EPA-660/3-75-014
MAY 1975
Ecological Research Series
ow Establishment and Initial
Entrapment of Heated Water
Surface Jets
National Environmental Research
Office of Research and Development
U.S. Environmental Protection Agency
Corvallis, Oregon 97330
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EPA-660/3-75-014
MAY 1975
FLOW ESTABLISHMENT AND INITIAL ENTRAINMENT OF
HEATED WATER SURFACE JETS
by
Heinz Stefan,
Loren Bergstedt,
and
Edward Mrosla
St. Anthony Falls Hydraulic Laboratory
University of Minnesota
Minneapolis, Minnesota
Grant R 800 435
Program Element 1BA032
ROAP/Task No. 21 AJH/15
Project Officer
Mostafa A. Shirazi
Thermal Pollution Branch
Pacific Northwest Environmental Research Laboratory
National Environmental Research Center
Corvallis, Oregon 97330
NATIONAL ENVIRONMENTAL RESEARCH CENTER
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
CORVALLIS, OREGON 97330
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ABSTRACT
Mathematical models to predict water temperature distributions
resulting from heated water surface discharges usually consider three
subregions of the flow field: (a) an outlet region or zone of flow
establishment (ZFE), (b) a zone of fully established jet flow, and (c)
a far field with mostly passive dispersion. Of these three regions,
zone (b) can be treated mathematically most readily using integral
techniques; zone (c) requires input specifying mean flow and turbulence
of the ambient flow field; and zone (a) depends essentially on the
geometry of the outlet and the discharge characteristics in terms of
the velocity and temperature of the water.
The results of an exprimental study dealing with zone (a) are
reported. The discharge channel had a rectangular cross section and led
into a deep, wide reservoir. The aspect ratio (width-to-depth ratio) of
the channel, the volumetric discharge rate, and the discharge temperature
were varied. A cross-flow was imposed in some of the experiments.
The length of the zone of flow establishment, XQ or s , was
measured in terms of mean excess temperature, mean velocity, and turbu-
lence intensity along the trajectory. The length, XQ or SD> was
related to channel aspect ratio A, outlet densimetric Froude number
F , and cross-flow-to-jet velocity ratio R. Total volumetric flow
rates during flow establishment were established as a function of dis-
tance along the jet axis and related to A and F . The results are
useful either for extension of existing mathematical models of fully
developed heated water surface jets of for verification of mathematical
models of the zone of flow establishment.
This report was submitted in fulfillment of Grant R 800 435 by the
St. Anthony Falls Hydraulic Laboratory, University of Minnesota, in Minneapolis
under the (partial) sponsorship of the Environmental Protection Agency. Work was
completed as of January 1975.
ii
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CONTENTS
Section
Page
I CONCLUSIONS x
II RECOMMENDATIONS 6
III INTRODUCTION 7
IV ZONE OF FLOW (RE-)ESTABLISHEMENT 9
V EXPERIMENTAL FACILITY AND DATA ACQUISITION 16
Basic Equipment ^5
Adaptation of Experimental Facility for Cross-Flow
Experiments 22
VI EXPERIMENTAL RESULTS 27
Pilot Studies 27
Main Experiments: Straight Jets 37
Length of Zone of Flow Establishment (ZFE) 37
Volumetric Flow Rates and Entrainment 76
Main Experiments: Curvilinear Jets 85
Photographs of Flow Features 85
Trajectories ........ 85
Length of Zone of Flow Establishment (ZFE) 87
VII ANALYTICAL EXPRESSIONS FOR ESTABLISHMENT LENGTH AND INITIAL
FLOW RATES 101
Length of Zone of Flow Establishment 101
Initial Entrainment for Straight Jets 107
VIII REFERENCES 110
IX PUBLICATIONS 112
X SYMBOLS AND UNITS 113
Appendix A - Isotherms T = constant and 5T = constant in Vertical
Sections along Jet Axis or Perpendicular to It,
A = 2.4, A - 9.6, R = 0 115
^Appendix B - Isotherms T = constant and 6T = constant and
Isovels V = constant in Vertical and Horizontal
Sections, A = 2.4, R = 0,41 132
iii
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FIGURES
No.
la Schematic showing zone of flow establishment (ZPE) near a
surface outlet 10
Ib Streaklines illustrating the surface flow pattern and flow
establishment -A=1.0, R=0, F =00, Exposure = 1/5 sec 11
2 Temperature and velocity distributions at the beginning and
end of the zone of flow establishment (schematic) 12
3a Schematic of experimental tank , 17
3b Schematic diagrams showing evolution of temperature and
velocity measuring and recording systems in the course of
the investigation 18
lj.a Schematic of adaptation of experimental tank for crossflow
experiments 23
i^b Experimental tank 25
l;c Crossflow velocity profiles 26
5a Surface centerline temperature versus axial distance,
semicircular outlet, low F 28
5b Surface centerline temperature versus axial distance,
semicircular outlet, high F 29
6 Definition of zone of flow establishment for non-buoyant,
fully submerged jets 31
7 Schematic showing range of alternative values of x
depending on definition ° 32
8a xo/dQ versus FQ at T = 0.98, 0.95, 0.90, and 0.80 and
A = 2.0 (semicircular pipe, positivg buoyancy) 3^
8b x /d versus F at T = 0.98 and visual observations,
o' o o
A = 2.0 (semicircular pipe, negative buoyancy) 35
9 Horizontal spreading angle, 0, of free shear zone
recorded photographically 36
lOa Streaklines illustrating surface flow patterns - A = 1.0,
R = 0, Exposure = 1/10 sec 38
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FIGURES
No. Page
10"b Streaklines illustrating surface flow patterns - A = 2.4»
R = 0, Exposure = 1/5 sec ................................... 59
lOc Streaklines illustrating surface flow patterns - A = 9«6»
R = 0, Exposure = 1/5 sec at F = o> and 1/10 sec at
F = 3-0 [[[ 40
11 Boundaries of visible core region for A = 1.0, 2.4» and 9«6
and various F
12 X0/d0 versus F at A = 1.0, 2.4* and 9>6 as determined
from photographs of surface tracers ..... . .......... ... ....... 42
13 Sample of time- averaged water temperatures along jet axis -
I4a
I4b
I4c
I4d
15a
15b
15c
l6a
I6b
l6c
I6d
*H» -Q
x /d versus F
o o o
A = 1
A = 2
x /d
o' o
A = 4
Vdc
A = 9
x/d
o' o
and
*A
and
x /d
o' o
and
.0, R
- n ..
versus F
o
.4, R
- n ..
versus F
o
8 R
__ {}
versus F
.6, R
- 0 ..
versus F
o
R - 0
versus F
o
R - 0
versus F
o
R - n .
§T versus
8T versus
§T versus
8T versus
x/do
x/d0
x/do
x/d
., j*\
at
at
at
at
at
at
at
-1*
T
T
T
T
T
T
T
along
along
along
along
-".?» v»*i
= 0.98,
= 0.98,
= 0.98,
= 0.98,
= 0.85;
= 0.90;
= 0.95;
jet axis
jet axis
jet axis
net axis
0.95, 0.90
0.95, 0.90
0.95, 0.90
0.95, 0.90
A = 1.0,
A = 1.0,
A = 1.0,
at A = 1.
at A - 2.
at A - 4«
at A - 9.
, and
, and
, and
0.80;
0.80;
0.80;
, and 0.80;
2.4,
2.4,
2.4,
0 an
4 an
8 an
4.8, and 9.6;
4.8, and 9.6;
4-8, and 9.6;
d R = 0
id R - 0
id R - 0
HH
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FIGURES
No. Page
17a x /d versus FQ at 5T = 0.02; A = 1.0, 2.4, l+.S, and
9.6; and R=0 62
17b x /d versus F at ST = O.Oi;; A = 1.0, 2.1i, k.8, and
O O O
9.6; and R = 0 63
18a §T versus y/d perpendicular to jet axis at A = 1.0,
x/dQ = 1.7, R = 0, and FQ = 2.2, 3.5, and 6.5 65
18b §T versus y/d perpendicular to jet axis at A = 1.0,
x/d =3-5, R = 0, and FQ = 2.2, 3*5, and 6.5 66
18c 5T versus y/d perpendicular to jet axis at A = 2.1+,
x/dQ = 2, R = 0, and FQ = 2.2, 3-8, and 15.8 67
18d 5T versus y/d perpendicular to jet axis at A = 9«6,
x/d =2, R = 0, and FQ = 2.1 and i*.l 68
19 Spreading angle and width of free shear zone at ST = 0.01
and 0.02 for A = 1.0, 2.1+, and 9.6; x/dQ = 2 and R = 0 .. 69
20a x/d versus F at 7=0.95, 0.90, and 0.80; A = 1.0;
o' o o
and R = 0 71
20b x /d versus F at 7 = 0.95, 0.90, and 0.80; A = U.8;
o o o
and H=0 72
20c xQ/d versus F at 7 = 0.98, 0.95, 0.90, and 0.80;
A = 9.6; and R = 0 73
21 x /d versus F at 7 = 0.9; A = 1.0, 1^.8, and 9.6; and
22a X0/<10 versus F at 57 = 0.05, 0.075, and 0.10; A = 1.0;
and R = 0 77
22b xQ/do versus FQ at 57 = 0.075 and 0.10, A = 1+.8, and
R = 0 78
23a Q/Q versus x/d at A = 1.0 and R=0 79
O 0
23b Q/Q versus x/d at A = ii.8 and R=0 80
o o
23c Q/QQ versus x/dQ at A = 9.6 and R=0 81
vi
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FIGURES
No.
25
26
27
28
Q/Q versus x/d for fully submerged, non-buoyant two- and
Q/QQ versus A at F = 1.8, F = 3.4, and FQ-* 0;
x/d 10; and R 0 ,
Photograph of vortex formation near the outlet at A = 2.4»
Photographs of discharge into a crossflow at A = 2.4,
R - 0.41, and F - 25.4 and 2.8
Typical centerline plume trajectory at A = 4-8, R = 0.4l>
and F = 3.25
Page
. 82
85
> u^
, 8k
i \Ji+
. 86
. RR
29a Plume trajectory at A = 1.0, R = 0.4l, and F =3.2 and CD 89
Plume trajectory at A = 2.4, R = 0.41, and F =3-2 and CD 90
Plume trajectory at A = 4»8» R = 0.41, and F =3.2 and CD 91
30a
30b
31
32
33a
33b
Observed trajectories, A = 1.0, 2.4>
Trajectories based on maximum velocity
4.8; F =3.3 and CD; and R = 0.41
s /d versus A at F =3.2, R = 0
0.90, 0.85, 0.80, 0.70, and 0.60
s /d versus A at F = 3«2, R = 0
o o o
0.03, 0.05, 0.07, and 0.10
so/dQ versus A at FQ = 3.2, R = 0
0.85, 0.80, 0.70, and 0.60
s /d versus A at F = CD, R = 0.
0.85. 0.80. 0.70. arid O.6O ._
and 4«8> F = 3*3 and
, A = 1.0, 2.4, and
.41, and T = 0.95,
.41, and $T = 0.02,
.41, and V = 0.90,
41, and V = 0.90,
34 Photograph of dye patterns near the outlet of a discharge into
a CtnRH-Plnw - A - 9_)i. IP — nn. T? — O.lil
35
36
92
93
95
96
97
98
100
105
106
vii
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TABLES
No. Page
1 Standard Deviations of x /w and x /d as given in
o' o o' o
Figs . ll^a, "b , and d .......................................... 53
2 Establishment Lengths x for Non-Buoyant, Fully Submerged
Jets ...... [[[ 5k
3 Comparison of Computed and Measured x /d Values
k Equivalent Values of T and ST ............................. 61
5 Ratios of x Values, (x ) /(x ) , derived from
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ACKNOWLEDGMENTS
Besides the authors, a number of research assistants and personnel
of the St. Anthony Falls Hydraulic Laboratory participated in this
study. The contribution, in the early phases of the experiments, made
by Manousos Katsoulis (now with the Minnesota Pollution Control Agency)
and the assistance provided by C. Shanmugham in the reducing and
plotting of the data are acknowledged. Dr. J. M. Killen and J.
Ferguson provided very essential help in the development and construc-
tion of the data acquisition systems. Shop help was provided by J.
Vandenberg and F. Thomas under the supervision of F. Dressel. The
manuscript was prepared for publication by Shirley Kii, editorial
assistant. The authors wish to express their gratitude to all these
people.
Dr. Mostafa A. Shirazi of the Thermal Pollution Branch, Pacific
Northwest Environmental Research Laboratory, acted as Project Officer.
His input was helpful in orienting the study toward those goals which
seemed most appropriate and timely. The authors are also indebted to
him for his review of the manuscript of the report.
IX
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SECTION I
CONCLUSIONS
1. Mathematical modeling of the zone of flow (re-)establishment (ZFE)
of heated water surface jets has been found to be difficult "because
of the complex dependence on outlet geometry, discharge velocity,
buoyancy, and ambient currents. Laboratory experiments have there-
fore been conducted to provide more observations and data on flow
patterns, temperatures, and flow velocities in the ZFE and some
distance beyond. Prom this information, relationships giving the
length x of the ZFE, the volumetric flow rates Q(X) versus
distance, and the initial spreading angle 0 have been derived.
The independent variables were outlet aspect ratio A, outlet
densimetric Froude number, and crossflow ratio R. The relation-
ships provided can be used to improve water temperature predictions
in the vicinity of heated water surface outlets.
2. In water temperature modeling, the length, x or s , of the zone
of flow establishment of heated water surface jets—also called the
outlet zone, core region, or zone of flow re-establishment—can be
defined in several different ways.
a. One can arbitrarily decide that the ZPE terminates where the time-
averaged surface temperature on the jet axis reaches a certain
magnitude relative to the outlet temperature.
b. On a log-log plot of axial surface temperature versus distance
one can extrapolate backwards from the fully established flow
data to the outlet temperature level line to define x . For
buoyant surface jets the procedure is not easily applicable, be-
cause the data do not plot on a straight line.
c. One can arbitrarily decide that the ZPE terminates where signif-
icant changes in turbulence intensity of temperature or velocity
occur along the jet axis.
