660/3-73-010
August 1973
Ecological Research Series
Dispersion In Hydrologic
And Coastal Environments
Office of Research and Development
U.S. Environmental Protection Agenc1
Washington, DC 20460
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RESEARCH REPORTING SERIES
Research reports of the Office of Research and
Monitoring, Environmental Protection Agency, have
been grouped into five series. These five broad
categories were established to facilitate further
development and application of environmental
technology. Elimination of traditional grouping
was consciously planned to foster technology
transfer and a maximum interface in related
fields. The five series are:
1. Environmental Health Effects Research
2. Environmental Protection Technology
3. Ecological Research
4. Environmental Monitoring
5. socioeconomic Environmental Studies
This report has been assigned to the ECOLOGICAL
RESEARCH series. This series describes research
on the effects of pollution on humans, plant and
animal species, and materials. Problems are
assessed for their long- and short-term
influences. Investigations include formation,
transport, and pathway studies to determine the
fate of pollutants and their effects. This work
provides the technical basis for setting standards
to minimize undesirable changes in living
organisms in the aquatic, terrestrial and
atmospheric environments.
EPA REVIEW NOTICE
This report has been reviewed by the Office of Research and
Development, EPA, and approved for publication. Approval
does not signify that the contents necessarily reflect the
views and policies of the Environmental Protection Agency,
nor does mention of trade names or commercial products consti-
tute endorsement or recommendation for use.
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EPA-660/3-73-010
August 1973
DISPERSION IN
HYDROLOGIC AND COASTAL ENVIRONMENTS
by
Norman H. Brooks
Grant No. 16070DGY
Program Element 1B1025
Project Officer
l
Richard J. Callaway, Chief
Physical Oceanography Branch
U.S. Environmental Protection Agency
Pacific Northwest Environmental Research Laboratory
Corvallis, Oregon 97330
For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 - Price $1.55
Prepared for
OFFICE OF RESEARCH AND MONITORING
U.S. ENVIRONMENTAL PROTECTION AGENCY
Washington, D.C. 20460
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ABSTRACT
This report summarizes the results of a five-year laboratory research
project on various flow phenomena of importance to transport and dis-
persion of pollutants in hydrologic and coastal environments. The
results are useful in two general ways: first, to facilitate the pre-
diction of ambient water quality from effluent characteristics in
various water environments; and secondly, to provide the basis for
design of systems (like outfalls) required to meet given ambient water
quality requirements.
The results for buoyant jets may be used for the design of waste-water
outfalls in oceans, reservoirs, lakes, and large estuaries. Particular
emphasis is given to line sources (or slot jets) which represent long
multiple-outlet diffusers, which are necessary for all large discharges
to get high dilutions.
For reservoirs which are density stratified, the results include formu-
lations for prediction of selective withdrawal, and a simulation pro-
cedure for predicting reservoir mixing by systems which pump water from
one level to the other.
For application to rivers and estuaries, laboratory flume experiments
were made to measure transverse mixing of buoyant or heavy tracer flows,
as well as for neutral-density flows.
Abstracts for all publications and reports resulting from the project
are given as an appendix to the report.
This report was submitted in fulfillment of Grant No. 16070DGY, by the
W. M. Keck Laboratory of Hydraulics and Water Resources, Division of
Engineering and Applied Science, California Institute of Technology,
under the sponsorship of the Environmental Protection Agency. Work was
completed as of December, 1972. This report has been distributed in
limited numbers as Report No. KH-R-29 of the Keck Laboratory of Hydraulics
and Water Resources, California Institute of Technology.
ii
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CONTENTS
Section Page
I CONCLUSIONS 1
II RECOMMENDATIONS ' 3
III INTRODUCTION 5
IV JET AND PLUME MIXING: PROBLEM DEFINITION AND
METHODS OF ANALYSIS 7
Definitions 7
General Assumptions 9
Assumptions for an Inclined Round Buoyant Jet
in a Stagnant Environment 11
Assumptions for an Inclined Two-Dimensional
Slot Buoyant Jet in a Stagnant Environ-
ment 13
Method of Solution 15
V RESULTS FOR TURBULENT BUOYANT JETS IN UNIFORM
ENVIRONMENTS 19
Round Buoyant Jet (Uniform Environment with
No Current) 19
Slot Buoyant Jet (Uniform Environment with
No Current) 29
Slot Buoyant Jet in a Current (Uniform
Environment) 33
VI RESULTS FOR TURBULENT BUOYANT JETS IN STRATIFIED
ENVIRONMENTS 39
Round Buoyant Jet (Linearly Stratified
Environment, No Current) 39
Slot Buoyant Jet (Linearly Stratified
Environment, No Current) 45
Approximate Solutions for Buoyant Jets in
Environments with Non-Linear
Stratification1 51
Reduction of Dilution by Blockage of Sewage
Field for Line Diffuser 54
Final Comment 57
VII RELEASE OF A SLUG OF DENSE FLUID INTO A TWO-LAYERED
ENVIRONMENT 59
VIII SELECTIVE WITHDRAWAL AND ARTIFICIAL MIXING IN
DENSITY-STRATIFIED RESERVOIRS 63
Selective Withdrawal 63
Artificial Mixing in Stratified Reservoirs 66
iii
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Section Page
IX TRANSVERSE "MIXING IN RIVERS AND OTHER SHEAR FLOWS 75
Transverse Mixing - No Density Difference 75
Transverse Mixing with Density Differences 82
Final Comment 91
X ACKNOWLEDGMENTS 93
XI REFERENCES 95
XII PUBLICATIONS, REPORTS AND TECHNICAL MEMORANDA 105
XIII APPENDIX - ANNOTATED LIST OF PUBLICATIONS,
REPORTS AND TECHNICAL MEMORANDA 109
A. JET AND PLUME MIXING 110
B. OCEAN OUTFALL DESIGN 120
C. SELECTIVE WITHDRAWAL AND ARTIFICIAL MIXING
IN DENSITY-STRATIFIED IMPOUNDMENTS 122
D. NATURAL DIFFUSION IN RESERVOIRS, LAKES, AND
OCEANS 124
E. MIXING IN TURBULENT SHEAR FLOWS 127
F. DISPERSION IN FLOW THROUGH POROUS MEDIA 133
G. GENERAL 135
iv
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FIGURES
PAGE
A buoyant jet in a laboratory tank, illustrating
how the ambient density stratification prevents
the jet from reaching the surface (from Fan and
Brooks, 1969, A-2). 8
Schematic diagrams of round buoyant jet problems
studied. ' ^Q
Schematic diagrams of slot buoyant jet problems
studied. ^4
Zone of flow establishment for an inclined round
buoyant jet. 17
•4
Centerline dilution of round buoyant jets in
stagnant uniform environments: 9 =0 (horizontal).
o
To get centerline dilution relative to the nozzle,
multiply S by 1.15 to adjust for the zone of flow
establishment; for average multiply by 2. (After
Fan and Brooks, 1969, A-2.) 20
Dilution of round buoyant jets in stagnant uniform
environments: 0 = 90 (vertical). To get dilution
relative to the nozzle, multiply S by 1.15 to
adjust for zone of flow establishment; for average
multiply by 2. (After Fan and Brooks, 1969, A-2.) 21
Trajectory and half-width b/b of round buoyant
jets in stagnant uniform ambient fluids: 6=0
(horizontal). (F and y/D scales based on a = 0.082
and X = 1.16.) (After Fan and Brooks, 1969, A-2.) 25
Half-width b/b of round buoyant jets in stagnant
uniform ambient fluids: 0 =90 (vertical).
Trajectories are all vertical lines. (F and y/D
scales based on a = .082 and X = 1.16.) (After
Fan and Brooks, 1969, A-2.) 26
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PAGE
9 Centerline dilution of slot buoyant jets in
stagnant uniform environments: 6=0 (horizontal).
(For average dilution, multiply by ^2.) (After Fan
and Brooks, 1969, A-2.) 30
10 Flow regimes for a slot jet in a current (keeping
9 = constant). The general flow situation in the
center of the graph is surrounded by the limiting
cases. The parameter P is the ratio between the
2/3
flow depth H and a characteristic length m/b
of the source. (After Cederwall, 1971, A-6.) 34
11 Observed flow regimes for a horizontal buoyant slot
jet in a co-flowing stream. Critical flow is defined
as the situation when the formation of a surface
wedge is incipient at the reference section. (After
Cederwall,' 1971, A-6.) 35
12 Observed flow regimes for a vertical buoyant slot
jet in a cross stream. Forced entrainment is
defined as a situation where the typical buoyant jet
flow pattern breaks up and there is efficient mixing
close to the source. (After Cederwall, 1971, A-6.) 36
13 Terminal height of rise £ for inclined round
buoyant jets with y = 0 to 0.01. (After Fan and
Brooks, 1969, A-2.) 40,
14 Terminal volume flux parameter y for inclined round
buoyant jets with y = 0 to 0.01. (After Fan and
Brooks, 1969, A-2.) 40
15 Terminal height of rise E for inclined slot
buoyant jets with y = 0 to 0.01. (After Fan and
Brooks, 1969, A-2.) 46
16 Terminal volume flux parameter y for inclined
slot buoyant jets with y = 0 to 0.01. (After Fail
and Brooks, 1969, A-2.) 46
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PAGE
17 Approximation of non-linear density profile by a
linear one. 52
18 Diagram for solving Equations 62 and 63 with
measured density profiles. (After Brooks, 1970, B-l.) 52
19 Schematic diagram of blocking of part of the water
column by the pollutant field. 55
20 Definition sketch for release of heavy fluid in
a two-layer environment. 60
21 Selective withdrawal from a reservoir through one
of several outlets at various levels in a dam.
(After Brooks and Koh, 1969, C-l.) 64
22 Summary of recommended formulas for selective with-
drawal. (After Brooks and Koh, 1969, C-l.) 64
23 Schematic diagram of a pumping system for mixing a
density-stratified reservoir. (After Ditmars,
1970, C-2.) 67
24 Measured and simulated density profiles for a
typical experiment. (After Ditmars, 1970, C-2.) 68
25 Typical non-dimensional density profiles by simulation
of reservoir destratification by pumping (for S = 500,
P = 2.5 x 10~4, F - 3). (After Ditmars, 1970, C-2.) 71
i
26 Summary of generalized simulation results for de-
stratifying reservoirs by pumping: fraction of
required potential energy increase (M) as a function
of P and t*. (After Ditmars, 1970, C-2.) 72
27 Definition sketch of plume geometry and coordinate
axes. (After Okoye, 1970, E-2.) 76
vii
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PAGE
28 Variation of the depth-averaged, dimensionless
mixing coefficient 9 with the aspect ratio' X "'for
all experiments performed in the present and
past studies. (After Okoye, 1970, E-2.) 78
29 The intermittency factor model for cross-wise
plume variation. (After Okoye, 1970, E-2.) 80
30 Growth of the geometric characteristics of the
region of intermittency: RUN 802. (After Okoye,
1970, E-2.) 81
31 Definition sketch from transverse mixing experi-
ments in laboratory channel. (After Brooks, 1970,
G-l.) 83
32 Overhead photograph of experiments in 110-cm wide
flume with 1-cm wide source. (Depth = 6.55 cm,
mean velocity = 45.2 cm/sec, shear velocity =
2.27 cm/sec.) (After Prych, 1970, E-l.)
(a) Exp. 116, Ap/p = 0 (no density difference).
(b) Exp. 128, Ap/p = -0.0158 (buoyant tracer). 84
33 Contours of equal relative concentration in cross-
sections downstream from a 1-cm-wide source which
discharged a fluid with a density the same as the
ambient fluid and with a relative concentration of
1.0. The crosses designate sampling points.(After
Prych, 1970, E-l.) 85
34 Contours of equal relative concentration in cross-
sections downstream from a 1-cm-wide source which
discharged a fluid with a density 0.0158 gr/cm3 less
than the ambient fluid and with a relative concentra-
tion of 1.0. The crosses designate sampling points.
(After Prych, 1970, E-l.) 86
viii
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PAGE
35 Variance - distance curves from flume experiments
with l-cm^wide source (smooth walls). (After Prych,
1970, E-l.) 87
36 The dimensionless excess variance, AV, as a function
of the dimensionless source strength, ft , and the
dimensionless source width, B. (After Prych, 1970,
E-l.) 89
37 The intercept, M^; as a function of the dimensionless
source width, B. (After Prych, 1970, E-l.) 90
ix
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SECTION I
CONCLUSIONS
The relationship of ambient water quality to effluent characteristics
depends on a complex interaction of physical, chemical, and biological
factors. This project has dealt with some fluid mechanics aspects of
dispersion and transport phenomena in water environments. This report
summarizes the results which are of the greatest practical value for
water quality analysis and design of outfalls and other hydraulic
structures for water quality control. The principal conclusions are
as follows:
1. A wide range of problems in buoyant jet mechanics in stratified
environments without ambient currents has been analyzed by the integral
analysis originally proposed by Morton, Taylor, and Turner (1). Both
two- and three-dimensional cases were solved. (See Sections IV, V, and
VI.)
2. The behavior of buoyant jets in a current can be described if
the ambient flow is of uniform density. Round buoyant jets in a cross
flow were modelled physically and mathematically by Fan (1967, A-l in
Appendix). Slot buoyant jets in a flume (two-dimensional) present a
more difficult problem because the whole regime of the flow field can
be strongly affected by upstream and downstream flow conditions (i.e.
whether the flow is well-mixed or two-layer, etc.). (See Section V.)
Thus jet and plume formulas are probably inaccurate for application to
large thermal discharges from line diffusers in restricted depths.
3. Graphs for the dilution in buoyant jets (both round and slot)
in uniform environment are given in Section V as functions of a depth
ratio and the Froude number. Slot jet solutions can be applied approxi-
mately for long multiport diffusers. Limiting cases of plume-like
behavior can often be used for sewage discharge in deep water from a
long diffuser.
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4. Results for buoyant jets in stratified environment are given
in abbreviated form in Section VI and in more detail in Fan and Brooks
(1969, A-2). Plume solutions are found to provide a conservative
approximation and are convenient for approximate solutions in cases of
non-uniform density gradients.
5. In the application of all buoyant jet and plume formulas it is
necessary to take account of the thickness of the pollutant field which
is generated at the top of the plume. The clear height of plume rise
is thus reduced, causing a lower dilution within the plume. (See
Section VI.)
6. For density-stratified reservoirs an approximate analysis has
been given for selective withdrawal (two-dimensional case) and for de-
stratification by pumping. In general for a wide range of system
parameters the time required for nearly complete destratification is ap-
proximately 0.2 times the reservoir volume divided by the pumping rate.
7. The dimensionless coefficient of transverse mixing of a
neutrally buoyant tracer fluid in a flume flow was found to be a function
of the depth-to-width ratio A (see Fig. 28). The range was:
5
-^T = 0.093 to 0.24
u*d
for A = 0.20 to 0.093
where D is the depth-averaged transverse mixing coefficient; u^ = bed
shear velocity; and d = depth. Values measured by others in the field
tend to be about twice as much. The mixing coefficient includes both
turbulent diffusion and lateral dispersion by secondary currents. (See
Section IX.)
8. In case of a tracer stream which was buoyant or heavy with
respect to the flume flow, the transverse mixing was accelerated by the
density-induced secondary currents. The effect was to produce a dis-
crete increase (Aa2) in the transverse variance of the tracer cloud at
the early stages of mixing, finally adjusting to the normal rate of
lateral spreading for a neutrally-buoyant tracer flow. (See Sec. IX.)
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SECTION II
RECOMMENDATIONS
Additional research is recommended on the following problems within the
scope of this report:
1. For the design and analysis of outfalls for effluent disposal,
a method is needed for predicting the character of the transition region
between the jet-mixing phase and the far-field current drift, in both
uniform and stratified cases.
2. The two-dimensional jet and plume formulas and analyses are
now being widely used for design of sewer outfall diffusers, but they
are poorly supported by laboratory experiments compared to the three-
dimensional case. In particular the entrainment coefficient (a) and the
spreading ratio (X) are subject to considerable uncertainty (+20%).
Comprehensive laboratory experiments are needed.
3. Laboratory research is needed to determine how the dilution
depends on the orientation of a line diffuser with respect to the current.
In outfall design it is necessary to evaluate performance for d.fferent
current directions and different possible diffuser layouts (with respect
to shore).
4. Detailed field observations of the hydrodynamic performance
of existing outfall diffusers are lacking and should be undertaken to
confirm laboratory results and to develop any needed "scale" correction
factors.
5. Experiments and analysis are needed for the case of buoyant
jets and plumes in a stratified ambient current. The combination of
currents and stratification is very common in the natural water bodies,
but very difficult to reproduce,along with a buoyant jet, in the
laboratory.
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6. The fluid mechanics of mixing of thermal discharges from
coastal diffusion structures in relatively limited depths needs to be
studied analytically and in the laboratory. Thermal discharges are so
large that they can modify the entire current and density structure in
the vicinity of the diffuser. The initial dilutions obtained by jet
mixing may be limited by the resistance of flow of diluting water
toward and away from the diffuser.
7. The theory for selective withdrawal is based on a scale-up
9
from laminar laboratory flows to turbulent flows in large reservoirs.
Good field observations are needed to establish better values of vertical
mixing coefficients and to check the validity of the theory. Three-
dimensional effects of reservoirs of irregular shape also need to be
studied.
8. Further research on transverse mixing of contaminants in
turbulent open channel flow is needed to document and explain the
differences in results between the laboratory and the field.
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SECTION IH
INTRODUCTION
This project, initiated in 1967, sought to bring together into a single
project several flow problems affecting water quality in hydrologic and
coastal environments. The fluid mechanics problems involved in dis-
persion in rivers, lakes, estuaries, coastal waters, and ground waters
have a number of features in common to more than one area — such as
turbulence, stratified flow, Taylor dispersion and jet-induced mixing.
We have tried to examine certain fundamental problems with this broad
point of view.
The contributions of the research have been on a variety of topics.
Since it is not feasible to report all the details on all subjects in
this single final report, this document will serve as a summary of the
most important practical results. The complete list of publications
and reports (32 in all) is given in Section XII in chronological order.
In addition, the Appendix presents abstracts for each item, arranged
by topics. They are cited in this report by author, date, and abstract
number in the Appendix. All items may be obtained either from the open
literature, or by ordering from the W. M. Keck Laboratory of Hydraulics
and Water Resources, 138-78, California Institute of Technology,
Pasadena, California, 91109.
The most significant results are presented in Sections IV-IX, as follows:
Jet and plume mixing (Sections IV-VII). The mathematical models,
and experiments on which they are based, provide the basis for the
design of outfalls for sewage disposal, whether of single or multiple-
jet type. The formulas give predictions of initial dilution, size of
plumes, and maximum height of rise in case of stratified environments.
Dumping of a heavy slug in a two-layer body of water is also discussed.
Examples are included.
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Reservoirs and lakes (Section VIII). Selective withdrawal from
density-stratified reservoirs has become a widespread practice in the
1960's, at dams which are equipped with outlets at various levels (or
adjustable). It is now possible to make reasonable predictions of the
thickness of the withdrawal layer from the observed density stratifica-
tion and the efflux rate, in order to predict the effectiveness of
selective withdrawal operations at dams.
A combination of the techniques of selective withdrawal and jet mixing
led to a mathematical model of a reservoir destratification technique,
based on pumping water from the epilimnion and discharging it in a
buoyant jet in the hypolimnion, where further mixing takes place.
: Transverse mixing in rivers and other shear flows (Section IX).
When, contaminants are introduced into rivers,as from outfalls, there is
spreading both vertically and transversely within the flow cross section.
For these analyses and experiments, the mixing is mainly induced by the .
natural turbulence in the river itself, rather than by jet action of the
discharge. The results allow for predicting the accelerated transverse
mixing caused by a discharge which is either heavy or buoyant with
respect to the surrounding flow.
* * *
The text in each section is not intended as a state-of-the-art report,
but rather a summary of the most usable results of this research project.
A full discussion of the pertinent literature appears in each of the
reports and publications. Citations to the literature (excluding items
supported by this grant) are given by numerals in parentheses and are
listed in Section XI. For jet and*-'plume mixing some additional
references are listed for work appearing after the project work was
done.
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SECTION IV
JET AND PLUME MIXING: PROBLEM DEFINITION AND METHODS OF ANALYSIS
Definitions
One of the first questions asked when an outfall sewer is being designed
is how to predict the dilution achieved initially by buoyant jet mixing
near the point of discharge. The initial dilution is defined as the
dilution of effluent with ambient fluid at the end of this mixing pro-
cess, driven by the buoyancy and momentum of the discharge. The initial
dilution may be attained at the surface, or at the point of maximum
height of rise in a stratified environment. If the concentration in the
jet at the source is called one unit, the local dilution is just the
reciprocal of concentration at any point in the flow field. The average
dilution within a buoyant jet may be shown to be 1.74.times the centerline
value in a'round jet, and \/2 times the centerline value for a two-
dimensional or line buoyant jet (assuming Gaussian profiles).
A buoyant jet is a turbulent free shear flow which has both momentum
and buoyancy at the source. There are two limits: if the initial mo-
mentum is small then it becomes a plume, whereas if the buoyancy is neg-
ligible it is a momentum jet. A typical buoyant jet from an outfall may
be jet-like near the outlet, but become basically plume-like at some
distance from the source. This is because the initial momentum gets
diffused across a plume of increasing size, whereas the buoyancy con-
tinually adds new momentum to (the plume as it keeps rising. Other
investigators, such as Morton, use the term "forced plume"^instead of
"buoyant jet". In this report the plume-like rising column of fluid
away from the source of a buoyant jet is often called a plume for sim-
plicity.
A dynamic model of turbulent buoyant jets was developed following the
basic integral technique of Morton, Taylor, and Turner (1) (see Fan,
1967, A-l, and Fan and Brooks, 1969, A-2). The flow pattern for a
buoyant jet in a stratified fluid in the laboratory is illustrated by
Fig. 1.
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oo
Figure 1. A buoyant jet in a laboratory tank, illustrating how the ambient density stratification
prevents the jet from reaching the surface (from Fan and Brooks, 1969, A-2).
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The basic definition sketches for a turbulent round buoyant jet and a
slot buoyant jet are shown in Figs. 2 and 3. In each case, the upper
drawing shows the case for uniform environment, wherein the plume rise
is stopped only at a free surface; the lower sketches show the linearly-
stratified cases where the stratification limits the height of rise of
the plumes. Only the assumptions will be carefully detailed here, with
the reader being referred to the original reports for the details of the
solutions.
