yt WATER POLLUTION CONTROL RESEARCH SERIES • 16130 DJU 02/71
AN ANALYTICAL
AND EXPERIMENTAL INVESTIGATION
OF SURFACE DISCHARGES OF HEATED WATER
ENVIRONMENTAL PROTECTION AGENCY • WATER QUALITY OFFICE
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WATER POLLUTION CONTROL RESEARCH SERIES
The Water Pollution Control Research Series describes the
results and progress in the control and abatement of pollu-
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Reports should be directed to the Head, Project Reports
System, Office of Research and Development, Water Quality
Office, Environmental Protection Agency, Washington, D.C. 20242,
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AN ANALYTICAL AND EXPERIMENTAL INVESTIGATION
OF SURFACE DISCHARGES OF HEATED WATER
by
Keith D. Stolzenbach
and
Donald R. F. Harleman
RALPH M PARSONS LABORATORY
FOR WATER RESOURCES AND HYDRODYNAMICS
Department of Civil Engineering
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
for the
WATER QUALITY OFFICE
ENVIRONMENTAL PROTECTION AGENCY
Research Grant No. 16130 DJU
February, 1971
For sale by the Superintendent of Documents, U.S. Government Printing Office
Washington, D.C., 20402 - Price $1.75
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EPA Review Notice
This report has been reviewed by the Water Quality Office,
EPA, and approved for publication. Approval does not signi-
fy that the contents necessarily reflect the views and poli-
cies of the Environmental Protection Agency, nor does mention
of trade names or commercial products constitute endorsement
or recommendation for use.
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ABSTRACT
The temperature distribution, induced in an ambient body of water by a
surface discharge of heated condenser cooling water must be determined for eval-
uation of thermal effects upon the natural environment, for prevention of recircu-
lation of the heated discharge into the cooling water intake, for improved design
of laboratory scale models and for insuring that discharge configurations meet
legal temperature regulations.
An analytical and experimental study of discharges of heated water is
conducted. The discharge is a horizontal, rectangular open channel at the sur-
face of a large ambient body of water which may have a bottom slope or a cross
flow at right angles to the discharge. Of interest is the dependence of the
temperature distribution in the receiving water as a function of the initial
temperature difference between the heated discharge and the ambient water, the
initial discharge velocity, the geometry of the discharge channel, the bottom
slope, the ambient cross flow, and the transfer of heat to the atmosphere through
the water surface.
The theoretical development assumes that the discharge is a three-
dimensional turbulent jet with an unsheared initial core and a turbulent region
in which the velocity and temperature distributions are related to centerline
values by similarity functions. Horizontal and vertical entrainment of ambient
water into the jet is proportional to the jet centerline velocity by an entrain-
ment coefficient. The vertical entrainment is a function of the local vertical
stability of the jet and thci buoyancy of the discharge increases lateral spreading.
A cross flow deflects the jet by entrainment of lateral momentum and a bottom slope
inhibits vertical entrainment and buoyant lateral spreading.
Experiments are performed in a laboratory basin in which all of the rele-
vant parameters, including the cross flow and the bottom slope, are varied and
three-dimensional temperature measurements are taken in the heated discharge.
The experiments verify that the theoretical model predicts the behavior
of heated discharges. The theory contains no undetermined parameters and the
comparison of the experimental and theoretical results does not involve any fitt-
ing of the theory to data. The rate of temperature decrease in the jet and the
vertical and lateral spreading are controlled by the initial densimetric Froude
number, the ratio of channel depth to width, and the bottom slope. A cross flow
deflects the jet but does not greatly affect the temperature distribution. Heat
loss does not significantly affect the temperature distribution in the heated
discharge within the region treated by the theory.
Application of the theory to prediction of temperatures in an actual
heated discharge is possible if the temperatures, velocities, and the geometry
of the discharge may be schematized by representative steady state temperatures
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and velocities and by an equivalent rectangular channel. If a model study is
necessary the theory indicates that temperature similarity requires an undistor-
ted model.
The theoretical model developed in this study may be extended by treating
a stratified ambient condition, by considering recirculation of the heated jet in
a finite enclosure and by development of a theory for the transition of the heated
discharge jet into a buoyant plume.
This report was submitted in fulfillment of Research Grant No. 16130 DJU
between the Water Quality Office, Environmental Protection Agency and the Massachusetts
Institute of Technology. .
/
Key words: thermal pollution; turbulent, buoyant jets; stratified flow; temperature
prediction; heated surface discharges.
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ACKNOWLEDGEMENT
This investigation was supported by the Water Quality Office,
Environmental Protection Agency, under Research Grant No. 16130 DJU as part
of a research program entitled "Prediction and Control of Thermal Pollution".
The project officer was Mr. Frank Rainwater, Chief, National Thermal Pollution
Research Program, Pacific Northwest Water Laboratory, at Corvallis, Oregon.
The cooperation of Mr. Rainwater is gratefully acknowledged.
Stone and Webster Engineering Corporation of Boston, Massachusetts, sup-
ported the purchase of experimental equipment in conjunction with another research
project.
The research program was administered at M.I.T. under DSR 71335 and 72324.
The staff and students at the Ralph M. Parsons Laboratory for Water Resources and
Hydrodynamics gave valuable suggestions throughout the study. The experimental
work could not have been performed without the particular help of Mr. Umit Unluata
Miss Alician Quinlan, Mr. Edward McCaffrey and Mr. Roy Milley. Typing was done
by Miss Kathleen Emperor. All computer work was done at the M.I.T. Information
Processing Center.
The material contained in this report was submitted by Mr. Stolzenbach
in partial fulfillment of the requirements for the degree of Doctor of Philosophy
at M.I.T.
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TABLE OF CONTENTS
Page
ABSTRACT 2
ACKNOWLEDGEMENT 4
TABLE OF CONTENTS 5
I. INTRODUCTION 9
1.1 Why Heated Discharges? 9
1.2 Effects of Heated Discharges 10
1.3 Temperature Prediction 12
1.4 Objectives of this Study 14
1.5 Summary of the Study 17
II. REVIEW OF PREVIOUS INVESTIGATIONS 18
2.1 Scope of the Review 18
2.2 Turbulent Jets 18
2.2.1 Physical Properties 18
2.2.2 Analytical Solutions 22
2.2.3 Deflected Jets 27
2.2.4 Confined Jets 27
2.3 Turbulent Flows with Density Gradients 28
2.3.1 Reduction of Turbulence 28
2.3.2 Submerged Buoyant Jets 32
2.3.3 Two Layer Flow 32
2.4 Surface Heat Exchange 33
2.5 Surface Discharge of Heated Water 35
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Page
III. THEORETICAL CONSIDERATIONS A2
3.1 Statement of the Problem 42
3.2 The Governing Equations 44
3.2.1 Basic Equations 44
3.2.2 Steady, Turbulent Flow Equations 45
3.2.3 Hydrostatic Pressure 47
3.2.4 Scale Variables 4g
3.2.5 Large Reynolds Numbers 40
3.2.6 Small Density Differences 5Q
3.2.7 Basic Scale Relationships 5^
3.3 Non-Buoyant Surface Jets 52
3.3.1 Governing Equations 52
3.3.2 Structure of the Jet 54
3.3.3 Boundary Conditions 58
3.3.4 Integration of the Equations ^
3.3.5 The Solution for Non-Buoyant Jets 55
3.4 Buoyant Jets 7^
3.4.1 Governing Equations 71
3.4.2 Structure of the Jet 73
3.4.3 The y Momentum Equation 75
3.4.4 Vertical Entrainment 73
3.4.5 Integration of the Equations 80
3.5 The Deflected Jet 80
3.5.1 The Governing Equations 80
3.6 Bottom Slopes 90
3.7 Solution Methods 96
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Page
IV. EXPERIMENTAL EQUIPMENT AND PROCEDURE 97
4.1 Previous Experiments 97
4.2 Experimental Set-up 101
4.3 Experimental Procedure 105
4.4 Data Analysis 106
V. DISCUSSION OF RESULTS 112
5.1 The Controlling Parameters 112
5.2 An Example of a Theoretical Computation 114
5.3 Comparison of Theory and Experiment 118
5.4 Comparison of Theory with Experimental Data 135
by other Investigators
5.5 Summary of Jet Properties 143
VI. APPLYING THE THEORY 152
6.1 Prototype Temperature Prediction 152
6.2 Modeling Heated Discharges 158
6.3 A Scale Model Case Study 162
VII. SUMMARY AND CONCLUSIONS 169
7.1 Objectives 169
7.2 Theory and Experiments 169
7.3 Results 171
7.4 Future Work 174
BIBLIOGRAPHY 176
LIST OF FIGURES AND TABLES 179
LIST OF SYMBOLS 182
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Page
APl'KNULX I. The Computer Program for Theoretical 187
Calculation of Heated Discharge Behavior
APPENDIX II. Deflected Jet in a Cross Flow: Integration 208
of the x and y Momentum Equations Over the
Entire Jet
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I. Introduction
1.1 Why Heated Discharges?
The discharge of heated water into rivers, lakes and coastal
regions has its roots in the growth of demand for electric power and
in the economics of power production. The demand for power generating
capacity in the United States is expected to continue doubling each
decade. As hydro power plant sites are developed to their full capa-
city, the demand will be met increasingly by fossil fuel and nuclear
steam plants. Regional economics and economics of scale result in an
increase in plant size; often fuel costs make nuclear plants more econ-
omic than coal or oil facilities (6 )•
The heat rejected by a steam power plant is related to the
thermodynamic efficiency of the power production process. Present lev-
els of efficiency are about 40% for coal and oil plants and about 30%
for nuclear plants. The nuclear plants reject significantly larger
amounts of heat: 10,000 BTU/kwh vs. 6,000 BTU/kwh (10). Present lev-
els of efficiency are determined by the economics of power plant design
and are mostly a function of the cost of capital items such as turbines
and condensers. Water is the only economic cooling fluid for large
power plants and at present the costs of its supply and of the heat dis-
posal system do not greatly influence the choice of economic efficiency.
Significant technological improvements in the efficiency of convention-
al steam cycles or the development of more efficient thermal power gen-
erating methods are not foreseen in the near future (6 ).
The total volume of cooling water used for all industries in
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the United States is fifty percent of all water use; the power industry
accounts for all but ten of that fifty percent 0.9). The large volumes
are dictated by the economic design of the cooling systems which require
low operating temperatures and high flows.
Thus the amount of heat rejected, the volume of water required,
and the temperature rise in the cooling system necessary to satisfy the
demand for power are determined largely by in-plant technological and
economic considerations not related to the ultimate heat disposal pro-
cess or its effect upon the environment.
The choice of heat disposal systems for a given power plant site
reflects the relatively fixed constraints on temperatures and flows.
Economic re-use of waste heat for domestic or industrial purposes is
inhibited by the low temperature uses and the large flows dictated by
the plant design. The rejected heat must in most cases be returned to
the natural environment. The present methods for heat disposal are
cooling towers, cooling ponds, and once-through cooling. For a 1000 MW
power plant unit the cost of cooling towers may be $10,000,000 and the
cost cf purchasing land for a cooling pond as high as $5,000,000. The
once-through cooling method, in which water is withdrawn from an adja-
cent waterway and returned after being heated, is the most economical
assuming the availability of the natural water (6 ).
1.2 Effects of Heated Discharges
Balancing the economic benefits of once-through cooling is the
impact which the discharge "has upon the natural environment and how
this in turn affects the power plant directly and the users of the
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power plant indirectly.
The most immediate undesirable effect of the heated discharge
upon the power plant is the recirculation of the heated water into the
intake, creating a feedback of heat which significantly raises the temp-
erature of the cooling water. The resulting reduction in plant effic-
iency is considered as a cost to the power company. Prevention of re-
circulation was the major concern in the design of a discharge system
before the era of large nuclear plants and public concern over thermal
effects. Separation of the intake and discharge structures or the de-
sign of the intake as a selective withdrawal device are the most common
techniques of preventing recirculation. A knowledge of the flow and
temperature distribution in the receiving waters near the heated dis-
charge is necessary for proper location and design of the intake for
the prevention of recirculation.
The indirect effects of heated discharges are related to the
cuanges in physical and chemical properties of water resulting from a
temperature rise. As the temperature of water increases the density,
viscosity and ability to dissolve gases decrease while the vapor pres-
sure of water increases. The effects of these changes may be striking:
density differences, although small, may produce stable stratified flows
which differ markedly from natural flow patterns; and evaporative water
losses are increased as vapor pressure rises. Dissolved oxygen, a prime
indicator of water quality, may decrease. Biochemical reactions which
consume oxygen and produce undesirable tastes and odors are promoted by
increases in water temperature. Higher level species such as fish or
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shellfish may not survive exposure to high temperatures or may have
vital life cycles inhibited by temperature rises (19).
The above effects have been lumped together under the title of
"thermal pollution" since the undesirable changes in the environment
are the result of discharges of heat. Prediction of the response of
the environment to increases in temperature is still at a very basic
level; a fundamental requirement for such an evaluation is always a
knowledge of the temperature distribution induced in the receiving
water by the heated discharge.
1.3 Temperature Prediction
Once-through cooling systems are characterized by an intake and
a discharge structure. The water velocity at the intake, for a given
pumping rate, is a function of intake design since converging flow
structures are not complicated by extreme flow separation. The minimum
possible discharge velocity, however, is usually closely related to the
in-plant velocity since special efforts must be made to de-accelerate
a water flow. Economic power plant design velocities are typically of
the order of 5 to 10 ft. per second. Thus even for a designed area
increase of ten times the discharge velocity is near 1 ft. per second.
This is in general higher than many ambient flow velocities in rivers
and coastal areas. At the point of discharge the momentum of the dis-
charge is significantly greater than that of the receiving fluid and
jet-like mixing of the heated and ambient water occurs. Ambient water
is entrained into the discharge jet and the average temperature and
velocity of the heated flow are reduced. Further from the discharge
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point the jet velocity is reduced to the magnitude of the ambient cur-
rents. In this region, entrainment does not occur and the temperature
distribution is determined by natural turbulent diffusion and convec-
tion. Ultimately, all of the rejected heat is passed to the atmosphere
through the water surface, a process driven by the elevation of the
water surface temperature. The density difference between the heated
discharge and the ambient water may inhibit mixing in the initial jet
region and may promote spreading of the discharge as a heated layer.
The temperature distribution induced by a heated discharge is
thus determined near the outlet by jet-like diffusion and in an outer
region by natural processes. Surface heat loss and density effects
occur throughout the flow. The interrelationship of these processes
is very complex. Some basic understanding of each phenomena alone
has been achieved: turbulent jet theory and solutions to convective
diffusion equations are well established and predictions of surface
heat loss rates are now possible. The effects of stratification are
the least understood. Analytical predictions of temperatures in the
vicinity of heated discharges are possible under the assumption that
the flow is either a pure jet, a plume, or a stagnant cooling region.
Often it is not possible to isolate the dominant physical pro-
cess and the flow may be significantly affected by density gradients
or the presence of solid boundaries. It is customary in such cases to
construct a physical model of the discharge and ambient waterway. Un-
fortunately model scaling laws must be based on an understanding of
the very process which the model is supposed to reproduce. It is
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often impossible, at the present level of knowledge about heated dis-
charges, to build a model which accurately replicates all of the import-
ant features of the temperature distribution.
For a given cooling water flow the performance of the discharge
structure is a function of its location, size and shape. Submerged
jets achieve greater mixing than surface discharges. The presence of
an ambient cross current increases natural heat dispersion while solid
boundaries inhibit mixing. Several small orifices of high velocity
produce more mixing than a single large one. The cost of the discharge
structure varies widely from the case of a simple pipe ending at the
water surface to a deeply submerged multiple port diffuser. Economical
design of the discharge requires the least expensive configuration which
will meet the restrictions imposed on the temperature distribution for
prevention of recirculation and thermal pollution. In particular fed-
eral and state standards are presently expressed in terms of the per-
missible area in which a certain temperature rise, usually from 1.5 to
5°F, is permitted. The power company must prove before construction
permits are granted that the design of the discharge structure will
meet the legal standard. The pressing demand for new power facilities
requires that accurate methods for relating discharge design to _the
temperature distribution induced by the heated discharge be developed.
1.4 Objectives of this Study
This investigation is motivated by the need, as discussed in
previous sections, for accurate temperature prediction techniques which
consider the effect of power plant condenser water discharge design
upon the temperature distribution in the ambient v;ater. Prevention
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of recirculation, evaluation of thermal effects, construction of reli-
able scale models, and economical design of discharge structures re-
quire an understanding of the physics of heated discharges.
The following general problem is considered: A horizontal sur-
face discharge of heated water from a rectangular open channel of arbi-
trary width and depth into a large unstratified body of water which
may be stagnant or flowing with an arbitrary velocity distribution at
right angles to the direction of discharge and which may be infinitely
deep or may have a sloping bottom. The temperature and velocity dis-
tjribution in the discharge are considered as functions of the discharge
channel geometry, the initial velocity and temperature rise, the mag-
nitude of the ambient cross jlow and bottom slope, and the surfa.ce
heat transfer rate. The discharge is considered only to the point
where jet-like behavior ceases and natural turbulence and convection
dominate the temperature and velocity distributions.
Figure 1-1 shows a schematic of the discharge scheme and the
relevant parameters. This problem is chosen because:
1) Often the allowable mixing zone, as defined by temperature
standards, is within the initial jet region.
2) Turbulent diffusion, density gradients, and surface
heat loss may be studied within the defined region.
3) The number of controlling factors is relatively few.
4) The discharge may be studied on a laboratory scale,
thereby allowing accurate measurements and close
control of all parameters.
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Surface Heat Loss
Heated Discharge
Side
View
Turbulent
Entrainment
Discharge
Channel
Ambient Cross Flow
Turbulent
Entrainment
Heated Discharge
Top
View
Turbulent
Entrainment
Bottom Slope
Figure 1-1. Schematic of Heated Discharge
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The region dominated by natural processes, commonly called the
"far field", is excluded because of the infinite number of possible
conditions and because of the difficulty in modeling natural phenomena
on a laboratory scale.
1.5 Summary of the Study
A guiding principle of this investigation is that an increased
understanding of the problem as defined is to be found in a synthesis
of existing knowledge of the physical phenomena of turbulent jets,
stratified flows, and surface heat loss. A review of the scientific
literature in these fields provides the framework for a three-dimen-
sional theory based on a turbulent jet model with modifications intro-
duced to include density gradients and surface heat loss. The theory
is developed carefully from the basic governing equations to a set of
integrated equations which may be solved numerically.
The discharge is modeled in a laboratory basin. All of the
parameters of interest are varied and three-dimensional temperature
measurements are taken. Comparisons of theory and experiment verify
the validity of the theoretical model.
The application of the theory to temperature prediction in actual
power plant discharges is discussed. The implications of the theoreti-
cal results for the design of scale models are illustrated by a case
study. Finally, some suggestions for future work are made.
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11. Re\djew__p_f__P_re^/io us Inves t i ga t ions_
2.1 Scope of the Review
The literature in the fields of turbulence and flows of non-
homogeneous fluids is vast. This review discusses those works which
are the major contributors to the theoretical analysis of this study.
Analytical and experimental investigations of turbulent jets, flows
with stable density gradients, and heat transfer in turbulent fluids
are examined. Whenever possible reference is made to the best avail-
able summary rather than to original publications.
Several recent investigations treat horizontal surface discharg-
es of heated water. These efforts are examined.
2.2 Turbulent Jets
2.2.1 Physical Properties
Abramovich (l ) and Townsend (34) discuss the general character-
istics of various kinds of submerged turbulent jets: (see Figure 2-1)
1) A turbulent shear flow region which increases in size away
from the jet origin. There is a distinct boundary between
the turbulent flow and a potential flow outside the jet.
The jet width, b, is defined as the mean distance from
the jet centerline to this boundary. In practice, however,
a half width b, ,„ is often defined as the point where the
mean x velocity, u, is one half the centerline mean velo-
city, u . For isolated submerged jets both b and b.. ,„ are
observed to grow linearly with x. In plane and axisyrmr.etric
jets, b = .22 x.
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I
l-
Turbulent Region
Figure 2-1. Structure of a Submerged Turbulent Jet
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2) An initial core region of unsheared fluid which is gradually
engulfed by the spread of the turbulent region.
3) Entrainment of ambient fluid into the turbulent region. The
entrainment process is characterized by small scale eddies
near the jet boundary under the overall control of larger
scale features such as the mean shear. The entrained fluid
velocity normal to the jet is v .
4) Heat, mass, momentum and kinetic energy are transported
within the jet by convective and diffusive processes. The
nature of the transport is different for each quantity.
5) Similarity of lateral profiles of velocity in the turbulent
region. The velocity may be expressed by:
u = uc f(y/b) (2.1)
where u is the centerline velocity and f is a similarity
function.
In heated jets, the lateral temperature profiles are similar
to each other but have a shape different from the velocity
profiles.
AT = AT t(y/b) (2.2)
where AT is the centerline temperature difference and t is
a similarity function.
Pearce (26) investigates the dependence of the jet characteris-
tics on the Reynolds number of the jet. For circular jets the Reynolds
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number is
u d
-2-° (2.3)
where u = initial velocity
d = initial diameter
o
v = kinematic viscosity
Pearce summarizes the changes in the structure of a circular water jet
as Uincreases:
IR < 500 - A long laminar cylindrical column, expanding
slightly. Some small stable wave instability but no
turbulence for x/d <100.
o
500
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the same magnitude in all directions.
2) The magnitudes of the turbulent fluctuations are small
compared to the maximum mean velocity at any section.
2.2.2 Analytical Solutions
The equations governing free turbulent jets are derived by
Schlichting (29). The assumptions common to all the cases considered
are:
1) Molecular transport terms are negligible.
2) Longitudinal diffusion is negligible with respect to
longitudinal advection.
3) Constant pressure throughout the jet.
The equations expressing conservation of mass, momentum, and heat ener-
gy for a steady state, two-dimensional flow are:
-> ______
3uv _ 5u'v'
.
(2<5)
3Tu 3Tv _ 3v'T'
"~ + = ~ <2'6)
where the quantities u'v1 and v 'T' are the time averaged fluxes of mom-
entum and heat in the y direction due to turbulent motions. A similar
set of equations may be formulated for axisymmetric flow.
The analytical solution of Equations 2.4 - 2.6 proceeds as
follows:
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1) The velocity and temperature distributions are assumed to
be given by similarity functions f and t as in Equations
2.1 and 2.2. The functions f and t are not specified, but
boundary conditions are stated. The most common boundary
conditions are that the jet is symmetrical about the center-
line axis, that the velocity is zero, the temperature con-
stant, and all turbulent transfer is zero at the jet boun-
dary.
2) The turbulent terms are related to the mean variables.
