Theory of Wing Sections - Including a Summary of Airfoil Data

338 Pages·1949·14.58 MB·English

by  Abbott

THEORY OF
WING SECTIONS
Including a Summary of Airfoil Data
By IRA H. ABBOTT
DIRECTOR OF AERONAUTICAL AND SPACE RESEARCH
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
and ALBERT E. VON DOENHOFF
RESEARCH ENGINEER, NASA
DOVER PUBLICATIONS, INC.
NEW YORK
Copyright © 1949, 1959 by Ira H. Abbott and
Albert E. von Doenhoff.
All rightsreserved.
This Dover edition, first published in 1959.is an
unabridged and corrected republication of the first
edition first published in 1949by the McGraw·Hill
Book Company, Inc. This Dover edition includes a
new Preface by the authors.
Library of CongressCatalog Card Number: 60-1601
International Standard BookNumber
ISBN-13: 978-0-486-60586-9
ISBN-lO: 0-486-60586-8
Manufacturedinthe UnitedStatesbyCourierCorporation
60586833
www.doverpublications.com
PREFACE TO DOVER EDITION
The new edition of this book originally published in 1949 results
from the continuing demand for a concise compilation of the sub
sonic aerodynamic characteristics of modern NACA wing sections
together with a description of their geometry and associated
theory. These wing sections, or their derivatives, continue to be
the most commonly used ones for airplanes designed for both
subsonic and supersonic speeds, and for application to helicopter
rotor blades, propeller blades, and high performance fans.
A number of errors in the original version have been corrected
in the present publication. The authors are pleased to acknowledge
their debt to the many readers who called attention to these errors.
Since original publication many new contributions have been
made to the understanding of the boundary layer, the methods of
boundary-layer control, and the effects of compressibility at super
critical speeds. Proper treatment of each of these subjects would
require a book in itself. Inasmuch as these subjects are only
peripherally involved with the main material of this book, and
could not, in any case, be treated adequately in this volume, it was
considered best to expedite republication by foregoing extensive
revision.
IRA H. ABBOTT
CHEVY CHASE, MD. ALBERT E. VON DOENHOFF
June, 1958
v
CONTENTS
PAGE
PREFACE TO DOVER EDITION . . . . V
PREFACE. . . . . . . . . . . . . . . . . vii
1. THE SIGNIFICANCE OF WING-SECTION CHARACTERISTICS 1
Symbols. The Forces on Wings. Effect of Aspect Ratio. Application of
SectionDatatoMonoplaneWings:a. BasicConceptsofLifting-lineTheory.
b. Solutions forLinearLift Curves. c. Generalized Solution. Applicability
ofSection Data.
2. SIMPLE TWO-DIMENSIONAL FLOWS 31
Symbols. Introduction. ConceptofaPerfectFluid. EquationsofMotion.
Descriptionof FlowPatterns. SimpleTwo-dimensional Flows: a. Uniform
Stream. b. Sources and Sinks. c. Doublets. d. Circular Cylinder in a
UniformStream. e. Vortex. f. CircularCylinderwith Circulation.
3. THEORY OF WING SECTIONS OF FINITE THICKNESS . 46
Symbols. Introduction. Complex Variables. Conformal Transformations.
TransformationofaCircleintoaWingSection. FlowaboutArbitraryWing
Sections. EmpiricalModification ofthe Theory. DesignofWingSections.
4. THEORY OF THIN WING SECTIONS . 64
I· •
Symbols. Basic Concepts. AngleofZeroLiftand PitchingMoment. De
signofMean Lines. EngineeringApplicationsofSection Theory.
5. THE EFFECTS OF VISCOSITY . 80
Symbols. Concept of Reynolds Number and Boundary Layer. Flow
around Wing Sections. Characteristics of the Laminar Layer. Laminar
Skin Friction. Momentum Relation. Laminar Separation. Turbulent
Flow in Pipes. Turbulent Skin Friction. Calculation of Thickness of the
Turbulent Layer. Turbulent Separation. Transition from Laminar to
TurbulentFlow. Calculation ofProfile Drag. Effect ofMach Number on
Skin Friction.
6. FAMILIES OF WING SECTIONS . . . . . . . . . . . . . . . . . 111
Symbols. Introduction. MethodofCombining Mean Linesand Thickness
Distributions. NACAFour-digitWingSections:a. ThicknessDistributions.
b. Mean Lines. c. Numbering System. d. Approximate Theoretical
Characteristics. NACA Five-digit Wing Sections: a. Thickness Distribu
tions. b. MeanLines. c. NumberingSystem. d. ApproximateTheoretical
Characteristics. ModifiedNACAFour-and Five-digitSeriesWingSections.
NACA J-SeriesWing Sections: a. Thickness Distributions. b. Mean Lines.
ix
x CONTENTS
c. Numbering System. d. Approximate Theoretical Characteristics.
