A numerical model of buoyant jet surface wave interaction

i
A NUMERICAL MODEL OF BUOYANT-JET SURFACE
WAVE INTERACTION
By
Francis Mitchell and David A. Chin
Technical Memorandum No. CEN-87-1
April 1987
M.S. Thesis
Department of Civil and Architectural Engineering
University of Miami
Coral Gables, Florida 33124

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UNIVERSITY OF MIAMI
A thesis submitted in partial fulfillment of
the requirements for the degree of
Master of Science in Civil Engineering
A NUMERICAL MODEL OF BUOYANT-JET SURFACE WAVE INTERACTION
Francis Mitchell
Approved:
__________________________ _________________________
David A. Chin Sidney L. Besvinick
Assistant Professor of Civil Associate Provost and Dean
Engineering for Research and Graduate
Chairman of Thesis Committee Studies (Interim)
__________________________ _________________________
Thomas D. Waite Samuel S. Lee
Professor of Civil Engineering Professor of Mechanical Engineering

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MITCHELL, FRANCIS (M.S., Civil Engineering)
A NUMERICAL MODEL OF BUOYANT-JET SURFACE WAVE INTERACTION. (May 1987)
Abstract of a master’s thesis at the University of Miami.
Thesis supervised by Professor David A. Chin.
A numerical model has been developed to simulate the effect of waves on the dilution of submerged
buoyant jets. Experiments were performed and results compared to the model predictions.
Experiments and model predictions agreed well. The model is useful in studying the performance of
shallow outfalls in a wave environment.

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Table of Contents
ACKNOWLEDGMENT............................................................................................................................. vii
LIST OF TABLES................................................................................................................................. viii
LIST OF FIGURES................................................................................................................................. ix
LIST OF SYMBOLS ............................................................................................................................... xi
CHAPTER I................................................................................................................................................1
INTRODUCTION.......................................................................................................................................1
1.1 OCEAN OUTFALLS.....................................................................................................................1
1.2 THE OCEAN ENVIRONMENT.....................................................................................................1
1.3 OBJECTIVE OF THIS STUDY .......................................................................................................2
CHAPTER II...............................................................................................................................................3
LITERATURE REVIEW...............................................................................................................................3
2.1 MIXING ZONES .........................................................................................................................3
2.2 EMPIRICAL RELATIONSHIPS......................................................................................................6
2.3 NUMERICAL FORMULATION ....................................................................................................9
2.3.1 Stagnant Environment ......................................................................................................9
2.3.2 Flowing Environments ....................................................................................................14
2.4 NUMERICAL MODELS.............................................................................................................15
2.4.1 Fan and Brooks’ Model (Fan, and Brooks, 1969)............................................................15

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2.4.2 Abraham’s Model (Abraham, 1970) ...............................................................................15
2.4.3 Robert’s Model (Roberts, 1977) .....................................................................................17
2.4.6 DKHPLM Model (Kannberg, and Davis, 1976) ................................................................17
2.5 WAVE CHARACTERISTICS .......................................................................................................17
2.6 OUTFALL CHARACTERISTICS...................................................................................................21
2.7 STUDIES CONCERNING WAVES EFFECTS ON BUOYANT JETS.................................................21
CHAPTER III............................................................................................................................................24
MODEL FORMULATION.........................................................................................................................24
3.1 ANALYTICAL FORMULATION ..................................................................................................24
3.2 NUMERICAL SOLUTION ..........................................................................................................30
3.3 MODEL LIMITATIONS .............................................................................................................36
CHAPTER IV ...........................................................................................................................................37
MODEL VALIDATION .............................................................................................................................37
4.1 CALIBRATION OF MODEL .......................................................................................................37
4.2 COMPARISONS WITH OTHER MODELS ..................................................................................37
4.3 COMPARISON WITH OUTPLM................................................................................................44
CHAPTER V ............................................................................................................................................60
SUMMARY AND CONCLUSION..............................................................................................................60
5.1 SUMMARY ..............................................................................................................................60

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5.2 CONCLUSION..........................................................................................................................61
APPENDIX..............................................................................................................................................62
A-1 RUNGE-KUTTA PROCEDURE.......................................................................................................62
A-2 PROGRAM LISTING.....................................................................................................................63
A-3 SAMPLE RUN..............................................................................................................................74
A-4 BIBLIOGRAPHY ...........................................................................................................................79

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ACKNOWLEDGMENT
I wish to express my appreciation for all the assistance that was given to me during these past few
months. I would like to thank Dr. David A. Chin whose friendship and scientific guidance has taught
me much.
My special thanks goes to the members of my committee for reading the manuscript. I am grateful
to my parents, Winchell Mitchell and Claudette Mitchell, my uncle Albert Chauvet, my aunt Mona
Chauvet, my project manager, David A. Chin, and my fiancée, Marlene Mitchell, for their
encouragement, help, and advice during the preparation of the thesis. Their guidance is greatly
responsible for any worthwhile contribution this thesis may have.

viii
LIST OF TABLES
Table 2. 1 Summary of Characteristics of Major Pacific Outfalls 23
Table 4. 1 Experimental Results 38
Table 4. 2 Predictions of Average Dilution for Boston MDC Outfall by Different Method 42

ix
LIST OF FIGURES
Figure 2. 1 Mixing Zone 4
Figure 2. 2 Zone of Flow Establishment 5
Figure 2. 3 Buoyant Jet Typical Conditions 10
Figure 2. 4 Buoyant-Jet in crossflow 16
Figure 2. 5 Wave Parameters 19
Figure 2. 6 Waves Classification 20
Figure 3. 1 Absolute Motion of Plume Element 25
Figure 3. 2 Relative Motion of Plume Element 26
Figure 3. 3 Process to Complete a Snapshot 32
Figure 3. 4 Experimental Snapshot 33
Figure 3. 5 Wave Effect on Plume Path 34
Figure 3. 6 Flow Chart 35
Figure 4. 1 Comparisons with Experimental Results 39
Figure 4. 2 Boston MDC Initial Conditions 41
Figure 4. 3 Plume Profile for Various Models 43
Figure 4. 4 Initial Conditions for Comparisons 45
Figure 4. 5 Dilution vs. Froude Number for OUTPLM and MODEL, Ua = 0 m/s, Unstratified 48

x
Figure 4. 6 Dilution vs. Froude Number for OUTPLM and MODEL, Ua = .05 m/s, Unstratified 49
Figure 4. 9 Dilution vs. Froude Number for OUTPLM and MODEL, Ua = 0 m/s, Stratified 52
Figure 4. 10 Dilution vs. Froude Number for OUTPLM and MODEL, Ua = .05 m/s, Stratified 53
Figure 4. 13 Trapped Level vs. Froude Number for OUTPLM and MODEL, Ua = 0 m/s, Stratified 56
Figure 4. 14 Trapped Level vs. Froude Number for OUTPLM and MODEL, Ua = .05 m/s, Stratified 57

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LIST OF SYMBOLS
A = cross sectional area of port
a = wave amplitude
B = buoyancy flux
B’ = local buoyancy flux
b = plume width
bo = initial plume width
c = contaminant concentration in plume
Cd = drag coefficient
cd = contaminant concentration in discharge
co = centerline plume concentration
D = discharge port diameter
F = discharge densymetric Froude Number
FD = drag force
Fo = discharge Froude Number
g = gravity
go = effective gravity
H = discharge port depth

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Hmax = maximum height of rise of trapped plume
h = total water depth
L = wave length
LQ, LM = length scales
M = discharge momentum flux
m, M’ = local momentum flux
n = snapshot number
N = number of snapshots
Q = discharge volume flux
Q’ = local volume flux
q = discharge per diffuser length
R = local Richardson number
Rp = plume Richardson number
r = radial coordinate
S = centerline dilutiom
SA = average dilution
Se = length of zone of flow establishment

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s = longitudinal coordinate
T = wave period
t = time
tf = final time
U = wave velocity vector
UA = ambient velocity
UAs = ambient velocity at port level
u = wave horizontal velocity component
v = relative velocity of plume centerline
ve = entrainment velocity
vo = plume centerline velocity
Vo = absolute velocity of plume element
w = wave vertical velocity component
x = horizontal coordinate
z = vertical coordinate
αj, αp = jet and plume entrainment coefficient
α = aspiration entrainment coefficient

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β = forced entrainment coefficient or local buoyancy flux
ϒ = angle between V and plume axis
∆t1 = Lagrangian time increment
∆t2 = time difference between snapshots
∆ρ = density deficiency
(∆ρ)d = density deficiency at discharge
∆ρo = centerline density deficit
θ = angle between Vo and X axis, also plume inclination
θ1 = port angle relative to X axis
θ2 = direction of ambient current
λ = turbulent Schmidt number
ρo = ambient density at discharge
ρa, ρꝎ = ambient density at specified elevation
superscripts:
* = non-dimensional quantity

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CHAPTER I
INTRODUCTION
1.1 OCEAN OUTFALLS
A common method of wastewater disposal for coastal communities is to discharge the waste at some
depth below the surface of a large body of water such as a lake or the ocean. In the case of ocean
outfalls, the discharge line terminates in a perforated section called a diffuser. Diffusers, located
generally 1 to 2 miles offshore, release wastewater through a number of exit ports spaced along the
pipeline. Because wastewater has a density less than that of the ambient, it rises. The mixing induced
by the rising plume is called near-field mixing and the region in which this mixing occurs is called the
mixing zone. Beyond this region, dilution is influenced primarily by ambient currents. In designing an
outfall, only the dilution in the mixing zone is under the control of the designer. In the mixing zone,
the parameters that affect the dilution are the port discharge, diameter and inclination, as well as
ambient conditions such as the depth, currents, density stratification, and possibly surface waves.
In the United States, ocean outfalls located on the west coast are typically much deeper than those
on the east coast, primarily due to the greater offshore slopes on the west coast. Outfalls situated off
the west coasts are at depth varying from 150 to 200 feet, while those on the east coast are at depths
varying from 50 to 100 feet.
1.2 THE OCEAN ENVIRONMENT
The ambient environment surrounding outfalls typically is density stratified and influenced by
currents and surface waves. All these factors affects plume dilution. For the case of ambient

2
stratification, since the rising plume entrains ambient seawater and becomes denser at it rises, the
density of the plume may equal the local density before the plume surfaces. If this happens, the
plume becomes trapped below the surface. Since dilution is directly related to the height of rise,
strong stratification has detrimental effect on the dilution. Robert (1977) showed that currents tend
to increase dilution, but the effect of waves on the performance of outfalls has not been previously
studied. Based on the water particle movements associated with waves, an increase in the plume
dilution is to be expected.
1.3 OBJECTIVE OF THIS STUDY
Using experimental results obtained at the University of Miami, a numerical model that simulates the
effect of waves on plume dilution has been developed and tested. The integral equations describing
jet behavior and waves motions are the basis of the model. The model simulations are compared
with the experimental results in order to assess its performance. Comparisons with other models
which apply when wave effects are small are also used to validate the model.