Abramovich ( 1 ), Pai (24 ), Schlichting (29 ) , and Hinze (17 )
discuss the well—known assumptions:
Prandtl's mixing length
u'v'
2 9u
17
3u
ay
u'T' = £ —
ay
3u
ay
(2.7)
Taylor's vorticity transfer
—
u'v
_3_u
ay
jhi
ay
u'T1 = 2iT
3y
3T
ay
(2.8)
Reichardt's inductive theory
TTTTr. , 3(u2)
ay
Prandtl's new theory
u'v'
9u -t^rr 3T
UT = e
(2.9)
(2.10)
In the above relationships the quantities X,, e , and e are functions
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of x only. The x dependence of these parameters is related
to the jet structure on dimensional grounds:
£ - b
ev * GT - Ucb
Careful experimental investigation has shown that none of
the local relationships between turbulent transfer and
mean gradients (Equations 2.7 - 2.10) are valid (34). For
many cases, however, the solutions based on these assumptions
do predict the gross behavior of the jet reasonably well.
3) The similarity functions (2.1 & 2.2) and the turbulent
hypotheses (one of Equations 2.7 - 2.10) are substituted
into the equations of motion (2.4 - 2.6). For jets, the
sufficient criteria that unique equations for the similarity
functions f and t exist, is found to be (34):
L = const. = e (2.12)
dx
where e is determined later by comparison of the solution
with experimental data. The functions f and t are determined
subject to the stated boundary conditions. A particular
failing of the mixing length hypothesis is that it results
in the same similarity function for both velocity and temp-
erature. Taylor's theory predicts that heat is diffused
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laterally faster than momentum and that;
t = Jl (2.13)
Prandtl's new theory gives the same result if e = 2e (see
Figure 2.2) (29)-
4) The behavior of u and AT is determined by integrating the
equations of motion (2.4 - 2.6) over the jet cross section
to yield expressions of constant momentum and heat flux in
the x direction. This completes the analytical solution.
K property of the solutions for plane and axisymmetric jets
is that
ve = auc (2.14)
i.e. the entrainment velocity is proportional to the center-
line velocity with the coefficient of proportionality called
an entrainment coefficient.
Analytical solutions which describe the jet behavior as accurat-
ely as those derived by the above process may be obtained by a consid-
erably less complicated procedure (24):
1) A specific form for the similarity functions, f and t, is
assumed.
2) Either the jet is assumed to spread linearly (Equation 2.12)
or the entrainment relationship (2.14) is assumed. The
appropriate constant, eor a, is determined empirically.
3) The integrated equations of motion are used as in (4) above.
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1.0
OCtO-Or
.75 --
.50
.25 --
-2.0
-1.5
-1.0 -.5
.5
1.0
1.5
y/b
2.0
l/2
Figure 2-2. Velocity and Temperature Distributions in Submerged Jets
Reichardt's Data: Velocity Q
Temperature Q
u/uc = (1 - .29 ;3/2)2
AT/AT = 1 - .29
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2.2.3 Deflected Jets
The case of turbulent jets discharged into a cross flow is dis-
cussed by Abramovich. The literature on the subject is well reviewed
by Fan (11). In all the cases considered the jet is deflected by two
processes:
1) Entrainment of the lateral cross flow momentum. The result-
ing force per unit length on the jet in the direction of
the cross flow is:
(2.15)
where V is the cross flow velocity and q is the total en-
trainment per unit length of jet.
2) A net pressure force caused by eddying of the ambient fluid
in the lee of the jet and by the distortion of the jet boun-
daries. This force per unit length is:
V2
FD = CD p r. D (2.16)
where C is a drag coefficient and D is the jet diameter.
2.2.4 Confined Jets
The problem of jets discharging over a solid boundary is re-
lated to diffuser flow as discussed by Schlichting (29). Abramovich
(1) also discusses the confined half jet. The effects of the boun-
dary are:
1) Pressure gradients in the lateral and longitudinal direc-
tion. Adverse longitudinal pressure gradients result in
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separation and eddying. For turbulent flows the point of
separation is determined only empirically.
2) A viscous boundary layer near the solid boundary. Longi-
tudinal momentum is decreased by wall shear.
2.3 Turbulent Flows with Density Gradients
2.3.1 Redaction of Turbulence
A discussion of turbulence in the presence of stable density
gradients is given by Phillips (27). Deissler (7 ) formulates a sim-
plified model of initially isotropic turbulence in the presence of a
vertical density gradient. Experimental work with turbulent, density-
stratified flow is reported by Webster (36). In all discussions the
parameter controlling the effect of the density gradient upon the flow
is the Richardson number given by:
£ 3p
(2.17)
3ul
As R. increases from 0 to 1 the vertical turbulent fluctuations dimin-
i
ish; vertical transfer of mass, heat and momentum decrease, mass and
heat transfer more so than momentum transfer. The region of maximum
density gradient becomes an "interface" between two layers of differ-
ent densities. Kato and Phillips (22) investigate the entrainment of
mass across such an interface, using a surface shear to generate tur-
bulence. They find the entrainment to be inversely proportional to the
Richardson number. Ippen and Harleiaan (20) report that viscous under-
flows are stable for IF<1.0 where IF is the densimetric Froude number
-2S-
-------
given by
IF = — - (2.18)
V is the average flow velocity, h is the flow depth and Ap is the den-
sity difference across the interface. Underflows of moderate turbulen-
ce are experimentally evaluated by Lofquist (23). Experiments over a
large range of Richardson numbers for fully turbulent shear flows are
treated by Ellison and Turner ( 9 ) . Their determination of entrainment
2
as a function of a gross Richardson number expressed as R. = I/IF is
given in Figure 2.3. The entrainment coefficient a is defined as the
ratio of the velocity across the interface, v , to the average velo-
city parallel to the interface, u. For R. = 0 the entrainment coeffi-
cient is equal to a , the value observed in homogeneous fluids.
Phillips (27) proposes a model for vertical transport of momen-
tum in a shear flow, u(z). The model is based upon a wind wave genera-
tion theory; after considerable manipulation the eddy viscosity, e,
defined by
T =pe|^ (2.19)
is given by
e ~ w'2 G (2.20)
o
where
00
G
° o w1
R(OdS (2.21)
-29-
-------
I
-------
is the integral time scale of the turbulent vertical velocity fluctua-
tions. The more persistent the vertical turbulent fluctuations, the
greater the vertical momentum transfer. Phillips suggests that in a
stratified flow the vertical velocity spectrum reflects the frequency
of the natural oscillation associated with the density gradient, the
Brunt-Vaisala frequency, N:
N
1/2
P
(2.22)
and that in the stratified flow.
G
oo
f R(O cos N WC < G (2.23)
• " °
Assuming that the Richardson number is not so large that vertical tur-
bulence has been completely damped i.e. that w1 is still the same mag-
nitude as the other components of the turbulent velocity fluctuations,
then the ratio of stratified shear to unstratified shear.
T e G
— = -5- = -2- (2.24)
T e G
ooo
The vertical shear is related to the vertical entrainment by
T ~ u v - au (2. 25)
e
Thus it must be that for a given shear flow u,
oo
f R(C) cos N£dC
— = - - < 1 (2.26)
j
•'
-31-
-------
providing a relationship between the reduction in entrainment and the
vertical density gradient.
2.3.2 Submerged Buoyant Jets
A number of analytical and experimental investigations have been
done for the case of submerged jets in which buoyancy acts on the jet
as a whole, producing a vertical acceleration but in which the turbu-
lent structure of the jet is not significantly altered by the density
gradients. Fan (ll) gives a review of the investigations which include
stratified and non—stratified environments, cross flows, and plane or
axisyrametric jets.
A major difference between the analysis of the buoyant and the
non-buoyant jets is that the linear spread of the turbulent region is
no longer generally valid because of the addition of buoyant forces.
In place of a linear spread assumption it is usually assumed that the
entrainment coefficient is constant along the jet. This assumption is
correct for non-buoyant jets as previously discussed, and gives gener-
ally good results for the buoyant jets considered. The value of the
entrainment coefficient is determined by comparison with experiment.
2.3.3 Two Layer Flow
If the local Richardson number is high or the overall densimetric
Froude number low, two layer flows of different densities may exist.
Harleman (12) discusses the equations of motion governing two layer
flow in open channels. Of particular interest is the case of a flow
issuing from a channel into a large body of denser fluid. If the den-
simetric Froude number based on the upstream depth, h , and velocity,
-32-
-------
u , is less than one, a "stagnant wedge" of dense water extends upstream
from the end of the channel beneath the flow of lighter fluid (see Fig.
2-4). The length of the wedge is given by:
l _!_ _2 + 3 F2/3-f IF4/3
fi I E
(2.27)
u
where IF = and f. is a friction factor for interfacial shear
i
P °
stress, averaged over the length L. Harleman and Stolzenbach (13) es-
timate the friction factor by using smooth turbulent friction laws in
which f. is a function only of the Reynolds number of the flow. At
the junction of the channel and the reservoir the interface rises
rapidly. The upper flow densimetric Froude number, based on the act-
ual layer depth and velocity, is unity at this point.
2.4 Surface Heat Exchange
A comprehensive review of surface heat exchange processes and
predictive equations is given by Brady (3 ). The prediction of natur-
al water temperatures requires a determination of all the natural heat
inputs to the water body. The steady state natural water temperature
for fixed heat inputs is referred to as the equilibrium temperature,
T . In a large body of water the response time for temperature changes
is longer than the characteristic oscillations in heat inputs. The
actual water temperature is never exactly equal to T for this reason.
In rivers, lakes, and reservoirs the difference between the actual and
equilibrium temperatures is usually less than 5°F. In the ocean it
may be as high as 10° - 20°F.
-33-
-------
1.0
Figure 2-4. Two Layer Flow in the Discharge Channel
-------
The "excess" temperature rise induced by the addition of power
plant heat rejection may be calculated by assuming that the "excess"
heat flux through the surface, q , is proportional to the difference
s
between the heated water surface temperature, T and the equilibrium
S
temperature, T .
pcp K (Tg - Te) (2.28)
where the coefficient of proportionality is a function of the water
surface temperature and meteorological conditions. Brady gives form-
ulas and graphs for determining the value of K given wind speed and
dewpoint temperature.
2.5 Surface Discharge of Heated Water
Experimental studies of three-dimensional surface discharges
of heated water have been done by Hayashi (16) , Wiegel (37), Jen (21),
Stefan (31) and Tamai (32). Discussion of the experiments is presented
in Chapter 4 and some of the results given in Chapter 5. Previous
attempts to treat the problem analytically are presented in this sec-
tion. Figure 2-5 shows the physical problem considered by each study.
Edinger and Polk present analytical solutions which assume that
the heated discharge is a point source of heat at the water surface on
the boundary of a uniform stream which is infinitely wide and deep.(8)
No lateral or vertical convection occurs, the temperature distribution
being determined entirely by vertical and lateral eddy diffusivities
and by convection in the direction of the uniform stream. Buoyant
-35-
-------
Uniform Cross Lurrenl
HUM;
L'n i I orm c ross Current
Heat Source
/ /
Isotherms
Infinitely Wide
and Deep Basin
Edinger and Polk
Constant Depth,
Infinitely Wide Basin
and i ross i urrent
No Vertical Entrainment
No Vert it a 1 Lnt rainment
Constant Depth Jet into
Infinitely Wide and Deep
Basin
HDopes
lun ninensional thannel
Hayashl
rigure 2-5. Schematics of Previous Theoretical Investigations
-------
effects are not considered. The results are fitted to field data by
adjusting the value of the vertical and horizontal diffusivities. No
correlation between the fitted values of the diffusivities and values
estimated a priori is found. This treatment is clearly valid only for
the region dominated by ambient turbulence as discussed in Chapter 1.
The difficulty in determining natural turbulent diffusivities is a sev-
ere limitation on the practical use of Edinger and Folk's theory.
Carter treats the case of a heated discharge from a channel into
an infinitely long and wide basin whose depth is uniformly the same as
that of the discharge channel and in which a uniform cross current is
flowing at right angles to the discharge. The jet velocity and tempera-
ture are assumed to be uniform across the width and depth of the jet,
varying only along the centerline coordinate. The jet is deflected by
pressure drag and entrainment of lateral momentum as expressed in inte-
gral momentum equations. Laboratory experiments are performed in which
the temperature field is measured. The entrainment into the jet is
determined from the experimental data and is input to the theory to ob-
tain predictions of the centerline trajectory which are compared with
the data. The scatter of the data is large but the points are near
the theoretical curves on the average. The neglect of buoyancy and
surface heat loss seems justified for this physical situation although
the spreading of the jet is not measured and may be dependent on buoy-
ant forces. Extrapolation of Carter's results to a deep receiving basin
are not possible since his measured entrainment curves would not be
valid. (5)
-37-
-------
Hoopes treats a physical problem very similar to Carter's. He
assumes that the effect of buoyancy is to make the vertical entrainment
zero and that the jet is vertically uniform and of constant depth even
though the receiving basin is infinitely deep. Hoopes assumes Gaussian
similarity functions for velocity and temperature and accounts for lat-
eral entrainment by defining a lateral entrainment coefficient. Inte-
grated continuity, momentum, and heat equations yield expressions for
jet deflection, width, and centerline temperature and velocity. The
theoretical model is compared with field measurements. Uncertainty in
the values of the ambient cross flows makes comparison of the theoreti-
cal and actual jet trajectories difficult. Also most of the temperature
data is taken far from the discharge where the temperature is nearly
the ambient value. This study illustrates the extreme difficulty in
using field data to check a theoretical model. (18)
Hayashi proposes a theory for a discharge from a channel of
finite width into a large, stagnant receiving body of water. He assumes
that the buoyant stability is so great that vertical entrainment is zero
and that all of the convective terms in the governing equations are neg-
ligible. The equations are integrated over the jet depth using a simi-
larity function to describe vertical temperature and velocity distribu-
tions. Analytical solutions for temperature and velocity are obtained
by assuming that horizontal eddy diffusivities are not functions of
space and taking into account heat loss from the water surface. Experi-
ments done by Hayashi do not agree with the theory and show a large
amount of scatter. The discrepancy between data and theory is a result
-38-
-------
ot initial mixing near the discharge point which the theory does not
account for. As discussed in Chapter 1, initial mixing is important
in most heated discharges. Also, the formation of stagnant cold water
wedges in discharge channels of low densimetric Froude number, IF, pre-
vents the local value of IF at the discharge point from being less than
unity. Thus Hayashi's theory, which is valid for 3F= 0, will not apply
to actual heated discharges. (15)
Wada considers a two-dimensional flow (vertical and longitud-
inal) of heated water and treats the full equations of motion in which
turbulent terms are represented by eddy coefficients. The vertical
eddy coefficients for heat and momentum have the form
(2.29)
where e , e.., and c are constants which are different for heat and
momentum and R is the local Richardson number. (35)
The equations of motion are transformed into a vorticity equa-
tion and solved numerically. The results are physically reasonable,
indicating an initial entrainment near the discharge and a stable two
layer system farther from the discharge. Wada states that the theory
is valid for discharges in which the ambient turbulence controls the
diffusion processes, making e and e, relatively constant in space.
The details of Wada's numerical scheme are not given in his
papers and a calculation for only one case is shown. It is doubtful
that many field conditions correspond to Wada's problem. If the dis-
-39-
-------
charge channel is wide, the jet may behave initially like a two-dimen-
sional flow with little lateral variations in temperature and velocity,
but the turbulent structure close to the discharge origin is jet-like
and is not constant in space as Wada assumes.
Table 2-1 summarizes the studies from the point of view of the
processes affecting the temperature distribution and the number of phys-
ical dimensions considered. Each of the authors presents a satisfact-
ory theory based on the physical situation he assumes. None of the
theories consider all of the factors discussed in Chapter 1 as being
important for the prediction of temperatures in the vicinity of a heat-
ed discharge.
-40-
-------
Direction of
Temperature
Variation
Factors Affecting Temperature Distribution
Discharge
Initial Channel Bottom Surface Ambient Ambient
Investigator x y z Mixing Buoyancy Geometry Slope Heat Loss Cross Flow Turbulence
Edinger yes yes yes no no
(8)
no no
no
yes
yes
Carter yes yes no yes
(5)
no
yes
no
no
yes
no
Hoopes
(18)
yes yes no yes
no
yes no
no
yes
no
Hayashi yes yes yes no yes
(15)
yes
no
yes
no
no
Wada
(35)
yes no yes yes yes
yes yes
no
no
yes
This
Study
yes yes yes yes yes
yes yes
yes
yes
no
Table 2-1. Summary of Previous Theoretical Investigations
-------
III. Theoretical Ccmsideratiions
3.1 Statement of the Problem
The preceding chapter reviews earlier theoretical investigations
of heated surface discharges. Each is severely limited: only Hayashi
considers surface heat loss. There is no attempt to include buoyant
convective motions and none of the theories reproduce the initial core
region in a turbulent jet. There is a need for a three-dimensional
theoretical development which considers all these factors.
This study treats a horizontal discharge of heated water at the
surface of a large body of ambient water. Figure 3-1 shows the rele-
vant geometrical characteristics of the discharge and the definition
of a Cartesian coordinate system. A theory for predicting the three-
dimensional velocity and temperature distribution within the discharge
jet is developed in the following sections. The initial temperature
difference between discharge and ambient water, the discharge velocity,
the channel geometry, surface heat transfer, the ambient cross flow,
and the bottom slope will determine the temperature and velocities in
the jet.
The theory is limited to the region of flow where jet turbulence
dominates ambient turbulence. The ambient cross flow is not necessar-
ily uniform but far from the jet the free surface is uniformly at z = Q
and the temperature is T .
cl
The basic hydrodynamic and thermodynamic equations governing the
flow are first transformed into usable form for the case of a non-buoy-
ant discharge into deep, stagnant ambient water. The effects of an
-42-
-------
Initial Velocity » UQ
Initial Temperature = TQ
t
Bottom Slope - S
Front View
u ,T
o' o
r
Top View
Ambient Temperature = T
Figure 3-1. Characteristics of the Discharge Channel
-------
initial temperature difference are then included in the theoretical
model. Finally, a buoyant jet discharged over a bottom slope or into
a cross flow is considered.
3.2 The Governing Equations
3.2.1 Basic Equations
A fluid flow is characterized by its velocity, density, pres-
sure, and temperature, all of which may be a function of space and
time. These four variables are related by equations expressing con-
servation of mass, momentum, and heat energy and an equation of state
which relates density, pressure, and temperature. In Cartesian coor-
dinates the complete equations for water flow, incorporating the Bouss-
inesq approximation, are:
=
J"L
3t 3x
ay
3u 3u
at ax
3uv 3uw _
-
2
3_u
3z
(3.2)
jiw
at
at
3uw
ax
3v
p 3y
3vw
3y
3y
3w _ 1 3p
az p az
1 awT t
I • — K
[32I
U2
fa2*
32T
3y2
U
,
3y
A
. 2
3z
32wl
(3.3)
(3.4)
(3.5)
3jp_
3T
a = 0
(3.6)
-44-
-------
where u,v,w = velocity components in the x,y,z direction
p = density
p = pressure
T = temperature
v = viscosity
g = gravitational acceleration
k = thermal conductivity
a = thermal expansion coefficient
= viscous generation of heat
In conjunction with initial and boundary conditions the fluid
flow is determined by these equations. There is no closed form solu-
tion to the governing equations. Approximations must be introduced
which simplify the equations to the point where solutions may be ob-
tained. The transformation of the governing equations will consist of
dropping certain negligible terms, of assuming the form of certain
variables, and of integrating the equations over a region in space.
3.2.2 Steady, Turbulent Flow Equations
In accordance with standard practice when dealing with turbulent
flows the velocity, pressure, density, and temperature are separated
into mean and fluctuating parts:
u - u + u'
v = v + v1
w = w + w' (3.7)
P = P + P1
P • P + P'
T = f + T1 __
-------
In general the mean values may be functions of time when con-
sidered over periods longer than the periods of the turbulent fluctua-
tions. This study considers only steady mean flows.
The result of substituting Equations 3.7 into Equations 3.1 -
3.6 and taking a time average of each t_en.: .; the well known set of
turbulent mass, momentum, and htac. equations.
3y
3x
3y
3z
(3.8)
3u 3uv 3uw
3uv . 3uw 1 3p , _ „ .
^ •+• -z— + -r— = - —r-*- + v I—^ +
3x 3y az p 3x I „ 2
A O
u . 3*u . 3"u| 3u'2 au'v1
3x
(3.9)
8uv 3v 9vw
—- -j- r-T-. ^ ^. „
1 3p , I 3 v
= -^ + V r-
P °y I ~ 2.
1 LdX
9— 9 —
-«- *.£• ./
3 v a v 3 v
2 32 3«2
'] ,3u'v' 3v' 3v'w'
(3.10)
3uw 3vw 3w2
3x 3y 3T
3u'w' 3v'w'
3x ' 8y
(3.11)
3uT . 3vT . 3wT
r -^ —
3x ' 9y
)2T 82T
"" 7^~ ' f\"
3u'T' 3v'T' 3w'T
3y
(3.12)
(3.13)
-46-
-------
Henceforth the mean variables are written without the bar. Fluctuating
qualities are still designated by a prime.
3.2.3 Hydrostatic Pressure
The mean pressure is considered to be the sum of two parts:
P = Ph + Pd (3.14)
where the hydrostatic pressure,? , satisfies the equation
3ph
-g (3.15)
The remaining component, p , is a dynamic pressure resulting from fluid
acceleration.
Equation 3.15 may be integrated:
P = - ( Pgdz + p (3.16)
where z is a point at which p = p . In this case the elevation of
the water surface, n, is a point where p = 0. Thus
Ph = - ( Pgdz (3.17)
'n
The density is separated into components:
pa - Ap (3.18)
d
where p is the density of the ambient water and is not a function of
ci
space. The deviation from the ambient density is -Ap and in a heated
-47-
-------
jet results from a temperature difference as defined by the equation
of state. If T is the ambient temperature corresponding to p , then
a a
-f
'T
and
P - Pa - I adT (3.19)
3 'l
a
(T
Ap = \ adT (3.20)
'T
If Equation 3.18 is substituted into the pressure Equation 3.17:
Ph = - I Pagdz + I Apgdz (3.21)
'n 'n
The pressure terms in the momentum equations are then:
1^ 5p _ _ 3n _ 3 f 3Ap , 1 "''d
rz .,.. i 3P,
dz -
p 3x 3x p -Ap J 3x
a n
1 IE - „ 3H g fZ 3Ap dz 1 3Pd ,.
•H — — ..T .. s: » o •'• - ^ — •- • •!*-• — •' I » vi^i ^ •• • • •' • ' — — i j y 4 j
p 3y s 3y Pa~AP ) 3y pa~AP Sy ' J
_ I IE = „ - -1—
p 3z g p -Ap
a
3.2.4 Scale Variables
The dominant terms in each equation are identified by estimating
the scale of each variable and the magnitude of spatial changes in each
variable. For this purpose the following "characteristic" or "scale"
-48-
-------
variables are defined:
x*,y*,z* = scale of changes in the x,y,z directions
n* = scale of changes in the water surface elevation
u*,v*,w* = scale of the mean velocity components
p * = scale of hydrostatic pressure changes
p * = scale of dynamic pressure changes
AT* = scale of mean temperature changes
Ap* = scale of mean density changes
u'*,v'*,w'* = scale of the velocity fluctuations u1 ,v' ,w'
AT1* = scale of the temperature fluctuations
Ap'* = scale of the density fluctuations
0* = scale of the viscous heat production
In discussing relative magnitudes it is useful to define a quan
tity 6 such that the following sequence of terms:
,21 1
6 , 6, 1, j , ~2
o
is increasing and each term is an order of magnitude different from
adjacent terms. The symbols « and » are stronger statements of rela
tive magnitude.