General Assumptions
The general assumptions underlying the analyses made in this investiga-
tion are listed as follows:
1. The fluids are incompressible.
2. Variations of fluid density throughout the flow field are
small compared with the reference density chosen. The variaton of
density can be neglected in considering inertia terms but it must be
included in gravity terms. As the variations in density are assumed small,
separate conservation equations can be written for volume flux and buoy-
ancy flux (or flux of the agent causing density change, i.e., heat or
salts). This is commonly called the Boussinesq assumption.
3. Within the range of variation, the density of the fluid is
assumed to be a linear function of either salt concentration or heat
content above the reference level.
4. The flow is fully turbulent. Molecular transport can be neg-
lected in comparison with turbulent transport. There is no Reynolds
number dependence.
5. Longitudinal turbulent transport is small compared with longi-
tudinal advective transport.
6. Pressure is hydrostatic throughout the flow field.
7. Curvature of the trajectory of the jet is small. In other
words, the ratio of the local characteristic width of the jet to the
radius of curvature is small. The effect of curvature will be neglected.
8. The velocity profiles are similar at all cross sections normal
to the jet trajectory. Similarity is also presumed for profiles of
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Jet Axis
ROUND JET GEOMETRY
i
g
-P
r
AMBIENT DENSITY
PROFILE
y* ?- -
Terminal
Point
0
\
t
xt
X
-P(y)
—=consl
cTy
ROUND JET GEOMETRY
AMBIENT DENSITY
PROFILE
Figure 2. Schematic diagrams of round buoyant jet problems studied.
Top: uniform environment; bottom: linear density
stratification.
10
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buoyancy and concentration of any tracer. The specific forms of the
profiles are given below. The analyses apply only to the zone of
established flow where all the profiles are fully developed. However,
for practical applications, the initial conditions must be adjusted to
take account of the zone of flow establishment. The results by
Albertson, Dai, Jensen and Rouse (2) for this region will be adopted in
application to practical problems.
Assumptions for An Inclined Round Buoyant Jet in a Stagnant Environment —
Uniform or with Linear Density-Stratification
In Fig. 2 the jets are issuing from the origin at an angle of inclination
6 with the horizontal. The axis of the jet is taken as a parametric
coordinate axis s. The angle between the s-axis and the horizontal is
denoted as 6. The radial distance to the s-axis at a normal cross
section A is chosen to be the r-coordinate. The angular coordinate (/?
is denoted as shown in the figures.
u* and p* are respectively local mean velocity and density, which ,are
functions of r and s, while u and p are the characteristic velocity and
density at the s-axis and are functions only of s. Axial symmetry is
presumed, allowing no dependence on if . The corresponding ambient
density values are similarly denoted as p * and p . In a uniform
a •. a
ambient fluid p * = p = constant.
a a
In a uniform environment, the jet axis is deflected upwards because of
the increase of vertical momentum flux due to the action of the buoyancy
force. The jet grows as it.rises and entrains ambient fluid.
In a linearly density-stratified environment (dp /dy = constant), the
jet axis is first deflected upwards because of the increase of
vertical momentum flux due to the action of the buoyancy force. Be-
cause of the turbulent mixing, the jet entrains the denser ambient fluid
and grows heavier with reduction of the driving buoyancy force. Since
the density of the ambient fluid decreases with height, the jet will
eventually become as heavy as, and then heavier than, the ambient fluid
at the same height. The buoyancy force thus reverses its direction and
11
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in the end will stop the rising of the jet at a. terminal- point (x ,y )
where the vertical momentum flux vanishes. The trajectory of the jet is
therefore in. general an S-shaped curve. After reaching the terminal
point the horizontal momentum flux (if any) will keep the jet moving in
the x-direction. But the flow cannot maintain the characteristics of a
turbulent jet after reaching the terminal level and collapses in the
vertical direction because of the suppression of vertical motion imposed
by the density stratification. The analysis does not cover that part.
Specific assumptions related to the analyses of round buoyant jet
problems are listed as follows :
1. The entrainment relation is given by the equation:
(1)
where Q is the volume .flux across the jet cross section A; a is a co-
efficient of entrainment for a round buoyant jet; b is the characteristic
length defined in Eq. 2; u is the characteristic velocity along s-axis.
2. Velocity profiles are assumed to be Gaussian:
u*(s,r) = u(s) e" (2)
where b = b(s) is a characteristic length defined by the velocity pro-
file. Commonly w = i/2 b is defined to be the nominal half-width of the
jet.
3. Profiles of density deficiency with respect to the ambient
density are assumed to be Gaussian:
Po-p*(S>r) po-p(s) _r2/(xb)2
po " po
(in a uniform environment) (3a)
p *(s,r) - p*(s,r) p(s) - p(s) -r2/(Xb)2
_§. --- §. - e
P P
*o o
(in a linearly density-stratified environment) (3b)
12
-------
where Xb is the characteristic length of the profiles; X2 is the turbulent
Schmidt number which is assumed to be constant and is usually found to be
somewhat larger than 1. Such profiles can also be regarded as buoyancy
profiles.
4. Concentration profiles for passive tracers (such as trace metals,
bacteria, etc.,which do not affect density) are also similar and assumed
to be Gaussian:
*t \ ' f \ -r2/(Xb)2 „.
c*(s,r) = c(s) e (4)
Assumptions for An Inclined Two-Dimensional Slot Buoyant Jet in a
Stagnant Environment — Uniform or.with Linear Density
Stratificat ion
In Fig. 3, the jets are issuing from the z-axis at an angle of inclina-
tion 9 with the horizontal. The axis of the jet is again taken as a
parametric coordinate axis s. The distance normal to the s-axis is
taken to be the n-coordinate as shown.
The flow configurations are entirely similar to those described above.
Specific assumptions related to the analyses of two-dimensional slot jet
problems are listed as follows:
1. The entrainment relation is given by the equation:
. = 2au (5)
ds
where q is the volume flux per unit length along z-axis; a is an entrain-
ment coefficient for a slot buoyant jet.
2. Velocity profiles are assumed to be Gaussians
2/1,2
u*(s,n) = u(s) e '° • (6)
T/2 b is again defined to be the nominal half width pf the jet.
3. Profiles of density deficiency with respect to the ambient
density are assumed to be Gaussian:
13
-------
Jet Axis
yi
g
f
"* "^3 ^>
p ~~ P
ro ra
SLOT JET GEOMETRY
AMBIENT DENSITY
PROFILE
wt
Terminal
Point
T
dy
s const.
SLOT JET GEOMETRY AMBIENT DENSITY
PROFILE
Figure 3. Schematic diagrams of slot buoyant jet problems studied.
Top: uniform environment; bottom: linear density
stratification.
14
-------
P -p(s)
e
po
(in a uniform environment) (7a)
p *(s,n) - p*(s,n) p (s)-p(s) __2//iM2
3. cl TL / \ Au)
= e
po po
(in a linearly density-stratified environment) (7b)
4. Profiles of passive-tracer concentration are also Gaussian:
c*(s}n) = c(s) e
Method of Solution
Using the above assumptions the following basic differential equations
are written:
1. Entrainment (conservation of volume)
2. Horizontal momentum flux (conserved)
3. Vertical momentum flux (changes due to jet buoyancy)
4. Buoyancy flux (conserved in uniform environment only)
5. Horizontal displacement
6. Vertical displacement
7. Passive tracer conservation
The equations are integrated across the plumes so that the system reduces
to only one independent variable, s, the distance along the axis. The
initial conditions include initial volume flux, momentum flux and
buoyancy flux. The environment may be uniform or linearly-stratified.
A large number of normalized computer solutions were made, as reported
in Fan and Brooks (1969, A-2). On the basis of these solutions, contour
graphs were prepared to summarize the results; some of the most useful
graphs are given in the next sections.
15
-------
All of the mathematical modelling applies only to the zone of established
flow, that is the zone beyond the distance of about 6D for round jets
(see Fig. 4). Within this short distance from the source, the boundary
layers at the edges of the jet are still growing; the similarity assump-
tions used in the analysis do not apply until these shear layers reach
the center of the jet. In this initial zone of flow establishment the
centerline velocity stays at u , the initial value and only starts de-
clining by jet diffusion beyond a distance of 6D.
However, the centerline concentration at the end of the zone of flow
establishment is already lower by the factor (1+X2)/2X2 for round jets,
because scalar quantities spread slightly faster. The ratio X is taken
as 1.16 (based on experiments); therefore (1+X2)/2X2 = 0.87. In all of
the following results it is necessary to make slight adjustments to
change the basis of the solution from the end of the zone of flow es-
tablishment to the beginning of the jet at the nozzle.
The values of the entrainment coefficient (a) and the spreading ratio
(X) used are as follows (Fan and Brooks, 1969, A-2):
a X
Round jets 0.082 1.16
Slot jets 0.14 1.00
In the original Fan and Brooks report, the results are given in a
general way, which permits the user to select values of a and X, although
they are restricted to constants.
The values given above for the slot jet (a = 0.14, X = 1.00) are revised
from those given by Fan and Brooks (1969, A-2) (a = 0.16, X = 0.89).
The new values come from a redrawing of the curves to fit the experi-
mental points better in the principal reference by Rouse, Yih, and
Humphreys (3). The previous values of a and X were based on the curves
drawn by them. The reader is cautioned that in any event the co-
efficients are poorly known and based on scanty evidence for the two-
dimensional case.
16
-------
VELOCITY
CONCENTRATION
8 BUOYANCY
Figure 4. Zone of flow establishment for an inclined round buoyant
jet.
List and Imberger (1973, A-12) have recently shown theoretically how
the two coefficients must be related to each other, and how a for a
buoyant jet is really a variable dependent on local Froude number, with
asymptotes of 0.057 for the pure axisymmetric jet and 0.082 for the
pure axisymmetric plume. The integrations made in the earlier report
did not take this variation into account, but the errors are not con-
sidered large unless the flow is more jet-like than plume-like. In
most practical cases of sewage discharge in the ocean, the buoyancy
effects are dominant, and the flows tend to be plume-like. Further
work is needed.
17
-------
SECTION V
RESULTS FOR TURBULENT BUOYANT JETS IN UNIFORM ENVIRONMENTS
This section gives the most important results for turbulent buoyant jets
discharging either horizontally or vertically into uniform ambient en-
vironments, without ambient currents. The full details of the solutions
are available in the report by Fan and Brooks (1969, A-2).
The case of discharge of a round buoyant jet into an ambient current is
not covered here but it is treated both analytically and experimentally
by Fan (1967, A-l). Some results by Cederwall (1971, A-6) for flow
regimes for slot jets into a current are included.
Further work is in progress for developing other simplified results for
use in design problems.
Round Buoyant Jet (Uniform Environment With No Current)
The analytical solutions for the dilution on the centerline of a round
buoyant jet in a uniform environment without current can be represented
(as shown in Figs. 5 and 6) by a function
SQ = f(y/d, F, 6) (9)
S = centerline (or minimum) dilution in the buoyant
o
jet (relative to concentration on centerline at
end of zone of flow establishment)
(
where y = vertical distance from center of jet at end of
flow establishment to the point of measurement
of S in center of plume
o
D = diameter of jet at source (at vena contracta if
there is jet contraction)
(10)
gD
19
-------
20
I
2
i i n i
4 6 810 20
F (fora-0.082,Xs1.16)
I I I I
40 6080100
Figure 5. Centerline dilution of round buoyant jets in stagnant
uniform environments: 0=0 (horizontal). To get
centerline dilution relative to the nozzle, multiply S
by 1.15 to adjust for the zone of flow establishment;
for average multiply by 2. (After Fan and Brooks,
1969, A-2.)
20
-------
O.30.4 0.60.8 I
2 4 6 8 10
m0» (2a2X~V/5F4/8-0.375F
I I I I s I I I i
4 6 810 20 40 6080100
F (fora-0.082.X=M6)
Figure 6. Dilution of round buoyant jets in stagnant uniform
environments: 6 = 90° (vertical). To get dilution
relative to the nozzle, multiply SQ by 1.15 to adjust
for zone of flow establishment; for average multiply
by 2. (After Fan and Brooks, 1969, A-2.)
21
-------
Q = discharge in initial jet
U = initial jet velocity
g = acceleration of gravity
p = reference density = density of ambient fluid
p = density on the centerline at the end of the zone of
flow establishment
6 = angle of discharge (a = 0 for horizontal;
6 = 90° for vertical).
Other parameters shown on the graphs (C^m~ and m ) relate to the theo-
o o
retical solutions and are not needed for application of the results in
terms of the above parameters.
The analytical results given here are based on a = entrainment coefficient
= 0.082, and X = spreading ratio = 1.16. The dilutions S are referred to
the end of the zone of flow establishment, and must be increased by the
factor 2X2/1+X2 = 1.15 (for X = 1.16) for practical application for re-
ferring the dilutions to the beginning of the jet (see Fig. 4). In other
words, if S , is the dilution referred to the concentration of the initial
od
discharge
s j " 1TTT S = 1.15 S for X = 1.16 (11)
od 1+XZ o o
To obtain the average dilution S , it is first necessary to find the
3,
average concentration c of a tracer across a plume weighted with flow
velocity to give a volumetric average:
u(s). e " '" c(s)
r , , -r2/b2 - . -rrb2
/ u(s)e 2irrdr
V 0
- X2 c(s)
s . - r- s • (12>
a X^ o
22
-------
For S , the average dilution referred to the original jet:
Sad ' 1 Sa - 2So
Thus to obtain the average dilution of the plume relative to the initial
jet, multiply the graph values of S by 2 for round jets.
The same type of adjustment for the zone of flow establishment applies
for the density difference [p - p(s) ] as for tracer concentration c(s),
because the spreading rate for the two is assumed to be the same by
Eqs. 3a and 4.
Note that the Froude number F used in this work is based on the relative
density difference, (p -p-^/p , at the end of the zone of flow establish-
ment. If the jet has an initial density of p at the point of discharge,
then the conservation equations for the zone of flow establishment yield
(see Fig. 4) :
_ . . n,
for X = 1.16 .
Alternatively , if the value of the source Froude number F, is known,
defined as
(15)
the conversion equation is
F = 1.07 F, . (16)
d
This difference is scarcely significant, nonetheless the distinction
is made for consistency. This distinction was not made in the Fan and
Brooks report, as they incorrectly referred to p-^as the initial jet den-
sity rather than the density at the end of the zone of flow establishment.
23
-------
The distance from the beginning of the jet to the end of the zone of
flow establishment is approximately 6D. For vertical discharges the
real dimensionless distance y/D must be reduced by 6 before entering
the graphs (Figs. 6 and 8).
In the theory the distance y is the vertical coordinate in the solutions
for buoyant jets which are presumed to rise indefinitely. In practice
we are often interested in the dilution at the point where the rising
plume reaches the water surface. Although the flow pattern is deflected
by the surface, the dilution calculated is often based on setting
y = total depth; however, a more conservative approach would be to use
the effective depth y somewhat less than the total depth (perhaps re-
duced by an amount equal to one quarter of the plume diameter at the
top).
Toward the upper left of both graphs (Figs. 5 and 6), the solutions be-
come identical, as the results for both cases are asymptotic to the
y/D
solution for a simple buoyant plume. For J-=- > 30, the plume solution
r
for centerline dilution may be used as follows :
S = 0.095 (y/D)5/3 (F)~2/3 . (17)
o
Referred to the beginning of the jet (multiplying by 1.15):
c/o T I "\ a ' ' v
SQd = 0.109 (y/vy F ' = 0.089 & - ^j— , (18)
P -p , -5/2
where g' = g - • Since F *\> D for given Q, S is actually inde-
pendent of D (and the initial velocity) for plume-like behavior. For
many applications (including extrapolation to values of y/D off the top
of the graphs), the plume solution, Eq. 18, is entirely adequate.
The half-width of the jets is defined as two standard deviations of the
transverse velocity distribution, or w = /2~ b , where b is defined in
Eq. 2 as
u*(s,r) = u(s) e~ . (19)
24
-------
to
Ul
0
50
2./2ax/D
80
Figure 7. Trajectory and half-width b/b of round buoyant jets in stagnant
uniform ambient fluids: 0 =0 (horizontal). (F and y/D scales
based on a = 0.082 and X =°1.16.) (After Fan and Brooks, 1969,
A-2.)
-------
200
b/b0=w/D
Figure 8. Half-width b/b of round buoyant jets in stagnant uniform
ambient fluids? 9 = 90° (vertical). Trajectories are
all vertical lines? (F and y/D scales based on a = .082
and X = 1.16.) (After Fan and Brooks, 1969, A-2.)
26
-------
At the end of the zone of flow establishment, b = b ; by conservation
of momentum, b may be related to the initial jet diameter D by the re-
lation b = D//2~.
o
Fig. 7 gives dimensionless trajectories for various Froude numbers,
overlaid with contours of w/D = b/b . For vertical buoyant jets, the
trajectories are all the same; therefore Fig. 8 gives only the widths
w/D for various F values. To obtain the full width (2w), multiply values
of w/D read from either graph by 2D.
The simple-plume result for width is
b = 0.102y (20)
or for total width, 2w
2w = 2>/2~ b = 0.29y . (21)
The reader is cautioned that the results given here are all based on the
entrainment coefficient a = 0.082. This value is appropriate for flows
which are plume-like and does not allow for smaller a-values near the
source where the flows may be more jet-like. List and Imberger (1972,
A-12) studied this problem at the very end of the grant period and de-
veloped a rationale for the transition of a-values from jet to plume
regions. Furthermore, even for pure plumes the a-value is only known
within + 15%. Therefore, the results obtained should not be considered
more accurate than + 15 to 20%.
Example. Consider a discharge of Q = 50 mgd of sewage effluent from a
diffuser at a depth of 65 feet in homogeneous sea water (no stratifica-
tion) . There is sufficient head to jet the sewage out at 15 fps. The
density of sea water is 1.025 gr/ml and sewage is 0.999 gr/ml. Compare
initial dilutions which can be obtained by a 50-port diffuser with a 5-
port one. Assume ports are rounded inside and produce no jet contrac-
tion, and are separated sufficiently to avoid interference. The jet
discharge is horizontal.
j . ,. . __ 50 x 1.55 cfs _ c ,-, f.2
Required total area of jets = —15 ft/sec 5.17 it"-
27
-------
a. For 50 ports:
1D2 = 5^17 = 0.1034 ft2
D = 0.362 ft = 4.35 in.
y/D = 65/0.362 = 180
15
/(.026) (32.2) (.362)
27
F = 1.07 FJ = 29
a
By Fig. 5,5 =100
S = 115 = dilution on centerline at top of plume
(by Eq. 11)
S d = 200 = average dilution (by Eq. 13)
b. For 5 ports:
f D == 1.034 ft2
D = 1.15 ft = 13.75 in.
y/D = 65/1.15 = 57
15
15.2
/(0.026)(32.2) (1.15)
F = 1.07 F. = 16.3
Q
By Fig. 5, SQ = 28;.
S , = 32
od
S . = 56 .
ad
All solutions assume no interference between rising plumes. For case (a)
(50 ports), Fig. 7 gives w/D = 35 at y/D =180 for F = 29; therefore,
the diameter at the head of the plume is approximately
2w = 2 x 35 (0.36 ft) = 25 ft. Similarly, for case (b) (5 ports), we
find w/D = 13 for y/D = 57 and F = 16; therefore, the diameter at the
top of the plume is predicted to be about 2w = 2 x 13 x 1.15 ft = 30 ft.
28
-------
Slot Buoyant Jet (Uniform Environment With No Current)
The analytical solutions for a slot buoyant jet are useful for multiport
line diff users in which a row of ports giving jet diameter D at spacing
s may be considered equivalent to a slot giving a jet of width B. If
the ports are spaced closely enough so that there is extensive inter-
ference, then the row of jets will produce a flow pattern similar to that
of a slot jet when viewed from a moderate distance. The equivalent width
of a slot jet is based on providing the same flux of momentum, volume and
buoyancy; therefore the areas must be equal or
2
f D = Bs,
B = ff. (22,
When designing a line diff user the important variable is jet area per
foot or B; the spacing s may be chosen small compared to the total depth
in order to realize the total benefit of a line source. However, excess-
ively close spacing or the use of -an actual slot is unnecessary. When
comparing different possible port spacings (s) , the port size must be modi-
fied in accordance with D « /s" in order to keep the same basic flux rates
per unit length of diff user. If the jets are so far apart that they act
as individual round buoyant jets without interference, then they are
farther apart than necessary; in other words, a better solution might be
to use smaller, closer jets which do interfere and thus better approxi-
mate a line source.
i
Fig. 9 gives the analytical solution for horizontal slot buoyant jets.
As for the round jets, the dilution can be represented as a function of
the Froude number and the depth ratio y/B. In this case, the Froude
number is
U
F = - 2 - ' (23)
o
But since we now estimate X = 1.0 for two-dimensional buoyant jets
(rather than X = 0.89 as given by Fan and Brooks), no correction is
29
-------
0.50.6 0.8 I
m = (a/X)"3F2/3=0.5l9F'
o
i I I I I
2 4 6 8 10
1 I 1 I I I
20 40 6080100150
F(for a =0.14,X=1.0)
Figure 9. Centerline dilution of slot buoyant jets in stagnant
uniform environments: 8=0 ^horizontal). (For
average dilution, multiply by fi .) (After Fan and
Brooks, 1969, A-2.)
30
-------
necessary for the zone of flow establishment, and (p -p,) = (p -p ) at
o -1- o d
the nozzle. The dilutions S given in the graphs are correct with
respect to the original jet.
The average dilution in the rising plume is again found from the weighted
average of the concentration:
r f x -n2/b2 , N -n2/b2
J u(s) e c(s)e
J
dn
f u(s)e-n
^ _™
dn
Therefore S = /2" S . (24)
cL O
The distance from the beginning of the jet to the end of the zone of flow
establishment is approximately 5B. This correction is rarely necessary
because B is small in most practical cases.
For the upper left portion of Fig. 9 the dilutions are asymptotic to
_/ In
those for a line plume. For (y/B)F ' > 20 (see Cederwall, 1971, A-6,
p. 28), the plume solution for centerline dilution may be used:
SQ = 0.38 (y/B)F~2/3 (25)
or SQ = 0.38 g'1/3yq~2/3 (26)
I
where q = discharge per unit length
For the plume approximation, the dilution is independent of B as it
should be; only the buoyancy flux is important and not the momentum.
