Table Of ContentProgress in
AERONAUTICAL
SCIENCES
Volume 3
Edited by
ANTONIO FERRI
Professor of Aerodynamics,
Polytechnic Institute of Brooklyn, U.S.A.
D. KÜCHEMANN
Royal Aircraft Establishment,
Farnborough, England
L. H. G. STERNE
Training Center for Experimental
Aerodynamics, Belgium
PERGAMON PRESS
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1962
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1962
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PREFACE
THE third volume in this series contains four articles dealing with
various topics mainly in the field of aerodynamics. It is certain that
the last word has yet to be written on any one of these topics and
certainly some of the information offered here, though not ephemeral,
will merit amplification in a few years' time. As is only to be expected
in a series of this kind, the individual articles are not interconnected
but report on fields where advances happen to have been made. As
such, they give perhaps some indication of the vigorous activity and
continuing growth of the aeronautical sciences. Certainly there would
appear to be no dearth of material for an annual volume in this series.
In fact, this third volume appears only a few months after the second
volume: what was originally intended to form one volume was split
into two for the greater convenience of the reader.
The first paper is a review of the aerodynamics of swept-winged air
craft and as such is a contribution to the solution of the problem of
present-day aerodynamic design which, as noted in Volume I, has
emerged as a consequence of developments which have gone beyond the
classical concepts of aircraft design. This paper is a first review of
part of the research work, which was a co-operative effort on a large
scale by British aircraft firms and research establishments, aimed at
the development of supersonic airliners. Not all the individual refer
ence papers concerned in the study have yet been published and it
is hoped to report on other aspects of this research work in later volumes
in this series.
The second paper of this volume is concerned with ducted propellers
which in recent years have attracted considerable interest owing to
their potential application to a variety of aircraft types, notably the
important category of those able to take off vertically or requiring a
short take-off run. The paper gives an exhaustive survey of the widely
dispersed literature in a way which does not ignore practical appli
cations.
The third paper, on hypersonic facilities, provides the reader with
a balanced review of the, to some, bewildering multiplicity of means
for experimenting with hypersonic flows. Even though development
still proceeds at a rapid pace, the overall outlines and the usefulness
of particular facilities for particular purposes begin to become more
firmly established and a survey is, therefore, both opportune and wel
come.
vii
vin PREFACE
The last paper gives an account of gust research—that fascinating
field of work where meteorology, fluid dynamics and aircraft dynamics
meet. The subject will be seen to be of great interest and prolific in
its many aspects, one of which is, of course, the comfort and safety of
ourselves as airline passengers.
SOME AERODYNAMIC PRINCIPLES FOR
THE DESIGN OF SWEPT WINGS
J. A. BAGLEY
Aerodynamics Department, Royal Aircraft Establishment, Farnborough
Contents
LIST OF SYMBOLS 2
1. INTRODUCTION 5
2. THE CLASSICAL AIRCRAFT AND THE EXTENSION TO SWEPT WINGS 10
3. THE SHEARED WING OF INFINITE SPAN 14
4. SWEPT WINGS OF FINITE ASPECT RATIO 22
5. THE DESIGN OF SWEPT WINGS 30
5.1. Aerodynamic features of the flow over swept wings 30
5.2. Design principles 38
6. CALCULATION METHODS AVAILABLE FOR DESIGN PURPOSES 43
6.1. Methods of designing the basic wing section 45
6.1.1. Incompressible flow 45
6.1.2. Compressible flow 48
6.2. Design methods for the finite wing, at subsonic speeds 51
6.2.1. Planform, camber and twist 51
6.2.2. Thickness 57
6.2.3. Compressible flow 61
6.2.4. Thickness and lift combined 62
6.3. Design methods for the wing-fuselage combinations at subsonic
speeds 63
6.4. Design methods for the wing-fuselage combinations at supersonic
speeds 67
6.4.1. Thickness problems 68
6.4.2. Lifting configurations 75
ACKNOWLEDGEMENTS 79
LIST OF REFERENCES 80
1
SOME AERODYNAMIC PRINCIPLES FOR
THE DESIGN OF SWEPT WINGS
J. A. BAGLEY
Aerodynamics Department, Royal Aircraft Establishment, Farnborough
Summary—The design of swept wings has now reached the stage
where a coherent set of aerodynamic principles has emerged. The pur
pose of this paper is to summarize these principles, and to indicate
methods of designing wings in accordance with them.
