Theory of Jets in Ideal Fluids

416 Pages·1966·18.359 MB·English

by  M. I. Gurevich

THE THEORY OF
JETS IN
AN IDEAL FLUID
by
M.I.GUREVICH
TRANSLATED BY
R.E. HUNT
TRANSLATION EDITED BY
E.E.JONES AND G.POWER
Department of Mathematics, University of Nottingham
PERGAMON PRESS
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Copyright © 1966
Pergamon Press Ltd.
This is a translation of the original Russian book
TeopnH CTpyii HfleaJibHoft >KHBKOCTH
published by Fizmatlit, Moscow 1961
Library of Congress Catalog Card No. 66-17247
2692/66
FOREWORD
THE theory of jets is an important and highly developed branch of hydro-
mechanics. The first problems in the theory were posed and solved by
Helmholtz and Kirchhoff in the middle of the last century. The most
important outcome of this theory was that the paradox of the absence of
resistance to a uniformly moving body in an ideal fluid could perhaps be
partially explained. Indeed, by using the theory of jets, success was achieved
in calculating the resistance of certain simple bodies in an ideal fluid.
However, the amount of resistance appeared considerably less than that
shown in experiments, and as a result the theory of jets was criticized.
Hydrodynamics tried to provide further patterns of flow about a body
(e.g. von Karman vortex street), but the theory of jets continued to develop
as a purely mathematical branch of hydromechanics, giving, nevertheless,
good results in the calculation of compression coefficients of jets flowing from
vessels.
During the years that hydromechanics has been developing, the speeds of
ships, propellers, turbines, etc., have been increasing, and hydroplanes and
seaplanes have come into existence. As speeds increased in the field of
marine engineering, it became clear that the results given earlier by the
theory of jets were relevant only to experiments at relatively low speeds.
These speed increases led to the study of conditions where the results of the
theory agreed with experiments, and the solving of numerous problems,
which earlier had seemed to be purely mathematical exercises, now acquired
a practical importance.
In this book the author has attempted to give a systematic account of the
modern theory of jets. The earlier chapters give a comparatively detailed
exposition of classical problems in the theory, some knowledge of the
fundamentals of hydromechanics and the theory of functions of a complex
variable being assumed. The following chapters gradually acquire a more
specialized character, many works being mentioned only briefly. During the
course of the survey, mention will be made of the important branch of the
theory concerning existence and uniqueness of solutions; only passing
mention will be made of supersonic jets, etc.
Such an approach is necessitated by the difficulty and extent of many
important works and the impossibility of giving any short and simple
account of them.
On the other hand, it has been considered useful to produce solutions to
general problems in the theory where these solutions are comparatively
simple and, especially, where they are determined numerically.
vii
viii THEORY OF JETS IN AN IDEAL FLUID
The author hopes that this book will give a general idea of the theory of
jets and as such serve as an introduction to further study of particular
problems.
The author would like to take this opportunity of expressing his sincere
thanks to Leonid Ivanovitch Sedov for supporting the idea of this book and
for his great scientific influence. He also wishes to thank G. A. Dombrovskii
for his help in writing § 45, N. N. Moiseev, Ya. I. Sekerzh-Zen'kovitch,
S. V. Fal'kovitch and L. A. Epstein for revising particular chapters and
paragraphs of the book and for their valuable observations, also N. A. Slez-
kin for the extensive bibliography on the theory of jets which he put at the
author's disposal. The author would like to thank just as sincerely all those
who sent him copies of their works, bibliographical information and contri-
buted other help of a similar nature.
CHAPTER I
INTRODUCTION TO THE THEORY OF JETS
IN PLANE STEADY FLOWS
§ 1. ON THE KINEMATICS OF PLANE FLOWS
Although this book is primarily intended for the reader acquainted with
the elements of hydrodynamics, a short account of the theory of plane,
steady flow of an ideal, incompressible fluid is included in the present
paragraph for the sake of convenience.