3.2.5 Large Reynolds Numbers
The ratio of any of the momentum terms to the viscous terms is :
(u*,v*.w*)(x*,y*.z*)
v
-49-
-------
where the appropriate choice of length and velocity scale is made.
This ratio is the Reynolds number, R. A condition for fully developed
turbulence is that the Reynolds number be very large i.e. H » 1. The
viscous terms are then negligible with respect to the convective terms.
Since k is the same order as v, the molecular heat transfer terms are
also negligible.
In the heat energy equation, the ratio of viscous heat produc-
tion to heat convection is
(u*2,v*2w*2)
'
Heat Convection R c AT*
P
2 ? ?
(u* v* w* )
where c is the specific heat of water. The quantity - - ' * - -
°P
is at most of order 1 in the jet. The viscous production of heat is
therefore neglected.
3.2.6 Small Density Differences
The deviations from the ambient temperature, T , under consider-
C*
ation are of the order 0 - 30°F. The coefficient of thermal expansion
is considered a constant over such a range and Equation 3.20 becomes
Ap = a (T - T ) = aAT (3.27)
3.
For T from 40°F to 100°F the value of - varies from 0 to ,0002°F~1.
a p
Thus the maximum value of Ap is .006 for AT = 30°F and T = 100°F. It
* 3.
Pa
may be said that
« 1 (3.28)
pa
-50-
-------
The pressure Equations 3.22 - 3.24 are simplified by noting that
—±— : -±- (3.29)
Pa-Ap pa
Since AT'* can be no greater than AT*, it must be that Ap'* is
less than or equal to Ap*. The mass conservation equation is simplif-
ied by dropping all terms scaled by Ap* or Ap1* since these are negli-
gible compared to terms scaled by p . The mass conservation then be-
3.
comes the familiar continuity equation for incompressible fluids.
3.2.7 jSasic Scale Relationships
The continuity Equation 3.30 implies that:
v* - ^-r u* and w* - -—- u* (3.31)
The scale for free surface changes, r\*, may be determined by
considering the momentum equations at very large depths i.e. as z -v -».
The effect of the jet is negligible and there is no motion at such
depths. Equations 3.22 and 3.23 become:
3 0 /-°° a. -, 3p,
0 = - g M _ _S_ I JZ^fi. dx - — —- at z = -oo (3.32)
'a ' n a
a n
-51-
-------
It must be that
_ (3.34)
pag
In a fully turbulent shear flow the turbulence is locally iso-
tropic. The scale of the fluctuating components, (u'*,v'*,w'*), are
the same order of magnitude. Furthermore the mean velocity in the dir-
ection of the flow is an order of magnitude greater than the fluctuat-
ing velocities:
'*
3.3 Non-Buoyant Surface Jets
3.3.1 Governing Equations
This section considers the case of a discharge of water horizon-
tally from a rectangular open channel into a large body of water at the
same temperature as the discharge. There is no bottom slope and no
ambient cross flow. With these conditions the equation of mass conser-
vation 3.30 is exact and the heat and state equations are not needed
since p and T are not variables.
The momentum equations with all negligible terras dropped are
u*2
shown below. The scale of each term is divided by —r— for convenience
X
in comparison of terms:
-52-
-------
o o
3u 3uv 3uw _ jh} _ _1_ pd _ 3u' 9u'v' _ 3u'w'
3x
1
3x
3y 3z 8 3x p 3x 3x 3y Sz
Si
* * (3-36)
n U* o U* ^
"a * a
2 •}_ o
+ 3v + 3vw 3n 1 Pd 3u'v' 3v' 3v'w'
3y 3z 8 3y p 3y 3x 3y 3z
a J
(3.37)
JL jt JL * J "Jt P J A «L JL.
x* x* x* ~T2 y* 72 y*" & 6 y* 6 "z*"
pu* pu*
a a
3uw + 3vw + 3w = J^_ ^P_d_ _ au'w' _ ly'w1 _ 3w'2
3x 3y 3z p 3z ~ 3x 3y 3z
S (3.38)
j^ j* z* ^d x*^ x* x*
x* x* x* 2 z~* 6 6 7* 6 7*
pau* y 2
If the turbulent terms in the x equation are to be the same
order as the convective terms it must be that:
£ ~ 6 and £ - 6 (3.39)
The first turbulent term is then negligible in all the momentum equa-
tions.
In the y and z equations the turbulent terms are of order 1 if
3.39 is true. The pressure term must also be of order 1 since the con-
vective terms are of order 6:
-53-
-------
Pd*
n U*
pa
(3.40)
The pressure term in the x equation is then negligible. The approxi-
mate equations become:
2
3u 3uv , 3uw _ 3u'v* ju'w'
~
0 . _ g in
g
3y p 3y 3y
a
3.3.2 Structure of the Jet
The set of equations is a well known result for turbulent shear
flows. Further assumptions about the nature of the jet are taker, from
established experimental and theoretical results in Chapter 2.
The gross features of the jet are defined in Figure 3-2. The
core region has a half-width s and depth r. The turbulent shear region
expanding horizontally has width b; the vertical shear region is of
depth h. Section A-A in Figure 3-2 shows four regions in the jet cross
-54-
-------
Side View
Jet Boundary
//////
Turbulent
Region
fop View
'Core Region
Regionfeeglon,
1 3 | 1 1
(RegionReglonl
1 * 1 2 1
b s
Section A-A
Figure 3-2. Geometrical Characteristics of the Jet
-------
section: an unsheared core region (//I) : a vertically sheared region
(//2) : a horizontally sheared region (#3); and a region sheared in both
directions (#4). The jet is assumed to be symmetric about the plane
y = 0. Thus only half the width of the jet will be considered in the
theoretical development.
The core regions are eventually engulfed by the spreading tur-
bulent zones. The distance at which r and s become zero are x and
r
x respectively. Ifx
-------
I
Ul
Centerline Velocity
i—-Center-line Temperature Difference
•Temperature Distribution
•Velocity Distribution
I _Reglpn_l
t 1 t T tl t T~t t 11 7
Kegion 2
M t M Ml T t f I.
I
Figure 3-3. Velocity and Temperature Characteristics of the Jet
-------
The form of the similarity function, f, is a basic assumption of this
theory. A function introduced by Abramovich (1 ) for the main region
of plane and axisymmetric jets is used
f = (1 - ;3/2) (3-47)
Abramovich uses a different function for the core region. The reason
for using 3.47 throughout the jet in this theory is discussed in a
later section.
3.3.3 Boundary Conditions
The conditions at x = 0 are assumed to be:
1 " ho
s = b
° (3.48)
b = h = 0
u = u
c o
These conditions imply that there is no significant turbulent boundary
layer in the discharge channel.
The theoretical development treats each of the four jet regions,
Thus boundary conditions must be specified on the boundaries between
regions. In the following development ose will be made of the kine-
matic boundary condition which states that particles of fluid on an
impermeable flow boundary remain at the boundary.
z = n: The kinematic boundary condition holds on the free surface:
-58-
-------
at z = n (3.49)
There is no turbulent momentum transfer across the free surface
w'2 = u'w1 = v'w' =0 at z = n (3.50)
y = 0: The jet is symmetrical about this plane. There is no net flux
of mass or momentum across y = 0.
2
v = v' = u'v' = v'w' =0 at y = 0 (3.51)
z = -r: The boundary of the vertical shear region is not a rigid sur-
face but there is no momentum transfer across this boundary:
w?2= u'w' - v'w' =0 at z - -r (3.52)
It is assumed that the vertical velocity at z = -r is given by:
w = w 0
-------
= s: This boundary is analogous to z = -r. The conditions are:
t 2
u'v* = v'w' = v = 0
v=v -r
-------
y a s + b; This surface is analogous to z = -r-h. The conditions are:
v = v
v - v f( C )
e z
-r < z < n
-h < z < -r
at y = s + b
(3.57)
v - 0 z < -h
v«2 = u'v' = v'w' = 0
3.3.4 Integration of the Equations
To solve the equation set (3.41 - 3.49) it is necessary to inte-
grate the equations over certain portions of the yz plane such that the
contributions of the turbulent transfer terms, about which no assump-
tion has been made, are zero.
If the y and z equations (3.43 and 3.44) are integrated over
the entire yz plane of the half jet the result is:
p
- — dz (3.58)
y=0
0 -
g(r+h)
S
=b~
V
0
)
-c
p J
Q
(r+h) pa
y
=b
Pd
pa
,s+b p,
0 = - \ -^
Jo pa
z=-n
dy
-(r+h)
Using Equation 3.40 it must be that
(3.59)
n* u*
Z* gz*
(2.60)
It is assumed that the discharge channel flow is subcritical so
that the water level at x = 0 is controlled by the level of the ambient
-61-
-------
u
fluid. This is true if t'ae channel Froude number, IFO = , is
less than unity. It then must be that
i* < « (3.61)
and water surface elevations are negligible in comparison to the scale
of vertical changes in the jet. Integration over the jet in the ver-
tical direction may begin at z = 0 without significant error.
Equations 3.58 and 3.59 imply that the total pressure within
the jet is constant from the interior to the boundary. This pressure
may be estimated by Equation 3,40.
Ph* + Pd* " & pu* (3.62)
Consider the flow of fluid outside the jet. Entrained fluid is trans-
ported from rest far from the jet, where the pressure difference in-
duced by the jet is zero, to the boundary of the jet where it has
velocity w or v . The flow outside the jet is nearly irrotational and
Bernoulli's theorem may be applied along a streamline.
t*\ _1_
£ G d
(3.63)
0 =-^pw + p,
2 e d
or using 3.62
-62-
-------
v - 6u*
e
w - 6u*
e
(3.64)
The entrainment velocities are proportional to the local longi-
tudinal velocity scale and are an order of magnitude less. This rela-
tionship is formalized by introducing entrainment coefficients:
v = a u
e y c
w = a u
e z c
(3.65)
where a and a are variables to be determined by the theory.
The x momentum and the continuity equations are integrated over
each region of the jet using (3.41 and 3.42) the similarity forms for
velocities, the boundary conditions, and the entrainment definition:
Region //I - continuity
du
rs - — + rv - sw =0
dx s r
Region # 2 - continuity
du h
-dx- + SUc d^ + Vbhll + S
Region // 3 - continuity
du b .
C. Q.S
I,r — j — + ru -j— - w, bl, - r [v -au] = 0
1 dx c dx hi s y cj
-63-
-------
Region //A - continuity
2 du hb .
*i ~~j +u *i [ b -; l-h-r-]+ [w -a u ]I,b - (v,-a u )I,h = 0
1 dx clL dx dxj hzcl bycl
Region //I - momentum
du
2 rs -~- + r v - s w =0
dx s r
Region it 2 - momentum
du 2h 2
s I, —1=— + s u -r^ + u v, hi. + u sw =0
2dx cdx cb2 cr
Region // 3 - momentum (3.66)
•) cont'd
duc b 2 ds
r I9 —, + r u -7^- - u w, bI0 - u rv =0
2dx cdx ch2 cs
Region // 4 - momentum
2 du hb
12 -ir- + \\ ^ If + b & + «ci2 (whb
r1 r1
= ( f(c) dc I, = \
'0 * JQ
2
where I = f(c) dc I = \ f
These eight equations are not^ sufficient to determine the eleven
variables. The following experimentally observed relationships are
assumed (1 ):
-64-
-------
S - E
Abramovich (1 ) notes that the value of e is different in the
core regions from that in the outer jet. He also uses a different
similarity function for the core regions and matches the inner and
outer regions by allowing h and b to change over a transition region.
This difficulty is bypassed by allowing e and f to be the same every-
where. The results for the outer region are the same as in Abramovich's
work and the differences in the core region are small. It is assumed
that e = .22 for all values of x.
3.3.5 The Solution for Non-Buoyant Jets
The solution of the Equations 3.66 - 3.68 is:
For all x,
v = v, = w = w, = 0
s b r h
h = b = ex
ho bo
Initial Region x < —— , x < —
u = u
c o
r = h - el_x
o 2
s = b - eI0x
o 2
-ay = az = eC^ - I,,)
-65-
-------
For h < b
o o
el,
el.
u = u
c o
r = 0
s = b - eI0x
o 2
For b < h
o o
el.
< x <
el.
u = u
c o
r =
(3.69)
-
For x >bQ/eI2 , x >
Jh b
• o o
u = u
C O I „ '-X
-66-
-------
s = 0
Defining the dimensionless variables:
x =
/h b
o o
fh b
o o
(3.70)
fh b
o o
f h b
o o
'h b
o o
and the constants:
1
el.
G> _i_
s >/h eI0
(3.71)
A = r—
-67-
-------
the solution for velocity and jet dimensions is:
b = h = ex x>0
u - 1.0
> ;<
Sk
x and x
s = 0
X < X < X
s r
r = 0
/A
X < X < X
r s
(3.72)
u
r = s = 0
x > x and x
s r
-68-
-------
For the assumed similarity function:
.450
(3.73)
.316
and thus since e « .22;
x - 14.A ft.
(3.74)
14.4
x =
A plot of these solutions is shown in Figure 3-4. The data
shown is from experiments by Yevdjevich (38) for turbulent air slot
jets with values of A from 1.0 to 94.0. Scaling x by /h b allows all
of Yevdjevich's data to be plotted on a single line, where he presents
separate curves for each value of h /b and scales x by b .
o o o
This section has considered a non-buoyant horizontal discharge.
The solution is obtained by proposing similarity forms for velocity
and temperature and by assuming linear jet spreading as discussed in
Chapter 2. The definition and determination of entrainment coefficients
are not necessary for the solution of the equations. However, in the
following section, which treats buoyant discharges, the linear spread
assumption is not valid and the entrainment coefficients determined
for non-buoyant jets will be used to give information about entrainment
-69-
-------
c
I
1000 2000
Figure 3-4. Non-Buoyant Jet Solutions: Data from Yevdjevich: 1< A< 94
-------
in the buoyant case.
3.4 Buoyant Jets
3.4.1 Governing Equations
This section considers the horizontal surface discharge of water
from a rectangular open channel into a large body of water. The temp-
erature of the discharge is greater than that of the ambient water.
There is no bottom slope and no ambient cross current, these effects
will be included later. The heat energy and state equations are coup-
led with the mass and momentum conservation equations.
The scaled momentum and heat equations (3.9 - 3.12) are:
3u + auv + auw _ in _ JL ( °°3Ao dz _ _i
3x 3y 3z ax p ) 3x p
a. T\ o.
Pa"
gz*
u*
ax
ay
(3.75)
<5 6 „* 6 T*
x*
z*
3uv H
ax
y*
av
ay
, 3vw an g 1
az ay p
a
">»*"
. p.*
y* d
9AD dz 1
) 9y p« 9i
n "
^8Z* x*
d 3u v
i ax
r
'
X*
y*
(3.76)
auv/
ax
z*
X*
3vw 3w 1 d au w
+
ay
z*
X*
T —
az
z*
X*
Pa 9Z 8X
n *
Pd x*
*2 z*
pu*
a
_ av w
ay
A X*
S*
r\\J
az
A x*
0 £
-71-
(3.77)
-------
3x 3y 3z 3x 3y 32
(3.78)
111 6 67$ <5 77T
If, in the x equation (3.75), the pressure term related to the
density difference is to be significant, it must be that:
Ap*
- 1 (3.79)
g Pa
u»
The y equation (3.76) indicates that if y*/x* - 6, there is no term of
sufficient magnitude to balance the y pressure terms which would be of
order 1/6 . The conclusion is that
y*/x* - 1 (3.80)
and as in the non-buoyant jet, for some turbulent term to be important
in the x equation,
z*/x* - 6 (3.81)
The z equation (3.77) implies that:
P/
-S- - 6 (3.82)
pu*
-72-
-------
The equations may now be written without the negligible terms.
The progression from the non-buoyant jet to the buoyant case is of in-
terest. For this reason all the terms appearing in the non-buoyant
equations are retained. In addition the free surface variation terms
are eliminated by using Equations 3.22 and 3.23. The density differ-
ence is replaced by the temperature difference using Equation 3.27.
H. + = 0 (3>83)
ax 9y 3z
3uw _ ag f
32 ~ P., )„
ML dz . inbLl _ inl^l (3.84)
3x 9y 3z p 9x 9y 9z
3. Z
9uv 9v2 3uv _ ag (~°° 3AT 9P '
3x 3y 3z p J 3y pa 3y 3y 3z
£1 Z 3
3z 3y 3z
(3>86)
3vAT 3wAT = _ 3v'AT' _
~ ~ ~
3x 3y 3z 3y 3z
3.4.2 Structure of the Jet
A comparison of the non-buoyant governing equations (3.41 - 3.44)
and those derived for buoyant jets (3.83 - 3.87) indicates the effects
of adding an initial temperature difference to the non-buoyant case:
-73-
-------
1) Buoyant forces in the x direction tending to accelerate
the jet.
2) Buoyant forces in the y direction which increase the lateral
convection.
These effects are incorporated into the jet structure as defined for the
non-buoyant jet. All the definitions of initial channel geometry, jet
regions, velocity distributions, and velocity boundary conditions are
assumed to be the same as those defined for the non-buoyant jet as in
Sections 3.3.2 and 3.3.3.
The temperature is assumed to be distributed as follows:
AT = AT T (y) T (2) (3.88)
i_ y L
where AT is the centerline temperature difference. The similarity
function for temperature is related to the similarity function for velo-
city by Equation 2.13:
T = 1.0 0 < y < s
y
T = t(£ ) s
-------
where
t = /f = 1 -C (3.90)
The boundary conditions on the turbulent transfer of heat are
as follows:
w'T' =0 z = -r -h
(3.91)
v'T' = 0 y = 0 and y = s + b
i.e. no transfer of heat is allowed from the jet into the ambient fluid.
w'T' = K(T - T ) at z = n (3.92)
3.
where K is the surface heat transfer coefficient as defined in Equation
2.28. It is assumed that the ambient temperature T is nearly the eq-
Q.
uilibrium temperature T .
3.4.3 The y Momentum Equation
The y momentum equation (3.43) formulated for a non-buoyant dis-
charge expresses a balance between turbulent momentum transfer and the
dynamic pressure gradient. The integrated form of Equation (3.43) im-
plies a relationship between the centerline jet velocity and the en-
trainment velocity which is formalized by defining a lateral entrain-
ment coefficient, a , as in Equation 3.65. The y equation in the case
of a buoyant discharge (3.85) contains lateral convective terms and a
-75-
-------
pressure gradient resulting from a lateral temperature gradient. It is
assumed that the non-buoyant balance between lateral turbulent entrain-
ment and the dynamic pressure as expressed in Equation 3.34 holds for
the buoyant jets and that the discussion in Section 3.3.4 is valid for
horizontal entrainment:
(3.93)
where a is assumed to be the same for each jet region as determined
for the non-buoyant jet (Equations 3.69).
Subtracting Equation (3.43) from Equation (3.85) yields a rela-
tionship batween the lateral temperature gradient and the lateral con-
vective terms
p 3y
3 Z
This equation may be integrated over the entire yz plane of the
half jet:
b+Sro T
dy
(b+Sro /s+b fo (
£ uv dzdy - *& \
'o '-h-r pa > o y-(r+h);
where ^ Is a dummy variable for z in the integration. In (3.94) a
lateral velocity, v, appears for which there is no previously assumed
distribution which will make the integration possible. This lateral
-76-
-------
velocity is a consequence of lateral pressure gradients and causes the
lateral jet spread to be greater than the non-buoyant linear spread,
It is assumed that the lateral spreading velocity is related
to the difference in the buoyant and non-buoyant spreading.
I db i*-*1-11 i /-» r»/• \
IS - feU" 0.96)
Since the spreading is induced by the lateral temperature difference in
the jet, the distribution of v is assumed to be proportional to the tem-
perature gradient in the y direction which is:
If = 0 0 (1 - Cy > Cy / s < y < b
v • 0 b < y (3.98)
-77-
-------
3.4.4 Vertical Entralument:
In Section 3.3.5 the solution of the non-buoyant jet equations
included vertical and horizontal entrainment coefficients, a and a .
y 2
In the previous Section 3.4.3 it is assumed that for buoyant jets the
lateral entrainment is not affect--" by the ; is the same as derived for the non-
buoyant jet. As discussed in Chapter 2, vertical entrainment is re-
duced in the presence of a vertical density gradient. In this section
a relationship expressing the reduction of entrainment as a function of
the local Richardson number (2.17) or alternately the local densimetric
Froude number (2.18) is formulated. It is assumed that the entrainment
for zero density gradient is, a , the value derived for non-buoyant
ti
jets.
The theory by Phillips presented in Chapter 2 is used to relate
the vertical entrainment to the local temperature gradient. Hinze indi-
cates that a reasonable form for double correlation functions such as
R in Equation 2.24 is:
2 2
- e~5 C (3.99)
Then it is true that:
G - |r (3.100)
o *•*-
and using (2.26) with a - a :
(3.101>
-78-
-------
where a is the reduced vertical entrainment coefficient.
sz
Hinze points out that for flow advected in the x direction with
velocity u, the integral time scale G is G = A/u where X is the inte-
gral space scale of the turbulence. Furthermore in free turbulent jets
A is proportional to the width of the turbulent region (34). It must
then be true that for vertical turbulent motions:
G ~ - - h/u
o c c
2
since N (see Equation 2.22) may be approximated by
(3.102)
N
_£.
P_
(3.103)
the entrainment relationship becomes:
sz
(3.104)
where IF is a local densimetric Froude number which corresponds very
LJ
closely to that defined by Ellison and Turner. Their data, see Figure
2-2, indicates that 6 ; 5.0. Thus it is assumed that in the buoyant
jet:
sz
a exp
z f
-5.0
P u
a c
(3.105)
-79-
-------
3.4.5 Integration of the Equations
The momentum and continuity equations are integrated separately
over the four jet regions shown in Figure 3-2 as in the case of the
non-buoyant jet. The heat equation is integrated over the entire jet.
The free surface elevation is estimated as
« 1 (3.106)
'"a
and may be neglected in the integration. The complete set of equations
are given in Table 3-1.
The equations governing the non-buoyant jet are a system of
non-linear first order differential equations. There is in general
more than one solution curve, the correct choice being determined by
the conditions at x = 0. Physically a jump from one solution to another
is possible as long as momentum, mass, and heat are conserved without
a gain in mechanical energy. The system of equations may also have
singular points where some or all of the derivitives are infinite.
This study will not treat the possibility of jumps in the jet solution
since downstream conditions must be specified.