3
CHAPTER II
LITERATURE REVIEW
2.1 MIXING ZONES
Consider an ocean outfall discharging sewage effluent through a diffuser. On entering the ocean,
there are three stages of the mixing process: (Fig. 2.1)
1. The zone of flow establishment.
2. The initial mixing zone or zone of established flow.
3. The zone of turbulent mixing in ocean current.
At the beginning of the zone of flow establishment (ZFE), the velocity profile is uniform. The liquid is
driven by momentum, so the ZFE is sometimes referred to as the jet-like region. At the boundary,
there is a shear layer between the jet and the ambient, thus the discharge velocity is slowed down.
The location where the velocity profile is no longer uniform, but Gaussian-like (Fig. 2.2), is where the
zone of established flow begins. In this region the wastewater is driven by both momentum and
buoyancy. Ambient fluid is entrained continuously and increase the width and the dilution of the
buoyant-jet. This regime continues until the plume density equals the ambient density, or the water
surface is reached. Subsequent dilution is accomplished by the action of wind, waves, and oceanic
currents.
The process of initial mixing (in the ZFE and zone of established flow) can be studied either by
empirical relationships or numerical models. In the initial mixing zone, the dilution is usually

5
Figure 2. 2 Zone of Flow Establishment

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referred to either as the average or the centerline dilution. It is related to the tracer
concentration profile which is, like the velocity profile, Gaussian. Empirical solutions are available
for buoyancy dominated vertical or horizontal discharges from a point or line source into an
unstratified or linearly stratified-stagnant environment whereby the maximum height of rise and
terminal dilution may be obtained. However, for more general cases, especially where discharge
is made at some angle into an arbitrarily stratified flowing environment, recourse must be made
to numerical techniques.
2.2 EMPIRICAL RELATIONSHIPS
Fisher et al. (1979) reported a series of empirical relationships characteristics of jets and plumes.
Those formulations, valid only for simple cases, are useful in engineering design.
When wastewater is discharged from an outfall into sea water it undergoes physical changes
relative to its density, temperature, and concentration of pollutants. This process called dilution
may be described in the functional form.
𝑆𝐴 = 𝑓(𝐷, 𝑣 𝑑, 𝑈𝐴, 𝑔 𝑜, 𝐻, 𝜃1, 𝜃1) (2.1)
Where SA is the average dilution, D is the port diameter, vd the discharge velocity, UA the
magnitude of the ambient current, go the effective gravity defined by
𝑔 𝑜 =
∆𝜌 𝑜
𝜌 𝑜
𝑔 (2.2)
Where ∆𝜌 𝑜 is the initial density difference between the discharge and the ambient, H the port
depth, 𝜃1 the port angle relative to the horizontal, 𝜃2 the direction of the ambient current relative
to the discharge direction, and 𝜌0 the ambient density at the discharge level.

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The parameters in equation 2.1 can be grouped together in various combinations. Three
parameters that are of prime importance in jet and plume dynamics are defined as follows:
The volume flux
𝑄 =
𝜋
4
𝐷2
𝑣 𝑑 (2.3)
The momentum flux
𝑀 = 𝑄 𝑣 𝑑 (2.4)
The buoyancy flux
𝐵 = 𝑄 𝑔 𝑜 (2.5)
Two length scales characteristics of jets and plumes can be deduced from the volume,
momentum, and buoyancy fluxes. They are:
𝐿 𝑄 =
𝑄
𝑀1/2
= 𝐴1/2
(2.6)
And
𝐿 𝑀 =
𝑀3/4
𝐵1/2
(2.7)
Where A is the initial cross sectional area of the port. The length scale 𝐿 𝑄 scales the length over
which the port geometry influences the effluent behavior, and 𝐿 𝑀 scales the distance from the
port at which buoyancy begins to dominate the flow.
As an approximation for the dilution for a point plume (pertaining to a single outlet discharge) in
a stagnant unstratified environment, Fisher et al. (1979) suggested,

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𝑆𝐴 = .126 [
𝐻
𝐿 𝑀
]
5/3
[
𝐿 𝑀
𝐿 𝑄
] (2.8)
And for the linearly stratified case
𝑆𝐴 = .10 [
𝐻 𝑚𝑎𝑥
𝐿 𝑀
]
5/3
[
𝐿 𝑀
𝐿 𝑄
] (2.9)
Where 𝐻 𝑚𝑎𝑥 is the maximum height of rise defined by:
𝐻 𝑚𝑎𝑥 = 3.98 [𝑄𝑔 𝑜]1/4
[−
𝑔
𝜌 𝑜
𝑑𝜌 𝑎
𝑑𝑧
]
−3/8
(2.10)
Where
𝑑𝜌 𝑎
𝑑𝑧
⁄ is the ambient density gradient.
Finally, for the case of a line source discharging into a stagnant-unstratified environment.
𝑆𝐴 = .537
𝑔 𝑜
1/3
𝐻
𝑞2/3
(2.11)
And for the linearly stratified case
𝑆𝐴 = .438
𝑔 𝑜
1/3
𝐻 𝑚𝑎𝑥
𝑞2/3
(2.12)
Where
𝐻 𝑚𝑎𝑥 = 2.84 [𝑞𝑔 𝑜]1/3
[−
𝑔
𝜌 𝑜
𝑑𝜌 𝑎
𝑑𝑧
]
−1/2
(2.13)
Where q is the discharge per unit length.
Equations 2.8 to 2.13 can be applied directly to estimate the dilution of a candidate design if all
the assumptions for those formulae are met. However, a numerical analysis will generally be

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necessary due to such ambient conditions as non-linear density stratification and currents. Also,
the pure plume assumption (H > 𝐿 𝑀) may be invalid, especially for shallow outfalls.
2.3 NUMERICAL FORMULATION
2.3.1 Stagnant Environment
When conditions are not ideal, empirical formulae applicable for simple cases are no longer
reliable. The use of a hydraulic model or a numerical model is then necessary to achieve a more
accurate solution.
The most general case to be considered is shown schematically in Fig. 2.3 where the discharge is
made at some angle into an arbitrarily stratified environment.
Experiments have demonstrated Gaussian similiraty profiles at all cross section for the density
deficiency, ∆𝜌, and the velocity, v, hence
𝑣(𝑠, 𝑟) = 𝑣𝑜(𝑠)𝑒𝑥𝑝 [−
𝑟2
𝑏2] (2.14)
∆𝜌(𝑠, 𝑟) = ∆𝜌 𝑜(𝑠)𝑒𝑥𝑝 [−
𝑟2
𝜆2 𝑏2
] (2.15)
Where r is the radial coordinate, b the characteristic jet width, Δ𝜌 𝑜 the centerline density
deficiency, 𝑣𝑜 the centerline velocity, and 𝜆 the turbulent Schmidt number which accounts for
the greater spread of buoyancy than mass. The density deficiency at any elevation, z, is defined
as the difference between the plume density and the ambient density at that elevation.
Defining the local volume flux Q’, momentum flux M’, and the density deficiency flux B’ by:
𝑄′
= ∫ 𝑣(𝑠, 𝑟)𝑑𝐴𝐴
(2.16)

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Figure 2. 3 Buoyant Jet Typical Conditions

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𝑀′
= 𝜌 𝑜 ∫ 𝑣2
𝐴
(𝑠, 𝑟)𝑑𝐴 (2.17)
𝐵′
= ∫ ∆𝜌𝑣(𝑠, 𝑟)𝑑𝐴𝐴
(2.18)
Conservation of mass yields
𝑑𝑄′
𝑑𝑠
= 2𝜋𝑏𝑣𝑒 (2.19)
Where 𝑣𝑒 is the entrainment velocity. It is generally assumed that 𝑣𝑒 is linearly proportional to
the plume centerline velocity, hence
𝑣𝑒 = 𝛼𝑣𝑜 (2.20)
Where 𝛼 is the entrainment coefficient.
Other equations governing the motion of buoyant-jets are:
The conservation of x momentum
𝑑(𝑀′ 𝑐𝑜𝑠𝜃)
𝑑𝑠
= 0 (2.21)
The conservation of z momentum
𝑑(𝑀′ 𝑠𝑖𝑛𝜃)
𝑑𝑠
= ∫ 𝑔∆𝜌(𝑠, 𝑟)𝑑𝐴𝐴
(2.22)
And the conservation of density deficiency
𝑑𝐵′
𝑑𝑠
=
𝑑𝜌 𝐴
𝑑𝑠
𝑄′ (2.23)
Combining equations 2.14 to 2.23 yields:

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𝑑𝑣 𝑜
𝑑𝑠
=
2𝑔𝜆2
𝑣 𝑜
Δ𝜌 𝑜
𝜌 𝑜
𝑠𝑖𝑛𝜃 −
2𝑣 𝑜 𝛼
𝑏
(2.24)
𝑑𝑏
𝑑𝑠
= 2𝛼 −
𝑏
𝑣 𝑜
2 𝑔𝜋2 ∆𝜌 𝑜
𝜌 𝑜
𝑠𝑖𝑛𝜃 (2.25)
𝑑∆𝜌 𝑜
𝑑𝑠
=
1+𝜆2
𝜆2
𝑠𝑖𝑛𝜃
𝑑𝜌∞
𝑑𝑧
− 2𝛼
𝑑∆𝜌 𝑜
𝑏
(2.26)
𝑑𝜃
𝑑𝑠
=
2𝑔𝜆2
𝑣 𝑜
2
Δ𝜌 𝑜
𝜌 𝑜
𝑐𝑜𝑠𝜃 (2.27)
𝑑𝑥
𝑑𝑠
= 𝑐𝑜𝑠𝜃 (2.28)
𝑑𝑧
𝑑𝑠
= 𝑠𝑖𝑛𝜃 (2.29)
At the end of the ZFE, where the velocity and the density profile start to be Gaussian, the initial
conditions are specified. From experiments, Albertson (1950) found the length of this zone to be
approximately 6.2D.
From conservation of momentum in the ZFE, the initial plume width, 𝑏 𝑜 is given by
𝑏 𝑜 =
𝐷
√2
(2.30)
And from the continuity of pollutant concentration
𝑐 𝑜 =
1+𝜆2
2𝜆2
𝑐 𝑑 (2.31)
And
Δ𝜌 𝑜 =
1+𝜆2
2𝜆2
[Δ𝜌] 𝑜 (2.32)

