Theory of Jets in Ideal Fluids

584 Pages·1965·16.463 MB·English

by  M. I. Gurevich (Auth.)

THEORY OF JETS
IN IDEAL FLUIDS
By M. I. GUREVICH
Translated from the Russian by
ROBERT L. STREET
STANFORD UNIVERSITY
KONSTANTIN ZAGUSTIN
UNIVERSIDAD CENTRAL DE VENEZUELA
1965
ACADEMIC PRESS New York and London
COPYRIGHT© 1965 BY ACADEMIC PRESS INC.
ALL RIGHTS RESERVED
NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM
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United Kingdom Edition
Published by
ACADEMIC PRESS INC. (LONDON) LTD.
BERKELEY SQUARE HOUSE, LONDON W. 1
Library of Congress Catalog Card Number: 65-27087
This book was originally published as:
Teoriya Strue IdeaVnoe Zhidkosti
Gosudarstvennoe IzdaleVstvo
Fiziko-Matematicheskoe Literatufy
Moscow, 1961
PRINTED IN THE UNITED STATES OF AMERICA
PREFACE
Jet theory is an important and extensively studied part
of hydrodynamics. Helmholtz and Kirchhoff were, in the 1850!s,
the first to formulate and solve jet problems. It was expected
that one of the results of the application of jet theory would
be an explanation of D'Alemberts' paradox, and the drag forces
on some simple bodies in an ideal fluid were computed using
Helmholtz's and Kirchhoff's results. However, the calculated
drag forces were considerably less than those measured experi­
mentally. As a result, the assumptions of jet theory were
criticized, and hydrodynamicists sought to create better models
of the flow around a body. On the other hand, jet theory did
accurately predict the contraction coefficients of streams
flowing from vessels.
In the following years, hydrodynamicists studied many
theories for the solution of jet problems. Meanwhile, the
speeds of ships, propellers, turbines, etc., increased steadily,
and hydroplanes appeared. With the increasing speed of objects
moving through the water it became clear that the results of
jet theory had previously been compared with experiments made
at relatively low velocities, where the basic conditions of
jet theory were, as a rule, not satisfied. This advent of
higher velocities brought regimes under study in which the
results of the jet theory coincided with the experimental
results, and the solutions of many problems of jet theory that
had previously seemed purely mathematical exercises now took
on practical importance.
v
PREFACE
In the present book the author has tried to give a sys­
tematic exposition of contemporary jet theory. The first
chapters give a relatively detailed exposition of classical
jet theory. It is assumed that the reader is somewhat familiar
with this theory, with basic hydrodynamics, and with the theory
of complex variables. The later chapters become more special­
ized, and many problems are perforce treated only super­
ficially. The very important part of jet theory dedicated to
the problem of the existence and uniqueness of solutions is
treated only briefly, and only superficial mention is made of
supersonic jets, etc
From another point of view, it was considered useful to
present the solutions to certain particular problems of jet
theory in those cases in which these solutions are relatively
simple and, particularly, for which numerical results are
obtained. This approach is required because of the complexity
of many aspects of jet theory and of the impossibility of making
short and simple presentations of these difficult areas. For
instance, to understand the well-known work of Leray it is, in
essence, necessary that one first study the extensive works of
Leray and Schauder on functional analysis.
In order to read the present book the reader need be
familiar only with the elements of complex variable theory,
except that in some problems reference is made to special
sections of mathematics (e.g., the theory of elliptic func­
tions). A basis for understanding the main part of this book
may be obtained from a study of Chapters I and II, and article
8 in Chapter III of Ref. [4] or Chapters I-V, VII, and article
1 in Chapter VIII and articles 1-3 of Chapter XI of Ref. [3].*
Vallentine, Applied Hydrodynamics, Chapts. 1, 2, 4, 5, and 6,
Butterworths, London, 1959 or Churchill, Complex Variables and
Applications, 2nd ed., McGraw-Hill Book Co., Inc., New York,
vi
PREFACE
The author hopes that the present book gives the general con­
cepts of jet theory and also serves as an entry to a more
extensive study of certain special problems.
The author takes this opportunity to express his gratitude
to L. I. Sedov for his support and for many stimulating dis­
cussions. The author sincerely thanks G. A. Dombrovskii for
his help in the writing of Section X.D and I. I. Moiseva,
A. I. Sekerzh-Zenkovich, S. V. Falkovich, and L. A. Epshtein
for their review of much of the book and their valuable com­
ments. Thanks are also due N. A. Slezkin for the large biblio­
graphy on jet theory which he put at the author's disposal.
With the same sincerity the author wants to thank all those
other persons who gave him reprints of their works, biblio­
graphic references, and other valuable help.
1960. Where appropriate, references to books more readily
available to the English-language reader have been added to
Russian references by the translators.
vii
TRANSLATORS' PREFACE
This book, Theory of Jets in Ideal Fluids, by M. I.
Gurevich, was first published in Moscow, Russia, in 1961,
although it has never been generally available to the English-
language reader. It is a comprehensive text on high-speed,
incompressible hydrodynamics with a brief but significant chap­
ter on compressible flow. The author1s definition of jet theory
is very broad, and he discusses a wide range of topics—e.g.,