3.5 The Deflected Jet
3.5.1 The Governing Equations
This section treats a jet discharging into an ambient cross
flow. Figure 3-5 defines a coordinate system (x, y, z) which is orien-
ted along the centerline of the jet. The angle between the x and y
axis is 8, the jet being discharged with an initial angle 9 . The
-80-
-------
Region //I — continuity
du
Q
rs —7— + rv - sw =0
dx s r
Region # 2 - continuity
duch dr
-dx~ + SUc dJ + vbhll + S Iwr -"sz
Region // 3 - continuity
ducb
I, r -,C + ru -p- - w.bl, -r [v -au] = 0
Idx cdx hi s y c
Region // 4 - continuity
du hb
I, —§— + u I, [b %L + h 5s-| [w -a u ]I.b - (v -a u )I1h - 0
1 dx c 1 L dx dxj l h szcj 1 b y c' 1
Region // 1 - momentum
j 9
pa
du 2h n J r dAT h2
r H r d 1
U |2rs -~ + r v - s w 1 + ags -r~AT -=- + I,,r — AT h = 0
cL dx s rj —~—Idx c ^ J dx c J
Region // 2 - momentum
du h _ , r QAI n , i
s I~ —§— + s u -r^- + u v, hi. + u sw + ags I. —-7=— + I,AT h -~\m
2dx cdx cb2 cr —'°— L 4 dx 3 c dxj
a
Region // 3 - momentum
duc2b 2 ds r13 dATcbr2 2 dATcbh
r I0 —3— + r u -7— - u w, bl,, - u rv +aa 1 +1 r —
2dx cdx ch2 c s —fi |_ 2 dx 3 dx
Table 3-1. Integrated Equations for Buoyant Jets
-81-
-------
Region // 4 - momentum
2 2
2 du hb o j H r dAT h b
X --+ u h + b ] + ( - vh) +
, - . dx
a
, AT h2 4s" + I,2 AT hb ~ | = 0
4 c dx 3
Chbfl .
c dxj
y-momentum
__
dx
heat:
~
where
^f
'n
hI7) (s + bI7)J -f KATc (s
= ( (1 -C3/2) d£ = .4500
- f
fn
f1 2 ^ 3/2 4
I2 = \ f (^)dc = \ (1 - C ' ) dC = .3160
'o
= \ t(5)d£ = f
»n 'n
(1 - C)d; = -6000
Table 3-1. Integrated Equations for Buoyant Jets (.cont'd)
-32-
-------
( [ t(c) dcdc = [ I
Jo >c 'o /
I = t(c) dcdc = I (1 - C)0
(I - I )e r ?0
r = 0 pauc
hn
1
e = spreading rate of a free turbulent region = .22
Table 3-1. Integrated Equations for Buoyant Jets (cont'd)
-83-
-------
db
dx
u
AT
o
= b = 0
= e
= u
o
= AT
Table 3-1. Integrated Equations for Buoyant Jets (cont'd)
-84-
-------
Cross Flow = V - V(x)
I
00
Top
View
Figure 3-5. Characteristics of a Deflected Jet
-------
cross flow velocity, V, may be a function of x but not y.
It is assumed that the deflection of the jet is sufficiently
gradual that locally a cylindrical coordinate system nay be defined in
ae ~1
which rde = dx, dr = -dy, and r = - ~
The governing equations for buoyant jets, (3.92 - 3.96) then
contain extra terms which account for the jet curvature:
3v + + v =0 (3.107)
By 3z 3x '
2 /•—00
t -Tt f\ _-w_?t- r\ ^.Ti *> rt — -_ f £\ AT1
f
(3.108)
jluy , _3v^ jyw -2 _3_6 ~2 ^6 = a£ ( °° 3AT . J^ _.^d. 3v]2
r\ T ""•*" n. "— "•" ^T^- ~"~U «*• i V n."' I 'i-, U ti - !5^r ^iv
3x 3x P ) ~ 3y P 3y 3y
UA "z a
8v>w' + a'2 li - v'2 |^ (3.109)
3z 3x ox
.
3uAT A 3vAT , 3wAT _ Sv'AT* 3w' AT
-- + ~~" "~ " " "
(
In the following treatment the effect of the cross flow on the
jet will be expressed in terms of integrations of the equations of
-86-
-------
motion over the entire jet. These results will be incorporated into
the structure of the theory for the undeflected buoyant jet. A major
assumption of the following development is that the cross flow velo-
cities are small compared with the jet centerline velocities i.e. defin-
ing a cross flow scale velocity y*:
^* - 5 (3-112)
If the x and y equations (3.108 and 3.109) are integrated over
the entire jet (see Appendix II),
o s+b
' dvdz
y
fd fS+b _ fo fs+b f- a-
-1 \ I u ^~z =M ( ^
dx J-(r+h) /-(s+b) pa J-(r+h) •'-(s+b)/z dx
+ q V cos 6 (3.113)
, /o /"S+b
a ( f
•'-(r+h) >-(s
where q is the total entrainment into the jet per unit distance along
the x axis. The derivation in AppendixII assumes 1) symmetry of the
jet about the y = 0 plane and 2) a potential flow outside the jet which
is equivalent to assuming that there is no pressure drag force on the
jet.
Equations 3.113 and 3.114 are valid for the whole jet. The
-87-
-------
effect of the cross flow is 1) to introduce an additional term in the
ic momentum equation to account for entrainment of cross flow x momen-
tum and 2) a bending of the jet so as to balance the entrainment of
cross flow y momentum. The cross flow has the result of making the
boundary conditions on velocity unsymmetrical around the jet boundary
so that strictly speaking Equations 3.113 and 3.114 do not hold for
any separate region of the jet. The jet will no longer be symmetrical
and the analysis of the preceding sections on buoyant and non-buoyant
jets cannot be applied.
It has been assumed, however, that for small cross flow velo-
cities relative to the jet velocity the distortion of the jet is small.
To apply the governing equations to the four separate jet regions the
following is assumed also:
1) Equation 3.113 which holds for the entire jet is assumed
to hold for each of the four jet regions separately,
i.e. the effect of the cross flow on the integrated x
momentum equation (3.108) is to add a. term which is
the product of Vcos6 and the entrainment per unit length
into the jet region being considered.
2) The similarity functions for velocity and temperature
within the jet are all the same as for the undeflected
jet except for u which is given by:
u = u F F + V cos 6
c y z
where F and F are as defined in (3.46),
y z
-88-
-------
3) The boundary conditions are the same as for the undeflected
jet except the conditions on the velocity at the jet boun-
dary which are:
w = w - V cos 6 -rr 0 < y < s
e dx J
w = w f(£ ) -V cos 6 -pr s < y < b
v = v + V cos 6 -rr -r < z < n
e dx
v-vf(z;)+V cos 6-7^ -h < z < -r
e z dx
z = -r-h
(3.116)
y = s + b
where w and v are related to the centerline velocity, u , by the
entrainment coefficients a and a as defined for the undeflected jet.
sz y
4) Equation 3.114 which holds for the entire jet is assumed
to hold for each half of the jet on either side of the
surface y = 0. If the y momentum equation (3.109) is then
integrated over the half jet as in Section 3.4.3, and 3.114
is used, the result is a lateral spreading equation iden-
tical with (3.95) in the deflected coordinate system. The
lateral spreading velocity is assumed to be distributed
as in Section 3.4.3 taking into account the different dis-
tribution for u i.e. for s
-------
(3.117)
and v = 0 elsewhere.
fab'
For deflected jets, the non-buoyant spreading rate,
is no longer a constant but is defined by the assumption that the en-
trainment coefficients a and a are the same, in the deflected non-
y sz
buoyant jet as in the non-deflected, non-buoyant jet. It is also as-
sumed that -r Fl is negligible i.e. that a non-buoyant, deflected
dx (dxJNB
jet boundary will not have a large curvature.
5) In the continuity Equation (3.107), the curvature term
is negligible with respect to the other terms. Equation
3.114 implies that:
d 6 *
^- - * S (3-118>
Thus this assumption is consistent with the basic premise (3.112).
With the above assumptions, the integrated equations governing
a deflected, buoyant jet may be obtained with (3.114) determining the
new variable 9. The coordinate system x, y, z may also be related to
the x,y,z system by geometrical considerations. The complete equation
set is given in Table 3-2.
3.6 Bottom Slopes
The presence of a solid boundary complicates the derivation of
-90-
-------
Region 1: continuity rs -r=- [u + Vcosfl] + rv - sw =0
dx c s r
Region 2: continuity s [— [h(u I + Vcos6)] + (u + Vcos6) -^ + w - a uj + v hi = 0
QX C J. C QX TL S Z Cr D X
Region 3: continuity r [— [b(u I + VcosS)] + (u + VcosG) -p - v +au] - w bl = 0
QX C J, C CtX S y C JT J_
Region 4: continuity -pr fhb(u I, + VcosO)] + (u I, + VcosO) |b ~ + h ~\ + (w, - a u )I,b
0 J dx L c 1 /J c 1 L dx dxj x h sz c' 1
T - (v. - a u ) I h = 0
b y c 1
d d r2 d
Region 1: x momentum (u + Vcos6)[2rs ~ [u + Vcos9] + rv - sw ] + —& s [-p- (AT —)+I r— (AT h) J
a
0
[jo 99 9 H T
-pr [h (u I. + 2Vcos6u I, + V cos 6)] + [u + VcosG] -£
dx c 2 c 1 c dx
dAT h2 -
+ w [u + Vcos9] - a u Vcos6 + ^*- [I, —T: + I0AT h -—• ]| + v,h (u 3
re szc p 4dx 3 c dx j b c
Table 3-2. Integrated Equations for Deflected Buoyant Jets
-------
Region 3: x momentum r -r=- [b (u !„ 4- 2Vcos6u I + V cos 0)] + [u + Vcose] -jl- -
LQ.A C fc C X C Q.3C
., r I dAT br2
v [u + VcosQ] + a u Vcose - w.b [u I0 + VcoselJ + ^ ~ —7^
sc yc J hc2 1 p U 2 dx
SL
dAT bh 2
. _. £ ^« | . fji t 1_ "f
j 92 222 2 99
Region A: x momentum -r=- Ihb(u I0 + 2Vcos6u I, -f V cos 9)] + [u I. + 2Vcos0u I,+V cos 6]
° dx c L cl Ic2 Cl
[ dr , . i~.
Jb -r^ + " "3T
L dx dx_
dAT h'b
.
p L 3 4 dx Ac dx 3 c dx
cl
Jet y momentum ~ fe - B|] ] Cu/bl^r + hI2) + 2Vcos6ucbI5(r + hlj_) + V2cos20b(r+h)]
• nJK •
- ^ AT ( \ +
Po C 2
Table 3-2. Integrated Equations for Deflected Buoyant Jets (cont'd)
-------
Jet heat
df f
ucATc(s + bV (r + hI7) + VcoseATc
(s + bI3) = 0
Jet bending
(s + bI2) (r + hI2) + 2Vcos6uc (s + bl^ (r
I
?
Jet x position
V2cos20(s-l-b) (r + h)J ^| - ucVsin0[-as2(s + bl^ + a (r + h^
Jet y position
•
dx
- cose - 0
where:
' I7 are as defined in Tatle 3-1
Table 3-2. Integrated Equations for Deflected Buoyant Jets (cont'd)
-------
db
— = non-buoyant spreading rate calculated from above equations
dx
NB
with AT = 0
s > 0
s = 0
i
VD
4>
a
- I)e
r = 0
a = a exp 1 -i.O
sz *
[-'•'
a c
e = spreading rate of a free turbulent region = .22
r = h
db
dx
u
u
X
o
y = h
AT = AT
e = e
I
J
x = 0
Table 3-2. Integrated Equations for Deflected Buoyant Jets (cont'd)
-------
the governing equations since a number of the basic assumptions made
for no slopes are no longer valid as discussed in Chapter 2. The jet
will remain attached to the bottom until adverse pressure gradients or
buoyancy effects cause separation, at which time vertical entrainment,
until then inhibited, will begin. This study will not attempt a theory
which predicts the point of separation due to adverse pressure grad-
ients. It is assumed that the jet will not separate from the bottom,
over the entire width of the jet, until the buoyant effects of lateral
spreading cause the slope of the bottom of the jet to be less than the
bottom slope i.e.
^ + ^ < S (3.119)
dx dx x
Before this point, vertical entrainment, w , is zero and the jet is
attached to the bottom:
• + ^=5 (3.120)
dx dx x
In addition, the presence of the bottom invalidates the expressions for
pressure gradients since the pressure gradient at z = -°° is no longer
known. It is assumed that until buoyant separation occurs that the
buoyant terms have no effect on the jet.
The formulation of Equation 3.120 in the pre-separation zone,
requires the relaxation of one of the other governing equations. It
is assumed that frictional losses are negligible and that the x momen-
tum equations are valid. Since the non-buoyant y-equations result in
-95-
-------
•^ = T~ which is known to be true only for jets free from solid
dx dx
boundaries, the y-equation is dropped.
The calculation of a jet over a bottom slope proceeds by testing
at each x step whether, if the bottom slope were not present, the ineq-
uality 3.119 would hold. If the jet is spreading less than the bottom
slope, calculation proceeds as if the bottom were not there. This
assumes that once separation occurs, the losses in the entrained flow
between the jet and the bottom are small. If the jet is spreading
more rapidly than the bottom allows, Equation 3.120 replaces the y-
equation and the calculation proceeds with that set of equations. This
formulation is valid with and without cross flows.
3.7 Solution Method
The solution of the governing equations is done using a fourth
order Runge-Kutta integration technique supplied in the IBM Fortran
Scientific Subroutine Package. The routine chooses a step size to meet
a specified error bound for each step. The calling program keeps track
of which region the calculation is in i.e. whether r or s are zero and
supplies the appropriate equations. The actual IBM routine was modified
slightly to prevent negative values of the physically positive vari-
ables. A listing of the program and an input format is given in Appen-
dix I.
The results of the calculations are output in dimensionless
form with u , AT and /h b being the normalizing velocity, tempera-
ture and length respectively.
-96-
-------
IV. Experimental Equipment and Procedure
4.1 Previous Experiments
As mentioned in Chapter 2, experimental studies of heated sur-
face discharges have been done by Wiegel, Jen, Stefan, Tamai, and
Hayashi. Some of the data from these experiments are compared with the
theory of this study in the next chapter. An examination of the experi-
mental techniques of the previous investigators is useful in designing
improved experiments and in avoiding duplication of effort. Table 4-1
gives a summary of the characteristics of each previous experimental
study:
1) Interest has clearly focused on low densimetric Froude
number jets. Only Stefan considers a case where
IF < 1.0 and a wedge of ambient water intrudes into the
discharge channel. Good control of discharge flow and
temperature is necessary if IF is to be held constant
during the course of an experiment. Wiegel1s and Jen's
experiments have a large variation in F for each case.
2) The magnitudes of the Reynolds numbers in Table 4-1
reflect the difficulty of achieving a large Reynolds
number in a laboratory experiment, especially when low
Froude numbers are required. Wiegel, Stefan and Hayashi
report values of Hclose to or below 3000, the minimum
value for fully developed turbulence in circular jets
as discussed in Chapter 2.
3) The dimensions of the receiving basins are typical of
-97-
-------
Investigator IF
Boundary
„, Maximum
Flow
Receiving Basin Control x/A b
o o
Measurements
Jen (21) 18-180 8300-21000
Thermocouples
26' x 15' x 1.5' none .010'
-------
university laboratory facilities except for Tamai who
uses a narrow channel. It is likely that the side walls
of Tamai's channel severely limit lateral mixing and
spreading, thus distorting the jet behavior. For this
reason, Tamai1s data is not discussed in Chapter 5. None
of the experiments considered cross currents in the re-
ceiving basin.
4) Only Stefan provides for a disposal of the heated discharge
at the boundary of the basin. Such control is necessary
if steady state conditions are to be achieved during an
experimental run. Boundary flow control may also reduce
eddying of the ambient water in the far corners of the
receiving basins.
5) Low densimetric Froude number discharges require large
channel dimensions if Reynolds numbers are to remain
high. The available scaled distance, x//h b , decreases
as IF decreases. Many of Stefan's results are for x//h b
less than 50. In such a limited distance the differences
in jet behavior between separate runs are small and it
is difficult to determine the effects of the controlling
parameters.
6) As noted by Wiegel, buoyant surface jets have a tendency
to meander in the horizontal plane about a mean centerline.
A fast scanning multi-probe temperature measurement system
-99-
-------
is best for obtaining an accurate three-dinensional
determination of temperature. Measurements with a sin-
gle movable probe such as Stefan uses must account for
the unsteady nature of the flow.
7) The velocity of the discharge in the laboratory is
usually less than 1 ft/sec and decreases rapidly away
from the jet origin. A three-dimensional velocity deter-
mination requires a probe capable of measuring velocities
less than .1 ft/sec. Only Stefan uses such a probe during
the experimental runs.
8) None of the previous experiments involve control of the
surface heat exchange from the water in the receiving
basin. This requires a room in which temperature and
humidity are regulated. This is usually difficult to
achieve in a room large enough to hold the experimental
basin necessary for jet studies.
In light of the above discussion of previous studies, the exper-
iments in this investigation are designed with the following objectives:
1) Close control of discharge flow and temperature.
2) Reynolds numbers greater than 10,000.
3) Boundary flow control which allows operation of the
experiment without a significant change in the steady
state behavior.
4) Values of x//h b greater than 50.
-100-
-------
5) An accurately positioned, fast scanning temperature
measurement system.
6) The capability for generating cross flows in the
receiving basin.
7) Addition of a sloping bottom in the receiving basin.
The measurement of velocity and the complete control of heat transfer
are considered beyond the resources of this study. However, the labora-
tory is air conditioned so the ambient temperature remains reasonably
constant.
4.2 Experimental Setup
Figure 4-1 shows the arrangement of the experimental equipment.
The receiving basin is 46' x 28' x 1.51. All the walls within the
basin, including the discharge channel, are constructed of marine ply-
wood. The adjustable sluice walls along the sides are held by c-clamps.
The probe platform rests on four leveling jacks which may be reached
from the sides of the basin. Point guages at each corner of the plat-
form assure accurate control of the platform level. The sloping bottom
is a lightweight aluminum frame over which industrial plastic sheeting
is stretched. The water level in the basin is monitored by a fixed
point guage.
The heated discharge is supplied from a steam heat exchanger
capable of delivering up to 60 gallons per minute of water at a cons-
tant temperature up to 150°F. The flow is introduced into the discharge
channel through a gravel filter to insure a uniform velocity distribu-
tion in the channel.
-101-
-------
Ba
1
O
NJ
1
t>" Pipe.
9
Sluice Wall
Side Channel
Hose End
Support
Frame
8' Manii Id 0 LI iy***^
1 — * u Pip| D'D "
1=^. To Pumping System
1
/i
B
B
• • : • 4- - :
' PVl Pipe
-•:••-?-• V\V« : __
tintlflrmmimyr " - " -^
U-*D ^-Hose Manifold
Wooden Probe Frame
y Sluice Wall
v ^*\
1
A
»— •
jBl""
Leveling Jack
^Aluminum
anne ^
c
Discharge Channel
travel Filter
C
j
anger
hannel Bottom Channel
> » ,M^£^ *i-
'
1
Adjustable Wai 1
/^xFixed Wall
HP-R B u
l...ftr
-rrrrrl^r
c-c
1
c
f-" Pipe
^
" L
^^ I.1 Pumping Svstem
•> It.
scale
B-B
Figure 4-1. Experimental Set-Up
-------
Boundary flow control is established at the sides and at the
rear of the basin. Water may be pumped into or out of each side chan-
nel to create a cross flow in the receiving basin. At the rear of the
basin there is a manifold consisting of 40 lengths of 3/4" hose with
one end mounted on the basin and the other end tapped into a 30' x 8"
PVC pipe outside the wall. The position and orientation of the hose
ends in the basin are adjustable along a frame suspended parallel to
the back wall.
A five horsepower centrifugal pump is used to provide two modes
of boundary control. Water may be withdrawn from the rear of the basin
through the. hose manifold and returned to the side channels or a cross
flow may be established in the basin by pumping from one side channel
to the other. A drain line and a cold water inflow line are connected
to the pumping system to permit replacement of heated water withdrawn
from the basin by new colder water.
All flows are monitored by Brooks rotameter flowmeters except
for the drain and cold water input flows which are measured with sim-
ple Venturi meters. Most of the pipe fittings and valves are schedule
80 PVC plastic. Water hose is used in place of pipe whenever possible.
The temperature measurement system consists of 67 Yellow Springs
Instrument #401 thermistor probes, 59 of which are mounted on the basin
platform (see Figure 4-2) and the remainder in the various inflow and
outflow pipes. The probes are connected to a Digitec (United Systems)
scanning and printing system capable of a scanning rate of 1 probe per
second. The temperature in degrees Fahrenheit is printed on a paper
-------
Figure 4-2. Mounted Thermistor Temperature Probes
(scale in inches)
-104-
-------
tape. The probes have an accuracy of about -f . 25°F with an interchange-
ability of + .05°F. The response time constant of the probes is 7 sec-
onds which filters out turbulent fluctuations. The probes are leveled
using the water surface as a reference plane.
The laboratory building is air conditioned and the temperature
is relatively constant. This has the advantage that surface heat loss
rates do not vary greatly. The wet and dry bulb air temperatures are
measured using two thermistor probes, one of which is covered by a
water saturated gauze strip, mounted in an air flow provided by a small
blower. The wet bulb and dry bulb temperatures are constant at about
b6°F and 73°F from day to day.
4.3 Experimental Procedures
An entire experimental run is completed in about one hour. Sur-
face temperatures are scanned initially to insure that no ambient temp-
erature gradients are present. The basin boundary flows are then es-
tablished. In the case of no cross flow, the back wall manifold is
primed and an equal flow to each side wall of 50 gpm is established in
all cases. The back manifold does not function well as a selective
withdrawal device and cannot withdraw from a surface layer without suc-
tion of air. It is used solely as a means for maintaining the basin
level and for preventing large eddies in the ambient fluid. The ver-
tical position of the hose ends is maintained at about .5 ft. from the
basin bottom, pointing toward the discharge. A very slow ambient cur-
rent from the side walls to the back is set up which reduces eddying
but does not distort the jet structure. The capability of the pumping
-105-
-------
system to replace the manifold withdrawal with cold water is not used
at all except to maintain the basin level by discharging into the drain
line a flow equal to the channel discharge.
The channel flow is established by first turning on a bypass
flow from the heat exchanger to a drain and adjusting the temperature
of this flow to the desired value. The flow to the discharge channel
is then gradually increased to the proper rate.
Table 4-2 gives a summary of the parameters of each run. The
Reynolds number of all runs is near 10,000 and the density differences
are all about .004, corresponding to a temperature difference of about
25°F. In Runs 1-10 only the densimetric Froude number and the channel
geometry are varied with no bottom slope or cross flow. In Runs 11-19
the bottom slope is placed in the basin and varied from 1/100 to 1/25
for several values of IF. The last runs, 20-25, investigate the case
of an ambient cross flow, without bottom slope, over the entire width
of the basin. To avoid difficulty in comparing the results of differ-
ent cross flow runs, the same cross flow was maintained continuously
during Runs 20-25. The distribution of cross flow velocity, measured
by timing submerged floats, is shown in Figure 4-3.