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Where [Δ𝜌] 𝑜 is the density deficiency of the effluent and 𝑐 𝑑 is the discharge concentration.
Laboratory experiments on pure plume yield 𝛼 = 𝛼 𝑝 = .082, and for pure jet 𝛼 = 𝛼𝑗 = .057.
(Rouse, et al., 1952)
While the behavior of a jet (a source of momentum flux only) and a pure plume (a source of
buoyancy flux only) can both be characterized by a constant entrainment coefficient, this cannot
be simply generalized to the case of a buoyant-jet, a source of mass, momentum, and buoyancy
fluxes. List and Imberger (1973), through experimental data, deduced that 𝛼 must be function
of the local Richardson number and suggested the following relation,
𝛼 = .057 + .083𝑅2
(2.33)
Where R is the local Richardson number, defined by
𝑅 =
𝜇𝛽1/2
𝑚5/4
(2.34)
Where m is the kinetic momentum flux
𝑚 =
𝜋
2
𝑏2
𝑣𝑜
2
(2.35)
𝜇 is the volume flux given by
𝜇 = 𝜋𝑏2
𝑣𝑜 (2.36)
And 𝛽 is the local buoyancy flux given by
𝛽 =
𝜋𝜆2
1+𝜆2
[
Δ𝜌 𝑜
𝜌 𝑜
]
2
𝑏2
𝑣0 𝑔 (2.37)

14
Continuing this study, List (1982) suggested a new version for the buoyant jet entrainment
coefficient.
𝛼 = 𝛼 𝑝 𝑒𝑥𝑝 [𝑙𝑛 [
𝛼 𝑝
𝛼 𝑗
] [
𝑅
𝑅 𝑝
]
2
] (2.38)
Where 𝑅 𝑝, the plume Richardson number, is given by
𝑅 𝑝 = .557 (2.39)
2.3.2 Flowing Environments
For the case of a buoyant jet in a uniform cross stream, the terminal dilution tends to increase
since the plume path becomes longer. The velocity and density profile are typically assumed to
be
𝑣(𝑠, 𝑟) = 𝑈𝐴 𝑐𝑜𝑠𝜃 + 𝑣𝑜(𝑠) 𝑒𝑥𝑝 [−
𝑟2
𝑏2
] (2.40)
Δ𝜌(𝑠, 𝑟) = Δ𝜌 𝑜(𝑠)𝑒𝑥𝑝 [−
𝑟2
𝜆2 𝑏2
] (2.41)
Where 𝑣𝑜 is the centerline velocity, and 𝑈𝐴 the ambient current as defined by Fig. 2.4 (Fan, 1967)
From mass conservation, equation 2.19 may be written as:
𝑑𝑄′
𝑑𝑠
= 2𝜋𝛼𝑏|𝑣𝑜 − 𝑈𝐴| (2.42)
The equations governing plume motion in this case are:
𝑑(𝑀′ 𝑐𝑜𝑠𝜃)
𝑑𝑠
= 𝜌 𝑎 2𝜋𝛼𝑏𝑈𝐴 |𝑣𝑜 − 𝑈𝐴| + 𝐹𝐷 𝑠𝑖𝑛𝜃 (2.43)
Where 𝐹𝐷 is the drag force assumed to be

15
𝐹𝐷 = .5 𝐶 𝐷 𝜌 𝑎 2√2 𝑏 𝑈𝐴
2
𝑠𝑖𝑛2
𝜃 (2.44)
𝑑(𝑀′ 𝑠𝑖𝑛𝜃)
𝑑𝑠
= ∫ 𝑔𝐴
∆𝜌(𝑠, 𝑟)𝑑𝐴 − 𝐹𝐷 𝑐𝑜𝑠𝜃 (2.45)
𝑑𝐵′
𝑑𝑠
=
𝑑𝜌 𝑎
𝑑𝑠
𝑄′ (2.46)
𝑑𝑐
𝑑𝑠
𝑄′
= 0 (2.47)
Where c is the concentration of pollutant.
2.4 NUMERICAL MODELS
Numerous computer models have been developed to calculate the initial dilution of jets and
plumes. The following discussion will describe the most popular models, among them, PLUME,
OUTPLM, and DKHPLM have been designated by the EPA as being acceptable for design. (Federal
Register, 1979)
2.4.1 Fan and Brooks’ Model (Fan, and Brooks, 1969)
This model developed by Ditmars (1969) calculate the dilution of buoyant jets discharged into a
stagnant, stratified or uniform density, ambient water. I is based on conservation of mass,
momentum, and buoyancy. A Gaussian similarity profile is assumed for both the velocity and
density deficiency.
2.4.2 Abraham’s Model (Abraham, 1970)
This model was developed to simulate the effect of a horizontal flow field on a buoyant jet in a
constant density ambient water. As in the Fan and Brooks’ model, velocity and tracer
concentration profiles are assumed to be Gaussian.

16
Figure 2. 4 Buoyant-Jet in crossflow

17
2.4.3 Robert’s Model (Roberts, 1977)
This model, using empirical relations, evaluates the initial dilution of a line source into a stratified
or unstratified ambient water with currents at an angle to the discharge. This models considers
only the buoyancy flux of the discharge and neglects the discharge momentum. Minimum
surface dilution is expressed as a function of the discharge per diffuser length, the magnitude of
the ambient current and its direction relative to the diffuser, the water depth, and the ambient
stratification.
2.4.6 DKHPLM Model (Kannberg, and Davis, 1976)
This model analyzes the discharge of a buoyant jet in a stratified, flowing ambient environment.
Aside from the conservation of mass, momentum, and buoyancy, this model carries the plume
analysis into three zones: the zone of flow establishment, the zone of established flow, and the
zone of merging flow. The Gaussian distribution is replaced by a 3/2 power approximation. This
model can be used to evaluate the dilution induced by a multiple port diffuser system.
2.5 WAVE CHARACTERISTICS
Waves in the ocean often appears as a confused and constantly changing sea of crests and
troughs on the water surface. Shore protection measures and coastal structure designs are
dependent on the ability to predict wave forms and fluid motion beneath waves, and on the
reliability of such predictions. Prediction methods generally have been based on simple waves
where elementary mathematical functions can be used to describe wave motion.

18
The most elementary wave theory, referred to as small amplitude or linear wave theory, was
developed by Airy (1845). The horizontal and the vertical component of the local fluid velocity
are given by the following equations:
𝑢 =
𝑎𝑔𝑇
𝐿
𝑐𝑜𝑠ℎ[
2𝜋
𝐿
(ℎ+𝑧)]𝑠𝑖𝑛[
2𝜋𝑥
𝐿
−
2𝜋𝑡
𝑇
]
𝑐𝑜𝑠ℎ[
2𝜋ℎ
𝐿
]
(2.48)
𝑤 = −
𝑎𝑔𝑇
𝐿
𝑠𝑖𝑛ℎ[
2𝜋
𝐿
(ℎ+𝑧)]𝑐𝑜𝑠[
2𝜋𝑥
𝐿
−
2𝜋𝑡
𝑇
]
𝑐𝑜𝑠ℎ[
2𝜋ℎ
𝐿
]
(2.49)
Where u is the horizontal velocity, w the vertical velocity, T the period, L the wave length, a the
wave amplitude, h the total depth, z and x the vertical and horizontal coordinates, and t the time.
Figure 2.5 illustrated those parameters.
Gravity waves may also be classified by the water depth in which they travel. The following
classifications are made according to the magnitude of h/L. For h/L < 1/25 the wave is classified
as a shallow water wave which induces a uniform horizontal velocity distribution. For 1/25 < h/L
< 1/2 the wave is a transitional water wave which induces a non-linear velocity distribution. For
h/L > 1/2 the wave is a deep water wave and the velocity becomes negligible on the bottom (Fig.
2.6). In the ocean where shallow outfalls are found (H < 50 feet), it is common for the waves to
feel the sea floor. Some kind of influence from the waves on the rising plume must then be
expected.

19
Figure 2. 5 Wave Parameters

20
Figure 2. 6 Waves Classification

21
2.6 OUTFALL CHARACTERISTICS
The primary goal of an outfall is to accomplish sufficient dilution of the effluent with the ambient
water. The waste water is released from the outfall either from a single open end or through a
number of exit ports spaced along the pipeline. The dilution obtained in a deep outfall diffusion
system depends primarily on the discharge, the length of the diffuser, the depth of discharge, the
port spacing, geometry, and the ambient currents and density stratification. Table 2.1 (Koh, and
Brooks, 1975) gives a summary of the characteristics of major outfalls on the west coast of the
United States. On the east coast, where the continental shelve is wide, outfalls are usually found
at shallower depths, usually less than 100 feet.
Overall, the ambient water quality requirements to be met are usually set by the states and cover
a variety of physical, chemical, and biological characteristics. For example, the state of California
requires that the initial dilution in the mixing zone must be greater than 100:1 for at least 50% of
the time and greater than 80:1 for at least 90% of the time during any one month period.
Furthermore, if possible, the outfalls should be long enough to meet shoreline bacterial
standards without disinfection. (California Water Resources Control Board, 1972)
2.7 STUDIES CONCERNING WAVES EFFECTS ON BUOYANT JETS
Shuto and Ti (1974) studied the effects of waves on vertically discharged buoyant jets. Their
experiments are characterized by
12.5 ≤
𝐻
𝐷
≤ 70
𝐻
𝐿 𝑀
= .66
ℎ
𝐿
≤ .05 (2.50)

22
It is apparent from equation 2.50 that the discharge behaved like a jet, and that the wave induced
motions were uniform with depth. This condition is not typical of outfalls where the effluent
becomes plume-like relatively quickly and the wave induced motions are not uniform with depth.
Ger (1979) studied also the influence of waves on buoyant jets. He found out the entrainment
coefficient to vary linearly with the wave characteristics
𝛼 = .05 + .63
𝑢
𝑣 𝑑
(2.51)
The experimental results are as follow.
𝐻
𝐿 𝑀
= .49 𝑡𝑜 2.1
𝐿 𝑀
𝐿 𝑄
= 2.1 𝑡𝑜 9.1
ℎ
𝐿
≤ .07 (2.52)
These conditions indicate again that the discharge was more in the jet-like region and also that
the wave induced velocities were almost uniform with depth. Conditions seldom encountered
in ocean outfalls.

