Theoretical Aerodynamics

474 Pages·1958·17.12 MB·English

by  Milne-Thomson

T H E O R E T I C A L
AERODYNAMICS
BY
L. M. MILNE-THOMSON, C.B.E.
EMERITUSP ROFESSORO F APPLID MATHEMATICSU. NIVERSITOYF ARIZONA
EMERITUPS ROFESSOORF MATHEMATICSI N THE ROYALN AVALC OLLEGE
Pno~wonO F APPLIEDM ATHEMATICISN BROWN UNIVERSITY
PROFESSOR IN THE MATHEMATICRSE SEARCH CENTER AT THEU NIVERSITY OF WlSCONSIV
VlSlTlNC PROFpssol AT THE UNIVERSITIES OF ROMEQ, UEENSLAND. CALGARY. mACO
FOURTH EDITION
Revised and enlarged
DOVER PUBLICATIONS, INC.
NEW YORK
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Copyright @ 1958 by L. M. Milne-Thomson.
All rights reserved.
This Dover edition, first published in 1973, is an
unabridged ant1 unaltered republication of the
fourth edition (1966) of the work originally pub-
lished by Mamillan and Company Limited in 1958.
International Standard Book Nioii her: 0-186-61980-X
Library of Congress Catalog Card Number: 73-85109
Manufactured in the United States by Courier Corporation
61980x1 1
www.doverpub1ications.com
SYMBOLS
THEf ollowing list is intended to show the most frequent conventional meaning
with which certain symbols are used in this book.
The list is not exhaustive nor doea it preclude some symbols being used in
other semw (which are always dehed).
The numbers in bracketa indicate the section where the meaning is iht used
or explained.
A aerodynamic force (1.01)
A aspect ratio (1.13)
a, a' slope of the (CL,a ) graph for finite aspect-ratio (11.24) ; inter-
ference factors (13.4)
a, slope of the (CL,a ) graph for two-dimensional motion (7.13)
b span (1.1)
C chord (1.11, 1-12) ; speed of sound (1.5)
C cross-sectional area of a wind tunnel (14.4)
lift coefficient, CD drag coefficient, etc. (1.73)
CL
D drag (1.02)
g, g acceleration due to gravity and its magnitude (20.01, 2-5)
i square root of - 1 (3.4)
i, j, k unit vectors along the 5-, y-, z-axea (11.1)
J rate of advance coefficient (13.42)
K circulation (5.5)
L lift (1.02)
1 typical length (1.71) : Ze in Joukowski transformation (6.1)
Y pitching moment (1.73) ; Mach number (1-71)
n Unit normal vector (9.31)
P preasure (1-4) ; aerodynamic pressure (2.13) ; angular velocity
of rolling (19.2)
Q engine torque (13-1)
q, p air velocity and speed (1.21); angular velocity of pitching (19.0)
R Reynolds' number (14'1)
r position vector (1.21)
1
7 angular velocity of yawing (19.2)
S plan area of wings (1.13)
T absolute temperature (2-5) ; propeller thrust (13.1)
t time
xviii SYMBOLB
u, v, w components of air speed (3.11)
V, V velocity of aircraft, and aircraft speed (1.01, 1-71)
W weight of aircraft (1.01)
UI complex potential (3.7) ; velocity of downwash (11.21) ; wing ;
loading (18.31)
X, Y,2 components of aerodynamic force (5.4)
+
x iy, complex variable (3.4)
incidence, angle of attack (1-15)
-
absolute incidence (7.13) ; angle of side-slip (18.33) ; J(1 M,z)
(15.4)
angle between Axes I and I1 (7.14) ; gliding angle (1-02)
dihedral angle (19.6) ; ratio of specific heats (15-01)
circulation (11-2)
aerofoil characteristic (11.53)
angle of downwash (11.24) ; complex variable (17.12)
wind tunnel interference angle (14.4)
vorticity vector (9.3)
+
it7 in conformal mapping (3.6)
efficiency of a propeller (15-1)
strength of a vortex (4.11) ; propeller characteristic (13.42)
k h an gle (16.1) ; relative aircraft density (20.2)
kinematic viscosity (1.6)
ratio of length of circumference of a circle to its diameter
impulsive preseure (3.31)
air pressure at infinity (5.32)
air density (1.3)
summation (1.71)
aerofoil characteristic (11.51) ; propeller characteristic (13-42) ;
unit of time (20.21)
velocity potential (3.31)
stream function (3.1)
angular velocity of aircraft (20.01)
angular speed of propeller (13.1) ; force potential (2.11)
surface vorticity (9.6) ; angular velocity (20.1)
magnitude of vorticity (3.21) ; angular speed
a a a
+ +
i - j - k - (9-1)
ax ay a2
REFERENCES TO LITERATURE
The following abbreviations are used :
Proc. Camb. Phil. Soc. Proceeding of the Cambridge Philosophical Sociely
Proc. Roy. Soc. (A) Proceedings of the Royal Society (Series A)
Proc. Roy. Soc. Edin. (A) Proceedings of the Royal Society oflidinburgh (Series A)
R. and M. Reports and Memoranda of the Aeronauticad Research
CmmiUee
N.A.C.A. ReForfs of fhe hrational Advisory Committee for Aero-
nauhcs (U.S.A.)