It is important to design wings so that the type of flow obtained in
practice is the same as that assumed in the design theory, and so that
it is a flow which is usable—i.e. which can be predicted and controlled.
It is shown that these requirements lead to the concept of a sub-critical
flow which can be obtained on certain swept wings. These are restricted
to a fairly narrow band of sweep angles, depending on the design Mach
number, and the aspect ratio and thickness of the wings are correspond
ingly limited.
Practical design methods are discussed in order to illustrate the
physical principles used in design, but a critical comparison of different
calculation methods is not attempted.
List of Principal Symbols
a Local speed of sound
a(y) = CL(y)l*e(y) Sectional lift slope on swept wing
a Speed of sound in free stream (except in Eq. 59A)
0
A Aspect ratio of wing
A a Aspect ratio of analogous wing
A\, A Constants in Eqs. 61 and 62
2
B = {l-Jfo2[(l-Opi)cos2 -Opi(l-|/i|)sin2 ]}*
Çt 9t
c(y) Wing chord
c Geometric mean chord
CD Overall drag coefficient
ÖD(y) Sectional drag coefficient
@DB(y) Sectional drag coefficient due to boundary layer effects
CD-F Drag coefficient due to skin friction (complete wing)
C = Do/qS
D0
^Di(y) Sectional drag coefficient due to thickness
ODV Vortex drag coefficient (complete wing, or sectional)
CW Wave drag coefficient due to lift
GL Overall lift coefficient
CL(2/) Sectional lift coefficient
CLCT Cruising lift coefficient
Value of CL for (L/D)
m
C Overall pitching moment of wing
m
2
Aerodynamic Principles for the Design of Swept Wings 3
Cp Pressure coefficient
Cp* Critical pressure coefficient, denned by Eq. (17)
Cpi Pressure coefficient at zero lift in incompressible flow
(Eq. 47)
(7 T Pressure coefficient at trailing edge
P
C u(%) Pressure coefficient on upper surface of wing
V
Cpi(x) Pressure coefficient due to wing thickness
Op2(#) Pressure coefficient due to wing lift
D Total drag of wing or wing-fuselage combination
DL Drag due to lift of wing or combination
Do = D —D . Drag independent of the lift distribution
L
DT Wave drag due to volume of configuration
Dw Wave drag due to lift of configuration
1 + sin φ
/(φ) =log- r—
1 — sin φ
Q(B) See Eq. (83)
Η(θ) See Eq. (84)
K = π^4((7z)v+Crz)w)/CrL2. Drag due to lift factor
K\(y) Spanwise interpolation factor used in Eqs. (68) and (76)
Ky = TTACDYJGL2' Vortex drag factor
K = TT.4Z2CW/2J32$2CL2. Wave drag due to lift factor
w
I Overall length of configuration, except in Eqs. (83) and
(84)
l(x), l(x, y) Local load coefficient, — Δ(7
Ρ
L Overall lift on wing or combination
L(x) Lift on section of combination intercepted by plane
x = constant
L(x, Θ) Lift on section of combination intercepted by oblique
Mach plane (see Eq. (84))
(L/D)m Maximum lift-drag ratio
(L/D)cr Cruising lift^-drag ratio
M = V/a Local Mach number
Mo = Vo/ao Free-stream Mach number
Merit Critical Mach number, i.e. value of Mo when M = 1 first
on body
Mes Sheaved wing critical Mach number, i.e. value of M when
