wing, however,because of natural twist and aerodynamically induced bending.

Lift and drag coefficients (CL, CD) were calculated from the resolved forces using thestandard equations:

L = yv*scL, D = yv*scD,where p is the air density (1-18 kg m~3), V is the wind velocity (m s"1), 5 is the pro-jected area (m2), L and D are the lift and drag force (Newtons) and CL and CD are thedimensionless lift and drag coefficients. The term £pF2 (the dynamic pressure head)was measured with the pitot tube and electric manometer.

Pressure distribution. The pressure distribution over the leading portion of the night-fcawk wing was measured at mid-span. Small (27-gauge) hypodermic needle tips wereforced through the rachis (central support of the fethers) in appropriate positions,

146 P. C. WITHERS

Table 1. Morphometric parameters for bird wings and vulture primary feather, andmorphometrics of wing cross-section at approximately mid-span

(Thickness ratio = maximum thickness/chord; camber ratio = maximum deviation of centre of wingfrom line connecting leading and trailing edges; nose radius ratio = approximate radius of wing atleading edge/chord; twist is base to tip twist angle.)

SwiftPetrelWoodcockWood duckQuailStarlingNighthawkHawkVulture primary

Length(m)

0-1410-2120-1710-257°'i450-1640-260O-3940-180

Projectedarea(m«)

0-0050-0116001370-02110-0109000880-01650-05220-00410

Wettedarea(m«)

0-01040-0240-0290-044002300350-0350-I I2

0-0086

AR

3 94'11 9

3-ii -8

3-o4-i3-0

7 9

Thicknessratio

0-0540-0480053o-ioo0-0360-0360-06200680-063

Camberratio

00540-0650-0810-069.O-IOI

O - I I 2

0-0690-0990039

Nose Radiusratio

0-012o-on0-019O-O2O

0-019003200350-032~ o

Twist(degrees)

597

11

513

51 0

15

with the blunt end of the needle flush with the upper (or lower) surface of the rachis.The bevelled end of the needle was pushed into thin-bore Polyethylene tubing (PE010 x 030), which was led under the secondary feathers to the base of the wing and outof the wind tunnel to the electric manometer. The needle was bent at the rachis toallow it to lie flush under the secondary feathers. This arrangement should result inminimal aerodynamic interference with the wing. The pressure at each orifice was thenmeasured, and the values expressed in the standard manner:

where p' is the measured static pressure head at the orifice, p is the ambient dynamicpressure head and q is the ambient static pressure head (Pope & Harper, 1966).Classic lifting-line theory was used to evaluate many of the aerodynamic properties ofthe bird wing, rather than lifting-surface theory. All units- are S.I.

RESULTS

Morphometrics. The ranges of various morphological parameters for the wingsstudied are shown in Table 1. Most of the wings were approximately elliptical inplanform, although the hawk wing was the most rectangular and the wood duck wingwas the most tapered. The surface areas ('wetted' areas) of the wings were about2-04-2-14 times greater than the projected areas.

Lift /Drag. The absolute values of lift and drag (in Newtons) were dependent uponwind velocity although the general form of the lift-drag polar and the maximumlift/drag ratio were independent of velocity. All lift and drag data presented here areconverted to the lift and drag coefficients (CL = zL/(pV2S)\ CD = zD/^V^S)) tocorrect for variation in dynamic pressure head (^pV2) and wing surface area (S). Thecorrection for dynamic pressure resulted in values of CL and CD which were independent of V over the narrow range of V used in this study. The aerodynamic forces al

Ta

ble

2.

Lin

ear

and q

uadra

tic

regre

ssio

n a

na

lyse

s fo

r th

e a

ero

dyn

am

ic p

rop

erti

es o

f b

ird

win

gs

and t

he

vult

ure

pri

mary

feath

er

tip

, a

s a

fun

ctio

n

of

the

an

gle

at

att

ack

(a

)

a =

co

nst

ant

(a0);

6 =

fi

rst

ord

er t

erm

(a

1);

c =

se

con

d o

rder

ter

m (

a1).

Val

ues

are

x ±

s.e.

, w

ith

th

e t-

valu

e fo

r th

e d

iffe

ren

ce f

rom

oin

par

enth

eses

. «i

is

calc

ula

ted

as

I/TT

AR

$, w

her

e 1

is t

he

slop

e of

th

e C

L-C

D r

egre

ssio

n,

and

e,

is c

alcu

late

d a

s 4

CD

{CJC

o)\»

xJ'n

AR

(se

e te

xt).

