Table Of ContentDEVELOPMENTS IN WATER SCIENCE, 5
advisory editor
VEN TE CHOW
Professor of Hydraulic Engineering
Hydrosystems Laboratory
University of Illinois
Urbana, Ill., U.S.A.
FURTHER TITLES IN THIS SERIES
1 G. BUGLIARELLO AND F. GUNTHER
COMPUTER SYSTEMS AND WATER RESOURCES
2 H.L. GOLTERMAN
PHYSIOLOGICAL LIMNOLOGY
3 Y.Y. HAIMES, W.A. HALL AND H.T. FREEDMAN
MULTIOBJECTIVE OPTIMIZATION IN WATER RESOURCES SYSTEMS:
THE SURROGATE WORTH TRADE-OFF METHOD
4 J.J. FRIED
GROUNDWATER POLLUTION
TURBULENT JETS
N. RAJARATNAM
Department of Civil Engineering
The University of Alberta
Edmonton, Alberta, Canada
ELSEVIER SCIENTIFIC PUBLISHING COMPANY
AMSTERDAM - OXFORD - NEW YORK 1976
ELSEVIER SCIENTIFIC PUBLISHING COMPANY
335 Jan van Galenstraat
P.O. Box 221, Amsterdam, The Netherlands
AMERICAN ELSEVIER PUBLISHING COMPANY, INC.
52 Vanderbilt Avenue
New York, New York 10017
Library of Congrecs Calaloging in Publication Data
Rajaratnm, N
Turbulent jets
(Developmentsin water science ; 5)
Includes bibliographical references and indexes.
1. Jets--Fluid dynamics. 20 Turbulence.
I. Title. 11. Series.
TA357.R3li 532'.517 76-860
ISBN C-4lCii -41372 -3
With 245 illustrations and 13t ables
Copyright 0 1976 Elsevier Scientific Publishing Company, Amsterdam
All rights reserved. No part of this publication may be reproduced, stored
in a retrieval system, or transmitted in any form or by any means, electronic
mechanical, photocopying, recording, or otherwise, without the prior written
permission of the publisher,
Elsevier Scientific Publishing Company, Jan van Galenstraat 335, Amsterdam
The Netherlands
Printed in The Netherlands
PREFACE
This book presents a detailed treatment of the mean flow characteristics
of incompressible turbulent jets for use by engineers. Regarding turbulence
characteristics, in most cases I have presented only the typical experimental
results connected with the similarity of the root-mean-square of the velocity
fluctuations and turbulence shear-stress profiles with the idea that these typi-
cal experimental observations will be useful to those readers who are in-
terested in constructing prediction models based on turbulence properties.
This book could be used by graduate students in engineering as well as senior
undergraduate students and engineers who have had a good introductory
course in Fluid Mechanics.
Chapter 1 discusses the plane turbulent jet followed by a discussion of the
simple circular jet in Chapter 2 and the radial jet in Chapter 3. Plane and
circular jets issuing into ambient fluids in motion in the direction of the jet
are discussed in Chapter 4. Plane shear layers in stagnant ambient fluid as well
as in moving surroundings are treated in Chapter 5. This is followed by a
study of axisymmetric shear layers in Chapter 6.
The effect of swirl on circular jets is discussed in Chapter 7. The behaviour
of jets in constant area ducts for axisymmetric as well as two-dimensional
configurations is analyzed in Chapter 8. Circular and plane jets in cross-flow
are studied in Chapter 9.
Turbulent wall jets, plane and axisymmetrical, issuing into stagnant
surroundings are discussed in Chapters 10 and 11. Plane wall jets in a con-
stant-velocity free stream are studied in Chapter 12. Chapter 13 presents a
preliminary treatment of three-dimensional free and wall jets. A list of about
230 references (referred to in the text) is given at the end of the book.
Looking at the available books in this field, Townsend’s excellent book on
‘The Structure of Turbulent Shear Flow’ (Cambridge University Press, 1956),
presents an excellent discussion on the similarity analysis as applied to plane
jets in a stagnant and varying velocity free stream and the circular jet in stag-
nant surroundings. ‘Boundary Layer Theory’ by Schlichting (McGraw-Hill,
New York, 1968) and ‘Fluid Dynamics of Jets’ by Pai (Van Nostrand, New
York, 1954) and ‘Jets, Wakes and Cavities’ by Birkhoff and Zarantonello
(Academic Press, New York, 1957) contain one chapter each on incom-
pressible turbulent jets. Hinze (McGraw-Hill, New York, 1959) devotes a
section in his book ‘Turbulence’ to jets.
