AN ABSTRTOT CF THE THESIS OF
RAYMOND GENE CAVOLA for the degree of MASTER
in MECHANICAL ENGINEERING
Title:
presented on
OF SCIENCE
APRIL 27, 1982
AN EXPERIMENTAL/ANALYTICAL INVESTIGATION OF
NEGATIVELY BUOYANT JETS DISCHARGED VERTICALLY UPWARD
INTO A CROSSFLOW CURRENT
Abstract approved:
Redacted for Privacy
Dr. Lorin R. Davis
An experimental and analytical study of negatively buoyant jets
discharged vertically upward into a crcssf low current was conducted
to obtain information on the fate of oil well produced water.
This
information could be used to validate and calibrate the short and
intermediate term portions of a computer model used to predict the
disposal. The experirriental results include the dilution arid plume
width, measured at selected incremental distances downstream ci the
discharge.
Independent parameters varied in this investigation were
the discharge deosimetric Froude Number, and the ambient to discharge
velocity ratio. Results indicate that the Frcucae NurrLber has the
greatest effect on dilution; decreasing the Froude Number increases
both dilution and plume width. Altering the velocity ratio has
little effect on dilution, but does affect plume width.
The analytical portion of this study includes a brief presenta-tion of the background of olume modeling, the mathematical develop-
ment of the produced water comouter model, and an outline of the pro-
cedure for tuning the coefficteucs within the model uslag the experi-
mental results of this investigation.
It is expected that the results presented here are sufficient for
validating and calibrating the initial phases of the computer model.
Upon calibration and validation, the model can be used as a predictive
tool for determining the fate of negatively buoyant jets.
AN EXPERIMENTAL/ANALYTICAL INVESTIGATION OF
NEGATIVELY BUOYANT JETS DISCHARGED VERTICALLY
UPWARD INTO A CROSSFLOW CURRENT
by
Raymond Gene Cavola
A THESIS
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
MASTER OF SCIENCE
Comnieted April 27, 1982
Coraip.encement June 1982
APPROVED:
Redacted for Privacy
Professor of Mechanical Engineering in charge of major
Redacted for Privacy
Head o/iePartment of Mechanica/Engineering
Redacted for Privacy
Dean of Graduate/Sichool
Date thesis is presented:
APRIL 27, 1982
Typed by Sandy Orr for RAYMOND GENE CAVOLA
Ac:oowLEDGEMENTs
The successful completion of this project would not have been
possible without the assistance of several people.
I owe many thanks
to:
Dr. Lorin Davis, my major professor, who unselfishly provided guidance, encouragement, and timely comments and
criticisms throughout the course of this work;
Hassan Bahrami, who provided assistance in conducting the
experiments; Jack Kellogg, for his help with the shop and
tools while I was constructing the apparatus;
A special note of thanks to my lovely wife, Nina, for her
encouragement, and many sacrifices, made so that this
thesis and degree could be
completed;
Sandy Orr, for typing this
document; and finally to Exxon
Production Research Company and the Offshore Operators
Committee, whose financial assistance made this study
possible.
TABLE OF CONTENTS
PAGE
INTRODUCTION
1
ANALYTICAL MODELING
3
Introduction
3
Model Development
6
Tuning the Mode1
25
EXPERIMENTAL MODELING
27
Modeling Parameters
27
Apparatus and Data Acquisition
30
Data Treatment
39
Uncertainty Analysis
46
Results
48
CONCLUSIONS
68
REFERENCES
69
APPENDICES
Appendix A - Graphs
70
Appendix B - Experimental Data
36
LIST OF FIGURES
FIGURE
1
2
PAGE
Idealized jet discharge described by mathematical
model. Cross sections are shown at three stages
of plume development
5
Coordinate system and definition sketch for a
round jet
7
3
Geometry of the collapsing jet plume
14
4
Plane of traverse of the probe
29
5
Schematic of the experimental apparatus and
electronic instrumentation
21
6
Schematic of the constant
reservoir
head
salt water
33
7
Enlarged view of conductivity probe
34
8
Typical calibration curve for the conductivity probe.
37
9
Section view showing lifter plate used for measuring
conductivity on the false bottom
10
Typical Visicorder print of conductivity and position
signals
41
11
Flowchart showing data treatment process
12
Example of typical excess salinity ratio data, as
a function of X/D, and its' representative curve.
.
44
Example of typical excess salinity ratio data, as
a function of Y/D, and its' representative curve.
.
