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

ii

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

iii

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

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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

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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

1

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.

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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

4

Figure 2. 1 Mixing Zone

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, 𝑣𝑑 , 𝑈𝐴, 𝑔𝑜 , 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

𝑔𝑜 = ∆𝜌𝑜

𝜌𝑜𝑔 (2.2)

Where ∆𝜌𝑜 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.

7

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

4𝐷2𝑣𝑑 (2.3)

The momentum flux

𝑀 = 𝑄 𝑣𝑑 (2.4)

The buoyancy flux

𝐵 = 𝑄 𝑔𝑜 (2.5)

Two length scales characteristics of jets and plumes can be deduced from the volume,

momentum, and buoyancy fluxes. They are:

𝐿𝑄 = 𝑄

𝑀1/2 = 𝐴1/2 (2.6)

And

𝐿𝑀 = 𝑀3/4

𝐵1/2 (2.7)

Where A is the initial cross sectional area of the port. The length scale 𝐿𝑄 scales the length over

which the port geometry influences the effluent behavior, and 𝐿𝑀 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

𝐿𝑀]

5/3

[𝐿𝑀

𝐿𝑄] (2.8)

And for the linearly stratified case

𝑆𝐴 = .10 [𝐻𝑚𝑎𝑥

𝐿𝑀]

5/3

[𝐿𝑀

𝐿𝑄] (2.9)

Where 𝐻𝑚𝑎𝑥 is the maximum height of rise defined by:

𝐻𝑚𝑎𝑥 = 3.98 [𝑄𝑔𝑜]1/4 [−𝑔

𝜌𝑜

𝑑𝜌𝑎

𝑑𝑧]

−3/8

(2.10)

Where 𝑑𝜌𝑎

𝑑𝑧⁄ is the ambient density gradient.

Finally, for the case of a line source discharging into a stagnant-unstratified environment.

𝑆𝐴 = .537 𝑔𝑜

1/3𝐻

𝑞2/3 (2.11)

And for the linearly stratified case

𝑆𝐴 = .438 𝑔𝑜

1/3𝐻𝑚𝑎𝑥

𝑞2/3 (2.12)

Where

𝐻𝑚𝑎𝑥 = 2.84 [𝑞𝑔𝑜]1/3 [−𝑔

𝜌𝑜

𝑑𝜌𝑎

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

9

necessary due to such ambient conditions as non-linear density stratification and currents. Also,

the pure plume assumption (H > 𝐿𝑀) 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) = ∆𝜌𝑜(𝑠)𝑒𝑥𝑝 [−𝑟2

𝜆2𝑏2] (2.15)

Where r is the radial coordinate, b the characteristic jet width, Δ𝜌𝑜 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′ = 𝜌𝑜 ∫ 𝑣2𝐴

(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)𝑑𝐴

𝐴 (2.22)

And the conservation of density deficiency

𝑑𝐵′

𝑑𝑠=

𝑑𝜌𝐴

𝑑𝑠𝑄′ (2.23)

Combining equations 2.14 to 2.23 yields:

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𝑑𝑣𝑜

𝑑𝑠=

2𝑔𝜆2

𝑣𝑜

Δ𝜌𝑜

𝜌𝑜𝑠𝑖𝑛𝜃 −

2𝑣𝑜𝛼

𝑏 (2.24)

𝑑𝑏

𝑑𝑠= 2𝛼 −

𝑏

𝑣𝑜2 𝑔𝜋2 ∆𝜌𝑜

𝜌𝑜𝑠𝑖𝑛𝜃 (2.25)

𝑑∆𝜌𝑜

𝑑𝑠=

1+𝜆2

𝜆2 𝑠𝑖𝑛𝜃 𝑑𝜌∞

𝑑𝑧− 2𝛼

𝑑∆𝜌𝑜

𝑏 (2.26)

𝑑𝜃

𝑑𝑠=

2𝑔𝜆2

𝑣𝑜2

Δ𝜌𝑜

𝜌𝑜 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, 𝑏𝑜 is given by

𝑏𝑜 = 𝐷

√2 (2.30)

And from the continuity of pollutant concentration

𝑐𝑜 = 1+𝜆2

2𝜆2𝑐𝑑 (2.31)

And

Δ𝜌𝑜 =1+𝜆2

2𝜆2[Δ𝜌]𝑜 (2.32)

13

Where [Δ𝜌]𝑜 is the density deficiency of the effluent and 𝑐𝑑 is the discharge concentration.