0
Eq. (15) is first satisfied
n Chordwise loading parameter, defined by Eqs. (57), (57A)
n = n(y , <p)
a & a
no Chordwise loading parameter, defined by Eq. (57)
q = %poVo2 Free-stream kinetic pressure
q(x) Source distribution on aerofoil
r Radial co-ordinate
r(x) Radius of fuselage
R Mean radius of fuselage
s Semi-span of wing (except in Eq. (64))
S Area of wing planform
S(x) Cross-sectional area of combination intercepted by plane
x = constant
S(x, Θ) Area of combination intercepted by oblique Mach plane
(see Eq. (83))
4 J. A. BAGLEY
SF(X) Cross-sectional area of fuselage
S(i)(x),SW(x),SM(x) Defined in Eqs. (19), (37), (38), and (41)
SW(x) = dz/dx
t
SW{x) = dzs/dx
t Wing maximum thickness
Vx, Vy> v, Vf, ν Velocity increments in x-, y-, z-, ξ- and ^-directions
z η
vx vy> Az Velocity increments in incompressible flow on analogous
9
wing
vq, v Velocity increments due to source distributions
X zq
νγ, νγ Velocity increments due to vortex distributions
χ ζ
V Local flow velocity
Vo Free-stream velocity; flight velocity of aeroplane
Vx* Vy Vzy Vf, V Components of V parallel to #-, y-, z-, f-, and η-directions
t v
Vo Vzo Components of Vo parallel to x- and z-directions
x f
TTp, Wi Final and initial weights of aeroplane
x Stream wise co-ordinate
xn End of pressure "plateau" on rooftop aerofoil section
y Spanwise co-ordinate
y & ~ ßy Spanwise co-ordinate on analogous wing
z Vertical co-ordinate
z(x) Aerofoil surface
Zs{x), z(x, y) Wing camber line or surface
s
zt{x)y zt{x, y) Aerofoil or wing thickness distribution.
a Wing incidence
o.(y) Sectional incidence
<£e(y) Sectional effective incidence, defined by Eq. (50)
oii(y) Sectional induced incidence, defined as ωαι(2/)
0
αΐο(2/) Downwash at infinity, defined by Eq. (49)
ß = {\l-M *\}i
0
ßx = {l-ifo2(co82 ~(7pi)P
9
ß = {l-lfo2cos2<p}*
2
y Ratio of specific heats, (y = 1-4 in numerical examples)
y{x)y γ(χ, y) Vortex distributions representing lift distribution on aero
foil or wing
Δ(7 Difference between pressure coefficients on upper and
Ρ
lower surfaces
Δα(ι/) Effective change of sectional incidence due to boundary
layer effects
η = x sin φ -f y cos φ Co-ordinate parallel to wing sweep lines
(except in Eq. (82))
Θ Inclination of oblique cutting plane to stream direction (in
Eqs. (83), (84))
Θ Defined by Eq. (71)
K(y) Spanwise interpolation factor in Eqs. (68) and (76)
\(y) Spanwise interpolation factor for load distribution defined
by Eq. (58)
K = A(Va)
M (y) Spanwise interpolation factor for velocity increments due
to thickness—see Eq. (67)
Aerodynamic Principles for the Design of Swept Wings 5
ξ = x cos φ — y sin φ Co-ordinate normal to wing sweep lines
p Air density (except in Eq. (79))
po Free stream air density
a Defined by Eq. (63)
r(y) Spanwise interpolation factor in Eqs. (68) and (76)
Ta = T(y )
φ Wing sweep angle
9a Sweep of analogous wing
<Pc/ a Sweep of mid-chord line
9t Sweep of thickness line
φο Leading-edge sweep
φι Trailing-edge sweep
Φ Velocity potential
Φ = ΒΦ/dx, Φ = δΦ/δι/, etc.