Depending on the definition used, the value of x varies signifi-
cantly, as is shown schematically in Fig. 7- It is the authors'
judgment that the 90 per cent level of time-averaged water temperatures
1
-------
measured along the centerline trajectory of the jet represents an
acceptable and realistic choice for the length of the ZFE, designated
in this report by x for straight jets and by s for curvilinear
jets. However, results at the 80, 95» and 98 per cent levels will
also be reported for comparison.
3. Actually, four different flow regions characterize the zone of flow
establishment. Going downstream, using water temperature (flow
velocity) on the axis (main trajectory) of the surface jet as an
indicator, the four regions are
a. The channel region (prior to discharge), characterized by
completely uniform water temperatures and usually a fully
developed velocity profile.
b. The core region just beyond the end of the channel, character-
ized by constant water temperatures and velocities with time
(no fluctuating components) along the jet axis.
c. The turbulent transition region, in which water temperatures or
velocities fluctuate intermittently, but turbulent shear layers
growing from the outer edge of the jet have not yet completely
penetrated each other.
d. The fully developed jet region, which is characterized by the
internally fully developed turbulence field, resulting in the
similarity of temperature and velocity distributions in cross
sections perpendicular to the jet axis.
The lengths of regions (b) and (c) are not the same in terms of tem-
perature or velocity, because mass and momentum are transported dif-
ferently in the shear layers. Practical definitions of the length
of the ZFE usually incorporate (b) plus a portion of (c).
it.. The dependence of x /d or s /d on the independent variables A
(= aspect ratio of the rectangular discharge channel), F (= outlet
densimetric Froude number), and R (= crossflow velocity ratio) has
been investigated. Of the parameters A, F , and R, the aspect
ratio A has the largest influence. It was found, from data ob-
tained for A = 1, A = 2.1±, A = U-8, and A = 9.6, that for
-------
straight jets (R = 0) the relationship
:r = KAn
do
with n = 0.1|6 and K = 6.0 gives a first estimate of s /d
at the 90 per cent level of time-averaged temperature concentra-
tion. The accuracy is on the order of +_ 25 per cent.
A significant deviation from the relationship given above was
found for A = 2.1^. This is attributed to a secondary motion when
the aspect ratio is favorable (i.e., near 2). However, the aspect
ratio is not a sufficient indicator for such secondary motion. A
semi-circular outlet channel (A = 2.0) did not show the same ef-
fect as a rectangular one (A = 2.i|) (see Figs. 8a and Hjib). Many
prototype channels are designed with an aspect ratio near 2, and
the question therefore deserves further attention.
5. The values of K and n in the above equation actually depend on
the temperature level T at which s /d is defined. For 1.0 <
T < 0.8 a second approximation equation was found by non-linear
curve-fitting. It is of the form
i. (16.0 - 12.8 T)l-2 A(°.85-O.U,T)
A further refinement and extension of the equation includes the
dependency of s /d on R. Values of R examined experimentally
were R = 0 and R = O.iil. The s /d values were found to be ap
^ o' o
proximately 1.14 times larger for the straight jet (R = 0). The
crossflow effect was expressed in the form of a coefficient exp
incorporated in the right-hand side of the equation. J? was found
to be on the order of -0.9.
The dependence of s /d on FQ is complex. Values of FQ
from 2 to 15 were experimentally investigated. In some cases there
is almost no dependency of s on F . In others a significant de
° €Po + P
pendency does exist. A function of the form 1 + /~.p \ has
been used to match the data, but it has been found to be only par-
tially satisfactory. The approximate values of the coefficients
-------
e, |3, and 8 are 0.5» -1.5> and O.ij. respectively. This function
has a maximum at F = 1/8 - /3/€ = 5.5.
6. The complete empirical equation proposed for s /d on the basis of
o o
the available data is of the form
-°-= (16.0- 12.8 T)^ A^'a^
d
o
The values of the coefficients were initially derived from graphs
and were improved upon using non-linear fitting. The equation was
derived from experimental data covering the following flow conditions:
1.0 £ A < 9.6, 0 £ R £ O.i^l, 2.0 £ PQ < 15, 0.8 < T < 0.98.
7. Except for very small aspect ratios (A < 2), it was observed that
the shear in the vertical shear layer and the piercing of cold water
from the bottom to the surface impose significant limitations on the
length of the ZFE, at least for straight jets. Horizontal mixing
appears secondary. The vertical mixing is presumably by large eddies
and sometimes breaking internal waves.
8. Surface flow patterns did not provide realistic information on the
length of the ZPE except for narrow outlet channels.
9. Total volumetric flow rates Q, were computed from measured veloci-
ties. Values of Q, increase with distance x from the outlet. The
rate of increase can be measured by comparing total volumetric flow
rates Q(X) with the initial flow rate Q . The ratio Q(x)/G was
found to depend more on aspect ratio A than on outlet Froude number 3?o«
The increase in total flow rate with increasing distance from the out-
let was cast into a form which would give a power law increase of vol-
umetric flows with the exponent depending on aspect ratio,
Using data for straight jets near the outlet and within 1.8 < 3?0 < CD
and 1.0 £ A £ 9.6 only, non-linear fitting gave
2.35+.75A
0.90 + A
r
«•* <$
-------
10. In jets with crossflow, large eddies with nearly vertical axes near
the surface "but bent over into the crossflow and reaching large
depths are the most outstanding mixing mechanism (Fig. 31+) . Flow
rates have not been computed for jets in crossflows.
11. The spreading of a heated surface jet near the outlet is known to de-
pend on the densimetric Froude number at the outlet. A spreading
angle can be defined and related to measurements and observations on
the water surface. Referring to the definition of <£ shown in
Fig. 9 and using photographs (streaklines) of surface tracer par-
ticles contained in the receiving tank to show the edge of the jet,
an empirical equation for ^ was derived having the form
225 + 10 F
o = 2.25 + FQ for Fo * 1-°
In a previous study , a tracer (lycopodium powder) was added to the
discharged warm water rather than to the tank. Using the edge of
the tracer cloud to measure the angle 0 yielded much larger
values of ^ • The results from Ref . 3 can be given in the form
360 + 18 F
Fo
o
The reason for the difference is that streaklines of tracers intro-
duced in the ambient water show that portion of the flow field in which
velocities are significant and therefore essentially identify the inner
region of the actual jet; by contrast, an outward spreading dye pattern
will more likely show the outer edge of the shear layer. This has sig-
nificance for dye and drogue studies near prototype outfalls.
12. Froude number effects on the length of the ZFE, the volumetric flow
rate, and the initial spreading angle were significant over the range
of Froude numbers investigated (0 < FQ < 15). It would appear,
therefore, that F values must be larger than 15 to eliminate
buoyancy effects within the ZFE.
-------
SECTION II
RECOMMEN.DATIONS
1. It is recommended that the experimental data and conclusions of this
study be used to verify existing mathematical models for water tem-
perature prediction in heated water surface plumes. The empirical
relationships can be incorporated into such models.
2. It is recommended that empirical relationships similar to those de-
rived for the length of the ZFB and the volumetric flow rate also be
developed for the width and the depth of the heated surface jet with-
in the outlet region. Sufficient data are available.
3. It is recommended that experiments similar to the ones described be
conducted with discharges over sloping bottoms and different angles
of discharge into the crossflow. The effects of these two param-
eters on the ZFE need further investigation.
k* It was found that at channel aspect ratios near 2, initial dilu-
tion of effluents is particularly strong at Froude numbers near
7.0, but quite sensitive to the shape of the channel (rectangular
versus circular). It is recommended that studies be initiated to
determine the optimum shape of surface outlet channels and the
beneficial effects of artificially induced secondary motion on
the initial dilution.
-------
SECTION III
INTRODUCTION
Analytical, numerical models for predicting temperature distribu-
tions resulting from surface discharges of cooling water into lakes and
reservoirs have been developed with increasing refinement over the past
several years. The models of Stolzenbach and Harleman , Motz and Bene-
p "3£ ) R
diet , Stefan, Hayakawa, and Schiebe , Prych , and Shirazi and Davis can
be cited. All these models are for steady-state jet flow; an integral
technique is used to describe the simultaneous flow of mass, momentum,
and heat along the main trajectory of the jet. The interaction with
ambient water and the atmosphere is described through various coeffici-
ents for turbulent mass exchange, entrainment, friction, surface heat
transfer, and other processes. Most of the models give a picture of the
temperature field near a surface outfall of cooling water which appears
to be realistic. It is uncertain, however, whether the models can pre-
dict actual temperature distributions with the accuracy which would be
desirable in view of the stringent temperature standards set by federal
and state agencies. Temperature predictions made by the models are inac-
curate for a number of reasons, including (a) inadequate formulations of
the fundamental equations (e.g., similarity of temperature and velocity
profiles), (b) inadequate numerical values of the coefficients used
(e.g., for turbulent entrainment from ambient currents), and (c) inade-
quate match between a real outfall situation and its idealized version as
used in the mathematical model.
Following classical, semi-empirical jet flow theories, mathematical
models of heated water surface jets consider separately a zone of flow
establishment and a fully established flow region. The zone of flow
establishment (ZFE), or more accurately, re-establishment, forms a
transition from the wall-shear-controlled flow in the outlet channel or
pipe to the free-shear flow in the receiving open body of water. There
is a lack of adequate information on the length of the zone of flow
establishment and the initial entrainment rates in the vicinity of the
-------
outfall and their relationship to outlet design and the operational char-
acteristics of the discharge flow. It was for this reason that an ex-
perimental study was conducted to obtain such information. The authors
have felt for some time that theoretical treatment of the ZFE would be
difficult because of the three-dimensional and transitory character of
the flow field. For this reason, only the fully established jet flow was
considered for mathematical modeling (see Stefan, Hayakawa, and Schiebe*).
The information provided through the study described herein can now be
used to determine where the fully established flow region begins and
what the volumetric flow rate at that point will be. This information
can be incorporated in the mathematical model to provide a realistic
picture of the temperature and flow field downstream of the cooling water
outfall.
8
-------
SECTION IV
ZONE OF PLOW (EE-)ESTABLISEiyiENT
The outlet region of a cooling water surface discharge usually
forms the transition from a channel flow to a free-jet flow as shown
schematically in Fig. la for discharge from a shallow rectangular chan-
nel into a deep pool. The initial core region and the shear layer, as
they appear on the water surface, are shown in Fig. lb. It is noteworthy
that the free-shear layer grows from the edges of the channel toward the
center of the jet. The zone of flow establishment or re-establishment
ends when the free-shear layer has reached the centerline or main tra-
jectory of the jet. This point will be designated by x (straight
jets) or s (curvilinear jets). Whether it is caused by the vertical
growth of the free-shear layer, as shown schematically in Fig. 2, or "by
the lateral growth is irrelevant to the definition of x and s , but
important for an understanding of the dependency of x and s on the
independent variables.
A zone of flow establishment is common to all jet flows. A de-
tailed description of it was given for circular, neutrally buoyant, sub-
merged jets by Albertson, Dai, Jensen, and Rouse , among others. A
theoretical analysis of the zone of flow establishment for fully sub-
•7
merged, round, buoyant jets was made by Hirst using the integral tech-
nique.
The heated water surface discharge is a buoyant half-jet due to
the presence of the water surface; in addition, the orifice from which
the discharge is released is often non-circular, and a cross-flow may
be superimposed. Some consideration was given to the zone of flow estab-
lishment in existing models. Information on its length was collected
Q
and synthesized by Shirazi . It was concluded, however, that not
enough information was available for adequate description and modeling
of this zone. An experimental investigation was therefore undertaken
to determine the characteristics of the flow inside and outside of the
-------
w
DISCHARGE
CHANNEL
SHEAR LAYER
SHEAR LAYER
Fig, 1° ~ Schematic showing zone of flow establishment (ZFE) near a surface outlet
-------
Fig Ib - Streaklines illustrating the surface flow pattern and flow establishment
A = UO, R = 0, FQ = OD, Exposure =
11
-------
x = 0
FLOW
•T<0,2)
SHEAR LAYER
T(x0,z)
x = 0
ox.
FLOW
SHEAR LAYER
Fig. 2 - Temperature and velocity distributions at the beginning
and end of the zone of flow establishment (schematic)
12
-------
zone of flow establishment in heated water surface jets. Specifically,
the length of the zone of flow establishment, the mechanism of dilu-
tion, and the amount of dilution were investigated. The information was
obtained using measurements of time-averaged temperature distributions,
turbulent temperature fluctuations, time-averaged velocity distributions,
turbulent velocity fluctuations, and flow visualization.
Discharges from rectangular channels into deep bodies of water as
shown schematically in Fig. la were investigated exclusively. The inde-
pendent variables were the channel width, w ; the discharge rate, Q ;
the temperature differential at the outlet, T - T , where T is the
o a o
discharge temperature at the outlet and T is the ambient water temper-
a
ature in the lake or reservoir; and the ambient cross-flow velocity, TT ,
3*
past the outlet. The primary dependent variables measured were local
water temperatures T., - T or flow velocities U, from which all other
Xt a
information was derived, including the length of the ZFE, x or s ,
and the entrainment expressed in terms of the flow rate Q. The flow rate
Q, was obtained by integration of local velocities over an area perpen-
dicular to jet trajectory.