Laboratory experiments on pure plume yield 𝛼 = 𝛼𝑝 = .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

2𝑏2𝑣𝑜

2 (2.35)

𝜇 is the volume flux given by

𝜇 = 𝜋𝑏2𝑣𝑜 (2.36)

And 𝛽 is the local buoyancy flux given by

𝛽 = 𝜋𝜆2

1+𝜆2 [Δ𝜌𝑜

𝜌𝑜]

2

𝑏2𝑣0𝑔 (2.37)

14

Continuing this study, List (1982) suggested a new version for the buoyant jet entrainment

coefficient.

𝛼 = 𝛼𝑝 𝑒𝑥𝑝 [𝑙𝑛 [𝛼𝑝

𝛼𝑗] [

𝑅

𝑅𝑝]

2

] (2.38)

Where 𝑅𝑝, the plume Richardson number, is given by

𝑅𝑝 = .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) = 𝑈𝐴𝑐𝑜𝑠𝜃 + 𝑣𝑜(𝑠) 𝑒𝑥𝑝 [−𝑟2

𝑏2] (2.40)

Δ𝜌(𝑠, 𝑟) = Δ𝜌𝑜(𝑠)𝑒𝑥𝑝 [−𝑟2

𝜆2𝑏2] (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= 𝜌𝑎 2𝜋𝛼𝑏𝑈𝐴 |𝑣𝑜 − 𝑈𝐴| + 𝐹𝐷 𝑠𝑖𝑛𝜃 (2.43)

Where 𝐹𝐷 is the drag force assumed to be

15

𝐹𝐷 = .5 𝐶𝐷 𝜌𝑎 2√2 𝑏 𝑈𝐴2 𝑠𝑖𝑛2 𝜃 (2.44)

𝑑(𝑀′𝑠𝑖𝑛𝜃)

𝑑𝑠= ∫ 𝑔

𝐴∆𝜌(𝑠, 𝑟)𝑑𝐴 − 𝐹𝐷𝑐𝑜𝑠𝜃 (2.45)

𝑑𝐵′

𝑑𝑠=

𝑑𝜌𝑎

𝑑𝑠𝑄′ (2.46)

𝑑𝑐

𝑑𝑠𝑄′ = 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

𝐿𝑀= .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

𝑣𝑑 (2.51)

The experimental results are as follow.

𝐻

𝐿𝑀= .49 𝑡𝑜 2.1

𝐿𝑀

𝐿𝑄= 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 𝑐𝑜𝑠𝛾

𝐴 [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

2𝐴

𝑐𝑜𝑠2𝛾 𝑑𝐴 𝑐𝑜𝑠𝜃 = 0 (3.3)

3. Conservation of 𝑧 momentum

𝑑

𝑑𝑠∫ 𝑣𝑜

2𝐴

𝑐𝑜𝑠2𝛾 𝑑𝐴 𝑠𝑖𝑛𝜃 = ∫ 𝑔 ∆𝜌(𝑠, 𝑟) 𝑑𝐴𝐴

(3.4)

4. Conservation of density deficiency

𝑑

𝑑𝑠∫ ∆𝜌 𝑣𝑜 𝑐𝑜𝑠𝛾 𝑑𝐴 =

𝑑𝜌𝑎

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 = 𝑐𝑜 𝑒𝑥𝑝 [− 𝑟2

𝜆2𝑏2] (3.7)

Where 𝑐𝑜 is the centerline concentration.

