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 '© ~949, 1959 by Ira H. Abbott and
Albert E. von Doenhoff.
All rights reserved under Pan American and Inter­
national Copyright Conventions.
Published in Canada by General Publishing Com­
pany, Ltd., 30 Lesmill Road, Don Mills, Toronto,
Ontario.
This Dover edition, first published in 1959. is an
unabridged and corrected republication of the first
edition fint published in 1949 by the McGraw-Hill
Book Company, Inc. This Dover edition includes a
new Preface by the authors.
Stand4,-d Book. Number: 486-60586-8
Library 01 Congress Catalog Card Number: 60-1601
Manufactured in the United States of America
Dover Publications, Inc.
ISO Varlek Street
New York, N. Y. 10014
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 ease, be treated adequately in this volume, it was
considered best to expedite republication by foregoing extensive
revision.
CHEVY CHASE, MD.
June, 1958
v
IRA H.
ABBOTT
ALBERT
E. VON DOENHOFF
PREFACE
In preparing this book an attempt has been made to present concisely
the most important and useful results of research on the aerodynamics of
wing sections at suberitical speeds. The theoretical and experimental
results included are those found by the authors to be the most useful.
Alternative theoretical approaches to the problem and many experimental
data have been rigorously excluded to keep the book at a reasonable
length. This exclusion of many interesting approaches to the problem
prevents any claim to complete coverage of the subject but should permit
easier use of the remaining material.
The book is intended to serve as a reference for engineers, but it should
also be useful to students as a supplementary text. To a large extent,
these two uses are not compatible in that they require different arrange­
ments and developments of the material. Consideration has' been given
to the needs of students and engineers with a limited background in
theoretical aerodynamics and mathematics. A knowledge of differential
and integral calculus and of elementary mechanics is presupposed. Care
has been taken in the theoretical developments to state the assumptions
and to review briefly the elementary principles involved. An attempt has
been made to keep the mathematics as simple as is consistent with the
difficulties of the problems treated.
The material, presented is largely the result of research conducted by
the National Advisory Committee for Aeronautics over the last several
years. Although the authors have been privileged to participate in this
research, their contributions have been no greater than those of other
members of the research team. The authors wish to acknowledge es­
pecially the contributions of Eastman N. Jacobs, who inspired and directed
much of the research. The authors are pleased to acknowledge the im­
portant contributions of Theodore Theodorsen, I. E. Garrick, H. Julian
Allen, Robert M. Pinkerton, John Stack, Robert 1'. Jones, and the many
others whose names appear in the list of references. The authors also
wish to acknowledge the contributions to the attainment of low-turbulence
air streams made by Dr. Hugh L. Dryden and his former coworkers at the
National Bureau of Standards, and to express their appreciation for the in­
spiration and support of the late Dr. George W. Lewis.
CHEVY CHASE, ~{D.
July, 1949
vii
IRA H.
ABBOTT
ALBERT
E.
VON DOENHOFF
CONTENTS
v
vii
PREFACE TO DOVER EDITION . . . . . . . . . . . .
PREFACE. . . . . . . . . . . . . . . . . . . . . . .
1. THE SIGNIFICANCE OF WING-SECTION CHARACTERISTICS . .
1
Symbols. The Forces on Wings. Effect of Aspect Ratio. Application of
Section Data to Monoplane Wings: a. Basic Concepts of Lifting-line T~eory.
b. Solutions for Linear Lift Curves. c. Generalized Solution. Applicability
of Section Data.
2. SIMPLE TWO-DIMENSIONAL FLOWS
31
Symbols. Introduction. Concept of a Perfect Fluid. Equations of Motion.
Description of Flow Patterns. Simple Two-dimensional Flows: a. Uniform
Stream. b. Sources and Sinks. c. Doublets. d. Circular Cylinder in a
Uniform Stream. e. Vortex. f. Circular Cylinder with Circulation.
3. THEORY OF WING SECTIONS OF FINITE THICI{NESS
46
.
Introduetiou. Complex Variables. Conformal Transformations.
Transformation of a Circle into & Wmg Section. Flow about Arbitrary Wing
SeetiODS. Empirical Modification of the Theory. Design of Wing Sections.
Symbols.
4. THEORY OF
TJIIN
WING SECTIONS . . . . . . . . . . . .
~.
64
Symbols. Basic Concepts. Angle of Zero Lift and Pitching Moment. De­
sign of Mean Lines. Engineering ApplicatioD8 of Section Theory.
5. THE EFFECTS OF VISCOSITY . . . . . . . . . . . . . .
80
Symbols. Concept of Reynolds Number and Boundary Layer. Flow
around Wmg Sections. Characteristics of the Laminar Layer. Laminar
Skin Frietion. Momentum Relation. Laminar Separation. Turbulent
Flow in Pipes. Turbulent Skin Friction. Calculation of Thickness of the
Turbulent Layer. Turbulent Separation. Transition from Laminar to
Turbulent Flow. Calculation of Profile Drag. Effect of Mach Number on
Skin Friction.
III
6. FAMILIES OF WING SECTIONS
Symbols. IDtroduction. Method of Combining Mean Lines and Thickness
Distributions. NACA F~igit Wing Sections: a. Thickness Distributions.
b. Mean Lines. c. Numbering System. d. Approximate Theoretical
Characteristics. NACA Five-digit Wing Sections: a. Thickness Distribu­
tions. b. Mean Lines. c. Numbering System. d. Approximate Theoretical
Characteristics. Modified NACA Four- and Five-digit Series Wing Sections.
N ACA l-Series Wing Sections: Q. Thickness Distributions. b. Mean Lines.
ix
CONTENTS
x
c. Numbering System.
d. Approximate Theoretical Characteristics.
NACA 6-Series Wing Sections: G. 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: 4. Angle of Zero Lift. b. Lift-curve Slope. c. ~faximum
Lift. d. Effect of Surface Condition on Lift Characteristics. Drag Charac­
teristics: G. 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 Charncteristics.
8. HIGH-LIFT DEVICES .
188
Symbols. Introduction. Plain Flaps. Split Flaps. Slotted Flaps: 4. De­
scription of Slotted Flaps. b. Single-slotted Flaps. c. Extemal-airfoil Flaps.
d. Double-slotted Flaps. Leading-edge High-lift Devices:a. Slats. b. Slots.
c. Leading-edge Flape. Boundary-layer Control. The Chordwise Load
Distribution ovcr Flapped 'Ving Sections.
9. EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS . . . .
247
Symbols.. Introduction. Steady Flow through a Stream Tube: 4. Adiabatic
Law. b. Velocity of Sound. c. Bernoulli's Equation for Compressible Flow.
d. Cross-sectional Areas and Pressures in a Stream Tube. e. Relations for
a Normal Shook. First-order Compressibility Effects: G. Glauert-Prandtl
Rule. b. Effect of Mach Number on the Pressure Coefficient. Flow' about
Wing Sections at IIigh Speed: 4. Flow at Subcritical Mach Numbers.
b. Flow at Supercritieal ~{ach Numbers.. Experimental Wing Characteris­
tics at High Speeds: 4. Lift Characteristics. b. Drag Characteristics.
c. Moment Characteristics. Wmgs for High-speed Applications.
REFERENCES
300
APPE~DIX
309
I. Basic Thickness Forms.
II. Mean Lines . . .
INDJ4~X
.
382
I I I. Airfoil Ordinates
406
IV. Aerodynamic Chnracteristies of \Ying Sections
449
CHAPTER 1
THE SIGNIFICANCE O:F WING-SECTION CHARACTERISTICS
1.1. Symbols.
A
An
CD
Co,
CL
CL ma x
CAl
CJI oe
D
E
E
G
H
J
L
L.
L,
M
8
V
X,"
a
a.
Go
ac
b
C
c
c'
Cd
Cd,
c,
Cl el
CI.
Cl max
c.
c....
C.
Cc
aspect ratio
coefficients of the Fourier series for the span-load distribution
drag coefficient
induced dra.g'"coefficient
lift coefficient
maximum lift coefficient
pitching-moment coefficient
pitching-moment coefficient about the aerodynamic center
drag
Jonesi' edge-velocity factor, equals ratio of the semi perimeter of the plan
form of the wing under consideration to the span of the wing
a factor (see Fig. 13)
a factor (see Fig. 14)
a factor (see Fig. 15)
a factor (see Fig. 9)
lift
" additional" loading coefficient
U basic" loading coefficient
pitching moment
wing area
speed
longitudinal distance between the aerodynamic center of the root section and
the aerodynamic center of the wing, positive to the rear
wing lift-curve slope
effective section lift-curve slope, 40/ E
section lift-curve slope
aerodynamic center
wing span
wing chord
mean geometric chord, 8/b
mean aerodynamic chord
section drag coefficient
section induced-drag coefficient
section lift coefficient
local U additional" section lift coefficient for a wing lift coefficient equal to unity
local U basic" section lift coefficient
section maximum lift coefficient
section-moment coefficient
section-moment coefficient about the aerodynamic center
root chord
tip chord
1
THEORY OF WING SECTIONS
2
d. section drag
f a factor (see Fig. 8)
k a spanwise station
l section lift
Z. " additional" section lift
It ",basic" section lift
m seetion moment
r an even number of stations used in the Fourier analysis of the span-load
distribution
u a factor (see Fig. 10)
v a factor (see Fig. 11)
tD a factor (see Fig. ~)
~ projected distance in the plane of symmetry from· the wing reference point
to the aerodynamic center of the wing section, measured parallel to the chord
of the root section, positive to the rear
71 distance alang the span
~ projected distance in the plane of symmetry from the wing reference point
to the aerodynamic center of the wing section measured perpendicu1U' to the
root chord, positive upward
ex angle of attack
act section angle of attack
a. effective angle. of attack
eli angle of downwash
section angle of attack for zero lift
a ... angle of zero lift of the root section
a. wing ang1e of attack measured from the chord of the root section
0.<1-0) angle of atteu:k of the root section for zero lift of the wiDi
fl angle of sweepbaek
e aerodyutUDic twist from root to tip
II. multiplier for obtaining the wing cbaracteristies
,_ multiplier for obtaining the span-load distribution
a,.
,
CX»J-l ( -
~)
multiplier for obtaining the inducecL-angle distribution
... ratio of the circumference of a circle to ita diameter
p mass density of air
)..1:
1.2. The Forces- on W-mgs. 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 of several motions as in the case of helicopters or omithopters.
Regardless of the type of lifting surface, its aerodynamic characteristics
will be stronglyatfected 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 the chordwise component of ,\relooity
is large compared with the spanwise component. Thus the wing-section
characteristics considered in this volume have a 1arge field of applicability.
A complete discussion of the application of section characteristics to the
THE 8IGNIFICANCE OF WING-8ECTION CHARACTERISTIC8
3
prediction of wing characteristics is beyond the scope of this work, but some
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
ponent of force acting in the plane of symmetry in & direction perpendicular
.12
2.4
J
40
I
.KJ
20
I
I
.08
1.6
I/
II
~
I " r
I ~
V" ~
~
'CL
~
J'I
I
.......
o
0
0
',­
/, ~D
'-/0
~~
i'
JV
f'
~
-­
o
~,
~
8
16
Angle of t1fIack,~."..
24
FIG. 1. Typical wing characteristics.
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
ten.d 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 of nondimensional coefficients that are functions primarily of the
attitude of_the wing. The lift and drag are given by the following ex­
pressjons:
~pV28CL
(1.1)
D == ~pVJ8CD
(1.2)
L ==
The lift and drag forces may be considered. to act at a fixed point 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 wing moving. 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
~ r--.. r"
jA~ VV
/
1.0
.8
~
V
V
~/ /
V
~
~
o
V
FIG.
V
~ ~/ ~ /t V
V
VI
V
~
~ V
6.
-.2
~
A
~
~
~
r
~
II. V V
~
V
/
V
V
/
v
/
V
0/0­
Angle ofo/fack, tt, (degrees)
2. Lift coefficients plotted as function of angle of attack for aspect ratios of 7 to 1.
the angle of attack of the wing. It is accordingly called the ., pitching
moment" and may be expressed as follows:
(1.3)
..~ convenient way of describing the aerodynamic characteristics of a
wing is to plot the values of the coefficients against the angle of attack,
which is the angle bet,veen the plane of the wing and the direction of
motion. Such a plot is shown in Fig. 1. The lift coefficient increases
.ilinost linearly with angle of attack until a maximum value is reached,
whereupon the wing is said to U stall." The drag coefficient has a minimum
value at a low lift coefficient, and the shape of the curve is approximately
parabolic at angles of attack below the stall. H the point about which
THE SIGNIFICANCE OF WING-BECTION CHARACTERISTICS
5
the moment is taken is properly chosen (the aerodynamic center), the
moment coefficient is essentially constant up to maximum lift. A measure
of the efficiency of the wing as a lifting surface is given by the lift-drag
1.4
j
1.0
A~ /
~V' V
V
V
'/
/
~
I
FIG.
~
·It:/~ /
(.)......6
-.2
~~
V
.8
o
------ --
A
1.2
1\~
V
~
V"
w~ V
V
v
/
'>:
~
V
V
V
Y
---
~
~
V
-
V
i/
r---.....
V
~
I
~
~
~
~
./0
.20
Drog'coefficiert, Co
3. Polar diagrams for seven wings with aspect ratios of 7 to 1.
ratio, which is also plotted in Fig. 1. This ratio increases from zero at
zero lift to a maximum value at a moderate lift coefficient, after which it
decreases relatively slowly as the angle of attack is further 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 wing drag is to reduce the wing area. This reduction of area
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
6
THEORY OF WING 8ECTION8
coefficient.combined with low drag coefficients for high-speed and cruising
flight. This combination of desirable qualities can be obtained only to a
limited extent by a single wing configuration. It is therefore customary
to use some retractable device such as flaps to improve the maximum lift
characteristics of the wing.
1.3. Effect of Aspect Ratio. Aspect ratio is defined as the ratio of the
span squared to the wing area (IJI/S), which reduces to the ratio of the
1.4
..
1.2
. . 0 -6
o
1.0
0
eJ~
."\
c
lA+
.8
~.
A
i"II'
~~
o
Vz~
0=4
61 J
-.2
~
-I----
• : 5
to
tI
-I--
I
+=2
8tf,.,.a
lUd'
o
I
e : 6
til :
,,'"
cl
T
-
+
G
p9
-·~/O-
0
10·
20­
Angle ofoltock, a,(degteeS)
FIG.
4. Lift coe8iciente as function of aqle of attack, reduced to aspect ratio of 5.
span to the chord in the case of & rectangular wing. Early wind-tunnel
investigations of wing characteristics showed that the rates of change of
the lift and drag coefficients with angle of attack were strongly affected
by the aspect ratio of the model. Wings of high aspect ratio were ob­
served to have higher lift-curve slopes and lower drag coefficients at high
lift coefficients than wings of low aspect ratio. The effect 9£ aspect ratio
on the lift curve is shown in Fig. 2.88 The wings of various aspect ratios
are shown to have about the same angle of attack at zero lift, but the
slope of the lift curve increases prograNively with increase of aspect
THE SIGNIFICANCB OF WING-8ECTION CHARACTERI8TICS
7
ratio. The effect of aspect ratio on the drag coefficient is shown in
Fig.3.88 Although the drag coefficients for all the models of various aspect
ratios are substantially equal at zero lift, marked reductions in the drag
1.4
1.2
••
A
0
1.0
•
0
•-
0
•
~
of
...
6
0
D
.8
+
•
:» •
A
C
411
+ -2
0
a~
A
~t
-
-
-3
o -4
c·
o·
• -5
I--
at
e -6
b
e -7
til
•
.2
I
~41~
~
-
J,
11
o
-.2
•
~
'·1)
ftA
0
•
0
•
.I
.2
Drag coefficienfl Cd
FIG. S. Polar diagrams reduced to aspect ratio of S.
coefficient occur at the higher lift coefficients as the 8Bpect ratio is in­
creased.
As a result of such observations, the Lanchester-Prandtl wing theory
was developed. This theory shows that, for wings having elliptical span­
wise distributions of lift, the following simple expressions relate the drag
8
THEORY OF WING SECTIONS
coefficients and angles of attack as functions of aspect ratio at constant
lift coefficients
1 1)
+ C (-!..
_!)
A' A
2
,
CL ( - - CD=CD+1["
A' A
a'
=a
L
1["
(1.4)
(1.5)
where CD and a' correspond, respectively, to the drag coefficient and
angle of attack (radians) of a wing of aspect ratio A'.
Application of Eqs. (1.4) and (1.5) to reduce the data of Figs. 2 and 3
to an aspect ratio of five results in the data of Figs. 4 and 5.88 These figures
show that the characteristics of a wing of one aspect ratio may be pre­
dicted with considerable accuracy from data obtained from tests of a wing
of widely different aspect ratio.
Equations (1.4) and (1.5) may be simplified by the concept of infinite
aspect ratio. If Cd and ao indicate the drag coefficient and angle of attack
of a wing of infinite aspect ratio, the characteristics of an elliptical wing
of aspect ratio A may be expressed as
(1.6)
(1.7)
A wing of infinite aspect ratio would have the same flow pattern in all
planes perpendicular to the span. In other words, there would be no
components of flow along the span, and the flow over the wing section
would be two-dimensional, Infinite aspect ratio characteristics are ac­
cordingly commonly called "section characteristics. II The section char­
acteristics are intrinsically associated with the shape of the wing sections
as contrasted with 'ling characteristics, which are strongly affected by
the wing plan form. The detailed study of wings is greatly simplified by
tile concept of wing-section characteristics because wing theory offers a
method for obtaining the properties of wings of arbitrary plan form from
a summation of the characteristics of the component sections.
1.4. Application of Section nata to Monoplane Vmgs. a. Basic Con­
cepts of Lifting-line Theory. The simplest three-dimensional wing theory
is that based on the concept of the lifting line. 88, ~ In this theory the wing
is replaced by a straight line. The circulation about the wing associated
with the lift is replaced by ~ vortex filament. This vortex filament lies
along the straight line; and, at each spanwise station, the strength of the
vortex is proportional to the local intensity of the lift. According to Helm­
holtz's theorem, a vortex filament cannot terminate in the fluid. The
variation of vortex strength along the straight line is therefore assumed to
THE SIGNIFICANCE OF WING-BECTION CHARACTERISTICS
9
result from superposition of a number of horseshoe-shaped vortices, as
shown in Fig. 6. The portions of the vortices lying along the span are
called the" bound vortices." The portions of the vortices extending down­
stream indefinitely are called the" trailing vortices."
The effect of trailing vortices corresponding to a positive lift is to induce
a downward component of velocity at and behind the wing. This down­
ward component is called the U downwash." The magnitude of the down­
wash at any section along the span is equal to the sum of the effects of all
the trailing vortices along the entire span. The effect of the downwash
FIG.
6. Vortex pattern representing a lifting wing.
is to change the relative direction of the air stream over the section. The
section is assumed to have the same aerodynamic characteristics with re­
spect to the rotated air stream as it has in normal two-dimensional flow,
The rotation of the flow effectively reduces the angle of attack. Inasmuch
as the downwash is proportional to the lift coefficient, the effect of the
trailing vortices is to reduce tile slope of the lift curve. The rotation of
the flow also causes a corresponding rotation of the lift vector to produce
a drag component in the direction of motion. This induced-drag coeffi­
cient varies as the square of the lift coefficient because the amount of
rotation and the magnitude of the lift vector increase simultaneously.
The problem of evaluating the downwash at each point is difficult be­
cause of the interrelation of the downwash, lift distribution, and plan form.
A comparatively simple solution was obtained by Prandtl'" for an elliptical
lift distribution. In this case the downwash is constant between the wing
tips and the induced drag is less than that for any other type of lift dis­
tribution. Equations (1.6) and (1.7) give the relation between the section
and wing characteristics for an elliptical lift distribution.
b. Solutions for Linear Lift Curves. Glauert" applied a Fourier series
analysis to the problem and developed methods for obtaining solutions for
wings of any plan form and twist. Anderson 10 applied Glauert's methods
THEORY OF WING SECTIONS
10
to the determination of the characteristics of wings with a wide range of
aspect ratio and straight taper and with a linear spanwise variation of
twist. Anderson considered the spanwise lift distribution for any typical
wing to consist of two parts. One part, called the U basic distribution"
,
is the distribution that depends principally on the twist of the wing and
occurs when the total lift of the wing is zero; it does not change with the
angle of attack of the wing. The second part of the lift distribution, called
the additional distribution," is the lift due to change of the wing angle
I(
120
~
10-­
r--..
-­
~
-
.........,;
~
r--..-l
r--­ ~-
la
c­ ~~
.... ~
~-.
~
r-.....
r-, r-;
"-­
<,
........
r-, ~
I ' "'
~
"" r-,"-.
",\
~
~
/,
o
a
-
~
~
r--....
~
r---.
2
--­
4
6
8
/0
/2
/4 /6
Distonce along semispon. feet
--1/
/8
20
Flo. 7. Typical wmiepan lift diatn1Jution. C£ - 1.2
of attack j it is independent of the twist of the wing and maintains the
same form throughout the reasonably straight part of the lift curve. A
typical distribution of lift over the semispan of a twisted, tapered wing
at a moderately high lift coefficient"is shown in Fig. 7.
Anderson presented his solutions in the form of tables and charts that
are easy to apply. In order to apply these solutions the wing geometry
must be known in the following terms:
8 wing area
II span
A aspect ratio
c,/e. taper ratio where Cc is the tip chord and c. is the root chord
E
aerodynamic twist in degrees from root to tip, measured between the sero lift
directions of the center and tip sections, positive for wash in
fj angle of sweepback., measured between the lateral axis aDd a line through the
aerodynamic centers of the wing sectiODS
Go section lift-eurve slope
THE SIGNIFICANCB OF WING-8ECTION CHARACTERISTICS
11
effective lift..curve slope, 04/B
B Jones" edge-velocity factor, equals ratio of the semiperimeter of the wing under
consideration to the span of the wing
CIe
c.. section-moment coefficient about the aerodynamic center
e
chord at any spanwise station
Anderson considered only the case of a linear distribution of twist between
the center section and the tip. The types of fairing ordinarily used for
wings result in nonlinear distributions of twist if the wing is tapered. The
departure from the assumed linear distribution may not be negligible if
appreciable twist is combined with a large taper ratio.
The data required to find the span'vise lift distribution are given in
Tables 1 and 2. The basic and additional loading coefficients L b and La
are presented at various spanwise stations for a wide range of aspect ratio
and taper ratio. The local basic section lift coefficient Cl, for a wing lift
coefficient equal to zero is given by the expression
Ea,$
Cit
== Ci) Lb
The local additional section lift coefficient c,.. for a wing lift coefficient
equal to unity is given by the expression
S
== Cb La
C,••
The actual or total local section lift coefficient C, for a wing lift coefficient
equal to C L is obtained from
c, == c,. + CV; lea
The lift-curve slope per degree a of the wing is obtained from
a;IIK J 1
a.
+ (57.3a./ ...A )
where the effective section lift-curve slope a~ is taken as the average for the
sections composing the wing and the factor J is given in Fig. 8 on page 16.
The Jones edge-velocity correctionj" which had not been derived when
Anderson!O obtained these results, has boon applied to Anderson's formulas
by the use of an effective section lift-curve slope. This effective slope a.
equals ao/E where E is· the ratio of the semiperimeter to the span of the
wing under consideration. It will be noted that this effective lift-curve
slope is not solely a property of the section but that it varies with the plan
form of the" wing.
The angle of attack of the wing corresponding to any value of the
wing lift coefficient CL is given by
a. = C
-
L
a
+ alo' + J E
THEORY OF WING SECTIONS
12
TABLE I.-BASIC SPAN LIFT-DISTRIBUTION DATA
Values of L. for tapered wings with rounded tips c'.
~
00110021003100'
I
0.6
~
Spanwise station 'II/(b/2)
-O.I~J
-0.121
-0.180
-0.192
-0.221
-0.248
-0.269
-0.288
-0.318
-0.342
-oaso -0.3M
-0.387 -0.380
-o.3St -0.399
-0.398 -0.411
-0.122
-0.162
-0.197
-0.224
-0.263
-0.275
-0.293
-0.322
-0.080
-0. lOS
-0.130
-0.14:8
-0.182
-0.178
-0.189
-0.207
-0.2"26
-0.234
-0.242
-0.2"7
-0.255
-0.082
-0.111
-0.135
-0.156
-0.173
-0.189
-0.200
-0.220
-0.239
-0.248
-0.256
-0.260
I -0.269
2
3
4
6
8
7
8
10
12
1-1
16
18
20
-0.118
-0.163
-0.183
-0.211
-0.235
-0.256
-0.274
-0.3CM
-0.329
2
3
-0.076
-0.098
-0.117
-0.131
-0.145
-6.1&6
-0.168
-0.182
-0.197
-0.206
-0.212
-0.219
-0.222
-0.163
-0.199
-0.228
-0.253
-0.276
-0.293
-0.323
-0.349
-0.370
-0.385
-0.403
-0.415
-0.3~
-0.370
-0.388
-0.405
-0.417
-0.122
-0.165
-0.199
-0.226
-0.252
-0.274
-0.291
-0.321
-0.3~
-0.368
-0.382
-0..00
-0.410
= ~ L.
-0.121
-o.lM
-0.199
-0.225
-0.2&2
-0.272
-0.290
-0.320
-0.34&
-0.365
-0.379
-0.393
-0.404
a:
0
-0.121
-o.IH
-0.198
-0.224
-o.2S0
-0.270
-0.288
-0.318
-0.341
-0.360
-0.37&
-0.387
-0.399
-0.121
-0.183
-0.1\17
-0.224
-0.247
-0.268
-0.285
-0.315
-0.337
-0.355
-0.370
-0.380
-0.392
-0.120
-0.162
-0.1.
-0.221
-0.244
-0.264
-0.282
-0.311
-0.331
-0.350
-0.362
-0.376
-0.386
-0.120
-0.161
-0.194
-0.219
-0.243
-0.261
-0.279
-0.306
-0.323
-0.342
-0.358
-0.388
-0.378
-0.085
-0.112
-0.137
-0.157
-0.175
-0.190
-0.205
-0.225
-0.237
-0.248
-0.256
-0.265
-0.272
-O.OM 1-00083
-0.110 -0.108
-0.135 -0.132
-0.166 -0.152
-0.172 -0.170
-0.190 -0.189
-0.204 -0.204
-0.225 -0.225
-0.237 -0.237
-0.248 -O.~48
-0.256 -0.2M
-00264 \-00262
-0.272 -0.2i0
-0.018
-0.017
-0.020
-0.021
-00016\-0.0111
-0.018 -0.018
-0.020 -0.021
-0.022 -O.O'J3
-0.024 . -0.026
-0.027 -0.029
-0.030 -0.030
-0.032 -0.032
-0.036 -0.038
-0.040 -0.00
-000'11-0.00
-0.043 -o.CM6
-o.<K8 -0.049
-0.120
-0.160
-0.192
-0.218
-0.242
-0.2M
-0.276
-0.299
-0.317
-0.33i
-o.M8
-0.360
-0.869
Spanwise station 'II/(b/2) == 0.2
"
6
8
7
8
10
12
14
16
18
20
-0.085
-0.112
-0.138
-0.159
-0.176
-0.192
-0.204
-0.224:
-0.240
-0.249
-0.258
-0.264
-0.271
-0.086
-0.113
-0.137
-0.159
-0.176
-0.192
-0.204
-0.225
-0.239
-0.248
-0.257
-0.265
-0.271
-0.086
-0.113
-0.137
-0.158
-0.176
-0.192
-0.205
-0.225
-0.238
-0.248
-0.2M
-0.265
-0.271
-0.086
-0.113
-0.137
-0.168
-0.176
-0.191
-o~205
-0.226
-0.238
-0.248
-0.256
-0.:..>&5
-0.272
Spanwise station y/(b/2) ==
2
3
4
5
6
7
8
10
12
14
18
18
20
-0.006
-0.002
0
O.OM
0.009
0.012
0.014
0.021
0.028
0.036
0.043
0.049
0.050
-0.011
-0.010
-0.006
-0.004
-0.002
-0.001
0
0.007
0.009
0.013
0.019
0.022
0.023
-0.013
-0.012
-0.011
-0.010
-0.008
-0.010
-0.008
-0.002
-0.001
0
0.002
0.004
0.006
-0.015
-0.015
-0.012
-0.012
-0.012
-0.013
-0.012
-0.010
-0.010
-0.010
-0.008
-0.008
-0.006
-0.016
-0.016
-0.016
-0.016
-0.018
-0.017
-0.017
-0.017
-0.017
-0.017
-0.016
-0.015
-0.014
-0.016
-0.018
-0.016
-0.018
-0.018
-0.018
-0.019
-0.020
-0.021
-0.021
-0.022
-0.022
-0.022
3
4:
6
6
1
8
10
12
1"
16
18
20
0.052
0.070
0.085
0.099
0.109
0.119
0.128
0.139
0.148
0.155
0.160
0.166
0.170
0.Q521
0.069
0.082
0.096
0.107
0.117
0.122
0.138
0.145
0.162
0.158
0.162
0.169
0.051
0.068
0.081
0.092
O.UM
0.114
0.121
0.135
0.141
O.UiO
0.154
0.160
0.165
0.050
0.068
0.080
0.091
0.102
0.112
0.120
0.132
0.140
0.148
0.151
0.158
0.159
00010
0.068
0.080
0.091 1
0.101
0.111
0.120
0.131
0.140
0.145
0.149
0.152
0.152
0.050
0.068
0.080
0.091
0.101
0.110
0.119
0.130
0.139
0.142
0.146
0.148
0.14&
0.4
-0.018
-0.016
-0.018
-0.020
-0.020
-0.020
-0.021
-0.022
-0.025
-0.028
-0.029
-0.031
-0.031
Bpanwise station 1//(b/2)
2
::z
-0.085
-0.113
-0.137
-0.158
-0.176
-0.191
-0.206
-0.226
-0.238
-0.248
-0.256
-0.265
-0.272
-0.016
-0.016
-0.019
-0.021
-0.021
-0.022
-0.025
-0.027
-0.029
-0.031
-0.034:
-0.038
-0.038
-O.o-?l
-0.025
-O.O".19
-0.030
-0.032
-0.035
-0.038
-0.041
-0.041
0.6
0.050
0.068
0.080
0.091
0.100
0.110
0.119
0.130
0.137
0.141
0.143
0.145
0.147
0.0li0
0.068
0.080
0.091
0.100
0.110
0:118
0.129
0.135
0.140
0.141
0.142
0.1.3
0.049
0.068
0.080
0.090
0.100
0.110
0.118
0.128
0.134
0.139
0.140
0.140
0.141
00049
0.068
0.080
0.090 1
0.100
0.109
(l117
0.126
0.132
0.138
0.139
0.139
0.140
0.048
0.068
0.080
0.090
0.100
0.108
0.116
0.124
0.130
0.135
0.136
0.138
0.140
THE SIGNIFICANCE OF WING-BECTION CHARACTERISTICS
13
TABLE I.-BASIC SPAN LIFT-DISTRIBUTION DATA.-(Continued.)
~I
0
I I I I
0.1
0.2
0.3
0..
LO.S
I I I I I
0.6
Spanwise station y/(b/2)
6
6
7
8
10
12
14
16
18
20
0.072
0.088
0.100
0.109
0.115
0.121
0.126
0.136
0.145
0.152
0.159
0.161
0.166
0.079
0.098
0.113
0.125
0.135
0.142
0.149
0.160
0.170
0.182
0.186
0.197
0.201
0.080
0.101
0.120
0.135
0.148
0.158
0.184
0.178
0.188
0.200
0.205
0.215
0.220
2
3
4
6
6
7
8
10
12
14
16
18
20
0.069
0.068
0.074
0.081
0.087
0.090
0.092
0.098
0.100
0.102
0.103
0.105
0.107
0.068
0.083
0.098
0.107
0.117
0.123
0.131
0.139
0.147
0.156
0.161
0.166
0.172
0.072
0.092
0.111
0.122
0.136
0.146
0.153
0.166
0.178
0.188
0.197
0.202
0.211
2
3
4:
0.082
0.102
0.123
0.138
0.152
0.163
0.174
0.188
0.200
0.210
0.216
0.224
0.232
0.083
0.104:
0.125
0.140
0.156
0.169
0.180
0.195
0.208
0.216
0.222
0.230
0.237
0.085
0.108
0.128
0.143
0.160
0.172
0.182
0.200
0.212
0.221
0.22H
O.2:J5
0.241
0.073
0.098
0.118
0.131
0.148
0.160
0.170
O.IM
0.198
0.208
0.219
0.228
0.233
0.075
0.099
0.121
0.138
O.IM
0.167
0.179
0.197
0.210
0.220
0.231
0.243
0.248
0.076
0.100
0.122
0.140
0.159
0.171
0.182
0.201
0.213
0.231
0.241
0.252
0.260
2
3
4
5
6
7
8
10
12
14:
16
18
20
0.038
0.044
0.050
0.052
0.0504
0.056
0.057
0.058
0.059
0.060
0.061
0.061
0.061
0.051
0.063
0.072
0.083
0.088
0.093
0.100
0.107
0.112
0.116
0.121
0.126
0.128
0.068
0.073
0.076
0.100
0.109
0.116
0.125
0.138
0.143
0.161
0.169
0.166
0.173
0.019
0.022
0.026
0.029
0.030
0.030
0.030
0.031
0.031
0.031
0.031
0.032
0.032
0.030
0.039
0.043
0.051
0.066
0.060
0.062
0.067
0.069
0.071
0.035
0.045
O.OM
0.065
0.071
0.078
0.081
0.090
0.095
0.102
0.111
0.121
0.128
0.059
0.078
0.092
0.107
0.119
0.130
0.140
0.152
0.165
0.174
0.184
0.194
0.203
0.060
0.079
0.095
0.110
0.122
0.135
0.146
0.162
0.179
0.190
0.203
0.213
0.225
0.060
0.080
0.097
0.112
0.128
0.140
0.152
0.171
0.189
0.202
0.218
0.229
0.239
0.9
1.0
0.086
0.110
0.130
0.148
0.162
0.173
0.183
0.202
0.216
0.227
O.~
0.2
0.248
U.086
0.110
0.130
0.143
0.163
O.li4:
0.183
·0.203
0.216
0.228
0.236
0.243
0.248
0.084
0.108
0.130
0.148
0.164
0.174
0.184
0.201
0.214
0.225
0.232
0.242
0.248
0.081
0.106
0.129
0.149
0.165
0.175
0.184
0.075
0.100
0.123
0.141
0.160
0.172
0.184
0.205
0.225
0.241
0.253
0.263
0.273
0.075
0.100
0.123
0.142
0.160
0.172
0.185
0.207
0.228
0.243
0.258
0.269
0.279
0.075
0.100
0.123
0.142
0.160
0.172
0.186
0.209
0.229
0.245
0.259
0.271
0.282
0.075
0.100
0.123
0.142
0.160
0.172
0.187
0.210
0.230
0.246
0.260
0.275
0.285
0.059
0.080
0.100
0.116
0.132
0.150
0.161
0.186
0.202
0.218
0.233
0.248
0.259
0.059
0.079
0.100
0.117
0.131
0.149
0.160
0.187
0.205
0.221
0.236
0.251
0.265
0.058
0.078
0.099
0.116
0.130
0.145
0.159
0.183
0.204
0.222
0.238
0.255
0.271
0.036
0.053
0.069
0.082
0.097
0.110
0.121
0.141
0.160
0.175
0.188
0.200
0.210
0.035
0.052
0.068
0.083
0.097
0.110
0.121
0.142
0.161
0.177
0.190
0.201
0.212
0.034
0.051
0.067
0.083
0.097
0.110
0.121
0.143
0.162
0.178
0.191
0.202
0.213
o.iss
0.210
0.220
0.229
0.238
0.247
= 0.9
0.075
0.100
0.123
0.141
0.160
0.171
0.183
0.203
0.221
0.238
0.249
0.260
0.268
Spanwise station 1I/(bj2)
0.8
= 0.8
0.085
0.109
0.128
0.147
0.160
0.173
0.182
0.201
0.214
0.223
0.232
0.239
0.245
Spanwise station y/(b/2)
0.7
= 0.95
0.060
0.080
0.099
0.113
0.130
0.144
0.158
0.178
0.198
0.211
0.222
0.236
0.245
0.060
0.080
0.100
0.114
0.132
0.148
0.160
0.182
0.200
0.216
0.229
0.241
0.251
Spanwise station 1I/(b/2) = 0.975
2
3
4:
5
6
7
8
10
12
14
16
18
20
OJ117
0.083
0.086
0.037
0.049
0.060
0.070
0.079
0.087
0.091
0.105
0.115
0.127
0.138
0.160
0.168
0.037
0.050
0.062
0.071
0.082
0.091
0.100
0.115
0.131
0.143
0.156
0.169
0.178
0.037
0.051
0.064:
0.075
0.088
0.098
0.107
0.124
0.141
0.155
0.169
0.182
0.193
0.037
0.052
0.068
0.078
0.091
0.101
0.112
0.132
0.149
0.163
0.178
0.191
0.202
0.036
0.054
0.069
0.081
0.094
0.107
0.120
0.138
0.153
0.171
0.182
0.197
0.208
THEORY OF WING SECTIONS
14
TABLE 2.-ADDITIONAL SPAN LIrr-DI8TRIBUTION DATA
V~ues of L. for tapered winp·with rounded tips, Cia.
~
o.t
-
~ L.
I I I I I
0.6
0.8
~
0.7
0.8
0.9
1.292
1.276
1.2C0
1.248
"1.237
1.282
1.229
1.222
1.219
1.214
1.208
1.203
1.199
1.200
1.283
1.20
1.221
1.211
1.203
1.198
1.187
1.180
1.172
1.161
1.180
1.162
1.287
1.263
1.228
1.2M
1.187
1.171
1.lea
I.W
1.143
1.188
1.127
1.118
1.109
1.282
1.2M
1.211
1.188
1.183
1.149
1.136
1.120
1.109
1.100
1.0lIO
1.0IJ0
1.070
1.211
1.2M
1.220
1.208
1.200
1.193
1.189
1.182
1.172
1.170
1.IM
1.180
1.166
1.263
1.228
l.209
1.1M
1.1M
1.174
1.188
1.118
1.148
1.144
1.135
1.130
1.123
1.280
1.221
1.198
1.181
1.1.
1.167
1.148
1.137
1.126
1.119
1.110
1.103
1.098
1.2a
1.214
1.188
1.1es
1.111
1.138
1.129
1.114
1.102
1.OM
1.087
1.078
1.089
1.169
1.118
1.148
1.138
1.128
1.121
1.111
Lt08
1.102
1.091
1.091
1.087
1.083
1.1.
1.117
1.147
1.1.
1.127
1.120
1.118
1.1M
1.099
1.090
1.080
1.080
1.078
LI88
1.J.a8
1.146
1.1M
1.128
1.119
1.111
1.102
1.094
1.087
1.081
1.076
1.071
1.188
1.111
1.014
1.018
1.014
1.015
1.013
1.012
1.011
1.008
1.008
1.003
1.011
l.0J3
1.023
1.02.
1.024:
1.024
I.OM
1.023
1.022
1.019
1.017
1.015
1.012
1.018
1.080
1.035
1.038
1.039
1.039
1.039
1.039
1.088
1.036
1.033
1.032
1.028
1.019
1.088
1.010
1.013
1.015
Spanwise station 1//(b/2) .. 0
2
3
4
a
e
7
8
10
12
14
18
18
20
1.489
1.489
1.127
1.169
1.585
1.809
1.829
1.881
1.188
1.708
1.726
1.741
1.76&
1.«10
1.430
1.412
1.473
1.492
1.610
1.1M
I.US
1.678
1.882
1.110
1.823
1.832
1.387
1.386
1.400
1.41"
1.428
1.440
1.319
1.'­
1.4M
1.419
1.477
1.491
1.502
1.513
1.520
1.527
1.632
1.639
I.M7
1.329
loMe
1.303
1.317
1.388
1.393
1.401
1.411
1.417
1.423
1.428
1.429
1.431
1.300
1.308
1.318
1.324
1.329
1.332
1.338
1.M7
1.349
1.3M
I.US
1.369
1.380
1.411
1.413
1.490
1.&02
1.&13
1.62&
1.631
1.339
1.3aO
1.380
1.369
1.378
1.388
1.392
1.409
1.420
1.429
1.433
1.441
1.'­
1.318
1.322
1.329
1..333
1.338
1.360
1.1""
1.361
1.881
1.386
1.388
1.310
1.312
1.301
1.302
1.302
1.301
1.300
l.aoo
1.300
1.301
1.308
1.309
1.309
1.308
1.307
I.M
1.288
1.279
1.272
1.287
1.2M
1.2M
1.2M
1.201
1.280
1.265
1.252
1.210
Spanwi8e station 1/1(b/2) .. 0.2
2
3
4
I
8
7
8
10
12
14
18
18
20
1.279
1.279
1.2M
1.288
1.290
1.291
1.294
1.299
1.302
1.307
1.308
1.309
1.311
1.217
1.2GO
1.2C1O
1.280
1.269
1.269
1.211
1.285
1.265
1.288
1.289
1.270
1.271
1.280
1.248
1.243
1.240
1.236
1.236
1.236
1.238
1.233
1.232
1.232
;.1.231
1.230
1.268
1.241
1.282
1.223
1.218
1.214
1.212
1.209
1.202
1.201
1.199
1.195
1.190
I
Spanwiae station 1I/(b/2) == 0.4
2
3
4
5
8
7
8
10
12
J"
11
18
20
1.217
1.220
1.223
1.228
1.229
1.229
1.229
1.228
1.228
1.228
1.228
1.228
1.228
1.190
1.191
LI92
1.193
1.1.
1.193
1.192
1.192
1.192
1.191
1.189
1.188
1.182
1.178
1.176
1.173
1.172
1.171
1.170
1.111
1.187
1.188
1.181
1.1.
1.1&2
1.140
1.172
1.181
1.182
1.159
1.1SS
1.152
I.lao
1.148
1.14&
1.138
1.131
1.129
1.127
Span~ise
2
a
t
s
8
7
8
10
12
14
18
18
20
0.970
0.950
0.932
0.920
0.909
0.900
0.891
0.881
0.812
0.888
0.881
0.818
0.851
0.978
0.982
O..MB
0.938
0.930
0.920
0.916
0.907
0.901
0.895
0.888
0.883
0.878
O.IM
0.976
0.982
0~9S3
0.949
0.940
0.938
0.929
0.923
0.918
0.912
0.908
0.898
1.172
1.161
1.1ae
1.149
1.146
1.ltO
1.118
1.132
1.125
1.111
1.112
1.111
1.110
O.9S6
0.947
0.941
0.937
0.931
0.925
0.920
1.170
1.169
1.149
1.140
1.132
1.124
1.120
1.113
1.107
1.100
1.097
1.092
1.089
1.1~
1.133
1.125
1.118
1.110
1.100
1.090
1.082
1.015
1.070
1.015
station 1//(b/2) .. 0.6
- 1.003
0.992
0.985
0.978
0.971
0.966
0.959
1.171
1.180
1.111
1.142
1.138
1.131
1.1.
1.121
1.111
1.104
1.101
1.100
1.098
0.991
0.992
0.988
0.981
0.976
0.972
0.911
0.958
0.9&3
O.MS
0.944
O.NO
1.010
I.OM
1.002
1.000
0.993
0.989
0.988
0.978
0.972
0.919
0.968
0.913
0.98
1.012
1.011
1.008
1.008
1.002
1.000
0.999
0.992
0.989
0.988
983
0.
0.981
0.978
1
1 0.998
.000
0.995
1.OM
1.063
1.052
1.0&1
1.OD
l.eNS
1.0&7
1.CN8
THE SIGNIFICANCE OF WING-8ECTION CHARACTERISTICS
15
TABLE 2.-ADDlTIONAL SPAN LIn-DISTRIBUTION DATA.-(Continued.)
I I I I I I
0.2
0,3
u
0.5
0.7
0.8
0.9
1.0
0.746
0.772
0.796
0.808
0.820
0.827
0.834
0.842
0.8·0
0.858
0.862
0.870
0.878
0.747
0.782
0.800
0.822
0.838
0.8M
0.868
0.877
0.887
0.894
0.901
0.909
0.747
0.790
0.816
0.834
0.851
0.861
0.872
0.887
0.899
0.911
0.921
0.930
0.937
0.748
0.799
0.824
0.846
0.S62
0.875
0.886
0.905
0.919
0.933
0.944
0.953
0.962
0.637
0.568
0.598
0.822
0.643
0.610
0.673
0.698
0.718
0.739
0.768
0.773
0.791
0.638
0.671
0.603
0.630
0.852
0.671
0.686
0.712
0.736
0.769
0.780
0.800
0.819
0.639
0.676
0.609
0.636
0.669
0.678
0.696
0.723
0.751
0.776
0.801
0.822
0.846
0.386
0.418
0.440
0.483
0.4.82
0.S02
0.621
0.553
0.583
0.609
0.635
0.858
0.680
0.388
0.418
0.444
0.469
0.490
0.610
0.629
0.666
0.598
0.628
0.8M
0.882
0.707
0.390
0.420
0.446
0.471
0.496
0.615
0.634
0.576
0.608
O.MO
0.671
0.702
0.730
0.279
0.298
0.316
0.333
0.3SO
0.386
0.383
0.416
0.448
0.478
0.610
0.639
0.570
0.281
0.300
0.319
0.338
0.367
0.373
0.391
0.428
0.481
0.495
0.629
0.680
0.693
0.282
0.301
0.322
0.342
0.361
0.381
0.400
0."38
0.473
0.610
0.M8
0.580
0.816
0.6
Spanwise station y/(b/2) == 0.8
2
3
4
6
6
7
8
10
12
14
16
18
20
0.460
0.«4
0.678
0.712
0.659
0.700
0.691
0.644
0.632
0.681
0.619
0.676
0.609
0.670
0.663
0.600
0.686
0.853
0.676
0.848
0.569 .0.641
0.664
0.638
0.669
0.636
O.MS
0.629
2
3
4
6
6
7
8
10
12
14
18
18
20
0.378
0.362
0.331
0.814
0.300
0.290
0.282
0.266
0.H3
0.246
0.239
0.234
0.231
0.486
0."7
0.436
0.424
0.416
0.410
0.403
0.383
0.376
0.370
0.381
0.361
0.388
0.61&
0.689
0.668
O.MS
0.631
0.617
0.1i04
0.486
0.472
0.462
O.~
0.131
0.726
0.723
0.720
0.717
0.713
0.710
0.704
0.702
0.699
0.698
0.698
0.698
0.740
0.743
0.746
0.748
0.748
0.748
0.748
0.748
0.74.8
0.748
0.748
0.750
0.753
0.746
0.7M
0.764
0.769
0.775
0.778
0.779
0.783
0.788
0.789
0.791
0.796
0.801
0.746
0.764
0.781
0.790
0.800
0.802
0.808
0.815
0.821
0.825
0.830
0.835
0.842
0.E45
Spanwise station 1I/(b/2) == 0.9
0.808
O.SOO
0.496
0.490
0.487
0.4M
0.481
0.472
0.469
0.418
0.468
0.410
0.473
0.626
0.628
0.632
0.631
0.631
0.636
0.636
O.Ml
0.M2
0.M6
0.M7
0.652
0.&6Q
0.631
0.M3
0.554
O.li8O
0.565
0.572
0.579
0.590
0.597
0.602
0.609
0.818
0.826
0.634
0.552
0.569
0.583
0.595
0.603
0.612
0.628
0.639
0.648
0.659
0.889
0.879
0.535
0.559
0.581
·0.600
0.615
0.628
0.638
0.856
0.869
0.684
0.898
0.710
0.722
8panwise station y/(b/?)
4.
6
6
1
8
10
12
1"
18
18
20
0.231
0.209
0.191
0.176
0.166
0.161
0.148
0.138
0.132
0.129
0.128
0.122
0.121
0.298
0.290
0.286
0.281
0.278
0.272
0.261
0.2M
0.2M
0.262
0.262
0.2M
0.2&8
0.834
0.339
0.362
0.1"
0.348
0.346
0.346
0.346
2
3
4
6
6
7
8
10
12
14
16
18
20
0.132
0.119
0.107
0.098
0.089
0.081
0.f117
0.069
0.088
0.088
0.084
0.083
0.082
0.172
0.166
0.163
0.168
0.168
0.168
0.168
0.168
0.181
0.183
0.186
0.189
0.171
0.207
0.210
0.214
0.217
0.219
0.222
0.228
0.233
0.242
0.2i8
0.265
0.288
0.271
2
3
O.MS
0.M9
0.361
0.367
0.3M
O.US
0.369
0.378
0.384
0.392
0.398
0.403
0.410
0.419
0.423
0.432
0.439
0."9
0.370
0.389
0.402
0.415
0.428
0.438
0.446
0.460
0.473
0.482
0.495
0.503
0.516
0.379
0.401
0.420
0.436
0.451
0.464
0.475
0.495
0.511
0.529
0.546
0.558
0.569
c::
0.536
0.664
0.590
0.613
0.631
0.646
0.858
0.679
0.698
0.716
0.729
0.743
0.759
0.95
0.381
0.407
0.428
0.449
0.486
0.481
0.495
0.620
0.M2
0.562
0.681
0.598
0.613
0.383
0.412
0.434
0.468
0.476
0.4904
0.510
0.638
0.686
0.588
0.610
0.829
0.648
8panwise station y/(b/2) -= 0.975
0.239
0.260
0.258
0.269
0.272
0.278
0.283
0.295
0.308
0.320
0.331
0.346
0.363
0.263
0.278
0.288
0.304
0.314
0.320
0.328
0.343
0.380
0.376
0.394
0.412
0.4.35
0.272
0.289
0.304
0.320
0.332
0.342
0.352
0.373
0.395
0.413
0.435
0.461
0.483
0.274
0.291
0.308
0.322
0.340
0.351
0.363
0.390
0.413
0.438
0.463
0.492
0.616
0.277
0.294
0.311
0.328
0.3"
0.359
0.374
0.403
0.430
0.468
0.488
0.618
0.544:
THEORY OF WING SECTIONS
16
where a.
aloe
= wing angle of attack measured from chord of root section
= angle of zero lift of root section
J = a factor presented in Fig. 9
The angle of zero lift for the wing a.(L-O) is obtained from
a.(L-O)
= ale. + J E
The induced-drag coefficient CD. is given by
CD,
C L2
== -A
+ CLEaJJ + (Eae)'tD
or u
where the effective section lift-curve slope a. is taken
lOO
---
Ctlc.-1.0)r r--.
--..;;::'
~
.98
r
~
.94
.92 roo.90
0
-
~
~
the average for
-- - - -1'6"
~~8
.--.-- ~ ~
.96
-1
- ....-.2
-
88
----
r'to
~~
,.....
~-
~ """"'""
",,-
Stroight- toper wing with rounded tips
- I I I ~111i?t~o/lwi,?gI I I I I I I I 1-
I
Z'
4
/2
10
Aspect rot/o
8
6
/4
/6
/8
~
o
2.
Flo. 8. Chart for determining lift,..eune elope.
G
a.
-= 11 + (57.3o,./ ...A )
the sections composing the wing and u, VJ and to are factors presented in
Figs. 10, 11, and 12, respectively.
The drag coefficient CD of the wing is obtained from
2 {bIt
CD = 8}o CdC dll~ CD,
where 11 = distance along span
Cd == local section drag coefficient corresponding to local section lift
coefficient C,
The wing pitching-moment coefficient C_. about an axis through the
aerodynamic center is given by
C.. = Ec.. - GEaeA tan fJ
17
THE BIGNIFICANCB OF WING-8ECTION CHARACTERISTICS
-.5
cJ ci=lO
-.4
~
..
.
8
-----­
--..
_t
.6
.4­
't2
­
Ito
-.3
J
-.2
-./
Sfraighf- toper wing with rounded tips
~
--I
7 7 £llipficol. wi79 I
o
4
6
8
1-­
I 1I I I I
m
~
m
M
Aspect rot/a
m
ffl
9. Chart for determining angle of attack.
CL
a. .. a«.
J«
a.(L-o) = a,
JE
FIG.
+
CI
/.0
a
~
-- -::::- --::::: ~
.96" /
",
0
/j 0
'.
+
~
~ :::::-..../
I;
.9.e / I~ /
Ifl
V
""
------
"'""-
.............
r<,
V-
~
........
<,
...............
"'--
I'---. ~ ~
"-
;;"7
-/;..
/0
--
-
h9-- .--..
..............
..............
...........
i'--..--;
~
~~~ /5 ~
Aspecf rafio ~ ':JO-"" ,~
r/f 1/
rl /
.84 l
II
.8 0
..............
....... ~
/~ If/I
.88
-- -
- r.:::::- --- ------ ----
vI VI
u­
+
z~
~
~/
o.
.........
~- -
-
./
.2
Sfroighf- toper wing with rounded tips
Elliptical wing
-
-
.71'6
o
.3
.4
.5
Toper
.6
ratio
.7
.8
FIG. 10. Chart for determining induced-drag fact.or u.
CDi
..
CLi
-A
+ CLfIJefJ + (<<J.)2tD
1r 1.1
.9
1.0
THBORY OF WING SBCTIONS
18
006
~
004
~
~
--.8
~
c,lc.-1.0 ~I'
/
002
/
j'/
/' / ""
0
""
.-. ~ ~
- .........
~
......
r-
1.6
.4
V
­.002
~
~
r--.
~ t--"
.2
.AI
....
Sfra,r- taper wing with rounded fips Jl..
10-
£//ip Col wIng. 11-0
006
""""-
008
o
l--~~
4
2
8
6
~
.-.~~
,......
m
~
Aspecf rafio
:FIG. 11. Chart for
~~
u.
~
/8
~
20
determiniDa inclueecl-drae factor IJ•
•ODS
... ~
A
.003
[if'
•.002
.001
o
,
.~1.8
.4
.004
~
.~
~.;:
:::-:
.t-O
~
~
c:--
-.... -r-- ,......
-
~
...
t,...-- ~-
~
~
-
"---
~~
1-0.
Ce:/c.-O
Straight-lope).. 'w~ng with rounded tips,
~ -;- T7·:~lliptico/,wi,?91 I ·f I I I l I
I"
Z
4
FIG.
6
8
10
Ie
Aspect ratio
/4
12. Chart for determining induoed-drag factor
/6
/8
~
-
~
EO
to.
where the section pitching-moment coefficient c,.., is constant along the
span. If the sections composing the wing have different pitching-moment
coefficients, a weighted average may be used. E and G are factors pre­
sented in Figs. 13 and 14, respectively.
19
THE SIGNIFICANCE OF WING-8ECTION CHARACTERISTICS
The longitudinal position of the aerodynamic center of the wing
given by
~
~b=HA tan{J
where X ac == longitudinal distance between aerodynamic center of root
section and aerodynamic center of wing, positive to the rear
H == a factor presented in Fig. 15
/.4
'-..
~,
~
l2
i'~
'--
~
to-
,....-
~
A spec t ratio 3
F===:= ~
- -
-'----
l---
lO
9
6
.8
E
.6
.4
'-
.2
o
~
-
-
-
Sfroighf- toper wing wifh rounded lips I - Elliptical wing
-
./ .e ..3
.4
Toper
.5
.6
ratio, C.,/cs
.7
.8
.9
1.0
FIo. 13. Chart for determining pitching moment due to section moment.
The maximum lift coefficient-for the wing may be estimated from the
assumption that this coefficient is reached when the local section lift c0­
efficient at any position along the span is equal to the local maximum lift
coefficient for the corresponding section. This value may be found con­
veniently by the process indicated by Fig. 16. Spanwise variations of the
local maximum lift coefficient Cz..a, and of the additional c~ and basic
c" lift distributions are plotted. The spanwise variation of Clmax - Cit is then
plotted. The minimum value of the ratio
Cr..s -
Cit
Clea
is then found. The minimum value of this ratio is considered to be the
maximum lift coefficient of the wing.
c. Generalized Solution. Although the Glauert" method used by Ander­
THEORY OF WING SECTIONS
20
son l O requires the assumption of a linear variation of lift with angle of
attack, methods of successive approximation have been developed by
Sherman109 and Tanills that permit application of the actual wing section
data. Boshar" applied Tani's method to the solution of this problem.
Sivells and Neelylll have developed procedures based on this method which
.Q3c
.
eo.
Vfs___ --
~I-
.OcB
~
./
.024
/
.OEO
l/
G
~
~
~
.0/6 pL/
.Ole
~
-----
~
.004!---
o
~
-c... .--
~
~
,.--
~
~
I...--
~
I--- ~
-
t!2 =::::- -
~
~
5
t::== '"'"""'"
-
poo--'
.--
~
-
-
./
FIG.
-
-
~
~ ~
-
-
I-
~
~
-
-r---
~
I--
-,.....
1-
-
-~
-
0-
-
- -
-
t---
po--
-
I---
1 - ~~
-
.~
-,...-- ASRtetjf ratio 2
-
-
1-~ I- ~
r-
.3
r---
'-'
,..........
~P'"'""
-
-
~
~
to--
4
,..--
-
"'"'"""'"
~
--- - 6
8_
~
~
I--
~
::::----
~
~~
-~
.008
~
V
V
~
- :7 k
~
.-.'" ~
~
t--
~
......,;".
-
-
1-
-
10-
-
!---
1-"-
~
-'"-
-
I--
l--
~
- '--
-
~
-P"-
r---
Straight-toper wing wifh rounded fips_
Ellipficol, wi19
t
I
I
I I I
I
I I
-
I
. .3
.2
14. Chart
fOT
.5
.4
.6
TODer ratio, c~/Cs
.7
.8
.9
1.0
determining pitching moment due to basic lift forces.
em". == - GtaoA tan fJ
can be carried. out by an experienced computer and which yield highly
satisfactory and complete lift, drag, and pitching-moment characteristics
of wings, This method will be presented in sufficient detail to permit such
calculations to be made.
The basic problem is the determination of the downwash at each point
along the span from summation of the effects of the trailing vortices. The
strengths of the trailing vortices, however, are unknown, but they are
intimately associated with the spanwise lift distribution which, in tum,
is dependent upon the downwash. The relation between the downwash
and the lift distribution as given by Prsndtl" is
04
!!.-.l
... 2...
= 180
b
/2 (d/d1l)(c,c/4b)d1l
-b/2
'1/1 - Y
(1.8)
21
THE BIGNI.FICANCB OF WING-8ECTION CHARACTERISTICS
°.24
I
c;jc.
~
e:::;;­
~~~
....­
l..---'" ~ l--­
~
~
­I.°
lt
~
~
-
-
I-­
~
,....­
~
,-­
L-­
I~
~6
- ... - --­ -
_1000.­
-
-
~
""-­
.EO
~
H
• /8
r--- r--... t--­
-­ --­
10­
I~C
~
.....
-
~
'YJ
./6
./ 4~
Sfroignf- toper wing with rounded tips_
I i I I I I I I -
,
- --;- --;- -;- Ellipficol,wi'791
1
FIG.
I
I
I I
8
6
4
10
IZ
Aspect ratio
:ib =
HA tan II
--
1.4
I---~-
ctma.:- c1b.. . ~
~-
~.
1"'-'­
¥ Cl mox•
-......
t - L-­
~
"
.... ~t-L36cIQj
1.0
~
/8
15. Chart for determining aerodynamic-center position.
1.6
1.2
I I
/6
14
1.-0- ~
.....
Cla;­
-
...
-
,
~\.
\
r---.
'~ \
I
"1\\,
\
.8
cz
Ctmoll.= 1.36
.6
,
.4
-
.2
~c1b
-
o
-.20
./
.2
.3
.4
.5
~~
.6
r-­ .......
.7
-
~
.8
2Jb
Flo. J6. EstimS.tiOD of CLcax for example winz,
.9
~
/.0
eo
THEORY OF WING SECTIONS
22
where a.
a= angle of downwash (in degrees) at spanwise position 1/1
'1111:1 spanwise position, variable in the integration and measured
from center line
The effective angle of attack of each section of the wing is
(1.9)
a.==a-a,
where a. == effective angle of attack of any section (angle between chord
line and local wind direction)
a == geometric angle of attack of any section (angle between chord
line and direction of free stream)
The effective angle of attack is a function of the lift coefficient of that
section !(c,) 80 that the equation for the effective angle of attack becomes
a. = !(c,)
=a
- 180
.
.!.1,,/2
(d/dy)(c,c/4b)dy
-"12
... 2...
(1.10)
1/1 - 1/
This general integral equation must be solved to determine the downwash
and span-load distribution. Equation (1.10) may be solved by assuming
a span-load distribution and solving for the corresponding downw• .
The load distribution corresponding to this calculated downwash is then
found using Eq. (1.9) and the section data. This load distribution is com­
pared with the assumed distribution, and a second approximation is made.
The process is continued until the load and downwash distributions are
compatible.
The span-load distribution is expressed, following Glauert," 88 the
Fourier series
(1.11)
where cos (J == - 2y /b
The development of this method will be limited to load distributions
symmetri-cal about the center line. Only odd values of n are therefore used.
The induced angle from Eq. (1.8) becomes
.
ISO
a, == --;:-
~nA"
sin nO
sin 6
(1.12)
If it is assumed that values of c,c/4b are known at an even number r
of stations equally spaced with respect to (J in the range 0 ~ 8 ~ ..., the
coefficients A. of the Fourier series
(Cit)
~1..
~..
b
r-l
==
.-1, ...
A
· n ­m1l"
• SID.
r
where m == I, 2, 3, . . . ,r - 1 may be found by harmonic analysis as
THE 8IGNIFICA1!CE OF WING-8ECTION CHARACTERISTICS
'ItA. =
23
r2.sm n rmr
Because of the symmetrical relations
(:). =(:)~
and
.
SID
mr
.
(r - m)r
n - = SIn n - - ­
r
r
for odd values of n, the summation of A" needs to be made only for values
of m from 1 to T /2. Therefore
(1.13)
where
4.
m1l"
11m"
== -smnr
r
11m"
z::
for m == 1, 2, 3, . . . , (r/2) - 1
and
r2.sin n '2
11"
for m == r/2
From these relations Sivells and Neely obtained multipliers "." which are
presented in Table 3 for r = 20. The coefficients Aft may be obtained by
multiplying the known value of clc/4b by the appropriate multipliers and
adding the resulting products.
By the substitution of the values of Aft from Eq. (1.13) into Eq. (1.12),
Sivells and N eel y ll1 obtained the following expression for the downwash
at the same stations at which c,c/b is known:
(1.14)
where k designates the station at which (Xi is to be evaluated, and m is also
a station designation that is variable in the summation. Values of 'Xmk
for r = 20 are given in Table 4.111 A similar table of values of (41r/I80)~k
was also given by Munk. 72 The induced-angle multipliers are independent
of the aspect ratio and taper ratio of the wing and thus may be used for
any wing whose load distribution is symmetrical.
Equation (1.14) permits the solution of Eq. (1.10) by successive ap­
proximations. For one geometric angle of attack of the wing, a lift dis­
~
TABLE a.-MULTIPLIERS " ..." FOR
A"
CoEFFICIENTS
•
m1l"
L A"slnn
20
•
19
An
A-l,3
2y
b
o
o.1564
o.3090
o.4540
.5878
.7071
o.8090
o.8910
.9511
o.9877
X
10
9
8
7
6
5
4
3
2
1
or
=
GENERAL FOURIER SERIES
L
m-l
10 ( CIC)
4b
'1m"
1ft
1
3
5
7
9
11
13
15
17
19
0.10000
0.197M
0.19021
0.17820
0.16180
0.14142
0.11756
0.09080
0.06180
0.03129
-0.10000
-0.17820
-0.11756
-0.03129
0.06180
0.14142
0.19021
0.19754
0.16180
0.09080
0.10000
0.14142
0
-0.14142
-0.10000
-0.09080
0.11756
0.19754
0.06180
-0.14142
-0.10021
-0.03120
0.16180
0.17820
0.10000
0.03129
-0.19021
-0.09080
0.16180
0.14142
-0.11756
-0.17820
0.06180
0.19754
-0.10000
0.03129
0.19021
-0.09080
-0.16180
0.14142
0.11756
-0.17820
-0.06180
0.19754
0.10000
-0.09080
-0.11756
0.19754
-0.06180
-0.14142
0.19021
-0.03129
-0.16180
0.17820
-0.10000
0.14142
0
-0.14142
0.20000
-0.14142
0
0.14142
-0.20000
0.14142
0.10000
-0.17820
0.11756
-0.03129
-0.06180
0.14142
-0.19021
0.19754
-0.16180
0.09080
-0.10000
0.19754
-0.19021
0.17820
-0.16180
0.14142
-0.11756
0.09080
-0.06180
0.03129
-0.20000
-0.14142
0
0.14142
0.20000
0.14142
~
~
~
~
~
s
~
~
?S-s
~
C
~
25
THE SIGNIFICANCE OF WING-BECTION CHARACTERISTICS
TABLE
3a.
.".
MULTIPLERS - '1t111 FOR DETERMINATION OF WING CoEFFICIENTS
4
2y
b
11"
- ".1
4
o
0.07854
0.15515
0.14939
0.13996
0.12708
0.11107
0.09233
0.07131
0.04854
0.02457
0.1564
0.3090
0.4540
0.5878
0.7071
0.8090
0.8910
0.9511
0.9877
tribution is assumed from which the load distribution c,c/b is obtained.
The corresponding downwash is calculated from Eq. (1.14) using values
of Ami: from Table 4. The downwash is subtracted from the geometric
angle of attack at each station to give the effective angle of attack of
the section [Eq (1.9)]. The section lift coefficients are obtained from
curves of the section data plotted against the effective angle of attack a e
where
ae
=
E(ao - ala)
+ alo
and E is the Jones" edge-velocity correction.
If the resulting "lift distribution does not agree with that assumed, a
second approximation to the lift distribution is assumed and the process
is repeated until the assumed and calculated lift distributions agree. The
entire process must be repeated for each angle of attack for which the
wing characteristics are desired. The amount of labor required to obtain
the complete characteristics of a wing may be reduced by using Anderson's
method" for the range of lift coefficients where the section lift curves are
substantially linear.
The wing lift coefficient is obtained by spanwise integration of the lift
distribution. This process'!' has been reduced to the following summation:
CL
=
~(CIC)
b ... 4: 71...
A~
11"
1
(1.15)
m==1
The multipliers r/4 ""'1 are equal to 11'"/4 times the multipliers for A and
are presented in Table 3a for r = 20.
The section induced-drag coefficient is equal to the product of the sec­
tion lift coefficient and the downwash in radians.
~
TABLE 4.-MULTIPLIERS ,,"'. FOR INDUCED ANGLE OF A'rrAClt
al. -
to ( e,f.)
~ b
",-1
~~
2y
b
0
0.1564
0.3090
0.4MO
0.3878
0.7071
0.8090
0.8910
0.9511
0.9877
~
k
m
10
9
8
7
6
6
4
3
2
1
0
0.1864
0.3090
0.4640
'"
0.5878
~"'~
0.7071
0.8090
0.8910
0.9611
0.9877
~
~
~
~
10
9
8
7
6
5
4
143.239
-116.624
0
-12.884
0
-4.051
0
-1.638
0
-0.459
-58.633
145.026
-64.802
0
-8.320
0
-67.298
IM.611
-82.917
0
-7.872
0
-2.871
0
-0.620
-6.980
0
-67.157
160.761
-61.803
0
-7.208
0
-2.016
0
0
-10.168
0
-72.472
177.054
-71.743
-2.866
0
-9.916
0
-82.083
202.571
-81.434
0
-7.899
0
0
-4.840
0
-10.926
0
-97.965
243.694
-96.962
0
-7.089
-2.880
0
-1.062
0
.... 0
0
-7.370
0
-1.491
3
2
-1.804
0
-3.394
0
-4.968
0
0
-6.812
-18.134
0
-17.888
0
-126.637
0
316.612 -180.628
-122.880
463.533
-167.045
0
1
-1.468
0
-8.768
0
-7.713
0
-26.68li
0
-329.976
915.601
~
s
~
~
~
o
~
THE SIGNIFICANCE OF WIJ.VG-8ECTIOlV CHARAC7'ERIS7'lCS
27
The wing induced-drag coefficient may be obtained by means of a span­
wise integration of the section induced-drag coefficients multiplied by the
local chord. AB in the case of the lift coefficient, this process has been
reduced to the following summation:
(1.16)
The section profile drag coefficient can be obtained from the section
data for the appropriate wing section and lift coefficient. The "ring profile
drag coefficient may be obtained by means of a spanwise integration of
the section profile drag coefficient multiplied by the local chord. This
process has again been reduced to the following summation:
(1.17)
where c == mean geometric chord Sib
The section pitching-moment coefficient can be obtained from the
section data for the appropriate wing section and lift coefficient. For
each spanwise station the pitching-moment coefficient is transferred to
the wing reference point by the equation
em == c... - ~ [cz cos (a. - ai)
C
+ Cdo sin
(a. - ai)]
- ~ [et sin (a. - a,) C
where
Cde cos
(a. - ai)]
c.. == section pitching-moment coefficient about wing reference point
a. == geometric angle of attack of root section
x == projected distance in plane of symmetry from wing reference
point to aerodynamic center of wing section, measured parallel
to chord of root section, positive to the rear
z == projected distance in plane of symmetry from wing reference
point to aerodynamic center of wing section, measured per­
pendicular to root chord, positive upward,
The wing pitching-moment coefficient may be obtained by spanwise in­
tegration by use of the multipliers previously used.
(1.18)
where c'
(b/2
= mean aerodynamic chord, 2/8 Jo
c2 dy
THEORY OF WING SECTIONS
28
1.6. Applicability of Section Data. The applicability of section data.
to the prediction of the aerodynamic characteristics of wings is limited by
the simplifying assumptions made in the development of wing theory. It
is assumed, for instance, that each section acts independently of its neigh­
boring sections except for the induced downwash. Strict compliance with
this assumption would require two-dimensional flow, that is, no variation
of section, chord, or lift along the span.
Such spanwise variations result in spanwise components of flow. H
these components are small, the sections act nearly independently. Com­
paratively small spanwise variations of pressure, however, tend to produce
large crossflows in the boundary layer. The air adjacent to the surface
has lost most of its momentum and therefore tends to flow directly toward
the region of lowest pressure rather than in the stream direction. These
crossflows become particularly marked under conditions approaching sep­
aration. The flow of this low-energy air from one section to another tends
to delay separation in some places and to promote it in others, with the
result that the lying characteristics may depart seriously from the cal­
culated ones.
In the development of the lifting-line theory, it was assumed that the
effect of the trailing vortices was to change the local angle of attack,
neglecting any change of downwash along the chord of the section. This
variation of downwash is not negligible for sections close to a strong trailing
vortex. Strong concentrations of the trailing vortices are obtained when­
ever a large span'vise variation of lift OCCUI'S. Consequently the sections
are operating in a curved as well. as a rotated flow field whenever the span­
wise variation of lift is not small. This curved flow field may be interpreted
as an effective change in camber which is not considered in ordinary lifting­
line tbeories. This difficulty is avoided by lifting-surface theory, the treat­
ment of which is outside the scope of this volume.
Experience has shown that usable results are obtained from lifting-line
theory if no spanwise discontinuities or rapid changes of section, chord,
or twist are present, and if the wing has no pronounced. sweep, These
conditions are obviously not satisfied near the wing tips, near the ex­
tremities partial span flaps or deflected ailerons, near cutouts and large
fillets, or if the lying is partly stalled. Since the assumed conditions are
not satisfied near the- wing tips, it is obvious that section data-are not
applicable to wings of low aspect ratio. In fact, an entirely different
theory" applies to wings of very low aspect ratio. The ordinary three­
dimensional wing theories are obviously not applicable at snpereritical
speeds when the velocity of sound is exceeded anywhere in the field of flow.
Despite these limitations, wing-section data have a wide field of ap­
plication to wings of the aspect ratios and plan forms customarily used
for airplanes flying at subcritical speeds. The lift, drag, and moment
of
29
THE SIGNIFICANCE OF WING-SECTION CHARACTERISTICS
characteristics of such wings may be estimated from the section data with
a fair degree of accuracy by Anderson's methods" and with a high degree
of accuracy by the method of Sivells and Neely.!" The order of accuracy
/.6
1.4
I/~
1.2
-
~
1f~
I "'--
r: ---
.8
.6
.2
,J
Calculated ~neralized method)
Colculofedflineuized method)
I/o..
CoIcUIoIed~~
method)
----- Calculated(linearized
method)
j
-
i
,,1/1
1I
/
t>
)"
\
-.2
~
L-::!,
CExperimenlo1
~
o
\
j
7
7
.4
"0
...,/
~ Experimental
1.0
o.
..e.-
to
oO.a?
.04 .06 .08
~
12
.10
d-4
.14
0
4
~
8
/2
16
..
0
~
.1
FlO. 17.
Calculated and measured characteristics of a wing of aspect ratio 10 with NACA
44-series sections.
1.4
1.2
c;,iQiiiled...
.8
. //
c
,
I
~
/
"'-"
1.0
.6
}
.4
~
"of{K~1<lsIs
sr,.
o
-.2
o
,
CDo
r--
0
-,
~
--- ~
ExptJl'imtJnlrJl- ~
--- - ColculaltJd -
i
~I
I
~
~,
..
7
I
1
JT
)
..
0
\
.i i\
·
~
~
.004.008 .012
~
---
~
I
t
J
I
~aNf*t
"•
.2
z..
I/'
/ ~
.02
.04 .06
.08.10
-4
/
0
4
CD
8
(&
/2
0
-./
em
18. Experimental and calculated characteristics of a wing having NACA- 64-210 airfoil
sections, 2-deg wasbout, aspect ratio of 9. and taper ratio of 2.5. R
4,400,000; M = 0.17.
FIG.
=
to be expected from predictions made by the latter methodl" 110 is indicated
by Figs. 17 to 20. The three-dimensional wing characteristics were obtained
from tests in the NACA 19-foof pressure tunnel. The predicted charac­
teristics were based on section data obtained in the N.L~CA two-dimen­
sional low-turbulence pressure tunnel.P' The agreement between the ex­
THEORY OF WING SECTIONS
30
-
1.4
1.2
1.0
.s
-.....
/"
CoJculoled-.,'/
/( fests
/t HtJ\e I
~Force
~
.l'''\
~
b(
I ~ r--.......
..- -~
....,
Experimenlol--r--- r--~
r-
-
r1
I
CoIculoted---
~a
,
t/
o
-.2.0
SIJfVf!YS
~
I
I
~
.)/
t
.D04.oos.fJ12
~~
t
o.a.D4.D6
0.8./0
-4
/
0
4
8
0
"12
-:/
~
~
4
~
Flo. 19. Experimental and calculated characteristics of a wing having NACA 65-210 airfoil
sections, 2-deg washout, aspect ratio of 9, and taper ratio of 2.5. R =.; 4,400,000; M = 0.17.
1.4
'"
/
1.0
.8
.6
•4
..2
o
•
J)....
1.2
rt
r ......... f
!l'·oFon:tltesls <;;j
t
,
- ..20 .000.DOS.W
..............
~
j
~
~
~~
~
;~
-~oEx~
~
4
~
--Ct1It:uItIItId----...:
r---.
j
v
,,,/
I
l
8
/2
0 -;/
a
~
FIG. 20. Experimental and caleulated characteristics of a wing having NACA 65-210 airfoil
aootions. o-deg washout, aspect ratio of 9, and taper ratio of 2.5. R == 4,400,000; J.\f = 0.17.
~
0
II
~~
V
)"
,I,
cQiaialed'*"i
p
-02 .04 .06 .08./0
-4
0
4
~
perimental and predicted characteristics is so good that the utmost care is
required in model construction and experimental technique to obtain em­
pirical wing characteristics as reliable as those predicted from the section
data.
CHAPTER 2
SIMPLE TWO-DIMENSIONAL FLOWS
2.1. Symbols.
H
K
8
V
X
Z
a
total pressure
nondimensional circulation, r /2ra V
pressure coefficient, (H - p)/~2PV2
resultant or free-stream velocity
component of force in the x direction
component of force in the y direction
component of force in the 8 direction
constant
a
~.;V
Y
b constant
c constant
e' constant
e" constant
e base of Naperian logarithm, 2.71828
Z section lift
In logarithm to the base e
m source strength per unit length
p pressure
r radial eoordinnte
• position of source on z axis
t time
u component of velocity in the ~ direction
u' radial component of velocity
" component of velocity Ia the 11 direction
,,' tangential component of velocity, positive counterclockwise
tD component of velocity in the • direction
z length in Cartesian coordinates
11 length in Cartesian coordinates
• length in Cartesian coordinates
r circulation, positive clockwise
8 angular coordinate, positive counterclockwise
• velocity potential
t/I stream function
II doublet strength, 2ms
... ratio of the circumference of a circle to its diameter
p mass density of air
6J
angular velocity, positive clockwise
2.2. Introduction. A considerable body of aerodynamic theory has
been developed with which it is possible to calculate some of the important
31
32
THEORY OF lVING SECTIONS
characteristics of wing sections. Conversely it is possible to design wing
sections to have certain desirable aerodynamic characteristics. The pur­
pose of this chapter is to review the basic fluid mechanics necessary for
understanding the theory of lying sections.
2.3. Concept of a Perfect FlUid. The concept of a perfect fluid is an
important simplification in fluid mechanics. The perfect fluid is considered
to be a continuous homogeneous medium within which no shearing stresses
can exist. For the purpose of this chapter the perfect fluid is also con­
sidered to be incompressible.
The assumption of zero shearing stresses, or zero viscosity, eliminates
the possibility of obtaining any information about the drag of wing sec­
tions or about the separation of the flow from the surface. This assump­
tion is very useful, nevertheless, because it simplifies the equations of
motion that otherwise cannot generally be solved and because the resulting
solutions represent reasonable approximations to many actual flows, In
all cases under consideration, the viscous forces are small compared with
the inertia forces except in the layer of fluid adjacent to the surface. The
direct effects of viscosity are negligible except in this layer, and viscosity
has little effect on the general flow pattern unless the local effects are such
as to make the flow separate from the surface.
The assumption of incompressibility also results in simplified solutions
that are reasonable approximations to actual flows except at high speeds.
Although gases such as air are compressible, the relative change in density
occurring in a field of flow is small if the variation of pressure is small
compared with the absolute pressure. This assumption leads to increas­
ingly important discrepancies as the local velocity anywhere in the field
of flow approaches the velocity of sound. The effects of compressibility
will be discussed in Chap. 9.
2.4. Equations of Motion. One of the fundamental conditions that
must be satisfied is that no fluid can be created or destroyed within the
field of flow considered. This condition means that the amount of fluid
entering any small element of volume must equal the amount of fluid
leaving the element. For an incompressible fluid the amount of fluid may
be measured by its volume. The equation of continuity, expressing this
condition, may be derived from the following considerations.
Figure 21 shows a small element of volume having dimensions dx, dy,
and dz. The components of velocity along each of the three axes x, 'Jj,
and z are u, u, and w. The volume of fluid entering across each of the faces
of the element of volume, perpendicular to the x, y, and z axes, respectively,
is
u dydz
entering yz face
v dz dx
entering zx face
wdxdy
entering xy face
SIMPLE TWO-D{/1fENSIONAL FWWS
33
To the first order of small quantities, the amount of fluid leaving the
element of volume across the corresponding opposing faces is
(u
+ ~: dx ) dy dz
(v + :;; d y) dz dx
(w
+ :: dz) dx dy
leaving yz face
leaving
zx face
leaving xy face
r
FIO.21.
Z
Element of volume considered in derivation of equation of continuity.
The condition of continuity requires that the volume of fluid leaving
the element must be equal to the volume entering. Hence the equation of
continuity for an incompressible fluid is
au + aV + ow = 0
ax
ay
iJz
(2.1)
The other .fundamental condition to be satisfied is that the motion of
the fluid must be in accordance with N ewton's laws of motion. This con­
dition may be stated as follows:
Du
Dt dx dy dz
X =
p
y=
p~
Z
=p
dxdydz
(2.2)
Dw
Dt ds: dy dz
where X, Y, Z == components of force on the element of fluid d» dy dz
in" the z, y, and z directions, respectively
t == time
and the differentiations are performed following the motion of the element.
THEORY OF WING SECTIONS
34
For axes stationary with respect to the observer, the components of veloc­
ity u, v, and ware, in general, functions of t, x, 'Y, and e. The total deriva­
tives given in Eqs, (2.2) may therefore be written
au
au
au
au Du
at + u ax + v ay + W iii == Dt
av
at
av
av
au"Dv
== D..t
+ u ax + v ay + 1D az
aw
at + U
aw
iJx
aw
8w
Dui
+ v iJy + w oZ == Dt
H gravitational forces are neglected, the only forces acting on the ele­
ment of volume are normal pressure forces. Let p dy dz be the force acting
on the face of the element (Fig. 21) in the yz plane. Then to the first order
[p + (apjox) dx] dy dz will be the force acting on the opposite face. The
resultant force on the element in the x direction is therefore
ap
X == .- ax dx dy dz
Similarly the resultant forces in the y and z directions are
y==-iJPdydzdx
Oy
Z == - ap dz dx dy
az
The equations stating that the motion of the fluid is in accordance with
Newton's laws may be written in the following form by substituting the
foregoing expressions for the forces and accelerations in Eqs. (2.2)
(au + u au + v au + to au)
at
ax ay
az
iJp == p (~ + u au + v av + w av)
_ lJp == p
ax
_
ay
- ap
- ==
iJz
at
p
ax
iJy
iJz
(2.3)
(ow
awax aw
aw)
-+'1.L-+v-+w­
at
ay
Bz
If we consider only steady motion, the derivatives with respect to
time are equal to zero. In this case the differential equations for the path
of an element are
dx
dy dz
(2.4)
-.=:-==U
f)
10
These equations merely state that the displacement along any axis is
proportional to the component of velocity along that axis." Preparatory to
BIJIPLB TWO-DI},(EN810NAL FWMTS
35
integrating Eqs. (2.3) along the path of motion for steady flow, we shall
multiply the equations for the x, 1/, and z components by dx, dy, and dz,
respectively. For example,
au)
ap
(u-dx+v-dx+tD-dx
au
au
--dx=p
ax
ax
ay
az
According to Eqs. (2.4), we have
v dx=udy
wdx=udz
Hence
- ap
- dx =
ax
p
(u au
- dx + u iJu
- dy + u au)
ax
ay
. -az dz =
p
d
(1.,)
2
~ u-
Similarly
- ap
dy = p d(!2 v2)
ay
- ap dz =
8z
pd(!ur)
2
Adding the three foregoing relations, we have
- dp = !!. d(u2 + tP + w) = !!- d(V2 )
2
2
where V = magnitude of velocity.
Integrating this equation gives Bernoulli's equation
p+%pV2 = H
(2.5)
where H, the constant of integration, is the total pressure.
The application of Eq. (2.5) is limited by the assumptions made in its
derivation, namely,
1. Perfect incompressible fluid.
2. Steady motion.
3. Integration along path of motion (streamline). It follows that this
equation may be applied only along streamlines in unaccelerated flow where
the effects of viscosity and compressibility are negligible.
2.6. Description of .Flow Patterns. It is evident from the foregoing
equations that the flow pattern would be completely determined if the
values of u, n, and to were known at every point. The theory of wing
sections assumes that the flow is two-dimensional, that is, W = o. Even
with this simplification, it is still necessary to specify both the components
of velocity u and v to define the flow, By the use of the equation ofcon­
tinuity (2.1) it is possible to simplify the problem further so that it is
necessary to specify only a single quantity at each point to determine the
flow pattern.
36
THEORY OF lVING SECTIONS
Consider the flow across the arbitrary line oba that connects the origin
o with a point a (Fig. 22). The amount of flow across this line will be the
same as that across any other line connecting the two points, for example,
line oea, provided that the equation of continuity holds for all points
within the region enclosed by the two lines. 'Vith a fixed origin 0 the
amount of fluid crossing any line joining the points 0 and a is therefore a
function only 'of the position of the point a for a given flow pattern. The
amount of fluid passing between the points 0 and a is given by the following
expression
1/1=
y
J: udy-vdx
or
elf = 11, dy - v dx
(2.6)
Because the value of 1/1 is independent of
the path of integration, Eq. (2.6) remains
valid even though dx and dy are varied inde­
pendently. The general expression for the
total differential of 1/1 is
d.I6
FlO. 22. Derivation of stream
function.
'I'
= at/! dx + iJl/I d
ax
ay Y
(2.7)
This equation is also valid for independent variations of dx and dy. The
coefficients of dx and dy in Eqs. (2.6) and (2.7) must therefore be equal,
or
:~~a~}
(2.8)
ax _"
A single function 1/1 has therefore been found by means of which it is
possible to define both components of velocity at all points in the field of
flow. This function'" is called the "stream function." Lines in the flow
along which t/t is constant are called Cl streamlines," and these lines are the
paths of motion of fluid elements for steady motion,
In general, the component of velocity in any direction may be obtained
by differentiating the stream function in a direction 900 counterclockwise
to the component desired. In polar coordinates, therefore, the expressions
for the radial and tangential components of velocity are
u
, =lay,
-­
roO
v, = - -~
­
dr
radial
t
tangential .\
(2.9)
SIMPLE TWO-DIMENSIONAL FWWB
37
Most flows of a perfect fluid are of the type known as II irrotational
motion." In irrotational motion the fluid elements move with translation
only. Their angular velocity is zero. This absence of angular velocity does
not, of course, prevent the ele­
ments from moving in curved
paths. To obtain expressions for
the angular velocity of an element
w ---,-.-..
in terms of the velocity deriva­ !?doting
tives, consider the motion of a
disc
solid disk. If ,ve assume clock­
wise motion to be positive, the
angular velocity of the disk may
be expressed as (Fig. 23)
CA)
=
!(au
_av)
2 ay ax
The expression
(~: -:=)
FIG.
23. Velocity derivatives for rotatiDg disk.
is called It vorticity." The vorticity is seen to be twice the angular velocity.
The integral of the tangential component of velocity around any closed
curve is defined as the circulation. A simple relation exists between the cir­
culation around a curve and the vorticity over the area enclosed by the
curve. Consider the small element shown in Fig. 24.
+ (u + ~; dy ) dx -
at: =
v dy
dr =
(auay _oxau) dx dy
or
(v
+ :~ d.'t) dy -
u dx
That is, the element of circulation is equal to the vorticity multiplied by
the element of area. For any finite area, the
u+/tdY
circulation is given by the following expression:
r
dy V
u
=
!!(au
ay -
av) dx dy
ox
(2.10)
The angular velocity of any element can be
changed only by the application of tangential
FIG. 24. Calculation of cireu(shearing) forces which, by definition, are ab­
lation about element of area.
sent in a perfect fluid. It follows that, if the
flow is once irrotational, it will rem~in irrotational and the circulation around
any closed path will equal zero. If the flow is not irrotational, the circu­
lation around any path moving with the fluid will remain constant.
ds
THEORY OF WING 8BC7.'IONS
38
H the flow is irrotational, it is possible to derive a second quantity
which, like the stream function, can be used to describe the flow pattern
completely. Consider the flow field indicated in Fig. 25. The line integral
of the velocity over the path oap must be equal to the line integral of the
velocity over the path obp if .the motion is irrotational in the region be­
tween the two paths. The value of this integral, called the U velocity
potential 4>," therefore depends only on the p0­
sition of the point p relative to the origin o, The
value of the velocity potential is
I
o
rP= J:udx+vdy
d;=Vcos a; ds
Fro. 25. Definition of ve­
locity potential.
or
d</>=1tdx+vdy
(2.11)
By a process of reasoning similar to that previously given for the stream
function, we obtain
1t
== a4>}
ax
(2.12)
aq,
v=­
iJy
In general, the component of velocity in any direction may be obtained
by differentiating the velocity potential in the direction of the desired
component. This property of the velocity potential makes it particularly
useful for the study of three-dimensional Bows. In polar coordinates, the
expressions for the radial and tangential components of velocity are
,aq,
ar
&dial}
U =
r
v' == ! oq,
tangential
roO
(2.13)
The equation of continuity for two-dimensional How
au + 8v == 0
ax ay
(2.14)
assumes a particularly simple form when expressed in terms of the velocity
potential
a2q, + atq, = 0
(2.15)
ax'- ay2
This equation is Laplace's equation in two dimensions.
The equation stating that the flow is irrotational is
au_ovc:O
ay
ax
SIMPLE TWO-DIMENSIONAL FWWS
39
When written in terms of the stream function, this equation becomes
a~
iJ21/I
a1/ + ax' = 0
(2.16)
In writing Eq. (2.15), it is implicit in the definition of q, that the motion
is irrotational and the equation itself states that no fluid is being created or
destroyed. In writing Eq. (2.16), it is implicit in the definition of '" that
no fluid ia being created or destroyed, and the equation itself states that the
flow is irrotational. Equations (2.15) and (2.16) impose rather general
conditions on functions chosen to represent actual flow patterns. Any
function of cP or t satisfying Eqs. (2.15) or (2.16) represents a possible flow,
It can be seen from Eqs. (2.8) and (2.12) that
ocP
oift
u=-=­
ax ay
aq,
at
V=-=-­
ay
ox
These equations indicate that lines along which ep is constant intersect lines
along which 1/1 is constant at right angles.
Because Eqs. (2.15) and (2.16) are linear, functions of ep or 1/1 repre­
senting possible flow patterns may be added to obtain new flow patterns.
This method of obtaining new flow patterns by superposition of known
flows is fundamental to the theory of wing sections because it leads to
simple solutions of complicated problems.
2.6. Simple Two-dimensional Flows. A few simple flows upon which
the theory of lying sections is based are described in this section.
a. Uniform Stream. Consider the functions
and
If=bY-ax!
(2.17)
ep == bz + ay
which satisfy Eqs. (2.15) and (2.16). The component of velocity along the
x axis is
iJ1/I iJep
u-=-=-=b
iJy
ox
The component of velocity along the y axis is
01/1 iJep
V:=:--=-==4
ax
ay
Equations (2.17) therefore represent a uniform field of flowhaving a velocity
V equal to v'a2 + b2 inclined to the x axis at an angle whose tangent is a/b.
b. Sources and Sinks. The -eoncept of sources and sinks is one of the
building blocks used to construct desired flow patterns. A source is con­
sidered to be a point at which fluid is being created at a given rate. A sink
40
THEORl" OF WING SECTIONS
is a point at which fluid is being destroyed. The flow about a point source
or sink is assumed to be uniform in all directions and to obey the equations
of continuity and of irrotational motion everywhere except at the POint
itself. In two-dimensional motion, the flow is assumed to be the same in
all planes perpendicular to the z axis. The point source in two-dimensional
motion is therefore a line parallel to the z axis from which fluid emanates at
a uniform rate throughout its infinite length. In accordance with these
assumptions, the radial component of velocity u' at any point at a distance
T from a source creating fluid at a rate m per unit length perpendicular to
the plane of flow may be expressed as
u'
~
2...r
=
.(2.18)
Because the sink is simply a negative source, the equation for the radial
component of velocity is the same as Eq. (2.18) except that the velocity is
directed toward the sink. The tangential component of velocity resulting
from a source or sink is zero.
.
Taking the origin at the source ,ve have
u == u' cos 8
v == u' sin (J
r=Vr+y2
where
cos (J ==
v'x2 + 11
• (J
SID
==
v'z! + y2
y
Substituting in Eq. (2.8), we have
ih/I
mx
u = iJy = 211'(x! + V)
Integration of this expression with respect to y results in the expression
1/1 = m tan-Ill
x
27
(2.19)
The same expression may be obtained by a corresponding process using
the expression for v and integrating with respect to x. It is seen that f/I is a
multiple-valued function of x and y. This peculiarity results from the
fact that the equation of continuity is not valid at the source itself.
The expression for the flow from a point source in terms of the velocity
potential <p may be obtained as follows:
oq,
mx
t t = - = -2- - ­
ax
cP = m In
2r
2r(x
+ y2)
vx + y2
2
(2.20)
SlJfPLE TWO-DIMENSIOlfAL FWWS
41
From Eqs. (2.19) and (2.20) it is seen that the streamlines for a source
situated at the origin are radial lines and that the equipotential lines are
circles about the origin.
c. Doublets. A doublet is the limiting case of the flow about a single
source and a sink of equal strength that approach each other in such a
manner that the product of the source strength and the distance between
the source and sink remains constant. The axis of the doublet is the line
joining the source andsink. To obtain expressions for the flow about a
doubtlet, consider the arrangement of "a source and a sink shown in Fig. 26.
The source and sink are situated on
y
the x axis at - 8 and 8, respectively.
The flow resulting from the combi­
nation is
1/1 = !!!.(tan-1 ~
2r
x
+8-
tan-1 _ Y- )
X -
8
Since
a-b
tan:" a - tarr? b = tan-1 - ­
I +ab
we have
l/I = m tarr:' - 2ys
2r
x 2 + y1. - 8 2
FlO. 26.
Streamlines of flow for a
source and sink.
For values of s small with respect to x and y,
1f =
If the product of m and
8
;:
C-; ~y:~)
remains constant as
8
approaches zero,
(2.21)
where p, = 2ms
d. Circular Cylinder in a Uniform Stream. The flow about a circular
cylinder in a uniform stream is obtained by superposing a uniform stream
[Eq. (2.17)] on the flow about a doublet [Eq. (2.21)] with the uniform
stream flowing from the source to the sink.
1f =
Y( IT -
21r(x:+
if»)
This equation may be written as follows in terms of polar coordinates:
'" = Y r
where
a2
r2
(1 - ~) sin 8
= JJ/21r l'
= x 2 + y2
(2.22)
THEORY OF WING SECTIONS
42
It is obvious from Eq. (2.22) that part of the streamline'" == 0 is the circle
r = a. At large distances from the origin the flow is uniform and parallel
to the x axis. Equation (2.22) therefore represents the flow about a
circular cylinder in a uniform stream. The streamlines for this flow pattern
are shown in Fig. 27.
The velocity distribution about the cylinder may be obtained by finding
the tangential component of velocity on the circle T == a.
11
= - : = - V ( 1 + ~) sin (J = - 2V sin (J
(2.23)
i
FIG.
27.
Streamlines for the flow about a circular cylinder in a uniform stream.
Because the circle r == a is a streamline, \ve may apply Bernoulli's equation
(2.5) to obtain the pressure distribution about the cylinder.
p
+ ~2P(4yt sin! 6) == H
If we define a pressure coefficient S as
S=
H-p
Mp VJ
(2.24)
the distribution of S over the surface of the cylinder is given by
S == 4 sin 2 8
(2.25)
Integration of the pressure distribution over the surface of the cylinder
will show that the resultant force is zero.
e. Voriex. A vortex is a flow pattern in which the elements of fluid
follow circular paths about a point, and the flow is irrotational at all points
except the center. The equations defining the flow pattern may be derived
directly from Laplace's equation for the stream function. Equation (2.16)
may be written 88 follows in terms of polar coordinatesr"
(2.26)
8IMPLE TWO-DIMEN8IONAL FWW8
43
Because the radial component of velocity is zero,
!ol/l = 0
ro8
and hence
lo~ =0
r afP
Equation (2.26) therefore becomes
lPt/t+!d1/l=O
dr
rdr
Integrating once with respect to r we obtain
d
ln 1/l + 1n r = c
dr
or
d1/l
dr =
I'
rc' = 2111
== -
,
v
(2.27)
where the constant of integration c' is assigned the value I' /21r. The veloc­
ity therefore varies inversely with the disy
tance from the center of the vortex. The
circulation I' about the vortex is equal to
- 2rrv'. The circulation is considered to be
positive in the clockwise direction, whereas v'
is positive in the counterclockwise direction.
The stream function may be obtained by in­
tegration of Eq. (2.27)
r
1/1 == 211" In r
+ e"
(2.28)
x
The pressure in the field of flow may be FIG. 28. Equilibrium conditions
for a vortex.
calculated from the condition that the pressure
gradients must maintain the fluid elements in equilibrium with the acceler­
ations (d'Alembert's principle). The centrifugal force acting on each ele­
ment must be (Fig. 28)
pv'"
--rd8dr
r
The pressure force acting on the element is
Equating the pressure and acceleration forces, we have
pV'2
r=
dp
dr
44
THEORY OF WIlvG SECTION8
Substituting the value of v' from Eq. (2.27) gives
p~
47N8
p= -
=
dp
dr
p~
srr+ H
where H = constant of integration
This equation may be written in the form
P+~pV'2==H
which shows that Bernoulli's equation may be applied throughout the
field of flow,
FIG. 29. Streamlines for the flow about a circular cylinder in a uniform stream with eireu­
lation corresponding to a wing-eection lift coefficient of 0.6.
f. Circular Cylinder with Circulation. The flow pattern represented by a
circular cylinder with circulation is the basic flow pattern from which the
flow about "ring sections of arbitrary shape at various angles of attack is
calculated. Such & flow pattern is obtained by superposing the flow pro­
duced by a point vortex upon the flow about a circular cylinder. Adding
the flows corresponding to Eqs. (2.22) and (2.28), we obtain
'" -=
vr(l - a~ sin 8 + 2r~ In !a
~J
(2.29)
where the constant e" in Eq. (2.28) has been taken equal to
-r 1n a
21r
A typical flow pattern for a moderate value of the circulation r is given
in Fig. 29.
The velocity distribution about the cylinder is found by differentiating
the expression for the stream function [Eq. (2.29)] as follows:
(ar.
2
&/I
r
- = V 1+-) sin8+­
ar
2rr
SIMPLE TTflO-DIMENSIONAL FWWS
45
The tangential component of velocity v' (positive counterclockwise) at the
surface of the cylinder is obtained from Eq. (2.9) and the substitution of
r == a.
Vi == -
2V sin 8 + ~
(2.30)
21ra
It is seen that the addition of the circulation I' moves the points of zero
velocity (stagnation points) from the positions 8 = 0 and 11" to the positions
8 = sin-1 -
r­
4raV
The pressure distribution about the cylinder may be found by applying
Bernoulli's equation (2.5) along the streamline 1/1 = o.
p
+ 21 P (4 V2 SIll
.