The independent variables were grouped into the following dimension-
less parameters:
Aspect ratio:
A = w /d (1)
o' o
Outlet densimetric Froude number:
y = ° (2)
^
Outlet Reynolds number:
U d
(3)
-------
and cross-flow velocity ratio:
In the four equations above,
d = depth of rectangular discharge channel
17 = average discharge velocity Q /(w d )
Ao =o - o = density differential between the density of the
o a o
ambient (cold) water, Q , and that of the discharged
cl
warm water, Q
g = acceleration of gravity
v = kinematic viscosity of the water at the discharge
The dependent variables were also reduced to dimensionless forms,
including the following:
Excess temperature ratio for time-averaged temperatures:
T(x,y,z) - T
T = T -T (5)
o a
Excess temperature ratio for fluctuating temperature
components :
(6)
Time-averaged velocity ratio:
V = - (7)
m,o
Fluctuating velocity component ratio:
ov(x,y,z)
SV = -^ (8)
i»o
-------
In the above relationships,
T(x,y,z) = a real, time-averaged local water temperature value
an,(x,y,z) = standard deviation of the fluctuating water tempera-
ture component at location (x,y,z)
U(x,y,z) = time-averaged local water velocity
U = maximum velocity at the outlet
mo
ov(x,y,z) = standard deviation of the fluctuating water velocity
component
When values of T, 5T, V, or 8V were plotted versus location,
the dimensions x,y,z were normalized using x/d , y/d , z/d , or
s/d , where d is the outlet depth. From the above information the
following relationships were specifically sought:
a) The length of the zone of flow establishment, x /d or
s /d
o' o
x
o
b) The dilution
_ / \
= f,(A, F , a, He ) (10)
15
-------
SECTION V
EXPERIMENTAL FACILITY AND DATA ACQUISITION
BASIC EQUIPMENT
The experimental facility consisted of a tank 1+0 ft (12.2 m) long,
16.5 ft (5-0 m) wide, and 1.5 ft (O.i|6 m) deep, thermally insulated all
around and with an instrumentation carriage riding on top. The heated
water was discharged from a rectangular channel with a maximum width of
2.32 ft (0.71 m) and a depth of up to 0.25 ft (0.076 m). A round stain-
less-steel pipe 0.369 ft (0.112 m) in diameter was used in some prelim-
inary experiments. The outlet was placed at right angles to the wall of
the tank. Average discharge velocities ranged from 0.1 to 1.5 ft/sec
(0.03 to 0.5 m/sec) and temperature differentials from 5 to 30 F (3 to
17 C). Figure 3a shows the overall dimensions of the tank. Maintaining
an unstratified stagnant pool while injecting heated water at the surface
is very difficult. A manifold withdrawal pipe placed across the tank 27
ft (8.25 m) from the outlet was used for the purpose. More water was re-
moved than was injected. To compensate for the difference and replenish
the cold water volume constantly, a feeder system at the downstream end
of the tank was used. A constant water level was maintained in the tank
using an overflow weir (Fig.
Velocities and temperatures in the flowfield near the outlet were
measured with several types of sensing devices which are schematically
represented in Fig. 3b:
a) Single YSI thermistors, types 1|27 and 406, with time constants
of 0.5 and 2.5 seconds respectively;
b) Rakes of YSI thermistors of the same types carrying eleven
sensors at 0.5 inch (0.0127 m) vertical intervals;
c) TSI hot film current meter, VT 1630T, 0-3 fps (0 - 0.9 m/s),
threshold velocity approximately 0.05 fps (0.015 ffl/s);
d) Delft propeller current meter, propeller diameter 15 mm,
range 1 to 120 cm/sec.
16
-------
Section A-A
Warm wafer
m)
Variable width
and depth -
warm water
inlet channel
Carriage
Temperature
and velocity
probes
Manifold withdrawal pipe
Weir
Plan View
(•6m)(«6m) (1 02m)
l_^I».l-Ol«.i-. jl J w4 .*•
(15.4m)
50 5*
(8.2m)
27'
(4.0 m)
»U. 13'
i)
M
Thermo-
stat
,
A
t
•
3
Warm
<
.. l
i
water
l
1
1 Heat 1
exch. t
\
-y
X
r+i
+y
t-
1
_i *-<> ^ ^ « , — _
•«•
«•
r
o
f
D
[Auxiliary
pump (cold
wafer
recirculation)
A
J
Auxiliary
pump
— txi — Cold
wafer
Drain supply
D
Flowmeters N. ^^2" line (.05 m) Main pump
^ — 1" line (.025 m) (warm water recirculation)
Steam
Fig. 3a - Schematic of experimental tank
-------
YSI Thermistor 406
-2000 n at 20°C
Time const. 2.5 sec
^
YSI
analog readout
te lethermometer
J
H
Subjective
time-averaged|
temperature
readout
(a) Preliminary temperature measuring instrumentation
CD
Yellowsprings Thermistor 427
-2000 Q at 20°C
Time const. 0.5 sec
^
P
Datascan amplifier
A - D Converter
(0 - 800 MV)
fe
Friden paper tape
punch
b
Objective
instantaneous
-p( temperature readings
for further
processing
(b) Intermediate temperature measuring instrumentation
Fig. 3b - Schematic diagrams showing evolution of temperature and velocity measuring and recording
systems in the course of the investigation
-------
Potentiometers
13
YSI Thermistors
427
Digital input
one channel
16 bit bytes
16
Analog inputs
Channel select
Convert/command
Multiplexer
and
A - to - D
converter
System
Formatter
16 bits
data
and
channel
Cassette
recorder
system
Phillips cassette
containing up to
120,000 16-bit
words
(c) Temperature data-logger system functional block diagram, System LPS-16
Fig. 3b (continued) - Schematic diagrams showing evolution of temperature and velocity measuring and
recording systems in the course of the investigation
-------
TSI
Hot film velocity
sensor
(VT 1630B,
0-3 V)
+
Magnetic tape
recorder
(Precision Instr.
PI 207S
+ 1.5 V)
(0.19 m/sec ->0.762 m/sec)
7-1/2 in./sec ->30 in./sec
w
RMS Meter
(2 sec integration
time for time-
smooth value)
/ Objective ^
/ time-averaged
*l and
y rms
\velocities S
(d) Intermediate velocity measuring instrumentation
Delft propeller
current meter
1 to 1 20 cm/sec
^
w
Interval times -
switch and
predetermined
counter
fc
w
Datascan amplifier
A - D converter
^
V
Friden
paper tape punch
^
w
Instantaneous
velocity readings
for further
processing
(e) Velocity data recording system
Fig. 3b (continued) - Schematic diagrams showing evolution of temperature and velocity measuring and
recording systems in the course of the investigation
-------
These sensors were used interchangeably.
Several types of metering and recording devices were used,
including
a) YSI direct readout telethermometer;
b) Direct readout Delft current meter;
c) Datascan A/D converter linked to a Priden model 2, 8-column
paper punch;
d) A MTEL digital data logging system LPS-16.
The Datascan/Friden system had only one channel. It was used initially
to record individual temperatures and later to record velocity read-
ings. Peripheral information could be added manually to the paper
tape.
The DATE! system scans and records 16 data points in digital form
on a cassette tape. One sweep of all 16 channels requires approximately
two seconds. The DATEL system was used primarily to record turbulent
temperatures sensed by the thermistor rake in the outlet region, where
the mixing of the effluent with ambient water is rather intense. The
data acquisition system was programmed to take three consecutive samples.
Each sample contained 15 individual measurements at each point. There-
fore a total of 1^.5 individual measurements were obtained over a 90~secon(i
period at each location. The following parameters were recorded on the
16 channels of the data acquisition system:
—a reference warm water outlet temperature, T
—a reference cold water (tank) temperature, T , taken at a
point with coordinates x = dQ, y = WQ + 2dQ, z = dQ
—the x, y, and z coordinates of the topmost temperature
sensor of the rake.
Prior to the sampling, auxiliary integer input giving experiment number
and date were recorded on the cassette. During the processing of the
data, additional identifiers for channel depth d , channel width w ,
21
-------
outlet densimetric Froude number F , and Reynolds number Re (com-
puted during the processing) are added to the output.
Raw data recorded on magnetic or paper punch tape were processed
on a Raytheon 703 data storage and processing unit. The first step was
a conversion of recorded digital values to real temperatures or real
velocities as required. The necessary calibration curves had been ob-
tained prior to the experiments.
Although temperature sensors were of the YSI i|00 interchangeable
series, each sensor used was calibrated individually. The calibration
curves were nearly linear, and third order polynomials were found to fit
the calibration data in the useful range of U5 to 95°F (7 to 35°C) to
within 0.2°F (0.1°C).
Velocities were sampled at a rate of 50 measurements per minute.
Fifty individual measurements were taken at each location. It was veri-
fied that the sampling period was long enough. The response of the Delft
current meter to turbulent velocity fluctuations has been investigated
by Schuyf . Velocity fluctuations with frequencies up to 2 Hz can be
recorded with little damping.
The velocity calibration was linear, and conversion of the raw data
was therefore quite easy.
Temperatures as well as velocities are turbulent in major portions
of the outlet region. From the sample data, all of which were obtained
under steady flow conditions, time-averaged values and standard devia-
tions were calculated. For this computation, 1;5 individual data points
were available at each location. With time constants on the order of
0.5 seconds, the sensors used were capable of sampling all the lower
frequencies present in the large-size, low-velocity flow field.
ADAPTATION OF EXPERIMENTAL FACILITY FOR GROSS-FLOW EXPERIMENTS
To generate a cross-flow in front of the discharge channel, two
manifold systems were installed as shown in Fig. l±a. The cross-flow was
produced by upstream discharge from one manifold and downstream withdrawal
22
-------
Section A-A
Pump
Downstream jet manifold
/ Suction manifold
ro
A
JT
Pumf
? / ^
II II 1.0' (0.3m)
*l K0.751 1
(0.23m) 5.5' (1.7m)
V
t C
1 *
2.2' (0.67m)
I
V 3.0' (0.9m)
\ t
x\
. -, -i . \^ Guide vanes
h* /.5 *H \
(2.3m) Upstream jet manifold
A
}
Fig. 4a - Schematic of adaptation of experimental tank for crossflow experiments
-------
through the other. A large vortex was generated in the tank. The flow
past the outlet channel was straightened with the aid of a series of
guide vanes spaced at nine-inch (23 cm) intervals. A view of the plume
in a cross-flow, looking down the discharge channel, is given in Fig.
4t>. An average cross-flow velocity U =0.22 fps (0.06y m/sec) was
cL
applied in all experiments. Local deviations from this average were
by up to +_0.06 fps, as is shown in Fig. 1+c. The average discharge veloc-
ity was maintained at U = 0.53 fps (0.162 m/sec), giving R = O.ij.1.
The upstream velocity profile had a gradient with depth as shown in
Fig. lj.c. This is typical of wind-driven lake currents as well as
gravity flows. The average cross-flow turbulence level measured at a
distance y/d =3.1 upstream from the axis of the outlet channel and
averaged over a depth 0 < z/d < 1.0 and a distance 0 < x/dQ < 28.8
was 26 per cent in all experiments. The outlet depth dQ was 0.2i;3 ft
-------
4b - Experimental tank. Heated
' 9* Crossflow from right to left
/ • I . \ - J—-J 4-^/^n
meter (right)
|nstrumeptation
quisition unit, current mete.
tape punch
25
-------
,02
VELOCITY, m/s
.04 .06
.08 .10
z/d
1.0
2.0
V = 0.0674
3.0
Average vertical velocity profile at y/d = 3
.10
.08
i
•%
t
8
LU
.06
.04
10
15
x/d
20 25
30
Fig. 4c - Crossflow velocity profiles
26
-------
SECTION VI
EXPERIMENTAL RESTJLTS
PILOT STUDIES
Before the investigation described herein was "begun, two series of
pilot studies were conducted. A circular pipe, J^.U inches (0.112 m) in
diameter, flowing half full was used as an outlet channel. In the first
series of experiments, water temperatures along the axis of the surface
jet were measured with a slow response thermistor (time constant = 2 sec).
In the second series, overhead photographs of surface tracers were made.
In Pigs. 5a and 5b the temperature results are reported in terms of
normalized excess temperatures above ambient versus axial distance. Re-
sults have been plotted separately for low (F < 4) and high (P >
7-5) Froude numbers to draw attention to the significant differences
between the results in these two categories. The differences are that
(a) a decrease in axial temperatures occurs closer to the outlet for low
densimetric Froude numbers than for high Froude numbers and (b) the rate
of temperature decrease with distance is higher for the higher Proude
numbers. The densimetric Proude numbers shown are based on the maximum
depth d of the outlet channel.
No physical reason for the difference was discovered at the time of
the experiments. It will be shown later that the buoyancy-induced later-
al spread of the plume enhances the vertical breakthrough of the free
shear zone if the aspect ratio w /d of the discharge channel is small.
This reduces the establishment length. On the other hand, higher outlet
Proude numbers are associated with higher discharge velocities and higher
turbulence and hence more rapid dilution of the discharge. A dependency
on Reynolds numbers, if it exists, could not be established because
Reynolds numbers could not be varied over a wide enough range.
For circular, fully submerged, non-buoyant jets the length of the
zone of flow establishment is generally accepted as 6.2 times the diam-
eter. This value was established by backward extrapolation from
-------
no
o»
1,0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.05
FULLY SUBMERGED
CIRCULAR JET
(after Albertson,
et al.6)
SYMBOL o
O 1.85
a 2.16
O 2899
A 3.78
Re
o
5620
5850
5350
8750
I I I I I I I I
I I I I I I I
8 10
20
40
60 80 100
Fig. 5a - Surface centerline temperature versus axial distance, semicircular outlet, low F
-------
ro
vo
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.05
SYMBOL
e
a
o
7.54
8.35
9.17
11.34
11.67
Reo
12,950
17,200
23,150
29,150
16,100
16.44 27,150
19.67 26,500-
I I
\ I I I I I J-J
-FULLY SUBMERGED
CIRCULAR JET ^
(after Albertson, et al.
* * -
a O
8 10
20
40
60 80 100
Fig. 5b - Surface centerline temperature versus axial distance, semicircular outlet, high F
-------
centerline velocity measurements as shown in Fig. 6 (see, for example,
Reference 6). A backward extrapolation cannot easily be applied to the
data of Fig. 5 because the data, particularly for the higher Froude num-
bers, do not fall on a straight line.