4.4 Data Analysis
The data from each experiment consists of flow rates, tempera-
tures in the jet, and temperatures of the various inflows and outflows
from the basin. The ambient temperature is defined as the initial basin
temperature. A variation in the surface temperature at the extremities
of the basin is determined to be about 0.5°F per hour but the sub-sur-
-106-
-------
Physical Variables
Run
No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
h (ft)
0
.290
(.417)
.258
.164
.452
.355
.079
.088
.081
.164
.083
.246
.246
.246
.355
.355
.355
.079
.079
.079
.164
.164
.164
.164
.164
.164
b (ft)
0
.380
.315
.188
.104
.063
.229
.104
.063
.042
.042
.313
.313
.313
.063
.063
.063
.043
.043
.043
.188
.188
.188
.042
.044
.042
ft
u
o sec
.24
(.14)
.19
.32
.35
.50
.48
.58
.72
.95
1.06
.20
.20
.20
.60
.60
.60
1.25
1.25
1.25
.33
.22
.21
.74
.98
.74
T °F
a
71.5
69.9
72.1
71.2
71.8
73.3
72.3
72.3
72.6
71.8
73.7
73.3
72.8
73.2
74.0
74.2
70.4
70.8
71.2
70.9
73.1
72.8
72.1
71.5
72.6
AT °F
o
25.3
25.6
24.3
25.8
24.1
27.4
26.1
27.0
23.7
27.6
23.6
23.9
24.3
26.3
25.6
25.1
28.0
27.6
27.2
32.2
36.1
15.4
34.1
31.2
15.5
Ap/p xlO3
3.
4.10
4.05
3.95
4.19
3.89
4.65
4.31
4.50
3.86
4.60
3.91
3.94
3.99
4.42
4.33
4.24
4.55
4.50
4.45
5.45
6.50
2.35
6.14
5.29
2.35
V ft
max sec
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
.085
.085
.085
.085
.085
.085
Dimensionless
S
X
00
00
CO
OO
OO
00
00
OO
OO
OO
.01
.02
.04
.01
.02
.04
.01
.02
.04
OO
OO
CO
OO
OO
OO
IRxlO
o
1.5
(1.1)
1.1
1.1
1.2
1.1
1.1
1.1
1.0
1.3
1.2
1.1
1.1
1.1
1.3
1.3
1.3
1.4
1.4
1.4
1.1
.8
.7
.9
1.3
.9
4 F
o
1.00
(.58)
1.03
1.83
1.42
2.35
4.40
5.22
6.53
6.60
9.55
1.13
1.13
1.13
2.70
2.70
2.70
11.60
11.60
11.60
1.16
1.20
1.88
4.10
5.85
6.60
Parameters
h /b
o o
.76
(1.10)
.82
.87
4.30
5.70
.35
.84
1.30
3.95
2.00
.78
.78
.78
5.70
5.70
5.70
1.90
1.90
1.90
.87
.87
.87
3.95
3.95
3.95
V
max
u
o
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
.26
.39
.40
.12
.09
.12
K n.5
— xlO
u
o
8.5
10.5
6.2
5.8
4.0
4.2
3.5
2.8
2.1
1.9
10.0
10.0
10.0
3.3
3.3
3.3
1.6
1.6
1.6
6.2
9.0
9.5
2.8
2.0
2.8
Note: a) All cross flows as in Figure 4-3. _5
b) Surface heat loss coefficient, K, assumed to be 2 x 10 ft/sec in all cases (see Chapter 5)
c) For Run 1 numbers in parenthesis indicate values upstream of cold water wedge.
d) Reynolds number, JR, defined as 4u
where
hydraulic radius
b h /(h + b ).
Table 4-2. Summary of Experimental Runs
-------
14
oo
•H
M
O
a
o
-------
face ambient temperature is constant over the time of the experiment.
Figures 4-4 and 4-5 are a sample of the three-dimensional temperature
distribution measured by the movable probes for Run No. 10 and Run No.
22. The data anlaysis resulted in centerline temperature rises and jet
widths and depths as follows:
1) The lateral profiles of surface temperature at all x are
plotted i.e. T vs. y at all x. Centerline temperature
rises above the ambient, AT , are immediately available
as in the position, y , of the centerline in the case
of deflected jets.
2) The half width of the jet, b^, is the distance between
the centerline and the point where the surface temperature
rise is half the centerline temperature rise. Actually
the average of the left and right half widths is deter-
mined in the data analysis.
3) At each value of x centerline temperatures at each depth
are determined and plotted i.e. AT vs. z at all x. The
half depth, h... ,„, is the depth at which the centerline
temperature rise is half the surface centerline temperature
rise.
The temperature rises and distances obtained by these processes are nor-
malized by the initial temperature rise, T - T , and the length,
O <1
/h b , respectively.
o o
-109-
-------
70.8 • 74.2
• -"7.4 • Bj!4 • 81.0 • 82.7 • 82.6 • 83.3 • 82.2
Surface Temperature Contours (z - 0)
.ertical Section at y = .25 ft.
Vertical
Scale
.? «71.7» 71.7» 71.6 «71.5
Section at x • 5.4 ft.
Figure 4-4. Temperature Distribution Measured for Run No. 3
Probe Position •
•76. V
70.6 • 70.9 • 74.0 • 77.2 • 77.7 W77
71.S • 71.7/» 81.8 • 80.4 • 80.5 • 80.5 • 81.8
»83.3 «82.5 «82.0
t> 0 «92 4/ • 87.5^/084.7 • 84.7 «83.5 «84.7
S5T5
•BQ.6
•80.3
. 3 • 80.6 • 80.5 • 81.0
^71.7 • 71.7 • 71.6 • 71.6 »71.7 • 71.
•71.7
-110-
-------
77.6
72
78.7 »78.6/ • 77.8
• 77.7
ft.
76.7
77.1
Figure 4-5. Surface (z = 0) Temperature Distribution Measured
for Run 22 - With Cross Flow
Probe Position •
-111-
-------
V. Discussion of Results
5.1 The Controlling Parameters
This chapter is a comparison of the results of the laboratory
experiments, in this study and others, with the calculations of the theor-
etical model developed in Chapter 3. The response of the characteris-
tics of the heated discharge to the dimensionless parameters describing
the discharge and the ambient conditions is examined.
The physical variables characterizing each discharge may be com-
bined into the following dimensionless parameters (see Table 4-2):
u u
IF = = ~ = densimetric Froude number
o
A = h /b = discharge channel aspect ratio
o o
— * surface heat loss parameter
u
o
S = bottom slope
x
— = cross flow parameter
The centerline velocity, centerline temperature and the shape of
the jet are functions of the above parameters and of the distance from
the origin:
-112-
-------
tu /\T , u
c c b h r ,
u~ » ~AT~~ ' ,-"'--" ' r~— ' / > ;••-•'-' = function
o o Jh b yh b /h b yh. b
voo oo oo oo
}-
A JL s _L _*
9 "• 1 ) ^ 9 )
o o v h b -
o o
The internal velocities v . w , v, and w. are not determined ex
s r b h
plicitly by the theory nor are they measured in the experiments. The
values of u and AT away from the centerline are calculated using the
similarity functions f and t as in Equations 3.46 and 3.89.
Other interesting dependent variables which are also functions
of x may be defined. In the buoyant jet the vertical entrainment is a
function of a local densimetric Froude number:
IF = = local densimetric Froude number
'agAT h
where u , AT , and h are local values at a given distance from the ori-
gin. The total flow in the jet may be determined by integrating the x
velocity over the jet cross section. The ratio of the flow at a given
x to the initial flow, u h b , is called the jet dilution:
o o o
-s+b
dzdy
/-Sf D fO
\ U
/o y-h-r
u h b u h b
o o o o o o
-113-
uc(r + IjhHs + Ijb)
= dilution
-------
where the similarity function for u CEquation 3.451 is used in the inte-
gration. The effect of surface heat loss upon the heated discharge may
be evaluated by calculating the excess heat flow in the jet where excess
heat is defined as heat which raises the water temperature above ambient
A parameter, analogous to dilution, may be defined as the ratio of the
local excess heat flow to the initial excess heat flow, pc AT u h b •
p o o o o"
H
ss+bro
PC uATdzdy _h)(s+I_b)
yo ^-h-r _ c c 7 7
PCP AToUohobo AToUohobo initial excess heat
5.2 An Example of a Theoretical Computation
Only b, h, and AT are determined in the experiments. Thus
values of UG, r, s, IF D and H cannot be compared with the theory.
Figures 5-1 and 5-2 show the calculated results for Run No. 6 (see Table
4-2) which includes all the variables and which illustrates the follow-
ing general structure of heated surface discharges.
1) A core region in which the centerline velocity and the
centerline temperature rise decrease very slightly. The lateral spread
of the jet is very large, requiring an initial rise of the jet bottom
boundary to satisfy mass conservation. The dilution, D, and the local
densimetric Froude number, F , do not vary greatly in this region. The
magnitude of E is much larger than F because the initial depth of the
turbulent region, h, is zero. There is no significant surface heat loss
in the core region.
2) An entrainment region in which the centerline velocity and
-114-
-------
Core Ke&iun
Entrapment Region Region Regie
Cor*
^, Region Entninaent Region
Vamt Lois
••lion
-trr
Figure 5-1. Theoretical Calculation of Jet Structure for Run 6.
IF •» 4.4 K/u - 4.2 x 10~5 V/u - 0
O O O
A - .35 S - 0
-115-
-------
90 100 0 10 20 30 40 SO 40
Figure 5-2. Calculated Isotherms of AT /AT for Run 6.
0 co
IF - 4.4
o
A - .35
K/u - 4.2 x 10
o
S - 0
-5
V/u - 0.0
-116-
-------
temperature drop sharply, approximately as 1/-X, as in the non-buoyant
jet. The jet spreads downward by turbulent processes at a rate of the
order dh/dx r .22. The lateral growth is dominated by gravitational
spreading at a much greater rate than the vertical turbulent spread.
Because of this large ratio of lateral to vertical spread, the jet reach-
es a maximum depth beyond which the bottom boundary rises to maintain
mass conservation. Local densimetric Froude numbers in this region de-
crease rapidly and the dilution rises sharply as a result of entrain-
ment. Surface heat loss is still negligible in this region i.e. H =
1.0.
3) A stable region in which vertical entrainment is inhibited
by vertical stability as indicated by the local densimetric Froude num-
ber which is of order one or less. The jet depth continues to decrease
because of lateral spreading. The small jet depths reduce the lateral
entrainment, the dilution and centerline temperature remaining relatively
constant in this region. The centerline velocity, however, drops sharply
as a consequence of the large lateral spread. The surface temperature
pattern, as shown in Figure 5-2, is dominated by the wide, constant temp-
erature stable region.
4) A heat loss region marking the end of the stable region.
The lateral spread is sufficiently large to allow significant surface
heat transfer and the temperature begins to fall again. Once surface
heat loss has begun to be important, the rate of temperature decrease
±3 very rapid and in conjunction with the low centerline velocity the
discharge may no longer be considered as a jet.
-117-
-------
5) A far field region dominated by ambient conyectiye and diffus-
ive processes which are beyond the scope cf this study. If ambient flow
velocities are large, the far field region may begin before the stable
region has formed. The excess discharge heat is ultimately lost to the
atmosphere in the far field.
5.3 Comparison of Theory and Experiment
In the previous section the general behavior of heated discharges
is discussed. Many of the characteristics of the discharges cannot be
generalized and must be determined by examining each case. This sec-
tion, in addition to comparing the data with the theory, examines the
behavior of the jet as a function of the controlling parameters discus-
sed in Section 5.1.
Chapter 4 includes a description of the determination of AT ,
h, ,? and b.. ,„ from the experimental data. It should be noted that the
half width, b1/2> and nalf depth, h-,-, are defined in terms of the temp-
erature distribution rather than by the velocity distribution as dis-
cussed in Chapter 2. The determination of the half width, b , is
difficult because of a) distortion of the surface temperature distribu-
tion in the lateral portions of the jet by recirculation of heated water
within the experimental basin and b) the small temperature differences
in the jet far from the discharge point.
In the theoretical computations for all the laboratory cases,
the surface heat loss coefficient is assumed to be 2 x 10 ft/sec.
2
corresponding to a heat transfer rate of 110 BTU/ft -day-F°. This is in
the low range cf values predicted from standard formulas (3 ) and is
-118-
-------
close to Hayashi's (.15) estimate for laboratory environments, 1.8 x 10
ft/sec.
In Runs 1-10 the densimetric Froude number, IF , and the aspect
ratio, A - h /b , are varied. There is no bottom slope or cross flow.
Figures 5-3 to 5-12 show the experimental data and the theoretical com-
putations. For low values of F the jet spreading is too large to per-
mit determination of the half width, b.. ,_.
Runs 1 and 2 illustrate the case of F -1.0 which is the lowest
o
possible value of IF . In Run No. 1 the channel densimetric Froude
number is actually less than one and a cold water wedge is observed in
the channel. As discussed in Chapter 2, the wedge will force the heated
flow to obtain f = 1.0 at the discharge point. The value of u and h
O of O O
to be used in the theoretical calculation is determined by this condi-
tion. There is little difference between the temperature distribution
for the wedge and non-wedge runs. In Run 2 the "toe" of the wedge is
positioned at the channel expansion and the vertical structure of the
jet is approximately the same as if the "toe" had intruded into the
channel. For such low IF discharges the entrainment region is very
small, the temperature of the stable region is high, and the heat loss
is promoted by large lateral spreading.
Runs 3 and 4 show the effect of the discharge channel aspect
ratio, A, on low IF discharges in which horizontal entrainment dominates
vertical entrainment. Large aspect ratios provide larger available area
for lateral entrainment and the temperature drops more quickly. Sur-
face heat loss behaves in an opposite manner, being promoted by the
-119-
-------
1.0
!i
wo
.5
CmurliM T«p*ntur« U««: 4T lit
J 1 I I I I 11
I I I I I I I
100
1—r
Too
-J
-5
100
»«H-D.pth; -t»
Theory -
Tot»l D«j>thi -(r
Thaory
200
Figure 5-3. Theoretical Calculations and Experimental Data for Run 1.
IF - 1.00 (wedge present) K/u •= 8.5 x 10 V/u « Q
A «• .76
S » «
x
-120-
-------
iT
c
4T
Centerline Temperature Rl»e:
Experiment •
J L
till
J I I I I I I I I
J I L
150
-2
-3
Half-Depth: -h. ,,/
LI t. o o
Exp«ri**nt ^
Theory —
Total Depth: -(r -f h)//h~b~
_. „ _ _ o o
Theory
Figure 5-4. Theoretical Calculations and Experimental Data for Run 2.
IF - 1.03 K/u - 10.5 x 10~5 V/u - 0
o o o
A - .82 S - »
-121-
-------
tTc
"o
.5
.1
^^oX.
^"Vnrv
Ccnterlin* Tecnpcrftturc Si»t: fil /4T
Experiment •
Theory . •
i 11,111 i iii
1 10
0
Ch.urxl
lotto»
-' 1
/TT
0 0
-2
-•)
100
«0
60
40
20
>
0 ,5^1
o ^ so 100
S i i
'V0^x'x
Half Dtpth:
-
^
-
1 1 1 1 1 1 II
100 '•'^\^ soo
ISO «/-'\^ 200
1
-h1/2//sx
ExptrlMnt •
Theory
Total Depth:
Theory
1 1
0 SO 100
/ I Half width:
.
o o
1
ISO n/-^^ JOO
1
"i/2//E?:
/
f j Exp«rl»*f)t 0
/ / • Theory .
~ I / Total Width
// •• Theory
Channal
Sid*
: (• + k)/r1i^
1
re 5-5. Theoretical Calculations and Experimental Data for Run 3
F - 1.83 K/u - 6.2 x 10~5
0 O
A = .87 S » »
X
-122-
V/u - 0
o
-------
Experiment
Theory ••
100
500
-3
Channel
Bottom
100
150 x//h b 200
o o
H.lf Depth: - h1/2'''h
-------
C«nt
-------
1.0
IT
c
ST
Experiment
Theory
I I I
50
—r
100
150
X//5TT" 200
o o
Channel
Bottom
77
Half Depth: -h, ,,/A b
II L o O
Experlnent Q
Theory i
Total Depth: -(r + h)/A b
o o
Theory
80
100
T-
150
—r
200
Half Width: b. .,/|h b
lit o o
Experiment 0
Total Width: <» + V>lJ\C\T
o o
Theory
Figure 5-8.
Theoretical Calculations and Experimental Data for Run 6.
-5
IF - 4.4
o
A - .35
K/u - 4.2 x 10
S - »
x
-125-
V/u - 0
-------
500
Channel
Bottom
-1
-2
-3
50
\
Experiment £)
Theory _^^«__
Total Deptu: -(r + h)/A b
Theory
200
Figure 5-9. Theoretical Calculations and Experimental Data for Run 7
-5
P = 5.22
o
A = .84
K/u = 3.5 x 10
o
V/u
-126-
-------
200
Half Width: b. .,//h b
i/z o o
Experiment
Channel
Side
Figure 5-10. Theoretical Calculations and Experimental Data for Run 8.
IF - 6.53
o
A - 1.30
K/u - 2.8 x 10
S • oo
X
-127-
-5
V/uo - 0
-------
C«nt«rlln* Tenpcracarc Rise: AT /AT
-3
lUlfUldU,: b /.VT
ltd*
Figure 5-11. Theoretical Calculations and Experimental Data for Run 9,
,-5
F - 6.60
o
A » 3.95
K/u - 2.1 x 10
o
X
-128-
V/u = 0
o
-------
Centerline Temperature Rise: AT /AT
0 0
60
Channel
Side
100
Half Width: b
. ,
iff 0 O
Experiment
Theory
Figure 5-12.
Theoretical Calculations and Experimental Data for Run 10,
-5
IF - 9.55
o
A - 2.00
K/u = 1.9 x 10
o
S - »
x
-129-
V/u - 0
o
-------
large horizontal surface areas in low aspect ratio discharges. The
theory does not reproduce the centerline temperature drop for large as-
pect ratios as well as for smaller values.
The same principle is illustrated in Runs 5 and 6. An increase
in aspect ratio may outweigh an increase in IF as shown in Figures 5-7
and 5-8. In Runs 7-10 (Figures 5-9 to 5-12) the effect of aspect ratio
and densimetric Froude number is diminished as IF increases. The jet
depths are much larger and heat loss is not as significant as in the
discharges of lower ]F .
Runs 1-10 indicate that the theory agrees well with the data
and is capable of predicting the structure of heated discharges as a
function of F and A. The experimental error is greatest in the deter-
mination of b, ,- an(* least for AT . Since the theory agrees best with
the data for ATc and worst for b, ,2 some of the discrepancy between the
data and the theory, especially for the half widths, can be attributed
to experimental error.
In the runs involving bottom slopes (Runs 11-19) (Figures 5-13
to 5-15) only surface temperatures are measured because of the diffi-
culty of positioning probes near the solid bottom. The bottom slope
has two counteracting effects upon the jet:
1) Vertical entrainment is inhibited by the presence of the
solid boundary.
2) The vertical core region of depth r persists as long as
the jet is attached to the bottom. This promotes lateral
entrainment over the full depth of the jet.
-130-
-------
1.0
ATc
AT"
o
.5
Centerline Temperature Rise: AT /ATQ
Experiment Run 11: •
Run 12:•
Run 13: 4
Theory
I I I
- 1/100
- 1/50
- 1/25
(value of S shown)
i i i i
i i i i i i
10
100
500
Figure 5-13. Theoretical Calculations and Experimental Data for Runs 11, 12 and 13.
F - 1.13
O
A - .78
K/u - 10.0 x 10
O
V/u - 0
o
"5
Run 11: S - 1/100
X
Run 12: S - 1/50
x
Run 13: S - 1/25
-------
1.0
sx - 1/100
Sf • 1/50
S, - 1/25
(value of S shown)
10
100
150
T-
x//iTT"
o o
200
Half Wldch: b,,-/Sh~b~
_ _ if & O O
Experiment Run 14:
Run 15:
Run 16:
Theory — — ^_ —
S - 1/100
S - 1/50
S^ - 1/2S
(value of S shown)
Figure 5-14.
IF - 2.7
o
A - 5.7
Theoretical Calculations and Experimental
Data for Runs 14, 15 and 16.
K/u - 3.3 x 10
o
V/u - 0
o
-5
Run 14: S - 1/100
X,
Run 15: S - 1/50
Run 16: S - 1/25
-132-
-------
.5
Centsrline Temperature Rise: AT /AT
__ c o
Experiment Run 17: 0 S - 1/100
Run IS: • Sx - 1/50
Run 1»: * Sx - 1/25
Theory ___^^_ (vslue of S shown)
__l 1 L
J 1 I I I I I I
I I'
100
50
ISO
200
80
60
Hslf Wtdth: b
. ,,
1/2 o o
Experiment Run 17: ^ S - 1/100
Run IB: B S^ - I/SO
Run 19: A S^ - 1/25
Theory — — _ (value of S, «horo)
Figure 5-15:
F - 11.6
o
A - 1.9
Theoretical Calculations and Experimental
Data for Runs 17, 18 and 19.
K/u - 1.6 x 10~5 Run 17: S - 1/K
o x
u
V/u - 0
o
Run 17: S - 1/100
Run 18: S - 1/50
x
Run 19: S - 1/25
-133-
-------
The resultant effect of these two phenomena is dependent upon the values
of ]P and A. Runs 11-13 indicate that for low IF the jet detaches from
the bottom almost immediately and that the centerline temperature rise
is nearly the same for all the slopes considered and is near the value
computed for no bottom slope. Runs 14-16 (Figure 5-14) show that for
large aspect ratios and low IF the bottom slope actually promotes higher
entrainment than without a slope. As with Runs 11-13 the data shows no
dependency upon the value of the bottom slope, indicating early separa-
tion of the jet from the bottom. In Runs 17-19 (Figure 5-15) the reduc-
tion of vertical entrainment increases as the slope decreases. This be-
havior is a consequence of the large densimetric Froude number and mod-
erate aspect ratio for these cases. The jet remains attached to the
bottom and the dilution is dependent upon the value of the slope.
The theory reproduces the trends in the data as discussed above.
Th'e low IFo computations (Runs 11-16) show more dependency on the value
of Sx than do the data, indicating that the theory is not predicting sep-
aration of the jet sufficiently close to the jet origin. For the large
IF runs (17-19) the theory underestimates the effect of the slope on the
reduction of entrainment. The theory fails to reproduce the lateral
spreading of the jet which is seen to be close to the calculated curve
for no slope rather than for the very small spread the theory predicts
when a slope is present. The assumption put forth in Section 3.6 that
the jet does not exhibit buoyant spreading is clearly not valid. The
jet must be spreading laterally at the edges and remaining somewhat
attached to the bottom near the centerline, distorting the jet beyond
-134-
-------
the capability of the theory to describe it with simple similarity func-
tions .
In Cases 20-25 a cross flow deflected the jet. Determination of
the jet centerline position from the temperature data is very difficult
since the total lateral temperature variation within the experimental
measurement area and far from the discharge origin is only a few degrees.
This is particularly true of the runs with low values of 3F . The agree-
ment of the experimental and calculated jet trajectories are scattered
but good on the average (see Figures 5-16 to 5-21). The measured center-
line temperature values are also close to the theoretical curves, being
not much different than the no-cross flow cases with the same IF and A.
o
As expected from the formulation of the equations governing deflected
y
jets, the jet deflection increases with IF , A, and — since the only
o
force turning the jet is entrainment of lateral momentum. The theoreti-
cal calculations are terminated when u = Vcos6 since the basic assump-
tion of small cross flow velocities does not hold beyond that point.