2
8
-
2Vr sin 8 +
ra
r
2
)
4ra2 ==
H
(2.31)
Setting
r
21Ta V = K
(2.32)
the pressure coefficient S [Eq. (2.24)] becomes
S == 4 sin 2 8 - 4K sin 8 + K2
(2.33)
The symmetry of Eq. (2.33) about the line 8 = 1f/2 shows that there can
be no drag force. The lift on the cylinder can be obtained by integration,
over the surface, of the components of pressure normal to the stream.
fo2rSa sin 0 dO
= 72PV fo2r (4a sin" 0 -
l = 72P V2
2
4aK sin" 9 + aJ(2 sin 8) dO
l == 72PV2ak[28 - sin 28]~r
l == 2pV 2akr == PlTr
It can be shown that the relation
l == pvr
is valid regardless of the shape of the body. 61
(2.34)
CHAPTER 3
THEORY OF WING SECTIONS OF FINITE TIDCKNESS
3.1. Symbols.
A., B.
H
8
V
a
coefficients of the transformation from x' to z
total pressure
pressure coefficient (H ­ p)/J-ipVt
velocity of free stream
radius of circular cylinder
Cr
section lift coefficient
e base of Naperian logarithms, 2.71828
i V=-i
In logarithm to the base e
m source strength per unit length
(~J
- (is
p
1-
p
local static pressure
radius vector of z (modulus)
component of velocity along x axis
component of velocity along y axis
local velocity at any point on the surface of the wing section
complex variable
Cartesian coordinate
real part of the complex variable % or
Cartesian coordinate
magnitude of the imaginary part of the complex variable z
complex variable
complex variable in the near-circle plane
complex variable for the flow about a circle whose center is shifted from the
center of coordinates
circulation, positive clockwise
section angle of attack
distance of the center of the circle from the center of coordinates
complex variable
mngnitude of the imaginary part of the complex variable r
angular coordinate of the complex variable z (argument)
angular coordinate of z' (see Fig. 32)
aeA is the radius vector of z
doublet strength per unit length
real part of the complex variable r
ratio of the circumference of a circle to its diameter
mass density of air
real part of the complex variable w
potential function
46
r
u
v
v
w
x
x
y
y
z
z'
z*
r
ao
E
r
"
8
fJ
~
p.
~
11'
p
q,
q,
r
THEORY OF WING SECTIONS OF FINITE THICKNESS
"
1/1
1/1
1/1
1/10
angular coordinate of z
magnitude of the imaginary part of the complex variable w
stream function
ae~ is the radius vector of z' (see Fig. 32)
average value of 1/1
00
infinity
47
3.2. Introduction. It was shown in Chap. 2 that the field of flow
about a circular cylinder with circulation in a uniform stream is known.
It is possible to relate this field of flow to that about an arbitrary "ring
section by means of conformal mapping. In relating these fields of flow,
the circulation is selected to satisfy the Kutta condition that the velocity
at the trailing edge of the section must be finite. Such characteristics as
the lift and pressure distribution may then be determined from the known
flow about the circular cylinder. The resulting theory permits the ap­
proximate calculation of the angle of zero lift, the moment coefficient, the
pressure distribution, and the field of flow about the section under con­
ditions where the flow adheres closely to the surface.
3.3. Complex Variables. Conformal mapping, which makes the cal­
culation of'wing-section characteristics possible, depends on the use of
complex variables. A complex number is a number composed of two parts,
a real part and a so-called imaginary part. The real part is just an ordinary
number that may have any value. The imaginary part contains the factor
v=!, which is given the symbol i. A complex variable z may therefore
be written in the form
z = x+ iy
where the symbols in the expression x + iy obey all the usual rules of
algebra. It should be remembered that~"2 is equal to - 1. Such a variable
z can be represented conveniently by plotting the real part z as the abscissa"
of a point and the magnitude y of the imaginary part as the ordinate.
By De Moivre's theorem, z may also be written in the form
z
=
reiIJ
where rand 8 are the polar coordinates.
Let us consider the complex variable
w=<I>+il/l
and let w
=
f(z) , that is,
4> + il/l
=
f(x
+ iy)
Then
:2q,2
+ i:~.. = /:"r(x + iy) == f'(z)
uX
u~t
and
cP
(j2
.a2;p ~ ( +.)
f"( )
iJ 2 + ~ a y 2 = JJl1J X
~y = z
y
THEORY OF WING SECTIONS
48
adding, we have
02</J + iJ2q, + i(iJ'-t/I + iJ~)
iJ y 2
ax2 iJ y 2
ax2
=
0
In any equation involving complex variables, the real and imaginary parts
must be equal to each other independently. Therefore
iJ2</J
a*lq,
or + iJ y
2
=
0
and
iJ~+ay=o
ar
iJlI
Because these equations are the same as those of Eqs. (2.15) and (2.16),
any differentiable function
w = j(z)
where
and
i = x+ iy
may be interpreted as a possible case of irrotational fluid motion by giving
the meaning of velocity potential and stream function, respectively.
The derivative dw/dz has a simple meaning in terms of the velocities
in the field of flow.
dw = d</>+ i a1/l
dz = dx + i dy
Further
a</J
at/>
d</J = - dx + - dy
</> and'"
ax
ay
a."
dl/l = - dx + - dy
ax
ay
81/1
Therefore
[(iJt/>/ax)dx
dw
dz
+ (o(j)loll)dy] + i [(iJ~/iJx)dx + (iJ1/I/iJy)dyJ
d.x+ i dy
=
In order for dw/dz to have a definite meaning, it is necessary that the value
of du: 'dz be independent of the manner with which dz approaches zero. If
dy is assumed to be zero, the value of the differential quotient dw/dz is
o</J + i01/! = u - iv
ax
a~l~
= dw
dz
(3.])
according to Eqs. (.2.8) and (2.12)./ Similarly, if dx is assumed to be zero,
the value of the differential Quotient duridz is
~ iJq, + 0'" = _ iv + 'It
toy
ay
THEORY OF WING SECTIONS
OJl~
FINITE THICKNESS
49
The expressions for simple two-dimensional flows given in Sec. 2.6 may
be expressed conveniently in terms of complex variables.
Uniform stream parallel to x axis
to
= Vz
(3.2)
Source at the origin
w=m ln z
2r
(3.3)
Doublet at origin with axis along x axis
w=~
(3.4)
2rz
Circular cylinder of radius a in a uniform stream
w=V(Z+:2)
(3.5)
Vortex at origin
10 =
ir
-In z
2r
(3.6)
Circular cylinder with circulation
w = V (z +
a2)ir
-z + -21r In -az
(3.7)
3.4. Conformal Transformations. .1\ conformal transformation con­
sists in mapping a region of one plane on another plane in such a manner
that the detailed shape of infinitesimal elements of area is not changed.
This restriction does not mean that the shape of finite areas cannot be
considerably altered. It has been ShO"·11 previously that the equipotential
lines and streamlines intersect at right angles, thus dividing the field of
flow into a large number of small rectangles. It has also been shown that
the equation
w = j(z)
represents a possible flow pattern. The equation
w = g(r)
represents another flow pattern where
~
r is the complex variable
+ i11
The coordinates in the z plane are considered. to be x and y, and those
in the r plane are ~ and 'I. If the equipotential lines and streamlines are
plotted in either of the planes, they will divide the plane into a large num­
ber of small rectangles. These rectangles will be similar at .corresponding
points in both planes. The corresponding points are found from the
relation
f(z) = g(t)
THEORY OF' WING SECTIONS
50
This equation accordingly represents the conformal transformation
from the z plane to the r plane or the converse. In practical use, the Ho\v
function in the z plane is known and the corresponding flow function in
the r plane is desired. To plot the flow known on the z plane on the r plane,
it is necessary to solve this relation for r and to obtain an equation in the
form
r == h(z)
In the transformations we shall consider, this relation will be given,
The velocities in the z plane, by Eq. (3.1), are
dw
_
dz ==u-~v
The corresponding velocities in the
dw
r plane are
dwdz
(3.8)
dt == liz dr
As a simple example of a conformal transformation, consider the re­
lations
W=V(Z+~)=Vt
These relations transform the flow about a circular cylinder on the z plane
[Eq. (3.5)] to uniform flow parallel to the ~ axis on the plane [Eq. (3.2)].
Corresponding points of both planes are obtained from the relation
r
r == z+­atz
" and are indicated in Fig. 30.
3.5. Transformation of a Circle into a Wing Section. A circle can be
transformed into a shape resembling that of a wing section by substitution
of the variable
a2
t==z+­z
(3.9)
into the expression for the flow about a circular cylinder having a radius
slightly larger than a, and so placed that the circumference passes through
the point x == a. If, in addition, the center of the larger cylinder is placed
on the x axis, the transformed curve will be that of a symmetrical wing
section (Fig. 31). In the present example, let the center of the larger
cylinder be placed at the point x = - E where E is a real quantity. The
radius of this cylinder will then be a + E. The equation of flow about the
larger cylinder with circulation is then
10
==
V(z* +
E
+ (a + E)2) +
Z*+E
ir In z* + E
211"
a+E
THEORY OF WIl{G SECTIONS OF FINITE THICKNES8
- - - - - - - ----4~-- + - - ~
51
--------­
Z PLANE
I
{=Z+-j-
2
I
.I;
( PLANE
FIG.
30. Conformal transformation of the flow about a circular cylinder to uniform flow.
FIo. 31. Conformal transformation of a circle into a symmetrical wing section.
THEOR1' OF WING SECTIONS
52
The more general expression for the flow about the circular cylinder with
the flow at infinity inclined at an angle aD to the oX axis is found by substitut­
ing the expression
Z + E = (z* + E)eiaro
1»
= V
[(Z + E)e-iao + (a ~~):eiao] + ;~ In (z ~ 2eE- iao
(3.10)
Substitution of Eq. (3.9) into Eq. (3.10) would result in the equation
for the flow about the wing section but would lead to a complicated ex­
pression. A simple way of obtaining the shape of the 'ling section is to
select values of z corresponding to points on the larger cylinder and find
the corresponding points in the r plane by the use of Eq. (3.9). The
velocity of any point on the wing section can be found from Eq. (3.8).
dw
dr
=
[v (e-
dw dz =
dz dr
Wzo _
(a
e
+ E)2
iae
)
(z + E)2
+
ir
] (~)
z~ - a2
2r(z + E)
It can be seen from this equation that the velocity at the point z = a
is infinite unless the first factor is zero. The point z = a corresponds to the
trailing edge of the wing section. The Kutta-Joukowsky condition states
that the value of the circulation is such as to make the first factor equal
to zero, which is the condition that ensures smooth flow at the trailing
edge. The value of the circulation satisfying this condition is found as
follows:
V
+ E)2eiaol + ir - 0
iao _
.
(a + E)2 J 2"..(a + E) ­
[e-
(a
ir = 211'"(a + E) V(e iao - e- iao)
Since
eiao - e- iao
•
•
••
2
== Sinh tao == t sin ao
or
I'
== 4"..(a + E)V sin
Qo
The leading edge of the wing section corresponds to the point
a2
r=-a-2E­ a + 2E
neglecting powers of
E
greater than one
r=
- 2a
Because the trailing edge corresponds to the point
r == 2a
THEORY OF WING
8ECTIO~V8
the chord of the wing section is 4a. The lift on the wing section is
and the lift coefficient is
Ct = 211" ( 1
53
OF FINITE THICKNESS
p
vr,
.
+ :) sin «XcI
In the limiting case as E/a approaches zero, the slope of the lift curve
dc,/da o is 2...per radian for small angles of attack. Detailed computations"
z-Pkne 1------....------.
z~Plme t - - - - - - f ! I i o - - - - - I
~=x+iy
32. Dlustration of transformations used to derive airfoils and calculate pressure dis­
tributions.
FIG.
will show that the thickness ratio of the wing section is nearly equal to
(3V3/4) (E/a). For wing sections approximately 12 per cent thick, the
theoretical lift-curve slope is about nine per cent greater than its
limiting value for thin sections.
3.6. Flow about Arbitrary Wing Sections.
vious section that the transformation
a2
s=z+­z
It was seen in the pre­
THEORY OF WING SECTIONS
54
transforms a circle in the z plane into a curve resembling a wing section in
the plane. Most wing sections have a general resemblance to each other.
If the aforementioned transformation is applied to a wing section, the
resulting curve in the z plane will therefore be nearly circular in shape.
Theodorsen recognized this fact and showed that the flow about the nearly
circular curve, and hence about the wing section, can be derived from the
flow about the true circle by a rapidly converging process. The basic
method is presented in reference 122, and a detailed discussion of the
method is given in reference 125. The derivation of Theodorsen's relation for
the velocity distribution about the wing is divided into three parts (Fig. 32).
1. Derivation of relations between the flow in the plane of the wing
section (r plane) and in the plane of the near circle (z' plane).
2. Derivation of relations between the flow in the z plane and in the
plane of the true circle (z plane).
3. Combination of the foregoing relations to obtain the final expression
for the velocity distribution in the r plane in terms of the ordinates of the
wing section.
The basic relation between the z' plane and the r plane is
r
r = r + --;~z
The coordinates of
(3.11)
r are defined by the relation
f=x+iy
and the coordinates of z' are defined by the relation
z' == tJt!I'+#}
By Eq. (3.11),
Since
e#}
and since
= cos () + i
sin
(J
r = a(~ + e~) cos 8 + ia(e1' - e-') sin ()
== 2a cosh '" cos (J + 2ia sinh '" sin 8
x
y
t == x+iy
= 2a cosh 1J' cos
= 2a
8}
sinh '" sin ()
(3.12)
Expressions for'" and fJ in terms of x and y are desired and may be obtained
as follows:
x
cosh'" == 2a cos 8
sinh '"
= 2a SID
~
6
,
I
THEORY OF WING SECTIONS OF FINITE THICKNESS
55
Since
l
cosh
l
sinh
'" -
'"
=1
(2a :OS S- (2a ~in 8Y= 1
!;
!
or
2 sin 8 = P +
l
~r + (~y
(3.13)
where
Because sin 8, and hence cos 8, are known in terms of the ordinates x and
y of the wing section, the values of sinh'" and cosh '" can be found from
Eq. (3.12).
The factor relating velocities in the z' plane to those in the r plane is
dz'/dr. From Eq. (3.11),
dr
dz'
=
1_
a2 = ,!(z, _ ~)
Z'2
=
-=
=
Z'
.!(aeY+ iB -
z'
~ a(eI' -
z
z~
Z'
ae-+- iB)
rI') cos 8 + ia(e" + e-I') sin 8
(2a sinh '" cos 8 + 2ia cosh '" sin 8)
(3.14)
The second step is to find the relation between the flows in the z' plane
and those in the z plane. The coordinates of z are defined by the relation
z = aeA+;'"
The transformation relating the z' to the z plane is the general transforma­
tion
i; (A.+iB.)(l/p)
z'=ze 1
By definition
Consequently
or
co
'"- }. + i«(J -
q» = "\'
4
(An + iBn)..!. (cos
rIO
1
where z has been expressed in polar form
z = r(cos
q>
+i
sin
q»
nq> -
i sin
nq»
THEORY OF WING SECTIONS
56
Equating the real and imaginary parts, we obtain the two Fourier ex­
pansions
~(A"
1/1 - ). = L,
-;:;- cos
n" + B..)
r" sm n"
1
and
00
(J-qJ=
1(Bn
(3.15)
A".)
-cosntp--Slnn'l'
rn
rn
1
These relations show that l/I - ~ and 8 - tp are conjugate functions. In order
for the deviation of the near circle from the true circle to be a minimum,
the value of ~ corresponding to the radius of the true circle is taken to be
1
2
1/10
). =
= -1
21r
.­1/1
0
df)
where the values of 1/1 correspond to points on the near circle. The required
transformation from the near circle to the true circle is then
(3.16)
If 1f/ is known as a function of tp, 8 - I{) can be found by a method de­
veloped by Naiman." The 1/1 function is assumed to be approximated by
a finite trigonometric series of the form
t/I(tp)
=
1/10 + Al cos
I{)
+ .. · + An-I cos (n - 1)1{)
+ A" cos <p + B 1 sin 'P + . · · + Ba-l sin
1
(n - l)cp
n-l
1/I(rp)
= 1/10 +
.--1
(A", cos m" + B", sin mrp) + A" cos
~
If '" is specified at 2n equally spaced intervals in the range 0 ~ cp ~21r:
that is, 0, 1r/n, 2r/n, ...., [(2n - l)1r]/n then
2n-l
\11o
where 1/Ir = value of.p at
f/' =
= 2~
1
\IIr
r-=O
T7r/n
2n-l
.4", =
! \" \IIr cos m!:!
nL,
n
rlCO
211-1
B", =
! " \IIr sin m nr
nL,
n
r-O
A" =
~
L(-
2n-l
r-=O
l)r\llr
B.=O
THEORY OF WING SECTIONS OF FINITE THICKNESS
57
Now, by Eqs. (3.15),
2:
1I-l(
2: 2: "'r
11-1
f{J -
(J
= E(Ip) =
(A.. sin nkp - B m cos mlp) + A .. sin nip
m-l
2n-l
1.
= 17:
mf{J
SIll
m:=1
cos m
r=O
rT
n
2n-l
- cos mlp~ "'r sin m r;
11-1 2n-l
=
~
LL
2n-l
)
2~"'t sin nf{J ~ (- 1)r"'r
+
2n-l
"'r sin m(f{J -
r:) + 2~ sin nf{J 2:(- 1)r"'r
m=l r=O
r=-O
Upon interchanging the order of summation, there is obtained
2n-l
E(Ip) =
~
.-1
2: 2:
"'r
r-O'
r:) + 2~ sin nip 2: (- 1)r"'r
2n-l
sin m(f{J -
m~l
r==O
If E is evaluated at the same points qJ at which 1f is given, that is, at the
points qJ = r'r/n, the variable qJ - (r7T/n) becomes [(r' - r)r]/n = - (Kr/n),
and the last term becomes zero.
"'n:
1fr = rr =,p
("r
n + nK 1r) = 1/1 (~ + Knr) ==.pK
2n-1
E(Ip) = -
11-1
1~
~.
n~ "'K ~
sm m n
K-O
](r
fI&~l
The summation over K may also be taken from 0 to 2n - 1 because of the
periodicity of 1/Ix. A simple expression can be obtained for the coefficients
of 1/IK
!(r
[( odd
.
[(r
cot -Slnm-=
2n
{
11,
0
Keven
and therefore
2:
2n-l
E(qJ) = - -1
n
.pK cot -K7r
2n
tc odd
E-l
or
Kodd
(3.17)
THEORY OF WING SBCTIONS
58
In most cases a value of n equal to 40 gives sufficiently accurate results.
Ordinarily 1/1 is known as a function of 6, and first approximations to the
values of and (J - f' are obtained by substitution of 6 for fJ in the relations
for IJ - fJ and
The factor relating the velocities in the true-circle plane to those in the
near-circle plane is dz' /dz. From Eq. (3.16),
"'0
"'0.
~'
dz = ~
{Iz+ d [(1/1 -1J'0) + i(t) dz
fP)]}
z' ~ [If + i(B - tp) + In z]
=
But, on the true circle,
z=~
from which
!s-== dz
!i In z = !i
(In a + 1/10 + if{J) == !i (kp)
dz
dz
Therefore
dz' = z'
dz
!!
[1/1 + i(O dz
+ ifP]
,,)
dz'
d
- = z' - (l/I + i8)
dz
dz
This expression may be written
dr
dz
=
z'
!i
(1/1 + is) dB
dJJ
dz
But
1
.df(J
z
dz
-=1­
or
dz = i d((J = i d(f(J ­ 8) + i dB
z
and
Hence
dz'
dz =
z'd.
1
zdO (- 1,1/1 + 8) 1 + (dEjdIJ)
where
or
z' 1 - i(tU/dD)
dz'
dz
=
z 1 + (dE/dB)
(3.18)
THEORY OF WING SECTIONS OF FINITE THICKNESS
59
The flow in the plane of the true circle of radius a,eh is described by the
following equation:
W==
a2e2h) if
Z
+-Inz
2r aeh
V ( z+--
(3.19)
where the value of a in Eq. (3.7) is here taken to be ae/". The velocities in
the true-circle plane are
2e2
dw
( 1 a- "")
-=V
dz
Z2
ir
+2rz
(3.20)
In order for the rear stagnation point of the circular cylinder to cor­
respond to the trailing edge of the wing section, the flow about the cylinder
is rotated through an angle ao equal to the angle of attack of the wing sec­
tion. This process corresponds to the application of the Kutta-Joukowski
condition. At zero angle of attack, the trailing edge corresponds to the
point
z = a,eh+~T
It is therefore necessary for the circulation r to have a value corresponding
to a rotation of the stagnation point by an amount ao + ET. This value
may be seen from Eq. (2.30) to be
I'
=
4r~V
sin (ao + ET)
Substituting the general value of z on the surface of the cylinder into
Eq. (3.20)
dw == V[I - e-2i (ae+.»
dz
+ 2i sin
(ao + ET)e-i<ao+.)]
from which the absolute value of dw/dz may be obtained
I~;I = 2V[sin
(an + q» + sin (ao + ET)]
(3.21)
The third step is to obtain the velocities in the plane of the wing section
from the expression for the velocities in the plane of the true circle. The
expression for the velocity in the plane of the wing section is
dw
dw dz dz'
dr = dz dz' dr
Multiplying Eqs. (3.14) and (3.18), we obtain
drdz'
dz' dz
=
1[I1-+ i(dl/l/dfJ)].
(dE/dO)
Z
.
·
(2a SInh ,pcos 8+ 2w cosh,p sm 8)
The absolute value of
Idrl == 2v'1 ++ (t#I/d8)2 v':sinh 1/1 + 8m
2
dz
eh[1
(de/dO)]
2
(J
THEORY OF WING SECTIONS
60
Dividing Eq. (3.21) by this expression,
Iddrwl= v =
where
v
V [sin (ao + tp) + sin (ao + Er)][l + (dejdO)]e"
-V(sinh2 '" + sinl 6)[1 + (dl/l/d8)I]
(3.22)
= local velocity at any point on surface of wing section
V = free-stream velocity
The necessary calculations to obtain the pressure distribution for a
given wing section are as follows:
1. The coordinates of the wing ~ction are found with respect to a line
joining a point midway between the nose of the section and its center of
curvature and its trailing edge for sections having sharp trailing edges. 122
The coordinates of these points are taken as (- 2a, 0) and (2a, 0), respec­
tively. It is usually convenient to let a have the value of unity.
2. sin 8 is found from Eq. (3.13).
3. sinh 11' is found from Eqs. (3.12).
4. '" is found from tables o( hyperbolic functions.
5. '" is plotted as a function of 6.
6. #0 is determined from the relation
"'0 =.1
(2" '" dB
2r Jo
7. A first approximation to
E
is obtained by conjugating the curve of
t/I plotted against 8 using Eq. (3.17).
8. From the curves of E and '" plotted against 8, dE/dO and th/I/dB
are found.
9. Determine F by the relation
F ==
[1 + (dE/d6»)eJ't
v(sinh2 '" + sin2 8)[1 + (dl/I/d8)2J
10. ; = F[sin (8 + ao + E)
+ sin (ao + ET)]
11. The pressure coefficient 8 == (V/V)2.
For most purposes the first approximation to E is sufficiently accurate.
If greater accuracy is desired, a second approximation may be found by
plotting 1/1 against 8 + E and repeating steps 6 and 7 before proceeding.
3.7. Empirical Modification of the Theory. Although the foregoing
theory for computing the pressure distribution about arbitary wing sections
is exact for perfect fluid flow, the presence of viscous effects in actual flows
leads to discrepancies. Even when the flow is not separated from the
surface, the thickness of the boundary layer effectively distorts the shape
of the section. One result of this distortion is that the theoretical slope of
the lift curve is not realized. A comparison of the actual lift and moment.
THEORY OF WING SECTIONS OF FINITE THICKNE8S
61
characteristics with the theoretical ones is made in Fig. 33 for the NACA
4412 wing section.a This comparison is typical in that the experimental
lift coefficient is lower than the theoretical value at a given angle of
attack and the theoretical value of the pitching-moment coefficient is not
realized.
Several methods suggest themselves for bringing the theory into closer
agreement with experiment. It is obvious that the lift coefficient can be
2.4
J
I
2.o~ lJsuoIlheofy
/
MbWned/~-------j
I---
Experiment
0
+
1/
J
/
V',l
1/
/1
/. e
~
It
~
.1! .8
§
~.4
J
......
i
l'
v
~
l'
//
0+
+
)
I
-.4
J
0
-. 8
r IJI
Cl",..-o
J.6
0
If
-. -
-+- ....-+
emc.
r--
.....
+
~
~
I
I
-16
-8
0
8
/6
24
Effecfke mgIe ofoIlock,ao,degrees
FIo. 33. Comparison between theoretical and measured lift and pitching-moment coefficients,
NACA 4412 wing section.
brought into agreement by reducing the theoretical angle of attack. This
method seriously alters the flow about the leading edge, leading to dis­
crepancies in the pressure distribution over the forward portion of the wing
section. Another method tried by Pinkerton'" is to reduce the circulation
to the required value by disregarding the Kutta condition. This method
leads to fair agreement over .the greater part of the wing section (Fig. 34)
but, as would be expected, leads to infinite velocities at the trailing
edge.
Pinkerton- found that fairly satisfactory agreement with experiment
could be obtained by effectively distorting the shape of the section (Fig.
34)~ This distortion is affected by finding an increment aET required to
avoid infinite velocities at the trailing edge when the eirculation is 00.­
THEORY OF WING SECTIONS
62
justed to produce the experimentally observed lift coefficient. The altered
function fa is arbitrarily assumed to be given by
E
fa
where
E
= E + aET
2 ( 1 - cos 8)
(3.23)
original value computed by methods of preceding section
modified value used to compute flow about distorted section
=
Ea =
I I I I
-3
«=8"
x
---r--2
Experiment
lJ5uoIlhtf!OtY
RedtK:ed Clrculolion
Modified theory
~ r-;
.. '='
~
p
-I ~
~-~
~ .... ~
-~ ~
~-.:::
~
~
~
I
~~
.~ s;
o
If'
o
~
.-
-~
...:::~-
50
--
I'.
J
,
~
,.\
100
Percenlchord
34. Compariaon between measured and various theoretical pressure distributions
NACA 4412 wing section.
FlO.
The agreement obtained by this method is indicated by Fig. 34 and the
moment coefficients of Fig. 33.
3.8. Design of WiDg Sections. Section 3.6 presents a method for
obtaining the pressure distribution for arbitrary wing sections. A knowl­
edge of the pressure distribution is desirable for structural design and for
the estimation of the critical Mach number and the moment coefficient if
tests are not available. The pressure distribution also exerts a strong or
predominant influence on the boundary-layer flow and, hence, on the sec­
tioJl. characteristics. It is therefore usually advisable to relate the aero­
dynamic characteristics to the pressure distribution rather than directly
to the geometry of the wing section. In the experimental development of
wing sections, it is accordingly desirable to have a method of determining
changes of shape of the section corresponding to desired changes in the
7'BEORY OF WING SECTIONS OF FINITE THICKNESS
63
pressure distribution. A method that has been used successfully for the
development of the newer NACA wing sections is presented in-reference 3.
Equation (3.22) gives the relation between the velocity distribution
over the wing section and the shape parameters 1/1 and 8. It is shown by
Theodorsen and Garrick12l that an alternate expression is
v
sin (ao + Ip)
+ sin
(ao + E'I')e4'o
V = v(sinh2 ~ + sin" 8) t[1 - (dE/d'P)]2 + [(dlY/d'P)]2}
(3.24)
Basic symmetrical shapes are derived by assuming suitable values of
dE/dtp as a function of tp. These values are chosen on the basis of previous
experience and are subject to the conditions that
(1r dE = ()
10 dqJ
and dE/dtp at qJ is equal to dE/dtp at - qJ. These conditions are necessary for
obtaining closed symmetrical shapes. Values of E{qJ) are obtained by in­
tegrating (dE/d'P) dtp. Values of 1/I(tp) are found by obtaining the conjugate
curve of E{tp) by means of Eq. (3.17) and adding an arbitrary value 1/10
sufficient to make the value of 1/1 equal to or slightly greater than zero at
'P = ".. This condition assures a sharp trailing-edge shape. If 1/1 equals
zero at qJ = ..., the wing section will have a cusplike trailing edge. Slightly
positive values of 1/1 result in more conventional trailing-edge shapes.
Theodorsen'P has shown that small changes in the velocity distribution
at any point on the surface are approximately proportional to 1 + (dE/dqJ).
The initially assumed values of dE/dip are accordingly altered by a process
of successive approximations until the desired type of velocity distribution
is obtained. After the final values of y, and E are obtained, the ordinates of
the symmetrical section are computed by Eq. (3.12).
Although a similar procedure may be used to design cambered sections,
it has been found convenient to design symmetrical sections which may
then be cambered by a method to be described in Sec. 4.5. This method is
satisfactory for thin or moderately thick sections. Very thick sections
must be designed directly with the desired camber. Goldstein17C1 has also
developed useful approximate methods for the design of cambered wing
sections.
CHA~TER
4
THEORY OF THIN WING SECTIONS
4.1. Symbols.
A
A
B
L
PR
S
U
V
a
C
c,
Clo
e'i
C.L E
c.e/4
e
k
1
In
1nLB
p
v
v"
l1v
J1v"
x
%1
'II
r
~
ai
a 1~ -0
fJo
../
8
'h
1£,)
r
p
~.FJ
coefficients of Fourier series
multiplier for obtaining the angle of zero lift
multiplier for obtaining the pitching-moment coefficient
lower surface ordinate in fraction of the chord
local load coefficient
pressure coefficient (11 - p) fJAp 'Jl1
upper surface ordinate in fraction of the chord
velocity of the free stream
mean-line designation; fraction of the chord from leading edge over which loading
is uniform at the ideal angle of attack
chord
section lift coefficient
value of c, corresponding to calculated value of ~t·4/V
ideal lift coefficient
section pitching-moment coefficient about the lending edge
section pitching-moment coefficient about the quarter-chord point
base of Naperian logarithm, 2.71828
multiplier for obtaining the angle of zero lift
section lift
logarithm to the base e
section pitching moment about the leading edge
local static pressure
local velocity over the surface of a symmetrical wing section at zero lift
component of velocity normal to the chon!
increment of local velocityover thesurface of a wing section associated with camber
increment of local velocity over the surface of a wing section associated with
angle of attack
distance along chord
fixed point on the %axis
ordinate of the mean line
circulation
section angle of attack
ideal angle of attack
angle of zero lift
quantity defined by Eq. (4.13)
circulation per unit length along chord
angle whose eosine is 1 - (2x/ c)
angle whose cosine is 1 - (2x1 / c)
quantity defined by Eq. (4.14)
ratio of the circumference of a circle to its diameter
mass density of air
infinity
THEORY OF THIN WING SECTIONS
65
4.2. Basic Concepts. Many of. the properties of wing sections are
primarily functions of the shape of the mean line, The mean line is con­
sidered to be the locus of points situated halfway between the upper and
lower surfaces of the section, these distances being measured normal to
the mean line. Although the mean line of an arbitrary lying section is
rather difficult to obtain, the construction of cambered lying sections from
given mean lines and symmetrical thickness distributions is relatively easy.
Among the properties mainly associated with the shape of the mean line
are
1. The chordwise load distribution.
2. The angle of zero lift.
3. The pitching-moment coefficient.
Other important results obtained from the theory of thin wing sections
y
~~-6---------"""'-X
11'
FlO.
35. Configuration for analysis of mean linea.
are a value of the slope of the lift curve and the approximate position of
the aerodynamic center.
The theory of thin lying sections was developed by Munk." Later
contributions to the theory were made by Birnbaum, Glauert," Theodor­
sen,t21 and Allen." The present treatment follows that given by Glauert."
For the purposes of the theory, the lying section is considered to be re­
placed by its mean line. The chordwise distribution of load is assumed to
be connected with a chordwise distribution of vortices.
Let 'Y be the difference in velocity between the upper and lower surfaces.
The total circulation around the section is then given by the relation
(4.1)
where c is the chord and the camber is assumed to be sufficieutly small for
distances along the chord line to be substantially equal to those along the
mean line. The configuration for the analysis is given in Fig. 35.
The vertical components of velocity caused by an element of the vortex
distribution along the mean line have the magnitudes [see Eq. (2.27)]
'Ydx
THEORY OF WING SECTIONS
66
where x is the position of the element and Xl is the point where velocity is
being calculated. The total vertical component of velocity at. any point
Xl is then
(4.2)
where v" =- component of velocity normal to chord line
The thin wing section must of course be a streamline. There is no flow
through it. The equation expressing this condition is
(4.3)
for small angles. Equations (4.2) and (4.3) are the fundamental relations
between· the shape of the mean line and its aerodynamic characteristics.
It is possible to obtain the characteristics of arbitary thin wing sections
from these relations. The pressure distribution or the chordwise load dis­
tribution may, however, be obtained more accurately and just as con­
veniently by the method presented in Sec. 3.6.
4.3. Angle of Zero Lift and Pitching Moment. Simple relations for
the angle of zero lift and the pitching moment of arbitrary thin wing sec­
tions may be obtained by the application of a Fourier method of analysis
to Eqs. (4.2) and (4.3).
Let
x
c
= - (1 ­
2
cos 6)
(4.4)
then
Assume that 'Y may be represented by a trignometric series as follows:
GO
'Y
=
2V(Ao cot ~ + LA. sin nO)
(4.5)
n-l
When "Y is expressed in this manner, it will be shown later that the coeffi­
cient Ao depends only on the angle of attack, and the coefficients Aftdepend
only on the shape of the mean line. Since
cot 2-
~ _ 1 + cos 8
"Y dz
sin s
= cV[Ao(l + cos 8) +
GO
LA. sin n8 sin 8]d8
1
THEORY OF THIN WING SECTIONS
67
The lift is
l
= J:pv-y dx
= 10r~pV2[Ao(l + cos 6) + ~
.. sin nO sin 6]dB
1
rcpV2(A o + MAl)
=
(4.6)
and the lift coefficient
C,
=
~:v2c = 2r(Ao+ ~ AI)
(4.7)
The expression for the pitching moment about the leading edge is given
by the relation
Since
sin 8(1 - cos 8) = sin 8 - sin 8 cos 8
= sin 8 - ~~ sin 28
mLB
1"'~ pV2 [ Ao(l -
=-
L
00
cos2 6) +
A .. sin nO(sin 6 -
~ sin 2tJ)] dB
1
Upon integration
(4.8)
Equations (4.7) and (4.8) provide simple relations between the lift and
moment and the first few coefficients of the Fourier series. The next step
is to determine the relations between the coefficients of the Fourier series
and the shape of the mean line.
Substituting Eqs. (4.4) and (4.5) into Eq. (4.2), the expression for the
vertical component of velocity becomes
V 111"{
Vn ()
81 = -
11"
0
A o(1 + cos 8)
COS 81 -
COS 8
HrA..
cos (n - 1)6 -'cos (n+ 1)61}d8
+ - -1 - [--------­
cos 61
-
cos 8
since
~[CQS
(n - 1)8 - cos (n + 1)8]
==
sin nO sin 8
It is shown by Glauert" that
r
cos nO dB =
Jo cos 8 - cos q,
'lI"
~ nq,
SIn
q,
(4.9)
THEORY OF WIJ.VG SECTIONS
68
eonsequently
and, dropping the subscript,
v
V
Ao +
= -
L Aft cos 118
00
(4.10)
1
The required relation between the Fourier coefficients and the shape of the
mean line is obtained by substitution of Eq. (4.10) in Eq. (4.3)
dy
dx
=
~
ao - A o + k A n cos n8
(4."11)
1
The standard expressions for the Fourier series coefficients are then
11'IT-dy dB
ao - Ao = r
0
dx
(4.12)
21'IT
An = -
r
0
dY
d cos nO dB
X
Equations (4.12) for A o, AI, and 411 2 can be transformed in such a manner
that the angle of zero lift and the pitching moment are expressed in terms
of the ordinates of the mean line rather than in terms of its slope. The
quantities {jo and Po defined as follows are convenient in effecting the change.
{jo -
and
11.0 =
~1"
ll.
dB
... 0 c 1+cos8
(4:.13)
1"
o
ll. cos 9 dB
(4.14:)
C
The value of fJo in terms of the coefficients of the original Fourier series
may be found by the following process.
Integrating Eq. (4.13) by parts we obtain
{J - 2[Y
0-; c
1cos 8]" 2 [" Idydx II - cos 9 d9
1+cos8 -;}o cdxd8V'l+cos8
0
(reference SO, formulas 296 and 578). The first term of this expression
equals zero if y approaches zero at 8 = 1(' faster than v'C=X. This is true
for nearly all wing sections. From Eq. (4.4),
dx = !:. sin 9 = !:. V(1
d8 2
2
+ cos 9)(1 -
cos 9)
THEORY OF THIN lVING SECTIONS
Hence
69
li ftd
J(1 - cos 8)d8
dx
{jo = - -
.". 0
and, from Eq. (4.12),
Po
= Ao -
ao
+ ~Al
(4.15)
The value of JJ.o in terms of the Fourier coefficients A" is found as follows:
Integrating Eq. (4.14) by parts,
JJo
. 6J1r = [·c~y mn
0
== -
lr
0
1 dy dx
.
- sm 8 dO
cdxd8
- -
t'.!.
dy (l - cos 28)dO
Jo 4dx
1)
(4.16)
+ Po)
(4.17)
1r(
4" Qo - Ao - 2 A 2
Comparing Eq. (4.7) with Eq. (4.15)
2".(ao
C, =
The angle of zero lift QL-O is therefore - {3o, and the slope of the lift. curve
according to Eq. (4.17) is 211" per radian.
The equation for the pitching-moment coefficient about the leading
edge is found by substituting Eqs. (4.15) and (4.16) into Eq. (4.8)
c",u
= (2#lO -
~ Po) -
i
Cl
(4.18)
The pitching-moment coefficient about a point one-quarter of the chord
behind the leading edge Cm./ equals Cwa L B + (%)c,. From Eq. (4.18),
4
Cwa
1r
~/t
= 2#lO - --{3
2 0
(4.19)
It should-be noted that the pitching-moment. coefficient about the quarter­
chord point is independent of the lift coefficient. The quarter-chord point
is therefore the aerodynamic center of thin wing sections. The aerodynamic
center for most commonly used wing sections is found to be approximately
the quarter-chord point at speeds where the velocity of sound is not reached
in.the field of flow.
By substituting the expression
x
c
= -
2
(1 - cos 8)
into Eqs. (4.13) and (4.14)
{30=
i
1y
o
-/1 (X)dX
­ C
C
C
(4.20)
THEORY OF "K'ING SECTIONS
70
and
i
IJo ==
1y (X)dx
- f2 - c c
(4.21)
o c
where
fl(~) = r[l- (x/c)] ~(x/e)[l f2
1 - (2x/c)
(Cx)= V(x/e)[l
- (x/c)]
(x/e)i
In order to avoid infinite velocities at the leading edge, it is necessary
that the coefficient A o of Eq. (4.5) be equal to zero. From Eq. (4.12),
Ao equals zero when
ao =
! ("dy d8
1f'Jo .d,x
Substituting the value of dx from Eq. (4.4)
1 {.
Qo
= ;}o
d dB
(e/2)Ysin 8 d8
and integrating by parts
ao == ![~y/cl·
sm S'Jo
1r
+.! rr2(y/c~ cos
8 dB
2
11"
Jo
sln 8
As previously discussed in connection with the integration of Eq. (4.13)
the first term of this expression equals zero for most wing sections. Sub­
stitution of the expression
x
= -2c (1 -
cos 8)
gives
(4.22)
where
fa (
X)
c
1 - (2x/c)
= 2-r{ (x/e)[l - (x/e)Jl ~
This angle of attack is called the "ideal angle of attack" by Theodorsen.P'
The lift coefficient corresponding to this angle of attack [obtained from
Eq. (4.1 i)] is called the "ideal" or ,., design" lift coefficient.
In general the angle of sero lift, the pitching moment about the quarter­
chord point, and the ideal angle of attack may be calculated by graphical
integration of Eqs. (4.20), (4.21.), and (4.22). For convenience in such
calculations, values of the functions /1, !2, and fa corresponding to several
values of x/c are presented in Table 5. Although the functions /1, j!, and
fa become infinite at both the leading and trailing edges, the integrands of
THEORY OF THIN WING SECTIONS
71
Eqs. (4.20) and (4.21) approach zero at the leading edge for mean lines
that approach zero faster than VXjC. Most mean lines satisfy this con­
dition. Similarly, the integrand of Eq. (4.21) approaches zero at the
trailing edge in most cases. In order to avoid the difficulties at the trailing
edge in Eq. (4.20) and at the leading and trailing edges in Eq. (4.22), parts
of these integrals are evaluated analytically." The analytical determinaft,
TABLE .5.-VALUES OF FUNCTIONS
z/c
fl(x/c)
0
0.0125
0.0250
0.0500
0.0750
/2,
f2(x/c)
CO
AND
fa
f.{x/c)
CO
00
2.901
2.091
1.537
1.306
1.179
8.774
6.085
4.131
3.226
113.15
39.73
13.84
7.403
4.716
2.447
1.492
0.980
0.662
0.1000
0.15
0.20
0.25
0.30
0.995
0.980
0.992
2.667
1.960
1.602
1.156
0.873
0.40
0.50
0.60
0.70
0.80
1.083
1.273
1.624
2.315
3.979
0.408
0
-0.408
-0.873
-1.502
0.271
0
- 0.271
- 0.662
- 1.492
-2.667
-4.131
- 4.716
- 13.84
1.049
0.90
0.95
10.61
29.21
1.00
00
00
-
00
tion of the increments at the critical points is accomplished by assuming
the mean line near the ends to be of the form
y
x
(X)2
-==A+B-+C
c
c
c
Evaluation of the increment of the angle of zero lift gives
(xC = 0.95 to 1.0)
dy
J1fJo == 0.964Yo.t, - 0.0954 dx
1
Where Yo." is the ordinate of the mean line at x/c = 0.95 and dy/dx1 is the
slope of the mean line at x/c = 1.0. This increment is added to the value
of 80 obtained by graphical integration from x/c = 0 to x/c = 0.95. The
increments for the ideal angle of attack are
dy
1:100i=
+ O.467Yo.05 + 0.0472 dx o
dy
{ - 0.467Yo.9S + 0.0472 dx
1
(~= 0 to 0.05)
(~ =
0.9,) to 1.0)
THEORY OF WING SECTIONS
72
These increments are added to the" values of Qi obtained by graphical
integration from x/c == 0.05 to x/c == 0.95.
By application of Gauss's roles for numerical integration to the expres­
sion for the angle of zero lift, Munk70 obtained a simple approximate
solution.
- ao' = klYl
+ kty2 + kaYa + k4'g4 + k,y,
where gl, gt, etc., are the ordinates of the mean line expressed as fractions
of the chord at the points Xl, X2, etc., as tabulated together with corre­
sponding values of the constants kl , k i , ete., calculated to give the angle of
zero lift in degrees.
kl
0.99458
0.87426
0.50000
0.12574
0.00542
Xl
X2
X,
%4
z,
1,252.24
109.048
32.5959
15.6838
5.97817
let
k,
~
k,
This method is useful for quick calculation of the angle of zero lift when
a high order of accuracy is not required.
Approximate solutions for the angle of zero lift and the moment
coefficient were also obtained by Pankhurst78 in the following form:
(XL-O
= 2;A(U + L)
c..1. == '1:B(U + L)
where U,L = upper and lower ordinates of wing section in fractions of
chord
A,B == constants given in following table
x
0
0.025
0.05
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.95
1.00
B
A
1.45
2.11
2.41
2.94
-
2.88
3.13
3.67
4.69
6.72
- 0.009
0.045
0.101
0.170
0.273
11.75
21.72
99.85
- 164.90
0.477
0.786
3.026
- 4.289
r.se
0.119
0.156
0.104
0.124
0.074
THEORY OF THIN WING SECTIONS
.78
Pankhurst's solution for the angle of zero lift is often more convenient
than Munk's because the constants are given for stations at which the
ordinates are usually specified. The solution is based on the assumptions
that the mean-line ordinates are represented satisfactorily by (U + L)/2
and that the tangent of the mean line at the trailing edge coincides with
the last t\VO points. The accuracy is accordingly expected to be limited
for sections with large curvatures of the mean line near the trailing edge,
or for thick, highly cambered sections.
4:.4. Design of Mean Lines. It is possible to design mean lines to
have certain desired load distributions by means of the theory of thin wing
sections. The fundamental relations are Eqs. (4.1), (4.2), and (4.3). A
simple example of the manner in which a mean line is designed may be
HI
demonstrated by the design of a mean line to have uniform load.
\
For a uniform chordwise load distribution, the value of -y in Eq. (4.2)
is a constant over the chord. Equation (4.2) may therefore be written
(4.23)
The lift is
and
Cl,
equals
C,
,
21
2-y
pV c
V
= - 2- = ­
or
Ve"
'Y=T
By substitution, Eq. (4.23) becomes
"ft(XI)
Cl'1 (x/c)d(x/c)
V C
411"
- (xI/c)
1
=
(4.24)
0
Special means must be used to integrate Eq. (4.24) because the inte­
grand becomes infinite as x approaches Xl. The integration is performed
as follows:
V"(Xl)
V C
=
=
cl'I·
4r ~
[l
wc
d(x/c)
>-,
0
(x/c) - (Xl/C)
+
r
d(x/c)
]
- (Xl/C)
}CZl/C)+e (x/c)
Cl, lim{[ln(~ - ~)~:Q-'
IJo + [In(~C - ~)Jl
411" e-+O
C
C
= ~~[ln(l-~)-ln(:l)~
C
}
St+e
THEORY OF WING SECTIONS
74
The angle of attack at which the uniform load distribution is realized is
zero because the assumed load distribution, and accordingly the mean line,
is symmetrical about the mid-chord point. Equation (4.3) becomes
~ = :; [ In(1-~) -In(~)J
(4.25)
Integration of this expression gives
y == -
~~[(l- ~)In(l- ~)+~ In ~J
(4.26)
where the constant of integration has been selected to make the ordinates
zero at the leading and trailing edges.
By similar but more complicated processes, the mean lines corre­
sponding to other types of load distribution may be calculated. In this con­
nection, it is interesting to note that the distribution of l' and the slope
of the mean line are conjugate functions at the ideal angle of attack [see
Eqs. (4.5) and (4.11)]. The slope of the mean line can therefore be obtained
from the assumed load distribution by the proper application of Naiman's
coefficientg8.73 as in Eq. (3.17). The mean-line ordinates can then be ob­
tained by graphical or analytical methods.
Mean lines have been caleulatedv corresponding to a load distribution
that is uniform from x/c = 0 to x/c = a and decreases linearly to zero at
x/c = 1. The ordinates are given bythe expression
~ - ~r(:'+ 1) {I ~ a[~(a - ~rln \a - ~I- ~(1 - ~rln(l -~)
+~(l-~r -~(a-~)1-~ln~+g-h~}
where
(4.27)
J) +-4IJ
- 1 [ 0)(12
g:::z-- a· - lna--
I-a
h=
4
n
1~ a (1 - aYIn(1 - a) - ~ (1 - a)] + 0
The ideal angle of attack for these mean lines is
Cl,}"
a;
= 21r(a + 1)
It \\;11 be noted that the ordinates of the mean lines are directly pro­
portional to the design lift coefficient. The fact that the load distributions
and the ordinates of the corresponding mean lines are additive may be
seen from examination of Eqs.. (4.5) and (4.11). This property is, of
course, a direct consequence of the linearizing assumptions made in writing
the fundamental expressions (4.1), (4.2), and (4.3).. This linearity is con­
venient because ordinates of a mean line having a load distribution made
THEORY OF THIN WING SECTIONS
75
up of the sums of load distributions for which the corresponding ordinates
are known can be obtained by simply adding the ordinates of the com­
ponent mean lines.
The linearizing assumptions lead to increasing errors as the slope of the
mean line increases. For geometrically similar mean lines, the slope in­
creases with the design lift coefficient, and the errors are consequently
greater for the larger design lift coefficients. In the case of the uniform­
load mean line [Eq. (4.26)], the slope is infinite at the leading and trailing
edges, and it is not to be expected that the assumed load will be maintained
2.0
-"-­
/
I(
°0
-~ ~
"
.2
.6
.4
r--. r---....
.8
1\
LO
~
FIo. 36. Typical basic load distribution at ideal angle of attack.
near these points. The region in which the slope is large is extremely small,
however, because of the manner in which the slope approaches infinity
[Eq. (4.25)].
4.6. Engineering Applications of Section Theory. The theory of wing
sections presented in Sec. 3.6 permits the calculation of the pressure distri­
bution and certain other characteristics of arbitrary sections with con­
siderable accuracy. Although this method is not unduly laborious, the
computations required are too long to permit quick and easy calculations
for large numbers of wing sections. The need for a simple method of
quickly obtaining pressure distributions with engineering accuracy has led
to the development of a method! combining features of thin and thick
wing-section theory. This simple method makes use of previously cal­
culated characteristics of a limited number of mean lines and thickness
forms that may be combined to form large numbers of related wing
sections.
The theory of thin wing sections as presented in Sec. 4.3 and reference
121 shows that the load distribution of a thin section may be considered
to consist of
1. A basic distribution at the ideal angle of attack (Fig. 36).
2. An additional distribution proportional to the angle of attack as
measured from the ideal angle of attack (Fig. 37).
The first load distribution is a function only of the shape of the thin
wing section or (if.the thin wing section is considered to be a mean line)
of the mean-line geometry. Integration of this load distribution along the
THEORY OF WING SECTIONS
76
chord results in a normal force coefficient which, at small angles of attack,
is substantially equal to the design lift coefficient. If, moreover, the camber
of the mean line is changed by multiplying the mean-line ordinates by a
constant factor, the resulting load distribution, the ideal or design angle of
attack as, and the design lift coefficient Ck may be obtained simply by
multiplying the original values by the same factor.
The second load distribution, which results from changing the angle of
attack, is designated the U additional load distribution," and the cor­
responding lift coefficient is designated the Ct additional lift coefficient."
5.0
4.0
~O
\
\
\
~
1.0
00
"~
.2
~ .............
.4
~
.6
-
r--. r--.-
r-,
.8
LO
~
Flo. 37.
Additional load distribution 8880Ciated with anale of attack..
This additional load distribution contributes no moment about the quarter­
chord point and, according to the theory of thin wing sections, is independ­
ent of the geometry of the wing section except for angle of attack. The
additional load distribution obtained from the theory of thin wing sections
is of limited practical application, however, because this simple theory
leads to infinite values of the velocity at the leading edge. This difficulty
is obviated by the exact theory of wing sections presented in Sec. 3.6,
which also shows that the additional load distribution is neither com­
pletely independent of the airfoil shape nor exactly a linear function of the
angle of attack. Suitable additional load distributions can be calculated
by the methods of Sec. 3.6 for each thickness form (symmetrical section)
and a typical value of the additional lift coefficient.. The variation of the
additional load distribution with the additional lift coefficient is arbitrarily
assumed to be linear.
In addition to the pressure distributions associated with these two load
distributions, another pressure distribution exists which is associated with
the thickness form, or basic symmetrical section, at zero angle of attack
and which is calculated by the methods of Sec. 3.6.
J
THEORY OF THIN WING SECTIONS
77
The velocity distribution about the wing section is thus considered to
be composed of three separate and independent components as follows:
1. The distribution corresponding to the velocity distribution over the
basic thickness form at zero angle of attack.
2. The distribution corresponding to the load distribution of the mean
line at its ideal angle of attack.
3. The distribution corresponding to the additional load distribution
associated with angle of attack.
The local load at any chordwise position is caused by a difference of
velocity bet,veen the upper and the lower surfaces. It is assumed? that
the velocity increment on one surface is equal to the velocity decrement
on the other surface. This assumption is in accord with the basic concept
of the distribution of circulation used in the theory of thin wing sections.
It can be shown that the relation between the local load coefficient P R 'and
the velocity increment ratios !:1v/V or !:1vo/V is
!:1v
V
~Va
or
1 PR
V =4v1V
where v/V is the velocity ratio at the corresponding point on the surface of
the basic thickness form. In actual practice the value of !:"v IV used is
that corresponding to the thin wing section or mean line where v/V equals
unity:
The velocity increment ratios !J.v/V and I1vo / V corresponding to com­
ponents 2 and 3 are added to the velocity ratio corresponding to
component 1 to obtain the total velocity at one point from which the
pressure coefficient S is obtained, thus
S = (::. ± .1v ± I1Va
V
V
)2
V,
,
(4.28)
This procedure is illustrated in Fig. 38. "Then this formula is used, values
of the ratios corresponding to one value of x/c are added together and the
resulting value of the pressure coefficient S is assigned to the surface of
the wing section at the same value of x/c. The values of !:,.v/V' and I1vo / V
are positive on the upper surface and negative on the lower surface for
positive values of the corresponding local loads.
When the ratio I1va/ V has the value of zero, integration of the resulting
pressure-distribution diagram will give approximately the design lift
coefficient Cli of the mean line. In general, however, the value of CI will
be greater than ct, by an amount dependent on the thickness ratio of the
basic thickness form. This discrepancy is caused by applying the values
of !:1v/V obtained for the mean line to the sections of finite thickness where
v/V is greater than unity over most of the surface.
The pressure distribution will usually be desired at some specified lift
coefficient not corresponding to cu: For this purpose, the ratio ava/V must
THEORY OF WIJtiG SECTIONS
78
1.2
E __---~
~ C A LYJN;
SECT1CJ'.J
ATZERO AtG...E OFATTACK
.8
v
V
.4
.4 [ -
MEAN l.f£ AT IlEAL
ANiLE OFATTACK
E
CM'SEREO YiN; SECTOJAT
lEAL ANiLE (F ATTACK
-
-
-
_
~
I~'
.4
0'------------­
~
E·_u===:::::'-­
VELOCITY IfCREMENT RESlLTING
FRQ.1 ANGLE rF ATTACK FOR
SYtAETRCAL ¥lING ~
-
0[
1.6
V
1.6
1.2
YJ..Ar.+- ~
V~
V -
V
.4
°0
.2
.4
.6
.8
LO
.Jt
FIG.
38. Synthesis of pressure distribution.
be assigned some value by multiplying the originally calculated values oi
this ratio by a factor J(a). For a first approximation, this factor may be
assigned the value
f{a) = C, - Cl,
Clo
where CI is the lift coefficient for which the pressure distribution is desired
and Clo is the lift coefficient for which the values of L1va / V were originally
calculated. If greater accuracy is desired, the value of J(a) may be ad­
THEORY OF THIN WING SECTIONS
79
justed by trial and error to produce the actual desired life coefficient as
determined by integration of the pressure-distribution diagram.
Although this method of superposition of velocities has inadequate
theoretical justification, experience has shown that the results obtained are
~
1.
c/
~
~
-~
.:
r- Upper .UI"t'aoe
~
~Lowr.Ul"1"_
·1'"
.o,
\
- r-, 1\
~
~
-~OrJ'
"~
~,
o Kx:perJ.-ent
o
.2
.,.
.&
.8
&.0
x/o
FIO.39. Comparison of theoretical and experimental pressure distributions for the NACA
66(215)-216 airfoil; Cl == 0.23
adequate for many engineering uses. In fact the results of even the first
approximation agree well with experimental data and are adequate for at
least preliminary consideration and selection of wing sections. A com­
parison of a first-approximation theoretical pressure distribution with an
experimental distribution isshown in Fig. 39. Some discrepancy naturally
occurs between the results of experiment and the results of any theoretical
method based on potential flow because of the presence of the boundary
layer. These effects are small, however, over the range of lift coefficients
for which the boundary layer is thin and the drag coefficient is low.
€ H APTER
5
THE EFFECTS OF VISCOSITY
6.1. Symbols.
A
F
F
H
H
K
L
£,
B
Be
R,.
Rz
Ra*
R,
U
U«
UtI
U.
Uz
V
c
Cd
ep
c/
d
"
l
l
In
log
n
p
q
go
r
u
u
v
v
11*
v.
constant
velocity gradient
skin friction
total pressure outside boundary layer
ratio of the displacement thickness to the momentum thickness
constant
characteristic length
length of mean free path of molecules
Reynolds number pVL/p
Reynolds number pVc/p.
Reynolds number pur/ p.
Reynolds number pVx/p.
Reynolds number pV6*/p. or pU6*/p.
Reynolds number pU6/p.
velocity outside boundary layer
maximum velocity
velocity outside boundary layer at the point Xo
velocity outside boundary layer at the laminar separation point
value of U at the point :z;
velocity of the free stream
chord
section drag coefficient
skin-friction coefficient
local skin-friction coefficient
diameter of pipe
base of Naperian logarithms, 2.71828
length
wave length
logarithm to the base e
logarithm to the base 10
exponent
local static pressure
dynamic pressure
dynamic pressure of the free stream ~p yt
radius of pipe
. component of velocity in boundary la.yer along the x axis
average velocity of flow in pipe
component of velocity in boundary layer along the y axis
mean molecular velocity
friction velocity v:;jp
velocity of slip
80
THE EFFECTS OF VISCOSIT1'
z
81
Cartesian coordinate (distance along surface)
z. effective length of flat plate
%0
11
a
position on surface at start of integration
Cartesian coordinate (distance normal to surface)
21r/l
fJi time rate of amplification of disturbance
a boundary-layer thickness
a* displacement thickness of the boundary layer
" nondimensional variable (y/x)Vifz
'1
8
80
~
~
p.
~
11"
p
.,.
q,
wit
1/11
GO
distance parameter plJ*y/ Il
momentum thickness of the boundary layer
momentum thickness at the point Zo
pressure-loss coefficient
laminar boundary layer shape parameter
viscosity of the fluid
coefficient of slip
ratio of the circumference of a circle to its diameter
mass density of the fluid
shearing stress at the wall (skin friction)
velocity parameter u/v·
stream function
nondimensional stream function
infinity
6.2. Concept of Reynolds Number and Boundary Layer. In order to
be able to compare directly the forces acting on geometrically similar
bodies of various sizes at various air speeds, it is customary to express the
forces in terms of nondimensional coefficients, as explained in Chap. 1.
For geometrically similar configurations, these coefficients tend to remain
constant, and they would remain exactly constant if all factors influencing
the flow about the body were properly accounted for. It is well known,
however, that some characteristics such as the drag and maximum lift
coefficients vary with the size of the wing for a given air speed, and with
the air speed for a given size of wing.
The two most important factors that are neglected in defining the,
coefficients are effects associated with the compressibility and viscosity of
air. An examination of the complete equations of motion- shows the
significant parameters to be the ratio of the air speed to the speed of sound
in air (the Mach number) and the Reynolds number pVL/p, where p is the
mass density of the fluid, V is the velocity of the free stream, L is the
characteristic length, and, "" is the viscosity of the fluid. It can be shown
that similarity of the flows about different bodies is obtained only if the
bodies are geometrically similar and if the Reynolds number and the Mach
number are the same. At speeds where the pressure variations around the
body are small compared with the absolute pressure (low Mach numbers)
the effects of compressibility are negligible, and the viscous effects may
be considered independently.
THEORY OF WING SECTIONS
82
Some concept of the physical significance of the Reynolds number may
be obtained by expressing this number in terms of the mean velocity and
the mean free path of the molecules, as suggested by von Karman. 137 The
kinetic theory of gases indicates that the coefficient of viscosity p, of a
homogeneous gas may be expressed by a formula of the type66
p, =
KtWL,
where K = constant
== mean molecular velocity
LI == mean free path of the molecules
Substituting this value of p. into the formula for the Reynolds number; we
obtain
1 V L
v
R=--K V t;
where R = Reynolds number
The Reynolds number is thus proportional to the product of the ratio
of the speed of the body to the mean speed of the molecules and of the
ratio of the size of the body to the mean free path of the molecules. For
bodies of ordinary size moving at low Mach numbers in air of ordinary
density, the ratio V (V is small and the ratio L/L/ is very large. Under
these conditions, the flow around the body corresponds to the conditions
under which this formula was derived, namely, that the velocities are not
large compared with the mean velocity and that the spatial dimensions of
the phenomenon are large compared with the mean free path.
The mean velocity v is of the same order of magnitude as the velocity
of sound, and consequently the ratio V IV is of the same order as the Mach
number. It
accordingly be inferred that the Reynolds and Mach
numbers are not independent parameters and that the scale effects ex­
perienced at high Mach numbers would be different from those at low
Mach numbers even though the Reynolds numbers were the same in each
case. Moreover, in a rarefied gas, as at extremely high altitudes, the mean
free path may be of the same order as, or even much larger than, the
characteristic length of the body. The concept of Reynolds number as
presented here presupposes a mean free path that is small compared with
the thickness of the boundary layer. Consequently the significance of this
concept as applied to flows in very rarefied gases or at high Mach numbers
is uncertain. Under such conditions, it is preferable to compare flows on
the basis of the independent parameters V IV and L/L,.
The effects of viscosity are of primary importance in a thin region near
the surface of the wing called the U boundary layer." Boundary layers,
in general, are of two types, namely, laminar and turbulent. The llow in
the laminar layer is smooth and free from any eddying motion. The flow in
the turbulent layer is characterized by the presence of a large number of
relatively small eddies. The eddies in the turbulent layer produce a trans­
may
THE EFFECTS OF VISCOSITY
83
fer of momentum from the relatively fast moving outer parts of the boundary
layer to the portions closer to the surface. Consequently the distribution
of average velocity is characterized by relatively higher velocities near the
surface and a greater total boundary-layer thickness in a turbulent than in
a laminar boundary layer developed under otherwise identical conditions.
Skin friction is therefore higher for turbulent boundary-layer flow than for
laminar flow.
Aerodynamically, the concept of the boundary layer is based on the
premise of a continuous homogeneous viscous fluid. According to this
concept the velocity within the boundary layer varies from zero at the
surface to the full local stream value at the outer edge of the layer. Such a
concept is valid only if the density of the gas is sufficiently great to limit
the mean free path of the molecules to a length very small compared with
the thickness of the boundary layer. The kinetic theory of gases" indi­
cates that the velocity at the surface is not exactly zero but that there
is a velocity of slip proportional to the velocity gradient.
du
v. = t dy
The coefficient of slip ~ has the dimension of length and may be con­
sidered as a backward displacement of the wall with the velocity gradient
extending effectively right up to the displaced wall where the velocity is
zero. It has been shown by Maxwell, Millikan, and others that, for most
surfaces, the coefficient of slip is very nearly equal to the mean free path of
the molecules. At ordinary altitudes, this distance is so small that it may
properly be neglected. At very high altitudes, this slip velocity may have
large effects even though the mean free path is still only a fraction of the
boundary-layer thickness. At still higher altitudes where the mean free
path is long, the entire concept of the boundary layer and viscosity as
presented here becomes invalid.
6.3. Flow around Wing Sections. When the pressures along the wing
surfaces are increasing in the direction of flow, a general deceleration takes
place. At the outer limits of the boundary layer, this deceleration takes
place in accordance with Bernoulli's law. Closer to the surface, no such
simple law can be given because of the action of the viscous forces within
the boundary layer. In general, however, the relative loss of speed is
somewhat greater for the particles of fluid within the boundary layer than
for those at the outer limits of the layer, because the reduced kinetic .energy
of the boundary-layer air limits its ability to flow against the adverse pres­
sure gradient. If the rise in pressure is sufficiently great, portions of the
fluid within the boundary layer may actually have their direction of motion
reversed and may start moving upstream. When this reversal occurs, the
boundary layer is said to be "separated." Because of the increased inter­
84
THEORY OF WING SECTIONS
change of momentum from different parts of the layer, turbulent boundary
layers are much more resistant to separation than are laminar layers.
Except under very special circumstances laminar boundary layers can
exist for only a relatively short distance in a region in which the pressure
increases in the direction of flow.
After laminar separation occurs, the flow may either leave the surface
permanently or reattach itself in the form of a turbulent layer. Not much
is known concerning the factors controllmg this phenomenon. Laminar
separation on wings is usually not permanent at flight values of the Rey­
nolds number except when it occurs on some wing sections neai the leading
edge under conditions corresponding to maximum lift. The size of the
locally separated region that is formed when the laminar boundary layer
separates and the flow returns to the surface decreases with increasing
Reynolds number at a given angle of attack.
The flow over aerodynamically smooth wings at low and moderate lift
coefficients is characterized by laminar boundary layers from the leading
edge back to approximately the location of the first minimum-pressure
point on both upper and Iower surfaces. If the region of laminar flow is
extensive, separation oecurs immediately downstream from the location
of minimum pressure and the flow returns to the surface almost immedi­
ately at flight Reynolds numbers as a turbulent layer. This turbulent
boundary layer extends to the trailing edge. If the surfaces are not suf­
ficiently smooth and fair, if the air stream is turbulent, or perhaps if the
Reynolds number is sufficiently large, transition from laminar to turbulent
flow may occur anywhere upstream of the calculated laminar separation
point. ­
For low and moderate lift coefficients where inappreciable separation
occurs, the wing profile drag is largely caused by skin friction, and the
value of the drag coefficient depends mostly on the relative amounts of
laminar and turbulent flow, If the location of transition is known or as­
sumed, the drag coefficient may be calculated with reasonable accuracy
from boundary-layer theory.
As the lift coefficient of the wing is increased by changing the angle of
attack, the resulting application of the additional type of lift distribution
moves the minimum-pressure point upstream on the upper surface, and the
possible extent of laminar flow is thus reduced. The resulting greater
proportion of turbulent flow, together with the larger average velocity
of flow over the surfaces, causes the drag to increase with lift coefficient.
At high lift coefficients, a large part of the drag is contributed by pres­
sure or form drag resulting from separation of the flow from the surface.
The flow over the upper surface is characterized by a negative pressure
peak near the leading edge, which causes laminar separation. The onset of
turbulence causes the flow to return to the S1ma~e LCJ a turbulent boundary
THE EFFECTS OF VISCOSITY
85
layer. High Reynolds numbers are favorable to the development of tur­
bulence and aid in this process. If the lift coefficient is sufficiently high or
if the reestablishment of the flow following laminar separation is unduly
delayed by low Reynolds numbers, the turbulent layer will separate from
the surface near the trailing edge with resulting large drag increases. The
eventual loss of lift with increasing angle of attack may result either from
relatively sudden failure of the boundary layer to reattach itself to the
surface following separation of the laminar boundary layer near the leading
edge or from progressive forward movement of turbulent separation.
Under the latter condition, the flow over a relatively large portion of the
surface may be separated prior to maximum lift.
5.4. Characteristics of the Laminar Layer. The characteristics of the
laminar boundary layer may be deduced from detailed consideration of the
general equations of motion. For two-dimensional steady motion neglect­
ing the effects of compressibility, these equations are28
u au +. v au
dX
u
ay
av + v iJv
ay
oX
= _
~ ap + ~ (a u2 + 02U)}
p
iJx
= _
! dp + J!'
2
(iJ V
2
pax
p iJy
d y2
dX2
p
+ iJ2v)
(5.1)
iJ y 2
These equations are known as the "K"avier-Stokes equations." Their
general solution is difficult. A simple integration can be performed only in
special cases. These equations, however, may be simplified by the use of
dimensionless variables and consideration of the order of magnitude of
each term. Prandtl" has shown that, if the viscous effects are assumed to
be confined to a thin layer over the surface, the following approximate
relation applies in this region
u au
ax
+ v au =
ay
_
! iJp + ~ a2u
p
ax
p
iJ y 2
(5.2)
This equation is a simplified form of the first relation of Eq. (5.1). Con­
sideration of the order of magnitude of the various terms of the second
relation of Eq. (5.1) indicates that op;iJy is small and hence that pressures
are transmitted unchanged through the boundary layer.
An interesting conclusion is reached by substitution of the following
nondimensional variables in Eq. (5.2).
X
Xl =-
c
U
UI
=-y
PI = 1­2
pV
Yl
=
'U.VR,
C
vl=VVR
THEORY OF WING SECTIONS
86
Equation (5.2) becomes
V 8(Ul V)
a(XIC)
11,1
+ VI V
a(UIV)
= _
VR a(Ylc/VR)
! a(PtPV 2 ) + a2 (Ul V)
p a (XIC)
a(Ylc/VR)"
and, upon simplification,
UI
au! + VI OUI = _ apl + a2ul
aXI
aYl
ax! aYl2
(5.3)
It will be noted that the viscosity does not appear directly in Eq. (5 3).
Solution of this equation will give ttl and VI in terms of Xl and 'Ut. Because
YI equals (y/c)vo" the effect of variations of the Reynolds number will be
merely to change y in a manner inversely proportional to the square root
u
y
FIG.
40.
Effect of pressure gradient on velocity distribution in laminar bounda.ry IS1'er.
of the Reynolds number. It follows directly that the thickness of the
laminar boundary layer varies inversely with the square root of the Rey­
nolds number. It may also be concluded that the shape of the velocity
distribution through the boundary layer is independent of the Reynolds
number, unless, of course, the pressure distribution is a function of the­
Reynolds number. It also follows that the position of the laminar separa­
tion point, characterized by the condition that au/ay is zero at the surface,
is independent of the Reynolds number.
A considerable amount of information concerning the velocity distribu­
tion in the laminar boundary layer can be derived from Eq. (5.3) without
actually attempting a solution. _~t the surface Ul and VI must both equal
zero. At the outer limit of the boundary layer Ul approaches a constant
value. Consequently near the outer edge of the layer a2u/all must be
negative and approach zero with increasing distance from the surface, as
shown in Fig. 40. At the surface, Eq. (5.3) simplifies to the form
82u1
apt
a yl2 = aXI
When apI/aXI is positive (increasing pressure in the direction of flow),
atujayt must be positive at the surface, and the shape of the velocity dis­
tribution will resemble that shown in Fig. 40 for this condition. Similarly
the velocity distribution must be linear near the surface for zero pressure
THE EFFECTS OF VISCOSITY
87
gradient, and the velocity profile will be continuously convex upward as
shown in Fig. 40 for decreasing pressures in the direction of HOlV.
5.6. Laminar Skin Friction. Blasius'vobtained a solution for Eq.
(5.2) for the case where op/ox is zero, that is, for a flat plate with uniform
velocity outside the boundary layer. Blasius found that Eq. (5.2) could
be greatly simplified by assuming a variable oi the form
== v.. v'Rz
x
11
where B; = pVx/p.
and by the introduction of the nondimensional stream function
"'1 =
fo~ Ul
dT/
= j(T/)
where Ul = u/V
The true stream function is
,p= fo"UdY
'" =
U
=
Vx
f"Ul
~.Io
dT/
at/! == at/! °11
oy
o"loy
=
=
Vx j(T/) =
v'R z
~J! xVj(,,)
P
[~J!p XVj/(T/)] (VRz)
=
x
Vj'(l1)
The velocity v perpendicular to the flat plate is given by
&p
01/1 0 71
v == - - == - - -.
ox
f)
071 ax
= ~ ~~["j/(T/)
- j(1I)J
Making similar calculations for au/dx, 8u/dy, and iJ2u/dy2 and substituting
in Eq. (5.2) with op/iJx zero, we obtain
2
2xLr
- V 11f ' (11)! ',/ (1I) + V2 1I}'(1I) - j(1I)Jl"(71) ==
2x
_
or
j(TJ)I"<',,)
+ 2/'''(1/) == 0
~pxp.
V e/"'("1)
2
(5.4)
The fact that neither x nor y appears explicitly in Eq. (5.4) indicates
that the initial assumption that 1fl is a function of TJ alone is correct, that is,
Ul is a function of "I alone.
The boundary conditions for the solution of Eq. (5.4) are that
J(fJ} == 0
!'(71) == 1
I" (11) --+0
for TJ == 0
for 71--+ 00
for 1/ ~ ex>
THEORY OF WING SECTIONS
88
Blasius obtained the solution of Eq. (5.4) by an approximate integration
in a Taylor series," The solution has not been given in explicit form but is
presented in the following table.
"
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
!'(.,,)
."
-
f'{.,,)
."
0
0.0664
0.1328
0.1990
0.2647
2.0
2.2
2.4
2.6
2.8
0.6298
0.6813
0.7290
0.7725
0.8115
4.0
4.2
0.3298
0.3938
0.4563
0.5168
0.5778
3.0
3.2
3.4
3.6
3.8
0.8461
0.8761
0.9018
0.9233
0.9411
5.0
5.2
5.4
5.6
5.8
6.0
I
I
I
4.4
4.6
4.8
I
I
!'(.,,)
0.9555
0.9670
0.9759
0.9827
0.9878
0.9916
O.9M3
0.9962
0.9975
0.9984
0.9990
A plot of these values is given in Fig. 41. These values define the
velocity distribution in a laminar boundary layer in the absence of a pres­
1.0
/
.8
/
~
~
~
/~
V
-6
:/
%
J
.4
JV
/
.2
)
V
-00
FlO.
41.
2
3
4
5
6
7
~AR;
Velocity distribution in the boundary layer over a flat plate according to Blasius.
sure gradient. The form of the variables in which this velocity distribution
is specified shows that the boundary-layer thickness increases as the square
root of the distance from the leading edgeand inversely as the square root
of the Reynolds number and that the velocity distribution is similar at all
points along a flat plate.
In addition to the velocity distribution through the boundary layer,
THE EFFECTS OF VISCOSITY
89
the solution of Eq. (5.4) gives a value of fl/('1) at the surface equal to
0.332, that is,
0.332 = /,,(.,,) = d(ujV) =
d(ujV)
d'1
d[(y/x)v'Rzl
For the skin friction, we have
.,. =
Cf
=
p,(~~)
= 0.332 p,V ~
dy *=-0
X
T I L . rnlL V 2 = 0.664 -IT v R%
72P
P x
}
0.654
= v- /R
(5.5)
%
The total drag coefficient for one side of a. fiat plate is then given by
11%
cr = -
x
CJ
0
r~4~dx = -l~xO~
. l!:- dx
z 0 VX pV
= !(1.323
x
fia.) = 1.328
~
"\}pV
(5.6)
The obvious definition of the boundary-layer thickness is the distance
from the surface to where the local velocity defect due to viscosity is zero,
that is, u/V == 1. Practically, this distance is not 'Yell defined because
u/V approaches unity asymptotically. A definition that has frequently
been used is that the boundary-layer thickness is the distance from the
surface to where u/V = 0.995. The preceding table indicates that, ac­
cording to this definition, o/x 5.3/v'R%. A thickness parameter that is
convenient for experimental 'York is the distance from the surface to the
point where (U/V)2 = ~2 or o/x'" 2.3/v'llz. For theoretical work it is fre­
quently desirable to obtain a measure of the effect of the boundary layer on
the outside flow. The reduced velocities within the boundary layer cause
a displacement of the streamlines around the body. This displacement is
called the U displacement thickness" of the boundary layer 0* and is
calculated from the relation
"'J
0* = -1 ~a (V - u)dy
1T
0
or in the general case
~. =
Y11 10t' (U -
u)dy
(5.7)
where U = velocity just outside boundary layer
For the laminar boundary layer over a flat plate with zero pressure gradient
~*
1.73
x= VR%
(5.8)
THEORY OF WING 8ECTION8
90
6.6. Momentum Relation. The KarmlLn integral relation27• 111 pro­
vides a method for relating the boundary-layer thickness to the pressure
distribution if the skin friction and the velocity distribution through the
boundary layer are known or assumed. The relation is useful because
good approximations to the skin friction and the velocity distribution can
often be obtained from consideration of local conditions. This relation is
applicable to both laminar and turbulent boundary layers. The Karman
integral relation is based on the principle of mechanics that, if any closed
surface is described within a steady stream of fluid, the time rate of in­
crease, within the surface, of momentum in any direction is equal to the
sum of the components in that direction of all the forces acting on the fluid.
H body forces such as weight are
absent, the only forces to be eon­
fI4
sidered are those acting on the surface,
dx fix
that is, normal forces caused by presu
sures and tangential forces caused by
_p6# ~'!dx viscous shearing.
Consider the equilibrium of the
element shown in Fig. 42. The mo­
mentum from left to right entering
/':;dx
the left face per unit time is
FlO. 42.
1"011/_. dy)
Forces on an element of the
Jo
boundary layer.
\flu,
while the momentum leaving the right face per unit time is
[ u(pu dy)
+
[:xl
u(pu dY)] d:c
The mass flow per unit time leaving the right face must be the sum of the
mass flow entering the left face plus the mass flow entering the upper
boundary, The mass flow per unit time through the upper boundary is
tx([
PUdy)d:c
This mass enters with the velocity U. The momentum entering the upper
boundary per unit time is then
u(:Xl'pu
dY)d:c
The total rate of change of momentum from left to right in the element is
then
mB EFFECTS OF VISCOSITY
91
The forces acting on the element are
pa
on the left face
- [PIS + ~(PIS)dx]
on the right face
P ::: da:
on the upper boundary
dx
on the lower boundary
-
T
The resultant force is
d
- - (p8)dx
dx
do
+ p -dx
dx -
T
dx
which reduces to
- {) dp dx dx
T
dx
Equating the resultant force and the rate of change of momentum
(:Xl'PU dY)d:x 2
Dividing through by
p
u(~l'pu dY)d:x =
-
IS:
dx -
T
dx
dx, we obtain
r
d
d f'
2
dx}o u dy - U dx}o u dy
0 dp
T
= - Pdx - -;;
This equation is not very convenient because the values of the integrands
do not approach zero at the upper limit of integration. -To avoid this
difficulty the integrands are expressed in terms of the variable (U - u). by
making the substitution
u=U-(U-u)
-d
l'[
dx o
L'[
J2
U-(U-u) dy-Ud
dx o
]
U-(U-u) dY=--..1!._!
0d
pdx
p
Expanding and collecting terms
dU
d
(0
d
t'
d
t'
UIS dx - dx U}o (U - u)dy + U dx}o (U - u)dy - tI.x}o (U.;... u)u dy
odp
T
==ptb;-p
At the outer limit of the boundary layer, Bernoulli's equation is valid;
hence
+ !2 pU2) = dH
= 0
dx
dx
and
dp
dU
dx = - pU dz
.!!(p
THEORY OF WING SECTIONS
92
Substitution of this relation in the previous equation gives
d
-dUl'
- (U-u)dy-dx o
l'
dx o
(U-u)udy=--T
p
(5.9)
This equation is known as the "!{arman integral relation." In this
form the integrals approach zero as the upper limit is approached. This
equation is suitable for the experimental determination of the skin friction,
although the accuracy of the determination may be affected considerably
by the form in which this equation is expressed.
Another form of Eq. (5.9) can be obtained by the use of the displace­
ment and momentum thicknesses. The displacement thickness is defined
by Eq. (5.7). The momentum thickness is related to the loss of momentum
of the air in the boundary layer and is defined as
8 = }o{'(I - ~)
1.£ dy = ~ r
U U
U 2 }0
(U -
1.£)1.£
dy
(5.10)
Substitution of tJ· and 8 in Eq. (5.9) gives
UlJ* dU +.!!:... U2fJ = i
dx
dx
UtJ. dU
dx
p
+ 2U8 dU + tr d8 = !
dx
dxp
dB +.!!...dU(tJ. +2)=~2
U dx (J
pU
dx
Introducing the shape parameter H =
= %plJ!
a*If}
and the dynamic pressure
q
dB +(/1 + 2)~dq =..:!.­
2
dx
q d»
(5.11)
2q
Equation (5.11) is in a convenient form for the calculation of the thick­
ness of the laminar boundary layer when the pressures are not constant, as
was assumed in the Blasius solution. 132 In many cases of interest, the pres­
sure gradients are sufficiently small so that the type of velocity distribution
through the boundary layer will not differ greatly from that in the bound­
ary layer of a flat plate.' Application of Eq. (5.10) to the Blasius distribu­
tion gives a value of 8/x equal to O.664/~. Consequently the value of
H for the Blasius distribution is 1.73/0.664 = 2.605. From Eq. (5.5),
T
=
p,U ~ ~
0.332- vRs
x
Substituting the value of 8
T
=
0.664 X 0.332
p,U
p,U
8 == 0.220 8
THE EFFECTS OF VISCOSITY
Substituting the values for Hand
dO +
dx
T
93
in Eq. (5.11)
(4.605)(_8_)
pU dU = O.220pU
2
dx
pU
~pU2
28
By use of the integrating factor 28CJ9·210, this expression becomes
iP = 0.4401L i" U 8 •210 dx
pUz 9.210j o
(5.12)
where U z = velocity at limit x
If U is known as a function of x, that is, if the pressure distribution is
known, Eq. (5.12) permits the boundary-layer thickness to be calculated
approximately if the pressure gradients are sufficiently small for the actual
distribution of velocity through the boundar)' layer to be approximated by
the Blasius distribution. For most wing sections at low and moderate lift
coefficients, the pressure gradients satisfy this condition back to the
position of minimum pressure. The skin friction may be assumed to be
that corresponding to the Blasius velocity distribution for the calculated
thickness of the boundary layer. A convenient way of performing these
calculations is to find an equivalent length of flat plate'l' 2 which is given by
i
X. =
.,.
..
.
~~.-.
..
(:.
{%
U",8.~lOJo
lj8.210
dx
(5.13)
where x, is the length of fiat plate in a stream of velocity Us that will
generate the same laminar boundary-layer thickness as actually exists on
the body in question at the point z.
5.7. Laminar Separation. It was pointed out in Sec. 5.3 that the
laminar boundary layer cannot ordinarily exist for long distances in a
region where the pressures are increasing ill the direction of flow, and the
extent of the laminar boundary layer under such conditions is limited by
separation. The Blasius solution does not, of course, give any information
about laminar separation because this solution is premised on the absence
of pressure gradients. Calculations of the location of laminar separation
require a solution of the boundary-layer equations in more general form.
One such attempt was made by Pohlhausen." He assumed that the
velocity distribution in the boundary layer could be expressed by a poly­
nomial of the form
0­
Y
(Y)2 + a3(Y)3
(Y)4
~ + a. ~
U = al ~ + at ~
'U
0
Using plausible assumptions regarding the physical characteristics of the
layer, Pohlhausen developed a method for calculating the velocity dis­
tribution in the boundary layer and the position of the separation point.
This method has only limited applicability to the calculation of the lam­
inar separation point. 105 Karman and Millikan-" obtained more accurate
94
THEORY OF WING SECTIONS
solutions of Eq. (5.2). The solutions are involved and require much cal­
culation, but the results appear to be satisfactory.
For most actual 'York on wing sections, the extent of the laminar
boundary layer in regions of adverse pressure gradients is too short to
require much detailed attention. It is frequently desirable, however, to
know the possible extent of the laminar layer as determined by laminar
separation. It has been possible to develop a method for applying the
I{arman-Millikan solution to the rapid estimation of the laminar separa­
tion point132 which is sufficiently accurate for most applications. More
detailed studies may, however, be required for the condition associated
with maximum lift.
The method for rapidly estimating the position of laminar separation
is based on the assumption that it is possible to replace the actual velocity
distribution effectively by a simplified velocity distribution consisting of a
region of uniform velocity followed by a region of linearly decreasing
velocity. The actual pressure distribution up to the point of minimum
pressure is used to calculate the equivalent length of fiat plate from
Eq. (5.13). Because dp/dx is zero at the point of minimum pressure, the
velocity distribution in the boundary layer at this point must approximate
the Blasius distribution very closely. The replacement of the actual pres­
sure distribution downstream of the point of minimum "pressure by a linear
velocity gradient is usually justified by the relatively small extent of the
laminar boundary layer in this region. Consequently, the laminar separa­
tion point may be estimated from positions calculated for a family of such
simplified pressure distributions.
The position of the laminar separation point was calculated by the
Kinnan-Millikan method for a series of pressure distributions of the fol­
lowing type.
u
- = 1
for 0 < !. < 1
Us
- z.­
and
u == 1 + F x - X e for 1 < ~
Uo
x.
- x.
where x == distance along surface from leading edge
x, == length of flat plate equivalent to length from leading edge to
position of minimum pressure
U = velocity outside boundary layer at any point x
U« == maximum velocity
F = velocity gradient (xc/Uo)(dU/dx) (constant for any given case)
Irrhe values of U8/ Ue, where U8 is the velocity at the laminar separation
POint, were calculated by the Kinnan-Millikan method for a series of
values of F, and the results are shown in Fig. 43.
In applying this method, the value of F is obtained from calculated
values of x,,/Uo and the value of ulj/dx correspondiug to the estimated
THE EFFECTS OF VISCOSITY
95
equivalent adverse velocity gradient. The corresponding value of UB/Uo
is found from Fig. 43 and is applied to the actual velocity distribution to
locate the position of laminar separation.
5.8. Turbulent Flow in Pipes. The earliest information on the nature
of turbulent flow was obtained from studies of the flow in pipes. Reynolds
found that, if the diameter of the pipe and the velocity were sufficiently
1.00
~
/
/
.96
/
~
~-
./
'/
V
I(
I
.92
.90 /
.880
1/
/
-.04
-.OS
-.16
-.12
-.20
-24
-.28
F
I""JG. 43. Velocity ratio for laminar separation as a function of the nondimensional velocity
gradient F.
small, the flow \'.&8 rectilinear or laminar, as ShO\VD by the behavior mf
colored filaments of fluid which did not mix with the surrounding fluid.
With larger diameters and greater velocities, the flow became turbulent, as
shown by diffusion of the colored filaments. Reynolds found that, if the
quantity pl'dlfJ. was less than about 2,000, the flow was always laminar.
The upper limit of this Reynolds number for laminar flow was indefinite
and depended mainly on the care taken to provide steady conditions and
smooth surfaces. Values of the upper limit as much as ten to twenty-five
times the lower limit have been found by various observers."
With laminar flow, the pressure gradient. along the pipe increases
linearly with the velocity; but, with turbulent flow, the gradient increases
nearly as the square of the velocity. The velocity distributions across the
diameter of the pipe for laminar and turbulent flows are shown in Fig. 44.
For turbulent flow, the velocity distribution is more nearly uniform than for
laminar flow, and the shearing stresses near the walls are considerably
higher. These conditions result from the increased transport of momentum
associated with the turbulent motion.
It has been observed that the nondimensional velocity distribution
THEORY OF WING SECTIONS
96
across the pipe is affected only slightly by the Reynolds number. On the
basis of dimensional reasoning similar to that used for the formation of
dimensionless coefficients for wings, Blasius" expressed the pressure loss in
coefficient form as follows:
r_
~ = Pi - 'P2_
l
~p:;z2
where PI - 112 = pressure drop in a pipe of length l and radius r with a fluid
of density p flowing at average velocity u.
For smooth pipes or pipes having geometrically similar roughness, the
coefficient ~ is a function only of the
:JIi!:~:~~~:~= =~~~:;:~:~~::E;:::::
in smooth pipes is
LAMINAR
:11
0.133
~= ~r
: ~£:~::iE:~1::~:~~:~~~:~
the pipe and the rate at which the
pressure-loss coefficient changes with
Reynolds number. The most im­
portant region in which to study the nature of the phenomena appeared to
be in the vicinity of the wall of the pipe where the velocity gradient is
high rather than in the middle of the pipe where the velocity is nearly
uniform. It appeared reasonable to assume that the velocity distribution
near the wall depended on the local quantities p and II- and the skin friction
T and was independent of the radius of the pipe. It was assumed, in ap­
proximate agreement with experimental evidence, that the velocity dis­
tribution across the entire pipe could be expressed as a function of y/r
alone where y is the distance from the wall of the pipe; that is, the velocity
distribution remains similar for various mean velocities and pipe sizes.
Equating the pressure loss to the skin friction
T\.J8JLENT
44. Velocity distribution across pipe
for laminar and turbulent flow.
FIG.
(PI - Pt)rr! = 2n-lT
2lT
PI-P2=-r
From Blasius' formula
(5.14)
THE EFFECTS OF VISCOSITY
97
If it is further assumed that the velocity distribution may be expressed as a
power function of y/r,
where V = velocity at center of pipe
or
3/
1/
= K p7 -&p.7 4 r-
]/
1
7/
7 4: ' 7 4
(r)(~)R
11
where K = a constant factor dependent on ratio of maximum velocity to
mean velocity
Applying the assumption that the velocity distribution near the wall and
hence the skin friction is independent of the radius of the pipe, the ex­
ponent of r in the foregoing expression must be zero,
or
~~+ ~n
=0
n= ~1
(5.15)
The assumption made in the derivation of this equation that the
velocity distribution is a function only of local parameters near the wall
would indicate that this equation would also apply to the velocity dis­
tribution in a turbulent boundary layer on a flat plate. In this case, r is
interpreted as the thickness of the boundary layer and V as the velocity
just outside the layer. The validity of the analogy between the flow in
pipes and that over a flat plate is shown by the data of reference 19, some of
which are presented in Fig. 45.
6.9. Turbulent Skin Friction. It is possible to calculate the skin
friction on ~fiat plate with turbulent flow by application of the one-seventh
power law [Eq. (5.15)] and the Blasius skin-friction formula [Eq. (5.14)] to
the local boundary-layer conditions. In the case of the flow of an incom­
pressible fluid in a pipe, the Reynolds number along the pipe remains
constant and the skin friction results in a loss of pressure. In the case of
the flat plate, the pressure is constant along the plate and the skin friction
causes a deceleration of the fluid particles, resulting in an increase of
the thickness of the boundary layer and a corresponding increase of the
boundary-layer Reynolds number, At any station along the plate, the
skin-friction drag over that portion of the plate upstream of the station
can be obtained from the rate of loss of momentum in the boundary layer.
It is assumed that, for any station, the velocity distribution in the boundary
layer corresponds to that in the pipe and that the local skin friction is the
same as that in a pipe at the same Reynolds number.
The particles in the boundary layer having an original velocity V up­
KI7$CMJ
CMftec
eooo
2000
-----
........
-
L.l .....
-
~-
L~
~
1000
~~
1000
~-
~~,.....
,....
-~
~
500
40,,-(:)
~
~
~
<:S
ti
CS
II
400
<::i
lI\Yv'M
-~-
-
,...,....
----
.fOOt"'
N
'"
......
--
~
...... .,.-
1000
_1000'
tS
I
~
I
I
"
a~
I
~
....
...--,
IIII
I
I
,
,
.....
~
~
II
500
,
,
, I I!'"
I
,
I
'~')CM
8
'''';
It)
~
a
~
...:
~
~
....:
a yl'M
t\i
~
---...... l'-'
~
....
~
~
400S?
ro:.
'"
~
~
~
~
r..:.
,..:.,..:.c::s
"....,....,
S;
C
~
.......
~
~
~
-~,.
--
~
~
~
~
~
-
..-...-'l .....
_
IIII
II1II1
~
() ()
~
ti
....
)I
-
~680CM
II
ij
-~
~
I
, ,
~
-~
...
~~
~
..-
x-I25CM
-
-'--
500
~
....
-~
2000
-~ -"I'"
,..--,
-
IioIIIOIl
CM/sec
-
2000
--~
~ ..--:
()
~
K'81$CM
CM,&tc
1000
500
8...:
...
~ 1""'""""1
~oIX·lOCM
and X Il 60CM
~
'to
--
-
~
'-
~
K'IOOCM
CM/sec
g
~
8 y"CM
...:...:t\i
FIG. 46. Logarithmic sua]o dj~"l\mA of tho vptocity in the boundary )nyer as a function 011/ at the sections z = 75, 87.5, 100, and
125 ern from tho leading edgo of the flt,t plate. The solid Imes represent thu one-seventh power law.
THE BFFECT8 OF VISCOSITY
99
stream of the plate are decelerated to a velocity u at station z, The rate of
loss of momentum at station z is given by the following expression and is
equal to the skin-friction drag of that portion of the plate upstream of
station x.
'
F
= p loa u(V -
u)dy
From the assumption that
V=(~)M
F
i2PV2c5
=
but
dF
dx =
If"
hence
7
2
dO
-r = 72 pV dx
A corresponding value of T may be obtained from Eq. (5.14) if the rels­
tion between ii and V is known. A commonly used experimental value of
ii/IT is approximately 0.8. Using this value Eq. (5.14) rna)' be written
in the form
r' = O.0225p~~p.~~o-3-~ TT~
Equating the
t\VO
expressions for
T
(5.16)
and integrating
(;Vy~ z
c5 = 0.37 (-tV)%x%
~ c5~ = 7; 0.0225
Solving for
a
(5.17)
Substituting this value of 8 into the expression for the drag
F = O.036pV%xG;x)%
In coefficient form
cr
=
lL F
7";.P
V2X
=
O.072R:r;-~
(5.18)
Equation (5.18) is a formula for the drag coefficient for one side of & smooth
flat plate of length x with turbulent flow over the whole surface. Figure
46 shows the skin-friction coefficient for turbulent flow from Eq. (5.18) and
for laminar flow from Eq. (5.6).
.
Nikuradse performed pipe experiments over a range of Reynolds num­
bers much larger than for previous investigations.76 • 7iG These experiments
THEORY OF WING SECTIONS
100
showed that the pressure losses departed from values predicted by the
Blasius formula above Reynolds numbers of about 50,000, the experi­
mental values decreasing less rapidly than the predicted values as the
Reynolds number was increased. As expected from the simplified analysis
of the relation between the pressure-loss coefficients and the velocity dis­
tributions, the velocity distributions at the higher Reynolds numbers are
approximated better by one-eighth, one-ninth, or smaller powers than
.0080 ~
.0070
~
.0060
.0050
f)()40
<,
~
~
=---
~
~
.-.
I!.......
-
~.- Turbulent
flI=I~-Eq.5.20
£q.5.18./'
f'~
"'~
"'i~
~ ~lt'"
~~
.........
............. .:::::::
,....
~LomintT(Eq.5.6)
:"r-...
0010
~
~
.0007
.0006
"" """
"" ""lIlli,
~~
~~
fJOO5
"'~
.0004
os
.ooaJ
~ ~ ""- .... ,.....
1..5 2 Z53
4 5 6 78906
1.5 2 25.3
4
5 6 78 9d
Rx
FIG.
46.
Laminar and turbulent skin-friction coefficients for one side of a fiat plate.
by the one-seventh power, On the basis of an elaborate and extended
dimensional analysis, a logarithmic expression was obtained28 • 128 for the
velocity distributions corresponding to Nikuradse's values for the pressure
losses.
q, == 2.5 In 1J + 5.5
(5.19)
where
u
<1>=v*
pt,*y
11=­
p.
v*={;
and is called the U friction velocity." The skin friction on a flat plate may
be calculated using this expression by a process using the same general
THE EFFECTS OF VISCOSITY
101
principles as that for the derivation of Eq. (5.18).28 The tabular solution to
this problem may be approximated by the expression
0.472
cp=---­
(5.20)
(log.1l:,;)2.ss
Skin-friction coefficients calculated from this formula are shown in Fig. 46.
Formulas (5.18) and (5.20) are in convenient form for the calculation of
skin friction on flat plates, For application to bodies having arbitrary
pressure distributions, it is more convenient to relate the local skin-friction
coefficient to a Reynolds number based on the momentum thickness of the
boundary layer. For the one-seventh power velocity distribution, such an
expression may be obtained from Eq. (5.16). For this case, according to
Eq. (5.10),
_T _
C/1 L U "2 72P
- ()
0 · 02~1
(-.!!-)~
;J
["'0
- . 02r-·lR-~
a,
p :
(5.21)
For the case of the logarithmic velocity distribution, approximate
formulas have been ohtuined relating t he local skin-friction coefficient to
the boundary-layer Reynolds number Ro• Thut given by Squire and
) . . oung!" is
er~ = [5.8HO log (-l.O~5Jlo)f
(5.22)
T
It is known that Eq. (5.21) is valid only for moderately small Reynolds
numbers, whereas Eq. (5.22) is a much better approximation- at large
Reynolds numbers. Equation (5.22) is, hO\VCVCf, inconvenient for use in
connection with the momentum equation (5.11). Interpolation formulas
have therefore been developed in the form
c, = .ilR,-n
(5.23)
where the coefficients and exponents nrc selected to correspond with the
desired range of Reynolds numbers.
For any particular application, the two constants of Eq. (5.23) may be
found by making Eq. (5.23) agree with the assumed skin-friction law at t\VO
values of R, including the range under consideration. If the subscript 1
corresponds to conditions at the low value of R s and the subscript 2 to
conditions at the high value of R" then
(T / Pl]2)l
log (T jp(]2)2
n-­
- log (R 8,./RIJJ
A
=
c/lRS1"
THEORY OF WING SECTIONS
102
The assumed skin-friction law may be Eq. (5.22) or the following formula
· 120
suggested by T etervm
T
1
(5.24)
2
pU = [
R
2.5 In 2.5(1 _ 5~) + 5.5
J2
The values of cJ corresponding to Eqs. (5.22) and (5.24) are plotted in
Fig. 47.
6.10. Calt;ulation of Thickness of the Turbulent Layer. A knowledge
of the thickness of the turbulent boundary layer is required for several
.007
.006
f){)5
I
-~~
~!"""o
.004
------
~.OO3
.......
......... ~I""-~
....
Eq.5.22 ~
Ef~·524
~"-
..
~
.002
2
s
4
5 6 78 9 f()4
--.
~~
2
....
3
~ 1-1-.
4
.-
r-
5 6 78 gl()5
R(J
FIG. 47. Local turbulent skin-Iriction coefficient as a function of the boundary-layer Reynolds
number.
purposes. The profile drag of the wing is, of course, intimately associated
with the thickness of the boundary layer at the trailing edge. It is also
desirable to know the thickness of the boundary layer at various points on
the wing surface in connection with the study of control surfaces, high-lift
devices, air intakes, and protuberances.
The thickness of the turbulent boundary layer can be calculated from
the momentum equation (5.11) if a relation between the local skin-friction
coefficient and the dependent variables 8, q, and H is knO\VD. Values of the
skin-friction coefficient have been found by assuming that the relation
between the boundary-layer Reynolds number R, and the skin-friction
coefficient
is the .same as the corresponding relation found from pipe
experiments. This relation has been plotted in Fig. 47. Although this
relation may not be very accurate, .particularly for values of the shape
parameter H differing considerably from those found in pipes, such marked
differences usually occur only when dqld» is large negatively. In this case
Eq. (5.11) shows that the contribution of the skin-friction term to the rate
of increase of the boundary-layer thickness dO/dx is relatively small. It
may accordingl~ybe assumed for practical calculations that CJ is independent
of the value of H.