Sp

ecie

s

Sw

ift

(n =

46

)

Pet

rel

(» =

45

)

Woo

dco

ck(n

=

8o)

Woo

d d

uck

(n =

40

)

Qu

ail

(n =

60

)

Sta

rlin

g(n

=

90

)

Nig

hth

awk

(n =

13

4)

Haw

k (

n =

00

)

Vul

ture

Pri

mar

y(n

= a

8)

L =

a +

ba

(a)

0-02

7±0-

025

(I-I

)(6

) 0-

099±

0-00

6 ('

6-7)

to

—(a

) 0

02

7±

00

14

(1-9

)(6

) 0-

084

±0

-00

5 U

T0 )

to

-(a

) 0-

049

±0

-03

3 (»

•«)

(b)

0-05

3 ±

0-0

03

(16-

8)to

-

(a)

— 0

-061

±0

-02

6 ( —

o-6

)(6

) 0-

061

±0

-00

4 (1

5-0)

to

—(a

) o-

ooi

±0

-01

5 (O

-I)

(6)

00

50

±0

00

1 (3

57

)to

-

(a)

0-0

78

±0

01

7 (4

-5)

(6)

o-o

6o

± 0

-003

(*4

'o)

to

—(a

) o

-o4

8±

o-o

io (

5-1)

(6)

0-0

78

±0

-00

2 (4

5-3)

to

-(a

) o-

o8o±

o-oi

2 (6

-6)

(6)

0-05

5 ±

00

02

(25

3)

to

-(a

) 0-

051

±0

-02

6 (2

-0)

(b)

0-0

75

±0

00

3 (2

40

)to

-

CD

= a

+ b

a +

ca*

0-05

95 ±

0-0

05

2 (1

1-4)

-00

03

7±

00

00

7 (-

5- a

)o

-oo

o6

±o

-oo

oi

(16-

6)

00

89

5 ±

00

05

8 (1

53

)0-

0049

± 0

-000

8 (6

-2)

o-o

oio

± o

-ooo

i (7

-3)

o-o

95

0±

o-o

o6

i (1

5-6)

0-00

26 ±

o-o

oio

(2-

5)0-

0004

± o

-ooo

i (4

'°)

0-09

78 ±

0-0

047

(2O"

9)—

00

02

1 ±

0-0

00

9 ( —

a- 4)0-

0009

±o

-oo

oi

(II-O

)

0-07

30 ±

0-0

04

6 (1

5-8)

— 0

-000

7 ±

0-00

03 (

—2 '5

)0

-00

04

±0

00

02

(16-

6)

0-1

43

71

0-0

04

4(3

2-4

)o-

oooi

±0

-00

05

(°'3

)o

-oo

o7

±o

-oo

oi

(12-

9)

O-O

622

± 0

-002

0 (3

I-8)

— 0

-004

3 ±

0-0

006

( —7- a)

O-O

OIO

±OO

OO

I (1

6-9

)

0-0

94

4±

00

02

8 (3

3-2)

o-o

oi8

±0

-00

05

(3'7

)o

-oo

o6

± o

-ooo

i (2

4-7)

0-05

18 ±

o-o

ioo

(5-2

)O

-OO

O2±

O-O

OO

I (O

-2)

o-o

oo

6±

o-o

oo

i (7

-4)

CD

=

O +

IIC

L +

C

CL

*

00

52

±0

-00

4 (1

47

)0

-18

2±

0-0

14

(—ia

-7)

0-2

80

±0

-02

4 (i

i'S

)

0-0

85

10

-05

5 (1

75

)0

-02

31

0-0

20

(— 1

-a)

0-38

4 ±

0-0

30

(9'6

)

0-08

4 ±

0-0

04

(ao-

6)0-

049

±0

-02

3 ( —

a-i)

0-36

9 ±

0-0

30

(12-

2)

0-10

5 ±

0-0

04

(24

'1)

Oii

4±

o-O

25

(-1

13

)o

-35

5io

-O4

i (8

-7)

0-09

6 ±

0-0

05

(20-

8)—

0-2

82 ±

0-0

25

( — 1

1 "3

)0-

482

±0

-03

5 (i

3-7

)

-oi5

7±

oo

i5

(-10

-5)

o-4o

o±o-

o25

(15-

8)00

63 ±

00

02

(33

5)

— O

-IO

I ±

0-0

09

(— 1

1 "7

)O

-2II

±O-O

II (

19-3

)0-

085

±0-0

04 (a

a-3)

-o-i

53

±o

-oi7

(-8

-8)

0-55

5100

33(2

5-7)

0-05

0 ±0

-008

(6-

4)o-

i73±

o-oi

8 ( —

9-7

)0-

284±

0-02

4 (n

'7)

«i

42

02

4

0-51

0-65

08

3

0-46

0-92

0-40

o-57

«• 0-56

04

3

o-66

05

5

1 "4

05

7

i-3

04

5

08

7

Bird to a

148 P. C. WITHERS

Table 3. Profile drag coefficient, maximum lift coefficient, maximum lift/drag ratio,of lift cwve, and i/nARe for bird wings and vulture primary feather (tvith angle ofattack at mid-chord in parentheses for data from present study)

Species CD, 1 (.CL/CD)M dd/da I/TT ARJ Reference

SwiftPetrelWoodcockWood duckQuailStarlingNighthawkHawkVulture

primaryThrushSparrowDuckSnipePigeon models

0-030 (+1 )0-070 (0°)0-082 ( + 2°)0-096 ( + 1°)0-055 ( + 3°)0-125 (°°)0051 ( + 30)O-074 ( + 2°)0-024 (°°)

0-050-16O - I I

O-II

0-06-0-13

o-8o ( + 8°)o-88(+i3°)o-9o(+i5°)0-90 ( + 20°)I I 0 ( + 25°)i-oo ( + 150)1-15 (+15°)i-o ( + 250)I-I5 (+12°)

o-8I-O-I-I0 9i-oO-8-I-2

17 ( + 5°)4-o( + 8°)3-5 ( + 8°)3-8 ( + 8°)60 ( + 8°)3-3 ( + 7°)9-o ( + 6°)3-8 ( + 6°)17 ( + 5°)

—32 - 8

o-i0 0 80-050-060-050-06o-o80-060-08

0-030-05o-o60-060-12-0-27

0-02 '0-320'330-16O-2I0-230 0 80-27007 /

Present study

°i3g 1 Nachtigall & Ken

0-34 Jo-330-1-0-7

Reddig (1978)Nachtigall (1979]

high wind speeds did not bend the wing sufficiently to alter the lift or drag coefficients.However, CL and CD would vary markedly over wider ranges of V (see Discussion).