Abramovich’s book on ‘The Theory of Turbulent Jets’ (M.I.T. Press,
Massachusetts, 1963) presents an extensive treatment of turbulent jets.
VI
Whereas Abramovich, most of the time, uses solution of integral equations
with assumed velocity profiles which results in complex formulas and numer-
ous charts, the present book uses a combination of similarity analysis of
equations of motions and integral equations and dimensional analysis with
the results of carefully chosen experimental results to develop a simpler
treatment of the subject. Further, the present book is more comprehensive in
treatment of the material covered in Chapters 4, 6, 7, 8, 9, 10, 11, 12 and 13.
I am thankful to my former graduate students, Dr. B.S. Pani, Dr. S.P. Rai,
Dr. S. Beltaos, my present graduate student, Mr. B.B.L. Pande, and Dr. T.
Gangadhariah for their help in this work. I am also thankful to Mrs. Julie
Willis for preparing an excellent final manuscript. Finally, I am grateful to
my wife and children for their tremendous help in many ways while I was
writing this book.
Thanks are expressed to the following publishers and editors for their per-
mission to reproduce figures from copyright publications:
Cambridge University Press (Journal of Fluid Mechanics),
American Society of Mechanical Engineers (Transactions ASME) ,
American Society of Civil Engineers (Proceedings and Transactions ASCE),
M.I.T. Press (Abramovich, 1963),
Royal Aeronautical Society (Aeronautical Quarterly and Journal of the Royal
Aeronautical Society),
Editor, Applied Scientific Research,
Butterworths (Combustion and Flame),
The Combustion Institute (Symposium on Combustion),
Springer Verlag (11 t h International Applied Mechanics Conference Pro-
ceedings),
Consultants Bureau (Fluid Dynamics),
Editor, Water Power,
American Institute of Aeronautics and Astronautics (Journal of A.I.A.A.),
Aeronautical Research Council,
Institute of Chemical Engineers,
Institution of Mechanical Engineers.
N. RAJARATNAM
Edmonton
CHAPTER 1
THE PLANE TURBULENT FREE JET
1.1S OME EXPERIMENTAL OBSERVATIONS
Let us consider a jet of water coming from a plane nozzle of large length
into a large body of water or a jet of air into a large expanse of air. Let the
height (or thickness) of the jet be 2b0 and let Uo be the uniform velocity in
the jet. If we use suitable flow visualization techniques, we will find that the
jet mixes violently with the surrounding fluid creating turbulence and the jet
itself grows thicker. Figure 1-1s hows a schematic representation of the jet
configuration discussed above, which is known as the plane turbulent free jet.
Experimental observations on the mean turbulent velocity field indicate that
in the axial direction of the jet, one could divide the jet flow into two distinct
regions. In the first region, close to the nozzle, known commonly as the flow
development region, as the turbulence penetrates inwards towards the axis or
centerline of the jet, there is a wedge-like region of undiminished mean veloc-
ity, equal to Uo.T his wedge is known as the potential core and is surrounded
by a mixing layer on top and bottom. In the second region, known as the
fully developed flow region, the turbulence has penetrated to the axis and as
a result, the potential core has disappeared. For a plane jet, the length of the
potential core is about 12bo, and in this chapter we will consider only the
fully developed flow region and we will discuss the flow development region
in Chapter 5.
In the fully developed flow region, the transverse distribution of the mean
velocity in the x-direction, i.e. the variation of u with y at different sections,
has the same geometrical shape as shown in Fig. 1-1.A t every section, u de-
creases continuously from a maximum value of urn on the axis to a zero value
at some distance from the axis. Let us now try to compare the distributions
at different sections in a dimensionless form. At each section, let us make
the velocity u dimensionless by dividing it by u, at that section and let b rep-
resent a typical length for that section. Let us take b as the value of y where
u is equal to half the maximum velocity. Let us now plot u/u, against y/b.
We will find that the velocity distributions at different sections fall on one
common curve. Figure 1-2(a and b) illustrate this aspect very vividly
(Forthmann, 1934). In Fig. 1-2, X denotes the axial distance from the nozzle.