Further simplification of the calculations is obtained by assuming a
c,
THE EFFECTS OF VISCOSITY
103
skin-friction law of the type indicated by Eq. (5.23).
(5.11) may be written in the form
de + (H +
dx
U
2) dU
dx
(J =
A~n
(pU)r&
e»
In this case, Eq.
. ' ,;,-."
The value of the factor (H + 2) is not greatly affected by the value of
H, which usually has values of about 1.4 or 1.5 with extreme variations
from about 1.2 to 2.5. Detailed calculations have S110\Vn that excellent
agreement can be obtained between experimental and calculated values
of 6 if H is assumed to be constant and to have a value"? between 1.4 and
1.6. "~ith H assumed constant, this equation is seen to he of the Bernoulli
type, and it can be integrated to give the value of 6 us follows:
(~)
C ",/c
=
1
(UjV)H+2
[(1
+
-t:
2Rc "
(U)(l/+l)
Xo!c
V.
(u+lH-l
~
dc
+ (~)"+l( ¥~)rH+2)("+I] 1/<1+..>
A
(5.25)
6.11. Turbulent Separation. To determine the turbulent separation
point, detailed information is needed concerning the velocity distribution
in the boundary layer and the effects on the -velocity distribution of such
factors as the pressure distribution and the skin friction. Although the
value of the shape parameter H can be found for any given velocity dis­
tribution, the value of H in itself does not define the velocity distribution.
Thedata collected by Grusehwitz'" and by Tetervin-" show experimentally
that the velocity distributions in turbulent boundary layers actually form
a one-parameter family of curves and that the velocity distributions can be
specified by the value of H. Velocity profiles for turbulent boundary
layersl 35 are presented for various values of H in Fig. 48. The value of H
increases as the separation point is approached. It is not possible to give
an exact value of H corresponding to separation, because the turbulent
separation point is not very well defined. The value of H varies so rapidly
near the separation 'point, however, that it is not necessary to fix accurately
the value of H corresponding to separation. Separation has not been
observed for a value of H less than 1.8 and appears definitely to have been
observed for a value of H of 2.6.
An empirical expression was found 135 for the rate of change of H along
the surface in terms of the ratio of the pressure gradient to the skin friction
and the shape parameter itself.' The following equation was derived to fit
the experimental data:
8 -dH = e4.680 (H dx
2.975)
Q2q - 2.03::>(H
[8d
,..
- 1.286) ]
- - - -
qdx
(5.26)
T
The skin friction corresponding to a given value of R6 may be obtained
from Fig. 47 or from Eqs. (5.22) or (5.24). Equations (5.11) and (5.26) are
THEORY OF WING SECTIONS
104
simultaneous first-order differential equations that can be solved by step­
by-step calculation. It is usually necessary to use such a method, although,
8 7
J/.~9
1.0
! ~.
d
.Q
r-
.
~
..
=­
'"U"'o
~
Q~
l7U
r'ii:Pc Cilt't
~
117'
-4
~
~~
~~
~
,
~~ ~
-.
.4
fP'~
..,
+
~
~1(
~
o-Q
r4-...~
.
~
~
-­
~ r--...
~~
.2
/2
/.6
1.4
.5
~
~
s-;
N
~
,r--..
..
~
r--­ B"'-­
'I
-~
~
~~
-
2.2
~
~AJ
K oht
r--
)(~2
~~
-
0' ~ ~
2.0
H
1.8'
r-­
IOoz
~~
---.... r---..
~
--.
0
o
..,~
... 1>
---v-
Q
~~
A
It:.
~
~
04:
v
0
6
........
..
h.
n
r ..
I\IIl
~A ~
~~ ~
.6
bh
''''V
ft_
.. ~
.....
"U
61.
2
........... ~
+ I
02
D3
~
N
~~
~
rw
v4
F--
.
2.4
~5
46
I--
2.6
q
7
08
~9
2.8
(0) • varia/ion 01 'rU wi/h H lor various values 01 ~
H
~r--"""--;
/.286 1---+--'--;---;r---1
A /.4
I:"
.,.....-..--....--.. 0
1--,~,r-,..,..e.,~'7"-:::~~~~--t--t--t--.-,t--t
1.5
a 1.6
0
tr
~~..,4.~~~~~,I'-+---+--t--t-~t-., + l8
x 1.9
1fU.~~~'#7~~-+--t---t---.,t---t---;r---t <J ~.O
~ 2./
~~~~~t----t--+--"""""'--t----t--'t---t
b.
2.2
~
2.3
~~o
'---ir---+--t---t--i---t---t---t---t 17
2.4
v 2.5
2.6
-0-2.7
6
~-+--t---+--t----+--t---+--t--t--1t-"""1-o-
1
2
4
~8
7
9
(b) - velocity profiles for turbulent botndory layers correspcnding
10 various KJlues of H
FIG.
48.
Velocity diatributions in turbulent boundary layers.
for SOme particular cases, the equations may be integrated directly. The
method of calculation is as follows. The values of the variables entering
into the computation at the initial station are substituted in the momentum
equation (5.11) and the equation for dH/dx (5.26). Values for d8/dx and
dH/dx are thus obtained at the initial station. An increment of the length
THE EFFECTS
or
VISCOSITY
105
along the surface of the body x is then chosen and multiplied by d8/dx
and dH /dx to give aO and 1::Jl, respectively. These increments of 8 and H
are added to the initial values and result in values of 8 and H for the ne-w
value of x, The process is repeated until the separation point is reached,
as shown by the value of H. For the purposes of the computations the
value of H corresponding to separation may be taken as 2.6. The choice of
the increments of x is a matter for the judgment of the investigator. AB a
general rule, the increment of x should be made small when d8/dx or dH/dx
changes rapidly from one value of x to the next.
5.12. Transition from Laminar to Turbulent Flow. Experience has
shown that transition from laminar to turbulent flow takes place at a Rey­
nolds number that depends upon the magnitude of the disturbances. For
viscous flows at very low Reynolds numbers, as in oil, all disturbances are
damped out by the viscosity, and the flow is laminar regardless of the
magnitude of any disturbance. .As the Reynolds -number is increased, a
condition is reached at which some particular types of disturbances are
amplified and eventually cause transition. This value of the Reynolds
number is called the U lower critical." Further increase of the Reynolds
number causes amplification to occur for a greater variety of disturbances
and increases the rate of amplification. Under these circumstances the
Reynolds number at which transit.ion occurs depends 011 the magnitude
of the disturbances. Transition can he delayed to high values of the
Reynolds number only by reducing all disturbances such as stream turbu­
lence, unsteadiness, and surface roughness to a minimum. For ftO\VS in a
pipe, the lower value of the critical Reynolds number based on the diameter
and tho mean velocity is about 2,300.98 The upper value appears to depend
only on the care taken in conducting the experiment, and values twenty
times as great as the lower value have been obtained.
I t has been found possible 99 • 100. 127,128 to compute the rate of amplification
or damping of disturbances of various frequencies and wave lengths. Such
computations's" permit. curves of constant amplification coefficient {:ji~* IV to
be plotted on coordinates of all· against the Reynolds number R,. = p 1l8*/p.
where a = 21rII and l is the wave length. Such a plot is given in Fig. 49.
Experimental points obtained by Schubauer and Skramstad'P are also
plotted in Fig. 49. The data of Fig. 49 were obtained for a Blasius profile
corresponding to flow over a flat plate. It will be noted that there is a
Reynolds number below which no disturbance will be amplified. This
value of the Reynolds number is the lower critical. The agreement between
the experimental and theoretical results is good considering the difficulties
of the experimental work and the simplifying assumptions made in the
development of the theory.
Similar calculations have been made by Schlichting and Ulrich'?' for
Pohlhausen's velocity profiles corresponding to various pressure gradients.
THEORY OF WING SECTIONS
106
The results are presented in Fig. 50, which is a plot of the lower critical
Reynolds number against the parameter X that defines the velocity distri­
bution through the laminar layer. According to the Pohlhausen method,
.44r--.....---.....--........----.---.....,.---,----.---,.---,--,-----...
I
.40
__9110-4
\
.36
\
\
-35
"
I
-/9
.32
. 28
.24­
a6*
.20
./6
./2
.08
.04
°0
400 800 /200
16fX)
2000 2400 2800
~ 3600 4000 4400
R
Contours of equal amplification according to Schlichting. V &lUC8 of fji~· i U v (aU
valuee to be multiplied by 10-4 ) opposite points are amplifications determined by experiment..
Faired experimental contour of zero amplification shown by broken curves.
FIG.
49.
the velocity distribution through the layer is given by the following
formula
and
A = p02dU
p,dx
Figure 5Q shows that decreasing pressures in the direction of flow are
favorable to the stability of the laminar boundary layer and that increasing
pressures are unfavorable..
The theory has not been developed to such a point that it can be used to
predict transition, although it is useful in pointing out the fundamental
reasons for the instability of the laminar layer at high Reynolds numbers.
The theory does not relate the boundary-layer disturbances to either the
THE EFFECTS OF VISCOSITY
107
.surface irregularities or the turbulence of the air stream. The theory also
does not relate the transition to the magnitude of the fluctuations of veloc­
ity in the boundary layer. Under these circumstances, it is necessary to
t--+--+--+--+--+--J~,--I 8
I
p(
I
V
t--~~--+---+I--"'I03jl+--+--+-~~-"""~~
I
I
I
I
I
-6 -4
-2
I
I
'f
1/
0
2
4
6
8
50. Critical values of lU- as 11 function of the shape parameter '" aceording to Schlichting
and Ulrich. I OI
FIG.
rely on empirical results to predict the location of transition. The empirical
data are discussed in Chap. 7.
6.13. Calculation of Profile Drag. In cases where it is known that the.
flow is not separated from the surface, the boundary-layer thickness at the
trailing edge of a wing section and the profile drag can be calculated from a
knowledge of the potential-flow pressure distribution and the location of
transition from laminar to turbulent flow, The momentum thickness of
the laminar boundary layer at the transition point can be calculated from
Eq. (5.12). The initial momentum thickness of the turbulent boundary
layer is taken to be the same as that of the laminar layer at the transition
point. The thickness of the turbulent boundary layer at the trailing edge
can be calculated from Eq. (5.25) . If the pressure at the trailing edge were
free-stream static pressure, as in the case of a flat plate, the profile drag
would be
THEORY OF WING SECTIONS
108
.0/6
.0/4
x
Experimenta! meoSU:;menIs
~
R-2.9xlO
T
R=6.0xlOs
~
-
lC/
Theoreticol colcu'r/J8"5
.0/2 -
--R=2.9x
----R-6'Ox106
I
/
.010
c>
/
,
~
~
ioo""~
"r;;'
/~/
iJl
x
\
+
\\
.006
J
J.
'X
'"
~'
~-- --.-iIk
1-
,~~
+
o
.2
-t
.-:l~
~t
.002
-.2
C,
.4
.8
.6
LO
FIG. 51. Comparison of calculated and experimentally measured polars for N.ACA 67,1-215
airfoil.
.014
.010
I
t
.012
'-
+~
~.008
~..........
~
.........
,
J
/
/'
¥
!~
"'--
...~
......
~+~- ... .-:. ~
---
-~
~
+ Experimedol
.004
measurement
-
.oO?:s
Theoretical
Or--'"
I---
f - - I---
colcu/olion
-.6
-.4
-.2
o
.2
Cz.
.4
.6
.8
1.0
/2
1.4
Comparison of calculated and experimentally measured polars for NACA ~015
airfoil at R = 5.9 X 10'.
FIG. 52.
where 8 = sum of momentum thicknesses on upper and lower surfaces
In the usual case, the pressure at the trailing edge is not the same as
free-stream static pressure, and some means must be found for finding the
effective momentum thickness at a point far downstream in the wake
THE EFFECTS OF VISCOSITY
109
where the pressure has returned to the free-stream static value. Squire
and Y oung'P derived the required relation by setting the value of T in
Eq. (5.11) equal to zero in the wake and finding an empirical relation
between Hand q/qo- The resulting expression for the profile drag is
(8)
ctl=2- (U~
- (Jl+fl)/2
c t 11 t
(5.27)
where the subscript t designates conditions at the trailing edge. The value
of H is the value used in calculating the boundary-layer thickness.
The agreement to be expected between experimental and calculated
profile drag coefficients is indicated
by Figs. 51 and 52. The calculated
14
--...r-­
r-­
drag coefficients presented in these
.... 8
"""'­ --.
..............
~
figures were obtained by a method"
r-----. ........... A
r---... r--­
fundamentally the same as that pre- ~ 12
r--­
senteel here.
¢
1.0
5.14. Effect of Mach Number
on Skin Friction. Solutions for the
velocity distribution through the
080
1
4
2
3
5
laminar boundary layer in corn­
Mach number
pressihle flow and for the correspond- FIG. 5~1. Skin-Iriction eoeffir-ienrs, ( ...4) No
ing skin friction have been obtained heat transferred to wall. (Ill wall tern perature
one-quarter of Iree-streurn temperature,
by von Karman and Tsien1ft for the
case of the flat plate. The results of the skin-friction calculations are pre­
sented in Fig. 53 for the case of no heat transfer to the plate and for the
case of a plate whose absolute temperature is one-quarter of that of the
free stream. These results show a moderate decrease in laminar skin
friction with increasing Mach number. Although these results indicate
that the laminar skin friction increases with beat transfer from the fluid
to the plate, this increase is small even for the extremely 10"" plate temper­
atures for which these results were obtained.
For the turbulent boundar)' layer, Theodorsen and Regier!" found ex­
perimentally that the skin friction is independent of the Mach number (at
least up to a value of 1.69). The experimental data are presented in Fig. 54.
These data were obtained by measuring the torques (moments) required
to rotate smooth disks in atmospheres of air and Freon 12 (CCI2F2) at
various pressures. Data obtained by Keenan and X eumannf for the skin
friction in pipes appear to confirm Theodorsen's conclusion for the fully
developed turbulent flow.
Analytical studies by Leeg6% on the effect of Mach number and heat
transfer on the lower critical Reynolds number for the case of the flat
plate indicate that increasing th~ Mach number has a destabilizing in­
fluence on laminar flow when there is no heat transfer to the plate. Heat
---
THEORY OF WING SECTIONS
110
transfer to the plate has a stabilizing effect, while the opposite is the case
for heat transfer from the plate to the fluid, This effect becomes stronger
I
-1.2
I
I I
Moch
P1essute
(in.HgobsJ
tvmber
Q24toQ62 r-­
1.8 fo.JO
Air
.4010 .96
Freon /2
30
t-­
14
.5J 10/.42
FI1!OII 12
to.77101.69
Fr«Jfl 12
5
I
I
Symbol
0
-1.4
04­
x
D
~ IIlloo....
~ ~ ~ ........ 100....
""""­
4.8
5.0
52
~
KDrrnDns
-Ys
r--. ........
"""­
~~
4
~ KcrnD1s /ominar-f/oW"'/ i" ~
-2.2
fomvlo, C",=.J.871r~
~,
-2·~.6
-
futtJu/enI-f/oN formub,
/"" Lm
CM=O.I46R
-~
"'"",,­
-2.0
[,I
I
60s
5.4
5.6
a
-"'1fI­
M=O.53 ~=l69
~
~ ~ ~.
~
5.8
6.0
6.2
6.4
....:..; ~ .......
+ it-+
6.6
6.8
-ZO
LDgI()R
FIG. M. Moment coefficient for disks as function of Reynolds number for several values of
Mach number with air and Freon 12 as mediums, Maximum Mach number, 1.69.
as the Mach number is increased, and, at moderate supersonic speeds,
comparatively small amounts of heat transfer to the plate stabilize the
laminar layer to very high values of the Reynolds number.
CHAPTER 6
FAMILIES OF WING SECTIONS
6.1. Symbols.
Ps resultant pressure coefficient
R radius of curvature of surface of modified NAC.A four-digit series symmetrical
sections at the point of maximum thickness
IT velocity of the free stream
a coefficient
a mean-line designation; fraction of the chord from leading edge over which loading
is uniform at the ideal angle of attack
e chord
ci. section design lift coefficient
c.e/. section moment coefficient about the quarter-chord point
d coefficient
k1 constant
m maximum ordinate of the mean line in fraction of the chord
p ehordwise position of 111
r leading-edge radius in fraction of the chord
r, leading-edge radius corresponding to thickness ratio t
maximum thickness of section in fraction of chord
v local velocity over the surface of a symmetrical section at zero lift
!:Av increment01local velocity over the surface of a wing section associated 'lith camber
Av.. increment of local velocity over the surface of a wing section associated with
angle of attack
%
abscissa. of point on the surfaee of a symmetrical section or a chord line
XL abscissa of point on the lower surface of tl. wing section
Xu abseisaa of point on the upper surface of a wing section
XC abscissa. of point on the mean line
YL ordinate of point on the lower surface of a wing section
Yu ordinate of point on the upper surface of a wing section
Yc ordinate of point on the mean line
y, ordinate of point on the surface of a symmetrical section
Qi
design angle of attack
8 tan-I (dyc/dzc )
T
trailing-cdge angle
6.2. Int:"oduction. Until recently the development of wing sections has
been almost entirely empirical. Very early tests indicated the desirability
of a rounded leading edge and of a sharp trailing edge. The demand for
improved wings for early airplanes and the lack of any generally accepted
wing theory led to tests of large numbers of wings with shapes gradually
improving as the result of experience. The Effie! and early RAF series
were outstanding examples of this approach to the problem.
111
THEORY OF WING SECTIONS
112
The gradual development of wing theory tended to isolate the wing­
section problem from the effects of plan form and led to a more systematic
experimental approach. The tests made at Gottingen during the First
World War contributed much to the development of modem types of wing
sections. Up to about the Second World War, most wing sections in
common use were derived from more or less direct extensions of the 'york
at Gottingen. During this period, many families of wing sections were
tested in the laboratories of various countries, but the 'York of the NACA
was outstanding. The NACA investigations were further systematized by
separation of the effects of camber and thickness distribution, and the experi­
y
./0
°u(xy,Yy)
e
-./0
'I
p!!_.LJ_-_~-=---
~(XLIYLJ
'Rodius fhrough end of chord
(mean line slopeOf a5 %
chord)
Xy=X-Jf sin9
XL=X"'t
sinS
Yu=~
+11 cos9
JL 7C -JI cos e
Sample calculationA lor derivation 01 the NACA 86.3-818 airfoil (0 -= 1.0)
-
tan'
- ­ ---1----1
sin"
FIG. 56. Method of combining mean lines and basic-thickness forms.
mental work was performed at higher Reynolds numbers than were generally
obtained elsewhere. The wing sections now in common use are either N .~CA
sections or have been strongly influenced by the NACA investigations.
For this reason, and because the NACA sections form consistent families.
detailed attention will be given only to modem NACA wing sections.
8.3. Method of Combining Mean Lines and Thickness Distributions.
The cambered wing sections of all NACA families of wing sections con­
sidered here are obtained by combining a mean line and a thickness
distribution. The process for combining a mean line and a thickness
distribution to obtain the desired cambered wing section is illustrated in
Fig. 55. The leading and trailing edges are defined as the forward and
FAMILIES OF WING SECTIONS
113
rearward extremities, respectively, of the mean line. The chord line is
defined as the straight line connecting the leading and trailing edges.
Ordinates of the cambered lying sections are obtained by laying off the
thickness distributions perpendicular to the mean lines. The abscissas,
ordinates, and slopes of the mean line are designated as Xc, Ye, and tan 8,
respectively. If Xu and Yu represent, respectively, the abscissa and ordi­
nate of a typical point of the upper surface of the wing section and y, is
the ordinate of the symmetrical thickness distribution at ehordwise position
%, the upper-surface coordinates are given by the following relations:
Xv
= :r -
y, sin 9 }
yv = Yc + y, cos "
(6.1)
The corresponding expressions for the lower-surface coordinates are
XL = X + Yt sin 8 }
YL = Ye - Yt cos 8
(6.1)
The center for the leading-edge radius is found by drawing a line through
the end of the chord at the leading edge with a slope equal to the slope of
the mean line at that point and laying off a distance from the leading edge
along this line equal to t.he leading-edge radius. This method of con­
struction causes the cambered wing sections to project slightly forward of
the leading-edge point. Beeause the slope at the leading edge is theoreti­
cally infinite for the mean lines having a theoretically finite load at the
leading edge, the slope of the radius through the end of the chord for such
mean lines is usually taken as t.he slope of the mean line at x/c equals
0.005. This procedure is ~ustified by the manner in which the slope
increases to the theoretically infinite value as x/c approaches o. The slope
increases slowly until very small values of x/c are reached. Large values
of the slope are thus limited to values
x/c very close to 0 and may be
neglected in practical wing-section design.
­
or
The data required to construct some cambered wing sections are pre­
sented in Appendixes I and II, and ordinates for a number of cambered
sections are presented in Appendix lll.
6.4:. NACA Four-digit Vmg Sections. a. Thickness Distributions.
"'hen the N ACA four-digit wing sections were derived.P it was found that
the thickness distributions of efficient wing sections such as the Gottingen
398 and the Clark Y were nearly the same when their camber was removed
(mean line straightened) and they were reduced to the same maximum
thickness. The thickness distribution for the NAC~4\. four-digit sections
was selected to correspond closely to that for these wing sections and is
\' given by the following equation:
±'Yt,=_t- (O.29G90·~-O.12600x-O.35160x2+0.28430r-O.l0150zC)(6.2)
0.20
THEORY OF WING SECTIONS
114
where t == maximum thickness expressed as a fraction of the chord
The leading-edge radius is
r, = 1.1019t2
(6.3)
It will be noted from Eqs. (6.2) and (6.3) that the ordinate at any point
is directly proportional to the thickness ratio and that the leading-edge
radius varies 88 the square of the thickness ratio. Ordinates for thickness
ratios of 6, 9, 12, 15, 18, 21, and 24 per cent are given in Appendix I.
b. Mean Line«. In order to study systematically the effect of variation
of the amount of camber and the shape of the mean line, the shape of the
mean lines was expressed analytically as two parabolic arcs tangent at
the position of maximum mean-line ordinate. The equations" defining
the mean lines were taken to be
m
Yt: :;: p;. (2px - x 2 )
forward of maximum ordinate
and
(6.4)
Yc
=
m
(1 _ p)t [(1 - 2p)
+ 2px -
r]
aft of maximum ordinate
where m = maximum ordinate of mean line expressed as fraction of chord
p == chordwise position of maximum ordinate
It will be noted that the ordinates at all points on the mean line vary
directly with the maximum ordinate. Data defining the geometry of
mean lines with the maximum ordinate equal to 6 per cent of the chord are
presented in Appendix II for chord,vise positions of the maximum ordinate
of 20, 30, 40, 50, 60, and 70 per cent of the chord..
c.. Numbering System. The numbering system for NACA wing sections
of the four-digit series is based on the section geometry. The first integer
indicates the maximum value' of the mean-line ordinate Yc in per cent of the
chord. The second integer indicates the distance from the leading edge
to the location of the maximum camber in tenths of the chord. The last
two integers indicate the section thickness in per cent of the chord.. Thus
the NACA 2415 wing section has 2 per cent camber at 0.4 of the chord
from the leading edge and is 15 per cent thick.
The first two integers taken together define the mean line, for example,
the NACA 24 mean line. Symmetrical sections are designated by zeros
for the first two integers, as in the case of the NACA 0015 wing section,
and are the thickness distributions for the family.
d. Approximate Theoretical Characteristic8. Values of (V/V)2, which is
equivalent to the low-speed pressure distribution, and values of v/V are
presented in Appendix I for the NACA 0006, 0009, 0012, 0015, 0018, 0021,
and 0024 wing sections at zero angle of attack. These values were cal­
culated by the method of Sec. 3.6. Values of the velocity increments
AIl.IV induced by changing angle of sttaok are also presented for an
FAMILIES OF WING SECTIONS
115
additional Iift coefficient of approximately unity. Values of the velocity
ratio v/V for intermediate thickness ratios may be obtained approximately
by_linear scaling of the velocity Increments obtained from the tabulated
values of v/V for the nearest thickness ratio; thus
(!!.V.)" = [(~)
1
II
1J ~ + 1
t1
(6.5)
Values of the velocity increment ratio ~vc/V may be obtained for inter­
mediate thicknesses by interpolation.
The design lift coefficient. Cl i , and the corresponding design angle of
attack oa, the moment coefficient Cm.lf) the resultant pressure coefficient P R ,
and the velocity ratio ~/V for the NAC.L~ &2, 63, 64, 65, 66, and 67 mean
lines are presented in Appendix II. These values were calculated by the
method of Sec. 3.6. The tabulated values for each mean line may be"
assumed to vary linearly with the maximum ordinate Ye; and data for
similar mean lines with different amounts of camber, within the usual
range, may be obtained simply by scaling the tabulated values. Data for
the NACA 22 mean line may thus be obtained simply by multiplying the
data for the NACA 62 mean line by the ratio 2:6, and for the X.~CA
44 mean line by multiplying the data for the N.t\CA 64 mean line by the
ratio 4:6.
Approximate theoretical pressure distributions may be obtained for
cambered lying sections from the tabulated data for the thickness forms
and mean lines by the method presented in Sec. 4.5.
6.6. NACA Five-digit Wlng Sections. 4. Thickne&8 Distributions. The
thickness distributions for the NAC.A~ five-digit wing sections are the same
as for the NACA four-digit sections (see Sec. 6.4a).
b. Mean. Lines. The results of tests of the N.&-\.CA four-digit series wing
sections indicated that the maximum lift coefficient increased as the posi­
tion of maximum camber was shifted either forward or a~t of approximately
the mid-chord position. The rearward positions of maximum camber were
not of much interest because of large pitching-moment coefficients. Be­
cause the type of mean line used for the NACA four-digit sections was not
suitable for extreme forward positions of the maximum camber, a new
series of mean lines was developed, and the resulting sections are the NACA
five-digit series.
. The mean lines are defined 46 by two equations derived so as to produce
shapes having progressively decreasing curvatures from the leading edge
aft. The curvature decreases to zero at a point slightly aft of the position
of maximum camber and remains zero from this point to the trailing edge.
The equations for the mean line are
Yc = ~kl[x3 - 3mx2 + m2(3 - m)x]
from x = 0 to z = ml
y.; = %k1m3(l - x)
from x = m to oX = c = l~
(6 6)
.
THEORY OF WING SECTIONS
116
The values of m were determined to give five positions p of maximum
camber, namely, 0.000, O.IOc, O.ISe. O.2Oc, and O.25c. Values of k1 were
initially calculated to give a design lift coefficient of 0.3. The resulting
values of p, m, and k1 are given in the following table.
Mean-line
designation
Position of
camber p
""
kl
210
0.05
0.10
0.15
0.20
0.25
0.0580
0.1260
0.2025
0.2900
0.3910
361.4
51.64
15.957
6.643
220
230
240
250
3.230
This series of mean lines was later extended" to other design lift
coefficients by scaling the ordinates of the mean lines. Data for the mean
lines tabulated in the foregoing table are presented in Appendix II.
c. Numbering System. The numbering system for wing sections of the
N..\ CA five-digit series is based on a combination of theoretical aerodynamic
characteristics and geometric characteristics. The first integer indicates
the amount of camber in terms of the relative magnitude of the design
lift coefficient; the design lift coefficient in tenths is thus three-halves of
the first integer. The second and third integers together indicate the
distance from the leading edge to the location of the maximum camber;
this distance in per cent of the chord is one-half the number represented by
these .-integers. The last two integers indicate the section thickness in
per cent of the chord. The NACA 23012 wing section thus has a design
lift coefficient of O.3 J has its maximum camber at 15 per cent of the chord:
and has a thickness ratio of 12 per cent.
d. Approximate Theoretical Characteristics. The theoretical aerody­
namic characteristics of the N ACA five-digit series wing sections may be
obtained by the same method as that previously described (Sec. 6.4<1) for
the NACA four-digit series sections using the data presented in Appen­
dixes I and II.
6.6. Modified NACA Four- and Five-digit Series Wing Sections. Some
early modifications of the NACA four-digit series wing section· included
thinner nosed and blunter nosed sections which were denoted by the
suffixes T and B, respectively. Some sections of this family49 with refiexed
mean lines were designated by numbers of the type 2R112 and 2R212, where
the first integer indicates the maximum camber in per cent of the chord and
the subscripts 1 and 2 indicate small positive and negative moments,
respectively.
Another series of N ACA five-digit wing sections" is the same as those
previously described except that the mean lines are reflexed to produce
theoretically zero pitching moment, These sections are distinguished by
FAMILIES OF JV[.;V'G SECTIONS
117
.he third integer, which is always 1 instead of o. These modified sections
lave been little used and will not be discussed further.
More important modifications common to both the NACA four-digit
md five-digit series wing sections consisted of systematic variations of the
hickness distributions. 1M These modifications are indicated by a suffix
-onsisting of a dash and two digits as for the NACA 0012-64 or the 23012-64
sections, These modifications consist essentially of changes of the leading­
edge radius from the normal value [Eq. (6.3)] and changes of the position
of maximum thickness from the normal position at 0.3Oc [Eq. (6.2)]. The
first integer following the dash indicates the relative magnitude of the
leading-edge radius. The normal leading-edge radius is designated by 6 and
j, sharp leading edge by o.
The leading-edge radius varies as the square of
this integer except for values larger than 8, when the variation becomes
]xbitrary.U6 The second integer following the dash indicates the position
of maximum thickness in tenths of the chord. The suffix -63 indicates sec­
tions very nearly but not exactly the same as the sections without the suffix.
The modified thickness forms are defined by the following two equations:
± y,
:=
aov'X + alX + ll2X2 + a~
ahead of maximum thickness
} () 7'
:l:Yt==do+dl(1-x)+d2(1-x)~d3(1-x)3 aft of maximum thickness (.)
The four coefficients d-o, dt , d~, and ds arc determined from the following
conditions:
1. Maximum thickness, t.
2. Position of maximum thickness. In.
3. Ordinate at the trailing edge x = 1, y, = O.Olt.
4. Trailing-edge angle, defined by the following table:
m
dl/
--:., (x
dx
0.2
0.3
0.4
0.5
0.6
= 1)
1.000t
1.170l
1.575t
2.325t
3.500t
The four coefficients 00, at, a2, and U3 are determined from the following
conditions:
1. Maximum thickness, t.
2. Position of maximum thickness. m,
3. Leading-edge radius, r, == ~2/2. T, = 1.1019(tI/6)2, where 1 is the
first integer following the dash in the designation and the value of I does
not exceed 8.
4. Radius of curvature R at-the point of maximum thickness
R=
1
2d: + 6d3(1 -
11t)
118
THEORY OF WING SECTIONS
Data" far some of the modified thickness forms are presented in
Appendix I. The data presented may be used together with data for the
desired type of mean line to obtain approximate theoretical characteristics
by the method presented in Sec. 4.5.
The family of N ACA wing sections defined by these equations has been
studied extensively by German aerodynamicists, who have applied designa­
tions of the following type to these sections:
NACA
1.8
25
14-1.1
30/0.50
where 1.8 = maximum camber in per cent of chord
25 = location of maximum camber in per cent of the chord from
the leading edge
14 = maximum thickness in per cent of the chord
1.1 = leading-edge radius parameter, r/t,2
30 = location of maximum thickness in per cent of the chord from
the leading edge
0.50 == trailing-edge angle parameter, (l/t)(tan T/2)
and
t = thickness ratio
T = leading-edge radius, fraction of chord
T = trailing-edge angle (included angle between the tangents to
the upper and lower surfaces at the trailing edge)
For wing sections having this designation, the selected values of the
leading-edge radius and of the trailing-edge angle are used directly with
the other conditions in obtaining the coefficients of Eq. (6.7)~ The selected
values for the maximum camber and location of maximum camber are
applied to Eq. (6.4) to obtain the corresponding mean line.
6.7. NACA l-series W-mg Sections. The NACA l-eeries wing sec­
tionsG- 113 represent the first attempt to develop sections having desired
types of pressure distributions and are the first family of NACA low-drag
high-critical-speed wing sections. In order to meet one of the require­
ments for extensive laminar boundary layers, and to minimize the induced
velocities, it was desired to locate th~ minimum-pressure point unusually
far back on both surfaces and to have a small continuously favorable
pressure gradient from the leading edge to the position of minimum pres­
sure. The development of these wing sections (prior to 1939) was ham­
pered by the lack of adequate theory, and difficulties ,,:ere experienced in
obtaining the desired pressure distribution over more than a very limited
range of lift coefficients. As compared with later sections, the N .~C.A.
I-series wing sections are characterized by sOla).lleading-edge radii, com­
paratively large trailing-edge angles, and slightly higher critical speeds
for a given thickness ratio. These sections have proved useful for pro­
pellers. The only commonly used sections of this series have the minimum
pressure located at 60 per cent of the chord from the leading edge, and data
"ill be presented only for these sections.
FAMILIES OF WING SECTIONS
119
o. Thickne88 Distributions. Ordinates for thickness distributions with
the minimum pressure located at 0.6c and thickness ratios of 6, 9, 12, 15,
18, and 21 per cent are presented in Appendix I. These data are similar
in form to those presented for the KACA four-digit series. These sections
were not developed from any analytical expression. The ordinate at any
station is directly proportional to the thickness ratio, and sections of
intermediate thickness may be correctly obtained by scaling the ordinates.
b. Mean Lines. The NACA l-series wing sections, as commonly used,
are cambered with a mean line of the uniform-load type [Eq. (4.26)].
Data for the mean line are tabulated in Appendix II. Thi, type of mean
line was selected because, at the design lift coefficient, it does not change
the shape of the pressure distribution of the symmetrical section at zero
lift. This mean line also imposes minimum induced velocities for a give~
design lift coefficient.
c. Numbering System. The N.A.CA l-series wing sections are designated
by a five-digit number as, for example, the N ACA 16-112 section. The first
integer represents the series designation. The second integer represents
the distance in tenths of the chord from the leading edge to the position of
minimum pressure for the symmetrical section at zero lift. The first Dum­
ber following the dash indicates the amount of camber expressed in terms
of the design lift coefficient in tenths, and the last t\VO numbers together
indicate the thickness in per cent of the chord. The commonly used
sections of this family have minimum pressure at 0.6 of the chord from the
leading edge and are usually referred to as the Nl\.CA 16-series sections.
d. Approximate Theoretical CharacleristiC8. The theoretical aerody­
namic characteristics of the NACA l-series wing sections may be obtained
by application of the method of Sec. 4.5 to the data of Appendixes I and II
by the same method as that described for the NACA four-digit wing
sections (Sec. 6.4d).
6.8. KACA &-series Wmg SectioDS. Successive attempts to design
wing sections by approximate theoretical methods led to families of wing
sections designated NACA 2- to 5-series sections." Experience with these
sections showed that none of the approximate methods tried was sufficiently
accurate to show correctly the effect of changes in profile near the leading
edge. Wind-tunnel and flight tests of these sections showed that extensive
laminar boundary. layers could be maintained at comparatively large
valdes of the Reynolds number if the wing surfaces were sufficiently fair
and smooth. These tests also provided qualitative information on the
effects of the magnitude of the favorable pressure gradient, leading-edge
radius, and other shape variables. The data a~ showed that separation
of the turbulent boundary layer over the rear of the section, especially
with rough surfaces, limited the extent of laminar layer for which the lying
sections should be designed. The wing sections of these early families
generally showed relatively low maximum lift coefficients and, in many
120
THEORY OF WING SECTIONS
cases were designed for a greater extent of laminar flow than is practical.
It w~ learned that, although sections designed for an excessive extent of
laminar flow gave extremely low drag coefficients near the design lift
coefficient when smooth, the drag of such sections became unduly large
when rough, particularly at lift coefficients higher than the design value.
These families of lying sections are accordingly considered obsolete.
The NACA 6-series basic thickness forms were derived by new and
improved methods described in Sec. 3.8, in accordance with design criteria
established with the objective of obtaining desirable drag, critical Mach
number, and maximum-lift characteristics.
a. Thick"neB8 Distributions. Data for NACA 6-series thickness dis­
tributions covering a wide range of thickness ratios and positions of
minimum pressure are presented in Appendix I. These data are com­
parable with the similar data for wing sections of the NACA four-digit
series (Sec. 6.4a) except that ordinates for intermediate thickness ratios may
not be correctly obtained by scaling the tabulated ordinates proportional
to the thickness ratio. This method of changing the ordinates by a factor
w ill, however, produce shapes satisfactorily approximating members of the
family if the change of thickness ratio is small.
b. Mean Lines. The mean lines commonly used with the NAC_.\.
6-series wing sections produce a uniform chordwise loading from the leading
edge to the point x/c = a, and a linearly decreasing load from this point to
the trailing edge. Data for NACA mean lines with values of a equal to
0, 0.1, 0.2, 0.3, 0.4, 0.5,0.6, 0.7, 0.8, 0.9, and 1.0 are presented in Appendix
II. The ordinates were computed from Eqs. (4.26) and (4.27). The data
are presented for a design lift coefficient e" equal to unity. All tabulated
values vary directly with the design lift coefficient. Corresponding data
for similar mean lines with other design lift coefficients may accordingly
be obtained simply by multiplying the tabulated values by the desired
design lift coefficient.
In order to camber NACA 6-series wing sections, mean lines are usually
used having values of a equal to or greater than the distance from the
leading edge to the location of minimum pressure for the selected thickness
distributions at zero lift. For special purposes, load distributions other
than those corresponding to the simple mean lines may be obtained by
combining two or more types of mean line having positive or negative
values of the design lift coefficient.
c. NtUmbering System. The NACA 6-series wing sections are usually
designated by a six-digit number together with a statement showing the
type of mean line used. For example, in the designation NACA 65,3-218,
a == 0.5, the 6 is the series designation. The 5 denotes the chordwise posi­
tion of minimum pressure in tenths of the chord behind the leading edge
for the basic symmetrical section at zero lift. The 3 following the comma
FAJIILIES OF n'IlVG SECTIONS
121
ives the range of lift coefficient in tenths above and below the design lift
oefficient in which favorable pressure gradients exist on both surfaces.
'he 2 following the dash gives the design lift coefficient in tenths. The
.LSt two digits indicate the thickness of the lying section in per cent of the
horde The designation a = 0.5 shows the type of mean line used. When
he mean-line designation is not given, it is understood that the uniform­
oad mean line (a = 1.0) has been used.
When the mean line used is obtained by combining more than one mean
ine, the design lift coefficient used in the designation is the algebraic sum
)f the design lift coefficients of the mean lines used, and the mean lines are
Jescribed in the statement following the number as in the following case:
{a
cs,
NAC.A. 65 3-218 = 0.5
= 0.3 }
,
a = 1.0
ci, = - 0.1
Wing sections having a thickness distribution obtained by linearly
increasing or decreasing the ordinates of one of the originally derived thick­
ness distributions are designated as in the following example:
N.~CA 65(318)-217
a = 0.5
The significance of all the numbers except those in the parentheses is the
same as before. The first number and the lust two numbers enclosed in
the parentheses denote, respectively, the low-drag range and the thickness
in per cent of the chord of the originally derived thickness distribution.
The more recent XACA 6-series wing sections are derived as members
of thickness families having a simple relationship between the conformal
transformations for 'ling sections of different thickness ratios but having
minimum pressure at the same chordwise position. These wing sections
are distinguished from the earlier individually derived wing sections by
writing the number indicating the Jew-drag range as a subscript, for
example,
N4~C.A. oor 218
a == 0.5
Ordinates for the basic thickness distributions designated by a subscript
are slightly different from those for the corresponding individually derived
thickness distributions. As before, if the ordinates of the basic thickness
distributions arc changed by a factor, the low-drag range and thickness
ratio of the original thickness distribution are enclosed in parentheses as
Iollows:
X.:\C..\ 65(311)-217
a = 0.5
For wing sections having a thickness ratio less than 12 per cent, the
low-drag range is less than 0.1 and the subscript denoting this range is
omitted from the designation, thus
N .t\C..-\ 65-210
or
122
THEORY OF WING SECTIONS
The latter designation indicates that the 11 per cent thick section was
obtained by linearly scaling the ordinates of the 10 per cent thick sym­
metrical section.
If the design lift coefficient in tenths or the thickness of the wing section
in per cent of the chord are not whole integers, the numbers giving these
quantities are usually enclosed in parentheses as in the following example:
NACA 65(318)-(1.5)(16.5)
a == 0.5
Some early experimental wing sections are designated by the insertion
of the letter x immediately preceding the dash as in the designation
66,22:-015.
Some modifications of the NACA 6-series sections are designated by
replacing the dash by a capital letter, thus
NACA 641A212
-In this case, the letter indicates both the modified thickness distribution
and the type of mean line used to camber the section. Sections designated
by the letter A are substantially straight on both surfaces from about
0.8c to the trailing edge.
d. Approximate Theoretical Characteri8ticB. Approximate theoretical
characteristics may be obtained by applying the methods of Sec. 4.5 to
the data of Appendixes I and II as described for the NACA four-digit series
sections in Sec. 6.4d. If t\VO or more of the simple mean lines are com­
bined to camber the desired section, data for the resulting mean line may
be obtained by algebraic addition of the scaled values for the component
mean lines.
8.9. NACA 7-series W-mg Sections. The NACA 7-series wing sections
are characterized by a greater extent of possible laminar flow on the lower
than on the upper surface. These sections permit low pitching-moment
coefficients with moderately high design lift coefficients at the expense of
some reduction of maximum lift and critical Mach number.
The N ACA 7-series 'ling sections are designated by a number of the
following type:
NACA 747...\315
The first number 7 indicates the series number. The second number 4
indicates the extent over the upper surface, in tenths of the chord from the
le&ding edge, of the region of favorable pressure gradient at the design
lift coefficient. The third number 7 indicates the enent over the lower
surface, in tenths of the chord from the leading edge, of the region of
favorable pressure gradient at the design lift coefficient. The significance
of the last group of three numbers is the same as for the NAC..~ 6-series
wing sections. The letter A which follows the first three numbers is 8
seria1letter to distinguish different sections having parameters that would
FAMILIES OF WING SEC7
1/ONS
123
correspond to the same numerical designation. For example, a second'
section having the same extent of favorable pressure gradient over the
upper and lower surfaces, the same design lift coefficient, and the same
thickness ratio 88 the original lying section but having a different mean­
line combination or thickness distribution would have the serial letter B.
Mean lines used for the NACA 7-series sections are obtained by combining
t\VO or more of the previously described mean lines. The basic thickness
distribution is given a designation similar to those of the final cambered
wing sections. For example, the basic thickness distribution for the NACA
747A315 and 747A415 sections is given the designation NACA .747A015
even though minimum pressure occurs at O.4c on both the upper and lower
surfaces at zero lift. Data for this thickness distribution are presented in
Appendix I.
The NACA 747A315 lying section is cambered with the following com­
bination of mean lines:
a == 0.4
{ a == 0.7
e'i == 0.763 }
eli == - 0.463
The NACA 747..\415 wing section is cambered with the following com­
bination of mean lines:
a == 0 4
. a == O~7
{
a == 1.0
C" == 0.763
}
ci, = - 0.463
c.; = 0.100
8.10. Special Combinations of Thickness and Camber. The methods
presented for combining thickness distributions and mean lines are suffi­
ciently flexible to permit combining any thickness distribution, regardless
of family, with any mean line or combination of mean lines. Approximate
theoretical characteristics of such combinations may be obtained by ap­
plication of the method of Sec. 4.5 to the tabulated data. In this manner,
it is possible to approximate a"'considerable variety of pressure distributions
without deriving new lying sections.
CHAPTER 7
EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS
7.1. Symbols.
R
U
V
Reynolds number, pVc/p.
velocity just outside the boundary layer
velocity of the free stream
a mean-line designation; fraction of the chord from leading edge over which loading
is uniform at the ideal angle of attack
ac aerodynamic center
c chord
Cd section drag coefficient
minimum section drag coefficient
c/ local skin-friction coefficient
c, section lift coefficient
ci, design lift coefficient
ClanR'I: maximum section lift coefficient
section pitching-moment coefficient about the aerodynamic center
c../4 . section pitching-moment coefficient about the quarter-chord point
k height
roughness
t time
~ abscissa measured from the leading edge
11 ordinate measured from the chord line
Go
section angle of attack
" angle of flap deflection
p. viscosity of air
p mass density of air
c.r.m.
e_.
of
7.2. Introduction. The theories presented in Chaps. 3 to 5 permit
reasonably accurate calculations to be made of certain characteristics of
wing sections, but the simplifying assumptions made in the development
of these theories limit their over-all applicability. For instance, the perfect­
fluid theories of Chaps. 3 and 4 permit the pressure distribution to be calcu­
lated," provided that the effects of the boundary-layer flo,v on the pressure
distribution are small. Similarly the viscous theories of Chap. 5 permit
the calculation of some of the boundary-layer conditions, provided that the
pressure distribution is known, Obvious interactions between the viscous
and potential flows are the relatively small effective changes of body shape
caused by. the displacement thickness of the unseparated boundary layer
and the seriously large changes caused by separation. N one of the wing
characteristics can be calculated with confidence if the flow is separated
124
EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS
125
over an appreciable part of the surface. Under these circumstances, wing­
section characteristics used for design are obtained experimentally.
Wind-tunnel investigations of wing characteristics were made before
airplanes were successfully flown and still constitute an important phase
of aerodynamic testing. Until recently wing characteristics were usually
obtained from tests of models of finite aspect ratio, The development of
lying theory led to the concept of wing-section characteristics that were
derived from data obtained from tests of finite-aspect-ratio wings, These
derived data were then used to predict the characteristics of wings of
different plan forms. The systematic investigations in the N'AC..~ variable­
density wind tunnel46, 48, 49 were examples of this type of investigation.
This method of testing is hampered by the difficulties of obtaining full-scale
values of the Reynolds number and sufficiently low air-stream turbulence
to duplicate flight conditions properly without excessive cost for equip­
ment and models. Other difficulties were experienced in properly correcting
the data for the support tares and interference effects and in deriving the
section characteristics from tests of models necessarily having varied span­
load distributions and tip effects.Go
In order to avoid some of these difficulties and to permit testing of
models that are large relative to the size of the wind tunnel, two-dimensional
testing equipment was built by the X.t\C.i\. The X.:\(~..\ two-dimensional
low-turbulence pressure tunneP33 provides facilities for testing wing sections
in two-dimensional flowat large Reynolds numbers in an air stream of very
)O\V turbulence, approaching that of the atmosphere. The wing-section
data presented here were obtained from tests in this tunnel.
This tunne}1D has a test section 3 feet wide and 7~~ feet high and is
capable of operation at pressures up to 10 atmospheres. The usual models
are of 2-foot chord and completely span the 3-foot width of the test section.
The lift is measured by integration of pressures representing the reaction
on the floor and ceiling of the tunnel. The drag is obtained from wake­
survey measurements, and the pitching moments are measured directly
by a balance. "~ing-section characteristics can be obtained from such
measurements with a high degree of accuracy.
The usual test.s were made over a range of Reynolds numbers from 3
t.o 9 million and at Mach numbers less than about 0.17.· This range of
Reynolds numbers covers the range where large-scale effects are usually
experienced between the usual low-seale test data and the large-scale
flight range. Individual tests have been made to provide some data at
lower Reynolds numbers applicable to small personal airplanes and at
much larger Reynolds numbers to indicate trends for very large airplanes­
The tunnel turbulence level is very low, of the order of a few hundredths of
1 per cent; and, although it is not definitely known that the remaining
turbulence is negligible, the tunnel results appear to correspond closely to
126
THEORY OF WING SECTIONS
eo
3 .6
A
..
~
Q
~.)
Serles
~.)
-V­
e
~
~
0
-
0
o
...
I'~
I)
•
•
-a
r..
..
~
2
20
8
16
12
Airfoil thIckness. percent or cbord
0
I)
2
(a)
NACA tour- and t l ve-d1s1t series.
-6
2
••
""o
•
~
~
o
••o
-
-2
~
....
.I'to.
0
-
-
-
~l
eo
o 0.2
>
V
A.
•c
::
-c
v
6
0.1.
" 0.6
to
::
..•
~
•
2
0
II
8
12
16
20
24
Airroil th1ckness, percent or cnord
(b)
FIG.
NACA
6~-
aeries.
56. Measured section angles of aero lift for a number of N..t\CA
127
EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS
-6
--
Cl
~
-2
/\.
o
(
..
s:::
o
~
()
A.
....
•.,
-
-.
"0
t)
~
•
:I
20
1&
8
12
16
20
.Alrtoll thickness. percent of chord
(c)
NACA 6J~- serIes.
eo
•
";
-6
'-4
rot
-4
0
s..
•"
'"
....•
--JI ~
-4
-2
*'If.~
5
0
.p
4~
~ t.
110
...c
A
A
0
0
0
-.
4-)
A.
v
V
-.
"
v
0
••
't:J
•
~
••
:I
II.
8
12
16
Alrtoil thickne.s. percent
(d)
IIACA
o
20
or
chord
65- eeriea.
airfoil sections of various thicknesses and cambers. R, 6 X 10'..
0.2
0.L.
" 0.6
A
-
0"
o
~
1
a 0
a 0.1
EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS
-6
f
•
.~
~ ~
It
CI I
....o
Q
.!f
-2
taO
c
C
e
(>
~
tI
.....,o
A..
o
0_
~
"..
-
••
-
.-
...,
•
~
•I
20
1&
8
12
16
20
.Airtoll thlckaess. percent of chord
(cJ
NACA 6~- serles.
eo
•
..,..
....
~
..
-6
ro4
0
•.
s.
--.ll ~
-4
I;.
~
0
...•
-2
110
~4·~
5
..
c
0
.p
0
A.
.. ~
..:tt
n
0
......
0
A..
.,;,.
V
-
'"
~
••
i
~
i
20
Ja.
8
12
16
A1rfoil tblckn•••• percent
(d)
BACA
or
20
chord
65- eerie••
airfoil eectiooa or various thicknesees and cambers. R, 6 X lOS.
0
a 0.1
0.2
6 O.L.
" 0.6
o
A.
24
127
THEORY OF WING SECTIONS
128
those obtained in flight. Application of these wing-section data to the
prediction of the characteristics of wings of finite span depends on the
adequacy of three-dimensional \V~g theory. These data are not applicable
at high speeds where compressibility effects become important.
7.3. Standard Aerodynamic Characteristics. The resultant force on a
"ring section can be specified by t"90 components of force perpendicular
and parallel to the air stream (the lift and drag, respectively) and by a
moment in the plane of these t\VO forces (the pitching moment). These
forces are functions of the angle of attack of the section. The standard
method of presenting the characteristics of 'ling Sections is by means of
plots of the lift, drag, and moment coefficients against angle of attack or,
alternately, plots of angle of attack, drag, and moment coefficients against
lift coefficient.
Plots of wing-section characteristics are presented in Appendix IV for a
wide range of shape parameters. On the left-hand side of each plot, the lift
coefficient and the moment coefficient about the quarter-chord point are
plotted against the angle of attack. On the right-hand side of each plot
the drag coefficient and moment coefficient about the aerodynamic center
are plotted against the lift coefficient. In most cases, the data indicated
in the following table are presented.
Surface
condition
Charneteristie
Split flap
deflection
degrees
Reynolds
number,
millions
Left-hand side
Lift
tift
Lift
.
.
.
I4Iift..•.........
Moment
.
~Io~ent
.
Smooth
Rough­
Smooth
Rough­
Smooth
Smooth
0
0
60
60
0
60
3,6,9
6
6
6
3,6,9
6
0
0
0
3,6,9
6
3,6,9
Right-hand side
Drag
Drag
Moment
.
.
.
Smooth
Rough"
Smooth
*O.011-ineh amin carborundum lipread tbinJy to cover 5 to 10 per cent of the area from the
leadinc ed«e to 0.0& alung boUa surfaces of a IleCtion wit.h a chOld of 24 inchee.
7.4:. Lift Characteristics. 4. Angle of Zero lift. As indicated in Chap.
4, the angle of zero lift of a wing section is largely determined by the
camber. The theory of wing sections provides a means for computing the
angle of zero lift from the mean-line data presented in Appendix II. The
agreement between the calculated and the experimental angles of zero lift
EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS
129
depends on the type of mean line used. Comparison of the theoretical
data given in Appendix II 'lith the experimental data of Appendix IV
shows that the agreement is good except for the uniform-load type (a = 1)
of mean line. The angles of zero lift for this type of mean line are generally
closer to 0 degrees than predicted.
The experimental values of the angles of zero lift for a number of NACA
four.. and five-digit and NACA C)-series wing sections are presented in
Fig. 56. The thickness ratio of the wing section appears to have little
effect on the angle of zero lift regardless of the type of thickness distribu­
tID
-3_ -6
~
'-t
..c
....
o
;
II
-4
'-'4
o
ol(.)
MACA
FIG.
-.
C)
~
8
12
16
Alrton thicknes.. percent
(e)
Q
-.
u
e
o
A..
v
6
or
v
20
chord
66- .eries.
56. (Cond1ldal)
tion or camber. For the NACA four-digit series wing sections, the angles
of zero lift are approximately 0.93 of the value given by the theory of thin
wing sections. For the NACA 23o-serics wing sections, this factor is
approximately 1.08; and for the X.\Cl\ f.)-~(~r ies sections with the uniform­
load type of mean line, this factor is approximately 0.74.
b. L~Jt-curve Slope. Lift-curve slopes fOI" a number of NACA four- and
five-digit series and KACA fJ-8Cri~ wing ~~·t.jons are plotted against thick­
ness ratio in Fig. 57. These values of the lift-curve slope were measured
for a Reynolds number of 6 million at vulues of the lift coefficient approxi­
mately equal to the design lift coefficient of the wing sections. This lift
coefficient is approximately in the center of the low-drag range for the
NACA f>-Series wing sections.
In the range of thickness-ratios from (i to 10 per cent, the NACA four­
and five-digit series and the XAC.-\ 64-series wing sections have values of
the lift-curve slope very close to the value given by the theory of thin wing
sections (2... per radian, or 0.110 per degree). Variation of Reynolds Dum­
THEORY OF WING SECTIONS
130
P1a~d .~bola 1ndlcat~
roUSb
condltl~n
..
•,
•
•ca.
•t
.12
.-
..
~
.10
...•
r--' . -
...
"'8aIootb
-- --:t t--
-.iii
-.
e
.06
A
"I
0
c-
-~ ~1~
~~
-
'\
--.
-.: ~
Bogp-
4 1g1t)
I
230 (5 4181t)
"
8
10
12
14
16
18
Alrtoll thickneas. percent or chore
hal
-33
at(4
e
,
---
00'
EI
o
:J
.. ~- ~
E
.~
:s
...•
t-~ 1 -
Berl••
22
20
NaCA tour- and f'lve-d1s1t aerie••
•
.lIa.
•
r 8lllooth_
.12
~
J
·'0
-
III
.Iw-- ~I'
--
,
--
·.!t- -
I
I
--~--4
-~ ~-,
\..acuab
~O
°1.1
eO
00.2
40.~
.08
vo.
,
,
.a
16
18
chord·
20
22
O£
.ACIl " - •• rlea.
I t
.12
..I.
~.)
~r
~~
1'f - --
....
~ -~
r- -
...•t
•
.10
°1.1
C
1
.08
e O.~
A o.
'9 o.
5
I
12
14
Alr1"o11 thlcknes.. percen1i
10
(b)
..
•
I
I
8
e
e
,
I
8
~
........J.~
....
---, r -
1\
I
- T
RouP
.,
aU
0
0.1
I I
10
12
14
Alrtoll. thicknesa. percerat
(0)
..
~
vre:-~- -
or
16
18
20
22
chord
lACA 6l,- .eriea.
FrG. 51. Variation of lift-eurve slope 1Ifith arfoil thickness ratio and camber for a
131
EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS
na...d
"1..0
31;;
•
••­ ....
-I2t •
s.
De
•
...8'•
,.
4~
J
4t
-
~J
1'--1 tt--
po--
10
1--
-., I-I"
0
08
6
V
:I · 064
,
•
8'
1"-_
-~
0
1
10
12
AlrtoU tb1c1cne••• percent or chord
(d J MACA 65-•• n •••
20
.12
22
'r 8IIIooth
11
A
4~
o.~
o.
o.
I
I
8
.....J.,
\ Roup
°'1
E>
I·
.•
.,
~J
...•
i•
r .ootb
I
.;
•
-Jllbo11 indicate rouF condition
14
~.)
~.)
,~
... ~
.10
J~
~
~.)- t-_
-=i.
I
rf
•
.s
cl
I
t
E)
e
A
l
0
0.2
•
/,
-
e
r'\, - -'
to)
~
r--Roup
....
~~
0.4
I I
6
8
10
Alrtoll
12
<e)
number of
N.~C.:'...
20
thiem....
percent or chord
MACA 66- aer1e••
airfoil sections in both thp smooth and rougb conditions. R, 6 X lOS.
22
THEORY OF WING SECTIONS
132
ber between 3 and 9 million and variation of camber up to O.04c appear to
have no systematic effect on the lift-curve slope. The thickness distribu­
tion appears to be the primary variable. For the NACA four- and five­
digit series wing sections, the lift-curve slope decreases with increase
of thickness ratio. For the NACA 6-series wing sections, however, the
lr---- -- 1'\
2.8
V)
/4-Series
(4-Digi1) ~ r---
2.4
(4-Digit)
)
r-,
I
I
I
OO-!:aies
- - (4-0igil) "
.-
~
~~
I4-Series
(4- Digif).~
~
~
',.. r-
..... ~
..
~
I
230-5eries
'/~
-: 7
If
~~
~ (5-DigiIJ
V
I
I
44-5eries
(4-DigifJ
--r ~ ~~
~~
l
v-:
QoIo."
~ ~ ~/
V~
-o
4r
Airfoil wiIh
split f/cJJ
I
,
~
~
!i4-DigiIJ
i'X r
~ r\)
r
~ L/
,r--
00-5eries~
rl~~
" 44-5eries
24-Series
(4-Digif) -
I
~
~
~j
/
PkJin
>airfoil
/ ~ ~~
---:
-
~)
--fbJgh
--5rnodh
I
J
I
Symbols with fJogs correspond 10
sinllbfed splil fbp deflected 60-
.4
I
00
4
8
I
I
/2
I
I
I
16
Airfoillhick~pen;:enI
20
24
of chotrI
(a) N ACA four- and five-digit series.
58. Variation of muimum section lilt coefficient with airfoil thickness ratio and camber
for .veral NACA airfoil sections with and without simulated split ftaps and standard rough­
ness. B, 6 X loa.
FIG.
lift-curve slope increases with increase of thickness ratio and with forward
movement of the position of minimum pressure on the basic thickness
form. The effect of thickness ratio is comparatively small for the NAC.\
66-series sections. The thick-wing-section theory (Chap. 3) shows that the
slope of the lift curve should increase with increasing thickness ratio in
.he absence of viscous effects. For wing sections with arbitrary modifica­
tions of shape near the trailing edge, the lift-curve slope appears to decrease
with increasing trailing-edge angle.
Some KACA6-series wing sections show jogs in the lift curve at the end
.
EXPERl}.{ENTAL CHARACTERISTICS OF WING SECTIONS
133
of the low-drag range, especially at lo,v Reynolds numbers. This jog be­
comes more pronounced with increase of camber or thickness ratio and
with rearward movement of the position of minimum pressure on the basic
thickness form. This jog decreases rapidly in severity with increasing
Reynolds number, becomes merely a change of lift-curve slope, and is
practically nonexistent at a Reynolds number of 9 million for most lying
I
-
...i
"6
-
-
_~o.'
2.8 -
.1,.
-'" 0
i , '"'\
-.
..."
~--
~ L-...... ~
'\ £ //
"'0 . 2 -
-
o
I
°Il
r--
~'i
~ ,...:3 ...
)\, -- :~
~,
~-
~):;J'
\ Y
~ M ~:~ r-,
7"
Cl 1
~
k:-: -_ .I' ~ r-, \.. 0.'J,.
,"F='="
- e
~" ~
I ,.
/ V~·V -&.-o ~ ......
JII 1/ -- :i'~
.4~
_J V,.)
~
~
~"O
~~
.... L:..-
~
~
ft
.2
~
"00
~
......,s::
o
o
:
C&l
~
1.2 l - - ~
...
6
.2~
e e L
L..
iIl.............
~
~--
90.6-
e
~~
~ .~
..0-
----
... ~,,~
..... ~
,'.
~-
~
·f ~-\~ ~;;
Plain
a1rt'o11
-6
~
\ \\ ~ '"'
\\ ~'( -vo.6-4
C~l
8IDOotb
\
\.- -6
~
-0 .2
00
I
ltouab
I--SJlDbola with 1"l-p correepood ~o
8!'-w..ated- ap11t 1'1ap :Serlect~d 600
o
o
A1rt'oll W1tb
ep11t. ~J.ap
~
8
12
16
A1rtoll th1ckDe... perceD t or
(6) NAC.A 63-series.
}"'IG. 58. (Continued)
20
Cbor4
sections that would he considered for practical application. This jog may
he a consideration in the selection of wing sections for small low-speed
airplanes. An analysis of the flow conditions leading to this jog is pre­
sen ted ill reference 134.
The values of the Iift-curve slopes presented are for steady conditions
and do not necessarily correspond to the slopes obtained in transient con­
ditions when the boundary layer has insufficient time to develop fully at
each lift coefficient. Some experimental resultglOG indicate that variations
of the steady value of the lift-curve slope do not result in similar variations
of the gust loading,
THEORY OF WING SECTIONS
134
c. Mazimum Lift. The variation of maximum lift coefficient with
thickness ratio at a Reynolds number of 6 million is shown in Fig, 58 for a
considerable number of NACA wing sections. The sections for which data
are presented in this figure have a range of thickness ratios from 6 to
24 per cent and cambers up to 4 per cent of the chord. From the data for
the NACA four- and five-digit wing sections (Fig. 58a), it appears that the
'.2
I
°Il
2.8
Ie
E
~
..
0
....
~~
-.",,~
.~",.
2.0
S
0
_~
1.6
....
......
0
••
i
-
°Il
1.2
o~
v
f'
'70.6- ..../ ~ 06=3
-
g
~,;~ ('.,
~ ~ ::::- ~
... ..J~•
~
...~
:,
or,
,
~ o>~ .,.,~
-1 ~ ,
" ~
,JI
,p
.......i
~
.4- -- -, ~ 0 .~ v
~ .2
"a .1' t l O - ~~.~ lS' ~.,~ h: :.II
..• 2.4­
0
--
~
.. 0.6
.4- -"'"
01
"Y-
~
- ~'V
.~ ,,"
00---,
:~ ... 1~,<:~
A
<>
~--
.2-
o .1
I---
~
.8
~--
Ji>-
iI_~ or~
-,
<,
=­
~
"9
0.6
.. .4
0
1:1
.2
.1
'00
~ .............
h
:
...
-... o~ ~
...... ',,:.~
~
..&-
~
~.
-
Airtoll wltn
split flap
'°1 1
PlAln
airfoil
-~ ~~: "-oE)
''0
\\
-:'0 \\-90.6.14
°Il
-<>
~ ~ -s
~A
.4
"ooth
I
-----R~
1
~.
•
00
with tlaS8 correspond to
ted .pllt flap detlected 600
"
.2
.1
-00
8
12
k
16
20
A1rto11 thiCknes., percent or chord
(c) NACA 64-eeriea.
FIG.
58.
(Continued)
maximum lift coefficients are the greatest for a thickness ratio of 12 per
cent. In general, the rate of change of maximum lift coefficient with
thickness ratio appears to be greater for sections having a thickness ratio
less than 12 per cent than for the thicker sections. The data for the KACA
6-seri~ sections (Figs. 58b to d) show a rapid increase of maximum lift
coefficient with increasing thickness ratio for thickness ratios less than
12 per cent. The optimum thickness ratio for maximum lift coefficient
increases with rearward movement of the position of minimum pressure
and decreases with increase of camber.
For wing sections having thickness ratios of 6 per cent and for wing
sections having thickness ratios of 18 or 21 per cent, the maximum lift
2.8.--......-.....-----...--..,.-.....,..-...---~ .......- -...........
i 2.4t--t--+---+-
0'
i­
....c::
"'•oo"
Airfoll with
spllt.flap
2.0 ..---+--f--I---4-~-+-~~
........~~~~~
'-4
.... 1.6
~
rot
g
::
o
••
J
Plain
airfoil
1.2
.8
i
t--4---t-~--+-~-+------t-~-.-_Smooth - - + - - + - - - + - - t
__
_ flougl1
.4 1---+--+--......-
SJlDbola with tlasa correspond to 0
.1aIula~e4 spllt rlap d.flected ·60
I&.
8
12
J.6
20
Airtoil thlCkne.-;, percent or chord
(tl) NACA 65-series.
FIG. 58. (Continued)
2.8
Cl
l
no.q- -'"
~
.
~
L.,./
~
0"
~
1.6­
01,1
O.~..-"
fi 1.2
..
...o
OO~ V
••
1IIl'-­
~
~ ...a­
.8
..
J;1"
.,.,,-"
~ oy-­
~_-4 .)--
,
~
~""'"
~
....~J ..............
Alrtol1. wltb
apllt flap
~11
~ -,
'a 0.4
<; -~
.2
...........-. -l:) 0
~'\.
-o
-0­
~
_J:>­
--..
..J!\
-,\_0lil
'\1:
-G
~
~
.)00­
.).--
--<7' ~,
-
P1a1n
ail'1'o!l
-e
~
'\~ ~A
Cl
l
0 .....
'""'0 .2_
~ ,-G 0
I
.~
»:
~ .,...-
() .2-..J,V j
4
.J---.
.~ ~
- ­
-V
....~
)
''T''
~
~
.~
~
.~ v:a ."~~ o~-
V) .V
"J¥'(~
--"
2.0
~
~
8o
~
~
.­
c:
~
.2­
bO ­
~U
Slooth
- - - - - Roup
i-­
I-
°0
SJIDbola witb 1"lap correspond to
a1aulatecl spIlt t"lap 4.nectecl 60°
8
12
16
20
A1rf'oll th1ckD.... perc.ati or chord
(e) NACA 66-eeriea.
FIG. 58. (C01&eluded)
J&.
135
~
THEORY OF WING SECTIONS
136
coefficients do not appear to be very much affected by the position of
minimum pressure on the basic thickness form. The maximum lift coeffi­
cients of sections of intermediate thickness, however, decrease with rear­
ward movement of the position of minimum pressure. The maximum lift
coefficients for the NACA 64-series wing sections cambered for "a design
lift coefficient of 0.4 are slightly higher than those for the NACA 44-series
2.0
..I 1.'
...•..
...'"'
0
a
C)
I0
1.2
...
~
...
.8
...I
.4­
Ii
~
ReJDo1dsD~
'­
Q
GJ
~
.....
~
1""""'
-- r--J
~ -rr
.p
6X~~
......
7
,Xl
0
••
••M
o
o
.,
2Jpe of o-.ber.
.8
a
59. Variation or maximum lift coefficient with type of camber lor some NACA 65a-418
airfoil sections from testa in the Langley two-dilnensional low-turbulcnce pressure tunnel.
FIG.
sections. The NACA 230-series sections, however, have somewhat higher
maximum lift coefficients.
The maximum lift coefficients of moderately cambered sections increase
with increasing camber (Fig. 58). For wing sections of about 18 per cent
thickness, the rate of increase of maximum lift coefficient with camber is
largest for small cambers. The effect of camber in increasing the section
maximum lift coefficient becomes progressively less as the thickness ratio
increases above 12 per cent. The variation of maximum lift with type of
camber is shown in Fig. 59 for one condition. No systematic data are
available for mean lines with values of a less than 0.5. It should be noted,
however, that lying sections such as the NACA 230-series with the maximum
camber far forward show large increments of maximum lift as compared
with symmetrical sections. Wipg sections with the maximum camber far
137
EXPERIAIENTAL CHARACTERISTICS OF lVING SECTIONS
forward and with normal thickness ratios stall from the leading edge with
large sudden losses of lift. A more desirable gradual stall is obtained when
2.0
..
'----.
-.... -a...
<, -......
1.6
~
-
1.2
...i
.8
...,
.4
.
-~
~R
~
~
--, ~ ~ t=rQB
l;Joo...;
~-!--
~~
e
~ A
~
k?'
"1~
0
MACA 24-serlea (4 digit)
c
~
0
w4
....
....
8
0
1.6
~ ~'
.p
R
G 9.0 x 106
ir
~
....
1.2
s::
....0
••
.8
i
1.6
V
.p
0
J
JlACA
e 6.0
Q
V
'.0
at. Stendard
rougbne,s
14-•• rl•• (4
6 )( lOb
dl&1t)
R
G 9.0 x
s
JV
1.2
I.{/
.r
1&.
--A
a 6.0
e
'.0
A StaDdard
l'OUSbDeaa
MACA CO-aeries
8
12
16
Alrto1l thickness. percent
(a)
106
(4-
d1a1t)
6 x
106
20
or chord
MACA tour-dlg1 t aeries.
60. Variation of maximum section lift coefficient wid. airfoil thickness ratio at several
Reynolds numbers for a number of NACA airfoil sections of different cambers.
FlO.
the location of maximum camber is farther back, as with the NACA 24-,
44-, and 6-series sections with normal types of camber.
The variations of maximum lift coefficient with Reynolds number (scale
effect) of a number of XACA wing sections are presented in Fig. 60 for a
range of Reynolds numbers from 3 to 9 million. The scale effect for the
NACA 24-, 44-, and 23D-series wing sections (Figs. 60a and b) having
THEORY OF WING SECTIONS
138
2.0
- ---" ... ~
P) ~ ~
l~
~
1.6
~
100­
Jl
I-
e Z_O
B
I
.. 1.2
~
e
I- 4.
0
..•­
1!
~.
.0
V
»c 1.06-/
/
~
'.0
r~.a
, x
i06­
V
~
Ill"'"
~~
ataDdard
~
~
~
...............
..
~
~~
.8
~ ~.
~~
~ r--......
I ' r-,
~
~
'"8
\.e
...
0
IfACA 2,O-a.rl•• (5 digit,
.4
...
~
...s..
2.0
0
••
I
R
1.6
.9.0 x 10'
B 6.0 ------.
G
'.0
~___JI~-+-M..-~:aiII'~-+-I~~
. . .--I
AStan4ard
ro\IIbDell8 -t-~t---+--~~--I-~--I----I
6 )( 106
1.2
XACA
4
~-.erl.8·(4
dlglt)
12
16
20
Airtoil thickness. percent of chord
(b)
8
RACA rour- and rl".-dlg1~ eerie••
FIG.
60. (Continued)
24
EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS
S,mbols with tlase correspond to
2.0
0'1
=0.4
aad
0.6
I'~
~
r
::::t~
R
~ r-;
e-, ~
~~
r"""
.........
= 0.6
~
~~
-
c~l
139
~
-.......
~
~
~
............
....
106
v
9.0 )(
A
Standard
rouatmeaa
e 0.0
e
~
'.0
6
)C
10 6
~ .8
.."
..•
o
.. 2.0.---.--......~.-.......-..-......- -.--..-....- ......- ...-.....
c
.p
~
::
•
9.0 )( 106
e 6.0
0
8
.p
::s
e
roushn•••
6 )( 106
c
4J
~
•g
'.0
4 Stand.ard
~ 1.2
...o
R
1.61--+~-+-~t---4--4.,...g~~:&-.d---4--+--~-I
.8 ......
......__a.--....._
......-....I_....a-
....... _~....
!
a 1.6 ...-....-........~......- -..-..-.............,....-............-..--......- -.
:c
R
e
9.0 x 106
6.0
1.2.I-----r-t--;--n~-r_;_-=-t_;~'__f::;~'a==:t()
0 ;.0
~+---t-~
.----+--t----1~~~-+----.\r'_-+--+_-___+_-~¥.ir_+_____1
.8 ..-........_ ......--.
o
--------I-""""'--.....
12
8
10
--a.-....... -~-01
20
Alrtol1 thickneas, percent or chord
(e) NACA 63-aerles.
Flo. 60. (Continued)
Standard
rougbnea 8
6
x 106
THEORY OF WING SECTIONS
140
SJlDbola nth flag e~reapond to et.i
2.0
0Il
=0.4 an4 0.6
=0.6
R
~-..............-....o 9.0
1-'--'86.0
x 106
1. 6 t---t""--t--t--+--t---+--~~"""'~rIl'-l.'---I-~
0 3.0
A
t----t---t--+--+--+--+---+-~~.a.----a.--'.....I-..'--I
~
...e
Standard
r~e'8
6
x
100
1.6
It
~ 9.0
,. 1.2 t---t"--t--'t"-~~-+--~.....+-a.
l?J 6.0
.....-...a-~;::a.-+:a~(:) 3.0
0
x 106
~
c
. -......-
~
t..e
~
4D
0
...A Standard
t--t--t-~~~r:::;;....-+--+--+--4---I-~--+---I
S
rougbne, a
6 x 100
.8
0
~
~
....
c
1.6
...
R
0
1'----+--......-.---..-.... <2> 9.0
.p
0
••
-,--~-+--4---+El
1.2
)C
10 6
03.0
Standard
I
.-a-~t--.......- - I - - - - L 6
rougbne,•
6 x 100
.-t
i
6.0
.8
1.6
R
e 9.0 x 106
fl 6.0
1.2 t---T--r--t---+-,.f"--+-~-4-~o-~~-I---I
1-.....;;;.--.--.... 0 3.0
A
t---t--t--'t--i'#--f--r,,,,+--+--01--4"':::'-+...--1
.8
Standard
roughne,s
6 x 111'
0
4
8
12
J.6
hlrrol1 thickness. percent
(dl
NACA
FIG.
20
or chord
64-serles.
60. (Continued)
24
EXPERIMENTAL CHARACTERISTICS OF WI}/G SECTIONS
8J11lbola with flap correspond to
01
1
= 0.6
4
.p
!...o
t
•oo
.p
....""...
C
...o
.p
o
••
J 1.6 r---,...-r--~r---"Ir---~.----...---~~"-'
U
=­
.8 "-......_~-.liil- ......_..a..---a._...a.---IIo.--...a._..&........._ - I
o
4
8
u
~
~
6
Airtoil thlckn•• s. percent at Chord
(e)
MACA
6S-8erl•• ~
FIG.60 (Continued)
141
THEORY OF WING 8BO'l'ION8
142
...I
o
1.2 ""-.........
--a_-a..._"--....... _.-.....~_....... _..a.___._..
,
J 1.6
a
9.0 x 106
'.0 .
rougme,_
8 &.0
1.2 .--r---,r-I---,r--o"'L..-;---t-;---,~::t::::::l0
t---+--+--t--­
.8.0
6. Standard
=EJ-+-+---t---+-~~c:;;p-~~-~
6 x 100
4
8
12
16
Air£oU thickne.s. percent
(r)
IIACA
o~
66- •• rl•••
FIG. 60. (Concluded)
20
chord
24
EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS
143
thickness ratios from 12 to 24 per cent is uniformly favorable and nearly
independent of the thickness ratio. Increasing the Reynolds number from
3 to 9 million results in an increase of the maximum lift coefficient of
approximately 0.15 to 0.20. The scale effect on the NACA 00- and 14­
series sections having thickness ratios of 0.12c and less is very small.
The scale-effect data for the NACA 6-series wing sections (Figs. 60c to e)
do not show an entirely systematic variation. In general, the scale effect
is favorable for these sections. For the NACA 64-series wing sections,
the increase of maximum lift coefficient with increase of Reynolds number
is generally small for thickness ratios less than 12 per cent, but it is some­
what larger for the thicker sections. The character of the scale effect for
the NACA 65- and 66-series wing sections is similar to that for the NACA
64-series, but the trends are not so well defined. The scale effect for the
NACA 6-series wing sections cambered for a design lift coefficient of 0.4 or
0.6 is greater than that for these sections with less camber. The data of
Fig. 61 show that the maximum lift coefficient for the NACA 63(420)-422
section continues to increase with Reynolds number up to values of at
least 26 million.
The values of the maximum lift coefficient presented are for steady con­
ditions. It is known that the maximum lift coefficient increases with the rate
of change of angle of attack. tOto The significant par~eteris (daoldt) (cjV).
Even such low rates of change of angle of attack as those encountered in
landing flares produce increases of the maximum lift coefficient.
d. Effect of Surface Condition on Lift Characteri8tics. It has long been
known that surface roughness, especially near the leading edge, has large
effects on the characteristics of lying sections. The maximum lift coeffi­
cient, in particular, is sensitive to leading-edge roughness. The effect on
maximum lift coefficient of various degrees of roughness applied to the
leading edge of the NACA 63(420)-422 wing section is shown in Fig. 62.
The maximum lift coefficient decreases progressively with increasing
roughness. For a given surface condition at the leading edge, the max­
imum lift coefficient increases slowly with increasing Reynolds number
(Fig. 63). Figure 64 shows that roughness strips located more than about
O.2Oc from the leading edge have little effect on the maximum lift coefficient
or lift-curve slope.
It is desirable to determine the relative effects of leading-edge rough­
ness on various wing sections. In order to make a systematic investigation
of this sort, it is necessary to select a standard form of roughness. The
standard leading-edge roughness selected by the NACA for 24-inch chord
models' consisted of 0.01I-inch carborundum grains applied to the surface
of the model at the leading edge over a surface length of 0.0& measured
from the leading edge on both surfaces. The grains were thinly spread to
cover 5 to 10 per cent of the area. This standard roughness is considerably
THEORY OF WING SECTIONS
144
2.8
2.0
1.6
... ...
A
.,.
1.~
.:s:::
~
...
.8
C)
;
"­
to.
~
0
()
1'4
~
0
II
J'
I
ri
C
~..r.,..
r
..."'"
.
..,.­
.JP
0
...
-- -
~~ ~
~"E)
-­
PJ '-'
-JJ
0
~
0
•
co
j
- .4
v
j
- .8
j
J:t
-1.2
~~
-~
:A
~
Il
G
6.0
lt0
x
106
a
•
0
A 20.0
• 26.0
If
r-fF
•
-1.6
~
-1'
0
,
J6
24­
or attack, Clo ' des
FIG. 61. Lift and drag e~acteristiC8 of the NACA 63(420)-422 airfoil at high Reynolds
number; TDT tests 228 and 255.
Section
-8
anel.
EXPERIMENTAL CHARACTERISTIC8 OF WING SECTION8
·032
.02 a
-
-:
-"""""
0
.2
- r--.
---...
<, ...........
-­
--­
.,
- ...-­
-4 zJ"
............... ............
~
.8
1.0
c
.02."
"d
o
R
,
I
6.0 x 106
0
B 10.0
.01
14.0
V
26.0
6
~
:~ ~
,
e
II J
'~-.6
20.0
l~j
~ r-,
~
~~
f!A V
4'1
~
~ .~
......
r:
~~
,
v
o
-1.2
-.8
-.1,.
o
&eo'lon lUt ooettlclent.
FIG. 61. (Concluded)
.8
cl
145
THEORY OF WING SECTIONS
146
I
I
1
1
I
B O.0Q4-1ftOh-sra1n
- X roUIbD...
..1.2 - roqJme..
Oft
L.S.
~~
O.ooe-1aOh~
•
i•
.:=
:
::.
...
-
L. L
~~
• O.Oll-hoh-ara b
ro~••• OIl
.8 ~ + lb.l1ao
~
~
-..­
CL..
~~
~ID
r
L.L
OIl 1,•••
~r
elllootil
~
,
,1
.~
j
...•=..
tI
OIl
('-­
~
II
0
/
­....
-)
-.8
r
Airfoil. DCA
~,
c
2' Ie 10'
" In.
T.ata. m!'
2'1) an4 262
III
Chord I
I I I
-1.'
~
"(WO)~a
-al.
0
8
16
01
4e.
FIG. 82. Lift characteriatica of a N ACA 63(420)-422 airfoll with varioua
at the leadiq edae.
-16
-8
I I
I
""loa
.'taok. .0..
I
•
dtcreee of
roughnea
2.0
...-01 ~
~
~
"
,..
I
...
.
~
\.
"
----- ­
..
.
'0 \\ ......
~ .~ 1-..
i-a. !~
~••
M'!It'=••
.8
'-­ -6
.p
-.om
o
o
o
8
12
16
20
~14a""r
••
FIG. 63. Effects of Reynolds number on maximum section lift coefIicient
63(420)-422 airfoil with roughened and smooth leading edge.
28
CI
of the NACA
EXPERIMENTAL CHARACTERISTIC8 OF WING SECTIONS
-
..
1.2
o
~
.
1; .8
::
:
44
44
I
-
-
I
I
I
o.~
G IaooUl
.
a
...
~
rw~
9Ill
I'
,
1
J(
~
...o
......-:.;;
~,
...
...I
~
rc
IV
LeE.
+ BoqbM.. atrip
OIl
....
o
r
at 0.200
• BoqbM.. atnp
at
?
I
I
1(1Iou. . . . .vip
147
0
I
/
-.'­
r ~o111
a. 2' x1,p
DCA ,,(!a20 J-Ja22
j
•• 8
-1.2
~~
CboN. ~ 111•
<r;
255
~.tl
)
'if
~
~
0
8
""loa aq1e of .tWk.
~
.0.
•
del
M. Lift characteriatice of a NACA 63(420)-422 airfoil with O.Oll-inch-grain rouahnesa
at various ehordwiae locatioDL
FIG.
I
.,j
.•­
DCA
a DCA
~ DCA
~
c
.012
.a,
..,r.•
.0OS
........
.,
tI
..,
&
.~
J
"2-215
'It.2-215
'5z-215
4
IlACA "~5
9
IACA
1~
~
••
J
e
67.1-215
r----, >---­
~
.
~
..J
c
a
.1
.,
Po~1tlon
.~
·5
or 111n1mUlll pressure,
.........
.,
.7
x/c
FIG. 65. Variation of minimum drag coefficient with position of minimum pressure for some
NACA 6-seriea airfoils of the same camber and thickness. R, 6 X I ()I.
148
THEORY OF WING SECTIONS
more severe than that caused by usual manufacturing irregularities or
deterioration in service, but it is considerably less severe than that likely to
be encountered in service as a result of accumulation of ice, mud, or damage
in military combat.
Maximum-lift-coefficient data at a Reynolds number of 6 million for a
large number of NACA wing sections are presented in Figs. 58 and 59.
The variation of maximum lift coefficient with thickness for the NACA
four- and five-digit series wing sections with standard roughness shows the
same trends as those for the smooth sections except that the values are
considerably reduced for all these sections other than the NACA OO-series
of 6 per cent thickness. The values of the maximum lift coefficient for
these rough sections with thickness ratios greater than 12 per-cent are sub­
stantially the same for a given thickness. Much less variation of maximum
lift coefficient with thickness ratio is shown by the NACA 6-series wing
sections in the rough condition than in the smooth condition. The varia­
tion of maximum lift coefficient with camber, however, is about the same
for the wing sections with standard roughness as for the smooth sections.
The maximum lift coefficients of rough wing sections decrease with rear­
ward movement of the position of minimum pressure; and, as in the
smooth condition, the optimum thickness ratio for maximum lift increases
with rearward movement of the position of minimum pressure. The NACA
64-series wing sections cambered for a design lift coefficient of 0.4 have
maximum lift coefficients consistently higher than the NACA 24-, 44-,
and 23o-series sections of comparable thickness when rough, with the
exception of the NACA 4412.
For normal wing sections, the angle of zero lift is practically unaffected
by the standard leading-edge roughness. The results presented in Ap­
pendix IV show that the effect of roughness is to decrease the lift-curve
slope for wing sections having thickness ratios of 18 per cent or more. The
effect increases with increase of thickness ratio. For lying sections less
than 18 per cent thick, tile effect of roughness on the lift-curve slope is
relatively small.
'1.6. Drag Characteristics. a. Minimum Drag of Smooth: Wing Sections.
The value of the minimum drag coefficient for smooth lying sections is
mainly a function of the Reynolds number and the relative extent of the
laminar boundary layer, and it is moderately affected by thickness ratio
and camber. If the extent of the laminar boundary layer is known, the
minimum drag coefficient may be calculated with reasonable accuracy by
the method presented in Sec. 5.13.
The effect on minimum drag of the position of minimum pressure that
determines the possible extent of laminar flow is shown in Fig. 65 for some
N J.~CA 6-series wing sections. The data show a regular decrease of drag
coefficient with rearward movement of minimum pressure. The variation
EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS
149
of minimum drag coefficient with Reynolds number for several wing sec­
tions is shown in Fig. 66. The drag coefficient generally decreases with
increasing Reynolds number up to Reynolds numbers of the order of
20 million. Above this Reynolds number, the drag coefficient of the NACA
65(Gl)-420 section remains substantially constant up to a Reynolds number
of nearly 40 million. The earlier increase of drag coefficient shown by
the NACA 66(2%15)-116 section may be caused by surface irregularities
because the specimen tested was a practical construction model. It may
be noted that the drag coefficient for the NACA 65,-418 lying section at
low Reynolds numbers is substantially higher than that of the NACA 0012
section, whereas at high Reynolds numbers the opposite is the case. The
higher drag of the NACA 65,-418 wing section at low Reynolds numbers is
caused by a relatively extensive region of laminar separation downstream
from the point of minimum pressure. This region decreases in size with
increasing Reynolds number. These data illustrate the inadequacy of low
Reynolds number test data either to estimate the full-scale characteristics
or to determine the relative merits of lying sections at flight values of the
Reynolds number.
The variation of the minimum drag coefficient with camber is shown
in Fig. 67 for a number of smooth 18 per cent thick N .A.CA 6-series wing
sections. These data show little change of minimum drag coefficient with
increase of camber. A considerable amount of systematic data is included
in Fig. 68 showing the variation of minimum drag coefficient with thickness
ratio for some NACA wing sections ranging in thickness ratio from 6 to
24 per cent of the chord. The minimum drag coefficient is seen to increase
with increase of thickness ratio for each series of wing sections. This in­
crease, however, is greater for the XACA four- and five-digit series lying
sections (Fig. GSa) than for the NACA 6-series sections (Figs. 686, c, and d).
b. Variation oj Profile Drag with Lift Coefficient. Most of the variation
of drag with lift for "rings of finite span results from the induced-drag
coefficient, which varies approximately as the square of the lift coefficient
for a given wing configuration. It is important to keep the induced drag in
mind when considering the variation of profile drag with lift, because the
variation of the wing drag coefficient with lift coefficient will be largely
determined by the induced drag, which is a function of the aspect ratio.
At low and moderate lift coefficients where there is no appreciable
separation of the flow, the drag is caused almost completely by skin fric­
tion. Under these circumstances, the value of the drag coefficient depends
upon the relative extent of the laminar boundary layer and the induced
velocities over the surfaces of the section and may be calculated by the
method of Sec. 5.13. As the lift coefficient increases, the average square of
the velocities over the surfaces increases, resulting in a. small drag increase
even though the relative extent of laminar flow was not affected. The
THBORY OF WING SECTIONS
150
.05
.05
J.ra
(roup 1e
b"(UO~
.
o
!:=
t:
.010
I
.008
J...I .­
i
.005
r----.- ~ ~ ....
.,
-
<,
.oct.
,.--r--..
--
,'"
r-,
.......
,
~ IIICA
.,....
~
I--~
.6
---
001Z
10--
-- r--...
DCA 25021
----.'-
r:-- r--~Iooo.
r--- r--
~t- ~>
r----.
~
'5(Je21)-420 .....-1 ~IVIIICA'S,-ua
1/
IIAC&
~
.8 1.0
......
r---....;.:
\
~
..... --l-I-1
~ III
~
~,
l~~
'-0,
l'~
.GCI1..
~
----.. ~-
)
eesae
DCA 25021
(a-oup 1etl41ll& edp)
A
~(
DCA "(2d5)-11& - '
~'"
2
I
I I
fN:[
.......
,..
~
,
8 10
20
50 ~o'
Flo. 66. Variation of minimum d.... coefficient with Reynolds number for several airfoils.
together with laminar and turbulent akin-friction coeflicienta for a flat plate.
~
.JI .016
o
I
I
6,-..
I
RACAalr.toU
e
rl••
-
£16­
~U65.,-818
A
9
..s
."
.004
~
o
••
g
!
a
o0
.2
.4
.6
.8
1. o
De.laB .eotion litt oo.~~lclint. 0li
FIG. 67. Variation of section minimum drag coefficient with camber for ee1.-eral NACA
8-:Ieriee airfoil eeetioDa of 18 per cent thickness ratio. R, 6 X 10'.
EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS
151
more important effect of lift coefficient is to change the possible extent of
laminar flow by moving the minimum-pressure points. In many of the
older types of wing section, the forward movement of transition is gradual
and the resulting variation of drag with lift coefficient occurs smoothly"
The pressure distributions for the XACA 6-series wing sections are such as
to cause transition to move forward suddenly at the end of the low-drag
range of lift coefficients. A sharp increase of drag coefficient to the value
corresponding to a forward location of transition on one of the surfaces
results. Such sudden shifts of transition give the typical drag curve for
these wing sections with a II sag" or ((bucket" in the low-drag range. The
same characteristic is shown to a smaller degree by some of the earlier
wing sections such as the NACA 23015 when tested in a low-turbulence
air stream.
The data presented in Appendix I for the NACA 6-series thickness
forms show that the range of lift coefficients for low drag varies markedly
with thickness ratio. It has been possible to design wing sections of 12 per
cent thickness with a total theoretical low-drag range of lift coefficients of
0.2. This theoretical range increases b)" approximately 0.2 for each 3 per
cent increase of thickness ratio. Figure 69 shows that the theoretical
extent of the low-drag range is approximately realized at a Reynolds num­
ber of 9 million. Figure 69 also shows a characteristic tendency for the
drag to increase to some extent toward the upper end of the low-drag range
for moderately cambered wing sections, particularly for the thicker ones.
All data for the KACA 6-serics wing sections show a decrease of the extent
of the low-drag range "pith increasing Reynolds number. Extrapolation of
the rate of decrease observed at Reynolds numbers below 9 million would
indicate a vanishingly small low-drag range at flight values of the Reynolds
number. Tests of a carefully constructed model of the NAC"'A 65(421)-420
section showed, however, that the rate of reduction of the low-drag range
with increasing Reynolds number decreased markedly at Re)·001<18 num­
bers above 9 million (Fig, 70). These data indicate that the extent of the
low-drag range for this wing section is reduced to about. one-half the
theoretical value at a Reynolds numberof 35 million.
The values of the lift coefficient for which low drag is obtained are de­
termined largely by the amount of cumber, The lift coefficient at the
center of the low-drag range corresponds approximately to the design lift
coefficient of the mean line. The effects on the drag characteristics of
various amounts of camber are shown in Fig. 71" Section data indicate
that the location of the low-drag range may' be shifted by even such crude
camber changes as those caused by small deflections of 8 plain flap,"
The location of the low-drag range shows SODle variation from that pre­
dicted from the simple theory of thin wing sections. This departure ap­
pears to be a function of the type of the mean line used and the thickness
THEORY OF WING SECTIONS
152
SerJ..ea
:
i}~-dlglt)
VI
2~O (5-d1S1t)
.016
....
• 012
i.
Rough
.
-"1 . -<:>--1
)-_..A,.
~- --1
e .008
.....
.,p
smooth 4.~ ~
I
o
- .0<:4
....
I
.)--"""11"
--It­ ~
.~--
'~
.--­ ~
--~.~
~
~
c
C)
....
()
:::
G
0
o
o
(a)
110
NACA four- and rive-d1git series.
cd
.;
.012
co
....
I
~
o
:
RO~J
.008
t--­ ...<>---~
..
_.J 1--...... ~
.-r
.~--
g
....ee
il
.004
o
-
o
~.
Salo )th
-t­
q
8
12
16
Airfoil thickness. percent
(b)
NACA
00
00.2
~ 5--­ ~
or
20
chord
A
a.It
w 0.6
24
63- series.
68. Variation of section minimum drag coefficient with airfoil thickness ratio for several
N.~CA airfoil sections of different cambers in both smooth and rough conditions. B, 6 X 1()6.
FIG.
153
EXPERIAfENTAL CHARACTERISTICS OF' n'ING SECTIONS
A
.008
I­
.o~ ~
Smooth
..r
(e)
MACA
64-
aeries •
• 012
.008
Jt-­ ...
~
.o~
Roush
-e-~ ~-~
~,)---
>--­
I
-­
ir:
I
~
-- ...... ~--
Cl
~
'-!\
Smooth
4\
~
~
..--
~
(d)
NACA
65- aerles.
A
.012
~.#Q
A
.008
~
Rough
1·~--
.o~
~oo7h
'r
.()-~ ).-~
.....
v
,)--­
__ ...4 ~.,~
'~
v
-
8
12
4
16
Alrrol1 thickness. percent or
(e) NACn 66- series.
FIG.6S. (Corduded)
c:> 0
0.2
~
A
V
o
--A
20
Cho~a
l
0.4
0.6
THEORY OF WING SECTIONS
154
ratio. The effect of thickness ratio is shown in Fig. 69 from which the
center of the low-drag range is seen to shift to higher lift coefficients with
increasing thickness ratio. This shift is partly explained by the increase of
lift coefficient above the design lift coefficient for the mean line obtained
when the velocity increments caused by the mean line are combined with
.032
G
a
.028
e
A
IfAOA 641 - Ja,12
RACA '42-~5
MCA 6~_~
.ADA 611:4-1;21
:'.~
..
.t
.p
I
o
.020
I
Z.016
...a
If tJ
W
v
IPI/
~
~
8
co
.012
~
.1
~
.008
'i:1
~
~~
""'i ~
.~
o
-1.'
~V
~V
~
-.
~
...... ~
~
-1.2
-.8
.
~ rJy
AI.~
--
_....
0
....
.8
8ectlOil 11ft coetftclent.O&
1.2
1.6
69. Drag characteristics of some NACA 64-series airfoil sections of various thicknesses,
cambered to a deei&n lift coefficient of 0.4. R, 9 X 10'.
FIG.
the velocity distribution for the thickness form according to the first­
approximation method of Sec. 4.5.
At the end of the low-drag range, the drag increases rapidly with
increase of the lift coefficient. For symmetrical and low-cambered wing
sections for which the lift coefficient at the upper end of the low-drag range
is moderate, this high rate of increase does not continue (Fig. 71). For
highly cambered sections for which the lift at the upper end of the low-drag
range is already high, the drag coefficient shows a continued rapid in­
crease.
Comparison of data for wing sections cambered with a uniform-load
EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS
2
..
..,;
8
V
~~
o
i
.:s
:=
'-t
i•
Ir--B
I
-.•
...
~
1 ' - - 1---
V
1J
....
...
~
na.
~e
I
0
.­
I
I
I
I
I
I
I I
Vppel'lJa1t of low-4rq . . . .
~J1 1JIdt of l_-4ras
I
-
155
~-
~
_.....
J
~
II
-
....~
"':"
1-·
o
2
o
(a)
8
~
20
2JJ
111,.148 . .bel'• •
12
Variation of upper and lower limits of low-drag range with Reynolds number,
(
J
•x J.o'
2.0
.0
e
.028
I I
II
~.o
•
61.'
~
" 25·0
~
'5.0
,
>
.~
~
J.,
',-
e~ ~
.008
~
.,...
.-.
e
-1.2
FIG.70
III
.
~
a; rw:...
-.8
-.4
0
---
~r.;)
-.~
-
.1,.
.8
"o~l_ 1U~ coer~~ol.at.
o~
1.2
L'
(b) Section drag characteristics at various Reynolds numbers.
Variation of low-drag range with Reynolds number for the NAC_4 65(421)-420 airfoil.
THEORY OF WING SECTIONS
156
mean line with data for sections cambered to carry the load farther forward
shows that the uniform-load mean line is favorable for obtaining low-drag
coefficients at high lift coefficients (Fig. 72).
Data for many of the wing sections given in Appendix IV show large
reductions of drag with increasing Reynolds number at high lift coefficients.
• 028
4~
.024
IA
tcJ
0
-
.p
cG)
is/
.020
V
if?
.....
...o
'-t
~
CD
.016
0
0
tQ
~
~
s:
0
.012
.....
.p
0
C)
It)
.008
l~
HACA 6~-4l8"
\ ~r\
~
~.jl~
.~
I
I
I
I
I
CA
:U
~.
"t:>:
~~
~.
~
.~
.oa,.
.",,:
I
65,-218
RACA
I
'"\
\
\
I
6 53-018,\
i\
.~
d~J
':> ~ ~
1\
I~ ~
~
~~
~
1.
L
-
~
p~
~.~~
.7
J
r 1M
.... Li. ~
....
~\
RACA 6~-618JlACA
).\
65.,-8J.8- W
0 ...........- ......- ............- ......- ...----......- .....----.....- .....- ............
o
1.2
.8
1.6
-1.2
-.8
-.4
Sectlon
~lrt
coefficient.
CI
FIG.71. Drag characteristics of some NACA. 65-seriea airfoil sections of 18 per cent thickness
with various amounts of camber. R, G X 10'.
This scale effect is too large to be accounted for by the normal variation of
skin friction and appears to be associated with the effect of Reynolds
number on the onset of turbulent flow following laminar separation near
the leading edge.1M
A comparison of the drag characteristics of the XACA 23012 and of
three NACA 6-series wing sections is presented in Fig. 73. The drag for
the N.~CA 6-series sections is substantially lower than for the NACA 23012
section in the range of lift coefficients corresponding to high-speed flight,
and this margin may usually be maintained through the range of lift
EXPERIl.{ENTAL CHARACTERISTICS OF rVING SECTIONS
157
coefficients useful for cruising by suitable choice of camber. The NACA
6-series sections show the higher maximum values of the lift-drag ratio.
At high values of the lift coefficient, however, the earlier NACA sections
.generally have lower drag coefficients than the N ACA 6-series sections.
c. Effect oj Surface Irregularities Oil Drag Characteristics. Numerous
measurements of the effects of surface irregularities on the characteristics
of wings have shown that the condition of the surface is one of the most
important variables affecting the drag. Although a large part of the drag
increment associated with surface roughness results from a forward move­
ment of transition, substantial drag increments result from surface rough­
ness in the region of turbulent flow." It is accordingly important to
maintain smooth surfaces even when extensive laminar flow cannot be
expected. The possible gains resulting from smooth surfaces are greater,
however, for wing sections such as the XA(~A G-series than for sections
where the extent of laminar flow is limited by a forward position of mini­
mum pressure,
No accurate method of specifying the surface condition necessary for
extensive laminar flow at high Reynolds numbers has been developed,
although some general conclusions have been reached, It may be presumed
that, for n given Reynolds number and chordwise location, the size of the
permissible roughness will vary direct ly with the chord of the lying section.