Since almost all practical cases fall within the plume approximation,
for which the angle of discharge is not a variable, the graph for a
vertical buoyant slot jet is not reproduced here (see Fan and Brooks,
1969, A^2).
31
-------
_ / In
In case (y/B)F < 20, but the solution point is still off the graph,
the solution may be obtained as follows. By similarity arguments, or by
examination of the functional relation in Fig. 9, the dilution may be
written approximately as
S F = f (J F-4/3> (27)
o is
Therefore a similar point in the graph may be found for use in evaluating
the function f (see example below) .
For the plume solution, the size may be taken as
b = 0.16y (28)
2w = 2-/2 b = 0.45y. (28a)
Example. Consider the previous example for round buoyant jets, case (a),
50 ports. Assume a line diffuser of 50 ports at 25-foot spacing, and
compute the plume dilution and width by slot- jet analysis.
B = * j>2 = 0.1034 ft* =
4s 25
y = 65 ft
y/B = 65/0.00414 = 15,700
F = 15 = 25.5 by Eq. 23.
/ (0.026) (32.2) (.00414)
Since this point is off the graph (Fig. 9), find the value of
(y/B)F~4/3 = 15,700/(255)4/3 = 9.7.
This ratio is less than 20, so we must use the similarity relation,
Eq. 27. Choose a similar point (y/B)1 = 200; solve for the similar
FT from the relation
(y/B)'(F')~4/3 =9.7
(F')4/3 = ^=20.6
F1 = 9.7
32
-------
For (y/B)' = 200, F1 = 9.7, Fig. 9 yields S ' = 20.
o
Finally S .
g£= (fr)2/3 = 8.85
o
S = 177
o
The average plume dilution is then
S = SI x 177 = 250
a
For the previous solution for 50 separate round jets it was found that
SQ = 115 (centerline) and S = 200 (average). Thus, it is indicated that
a closer approach to a line source by using more ports of smaller
diameter and closer spacing could improve the dilution from about 200 up
to a maximum of about 250; this presumes that the overall diffuser length
is kept constant (50 x 25 = 1250 ft) and that the discharge velocity is
not changed. To increase the dilution beyond 250 would require
increasing the total diffuser length in order to reduce the unit dis-
charge q (cf. plume formula, Eq. 26).
The validity of the lower right part of Fig. 9 (where the S contours
slope down) is open to serious question. When the depth is small and
the jetting strong along a line source, the limiting factor may be
access for the water supposedly entrained. The dilution in this case
may be governed by the overall flow-field dynamics, and is not well
represented by jet mechanics based on the assumption of an infinitely
deep flow field.
Slot Buoyant Jet in a Current (Uniform Environment)
Under this project, Cederwall (1971, A-6) studied the problem of a two-
dimensional buoyant jet by laboratory experiments and dimensional
analysis, including cases with and without ambient current. One of his
principal objectives was to identify the important flow regimes, as
shown by the next three figures (Figs. 10, 11, 12) taken from his
report.
The flow regime for a buoyant slot jet discharging near the bottom is
characterized by the following variables for cases of relatively small
volume flux (following Cederwall's notation):
33
-------
The Source Froude number
100
Simple jets I0
ua»0
1.0
0.1
0.01
Simple jets in
ambient flow
"Nothing"
(Diffusion of "leaking11
tracer from source due
to ambient turbulence)
Buoyant j<
ambient f
Simple plumes in
ambient flow
(Rouse's solution)
The momentum flux ratio
0.01
10
100
ufB
Buoyant jets
Simple plumes
ua«0
Figure 10.
Flow regimes for a slot jet in a current (keeping 6 = constant).
The general flow situation in the center of the graph is surrounded
by the limiting cases. The parameter P is the ratio between the
flow depth H and a characteristic length m/b^/3 of the source.
(After Cederwall, 1971, A-6.)
-------
w
'jxs
II
u.
* «.oo
E
3
C
0>
•o
o
u.
O
U
O
10
0
0.10
0.05
0
i ! | 1 1 1 1 1 1 1 1 1 | 1 1 1 1 I 1 | i i i
Legend: Critical flow •
o Maintained buoyant a v
9 = 0 jet flow pattern —•-
**s&mijil&
Thermal waste m Supercritical flow
w discharge in river * ^
'-
Subcritical flow
"
jet or plume -like n D
patterns DC Q
a
I ^jr~*:* Ocean disposal D
of sewage Q
.1 0.5 1.0 5 10 50
III
— —•
i i i
10
V
The momentum flux ratio -r- »
k
Figure 11. Observed flow regimes for a horizontal buoyant slot jet in a
co-flowing stream. Critical flow is defined as the situation
when the formation of a surface wedge is incipient at the
reference section. (After Cederwall, 1971, A-6.)
-------
•yn
10.0
5.0
Co
II
u_
v.
a>
JQ
E
3
C
a>
•o
3
O
UL
o>
o
3
o
CO
0)
1.00
0.10
0.05
I I I
T I I IT I I I
I I ill
Legend: No intrusion by wedge O
Wedge intrusion n
Deflected wedge A
Jet or plume-like patterns: Unfilled sym.
Forced entrainment: Filled symbols
Supercritical flow
A A
jet or plume-like
patterns
Subcritical flow
Deflected surface
wedge
a
i I I I I I
0.1
0.5 1.0 5 10
50 100
2i i
500 1000
The momentum flux ratio -r =
Figure 12. Observed flow regimes for a vertical buoyant slot jet in a cross
stream. Forced entrainment is defined as a situation where the
typical buoyant jet flow pattern breaks up and there is efficient
mixing close to the source. (After Cederwall, 1971, A-6.)
-------
m = u.2B = kinematic momentum flux of the source
Po~pl
b = g u.B = kinematic buoyancy flux
6 = angle of injection (9 = 0 is horizontal in
downstream direction)
u = ambient current velocity
H = total depth of flow.
The flow pattern is just characterized by the following three dimension-
less groups: 3
o
1. F = —r— = Froude number (strength of current relative
to buoyancy flux at source)
u 2H u 2H
3. 3.
2. = —£=• = momentum flux ratio (current/source)
Uj
3. 9
The source Froude number represents the strength of the ambient current
relative to the buoyancy of the source: the greater F is, the more the
plume will be "bent over" by the current. Similarly the ratio
u 2H/u.2B indicates how strong the current is relative to the source
a 3
momentum.
For a given angle of injection 9, the flow regions can be classified as
shown schematically in Fig. 10, with indication of various limiting
cases. Arrows indicate which way one moves on the graph when certain
variables are changed; for example, when the ambient velocity is in-
creased, one moves along lines of 3/2 slope on a log-log graph. The
extreme case in the upper right is for very small source buoyancy and
momentum, in which case there is no dynamic effect of the source but
only passive diffusion in the boundary layer due to ambient turbulence.
Rouse's solution (4) is the limiting case of small buoyancy input and
negligible source momentum.
In Fig. 11, the principal flow regimes for horizontal buoyant jets are
defined on the basis of flume experiments; Fig. 12 gives the same for
37
-------
vertical jets. The purpose of these figures is to show from exploratory
experiments the range and complexity of possible flow regimes. The
solutions for slot buoyant jets obviously apply only in the lower left
corners of Figs. 11 and 12 as the asymptote for u -»• 0. It is not possible
3>
to give a definitive quantitive criterion at this time for how small u
cL
must be for Fig. 9 to apply, although Fig..11 gives some guidance.
In the course of Cederwall's experiments it was found that these two-
dimensional stratified flows are sensitive to boundary conditions at the
end of the flume and the duration of an experiment. The diluting water
on the downstream side of a buoyant slot jet in a current can only come
from downstream because the rising jet itself (from wall to wall of the
flume) prevents the cool water from upstream from getting underneath.
However, in prototype conditions all line sources are finite, and there-
fore some flow of diluting water from the sides to the downstream side of
the jet is always possible. In two-dimensional flume experiments' it is
not clear how the downstream condition should be modelled. Consequently,
the results of Figs. 11 and 12 should be regarded as preliminary until
there is further investigation of the sensitivity to downstream boundary
conditions, and their relation to three-dimensional aspects of the far field.
In case of a light current and deep water, there may develop a thick
pollutant field over a line diffuser. Although the dynamics of the
rising plume may be little affected, nevertheless the height of rise
through uncontaminated water is reduced. The entrainment of clean
diluting water is reduced, and a correction to the initial dilution is
required. An empirical procedure is given at the end of the next
section for making this correction (p. 54).
38
-------
SECTION VI
RESULTS FOR TURBULENT BUOYANT JETS IN STRATIFIED ENVIRONMENTS
This section summarizes important results for three- and two-dimensional
buoyant jets into quiescent density-stratified environments. An approxi-
mate graphical method is also given for determining the maximum height
of rise in environments with irregular density profiles. Exact solutions
may be made by programs developed by Ditmars (1969, A-3) and Sotil
(1971, A-ll).
Round Buoyant Jet (Linearly Stratified Environment, No Current)
The basic flow pattern for this case is shown in Fig. 2 (lower half).
It is necessary to assume a linear density profile in order to get
generalized solutions. Non-linear or irregular profiles can often be
approximated as linear with reasonably good results , or numerical
solutions may be generated specially on a case-by-case basis.
The important characteristics of the solutions are the maximum height of
rise y and the dilution of a tracer at the top of the rising plume
•'max r e r
(assuming constant background concentration of the tracer over depth).
Figs. 13 and 14 from Fan and Brooks (1969, A-2) give the basic results
in dimensionless form. The definitions are as follows:
6 = angle of discharge (0 = horizontal, 90 =
vertical' in direction of buoyancy)
y = height of rise from source to top of plume
'max
5 = dimensionless height of rise « y
m = dimensionless momentum flux parameter at the end of
0 the zone of flow establishment
y = dimensionless volume flux parameter at the end of
0 the zone of flow establishment
u = dimensionless volume flux parameter at the top of plume
S = centerline dilution at the top of the plume (relative
to end of zone of flow establishment)
39
-------
2.82
90
Terminal volume flux parameter
y for inclined round buoyant
jets with yQ = 0 to 0.01. (After
Fan and Brooks, 1969, A-2.)
60
90
Figure 13. Terminal height of rise £ for
inclined round buoyant jets with
y = 0 to 0.01. (After Fan and
Brooks, 1969, A-2.)
-------
The variables may be related to two other variables F and T as follows:
U
F = —— = densimetric Froude number (29)
PO~PI
T = - - — = a stratification parameter (30)
where p. is the centerline density at the end of the zone of flow estab-
lishment, p is the reference density (ambient fluid at source level),
and p is the ambient density. A small adjustment (-13%) is necessary
3.
to convert the discharge values of (p -p,) to (p -p ) , as given by Eq. 14.
o d o 1
With the entrainment coefficient a = 0.082 and the spreading ratio
X = 1.16:
m = 0.324 F2 JT1 (31)
o
y
o
= 2.38 F1/4!-5/8 (32)
S , =1.15 y /y (centerline, relative to discharge) (33)
S , = 2 y /y (average, relative to discharge) (34)
ad t o
3/8
= 1.37 ? F T (35)
5 and y are solutions read from Figs. 13 and 14, or from more detailed
figures in Fan and Brooks (1969, A-2 , Ch. IV).
For most practical cases it is sufficient to use the plume solution
(m ^ 0, y 'VO); this will overestimate the height of rise slightly and
underestimate the dilution. Therefore we may simplify Eq. 35 by
putting in g = 2.82, and rearranging:
= 3.86 FT*8 (36)
°r dp
41
-------
pl~pd
where Q = irU D2/4, and g' = - g. For this case a better value of a
o a p^
is 0.093 according to Morton, Taylor and Turner (1); with this value the
coefficient above in Eq. 36 becomes 3.75. The difference (less than
6%) is well within the limits of accuracy.
The centerline dilution for the limit.ing plume case is found by putting
Eq. 32 and y =1.71 (solution for m = 0, y =0) into Eq. 33:
Std = 1.15 y /vQ = 1.15 (1.71)(2.38)~1F~1/4T5/8
S , = 0.83 F~1/4T5/8 (38)
td
S-0. 70
For another useful form, combine Eq. 38 with Eq. 36
y
max S . = 0.826 (3.86)T = 3.19T
D td
p -p
= 3.19 ° X
max td " dp
-------
Another useful expression for the dilution at the top of the plume is
obtained by eliminating the density gradient (or T) in combining Eq. 36
and 38.
(3.86)-5/3
0.0872 (g-^—:
p " max ' o
o
-2/3
= °-071
It is interesting to compare this result with Eq. 18 for non-stratified
environment with height of rise y; the form of the equation is the same,
but the coefficient 0.089 for the non-stratified case is 25% more than
the coefficient (0.071) in Eq. 41. The reason for the difference is
not readily apparent, but intuitively it may be attributed to the lower
velocity of the rising plume as it approaches its terminal height in a
stratified environment compared with the non-stratified case in which
the flow reaches the level y (or surface) with appreciable residual
velocity.
Example . Consider a discharge of 50 mgd (78 cfs) from a multiple-port
diffuser at 100 ft depth when there is a density gradient in the ocean
as follows: }
Seawater, bottom (-100 f t) , p = p. = 1.02580
3. A
Seawater, top p = 1.02460
3.
a_ .00120 _ -5 -1
PI dy " 1.026(100)
(This density difference is approximately equivalent to a temperature
difference of 6°C.) The density of disch
8d = g(pl~pd)/Pl = <02568 = °'826 ft/sec2-
difference of 6°C.) The density of discharge is p = 0.99950, and
43
-------
Determine whether the sewage field will be submerged or not for the case
of (a) 50 ports and (b) 5 ports. For approximate analysis on the safe
side, use the buoyant plume approximation and neglect the horizontal
momentum and extra mixing induced near the bottom. Assume no interfer-
ence between jets.
By Eq. 37, with Q per port = 78/50 cfs/port:
y = 3.98 (||x 0.826)1/4(32.2 x 1.16 x 10~5)~3/8
H13.X j U
y = 82 ft
'max
The centerline dilution is found by Eq. 39 to be
.0263
td 2>8 .0012x( 82/100) 75'
The average is by Eq. 40:
, ° 128
ad
For part (b) , 5 ports, we obtain by ratios
Q = 15.5 cfs/port
y = 82 ft (T-) = 146 ft.
max 1.55
This result exceeds the total depth, and therefore the plume rises to
the surface (y = 100 ft) and is not kept submerged by the stratification.
From this example, it is apparent that multiple jet diff users greatly
enhance the possibility of generating a submerged sewage field.
44
-------
Slot Buoyant Jet (Linearly Stratified Environment, No Current)
The basic flow pattern for this case is shown in Fig. 3 (lower half).
The development for slot buoyant jets parallels that for round buoyant
jets described in the preceding section. The basic results from Fan
and Brooks (1969, A-2) are shown in Figs. 15 and 16. The definitions
are as follows:
B = slot jet width or equivalent (= —• — )
^f S
u
o
= densimetric Froude number (42)
PQ-P1
T = - j - = stratification parameter (43)
dp
Assuming a = 0.14 and X = 1.00 (slightly changed from Fan and Brooks),
the solution parameters are:
m = 0.500 F2 T'1 (44)
o
y =1.85 F^V1 (45)
o
= 0.96 g F1/3T1/2 (46)
The dilution at the top of the plume S needs no correction for the zone
of flow establishment because X = 1.00 (Pi=P,):
1 d
S = vyTy (centerline) (47)
t 'to
S = \] 2y /y (average) (48)
a i t o
The values of £ and y are the solutions read from Figs. 15 and 16, or
from more detailed figures in Fan and Brooks (1969, A-2, Ch. VI).
For most practical cases it is sufficient to use the plume solution
(m ~ 0, y ~ 0); this will overestimate the height of rise slightly and
45
-------
2.96
ON
Figure 16. Terminal volume flux parameter
y for inclined slot buoyant
jets with y = 0 to 0.01. (After
Fan and Brooks, 1969, A-2.)
90
Figure 15. Terminal height of rise £ for
inclined slot buoyant jets with
(After Fan and
y = 0 to 0.01.
Brooks, 1969, A-2.)
-------
underestimate the dilution. By putting 5 = 2.96 in Eq. 46, and re-
arranging
C49)
dp -1/2
ymax= 2.84 (qg') '-(-f-df) (50)
1
P ~Pd
wherein gf = —* g and q = discharge per unit length = U B. The final
coefficient in this case has not been directly confirmed by experiments;
the above result is entirely theoretical, based on a and A for buoyant
plumes in a non-stratified fluid. No experiments were made for this
case during this project.
The centerline dilution for the limiting plume case is obtained from the
solution y = 1.41 (for m ~ 0, y ~ 0) and the substitution of Eq. 45
t o o
into Eq. 47:
1741 ,-1/3 Tl/2
S -0.87 F~1/3 T 1/2 (51)
An
o /o _-i „ "(•> _i /o
S/*\ /I —T f I \ ^* / ••' "*• / ft &*\ * / *• / C1 \
t = 0.87 (qg ) q (- T— TTT") (52)
1
For the average dilution multiply Eq. 52 by /2" .
In terms of Ap over the height of rise, defined as
we obtain from Eqs. 49 and 51:
' St = 2.84(0.87)1
S. = 2.5 BT
ymax
Pi-p,
^ = 2.5 -i—£• > (53)
t Ap
47
-------
and the average dilution by Eq. 48 is
p~p
(54)
The centerline dilution at the top is also conveniently expressed in
terms of y by eliminating the density gradient (or T) from Eq. 49
and 51:
St B = 0.87 -2/3
y 2.84
max
S = 0.31 g|1/3y q"2/3 (55)
t max
This may be compared with the corresponding line plume result for the
non-stratified case, Eq. 26, which is of the same form but with co-
efficient of 0.38 instead of 0.31, and y replacing y . The coefficient
TQ.SLX
is 23% larger for the non-stratified case; a nearly similar percentage
difference (25%) was found for the round plume dilution formulas (non-
stratified vs. stratified). Again the difference might be attributed to
the slower rate of rise in the stratified case because the plume reaches
the top at zero velocity compared to "bumping" the free surface with
significant residual velocity. (The basic entrainment equation pre-
sumes that entrainment is proportional to plume velocity, Eq. 5.)
The formulas for height of rise (Eq. 50) and for dilution (Eq. 52,
53, 54 and 55) are for the limiting case of a buoyant plume, and are
thus conservative for design; in other words, the height of rise is
overestimated and the dilution underestimated. More realistic co-
efficients in the equations may be found by calculating m and y for a
typical line diffuser, and determining values of £ and y from Figs. 15
and 16. Representative values for a line diffuser for sewage discharge
in the ocean might be as follows:
U = discharge velocity = 10 ft/sec
2
q = discharge per unit length = 0.07 ft /sec
48
-------
B = q/U = 0.007 ft
Q
Ap = p -pd = 0.026
dpa -5 -1
3-^ = 0.001/100 ft = 10 ft
dy
p = 1.025
o
g' = AP g = 0.84 ft/sec2
o
F = °- = 10 = 131
\/g"rB Y. 84x0. 007
0.007 x
mQ = 0.50 F2T -1 = 0.023 (Eq. 44)
y =1.85 F2'3T-1 = 1.3 x 10~4 (Eq. 45)
The purpose of the above calculation is only to establish orders of
-4
magnitude; we find that y is so small (-10 ) that it has negligible
effect on the plume dynamics, while the magnitude of m (-0.02) does
affect the solution slightly. We find from Figs. 15 and 16 (or Ref.
A-2): Buoyant slot jet (m - 0.02)
VerticalHorizontal Plume
discharge discharge values
? , 2.78 2.61 2.96
yt 1.46 1.48 1.41
Often a horizontal discharge is used, but with the flow split between
the two sides of the pipe (ports on both sides). The initial momentum
theoretically cancels out for the whole flow pattern, but there is still
the benefit of high dilution with "heavy" bottom water. In this case,
it is recommended that the flow be regarded as-a single line, but that
credit be allowed for the momentum by using £ = 2.61 and u = 1.48 for
horizontal discharge. In other words we are saying that the dilution
49
-------
obtained with ports on both sides is presumed to be as good as it
would be if they were all on one side, jetting horizontally.
The revised equations, recommended for design purposes for line multi-
port diff users in a stratified environment are as follows:
Height of rise:
dp ~
1/3
' ' -
Centerline dilution at top:
St = 0.89 (qg'r'J q " (- *- ^ ) (57)
1
P,-PJ
St - 2.2 -L-4 (58)
t = 0.36 g'1/3 ymax q- (59)
For average dilution, multiply above values by i/2~ .
The numerical coefficients given in the formulas above differ somewhat
from those given previously (Brooks, 1970, B-l; Fischer and Brooks,
1970, B-2); the values of a and X have been adjusted and some arithmetic
errors corrected.
Example. For the data given immediately above we may make the calcula-
tion of y and S from Equations 56 and 59 respectively, as follows:
H13.X t
ymax = 55 ft'
St = 0.36 (0.84)1/3 (55) (0.07)~2/3
S = 110 (centerline)
S = 155 (average)
CL
50
-------
Approximate Solutions for Buoyant Jets in Environments with
Non-Linear Stratification
In practice, the ambient density stratification will never be exactly
linear; to solve buoyant jet problems one may either make a direct
computer solution, or resort to a linear approximation of the measured
profile, using the formulas of the preceding subsections.
For direct computer solutions, the reader is referred to programs
developed for this project: for the round buoyant jet, Ditmars (1969,
A-3) and for the slot buoyant jet, Sotil (1971, A-ll). The program
input for each includes the measured density profile for the environ-
ment, represented by the density values at selected elevations in the
fluid. The solution proceeds from the same assumptions and equations
as used for the cases of constant density gradients.
For most problems, a reasonable approximate solution (Brooks, 1970, B-l)
is obtained by assuming an equivalent uniform density gradient over that
part of the total depth through which the buoyant plume rises. In
Fig. 17 the point of discharge is at 0 (neglect the size of the zone of
flow establishment), with y measured upward from this point. The
measured density profile is plotted from the water surface down, and
the point 0 may be shifted up or down along the profile in the process
of testing different designs. Between 0 and any point A the mean
gradient is
Apo
dy
(60)
ave
For simplicity Ap is taken as positive for a stable profile, thus making
the Ap-axis read backwards (measuring from right to left). To find the
height of rise, y , let the point A be at the level y = y , and
H13X <* aidA
substitute Eq. 60 into Eq. 37 for round buoyant jets:
}"3/8 C61)
51
-------
Figure 17. Approximation of non-linear density profile by a linear one.
max
Point of
discharge
Figure 18. Diagram for solving Equations 62 and 63 with measured
density profiles. (After Brooks, 1970, B-l.)
