WATER POLLUTION CONTROL RESEARCH SERIES •
16130 FSU 12/71
SURFACE DISCHARGE OF HEATED WATER
U.S. ENVIRONMENTAL PROTECTION AGENCY
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WATErt POLLUTION CONTROL RESEARCH SERIES
The Water Pollution Control Research Series describes the
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Protection Agency, Washington, D.C. 20460.
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SURFACE DISCHARGE OF HEATED WATER
by
H. Stefan,
N. Hayakawa,
and
F.R. Schiebe
University of Minnesota
St. Anthony Falls Hydraulic Laboratory
Mississippi River at 3r
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EPA-Review Notice
This report has been revieved by the Environmental Protection
Agency and approved for publication. Approval does not
signify that the contents necessarily reflect the vievs and
policies of the Environmental Protection Agency nor -does
mention of trade names or commercial products constitute
endorsement or recommendation for use.
ii
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PREFACE
The dissipation of heat from cooling water rejected by power generating
plants has become a problem of great concern because of its ecological
impact. Research efforts _in the area of thermal pollution, thermal ad-
dition, calefaction, or whatever name may be used have been tremendously
increased over the past decade. The St. Anthony Falls Hydraulic Labo-
ratory has participated in these efforts, and several reports on related
topics have preceded this one.
This volume is divided into three parts, each of which contains its own
conclusions and recommendations, table of contents, series of figures
and tables, and list of bibliographical references. The parts are as
follows:
Part I: "Three-Dimensional Jet-Type Surface Plumes in Theory
and in the Laboratory," by H. Stefan (Project Report
No. 126, St. Anthony Falls Hydraulic Laboratory,
University of Minnesota, December 1971). pages I-i
through 1-129 plus Appendices A and B.
Part II: "The Two-Dimensional Buoyant Surface Jet and the
Internal Hydraulic Jump," by H. Stefan and N. Haya-
kawa (Project Report No. 127, St. Anthony Falls Hy-
draulic Laboratory, University of Minnesota,
August 1971), pages Il-i through 11-55-
Part III: "Field Measurements in a Three-Dimensional Jet-Type
Surface Plume," by H. Stefan and F. R. Schiebe
(Project Report No. 128, St. Anthony Falls Hydraulic
Laboratory, University of Minnesota, December 1971)»
pages Ill-i through 111-35-
From these titles it is clear that the report deals with one particular
aspect of the very complex thermal pollution problem. Emphasis is on
flow, entrainment, and dilution of heated water effluents near surface
outlets. The results are of significance for open-cycle operating
systems and may also find application in the operation of cooling ponds
and mixing chambers.
iii
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Part I: Three-Dimensional Jet-Type Surface Plumes
in Theory and in the Laboratory
by
H. Stefan
December 1971
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ABSTRACT
The steady flow of heated water from a channel into a reservoir or lake
has been studied analytically and experimentally. A three-dimensional
buoyant-jet-type model has been developed to predict the main trajec-
tory, velocity, and temperature distributions in that portion of the
plume in which the flow is dominated by the momentum and the buoyancy
of the discharge and has free boundaries. The interaction between
turbulent mixing, buoyant spreading, and surface cooling, which are
crucial for the development of any thermal plume, can be illustrated
with the aid of the model. The effects of weak cross currents and weak
wind on the development of a surface plume are also incorporated into
the model. The model does not apply to heated water discharges which
cling to a shoreline due to a particular shoreline configuration or to
wind or current conditions. The effects of cold water wedge penetra-
tion into an outlet channel can be accommodated by the model.
Experimental results on temperature and velocity distributions in free-
jet-type three-dimensional thermal plumes have been used to verify some
of the assumptions made in the numerical model, particularly those re-
garding Gaussian velocity and temperature distributions and lateral
spread coefficients. In addition, the measurements have been used to
illustrate changes in total flow rate, total heat storage, and dimen-
sions of a surface plume. The distribution of temperatures, velocities,
and Richardson numbers in an experimental surface plume has been illus-
trated using different types of contour plots. There is reasonable agreement
between the results of the experiments and the proposed analytical
prediction method.
I-iii
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CONTENTS
Section page
List of Figures and Tables I-vi
List of Symbols and "Units I-x
I CONCLUSIONS 1-1
II RECOMMENDATIONS 1-3
III INTRODUCTION 1-5
IV PREVIOUS STUDIES ON THERMAL PLUMES 1-7
Theoretical Studies 1-7
Experimental Studies 1-7
Field Measurements 1-7
V THEORETICAL STUDY 1-9
Concept of Analytical Plume Model 1-9
Basic Equations 1-11
Solutions of the Basic Equations 1-27
Non-Dimensional Analysis and Results 1-31
VI EXPERIMENTAL STUDY 1-1$
Apparatus and Data Acquisition 1-1$
Data Preparation I-lj-9
Data Processing 1-53
Distribution Functions J 1-53
Spread 1-63
Spread Velocities 1-63
Spread of Isotherm Patterns 1-66
Spread of Iso-Velocity Concentrations 1-81
Visual Spread Angles 1-99
Entrainment and Mixing near the Outlet 1-105
Volumetric Fluxes, Heat Fluxes, and Heat
Contents 1-105
Entrainment Mechanisms 1-107
Flow Regime o'f Thermal Plume 1-119
VII ACmMLEDGMENTS 1-123
VIII REFERENCES 1-125
*"**"'*********"'*******•**"*******•*** —• ^*-^
Appendix A. Computer Program for Analytical Model (Fortran IV) ... I-A-1
Appendix B. Zone of Flow Establishment I-B-1
I-v
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LIST OF FIGURES AND TABLES
Page
1 DEFINITION SKETCH - PLAN VIEW OF BUOYANT SURFACE JET 1-12
2 DEFINITION SKETCH - CROSS SECTION OF BUOYANT SURFACE JET 1-13
3 DEFORMATION OF CROSS SECTION OF PLUME BY (a) CURRENT, (b)
BUOYANCY, (c) WIND - SCHEMATIC 1-15
k FIT OF EQUATION FOR VERTICAL ENTRAINMENT COEFFICIENT WITH
EXPERIMENTAL DATA FROM ELLISON AND TURNER [l|2] 1-19
5 PLUME TRAJECTORY IN CROSS-WIND PERPENDICULAR TO DISCHARGE.
ao = 90°, p = 0, FQ = 3-75, Ug = 0, Kg = 0 1-33
6 TOTAL VOLUMETRIC FLOW RATE VERSUS DISTANCE OF TRAVEL. PLUME
I-3U
IN CROSS-WIND PERPENDICULAR TO DISCHARGE, a =90°, |3 = 0,
7 EXCESS TEMPERATURE ABOVE AMBIENT ALONG MAIN TRAJECTORY. PLUME
IN CROSS-WIND PERPENDICULAR TO DISCHARGE, a =90°, p = 0,
F = 3-75, U = 0, K = 0 ° 1-35
O Bo
8 DEPTH OF THERMOCLINE BELOW WATER SURFACE ALONG MAIN TRAJEC-
TORY. PLUME IN CROSS-WIND PERPENDICULAR TO DISCHARGE.
ao = 90°, |3 = 0, FQ = 3.75, Us = 0, Kg = 0 1-36
9 PLUME TRAJECTORY IN CROSS-WIND PERPENDICULAR TO DISCHARGE.
a = 90°, p = 0, F = 15.0, U = 0, K = 0 1-37
\j \j o a
10 TOTAL VOLUMETRIC FLOW RATE VERSUS DISTANCE OF TRAVEL. PLUME
IN CROSS-WIND PERPENDICULAR TO DISCHARGE, a =90°, p = 0,
FQ = 15.0, Ug = 0, Kg = 0 ° I-58
11 EXCESS TEMPERATURE ABOVE AMBIENT ALONG MAIN TRAJECTORY. PLUME
IN CROSS-WIND PERPENDICULAR TO DISCHARGE, a =90°, p = 0,
FQ = 15.0, Us = 0, Ks = 0 ° 1-39
12 DEPTH OF THERMOCLINE BELOW WATER SURFACE ALONG M&.IN TRAJECTORY.
PLUME IN CROSS-WIND PERPENDICULAR TO DISCHARGE, a =90°,
p = 0, F = 15.0, U = 0, K = 0 ° 1-1+0
O S S
13 PLUME TRAJECTORY IN CROSS-CURRENT PERPENDICULAR TO DISCHARGE.
a = 90°, Fn = 3.75, W = 0, K = 0 1-^1
U U a
11+ TOTAL VOLUMETRIC FLOW RATE VERSUS DISTANCE OF TRAVEL. PLUME IN
CROSS-CURRENT PERPENDICULAR TO DISCHARGE, a =90°, F =3.75,
W = 0, K = 0 ° ° 1-^2
8
I-vi
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Page
15 EXCESS TEMPERATURE ABOVE AMBIENT ALONG MAIN TRAJECTORY.
PLTME IN CROSS-CURRENT PERPENDICULAR TO DISCHARGE.
ao = 9°°' Po = 5'75' W = °' Ks = ° I'1*3
16 DEPTH OP THERMOCLINE BELOW WATER SURFACE ALONG MAIN TRAJEC-
TORY. PLTME IN CROSS-CURRENT PERPENDICULAR TO DISCHARGE.
a = 90°, F = 3-75, W = 0, K = 0 1-144.
U Q D
17 PLUME TRAJECTORY IN CROSS-CURRENT PERPENDICULAR TO DISCHARGE.
an = 90°, F^ = 15-0, W = 0, K = 0 1-1+5
U O D
18 TOTAL VOLUMETRIC FLOW RATE VERSUS DISTANCE OF TRAVEL. PLUME
IN CROSS-CURRENT PERPENDICULAR TO DISCHARGE, a = 90 ,
F^ = 15.0, W = 0, K = 0 ° 1-1+6
o s
19 EXCESS TEMPERATURE ABOVE AMBIENT ALONG MAIN TRAJECTORY.
PLTME IN CROSS-CURRENT PERPENDICULAR TO DISCHARGE.
a = 90°, F^ = 15.0, W = 0, K = 0 I-ltf
LJ O D
20 DEPTH OF THERMOCLINE BELOW WATER SURFACE ALONG MAIN TRAJEC-
TORY. PLTME IN CROSS-WIND PERPENDICULAR TO DISCHARGE.
a = 90°, F = 15.0, W = 0, K =0 1-1+8
o o s
21 DEFINITION SKETCH - SURFACE DISCHARGE INTO EXPERIMENTAL
TANK 1-52
22 VARIATION OF VELOCITY COMPONENT u WITH DEPTH AT VARIOUS
DISTANCES FROM THE OUTLET ALONG CENTERLINE OF TANK. 1-51+
EXP. 215
23 STANDARD DEVIATIONS a and a- OF VELOCITY COMPONENT u
AND TEMPERATURE T RESPECTIVELY AS DERIVED FROM VELOCITY
AND TEMPERATURE PROFILES MEASURED AT SELECTED LOCATIONS
ALONG MAIN TRAJECTORY OF THE HEATED WATER SURFACE JET.
COMPARISON WITH PREDICTIONS OF ANALYTICAL MODEL (SOLID
LINES) 1-56
2ij. SIMILARITY PARAMETER X VERSUS DISTANCE ALONG MAIN TRAJEC-
TORY FROM DISCHARGE POINT 1-57
25 SIMILARITY PARAMETER A. VERSUS DISTANCE FROM MAIN TRAJEC-
TORY v 1-59
26 STANDARD ERROR BETWEEN NORMAL DISTRIBUTION FUNCTIONS AND
EXPERIMENTAL VELOCITY PROFILES MEASURED ALONG MAIN TRAJEC-
TORY 1-60
27 STANDARD ERROR BETWEEN NORMAL DISTRIBUTION FUNCTIONS AND
EXPERIMENTAL TEMPERATURE PROFILE MEASURED ALONG MAIN
TRAJECTORY 1-62
I-vii
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Page
28 LATERAL SPREAD VELOCITIES v AT TWO DEPTHS (z = 0 and
z = 0.1 ft) FOR EXP. 207 1-61*
29 VARIATION OF VELOCITY COMPONENT v WITH DEPTH AT VARIOUS
DISTANCES FROM THE OUTLET AND THE MAIN TRAJECTORY.
EXP. 20? 1-65
30 MAXIMUM SPREAD VELOCITIES VERSUS CENTERLINE DENSIMETRIC
FROUDE NUMBERS FOR INDIVIDUAL CROSS SECTIONS PERPENDICU-
LAR TO THE MAIN TRAJECTORY I- 67
31a - 31,g ISOTHERMS FOR SEVERAL EXPERIMENTS. HORIZONTAL
SECTIONS AT VARIOUS DEPTHS BELOW THE WATER SURFACE 1-68 - 1-79
32 DEPTH OF PENETRATION (L OF HEATED WATER SURFACE JET AS
DERIVED FROM ISOTHERM PATTERNS I_80
33a - 33k ISO-VELOCITY CONCENTRATIONS OF VELOCITY COMPONENT
u FOR SEVERAL EXPERIMENTS. HORIZONTAL SECTIONS AT
VARIOUS DEPTHS BELOW THE WATER SURFACE 1-82 - 1-92
LENGTH x™
VELOCITY-C
LENGTH xu, OF IDENTIFIABLE JET TYPE FLOW AS DERIVED FROM
CONCENTRATION PATTERNS 1-80
35a - 35d ISO-VELOCITY CONCENTRATIONS OF VELOCITY COMPONENT
u FOR TWO EXPERIMENTS. VERTICAL SECTIONS AT VARIOUS
DISTANCES FROM THE POINT OF DISCHARGE PERPENDICULAR TO
MAIN TRAJECTORY 1-94 - 1-97
36 ISO-VELOCITY CONCENTRATIONS OF VELOCITY COMPONENT u FOR
EXP. 207. VERTICAL SECTION THROUGH MAIN TRAJECTORY
(y = o) i-98
37 LATERAL SPREAD OF HEATED WATER JET ON WATER SURFACE AS
ILLUSTRATED BY VALUES OF STANDARD DEVIATIONS b(x)
PLOTTED AGAINST DISTANCE x FROM POINT OF DISCHARGE.
ANALYTICAL MODEL PREDICTIONS (SOLID LINES) AND EXPERI-
MENTAL DATA (DOTTED LINES) 1-100
38 VISUAL SURFACE SPREADING PATTERN (TIME EXPOSURE OF TRACER
PARTICLES) I-101
39 VISUAL SURFACE SPREADING ANGLE 0 OF HEATED WATER SURFACE
JET. EXPERIMENTAL DATA ° 1-102
1|0 SURFACE SPREADING OF VISUAL CONTOUR OF HEATED WATER SURFACE
JET. EXPERIMENTAL DATA I
I-viii
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Page
Ul TOTAL VOLUMETRIC PLOW VERSUS DISTANCE FROM SOURCE OP DISCHARGE 1-106
U2 SURFACE WATER TEMPERATURE DECLINE ALONG MAIN TRAJECTORY OP
HEATED SURFACE JET 1-108
143 HEAT CONTENT VERSUS DISTANCE PROM SOURCE OP DISCHARGE 1-109
1M - e ISO-LOGARITHMS OF RICHARDSON NUMBERS (INCREMENTED BY 1*.0)
FOR TWO EXPERIMENTS. HORIZONTAL SECTION AT VARIOUS DEPTHS 1-110 - 1-115
l*5a - c ISO-LOGARITHMS OF RICHARDSON NUMBERS (INCREMENTED BY i+.O).
VERTICAL SECTION THROUGH CENTERLINE OF TANK 1-116 - 1-118
1*6 DENSIMETRIC FROUDE NUMBERS VERSUS DISTANCE ON MAIN TRAJECTORY.
ANALYTICAL MODEL PREDICTIONS (SOLID LINES) AND EXPERIMENTAL
DATA 1-121
U7 ENTRAINMENT COEFFICIENTS VERSUS DISTANCE ALONG MAIN TRAJEC-
TORY FOR SEVERAL EXPERIMENTAL CONDITIONS AS FOUND FROM THE
ANALYTICAL MODEL 1-122
Table 1 - SUMMARY OP EXPERIMENTS 1-50 -1-51
I-ix
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LIST OF SYMBOLS AM) UNITS
a Standard deviation of velocity distribution in vertical plane
through main trajectory [ft]
b Standard deviation of velocity distribution at water surface
[ft]
C Drag coefficient [-]
C Heat content [BTU]
Coefficients related to various functions for entrainment
°2 coefficient [-]
°3
Cj Coefficient designating turbulent spread of non-buoyant,
^ circular, submerged jet [-]
c- Coefficient designating buoyancy induced lateral spread of
^ plume [-].
Cg = exp (-0.5)
c_ Coefficient related to initial spread of contour of plume
c Specific heat of water [BTU slugs' °F~ ]
D Drag force [ibs]
D Hydraulic diameter of discharge channel [ft]
d Depth [ft]
d Depth at discharge channel [ft]
F densimetric Froude number [-]
P Force [ibs]
fl
f2
f, Values of specified integrals [-]
I-x
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g Acceleration of gravity (32.2 ft sec" )
H Heat flux [BTU sec"1]
K Entrainment coefficient [-]
KS Surface heat transfer coefficient [BTU ft"2 °F"1 hr"1]
k Constant concentration [-]
M Momentum flux [ibs]
m Number limiting depth of plume [-]
N Number of steps
N Number of positive values in a profile [-]
n Number limiting width of plume [-]
3 _]_
Q, Volumetric flow rate [ft sec ]
R Radius of curvature of plume in horizontal plane [ft]
Re Reynolds number
Ri Richardson number
r Distance from and perpendicular to main trajectory [ft]
s Distance from virtual origin along main trajectory [ft]
T Temperature
t Time [sec]
U Average discharge velocity [ft sec" ]
o
s
Constant stream or current velocity [ft sec ]
-1-
u Flow velocity tangent to main trajectory [ft sec~ ]
v Flow velocity perpendicular to main trajectory [ft sec" ]
W Wind velocity [ft sec"1]
WQ Width of discharge channel [ft]
x Coordinate in direction perpendicular to current [ft]
y Coordinate in direction of current [ft]
a Depth [ft]
I-xi
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a Angle "between plume trajectory and current [degrees]
P Angle between wind and current [degrees]
8 Halfwidth to depth ratio of plume [-]
€ Error
\ Similarity parameter [-]
r -2i
\i Dynamic viscosity [Ib sec ft J
Q Density of water [slugs ft~^]
n
T Wind shear stress [ib ft~ ]
v Spread angle of contour [degrees]
Subscripts
a air
c contour
D drag
E equilibrium
e experimental
f friction
h horizontal
i control volume
i element along main trajectory
j element along vertical profile
& lake or ambient
n net value
o initial value
o neutrally buoyant (for K only)
r in r-direction
s in s-direction
T temperature
th theoretical
u velocity
v vertical
w wind
Superscript * differential with reference to ambient water
I-xii
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SECTION I
CONCLUSIONS
The study reported in Peart I deals with the steady flow of heated water
from a channel into a reservoir or lake. A three-dimensional buoyant-
jet-type analytical model has "been developed and compared with labora-
tory data. The model is based on the observation that a heated water
surface discharge is initially dominated by the momentum and the buoy-
ancy of the discharge. The absence of solid boundaries and the un-
hindered spread of the heated water are essential. The model will
predict the main trajectory, the depth, and the width of the plume and
the centerline velocities and temperatures in the plume. The model
includes the effect of cold water wedge penetration into an outlet
channel.
The method of calculation is such that the geometrical extent of the
flow field which can be legitimately covered by a jet-type model (the
nearfield) becomes apparent from the results. The experimental data
which are reported and analyzed have been valuable as a guide to a
physically meaningful analytical description of the flow. A spread
parameter was used and its possible magnitude derived from experimental
data.
Results obtained with the analytical model and from experiments show
substantial initial dilution of effluents even when buoyancy and
stratification are strong.
1-1
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SECTION II
RECOMMENDATIONS
It is recommended that the effects of solid boundaries and strong winds
on thermal plumes "be further investigated both analytically and experi-
mentally. A substantial change in the flow and turbulent mixing proc-
esses is induced when heated water is injected into strong cross currents
or onshore winds. Further attention should be given to these circum-
stances.
1-5
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SECTION III
INTRODUCTION
The possibility of ecological stresses on the aquatic environment
produced by discharges of large quantities of heated water from indus-
trial plants, particularly power generating plants, has been discussed
by numerous authors. References [l,2,3,U»5]* are examples of such dis-
cussions. A large amount of information on the effects of heat on
organisms is becoming available, and it indicates that future site
selection and operating policy for power plants will be much more com-
plex in the light of their variable impact en water ecology.
Temperature standards have been set by individual states under the
guidance and supervision of the federal government. These standards
have an important bearing on power plant site selection and mode of
operation, especially when the standards are very restrictive. Mixing
zones are often allowed near the heated water outfalls. Temperature
standards are frequently defined in terms of a temperature increment
between an ambient water temperature and the outlet temperature and
in terms of a maximum discharge temperature, both of which must be met
at the end of the mixing zone. It is therefore necessary to be able
to predict in the planning stage the temperature decay in a mixing
zone. The present investigation was concerned with that problem.
Heated water can be discharged into a natural body of water in many
different ways. Frequently the discharge is from an open channel, and
attention will be focused on this case.
References for Part I are listed on pages 1-125 through 1-129.
1-5
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SECTION IV
PREVIOUS STUDIES ON THERMAL PLUMES
The problem of heat dispersion in a body of water has been attacked
theoretically, experimentally, and through site studies. A general
solution method according to which point temperatures and velocities
in all thermal plumes could be calculated has not been found. A few
theoretical approaches have been tried, with some success, and labora-
tory and field measurements have been accumulated and interpreted.
More such data will certainly be forthcoming.
Theoretical Studies
The surface plume problem has generally been treated theoretically as
either a passive dispersion problem, as in Refs. [6,7>8], or a two-
dimensional turbulent jet problem as in Refs. [9,10,11,12]. The plume
has also been treated theoretically as a purely buoyant surface jet
[13] and as a three-dimensional surface jet [ll|]. These theories apply
to the near field only, where turbulence is generated by shear between
discharge and ambient fluid. Both jet-type dispersion and passive dis-
persion in a free turbulent field are treated with various simplifying
assumptions in Ref. [15]. There have also been a number of studies
dealing with non-buoyant two- or three-dimensional jet discharges into
cross-currents or co-current streams; Refs. [l6] and [l?] are examples.
Experimental Studies
Experiments with three-dimensional surface discharges of heated water
into cold water tanks have produced temperature data in plumes for a
great variety of outlet conditions. Numerous case studies have been
made of specific power plants, as a recent compilation [18] shows, and
there have also been a few investigations dealing with idealized geo-
metrical boundary conditions and more fundamental aspects of plume
formation [lj,11^,16,17,19,20,21,22]. In this context it is also per-
tinent to cite studies concerned with fresh water flows into saline
bodies of water [23,214].
Model studies of specific power plant sites are, of course, useful in
dealing with complicated geometrical boundary conditions. Although
scale effects on turbulent mixing and entrainment are beyond control,
each experimental plume can be considered a plume in its own right.
Therefore, laboratory data are as useful as field measurements when
they are used to formulate and examine deterministic prediction methods*
Field Measurements
Field measurements usually consist of sets of temperature data [25,26,
27,28,29]. In a few instances, current measurements have been carried
out simultaneously [29]. Such measurements are, of course, very useful
in the development of prediction techniques, especially if the discharge,
ambient water, and meteorological conditions are measured simultaneously.
1-7
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SECTION V
•THEORETICAL STUDY
Concept of Analytical Plume Model
Power plants operating with open cooling water supply systems withdraw
the cooling water from natural sources such as rivers, lakes, and the
ocean. They use it once and discharge it at elevated temperatures back
into the environment. The present study was concerned with the flow and
dissipation of heat in the receiving body of water, with emphasis on
lakes. The effects of currents and wind, surface cooling, and mixing
are combined in a comprehensive numerical model.
Condenser cooling water temperatures generally range from 10° to 35 P
above intake temperatures. Flow rates are of the order of from several
hundred to several thousand cfs for the largest power plants presently
in operation or under construction. When such large quantities of heated
water are discharged into a lake, a so-called thermal plume is formed.
If the environmental impact of heated water discharges is to be assessed,
it is necessary to be able to predict the size of the plume, its tempera-
ture, and its flow characteristics.
At the point of discharge, usually at the end of an outlet channel, a
heated water plume is basically a turbulent, incompressible jet of very
large dimensions4 It is characterized by an initial mass flux, momentum
flux, buoyancy flux, and energy flux. As the water moves away from the
surface outlet into a larger and colder body of water it will display
some features which are familiar from high speed jet flows, such as
entrainment of surrounding fluid by turbulent mixing. In the absence of
both solid boundaries near the jet and external forces, the momentum flux
will be conserved. Sometimes the effects of solid boundaries on the jet
are felt rather strongly, however, and the conservation of momentum as-
sumption does not apply. As the flow volume increases and the velocities
in the jet decrease, the characteristics of the receiving body of water—
in particular its currents and its turbulence—will be imprinted more
and more on the jet. Eventually the jet flow will be altered to such an
extent that the initial characteristics will be unrecognizable.
Simultaneously with the jet's development, the atmosphere has its effect
on the thermal plume. There is an energy flux, reversible in direction,
across any air-water interface. Usually heat energy is transmitted by
evaporation, conduction, convection, and long-wave backradiation from
the plume to the atmosphere, with the result that the heat flux carried
by the plume decreases with distance from its point of discharge. Some-
times the direction of the heat flux is reversed. The water surface is
also where the transfer of mechanical energy provided by wind takes
place. Shear stresses at the water surface, wave generation, currents,
and turbulence are very closely associated with the action of wind. Prom
field studies it is well known that wind has an influence on the shape
and extent of thermal plumes.
1-9
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A third and very important aspect of thermal plumes is that of buoyancy
and stratification. Since the density of fresh water decreases when the
temperature is raised above 59-2°F, heated water will generally float
when surrounded by colder water. This is also the case when sea water
is used as a source of cooling water. Buoyancy will cause lateral spread
of a plume beyond that produced by turbulent mixing. It will also tend
to reduce or eliminate vertical mixing between a thermal plume and the
underlying water. The thermal plume will thus display certain features
which are typical for stratified flows.