It is known, at one extreme, that the surfaces do not have to be polished
or optically smooth. Such polishing or waxing hus shown no improvement
in tesu,.3 in the Kf\CA two-dimensional low-turbulence tunnels when ap­
plied to satisfactorily sanded surfaces. Polishing or waxing a surface that
is not aerodynamically smooth ,,;U, of course, result in improvement, and
such finishes may be of considerable practical value be cause deterioration
of the finish may be easily seen and possibly postponed. Large models
having chord lengths of 5 to 8 feet tested in the NACA two-dimensional
low-turbulence tunnels are usually finished by sanding in the ehordwise
direction with Xo, 320 carborundum paper when an aerodynamically
smooth surface is desired," Experience has shown the resulting finish to be
satisfactory' at flight values of the Reynolds number. Any rougher surfuee
texture should be considered as a possible source of transition, although
slightly rougher surfaces have appeared to produce satisfactory results in
some cases..
l.;oftin6S 8ho\\'00 that small protuberances extending above the general
surface level of an otherwise satisfactory surface are more likely to cause
transition Ulan are small depressions. Dust particles, for example, are
more effective than small scratches in producing transition if the material
at the edges of the scratches is not forced above the general surface level.
Dust particles adhering to the oil left on wing surfaces by fingerprints may
be expected to cause transition at high Reynolds numbers.
THEORY OF WING SECTIONS
158
Transition spreads from an individual disturbance with an included
angle of about 15 degrees. H • 42 A few scattered specks, especially near the
2.8
..
I
I
2.0
I
o
e
DCA 6~-418
D DCA
~-
t----
r---
1.2
65,-418, a
---..
=0.5 - r-.
~~~
..........
-~--
~.
rJ) ~
~.-
I
r:'
'r
II
.8
r
;
l~
~
o
I
o
-rwlr
--4
sa
•o
.....
-.2
J
•
- .8
7
1
~...I
-
--
~
-
~
"
(.;1
'-t
~
S
o
-.~
~
I
=­ -.4
-16
-8
8tQ~lOD
0
a
aDS1. or attack.
~b
(10.
deg
24
FIo.72. Comparison of the aerodynamic characteristics of the NACA 65,-418 and NAC_~
66,-418, CI ==.0.5 airfoils. B, 9 X 10'.
leading edge, will cause the flow to be largely turbulent. This fact makes
necessary an extremely thorough inspection if low drags are to be realized.
Specks sufficiently large to cause premature transition can be felt by hand.
The inspection procedure used in the N ACA two-dimensional low-turbulence
EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS
159
tunnels is to .feel the entire surface by hand, after which the surface is
thoroughly wiped with a dry cloth.
0J2
028
o
td
0- •
G24
...•
020
.
1i
...o
•
.~ ~.....'r---.....-_.8
I
.2
or-......._
1.0
.. _ + - - - t - _ t - - - t -......._+---I
rI
'-4
'-4
8
0
.ou
:: .
a
:
.b
a0
012
0
If
/J
008
AftL
"T
.
o· 0
....-•..
-
0
4
1;
a.o. position
0
""8
....
..1
Q
-.2
JUOA-'5,~8
-II IIICA 653-418,
0
I -.,
011
=0.5
-.060
.267
-·Qq1
I I
•
-1.6
•
zlo
.2'5
-1.2-
-.8
-.J..
0
.k
"etlon 11ft coeff1cient.
FIG. 72. (Ccmcluded)
7/0
I
•J
1.2
1.6
z.0
c&
It has been noticed that transition caused by individual sharp pro­
tuberances, in contrast to waves, tends to occur at the protuberance.
Transition caused by surface waviness appears to move gradually up­
stream toward the wave as the Reynolds number or wave size i~ increased.
The height of a small cylindrical protuberance necessary to cause
THEORY OF WING SECTIONS
160
transition when located at 5 per cent of the chord with its axis normal
to the surface" is shown in Fig. 74. These data were obtained at rather low
2.8
I 1
T
II~
e
DCA
EI lIACA
.1IACA
""""-
I I I I
I
I
I
I
-l&J.~
~~
I--
.
I
Jl
x
106
6
6 lIACA 65~~5
9 DCA " 215)~,
o
.: 1. 6
I
!012
9
9
roupn... 9
8taJdaiid
~. Q~
...
I
9
F
nJ. L.
~~
e:
....•u
.......
I
Ih
,1. 2
.p
.......
"­
4"1
,
g
....
.p
o
~
~.
~
ro
IJ'
o
a
~
~
II
~
~t\
J
"
."
)
o
~
.~
.Art
t·
0
~
--. '9
~
-
~
...
,[)
"'"'"
~lfl
\,.I
.......
Ul.Y
~
,1--
f\
<JsV
j3
.~
rv
7
-1 .2
~
e
•
!
-.4
.-1. .6
-2J,.
-16
seotion
-8
8D&1~
0
of attack.
8
l'
"0' dec
24
FIG. 7~. Comparison of the aerodynarnie characteristics of SOD1C N.ACA airfoils from tests
in the Langley two-dimensional low-turbulence pressure tunnel.
values of the Reynolds number and show a large decrease of the allowable
height with increase of Reynolds number.
Analysis'S of these data showed that the height of the protuberance
that caused transition depended on the shape of the protuberance and on
the Reynolds number based on the height of the protuberance and the local
EXPERIMENTAL CHARACTERISTICS OF It''lNG SECTIONS
161
velocity at the top of the protuberance. This Reynolds number is plotted
against the fineness ratio of the protuberance in Fig, 75 for protuberances
located at various chordwise positions on two wing sections.
•0,2
I I I
t ·02"
i•
9 Ie 10' .....
.9 ~"' r-,
$-ur;
-415
~ IlACA
o •
"
6 ~ ,~
StaDel
rougbne••
• DCA. '52-415
" DCA 66(215)~
Iu
1
It
~
G DCA
B DCA
.;: •
...
. .-­-­
, ,I
1
9 -_.~
\
\
.
Ol~
.I
012
~
'" ~
008
,
~
f\
',-""
..... '
-v"f..:J
~
~
.
o
.....
l~" f\Vp
) ~W
,
J~ ~v
I
j /)1
/ I/~ V
J~ /
_llt V
~I
I'oT
V
1_ 1"1
~
~
V
0
o
.....
..;.
:-or
o
...-
....:. -
...~
­ .2
...o
Ii
CI.
1­
·m
0
I
I
....
W
...
-
v
v
--..
I
-.~
-.~o
.2
D ~.o x 1
4
.0 x 10
" 9.0 x 10
I
...
..c.
poaIt}On
7./c
7. c
o 9.0 " l~
. ­ .,
-
-­
...
R
.
~
~
~
-
.1
-1.6
r
:1
i"'-.~
~
v
~" ~\ ,1\
2
o
I
\/ 1
~ ~ ~ ~ ~J
to
!...
I
.26
-.
.265
I
I
2
-.105
I
I
1
I
-1.2
-.8
-.If.
0
-fa.
sectIon 11ft coefficient.
FIG.
.8
c~
1.2
1.6
2. o
73. (Cuncluded)
The effect of Reynolds number on permissible surface roughness' is also
indicated in Fig. 76, in which a sharp increase of drag at a Reynolds number
of approximately 20 million occurs for the model painted with camouflage
lacquer. Experiments with models finished with camouflage paint" in­
162
THEORY OF WING SECTIONS
dicate that it is possible to obtain a matte surface without causing pre­
mature transition but that it is extremely difficult to obtain such a surface
sufficiently free from specks without sanding.
~
.050
i .*
~:i
I .
"',­ ',,­
~i·*
S'"
",I
....
~:.Q20
<, .......
OCto
~ ..........
~
i
.010
i
°0
,
2
1
5
---... t"-­
,
-- -- -8
,
-10
uJ
-JDs BaJD014a m.ber. ..
FIo. 74. Variation with wing Reynolds number of the miDimum height of a cylindrical
protuberance neceusry to cause premature transition. Protuberance has O.035-incb di­
ameter with axis nonnal to the wing surface and is located at 5 per eent chord of a 9O-ineh­
chord symmetrical6-series airfoilaection of 15 per cent thickness and with minimum preseure
at 70 per cent chord.
Low-drog airfoil numIJer 2
~
0.035 in.
.20
.035 iii.
015 in.
.20
.50
OJ5 in.
Low-dtog oirfrJllllllnber I
+
JC
~ ~+
D
x.~
'"
~... e
~
<>
~~
~
~
+
~
DiomeIer
0.035 in..
.035 in•
.035 in.
035 in.
.035 in.
.35
.50
V
V
~~~
~
a058
•20
A
~
Diameter
a05
0
'q
•65
~
............... r---~
1
~~
1.0
2.0
sa
40
Projeclion fineness ratio, ~
50
6.0
lO
75. Variation of boundary-layer Reynolds number factor with projection fineness ratio
for two low--drag airfoils.
FlO.
The magnitude of the favorable pressure gradient appears to have a
small effect on the permissible surface roughness for laminar flow. Figure
77 shows that the roughness becomes more important at the extremities of
163
EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS
the low-drag range where the favorable pressure gradient is reduced on one
surface.' The effect of increasing the Reynolds number for a surface of
...
<IlL
12
/6
20
24
28
32
36
40
..
-
-­
44
48
52
56
6OxKJ4
~
W
#
~
~olei
Reynolds number, R
(o)SMOQTH CONDITION
4
8
~
f
~
-
OC
"'"
k70
~
$
~
N
M
R
Reynolds nmoe; R
(b') LACQUER CAMO..FLAGE UNIMPROVED AFTER ~INTING
76. Variation of drag coefficient with Reynolds number for a 6O-inch-chord model of
the NAC.4 e.S(m)-420 airfoil for two surface conditions.
FIG.
Smooth cond1 tion
enamel ca:nourlase
wIth all specl:a ~~lt orr
with bl&.de
S~thetlc
~
..
o
.012
..•
.p
s::
.: .0G8 t--1--~--+---+---+---I----b~~'./-----4-~~---1
~
'-t
8o
~
2
'0
a
o
...
.p
o
cI
section lUt coefficient. c
1
FIG. 77. Drag characteristics of NACA 65 l 4n )-420 airfoil for two surface conditions.
marginal smoothness, which has an effect similar to increasing the surface
roughness for a given Reynolds number, is to reduce rapidly the extent of
the low-drag range and then ~o increase the minimum drag coefficient
THEORY OF WING SECTIONS
164
(Fig. 77). The data of Fig. 77 were especially chosen to show this effect.
In most cases, the effect of increasing the Reynolds number when the sur­
faces are rough is to increase the drag over the whole low-drag range. The
effect of pressure gradient as shown in Fig. 77 is apparently not very
powerful and tends to be evident only in cases of surfaces of marginal
smoothness. 17
.;,5
I---~
0Z8 io--- ~
.
.....•"
..
o
~
-
~
I I I I I I I
Alrf"o11s DCA
26 x
lit
Chords
.... -Test.s
.
In.
262
O.OI1-J.aoh-sralD
rowdal••• on L.L
~ ...a O.OQb-laob-gra1n
;J
~rJJ
... ! l\
~.,
l
~
./
r,.
~
• 016
012
~.
A
l
o
.I
~
'1'D'l' 21jC; aIl4
l'f
10
:J
(lao)-!a22
•
Io . 020
ia
6,
lr!'
~
lit:
008
~~ r-,
~ r-, r--- L::: :r.A.
~
r-----. ~
~
"'.
~ ~-V"
-
~
t-
,.
OD
L.B.
x O.OO2-1Dah-sraua
rousJm•••
OIl
L. L
V If JJ
:",A
N'
~ t""j; ."
~
~
~
rOQBbD...
_
1:7
t.
rA..... +She11&o
Oft
L.E.
~~Ga.oo~
"V
Gala.
0
-1.6
FIG.
ne88
•
-1.2
-.8
-.1&.
0
...
.8
seotlon 11ft ooern.leat.
0,1.2
1.6
2.0
2.4
78. Drag characteristics of a NACA 63(420)-422 airfoil with various degrees of rougb­
at the leading edge.
More difficulty is generally encountered in reducing the waviness to
permissible values for the maintenance of laminar flow than in obtaining
the required surface smoothness. In addition, the specification of the
required fairness is more difficult than that of the required smoothness.
The problem is not limited to finding the minimum wave size that will
cause transition under given conditions because the number of waves and
the shape of the waves require consideration.
H the wave is sufficiently large to affect the pressure distribution in such
a manner that laminar separation is encountered, there is little doubt that
such a wave will cause premature transition at all useful Reynolds num­
bers. Relations between the dimensions of a wave and the pressure dis­
tribution may be found by the method suggested by Allen." If the pressure
EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS
165
distribution over the wave is known, the criteria for laminar separation
discussed in Sec. 5.7 may be applied. The size of the wave required to
reverse the pressure gradient increases with the pressure gradient. Large
negative pressure gradients would therefore appear to be favorable for
+
.o"
A1rtol1 I
DCA
6,lLzO)-422
III 2& x 106
ChoN, ~ 111.
Teat.
'1' 255
.052
A
,
I
~
¥
.012
f .....
~'"
~~ ~
~
.008
..---0
Jloastm...
.trip
on L.s.
" "r-,
I
~ ~ r­ '---.
""""--. :..­
rv­
'--<~ ~ ....
~
~
""""­
o
-1.6
.1.2
.. 8
... Il
f
-
~
.
~
V
..,..
-
....
/1
...-..4'
~
-
../
~AC.A
Jl1.
"Jl1
~~
~
"
~ '"
4­
"
-'
0
....
.8
aeetlan 11ft coe1"l'lClent.
FIG. 79. Drag characteristics of a
at various chordwise Iocatione,
~
V1
K.
I(
Ioastme••
atrlp­
at 0.200
_
+ Bouatme••
at.rip
at 0""0
_
G 8IIooUa
1.2
01.
1.6
2.0
2....
63(420)-422 airfoil with O.Oll-inch-grain roughness
Experimental results have shown tills conclusion to be
qualitatively correct.
For the t}1JCS of waves usually found on practical-construction wings,
the test of rocking a straightedge oyer the surface in a ehordwise direction
is a fairly satisfactory criterion." The straightedge should rock smoothly
without jarring or clicking. The straightedge test will not show the exis­
tence of waves that leave the surface convex. Tests of a large number of
practical-construction models, however, have shown that those models
wavy surfaces.
THEORY OF WING SECTIONS
166
which passed the straightedge test were sufficiently free from small waves
to permit low drags to be obtained at flight values of the Reynolds number.
It does not appear feasible to specify construction tolerances on or­
dinates of wings with sufficient accuracy to ensure adequate freedom from
dt
~
"'
~'
.,.
-
_a
·.'"......
~
..
.­
a
...::•
o • 02-
+
•oo
..,.016
S
.. .
t:
~
OU
~~+ IlACA '5(22')~2 (-at'le4)
0...1
I
,8 IUCA " ( i20)~2Z
~~
~
_
~A
-
-1.2
I
I
~~
-F:=o.
~
(/
_.... ~~~
~
~
""
a.
-.8
......
0
26 •
....
ltJ
.8
1.2
1.6
"otloa l1tt oCMt1'lol_t. . ,
FIG. SO. Drag characteristics of two NACA 6-...~rics airfoils with O.Oll-ineh-grain roughnese
at O.3Oc.
waviness. If care is taken to obtain fair surfaces, normal tolerances may
be used without causing serious alteration of the drag characteristics.
If the wing surface is sufficiently rough to cause transition near the
leading edge, large drag increases are to be expected even if the roughness
is confined to the region of the leading edge. Figure 78 shows that, although
the degree of roughness has some effect, the increment of minimum drag
coefficient caused by the smallest roughness capable of producing transi­
tion is nearly as great as that caused by much larger grain roughness when
the roughness is confined to the leading edge.! The degree of roughness has
a much larger effect on the drag at high lift coefficients. If the roughness
is sufficiently large to cause transition at all Reynolds numbers considered,
EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS
167
the drag of the wing with roughness only at the leading edge decreases
with increasing Reynolds number.'
The effect" of fixing transition by means of a roughness strip of car­
borundum of O.011-inch grains is shown in Fig. 79. The minimum drag
32 x 10'
28
.....
0:
•
'Ia
...-3
0
~
20
16
12
!
8
.1'
0"----11.----........-.-....-.........-
o
.08
........- - ' - -.......-
...
.2k .~ .~ .48 .56 .'4 .72
Sectloo l1tt
coetrl~1ent.
01
.56 .'4 .72
o
8ec~1oa
lift
coet~lcleDt.
01
FIG.81. Comparison of section drag coefficients obtained in flight on various airfoils. Tests
of NACA 27-212 and 35-215 sections made on gloves.
increases progressively with forward movement of the roughness strip.
The effect on the drag at high lift coefficients is not progressive; the drag
increases rapidly when the roughness is at the leading edge. Figure 80
shows that the drag coefficients for the X.A.C.-\. 65(223)-422 and 63(420)-422
lying sections' are nearly the same throughout most of the lift range when
the extent of the laminar flow is limited to O.30c. These data indicate that,
THEORY OF WING SECTIONS
168
for wing sections of the same thickness ratio and camber, the drag coeffi­
cients depend almost entirely on the extent of the laminar boundary layer
if no appreciable separation occurs.
The variation of minimum drag coefficient with thickness ratio for a
number of NACA four-digit, five-digit, and 6-series wing sections with
.0/2
I
o Surfaces 05 received excepllighllysmded
I- 0
Both surfocespointedandfinished10 rear spar
-..
0/2
/6
20
24
28
32
36x/0 6
Rf!I'1OIds number, R
FIG. 82. Drag scale effect on l00-inch-ehord praetieal-eonstruction model
65(216)-3(16.5)(approx.) airfoil section. C1 = 0.2 (approx.).
or
the NACA
leading edges rough is shown in Fig. 68. The minimum drag coefficients
increase with thickness ratio but are substantially the same for all the
sections of equal thickness ratio. The increments of drag coefficient caused
by leading-edge roughness are correspondingly greater for the sections
having the lower drag coefficients when smooth.
The section drag coefficients of several airplane wings have been
measured in flight l 60 by the wake-survey method, and a number of practical­
construction wing sections have been tested in the NACA two-dimensional
low-turbulence pressure tunnel! at flight values of the Reynolds number.
Some of the flight data obtained by Zalovcik l 60 are summarized in Fig. 81.
EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS
169
All wings for which data are presented in Fig. 81 were carefully finished to
produce smooth surfaces. Care was taken to reduce surface waviness to a
minimum for all the sections except the NACA 2415.5, N-22, Republic
r
~
.ex»'
t
·005
•S
f
..s
..
i
-
.our
.:
~
0.910
( 101 .12• 58
:= .010
,,~
Cd"
•
I--
+I
-
~
....
-
./
!\lrtnIl..,t
!
.~
I
t
.003
<,
.002
k'
Cd
r-, r-,
1Aa1nC"
I I
.001
1
•
2
2.6511'"
,....
c
CondlUon of lIodel
1
0
0.°79
"
0.07~
AJJ recelnd
PalMed with gray pr1Jller
aurracer
C...ourlase p&1nced with
almulatad door jolnta
• o.O"n
II
10
! I I I I I " Ii I I I
20
IIaJD014. n"","er.
B
FIG. 83. Variation of the drag coefficient with 1I"Y1101<l8 nUll11JCr for the ~ACA 2301li airfoil
IIOCtion together with laminar and turbulent ski ll-fricti on cocffieioute for a flat plate.
8-3, 13, and the XACA 2i-212. Curvature gauge measurements of surface
waviness for some of these sections are presented by Zaloveik.P" These
data show that the KACA 2-,3-, and 6-sel;es sections that permitted ex­
tensive laminar flow had substantially lower drag coefficients when smooth
than did the other sections.
Data obtained in the XACA two-dimensional low-turbulence pressure
tunnel' for typical practical-construction sections are presented in Figs, 82
THEORY OF WING SECTIONS
170
to 86. Figure 87 presents a comparison of the drag coefficients obtained
in this wind tunnel for a model of the NACA 0012 section and in fiight for
the same model mounted on an airplane.' For this case, the wind-tunnel
and flight data agree to within the experimental error.
The wind-tunnel tests of practical-construction wing sections as de­
livered by the manufacturer showed minimum drag coefficients of the
order of 0.0070 to 0.0080 in nearly all cases regardless of the type of section
.016
' 1
I
I
I
•
~OIlACA 6~(216)-4l7(~F.' .. bull'
~OIlACA 2 011)
\\
\\
~
l..n----""
~
o
-.2
~
~
~
(.pprox.
~
V"
/
.. bull t
~
o
.8
sect10n 11rt coefficient.
c~
Flo. 84. Drag characteristics of the NACA 65.2-417 (approx.) and NACA 23015 (apllrox.)
airfoil eectiODB built by praetical-conatruction methods b)~ the same manufacturer. R,
10.23 X lOS.
(Figs. 82 to 86). Such values may be regarded as typical for good American
construction practice during the Second World War.
Finishing the 800­
\
tions to produce smooth surfaces always resulted in substantial drag re­
ductions, although considerable waviness usually remained. None of the
sections tested had fair surfaces at the front spar. Unless special care is
taken to produce fair surfaces at the front spar, the resulting wave may
be expected to cause transition either at the spar location or a short
distance behind it. One practical-construction specimen tested with
smooth surfaces maintained relatively low drags up to Reynolds numbers
of approximately 30 million [NACA 66(2%15)-116 wing section of Fig. 66].
This specimen had no spar fonvard of about 35 per cent chord from the
leading edge and no spanwise stiffeners forward of the spar. This type of
construction resulted in unusually fair surfaces.
Few data are available on the effect of propeller slipstream on transition
or wing drag; the data that are available do not show consistent results.
0 11&0&
-
- a
r- 0
l\1 ~
I
~
o
,\
•• 012
....11•
1.008
66(21S)-U6. lOO-lDch
Obor4.
0&
80A
25016.
1OO-lDeb 60N.
0, •
:I
&!Ii
IllaA "C211i )-U' (n1Nllt . .1 J.
0.11)
laoh obol'4.
0.19
IIICA 25016 ("bullt .04(:1), l00-1aob chON.
0, • 0.19
\~~\
C)
- ~' I\.
~~
o
eo
.
~
..,•
I~
!.­
.,
\
~
"~
.A.~
.A
~
o
&
o
FIG.
o
28
8
85.
Scale effect on drag of the NAC..-\. 66(215)-116 and NACA 23016 airfoil sections
88 received.
built by practical-eonstruction methods by the same manufacturer and tested
•016
I
~
o
.012
0
a
I
D...l. wins. or1g1Dal concU tlon
DCA 65__erle. -ins, original coD41~10D_
~
~
o0
'"
8
~~~
.....
12
"JDOu.
~
20
D1aber.
-
21,.
28
a
~
)C
10'
86. Drag scale effect for a model of the N AC.A... 65-serics airfoil section 18.27 per cent.
thick and t.he Davis airfoilllOOtion 18.27 per cent thick, built by practical-construction nletbods
by the same manufacturer. c, == 0.46 (approx.) .
FIG.
0
u
• 012
-
.p
s:
....•0
...
0
0
.008
W1ncl tunnel
FlIght
'-4
~
80
·~.t
Z
"G
.....
r-""U
eo
....
-
""'
~
• ad,.
s
:I
0
cI
0
o
8
16
12
Re,nolds number,
20
28 )( lrP
R
FIG. 87. Comparison of drag coefficients measured in flight and wind tunnel for the NACA
0012 airfoil tteCtiOll at zero ilit.
171
THEORY OF WING SECTIONS
172
This inconsistency may result from variation of lift coefficient, surface con­
dition, air-stream turbulence, propeller advance-diameter ratio, and num­
t
A1rtoU ...'1. .
Boot. KACA "(2X15)-en.8
!!P. ileA 67.1-(1.,)15
.....\ ..~, 5.98
II
~
n4J._---.
-.Uer
'ip
~.012
0"
I
~l'
1d.D'-',11DI
!'rope11al'
......a
•
.---..,."~
I.~
!
....-..;
~r
~ ~~
a-­
~
o
I
°8
1
,
5
Dlat. . . tfta
4
~1
3
2
1
0
oater 11De. tt
88. The effect of propeller operation on section drag coefficient of a fighter-type airplane.
from tests of a model on the Langley 19-foot pressure tunnel. CL = 0.10; R, 3.7 X 10'.
FIG.
ber of blades. Some early British investigation Sl51 , 158. 159 showed that
transition occurred at 5 to 10 per cent of the chord from the leading edge
in the slipstream. Similar results were indicated by tests in the N..~CA
8-foot high-speed tunnel. 4I Drag measurements in the N.~CA 19-foot
pressure tunnels (Fig. 88) indicated that only moderate drag increments
EXPERIJIENTAL CHARACTERISTICS OF WING SECTIONS
173
resulted from a windmilling propeller. Although these data are only quali­
tative' because of the difficulty of making wake surveys in the slipstream,
they seem to preclude very large drag increments such as would result
from a movement of transition to a position close to the leading edge. A
I
20 x 106
........
I~
a
10
Lef't wing .eC1;iOl1}/
in slipatream
1/r---.. . . .
f=:-- r-
-r---
J.. . . . . V
Bight wing .ection
out.1~e a11pstream
f
o
Ie
.p
e,
..s
-50
o'-~
~
.-of
•
=a .40
e Bight
outaide allpatream
".
~
""
.30
::
.20
0
..s•
..•
o
CIt
t)
~
Jt
o
6
.10
wing .ection}
til
I
I
I
x.ttt wing section
1D al1patre8lll
D
Power on-----
,
o Power or~---I
.1
~-
V
10--
~-
~
r> ~>
""
..,Ito...
I
.2
section 11ft coe1'£lclent.
Fro. 89. Flight measurements of transition on a NACA
............
~
c~
f)()"~rics wing
within and outside
the slipstream.
similar conclusion is indicated by X ..\.C..\ flightdata/ (Fig. 89), which show
transition as far back as 20 per cent of the chord in the slipstream.
Even fewer data are available on the effects of vibration on transition.
Tests in the X . .\ CA 8-foot high-speed tunnel 43 showed negligible effects,
but the range of frequencies tested may not have been sufficient.ly wide to
represent conditions encountered on airplanes. Some K .~(~.~ Hight data"
showed small but consistent rearward movements of transition outside the
slipstream when the propellers were feathered. This effect was noticed
even when the propeller on the opposite side of the two-engined airplane
was feathered, and it was accordingly attributed to vibration or noise. In
THEORY OF WING SECTIONS
174
some cases, increases of drag have been noted in wind tunnels when the
model or its supports vibrated. A tentative conclusion may be drawn
from the meager data that vibration may cause small forward movements
of transition on airplane wings,
The skin friction associated with a turbulent boundary layer on a
smooth surface decreases with increase of Reynolds number, as shown in
Chap. 5. This favorable scale effect is not obtained at high Reynolds
,
,
.­
e;
.IS~
o
..,
~
.s
'\
....
~
~
..
.5
<,
~
~
r
\
FIG.
a
3.IJ
J.1
.I.,
• .0
---
~~ ~
~
-
~
-
[~'"
~
~ V''''
~ .......
~
\
J.6
U
to
U
ff
--..••¥
-
'Wl,,­
~
~~
..
~
~---
---
f6
-
~-..,
~
•f
.3
IIU
1Sl.JJ
~~
~
,
,;'
~~
.~,
.'
0
••
•
~
-...
....
.....
...-­
~
r-, r-w­
S/J
-
0
~ i!b.a
II
-
U
-""1"­
~
s.,
1­
--
..-.a.IIIl
~r-
SI
II
u
90. Coefficient of resistance of rough and smooth tubes dependent on Reynolds number.
numbers on rough surfaces. An explanation of this effect is that the pres­
sure drag of the individual protuberances constituting the roughness con­
tribute to the skin friction when they project through the laminar sublayer,
The shapes of the individual protuberances are usually such that the scale
effect on the pressure drag should be small. The result at large Reynolds
numbers for generally roughened surfaces is that the skin-friction coeffi­
cient is essentially constant.
The general nature of the scale effects for rough surfaces has been
indicated b)' experiments in pipes." These data (Fig. 90) show that at lo"r
Reynolds numbers the pipe-loss coefficient decreases with increasing Rey­
nolds number at the rate expected for laminar flow, When transition
occurs, the pipe-loss coefficient increases to the value expected for tur­
bulent flow over smooth surfaces. For very small roughness, the pipe-loss
coefficient decreases with increasing Reynolds number along the turbulent
skin-friction curve for smooth surfaces until large values of the Reynolds
number are reached. For any given size of roughness, however, there ap­
EXPERIMENTAL CHARACTERISPICS OF liTING SECTIONS
175
pears to be a Reynolds number beyond which the coefficient increases
slightly to a constant value. The value of the Reynolds number decreases
and the constant value of the coefficient increases as the grain size of the
roughness increases. At large values of the Reynolds number, the rough­
ness causes large increments of skin friction over the values corresponding
to smooth surfaces.
On the basis of the pipe experiments and reasoning similar to the fore­
going, von KarmAn139 obtained a formula for the grain size just sufficiently
large to affect turbulent skin friction. This formula may be written
pl:k=3.<L
p.
f!
V4
Reynolds number based on grain size of roughness and local
'vhere -pUk
- ::: velocity outside boundary layer
JI.
c,
= local skin-friction coefficient
For a 'ling of approximately 9-foot chord at a speed of 300 feet per second,
the limiting grain size is approximately 0.0004 inch and varies only slightly
over the surface, increasing toward the trailing edge.
The effect of roughness on the drag of wing sections with largely turbu­
lent flow is analogous to the effect in pipes as ShO\\·113 by Fig. 83. At the
lower Reynolds numbers, the drag coefficient decreases with increasing
Reynolds number at about the rate expected for turbulent flow over
smooth surfaces, In this case, the drag remains essentially constant at
Reynolds numbers above about 15 million. ...~t a Reynolds number of
70 million the increment of drag caused by the moderate roughness was
of the order of one-quarter of the drag that would have been obtained had
the favorable scale effect continued. The sensitivity of the turbulent
boundary layer to roughness is so great that airplanes should not be
expected to show favorable scale effects at large Reynolds numbers unless
considerable care is taken to obtain smooth surfaces.
It should be noted that, in contrast to the situation with respect to the
laminar layer, it is relatively easy to obtain reductions of drag by attention
to the surface conditions for turbulent flow. Even though the allowable
size of the roughness is very small, each speck of roughness presumably
contributes onI)" the drag of itself and does not have any appreciable effect
on the skin friction over the surface downstream. Consequently favorable
effects may be expected from careful finishing of the general surface of
modem high-performance airplanes even though imperfections such as
rivets and seams may be present.
d. Unconsenxuioe li'ing Sections. The need for low drags in order to
obtain long range for large airplanes flying at speeds below the critical
Mach number leads to designs having high wing loadings to reduce the
wing area and profile drag together with relatively )O\V span loadings to
THEORY OF WING SECTIONS
176
avoid high induced drags. These tendencies result in "rings of high aspect
ratio that require large spar depths for structural efficiency. The large
spar depths require the use of thick root sections. The comparatively high
lift coefficients corresponding to the cruise condition for such designs leads
to the use of large cambers to obtain low profile drags. Such designs are
encouraged by the drag characteristics of modem smooth wing sections
~
..........
.yr
r=­
/
4
~
o
•
OOO} (l~ cUBit)
",,\~
:~
I
644
v2,o
8
A1rf'oU
(a)
~ r-,
,,~
lV
....s
~
v
~~
I
I
r-,
MeA
12
16
perceDt of cho1"4
tblokD....
to~
I
(5 cU81t)
20
ao4 tlve-4161\ serle••
'f4
FIG. 91. Variation of the lift coefficient eorresponding to a drag coefficient of 0.02 with thick­
ness and camber for a number of NACA airfoil sections with roughened leading edges.
R. 6 X lOS.
which show relatively small increases of drag coefficient with increasing
thickness ratio and camber (Sec. 7.00).
Unfortunately airplane 'lings are not usually constructed with smooth
surfaces, and, in any case, the surfaces cannot be relied upon to stay
smooth under all service conditions. The effect of roughening the leading
edges of thick wing sections is to cause large increases of the drag coefficient
at high lift coefficients. The resulting drag coefficient may be excessive
at cruising lift coefficients for heavily loaded high-altitude airplanes. Wing
sections that have suitable characteristics when smooth but have exces­
sive drag coefficients when rough at lift coefficients corresponding to cruis­
ing or climbing conditions are called "uneonservative."
-The decision as to whether a given lying section is conservative will
depend upon the power and wing loadings of the airplane. The decision
may be affected by expected service and operating conditions. For ex­
EXPERIMENTAL CHARACTERISTICS OF U·ING SECTIONS
177
ample, the ability of a multiengine airplane to fly with one or more engines
inoperative in icing conditions or, in the case of military airplanes, after
1.1,
<,
1.2
1!C
o
/
1.0
/
...•
..
......•a
t
<~
8
/
o
•
V
o
o
II
•
~
.>
-:
~
/
..
c
~
~
~
~
1
00
() .2
A .~
v •
V
41
~
Ct
/
L
~
~
V
o
,.a
o
••
s:
o
NACA 63-
(b)
i..
ler1~ae
..
........•
~
1.2
r---
o
...•o
~~
o 1.0
;....
..
~
V
~
......
~
5
a
e
)
1
.8
~/
)
,..
~)
.J"
\
'I\,
el
t
00
00.1
00.2
.Oe2
c/
4
~
<,
V
/
t
'"
-, -,
"-----
".-..:.r
/0
!
0
//
/
~
........6
A
VO•
8
16
12
A:'rCol1 thlckne:.s, percent
(C )
FIG.
i;ACA
64-
or
ZO
2~
c.hord
se rlea.
91. (Continued)
suffering damage in combat may be a consid~ration. As an aid in judging
whether the sections are conservative, the lift coefficient corresponding to
a drag coefficient of 0.02 was determined from the figures of Appendix IV
for a number of NACA wing sections with roughened leading edges. The
variation of this lift coefficient with thickness ratio ar.d camber' is shown
THEORY OF WING SECTIONS
178
in Fig. 91 for a Reynolds number of 6 million. These data show that, in
general, the lift coefficient at which the drag coefficient is 0.02 decreases
with rearward movement of the position of minimum pressure and with
.1.2
~~
~
o
~
1.0
~~
•
...Ii
ef
..-=
.8
~7
t
Z
o
,
~
.6
I
/
..V /
~~
~
V
~l
r-, ~
~
<,
o,!.
GO
~
~O.2
AO.~
jO.
II
.c
...o
.:•
.~
~
(4)
i
MACA
65- .erie.
CJ,l
-
-00
e
0.2
/
40.4
~
/
&
::
.8
o
•
O'J
(
/
/
7
----
/
~ i':
-e1.
~~
r-, 2\
~~
\
~
0
•
n
12
1)
20
2"
Alrtol1 thickness. percent or chord
(e)
FIG.
NACA 66- aeries.
91. (Concluded)
increased thickness above thickness ratios of about 15 per cent. For wing
sections thinner than approximately 18 per cent, the effect of camber is to
increase this lift coefficient. For the thicker sections, however, increasing
the camber becomes relatively ineffective and may even be harmful in
extreme cases. The highest values of this lift coefficient for wing sections
having thickness ratios greater than 15 per cent are obtained with the
NACA 64-series sections having a design. lift coefficient of 0.4.
EXPERIltfENTAL CHARACTERISTICS OF WING SECTIONS
179
7.6. Pitching-moment Characteristics. The variation of the quarter­
chord pitching-moment coefficient at zero angle of attack with thickness
SlDa1e
•1
n.aAId
•pIbola
o
,
aN t or (,tf' dada eel .p,1It
......
aerlea
......­
....
~
ioS
:I
...""I
-.1
,.o
-
-
-
r---Ii
OOr 41&1t)
e
:.~
A J,.Ii
• 250 (5 41&1t)
..
it
-.2
t
~ .l..
i -.,
iI
~
4
(a)
tbl.....
8
AutoU
MACA
12
to---.. r---,
k
1&
20
peroent or c:bor4
tour- and tlYe-dl81t aerlea •
•1
--..
"...
i
...
.....o
t
:
o
-v
--&
:r­
-.1
V­
°ls.
~~
00
.0.2
~ ..........
-.2
r- ~ h
""c
I
..,
h
'h_
-·5
··!JO
~ Io.......~
-
~
~
~
""V­ -.::Ill
4
8
12
Alr1"o11 thiekn•••.•
16
pe~ent
\..
-,
I...-.­
f'---.
Ao.h
.0.'
~
--
20
24
or chord
(b) HAC. 63- eerlell.
92. Variation of section quarter-chord pitching-moment coefficient (measured at an
angle of attack of zero degrees) with airfoil thickness ratio for several NACA airfoil sections
of dift'erent camber. R. 6 X I ()I.
FIG.
ratio and camber is presented in Fig. 92 for a large number of NACA wing
sections," The pitching-moment coefficients of the NACA four- and five­
digit series sections become more negative with decreasing thickness ratio.
Comparison of the experimental data in Fig. 92 with the theoretical values
obtained from the theory of thin wing sections shows that the absolute
THEORY OF WING SECTIONS
180
magnitudes of the pitching-moment coefficients for the NACA four- and
five-digit series sections are somewhat less than those indicated by the
theory. The pitching-moment coefficients for the NACA 6-series wing
.1
-
.
~
......•
~
,,:..
~
.A.
.A.
..,;,.
..;.
......
- - --- -
-.1
o
I
o
...
~.~
-.2
~o~
~ '"--
B 0.1
----
.~:~-L I-&-
i -.,
.....
64-
oJ--
-
0.2
o.L.
" 0.6
H1
~ ~~
~
8
12
16
Alr1'o11 tblClmeaa. percent
lIACA
-
~
.A
'V
"V"'" ~
4J-- ~
(c)
c&l
eo
'-
20
or cborcl
aeries.
$lasle t"lasP4 sJIIIbols are £or 6fP siMulated split 1'lap
.1
1
.....j
t
..
...
0
• • o. 5
~.aa
-.
....
~
'"
---
-
.....
~~
C
~
~
0
«-:
oJ-.....
a
I -.,
,,:..
r«: ~
o~
.~
~h
--*
A..
-.1
I0 -.2
.l1De,
~
~
~
......, f'--- -<J
.....
L.
8
12
16
20
Airfoil thlckness. percent ol chord
(d)
65- aeries.
92. (Continued)
MACA
FIG.
sections show practically no variation with thickness ratio or position of
minimum pressure. As indicated by the theory of thin wing sections, in­
creasing the amount of camber causes a nearly uniform negative increase
of the pitching-moment coefficient. In general, the absolute magnitude of
the quarter-chord pitching-moment coefficients for the K AC.~ 6-series wing
sections having mean lines of the type a = 1.0 are approximately three­
EXPERIltlENTAL CHARACTERISTICS 0// WING SECTIONS
181
.1
~
0
I
..
~
c;
.:
7"
"':"
"":"
~
A.
~
h­
~~
-.1
0
::
'-c
•a
0
-.2
~R Nt...
~ ~
.pi
c:
I
:.I
-.,
~
~h
r«.. r---., .h­
-­rc;r--...
-..,
"'"
--1&.0
~
~~
4
8
12
16
20
AlrtoU thlcJ.:nes.•, percent or chorcl
(e)
HACA
FIG.
66- aeries.
92. (Concluded)
.­ •.
c:
.:
V
~
"s...
o
o
-­
V
..•
v
65,-618/
I»
­
~
•
~
o
I
.
V
c
,i
lL
-'1--66(21 «; )-216 • •
2'012
-.02
-.oL
I
j
-.06
= 0.6
-.08
....10
-.12
-.l~
-.16
1beoretlcal IDoment coefficient tor ~t.e airfoIl
. . .n liae about quarter-chord point
FIG. 93.
Comparison of theoretical and measured pitching-moment coefficients for eome
NACA airfoils. R, 6 X l()S.
182
THEORY OF WING SECTIONS
quarters of the theoretical values. The pitching-moment coefficients for
sections having mean lines of the type a < 1.0 are equal to or slightly more
negative than the theoretical values as shO\VD in Fig. 93. Consequently,
changing the type of mean line from a = 1.0 to a < 1.0 to reduce the
magnitude of the pitching-moment coefficient is relatively ineffective un­
less the value of a is reduced to a small value.
The variation of chordwise position of the aerodynamic center at a
Reynolds number of 6 million for a large number of NACA wing sections
is presented in Fig. 94. From the data presented in Appendix IV, there
appears to be no systematic variation of ehordwise position of the aero­
dynamic center with Reynolds number between 3 and 9 million. For the
N ACA 24-, 44-, and 23o-series wing sections with thickness ratios ranging
from 12 to 24 per cent, the chordwise position of the aerodynamic center is
ahead of the quarter-chord point and moves forward with increases of
thickness ratio. The data for the NACA 00- and 14-series sections show
that the aerodynamic center is at the quarter-chord point. The aero­
dynamic center is aft of the quarter-chord point for the NACA 6-series wing
sections and moves rearward with increase of thickness ratio. There ap­
pears to be little systematic variation of the chordwise position of the aero­
dynamic center with camber or position of minimum pressure for these
sections. For the thick cambered sections, the chordwise position of the
aerodynamic center appears to move forward as the design position of
minimum pressure moves back. For wing sections with arbitrary modifica­
tions of shape near the trailing edge, the trailing-edge angle appears to be
an important parameter affecting the chord,vise position of the aero­
dynamic center. For such sections, the aerodynamic center moves forward
as the trailing-edge angle is increased."
EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS
183
.26
.24
~
HACA
r---
r---.­
~~
2~- •• rl.a
"'"
~~
.22
.2&
~~~
IlACA
~ ~.)
-..... re.-.
---­ ---­
_G
e
~ a,
~
.
JIACA 14-aerl••
.
MCA OO-aerle.
.22 0
I
~
I
I
.
8
12
16
Airfoil thieme.a. percent ot
(a)
r-,
4·)
Hl.CA 24... erlea
.22
~
44-aerl••
20
ch~rd
NACA four- and five-digit series.
94. Variation of section chordwise position of the aerodynamic center "pith airfoil
thickness ratio for several NACA airfoil sections of different cambers. R, 6 X 10'.
FIG.
THEORY OF WING SEC7'IONS
184
SJIIlbo18 with f'lap corre8poDd to
~
c11
~
~
.
=0.6
~
~
~~
G
Cl.,
=0.4 and 0.6
A
.
V~
V
.-...
C11
W
~
0
~
-21,­
l~
~
=0.2
.
4~
....­
~~
.......
~
L..e
~
G
ell =0
o
8
12
16
20
Alrl'o11 thicknesa. percent ot chord
J,.
(b)
FIG.
NACA 63-series.
94. (Continued)
2JI.
EXPERIMENTAL CHARACTERISTICS OF IVING SECTIONS
.~
=0.6
Bplbo18 with tlas8 correspond to 01
1
.--t.~
o
:::i--~
•
•24"-.......­ ......-----......~-------.a..-...a.---.l~~
~ .28
c.~
.-......~ ~
.~.
01
1
~.,
--­
= 0.2
..",J0)
~ ..",-
Cl
-­
c·,.........
---­ ----­
Cl
4
8
=0.1
l
l
o ~..,­
,...".,.....
~.
-
=0
12
16
20
Alrroll th1ckness, percent or chord
(c) N ACA 6·l-serics.
FIG.
94. (Continued)
185
THEORY OF WING SECTIONS
186
8Jmbo18 with
nasa
cCll'reapord to 0Il
.28
=0.6
~ .~
··L
~ .26
-
...
•
•
01
~ .~
1
= 0.4
and
~.)
0.6
0
0
1 .28
f:.~
c
..•
.2&
...
.24
0
.28
0
.p
a.
e
e,....­
a
~
j.:L..-­
.,~
~Il
=0.2
.~
••
~
~
~
CI
~~
to­
e
.2&
~
0Il
~
w
4-)
4·)
8
=0
-
-­
.­
v
~
.
12
16
A1r.toU thlolme••• perOeDt
(d) NACA 65-series.
FIG.
•.)
....
94_ (Continued)
20
~
chord
EXPERIlYfENTAL CHARACTERISTICS OF WING SECTIONS
.28
l...--J.)
\;I
s:• .24
c, 1 = 0.4
.p
i
o
o
1.28
~
2
: .26
~
o
--.
4r
c~l
5
.....24
.p
..•
...:..
v
t:\
=0.2
8. .28
....••
t• .26
a
G
:.-.--­ ~
~
r--­
~
...;..
,...­
ct.
l
=0
4
8
12
16
20
AirtoU thickness. percent or chord
(e) NACA 56-series.
FIG.
94.
(Concluded)
187
CHAPTER 8
mGH-LIFT DEVICES
8.1. Symbols.
CD
C DOlin
CLmax
/i.C L DUlX
drag coefficient
minimum lift coefficient
maximum lift coefficient
increment of maximum lift coefficient
C N normal-force coefficient
eN/flap normal-force coefficient based on the area. of the flap
CQ volume-flow coefficient, Q/VS
E flap-chord ratio, cllc
G moment arm, in terms of the chord, of the basic normal force about the quarter­
chord point
u ~-Iach number
P Q& incremental additional load distribution associated with flap deflection
P b" incremental basic load distribution associated with flap deflection
Ps incremental load distribution associated with flap deflection
Q volume of flow through slot
R Reynolds number, pVcJp.
S wing area
V velocity of the free stream
c chord
Cd section drag coefficient
CI flap chord
CA section flap hinge-moment coefficient, h/~pytcr
Cl
section lift coefficient
e'l section flap lift coefficient, ll/>~pVtcl
Cl mas maximum section lift coefficient
~Imax increment of maximum section lift coefficient
em section pitching-moment coefficient, m/~p lltc2
Cm~/, section pitching-moment coefficient about the quarter-chord point
c.1 section pitching-moment coefficient about the quarter-chord point with flap
neutral
~ section pitching-moment coefficient about the quarter-chord point with flap
deflected
~c". incremental section pitching-moment coefficient about the quarter-chord point
Cal section normal-force coefficient with flap neutral
Cnt section normal-force coefficient with flap deflected
~a incremental additional normal-force coefficient associated with flap deflection
Ca" incremental basic normal-force coefficient associated with flap deflection
C-n/a incremental flap normal-force coefficient
At:. incremental section normal-force coefficient
(cp)J center of pressure of the load on a flap measured from the leading edge of the
flap
k section flap hinge-moment coefficient (positive in direction of positive flap
deflection)
188
HIGH-LIFT DElT/CES
l,
m
n/ a
q
r
v
~I'a
x
y
a
ao
&to
"Yael' "Yb&
8
8/ a
a/_
~cl
p
T
Tn. T..
189
lift force acting on the flap per unit span
section pitching moment
incremental flap normal force per unit span
dynamic pressure, ~p VI
leading-edge radius
thickness of wing section
local velocity over the surface of a symmetrical section at zero lift
increment of local velocity over the surface of a wing section associated with
angle of attaek
distance parallel to chord
distance normal to chord
angle of attack
section angle of attack
increment. of section angle of zero lift
ratios of the flap normal force to the section normal force for the incremental
additional and the incremental basic normal forces
flap deflection
deflection of fore flap or vane
deflection of main flap of a double-slotted flap configuration
increment of flap deflection
mass density of air
turbulence factor, effective Reynolds number /tcst Reynolds number
see Eq. (8.8)
8.2. Introduction. The auxiliary
devices discussed in this chapter are
essentially movable elements that per­
mit the pilot to change the geometry
and aerodynamic characteristics of the
wing sections to control the motion
of the airplane or to improve the per­
formance in some desired -manner.
The desire to imp ove performance
by increasing the wing loading while
maintaining acceptable landing and
take-off speeds led to the developmen t
of retrae able devices to improve the
maximum lift coefficients of wings
without changing the characteristics
for the cruising and high-speed flight
conditions. Some typical high-lift de­
vices are illustrated in Fig. 95. The
aerodynamic characteristics of some
typical high-lift devices are presented
and discussed in the following sec­
tions. Primary emphasis is given to
the capabilities and relative merits
SPlIT FLAP
C----~:::--=­
~
EXTERNAl AIRFOIL RAP
c:
~
DOUBLE· SLOTTED FlAP"
/
~-----:::-::--==LEADING EDGE SlAT
FIG.
95. Typical high-liit devices.
THEORY OF WING SECTIONS
190
of the various devices. Numerous references to the literature are given to
provide design information.
•8
I
I
I
I
I
.,
~
/~
1<aJ
./
4
..
CD
to
>
.,.f
~
0
CD
Cf-t
ft-t
CD
D.
as
.-l
fH
~
/
V
./
~,
.2
CD
0
~
/
.~
s::CD
.L
~"
ri.7
,-..
~
10-'"
r\
I
BxperlaenU.l
~ ~
~
C
~!J
j~
/rfJ
!/
r
0
(a)
.8
range f'rom 0° to 10°.
I I I I I I
.,
4fI#'
~~
-:lheoretlcal
l
~",
-'~
AI ~ ~
,/
/~
.4
IAl
,
"L
I
I
y
~
'Iii'
\
I
BxperSmental -
-
~
_l/~
II
.2
-
~~
itf
/
til
~ ~,
.JIll'
...,~
",~
~
0
0
CD
,.
~
....... ~
"'" V
/- ~
s::
......,
"'~
."
",.
r-,
,-~
dOloca
I
'1heoret1cal
/~
~~
,
J~
~
o 0V
.1
.2
.,
.0
•
flap-chord ratio, E
(b)
6
range from 0 0 to 200
•
Flo.96. Variation of section flap effectiveness with flap chord ratio for true-airfoll-eontour
flaps without exposed overhang balance on a number of airfoil sections; gaps sealed; Cz -= o.
8.3. Plain Flaps. Plain trailing-edge flaps are formed by hinging the
rearmost part of-the wing section about a point within the contour. Down­
ward deflections of the trailing edge are called "positive-flap deflections."
Deflection of a plain flap with no gap effectively changes the camber of the
wing section, and some of the resulting changes of the aerodynamic ehsrae­
Sym­
hoi
-
0
+­
X
0
-
o
-­
V
Air-flow characteristics
Type of flap
Basic airfoil
NACA 0009
NACA0015
NACA 23012
NACA
66(2x15)-009
NACA
66-009
NACA-
-­
63,4-4(17.8) approx,
~
66 (2xl5)-216,
a = 0.6
-­
<1
E>
-­
~
-­
b.
-­
~
17
$)
-­
JJ
-­
-0
-­
b
S>
-­
~
-­
<f
Q
M
R
Plain
Plain
Plain
1.93
1.93
1.60
0.08
0.10
0.11
1.4 X lOS
2.2 X lOS
Plain, straight contour
1.93
0.10
1.4 X 10'
Plain
1.93
0.11
1.4 X 10'
Internally balanced
Approach­
ing 1.00
0.17
2.5 X 10'
Internally balanced
Approach­
ing 1.00
0.18
5.3 X 10'
Intemally balanced
Approach­
ing 1.00
0.14
6.0 X 10'
NACA
NACA
Piain
64,2-(1.4) (13.5)
NACA
65,2-318 approx,
NACA
63(420)-521 approx,
NACA
66(215)-216
a ,. 0.6
NACA
66(215)-014
NACA
66(215)-216
a == 0.6
NACA
NACA
8.0 X IO'
Approach­
ing 1.00
0.20
to
0.48
2.8 X lOS
Plain
1.93
0.09
1.2 X lOS
Plain
Approach­
ing 1.00
.....
6.0 X 10'
0.13
6.0 X 10'
0.13
6.0 X 10'
0.13
6..0 X 1()6
0.14
8.0 X lOG
0.13
6.0 X 106
0.13
6.0 X 106
Internally balanced
Internally balanced
NAC.A
Internally balanced
Internally balanced
KAC4~
64,3-1 (15.5) approx.
Approach­
ing 1.00
Approach­
ing 1.00
....
Internally balanced
12)-213
745A317 approx.
NACA
~3-013 approx.
.... 13.0 X loa
mg 1.00
6..0 X 10'
Plain
NACA
Approach­
....
0.14
Plain
65.-421
65 (
Internally balanced
Plain
65r415
NACA
65r418
-
.,.
NACA
66(2x15)-116,
a == 0.6
-
Intemally balanced
Approach­
ing 1.00
Approach­
ing 1.00
Approach­
ing 1.00
Approach­
ing 1.00
Approach­
ing 1.00
Approach­
ing 1.00
A:pproach109
(c) Supplementary information.
Fla. 96.
(CQTJditded)
191
1.00
to
6.8 X 106
I 0.13
I
6.0 X lQ6
•
THEORY OF WING SECTIONS
192
teristics may be calculated from the theory of thin wing sections (Chap. 4)
if the flow does not separate from the surface. For most commonly used
wing sections, this condition is satisfied reasonably well for flap deflections
of not over 10 to 15 degrees. The theory permits calculation of the angle
of zero lift, the pitching-moment coefficient, and the ehordwise load dis­
tribution with reasonable accuracy. The flap loads and hinge moments
may also be obtained, but the accuracy of these quantities is relatively poor
because the effects of viscosity are particularly pronounced over the aft
portion of the section. The effectiveness of the flap in increasing the
to
maximum lift coefficient cannot be
(6 in rOdor/s)
calculated.
~'Ch I
Glauert" calculated the effect of
.8
do
plain flaps in changing the angle of
zero lift, the pitching moment, and
dcm e
/ ' ~ <,
the flap hinge moments. The cal­
~_~4
.6
If
d4
r-,
I
culated effect of flap deflection on
the angle of zero lift is shown in Fig.
'~
"",
dch _
96, where the flap effectivenessJ
l1aol
alJ is plotted against the ratio
I
~ -, dcz~
.2
of
the
flap chord to the section chord
~~
~ -,
L.-­ ~
cllc.
Numerous
experimental points
~
r-, are also shown for flap deflections of
6
.2
.4 c·
.8
I.IJ 0 to 10 degrees and 0 to 20 degrees.
N -Z-
"
/
I
-:
'" I'< -,
-,
~,.....
In general, the flap effectiveness is
less than that indicated by the the­
ory, and the discrepancy increases
with increased flap deflection for small chord flaps. The calculated ef­
fectiveness of a plain flap in changing the section pitching-moment coeffi­
cient about the quarter-chord point dc./dO is shown in Fig. 97. The
calculated values of the rate of change of hinge moment with lift coeffi­
cient dch/dc, and the rate of change of hinge moment with flap deflection
dcA/tU are also shown in Fig. 97. For symmetrical wing sections, the theo­
retical pitching-moment and hinge-moment coefficients are
97. Theoretical hinge moment and
pitching moment characteristics of plain
trailing-edge flaps.
FIG.
em =
a(~)
_ (den) + ~ (dC\)
lU
en. - C, -
0­
dCI
The linearity of the theory of thin wing sections permits the applica­
tion of the values obtained from Figs. 96 and 97 to cambered as well as
Symmetrical sections. In the case of cambered sections, these values are
applied as increments to the corresponding values for the unflapped
section.
HIGH-LIFT DEVICES
193
Pinkerton" calculated the flap loads that are presented in Fig. 98. The
rate of change of the flap lift coefficient with the section lift coefficient
dell/dc l and with flap deflection dc,l/dJ are plotted against the ratio of flap
chord to section chord.
3.0
2.8
1
I
W/Ild
dir«:lion
.....
L.I
1
~''''''''''':
I
\oc
2.6
I
2.4
2.2
ik
CII:;;;'
r\..
~
2.0
"
1.8
~~
~I~
a
"c, +::;"6
de:
I\.
1.6
1.4
'\.
1.2
"r\.-, !f:JL
1,,<6
1.0
~
.6
.4
.2
)
~
ooV ./
~
~
~
~
~
/
V
~ t<~
r\
.2 .3
.4
.5 .6
.7
.8
£, Ratio of flap chord 10 totatchord
l_
'\~
-,
.9
1.0
FlO. 98.
Theoretical variation of flap lift coefficient with section lift coefficient and 1lap
deflection.
The pressure distribution over a lying section with a deflected plain flap
may be calculated by the "ring-section theory of Chap. 3, but such calcula­
tions for a number of flap deflections would be unduly laborious. Some
knowledge of the ehordwise load distribution is given by the theory of thin
lying sections. The change of load distribution caused by flap deflection
can be resolved into t\VO components, one of which is simply the additional
load distribution associated with angle of attack. The other component
is the difference bet,veen the load distribution on the original and deflected
mean lines at their respective ideal angles of attack, These load distribu­
THEORY OF WING SECTIONS
194
tions may be calculated by the theory of thin wing sections (Chap. 4) as
explained by Allen,' but the results are not very accurate. For example,
the simple theory predicts infinite load concentrations at the leading edge
and hinge location. The infinite load concentration at the leading edge
may be avoided by use of the additional types of load distribution associ­
ated with angle of attack as obtained from thick-wing-section theory and
presented in Appendix I. Allen' obtained empirical load distributions
analogous to those associated with changes of the camber that permit
calculation of reasonable approximations to the load distributions of flapped
6
- - - R.N. .356 x 10
- - R.N.6.10 x/OS
-3
6=50­
8·:40·
8=30 0
+/
0
4=20
. . 4=/0 0
ConsIont angle of o~/ack
20%cflop. art"
FIG.
99.. Distribution of pressure over a wing section with a plain flap.
wing sections. Allen's method is presented in Sec. 8..8. Once the load
distribution is obtained, the pressure distribution may be calculated by the
method of Sec.. 4.5. Some typical pressure distributions for sections with
plain flapsfl are presented in Fig. 99.
The lift, drag, and moment characteristics of a typical NACA 6-series
wing section with a O.2Oc plain ft ap3 are presented in Fig. 100 for several
flap deflections up to 65 degrees. These and other data" show that the
angle of maximum lift coefficient with the flap deflected is generally some­
what less than for the plain wing section. Curves of the maximum lift
coefficient plotted against flap deflection for two sections" are presented
in Fig. 101. Plain flaps of 0.2Oc appear to be capable of producing incre­
ments of the section maximum lift coefficient ranging up to about 1.0 and
are more effective when applied to sections with small amounts of camber.
Addit~onal data on plain flaps are presented byJacobs," Abbott," and
Wenzmger. l 48, 1., 116
HIGH-LIFT DEVICES
:~::
~
e ••
.... ... ..
i!::
I~:: :i:·
i~"
. . .•.
:::: :::i
;.. *:=! :i::
;.!:....
:E: :r-: !::: .•.
195
e ••••••
;!~:
eo.
:e:: :::: ;~!:
~~g
. 'mm i ! :::~
m= ~j:~ m:
.., . .
.
2.8
i::;
;:
.... .. ... .... :~~
2.4
':::
••
:
_
• • e o••
....... _
2.0
':i:
..
• •••
e
..
.
~:.:..
.... m~
:
...•.­
.....
i;;..
....
.
:
..
'
.
:­
~~:fiI;1.'
J~~'J
......
;i:;
.
.
~:~:
~:: •
, ... ..
••.••••
.
.):
.;:4 .. (:
: :.: : : :. : :.: . .: . :i:~: ~ I~ ~~ .~ ~
'.e
:1::..
'::Jf:j
..
::::
.
..
..
....
f: I(.
~!!'JYI\"
,)
:~' •.. ~J~j,~
(iii
..:
..
:.
i.~ ~j i~ ~ilih~~,
:/·r. ~J 'i: II."'·· •
Ji ".J:l.!;i!.i.i
:;:~..
~:
" ••,
i· ~ii:·
~
'e'
'"
:;::::
. , ~.::. ':. :.: :' ;~~~ ~~; ~p~ ~~: (. . :::~~'.:'
lJ.i!.llI~
1 i /1: ­
~ -:
~J~' ).J:::
":.
... ...
.. ~:~:'1.
0.4
o
-0.4
I~
I
- " ..
0
I
::iIJ~;:·~j:tfaj~.-j..jl
',oi: II. 0."/ F' r~1 r t.
~ Jilt:· t-.I P J -; I
-. ;.. , - .~ .J:/ 1/' I ( ~ "
.....
: If· (;~ ~/t: I Ifill "
0'0 ._;
..
..... .~;
....
::
.
..
: ..
;;!~ ;=:
.. : ..•~:
.'-.'
~: ~.~. ~~:~
•.
...
-1.2
0
J'
(4es)
:':-::<::"~_:::~~;;~:;~;~:J':~~~~.g fi '.: ':'.
.. ~:
-0.8
If.
~-
..
-24
•.
.~ .:.
:·f
:" ..
~.• ~
.
.:. (.
~."
... ...
. ~.
.
~~
=r~
-
....
..16
;;:
-8
,
1+_~-+-+-tf-o+""'+-~
~
I
I
6
(4es)
...
.... :
--
& -i~
i"'~ ~ -15
:i
;.-I: :
..
:":.
;&;:
~.
•. ..
0
'.
~"l'
L
'~'I'l;
8
I
16
I .. :
24
32
Section angle of tJtt4ck. «0' U6
100. Aerodynamic characteristics of the NACA 66(215)-216 airfoil section with O.20c
sealed plain flap.
FlU.
THEORY OF WING SECTIONS
196
....;g;
:ii
:
.
~~:: " .
"
0.028 ............ :::!................•............................. : .... ...
!
!
...
·1 I I
t
........
i
~
..
~-'
0.024
!.
~
i
I
O.ORO
I
1
.$1 0.016
t
l
0.012
j
!
'" :::...
:::.::: :::.::
... :=::..
.,.
....
l;"
=, "fI :~ :/i/, I
:
i
~
0.008
0.004
.
0
~-O.2
Co»
if
.!
.!l -0.4
~
§
1
::~1,15~:~;~f:j;:~: ;~~~ . . :' ::: ~.~~ r:~ ~~\
r
.. ..
.. ::: ~:
:
-0.6 ........................:
~
.. r:'
,
~
:.' ..
.
_m:
t~
:'
:
._ . ..
;~
:..
I
I
!
i
I
:
: ..
., ..:
i
i
-0.8
-0.8
-0.4
o
FIG..
0.4
0.8
1.2
Section li/t coefficient, c,
100. (Concluded)
1.6
2.0
2.4
HIGH-LIFT DE1'ICEB
197
When the turbulent boundary layer over the aft portion of the wing
section is thin and resistant to separation (as when extensive laminar flow
is obtained), small deflections of a plain flap do not cause separation but
shift the range of lift coefficients for which low drag is obtained.' If the
extent of laminar flow is not large or if the flap deflection is sufficiently
2.8
J3
2.~
DCA
~
..It
..
2.0
~
s:
~
c...
~
1.6
3
0
~
~
c
1.2
0
...
V
J~
~
~~
'-4
......
~ DCA 65,3-618 ~.
0
0
66(215)-216 ~ ~~~
....... >
; JI
e-Vc?";S
-t
..g•
0
.!S
••
~
=­
.k
o
~
~
~
~
~
Plap detlection. 6 . des
FIG. 101. Aiaxirnuill lift coefficients for the NACA 65,3-618 and NACA
foils fitted with O.20..airfoil-chord plain fiaps. R. 6 X 1(;'.
.
()6(215)-~16
air­
large to be of interest as a high-lift device, the flow over the flap separates
and large drag increments result. .A. typical variation of drag with lift
coefficient for the NAC..\ 23012 wing section with a O.20c plain flap de­
flected the optimum amount at each lift coefficient/ is shown in Fig. 102.
8.4. Split Flaps. The split flap is one of the simplest of the high-lift
devices. The usual split flap is formed by deflecting the aft portion of the
lower surface about a hinge point on the surface at the forward edge of the
deflected portion (see Fig. 95). One variation of the simple split flap has
the hinge point located forward of the deflected portion of the surface in
THEORY OF WING SECTIONS
198
such a manner as to leave a gap between the deflected flap and the wing
surface. In another variation the leading edge of the flap is moved aft as
the flap is deflected, either with or without a gap between the flap and
the wing surface. Split flaps derive their effectiveness from the large
increase of camber produced and, in the case of some of the variations,
from the effective increase of wing area.
,
I
I
II
,
J
I
I
l
J
~
I
I
IPlDfnf_~
I"
I~~ I
I
,,~~~
.04
~
~
~
~~ ~
.-.~ ~
'I
1/
r-,
~
... ~
I
J
~
'Y~
t>
U
~
~
~
StJditJn /lfl Df)tI/Iit:itlnI,e"
FIG. 102.
Comparison of profile-drag envelope polars for the NACA 23012 airfoil with
O.2Oc plain and split Saps.
A
~
The lift and moment characteristicsl for typical NACA 6-series wing
sections with O.2Oc split flaps deflected 0, 40, 50, 60, and 70 degrees are pre­
sented in Figs. 103 and 104. Similar data for a flap deflection of 60 degrees
are presented for most of the wing sections of Appendix IV. The lift-curve
slope with split flaps is higher and the angle of maximum. lift is somewhat
lower than for the plain section. There appears to be a tendency for the
lift-curve slope for large flap deflections to be less than that for moderate
deflections.v 154
Inspection of Figs. 103 and 104 shows that the split flap is more effective
in increasing the maximum lift coefficient when applied to the thick than
when applied to the thin "ring sections. Figure 58 shows that the maximum
lift coefficients of NACA 6-series wing sections with O.20c split flaps de­
flected 60 degrees increase with thickness ratio up to ratios of at least
18 per cent. Figure 58 shows comparatively little variation of maximum
lift coefficient for NACA four-digit wing sections with O.2Oc split flaps for
thickness ratios greater than 12 per cent, but the maximum lift coefficient
HIGH-LIFT DEVICES
199
•.••• "I~r-
2.4
+_+,-,=
............. ~i)1
:.~ • . .
AV-Vi
e
J.
, 'J
.~~
_.
r/
: \
~
1
}. 'II
II 1,
,
r,,'/
/
"I 0.8 ~-;.-...I--+-f--l~oIltH-II-~-+-I-""""'"
r. rl.
.. ..J"';
i
.1
~
0.4
I/r~/
:~~:.
Ifl/.:
... ~ - ..'­
..
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-
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"'- ....
(deS)",-":';':
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;..
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.. J.. ..• ' .., .. ._ . - ... :;
~"--r-~..;""",,,-~~-+--&o-~~~~~~
'il
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.J
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,J/J.l
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&dioII G1I6k of atIGei, ...,. . .
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.. .i
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;
.....-t.. .T-lt-+-.......-+-I.....:..r.......
Cj
:I
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:
=-­
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&i8~+t+~RmP+4++++tt~~:++:t=d=~~
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.'
A
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:
I -O.8ltttJet!:tt!lt:tJjtt:1:ttttrnm~!eI!~~~tt1
~
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- ....
:
I-'i~~-
·i:··· ~:!:
..•
: •.... l'.:..
0
.- . . • . • • • . ··1·.;..
-0.4 ...... I:... , ...
-0.8 -0.4
: ': ... ::'-
o
'"
'"
...
0.4
:j'
i
--:;!
_.'
0.8
i
t
~
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eo
,-;
.T.! IJ·
.. II ~.
l~
:.'
1.6
Set:tita lift t:«/fit:iDII. c,
FIo. 103. Lift and moment characteristics of the NACA 66(215)-216 airfoil
O.2Oc split flap. R. 6 X lOS.
eectiOD
with
THEORY OF WING SECTIONS
200
o
e
6
V
~/:I
~-t-+--+-+~~ ~ ~~;"""'-'-J-""""""""""~-+-.f--I.--I-~-i
(
)
tIl}
iJ
I
i'
i7t
--.'f "I
j
-
~-:-.
i .. r·· _...... I ..
~
-16
- 8
0
8
~4
16
Section ",.". of atled. «0. . .
o
· -j
:
joii""
1 •
r­
:
:...!
tin
· .iV-. i\
,,~. N\.
....
..
-
&ft
-,
I
~
!
·
-
r-
-0.4
-0.8
~
:
!
-
..
l
-0.4
1---_
I
.
..
-~.~
-
0
-­ -- ...,.
~-
-­
:
--­
.O~
.
0.4
.
.
~_.
_.
0.8
~-
I­ -
~.-
..
Q'
.,
1.2
8edioIIli/t eoeffidat.
c,
t,
;
~o-
1.6
­
.. _.
- -_.
....
-:::.
-u
-. ._­
-
1
~.~ ~
_.
_0.
-
··1
.­ -­
~ ~
.~
1
.-.
-~ ~
'I
I
1-.
~.
,
~ ~..J
!
.­
~-.:/
,. ~ - -
- .~ ~:..
:
.....
1-­
_.­
i
-2.0
2.4
104. Lilt and moment eharacteristiea of the NACA 651-212 airfoil eectiOD with O.2Oc
split flap. R, 6 X 10'.
FIG.
HIGH-LIFT DEli''lCE8
~1
decreases for smaller thickness ratios. These results are in essential agree­
ment with those1M of Fig. 105, which shows that, for the NACA 23o-series
sections, the maximum lift coefficient generally increases with thickness
ratio for large-chord split flaps but decreases for small-chord flaps.
3.2
I
.
ef·
~
2.8
[~
2.4
~
r---­
---
~
::z­
•4,0c
....., .- .30c
~
~
-
.""..... ~
s:I
~
~
...... 2.0
j
I
r----c ~ .IOc
C­
~ 1.6
~
I
- .2Oc
i
1
I
.....
...-.. .............
1.2
.~
~
----
-...........­
~
~,
~ '-..
~PIoin airfoil
~
i
..
.8
a
12
16
20
24
28
AirIo1II1icknt1ss,prctJnI e (NACA. 230 u,itJs)
32
FIG. 105. Effect of airfoil thiCknC88on maximum lift coefficient of N AC~\ 2.1O-8eriea airfoils
1\ith and without split flaps.
The variation of tile increment of maximum lift coefficient resulting
from deflection of split flaps of to, 20, 30, and 40 per cent chord 1i4 is shown
in Fig. 106. For wing sections of normal thickness ratios, substantially
full benefit is obtained for flap deflections of 60 or 70 degrees. The effect
of the chord. of split flaps on the increment of maximum lift coefficientl M is
shown in Fig. 107. For wings of normal thickness ratio, comparatively
little benefit is obtained by increasing the flap chord to more than 20 or
25 per cent of the section chord, although large flap chords are more effec­
tive on thick than on thin sections.
Deftection of a split flap produces a bluff body, and large drag incre-
THEORY OF WING SECTIONS
202
ments are to be expected. Figure 102 shows that the drag of the NACA
23012 section with a 0.2Oc split flap deflected the optimum amount at each
lift coefficient is about the same or less than that for a plain flap.!
The pressure distributions over wing sections with highly deflected split
ftaps are similar to those for plain flaps (Fig. 99). The pressure distribu­
tions with deflected split flaps may be predicted by the semiempirical
method of Allen.'
v
j//
/1/
....,->-­
7/AI
1//
lb
20
40
60
80
(a)-NACA23012 ARFOIL
FIG.
020406080
Flog delleclion,66 deg.
(b)-MACA 23021 AIRFOIL
r
o
.20
40
60
80
sx
(c)-NACA 23030AIRFOIL
106. Effect of split-flap deflection on increment of maximum lift coefficient for various
airfoils and flaps.
Data required for the structural design of split flaps are given by
Wenzinger and Rogallo.l11 The normal force coefficients and centers of
pressure for a 0.2Oc split flap at various deflections are shown in Fig. 108.
The data of Wenzinger and Rogallo show that the chordwise position of
the forward edge of the flap has only a small effect on the flap loads.
The effect of a gap between the wing surface and a split flap with a
nominal chord of 0.2Oc hinged at O.SOc from the leading edge of the sec­
tionllO is shown in Fig. 109. The data show that the lCBJ of maximum lift
coefficient associated with the gap is considerably greater than that caused
by removing the same area from the trailing edge of the flap.
Figure 110 summarizes the effect 1," on maximum lift coefficient of
moving a O.2Oc split flap toward the trailing edge from its normal position.
These data show that the percentage increment of maximum lift coefficient
obtained by moving the split flap to the rear is about the same as the
percentage increase of wing area measured as projected onto the original
HIGH-LIFT DEVICE8
203
chord line. Similar results have been obtained!" for O.3Oc and O.4Oc split
flaps.
8.6. Slotted Flaps. G. Description oj Blotted Flaps. Slotted flaps pro­
vide one or more slots between the main portion of the wing section and the
2.0
I
6: 75·
fJO°
1.8
1.6
9fJ.)r
N.AC.A.
-
23OJ()
2102/
V
/
~
~ ~.I
~
75-
/SO·
I
/ /
g5*'
/
V
V
~
1/
/785·7
7/
VI V
J,~ I BO
D17
[,.......- ..-75-
60·
55P-23012
c
u~
.4
r/
I
.2
a
20
30
40
50
Flap chotrJ,pen:enI C
Flo. 107. Effect of chord of split flap on inerement of maximum lilt coefficient for three airfoil
thicknea8eL
deflected flap; and they derive their effectiveness from increasing the
camber and, in some cases, from increasing the effective chord of the sec­
tion. The slots duct high-energy air from the lower surface to the upper
surface and direct this air in such a manner as to delay separation of the
flow over the flap by providing bdundary-layer control.
The numerous types of slotted flap are classified by their -geometry.
Several types are shown in Fig. 111. The primary classification is the
number of slots. The single-slotted flap is the simplest and most generally
THEORY OF WING SECTIONS
204
used type. Double-slotted flaps have been used to some extent, and
multiple-slotted or venetian-blind flaps have been investigated. An im­
portant consideration in the design of slotted flaps, especially of the single-
r-
i
s:
~I
c
~
.20~
.SOc
Hnge
axis
l8­
;:::
t
20
c
-
40 ~-
~
,no.....
-----...~
i
6.
41\
.............. 60
Ci
~
deg.
0
20
A
40
60
a
..
~ I-""""
1.20
0-""'"'"
~.80
",.,-
~
.-a-~
"......- ~
......
~
- --
:I
r-a
\S
----
~
c:,.,...--
.40
-----
ro--..-~
,-
0
o
40
.80
1.20
eN
1.60
~
\
LOO
FIo.108. Normal force coefficients and centers of pressure of a O.2Oc split flap at O.8Oc on a
NACA 2212 wing.
slotted type, is the extent to which the flap moves aft as it is deflected.
The movement of the flap may vary from a simple rotation about a fixed
point to a combined rotation and translation that moves the leading edge
of the flap to the vicinity of the normal trailing-edge position. Rearward
movement of the flap requires an extension of the upper surface over some
or all of the flap in the retracted position. This extension of the upper
surface serves to direct the flow of air through the slot in the proper diree­
HIGH-LIFT DEVICES
205
tion and is called the "lip." The external-airfoil Hap (Fig. Ill) may be
considered as a special case of the single-slotted flap with the distinguishing
feature that it does not retract within the section.
The flow about a wing section with a deflected slotted flap is very
complicated, and no adequate theory has been developed to predict the
aerodynamic characteristics. Consequently the information required for
(0 )
,...---------c=IO.00"---------"'=---~
(b)
(e)
Y.IOC
60°
0.20';(
\:~
FIG.
109. (4) Details of split flaps with gaps tested on Clark Y wing.
design is obtained entirely by empirical methods. Although many experi­
mental data have been accumulated, the large number of configurations
possible and the sensitivity of the chacteristies to small changes in the slot
configuration make the design problem a difficult one.
b. Single-sloUed Flaps. One important parameter in the design of single­
slotted flaps is the chordwise position of the lip. Although completely
comparable data are not available for configurations with varied positions
of the lip, the maximum lift coefficients appear to increase as the lip
position approaches the trailing edge for lying sections of moderate thick­
ness. This effect19• 153 is shown in Fig. 112, where increments of maximum
lift coefficient are presented for the NAC£.\. 23012 section with single­
slotted flaps having lips located at O.82ic, O.9OOc, and 1.000c. The con­
THEORY OF WING SECTIONS
figuration with the lip located at 0.827c is not exactly comparable with the
others because the chord of this flap is 0.2566c as compared with 0.3Ocfor
the others. It is apparent, nevertheless, that the increment of the max­
imum lift coefficient is considerably higher for the configuration having the
lip at the normal trailing-edge position than for those with a more forward
lip location. It is uncertain whether any of the difference between the
2.4
I
I
I
I
I
I
~No gqJ(remating flt:p Itdlhg edge)
I
A",
~
2.0
.....
.........
---- ----.....
~
''x
8 . . . ,...
I
<-s­
1.6
>--
~
r-,
'''" r..,.,
I"-
Witlrgrp_V
(temrNi7g flop
Ieodilg edge)
C
1'.-
~
,,~
i'o"' .....
~
..........
to--.............
1.2
..~4i
AtJfJop/
.8
--~ "---
~--::"'~
r-- ::::::-::- t::-o--. .......
.~I--":::
oo
I
~
~
I
~
~
w
Reduclial ofchotrJ,percenl flcp chad
FIo. 109. (Conduded) (b) Effect on CLmas and on CD at CLmaz of reducing the chord of a
0.2Oc aplit flap. a.. 60 dep'8e8.
other two configurations should be attributed to the difference in lip posi­
tion. Envelope polars for these flaps are presented- in Fig. 113.
When the lip is located at or near the normal trailing-edge position,
the thickness of the flap is necessarily less than that for a more forward
location of the lip20 (Fig. 114a). In the case of thin wing sections, especially
of the NACA 6-series type, the flap thickness may become too small with a
rearward location of the lip to permit favorable slot configurations. Under
such conditions, the favorable effect on the maximum lift coefficient of
moving the lip toward the normal trailing-edge position may not be
realized. CahillJO shows (Fig.. 114b) that the maximum lift coefficients for
HIGH-LIFT DEVICES
207
c
FIo. 110. Contours of
eL..... for various positions of trailing edge of 20 per cent flap.
c::::
~
SlDTTED FLAPWITH LONG LIP
c
~-----
c
~
Ell1£RNAL-AIAfllIL FLAP
,
1"'10. Ill. Several types of slotted flaps.
~
1.8
~~~
/
1.6
7
00
\
",
II"
/
1.4
I
~
V·"
1"2
,",I
~ 1.0
·~1
~ .6
~/
/
V
.2
,I
~
I
,,
.... ~
~
,,"
......
1-0-­
---'"
-­ -­
,
I
~
V~~ /
"'<
C
I
~
~
~
~
I(
~
"
~
~
~.
J4
/
I
I
j
/
7
V
)
"","--.
~
~
/
I
~
/"
,I "
j
J
I
V I
i
.4
,"l
J7
11
.8
i
~
~,
,I
~
O.30c FOWLER FLAP;GAP-O.0I5c
.\
I
-s
---­ Q2!J66c..sldled1/q'J 2-h (ref.153) 1 0 ­
_.- Q3Oc..... Fowler l/(pj gcrJ·01Jl5c
- ­ Q3Oc••... slolled1/q)"!11 ex/ended I - ­
IlPIgq:J-O.02c
~
»
~
~
Flopdeflection, 4, deg.
Flo. 112. Comparison of increments of seetlon maximum lilt cneffi~ieut for three flaps on a NACA 23012 airfoil.
~
C
~
HIGH-LIFT DEl"ICEB
209
the NACA 65-210 section are essentially the same for lip positions of O.84c,
0.9Oc, and 0.975c. The structural difficulties presented by a long thin lip
extension and the mechanism necessary for the corresponding large rear­
ward movement of the flap are such as to discourage the use of rearward
locations of the lip unless such configurations result in substantial improve­
ment of the maximum lift coefficient.
:>
Q30c sIoIfed llop with extended lip; -w---+--+-t--+---+----f
gop=O.02c
.20~
~
--
_
----
\
asoc Fowler Ilop;gop'O.o15c
o.2566c s/olted flop 2-h
(reference J53)
I
Plain airfoil
'V I
I
t
I
I
.4
.8
1.2
1.6
2.0
2.4
2.8
3.2
$«lion liftcoefIJc/enl1cz
FIG.
113.
Envelope polar curves for three slotted flaps on a NACA 23012 wing section,
The effect of flap chord on the increment of maximum lift~· U3 is in­
dicated by Fig. 115. The data presented .for the O.. 25(j(ic and O.4Oc flaps
are reasonably comparable in that the shapes of the slots are generally
similar. Figure 115 shows that larger increments of maximum lift coeffi­
cient are obtained with the larger chord flap, but the increased effectiveness
is small compared with the increase of flap chord, The slightly higher
maximum lift coefficients obtainable with large chord flaps do not appear
to justify the structural difficulties encountered with such flaps, and flap
chords in excess of O.25c to 0.3Oc are seldom used.
The maximum lift coefficients obtaineiP" 96. 1&3 with various arrange­
ments of slotted 8ap on NACA 23012, 23021, and 23030 wing sections are
plotted in Fig. 116. The flap chord was 25.60 per cent of the section chord
in all cases. These data indicate little variation of the maximum lift coeffi­
cient with thickness ratio from 12 to 30 per cent for this type of wing
section. A few data3, . are also shown in Fig. 116 for comparable slotted
flaps on NACA 6-series wing sections, In this case, the flap chords are 25
or 30 per cent of the section chord. These limited data indicate that, for
the NAC.A. 6-8eries sections, the maximum lift coefficients obtainable with
~
---O.84c
~
o""'"
2.,8
2.,4
Slotled flap I
-
~ :;;;...00"""
~~~
-~ ~
J-~~
(.)
~
I '.6
- - O.90c
~
~
:f
§
s
.6 1.2
5/011«1 flop 2
-~
~
>---- l---'
J
.4
~
gj
es
c1
(degJ
~
o Ph/no/rfoI1.
---~
e SbHfd fk¥J 1_ _ 45
A
'1
~
SIdled/Iff' 2 _ _ 41.3
SIdled IlqJ 3._ _ 35
I
0
-
Sio/led flop 3
FlO.
~
~
i
.8
(0) CONFIGURATICWS
"'<
.--
-
..
_.-:
-
Reynolds trr!1Iw, R
(b)
MAXI~
LFT OlTA
114. Variation of maximum eect.ion lift cooflicjent. with .RM)Onoldi' number Cor several slotted flaps on the NACA
66-210 whig 8(wtion.
HIGH-LIFT DEVICBS
211
1.4
I'
c
/
D.2566cflap
V
f
/
I
,
...........
.-'
-,'
-
-­ -­
~-
-­
~'
~o.4Oc flap
i.
i
I
I ,,
I
c-------­
I
./
'I
l
"" O.2566c flop
I
I'J "
l'
If
00
(0) CONFIGURATIONS
D
20
.JO
40
Flop *,~,dtJg.
!JO
60
(b) INCREMENTS OFMAXIMUM UFT COEFFICIENT
FIG. 115. Effect of flap chord on increments of section maximum lift coefficient for the N ACA
23012 wing section.
3.2
I
+
--- ---..
§
1
.............,
4
E
.~
2.0
~
.~
~
1.6
-
Symbol Kfng section series
0
x
...
~
I
1.2
6
10
,
N.A&.A.230N.AC.A. 6 N.A.CA 6 -
14
Flop chord ratio
O.2566c
--
o.ese
1 i
,
B..
22
O.JOc
1
--
I
26
30
section Ihickness rolio, Pc, percent
FIG.
116. Maximum lift coefIiciente for various arrangements of slotted flaps.
212
THEORY OF WING SECTIONS
slotted flaps on 10 per cent thick sections are appreciably less than those
obtainable with thicker sections.
The effects on the maximum lift coefficient of some variations of shape
of the slot are illustrated'" in Fig. 117. In configurations l-a and l-e, the
slot is only slowly converging, if at all, at the end of the lip, and these COD­
[
Skilled flop 1-0
8,640~.
62.~6
Czm..
[
SIoIIedflop I-b
6,6SOdtlg.
clmax.· 2.76
l
SJolltJd flop t-c
6,.:55 de9.
"1mcr.'X. ·2.75
[
SIrJIIedfkJp '-ee
6i 645 t!tJ9.
'f"...•2.il9
l
SIdled f/cfJ e-«
6,.:50~.
'ir..
·2.81
l
SbIt«J fltJp2- i
4,·60dtlg.
Czm... • 2.674
[
SItJIIt1d fltJp 3-1
t1,.50~
'"'-w. .:2
I
SItJIItId fltJp 3-g
6,.55~
c'--.·
.60
FIG. 117. l\laximum lift coefficients attainable with various ammgements of slotted flaps
on the NACA 23012 wing section.
figurations have the lowest maximum lift coefficients of the I-series
configurations. A short extension of the lip as in configuration I-b, which
makes the slot definitely convergent and directs the air downward toward
the flap surface, is effective in increasing the maximum lift coefficient. It
may be concluded from these and other data that the slot should be
definitely convergent in the vicinity of the lip and shaped to direct the air
downward toward the flap. The effects of changing the radius of curvature
at the entry to the slot from the lower surface arc shown by configurations
I-b and l-c of Fig. 117 for a flap having a comparatively small rearward
displacement when deflected. Decreasing the radius of curvature from
about 0.0& to O.04c did not produce a significant difference in the maximum
HIGH-LIFT DEV"ICES
213
lift coefficient. Other data indicate that this radius of curvature is of little
importance when the flap is displaced rearward enough to produce 8. large
area for the entry of air into the slot.
It is difficult to draw general conclusions about the proper shape of a
slotted flap. Figure 117 shows that the highest maximum lift coefficients
were obtained with flap 2, which is shaped more like a good wing section
than flaps 1 or 3. The difference between the maximum lift coefficients
produced by flaps 1 and 2 is small and may be caused by the difference in
slot shape and lip extension rather than by the difference in flap shape.
FlO. 118. Contours of flap location for
23012 wing eeetion.
Q lllar
Slotted flap 2-11., &, = 60 degree. on NACA
Flap 3, however, appears to be too blunt with a too small radius of curva­
ture on the upper surface aft of the lip in the deflected position.
Typical cont.ourst61 of the maximum lift obtainable with various flap
positions at one flap deflection are shown in Fig, 118. In' general, the
optimum flap position for good flaps at large deflections appears to he that
which produces a slot opening of the order of O.. Olc or slightly more and
which locates the foremost point of the flap about O.Olc forward of the lip.
The maximum lift coefficient, however, is frequently sensitive to the flap
position, and the optimum position is best determined by test.
A complete set of section eharacterisrics'P for a typical single-slotted
flap configuration is shown in Fig. 119. This figure illustrates the charac­
teristic ability of slotted flaps to produce high lift coefficients with com­
paratively small profile drag coefficients.
The increment of moment coefficient associated with the use of single­
slotted flaps" is illustrated by Fig. 120. This figure shows the ratio of the
increment of the section pitching-moment coefficient" to the increment of
the section lift coefficient at an angle of attack of 0 degree for three wing
sections and several flaps. The moment coefficients used in this case are
~
o
~
t9
~
-
~~
0,
d6g.
o
.20
,
--b.
~
2O--v
30--0
50--1>
60--6
--
d
I.
_nose
-
[ ~
I:lJ"
/
~
I'V
for deflecfed flop
II
~
J~v
~
.A~
rr
....
J~
~
lD'"~!o"""'"
10-~1IIr'
-
o
~~ ~
" - ID-
~
~
-'
~ ~v
l--"""
l-=== ~~
I
/6
~
~
~~
~-
~I-"""
L...;l~
~I""'"
~~
lJW-­
-~
IP"
o
.4
~v
l/~
I...oo"'~
L..--'~
10<'"
1.2
.8
..... ~
"J"
N''''
~ 'C
l...oo"'~ ....~
~t,I' .~
~,;IIiT
;/..".
l/~
~Io"""
"
1/ ......
1.6
20
2.4
2.8
Section lifl coefficienll cl.
Path of flap nose for various fiap deflections.
lip in per cent airfoil chord c.
81,
degrees
0
10
20
30
x
y
8.36
5.41
3.83
2.63
3.91
3.63
3.45
3.37
Distances measured from lower edge of
If,
degrees
x
y
40
50
60
1.35
0.50
0.12
2.43
1.63
1.48
I
FIG. 119. Section aerodynamic characteristics of NACA 23012 airfoil with slotted flap 2-A.
214
HIGH-LIFT DElTICB8
215
based on the total chord with the flap deflected. The ratio plotted appears
to be fairly constant at yaIues between about - 0.17 and - 0.21, instead
of varying with flap-chord ratio as in the case of plain flaps at small
deflections.
The normal-force coefficient, pitching-moment coefficient, and center
of pressure for the flap of the configuration of Fig. 119 (configuration 2-h of
reference 153) are shownlU in Fig. 121. All the coefficients are based on
the flap chord. The pitching moments are taken about the quarter-chord
point of the flap, and the center of pressure is given in per cent of the flap
o
-.04
1'---
-.08
"~I"
~
~
Airfoi/ sections
~-
o
a
1-0---
o
NAG4 23012
/
N4CA 2J02I
N4G4 65-2/0
~
/'
<3-./2
/V
-./6
V
•10
~o()
.20
~
~/
~T/le(relcol
(plain flop)
V
0
~)
V-
V
i-'-
-
0
.
.30
.40
.50
.60
F/q) -chotdratio, £
FIG. 120. Variation of ratio of increment of aection pitching-moment eoefIicienf, to increment
of section lift coefficient with flap-ehord ratio for several sections with elotted flaps. GO
= 0 degrees.
chord from the leading edge of the flap. These data are useful in determin­
ing the loads on the flap and the flap linkages, and they show that the
normal force coefficients on the flap are less than those for the wing section.
c. External-airjoil Flaps. The external-airfoil flap investigated by
Wragg: 56GPlatt,838.f86 and Wenzinger161 may be considered as a special case
of the single-slotted flap in which the ftap does not retract within the wing
section. The maximum lift coefficients'4 obtained at an effective Reynolds
number of 8 million for a N ACA 23012 wing section with & O.2Oc external­
airfoil flap of the same section are shown plotted against flap deflection in
Fig. 122. These maximum lift coefficients are based on the combined areas
of the wing and flap. The values presented were obtained from tests of a
...
~
0)
§I
L.E.
~~20
't5 '.-: et
i i~
~t
40
0
~
-./
~
~
j{-.2
I)
~
~3
....
j'~
2.0
(j
~8
1.6
O.i
~
~
~~
21·w
1.2
Cb~
~~~ .8
4~
~.~
l~ .4
~ §' °0
.6'8
~
oJ::.
.4
.8
1.2 1.6 2.0
.4
.8
1.2 1.6 2.0 2.4
0
.4
Norma/-force coefficient of-combination,cn
.8
1.2 1.6 2.0
0.f =40()
(c) is =50 0
I
Seolion chnrBcteristiclt of the flap alone or the NACA 23012 airfoil with a 0.25660 slotted flap.
(0)
FlO. 121.
2.4 0
0.f
=30 D
(b)
2.4 28 '3
~
~
~
~
~
S
~
HIGH-LIFT DEVICES
217
finite span model and were only partly corrected to section data. The
corresponding section maximum lift coefficients are judged to be about
4 per cent higher than those presented. If these maximum lift coefficients
were based on the chord of the wing section, as is customary for other types
of flap, the resulting values would be about the same as those for single­
slotted flaps. Although the extemal-airfoil flap appearedu, 8& to offer some
2.4
..:­
a/f""
.i
/
V
./
'/
i-r
~~
~
\)
~ 1.6
1
~
C",
~ 1.2
~
1§
~
.§ .8
~
P,iKJ/~
f
4
o
JO
20
XJ
Flop defleclion l degrees
40
50
FIG. 122. Variation of maximum section lift coefficient. based on chord of wing section plus
flap, with flap deflection for the NACA 23012 wing section with 0.20 c. external-airfoil flap.
advantages as a full-span flap, it has not been used extensively because of
its failure to show definite advantages over retractable Haps and because
its use would probably aggravate the icing hazard.
d. Double-slotted /laps. Double-slotted flaps3 (Fig. 123) produce sub­
stantial increments of maximum lift coefficient over that obtainable with
single-slotted flaps. The fore flap or vane of the double-slotted flap assists
in turning the air downward over the main flap, thus delaying the stall of
the flap to higher deftections. It has frequently been found possible to
develop flap-vane combinations that can be retracted into the wing section
without relative motion between the vane and the flap. Such an arrange­
ment is desirable because the linkage system is much less complicated
than when relative motion is required between the flap and vane.
218
THEORY OF WING SECTIONS
Investigations by Harris,40 Purser," and Fischel' ! of approximately O.3Oc
and 0.4Oc double-s1otted flaps on the NACA 23021 and 23012 wing sections
indicated maximum lift coefficients of the order of 3.3 and 3.5 for the two
sizes of flap. Typical results for the NACA 23012 section are shown in
Figs. 124 and 125. The fore flap and the main flap did not deflect together
as a unit for these configurations. Aerodynamic characteristics' are pre­
sented in Fig. 126 for the NACA 65,-118 section with the O.309c double­
slotted flap shown in Fig. 123. For this configuration, the vane and flap
.....-------lOCLf----------..I
...----.78/----------1...
moved together as a unit up to deflections of 45 degrees. At higher angles,
the flap rotated about a pivot and the vane remained fixed.
The variation of maximum lift obtained for thin NACA 64-series
seetions" with approximately 0.3Oc double-slotted flaps is shown in Fig.
127. The type of fiap used for these tests is illustrated in Fig. 128. The
:8ap and vane deflected as a unit. These data. (Fig. 127) show that the
maximum lift coefficient decreases rapidly 88 the thickness ratio of the wing
section is decreased to values below 10 or 12 per cent. The maximum lift
coefficient obtained for the NACA 1410 wing section21 is also plotted to
indicate the effect of type of section. The maximum lift coefficient ob­
tained for the NACA 65r118 wing section with a flap deflection of 45 de­
grees (Fig. 126) is also plotted to indicate the effect of larger thickness
ratios. The NACA 65,-118 data are not exactly comparable, but the
indicated gradual rise of the maximum lift coefficient as the thickness rano
is increased to 18 per cent is believed to be representative.
The effect of the design position of minimum pressure on the maximum
lift coefficients obtained 'lith double-slotted flaps on 10 per cent thick
NACA 6-series wing sectionsD is shewn !D Fig. 129. The type of flap used
HIGH-LIFT DEVICES
~
I-
-
~
/Chad
-
.
219
~~'I
I
~~
~'''~~
-
v~~ +~
40'i;~
-
~
I-I--
I--
--~~,...
.28
6~~
-~ --
~
X2
40 50 60 70
0.60 - 40 /.60 1.60
Y2
.05 205 2.05 1.05
'--
..........
/
~
·'1
11.
.24
~
Of2'd!g.
.
D
~.20
70~~
~
.~
~ ./6
§
t·/2
:tl
6fV ~
I
V
•
l~
1..os
-~
.~
~.04
~
"'50/ V
V tJ
~
~,
A
.<.J
)
VI
P
..-10~ VO
~
l
o
t
/6
i
8
~
6'2,deg
".;;
~
\ .\ .\ .V' ~ ~
\J~ v
ot'r~
40 50 60 70
li
~
; I
0
~
~
~ -8 -.4
o
.4
.8
1.2
1.6
2.0
2.4
2.8
,
)~ J
3.2
3.6
5ec/lon lifl coefficient, 'i
FIG. 124. Aerodynamic eection characteristics of the NACA 23012 airfoil with a O.3Oc
double-elotted flap. ala == 25 degrees; %1 :: 0.41; III == 1.72. (Values of %1, 111. %It and 1/1 are
given in per cent of airfoil ehord.)
HIGH-LIFT DEVICES
221
was similar to that shown in Fig. 128. These data show that the highest
maximum lift coefficient was obtained for the NACA 64-series sections and
that the maximum lift coefficient decreases rapidly as the minimum-pressure
I~ ~
3.2
11) ~n r-, 45
6
(degJ
/ ~J V >55
'~
6~1 ) I I
~
'(
/ V;
.\
'p
~J
I ~ 1ft
/ /J /35!?o ~ '~
V) / I
.\
1
~ I---
.A
2.8
;x
2.4
1/
)'
2.0
,
/ I, I f lJ 'J0d ~
Vj / I
! III o~ ~ ~
1.6
.r
)'f'
I I / I If
il ~ ~J ) j I~ V 6
// f
I~ j
(deg)
fI
I{
~l
/ /1
If
t I f 'P~It j
.4
~I / T
,V If
~
I
j
7 )
o II
J r if ,I
,
r
1
~
,~
~
.~
~
Ci)
(
~.
1
-.4
-.8
~)
}
)
I JPI
~ 4
-/·?'24
'Ii-
V
j
r!
/
-/6
J
I
-8
0
8
/6
2~;
Section angle of oIl~ tXo,deg.
FIG.
126. NACA 653-118 airfoil section with O.309c double-slotted flap.
position moves farther aft. It is thought that the rather large variations
shown in Fig. 129 are associated with the fact that the thickness ratio of
the sections \\"88 in a critical range, as indicated by Fig. 127.
Load data for typical double-slotted flaps are given by Cahill.!! These
data show that a disproportionately large part of the load is carried by the
vane. Normal-force coefficients for the vane based on the vane chord and
the dynamic pressure of the free stream reached a value in excess of 5.0.
THEORY OF WING SECTIONS
222
r­
I
.028
e
-/
V
6
.020
)
~
V 1,/
....
...... ro-..
'~
~
'/ I
~ -....
,~
1,
~
~ r-,
~
~P-- ~ p.­ ~
'"
........tI.
)
~
V~
J
LJ
...-It!
~
v
I ~
~~~
)Jf
V
-
/ I
10
)~
-
1;.35
J
J
(dtJg)
r--­
20
p
.". V
r-­
~
~
r-­
r;JJ1
-
/
'I
-
J
~V
~
J
.004
o
,
.....
f--45
~
r-,
r........ >-­~
.....
~
o
IO
"V'll
,.....
~
~ 20
16-- r---­
.4
t'­
.-...,
~
~ ~.
~
.­
.....
-
-.
~ l...d..
/2
.8
/.6
Seclion lifl txJeffici~Ct
FIG.
I
~
~
..........
-
fd8g.)
~ ~ ......
~ ~ .....
r--c:... ~ ~
-.4
-
.. ---­
~
lIo.... ....
65~ ~
-~8
..D..