The CL and CD of the various bird wings and the vulture primary feather alteredmarkedly at differing angles of attack (a) at mid-span; the CD-CL curves, CL, CD andCfJCD data are summarized in Figs. 1-3. The value of a was arbitrarily measuredonly at mid-span, although the local a varied along the wing span because of inherentwing twist (see Table 1) and due to aerodynamic forces deforming (including twisting)the wing.

Aerodynamic parameters for the bird wings and vulture feathers were generallysimilar (Figs. 1-3; Tables 2, 3) although there were many qualitative differences. Therelationship between CD and CL was generally U-shaped except at high a when CL

levelled off or even declined. Quadratic, best-fit equations were fitted to the CD-CL

data (excluding data at high a) since the relation is, in theory, parabolic. The curveswere often quadratic rather than parabolic (i.e. parabolic but shifted from the originsince the minimum CD was not at zero lift; Figs. 1-3, Table 2). The CD varied with ain a parabolic fashion, and the quadratic best-fit equations are presented in Table 2.The CL was usually linear at intermediate a and relatively constant at lower and highera. The best-fit linear equations for CL at intermediate a are summarized in Table 2.

The profile drag coefficient (ideally determined as the minimum drag which occursat zero lift) was from 0-024 t 0 °'I2S but generally did not occur at the zero lift angle.The maximum CL values varied from about 0-7 to 1-2 at a of 15-200 (Table 3). TheCL/CD ratio was often quite variable, reflecting the errors involved in lift and dragmeasurement, but CL/CD always increased from negative values at low a to maximalvalues at a of 5-80, then declined markedly at higher a. This is the expected form of therelation, but there is no theoretical equation so best-fit curves were obtained by using10th degree polynomial equations to indicate the approximate relationship. Themaximum values for CL/CD (= L/D) varied from 3-3 to 17 for the different wingsand vulture feather.

The aerofoil efficiency factor («) was calculated in two ways. Firstly, e was calculated

Co

0-5-

n-4-

0-3-

0-2-

0 1 -

(a) Starling •

• • •

*• • • "• * J• *

•• •

.t" . . . -r'• M • •> • •

• • M • • •• * • • • •

-0-3 0 0 0-3 0-6 0-9

1-6

1-2 -

CD

0 - 8 -

0-4 -

(b) Hawk

J-"i r-0-4 0

1—0-4

1—0-8 1-2

CD

0-75 -

0-60 -

0-45 -

0-30 -

015 -

0-00 -

(c)QuaU

*

*•*

i—

•

;

•

• ••* •

—1 1 1

Co

0-30 •

0-25 •

0-20 •

015 •

0 1 0 •

CL

• 1 -2

0

Y- 00

- lo>>

t# 7 •'-r '•/•'

vf 'J -»~«J ° g °• • OO

- 3

-14 - 7 0 7 14

- 0

- - 1

CL/CD

-4-5

- 3 0

-1-5

- -1-5

15

C6

0-90k-

0-72

0-544-

0-36 •

0 1 8 -

0-00-

CL

• 1 -2

• 0-8

• 0-40

o •o V• 0 m °

-*r- +•• -0-4

5

>

i' V

•]/

i o 9*oo i%6o

^ ^

^ 4

•'hit

9

O

K. Ci

/ ^ *

xx^

- 7-2

- 5-4

- 3-6

- 1-8

- 0 0

- - 1 -

-0-30 000 0-30 0-60 0-90 1-20 -22-5

CL

0-0 22-5 37-5

Fig. 1. For starling, hawk and quail wings. Left: lift coefficient (Cx,) as a function of dragcoefficient (CD). Right: lift coefficient (Ci), drag coefficient (C D), and lift/drag ratio (CL/CD)at differing angles of attack (a).

0-75 -

0-60 -

CD

0-45 -

0-30 -

015 -

(a) Wood duck

CD

0-75 -

0-60 •

0-45 -

0-30 -

0-15 -

CL

• 1-2

• 0 - 9

• 0 - 6

• 0 - 3

•

•

• / N ^1 ' t.V*

-!- t.'hf'f

* * -r

c,+ > { /

V /\ °/cjcd

-0-25 0O0 0-25 0-50 0-75 1-00

CLICD

- 4 - 5

30

- - 1 - 5

- 0 0

-1-5

CLICD

0-90 -

0-72 -

0-54 -

0-36 -

018 -

0-00 i

•

(b) Woodcock "

••

•• •••

• •• •• • • •* • •

•• • • • •••• . .itr. ••

•

0-3 0-0 0-3 0-6 0-9 1-2

CL

-10 0 10 20 30

-10 0 10 20 30

CD

0-40 -

0-32 -

0-24 -

016 -

008 -

(c) Nighthawk

•• •• •• •• •

• •• • •

K . *•* :•

• • • •

0-40 - _

0-32 -

0-24 -

016 -

008 --0-4

I I-0-35 0 0-35 0-70 105

**L

- 1 - 2

* •

- 0-8 ''£

• ••

J j /

/ • cjcd

' £ c"

- 9

- 6

- - - 3

- 0

- - 3

I I I 1 r-6 0 6 12 18

aFig. 2. For woodcock, wood duck and nighthawk wings. Left: lift coefficient (Cj,) as a functionof drag coefficient (CD)- Right: lift coefficient (Ct), drag coefficient (CD), and lift/drag ratio(CL/CD) at differing angles of attack (a).