Let there be a plane steady flow in an ideal, incompressible fluid. It is said
that the flow possesses a velocity potential φ, if
F = gradç?, (1.1)
where V is the velocity vector. We take in the plane of flow the fixed system
of Cartesian coordinates x, y, and referred to these coordinates, the continu-
ity equation takes the form [43], [51]
d\ ÖV _ x
0 Q 2
+ (L2)
ä^" ä^-°·
Proceeding from this equation, we may calculate the velocity poten-
tial ([67], [49]) as the real part of the function of a complex variable
w(z) = φ + iip, where z = x + iy. The function w(z) is called the character-
istic function or complex potential. The imaginary part ψ of the character-
istic function is called the stream function.
The conjugate functions φ and ψ satisfy the Cauchy-Riemann conditions
[67]:
dçp dip dçp dip
(1.3)
dx dy dy dx'
which ensure that the value of dwjdz at a point z is independent of the
direction of differentiation.
Let 0 be the angle that the velocity vector at any point of the flow-plane z
makes with the #-axis. Then the axial components v v of the velocity
X9 y
vector are given by
dtp dtp
v = v cos Θ = -z- = -t-,
x
dx dy (1.4)
d<p dxp
dy
dx
la TJ 1
2 THEORY OF JETS IN AN IDEAL FLUID
Now consider a displacement in the streamline direction, i.e. in the
direction of the velocity vector. Here, the infinitesimal increases of the
coordinates are dx — cos Θ ds, dy = sin Θ dsç,, where ds is the curve
v 9
differential. From (1.4) it follows that
d<P = ττ-dx + rr-dy = v cos2 Θ ds + v sin2 Θ ds ,
dx dy ψ œ ψ œ
dw = ττ- dx + rr- dy = — v sin Θ cos 0 efo^ + υ cos 0 sin Θ ds ,
r (to dy ψ ψ œ
that is
d(p = v άδ , dip = 0. (1.5)
φ
Thus, along each streamline the stream function ψ has a constant value,
whereas φ increases in the direction of the velocity. Let us now see how ψ
will vary along any line orthogonal to the streamline. The differentials of the
coordinates along a line orthogonal to the streamline can be represented by
dx = cos (0 + — 1 dSy, = —sin Θ ds ,
y}
dy = sin ίΘ + γ\ ds = cos Θ ds
v wi
where άβ is the curve differential. Then from (1.4) we find
ψ
άφ = 0, dtp^vdSy,. (1.6)
From (1.5) and (1.6) it follows that the lines φ = const, (equipotential lines)
are orthogonal to the streamlines ψ = const. The second formula of (1.6)
shows that ψ measures the discharge of the fluid, and increases along the
equipotential line obtained by rotating the velocity vector anticlockwise
through an angle π/2. In this way we see that the flux of fluid between two
stream lines is equal to the difference in values of the stream function on
these lines.
Let us now consider the function dw/dz, called the complex velocity.
In accordance with (1.4),
dw dœ . dw . . , _,
a y
— = -Χ + *-2ΐ=ν— iv = νβ~ιθ. (1.7)
y
dz dx dx x y v '
Obviously the complex velocity dw/dz is the mirror image with respect to
the #-axis of the velocity vector
dw . ._ .,
-7- = Όχ + *V t1·8)
We quote several very simple well-known examples of complex potentials,
which will be referred to subsequently.
THEORY OF JETS IN PLANE STEADY FLOWS 3
1. The Complex Potential of a Uniform, Continuous Flow, Parallel to
the «-axis, is expressed thus
dw ,,
w = vz or —- = v, (1 .Λ9ν)
dz
where v is a real constant, equal to the velocity. By comparison with (1.7)
we see that everywhere in the z-plane
v = v; v = 0.
x y
2. A Source Situated at a Point z . The complex potential due to such a
0
source is
w=£-ln(z-z ). (1.10)
0
For a single closed circuit of the flow in a counter-clockwise direction round
the point z , for example along a circle of radius r with centre at z , the
0 0
imaginary part of w, obviously increases by the constant q, where
w(re2ni) — w(re6i) = ■£- (In re2ni — In r) = qi,
therefore, in accordance with (1.6), q > 0 suggests that the discharge is due
to a source. If q < 0, then a sink exists.