The difficulty in applying the extrapolation procedure is, of
course, that a jet with a high densimetric Froude number at the outlet
tends toward smaller local densimetric Proude numbers (for which buoy-
ancy effects are important) as it moves away from the outlet. This was
shown by Stefan, Hayakawa, and Schiebe'* and explains many of the heated
surface jet's characteristics, including the observation that a log
temperature concentration versus log distance plot as shown in Pig. 5
does not give a straight line in the fully established flow region.
This has also been documented by other investigators, including Dornhelm,
9 10
Nouel, and Viegel , Tamai, Wiegel, and Tornberg , and Jen, Wiegel, and
Mobarek . It is therefore quite difficult to use the standard procedure
of backward extrapolation illustrated in Pig. 6 to define the length of
the zone of flow establishment. A number of more general alternatives
for specifying the length of the ZFE had to be considered.
It is important to emphasize that the real instantaneous length of
the zone of flow establishment fluctuates considerably in time because
the jet discharge produces a highly turbulent flow and eddies in the free
shear layer are of considerable size relative to the size of the outlet.
The length of the ZFE must therefore be defined on the basis of either
time-averaged measurements or intensities of fluctuating components.
Measurements of water temperature, flow velocity, or tracer materials
can be used. Pigure 7 illustrates this approach using instantaneous
temperature measurements or velocity measurements. It is apparent that
a definition of the ZFE based on time-averaged values is quite sensitive
to the absolute temperature level at which the end of the ZPE is marked.
Normalized excess temperature values T and centerline velocity ratios
V on the order of 0.98, 0.95, 0.90, 0.85, and 0.80 will be used as
cutoff points. Fluctuating components of temperature and velocity are
also suited to defining the establishment length, because they are
-------
FREE SHEAR
ZONE
FLO
CORE
LOW , ^X^.
—^ | —
Log Uc
9999m
Log
Fig. 6 - Definition of zone of flow establishment
for non-buoyant, fully submerged jets
(after Albertson,
tully sut
et al.6)
-------
FLOW
8T
h—H
'RANGE'
Fig. 7 - Schematic showing range of alternative values
of x depending on definition
-------
nearly independent of ambient temperature controls. Values of T and
V on the order of 0.02, O.OI^, and 0.10 will be used. With any of the
definitions used there will be significant spread, as is shown in Pig.
7-
A reduction of the data of Figs. 5a and 5b in terms of T versus
x/d is given in Fig. 8a. Some data for a negatively buoyant jet
are shown in Fig. 8b. Buoyancy effects for the negatively buoyant jet
begin to show up at F < 50» whereas positive buoyancy appears to be
negligible at F > 20. This could be interpreted as a surface effect;
the free surface does not significantly hinder the sinking plume, but it
is a boundary for the rising (positively buoyant) one.
Because of the aforementioned difficulties, another series of pilot
experiments was run. Tracer particles on the surface of the tank were
used to show the surface spreading and mixing pattern near the outlet.
Because the water discharged from the channel was clear, a visual impres-
sion of the extent and shape of the zone of flow establishment on the
water surface was obtained. A graduated marker suspended over the water
surface was used to estimate x . The length of the ZFE could not be
measured easily in this way, because the end of the zone is usually
marked by the convergence of large eddies which form in the free shear
zone between the free surface jet and the ambient reservoir. Since
these eddies move downstream apparently randomly, the length XQ also
varies randomly with time around some mean value. It was therefore de-
cided to use the average of several photographs to record visual values
of x . These data are of some interest for winter conditions; they are
reproduced in Fig. 8b and supplement Fig. 8a.
The photographic records can also be used to derive lateral spread-
ing angles for the shear zone 0 and spreading angles for the core
region. A set of values of is given in Fig. 9. The strong depend-
ence of ^ on F and the nearly complete lack of dependence on A,
the aspect ratio, are quite obvious. This means essentially that the
buoyancy-induced initial lateral spreading is quite similar in all
situations in which the discharge Froude number is the same. A
33
-------
50
40
30
20
10
7
6
5
4
3
T I I I I I I I
T = .98 .95 .90 .80
I I I I I I I I
5 6 7 8 9 10
20
30
Fig. 8a - x^/do versus FQ at T = 0.98, 0.95, 0.90, and 0.80 and
A = 2.0 (semicircular pipe, positive buoyancy)
-------
50
40
30
20
1 TT
10
8
7
6
5
4
3
A Visual observation (surface tracer)
O T = 0.98
I I I I I I I I
Fig. 8b -
5
*/<*.
6 7 8 9 10
20
30
x /d versus
a o
F at T = 0.98 and visual observations/
o
A = 2.0 (semicircular pipe, negative buoyancy)
35
-------
VISIBLE EDGE OF
FREE SHEAR ZONE
90C
60C
30°
A
o 1.0
A 2,4
o 10.3
225+ 10F
APPROX. ASYMPTOTIC
VALUE FOR N ON-BUOYANT JETS -,
456
F
8
Fig. 9 - Horizontal spreading angle, #, of free shear
zone recorded photographically (average value
of co not shown)
-------
dependency on aspect ratio need not be included initially. Mathematical
modeling of the width of thermal surface plumes has been a problem, and
relationships for surface spreading have mostly ignored the effects of
the aspect ratio. The information provided should be helpful in re-
solving some of the difficulties.
MAIN EXPERIMENTS: STRAIGHT JETS
Length of Zone of Flow Establishment (ZFE)
Photographs of Surface Flow
White, buoyant, nearly spherical particles with a specific gravity
of approximately 0.95. approximately 3/16 inch (5 mm) in diameter, were
spread on the surface of the experimental tank and retained there by a
boom which prevented their entrainment into the recirculating flow.
Water discharged from the outlet channel entrained the surface par-
ticles. The core region and the shear layers, defined schematically
in Pig. la, were made visible by the behavior of the particles on the
water surface of the tank. The core region was represented by a dark
triangular area and the shear layer by the random path of the particles,
which contrasted with the particle path in the ambient, more quiescent
pool. Figures lOa, lOb, and lOc give illustrations for outlet channel
aspect ratios A = 1, 2.1^, and 9«6 and Froude numbers F ranging from
oo (isothermal) to 2.5. A closer inspection of the photographs reveals
the following features:
a. Under isothermal conditions, the jet discharge has a tendency
to meander slightly around the main trajectory. This phe-
nomenon is caused by the vortex shedding in the free shear
zone and is more apparent for outlet channels of small aspect
ratio than for those of wide aspect ratio; increased buoyancy
of the discharge seems to suppress the phenomenon. The mean-
dering phenomenon was also mentioned by Jen, Wiegel, and
Mobarek .
b. Under isothermal conditions, fluid particles entrained on the
tank surface reach the center of the jet regardless of aspect
37
-------
Fig. 10a - Streaklines Illustrating surface flow patterns - A = 1.0, R = 0, Exposure = 1/10 sec
-------
3.5 ft
(1.07 m)
LO
•
CN
-H
3.5 ft
(1.07 m)
u
0)
Ul
IO
0)
D
CO
o
CL
X
0>
1-8
E-n
w>
O ">
.- O
is
i-o
8 l|
o-o
Si o
i
o
39
-------
5.0 ft
(1.52 m)
-H
o
•
CO
O -Q
05 O
§.
3
8
O"
-------
.4 .3
A - 1.0
— 4
A - 2.4
A = 9.6
— 5
25
Fig. 11 - Boundaries of visible core region for A = 1,0, 2.4, and 9,6 and various F
-------
50
40
30
1 I I I I I I
A = 1.01
20
10
9
8
7
6
5
4
J I I I I I I I
567
-------
ratio. This does not mean, however, that the point of con-
vergence marks the end of the ZFE.
c. When Froude numbers F are as low as 2 or 3, fluid particles
o
entrained from the tank surface do not reach the centerline of
the jet; this implies that the end of the ZFE is not controlled
by convergence of the lateral shear layers, but must "be
reached when the vertical shear layer reaches the water
surface.
d. Lateral entrainment velocities have a "pinching effect" on the
jet discharge near the outlet, apparently resulting in faster
convergence of the surface-entrained particles; the pinching
effect is caused by the jet-generated momentum of the later-
ally entrained flow, and it is more apparent for the wide
channel than for the narrower one.
Approximate boundaries of the visual (surface) core region as
derived from the photographs are shown in Pig. 11. A normalized repre-
sentation is given. Each line is an average "based on three to five
instantaneous pictures. Changes that occur in the visual core length are
consistent with Proude number, but are not proportional to the width of
the outlet channel. The buoyancy effect appears to be much stronger for
the narrow channel than for the wide one. Por the narrower channels the
visible core region seems to be independent of width. This is illustrated
more directly in Pig. 12. The dependency on aspect ratio appears to be
much stronger than that on Proude number.
Vater Temperature Data
Time-Averaged Water Temperatures—Time-averaged water temperatures or
velocities measured along the axis of the jet (on the surface of the tank)
decline gradually with distance from the point of discharge. Figure 15 gives
an illustration of the effect of Froude number on the temperature distribu-
tion. Data similar to those of Pig. 13 have previously been reported in
the literature by several experimenters. A type of summary was provided
Q
by Dornhelm, Nouel, and Wiegel . Figure 13 shows normalized excess
-------
1 I I I I I
0.20
Fig. 13 - Sample of time-averaged water temperatures along jet axis - A = 2.4, F = 2.2, 3.8,
3.9, 6.4, and 15.8 °
-------
temperature T versus normalized distance, x /w . The plot given is
useful for illustrating a number of points, including the following:
a) It is difficult to find the length of the ZFE by extrapolat-
ing backwards from the initial temperature decline curve.
b) An excess temperature concentration value between 0.98 and
0.80 can be chosen arbitrarily to define the length of the
ZFE; it can be seen that it is easy to commit major errors in
doing this.
Yalues of x /d derived for different values of T are given in Figs.
lUa» b, c, and d for aspect ratios of 1.0, 2.14., 1+.8, and 9-6 respectively.
The results can be interpreted as follows:
There exist two virtually opposite effects of buoyancy on the
length of the ZFE:
a) The buoyancy-induced spread of the plume (illustrated in
Pig. 9) is associated with outward velocities which prevent
the horizontal convergence of the shear layers and thereby
lengthen the ZFE.
b) The buoyancy-induced lateral spread of the plume also reduces
the depth of the plume and brings the vertical shear layer
closer to the surface, thereby promoting the shortening of
the ZFE.
Figures ll+a through d show that effect (a) typically appears at FQ> 5;
effect (b) seems to be dominant at F < l±. There is also some dependency
on aspect ratio; for example, at A = 1, effect (b) seems to be particu-
larly strong.
The above statements can be carried one step further by saying that
the length of the ZFE seems to be controlled primarily by the growth of
the vertical shear layer regardless of F values except when the aspect
ratio is small (A < 2). The effect of the aspect ratio on the ZFE is
shown explicitly in Figs. 15a, b, and c. The trend from an aspect ratio
of 1 to lj.,8 to 9«6 is fairly consistent. For a reason unknown to the
authors, it was more difficult to obtain consistent data for A = 2.1^
-------
50
40
I I I I I I I I
30
20
T= .98.95.90 .80
10
9
8
7
6
5
4
i I I I I I I I
Fig. 14a -
4 5 6 7 8 910
*y<*
or o
A = 1.0, R = 0
20
30
versus F at T = 0.98, 0.95, 0.90, and 0.80;
o
-------
50
40
30
20
F
o 10
9
8
7
6
5
4
I I I I Mill
T= .98 .95 .90
I L I I I I J I I
2 3 45678910
x/d
or o
20 30
Fig. 14b - Xo/do versus FQ at T = 0.98, 0.95, 0.90, and 0.80;
A = 2.4, R = 0
-------
50
40
30
20
10
9
8
7
6
5
4
1 I I I I I I I
T=.98 .95 .90
I I I I I I I I
.80
5 6 7 8 9 10
20
30
Fig. 14c - x /d versus F at T = 0.98, 0.95, 0.90, and 0.80;
o> o o
A = 4.8, R = 0
-------
50
40
30
20
o 10
9
8
7
6
5
4
T~T
T=.98 .95 .90
5 6 7 8 9 10
20
30
Xc/d0 versus F °* T = °°98' °'95/ Oo90/ and °'80'
A = 9.6, R = 0
Fig. 14d - *d/dQ versus F
-------
50
40
30
20
10
7
6
5
4
3
1 I I I I ! I
A = 1.0
I L I I I I I I
4 5 6 7 8 9 10
20
30
Fig. 15a - x /d versus F at T = 0.85; A = 1.0, 2.4, 4.8, and
a o o
9.6; and R = 0
50
-------
50
40
30
20
o 10
9
8
7
6
5
4
COMPUTED
3-DIM. 12-DIM.
(A = 1) j
A=1.0 2.4
I I I I I I I I
4.8 / 9.6
4 5 6 7 8 9 10
x /d
o' o
20
30
Fig. 15b - x /d versus F at T = 0.90; A = 1.0, 2.4, 4.8, and
9.6; and R = 0
51
-------
50
40
30
20
0 10
9
8
7
6
5
4
I 'I I I I I
COMPUTED!
3-D'M- 12-DIM
Fig. 15c - xo/do versus FQ at T = 0.95; A = 1.0, 2.4, 4.8, and
9.6; and R = 0
52
-------
than for any other value of A. This is indicated "by the standard devia-
tions of the x /d values given in Table 1. A strong secondary motion
causing upwelling along the centerline of the jet relatively close to the
outlet seemed to be one possible source of the peculiarity.