Experimental studies and field measurements have revealed quite clearly
the dual character of thermal plumes as turbulent jets and stratified
flows. A simplified mathematical formulation will be given for the
heated water plume problem in terms of a buoyant jet flow into a slow
cross current under the effects of wind and surface cooling. The model
will predict the main trajectory, the depth, the width, and the tempera-
ture distribution of the plume as well as the rate of heat dispersion.
Since the geometry of heated water outlet channels and shoreline config-
urations varies from one discharge to another, it is often advantageous
to consider separately an outlet region, a near field, and a far field.
In the outlet region, channel and lake shore geometries have a major
effect on the flow and mixing in addition to the source characteristics
of momentum and buoyancy flux. If the outlet and shore geometries are
complicated, it may be necessary to study this zone separately using
analytical or numerical methods or even physical model studies. The
model which will now be discussed begins at the point from which solid
boundaries have a negligible effect on the plume and considers essen-
tially the flow in the near field.
The near field is that portion of the plume in which the dynamics are
controlled by the momentum of the discharge and by external forces such
:as wind, shear, and well defined currents. The far field is that por-
tion of the plume where the heated effluent is dispersed essentially
by turbulent mixing and heat transfer mechanisms which are controlled by
the overall hydrodynainic and thermodynainic characteristics of the re-
ceiving body of water, including specifically large-scale turbulence,
existing stratification, and overall velocity patterns.
The theory to be presented considers a three-dimensional half-jet and the
essential forces and processes contributing to its development. Since at
the present time turbulent flow and transport processes are not under-
stood completely enough to permit an exact solution for the three-dimen-
sional turbulent flow in a thermal plume, the problem is formulated in
terms of experimentally supported and mostly well documented semi-empirical
relationships. Several of these are drawn from previous studies, but
some new ones are also proposed. All relationships together constitute
a mathematical model of the thermal plume.
1-10
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Basic Equations
The theory approximates the real plume by a three-dimensional horizontal
half-jet discharging at some angle a into an existing current as
shown in Pig. 1. A finite shear stress produced by wind is assumed to
exist at the water surface. The behavior of the heated water jet in a
body of water of infinite extent will be examined. A real outlet is
usually located at the shore, and the last assumption will be satis-
fied only if the current and wind patterns allow the jet to move away
from the shoreline. A strong initial momentum and an offshore wind
will generally help to produce such a situation.
In the analysis the thermal plume is described in terms of its main
trajectory and its velocity and temperature distributions in cross sec-
tions perpendicular to the main trajectory. The main trajectory is the
streamline through all points with maximum velocity at the water surface.
The plume in this study is characterized by velocities and temperatures
in excess of those of the surrounding water. The situation is shown
schematically in Fig. 2.
The excess velocity and excess temperature distributions in a plane per-
pendicular to the main trajectory are assumed to be similar to each other
when normalized with respect to centerline values (for velocity and temper-
ature) and standard deviations (for distances from the main trajectory),
respectively. The practical evaluations will be carried through with Gaus-
sian distributions in both vertical and horizontal directions, although
other distributions can be used as well without changing the essence of
the procedure. Gaussian profiles approximate experimental data on sub-
merged jets quite well [JO,31,52] and have been used before in numerical
models [9,11,30], The velocity distribution is shown schematically in
Fig. 2. Temperature and velocity measurements in surface plumes [20,21,22]
support to some degree the use of Gaussian distributions in the description
of three-dimensional plumes. An analysis of experimental data to this ef-
fect will be presented in a later section. Because of the complicated
interaction between buoyancy and turbulence in a horizontally stratified
flow, it is thought that under close scrutiny the similarity principle
does not hold rigorously for the vertical distributions. This serious
theoretical defect is fully realized, but appears not to be of overwhelm-
ing significance for the overall picture of a plume.
The spread of these Gaussian distributions, as measured by standard
deviations, will be different in the horizontal and vertical directions
because of buoyancy as evidenced by experimental and field data. The
relative velocity and temperature distributions are therefore described
by the equations
u*
u*
T*
1-11
-------
no
WIND SHEAR
STRESS T
CURRENT
VELOCITY Us
FLOW
ORIGIN
OF PLUME y
u (s,o,o)
Fig, 1 - Definition Sketch - Plan View of Buoyant Surface Jet
-------
M
.1.
FLOW
MAIN TRAJECTORY
X
Fig. 2 - Definition Sketch - Cross Section of Buoyant Surface Jet
-------
.
where u (s,r,z) is the velocity at a point with the coordinates (s,r,z)
relative to the surrounding fluid; u* = u - U cos a, where u is the
absolute local velocity in the plume, U is fhe velocity of a uniform
current surrounding the jet, and a is Ihe jet angle relative to the
current; the thermal plume is assumed to move imbedded in the current;
s is the distance from the beginning of the buoyant jet field which is
usually located at or downstream from the actual outlet; r is the
normal distance from the main trajectory; z is the depth below the
water surface; and T*(s,r,z) is the water temperature at the same
point as u* relative to the surrounding water. Thus u*(s,0,0) and
T*(s,0,0) are the relative velocities and temperatures, respectively,
on the main trajectory and a(s) and b(s) are standard deviations
indicating the lateral or vertical spread of the distributions;
T* = T - T . where T. is the water temperature in the impoundment.
All velocities in a cross section perpendicular to the main trajectory
are assumed to be parallel to the tangent on the main trajectory, and
A, and A, are similarity parameters. In the absence of buoyancy and
for fully submerged axisymmetric jets A = A,= 1.16 has been used in
Ref. [jOJ. In buoyant surface jets the Horizontal spread of momentum
is produced in a somewhat different way than the vertical spread. The
vertical spread is essentially by turbulent mixing, including the ef-
fects of buoyancy; the horizontal spread is by turbulent mixing plus
buoyant spread. In a later section an analysis of some experimental
data will be given. As will be explained, there is not enough justifi-
cation at this time for choosing values of A, and A different
from each other or much different from unity. It is therefore tenta-
tively proposed to use A. = A,, = 1.05«
The local buoyancy force per unit weight will follow a distribution
quite similar to that of the temperature. If the temperature range
covered by a specific temperature profile is not too wide, the density
relationship Q = Q(T) can be approximated by a straight line and the
density deficit distribution is then
(3)
#
Here Q = Q» - Q where Q is the actual local density and Q . is
the density of the surrounding colder fluid (lake).
The density difference between the heated water in the jet and the
cold water in the lake produces a loss in axisymmetry in a plane per-
pendicular to the main trajectory. In addition, the curvature of the
main trajectory of a high-speed turbulent jet due to a cross current
should result in a loss of symmetry with respect to a vertical plane
(Fig. 3a) as shown in Refs. [3^] and [35]. In buoyant and slowly
moving surface jets (half- jets) with large radius of curvature of the
main trajectory, however, the latter effect should be small. The
-------
MAIN TRAJECTORY
CURRENT
WIND
H C
(c)
Initial Cross Section
Resultant » it
Fig. 3 - Deformation of Cross Section of Plume by (a) current,
(b) buoyancy, (c) wind - Schematic
1-15
-------
following analysis will therefore "be concerned mainly with situations
without vortex formation in the wake of the surface jet. This means
in part that there is enough turbulence at a small enough scale remain-
ing from the initial momentum of the jet or produced by external cur-
rents, as well as enough buoyancy at any cross section, to compensate
for the current- or wind-produced destruction of the symmetry of a
plume.
Because a thermal plume is also a density stratified flow, it may have
features which are well known from open channel flow. In particular it
is possible that the thickness of the plume is controlled from down-
stream rather than upstream. The conditions under which such a control
exists in a two-dimensional plume have been discussed in Refs. [15] and
[56]. It has been shown [36] that the value of the densimetric Proude
number in a given cross section is the controlling dimensionless
parameter. If the value of this number is larger than the critical
value, a downstream control does not exist. The present model applies
to flows for which the densimetric Froude number, as defined later in
Eq. (ij.), remains larger than critical. This is likely to be the case
under various natural conditions if there is a current, substantial
wind drift, or simply sufficient heat transfer through the water sur-
face. It may not hold true in a lake in which the surface cooling is
temporarily very slow or if the wind drift is in direct opposition to
the direction of discharge. An internal backwater effect will then
become apparent as shown by Pig. 9c of Ref. [36] and by Pig. 12 of
Ref. [20J. The internal Proude number can be defined as
P = P + PS
where
*. *t n n\ V cos a
* u (s.0,0) , _ 8 f]\
P = —5 \~i~i i -- and P = — - =-5- (14)
To arrive at the basic equations it is necessary to examine the volu-
metric flux Q, the heat flux H, and the momentum flux II in cross
sections perpendicular to the main trajectory of the plume. Since the
fluid is incompressible and since the Boussinesq assumption can also be
made because temperature-induced density differences are very small,
the volumetric flux Q is used instead of the mass flux.
oo CD
Q(s) = I i u(s,r,z) dzdr (5)
1-16
-------
OD OD
H(s) = / /u(s,r,z) cpQT*(s,r,z) dzdr (6)
-ot)0
00 00
M(s) = f /*Qu2(s,r,z) dzdr (7)
•SOD-T)
These fluxes are of course absolute values. The limits of integration
of Eqs. (5), (6), and (7) must match the velocity and temperature dis-
tributions of Eqs. (l) and (2), which provide for infinite width and
depth of the jet. The real jet is limited in width, and therefore the
limits of integration should be replaced by -nb and +nb for the
r-direction and by zero and ma for the z-direction where n and m
are finite numbers.. The visual spreading angle of non-buoyant axi-
symmetric jets is of the order of li| to 19° according to Refs. [37]
and [38] . It is therefore physically meaningful to limit n and m
values to n = m = 3.
The volumetric flux increases in the flow direction as a consequence
of entrainment. The process is very complex. In the initial stage of
the plume's development, turbulence is produced mostly by the shearing
motion between the heated water jet and the surrounding cold water. In
such a flow it has been shown [32] that the rate of entrainment is pro-
portional to the velocity of the main jet relative to its surroundings.
Application of the entrainment principle to the buoyant half- jet yields
. K u*(s,0,0) <• + (8)
The eddies which produce this kind of mixing are on the same geometrical
scale as the jet itself. For a neutrally buoyant, fully submerged,
axisymmetric turbulent air jet K = 0.057 was found to match experi-
mental data [33] best, while a study with liquid jets [38] resulted in
K = 0.059. K = 0.082 is adequate for axisymmetric buoyant plumes
with vertical axes [31]. The same value was used in Ref. [33] to match
experiments on buoyant plumes with curvilinear main trajectories.
K = 0.089 has been used for buoyant slot jets [30]. In both two- and
three-dimensional situations K was independent of distance from the
source, and a comparison of experimentally measured flow rates with the
predicted ones was used in support of this hypothesis. In addition, the
same amount of entrainment occurred per unit length of jet contour in a
given cross section because of the axisymmetry of the flow. Neither of
these assumptions is applicable to the horizontal buoyant surface jet,
because its entrainment is a function not only of the centerline veloc-
ity, but also of the degree of stratification. It is believed that the
entrainment principle is still useful, but it must be applied in a
1-17
-------
modified form. A variable entrainment coefficient along the contour of
the jet must be used. Equation (8) specifies a semi-elliptical contour
of the half-jet. Vertical entrainment through the bottom part of the
surface jet is inhibited by buoyancy, while horizontal entrainment
through the sides of the half-jet is not. Since the real distribution
of K values along the contour is unknown, an approximation is used.
It is proposed to use a constant horizontal entrainment parameter
K, = 0.059. Because density stratification inhibits turbulence, ver-
tical entrainment must be equal to or smaller than the horizontal entrain-
ment, dependent on some stratification parameter. It has been shown
[39»^0»1A] that an overall Richardson number in terms of an overall depth,
average velocity, and average density differential is an adequate strat-
ification parameter. Since this number, raised to the power (-2), is
equivalent to a densimetric Froude number, the dimensionless parameter
F* as defined in Eq. (4) has been retained as a stratification param-
eter in this model. Experimental data on the effects of stratification
on vertical entrainment are relatively scarce. The field data reported
in [i|l] and the laboratory data reported in [ij.2] and [143] are probably
the best known, and various ways of fitting these have been proposed.
A short review made in 1969 [UU] shows that relationships of the form
K /K =
v7 o
(9)
-c3/(F*)
^
have been used as best fits with available measurements. Here a^,
a , c,, c^, and c, are positive real numbers and KQ is the
eady exchange coefficient in neutrally buoyant flow. A restriction
on Eq. (9) is, of course, that 0 < K/K < 1.0. The relationship
retained in this study, which fits experimental data from Ref. [U2],
is, as shown in Fig. 1±,
=£= 1.0 - 1.33 log (
(10)
where K. is the horizontal entrainment parameter. It is applied in the
range 1Y12 < F* < 6.32 and covers the transition of the entrainment co-
efficient in the range 0 < K/K < 1.0. In Fig. k ^^Ah was Pl°tted
on a linear scale because it is Believed that the customary logarithmic
1-18
-------
1.0
.8
.6
Kv/Kh
vo
.4
.OI
.03 .05 .1
.3 .5
!/(F*)2
10
LO
3.0
Fig. 4 - Fit of Equation for Vertical Entrapment Coefficient with Experimental Data from Ellison and Turner [42]
-------
distortion of that scale tends to result in too much emphasis on the low
values in the fitting process. It is believed that the horizontal
entrainment also depends to some extent on the degree of stratification.
For lack of experimental data it was not found possible to express this
relationship in quantitative form.
If the driving force producing vertical entrainment is not the momentum
of the jet, but the shear stress produced by wind blowing over the
water surface, Eq.. (10) can no longer be applied. It has been shown in
Refs. [14.3] and [^5] how an entrainment coefficient can be obtained in
that particular situation.
Another change must be made in the horizontal entrainment coefficient
to accommodate the turbulence inherent in the ambient current in which
the thermal plume is embedded. An entrainment parameter related to the
characteristics of the ambient current must then be used. Experiments
on jet discharges into co-current streams indicate that entrainment is
larger than for jets discharged into stagnant environments, but it ap-
pears very difficult at this time to express observations in quantita-
tive form.
Circular jets discharged into a cross current have been found to entrain
much more ambient fluid than jets discharged into stagnant fluid.
Entrainment coefficients as high as 0.1| and 0.5 have been reported [1+6].
The increase is caused by a pair of large vortices which form in the
cross section of the plume perpendicular to the main trajectory as
shown, for example, in Ref. [3U]« In a surface jet only one vortex
exists, as shown in Fig. 3a. The cross current, which causes the
deflection of the main trajectory of the plume, is responsible for the
internal vortex- type circulation in the plume. A horizontal entrainment
coefficient variable with the radius of curvature of the main trajectory
may therefore be used. Based on available experimental and field data,
an approximate relationship K, = K, (R) may be formulated. When such
a relationship is applied to a surface plume, however, there is an ad-
ditional complication, because buoyancy will oppose the onset of an
internal vortex-type circulation pattern. After the internal circula-
tion occurs, the assumed Gaussian distributions of temperature and
velocity will be destroyed. This analysis is therefore restricted to
situations with sufficient buoyancy or large enough radii of curvature
to prevent internal circulation. Tentative bounds are F* < 6. 3 and
R/b > 100, with smaller values of R/b acceptable as F* decreases.
It is proposed to calculate an average entrainment coefficient K for
each cross section from the relationship
a + K b
where a(s) and b(s) are used as measures of the relative depth and
width of the plume, respectively.
1-20
-------
As the plume moves away from the outlet, the total flow rate increases
by turbulent entrainment, and the plume dimensions also increase. It
is proposed to use the relationships
*/
fdb\ u (8,0,0)
Sis' = ck ~, \
T H u (s,0,0) + U cos a
3
K '
T n S T
to describe the increase in geometrical size of the plume cross section.
The first equation implies that the lateral, turbulence-induced spread
of the plume is the same as that for non-buoyant jets except that the
absolute spreading angle is distorted by the velocity of the surrounding
current into which the plume is being injected. The second equation
implies that the vertical and horizontal spreading ratio is closely
related to the vertical and horizontal mixing ratio. This hypothesis
is an extension of non-buoyant jet results to buoyant surface jets. The
coefficient c, is the spreading angle of a jet in a stagnant reservoir
without buoyancy effects. In a three-dimensional jet c, = 0.081 [37]-
The thermal plume will spread laterally not only by turbulent mixing,
but also under the effect of buoyancy, as shown in Fig. Jb. In this
process some fluid is removed from the center of the plume and moved
laterally toward the edges of the plume. An internal circulation pat-
tern with a. horizontal peak velocity at some distance from the main
trajectory will result (Fig. 3b)» Buoyancy may tend to restore the
symmetry of the flow lost due to current or wind action. The rate of
lateral spread due to buoyancy can be approximated using an equation
of the form
which is analogous to that used in the analysis of unsteady density cur-
rents [L7],
spread [12].
~" -"•"-" »«***.c*-i-wg>WW.Q UW UllGli U L4OCU. J-X1 UllC CULICUL.^ OJ-O t-/J- LUXCt UOCLU^y U.GJ,iOJ- UJ OU.J--
rents [1^7]. This equation has been used before to describe the lateral
At a distance b from the main trajectory the forward velocity is
u(s,b,0). The horizontal buoyancy- induced angle of spread at this point
should be
u(s,b,0)
1-21
-------
jt
Since u(s,b,0) = cg u(s,0,0) = Cg(u (s,0,0) + FS cos a) (17)
where Cg = exp (-0.5) is a coefficient related to the Gaussian
velocity distribution, the above equation reduces, using Eq. (i|), to
±i
Conservation of mass in a cross section requires that as the width of
the plume increases at the rate given "by Eq. (l3)» the depth decreases;
this requirement is satisfied by the relationship
^ T,
B B
Equations (18) and (19) represent, respectively, the lateral and the
vertical spread due to buoyancy.
The interaction between spread by buoyancy and spread by mixing is com-
plicated. It is proposed to consider the possibility that the effects
of turbulence and buoyancy on spreading are additive. It is conceivable
that the lateral spreading process also influences the formation of tur-
bulence in the shear zone between the jet fluid and the surrounding
fluid, but if this is the case it is impossible to take it into consid-
eration at the present time. The total spread of the jet in the sur-
rounding fluid must be of the order of
ft • «ft>B + T
ft "
-------
rate of change of the heat flux is equal to the rate of heat loss
to the atmosphere. The unit rate of surface heat transfer can be
found [14.8] from a relationship of the form
H = ZQ[T(s,r,0) - Tj (23)
where the coefficient of surface heat transfer K and the equilibrium
temperature T,., are essentially functions of the prevailing meteor-
ological conditions: solar radiation, wind velocity, air temperature,
and relative humidity. The gradient of the total heat flux in the main
direction of the trajectory must be balanced by the heat flux through
the water surface:
nb nb
= / Hn dr = Kg / [T(s,r,0) - TE]dr (2iO
-nb -nb
In addition to mixing, spreading, and cooling, the momentum flux must
be considered separately with regard to its components in the x- and
the y-direction or in the s- and r-direotions, all of which are shown
in Fig. 1. The rate of change of momentum flux in the s- and r-direc-
tions is equal to the force components (forces per unit length) acting
on the control volume.
dM
(25)
dM
-r^- = 0 = ^
dr r'
External forces which cause changes in momentum flux are produced by
wind shear stress at the water surface, dynamic pressures due to the
ambient current, and frictional forces which do not result in entrain-
ment. The latter restriction is necessary, as.illustrated by the dif-
ferent effects of free and solid boundaries on submerged jets. On a
free boundary, frictional forces are translated into entrainment, and
no loss in total momentum results from frictional action. A solid
boundary does not allow entrainment and will cause a loss in total
momentum flux.
A plume without any turbulent entrainment must be laminar in character
on all its boundaries. If the resultant shear force in this case is
Df 1 , the shear force for the partially mixing plume is approximately
1-23
-------
Df = Df lam -V- <26a>
The lower and upper bounds for D~ are thus zero and D,, n . The
i l lam
value of D.p ., can be approximated using Newton's shear law in con-
junction with the maximum velocity gradients perpendicular to the con-
tour of the plume to find local shear stresses. Integration of the
shear stresses along the contour will result in a force per unit length
Df lam = -
Most often this force will be small compared to other external forces.
The external force resulting from the wind shear stress T on a sur-
face strip of unit length is found from the ' relationship w
F = 2nb T (27)
Numerous empirical relationships have been reviewed for the calculation
of wind shear stress from wind velocity data [1*9, 50], and these can be
selectively incorporated into the model. A surface stress due to wind
transverse to the main trajectory will also tend to stretch the width
of the plume. A large-scale eddy with water velocities at the surface
in the direction of the wind shear stress will be produced as schemat-
ically shown in Fig. 3c. If such a circulation appears, it will tend
to destroy the assumed temperature distribution in the model. It is
therefore not included in this analysis. It should be mentioned that
the wind may blow at any angle |3 with reference to the x-axis, as is
shown in Pig. 1.
Equation (27) does not include wind effects on the cold water, because
these are contained in the specifications of the external current into
which the heated water is discharged. This is meaningful for two
reasons. First, the cold water currents in a body of water receiving
thermal effluents frequently depend not only on wind, but elso on the
shore and bottom topography. In fact, the relationship between currents
and wind is quite complex and must be investigated apart from the ther-
mal plume problem. The plume itself is considered to be embedded in
the cold water and subject to separate wind action. The second reason
is the observation that the cold water current sometimes represents a
much larger mass flux than the heated water discharge. The longshore
currents in the Great Lakes and the flow in impounded rivers can be
cited as examples. These masses of water are less susceptible to
motion by wind than the buoyant surface jet which frequently spreads
out as a thin layer on top of such a current. A substantial amount of
-------
time is required to reach a steady wind generated current in a large
lake.
The current into which a plume discharges produces a dynamic pressure
normal to the direction of the main trajectory and also a shear stress
parallel to it. If the thermal plume is approximated by a solid body
somewhat similar to a bent half-cylinder, the dynamic pressure forces
will produce a form drag
2
p(U sin oc)
' °D
perpendicular to the main trajectory and a friction drag
o
(b + a)
parallel to the main trajectory. The friction forces given in Eqs.
(26a) and (29) are of course the same. Equation (26a) may be easier
to use than Eq. (29) because of the friction coefficient C,,. The
form drag coefficient C in Eq. (28) was taken as equal to unity
because of the semi-elliptical shape of the plume.
The change in momentum flux must also account for the original momen-
tum of the entrained fluid, meaning -r^ QU cos a in the s-direction
dft as s
and -T* QU sin a in the r-direction.
Pressure (buoyancy) forces in the direction of the main trajectory are
not included in the analysis. The buoyancy force in any cross section
ma nb
ls Si I Q*(s,r,z) zdz and can be calculated from the density
0 -nb
deficit distribution given in Eq. (3). It can be shown that this value
is also equal to 0.33 g^ fc Q*(s,0,0) ab. The value of this quantity
is usually less than one per cent of the momentum flux of the jet. The
net longitudinal buoyancy force acting on a control volume in the plume
is 0.33 gA fc dr9*(sA°)ab1 which is usuaiiy negligible in relation
v 5 ds
to the momentum flux.
1-25
-------
Summarizing, the momentum equations in terms of all external forces
are
= D + F cos (p - a) + Q U cos a (JO)
_
dr ds * s R D w
Us sin a = - •£ + Dp - FW sin (|3 - a)
The radius of curvature of the main trajectory is
3/2
where y(x) is the equation of the main trajectory in the x-y-plane.
The curvature of the main trajectory is also associated with a change
of the flow direction a. Considering a length element ds,
da = f (55)
will be a good approximation for the change in angle.
All secondary circulation patterns shown in Fig. 5 tend to destroy the
Gaussian velocity and temperature distributions. It can be speculated
that the result will often be a more uniform temperature distribution
within the plume, but there is no proof of this. Certainly, all the
above circulation patterns are superimposed on the main motion of the
plume along the main trajectory, and frequently on each other as well.
Thus a theoretical argument for the use of symmetrical temperature and
velocity profiles such as those used in the preceding section cannot be
given. However, experiments in the laboratory and in the- field support
the assumption.
1-26
-------
Solutions of the Basic Equations
The basic equations of the previous sections are solved by an explicit,
finite-difference, forward-stepping method. Calculations of the
volumetric, heat, and momentum fluxes as defined previously were made
using the following formulations:
TJg cos o^
Hi = Tiui8ibif2 + Ti Us COS ai
2 TT 2 2
i a-t)> ^2 + 2Q u cos ai ui aib- f i + Q "2 ^s cos
where the subscript i refers to a particular cross section at dis-
tance s from the origin of the plume, u. is equal to the excess-
centerline velocity u*(s,0,0), T. is equal to the excess-center-
line temperature T*(s,0,0), and feie coefficients f., through
are consi;an"t;s equivalent to the following definite integral
ues :
nb. ma.
H- 1 (t)2] exp[" * (t)
drdz
= TT erfn (-) erfn ( (37)
1-2?
-------
rib. ma.
i
- ()] exp[-
v i ' v h
-^ erfn(n) erfn(m) (39)
2
nb. ma.
j. j* _L
Values of f, through f, can also be calculated if other than
Gaussian velocity and temperature distributions are used.
Basic input parameters include the initial flow rate Q , the initial
depth a , the initial momentum flux M , the outlet temperature T ,
the lake0temperature T», the current velocity U , and the outlet °
angle a with respect to the current direction. If there is wind,
its direction [3 and the shear stress on the water surface T which
it produces are specified. Since T must be calculated fromwwind
speed, it is possible to include sucn calculations in the program.
The wind speed W at a specified altitude (e.g., 30 ft) is then an
input variable.
Prediction of surface heat loss requires specification of the equi-
librium temperature T_ and the surface heat transfer coefficient
K . These can be calcinated from semi-empirical relationships [5l]«
If the actual calculations are again transferred into the program,
the necessary input values must include solar radiation H , dew
point temperature T-., and wind speed W.
1-28
-------
The step length d and the number of steps must be chosen. Values
approximately equivalent to the outlet depths were chosen for the step
length. The number of steps depends, of course, on the size of the flow
field to be investigated.
The set of equations can be solved numerically in a meaningful manner
and at little cost in computer time by extrapolating forward from one
cross section to another starting at the outlet. The process is the same
as that used in backwater profile computations in supercritical channel
flows. The procedure is justified by the fact that a heated water surface
jet as described herein is an internally supercritical (or at least crit-
ical) three-dimensional flow. The heated water surface jet may lose this
property at some distance downstream from the outlet. The centerline
densimetric Froude number defined in Eq. (i^) can be used as an indicator.