The theoretical calculations used an analytical expression for the cross
flow velocity as a function of x (see Figure 4-3).
5.4 Comparison of Theory with Experimental Data by Other Investigators
As mentioned in Chapters 2 and 4 a number of other studies
collected field and laboratory data for three-dimensional heated surface
discharges. These results are compared with the theory in this section.
Data from Stefan (31), Hoopes (18) and Tamai (32) are not used. Stefan's
data points are too close to the jet origin in most cases to permit exam-
ination of trends in the data; Hoopes1 field temperature measurements
-135-
-------
1.0
AT
c
AT
Center line Temperature Rift: ATc/
Experiment
Theory —
J L
' ' ' '
J I I I I I I I
1 T
100
50
—T
100
—r~
150
Foiitton of Centerllne
Experiment 0
200
Figure 5-16. Theoretical Calculations and Experimental Data for Run 20.
F - 1.16
o
S - .87
x
K/u - 6.2 x 10"
o
S - <*>
x
Vmax/uo - '26
•J36-
-------
4T
£
4T
Experiment
Theory
I 1 1 I 1 I I 1
80
A\ b
o o
Channel
Side
50
100
150 X/.VTT 200
Position of Centcrline
Experioeot 9
Theory ———
Figure 5-17. Theoretical Calculations and Experimental Data for Run 21.
_ » j.. ^u K./U • y.u x J.T
o
A - .87
K/UQ - 9.0 x 10"
V - u - .39
max o
x
-137-
-------
1.0
n
.5
Centerllne Temper Cure MM:
Experiment «
J L
I I I I I I
J 1 I.
10
100
300
100
10
to
*0
50
~r
150
T~
Position of C«nt«rllD«
Theory
Figure 5-18. Theoretical Calculations and Experimental Data for Run 22.
IF - 1.88
o
A - .87
K/u - 9.5 x 10
o
S - »
x
-5
Vmax/uo * •*
-138-
-------
Centerllne T«np*rature Rlae: AT /AT
10
500
100
SO
60
Po«ltlon of C«nt«rllg«
Theory
200
ChttMl
S14*
Figure 5-19. Theoretical Calculations and Experimental Data for Run 23.
V - 4.10
o
A - 3.95
K/u - 2.8 x 10
o
-5
V /u - .12
max o
-139-
-------
SOO
100
/h b
o o
60
20
50
T
Poiition of Ccnt«rlint
Ixpcriiwnt %
Theory ^—_
Chaaiul
Slit
Figure 5-20. Theoretical Calculations and Experimental Data for Run 24.
F - 5.85
o
A - 3.95
K/u - 2.0 x 10
o
S - »
x
-5
Vmax/uo - -09
-140-
-------
1.0
.5
Cmttrllm Ttap«ntur> Rlit: AT /AT,
20
Figure 5-21. Theoretical Calculations and Experimental Data for Run 25.
TF - 6.60
o
A - 3.95
K/u - 2.8 x 10
o
S « co
X
-5
V /u - .12
max o
•441-
-------
are too close to ambient values to be useful; and Taniai's data is sever-
ely affected by the fact that the experimental basin is no more than 40
times as wide as the discharge orifice. This inhibits the lateral en-
trainment which is so important in three-dimensional heated discharges.
Jen's (21) experiments are >'one wit circular orifices and values
of IF from 18 to 180. He found that the centerline temperature drop,
beyond the core region, is independent of IF and given empirically by
AT 14R
where R is the initial jet radius. For IF = °° , the temperature rise
o o
beyond the core region predicted by the theory of this study is:
AT 10.7,/h b
- —^ «•«
which is valid for rectangular discharge channels. The experimental
data for Jen indicates that the theoretical expressions for I" = »
are approximately valid for IF > 15. The quantity v/hb is the square
root of half the discharge channel area. If it is assumed that for a
fully submerged circular discharge pipe the proper scaling is also the
square root of half the discharge area, then the quantity >/h^b~ is re-
placed by R /•£ = 1.25 R . Then the expression for the centerline temp-
r J o V2 o r
erature rise beyond the core region (IF = <=°) and a fully submerged circu-
lar discharge pipe becomes, according to the theory of this study:
-142-
-------
AT 13.4 R
^. - 2. (approximately valid for I" ?> 15) C5.3)
AJ- X O
which is almost identical with Jen's empirical result (5.1). (see Fig-
ure 5-22) For circular discharges the aspect ratio should be taken as
2.0.
Wiegel (37) studied rectangular orifices with aspect ratios of
1/2 and 1/5 and W from 20 to 30. Figure 5-23 shows that his data, when
normalized by /h b falls close to the line given by Equation 5.2.
Figure 5-23 also shows Wiegel's data for a discharge of F from 20 to 30,
aspect ratio of 1/5, and bottom slopes from 1/200 to 1/50. The beginn-
ing of the entrainment region is delayed by the inhibition of vertical
entrainment. Later dilution is seen to be a function of the slope value.
The theory of this study satisfactorily reproduces the delay in initial
entrainment but not the bahavior farther from the origin.
Hayashi (16) studied heated discharges from rectangular orifices
into stagnant receiving water with no cross flow. His data and the
corresponding theoretical calculations are shown in Figure 5-24. The
data scatter is large and the agreement mostly qualitative. The results
illustrate the property of heated discharges that a lower densimetric
Froude number produces less dilution in the entrainment region but higher
eventual heat loss far from the discharge point.
5.5 Summary of Jet Properties
The previous sections have considered the general structure of
-143-
-------
fe
r
1.0
AT
AT -
.5 _
I I I I I I I
1 I
Centerline Temperature Rise: AT /AT
I I I I I I
10
100 x/R /»/2
o
500
Figure 5-22. Theoretical Calculations and Experimental Data for Circular Orifices -
Jen (21)
IF - 18 - 180 K/u =* .5 x 10~5
o
A - 2.0
V/u - 0
o
S -
X
-------
fe
r
— Centerline Temperature Rise: AT /AT
(values of A and S shown)
.1
100 x//h b
o o
500
Figure 5-23. Theoretical Calculations and Experimental Data for Rectangular Orifices -
Wiegel (36)
K/u - 1.0 x 10~5 V/u = 0
F - 20-30
o
• o
-------
1.0
AT
AT"
o
.5
Theory
This Study
Hayashl
I I
Centerllne Temperature Rise: AT /AT
Experiment O IF * 1.
F - 2.6
o
(value of ff shown)
o
I I I I I I
I I
F » 1.4
I I I •"! I '
I • 2.6
o
I I
10
100
h b
o o
500
Figure 5-24. Theoretical Calculations and Experimental Data for Rectangular Orifices -
Hayashi (16)
- 1.4 and 2.6
o
A - 2.0
K/u - 10.& x 10
o
S « eo
-5
V/u - 0
o
-------
heated discharges, the dependence of the detailed structure of the dis-
charge on the dimensionless parameters Tf , A, K/u , S and 7/u , and
O O JC O
the ability of the theory of this study to reproduce experimental res-
ults. The theory successfully describes the jet behavior except for
lateral spreading in the presence of a bottom slope. In this section
the theoretical calculations are used to quantify some of the general
statements in the previous sections.
The dilution and corresponding centerline temperature in the
stable region are of interest since the stable region is a significant
portion of the surface temperature distribution as previously discussed.
The centerline temperature rise may be related to the dilution by assum-
ing that no heat has been lost i.e. H = 1.0 and considering a position
beyond the core region i.e. r = s - 0. The definitions of D and H may
then be combined to yield:
AT I -i i i -, c
c - 1 ! - i-5 /c ,,
I~J n ~ IT (5.*)
Figure 5-25 is a plot of the dilution in the stable region vs.
T with values of A indicated. The points are the theoretical calcula-
o
tions for Runs 1-10 and for several of the previous investigations, all
with no bottom slope or cross flow. Some tentative lines indicating
the curves for constant A are drawn but many more calculations are
needed to improve the reliability of the plot. Nevertheless it is clear
that the ultimate stable dilution in a buoyant jet depends primarily
upon W and to a lesser extent upon A.
-147-
-------
u
Stable
15
10
1.3
84
35
A-1.0
10
Figure 5-25. Dilution in the Stable Region of Buoyant Discharges
(Points are calculated by the theoretical model -
Values of A indicated)
-148-
-------
h
The maximum depth of the discharge jet, -5si. j ia aiso o/f inter-
/h b
o o
est and may be determined from the theoretical calculations. Figure
5-26 indicates that the maximum vertical penetration of the jet is a
function of ]F with a very small dependence upon A. The results are
approximated by the following relationship:
.5 P (5.5)
o
ftv b
o o
Other interesting features of the heated discharges treated in
this study are the values of x at which the various jet regions begin.
Unfortunately the boundaries of the jet regions are not well defined,
transition zones being present between the regions. The lengths of the
core regions, x and x , are not the same as for a non-buoyant jet, in
S r
which they are functions only of A, but are rather a function of both
IF and A in the buoyant discharge. In the absence of a great many cal-
o
culations from which -to make general plots, the detailed structure of
each specific discharge is best determined by a separate theoretical
calculation for each case.
In the previous section, little was said about the effect of the
heat loss parameter, K/u , upon the jet behavior. For all but the low-
est values of 3F and A, the theoretical calculations indicate that heat
loss has no significant effect until after a stable region has formed and
the jet has undergone considerable spreading. This is also indicated by
the experimental data. Thus the value of K/u will affect only the rate
-149-
-------
max
'h b
o o
10
IF
Figure 5-26. Maximum Jet Depth Vs. F as Calculated by Theory
-150-
-------
at which the stable region loses heat. Since jet-like behavior ceases
at this point, the value of K/u may be said to have no effect upon the
region of the discharge considered in this study. Surface heat loss may,
however, be important in the far field region as discussed in the follow-
ing chapter.
-151-
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VI. Applying the Theory
This chapter considers the use of the theoretical .model for sur-
face discharges of heated water in meeting the objectives outlined in
Chapter 1 with respect to prototype temperature prediction and the de-
sign of scale models of heated discharges.
6.1 Prototype Temperature Prediction
Prediction of temperatures in the vicinity of a specific dis-
charge configuration requires the parameters IF , A, S , K/u , and V/u
o x o Q
be specified in addition to AT , T , and the discharge channel area.
O £L
The geometry of actual field configurations are rarely as simple as
those assumed in the theoretical development and judgement must be exer-
cised in the schematization of the discharge to a less complex form
for which the dimensionless parameters may be given. This section is
a discussion of the data requirements for treatment of a heated surface
discharge by the theory of this study and of some schematization tech-
niques.
The following are the physical data needed as input to the
theoretical calculations:
The ambient temperature, T , is assumed in the theory to be
constant in space and time. Actual ambient receiving water temperatures
are often stratified vertically or horizontally and may be unsteady due
to wind, tidal action or diurnal variations in solar heating. The nat-
ural stratification of the ambient water may be increased by accumula-
tion of heat at the water surface if the discharge is in a semi-enclosed
region. The value of the ambient temperature, for a given initial temp-
-152-
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erature rise, &T , determines the initial density difference between the
discharged and ambient water. Also tbe effectiveness of the entrainment
of ambient water in reducing the discharge temperatures is a function
of the ambient stratification.
If the temperature differences resulting from ambient stratifica-
tion in the vicinity of the discharge are the same magnitude as the init-
ial discharge temperature rise, the theoretical model of this study
should not be applied without further development to account for the
stratification. The theory is valid if an ambient temperature, T , may
cl
be chosen which is representative of the receiving water temperatures,
that is, if the temporal or spatial variations in ambient temperature
do not differ from T by more than a few degrees Fahrenheit.
&
The initial discharge temperature rise, AT , is determined from
the discharge temperature, T , and the chosen ambient temperature, T .
The value of AT will be equal to the temperature rise through the con-
densers only if the intake temperature is equal to T . Actual discharge
temperatures should be relatively steady unless the power plant has sev-
eral different condenser designs in which case the discharge temperature
will vary with the plant load. Use of the theory requires that a con-
stant value of ATQ be specified, the choice being based on the likely
steady value of the discharge temperature.
The discharge channel geometry is important since the theory
uses one half the square root of the discharge channel flow area as a
scaling length. Calculation of the discharge densimetric Froude number,
JF , requires specification of the initial depth, h , and the aspect
-153-
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ratio, A, requires both, h and the initial width, b . Rectangular and
circular discharge geometries are discussed in Chapter 5. The following
procedure is suggested for any channel shape:
a) Let h be the actual maximum discharge channel depth
so that calculation of F is not affected by the schema-
tization.
b) Let b be such that the correct discharge channel area
is preserved:
, channel area
bo " 2h
o
Then the aspect ratio is given by:
h 2hQ2
A " ~
b channel area
o
The discharge channel geometry may vary with time if the eleva-
tion of the receiving water changes due to tidal motion or other causes.
In this case separate calculations for each elevation of the receiving
water must be made, assuming that the steady state theory predicts the
instantaneous temperature distribution.
The initial condenser water discharge velocity, u is a function
of the power plant condenser water pumping rate and the channel area.
The bottoir. slope, S is usually not uniform as the theory assum-
es. The average value used as input to the theory should be close to
the value of the slope near the discharge point since it is the initial
-154-
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portion of the discharge jet which is most affected by the bottom slope.
The surface heat loss coefficient, K, may be determined from
meteorological data as discussed by Brady (3) who gives several useful
diagrams for estimation of K.
The cross flow velocity, V, in the receiving water may be meas-
ured by drogue surveys or estimated from current records or flow measure-
ments in the case of a river. The theory accepts as input (see Appen-
dix I) any analytical function which gives V as a function of x.
The dimensionless parameters IF , A, S , K/u , and V/u are func-
O * ' X* O (3
tions of the above physical variables. The initial density difference
Ap does not appear explicitly in the theory but is represented by a T .
In calculating the initial densimetric Froude number, F , the actual
value of Ap should be used to minimize the error in the assumption that
the coefficient of thermal expansion, a, is constant. This choice of
Ap is equivalent to choosing an average value of a over the temperature
range T to T + AT .
3. ai O
When the discharge channel densimetric Froude number, IF , is
less than unity, a wedge of ambient water intrudes into the discharge
channel as discussed in Chapter 2 and the heated flow will be forced to
obtain a densimetric Froude number of unity at the discharge point (see
Figure 2-4). The depth of the heated flow in the presence of a wedge,
h *, is given by: (12)
o
h *
-155-
-------
where h and IF are based on the channel dimensions. A flow area may he
o o v
calculated based on the depth h * and the aspect ratio calculated as des-
cribed previously in this section. The actual discharge may be directed
parallel to a wall which will effectively cut off lateral entraintnent
from one side of the jet. In this case the discharge may be schematized
by assuming that the solid boundary is the centerline of a jet whose
discharge channel is twice the width of the actual discharge (see Figure
6-1). The width b is then twice the width calculated by the procedure
discussed previously.
More complicated arrangements of the discharge channel relative
to the boundaries of the ambient receiving waters are beyond the capab-
ilities of the theory of this study. If the discharge channel is nearly
at right angles to the solid boundaries, the theory may be used as dev-
eloped, or if the discharge is directed parallel to a straight boundary,
the schematization described above may be used (see Figure 6-1). How-
ever, if it is not clear whether the jet will entrain water from one or
both sides or if irregularly shaped solid boundaries will deflect the
jet or distort it from the form assumed in the theory, a meaningful
schematization is not possible.
Once the schematization of the discharge configuration is achiev-
ed, the theoretical calculation is performed by the computer program des-
cribed in Appendix I. The inputs to the program for each calculation
are F , A, S , K/u and V/u (as a function of x/Vh b ). The program
outputs the values of ATc/ATQ, UC/UQ, b/J^Tb^, h/y^Tb^, r/y^hlT, s/v^TT",
6, and the position of the jet centerline as a function of the distance
along the centerline.
-156-
-------
7 7 III
/ r i f i
90°
~ near 90°
o
Discharges with Entrainment from Both Sides
J
1
b
, 0
T f 1 t Tl I i I i • i i i i
Discharges with Entrainment from One Side
1 r
' ' ' i
1
f
i
\ /
f ' ' ' '
f
f
f
™
Schematization Possible
Schematization not Possible
Because of Irregular Geometry
Discharges with Obstructions
Figure 6-1. Discharge Channel Schematization
-157-
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6.2 Modeling Heated Discharges
Laboratory scale models are used when the determination of temp-
eratures in the vicinity of a prototype heated discharge is beyond the
capability of analytical temperature prediction techniques. The theory
of buoyant discharges developed in this study makes the construction of
a model unnecessary if the region of interest is within the scope of the
theory and if the prototype may successfully be schematized as discussed
in the previous section.
Many prototype situations may be divided into a near field and
a far field region. In the near field region the temperature distribu-
tion is controlled by jet entrainment, buoyant spreading and the local
ambient cross flow. The theory of this study is a treatment of the
near field. In the far field region ambient turbulent dispersion, con-
vection, and surface heat loss determine the temperature distribution.
The far field and near field are not independent of each other; the
near field local ambient conditions are a function of the far field
temperature and flow distribution and the input of heat to the far field
is controlled by the near field characteristics.
Information may be needed about the near field or the far field
or about both. Thermal pollution regulations may be such that maximum
temperature levels must be met within the near fiej.d. On the other hand
an evaluation of the effecLs of the heated discharge upon the natural
environment requires information about the far field temperature dis-
tribution. Recirculation studies may involve either of the regions
depending upon the relative location of the intake and discharge.
-158-
-------
The theory of this study may he used to analyze the prototype
temperature distribution if the near field temperatures are of interest.
Schematization of the prototype near field is possible within the limita-
tions discussed in the previous section. In general the theory may be
applied if the far field ambient temperature is independent of the near
field structure and if the geometry in the vicinity of the discharge is
relatively simple.
In the case that the prototype does not yield to schematization
or the far field conditions may not be estimated and may be a desired
result of the analysis, a model study is required. If the far field
temperature distribution is of interest the model must reproduce not
only the far field physical phenomena but also the manner in which the
near field inputs heat to the far field. In a heated discharge, this
input is characterized by the dilution and corresponding temperature
of the stable region. A model which is to reproduce the prototype far
field temperature distribution must as a minimum requirement have a
near field stable region with the same temperature rise and total flow
at the same location as in the prototype.
The physical dimensions and fluid flows in a model of a heated
discharge must be scaled from the prototype in a manner which insures
similarity of the temperature distribution between model and prototype.
Analytical statements of the proper scaling are commonly called "model
laws" and will be different depending upon which physical process dom-
inates the distribution of heat.
The characteristics of the near field behavior of buoyant jets
-159-
-------
are discussed in Chapter 5. The diraensionless centerline controlling
parameters are IF , A, S and y/u f surface heat loss being unimportant
in the near field. For similitude of model and prototype near field
temperature distribution it is sufficient that IF , A, S and V/u be
O X o
the same in model and prototype. If the following scale ratios are de-
fined:
z • ratio of vertical lengths in model and prototype
L « ratio of horizontal lengths in model and prototype
u - ratio of velocities in model and prototype
AT • ratio of temperature rises in model and prototype
Ap * ratio of density differences in model and prototype
then the similarity requirements for the near field become:
1/2
1.0 or ur = (Aprzr) (6.4)
Ar " Sxr = 1<0 or Lr = Zr
Thus the model must be undistorted (Equation 6.5) with the velo-
city ratio related to the length scale ratio and the density ratio by
the well-known densimetric Froude model law (Equation 6.4). In most
cases Ap is not much different from unity because of the problems of
working with very large or very small temperature differences in the
laboratory.
The temperature and dilution in the stable region of a heated
discharge are functions of IF and A (see Figure 5-25). Thus even if
-160-
-------
the detailed structure of the near field region is not of interest, sim-
ilarity of near field heat input to the far field requires an undistorted,
densimetric Froude model.
Distorted models have some distinct advantages over undistorted
models especially in regard to laboratory space requirements. For a
model to reproduce a large horizontal area of the prototype, making
z > L results in more workable water depths and higher model Reynolds
numbers than if z = L . Distortion may be required by model laws for
similarity of heat loss (14) or frictional effects (13) in the far
field. If a distorted model is constructed, the aspect ratio and bottom
slope will be different in the model than in the prototype even if IF
is the same. If this were the only effect of distortion, one might
attempt to adjust the model values of IF to compensate for the distor-
tion of S and A. However, the ratios of dimensionless horizontal and
X
vertical distances in model and prototype are:
0 O
i , i .1 „ (6.6)
y£ir
(6.7)
The distortion thus tends to make the dimensionless x and y values of any
physical position less in the model and the dimensionless z distances
more. The model temperature distribution would be wider, longer, and
-161-
-------
shallower than the prototype by a factor of Vz^/L . Varying IF in the
model arbitrarily may produce the sane stable temperature or ,flow as the
prototype but will not counteract the vertical and horizontal distortion
of the temperature field. To compensate for a larger A in the model the
value of F must be lowered in the model, which will produce an even
wider and shallower model temperature distribution. It is concluded that
it is impossible to build a distorted scale model which reproduces cor-
rectly the minimum required characteristics of the near field region of
a heated surface discharge.
This result has been generally realized in the past. In certain
cases a separate undistorted near field model has been constructed and
the results used to adjust the discharge configuration in a distorted
far field model to reproduce the stable region as observed in the near
field model. The theory of this study should make it possible to cir-
cumvent the construction of near field models except in cases where
geometrical complications are extreme and the theory cannot be used.
6.3 A Scale Model Case Study
A laboratory study of a heated discharge was described by
Harleraan and Stolzenbach (14) in 1968. The power plant is located on
the coast of Massachusetts near Plymouth (see Figure 6.2). The primary
discharge configuration under consideration was a surface discharge from
a trapezoidal open channel into a coastal region having a relatively uni-
form bottom slope. The ambient cross current in the coastal zone was
primarily due to wind-driven circulation. The intake was directly down-
wind of the discharge and was protected by two breakwaters (see Figure
6-2).
-162-
-------
N
1/2 MILES
CAPE COD BAY
SCALE
-30
Dischorge Channel
Pilgrim
Station Nuclear
Power Plant Site
MANOMET
FIGURE 6.2 LOCATION OF POWER PLANT
-------
The purpose of the model study was twofold; to determine the
extent of far field recirculation of the heated water into the intake
and to evaluate the near field temperature distribution. Thus both far
and near field regions were of interest. Any analytical approach to
predicting the near field temperatures would have been frustrated by the
impossibility of determining whether the discharge jet entrainment would
be cut off laterally by the breakwaters protecting the intake. This
was a particularly strong possibility because the prevailing current
would deflect the discharge toward the breakwaters.
In the model study (AT ) was determined from the arbitrarily
chosen condition (Ap) = 1.0. The horizontal length scale was L = 1/240
to allow reproduction of a sufficient length of coastline in the experi-
mental basin. A vertical scale ratio was chosen as z = 1/40 on the
basis of far field surface heat loss similarity (see Ref. 14 ). Harleman
and Stolzenbach justified the 6 to 1 distortion in the model on the
basis of Wiegel's data for bottom slopes with A = 1/5 (see Figure 5-23).