52
-------
Simplifying:
" <«>
Similarly for line buoyant jets, Eqs. 56 and 60 yield
2/3 APP -1
= 6'25 <«'> ' -C «> (63)
Equations 62 and 63 may now be plotted on a graph of y vs. Ap; each
represents all possible solutions (y ,Ap ) for the given buoyancy flux
IH3.X 3.
of the source (point or line) . If these curves are placed on a trans-
parent graph overlying measured profiles, as shown in Fig. 18, the
solutions are represented by the intersections (i.e. we find the solution
values of (y , Ap ) which satisfy the measured profiles).
max a
Note that for high values of Ap the point source solution gives higher
values of y , whereas for small Ap the line source solution gives the
'max' F 6
higher values of y .It is recommended that in either case the higher
max — fi -
value of the solution y always be used as a best estimate. For large
-, 'max J 6
heights of rise, the flow pattern over a multiple-port diffuser will
become essentially a line plume (with interference between jets in a
row) , whereas for low rises the diffuser may generate essentially a
series of independent round plumes, which stop rising before significant
overlapping occurs. For the three ambient profiles in Fig. 18, the
solutions are shown as y , y? , y . Note that profile 3 allows the plume
to reach the surface. With a transparency for the curves representing
Equations 62 and 63, it is easy to investigate different possible depths
of discharge at maiiy times of year (varying density profiles) .
This procedure can not be expected to be highly accurate , but nonetheless ,
it clarifies the relationships between the variables, and provides a
rational basis for selecting a diffuser length and depth for achieving
submergence various percentages of the time.
53
-------
The dilutions are now readily obtained by applying Eq. 39 for round
buoyant plumes; or Eqs. 53 or 58 for buoyant slot plumes or jets. For
a given effluent density, the dilution is simply proportional to (Ap )
3.
Alternately the dilutions may be determined from y values by Eq. 41
for round plumes, Eq. 55 for slot plumes, or Eq. 59 for typical slot jets.
It is important to note that when the density gradient is strong the
dilution is decreased because y is less. The desire for submergence
must be balanced with the need for high dilutions in developing an opti-
mum design for an outfall diffuser.
This procedure has been applied successfully to several outfall designs,
the first of which was a 27,400-foot long outfall, 10 feet in diameter,
with a 6,000-foot long diffuser, built for the Orange County Sanitation
District, California (5).
Reduction of Dilution by Blockage of Sewage Field for Line Diffuser
The dilution formulas may substantially over-estimate the dilution over
line diffusers in case of light currents. In this case the sewage field
may grow to such thickness over the diffuser that the height of rise
through uncontaminated water may be reduced. Thus the entrainment of
new diluting water is reduced, while continued plume rise within the
sewage field itself does not increase the net dilution. The internal
dynamics remain about the same.
The following approximate analysis may be used for cases with or without
ambient stratification (see Fig. 19). As an approximation the "fallback"
from the top of the plume to the top of the field is neglected.
Let b = average width of cloud
L = diffuser length
y = maximum height of rise above diffuser
(may be taken as total depth if unstratified)
h = thickness of sewage field
U = current speed
y = y_0 -h = height frpm diffuser to bottom of sewage field
max
54
-------
u
Diffuser pipe
ELEVATION
SECTION A-A
Figure 19. Schematic diagram of blocking of part of the water column
by the pollutant field.
55
-------
Q = sewage discharge
s
S = centerline dilution at top of plume as calculated by
formulas assuming no blockage
S _ = centerline dilution at height y (bottom of sewage field)
* which is taken to be the reduced dilution for the
plume because of blockage
S = average dilution in sewage field over diffuser
3.
For all buoyant plumes (with or without stratification), the dilution is
approximately linear with height, that is:
/=/- -(64)
t max
The average dilution in the field is v2 times the centerline value, or
S = T/2 S . (65)
ay
But the efflux from the outfall site may be expressed by the continuity
equation
Hence,
SaQg = Ubh = Ub(ymax-y) (66)
Ub(y - y)
S = max (67)
Dividing by S
S Uby
(68)
•y,
Let P be a factor defined as
t /2 Q S max
S L
max
and using Eqs. 64 and 69 in 68, we get
56
-------
s s
(70)
or
Sy = St
This is the desired result. One computes the dilution S without con-
t-i
sidering blockage, and then finds the correction factor •JTJJ for blockage
based on known quantities. The average field dilution is finally given
by Eq. 65.
If P = 2 the analysis indicates that the sewage cloud has filled the
upper 2/3 of the space with only 1/3 left at the bottom for entry of new
water. For P > 2 , the above analysis is not to be trusted, and the
whole dilution phenomenon is probably limited by far-field phenomena
rather than near-field (i.e. diffuser details). This appears to be es-
pecially so for thermal outfalls, and additional research is underway.
Final Comment
The tesults of calculations for buoyant jets in a stratified environment
are probably no more accurate than +_ 20%. The results are primarily
theoretical, with quantitative confirmation only for round buoyant jets
in laboratory tanks. Field data check these results qualitatively for
outfall diffusers recently built, but this writer knows of no quantita-
tive studies of submerged sewage fields which are sufficiently detailed
i
to evaluate the formulas above. At the field scale, measurements are
extremely difficult; however, more laboratory work on line buoyant jets
is badly needed.
57
-------
SECTION VII
RELEASE OF A SLUG OF DENSE FLUID
INTO A TWO-LAYERED ENVIRONMENT
Problem
When a slug of sludge or dredge solids is dumped into the sea or any
other body of water, it may sink only to the level of a density inter-
face such as a thermocline. Although the initial slug is heavier than
the upper layer and starts sinking, the entrainment of the upper layer
may so reduce the effective density of the slug that it does not pene-
trate into the lower more dense layer. Sullivan (1972, A-8) investigated
this problem with dimensional analysis and small-scale laboratory experi-
ments. First he considered the case of a slug without initial momentum,
and then with initial momentum. In the latter case, distinct vortex
rings formed. This problem is similar to the idea of puffing smoke out
of a smokestack in an attempt to penetrate atmospheric inversion layers.
Results
Only the case of zero initial momentum will be considered here. The im-
portant variables are defined as follows (see Fig. 20):
V = volume of heavy fluid initially released
p. = initial density of heavy fluid injected
p = density of upper layer
p = density of lower layer
g = gravity
Z = depth from release point to interface
Under the assumption that all the density differences are small compared
to p, the variables of physical significance are:
59
-------
t
I
t
Density,/?.
Volume, V
Figure 20. Definition sketch for release of heavy fluid in a two-layer
environment.
g(p.-p1)V = submerged weight (the driving force of
the flow pattern)
Z = distance of travel to the interface
g(po~Pi) = difference in unit weight between the layers
These three variables may be combined into just one dimensionless group
which must characterize the flow pattern:
A
(p±-Pl) V
(72)
60.
-------
Sullivan found the following criteria experimentally:
A > 29 Less than 10% of injected slug
penetrates the lower layer
A < 1.5 More than 90% of injected slug
continues into the lower layer
These numbers should be considered tentative because the experiments
were conducted at low Reynolds number for which the flow was partly
laminar.
Example. A load of 1,000 ft3 of sludge at p = 1.0300 is to be suddenly
dumped in the ocean (p = 1.0245); at 40 feet depth there is a thermo-
cline below which p = 1.0255. Find whether the sludge sinks into the
lower layer.
Compute A by Eq. 72:
= (1-0255 - 1.0245)403
(1.0300 - 1.0245)1000
= 11.6
As this number is below 29 but above 1.5, there will be substantial but
not complete penetration of the thermocline by the sinking fluid mass.
The outer portions of the sinking mass will entrain enough of the fluid
of the upper layer to make the mixture density less than that of the
lower layer.
If the sludge is discharged with some initial velocity (for instance,
if dropped on the water surface) then the penetration would be greater.
(See Sullivan, 1972, A-8 for details of analysis.) On the other hand,
if the volume V were ten times smaller, then it would not penetrate the
interface. A slow continuous release would also have less penetration.
Of course, whether penetration of the thermocline is desirable or not
depends on the circumstances of a particular problem.
61
-------
SECTION VIII
SELECTIVE WITHDRAWAL AND ARTIFICIAL MIXING
IN DENSITY-STRATIFIED RESERVOIRS
Selective Withdrawal
Another important stratified flow pattern occurs at the outlet of a dam
in a stratified water reservoir as depicted in Fig. 21. A withdrawal
layer forms at the level of the outlet because lighter fluid above and
heavier fluid below cannot be drawn through the outlet.
Brooks and Koh (1969, C-l) have provided a detailed review of this
problem and have recommended an analysis for extending the earlier work
of Koh (6) to apply to large-scale turbulent reservoirs. A linear
approximation is used for the ambient density profile, and the reservoir
flow pattern is considered to be two-dimensional (i.e. uniform across
the reservoir). The remainder of this sub-section is extracted from
Brooks and Koh (1969, p. 1396-7).
For turbulent withdrawal flows away from immediate vicinity of the out-
let, it is hypothesized that Koh's viscous diffusive experiments and
analysis can be applied by replacing v and D by E , the vertical eddy
diffusivity. For self-generated turbulence, it is predicted that the
r— 1/2
proper characteristic length is a = (q/v/gT) , where q is the unit dis-
charge and e = - — -£-. This same length a is useful for the inviscid
PQ dy
case. 1
For steady withdrawal flows from a linearly stratified reservoir the
following formulas are recommended for the thickness of the flowing
layer (5) in terms of the characteristic length a at distance x from
the outlet:
a. Inviscid flow result (very close to the dam):
- = 2.7 + 0.2
a —
r— 1 /9
where a = (q/v^je)
63
-------
DAM
OUTLETS ic*-"
OR
PENSTOCKS
WITHDRAWAL
LAYER FLOW
Figure 21. Selective withdrawal from a reservoir through one of several
outlets at various levels in a dam. (After Brooks and Koh,
1969, C-l.)
KOH EXPERIMENTS
EQ.74
O FONTANA
RESERVOIR
INVISCID RESULT:-|=2.7
(qz/ge)
Figure 22. Summary of recommended formulas for selective withdrawal.
(After Brooks and Koh, 1969, C-l.)
64
-------
b. Turbulent flow, moderate distances (estimated on basis of
Koh's experiments):
(74,
for 2.7 < - < 13.7
3.
c. Turbulent flow, large distances (estimated on basis of
Koh ' s theory) :
•^ = 7.14 (k, -)1/3 (75)
a /a
for - > 13.7
Si
The relationship of these formulas is shown in Fig. 22.
The parameter k_ = E /q can only be determined from field experiments.
Z. IQ. A
It is anticipated that k» is of the order 10 .
Transients in the withdrawal currents caused by fluctuating power re-
leases are predicted to be very persistent, taking many hours to die
out. Transients may be thought of as very slow-moving internal waves.
They will undoubtedly greatly complicate field measurements and manage-
ment techniques in the future.
As a technique for water quality management, selective withdrawal has
prob.ably been somewhat overrated for continuously stratified flows.
The temperature of water withdrawn at an outlet may be that of the water-
column at that level, but, nonetheless, the discharge is a blend of
water from the entire withdrawal layer thickness, 6. Typical values of
-------
Artificial Mixing in Stratified Reservoirs
A stratified water reservoir may be mixed by pumping water from one level
to the other as shown in Fig. 23. At the intake there is essentially a
selective withdrawal pattern while at the outlet a buoyant jet rises to
its equilibrium level in the ambient stratification. As this jet rises
it entrains bottom water and lifts it to the level where the plume
spreads out laterally.
Ditmars (1970, C-2) developed a computer simulation for this process
using short time steps. He assumed the selective withdrawal and buoyant-
jet parts of the flow could be analyzed separately and that they were
occurring in a quasi-static reservoir. In other words after a time step
of pumping, the density profile would be adjusted to reflect the dis-
placements resulting from the pumping. The constant density surfaces
in the reservoir were assumed horizontal. The models for simulating the
withdrawal and jet discharge were those developed during this project
(primarily Brooks and Koh, 1969, C-l, and Ditmars, 1969, A-3).
The results of such a simulation are illustrated in Fig. 24 and compared
with a laboratory experiment in a tank 9 m long, 45 cm deep and 61 cm
wide. The levels of the intake and discharge pipes are shown at the
right of the graph. The variable T is the characteristic time defined
as T = V/Q, where V is the volume of tank between levels of intake and
discharge and Q is the pumping rate. The real time elapsed from the
start of pumping (t) is normalized by dividing by T. The agreement
between theory and experiment is reasonably good. (The measured density
profile at t = 0 is taken as the initial condition for the simulation,
so perfect agreement is indicated at t = 0.) The effect of the jet is
seen to be an ever-increasing uniformly mixed layer at the bottom. The
rising plume generally stops rising and spreads out where the density
curves suddenly change slope.
66
-------
AMBIENT DENSITY
Figure 23. Schematic diagram of a pumping system for mixing a density-stratified reservoir.
(After Ditmars, 1970, C-2.)
-------
EXPER. NO. 14
STA. 8.10 m
T=22.2hr
t/T
0
0.135
0.270
Corresponding
simulation results
2.5 3.0 3.5 4.0 4.5
(DENSITY-I)X1000 - g/ml
1 = 0.99841 g/ml
Figure 24. Measured and simulated density profiles for a typical
experiment. (After Ditmars, 1970, C-2.)
68
-------
The simulation analysis can be made for a reservoir of any geometric
shape and any stable density profile at the start. However, to under-
stand the sensitivity of the procedure to various parameters, it was
found desirable to develop some generalized results for linear density
profiles and reservoirs of constant surface area (vertical walls). The
principal variables are as follows:
V = volume of water between levels of pump intake
and discharge
d = depth, intake to discharge
gAp = difference in ambient weight density between outlet
and inlet levels
= initial buoyant unit weight of jet discharge
p = reference density (at intake)
Q = pumping rate
D = discharge jet diameter
y = elevation above the discharge pipe
t = time since beginning of pumping
g(p-p ) = buoyant weight at any y and t
The density profile may now be expressed in dimensionless form as
p* = f(y*, t*, F, S, P) (76)
wherein
p-p
P
0
(77)
Ap
y* = y/d (78)
t* = -E2. (79)
V
F = Q = Froude number of the jet (80)
69
-------
S = (81)
cT
F - Q(g f r1/2,T5/2 C82)
0
The variables F, S, and P are fixed parameters for each flow.
A typical result of these non-dimensional simulations is shown in Fig. 25,
r
From various simulation runs it was found that the results were not very
sensitive to F, the jet Froude number. The mixing was slightly more
effective for higher F values, but only at greatly increased energy input
and lower efficiency. Thus most of the simulations to investigate the
importance of P and S were made at a low Froude number, F = 3.
A convenient measure of the degree of dastratification is the increase
of potential energy associated with the redistribution of mass. In the
stratified condition the center of mass is slightly below the center of
volume, to which it is raised after complete mixing. The work required
to do this is called "required potential energy increase." The success-
ive steps of destratification may then be characterized by the fraction
of this potential energy increase achieved at any time t*. Figure 26
gives the results to M as a function of t* and P for three values of
S?and F = 3. The time required to reach a given degree of mixing (such
as M = .90) is virtually independent of S, the shape parameter; for
large S-values the reservoir has a very large area compared to the depth
squared. The real time for pumping, t = t*V/Q, is directly proportional
to the volume, but otherwise the shape has no direct effect because we
assumed density surfaces to be flat (which cannot be exactly true for
large reservoirs).
The sensitivity of M to P is also small. The parameter P indicates the
local strength of the jet in relation to the density stratification: for
small P the maximum height of rise of the buoyant jet (at t = 0+) is
small compared to the total depth, and the converse for large P.
70
-------
1.25
1.00 -
0.75 -
I
X
Q_
UJ
O
0.50 -
0.25 -
0.00
= 3.0
S = 500
P=2.50XIO~4
Profiles at intervals of t
0.00
0.25
0.50
0.75
1.00
Figure 25. Typical non-dimensional density profiles by simulation
of reservoir destratification by pumping (for S = 500,
P - 2.5 x 10-1*). (After Ditmars, 1970, C-2.)
71
-------
VI
NJ
UJ
\J.3V
0.28
0.26
0.24
0.22
0.20
0.18
0.16
0.14
ni9
'::
::
:":
;•.•;:;
: HI;
i •' : : :
:::::
.
o
. •." ;
::: . ;:
~i~:
'
'•'. :':'. :
-
\
jj
1
;-.
';
: ;
::::
. ' .
::B:
'
:;
3
•
~
».
... . -
1
-;
•-
o
1
JM
T
I!:
::: :
:: -
::.
-"
: : E : :
: : : .
•-
•;
I
I
-'
:
;i'*f
i »
•
ii
::
'.: :':.
Af
f
M
f
-
a
S |
£ |
= f
<
?.<
S
.;
90
ji:
•:::
::::
: : : : :
::::
!:i!;=
:::::
: : • :
3.?0 h
t '::
i
__:;
1 :::
:|
::: :
::::::
:-::::::
::: ..
::: C
: : :
::::
::::
::::
iilli
:...:. '
::: : :
-- JB-4
;;
.LUI iii
/
ft
Kll
W
I
M = FRACTION OF REQUIRED
POTENTIAL ENERGY INCREA
S. F=3.0
o 500 FOR ALL
« 2930 SIMULATIONS
e 50,000
>
j
;.;
:: '
:: ..
:;
|
::
'-'-
t
'-[
|
:•
-;
•
:
t
: '
. !._
tl
:
' "
^
^
I
1
~:
;:
'
f -
i
[
O
>>
i
**
1
.
SE
fl |
- ;
-
i
!
T
'
'••
10"
10
,,-B
10
.-4
10"
10
-2
10"
10"
Figure 26. Summary of generalized simulation results for destratifying reservoirs by pumping;
fraction of required potential energy increase (M) as a function of P and t*.
(After Ditmars, 1970, C-2J
-------
In general, we may say that Fig. 26 implies that it takes
t* % 0.21 + 0.02 to achieve M = .80 (or close to complete mixing) re-
gardless of the values of the parameters S, P, and F. The real time
required is thus
t = 0.21T = 0.21V/Q
Therefore, any pumping system has to be designed to pump at least 21%
of the entire reservoir volume through the system in the time set to
accomplish the mixing. The remaining water is moved by entrainment or
displacement.
The mixing may be accomplished by one or several pumping systems, inas-
much as the result is insensitive to the parameter P. Even the provision
of multiple-port diffusers would probably be of little use after the
bottom water becomes fairly uniform. Furthermore jetting the water at
high velocity is of little value. The lack of success of some previous
field attempts (see Ditmars, 1969, C-2, Ch. 5) seems to be due to lack
of adequate pumping capacity. However, for very large lakes the pro-
vision of very large pumping systems becomes very costly, as the required
pumping rate may easily exceed the mean river flow.
This simulation does not include the effects of heat additions or losses
during the mixing process. These factors can be added to the simulation
as desired.
Another way to mix a stratified reservoir, or to reaerate it is to bubble
air or oxygen. Cederwall and Ditmars (1970, A-5) made an analytical
study of air bubble plumes, using the same integral techniques as for
buoyant plumes treated previously. The mixture of water and bubbles
was considered equivalent to a fluid of lighter density (for short time
period). The results showed only fair agreement with previously published
experimental results by others.
73
-------
SECTION IX
TRANSVERSE MIXING IN RIVERS
AND OTHER SHEAR FLOWS
In rivers and estuaries it is often important to know how rapidly
pollutants will spread laterally across a flow. For example, if an
outfall discharges near the middle of a river, how far downstream will
it be before the pollutant is well mixed across the river (according to
some set criterion)? And what is the pollutant concentration along the
bank at various distances downstream? It is possible to get approxi-
mate answers to these questions by the classical diffusion equation,
provided we can predict a good value of the diffusion coefficient for
transverse mixing. The case of a pollutant of different density from
the river requires special attention because of the temporary secondary
circulations induced by the discharge.
This research project included several studies of transverse mixing,
as reported in the Appendix, Section E. In the following sections,
some important results are given.
Transverse Mixing — No Density Difference
A major part of Okoye's work (1970, E-2) was concerned with measurement
of the lateral diffusion coefficient in open channel flows in the 40-
meter- and the 18-meter-long tilting flumes in the Keck Laboratory.
The transverse mixing coefficient was determined from measurements of
I
the distribution of a salt tracer downstream from a continuous source
(small tube near middle of cross section). The salt solution was made
neutrally buoyant by the addition of methanol. Fig. 27 shows the flow
pattern schematically.
75
-------
-u(y)
_7
Velocity
Distribution
-solid bottom
boundary
(a) Section
solid bottom
boundary layer
(near source)
Vertical Concentration
Profile
w
— u(z)-
Velocity
Distribution
-side wall
-Point Source
-Diffusing
plume
side wall
boundary layer
side wall
boundary layer
(b) Plan
Transverse
Concentration Profile
side wall
Figure 27. Definition sketch of plume geometry and coordinate axes. (After Okoye, 1970, E-2.)
-------
The depth- averaged coefficient of transverse mixing 5 was defined and
z
calculated by the relation:
where u = mean flow velocity in the flume cross section; a2 = variance
of the concentration profile in transverse direction at any level in the
flow; a2 is the value of a2 averaged over the depth; and x is the dis-
tance downstream from the source. The theoretical basis for this
relation in a flow with both vertical and horizontal mixing is given in
detail by Okoye (1970, E-2) . The coefficient thus defined includes both
turbulent diffusion and transverse dispersion (Taylor type) associated
with the secondary currents. The value of a2 increased at a linear
rate, with distance downstream, thereby making D independent of x.
z
The mixing coefficient is normalized by dividing by the product of u ~ ,
the shear velocity for the bottom, and d the depth:
D
0 = -~r (84)
u*bd
Fig. 28 summarizes the results of the present study by Okoye and com-
pares them with other laboratory and field measurements. It was found
that the value of 9 is a function of the depth-width ratio, d/w = X. The
values of 6 ranged from 0.24 (at X = 0.015) to 0.093 (at X = 0,20) in the
laboratory. The few values measured in the field by others are about
twice as large, presumably because of stronger secondary currents
caused by bends or channel irregularities.