0-8 -

0-6 -

0-4-

0 -2 -

-0-25 0 0-25 0-50 0-75

0-8 -

0-6 -

0-4 -

0-2 -

I I I I-0-25 0 0-25 0-50 0-75

0 - 5 -

0-4 -

0-3 -

0-2 -

01 -

0 -

(c) Vulture feather

•

•>

•• •• •

• •

• • • •

\ i r-0-35 0 0-35 0-70 105

1-25 -

1-00 -

0-75 -

0-50 -

0 - 2 5 - -

0 -

30 45

CD

0 - 8 -

0-6 -

0-4 -

0 - 2 -

0 -

a

CL

-1-2

- 0-8 *P*\IF

-0-4 U

-o J •;-/'

0-4

' cjcd

CJCD

- 5-4

-3-6

-1-8

- 0

- -1-8

i i i r-12 0 12 24 36 48

a

CD0-6-

04 -

0-2 -

0 -

CL

- 10

- 0-5

-o\

\

- - 1 0 •

V +

'A:/\cjcd

/ XJi \V'"' JCd

CLICD

- 18

- 12

- 6

- 0

- -6

i i i r-24 -16 -8 0

a8 16

Fig. 3. For swift and petrel wings, and the tip of the vulture primary feather. Left: lift co-efficient (C[) as a function of drag coefficient (CD). Right: lift coefficient (CL), drag coefficient(CD), and lift/drag ratio (CL/CD) at differing angles of attack (a).

152- 1 - 6 -

- 1 - 2 -

-0 -8 -

- 0 4 -

P. C. WITHERS

- 4 0

01 0-2I

0-3 04 OS x/c- 9 O

0-8 -

04 -

Bottom

x/c

- 1 O

-04 -

-0-8 -

Fig. 4. Pressure distribution for the leading one half of the nightjar wing at mid-span, withdiffering angles of attack (<x). x/c indicates position of pressure measurement relative to thewing chord (c). P is the pressure at the flush orifice corrected for ambient static and dynamicpressure heads (see text).

Bird wings as fixed aerofoils

Table 4. Lift and pressure drag coefficients calculated for the nighthawk icing from thepressure distributions at differing angles of attack (a), compared to values measured for thewing using force transducers assuming skin friction drag coefficient was 0-02, and calcu-lating induced drag as measured CL/TTKK (0-9)

a

- 8- 4+ a+ 7+ 20

CalculatedcL

— 0-46-013

o-44076066

Measured

cL— 0-40— 0-18

0-29o-6oI-I

Calculated

0-060-040-030-060-18

MeasuredCi>,m

o-o80-060-040-040-28

from a regression of CD and CL since the theoretical slope of this relationship isI/(TTAR<?) and the intercept is the profile drag coefficient (Greenewalt, 1975). Thevalues of e were also calculated from the (CL/CD)max value since e = \CD, pro

(Ci/CD)2max/(7r AR) (Greenewalt, 1975). The values of e were in reasonable agree-

ment using the two different methods, and generally ranged from 0-3 to o-8 (ignoringthe unreasonable values > 1).

Pressure distribution. The pressure distribution (expressed as P, corrected for dyna-mic and static pressure head) over the mid-span of the nightjar wing altered markedlyat differing a (Fig. 4). The values for CL and CD can be calculated from these distri-bution data (Pope & Harper, 1966). Only the pressure for the leading one-half of thewing section was measured, but this is the most important in terms of lift and drag, soapproximate calculated values for CL and CD are presented in Table 4. Theseestimated CL and CD values are in general agreement with CL and CD values measuredfor the whole wing.

DISCUSSION

The lift and drag characteristics of bird wings and the vulture feather resemble, ingeneral form, those of conventional (man-made) aerofoils (e.g. Von Mises, 1959;Hoerner, 1965; Pope & Harper, 1966; Hoerner & Borst, 1975), other bird wings(Nachtigall & Kempf, 1971; Reddig, 1978; Nachtigall, 1979) and insect wings (Jensen,1956; Vogel, 1966; Nachtigall, 1976). However, there are significant quantitativedifferences between the aerodynamic performances of conventional aerofoils, birdwings, and insect wings. The following discussion relates how these differences areattributable to variation in wing morphometrics and air-flow regime (i.e. Re).