The velocity profiles at different sections which could be superposed in this
manner are said to be ‘similar’. The two non-dimensionalizing quantities are
called, respectively, the velocity scale and the length scale. A very large
number of flows in the field of turbulent jets exhibit this property of
2
(0) ,
PCOOTREENJT IAL
Fig. 1-1.D efinition sketch of plane turbulent free jets.
similarity. In order to use these similarity profiles for predicting the mean
velocity field in any particular problem, we have to be able to predict the
manner of variation of the velocity and length scales.
1.2 EQUATIONS OF MOTION
In this section we will develop the equations of motion for the plane
turbulent free jet. The Reynolds equations in the Cartesian system are
written as (Schlichting, 1968, Chapter 18):
au
-+u---a+uv --+awtl- au = ---1+v a P
ax aZ
at ay P ax
ajP au" am)
-~ I I
[I-11
ax ay az
-av+ (,-av+ v-+avw - aavZ = ---1+ uap -ax2 +aa2yv2 aa+2Zv )2
at ax ay P aY
3
409
A;=O cm
o = 10 cm
A : 20 cm
u 0 = 35 cm
-5201
7
10
-
-10 -5 Ocm 5 10 25
-Y-
Fig. 1-2. Velocity distribution for plane turbulent free jets (Forthmann, 1934).
aw
aw aw aw 1 aP
and:-+u-+v-+w- = ----+v
at ax ay a2 P
avrwr awl2
aulw)
[I-31
The continuity equation is written as:
au
-+--a+v- aw = 0
ax ay az
where the X-axis defines the axial direction of the jet, the Y-axis is normal
to the X-axis and is in the direction of the height of the nozzle and the Z-axis
4
is the third axis of the coordinate system; u, v and w and u', v' and w' are
the turbulent mean and fluctuating velocities in the X-, Y- and 2-coordinate
directions, p is the mean pressure at any point, v is the kinematic viscosity,
and p is the mass density of the fluid and t is the time variable.
Because the mean flow is two dimensional, w = 0, a/az of any mean quan-
-
tity is zero; u'w'= 0; v~ 'w' = 0 and since the mean flow is steady aulat = 0
and &/at = 0. Further, since the transverse extent of the flow is small, u is
generally much larger than v in a large portion of the jet and velocity and
stress gradients in the y-direction are much larger than those in the x-direc-
tion. With these considerations, the equations of motion could be shown to
reduce to the form:
au av
-+-=
0
[I-71
ax ay
Integrating [ 1-61 with respect to y from y to a point located outside the jet,
we obtain:
-
p = p_ -pd2
where p, is the pressure outside the jet. Differentiating the above equation
and substituting in [ 1-51, we get:
-
u-aa+xu v- aayu = --p1_ d_dpx _ +v7aay2- u- --a(auuy'd aax r24 -3 [I-81
The last term in the above equation is smaller than the other terms and could
be dropped. Hence we obtain the reduced equations of motion as:
where p- is simply written as p for convenience. In [l-91, we could rewrite
the last two terms as:
where 71 and T~ are, respectively, the laminar and turbulent shear stresses and
1-1 is the coefficient of dynamic viscosity. In free turbulent flows, due to the
5
absence of solid boundaries, rt is much larger than r1 and hence it is reason-
able to neglect r1a nd rewrite [l-91 as:
[l-101
Further, because in a large number of practical problems the pressure gradient
in the axial direction is negligibly small and also to study the jet under rela-
tively simpler conditions, let us set dpldx = 0. Then [l-101 and [l-71 be-
come:
[ 1-11]
[ 1-12 1
which are the well-known equations of motion for the plane turbulent free
jet with a zero pressure gradient in the axial direction. For the sake of con-
venience, in this book, rt is often written simply as r.
1.3 THE INTEGRAL MOMENTUM EQUATION
For the plane turbulent jet issuing into a large stagnant environment and
expanding under zero pressure gradient, since there is no external force in-
volved, it is easy to see that the momentum of the jet in the axial direction is
preserved. Let us now derive this criterion in an elegant manner, and this pro-
cedure will be helpful when we study more complex situations.
Multiplying [l-111 by p and integrating from y = 0 to y = 03, we have:
[ 1-13 1
Let us now consider the different terms of the above equation.
i a
au 1 d
p u z d y = - rz(pu2)dy = -2 -dx rpu2dy (by Liebnitz rule*)
2
0 0
A general statement of the Liebnitz rule can be given as:
db
yx, b) --
dx
For proof of the above rule, see 'Advanced Mathematics for Engineers' by H.W. Reddick
and F.H. Miller. Wiley, New York 1962, third edition, p. 265.