Combining equation 3.2 to 3.7 after further manipulation give:

𝑑𝑣𝑜

𝑑𝑠=

2 𝑔 𝜋2

𝑣𝑜𝑐𝑜𝑠2𝛾 Δ𝜌𝑜

𝜌𝑜𝑠𝑖𝑛𝜃 −

2𝑣𝑜

𝑏[𝛼 +

𝛽

𝜋𝑡𝑎𝑛𝛾] (3.8)

𝑑𝑏

𝑑𝑠= 2 [𝛼 +

𝛽

𝜋𝑡𝑎𝑛𝛾] −

𝑏 𝑔 𝜋2

𝑣𝑜2𝑐𝑜𝑠2𝛾

Δ𝜌𝑜

𝜌𝑜 𝑠𝑖𝑛𝜃 (3.9)

𝑑𝜃

𝑑𝑠=

2 𝑔 𝜆2

𝑣𝑜2𝑐𝑜𝑠2𝛾

Δ𝜌𝑜

𝜌𝑜 𝑐𝑜𝑠𝜃 (3.10)

𝑑Δ𝜌𝑜

𝑑𝑠=

1+𝜆2

𝜆2 𝑠𝑖𝑛𝜃 𝑑𝜌 ∞

𝑑𝑧− 2 [𝛼 +

𝛽

𝜋𝑡𝑎𝑛𝛾]

Δ𝜌𝑜

𝑏 (3.11)

28

𝑑𝑐𝑜

𝑑𝑠= −

2𝑐𝑜

𝑏 𝑑𝑏

𝑑𝑠−

𝑐𝑜

𝑣𝑜 𝑑𝑣𝑜

𝑑𝑠 (3.12)

𝑑𝑧

𝑑𝑠= 𝑠𝑖𝑛𝜃 (3.13)

𝑑𝑥

𝑑𝑠= 𝑐𝑜𝑠𝜃 (3.14)

Written in non-dimensional form

𝑑𝑣∗

𝑑𝑠∗=

2𝜆2𝑠𝑖𝑛𝜃 Δ𝜌∗

𝑣∗𝐹𝑜2 −

2𝑣∗

𝑏∗ [𝛼 +

𝛽

𝜋 𝑡𝑎𝑛𝛾] (3.15)

𝑑𝑏∗

𝑑𝑠∗ = 2 [𝛼 + 𝛽

𝜋 𝑡𝑎𝑛𝛾] −

𝜆2𝑏∗

[𝑣∗𝐹𝑜∗]2 Δ𝜌𝑜

∗𝑠𝑖𝑛𝜃 (3.16)

𝑑𝜃∗

𝑑𝑠∗ = 2𝜆2

[𝑣∗𝐹𝑜∗]2 Δ𝜌𝑜

∗𝑐𝑜𝑠𝜃 (3.17)

𝑑Δ𝜌∗

𝑑𝑠∗ = 1+𝜆2

𝜆2 𝑑𝜌∞

∗

𝑑𝑧∗ 𝑠𝑖𝑛𝜃 − 2

𝑏∗ [𝛼 + 𝛽

𝜋 𝑡𝑎𝑛𝛾] Δ𝜌𝑜

∗ (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∗ = 𝑣𝑜 𝑐𝑜𝑠𝛾

𝑣𝑑 (3.22)

𝑠∗ = 𝑠

𝐷 (3.23)

Δ𝜌𝑜∗ =

Δ𝜌𝑜

𝜌𝑜 (3.24)

29

𝑏∗ = 𝑏

𝐷 (3.25)

𝑐∗ = 𝑐

𝑐𝑑 (3.26)

Where 𝑐𝑑 is the concentration at the discharge level

𝐹𝑜 = 𝑣𝑑

√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+ 𝑣𝑜 𝑠𝑖𝑛𝜃

𝑢+ 𝑈𝐴+ 𝑣𝑜 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)

𝑆𝑒∗ = 2.8 𝐹2/3 𝐹 < 2 (3.31)

𝑆𝑒∗ = .113 𝐹2 + 4 2 ≤ 𝐹 ≤ 3.2 (3.32)

𝑆𝑒∗ =

5.6 𝐹2

√𝐹4+18 𝐹 > 3.2 (3.33)

Where 𝐹 is the relative densymetric Froude Number defined by:

30

𝐹 = 𝑣𝑑− 𝑈𝐴𝑠𝑐𝑜𝑠𝜃1

√Δ𝜌𝑜 𝑔 𝐷

𝜌𝑜

(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 𝑈𝐴𝑠𝑐𝑜𝑠𝜃1. The initial conditions

described by 𝑣𝑜∗, 𝑐𝑜

∗ , Δ𝜌𝑜∗ , 𝑏𝑜

∗ are:

𝑣𝑜∗ = 1 (3.35)

𝑏𝑜∗ =

1

√2 (3.36)

𝑐𝑜∗ =

1+𝜆2

2𝜆2 (3.37)

Δ𝜌𝑜∗ = 𝑐𝑜

∗ (Δ𝜌)𝑑

𝜌𝑜 (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, Δ𝜌𝑜 , 𝑐𝑜,

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, Δ𝜌𝑜 , 𝑐𝑜 , 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

35

Figure 3. 6 Flow Chart

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 = [∑ 𝑆𝑖𝑁𝑖=1 ]

√2

𝑁 (3.40)

Where 𝑆𝑖, the centerline dilution for a snapshot i, is given by

𝑆𝑖 = 𝑐𝑑

𝑐𝑜 (3.41)

Where 𝑐𝑜 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/cm3sec. 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 = 𝑑𝜌𝑎

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

𝑑𝑋𝑖

𝑑𝑠= 𝑓𝑖 (𝑋1, 𝑋2, … … … … … , 𝑋𝑛) 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, … … … … … , 𝑋𝑛) (A-1.3)

𝐾𝑖2 = ∆𝑠 𝑓𝑖 (𝑋1 + 1

2𝐾11, 𝑋2 +

1

2𝐾21, … … … 𝑋𝑛 +

1

2𝐾𝑛1) (A-1.4)

𝐾𝑖3 = ∆𝑠 𝑓𝑖 (𝑋1 + 1

2𝐾12, 𝑋2 +

1

2𝐾22, … … … 𝑋𝑛 +

1

2𝐾𝑛2) (A-1.5)

𝐾𝑖4 = ∆𝑠 𝑓𝑖 (𝑋1 + 𝐾13, 𝑋2 + 𝐾23, … … … 𝑋𝑛 + 𝐾𝑛3) (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)) INPUTROW = Replace(INPUTROW, " ", "") 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)) INPUTROW = Replace(INPUTROW, " ", "") Range(INPUTROW).Value = "SNAPSHOT # " & II End If

'PLUME POSITION ALONG THE X AXIS INPUTROW = ("C" + Str(KK + I)) INPUTROW = Replace(INPUTROW, " ", "") Range(INPUTROW).Value = X

'PLUME POSITION ALONG THE Y AXIS INPUTROW = ("D" + Str(KK + I)) INPUTROW = Replace(INPUTROW, " ", "") Range(INPUTROW).Value = Y

'PLUME CENTERLINE VELOCITY INPUTROW = ("E" + Str(KK + I)) INPUTROW = Replace(INPUTROW, " ", "") Range(INPUTROW).Value = VM

71

'PLUME RADIAL WIDTH INPUTROW = ("F" + Str(KK + I)) INPUTROW = Replace(INPUTROW, " ", "") Range(INPUTROW).Value = WPL

'PLUME CENTERLINE ANGLE INPUTROW = ("G" + Str(KK + I)) INPUTROW = Replace(INPUTROW, " ", "") Range(INPUTROW).Value = (TH2 * 180 / PI)

'PLUME CENTERLINE DENSITY DIFFERENCE INPUTROW = ("H" + Str(KK + I)) INPUTROW = Replace(INPUTROW, " ", "") Range(INPUTROW).Value = DR

'PLUME CENTERLINE DILUTION INPUTROW = ("I" + Str(KK + I)) INPUTROW = Replace(INPUTROW, " ", "") Range(INPUTROW).Value = DL

'PLUME SNAPSHOT TIME INPUTROW = ("J" + Str(KK + I)) INPUTROW = Replace(INPUTROW, " ", "") Range(INPUTROW).Value = TM1