3. A Doublet (Dipole) at the Point z = z . If a source and sink of equal
0
intensity are set at a small distance apart, and then brought together,
simultaneously increasing the intensities of the source and sink inversely
proportionally to the distance between them, then the limiting flow will be
that due to a doublet (dipole). The straight line from sink to source is known
as the axis of the doublet. The complex velocity potential of a dipole situated
at point z , with its axis making an angle oc with the #-axis, is
0
Meia
"--εΐτπ3· <""
The quantity M is called the moment of the dipole.
4. A Vortex at a Point z = z . The complex potential of a vortex situated
0
at the point z is expressed by the formula
0
w== ln(z Zo) (L12)
éï ~ -
For a single circuit of the flow about the point z in a counter-clockwise
0
direction the real part of w increases by the constant Γ. The circulation [43],
[51] about the vortex is equal to
φυ dx + Vy dy = φ-£ dx + ^ dy = ψαψ = Γ.
χ
5. Sources and Vortices in a Fluid Bounded by a Rectilinear Wall. Let a
fluid occupy the upper half-plane y > 0, whilst the a;-axis represents a solid
wall. In addition, let there be at the point z = x + iy a source with a
x ± x
la*
4 THEORY OF JETS IN AN IDEAL FLUID
discharge strength q. In order to obtain the velocity potential of such a flow
we can temporarily assume that there is also a flow of fluid in the lower half-
plane. This flow must be chosen so that the #-axis is a streamline. The
velocity potential of the flow thus obtained will give the result, since we can
always substitute a streamline in an ideal fluid for a wall. In order to form
such a flow we place at the point z = x — iy a source of strength q. The
x 1
complex potential of the flow from the sources at points z and z is
x x
w = J^ln (z - ) + i-ln (z - z\). (1.13)
Zl
This is the required complex potential.
Indeed the complex potential
w = γ- In [z2 - z{z -f z\) + Z&] = ψ In (z2 - 2zx + x\ + y\)
x x
is real when z = #, so that the #-axis is the streamline y = 0. It is obvious
that w in the region of the flow possesses a special logarithmic character not
only at point z , but also at infinity where there is a sink. Of course it could
x
not be otherwise, since the fluid flowing from the source cannot just dis-
appear. It is not always necessary for a sink to exist at infinity, for example,
a sink may exist at the point 2 = z , at a finite distance. The complex
2
potential function for such a flow is
w = J-ln (z - z ) + Jj-ln (z - z\) - J^ln (z - z ) - J-ln (z - z )
x 2 2
q In (* ~ Zl) (g ~ ^ (1.14)
2π (z — 2) (z — z ) '
2 2
It is also easy to find the complex potential of a vortex at point z with
x
circulation Γ, provided that the #-axis is the solid wall. For this, the flow
should be continued into the lower half-plane, placing at z a vortex with a
x
circulation —Γ, so that
w in [in (s zi) in (z (Li5)
=L· j^f=L· - - - '^ ■
Indeed, during the counter-clockwise flow about the point z along an
1
infinitesimal circle, In (z — z ) does not change, whereas In (z — z) is
± 2
increased by 2πί; hence w receives an increase Γ, so that at the point z
x
there is a vortex with circulation Γ. Furthermore, the #-axis is a streamline,
because on the a>axis
1η1
\x-Zx\ -έ =°·
Im w = — — In
2π
From a mathematical point of view this method of obtaining the complex
potentials of sources and vortices in the presence of a wall by means of
THEORY OF JETS IN PLANE STEADY FLOWS 5
placing sources and vortices at points symmetrical in relation to the wall,
is a particular case of the use of the Riemann-Schwarz principle of symmetry
(cf. books on the theory of functions [67], [49]). We will apply this principle
to one example.
