Table Of ContentHMT
THE SCIENCE & APPLICATIONS OF HEAT AND MASS TRANSFER
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Editor-in-Chief:
D. BRIAN SPAIDING
Imperial College of Science and Technology, London, England.
ALSO IN THIS SERIES
SPALDING
GENMIX: A General Computer Program for Two-dimensional
Parabolic Phenomena
KHALIL
Flow, Mixing and Heat Transfer in Furnaces
REZK
Heat and Fluid Flow in Power System Components
CHEN and RODI
Vertical Turbulent Buoyant Jets—A Review of Experimental Data
JALURIA
Natural Convection Heat and Mass Transfer
Pergamon Related Journcds
CHEMICAL ENGINEERING SCIENCE
INTERNATIONAL JOURNAL OF HEAT AND MASS TRANSFER
LETTERS IN HEAT AND MASS TRANSFER
PHYSICOCHEMICAL HYDRODYNAMICS
TURBULENT BUOYANT
JETS AND PLUMES
Edited by
WOLFGANG RODI
Universität Karlsruhe, Karlsruhe, Germany
PERGAMON PRESS
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Copyright © 1982 Pergamon Press Ltd.
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First edition 1982
Library of Congress Cataloging in Publication Data
Main entry under title:
Turbulent buoyant jets and plumes.
(HMT : the science & applications of heat and
mass transfer ; v. 6)
Including bibliographies.
Contents: Mechanics of turbulent buoyant jets
and plumes/by EJ. List—Turbulent buoyant
jets in shallow fluid layers/by G.H. Jirka—
A turbulence model for buoyant flows and its
application to vertical buoyant jets/by M.S.
Hossain and W. Rodi.
I. Jets—Fluid dynamics. 2. Plumes (Fluid
dynamics) 3. Turbulence. I. Rodi, Wolfgang.
II. Series.
TA357.T885 1982 628.5 82-5258
British Library Cataloguing in Publication Data
Turbulent buoyant jets and plumes.
—(HMT, the science & applications of heat and
mass transfer; v.6)
1. Jets—Fluid dynamics 2. Plumes (Fluid dynamics)
3. Turbulence
I. Rodi, W. II. Series
532'.0527 QC158
ISBN 0-08-026492-1
in order to make this volume available as economically
and as rapidly as possible the authors' typescripts have
been reproduced in their original forms. This method un
fortunately has its typographical limitations but it is
hoped that they in no way distract the reader.
Printed in Great Britain by A. Wheaton & Co. Ltd., Exeter
Preface
The discharge of waste fluid from industrial, agricultural or domestic sources into
the environment, be it the hydrosphere or the atmosphere, usually leads to the
formation of turbulent jets and plumes. The dispersion of the waste and the related
dilution of pollutants are governed by the mean-flow and turbulence characteristics
of the resulting jets or plumes, which themselves depend on the environmental
conditions. In many cases, the density of the discharge fluid is different from
that of the environment, either due to different temperature or chemical composi
tion or due to suspended particles, and the resulting buoyancy forces can have a
great effect on both the mean-flow and mixing ch iracteristics and hence on the
dispersion of the rejected pollutants. In order to control and reduce the impact of
waste emissions, one needs to understand the basic physical mechanisms governing
turbulent buoyant jets and plumes, and one also needs methods to predict these
flows. The present volume aims to foster these needs as it discusses the basic
mechanisms involved in some detail and also presents formulae to estimate,
and a mathematical model to calculate, the behaviour of turbulent buoyant jets and
plumes under various conditions.
Volume 4 of the HMT-Series is closely related with the present one as it reviews
critically experimental data for the subgroup of vertical buoyant jets and plumes.
Indeed it was intended to include this work in the present volume, but that
review was completed so much earlier than the other contributions that prior
publication as a separate volume appeared more opportune. The first contribution to
this volume, by List, complements the previously published review by including new
experimental data but mainly by using the data as basis for a detailed discussion
of the physics of turbulent buoyant jets and plumes, including those in cross
flows. The mechanism of jets is described by following from the initial discharge
at an orifice with shear layer instability, to the development of large-scale
vortices, through to the subsequent fully developed turbulence that ensues. The
influence of body forces on the jet development and in particular on the entrain-
ment that controls dilution is discussed. The turbulence structure within jets is
examined using published experimental data, and the influence of buoyancy on this
structure is described.The effects of ambient density stratification and cross flow
on the development of turbulent jets and plumes are summarized briefly. With the
aid of dimensional analysis, the experimental results are condensed into simple
formulae for describing the main integral parameters like jet width and entrain-
ment. These formulae are often sufficient for an estimation of jet and plume
vu
viii Preface
behaviour for practical purposes.
While J,ist’s contribution is concerned solely with jets and plumes in an infinitely
large receiving fluid, Jirka’s article deals specifically with the interaction of
jets and plumes with fluid boundaries such as horizontal walls, free surfaces or
interfaces. First, the characteristics of horizontal buoyant surface jets jn a
semi-confined environment are reviewed, with particular emphasis on thc influence
of buoyancy on jet development. An entrainment relationship is derived and compared
with experimental data, and the possible formation of hydraulic jumps is discussed.