Aerodynamic drag. The minimum drag coefficients measured here for the bird wingswere from 0-03 to 0-13, with the primary feather tip of a black vulture having a dragcoefficient of 0-024. These drag coefficients are similar to those measured for otherbird wings, or parts of bird wings, or models of bird wings. Minimum drag coeffi-cients for insect wings are similar or higher: locust forewing 0-024; locust hind wing,0-06 (Jensen, 1956); Drosophila wing 0-3 (Vogel, 1966). These values for bird andinsect wings are considerably higher than those for conventional aerofoils, whichtypically have minimum drag coefficients of less than o-oi. As will be shown below,

se marked variations in minimum drag coefficient for insect, bird and man-madeigs do not necessarily reflect different wing morphology or function but simply the

Re at which they operate.

154 P. C. WITHERS

l - l

0 1 -

0 0 1 -

0001

102 104I

10*1

10'Re

Fig. 5. Drag coefficient (CD) for bird wings and vulture primary feather (solid circles), insectwings (open circles, below Re = io') and other aerofoils (open circles) expressed relative to'wetted area' (see text). Solid lines are for calculated skin friction drag coefficients, with tran-sition from laminar to turbulent shown by broken lines. Vulture primary feather is solid circleclosest to line for laminar skin friction drag coefficient. Insect wing data are from Jensen (1956)and Vogel (1966); other aerofoil data from Tucker & Parrott (1970).

There are three basic components of aerodynamic drag: skin friction drag (dragcoefficient = CDJ), pressure drag (CDiVTe) and induced drag (CDiln). The first twodrag terms are generally combined into a single term, profile drag (CDt pro).

Skin friction depends upon the Re, and whether flow is laminar or turbulent. Forlaminar flow,

CDif= 1-33/ yjRe (Blasius' equation),

whereas the relationship for turbulent flow is

log (Re.CDtf) = 0-24/ *JCDtf (Schoenherr's equation)

(see Hoerner, 1965). Transition from laminar to turbulent flow usually occursspontaneously at Re of IO ' - IO 6 but transition can be forced to occur at lower Re (andalso occurs immediately after the thickest section of an aerofoil at low Re). LaminarCDtf ~ turbulent CDf in the range of Re applicable to bird wings (Fig. 5). Theminimum values for CD of bird wings are generally greater than the estimated skinfriction drag coefficient, although the drag of the vulture primary feather is onlyslightly greater than the predicted skin friction drag (Fig. 5). (Note that the CD valuesin Fig. 5 are expressed relative to 'wetted' area, rather than the projected area becauseit is 'wetted' area, not projected area, that determines CDtf.) The vulture feather, notunexpectedly, had the drag coefficient most similar to its skin friction drag because it isthe 'cleanest' aerodynamic shape. The CD values of insect wings are also only a littlegreater than the predicted skin friction CD (Fig. 5). Aeroplane wings, which opera t^much higher Re, have correspondingly lower CDif (Fig. 5).

Bird zvings as fixed aerofoils

01 -

CD

0-01 -

0-001

104 10s 10*I

107 10*

Re

Fig. 6. Drag coefficient (Cz>) for bird wings and vulture primary feather tip at differingReynold's numbers, expressed relative to projected area, compared with values for conventionalaerofoils of thickness ratios from 006 to 0-5, and compared to calculated skin friction dragcoefficients for laminar and turbulent flow. Aerofoil data from Hoerner (1965).

0-4 - ,

0-3 -

CD

Sub

Aerofoil

0-2 -

01 - Super

0 1 0-2 0-3 0-4 0-5

Thickness, Camber

Fig. 7. Total drag coefficient (Cp) for bird wings as a function of camber, and total drag co-efficients for conventional aerofoils at subcritical and supercritical Re as a function of thickness.Magnitude of skin friction drag is indicated by stippled area. Aerofoil data are from Hoerner(196s).

The pressure drag coefficient is determined by the magnitude of the turbulent wakeleft behind an object, and hence is lowest for streamlined bodies with delayed separ-tion. The CDi pre is experimentally determined as CDiPT0 — CDif. Pressure dragbecomes relatively more important than skin friction to total drag at high Re since skinfriction is reduced. Separation of air flow is delayed at high Re by boundary layerturbulence, and this is reflected by a dramatic decrease in CDt pre (or CD pro) at Reabove a critical level, about io8 (Fig. 6). Conventional aerofoils, which usually operateabove this critical Re, have a much lower drag than bird and insect wings. However,

wings have higher drag coefficients (Fig. 7) than aerofoils at equivalent Re (belowcritical range), perhaps reflecting the surface roughness of bird wings and the

tendency of individual feathers to flutter and increase drag, or wing twist (see below).

156 P. C. WITHERS

Table 5. Multiple linear regression analysis for aerodynamic parameters as a function ofmorphometric parameters, showing significantly correlated parameters {and r-values) andpresenting the multiple linear regression equation (n = 9 for all analyses)

Aerodynamic Multiple linear regressionparameter Correlated morphometric parameters equations

CD, pro A R = aspect ratio (r = -0-54) CD pro = —00074 + 0-969 (j/c)f/c = camber (r = 0-71)r/c = nose radius (r = 0-57)

CL, mi! */ = position max. camber (r = 0-67) Ct , M I = 1-24 — 0-55 (xf)

L/D nun AR = aspect ratio (r = 0-72) L/D mMI = 21-4 — 181 (f/c)f/c = camber (r = 0-74)r/c = nose radius (r = 0-57)

dCL,/dct f/c = camber (r = 0-70) dCh/da. = 0-105-0-48 (f/c)AR = aspect ratio (r = 052)

dCD/dCL' AR = aspect ratio (r = 0-62) dCD/dCLx = 0-326-0032 (AR)