Table 1. STANDARD DEVIATIONS OF x /w and x /d
o' o o' o
AS GIVEN IN FIGURES ll+a, b, and d
Aspect Ratio A = 1 2.1+ 9.6
Standard ) ( O(XO/WQ) = 0.60 0.77 0.172
Deviation | \ a(xo/dQ) = 0.60 1.85 1-72
How do the results of Figs, llj. and 15 compare with those for non-
buoyant, circular, fully submerged jets? There are three main differ-
ences between the flows leading to the results of Figs. 11; and 15 and
the non-buoyant, circular, fully submerged jets investigated, for example,
by Albertson, Dai, Jensen, and Rouse : (l) initial jet shape; (2) buoy-
ancy; and (3) free surface.
To relate circular and non-circular jets one might use the hydrau-
lic diameter as shown in Table 2. The Albertson, et al., definition of
x for three-dimensional jets corresponds approximately to the 85 per
cent value of their measured velocity concentration, and their defini-
tion for two-dimensional jets corresponds approximately to the 90 per
cent value. Comparisons must therefore be made at those concentration
levels. A comparison of non-buoyant and buoyant jet results (based on
time-averaged temperatures) after elimination of shape effects is given
in Table 3. The results are also incorporated in Fig. 15-
As the buoyancy decreases in significance, values of xo/do "t611*
toward the values given in the second column of Table 3- Buoyancy
appears to reduce the length of the ZFE when the aspect ratio is small
(A < 3) and to lengthen the ZFE when the aspect ratio is large (A > 3)•
The greatest lengthening effect appears at F values in the vicin-
ity of 5. By extrapolation as shown in Fig. 15a it is found that
53
-------
Table 2. ESTABLISHMENT LENGTHS XQ COMPDTED FOR NON-BUOYANT, POLLY
SUBMERGED JETS (after Albert son, et al. )
x
For a two-dimensional, non-buoyant jet, -- = 5-2
For a three-dimensional, non-buoyant, axisymmetric jet, jj~ = 6.2
o
applied to rectangular channels of different aspect ratios A,
A =1.0:
A = 2.k: \ = 2.18 dQ -£• = 13-5
The limiting aspect ratio at which x^d =10.1^ for both two- and three-
dimensional jets is A = l.^i'
Table J. COMPARISON OF COMPDTED AND MEASURED x /dQ VALUES
_ Vfo _
Computed Measured
Non-Buoyant Buoyant
A _ B _ Max. Min. T
1.0 8.3 (thrftfi-dimansional) 7-U 5-U 0.85
2.4 10. U (two-dimensional) 7.2 U.8 0.90
4.8 10.4 (two-dimensional) 15.2 10.3 0.90
9.6 lO.U (two-dimensional) 20.5 15.0 0.90
51*
-------
at approximately FQ > JO the effects of buoyancy on the ZPB became
negligible.
A more direct comparison of shape effects is possible using the
results obtained with a semicircular outlet channel (A = 2), reported
in Pig. 8a, and the data for the rectangular outlet channel (A = 2.1;)
reported in Pig. ll^b. It is most intriguing to observe that the trends
reported on the two graphs are opposites. Furthermore, the order of
magnitude of the x_/d values is significantly higher for the semi-
circular outlet than for the rectangular one. The buoyancy effect
manifests itself for the semicircular outlet by reducing the length
of the ZFE (minimum of x /d at F =3); for the rectangular out-
let, xo/dQ has a maximum at PQ = 3. In attempting to resolve or
explain this apparent discrepancy, one must consider (a) the definition
of the Proude number in both cases and (b) the secondary currents.
For the semicircular channel d represents the maximum depth,
whereas for the rectangular channel it is the average depth. Proude
numbers for the semicircular channel should therefore be increased
before comparisons are made. (The factor would be approximately 1.2.)
The secondary motion referred to is documented only indirectly through
plots such as those shown in Figs. A-7 and A-9, to be discussed later.
The differences between Pigs. 8a and ll^b represent a shape effect other
than that expressed through the aspect ratio A.
A comparison of Pigs. 11 and 15 shows that surface tracers do not
give a good indication of the length of the ZFE, because they rely
primarily on the horizontal shear layer at the surface, whereas the
length of the ZFE depends more strongly on the vertical shear layer.
At small aspect ratios the visually determined values of XQ are
considerably lower than those found from time-averaged temperature data.
At large aspect ratios the results are more similar.
Fluctuating Water Temperature Components—The standard deviation a^
of the fluctuating water temperature component was computed from the rec-
ord obtained at each point. The standard deviation was normalized in the
55
-------
same way as the time- averaged water temperature measurements as defined
in Eq. (6). The resulting dimensionless parameter §T was plotted in
several formats.
Centerline values of ST versus x/d, are given in Figs. l6a
through I6d. The following observations were made:
a. The background level of the water temperature fluctuations as
determined by ST is less than one per cent in the core
region of the discharge. This value is a measure of the
accuracy with which water temperatures in the discharge
channel were controlled by the heat exchanger-thermostat-
buffer tank system.
b. There is a gradual transition from the near isothermal core
region to the region in which water temperature fluctuations
are more significant. This reflects the presence of turbu-
lent eddies penetrating from the free shear layer into the
core region.
c. Peak values of ST reached were on the order of 5 to 13 per
cent. A dependence of peak values on outlet Froude number,
but not on aspect ratio is apparent. Values of F between
2.0 and 3-0 consistently gave the lowest measured peak values
of 5T. In general .terms this can be interpreted as the effect
of increased stratification stability on turbulence.
d. The point at which peak values of ST occur was generally
found to be farther from the outlet when aspect ratios were
large. This reflects a lengthening of the ZFE which is in
accordance with earlier findings based on time-averaged tem-
peratures .
e. It should be remarked that results for aspect ratio
A = 2.1; again showed a much larger spread of results
at nearly identical F values than those for any
other aspect ratio. No clear explanation for this
phenomenon has been found, but it is suspected that it may
56
-------
.10
Ul
.08
.06
ST
.04
.02
2.40
3.30
4.93
8.92
11.50
I I I I
Reo
11,000
17,000
20,000
29,000
27,600
I I I I I I
789 10
20
30
u
Fig. 16a - 5T versus x/d along jet axis at A = 1.0 and R = 0
-------
.10
1 I I I 1 I I
.06
VJl
00
.04
.02
1.97
3.85
6.90
9.92
15.80
9,200
10,600
24,000
21,000
29,700
I I I I I I
5
x/d
6 7 8 9 10
20
30
Fig. 16b - 8T versus x/d along fef axis af A = 2.4 and R = 0
-------
VJI
.10
.08
.06
ST
.04
.02
o
2.04
3.22
1 I \ I I I I
Re
o
5,730
10,800
8.00 17,300
I I I I 1 1
7 8 9 10
20
30
Fig. 16c - 5T versus x/dQ along jet axis at A = 4.8 and R = 0
-------
Fig. 16d - Sj versus x/d along jet axis at A = 9.6 and R = 0
-------
be related to the secondary flow mechanism referred to pre-
viously.
The length of the ZPE was derived from Pigs l6a through l6d "by
arbitrarily selecting the two and four per cent levels of ST. The re-
sults are accumulated in Pigs. 17a and l?b. The significant features of
the plots are
a. the dependency of x /d on the aspect ratio at low Froude
numbers;
b. the possible existence of an upper envelope curve that encom-
passes all aspect ratios and Froude numbers.
Figures 15 and 17 both give lengths of the ZFE, but they are based on
different definitions. Although the graphs are quite dissimilar, they
show similarly strong effects of aspect ratios on x /d values. To
establish a more direct relationship between Figs. 15 and 17, the T
levels which would give the same values of x /d as were found at the
T = 0.85, 0.90, and 0.95 levels were determined. Average values of 5T
and T for all Froude numbers were used, and the results are given in
Table Ij. Through this table a relationship is established between the
ZPE data derived from time-averaged temperatures and those derived from
fluctuating components. The scatter of the results is represented by
the standard deviation a~_ around the weighted mean for all aspect
ratios. It is quite evident that the results from time-averaged and
fluctuating temperatures correspond less consistently as the aspect ratio
goes up.
Table k- EQUIVALENT VALUES OP T and 5T
T = 0.85 T = 0.90 T = 0.95
A
1.0
2.4
4.8
9.6
5T%
5.9
5.4
8.2
4.1
o/
°ST ^
1.0
2.0
3.0
3.1
5T°/6
4.7
U-5
6.6
3.2
CT* >C
AT
0.9
2.0
2.3
3-5
i ST°/>
3-2
3-4
4.2
2.2
o/
0.6
2.1
1.6
2.6
61
-------
50
40
30
20
10
9
8
7
6
5
4
TT
A = 2.4
I I I I I I I I
4 5 6 7 8 9 10
x/d
o o
20 30
Fig. 17a -x/d versus F at 5T = 0.02; A = 1.0, 2.4, 4.8, and
o o o
9.6; and R = 0
62
-------
50
40
30
20
10
9
8
7
6
5
4
1 I I I I I I I
A= 1.0
I I
I I I I I
4 5 6 7 8 9 10
x /d
Of O
20
30
Fig. 17b - x /d versus F at ST = 0.04; A = KO, 2.4, 4.8, and
o o o
9.6; and R = 0
-------
Measurements of the fluctuating water temperature components are
also useful for showing the entrainment and mixing mechanisms near the
outlet. To obtain this information, water temperature profiles were
measured not only along the centerline of the jet, "but also in two or
three cross sections perpendicular to it. Although it is possible to
present only a selection from the data obtained, the following observa-
tions merit being listed and illustrated:
a. The lateral spreading and mixing by large eddies formed in the
free shear layer between the discharged jet and the ambient
stagnant water appears to be more or less independent of the
aspect ratio, as was inferred previously from visual observa-
tions of spreading angles (Pig. 9). Figures 18a, b, c, and d
show typical results for aspect ratios of 1.0, 2.1±, and 9.6.
b. The effect of buoyancy (low F ) is to increase the spreading
angle, as is well known. This is illustrated in Pig. 19.
c. Figure 19 also shows that the width of the free shear layer,
as measured by 5T values near the surface, is wider for the
wider outlet channel. This is related to the dimensions of
the horizontal eddies (vertical axis), which are apparently
larger for the wider channel.
A series of graphs showing iso-temperature fluctuation lines is
given in Appendix A. These graphs illustrate the effects of aspect ratio
and Proude number in more detail; they identify areas of high fluctua-
tion intensity. It can be seen, for example, that at a high degree of
stability of stratification (low P values), the measurements identify a
surprisingly wide vertical area of high 5T values. This is believed to be
the result of internal waves forming near the point of discharge where the
discharged warmer jet flow meets the colder ambient as observed in dye ex-
periments. The existence of such waves (breaking or non-breaking) has
previously been reported by Stefan, Hayakawa, and Schiebe . Larger
areas of high $T values are not necessarily associated with more
entrainment of ambient colder water. They may merely reflect larger
-------
ON
VJl
Fig. 18a - 5T versus y/d0 perpendicular to jet axis at A = 1,0,
x/d0 =1.7, R = 0, and F0 = 2.2, 3.5, and 6.5
-------
ON
Fig. 18b - ST versus y/d0 perpendicular to jet axis at A = 1.0, x/d = 3.5, R = 0,
and FQ = 2.2, 3.5, and 6.5
-------
o
cr\
Fig. 18c - 5T versus y/d perpendicular to jet axis at A = 2.4, x/d = 2,
R = 0, and F° = 2.2, 3.8, and 15.8 °
-------
Fig. 18d - ST versus y/cL perpendicuJar to jet axis at A = 9.6, x/d0 = 2,
R = 0, and Fft = 2.1 and 4.1
-------
0\
vo
Dork symbols indicate
ST = 0.02.
NOTES: 1.
Open symbols indicate
5T = 0.01.
Projection of the
channel wall
Fig. 19 - Spreading angle and width of free shear zone at 5T = 0.01 and 0.02 for
A = 1.0, 2.4, and 9.6; x/dQ =2 and R = 0
-------
amplitudes of the vertical motions, and only if these waves are breaking
waves or eddies will the entrainment be increased.
Upwelling of isotherms can be observed in Figs. A-7, A-8, and A-9
along the centerline of the jet. This is caused by an inward flow of
cold water underneath the jet. As the flow converges toward the center-
line of the jet, upwelling is caused through a stagnation effect. The
phenomenon is much weaker for A = 9-6 than for A = 2.4-
In all longitudinal sections the iso-concentration lines of time-
averaged water temperatures are quite similar in shape to the iso-fluc-
tuation lines.
Velocity Data
Time-Averaged Velooitiea—Velocities were measured as local samples
consisting of 50 instantaneous measurements spread over a one-minute
period. Because of the size of the Delft current meter (diameter of the
rotor = l^ mm), the measurement cannot be referred to as a point measure-
ment; the results are nevertheless presented because they supplement the
temperature data and are also used to calculate volumetric flow rates
through integration of velocities in cross sections perpendicular to the
jet axis.
Samples of instantaneous velocity measurements were used to compute
time-averaged local velocities which were subsequently normalized using
the centerline discharge velocity as a reference value. The resulting
velocity concentrations, defined by Eq. (7)» were first plotted versus
distance along the axis of the jet, as were the excess temperatures in
Fig. 13. Distances x /dQ were read at the 0.98, 0.95, 0.90, and 0.80
levels and plotted versus Froude number F for different aspect
ratios. The results are shown in Figs. 20a, b, and c and 21.
The results derived from time-averaged temperatures in Figs. 11* and
15 can be compared with those obtained from time-averaged velocities in
Figs. 20 and 21. It is apparent that increases in outlet channel aspect
ratio result in longer ZFE lengths in terms of both temperature and
velocity. The variations with Froude number are not consistent, however.