23
Table 2. 1 Summary of Characteristics of Major Pacific Outfalls
a. Outfalls are numbered as follows: 1. Sanitation Districts of Los Angeles County, Whites Point No. 4
2. Metro Seattle (West Point) 3. Sanitation Districts of Orange County, California. 4. Honolulu (Sand Island)
Outfallsa Port Diameter
d
(inches)
Pipe Diameter
D
(inches)
Port Spacing
L
(ft)
Depth of Discharge
h
(ft)
Port Discharge
Q
(ft3/s)
Height to Plume
Regime Zb
(ft)
Merging
Height Zm
(ft)
1 2 -3.6 120 12 165 -190 .38 14 36
2 4.5 – 5.75 96 6 210 - 240 .86 10 18
3 2.96 – 4.13 120 24 175 - 195 .89 26 72
4 3.00 – 3.53 84 24 220 - 235 .58 18 72

24
CHAPTER III
MODEL FORMULATION
3.1 ANALYTICAL FORMULATION
When a buoyant jet is discharged into water where wave motion exists, the axis of the plume is
advected horizontally and vertically. The dilution rate of the plume is affected by this motion.
Consider Fig. 3.1, 𝑈 is the magnitude of the ambient velocity, 𝑣𝑜is the plume centerline velocity
relative to the ambient, 𝑉𝑜 is the resultant plume velocity, 𝜃 is the angle the resultant, Vo , makes
with the horizontal axis and 𝛾 is the angle the relative velocity 𝑣𝑜 makes with the plume axis. The
relative velocity, 𝑣, may be assumed to have a Gaussian profile,
𝑣(𝑠, 𝑟) = 𝑣𝑜(𝑠)𝑒𝑥𝑝 [−
𝑟2
𝑏2
] (3.1)
Where r is the radial coordinate, b the characteristic jet width, vo the jet centerline velocity.
The relative velocity as shown in Fig. 3.2 has two components which will induce different
entrainment mechanisms. The component in the 𝑠 direction will induce radial entrainment,
while the component in the 𝑟 direction will induce forced entrainment of the ambient fluid. The
basic equations governing the behavior of a buoyant conservative contaminant are
1. Conservation of mass
𝑑
𝑑𝑠
∫ 𝑣(𝑐𝑜𝑠𝛾)𝑑𝐴 = 2 𝜋 𝑟 𝛼 𝑣𝑜 𝑐𝑜𝑠𝛾 + 2 𝑟 𝛽 𝑣𝑜 𝑠𝑖𝑛𝛾 = 2 𝜋 𝑟 𝑣𝑜 𝑐𝑜𝑠𝛾𝐴
[𝛼 + 𝑇𝑎𝑛 𝛾
𝛽
𝜋
] (3.2)
where 𝛼 is the radial entrainment coefficient and 𝛽 the forced entrainment coefficient.

25
Figure 3. 1 Absolute Motion of Plume Element

26
Figure 3. 2 Relative Motion of Plume Element

27
2. Conservation of 𝑥 momentum
𝑑
𝑑𝑠
∫ 𝑣𝑜
2
𝐴
𝑐𝑜𝑠2
𝛾 𝑑𝐴 𝑐𝑜𝑠𝜃 = 0 (3.3)
3. Conservation of 𝑧 momentum
𝑑
𝑑𝑠
∫ 𝑣𝑜
2
𝐴
𝑐𝑜𝑠2
𝛾 𝑑𝐴 𝑠𝑖𝑛𝜃 = ∫ 𝑔 ∆𝜌(𝑠, 𝑟) 𝑑𝐴𝐴
(3.4)
4. Conservation of density deficiency
𝑑
𝑑𝑠
∫ ∆𝜌 𝑣𝑜 𝑐𝑜𝑠𝛾 𝑑𝐴 =
𝑑𝜌 𝑎
𝑑𝑠𝐴
∫ 𝑣𝑜𝐴
𝑐𝑜𝑠𝛾 𝑑𝐴 (3.5)
5. Conservation of pollutant
𝑑
𝑑𝑠
∫ 𝑣𝑜𝐴
𝑐𝑜𝑠𝛾 𝑐 𝑑𝐴 = 0 (3.6)
The concentration profile is assumed to be Gaussian, then
𝑐 = 𝑐 𝑜 𝑒𝑥𝑝 [−
𝑟2
𝜆2 𝑏2
] (3.7)
Where 𝑐 𝑜 is the centerline concentration.
Combining equation 3.2 to 3.7 after further manipulation give:
𝑑𝑣 𝑜
𝑑𝑠
=
2 𝑔 𝜋2
𝑣 𝑜 𝑐𝑜𝑠2 𝛾
Δ𝜌 𝑜
𝜌 𝑜
𝑠𝑖𝑛𝜃 −
2𝑣 𝑜
𝑏
[𝛼 +
𝛽
𝜋
𝑡𝑎𝑛𝛾] (3.8)
𝑑𝑏
𝑑𝑠
= 2 [𝛼 +
𝛽
𝜋
𝑡𝑎𝑛𝛾] −
𝑏 𝑔 𝜋2
𝑣 𝑜
2 𝑐𝑜𝑠2 𝛾
Δ𝜌 𝑜
𝜌 𝑜
𝑠𝑖𝑛𝜃 (3.9)
𝑑𝜃
𝑑𝑠
=
2 𝑔 𝜆2
𝑣 𝑜
2 𝑐𝑜𝑠2 𝛾
Δ𝜌 𝑜
𝜌 𝑜
𝑐𝑜𝑠𝜃 (3.10)
𝑑Δ𝜌 𝑜
𝑑𝑠
=
1+𝜆2
𝜆2 𝑠𝑖𝑛𝜃
𝑑𝜌 ∞
𝑑𝑧
− 2 [𝛼 +
𝛽
𝜋
𝑡𝑎𝑛𝛾]
Δ𝜌 𝑜
𝑏
(3.11)

28
𝑑𝑐 𝑜
𝑑𝑠
= −
2𝑐 𝑜
𝑏
𝑑𝑏
𝑑𝑠
−
𝑐 𝑜
𝑣 𝑜
𝑑𝑣 𝑜
𝑑𝑠
(3.12)
𝑑𝑧
𝑑𝑠
= 𝑠𝑖𝑛𝜃 (3.13)
𝑑𝑥
𝑑𝑠
= 𝑐𝑜𝑠𝜃 (3.14)
Written in non-dimensional form
𝑑𝑣∗
𝑑𝑠∗
=
2𝜆2 𝑠𝑖𝑛𝜃 Δ𝜌∗
𝑣∗ 𝐹𝑜
2 −
2𝑣∗
𝑏∗
[𝛼 +
𝛽
𝜋
𝑡𝑎𝑛𝛾] (3.15)
𝑑𝑏∗
𝑑𝑠∗
= 2 [𝛼 +
𝛽
𝜋
𝑡𝑎𝑛𝛾] −
𝜆2 𝑏∗
[𝑣∗ 𝐹𝑜
∗]2
Δ𝜌 𝑜
∗
𝑠𝑖𝑛𝜃 (3.16)
𝑑𝜃∗
𝑑𝑠∗
=
2𝜆2
[𝑣∗ 𝐹𝑜
∗]2
Δ𝜌 𝑜
∗
𝑐𝑜𝑠𝜃 (3.17)
𝑑Δ𝜌∗
𝑑𝑠∗ =
1+𝜆2
𝜆2
𝑑𝜌∞
∗
𝑑𝑧∗ 𝑠𝑖𝑛𝜃 −
2
𝑏∗ [𝛼 +
𝛽
𝜋
𝑡𝑎𝑛𝛾] Δ𝜌 𝑜
∗
(3.18)
𝑑𝑐∗
𝑑𝑠∗ = −
2𝑐∗
𝑏∗ [𝛼 +
𝛽
𝜋
𝑡𝑎𝑛𝛾] (3.19)
𝑑𝑧∗
𝑑𝑠∗ = 𝑠𝑖𝑛𝜃 (3.20)
𝑑𝑥∗
𝑑𝑠∗
= 𝑐𝑜𝑠𝜃 (3.21)
Where the dimensionless parameters are defined by
𝑣∗
=
𝑣 𝑜 𝑐𝑜𝑠𝛾
𝑣 𝑑
(3.22)
𝑠∗
=
𝑠
𝐷
(3.23)
Δ𝜌 𝑜
∗
=
Δ𝜌 𝑜
𝜌 𝑜
(3.24)

29
𝑏∗
=
𝑏
𝐷
(3.25)
𝑐∗
=
𝑐
𝑐 𝑑
(3.26)
Where 𝑐 𝑑 is the concentration at the discharge level
𝐹𝑜 =
𝑣 𝑑
√𝑔 𝐷
(3.27)
𝑥∗
=
𝑥
𝐷
(3.28)
𝑧∗
=
𝑧
𝐷
(3.29)
The relationship between 𝛾 and 𝜃 is given by
𝛾 = 𝜃 − 𝑡𝑎𝑛−1
[
𝑤+ 𝑣 𝑜 𝑠𝑖𝑛𝜃
𝑢+ 𝑈 𝐴+ 𝑣 𝑜 𝑐𝑜𝑠𝜃
] (3.30)
Where 𝑢 and 𝑤 are the velocity component of the wave and 𝑈𝐴 the velocity of the ambient
current
Computations begin at the downstream end of the ZFE. The length of the ZFE, obtained from the
relative discharge Froude Number, is based on the following equations (Abraham, 1963)
𝑆 𝑒
∗
= 2.8 𝐹2/3
𝐹 < 2 (3.31)
𝑆 𝑒
∗
= .113 𝐹2
+ 4 2 ≤ 𝐹 ≤ 3.2 (3.32)
𝑆 𝑒
∗
=
5.6 𝐹2
√𝐹4+18
𝐹 > 3.2 (3.33)
Where 𝐹 is the relative densymetric Froude Number defined by:

30
𝐹 =
𝑣 𝑑− 𝑈 𝐴𝑠 𝑐𝑜𝑠𝜃1
√
Δ𝜌 𝑜 𝑔 𝐷
𝜌 𝑜
(3.34)
Where 𝑈𝐴𝑠 is the ambient horizontal velocity at the source, and 𝜃1 the discharge angle. The zone
of flow establishment varies then according to the magnitude of 𝑈𝐴𝑠 𝑐𝑜𝑠𝜃1. The initial conditions
described by 𝑣𝑜
∗
, 𝑐 𝑜
∗
, Δ𝜌 𝑜
∗
, 𝑏 𝑜
∗
are:
𝑣𝑜
∗
= 1 (3.35)
𝑏 𝑜
∗
=
1
√2
(3.36)
𝑐 𝑜
∗
=
1+𝜆2
2𝜆2
(3.37)
Δ𝜌 𝑜
∗
= 𝑐 𝑜
∗ (Δ𝜌) 𝑑
𝜌 𝑜
(3.38)
Where (Δ𝜌) 𝑑 is the initial density difference at the discharge.
3.2 NUMERICAL SOLUTION
A fortran program has been developed to solve the problem posed by the discharge of
wastewater from a single port at an arbitrary angle, velocity, density, and diameter into either a
stagnant, flowing or wavy, non-linearly stratified environment.
The method of solution is to first determine the initial conditions at the ZFE for 𝑣𝑜, 𝑏, 𝜃, Δ𝜌 𝑜, 𝑐 𝑜,
and in the case of a moving environment, the value of the horizontal and vertical velocity at that
location. The solution then proceeds using a fourth order Runge-Kutta integration method to
solve equations 3.15 to 3.21 using a small fixed increment Δ𝑠 along the jet trajectory. The
increment Δ𝑠 should be small enough such that the resulting dilution does not change when Δ𝑠

31
is made smaller. The variables 𝑣𝑜, 𝑏, 𝜃, Δ𝜌 𝑜, 𝑐 𝑜, 𝑥, 𝑧, are evaluated. Based on the relative
centerline velocity calculated at 𝑠 + Δ𝑠, the time required, Δ𝑡, for a jet element to travel the
distance Δ𝑠 is found. For a moving environment, the local ambient velocity and Δ𝑡 give the
additional advection experienced by the jet element. Following this process, a new position and
inclination of the jet element is found. This procedure is repeated and computations cease when
the surface is reached, or when there is no density difference between the ambient and the
plume (zero buoyancy), or when the jet inclination changes sign.
To determine a snapshot, a plume element is released at time t and calculations proceed until
stopping criteria are met. The time at which this happens is set to be the final time tf1. Another
plume element is released and the same process continues and so on until the time of release of
one element equals the final time (Fig. 3.3). the end result is a representation of the plume
trajectory as it will look if a picture was taken at time tf1. Figure 3.4 shows a physical
representation of the experiment while Fig. 3.5 is a computer simulated snapshot. They show
close similarities.
To represent the next snapshot, a plume element is released at time 𝑡 + Δ𝑡2. If the time it
reaches the surface is equal to
𝑡𝑓𝑛 = 𝑡𝑓𝑛−1 +
𝑇
𝑁−1
(3.39)
Where N represents the number of snapshots wanted, and T the wave period; the release time
is incremented and calculations proceed to the next plume element in that snapshot. Otherwise,
a new release time is tried. This method will calculate as many snapshots as required. The
simplified flow chart in Fig. 3.6 outlines this procedure.

32
SNAPSHOT COMPLETED
Figure 3. 3 Process to Complete a Snapshot

33
Figure 3. 4 Experimental Snapshot

34
Figure 3. 5 Wave Effect on Plume Path

36
Because of the unsteady conditions induced by waves, the plume shape is transient. Sometimes
it becomes shorter or longer. This results in decrease or increase of the dilution over a wave
period. So, in order to obtain the average dilution, several snapshots must be calculated. The
average dilution over a period is defined by:
𝑆𝐴 = [∑ 𝑆𝑖
𝑁
𝑖=1 ]
√2
𝑁
(3.40)
Where 𝑆𝑖, the centerline dilution for a snapshot i, is given by
𝑆𝑖 =
𝑐 𝑑
𝑐 𝑜
(3.41)
Where 𝑐 𝑜 is the centerline concentration at the terminal point of the plume snapshot.
3.3 MODEL LIMITATIONS
The primary limitation of this model is the case where the magnitude of the vertical wave induced
velocity is so large that it forces the plume to curve down into itself. This occurs near the surface
and causes re-entrainment which is not accounted for in the calculations. Another problems
involves the case where the relative exit velocity approaches zero. This is a problem because the
relative Froude number is used to calculate the initial conditions. Fortunately, in a typical case,
this will require a very large wave, or a strong current which is unrealistic in typical ambient
conditions at outfall locations.
This model uses two empirical entrainment coefficient, 𝛼 and 𝛽. For plume-like conditions, 𝛼 is
assumed in the order of .083, (Rouse, et al., 1952) while for 𝛽 experimental results are needed
to estimate its value. The next chapter compares the model results with experiments and from
there the value of 𝛽 is estimated.

37
CHAPTER IV
MODEL VALIDATION
In order to validate the model, it was first tuned for various values of 𝛽 until it showed good
agreement with the experimental results. Then, the wave amplitude was put equal to zero and
the model predictions for the cases of arbitrary stratification, and stagnant or flowing ambient
were compared with the results of previously developed models.
4.1 CALIBRATION OF MODEL
Experiments performed at the University of Miami (Chin, 1986), the results of which are listed in
Table 4.1, were the only appropriate data available for this study. The h/L ratio describing wave
type, ranges from .2 to .4, which are the conditions experienced by shallow outfalls. Various
values of 𝛽, the forced entrainment coefficient, were tried until the model showed good
agreement with the experiments. A 𝛽 value of .27 in the model was found to give results close
to the experimental measurements. This compares favorably with the results of Winiarski and
Frick (1976) who found 𝛽 to be in the range of .1 to 1. The calibration results shown in Fig. 4.1
demonstrate that the assumption about the forced entrainment coefficient, 𝛽, produces good
agreement with experimental data.
4.2 COMPARISONS WITH OTHER MODELS
Since the only published comparison of previously developed numerical models was performed
for the Boston outfall, the model predictions for the dilution of wastewater were compared with

38
Table 4. 1 Experimental Results
Q = 21.8 cm3
/s
D = 6.4 mm
H= 85.3 cm
h = 91.4 cm
Exp. No. ρa ρo T a h/L Uas/V S S S/So S/So
g/cm3
g/cm3
sec. cm exp. model exp. model
1 994.2 963.8 0 0 0 0.0000 70 68.3 1.00 1.00
2 993.6 963.2 0 0 0 0.0000 70 68.3 1.00 1.00
3 991.4 962.9 1.9 1.5 0.194 0.0469 80 83.5 1.14 1.22
4 994.3 958.3 1.9 2.1 0.192 0.0657 90 90.8 1.29 1.32
5 994.4 962.9 1.4 2.3 0.310 0.0440 90 81.4 1.29 1.19
6 993.4 964.0 1.4 3.0 0.307 0.0580 90 87.6 1.29 1.28
7 993.4 965.0 1.2 4.8 0.417 0.0529 80 100.8 1.14 1.47
8 994.7 964.7 1.4 7.2 0.313 0.1350 125 136.6 1.79 2.00
9 994.5 963.8 1.4 2.6 0.311 0.0480 95 85.1 1.36 1.25
10 994.5 963.8 1.4 4.9 0.311 0.9170 110 114.0 1.57 1.66
11 994.2 963.8 1.2 2.8 0.411 0.0319 80 82.5 1.14 1.21
12 994.8 964.2 1.9 1.6 0.194 0.0500 90 85.8 1.29 1.26
13 993.5 966.0 1.9 2.6 0.194 0.0813 90 100.2 1.29 1.47
14 992.9 963.9 1.9 4.0 0.194 0.1234 120 128.0 1.71 1.87
15 993.9 964.6 1.9 3.3 0.194 0.1030 105 113.6 1.50 1.66
16 994.1 965.6 1.4 3.7 0.311 0.0699 100 96.7 1.43 1.41
17 994.4 964.1 1.4 6.0 0.311 0.1136 110 139.5 1.57 2.04
18 993.6 964.3 1.4 1.9 0.311 0.0359 85 80.0 1.21 1.17
19 994.3 964.5 1.2 4.2 0.411 0.0480 95 95.1 1.36 1.39

39
Figure 4. 1 Comparisons with Experimental Results

40
those obtained in this comparative study (Alam et al. 1982). The data used are shown graphically
in Fig. 4.2.
The model results for both the stagnant-unstratified and stagnant-stratified ambient conditions
are within the range of the predictions of the other models (Fig. 4.3, Table 4.2). The model
predictions for the stagnant-unstratified conditions were 27% lower than the highest estimate
(PLUME) and 44% higher than the lowest estimate (Roberts). For the stagnant-stratified case,
again the same consistency occurs. The model predictions were 23% lower than the highest
estimate (Fan & Brooks) and 26% higher than the lowest estimate (Wright et al. 1982).
For the flowing-unstratified or flowing-stratified ambient conditions, the model predictions were
comparable to those obtained using previously developed models. For the flowing-unstratified
case, the model estimations were 33% to 85% lower than the highest estimate, and 2% to 48%
higher than the lowest estimate. Roberts (1982) in his discussion maintained that for the flowing-
unstratified case, the experiments of Buhler (1974) are probably the most reliable. He further
explained that Roberts’ model underestimated the dilutions by 23% to 42%, while DKHPLM and
Abraham over estimated by 58% to 613%. The predictions of DKHPLM and Abraham become
worse as the current speed increases. So based on Roberts’ discussion, the model developed in
this study would closely match the results of Buhler. The difference in predictions varies from
2% to 17% with Buhler having the lower estimates.
For the flowing-stratified ambient cases, following Roberts’ discussion, DKHPLM, OUTPLM and
Abraham’s models predicted as in the unstratified case, extremely high dilution. In the stratified
case, there are no systematic experiments available for prediction although the field experiments

41
Figure 4. 2 Boston MDC Initial Conditions

42
Table 4. 2 Predictions of Average Dilution for Boston MDC Outfall by Different Method
Current
m/s
Buhler Wright, et al. Roberts Fan & Brooks PLUME DKHPLM Abrahams OUTPLM MODEL
0 205 - 132 309 320 205 - 214 235
0.027 205 - 132 - - 383 323 215 255
0.078 252 - 146 - - 828 1014 364 298
0.146 354 - 273 - - 1471 2524 2573 363
0 - 71 74 83 78 91 - 89 97
0.027 - - 74 - - 144 150 90 104
0.078 - - 81 - - 204 389 130 118
0.146 - - 109 - - 264 958 374 139
Unstratified (March)
Stratified (August)