Anticipating results to be shown later, it appears that the critical densi-
metric Froude number value is in the vicinity of J>.Q. The analysis of the
thermal plume using a heated jet type model and the procedure described
herein should not be carried on if the densimetric Froude number values
are less than critical. The physical significance of this statement is
relatively simple: The flow should not be considered a jet flow if its
momentum has become small compared to the buoyant forces. The magnitude
and direction of external forces thus have great influence on the size of
the area in which the analytical jet-type flow model can be applied. By
definition, the area is considered equivalent to the nearfield. Onshore
wind or currents may reduce the nearfield considerably, while longshore
currents and offshore winds may stretch it.
Effects of the farfield or downstream controls on heated water surface
jets have been reported for two-dimensional flows [15,36,52] and will be
further investigated in Part II of this report.
Details of the numerical computations can be gathered from the computer
program given in Appendix A. The following explanations will be helpful:
The initial conditions as described above are used to compute an initial
plume width b and centerline velocity u such that the initial flow
rate and the initial momentum flux are as specified. Equations (ji|.) and
(36) are used for that purpose. Then the equations given earlier are
solved in a specific order. First the entrainment is found from Eq. (8)
with the aid of Eqs. (10) and (ll). Then ft. _ = ft. + aft.. The heat
loss dlL is found from Eq. (2iO in the form"1"1 1 1
K ds
-dH. = [T.b.f,. + 2(T^ - TE)nb.] __
where f = -±- / exp[- %(T^) ] dr . ^ Vgff erfn (^) (U2)
1-29
-------
Then E. _ = H. + dH.. Equation (50) is then used to evaluate the change
in momentum flux in the direction of the main trajectory and Eq. (31) to
find the radius of curvature of the main trajectory R.. Then
M. . = M. + dM. and with the radius of curvature from1Eq. (51). the
T i-i ~\ ^
change in the trajectory angle da. can be calculated using Eq.. (53)•
dc^ = R± ds (1+3)
Then a, , = a. + da..
i+l i i
The plume characteristics in the (i+l)-th cross section—in particular
the standard deviations a. , and b. .., the centerline velocity
u. _ , and. the centerline temperature excess T. 1—can be found from
tfee" previously calculated fluxes Q. ,, M. ., """and H. , in conjunc-
tion with the spread characteristics of Eqs. (20), (2l)t and (22).
Simultaneous solution of Eqs. (3U) and. (36) for the (i+l)-th values
instead of the i-th gives u. , and the area (a. , b. .,), and
the individual values of the (i+l)-th standard deviations are then
found using Eq. (22). Prior to this, Eqs. (12) through (21) must be
solved for each individual step. The centerline temperature T.
can then be found directly from Eq. (35)* After completion of
all these calculations, the orientation of the main trajectory and
the velocity and temperature distributions in the (i+l)-th cross sec-
tion are known. If the calculations are repeated for N steps, the
characteristics of a whole plume can be found. The described process
is simple and explicit, and changes in step lengths have little effect
on the results if the step length remains of the order of the initial
depth.
The limitations of the model described lie, of course, in the assump-
tions made initially: similarity of all profiles, specific lateral
spreading patterns regardless of wind and current conditions, and
absence of solid boundaries. The coefficients used are derived from a
rather small number of experiments.
To facilitate interpretation of the output, the numerical program was
extended by a contour plotting program which is a modified version of
program CONTOUR from the University of Michigan. As a result, an
output consisting of isotherm patterns can be obtained; Pig. 31 is an
example. The point of discharge is at the center bottom of the picture,
and there is no wind or current. The plot shows surface temperatures.
Near the outlet the contour interpolation is slightly erroneous because
of the large grid sizes and the interpolation procedure in the contour
plotting package.
To determine the accuracy of the method, predicted temperatures were
compared with measurements in the field and under laboratory conditions.
Results and comparisons with laboratory data will be given in the
following section.
1-30
-------
Non-Dimensional Analysis and Results
The procedure described in the previous section will produce results
for an individual plume under individual ambient conditions. However,
dimensionless quantitative information on, for example, temperature
decay, spread, and flow velocities in a plume in terms of all the
relevant parameters influencing the behavior of the thermal plume is
also desirable. Such information would greatly aid in making general-
ized predictions of the character of the plume for different field
conditions. Attempts to provide such general information have failed
owing to the large number of factors affecting the plume characteris-
tics as well as the interaction and mutual interference among these
parameters. If the initial plume characteristics u = u(0,0,0) and
a = a(0) and the initial water temperature excess T - T. =
T°0,0,0) - T. are chosen as reference values, the indipendent vari-
ables can be written as
-' K(T -
V °' V V To-V uo«°P '
in which T is the initial densimetric Froude number. The dependent
variables are
rji _ m
JL * _£_ A JL JL -2L JL JL
u ' T - T/ a ' b ' a ' H ' a ' a ' a
00,60000000
where the subscript zero again refers to the initial cross section at
(s = 0). It has not been found possible to produce dimensionless
results for the dependent variables in terms of all the independent param-
eters. However, it was found that the initial behavior of most of the
heated water surface jets investigated is controlled mainly by the hydro-
dynamic and turbulent convective processes. For purposes of illustra-
tion it is therefore justifiable and useful to eliminate those independ-
ent variables which refer to heat transfer through the water surface by
setting T = TE and K =0. The results will then refer to the mixing
zone near the point of discharge only. The remaining independent vari-
ables then are
T-* V T- «* £-• V P
o o o
The effects of several combinations of these dimensionless parameters
on dependent variables were investigated in two groups: plumes in
wind without cross currents (U = 0) and plumes in cross currents
without wind (W = 0). The win! or current was assumed to be at right
1-31
-------
angles to the discharge (a = 7J/2 or p = 0). The results of those
investigations are shown in Pigs. 5 through 20.
Figures 5 through 20 are largely self-explanatory. The first four
figures refer to a low Froude number discharge (P = 3.75) into wind.
The deflection of the main trajectory as shown in Pig. 5 is more sub-
stantial for the semicircular jet (la /a = l) than for a wide semi-
elliptical one (b /a = U) • The reason is that the circular discharge
has a smaller initial flow rate and initial momentum than the wider one
There is substantial dilution of the discharge (Pig. 6), resulting in
a significant temperature drop (Fig. 7) despite significant stratifica-
tion. The second set of four figures refers to a higher Froude number
discharge (F = 15). The deflection of the main trajectory is larger
than "before for identical W/u ratios (Pig. 9)« Tne reason is the
larger amount of entralnment o? the faster jet, which is associated
with larger plume dimensions, including surface areas exposed to wind.
There is a substantial difference in the dilution of the circular
(b /a = l) and the wide elliptical (b /a = i|) discharge. The
centerline temperature decay by turbulent mixing with ambient lake
water is overwhelming. Figure 12 shows very nicely the initial jet
regime followed by a stratified flow process. It can also be noted
that wind will tend to sustain turbulent entrainment (Pig. 10) and
further growth of the plume's thickness (Fig. 12). The third and
fourth sets of figures refer to heated surface jet discharges at right
angles into cross-currents at a low densimetric Froude number (P =
3.75) and a high densimetric Froude number (F = 15.0), respectively
1-32
-------
250r
200-
nl , , , ,
250
200
150
100
50
0
" /4.4 /
i *
l ••'/%
_ l ,$o = 8.8
i /
l /
-i /
I /
li
i
i
I bo
(Tj™"s 4
do
•
1 I 1 1 J
50 100., 150 200 250
'do
Fig. 5 - Plume Troiectory in Cross-wind Perpendicular ro Discharge.
ao=90°, P = 0, F0 = 3.75, U$ = 0, K$ = 0
1-33
-------
2-
JL 2
Qo
*•«
w
0 50 100 150 200 250 300
S,
Fig. 6 - Total Volumetric Flow Rate versus Distance of Travel. Plume
in Cross-wind perpendicular to Discharge. a = 90°, P = Oi
F = 3.75, U = 0, K = 0 °
o s s
-------
.8
.6
.4
.2
\
_\
0
^^^r^.4
Figt 7 -
50 100 150 200 250 300
S/do
Excess Temperature above Ambient along Main Trajectory* Plume
in Cross-wind Perpendicular to Discharge, a = 90 , P = 0,
FQ = 3.75, Us = 0, Ks = 0
1-35
-------
Q
do
1.0
.8
.6
4
1.0
o. 8
_ «w
.6
8.8
4.4
W""1:?
17=0
50
100 150 200 250 300
Fig. 8 - Depth of Thermocline below Water Surface along Main
Trajectory, Plume in Cross-wind Perpendicular to
Discharge, a = 90°, P = 0, F = 3.75, U = 0,
ix _ /N O O S
1-36
-------
H
300
250
200
300
250
200
150
100
50
=3-3
50 100 ISO
50 100 150
Fig. 9 - Plume Trajectory in Cross-wind Perpendicular to Discharge.
a= 90°, f) =0, F= 15,0, Us = 07 K$ = 0
-------
f4
12
10
8
W _
0 50 100 150 200 250 300
s/ao
Fig. 10 - Total Volumetric Flow Rate versus Distance of Travel, Plume
in Cross-wind Perpendicular to Discharge. a = 90 , P = 0,
Fo = 15.0, Us = 0, Ks = 0 °
1-38
-------
50
100
150
%0
200 250 300
Fig. 11 - Excess Temperature above Ambient along Main Trajectory*
Plume in Cross-wind perpendicular to Discharge, a =90°,
P = 0, _ F = 15.0, U = 0, K = 0 °
1-39
-------
50 100 150 200 250 300
Fig. 12 - Depth of Thermocline below Water Surface along Main
Trajectory. Plume in Cross-wind perpendicular to Discharge.
a =90, P = 0, F =15.0, U = 0, K = 0
-------
300r
250-
200-
x * - IR
s Uo '•l*
.30
100 150 200 250 300
Fig. 13 - Plume Traject'ory in Cross-current Perpendicular to Discharge.
°o = 90°' Fo = 3-75' W = °' Ks = °
-------
Q_
Qo
.30
50 100 150 200 250 300
s/a<
Fig. 14 - Total Volumetric Flow Rate versus Distance of Travel. Plume
in Cross-current Perpendicular to Discharge. a = 90 ,
FQ = 3.75, W = 0, K$ = 0 °
1-42
-------
= ,30
0
Ffg. 15
200 250 300
s/a,
Excess Temperature above Ambient along Main Trajectory.
Plume in Cross-current perpendicular to Discharge.
a = 90°, F = 3.75, W = 0, K = 0
-------
2r
±li = 3
Uo -°
0 50 100
150 200 250 300
s/do
2r
ib.
Uo"
.15
_
do
100 150 200 250 300
Fig. 16 - Depth of Thermocline below Water Surface along Main
Trajectory. Plume in Cross-current perpendicular to
Discharge. « = 90°, Pn = 3.75, W = 0, K = 0
-------
250
200
150
'do
100
50
u
0
50 100 150 200 250
y/o,
Fig. 17 -
Plume Trajectory in Cross-current perpendicular to
Discharge, a = 90°, F = 15.0, W = 0,
Ks=0
1-1*5
-------
150 200 250 300
Fig. 18 - Total Volumetric Flow Rate versus Distance of Travel., Plume
in Cross-current Perpendicular to Discharge. a = 90 t
FQ = 15.0, W = 0, K$ = 0 °
1-1+6
-------
I!
To*
100 150
s/a«
200 250 300
Fig. 19 - Excess Temperature above Ambient along Main Trajectory,
Plume in Cross-current perpendicular to Discharge.
a = 90°, F = 15.0, W = 0, K = 0
I-ltf
-------
do
^.^^ U0 "*
50 100
150 200 250
s/o.
300
Fig. 20 - Depth of Thermocline below Water Surface along Main Trajectory*
Plume in Cross-wind Perpendicular to Discharge. a = 90 ,
FQ = 15.0, W = 0, K$ = 0
-------
SECTION VI
EXPERIMENTAL STOUT
Apparatus and Data Acquisition
The experimental apparatus consisted of a large rectangular tank 17 ft
wide, i;0 ft long, and 2 ft deep. Hot water was discharged from a chan-
nel 6 inches wide and approximately 1.6 inches deep at right angles
into the tank, which was filled with cold water. Point measurements of
temperature and velocity were taken in the vicinity of the outlet with
the aid of a combined temperature-velocity probe described in Ref. [22],
To assure steady-state conditions, cold water was fed into the tank con-
stantly at very low velocity. The excess flow drained over a weir at
the downstream end of the tank. Cross-currents and wind were absent in
the experiments. More information on the experimental apparatus is given
in Ref. [22].
Data Preparation
The experimental data obtained by the procedures outlined in Ref. [22]
consisted of velocity and temperature measurements at variable depth z
at a given location (x,y) in the tank. Figure 21 will serve as a
definition sketch. Telocity components in the x-direction (u-component)
and in the y-direction (v-component) were measured. Usually fifteen to
thirty measurements per profile were available. Temperature and veloc-
ity were usually not measured at the same depth z because of the par-
ticular configuration of the tethered sphere probe used. Therefore 51
equidistant data points at 0.01 ft vertical spacing were obtained by
interpolation for each vertical profile. Temperature profiles were
approximated using a third order polynomial between any three given
measurements, and the interpolation was carried out numerically. For
the velocity profiles a graphical interpolation was preferred because
the data had more scatter and smoother velocity distributions were
desirable.
Table 1 summarizes the conditions for which experimental data were avail-
able. "FLTTSH" signifies that the discharge channel terminated in the
wall of-the tank, while "PROJ." means that the outlet channel projected
4.8 ft into the tank. Q is the heated water discharge rate, d is
the depth of the outlet Shannel, T is the outlet water temperature,
T. is the cold water temperature i& the tank (lake), T is the air
temperature, and Re is the outlet Reynolds number defined as
oo
I-
-------
Table 1 - SUMMARY
Q a T T. T Re F
o o o Z a oo
OUTLET TANK AIR
d T T. T
) o o Z a
f -Q •(* cs j f "P4" ) ( Tp j f TT ] ( "P I II i ^
215 FLUSH .0061|5 .162 91.0 70.0 78 1,590 .62
207 FLUSH .0131 .165 76.5 51.5 78 2,700 1.38
216 FLUSH .0376 .162 92.0 73.0 81 9,370 3.73
211* PROJ. .0061*5 .155 87.0 68.0 80 1,520 .72
211 PROJ. .011*1 .159 81*.0 60.0 76 3,190 1.1*3
217 PROJ. .0376 .155 92.5 73-5 81 9,1*00 3.98
1-50
-------
OF EXPERIMENTS
T -T
o a
VTa VT.« V*e Ho Co
(-) (ft5°F/Seo) (°F ft2) (ft)
13 21 .619
-1.5 25 -.060
11 19 .578
7 19 .369
8 21* .333
11-5 19 .605
.135
.328
• 715
.123
• 339
.715
1.70
2.06
1.51+
1.1*7
1.91
1.1*7
•^•^^•a
.1*0
.1*0
.40
. liO
.1*0
• liO
1-51
-------
Heated water channel
v(x,y,z)
T(x,y,z)
u(x,y,z)
Fig0 21 - Definition Sketch - Surface Discharge into Experimental Tank
1-52
-------
where U is the average discharge velocity, w = 0.5 ft is the width of
the outlet channel, and v is the kinematic viscosity at the outlet.
F ' is the outlet densimetric Froude number, defined as
= Q (g'w d O / (U5)
~o ~ov" ~o' ^ove o o ' ^JJ
AQ
where g' = —-°-g is a reduced acceleration of gravity. The outlet
heat discharge (heat flux) into the tank is
H =oc Q (T - T,)
O P O O ^v
and the relative heat content in the outlet cross section is
C = oc d w (T - T.)
O P 0 O O Xf
D is the hydraulic diameter of the outlet cross section defined as
Data Processing
Distribution Functions
Each temperature or velocity profile measured in a vertical section
was graphed using a scaleplot subroutine (SCLPLT) from the CDC 6600
system library of the University of Minnesota Computer Center. The
graphs obtained were of course nearly identical to those obtained with
the raw data and reported in Refs. [22] and [55]. Mere inspection of
the output showed that the measured profiles were similar to each
other and to normal distribution functions only in first approximation.
As stated in Ref. [53], there appears to be no unique function which
rigorously describes all distributions in a plume. A sample of meas-
ured velocities in Fig. 22 may illustrate this point.
The standard deviations identified as a and Aa in Eqs. (l) and (2),
respectively, were calculated for each temperature and velocity profile.
The standard deviation for the velocity profiles was found from the
relationship
1-55
-------
u, f^sec
.2
Nl
.3
.4'
u, ft/sec
X=0,5ft
X=l.5ft
= 3.0ft
= 8.0ft
1
Variafion of Velocity Component u with Depth at Various Distance*
from the Outlet along Centerline of tank. Exp. 215
I-5U
-------
= a =
'N
.1=1
1/2
(U9)
where u. is the velocity component in the x-direction at a point
and N ^is the number of positive measurements in a profile. A
vertical profile consisted of no more than N = 51 points. Simi-
larly, the standard deviation for temperatureppro files was calcu-
lated from
V =
-N
f
.1=1
1/2
- *,) »V
(50)
The calculated values of a- and a showed dependence on location.
Figure 23 gives values for sill six experiments as a function of dis-
tance from the outlet. Predictions made with the aid of the analyt-
ical model are shown as solid lines. There is a substantial amount
of scatter in the experimental data, particularly those derived from
velocity measurements. The analytical model predicted the measured
standard deviations, especially those for temperatures, reasonably
well.
The vertical distortion parameter \ = a™/a was assumed to be cal-
culable from the standard deviationsvfor temperature and velocity. It
was found to vary considerably from one location to another. This is
shown in Pig. 21;. Yalues of A, less than unity mean that the tem-
perature profile reached smaller depths than the velocity profile, a
situation which characterizes fast momentum transfer and slow heat
transfer, as would be typical for a very stably stratified shear flow.
Values of \ larger than unity indicate that the temperature profile
covered a larger depth than the velocity profile, a situation which
could be produced by the accumulation of warm water in the experimental
tank before it flowed over the weir at the downstream end of the tank.
It is known that this condition existed in the downstream portion
of the tank for those experiments using the largest flow rates. Heat
conduction could also be responsible for the temperature distributions'
reaching larger depths than the velocity distributions. Considering
the residence times of the heated water in the upstream portion of the
1-55
-------
o
£ J
o
•o
c
o
X
o
A
•
°u
O
A
a
Exp
215
207
216
468
x, In feet
0 207
10 12 14
6 8 10 12
x. in feet
Fig. 23 - Standard Deviations a and
14
16
j of Velocity
Component u and Temperature T respectively
as derived from Velocity and Temperature Profiles
measured at Selected Locations along Main
Trajectory of the Heated Water Surface Jet.
Comparison with Predictions of Analytical Model
(solid lines)
1-56
-------
-i r
a
A
a
A
A 207
o2IS
0216
A * O
a
A a
o o
0 5 tO 15 20 25 30 35
'Do
X/r
o 211
v 217
V » 7
_| 1 | I
10 IS 20 25 30 35 4O 45
'0.
Fig. 24 - Similarity Parameter K versus Distance
along Main Trajectory from Discharge Point
1-57
-------
tank, however, this is only a remote possibility. While the above mech-
anisms can be cited to explain whatever trend can be detected in the
data, much of the scatter in Fig. 2l± is due to experimentation, particu-
larly the difficulties encountered in measuring small velocities in a
thin layer of heated water.
The way in which the similarity parameter A. varied in cross sections
perpendicular to the main trajectory was also examined. The results
presented in Fig. 25 show that there is a slight tendency for A
values to increase toward the edges of the jet. This is the result
of an existing temperature stratification in the tank outside the heated
water jet, as illustrated by Fig. JI£. It may also be noted that A, is
usually in the range 1 < A < 2. For the above reasons it was notv
really possible to accurately assess the value of A , but it appears
justified to use a value of A close to 1 for a heated jet into a non-
stratified environment.
To determine how well the measured temperature and velocity profiles
agreed with assumed Gaussian distributions, a standard error was calcu-
lated for each vertical profile. The error is defined for a velocity
profile as
N^ "11 /?
1 *-? 2 '
f £ (u (x,y,zj - uth(x,y,z )) (51)
!pj=l J
where u and u,, refer to experimental (measured) and theoretical
(calculafed) velocities, respectively. The theoretical value u., was
calculated from Eq.. (l), in which x and y were substituted for s
and r, respectively, because of the absence of wind and cross-currents»
and a (x,y) was used as the standard deviation. Thus
= ue(x,y,0) exp[- \ (jy) ] (52)
For easier interpretation a dimensionless standard error e *(x,y) was
actually calculated
u x,y,0)
(55)
6
The results given in Fig. 26 show no specific effect of distance.
Standard errors average out to nearly 0.1. This is a good result con-
sidering that the standard error computation has nothing to do with a
best fit, but is rather an evaluation of the shape of the velocity dis-
tribution in which the measured surface value is used as a reference.
1-58
-------
4
3
Xv 2,
1
o
1 i 1 1 —
Run 207 o
o
A
Q
0
O
r * o 1
o
O.oo
A OO A ^
1 1 1 .
X= .5 ft.
1.5
3.0
6.0
8.0
0
a
i
-3 -2
-I
y (ft)
t
3
Xv 2
1
0
1 1 1 , , 1 , ,
Run 216
•
A
o X» .5 ft.
o 1.5
A 3.O
> 5.0
0 8.0
0 0£0 0
0 6 t> >
A 0 >
o
1 1 1 1 1
•4 -3 -2 -I
y (ft)
Fig. 25 - Similarity Parameter A. versus Distance from Main
Trajectory
1-59
-------
.3
-------
Standard errors were also calculated for the temperature data or, more
precisely, the excess temperature T*(x,y,z.) above the cold water
temperature. The final equation used is ^
Te*[x7y7o7
N
N
PJ=1
T *(x,y,z
0? *(x,y,6) " exp
1/2
(510
where T * = T - T.. The results were plotted in Pig. 27. They
are quit! consistent with those in Fig. 26. The results for the
"temperature data seem to suggest certain trends with distance
(plotted in dimensionless form with the hydraulic diameter D = O.k ft
as a reference length), but there are not really enough data to warrant
analysis of this detail.
previous discussions centered around vertical distributions of
"temperature and velocity components in the x-direction. The analyt-
ical model described also specifies the horizontal distributions as
Gaussian. There were not enough data in any horizontal section to
really check this hypothesis. No more than seven measurements were
^aken in a horizontal cross section at a constant depth. The raw data
^ported in Ref. [53] suggest that normal distribution functions should
^3 descriptive of what was observed. No further analysis is possible.
^ was noted that the widths of the velocity and excess temperature
Distributions were very similar, suggesting that the horizontal simi-
larity parameter A_ must also have a value close to 1.
1-61
-------
^ v
! , 1 1 ,
10 15
20 25 30 35 40
Fig. 27 - Standard Error between Normal Distribution Functions
and Experimental Temperature Profile measured along
Main Trajectory
1-62
-------
Spread
A second major objective was to obtain information on the horizontal and
vertical spread of the plume after discharge. Spread was evaluated in
several different ways using
Velocity
Spread of isotherm patterns
Spread of velocity distributions
Visual spread
Spread Velocities — The velocity component v perpendicular to the
main trajectory is probably the most direct measure of lateral spread.
In the analytical model it was assumed to be related to a densimetric
Proude number for the centerline section. Plots of lateral (spread)
velocity components v are given in Fig. 28 as an example. Apparently
surface lateral spread velocities reach a maximum at a distance approxi-
mately one-third of the total width of the plume from the main trajec-
tory. In the analytical model this point would be represented by y = b,
where b is the standard deviation defined in Eq. (l). The lateral
spread velocity changes with depth. It reverses direction at some
distance from the water surface. Outward spread velocities (away from
the main trajectory) are found near the water surface and inward flow
at some depth below. Figure 29 shows examples of measurements. The
reversal in flow direction occurs at a depth approximately equal to the
"thermocline or standard deviation a in Eq. (l). The inward flow ap-
pears to provide the necessary fluid for entrainment and also for the
replacement of warm water when lateral spread reduces the depth of the
thermocline. The observation of an inward flow thus supports the dis-
placement (spread) equation (19) in the analytical model.
Equation (18) suggests that the buoyancy-induced surface lateral spread
velocity at a distance y = b from the main trajectory is related to
the centerline densimetric Froude number. The spread velocities v
Deported in Figs. 28 and 29 are totals produced by turbulence and
buoyancy. According to Eq. (2l) the two effects are additive. In
Ref. [37] it has been shown that the centerline velocity u in a fully
Developed non-buoyant, axisymmetric jet varies according to
=6.2 (*>6.2D) (55)
Using the notations of this paper. The lateral spread velocity was
found [37] to have a maximum of
v(x.b,0) x ^
o
1-63
-------
-3
Fig. 28 - Lateral Spread Velocities v at Two Depths (z = 0
and z = 0.1 ft) for Exp. 207
-------
v. ft/sec
-.06 -.04 -.02 00 O2 .04 O6
.2
.4
A
A
A
A
O
a
a
a
x* 0.5 ft
a y = -0.5 ft
A 0.5 ft
-.06 -.04 -.02
v, ft /sec
00 .02
.04 .06
,8
.5 3
rf
4
°o «
a<>
a
0
o*
o
a o
3 o
1.5 ft
o y = -1.0 ft
a -0.5 ft
A 0.5 ft
•» 1.0 ft
Ffg. 29 - Variation of Velocity Component v with Depth
at Various Distances from the Outlet and the
Main Trajectory. Exp. 207
1-65
-------
Thus the turbulence -induced spread in dimensionless form has a value of
(db/dt)_
The buoyancy- induced spread according to Eqs. (l^) and (k) is
/dbx _ u(x?Q?0) ,,-g)
WT, ~ C5 F* ^ '
is
which in the absence of both a cross current and wind is also equal to
(db/dt) c
f __ 2.
u(x,0,0) ~ F
To evaluate the coefficient c^, measured maximum spread velocities
(Fig. 28) were reduced to dimensionless form and a value of 0.0565 sub-
tracted. The resulting values were plotted in Pig. 30 versus 1/F.