70
-------
50
40
30
20
o 10
9
8
7
6
5
4
TTTT
I I I I I I I I
56789 10
20
30
Fig. 20a - x /d versus F at V = 0.95, 0.90, and 0.80; A = 1.0;
O O 0
and R = 0
71
-------
50
40
30
20
o 10
9
8
7
6
5
4
i—i—i i i in 11
V= .95
I I I I I I I I I
4 5 6 7 8 9 10
x /d
G O
20 30
Fig. 20b - x/d versus F of V = 0.95f 0.90, and 0.80; A = 4.8;
and R = 0
72
-------
50
40
i i i 11 i n
301
V= .981
.95
.90
20
10
9
8
7
6
5
4
i i I I I I I I
5 6 7 8 9 10
20
30
Fig. 20c - xydo versus FQ at V = 0.98, 0.95, 0.90, and 0.80;
A = 9.6; and R = 0
-------
50
40
30
20
0 10
9
8
7
6
5
I I I | I
I 1 I I I I I I
4 5 6 7 8 9 10
20 30
Fig. 21 - xo/do versus FO at V = 0.9; A = 1.0, 4.8, and 9.6;
and R = 0
-------
The Froude number effect is more subtle than that of the aspect ratio;
Proude number trends are similar for aspect ratios A = 1.0 and A = 14.8,
but dissimilar for A = 9.6 "below approximately F = J. Absolute
values of x /dQ at the 0.90 level or the 0.80 level do not match.
Transfer of mass and transfer of momentum are somewhat different, but
there are also differences in the spatial and temporal resolution of
the temperature and velocity instrumentation, as was pointed out earlier;
in addition, temperatures are initially uniform in the outlet channel,
whereas velocities are not. (it should be remembered that the veloci-
ties have been normalized using the maximum velocity at the outlet.)
Typical normalized average outlet velocities range from 1.156 at A = 1.0
to 1.095 at A = i^.8 and l.Olj. at A = 9-6. Time-averaged percentage
ratios of velocities should be increased by these factors before a
comparison with time-averaged temperature data is made. But even
then it will be difficult to interpret the results of such a com-
parison.
When the above procedure is applied, it is found that the lengths
of the ZFE based on time-averaged velocities are still significantly
shorter than those derived from time-averaged temperatures. Table 5
gives the ratios.
Table 5. EATIOS OP XQ VALUES, t ^ DERIVED PROM
_ ___ ____ _
TEMPERATURE AND VELOCITY
A = 1.0 A = lj.8 A = 9.6
T=T= T=T= _ T=T =
,80 o .90 .80 o .90 .80
2.0 2.7 2.2 2.0 1.6 1.5 1.7 0.93 0.9
3.0 2.7 2.1* - -
3.1* 1.9 1-7 3.3 1.1* 1.3 3-2 1.2 1.0
k-k 2.0 1.8 7.8 1.1* 1.1 7.0 1.3 1.3
11.6 1.5 1.5 10.5 1.1 1.0 10.0 l.i 1.2
75
-------
Fluctuating Velocity Components—Interpreting the turbulent veloc-
ity data is even more difficult than interpreting the time-averaged
values, because the exact response characteristics of the Delft current
meter are not known. Sample results are presented in Figs. 22a and 22b.
Volumetric Flow Rates and Entrainment
Integration of velocity profiles in cross sections perpendicular to
the jet axis provided information on the volumetric flow rates, specific-
ally on their variation with distance from the outlet. This information
gives both insight into the entrainment mechanism and data which can be
compared directly with those obtained using mathematical models. This
information is contained in Figs. 23a, b, and c. The effects of the as-
pect ratio as well as those of the Froude number are clear. For further
comparison, Fig. 21± shows the computed flow ratio for non-buoyant, fully
submerged jets after Albertson, et al. . The results have been obtained
by applying the three-dimensional theory to outlets of small aspect ratio,
A = 1 and A = 2.1; (a hydraulic diameter was used instead of a real di-
ameter), and the two-dimensional theory to A->o> . The results of Fig.
2\4 match those of Fig. 23 qualitatively. To make the effects of A
and F even clearer, Fig. 25 was prepared. It is noteworthy that
at the larger aspect ratios, changes in the Froude number reduce the
flow ratio by no more than 20 per cent below that for isothermal
(F ->oo) conditions. This means that an increase in stability due to
density stratification does not reduce the initial entrainment excess-
ively. It might be argued that buoyancy increases the contact area be-
tween warm and cold water through lateral spreading and thereby counter-
acts the stability effects to some degree. But the main reason for the
relatively high degree of entrainment in the presence of buoyancy seems
to be the vortex and internal wave formation near the outlet due to
viscous shear regardless of Froude number. Figure 26 illustrates this
vortex formation. The photograph was obtained by injecting blue dye
at the wall of the outlet channel near the water surface and red dye at
the bottom of the outlet channel. Even in the black-and-white reproduc-
tion, the vortex (and internal wave) shedding shows up very clearly.
76
-------
50
40
30
20
10
9
8
7
6
5V
NT
I I I I I I I I
5 6 7 8 9 10
20
30
Fig. 22a - x
-------
50
40
30
20
10
9
8
=o 7
6
i—r
5V =
3 4 56789 10
20 30
Fig. 22b - xo/dQ versus FQ at 5V = 0.075 and 0.10, A = 4.8,
and R = 0
78
-------
Q/Qr
VD
SYMBOL F0
Re,
o
O oo 10,500
O 4.4 12,400
A 3.2 14,000
O 2.0 8,000
A
8
10
15
20
25
Fig. 23a - Q/Q0 versus x/d0 at A = 1.0 and R = 0
-------
SYMBOL
O
o
A
o
CO
3.50
2.04
13,100
9,500
7,300
I
10
15
x/d,
20
25
30
Fig. 23b - Q/Q0 versus x/cL at A = 4.8 and R = 0
-------
00
SYMBOL F
o
e oo
a 3.5
o 1.8
Reo
6100
8200
9500
10
15
x/cL
20
25
30
Fig. 23c - Q/QO versus x/cfe at A = 9.6 and R = 0
-------
CO
ro
Q/G>
ASPECT RATIO = 1
10
15
x/d
20
25
30
Fig. 24 -
Q/Q0 versus x/dQ for fully submerged, non-buoyant two- and three-dimensional
fets (computed after Albertson, et al.6 to illustrate effect of aspect ratio)
-------
CD
Q/Q,
Fig. 25 - Q/Q
4 6
ASPECT RATIO, A
versus A at
8
10
FQ = 1.8,
x/d = 10; and R = 0
o
F = 3.4,
o
12
and F -*• CD;
o
-------
Outlet channel
Flow
Fig. 26 - Photograph of vortex formation near the outlet at
A = 2.4, FQ = 3.46, and R = 0
-------
MAIN EXPERIMENTS: CURVILINEAR JETS
Photographs of Flow Features
Photographic data on surface flow patterns were not acquired in a
systematic manner because they are of questionable value with regard to
determining the length and trajectory of the zone of re-establishment.
However, they do provide useful illustrations of some characteristic flow
features. Figure 27, for example, shows the discharge from a channel
with A = 2.1; at F =25.8 and F =2.8. Significant features are
a. the approximate surface extent of the plume as shown by the
streaks of floating particles;
b. the deflection of the jet by the cross-flow;
c. the vortex shedding and undulation of the jet trajectory at
the higher F value;
d. the area of upwelling of the cross-flow marked by the absence
of particles on the downstream side of the plume;
e. the core region, a triangular region without particles in
front of the outlet.
Trajectories
The plume trajectory was determined by several different methods at
each aspect ratio. The plume axis location was first estimated by in-
jecting dye into the outlet channel and observing the path of the most
rapidly moving dye. The flow direction was then measured with a vane
suspended by a taut mono filament thread. A 1/1^-in.-thick brass disk was
also attached to the vane shaft to provide inertia to dampen the high-
frequency oscillations. These data were then used to supplement the dye
measurements to determine the plume trajectory to be used as a baseline
for the lateral velocity and temperature profiles. The velocity and tem-
perature measurements were taken in planes perpendicular to the baseline
plume trajectory. The same procedure was followed at each aspect ratio
for the isothermal case and with a Froude number of approximately 3.25-
85
-------
CD
ON
FQ = 25.8
F = 2.8
o
Fig. 27 - Photographs of discharge into a crossflow at A = 2.4, R = 0.41, and
F = 25.4 and 2.8
-------
A typical plot of the baseline trajectory for an aspect ratio of 1+.8 at
a Froude number of 3,25 is shown in Fig. 28. The dashed lines show the
location of the lateral profiles. Note also in Fig. 28 that the plume
centerline lies very near the shear zone "between the outlet flow and the
cross-flow. The plume seems to decay mostly from the upstream side,
where the warm water is swept down and under the plume and then down-
stream "by the cross-flow.
The plume trajectory can also be determined from the maximum
velocities and temperatures measured in the lateral surveys. These are
shown for aspect ratios of 1.0, 2.1^, and i|.8 in Figs. 29a, b, and c
respectively. Note that the effect of buoyancy is to straighten out the
trajectory. The observed centerline coincides quite well with that deter-
mined from the maximum velocities for both the buoyant and non-buoyant
cases. The line of maximum temperature lies downstream of the maximum
velocity for both of the narrower channels (A = 1.0 and 2.i|) and coin-
cides with the maximum velocity for the wide channel (A = 1+.8).
Figure J>0a shows a summary plot of the observed plume centerlines
for all three channels for both the buoyant and non-buoyant cases. A
summary plot giving our best estimate of plume trajectory based on maxi-
mum velocities from the lateral surveys is given in Fig. 30b. The effect
of both aspect ratio and Froude number can be seen clearly. As the chan-
nel width increases, the plume trajectory tends to flatten out. The
effect of changing the Froude number from non-buoyant to F =3-25 is
much larger than the effect of the aspect ratio. The Froude number was
varied by changing only the outlet temperature. The outlet velocity was
held constant for all the crossflow experiments. The average outlet
velocity was 0.53 fps (0.162 m/sec) and the crossflow velocity ratio, R,
was Q.1^15. The outlet Reynolds number, Re , varied from about 11,000
to 15,000 depending on the outlet water temperature.
Length of Zone of Flow Establishment (ZFE)
The development length s /d has been measured using three dif-
ferent techniques: (l) outflow velocity decay, (2) temperature concentra-
tion, and (3) temperature fluctuation. The velocity fluctuations were
8?
-------
10
x/d
15
20
25
OUTLET
CHANNEL
0
y/d
o
-5
-10
I
I
NOTES: 1. O Observed center-line from
dye Injection.
2. Flag orientation shows flow
direction from vane measurements.
TT \3> Dashed lines indicate lateral
I \ profile locations. —
\
CROSSFLOW
\ \
CENTERLINE
TRAJECTORY
\ \ \
I
I
I
Fig. 28 - Typical cenferline plume trajectory at A
R = 0.41, and FQ = 3.25
88
= 4.8,
-------
-10 —
00
VO
O Maximum Velocity, F = 3.2
o
A Maximum Temperature, F =3,2
O Maximum Velocity, F = a>
—— Observed Centerline, F =3.2
— —. ——Observed Centerline, F = ao
o
15
20
25
x/d
Fig. 29a - Plume trajectory at A = 1.0, R =
0.41, and FQ= 3.2 and oo
-------
-10 —
vo
o
Observed Center line,
F0=3.2
__ m^m m^mm Observed Center! ine,
F0 = 00
O Maximum Velocity, F = 3.2
A Maximum Temperature, F =3.2
O Maximum Velocity, F = co
10
15
20
25
Fig. 29b - Plume trajectory at A = 2.4, R = 0.41, and Fo= 3.2 and co
-------
-10
-5
vo
Observed Centerline, F =3.2
^~~— ^~ Observed Center line. F = o>
o
O Maximum Velocity, F = 3.2
A Maximum Temperature, F =3.2
D Maximum Velocity, F = co
5 10
o
Fig. 29c - Plume trajectory at A = 4.8, R
15
20
0.41, and F = 3.2 and oo
25
-------
VO
ro
-10
-5
A =
10
15
— Fo = co
Fo = 3.2
20
25
Fig. 30a - Observed trajectories, A = 1.0, 2.4, and 4.8; FQ= 3.3 and co; and R =
0.41
-------
vo
-10
-5
A = 1.0
10
15
Fo=co
F = 3.2
o
20
25
Fig. 30b - Trajectories based on maximum velocity, A
and R = 0.41
= 1 .0, 2,4, and 4.8; F = 3.3 and co;
-------
not used to determine the ZFE length "because of the high background tur-
bulence levels both in the outlet channel and in the cross-flow. It
should be recalled that the crossflow turbulence level is 26 per cent.
Results are given in Tigs. 31 through 33b. The development length has
been taken as the distance along the plume centerline (trajectory).
Note the decrease in s /d for an aspect ratio of 2.4 relative to
its value for A = 1.0 near the outlet. This initial discrepancy
disappears farther from the outlet and by the time the 80 to 85 per cent
level is reached, the variation of s /d with the aspect ratio is
nearly linear on log-log paper; i.e., s /d is proportional to (A)n.
The dropoff in x /d values near A = 2.1). was observed previously
for the straight discharges. It reflects a rather effective penetration
of ambient cold water into the discharged warm water caused by a second-
ary motion around a horizontal axis, which is apparently favored by the
aspect ratio. Isotherm and isoturbulence lines presented in Pigs. B-7
through B-9 strongly suggest the existence of such a mechanism, at
least in the case of a straight jet discharge. In the presence of a
cross-flow, the isotherm pattern is significantly different, but a vor-
tex again forms around a horizontal axis.
The effect of this early mixing at a favorable aspect ratio fades
out with increasing distance from the outlet, due to the superposition
of other effects, so that at lower excess temperature concentrations of
70 and 60 per cent the channel with A = 2.1* no longer shows a trend
which is different from those of the other aspect ratios.
It is of some interest to explore whether the end of the ZPE is
dominated by vertical or horizontal entrainment of ambient water. As the
length of s /d increases rather drastically with the aspect ratio,
one may also wonder whether this trend would come to a halt at a large
enough aspect ratio.