This data indicated that the effects of the bottom slope and aspect
ratio distortions tended to counteract each other with respect to AT .
c
Figure 6-3 shows the temperature distribution measured in the model for
a typical tide and current condition. The temperature distribution and
observed current patterns in the model indicated that the discharge jet
does entrain water from both sides.
With the knowledge that the entrainraent in the jet is not later-
ally inhibited and using the theoretical results of the present study,
the model and prototype may be schematized as discussed in the previous
-164-
-------
z = 0 ft.
Scheme T
QQ" 650 cfs
Surface Current
.6 knots
z - -1.7 ft.
z - -3.3 ft.
L
Figure 6-3. Temperature Rise Contours Measured
in the Pilgrim Plant Model Study
-------
section. The relevant parameters are:
a
AT
u
c
V
K
IF
o
A
K/u
V/u
Prototype
65°F
28.7°F
720 cfs
6.7 ft
26.0 ft
2.0 ft/sec
1.0 ft/sec
3.6 x 10~5 ft/sec
(200 BTU/ft2-day°F)
1/60
2.0
,26
1.8 x 10~5
.5
Modal
77°F
25"F
5.2 gpm
.17 ft
.11 ft
.32 ft/sec
.16 ft/sec
2 x 10~5 ft/sec
(110 BTU/ft2-daycF)
1/10
2.0
1.5
6 x 10~5
.5
The centerline surface temperatures observed in the model are
shown in Figure 6-4. The model data is plotted in prototype distances
(ft.) by scaling the model distances with the horizontal scale ratio
L = 1/240. A theoretical calculation using the theory of this study
with the tabulated model parameters is also shown. In the theoretical
calculation V/u = 0 and S = « since the cross flow in the model was
O J»
not significant in the near field region and the large value of S (1/10)
A
in the distorted model had little effect on such a lev; densiraetric
-166-
-------
1.0
i i i i I \ i
500
1OOO
5000
x - Dlcttnce Along Jet Centerline
In Prototype (ft.)
Figure 6-4. Theoretical Calculations and Model Data for Pilgrim Station
Power Plant Model Study
-------
Froude number discharge. The agreement between the model data and the
theoretical calculation plotted in prototype distances Oft.) is good.
It should be kept in mind that the scheraatization of the model for in-
put to the theory is based upon the observed two-sided entrainment in
the model. Figure 6-4 also shows the theoretical calculation for the
prototype parameters. The model and prototype curves are close together
up to about 500 ft. from the discharge, but the curves diverge from that
point on. The argument based on Wiegel's data is obviously limited by
the fact that the F in Wiegel's runs was 25 rather than 2. The model
case shows the formation of a stable region which is apparently not
present in the prototype although the limitations of the theory with
respect to discharges with bottom slopes must be kept in mind. The
model temperature distribution extends farther from the discharge than
that of the prototype by roughly a factor of 2 to 3 which is what Equa-
tion 6.6 predicts for z = 6L as was the case for this model.
This case study indicates that the distorted model does not re-
produce the prototype temperature distribution although the results are
conservative since the model showed higher temperature rises at corres-
ponding distances from the discharge channel. Also the agreement of
model results and theoretical calculations is a verification that the
theory could be substituted for the construction and operation of the
model as far as near field temperatures are concerned.
-168-
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VII. Summary and Conclusions
7.1 Objectives
The use of natural waterways for disposal of power plant rejected
heat necessitates prediction of the induced temperature distribution in
the receiving water for:
a) Evaluation of thermal effects upon the natural environment.
b) Prevention of recirculation.
c) Design of discharge structures to meet temperature standards.
d) Scale model design.
This study treats a three-dimensional horizontal surface dis-
charge of heated water from a rectangular open channel at the surface
of a large ambient body of water which may have a bottom slope and a
cross flow at right angles to the discharge. Of interest is the de-
pendence of the temperature distribution in the vicinity of the dis-
charge point upon the initial temperature difference between the ambient
and discharged water, the initial discharge velocity, the discharge
channel dimensions, the bottom slope, the ambient cross flow, and sur-
face heat loss to the atmosphere. The primary objective is the predic-
tion of the temperature distribution within the near field region domin-
ated by turbulence generated by the discharge. Temperature distributions
in the far field region where ambient processes dominate are beyond the
scope of the study.
7.2 Theory and Experiments
Previous knowledge related to heated discharges consists of
established theories for isothermal turbulent jets, measurements of the
-169-
-------
physical properties of turbulent jets, experimental work on stratified
turbulent flows, and a sraall number of laboratory experiments on heated
surface jets. Previous attempts by others to develop a theory for heat-
ed three-dimensional surface discharges have been limited by failure to
consider all the factors affecting the temperature distribution in the
discharge.
The theory of this study has the following structure:
a) The discharge is basically a three-dimensional turbulent
jet with a well defined turbulent region in which velocity
and temperature are related to centerline values by simil-
arity functions. Entrainment of ambient fluid into the
turbulent region is proportional to the centerline velo-
city by an entrainment coefficient determined from non-
buoyant jet theory. Unsheared core regions are accounted
for in the theory.
b) The effect of buoyancy on the discharge is to reduce ver-
tical entrainment and to promote lateral spreading of the
of the jet as a density current. A major contribution
of this study is the incorporation of these buoyancy
effects into the turbulent jet theory.
c) Surface heat loss is assumed to be proportional to the
surface temperature rise above ambient in the discharge
and is accounted for throughout the entire discharge
length.
d) The jet is deflected in the presence of an ambient
-170-
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cross flow by entrainment of lateral momentum.
e) A bottom slope is assumed to inhibit vertical entrainment
and buoyant lateral spreading as long as the discharge
jet remains attached to the slope.
The theoretical development synthesizes the results of previous
investigations of turbulent jets and stratified flows. 'No new empirical
constants are introduced in the theory and the comparison of the theory
with experiments does not involve any "fitting" of the theory to the
data.
Experiments are performed in which all the controlling parameters
are varied. Care is taken to insure establishment of fully turbulent
steady state conditions in the discharge and accurate three-dimensional
temperature measurements are taken. The data reduction results in cen-
terline temperatures, discharge depths and widths, and centerline posi-
tion in the case of an ambient cross current.
7.3 Results
A comparison of experimental data from this study and others
with the theoretical calculations indicates that the theory reproduces
the centerline temperature rise, the jet width and depth, and the cen-
terline position. When a bottom slope is present good agreement is
achieved for centerline temperature rises although the theory does not
predict correctly the lateral spreading of the heated discharge.
Particular theoretical results are:
a) For nonbuoyant jets (F = °°) the discharge spreads linearly
with distance from the origin. The centerline temperature
-171-
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beyond the core region is giyen by;
AT 1/2
£ _ , (1/2 x Area of Discharge Channel)^
AT ~ 1J<* x
This expression is valid for discharges in which IF > 15.
b) Buoyant jets (1 < IF < 10) consist of a core region, an
entrainment region, a stable region, and a heat loss
region which marks the end of the jet and the beginning
of the far field which is dominated by ambient processes.
c) Entrainment increases with increasing initial densimetric
Froude number, IF , and with discharge channel aspect
ratio, A « h /b . The temperature and diluted flow in
o o
the jet in the stable region are functions of F and A
(see Figure 5-25).
d) The vertical spread of the jet is inhibited and the lateral
spread increased by the buoyancy of the discharge. Spread-
ing increases with decreasing IF and the maximum vertical
penetration of the jet is a function of IF (see Figure 5-26).
e) Surface heat loss has no significant effect on the behavior
of the discharge in the core region, the entrainraent region
or the stable region.
f) Ambient cross flows whose velocities are small relative to
the initial discharge velocity deflect the discharge center-
line but do not greatly affect the temperature distribution.
-172-
-------
g) Bottom slopes tend to inhibit vertical entrainment for
heated discharges in which IF is greater than about 5.
Discharges with smaller values of F tend to separate
from the bottom close to the discharge point and show
little dependence on S , the value of the slope. There
X
is some indication that in the region where the jet is
attached to the bottom, the increased lateral area actually
promotes increased entrainment in low 3F discharges.
o
This study of heated surface discharges satisfies its broad ob-
jectives to the following extent:
a) Evaluation of thermal effects: The theoretical model pre-
dicts the limits of the near field temperature distribution
in which temperature rises are significant. The diversion
of natural flows by jet entrainment is also a result of
the theoretical calculations.
b) Recirculation: The dependence of the maximum vertical
discharge depth on ]F (Figure 5-26) enables design of
skimmer wall structures which require a knowledge of the
depth of the heated layer at the intake. The calculated
dimensions and position of the discharge jet aids in the
location of an intake for prevention of recirculation.
c) Discharge design; The temperature and shape of the heated
discharge is directly related to the densimetrie Froude
number, IF , and the aspect ratio, A, of the discharge
channel by the theory. Theoretical calculations may be
-173-
-------
used to evaluate the success of discharge configurations
in meeting temperature standards.
d) Scale models; The theory indicates that near field temp-
erature distributions are not reproduced in distorted models.
If the prototype discharge configuration may be schematized
as input to the theory, the theoretical calculations will
make construction of an undistorted near field model unneces-
sary.
7.4 Future Work
The immediate need is for incorporation into the theory of some
of the more complex aspects of heated discharges:
a) Stratified receiving water: This requires more complicated
assumptions for heat transfer within the jet and for buoy-
ant spreading rates.
b) Accumulation of heat in the far field: If the far field is
of finite size, the rejected heat will cause the far field
surface temperature to rise until equilibrium with surface
heat loss is achieved. The resulting stratification must
be described and its interaction with the discharge entrain-
ment and buoyant spreading accounted for as in (a) above.
c) A far field model: Buoyant plume theories (4 ) (28) have
already been postulated. A complete picture of the temp-
erature distribution in the near and far field regions awaits
the development of a theory for the transition region between
the near and far fields.
-174-
-------
Much, of the field data on temperatures in the. vicinity of actual
prototype discharges is either unavailable to researchers or is of such
poor quality that it is unusable. Improvements in the theory of this
study and the development of extensions as described above depend cruc-
ially upon the availability of accurate, complete field data for power
plant heated discharges.
-175-
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BIBLIOGRAPHY
1. Abramovich, G. N. , The Theory of Turbulent Jets, TheM.I.T. Press,
M.I.T. Cambridge, Massachusetts, 1963.
2. Batchelor, G. K., An Introduction to Fluid Dynamics, Cambridge
University Press, 1967.
3. Brady, D. K. and others, "Surface Heat Exchange at Power Plant
Cooling Lakes", Edison Electric Institute, Publication No. 69-901,
November 1969.
4. Brooks, N. H., "Diffusion of Sewage Effluent in an Ocean Current",
in Waste Disposal in the Marine Environment, Pergamon Press, 1960.
5. Carter, H. H., "A Preliminary Report on The Characteristics of a
Heated Jet Discharged Horizontally into a Traverse Current Part I
- Constant Depth", Technical Report 61, Chesapeake Bay Institute,
Johns Hopkins University, November 1969.
6. Cootner, P. H. and G. 0. G. Lof, Water Demand for Steam Electric
Generation, Resources for the Future, Inc. , The Johns Hopkins
Press, Baltimore, Maryland, 1965.
7. Deissler, R., "Turbulence in the Presence of a Vertical Body Force
and Temperature Gradient", Journal of Geophysical Research, Vol.
67, No. 8, July 1962.
8. Edinger, J. E. and E. M. Polk, Jr., "Initial Mixing of Thermal
Discharges into a Uniform Current", Report No. 1, Department of
Environmental and Water Resources Engineering, Vanderbilt Univer-
sity, October 1969.
9. Ellison, T. H. and J. S. Turner, "Turbulent Entrainment in
Stratified Flows", Journal of Fluid Mechanics, Vol. 6, Part 3,
October 1959.
10. "Environmental Effects of Producing Electric Power", Hearings before
the Joint Committee on Atomic Energy, Congress of the United States,
U. S. Government Printing Office, 1969.
11. Fan, Loh-Kien,"Turbulent Buoyant Jets into Stratified or Flowing
Ambient Fluids", W. M. Keck Laboratory of Hydraulics and Water
Resources, California Institute of Technology, Report No. KH-R-15,
June 1967.
12. Harleman, D. R. F., "Stratified Flow", in Handbook of Fluid Dynamics.
V. L. Streeter, Ed., McGraw Hill Book Company, Inc., 1961.
-176-
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13. Harleman, D. R. F. and K. D. Stolzenhach, "A Model Study of Thermal
Stratification Produced by Condenser Water Discharge", M.I.T. Hydro-
dynamics Laboratory Technical Report No. '107, October 1967.
14. Harleman, D. R. F. and K. D. Stolzenbach, "A Model Study of Proposed
Condenser Water Discharge Configurations for the Pilgrim Nuclear
Power Station at Plymouth, Massachusetts", M.I.T. Hydrodynamics
Laboratory Technical Report No. 113, May 1969.
15. Hayashi, T. and N. Shuto, "Diffusion of Warm Water Jets Discharged
Horizontally at the Water Surface", IAHR Proceedings, Ft. Collins,
Colorado, September 1967.
16. Hayashi, T. and N. Shuto, "Diffusion of Warm Cooling Water Dis-
charged from a Power Plant", Proceedings of the llth Conference
on Coastal Engineering, London, 1968.
17. Hinze, J. 0., Turbulence. McGraw Hill Book Company, Inc., 1959.
18. Hoopes, J. A. and others, "Heat Dissipation and Induced Circula-
tions from Condenser Cooling Water Discharges into Lake Monona",
Report No. 35, Engineering Experiment Station, University of
Wisconsin, February 1968.
19. Industrial Waste Guide on Thermal Pollution. U. S. Department of
the Interior, Federal Water Pollution Control Administration,
Corvallis, Oregon, September 1968.
20. Ippen, A. T. and D. R. F. Harleman, "Steady-State Characteristics
of Subsurface Flow", U. S. Department of Commerce, NBS Circular
521, "Gravity Waves", 1952.
21. Jen, Y. and others, "Surface Discharge of Horizontal Warm Water
Jet", Technical Report HEL-3-3, University of California, December
1964.
22. Kato, H. and 0. M. Phillips, "On the Penetration of a Turbulent
Layer into Stratified Fluid", Journal of Fluid Mechanics, Vol. 37,
Part 4, July 1969.
23. Lofquist, K., "Flow and Stress Near an Interface Between Stratified
Liquids", The Physics of Fluids, Vol. 3, No. 2, March-April 1969.
24. Pai, Shih-I, Fluid Dynamics of Jets, D. Van Nostrand Company, Inc.,
1954.
25. Parker, F. L. and P. A. Krenkel, Engineering Aspects of Thermal
Pollution, Vanderbilt University Press, 1969.
-177-
-------
26. Pearce, A. F., "Critical Reynolds Number for Fully-Developed Turbu-
lence in Circular Submerged Water Jets", National Mechanical Engin-
eering Research Institute, Council for Scientific and Industrial
Research, CSIR Report MEG 475, Pretoria, South Africa, August 1966.
27. Phillips, 0. M., The Dynamics of the Upper Ocean, Cambridge Univer-
sity Press, Cambridge, England, 1966.
28. Prych, E. A., "Effects of Density Differences on Lateral Mixing
in Open-Channel Flows", W. M. Keck Laboratory of Hydraulics and
Water Resources, California Institute of Technology, Report No.
KH-R-21, May 1970.
29. Schlichting, H. , Boundary-Layer Theory, Translated by J. Kesten,
Sixth Edition, McGraw Hill Book Company, Inc., 1968.
30. Stefan, H. and F. R. Schiebe, "Experimental Study of Warm Water
Flow Part I", Project Report No. 101, St. Anthony Falls Hydraulic
Laboratory, University of Minnesota, December 1968.
31. Stefan, H. and F. R. Schiebe, "Experimental Study of Warm Water
Flow into Impoundment? Part III", Project Report No. 103, St.
Anthony Falls Hydraulic Laboratory, University of Minnesota,
December 1968.
32. Tamai, N. and others, "Horizontal Surface Discharge of Warm Water
Jets", ASCE Journal of the Power Division, No. 6847, PO 2,
October 1969.
33. Townsend, A. A., "The Mechanism of Entrainment in Free Turbulent
Flow", Journal of Fluid Mechanics, Vol. 26, Part 4, 1966.
34. Townsend, A. A., The Structure of Turbulent Shear Flow^ Cambridge
University Press, Cambridge, England, 1956.
35. Wada, A., "Studies of Prediction of Recirculation of Cooling Water
in a Bay", Proceedings of the llth Conference on Coastal Engineering,
London, 1968.
36. Webster, A. G., "Experimental Study of Turbulence in a Density-
Stratified Shear Flow", Journal of Fluid Mechanics, Vol. 19, Part
2, June 1964.
37. Wiegel, R. L. and others, "Discharge of Warm Water Jet over Sloping
Bottom", Technical Report HEL-3-4, University of California,
November 1964.
38. Yevdjevich, Vujica M., "Diffusion of Slot Jets with Finite Length-
Width Ratios", Hydraulic Papers, Colorado State University, Ft.
Collins, Colorado, No. 2, 1966.
-178-
-------
LIST OF FIGURES AND TABLES
Figure Page
1-1 Schematic of Heated Discharge 16
2-1 Structure of a Submerged Turbulent Jet 19
2-2 Velocity and Temperature Distributions 26
in Submerged Jets
2-3 Reduction of Entrainment as a Function of 30
Richardson Number
2-4 Two Layer Flow in the Discharge Channel 34
2-5 Schematics of Previous Theoretical Investigations 36
3-1 Characteristics of the Discharge Channel 43
3-2 Geometrical Characteristics of the Jet 55
3-3 Velocity and Temperature Characteristics of the Jet 57
3-4 Non-Buoyant Jet Solutions 70
3-5 Characteristics of a Deflected Jet 85
4-1 Experimental Set-Up 102
4-2 Mounted Temperature Probe 104
4-3 Cross Flow Velocity in Experimental Basin 108
4-4 Temperature Distribution Measured for Run 3
4-5 Surface Temperature Distribution Measured for
Run 22
5-1 Theoretical Calculations of Jet Structure for 115
Run 6
5-2 Calculated Isotherms of AT /AT for Run 6 11 A
c o •LJ-°
5-3 Theoretical Calculations and Experimental Data 120
for Run 1
5-4 Theoretical Calculations and Experimental Data
for Run 2
-179-
-------
Figure Page
5-5 Theoretical Calculations and Experimental 122
Data for Run 3
5-6 Theoretical Calculations and Experimental 123
Data for Run 4
5-7 Theoretical Calculations and Experimental 124
Data for Run 5
5-8 Theoretical Calculations and Experimental 125
Data for Run 6
5-9 Theoretical Calculations and Experimental 126
Data for Run 7
5-10 Theoretical Calculations and Experimental 127
Data for Run 8
5-11 Theoretical Calculations and Experimental 128
Data for Run 9
5-12 Theoretical Calculations and Experimental 129
Data for Run 10
5-13 Theoretical Calculations and Experimental 131
Data for Runs 11, 12 and 13
5-14 Theoretical Calculations and Experimental 132
Data for Runs 14, 15 and 16
5-15 Theoretical Calculations and Experimental 133
Data for Runs 17, 18 and 19
5-16 Theoretical Calculations and Experimental 136
Data for Run 20
5-17 Theoretical Calculations and Experimental 137
Data for Run 21
5-18 Theoretical Calculations and Experimental 138
Data for Run 22
5-19 Theoretical Calculations and Experimental 139
Data for Run 23
5-20 Theoretical Calculations and Experimental 140
Data for Run 24
-180-
-------
Figure Page
5-21 Theoretical Calculations and Experimental
Data for Run 25
5-22 Theoretical Calculations and Experimental
Data for Circular Orifices - Jen (21)
5-23 Theoretical Calculations and Experimental
Data for Rectangular Orifices - Wiegel (37)
5-24 Theoretical Calculations and Experimental
Data for Rectangular Orifices - Hayashi (15)
5-25 Dilution in the Stable Region of Buoyant
Discharges
5-26 Maximum Jet Depth Vs. IF as Calculated by
the Theory °
6-1 Discharge Channel Schematization
6-2 Location of Power Plant
6-3 Temperature Rise Contours Measured in 165
the Pilgrim Station Model
6-4 Theoretical Calculations and Model Data 167
for the Pilgrim Station Power Plant
II-l Generalized Cross Section of a Deflected Jet 21°
Table Page
2-1 Summary of Previous Theoretical Investigations ^1
3-1 Integrated Equations for Buoyant Jets 81
3-2 Integrated Equations for Deflected, Buoyant Jets 91
4-1 Summary of Previous Experiments 98
4-2 Summary of Experimental Runs 107
-181-
-------
LIST OF SYMBOLS
A — the discharge, channel aspect ratio = h /b
o o
a - the coefficient of thermal expansion = - ~p
31
b - horizontal distance from the jet centerline to
the jet boundary
b - one half the width of a rectangular discharge channel
b. ,„ - horizontal distance from the jet centerline to the point
where the temperature or velocity, as defined, is one
half the centerline value
D - the ratio of flow in the jet to the initial discharge
flow = dilution
D ,. - the dilution in the stable region of a heated discharge
db
-j—• - the spread of a non-buoyant jet
NB
F - the densimetric Froude number =
P
IF - the densimetric Froude number of the discharge channel
u
Ep
™ * gn
D O
a
u
It' - the local densimetric Froude number in the jet = -c-
[3/2T
1 - (j;) J
G - the turbulent integral time scale in a homogeneous flow
G - the turbulent integral time scale in a stratified flow
s
H - the ratio of heat flow in the jet to the initial discharge
heat flov;
h - vertical distance from the jet centerline to the jet
bound :irv
-182-
-------
h - depth, of the discharge channel
h * - depth of the heated flow at the point of discharge if a
cold water wedge is present
h^ 12 - vertical distance from the jet centerline to the point
where the temperature or velocity, as defined, is one
half the centerline value
h - the maximum value of h obtained in a heated discharge
,1
= .4500
-I
-t
-c
-tt
= .3160
.6000
I, - t(Od?d? = .2143
.2222
I6 - I f UK"" - .1333
C
£
I - \ f(C)t(OdC - .3680
K - the surface heat loss coefficient
k - the thermal conductivity of water
L - the scale ratio for horizontal lengths between
model and prototype
r 3 i 1/2
N - the Brunt-Vaisala frequency - \-~ -^H
p - the pressure
p, - the dynamic pressure
P_ - the hydrostatic pressure
d.