A
r­
5~ rd:::: r---..
-
0
'j'U
1,35~l---...
I--
~
-.
In
126. (Condtule4)
....
2.0
-
--::: ~
1
r-i
l:.,,;R
-
2.4
,
... ~f>I
2.8
_d
3.2
.,
HIGH-LIFT DEVICES
223
I
I
',,..A)
N.A.CA. 65., -/18
A!A.aA./4/0~
)
(~
4
FIG.
I
l.­
--N.A.C.A.64-series
(f
_
8
20
/2
/6
SecIiaIlhickness lf1fio, ~ lpercenf
127. Effect of thickness ratio and type of wing section on maximum lift with double.
slotted 1laps.
Ftpdud
line
~---------
. T 1 ' 5 c - - - - - - - - -....
(o)AIRFOL WITHRAP
XI
c::
Airfo7chord line~
,,;
r
12·
8~~
FIG.
128. Typical airfoil and flap configuration.
THEORY OF WING SECTIONS
224
3.2
I
I
I
I
Double slottedflop
wiIh
fore tip
.~
r---...... ......~
01t75c
2.8
J
.>---
2.4
~--
r----... ~
~_J.
---,---
~)
----4)
~
J...
~ 2.0
i
Plain wing section
fA
~ 1.6
~
.~
/1
~
..............
11
:~
~
...............
J
1.2
~--
p-' ......
)-_1
-
."'4
.8
--Smooth
---- Leading edge rough
.4
':'2
.3
.4
.5
.6
Position ofmimil1lU71 pteSSIIe, %
maximum section lift coefficient with position of
.7
129. Variation of
minimum pressure
for 80rne NACA tHeries wing aectiODS of 10 per cent tbiclmees and a design lift coefficient
of 0.2 .. R, 6 X 10'..
FIG.
FIXED AUXILIARY WING SECTION (AXED SLAT)
~<C-------------LEADING EDGE RETRAC'1MlE SlAT
FIG..
130.. Examples of fixed and retractable slats.
225
HIGH-LIFT DE"VICES
8.6. Leading-edge High-lift Devices. a. Slats. Leading-edge slats are
airfoils mounted ahead of the leading edge of the wing in such an attitude
as to assist in turning the air around the leading edge at high angles of
attack and thus delay leading-edge stalling. They may be either fixed in
I.lir--r-.---,r--...,..-...,...-..~-....-..-----.-------
~
2/.2t----r--t---+-~~-+-_I_______1.~~~-_J___J,.-__1
~
10
20
30
40
50
60
'Flop deflection, 6, ,deg.
FIG. 131.
efficient.
Eft"eet of flaps and leading-edge slat on increment of maximum section lift co­
position or retractable (Fig. 130). The fixed leading-edge slat consisting
of an auxiliary airfoil mounted ahead of the wing leading edge has been
investigated in detail by Weick and Sanders.s" This investigation showed
that leading-edge slats of this type with chords varying from 7.5 to 25 per
cent of the wing chord and with various sections all produced substantially
the same maximum lift coefficient when located in the optimum position
for the ratio
CLmax2
( Dmin
THEORY OJ! WING SECTIONS
226
The value of this maximum lift coefficient was about 1.64 for the rather
low Reynolds numbers of the tests. It is doubtful whether such configura­
tions would experience much beneficial scale effect.
The effectiveness of the retractable leading-edge slat107 shown in Fig.
130 in increasing the maximum lift coefficient and the angle of attack for
24
I I I I I I I I I
- - ­ Q2566c sidled fk:p
---O.20c
- ­ O.2566c
--Q2Oc
split fkp
stxled f~arJB:xJing. sial
splif flopmd 1eodiIg. sbf
6
2
.JI"'-
8~
~
~
~
--
_...­
~---
-~--.
........
­
,,/
4
n.
-
.......... ~~
'--r--­-... ------......!
4
o
10
20
30
..
---..IL_
~~
40
~
50
r-,
60
Fkp deflechOnl 6 , ,deg.
FIG.
132. Effect of 1lape and leading-edge slat on angle of attack for maximum lift.
maximum lift is shown in Figs. 131 and 132. These data, which were
obtained on a K.~CA 23012 lying section, indicate an increment of about
0.5 for the maximum lift coefficient and about 8 degrees for the angle of
attack for maximum lift. The increment of maximum lift decreased to
about one-half of its value for the unflapped section when either a split
or slotted flap was deflected to optimum deflections despite readjustment
of the slat to optimum positions with flap- deflected. Weick and Platt!·
obtained considerably larger increments of maximum lift coefficient with
a special retractable slat (Fig. 133) having a shape providing a rounded
entrance into the slot. The increments of maximum lift coefficient with
HIGH-LIFT DEVICES
227
this configuration were 0.81 for the unflapped section and 0.45 with a
deflected slotted flap.
b. Slots. Slots to permit the passage of high-energy air from the lower
surface to control the boundary layer on the upper surface are common
features of many high-lift devices. The most common application is the
slotted flap. When the slot is located near the leading edge, the con­
figuration differs only in detail from the leading-edge slat. Additional
slots may be introduced at various chordwise stations. Weick and Bhortal'"
made a systematic study of slots on a Clark Y airfoil. The results of
this investigation are summarized in Fig. 134. For the unflapped section,
the most effective position for a single slot is near the leading edge, and
the effectiveness decreases as the slot is moved aft. Multiple slots are
relatively ineffective on the plain airfoil unless they include a slot near the
FIG.
133. Special retractable slat on Clark Y wing section.
leading edge, in which case a total of three slots, all located well forward,
is optimum. For the flapped section, the slot located near the leading edge
was effective. A single or double slot at the flap changed the type of flap
from plain to slotted with- a corresponding increase of the maximum lift
coefficient. Load data for the leading-edge slot are given by Harris and
Lowry." These data show resultant-force coefficients as large as about
7.5 for that portion of the wing section ahead of a slot near the leading edge.
If slots are considered as a fixed high-lift device, the profile drag in the
high-speed. flight attitude is an important characteristic. Figure 134 shows
that any of the slots investigated cause large increments in the minimum
profile drag. This increment increases with the number of slots and de­
creases with rearward movement of the slots. Attempts have been made
to maintain low drags with slots open by locating the slots so that there
would be no flow through them in the high-speed. condition. lOS Such con­
figurations have failed to improve the ratio of maximum lift to minimum
drag over that for the plain wing section.
c. Leading-edge Flap8. A leading-edge flap may be formed by bending
down the forward portion of the wing section in a manner similar to that
in which the trailing edge is deflected in the case of plain flaps. Other
types of leading-edge flap are formed by extending a surface downward
and forward from the vicinity of the leading edge. As shown in Fig. 135,
such flaps may extend smoothly from the upper surface near the leading
edge, may be hinged at the center of the leading-edge radius, or may be
hinged on the lower surface somewhat aft of the leading edge. Although
THEORY OF WING SECTIONS
228
Slot combination
CLma x
CDmi D
CLma'!C.
a
CDm iD
degrees
CL
max
«:
=:=::--.-....
1.291
0.0152
85.0
15
rr:
:::--=----..
1.772
0.0240
73.8
24
c:7~
1.596
0.0199
SO.3
21
C
7c:=::::..-..
1.548
0.0188
82.3
19
c:
:Jc::::,...
1.440
0.0164
87.8
17
r~~
1.902
0.0278
68.3
24
7~
1.881
0.0270
69.7
24
:J~
1.813
0.0243
74.6
23
~L7C?c:::=::--..-
1.930
rr: ­
rr: -
rL7c=---r~
I
i
0.0340
I
I
56.8
I
25
I
1.885
0.0319
59.2
24
rC7C7oc:::::::::.
-z:':70~
c:7C?c:=::::..-..
1.885
0.0363
51.9
25
1.850
0.0298
62.1
24
1.692
0.0228
74.2
22
c:7t:==7c::::,...
1.672
0.0214
c: -
1.510
0.0208
7a~
c:7C7oc::::,...
II
I
I
78.2
72.6
I
1.662
I
I
0.0258
r
I
64.4
I
i
Ii
I
II
22
19
I
22
I
(a) Multiple fixed slots.
FIG.
134. Aerodynamic characteristics of a Clark Y wing with slots and flaps.
HIGH-LIFT DEVICES
Slot combination
CD mi u*
CLms x
229
II
-----------1----1----1----11----1
I
1.950
0.0152
2.182
!
128.2
12
0.0240
91.0
19
2.235
0.0278
80.3
20
2.200
0.0340
64.7
21
2.210
0.0270
81.8
20
1.980
0.0164
120.5
12
1.770
0.0164
IOS.O
14
0.0208
117.5
16
2.500
0.0258
96.8
18
2.185
0.0214
102.0
18
0.0243
93.2
19
2.320
0.0319
72.7
20
2.535
0.0363
69.8
20
0.0298
87.3
20
0.0298
68.3
21
I
2.-1-12
I
l
2.261
I
2.600
t
2.035
I
II
i
• CD' with flap neutral.
man
(b) Multiple fixed slots and a slotted flap deflected 45 degrees.
FIG.
134. (Concluded)
230
THEORY OF WING SECTIONS
none of these devices is as powerful 88 trailing-edge flaps, they may be
used full span without mechanical interference with lateral-control de­
vices and they are effective when combined with trailing-edge high-lift
devices. Leading-edge flaps reduce the severity of the pressure peak
ordinarily associated with high angles of attack and thereby delay separa­
(0) DROOPED LEADING EDGE
]
(b) UPPER SURFACE LEADING EDGE FLAP
(c)LOWER Sl,eqfACE LEAOINGEDGEFLAP
(d) FLAP HINGED ABOUT LEADING EDGE RADIUS
FIG.
135. Various types of 1eading-edge flaps.
tion. Leading-edge flaps received little consideration until German in­
vestigators became interested in them during the Second '\\"'orld War.
Krueger,.,60 Lemme,l3· M, 65 and Koster" showed increments of the
maximum lift coefficient of 88 much as 0.7. These increments were, how­
ever, applied to maximum lift coefficients for the plain wings of the order
of 0.72. These low maximum lift coefficients resulted from the small
leading-edge radii of the wing sections usually used for these investigations
and the very low Reynolds numbers of the tests. These investigations in­
dicated that the effectiveness of leading-edge Haps increased with decreasing
leading-edge radius. Typical rcsultsH are shown in Fig. 136. The inere­
HIGH-LIFT DEVICES
231
moot of maximum lift coefficient, deLmasJ is plotted against the leading­
edge-radius parameter {r/c)/{t/c)2 where r is the leading-edge radius. The
value of this parameter for NACA four- and five-digit wing sections is 1.1.
FullmerU investigated t\VO types of leading-edge flap on the NACA
641-012 section at a Reynolds number of 6 million. The chord of the flap
was 10 per cent of the section chord, and the configurations corresponded to
........
.6
o Withou/ split fbp
.~
.8
~
I
<,
""",-re
?J
-R=0.57x 10'
a Corf(c~fig.135";lh split flop
~
rI
~~ rr R=2.I Xd
<,
"',
r-,
n Corf(bJ,fig.l35
-
<>
-
<,
lei
~
o
Ccd(bJ,"~r.135wifh splitfbp
r-,
r-,
~
.2
=60
cf Corf(c~fig.135
I
\
.2
s
~~
itR=O.72x 106
-
~=a20
(UBM3067)
I
~
-
Withsplit flop
r-R=2.2 x KJS
t- r-, ~
/11/
r:,.
.6
.8
'"
<,
1.0
~
'" <,
"-
£4
~
~
(~;2
FlO. 136. Effeetiveneea 01 leadi11ll-4M1ge flaps for various wing sections. Increment of lift
resulting from leading-edae flap, c,
88 a function of the leading-edge radius coefficient.
(r/c}/{t/c)t.
.al:
b and c of Fig. 135. The increments of maximum lift coefficient and of the
angle of attack for maximum lift are shown in Fig. 137. The leading-edge
radius parameter for the NAC...\ 641-012 section is 0.72. The maximum
increments of Fig. 137 are shown plotted on Fig. 136 for comparison with
the German results. The maximum lift coefficient of the plain wing sec­
tion for these tests was 1.42.
8.7. Boundary-layer Cpntrol. The idea of removing the low-energy
air of the boundary layer, or of adding kinetic energy to the boundary layer,
as a means for increasing the maximum lift has been obvious since the basic
mechanism for separation was first understood." The kinetic energy of
the layers of air close to the surface may hi increased by removing low­
energy air through suction slots or a porous surface. Another common
method is to blow high-energy air through backward-directed slots. The
232
THEORY OF lYING SECTIONS
air handled through either the suction or blowing slots may be carried
through the interior of the wing and the necessary energy supplied by a
blower. Alternately, the pressure difference required may be obtained
from the variation of pressure about the wing section itself. This method
I!!«JJ:...
l.J:Mer 5lIfoce~. '-e4e flop
}
o t.ower:5lIfoce
flop willi
a ftrJilhg-edoe flqp
lJppei-sufoce leodirJg -e4e flop
}
£fiper-Sllfoce Ieoditig-edge fbp with
Imiling-e4e ftp
Fig./35,
calle)
A- /.'71:
/{/. ..,~
conl{b)
8
A
-.
4
I
-
- ----
~
~
~
/
-./'"
'¥
~
rA
8
/ "'" -,
Q,.-
.4
~
Do..
A
lJ()
~
~
/
I
~.
y"
/00
/20
140
/60
l.I!odtY;-edge f/qJ deflecfionl 6,LE!deg.
FIG. 137. Variation of the increment of maximum section lift coefficient and the increment
of section angle of attack for maximum section lift coefficient with leading-edge flap deflec­
tion. NACA 64t..Q12 airfoil section with leading- and trailing-edge split flaps. R, 6.0 X U)6.
has been used successfully only in the case of blowing slots on the upper
surface using air taken from the lower surface. Examples of such arrange­
ments are the previously discussed slotted flaps, slots, and leading-edge
slats.
It is obvious that, if boundary-layer control is applied at sufficiently
close intervals along the upper surface of a wing section, separation can be
avoided up to very high values of the lift coefficient. The increments of
maximum lift coefficient are obtained as an extension of the lift curve to
HIGH-LIFT DEVICES
233
higher angles of attack as contrasted with the displacement of the lift
curve resulting from deflection of trailing-edge flaps. The problem con­
fronting the designer is to apply sufficient boundary-layer control to obtain
the desired values of the maximum lift coefficient without increasing the
weight and complexity of the airplane to such an extent as to nullify the
gains in performance expected from the higher lift coefficients. These
considerations of weight and complexity have prevented extensive use of
6.0
~
5.0
/
~4.0
......"
i
Theory.. . "
.~
.~
~
)
§ .10
.~
~
(j)
f
/r
/
/
v
l(
/'
~.
)C
/
~/
~.
J
~
I
2.0
V» '--- Experiment _
r--­
CO=Q038
/
V)'­
1/
1.0
a
I
V·
JV
.-20.
-/0
0
10
20
Angled ottoclt1t%, deg.
138. Variation of lift coefficient with angle ot attack for a thick wing section with
boundary-layer control.
FlO.
any except the simple types of boundary-layer control such as slotted flaps
and leading-edge slots and slats.
Successful application of boundary-layer control must delay both tur­
bulent separation over the aft portion of the wing and permanent laminar
separation over the forward portion (see Sec. 5.3). The prevention of
permanent laminar separation for thin or even moderately thick sections
by means of suction slots is a difficult problem involving critically located
slots in a region of greatly reduced pressure that requires high-pressure
blowers. Nearly all the early investigations avoided this problem by using
very thick sections with large leading-edge radii. 102. 101, 104 Using such sec­
tions, Schrenk103 obtained lift coefficients of over 5.0 using a single suction
slot (Fig. 138) with a volume-flow coefficient Co of 0.038.
The high drag coefficients of such thick wing sections in the high-speed
flight condition led to investigations of boundary-layer control on thinner
THEORY OF WING BECTIONS
sections.II,u,WI Knight and Bamber" obtained maximum lift coefficients-­
of about 3.0 (Fig. 139) for the NACA 84-M wing section with a single
backward-opening blowing slot.
NACA 84-Mwingsecfion. SloIwidfh=Q667%c
Slotof 53. 9 ~ c from leading edge. Inlemo/
pressure /2 q higher than free- sllfJOlTl sialic
pressure
I
~
,
I
I
I
I
c, s/olled Ming sediJn
JV
2.4
L'
/
V
7
If
I
7
\
\
y
\
./
.~ ,V
8/
FIG.
.rl.
I) ~
~
~
0
-:
/
Il
o
.L
~ ~c, pbiI wing section
I
I · ••
I
II'-lfdstnIt/'wing ssdion
~
II .I
7.
~J
I
/ /
1/ Jif
1/
..4
~
~"",. -.-...... ~
~
8
K:.~~~~_
~
-~
~
~
16
24
32
40
48
~ofolfodr,f4deg.
139. Effect of boundary-layer control by means of a baekward-opening blo'triDc 1101.
Much of the more recent work'" lion boundary-layer control has com­
bined suction slots with flaps and leading-edge slots. Quinn" obtained
maximum lift coefficients exceeding 4.0 for the NACA 65a-418 section with
a double-elotted flap and a single suction slot (Fig. 140). The maximum
tift coefficient for the unfIapped section increased rapidly with small
HIGH-LIFT DEVICES
235
~-----.45c----~-a
44
4.0
-V "'--V
0, =65°
~
-"
L-----' ~ ~ ~
/
l-
"""'"
~.6
~
~~
~
~
~
~
"6f=4S~
/
0,=0°"",,,,V~
~
~
~
10-"""
----
~
~
~
1.6 I(
/
......
1.20
.008
••016
.024
.032
.040
Flow coefficient, CQ
FIG.I40. Configuration and maximum lift characteristics of the NACA 65,-418 wing aection
with double-elotted flap and boundary-layer control by suction. R, 6.0 X 10'.
236
THEORY OF WING 8ECTIONS
amounts of flow through the slot, but the variation of maximum. lift coeffi­
cient for the Happed section was nearly linear with the flow coefficient up
to a value of 0.040. Quinn93 has concluded that the maximum effectiveness
of suction slots is nearly reached when the quantity of air removed is
equal to that which would pass through the displacement thickness of the
boundary layer at the slot location with the local velocity outside the
boundary layer. Quinn found that the maximum lift was limited by
leading-edge stalling for the 18 per cent thick sections. He used a leading­
L..,....o-" ~
~V
~
V
.01
).---"
~
.02
.03
.04
FkM coefficienf,CQ
FIo. 141. Configuration and maximum lift characteristics for the NACA M 1A212 wiug
section with leading-edge slat, double-slotted flap and boundary-layer control.
edge slatt 5 to control the leading-edge stalling of the NACA 641A212 section
with a suction slot and a double-slotted flap. This combination of high-lift
devices gave a maximum lift coefficient of nearly 4.0 for this thin section95
(Fig. 141).
8.8. The Chcrdwise Load Distribution over Flapped W'mg SectiODS.
Allen' obtained empirical load distributions that permit calculation of
reasonable approximations to the load distributions of "ring sections with
deflected plain or split flaps. These load distributions are analogous to
those associated with changes of camber in that the ratio Pb&/c~ is inde­
pendent of the angle of attack where Pili is the incremental basic load
distribution associated with flap deflection and en., is the basic normal­
force coefficient associated 'lith flap deflection. The total load distribu­
tion is the sum of the incremental basic distribution P"" the incremental
additional load distribution Po" and the load distributionof the unflapped
section at the same angle of attack. The incremental additional load
HIGH-LIFT DEVICE8
237
distribution P a & is taken to be the same as the additional load distribution
associated with angle of attack. That is, for a normal additional force
coefficient of unity,
P = 4 dV a -!
GI.
V V
where the values of dVa/V and vIV are tabulated in Appendix I for various
wing sections.
Allen also found empirically that the values of P",/~ at any point
(x/c)/(I - E) (points ahead of the flap hinge) and [I - (x/c»)/E (points
aft of the hinge) were independent of the flap-chord ratio E for any given
angular deflection of the flap. Values of P"./c.., are given in Table 6 for
both plain and split flaps. The incremental load distribution associated
with flap deflection is thus defined in terms of two known types of distribu­
tions. The remaining problem is to determine the magnitudes of these
two types of distributions associated with a given flap deflection.
The magnitudes of the load distributions are determined from force
test data obtained at the desired angle of attack with the flap neutral and
deflected. The additional load distribution contributes no moment about
the quarter-chord point. The incremental basic distribution is therefore
selected to produce the measured increment of moment coefficient between
the flap-neutral and flap-deflected conditions. The remainder of the differ­
ence between the normal-force coefficients with the flap neutral and de­
flected is equal to the incremental additional load distribution.
The normal-force distribution is assumed to act as shown in Fig. 142.
From force tests of the wing section with flap neutral, C1ft1 (quarter-chord
pitching-moment coefficient) and ella (normal-force coefficient) correspond­
ing to the normal-force distribution shown in Fig. 142a are obtained.
From force tests of the wing section at the same angle of attack with the
flap deflected, ems and CftS corresponding to the normal-force distribution
shown in Fig. 142b are obtained. Let
aem =
Cm: -
em l , }
=
en: -
C ra.,
deft
(8.1)
where em.' and Cfta' are the pitching-moment and normal-force coefficients
corresponding to the normal-force distribution for the section with flap
neutral when plotted normal to the chord of the section with flap deflected
as shown in Fig. 142c. Then ~Cm and .1c,. are the pitching-moment and
normal-force coefficients of the incremental normal-force distribution
when the incremental distribution is plotted normal to the chord of the
section with the flap deflected as ShO"l1 in Fig. 142d.
In many cases, the approximation
(8.2)
THEORY OF WING SBCTIONS
238
is sufficiently exact except when the flap-chord ratio E and the flap deflec­
tion 8 are simultaneously large or when the shape of the mean line is such
that the load is large near the trailing edge. In such cases, values of Crat'
and Cat' may be obtained graphically from the theoretical load distribution
for the unflapped section adjusted to agree with the experimental data.
Let Ac.' and Ac.' be the pitching-moment and the normal-force coeffi­
cients of the incremental normal-force distribution plotted normal to the
flap-neutral chord as shown in Fig. 142e. Because the incremental basic
em,'
h=~n=~/====-__
~_cn_/.....;:' ;;;;:;::::==--..
(a) NORMAL-FORCE DISTRBI1lON
(e) DISTRI8UTION SHOWN IN(a)
(b) NORMAL-FORCE DISTRIBUTION
FOR AIRR)IL WITH FLAPNEUTRAL. FOR AIRFOL WITH FLAPDEF'l.B:T.
ED~
f'::
~~
6Cn
"
WITH FLAPNORMAL FORCE
0ISTR8UT1ON PLDTTED NORMAL
lOFLAP DEFLECTED CHORD.
/l(j
(d) INCREMENlJ'L NORMAL -FORCE
DISTRBITION CMJSED BYFLAP
AC~
~
(e) DISTRI8UTIDN SHOWN'IN (d)
PLDnED NORMAL TOFLAP-NElITRAL
DEFLECTION.
CHORD.
FIG. 1.(2.
Normal-force distribution and incremental normal-force diBtribution for flaps
neutral and deflected.
normal-force distribution is responsible for the increment of the quarter­
chord pitching moment, then, if G is the moment arm in terms of the chord
of the basic normal force about the quarter-chord point,
ac.' == Gc..,
t:.e,., == e.., + en.,
wliere en _, = additional normal-force coefficient associated with flap de­
flection
c...,=~'
c,,~.
}
LAc.'
=
~c",
-
(8.3)
G
The value of G is a function of E and 8. Values of G are given in Table 7.
The correlation between the fictitious values of ~c",,' and ACta, and the
measured values of de,. and ~CR must be established in order to determine
c... and cfteI from force tests. The incremental flap normal-force coefficient
is
HIGH-LIFT DEVICES
I, -
nl,
239
(8.4)
Ca -gEc
­
where nl, is the incremental flap normal force per unit span.
Then
J1c.' =z a~ + Ec"" (1 - cos 8)
}
and
&:." = &:., - Ee. ' (1 - cos 6)(% - E)
'I,
(8.5)
The incremental1lap normal force may be considered as a combination
of two components due to the incremental additional and the incremental
basic normal-force distributions. Let 1'0, and "rll, be the ratios of the flap
normal force to the section normal force for the incremental additional and
the incremental basic normal forces, respectively. Then
(8.6)
or
(8.7)
The contribution to the flap load of the additional normal-force dis­
tribution is small compared with the basic contribution; and, for the
purpose of determining &:." and J1Ca', the following approximation may
therefore be used:
and Eqs. (8.5) become
4c.' = J1c.
&:."
/ii;w
+ En, G
(1 -
= &:., -
E-y", ~' (1 - cos 6)(~ - E)
so that
At;.,
~c.'
where
cos a)
= 4c. + T" ~c.}
= 'T'" ~c.
(8.8)
E(1 - cos 8)("Yb,/G)
cos 8)(~ - E)('YbtJ/G)
T"
= 1 + E(l -
T.
= 1
+ E(l -
1
cos 8)(% - E)('Y6,/G)
The values of'T. and 'T.. as determined by Allen are given in Tables 8 and 9.
In using this method, the values of J1c. and tic", are obtained from force
tests of the flapped and unflapped section at the same angle of attack.
From Eq. (8.8)~ values of 11c"" and &:." are obtained using Tables 8 and 9.
~
TABLE
6.-P",/C"6,
DISTRIBUTION
b. Plain flap at 8 • 20 degrees
a. .Plain flap at 8 - S, 10, and 16 degreel
0.05 0.10 0.15 0.20 0.25 0.30 0.31 0.40
B
o.~
0.&0 0.55 0.80 0.65 0.70 0.05 0.10 0.16 0.20 0.25 0.30 0.36 0.40 0.46 0.50 0.66 0.60
--0 -0 -0 -0 -0 -0 --0 -0 ---0 -0 --0 -0
0
0
0
0
0
0
-- -- 0
0.18
0.26
0.39
0.50
0.62
0
0.19
0.27
0.40
0.62
0.64
0.15
0.23
0.34
0.44
0.55
0.16
0.24
0.36
0.46
O.M 0.67
0.17
0.25
0.36
0.47
0.68
0.17
0.25
0.38
0.49
0.60
0.18
0.26
0.39
0.&0
0.62
0.20 0.21 0.22 0.23
0.29 0.30 0.32 0.34
0.42 0." 0.46 0.49
O.M 0.67 0.60 0.63
O.M 0.67 0.70 0.73 0.77
0.25
0.36
0.52
0.67
0.82
0.27
0.39
0.56
0.72
0.88
0.15
0.23
0.34
0.44
O.M 0.55
0.16
0.23
0.34
0.46
0.66
0.16
0.24
0.35
0.46
0.67
0.17
0.26
0.36
0.47
0.68
0
0.17
0.26
0.38
0.49
0.60
0.66
0.80
0.98
1.24
1.74
0.67
0.82
1.00
1.26
1.75
0.69
0.83
1.01
1.27
1.7J
0.71
0.8&
1.03
1.29
1.77
0.73
0.88
1.06
1.3l
1.79
0.75
0.90
1.08
1.34
1.81
0.77
0.93
1.11
1.38
1.84
0.92
1.10
1.30
1.59
2.06
0.98
1.17
1.39
1.67
2.16
1.05
1.25
1.48
1.78
2.29
0.66
0.79
0.97
1.23
1.73
0.66
0.80
0.98
1.24
1.74
0.67
0.82
1.00
1.26
1.75
0.69
0.83
1.01
1.27
1.76
0.71
0.85
1.03
1.29
1.77
0.73
0.88
1.05
1.31
1.79
0.76
0.90
1.08
1.34
1.81
0.70
0.60
0.60
8.74
6.45
4.96
3.86
3.09
2.48
6.04
4.49
3.51
2.72
2.18
1.72
4.83
8.78
2.92
2.23
1.78
1.42
4.40
3.32
2.67
1.97
1.57
1.M
4.01
2. •
2.31
1.77
1.41
1.11
3.71
2.77
2.12
1.84
1.30
1.02
3.&0 3.35 3.23 3.15 3.11 3.06 3.04
2.80 2." 2.39 2.32 2.26 2.22 2.19
2.00 1.90 1.81 1.74 1.69 1.84 1.60
1.63 1.44 1.37 1.32 1.27 1.22 1.19
3.02
2.16
1.57
1.16
1.21 1.14 1.08 1.04 1.00 0.96 0.93 0.91
0.96 0.89 0.86 0.81 0.77 0.7& 0.72 0.70
5.S3
1.08
4.23
3.71
3.33
2.98
4.05
3.&3
2.99
2.63
2.3&
2.00
3.38
2.98
2.60
2.16
1.92
1.73
3.02
2.61
2.19
1.90
1.89
1.60
2.83
2.36
1.97
1.71
1.62
1.35
2.70
2.18
J.81
1.68
1.40
1.24
0.40
(back of 0.30
hinge) 0.20
0.10
0.05
0
1.90
1.40
0.92
0.48
0.32
0
1.34
0.09
0.63
0.34
0.19
0
1.10
0.81
0.&3
0.27
0.16
0
0.98
0.70
0.46
0.24
0.14
0
0.88
0.63
0.41
0.21
0.12
0
0.78
0.67
0.38
0.20
0.11
0
0.73
0.63
0.35
0.18
0.10
0
2.85
2.32
1.93
1.39
1.16
0
1.87
1.63
1.3&
0.98
0.70
0
I.M
1.34
1.11
0.79
0.68
0
1.31
1.18
0.96
0.69
0.&0
1.19
1.04
0.86
0.62
0.4&
0
1.09
0.9&
0.79
0.67
0.40
0
~/e
0
0.05
0.10
0.20
0.30
0.40
0
0.16
0.22
0.33
0.43
0.54
1-8
(ahead 01 0.60 0.66
hinge)
0.60 0.79
0.70 0.97
0.80 1.23
0.90 1.73
1.00
--- 0.90
0.80
1 - ~/c
B
0.16
0.23
0.34
0.45
0.19
0.27
0.40
0.&2
0.89
0.60
0.33
0.17
0.10
0
0.80
0.96
1.15
1.42
1.88
0.81
0.47
0.31
0.18
0.09
0
0.84
1.00
1.19
1.46
1.92
0.62
0.46
0.29
0.13
0.08
0
0.88
1.0j
1.24
1.52
1.98
0.69
0.43
0.28
0.16
0.08
0
0.57
0.42
0.27
0.14
0.08
0
0.&6
0.40
0.26
0.13
0.07
0
0.&3
0.39
0.25
0.13
0.07
0
0.16
0.22
0.33
0.43
a
- --0 -0
0
0
0.20
0.29
0.42
0.14
0.67
0.21
0.30
0.44
0.57
0.70
0.22
0.32
0.46
0.60
0.73
0.23
0.34
0.49
0.63
0.77
~
~
~
0.77
0.93
1.11
1.38
1.84
0.80
0.96
1.15
1.42
1.88
0.8J
1.00
1.19
1.46
1.92
0.88
1.05
1.24
1.52
1.98
0.92
1.10
1.30
1.59
2.06
~
2.63
2.05
1.70
1.48
1.30
J.15
2.58
1.96
1.62
1.39
1.22
1.08
2.56
1.88
1.65
1.32
1.17
1.03
2.56
1.83
1.49
1.27
1.12
0.98
2.58
1.78
1.44
1.22
1.08
0.94
2.62
1.75
1.40
1.18
1.04
0.91
1.02
0.88
0.73
0.53
0.38
0
0.96
0.82
0.68
0.50
0.35
0
0.91
0.78
0.65
0.47
0.33
0
0.88
0.73
0.62
0.45
0.31
0
0.83
0.72
0.69
0.43
0.30
0
0.80
0.89
0.56
0.41
0.28
0
~
s
~
~
C':S
~
~
d. Plain or .pUt flap at a- 40 degrees
c. Plain flap at' .. 30 deereee
-
I
0.05
E
z/c
1-8
(ahead of
hinge)
--­
!-.=. ~/c
B
(bank of
hinge)
-­
0
0.05
0.10
0.20
0.30
0.40
0
0.15
0.22
0.33
0.43
O.M
0.&0
0.60
0.70
0.80
0.00
0.66
0.70
OJ.7
1.23
1.67
1.00
O.DO
".rJO
0.80
0.70
0.60
O.SO
0.40
0.30
0.20
0.10
0.05
0
4.61
4.34
4.08
3.82
3.56
3.25
2.89
2.44
1.81
1.53
0
0.16
0.20
0.26
0.30
0.36
0
0.18
0.23
0.34
0.4&
0.56
0
0.16
0.24
0.3&
0.46
0.67
0
0.11
0
0.t7
0.6H
O.US
0.67
0.82
I.OU
1.24
1.26
1.68
1.69
:4.12
3.20
2.60
2.70
2.56
2.37
2.20
0.10
-­
0
0.16
0.23
0.34
0.44
0.&5
0.66
O.RO
3.07
2.89
2.60
2.4.9
2.29
2.0'
1.72
1.27
0.92
0
0.40
0.45
0.60
0.06
0.10
0.16
0.20
0.25
0.30
0.35
0.40
0.19
0.27
0.40
0.52
0.64
0
0.20
0.29
0.42
0.54
0.67
0
0.21
0.30
0.44
0.67
0.70
0.16
0.22
0.33
0.43
0
0.16
0.23
0.34
0.44
0.66
0.16
0.23
0.35
0.46
0.56
0
0.16
0.24
0.36
0.46
0.67
0
0.11
0.26
0.38
0.47
0.38
0.17
0.23
0.39
0.49
0.60
0
0.18
0.26
0.40
0.61
0.82
0
0.19
0.27
0.42
0.62
O.M
tt:
0.77
O.!l:l
1.11
1.38
1.74
0.80
O.!J(\
0.84
1.00
r.rs
1.IH
1.42
1.77
0.89
0.83
1.01
1.26
1.61
0.71
0.86
1.0:1
1.28
1.61
0.73
0.88
1.05
1.30
1.62
0.7&
0.00
1.08
1.33
1.62
0.77
0.93
1.11
1.36
1.63
:;;~
2.01)
1.8.'1
1.66
1.62
1.40
1.29
2.01
1.17
1.11
0.98
0.82
0.61
0.44
0
- ­ - ­ -­ -­ - ­
-­ - ­ -0 ­ - ­ -­
-­ - ­ -­ -­ -­ -­
0
0
0
0.1':4
1.01
1.27
1.70
2.36
2.:~7
2.0~
2.20&
2.0tJ
1.94
1.79
1.88
1.64
1.GS
1.45
1.41
1.03
0.76
0
O.~
0.25
0.38
0.47
0.68
0.38
0.49
0.60
0
0.18
0.26
0.39
0.60
0.62
0.71
0.145
0.73
0.75
n.AA
o.no
1.05
1.0M
1.34
1.73
I.oa
1.2U
r.ai
1.71
1.72
2.22
2.15
2.02
1.88
1.74
1.61
2.13
2.01
1.86
1.73
1.60
1.47
2.08
1.90
1.46
1.29
1.34
1.18
1.00
0.74
1.25
1.10
0.93
0.60
0.49
0
1.22
1.09
0.90
0.81
0.86
o
I 0O,6'J
0.53
0
l.i6
1.62
1.49
1.37
1.03
0.87
0.65
0.48
0
1.77
1.59
1.45
1.34
1.22
O.M
0.66
0.80
1.46
1.80
0.88
0.79
O.n7
1.22
1.68
1.23
1.&9
0.68
0.82
1.00
1.2&
1.60
1.UO
1.73
1.S2
1.40
1.28
1.17
4.10
4.24
•. 29
4.27
4.14
3.92
2.90
3.01
3.04
3.02
2.91
2.76
2.42
2.52
2.&3
2.48
2.39
2.27
2.12
2.23
2.22
2.18
2.10
1.98
1.90
2.02
2.00
1.96
1.89
1.77
1.88
1.88
1.84
1.81
1.74
1.62
1.81
1.78
1.73
1.70
1.62
1.61
1.76
1.72
1.66
1.69
1.62
1.42
1.06
3.64
3.28
2.82
2.11
1.79
0
2.&6
2.31
1.98
1.48
1.08
0
2.11
1.90
1.62
1.20
0.89
1.83
1.04
1.40
1.0&
0.77
0
1.64
1.47
1.26
0.96
0.89
0
1.50
1.34
1.15
0.87
0.63
0
1.40
1.24
1.07
0.81
0.58
0
1.31
1.17
1.00
0.76
0.93
0.78
0.68
0.41
0
O.OM
0
:::~
-s
~
tlj
~
~
t'I.)
O.M
0
~
~
~
TABLE
8.-P.,Ie"",
DISTRmUTIoN.-(Conlinued)
I. Plaln or epUt Sap at' -
,. Plain or 8PUtSap at' - 80 deere-
•
0.08
----~
ale
1-8
[ahead of
billie)
--1 -ale
B
(back of
billie)
0
0.05
0.10
0.20
0.80
0.10
0.16
0.20
0.26
0.30
0.36
0.40
0.06
0.10
0.15
0.20
0.26
eo d..-.
0.30
--- - --- -- - - - - - - - - - - - - - - - - - - - 0
0.18
0.22
0.33
0.43
0
0.18
0.23
0.36
0.46
O.M
0
0.18
0.24
0.34
0.48
0.67
0
0.11
0.25
0.38
0.47
0.&8
0
0.17
0.26
0.38
0.49
O.~
O.M
0
0.18
0.23
0.34
0.44
0.66
0.60
0.80
0.70
0.80
0.90
0.88
0.'19
0.97
1.21
1.M
0.86
0.80
0.98
1.22
1.66
0.68
0.82
1:00
1.23
I .•
0.89
0.83
1.03
1.26
1.67
0.71
1.00
0.90
0.80
0.70
0.80
0.50
3.81
4.18
4.26
4.27
4.21
4.07
2.88
2.90
8.00
3.02
2.97
2.88
2.21
2.46
2.48
2.43
2.38
2.00
2.16
2.20
2.18
2.14
2.06
1.88
1.96
1.98
1.98
1."
I.M
1.81
1.77
1.89
0.40
0.30
0.20
0.10
0.05
0
3.88
3.62
3.08
2.34
2.02
0
2.71
2.49
2.16
1.64
1.22
0
2.24
2.M
1.78
1.33
1.00
0
1.96
1.78
1.63
1.18
0.87
0
1.73
1.68
1.37
1.06
0.78
0
1.59
1.44
1.28
0.06
0.70
0
2.80
O.U
1.OS
1.28
1.61
0.60
0.73
0.88
1.08
1.29
1.67
1.77
1.81
1.82
0
0.18
0.28
0.39
0.61
0.82
0
0.19
0.21
0.40
0.62
0.84
0
0.13
0.22
0.33
0.43
0
0.16
0.23
0.34
0.44
O.M
0.73
0.90
1.11
1.32
1.67
0.77
0.93
1.14
1.36
1.67
1.71
1.72
1.72
1.70
1.83
1.37
1.48
1.34
1.18
0.90
0.66
0
0.36
~
O.~
-0 -
O.M
0
0.18
0.23
0.33
0.43
0.68
0
0.18
0.24
0.36
0.48
0.67
0
0.17
0.26
0.38
0.47
0.68
0
0.17
0.23
0.38
0.49
0.80
0
0.18'
0.28
0.39
0.61
0.82
0.88
0.79
0.97
1.20
1.62
0.87
0.80
0.98
1.21
1.63
0.68
0.82
0.99
1.22
1.M
0.89
0.83
1.01
1.24
1.66
0.71
0.86
1.02
1.26
1.M
0.73
0.88
1.05
1.28
1.63
0.76
0.90
1.07
1.29
1.61
0.77
0.93
1.10
1.32
1.80
1.87
1.68
1.M
1.69
1.M
1.48
3.8l
4.M
4.18
2.22
2.38
2.43
2.48
2.46
2.42
1.97
2.08
2.13
2.18
2.17
2.11
1.81
1.88
1.94
1.98
1.96
1.89
1.71
1.78
1.78
1.81
1.79
1.73
1.88
1.88
1.88
1.70
1.87
1.82
1.69
4.27
4.18
2.69
2.80
2.M
3.02
3.01
2.93
1.82
1.69
1.69
1.87
1.62
1.39
1.26
1.09
0.84
0.81
0
4.01
3.71
3.27
2.&1
2.18
0
2.82
2.82
2.29
1.78
1.31
0
2.33
2.11S
1.88
1.43
1.08
0
2.02
1".88
1.83
1.26
0.93
0
1.80
1.86
1.48
1.66
1.62
1.33
1.03
0.78
0
1.M
1.41
. 1.24
0.98
0.71
0
1.46
1.32
1.18
0.90
0.88
0
4.27
1.1~
0.84
0
0.19
0.27
0.40
0.62
O.M
~
~
~
~
~
~
~
~
~
~
~
1/.
-
%/c
1-E
(ahead or
hiDe.)
--­
1 - %/c
B
(baok of
hinge)
..
0.06
E
A. Split Sap at a - 30 d. . . .
Split flap at 3 • 20 del....
-_. -­ - ­ - ­
-­ -­
-­-­
-­
-­
-­
-­
-­
-­
--­
-­
-­
-­
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.10
0.16
0.20
0.24
0.30
0.36
0.40
0.46
0.60
0.0&
0.10
0.16
0.20
0.21
0.30
0.3&
0.40
0.46
0.80
0.16
0.23
0.34
0."
0.&&
0.18
0.23
0.36
0.46
0.66
0.18
0.24
0.36
0.46
0.67
0.17
0.21
0.17
0.21
0.36
0.47
0.68
0.38
0.49
0.80
0.18
0.28
0.39
0.61
0.82
0.19
0.27
0.40
0.62
O.H
0.20
0.29
0.42
0.66
0.87
0
0.21
0.30
0.'"
0.&7
0.70
O.lS
0.22
0.33
0.'3
0.14.'
0.16
0.23
0.34
0.44
0.11
0.18
0.23
0.81
0.41
0.68
0.18
0.24
0.3&
0.48
0.67
0.17
0.26
0.38
0.47
0.18
0.17
0.28
0.38
0.49
0.80
0.18
0.26
0.39
0.61
0.82
0.19
0.27
0.40
0.62
0.84
0.20
0.29
0.42
0.&1
0.87
0.21
0.30
0."
0.67
0.70
0.71
0.86
1.03
1.29
1.73
0.73
0.88
1.05
1.31
1.73
0.76
0.00
1.08
1.34
1.73
0.77
0.93
1.11
1.38
1.73
0.80
0.90
1.16
1.42
1.74
0.84
1.00
1.19
1.46
1.74
0.68
0.79
0.97
1.23
1.M
0.66
0.80
0.98
1.24
1.86
0.88
0.82
1.00
1.28
1.86
0.69
0.83
1.01
1.27
1.67
0.71
0.86
1.03
1.29
1.88
0.73
0.88
1.06
1.31
1.88
0.76
0.90
1.08
1.34
1.88
0.17
0.93
1.11
1.38
1.89
0.80
0.96
1.16
1.42
1.70
O.M
1.90
1.88
1.90
1.78
1.81
1.46
1.28
1.83
1.84
1.79
1.63
1.48
1.32
1.16
4.&&
4.60
4.49
<&.21
•. 02
3.71
3.15
3.19
3.18
3.02
2.83
2.80
2.83
2.69
2.16
2.14
2.14
2.09
1.91
1.83
1.88
1.98
1.99
1.89
1.88
1.81
1.70
1.57
1.43
1.83
1.80
1.72
1.59
1.48
1.36,
1.78
1.73
2.14
2.32
2.38
2.32
a.19
2.04
1.87
1.73
1.68
1.18
1.48
1.36
1.22
0.99
3.36
2.96
2.38 I.M
2.08. 1.71
1.89
1.48
2.88
1.88
1.84
0.98
0
1.14
1.88
0.9&
0.70
0
1.29
1.12
1.01
0.73
0.13
0
1.21' 1.15
1.05
1.00
0.96 I 0.90
1.08
1.11
1.32
1.19
0. •
0.88
0
0
0.05
0.10
0.20
0.30
0.40
0
0.16
0.22
0.33
0.43
0.14.
O.SO
0.60
0.70
0.80
0.90
0.66
0.79
0.97
1.23
1.69
0.88
0.80
0.98
1.24
1.70
0.88
0.82
1.28
1.71
0.89
0.83
1.01
1.27
1.72
1.00
0.90
0.80
0.70
0.60
0.60
4.79
4.84
4.65
4.32
3.04
3.61
3.33
3.36
3.29
3.06
2.77
2.46
2.79
2.83
2.74
2.11
2.27
2.03
2.• 4
2.69
2.•0
2.21
2.00
1.77
2.22
2.28
2.18
1.99
1.80
1.69
2.08
2.09
1.99
1.84
1.86
1.<&8
1.97
1.98
1.81
1.12
1.64
1.36
0.40
0.30
0.20
0.10
0.06
0
3.06
2.80
2.13
1.14.
1.29
0
2.16
1.77 ,1.14.
1.61
1.30
1.23 \ 1.08
0.88
0.77
O.M 0.66
0
0
1.37
1.26
1.18
1.06
0.96
0.69
0.50
0
0.87
0.83
0.t5
0
1.17
0.98
0.81
0.69
0.42
1.83
1.&0
1.08
0.78
0
1.00
0
1.10
0.92
0.76
0.11
0.89
0
1.70
1.M
1.38
1.21
1.04
0.88
0.72
0.52
0.37
0
O.M
0.68
0.49
0.36
0
1.91
1.63
0
2."
2.32
0.81
0
1.92
1.82
1.69
1.14
J.M
1.12
1.41
1.28
1.00
1.19
1.46
1.70
~
~
~
~
-s
o~
~
~
C';J
I
1.38
1.21
1.09
0.79
0.17
0
O.es
0.49
o
I
O.M
0.47
0
1.09
0.96
0.81
0.81
0.44
0
~
THEORY OF WING SECTIONS
244
The incremental basic eft" and additional en., normal-force coefficients are
obtained from Eqs. (8.3) and Table 7. The appropriate values of the
TABLE 7.-VALUES OF
~Eaeg
~~I
5,10,15 I
30
20
4.
-0.474
0.05
0.10
0.15
0.20
0.25
-0.448
-0.423
-0.397
-0.372
-0.347
0.30
0.35
0.40
0.45
0.50
-0.294
-0.268
-0.242
0.55
0.60
0.65
0.70
-0.215
-0.189
-0.163
-0.136
-0.320
G
40
50
60
Plain flaps
-0.476
-0.451
-0.428
-0.404
-0.380
-0.477
-0.453
-0.431
-0.408
-0.387
-0.478
-0.455
-0.434
-0.411
-0.392
-0.479
-0.456
-0.435
-0.414
-0.395
-0.4i9
-0.456
-0.435
-0.415
-0.396
-0.357
-0.334
-0.311
-0.288
-0.265
-0.366
-0.346
-0.325
-0.304
-0.283
-0.372
-0.352
-0.332
-0.375
-0.357
-0.339
-0.377
-0.360
-0.342
-0.242
-0.220
b. Split flaps
......
. . . ...
......
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
~
......
......
......
I
I
......
......
......
......
incremental basic
Table 6 as follows:
I
P",
-0.476
-0.452
-0.430
-0.407
-0.385
-0.477
-0.454
-0.432
-0.409
-0.388
-0:478
-0.455
-0.434
-0.411
-0.392
-0.479
-0.456
-0.435
-0.414
":'0.395
-0.479
-0.456
-0.436
-0.415
-0.396
-0.363
-0.342
-0.320
-0.299
-0.278
-0.367
-0.347
-0.327
-0..372
-0.352
-0.375
-0.357
-0.339
-0.3i7
-0.360
-0.342
-0.30i
-0.287
-0.333
I
normal-force distribution are obtained by use of
The appropriate value of the incremental additional Paa normal-force dis­
ttibution is obtained from the data of Appendix I as follows:
(8.10)
HIGH-LIFT DEVICES
245
TABLE 8.-VALUES OF T'.
~
10
0
0.05
0.10
0.15
0.20
0
-0.00
-0.01
-0.01
-0.01
0.25
0.30
0.35
0.40
0.45
-0.01
-0.02
-0.02
-0.03
0.50
0.55
0.60
0.65
0.70
-0.03
-0.04
-0.05
-0.06
-0.07
~I
40
30
50
60
~--_.:...--_-_..:.-.._---=------~--_---:.._---
o
0.05
0.10
0.15
0.20
4.
~0.02
0
-0.02
-0.03
-0.04
-0.05
-0.06
-0.07
-0.08
-0.09
-0.11
-0.12
-0.14
-0.17
I
I
Plain Haps
0
-0.05
-0.0;
-0.09
-0.11
0
-0.09
-0.14
-0.18
0
-0.14
-0.23
-0.30
-0.23
-0.38
-0.14
-0.16
-0.19
-0.21
-0.25
-0.27
-0.32
-0.37
-0.42
-0.46
-0.54
-0.63
-0.71
I
0
-0.21
-0.34
-0.47
-0.59
I
I
-0.72
-0.85
-0.98
-1.11
-0.28
,
I
-..
----------··--0
b. Split
--r----O-
-0.02
f1~lPS
-0.04
-0.05
-0.05
-0.07
-0.10
-0.12
0.25
0.30
0.35
0.40
0.45
-0.06
-0.07
-0.08
-0.10
-0.11
-0.15
-O.li
-0.20
-0.23
-0.26
0.•50
-0.13
-O.2H
-0.03
­
-~.09 1--~.14···
-0.14
-0.18
-0.23
-0.2.
-0.32
-0.37
-0.42
o
;
-0.23
-0.30
-0.38
-0.21
-0.34
-0.47
-0.59
I
-0.46
-0.54
-0.63
-0.71
-0.72
-0.85
-0.98
-1.11
t
t
i1
!
I
i
The entire incremental normal-force distribution is obtained by adding the
incremental basic and additional distributions
(8.11)
The final load distribution is obtained by adding this incremental dis­
tribution to the load distribution for the unflapped lying section at the same
angle of attack (see Sec. 4.5).
Approximate hinge-moment coefficients may be obtained by an exten­
sion of this analysis." Such hinge-moment coefficients are nol considered to
be reliable because the discrepancies between the predicted and actual
THEORY OF WING SECTIONS
246
TABLB
~IIO
9.-VALUB8
20
01' 'F'.
30
40
50
60
1.00
1.06
1.09
1.11
1.12
1.00
1.10
1.15
1.18
1.21
1.00
1.15
1.22
1.28
1.32
1.14
1.14
1.15
1.15
1.23
1.36
1.38
1.39
1.39
a. Plain flaps
0
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
1.00
1.00
1.00
1.01
1.01
1.80
1.01
1.02
1.02
1.03
1.00
1.03
1.05
1.06
1.06
1.01
1.01
1.01
1.01
1.01
1.03
1.03
1.03
1.03
1.03
1.07
1.07
1.07
1.08
1.07
1.01
1.01
1.01
1.01
1.00
1.03
1.03
1.03
1.07
1.24
1.25
1.25
b. Split flaps
0
0.05
0.10
0.15
0.20
....
0.25
0.30
0.35
0.40
.....
....
....
.....
....
....
....
....
0.45
....
0.50
....
1.00
1.01
1.02
1.02
1.03
1.00
1.03
1.05
1.06
1.07
1.00
1.06
1.09
1.11
1.12
1.00
1.10
1.15
1.18
1.21
1.03
1.03
1.03
1.14
1.14
1.15
1.15
1.23
1.24
1.25
1.25
1.03
1.07
1.08
1.08
1.08
1.08
1.03
1.07
1.03
1.00
1.15
1.22
1.28
1.32
1.36
1.38
1.39
1.39
load distributions tend to be appreciable at the trailing edge where the
contribution to the hinge-moment coefficient is the largest.
In applying this method, it should be remembered that, although the
tabulated characteristics are differentiated by flap deflection, the important
variable is the degree to which the flap is stalled. The tabulated charac­
teristics for plain flap deflections up to 15 degrees are for unstalled condi­
tions, whereas those for larger flap deflections represent progressively
increased separation. Consideration should be given to the selection of
the characteristics to be used in order to represent the actual flow condi­
tions properly.
CHAPTER 9
EFFECTS OF COMPRESSmILITY AT SUBSONIC SPEEDS
9.1. Symbols.
CII pressure coefficient, (p - PflO)/ Jip 1f1
CIIe critical pressure coefficient corresponding to a loeal Maeh number of unity
B
to~ energy
K
constant
per unit ID888
M }'{ach Dumber, V fa
R universal gas constant, e" -
c~
8
croes-eeetional area of a stream tube
T absolute temperature
V velocity of the free stream
component of velocity along the span
V. component of velocity normal to span
(J
speed of sound
a lift-curve slope
v.
b span
c, section lift coefficient
Cp specific heat at constant pJ'e8SU1'e
specific beat at eoostant volume
Cr
e base of Naperian loprith~ 2.71828
In logarithm to the base e
m Jn888 flow per unit area
loeal static pressure
Po total pressure
PflO static pressure of the free stream
t time
t thickness ratio
u component of velocity parallel to the % axis
v volume
component of velocity parallel to the y axis
V
component of velocity parallel to the z axis
to
z; y, Z Cartesian coordinates
a angle of attack
fJ angle of sweep
.., ratio of the specific beats, c./c.
.,
P.
11"
P
1/ v''! - l\ft
ratio of the circumference of a eirele to its diameter
density of the fluid
ID888
9.1. Introduction. The wing-section theory and experimental data
presented in the preceding chapters are applicable to conditions where the
variation of pressure lR small compared with the absolute pressure. This
247
248
THEORY OF WING SECTIONS
condition is well satisfied when the speed is low compared with the speed
of sound. At a flight speed of 100 knots at sea level, the impact pressure is
only about 34 pounds per square foot as compared with an ambient static
pressure of 2,116 pounds per square foot. At 300 knots, the impact pressure
increases to approximately 321 pounds per square foot; and, at 600 knots,
the impact pressure is 1,494 pounds per square foot. At the higher speeds,
the pressure and corresponding 'volume changes are obviously not neg­
ligible. It is to be expected, therefore, that wing-section characteristics at
high speeds will not agree with those predicted by incompressible Bow
theory or with experimental data
obtained at low speeds.
Examination of the complete
equations of motion89 shows that the
parameter determining the effect of
speed is the ratio of the speed to the
speed of sound. This ratio is called
the "Mach number." The physical
significance of the speed of sound
becomes evident in considering the
difference between subsonic and sup­
ersonic flows, The speed of sound
is the speed at which pressure im­
pulses
are transmitted through the
FlO. 143. Significance of Mach angle.
air. At SIOlV flight speeds, the pres­
sure impulses caused by motion ~t the wing are transmitted at relatively
high speed in all directions and cause the air approaching the wing to
change its pressure and velocity gradually. Slow speed flows are accord­
ingly free of discontinuities of pressure and velocity. At supersonic speeds,
no pressure impulses can be transmitted ahead of the wing, and the pres­
sure snd velocity of the air remain unaltered until it reaches the immedi­
ate vicinity of the wing. Supersonic ft.ow is thus characterized by
discontinuities of pressure and velocity. As shown in Fig. 143, the
pressure impulses are propagated in all directions at the speed of sound
a while the 'ring moves through the air at the velocity V. The envelope
of the pressure impulses is a straight line (for small impulses) which
makes a slope sin-1a/V or sin-t M with the direction of motion, where
M is the Mach number. This line is called the" Mach line." The air
ahead of the Mach line is unaffected by the approaching wing.
The present discussion of the characteristics of wing sections in com­
pressible flow is limited to subsonic speeds. The pressure-velocity relations
for a stream. tube will be developed, and a brief summary will be presented
of compressible flow theory as applied to wing sections. The theory of
wing sections in subsonic flow generally consists of the determination of
EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS
249
the first-order compressibility effects on the incompressible flow solutions
previously developed. The theory is generally limited to speeds at which
the velocity of sound is not exceeded in the field of flow. A brief discussion
of experimental data will, however, include data in the lower transonic
region where the speed of sound is exceeded locally.
9.3. Steady Flow through a Stream Tube. a. Adiabatic Law. The
relation among pressure, density, and temperature for a perfect gas is
p = pRT
where (in consistent units)
p = pressure
p = density, moles per unit volume
T ~ absolute temperature
R = universal gas constant = Cp - c.
Cp = specific heat per mole at constant pressure
Ct, = specific heat per mole at constant volume
This equation may be written
(9.1)
or
PIl't
r;=
112"2
T2
where the subscripts 1 and 2 refer to any two conditions of/the gas. When
the temperature is constant (isothermal changes), the relation between
the pressure and the density is obviously
P1
PI
-=­
'P2 P2
Except for this special case, pressure and volume changes are accompanied
by temperature changes. In the high-speed ftO\VS of primary interest,
these changes occur so quickly that individual elements of the gas generally
do not lose or gain any appreciable heat. Changes in state of the gas are
thus assumed to be adiabatic. In this case,
(9.2)
where v = ratio of specific heats, cp/cv
From Eqs, (9.1) and (9.2), the corresponding temperature relation is
E! =
(Tl)"Y/{Y-l)
(9.3)
T2
b. Velocity of Sound. Expressions for the velocity of sound may be
derived from consideration of the equations of motion and of continuity.
The equation of continuity for a compressible fluid may be developed in a
manner parallel to that for Eq. (2.1) and is
P2
THEORY OF WING SECTIONS
250
a(pu)
ax
+ iJ(pv) + a(pw) == _ ap
ay
az
at
(9.4)
For motion in one dimension, this equation reduces to
iJ(pu)
ap
ax== - at
or
ap
au
iJp
-+p-+uat
ax
ax ==0
dividing 1;ly p
lap + au+~ap = 0
pat
(9.5)
pax
a2:
In the concept of a velocity of sound, the disturbances are considered to
be small 80 that the changes in density are small compared with the density
and the corresponding velocities are small. Retaining only the first powers
of small quantities, Eq. (9.5) becomes
! ap + au.... 0 = a In p + au
p at
ax
at
ax
(9.6)
For one-dimensional flow, Eq. (2.3) becomes
_ 8p == p a~+ pu8u
ax
at
ax
Again retaining only the first power of small quantities and dividing by p,
webave
au
lap
at -=-j;8x
(9.7)
It is known from the equation of state that p is a function of
(9.7) can then be written
au
at == -
Idp8p
; dp az ==
dp
a In
p.
Equation
p
- dp ax
(9.8)
Difierentiating Eq. (9.6) with respect to z and Eq. (9.8) with respect to t,
and combining, we have
h
dpatu
(9.9)
at' == dp8r
The solution of this equation is
u
=f{~ -
Ji, t) + ft(X+ Ji,t)
These functions represent disturbances traveling in opposite directions
with the veloc~ty V dp/dp. This velocity is termed the velocity of sound a.
As stated previously, the changes in pressure occur so rapidly that no
heat is considered to be conducted to or away from the elements of gas.
EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS
251
The relation between the pressure and density is therefore considered to
be adiabatic [Eq. (9.2)]. Then
fiiP = FYP
vdP
v-;
a=
(9.10)
For a perfect gas,
a == K-v'i'
(9.11)
where T is the absolute temperature, and the value of K is given in the
following table for various English units, and for dry air, -y == 1.4.
Units of temperature
Units of velocity
Feet per second
Miles per hour
Knots
OF. absolute
°C. absolute
49.02
33.42
29.02
65.76
.
.
.
44.84
38.94
In applying the concept of the velocity of sound to various problems,
it should be kept in mind that the concept as derived is valid only for small
disturbances. Very strong shock waves and blasts should not be expected
to propagate at speeds corresponding to the local velocity of sound.
c. Bernoulli's Eqootion for Comprem.1Jle Flow. In deriving Bernoulli's
equation for incompressible flow [Eq. (2.5)]J the followingexpression was
obtained: \
Writing this equation in integral form, we have
!2 JT2 + fdpP = E
(9.12)
Relating the pressure and density by the adiabatic law of Eq. (9.2) and per­
forming the indicated integration,
!
2
V2 + _'1_. ~
'Y-1p
zr
E
=
! V! + c T
2
"
(9.13)
This equation can also be derived from thermodynamic reasoning.
The thermodynamic reasoning indicates that the quantity E in Eq. (9.13)
is a constant along any stream tube for adiabatic changes whether reversible
or not. For example, E is constant throughout & stream tube containing a
normal shock. For reversible changes, Eqs. (9.2) and (9.13) together
with a knowledge of the initial conditions permit the calculation of the
variation of pressure and density with velocity.
252
THEORY OF WING SECTIONS
d. Cro8s-sectional Areas and PreBSUTe8 in a Stream Tube.
Along a
stream tube, the equation of continuity for steady flow is
pSV = constant
where S = cross-sectional area of tube
It is interesting to contrast the manner in which the cross-sectional area
varies with velocity for compressible and incompressible flow, For the
latter case, the area is seen to vary inversely with the velocity. The
determination of the variation of area with velocity for the compressible
case requires more extensive analysis. Differentiating the equation of con­
tinuity, and dividing by pSV, we obtain
! dB +! dp +.!..
8dV pdV V
=0
(9.14)
Differentiating Bernoulli's equation with respect -to the velocity,
V
or
+ _'Y_ (.! dp _ 1!.) dp = 0
'Y - 1 p dp
v + _1_ (1. a
2 _
'Y - 1 p
or
p'- dV
2
a ) dp = 0
p dV
a2 dp
V+ pdV =0
(9.15)
From Eq. (9.14),
dS=_~(l+Vdp)
dV
V
pdV
Substituting the value of dp/dV from Eq. (9.15), we obtain
dB
dV
=_
§.(lV2) = V
a2
~(1- W)
V
(9.16)
This relation shows that the stream. tube contracts as the velocity
increases for Mach numbers less than unity. The area of the stream tube is
a minimum for a Mach number of unity and increases with Mach number
for supersonic flow.
From consideration of Eqs, (9.2), (9.3), and (9.11), it may be seen that
the pressures, densities, temperatures, and velocities of sound are con­
nected by the equations
(9.17)
where the subscript m may denote the condition at any point but, in the
follo"ring development, the subscript will denote the condition at the
minimum cross-sectional area of the stream tube where the local Mach
number equals unity.
EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS
253
In order to obtain an expression for the cross-sectional area of the
stream tube in terms of the local Mach number M, it is convenient to
express Bernoulli's equation (9.13) as
!
2
l\I2 + _1_
'Y - 1
= ~+
2
1 am
2( 'Y - 1) a2
(9.18)
where a and am represent, respectively, the velocities of sound at any point
along the stream tube and at the point of minimum cross-sectional area.
From continuity, we can write
s;
pV
8 = Pma..
1.0
J
.8
I
'fI"
,/
~
<,
r-,
~
/
. J
.6
s"'~
/
.4
-,
/
r-,
r-,
"'r-;
J
.2
/
I
f
~ <,
-....... ---....
~
o
.4
.8
1.2
2.4
2.0
1.6
2.8
3.2
II
FIo. 144. Variation of etream-tube area with local Mach number.
where Sand S. are, respectively, the cross-sectional areas of the stream
tube at any point and at the point of minimum area. Substituting the
value of piPa from Eq. (9.17),
~=
(:j2/(Y-U
(a9
=
M
(0:)
(7+U/(y-U
and, from Eq. (9.18),
s. _. (
S -
l\{
~
+ .1
) ('Y+U/2<r-t>
(~- 1)M2 + 2
(9.19)
A plot of 8 m/S against M is given in Fig. 144.
The pressure relations along the stream tube may be obtained by sub­
stituting the value of am/a from Eq. (9.18) into Eq. (9.17)
p (
"y + 1
pm = (-y - l)M2
)y/(y-t>
+2
(9.20)
254
THEORY OF WING SECTIONS
where p == pressure at any point along stream tube.
It is often more convenient to express the local pressure in terms of the
pressure at the stagnation condition,
p (
2
)-r/<..,-I.)
Po == ('Y - 1)M2+ 2
(9.21)
where Po is the pressure of the fluid at rest.
The temperature relations may be obtained from Eqs. (9.17) and (9.18)
as
T
'Y+ 1
T,. == (y - I)M2 + 2
or
To == 1 + y - 1 MI
(9.22)
T
2
It is interesting to note that there is a limiting velocity which is reached
when the gas is expanded indefinitely into a vacuum. Bernoulli's equation
(9.13) may be written
111-+
20.2
1
'Y- 1
(a)'
a..
'Y+l
==2(-y-l)
With infinite expansion, the temperature approaches zero and the local
velocity of sound a approaches zero. Accordingly
V ....... =
a.
~'Y + 1
'Y-l
(9.23)
The local Mach number M, of course, continues to increase indefinitely as
the expansion is increased.
e. Relations for a N ormol8hock. In deriving the relations for the cross­
sectional areas, pressures, and temperatures existing in a stream tube, the
entire process was considered to be adiabatic and reversible. Experience
has shown that this condition is satisfied whenever the velocity is increasing
in the direction of flow. However, when an attempt is made to decelerate
supersonic flows, discontinuities in velocity, pressure, and temperature
generally occur. These discontinuities occur in a very short distance along
the direction of motion and are termed U shock waves." The velocity and
total pressure of the air decrease and the temperature and static pressure
increase in going through the shock. The process, although adiabatic, is
irreversible and characterized by an increase of entropy. The shock is
stationary for a tube of fixed configuration with constant pressures at the
inlet and exit. Since the velocity on the upstream face is supersonic, the
shock can hardly be considered to propagate at the velocity of sound.
Similar discontinuities occur forward of the nose of a blunt body traveling
at supersonic speeds. In the case of a simple tube or of the shock im­
EFFECTS OF COMPRES8IBILITY AT SUBSONIC SPEEDS
255
mediately forward of the nose of a blunt body, the plane of the shock is
normal to the direction of flow. This type of shock is a fundamental one
and is termed a "normal shock," as distinguished from the oblique shocks
shown in Fig. 143.
Relations for determining the conditions across a normal shock wave
from the conditions on one side are derived'" from considerations of mass,
momentum, and energy. If the subscripts 1 and 2 denote, respectively,
the conditions on opposite sides of the shock and if m is the mass flow per
unit area and p is the static pressure, the condition for continuity for a
shock stationary with respect to the observer is
m
The equation of conservation of momentum is
PIUl
=
P2~
(9.24)
=
P2 - PI = m(ul - U2)
(9.25)
The equation for conservation of energy is
PIUl -
PtU2
_!2 m (u~! -
Ut!)
=~
"Y- 1
(l!: _PI)
[.J2
PI
(9.26)
The solution" for these conditions is
P2
P; =
(P2/PI) (1' + 1) / (1' - 1) + 1
(p,./Pl) + (1' + 1)/ ()' - 1)
(9.27)
This equation relates the density and pressure ratios across the shock when
This equation corresponds to a
change in entropy except in the case where PI equals P2. It can be shown"
that P2 must be greater than PI for the change in entropy to be positive.
It would be convenient to relate the pressure and density rat.ios across the
normal shock to the upstream Mach number. Such an expression may be
obtained by expressing Eqs.. (9.24) and (9.25) in terms of the local Mach
number and combining to give
PI and PI are the upstream conditions.
P2 - 1 =
PI
Substituting the value of
1\'112 =
,.
Mt~'Y(l -~)
PI/ P2 from
'Y [1 -
\
P2
Eq. (9.27) gives
" (P2/Pl) -:- 1 ,
_
<P!lpl) + ("Y 1""" 1 C:)-y - 1) ]
<P:. pd (1' + 1);' (1' - 1) 1
(9.28)
+
From Eq. (9.24), we can write
M~ = ~~ ~
l\I 1
P2 P2
(9.29)
Values of the ratio Pot/Po. may be calculated from Eqs. (\J.23) , (9.27),
(9,,28), and (9.29) where po is the total pressure. This ratio is an index of
THEORY OF WING SBCTIONS
256
the useful energy remaining in the air downstream of the shock. Values
of Mt/M1, p,./pl, PI/PI, and PoJPo. are plotted against M in Fig. 145.
9.4. First-order Compressibility Effects. a. Glauert-Prandtl rule. The
Glauert-Prandtl rule36 relates the lift coefficient or slope of the lift curve
of a wing section in compressible flow with that for incompressible flo,v.
This relation was derived for the case of small disturbance velocities and
20 5 1.0
16 4
~
~
\"
.8
<,
M2
J2 3
r-,
"-
~
.6
I
PD2
r-,
,/
<,
~~
............
./
8
2
.4
/t,I"
4
I
/'
./
~
/
~
~
V
~
"
:>< ~
./
~
.2
~ PDI
V
?~ V
~
~
/
~
'"
V
~
~
V
7
./
."".,.,.,...,
~~
.c~2
P,
/'
---r-,
~
'-.
~
~
-
~ -..............
v~
r-----.. ~
r----.-
~~
L4
1.8
26
3.4
4.2
M,
FlO.
145. Pressure, density, and Mach-number relations for a normal shock.
low Mach numbers. These conditions are approximated for thin wing
sections with small amounts of camber at low lift coefficients at speeds well
below the speed of sound. The Glauert-Prandtl relation is
Cl c
at;
ci,
a;
1
vI-M2
-=-=-==~
(9.30)
where the subscripts c and i denote, respectively, the compressible and
incompressible eases. In deriving this formula, Glauert considered all the
velocities over the wing section to be increased by the same factor. Equa­
tion (9.30) may therefore be applied equally well to the moment coefficient.
The Glauert-Prandtl rule agrees remarkably well with experimental
data, considering the assumptions made in its derivation. A comparison
of this rule as applied to the slope of the lift curve for three wing sections
of 6, 9, and 12 per cent thickness is given" in Fig. 146.
KapianM obtained a first-step improvement of the Glauert-Prandtl rule
EFFECTS OF COMPRESSIBILITY AT ~UBSONIC SPEEDS
257
for the lift of an elliptical cylinder and extended it to arbitrary symmetrical
profiles, The Kaplan rule is
~~ = I' + 1 ~ t [1'(1' - 1) + ~('Y + 1)(1'2 - 1YJ
where
Il
(9.31)
1
vl-M2
= -_""'/===
;
t = thickness ratio
.24.-----..-----or----or----..,.--....--....
.201-----t-------t------+------1,........--+----1
.'61-----+-----+------I----I-oI-~---&.--_t
dCt
da,
.12....------4-----+-­
~-~~-+-_+__-t----_t
.081-----'-------1-----+------;------1
.2
.6
.4
.8
·ID
N
FlO. 146. Lift-eurve slope variation with Mach number for the NACA 0012-03. 0009-63.
and 0006-63 airfoils.
Equation (9.31) approaches the Glauert-Prandtl rule as the thickness ratio
approaches zero. It will also be noted that the Kaplan rule shows a "peak
dependence on the ratio of the specific heats 1'. The Kaplan rule for wing
sections 5, 10, 15, and 20 per cent thick in air together with the Glauert­
Prandtl rule are plotted in Fig. 147.
b. Effect oj j{ach number on tile Pressure CoejJicicnt. The Glauert­
Prandtl rule supplies a first approximation to the variation of pressure
coefficient with Mach number.
C~l
Cp ,
1
=
vI - ]\{2
THBORY OF WING SECTIONS
258
Numerous attempts have been made to obtain more accurate expressions
for the variation of pressure with Mach number, notably by Chaplygin,15
Temple and Yarwood, 119 and von Karman and Tsien. 129,14O Garrick and
Kaplan" succeeded in unifying these results 88 approximate solutions of
It',l
0.20
~
J5
2.5
rule
./0
1.05
I
I
I
~Kaplon
J
I I
1.Pra1dt1-G!aJerf
I
I
Ir'
III
rule
JI
2.0
flI
'1 /
J
1.5
~
1·°0
-----ae
~
0.4
0.6
0.8
1.0
M1
FIG. 147.
Ratio of lifts for compressible and incompressible fluids as function of stream
Mach number.
the general problem and presented two other approximations. No simple
general solution of the problem is known.
The I{arman-Tsien relation is widely used in the United States. Ex­
perimental evidence appears to indicate that this relation is as applicable
as any of the solutions. This relation is
CPM
1
CPt = viI - M2 + [M2/(I + viI - M2)J(C p J 2)
(9.32)
EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS
259
A plot of the pressure coefficient as a function of the free stream Mach num­
ber AI is given in Fig. 148 for a number of values of the incompressible
/.0
.9
.8
1
.~
i· 7
~
t~
~
~
.6
~
u~
.5
.4
.3
.2
.­
~
./
°0
./
.4
.5
.6
.7
Free-stream Mach 1III11ber,Mo
148. Variation of local preesure coefficient and local Mach number with free stream
Mach number according to KArm8.n-Tsien.
FIG.
pressure coefficient. Also shown in Fig. 148 are curves of the local Mach
number ML.
A special use for relations such as Eq. (9.32) is the prediction of the
stream Mach number at which the velocity of sound is reached locally
THEORY OF WING 8ECTIONS
260
over the wing section. Jacobs" applied the Glauert-Prandtl rule to this
problem. It is now customary in the United States to use the Karman­
3.5
./
.4
,,5
.6
.7
Free -stream MochnumberlMo
FIG.
.8
.9
148. (Concluded)
Tsien relation [Eq. (9.32)]. This problem may be solved from Fig. 148 by
following the curve for the highest incompressible pressure coefficient on
the wing section to the line ML -= 1 and reading the corresponding stream
EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS
261
Mach number. Figure 148 may be used conveniently when the maximum
local pressure coefficient is known experimentally at some subcritical Mach
number. If the incompressible pressure coefficient is known theoretically,
it is more convenient to use Fig. 149, which gives the corresponding criti­
cal Mach number directly.
1.0
.9
.8
1
\
\
I
\
II
~
-,
r-,
<,
~
~:--
~
.4
I
I
.~
.4
I
l6
2.0
LDw-spetJd presstn coefficient, Cp
.8
/2
2.4
2.8
Flo. 149. Critical Mach number chart from KarmAn-Tsien relat.ion.
9.6. Flow about Wing Sections at High Speed. Consideration of the
effects of compressibility on the flow over wing sections may be conveniently
divided into two parts" The first part deals with the characteristics in
the range of subcritieal Mach numbers. 111C second part deals with the
characteristics at supercritical Maeh numbers. The critical Mach number
dividing these t\VO ranges is defined us the stream Mach number at which
the local velocity of sound is just attained at any point in the field of flow.
a. Flow at Subcritical Moen Numbers. The first-order effects of com­
pressibility on the potential flow at suberitical Mach numbers are well
described by the theoretical relations of Sec. 9.4. An indication of the
accuracy of the Karman-Tsien relation is given!" in Fig. 15Ob. This
figure shows the experimental pressure distribution for the NACA 4412
section at a Mach number of 0.512 and a pressure distribution predicted
from the experimental data obtained at a Mach number of 0.191 (Fig.
THEORY OF WING SECTIONS
262
-1.6
-1.2
~-
-.8
-A
c'P
0
-- ---­
~
..........
/
........... ~
.
r
~
--
~
-.......
~
~~
...
r
~
........
~ .......
~~
-­
~
~
(b) M=0.5/2
(a) M=0./9/
>
~~
(
.8
--/.6
-­ ~
-1.2
~
.......~
./
'
I
-.8
f
,..
r
4
-.4
Cp 0
--
""'­
.......
.......
-I. 6
~
-I. 2
\
\
.... ~,
I
r----."",,-:
~
1',
--.
rr
-=:-....
~~
817 ~
<,~......
'i~
~
-
\
"\
/
J
"<, ......
4y-. ~~
""lr
""'-..... ......
°T
....
:
4
Iii
~
~
~
"<,
\
\
Cpc
~
-'i'
......1'.
~...... I'-o..
~~
(f)
(e) M=Q664
8
~
(d) M=0.640
"'
Cp -.
~,
-­I tp;
(c) M=0.596
.8
,
~
M:Q690
;
I. 2
-I.2
-. 8"....
-.4
t
i
I.........
~
~.-
~
-._~
,
\
.'"
1Jc
~
~~
OJ
~-
,
.......
'~ ......
..
Dotted line in(b)represents application of Kormon-Tsien
.......~
.4
-
-~
-~
(g) M=O.735
8
..
reloliJn 10 doloin tal
x t,pper surface
o Lower surface
~
I. 0
20
40
60
Chor~percenf
FIG.
80
100 0
20
40
60
80
Chord,percefl'
150. Pressure distribution for the NACA 4412 airfoil. a=: 1°52.5'.
100
EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS
263
15Oa). The accuracy appears to be satisfactory for such applications.
The larger area of the pressure diagram at the higher Mach number corre­
sponds to an increased lift coefficient at the same angle of attack. The
resulting change of lift-curve slope has been shown (Fig. 146) to agree
fairly well with the Glauert-Prandtl rule.
The applicability of these theoretical relations is limited to cases where
the effects of separation are not marked. If separation is present, the
increased pressure gradients associated with higher Mach numbers tend to
increase the separation and thus change the whole field of flow, Moreover,
the absence of appreciable separation at low speeds is not necessarily a
sufficient criterion, because the increased pressure gradients at the higher
Mach numbers may cause separation in marginal cases. The theoretical
relations may therefore be applied with confidence only to wing sections
of normal shape at low lift coefficients.
b. Flmo at Supercritical Mach Numbers. The flow at supercritical
Mach numbers is partly subsonic and partly supersonic and is characterized
by the presence of shocks. No theory dealing with these mixed subsonic
and supersonic flows has been developed to the extent of being useful in
the prediction of wing characteristics. The almost complete lack of theoret­
ical treatment requires reliance on experimental data at supercriticaI Mach
numbers.
For the typical case of a wing section operating at a small positive lift
coefficient, the velocity of sound is reached first on the upper surface, as is
shown in Fig. 150 by the pressure coefficient increasing negatively beyond
the critical value CPc_ Figure 150c shows no drastic change in the pressure
distribution when the local velocity' of sound is exceeded by a small amount.
Many experimental data indicate that drastic changes in the forces on wing
sections do not occur until the critical Mach number is exceeded by a
small but appreciable margin. There is some doubt as to whether a shock
necessarily occurs when the velocity of sound is locally exceeded by a small
margin. DC In any case, the losses associated with a shock from very low
supersonic velocities are very small, as shown b)" Fig. 145. Such small
losses would not be expected to produce drastic changes in the field of
flow.
A shock occurs when the critical Mach number is exceeded appreciably,
as shown by Fig. 150d to g. The shocks act, at least qualitatively, like
a normal shock to reduce the velocity to subsonic values. The resulting
rather sudden changes in the pressure distribution are shown dotted in
Fig. 150. Schlieren pictures of the shocks corresponding to the diagrams
of Figs. 150! and g are shown-" in Fig. 151. The positions of the shocks
as shown by the photographs correspond closely with the dotted regions of
the pressure diagrams. Figure 152 presents another series of schlieren
photographs showing the shocks on a NACA 23015 wing section at an
THEORY OF WING SECTION8
264
angle of attack of 3 degrees. These pictures.!" which were taken in the
NACA rectangular high-speed wind tunnel, show the progressive intensi-
FIG.
151. Schlieren photograph of flow for the NACA 4412 f!,irfoil. a
=
1°52.5'.
fication and rearward movement of the shock with increasing Mach num­
ber.
The character of the flow with shock is greatly affected by the inter­
action of the shock and the boundary layer. It is apparent that the shock
cannot extend to the surface through the region of Iow velocity in the
EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS
265
boundary layer. If the plane of the shock is perpendicular to the flow,
the increase of static pressure through the shock will correspond to that
shown in Fig. 145. The relatively high pressure downstream of the shock
will tend to propagate upstream through the boundary layer. The pres­
M:O.40
M=O.60
M=O.550
M:O.650
FlO. 152. Schlieren photographs of flow. N.:\CA 2:J015 airfoil. NACA rectangular high­
speed wind tunnel. AiI"foil chord, 2 inches. Angle of attack. 3 degrees.
sure distribution measured at the surface cannot , therefore, be expected
to show a discontinuity. Figure 150 shows that the sharp pressure rise is,
in fact, shown as a steep gradient at the surface. Such propagation of the
pressure along the boundary layer necessarily leads to pressure gradients
across the boundary layer in the vicinity of the shock. This situation is
quite different from the usual boundary-layer conditions at low l\Iach
numbers.
The large adverse pressure gradients associated with the shock always
produce a large increase of boundary-layer thickness and frequently cause
266
THEORY OF WING SECTIONS
separation, as shown U 4 in Figs. 152 and 153. This separation is an im­
portant or even predominant factor in causing the large force changes
occurring at supercritical speeds. The relative importance of the shock
losses and the effects of separation on the drag of a typical wing section
M=O.750
M:O.70
M:O.821
M=O.80
FIG.
152. (Concluded)
are shown-" by Fig. 154. This figure shows the total pressure loss across
the wake of the section. These losses are intimately associated with the
drag, and, to the first order, the drag coefficient is proportional to the area
under the curve. Intense shock has not occurred at the lower Mach num­
ber, and the diagram has the typical shape obtained at lo,v speeds. Intense
shoek has occurred at the higher Mach number, and the drag coefficient
has obviously increased. This increase may be considered to be of t\VO
parts. The very large losses of pressure in the center of the diagram are
similar in shape to the lower speed diagram and are attributed to the usual
type of loss associated with skin friction and sep RXat,i on. The smaller
267
EFFECTS OF COMPRE88IBILITY AT 8UBSONIC SPEEDS
pressure losses extending far out into the stream are attributed to the
losses across the shock. It is apparent that the drag rise associated with
the usual type of loss is as important as that directly associated with the
losses across the shocks.
The pressure rise measured'P on the surface of the wing section in the
vicinity of the shock is smaller than that corresponding to a normal shock
with the losses of total pressure indicated by
the wake surveys. The pressure rise correspond­
ing to a normal shock may also be obtained
by means of Fig. 145 from the local Mach num­
ber ahead of the shock as indicated by pres­
sure measurements on the surface. In this case,
also, the computed pressure rise is greater than
that shown by the surface pressures. The com­
plete explanation of this discrepancy is not
known, but two factors could tend to reduce
l\>1 0.691
the pressure rise on the surface below that FIG. 153. Schlieren photograph
of separated flow for rear portion
corresponding to a normal shock.
of NACA 23015 airfoil. XACA
The first factor is the pressure gradient rectangular biJ!h-slx-t'd wind
through the boundary layer previously dis­ tunnel. Airfoil ehord, [; inehes.
Angle of ut tar-k, (. dt~~rN'~'
cussed. The importance of this factor may he
inferred from Figs. 152 and 153. The lines in the schlieren photographs
correspond to density gradients. Dark Jines are 8ho''''11 in the vicinity of
81:
()
~.4-
~
~
.s-:
M
Q810
t1
~ .2­
15
~
.708
./­
.~
Col)
§
0 240 200
/60
/20
80
40
0
40
80
/20
160
200
240
Distance across KOte.l perceri chord
FIG. 154. '''''ake shape and total pressure defect as influenced by Mach number,
0012 airfoil, a = 0 degrees; NACA rectangular high-speed wind tunnel.
~ACl~.
the shock making a small angle with the surface. Although these density
gradients may be partly associated with temperature gradients, the sharp­
ness and intensity of these lines are thought to indicate the presence of
appreciable pressure gradients.
The second factor is the possible reaction of the boundary layer on the
shock. The thickening or separation of the boundary' layer tends to pro­
duce oblique shocks, Such shocks would produce some pressure rise ahead
THEORY OF WING SECTIONS
268
of the normal shock and would accordingly reduce the intensity of the
normal shock, especially close to the surface. The extent to which the
normal shock is locally softened in this manner is uncertain, but Figs. 152
and 153 suggest that this mechanism may be of considerable importance.
Investigations made by the NACA, by Liepman,· and by Ackeret4 and
his associates emphasize the complexity of the interaction of the shock and
the boundary layer. These investigations indicate important differences
between the shock phenomena for laminar and for turbulent boundary
layers. Although exceptions have been noted, laminar boundary layers
.4
CD
em
-·~I
.2
.3
.4
.5.6
.8
Mach numbtJr,M
FIG. 155.. Force and moment coefficient variation with Mach number. NACA 23015 airfoil.
a == 0 degrees; NACA rectangular high-speed wind tunnel.
tend to produce multiple shocks of lower intensity, or U lambda-type"
shocks, while those associated with turbulent boundary layers tend to
resemble intense normal shocks more closely. The understanding of these
phenomena awaits further theoretical and experimental investigations.
The typical changes of forces experienced at supercritical speeds by a
wing section of moderate thickness are shown!" in Fig. 155. The Mach
number for force break is seen to be appreciably higher than the critical
Mach number. At" Mach numbers higher than that for force break, the
lift coefficient decreases, the drag coefficient increases, and the moment
coefficient usually increases negatively.
9.6. Experimental Wing Characteristics at High Speeds. The existing
three-dimensional lying theory, which is the basis for the concept of section
characteristics, is not valid for supercritical speeds where part of the flow
is supersonic. Consequently section data cannot be applied quantitatively
to the prediction of the characteristics of wings at such speeds. Moreover,
present trends for wings designed for efficient operation at supercritical
speeds are toward low aspect ratios and large amounts of sweep. As
EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS
269
pointed out previously, section data are of doubtful quantitative signif­
icance for such plan forms at any speed. Under these circumstances, the
detailed consideration of section characteristics as design data is not
warranted.
Nevertheless, the characteristics of wings at supercritical speeds de­
pend to some extent on the lying sections employed. The variation of the
wing characteristics with wing section appears to be qualitatively similar
to the variation of the wing-section characteristics- if the lying plan form
is such as to permit any semblance to two-dimensional flow, Consequently,
the purpose of this section is to present typical data showing qualitatively
the major effects of variation of the wing profile at high subsonic Mach
numbers.
a. Lift Characteristic«; One of the first and most severe effects of com­
pressibility encountered in flight was the tendency of many airplanes to
"tuck under" in high-speed dives. This U tucking under" tendency con­
sisted of a large negative shift in the angle of trim tbgether with a large
increase of stability that resisted the efforts of the pilot to trim at the
desired positive lift coefficients required for recovery from the dive. The
resulting large elevator motions required to recover from the dive, or even
to prevent the dive from becoming steeper, corresponded to excessive con­
trol forces, thus leading to the impression that the "~t.ick became frozen."
These changes in longitudinal stability limited the maximum safe flying
speed for many airplanes.
These effects are intimately associated with tile changes in lift charac­
teristics of wing sections at Mach numbers above that for force break.
The relatively thick cambered wing sections commonly used during the
Second ,Vorld ar experienced a positive shift of the angle of zero lift and
a reduction of lift-curve slope. Although ID3ny effects associated with
other characteristics of the airplane are present, the change in angle of
zero lift directly affects the angle of trim and the reduction of lift-curve
slope directly increases the longitudinal stability.
The shift in the angle of zero lift is associated with the camber of the
wing section. ..~ symmetrical section shows, of course, no change in the
angle of zero lift 116 (Fig. 156). The rather large positive shift in the angle
of zero lift shown!" in Fig. 157 for the X.~(~.:\ 2409-34 section is typical
for cambered sections without reflex. In this case, the change amounts to
about 2 degrees for a change of Mach number from 0.80 to only 0.83.
The effect of increasing the camber is showu'" by Fig. 158. Doubling the
amount of camber causes a similar shift to occur at a lower Mach number.
The effects of thickness on the lift characteristics of"symmetrical sec­
tions are shown" in Figs. 159 to 161 for the N..~CA 0006-34, 0008-34, and
0012-34 sections. The 6 per cent thick section shows a large increase of
lift coefficient at a given angle of attack with increasing Mach number at
"r
THEORY OF WING SECTIONS
270
values below about 0.85. At Mach numbers above the force break, the
lift coefficient decreases rapidly, but it never goes below the low-speed
values, at least at Mach numbers up to 0.95. The 12 per cent thick section,
however, fails to show much increase of lift coefficient at subcritical Mach
numbers, but it shows a large decrease of lift coefficient at Mach numbers
just above the force break. At a Mach number of 0.85, the lift coefficient,
c::
----~ -====­
I/.
4
,
~
....-:"'~
~
~~
~
~~
~~
,
t
.06
,,
\
-.
,
I
.... ...
f~-~ f'... .. ~---~
---- ,
,,
-.2
I
,
,,
f--
--
o
~
:~
,
,,
,,
,
,,
,/
l,/
,
l
"
M=
~
.40-.60-.70--.80 ---- +-.83 ---,-- t
r
V
V
~
_......... :,...-' ~ ,~.........
~
-- --.2
I I
.- ---<o---_..J ___
-- -
-0:::--
.4
I
t
--
\
.6
1
.B
-~_.-
- --
.~
to
Lifl coefficienfl Ct
FIG.
156. Aerodynamic characteristics of the NACA 0009-34 airfoil.
and accordingly the lift-curve slope, are only about two-thirds of the low­
speed value. At higher Mach numbers, the lift coefficients again increase.
The effects of Mach number on the lift characteristics of the 8 per cent
thick section are intermediate. These data show that the thin sections
experience force breaks at higher Mach numbers than the thicker sections
and that the lift force breaks are much less severe for the thin than for the
thicker sections.
The effects of thickness for a series of cambered sections are shown" in
Figs. 162 to 165. These data indicate to an even greater degree the adverse
effect of thickness on the character of the lift force break. Comparison of
these data with those for the symmetrical sections (Figs. 159 to 161) shows
the adverse effects of camber on the Mach number for lift force break anti
on the shift in the angle of zero lift.
EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS
/2
01/./"
V')
8
~~~ ~;'
.-4#
4
~
,"
---
.., '
~
~~
""
,'"
,
~
".
~
~
,,~
,
~~
~:;::; 7'/
"/
~~
If'
~
.. "
V"
I
,
I
,
./2
.40-.60-.70--BO-··.8.3 -- ---
./0
I
,
,,
<,
.02
~
./
~
r--
-.2
.. - ---
"'~
.-
,. ,
,
,
I
,
I
~.
--r--'
o
J
'1
1/'I
I
j
,I
t: I
I
I
I
/ ')
I
,
I)
hI
,
~
-~ ~
.~
r-
s:
--- ---. -- --
..2
J
I
"
I
v
,
",,;
~-
Ii
II
/1
-,'
i
I
,I
,
, ,"
-"
I
1
I
,
--.,
J
/
I
,,
i', -..
,
I
M=
.04
/
-:~
~
:;::;--
1-
~~
-",
.4
.6
.8
LO
Lift coefficientI Ct
FlG.la7. Aerodynamic cbaractBistics of the NACA 2409-34 airfoil.
271
THEORY OF WING SECTIONS
272
Comparatively few data are available to show the effect of thickness
distribution on the characteristics above the force break. The trends
indicated by Stack and von Doenhoff116 cannot be considered representa­
tive for properly designed families of "ring sections because of the arbitrary
,
.--
4
",-
,/
../'
./
,
I
.- , /
-
--~
~~
~
~.'"
~~
9
d
."
~ rt/
:
..........
17
,.1
./0
If
M=
,I
.40-.60-.70--.80 ----
V
.08
-, --
/"
\
\
,~
/'
-,
!
!
,/
~'
~-
\ ,
\
...... ~',
.02
~ ""-
f--
.---
-
.~
".,.
...
v·/
-
....=:::::: :;;;0.0-
~
,/
~
./
--- -.2
o
.2
Lifl
FIG.
.--.6
.8
T-
.4
/.0
coefficien~ c&
158. Aerodynamic characteristics of the NACA 4409-34 airfoil.
geometrical manner in which the sections were varied. If the thickness
distributions are varied in such a manner as to avoid peaks in the low­
speed pressure distributions, as in the case of the N ACA 6-series sections,
the indications are that the ~ffect of thickness distribution is minor com­
pared with the effects of thickness ratio and camber. There is some indica­
tion 1l 6 that the trailing-edge angle should be kept small to avoid adverse
effects on the lift characteristics similar to those for thick sections.
Figures 166 to 168 show the17b lift and drag characteristics of the NAC..~
66-210 wing section. The most outstanding feature of these data as com­
EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS
.7
.6
1
a."
.5
/
fdegJ
5
.....
4
~
3
2
./
~
;..­ ~
L----~
~
~~
-­
--"
v
V
~
~~
jf
~~
~
_L.-J ~
lla-- ~
I/'
/
~ ~~
\~
1/ r\~
, -...a
1.1 ~~
\I
d
~""'"
~~
,..,. ~
~
~~
,-oc:
~ r<\
o
-:1
o
.1
.2
.s
.5
.4
Moch
.6
.7
.8
.9
1.0
ooroe«, M
FIG. 159. EfI'e("t of compressibility on the lift of the N ACA OOOH-34 airfoil.
.7
..
)V
.6
ao
Itdegl
.5
6
.4
/
~V
1_ ~ V
5
"IIl"'­ ~ .....
4
-"~
~
~
~
~
......
v:
l,I
~
I' I"
~,
V
~
'r'( ~
W'
L".....-ol
1\ ~
.JIn \
W
\ '~
~
.3
3
...-
~
~
j.....­
~,..-
'\
....-I~
2
t-­ ~ ~
./
0
~~
~
~
~
~..-
~~
1
L.P
'"""
~
~
~
0
./
-I
...
w
.....
~
-
~
~KI"
.2
o
./
.2
.3
.4
.5
.6
.7
.8
.9
1.0
Mach number~ M
Flo. 160. Effect of compresaibilit:r on the lift of the NAC.A 0008-34 airfoil.
273
.7
.6
.5
.4
.1'
.3
.6
~
...2
(d':;J
..
L­
6
.6
~ ~
~tA
-
5
I
.~
4
~
3
1.3
Set>
:t: .1
I~
~
§
SO
J'..
u
-:1
c ......
.,~
-
I
o
~
~
1\ ~, rr
u... lrl 1/. ~
~ l't'
~
I\' ']
~
e
./
1.$I!
'"
....
~L.
~
-.2
~
...J~
-:~
"""'ll'­
A
0
-A
-./
0
./
.2
.3
.4
.~
.4
.~
.~ l
• J
Jr,o
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FlO.
161. Ettect of compressibility on the lilt of the NACA 0012-34 airloil.
.8
.9
1.0
.Mach ntI'IIber 1 M
FIG.
162. Eft'eot of compressibility on the lilt ot the NACA 2306 airfoil.
.8
.1
I I J J_ J
",JTIII
'Ir­
-
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163. EJleot of eompre88ibility on the lift of the NACA 2309 airfoil.
1
I
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L~
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-,....-
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to-­
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Mach IHJfTIber, M
FIG.
_
I
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er.
->-(deg)-,--
~_
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.9
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164. Effect of compressibility on the lift of the NACA 2312 airloil.
THEORY OF WING SECTIONS
276
pared with th~ for the NACA
f~ur-digit series secti~ns is the ability of
the 6-series seefion to carry large lifts at moderately high angles of attack
and high Mach numbers. The lift-curve slope for the NACA 66-210
section at a Mach number of 0.75 is high and substantially constant up to
a lift coefficient of about 0.8 (Fig. 168). At the same Mach number, the
NACA 2309 section (Fig. 163) shows a reduced lift-curve slope at a lift
.8
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Mach IVfTIIJer. M
FIo. 165. Effect of compressibility on the lift of the NACA 2315 airfoil.
coefficient of only about 0.55 and almost no increase of lift for a change of
angle of attack from 4 to 5 degrees.
Figure 169 ShO\VS376 the predicted critical Mach numbers and the Mach
numbers for force divergence as functions of the lift coefficient for the
NACA 66-210 section. The outstanding characteristic of these data is the
wide range of lift coefficients over which high Mach numbers for force
divergence are realized as compared with the range for high critical Mach
numbers. The predicted critical Mach numbers are lower than those for
force divergence but approximate the latter values over the range of lift
coefficients where the speed of sound is first reached near the location of
minimum pressure at the design lift coefficient. It is apparent that critical
Mach numbers predicted on the basis of the attainment of the velocity of
EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS
277
sound near the leading edge are useless as an indication of the force charac­
teristics of NACA 6-series sections.
The general effects of Mach number on the maximum lift coefficients of
wing sections may be inferred from the data'19 of Fig. 170. These data were
....- Symbol a
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FIG. 166. The variation of section lift coefficient with ~lach number at various angles of
attack for the NACA 66-210 airfoil.
obtained from tests in the NACA Langley Hi-foot high-speed wind tunnel
of a lying having an aspect ratio of Gand having X.•\CA 23016 sections at
the root and NACA 23009 sections at the tip. At Mach numbers above
0.30, the maximum lift coefficient decreases rapidly with Increasing Mach
number up to values of about 0.55 where the maximum lift coefficient is
approximately 1.0. At higher Mach numbers, the actual maximum lift
coefficient decreases more slowly; but the angle of attack, or lift coefficient,
at which the lift curve departs radically from its normal slope continues to
THEORY OF WING SECTIONS
278
decrease to the limits of the tests. The resulting variation of maximum
lift coefficient with Mach number is shown" in Fig. 171.
'
The increase of maximum lift coefficient with Mach number at values
below 0.30 is associated with the variation of Reynolds number for these
.20
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Mach fJII'TJberlM
Flo. 167. The variation of section drag coefficient with Mach number at various angles of
attack for the NACA 66-210 airfoil.
tests. This effect was studied more extensively in the NACA 19-foot­
pressure wind tunnel where the effects of Mach and Reynolds numbers
could be separated" for low Mach numbers. These data indicated, at least
qualitatively, that the peak maximum lift coefficient is determined by the
critical Mach number at maximum lift which, of course, is attained at
relatively low free-stream Mach numbers. At subcritical Mach numbers,
the maximum lift coefficient is primarily a function of the Reynolds num­
ber.&8 at very low Mach numbers, although the effect of Mach number is
279
EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS
l2
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700
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FIG. 168. The variation of section lift coefficient with angle of attack at various Mach nurn­
hera for the NACA 66-210 airfoil,
.300
.500
.9
,
I
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169. Critical Mach numbers for the NACA 66-210 airfoil.
""
t-,
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5ecIion lif/ coefficien/)c1
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.8
THEORY OF WING SECTIONS
280
not negligible. At supercritical Mach numbers, the value of the maximum
lift coefficient decreases rapidly with increasing speed, and the effect of
Reynolds number is secondary. For the wing tested, the critical Mach
1.6
I
M=O.20 ..... <:»
.30-".
/.4
~
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FIG.
~
-8
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0
4
8
/2
/6
20
24
28
a1deg.
170. Variation of wing lift coefficient 'with angle of attack for several Mach numbers.
"-88
number
of the order of 0.25 to 0.30 for the flaps-retracted condition
and of 0.17 to 0.20 with O.2Oc split flaps deflected 60 degrees.