43
Figure 4. 3 Plume Profile for Various Models

44
of Hendricks (1977) who measured dilutions within 95% to 135% of those predicted by Roberts’
model may be cited in support of this model. The model in this study predicted dilutions 28%
higher than Roberts’ model.
Figure 4.3 and Table 4.2 list the predictions of the average dilutions for Boston MDC outfall by
the different methods discussed in this section. Since the model predictions were comparable
to the other models, this gives further credence to the numerical formulation, and the calibration
value of 𝛽 estimated from comparisons with the wave experiments.
4.3 COMPARISON WITH OUTPLM
In order to provide a more detailed comparison with previous developments in simulating the
behavior of buoyant jets, the model was compared in detail with a typical model, OUTPLM.
Because so many variables are involved in the plume behavior, a complete comparisons with
OUTPLM is therefore unwieldy. In order to provide a more detailed comparisons, some
parameters were kept constant. Since the main application of this model is in the range of
shallow outfalls, a port depth of 30 meters was selected. Two density profiles characteristic of
the stratified and unstratified cases were selected. Because most outfalls are subject to ambient
currents, velocities ranging from 0 m/s to .2 m/s were used. The port diameter and effluent
density were kept constant, only the port discharge, hence Froude number, was allowed to vary
(Fig. 4.4).
For the stagnant-unstratified case (G = 0 g/cm4, where 𝐺 =
𝑑𝜌 𝑎
𝑑𝑧
⁄ the ambient density
gradient), and for a small current of .05 m/s, both OUTPLM and the model predicted dilutions
differ by no more than 15% from each other. At higher Froude numbers, both models show the
same asymptotic behavior. They approach a constant value of around 80 for the dilution. At

45
Figure 4. 4 Initial Conditions for Comparisons

46
lower Froude numbers, OUTPLM predicted lower dilutions, especially in the case of the flowing
ambient (Fig. 4.5, 4.6). These graphs clearly show how the discharge Froude numbers, or the
ambient currents influence the dilution.
For higher currents, .1 m/s to .2 m/s, OUTPLM, like in the case of Boston MDC outfall, predicted
excessively high dilution while the model is more on the conservative side (Fig. 4.7, 4.8). again,
at higher discharge Froude number, both models give similar results, and the same asymptotic
behavior is observed.
The next step was to evaluate the model predictions for the stratified case (G = .086 g/cm4). For
both the stagnant case and a small current of .05 m/s, the models, either at higher or lower
Froude numbers, predicted dilutions within 15% of each other (Fig. 4.9, 4.10). for a Froude
number of around 20, and for both ambient cases, both models yield almost identical results. At
Froude numbers less than 20, the model predictions are higher than those predicted by OUTPLM,
but at Froude numbers higher than 20, OUTPLM predictions are higher.
For higher currents, .1 m/s to .2 m/s, and ambient stratification (G = .086 g/cm4), OUTPLM
predicted higher dilution for low discharge Froude numbers. However, at high Froude numbers,
both OUTPLM and the model predictions converged (Fig. 4.11, 4.12).
It is interesting to note for the stratified case, the trapping height in both the model and OUTPLM
closely match each other. In all cases less than 30% difference was observed (Fig. 4.13 to 4.16).
These comparisons show that the model at higher discharge Froude numbers agree closely with
OUTPLM predictions. While at lower discharge Froude numbers and small ambient currents the

47
model predicted higher dilutions, for increasing current speed, OUTPLM overestimate the
dilution.

48
Figure 4. 5 Dilution vs. Froude Number for OUTPLM and MODEL, Ua = 0 m/s, Unstratified

49
Figure 4. 6 Dilution vs. Froude Number for OUTPLM and MODEL, Ua = .05 m/s, Unstratified

50

51

52
Figure 4. 9 Dilution vs. Froude Number for OUTPLM and MODEL, Ua = 0 m/s, Stratified

53
Figure 4. 10 Dilution vs. Froude Number for OUTPLM and MODEL, Ua = .05 m/s, Stratified

54

55

56
Figure 4. 13 Trapped Level vs. Froude Number for OUTPLM and MODEL, Ua = 0 m/s, Stratified

57
Figure 4. 14 Trapped Level vs. Froude Number for OUTPLM and MODEL, Ua = .05 m/s, Stratified

58

59

60
CHAPTER V
SUMMARY AND CONCLUSION
5.1 SUMMARY
Several investigators, through experimental results and numerical models, have been able to
simulate the effect of various ambient conditions on plume dilution. Ambient conditions such as
stratification and currents have been the major area of interest. Although outfalls are subject to
wave actions, no attempt had been made to simulate this ambient case. Experiments performed
by Shuto and Ti (1974), and Ger (1979) to investigate wave effects on plume dilution were
characterized by wave induce velocities which were uniform with depth and a discharge were
dilution was influenced more by momentum than buoyancy. These conditions were not typical
of outfalls. Recent laboratory experiments performed at the University of Miami are the only
source of data available for this study. A numerical model has been developed to simulate the
effect of waves on the dilution of a submerged buoyant jets, based on the assumption that the
classical momentum, continuity, and buoyancy equations remain valid in a Lagrangian coordinate
system which moves with the wave induced velocities. The process of solving the basic equations
describing the jet dynamics is performed using a fourth order Runge-Kutta procedure in the
Lagrangian coordinate system. The terminal dilution is averaged over a wave period. To obtain
the best estimate of dilution, several snapshots of the plume profile during each wave period are
created. The model contains two empirical coefficients; the entrainment coefficients associated
with aspiration and impingement. It is assumed that the aspiration coefficient in the Lagrangian

61
coordinate is identical with that in the stagnant ambient case, while the impingement coefficient
is required to be within the range found in previous investigations, with its exact value being
estimated by comparing model predictions with experimental results. The value obtained from
these comparisons is consistent with those found by other investigators. The model is able to
handle such ambient conditions as currents and stratification. As wave effects become negligible
the model should asymptotically produce results consistent with previously developed models
which neglect wave effects. This was demonstrated by comparing the predictions of this model
with previously developed models for selected ambient environments.
5.2 CONCLUSION
A model has been developed that simulates the effects of waves, currents and ambient
stratification on plume dilution. Previous experimental results have shown that waves do
influence the dilution induced by hallow outfalls. The model developed in this study can be used
as a tool to quantify this effect. If, at an outfall location, the prevalent waves significantly affect
the dilution, then one may be justified in considering these waves in the design of the outfall.
This approach may be economical since outfalls could then be located at shallower depths than
indicated by neglecting waves.

62
APPENDIX
A-1 RUNGE-KUTTA PROCEDURE
The Runge-Kutta procedure is an explicit method used to obtain accurate solutions to a system
of differential equations of the form
𝑑𝑋 𝑖
𝑑𝑠
= 𝑓𝑖 (𝑋1, 𝑋2, … … … … … , 𝑋 𝑛) 𝑖 = 1, 𝑛 (A-1.1)
Starting from the initial values of the variables at a point 𝑠, 𝑋𝑖(𝑠); the functions are evaluated at
a point 𝑠 + ∆𝑠 using
𝑋𝑖(𝑠 + ∆𝑠) = 𝑋𝑖(𝑠) +
1
6
(𝐾𝑖1 + 2𝐾𝑖2 + 2𝐾𝑖3 + 𝐾𝑖4) (A-1.2)
Where 𝐾𝑖1, 𝐾𝑖2, … … … 𝐾𝑖4 are defined by:
𝐾𝑖1 = ∆𝑠 𝑓𝑖 (𝑋1, 𝑋2, … … … … … , 𝑋 𝑛) (A-1.3)
𝐾𝑖2 = ∆𝑠 𝑓𝑖 (𝑋1 +
1
2
𝐾11, 𝑋2 +
1
2
𝐾21, … … … 𝑋 𝑛 +
1
2
𝐾 𝑛1) (A-1.4)
𝐾𝑖3 = ∆𝑠 𝑓𝑖 (𝑋1 +
1
2
𝐾12, 𝑋2 +
1
2
𝐾22, … … … 𝑋 𝑛 +
1
2
𝐾 𝑛2) (A-1.5)
𝐾𝑖4 = ∆𝑠 𝑓𝑖 (𝑋1 + 𝐾13, 𝑋2 + 𝐾23, … … … 𝑋 𝑛 + 𝐾 𝑛3) (A-1.6)

63
A-2 PROGRAM LISTING
Dim H1(100), D(100), ZK(7, 4), K, L, L0, L1, L2, LA, TF1(20), TW1(20), PI As Double
Dim U, W, REF1, REF2, X, Y, SIGMA, RVX, RVY, RV, FO, FO1, SI, TM, TM1 As Double
Dim TH, DIA, DV, Z, DD, UA, XI, YI, A, T, H, DS, RX, RY As Double
im VMS, BS, B, DRS, DI, TF, TH2, DL As Double
Dim I, J, N, N1, N2, KK, II As Integer
Public Sub OUTFALL_BUOYANT_JET()
'
'THIS PROGRAM WAS WRITTEN AS PART OF A REQUIREMENT FOR A MASTER'S THESIS
'THIS PROGRAM CALCULATES THE INITIAL DILUTION OF BUOYANT JETS AND PLUMES
'RELEASED INTO A STRATIFIED OR UNSTRATIFIED ENVIRONMENT, AND UNDER WAVES
'OR CURRENT ACTIONS.
'Master degree thesis by Francis Mitchell
'Update
'This code is the visual basic version of the original code written in fortran
'This code is run as a visual basic excel macro
'RESET TABLE FOR NEW ANALYSIS
RESET_TABLE
'READ PORT ANGLE
TH = Range("C3").Value
'READ PORT DIAMETER
DIA = Range("C4").Value
'READ DISCHARGE VELOCITY
DV = Range("C5").Value
'READ PORT DEPTH
Z = Range("C6").Value
'READ DISCHARGE DENSITY
DD = Range("C7").Value
'READ AMBIENT CURRENT
UA = Range("C8").Value
'READ POSITION ALONG X AXIS
XI = Range("C9").Value
'READ WAVE AMPLITUDE
A = Range("C10").Value
'READ WAVE PERIOD
T = Range("C11").Value
'READ TOTAL DEPTH
H = Range("C12").Value
'READ UNIT DISPLACEMENT