The profile drag coefficient, although markedly influenced by Re, is also dependentupon the thickness of the wing, particularly at low Re when flow separation occursimmediately after the thickest or most cambered portion of the wing. This is clearlyshown by the relationship between COj pro and thickness (or camber) for bird wingsand aerofoils at subcritical and supercritical Re (Fig. 7). The CDt pro of different birdwings was correlated with aspect ratio, camber and nose-radius (Table 5). Camberwould be expected to have the same effect as thickness in determining the point offlow separation, the magnitude of the turbulent wake, and the drag coefficient. Indeed,camber was the morphological parameter which best predicted CDt pro using multiplelinear regression analysis (Table 5).

Induced drag arises from the deflection of air by a wing; CDt ln is related to the liftcoefficient and aspect ratio as:

Induced drag is dependent upon aspect ratio since the angle through which air mustbe deflected to produce a given change in momentum (hence lift) increases as aspectratio decreases. Induced drag is also dependent upon morphological parameters suchas wing planform (elliptical wings have the lowest CDi ln) and wing twist. Wing plan-form only slightly alters CDt ln and can be ignored for bird wings, which tend to beelliptical in planform. Wing twist, however, can markedly increase CDt ln by simul-taneously producing positive and negative lift (which algebraically cancel out so CL islow) but drag is high since the induced drags from positive and negative lift addtogether, rather than cancel. Such an effect of wing twist on CDi In is seen in the aero-dynamic data not as induced drag, but as an apparently high profile drag coefficient.The high twist of bird wings thus contributes to their high CDi pro and probablyexplains why the minimum drag coefficients were often not observed at the angle ofzero lift.

Induced drag of wings is usually greater than calculated as Cjf/nAR even fqjelliptical, untwisted wings because a wing is not a perfectly efficient device. An aer^

Bird ivings as fixed aerofoils 157

roil efficiency factor (e) is therefore introduced into the equation relating CDi ln and

^ D, In "— *- £ / / /x i lxc.

Values for e calculated for bird wings generally were from 0-3 to o-8, with valuesgreater than 1 discounted as experimental error or improper calculation. For conven-tional aerofoils, e is typically 0-9-0-95. The interpretation of e for bird wings is some-what different from that for aerofoils at high Re (see below).

In summary, the drag of bird wings is considerably greater than that of conventionalaerofoils for a variety of reasons. Skin friction drag is higher because of the lower Re,and whether air flow over bird wings is laminar or turbulent makes little difference tothe value for skin friction drag. Pressure drag is high for bird wings because the Re issubcritical and so separation occurs immediately after the thickest part of the wing; italso appears to be high because bird wings are twisted. Feathers may also result insome additional drag compared to a smooth aerofoil. Induced drag appears to begreater for bird wings, which have lower aerofoil efficiency factors. Insect wings,which operate at lower Re than bird wings, have even higher drag coefficients despitetheir 'cleaner' aerodynamic shapes, because of higher skin friction drag.

Aerodynamic lift. The lift coefficient for bird wings varied from about —0-4 toabout +1-2 for the various species, depending upon the angle of attack. This issimilar to the CL values noted elsewhere for bird wings, parts of wings, or models(Table 3), and also for insect wings (Jensen, 1956; Vogel, 1966). The slope of therelationship between CL and a (i.e. the lift curve slope, dCL/da) was about 0-05-0-10for the bird wings. Other values for bird wings are from 0-03 to 0-23 (Table 3) anddata of Jensen (1956) and Vogel (1966) yield values of about 0-04 for insect wings. Thelift coefficient of a thin aerofoil is, in theory,

CL = 2?r sin a.,

where a is the angle of attack (Hoerner & Borst, 1975). The lift curve slope is calcu-lated as:

dCL/da = 27r2/i8o = o-n

(Hoerner & Borst, 1975). However, many factors influence boundary layer adherenceto a wing, and hence alter the lifting characteristics of real (and non-thin) wings. Suchfactors include cross-sectional shape of the wing, shape of the leading and trailingedges, thickness, camber and Re.

Lateral flow of air around wing tips is also of significance to aerofoils of low aspectratio (< 5) such that dCL/da is higher (= 0-5 x AR up to a maximum of about 0-27;Hoerner & Borst, 1975). However, values for dCJda of the bird wings were usuallyless than or equal to o-i (Table 2). The dCL/da for bird wings may have been under-estimated slightly since the CL-a relationship often appeared to be S-shaped (sig-moidal) but a linear regression was used to estimate the mean slope.

The maximum lift coefficient of aerofoils is predicted to be:

fcloerner & Borst, 1975). However, actual values of CLi max are only 10-20% of thepredicted values, even for conventional aerofoils with extensive boundary layer con-

158 P. C. WITHERS

trol. Bird wings had even lower CLi max than conventional aerofoils. The CLi max OTaerofoils decreases at Re less than about io6 because the progressive increase in thick-ness of the boundary layer along the wing chord promotes flow separation near theleading and trailing edges (Hoerner & Borst, 1975). This propensity for boundarylayer separation limits CLt max to low values at Re < io5 for both bird wings (Figs.1-3) and other aerofoils.