For comparison it may be noted that the value of D /u^d, the normalized
vertical mixing coefficient, is K/6 (K = von Karman constant) or 0.07.
Thus in the flume the transverse mixing coefficient is of the order of
1.3 to 3 times larger than the vertical mixing coefficient.
The original report by Okoye also gives results for variation of D in
the vertical; near-source spreading in the vertical direction; cross-
sectional maps of tracer concentration; decay of maximum concentration;
and position of point of maximum concentration.
77
-------
OQ
o
u_
u_
o
8
U
cn
cc
CO
LU
1.00
0.40
0.10
0.04
Laboratory Measurements
« Elder (I)
o Sullivan (6)
a Kalinske and Pi en (30)
Prych(39)
Flume SI '
Flume SZ
Flume R2
A
Present
Study
Field Measurements
o Yotsukura et a I (8)
H Fischer (7)
® Glover (42)
I I
0.01
0.04 0.10
ASPECT RATIO, X
I I
0.40
1.00
Figure 28. Variation of the depth-averaged dimensionless mixing coefficient 8 with the aspect
ratio X for all experiments performed in the present and past studies. (After Okoye,
1970, E-2.)
-------
The second phase of Okoye's work dealt with phenomena associated with
fluctuations in the spreading plume phenomenon. For a fluctuating
plume, Fig. 29 illustrates the concept of regions of intermittency.
Points within these regions are within the plume part of the time;
the fraction of the time within the plume is called the intermittency
factor,! . Within the plume there is an inner core for which 1=1,
meaning that the region is always within the plume.
The growth of these zones is shown in Fig. 30 for a typical experiment;
W,. is the extreme limit (I_ = 0) measured from the central axis; A is
the edge of the continuous inner core (I = 1); and 2 is the mean
position of the edge of the plume (If ~ 0.5). For all the flume experi-
ments (wide rectangular channels), these zone widths fit the following
dimensionless relations:
2=3-6Vi> ^ <85>
u
|-3.3Rz(^)2/3 ^ (86)
U
A = 2Z - Wf (87)
where R = (f /f )1/4
w s r
R = (f /f )1/3
z s r
f = bed friction factor (smooth)
s
f = bed frictidn factor (observed)
X = value of x corrected slightly for virtual origin
of Wf
Uj., = shear velocity for the bottom
«D
u = mean velocity
d = depth
Also investigated were the fluctuation frequencies for the plume as a
whole, and the characteristics of concentration fluctuations with time
at a point.
79
-------
Source
extreme limit
of plume
boundary
instantaneous
plume boundary
extreme limit of
plume boundary
Region of
Intermittency
Core of
_. Continuous Record
Region of
Intermittency
(a) Geometric Features of the Physical Model (plan)
If(x,,y,;z)
1.00
0.50
0.00,
Note:
If(x,,y,;z) = Intermittency Factor
at section x, and
level y(.
?= Mean position of the
plume front.
-2
Region of
Intermittency
Core of
Continuous
Record
Region of
Intermittency
(b) Distribution of the Intermittency Factor Across the Plume
Figure 29. The intermittency factor model for cross-wise plume variation.
(After Okoye, 1970, E-2.)
80
-------
00
UJ
a
o
40 -
E
o
IN
8 30
CO
£C
UJ
O
O
O
DC
10
Wf: extreme limit of the plume boundary (If=0)
Z : mean position of plume front {!.»0.50)
A : outer edge of the inner core (I =4.0)
= 5.36cm,u = 43.7cm/sec
2.17cm/sec, R «I.I73XI05
200 400 600 800 1000 1200
DISTANCE FROM THE SOURCE, x,cm
1400
1600
1800
Figure 30. Growth of the geometric characteristics of the region of intermittency: RUN 802.
(After Okoye, 1970, E-2.)
-------
Transverse Mixing with Density Differences
If the pollutant flowing into a river or estuary is either heavier or
lighter than its surroundings, it will induce a strong secondary flow
pattern by sinking (or rising). This effect will accelerate the trans-
verse spreading of the pollutant, but gradually vertical mixing will
spread the pollutant uniformly over the depth and the driving force of
the density difference approaches zero. The transverse mixing co-
efficient approaches that for a normal shear flow without density
difference.
Prych (1970, E-l) presented the results of a comprehensive study on
this subject. The basic flow pattern in shown in Fig. 31: within the
40-meter tilting flume (110 cm wide) a tracer stream was introduced
through a slot of width b at approximately the same velocity as the
stream flow.
Fig. 32(a) is an overhead photograph of the spreading of the tracer in a
flume experiment when the density difference was zero (—— = 0). Point
P
measurements of concentration were made at different stations down-
stream as shown in Fig. 33. Measurements of the transverse mixing co-
efficient without density difference were made for all runs (by Eq. 83)
to establish a basis for comparison with density runs; the results
agreed well with Okoye's results (see Fig. 28).
For the case of a buoyant tracer stream (— = - 0.0158), the spreading
is much more rapid especially near the source as shown by Figs. 32(b) and
34. Note especially the evidence of the strong transverse surface
current away from the source caused by the density difference.
For a quantitative description of the phenomenon, Prych computed the
second moments of the transverse concentration distributions (averaged
over the depth), and plotted these against distance downstream, x
(see Fig. 35). The a2 for flow with hot water (Exp. 128) grew very
rapidly at first, but then slowed to the same growth rate (-:—) as the
QjC
uniform density case (lowest curve, Exp. 116); other curves for heavy
tracer fluid do not show such rapid spreading.
82
-------
SECTION
A-A ^
— »--.
O
** —-..
O
PLAN
concentration
or
density-
u
density-induced
secondary circulation
Figure 31. Definition sketch from transverse mixing experiments in
laboratory channel. (After Brooks, 1970, G-l.)
83
-------
(a)
(b)
Figure 32. Overhead photograph of experiments in 110-cm wide flume
with 1-cm wide source. (Depth = 6.55 cm, mean
velocity =45.2 cm/sec, shear velocity =2.27 cm/sec.)
(After Prych, 1970, E-l.)
(a) Exp. 116, Ap/p = 0 (no density difference).
(b) Exp. 128, Ap/p = -0.0158 (buoyant tracer).
84
-------
oo
EXPERIMENT 116, -^-=
t
-55 -40
-30
-20
-10 0
LATERAL COORDINATE, z, IN CM
40 55
Figure 33. Contours of equal relative concentration in cross-sections downstream from a 1-cm
wide source which discharged a fluid with a density the same as the ambient fluid
and with a relative concentration of 1.0. The crosses designate sampling points.
(After Prych, 1970, E-l.)
-------
00
-0.0158
n-ZOO cm
x=500cm § § § §
UJ
I
8
o
m
8
= 1000 cm o'
§
o
§
§
o
i
o
d
1500 cm
o
§
in
§
8
I
0
2000 cm o
'.U,
1 \ M \
-40 -30
55 §
• \ ) ; ^
: ( (:
•=! S \l ! M
i \ m \ \ i /// i M | i | i m | i |
-20 -10 0 10 20 30 40
LATERAL COORDINATE, z, IN CM
Figure 34. Contours of equal relative concentration in cross-sections downstream from a 1-cm
wide source which discharged a fluid with a density 0.0158 gr/cm3 less than the
ambient fluid and with a relative concentration of 1.0. The crosses designate
-sampling points. (After Prych, 1970, E-l.)
-------
400
DISTANCE DOWNSTREAM,x,IN CM
SYMBOL EXP.
Figure 35. Variance - distance curves from flume experiments with
1-cm-wide source (smooth walls). (After Prych, 1970, E-l.)
87
-------
The excess variance (Aa2) for density-stratified cases compared to
uniform cases was summarized for many runs in a suitable dimensionless
graph relating the important parameters.
The dimensionless variables used in Fig. 36 are:
: dimensionless excess variance (88)
Aa2 = excess of transverse a2 for pollutant cloud
over what it would be without
density difference (depth-averaged)
B = -r = dimensionless source width
d
b = source width
d = depth
|M | = -LJLL gb/(ctu.,.)2 = dimensionless source strength (89)
Pa
Ap = density difference between injected and ambient
fluids
p = ambient density
a
g = gravity
UA = shear velocity
a = D /u.d = dimensionless transverse mixing coefficient
z
The curve in the upper left corner (dashed) is for buoyant tracers
which spread faster than tracers which sink to the bottom and are re-
tarded by bottom friction.
The straight line portions of the graph may be described by a single
equation
AV=('^T-L)3/2 (90)
where M* is the intercept plotted in Fig. 37. For narrow sources (small
B) the curve for buoyant injected fluid (Ap/p < 0) is different from
3.
the one for heavy injected fluid. Both curves are horizontal for narrow
88
-------
0.247
0.1 S3
0.090
0258
0.156
60 100
300
600 1000
IQpOO
DIMENSIONLESS SOURCE STRENGTH,|Md|=l^gd/(au/
Figure 36. The dlinensionless excess variance, AV, as a function of
the dimensionless source strength, M^, and the dimension-
less source width, B. (After Prych, 1970, E-l.)
89
-------
3000
vO
O
Lp/p > 0
'a
AyO/yO < 0
7 a
100
60
30
0.3 0.6 I 36 10
DIMENSIONLESS SOURCE WIDTH,B
Figure 37. The intercept, M[j*> as a function of the dimensionless source width, B.
(After Prych, 1970, E-l.)
60 100
-------
sources (B = b/D < 0.5), because the lateral spreading depends only on
the total strength of the buoyancy source without regard to the geometric
ratio B = b/d. At the other limit, the M/ « B, and M, /M/ becomes inde-
pendent of b; in other words the source is so wide that its dynamics is
controlled by the depth.
The excess variance is a one-time additional spreading of the pollutant
cloud which occurs within an approximate distance of
xau*
X = -=— < 1.5 (91)
ud
or
f < £
The dimensionless rate at which the excess AV grows is also given by
Prych (1970, E-l).
Final Comment
The transverse spreading of a buoyant or heavy pollutant stream can be
predicted from the results of this section. First the ordinary mixing
coefficient D is obtained from Fig. 28 and Eq. 84; secondly the value
z
of AV is determined from Fig. 37 and Eq. 90. Then
2xD
a2 = -s-5- + d2AV + a2 (93)
u o
when a2 is the initial valbe. Furthermore if X < 1.5, AV must be re-
o
placed by rAV where r(
-------
Coudert (1970, E-4) provided a very useful numerical program for solving
the problem of diffusion in two dimensions (vertically and downstream)
from a point source in two dimensions (i.e. line source across the
stream in z-direction). With it he examined the near-source effects of
non-uniform profiles of velocity and diffusion coefficient. For
example, the locus of y-values for the peak concentration at each down-
stream station initially sinks below the level of the source. Examina-
tion of these details proved helpful in the interpretation of Okoye's
results (1970, E-2).
92
-------
SECTION X
ACKNOWLEDGMENTS
The writer wishes to acknowledge with appreciation the support of this
broad long-term grant by the EPA (and predecessor agencies) under
Grant No. 16070 DGY, for which Mr. Richard Callaway served as EPA
Program Officer.
The contributions of all the other research collaborators whose names
appear as authors on the publication list in Section XII were essential
to the success of this project. To them I express my special gratitude
for their capable work which made the project possible.
Staff assistance in the Keck Laboratory is acknowledged in each of the
individual publications as appropriate; however, the writer wishes to
express special thanks to Mrs. Pat Rankin for typing this report and
preparing it for publication.
93
-------
SECTION XI
REFERENCES
The list below does not include those references which are included in
the list of Publications, Reports, and Technical Memoranda of this
project (see Section XII). Many more references were given in the
literature review of each of the publications listed in Section XII.
However, some additional references to work published on jet and plume
mixing are listed below. By and large these items appeared during the
interval between the preparation of research reports and the prepara-
tion of this final summary report.
References Cited
1. Morton, B. R. , Taylor, G. I. and Turner, J. S., "Turbulent
Gravitational Convection from Maintained and Instantaneous
Sources," Proc. Roy. Soc. London, A234(1956), pp. 1-23.
2. Albertson, M. L., Dai, Y. B., Jensen, R. A., and Rouse, H.,
"Diffusion of Submerged Jets," Trans. ASCE, Vol. 115,
1950, pp. 639-697.
3. Rouse, H., Yih, C. S. and Humphreys, H. W., "Gravitational
Convection from a Boundary Source," Tellus, 4^1952),
pp. 201-10.
4. Rouse, H., "Gravitational Diffusion From a Boundary Source in
Two-Dimensional Flow", Jour, of App. Mech., ASME, Sept. 1947,
pp. A225-A228.
5. John Carollo Engineers (Lafayette, Calif.),Final Report on Design
of Ocean Outfall No. 2 for County Sanitation Districts of
Orange County, P. 0. Box 8127, Fountain Valley, Calif., 1970.
6. Koh, Robert C. Y., "Viscous Stratified Flow towards a Sink,"
Journal of Fluid Mechanics, Vol. 24, 1966, pp. 555-575.
95
-------
Recent References on Buoyant Jets and Plumes
and Applications to Mixing Problems
Abraham, G., "The Flow of Round Buoyant Jets Issuing Vertically into
Ambient Fluid Flowing in a Horizontal Direction," Proceedings, 5th
International Water Pollution Research Conference, San Francisco, July -
August 1970, Paper 111-15, 7 pp.
Abraham, G., "Jets and Plumes Issuing into Stratified Fluids," Proceedings,
International Symposium on Stratified Flows, International Association for
Hydraulic Research, Novosibirsk, 1972.
Abraham, G., and Eysink, W. D., "Jets Issuing into Fluid with a Density
Gradient," J. of Hydraulic Research, vol. 7, no. 2, 1969, pp. 145-175.
Anwar, H. 0., "Behavior of Buoyant Jet in Calm Fluid," J. Hydraulics
Division. ASCE. vol. 95, no. HY4, July 1969, pp. 1289-1303.
Anwar, H. 0., "Experiment on an Effluent Discharging from a Slot into
Stationary or a Slow Moving Fluid of Greater Density," J. of Hydraulic
Research, vol. 7, no. 4, 1969, pp. 411-431.
Anwar, H. 0., "Measurements on Horizontal Buoyant Jet in Calm Ambient
Fluid, with Theory Based on Variable Coefficient of Entrainment Deter-
mined Experimentally," La Houille Blanche, April 1972, pp. 311-319.
Anwar, H. 0., "Appearance of Unstable Buoyant Jet," J. Hydraulics Div.,
ASCE. vol. 98, no. HY7, 1972, pp. 1143-1156.
Ayoub, G. M., "Buoyant Jets in a Co-flowing Current; an Experimental
Investigation," Grenoble, Soc. Hydro-techn. de France, 1972, Document R22.
Baines, W. D., and Turner, J. S., "Turbulent Buoyant Convection from a
Source in a Confined Region," J. Fluid Mechanics, vol. 37, 1969, pp. 51-80.
Baumgartner, D. J., and Trent, D. S., "Ocean Outfall Design Part I:
Literature Review and Theoretical Development," U. S. Dept. of Interior,
Federal Water Quality Administration, 1970, 127 pp.
Baumgartner, D. J., Trent, D. S., and Byram, K. V., "User's Guide and
Documentation for Outfall Plume Model," Environmental Protection Agency,
Pacific Northwest Water Laboratory, Working Paper #80, May 1971, 29 pp.
Becker, H. A., Hottel, H. C., and Williams, G. C., "The Nozzle-Fluid
Concentration Field of the Round, Turbulent, Free Jet," J. Fluid Mechanics,
vol. 30, no. 2, 1967, pp. 285-303.
96
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Benedict, B. A., Polk, E.M. Jr., Yandell, E.L., Jr., and Parker, F. L.,
"Surface Jet and Diffusion Models for Discharge of Heated Water," Proc.,
14th Congress IAHR, Paris, 1971, vol. 1, Paper A 22, pp. 183-190.
Bourodimos, E. L., "Turbulent Transfer and Mixing of Submerged Heated
Water Jet," Water Resources Research, vol. 8, no. 4, pp. 982-997, August
1972. ~
Braun, W. and Goldstein, M. E., "Analysis of the Mixing Region for a Two-
dimensional Jet Injected at an Angle into a Moving Stream," NASA Technical
Note D-5531, Nov. 1969.
Briggs, G. A., "Plume Rise," USAEC Division of Technical Information,
1969, 81 pp.
Bringfelt, B., "A Study of Buoyant Chimney Plumes in Neutral and Stable
Atmospheres," Atmospheric Environment, vol. 3, 1969, pp. 609-623.
Bringfelt, B., "Plume Rise Measurements at Industrial Chimneys," Atmos-
pheric Environment, vol. 2, 1968, pp. 575-598.
Brock, R. R., "Power Law Solutions for Vertical Plumes," J. Hydraulics
Division. ASCE. Sept. 1970, HY9, pp. 1803-1817.
Bryce, J. B., and Elliott, R. V., "Thermal Plume Measurements in Lake
Ontario and Resulting Phenomenological Model," Proc., International
Symposium on Stratified Flows, International Association for Hydraulic
Research, Novosibirsk, 1972.
Cederwall, Klas, "Dimensional Considerations Applied to Some Diffusion
Problems," Sartryck ur Vatten 2/72, pp. 137-151. (Also Symposium on
"The Physical Processes Responsible for the Dispersal of Pollutants in
the Sea with Special Reference to the Nearshore Zone", Denmark, 1972.)
Csanady, G.,T., "Bent-over Vapor Plumes," J. of Applied Meteorology,
vol. 10, Feb. 1971, pp. 36-42.
i
Dornhelm, R., Nouel, M., and Weigel, R. L., "Velocity and Temperature in
Buoyant Surface Jet," J. Power Division, ASCE, vol. 98, no. P01, June
1972, pp. 29-47.
El Mahgary, Y. S., "Thermal Diffusion of the Warm Water of Power Plants
into a Sea Basin." Proc. 14th Congress IAHR, Paris, 1971, vol. 1, Paper
A 40, pp. 333-340.
Fay, J. A., Escudier, M., and Hoult, D. P., "A Correlation of Field
Observations of Plume Rise," J. Air Pollution Control Association, vol.
20, no. 6, 1970, pp. 391-397.
Fay, James A., "Buoyant Plumes and Wakes," Annual Review of Fluid Mechanics,
vol. 5, 1973, pp. 151-160.
97
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Fox, D. G., "Forced Plume in a Stratified Fluid," J. Geophysical Research.
vol. 75, no. 33, 1970, pp. 6818-6835.
Gunwaldsen, Ralph W., Brodfield, Bruno, and Hecker, George E., "Cooling
Water Structures for FitzPatrick Nuclear Plant," J. Power Division, ASCE,
vol. 97, no. P04, Proc. Paper 8572, December 1971, pp. 767-781.
Gustafson, B., and Larsen, Ian, "Jet Diffusion in Stagnant Stratified
Waters," Water Research, vol. 4, no. 5, May 1970, pp. 353-361.
Hansen, Jens and Schroeder, Hans, "Horizontal Jet Dilution Studies by
Use of Radioactive Isotopes." Acta Polytechnical Scandinavica, Civil
Engineering and Building Construction Series, no. 49, Copenhagen, 1968,
23 pp.
Harleman, D. R. F., "Submerged Diffusers in Shallow Coastal Waters,"
Presented at Coastal Zone Pollution Management Symposium, Charleston,
S. C., February, 1972, 19 pp.
Harleman, D. R. F., "Thermal Stratification due to Heated Discharges,"
Proc. International Symposium on Stratified Flows, International Associa-
tion for Hydraulic Research, Novosibirsk, 1972.
Harleman, D. R. F., Jirka, Gerhard, and Stolzenbach, K. D., "A Study of
Submerged Multi-port Diffusers for Condenser Water Discharge with Applica-
tion to the Shoreham Nuclear Power Station," Ralph M. Parsons Laboratory,
MIT Report No. 139, July 1971, 121 pp.
Harleman, D. R. F., Jirka, G., Adams, E., and Watanabe, M., "Investiga-
tion of a Submerged, Slotted Pipe Diffuser for Condenser Water Discharge,
from the Canal Plant, Cape Cod Canal," Ralph M. Parsons Lab., MIT, Report
no. 141, Oct. 1971, 58 pp.
Harleman, D. R. F., and Stolzenbach, K. D., "Fluid Mechanics of Heat
Disposal from Power Generation," Annual Review of Fluid Mechanics, vol. 4,
1972, pp. 7-32.
Harleman, D. R. F. (ed.), "Engineering Aspects of Heat Disposal from Power
Generation," MIT Summer Session, June 26-30, 1972, Ralph M. Parsons Lab.,
approx. 800 pages.
Hayashi, T., "Dynamical Similitude on the Diffusion of Warm Water Jet
Discharged Horizontally at Water Surface," Proc. U. S.-Japan Seminar on
Similitude in Fluid Mechanics, Sept. 21-28, 1967, pp. 71-74.
Hayashi, T., "Turbulent Buoyant Jets of Effluent Discharged Vertically
Upwards from an Orifice in a Cross-Current in the Ocean," International
Association for Hydraulic Research, Proceedings, 14th Congress, Paris,
1971, vol. 1, no. A-19, pp. 157-166.
98
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Hetsroni, G., and Sokolov, M., "Distribution of Mass, Velocity and Inten-
sity of Turbulence in a Two-Phase Turbulent Jet," J. of Applied Mechanics,
ASME. vol. 38, E, no. 2, June 1971, pp. 315-327.
Hewett, T. A., Fay, J. A., and Hoult, D. P., "Laboratory Experiments of
Smokestack Plumes in a Stable Atmosphere," Atmospheric Environment, vol. 5,
pp. 767-789, Sept. 1971.
(also MIT Fluid Mechanics Lab., Publication 70-9, 1970.)
Hill, B. J., "Measurement of Local Entrainment Rate in the Initial Region
of Axisymmetric Turbulent Air Jets." J. of Fluid Mechanics, vol. 51, part
4, 1972, pp. 773-779.
Hirst, E. A., "Analysis of Round, Turbulent, Buoyant Jets Discharged to
Flowing Stratified Ambients," Oak Ridge National Laboratory, ORNL-4685,
June 1971, 37 pp.
Hirst, Eric, "Analysis of Buoyant Jets Within the Zone of Flow Establish-
ment," Oak Ridge National Laboratory, ORNL-TM-3470, August 1971, 41 pp.