The results suggest that Cp. «: 0.5.
Spread of Isotherm Patterns — Another, more descriptive way to illus-
trate the spread of the heated water surface jet is to graph isotherms
in various sections. Isotherm patterns in vertical planes along the
main trajectory and perpendicular to it have been given in Refs. [22]
and [53]- The results indicated in Ref. [53] are also reflected in
the data of Pig. 23 of this report. The standard deviations of temper-
ature profiles (and velocity profiles) as plotted in Pig. 23 can be
used as measures of vertical spread. It is quite apparent that temper-
ature induced buoyancy caused the heated water to spread on the surface
of the tank without significant penetration into it.
To obtain a picture of the horizontal spread of the surface jet a
different kind of graph was used. Figure 31 gives examples. Program
CONTOUR was used to do the plotting. The outlet is at the bottom cente*
of each map and the flow is from the bottom to the top of the map. Co-
ordinates are given in feet with reference to the upstream end of the
tank (bottom of picture) and the centerline of the tank, respectively.
Alternate contours are printed. The numbers inside the map are dimen-
sionless excess temperatures of the form T*(x,y,z)/T*(0,0,0) multi-
plied by a factor of 10. Examination of the maps reveals a number of
features. First, in each run a "typical" plume temperature distributioJj
is formed around the outlet. The depth d™ at which the pattern dis-
appears has been plotted versus the densimetric outlet Proude number
in Pig. 32. The results can be approximated by the relationship
do ~
1-66
-------
H
in
in
>i
.6
.5
,4
.3
.2
.1
.1
.2
.3
4
'/F
.5
.8
.7
.8
Fig, 30 - Maximum Spread Velocities versus Centerline Densimetric Froude Numbers for
Individual Cross Sections Perpendicular ro rhe Main Trajectory
-------
-5.000
-4.000
-3.000
"2.000
-1.000
0.000
1.000
2.000
3.000
4.000
M
2<:22
222222222222224«222422
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2222222222222225>222?«:2222
-22222«2222222«!22222<:2Z2222
22222222222222^22222222222
22222^222222221222221
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44444
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44444
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44444444444444444444
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44.4444444*4444444444 444444
444444444*444444444
444444444444444444
444444444444444444
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444444444444444444
444444444444444444
44444444444444444444
4444444
44444
66
6666666
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-------
+.OOO
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4.000
I
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22222*22222222*
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44*4444444444444444444*444444444444444444444444444
4444444444444444444444*4444444444444444444444444
4t**4ft*4444444444444«.44444444444444444*44444444
4444444444444444444444444444«44444444444444444
«444Mi.4444444444444444444444444444444444444
«44444444444444444442222222222222222222222
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2?2222222222?2?2
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22222*22222222*22222*2222*2222 22*22
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222Z2*22222222*22222*2222?222?222j
-22222*22222222*22222*22222222^2*2
22222*22222222*22222*222222222222
22222222222222*22222*22222222*222
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22222 2222222222222222Z22222222222
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22?222 22222222222222*222222222222222
22?222 222222222 2?2222?2*222222222
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4 22222 22222222222222222
44*4444 2222222222222222222222 2?22222?222222222
4444* 444 222222*22222222222222 222222222222J222
4444 6 444 22222222222222222222 222222222222222
4444 4666 44 222222222222222222 222222222222
22222222222222222222 2222222222222222
22222222222222222222222222222722222222222
22222222222222222222222222222222222222222
2222
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4444
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<,t>W>
«,66b
t,66f>
44
44
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2222222Z22222222222222222?222222222222222
3.000
10.000
9.000
8.000
7.000
6.000
5.000
4.000
3.000
2.000
1.000
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Fig. 31b -
Isotherms for Run 215
(Horizontal section
fordepfh z= 0.03ft):
-------
1
-J
o
-5.0UO -4.000 -3.000 -2.000 -1.000
-1
1
1
1
1
1
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1
1
1
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1
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ZZ2Z22222ZZ222222Z2ZZ?.ZZ-?.Z?.2?.ZZ22?Z222Z?ZZZ
Z~tZZZ2.te.fZ2ZZtZm.iZZt.fti.ZstZZ2Zl.t2
zztzzzztztzzz Zfttttzfy.ztztztzzt?zzz
zzzzzzttztzz zzzztzzttztzttzzzzzzzz
ZZZZZZZH7ZZZ ZZZtiZZZtZiZiZZZttZtZZ
zzzzzzztzz tzzzzzzzztztzzzzzztz
zzzzzztz iitzzaztzzzzzzz
ZZ ZZttZiZ
uooo
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00OOOOU000000000000000
0000000010000000000000
000 00000 01(10 0000 OOUOOOO
OOOOOOOOOOOOOOOOOOOCOOO
0000000000nOOOooO0000000
000000000UOU000000000000
00000OOOOOOnOOOOOOOOOOOOO
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000 06 00 OOOilO 000 00000 000
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00000
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ztztzzzz
22222*222222
zzzzzzzz 22222?
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ZZZZZi ZZZZt
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zzztzz zzzzt
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2.000 3.000 4.000 5.000
I I I I
-,~_-^..??22;»2?22a272?22222?2 10.000
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2?22222?2?22?222222?22222?222
9.000
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t.ooo
22?222222222222222?222
2222222222222Z222222222
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Fig. 3lc -
Isofherms for Run 215
(Horizontal section
for depth z = 0.05 ft)
-ntiiiiiiiiinniiiuiuiuiii inn inni 12222223J33322222H in iniimuiiiiiuniiinmiiiiiiiiiii -.000
-------
' 0.000 1.000
ll j
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aouuuoo oou otico ooou ooouooooooooo oo oooooooooooooooooo ooooooo ooo ooonoo
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QOOUOOUOOUOOOOOHOOOOOOOOOOOOOOOOOOOUOOOOOOOUOOOOOOOOoOOOOOOOOOOOOOOflO
000000000 OOOO 000 OOOOOOOOOOOOOOO 0001)0000 OOOOUOOOOOOOOflOO OOOOO
OOOOOOOOUOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 000
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0
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^ I 1 I I I I - I I I
1
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Fig. 31 d -
Isotherms for Run 215
(Horizontal section
for depth z = 0.07 ft)
-------
f. -*.000 -3.000 -2.000 -U006 0.000 1.000. ?.00n 3.000
fi I i i I r i I i
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2.000
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Fig0 31e -
isotherms for Run 207
(Horizontal section
for depth z = 0,01 ft)
-------
fs -4.00g -3.000 -?.000 "1,009 0.000 J.OOO '2.000 3.000
I * I I 1 I t I I I
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-------
L' -4.006
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222222222222222222222222222222222222222222222 9.000
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2222222222222222222222222222222222222222
Z2ZZZ22ZZ222222222222222222Z22222222Z2
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22222222222222222222222222222222 7.000
222222222222222222222222222222?
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2222222227222222222222222222 6.000
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4
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Fig. 31 g -
Isothenns for Run 207
(Horizontal section
for depth z = 0.10 ft)
-------
vn
f -+.9uo -j.uuo -z+ooo -i.ooo 0,000 l.oon . • ?.oao 3.000 **ooa
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00000000000000OOOOOOU00OOOOOOOOO00000000000000000OOOOOOOO000000000000000000000000
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OOOOO UOnoOOOOOOnOOOODOOOOOOOnriO 00 00000000000000010OOOOOOO OOOOOOOOOOOOOOOOOOOOoOOO
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OOOOOWOOOOOOOOOHOOOUUUO OOOOO Jf>000 OOOOOOOO 00000001 OOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOO
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I I I t I 1 I I I
iFig. 31h -
I Isotherms for Run 207
(Horizontal section
for depth z = 0.16 ft)
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Isotherms for Run 216
(Horizontal section
for depth z = 0001 ft)
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Fig8 31 j -
Isotherms for Run 216
(Horizontal section
for depth z = 0.05 ft)
-------
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Fig. 31k -
Isotherms for Run 216
(Horizontal section
for depth z = 0,10 ft)
-------
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3
3
3
3
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3
3
3
3
3
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3
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«4444444444444444444444444444 444»44*44444<»4/» 4444444
<><>444»4444444444 4444444444
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4444444444444444444444444444444444444444
44444»44444«4'V4 44444444*44444 44 4444444444
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44 44H444444444444444444444444 4444 44444444^
44444444<>444444444044444
-------
2.0
Fig. 32 - Depth of Penetration dj of Heated Water Surface Jet as derived from
Isotherm Patterns
50
25
Fig. 34 - Length x- of Identifiable Jet Type Flow as derived from Velocity-
Concentration Patterns
1-80
-------
0.7 < F < U.O. The depth cL can "be considered a measure of
depth to which the outlet (jet) flow penetrates into the ambient
vater.
second observation pertains to the temperatures in the water Bur-
bunding the surface jet pattern. The ambient surface temperatures are
Bot those of the cold water, but have some intermediate value. The
heated water is discharged into a tank with an unintended existing (weak)
temperature stratification which has been produced by flow conditions
established through the boundaries of the tank and by the heat loss from
the tank surface to the atmosphere. The tank had an overflow weir placed
at the downstream end and vertical walls on either side. In the absence
°f any currents in a prototype a similar situation can exist.
^ third observation regards the shape of the plume as shown by the iso-
therm patterns. The near surface patterns (z = 0.01 ft) generally
consist of a very nearly uniform flow over a short distance (one foot)
°eyond the outlet followed by a gradual lateral expansion. The angle
lateral spread can be seen to depend somewhat on the outlet densi-
Froude number.
^ fourth observation regards the randomness in the temperature distribu-
tion patterns. From a theoretical point of view all temperature dis-
^ibutions should be symmetrical with reference to the centerline of the
tank. But except for a small area near the outlet, measurements do not
ahow such perfect symmetry, because the tank is not a perfectly quies-
^nt pool. The low- velocity inflow of cold water at the bottom of the
tank, the free surface flow over the weir at the end of the tank, and the
Sair of very large and very slow eddies forming frequently on either
aicle of the outlet all contributed to some very slow and barely detect-
able flow patterns on which the heated water jet flow and the free shear
^bulence associated with it were superimposed. Near the outlet the jet
^chanism was dominant, but at larger distance from the outlet more
andomness in the patterns became clearly apparent. This investigation
only with the buoyant jet- type flow mechanism near the outlet and
not cover anything beyond. Motion of the ambient fluid exists in
bounded experimental tank with throughflow. The relevance of
^asurements very far downstream from the outlet must therefore be
Rationed.
T*ototype plumes also break up into complex velocity and temperature pat-
usually by passive diffusion in currents. Wind frequently con-
s to the disappearance of the typical plume patterns.
of ISO-Velocity Concentrations —Rather than temperature meas-
bs, velocity measurements could be used to obtain maps similar
those of Fig. 31. Velocity components in the x-direction (u-compo-
were used for that purpose. Local measurements were reduced to
ansionless velocity concentrations using the average discharge
8locity as a reference value. Thus velocity concentrations are
*fined as u(x,y,z)/[Jo, where UQ - Qo/(wQdo). Figure 33 shows
1-81
-------
H
i
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ro
r .TS.OOO -*.ooo -3.000 -2.ooo~ -i.ooo o.noo i.ooo ~ z.iro* 3. or
I 1 I I I I I * T
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nooooooooooooono
nooooooooooooooo
Value
below 0.0s
0.0 to 0.2v
0.2 to 0.4>
0.4 to 0.6'
0.6 to 0.8
0.8 to KO:
1.0 to 1.2
1.2 to U4
1 A *« 1 A '
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1.6 to 1.8
above 1.8
.. - - — - — '^
I T
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A. 000
5.000
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2.000
1.000
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Fig, 33a -
Iso-velocity
concentrations of
u-component For
Run 207
(horizontal section
for depth
z = 0.01 ft)
-------
r Zs.aoo -'47000r-j*"ood" -2.000 -i.ooo o.ooo i.ooo 2.000 3.000 4.000
, i i i ii i l l T T T
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i
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-------
-~5.000 -*.000 -3.000 -2.000 -1.000 0.000 1,000 2.000 3.000 *.000 5,000
I II I I I I 1 t I T .
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22 OOOOOOOOOOOOOO--
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222 222 000
222 232 000
222 222 000
222 22 000—
222 22 000
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3222222222222222
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232222223322222
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223222 2222
322222 2232
22232 3233
2223 2222
2223 322
2222 222
4.000
3.000
2.000
1.000
Fig., 33c -
Iso-velocity
jconcentrations of
u-component for
Run 207
(horizontal
section for depth
z = 0.07 ft)
.00011122223222111000—
I I 1
-.000
\
-------
rr--
.000
~-2'.000
-1*000
0.000
1.000
3.000
4.000
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— — — — .OOOOOOOOOOOOOUOOOOOOOOOOOOOOOOOOOOOI
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) 0000 000 00 00 0000 00 00000 000 00--
00000000000000000000000000 —
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10 000 00 00000 00 00 000 00 000
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I Fig. 33d -
| Iso-velocity
I concentrations of
u-component for
Run 207
j (horizontal section
for depth
7. = 0.10 ft)
-------
j
H j
00 •
"^5.000 -4.000 -T.OUO -2.000 -1.000 0.000 1.00(1
1 I I I I I I
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— — — OOOOUOOUOOOOUOuOOOOOOOOOOOOOOOdOOOOOOOOO--- — — — — — — «
— — — oooooooooooooooooooonoooooooouoooouoo— — -————.
..... --- . --- OOOOOUOOOUDUOOOOOOOU 000 000000000 000— --------- — ------
.. ---------- . — uoOOOOO 00 OOOOOOOOU 00000000000 OOnO -------------- - ------
. -------------- 0000 1*0 UO 00000000000000 OUOOOOOOnO -------- -_..—-_ ----
— — — — — UOOooOOOUQOOOOCOOdOOOOUOOOOOoOOO — —_———-—..
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I Fig. 33e -
I Iso-velocity
(concentrations of
o-component for
Run 207
(horizontal
section for depth
z = 0.14 ft)
.000000111000000.
I
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-------
> i
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Fig. 33F -
Iso-velocity
| concentrations of
1 u-component for
Run 207
(horizontal
| section for depth
j z - 0.20 ft)
-------
r-?,
11
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;Fig. 33g -
ilso-velocity
{concentrations of
u-component for
Run 215
(horizontal section
For depth
2-0.01 Ft)
,000
I
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I Fig, 331 -
I Iso-velocity
j concentrations of
j u-componenf for
Run 215
(horizontal section
for depth
z = 0.05 ft)
-------
-5.000
-3.000
-2.000
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—OOOOOOOOOOOO000000000000000000000000000000000000000000OOOOOOOOOOOOOO000000000000000000000000000000
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I Fig* 33j -
] Iso-velocity
I concentrations of
I u-component for
j Run 215
i (horizontal section
for depth
z = 0.07 ft)
-------
-5.000 -<».000
I I
T I
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•0000 uOOiJOO 0 0000 OlMIO '10(10 DOO,"OflOOOOOOnOOO OOOOOOOOOOOOO 000 0000') 0 ---
• uovu Duotjuoooot; ooi> ot><>ooooonnoo'"|ooooooooooooooooooooooooonooon--—-...noonoioooonoo
-in; on juoooouo!)uuo''o oj oi)oon j ooo ouvooonoooouooooooonoooooonoooon--.--—..ooooooooonooo
-OdOO'iOChlOOoOOoOOJOOOOOOO OOOOOOO 0000 000000000000 000000000 00000—.-.--...000000000000.0
._.—i.ooooouooounoooooooonouooKOUUOOOOooooooooooooooooonoooono-—— -——--
....— cooiioooo onoo oo no oooooooooo ouoooo ooooooooooooooo-------- .......—
... .. ._ u<>00 00 00 00 0 J OOU 00 OOOilOOO 00 Q(J 00 U 000 000 00 00000. ......... ...........
... . --OOMOOJOOOOOOCOOOoooooo oo ooooo 100000 "oo oo—-— —-—--—-
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. OOOOO OOOOOtl 0(1 OOOUO 0000—--—-.-- -oooUOn 0000000000 000000000000 00000000 0000000(10110 00000000
.. .-(lOOOQUOOOOOOOOuUuuOOOUuOUOOOOOJ I' 00 'Ml 00 il 0000000000 OOOOOQCI000100 000000000000 (100000000 000000 'I OOOOO.-.
-oil oooooo uuoooooouooooooooou no it j ojooG no ooo otjij oo i) ooooo ooooooooooooonooo ooonnoo o nnoooononooonnooooonon-
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oooooooooo (i ooouooooooo 41 oojooooooooooo..-.oooooouoooooooooooooooooooooooooooooooonoonnnnnoonooooooooon
oooooooouooooouoouij'JOfi(|uouoo(|ot)OOQiioo-_--.no')oo'joouOQoooooooooooQOOooooor)oooooo(ioooooiionooooooooooooo
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OQOUOCOOCiOCOOOOOOOOOuooOOUOOOO-----—-----nOOOOOOOOOOOOOQOOOOOOOOOOflOOOOOflOOOnoOOOOnoflfiOOOnoOOOOOOOOO
OOOOOOOOtOOOOO'JOOoOOl)0'''0"OOUU- -- ——- — 00OOOOO000000OOOOOOOOOOOOO 000000000000000000OOOOOnOOOOOOOOO
000000JOUOOOOO0000000000000--------—.---—000000000000000000000000000000OOOOOOO000000100000000000000
00000030(1 OOOOOOOOOOOOO0000——-—...... 000000OOOOOOOOOOOQOOOOOOOOOOOOOOOOOOOOOOnOOOOOOO00000000
-oooooooooooououoooouuoooo—•.--»_.._.__---.-.-._---.--ooooooooooooooooooooooonooooooooooooooooooooonono
OOOOOOOOtOOOOoUOOUOOOflOO------•- • —_.--. 00 OOt)OOOOOOOOOOOOOOPOOOO"()OOOOOOOOOOnonOOOO 000000000
OOOOOOOOOOOOOOQOOOOOOO--""""————•—-------oooooooooooooooooooooonoooooooooooooftf>ooooooooooonooo
ooooooooooooooooouou------------------—--------oouooooooooooogoooooooooooonooooooooooooooonoooooooooo
000000001/0 OOOOOOO 0-—_--«-——-- — - — - — - —000000000000000000000000 000 000 OOOOOOOOOOOOOOOOOOOOOOon
ooooooy ooooooooo-——————-.-..-.- ......vooooooooooooooooooooooooooooooooonooooonooonoooooooono
.000000000000000.—————-—.—. .._._—.-—b oouo oooooooooooooooooooooooooooooooonooooooooooooooooo
I I I I I I I I f T T
P.OOO
"'.OOO
?.on
-------
velocity concentrations plotted by CONTOUR for horizontal planes. Again
the discharge is at the bottom center of each figure and flow is from
the bottom to the top. The symbol table in Fig. 33a gives the signifi-
cance of each printed number.
The spreading patterns described by iso-velocity-concentration lines
axe quite similar to those found for iso-temperature concentrations.
The following observations can be made:
Near the outlet the surface jet is surrounded by a flow
field with negative velocity. These velocities are part of
a large eddy which forms on either side of the warm water
jet. Lateral spread appears to depend on outlet densi-
metric Froude numbers. Smaller numbers are associated with
faster spread.
The horizontal shear stresses which the jet exerts on the
underlying temperature-stratified fluid appear to produce
a sort of low-amplitude standing wave pattern at certain
depths. The phenomenon was also apparent in some of the
temperature concentration patterns shown in Fig. 31- The
distance x over which the velocity distribution could be
clearly identified as a jet-type flow was estimated and
graphed versus F in Fig. 3U- The results are approxi-
mated by the relationship
r=22-5 /o" <61)
0 ™
Although the measurements leading to Figs. 32 and 3^4- a^e
only very crude estimates, it is worthwhile to note that
the depth of penetration did not exceed two times the out-
let depth and the spreading jet-type flow patterns per-
sisted only over a relatively short distance of fifty
times the outlet depth or less.
Iso-velocity concentrations have also been plotted in vertical cross
sections perpendicular to the main axis of the experimental tank.
Again only u-components of velocities were used. Examples are given
in Fig. 35. The profiles show quite nicely the decrease in depth and
the lateral spreading of the surface jet with distance. The shape of
the profiles justifies Eq. (l).
A velocity concentration map for a vertical section through the main
trajectory is given in Fig. 36. Flow is from left to right and dis-
tances are given in feet. The stratification is quite apparent.
To obtain velocity concentrations it is necessary to reduce each
measurement by a reference velocity. The average discharge velocity
at the outlet was chosen as such a value. The resulting concentration
1-93
-------
•$.000
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I 1 1 1 I I II T T 1
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Fig. ^
Iso-velocity
concentrations
for Run 215„
(Vertical section
at x = K5 ft)
-------
.5.000 -4.000 -3.000 -2.000 -1.000 0.000 1.000 2.000 . 3.000
I I I 1 I 1 1 1 1
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.......... ..................... ..........00000———————
. . ._..__...._-........ 000——— — ——**.——,
4.000 S.OOA
I I ....
............. ..030
, ., ..joe
...... ..ISO
...—...—. -.266
... — -.250
— . ..... ..300
.... . .. -.390
Fig. 35c -
Iso-velocity
concentrations
for Run 207.
(Vertical section
for x=0.5ft)
.400
.450
-------
-S.600
-3.000 •2.000 -1.000 0.000 1.000 2.000
I I I I I 'I
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C 000 000
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— .... ------ .— ..Qoooaonooao oooooooo
0000000000000000-
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. — ..... — ... — ... — ..aonoaooooooooo--
... — . — ... ---- . — — onoootfoooono—
... — --—-.--.030 000000 — --<
—.«.-..--..-......._._._... no — — .-—
44444444 22222 OOflOOO-
44444444 22222 OOnOOO—
22222 000000—
22222 000000—
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2?222 000000—
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2??222 00(1000 —
2?222 000000--
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22*222 OOflOOO—
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Fig. 35d -
Iso-velocity
concentrations
for Run 207.
(Vertical section
for x = 3.0 ft)
-------
0.000
I
3
3
3
3
3
-3
3
3
3
22
222
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22222
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1 22222
1 2222
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000 222
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000 222
— 00 22*
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tr1 -000 222222222222222
vi -000 ^2222222222
CO -0000 2222222
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4*44444,4*44444444444444444»44444444444444444444444*4444444
44*444*4444444444444444444*44444444444444444444444444444
44444444*444444444*444444444444444444444444444444444444
44444*44*4*4*4*44444444444*444444444444444444444444444
4*4444*4444444*444*4444444*44444444444444444444444444
4*44444444444444*444444444*44444*44444444444444444*44
444*4444444444444*4444*444*444444444444*444444*444444
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444444*4444*4444*444444*44*4444444444444444444444444
4444444*444444444444444*4444444444444444444444444444
44444444*444*44*4444444*44*4444444444444444444444444 ??.2
444 444444*4*44444444444 4 44't 4444444 44 444 4444444 *4444& ?J2
4*4444444444444*444444444444 44444444444 222 0
44444*4*4444444-»444444 4 222 00
4444*44*4444444*4* ?22 0000
444*4444*4*44 ?22 00000
4*4444*4 22222222222?22?222
-------
values show the velocity decay with distance from the outlet quite well.
To illustrate the actual spread of the plume it is helpful to reduce all
velocity data with the velocity on the main trajectory as a reference
value. Such velocity concentrations then are equal to u(x,y,z)/u(x,0,0).
Plots obtained with these concentrations for zero depth were used to
find the horizontal standard deviations b(x) "by locating the points
at which u(x,"b,0)/u(x,0,0) = 0.606. The results are plotted for three
tests in Fig. 37 » and a comparison with values predicted by the analyt-
ical model is given.
Visual Spread Angles — In the previous sections the vertical and hori-
zontal spreading of the surface plume were defined quantitatively in
terms of spread velocities or the change in standard deviations for
temperature and velocity profiles with distance. It is also possible
to use visual contours of the buoyant surface jet to measure spread.
To define the contour one must use a tracer in the experiment as de-
scribed in Ref. [53]- Figure J>8 is an example of a visual spreading
record near the outlet obtained using floating particles and time-expo-
sure photography. More examples can be found in Ref. [53] • The total
width of the surface plume measured perpendicular to the main trajectory
can easily be recorded as a function of distance from the outlet.
If the shapes of the velocity and temperature distributions in vertical
cross sections are approximated by normal distributions such as are
specified in Eqs. (l) and (2), the spread of the contour and the spread
measured by standard deviations are correlated. If the contour repre-
sents a line of iso-velocity concentration of magnitude k, the lateral
contour spread at the surface drc/ds or dyc/dx is
, 1/2
Where db/dx is the standard deviation spread. Contour spreads were
measured on plots of visual tracers. However, they are difficult to
Use because it is uncertain how small a value of k should be used to
describe the contour of the plume. Therefore visual records of tracers,
isotherms, and iso-velocity concentrations were used only to find the
initial angle of spread. The tangents drawn to the contours of avail-
able surface spreading records define a horizontal spreading angle
as shown in Fig. 39. This angle was measured on available records
in Ref. [53]. Part of the scatter is due to the fact that the experi-
mental tank into which the water was discharged was partially tempera-
ture stratified. The spread of the jet will depend on that stratifica-
tion, and an additional parameter is needed to take it into considera-
tion. Considering largely the velocity data, a very approximate rela-
tionship between the outlet spreading angle and the outlet densimetrio
number appears to be
1-99
-------
1.5
1.0
215
O
0>
H
O
O
O Q QD
207
Exp left- right
215
207
Fiq 37 - Lateral Spread of Heated Water Jet on Water Surface as Illustrated
by Values of Standard Deviations b(x) plotted against Distance x
from point of Discharge. Analytical Model Predictions (solid lines)
a\\d Ex$>ev\n\enVa\ Dcrta (dotted \mes>
-------
Fig. 38 - Visual Surface Spreading Pattern (Time Exposure
of Tracer Particles)
1-101
-------
H
4->
O
from tracer records •
from velocity records T
from Reference [l5| a
from Ref. [24]
approx. assymptotic-
value for non-buoyant jets
8
10
Fig. 39 - Visual Surface Spreading Angle of Heated Water Surface Jet.