The values of CL> max of bird wings measured in the wind tunnel were quite low(^ 1-2). Similar values of CLmax for bird wings are reported elsewhere (Table 3).This is in marked contrast to calculated or estimated values of CLt max for birds orbird wings for gliding or flapping flight (Tucker & Parrott, 1970; Weis-Fogh, 1973;Norberg, 1975). However, Norberg (1976) calculates lower values which are moreconsistent with steady state for hovering bats, of about 1-4-1-6. Such high Ci>maxvalues are not consistent with steady-state aerodynamics, either for the measured birdwings or conventional aerofoils at low Re. The body and tail of birds probably con-tributes some lift (hence CL based only on wing projected area is overestimated).Separation of primary feathers during the wing stroke may increase the effectiveaerofoil area, hence use of the wing projected area would also overestimate the actualCL (see P. C. Withers, in preparation). Further, non-steady state aerodynamics havebeen shown to yield higher CLt max values in some instances (Weis-Fogh, 1973, 1976).

Morphological parameters, such as camber and leading-edge shape, can also affectCL, max- The CL. max f°r bird wings was correlated with the position of maximumthickness, such that wings with the point of maximum thickness near the leading edgehad the lowest CLt max. The CLi max of bird wings (but not the vulture primary feathertip) was also correlated with camber (r = 0-52), as would be expected.

The CL of conventional aerofoils decreases dramatically at high a, but this is not sofor conventional aerofoils or bird wings at low Re. A high CL at large a is of signifi-cance to birds during landing, when the wing may be ' stalled' at high a, with a con-sequently high drag to decelerate the bird, without reducing CL. No special propertiesof bird wings are required to explain this lack of wing stall at high a, although Vogel(1966) suggests that insect wings are specially adapted to maintain high CL after'stall'.

Lift/drag ratio. The CL of bird wings determines the angle of attack and windvelocity required for horizontal flight. The CD determines the power required forflight (power = drag force x velocity). The lift/drag ratio (CL/CD = L/D) reflectsthe performance of the aerofoil, i.e. its aerodynamic 'cleanness' and glide angle. TheCL/CD of bird wings was maximum at small, positive a (and at low CL values);(CL/CD)max ranged from 3-3 for the starling wing to about 17 for the swift wing andvulture primary feather. Conventional aerofoils typically have CL/CD ratios of 20 ormore, at high Re, but (CL/CD)mStX is much lower at Re's similar to those of the birdwings (Pope & Harper, 1966). The (CL/CD)m&x of moth, butterfly and Drosophilawings is even lower (2-4) than that of bird wings because of their very low Re (Vogel,1966; Nachtigall, 1976), although it is about 8-10 for locust wings (Jensen, 1955). It isclear that the (C£/CD)max, like many other aerodynamic parameters, is markedlydependent upon Re.

Aerofoil efficiency factor. The aerofoil efficiency factor, e (or Monk's span factor,

Bird wings as fixed aerofoils j eg

Woerner and Borst's factor, a'; Tucker's correction factor, R) is a correction factorapplied to the induced drag coefficient;

The value of e indicates the efficiency of the wing in deflecting air in order to producelift. The theoretical value of e i9 i for elliptical, untwisted wings, but is generally0-9-0-95 for conventional aerofoils. The value of e is often assumed to be high for birdwings (i-o by Pennycuick, 1969; 0-9 by Tucker, 1973) although estimates for birdsare from 0-5-0-7 (Greenewalt, 1975).

It proves to be much more difficult to determine e for a bird wing than for a con-ventional aerofoil, and the meaning is quite different. Greenewalt (1975) states that iti9 possible to determine both e and CDi pro by plotting CL*/n AR against CDj total sincethe result is a straight line whose slope is i/e and intercept is CDpT0. However, thisassumes that CDt pro is independent of Ct, CD and ct. In fact, the relationship betweenCJ^/TTKR. and CDi t^ i 1 S clearly not linear and great care is required in selecting therange of CDi total u s e d t 0 estimate e since the slope is low (and e is high) at low a, andthe slope is high (and e is low) at greater a.

Values of e calculated in this study by two different methods were generally from0-3 to o-8, but some values were unreasonably high. Furthermore, values of e for birdwings, are not simply an efficiency factor for the following reason. CDi pr0 is a signifi-cant portion of total drag, for bird wings, and CDi pro will change at differing a. Thus,CDi wtai alters by more than the change in induced drag, and includes not only themechanical efficiency term but also the change in CD< pro at differing a. This is not thesituation for conventional aerofoils at high Re since CDf pro is a less significant pro-portion of CDi total an<^ ' more closely approximates a mechanical efficiency factor(CDtPro also changes with a for conventional aerofoils, but it is still relatively in-significant relative to CDiln). Values of e for insect wings range from about 02(Drosophila) to 0-5 (Schistocerca) (calculated from data of Jensen, 1956; Vogel, 1966).Similarly, the value of e for a flat plate at Re = 4 x io4 is low, at about 0-4 (calculatedfrom data in Hoerner, 1965). It is therefore of little intrinsic value to calculate e forbird wings for this reason, except for the sake of having a value for substitution intovarious equations.