NACA6-8eriesWingSections: a. Thickness Distributions. b. Mean Lines.
c. Numbering System. d. Approximate Theoretical Characteristics.
NACA 7-8eries Wing Sections. Special Combinations of Thickness and
Camber.
7. EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS. 124
Symbols. Introduction. Standard Aerodynamic Characteristics. Lift
Characteristics: a. Angle of Zero Lift. b. Lift-curve Slope. c. Maximum
Lift. d. Effect ofSurface ConditiononLift Characteristics. DragCharac
teristics: a. Minimum Drag of Smooth Wing Sections. b. Variation of
Profile Drag with Lift Coefficient. c. Effect of Surface Irregularities on
Drag Characteristics. d. Unconservative Wing Sections. Pitching
moment Characteristics.
8. HIGH-LIFT DEVICES. 188
Symbols. Introduction. Plain Flaps. Split Flaps. Slotted Flaps: a. De
scriptionofSlottedFlaps. b. Single-slottedFlaps. c. External-airfoilFlaps.
d. Double-slottedFlaps. Leading-edgeHigh-liftDevices:a. Slats. b. Slots.
c. Leading-edge Flaps. Boundary-layer Control. The Chordwise Load
Distributionover FlappedWingSections.
9. EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS . . . . 247
Symbols. Introduction. SteadyFlowthroughaStreamTube:a. Adiabatic
Law. b. VelocityofSound. c. Bernoulli'sEquationforCompressibleFlow.
d. Cross-sectionalAreasand Pressures in a Stream Tube. e. Relations for
a Normal Shock. First-order Compressibility Effects: a. Glauert-Prandtl
Rule. b. Effect ofMach Numberon the Pressure Coefficient. Flow about
Wing Sections at High Speed: a. Flow at Subcritical Mach Numbers.
b. Flow at Supercritical Mach Numbers. Experimental Wing Characteris
tics at High Speeds: a. Lift Characteristics. b. Drag Characteristics.
c. MomentCharacteristics. Wingsfor High-speed Applications.
REFERENCES 300
APPENDIX I. BasicThickness Forms. 309
II. Mean Line" . 382
III. AirfoilOrdinates 406
IV. Aerodynamic Characteristicsof WingSections 449
INDEX. 689
CHAPTER 1
THE SIGNIFICANCE OF WING-SECTION CHARACTERISTICS
1.1. Symbols.
A aspect ratio
An coefficientsofthe Fourierseriesfor thespan-loaddistribution
CD drag coefficient
CD, induced dragcoefficient
CL lift coefficient
CLmax maximum lift coefficient
C pitching-momentcoefficient
M
CM"" pitching-momentcoefficientabout the aerodynamic center
D drag
E Jonesw edge-velocity factor, equals ratio of the semiperimeter of the plan
form ofthe wingunder considerationto the spanofthewing
E a factor (seeFig. 13)
G a factor (seeFig. 14)
H afactor (seeFig. 15)
J a factor (seeFig. 9)
L lift
La "additional" loading coefficient
L "basic" loading coefficient
b
M pitchingmoment
S wingarea
V speed
X longitudinal distance between the aerodynamic centerofthe root section and
ac
the aerodynamic centerofthe wing,positiveto therear
a winglift-curveslope
a. effectivesection lift-curveslope, aolE
ao section lift-curveslope
ac aerodynamic center
b wingspan
C wingchord
c
mean geometric chord, Sib
cf mean aerodynamic chord
Cd section drag coefficient
Cd, section induced-dragcoefficient
CI section lift coefficient
Clal local"additional"sectionliftcoefficientforawingliftcoefficientequaltounity
Clb local"basic" section lift coefficient
C/max section maximum lift coefficient
Cm section-momentcoefficient
Cm"" section-momentcoefficientabouttheaerodynamiccenter
c. root chord
c/ tipchord
1
2 THEORY OF WING SECTIONS
d section drag
f a factor (seeFig. 8)
k a spanwise station
l section lift
la "additional" section lift
lb "basic" section lift
m section moment
r an even number of stations used in the Fourier analysis of the span-load
distribution
u a factor (seeFig. 10)
v a factor (seeFig. 11)
w a factor (seeFig. I:?)
x projected distance in the plane of symmetry from the wing reference point
to theaerodynamic centerofthe wingsection, measured parallelto the chord
ofthe root section, positive to the rear
y distance along the span
z projected distance in the plane of symmetry from the wing reference point
to the aerodynamic center ofthe wingsection measured perpendicular to the
root chord, positiveupward
a angle ofattack
ao section angleofattack
a. effectiveangle.ofattack
ai angle ofdownwash
alo section angle ofattackfor zerolift
alOe angleofzerolift oftheroot section
as wingangle ofattack measured from thechord ofthe root section
a.(L_O) angle ofattackofthe root section forzerolift ofthe wing
fJ angle ofsweepback
aerodynamic twist from root to tip
E
multiplierfor obtainingthe wingcharacteristics
11m!
l1mn multiplierforobtainingthe span-loaddistribution
o cos-1(_ 2:)
Amk multiplierforobtainingthe induced-angledistribution
ratio ofthe circumference ofa circleto its diameter
7r
p massdensity ofair
1.2. The Forces on Wings. The surfaces that support the aircraft by
means of dynamic reaction on the air are called wings. An aircraft may
have several wings which may either be fixed with respect to the fuselage
or have any ofseveralmotionsas in the case ofhelicoptersor ornithopters.