Experimental Data °
-------
where 0 is given in degrees. The data were converted from contour
spread values using the simple relationship
= 2 arctan('
^ - _ ^.^ w—iv ^ ,
The present data and also those taken from another source show a great;
deal of scatter, the source of which is believed to be an existing stnati-
fication in the tank which was not given sufficient attention. The buoy-
ancy experienced by a jet discharged into a tank will be strongly influ-
enced by an existing temperature stratification.
The same data were also plotted on tan /2 versus 1/F in Fig. ij.0,
and again considerable scatter was obtained. The velocity data appear
to fit a straight line with the description
n 7
(~to> = F + °'15 for Fo > 1'°
0 o
boundary or contour of a surface plume was previously estimated to
y « mb, with m « J>. Thus
c
(66)
By comparison with Eq. (20) one finds that the second part of the
^ight-hand side of the equation is approximately the turbulent spread
the first part is the buoyancy induced spread. By comparing with
(18) it is found that
n
Here c is of the order of 1.25 to 2.5> and hence c^ is of the order
°f from' 0.25 to 0.50. These values agree reasonably well with the values
found from the lateral velocities in one of the previous sections. A
value of Cp. = O.ijO was used in the analytical model.
1-103
-------
H
H
1 I
o from temperature contours
v from velocity contours
^dxj
J I
.2 4
.8
1.8 2JO
1.0 1.2 14 1.6
_L
Fo
Fig. 40 - Surface Spreading of Visual Contour of Heated Water Surface Jet. Experimental Data
-------
Entrainment and Mixing near the Outlet
Volumetric Fluxes, Heat Fluxes, and Heat Contents — The volumetric
flux, the heat flux, and the heat content along the main trajectory
(main axis of the tank) were calculated by integration of the meas-
ured velocity and temperature profiles in cross sections perpendicu-
lar to the axis of the tank. The theoretical values of the fluxes are
given in Eqs. (5), (6), and (68) with the provision x = s, y = r.
oo oo
C(x) = Qcp F /*T*(x,y,z) dxdy (68)
-CD
The limits of integration were changed. Instead of an infinitely large
cross section, a finite one defined as that area perpendicular to the
main trajectory in which the u-velocity components were all positive was
used. The integration was carried out in three steps: integration of
individual profiles of velocity (or excess temperature or products of
these) with respect to depth; multiplication of each result "by distance
between midpoints of adjacent profiles; and addition of several of
these results for an individual cross section x = const. The results
of all integrations were reduced to dimensionless form by using the
corresponding outlet values for reference; e.g., Q for volumetric
flux, pQ c (T - T ) for heat flux, and d w (T - T ) for heat con-
tent. The Results referring to volumetric f?uxes°are plotted in Pig. 1+1
versus distances from the outlet.
Primarily because of the small number of vertical profiles measured in
each cross section perpendicular to the x-axis, the experimental results
show scatter. They show conclusively, however, that there is a sub-
stantial amount of entrainment of surrounding water by the heated water
jet even if the vertical stratification is strong. Values predicted by
the analytical model are also shown in Pig. ijl. For further comparison,
the theoretical distribution of volume flux for a semi-circular non-buoy-
ant jet is also given. It appears that the measured entrainment was
initially larger than that predicted for non-buoyant jets. This observa-
tion is quite reasonable considering that the experimental jet had a
fully developed velocity distribution at the point of discharge, while
the non-buoyant jet theory assumes that the discharge velocity is uniform.
The actual zone of flow establishment is therefore much shorter in the
experiments than in that theory. For the same reason the analytical
model also predicts more initial entrainment than the non-buoyant jet
theory. On the other hand, a fully developed non-buoyant circular jet
has an entrainment characteristic equal to
t't (69)
1-105
-------
0.
Non-buoyant
circular jet-^
(diameter Do)/
af»er
215
207
0
x in feet
10
12
14
°fstance
19
Source of -
1-106
-------
In the experiments reported D =0.1)., and the resulting relationship
is shown in Fig. IfL. It represents a limiting case of the analytical
model (F — •+- CD ) and is therefore tangent to the result for Exp. 216
(F = 3«75)« It is also evident from Fig. 1|1 that buoyancy essentially
reduces entrainment further downstream from the outlet. Differences in
total flow rate between a non-buoyant and a buoyant jet become quite
marked.
Heat fluxes, calculated and measured, show only small significant
variations with distance. The experiments were carried out in an
enclosed and fairly well insulated space. In steady state operation
the rate of surface heat transfer was therefore small.
Entrainment and the resulting increase in flow rate as shown in Fig. 14!
produce a substantial dilution of the heated effluent and hence a sub-
stantial temperature drop. Measured centerline water surface tempera-
tures reported in Fig. 1^2 show this quite clearly. For comparison,
computed temperatures are also shown. The fit between measurements and
calculations is strikingly poor, and there are two good reasons for
this: first, the failure to account for a zone of flow establishment
and second, the stratification of the water in the receiving experi-
mental tank.
It is believed that the length of the zone of flow establishment for
Velocity distributions is different from that for temperature distribu-
tions, because in the discharge cross section the velocity distribution
is non-uniform, but the temperature distribution is uniform. The zone
°f flow establishment for temperature is therefore longer and of the
°3?der indicated in Ref . [37]. The zone of flow establishment for
Velocities is relatively short, if not negligible. The basic analyt-
ical model as described does not include a zone of flow establishment.
second source of disagreement is the temperature stratification in
tank as illustrated by Fig. 31. The water entrained by a surface
from a temperature stratified reservoir is not of constant tempera-
ture. Water entrained laterally is warmer than water entrained ver-
tically. The coldest (bottom) water was taken as a reference, and
hence stronger temperature drops have been calculated than were ac-
tually measured.
heat content in a cross section of the plume, as defined by Eq. (68),
a -measure of the amount of heat retained in the receiving body of
er as a function of location — i.e., distance from the point of dis-
charge. An example of measured results is given in Fig. ,i£.
JSStrainment Mechanisms — Vertical turbulent mass and momentum transport
Processes in density stratified 'fluids depend, as is well known, on the
Degree of gravitational stability which the fluid shows in the presen6e
of flow induced shear stresses. Generally the Richardson number
1-107
-------
o
o>
60-
Fig. 42 - Surface Water Temperature Decline along Main Trajectory
of Heated Surface Jet
-------
Fig. 43 - Heal- Content versus Distance from Source of Discharge
1-109
-------
(70)
ia used as a local stability parameter. The heated water flow from a
channel into a reservoir is a density stratified, turbulent flow, and
information on the local values of Richardson numbers may shed some
light on the vertical mixing process in a plume.
A word of caution is necessary at this point, however, because inherent
in the derivation of the Richardson number is the assumption that local
turbulence is a function of local mean velocity gradients. This assump
tion does not necessarily apply to a plume in which the geometrical
(depth and width) scale can change drastically with distance from the
outlet. There is also turbulence production on the sides of the plume
as well as on the bottom. Results given in Ref . [5k] suggest caution.
Richardson numbers were calculated numerically. For uniform vertical
spacing of measurements, the Richardson number for point i is approxi
mately
-
+
- ui\
- -J
Since densities were not measured directly, conversion of the tempera-
ture measurements to densities was made using the empirical relationship
Q = 62.U3[1 - (6T2 - 56T + Itf)(0. 000001 )] (72)
where Q is obtained in lb/ft* if T is given in centigrades. Dif-
ficulties arose, of course, in the numerical computations when the
velocity gradient 3u/3z took a value of zero. It was therefore
decided to have the Richardson number assume special designated values
if 3u/3z = 0 or 3Q/3z = 0 or both. All results were obtained in
tabular form. To facilitate interpretation a graphical presentation
of the numerical results with the aid of program COM'OTJR is given in
Fig. 14|. Since the actual values of Richardson numbers covered a very
wide range, and the contour plot could differentiate only between ten
different symbols, logarithms (base 10) of Richardson numbers instead
of actual values were plotted. The numbers actually printed in Pigs.
i4j. and U5 are equal to loginRi + if. Using logarithms was desirable
also from another point of view: Velocity measurements with the aid
of the tethered sphere probe are probably not precise enough to give
accurate values of velocity gradients. This problem exists in part
also when measurements are being made with other velocity measuring
1-110
-------
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Fig. 44a -
Iso-logarUhms of
Richardson num-
bers for Run 215.
(Horizontal section
at z = .05 ft)
-------
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Richardson num-
bers for Run 215.
(Horizontal section
at z= .14ft)
-------
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44444444444444*4444444444444444444444444444
44444444444444444444444444444444444444444444
44444444444444444444444444444444444444444444
44444444*4444444**4444«444444*44444*44444444
444444444444444*4444444 444444*44444444444444
444444444444444444.444 444444444444<,444444
44444444444444444444 44444444444444*444
4444444444444444444 4444444444444V444
4444444*444444 444444*4*4*44
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6666bt>66bbb66bbb66666666b66bb6bb6 444444444444444444
6b66b66bbbb66(jbb6bb66bbbb666b6666b6 444444444
6666b666bbbbbbbb6bb6bbb666bbbb6b666666666
6b66666bbbbb66666bbbbb66bb6bbb666666666666666
6666b666bbb6bb6b6bbbbbbb66bbb66b6666666666b66b6666
-66S6b6b6bbbbb6bb66bbbbbbbbbbbbbb666666
66666666tbbbbbbb6b666bbbb6bbbbbb66666
66b6b666bbbbbebbbbbbbbb6ebbbbb6b66
6b66bb66bbbbbbbbbbh6bbt>bb6bbb6
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666 66666666666
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I I I I I I I I ! I T
6.000
4.000
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Fig. 44c -
Iso-logarithms •
Richardson nurr
bers for Run 20
(Horizontal se<
at z = .05 ft)
-------
M
-5.000 -4.000 •3.000 -1.000 -l.OOO 0.000 1.000 2.000 3,000 4.000
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1 I X 11 1 1 1 T 1 1
7.000
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Fig. 44d -
Iso-logarithms of
Richardson num-
bers for Run 207.
(Horizontal section
at z= .H ft)
-------
s
5
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S
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S
5
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5
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z i i r z i i i r i r
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I I I I I I I I 1 1 I
9.000
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Fig. 44e -
Iso-logarithms of
Richardson num-
bers for Run 207.
(Horizontal section
at z= .20 ft)
-------
5.000 6.000 7.000 8.000 9.000 10.000 11.000 12.000
1 I ' I I I I j i f
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0.000
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Fig0 45a -
.400 Iso-logarithms of
Richardson numbers
for Run 214.
(Vertical section
_ for y = 0)
.460 7 '
-------
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88888 880
88RS . 8888
8888 848
888 8Q8
888 888
AR8 8RR
BBS 8%8
7
7
7
7
66666666666
66666666666
666666666666
666666666666
666666666666
666666666666
666666666666
6666666666666
6666666666666
,00ft
n.ooo
..ftSO
-.TOO
-.150
-,?00
..300
-.350
..400
..450
<&
-------
devices. The logarithm of the Richardson number indicates essentially
the order of magnitude and thereby eliminates some of the measurement
errors except in those areas in which gradients are very small.
By the above rule Richardson numbers in the range 0.1 < Ri < 1
obtain the symbol 3» It is generally accepted that a density strati-
fication is stable—i.e., turbulent mixing is absent—when Ri > 1.
Therefore areas with numbers larger than 3 should represent stably
stratified flow regions with little or no vertical mixing. Alternate
contour bands have been plotted. Figure l^ represents horizontal cross
sections. The outlet is at the middle of the bottom. All scales are
in feet.
Figure 1|5 represents vertical cross sections. The outlet is at the
Upper left. The first impression of Pig. [4.5 is of a rather disorganized
pattern. A closer examination shows, however, the following typical
features:
Near the surface is a thin layer of weak stratification; under-
neath, a much thicker layer with fairly stable stratification
(l < Ri < 10). The thickness of that layer appears to in-
crease with distance from the outlet; underneath that layer
more stably stratified fluid and finally the cold water in
the tank with Richardson numbers varying rather randomly.
In that last layer the velocity and temperature gradients
are very small and Richardson numbers are probably meaning-
less for that reason, because of errors in temperature and
velocity measurements.
Near the outlet the patterns differ depending on the outlet
Froude number. In Run 21ij. (P < l) a cold water wedge is
formed in the channel and its effect on Richardson numbers is
clearly apparent. The flow as it leaves the outlet has a
stable thermocline at approximately one-third of the depth,
but both the top and the bottom are unstable. In Run 211
the stability near the outlet increases from top to bottom
of the outlet channel. The lower two-thirds are stably
stratified. In Run 21? (F - i|) the major portion of the
outlet flow is unstably stratified.
JLlow Regime of Thermal Plume
It has: been shown that the heated water surface jet is subject to
inertial and buoyant forces, the interplay of which largely deter-
mines the behavior of the jet. This interaction is uninhibited by
Solid walls, and one might therefore expect that the flow regime
Which the surface jet assumes is one in accordance with the minimum
energy principle. In using the analytical model described it was
found that the buoyant surface jet tends toward a regime with a
Specific local densimetric Froude number as defined in Eq. (i|).
In the absence of currents and wind this number was found to be
1-119
-------
close to 3»0« Figure 1^6 illustrates the situation for the laboratory
experiments. The value of 3«0 must be considered representative for an
internally critical flow. Usually critical, gravity controlled flows
are associated with Froude numbers equal to unity. The value 3.0
arises because of the definition of F* (on the main trajectory) as
given in Eq_. (1|). If instead of F* the ratio of inertia! to buoyant
forces in a cross section is used, one finds that with
I = f u/ a, t,. fj (73)
for the inertia! forces and
nb ma
B = / / Q*(x,y,z) zgdzdr
-nb ''O
2 1m2
= gQ (x,0,0)\ a. b fJl - exp[- j (—) ][ (7^)
-L V J. J. ^ t /Vy.
for the buoyancy forces, the ratio is equal to 0.51^5 F*. The critical
value of the ratio then is close to 1.6. Values of A. = A. = 1.05 and
m = n = 3 were used to arrive at this result.
Experimental data on densimetric Froude numbers show them to be somewhat
smaller than predicted, as is also shown in Fig. 1^6. A sharp dropoff
in experimental values occurs at approximately 6 ft from the outlet,
possibly because of density stratification in the tank.
In the absence of wind and cross-currents the heated water jet thus
appears to tend toward a very specific flow regime. In the process
the flow appears to reduce its overall entrainment coefficient dras-
tically, as shown in Fig. Itf'•
When a cross-current or wind is present, it is not clear whether the
flow regime should be defined in terms of the total densimetric Froude
number F or the relative densimetric Froude number F* as defined
in Eq. (1^). It is also questionable whether a specific regime is
reached in all situations. Therefore the conclusions of the preceding
paragraphs should be applied only to situations without wind or cur-
rent.
1-120
-------
F" 3
215
•„•
A A
' '
4 6
x, in feel
A
a
8
a
o
10
I
4
Fig. 46
0 a
o
211 a
214 0
8 10 12
x, in feet
14
16
Densimetric Froude Numbers versus Distance on
Main Trajectory. Analytical Model Predictions
(solid lines) and Experimental Data
1-121
-------
H
ro
K
Fig. 47 - Entrainment Coefficients versus Distance along Main
Trajectory for several experimental conditions as
found from the Analytical Model
-------
SECTION VII
ACZNOTrfLESXJMENTS
In the course of this study Mr. P. Vaidyaraman, Mr. W. Geiger, Mr. Glen
Martin, and Mr. K. Streed provided valuable assistance. Their partici-
pation in various phases of the project is gratefully acknowledged. The
manuscript of this report was edited and typed by Mrs. Shirley Kii.
The help and encouragement of Dr. Mustafa Shirazi, the Project Officer,
are gratefully acknowledged.
1-123
-------
SECTION VIII
REFERENCES
[l] "Biological Aspects of Thermal Pollution," edited by P. A. Krenkel
and F. L. Parker, Proc. of the National Symposium on Thermal
Pollution, Portland, Oregon, June 1968.
[2] Presentation prepared "by the staff of the National Water Quality
Laboratory of the FWQA, Duluth, Minnesota, for the Joint Commission
on Atomic Energy, November 1969*
[5] Jensen, L. D. j Davies, R. M. ; Brooks, A. S., and Meyers, C. D.
The Effects of Elevated Temperature upon Aquatic Invertebrates, A
Review of the Literature, Res. Rep. No. 1*, Res. Proj. RP-U9,
Department of Geography and Environmental Engineering, The Johns
Hopkins University, September 1969.
[i|] Parker, P. L. and Krenkel, P. A., Thermal Pollution, Status of the
Art, Rep. No. 3» Dept. of Environmental and Water Resources Engi-
neering, Vanderbilt University, December 1969.
[5] Thermal Pollution - 1968, hearings before the Subcommittee on Air
and Water Pollution of the Committee on Public Works, U.S. Senate,
Parts 1, 2, 3» and I)..
[6] Edinger, J. E. and Polk, E. M. , Initial Mixing of Thermal Discharges
into a Uniform Current, Rep. No. 1, Department of Environmental and
Water Resources Engineering, Vanderbilt University, October
[7] Wada, A., A Study on Phenomena of Flow and Thermal Diffusion
Caused by Outfall of Cooling Water, Tech. Rep. C-66006, Central
Research Institute of Electric Power Industry, Tokyo, January
[8] Wada, A. , Numerical Analysis of Distribution of Flow and Thermal
Diffusion Caused by Outfall of Cooling Water, Tech. Rep. C-670014,
Central Research Institute of Electric Power Industry, Tokyo,
January 1969.
[9] Hoopes, J. A.; Zeller, R. W. ; and Rohlich, G. A., Heat Dissipation
and Induced Circulation from Condenser Cooling Water Discharges
into Lake Monona, Rep. No. 35» Eng. Exp. Station, Univ. of Wis-
consin, February 1968.. (See also Zeller, R. W. , Hoopes, J. A. and
Rohlich, G. A., "Heated Surface Jets in Steady Cross Current,"
Proc. ASCE. Jrl. Hydr. Div. . September 1971, pp.
[lO] Carter, H. H. , A Preliminary Report on the Characteristics of a
Heated Jet Discharged Horizontally into a Traverse Current - Part I
Constant Depth, Tech. Rep. No. 6l, Chesapeake Bay Institute, Johns
Hopkins University, November 1969.
1-125
-------
[ll] Motz, L. H. and Benedict, B. A., Heated Surface Jet Discharged
into a Flowing Ambient Stream, Rep. No. k> Dept. of Environmental
and Water Resources Engineering, Vanderbilt University, August 1970-
[12] Cederwall, K., "Dispersion Phenomena in Coastal Environments,"
reprint from the Journal of the Boston Society of Civil Engineers,,
January 1970.
[13] Hayashi, T. and Shuto, N., "Diffusion of ¥arm Water Jets Discharged
Horizontally at the Water Surface," Proc. 12th Congr., IAHR,
Colorado, Yol. i|, September 1967.
[ill] Stolzenbach, K. D. and Harleman, D. R. F., An Analytical and Ex-
perimental Investigation of Surface Discharges of Heated Water,
Rep. No. 155, Ralph M. Parsons Laboratory for Water Resources
and Hydrodynamics, Mass. Inst. of Tech., February 1971.
[15] Koh, R. C. Y. and Fan, L. N., Mathematical Models for the Predic-
tion of Temperature Distributions Resulting from the Discharge of
Heated Water into Large Bodies of Water, 16130 DWO 10/70, Water
Pollution Control Research Series, EPA, October 1970.
[16] Flatten, J. L. and Keffer, J. F., Entrainment in Deflected Axi-
symmetric Jets at Various Angles to the Stream, Mech. Eng. T.P.
6808, TTniv. of Toronto, June 1968.
[17] Bowley, W. W. and Sucec, J., "Trajectory and Spreading of a Turbu-
lent Jet in the Presence of a Cross Flow of Arbitrary Velocity
Distribution," Tech. Paper 69-GT-35. ASME, March 1969.
[18] Silberman, E. and Stefan, H., Physical (Evdraulic) Modeling of
Heat Dis-persion in Large Lakes; A Review of the State of the Art,
Report ANL/ES-2, Argonne National Laboratory, August 1970.
[19] Wiegel, R. L.; Mobarek, I.; and Jen, Y., Discharge of Warm Water
Jet over Sloping Bottom. Rep. EEL 3-U» Hydraulic Engineering
Laboratory, Univ. of California, November 1961;.
[20] Jen, Y.; Wiegel, R. L.; and Mobarek, I., "Surface Discharge of
Horizontal Warm Water Jets," Proc. ASCE. Jrl. Power Div.,
April 1966.
[21] Tamai, N.j Wiegel, R. L.; and Tornberg, G. F., "Horizontal Dis-
charge of Warm Water Jets," Proc. ASCE. Jrl. Power Div.,
October 1969.
[22] Stefan, H. and Schiebe, F. R., "Heated Discharge from Flume into
Tank," Proo. ASCET Jrl. Sanitary Eng. Div., December 1970.
1-126
-------
[23] Kashiwamura, M. and Yoshida, S., "Outflow Pattern of Fresh Water
Issued from a River Mouth," Coastal Engineering in Japan, Vol. 10,
1967.
Wood, I. R. and Wilkinson, D. L. , discussion of "Surface Discharge
of Horizontal Warm-Water Jet," by Jen, Wiegel, and Mobarek, Proo.
ASCE, Jrl. Power Div. , Vol. 93, No. P01, March 1967, pp.
[25] Cheney, W. 0. and Richards, G. V., Ocean Temperature Measurements
for Power Plant Design, Pacific Gas and Electric Company,
date not shown.
[26] Fitch, N. R. , Temperature Surveys of St. Croix River, 1969-1970.
Environmental Monitoring Program, Northern States Power Company,
December 31, 1970.
[27] Squire, James L. , Jr., Surface Temperature Gradients Observed in
Marine Areas Receiving Warm Water Discharges, Tech. Paper No. 11,
Bureau of Sport Fisheries and Wildlife, January 1967.
[28] Ackers, P., "Modeling of Heated Water Discharges," Chapter 6 of
Engineering Aspects of Thermal Pollution, edited by F. L. Parker
and P. A. Krenkel, Proc. of the National Symposium on Thermal
Pollution, Vanderbilt University, 1968.
[29] Asbury, J. G. ; Grench, R. E.; Nelson, D. M. j Prepejchal, W. ;
Romberg, G. P.; and Siebold, P., A Photographic Method for Deter-
mining Velocity Distributions within Thermal Plumes, Rep. ANL/ES-i;»
Argonne National Laboratory, February 1971.
[30] Fan, L. N. and Brooks, N. H. , Numerical Solutions of Turbulent
Buoyant Jet Problems, Rep. Ho. KH-R-18, W. M. Keck Laboratory of
Hydraulics and Water Resources, January 1969.
[31] Rouse, H.; Yih, C. S.; and Humphreys, H. W. , "Gravitational Con-
vection from Boundary Source," Tellus , Vol. i^, No. 3, August
1952, pp. 201-210.
[32] Abraham, G. , "Horizontal Jets in Stagnant Fluid of Other Density,"
Proo. ASCE, Jrl. Evdr. Div., July 1965, pp. 139-15U-
[33] Fan, L. N. and Brooks, N. H. , discussion of "Horizontal Jets in
Stagnant Fluid of Other Density," Proo. ASCE, Jrl. Hydr. Div..
March 1966.
Abramovich, G. N. , The Theory of Turbulent Jets, Massachusetts
Institute of Technology Press, Cambridge, Mass., 1963.
[35] Gordier, R. L., Studies on Fluid Jets Discharging Normally into
Moving Liquid. Technical Paper No. 28-B, St. Anthony Falls Hy-
draulic Laboratory, Univ. of Minnesota, 1959-
1-127
-------
[36] Stefan, H., "Stratification of Flow from Channel into Deep Lake,"
Proc. ASCE, Jrl. Hydr. Div., July 1970.
[37] Albertson, M. L.; Dai, Y. B.; Jensen, R. A.; and Rouse, H.,
"Diffusion of Submerged Jets," Trans* ASCE, Vol. 115, 1950,
pp. 639-697.
[38] Folsom, R. G. and Ferguson, C. K., "Jet Mixing of Two Liquids,"
Trans. ASMS, January 1949» PP- 73-77-
[39] Rossby, C. G. and Montgomery, R. B., "The Layer of Frictional
Influence in Wind and Ocean Currents," Papers in Physical Ocean-
ography and Meteorology, Vol. 3> No. 3, 1935«
[40] Parr, A. E., "On the Probable Relationship between Vertical Sta-
bility and Lateral Mixing Processes," Conseil Permanent Interna-
tional pour 1'exploration de la Her, Jrl. du Conseil, Vol. 11,
No. 3, 1936.
Munk, W. H. and Anderson, E. R., "Notes on a Theory of the Thermo-
cline," Jrl. of Marine Research, Vol. 7» 1948'
[42] Ellison, T. H. and Turner, J. S., "Turbulent Entrainment in Strati-
fied Flows," Jrl. of Fluid Mechanics, Vol. 6, Part 3, October 1959-
Kato, H. and Phillips, 0. M., "On the Penetration of a Turbulent
Layer into a Stratified Fluid." Jrl. of Fluid Mechanics. Vol. 37,
Part 4, July 1969.
[44] Wada, A., Numerical Analysis of Distribution of Flow and Thermal
Diffusion "Caused by Outfall of Cooling Water, Tech. Rep. G 67QQL.
Central Res. Inst. of Electric Power Industry, January 1969.
[45] Sundaram, T. R. and Rehm, R. G., "Formation and Maintenance of
Thermoclines in Stratified Lakes Including the Effects of Power-
Plant Thermal Discharges," AIAA 8th Aerospace Science Meeting,
New York, January 1970.
Fan, Loh-Nien, Turbulent Buoyant Jets into Stratified or Flowing
Ambient Fluids, Rep. No. KH-R-15, W. M. Keck Laboratory of Hydrau-
lics and Water Resources, June 1967-
[47] Abbott, M. B., "On the Spreading of One Fluid over Another," La
Houille Blanche, Vol. 16, No. 6, 1961, pp. 827-846.
[48] Edinger, J. E.} Duttweiler, D. W. 5 and Geyer, J. C., "The Response
of Water Temperatures to Meteorological Conditions," Water Resource**
Research, Vol. 4, No. 5, October 1968, pp. 1137-1143.
[49] Roll, H. U., Physics of the Marine Atmosphere. International Geo-
physics Series, Vol. 7» Academic Press, New York, 1965.