6. A Source and Sink Inside a Circle. Let there be a solid wall in the shape
of the circle \z\ = 1, and inside this circle at the points z = rë°x and z = r ew*
x x 2 2
let there be respectively a source and a sink of equal intensity Q. The complex
potential w(z), continued on to the whole plane of the variable z, must
possess logarithmic singularities corresponding to the source and the sink
at the points symmetrical to z and z in relation to the circle \z\ — 1. The
x 2
points P and P' are said to be (cf. [67], [49]) reciprocally symmetric with
respect to the circle C, if they lie on the same radius drawn from the centre
of C, so that the product of their distances from the centre of the circle C
is equal to the square of its radius. In this way the points elcTi/ri an(I el(rVr2
will be symmetric to the points r-^e101 and r exa* respectively. Hence we
2
obtain
(1.16)
The function w(z) obviously possesses the appropriate singularities inside
the circle \z\ = 1. We can ensure the correctness of the choice of w(z) if we
show that on the circle \z\ = 1 the imaginary part of w(z) is constant, so
that this circle is a streamline. For a point on the circle \z\ = 1 we can put
z = eia, where the quantity σ is real. Then
q , (eia - r,e'"ffl) (rei<7 - eiaA r
Im w(z) = Im~-ln- i ' * 1i >2 *
2π (eia — r eia*) (r eia — e·*»)^
2 2
q (eia — reiai) (r^-"* — e~ia) eia^eiar
x 2
~ m2n Π (eia - r eia*) (r e~ia — e~io) e,(T»ei(rr
2 2 3
q
= — (a — a ) = const.
1 2
Observation on the Principle of Symmetry. The principle of symmetry is
concerned with single-valued holomorphic functions, although we have
applied it to those with singularities. The justification for this is found by
direct verification. Although this is not the place to study a purely mathe-
matical problem concerning the broadest formulation of the principle of
symmetry, we will make an observation that is necessary for what follows
later.
Let AB be a segment of a straight line or an arc of a circle, and let it at the
same time be a part of the boundary of a region D of the independent
complex variable z, on which is defined the function f(z), real on AB.
Furthermore, let the function f(z) at the point z of the region D have the
0
6 THEORY OF JETS IN AN IDEAL FLUID
singularity (z — z )a or, in other words, in the neighbourhood of the point
0
z let there exist an expansion
0
f(z) = {z - z Y [a + a(z - z) + a(z - z )2 + ...],
Q 0 x 0 2 0
where <x is a real quantity. Then we can continue the function f(z) through
AB into the region D*, which is symmetric with D in relation to AB. At the
same time we can introduce a singularity (z — z* )" at the point z* which is
symmetric as regards AB with the point z . If, however, the modulus \f(z)\
0
of the function on AB is constant, then there will be a singularity (z — z*)~a
at ZQ .
By using a bilinear transformation of the independent variable we can
always manage to make AB a segment of the real axis, whilst symmetrical
points remain symmetrical. Let us suppose that this transformation has
already been completed, then z* = z . First, let Im [/(z)] =0on AB. We
0
will represent /(z) as
ί(ζ) = (ζ-ζ0)«(ζ-ζ0)«Ρ(ζ),
where the function P(z) is simple and holomorphic in D. It is obvious that
(z — ζ^Υ (Z — z Y = [z2 — z(z -f- z) + ζζ]Λ is real on AB, where Im [z] = 0.
Q 0 0 0 0
Then for /(z) to be real on AB, the function P(z) must be real on this seg-
ment. But in this case P(z) can be continued by means of the principle of
symmetry into the region D*. As (z — ζ )α (ζ — ζ)Λ takes symmetric
0 0
(conjugate) values at symmetric (conjugate) points, then the whole function
f(z) continues symmetrically into D*.
By analogy, if |/(z)| = 1 on AB, then f(z) can be represented as
Q(zh (L17)
f^-(BfJ
where the function Q(z) is simple and holomorphic in D. Since on AB
I \z - v I
then also |Q(z)| = 1 on AB. Hence the function Q(z) can be continued into
region D* in accordance with the principle of symmetry. At the same time,
in accordance with the above-mentioned concept of symmetry relating to
the circle \f(z)\ = 1, the values of /(z) at points symmetric with respect to
AB will have reciprocal modulii and identical arguments. The reader will
have no difficulty in verifying for himself that the function I — 1
\z — z 1
0
possesses such a kind of symmetry with respect to the real axis. From (1.17)