It is possible that the manner of mounting the wing used here (no end-plate or wallboundary as a reflecting plane) overestimated the CD at high a. However, the values ofi/nARe calculated for bird wings from other studies are also quite low (Table 3;i/nXRe could not be converted to e since AR was often not given). Furthermore,Nachtigall (1979) used a double end-plate system which approximates a wing ofinfinite span, so induced drag should be o (Von Mises, 1959). Nevertheless, there wasclearly a marked increase in CD at high a, indicating that CDt ln is a small fraction ofthe increase in CD\ CDt pro being the most important.

Pressure distribution. The pressure distribution over the nightjar wing (Fig. 4) wassimilar to that over a conventional aerofoil (Harper & Pope, 1966; Hoerner & Borst,19-75), with a pronounced suction peak forming on the upper, leading edge at a ofc-90. This suction peak diminished at lower and higher a. A positive pressure^veloped on the lower wing surface at moderate to high a (^ 20) whereas a suction

6-2

i6o P. C. WITHERS

-2°> • —i

+7°

+20°

Fig. 8. Suggested pattern of air-flow over a bird wing at differing angles of attack, showingdirection and magnitude of pressure distribution (solid arrows) as measured for the nighthawkwing at mid-span, the likely position of boundary layer separation (•) and regions of turbulentair flow (small arrows).

pressure formed at lower a. The relative magnitude and sign of the pressure distri-bution indicated the development of negative lift at a. < o°, maximum positive lift atmoderate a, and a decline in lift at higher a. In fact, the actual lift and drag coefficientscan be calculated from the pressure distribution (Pope & Harper, 1966). The calcu-lations for the mid-span of the nighthawk wing, although only approximate, are con-sistent with the measured lift coefficients and with the pressure drag coefficientestimated from the measured CD (the pressure distribution allows the estimation onjjfcof the pressure drag coefficient, not skin friction drag or induced drag coefficient^

Bird zoings as fixed aerofoils 161

Rve deviation of the CL and CDt pre values from measured CL and estimated CD pre

reflect not only potential experimental errors but also possible variation in CL andCDt pre along the wing span.

The pressure distribution not only enables the estimation of local CL and CD< pre,but also indicates the air flow pattern over the wing. Both the pressure distribution forthe nighthawk wing and basic aerodynamic considerations (Schlicting, 1966) indicatea transition from laminar to turbulent flow immediately after the thickest part of thewing, and flow separation (Figs. 4, 8). Such a flow pattern is in marked contrast to thatover a conventional aerofoil, where the higher kinetic energy of the air delays turbu-lence and separation to a point much closer to the trailing edge than the thickest partof the wing. This difference in flow pattern for wings at low and high Re results in thehigher CDw pre for the former. Birds, by virtue of their small wings and low flightvelocity, are generally restricted to Re less than about io5, although the largest flyingbirds may have Re's into the transition zone and greatly benefit in terms of reduceddrag and also increased lift coefficients. However, the flow pattern for a flapping birdwing may bear little resemblance to that measured in this study for a stationary wing.It would be of great interest to measure the pressure distribution or air flow of aflapping wing. Perhaps the standard boundary layer visualization techniques used foraerofoils could also be applied to bird wings.

Lift/drag relation. There is clearly an inverse correlation between the lift and dragperformance of bird wings (Table 3). For example, the most streamlined wing (swift)had the lowest CDt pro and CL< max, and highest (CL/CD)mix. The starling wing had thehighest CDtJ)T0, a high Ci j m t t x and the lowest (CL/CD)m?LX. The morphology of thewings clearly contributed to these differing lift and drag performances (Table 5). Thewing loading of the birds (kgm~2; from Poole, 1938, and Hartman, 1961) was notsignificantly correlated with CD<min, Ci > m a x or (L/Z))max but was significantlycorrelated with the lift curve slope (dCL/da = 0086 — 0-004 (wing loading); P <0-05). The aerodynamic performance of different bird species would seem to bedetermined by their wing morphology, which in turn would reflect many ecologicaland physiological constraints. The great variety of shape, size and aerodynamic per-formance of bird wings testifies to the differing trade-offs between lift and dragperformance required by various bird species.

I am greatly indebted to E. Shaughnessy of the Engineering Department at DukeUniversity for use of the wind tunnel, pitot tube and pressure sensor, and for valuablediscussion. I thank P. L. O'Neill for valuable discussion. I also thank V. Tucker andK. Schmidt-Nielsen for loan of equipment.

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357-387-SCHLICTING, H. (1966). Boundary Layer Theory. New York: McGraw-Hill.TUCKER, V. A. (1968). Respiratory exchange and evaporative water loss in the flying budgerigar. J. exp.

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34, 841-846.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. 5a, 345-367-VOOEL, S. (1966). Flight in Drosophilia. III. Aerodynamic characteristics of fly wings and wing models.

J. exp. Biol. 46, 431-443.VON MISES, R. (1959). Theory of Flight. New York: Dover Publications.WEIS-FOGH, T. (1973). Quick estimates of flight fitness in hovering animals, including novel mechanisms

for lift production. J. exp. Biol. 59, 169-230.WEIS-FOGH, T. (1976). Energetics and aerodynamics of flapping flight: a synthesis. In Insect Flight

(ed. R. C. Rainey). Blackwell Scientific Publ. Symp. Royal Ent. Soc. 7, 48-72.