1-128
-------
[50] Wada, A., Effect of Winds on a Two-Layered Bay, Tech. Rep. C-66002,
Central Research Institute of the Electric Power Industry, Tokyo,
September 1966.
[5l] Brady, D. K.; Graves, W. L., Jr.; and Geyer, J. C., Surface Heat
Exchange at Power Plant Cooling Lakes. Report No. 5, Project RP-1*9,
Department of Geography and Environmental Engineering, Johns
Hopkins University, November 1969.
[52] Wilkinson, D. L. and Wood, I. R., "A Rapidly Varied Flow Phenomenon
in a Two-Layer Plow," Jrl. of Fluid Mechanics, Vol. U7, Part 2,
May 1971, pp. 2i|l-256.
[53] Stefan, H. and Schiebe, F. R., Experimental Study of Warm Water
Flow into Impoundments. Part III; Temperature and Velocity Fields
Near a Surface Outlet in Three-Dimensional Flow. Project Report
No. 103, St. Anthony Falls Hydraulic Laboratory, Univ. of Minne-
sota, December 1968.
Wada, A., "A S,tudy of Mixing Process in the Sea Caused by Outfall
of Industrial Warm Water," Coastal Engineering in Japan. Vol. XII,
1969, PP. 11+7-158.
1-129
-------
Appendix A
COMPUTER PROGRAM FOR ANALYTICAL MODEL
(FORTRAN iv)
I-A-l
-------
PLUME ( INPUT»QUTPUT)
C ST C^OIX FI&UU DATA, SEPTEMBER 4»l969
UIMENSION UtzOO) «T (gOO) iU'(200) f AtgOO) »H<2°0> »OELRO(200> *H<200) t
1 ALFA<200) »X(200> iY<200) »XL(200) ,Yt<200) »XR{?00) »YR<200) tK(200)
2 M0<200> ,F'<200) ,AR£A(200)
HEAL MtN,L»Kl»KS»KH»K»KV,MO»MOO»KRATlO
N=3.0
M=3.0
L=1.05
COEFFICIENTS
Fl=3.1
-------
TO77.5
DPO=10.
C INITIAL SURFACE HEAT TRANSFER PARAMETERS
FUWl=70.+.7*WlND«*2.
KS=(15.7*(BET*.26)«FUwI)/(3600.*24.)
£QU=TO+HS/KS
PRINT 31
51 FORM.AT(40Xi#KS (BTU/SQFT SEC OEGF)«»5X »#EtfUILl9RlUM TEMP (DEG
PRINT 60«KS»EQU
60 FORMAT l/»33x»2G25.4,//)
c COLO WATER DENSITY
TCC=(5./9.)*(TC-32.)
HHoTCsRfioCC»62.43/32.17
C INITIAL DENblTY DIFFERENTIAL
C INITIAL PARAMETERS
96 UO=(JO/(DPO*WIUO)
FO sUO/i ((UEL«00/1.94)»32.2«OPO)«*«5)
C STARTING CONDITIONS
ALFA(l)
T(D*TO-TC
C INITIAL STANDARD DEVIATIONS
CC=0.t>*1.94tt3tl4l5*UStt*2*{CoS(ALFA(l) ) ) #»2«M»N*Q (1) »0.5#MOO*3.1415
1*US*COS(ALFA(1))*M#N
cc
)/(2.*AA )
o(l)=Q(l)/(A(l)«(U(l)»Fl*0.5»3.14l5»uS»CoS(ALFA(l))«M«M ))
C INITIAL COORDINATES
X (1) = . 0
C INITIAL HEAT AND MOMENTUM FLUXES
*T(1)*A<1)«8(1
C DENSITY DIFFERENTIAL
IbO rCEL=(5./9.)»(T(I)*TC-32.)
rtnOC»i.-(6.«TiEL**2«-36,»TCEL+47.)*»oOOOOl
C DENSiMETRIC FROUOE NUMBER
F(I)=U(I)/(((DEL«0(I)/1,94)*32,2«L*A(I)
IF (F d> .GE. 3.0) GO TO 9^
BO TO 96
9b IF (I .GT. 1) GO TO 97
I-A-3
-------
PRINT 501
501 FORMAT <5QX,*UO(FT/SEC)*•VOX»*FO*»/)
PRINT SOO»UO»FO
500 FORMAT (47X,2G15.3,/)
C ENTRAPMENT COEFFICIENTS
97 IF(0(I)/(US»COS(ALFA(I»>«LE.O»1) KHsKHAMb
IF (KV.t.T.1.0) KV*1.0
IF(KV.LT.OiO) KVS0«0
KHs.059
K\/=KH*KV
K(I)«(KH*A(I)+KV*B(I))/
C PRINTOUT OPTION 1
IF (I .GT.l) GO TO 555
31
A(F,6x*8(FT)»,6
40 FORMAT (1X,I3,3X,9F11.2»1F11.4.1F11.2)
NEW FLOW RATE
UQ=K(I)*U(I)*(A*B(I»**.l/2.
C SURFACE HEAT TRANSFER CoEFFICIENTS
TMs(T(I)*TC*TD)/2.
bET=.255-,0085»TM*.000204»TM»*2,
EQU=TO*HS/KS
H(I*1>=H(1)-HLOSS
)#DFLAM/KH*(-I.)
C FIND WIND FORCE FW
C NEW MOMENTUM AT 1*1
GAMAs8ETA-ALF«(I)
M0( I*l)BMO(I) *OMO
C NEW ANGLE OF *AIN TRAJECTORY AT 1*1
ALFA(I*1)»ALFA(I)-OALFA
C FIND NEW ARtA AT 1*1
Kl*US«COS(ALFA(I*l) )»2t
F1»/C1.94»Q>
3.1415*K1»(K1/(8.«N»M)-MO(I*1)/U.»1.
IF(ZZ.LE.u«0) GO TO 4
U(I*l)«t-Z2*SQRT(ZZ))/(2.»ZD
AR£A(I*l)»Q(I*l)/(U»Fl*Kl*3.lM5/4.)
-------
GO TO 201
4 PRINT 5
5 FORMAT<1OX«*VALUE OF ZZ IS NEGATIVE*)
201 CONTINUE
C RATIO OF NEW STANDARD DEVIATIONS AT I+1
DBB=(C5/(F6*U+US*COS(ALFA(I))))*C32«2*A*L*DELRO#DBB)/CB( I ) )
DBT=C4*(U( I )/+DB
AXRAT=BPROX/APROX
C NEW STANDARD DEVIATIONS
A< 1 + 1 )=(AREA(1+1 )/AXRAT>**«5
B< i-f i )=AXRAJ*A( i-n >
C NEW TEMPERATURE
T+DELX
Y(I+I)=Y
-------
Appendix B
ZOUE OF PLOW ESTABLISHMENT
The analytical method of plume analysis described previously starts with
a fully developed jet. Obviously, this condition will not be found at
the real point of discharge. The starting point of the analysis presented
has therefore been called the virtual origin (s = 0). Between the real
origin and the virtual origin of the jet is the outlet region or zone of
flow establishment. An analytical treatment of the outlet region is
possible only if the geometry is not too complicated. Structures such as
levees, as well as the bottom topography in shallow receiving waters,
may interfere with the formation of a zone of flow establishment. In
many cases it may be necessary to resort to physical experimentation to
find the location of the virtual origin and the initial flow conditions
of the analytical model. Only if the outlet geometry is fairly regular
will it be possible to treat the outlet region analytically. In that case
it is possible to draw to some extent on information obtained through
studies of non-buoyant jet flows. Buoyancy does introduce some important
modifications, however. The zone of flow establishment for neutrally
buoyant submerged circular jets with initially uniform velocity and tem-
perature has been well documented [j?]- The length of the zone of flow
establishment was found to be of the order of 6.2 times the diameter of
the discharge nozzle. This value can be found from experimental data;
for example, by plotting centerline velocities versus distance. Over the
length of the zone of flow establishment the velocity will be constant.
The heated water half-jet under consideration is non-circular, buoyant,
^d non-uniform with respect to velocity at the point of discharge.
typically it may have the features of a fully developed channel flow
before entering the reservoir. The initial conditions for the zone of
flow establishment are a temperature distribution of the form
T*(0,r,z) = T*(0,0,0) = const (75)
w
for r < — and z < d^
— z o
^ a velocity distribution
u(0,r,z) = u(0,0,0) f(r,z) (76)
where f(r,z) may be any function suitable for describing the discharge
Velocity profile, for example
I-B-1
-------
f(r.,
for a rectangular channel, with n, and n~ being constants.
In the zone of flow establishment the temperature profile will transit
from the form given by Eq.. (75) to that given by Eq.. (2), while the
velocity distribution will transit from Eq. (76) to Eq. (l). The
details will be governed largely by the free-shear turbulence gener-
ated by shear stresses between discharged and ambient water.
The length of the zone of flow establishment (XQ) for buoyant surface
jets cannot be measured as easily as for non-buoyant ones, because
buoyancy induces an acceleration in the longitudinal direction near the
outlet which masks the end of the zone of flow establishment. To obtain
information on (x ) one must therefore examine velocity distributions
rather than center?ine velocities. Figure 33 illustrates some of the
velocity distributions obtained by computer plotting. For the evalua-
tion of x these are not accurate enough due to interpolation errors.
Instead, graphs of actually measured velocities such as are given in
Figs. 29 through 3U of Ref. [53] must be used. These measurements,
all of which refer to strongly buoyant jets, indicate that it takes
less than six times the width of the outlet channel to develop a hori-
zontal velocity profile and less than six times the depth to develop
a vertical velocity profile. The measurements were not sufficiently
detailed to enable evaluation of a dependence between densimetric
Froude number at the discharge F , as defined in Eq.. (1^5)» and x .
It appears that for experiments 2?5, 207, 211;, and 211, summarized in
Table 1, x was very approximately equal to three times the outlet width.
The mechanism which produces this result is probably not horizontal
momentum transfer by eddy diffusivity, but rather a secondary effect
of the buoyancy-induced horizontal currents shown in Fig. 28. The
currents move high-momentum fluid from the center of the plume toward
the edges, where the shallow thickness of the flow activates shear
stresses in horizontal planes which reduce the momentum of the flow
rapidly. The same horizontal transport mechanism applies in part also
to heat. Thus the zone of flow establishment for velocity and tempera-
ture is shortened by buoyancy effects, as stated earlier.
From some of the data on spreading angles at the outlet presented in
Fig. 39 it might be gathered that buoyancy effects become small
if F > 10. Zones of flow establishment for temperature may be
somewnat longer than those for velocity. All the above results
apply to straight jets rather than curved ones.
Once the length of the zone of flow establishment has been determined,
an approximate analytical treatment of the zone of flow establishment
can be given analogous to that found in treatments of non-buoyant jet
I-B-2
-------
flows [57]« For high values of F (F > 10 perhaps) results regarding the
flow rate and the dimensions of thi jet°at the end of the zone of flow
establishment can be taken directly from the literature [37]. In the
intermediate range (l < F < 10) the velocity distribution can be ap-
proximated by a function 01 the form
u*(s,r,z) = u*(0,0,0) = UQ (78)
for the inner core region (r < w/2 and z < d) and
u*(s,r,z) = u*(0,0,0) expRz " d)21 expRr " w)2 (79)
for the turbulent diffusion region (r > w/2 and z > d). The dimen-
sions of the core region can be assumed to decrease linearly with
distance such that
-S- = 1 - -£- (80)
wo ^
and f- = 1 - -S- (81)
d """
o
where x and x , are the lengths of the zone of flow establishment
for widtn and deptn, respectively, for velocity. Turbulent entrainment in
the zone of flow establishment is different from that in fully developed
flow. In a turbulent, non-buoyant, circular jet the entrainment coef-
ficient varies linearly from virtually zero to K = 0.059 over the
length of the zone of flow establishment. A similar variation can be
introduced for the zone of flow establishment for buoyant surface jets.
In the total picture the computational scheme developed for fully
developed jets can be extended to include a zone of flow establishment.
For very low densimetric Froude numbers (tentatively F < 3) the zone
of flow establishment is very short and the computation!!! scheme devel-
oped is applicable without further extension. For high densimetric
Froude numbers (tentatively F > 10) the zone of flow establishment can
be treated as non-buoyant. Only in the intermediate range would the
extension of the analytical model as presented be of value.
I-B-3
-------
PART II: The Two-Dimensional Buoyant Surface Jet
and the Internal Hydraulic Jump
H. Stefan
and
N. Hayakawa
August 1971
-------
ABSTRACT
A theoretical and experimental study of the two-dimensional flow of
heated water from a channel into a deep reservoir filled with cold
water was made. Only flows producing a dilution of warm water by en-
trainment of cold water were investigated. These flows consisted of a
mixing zone near the outlet followed downstream by a stratified flow.
The outlet flow characteristics, the type of downstream control, and the
geometry of the transition were identified as the essential controlling
parameters for the entrainment. The amount of vertical entrainment in
the mixing zone at the outlet was calculated and measured. The flow in
the mixing zone where the entrainment is produced was found to be a
combination of a buoyant half jet with an inverted internal hydraulic
jump. The simultaneous presence of both inertial and buoyant forces
was responsible for this occurrence. Theoretical and experimental re-
sults show that substantial dilution occurred in the outlet mixing zone,
but actual rates of entrainment were quite sensitive to changes in flow
conditions.
This report was submitted in fulfillment of Project Number 16130 FSU
under the sponsorship of the Water Quality Office, Environmental
Protection Agency.
-------
CONTENTS
Section Page
List of Figures Il-vi
List of Symbols and Units II-viii
*
I CONCLUSIONS II-l
II RECOMMENDATIONS II-3
III INTRODUCTION II-5
IV BASIC EQUATIONS II-9
V THEORETICAL RESULTS 11-13
Steep Beach 11-13
Plat Beach 11-31
VI EXPERIMENTAL FACILITY 11-33
VII EXPERIMENTAL RESULTS 11-35
VIII APPLICATION OF THEORETICAL AND EXPERIMENTAL
RESULTS 11-51
IX ACKNOWLEDGMENTS 11-53
X REFERENCES 11-55
II-v
-------
LIST OF FIGURES
PAGE
1
2
3
ka
1*
1(0
5a
5b
5c
6
7a
7b
8a
8b
9
10
11
12
13a
13b
DEFINITION SKETCH - TWO-DIMENSIONAL, HEATED WATER
DEFINITION SKETCH - INTERNALLY SUBMERGED OUTLET CHANNEL
SCHEMATIC STREAMLINE PATTERN NEAR OUTLET
E1 VERSUS F ' FOR H2 = o>
H1 VERSUS FQ' FOR H2 = 8
H1 VERSUS F ' FOR H2 = 1
Q VERSUS F ' FOR H0 = CD
O d.
Q VERSUS F ' FOR H0 = 8
O f—
Q VERSUS F ' FOR H0 = 1
O c.
Q FOR INFINITELY LARGE F ' AS A FUNCTION OF
H1 AND H2 °
AE/E VERSUS F ' FOR H0 = CD
' o o 2
AE/E VERSUS F ' FOR H0 = 8
' O O d.
-w,/rh VERSUS F FOR H_ = CD
r O O 2
-w./rhrt VERSUS F ' FOR H0 = 8
TO O 2.
E, VERSUS F ' FOR H2 = 8 UNDER THE ASSUMPTION
w1 = 0
Q VERSUS F ' FOR H? = 8 UNDER THE ASSUMPTION
w = 0
SCHEMATIC REPRESENTATION OF OUTFLOW OVER FLAT BEACH
EXPERIMENTAL APPARATUS
PHOTOGRAPH OF BUOYANT SURFACE JET FLOW DOWNSTREAM
FROM OUTLET CHANNEL
PHOTOGRAPH OF INTERNAL HYDRAULIC JUMP DOWNSTREAM
FROM OUTLET
II-6
II-7
11-10
11-16
11-17
11-18
11-19
11-20
11-21
11-23
11-25
11-26
11-27
11-28
11-29
11-30
11-32
II- 3k
11-36
11-37
Il-vi
-------
LIST OF FIGURES (Cont'd) PAGE
!3o PHOTOGRAPH OF INTERNALLY SUBMERGED OUTLET FLOW 11-38
13d PHOTOGRAPH OF BREAKING OF INTERNAL WAVES NEAR THE
OUTLET II-39
Ik SCHEMATIC STREAMLINE PATTERN NEAR INTERNALLY SUBMERGED
OUTLET 11-1$
15a SELECTED VERTICAL TEMPERATURE PROFILES ASSOCIATED WITH
FLOW OF FIG. 13a II-U
15b SELECTED VERTICAL TEMPERATURE PROFILES ASSOCIATED WITH
FLOW OF FIG. 13b 11-^2
15c SELECTED VERTICAL TEMPERATURE PROFILES ASSOCIATED WITH
FLOW OF FIG. 13o 11-10
15d SELECTED VERTICAL TEMPERATURE PROFILES ASSOCIATED WITH
FLOW OF FIG. 13d 11-14;
16 PHOTOGRAPH OF FLOW OVER A FLAT BEACH II-U5
17 EXPERIMENTAL AND THEORETICAL DATA OF H, VERSUS F ' II-1|6
1 o
18 EXPERIMENTAL AND THEORETICAL DATA OF Q VERSUS F ' II-U7
19 DIMENSIONLESS LENGTH OF OUTLET MIXING ZONE VERSUS FQ' II-U9
-------
LIST OP SYMBOLS AND UNITS
E Initial energy flux of warm water discharge [ib ft/sec]
E. Final energy flux after mixing [ib ft /sec]
E_ Additional energy flux due to entrainment [ib ft/sec]
F Initial outlet densimetric Froude number [-]
o L J
F. Final densimetric Froude number [-]
F Minimum possible outlet densimetric Froude number for given
H.J [-]
g Acceleration of gravity [ft/sec ]
h Initial depth of warm water discharge [ft]
h.. Final depth of warm water layer in reservoir after mixing [ft]
hp Depth of cold water layer [ft]
X, Approximate length of mixing zone [ft]
p(s) Static pressure on beach [ib/ft ]
q. Initial volumetric warm water flux [ft^/sec ft]
q... Final volumetric warm water flux after mixing [ft^/sec ft]
q.p Cold water volumetric flux [ft^/sec
Q ^"1 / r n
— , flux (entrainment) ratio |_-J
0 , density deficit ratio [-]
s Coordinate along beach [ft]
u Average discharge velocity [fps]
U. Average flow velocity after mixing [fps]
u~ Average cold water flow velocity [fps]
-------
w,j Rise of water surface between end of mixing zone and outlet [ft]
x Horizontal coordinate, starting at outlet cross section [ft]
2 Vertical coordinate, starting at water surface [ft]
a , a- , ct Energy flux correction coefficients
/} , /8 , ft Momentum flux correction coefficients
y Slope of "beach
AE E - E.., energy loss in mixing zone
Pp - P , density differential-between warm and cold water at
point of discharge [slugs/ft^]
P2 -^*J\ » density differential between warm and cold water
after mixing [slugs/ft-3]
Q Initial (discharge) warm water density [slugs/ft^]
Final warm water density after mixing [slugs/ft^]
2 Cold water density [slugs/ft3]
(s) Shear stress along beach [ib/ft ]
Il-ix
-------
SECTION I
CONCLUSIONS
Theoretical analysis and experimental verification of vertical mixing
near a two-dimensional, heated water surface outlet were made. As
the heated water is discharged on top of cold water it may entrain
and become mixed with cold water near the outlet. If sufficient depth
in the receiving reservoir is provided, a stratified flow will result.
The following conclusions were reached:
1. The type of flow near the outlet and the amount of entrainment depend
on three groups of parameters:
a. The outlet flow conditions and in particular the outlet densi-
metric Froude number
b. The control of the stratified flow from downstream, in particular
the depth or thickness of the warm water layer
c. The geometry of the transition from the outlet channel to the
reservoir, in particular the depth of the reservoir and
the slope of the beach.
2. The mixing zone near the outlet may be a buoyant half-jet followed
by an internal hydraulic jump, a free internal hydraulic jump or a
submerged outlet flow as defined in Ref. [2]^ The conditions for
the occurrence of these flows and the associated entrainment ratios
have been derived and are shown in Pigs. i|.a, Ijb, and i|.c and 5a» 5b»
and 5c for a vertical beach.
3. Theoretical results were derived for a vertical beach face (90° slope)
and agree reasonably well with measurements. There is experimental
evidence that the results are insensitive to changes in slope angle
if the angle varies in the range 23 < a < 90°.
k» A flat beach of the order of 5 or less was observed to inhibit the
formation of any surface jet or internal jump and to prevent any
mixing.
5. A free mixing internal hydraulic jump requires an outlet densimetric
Proude number F ' > 2.25 or E, > 2.73 if the reservoir has in-
finite depth. As" the reservoir aepth is reduced these limiting
values shift toward 1.0. A free internal mixing hydraulic jump is
characterized by the separation of the interface (thermocline)
from the bottom of the outlet channel and the existence of an
internal inverted roller.
*
References for Part II are listed on page 11-55-
II-l
-------
6. A limiting densimetric Frou.de number for the onset of submergence of
the outlet was established and called F ' . It was shown to "be a
function of the dimensionless warm water Septh H^ and the cold
water depth H? (Figs. i|a, i*b, and he and Figs. £a, 5>b, and £c).
If an actual F ' is less than F* , the outlet will be internally
. , o om
submerged.
7. Theoretical analysis predicts that the dimensionless downstream
depth H1 of the free mixing internal hydraulic jump has an upper
and a lower bound. The upper bound is associated with F'omf ^e
lower bound with Q = 1.
8. The rate of entrainment of cold water by the warm water was found
to be very sensitive to changes in both the outlet densimetric Froude
number F and the downstream depth E^ , as illustrated by Fig. 18.
The achievement of prescribed entrainment rates will therefore re-
quire accurate operational controls.
9. Changes in water surface elevation over the length of the mixing
zone are quite small. They are, however, essential in the control
of buoyancy effects. Consequently, the assumption of a horizontal
water surface cannot be accepted in the analysis of the phenomenon.
10. Theoretically it is possible to entrain any amount of cold water,
but Figs. 5a, £b, and 5c indicate that dilution ratios beyond Q = 2
(one part warm water to one part cold water) may not be very practi-
cal.
11. Lengths of the mixing zone for non-submerged outlets were measured
and reported in Fig. 19. In first approximation the length changes
linearly with the densimetric outlet Froude number F^ .
12. Turbulent entrainment and mixing by the buoyant jet-internal hydraulic
jump mechanism requires considerably less energy than the use of
conventional diffuser arrangements.
II-2
-------
SECTION II
HECOMMENDATIOM3
It is recommended that the results presented herein "be considered for
application to large-scale heated water mixing "basins and cooling ponds.
The internal hydraulic jump appears to "be a mechanism which can be pro-
duced easily. It is associated with substantial mixing between warm
and cold water resulting in lower discharge water temperatures. Ad-
mittedly the rate of entrainment is quite sensitive to changes in the
flow conditions, but adequate control of the outlet flow should be
possible.
It is also recommended that the study undertaken for a two-dimensional
flow herein be extended to the three-dimensional situation.
II-3
-------
SECTION III
INTRODUCTION
One of the many problems associated with "thermal pollution control" is
the rate of dilution of heated water discharges "by turbulent mixing
with ambient water. Accurate water temperature predictions in the^
vicinity of cooling water outfalls of power generating plants require
among other things an understanding of the flow and mixing processes which
occur at or near the source of discharge. Frequently the outlet is an
open channel discharging the heated water at the surface of the receiving
body of water as shown in Pig. 1 . Because such a discharge has not only
momentum but also buoyancy, a variety of flows associated with differing
amounts of entrainment and mixing may occur. A classification of these
flows for a two-dimensional situation has been given in Refs.LlJ and [2J.
The classification is similar to that given for water surface profiles
in textbooks for open channel flow. It hinges on the recognition of the
fact that the flow pattern produced by a two-dimensional heated water
surface discharge is controlled by the discharge characteristics at the
outlet as well as by the downstream flow conditions. Downstream flow
control is effective and determines, for example, the maximum thickness
of the heated water layer floating on top of the cold water. This down-
stream control is physically imposed either by a structure (i.e. gate
or weir), by a particular geometry of the channel (i.e.. sudden channel
expansion), or, in the absence of all of these, by the cooling process
on the water surface. Near the outlet the upstream controlled flow from
the channel joins the downstream controlled internal flow in the receiv-
ing body of water. It is at this transition that a variety of flows are
observed. A classification and explanation of such two-dimensional patterns
was given in Ref. [2] under the assumption of a constant flow rate in the
heated water layer. This is well justified for some types of discharge
but not for others. A discussion by Wilkinson L3J extended the original
paper but excluded free surface effects and finite depths of the receiv-
ing reservoir.
This study is concerned with the vertical entrainment and mixing between
warm and cold water which occurs in the transition region near the out-
let , particularly when the flow resembles that of a buoyant turbulent
jet or an internal hydraulic jump. Experiments show that in the two-
dimensional situation the initial jet-type flow from the outlet channel
usually produces some sort of internal roller at the interface. That
portion of the flow associated with the roller is called an irternal
mixing hydraulic jump. Between the outlet channel and the roller the
flow is strictly a jet flow (half- jet). If the upstream toe of the
internal roller reaches the outlet channel, the outlet begins to be
internally submerged as schematically shown in Pig. 2. For more Details
on the flow patterns described the reader may refer to Ref a LU and L^J«
The internal hydraulic jump was previously analyzed by Yih and Guha
under the assumption that there is no entrainment across the interface
separating the two layers. The momentum equations of the two-layered,
II-
-------
n
H
Discharge
canal
Reservoir
HEATED WATER
Thermo
cline
COLD WATER
Pb)
Fig. A - Definition Sketch - Two-dimensional, Heated Water Surface Discharge
-------
H
M
I
r=9o
hr
Thermocline
1
hz
Fig,, 2 - Definition Sketch - Internally Submerged Outlet Channel
-------
non-miscible flow system were derived and used to demonstrate the exis-
tence of several solutions. A good correlation between experimental data
and theoretical predictions for cases with one layer in motion and the
other at rest was demonstrated. Hayakawa [0] gave a mathematical refine-
ment of the solutions relating to their physical interpretation.