Regardless of the type of lifting surface, its aerodynamic characteristics
willbe strongly affected by the shape of the wing section. The wing char
acteristics may be predicted from the known aerodynamic characteristics
of the wing section if the span is large with respect to the chord, if the
Mach numbers are subcritical, and if thechordwise component of velocity
is large compared with the spanwise component. Thus the wing-section
characteristics consideredin thisvolume have a large fieldof applicability.
A complete discussion of the application of section characteristics to the
THE SIGNIFICANCE OF WING-SECTION CHARACTERISTICS 3
predictionofwingcharacteristicsisbeyondthescopeofthis work, butsome
indication of the methods is given for the case of the monoplane wing
in steady straight flight without roll or sideslip.
The monoplane wing supports the airplane by means of a lift,force
generated by the motion through the air. This lift is defined as the com
ponentofforceactingintheplane ofsymmetryinadirectionperpendicular
./2 2.4
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./0 2.0
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3M
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o 8 /6 24
Angleofollack,a,degrees
FIG. 1. Typical wingcharacteristics.
to the line of flight. In addition to the lift, a force directly opposing the
motion of the wing through the air is always present and is called the
"drag." For a given attitude of geometrically similar wings, the forces
tend to vary directly with the density of the air, the wing area, and the
square of the speed. It is accordingly convenient to express these forces
in terms ofnondimensional coefficientsthat are functions primarily ofthe
attitude of.the wing. The lift and drag are given by the following ex
pressions:
L =7~pV2SCL (1.1)
D =~pV2SCD (1.2)
The lift and drag forces may be considered to act at a fixedpoint with
respect to the wing. A complete specification of the forces acting on the
4 THEORY OF WING SECTIONS
wing requires a knowledge of the moment about this fixed point. For a
symmetrical wingmoving with translation only in the plane of symmetry,
the side force perpendicular to the lift and drag is equal to zero, and the
moment acts in the plane of symmetry. This moment tends to change
1.4
-
1.2
i~~~
V >.--r
u / 1/
1.0
~~ V
~ V
/
.8
b~V V l-1
V //
~/ V V
V
A V V
#~/ V
V
/'
~~,/
o v~
~
V
//
-.2rt
O· 10·
Angle ofonock,et,(degrees)
FIG.2. Liftcoefficientsplottedas function ofangle ofattackfor aspectratiosof7to 1.
the angle of attack of the wing. It is accordingly called the "pitching
moment" and may be expressed as follows:
M = 72PV2ScCM (1.3)
A convenient way of describing the aerodynamic characteristics of a
wing is to plot the values of the coefficientsagainst the angle of attack,
which is the angle between the plane of the wing and the direction of
motion. Such a plot is shown in Fig. 1. The lift coefficient increases
almost linearly with angle of attack until a maximum value is reached,
whereupon thewingissaidto"stall." Thedrag coefficienthas aminimum
value at a lowlift coefficient, and the shape ofthe curve is approximately
parabolic at angles of attack below the stall. If the point about which
THE SIGNIFICANCE OF WING-SECTION CHARACTERISTICS 5
the moment is taken is properly chosen (the aerodynamic center), the
moment coefficientisessentiallyconstantup to maximum lift. Ameasure
of the efficiencyof the wing as a lifting surface is given by the lift-drag
104
-
A
1.2 ~~ v '"
v
I;} ~ V- .-/
~ k'
,/'
1.0
~/ / ~
'~ V
/ V .,..,
./ ./
.8
~/ /1
~/ / V-
,/'"
/ y
'I / V
/'
~
/
V;(
o
~
~
s-,
...2
~
~
~~ ~ .~
Drag/coefficient, CO
FIG.3. Polardiagramsfor seven wingswithaspectratiosof7to 1.
ratio, which is also plotted in Fig. 1. This ratio increases from zero at
zerolift to a maximum value at a moderate lift coefficient,after which it
decreasesrelatively slowly as the angle of attack isfurther increased.
It is desirable for the wing to have the smallest possible drag. Inas
much as the high-speed lift coefficient is usually substantially less than
that corresponding to the best lift-drag ratio, one of the best ways of
reducing the wingdrag is to reduce the wingarea. This reduction ofarea
is usually limited by considerations of stalling speed or maneuverability.
These considerations are directly influenced by the maximum lift coeffi
cient obtainable. The wing should therefore have a high maximum lift