'PLUME LEFT SIDE POSITION ALONG X AXIS XLEFT = X - (0.5 * WPL * Sin(TH2)) INPUTROW = ("K" + Str(KK + I)) INPUTROW = Replace(INPUTROW, " ", "") Range(INPUTROW).Value = XLEFT

'PLUME RIGHT SIDE POSITION ALONG X AXIS XRIGHT = X + (0.5 * WPL * Sin(TH2)) INPUTROW = ("M" + Str(KK + I)) INPUTROW = Replace(INPUTROW, " ", "") Range(INPUTROW).Value = XRIGHT

'PLUME LEFT SIDE POSITION ALONG Y AXIS YLEFT = Y + (0.5 * WPL * Cos(TH2)) INPUTROW = ("L" + Str(KK + I)) INPUTROW = Replace(INPUTROW, " ", "") Range(INPUTROW).Value = YLEFT

'PLUME RIGHT SIDE POSITION ALONG Y AXIS

72

YRIGHT = Y - (0.5 * WPL * Cos(TH2)) INPUTROW = ("N" + Str(KK + I)) INPUTROW = Replace(INPUTROW, " ", "") Range(INPUTROW).Value = YRIGHT

End Sub

Public Sub INITIAL()

'PLUME POSITION ALONG THE X AXIS INPUTROW = ("C" + Str(KK + I)) INPUTROW = Replace(INPUTROW, " ", "") Range(INPUTROW).Value = XI

'PLUME POSITION ALONG THE Y AXIS INPUTROW = ("D" + Str(KK + I)) INPUTROW = Replace(INPUTROW, " ", "") Range(INPUTROW).Value = YI

'PLUME CENTERLINE VELOCITY INPUTROW = ("E" + Str(KK + I)) INPUTROW = Replace(INPUTROW, " ", "") Range(INPUTROW).Value = DV

'PLUME RADIAL WIDTH INPUTROW = ("F" + Str(KK + I)) INPUTROW = Replace(INPUTROW, " ", "") Range(INPUTROW).Value = DIA

'PLUME CENTERLINE ANGLE INPUTROW = ("G" + Str(KK + I)) INPUTROW = Replace(INPUTROW, " ", "") Range(INPUTROW).Value = TH

'PLUME CENTERLINE DENSITY DIFFERENCE INPUTROW = ("H" + Str(KK + I)) INPUTROW = Replace(INPUTROW, " ", "") Range(INPUTROW).Value = (DI - DD)

'PLUME CENTERLINE DILUTION INPUTROW = ("I" + Str(KK + I)) INPUTROW = Replace(INPUTROW, " ", "") Range(INPUTROW).Value = (DI - DD) / DI

73

'PLUME SNAPSHOT TIME INPUTROW = ("J" + Str(KK + I)) INPUTROW = Replace(INPUTROW, " ", "") Range(INPUTROW).Value = TM1

'PLUME LEFT SIDE POSITION ALONG X AXIS XLEFT = XI INPUTROW = ("K" + Str(KK + I)) INPUTROW = Replace(INPUTROW, " ", "") Range(INPUTROW).Value = XLEFT

'PLUME RIGHT SIDE POSITION ALONG X AXIS XRIGHT = XI INPUTROW = ("M" + Str(KK + I)) INPUTROW = Replace(INPUTROW, " ", "") Range(INPUTROW).Value = XRIGHT

'PLUME LEFT SIDE POSITION ALONG Y AXIS YLEFT = YI + (0.5 * DIA) INPUTROW = ("L" + Str(KK + I)) INPUTROW = Replace(INPUTROW, " ", "") Range(INPUTROW).Value = YLEFT

'PLUME RIGHT SIDE POSITION ALONG Y AXIS YRIGHT = YI - (0.5 * DIA) INPUTROW = ("N" + Str(KK + I)) INPUTROW = Replace(INPUTROW, " ", "") 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

74

A-3 SAMPLE RUN

75

76

77

78

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

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Pacific Northwest Lab, Corvallis, Oregon. (1971).

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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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