It should be noted that the maximum usable lift coefficient for an air­
plane at high Mach numbers may not be the same as the actual maximum
lift coefficient. Referring to Fig. 170, the airplane may be unable to fly on
the relatively flat portions of the lift curves above the force breaks, be­
cause of the serious buffeting associated with the partly separated flow and,
perhaps also, because of stability and control difficulties.
EFFECTS OF COJIPRESSIBILITY AT SUBSONIC SPEEDS
281
At supercritical speeds, changing the camber near the trailing edge, as
by small deflections of a flap or aileron, cannot affect the flow over the
front part of the section because pressures cannot propagate upstream
through the supersonic region. Consequently, the flap effectiveness must
be greatly reduced at supercritical Mach numbers. This effect is clearly
shown!" by Fig. 172. This loss of flap effectiveness is related to the positive
1.6
1.4
V
~
_
......
............. """"'"
<,
,
r-,
1.2
\.
\
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~/.O
Maxinun lifl
coefficient
.\
-, \
~
rsa
Lift coefficient ob/oinedot 2°
10 3°0b0ve the angle of ottoc«
01 whichinitiol separation ofthe '
flow from the wing occurred
.8
r---
--t,
V" "
",
"
.6
_2
.3
.4
.5
.6
.7
M
FIG.
171. Variation of wing maximum lift coefficient with 1\tlach number,
shift of the angle of zero lift for cambered sections. The rearward port.ion
of a cambered section rna)" be considered to be a special case of a deflected
flap. The loss of flap effectiveness at the force break leaves the forward
portion of the section at an effectively lower angle of attack, and the
resulting change in lift coefficient corresponds to a positive shift of the
angle of zero lift. Conversely, a strongly reflexed section would show the
opposite effect. It follows that the change in lift characteristics of cam­
bered sections may be reduced by incorporation of reflex or of an upwardly
deflected flap.
b. Drag Characteristics. The effect of thickness ratio on the drag
characteristics of symmetrical wing sections is illustrated" by Figs. 173 to
175. These data show the typical rapid increase of drag coefficient at high
Mach numbers. This rapid rise (force hreak) occurs at smaller Mach
numbers as the angle of attack and thickness are increased. Similar data
THEORY OF n'ING SECTIONS
282
6,0·
-p~
M,O.44
-I
-0
\,)q"
...:
.~
I
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~
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I
"M,073
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,
,
50
100
-----lJpper surface
o
50
,00 0
50
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Percent chord
- - Lower surface
FIG. 172. Variation of aileron action with Mach number.
tribution results. Symmetrical airfoil, 19 per cent thick, a
high-speed wind twme1 ..
Two-dimensional pressure dis­
0 degrees; NACA rectangular
==
EFJt'ECTB Oil COMPRESSIBILITY A7' SUBSONIC} SPEEJJS
.08
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Mach number~M
FIG.
173. Effect of compressibility on the drag of the NACA 0006-34 airfoil.
./0
.09
IJ
V
.08
I
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V,
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FIG. 174. Etfe~t of Mmpre..."Sibility on the drag of the NACA 0008-34 airfoil.
283
THEORY OF WIJ.VG SECTIONS
284
,.
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FIG.
./
.2
.3
.4
.5
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.7
.8
.9
to
Mach ntmber, M
175. Effect of compressibility on the drag of the NACA 0012-34 airfoil.
./2
I
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.101
I I
I
1 1 I
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11
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to
Mach fNJmber,M
Flo. 176. Effect of compressibility on the drag of the NACA 2306 airfoil.
I
.03
.01
00
~
a
~
~ .()6
~
,
.09
i· O?
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Mach number, M
~
.r
.8
.9
1.0
177, F.Jrect of compressibility on the drag of the NACA 2309 airfoil.
~
~
~
THEORY OF Ti'ING SECTIONS
286
for the cambered sections, NACA 2306, 2309, 2312, and 2315, are shown lO
in Figs. 176 to 179. The effect of camber is to reduce the Mach number at
which the drag force break occurs as compared with the data of Figs. 173
to 175. Although the thickness distribution is not the same for these
1
.121-~~---t--t---+---+-+---f----t---+---+---+---t---+--+--+-~-I--I
.111--~1--t---t--t---+-+--+--f---+---+---+--+---t--+--+--1-"""
,
r
J
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.......... .07 J---.ot---lf--+--+-+--+---+---+--+--+--+--I--+---+--+-+-Io+-fM.~1---I
I
.~
.~
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~ • 061---'~~--+--t--+-+--+--+--+----t----t---+--t-~"""'t-'l-fW--+----t--t
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a
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) I J 'JJ
V it r~(1
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~ .05 ~t---I--t---t---t---t---+--f---+---t--+---+---t---+-tl""-+-lll~""'Il--+--+--t
cs
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4
3
2
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./
.2
I
.3
.4
.5
S
.7
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Moch f'lU77ber,M
J4·IG. 178. Effect of compressibility on the drag of the NACA 2312 airfoil.
symmetrical and cambered sections, the differences attributable to this
fact are very small, at least for the lower thickness ratios.
Comparisons of the predicted critical Mach numbers and the Mach
numbers for drag force break are presentedf" in Fig. 169 for the NACA
66-210 wing section. For the range of lift coefficients corresponding to
high predicted critical speeds, the Mach number for drag force break is
about 0.01 to 0.03 greater than the critical. The Mach number for drag
force break is intennediate between that for lift force break and the critical
EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS
287
value. The slope of the boundary for drag force break is similar to that
for the critical Mach number, but high Mach numbers without force break
r
I,
I
I~
"l
v I
f
~
If
I I j
I I :/
I/~·
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~
.04 ---....--+-.....--4-(deg}
I---I---"--....-..+- 5
j
It
--+--+--I--~-+---+-;J-~~""""'--"""--'---+---t
4
3
2
i
.4
.5
.6
.7
.8
,9
1.0
Mach nunber, M
179. Effect of compresaibility on the drag of the NACA 2315 airfoil.
.010
FlO.
/
-I
.I
.2
.3
may be realized over a much wider range of lift coefficient than is indicated
by the theoretically predicted critical speed.
c. M omen: Characteristics. Figures 157 and 158 illustrate116 the nega­
tive shift of the moment coefficient with increasing Mach number that is
THEORY OF WING SECTIONS
288
typical for cambered wing sections. In addition to this shift, there is a
tendency at the highest Mach numbers shown for the moment coefficient
to increase negatively with increasing lift coefficient as is shown37b more
.3
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.9
Mach number, M
FIG. ISO. The variation of section moment coefficient with Mach number at various angles
of attack for the NACA 66-210 airfoil.
clearly for the N ACA 66-210 section in Fig. 180. This rearward shift of the
aerodynamic center at high Mach numbers is stabilizing, and the more
negative values of the moment coefficient tend to make the airplane trim
at lower angles of attack. These moment changes thus add to the pre­
viously discussed effects of the lift coefficient in producing the "tucking
under" tendency of airplanes with thick cambered wing sections.
EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS
289
The moment characteristics of the NACA 0006-34, 0008-34, 0012-34,
2306, 2309, 2312, and 2315 wing sections are shown" in Figs. 181 to 187.
These data indicate that the changes in moment characteristics with Mach
/
o
..lli
~
_. Cz =QI
-./
./
-­
o
Ct
1"'"'04kt...
-.....I:t
~-
=Q2
...
'Vll
~~
we
Ct =0.3
o
,
./
Ct
FIG.
181.
1
I
-./
./
2
3
.4
i '~~
I
{
=0.5
.5
~~
hJ
I
o
o
"'
Ct =Q4
-./
I
.6
7
i
8
.9
to
Mach number, M
Effect of compressibility on the pitching moment of the N ACA 0006-34 airfoil.
number are minimized for symmetrical sections, although a rearward shift
of the aerodynamic center at high Mach numbers persists,
9.7. Wings for High-speed Applications. As discussed in the previous
section, all wing sections experience undesirable variations in their charac­
teristics at high subsonic Mach numbers. Although these undesirable
variations can be minimized by the use of thin symmetrical sections, it
does not appear to be possible to eliminate them in two-dimensional flow.
In three-dimensional flow, however, it is obvious that the shocks cannot
extend tothe wing tips without modification. The presence of the shock at
THEORY OF WING 8ECTIONS
290
./
o
Ct =QI
-./
.1
I
I
o
-
T
ct. =0.2
t-· I
J
~ .1
TTl T
.~
~
§
i
~
1I 1I
0
.,..~
C =Q3
t
--
... 1
.1
o
~
Ct
=0.4
-~
-.1
./
1 I I
11 I
o
-.1
FIG.
r I
ct. =0.5
1
-",~
~~
.4
.5
.1
.4
.6
l '.0
9
Mach numbe'l M
182. Effect of oompre88ibility on the pitching moment of the NACA 0008-34 airfoil.
0
1
.2
.3
EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS
291
./
"­
o
~
Ct =0./
-./
.,
-..W-O
""'" i'D«l
c" =0.2
~
~
-./
./
I
-
o
j
T
ct. =Q4
-"0
FIG.
~
ct. =Q3
./
.2
.s
.4
.5
.6
ra...
I
.7
.8
--.
~
.9
1.0
Mach rvmber,M
183. Etlect of compreeaibility on the pitching moment of the NACA 0012-34 airfoil
THEORY OF WING SECTIONS
292
o
-.1
o
-.1
~
T r-­
I I I
I
Ct
~~
T
I
r-.r ~
I
I I I
J
.§
~
~
-.~
~
.A.
1
~
I
~'""-
I I I
Ct
=0
I
1
I
I""--. I...A..
I
fI-·1
- I~ ~
=-0./
Ct
I
o
)~
>--. ~
=-0.2
o
-.1
~
I I I
.,.,
ct. =Q/
~
V
~~
-.-' 0
a
j
I
r
r
Ct
=0.2
0
1
I
.L
o
-.1
1
I
~
Ct
I
I
Ct
'-"'" '-c
~ L­
~
~ l-ft..
=Q3
.1
rv""
I
1"'-.-..
~~l4.
=0.4
•
i.
T
I
~
Ct
I
_a.
1~ ~
I I I
=Q5
o
FIG.
~
I I I
I
-:.1
....
I I I
o
-:/
~~
I I I
-:1
-.1
""'-t ~~
'T
I
I I r
1\.e
.............
'"' ~
~~
Ct=Q6
o .1 .2 .3
.4
.5
.6
.7
.8
.9
I,{)
Mach ntmberl M
184. Effect of compressibility on the pitching ~oment of the NACA 2306 airfoil.
EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS
o
)~
I I I
.......
ITt
-./
Ct
o
=-Q2
I I I
I T
-./
I
.-.'1'
"""'117'
.)~
.-
~
",,,,,,,,,. ~
CZ=-O./
o
I I I
~t lo'
t
I I
I T I
Cl =0./
j~
""""'Il~
~~
LA
I I I
r I
-1
-./
..
~
~
~
~
~
~
~
c t =0.3
o
I I I
p--r 1
-./
FIG.
."
ct =0.2
o
W~
-­ r.ea., ~!r ..
I I I
1 1 f
-./
o
293
Ct
./
l
=Q4
.,...
"
~
.4
.5
.6
.7
.8
.9
/.0
Mach numbet; M
185. Effect of conlpressibility 011 the pitching moment of the NACA 2309 airfoil.
.2
.J
THEORY OF WING SECTIONS
294
o
I I I
I 1 I
-./
)"'" ~
---.~
Ct=O
o
I
I
1
I
.­
./ N ~
~
Ct.= 7.1
I I I
~ ~~
I I I
c" =Q2
1 I 1
-./
I 1 I
.­
""'lJIIIl
I I I
-./
FIG.
~
~
~
ct =0.3
o
o
-A.
I I I
./
.2
.3
c,,=Q4
.6
.5
Ma:h number. M
.,
.7
.8
-
rw
~~
.
(
o
L
186. Effect of compressibility on the pitching moment of the NACA 2312 airfoil.
EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS·
o
295
". ~
~
ll(
~ ~
t ;,0
-J
C
-
o
-./
Ct
~r
~
-J't'"
~~
~
""
-­
ct.~a2
o
-
:.Mil
'L
~
-./
ct - Q3
o
...-III
~
.1
.~
..
.4
.5
~
~
Ct=a4.
-:/
0
".".
.6
.T
,8
.9
I.
Mach number, M
FlO.
187. Effect of compressibility on the pitching moment of the NACA 2315 airfoil.
296
THEORY OF WING SECTIONS
the lying tip with the accompanying sudden pressure rise would induce a
three-dimensional flow that would tend to reduce the severity of the shock.
The existence of such H tip relief" became apparent in early tests of propellers
at high tip speeds. The data showed that losses of efficiency did not OCcur
until tip Mach numbers were reached that were well in excess of those at
which the sections used at the tips suffered large increases of drag in two­
dimensional flow, Corresponding advantages for "rings may be expected
if the aspect ratio is sufficiently low, Experimental data have confirmed
the advantages of low aspect ratio at high Mach numbers.
The theoretical approach for wings of very low aspect ratio is entirely
different from that for normal wings of high aspect ratio, and the concept
of wing-section characteristics is not applicable. Jones" showed that, for
wings with aspect ratios approaching zero, the lift depends on the angle
of attack and on the positive rate of increase of span in the direction of the
air flow, or
db
C, = ret dx
where C, = local lift coefficient of section perpendicular to direction of air
flow
a = angle of attack
b = span
x = distance in the direction of air flow
db/dx is positive
This simplified theory indicates that C, = 0 for those portions of the wing
where db/d,x is zero or negative. This theory shows that the center of lift
for a triangular wing with the apex forward is at the center of area. The
theory indicates that, for the particular ease of a slender triangular plan
form, the lift coefficient and center of pressure are independent of Mach
number at both subsonic and supersonic speeds so long as the wing lies
well within the Mach cone. The applicability of the theory at a Mach
number of 1.75 is shown by Fig. 188. Low-aspect-ratio wings of triangular
plan form thus appear to provide one solution to the stability difficulties
in the transonic speed range.
Another solution to the problems of the transonic speed range is the use
of large amounts of sweep. The manner in which sweep is effective is
clearly seen from consideration of a wing of infinite span with sideslip
(Fig. 189). It is obvious that the spanwise component of velocity V. will
produce no forces on the wing, neglecting viscosity, and that the force on
the wing will be determined by the normal component of velocity V".
The- resulting velocity l' may be transonic or even supersonic, while the
normal component V ft is less than the critical speed if the angle of sideslip,
or sweep, fJ is sufficiently large. This effect of sweep in avoiding compres­
EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS
297
I
.os"
""
li <,
' ..
~
~/
/
/v
~
/
.02
)
Cm
o
)
/
~ TheoreIicol
(slope
~a02J
V
-
-
./Theorelical
,V
.~
-.02
V
/
.l'
V
V
/V
-.04
-.J
,
/V
-2
-I
0
I
2
3
«,dig
188. Test of triangular airfoil in Langley model supersonic tunnel 1\lach number, 1.75;
Reynolds number, 1,600,000.
FIG.
veIOCi/,ry component normal
( 10 wingtVcosjJ)
--,----~
117
streon veJocily..... V
Angleof
sweea-«
fi
Spamvise velocity
component (VsinjJ)
FIG.
189. Velocity components on a swept wing.
THEORY OF WING SECTIONS
298
10------------ 80mm - - - - - - - - - - -..
-k::
~---------------
Profile R-4009
I
I
1 /
1/'
/
~ol
rnetJSUI'1!fTI(
400
<,,/1
(.,'-1'
-..tso~
FIG.
190. Wing placed obliquely across a two-dimensional wind tunnel.
-qs
x
D,S
~O
o
0
u
P
1
- e - /J-'Oj• ,......
cos' 0 -O""'i ~c'tiS'ZlF
H=(J
FIo. 191. Comparison of pressure distribution over normal and oblique wing.
EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS
299
sibility troubles was independently discovered by German investigators
and by JonesU during the Second World War.
Application of this principle results in wings with large amounts of
sweepback or sweepforward. Such plan forms obviously violate the simple
two-dimensional considerations at the root and tip even if the wing has no
taper. These violations, however, do not appear to be sufficient to destroy
the value of sweep if the wing is of reasonably high aspect ratio, as in­
dicated by tests'" of a wing placed obliquely across a two-dimensional wing
tunnel (Fig. 190). Figure 191 shows that the pressure distribution over
the center section of the oblique wing is reasonably close to that expected
from the simple theory, despite the fact that the tunnel w8JIs violate the
assumed end conditions. Experimental studies have shown that swept
wings of all useful aspect ratios are effective in postponing and alleviating
compressibility troubles. Consequently, the theory of swept wings of
arbitrary plan forms is being intensively investigated. 1I • u , lG. U7
It thus appears that the wings of the high-speed airplanes of the future
may be quite different from those of comparatively slow subsonic air­
planes. These highly swept low-aspect-ratio "rings, however, have serious
disadvantages for the low-speed flight necessary for landing and take-off.
Some of these disadvantages are lo\v lift-curve slopes, high angles of stall,
and poor stability characteristics. These disadvantages seriously com­
promise the design of airplanes using such plan forms and may force the
use of more conventional plan forms with very thin supersonic sections
for some applications. In anycase, these disadvantages at slow speeds are
expected to dictate the use of conventional plan forms for airplanes de­
signed for such speeds that it is possible toavoid serious compressibility
difficulties by the use of thin wing sections.
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6. ALLEN, H. JULIAN: Calculation of the Chordwise Load Distribution over Airfoil
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7. ALLEN, H. JULIAN: A Simplified Method for the Calculation of Airfoil Pressure
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17. BRASLOW, .ALBERT L.: Investigation of Effects of Various Camouflage Paints and
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300
REFERENCES
22.
301
F.: Aerodynamic Data for a Wing Section of the Republic XF-12
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23. CAHILL, JONES F.: Summary of Section Data on Trailing-edge High-lift Devices.
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25. CHAPLYGIN, SERGEI: Gas Jets. NACA TM No. 1063, 1944 (from Scientific Mem­
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26. CIlABTEBS, ALEX C.: Transition between Laminar and Turbulent Flow by Trans­
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27. DRYDEN, H. L., and KUETHE, A. l\{.: Effect of Turbulence in Wind Tunnell\feasure­
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CAHILL, JONES
302
THEORY OF WING SECTIONS
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61.
62.
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303
NACA. Tl\1 No,
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218-234, October 1945; and Part III, vol. 3, No.4, PP. 277-301, Januar.)", 1946.
67. LIPPISCH, A., and BEUSCHAUSEN, 'V.: Pressure Distribution Measurements at
High Speed and Oblique Incidence of Flo,,·. NACA Tl\1 No. 1115, 1947.
68. LoFTIN, LAURENCE K., JR.: Effects of Specific Types of Surface Roughness on
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69. LoWRY, JOHN G.: "rind-tunnel Investigation of an NACA 23012 Airfoil with Several
Arrangements of Slotted Flap with Extended Lips. NACA TN No. 808, 1941.
70. MUNK, l\IAX !\'1.: The Determination of the Angles of Attack of Zero Lift and
Zero Moment, Based on Munk's Integrals. NACA TN No. 122, 1923.
71. l\IUNK, !\{AX M.: Elements of the \Ving Section Theory and of the Wing Theory.
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72. :l\{UNJt, Max 1\1.: Calculation of Span Lift Distribution (Part 2), Aero Digest,
vol. 48, No.3, p. 84, Feb. 1, 1945.
73. NAlMA.N, IRVEN: Numerical Evaluation of the e-Integral Occurring in the Theo­
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1944 (Wartime Rept. No. L-136).
73a. NAD1AN, IRVEN: Numerical Evaluation by Harmonic Analysis of the e-Funetion
of the Theodorsen Arbitrary Potential Theory. NACA ARR No. LSRI8, Septem­
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14. NEELY, RoBERT H., BOLLECH, THOMAS V., WESTRICK, GERTRUDE C., and GRAHAM,
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LAMB, HORACE:
LEES,
304
77.
THEORY OF WING SECTIONS
W.: Wind-tunnel Tests of a Wing with a Trailing-edge Auxiliary
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81. PINKERTON, RoBERT ~'I.: Analytical Determination of the Load on a Trailing
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82. PINKERTON, ROBERT 1\1.: Calculated and Measured Pressure Distributions over
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83. PLATT, ROBERT C.: Aerodynamic Characteristics of Wings with Cambered Ex­
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84:. PLATr, ROBERT C., and ASBOTr, IRA H.: Aerodynamic Characteristics of NACA
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85. PLATT, H.OBERT e., and SHORTAL, JOSEPH A.: Wind-tunnel Investigation of "rings
with Ordinary Ailerons and Full-span External Airfoil Flaps. NACA Rept, No.
603, 1937.
86. POHLIIAUSEN, 1\:.: Zur niiherungsweisen Integration der Differentialglcichung der
laminaren 'Grenzschicht, Abhandl. aero. Inst. Aachen, 1. Lieferung, 1921. See also
z. angelO. ,,"tath. !alech., Btl. 1, Heft 4, pp. 252-258, 1921.
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1928 (originally presented to the Third International Mathematical Congress,
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88. PRANDTL, L.: Applications of Modern Hydrodynamics to Aeronautics. NACA
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90. PUCKETT, Allen E.: Supersonic Wave Drag of Thin Airfoils, J. Aeronaut. Sci.,
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91. PURSER, PAUL E., FISCHJ<;L, JACK, and RIEBE, JOHN l.f.: Wind-tunnel Investigation
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92. PURSER, PAUL E., and JOHNSON, IIAROLD S.: Effects of Trailing-edge Modifications
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93. QUINN, JOHN II., JR.: Tests of the NACA. 65,-018 Airfoil Section with Boundary­
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94. Qt:IN~, JOHN H., JR.: Wind-tunnel Investigation of Boundary-layer Control by
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95. ,QUINN, JOHN H .• JR.: Tests of the NACA 641A212 Airfoil Section with a Slat, a
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1293, 1947.
NOYES, RICHARD
REFERENCES
305
96. RECANT, I. G.: Wind-tunnel Investigation of an NACA 23030 Airfoil with Various
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97. REID, E. G., and BAMBER, M. J.: Preliminary Investigation on Boundary Layer
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98. ScHILLER, L.: Die Entwicklung der laminaren Geschwindigkeitsverteilung und ihr
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1935.
100 ScHLICHTING, H.: Zur Entstehung der Turbulenz bei der Plattenstromung, Nachr.
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102. ScHRENK, OSKAR: Experiments with a \Ving Model from Which the Boundary
Layer Is Removed by Suction. NACA T~I No. 534, 1929.
103. ScHRENK, OSKAR: Experiments with a Wing from \Vhich the Boundary Layer Is
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104. SCHRENK, OSKAR: Experiments with Suction-type Wings, NACA TM No. 773,
1935.
105. ScHUBAUR, G. B.: Air Flow in the Boundary Layer of an Elliptical Cylinder.
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106. ScHUBAUR, G. B., and SKRAMSTAD, If. 1\:.: Laminar-boundury-layer Oscillations
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107. ScIIULDENFllEI, l\{ARVIN J.: \Vind-tunnel Investigation of an NACA 23012 Air­
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lOS. SHERMAN, ALBERT, and HARRIS, ''1''. A.: The Effects of Equal Pressure Fixed Slots
on the Characteristics of a Clark-Y Airfoil. :KACA rrN No. 507, 1934.
109. SHElWAN, ALBERT: A Simple l\lcthod of Obtaining Span Load Distribution.
NACA TN No. 732, 1939.
109a. SILVERSTEIN, ABE, KATZOFF, SAMUEL, and lIooTMAN, JAMKS: Comparative
Flight ar.J Full-scale Wind-tunnel Meusureruents of the :\In.xinluDl Lift of an
Airplane. NACA Rept. Ko. 618, 1938.
110. SIVELLS, JAMES C.: Experimental Characteristics of Three \Yinb"S of NACA
64-210 and 65-210 Airfoil Sections with and without 2° \Yashout. !\.~Ci\ 'l"X No.
1422, 1947.
Ill. SIVELLS, JAMES C., and KEELY, ROBERT II.: Method of Calculating Wing Char­
acteristics by Lifting-line Theory Using Xonlinear Section Lift Data, K.ACA 1'N
No. 1269, 1947.
112. SQUIRE, H. B., and YOUNG, A. D.: The Calculation of the Profile Drag of Aero­
foils. R. & l\{. No. 1838, British AIlC, 1938.
113. STACK, JOHN: Tests of Airfoils Designed to Delay the Compressibility Burble.
NACA TN No. 976, 1944.
114. STACK, JOHN: Compressible Flow's in Aeronautics, Eighth Wright Brothers Lec­
ture, J. Aeronaut. s«; vol. 12, No.2, April, 1945.
115. STACK, JOHN, LINDSAY, \V. F., and LI'M'ELL, ROBI-.:RT E.: The Compressibility
Burble and the Effects of Compressibility on Pressures and Forces Acting on an
Airfoil. NACA Rept. No. 646, .1938.
306
THEORY OF WING SECTIONS
116. STACK, JOHN, and VON DOENHOFF, ALBERT E.: Tests of 16 Related Airfoils at
High Speeds. NACA Rept. No. 492, 1934.
117. STEWART, H. J.: Lift of a Delta. Wing at Supersonic Speeds, Quart. Appl. Math.,
vol. IV, No.3, pp. 246-254, October, 1946.
118. TAm, ITIBO: A Simple Method. of Calculating the Induced Velocity of a Mono­
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August, 1934­
119. TEMPLE, G., and YARWOOD, J.: The Approximate Solution of the Hodograph
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120. TETERVIN, NEAL: A Method. for the Rapid Estimation of Turbulent Boundary­
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1944 (Wartime Rept. No. L-16).
121. THEODORSEN, THEODORE: On the Theory of Wing Sections with Particular Refer­
ence to the Lift Distribution. NACA Rept. No. 383, 1931.
122. THEODORSEN, THEODORE: Theory of Wing Sections of Arbitrary Shape. N.ACA
Rept. No. 411, 1931.
123. TuEODORSEN, THEODORE: Airfoil Contour Modification Based 011 e-Curve Method
of Calculating Pressure Distribution. NACA ARR No. IAG05, 1944 (Wartime
Rept. No. L-135).
124. THEODORSEN, THEODORE: ..~ Condition on the Initial Shock. NACA TN No.
1029, 1946.
125. THEODORSEN, THEODORE, and GARRICK, 1. E.: General Potential Theory of Arbi­
trary 'Ving Sections. NACA Rept. No. 452, 1933.
126. THEODORSEN, THEODORE, and REGIER, ARTHUR: Experiments on Drag of Re­
volving Disks, Cylinders and Streamline Rods at High Speeds. NACA .A.CR
No. IAFI6, 1944 (Rept. No. 793).
127. TOLLMIEN, "rIO: The Production of Turbulence. NACA T1"1 No. 609, 1931.
128. TOLLMIEN, W.: General Instability Criterion of Laminar Velocity Distributions,
NACA T!\l No. 792, 1936.
129. TSIEN, Hsue-Shen: Two-dimensional Subsonic Flow of Compressible Fluids, J.
Aeronaut. s«, vol. 6, No. 10, pp. 399-407, August, 1939.
132. VON DoENHOFF, ALBERT E.: A Method of Rapidly Estimating the Position of the
Laminar Separation Point. NACA TN No. 671, 1938.
133. VON DOENHOFF, ALBERT E., and ABBOTr, FRANK T., Ja.: The Langley Two­
dimensional Low.- turbulence Pressure Tunnel. NACA TN No. 1283, 1947.
134. VON DoENHOFFJ ALBERT E., and TETERVIN, NEAL: Investigation of the "aria­
tion of Lift Coefficient with Reynolds Number at a Moderate Angle of Attack on
a Low-drag Airfoil. NACA CB, 1942 (Wartime Rept. No. L-661).
135. VON DOENHOFF, A. E., and TETEBVIN, NEAL: Determination of General Relations
for the Behavior of Turbulent Boundary Layers. NACA ACR No. 3G13, 1943
(Rept. No. 772).
136. VON KARMAN, T.: tJber laminare und turbulente Reibung, Z. angew. Alath. It/ech.,
Bd. 1, Heft 4. pp. 233-252, 1921.
137. VON KAmIAN, T.: Gastheoretische Deutung der Reynoldssehen Kennzahl, Abhandl.
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138.. VON K.AB.MAN, T.: Mechanical Similitude and Turbulence. NACA Tl\{ No. 611,
1931.
139. VON KA.mUN, T.: Turbulence and Skin Friction, J. Aeronaut. Sci., vol. 1, No.1,
pp. 1-20, January, 1934.
140. VON K!RM:!N, T.: Compressibility Effects in Aerodynamics, J. Aeronaut. Sci.,
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REFERENCES
307
141. VON KAmfIN, T., and MILLIKAN, C. B.: On the Theory of Laminar Boundary
Layers Involving Separation. NACA Rept. No. 504, 1934.
142. VON KAmfAN, T., and TSIEN, H. B.: Boundary Layer in Compressible Fluids,
J. Aeronaut. s«, vol. 5, No.6, April, 1938.
143. WARnELD, CALVIN N. for the NACA Special Subcommittee on the Upper
Atmosphere: Tentative Tables for the Upper Atmosphere. NACA TN No. 1200,
1947.
144. WEICK, FRED E., and ILuuus, THOMAS A.: The Aerodynamic Characteristics of
a Model Wing Having a Split Flap Deflected Downward and Moved to the Rear.
NACA TN No. 422,1932.
145. WEICK, FRED E., and PLATr, RoBERT C.: Wind-tunnel Tests on a Model Wing
with Fowler Flap and Specially Developed Leading-edge Slot. NACA TN No.
459, 1933.
146. WEICK, FRED E., and SANDERS, ROBERT: Wind-tunnel Tests of a Wing with Fixed
Auxiliary Airfoils Having Various Chords and Profiles. NACA. Rept. No. 472,
1933.
147. 'WEICK, FRED E., and SHORTAL, JOSEPH 1\.: The Effect of Multiple Fixed Slots
and a Trailing-edge Flap on the Lift and Drag of a Clark Y.Airfoil. NACA Rept.
No. 427, 1932.
148. WENZINGER, CARL J.: Wind-tunnel Investigation of Ordinary and Split Flaps on
Airfoils of Different Profile. NACA Rept. No. 554, 1936.
149. WENZINGER, CARL J.: Pressure Distribution over an Airfoil Section with a Flap
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150. WENZINGER, CARL J.: "rind-tunnel Tests of a Clark Y \Ving Having Split Flaps
with Gaps. NACA TN No. 650, 1938.
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152. WENZINGER, CARL J., and DELANO, JAMES B.: Pressure Distribution over an
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153. 'VENZINGEB, CAllL J., and HARRIS, THOMAS A.: Wind-tunnel Investigation of
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154. WENZINOER, CARLJ., and HAlUlIS, THOMAS l\.: Wind-tunnel Investigation of NACA
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155. WEN ZINGER, CARL J., and H.O\RRIS, THOMAS A.: Wind-tunnel Investigation of an
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No. 677, 1939.
156. WENZINGER, CARL J., and RoGALLO, FRANCIS M.: Resume of Airload Data on
Slats and Flaps. NACA TN No. 690, 1939.
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1921, PP4 32-34.
157. YOUNG, A. D.: Note on the Effect of Slipstream on Boundary Layer Flow. Rept.
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158. YOUNG, A. D.: Further Note on the Effect of Slipstream on Boundary Layer
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159. YOUNG, A. D., and ~{ORJUS, D. E.: Further Note on the Eff~t of Slipstream on
Boundary Layer Flow. Rept. No. B.A. 1404h, British RAE, September, 1939.
160. ZALOVICIt, JOHN A.: Profile Drag Coefficients of Conventional and Low-drag
Airfoils as Obtained in Flight. NACA ACR No. 1AE31, 1944 (Wartime Rept.
No. L-139).
APPENDIXES
APPENDIX I
BASIC THICKNESS FORMS
CoNTEN)'S
NACA Designation
0006
0008
0008-34
0009
Page
311
312
313
314
315
316
317
0010
0010-34
0010-35
0010-64
318
0010-65
319
0010-66
0012
320
0012-64
321
322
323
0015
324
0018
325
326
0012-34
0021
327
0024
16-006
16-009
16-012
16-015
.
.
.
.
331
16-018 .
332
328
329
330
16-021 .
333
63-006 .
334
63-009 .
335
63-010 .
63r.Q12.
63r015.
63r018.
6&-021.
336
337
338
339
340
341
63AOO6
342
63A008
63AOIO
631A012
344
~A015
345
343
346
64-006 .
309
THEORY 0// WING SECTIONS
310
Page
347
348
N ACA Designation
64-008 .
M-009 . . . . · .
64-010 .
641-012.
64:r01S . . • · • • • • . .
64:r018. . . . . . . . .
64.-021. .
349
350
351
352
353
. ....
64AOO6
64AOOS
354
65-009 . .
355
356
357
358
359
360
361
66-010 .
65t-012.
363
65r015. .
3M
MAOI0
64tAOI2
MsA015 .
66-006 . .
65-008 . .
382
65r018. .
65A21..
365
. .....
366
65AOO6
867
368
65AOO8
65AOIO
65 1A012 .
369
370
371
65tA015 .
66-006 .
66-008 . .
66-009
66-010 . .
~~12. . . . . . . .
66r015 . .
66r018. .
372
.
373
374
375
376
66r021 ..
377
378
379
67,1-015 .
747A015 .
380
381
APPENDIX I
311
I••
,_ -----......
~
r--- '---
r---
h
.8
0006
-MICA
\
-
.4
.",.-
~
D
.4
I
x
I
y
(per cent c) __
(pe
__
r _OO_Il_t_c}_.
o
o
I
(t' /1')2
.
1.0
.8
.6
%C
1'/l'
0.5
1.25
2.5
5.0
.....
0.947
1.307
1.777
0
0.880
1.117
1.186
1.217
0
0.938
7.5
2.100
2.341
2.673
2.869
2.971
,_._
I
.1J.'./ V
\----1
3.992
2.015
1.057
1.364
1.089
1.103
0.984
0.696
1.225
1.212
1.206
1.190
1. t 79
1.107
1.101
1.098
1.091
1.086
0.562
0.478
0.378
0.316
0.272
50
60
70
3.001
2.902
2.647
2.282
1.832
1.162
1.136
1.109
1.086
1.057
1.078
1.066
1.053
1.042
1.028
0.239
0.189
0.152
0.123
O.(l97
80
1.312
1.026
1.013
0.073
~:~
;::~
10
15
20
25
30
40
~.
90
95
100
__.. _ ;:~_l_J:
I~.E.
radius: 0.40 per cent c
7'HEORY OF lVING SECTIONS
312
1.6
1.2
r:
I
-- t---
100....
---.. r---.
r--- r---. . . .
.8
/MCA OOOB
.4
L,-.---
I'---
-
-",
L.--- ~
o
.4,
.8
.6
1.0
S,C
z
11
(V/V)t
V/V
At'./V
1.263
1.743
2.369
0
0.792
1.103
1.221
1.272
0
0.890
1.050
1.105
1.128
2.900
1.795
1.310
0.971
0.694
7.5
10
15
20
25
2.800
3.121
3.564
3.825
3.961
1.284
1.717
1.272
1.259
1.241
1.133
1.130
1.128
1.122
1.114
0.561
0.479
0.379
0.318
0.273
30
4.001
3.869
3.529
3.043
2.443
1.223
1.186
1.149
1.111
1.080
1.106
1.089
1.072
1.054
1.039
0.239
0.188
0.152
0.121
0.096
1.749
0.965
0.537
0.084
1.034
0.968
0.939
1.017
0.984
0.969
0.071
0.047
0.031
0
(per cent c) (per cent c)
0
0.5
1.25
2.5
5.0
40
50
60
70
80
90
95
100
-_
.._---.
0
.....
.....
. ....
L.E. radius: 0.70 per cent e
NACA 0008 Basic Thieknees Form
313
.APPENDIX I
I. 6
2/_
----
I.
---..,~
J
"""--
(~)' o.
1\,
I
o.
~
i
,.---
I
,
'---
o
o
,
0.2
0.4
I
~
~
f
0.6
08
a
1.0
C
%
11
(per cent c) (per cent c)
(v/V)!
,,/1"
~'./V
2..089
0
0.917
1.023
1.092
1.137
0
0.958
1.011
1.045
1.066
4.839
1.338
0.966
0.691
0.564
1.162
1.188
1.206
1.217
1.202
1.078
1.090
1.098
1.103
40
2.436
2.996
3.396
3.867
4.000
0.485
0.387
0.326
0.248
0.197
50
60
70
80
90
3..884
3.547
2.987
2.213
1.244
1.185
1.163
1.127
1.067
0.993
95
0.684:
0.080
0.932
0
_.­
0
1.25
2.5
5.0
7.5
10
15
20
30
100
0
0.756
1.120
1.662
1.096
1.089
1..079
t
L062
1.033
0.996
0.965
0
L.E. radius: 0.174 per cent c
NACA 0008-34 Basic Thickness Form
0.157
0.128
0.100
0.074
0.047
0.031
0
THEORY OF "K'ING BECTION.r;
314
I.'./I
_r _....
r--­ ~
-,
I.
-r----. ---......
r----. r--......
'\
--
\
IMCA 0009
1
..­
~
~ ......
< :,
II
%
Y
(peI: cent c) (per cent c)
0
0.5
1.25
2.5
5.0
0
.....
1.420
1.961
2.666
.8
.6
%,C
(V/V)2
tJ/V
lJJJ./V
0
0.750
1.083
1.229
1.299
0
0.866
1.041
1.109
1.140
0.595
1.700
1.283
0.963
0.692
0.560
0.479
0.380
0.318
0.273
i
7.5
10
15
20
25
3.150
3.512
4.009
4.303
4.456
1.310
1.309
1.304
1.275
1.145
1.144
1.142
1.137
1.129
30
4.501
4..352
3.971
3.423
2.748
1.252
1.209
1.170
1.126
1.087
1.119
1.100
1.082
1.061
1.043
0.239
0.188
0.151
0.120
0.095
1.967
1.086
0.605
0.095
1.037
0.984
0.933
0
1.018
0.982
0.966
0
0.070
0.046
0.030
0
40
50
60
70
80
90
95
100
~~,
..
I...E. radius: 0.89 per cent r.
NACA 0009 Basic Thieknees Form
_---­
315
APPENDIX I
2.D
.­
1.2
(v)'
rf ----
:---.­
,
r---­
r--...
---... r---­
~ .......
.8
NACA 0010
--"\
1
- ---­
-­
~~
'--.. "--­
.4,
ste
X
i
Y
I
I
0
0.5
1.25
2.5
5.0
0
0
.....
0.712
1..061
1.237
1.325
0.844
1.030
1.112
1.151
2.962
1.341
1.341
1.341
1.329
1.309
3..500
3.902
4.455
4.782
4.952
25
~D./lr
0
1.578
2.178
7.5
10
15
20
l?/V'
(r,/V)2
(per cent c) (per eeut c)
1.0
.8
.6
~
2.372
1.618
1.255
0.955
0.690
0.. 559
0.479
iO.380
0.318
0.273
1..158
1.158
1.158
1.153
1.144
I"
30
40
5.002
4.837
4.412
3.803
3.053
50
60
70
80
90
95
100
I
I
2.187
1.207
0.672
0.105
1.284
1~237
1.190
LI38
1.094
I
!
II
I
1.040
0.960
0.925
.....
f
I
I
t
l
I
1.133
1..112
1.091
1.067
1.046
1.020
0.980
0.962
. ....
L.E. radius: 1.10 per cent c
NACA 0010 Basic Thickness Form
0.239
0.188
0.150
0..119
0.094
0.069
0.045-'
I
0=1
0.030
THEORY OF WING SECTIONS
316
i6
-
~
L2
f
""----­
............
~
(vY.8 ~
r-,
,
['\
NACA 00/0--.14
.4
~
--­----­
--­-
r---.
o
.4
~
.8
.6
~
~~
1.0
x/c
--x
y
(per cent c) (per cent c)
(u/V)t
vv
~Va/V
0
0.944
1.400
2.078
2.611
0
0.892
1.011
1.113
1.167
0
0.944
IJ)05
1.055
1.080
3.857
1.282
0.950
0.688
0.564
40
3.044
3.744
4.244
4.&13
5.000
1.200
1.238
1.256
1.265
.1.253
1.095
1.113
1.121
1.124
1.119
0.486
0.389
0.327
0.249
0.197
50
60
70
80
90
4.856
4.433
3.733
2.767
1.556
1.235
1.205
1.157
1.089
0.990
. 1.111
1.098
1.076
1.044
0.995
0.856
0.100
0.910
0
0.954
I
0
I
0
1.25
2.5
5.0
7.5
10
15
20
30
95
100
i
I
I
0.159
0.127
0.100
0.073
0.045
I
L.E. radius: 0.272 per cent c
NACA 0010-34 Basic Thickness Form
0.030
0
317
APPENDIX I
1.6
l2
~
,r
""".-­
r-,
\
NACA 0010"J5
l----"'
~
-
r---. ---.....
o
.4
.2
xlc
r---. .............
~
.8
6
v/V
---~.
,
---­
10
bv./V
I
__.- -_.i-
-~.068-
o
0
o
1.25
2.5
5.0
7.5
0.878
1.267
1.844
2.289
0.954
1.032
1.087
0.977
1.016
1.043
1.122
1..059
0.679
0.. 555
10
15
20
30
40
2.667
3.289
3.789
4.478
4.878
1.141
1.172
1.194
1.214
1.229
1.068
1.083
1.093
1.. 102
1.109
0.476
0.382
0.323
0.247
0 . 198
50
5.000
60
70
4.867
4.389
80
90
I
3.500
2.100
1.235
1.240
1.227
1.176
1.046
1.111
1.114
1.108
1..084
1.023
0.. 162
0.131
0.104
0.076
0.048
1.178
0.100
0.920
0.959
j
0.030
0
95
100
I
I-----~---
._._ .. ...
~
I
j
o
o
1.309
0..952
_---~_._-~--_.~._._-
L.E. radius: 0.272 per cent c
NACA 0010-35 Basic Thickness Form
THEORY OF WING SECTIONS
318
1.6
___ L
, ,NA1CAoh,o
r- .............
1""Ia......
r
~I .........
.....--......
......
::::::
~
R
NACA 0010-64-
r--- r---.
~
-
~-
o
:&
.~
10
15
20
30
40
50
60
70
80
90
95
100
~
to
.6
x/c
(v/V)t
»rv
~../V
0
1.511
2.044
2.722
3.178
0
1.108
1.245
1.286
1.277
0
1.053
1.116
1.134
1.130
2.324
1.286
0.966
0.690
0.556
3.533
4.056
4.411
4.856
5.000
1.269
1.261
1.248
1.244
1.242
1.127
1.123
1.117
1.116
1.115
0.475
0.377
0.316
0.241
0.193
4.856
4.433
3.733
1.556
1.231
1.211
1.155
1.089
0.980
1.110
1.101
1.074
1.043
0.990
0.155
0.126
0.098
0.072
0.045
0.856
0.100
0.912
0
0.955
0
0.030
0
Y
(per cent c) (per cent c)
0
1.25
2.5
5.0
7.5
.4
.....--
2..767
L.E. radius: 1.10 per cent r.
NACA 0010-64 Basic Thickness Form
I
--
APPENDIX I
319
1.6
12
(v!
~~
.........
f
.8
WACA 0010-65
.4
- ---....
~ """"""""
---­
'-- .......
o
"',­\
.2
~
~
.8
.6
A-
,
LO
x-/c
II
x
y
(lIIV)!
(per cent c) (per cent c)
,
0
1.25
2.5
5.0
7.5
I
i
10
.15
20
30
40
50
60
70
80
_
..
I
I
I
I
0
1.068
2.584
1.295
0.970
0.684
0.551
0
1.140
1.273
1.271
1.252
3.300
3.756
4.089
4.578
4.889
1.236
1.213
1.200
1.196
1.212
1.112
1.101
1.095
1.094
1.101
0.470
0.372
0.312
0.239
0.193
1.229
1.234
1.226
1.173
1.049
1.109
1.111
1.107
1.083
1.024
0.158
0.128
0.103
0.076
0.046
0.915
0
0.957
0
5.000
4.867
4.389
3.500
I
2.100
I
1.178
0.100
---- l
41Ja / V
1.467
1.967
2.589
2.989
90
_-~.,
»rv
0
95
100
t
I
... _.
t
I
1
tI
j
__
!
._...._.­
1.128
1.127
1.119
I
I
I
II
I
'-'
L. E. radius: 1.10 pet" cent r
NACA 0010-65 Basic Thickness Form
0.029
0
,~
..
,­
-..
THEORY OF WING SECTIONS
320
l6
l2
r­ ...............
~
f
---­ '"
.....
,
,
\
NACA0010-66
.4
~
r---...
o
--..... ...............
-
--­ ,.,,--­
.2
.4
x/c
z
11
(per cent c) (per cent c)
.6
lO
(V/V)I
v/V
4v./V
0
1.489
2.011
2.656
3.089
0
1.130
1.246
1.286
1.282
0
1.063
1.116
1.134
1.132
2.434
1.289
0.959
0.687
0.554
40
3.400
3.856
4.178
4.678
4.822
1.258
1.225
1.209
1.189
1.178
1.122
1.107
1.100
1.090
1.085
0.471
0.372
0.310
0.236
0.190
SO
60
70
80
90
4.956
0.000
4.889
4.300
2.833
1.184
1.214
1.265
1.278
1.135
1.088
1.102
1.125
1.130
1.065
0.153
0.129
0.104
0.080
0.049
95
100
-------
1.656
0.100
0.960
0
0.980
0
0.030
0
0
1.25
2.5
5.0
7.5
10
15
20
30
L.E. radius: 1.10 per cent c
NACA OOlO-M Basic Thickness Form
APPENDIX 1
321
1.6
1.2
-
(
..............
I
~
~-
--~
~ ......
..........
.8
,
'\
MACA 0012
~
i'--­
-
-r--- r--­
-
~~
.8
D
%
11
(per cent c) (per cent c)
0
0.5
1.25
2.5
5.0
7.5
10
15
20
25
~
~
-
..­
1.0
(fJ/V)1
v/V
Avo/V
0
0.640
1.010
1.241
1.378
0
0.800
1.005
1.114
1.174
1.988
1.475
1.199
0.685
4.200
4.683
5.345
5.737
5.941
1.402
1.411
1.411
1.399
1.378
1.184
1.188
1.188
1.183
1.174
0.558
0.479
0.381
0.319
0.273
1.350
1.288
1.228
1.166
1.109
1.162
1.135
1.1OS
1.080
1.053
0.239
0.187
0.149
0.118
0.092
1.022
0.978
0.952
0
0
.....
1.894
2.615
3.555
30
6.002
40
50
60
70
5.803
5.294
80
90
95
100
2.623
1.044
1.448
0.807
0.956
0.906
0
4.563
3.664
0.126
I~E.
radius: 1.58 per cent c
NACA 0012 Basic Thickness Form
0.934
I
I
0.068
0.044
0.029
0
THEORY OF WING BECTIONS
322
L6
-'--­
V
- r---.
7
~ ..........
~
<,
f
~
,
~
r---­ r---..
!'--.. t--.........
o
.2
%
11
(per cent c) (per cent c)
0
1.25
2.5
0.0
7.5
10
15
20
30
40
50
60
70
80
90
~5
100
,
r\
NACA 0012-34
~
.4
-x./c
.8
.6
I'--.......
~~
1.0
(v/V)t
fJ/V
I:.fJ./V
0
1.133
1.680
2.493
3.133
0
0.865
0.997
1.122
1.186
0
0.930
0.999
1.069
1.089
3.154
1.251
0.933
0.683
0.560
3.653
4.493
5.093
5.800
6.000
1.229
1.282
1.310
1.329
1.311
1.109
1.132
1.145
1.153
1.146
0.484
0.389
0.329
0.250
0.198
5.827
5.320
4.480
3.320
1.867
1.284
1.249
1.192
1.112
0.985
1.133
1.11'8
1.092
1.055
0.992
0.158
0.128
0.098
0.071
0.045
1.027
0.120
0.894
0
0.946
0.029
0
0
L.E. radius: 0.391 per cent c
NACA 0012-34 Basic Thic1mess Form
APPENDIX I
323
L6
,r:
r----. ~
~
.8
r-,
1\
\1
NACA 00/2-64­
.4
-
----.
---­
-
~
~ lo-.........
~
l---­ ~
o
.4-
%
11
(per cent c) (per cent c)
.8
.6
xle
J,O
(V/V)2
»rv
~v./V
0
1.813
2.453
3.267
3.813
0
1.072
1.270
1.330
1.325
0
1.035
1.127
1.153
1.151
2.019
0.952
0.685
0.554
1.322
1.313
40
4.240
4.867
5.293
5.827
6.000
1.150
1.146
1.141
1.139
1.140
0.474
0.372
0.315
0.241
0.199
SO
60
70
80
90
5.827
5.320
4.480
3.320
1.867
1.280
1.244
1.189
1.102
1.131
1.115
1.090
0.993
0.996
0.154
0.126
0.096
0.070
0.044
95
1.027
0.120
0.889
0
0.943
0
1.26
2.0
S.O
7.5
10
15
20
30
100
1.303
1.297
1.300
1.050
0
L.E. radius: 1.582 per cent c
NACA 0012-64 Basic Thickness Form
1.236
0.028
0
-
THEORY OF WING SECTIONS
324
1.6
I.Z
r
I
~
--......... i"'-<,
~
I
..............
~ --;
''\
.8
\,
NACA 0015
<,
o
-
.--
r
---
---
:---~
----
---
0
0.5
1.25
2.5
5.0
I--'
1.0
I
Av./V
-
2.367
3.268
4.443
0
0.546
0.933
1.237
1.450
0
0.739
0.966
1.112
1.204
1.600
1.312
1.112
0.900
0.675
7.5
10
15
20
25
5.250
5.853
6.682
7.172
7.427
1.498
1.520
1.520
1.510
1.484
1.224
1.233
1.233
1.229
1.218
0.557
0.479
0.381
0.320
0.274
30
7.502
7.254
6.617
5.704
4.580
1.450
1.369
1.279
1.206
1.132
1.204
1.170
1.131
1.098
1.064
0.239
0.185
0.146
0.115
0.090
3.279
1.810
1.008
0.158
1.049
0.945
0.872
0
1.024
0.972
0.934
0
0.065
0.041
0.027
0
40
50
60
70
80
90
95
}OO
0
~V
(V/V)t
~
.s
z/c
I
X
Y
(per cent c) (per cent c)
r---. ~ .
.....
.....
L.E. radius: 2.48 per cent c
NACA 0015 Basic Thickness Form
_.-
_
..
_~
...­
325
APPENDIX I
1.6
1.2
~
(
..........
I
I
~~
<,
~ .........
.........
~ "­
<,
1\
\,
NACA DOl.
.4
-:
<,
-
~
..----
I"--.
.4
()
7.5
10
15
20
25
30
40
50
60
70
80
90
95
100
--­
---­
~
~
.6
r--­
~
.8
~
~
1.0
(,'1V)2
__v~1
4v./IT
0
0.465
0.857
1.217
1.507
0
0.682
0.926
1.103
1.228
1.342
1.178
1.028
0.861
0.662
6.300
7.024
8..018
8.606
8.912
1.598
1.628
1..633
1.625
0.555
1.592
1.264
1.276
1.278
1.275
1..262
9.003
8.705
7.941
6.845
5.496
1.556
1.453
1.331
1.246
1.153
1.247
1.205
1.154
1.116
1.074
0.238
0.184
0.144
0.113
0.087
1.051
0.933
0.836
0
1..025
0.966
0.914
0
0.063
0.039
0.025
0
v
x
t
(per cent c) (per cent c)
0
0.5
1.25
2.5
5.0
~
0
.....
2.841
3.922
6.332
3.935
2.172
1.210
0.189
I
I
L.E. radius: 3..56 per cent c
NACA 0018 Basic '£hickness Form
0.479 .
0..381
0.320
0.274
THEORY OF 'A,'IING SECTIONS
326
I
~
r-, <,
7
1.2
"' r-,
1
-"
11
(per cent c) (per cent c)
0
0.5
1.25
2.5
5.0
0
r--­ ---..
(V/V)t
~
I
\,
r--- r---.
~
.8
.6
X/C
-,
\
L.---­ ~
.4
<,
(J(J2/
---
~
V
r-, ...-...
()
X
r-,
~~
MeA
.4
r-,
~
1.0
fJ/V
li.v./V
0
0.630
0.887
1.087
1.242
1.167
1.065
0.946
0.818
0.648
6.221
0
0.397
0.787
1.182
1.543
7.5
10
15
20
25
7.350
8.195
9.354
10.040
10.397
1.682
1.734
1.756
1.742
1.706
1.297
1.317
1.325
1.320
1.306
0.550
0.478
0.381
0.320
0.274
30
10.504
10.156
9.265
7.986
6.412
1.664
1.538
1.388
1.284
1.177
1.290
1.240
1.178
1.133
1.085
0.238
0.183
0.142
0.111
0.084
4.591
2.534
1.412
0.221
1.055
0.916
0.801
0
1.027
0.957
0.895
0
0.061
0.037
0.023
0
40
50
60
70
80
90
95
100
......
3.315
4~576
L.E. radius: 4.85 per cent c
NACA 0021 Basic Thickness Form
327
APPENDIX I
rr-«:
7
I
r-.
,,~
" r-, <,
I
r-,
'"
.s
M4CA 0024
. .4
o
V --r-,
.j.
y
x
(per cent c) (per cent c)
-
r---.
r----.. r----..
~
x/c
L..--- ~
.8
..8
~
\
\
t---.
1
~
I. D
(V/V)!
V/V
4D./V
3.788
5.229
7.109
0
0.335
0.719
1.130
1.548
0
0.579
0.848
1.063
1.244
1..050
0.964
0.870
0.771
0.632
7.5
10
15
20
25
8.400
9.365
10.691
11.475
11.883
1.748
1.833
1.888
1.871
1.822
1.322
1.354
1.374
1.368
1.350
0.542
0.476
0.383
0.321
0.274
30
50
60
70
12J)04
11.607
10.588
9.127
7.328
1.777
1.631
1.450
1.203
I
1.333
1.277
1..204
1.151
1.097
0.238
0.181
0.140
0.109
0.082
80
90
95
100
5.247
2.896
1.613
0.252
1.065
I
I
1.032
0.944
0.879
0
0..059
0.035
0.022
0
0
0.5
1.25
2.5
5.0
0
.........
40
I
1.325
0.891
0.773
0
j
L.E. radius: 6.33 per eent c
NACA 0024 Basic 'I'hickness Form
THEORY OF WING SECTIONS
328
2.0
1.6
1.2
---.
r:
~
\
.B
NACA 16-006
-­
e---'
11
10
15
20
30
40
lit)
60
10
80
90
95
100
!
(fl/V)t
v/V
0
0.646
0.903
1.255
1.516
0
1.059
1.085
1.097
1.105
0
1.029
1.042
1.047
1.051
5.471
1.376
0.980
0.689
0.557
1.729
2.067
2.332
2.709
2.927
1.108
1.112
1.116
1.123
1.132
1.053
1.055
1.057
1.060
1.064
0.476
0.379
0.319
0.244
0.196
3.000
2.917
2.635
2.099
1.259
1.137
1.141
1.132
1.104
1.035
1.066
1.068
1.064
1.051
1.017
0.160
0.130
0.104
0.077
0.049
0.707
0.060
0.962
0
0.981
0
0.032
0
(per cent c) (per cent c)
0
1.25
2.5
5.0
7.5
,
/.0
.8
%
\
\
L.E. radius: 0.176 per cent c
NACA 16-006 Basic Thickness Form
l1v./V
APPENDIX I
329
..
,
1.2
,­
...........
~~
.8
\
NM:A 16-00g
1
.4
---
~-
r---­
-
~
.4
(J
%
11
Z/c
\
.8
.6
"""""""­
~
/.0
(V/V)2
V/V
J1v./V
0
0.969
1..354
1.882
2.274
0
1.042
1.109
1.139
1.152
0
1.021
1.053
1.067
1.073
3.644
1.330
0.964
0.684
0.554
10
15
20
30
40
2.593
3.101
3.498
4.063
4.391
·1.158
1.168
1.177
1.190
1.202
1.076
1.081
1.085
1.091
1.096
0.475
0.378
0.319
0.245
0.197
50
60
70
80
90
4.500
4.376
3.952
3.149
1.888
1.211
1.214
1.206
1.156
1.043
1.100
1.099
1.075
1.022
0.160
0.. 131
0.103
0.076
0.047
95
1.061
0.090
0.939
0
0.969
0
0.030
0
(per cent c) (per cent c)
0
1.25
2.5
5.0
7.5
tOO
I
1~106
L.E. radius: 0.396 per cent c
NACA 16-009 Basic Thickness Form
THEORY OF WING SECTIONS
330
1.6
I-Z
T ----
to--"'"
~
~
""r\.\
.8
NACA 16-DI2
.4
~
I'---
-
~
100-0-..
D
!.........-.
.t
.4
\
t--- t--.......
L--- ~
I.D
.6
SIC.
x
y
(v/V)t
V/V
I1v./V
0
1.292
1.805
2.509
3.032
0
1.002
1.109
1.173
1.197
0
1.001
1.053
1.083
1.094
2.624
1.268
0.942
0.677
0.561
10
15
20
30
40
3.457
4.135
4.664
5.417
5.855
1.208
1.223
1.237
1.257
1.271
1.099
1.106
1.112
1.121
1.128
0.473
0.378
0.319
0.245
0.197
50
60
70
80
90
6.000
5.836
5.269
4.199
2.517
1.286
1.293
1.275
1.203
1.051
1.134
1.137
1.129
1.097
1.025
0.161
0.131
0.102
0.075
0.045
95
100
1.415
0.120
0.908
0
0.953
0
0.027
0
(per cent c) (per cent c)
0
1.25
2.5
5.0
7.5
L.E. radius: 0.703 per cent c
NACA 16-012 Basic Thickness Form
331
APPENDIX I
1.6
:",.--­ ~
/
~
-
--'" -,
"'"
{
\
M4CA /6-0/5
~
i'---..
D
x
~
---­
-
10
15
20
--­ ---­
...---­
.8
/.0
0
1.615
2.257
3.137
3.790
0
0.956
1.105
1.200
1.239
0
0.978
1.051
1.095
1.113
0.916
0.668
0.547
4.322
1.256
1.278
1.297
1.327
1.349
1.121
1.130
1.139
1.152
1.161
0.471
0.377
0.318
0.245
0.197
0.161
0.131
0.102
0.074
0.043
40
SO
60
7.500
7.293
1.364
70
6.. 587
80
90
5.248
3.147
1.348
1.254
1.053
1.168
1.172
1.161
1.120
1.026
95
1.768
0.150
0.875
0
0.935
0
100
~
v/V
6.168
5.830
6.772
7.318
30
...............
(V/V)2
(per cent c) (per cent c)
0
1.25
2.5
5.0
7.5
r--­
~
v
\,
1.374
L.E. radius: 1.100 per cent c
NACA 16-015 Basic Thickness Form
I
~v./V
2.041
1.209
I
0.025
0
THEORY OF WING SECTIONS
332
-
.w
-
/'
~
-
- --­
..........
<,
-7
t-,
\
I
\
.8
V
~
~ t----
- r--­
~
.6
I--­
(J
%
'Y
(per cent c) (per cent c)
,
\
NACA /S-OI8
.4
~
~
L.---­ ~
.8
~
~
/.0
(V/V)I
fJ/V
~fJ./V
0
1.938
2.708
3.764
4.548
0
0.903
1.092
1.217
1.271
0
0.950
1.045
1.103
1.128
1.744
1.140
0.883
0.657
0.541
5.186
6.202
6.996
8.126
8.782
1.302
1.332
1.357
1.399
1.426
1.141
1.154
1.165
1.183
1.194
0.468
0.376
0.318
0.245
0.198
70
80
90
9.000
8.752
7.904
6.298
3.776
1.447
1.452
1.421
1..306
1.051
1.203
1.205
1.192
1.143
1.025
0.162
0.131
0.102
0.073
0.042
95
100
2.122
0.180
0.837
0
0.915
0
0.024
0
0
1.25
2.5
5.0
7.5
10
15
20
30
40
SO
60
f
L.E. radius: 1.584 per cent c
NACA 16-018 Basic Thickness Form
APPENDIX 1
.8
~
V
- I
-/
,
---- --­
--.... <,
~
I.
\
~
NACA /S-(J21
.4
V
---­
~ r--.-.
...-­
-~
. ---.--.... ~
_l.---' ~
~
D
.4
z
11
(per cent c) (per cent c)
1\
'\
\
zjc
.8
B
~
to
(V/V)t
tJ/V
!lv.IV
0
2.261
3.159
4.391
5.306
0
0.826
1.062
1.221
1.295
0
0.909
1.031
1.105
1.138
1.574
1.069
0.828
0.640
0.534
10
15
20
30
40
6.050
7.236
8.162
9.480
10.246
1.342
1.391
1.419
1.474
1.506
1.159
1.179
1.191
1.214
1.227
0.463
0.374
0.317
0.245
0.198
50
60
70
10.500
10.211
9.221
7.348
4.405
1.535
1.536
1.495
1.361
1.039
1.239
1.166
1.019
0.162
0.131
0.102
0.072
0.041
2.476
0.210
0.801
0.895
0
0.023
0
0
1.25
2.5
5.0
7.5
80
90
95
100
I
0
1.239
1.223
ThE. radius: 2.156 per cent c
NACA 16-021 Basic Thickness Form
THEORY OF WING SECTIONS
334
1.8
,0
1.2
(f)~
I
t
.....
I
~CI,,03 ,..,.,.. $Ur"~
~03 L~
??
J
."foce
r-
r--..
~
~
~
.8
NACA t13-D06
~
...............
D
.4
x
y
(per cent c) (per cent c)
'/.
S,C
1.0
.8
.6
(f)/V)2
V/V
J1v./V
0
0.503
0.609
0.771
1.057
0
0.973
1.050
1.080
1.110
0
0.986
1.025
1.039
1.054
4.483
2.110
1.778
1.399
0.981
5.0
7.5
10
15
20
1.462
1.766
2.010
2.386
2.656
1.130
1.142
1.149
1.159
1.165
1.063
1.069
1.072
1.077
1.079
0.692
0.562
0.484
0.384
0.321
25
30
2.841
2.954
3.000
2.971
2.877
1.170
1.174
1.170
1.164
1.151
1.082
1.084
1.082
1.079
1.073
0.279
0.245
0.218
0.196
0.176
2.723
2.517
2.267
1.982
1.670
1.137
1.118
1.096
1.074
1.046
1.066
1.057
1.047
1.036
1.023
0.158
0.141
0.125
0.111
0.098
1.342
1.008
0.683
0.383
0.138
1.020
0.994
0.965
0.936
0.910
1.010
0.997
0.982
0.967
0.954
0.085
0.073
0.060
0.047
0.032
0
0.886
0.941
0
0
0.5
0.75
1.25
2.5
35
40
-is
50
55
60
65
70
75
80
85
90
95
100
LE. radius: 0.297 per cent c
-NACA 63-006 Basic Thickness Form
APPENDIX.]
I.IJ
ro..
1.2
f/<
.8
,
c, •.
/
33~
I
~ ~ slrfaet!J
~
~
~
I ~
r--.08 Lower .•ur'f«:e
~~
V
~
~
MACA 63-otJ!J
.4
[,--­
~
r--­ " - ­
_L--- ~ -
"'""-­
.z
a
%
.4
Y
{11/V)2
(per cent c) (per cent c)
0
0.5
0.75
1.25
2.5
5.0
7.5
10
15
20
0
0.749
0.906
1.151
1.582
2.196
%/c
0
0.885
i
I
I
1.002
l.OSI
1.130
lO
.6
fJ/V
,
J1tJ./V
0
0.941
1.001
IJ)25
3.058
1.889
1.063
1.647
1.339
0.961
1.180
1.205
1.221
1.241
1.255
1.08B
1.105
1.114
1.120
0.689
0.560
0.484
0.386
0.324
4.275
1.264
1.124
0.281
4.442
4.500
4.447
4.296
1.269
1.265
1.255
1.235
1.126
1.120
1.111
0.248
0.220
0.196
1.208
1.099
1.175
65
4.056
3.739
3.358
2.928
70
2.458
1..084
1.068
1.051
1.032
0.156
0.140
0.124
0.109
0.095
75
80
1.966
1.471
85
90
0.550
95
0.196
0.984
0.942
0.903
0.868
1.012
0.992
0.971
0.950
0.932
0.082
0.069
0.057
0.044
0.030
0
0.838
0.915
0
25
30
35
40
45
50
55
60
100
2.655
3.024
3.591
3.997
0.990
!
1.141
1.104
1.065
1.025
1.098
1.125
L.E. radius: 0.631 per cent c
N.~CA
63-009 Basic Thickness Form
0.175
THEORY OF WING SECTIONS
336
ttl
/'
I.e
.8
c, ..
I
10 , . . , . ..,-'!ot»
ro;: ~~~
/r
V
v
I
1'--.10 LOWf1r
~
I
.".,~
~~
~~
~
NACA 63-010
.4
-r --­ r---­ t---..­
~
L----­
I"-­ "'-­
-­
o
.4
I
~-
/.0
.8
.6
%/C
x
JI
(per cent c) (per cent c)
0
0.5
0.75
1.25
2.5
5.0
7.5
10
15
20
25
30
35
40
45
50
55
60
65
70
(,,/V)2
fJ/V
11v./V
0.841
0.978
1.037
1.131
0
0.917
0.989
1.018
1.063
2.775
1.825
1.603
1.316
0.952
1.193
1.223
1.245
1.270
1.285
1.092
1.106
1.116
1.127
1.134
0.687
0.560
0.484
0.386
0.325
4.753
4.938
5.000
4.938
4.766
1.296
1.302
1.299
1.286
1.262
1.138
1.141
1.140
1.134
1.123
0.282
0.248
0.220
0.196
0.175
4.496
4.140
3.715
3.234
2.712
1.231
1.193
1.154
1.113
1.069
1.110
1.092
1.074
1.055
1.034
0.156
0.139
0.123
O.lOS
0.094
1.025
1.012
0.989
0.967
0.945
0.924
0.081
0.069
0.056
0.043
0.030
0.907
0
0
0
0.829
1.004
1.275
1.756
2.440
2.950
3.362
3.994
4.445
I
75
80
85
2.166
1.618
90
95
0.604
0.214
0.979
0.935
0.893
0.853
0
0.822
100
1.088
L.E. radius: 0.770 per cent c
NACA *':>.-1)10 Basic Thickness Form
APPENDIX I
l8
,
~
_C,..
337
14 Lt:Per surfQCfl
.............
I
loy ~
~
£2
<.J
(/ .14 Lo",.,- surfoct!!I """ ~ ~
~
---­
L.--.
I{
.8
~
,
'-
~
NACA 63J-a2
.4
~
"--.....
~
- r---
~
-
a
.P
.4
I
~
.6
-----­
--­--
v
(vIV)2
"/1'
0
0.985
1.194
1.519
2.102
0
0.750
0.925
1.005
1.129
0
0.866
0.962
1.003
5.0
7.5
10
15
20
2.925
3.542
4.039
4.799
5.342
1.217
1.261
1.294
1.330
1.349
1.103
1.123
1.138
25
5.712
5.930
6.000
5.920
5.704
1.362
1.370
5.370
4.935
4.420
3.840
3.210
2.556
1.902
1.274
0.707
0.250
1.023
0.969
0.920
0.871
0.826
x
(per cent c) (per cent c)
0
0.5
0.75
1.25
2.5
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
---­
I
i
0
1.063
I
I
Jlv.IV
2.336
1.695
1.513
1.266
0.933
1.161
0.682
0.559
0.484
0.387
0.326
1.317
1.167
1.170
1.169
1.161
1.148
0.283
0.249
0.221
0.196
0.174
1.276
1.229
1.181
1.131
1.076
1.130
1.109
1.087
1.063
1.037
0.155
0.137
0.121
0.106
0.091
1.011
0.984
0.959
0.933
0.909
0.079
0.067
0.055
0.042
0.029
0.889
0
1.153
1.366
1.348
II
1.0
.8
%C
0.791
I
I
L.E. radius: 1.087 per cent c
NACA 631-012 Basic Thickness Form
THEORY OF WING SECTIONS
~ ,•.22 ~ !IUrlI:It»
.........
(
'.I
r
III
I
- I
(v)'
--........ r-,
.J1I'"
~
/
~~~~ r-,
~~22
~
Loww-
"'oce . . ~ ~
~
A
~~............
NACA tJ3z-tJl5
.4­
V
~
-r---- r---
<, t---.
L.---- J---­
r--- t--­
~
a
~,...
.8
%
11
1.0
(vlv)!
V/V
4v./V
0
1.204
1.462
1.878
2.610
0
0.600
0.822
0.938
1.105
0
0.775
0.907
0.969
1.051
1.918
1.513
1.379
1.182
0.903
5.0
7.5
10
15
20
3.648
4.427
5.055
6.011
6.693
1.244
1.315
1.360
1.415
1.446
1.115
1.147
1.166
1.190
1.202
0.674
0.557
0.484
0.388
0.330
25
30
7.155
7.421
7.500
7.386
7.099
1.467
1.481
1.475
1.446
1.401
1.211
1.217
1.214
1.202
1.184
0.286
0.251
0.222
0.196
0.174
6.665
6.108
5.453
4.721
3.934
1.345
1.281
1.220
1.155
1.085
1.160
1.132
1.105
1.075
1.042
0.153
0.135
0.118
0.102
3.119
2.310
1.541
0.852
0.300
1.019
0.953
0.894
1.009
0.789
0.976
0.946
0.916
0.888
0.076
0.063
0.051
0.039
0.026
0
0.750
0.866
0
(per cent c) (per cent c)
0
0.5
0.75
1.25
2.5
35
40
45
50
55
60
65
70
75
80
85
90
95
100
0.83~
L.E. radius: 1.594 per cent c
NACA 63r015 Basic Thickness Form
0.088­
APPENI)[.\'- I
eo
~,-.32
L
/.6
0,.
1.2
/"
/ /
I,l V'
J
~
"
r-,
~~
<,
~~
~
I
I
~
~
NfCA 633-018
--
~
-
r--­
o
%
<,
~32 LOflllf!lr ""acll
I
,
..............
V
I
V
lIppe". surfloc.
<,
(
339
'"
I
I---­ ~
ZIt:
~
r--­
~
1.0
.B
6
(v/1T ) 2
y
(per cent e) .(per rent c)
r--.. r---
"IV
/i,v./V
1.639
0.848
0
0.664
0.837
0.921
i
0
!
i
0.5
0.75
I
1.25
f
2.5
f
5.0
7.5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
I
I
I
I
I
I
0
0
1.404
1.713
2.217
3.104
0.441
0 ..700
1.065
1.032­
4.362
5.308
6.068
7.225
1.260
1.360
1.424
1.122
1.166
1.193
1.225
1.244
1.500
8.048
1.547
8.600
8.913
9.000
8.845
8.482
1.579
1.598
1.585
1.550
1.490
1.361
1.258
1.105
0.871
0.663
0.553
0.484
0.390
0.333
1.257
0.289
1.264
0.253
1.259
1.245
1.221
0.223
0.197
0.173
1.188
1.153
1.119
1.082
0.152
0.133
0.115
0.099
1.043
0.084­
1.004
7.942
7.256
6.455
5.567
4.622
1.252
1.170
1J187
3.650
2.691
1.787
0.985
0.348
1.009
0.933
0.868
0.807
0.753
0.966
0.932
0.898
0.868
0.072
0.059
0.048
0.036
0.024
0
0.712
0.844
0
1.411
1.330
I
~--
L.E. radius: 2.120 per cent c
N.~CA
63...018 Basic Thickness Form
THEORY OF WING SECTIONS
340
a
.z
z
71
(per cent C) (per cent C)
.4
Z/c
.6
(V/V)2
vlV
Av.IV
0
1.583
1.937
2.527
3.577
0
0.275
0.564
0.7'5
1.010
0
0.524
0.751
0.851
1.005
1.439
1.236
1.156
1.034
0.842
5.065
6.182
7.080
8.441
1.260
1.394
1.487
1.592
20
9.~O
1.655
1.122
1.181
1.219
1.262
1.286
0.653
0.550
0.484
0.392
0.335
25
10.053
10.412
10.&00
298
9.854
1.698
1.578
1.303
1.312
1.307
1.286
1.256
0.291
0.255
0.225
0.198
0.173
50
9.206
55
60
65
8.390
1.479
1.380
1.281
1.180
1.084
1.216
1.175
1.132
1.086
1.041
0.150
0.130
0.112
0.096
0.081
0.997
0.954
0.916
0.880
0.849
0.068
0.057
2.021
1.113
0.392
0.994
0.911
0.839
0.774
0.721
0..046
0
0.676
0.822
0
0
0.5
0.75
1.25
2.5
5.0
7.5
10
15
30
35
40
45
to.
70
7.441
6.396
&.290
75
4..160
80
85
90
96
100
a.OM
1.721
1.709
1.654
. L.E. radius: 2.6liO per cent c
r~ACA 63M}21
Basic 'Thickness Form
0.035
0.023
APPENDIX I
0
1.2.
r-l
IlL"
r:
341
c/ =.01Uppersurface
."
I-+-- r---.
"".01 lower surrace
I
.8
I
-- r-.
NACA 6JAOO6
.4
.....-
r--- "---
o
%
.2
11
1.25
2.5
x/c
.6
vlV
J1v./I··
0
0.495
0.595
0.7M
1.045
0
0.900
1.063
1.086
1.112
0
0.949
1.031
1.042
1.055
4.560
1.447
1.134
1.747
1.989
1.142
1.1SO
1.159
1.165
1.065
1.069
1.072
1.077
5.0
7.5
10
15
2.362
20
2.631
25
30
35
4{)
2.820
2.942
2.996
2.985
45
2.914
50
55
60
2.788
1.168
1.079
1.370
0.976
0.693
0.563
0.485
0.383
0.321
1.162
1.151
1.067
1.058
1.049
1.039
1.028
0.. 140
0.126
0.112
0.085
0.072
0.060
0..047
0.033
0
1.169
1.859
75
1.556
1.035
80
1.248
1.010
85
0.939
90
95
0.. 630
0.986
0.96-1
0.322
0.939
1.017
1.005
0.993
0.982
0.969
100
0.013
0
0
2.613
2.396
2.143
2.079
1.794
1.081
1.082
1.081
1.078
1.073
1.170
1..138
1.120
1.100
1.079
1.057
65
70
to
.8
(fllV)t
(per cent c) (per cent c)
0
0.5
0.75
A-
L.E. radius: 0.265 per cent c
T.E. radius: 0.014 per cent c
NACA 63AOO6 Basic Thickness Forol
0.278
0.24:4
0.217
0.195
0.175
0.158
0.098
THEORY OF WING SECTIONS.
342
l6
, c/
,
=.05Upper surface
I
0
t
~
?(
-r- ~
~
~
'.OS lower surface
.............
~
NACA6.1AOO8
~
..............
o
--
.2.
.4
.6
lO
.8
X/I!
%
(V/V)2
»rv
AIl./V
0
0.658
0.791
0
0.850
1.034
0
0.922
1.017
11
(per cent c) (per cent c)
1.003
1.080
1.039
1.391
1.132
1.064
3.465
1.961
1.674
1.344
0.967
1.930
2.332
2.656
3.155
3.51&
1.168
1.185
1.198
1.212
1.221
1.081
1.089
1.095
1.101
1.105
0.689
0.562
0.484
0.383
0.322
3.766
3.926
3.995
3.978
3.878
1.227
1.230
1.228
1.219
1.204
1.108
1.109
1.108
1.104
1.097
0.279
0.246
0.218
0.195
0.174
3.705
3.468
3.176
2.837
2.457
1.183
1.159
1.132
1.104
1.073
1.088
1.077
1.064
1.051
1.036
0.156
0.138
0.123
0.109
0.096
90
95
2.055
1.647
1.240
0.833
0.425
1.042
1.010
0.980
0.951
0.919
1.021
1.005
0.990
0.975
0.959
0.083
0.070
0.058
0.045
0.030
100
0.018
0
0
0
0
0.5
0.75
1.25
2.5
5.0
7.5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
L.E. radius: 0.473 per cent c
T.E. radius: 0.020 per cent c
N ACA 63AOOS Basic Thickness Form
APPENDI..Y. I
343
l6
.~". cl =.09 Upper surface
....
r/L
~
1.2
(vY 1fA
--..;~
,...-
~
~
~~
""~
'.09 lower surface
.8
NACA6JAOIO
~
-
~
---..;
'--...........
o
, .6
.8
x/c
x
11
(per cent c) (per cent c)
0
0.5
0.75
o
~
2.5
0.816
0.983
1.250
1.737
5.0
7.5
10
15
20
2.412
2.917
3.324
3.950
4.400
25
4.714
30
4.913
35
40
4.995
4.968
4.837
!I
1.25
45
50
55
60
65
70
75
80
85
90
95
100
I
I
i
I
II
I
i
I
II
,
t
1
!
4.613
4.311
3.943
3.517
3.044
2.545
2.040
1.535
1.030
0.525
0.021
I
(V/V)2
I
i
1.043
i
1.140
1.068
I
1.200
1.095
1.107
1.116
1.126
1.132
0
i
0.774
0.985
!
I
II
I
II
Ii
II
I
1.225
1.245
1.268
1.282
1.290
1.294
1.291
1.279
1.258
LO
&./"..
'II V
0
0.880
0.992
1.021
~
,
1.136
1.138
1.136
1.131
1.122
I
I
I
I
I
0.280
0.247
0.218
0.195
0.174
I
I
1.048
1.009
1.02-1
1.004
I!
0.972
0.938
O.9CiO
0.969
0.949
I
j
0
0
I
I
1.078
1.061
1.042
0.684
0.560
0.483
0.383
1.109
1.094
2.805
1.833
1.594
1.307
0.957
I
1.230
1.196
1.162
1.125
1.086
Ii
-
i
I
I
,
0.324
0.155
0.137
0.122
O.lOS
0.09-1
1
0.986
L.E. radius: 0.742 per cent c
T.E. radius: 0.023 per cent c
NACA 63AOI0 Basic Thickness Form
I
I
0.081
0.068
0.057
0.044
0.030
0
THEORY OF WING SECTIONS
344
16
IJI' cl :./2 Vpp~r surface
rl)Lt
~
~
~/~
(fit rr
.B ,
~
"'-o
~
~
............
'.It Lower surface
~
NACA6~AOI2
--
~
~
..4
~
~Iooo..:.
t - ~-
.6
.8
x/c
z
11
(per cent c) (per cent c)
LO
(vlv)'J
plV
i1v./V
2.361
1.701
1.515
0
0
0.5
0.973
1.173
1.492
2.078
0
0.686
0.924
0.985
1.136
0
0.828
0.961
0.992
1.066
0.935
2.895
3.504
3.994
4.747
5.287
1.229
1.265
1.291
1.324
1.344
1.109
1.125
1.136
1.151
1.159
0.679
0.559
0.482
0.384
0.325
25
30
35
5.664:
40
45
5.957
5.792
1.355
1.360
1.357
1.340
1.312
1.164
1.166
1.165
1.158
1.145
0.281
0.248
0.219
0.196
0.174
50
5.517
5.148
4.700
4.186
3.621
1.275
1.234
1.191
1.145
1.098
1.129
1.111
1.091
1.070
0.154
0.136
0.120
0.106
0.092
85
90
1.826
1.051
1.007
0.964
1.025
80
3.026
2.426
1.225
0.925
9S
0.625
100
0.025
0.75
1.25
2.5
5.0
7.5
10
15
20
55
60
65
70
75
"-­
~
............ ~
jIlII""""
5.901
5.995
1.048
1.258
0.880
1.003
0.982
0.962
0.938
0.079
0.066
0.055
0.042
0.029
0
0
0
L.E. radius: 1.071 per cent c
T.E. radius: 0.028 per cent c
NACA 6&A012 Basic Thickness Form
APPENDIX I
345
,~t:l =.18 Upper surface
1.6
i2
...............
f
O~ ~- .................. ~
I
't"
~
V
f----
II ~ ,
r--:::: ~ e-,
~~
\
.8
r/
J8 lower $ur'Fact!
I
~
r--... ~
NACA 6:12 AO/5
- r---
~"""""
<, r---..
o
.2
.4
I'--- ~
~~
.,.....- "---~
.6
J.()
x/c
z
11
(per cent c) (per cent c)
(V/V)2
"IV
&J./V
0
0.550
0.825
0.882
1.120
0
0.742
0.908
0.939
1.058
1.504
1.370
1.176
0.005
1.257
1.121
1.150
1.167
1.187
1.199
0.669
0.555
0.482
0.384
0.326
1.396
1.206
1.210
1..207
1.198
1•.182
0.282
0.250
0.220
0.196
0.174
6..858
6.387
5.820
5.173
4.468
1.349
1.296
1.237
1.175
1.115
1.161
1.138
1.112
1.084
1..056
0.152
0.134
0.118
0.104
0.090
90
95
3.731
2.991
2.252
1.512
0.772
1.055
1.000
0.950
0.900
0.850
1.027
1.000
0.975
0.949
0.922
0.077
0.064
0.052
0.040
0.028
100
0.032
0
0
0
0
0.5
0.75
1.25
2.5
5.0
7..5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
0
1.203
1.448
1.844
2.579
3.618
4.382
4.997
5.942
6.619
7.091
7.384
7.496
7.435
7.215
1.323
1.361
1.408
1.437
1.455
1.464
1.458
1.435
1.930
I
L.E. radius: 1.630 per cent c
T.E. radius: 0.037 per cent c
NACA ~015 Basic Thickness Form
THEORY OF WING SECTIONS
346
1.2
ke,..oe ~ - * ­
,D
,r..,.
~
I~
~oe Lower
"""ot:e
r----... r----.
..............
~
.B
!MeA M-008
.4
L..­
~
o
.2
.4
,I
.s-
.11
1.0
%,C
'II
%
(per cent c) (per cent c)
(vlV)t
fJIV
&D./1'
0
0.5
0.75
1.25
2.5
0
0.494
0.596
0.754
1.024
0
0.995
1.058
1.085
1.1OS
0
0.997
1.029
1.042
1.053
4.623
2.175
1.780
1.418
0.982
5.0
1.405
1.692
1.928
2.298
2.572
1.119
1.128
1.134
1.146
1.154
1.058
1.062
1.065
1.071
1.074
0.692
0.560
0.483
0.385
0.321
2.772
2.907
2.981
2.995
2.919
1.160
1.1M
1.168
1.171
1.160
1.077
1.079
1.081
1.082
1.077
0.279
0.246
0.220
0.198
0.178
2.775
2.575
2.331
2.050
1.740
1.143
1.124
1.102
1.079
1.054
1.069
1.060
1.050
1.039
1.027
0.1fi8
0.142
0.126
0.112
0.098
1.412
1.072
0.737
0.423
0.157
1.028
1.000
0.970
0.939
0.908
1.014
1.000
0.985
0.969
0.953
0.085
0.072
0.060
0.047
0.031
0
0.876
0.936
0
7.5
10
15
20
25
30
35
40
45
50
55
60
85
70
75
80
85
90
95
100
L.E. radius: 0.256 per cent c
NACA 64-006 Basic ThicJmess Form
347
APPENDIX 1
.6
(
/'
~
~C',
•.04
Up~
$urfoce
I
~~ ~ ~IU Lower
surlOCtfI ~
~
r
r---..
~
~
~
~
NACA 84-008
.4
~
r--o
x
-
....
10--.
.2
I
.4
y
(V/V)2
I
0
0
II
0.658
0.794
1.005
1.365
1.875
/.0
.8
.6
xjc
I--­
V/V
fJ.V./V
0.912
1.016
1.084
1.127
0
0.955
1.008
1.041
1.062
3.544
l.994
1.686
1.367
0.969
1.152
1.167
1.179
1.195
1.208
1.073
1.080
1.086
1..093
1.099
0.688
0.560
0.480
0.385
0.323
1.217
1.225
1.230
1.235
45
3.704
3.884
3.979
3.992
3.883
1.103
1.107
1.109
1.111
1.105
0.279
0.246
0.220
0.198
0.176
50
55
60
65
70
3.684
3.411
3.081
2.i04
2.291
1.191
1.163
1.133
1.102
1.069
1.091
1.078
1.050
1.034
0.158
0.141
0.125
0.110
0.096
75
80
85
90
95
1.854
1.404
0.961
0.550
0.206
1.033
0.995
0.957
0.918
0.878
I.Q16
0.997
0.978
0.958
0.937
0.083
0.071
0.059
0.046
0.031
0
0.839
0.916
0
(per cent c) . (per cent c)
0
0.5
0.75
1.25
2.5
5.0
7.5
10
15
20
25
30
35
40
I
2..259
2..574
3.069
3.437
1.220
1.064
I
I
t
100
I
L.E. radius: 0.455 per cent c
NACA 64-008 Basic Thickness Form
THEORY OF WING SECTIONS
348
/.6
{
--V
~
",.-
K
~
~
c,•.
06 UpptIr surftXVI
-.l
t--""_ ~
~
~
T
r--:otJ Lower .urf'DCfI ~
T
~
~
.8
~
HACA 1U-IJ09
~
--
,...-
r-----. .........
o
.2
x
'Y
(per cent c) (per cent c)
0
0.5
0.75
1.25
2.5
5.0
7.5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
.4
%/c
II--..
I - - to--"
.6
.8
to
(V/V)2
V/V
~./V
0
0.739
0.892
1.128
1.533
0
0.872
0.990
1.131
0
0.934
0.995
1.037
1.063
3.130
1.905
1.637
1.340
0.963
2.109
2.543
2.898
3.455
3.868
1.166
1.186
1.200
1.221
1.236
1.080
1.089
1.095
1.105
1.112
0.686
0.560
0.479
0.383
0.323
4.170
4.373
4.479
4.490
4.364
1.246
1.255
1.262
1.267
1.246
1.116
1.120
1.123
1.126
1.116
0.281
0.248
0.221
0.198
0.176
4.136
3.826
3.452
3.026
2.561
1.217
1.. 183
1.149
1.112
1.073
1.103
1.088
1.072
1.055
1.036
0.158
0.140
0.125
0.109
0.095
2.069
1.664
1.069
0.611
0.227
1.033
0.992
0.950
0.907
0.865
1.016
0.996
0.975
0.952
0,930
0
0.822
0.907
0.082
0.070
0.057
0.044
0.030
()
1.075~
L.E. radius: 0.579 per cent c
NACA M-009 Basic Thickness Form
APPENDIX I
I. 6
349
I
I
ve, -.08 ~
I
I
surf.:.
L
.--­ ~ /. .. rti:
~
-~
~
r--.D9
I
-~
~
I~
L...-
""oc. ~ ~
8
r-, r-,
NACA 64-010
"
~
r - ­ ~ I"'---­
~
-
I'---­ _
()
z
5.0
7.5
10
15
20
25
30
~
I.D
.8
.6
1/
(v/l')2
v/l"
l1v./V
0
0.820
0.989
1.250
1.701
0
0.834
0.962
1.061
1.130
0
0.913
0.981
1.030
1.063
2.815
1.817
1.586
1.313
0.957
2.343
2.826
3.221
3.842
1.181
1.206
1.221
1.245
1.262
1.087
1.098
1.105
1.116
1.123
0.684
0.559
0.480
0.386
0.325
1.129
1.134
1.138
1.140
1.131
0.280
0.246
0.220
0.199
0.176
(per cent c) (per cent c)
0
0.5
0.75
1.25
2.5
~
4.302
4.639
4..864:
4.980
4.988
45
4.843
1.275
1.286
1.295
1.300
1.279
50
4.586
4.238
3.820
3.345
2.827
1.241
1.201
1.161
1.120
1.080
1.114
1.096
1.077
1.058
1.039
0.158
O.lS9
0.124
0.109
0.095
2.281
1.722
1.176
0.671
0.248
1.036
0.990
0.944
0.900
0.850
1.018
0.995
0.972
0.949
0.922
0.081
0.069
0.057
0.044
0.030
0
0.805
0.897
0
as
40
55
60
65
70
75
80
85
90
95
100
L.E. radius: 0.720 per cent c
NACA 64-010 Basic Thickness Form
THEORY OF WING SECTIONS
350
I
-
-~
1.2
.8
1
I
I
,-J2 Upper .8Urface
~~
r~ ~ ~
~
~
~
1(/ ~2 Lower .".r«11
r/
~~
l'.
"­
MCA 841-tJ12
t->
- r-­
~
i'--­ t - - .
-
~ """"'-­
~~
~
1.0
.8
(J
y
x
II
(V/V)2
»rv
0
0.978
1.179
1.490
2;035
0
0.750
0.885
1.020
1.129
0
0.866
0.941
1.010
1.063
5.0
7.5
10
15
20
2.810
3.394
3.871
4.620
5.173
1.204
1.240
1.264
1.296
1.320
1.097
1.114
1.124
1.139
1.149
0.685
0.559
0.482
0.388
0.328
25
30
35
40
45
5.576
5.844
5.978
5..981
5.798
1.338
1.351
1.362
1.372
1.335
1.156
1.162
1.167
1.171
1.156
0.281
0.247
0.221
0.199
0.177
50
55
60
5.480
5.056
4.548
3.974
3.350
1.289
1.243
1.195
1.144
1.091
1.136
1.115
1.093
1.070
1.044
0.158
0.138
0.122
0.103
0.088
2.695
2.029
1.382
0.786
0.288
1.037
0.981
0.928
0.874
0.825
1.018
0.990
0.963
0.935
0.908
0.074
0.063
0.052
0.045
0.028
0
0.775
0.880
(per cent c) (per cent c)
0
0.5
0.75
1.25
2.5
65
70
75
80
85
90
95
100
L.E. radius: 1.040 per ce~t c
NACA 041-Oi2 Buic Thickness Form
I
~Vtl./V
2.379
1.663
1.508
1.271
0.943
II
I
0'
.--
~
~"
APPENDIX I
351
.22 LtlWt!r ."(Dce
.2
.4
.6
xjc
.8
1.0
I
X
Y
1---_
(v/l'F
I
tJ/V
-----I-----!-----:­
(per cent c) (per cent c)
~o/l'
I
o
o
o
o
0.5
0.75
1.208
1.456
1.842
2.528
0.670
0.762
0.896
1.113
0.819
0.873
0.947
IJJ55
1.231
1.109
1.284
15
3.504
4.240
4.842
5.785
20
6.480
1.323
1.375
1.410
1.133
1.150
1.172
1.187
0.670
0.559
0.482
0..389
0.326
25
6.985
7.319
7.482
7.473
7.224
1.454
1.470
1.485
1.426
1.198
1.206
1.213
1.218
1.195
0.285
0.250
0.225
0.202
0.179
1.25
2.5
5.0
7.5
10
30
35
40
45
50
70
6.810
6.266
5.620
4.895
4.113
75
80
3.296
2.472
55
60
65
85
1.677
90
95
0.950
0.346
1.434
1.365
1.300
1.233
1.167
1.101
1.033
0.967
0.902
0.841
0.785
1.168
1.140
1.110
1.080
1.049
1.016
0.983
0.950
0.917
0.886
1.939
1.476
1.354
1.188
0.916
i
I
I
I
i
I
i
I
"
0.158
0.135
0.121
0.105
0.090
0.078
0.065
0.054
0.041
0.031
100
o
0.730
o
-----~-----_...:..-----_I~~._I_
-----1
L.E. radius: 1.590 per cent c
NACA 64 r015 Basic Thickness Form
THEORY OF WING SECTIONS
352
()
.4
y
x
(per cent c) (per cent c)
0
0.5
0.75
1.25
2.5
5.0
7.5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
0
1.428
1.720
2.177
3.005
4.186
5.076
5.803
6.942
7.782
8.391
8.789
8.979
8.952
8.630
(V/V)2
0
0.54.6
0.705
0.862
1.079
I
I
1.244
1.327
1.380
1.450
1.497
1.535
1.562
1.585
vii'
1.518
J1fJ./l'
0.739
0.840
0.920
1.039
1.646
1.360
1.269
1.128
0.904
1.115
1.152
1.175
1.204
1.224
0.669
0.558
0.486
0.391
0.331
1.239
1.259
1.265
1.232
0.288
0.255
0.228
0.200
0.177
0
1.250
1.600
1.0
.8
.6
Z/c
8.114
7.445
6.658
5.782
4.842
1.354
1.272
1.190
1.109
1.198
1.164
1.128
1.091
1.053
0.154
0.134
0.117
0.102
0.088
3.866
2.888
1.951
1.101
0.400
1.028
0.952
0.879
0.812
0.747
1.014
0.976
0.937
0.901
0.864
0.074
0.063
0.051
0.039
0.027
0
0.695
0.834
0
1.436
i
L.E. radius: 2.208 per cent c
NAC.A 64r018 Basic Thickness Form
APPENDIX I
353
(~)z
.4
-I.
1.0
.8
.6
%,C
I
I
(per ::nt C) (per
0
0.5
0.75
1.25
2.5
5.0
7.5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
~nt C)
I
()
j
1.646
I
\
{V!V)2_
3.485
0
0.462
C.603
0.759
1.010
4.871
5.915
6.769
8.108
9.095
1.248
1.358
1.431
1.527
1.593
9.807
10.269
10.481
10.431
10.030
1.654
I
I
1.985
!
2.517
I
t
I
1.681
1.712
1.709
1.607
9.404
8.607
7.678
6.649
5.549
4.416
3.287
2.213
1.245
0.449
0
1.117
1.165
1.196
1.236
1.281
1.297
1.308
1.307
1.268
1.228
0.932
0.851
0.778
0.711
1.010
0.965
0.923
0.882
0.844
0.653
0.808
1.020
1.458
;
1.274
1.203
1.084
0.878
I
i
1.262
1.186
1.143
1.099
1.055
1.112
I
I
0.680
0.776
0.871
1.005
~./V
_ _-­
..
0
1.507
1.307
!
----
1.406
1.209
I
v/lf"
I
I
I
1
I
I
i
!
1I
II
I
I
I
I
L.E. radius: 2.884 per cent c
NACA 64.-021 Basic Thickness Form
0.665
0.557
0.486
0.395
0.335
0.293
0.259
0.232
0.202
0.178
0.155
0.134
0.116
0.099
0.084
0.071
0.059
0.047
0.036
0.022
0
THEORY OF WING SECTIONS
354
0
l2
(vt
..-I
P:-rJf
II
/ "I =.02 Upper surface
H--
~02 Lowersurface
...............
.8
...........
~
NACA 64AOO6
.--~
o
.2
y
x
(per cent c) (per cent c)
.4
lO
.6
x/c
(V/V)2
v/V
t.Vo/V
0
0.485
0.585
0.739
1.016
0
1.019
1.046
1.076
1.106
0
1.009
1.023
4.688
2.101
1.798
1.037
1.052
0.980
5.0
7.5
10
15
20
1.399
1.684
1.919
2.283
2.557
1.118
1.126
1.132
1.141
1.149
1.057
1.061
1.064
1.072
0.694
0.564
0.482
0.382
0.321
25
2.757
2.896
2.977
2.999
2.945
1.154
1.158
1.162
1.165
1.156
1.074
1.076
1.078
1.079
1.075
0.278
0.246
0.219
0.197
0.177
70
2.825
2.653
2.438
2.188
1.907
1.142
1.125
1.107
1.087
1.066
1.069
1.061
1.052
1.043
1.032
0.159
0.143
0.126
0.112
0.099
75
80
85
90
95
1.602
1.285
0.967
0.649
0.331
1.043
1.018
0.992
0.964
0.935
1.021
1.009
0.996
0.982
0.967
0.087
0.074
0.061
0.047
0.033
100
0.013
0
0
0
0
0.5
0.75
1.25
2.5
30
35
40
45
50
55
60
65
1.068
L.E. radius; 0.246 per cent c
T.E. radius: 0.014 per cent c
NACA 64A.OO6 Basic Thickness Form
1.422
355
APPENDIX I
/ Cl
= .046 Upper surface
/
~
l2
(vt K
J
I
~r--
~
r---. ~ ...
'.tJ4.6 Lower sarfoce
.............
.8
~
NACA 64,4008
.4
~
,........
-.-.,
r-­ ..........
o
.2
.6
.8
LO
xle
x
y
(per cent c) (per cent c)
0
0.5
0.75
1.25
0
0.646
0.778
0.983
2.5
1.353
(V/V)2
v/V
lw./V
0
0.947
1.005
1.068
1.122
0
0.973
1.033
1.059
3.546
1.972
1.697
1.352
0.971
0.692
0.564
0.481
0.382
0.323
1.002
5.0
1.863
7.5
2.245
2.559
3.047
3.414
1.151
1.165
1.176
1.191
1.201
1.073
1..079
1.084
1.091
1.096
3.681
3.866
3.972
3.998
3.921
1.2\>9
1.217
1.221
1.225
1.211
1.100
1.103
1.105
1.107
1.100
3.757
3.524·
1.191
1.167
1.141
1.113
1.084
1.091
1.080
1.068
1.055
1.041
0.158
0.141
0.125
0.111
0.098
1.026
1.010
0.993
0.975
0.956
0.084
0.072
0.059
0.045
0.032
0
0
10
15
20
25
30
35
40
45
50
55
60
65
3.234
70
2.897
2.521
75
2.117
80
85
1.698
90
95
1.278
0.858
0.438
1.053
1.020
0.987
0.951
0.914
100
0.018
0
L.E. radius: 0.439 percent c
T.E. radius: 0.020 per cent c
NACA 64AOOS Basic Thickness Form
I
I
I
0.279
0.247
0.221
0.198
0.177
THEORY OF WING SECTIONS
356
L6
/C,
0,
= .08 Upper surface
Ii
12
r t. ~ '-::=:
t;(
r
-
~loo...-
~
"""'iiillIII
~
~~
~
t', .08 Lower 6urface
~
NACA 64AOIO
A
--­
~-
r---- I--....
o
~
.6
.4
t - - - .....