Hirst, E. A., "Buoyant Jets Discharged to Quiescent Stratified Ambients,"
J. of Geophysical Research, vol. 76, no. 30, Oct. 20, 1971, pp. 7375-7383.
Hirst, E. A., "Zone of Flow Establishment for Round Buoyant Jets," Water
Resources Research, vol. 8, no. 5, October 1972, pp. 1234-1246.
Hirst, E. A., "Buoyant Jets with Three-Dimensional Trajectories," J. of
the Hydraulics Division, ASCE, vol. 98, no. HY11, Nov. 1972, pp. 1999-2014.
Holly, Forrest M., Jr., and Grace, John L., Jr., "Model Study of Dense Jet
in Flowing Fluid," J. of the Hydraulics Div., Proceedings, ASCE, vol. 98,
no. HY11, Nov. 1972, pp. 1921-1933.
Hoult, D. P., Fay, J. A., and Forney, L. J., "A Theory of Plume Rise Com-
pared with Field Observations," J. Air Pollution Control Association, vol.
19, 1969, pp. 585-590.
Hoult, D. P., and Weil, J. C., "Turbulent Plume in a Laminar Cross Flow,"
Atmospheric Environment, vol. 6, 1972, pp. 513-531.
(Also MIT Fluid Mechanics Laboratory Pub. no. 70-8, 1970.)
Jain, S. C., Sayre, W. W., Akeyampong, Y. A., McDougall, D., and Kennedy»
J. F., "Model Studies and Design of Thermal Outfall Structures, Quad-
Cities Nuclear Plant," IIHR Report no. 135, University of Iowa, Sept.
1971, 101 pp.
Koh, Robert C. Y., "On Buoyant Jets," International Association for
Hydraulic Research, Proc. 14th Congress, Paris, vol. 1, no. A-18, 1971,
pp. 145-156.
99
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Koh, R. C. Y., and Fan, Loh-Nien, "Mathematical Models for the Prediction
of Temperature Distributions Resulting from the Discharge of Heated
Water into Large Bodies of Water," EPA, Water Quality Office, report
16130 DWO 10/70, Oct. 1970, 219 pp.
Krystantos, R., "The Turbulent Jet from a Series of Holes in Line,"
Aeronautical Quarterly, vol. 15, no. 1, 1964, pp. 1-28.
Larsen, J., and Hecker, G., "Design of Submerged Diffusers and Jet Inter-
action," ASCE National Water Resources Engineering Meeting, Jan. 24-28,
1972, Atlanta, Georgia, Me'eting preprint 1614, 24 pp.
Liseth, Paul, "Mixing of Merging Buoyant Jets from a Manifold in Stagnant
Receiving Water of Uniform Density," Hydraulic Engineering Lab., U. C.
Berkeley, Report HEL 23-1, Nov. 1970.
Ljatkher, V. M., "Hydrothermal Modelling and Design of Density Flows in
Cooling Systems of Thermal and Nuclear Power Plants," Proc., International
Symposium on Stratified Flows, International Association for Hydraulic
Research, Novosibirsk, 1972.
Mahajan, B. M., and John, J. E. A., "Mixing of Shallow Submerged Heated
Water Jet with an Ambient Reservoir," AIAA Journal, vol. 2, no. 11, Nov.
1971, pp. 2135-2140.
Margason, R. J., "The Path of a Jet Directed at Large Angles to a Subsonic
Free Stream," NASA Technical Note D-4919, 1968.
Maxwell, W. h. C., "Flux Development Region in Submerged Jets," J. Engi-
neering Mechanics Division, ASCE, vol. 96, EM 6, paper 7756, 1970, pp.
1061-1079.
Maxwell, W. H. C., and Pazwash, H., "Boundary Effects on Jet Flow Patterns
Related to Water Quality and Pollution Problems," WRC Research report no.
28, University of Illinois, Jan. 1970, 84 pp.
Maxworthy, T., "Experimental and Theoretical Studies of Horizontal Jets in
a Stratified Fluid," Proc. International Symposium on Stratified Flows,
International Association for Hydraulic Research, Novosibirsk, 1972.
Mih, W. C., and Hoopes, J. A., "Mean and Turbulent Velocities for Plane
Jet," J. Hydraulics Division, ASCE. vol. 98, no. HY7, July 1972, pp. 1275-
1294.
Morton, B. R., "The Choice of Conservation Equations for Plume Models,"
J. of Geophysical Research, vol. 76, no. 30, Oct. 20, 1971, pp. 7409-
7416.
Motz, L. H. and Benedict, B. A., "Heated Surface Jet Discharged into a
Flowing Ambient Stream," U. S. Environmental Protection Agency, 16130 FDQ
03/71, March 1971, 207 pp.
100
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Motz, L. H., and Benedict, B. A., "Surface Jet Model of Heated Discharges,"
J. Hydraulics Division. ASCE. vol. 98, no. HY1, Jan. 1972, pp. 181-199.
Muraoka, K., "Turbulent Diffusion of Free Jet by Tracer Test," Osaka Uni-
versity Technology Report no. 971, Oct. 1970, pp. 809-822.
Neale, L. C. and Hecker, F. E., "Model Versus Field Data on Thermal
Plumes from Power Stations," Proc. International Symposium on Stratified
Flows, International Association for Hydraulic Research, Novosibirsk,
1972, 7 pp.
Flatten, J. L. and Keffer, J. F. , "Entrainment in Deflected Axisymmetric
Jets at Various Angles to the Stream," Univ. of Toronto Mechanical
Engineering Dept., UTME-TP-6808, 1968, 51 pp.
Flatten, J. L. and Keffer, J. F., "Deflected Turbulent Jet Flows,"
J. Applied Mechanics, ASME. vol. 38, Series E, no. 4, Dec. 1971, pp. 756-
758.
Policastro, A. J. and Tokar, J. V., "Heated-Effluent Dispersion in Large
Lakes: State of the Art of Analytical Modeling. Part I: Critique of
Model Formulations," Argonne National Lab., ES-11, January 1972, 374 pp.
Pratte, B. D., and Baines, W. D., "Profiles of the Round Turbulent Jet
in a Cross Flow," J. Hydraulics Division, ASCE, vol. 93, no. HY6, 1967, pp.
53-64.
Pritchard, D. W. and Carter, H. H., "Design and Siting Criteria for Once-
through Cooling Systems Based on a First Order Thermal Plume Model,"
Chesapeake Bay Institute, Johns Hopkins University, Technical report no.
75, April 1972, 51 pp.
Rao, K. V., "The Buoyant Plume Model Above a Heat Source," Atmospheric
Environment, vol. 4, 1970, pp. 557-575.
Robideau, R. F., "The Discharge of Submerged Buoyant Jets into Water of
Finite Depth," General Dynamics, Electric Boat Division, Groton, Conn.,
Report No. U440-72-121, Nov. 1972, 57 pp.
Sami, S., Carmody, T., and Rouse, H., "Jet Diffusion in the Region of
Flow Establishment," J. Fluid Mechanics, vol. 27, no. 2, pp. 231-252, 1967.
Schwartz, J., and Tulin, M. P., "Chimney Plumes in Neutral and Stable
Surroundings," Atmospheric Environment, vol. 6, no. 1, January 1972, pp.
19-36.
Silberman, E., "Warm .Water Discharges into Lakes and Reservoirs," Proc.
International Symposium on Stratified Flows, International Association
for Hydraulic Research, Novosibirsk, 1972.
Sharp, James J., "Physical Interpretation of Jet Dilution Parameters,"
J. Sanitary Engineering Div., ASCE, vol. 94, no. SA1, Feb. 1968, pp. 55-64.
101
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Sharp, James J., "Spread of Buoyant Jets at the Free Surface," J. of the
Hydraulics Division. ASCE, vol. 95, no. HY3, May 1969, pp. 811-825.
Sharp, James J., "Unsteady Spread of Buoyant Surface Discharge," J. of the
Hydraulics Division, ASCE, vol. 97, no. HY9, September 1971, pp. 1471-1492.
Shirazi, M. A., and Davis, L. R., "Workbook of Thermal Plume Prediction,
vol. I, Submerged Discharge," U. S. Environmental Protection Agency,
Report no. EPA-R2-72-005a, August 1972, 228 pp.
Shuto, N., "Buoyant Plume in a Cross Stream," Coastal Engineering in
Japan, vol. XIV, Dec. 1971, pp. 163-173.
Slawson, P. R., and Csanady, G. T., "The Effect of Atmospheric Conditions
on Plume Rise,," J. of Fluid Mechanics, vol. 47, 1971, pp. 33-49.
Sonnichsen, J. C., "Lateral Spreading of Heated Discharge," J. Power
Division. Proc. ASCE, vol. 97, no. P03, July 1971, pp. 623-630.
Stefan, H., "Modeling Spread of Heated Water Over Lake," J. Power Division,
ASCE, vol. 96, no. P03, June 1970, pp. 469-482.
Stefan, H., "Dilution of Buoyant Two-Dimensional Surface Discharges,"
J. Hydraulics Division, ASCE, vol. 98, no. HY1, January 1972, pp. 71-86.
Stefan, H., "Spread and Dilution of Three-Dimensional Rectilinear Heated
Water Surface Jets," Proc. International Symposium on Stratified Flows,
International Association for Hydraulic Research, Novosibirsk, 1972.
Stefan, H., and Schiebe, F. R., "Heated Discharge from Flume into Tank,"
J. Sanitary Engineering Division, ASCE, vol. 96, no. SA6, 1970, pp. 1415-
1433.
Stefan, H. and Vaidyaraman, P., "Jet Type Model for the Three-Dimensional
Thermal Plume in a Crosscurrent and under Wind," Water Resources Research,
vol. 8, no. 4, Aug. 1972, pp. 998-1014.
Stephens, N. Thomas and McCaldin, Roy 0., "Attenuation of Power Station
Plumes as Determined by Instrumented Aircraft," Environmental Science
and Technology, July 1971, pp. 615-621.
Stolzenbach, K. D. and Harleman, D. R. F., "An Analytical and Experimental
Investigation of Surface Discharges of Heated Water," Ralph M. Parsons Lab.,
MIT, Report no. 135, 1971, 212 pp.
Tamai, N., Wiegel, R. L., and Tormberg, Gordon I., "Horizontal Surface
Discharge of Warm Water Jets," J. Power Division, ASCE, vol. 95, no. P02,
October 1969, pp. 253-276.
102
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Telford, J. W. , "Convective Plumes in a Convective Field," J. of the At-
mospheric Sciences, vol. 27, no. 3, 1970, pp. 347-358.
Tennessee Valley Authority, "Full Scale Study of Plume Rise at Large
Generating Stations," TVA, Division of Health and Safety, Muscle Shoals,
1968.
Thomas, F. W., Carpenter, S. F., and Colebaugh, W. C., "IV. Recent
Results of Measurements, Plume Rise Estimates for Electric Generating
Stations," Phil. Trans., Royal Society. London. Series A, 265, 1969,
pp. 221-243.
Tokar, J. V., "Thermal Plumes in Lakes: Compilations of Field Experience,"
Argonne National Laboratory, Environmental Sciences, Argonne, Illinois,
Report no. ES-3, August 1971, 170 pp.
Tomai, N., "Diffusion of Horizontal Buoyant Jet Discharged at Water
Surface," Proc., 13th Congress IAHR, Kyoto, Japan, 1969, vol. 3, paper
24, pp. 215-222.
Tsang, G. , and Wood, I. R. , "Motion of Two-Dimensional Starting Plume,"
J. of Engineering Mechanics Division, ASCE, vol. 94, no. EM6, December
1968, pp. 1547-1561.
Tsang, G., "Laboratory Study of Line Thermals," Atmospheric Environment,
vol. 5, 1971, pp. 445-471.
Tsang, G., "Entrainment of Ambient Fluid by Two-Dimensional Starting
Plumes and Thermals," Atmospheric Environment, vol. 6, 1972, pp. 123-132.
Weil, J., "Mixing of a Heated Surface Jet in Turbulent Channel Flow,"
Hydraulics Laboratory, University of California at Berkeley, Report no.
WHM-1, June 1972, 166 pp.
Weil, J. C., and Hoult, D. P., "Effective Stack Heights for Tall Stacks,"
MIT Fluid Mechanics Lab. Publication 71-14, October 1971, 42 pp.
Wygnanski, I., and Fiedler, H., "Some Measurements in the Self Preserving
Jet," J. of Fluid Mechanics, vol. 32, no. 2, 1968, pp. 577-612.
Zeller, R. W., Hooper, J. A., and Rohlich, G. A., "Heated Surface Jets in
Steady Cross-current," J. Hydraulics Division. ASCE, vol. 97, no. HY9,
paper 8385, Sept. 1971, pp. 1403-1426.
103
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SECTION XII
PUBLICATIONS, REPORTS, AND TECHNICAL MEMORANDA
(W. M. Keck Laboratory of Hydraulics and
Water Resources, California
Institute of Technology)
Dispersion in Hydrologic and Coastal Environments
(EPA Grant No. 16070 DGY)
May 1967-December 1971
(Note; Abstracts for all items are given in the Appendix,
by subject shown in right-hand column.)
Abstract
No.
1967 1. List, E. John, and Brooks, Norman H., "Lateral
Dispersion in Saturated Porous Media,"
J. Geoph. Research, Vol. 72, No. 10,
May 15, 1967, pp. 2531-2541. F-l
2. Fan, Loh-Nien, "Turbulent Buoyant Jets into Stratified
or Flowing Ambient Fluids," W. M. Keck Lab. of
Hydraulics and Water Resources, Report No. KH-R-15,
June 1967, 196 pp. A-l
3. Yudelson, Jerry M., "A Survey of Ocean Diffusion Studies
and Data," W. M. Keck Lab. of Hydraulics and Water
Resources, Tech. Memo No. 67-2, Sept. 1967 D-l
1968 4. List, E. John, "A Two-Dimensional Sink in a Density-
Stratified Porous Medium," Journal of Fluid
Mechanics, Vol. 33, Part 3, Sept. 2, 1968,
pp. 529-543. F-2
5. Fan, Loh-Nien, and Brooks, Norman H., Discussion of
"Physical Interpretation of Jet Dilution
Parameters," by James J. Sharp, J. of San. Eng.
Div., ASCE, Vol. 94, No. SA6, Dec. 1968,
pp. 1295-1299. A~4
1969 6. Fan, Loh-Nien, and Brooks, Norman H., "Numerical Solutions
of Turbulent Buoyant Jet Problems," W. M. Keck
Laboratory of Hydraulics and Water Resources,
Report.No. KH-R-18, January 1969, 94 pp. A-2
105
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Abstract
No.
7. Ditmars, John D., "Computer Program for Round Buoyant
Jets into Stratified Ambient Environments,"
W. M. Keck Lab. of Hydraulics and Water
Resources, Tech. Memo No. 69-1, March 1969, 27 pp. A-3
8. Brooks, Norman H., and Koh, Robert C. Y., "Selective
Withdrawal from Density-Stratified Reservoirs,"
J. Hyd. Div., ASCE, Vol. 95, HY4, July 1969,
pp. 1369-1400. (Also in Proceedings, ASCE
Specialty Research Conference on "Current Re-
search into the Effects of Reservoirs on Water
Quality," Portland, Oregon, Jan. 22-24, 1968,
published by Vanderbilt University, Dept. of
Environmental and Water Resources Engineering,
Technical Report No. 17, pp. 169-214.) C-l
9. Brooks, Norman H., Discussion of paper "The Mechanics
of Thermally Stratified Flow," by D.R.F. Harleman,
Proc., National Symposium on Thermal Pollution (1968),
Vanderbilt Univ. Press, Nashville, Tenn., 1969,
pp. 165-172. G-2
10. Prych, Edmund A., Discussion of "Numerical Studies of
Unsteady Dispersion in Estuaries," by D, R. F.
Harleman, Chok Hung Lee, and L. C. Hall, J. of San.
Eng. Div., ASCE, Vol. 95, SA5, Oct. 1969, E-5
pp. 959-964.
11. Wooding, R. A., "Growth of Fingers at an Unstable Diffusing
Interface in a Porous Medium or Hele-Shaw Cell," J. of
Fluid Mechanics, Vol. 39, Part 3, 1969, pp. 477-495.
(Also in slightly more detail as W. M. Keck Laboratory
of Hydraulics and Water Resources, Tech. Memo No. 69-5,
March 1969, 38 pp.) F-3
1970 12. Brooks, Norman H., "Lecture Notes on Conceptual Design of
Submarine Outfalls," Univ. of Calif., Berkeley, Water
Resources Engineering Educational Series, Program VII,
Pollution of Coastal and Estuarine Waters, Jan. 29-30,
1970 (Part I, 25 pp., Part II, 12 pp.). Available as
Tech. Memos 70-1 and 70-2, W. M. Keck Laboratory of
Hydraulics and Water Resources, Caltech. B-l
13. Cederwall, Klas, "Dispersion Phenomena in Coastal Waters,"
John Freeman Memorial Lecture, J. of Boston Society
of Civil Engineers, Vol. 57, No. 1, Jan. 1970,
pp. 34-70. A-9
106
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Abstract
No.
14. Prych, E. A., "Effects of Density Differences on
Lateral Mixing in Open-Channel Flows,"
W. M. Keck Laboratory of Hydraulics and
Water Resources, Report No. KH-R-21, May 1970,
225 pp. E-l
15. Coudert, Jean F., "A Numerical Solution of the Two-
Dimensional Diffusion Equation in a Shear
Flow," W. M. Keck Laboratory of Hydraulics
and Water Resources, Tech. Memo 70-7, June 1970,
38 pp. E-4
16. Ditmars, John D., "Mixing of Density-Stratified Im-
poundments with Buoyant Jets," W. M. Keck
Laboratory of Hydraulics and Water Resources,
Report No. KH-R-22, Sept. 1970, 203 pp. C-2
17. Cederwall, Klas, and Ditmars, J. D., "Analysis of Air-
Bubble Plumes," W. M. Keck Laboratory of Hydraulics
and Water Resources, Report No. KH-R-24, Sept.
1970, 51 pp. A-5
18. Brooks, Norman H., "Thermal Power — New Crisis for
the Environment," Proc. Symposium on Water En-
vironment and Human Needs, Parsons Water Re-
sources and Hydrodynamics Lab., M.I.T., Oct.
1970, pp. 146-177. G-l
19. Rumer, R. R., and Hoopes*, J. A., "Modelling Great
Lakes Circulation," Proc. Symposium on Water
Environment and Human Needs, Parsons Water Re-
sources and Hydrodynamics Lab., M.I.T., Oct.
1970, pp. 212-247.
(*U. Wisconsin) D-3
i
20. Okoye, Josephat K., "Characteristics of Transverse
Mixing in Open-Channel Flows," W. M. Keck
Laboratory of Hydraulics and Water Resources,
Report No. KH-R-23, November 1970, 269 pp. E-2
21. Fischer*, Hugo B., and Brooks, Norman H., "Technical
Aspects of Waste Disposal in the Sea through
Submarine Outlets," FAQ Technical Conference on
Marine Pollution and Its Effects on Living Re-
sources and Fishing, Rome, Paper FIR: MP/70/R-4
(16 pp.).
(*U. California, Berkeley.) B-2
107
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Abstract
No.
1971 22. Cederwall, Klas, Discussion of "Horizontal Surface
Discharge of Warm Water Jets," J. of the Power
Division, ASCE, Vol. 97, No. P01, Jan. 1971,
pp. 229-234. A-10
23. List, E. J., "Laminar Momentum Jets in a Stratified
Fluid," J. of Fluid Mechanics, Vol. 45,
February 15, 1971, pp. 561-574. A-7
24. Cederwall, Klas, "A Buoyant Slot Jet into Stagnant or
Flowing Environment," W. M. Keck Laboratory of
Hydraulics and Water Resources, Report No. KH-R-25,
March 1971, 86 pp. A-6
25. Sullivan, Paul J., "Some Data on the Distance-Neighbour
Function for Relative Diffusion," Journal of Fluid
Mechanics, Vol. 47, Part 3, June 14, 1971,
pp. 601-607. D-2
26. Sotil, C. A., "Computer Program for Slot Buoyant Jets
into Stratified Ambient Environments," W. M. Keck
Laboratory of Hyeraulics and Water Resources,
Tech. Memo 71-2, June 1971, 35 pp. A-ll
27. Rumer, Ralph R., Jr., "Internal Seiches and Interfacial
Mixing in Stratified Lakes," W. M. Keck Laboratory
of Hydraulics and Water Resources, Tech. Memo 71-3,
July 1971, 39 pp. D-4
28. Cederwall, Klas, "Float Diffusion Study," Water Research,
Vol. 5, pp. 889-907, Nov. 1971. (Also W. M. Keck
Laboratory of Hydraulics and Water Resources, Tech.
Memo 71-1, April 1971.) E-3
29. Sullivan, Paul J., "Longitudinal Dispersion within a Two-
Dimensional Turbulent Shear Flow," Journal of Fluid
Mechanics, Vol. 49, Part 3, Oct. 15, 1971, pp. 551-
576. . E-6
1972 30. Sullivan, Paul J., "The Penetration of a Density Inter-
face by Heavy Vortex Rings," Water, Air, and Soil
Pollution. !_, 3,-pp. 322-336, July 1972. A-8
1973 31. List, E. J., and Imberger, Jorg, "Turbulent Entrainment
in Buoyant Jets and Plumes," J. of Hydraulics Div.,
ASCE. 1973 (in press). A-9
32. Rumer, Ralph R., Jr., "Interfacial Wave Breaking in
Stratified Liquids," J. of Hydraulics Div.. ASCE.
Vol. 99, No. HY3, March, 1973, pp. 509-524. D-5
108
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SECTION XIII
APPENDIX
ANNOTATED LIST OF
PUBLICATIONS, REPORTS AND
TECHNICAL MEMORANDA
The following documents describe in greater detail the research
supported entirely or partially by this grant (1607ODGY) during the
period May 1, 1967 to December 31, 1971. In some instances, part of
the research was done under previous grants of the National Institutes
of Health, USPHS, with final publication during this period.
Individual copies of most of these documents may be obtained on
request to the Secretary, W. M. Keck Laboratory of Hydraulics and
Water Resources, California Institute of Technology, Pasadena,
California, 91109.