In the case of hot water flowing on top of cold water a certain amount
of flow entrainment exists from one layer to another across the inter-
face, resulting in a more complicated situation than was investigated
by Yih and Guha [li] . A more recent short theoretical discussion on the
bounds of possible downstream depths of mixing internal hydraulic jumps
was given by Wilkinson [3]•
In this study the one-dimensional momentum equation for the two-layered
transition region from the channel to the downstream stratified flow is
given. Theoretical analysis of this equation includes the internal
hydraulic jump and the buoyant jet type flow. Laboratory measurements
of the vertical entrainment are compared with the theoretical predictions*
II-8
-------
SECTION 17
BASIC EQUATIONS
The geometrical boundary conditions of the flow problem investigated
are shown in Fig. 1. The flow is from an outlet channel into a reservoir,
or more precisely into a surface layer (epilimnion) which must form as
heated water continues to be discharged under steady flow conditions.
The epilimnion is separated from the colder reservoir water by a ther-
mocline which will be replaced by a stable interface in the ensuing cal-
culations. The situation is thus approximated by a two-layered flow.
Between the outlet and the stable thermocline there is frequently a
mixing zone through which colder water is entrained from beneath. Fig. 3
shows a schematic streamline pattern. Since the flow in both layers is
usually a turbulent one, the entrained cold water is rapidly mixed with
the upper layer,
The flow in the entrainment zone can be considered either as a buoyant
surface jet or as an internal mixing hydraulic jump. Submerged jets and
hydraulic jumps can be analyzed using the impulse-momentum theorem. The
problem is essentially a hydrodynamic one. Only convective turbulent
heat transfer is considered. Conduction, radiation, and evaporation
effects are ignored because of the usually short residence time the water
is in the mixing zone.
The continuity, momentum, and energy equations can be written for a
control volume limited by the following lines: a vertical cross-section
through the outlet channel upstream, a vertical cross-section through
both layers in the reservoir a short distance downstream from the mixing
zone, the free water surface, and the bottoms of channel and reservoir.
The steady flow continuity equation is
where P , p7, and PT designate the mean densities and qQ , q1 , and
IP designate -che flow rates in the outlet cross section, the warm water
layer and the cold water layer downstream, respectively.
The volumetric flow rates are
-o
10 4C
•I'/l
(2a)
dz =
II-9
-------
Outlet
channel
M
I
Surface
Heated Water
Thermocline
Cold Water
Bottom
Fig. 3 - Schematic Streamline Pattern Near Outlet
-------
^H-.
/2 1 __
Ug dz = Ug h2 (2o)
u , u, , and u~ designate velocities and h , h. - w, , and h_ designate
depths at the outlet and in the warm and cold water layers, respectively.
Since the flow is considered incompressible, the volumetric flow rates
give
(3)
From the previous equations a density deficiency flux equation can "be
derived:
where AP = P - P and
O C 0 \
The impulse-momentum equation is
(h1 - V - *o
Ps + Po + P1
irtiere /8 , ^9 , and /8 are momentum correction coefficients; P , PQ,
and P1 , are static pressure forces ; and F is a force due to shear
stress along the sloping beach. Further difinit ions are:
£
P = Biny/p(s) ds (6a)
s /
T)
where p(s) is the static pressure distribution on the sloping beach
&
Fa = cos y |T(S) ds (6b)
Jo
where r(s) is the shear stress distribution on the sloping beach,
P = lgph2 (60)
-------
P1 = - \ g P1 (h1 - Wl )2 - g P1 (h1 - Wl) h2 - 1 e P2 h22 (6d)
(Ehe energy fluxes into or out of the control volume are
Eo = I «o po
+ P h1 - V - 2 * P1
2 1
U1
E2 =
(7o)
Ihe water stirface at the outlet was taken as the datum level.
Since the flow within the control volume is highly turbulent, an energy
loss of substantial magnitude must be expected. It can be defined as
AE = EQ = E2 - EI (8)
but its value cannot be predicted a, priori .
The control valume as used in the above equations comprises both the
heated water and the cold water layers.
11-12
-------
SECTION V
THEORETICAL EESTJLTS
The set of equations derived in the previous section cannot be solved,
for example to find the unknown downsteam flow rate q^, without infor-
mation on the static pressure and shear stress distributions along the
sloping beach. Such information, it appears, requires the solution of
the equations of motion for the described boundary conditions. Since
the flow within the control volume is a highly turbulent shear flow,
partly affected by buoyancy, a great many assumptions are necessary to
find T(S) and p(s).
Instead of proceeding in this direction it was decided to examine two
special cases for which the static pressures and shear stresses on the
beach could be found more easily: very steep and very flat beaches.
Steep Beach
On a 90° beach the shear stresses caused by the entrained cold water flow
are likely to be small and directed vertically. As the slope angle
goes from zero to 90 » "the shear force F in the momentum equation
therefore goes to zero. At the same time the static pressure force P
is likely to become a nearly hydrostatic force, at least for the flow
schematically shown in Pig. 1 , because flow velocities along the beach
slope become very small.
PQ h0 + 1 g J~ (^ + h2 - ho)] (hl + h2 - ho) (9)
With these simplifications, substitution of Eqs. (2a), (2b), and (2c) in
the momentum equation (5) yields the relationship
O + 7 g po(hi - h )(h1 - h0 + 2hJ - 1 g P.
2 21 o1 o 2 1
+ 2h2) (10)
stream
a given set of outflow conditions (q , h , p ) and a specific down-
am control (h.,, h2) Eqs. (3)j_(U)» arid (TO) Still contain four un-
knowns, namely ql, qf , w1 and P* . The conservation of volume and mass
equations (Eqs. 3 and U) may be used to remove q2 and P.. from Eq. (10),
11-13
-------
"but one additional condition is needed for a complete solution.
relationship grows out of the following considerations.
This
The buoyant surface jet is in many respects similar to the turbulent,
fully submerged non-buoyant half-jet, but there are also two significant
differences: the presence of a free water surface and buoyant forces.
The non-buoyant turbulent submerged jet is generally assumed to develop
in a hydrostatic pressure field and the flow leaves the pressure field
unaffected. It is believed that this hypothesis, well supported by ex-
perimental evidence, also applies to the buoyant surface half-jet dis-
charged over a steep beach. This means essentially that the static
pressure at all points located in a horizontal plane in the cold water
layer must be the same. The following equation is then obtained:
w.,
or
F2 ^
(11)
when Eq. (Ij.) is substituted.
Using Eq. (11), Eq. (10) was reduced to the following dimensionless forms
i2
1 -r)[(Hl-r)2 -
- r)]
(l -r)H2-
- r)'
(12)
in which F1 is the densimetric Froude number defined as
o
1
, r 2
and q = <1.|/<10» H.J = ^/hQ, H2 = h2/hQ and r = A^O/P2 • Because the
free water surface is nearly horizontal, it is tempting to assume
w.. = 0, making the use of Eq. (11) superfluous. This alternate solution '
-------
will also be pursued, although it is not likely to yield as good a re-
sult as Eq. (12). When Eq. (10) is reduced to the dimensionless form
under the assumption of a horizontal water surface or w., = 0 one
obtains
H., [H.J + 2(H2 -
2Q[{(Q-02 - 9- (1-r) H2| EL +§1 H, Q(Q-r)]
P2 i P2 *
Conditions which satisfy this equation will be compared with those
which satisfy Eq. (12)« The density differential r is a small posi-,
tive quantity within the context of this work. It is of the order 10 J
for a temperature difference of £ ^10 F. H and EL are positive quan-
tities. Because the flow under consideration is two-dimensional, !L
is expected to exceed unity. The ratio Q of flow rates in the warm
layer, measured downstream and upstream from the internal jump, is equal
to or greater than unity by definition of the problem.
Figs. Ua, l|.b, and I+c show plottings of EL versus F ' as obtained
from Eq. (12) for three reservoir depths EL = GO, 8 and 1, respectively.
In Figs. £a, £b, and £c the same results are displayed in the form of
Q, versus F1 for the same values of EL, respectively. Comparison of
Figs, lia and0Ub (or Figs. !?a and £b) indicates that the flow is not
very sensitive to a change in Hg (the dimensionless cold water layer
thickness) as long as EL is considerably larger than one. Tinder this
condition an increase in EL produces a small increase in the rate of
entrainment Q. B = & = B0 = 1»® was assumed in the numerical evaluation.
o 1 *-
Figs. I+a, i^b, and 1+c further indicate that for a given F1 there exist
two EL values that belong to the same Q. The two solutions have
different physical meaning. The larger of the two EL values refers
to the internal hydraulic jump, the smaller to a buoyant jet. This can
"be seen by examination of the lines representing constant downstream
densimetric Froude numbers, Fj , which are defined as
•=f *(*, - "-,
and which are also plotted in Figs. Ua, Ijto, and lie. It can be shown
that the minimum value of F1 for any curve Q = const in Figs, l^a,
i|b, and lie corresponds to a downstream densimetric Froude number F^
equal to unity, which by analogy to the ordinary open channel hydraulic
jump may be considered as representative of an internally critical flow
-------
H
H,
Fig« 4a - H, versus F for H~= oo
-------
H,
F.'
Fig. 4b - H, versus F ' for H0 = 8
I o 2
-------
H
H
-L
H,
Fig. 4c - H^ versus F for H2=
-------
M
H
Fig. 5a - Q versus F for H2 - oo
-------
H
H
8
12
16
20
Fig. 5b - Q versus F ' for H0= 8
° o ^
-------
K-IO
I I
o
8
Fo'
12
16
20
Fig. 5c - Q versus F ' for
-------
condition. The connection of all minimum values of F for different Q,
values is identical to the line F.' = 1 and represents a separation line
between internal hydraulic jumps and surface jets. Considering the two
possible values of IL for a given set of values of Q, F1 and IL,
one value of IL always represents a flow with a downstream densimetric
Froude number F! > 1 and is a buoyant (internally supercritical)half-
jet. The other value of |H1 represents a flow with a downstream den-
simetric Froude number F. < 1 and is an internal hydraulic jump.
A surface buoyant jet is an internally supercritical flow and uniquely
controlled by the source (outlet) conditions. The internal hydraulic
jump is an internally supercritical flow on the upstream side and in-
ternally subcritical on the downstream side. In that case, depth IL
is controlled from downstream. It appears that under steady flow condi-
tions the two-dimensional warm water surface flow will always have a
downstream control and therefore an internal hydraulic jump must occur
somewhere downstream from the outlet.
Figures i^,, i|b, and i^c show furthermore that for constant entrainment,
high values of F' are associated with lower values of IL. For an in-
finitely large outlet densimetric Froude number F' , asymptotic re-
lationships between IL and Q were obtained for different H? values.
Fig. 6 shows the results and in particular the special solution
Q = JnT (1£)
if the reservoir is infinitely deep (H2 — *>OD). This is precisely the
relationship arrived at by Albert son et al [6] for non-buoyant, two-
dimensional submerged turbulent jets discharged into an infinite fluid
medium. Fig. 6 substantiates the assertion that the lower branches of
the IL versus F1 curves in Figs. Ija, i|b, and l±c represent a jet-
type flow.
To the left of the line F* = 1 in Figs, ita, I4.b, and l±c, two particular
lines are of special importance. One is the line representing Q = 1
which is given as
,o (H- - r)
2F 2 = -1 - - (E + 1 - 2r) (16)
(1 - r)
Eq. (16) represents the non-mixing internal hydraulic jump and is essen-
tially identical to the relationship given by Yih and Guha [ij.
The other line is an envelope of the family of IL versus F1 curves which
gives the minimum densimetric Froude number, F , for a given IL .
The curve connecting F values is also plottiS in Figs. £a, 5b, and
5om
c.
11-22
-------
0 2
H,
8 10
Fig. 6 - Q for infinitely large FQ as a function of H, and r
n-23
-------
It is thought that mixing internal hydraulic jumps require larger down-
stream depths H, than non-mixing ones for the same P ' and H values.
The difference is of course in the entrainment of fluid? Therefore, it
is believed that the mixing internal hydraulic jump is seen in the region
bounded by the lines Q = 1 and F '. In this region, however, there
exist two Q values for every given pair of H, and F ' values. Pigs.
5a and 5b show this even more clearly. Considering the ihape of the
curves of E^ = constant in Pigs. 5a, 5b, and 5c, it can be seen that
the smaller Q value approaches Q = 1 as P ' is increased, while the
larger one grows to a finite value which is given by the non-buoyant
mixing jet theory. It is thought that only the smaller Q value is
physically meaningful, but no proof has been established.
To the left of the P ' line in Pigs, i^a and 1+b is a region where the
submerged internal hyaraulic jumps should be expected to occur. In the
case of the submerged internal hydraulic jump the degree of submergence
(h^ in Fig. 2) is a new and unknown variable which must be included in
the momentum equations. Consequently these equations cannot be solved
without an additional relationship.
Energy losses associated with the entrainment and mixing process have
been evaluated using Eqs. (?a), (?b), (?c), and (8).
AE "2 (6-1)5 2
_
Eo " °o (l-r)H22 % (l-r)Fo'2Q ao
"o
Results are given in Pigs. 7a and 7b for values of H = o> and 8 which
show a minimum positive value of the energy loss at the conjectured transi
tion F, ' = 1; a = a, = ap = 1.0 has been assumed.
The slope of the free water surface was found to be upward in the flow
direction, as indicated in Pig. 1, and small. A plot of the dimensionless
-w
water surface elevation r- — is given in Pigs. 8a and 8b. A strong
o
dependence of the results on the thickness of the upper layer, H, , is
apparent .
Pigs. 9 and 10 give results of H, and Q, respectively, versus P '
using Eq.. (lj) instead of Eq,. (127. Eq,. (lj) implies that the water
surface is horizontal. Comparison of Pigs. 9 and 10 with Pigs. 1+ and 5
-------
Fig. 7a - A^/EQ versus FQ for H2 = oo
11-25
-------
LO
.8
.6
EC
4
.2
8 10
Fo'
Fig. 7b - AE/E versus F ' for H0 = 8
n-26
-------
12
"wl
Fig. 8a - Dimensionless water surface elevation —r-— versus F ' for H0 = oo
0 n
11-27
-------
F.'
Fig. 8b - Dimensionless water surface elevation -^— versus FQ' for
11-28
-------
M
M
F.'
Fig. 9 - H| versus FQ* for H2 = 8 under the assumption w, = 0
-------
H
H
CO
8
12
16
R.'
Fig. 10 - Q versus F for W~- 8 under the assumption w-i = 0
-------
reveals large differences in results, particularly in the lower Froude
number range. Eq_. (13) does not give Yih and Guha's internal hydraulic
jump equations when Q is set equal to unity. It may be concluded that
the solutions in Figs. 9 and- 10 are valid and useful only when the cru-
cial assumption w^ = 0 is physically imposed by a solid wall such as
proposed in Ref. [3]. w. = 0 is not a valid assumption for the free
surface flow under consideration.
Flat Beach
The two-dimensional flow over a flat beach is very similar to that in a
diffuser. "Plat" in this context means that there is no separation of
the streamlines at the outlet. Further downstream the flow will separate
from the beach smoothly because of its buoyancy. This will occur at a
depth usually established by downstream conditions. Fig. 11 shows sche-
matically the flow over a flat beach. The entrainment can be expected
to be zero or near zero; experimental evidence to this effect will be
presented later. Because of this, there was relatively little reason
to investigate this flow. The only unknown quantity of interest is the
slope of the water surface, and its value was not a prime objective. If
it is to be calculated, nevertheless, one can use the momentum equation
and a constant flow rate in the upper layer. The pressure distributions
are hydrostatic, and the shear stress along the beach can be found from
boundary layer theory or from one of the semi-empirical open channel
flow equations.
11-31
-------
Channel
Surface
wd
M
CO
ro
1
^^^^^^^^^^^^^ 1
1
h
i
— j
q
ill _
i i
E
hd
Thermo
Bottom
Fig. 11 - Schematic representation of outflow over flat beach
-------
SECTION VI
EXPERIMEETAL FACILITY
For the experimental study a 6 in. wide, 1f? in. deep, 1^0 ft long glass
walled channel was used. The experimental apparatus is shown schemati-
cally in Fig. 12. Heated water was discharged horizontally at the
surface. The downstream depth (withdrawal), the heated water discharge
rate, the temperature of the heated water, and the temperature of the
cold water supply were variable. The flow of the warm water on top of
the cold water could be observed over a distance of 13 ft with the aid
of minute quantities of dissolved potassium permanganate. Vertical
temperature profiles were recorded with a thermistor temperature probe.
All measurements were taken in the centerline of the flume. Downstream
from the outlet the heated water layer was withdrawn selectively by
adjusting the withdrawal plate and the flow rate. The withdrawal flow
was metered.
The channel bottom and the flume walls were hydraulically smooth. The
downstream gate was adjustable and allowed variation of the inflow depth
h. . Heated water temperatures were only a few degrees Fahrenheit above
equilibrium temperature—that is, the water temperature at which the net
heat exchange through the water surface was zero. Surface heat loss
effects on the temperature were small compared to mixing effects mainly
because of short residence time.
II-33
-------
worm
infl
i
H
H
fc
water dei
OW gc
i i
!!
1
1
1
i
pth depth
ige tempera! i
1 .'
v
—
/ *
1 V . 7- .
^S^^--^
i and upper layer cold wate
jre gage withdrawal inflow
, a ft
-4—0 //
1
_J
>^
« II1.. - ,., .- .^^ A* •. ^ C1 - to
.
i! t
L— A'-J
Fig. 12 - Experimental Apparatus
-------
SECTION VII
EXPERIMENTAL RESULTS
Figs. 13a through 13d show several typical flow patterns for a 90° beach.
The first example in Pig. 13a is a typical case of buoyant surface jet
type mixing. The spread in depth is nearly linear over the first foot
or so. Then buoyancy forces begin to bend the interface upward. The
second example in Fig. 13b is a typical internal hydraulic jump pattern.
The flow along the interface is directed toward the outlet, at least
over the first 2 ft.
An example of less violent mixing and internal submergence of the outlet
is given in Fig. 13c. It can be seen, although not very clearly, that
the discharged heated water spreads jet-like until it reaches the inter-
face. The flow is not exactly a jet, however, because there is a roller
on its lower boundary. The roller causes some entrainment of cold water
and also generates an eddy beneath the outlet. Fig. 1i|. shows the asso-
ciated streamline pattern schematically. The last example in Fig. 13d
shows a strong stratification at the outlet. At a short distance from
the outlet, the interface becomes unstable and breaks. A not very well
defined kind of hydraulic jump follows further downstream. The outlet
is slightly submerged. Fig. 13d illustrates the complexity of some flow
and mixing patterns at a heated water surface outlet and the problem of
interfacial mixing by waves, which .will not be pursued further in this
study.
Figs. I5a through l£d are temperature profiles which correspond to
Figs. 13a through 13d, respectively. All profiles show the well-mixed
character (uniform temperature) of the upper layer. Temperature profiles
(shown in Figs. 1$c and l£d) which are close to the outlet also demon-
strate the existence of an intermediate layer between the upper warm
layer and the lower cold layer. This intermediate layer is a consequence
of the submergence of the outlet as can be seen in Figs. 13c and 13d.
The flow down a flat beach is much less spectacular than for a steep
beach, as can be seen in Fig. 16. No measurable amount of entrainment
was detected. The separation from the beach is very smooth.
Figs. 17 and 18 show different kinds of plottings of experimental data
for a 90° beach. The dimensionless depth of the lower layer Eg in the
experimental channel ranged from 3*5 to 13.$. Theoretical values for
H_ = 8 were chosen for comparison. All experimental data points lie
aiove the line Q = 1 (Eq. 16) because of entrainment into the moving
upper layer. Fig. 1? also displays values of F , the minimum F
value that is expected for a free mixing interxiaThydraulic jump, for
H =8. Flow conditions to the left of that line should represent in-
tlrnally submerged outlets. The experimental data of the observed free
hydraulic jumps were generally found on or below the FQm line and
above the Q = 1 line.
11-35
-------
M
M
Fig. 13a - Photograph of buoyant surface jet flow downstream from outlet channel.
Flow is from left to right. Heated water is dyed and appears darker than
cold water. F ' = 11.0, H, = 19.0, H- = 5.7, Q= 1.58
o *
-------
Fig. 13b
- Photograph of internal hydraulic jump downstream from outlet.
FQ' = 4.0, H1 = 5.6, H2 = 6.6, Q = 1.20
-------
Fiq. 13c - Photograph of internally submerged outlet flow.
F1 =3.4, H, = 6,\r H9= 5.0, Q = U06
-------
H
H
CO
Fig, 13d - Photograph of breaking of internal waves near the outlet,
Fo' = 1.8, H, = 2.9, H2 = 4.6, Q = 1.08
-------
Channel
M
M
Warm water
Cold water
Fig« 14 - Schematic streamline pattern near internally submerged outlet
-------
H
H
Temperoture
50 70
50
70
50
70
X=0
X=3.25'
50
70
Fig* 15a - Selected vertical temperature profiles associated with flow of Fig. 1 3a
-------
Temperature °F
50 70 50
E 0
o
o
.a
o
o
^o
"o
Z
S. .6
0)
O
X=0'
50 70
= 21'
X=I2.6'
. \5b - SeVected vertical temperature profiles associated with flow of Fig. 13b
-------
Temperature °F
50 70 50 70
50
M
Visible
interface
50 70
X*I2.6'
Fig. 15c - Selected vertical temperature profiles associated with flow of Fig. 13c
-------
Temperature °F
50 70 50
70
77/77
50 70 50 70
= I26'
Fig,, A5d - Se\ected vertical temperature profiles associated with flow of Fig0 13d
-------
M
ft
Fig. 16 - Photograph of flow over a flat beach
-------
25t
20
H,
M
10
, Q=l
(Theoretical)
Experimental Q
O I -I.I
• 1.1 -1.2
A 1.2-1.3
A 1.3—1.4
D 1.4-1.5
• 15-1.6
8
10
12
14
Fig0 17 - Experimental and theorefical data of H, versus F ' «
(A. flag on a symbol designates an internally submerged
outlet. The present theory does not apply to this situation,)
-------
M
H
Experimental H,
O 0—3
3—6
6—9
12
A
A 9
D 12—15
• 15—18
O 18 — 21
14
Fig0 18 - Experimental and theoretical data of Q versus F ' »
o
(A flag on a symbol designates an internally submerged
outlet,, The present theory does not apply to this situation.)
-------
The rate of ent r aliment , measured as Q, is shown in Pig. 18 as a func-
tion of F '. All data points lie below the P ' line for H = 8.
Theoretical predictions of flow conditions in terms of R. and P , for
H = 8 are also shown in Pig. 18. Good agreement with observed experi-
mental data is found. Pig. 18 reveals that Q, changes rapidly as either
P ' or H, is changed while the other parameter is fixed. Because of
tnis high sensitivity of Q values to the parameters P ' and E. it
may be difficult to accurately control Q in practical cases. Submerged
internal hydraulic jump cases, as can be seen in Pigs. 17 and 18, are
observed to have larger H, and smaller Q, values.
Pig. 19 shows the dimensionless length for the internal mixing zone.
Actual lengths, -0, of the internal mixing zone could only be measured
with very low accuracy and were frequently close to three feet.
=10F' (10)
may serve as a first approximation of the results.
Given a choice between different P ' values for given H, , a maximum
of mixing will occur when P ' = P ' . This condition also represents
the limits between internally free and submerged outlet conditions, as
noted earlier.
-------
IOO
O
80
60
40
20
cr
Experimental
H,
O 0-
• 3
A 6
A 9
O 18
21
3
6
9
12
•21
24
8
10
12
Fig, 19 - Dimensionless length of outlet mixing zone versus F '
(A flag on a symbol designates internally submerged
outlet flowe)
-------
SECTION VIII
APPLICATION OP THEORETICAL AND EXPERIMENTAL RESTJLTS
The results given in the previous sections may find application in the
design of mixing chambers and prediction of outfall temperatures. In
this study the flow conditions at the outlet and the depth of the warm
water layer downstream from the transition and mixing zone have been
treated as independent variables. This requires that no cold water en-
croaches into the outlet channel, a condition which is easily met at
high outlet densimetric Froude numbers, in particular if F ' > 1. The
downstream depth H-, is, however, not completely independent of the out-
let flow. The thicKness of the warm water layer H, depends on the type
of control exerted from downstream as explained in the introduction. It
also depends on the flow rate after mixing, Q.., and hence on the entrain-
ment rate itself. Application of the results derived in the previous
section will therefore usually require simultaneous consideration of the
type of downstream control and entrainment. This point has been clearly
made in several references, e.g. [l,2,j]. It has also been shown in
those references how the depth H, can be found if the downstream
control is a weir [2], or a submerged sluice gate [5], or the surface
cooling process [lj. Other types of downstream controls could be
investigated in a similar manner.
The solution of the entraining jump problem for a given outlet condition
as investigated in this study and the solution of the stratified flow
problem for a given type of downstream control must match where the two
flow regions join (cross section 1 in Pig. l). The matching has to be
done essentially in terms of the depth H^ and the flow rate Q at the
junction of the two flow regions. This will usually require a trial and
error process, but will not impose large difficulties if the necessary
separate computations for each zone are carried out numerically or if a
set of solutions for each zone is given in graphical form. Figs. l*a, ljb,
Uc, 5a, 5b, and 5c may serve as solutions for the entraining jump or
outlet mixing region.
The interaction between the outlet entrainment problem and the downstream
stratified flow problem suggests that the selection of a desired amount
of mixing is possible only if the downstream control is variable. En-
trainment is most effectively increased by increasing the downstream depth.
A maximum entrainment and mixing rate is achieved at the point of begin-
ning submergence of the outlet. Actual ^ submergence will substantially
reduce the amount of entrainment as indicated by experimental data. There
will be no entrainment (Q = l) if PQ' < 2.25 and the reservoir is
infinitely deep.
A reduction of the depth H of the receiving channel adversely affects
entrainment, which is important in the design of mixing chambers. The
reduction in entrainment is significant when E^ takes values of less
than 8.
11-51
-------
The effect of beach slope on entrainment was investigated only experi-
mentally. Results obtained with a = 23 slope angle differed little
from those obtained for a = 90 • Results presented in this study can
therefore be applied with some confidence to beach slopes in the range
23 < a < 90 . At a = 5-75 "the beach effect was clearly to produce
a flow similar to that in a diffuser and to suppress all mixing.