The main portion of Jirka’s contribution is concerned with the development of
buoyant jets discharging into shallow, vertically confined fluid layers. Various
configurations of practical importance are considered like submerged discharges
interacting with the surface as well as discharges at the surface or at a density
interface. Of major concern is the development of stable or unstable flow in the
confined layer, the former being associated with buoyant spreading motion along the
bounding surface and the latter with the formation of a recirculation cell.
Integral analysis for different flow regimes leads to simple formulae for describ-
ing the flow development, including stability criteria for the layer flow. The
formulae also allow the calculation of dilution and are often sufficient for
estimating the main parameters of practical interest.
In the last contribution, by Hossain and Rodi, a significantly more complex
mathematical model is described which allows detailed calculations of the flow,
including not only the integral parameters but also the distributions of velocities,
temperature, etc. These distributions are governed by basic differential equations
like the time-averaged Navier-Stokes and temperature equations, which contain
turbulent transport terms. The latter may be influenced strongly by buoyancy and
require the introduction of a suitable turbulence model before the equations can be
solved. Hossain and Rodi describe such a model, which is derived by simplification
of a second-order model involving differential transport equations for the
turbulent transport terms (Reynolds stresses and turbulent heat or mass fluxes).
The resulting algebraic stress/flux model is suitable for general buoyant flows,
except those with extended regions in which the turbulent transport is against the
gradient of the transported quantity (counter gradient transport). The model is
tested by application to those vertical-buoyant-jet cases for which data were
reviewed in HMT-Volume 4, while the application to the horizontal surface jet
discussed in Jirka’s contribution is described elsewhere. Judging from these
verifications, the model appears to simulate realistically the most important
characteristics of turbulent buoyant jets and plumes and should be a useful tool
for simulating in detail flows of this type.
Karlsruhe, January 1982 W. Rodi
Mechanics of Turbulent Buoyant
Jets and Plumes
E. J. LIST
California Institute of Technology
Pasadena, California 91125
1
Acknowledgements
This article was written at the California Institute of Technology in Pasadena,
California. Without the support of Caltech, faculty colleagues, students, and
staff it could not have appeared. Over a period of years many sponsors have
supported the research program in the W. M. Keck Laboratories and this article
has drawn substantially on work performed under those research grants. In
particular, the support of the U.S. National Science Foundation, the Southern
California Edison Company, and the Ford Energy Program at Caltech are gratefully
acknowledged. The author is particularly appreciative of the assistance received
from Joan Mathews and Melinda Hendrix-Werts in the preparation of this article.
2
List of Symbols
A jet orifice cross-sectional area
a non-dimensional constant
B specific buoyancy flux
b(x) jet lateral dimension at boundary
b* jet lateral dimension where heat flux is 37% of maximum mean
b jet lateral dimension where velocity is 37% of maximum mean
u
b jet lateral dimension where concentration is 37% of maximum mean
n
Θ J
C-JCLJC^JC, non-dimensional constants in jet trajectory equations
C ,C. plume and jet invariants
P J
c specific heat at constant pressure
P
D jet dimension at orifice
D-,D non-dimensional constants in jet dilution equations
?
E energy released from heat source
F^ buoyancy
Gr Grashof number
g gravitational acceleration
H energy release rate
h^ length scale for jet in density-stratified environment
h length scale for plume in density-stratified environment
4 Turbulent Buoyant Jets and Plumes
non-dimensional constants
horizontal length scale in density-stratified crossflows
buoyant jet length scale
RM
R jet orifice scale (= A’/’)
Q
M specific momentum flux of jet at the orifice
m local specific momentum flux
N Brunt-VBiskIlkI frequency
-
P mean pressure inside jet
mean pressure in environment
Q jet specific mass flux at the orifice
q2 turbulent kinetic energy/unit mass
R local buoyant jet Richardson number
autocorrelation function
R plume Richardson number (invariant)
P
r lateral dimension in cylindrical polar coordinates
lateral position in jet or plume where velocity is half
maximum velocity
S dimensionless buoyant jet parameter in stratified environment
t time
reference time
T’ temperature fluctuation
maximum mean temperature on jet axis
Tm
tempera t ure in the environment
U mean crossflow velocity
mean velocity at jet orifice
uO
maximum mean velocity on jet axis
-
U mean velocities in axial direction on jet or plume
U’ local velocity fluctuation from mean in x-direction
-
V transverse mean velocity
V’ local velocity fluctuation from mean in transverse direction
Mechanics of Turbulent Buoyant Jets and Plumes 5
mean vertical velocity on jet axis in a crossflow
m'
X streamwise Cartesian coordinate
Y tracer mass flux in jet or plume
Y transverse coordinate in plane jet or plume
lateral position in plane jet or plume at which velocity is
y4
half maximum velocity
vertical coordinate
jet trajectory function in crossflow
characteristic length scale for plume in crossflow
characteristic length scale for transition in jet in crossflow
characteristic length scale for jet in crossflow
Greek Symbols
a entrainment coefficient
non-dimensional constants
jet and plume entrainment coefficients
local specific buoyancy flux
turbulent momentum flux coefficient
pressure force coefficient
E specific turbulent kinetic energy dissipation rate
dimensionless coordinate
Ic thermal diffusivity
e
species concentration
maximum mean species concentration
mean species concentration
8 flux-weighted mean concentration
local deviation from mean species concentratiou
ratio of b e /b dimensionless constant
U'
microscales
integral scales