The documents are arranged by subject, with an abstract for
each. The writer acknowledges the individual authors for most of
the following abstracts, although some editorial revisions and
additions have been made by this writer.
The major subject headings are as follows:
A. JET AND PLUME MIXING
B. OCEAN OUTFALL DESIGN 120
C. SELECTIVE WITHDRAWAL AND ARTIFICIAL MIXING IN
DENSITY-STRATIFIED IMPOUNDMENTS 122
D. NATURAL DIFFUSION IN RESERVOIRS, LAKES, AND OCEANS 124
E. MIXING IN TURBULENT SHEAR FLOWS 127
F- DISPERSION IN FLOW THROUGH POROUS MEDIA 133
G. GENERAL 135
109
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A. JET AND PLUME MIXING
A-l. FAN, LOH-NIEN, "Turbulent Buoyant Jets into Stratified or Flowing
Ambient Fluids," W. M. Keck Laboratory of Hydraulics and
Water Resources, Report KH-R-15, Caltech, June, 1967, 196 pp.
Theoretical and experimental studies were made on two classes
of buoyant jet problems, namely:
1) an inclined, round buoyant jet in a stagnant environment
with linear density-stratification;
2) a round buoyant jet in a uniform cross stream of homogeneous
density.
Using the integral technique of analysis, assuming similarity,
predictions were made for jet trajectory, widths, and dilution ratios,
in a density-stratified or flowing environment. Such information is
of great importance in the design of disposal systems for sewage
effluent into the ocean or waste gases into the atmosphere.
This study of a buoyant jet in a stagnant environment extended
the Morton type of analysis to cover the effect of the initial angle
of discharge. Numerical solutions have been presented for a range
of initial conditions. Laboratory experiments were conducted for
photographic observations of the trajectories of dyed jets. In
general the observed jet forms agreed well with the calculated
trajectories and nominal half widths when the value of the entrain-
ment coefficient was taken to be a = 0.082, as previously suggested
by Morton.
The problem of a buoyant jet in a uniform cross stream was
analyzed by assuming an entrainment mechanism based upon the vector
difference between the characteristic jet velocity and the ambient
velocity. The effect of the unbalanced pressure field on the sides
of the jet flow was approximated by a gross drag term. Laboratory
flume experiments were performed with sinking jets, which are directly
analogous to buoyant jets. Salt solutions were injected into fresh
water at the free surface of the flow in a large flume. The jet
trajectories, dilution ratios and jet half widths were determined by ;
conductivity measurements. The entrainment coefficient, a, and / /
drag coefficient, C^, were found from the observed jet trajectories
and dilution ratios. In the ten cases studied where jet Froude
number ranged from 10 to 80 and velocity ratio (jet: current) k from
4 to 16, a varied from 0.4 to 0.5 and Cd from 1.7 to 0.1. The jet
mixing motion for distances within 250D (D = initial diameter) was
found to be dominated by the self-generated turbulence, rather than
110
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the free-stream turbulence. Similarity of concentration profiles
has also been discussed.
A-2. FAN, LOH-NIEN, and BROOKS, NORMAN H., "Numerical Solutions of
Turbulent Buoyant Jet Problems," W. M. Keck Laboratory of
Hydraulics and Water Resources, Report KH-R-18, Caltech,
January 1969, 94'pp.
Theoretical solutions were obtained on four classes of turbulent
buoyant jet problems, namely:
1) an inclined, round buoyant jet in a stagnant, uniform
ambient fluid;
2) an inclined, round buoyant jet in a stagnant ambient fluid
with linear density-stratification;
3) an inclined, slot buoyant jet in a stagnant, uniform
ambient fluid;
A) an inclined, slot buoyant jet in a stagnant ambient fluid
with linear density-stratification.
This report is a summary of the numerical solutions on buoyant
jets in stagnant environments carried out in connection with previous
investigations at Caltech by Fan, Brooks, and Koh. Using the integral
type of analysis assuming similarity, predictions can be made for
jet trajectory, widths, and dilution ratios, in a uniform or density-
stratified environment without ambient currents. Numerical solutions
have been presented in dimensionless form for a wide range of initial
conditions, including the effect of the initial angle of discharge.
Since the integral analysis is only for the zone of established flow,
adjustments are given(for the effects of the zone of flow establish-
ment for finite jets (i.e. the initial development of self-similar
profiles).
Problems with non-linear density profiles are not readily
treated in generalized non-dimensional form. Rather it is more
feasible to make case-by-case calculations using dimensional
variables. A program for such calculations for a round jet is
available in a technical memorandum by Ditmars.
These solutions are useful in the design of submerged jet
diffusers for disposal of sewage effluent into the ocean or cooling
water into a lake.
Ill
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A-3. DITMARS, JOHN D., "Computer Program for Round Buoyant Jets into
Stratified Ambient Environments," W. M. Keck Lab. of Hydraulics
and Water Resources, Tech. Memo. No. 69-1, Caltech, March 1969, 27 pp.
The gross behavior of an inclined round turbulent buoyant jet
in a stratified ambient environment is determined by quadrature of
the governing differential equations. The FORTRAN IV (level G)
language is used for the program, which has been run on an IBM 360/75
digital computer. The essential input to the problem includes the
location of the jet; the initial values of jet velocity, jet diameter,
and angle of inclination; the density of the discharged fluid; and
the density profile of the ambient environment. The density profile
may have any gravitationally stable shape. It is provided as input
to the program by supplying the density at arbitrarily selected
points (which best fit the actual profile) and calculated as linear
between these points. The output consists of the jet trajectory in
rectangular coordinates; the nominal width; the centerline velocity,
density difference, and dilution ratio. All of these parameters are
printed out at uniform intervals along the jet trajectory. Calcula-
tions are stopped at the maximum height of rise or greatest depth of
sinking of the jet or at any predetermined vertical or horizontal
coordinate.
The program is applicable to several limiting cases of round
turbulent jets. Horizontal, vertical, and inclined jets (-90° £ Q
< 90°) with initially positive or negative buoyancies can be handled
in stratified or uniform ambient environments. The jet may be
placed at any elevation in the ambient environment. Simple momentum
jets can be handled, although simple plumes are not amenable to this
technique.
For cases of linear density profiles in the environment a
variety of numerical computer solutions have been presented in
generalized non-dimensional coordinates by Fan and Brooks. However,
for arbitrary non-linear density profiles, generalized non-dimensional
solutions are not feasible; it is for this reason that the computer
program is being made available for case-by-case solutions.
A-4. FAN, LOH-NIEN, and BROOKS, NORMAN. H., Discussion of "Physical
Interpretation of Jet Dilution Parameters," by James J. Sharp,
Jour, of Sanitary Eng. Div., ASCE,Vol. 94, No. SA6, Dec. 1968,
pp. 1295-1299.
This discussion explains the effect of the geometry of buoyant
jet trajectories on the shape of the curves in the usual dilution
graph (dilution contours plotted as function of Froude number, F,
112
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and normalized vertical coordinate, y/D). Dilution graphs are given
for horizontal and vertical buoyant jets in uniform environment,
according to the Fan and Brooks analysis. For low Froude numbers,
both graphs approach the buoyant plume solution, whereas for high
Froude numbers (high jet efflux velocities) the results are markedly
different because of the much longer trajectories developed for the
horizontal jets than for the vertical jets in rising a prescribed distance.
A-5. CEDERWALL, KLAS, and DITMARS, JOHN D., "Analysis of Air-Bubble Plumes,"
W. M. Keck Laboratory of Hydraulics and Water Resources,
Report No. KH-R-24, Caltech, Sept. 1970, 51 pp.
The air-bubble plume induced by the steady release of air into
water has been analyzed with an integral technique based on the
equations for conservation of mass, momentum and buoyancy. This
approach has been widely used to study the behavior of submerged
turbulent jets and plumes, as for example by Fan and Brooks (see
above). The case of air-bubble induced flow, however, includes
additional features. In this study the compressibility of the air
and the differential velocity between the rising air bubbles and the
water are introduced as basic properties of the air-bubble plume in
addition to a fundamental coefficient of entrainment and a turbulent
Schmidt number characterizing the lateral spreading of the air bubbles.
Theoretical solutions for two- and three-dimensional air-bubble
systems in homogeneous, stagnant water are presented in both dimen-
sional and normalized form. Comparison with published data indicates
fairly good agreement between theory and experiment. The further
complication of a stratified environment is briefly discussed since
this case is of great practical interest.
The paper is to be considered as a progress report, as future
experimental verification of various hypotheses is needed.
A-6. CEDERWALL, KLAS, "A Buoyant Slot Jet into Stagnant or Flowing
Environments," W. M. Keck Laboratory of Hydraulics and Water
Resources, Report KH-R-25, Caltech, March 1971, 86 pp.
The diffusion following the release of a buoyant slot jet into
a confined, uniformly flowing environment has been studied. A
dimensional analysis reveals the complexity of the problem; there
are in general four governing dimensionless numbers. For small mass
flow from the source (the source then being characterized by initial
flux of momentum, m, and buoyancy, b, only) the governing flow para-
meters reduce to:
113
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2
1. Source Froude number F = u /b
a
2
2. Momentum flux ratio = u H/m
Si
3. Angle of discharge, 0
wherein u& is the ambient velocity and H is the depth of the
ambient flow.
Experiments were conducted first with a horizontal, buoyant
slot jet into stagnant, ambient fluid. Observed trajectories and
centerline dilutions were in good agreement with existing theories.
Next two sets of experiments were performed with a vertical and a
horizontal buoyant slot jet issuing into a uniformly flowing stream.
A two-layer flow analysis provided the rationale for a class-
ification of flow regimes. It was found that the jet effluent
cloud would penetrate upstream from the jet outlet whenever the
source Froude number was less than about unity. For high source
Froude numbers the jet is swept downstream, either as a wall jet
(in spite of its buoyancy) or in an unstable highly turbulent
pattern.
The results are applicable to the discharge of cooling water
(or other buoyant waste water) from multi-port line diffusers
into rivers, homogeneous estuaries, or shallow reservoirs wherein
the jet diffusion is strongly influenced by the limited depth of
flow.
A-7. LIST, E. J., "Laminar Momentum Jets in a Stratified Fluid,"
J. of Fluid Mechanics, Vol. 45, February 15, 1971, pp. 561-574,
Solutions are presented for creeping flows induced by two-
and three-dimensional horizontal and vertical momentum jets in a
linearly-stratified unbounded diffusive viscous fluid. These
linear problems are solved by replacing the momentum jet by a
body force singularity represented by delta functions and solving
the partial differential equations of motion by use of multi-
dimensional Fourier transforms. The integral representations for
the physical variables are evaluated by a combination of residue
theory and numerical integration.
The solutions for vertical jets show the jet to be trapped
within a layer of finite thickness and systems of rotors to be
114
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induced. The horizontal two-dimensional jet solution shows
return flows above and below the jet and a pair of rotors. The
three-dimensional horizontal jet has no return flow at finite
distance and the diffusive contribution is found to be almost
negligible in most situations, the primary character of the
horizontal flows being given by the non-diffusive solution.
Stokes's paradox is found to be non-existent in a density-
stratified fluid.
The results obtained for laminar momentum jets in a stratified
fluid will be helpful in understanding turbulent jets in stratified
fluid. This work differs from that previously discussed in this
section in that: (a) there is no assumption of self-similarity of
velocity profiles, and hence the dynamics of jet collapse and
induced currents can be studied; and (b) the buoyancy or mass
fluxes at the source are negligible, and the flow is driven
solely by the momentum input.
A-8. SULLIVAN, PAUL J., "The Penetration of a Density Interface by
Heavy Vortex Rings," Water, Mr. and Soil Pollution, !_, 3,
pp. 322-336, July 1972.
This paper describes experiments in which small volumes of
heavy fluid were released in the uppermost of two uniform layers
of fluid and the degree of penetration into the lower layer
determined. When the inj ected fluid had no initial momentum,
less than ten percent of the released fluid continued into the
7 T 3p
lower layer if g —=r- was greater than 29; more than ninety
z 13p
percent continued into the lower layer if g —=— was less than
1.5. (ZT is the distance from release to the interface,
P2 ~ Pi
P = —: where p, and p0 are the densities of the upper and
Pi -1- ^
(P -Pi )
the lower fluid and F = g '— V where V is the volume and p
P!
is the density of the injected fluid.)
When initial momentum was given to the heavy fluid, a vortex
ring then formed, penetrated into the lower layer, broke up, and
more than ninety percent of this fluid remained in the lower layer
115
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i 9
Z' •'P K 3
if g —— was less than 0.30 (—) . The vortex ring was found to
2«3p
remain intact during its travel in the lower layer if g —— was less
2
than 0.17 (i~)3. (K is the initial circulation of the injected fluid.)
r
The results may tentatively be applied to the sudden dumping
of sludge in the ocean, when the ocean density structure can be
adequately approximated by two homogeneous layers of slightly
different density. There is uncertainty, however, regarding the
effect of Reynolds number on the critical dimensionless numbers
cited above.
A-9. CEDERWALL, KLAS, "Dispersion Phenomena in Coastal Environments,"
J. of Boston Soc. of Civil Eng., Vol. 57, No. 1, Jan. 1970,
pp. 34-70.
This paper, presented as the 1970 Freeman Memorial Lecture
before the Boston Society of Civil Engineers, reviews a wide range
of topics relating to dispersion of both sewage and thermal
effluents in coastal waters with special emphasis on Swedish
practice.
There are several alternative methods for the prediction of
the dispersion pattern in the receiving water area. Four main
approaches may be distinguished.
a) A purely theoretical analysis supported by general
experience on the diffusive properties, circulation
and exchange conditions of the water area in question.
b) The same as under a), but with supplementary field
surveys to establish characteristic prototype
behavior.
c) Tracer technique for direct in-situ simulation of
transport and mixing of the waste effluent.
d) Scale tests by means of a hydraulic model.
For thermal discharges, the author describes two phenomena,
diffusion of a submerged jet and surface spreading due to buoyancy.
If a surface discharge Jet is dominated by its momentum, it may be
116
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considered as approximately equivalent to half of a complete
submerged jet. On the other hand a low-speed channel discharge,
dominated by buoyancy rather than momentum, will spread laterally
as a wave front. Other cases are intermediate, because they involve
both turbulent entrainment and gravitational spreading. The effects
of limited depth are also discussed, with the limiting situation
being similar to a wall jet.
For sewage discharges into confined water bodies the initial
mixing and diffusion may not be as important as the overall flushing
of the water body, such as fjords and archipelagos. Flushing may
occur naturally by density-induced circulation caused by fresh
water runoff, by wind-driven currents, by tidal currents, or a
combination. « In each case the density structure and the vertical
mixing are important. Because of the complexity of the phenomena,
there is need for considerable input of field data into any
theoretical models. Therefore, it is often more effective to make
tracer studies of overall flushing of confined bodies. The paper
describes the uses of continuous, intermittent, and instantaneous
injections of tracers. Both radioactive and fluorescent tracers
have been used in Swedish coastal studies.
A-10. CEDERWALL, KLAS, Discussion of "Horizontal Surface Discharge of
Warm Water Jets," by N. Tamai, R. L. Wiegel, and G. F. Thornberg,
Power Division, A.S.C.E., Vol. 97, No. P01, Jan. 1971,
pp. 229-234.
This discussion is essentially the section on thermal
discharges from the preceding paper, "Dispersion Phenomena in
Coastal Environments."
A-ll. SOTIL, C. A., "Computer Program for Slot Buoyant Jets into
Stratified Ambient Environments," W. M. Keck Laboratory of
Hydraulics and Water Resources, Tech. Memo. No. 71-2,
June, 1971, 35 pp.
This technical memorandum closely parallels that by Ditmars,
abstracted earlier in this section ("Computer Program for Round
Buoyant Jets into Stratified Ambient Environments," Tech. Memo. 69-1)
117
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The principal change is from axisymmetric jets ("round jets") to
two-dimensional jets ("slot jets"). Both programs allow the
solution of buoyant jet problems for irregular ambient density
profiles, rather than being constrained to linear profiles as is
the case for the generalized solutions of Fan and Brooks.
This paper gives examples for multi-port outfall diffusers
(for waste disposal in the ocean). In deep water a row of ports
can be idealized as an equivalent slot jet. In one example, it
is shown that the maximum height of plume rise computed by this
program is only slightly different from that given by simple
plume theory. In another example, the relationship of diffuser
length and depth to dilution and submergence is explored. Finally,
the sensitivity of the solutions to the value of the entrainment
coefficient is studied for one case.
A-12. List, E. J., and Imberger, Jorg, "Turbulent Entrainment in Buoyant
Jets and Plumes," J. Hydraulics Div., ASCE, 1973 (in press).
Dimensional reasoning, coupled with published experimental
results, is used to show that there is no unique entrainment co-
efficient for turbulent buoyant jets directed vertically upwards.
The behavior of such jets in uniform density environments is found
to be governed by an entrainment function and a buoyancy function,
both of which are functions of the local jet densimetric Froude
number and the local jet spreading rate. For plumes, Batchelor's
analysis is used to show that the local Froude number and jet
spreading angle are constant. These two facts are then used to
show that the buoyancy and entrainment functions in plumes are also
constant and that their values can be deduced from an experimental
velocity profile alone. The value of the buoyancy function so ob-
tained shows remarkable agreement with the value computed directly
from the density profile.
Experimental results are also used to show that the spreading!
angle of a round vertical buoyant jet is virtually independent of
elevation, and this result leads to the conclusion that the entrain-
ment function is linearly dependent on the inverse of the local j.et
Froude number thus confirming the result obtained by Priestley and
Ball (1955). The buoyancy function is found to be constant with the
value 1.16 for axially symmetric buoyant jets.
118
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Similar results are derived for two-dimensional buoyant jets,
although existing experimental data is inconsistent and therefore
inconclusive with respect to predicting the form of the entrainment
and buoyancy functions.
These results will allow for improved modelling of round
buoyant jet behavior under less restrictive conditions (non-vertical
discharge, stratified ambient fluid) by allowing an appropriate
transition in the entrainment coefficient (a) from jet-like
behavior near the source (a ~ 0.057 for round jets) to plume-like
behavior far from the source (a „ 0.082 for round plumes).
119
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B. OCEAN OUTFALL DESIGN
B-l. BROOKS, NORMAN H., "Lecture Notes on Conceptual Design of Submarine
Outfalls," Univ. of Calif., Berkeley, Water Resources Engin-
eering Educational Series, Program VII, Pollution of Coastal
and Estuarine Waters, Jan. 29-30, 1970 (Part I, 25 pp.,
Part II, 12 pp.) Available as Tech. Memos 70-1 and 70-2,
W. M. Keck Laboratory of Hydraulics and Water Resources,
Caltech.
Part I deals with the prediction of initial dilution for an
ocean outfall by use of various formulas for buoyant jets and
plumes. For uniform environment, the cases covered are single
round horizontal buoyant jets (open-end pipe); simple round plume
(limiting case of low jet velocity for round buoyant jet) ; and
simple line plume (multiport diffuser in deep water). For linearly
stratified ambient water, the cases of round and line buoyant plumes
are presented, with formulas for predicting height of rise and
dilution. An empirical procedure for non-linear environments is
derived. For most disposal problems through multiport diffusers in
deep water the plume approximations are adequate. Examples are
given, with the presumption that the currents are weak.
Part II explains the procedure for designing a multiport
diffuser which will distribute the flow well without undue head
loss or sea water intrusion. Effects of density difference, bottom
slope, and flow variability are included. The procedure is essen-
tially that given by Rawn, Bowerman and Brooks (Trans. ASCE, 126,
III, 1961, 344-388), except for the port discharge coefficients
"which are based on more recent laboratory experiments (not part of
this project). The calculation procedure may be easily programmed
for computer solution.
B-2. FISCHER*, H. B., AND BROOKS, N. H., "Technical Aspects of Waste
Disposal in the Sea through Submarine Outlets", FAQ Technical
Conference on Marine Pollution and Its Effects on Living
Resources and Fishing, Rome, Dec. 1970, Paper FIR: MP/70/R-4
(16 pp.). (*Univ. of California, Berkeley.)
This paper covers the same material on initial dilution as in
Brooks (B-l above). In addition a model is given for predicting
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further natural dilution and dieoff between outfall and shore. At
the end of the paper there are comments on thermal discharges,
raising the question of whether it is preferable to have a warm
thinner layer than a thick well mixed layer of smaller temperature
rise.
121
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C. SELECTIVE WITHDRAWAL AND ARTIFICIAL MIXING
IN DENSITY STRATIFIED IMPOUNDMENTS
C-l. BROOKS, NORMAN H., and KOH, ROBERT C. Y., "Selective Withdrawal
from Density-Stratified Reservoirs," J. of the Hydraulics
Division, ASCE. Vol. 95, No. HY4, Proc. Paper 6702,
July, 1969, pp. 1369-1400. (Also in Proc., ASCE Specialty
Research Conference on "Current Research into Effects of
Reservoirs on Water Quality," Portland, Oregon, Jan. 22-24,
1968, published by Vanderbilt University, Dept. of Environ-
mental and Water Resources Eng., Tech. Report No. 17,
pp. 169-214.)
Analyses and experiments for selective withdrawal flows
from linearly stratified fluids, including inviscid, viscous, and
turbulent cases have been presented. A review of discrete layer
systems is also included. These thin jet-like flows have been
observed both in the laboratory and in large reservoirs, and
boundary layer assumptions appear to be justified.
For turbulent withdrawal flows away from the immediate
vicinity of the outlet, it is hypothesized that Koh's viscous
diffusive experiments and analysis can be applied by replacing
the kinematic viscosity, v, and the molecular diffusivity, D,
by the vertical eddy diffusivity, E^ For self-generated
turbulence, it is predicted that the proper characteristic
length is a = (q/v/ge)^/2j in which q = the unit discharge and
e = -(1/po) dp/dy.
Equations and a graph are given for predicting the
thickness of the withdrawal layer under various flow assump^
tions from real reservoirs. Transients are also presented.
The results of this paper may be used in predicting the
performance of a multioutlet system in dams for water quality
control by selective withdrawal.
C-2. DITMARS, JOHN D., "Mixing of Density-Stratified Impoundments
with Buoyant Jets," W. M. Keck Laboratory of Hydraulics
and Water Resources, Report No. KH-R-22, Sept. 1970, 203 pp.