11-52
-------
SECTION IX
ACKNOWLEDGMENTS
The authors wish to acknowledge the assistance of Dipl. Ing. w". Geiger
in the preparation of the experimental apparatus.
11-55
-------
SECTION X
REFERENCES
[l] Koh, R. C. Y. "Two-dimensional Surface Warm Jets," Journal of the
Hydraulics Division, Proc. of ASCE, Vol 97, No. HT6, 1971, pp 819-
836.
[2] Stefan, H. "Stratification of Plow from Channel into Deep Lake,"
Journal of Hydraulics Division, Proc. of ASCE, Vol 96, No. HY7,
1970, pp 1U17-143U-
[3] Wilkinson, D. L. Discussion of "Stratification of Flow from Channel
into Deep Lake," Journal of Hydraulics Division. Proc. of ASCE,
Vol 97, 1971, pp 602-609.
[k\ Yih, C. S. and Guha, C. R. "Hydraulic Jump in a Fluid System of
Two Layers," Tellus. Vol 7, No. 3, 19#» PP 358-366.
[f>] Hayakawa, N. "Internal Hydraulic Jump in Co-Current Stratified
Flow," Journal of the Engineering Mechanics Division. Proc. of ASCE,
Vol 96, No. EM5, 1970, pp 797-800.
[6] Albertson, M. L.; Dai, Y. B.; Jensen, R. A.; and Rouse, H. "Diffu-
sion of Submerged Jets," Transactions of ASCE. Paper No. 2^09,
Vol 115, 1950.
H-55
-------
Part III: Field Measurements in a Three-Dimensional
Jet-Type Surface Plume
H. Stefan
and
. R. Schiebe
December 1971
-------
ABSTRACT
Field tests were conducted to measure temperature,
velocity, and current direction with depth at
several locations in the cooling water plume of the
Allen S. King power generating plant on Lake St.
Croix. The results of the study compare favorably
with data from laboratory and analytical models
operated under similarity criteria, specifically
geometry and densimetric Proude number similarity.
-------
CONTENTS
Section
List of Figures Ill-vi
List of Tables III-vii
List of Symbols and Units Ill-ix
I CONCLUSIONS III-l
II RECOMMENDATIONS '. III-5
III INTRODUCTION III-5
IV THE SITE III-7
V HYDROLOGICAL AND METEOROLOGICAL CONDITIONS III-9
VI INSTRUMENTATION III-ll
VII RESULTS 111-13
VIII SUMMARY 111-51
IX ACKNOWLEDGMENTS 111-33
X
III-V
-------
LIST OF FIGURES
PAGE
1 COOLING WATER DISCHARGE CHANNEL AT ALLEN S. KING POWER
GENERATING PLANT. LOCATION OF MEASURING STATIONS III-8
2 INSTRUMENT ARRANGEMENT. CURRENT METER (LEFT), THERMISTOR
(CENTER), AND CURRENT DIRECTION VANE (RIGHT) 111-12
3a - Jf PROFILES OF MEASURED VELOCITIES AND TEMPERATURES III-ll| - 111-19
k ISOTHERMS IN VERTICAL CROSS SECTION THROUGH CENTERLINE OF
DISCHARGE CHANNEL 111-20
5a - 5"b ISOTHERMS IN VERTICAL CROSS SECTION THROUGH CENTERLINE
OF DISCHARGE CHANNEL AS DERIVED FROM NSP MEASUREMENTS IN
1969 AND 1970 111-21 - 111-22
6a - 6c VELOCITY COMPONENTS u AND v IN DIRECTION OF
DISCHARGE AND PERPENDICULAR TO IT, RESPECTIVELY 111-21^-111-26
7 RICHARDSON NUMBER DISTRIBUTION WITH DEPTH FOR SELECTED
LOCATIONS 111-29
Ill-vi
-------
LIST OF TABLES
1 Hydrological, Meteorological, and Plant Operating Conditions III-9
2 Summary of Outlet Conditions III-IJ
3 Summary of Conditions within the Plume 111-27
-------
LIST OF SYMBOLS AND UNITS
b Channel width at outlet [ft]
F Local densimetric Froude number based on local absolute
velocity
F* Local densimetric Froude number based on excess velocity
F Outlet densimetric Froude number
o
^o
g Gravitational acceleration [ft sec ]
h Depth of thermocline at outlet [ft]
n Depth constant [ft]
Q Discharge [ft* sec" ]
Ri Local Richardson number
T Temperature [degrees F]
U Average velocity [ft sec ]
7 Excess velocity [ft sec ]
z Depth [ft]
(3
c
Q
tf
Slope of density-temperature curve (3Q/3T) [slugs ft~5 °F~1]
Ratio of inflection depths of temperature and velocity
profiles
Density [ibs sec ft" ]
Depth of inflection of velocity profile in plume [ft]
Ill-ix
-------
SECTION I
CONCLUSIONS
Temperature and velocity distributions resulting from jet-type surface
discharges of heated water into near quiescent lakes appear to be
little affected by the geometrical scale of the flow field. This
implies also that the Reynolds number effect on the flow must be
small. Vertical and horizontal spread of the heated water discharge
depends primarily on the relative magnitudes of buoyant and inertia!
forces as conveniently expressed by densimetric Froude numbers speci-
fied at the outlet. These observations validate the assumptions
made in the analytical model described in Part I of this report.
III-l
-------
SECTION II
RECOMMENDATIONS
To further improve the predictive ability of either analytical or labora-
tory models, more extensive and accurate field measurements using im-
proved techniques should be obtained on the same site. The main objec-
tive of such investigations would be to more accurately evaluate mixing
and entrainment coefficients in both near and far fields of the plume
under a variety of hydrological and meteorological conditions.
The processes which control the time-dependent fluctuations of tempera-
ture, velocity, and other related mixing and spreading parameters are
inadequately understood. Diurnal effects deserve particular attention.
III-3
-------
SECTION III
INTRODUCTION
In order to verify some of the analytical and experimental findings
described in Part I of this report, it appeared desirable to make com-
parisons with field data. Temperature fields near cooling water outlets
have been measured at various sites, and more data are forthcoming.
References [25] through [29] in Part I are examples of such measure-
ments. However, few measurements have been taken in thermal plumes
with low densimetric Froude numbers at the outlet. For this reason,
and in order to develop a better appreciation of prototype problems,
it was considered worthwhile to obtain some field measurements. A
location particularly suitable for these measurements was found at the
cooling water outlet of the A. S. King power generating plant on Lake
St, Croix, approximately JO miles east of Minneapolis. Two surveys
were taken, on June 18 and July 30, 1971-
III-5
-------
SECTION IV
THE SITE
With one unit installed the A. S. King plant has a capacity of 550 MW.
When operating at full capacity the cooling water flow rate is of
the order of 630 cfs. The intake is through a skimmer wall from the
bottom of the lake and the discharge through a canal 10 to 12 ft deep,
approximately 1300 ft long and i|5 ft wide at the bottom, with side
slopes of 3s!• Before entering the lake the channel widens to approxi-
mately li;5 ft at the bottom. The discharge into the lake is at an
angle of nearly 90 degrees to one shoreline and J+5 degrees to the
other as shown in Fig. 1. The lake is actually a natural basin of the
St. Croix River. Near the cooling water outlet the basin is approxi-
mately 3000 ft wide and up to 1*0 ft deep. A major portion of the lake
is 25 to 30 ft deep. The shore is moderately steep with a slope of the
order of 5:1 near the cooling water outlet channel. Figure 1 shows the
channel and shoreline configurations. Because the discharge is from
an exposed point on the shoreline and because the slope of the shore
is not too flat, the heated water discharge forms a buoyant surface
jet very similar to that described in the analytical and experimental
studies.
III-7
-------
M
£
Fig. 1 - Cooling water discharge channel at Allen S. King power generating plant.
Location of measjrine stations
-------
SECTION T
HXDROLOGICAL AMD METEOROLOGICAL CONDITIONS
At the time of the survey the lake cooling water and meteorological con-
ditions were as shown below in Table 1.
Table 1
HYDROLOGICAL, METEOROLOGICAL, AND PLANT OPERATING CONDITIONS
J&ie 18 July 50
Elevation
Surface water temp.
upstream from discharge
Surface flow velocity
upstream from discharge
Cooling water flow rate
Plant load
Cooling water dis-
charge temperature
Solar radiation
Wind velocity at 35 ft
Wind direction
Dry bulb temp
Wet bulb temp
Sky
Duration of survey
Sources of information:
close to 675 ft 675
583
560
80
0 to
,o(2)
o(D
81° to 86'
olear^2'
13:30 - 17:00
(1)
NSF,
(2).
O.J+
560
(1)
to 1*00 MW
79-1° to 80.l4°P
-2
min
1.1 to 1.3 oal cm
k to 7
NNW to
6k° to
57.5° to 58.5°P(2)
clear to 50^6 cover^ '
- 16:00
measurement,
(3)
estimate
III-9
-------
SECTION VI
INSTRUMENTATION
Water temperature, flow velocity, and direction of flow were measured
at selected stations shown in Fig. 1 and depths shown on the graphs
to follow. Locations were determined Toy triangulation with transits
from a base line on shore between points D and G in Fig. 1.
Several reference points, such as the tower Tl, were available.
Depths were measured by distance from the water surface. A Yellow
Spring Tele-Thermometer (thermistor), a Gurley current meter, and a
current direction vane similar to that described in Ref. [l]* were
used to find water temperatures, current magnitudes, and current
directions, respectively. Only horizontal current velocities were
measured.
The instruments were assembled on a triangular base as shown in Fig. 2
and lowered from a boat. The orientation of the instrumentation
assembly was fixed with the aid of a telescope and a solid support
system. The electrical signals of all meters were transmitted to the
water surface, read, and manually recorded.
References for Part III are listed on page III-35-
III-ll
-------
Fvg. 2 - Instrument arrangement. Current meter (left), thermistor (center), and current direction vane (right)
-------
SECTION VII
RESULTS
The measured temperature profiles with depth are shown in Fig. 3. A
strong temperature stratification exists near the outlet which can be
made even more apparent by plotting isotherms in a vertical cross sec-
tion through the centerline of the discharge channel such as is shown
in Fig. 4« For comparison, similar plots derived from much more ex-
tensive measurements carried out by Northern States Power Company (NSP)
in previous years [2] are shown in Fig. 5- Figures 14 and 5 illustrate
the type of stratification which develops when heated water is dis-
charged at very low densimetric Froude numbers. A cold water wedge
penetrates or attempts to penetrate into the outlet channel. The
situation is very similar to the arrested salt water wedge which is
encountered in rivers and channels discharging fresh water into the
sea. Analyses of the salt (cold) water wedge phenomenon are given
in Refs. [3], [L], and [5]. Observations of the phenomenon are re-
ported in Ref. [6J.
A mixing layer (metalimnion) of substantial thickness between the cold
water and the heated water effluents can be observed in Figs. 4 and 5.
This layer is probably caused by entrainment from the upstream toe of
the cold water wedge. Although the outlet flow is not exactly two-
layered, it is of interest to calculate densimetric Froude numbers at
the end of the outlet channel for the field conditions using the depth
from the water surface to the thermocline, h, defined as the inflection
point of the vertical temperature profile, as the characteristic depth.
The results are shown in Table 2.
Table 2
SUMMARY OF OUTLET CONDITIONS
Date
6-18-71
7-30-71
6-18-69
9-4-69
8-13-70
Q
(cfs)
583
560
650
660
62k
h
(ft)
6.
7-
U-3
4.0
3.7
b
(ft)
199-
196.
195.1
205
205.9
T
max
(°:
85
80
81
88
93
F)
•
.8
•
.2
• U
T
min
(°F
79.
70.
68.
72.
79.
0
2
U
5
0
0.
0.
0.
0.
0.
.££
Q
(-)
00111
00139
00159
00237
00242
(
1
0
1
1
1
Po
-)
.06
• 73
.65
• U5
• 50
111-13
-------
0
Vel (fps) (o)
3 .4 5 .6 .7 B 9
•• p ' • T' """ T" i T I
0
2
o.
•
O
8
Position I
84"
79 80 8i 82
Temp (°F) (a)
83
85
Vel (fps) (o)
4
77 78 79
80 81 82
TempCF) (o)
83 84
Fig. 3a - Profiles of measured velocities and temperatures
6-18-71
-------
Vel (fps) (o)
.4 .5 .6
80 81 82
Temp (°F)
.1
.2
Vel (fps) (o)
4 .5 .6 .7 .8 .9
fe 4
a.
o
8
10
Position I3C
77 78 79 80 81 82 83 84 85
Temp l*F) (o)
Fig. 3b - Profiles of measured velocities and temperatures (continued)
6-18-71
111-15
-------
0
2
1 4
H
o 6
8
10
Vel (fps) (o)
.1 .2 .3 .4 .5 .6 .7 .8
Vel
Temp
Position 3C
78 79
0 8 82
T«mp (°F) (o)
83 84 85
Fig. 3c - Profiles of measured velocities and temperatures (continued)
6-18-71
111-16
-------
(fpi) (o)
6
LQ L2
Vel (fps) (e)
T«mp (*F) (o)
70 72 74 76 78 80 82
0 .2 4 .6 .8 1.0 1.2
74 76 78 80 82
0 .2 .4
(fp«) (o)
1.2.
10
12
-4-
Position 4
1.2
70 72 74 76 78 80 82
T.mp (V) (o)
Fig. 3d - Profiles of measured velocities and temperatures (continued)
7-30-71
lll-l?
-------
12
70 72 74 76 78 80
Temp (eF) (o)
70 72
74 76 78
Temp CF) (o)
80 82
,0 .2
Vel (fps) (o)
4 .6 .8 1.0 1.2
CL
o>
O
8
10-
12
Position 7
10
12
Vel (fps) to)
•H .4 6 .8 1.0 1.2
Position 9
70 72 74 76 78 80 82
Temp CF)(a)
70 72
74 76
Temp (•
78 80 82
Fig. 3e - Profiles of measured velocities and temperatures (continued)
7-30-71
111-18
-------
f 6
o.
O
8
10
12
Vel (fp»)(o)
=
Temp)
Position 10
el
70 72
74 76 ' 78
Temp (»F)(o)
80 82
(fps)(o)
0
2
4
<•*
£ 6
0.
o
8
10
12
r
.
f
T Position II
i
1
Temp CF) (o)
82
Fig. 3f - Profiles of measured velocities and temperatures (continued)
7-30-71
111-19
-------
400
800
Distance (ft)
1200 1600
2000
.2400
DISCHARGE CHANNEL
400
800
Distance (ft)
1200 1600
2000
2400
10
•*
*
: l5
L
>
i
20
25
30
40-
DISCHARGE CHANNEL
7-30-71
Fig. 4 - Isotherms in vertical cross section through center!ine of
discharge channel
III-20
-------
400
800
Distance (ft)
I2g0 1600
2000
M
M
M
2400
DISCHARGE CHANNEL
Fig. 5a - Isotherms in vertical cross section through centerline of
discharge channel as derived from NSP measurements in
1969 and 1970
-------
Distance (ft)
400
1000
4000
H
M
H
ro
35U
4OQ
1090
2000
Di*tance (ft)
4000
Fig. 5b - Isotherms in vertical cross section through center! ine af discharge channel as derived from
NSP measurements in 1969 and 1970 (continued)
-------
The densimetric Froude number, F , is defined by
0
(1)
where Q is the heated water discharge rate, b is the width of the
heated water layer at the outlet, h is the depth of the thermocline
below the water surface, and AQ is the density difference between the
warmest and the coldest water in the outlet cross section. Isothermal
patterns at the water surface were not derived from the present experi-
ments because of the small number of measurements. Measurements of these
patterns in previous years have been well documented by Northern States
Power Company in Ref. [2].
The temperature stratification measured is also reflected in the
velocity measurements. Although velocities below 0.10 fps could not
be measured satisfactorily, it is already apparent from the results
in Pig. 3 that the heated water forms a surface shear flow with a
strong velocity gradient near the thermocline.
The resultant velocity vectors were decomposed into components which are
respectively parallel (u) and perpendicular (v) to the discharge
channel for the second survey (7-30-71); u, and v are positive in
easterly and southerly directions, respectively (see Pig. l). ^he re-
sults given in Fig. 6 show strong gradients of both components with
depth. The buoyant spread perpendicular to the main direction of the
flow is evident in the velocity profiles measured at positions 2, 3, and
i^. The plume is injected into a weak cross-current which is essentially
produced by the breeze blowing from the north along the main orientation
of the lake. At positions 2, 3, and 1± there is a reversion of the flow
direction of the v-velocity component with depth. This was also observed
at positions 9 and 10, but not at 7-
Since the v-velocity component is essentially a spread velocity perpendic-
ular to the main direction of the flow, the measurements at stations 2 3
ij., 7, and 9 illustrate the buoyancy-induced lateral spread. The distribu-
tions of the spread velocities with depth are very similar to those found
in laboratory experiments in a tank and reported in Part I of this report.
The data shown in Pig. 3 can also be used to calculate Richardson numbers
at various depths. The velocity and temperature measurements are approxi-
mated by functions of the form
T* = T * exp[- -i_] (2)
*** /^A '"_••" * *
cA,
-------
Position I
M
M
I
Position 2
= 5
10
Position 3
-4 -.2
Vtl (fps)
Position 4
T
0 .2 4
Vtl (fps)
Fig, 6a - Velocity components u and v in direction of discharge and perpendicular to It, respectively
-------
Position 5
10
M
ro
ui
IO
-2.
Vtl (fpt)
Position 6
.2 4 .6 .8 1.0
Vol (fpt)
£ 5
10
-4 -.2
Position 7
Position 9
0 .2
Vel (fps)
.8
Fig. 6b - Velocity components u and v in direction of discharge and perpendicular to it/ respectively
(continued)
-------
Position 10
C 5
a.
o|0
-.4
Position II
-v
l-u
I
0 .2 4
Vel (fp$)
.6 B
Fig. 6c - Velocity components u and v in
direction of discharge and perpendicular
to it, respectively
111-26
-------
.., ...
V X 2*o (W
where the density-temperature relationship in the range under con-
sideration has been assumed, to be linear. In these equations
do/dT = |3 (slugs ft °P" ), T* is the temperature excess above
the cold Jake temperature, V is the excess velocity above the
ambient lake current velocity at large depths below the surface
layer of heated water, subscript m refers to the maximum values of
these variables in a given vertical profile, and Ka and a are
the distances from the water surface to the inflection points of the
temperature and the velocity profiles, respectively. These constants
cannot be determined with great accuracy from the available data.
Nevertheless some crude estimates for three stations are given in
Table J>. Richardson numbers were then computed from Eq.. (j^). The
results obtained were not satisfactory. The lowest Richardson numbers
(ranging from 0.1 to 2.0) were found near the thermocline, while very
high values occurred near the free surface and below the thermocline.
This is an unacceptable result because it would mean that the stability
of the stratification is weakest near the thermocline and stronger above
and below. It can be shown that this result is due to the description
of the temperature and velocity profiles by normal distribution func-
tions. It must therefore be concluded that such functions adequately
describe the overall features of the thermal plume, but are not satis-
factory for the computation of local properties.
Table 5
SIJMMRY OP CONDITIONS WITHIN THE PLUME
* 13 V
m c m
Posi-
tion
2
3
1*
9
10
°F
10
9.2
8.7
l*.o
U.8
OJ.UQB
ft'^sec"1
2.1+8x10'^
2.1*8x10"^
2.1*8x10'^
2.1*8x10"^
2.1*8x10"^
ft sec~
.65
•51*
• 36
• 32
.29
(-)
1.7
1.7
1.0
1.5
1.5
ft
1.1*
1.8
l*.o
2.0
2.0
ft
6.2
6.2
6.2
6.2
6.2
C-V
2.071*
1.581*
0.950
1.1*38
1.190
(-)
2.1*59
2.11+1
1.529
2.781
2.380
111-27
-------
The selection of an exponential function to describe the measured veloc-
ity distribution with depth will remedy the above discrepancy. An ex-
ponential function of the form
V = V exp[—] (5)
m L n ^
fits experimental data almost as well as the normal distribution func-
tion. It results in a Richardson number distribution of the form
g T * |3 z 2
m
Numerical results given in Fig. 7 appear to be reasonable. Richardson
number values are obviously extremely sensitive to minor changes in
input data. The numerical field data must therefore be regarded and
used with great caution. Calculated Richardson numbers give orders of
magnitude at best.
Considering that a stable stratification, inhibiting vertical mixing,
forms at a Richardson number value near unity, the results obtained
from the field measurements suggest that the horizontal stratification
was quite strong at all five locations for which data are shown. This
result is in agreement with findings from the laboratory experiments
and from the analytical model.
It is also of interest to calculate the overall densimetric Froude
numbers from the data. In accordance with Eq. (j^) in Part I of this
report, the number is defined as
\
,VPO .
1/2
Selected values are shown in Table 3.
Another set of densimetric Froude numbers can be calculated by consider
ing measured absolute flow velocities U instead of excess velocities
V If Um is the measured absolute waller surface velocity at a
station, tne new densimetric Froude number is
U
m
m
111-28
-------
100
H
M
H
&
VO
Fig. 7 - Richardson number distribution with depth for selected locations
-------
Results axe shown in the last column of Table 3« The average value is
2.26, which is close to the values obtained in laboratory experiments
and shown in Fig. 1*6 of Fart I.
111*30
-------
SECTION VIII
SUMMARY
A small number of temperature and velocity measurements were taken near
the cooling water outlet channel of the A. S. King power generating
plant on Lake St. Croix on two occasions during the summer of 1971. The
results were analyzed in terms of
1. Temperature profiles with depth
2. Velocity components in direction of discharge with depth
5. Velocity components perpendicular to direction of discharge
(spread velocities)
1|. Isotherms in a vertical plane through the center of the dis-
charge channel
5. Densimetric Froude numbers for individual stations
6. Richardson numbers as a function of depth
All the field results agree within reason with experimental laboratory
data obtained for the same outlet densimetric Froude numbers. For
instance, temperature, velocity, and Richardson number profiles show the
same type and degree of stratification; the horizontal spread is of the
same nature as in the laboratory; and densimetric Froude numbers at
locations near the outlet are of the same magnitude as found through
laboratory experiments and analysis.
111-31
-------
SECTION IX
ACKNOWLEDGMENTS
The authors were assisted in the field surveys by students from the
Department of Civil and Mineral Engineering, in particular Messrs.
Glen Martin, David Ford, and Loren Bergstedt. Mr. Earl Streed
prepared some of the figures. Professor E. Silberman helped in
making the arrangements for the surveys. Northern States Power
Company kindly provided a "boat and operator. The manuscript of
this report was edited and typed by Mrs. Shirley Kii.
Ill-33
-------
SECTION X
REFERENCES
[l] Csanady, G. T. and Mekinda, M., "Rapid Fluctuations of Current
Direction in Lake Huron," Proceedings of the Thirteenth Confer-
ence on Great Lakes Research, Part I, International Association
for Great Lakes Research, 1970.
[2] Pitch, N. R., Temperature Surveys of St. Croix River, 1969-1970,
Northern States Power Company, December 1970.
[3] Ippen, A. T. and Harleman, D. R. P., Steady-State Characteristics
of Subsurface Flow, Circular 521, U.S. National Bureau of Stand-
ards, 1952, pp. 79-93.
[U] Harleman, D. R. P., "Stratified Plow," Section 26 of Handbook of
Fluid Dynamipp, edited by V. L. Streeter, McGraw-Hill Book
Company, 1961.
[5] Keulegan, G. H., "The Mechanism of an Arrested Saline Wedge,"
Section 11 of Estuary and Coastline aydrodynamios, edited by A.
T. Ippen, McGraw-Hill Book Company, 1966.
[6] Keulegan, G. H., An Experimental Study of the Motion of Saline
Water from Looks into Fresh Water Channels. Report 5168. U.S.
National Bureau of Standards, March 1957*
IH-35
-------
« 1 Access/on Number
w
A 1 Subject Field & Group
05G
SELECTED WATER RESOURCES ABSTRACTS
INPUT TRANSACTION FORM
Organization
Minnesota Univ., Minneapolis, St. Anthony Falls
Hydraulic Lab.
Title
SURFACE DISCHARGE OF HEATED WATER
JQ \ Authors)
Stefan, H.
Hayakawa, N.
Schiebe, F. R.
|£ I Project Deat&iation
Project No. 161JO FSU
21 1 Note
Citation
23
Descriptors (Starred First)
*0utfalls, *Thermal Pollution, *Mathematical Model, Thermal Plume Data
25
Identifiers (Starred First)
*Thermal Outfalls, ^Surface Discharge
27
Abstract
A comprehensive analytical model has been developed to describe the flow of heated water
from a channel onto the surface of a lake or reservoir. This analytical tool can be
used to predict depth, width, temperature, and flow velocity in a heated water surface
jet. Weak cross-currents and winds are included. The model also predicts the total
amount of heat actually lost to the atmosphere and the amount of ambient water entrained
As presented, the analytical method is simple and inexpensive to apply. It assumes
fully established buoyant jet flow into a homogeneous environment. It can be extended
to include, for example, an outlet zone (zone of flow establishment) or stratification
in the ambient water. Experimental (laboratory) data have been analyzed for comparison
with the analytical method and also to illustrate the physical features of heated water
surface discharges, particularly those with low densimetric Froude numbers. The two-
dimensional mixing internal hydraulic jump has been analyzed theoretically and experi-
mentally. Criteria for the existence of the phenomenon and the rates of entrainment
which it may produce have been established theoretically and verified with a limited
number of experimental data. The results of two field surveys in a thermal plume are
presented. Comparisons with laboratory data illustrate the outstanding significance
of the densimetric Froude number and the relative insignificance of the Reynolds number
as modeling parameters. (Stefan-Minnesota)
Abstractor
G. Stefan
Institution
St. Anthony Falls Hydr.
Lab., U. of Minn.. Minneapolis
WR;I02 (REV. JULY lOfiO)
WRSIC
SEND. WITH COPY OF DOCUMENT. TO!
**J.S. GOVERNMENT PRINTING OFFICE: 1972-484-484MS8 1-3
RESOURCES SCIENTIFIC INFORMATION CENTER
ARTMENT OF THE INTERIOR
OPO1 IS7O - 4O7 -8«1
WASHINGTON, D. C. 20240
------- |