~
....-­
.B
LO
x/c
%
11
(V/V)2
vtV
&J./V
0
0.804
0.969
1.225
1.688
0
0.868
0.952
1.042
1.130
0
0.932
0.976
1.021
1.063
2.868
1.845
1.603
1.300
0.957
2.327
2.905
3.199
3.813
4.272
1.178
1.201
1.217
1.238
1.254
1.085
1.096
1.103
1.113
1.120
0.280
0.248
0.221
0.199
0.177
(per cent c) (per cent c)
0
0.5
0.75
1.25
2.5
5.0
7.5
10
15
20
I
I
II
0.688
0.562
0.480
0.382
0.324
25
4.606
1.266
30
4.&17
25
4.968
4.995
4.894
1.275
1.282
1.268
1.125
1.129
1.132
1.135
1.126
4.684
4.388
4.021
3.597
3.127
1.240
1.208
1.174
1.139
1.102
1.114
1.099
1.084
1.067
1.050
0.158
0.140
0.124
0.109
0.096
1.063
95
2.623
2.103
1.582
1.062
0.541
0.981
0.938
0.893
1.031
1.011
0.990
0.969
0.945
0.083
0.070
0.058
0.044
0.031
100
0.021
0
0
40
45
50
55
60
65
70
75
80
85
90
1.288
1.023
L.E. radius: 0.687 per cent c
T .E. radius: O'()?-3 per cent c
I
II
0
APPENDIX I
l6
,
0
#
357
/ c1 =./3 Upper surFQ(Y
- ~........ ~
~~~
~~~
~
1.2.
L
{~
(vt V
.8
~
........
~
"J~ lower surFace
~~
~
I
~
NACA 64,AO/2
~
-r--- !---
I'"""""""
f'-.-. ~
o
.2
.4
.6
x/c
z
I
r--. .--­
........
L.-­ ~
LO
,._­
i
l7/ v
y
(V/V)2
I
0
0.792
0.893
1.25
0
0.961
1.158
1.464
1.006
2.5
I
2.018
1.127
l.Oti2
0.941
5.0
7.5
10
15
1.201
1.000
1.235
1.111
1.257
1.121
20
2.788
3.364
3.839
4.580
5.132
25
30
5.534
5.809
35
40
5.965
5.993
45
5.863
50
55
60
5.605
(per cent c) (per cent c)
0
0.5
0.75
65
70
75
80
85
5.244
4.801
4.289
3.721
I
1.135
1.144
1.324
1.151
1.156
1.160
1.164
1.152
0.281
0.249
0.221
0.199
0.177
1.135
1.118
1.099
1.079
1.057
0.157
0.139
0.123
0.108
0.094
0.873
1.035
1.011
O.98i
0.962
0.934
0.080
0.068
0.056
0.042
0.029
0
0
0
1.34fi
1.289
1.250
1.207
1.164
1.118
0.974
1.263
0.925
0.644
100
0.025
I.U71
1.023
i
1.254
1.308
3.118
2.500
1.882
90
95
!
2.408
1.720
1.515
1.2..~
1.354
1.326
I
0
0.890
0.945
1.003
0.681
0.560
0.478
0.383
0.325
1.336
I
i
~l'CI/V
L.E. radius: 0.994 per cent c
"f.E. radius: 0.028 per eent c
NAC.A.. 64 1A012 Basic Thickness Form
THEORY OF U'ING SECTIONS
358
,
L6
/c, = .2/ Upper surface
.-- r----- ~~
~
~
,0
l2
/
~
If J K,
(vt '/
.8
~~
~
~
~
I
~~
~
".2/ lower surface
r-,
NACA 64-2 A O/5
I
.4
- r--- r---
~
r
'-- r--- -
o
.2
- .6
~~
.4
-------
,......
~
LO
.8
x/c
x
11
(per cent c) (per cent c)
(VJV)2
vJV
.:1t·a / l'
0.5
0.75
1.25
2.5
0
1.193
1.436
1.815
2.508
0
0.678
0.789
0.936
1.110
0
0.823
0.888
0.967
1.054
1.956
1.552
1.404
1.189
0.912
5.0
7.5
3.477
4.202
1.226
1.280
4.799
5.732
6.423
1.314
1.390
1.107
1.131
1.146
1.166
1.179
0.671
0.552
0.478
0.384
0.326
40
45
6.926
1.270
7.463
7.481
7.313
1.413
1.430
1.445
1.458
1.414
1.189
1.196
1.202
1.207
1.189
0.283
0.249
0.222
0.201
0.177
50
55
60
65
70
6.978
6.517
5.956
5.311
4.600
1.364
1.311
1.255
1.198
1.139
1.168
1.145
1.120
1.095
1.067
0.156
0.137
0.121
0.106
0.091
75
80
1.079
1.020
0.961
0.901
0.843
1.039
1.010
0.980
0.949
0.918
0.078
0.065
0.053
0.041
95
3.847
3.084
2.321
1.558
0.795
100
0.032
0
0
0
0
10
15
20
25
30
35
85
90
.1.360
LE. radius: 1.561 per cent c
T.E. radius: 0.037 per cent c
NACA 64tA015 Basic Thickness Fonn
0.027
APPENDIX I
~,Cls.OJ
.I.e 10
359
I
I .............
~r:L
r--:Ol Lower""'or:e
.8
I
I
Lt;per surfoce
r----.. r-----...
-.......
r-.
I
NACA 65-006
.4
--­
~
D
.4
x
y
5.0
7.5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
/.0
.8
(V/V)2
v/V
~Va/V
0
0.476
0.574
0.717
0.956
0
1.044
1.055
1.063
1.081
0
1.022
1.027
1.031
1.040
4.815
2.110
1.780
1.390
0.965
1.310
1.589
1.824
2.197
2.482
1.100
1.112
1.120
1.134
1.143
1.049
1.055
1.058
1.065
1.069
0.695
0.560
0.474
0.381
0.322
2.697
2.852
2.952
2.998
2.983
1.149
1.155
1.159
1.163
1.166
1.072
1.075
1.077
1.078
1.080
0.281
0.247
0.220
0..198
0.178
2.900
2.741
2.518
2.246
1.935
1.165
1.145
1.124
1.100
1.073
1.079
1.070
1.060
1.049
1.036
0.160
0.144
0.128
0.114
0.100
1.594
1.233
0.865
0.510
0.195
1.044
1.013
0.981
0.944
0.902
1.022
1.006
0.990
0.972
0.950
0.086
0.074
0.060
0.046
0.031
0
0.858
0.926
0
(per cent c) (per cent c)
0
0.5
0.75
1.25
2.5
.6
L.E. radius: 0.240 per cent
c
NACA 65-006 Basic Thickness Form
f'HEORY OF WING SECTIONS
360
~
1
,
1/' -.001 lJp~r sur'a~
0
1.2
~
I I """'ilIlIiiii
~IH LtItIWr ~1ICtI
~
"'" r-,
.1
NACA 85-008
·4
~
~
-
r---........
.....
~
a
.4
x
11
(per cent c) (per cent c)
Z/c
1.0
.8
.11
(V/V)2
V/V
Avo/V
0
0.627
0.756
0.945
1.267
0
0.978
1.010
1.043
1.086
0
0.989
1.005
1.021
1.042
3.695
2.010
1.696
1.340
0.956
5.0
7.5
10
15
20
1.745
2.118
2.432
2.931
3.312
1.125
1.145
1.158
1.178
1.192
1.061
1.070
1.076
1.085
1.092
0.689
0.560
0.477
0.382
0.323
25
30
35
40
45
3.599
3.805
3.938
3.998
3.974
1.203
1.210
1.217
1.222
1.226
1.097
1.100
1.103
1.105
1.107
0.281
0.248
0.221
0.199
0.178
50
55
60
65
70
3.857
3.638
3.337
2.971
2.553
1.222
1.193
1.163
1.130
1.094
1.105
1.092
1.078
1.063
1.046
0.160
0.145
0.128
0.113
0.098
75
80
85
2.096
1.617
1.131
0.664
0.252
1.055
1.014
0.971
0.873
1.027
1.007
0.985
0.961
0.934
0.084
0.072
0.059
0.044
0.031
0
0.817
0.904
0
0
0.5
0.75
1.25
2.5
90
95
100
(i923
L.E. radius: 0.434 per cent c
NACA 65-008 Basic Thickness Form
APPENDIX I
I.G
I
/e I -.IM
1.2
361
r
I
I
tP/HIr ",'act1
f
C\
,...-r
~
~~
I'-
~.D6
I
-Low.- .ur'ac.
f'
V
~
.8
/MeA
ts5-009
~
r----
I000o......
o
L---
--
r--- I....
-
~
1.0
.8
.2
x
y
(per cent c) (per cent c)
"' r-,
(vlV)2
vjV
I1v.IV
0
0.700
0.845
1.058
1.421
0
0.945
0.985
1.037
1.089
0
0.972
0.992
1.018
1.044
3.270
1.962
1.655
1.315
0.950
5.0
7.5
10
15
20
1.961
2.383
2.736
3.299
3.727
1.134
1.159
1.177
1.200
1.216
1.065
1.077
1.085
1.095
1.103
0.687
0.660
0.477
0.382
0..323
25
30
35
40
4.050
4.282
4.431
4.496
4.469
1.229
1.238
1.246
1.252
1.258
1.109
1.113
1.116
1.119
1.122
0.280
0.248
0.220
0.198
0.178
4.336
4.086
3.743
3.328
2.856
1.250
1.220
1.185
1.145
1.103
1.118
1.105
1.089
1.070
1.050
0.160
0.144
0.128
0.111
0.097
2.342
1.805
1.260
0.738
0.280
1.059
1.013
0.963
0.912
0.856
1.029
1.006
0.981
0.955
0.925
0.084
0.071
0.059
0.044
0.030
0
0.797
0.893
0
0.5
0.75
1.25
2.5
45
50
55
60
65
70
75
80
85
90
ss
100
L.E. radius: 0.552 per cent c
NACA 65-009 Basic Thickness Form
I
0
THEORY OF WING SECTIONS
362
1.4
1.2
.8
I
--­
--­
(/
)\
r
t
--::: :::-­
~ \08
I
J
I
/ ', •.08 IJpptIr 6Ur'ace
,
~
~
LDWf!r
Sur~DC4P
~
""'--,
NACA 65-010
,--.............
- r--­
­­
.z
(J
y
x
.4
--­
~
.s
I"'---­
~~
.8
/.
(V/V)2
v/V
l:.VA/V
2.5
0
0.772
0.932
1.169
1.574
0
0.911
0.960
1.025
1.085
0
0.954
0.980
1.012
1.042
2.967
1.911
1.614
1.292
0.932
5.0
7.5
10
15
20
2.177
2.647
3.040
3.666
4.143
1.143
1.177
1.197
1.224
1.242
1.069
1.085
1.094
1.106
1.114
0.679
0.558
0.480
0.383
0.321
25
4.503
4.760
4.924
4.996
4.963
1.257
1.268
1.277
1.284
1.290
1.121
1.126
1.130
1.133
1.136
0.280
0.248
0.222
0.199
0.179
70
4.812
4.530
4.146
3.682
3.156
1.284
1.244
1.202
1.158
1.112
1.133
1.115
1.096
1.076
1.055
0.160
0.141
0..126
0.110
0.097
75
80
85
90
95
2.584
1.987
1.385
0.810
0.306
1.062
1.011
0.958
0.903
0.844
1.031
1.005
0.979
0.950
0.919
0.082
0.070
0.058
0.045
0.030
0
0.781
0.884
0
(per cent c) (per cent c)
0
0.5
0.75
1.25
30
35
40
45
50
55
60
65
100
L.E. radius: 0.687 per cent c
NACA 66-010 Basic Thickness Form
APPENDIX I
It---+--t-----i~
363
NACA 6~-OI2
.41----+---+-~f--__+_-_+_____f-__+_-_+___t-"'"""1
o
.2"
x
y
.4
1.0
.8
.6
X/C
(t,/lr)~
v/1T
&toll"
0
0.923
1.109
1.387
1.875
0
0.848
0.935
1.000
1.082
0
0.921
0.967
].000
1.040
2.444
1.776
1.465
1.200
0.931
5.0
7.5
10
15
20
2.606
3.172
3.647
4.402
4.975
1.162
1.201
1.232
1.268
1.295
25
30
35
40
5.406
1.316
5.;16
1.332
5.912
5.997
5.949
1.343
1.350
1.357
(per cent c) (per cent c)
0
0.5
0.75
1.25
2.5
45
50
55
iI
I
1.078
1.096
1.110
1.126
1.1.38
1.147
1.154
1.159
1.162
1.165
0.282
0.251
0.223
0.204
0.188
1.159
1.138
1.115
60
65
70
3.743
1.295
1.243
1.188
1.134
75
80
85
90
95
3.059
2.345
1.630
0.947
0.356
1.073
1.010
0.949
0.884
0.819
1.036
1.005
0.974
0.940
0.905
0
0.748
0.865
100
I
0.702
0.568
0.480
0.389
0.326
5.757
5.412
4.943
4.381
1.343
1
1.065
0.169
0.145
0.127
0.111
0.094
0.074
0.062
0.049
0.038
0.025
1.090
L.E. radius: 1.000 per cent c
NACA. 65 1-012 Basic Thickness Form
I
0
THEORY OF WING SECTIONS
364
c, •. 22 ~ surl'oce
o
.2
x
'Y
8
6
1.0
(V/V)2
V/V
Av./l
0
1.124
1.356
1.702
2.324
0
0.654
0.817
0.939
1.063
0
0.809
0.904
0.969
1.031
2.038
1.729
1.390
1.156
0.920
5.0
7.5
10
15
20
3.245
3.959
4.555
5.504
6.223
1.184
1.241
1.281
1.336
1.374
1.088
1.114
1.132
1.156
1.172
0.682
0.563
0.487
0.393
0.334
25
6.764
7.152
7.396
7.498
7.427
1.397
1.4:18
1.438
1.452
1.464
1.182
1.191
1.199
1.205
1.210
0.290
0.255
0.227
0.203
0.184
7.168
6.720
6.118
5.403
4.600
1.433
1.369
1.297
1.228
1.151
1.197
1.170
1.139
1.108
1.073
0.160
0.143
0.127
0.109
0.096
3.744
2.858
1.977
1.144
0.428
1.077
1.002
0.924
0.846
0.773
1.038
1.001
0.961
0.920
0.879
0.078
0.068
0.052
0.038
0.026
0
0.697
0.835
0
(per cent c) (per cent c)
0
0.5
0.75
1.25
2.5
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
L.E. radius: 1.505 per cent c
X ACA. 65r015 Basic Thickness Form
Y
APPENDIX I
(J
.4
x
JI
(per cent c) (per cent c)
0
0.5
0.75
1.25
2.5
0
1.337
1.608
2.014
2.751
365
.s
z/c
14
(v/V)~
v/1T
j.r:./V
0
0.625
0.702
0.817
1..020
0
0.i91
0.838
0.. 904
1..010
1.746
1.437
1.302
1.123
0.858
5.0
7.5
10
15
20
3..866
4.733
5.457
6.606
7.476
1.192
1.275
1.329
1.402
1.452
1..092
1.129
1.153
1.184
1.205
0.650
0.542
0.474
25
1.488
1.515
1.539
1.. 561
1.578
1.220
45
8.129
8.595
8.886
8.999
8.901
0.285
0.251
0.225
0.203
0.182
50
8..568
55
1.526
1.440
1.353
1.262
1.235
1.200
65
70
8.008
7.267
6.395
5.426
75
4.396
80
30
35
40
60
85
90
95
100
1.170
0.. 385
0.327
1.231
1:241
1.249
1.256
0.157
0.137
0.118
0.104
1.163
1.123
1.082
3.338
2.295
1.319
0.490
1.076
0.985
0.896
0.813
0.730
1.037
0.992
0
0.657
0.811
0.087
0.074
0.062
0.050
0.039
0.026
0.947
0.902
0.854
L.E. radius: 1.96 per cent c
NACA 65r 0 18 Basic Thickness Form
I
0
THEORY OF ,"lING SECTIONS
366
2.0
1.2
.8
»:
'I
I ...--- IV
~
x
I
----
[\
/~ ~
~ r--"\
-:
or / v
/ / \.J/4
I IV
a
I
I
\ . / 1 :1 -.-I/. ~ svr'oce
I
1.6
I
I
r:
<,
LDWtlr
~
'" ~~~
NACA 65. -OZI
- --- ---.r---
-- ---
~
y
"~
$urf"~ <,
{U/V)2
l---'
~
L-----
1.0
.8
f'/l~
(~r~nt~ 1_(p_e_r_~_n_t_~_I_~~~_~~~~~I!~~~~1
o
o
o
o
0.5
0.75
1.25
2.5
1.522
1.838
2..301
3.154
0.514
0.607
0.740
0.960
0.. 717
0.779
0.860
0.980
5.0
4.472
5.498
6.352
7.700
8.720
1.186
1.293
1..371
1..469
1.533
1.137
1.171
1.212
9.487
10.036
10.375
10.499
10.366
1.580
1..621
9.952
9.277
8.390
7.360
6.224
1.633
5.024.
1.073
3.800
2.598
1.484
0.546
0.970
0.872
0.778
0.694
o
0.616
7..5
10
15
20
25
30
35
40
45
50.
55
60
65
70
75
80
85
90
95
100
1.654
1.680
1..700
1.508
1.397
1.286
1.177
1.531
1.333
1.215
1.062
0.838
1.238
0.649
0.544
0.478
0.388
0.330
1.257
1.273
1.286
1.296
1.304
0.289
0.255
0.229
0.206
0.184
1.278
1.228
1.182
1.134
1.085
0.158
0.139
0.120
0.101
0.087
1.036
0.985
0.882
0.833
0.073
0.058
tl.047
0.035
0.020
0.785
o
1.089
0.934
L.E. radius: 2.50 per cent c
NACA 65.-021 Basie-Thickness Form
APPENDIX I
367
1.6
0
I
L2.
Cl
/
=.01 Upper surface
,!,
~
'""""IIIIl
~-
r----. __
r-,
"a/lower surface
I
.8
NACA6SAOO6
~
r--..
.4-
()
LO
.8
.6
x/c
%
Y
!
(per cent c). (per cent c)
I
(v/1T )2
!
v/V
0
0
0
0.5
0.75
1.25
2.5
0.718
0.981
5.0
1.313
7.5
1.591
10
15
20
25
0..164
O.5ft3
40
45
.2.992
50
2.925
2.793
2.602
2.364
2.087
35
55
60
65
70
75
80
85
1.775
1.029
1.039
1.139
1.145
1.149
1.153
1.157
1.159
1.070
1.072
1.074
1.076
1.077
1.157
1.141
1.07(;
1.131
-
II
I
4.879
2.145
1.763
1.365
0.966
0.688
0.562
0.480
0.382
0.323
I
t
I
!
ii
!
0.278
0.246
0.219
0.198
0.178
o.isu
I
1.124
1.()(i8
1.06U
,I
1.106
1.083
1.052
1.041
0.143
0.127
0.112
0.099
i
1.059
1.032
0.973
0.936
1.029
1.016
1.001
0.986
0.967
0.076
0.061
0.047
0.033
0
0
0
I
95
1.437
l.OS3
0.727
0.370
100
0.013
I
90
0
1.017
1.021
1.049
1.055
1.058
1.063
1.067
1.101
1.112
1.120
1.824
2.194
2.474
2.687
2.842
2.945
2.996
30
1.034
1.043
1.058
1.080
4v./V
I
-~_._---
i
1.003
L.E. radius: 0.229 per cent c
T.E. radius: 0.014 per cent c
NACA 65AOO6 Basic Thickness Form
OJl87
THEORY OF WING SECTIONS
368
L6
0\
,'" cl-.05 Upper surface
I T .........
.8
t?~5 lower surface
r
~
......... ...............
r-,
WACA65AOO8
A
---
~
r---- ............
o
.4
I
I
.6
lO
~/c
i
%
Y
(per cent
c) (per cent
c)
0
0.5
0.75
I---...
(V/V}2
v/V
I
Av./V
!
0
0.973
1.001
1.038
1.088
0
0.986
2.5
0
0.615
0.746
0.951
1.303
5.0
7.5
10
15
20
1.749
2.120
2.432
2.926
3.301
1.127
1.145
1.157
1.175
1.186
1.062
1.070
1.076
1.084
1.089
25
3.585
3.791
3.928
3.995
3.988
1.195
1.202
1.207
1.213
1.217
1.093
1.096
1.099
1.101
1.103
0.279
0.247
0.219
0.198
0.178
3.895
3.714
3.456
3.135
2.763
1.214
1.191
1.167
1.108
1.102
1.091
1.080
1.067
1.053
0.161
0.144
0.128
0.112
0.098
95
2.348
1.898
1.430
0.960
0.489
1.076
1.041
1.002
0.961
0.916
1.001
0.980
0.957
100
0.018
0
0
1.25
30
35
40
45
50
55
60
65
70
75
80
85
90
i.iss
1.000
~
1.019
1.043
1.037
1.020
L.E. radius: 0.408 per cent c
T.E. radius: 0.020 per cent c
i
I
I
3.698
2.010
1.693
1.333
0.954
0.685
0.561
0.479
0.382
0.322
0.086
0.073
0.060
0.046
C
NACA 6SAOOS Basic 'rhickness Form
APPENDIX I
369
L6
,e, = ./0 Upper surface
I
~
0
,
0.V-
r[
~
.-. """"""'"'
~
~
~ .......
"'lIIIIIIlI
"'JO lower surf"ace
~
INAcA 6SA010
-
~~
~ t--...
o
%
.2
.4
x/c
II
y
(per cent c) (per cent c)
o
0
0.5
0.75
1.25
2.5
0.765
0.928
1.183
1.623
5.0
7.5
10
15
20
2.182
2.650
3.040
3.658
4.127
25
35
40
4.483
4.742
4.912
4.995
45
4.983
50
55
4.863
4.632
4.304
3.899
3.432
30
60
65
70
90
95
2.912
2.352
1.771
1.188
0.604
100
0.021
75
80
85
I
I
I
I
I
,I
I
Ii
I
i
II
II
I
I
I
I
r--- ~-
-
.6
.8
(V/V)2
v/V
0
0.897
0.948
1.010
1.089
0
0.947
0.974
1.148
1.176
1.194
1.218
1.234
1.071
1.247
1.257
1.265
1.272
1.277
1.117
1.121
1.125
1.128
1.130
1.271
1.241
1.208
1.127
1.114
1.099
1.083
1.172
J.O
,
Av./V
I
2.987
1.878
1.619
1.303
0.936
1.084
1.093
I
1.104
1.111
I
0.679
0.559
0.478
0.382
0.323
I
I
1.005
1.044
1.133
1.064
1.091
0.999
0.949
0.893
1.045
1.023
0.999
0.974
0.945
0
0
1.047
~~
L.E. radius: 0.639 per cent c
T.E. radius: 0.023 per cent c
NACA 65AOIO Basic Thickness Form
I
!
II
1
I
0.281
0.249
0.222
0.198
0.178
0.161
0.144
0.127
0.111
0.097
0.084
0.071
0.058
OJM5
0.029
0
THEORY OF WING SECTIONS
370
L6
u
.-I C, ·./S Upper surface
- <.
.
(/
,...O~
r(
~
~
~
~
~~
~
.8 I
~ ....
~
" ./5 lower surface
-,
NACA6~AOI2
~
r'----
o
~
.2
--- -------- ----------
-
I--""""'"
.6
A
1.0
.8
-x/a
I,
y
(V/V)2
v/V
AVa/I'
0
0.913
1.106
1.414
1.942
0
0.824
0.883
0.969
1.081
0
0.908
0.940
0.984
1.040
2.520
1.757
1.543
5.0
7.5
10
15
20
2.614
3.176
3.647
4.392
4.956
1.166
1.204
1.228
1.263
25
30
5.383
5.693
5.897
5.995
5.977
%
(per cent c) (per cent c)
0
0.5
0.75
1.25
2.5
i.ses
0.914
1.285
1.097
1.108
1.124
1.134
0.672
0.557
0.477
0.382
0.324
1.301
1.313
1.324
1.332
1.338
1.141
1.146
1.151
1.154
1.157
0.281
0.250
0.224
0.198
0.178
5.828
5.544
5.143
4.654
4.091
1.329
1.292
1.251
1.156
1.153
1.137
1.118
1.097
1.075
0.161
0.143
0.126
0.111
0.096
90
95
3.467
2.798
2.106
1.413
0.719
1.104
1.051
0.994
0.936
0.871
1.051
1.025
0.997
0.967
0.933
0.082
0.069
0.057
0.043
0.027
100
0.025
0
0
0
35
40
45
50
55
60
65
70
75
80
85
1.204
1.080
L.E. radius: 0.922 per cent c
1'.E. radius: 0.029 per cent c
NACA 65 1A012 Basic Thickness Form
APPENDIX I
_/~ Cl
L6
J,:
r
~
~
-r-,
~
l,...---­
~
7 i>
.4
~~
"" ~
1A
fl
~
-r--- r---.
L-.­ ~
.2
~
11
0
1.131
1..371
1.750
2.412
r---- "'"-­
~
~
lO
.8
x/c
(per cent c) (per cent c)
0
0.5
0.75
1.25
2.5
~
WACA 652 .40 15
~,....
o
~
' ... .22 lower surrac«
[,....­
~ I"----........
%
=.22Uppersurface
'"--.
/
.8
371
(v/V)%
vlV
&J./V
0
0.714
0.781
0.891
1.059
0
0,,845
0.884
0.944
1.029
2.048
1.586
1.417
1.195
0.880
1.089
1.115
1.131
1.152
1.166
0.660
0.553
0.476
0.282
0.252
0.227
5.0
7.5
10
15
20
3.255
3.962
4.553
1.187
1.243
5.488
1.328
1.359
25
45
6.734
7.122
7.376
7.496
7.467
1.401
1.416
1.427
1.437
1.176
1.184
1.190
1.195
1.199
50
55
60
65
70
7.269
6..903
6.393
5.772
5.063
1.419
1.368
1.311
1.249
1.186
1.191
1.170
1.145
1.118
1.089
0.161
0.142
0.124
0.109
75
4.282
3.451
2.598
95
0.887
1.123
1..056
0.986
0.913
0.841
1.060
80
85
90
1.028
0.993
0.956
0.917
0.080
0.067
0.055
0.041
0.026
100
0.032
0
0
0
30
35
40
6.198
1.743
1.280
1.383
0.382
0.326
0.204­
0.181
0,(194
L.E. radius: 1.446 per cent c
T.E. radius: 0.038 per cent c
....,.. • ,....
,. . . . n . . . " ' ..
~
_
~
:_1
'T."
_
THEORY OF WING SECTIONS
372
/.6
/C'i 01
1.2
uprr
Surface
JO
~-
--~,
/ Lower .urface
~'"
.8
NACA 68-006
.4
--­
-~
o
.2
%
11
(per cent c) (per cent c)
0
0.5
0.75
1.25
2.5
5.0
7.5
10
15
20
,
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
tOO
I
I
I
I
.4
X/C
(V/V)2
1.0
8
6
v/IT
Av./V
0
0.461
0.554
0.693
0.918
0
1.052
1.057
1.062
1.071
0
1.257
1.524
1.752
2.119
2.401
1.086
1.098
1.107
1.119
1.128
1.042
2.618
2.782
2.899
2.971
3.000
1.133
1.138
1.142
1.145
1.148
1.064
1.067
1.069
1.070
1.071
2.985
2.925
2.815
2.611
2.316
1.151
1.153
1.155
1.154
1.118
1.073
1.074
1.075
1.074
1.057
0.161
0.145
0.130
0.116
0.102
1.953
1.543
1.107
0.665
0.262
1.081
1.040
0.996
0.948
0.890
1.040
1.020
0.998
0.974
0.943
0.089
0.075
0.061
0.047
0.030
0
0.822
0.907
0
1.026
1.028
1.031
1.035
I
I
I
1.048
1.052
1.058
1.062
L.E. radius: 0.223 per cent c
NACA 66-006lJasic Thickness Form
I
I
I
!
/
I
4.941
2.500
2.020
1.500
0.967
0.695
0.554
0.474
0.379
0.320
0.278
0.245
0.219
0.197
0.178
373
APPENDIX I
'41'!1
~~,.
/0
(;)' I(~
~
1~03 LO~
03 LPPer surfoce
.s:
"""lIIIlIil
~~
-=
"' <,
MACA 66-D08
--
l...-­
r--- ........
.4
%
11
X/C
r-­
~
.....
.~
1.0
j
{V/V)2
»rv
0
0.610
0.735
0.919
1.219
0
0.968
0
0.984
1.011
1.023
1.038
6.0
7.5
10
15
20
1.673
2.031
2.335
2.826
3.201
1.107
1.128
1.141
1.1SS
1.171
1.052
1.062
1.076
1.082
0.552
0.474
0.379
0.321
25
3.490
30
36
3.709
3.865
40
3.962
46
4.000
1.178
LI86
1.191
1.196
1.201
1.085
1.089
1.091
1.094
1.096
0.278
0.246
0.220
0.198
0.178
1.098
0.161
0.145
0.130
0.115
0.101
(per cent c) (per eent e)
0
0.5
0.75
1.25
2.5
1.023
1.046
1.078
1.068
liO
3.978
55
3.896
80
3.740
3.459
1.205
1.208
1.213
1.202
3.062
1.156
1.075
2.574
2.027
1.447
0.864
0.338
1.103
1.048
0.989
0.926
0.855
1.050
1.024
0.994
0.962
0.925
0
0.768
0.876
6S
70
75
80
85
90
IS
100
i
Av./J-·
3.794
2.220
1.825
1.388
0.949
I
I
1.099
1.101
1.096
0.689
0.087
I
-_1--- ______
L.E. radius: 0.411 per cent c
NACA 66-008 aasic Thickness Form
0.073
0.058
0.045
0.029
0
THEORY OF WING SECTIONS
374
zo
f.Z
¥,C, •• OS ttJper eurface
I'
~
(t)2 ~
V
.8
~
I
~~
~05 tppM­ .ur'~
~
"" r-,
f44C4 66-oot/
.4
-r----
l"...--'
-
r-­ '""--­
a
.4
x
y
~/C
~
--­---­
.8
.6
1.0
wv»
fJ/V
~./V
0
0.687
0.824
1.030
1.368
0
0.930
0.999
1.036
1.079
0
0.964
0.999
1.018
1.039
3.352
2.100
1.750
1.340
0.940
5.0
7.5
10
15'
20
1.880
2.283
2.626
3.178
3.601
1.119
1.142
1.159
1.178
1.190
1.058
1.069
1.077
1.085
1.091
0.686
0.552
0.473
0.379
0.323
2S
30
35
40
45
3.927
4.173
4.348
4.457
4.499
1.201
1.210
1.217
1.221
1.228
1.096
1.100
1.103
1.105
1.1OS
0.280
0.246
0.220
0.197
0.178
50
4.475
4.381
4.204
3.882
3.428
1.232
1.237
1.240
1.230
1.172
1.110
1.112
1.114
1.109
1.083
0.161
0.145
0.130
0.116
0.100
2..877
2.263
1.611
0.961
0.374
1.113
1.050
0.985
0.915
0.839
1.055
1.025
0.992
0.957
0.916
0.085
0.071
0.057
0.043
0.028
0
0.747
0.864
0
(per cent c) (per cent c)
0
0.5
0.75
1.25
2.5
55
60
65
70
75
80
85
90
95
100
L.E. radius: 0.530 per cent c
NACA 66-009 Basic Thickness Form
APPENDIX 1
. .........'-~
I. ~
I.
I"
I
375
I
I
/~, -.07 ~ Arf~
~
~
K
-­I
I
r-.o71.-wr " " c .
I
~
f'~
'" r-,""
,~
"-.­
M-OJ(}
.4
L.o-- ....-­
r--- r--..
,........
.-- I - -
r---- .........
I.D
.B
%
11
(per cent c) (per cent.c)
(rJ/V)2
v/IT
~.IV
0
0.5
0.75
1.25
2.5
0
0.759
0.913
1..141
1.616
0
0.896
0.972
1.023
1.078
0
0.947
0.986
1.011
1.038
3.002
2.012
5.0
7.5
2.087
1.125
1.154
1.174
1.198
1.215
1.061
1.074
1.084:
1.095
1.102
0.682
0.551
0.473
0.379
0.322
4.363
1.226
4.6.36
1.236
1.243
1.249
1.255
1.107
1.112
1.115
1.118
1.120
0.279
0.246
0.220
0.198
0.178
1.261
1.265
1.270
1.250
1.190
1.123
1.125
1.127
1.118
1.091
0.161
0.146
0.130
0.114
0.099
to
IS
20
25
30
35
40
45
SO
55
60
65
70
75
80
85
90
95
100
2.536
2.917
3.630
4.001
4.&12
4.953
S.OOO
4.971
4.865
4.665
4.302
3.787
1.686
1.296
0.931
3.176
2.494:
1.773
1.054
0.408
1.121
1.059
1.052
1.026
0.979
0.904
0.821
0.989
0.951
0.906
0.085
0.070
0.056
0.043
0.027
0
0.729
0.854
0
L.E. radius: 0.662 per cent c
NACA 66--010 Basic Thickness Form
THEORY OF WING
376
I. 6
I. 2
8ECTIONl~
I
I
I
_/~, -./Z ~psr l~urFac.
--­r: ---,(J
IF
~
V< 'r--:IZ ILow."I I
~
-~
,~
:wr,~
~
)1I
~
'"
/MCA tiIia-O/2
~
-
~
~ t---..L...
o
-4
x
(per cent c)
11
5.0
7.5
10
15
20
I
I
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
I
t--.....
I
xc
1.0
.8
.6
(V/V)2
V/V
s1v./V
0
0.906
1.087
1.358
1.808
0
0.800
0.915
0.980
1.078
0
0.894
0.957
0.990
1.036
2.569
1.847
1.575
1.237
0.913
2.496
3.037
3.496
4:.234
4.801
1.138
1.177
1.204
1.237
1.259
1.067
1.097
1.112
1.122
0.674
0.549
0.473
0.380
0.323
5.238
5.568
5.803
5.947
6.000
1.275
1.287
1.297
1.303
1.311
1.129
1.134
1.139
1.142
1.145
5.965
5.836
5.588
5.139
4.515
1.318
1.323
1.331
1.302
1.221
1.148
1.150
1.154
1.141
1.105
3.767
2.944
2.083
1.234
0.474
1.139
1.053
0.968
0.879
0.788
1.067
1.026
0.984­
0.938
0.888
0
0.687
0.829
(per
0
0.5
0.75
1.25
2.5
I'--- r---
_I----­ ~ ~
cent c)
1.085
f
!
i
0.280
0.246
0.221
0.197
0.176
0.162
0.147
0.132
0.113
0.098
0.084
0.069
0.053
0.040
0.031
I
100
L.E. radius: 0.952 per cent. c
NACA 66t-012 Basic Thickness Form
I
0
APPENDIX I
377
-.,
/.~
M
l,...-C",•• '
I./pp#Ir :!IU"fOC.
1
~
1
~
1.1
-
0/
~J
(
I
21#:Per~
~II
NACA
~
~~
~~
I
,
,~
-r/,
i->
--"-
661 -0 15
r--- to---
1----
t---
o
~
.2
I
.4
z c) (per cent
Y c)
(per cent
i
X/C
.6
----
~
r--- "--~
~
1.0
.8
I
("'IF):
to/I'
0
0.760
0.840
0.929
1.055
0
2.139
0.872
0.916
0.9641.027
1.652
j
..1"./1'
0
0.5
0.75
1.25
2.5
0
1.122
1.343
1.675
2.235
~.O
7.5
10
15
3.100
3.781
4.358
5.286
20
~.995
1.288
1.317
1.078
1.099
1.114
1.134
1.148
25
6.543
6.956
7.250
7.430
7.495
1.340
1.356
1.370
1.380
1.391
1.158
1.164
1.170
1.175
1.179
7..450
7.283
6.959
6.372
5.576
1.401
1.411
1.420
1.367
1.260
1.184
1.188
1.192
1.169
1.122
4.632
3.598
2.530
1..489
0.566
1.156
1.053
0.949
0.841
0.744
1.075
0.974
0.920
0.863
0.080
0.065
0.051
0.039
0.025
0.639
0.799
0
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
1.163
1.208
1.242
0
~~---_
... -_._......-- ..... _- ­
1.026
L.E. radius: 1.435 per cent c
N ACA 66::-015 Basic Thickness Form
j
I
I
I
I
I
!
t
I
i
1.431
1.172
0.895
0.663
0.047
0.473
0.381
0.322
0.280
0.248
0.222
0..200
0.180
0.163
0.146
0.131
0.113
0.096
7.'HBORY OF WINO .SECTIONS
378
%
11
(per oeot c) (per cent c)
(fJ!V)I
fJ/V
MJ./v
0
0.806
0.857
1.773
1.456
1.312
1.121
0
0.5
0.15
1.25
0
1.323
1.571
1.952
2.5
2.646
1.0
7.5
10
3.690
4:.613
5.210
16
20
8.333
7.las
1.393
1.134:
1.182
1.1SO
25
30
3&
40
4&
7.848
8.348
8.101
8.918
8.998
1.423
1.193
1.4:45
1.202
1.484
1.481
1.210
1.217
1.223
SO
IS
60
8.942
8.733
8.323
65
70
7.580
6.197
1.S09
1.522
1.534
1.438
1.302
75
80
5.4:51
8S
90
95
100
0
0.650
0.735
0..850
1.005
1.114
1.234
1.285
1.350
1.496
0.897
1.002
1.074
1.111
1.228
1.234
1.238
LI99
1.141
0..858
0.649
0.645
0.472
0.381
0.323
0.282
0.250
0.223
0.201
0.181
0.163
0.147
0.131
0.114
0.095
0.848
1.172
1..046
0.922
0.803
0.692
0.896
0.832
0.077
0.061
0.048
0.037
0..022
0
0.S87
0.766
0
4.206
2.934
1.714
1.083
1.022
0.9S0
L.E. radius: 1.955 per cent c
NACA 66r018 Basic Thickness Form
APPENDIX I
.4
z
JI
(per cent c) (per cent c)
0
0
0.5
0.75
1.25
2.5
1.525
1.804
5.0
7.5
10
15
20
25
30
35
40
45
SO
55
60
65
70
75
2.240
3.045
4.269
Z/c
(V/V)J
0
0.580
0.635
0.755
0.952
0
0.~1
0.797
0.869
0.976
&1.1 V
1.547
1.314
1.218
1.054
0.828
6.052
7.369
8.376
1.148
1.185
1.208
9.153
9.738
10.1M
10.407
10.500
1.499
1.528
1.551
1.574
1.594
1.224
1.236
1.245
1.255
1.263
.0.251
0.224
0.202
0.183
1.611
1.629
1.648
1.269
1.276
1.284
10.434
10.186
9.692
8.793
7.610
6.251
4.796
3.324
1.069
1.116
0.283
1.508
1.228
1.335
1.155
0.165
0.148
0.132
0.114
0.093
1.084
0.073
1.015
0.944
0.873
0.805
0.058
0.046
0.034
0.020
0.734
0
90
1.924
0.717
0
0.539
100
_IV
0.635
0.542
0.472
0.381
0.324
95
85
1.0
.•
1.143
1.2-&6
1.318
1.405
1.459
5.233
1.176
1.031
0.891
0.763
0.648
80
379
L.E. radius: 2.550 per cent c
NACA 66r021 Basic Thickness Form
THEORY OF WING
380
~ECTIONS
2.D
I
/e, • .J2
1.6
1.2
(v)'
8
(/ e­
t.
/
~
~
I---~
t=-­
~
-
~
~
~
,~
/MeA
t->
I
svrFac»
i"'"-./2 LorNtN'" .,ftl~
r/
I
.4
o
~
I
Uppttr
-----­ --­
.z
x
'!J
(per cent c) (per cent c)
-,
67,1-015
r---~
., x/c'
(V/V)2
-----.. r--.....
~
~
/.0
.8
6
»rv
~G/V
I
0
0.5
0.75
1.25
2.5
5.0
7.5
10
15
~
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
I
0
1.167
1.394
1.764
2.395
0
0.650
0.970
1.059
0
0.806
0.985
1.029
1.140
1.068
2.042
1.560
1.370
1.152
0.906
3.245
3.900
4.433
5.283
5.940
1.209
1.100
1.113
1.122
1.134
1.142
0.667
0.548
0.470
0.370
0.312
6.404
6.854
7.155
7.359
7.475
1.318
1.330
1.341
1.851
1.360
1.148
1.153
1.158
1.162
1.166
0.276
0.248
0.221
0.201
0.180
7.497
7.421
7.231
6.905
6.402
1.368
1.375
1.381
1.388
1.390
1.170
1.173
1.175
1.178
1.179
0.160
0.142
0.124
0.111
0.108
5.621
4.540
3.327
2.021
0.788
1.321
1.176
1.018
0.864
0.712
1.149
1.084
0.930
0.844
0.094
0..071
0.060
0..045
0.025
0
0.570
0.755
0
1.239
1.259
1.285
1.304
1.009
L.E.. radius: 1.523 per cent c
NACA 67,1-015 Basic Thickness Form
II
APPENDIX I
2.0
381
r--....,.-...,...-.....--r--.......­ .......
-....-~
~l
............_ ....
I aurtlce
=.22 Upper
,r-oIo---+--I-L---­ ......K~
•8
~
II'(
.. . . .
I
.IJ--;---t----ir--f--+---+--+--J----+_-I
----I--­
'-........r--­ r----I-_....
_
_I---,-­
.2
.4
xlc. 6
~
.8
1.0
j
%
11
i
(per cent c) (per cent r) i
- - - - - -.._ -­
~_._.---
(I'll')"!
--..
-"--~"---
~t~CI/l!
1,'/1"
- _.._ ­
..
---
---
()
0
()
0.5
0.75
1.199
1.25
1.801
2.462
0.660
0.799
0.942
1.100
0
0.812
0.894
0.971
1.049
2.028
1.680
1.560
1.325
0.990
6.384
1.201
1.259
1.295
1.339
1.369
1.096
1.122
1.138
1.156
1.170
0.695
0.551
0.465
0.383
0.324
6.898
7.253
7.454
7.494
7.316
1.390
1.409
1.42"&
1.435
1.391
1.179
1.187
1.193
1.198
1.179
0.283
0.252
0.224
0.199
0.176
7.003
6.584
6.064
5.449
4.738
1.348
1.306
1.265
1.221
1.178
1.161
1.143
1.125
1.105
1.085
0.156
0.138
0.122
0.108
3.921
3.020
2.086
1.193
0.443
1.115
1.027
0.938
0.852
0.774
1.056
1.013
0.969
0.923
0.880
0
0.703
0.838
1.435
2.5
5.0
7.5
10
15
20
25
30
35
40
3.419
4.143
4.743
5.684
I
I
I
45
50
55
60
65
70
75
80
85
90
95
100
I
i
!
i
I
t
I
I
I
---_._-
I
I
0.093
0.079
0.065
0.052
O.MO
II
L.E. radius: 1.544 per cent. c
NACA 747A015 Basic Thickness Form
0.028
0.018
APPENDIX II
MEAN LINES
CONTENTS
NACA Designation
62
63 . . . . .
Page
383
384
64
65
66
385
386
387
388
389
67
210 .
220 .
390
230 .
240
250 .
a91
.
392
393
394:
4==0
4
== 0.1
4
== 0.2
a == 0.3
a -= 0.4
a::s= 0.5
4 == 0.6
a == 0.7
a == 0.8
. . . . .
. ....
a := 0.8 (modified for use with N ACA 6A-eeries sections)
a -= 0.9
a -= 1.0
.
382
395
396
397
398
399
400
401
4<rJ
.
403
404
405
APPENDIX II
)~
I.0
1
I
I~
383
~
i""---. r--­
r--- r--.....
-....
....
r-,
NACA 62
f1ItJt:1nJine
~
.2
D
Cl.
-
...-­
== 0.90
.4
(Xi
== 2.81
x/c
0
~
Ca e/
.
r - - to--...
1.0
.8
.6
=:I
-
0.113
I
%
I/e
(per cent c)
(per cent c)
0
0
0.726
1,.406
2.625
3.656
0.60000
0.56250
0.52500
0.45000
0.37500
0
0.682
1.031
1.314
1.503
0
0.171
0.258
0.328
0.376
10
4.500
15
5.625
1.651
1.802
20
25
6.000
5.977
5.906
0.30000
0.15000
0
- 0.00938
- 0.01875
1.113
0.413
0.451
0.383
0.318
0.279
5.625
5.156
4.500
3.656
2.625
-
0.03750
0.05625
0.07500
0.09375
0.11250
0.951
0.843
0.741
0.635
0.525
0.238
0.211
0.185
0.159
0.131
90
1.406
95
100
0.727
0
- 0.13125
- 0.14062
- 0.15000
0.377
0.261
0
1.25
2.5
5.0
7.5
30
40
50
60
70
80
I
dyc/d.c
Pa
t:J.vlV '"'" P./4
1.530
1.273
Data for NACA Mean Line 62
I
I
0.094
0.065
0
THEORY all It·ING SECTIONS
384
1.0
o
I
v- -........... <,
r--­
~
---­ ----r---_ ...
~
NACA 63
",.", line
~
~
o
.2
e'i
x
(pe~~ent c)
I
=
.4
0.80
ai
'/
X,C
= 1.60°
---
t--- r--­
.8
.6
Ca e/
4
~
1.0
= - 0.134
I
y.
PR
dYe/dx
(per cent c)
, .It'jV
=
PR/4
---~~~.-
0
0.489
0.958
1.833
2.625
0.40000
0.38333
0.36667
0.33333
0.30000
10
15
3.333
20
5.333
5.833
6.000
0.26667
0.20000
0.13333
0.06667
0
()
1.25
2.5
5.0
7.5
4.500
25
30
50
60
70
80
5.878
5.510
4.898
4.041
2.939
-
00
95
100
1".592
0.827
0
- 0.14694
- 0.15918
- 0.17143
40
I
I
I
I
1
0.02449
0.04898 .
0.07347
0.09796
0.12245
0
0.389
0.553
0.788
0.940
0
0.097
0.138
0.197
0.235
,
1.259
1.233
1.160
0.267
0.305
0.315
0.308
0.290
0.949
0.850
0.762
0.673
0.560
0.237
0.213
0.191
0.168
0.140
0.406
0.291
0
0.102
0.073
1.066
1.220
i
I
I
I
i
I
i
Data for NACA Mean Line 63
0
385
APPENDIX II
2.0
to
/
--
V
----r--­
r--­ --...........
............
(
(J
r-,
NACA 64
mean line
---- ---
-
~
II
.2
e'i = 0.76
X
( per
cent c
o
)
.4
ai'
= 0.74°
I(per cent
Yc
)
c
-------1------ ------- I
I
d!l</ti.r
I
I
0
1.25
0.369
2.5
0.726
5.0
1.406
7.52.039
I
10
15
20
25
30
2.625
3.656
4.500
5.156
5.625
40
50
60
70
80
6.000
5.833
5.333
4.500
3.333
90
1.833
0.958
I -
o
J
95
100
I
%C
I
Data for
­
-
.6
C.r/ t
------
=-
I!
pR
I--.
1.0
0.157
,j--~V-/-V-=-P-R/--4
i
--.-- ----- ---.------ ;------_._­
O.3()()OO
0
.
0
0.29062
0.28126
0.26250
0.24375
0.257
0.391
0.546
0.668
_
I
0.064
0.098
0.137
0.167
0.22500
0.18750
0.15000
0.11250
0.07500
O.i48
0.871
0.966
1.030
1.040
I
0
0.03333
0.06667
0.10000
0.13333
0.999
0.910
0.827
0.750
0.635
0.16667
0.18333
0.20000
0.466
0.334
0
X.~C.:\
l\IC1Ul Line 64
'I
I
I
I
1
0.187
0.218
0.242
0.258
0.260
0.250
0.. 228
0 ..207
0.188
0.159
0.117
0.084
0
THEORY OF WING SECTIONS
386
3D
2.0
1.0
~~
D
V
----- --
- ----..
~
'\
NACA 65
mean . .
o ----
.8
e'i == 0.75
'l/c
%
(per cent c) (per cent c)
0
1.25
2.5
5.0
7.5
10
16
20
25
30
40
50
60
70
80
90
95
100
r--- r---
L.--- ~
0
Qi
a:
00
d1lo/dz
~
1.0
c.c/4 == - 0.187
PB
AvIV
c:
Pa/4
0.296
0.S85
1.140
1.665
0.24000
0.23400
0.22800
0.21600
0.20400
0
0.205
0.294
0.413
0.502
0
0.051
0.074
0.103
0.126
2.160
3.060
3.840
4.500
5.040
0.19200
0.16800
0.14400
0.12000
0.09600
0.671
0.679
0.760
0.824
0.872
0.14.3
0.170
0.190
0.206
0.218
5.760
6.000
5.760
5.040
3.840
0.04800
0
- 0.04800
- 0.09600
- 0.14400
0.932
0.951
0.932
0.872
0.760
0.233
0.238
0.233
0.218
0.190
2.160
1.140
0
- 0.19200
- 0.21600
- 0.24000
0.571
0.413
0
0.143
0.103
0
Data for NACA Mean Line 65
APPENDIX 11
387
.0
I .IJ
tJ
~
V
...----
~
---- --
:-....
'"
r-,
\
NACA 66
I'I'N!'GJ . .
-
41!11
.--- ~ ~
.2
D
c'- == 0.76
----
~
.4
a, == -
'/
Z,t:
0.74°
t--- ~
1.0
.8
.6
Ca e/ .
== - 0.222
---~
%
1/1:
(per cent c)
(per cent c)
dy,:!dx
1
PR
4v/V = PR/4
~._-----_.-...-----
0
1.25
2.5
5.0
7.5
0
0.247
0.490
0.958
1.406
0.20000
0.19583
0.19167
0.18333
0.17500
0
0.135
0.244
0.334
0.408
0.034
0.061
0.084
0.102
20
25
30
1.833
2.625
3.333
3.958
4.500
0.16667
0.15000
0.13333
0.11667
0.10000
0.466
0.557
0.635
0.700
0.750
0.117
0.139
0.159
0.175
0.188
40
50
80
70
80
5.333
5.833
6.000
5.625
4.500
0.06667
0.03333
0
- 0.07500
- 0.15000
0.827
0.910
0.999
1.040
0.966
0.207
0.228
0.250
0.260
0.242
90
2.625
95
1.406
- 0.22500
- 0.26250
- 0.30000
0.748
0.546
0
0.187
0.137
0
10
15
100
0
,
Data for NACA Mean Line 66
0
I
THEORY OF WING SECTIONS
388
A
.W'
I
-­...
./
oV
~
~
~
~
~
r-,
~~
\
,
NACA 67
nwon line
~
-
"
~~
x
..-­
r--...... ~
./J
cu == 0.80
(per cent c)
~
(Xi
Yc
(per cent c)
== -
1.60 0
dyc/dx
c.c/.
== - 0.266
PB
J11J/V == PB/4
0
0.212
0.421
0.827
1.217
0.17-143
0.16837
0.16531
0.15918
0.15306
0
0.137
0.195
0.291
0.356
0
0.034
0.049
0.073
0.089
10
15
20
25
30
1.592
2.296
2.939
3.520
4.041
0.14694
0.13469
0.12245
0.11020
0.09796
0.406
0.483
0.560
0.616
0.673
0.102
0.121
0.140
O.IM
0.168
40
50
60
70
80
4.898
5.510
5.878
6.000
5.333
0.07347
0.04898
0.02449
0
- 0.13333
0.762
0.850
0.949
1.160
1.259
0.191
0.213
0.237
0.290
0.315
90
95
100
3.333
1.833
0
- 0.26667
- 0.33333
- 0.40000
1.066
0.788
0
0
1.25
2.5
5.0
7.5
I
I
Data for NACA Mean Line 67
I
I
0.267
0.197
0
APPENDIX II
389
20
to
"
~
\
\ ...
...........
------.
o
--­ NACA2/0
mean/ina
2.e
c
o
_
....•..
.2
_----,eli
x
)
(per rent c
__ _,--_.= 2.09°
= 0.30
.---.
X/c
.4
__.
a;
.. ..
__ ....
-
-
-_._.==__-
CwL~/.
...
_~
y~ e
) !
dlleld..r
!I (per cent
ll
1.0
.8
.6
O.OO()
.~_.--;-_
Pit
..
_.-.
-_ ..... _-_ .._.-._.
i ~v/l/" == PR/4
-- - - - - ~'-_._---.:----....---- -_. 1-"·- ---1--------····­
o
t .25
2.5
5.0
7.5
10
15
20
25
30
!
i
I
!
-to
50
00
70
80
90
95
100
I
f
0
0.596
0.928
1.114
1.087
.
II -
I
0.59613:
0.36236
0.1 S5O-l
0.00018
1.058
0.999
0.940
0.881
0.823
I
I
0.705!
0.588
I
Q4W
'I
0.353
0.235
0.118
0.059
0
I
IJ
_ 0.01175
o
0
1.381
1.565
1.221
0.781
0.345
0.391
0.305
0.195
0.626
0.489
O.-lO8
0.348
0.302
0.156
0.122
0.102
0.242
0.198
0.061
0.049
0.040
0.032
0.025
'
Ql00
0.128
0.098
0.065
!
0.044
0
Data for N ACA :\lean Line 210
0.087
0.075
i
O.(}16
0.011
o
THEORY OF WING SECTIONS
300
0
1.0
T' r-,
I'---.... " - ­
IJ
NACA 220
ItI«JI7line
..
""
D
.4
Cl.
== 0.30
x
1/e
(per cent c) (per cent c)
0
ai
,I
sIC
== 1.86 0
dYe/dx
.8
.6
1.0
c.c/t -= - 0.010
P.
4v/V == Pa/4
0
0.442
0.793
1.257
1.479
0.39270
0.31541
0.24618
0.13192
0.04994
0
0.822
1.003
0.988
0.900
0
0.206
0.251
0.247
0.225
10
15
20
25
30
1.535
1.463
1.377
1.291
1.205
0.00024:
0.801
0.615
0.465
0.378
0.326
0.200
0.154
0.116
0.095
0.082
40
SO
60
70
80
1.033
0.861
0.689
0.516
0.344
0.253
0.205
0.169
0.135
0.100
0.063
0.051
0.042
0.034
0.025
90
95
100
0.172
0.086
0
0.064
0.040
0
0.016
0.010
0
1.25
2.5
5.0
7.5
- 0.01722
Data for NACA Mean Line 220
APPENDIX II
391
(J
I.D
1/r-,
--
...............
r---­
~
NACA 23IJ
lin.
IIWJt1n
,.
~
II
.
Cit
.2
== 0.30
z
(per cent c)
0
(per
Yc
cent c)
.4
a, ==
sIc
1.650
0.30508
0.26594
0.22929
0.16347
0.10762
10
15
20
25
30
1.701
1.838
1.767
1.656
1.546
0.06174
- 0.00009
- 0.02203
40
50
60
1.325
1.104
0.883
0.662
0.. 442
2.5
5.0
7.5
70
80
90
95
100
0.221
0.110
0
1.0
c.c/t == - 0.014
Pa
dyc/dx
0
0.357
0.666
1.155
1.492
1.25
.11
0
0.853
0
0.132
0.168
0.198
0.213
0.859
0.678
0.519
0.419
0.361
0.215
0.170
0.130
0.105
0.090
0.274
0.217
0.. 177
0.144
0.105
0.069
0.054
0.044
0.036
0.026
0.069
0.042
0
0.017
0.011
0
0.528
0.673
0.791
- 0.02208
I
AviV == PRI4
Data for NACA l\fean Line 230
THEORY OF WING SECTIONS
392
Zeo
-: ~
V
o
.............
r---. ~
-­
P-............
~C;l240
m~
o
.2
ci,
x
= 0.30
Ye
(per cent c)
(per cent c)
.4
ai
=
'l"e
.1
%,C
1.45°
dyc/ch
.8
.6
Cr../.
=-
1.0
0.019
..lv/II' = PR/4
PR
-
------­
0
0.301
0.572
1.035
1.397
0.25233
0.22877
0.20625
0.16432
0.12653
0
0.377
0.491
0.625
0.718
0
0.094
0.123
0.156
0.180
10
15
20
25
30
1.671
1.991
2.079
2.018
1.890
0.09290
0.03810
- 0.00010
- 0.02169
0.750
0.677
0.566
0.477
0.410
0.188
0.169
0.142
0.119
0.103
40
50
70
80
1.620
1.350
1.080
0.810
0.540
0.304
0.234
0.186
0.150
0.110
0.076
0.059
0.047
0.038
0.028
90
95
100
0.270
0.135
0
0.071
0.047
0
0.018
0.012
0
0
1.25
2.5
5.0
7.5
60
I
- 0.02700
Data for NA.CA Mean Line 240
393
APPENDIX II
2.0
1.0
.-1_
--..
r-............ r--­
V
a
r-­
~
-~
NACA 250
",.an "".
.!.!...2
c
o
-
.4,
.8
.6
1.0
SIC
--- ._..
cr,
_~~-_.-
== 0.30
at
=
1.26°
- _._--------- ----------.
..
X
Yc
(per cent c)
(per cent c)
----~
....
_---­
1
dYc/dx
_ _ _...._
.. _ .. __ -4-..--_ _,_
0
0.258
0.498
0.922
1.277
0.21472
0.19920
0.18416
0.15562
0.12909
10
15
20
25
30
1.570
1.982
2.199
2.263
2.212
40
50
60
70
80
1.931
1.609
1.287
0.965
0.644
0
1.25
2.5
5.0
7.5
90
95
100
I
I
0.322
0.161
0
c.. ~/.
=-
0.026
-------------.- f--------.- -­ --­
l~H
i ~/::'1~ = P,,/4
I
- ­ ­ -­ . --------- ._-­
-_.-~-
- _.-......
0
0.281
0.369
0.477
0.552
0
0.070
().092
0.119
0.138
0.10458
0.06162
0.02674
- O.OOOOi
- 0.01880
0.592
0.624
0.610
0.547
0.470
0.148
0.156
0.153
0.137
0.117
- 0..03218
0.346
0.255
0.197
0.154
0.119
0.087
0.064
0.049
0..038
0.030
0.076
0.051
0
0.019
0.013
0
!
i
I
i
:
Data for KAC.-\. Mean Line 250
!
THEORY OF WING SECTIONS
394
2.0
-..............
~'"
.........
<,
/.0
~
~
r-, ""­
........
~
~
.-.
v
MACA 0-0
IIIt1t1I'I
'M
2
D
Cia sa
1.0
%
1/c
(per cent c)
(per cent c)
0
0
0.460
0.641
0.964
1.641
0.5
0.75
1.25
2.5
-
~
V
CZi
=­ 4.56 0
c.c/.
t--- t - -........
a:
-
0.083
p.
4v/V == PR/4
0.75867
0.69212
0.60715
0.48892
1.990
1.985
1.975
1.950
0.498
0.496
0.494
0.488
1.900
1.850
1.800
1.700
1.600
0.475
0.463
0.450
0.425
0.400
d1lc/d%
5.0
2.693
7.5
10
3.507
4.161
6.124
6.747
0.36561
0.29028
0.23515
0.15508
0.09693
6.114
6.277
6.273
6.130
5.871
0.05156
0.01482
- 0.01554
- 0.04086
- 0.06201
5.516
5.081
4.581
4.032
3.445
-
0.07958
0.09395
0.10539
0.11406
0.12003
0.900
0.800
0.700
0.600
0.250
0.225
0.200
0.175
0.150
75
80
85
2.836
2.217
90
95
1.013
0.467
-
0.12329
0.12371
0.12099
0.11455
0.10301
0.500
0.400
0.300
0.200
0.100
0.125
0.100
0.075
0.050
0.025
·0
- 0.07958
0
0
Ui
20
25
30
35
40
45
50
55
60
65
70
100
1.604
1.500
1.400
1.300
1.200
1.100
1.000
Data for NACA Mean Line a
=0
0.375
0.350
0.325
0.300
0.275
APPENDIX II
395
3JJ
-
20
r-; ........
~'" r-;
1.0
-..........
"'~ -.......
'"
r-;
o
NACA a-al
mtIQI7
IinII
.!.L.e
c
.2
eli -
z'
(per cent c)
0
0.5
0.75
1.25
2.5
5.0
7.5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
-r---- r--­
~
V
()
1.0
«i sa
1/e
4.43°
dYe/dx
(per cent c)
0
0.440
0.616
0.933
1.608
2.689
3.551
4.253
5.261
5.906
0.38235
0.31067
0.25057
0.16087
0.09981
6.282
6.449
0.05281
0.01498
- 0.01617
- 0.04210
- OJ)6373
5.664
5.218
4.706
4.142
c.c/.
,
a:
-
to
0.086
PB
AviV = Pll/4
1.818
0.455
1.717
1.616
0.429
0.404
1.515
0.379
0.354
I
i
I
!
0.73441
0.67479
0.59896
0.49366
6.443
6.296
6.029
r--..~
.8
.6
3..541
-
0.08168
0.09637
0.10806
0.11694
0.12307
2.916
2.281
1.652
1.045
0.482
-
0.12644
0.12693
0.12425
0.11781
0.10620
0
- 0.08258
h
11
II
; I
I}
1
.j
jJ
lJ
;
!
I
!
1.414
1.313
1.212
1.111
0.328
0.303
0.278
1.010
0.909
0.808
0.707
0.606
0..253
0.227
0.202
0.177
0.152
0.505
0.404
0.303
0.202
0.101 ..
0.126
0.101
0.076
0.050
0.025
0
0
Data for N ACA Mean Line a == 0.1
THEORY OF WING SECTIONS
396
~
~
1.0
~
.............
o
""'
..........
~" r-;
/MeA G-D.2
'e
C
mtH1f7/ine
.2
~
r - - t---
~
D
.2
.4
./.
r--- t - - -
t--
1.0
.8
.6
%,C
e'i -= 1.0
x
(per cent c)
0
0.5
0.75
5.0
7.5
10
15
20
I
I
I
I
45
60
65
70
I
I
80
85
90
95
100
I
=: -
0.094
PR
l!JJ/IT == PB/4
0
0.414
0.581
0.882
1
1.530
0.69492
0.64047
0.57135
0.47592
2.583
3.443
4.169
5.317
6.117
0.37661
0.31487
0.26803
0.19373
0.12405
1.667
0.417
6.572
6.777
6.789
6.646
6.373
0.06345
0.02030
- 0.01418
- 0.04246
- 0.06588
1.563
1.459
1.355
1.250
1.146
0.391
0.365
0.339
0.313
0.287
5.994
5.527
4.989
4.396
3.762
-
0.08522
0.10101
0.11359
0.12317
0.12985
1.042
0.938
0.834
0.729
0.625
0.260
0.234
0.208
0.182
0.156
3.102
2.431
1.764
1.119
0.518
-
0.13363
0.13440
0.13186
0.12541
0.11361
0.521
0.417
0.313
0.208
0.104
0.130
Q.I04
0.078
0.052
0.026
0
- 0.08941
0
0
I
75
c. e/ .
d1lc/dx
(per cent c)
25
30
35
40
50
55
== 4.17°
1/e
1.25
2.5
(Ii
Data for NACA Mean Line a
= 0.2
397
APPENDIX II
~~
<,
I.:
'"'~ .........
r-, <,
'"'
A
NACA 4-0.3
<,
mean line
.­
A
~
--
~
o
,l
.4
:-- r---
r---
.8
%/C
e'l
(per
== 1.0
3.84
(Ii -
0
o
0.5
0.75
1.25
2.5
5.0
7.5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
(per
' __ - .
o
0.389
to
C.ri. == - 0.106
~~ I :nt_:~Ldy./~~~
c)
­
_~~~~~=
1'_/4
0.832
1.448
0.65536
0.60524
0.54158
0.45399
2458
3.293
4.008
5.172
6.052
O.363-t-l
O.3078U
O.26(i21
O.2tr24U
0.15068
6.685
7.072
7.175
7.074
6.816
0.10278
0'-»833
- 0.00205
- OJlfH92
1.319
1.200
6.433
5.949
5.383
4.753
4.076
- 0.()8746
- O.l()567
U.989
- 0.1201-4
- 0.13119
- 0.13901
0.879
0.769
O.fl59
U.275
0.2-1.
0.220
0.192•
0.165
3.368
2.645
1.924
1.224
0.570
-
0.14365
0.14500
0.14279
0.549
0.440
0.330
0.220
0.110
0.137
0.110
0.082
0.055
0.028
o
- 0.09907
0
o
0.5-16
-
O.0371()
1.538
0.385
1.429
0.357
0.330
fl. 302
1.099
O.1363R
0.12430
j
Data for NAC..A.. Mean Line a
=
0.3
THEORY OF WING SECTIONS
398
:uJ1_ _-.....-~--,--r---r-.-....,.-.,.--,
- r---- r---.
.r---",",",-­
o
.4
ct; -.:
1.0
3.46°
(J[i -
x
fie
(per cent c) (per cent c)
0
0.5
0.75
1.25
2.5
0
0.366
0.514
0.784
1.367
6.0
7.&
10
15
20
"'2.330
z/c
c..clt
1.0
.8
.6
-= - 0.121
p.
4v/V - P/f/4
1.429
0.357
dYc/tk
0.61759
0.57105
0.61210
0.43106
3.131
3.824
4.968
5.862
0.34764
0.29671
0.25892
0.20185
0.Ui682
6.&46
7.039
7.343
7.439
7.275
0.11733
0.07988
0.04136
- 0.00721
- 0.05321
1.310
0.327
70
6.929
6.449
5.864
6.199
4.475
-
0.08380
0.10734
0.12567
0.13962
0.14963
1.190
1.071
0.952
0.833
0.714
0.298
0.268
0.238
0.208
0.179
76
80
8&
90
95
3.709
2.922
2.132
1.361
0.636
-
0.15589
0.15837
0.15683
0.lli062
0.13816
0.695
0.476
0.357
0.238
0.119
0.149
0.119
0.089
0.060
0.030
0
- 0.11138
0
0
25
30
35
40
4S
50
86
60
65
100
Data for NACA Mean Line a == 0.4
APPENDIX II
399
r-. <,
r-,
<,
o
r-,
NACA 1:1=0.5
",.., lin.
~
~
r---- t---
~
()
(per
:ent
C) , (per
:nt
C)
I
d~:dz_1
o
o
0.5
0.75
1.25
2.5
0.345
0.485
0.735
1.295
0.58195
0.53855
0.48360
0.40815
0.33070
0.28365
0.24890
0.19690
0.15650
5.0
7.5
10
15
2.206
2.970
3.630
4.740
20
5.620
25
30
6.310
6..840
7.215
35
40
7.430
45
7.490
50
7.350
6.965
6.405
5.725
4.955
55
60
65
70
70
4.130
80
85
90
95
3.265
100
t--- t-....
.8
.6
1.0
4IJor -
p.
P./4
1
I
1.333
0.333
0.12180
0.09000
0.05930
0.02800
- 0.00630
- 0.05305
- 0.09765
- 0.12550
- 0.14570
- 0.16015
2.395
1.535
0.720
-
I -
0.16960
0.17435
0.17415
0.16850
0.15565
o
II -
0.12660
1.200
I
I
1.067
0.933
0..800
I
0.300
0.267
0.233
0.200
0.667
0.533
0.400
0.267
0.133
0.167
0.133
0.100
0.067
0.033
o
o
Data for NACA Mean Line a
=-:
0.5
rHEORY OF WING SECTIONS
400
zo
1"-
1.0
"',
<,
o
r-,
NACA 41:0.6
If'IINIn Ii,.
o
~
l.---­ ~
r-- r---­
I"""""""'"
.4
eli
-= 1.0
G, == 2.58
%
1/c
(per cent c)
(per cent c)
0
0
0.5
0.75
0.325
0.455
0.695
,I.
%,C
0
2.5
1.220
5.0
7.5
10
15
20
2.080
2.805
3.435
4.495
5.345
0.31325
0.26950
0.23730
0.18935
0.15250
25
30
6.035
6.570
6.965
7.235
7.370
0.12125
0.09310
0.06660
0.04060
0.01405
35
40
45
== -
t-­
1.0
0.158
p.
dllt:/dz
0.54825
0.50760
0.45615
0.38555
1.25
c.c/4
r---
.8
.4
~v/V·
= PH/4
1.250
0.312
70
7.370
7.220
6.880
6.275
5.505
-
0.01435
0.04700
0.09470
0.14015
0.16595
1.094
0.938
0.273
0.234
75
80
85
90
95
4.630
3.695
2.720
1.755
0.825
-
0.18270
0.19225
0.19515
0.19095
0.17790
0.781
0.625
0.469
0.312
0.156
0.195
0.156
0.117
0.078
0.039
0
0
50
55
60
65
100
0
I
- 0.14550
nata for NACA Mean Line 4 == 0.6
APPENDIX II
401
(Ji
20
"­
1-0
-...
'"' -, -,
NACA a-47
",." /WM
~
r--
I
----- ---.
...--- ---­
Cia =-:
1.0
I
·--I~-d I~
I
yc
Ii
0.305
0.425
0.655
0.51620
0.47795
0.42960
2.5
1.160
0.36325
5.0
7.5
1.955
2.645
0.29545
0.25450
10
15
20
3.240
4.245
5.060
0.22445
0.17995
0.14595
25
30
35
40
45
5.715
0.11740
O.092()()
O.On840
OJ)4570
0.02315
50
7.155
7.090
0
- 0.02455
6.900
6.565
6.030
- 0.05185
- 0.08475
- 0.13650
X
(per :nt c)
0.5
0.75
1.25
(pe: rent c)
y,
6.240
6.635
6.925
7.095
55
60
65
70
75
5.205
- 0.18510
80
85
90
95
4.215
3.140
2.035
0.965
-
100
I
0
/.IJ
c...j. -= - 0.179
2.09 0
ai -
~
0.20855
0.21955
0.21960
0.20725
, - 0.16985
Data for XACA
~ienn
t
I
} 1.176
0.294
i
!
t
I
1
II
I
J
0.980
0.784
0.588
0.392
0.196
o
Line a -- 0.7
I
I
I
0.245
0.196
0.147
0.098
0.049
o
THEORY OF WING 8ECTION8
402
..---­
....,
-
<,
"
(J
NACA -e/J.8
",..,
n,.
.2
a
,-------I.----
.4
e'i
-= 1.0
1.64
eli -
fie
%
2.5
0
0.287
0.404
0.616
1.077
5.0
7.5
10
15
20
1.841
2.483
3.043
3.986
4.748
25
30
5.367
5.863
6.248
6.528
6.709
1.25
35
40
45
50
55
60
65
70
75
80
85
90
95
100
SIC
0
.8
.6
""""""--­
/.0
c../. - - 0.202
p.
dl/c/a
(per cent e) (per cent e)
0
0.5
0.75
--- r---....
to-"""
AviV
== PIl/4
0.48535
0.44925
0.40359
0.34104
0.27718
0.23868
0.21(}lj()
0.16892
0.13734
0.11101
0.08775
0.06634
0.04601
0.02613
6.790
6.770
6.644
6.405
6.037
-
0.00620
0.01433
0.03611
0.06010
0.08790
5.514
4.771
3.683
2.435
1.163
-
0.12311
0.18412
0.23921
0.25583
0.24904
0
- 0.20385
1.111
0.278
0.833
0.556
0.278
0.208
0.139
0.069
0
0
Data for NACA MeaD Line a .. 0.8
APPENDIX 11
403
2.0
1.0
""
,,~
o
\
WACA • - O.B (modlfl_d)
Mean II".
2
o t-.--­
o
~
- r--- r----.
~
.2
.4
.6
~
.8
1 0
x/e
eli
-= 1.0
Gr, -=
'lie
%
1.25
2.5
5.0
7.5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
I
c.ct.
0
0.281
0.396
0.603
1.055
0.47539
0.44004
0.39531
0.33404
1.803
2.432
2.981
3.903
4.651
0.27149
0.23378
0.20618
. 0.16546
0.13452
5.257
5.742
6.120
6.394
6.571
0.10873
0.08595
0.06498
0.04507
0.02559
6.651
6.631
6.508
6.274
5.913
-
0.00607
0.01404­
0.03537
0.05887
0.08610
5.401
4.673
3.607
2.452
1.226
-
0.12058
0.18034
0.23430
0.24521
0.24521
0
- 0.24521
:II:
0.219
------
PIl
AfJ/V .. P./4
1.092
0.273
11.~
0.274
}1.100
0.276
} 1.104
0.276
1.108
1.108
0.277
O.Z17
1.112
1.112
0.840
0.588
0.368
0.278
0.278
0
0
dy./a
(per cent e) (per cent c)
0
0.5
0.75
1.400
Data for NACA Mean Line a -= 0.8 (modified)
0..210
0.147
0.092
THBoRY OF WING SECTIONS
404
D
I. D
--
.,
1\
\
- r--­
~
NACAa.O,9
meon/i/'ltl
L"....--­ ~
~
IJ
t(J
ci, == 1.0
x
(per cent c)
0
0.5
0.75
1.25
2.5
Gi
== 0.90 0
Yc
0
0.269
0.379
0.577
1.008
5.0
7.5
10
15
20
1.720
2.316
2.835
3.707
4.410
0.25786
0.22153
0.19liOO
0.15595
0.12644
2S
4.980
5.435
5.787
6.M5
6.:212
0.10196
0.08047
0.06084:
0.04234
0.02447
40
45
50
55
60
65
70
75
80
85
90
95
100
6.290
6.279
6.178
5.981
5.681
-
0.00678
0.01111
0.02965
0.04938
0.07103
2.984:
1.503
-
0.09583
0.1260:)
0.16727
0.25204:
0.31463
0
- 0.26086
5.265
4.714
3.987
0.225
z= -
Pa
dyc/dz
(per cent c)
0.45482
0.42OM
0.37740
0.31821
30
35
c.c/.
~/V
=
PR/4
1
1.05:1
0.263
0.526
0.132
0
0
Data for NACA Mean Line (I
:II:
0.9
APPENDIX II
405
:J.O
.0
.0
D
NACA ••/.0
mean /;n.
~
- r--- r--­
..",..-- ~ -
D
.4
e'i
1.25
2.5
0.930
5.0
7.5
10
1.580
i
15
20
25
30
35
40
45
65
70
75
80
85
90
95
100
00
c..,J.
I
J
2.120
2.585
3.365
3.980
4.475
4.860
5.150
5.355
5.475
0.08745
0.00745
0.04925
0.(13225
0.01595
5.515
5.355
5.150
4.860
4.475
3.980
3.365
2.585
11&:
-
0.250
PR
j
0.23430
0.19995
0.17485
0.13805
0.11030
5.475
,
dyc/dx
0.42120
0.38875
0.34770
0.29155
I
Ii
I
-
0
0.01595
I,II
1.580
!
- 0.23430
0
ti
i
II
I
I
II I
If urn
i
1(
t
f;
!
I
II
II
- 0.08745
- 0.11030
- 0.13805
= PR/4
Ii
I'
II
I~V/V
II
:I
OJ>3225 I:
0.04925
0.06745
UJ
.8
.6
0
!
0.250
0.350
0.535
50
55
60
a, =
Yc
(per cent c)
X
(per cent c)
0
0.5
0.73
= 1.0
sIc
I
I
I
i
I
I
j
I - 0.17485
I
I
Data ior N ACA :\IcuD Line Il = 1.0
j
I
0.250
APPENDIX III
AIRFOIL ORDINATES
CoNTENTS
NACA Designation
Page
408
0010-34 a == 0.8 (modified) e'i II: 0.2 .
0012-64 a == 0.8 (modified) e'i == 0.2 .
1408
1410
1412
2408
2410 . . . . .
2412
2415 . . . . .
408
408
409
409
409
410
410
410
2418
2421
2424
411
.....
411
411
412
412
4412
4415
.....
4418
4421
4424
23012
23015
23018 .
23021
23024
63-206 .
63-209 .
63-210 .
63 1-212
63.-412.
63r215 . .
412
413
413
413
414
414
414
415
415
416
416
417
.
417
418
418
63r415. '. . . . . .
63t-615. . . . . .
63,-218. .
63r418.
419
419
63,-618. .
420
420
63r221 . .
63.-421.
421
421
63A21 0
64-108 . . . .
64-110 .
64-206 . .
422
423
422
423
424
64-208 . . . .
64.-209. . . . .
424
406
APPENDIX III
NACA Designation
64-210
.
641-112 .
. ....
641-212 .
641-412. .
642-215. .
64r415. . . . . . .
64,-218 . .
64r418.
64r618 . .
64 221
407
.
425
426
426
427
427
428
428
429
.
64.-421 ..
Page
425
. ....
429
430
430
431
431
432
64A210
64A410
64IA212 .
642.~215 .
65-206 . .
65-209 . .
65-210 .
65-410 .
651-212 . .
65.-212 a == 0.6 . .
65.-412
.
65r215. . . . . . .
65r415. . . . .
65r415a == 0.5 . .
65r 218 . . . . . . .
65,..418. . . . .
65r418 a - 0.5.
­
65r618. . . . .
65,-618 a == 0.5 .
65...221
.
65r421
.
65.-421 a == 0.5 .
66-206 . .
66-209 . .
66-210 . .
66.-212.
432
433
433
434
434
435
435
436
436
437
437
438
438
439
439
440
440
441
441
442
442
443
66r215.
443
66r415.
444
66r218.
66r418.
66...221.
67,1-215 .
747A315 .
747A415 . . • • • • .
. ....
444
445
445
446
446
447
(Stations and ordinates given in
NACA 0012-64
a = 0.8 (modified)
e'i = 0.2
(Stations and ordinates given in
per cent of airfoil chord)
per cent of airfoil chord)
NACA 0010-34
a = 0.8 (modified)
c" == 0.2
_ .._.
Upper surface
Lower surface
Upper surface
Ordinate Station
Ordinate
0
1.928
2.659
3.623
4.295
0
1.393
2.664
5.177
7.678
-
0
1.686
2.237
2.901
3.323
4.832
6.221
6.974
7.279
10.175
15.161
20.142
30.100
40.054
-
3.640
4.083
4.361
4.678
4.721
49.993
60.038
70.077
80.120
90.091
7.157
6.622
5.662
4.253
2.355
50.007
59.962
69.923
70.880
89.909
-
4.497
4.018
3.296
2.383
1.375
95.050
100.000
1.271
0.120
04.950
100.000
Ordinate Station
Ordinate
Station
0
0.813
1.324
2.593
5.113
-
0
0.632
0.820
1.186
1.714
0
1.107
2.336
4.823
4.887
0
0.790
1.062
1.608
2.436
7.378
9.875
14.876
19.886
20.017
3.094
3.637
4.523
5.172
5.080
7.622
10.125
15.124
20.114
30.083
-
2.122
2.445
2.961
3.312
3.684
9.825
14.839
5.645
19.858
29.000
39.946
39.955
49.994
(lO.O31
70.ntH
80.100
6.279
6.186
5.735
3.700
40.045
50.006
59.960
OH.H3H
79.900
-
3.721
3.526
3.131
2.5"Ul
1.83U
00.076
95.042
100.000
2.044
1.100
0.100
89.92-1
94.958
100.000
- 1.004
- 0.610
- 0.100
Station
0
0.687
1.176
2.407
tj.015
L.E. radius: 0.272
Slope of rudius through L.E.: 0.095
7.322
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Lower surface
- 0.781
- 0.120
L.E. radius: 1.582
Slope of radius through L.E.: 0.095
~
00
NACA 1408
Station
Ordinate
Lower surface
Station Ordinate
0
1.189
2.418
4.896
7.386
0
1.324
1.862
2.602
3.138
0
1.311
2.582
5.104
7.614
9.883
14.889
19.904
24.926
29.950
3.558
4.171
4.574
4.819
4.939
10.117
15.111
20.096
25.074
30.050
-
2.682
2.953
3.074
3.101
3.063
40.000
50.020
60.034
70.041
SO.03D
4.869
4.502
3.931
3.193
2.305
40.000
49.980
59.966
69.959
79.961
-
2.869
2.556
2.153
1.693
1.193
90.027
95.016
100.000
1.271
0.698
0.084
89.973
94.984
100.000
0
- 1.200
- 1.620
- 2.134
- 2.458
- 0.659
- 0.378
- OJJ84
L.E. radius: 0.70
Slope of radi us through L. E.: 0.05
~
~
~
~
~
~
~
~
~
C
~
NACA 1410
NACA 1412
NACA 2408
(Stations and ordinates given in
per cent of airfoil chord)
(Stations and ordinates given in
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Station
()
Station
0
1./i15
0
t.tuM
2.378
4.845
0
1.ll30
2.398
4.870
7.358
2.297
3.194
3.837
9.854
29.937
4.338
5.062
5.531
5.809
5.940
40.000
50.025
00.042
70.051
80.049
5.836
5.385
4.692
3.804
2.741
90.034
95.021
100.000
1.513
0.832
0.105
24.907
Upper surface
Lower surface
Upper surface
Lower surface
..
Ordinate Station Ordinate
1.174
14.861
19.880
_­
Lower surface
--
per cent of airfoil chord)
0
1.320
2.002
7.642
-
2.055
2.726
3.157
10.146
15.139
20.120
25.093
30.063
-
3.462
3.844
4.031
4.091
4.064
5.130
I
I
I 40.000
49.975
I
- 3.836
- 3.4:i9
59.058
69.949
- 2.914
- 2.30,1
79.951
- 1.629
89.966
0.901
94.979 , - 0.512
I
100.000
-
!-
0.105
L.E. radius: 1.10
Slope of radius through L.E.: 0.05
Ordinate Station
7.330
0
0
l.or)t1
2.733
3.786
4.537
1.:\tI2
2.U22
5.118
5.951
6.486
6.799
6.040
9.82'.
14.833
19.857
24.889
29.925
40.000
50.029
60.051
70.061
80.0SS ,
6.803
5.155
7.670
lO.l7G
15.167
20.143
ss.iu
30.075
40.000
49.071
0.267
5.453
4.413
3.178
60.930
79.942
1.753
90.040 1
0.966
95.025
0.126
100.000
89.960
94.975
100.000
I
5fi.940
Ordinate
0
Station
Ordinate
Station
0
1.372
2.663
5.206
7.273
0
1.3140
1.977
2.829
3.471
0
- 1.8..10
1,12R
- 2.491
- 3.318
4.794
- 3.857
2.337
7.727
Ordinate
0
- 1.134
- 1.493
- 1.891
- 2.111
-
4.242
4.733
4.986
5.081
5.064
9.768
14.778
19.809
24.852
29.900
3.987
4.776
5.320
5.677
5.875
10.232
15.222
20.191
25.148
30.100
-
2.237
2.338
2.320
2.239
2.125
-
4.803
4.321
3.675
2.913
2.066
40.000
50.039
70.081
80.078
5.869
5.473
4.820
3.942
2.858
40.000
49.961
59.932
69.919
79.922
-
1.869
1.585
1.264
0.942
0.636
- 1.141
- 0.646
- 0.126
90.054
95.033
100.000
1.575
0.855
0.084
89.946
94.967
100.000
L.'E. radius: 1.58
Slope of radius through L.E.: 0.05
no. 008
~
~
t:::J
""-c
~
""-c
'"'-..
""-c
- 0.353
- 0.217
- 0.084
L.E. radius: 0.70
Slope of radius through L.E.: 0.1
....
o
co
NACA 2410
(Stations and ordinates given in
per cent of airfoil chord)
Upper eturface
Station
Lower surface
Ordinate Station Ordinate
1.098
2.297
4.742
7.217
0
1.694
2.411
3.420
4.169
0
1.402
2.703
6.258
' 7.783
9.710
14.722
19.761
24.814
29.875
4.766
5.665
6.276
6.668
6.875
10.290
15.278
20.239
26.186
30.126
40.000
lj().049
60.085
70.102
80.097
90.067
96.041
100.000
6.837
6.366
5.lS8O
4.651
3.296
0
1.816
0.990
0.105
-
0
1.448
1.927
2.482
2.809
- 3.016
- 3.227
- 3.276
- 3.230
- 3.123
- 2.837
- 2.468
- 2.024
-l.Ml
- 1.074
- 0.594
- 0.352
40.000
49.951
59.915
69.898
79.903
89.933
94.959
100.000 - 0.105
L.E. radius: 1.10
Slope of radius through L.E.: 0.1
NACA 2412
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Lower surface
Station Ordinate Station Ordinate
0
1.2a
2.5
6.0
7.5
.. ....
2.15
2.99
4.13
4.96
10
16
20
25
30
6.63
6.61
7.26
7.67
7.88
40
7.80
7.24
6.36
6.18
3.75
so
60
70
80
90
95
100
100
2.08
1.14
(0.13)
.. ....
2.5
5.0
7.6
10
15
20
25
30
40
50
60
70
80
90
95
100
100
Upper surface
0
1.66
2.27
3.01
3.46
0
......
1.25
2.71
3.71
5.07
6.06
-
3.76
4.10
4.23
4.22
4.12
10
15
20
25
30
6.83
7.97
8.70
9.17
9.38
-
3.80
3.34
2.76
2.14
l.l5O
40
- 0.82
- 0.48
(- 0.13)
0
L,E. radius: 1.58
Slope of rArdius through L.E.: 0.10 •
•.o....
Lower surface
Station Ordinate Station Ordinate
-
0
1.25
NACA 2415
(Stations and ordinates given in
per cent of airfoil chord)
2.5
5.0
7.5
50
60
70
80
90
95
100
100
0
1.25
2.5
5.0
7.6
-
0
2.06
2.86
3.84
4.47
10
15
20
25
30
-
4.90
5.42
5.66
5.70
5.62
9.25
8.57
7.50
6.10
4.41
40
50
60
70
80
- 5.26
- 4.67
-3.90
- 3.05
- 2.15
2.45
1.34
(0.16)
90
- 1.17
- 0.68
(- 0.16)
0
......
95
100
100
L.E. radius: 2.48
Slope of radius through L.E.: 0.10
~
~
~
e.c:
~
s
~
~
~
~
~
r,j
NACA 2418
NACA 2421
NACA 2424
(Stations and ordinates given in
per cent of airfoil chord)
(Stations and ordinates given in
per cent of airfoil chord)
(Stations and ordinates given in
Upper surface
Station Ordinate
0
1.20
2.5
6.0
7.5
10
I
16
20
2S
30
40
50
60
70
80
90
95
100
100
••
I
••••
3.28
4.45
6.03
7.17
8.05
9.34
10.15
10.65
10.88
10.71
9.89
8.66
7.02
5.08
2.81
1.55
(0.19)
..... ...
Lower surface
Station
Ordinate
0
0
1.25
2.5
5.0
7.5
Station Ordinate Station Ordinate
0
1.25
2.5
5.0
7.5
-
2.45
3.44
4.68
5.48
-
6.03
6.74
7.09
7.18
7.12
-
6.71
5.99
5.04
3.97
2.80
40
100
- 1.63
- 0.87
(- 0.19)
90
95
100
100
0
100
10
15
20
25
30
40
50
60
70
80
90
95
L.E. radius: 3.66
Slope of rndius through L.E.: O.to
Lower surface
Upper surface
10
15
20
25
30
50
60
70
80
••••
t
••
1.25
2.5
5.0
8.29
7.5
9.28
10.70
11.59
12.15
12.38
10
15
20
25
30
12.16
11.22
9.79
7.94
40
5.74
80
3.18
1.76
(0.22)
90
95
100
100
.......
50
60
70
Upper surface
Lower surface
Station Ordinate Station Ordinate
0
3.646
4.965
6.614
7.692
-
0
2.82
4.02
5.51
6.48
0
0.885
2.012
4.380
6.820
0
3.892
5.449
7.502
9.052
-
7.18
8.05
8.52
8.67
8.62
9.300
14.333
19.427
24.555
29.700
10.215
11.888
12.959
13.593
13.874
10.700 - 8.465
15.667 - 9.450
20.573 - 9.959
25.445 - 10.155
30.300 - 10.124
-
8.16
7.31
6.17
4.87
3.44
40.000
50.118
60.203
70.244
80.233
13.606
12.532
10.903
8.824
6.352
40.000
49.882
59.797
69.756
79.767
- 1.88
90.161
95.098
100.000
3.502
1.930
0
3.87
5.21
7.00
per cent of airfoil chord)
- 1.06
(- 0.22)
0
L.E. radius: 4.85
Slope of radius through I".E.: 0.10
.....
0
1.615
2.988
5.620
8.180
-
-
9.606
8.644
7.347
5.824
4.130
~
~S2
><
~
~
.....
89.839 - 2.280
94.902 - 1.292
0
100.000
L.E. radius: 6.33
Slope of radius through L.E.: 0.10
~
.....
.....
....
~
NACA 4412
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Station
Ordinate
0
0
1.25
2.5
2.44
3.39
4.73
5.76
5.0
7.5
10
15
20
25
30
40
50
60
70
80
90
95
100
100
Lower surface
Station Ordinate
0
1.25
2.5
5.0
7.5
NACA 4415
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Station
0
1.25
2.5
5.0
-
0
1.43
1.95
2.49
2.74
9.41
9.76
10
15
20
25
30
-
2.86
2.88
2.74
2.50
2.26
9.80
9.19
8.14
6.69
4.89
40
50
60
70
80
-
1.80
1.40
1.00
0.65
0.39
60
70
80
2.71
1.47
(0.13)
90
95
100
- 0.22
- 0.16
(- 0.13)
0
90
95
100
100
6.59
7.89
8.80
..... .
100
IJ.E. radius: 1.58
Slope of radius through L.E.: 0.20
Ordinate Station Ordinate
t
••••••
- 3.27
6.91
7.5
- .3.71
Station
0
1.25
2.5
5.0
7.5
.......
3.76
5.00
6.75
8.06
3.98
4.18
4.15
3.98
3.75
10
15
20
25
30
9.11
10.66
11.72
12.40
12.76
11.25
10.53
9.30
7.63
5.55
40
-
3.25
2.72
2.14
1.55
1.03
40
50
60
70
80
12.70
11.85
10.44
8.55
6.22
3.08
90
95
100
100
- 0.57
- 0.36
(- 0.16)
0
90
95
100
100
3.46
20
9.27
1.67
(0.16)
II
•••••
50
60
70
80
L.E. radius: 2.48
Slope of radius through L.E.: 0.20
~
Lower surface
Ordinate Station Ordinate
-
10.25
10.92
11.25
50
Upper surface
10
15
20
25
30
7.84
40
0
- 1.79
- 2.48
5.74
10
15
30
0
1.25
2.5
5.0
3.07
4.17
7.5
25
Lower surface
NACA 4418
(Stations and ordinates given in
per cent of airfoil chord)
1.89
(0.19)
••••••• I
0
0
2.11
- 2.99
- 4.06
- 4.67
~
10
- 5.06
15
-
~
1.25
2.5
5.0
7.5
20
25
30
40
50
60
70
~
5.49
5.56
5.49
5.26
- 4.70
- 4.02
-3.24
80
- 2.45
- 1.67
90
95
100
100
- 0.93
- 0.55
(- 0.19)
0
L.E. radius: 3.56
Slope of radius through L.E.: 0.20
~
~
~
~
~
~
~
~
C
~
NACA 4421
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Lower surface
Ordinate Station Ordinate
Station
..... . -
0
1.25
2.5
5.0
4.45
5.84
7.82
9.24
7.5
0
1.25
2.5
5.0
7.5
0
- 2.42
- 3.48
- 4.78
- 5.62
15
20
10.35
12.04
13.17
10
15
20
- fi.t5
- (}.75
25
13.88
25
- 6.92
30
14.27
30
- 6.76
40
50
60
14.16
13.18
11.60
9.50
6.91
40
-
10
70
80
00
95
100
100
I
3.85
2.11
(0.22)
..... ..
50
60
70
80
90
95
100
100
- 6.98
6.16
5.34
4.40
3.35
2.31
- 1.27
- 0.74
(- 0.22)
0
L. I~. radius: 4.85
Slope of radius through 1.1. E.: 0.20
NACA 4424
NACA 23012
(Stations and ordinates given in
(Stations and ordinates given in
per cent of airfoil chord)
per cent of nirfoil chord)
I .ower surface
Upper surface
Station
1.536
0
3.964
5.624
3.775
6.153
7.942
9.651
8.611
13.674
18.858
24.111
29.401
11.012
13,<)45
14.416
15.287
0
0.530
I5.nOG
50.235
14.47·'
12.674
10.312
00.405
70.487
80.464
7.447
i
4.099
2.240
•
It
••
••
L.E. radius: 6.33
0
0
1.970
- 3.472
3.4M
6.225
8.847
.io.ooo
I
··JU.7G5
59.505
00.513
iO.536
80.680
94.804
lOOJ)(X)
I
I
Station
0
1.25
- 4.656
2.5
- 6.0f>6
- 6.931
5.0
7.5
11.389 - 7.512
16.326 - 8.I09
21.142 - 8.416
25.&~9
- 8.411
30.500 - 8.238
15.738
40.000
90.320
95.196
100.000
Station Ordinate
Ordinate
Upper surface
10
15
20
25
30
- 7.006
- 0.698
40
- 5.562
Ordinate Station Ordinate
......
2.67
3.61
4.91
5.80
6.43
7.10
7.53
7.60
7.55
60
7.14
6.41
5.47
- 4.312
70
4.36
- 3.003
80
3.08
- 1.655
90
95
-
0.964
0
Slope or radius through L.E.: 0.20
50
100
100
Lower surface
1.68
0.92
(0.13)
......
0
1.25
2.5
5.0
7.5
10
15
-
0
1.23
1.71
2.26
2.61
20
- 2.92
- 3.50
- 3.97
25
- 4.28
30
- 4.46
40
50
60
70
80
-
4.48
4.17
3.67
3.00
~
~
~
~
~
.......
><
.......
.......
.......
- 2.16
90
95
- 1.23
- 0.70
100
100
(- 0.13)
0
L.E. radius: 1.58
Slope of radius through L. E.: 0.305
~
~
~
NACA 23015
(Stations and ordinates given in
per cent 01 airfoil chord)
Upper surface
Lower surface
Station Ordinate Station Ordinate
0
1.20
2.5
5.0
7.5
.... .,
3.84
4.44
5.89
6.90
0
1.25
2.5
6.0
7.5
-
0
1.54
2.25
3.04
3.61
NACA 23018
(Stations and ordinates given in
per cent of airfoil chord)
Lower surface
Upper surface
Station
0
1.25
2.5
5.0
7.5
Ordinate
•••••
t
10
7.64
10
- 4.09
10
8.83
15
8.52
20
26
30
8.92
15
20
26
15
20
25
30
-
9.86
10.36
10.56
10.OS
40
- 5.92
50
60
70
80
-
40
50
60
70
80
90
95
100
100
9.08
9.06
8.59
7.74
6.61
5.25
3.73
2.04:
1.12
(0.16)
.. ....
90
95
100
100
4.84
6.41
6.78
5.96
5.SO
4.81
3.91
2.83
- 1.89
- 0.90
(- 0.16)
0
L.E. radius: 2.48
Slope of radius through L.E.: 0.305
30
40
50
60
70
80
10.04
9.05
7.76
6.18
90
2.39
1.32
(0.19)
ss
100
100
•
4.09
6.29
6.92
8.01
4.40
.... ...
Station
Ordinate
0
1.25
2.5
5.0
- 3.80
7.5
- 4.60
0
- 1.83
- 2.71
NACA 23021
(Stations and ordinates given in
per cent of airfoil chord)
0
1.25
2.5
5.0
7.5
Ordinate
•••
It
••
4.87
6.14
7.93
9.13
10
15
20
25
30
- 5.22
- 6.18
- 6.86
-7.27
-7.47
10
15
20
25
30
10.03
11.19
11.80
12.05
12.06
40
50
60
70
80
-7.37
- 6.81
- 5.94
- 4.82
- 3.48
40
50
60
70
SO.
11.49
10.40
8.90
7.09
5.05
- 1.94
90
-
1.09
95
(- 0.19)
100
100
2.76
1.53
(0.22)
00
95
100
100
0
L.E. radius: 3.56
Slope of radius through L.E.: 0.305
~
Lower surface
Upper surface
Station
~
.....
.......
Station Ordinate
0
1.25
2.5
5.0
7.5
10
15
20
25
30
40
50
60
70
80
90
95
100
100
-
0
2.08
3.14
4.52
5.50
-
6.32
7.51
8.30
8.76
8.95
-
8.83
8.14
7.07
5.72
4.13
- 2.30
- 1.30
(- 0.22)
0
L.E. radius: 4.85
Slope of radius through L.E.: 0.305
~
~
~
~
~
~
~
~
~
~
~
~r;,j
APPENDIX III
NACA 63-206
(Stations and ordinates given in
percent of airfoil chord)
NACA23024
(StatiODS and ordinates given in
per cent of airfoil chord)
Upper surface
I
Lower surface
Station Ordinate Station Ordinate
o
415
Upper Surface
Station Ordinate
Lower Surface
Station
Ordinate
1
0
0
0
.458
.703
1.197
2.438
.551
.677
.876
1.241
.542
.797
1.303
2.562
4.932
0
-.451
0.?:l7
1.331
3.853
6.601
0
4.017
5.764
8.172
9.844
9.423
15.001
20.253
25.262
30.265
11.049
12.528
13.237
13.535
13.646
10.577 - 7.647
14.999 - 8.852
19.747 - 9.703
24.738 - 10.223
29.735 - 10.454
7.429
9.930
14.934
19.941
1.776
2.189
2.526
3.058
3.451
5.068
7.571
10.070
15.066
20.059
-
1.144
1.341
1.492
1.712
1.859
40.256
50.235
60.202
70.162
80.116
12.928
11.690
10.008
7.988
5.687
39.744 - 10.278
49.766 - 9.482
59.798 - 8.242
69.838 - 6.664
79.884 - 4.803
24.950
29.960
34.970
39.981
44.991
3.736
3.926
4.030
4.042
3.972
25.050
30.040
35.030
40.019
45.009
-
1.946
1.982
1.970
1.900
1.7R2
90.064
95.036
3.115
1.724
89.936
94.964
50.000
55.008
60.015
65.020
70.023
3.826
3.612
3.338
3.012
2.642
50.000
54.992
59.985
64.980
69.977
- 1.620
75.023
80.022
85.019
90.013
95.006
2.237
1.804
1.356
.900
.454
74.927
79.978
84.981
89.987
94.994
100
......