122
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This study was an investigation of the mixing of density-
stratified impoundments by means of buoyant jets created by a
pumping system. The deterioration of water quality which often
occurs in density-stratified lakes and reservoirs may be counter-
acted by mixing. The physical aspects of the mixing process are
the primary concern of this study, although several implicatipns
regarding changes in water quality are indicated.
A simulation technique is developed to predict the time-
history of changes in the density-depth profiles of an impound-
ment during mixing- The simulation model considers the impound-
ment closed to all external influences except those due to the
pumping system. The.impoundment is treated in a one-dimensional
sense, except for the fluid mechanics of the three-dimensional
jet and selective withdrawal of pumping system. The numerical
solution to the governing equations predicts density profiles at
successive time steps during mixing, given the initial density
profile, the area-depth relation for the impoundment, the
elevations of intake and jet discharge tubes, and the jet
discharge and diameter. The changes due to mixing in the profiles
of temperature and of a conservative, non-reacting tracer can be
predicted also.
The results of laboratory experiments and two field mixing
experiments in which density-stratified impoundments were mixed
using pumping systems show that the simulation technique predicts
the response of the impoundment reasonably well.
The results of a series of simulated mixing experiments for
impoundments which have prismatic shapes and initially linear
density profiles are given in dimensionless form. For these
special conditions, the efficiency of the pumping system increased
as the jet densimetric Froude number decreased, and the time
required for complete mixing was a fraction of the characteristic
time, T i ¥/Q (where -¥• is the impoundment volume included between
intake and jet elevations and Q is the pumped discharge).
Recommendations are made for the application of the
generalized results and for the use of the simulation technique
for lakes and reservoirs which are not closed systems.
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D. NATURAL DIFFUSION IN RESERVOIRS.
LAKES. AND OCEANS
D-l. YUDELSON, JERRY M., "A Survey of Ocean Diffusion Studies and Data,"
W. M. Keck Lab. of Hydraulics and Water Resources, Tech. Memo
No. 67-2, Caltech, Sept. 1967, 126 pp.
A study was made of eddy diffusion in the ocean, with emphasis
on the dispersion of sewage waste fields.
A thorough search of the literature on ocean diffusion was made,
special emphasis being placed on publications of the 1959-1967 period.
The object, of the search was to review recent theoretical developments
and to procure as much observational data as possible, for comparison
with several theoretical models of ocean diffusion.
Equations for the dispersion of sewage fields, developed by
N. H. Brooks, were compared with many observations of ocean diffusion
and found to give good agreement with observed fact. The validity
of other equations for ocean diffusion was also discussed.
D-2. SULLIVAN, PAUL J., "Some Data on the Distance-Neighbour Function for
Relative Diffusion," Journal of Fluid Mechanics, Vol. 47, Part 3,
June 14, 1971, pp', 601-607-
Repeated observations of dye plumes on Lake Huron are interpreted
according to the theoretical proposals of Richardson (1926) and
Batchelor (1952) about the characteristics of a dispersing cloud of
marked fluid within a field of homogeneous turbulence. The results
show the average of several instantaneous, concentration distributions
about their centre of gravity to be approximately Gaussian and the
distance-neighbour function to be of approximately Gaussian form.
The data are consistent with the theoretical description given by
Batchelor, namely,
q(y,t) = (2Tr"yV1/2exp(-y2/2y2),
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where q(y,t) is the distance-neighbour function and a is the constant
of the '4/3-power law1. The average value of a is estimated to be
0-12 cm2/3 sec"1. The rate of turbulent energy dissipation in the
near-surface currents of Lake Huron is estimated as e ^ 2-1 x 10~3
cm2 sec~3.
(The field work was performed as a thesis project by the author
at the University of Waterloo; at a later date the preparation of
this paper for publication was undertaken as an activity under this
grant.)
D-3. RUMER, RALPH R., JR., and HOOPES, JOHN A., "Modelling Great Lakes
Circulation," The Water Environment and Human Needs, Symposium,
October 1-2, 1970, Parsons Laboratory for Water Resources and
Hydrodynamics, M.I.T., pp. 212-247.
Better knowledge of circulation processes in large lakes would
enable better predictions of the fate of pollutants discharged into
these water bodies. The use of rotating hydraulic models of large
lakes in conjunction with the study of mathematical and numerical
models and the collection of prototype field data has proven to be a
useful method for studying these circulation processes.
This paper reviews the theoretical justification and limitations
of rotating laboratory models. Model studies can provide significant
information on the large-scale current and seiche motions generated
by inflows and outflows to the lakes and by wind stresses imposed at
the surface, and on the residence periods for conservative tracers
introduced at various points.
(Only the preparation of Rumer's part of this review paper was
supported by this research project.)
125
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D-4. RUMER, RALPH R., JR., "Internal Seiches and Interfacial Mixing in
Stratified Lakes," W. M. Keck Laboratory of Hydraulics and
Water Resources, Tech. Memo 71-3, Caltech, July 1971, 39 pp.
The horizontal velocities of the upper and lower layers associated
with an internal seiche episode in a stratified lake are examined in
relation to critical shear gradients necessary for the growth of
unstable short period interfacial waves with frequency close to the
so-called Brunt-Vaisala frequency. Experimental results are presented
which help to clarify the conditions for the occurrence of the short
period waves. Charts summarizing the findings of this study are
presented which should be of help in predicting the occurrence of
internal wave breaking in closed basins. It is believed that inter-
facial wave breaking in a stratified lake increases vertical mixing
between the epiliminon and hypoliminon, and therefore may be a factor
in the seasonal development of the thermocline and in the distribution
of dissolved and suspended substances.
D-5. RUMER, RALPH R., JR., "Interfacial Wave Breaking in Stratified Liquids,"
J. of Hydraulics Div., ASCE. vol. 99, No. HY3, Mar. 1973,
pp. 509-524.
This published paper is a slightly revised version of D-4 above.
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E. MIXING IN TURBULENT SHEAR FLOWS
E-l. PRYCH, EDMUND A., "Effects of Density Differences on Lateral Mixing
in Open-Channel Flows," W. M. Keck Laboratory of Hydraulics
and Water Resources, Report No. KH-R-21, Caltech, May 1970,
225 pp.
This study investigated lateral mixing of tracer fluids in
turbulent open-channel flows when the tracer and ambient fluids
have different densities. Longitudinal dispersion in flows with
longitudinal density gradients was also investigated analytically.
Lateral mixing was studied in a laboratory flume (40 meters
long and 110 centimeters wide), by introducing fluid tracers at
the ambient flow velocity continuously and uniformly across a
fraction of the flume width and over the entire depth of the
ambient flow. Fluid samples were taken to obtain concentration
distributions in cross-sections at various distances, x, downstream
from the tracer source. The data were used to calculate variances
of the lateral distributions of the depth-averaged concentration.
When there was a difference in density between the tracer and the
ambient fluids, lateral mixing close to the source was enhanced
by density-induced secondary flows; however, far downstream where
the density gradients were small, lateral mixing rates were
independent of the initial density difference. A dimensional
analysis of the problem and the data show that the normalized
variance is a function of only three dimensionless numbers, which
represent: (1) the x-coordinate, (2) the source width, and
(3) the buoyancy flux from the source.
A simplified (set of equations of motion for a fluid with a
horizontal density gradient was integrated to give an expression
for the density-induced velocity distribution. The dispersion
coefficient due to this velocity distribution was also obtained.
Using this dispersion coefficient in an analysis for predicting
lateral mixing rates in the experiments of this investigation gave
only qualitative agreement with the data. However, predicted
longitudinal salinity distributions in an idealized laboratory
estuary agree well with published data.
The results are applicable to discharges of cooling water
(or other buoyant or heavy waste waters) from canals or open-end
outfalls into turbulent shear flows such as rivers or homogeneous
estuaries. For example, it is possible to predict how rapidly
hot-water from a low-velocity canal outlet from a power plant on
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the bank of a river will spread across the river and be diluted
with the river flow.
E-2. OKOYE, JOSEPHAT K., "Characteristics of Transverse Mixing in Open-
Channel Flows," W. M. Keck Laboratory of Hydraulics and Water
Resources, Report No. KH-R-23, Caltech, November 1970, 269 pp.
The transverse spreading of a plume generated by a point
source in a uniform open-channel flow was investigated. A neutrally-
buoyant tracer was injected Continuously at ambient velocity through
a small round source at a point within the flow. Tracer concentra-
tion was measured in situ at several points downstream of the source
using conductivity probes. Most of the experiments were conducted
in the tilting flume which is 40 meters long and 110 centimeters
wide.
Tracer concentrations were analyzed in two phases.
In Phase I, the time-averaged concentration was evaluated,
its distribution within the plume determined, and characteristic
coefficients of transverse mixing calculated. It was shown that
the transverse mixing coefficient varied with the distance from
the bed and was highest near the water surface where the flow
velocity was greatest. In contrast to previous speculation, the
ratio of the depth-averaged coefficient of transverse mixing "Dz
to the product of the (bed) shear velocity u^ and the flow depth
d was not a constant but depended on the aspect ratio X = d/W,
where W = flume width. For laboratory experiments "Dz/u^d
decreased from 0.24 to 0.093 as X increased from 0.015 to 0.20.
In Phase II, the temporal fluctuation of tracer concentra-
tion was studied in three sections. In the first, the inter-
mittency factor technique was used to delineate three regions of
the plume cross section: an inner core where the tracer concen-
tration c(t) was always greater than the background C^; an inter-
mittency region where c(t) was only intermittently greater than
%; and the Outer region where C^ was never exceeded. Dimensional
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analysis furnished universal curves for prediction of the geometric
characteristics of the three regions. In the second section, the
entire plume, at a fixed station, was treated as a fluctuating cloud,
Variances characterizing the fluctuation of the plume centroid and
the variation of the plume width were calculated and compared. In
the third section, the intensity and probability density of the
concentration fluctuations at fixed points were calculated. The
distribution of the peak-to-average ratio was also determined.
Finally the results of the two phases of study were inter-
related' to evaluate their contributions to the transverse spreading
of the plume.,
E-3. CEDERWALL, KLAS, "Float Diffusion Study," Water Research, Vol. 5,
pp. 889-907, Nov. 1971. (Also W. M. Keck Laboratory of
Hydraulics and Water Resources, Tech. Memo. 71-1, April, 1971.)
This paper reviewed the literature on lateral diffusion of
solutes and discrete particles in open-channel flow, and presented
the results of a new set of experiments with floats. The floats
were each constructed of four vanes at right angles to each other
and attached to a vertical axis; flotation was arranged so that the
vanes were submerged 6 cm below the surface. In contrast to
previous float studies, these floats measured the diffusion within
the flow, rather than at the free surface. Three sizes were used,
measuring 4, 8, and 16 cm respectively for the outside lateral
dimensions (tip-to-tip of vanes).
From fifty repeated drops for each of three floats in two
flow conditions, the variance of the lateral position of each
group was determined as a function of distance downstream; the
diffusion coefficient was determined from the rate of change of
the variance. It was found that the diffusion coefficient for the
particles decreases as the size increases, becoming for the largest
floats only 40% of the value for solutes as measured by Okoye in
the same flume.
The total rotations of the floats (summed in an absolute
sense) were also observed and related to float size. The response
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of the floats to the vorticity field also decreased with increasing
float size, indicating that the larger floats are not affected by
the smaller scales of vorticity.
E-4. COUDERT, JEAN F., "A Numerical Solution of the Two-Dimensional
Diffusion Equation in a Shear Flow," W. M. Keck Laboratory
of Hydraulics and Water Resources Tech. Memo 70-7, Caltech,
June, 1970, 41 pp.
A numerical method developed in the last two years was used
to solve the two-dimensional diffusion equation with a Dirac 6-function
as boundary value, representing a steady line source across a stream.
The velocity profile is taken as logarithmic and the diffusivity
profile as parabolic, although the method can be used for any
measured distribution of velocity and diffusivity. The source can
be set anywhere from the bed up to the free surface. The results
for each problem are given in a dimensionless form and a significant
set of plots is automatically produced. The method can be easily
applied to more complicated cases (such as unsteady, point sources).
The numerical technique may be applied to the practical
problem of predicting the distance along a river that it takes
for a contaminant introduced at the bed to become essentially
fully mixed over the depth; the detailed transitional concentration
profiles are also developed.
E-5. PRYCH, EDMUND A., Discussion of "Numerical Studies of Unsteady
Dispersion in Estuaries," by D. R. F. Harleman, Chok Hung Lee,
and L. C. Hall, Jour, of Sanitary'Ehg. Div., ASCE, Vol. 95,
No. SA5, pp. 959-964.
In this discussion it was shown that some numerical
estuary models, such as given by the original paper, can accumulate
errors which have the same effect as increased dispersion.
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Prych's analysis discloses that deficiencies in the finite
difference approximation for the convective term in the mass-transport
equation may cause much more dispersion in the computations than
does the actual dispersion term.
The actual longitudinal dispersion coefficient for the Potomoc
Estuary analysis presented by the authors may be ten times what they
inferred from their model which inadvertantly included large
pseudo-dispersion due to numerical errors. Even verifying an
estuary model against field data does not reveal this kind of error,
because it is compensated for by reduction in the derived natural
dispersion coefficient.
E-6. SULLIVAN, PAUL J., "Longitudinal Dispersion within a Two-
Dimensional Turbulent Shear Flow," Journal of Fluid^
Mechanics, Vol. 49, Pt. 3, 15 Oct. 1971, pp. 551-576.
This paper describes some laboratory and numerical
experiments made on the longitudinal dispersion in an open channel
flow. Particular attention has been paid to the initial stages
of the process.
Physical arguments suggest that the streamwise dispersion of
a line of marked fluid elements across a two-dimensional turbulent
shear flow occurs in three distinct stages. These stages are
identified by a change in the form of the distribution of marked
fluid elements in the flow direction. The skewed distribution of
the first stage is readily identified by a constant value (approx-
imately 1.1 for the ratio of the peak velocity (V^) of the
distribution to the mean-flow velocity U; experiments using dyed
fluid, made at this stage of the process, have revealed six
identifiable features of the suggested distribution. The dis-
tributions suggested for the second and the third stage are
consistent with the experimental findings of Elder (1959) for
the second stage and Taylor (19540 for the third stage.
An attempt has been made to simulate the process numerically
using a Markovian model. The results of the simulation confirm
features suggested by physical arguments and are in agreement with
the open channel experiments.
131
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The results of an experiment, in which the three-dimensional
motion of small neutrally buoyant spheres was recorded in many
small discrete time intervals, corroborate the theoretical
suggestions and simulation results.
(This research was done by the author as a doctoral thesis
at Cambridge University prior to his residence at Caltech; the
EPA/WQO research grant supported the preparation of this paper for
publication.)
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F. DISPERSION IN FLOW THROUGH POROUS MEDIA
F-l. LIST, E. JOHN and BROOKS, NORMAN H. , "Lateral Dispersion in
Saturated Porous Media," J. Geophysical Research, Vol. 72,
No. 10, May 15, 1967, pp. 2531-2541.
An analysis of experimental results from a series of
lateral dispersion experiments is presented. It is shown that
lateral dispersion for low molecular Peclet numbers is adequately
described by Saffman's capillary model but that velocity power
laws are limited in their application. The dynamic P6clet
number is shown to obtain a maximum at molecular Peclet numbers
of 0(104).
(This work was completed under a preceding grant (USPHS/NIH
WP-00680) although finally published during this grant period.)
F-2. LIST, E. JOHN, "A Two-Dimensional Sink in a Density-Stratified
Porous Medium," J. of Fluid Mechanics, Vol. 33, Part 3,
Sept. 2, 1968, pp. 529-543.
A solution is offered for the flow induced by a two-
dimensional line sink in a saturated, density-stratified porous
medium. It is found that fluid is selectively withdrawn from a
thin layer at the elevation of the line sink and not from the
entire medium. The velocity distributions predicted by the
theory are checked by experiments in a Hele-Shaw cell and good
agreement found.
The results of this research may be applied to the
problems of selective extraction of one lens of ground water from
a layered system, wherein the different layers may have different
water qualities and densities.
(This experimental work was completed under a preceding
grant (USPHS/NIH WP-00680) and the theoretical development was
done by the author while a staff member at Univ. of Auckland;
however, the final publication occurred during this grant period.)
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?~3. WOODING, ROBIN A., "Growth of Fingers at an Unstable Diffusing
Interface in a Porous Medium or Hele-Shaw Cell," J. of Fluid
Mechanics, Vol. 39, Part 3, 1969, pp. 477-495. (Also in
slightly more detail as W. M. Keck Laboratory of Hydraulics
and Water Resources Tech. Memo No. 69-5, Caltech, March 1969,
38 pp.)
Waves at an unstable horizontal interface between two
fluids moving vertically through a saturated porous medium are
observed to grow rapidly to become fingers (i.e. the amplitude
greatly exceeds the wavelength). For a diffusing interface, in
experiments using a Hele-Shaw cell, the mean amplitude taken over
many fingers grows approximately as (time) , followed by a
transition to a growth proportional to time. Correspondingly,
the mean wave-number decreases approximately as (time)"-*-' .
Because of the rapid increase in amplitude, longitudinal
dispersion ultimately becomes negligible relative to wave growth.
To represent the observed quantities at large time, the
transport equation is suitably weighted and averaged over the
horizontal plane. Hyperbolic equations result, and the ascending
and descending zones containing the fronts of the fingers are
replaced by discontinuities. These averaged equations form an
unclosed set, but closure is achieved by assuming a law for the
mean wave-number based on similarity. It is found that the mean
amplitude is fairly insensitive to changes in wave-number.
Numerical solutions of the averaged equations give more detailed
information about the growth behaviour, in excellent agreement
with the similarity results and with the Hele-Shaw experiments.
The results are applicable to the prediction of convective
instabilities in ground water masses when heavier water (colder
or more saline) is introduced or recharged above a slightly
lighter layer. In large open bodies of water the convective
mixing proceeds fairly rapidly when fluids are hydrostatically
unstable, because there is little internal resistance. On the
other hand, in ground water, there is considerable resistance to
convective overturning, and rates are much slower.
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G. GENERAL
G-l. BROOKS, NORMAN H., "Thermal Power — New Crisis for the Environ-
ment," Proc. Symposium on Water Environment and Human Needs,
Parsons Water Resources and Hydrodynamics Lab., MIT,
Oct. 1970, pp. 146-177.
The rapid growth of electric power demand has led to a crisis
in power-plant siting because of the substantial environmental
effects of large thermal power stations. The water environment has
been the most convenient receptor for the huge discharges of waste
heat. Inland JLakes and rivers have already been used or committed
to their tolerable limits of temperature, and marine and estuarine
discharges are raising difficult questions of long-range strategies
and effects.
This survey paper will cover the following topics: magnitude
of the waste heat problem, now and projected for the future;
hydraulic and hydrologic aspects of thermal discharges; alternative
strategies and costs for power-plant siting (e.g. inland plants
with cooling towers vs. coastal plants); some broad issues for
society relating to choices between environmental alternatives and
stopping the rapid growth of energy usage; and implications for
urban planning.
Some research results by Prych (E-l above) on accelerated
transverse mixing of a buoyant fluid in an open channel flow are
included.
G-2. BROOKS, NORMAN H., Discussion of paper "The Mechanics of Thermally
Stratified Flow" by D. R. F. Harleman, Proc. National Sym-
posium on Thermal Pollution (1968), Vanderbilt Univ. Press,
1969, pp. 165-172.
The anology of thermal outfalls to sewer outfalls is dis-
cussed, especially with reference to the possibility of using
multiple-jet diffusers to increase the initial dilution for dis-
charges from thermal outfalls (lower AT). Submergence of waste
heat below thermoclines is also possible in large bodies of water.
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This strong mixing approach can be expected to be used in the ocean
because of the large heat capacity and less danger of interfering
with annual cycles of thermal stratification and overturning.
The reader is cautioned against using the common jet and plume
formulas, which presume infinite flow fields, in cases of restricted
depths, i.e. situations where restricted access of diluting water
may limit dilutions obtained. Hydraulic model studies, as described
by Harleman, are needed in such cases.
136 «US. GOVERNMENT PRINTING OFFICE: 1973 546-3U/UO 1-3
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SELECTED WATER
RESOURCES ABSTRACTS
INPUT TRANSACTION FORM
J. Accession No.
w
4. Title
Dispersion in Hydrologic and Coastal Environments
Author(s)
Norman H. Brooks
Organization
W. M. Keck Laboratory of Hydraulics & Water Resources
Division of Engineering and Applied Science
California Institute of Technology, Pasadena, Calif. 91109
Environmental Protection Agency
Dec. 1972
KH-R-29
10. Projectfo.
EPA 16070DGY
11. Coatrtct/Grtwtffo.
Typ
I-'
IS. Supplementary flotes
.'«!
Final, through Dec. 1972
Environmental Protection Agency report number,
EPA-660/3-73-010, August 1973.
If. Abstract
This report summarizes the results of a five-year laboratory research project on
various flow phenomena of importance to transport and dispersion of pollutants in hydro-
logic and caostal environments. The results are useful in two general ways: first, to
facilitate the prediction of ambient water quality from effluent characteristics in vari-
ous water environments; and secondly, to provide the basis for design of systems (like
outfalls) required to meet given ambient water quality requirements.
The results for buoyant jets may be used for the design of waste-water outfalls in
oceans, reservoirs, lakes, and large estuaries. Particular emphasis is given to line
sources (or slot jets) which represent long multiple-outlet diffusers, which are neces-
sary for all large discharges to get high dilutions.
For reservoirs which are density stratified, the results include formulations for
prediction of selective withdrawal, and a simulation procedure for predicting reservoir
mixing by systems which pump water from one level to the other. For applications to
rivers and estuaries, laboratory flume experiments were made to measure transverse mixing
of buoyant or heavy tracer flows, as well as for neutral-density flows.
Abstracts of all publications and reports resulting from the project are given as
an appendix to the report.
27a. Descriptors
Buoyant jets*, stratified flow, reservoir withdrawal, turbulent mixing*, diffusers.
17b. Identifiers
Outfalls*, waste dilution, thermal pollution
/7c. COWRR Field & Group Q5B
18. Availability
19.
->ort;
Unclassified
..
Pages
22. P'ice
Send To:
WATER RESOURCES SCIENTIFIC INFORMATION CENTER
U.S. DEPARTMENT OF THE INTERIOR
WASHINGTON. D. C. 2O24O
Abstractor
Norman H. Brooks
I institution California Institute of Technology
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