Z.a.M.M. Zehchrij ,fur angewandte Mdhematik und M&nik
Milne-Thomaon Theure&ul Hydrodynamics. References are to the 4th edition.
alpha a A nu
beta P B xi
gaminn Y r omicron
delta 6 0
Pi
epsilon e l 3 rho
zeta sigma
eta r l H tau
theta 8 0 upsilon
iota L J Phi
I<
kappa K Chi
lambda X I 1
psi
n1u P M omega
PREFACX TO THE FIRST EDITION
THE airflow round an akcrnft is a phenomenon of high complexity. To
study it, in the present state of our knowledge, demands simplifying assump
tions. Theae must be largely based on experimental obeervation of what
actually happens ; that is one aepect of the practical side of aarodpamk
To make mathematical deductions and predictions belong to the theoretical
aide and it is the theoretical* side with which thie book is concerned.
The aim is therefore.to lay bare the assumptions, to bring them to explicit
statement so that the reader may be comiously aware of what is mwd,
and then to examine what 089 be deduced from the assumptions aa a first
approximation.
The treatment is based on my lecturea to junior members of the Royal
Corps of Naval Conetructors at the Royal Naval College during the past ten
Y-.
The mathematical equipment of the reader is preaumed not to extend
beyond the elementa of the differential and integral calculus. What further
is needed is mostly developed in the course of the exposition, which is thus
reasonably wlf-con&ed. It is therefore hoped that the book will provide a
solid introduction to the theory which is the indispensable basis of practical
applications.
Since the use of vectors, or in two-dimensions the complex variable, intro-
duces such notable simplifications of physical outlook and mathematical
technique, I have had no hesitation in using vector methods. On the other
hand the subject has been presentad in such fashion that the reader who prefers
Cartesian notations should encounter little difficulty in adapting the vedor
arguments to a carteaian presentation. Chapter XXI on vectors has been
added for the benefit of those with little or no previous acquaintance with
vector methods. This chapter may be read first, or just before Chapter IX,o r
merely used as a compendium for reference..
Apart from Chapters I and I1 which are of a preliminary general character,
and Chapter XXI on vectors, the work falls into four fairly well-defined p-.
Chapters 111 to VIII con& the theory of two-dimensional, chaptens IX to
XIV that of three-dimensional aarofoils, including propellers and wind tunnel
corrections. Chapters XV, XVI, XVII deal with the effect of the compres-
sibility of air in subaonic and supersonic flow. Chapters XVIII to XX are
concerned with the aircraft aa a whole.
The chaptere are divided into sections numbered in the decimal notation.
The equations are numbered in each section independently. Thus 7-14(3)
refers to equation (3) in section 7-14 which, as the integer before the decimal
point indiwtea, occurs in Chapter VII. Baokwsrd and forward referenma am
~~ ~
*TO "the aninetrnated and popular world" padied and betied am antonyme; a
palpably falea propition.
vi PREFAOE
freely used to aid the reader in following an argument or in comparing sirmlar
situations. Each diagram, of which there are 260, bears the number of the
section to which it belongs and may therefore be traced without delay or
exasperation.
About 300 exercises have been provided, collected into sets of Examples
at the ends of the chapters. Some of these are very easy, others quite =cult,
and many supplement the text. The majority were composed specially for
this book.
References to literature are given where they appear to me to be appropri-
ate or useful, but no attempt has been made to give systematic citations. I
have made absolutely no endeavour to settle or assign priority of discovery.
The coupling of a particular name with a theorem or method simply indicatea
an association in my own mind. The proper historical setting I must leave to
those who have the time and the taste for such research.