"true" jets, hydrofoils, separated flow, and free-surface flow.
Professor Gurevich. is well-known in Russia and throughout the
world as a mathematician and hydrodynamicist; he has published
many papers on the theory of ideal fluid flow.
A particular strength of this book is its presentation of
complete analyses with clear exposition of principles, and many
examples, tabulated results, and comparisons of the theoretical
results with experimental data. The book is basically theo­
retical and makes extensive use of complex-function theory.
We believe that this book is both a good reference work of
lasting value and an excellent advanced theoretical-hydro­
mechanics textbook. In addition, the literature list is exten­
sive and current through 1960. The listed Russian works are,
in general, available in either well-known Russian journals or
in translated form. We have replaced original Russian refer­
ences with their translated versions when possible.
In undertaking this translation, we were materially assisted
by Professor Gurevich1s transmittal of three original copies of
the book and his personal list of errata. During the transla­
tion, our objective has been to present the translation in the
ix
TRANSLATORS' PREFACE
form it would have taken were English Professor Gurevichfs
native tongue. Thus, while the basic organization and point
of view have not been changed, the result is a free rather than
a literal translation. Where our personal experience and knowl­
edge permitted, we added English-languare references to aid the
reader; however, none of the references in the original text
were omitted.
For the reader seeking work accomplished in the field since
this book was originally published in 1961, we suggest referral
to the Journal of Fluid Mechanics, Cambridge University Press,
London or New York, and the Journal of Ship Research, Society
of Naval Architects and Marine Engineers, New York. Finally,
we took the liberty of adding an index, which was not present
in the Russian original.
We gratefully acknowledge the work of the following persons
whose efforts were essential to the successful completion of the
translation: Mrs. Robert Street, for editing and for typing
part of the draft; Mrs. Byrne Perry, for typing the remainder
of the draft; Mrs. Janet Gordon Berg, for preparing the draft
for final typing and for layout of the illustrations; Mrs.
Cathryn Adams for typing the final manuscript; and Professor
Byrne Perry, for his review and comments on the translation.
Finally, we acknowledge our deep appreciation to the U. S.
Office of Naval Research, Fluid Dynamics Branch, and its Head,
Mr. R. D. Cooper, for their support of our work through Con­
tract Nonr 225(71).
Stanford, California R. L. Street
June 1965 K. Zagustin
x
AUTHOR'S PREFACE TO THE ENGLISH EDITION
In the present book an attempt is made to set forth system­
atically the theory of jets in ideal fluids, an important area
in hydromechanics with practical and theoretical significance.
Jet theory has been developed in various countries, with signi­
ficant contributions by Russian scholars. It is probable that
the English-language reader will find references to publications
that were heretofore unavailable to him because of the language
barrier.
Some interesting developments on jet theory have appeared
since 1961, but it was not generally possible to refer to them
in the translation. However, the present book does provide a
sufficient basis for understanding the more recent publications.
A number of minor errors in the original Russian book have
been corrected in this English translation.
I wish sincerely to thank the translators for their efforts
in preparing the English edition of this book.
Moscow, 1965 M. I. Gurevich
xi
CHAPTER I. INTRODUCTION TO THE THEORY OF PLANE,
STEADY JET FLOWS
A. SOME INFORMATION ON KINEMATICS
This book is meant for a reader who is familiar with
the elements of hydrodynamics; however, for the reader's con­
venience, some basic reference material on the theory of the
plane, steady flow of an ideal, incompressible fluid is given
here.
Consider a plane, steady flow of an ideal, incompress­
ible fluid. It is said that the flow possesses a velocity
potential cp, if
V = grad cp , (l.l)
where V is the velocity vector. If we establish a fixed
system of Cartesian coordinates x,y in the plane of the
flow, then the continuity equation has the form [l_,2]
(1.2)
From this equation, we see that the velocity potential can be
considered [3,4] as the real part of the complex function
w(z) = cp + i^|r, where z = x + iy. The function w(z) is
called the characteristic function or the complex potential,
and its imaginary part \|r is called the stream function.
1
THEORY OF JETS IN IDEAL FLUIDS
The conjugate functions cp and \|r satisfy the Cauchy-Riemman
conditions [3]
(1.3)
which guarantee that, at a point z, the derivative dw/dz
is independent of the direction along which the differentia­
tion is performed. When the angle 6 is defined as the angle
between the velocity vector and the positive x-axis at some
point z, the components and v^ of the velocity vector
along the coordinate axes are
v = v cos 0
x
(1.4)
v = v sin 0
y
For motion along a streamline--i.e., in the direction
of the velocity vector--differential increments of the coor­
dinates are given by dx = cos 0 ds , dy = sin 0 ds , where
cp cp
ds is the differential arc distance measured perpendicular
cp
to equipotential lines. From Eq. (l.4) it follows that
dy = v cos 0 ds^ + v sin 0 ds^ ,
dy = -v sin 0 cos 0 ds^ + v cos 0 sin 0 ds^ ,
i.e. ,
dcp = v ds , dt = 0 . (1.5)
Thus, along every streamline, the stream function \|r is con­
stant, and cp increases in the direction of the flow.
2