It is obvious from the results obtained and from direct observa-
tion that at small aspect ratios, horizontal entrainment of cold water
by large eddies shed in the free shear zone between the discharged jet
and the ambient current is responsible for the ZFE. Figure 27 shows
9k
-------
VJ1
I III
5 6 7 8 9 10
20
30
Fig. 31 - So/d0 versus A at FQ= 3.2, R = 0.41, and T = 0.95, 0.90, 0.85, 0.80, 0.70,
and 0.60
-------
vo
I I I I I I I
5 "67 8 9 10
Fig. 32 - So/d0 versus A at Fo= 3.2, R = 0.41, and ST = 0.02, 0.03, 0.05, 0.07, and
0.10
-------
VD
10
9
8
7
6
I I I I I I I
1
V
V 98
O 95
a 90
A 85
O 80
$ 70
O 60
20
30
2 3 4 56789 10
$/d
a o
Fig. 33a - Sj/dj, versus A at Fo = 3.2, R = 0.41, and V = 0.90, 0.85, 0.80, 0.70, and
0.60
-------
vo
CD
I I I I I I
5 6 7 8 9 10
Fig. 33b - SO/JQ versus A at FQ = oo, R = 0.41, and V = 0.90, 0.85, 0.80, 0.70, and
0.60
-------
the formation of these eddies. The axis of these eddies is not vertical,
but bent over and entrained by the cross-flow underneath the discharged
jet as shown in Pig. Jl| (photo), so that horizontal and vertical entrain-
ment and mixing are linked by the same eddies. It can therefore be
expected that at small aspect ratios and closer to the outlet—i.e., for
T = 0.95 or 0.90 or 0.85—the horizontal entrainment is the limiting
mechanism. Farther downstream, vertical entrainment is very important
as well. Whether there is a limit to the value of s /d cannot
o' o
really be determined on the basis of the available data. It should
also be pointed out that the cross-flow has two opposing effects on
s ; one is to lengthen the ZFE by adding a transverse (lateral)
velocity component to the flow, and the other is to increase the
in-i Ting intensity. The end of the ZIE usually falls into the fairly
straight portion of the trajectory, and therefore the effect of the
cross-flow on the length of the ZEE will be seen to be essentially a
slight shortening effect.
99
-------
Outlet
channel
Dye injected
at surface
I
u
Dye injected at
bottom of outlet
Fig. 34 - Photograph of dye patterns near the outlet of a discharge
into a crossflow - A = 2.4. F = co, R = 0.41
o
100
-------
SECTION VII
ANALYTICAL EXPRESSIONS FOR ESTABLISHMENT LENGTH
AND INITIAL FLOW RATES
LENGTH OF THE ZONE OF FLOW ESTABLISHMENT
The relationship sought is of the basic form
s
£(•£-, FQ, A, T, R) = 0 (11)
o
The function f has teen studied by several investigators. Motz and
2
Benedict expressed it in the form
(12)
where x is the x-projection of the distance s measured along
o I o
the trajectory. Prych4 simply used the relationship
xo = H>
where D is the equivalent diameter of a semicircular outlet channel.
The relationship was rewritten by Shirazi and Davis as
=6. 38 A1/2 (1U)
5
Shiraai and Davis proposed the relationship
- 5-1. A2/5 1/3
(15)
Equation (15) can be used only for F < 10. it gives
x = 0 when F -> CD.
o o
101
-------
12
The effect of a cross-flow on s was expressed by Pan for
circular, fully submerged jets, previously investigated "by Gordier1',
ae
s
~= 6.2 exp(-3.32 H) (16)
o
Shirazi ^ specified a function f of the form
f (o) = expa (f-) R° Pod A6 X* a* (17)
o
where f(o) is a desired plume characteristic— for example, T; K is
the slope of the beach over which the discharge is made; and a is the
discharge angle with respect to the current. Laboratory and field
data, taken from different sources and covering some of the independent
variables in a not-very-coherent way, were assembled. Multiple regres-
sion analysis was applied to individual data separately and to one or
more sets in combination to obtain the constants. Results based on
Q
excess temperature T were as follows t
Group I: Discharge into deep stagnant pool
(f- f- R° F0-°'* A°'55
O
-0.1 < a < 2.5
Group II: Discharge into a stagnant pool over a sloping shore
(f-)'- E° I"0' 22 A0'
O
0.77 < a < 1.1+6
Group III: Discharge into a deep moving flow
T = «pa (f ,°-» I -«•* A0'2' «-°-» (20)
0'2'
O
0.36 < a < li.06
102
-------
The data previously obtained in the same tank as was used for this study
and repor-
A= 3.07,
and reported "by Stefan, Hayakawa, and Schiebe^ gave, for Group I and
T = e^p* (f-) *o°'°5
o
In the analysis of the experimental data collected under the pres-
ent study it seemed advisable to again use separation of the variables.
However, enough data had been collected to attempt formulations which
would fit a scheme of the form
fl(^ f2 f3(A) Vo) f5(R) = ° (22)
o
or even more complicated functions of several independent variables. In
pursuing this idea, it was found that the strongest dependence of
s /d values at specified levels of T seemed to be with respect to
A. A plot of log A versus log(sQ/d ) such as that shown in Pig. Jl
for R = 0.1(1 shows an approximately linear relationship, particularly
at T = 0.8 and below. A similar plot for the straight jet data
(R = 0) was made using the data from Figs. ll±a, b, c, and d, and the
result was found to be nearly the same. An appropriate functional
relationship therefore seemed to be
f = KA* (23)
o
Prom the graphs n =0.5 was found to best fit time-averaged
water temperature data for both straight and curvilinear jets at
T = 0.9; rL-O.B for T = 0.80. Water temperature fluctuation
data at T = 0.10 produced a value of n = 0.6. While values of n
seemed fairly independent of R, K values differed by a factor of
approximately 1.1; when R was changed from 0 to O.lil. The straight
flow produced the larger E, and thus tne greater development length.
Subsequently it was found that for all the data collected, a relation-
ship existed between K and n that could be cast into the form
103
-------
K1/"' = r/n
The data are shown in Fig. 35- a was found to be nearly constant and
to equal approximately 1.3« An approximate linear fit for y(T) was
y = + ?R), with 17 being on the order of -0.9 for
0 < R £ O.J|1. The form of the function f, will be derived from the
data for A = 1.
The coefficients e= 0.5, (3 = -1.5t and S = 0>k provide an accept
able fit to the data. Equation (27) has a maximum at F = -f +•*.
€ + B
The asymptotic values of f, are f, (l) = 1 + — jr*- and f, (CD) = l.
6
The format of the complete empirical equation, then, is
derived from data in the following ranges:
0.8 < T < 0.98 2.0
-------
n
I I I I
Flagged symbols indicate
velocity data
Open symbols indicate R = 0
Dark symbols indicate R = 0.41
II!
6 7 8 9 10
Fig. 35 - n versus K
105
-------
o
ON
y 2
y = 8.2 - 7.0 T —
0.5 0.6
0.7 0.8
T
Fig. 36 - y versus T
0.9
1.0
-------
<£= 16.0, i// = -12.8, n = 0.1+6, a = 1.2, (3 = -1.50, « = 0.50,
8 = 0.1^0, and 17 = -0.9* Here n is not really a constant, but is
somewhat dependent on the temperature level T. A modified version of
Eq. (28) in which n = constant was replaced by
n =
was also investigated. The numerical coefficients obtained by non-
linear curve-fitting for the modified Eq. (28) were n. = 0.85 and
n2 = -0.10;.
The modified Eq. (28) no longer fits the general format of Eq. (22)
At that level of approximation, many different solutions are possible.
Substituting the above numbers in Eq. (28) gives the final dimensionless
equation for the length of the zone of flow establishment:
. (16.0-12.8
A value of T = 0.90 is thought to represent a meaningful choice to
designate the end of the ZFE. For this value the above equation becomes
(50)
Approximations beyond Eq. (28) produce only minor improvements unless
vast amounts of data are available. The non-linear curve-fitting program
was conducted with 85 data points. Such a number does not justify higher
order approximations beyond Eqs. (22) and (28).
INITIAL ENTRAINMENT FOR STRAIGHT JETS
Empirical flow rate equations are needed only for the ZFE, since the
fully developed flow region can be handled readily by mathematical models.
The available data justify the derivation of empirical equations only for
straight jets. The relationships given by Albert son, et al. may serve
as a guide in formulating the equations. Outlet shape (aspect ratio A)
10?
-------
and outlet densimetric Froude number (FQ) must be incorporated into the
formulations. For fully submerged axisymmetric and slot jets, respec-
tively, Albertson, et al. have given the following relationships for
the volumetric flow rates:
2
within the ZFE: 7*- = 1 + 0.083 ^~ + 0.0128 (•—-) (jia)
So oo
0.080^- (jib)
o o
in the fully established flow region:
(32a)
o
°'5
(32b)
For rectangular outlets of variable aspect ratio A, a format of the
equation is sought that will degenerate into those given above when
A -» 2 or A-^OD. Because of free surface and shape effects, the coef-
ficients should be expected to be different from those given in the
above equations.
The F effect can be cast into a coefficient of the form
c, °
(1 - — •*-) where c_ = 0.5 and q = Q.k « With this provision, the
V
data shown in Fig. 25 will replot on a single line. The effect of A
will be incorporated in the coefficients c,, c-, and p in the
equation
(33)
F ^
O
Values of c.. , c«, and p must be derived from the data. Using a
non-linear curve- fit ting technique, the following coefficients were
found to provide a good match of the data: c, =0; Cp = 0.087; and
p = *35 + °*'^. The final empirical equation for volumetric flow
0. y\J 4" A
108
-------
rates of straight jets near the outlet is
0.90-I-A
Experimental data from whioh the above relationship was derived covered
the ranges 1 < A £ 9.6 and FQ > 1.8. The eq
outlet f within the ZFE, and somewhat beyond the
the ranges 1 < A £ 9.6 and FQ > 1.8. The equation is valid near the
109
-------
SECTION VIII
REFERENCES
1. Stolzenbach, K. D. and D. R. F. Harleman. An Analytical and Experi-
mental Investigation of Surface Discharges of Heated Water. Water Pol-
lution Control Research Series, U.S. Environmental Protection Agency,
16130 DJV February 1971. (Also: Stolzenbach, K. D. and D. R. F. Harle-
man. Three Dimensional Heated Surface Jets. Water Resources Research
9(1):129-137, February 1973.)
2. Motz, L. H. and B. A. Benedict. Heated Surface Jet Discharged into a
Flowing Ambient Stream. Water Pollution Control Research Series, U.S.
Environmental Protection Agency, 16130 FDQ, March 1971.
3. Stefan, H., N. Hayakawa, and F. R. Schiebe. Surface Discharge of
Heated Water. Water Pollution Control Research Series, U.S. Environ-
mental Protection Agency, 16130 FSU December 1971. (Also: Stefan, H. and
P. Vaidyaraman. Jet-Type Model for the Three-Dimensional Thermal Plume
in a Cross-Current and Under Wind. Water Resources Research 8(4):998-
1014, 1972.)
4. Prych, E. A. A Warm Water Effluent Analyzed as a Buoyant Surface Jet.
Sveriges Meteorologlska och Hydrologiska Institut, Stockholm, Report
No. 21, 1972.
5. Shirazi, M. A. and L. R. Davis. Workbook of Thermal Plume Prediction -
Vol. 2 - Surface Discharge. Environmental Protection Technology Series,
U.S. Environmental Protection Agency, R2-72-005b, May 1974.
6. Albertson, M. L., Y. B. Dai, R. A. Jensen, and H. Rouse. Diffusion of
Submerged Jets. Transactions, ASCE 115:639-697, 1950.
7. Hirst, E. Zone of Flow Establishment for Round Buoyant Jets. Water
Resources Research 8(5):1234-1246, October 1972.
8. Shirazi, Mostafa A. A Critical Review of Laboratory and Some Field
Experimental Data on Surface Jet Discharge of Heated Water. Pacific
Northwest Environmental Research Laboratory, Corvallis, Oregon,
Working Paper No. 4, 1973.
9. Dornhelm, R., M. Nouel, and R. L. Wiegel. Velocity and Temperature in
a Buoyant Jet. Jrl. Power Div. ASCE 98(P01):29-47, June 1972.
10. Tamai, N., R. L. Wiegel, and G. F. Tornberg. Horizontal Surface
Discharge of Warm Water Jets. Jrl. Power Div. ASCE 95:253, October 1969.
11. Jen, Y., R. L. Wiegel, and I. Mobarek. Surface Discharge of Horizon-
tal Warm Water Jet. Jrl. Power Div. ASCE 92(P02):l-28, April 1966.
110
-------
12. Fan, Loh-Nien. Turbulent Buoyant Jets into Stratified or Flowing
Ambient Fluids. W. M. Keck Laboratory of Hydraulics and Water
Resources, California Institute of Technology. Technical Report No.
KE-R-15, June 1967. 196 p.
13. Gordier, R. L. Studies on Fluid Jets Discharging Normally into
Moving Liquid. St. Anthony Falls Hydraulic Laboratory, University
of Minnesota, Technical Paper No. 28-B, 1959- M* P-
114.. Shirazi, M. A. Some Results from Experimental Data on Surface Jet
Discharge of Heated Water. Proceedings, First World Congress
on Water Resources, International Water Resources Association,
Chicago, October 1973.
15. Schuyf, J. P. The Measurement of Turbulent Velocity Fluctuations
with a Propeller-Type Current Meter. Journal of Hydraulic Research
4(2)* 1966.
Ill
-------
SECTION IX
PUBLICATIONS
RESULTING PUBLICATIONS
1. Stefan, H., E. Mrosla, and L. Bergstedt. Experimental Heated Surface Jet
Studies. Presented at 55th Annual Meeting of the American Geophysical
Union, Washington, D.C., April 1974 (oral presentation only).
PAST PUBLICATIONS RELATING TO THE SUBJECT
1. Stefan, H. Modeling Spread of Heated Water over Lake. Jrl. Power Div.
ASCE. 96(P03):469-482, June 1970.
2. Stefan, H. Stratification of Flow from Channel into Deep Lake. Jrl.
Hydr. Div. ASCE. 96(HY7):1417-1434, July 1970.