This book was intended to appear long since, but other preoccupations
during the war years prevented that. The delay has, however, allowed the
presentation of matter which has appeared in the interim, and has also given
me the fortunate opportunity of having the whole book read in manuscript by
my colleague Mr B. M. Brown, to whom I owe a great debt for his Criticism
and improvements. Another friend, Mr A. C. Stevenson, has likewise rendered
invaluable crervice by his diligent help in proof reading and by important
suggestions. To both these friends I wish to express my lively gratitude and
appreciation. I also take this opportunity of expressing my thanks to the
officials of the Glasgow University Press for the care and attention which they
have given to the typography, and for maintaining a standard of excellence
which could scarcely have been surpassed in the pre-war years.
L. M. MILNE-THOMSON
Royal Naval College
Greenwich
May 1947
PREFACE TO THE FOURTH EDITION
THE gratifying reception accorded to this work has encouraged me to strive
for improvements. Opportunity has been taken to make several corrections, to
revise certain passages and to carry out extensive rearrangements. Additional
matter has been introduced particularly in connection with supersonic flow.
Moreover a new circle theorem, here called the second circle theorem ”, to
‘I
deal with flow of constant vorticity is given for the h t ti me.
L. M. M.-T.
Mathematics Department
The University of Arizona
Tucson, Arizona
Ap’ll966
CONTENTS
1'.1.0.
SYMBOLS - xvii
REFERENCES TO LrrERATURE - xix
GREEK ALPHABET - xx
CHAPTER I
PRELIMINARY NOTIONS
1·01. Aerodynamio force 1
1-02. Lift and drag 2
1·1. Monoplane aircraft 3
1-11. Chord of a profile 4
H2. Chord of an aerofoil 5
1·13. Aspect ratio 5
H4. Camber 6
H5. Incidence 6
1·2. Fluids 7
1·21. Velocity 8
1·22. Streamlines and paths of partioles 9
1·23. Stream tubes and filaments 10
1·3. Density 11
1·4. Pressure 11
1·41. Thrust due to pre88ur6 12
1·5. The speed of sound 13
1·6. Maxwell's definition of viscosity 15
1·7. Physical dimensions - 16
1·71. Aerodynamio force; dimensional theory 17
1·72. Similar systems; scale effect 19
1·73. Coeffioients 19
1·8. The boundary layer 20
1·9. Approximations- 23
EXA.MPLES I 26
CHAPTER II
BERNOULLI'S THEOREM
l!·I. Bernoulli's theorem - 29
2·11. Incompressible fluid in the gravitational field 30
2·12. The constant in Bernoulli's theorem - 31
2·13. Aerodynamic pre88ur6 31
2·2. The Pitot tube - 32
2·3. The work done by air in expanding 33
viii CONTENTS
PAGII
2·31. Bernoulli's theorem for compressible flow 34
2·32. Application of Bernoulli's theorem to adiabatio expa.nsion • 34
2·4. The Venturi tube 36
2·41. Flow of air measured by the Venturi tube • 37
2·5. Standard atmosphere 37
EXAMPLES II 39
CHAPTER III
TWO-DIMENSIONAL MOTION
3·0. Motion in two dimensions - 42
3·1. Stream function - 42
3·11. Velocity derived from the stream function - 44
3·12. Rankine's theorem 44
3·13. The stream function of a uniform wind 45
3·14. Circular cylinder 46
3·15. The dividing streamline 47
3·2. Circulation - 47
3·21. Vorticity - 49
3·22. Motion of a fluid element 50
3·3. Irrotational motion 51
3·31. Velocity potential 52
3·3ll. Laplace's equation 53
3·32. Cyclic motion 54
3·4. Complex numbers 54
3·41. The argument 65
3·42. Differentiation - 57
3·43. Holomorphic functions 57
3·44. Conjugate funotions - 57
3·45. The functionJ(z) 58
z
3·47. The coordinates z and 59
3·5. Cauchy's integral theorem - 60
3·51. Singularities 60
3·52. Residues 61
3·53. Cauchy's residue theorem - 61
3·6. Conformal mapping - 62
3·7. Complex potential 63
3·71. The complex velocity 65
3·8. Application of conformal mapping 66
ExAln>LBs ill - 67
CHAPTER IV
RECTILINEAR VORTICES
'·0.
Two-dimensional vortioee 70
4·1. Circular vortex • 70