0
2.223
3.669
6.147
8.399
100
-
0
3.303
4.432
5.862
6.860
- 2.673
- 1.504
0
L.E. radius: 6.33
Slope of radius through LE.: 0.305
100.000
0
100.000
- .537
- .662
- .869
-- 1.4221
1.196
- .952
- .698
- .447
- .212
- .010
.134
.178
0
L.E. radius: 0.297
Slope of radius through L.E.: 0.0842
THEORY OF WING SECTIONS
416
NACA 63-209
(Stat.ions and ordinates given in
per cent of airfoil chord)
Upper surface
Lower surface
Station Ordinate Station
0
0.436
0.680
1.170
2.408
0
0.796
0.973
1.255
1.765
0
0.563
0.820
1.330
2.592
4.897
7.394
9.894
14.901
19.912
2.510
3.077
3.539
4.263
4.792
24.925
29.940
34.956
39.971
44.986
Ordinate
NACA 63-210
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Station
Lower surface
Ordinate Station Ordinate
-
-
0
0.696
0:833
1.041
1.393
0
0.430
0.669
1.162
2.398
0
0.876
1.107
1.379
1.939
0
0.570
0.831
1.338
2.602
-
0
0.776
0.967
1.165
1.567
5.103
7.606
10.106
15.099
20.088
-
1.878
2.229
2.505
2.917
3.200
4.886
7.382
9.882
14.890
19.902
2.753
3.372
3.877
4.665
5.240
5.114
7.618
10.118
15.110
20.098
-
2.121
2.524
2.843
3.319
3.648
5.169
5.414
5.530
5.518
5.391
25.075
30.060
35.044
40.029
45.014
-
3.379
3.470
3.470
3.376
3.201
24.917
29.933
34.951
39.968
44.985
5.647
5.910
6.030
6.009
5.861
25.083
30.067
35.049
40.032
45.015
-
3.857
3.966
3.970
3.867
50.000
55.012
60.022
65.029
70.033
5.159
4.834
4.429
3.958
3.430
50.000
54.988
59.978
64.971
69.967
-
2.953
2.644
2.287
1.898
1.486
50.000
55.013
60.024
65.032
70.036
5.599
5.235
4.786
4.264
3.684
50.000
54.987
59.976
64.968
69.964
-
3.393
3.045
2.644
2.204
1.740
75.034
80.032
85.027
90.019
95.009
2.861
2.267
1.663
1.067
0.512
74.966
79.968
84.973
89.981
94.991
-
1.071
0.675
0.317
0.033
0.120
75.038
80.036
85.030
90.021
95.010
3.061
2.414
1.761
1.121
0.530
74.962
79.964
84.970
89.979
94.990
-
1.271
0.822
0.415
0.087
0.102
100.000
I
0
100.000
0
L.E. radius: 0.631
Slope of radius through L.E.: 0.0842
100.000
0
100.000
~.671
0
L.E. radius: 0.770
Slope of radius through L.E.: 0.0842
APPENDIX III
NACA 63 1-212
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Station
Upper surface
Station
0
0.336
0.567
1.041
0
1.032
1.260
1.622
2.284
0
0.583
0.843
1.355
2.622
-
0
0.932
1.120
1.408
1.912
4.863
7.358
3.238
3.963
4.554
5.470
6.137
5.137
7.642
10.141
15.132
20.118
-
2.606
3.115
3.520
4.124
4.545
6.606
6.901
7.030
6.991
-
4.816
4.957
4.970
4.849
4.609
9.859
N ACA 63 1-412
(Stations and ordinates given in
per cent of airfoil chord)
Ordinate Station Ordinate
0
0.417
0.657
1.145
2.378
14.868
19.882
Ordinate
2.257
4.727
7.218
9.718
14.735
:::1
3.544
5.273
7.782
10.282
15.265
20.235
-
25.200
30.160
35.118
- 3.919
4.379
5.063
6.138
6.929
7.499
7.872
8.059
8.062
7.894
50.000
55.016
60.029
65.038
70.043
6.473
6.030
5.491
4.870
4.182
50.000
54.984
59.971
64.962
69.957
-
4.267
3.840
3.349
2.810
2.238
7.576
50.000
55.031 1 7.125
6.562
60.057
65.076
5.899
70. 087 1 5.153
75.045
3.451
2.698
74.955
79.958
84.965
89.975
94.988
-
1.661
1.106
0.601
0.190
0.066
75.089
80.084 i
85.0iO I
90.025
95.012
100.000
1.947
1.224
0.566
0
100.000
0
I
4.344
3.492
2.618
90. 049 1 1.739
0.881
95. 023 1
0
100.000
•• _ . _ _ •• _ _ _ _ ... _
L.E. radius: 1.087
Slope of radius through L.E.: 0.0842
Ordinate
-
6.799
I
Station
0
0.664
0.933
1.459
2.743
25.100
30.080
35.059
40.038
45.018
29.840
882
34.
39.924 1
44.9H4 .
Lower surface
0
1.071
1.320
1.719
2.460
24.900
29.920
34.941
39.962
44.982
80.042
85.035
-
Lower surface
417
...... _ _ •• r
0
0.871
1.040
1.291
1.716
2.280
2.685
2.995
3.446
3.745
- 3.984
45.036
- 3.939
- 3.778
- 3.514
50.000
54.969
59.943
64.9201
69.913
- 3.164
- 2.745
2.278 1
1.i79 I
- 1.265
40.076
I
I-
I-
I
74.911 I - 0.764
79.916 t - 0.308
84.930
0.074
89.951
0.329
94.977
0.383
I
I
ioo.ooo
._
...
--
I __0
_..
.-
.._ ...­
I.,.E. radius: 1.087
Slope of radius through L.E.: 0.1 sss
THEORY OF WING SECTIONS
418
NACA63r215
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Station
Ordinate Station Ordinate
0
0.601
0.863
1..380
Upper surface
Station
Lower surface
Ordinate Station
Ordinate
2..652
-
0
1.150
1..388
1..766
2.420
0
0.300
0.525
0.991
2.198
0
1.287
1.585
2.074
2.964
0
0.700
0.975
1.509
2.802
5.171
7.677
10.177
15.166
20.148
-
3.328
3.999
4.535
5.336
5.895
4.660
7.147
9.647
14.669
19.705
4.264
5.261
6.077
7.3'8
8.279
5.340
7.853
10.353
15.331
20.295
- 3.000
- 3.565
- 4.009
25..125
30.100
35.074
45.023
-
6.259
6.448
6.470
6.315
6.004
24.760
29.800
34.852
39.905
44.955
8.941
9.362
9.559
9.527
9.289
25.250
30.200
35.148
40.095
45.045
-
5.361
5.474
5.439
5.243
4.909
7.768
7.203
6.524
5.751
4.906
50.000
54.981
59.965
64.953
69..947
-
5.562
5.013
4.382
3.691
2.962
50.000­
55.039
60.070
65.093
70.106
8.871
8.298
7.595
6.780
5.877
50.000
54.961
59.930
64.907
69.894
-
4.459
3.918
3.311
2.660
1.989
4.014
3.106
2..213
1.368
0.616
74.945
79.949
84.957
89.970
94..986
-
2.224
1.513
0.867
0.334
0.016
75.109
80.102
85.085
90.059
95.028
4.907
3.900
2.885
1.884
0.931
74.891
79.898
84.915
89.941
94.972
- 1.327
- 0.716
- 0.193
0.184
0.333
0
0.399
0.637
1.120
2.. 348
0
1.250
1.528
2.792
4..829
7.323
9.823
14.834
19..852
3.960
4.847
5.569
6.682
7.487
24.875
29.900
34.926
39.952
44.977
8.049
50.000
55.019
60.035
65.047
70..053
75.055
SO.051
85.043
90.030
95.014
100.000
Lower surface
NACA63r415
(Stations and ordinates given in
per cent of airfoil chord)
1.980
8.392
8.530
8.457
8.194
0
40..048
100.000
0
L.E. radius: 1.594
Slope of radius through L.E.: 0.0842
100.000
0
100.000
-
0
1.087
1.306
1.646
2.220
- 4.656
- 5.095
0
L.E. radius: 1.594
Slope of radius through L.E.: 0.1685
APPENDIX III
NACA 63r615
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Station
Ordinate
Lower surface
Station
419
NACA 63a-218
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Lower surface
Station
Station
-
0
1.017
1.214
1.517
2.013
0
0.382
0.617
1.096
2.319
0
1.449
1.778
2.319
3.285
0
0.618
0.883
1.404
2.681
-
0
1.349
1.638
2.105
2.913
2.664
3.123
3.476
3.972
4.290
4.796
7.288
9.788
14.801
19.822
4.673
5.728
6.581
7.895
8.842
5.204
7.712
10.212
15.199
20.178
-
4.041
4.880
5.547
6.549
7.250
25.150
30..120
40.057
45.027
-
7.704
7.940
7.970
7.774
7.387
-
6.839
6.161
5.384
4.537
3.650
-
2.754
1..894
1.113
0.467
0.032
Ordinate
0
0.205
0.418
0.866
2.0li0
0
1.317
1.634
2.159
3.129
0
0.795
1.082
1.634
2.950
4.492
6.973
9.473
14.504
19.558
4.560
5.667
6.578
8.010
9.066
5.508
8.027
10.527
20.442
-
24.625
29.700
39.857
44.932
9.830
10.331
10.587
10.598
10.384
25.37S
30.300
35.222
40.143
45.068
-
4.460
4.499
4.407
4.172
3.814
24.850
29.880
34.911
39.943
44.973
9.494
9.884
10.030
9.916
9.577
50.000
55.058
60.105
65.139
70.159
9.974
9.393
8.665
7.809
6.847
50.000
04.942
59.895
64.861
69.841
- 3.356
- 2.823
- 2.239
9.045
8.351
7.526
- 1.015
50.000
55.023
60.042
65.055
70.062
5.594
50.000
64.977
59.958
64.945
69.938
75.163
5.800
74.837
79.847
84.873
89.911
94.958
- 0.430
0.083
0.483
0.704
0.651
75.064
80.059
85.049
90.034
95.016
4.544
3.486
2.459
1.501
0.664
74.936
79.941
84.951
89.966
94.984
34.778
so. 153
85.127
90.089
95.042
100.000
4.693
3.555
2.398
1.245
0
1~.496
100.000
Ordinate
Ordinate
-
1.629
0
100.000
6.597
0
35.()89
100.000
0
~_._--
L.E. radius: 1.594
Slope of radius through L.E.: 0.2527
L.E. radius: 2.120
Slope of radius through L.E.: 0.0842
THEORY OF WING SECTIONS
420
NACA 63r418
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Station
Ordinate
Lower surface
Station
NACA 63.-618
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Ordinate
Station
Ordinate
Lower surface
Station
Ordinate
0
0.267
0.487
0.945
2.140
0
1.484
1.833
2.410
3.455
0
0.733
1.013
1.555
2.860
-
0
1.284
1.553
1.982
2.711
0
0.156
0.361
0.797
1.965
0
1.511
1.878
2.491
3.616
0
0.844
1.139
1.703
3.035
-
0
1.211
1.458
1.849
2.500
4.593
7.077
9.577
14.602
19.645
4.975
6.139
7.087
8.560
9.632
5.407
7.923
10.423
15.398
20.355
.;..
-
3.711
4.443
5.019
5.868
6.448
4.393
6.868
9.367
14.404
19.469
5.268
6.542
7.586
9.219
10.418
5.607
8.132
10.633
15.596
20.531
-
3.372
3.998
4.484
5.181
5.642
24.699
29.760
34.823
39.886
44.949
10.385
10.854
11.058
10.986
10.672
25.301
30.240
35.177
40.114
45.054
-
6.805
6.966
6.938
6.702
6.292
24.549
29.640
34.734
39.829
44.919
11.273
11.822
12.086
12.056
11.767
25.451
30.360
35.266
40.171
45.081
-
5.903
5.990
5.906
5.630
5.197
50.000
55.046
60.083
65.110
70.125
10.148
9.446
8.596
7.626
6.564
50.000
54.954
59.917
64.890
69.875
- 5.736
-
5.()66
4.312
3.506
2.676
50.000
55.069
60.125
65.164
70.187
11.251
10.541
9.667
8.655
7.534
50.000
54.931
59.875
64.836
69.813
-
4.633
3.971
3.241
2.475
1.702
75.128
80.119
85.099
90.069
95.032
5.438
4.280
3.130
2.017·
0.978
74.872
79.881
84.901
89.931
94.968
- 1.858
- 1.096
- 0.438
0.051
0.286
75.191
80.178
85.147
90.103
95.048
6.330
5.073
3.800
2.531
1.293
74.809
79.822
84.853
89.897
94.952
- 0.960
- 0.297
0.238
0.571
0.603
100.000
I
0
100.000
0
L.E. radius: 2.120
Slope of radius through L.E.: 0.1685
100.000
0
100.000
0
L.E. radius: 2.120
Slope of radius through L.E.: 0.2527
421
APPENDIX III
NACA 63r221
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Station
NACA 63.-421
(Stations and ordinates given in
per cent of airfoil chord)
Lower surface
Upper surface
Ordinate Station Ordinate
Station
1.425
0
- 1.527
- 1.861
- 2.414
0.237
0.452
0.902
2.708
- 3.385
o
Lower surface
Ordinate
Station
Ordinate
0
0.7fl3
1.048
1.598
2.914
0
- 1.461
- 1.774
- 2.289
3.181
0
0.367
0.600
1.075
2.292
0
1.627
2.001
2.628
3."57
0
0.633
0.900
4.763
7.253
9.753
14.767
19.792
5.375
6.601
7.593
9.111
10.204
5.237
7.747
10.24i
11).233
20.208
-
4.743
5.753
6.559
7.765
8.612
4.5271 5.675
5.473
7.007 I 7.010
7.993
9.506 I 8.097
10.494
14.535 1 9.774 t 15.465
19.5851 10.993
20.415
24.824
29.860
34.897
39.934­
44.969
10.946
11.383
11.529
11.369
10.949
25.176
30.140
35.103
40.066
45.031
-
9.156
9.439
9.469
9.227
8.i59
24.6491 11.837
29.719
12.352
34.;93 J 12.558
39.8()7: 12.439
4-1.937; 12.044
50.000
55.027
60.048
65.063
70.071
10.309
9.485
8.512
7.426
6.262
8.103
50.000
55.054
11.412
10.580
60.096
9.582
65.126,
70.1-l3 i
8.455
7.232
75.073
5.054
3.849
2.693
1.629
0.708
75.1451
80.135 i
5.947
74.855
2.367
4.643 I 79.865 I - 1.459
85.1·11
3.364
so.067
85.056
90.039
95.018
100.000
0
50.000
54.9i3
59.95~
I-
I-
- ;.295
I- 6.370
5.366
64.931 i -
69.929\74.927 79.933 l 84.944 1 89.961
94.982 -
il
4.318
3.264
2.257
1.347
0.595
0.076
__
.100.000 L_.~
.
-.--~
L.E. radius: 2.650
Slope of radius through L. E.: 0.0842
I
{i
2.086!
9(!.07~
i
I
0
1.661
2.054
2.717
3.925
-
4.U!
5.314
6.029
7.082
7.809
25.351 30.281 35.207 40.133 I 45.OG3 1 -
8.257
8.464
8.438
8.155
7.60-1
5O.0on
7.000
6.200
54.946
I
11
I'
J -
!-
I-
59.9t).J
5.298
6-l.8~: I - 4.335
69.851
3.344
II-
8-4.889 I - O.G72
2.144 , 89.922
I- O.01H
l;:;;L..~.l~ l;:~ L_.~·242
1..E. radius: 2.650
Slope of radius through L.E.: O.l6S5
THEORY OF WING SECTIONS
422
NACA 63A210
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Station
Ordinate
Lower surface
Station
NACA 64-108
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Ordinate
Station
Lower surface
Ordinate Station Ordinate
0
0.423
0.664
1.151
2.384
0
0.868
1.058
1.367
1.944
0
0.577
0.836
1.349
2.616
-
0
0.756
0.900
l.l25
1.522
0
0.472
0.719
1.215
2.460
0
0.682
0.828
1.058
1.457
0
0.528
0.781
1.285
2.540
-
0
0.632
0.758
0.950
1.271
4.869
7.364
9.863
14.869
19.882
2.769
3.400
3.917
4.729
5.328
5.131
7.636
10.137
15.131
20.118
-
2.047
2.428
2.725
3.167
3.468
4.956
7.455
9.955
14.958
19.962
2.032
2.471
2.832
3.405
3.835
5.044
7.545
10.045
15.042
20.038
-
1.716
2.047
2.316
2.733
3.039
24.898
29.916
34.935
39.955
44.975
5.764
6.060
6.219
6.247
6.151
25.102
30.084
35.065
40.045
45.025
-
3.662
3.764
3.771
3.689
3.523
24.968
29.974
34.980
39.987
44.994
4.152
4.370
4.494
4.528
4.431
25.032
30.026
35.020
40.013
45.006
-
3.256
3.398
3.464
3.456
3.335
49.994
55.012
60.028
65.041
70.052
5.943
5.637
5.245
4.772
4.227
50.006·
54.988
59.972
64.959
69.948
-
3.283
2.985
2.641
2.262
1.861
50.000
55.005
60.010
65.013
70.015
4.236
3.959
3.617
3.219
2.777
50.000
54.995
59.990
64.987
69.985
-
3.132
2.863
2.545
2.189
1.805
75.061
80.074
85.072
90.050
95.026
3.624
2.974
2.254
1.519
0.769
74.939
79.926
84.928
89.950
94.974
-
1.464
1.104
0.812
0.539
0.279
75.016
80.015
85.013
90.010
95.005
2.302
1.802
1.297
0.808
0.364
74.984
79.985
84.987
89.990
94.995
-
1.406
1.006
0.625
0.292
0.048
100.000
0.021
100.000
- 0.021
100.000
L.E. radius: 0.742
T.F. radius: 0.023
Slope of radius through L.E.: 0.095
0
100.000
0
L.E. radius: 0.455
Slope of radius through L.E.: O.()4-2
APPENDIX III
NACA 64-110
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Station
0
0.465
Ordinate Station Ordinate
1.207
2.450
0
0.844
1.023
1.303
1.793
4.945
7.443
9.944
14.947
19.953
2.500
3.037
3.479
4.178
4.700
24.959
29.967
34.175
39.984
44.992
5.087
0.71~
50.000
55.007
60.012
65.016
70.019
75.020
80.019
85.016
90.012
95.006
100.000
Lower surface
0
o.sss
0.788
1.293
2.550
D.055
1.557
10.056
1"5.053
20.04:7
25.04:1
30.033
-
0.442
0.524
0.645
0.836
-
2.184
2.613
2.963
3.506
3.904
4.934
7.432
9.933
14.937
19.943
1.719
2.115
2.444
2..970
3.367
5.066
7.568
10.067
15.063
20.057
-
1.087
1.267
1.410
1.624
I.nS
4.191
4.378
4.465
4.452
4.295
24.952
29.961
34.971
39.981
25.048
30.039
35.029
40.019
45'(J09
- 1.877
- 1.935
- 1.951
44.991
3.667
3.879
4.011
4.066
4.014
4.034
3.690
3.284
2.830
2.341
50.000
55.008 !
60.015 I
65.020
70.023 I
3.878
3.670
3.402
3.080
2.712
50.000 - 1.672
54.992 - 1.480
59.985 - - 1.260
64.980 - 1.020
69.977 - 0.768
50.000
54.993
59.988
64.984
69.981
-
74.980
79.981
84.984
89.988
- 1.833
- 1.324
I
94.994
100.000
- 0..840
- 0.413
I -
I
Ordinate
0
0.541
0.796
1.302
2.560
5.138
4.786
4.356
0
Ordinate Station
0
0.542
0.664
0.859
1.208
40.016
45.008
2.729
2.120
1.512
0.929
0.406
Station
Lower surface
0
0.459
0.704
1.198
2.440
35.025
3.313
Upper surface
0
0.794
0.953
1.195
1.607
5.495
5.524
5.391
3.860
NACA 64-206
(Stations and ordinates given in
per cent of airfoil chord)
-
-
5.350
423
0.090
0______
L.E. radius: 0.720
Slope of radius through L.E.: 0.042
I
i
2.307
800024
1.868
85.020 1.410
90.015 1I 0.940
0.473
95.007
75.025
100.000
1
0
0
-
1.924
- 1.824
I 74.975
I
I
i
- 0.517
79.976 - 0.276
0.064
84.980
0.094
89.985 I
94.993l
0.159
Ii
100.000 t
0
LE. radius: 0.256
Slope of radius through L.E.: 0.084
THEORY OF WING SECTIONS
424
NACA 64-208
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Station
Ordinate
Lower surface
Station
NACA 64-209
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Ordinate
·Station
Ordinate
Lower surface
Station
Ordinate
0
- 0.686
- 0.819
- 1.018
0
0.445
0.688
1.180
2.421
0
0.706
0.862
1.110
1.549
0
0.555
0.812
1.320
2.579
-
0
0.606
0.722
0.896
1.177
0
0.438
0.680
1.172
2.411
0
0.786
0.959
1.232
1.716
0
0.562
0.820
1.328
2.589
4.912
7.410
9.909
14.915
19.924
2.189
2.681
3.089
3.741
4.232
5.088
7.590
10.091
15.085
20.076
-
1.557
1.833
2.055
2.395
2.640
4.901
7.398
9.899
14.905
~9.915
2.423
2.965
3.413
4.127
4.663
24.935
29.948
39.974
44.988
4.598
4.856
5.009
5.063
4.978
25.065
30.052
35.039
40.026
45.012
-
2.808
2.912
2.949
2.921
2.788
24.927
29.941
34.956
39.971
44.986 ,
50.000
55.011
60.020
65.027
70.031
4.787
4.506
4.152
3.733
3.263
50.000
54.989
59.980
64.973
69.969
-
2.581
2.316
2.010
1.673
1.319
50.000
55.012
60.022
65.030 1
70.035 ,
75.032
2.749
2.200
1.634
1.067
0.522
74.968
79.969
84.973
89.981
94.990
-
0.959
0.608
0.288
0.033
0.110
34.961
~.031
85.027
90.019
95.010
100.000
0
100.000
0
I,.E. radius: 0.455
Slope of radius through L.E.: 0.084
-
1.344
5.099
7.602
10.101
15Jl95
20.085
-
1.791
2.117
2.379
2.781
3.071
5.064
5.345
5.509
5.561
5.459
25.073
30.059
35.044
40.029
45.014
-
3.274
3.401
3.449
3.419
5.239
4.921
4.523
4.056
3.533
50.000
54.988
59.978
64.970
69.965
-
3.033
2.731
2.381 .
1.996
1.589
75. 036 1 2.964
2.360
80.035
85.030 ; 1.742
90.021 i 1.128
0.543
95.011
74.964
79.965
84.970
89.979
94.989
-
1.174.
I
I
I
100.000
0
100.000
- 3.269
- 0.;68
- 0.396
- 0.094
0.089
0
L.E. radius: 0.579
Slope of radius through L.E.: 0.084
t
APPEl·lDIX III
NACA 64-210
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Station
Lower surface
Ordinate Station Ordinate
425
NACA 64 1-112
(Stations and ordinates given in
per cent of airfoil chord)
t· pper surface
Station
Ordinate
o
0
0.431
0.673
1.163
2.401
0
0.867
1.056
1.354
1.884
4.890
7.387
9.887
14.894
19.905
2.656
3.248
3.736
4.514
5.097
2.024
5.110
2.400
7.613
10.113 I - 2.702
15.106 - 3.1~
20.095 1 - 3.50il
2.967
3.605
4.128
4.956
5.571
24.919
29.934
34.951
39.968
44.985
5.533
5.836
6.010
6.059
5.938
25.081
30.066 i 35.049140.032 45.015 -
3.743
3.Sg-1
3.950
3.917
3.748
6.024
6.330
6.493
6.517
6.346
50.000
55.014
60.025
65.033
70.038
5.689
5.333
4.891
4.375
3.799
50.000 54.987 1
59.975 I 64.967169.962 -
3.483
3.143
2.749
1.855
6.032
5.604
5.084
4.489
3.836
75.040
80.038
85.033
90.024
95.012
3.176
2.518
1.849
1.188
0.564
74.960 I
79.962 ,
84.968
89.977
94.988
- 1.386
- O.!2?
- O.D03
- 0.15-1
0.068
3.143
2.427
1.718
1.044
0.446
100.000
0
0
0.569
0.827
1.337
2.599
0
0.767
0.916
1.140
1.512
-
Ii-
I-
!
100.000
1
2.31~
0
----
L.E. radius: 0.720
Slope of radius through L.E.: 0.084
1.002
1.213
1.543
2.127
o
L.E. radius: 1.040
Slope of radius through I ... I~.: 0.042
'l'HEORY OF WING SECTIONS
426
NACA 64 1-212
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Station
Lower surface
Ordinate Station Ordinate
NACA 641-412
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Lower surface
Station Ordinate Station
-
Ordinate
0.841
1.353
2.618
-
0
0.925
1.105
1.379
1.846
0
0.338
0.569
1.045
2.264:
0
1.064
1.305
1.690
2.393
0
0.662
0.931
1.455
2.736
3.123
3.815
4.386
5.291
5.968
5.132
7.636
10.135
15.128
20.114
-
2.491
2.967
3.352
3.945
4.376
4.738
7.229
9.730
14.745
19.772
3.430
4.231
4.896
5.959
6.760
5..262
7.771
10.270
15.255
20.228
- 2.166
- 2.535
-2.828
- 3.267
- 3.576
24..903
29.921
34 . 941
39.961
44.982
6..470
6.815
7.008
7.052"
6.893
25..()97
30.079
35.. 059
40.039
45.018
-
4.680
4.871
4.948
4.910
4.. 703
24.805
29.842
34.882
39.923
44.963
7.363
7.786
8.037
8.123
7.988
25.195
30.158
35.118
40.077
45.037
- 3.783
- 3.608
50.000
55.016
60.029
65.039
70.045
6.583
fi.151
5.619
5.004
4.322
50.000
54.. 984
59.971
64.961
69.955
-
4.377
3.961
3.477
2.944
2.378
50.000
55.032
60.059
65.078
70.090
7.686
7.246
6.690
6.033
5.293
50.000
54.968
59.941
64.922
69.910
-
75.047
80.045
85.038
90.027
95.013
3.590
2.825
2.054
1.303
0.604
74.953
79.955
84.962
89.973
-
1.800
1.233
0.708
0.269
0.028
75.094
80.089
85.076
90.055
95.027
4.483
3.619
2.722
1.818
0.919
74.906
79.911
84.924
89.945
94..973
- 0.903
- 0.435
- 0.038 1
0.250
0.345
0
0.418
0.659
1.147
2.382
0
1.025
1.245
1.593
2.218
4.868
7.364
9.865
14.872
19.886
100.000
0
0
0~582
94.987
100.000
0
L.E. radius: 1.040
Slope of radius through L.E.: 0.084
100.000
0
100.000
-
0
0.864
1.025
1.262
1.849
- 3..898
- 3.917
- 3.839
3.274
2.866
2.406
1.913
1.405
0
L.E. radius: 1.040
Slope of radius through L.E.: 0.168
APPENDIX III
NACA 64t-215
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Station
Lower surface
Ordinate Station
427
NACA 64r415
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Ordinate
Station
Ordinate
Lower surface
Station
Ordinate
0
0.399
0.637
1.122
2.353
0
1.254
1.522
1.945
2.710
0
0.601
0.863
1.378
2.641
-
0
1.154
1.382
1.731
2.338
0
0.299
0.526
0.996
2.207
0
1.291
1.579
2.038
2.883
0
0.701
0.974
1.504
2.793
4.836
7.331
9.831
14.840
19.857
3.816
4.661
5.356
6.456
7.274
5.164
7.669
10.169
15.160
20.143
-
3.184
3.813
4.322
5.110
5.682
4.673
7.162
9.662
14.681
19.714
4.121
5.075
5.864
7.122
8.066
5.327
7.838
10.338
15.319
20.286
-
2.857
3.379
3.796
4.430
4.882
24.878
29.901
34.926
39.952
44.977
7.879
8.290
8.512
8.544
8.319
25.122
30.099
35.074
40.048
45.023
-
6.089
6.346
6.452
6.402
6.129
24.756
29.803
34.853
39.904
44.954
8.771
9.260
9.541
9.614
9.414
25.244
30.197
35.147
40.096
45.046
-
5.191
D.372
5.421
5.330
5.034
50.000
55.020
60.036
65.048
70.055
7.913
7.361
6.691
5.925
5.085
50.000
54.980
59.964
64.952
69.945
-
5.707
5.171
4.549
3.865
3.141
50.000
9.016
8.456
7..762
6.954
6.055
50.000
54.960
59.928
64.904
69.889
- 4..604
- 4.076
- 2.834
- 2.167
15.058
4.191
3.267
2.349
1.466
0.662
74.942
79.945
84.954
89.967
94.984
- 2.401
- 1.675
5.084
4.062
3.020
1.982
0.976
74.885
79.891
84.908
89.934
94.968
- 1.504
- 0.878
- 0.328
0.086
0.288
100.000
0
80.055
85.046
90.033
95.016
100.000
0
1 100 .000
- 1..003
- 0.432
- 0.030
0
L.E. radius: 1.590
Slope of radius through L.E.: 0.084
55.040
60.072
65.096
70.111
75.115
so. 109
85.092
90.066
95.032
100.000
0
0
-
1.()91
- 1.299
- 1.610
- 2.139
-
3.478
L.E. radius: 1.590
Slope of radius through L.E.: 0.168
THEORY OF VlING SECTIONS
428
NACA 64r218
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Station
Ordinate
Lower surface
Station
NACA 64,-418
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Ordinate
Station
Lower surface
Ordinate
Station
-
Ordinate
0
0.380
0.617
1.099
2.325
0
1.473
1.785
2.279
3.186
0
0.620
0.883
1.401
2.675
-
0
1.373
1.645
2.065
2.814
0
0.263
0.486
0.950
2.152
0
1.508
1.840
2.370
3.357
0
0.737
1.014
1.550
2.848
4.804
7.297
9.797
14.808
19.828
4.497
5.496
6.316
7.612
8.576
5.196
7.703
10.203
15.192
20.172
-
3.865
4.648
5.282
6.266
6.984
4.609
7.095
9.595
14.617
19.657
4.800
5.908
6.823
8.277
9.366
5.391
7.905
10.405
15.383
20.343
-
3.536
4.212
4.755
5.585
6.182
24.853
29.881
34.912
39.942
44.972
9.285
9.760
10.009
10.023
9.725
25.147
30.119
35.088
40.058
45.028
-
7.495
7.816
7.949
7.881
7.535
24.707
29.763
34.823
39.885
44.9-15
10.176
10.730
11.037
11.093
10.820
25.293
30.237
35.177
40.115
45.055
-
6.596 .
6.842
6.917
6.809
6.440
50.000
55.024
9.217
8.540
7.729
6.812
5.814
50.000
54.976
59.957
64.943
69.935
-
7.011
6.350
5.587
4.752
3.870
50.000
55.047
60.086
65.114
70.131
10.320
9.635
8.799
7.841
6.784
50.000
54.953
59.914
64.886
69.869
-
5.908
5.255
4.515
3.;21
2.896
4.760
3.683
2.623
1.617
0.716
74.932
79.936
84:.946
89.962
94.981
-
2.970
2.091
1.277
0.583
0.084­
7~.135
5.654
4.477
3.294
2.132
1.030
74.865
79.873
84.892
89.923
94.963
-
2.074
1.293
0.602
0.064
0.234
60.043
65.057
70.065
75.068
so.064
85.054
90.038
95.019
100.000
0
100.000
0
L.E. radius: 2.208
Slope of radius through L.E.: 0.084
80.127
85.108
I
90.0771
95.037
100.000
I
0
,
100.000
0
- 1.308
- 1.560
- 1.942
- 2.613
0
L.E. radius: 2.208
Slope of radius through L.E.: 0.168
APPENDIX III
NACA 64r618
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Station
429
N ACA Mc-221
(Stations and ordinates given in
per cent of airfoil chord)
Lower surface
L pper surface
Ordinate Station Ordinate
Station
Lower surface
I
Ordinate Station Ordinate
0
0.150
0.359
0.805
1.982
0
1.534
1.885
2.452
3.518
0
0.850
1.141
1.695
3.018
-
0
1.234
1.465
1.810
2.402
0
0.362
0.596
1.075
2.297
0
1.690
2.049
2.618
3.665
0
0.638
0.904
1.425
2.703
4.417
6.895
9.395
14.427
19.486
5.093
6.312
7.322
8.937
10.153
5.583
8.105
10.605
15.573
20.514
-
3.197
3.768
4.220
4.899
5.377
4.772
7.264
9.763
14.776
19.799
5.182
6.334­
7.282
8.778
9.889
5.228
7.736
10.237
15.224
20.201
-
24.560
29.645
34.735
39.827
44.917
11.065
11.698
12.065
12.163
11.915
25.440
30.355
35.265
40.173
45.083
-
5.695
5.866
5.885
5.737
5.345
24.829
29.861
34.897
39.933
10.701
11.240
11.510
11.502
11.125
25.171
30.139
35.103
40.067
45.032
-
50.000
55.071
60.129
65.171
70.196
11.423
10.730
9.870
8.870
7.754
50.000
54.929
59.871
64.829
69.804
-
4.805
4.160
3.444
2.690
1.922
50.000
55.027
60.050
65.065
70.0i5
10.507
9.702
8.749
7.679
6.521
54.9731-
50.000
59.950
64.935
69.925
- 8.301
7.512
- 6.607
- 5.619
- 4.577
75.203
80.191
85.161
90.115
95.056
6.544
5.270
3.963
2.646
1.344
74.797
79.809
84.839
89.885
94.944
- 1.174
- 0.494
0.075
0.456
0.552
75.077
80.073
85.001
90.044
95.021
74.923
79.927
- 3.520
- 2.490
M.939
. 89.956
94.979
- 0.727
100.000
0
100.000
!- _. __ 0
..
.-._-­
L.E. radius: 2.208
Slope of radius through L.E.: 0.253
4-1.
008
100.000
--
--~
i
1
5.310
4J)82
2.885
1.761
0.765
I
0
Il~.OOO 1
L.E. radius: 2.884
Slope of radius through
0
- 1.590
-
1.909
- 2.404
- 3.293
4.550
5.486
6.248
7.432
-8.~7
8.911
9.296
9.450
9.360
8.935
- 1.539
- 0.133
0
I~.I~.: O.OSl
-
THEORY 0/;' WING SECTION8
430
NACA 64r421
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Ordinate
Station
Lower surface
NACA 64A210
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Station
Ordinate
Station
0
0.424
0.665
1.153
2.387
Ordinate
-
Lower surface
-
Station
Ordinate
0
0.856
1.044
1.342
1.895
0
0.576
0.835
1.347
2.613
0
- 0.744
-
0
0.227
0.445
0.903
2.096
0
1.723
2.101
2.707
3.834
0
0.773
1.055
1.597
2.904
-
0
1.523
1.821
2.279
3.090
4.545
7.028
9.528
14.553
19.599
5.482
6.744
7.786
9.442
10.678
5.455
7.972
10.472
15.447
20.401
-
4.218
5.048
5.718
6.750
7.494
4.874
7.369
9.868
14.874
19.885
. 2.685
3.288
3.792
4.592
5.200
5.126
7.631
10.132
15.126
20.115
- 1.963
24.657
29.723
34.794
39.865
44.936
11.591
12.209
12.539
12.572
12.220
25.343
30.277
35.206
40.135
45.064
-
8.011
8.321
8.419
8.288
7.840
24.900
29.917
34.935
39.955
44.975
5.656
5.984
6.192
6.274
6.208
25.100
30.083
35.065
40.045
45.025
- 3.554
- 3.688
- 3.744
11.610
10.797
9.819
8.708
7.491
50.000
54.945
59.901
64.869
69.850
-
7.198
6.417
5.535
4.588
3.603
49.994
55.012
60.028
65.042
70.054
6.014
5.714
5.323
4.852
4.310
50.006
54.988
59.972
64.958
69.946
6.203
4.876
3.556
2.276
1.079
74.846
79.855
84.878
89.913
94.958
-
2.623
1.692
0.864
0.208
0.185
75.063
80.076
85.074
90.052
95.027
3.702
3.037
2.301
1.551
0.785
74.937 - 1.542
1.167
79.924
84.926 I - 0.859
89.948 - 0.571
94.974 - 0.295
100.000
0.021
50.000
55.055
60.099
65.131
70.150
75.154
80.145
85.122
90.087
95.042
100.000
I
0
100.000
0
L.E. radius: 2.884
Slope of radius through L.E.: 0.168
- 0.886
- 1.100
- 1.473
- 2.316
- 2.600
- 3.030
- 3.340
- 3.716
- 3.080 I
354 1
-- a.3.062
- 2.719
- 2.342
- 1.944
I-
100.000
-
- 0.021
LE. radius: 0.687
T.E. radius: 0.023
Slope of radius through L.E.: 0.095
I
APPENDIX 111
NACA 64A4:10
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
I
i
0
0.350 :
0.582
1.059
2.276
NACA 64 1A212
(Stations and ordinates given in
per cent of airfoil chord)
Lower surface
Station Ordinate Station Ordinate
431
Upper surface
Station
Lower surface
Ordinate Station
Ordinate
0
0.902
1.112
1.451
2.095
0
0.650
0.918
1.441
2.724
-
0
0.678
0.796
0.969
1.251
3.034
3.865
4.380
5.366
6.126
5.251
7.770
10.263
1'5.252
20.230
-
1.592
1.919
1.996
2.244
2.406
4.849
7.343
9.842
14.849
19.862
3.145
3.846
4.432
5.358
6.060
5.151
7.657
10.158
15.151
20.138
6.705
7.131
7.414
7.552
7.522
25.200
30.166
35.129
40.090
45.050
-
2.499
2.537
2.518
2.436
2.266
24.880
29.900
34.922
39.946
44.970
6.584
6.956
7.189
7.272
7.177
25.120 / - 4.482
30.100 - 4..660
35.078 f - 4.741
40.054
4.714
45.030
4.549
49.989 . 7.344
55.025 ' 7.040
60.057 . 6.624
65.085 j a.l06
70.108 J 5.490
50.011
2.O'M
49.993
55.015
60.034
65.050
70.064
6.935
6.570
6.103
5.544
4.903
50.007154.985 59.966 i 64.950
69.936 -
75.075
80.090
74.925 I - 2.037
79.910 - 1.563
84.912 - 1.159
89.938 - 0.771
94.968 - 0.398
4.749
7.230
9.737
14.748
19.770
24.800
29.834
34.871
39.910
44.950
;
;
:
.
I
I 4.780
54.975
59.943
64.915
69.892
3.967
3.018
2.038
1.028
74.874
79.849
84.852
89.896
94.947
100.000 1 0.021
100.000
75.126
SO.151
85.148
90.104
95.053
J
i
;
j
j
1~
0
0.409
0.648 II
1.135
2.365
0
1.013
1.233
1.580
2.225
-
1.736
1.418
1.086
0.760
-
0.460
0.229
0.132
0.076
0.048
90.062
95.032
4.197
3.433
2.601
1.751
0.888'
- 0.021
100.000
0.025
L.E. radius: 0.687
T.E. radius: 0.023
Slope of radius through L.E.: 0.190
85.088
-
I
0
0.591 ·0.852 I 1.365
2.635 I -
I-
0
0.901
1.075
1.338
1.803
- 2.423
- 2.874
3.240
3.796
- 4.200
1I
­
II-
1-
I
IO?OOO
I-
4.275
3.918
3.499
3.034
2.537
0.025
L.E. radius: 0.994
r.s, radius: 0.028
Slope of radius through L.E.: 0.095
THEORY OF WING SECTIONS
432
NACA 64JA215
(Stations and ordinates given in
per cent of airfoil chord)
L pper surface
Lower surface
NACA 65-206
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Ordinate Station
Ordinate
Station
0
1.243
1.509
1.930
2.713
0
0.612
0.876
1.393
2.667
0
1.131
1.351
1.688
2.291
0
0.460
0.706
1.200
2.444
4.811
7.304
9.802
14.811
19.827
3.833
4.683
5.391
6.510
7.351
5.189
7.696
10.198
15.189
20.173
- 3.111
- 3.711
- 4.199
24.849
29.875
34.903
39.933
44.963
7.975
8.417
8.686
8.766
8.627
25.151
30.125
35.()97
40.067
45.037
49.992
55.018
60.042
65.063
70.079
8.308
7.843
7.258
6.566
5.782
50.008
54.982
59.958
64.937
69.921
75'()93
80.111
85.109
90.076
95.039
4.926
4.017
3.039
2.046
1.039
74.907
79.889
84.891
89.924
94.961
100.000
0.032
100.000
Station
0
0.388
0.624
1.107
2.333
I
Lower surface
-
Ordinate Station
Ordinate
0
0.524
0.642
0.822
1.140
0
0.540
0.794
1.300
2.556
0
0.424
0.502
0.608
0.768
- 5.491
4.939
7.437
9.936
14.939
19.945
1..625
2.012
2.340
2.869
3.277
5.061
7.563
10.064
15.061
20.055
-
5.873
6.121
6.238
6.208
5.999
24.953
29.962
34.971
39.981
44.990
3.592
3.824
3.982
4.069
4.078
25.047
30.038
35.029
40.019
45.010
-
5.648
5.191
4.654
4.056
3.416
50.000
55.009
60.016
65.022
70.<Y>-6
4.003
3.836
3.589
3.276
2.907
50.000
54.991
59.984
64.978
69.974
-
-
2.766
2.147
1.597
1.066
0.549
75.028
80.027
850024
90.018
2.489
2.029
1.538
1.027
1
95'(x)9 ! 0.511
74.972
79.973
84.976
- 0.699
- 0.437
- 0.192
0.007
0.121
-
-
4.948
i
- 0.032
L.E. radius: 1.561
T.E. radius: 0.037
Slope of radius through L.E.: 0.095
100.000 t
0
89.982
94.991
I 100.()()()
-
-
- 0.993
- 1.164
- i.306
- 1.523
- 1.685
- 1.802
- 1.880
- 1.922
- 1.927
- 1.888
1.797
1.646
1.447
1.216
0.963
0
L.E. radius: 0.240
Slope of radius through l..E.: 0.084
433
APPENDIX III
NACA 65-209
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Ordinate
Station
Lower surface
Station
Ordinate
0
0.441
0.684
1.177
2.417
0
0.748
0.912
1.162
1.605
0
0.559
0.816
1.323
2.583
-
0
0.648
0.772
0.948
1.233
4.908
7.405
°2.275
2.805
3.251
3.971
5.092
7.595
10.096
15.091
20.082
-
25.071
30.058
35.044
40.029
45.014
-
~.904
14.909
19.918
4.522
24.929
4.944
29.942
5.254
5.4:61
34.956
39.971
44.986
I
5.567
5.564
I
5.439
50.000
5.181
013
55.
1 4.814
60.024
4.358
65.033
3..828
70.039
3.237
75.041
2.601
80.040
1.933
035
85.
90.026 1 1.255
0.596
95.013
100.000
I
0
Upper surface
Station
Lower surface
I
Ordinate Station Ordinate
o
0.435
0.678
1.169
2.408
0
0.819
0.999
1.273
1.757
-
0
0.719
0.859
1.059
1.385
1.643
1.957
2.217
2.625
2.930
4.898
7.394
9.894
14.899
19.909
2.491
3.069
3.555
4.338
4.938
5.102
7.606
10.106
15.101
20.091
-
1.859
2.221
2.521
2.992
3.346
3.154
3.310
3.401
3.425
3.374
24.921
30.0641- 3.788
25.079
- 3.607
34.951
5.397
5.732
5.954
39.9()s
6.067
40.032 t - 3.~
6.058
45.016
50.000 I - 3.233
I 54.9871 - 2.991
~9.976 I - 2.672
ti4 .967
- 2.298
69.961 - 1.884
74.959 79.960 84.965 89.974 987
94.
1
100.000
NACA 65-210
(Stations and ordinates given in
pel" cent of airfoil chord)
1.447
1.009
0.587
0.221
0.036
0
L.E. radius: 0.552
Slope of radius through L.E.: O.OSl
29.936
44.984 I
i
5.915
5.625
5.217
4.712
4wl28
75.045
8l!.044 ,
85
90.028
95.01-l,
3.479
2.783
2.057
1.327
0.622
1\
Jl38I'
II-
35.~~
-
I-
3.894
- 3.868
I
50.000 i
55.014
OO.tl27 j
(\5.03()
70.043
I
0
i
0.565
0.822
1.331 I
2.592 I
50.000 I 54.986
SU.973 I 64.96!
69.951 i -
3.709
3.435
3.075
2.652
2.184
74.955
Ii -
79.9~6
I -
1.689
1.191
0.711
0.293
0.010
I-
1°
I
II!
!-
1-
1 8-1.062 I
89. 972
94.986
I!
I_~:~~J_~ i 100.000
-
0__
1..E. radius: 0.687
Slope of radius through L.E.: 0.084­
THEORY OF WING SECTIONS
434
NACA 65-410
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Station
Lower surface
Ordinate Station
NACA 651-212
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Lower surface
Ordinate Station Ordinate
Ordinate
Station
0
0.661
0.781
0.944
1.191
0
0.423
0.664
1.154
2.391
0
0.970
1.176
1.491
2.058
0
0.577
0.836
1.346
2.609
0
0.372
0.607
1.089
2.318
0
0.861
1.061
1.372
1.935
0
0.628
0.893
1.411
2.682
4.797
7.289
9.788
14.798
19.817
2.800
3.487
4.067
5.006
5.731
5.203
7.711
10.212
15.202
20.183
- 1.536
-1.791
- 1.999
- 2.314
- 2.547
4.878
7.373
9.873
14.879
19.890
2.919
3.593
4.162
5.073
5.770
5.122
7.627
10.127
15.121
20.110
-
24.843
29.872
6.290
6.702
6.983
7.138
7.153
25.157
30.128
35.()97
2.710
2.814
2.863
2.854
2.773
24.906
29.923
34.942
39.961
44.981
6.300
6.687
6.942
7.068
7.044
25.094
30.077
35.058
40.039
45.019
- 4.510
- 4.743
45.032
-
7.018
6.720
50.000
54.971
59.947
64.927
69.915
-
2.606
2.340
2.004
1.621
1.211
50.000
55.017
60.032
65.043
70.050
6.860
6.507
6.014
5.411
4.715
50.000
54.983
59.968
64.957
69.950
74.910
79.912
84.924
89.943
94.971
- 0..792
75.053
80.052
85.045
90.033
95.017
3.954
3.140
2.302
1.463
0.672
74.947
79.948
84.955
89.967
94.983
34.903
39.936
44.968
50.000
55.029
60.053
65.073
70.085
75.090
80.088
85.076
90.057
95.029
100.000
6.288
5.741
5.099
4.372
3.577
2.129
1.842
0.937
0
40.064
100.000
-
- 0.393
- 0.037
0.226
0.327
0
L.E. radius: 0.687
Slope of radius through L.E.: 0.168
100.000
I
0
100.000
0
- 0.870
- 1.036
- 1.277
- 1.686
2.287
2.745
3.128
3.727
4.178
- 4.882
- 4.926
- 4.854
- 4.654
- 4.317 I1
- 3. 872
- 3.351
- 2.771
-
2.164
1.548
0.956
0.429
- 0.040
0
L.E. radius: 1.000
Slope of radius through L.E.: 0.084
1
APPENDIX III
NACA 65.-212
(J == 0.6
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Lower surface
435
NACA 65t-412
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Lower surface
Station Ordinate Station Ordinate
Station Ordinate Station Ordinate
0
0.399
0.638
1.124
2.356
0
0.982
1.194
1.520
2.113
4.837
7.329
9.827
14.833
19.848
0
0.601
0.862
1.376
2.644
-
0
0.852
1.012
1.242
1.625
3.017
3.728
4.330
5.298
6.042
5.163
7.671
10.173
15.167
20.152
-
2.185
2.606
2.956
3.500
3.904
24.869
29.894
34.921
39.951
44.983
6.611
7.029
7.304
7.444
7.423
25.131
30.106
35.079
40.049
45.017
-
4.197
4.401
4.518
4.550
4.475
50.017
65.051
60.094
65.123
70.124
7.231
6.856
6.318
5.634
4.842
49.983
54.949
59.906
64.877
69.876
-
4.283
3.968
3.566
3.124
2.640
75.112
80.090
85.064
90.036
95.013
3.983
3.082
2.173
1.297
0.521
74.888
79.910
84.936
89.964
94.987
-
2.131
1.604
1.085
0.595
0.191
100.000
0
1100.000
0
L.E.. radius: 1.000
Slope of radius through L.E.: 0.110
0
0.347
0.580
1.059
2.28.1
0
1.010
1.236
1.588
2.234
0
0.653
0.920
1.441
2.717
-
0
0.810
0.956
1.160
1.490
4.757
7.247
9.746
14.757
19.781
3.227
4.010
4.672
5.741
6.562
5.243
7.753
10.254
15.243
20.219
-
1.963
2.314
2.604
3.049
3.378
24.811
29.846
34.884
39.923
44.962
7.~93
7.658
7.971
8.139
8.139
25.189
30.154
35.116
40.077
45.038
-
3.613
3.770
3.851
3.855
3.759
50.000
55.035
60.064
65.086
70.101
I
7.963
7.602
7.085
6.440
5.686
50.000
54.965
59.936
64.914
69.899
-
3.. 551
3.222
2.801
2.320
1.798
75.107 I
80.103
85.090 ,
90.066
95.033
4.847
3.935
2.974
1.979
0.986
74.893
79.897
84.. 910
89.934
94.967
- 1.267
- 0.751
- 0.282
0.089
0.278
I
100.000
0
100.000
_.
--
0
.-.-._.­
I~.E. radius: 1.000
Slope of radius through L, E.. : 0.168
THEORY OF WING SECTIONS
436
NACA 65r215
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Station
Ordinate
Lower surface
Station
NACA 65r415
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Ordinate
Station
Lower surface
Ordinate Stati~n Ordinate
0
0.406
0.645
1.132
2.365
0
1.170
1.422
1.805
2..506
0
0.594
0.855
1.368
2.635
-
0
1.070
1.282
1.591
2.134
0
0.313
0.542
1.016
2.231
0
1.208
1.480
1.900
2.680
0
0.687
0.958
1.484
2.769
4.848
7.342
9.841
14.848
19.863
3.557
4.380
5.069
6.175
7.018
5.152
7.658
10.159
. 15.152
20.137
-
2.925
3.532
4.035
4.829
5.426
4.697
7.184
9.682
14.697
19.726
3.863
4.794
5.578
6.842
7.809
5.303
7.816
10.318
15.303
20.274
- 3.510
- 4.150
- 4.625
24.882
29.904
34.927
39.952
44.976
7.658
8.123
8.426
8.569
8.522
25.118
30.096
35.073
40.048
45.024
-
5.868
6.179
6.366
6.427
6.332
24.764
29.807
34.854
39.903
44.953
8.550
9.093
9.455
9.639
9.617
25.236
30.193
35.146
40.097
45.047
-
4.970
5.205
5.335
5.355
5.237
50.000
55.021
60.039
65.053
70.062
8.271
7.815
7.189
6.433
5.572
50.000
54.979
59.961
64.947
69.938
-
6.065
5.625
5.047
4.373
3.628
50.000
55.043 ,
60.079 I
65.106 ~
70.124 i
9.374
8.910
8.260
7.. 462
6.542
50.000
I 54.957
59.921
64.894
69.876
-
4.962
4.530
3.976
3.342
2.654
75.065
80.063
85.055
90.040
95.020
4.638
3.653
2.649
1.660
0.744
74.935
79.937
84.945
89.960
94.980
-
2.848
2.()61
1.303
0.626
0.112
75.131 II
80.126 1
85.109\
90.080 j
95.040 I
5.532
4.447
2.175
1.058
74.869
79.874
84.891
89.920
( 94.960
-
1.952
1.263
0.628
0.107
0.206
0
1100.000
I
100.000
0
100.000
0
L.E. radius: 1.505
Slope of radius through L.E.: 0.084
I
I
100.000
I
3.320
I
-
0
1.008
1.200
1.472
1.936
- 2.599
- 3.098
0
L.E. radius: 1.505
Slope of radius through L.E.: 0.168
APPENDIX III
NACA 65r415
(J == 0.5
(Stations and ordinates given in
per cent of airfoil chord)
NACA 65.-218
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Lower surface
Lower surface
Ordinate Station Ordinate
Station
Station Ordinate Station Ordinate
4.099
5.122
5.985
7.383
8.459
5.426
7.946
10.451
15.432
20.389
- 2.335
- 2.746
-.3.081
- 3.591
- 3.963
24.671
29.743
34.825
39.916
45.019
9.280
9.883
10.280
10.470
10.423
25.329
30.257
35.175
40.084
44.981
- 4.232
- 4.411
I- - 4.508
- 4.526
- 4.431
50.152
10.106
55.262
9.501
60.307
8.672
65.314
7.684
70.294 , 6.573
49.848
54.738
59.693
64.686
69.706
- 4.226
- 3.929
- 3.548
75.253
80.199
74.747
79.801
84.863
89.923
94.973
-
90.077
95.027
100.000 ,
-_._­
),.~~.
0
100.000
- 2.609
1
radius: 1.505
Slope of radius through L.E.: 0.233
0
0.612
0.875
1.390
2.660
4.819
7.311
9.809
14.818
19.835
4.178
5.153
5.971
7.276
8.270
5.181
7.689
10.191
15.182
20.165
-
24.858
9.023
9.566
9.916
10.070
9.996
25.142
30.116
I-
40.058
45.028
!-
I
I
J
75.077 1
80.074 I
85.(163 f
90. 046 1
95.023 I
2.083
1.545
1.014
0.527
0.139
0
0
1.382
1.673
2.116
2.932
50.000
55.026
60.047
65.063
70.073
-3.1~
-
0
0.388
0.625
1.110
2.340
29.884_
34.912
39.942
44.972
100.000 ;
...
~
85.13i
5.387
4.157
2.930
1.755
0.715
-
-
I
0.755
1.036
1.573
2.874
0
0.957
1.132
1.3i7
1.776
0
- 1.282
- 1.533
-
I
I
I
1.902
- 2.560
- g.546
-
35.0881 -
4.305
4.937
5.930
6.676
7.233
7.622
7.856
7.928
7.806
!-
9.671
9.103
8.338
7.425
6.398
50.000
7.4ti5
54.974 f - 6.913
59.953 j - 6.196
U4.937 : - 5.365
69.927
5.290
4.133
2.967
1.835
0.805
I
0
1
I
I
- 4.454
74.923 : 79.920 84.93i 89.95·1 94.977 100 .000 .
_ _ _· " _ " ' ·_ _ .W __ ". _ _ _ _ '
-
4.574
7.054
9.549
14.568
19.611
0
-
0
1.233
1.520
1.965
2.812
~
0
0.245
0.464
0.927
2.126
~
-
Upper surface
437
3.500
2.Ml
1..621
0.801
0.173
0
. . . . . . _ _ _ - . . . , _... _
L.E. radius: 1.96
Slope of radius through L.E.: 0.084
THEORY OF WING SECTIONS
438
NACA 65a-418
(Stations and ordinates given in
per cent of airfoil chord)
Lower surface
Upper surface
Station
Ordinate Station
Ordinate
NACA 65,-418
a == 0.5
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Station
0
0.278
0.503
0.973
2.181
0
1.418
1.729
2.209
3.104
0
0.722
0.997
1.527
2.819
4.639
7.123
9.619
14.636
19.671
4.481
5.566
6.478
7.942
9.061
5.361
7.877
10.381
15.364
20.329
-
3.217
3.870
4.410
5.250
5.877
24.716
29.768
34.825
39.884
44.943
9.914
10.536
10.944.__
11.140
11.091
25.284
30.232
35.175
40.116
45.057
-
6.334
6.648
6.824
6.856
6.711
50.000
55.051
60.094
65.126
70.146
10.774
10.198
9.408
8.454
7.368
50.000
M.949
59.906
64.874
69.854
-
6.362,
5.818
5.124
4.334
3.480
75.154
80.147
85.127
90.092
95.046
6.183
4.927
3.638
2.3liO
1.120
74.846
79.853
84.873
89.908
94.954
-
2.803
1.743
0.946
0.282
0.144
100.000
J~.E.
0
100.~
__
0
- 1.218
- 1.449
-1.781
- 2.360
0
radius: 1.96
Slope of radius through L.E.: 0.168
Lower surface
Ordinate Station
Ordinate
0
0.197
0.411
0.868
2.057
0
1.440
1.766
2.271
3.233
0
0.803
1.089
1.632
2.943
-
0
1.164
1.378
1.683
2.197
4.493
6.966
9.459
14.481
19.533
4.715
5.891
6.882
8.482
9.709
5.507
8.034
10.5041
15.519
20.467
-
2.951
3.515
3.978
4.690
5.213
24.604
29.691
34.789
39.899
45.022
10.643
11.325
11.770
11.970
11.897
25.396
30.309
35.211
40.101
44.978
-
5.595
5.853
5.998
6.026
5.905
50.182
55.313
60.364
65.372
70.347
11.506
10.788
9.820
8.674
7.397
49.818
54.687
59.636
64.628
69.653
-
S.626
5.216
4.696
4.094
3.433
75.298
80.232
85.159
90.089
95.030
6.038
4.636
3.247
1.930
0.777
74.702
79.768
84.841
89.911
94.970
100.000
0
- 1.331
- 0.702
- 0.201
_-_
100.000
-_...
- 2.734
- 2.024
0
... -
L.E. radius: 1.96
Slope of radius through Js; E.: 0.233
APPENDIX III
NACA 65.-618
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Station
Ordinate
Lower surface
Station
Ordinate
0
1.146
1.356
1.651
2.152
NACA 65,-618
a = 0.5
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Station
0
0.172
0.385
0.839
2.026
0
1.446
1.776
2.293
3.268
0
0.828
1.115
1.661
2.974
4.462
6.936
9.431
14.455
19.506
4.776
5.971
6.978
8.602
9.848
5.538
8.064
10.569
15.545
20.494
24.574
29.652
34.738
39.826
44.915
10.803
11.504
11.972
12.210
12.186
25.426
30.348
35.262
40.174
45.085
-
5.433
5.672
5.792
5.784
5.616
50.000
65.077
60.141
65.189
70.219
11.877
11.293
10.479
9.482
8.338
50.000
54.923
59.859
64.811
69.781
-
5.259
4.723
4.063
3.302
2.506
75.230
80.220
85.189
90.138
7.075
5.719
4.306
2.863
1.433
74.770
79.780
84.811
89.862
94.932
- 1.705
- 0.943
- 0.268
0.239
0.463
95..068
100.000
0
100.000
-
- 2.880
- 3.427
- 3.876
- 4.564
- 5.072
0
L.E. radius: 1.96
Slope of radius through L.E.: 0.253
439
Lower surface
Ordinate Station
Ordinate
II 0
0
0.941 . - 1.055
1.244 - 1.239
1.811 - 1.493
3.154 - 1.895
0
0.059
0.256
0.689
1.846
0
1.469
1.821
2.375
3.449
4.248
6.706
9.194
14.225
19.301
5.115
6.448
7.575
9.404
10.815
5.752
8.294
10.806
15.775
20.699
24.407
29.537
11.893
12.687
13.209
13.456
13.395
25.593 - 4.321
30.463 . - 4.. 479
35.3161 - 4.551
40.151 - 4.540
44.966 - 4.407
12.974
12.173
11.090
9.806
8.374
49.727
54.532
59.454
64.443
69.481
34.684
39.849
45.034
50.273
55.468
60..546
65.557
70.519
75..445
so.347
85.239
90.133
95. 046 1
100.000 I
i
!
I
I
- 2..469
- 2.884
- 3.219
- 3.716
- 4.071
- 4.154
- 3.815
- 3.404
- 2.936
- 2.428
6.851 i 74.555 . - 1.895
5.279 I 79.653 t - 1.361
3.720
84.761 1 - 0.846
2..233
89.867 - 0.. 391
0.920
94.954 - 0.055
I
1
0
I lOO.()~:lO________~ _~ ___
L.E. radius: 1.96
Slope of radius through L. e.: 0.349
THEORY OF WING SECTIONS
440
NACA 65r 221
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Station
Ordinate
Lower surface
Station
Station
0
0.247
0.468
0.933
2.135
0
1.567
1.902
2.402
3.335
0
0.628
0.892
1.410
2.684
-
0
1.467
1.762
2.188
2.963
4.791
7.280
9.778
14.787
19.808
4.783
5.918
6.865
8.370
9.514
5.209
7.720
10.222
15.213
20.192
-
24.834
29.865
34.898
39.932
44.967
10.381
11.007
11.404
11.570
11.461
25.166
30.135
35.102
40.068
45.033
-
50.000
55.030
60.054
65.072
70.084
11.055
10.372
9.461
8.390
7.195
75.088
80.084
85.072
90.052
95.026
5.918
4.595
3.270
2.000
0.861
I
0
Upper surface
Ordinate
0
0.372
0.608
1.090
2.314
100.000
NACA 65r421
(Stations and ordinates given in
per cent of airfoil chord)
Ordinate
Lower surface
Station
Ordinate
0
1.601
1.956
2.493
3.505
0
0.753
1.032
1.567
2.865
-
0
1.401
1.676
2.065
2.761
4.151
5.070
5.831
7.024
7.922
4. 582 1 5.085
7.062
6.329
9.557
7.371
14.575
9.034
19.616
10.304
5.417
7.938
10.443
15.425
20.384
-
3.821
4.633
5.303
6.342
7.120
8.591
9.063
9.344
9.428
9.271
24.668
29.729
34.796
39.865
44.934
11.271
11.976
12.433
12.640
12.556
25.332
30.271
35.204
40.135
45.066
-
7.691
8.088
8.313 ·
8.356
8.176
50.000 ~54.970 ~ 59.946 64.928 69.916 -
8.849
8.182
7.319
6.330
5.251
50.000
55.059
60.108
65.145
70.168
12.158
11.467
10.531
9.419
8.166
50.000
54.941
59.892
64.855
69.832
-
7.746
7.087
6.247
5.299
4.278
74.912
79.916
84.928
89.948
94.974
4.128
3.003
1.924
0.966
0.229
75.176
80.167
85.143
90.104
95.051
6.811
5.388
3.940
2.514
1.176
74.824
79.833
84.857
- 3.231
- 2.204
- 1.248
100.000
-
0
L.E. radius: 2.50
Slope of radius through L.E.: 0.084
I
100.000
0
89.896\- 0.446 1
94.949
100.000 I.
0.088 ,
0
L.E. radius: 2.50
Slope of radius through L.E.: 0.168
1
APPENDIX III
NACA65r421
a = 0.5
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Station
Lower surface
0
o
0.155
0.363
0.813
1.992
1.620
1.991
2.553
3.631
0.845
1.137
1.687
3.008
-
1.344
1.603
1.965
2.595
4.414
6.880
9.371
14.395
19.455
5.315
6.651
7.773
9.572
10.951
5.586
8.120
10.629
15.605
20.545
-
3.551
4.275
4.869
5.780
6.455
24.538
29.639
34.754
39.882
45.026
12.000
25.462
12.765
30.361
13.258 1 35.246
40.118
13.470
13.362
44.974
-
6.952
7.293
7.486
7.526
7.370
50.211
55.362
60.421
65.428
70.398
12.890
12.056
10.942
9.637
8.193
49.789
54.638
59.579
64.572
69.602
-
7.010
6.484
5.818
5.057
4.229
75.340
80.264
85.181
6.664
5.09i
3.550
74.660
79.736
84.819
89.900
94.966
-
3.360
2.485
1.634
0.867
0.257
95.034
100.000
NACA66-206
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Station
Lower surface
Ordinate Station
Ordinate
Ordinate Station Ordinate
o
90.IOO
441
I
2.095
0.833
0
1
1100.000:
0
0
L.E. radius: 2.50
Slope of radius through L.E.: 0.233
0
0.461
0.707
2.447
0
0.509
0.622
0.798
1.102
0
0.539
0.793
1.298
2.553
-
0
0.409
0.482
0.584
0.730
4.941
7.439
9.939
14.942
19.947
1.572
1.947
2.268
2.791
3.196
5.059
7.561
10.061
15.058
20.053
-
0.940
1.099
1.234
1.445
1.604
3.513
3.754
3.929
4.042
4.095
25.046
30.038
1.202
24.954
29.962
:H.971
39.981
44.990 1
I
I
50.000 t
- 1.723
- 1.810
35.029 - 1.869
40.019 f - 1.900
1.905
45.010
1-
4.088
55. 009 1 4.020
60.018 I 3.886
65.026 I 3.641
70.031 , 3.288
I
75.034
I
I
SOo0341
85.031
90.023 t
95.012 I
100.000
I
2.848
2.339
1.780
1.182
0.578
l)
50.000
54.991
59.982
64.974
69.969
, - 1.882
1.830
t - 1.744­
! - 1.581
t - 1.344
Ii
74.966
i - 1.058
84.969
89.977
I 94.988
- 0.434
- 0.148
0.054
I 7909661 - 0.747
!
1 100 .000
0
L.E. radius: 0.223
Slope of radius through L.E.: 0.084
THEORY OF WING SECTIONS
NACA 66-209
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Station
Ordinate
Lower surface
Station
Ordinate
Upper surface
Station
Lower surface
-
Ordinate Station Ordinate
0
0
0
0.442
0.686
1.179
2.420
0.735
0.892
1.135
1.552
0.558
0.814
1.321
2.580
-
0.635
0.752
0.921
1.180
0
0.436
0.679
1.171
2.412
0
0.806
0.980
1.245
1.699
4.912
7.409
14.912
19.921
2.194
2.705
3.141
3.850
4.396
5.088
7.591
10.092
15.088
20.079
-
1.562
1.857
2.107
2.504
2.804
4.902
7.399
9.898
14.903
19.912
2.401
2.958
3.432
4.202
4.796
588
2.
5.098
7.601
10.102
15.097
20.088
24.931
29.944
34.957
39.971
44.986
4.821
5.145
5.378
5.528
5.594
25.069
30.056
35.043
40.029
45.014
-
3.031
3.201
3.318
3.386
3.404
24.924
29.937
34.952
39.968
5.257
5.608
5.862
6.024
6.095
25.076
30.063
35.048
40.032
45.016
50.000
55.014
60.027
65.038
70.046
5.578
5.476
5.275
4.912
4.400
50.000
54.986
59.973
64.962
69.954
-
3.372
3.286
3.133
2.852
2.456
50.000
55.016
60.030
65.042
70.051
6.074
5.960
5.736
5.332
4.759
50.000
54.984
59.970
64.958
69.949
75.050
3.772
3.058
2.283
1.477
0.690
74.950
79.950
84.9&6
89.965
94.982
-
1.982
1.466
0.937
0.443
0.058
75.056
80.055
85.049
90.037
95.019
4.071
3.289
2.445
1.570
0.724
74.944
79.945
84.951
89.963
94.981
9.~
SO.05O
85.044
90.034
95.018
100.000
0
0
NACA 66-210
(Stations and ordinates given in
per cent of airfoil chord)
100.000
L.E. radius: 0.530
Slope of radius through L.E.:
0
0.084
44.984­
100.000
0
0
0.564
0.821
1.329
100.000
0
- 0.706
- 0.840
1-
- 1.031
1.327
- 1.769
- 2.110
- 2.389
- 2.856
- 3.204
1
- 310467!
- 3.664
- 3.802
- 3.882
=~:I
- 3.770
- 3.594 1
- 3.272
- 2.815 1
- 2.281
-
1.697
1.099
0.536
0.092
0
L.E. radius: 0.662
Slope of radius through L.E.: O.~
I
APPENDIX III
NACA~-212
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Station
Lower surface
Ordinate Station Ordinate
NACA66r215
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Station
0
0.424
0.666
1.156
2.395
0
0.953
1.154
1.462
1.991
0
0.576
0.834
1.344
2.605
-
0
0.853
1.014
1.248
1.619
0
0.406
0.646
1.134
2.370
4.883
7.379
9.878
14.883
19.894
2.809
3.459
4.011
4.905
5.596
5.117
7.621
10.122
15.117
20.106
-
2.177
2.611
2.977
3.559
4.004
4.855
7.349
90848
14.854
190868
24.908
29.925
34.943
39.962
44.981
6.132
6.539
6.833
7.018
7.095
25.092
30.075
35.057
40.038
45.019
-
50.000
&5.019
60.036
65.051
70.061
7.068
6.931
6.659
6.169
5.487
50.000
54-.981
59.964
64.949
69.939
75.066
80.065
85.057
90.043
95.022
4.661
3.739
2.755
1.750
0.189
74.934
79.935
84.943
89.957
94.978
100.000
0
100.000
443
Ordinate
Lower surface
Station Ordinate
0
1.168
1.409
1.778
2.417
0
0.594
0.854
1.366
2.630
3.413
4.202
4.872
5.957
6.790
5.145
7.651
10.152
15.146
20.132
4.342
4.595
4:.773
4.876
4.905
7.437
24.886
29.906 1 7.9Z1
34.929 ( 8.280
39.952 I 8.501
44.976
8.590
25.114
30.094
35.071
40.048
-
4.862
4.741
4.517
4.109
3.543
50.000
55.023
6O.0f5
65.063
70.075
8.553
8.378
8.030
7.402
6.547
50.000
54.977
59.955
-
2.871
2.147
1.409
0.716
0.157
75.081
80.079
85.070
90.052
95.026
5.526
4.393
3.202
2.005
0.881
0
L.E. radius: 0.952
Slo~ of radius through L.E.: 0.084
f
1
j
100.000
I
0
-
f -
2.781
3.354
3.838
4.611
5.198
-
5.647
5.983
6.220
6.359
6.400
69.925
-
6.347
6.188
5.888
5.342
4.603
74.919
79.921
84.930
89.948
94.974
-
3.736
2.801
1.856
0.971
0.249
45.024
64.937
100.000
-
0
1.068
1.269
1.564
2.045
0
L.E. radius: 1.435
Slope of radius through L.E.: 0.084
THEORY OF WING SECTIONS
444
NACA 66r415
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Ordinate Station Ordinate
Station
0
0.314
0.544
1.019
2.241
I
4.711
7.199
9.696
14.709
19.736
24.771
29.812
34.SS7
39.904
44.952
Lower surface
I
Upper surface
Station
Lower surface
Ordinate Station
Ordinate
0
1.206
1.467
1.873
2.592
0
0.686
0.956
1.481
2.759
-
0
1.006
1.187
1.445
1.848
0
0.389
0.628
1..115
2.. 346
0
1.368
1.636
2.054
2.828
0
0.611
0.872
1.385
2.654
3.718
4.617
5.381
6.624
7.581
5.289
7.801
10.304
15.291
20.264
-
2.454
2.921
3.313
3.. 932
4.397
4.827
7.320
9.818
14.825
19.841
4.002
4.933
5.724
7.004
7.982
5.173
7.680
10.182
15.175
20.159
-
3.370
4.085
4.690
5.658
6.390
8.329
8.897
9.309
9.571
9.685
25.229
30.188
-
4.749
5.009
5.189
5.287
5.305
24.863
29.887
34.914
39.942
44.971
8.742
9.31,'
9.731
9.989
10.093
25.137
30.113
35.086
40.058
45.. 029
-
6.952
7.373
7.671
7.847
7.903
50.000
55.. 028
60.054
65.075
70.()89
10.045
9.828
9.394
8.610
7.568
50.000
54.972
59..946
64.925
69.911
-
7.839
7.638
7.252
6.550
5.624
75.095
6.345
5.001
3.606
2.230
0.961
74.905
79.907
84.919
89.940
94.970
- 4.555
- 3.409
- 2.260 .
35.143
40.096
45.048
50.000
55.046
60.090
65.126
70.150
9.656
9.473
9.100
8.431
7.518
50.000
- 5.244
54.954
59.910
64..874
69.850
-
5.093
4.816
4.311
3.630
75. 162 1
80.159 I
85.139 t
90.104
95.. ~'i3
6.419
5.187
3.872
2.519
1.196
74.838
79.841
84.861
89.896
94.947
-
2.839
2.003
1.180
0.451
0.068
100.000
NAC.A 66.-218
(Stations and ordinates given in
per cent of airfoil chord)
0
100.000
0
LE. radius: 1.. 435
Slope of radius through L.E.: 0.168
80.093
85.. 081
90.060
95.030
100.. 000
0
100.000
0
- 1.268
-
1.496
- 1.840
- 2.. 456
i
96
- 1.1 1
- 0.329
0
L.E. radius: 1.955
Slope of radius through L.E.: 0.084
I
THEORY OF WING SECTIONS
446
NACA 67,1-215
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Station
Lower surface
NACA 747A315
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Ordiilate Station Ordinate
Station
Lower surface
-
Ordinate Station Ordinate
0
0.402
0.642
1.128
2.361
0
1.213
1.460
1.867
2.577
0
0.598
0.858
1.372
2.639
-
0
1.113
i.320
1.663
2.205
0
0.. 229
0.449
0.911
2.109
0
1.305
1.599
2..065
2..935
0.771
1.051
1.589
2.891
-
1.927
4.848
7.344
9.845
14.854
19.869
3.557
4.321
4.947
5.954
6.735
5.152
7.656
10.155
15.146
20.131
-
2.925
3.473
3.913
4.808
5.143
4.564
7.053
9.558
14.599
19.668
4.264
5.286
6.140
7.497
8.503
5.436
7.947
10.442
15.401
20.332
-
2.518
2.952
3.304 1
3.843
4.247
24.887
29.908
34.930
39.953
44.976
7.348
7.825
8.185
8.430
8.570
25.113
30.092
35.070
40.047.
45.024
-
5.558
5.881
6.125
6.288
6.380
24.758
29.867
35.001
40.200
45.375
9.242
9.731
9.982
9.962
9.572
25.242
30.133
34.999
39.800
44.625
-
4.546
4.773
4.926
5.020
5.040
50.000
55.024
60.047
65..068
70.086
8.600
8.516
8.302
7.935
7.373
50.000
54.976
59.953
64.932
69.914
-
6.394:
6.326
6.160
5.875
5.429
50.447
55.463
60.435
65.366
70.241
8.964
8.206
7.324
6.365
5.354
49.553
54.537
59.565
64.634
69.759
-
5.014 I
4.930 I
4.772 1
4.509
75.098
80.100
85.092
90.071
95.037
6.515
5.335
74.902
79.900
84 ..908
89.929
94.963
-
4.725
3..743
2.653
1.. 503
0.471
75.130
80.073
85.038
90.016
4.336
3.295
2.257
1.289
95.004:
0.481
74.870
79.927
84.962
89.984
94.996
100.000
3..999
2.537
1.103
0
100.000
0
100.000
0
0
I 100.000
-­
L.E. radius: 1.523
Slope of radius through L.E.: 0.084
0
- 1.031
- 1.207
- 1.473
- 4.110 I
- 3.502
- 2.743
- 1.91sl
- 1.097
- 0.405
0
- .._---,_.­
L.E. radius: 1.544
Slope of radius through L.E.: 0.232
445
APPENDIX III
NACA 66,-418
(Stations and ordinates given in
per cent of airfoil chord)
Upper surface
Station
0
1.205
1.412
1.719
2.256
0
0.372
0.610
1.095
2.323
0
1.570
1.869
2.342
3.226
0
0.628
0.890
1.405
2.677
-
3.042
3.651
4.163
4.977
5.589
4.800
7.291
9.788
14.797
19.815
4.580
5.653
6.565
8.039
9.170
5.200
7.709
10.212
15.203
20.185
- 6.053
- 6.399
24..840
4.306
5.347
6.231
7.669
8.773
9.f133
10.287
34.829 ! 10.759
I
885
11.059
39.
1
11.188
44.943
I
I
75.191
80.185
85.162
90.120
95.000
100.000 I
-
Ordinate Station Ordinate
-
4.656
7.140
9.636
14.651
19.683
I
Lower surface
Station
0
1.405
1.692
2.147
3.000
50.000
55.056
ftO.l07
65.149
70.178
Upper surface
Ordinate Station Ordinate
0
0.280
0.509
0.981
2.194
24.726
29.775
Lower surface
NACA66r221
(Stations and ordinates given in
per cent of airfoil chord)
11.148
10.923
10.464
9.639
8.539
7.238
5.794
4.276
2.744
1.275
0
0
~720
0.991
1.519
2.806
5.344
7.860
10.364:
15.349
20.317
40.11~ - 6.i;~ f
45.051 1- 6.80~
10.047
10..709
29.869
11.183
34.900
11..478
39.933
44.967. 11.595
50.000
54.944
59.893
M.SSI
69.822
50.000
55.032
60.063
65.087
70.103
25.274
30.225
35.171
- 6.6.39
-
I-
6.736
6.543
6.183
5.519
4.651
74.809 . - 3.658
79.8151 - 2.610
84.838 - 1.584
89.880 - O.6i6
94.940 - 0.011
100.000
0
L.E. radius: 1.955
Slope of radius through L.E.: 0.168
J
75.109
so. 106
85.092
90.067
95. 034 1
100.000 t
i
11.537
11.281
10.763
9.823
8.581
7.145
5.591
3.996
2.440
1.032
0
-
0
1.470
1.729
2.128
2.8M
- 3.948
- 4.805
- 5.531
6.693
- 7.578
I-
1-
25.160 f - 8.257
8.765
30.131
35.100 I - 9.123
40.06719.~
45..033 - 9.405
5O.0~
I-
9.331
54.968
! -
9.091
59.937l - 8.621
64.913
69.897
- 7.763
I - 6.637
1-
74.891
5.355
79.894 ; - 3.999
84.908
2.f>50
I-
89.933
i-
1.406
94.966 , - 0.400
lOO.ooo 1
0
L.E. radius: 2.550
Slope of radius through L.E.: 0.084
APPENDIX III
447
NACA 747A415
(Stations and ordinates given in
per cent of airfoil chord)
Lower surface
Upper surface
Station
Station Ordinate
Ordinate
0
0.183
0.398
0.852
2.041
0
1.318
1.622
2.106
3.016
0
0.817
1.102
1.648
2.959
4.487
6.972
9.476
14.521
19.598
4.411
5.488
6.390
7.827
8.897
5.513
8.028
10.524
15.479
20.402
- 3.501
- 3.845
24.698
29.818
34.964
40.176
45.364
9.687
10.216
10.497
10.499
10.121
25.302
30.182
35.036
39.824
44.636
-
50.447
55.474
60.454
65.393
70.273
9.516
8.753
7.859
6.878
5.838
49.553
54.526
59.546
64.607
69.727
- 4.462
4.381
4.235
3.992
- 3.622
75.164
80.107
4.783
3.692
2.592
1.546
0.639
85..066
90.037
95.015
ioo.ooo
0
f
t
-
0
0.994
1.160
1.406
1.822
- 2.349
- 2.730
- 3.038
4.095
4.286
4.411
4.485
4.~93
II
I-
74.836 79.893 84.934 89.963 94.985\100.000
3.053
2.344
1.578
0.838
0.247
0
L.E. radius: 1.544
Slope of radius through L.E.: 0.274
APPENDIX IV
AERODYNAMIC CHARACTERISTICS OF WING SECTIONS
C.aNTENTS
NACA Designation
0006 .
()()()9
.
.
•
.
.
Page
452
.
0016-34:
.....
0010-35
. . . . ..
0016-34: a == 0.8 (modified) eli = 0.2 . .
0012 . . . . . . . . . . . . . .
0012-64
0012-64 a
1408
.
== 0.8
1410
(modified) C'l, == 0.2 .
.
.
462
464
466
468
470
472
1412
2408
474
476
478
2410
2412
2415
2418
2421
454
456
458
460
480
482
.
.
484
2424
4412
486
488
4415
490
4418
4421
4424
23012
23015
23018
23021
23024
63-006
492
494
496
498
500
502
504
506
508
510
.
.
..
.
.
63-009 .
63-206 .
63-209 . .
63-210 .
631-012 . .
631-212
.
520
6&.-412
.
522
524
63r015..
63r215 . . .
63r415.
63z-615. .
512
.514
516
518
.
526
528
530
THEORY OF WING SECTIONS
450
Page
532
534
NACA Designation
63r018.
63r218.
63r418.
63,-618.
536
538
540
63.-021 . .
542
544
546
634-221 . .
6&-421.
63AOI0
63A210
64-108
64-110
64-206
64:-208
64:-209
548
.....
64-()(M). . • • • • . •
64-()()9 •
. .
. . .
..
.
.
.
..
558
560
562
liM
64-210 . .
64.~12. . . . . . . .
566
568
570
64t-112. .
Mt-212 . .
64.-412 . .
64r015.
64r215
.
64r415
.
64r018
.
572
574
576
578
580
582
584
586
588
590
592
64r218 . .
64r418.
64r618 . .
64r021 . .
64.c-221.
64r421.
MAOIO
594
596
li98
600
602
604
606
608
610
612
614
616
64A210
MA410
64 1A212 . .
64,A215 .
85-()()6.
. .....
. . . . .
65--009 .
66-206 . . .
65-209 .
65-210 . .
65-410 .
6&-012.
65t-212
65s-212 II == 0.6 .
~-412.
550
652
554
556
. . . . .
6Sr015. . . . . . . .
65r215. . . . . . . .
66r415. . . . . .
65.t-410 II :II: 0.5 • .
.
618
620
622
624
626
628
630
APPENDIX IV
NACA Designation
65r018
.
65,-218
.
65r418. . . . .
65r418 a - 0.5 .
65.-618. . . . .
65r618 a :=II 0.5 .
65.-021
.
65.-221
.
65r421
.
65r421 a - 0.5 .
66-006 . . .
66-009 . .
~206
•..
66-209 . .
66-210 . .
66r012.
66,-212.
66:-015.
602-215.
66r415.
66,-018. .
66,-218. .
66,.418.
66c-021 ..
66r221.
67,1-215 . .
747A315.
147A415.
451
Page
632
634
636
638
640
642
644
646
648
650
652
8M
656
6li8
660
662
664
666
668
670
672
674
676
678
680
682
684
686
THEORY OF WING SECTIONS
452
3. 6
,­
3. "!:II
S
2. 4­
2J)
I,..'"
a;
.,
~J~
I .6
'I.
1'\'5
,
1'4
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16
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16
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.i
't.;;
t
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rtt
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1,
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~
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8
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-8
0'
Section D!'19.
NACA 0006 Wing
0
8
/6
Dffack, «0, dsq .
SectiOI'
32
453
APPENDIX IV
.
~
0
~
,,
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I
,
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I
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lP" L&I
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1-.:
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8
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o 3.lh"'~~250
0
R
ao ~t~~250
90
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00
~ 6D
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3
-.
-.5
-1.6
-1.2
-.8
Q2(Jc
eil1vJa1'«l8plt flap defected ISO· :
,,6.0
• &0
S~" rouq/1ne$s
-.4
-1111
-
0
.4
St!!Cfb7 lift COtI'f';cient,
·c,
N ~C.A 0006 Win« Section (Continued)
.8
/.2
~~
~I-
1.6
THEORY OF WING SECTIONS
454
3.8
3.2
_....
......,
-
24
~.~
8f/
J
IJ
1.6
fJ
r­
~
.
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.-.5
-2.IJ.
-~
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-8
0
8
/6
Section ongIe of Q/~ . . deg
NACA 0009 Wing Section
24
455
APPENDIX IV
.IJ3,
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a 6.01 I
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c 9.0~
250
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24
APPE.VDIX IV
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APPENDIX IV
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THEORY OF WING SECTIONS
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APPENDIX IV
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8
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16
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THEORY OP WING SECTIONS
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24
32
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24
32
APPENDIX IV
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1.2
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APPENDIX IV
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THEORY OF WING SECTIONS
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THEORY OF WING SECTIOIVS
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THEORY OF WING SECTIONS
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APPENDIX IV
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APPENDIX IV
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APPENDIX IV
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THEORY OF WING SEOTIONS
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APPE.YDIX IV
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535
A.PPENDIX IV
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THEORY OF WING SECTIONS
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24
32
APPENDIX IV
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NACA
f);J~18
\Vlng Section (Continued)
.8
1.2
1.6
538
THEORY OF WING 8ECTION8
3.6
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D'
24
32
APPENDIX IV
539
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NACA 63,..618 Wing Section (Continued)
1.2
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THEORY OF WING SECTIONS
540
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28
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24
32
APPENDIX IV
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542
THEORY OF WING 8ECTIONS
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24
J2
APPENDIX IV
543
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rr
lP
THEORY OF WING 8ECTIONS
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24
32
APPENDIX IV
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63.-421 ~·in;; Section (Continued)
l2
1.6
546
THEORY OF WING SECTIONS
3.6
3.2
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NACA 63AOIO Wing Section
8
do, deg
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24
32
547
APPENDIX IV
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NAPA 63AOIO Wing Section (Continued)
1.2
1.6
THEORY OF WING SECTIONS
548
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NACA 63A210 Wing Seet10n
8
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ffl
deg
I
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24
32
549
APPENDIX IV
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(Continued)
THEORY OF WING SECTIONS
2.8
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NACA 64-006 Wing Section
24
..32
551
APPENDIX IV
,
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NACA 64-006 Wing Section (Continued)
.8
I.Z
1.6
552
THEORY OF WING SECTIONS
36
3.2
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NACA fl4-OO9 Wing Section
24
JZ
553
APPENDIX 11'
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1.6
THEORY OF WING SECTIONS
554
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APPENDIX IV
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APPENDIX IV
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APPENDIX IV
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APPENDIX IV
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APPENDIX IV
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APPENDIX IV
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32
APPENDIX IV
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APPENDIX IV
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24
APPENDIX IV
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THEORY OF WING SECTIONS
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APPENDIX IV
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NACA 64,-418 Wing Section
24
APPENDIX IV
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1.6
THEORY OF WING SECTIONS
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16
24
APPENDIX IV
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APPENDIX IV
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APPENDIX IV
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THEORY OF WING ljECTION8
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24
32
APPENDIX IV
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597
THEORY OF WING SECTIONS
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APPENDIX IV
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THEORY OF WING SECTIONS
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THEORY OF WING SECTIONS
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621
APPENDIX IV
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24
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623
APPENDIX IV
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APPENDIX IV
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THEORY OF WING SECTIONS
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0
8
Section angle of attack,
NACA 65!-415 Wing Section
«OJ deg
/6
24
32
APPENDIX IV
629
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t
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THEORY OF WING SECTIONS
630
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NACA 65z-415 Wing Section
a == 0.5
24
631
APPENDIX IV
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NACA 00:-415 'Ving Hect.j()n
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THEORY OF WING SECTIONS
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24
32
633
APPENDIX IV
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THEORY OF WING SECTIONS
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24
32
635
APPENDIX IV
2­
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THEORY OF WING SECTIONS
636
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NACA 653418 Wing Section
24
637
APPENDIX IV
.1)16
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.8
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1.6
THEORY OF WING SECTIONS
638
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a= 0.5
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24
32
APPENDIX IV
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NACA 651-418 'Ving Section
a = 0.5 (Continued)
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1.2
1.6
THEORY OF lVING SECTIONS
640
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NACA 653-618 Wing Section
24
32
641
APPENDIX IV
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NACA 65:r-618 \Ving Section (Continued)
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1.2
1.6
THEORY OF WING SECTIONS
642
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NACA 65a-G18 Wing Section
a = 0.5
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8
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24
32
643
APPENDIX IV
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1.2
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32
645
APPENDIX IV
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THEORY OF WING SECTIONS
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J2
APPENDIX IV
647
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APPENDIX IV
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THEORY OF WING SECTIONS
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32
APPENDIX IV
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THEORY OF WI.LVG SECTIONS
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APPEA·DIX IV'
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654
THEORY OF WING SECTIONS
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32
APPENDIX IV
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APPENDIX IV
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THEORY OF lVING SECTIONS
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24
659
APPENDIX IV
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NACA 66-209 WiOR Section (Continued)
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660
THEORY OF WING SECTIONS
8.6
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THEORY OF WING SECTIONS
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APPENDIX IV
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THEORY OF WING SECTIONS
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APPENDIX IV
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NACA 67,1-215 Wing ~ction (Continued)
1.2
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THEORY OF WING SECTIONS
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24
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APPENDIX IV
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1.2
1.6
THEORY OF WING SECTIONS
686
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~i ~ ~:.::' ;~: ~~~~~ ~ ~'I~~ ~;
-8
0
8
16
Section tmgle 0/ tJttGck, ~, deg
NACA 747A415 Wing Section
24
APPENDIX IV
687
INDEX
A
Ackeret, J., 268
Adiabatic law, 249
Aerodynamic center, experimental, 182
theoretical, 69
wing, 5, 19
Aerodynamic characteristics, experimen­
tal, 449
Airfoils (see Wing sections)
Allen, H. Julian, 65, 164, 194, 202, 236
Anderson, Raymond F., 9
Angle of attack, 4
effective, 22, 25
ideal, 70
induced, 20
wing, 8,11
Angle of zero lift, experimental, 128
theoretical, 66, 69
approximate methods, 72
variation with flap deflection, 192
wing, 16
Aspect ratio, 6
at high speeds, 296
B
Bamber, Millard J., 234
Bernoulli's equation, 35, 251
Birnbaum, Walter, f~
Blasius, H., 87, 96
Boshar, John, 20
Bound~layer, 80
compressibility effects, 109
concept, 81
crossftows, 28
ht.minar,85
Blasius solution, 87
characteristics, 85
separation, 83, 86, 93
skin friction, 87
thickness, 89, 92
transition, 105, 157
velocity distribution, 87
Boundary layer, momentum relation, 90
slip, 83
thickness, displacement, 89
momentum, 92
turbulent, 95
pipe flow, 95
separation, 83, 103
skin friction, 97, 174
thickness, 102
velocity distribution, 97, 100
Boundary-layer control, 231
c
Cahill, Jones F., 206, 221
Camber, 65
application to thickness forms, 75, 112
design of mean lines, 73
effect of, on aerodynamic center, 182
on angle of zero lift, 128
on lift curve, 132
on maximum lift, 136
on minimum drag, 148
on pitching moment, 179
on profile drag, 151, 178
induced, 28
shapes and theoreti~l characteristics,
382
Chaplygin, Sergei, 258
Chord, 113
mean aerodynamic, 21
Circular cylinder in uniform stream, 41,
49
with circulation, 44, 49
Circulation, 37
lift, 45
thin wing sections, 65
vortex, 43
Complex number, 47
Complex variable, 47
Compressibility effects, 247
boundary layer, 109
drag characteristics, 281
first order, 256
689
7'REORY OF lVING SECTIONS
690
Compressibility effects, flap effectiveness,
281
lift characteristics, 269
pitching-moment characteristics, 287
pressure coefficient, 257
wing characteristics, 268
Conformal mapping, 47, 49
Conformal transformations, 49
Continuity, 32, 38, 249
D
Design of wing sections, 62
Designation of wing sections, III
German designations of NACA sectioDS,118
KACA four-digit sections, 114
XACA five-digit sections, 116
X.ACA modified four- and five-digit
sections, 116
XACA L-series sections, 119
X A.CA 6-series sections, 120
K.-\CA. 7-series sections, 122
Doublet, 41, 49
Downwash, 9, 20
Drag characteristics, 148
calculation, 107
compressibility etIects, 281
minimum, 148
roughness effects, 157, 175
section induced, 25
variation with lift, 149
wing induced, 16, 27
E
Experimental techniques, 125
Flow around wing sections, 83
at high speeds, 261
Fluid, perfect, 32
Fullmer, Felicien F., 231
G
Garrick, I. E., 63, 258
Glauert, H., 9, 65, 67, 192
Glauert-Prandtl rule, 256
Goldstein, Sidney, 63
Gruschwitz, E., 103
H
Harris, Thomas A., 218, 227
High-lift devices, 188
I
Irrotational motion, 37
J
Jacobs, Eastman N., 194, 260
JQDes, Robert T., 296, 299
Jones edge-velocity correction, 11, 25
l~
Kaplan, Carl, 256, 258
Kaplan rule, 257
}{&nnaD integral relation, 90
}{arman-Tsien relation, 258
Keenan, Joseph H., 109
}\:night, Montgomery, 234
Koster, H., 230
Krueger, W., 230
Kutta-Joukowsky condition, 52
F
L
Families of wing sections (see Wing sec­
tions)
Fischel, Jack, 218
Flaps, 188
external airfoil, 215
leading-edge,7Zl
plain trailing-edge, 190
split trailing-edge, 197
slotted trailing-edge, 203
Laminar flow (see Boundary layer)
Lanchester-Prandtl wing theory, 7
Leading edge, 113
Lees, Lester, 109
Lemme, H. G., 230
Liepman, n. \V., 268
Lift characteristics, compressibility effects,
269
design lift, 70
691
INDEX
Lift characteristics, experimental, 128
ideal lift, 70
maximum lift, 134
theoretical, 53
wing, 3, 8
maximum, 19
Lift-curve slope, effective, 11
experimental, 129
theoretical, 53, 69, 256
wing, 11
Limiting speed, 254
Loads, chordwise, 53, 60, 73, 75
flaps, 236
plain trailing-edge, 192, 236
slotted trailing-edge, 215, 221
split trailing-edge, 202, 236
span\\?ise,9, 19, 25
Loftin, Laurence K., Jr., 157
Lowry, John G., 227
M
Mach line, 248
Mach number, 81, 248
critical, 259, 261
(See also Compressibility effects)
Mean lines, 382
(See also Camber)
Millikan, C. B., 93
Moment (see Pitching-moment charac­
teristics)
Motion, equations of, 32, 85
irrotational, 37
Munk, Max ~I., 23, 65, 72
N
Naiman, Irven, 56, 74
Navier-Stokes equations, 85
Keely, Robert II., 20
Neumann, Ernest P., 109
Xikuradse, J., 99
Normal shock, 254
Noyes, Richard W., 304
o
Ordinates of cambered wing sections,
406
p
Pankhurst, R. C., 72
Pinkerton, Robert M., 61
Pitching-moment characteristics, 179
compressibility effects, 287
theoretical, 66, 69
approximate, 72
wing, 4, 16, 27
Platt, Robert C., 215, 226
Pohlhausen, K., 93, 106
Prandtl, L., 9, 20, 85, 96
Prandtl-Glauert rule, 256
Pressure coefficient, 42
Pressure distribution, theoretical, 53
approximate, 75
empirically modified, 60
Propeller slipstream, effect on drag, 170
Purser, Paul E., 218
Q
Quinn, John H., Jr., 234, 236
R
References, 300
Regier, Arthur, 109
Reynolds number, 81
(See also Scale effect)
Reynolds, 0., 95
Rogallo, Francis l\{., 202
s
Sanders, Robert, 225
Scale effect, 81
aerodynamic center, 182
lift-curve shape, 133
lift-curve slope, 129
maximum lift, 137
minimum drag, 148
profile drag, 151
Schlichting, H., 105
Schrenk, Oskar, 233
Schubauer, G. n., 105
Sections (see \Ving sections)
Sherman, Albert, 20
Shock, normal, 254·
on wing sections, 263
Shortal, Joseph A., 227
THEORY OF WING SECTIONS
692
Sinks, 39
Sivells, James Co, 20
Skin friction (Bee Boundary layer)
Skramstad, H. x., 105
Slats, leading-edge, 225
Slots, 2:1.7
Sound, velocity of, 249
Source, 39, 49
Speed, limiting, 254
Squire, H. s., 101, 109
Stack, John, 272
Stagnation points, 45
Stream function, 36
Stream tube, compressible flow in, 249, 252
Streamlines, 36
Superposition, 39, 75
Surface condition, 143, 157
effect of, on drag, 157, 175
on lift, 143
on transition, 157
standard leading-edge roughness, 143
SWL-ep, 296
T
Taper ratio, 11
Tani, Itiro, 20
Techniques, experimental, 125
Temple, G., 258
Tetervin, Neal A., 102, 103
Theodorsen, Theodore, 54, 63, 65, 70, 109
Thickness distributions, 309
effect of, on aerodynamic center, 182
on angle of zero lift, 129
on lift-curve shape, 133
on lift-curve slope, 132
on maximum lift, 134
on minimum drag, 148
on pitching moment, 179
on profile drag, 148, 178
Thickness forms and theoretical data; 309
Thickness ratio, effect of, on aerodynamic
center, 182
on angle of zero lift, 129
on lift-curve shape, 133
on lift-curve slope, 53, 132
on maximum. lift, 134
on minimum drag, 148, 178
on pitching moment, 179
on profile drag, 151
(See also Wing sections)
Trailing edge, angle, 117
effect of, on aerodynamic center, 182
on lift-curve slope, 132
definition, 113
Transformation of circle in to wing sec­
tion, 50
Transient conditions, 133, 143
Tsien, H. S., 109, 258
Turbulent flow (Bee Boundary layer)
Twist, 11
u
firich, A., 105
U nconservative sections, 175
Uniform stream, 39, 49
v
Velocity, limiting, 254
Velocity potential, 38
Vibration, effect on drag, 173
Viscosity (see Boundary layer)
von Kdrmdn, To, 82, 90, 93, 109, 175, 258
Vortex, 42, 49
system on wings, 9
Vorticity, 37
w
Weick, Fred E., 225-227
Wenzinger, Carl J., 194, 202, 215
Wing sections, combinations of NACA
sections, 123
concept, 8
experimental characteristics, 124, 449
families, 111
N ACA four-digit, 113
NACA five-digit, 115
N ACA modified four- and five-digit,
116
NACA T-series, 118
NACA 6-scri~,s, 119
NACA 7-series, 122
ordinates, 406
unconservativc sections, 175
Wings, 1
aerodynamic center, 5, 19
angle of attack, 8, 11
angle of zero lift, 16
applicability of section data, 28
693
INDEX
Wings, aspect ratio, 6
drag, 3, 16, 27
induced, 8, 16, 27
forces, 2
lift, maximum, 19
lift-curve slope, 11
'lift distribution, 9, 25
pitching moment, 16, 27
Wragg, C. A., 215
y
Yarwood, J., 258
Young, A. D., 101, 109
z
Zalovick, John A., 168
\. '-v
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