3. Stefan, H. and F. R. Schiebe. Heated Discharge from Flume into Tank.
Jrl. San. Engr. Div. ASCE. 96(SA6):1415-1433, December 1970.
4. Silberman, E, and H. Stefan. Physical (Hydraulic) Modeling of Heat
Dispersion in Large Lakes, A Review of the State of the Art. St. Anthony
Falls Hydraulic Laboratory, University of Minnesota. Project Report
No. 115, 1970.
5. Stefan, H. Dilution of Buoyant Two-Dimensional Surface Discharges.
Jrl. Hydr. Div. ASCE. 98(HYl):71-86, January 1972,
6. Stefan, H. and N. Hayakawa. Mixing Induced by an Internal Hydraulic Jump.
Water Resources Bulletin, American Water Resources Association 8(3):531-
545, June 1972.
7. Stefan, H. and P. Vaidyaraman. Jet Type Model for the Three-Dimensional
Thermal Plume in a Cross-current and Under Wind. Water Resources Research
8(4):998-1014, August 1972.
8. Stefan, H., C. S. Chu, and W. Ho. Impact of Cooling Water on Lake Temperatures,
Jrl. Power Div. ASCE. 98(P02):253-272, October 1972.
112
-------
SECTION X
SYMBOLS AKD UNITS
A = aspect ratio
cl
,« = coefficients related to various functions for entrainment
coefficient
EL = hydraulic diameter of discharge channel [ft]
D = diameter of outlet channel [ft]
d = depth of discharge channel [ft]
FO = densimetrio Fxoude number
O
g = acceleration of gravity [32.2 ft sec~ ]
Q = volumetric flow rate [ft3 sec ]
R = cross-flow velocity ratio
eQ = Reynolds number
r = distance from and perpendicular to main trajectory [ft]
s = distance from outlet along main trajectory [ft]
s = length of zone of flow establishment along main trajectory [ft]
T = time-averaged excess temperature ratio
T = discharge temperature at the outlet [ F]
T = ambient water temperature in the lake or reservoir [ P]
8-
TJ = average ambient crossflow velocity [ft sec" ]
a
-1-
U = average discharge velocity [ft sec" ]
U = flow velocity [ft sec" ]
V = time-averaged velocity ratio
w = width of discharge channel [ft]
x = coordinate parallel to outlet centerline [ft]
x = length of zone of flow establishment [ft]
y = coordinate perpendicular to outlet centerline [ft]
z = depth [ft]
113
-------
SYMBOLS AND UNITS (Continued)
a = discharge angle with respect to cross-flow
a = standard deviation
r\ -i
v = kinematic viscosity [ft sec~ ]
X = slope of beach
Q = density of water [slugs ft]
<#> = lateral spread angle of shear zone [degrees]
= coefficients related to various functions for length of zone of
flow establishment
Subscripts
a = ambient water
c = cross-flow
& = local
m = maximum
o = initial value, outlet
-------
APPENDIX A
ISOTHERMS T = constant and 8T = constant
IN VERTICAL SECTIONS ALONG JET AXIS OR PERPENDICULAR TO IT,
A = 2.1±, A = 9-6, R = 0
115
-------
FIGURES — APPENDIX A
No.
A-l Vertical Temperature Profile along Jet Axis - A = 2.1^,
F =2.2, R=0 117
o
A-2 Vertical Temperature Profile along Jet Axis - A = 2.J+,
F =3.85, R=0 118
A-3 Vertical Temperature Profile along Jet Axis - A = 2.1;.
FQ = 15.8, R=0 119
A.-1; Vertical Temperature Fluctuation Profile along Jet Axis -
A = 2.U, F =2.2, R=0 120
A-5 Vertical Temperature Fluctuation Profile along Jet Axis -
A = 2.U, F =3.85, R=0 121
A-6 Vertical Temperature Fluctuation Profile along Jet Axis -
A = 2.U, FQ = 15.8, R=0 122
A-7 Vertical Temperature Profile Perpendicular to Jet Axis -
A = 2.U, FQ = 2.2, R = 0, x/dQ = 1.8 123
A-8 Vertical Temperature Profile Perpendicular to Jet Axis -
A = 2.1;, F =3.85, R = 0, x/d =1.8 12U
o o
A-9 Vertical Temperature Profile Perpendicular to Jet Axis -
A = 2.1;, FQ = 15.8, R = 0, x/dQ = 1.8 125
A-10 Vertical Temperature Profile along Jet Axis - A = 9.6,
F =2.15, R = 0 126
o
A-ll Vertical Temperature Profile along Jet Axis - A = 9.6,
FQ = 1;.!, R= 0 12?
A-12 Vertical Temperature Fluctuation Profile along Jet Axis -
A = 9-6, FQ = 2.15, R = 0 128
A-13 Vertical Temperature Fluctuation Profile along Jet Axis -
A = 9.6, FQ = i|.l, R=0 129
A-ll; Vertical Temperature Profile Perpendicular to Jet Axis -
A = 9.6, FQ = 2.15, R = 0, x/do = 1.9 1JO
A-15 Vertical Temperature Profile Perpendicular to Jet Axis -
A = 9.6, FQ = U.I, R = 0, x/dQ = 1.9 131
116
-------
8
)0
12 14
0.5 —
1.0
1.5
2.0
c
,.
Fig. A-l - Vertical temperature profile along jet axis - A = 2.4, F =2.2, R = 0
-------
00
Fig. A-2 - Vertical temperature profile along jet axis - A = 2.4, F = 3.85, R = 0
-------
Fig. A-3 - Vertical temperature profile along jet axis - A = 2.4, F = 15.8, R = 0
-------
2.0
5T = TEMPERATURE FLUCTUATION
2.5
Fig. A-4 - Vertical temperature fluctuation profile along jet axis - A = 2,4, F =2.2, R = 0
-------
ST = TEMPERATURE FLUCTUATION
Fig. A-5 - Vertical temperature fluctuation profile along jet axis - A - 2.4, FQ - 3.85, R - 0
-------
ro
Ffg. A-6 - Vertical temperature fluctuation profile along jet axis - A = 2.4, F =15.8, R = 0
-------
ro
0.5
0.5
1.0
1.5
2.0
2.5
K5
T=0
2,5 ______ 3.0
Fig. A-7 - Vertical temperature profile perpendicular to jet axis - A - 2.4, F - 2.2, R - 0,
*/d0-i.e
-------
ro
0.5
1.0
1.5
2.0
2.5
y/do
0.5 10 1.5 2.0 2.5 3.0 3.5
"I I
T= .90
T= .20
Fig. A-8 - Vertical temperature profile perpendicular to jet axis - A = 2.4, F = 3.85, R = 0,
x/d = 1.8 °
o
-------
ro
0.5
0.5
1.0 F*
1.5
2.0 r—
2.5
2.0 2.5
3.0
3.5
Fig. A-9 - Vertical temperature profile perpendicular to jet axis - A = 2.4, F =15.8, R = 0,
x/d. =1.8 °
-------
oP-
8
10 12 14
ro
cr\
0.5
1.0
1.5
T= .20
T = 0
2.0
2.5
Fig. A-10 - Vertical temperature profile along jet axis - A = 9.6, F =2.15, R = 0
-------
8
10
12
14
0.5
1.0
1.5
2.0
2.5
T = 0
T= .20
Fig. A-ll - Vertical temperature profile along jet axis - A = 9.6, F = 4.1, R = 0
-------
CD
0
0.5
1.0
1.5
2.0
K/d
8
10
12
14
2.5
5T= .07
5T = .05
8T = .03
5T= .01
ST = TEMPERATURE FLUCTUATION
I I J I
Fig. A-12 - Vertical temperature fluctuation profile along jet axis - A = 9.6, F = 2.15,
R = 0 °
-------
ro
vo
TEMPERATURE
FLUCTUATION
2.5
Fig. A-13 - Vertical temperature fluctuation profile along jet axis - A = 9.6, F = 4.1,
R = 0
-------
o
8
10
12
14
0.5
1.0
1.5
2.0
2.5
T = 0
Fig. A-14 - Vertical temperature profile perpendicular to jet axis - A = 9.6, F =2.15, R = 0,
x/d =1.9 °
-------
y/d
8
10 12
0.5
1.0
1.5
T= .10
T = 0
14
2.0
2.5
Fig. A-15 - Vertical temperature profile perpendicular to jet axis - A = 9.6, F =4.1,
R = 0, x/dQ =1.9 ° .
-------
APPENDIX B
ISOTHEEMS T = constant and ST = constant
AND ISOVELS V = constant
IN VERTICAL AND HORIZONTAL SECTIONS,
A = 2.4, R = O.Ul
132
-------
APPENDIX B
FIGURES
No.
B-l Vertical Temperature Profile Perpendicular to Jet Axis -
A = 2.1+, FQ = 3-35, R = 0.1+1, s/dQ = l+.l 131+
B-2 Vertical Temperature Profile Perpendicular to Jet Axis -
A =2.1+, FQ = 3-35, R= 0.1+1, s/dQ = 8.5
B-3 Vertical Temperature Profile Perpendicular to Jet Axis -
B-l|
B-5
B-6
B-7
B-8
Vertical Temperature Fluctuation Profile Perpendicular to
Jet Axis - A 2.1+, ' F - 3. 35, R 0.1+1, s/d !+•!
Vertical Temperature Fluctuation Profile Perpendicular to
Jet Axis - A - 2.1+, F -3*35, R - 0.1+1, s/d - 8.5 ,
Vertical Temperature Fluctuation Profile Perpendicular to
T«+ A-Hs - A ? )i T? 3.^5 E 0 Jil a/A 13.3 ....
Temperature and Velocity Profiles at Water Surface -
A 2.1i, F 3.35, R O.iil ,
Temperature Fluctuation Profile at Water Surface -
A - 2.k. F -3.35. R - O.iil
. . . 137
, .. 138
. .. 13<3
, . . lliO
... llil
133
-------
Fig. B-l - Vertical temperature profile perpendicular to jet axis - A = 2.4, F = 3.35,
R = 0.41, s/d = 4.1 °
-------
-2.0
Fig. B-2 - Vertical temperature profile perpendicular to iet axis - A = 2.4, F = 3.35,
R = 0.41, $/do= 8.5 °
-------
-2.0
Fig. B-3 - Vertical temperature profile perpendicular to jet axis - A = 2.4, F = 3.35,
R = 0.41, s/d = 13.3 °
o
-------
VJ4
0.5
1.0
1.5
2.0 —
2.5
I I
I I I
2.0 1.5 1.0 0.5
Fig. B-4 - Vertical temperature fluctuation profile perpendicular to fet axis - A - 2.4,
FQ = 3.35, R = 0.41/ s/dQ = 4.1
I I
0 -0.5 -1.0 -1.5 -2.0 -2.5 -3.0
o
-------
00
5T= .19
5T= .15
1.0 0.5
-0.5 -1.0 -1.5 -2.0 -2.5 -3.0
Fig. B-5 - Vertical temperature fluctuation profile perpendicular to jet axis - A = 2.4,
Fo = 3.35, R = 0.41, s/dQ = 8.5
-------
VM
VO
Fig. B-6 - Vertical temperature fluctuation profile perpendicular to jet axis
FQ = 3.35, R = 0.41, s/dQ = 13.3
-5
A = 2.4,
-------
-10 r-
VELOCITY
TEMPERATURE
25
Fig. B-7 - Temperature and velocity profiles at water surface
R = 0.41
A = 2.4, FQ = 3.35,
-------
-10
-5
6T= .13
10 15
o
20
25
Fig. B-8 - Temperature fluctuation profile at water surface - A = 2.4. F = 3.35.
R = 0.41 °
-------
ELECTED WATER
IESOURCES ABSTRACTS
NFUT TRANSACTION FORM
1. Report No.
EPA-660/3-75-014
2-.
3. Accession No.
W
1|. Title: FLOW ESTABLISHMENT AND INITIAL ENTRAINMENT
OF HEATED WATER SURFACE JETS
5. January 1975
6.
8.
7. Author(s): Stefan, H.; Bergstedt, L.; Mrosla, E.
10. Project No.
Organization:
St. Anthony Falls Hydraulic Laboratory
University of Minnesota
Minneapolis, Minnesota
11. Contract/Grant No.
R 800 U35
.2. Sponsoring Organization; Environmental Protection Agenc.v
.5. Supplementary Notes:
Environmental Protection Agency Report Number
Type of Report and
Period Covered
Final
16. Abstract:
Mathematical modeling of the zone of flow (re-)establishment (ZFE) of heated
water surface jets has been found to be difficult because of the complex dependence
on outlet geometry, discharge velocity, buoyancy, and ambient currents. Labo-
ratory experiments have therefore been conducted to provide more observations and
data on flow patterns, temperatures, and flow velocities in the ZFE and some
distance beyond. From this information, relationships giving the length x of
bhe ZFE, the volumetric flow rates Q(X) versus distance, and the initial
spreading angle have been derived. The independent variables were outlet
suspect ratio A, outlet densimetric Froude number, and cross-flow ratio R.
foe relationships provided can be used to make more accurate temperature
>redictions in the immediate vicinity of a heated water surface channel.
17&.Descriptors:
Thermal pollution, Discharge (water), Water quality, Water quality control,
Water pollution, Water pollution control, Outfalls, Experimental data, Flow
establishment, Mixing zone, Dilution, Heated water, Jet flow, Buoyant jet
L7b.Identifiers:
Thermal plume, Surface discharge, Heated water jet
18. Availability:
19. Sec. Class
(Report)
20. Sec. Class
21. No. of
pages: 11^2
22. Price:
Send to:
WRSIC
USDI
— . , _IMaM^^Mfc-jaa4J__^A^JL__jh^^a
Institution; TJniv. of Minnesota, SAFHL
* U.S. GOVERNMENT PRINTING OFFICE; )975-698- 268)120 BEGION 10
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