-183-
-------
Q - the discharge channel flow
q - the entrainment flow into the jet per unit length of jet
R - the Eulerian time coefficient of a turbulent flow
£_ 5p
R. - the Richardson number = p §z
R,. - the hydraulic radius of the discharge channel flow
R - the radius of a circular discharge pipe
° AuoRH
H - the jet Reynolds number =
r - the vertical distance from the jet centerline to the
boundary of the core region
S - the bottom slope
x
s - the horizontal distance from the jet centerline to the
boundary of the core region
T - the temperature
T - the ambient temperature
3
T - the equilibrium temperature
T - the jet surface centerline temperature
T - the temperature of the heated flow in the discharge
channel
T - the temperature at the water surface
S
AT - temperature rise above ambient in the jet - T - T
£1
AT - the surface temperature rise above ambient at the jet
centerline T - T
c a
AT - the temperature difference between the discharge and
the ambient water = T - T
o a
AT - the scale length for temperature rises between model
and prototype
3/2
t - the similarity function for temperature = 1 - £
-184-
-------
u,v,w - the velocity components in the fixed coordinate system
UjV^w - the velocity components in the coordinate system relative
to the centerline of a deflected jet
u - the velocity in the discharge channel
u * - velocity of the heated flow at the point of discharge
if a cold water wedge is present
u - the surface centerline jet velocity
u - the scale ratio for velocities between model and prototype
V - the ambient cross flow velocity
v - the lateral velocity of the entrained flow at the jet
e boundary
v, - the lateral velocity in the jet at y » s and -h s and -r < 2 < n
S
w, - the vertical velocity in the jet at z » -r and s < y < b
w - the vertical velocity of the entrained flow at the jet
boundary
w - the vertical velocity in the jet at z * -r and 0 < y < s
x,y,z - the fixed coordinate directions
x,y,z - the coordinate directions relative to the centerline
of a deflected jet
x - the distance from the jet origin to the point where s » 0
S
x - the distance from the jet origin to the point where r « 0
z - the scale ratio for vertical lengths between model and
prototype
a - the entrainment coefficient
a - the entrainment coefficient in a homogeneous flow
a - the entrainment coefficient in a stratified flow
S
a - the lateral entrainment coefficient in non-buoyant and
buoyant jets
-185-
-------
a - the vertical entrainment coefficient in a non-buoyant jet
a - the vertical entrainment coefficient in a buoyant jet
sz
6 - a scaling quantity one order of magnitude less than unity
e - the spread, -:— , of the turbulent region in an undeflected
non-buoyant jet
n - the water surface elevation
9 - the angle between the jet centerline (x axis) and the y axis
6 - the angle between the discharge channel centerline and the
y axis
p - the density of water
p - the density of the ambient water
3i
p - the density of the heated discharge
Ap - the difference between the density and the ambient water
density = p - p
£L
Ap - the difference between the density of the heated flow in
the discharge channel and the ambient density = p - p
o a
Ap - the scale length for density differences between model
and prototype
$ - viscous generation of heat
v - the kinematic viscosity of water
Superscript * - indicates a characteristic magnitude of the
variable
Superscript - indicates a turbulent fluctuating quantity
-186-
-------
Appendix I. The Computer Program for Theoretical
Calculations of Heated Discharge Behavior
The computer program is written in Fortran IV, G level, Mod 3
and the calculations for this study are done on an IBM 360-65 computer.
The program consists of a main program and five subroutines:
MAIN; This program reads the input data for the theoretical calcula-
tions, sets up the initial conditions for each calculation, and calls
the subroutine SRKGS which performs the actual calculations.
Input Data Format
Number of Calculations 13
One Set [IF , S , A, K/u , Q * ERR, b. 8F10.5
I o x O O r i
for Eachj I]L, 1^ T.y 1^, 1^, Ig, I?, e 8F10.5
Calcula-[vi, V2, V3> V4, V5§ Vfi, V , Vg 8F10.5
tion
u
where: IF = initial densimetric Froude number « —
o
I
S = bottom slope Jp 8 o
A * channel aspect ratio » h /b
o o
K/u = surface heat loss coefficient parameter
6 = initial angle of the discharge with the
boundary of the ambient region
XF * The value of x/Vh b at which the program should
terminate for this calculation. In this study,
x., varies from 50 to 200.
i
-187-
-------
ERR = Maximum allowable numerical truncation error in
each dependent variable, at each time step. In this
study a value of ERR = .01 is used.
b. • Initial value of h/yh b and b/>^i b to be used
i o o o o
in this calculation. The mathematical formulation
of the theory sets h=b=0atx=0 but in the
numerical scheme a finite value must be given. A
—8
value of b. » 10 is used in this study. The
calculations are found to be independent of the
choice of b. as long as it is small.
I - I = Integrations of similarity functions as given
in Table 3-1.
e = Spreading rate of the turbulent region of a non-
buoyant jet.
Vn - V0 = Input constants describing the cross flow - (see
J. o
description of subroutine CROSS).
SRKGS; This subroutine is a modified version of the IBM Scientific
Subroutine Package routine DRKGS. It is a fourth order Runge-Kutta
solution to a system of first order differential equations.
The form of the equation is;
dx~ ' ci
where a is a coefficient matrix, y. is a vector of the independent
-LJ J
variables and c. is a vector of the constants in the i equations.
-188-
-------
The program MAIN provides the initial values y. and the sub-
routine SRKGS advances the solution by calling subroutine FCT (described
dy
below) which calculates the values of a..> c., and -r-*- at each step.
The solution is terminated when x «= x,. The results of each step are
output by subroutine OUTP.
FCT; This routine takes the current values of x and y. and computes
J
the values a., and c according to the theory of this study, the system
of equations in (I--1) are then solved by an IBM Scientific Subroutine
DGELG which returns the values of 9y. /3x to FCT which returns them to
SRKGS.
OUTP: The independent variables y. and u/u , AT /AT , h//h b , b//h b ,
J OCO OO OO
r//h b » s//h bQ, db/dx, x//h b , y//h b , and e • The values of these
variables are output after each step of the calculation. The value of
the dilution D and the heat flow H (see Chapter 5) are also calculated
and printed. Each page of output displays the input variables for that
particular calculation on the page heading.
CROSS_ This subroutine is called whenever the velocity of the cross flow
is required. The subroutine uses the values of V- - Vg to compute the
cross flow as a function of x//h b . For each particular application
o o
of the theory, the subroutine may be different. The only requirements
are as follows :
a) The subroutine returns the value of V/u at the given x/ >h b .
o
b) V8 is reserved for return of ^-j- ,^-j to the main
° " o o
program.
-189-
-------
In this study the following algorithm is used;
(1-2)
00
(1-3)
o o
Note; The program may generate numerical underflows in certain calcu-
lations and corresponding error messages may be printed out. The re-
sult of an underflow calculation is set to zero automatically.
-190-
-------
C MAIN PROGRAM
REAL II, 12, IT, 15, 16
DOUBLE PRECISION PRMT ,Y , AUX , DERY, TH, XP, YP, 8, U , G, H, BX, S ,R, X, XL IM
INTEGER RR
EXTERNAL FCT,OUTP
EQUI VALE MCE ( Y( 1 ) , TH) , ( Y(2 ) ,XP) , ( Y( 3 ) ,YP ), (Y (41 , B) , (Y(5),U)
FSinVALEf^CF (Y(6),G),(Y(7),H),
-------
DERY(N)=..l
2 CONTINUE
CALL SRKSS(PRMT,Y,DERYfNDIM,IHL F, FCT, CUJTP, AUX)
1 CONTINUE
CALL EXIT
FND
-------
c
c
c
c
u>
c.
c
SUBROUTINE SRKGS(PRMT,Y,DERY,NDIM,THLF,FCT,OUTP,AUXl
DOUBLE PRECISION PRMT,YfDERY,AUX,A,R,C,X,XEND,H, AJ,BJ,CJ, Rl ,R2,
1DELT,DABS
DIMENSION Y(!4),OCRY(14),AUX(8tl^)»A(4),B(4)fC(4),PRMT(5)
DP 1 1=1 ,NDIM
1 AUX( 8»I )=.'^6666666666666666700* DERY( I)
X=PRMT(1)
H=PRMT( 3)
PPMT (5) = "«.D'i
CALL FCT(X,Y,DERY)
ERROR TEST
IFf.781186547«5Dn
A{4) = . 1666f>666666666667D^
B(l ) = 2.D'1
B(2)=1.D"
3(3 )=1,D.°
B(4)=2,D~
C( 1 ) = .5DA
C(2 )=. 292893 2 1881 34574800
C (3) = 1,7' 71 "^6781186547500
PREPARATIONS OF FIRST RUNGE-KUTTA STFP
DO 3 I=1,NDIM
AUXd ,!) = YU)
AUX(2tI )=
DRKS1130
DRKG1140
DRKG115P
DRKG1160
DRKG1170
DRKG1180
DRKG1190
DRKG1210
DRKG1220
DRKG1230
DRKG1240
DR
-------
c
c
c
c
c
c
c
H=H+H
IHLF=-1
ISTEP=0
A IF( ( X+H-XEN[5)*H)7,6,5
START OF A 1UNGE-KUTTA STEP
5 H=XENO-X
6 IEND=1
RECORDING OF INITIAL VALUES OF THIS STEP
7 CALL OUTP(X,Y,DERY,IREC,NDIM,PRMT)
IF( PRHT(5) )A"!t 8, AO
8 ITEST=0
9 ISTEP=ISTEP+!
START OF INNERMOST RUNGE-KUTTA LOOP
J = l
DO 91 I=1TM3IM
AUX( 3, 11=^,0^
91 AUX{ 6, I )=0,r)^
10 AJ=A(J)
BJ=B(J)
CJ=C( J)
00 11 I=1»NOIM
R1=H =DERY(I )
R2=AJ*'(R1-BJ'-AUX( 6, I ) )
Y( I )=Y(I
DRKG14-60
11 AUX(6, I) = AUX(6, I)+R2-CJ*Rl
IF
-------
c
c
c
c
c
c
VO
Ln
C
c
c
c
14 CALL FCT(X,Y»DERY)
GOTO 1<>
END 3F INNERMOST RUNGE-KUTTA LOOP
TEST OF ACCURACY
15 IFUTESTI16tl6t2i
IN CASE ITEST="> THERE IS NO POSSIBILITY FOR TESTING OF ACCURACY
16 DO 17 1=1 ,NOIM
17 AUX(4,I»=Y(I)
ITEST=1
ISTEP=ISTEP+ISTEP-2
18 IHLF=IHLF-H
X=X-H
H=.500*H
DO 19 I=1,NDIM
Y(I)*AUX(lfII
DERY( I)=AIIX(?tI)
19 AUX(6,I)=AUX(3tI)
GOTO 9
IN CASE ITEST=1 TESTING OF ACCURACY IS POSSIBLE
2r IMOD=ISTEP/2
IF(ISTEP-IMOD-IMOD)21,23,21
21 CALL FCT(X,Y,DERY)
DO 22 I=1,NOIM
AUX(5,I)=Y(I)
22 AUX(7,I)=DERY(I)
GOTO 9
COMPUTATION OF TEST VALUE DELT
DO 24 I=!,NDIM
?4 DELT=OELT*AUX(8, I)*DABS(AUX( 4,1 )-YU ) )
IF(OELT-PRMT(4) )?8,28,25
DRKG1780
DRKG1790
DRKG1800
DRKG1810
DRKG1820
DRKG1839
DRK31840
DRKG1850
DRKG1853
DRKG1870
DRKG1880
ORKG1890
DRKG1900
DRKG1913
DRKS1920
DRKG1930
DRKG1940
DRKG1960
DRKG1970
DR
DRKG2123
DRKG2130
-------
C DRKG214D
C ERROR IS TOO GREAT DRKG215O
?5 GO TO 26
?6 00 27 1=1, NO I M DRKG2170
27 AUX(4,I)=AUX(5 ,1) DR
-------
35 IHLF = IHLF-1
ISTEP=ISTFP/? DRKG2490
H=H+H DRK3250O
GOTO 4 DRKG7510
C DRKG2520
C DRKG2530
C RETURNS TO CALLING PROGRAM DRKG2540
36 THLF=ll DRKG2550
CALL FCT(X,Y,DERY) DPXG2560
GOTO 39 ORKG257?
37 IHLF=12 DRKG2580
GOTO 39 DRKG2590
38 IHLF = 13 ORKG26CO
39 CALL OUTP
-------
SUBROUTINE FCTr H=l.^D-7
7^1 CONTINUE
SLPPE=1
FLAG=1
CAL L CRnSS(XP,V)
IF(V) 7=1,75,76
75 EPSB=EPS
FLAG=2
76 IF (R-.OTO'U) 8^,80,81
RP R=«~'.
WINF=Il-"U*EPS/2
GO TO 811
81 WINF=(I1-I2) *.EPS<:U
811 IF (S-.000^1) 82,82,83
82 S=r%
VINF=-IlfcUJ EPS/2
-------
GO TO 831
83 VINF=-U1-I2)*EPS*U
831 VCT=OCOS(TH)*V
WINFS = WINF
FIS=FI
B2=S+6*T2
83=8*16
B4=B*I5
BT=S+B*IT
H? = R-«-H*I2
HGG=R*R 3 5+Tl*HfeR+T2*H*H
HT=R-l-H*IT
VO
VO
UM1= U* 12+2. VCT*I1
888 DO 1 N=l,14
DP 1 NN=lf14
1 CONTINIJF
GO TO
FLAG
GO TO 6
301 ARG= 53':G*H/(U*U*FR*FR)
312 ARG=0.
3^3 IF (ARG-1H, )
GO TO 6
3C WINF=WINF*EXP(-ARG)
5 A(l,l ) = U*U
A (2, 2 I *1,1
A(3,3)=l,n
VCT*VCT*(S+B)*{R+H)
-------
A<4,4) = 1,0
OERY<4)= 8X
A(5,1)=-H*B*VST
A (5, A )= Il*U*H*Il + H
ACS, 5 )=I1*H*B*I1
A( 5,7)=I1*U*B*I1*B*VCT
A(5,9)=UtH*IH-H*VCT
At 5,10) = U*8*IH-B*VCT
A(5,11)=-H*I1
A(5,12)=8*I1
A(6,1)=-2.*H-B*(U*I1*II+VCT)*VST
A(6,4 ) = U" H^UM2 + TlvT2* H^H*G1' F
A(6, 5)=?
A(6, 7) =
A(6,9I=
A(6,10)
A(6,ll >=-H*UMl
A(6,12)«B*UMl
A(7,l )=-G*HG BG'VST
A ( 7f 4)=G* ( U*I T*HT+T1*VCT*HG
A(7,5) = GtHT^BT
A (7, 7) = GMU*
A { 7 ,9 ) =G** I U* HT+VCT*HG )
A(7,1C)=G*( U«-BT*VCT*BG)
GO TO (3*4,3^5), FLAG
304 A(8,8)=l,0
OERY(4)=0.
GO TO 3f>6
305 GO TO (3^51,3052), SLOPE
A( 8,1)=-(BX-EPSBJ*VST*2. *( U*H1* B4+VCT* (R+H)*B)
A(8,4)= (BX-EPSB)*(U*U*I6*H2+2.*VCT*U*H1*I5+VCT*VCT*'
-------
(BX-EPSB)MU*UM2*B3+2.*VCT'B4*I1*U+VCT*VCT*B)
. *VCT*B4*H1+VCT*VCT* - U^R-^R
Adi, 12)= -I1*B
A(ll,14)= -R
A(12tl)*-2. «B*R*(U*Il-»-VCT)*VST
AC 12,4) = R MJ*U*!H-R*VCT*VCT+T1*G*FI*< . 5*R*R4-R*H*T1I
A(12t6)= T1«B*<, «5*R^R+R*H-*T1 )*FI
A(12,7)=T1"T1*B*R*G*FI
A ( 1 2 ,9 ) =R* ( U -U*2, *U«-VCT-K R*. 5+H* Tl ) *G*F I ) *R*VCT* VCT
A(12,lr>)=A(12,7)/Tl
A(12,14)= -R (U*VCT)
6' IF (S» 61,61,62
61 A(9,9)*l,'>
-------
A(14,14)=l
GO TO 70
62 A<9, 1 )=-
A(9,7) = I!*S*U+S*VCT
A(9,10)=S*U+S*VCT
A(9,11)=I1*H
A( 9,13)=S
A(11,1}=-2,*S*H*
-------
( 1?)=
OERY(13)=
IF (V) 7?, 73, 71
71 VR=VCT-VEL(8)/VST
00 72 1=2,14
DEPY( I^DE^Yd
72 CONTINUE
73 CONTINUF
on 2^ 1=1 ,14
no 21 j=i,i*
OA=OABS(A( I, J) )
IF 1DA-AMAX) 21,21,22
22 AMAX=DA
21 CONTINUE
IF (AMAX) 2% 2*^,24
^ 24 DP 23 J=l,14
g A( I,J)=A( I, JJ/AMAX
1 23 CONTINUE
OERY(I)=DERY(I)/AHAX
2r CONTINUE
EPS = .10-16
CALL OGELGCOHRY, A, 14, 1 , ERS ,1 ER)
IF (IER) 9,8,7
9 WRITE (P,5'^> IER, FLAG, SLOPE
WRITE (P,5H1) < (A(I,J) ,J=l,14),OERYm,I=l,14)
CALL EXIT
7 WRITE (P^Oi) IER.FLAG, SLOPE
FORMAT (]X,18HERRnR IN MATRIX = ,12,14,14)
FORMAT (15(08.2))
GO TO (32%321 ), FLAG
FLAG=2
FPSB=DERV(4)
FI=FIS
GO TO 888
-------
3?1 GO TO (322, 323), SLOPE
322 SJ=DERY(7) + DERY(ir\)
SXC=SX"DSIN( TH)
IF (SXC-SJ) 3221*3221,323
3221 SLOPE=2
WINF=0.
GO TO 88B
323 FI = FIS
VEL<8)=0,
RETURN
END
O
•o
-------
SUBROUTINE OUTP(X,Y,OERY , I HLF,NDIM, PRMT )
INTEGER P
DOUBLF PRECISION DERY, Y, PRMT, X, TH,XP , YP ,U ,B ,G ,H,B X, S, R, XUM
REAL IItI?»TTf15,16,M
DIMENSION DERY(14),PRMT(5) ,Y(14) ,VEL( 8)
COMMON NPAGE,NLINE, FR,FI,AS,E,I1, 12,1T,T1,T2,EPS,I 5,16,SX,THI,VEL
COMMON FRR
TH=Y(1)
XP=Y(2)
YP = Y{ 3»
RX=Y(8)
r!, S=Y(9)
g R=Y(10)
1 P = 6
CALL CROSS«P,V)
IF (NLINE-«5-U ^,11,11
11 NPAGE = NPAGE +1
NLINE = "
WRITE 2) NPAGE
23? FORMAT (1H1,24HBUOYANT JET C ALCULAT IONS, ir»X, 5HPAGE ,12)
WRITE
-------
FC=ABS( FC)
FC=FC*':.5
FC=FR/FC
Q=U*( S+BM1)1 <*+H*Il)-i-V*DCOSf TH)*(S-t-B)* FORMAT ( 3 < IX ,08 .3 ), IX, 09. 3, I'M 1 Xt 08. 3) » IX, I 3)
IF (U-V-DCOS(TH) ) 500,500,501
500 PRHT(5)=1
5D1 RETURN
END
o
T
-------
SUBROUTINE CROSS(XP,V)
REAL lit 12,IT,15,16
DOUBLE PRECISION XP
DIMENSION VELC8)
COMMON NPAGE,NLINE,FR,FI,AS, E, II, 12, IT,T1,T2 ,EPS, 15,1 6, SX ,THI ,VEL
A=XP*VEL(4I
A= A-VELC1)
VEL(8) = -A*VEL (2 )'2.*' VEL ( A)
A=A*A
A=VEL (2)*A
V=VEL(3)*EXP(-A)
RETURN
END
-------
Appendix II. '"'of! crteri Jet TO a Cross Flow:
Integration. of_ the x and y'Jj orient urn
Equations over the Entire Jet
This appendix considers the x and y momentum equations (3.108
and 3.109) for a deflected jet as given in Section 3.5.1. These equa-
tions will be integrated over the entire jet to give equations 3.122
and 3.123 which are used in Section 3.5.1 to develop the governing equa-
tions for deflected jets. The basic assumptions of the derivation are:
1) The velocity distributions within the jet are symmetrical
about the y = 0 plane i.e. the cross flow does not greatly
distort the jet from the structure defined for the undeflec-
ted jet in Chapter 3.
2) The flow outside the jet is irrotational.
Both of the above assumptions require that the cross flow velocity is
small with respect to the initial jet velocity as discussed in Section
3.5.1.
The x and y equations (3.108 and 3.109) are:
_2 „_, „.,__ .
/i ~ •. (? t) 3.2 \ o & 1 * ~ gll v o u W «•* ** i — t do
2uv —- = —a \ —z— az =— - • vn'-u' -i"i
3x 3y 3z 3x p ), 3x 3y 32
3 Z
(II-l)
3uv 3v 3vw _ 38 2 _ ~2. _ ag_ ( 8AT ,-__!_
Sx 3y 3z 9x p / - 39 p
a z a
v'2) (II-2)
-208-
-------
Figure II-l shows a local section of the jet in which the shape
of the jet cross section, A, is arbitrary except for symmetry about the
y = 0 plane and in which the distribution of entrainment flow around the
jet section circumference, C, is also arbitrary.
Equations II-l and II-2 are integrated over the entire jet area,
A, taking advantage of basic scale relationships introduced in Chapter
3, the symmetry of the jet, and the boundary conditions at the jet boun-
dary as given in Chapter 3.
-pr $ u2 dydz = -^$1 dz'dydz + u(wdy-vdz)
A a A ' 2 C
de Lf ,2 ,,,, jL pd
A C pa C
v(wdy-vdz)
It is assumed that everywhere on the jet boundary (see Figure
II-l):
u
V cos 9 (II-5)
Noticing that the total entrainment per unit length of jet, q , is given
by:
q = 9 (wdy - vdz) (II-6)
then the integrated ic equation (II-3) becomes:
-209-
-------
VcosG / \Vsin6
Cross Flow
Top View of Deflected Jet
1
VsinQ
Cross Flow
Cross Section of Jet
Total Entrainment/Unit Length = q
Figure II-l. Generalized Cross Section of a Deflected Jet
-210-
-------
-£ <$ u2 dydz - Sfi. ^ j ^|i- dz'dydz + q£ V cos 0 (II-7)
pa A ' z
The above equation, when the integrations are written in terms of the
jet geometry defined in Chapter 3, is the desired result i.e. Equation
3.113.
The y momentum Equation (11-4) is not so easily dealt with since
the velocities on the boundary are not known without determining the
flow of the ambient fluid around the jet. The assumption of irrotation-
al flow makes it possible to use several properties of potential flows
to evaluate the terms in II-4.
A. Bernoulli equation is written from a point far from the jet
to the jet boundary:
pv2
a
+
1 r.2 . -2 , -2,
Pd + 2 pa [U + V + W ] r
on C
(11-8)
where p, is the dynamic pressure far from the jet. This may be
written:
P = H - p [v2 + w2] on C (II-9)
ir O O
where H - £^- + p - p V cos 6 (11-10)
is constant and will not contribute to the pressure integral over the
-211-
-------
circumference of the jet.
Then Equation II-4 becomes:
<** A C 2
Defining a complex coordinate system in the y-z plane
C = y + i z (II-12)
and a complex potential, Q, for which the velocities v and & are given
by
» v - i w (11-13)
Equation 11-11 becomes
u2 dyd2 - Real ±
Batchelor (2) shows that fot an arbitrary shaped cylinder in the pres-
ence of a cross flow, V sin 6, and having a total mass flux per unit
length into the cylinder q the above integral may be easily evaluated
and the integrated y equation becomes simply:
H2 dydZ = - q V sin 0 (11-15)
A
which is the desired result.
-212-
------- |