64
DS = Range("C13").Value
'READ NUMBER OF SNAPSHOTS PER PERIOD
N2 = Range("C14").Value
PI = 4 * Atn(1)
'FIND WAVE LENGTH
L0 = 1.561 * T ^ 2
L1 = L0
45 L2 = L1 - F(L1, L0, H) / FPRIME(L1, L0, H)
If Abs(L2 - L1) < 0.0001 Then
L = L2
Else
L1 = L2
GoTo 45
End If
K = 2 * PI / L
SIGMA = 2 * PI / T
LA = 1.19
CO = (1 + LA ^ 2) / (2 * LA ^ 2)
DT1 = DS / DV
'READ AMBIENT WATER DENSITY
70 For N = 1 To 5
INPUTROW = ("C" + Str(18 + N))
INPUTROW = Replace(INPUTROW, " ", "")
H1(N) = Range(INPUTROW).Value
INPUTROW = ("D" + Str(18 + N))
D(N) = Range(INPUTROW).Value
Next
CT1 = (1 + LA ^ 2) / LA ^ 2
CT2 = 2 * LA ^ 2
CT3 = 2 * LA
YI = H - Z
'FIND AMBIENT DENSITY AT DISCHARGE LEVEL
For N = 1 To 5
If YI >= H1(N) And YI <= H1(N + 1) Then
DI = (YI - H1(N)) * (D(N + 1) - D(N)) / (H1(N + 1) - H1(N)) + D(N)

65
CT4 = (DI - DD) / DI
DRAI = ((D(N + 1) - D(N)) / (H1(N + 1) - H1(N))) * (DIA / DI)
End If
Next N
DLT = 0
'PERFORM ANALYSIS FOR EACH SNAPSHOT
'SET STARTS OF RESULTS AT SPECIFIED ROW VALUE
KK = 32
For II = 1 To N2
NREF = 0
If (II = 1) Then GoTo 95
TF = TF1(1) + (II - 1) * DT1
GoTo 100
95 TF = 0
100 TW = TW1(II - 1)
If (II = 1) Then
TW = 0
End If
I = 1
'FIND FROUDE NUMBER AND LENGTH OF ZONE OF FLOW ESTABLISHMENT
GoTo 125
120 TW = TW + 0.01
125 REF3 = K * (H - Z)
REF4 = K * H
U = A * 9.81 * K * COSH(REF3) * Sin(-SIGMA * TW) / (COSH(REF4) * SIGMA)
W = -A * 9.81 * K * SINH(REF3) * Cos(-SIGMA * TW) / (COSH(REF4) * SIGMA)
RVX = DV * Cos(TH * PI / 180) - (U + UA)
RVY = DV * Sin(TH * PI / 180) - W
RV = (RVX * RVX + RVY * RVY) ^ 0.5
FO = RV / (9.81 * DIA) ^ 0.5
FO1 = RV / (9.81 * CT4 * DIA) ^ 0.5
If FO1 < 2 Then GoTo 130
GoTo 135
130 SI = 2.8 * FO1 ^ (2 / 3)
GoTo 150

66
135 If FO1 <= 3.2 Then GoTo 140
GoTo 145
140 SI = 0.113 * FO1 ^ 2 + 4
GoTo 150
145 SI = 5.6 * FO1 ^ 2 / (FO1 ^ 4 + 18) ^ 0.5
150 TM = TW + (SI * DIA / DV)
'FIND INITIAL CONDITIONS
BSI = 1 / Sqr(2)
BS = BSI
DRSI = CT4
DRS = DRSI
CI = 1
CM = CI
DRA = DRAI
TAI = TH * PI / 180
TA = TAI
Y = YI + (SI * DIA * Sin(TH * PI / 180))
X = XI + (SI * DIA * Cos(TH * PI / 180))
VMSI = 1
VMS = VMSI
J = 1
155 JJ = 1
C = 2
'FIND ENTRAINMENT COEFFICIENTS
RY = W + VMS * DV * Sin(TA)
RX = U + UA + VMS * DV * Cos(TA)
TH2 = ATan2(RY, RX)
TH3 = Abs(TA - TH2)
ALF1 = 0.0833 + 0.27 * (Abs(Tan(TH3)) / PI)
FCTR = Cos(TH3)
FO2 = FO * FCTR
'RUNGE KUTTA PROCEDURE
160 ZK(1, JJ) = DS * ((CT2 * DRS * Sin(TA) / (VMS * FO2 ^ 2)) - (2 * VMS * ALF1 / BS))
ZK(2, JJ) = DS * (2 * ALF1 - (BS * LA ^ 2 * DRS * Sin(TA) / (VMS ^ 2 * FO2 ^ 2)))
ZK(3, JJ) = DS * 2 * LA ^ 2 * DRS * Cos(TA) / (VMS ^ 2 * FO2 ^ 2)
ZK(4, JJ) = DS * (CT1 * Sin(TA) * DRA - 2 * ALF1 * DRS / BS)
ZK(5, JJ) = DS * Cos(TA)
ZK(6, JJ) = DS * Sin(TA)

67
ZK(7, JJ) = DS * (-2 * CM * ALF1 / BS)
If JJ = 3 Then GoTo 165
GoTo 170
165 C = 1
170 If (JJ = 4) Then GoTo 175
VMS = VMSI + ZK(1, JJ) / C
BS = BSI + ZK(2, JJ) / C
TA = TAI + ZK(3, JJ) / C
DRS = DRSI + ZK(4, JJ) / C
XS = (X / DIA) + ZK(5, JJ) / C
YS = (Y / DIA) + ZK(6, JJ) / C
JJ = JJ + 1
GoTo 160
175 VMS = VMSI + (ZK(1, 1) + 2 * ZK(1, 2) + 2 * ZK(1, 3) + ZK(1, 4)) / 6
BS = BSI + (ZK(2, 1) + 2 * ZK(2, 2) + 2 * ZK(2, 3) + ZK(2, 4)) / 6
TA = TAI + (ZK(3, 1) + 2 * ZK(3, 2) + 2 * ZK(3, 3) + ZK(3, 4)) / 6
DRS = DRSI + (ZK(4, 1) + 2 * ZK(4, 2) + 2 * ZK(4, 3) + ZK(4, 4)) / 6
DX1 = ((ZK(5, 1) + 2 * ZK(5, 2) + 2 * ZK(5, 3) + ZK(5, 4)) / 6) * DIA
DY1 = ((ZK(6, 1) + 2 * ZK(6, 2) + 2 * ZK(6, 3) + ZK(6, 4)) / 6) * DIA
CM = CI + (ZK(7, 1) + 2 * ZK(7, 2) + 2 * ZK(7, 3) + ZK(7, 4)) / 6
If DY1 <= 0 Or DRS <= 0 Then GoTo 180
GoTo 185
180 TF = TM
GoTo 210
185 X = X + DX1
Y = Y + DY1
DL = 1 / (CM * CO)
DT = DS * DIA / (VMS * RV)
TM = TM + DT
'FIND WAVE OR CURRENT AVECTION
U = A * 9.81 * K * COSH(K * Y) * Sin(K * (X - XI) - SIGMA * TM) / (SIGMA * COSH(K * H))
W = -A * 9.81 * K * SINH(K * Y) * Cos(K * (X - XI) - SIGMA * TM) / (SIGMA * COSH(K * H))
DX2 = (U + UA) * DT
DY2 = W * DT
X = X + DX2
Y = Y + DY2
'STOPPING PROCEDURE
ZREF = (2 * Z / 3) + A * Sin(K * (X - XI) - SIGMA * TM)
DREF = DY1 + DY2
If I > 1 And TM >= TF Then GoTo 210

68
If (DREF < 0 And Y > (Z / 3) And I = 1 And NREF = 0) Then
YREF = Y
DILREF = DL
NREF = 1
Else
End If
If Y >= ZREF And I = 1 Then GoTo 190
GoTo 195
190 If (II > 1 And (TM - TF) < 0) Then GoTo 120
TF = TM
TF1(II) = TF
TW1(II) = TW
GoTo 210
195 For N = 1 To 5
If Y >= H1(N) And Y <= H1(N + 1) Then GoTo 200
GoTo 205
200 DRA = ((D(N + 1) - D(N)) / (H1(N + 1) - H1(N))) * (DIA / DI)
205 Next
VMSI = VMS
BSI = BS
TAI = TA
DRSI = DRS
B = BS * DIA
VM = VMS * RV
CI = CM
J = J + 1
GoTo 155
210 RESULTS
TW = TW + DT1
If TW >= TF Then GoTo 225
If I = 1 Then GoTo 215
GoTo 220
215 DLT = DLT + DL
220 I = I + 1
GoTo 125
225 I = I + 1
INITIAL
Next II
'PRINT PLUME AVERAGE DILUTION
DLA = Sqr(2) * DLT / N2
Range("C26").Value = DLA

69
End Sub
Public Function F(L1, L0, H)
REF1 = 2 * PI * H / L1
F = L1 - L0 * TANH(REF1)
End Function
Public Function FPRIME(L1, L0, H)
REF1 = 2 * PI * H / L1
REF2 = -2 * PI * H / (L1 ^ 2)
FPRIME = 1 - L0 * SECH(REF1) ^ 2 * REF2
End Function
Public Function TANH(X)
TANH = (Exp(X) - Exp(-X)) / (Exp(X) + Exp(-X))
End Function
Public Function SECH(X)
SECH = 2 / (Exp(X) + Exp(-X))
End Function
Public Function COSH(X)
COSH = (Exp(X) + Exp(-X)) / 2
End Function
Public Function SINH(X)
SINH = (Exp(X) - Exp(-X)) / 2
End Function
Function ATan2(Y, X)
Select Case X