ζ ν
ω Downwash factor, see Section 6.2.1
1. Introduction
AEBOPLANES with swept wings have been designed to meet various
requirements since the earliest days of powered flight, but during the
last twenty years swept wings have been employed primarily for high
speed aircraft. During this period, the problem of designing wings to
have high aerodynamic efficiency in the cruising condition has been
studied, initially for subsonic speeds and subsequently for supersonic
speeds. At the present stage, a coherent set of aerodynamic principles
has emerged from this work which seem to be generally accepted, and
the time appears to be opportune for a survey of this field. The purpose
of this paper is to summarize these aerodynamic principles, and to
bring into perspective methods of designing wings in accordance with
these ideas.
The basic premise of the design approach described here is that to
attain high aerodynamic efficiency it is necessary to aim for a flow
pattern which will be stable and predictable, and will have a lowjdrag.
A type of flow which meets these requirements, and can apparently be
obtained on sweptback wings of the kind considered here, is that found
on a two-dimensional aerofoil at low speeds. This type of flow is utilized
on the '"classical" aircraft, and its essential features have been described
by Maskell1. Flow separation takes place only along the trailing edge, so
that extensive regions of separated flow on the aerofoil surface itself
are avoided, and viscous effects are confined to a thin boundary layer.
Furthermore, the flow is smooth everywhere and there are no discon
tinuities such as shock waves.
Maskell1 and Küchemann2 have shown how the prescription of a
particular type of flow pattern is in itself a powerful factor in defining
the shape of a practical aeroplane. This approach is adopted in Section
2 of this paper in order to define the class of wings to be considered.
6 J. A. BAGLEY
The requirement for high aerodynamic efficiency in the cruising condi
tion is interpreted as a requirement for a specified lift-drag ratio, and
it is shown that practical aeroplanes to meet this requirement fall into
a fairly limited range of geometries. Consequently, the later sections
are concerned mainly with the drag of wings and wing-fuselage com
binations which fall into this limited family.
The drag forces on any flying body can be separated into those due
to skin friction and those arising from the summation of the pressures
on the surface of the body. Methods of estimating these normal-pressure
forces, or of designing body shapes on which they will be small, can be
divided into two categories. One group depends on the calculation of the
surface pressure distribution itself, which can then be integrated to
give the pressure drag; this may be categorized as the "near field"
approach to design. The second group utilizes what may be called the
"far field" approach, where the drag is identified with the transport of
momentum through a large control surface surrounding the body. The
momentum transport can then be related to the distributions of volume
and lift on the body, without considering the details of the surface pres
sure distribution.
By considering as the control surface enclosing the configuration a
cylinder of large radius with its axis parallel to the stream, it can be
shown that at subsonic speeds the only transport of momentum is
through the downstream end of the cylinder (see, for example, Heaslet
and Lomax, Section D.14 of Ref. 3). This surface can be taken infinitely
far downstream of the body ; it then becomes the so-called Trefftz plane,
and the drag corresponding to the momentum transport through this
surface in in viscid flow is the vortex drag of the configuration associated
with the lift distribution. (In a real viscous flow, there may be an addi
tional drag term, due to the influence of the boundary layer. See page
32). The "far field" approach thus leads, at subsonic speeds, to the well-
known relations between the spanwise distribution of lift, the down-
wash behind the wing, and the vortex drag. In particular, if the trailing
vortex sheet is assumed to be planar, it leads to the familiar theorem
which states that the vortex drag has its minimum value, for given total
lift and span, when the spanwise load distribution is elliptical.
At supersonic flight speeds, there is an additional transport of mo
mentum through the curved surfaces of the cylinder, which gives rise
to the so-called "wave drag" of the configuration. It is manifested
physically as a change in the flow pattern, usually giving rise to shock
waves near the nose and tail of the configuration, and frequently at
other places also. The wave drag might be evaluated, in principle, by
considering the energy losses through these shocks, or by integration
of the pressure distribution over the whole surface of the configuration,