70
Case Is > 0
ATan2 = Atn(Y / X)
Case Is < 0
ATan2 = Atn(Y / X) + PI * Sgn(Y)
If Y = 0 Then ATan2 = ATan2 + PI
Case Is = 0
ATan2 = 0.5 * PI * Sgn(Y)
End Select
End Function
Public Sub RESULTS()
If DRS < 0 Then GoTo 230
GoTo 235
230 DRS = 0
235 VM = VMS * RV
B = BS * DIA
WPL = 2.828 * B
DR = DRS * DI
TA1 = TH2
TM1 = TF
'PRINT SNAPSHOT NUMBER
If I = 1 Then
INPUTROW = ("B" + Str(KK + I))
Range(INPUTROW).Value = "SNAPSHOT # " & II
End If
'PLUME POSITION ALONG THE X AXIS
INPUTROW = ("C" + Str(KK + I))
Range(INPUTROW).Value = X
'PLUME POSITION ALONG THE Y AXIS
INPUTROW = ("D" + Str(KK + I))
Range(INPUTROW).Value = Y
'PLUME CENTERLINE VELOCITY
INPUTROW = ("E" + Str(KK + I))
Range(INPUTROW).Value = VM

71
'PLUME RADIAL WIDTH
INPUTROW = ("F" + Str(KK + I))
Range(INPUTROW).Value = WPL
'PLUME CENTERLINE ANGLE
INPUTROW = ("G" + Str(KK + I))
Range(INPUTROW).Value = (TH2 * 180 / PI)
'PLUME CENTERLINE DENSITY DIFFERENCE
INPUTROW = ("H" + Str(KK + I))
Range(INPUTROW).Value = DR
'PLUME CENTERLINE DILUTION
INPUTROW = ("I" + Str(KK + I))
Range(INPUTROW).Value = DL
'PLUME SNAPSHOT TIME
INPUTROW = ("J" + Str(KK + I))
Range(INPUTROW).Value = TM1
'PLUME LEFT SIDE POSITION ALONG X AXIS
XLEFT = X - (0.5 * WPL * Sin(TH2))
INPUTROW = ("K" + Str(KK + I))
Range(INPUTROW).Value = XLEFT
'PLUME RIGHT SIDE POSITION ALONG X AXIS
XRIGHT = X + (0.5 * WPL * Sin(TH2))
INPUTROW = ("M" + Str(KK + I))
Range(INPUTROW).Value = XRIGHT
'PLUME LEFT SIDE POSITION ALONG Y AXIS
YLEFT = Y + (0.5 * WPL * Cos(TH2))
INPUTROW = ("L" + Str(KK + I))
Range(INPUTROW).Value = YLEFT
'PLUME RIGHT SIDE POSITION ALONG Y AXIS

72
YRIGHT = Y - (0.5 * WPL * Cos(TH2))
INPUTROW = ("N" + Str(KK + I))
Range(INPUTROW).Value = YRIGHT
End Sub
Public Sub INITIAL()
'PLUME POSITION ALONG THE X AXIS
INPUTROW = ("C" + Str(KK + I))
Range(INPUTROW).Value = XI
'PLUME POSITION ALONG THE Y AXIS
INPUTROW = ("D" + Str(KK + I))
Range(INPUTROW).Value = YI
'PLUME CENTERLINE VELOCITY
INPUTROW = ("E" + Str(KK + I))
Range(INPUTROW).Value = DV
'PLUME RADIAL WIDTH
INPUTROW = ("F" + Str(KK + I))
Range(INPUTROW).Value = DIA
'PLUME CENTERLINE ANGLE
INPUTROW = ("G" + Str(KK + I))
Range(INPUTROW).Value = TH
'PLUME CENTERLINE DENSITY DIFFERENCE
INPUTROW = ("H" + Str(KK + I))
Range(INPUTROW).Value = (DI - DD)
'PLUME CENTERLINE DILUTION
INPUTROW = ("I" + Str(KK + I))
Range(INPUTROW).Value = (DI - DD) / DI

73
'PLUME SNAPSHOT TIME
INPUTROW = ("J" + Str(KK + I))
Range(INPUTROW).Value = TM1
'PLUME LEFT SIDE POSITION ALONG X AXIS
XLEFT = XI
INPUTROW = ("K" + Str(KK + I))
Range(INPUTROW).Value = XLEFT
'PLUME RIGHT SIDE POSITION ALONG X AXIS
XRIGHT = XI
INPUTROW = ("M" + Str(KK + I))
Range(INPUTROW).Value = XRIGHT
'PLUME LEFT SIDE POSITION ALONG Y AXIS
YLEFT = YI + (0.5 * DIA)
INPUTROW = ("L" + Str(KK + I))
Range(INPUTROW).Value = YLEFT
'PLUME RIGHT SIDE POSITION ALONG Y AXIS
YRIGHT = YI - (0.5 * DIA)
INPUTROW = ("N" + Str(KK + I))
Range(INPUTROW).Value = YRIGHT
'RESET BEGIN OF ROW FOR NEXT SNAPSHOT
KK = KK + I + 1
End Sub
Sub RESET_TABLE()
'
' RESET_TABLE Macro
'
'
Range("C33:N1000").Select
Selection.ClearContents
End Sub

79
A-4 BIBLIOGRAPHY
1. Abraham, G., “The flow of Round Buoyant Jets Issuing Vertically Into Ambient Fluid
Flowing in a Horizontal Direction,” Advances in Water Pollution Research. 5th paper
III-15, 7 pp. (1970).
2. Alam, M.Z., Harleman, R.F., and Colonell, M.J., “Evaluation of Selected Initial Dilution
Models,” Journal of the Environmental Engineering Division, ASCE, Vol. 108, No.EE1,
pp.159-186. (1982).
3. Albertson, M.L., Dai, Y,B., Jensen, R.A., and Rouse, H., “Diffusion of Submerged Jets,”
Transactions, ASCE, Vol. 115, pp.639-697. (1950).
4. Airy, G.B., “On Tides and Waves,” Enciclopaedia Metropolitana. (1845).
5. Baumgartner, D.J., Trent, D.S., and Byran, K.V., “User Guide and Documentation for
Outfall Plume Model,” Working paper No. 80, U.S. Environmental Protection Agency,
Pacific Northwest Lab, Corvallis, Oregon. (1971).
6. Buhler, J., “Model Studies of Multiport Outfalls in Unstratified, Stagnant or Flowing
Receiving Water,” Ph.D. Thesis, University of California at Berkeley. (1974).
7. State of California, Water Resources Control Board, “Water Quality Control Plan for
Ocean Waters of California,” 15 pp. (1972).
8. Chin, D.A., “A Study of the Influence of Surface Waves on Ocean Outfall Dilution,”
Technical Report No. CEN-87-1, Department of Civil and Architectural Engineering,
University of Miami, Coral gables, Florida. (1986).

80
9. Ditmars, J.D., “Computer Program for Round Jets into Stratified or Flowing Ambient
Fluids,” W.M. Keck Laboratory of Hydraulics and Water Resources, California Institute
of Technology, Pasadena, California. (1969). 27 pp.
10. “Rules and Regulations of the Environmental Protection Agency,” Federal Register,
Vol. 44, No. 117, Washington, D.C.. (1979).
11. Fan, L.N., “Turbulent Buoyant Jets into Stratified or Flowing Ambient Fluids,” Report
No. KH-R-15, W.M. Keck Laboratory of Hydraulics and Water Resources, California
Institute of Technology, Pasadena, California. (1967). 196 pp.
12. Fan, L.N., and Brooks, N.H., “Numerical Solution of Turbulent Buoyant Jet Problems,”
Report No. KH-R-18, W.M. Keck Laboratory of Hydraulics and Water Resources,
California Institute of Technology, Pasadena, California. (1969).
13. Fisher, H.B., List, E.J., Koh, R.C.Y., Imberger, J., and Brooks, N.H., “Mixing in Inland and
Coastal Waters,” Academic Press, New-York. (1979).
14. Ger, A.H., “Wave Effects on Submerged Buoyant Jets,” Proceeding of the Eight
Congress, International Association of Hydraulic Research, pp. 295-300. (1979).
15. Hendricks, T.J., “In Situ Measurements of Initial Dilution,” Southern California Coastal
Water Research Project, Annual Report. (1977).
16. Kannberg, L.D., and Davis, L.R., “An Experimental/Analytical Investigation of Deep
Submerged Multiple Buoyant Jets,” Working Paper No. 600/3-76-101, U.S.
Environmental Protection Agency, Corvallis Environmental Research Laboratory,
Corvallis, Oregon. (1976).

81
17. Koh, R.C.Y., and Brooks, N.H., “Fluid Mechanics of Wastewater Disposal in the Ocean,”
Annual Review of Fluid Mechanics, Vol. 7, pp. 187-211. (1975).
18. List, E.J., and Imberger, J., “Turbulent Entrainment in Buoyant Jets and Plumes,”
Journal of the Hydraulics Division, ASCE 99, pp. 1461-1474. (1973).
19. List, E.J., “Mechanics of Turbulent Buoyant Jets and Plumes,” HMT, The Science and
Application of Heat and Mass Transfer, Wolfgang Rodi, Ed., Vol. 6, Pergamon Press,
London. (1982).
20. Roberts, P.J., “Dispersion of Buoyant wastewater Discharged from Outfall Diffusers of
Finite Length,” Report No. KH-R-35, California Institute of Technology, Pasadena,
California. (1977).
21. Roberts, P.J., “Evaluation of Selected Initial Dilution Models: Discussion,” Journal of
Environmental Engineering, ASCE Vol. 109, No. 5, pp. 1218-1222. (1983)
22. Rouse, H., Yih, C.S., and Humphreys, H.W., “Gravitational Convection from a Boundary
Source,” Tellus 4, pp. 201-210. (1952).
23. Shuto, N., and Ti, L.H., “Wave Effect on Buoyant Plumes,” Proceedings of the
Fourteenth Coastal Engineering Conference, pp. 2199-2209. (1974).
24. Winiarski, L.D., and Frick, W.E., “Cooling Tower Plume Model,” Report No. EPA-600/3-
76-100, Environmental Protection Agency, Corvallis, Oregon. (1976)
25. Wright, S.J., “Outfall Diffuser Behavior in Stratified Ambient Fluid,” Journal of
Hydraulics Division, ASCE, Vol. 108, No. HY4. (1982).

82
VITA
Francis Mitchell was born in Port-au-Prince, Haiti on November 8, 1960. His parents are Winchell
Mitchell and Claudette Mitchell. He received his elementary education in Saint Louis de
Conzague school, Delmas. In January 1981 he entered the college of Arts and Sciences of Miami-
Dade Community College South Campus from which he transferred to the School of Engineering
of the University of Miami and graduated with the B.S. degree in August 1985.
In January 1986 he was admitted to the Graduate School of the University of Miami. He was
granted the degree Master of Science in May 1987.
Permanent Address: 8365 SW 112th Street, Miami, Florida 33156

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