45
13
14
A repeat of Figure 12 with 90% confidence interval
included to indicate the quality of the experimental
data
i17
Excess salinity ratio as a function of velocity
ratio
49
16
Excess salinity ratio as a function of Froude Number
51
17
Plume half-width as a function of X/D and R for
15
18
FR05
52
Plume half-width as a function of X/D and R for
FR1O
53
LIST CF FIGURES (continued)
FIGURE
19
PAGE
Plume half-width as a function of X/D and R for
FR15
54
20
Comparison of regression model with data averages
curve for the conditions:
FR = 0.5, R = 0 5
21
Comparison of regression model with data averages
curve for the conditions:
FR = 0.5, F = 0.99
.
57
Comparison of regression model with data averages
curve for the conditions: FR = 0.5, R = 1.47
.
58
Comparison of regression model with data averages
curve for the conditions: FR = 1.01, F = 0.48.
.
59
Comparison of regression model with data averages
curve for the conditions:
FR = 1.01, F = 1.01.
-
60
Comparison of regre.sion model with data averages
curve for the conditions:
FR = 1.01, F = 1.48.
.
.
.
22
.
.
23
.
24
.
25
.
.
61
26
Comparison of regression model with data averages
curve for the conditions:
FR = 1.44, F = 0.52.....62
27
Comparison of regression model with data averages
curve for the conditions: FR = 1.44, R = 1.02,
.
.
Comparison of regression model with data averages
curve for the conditions: FR = 1.44, R = 1.49.
.
28
.
29
30
31
63
*
64
Velocity distribution in the boundary layer above
the false bottom for R = 0 5
65
Velocity distribution in the boundary layer above
the false bottom for F = 1 0
S6
Velocity distribution in the boundary layer above
the false bottom for R = 1.5 .........67
A-i
A- 2
A- 3
Plume centerline excess salinity ratio versus X/D,
for FR = 0.50, P. = 0.50, and Y/D
0 1
71
Plume centerline excess salinity rario versus X/D,
for FR = 0.S0, F = 0.99, and. Y/D
0 1
72
Plume centerline excess salinity ratio versus X/D,
for FR = 0,50, P. = 1.47, and Y/D
0 1
73
LIST OF FIGURES (continued)
PAGE
FIGURE
A-4
Plume centerline excess salinity ratio versus X/D,
0.1
for FR = 1.01, R = 0.48, Y/D = 0, and Y/D
.
Plume centerline excess salinity ratio versus Y/D,
for FR = 1.01, R = 0.48, and X/D = 2.5, 5 and 10.
.
.
A-5
A-6
Plume centerline excess salinity ratio versus X/D,
0.1
for FR = 1.01, R = 1.01, Y/D = 0, and Y/D
A-8
.
.
Plume centerline excess salinity ratio versus X/D,
0.1
for FR = 1.01, R = 1.48, Y/D = 0, and Y/D
.
78
.
80
.
81
.
82
Plume centerline excess salinity ratio versus Y/D,
for FR = 1.01, R = 1.48, and X/D = 2.5 and 5
A-la
Plume centerline excess salinity ratio versus X/D,
0.1
0, and Y/D
for FR = 1.44, R = 0.52, Y/D
.
Plume centerline excess salinity ratio versus Y/D,
for FR = 1.44, R = 0.52, and X/D = 2.5, 5 and 10.
Plume centerline excess salinity ratio versus X/D,
0.1
0, and Y/D
for FR = 1.44, R = 1.02, Y/D
.
A-l3
A-l4
Plume centerline excess salinity ratio versus Y/D,
for FR = 1.44, R = 1.02, and X/D = 2.5, 5 and 10.
Plume centerline excess salinity ratio versus X/D,
0.1
for FR = 1.44, R = 1.49, Y/D = 0, and Y/D
.
A-15
76
Plume centerline excess salinity ratio versus Y/D,
for FR = 1.01, R = 1.01, and X/D = 2.5, 5 and 10.
A-9
A-12
75
.
.
A-ll
74
.
.
A-7
.
Plume centerline excess salinity ratio versus Y/D,
1.44, P. = 1.49, and x/D = 2.5, 5 and 10.
for FR
.
.
77
83
.
84
.
85
NOMENCLATURE
a
a
0
semi-minor axis of collapsing element
radius of element at end of convective descent
semi-major axis of collapsing element; also plume
radius in convective descent phase
B
relative buoyancy
C
plume salt concentration
ambient salt concentration
C
Carag
CD
drag coefficient for a.n elliptical cylinder edge
drag coefficient for a two-dimensional cylinder
drag coefficient for a two-dimensional wedge
CD
3
drag coefficient for a two-dimensional plate normal
4
C
fric
D
toflow
skin friction coefficient
drag force; also discharge port diameter
form drag on collapsing element
E
E
in
entrainment function
momentum jet entrainment
entrainment by a two-dimensional thermal
ET
F
Fbf
F
,
1
F
total entrainment
buoyancy force
bed friction force
FD
driving force of collapse; also drag force
Ff
skin friction drag of collapsing element
rictn
bottom friction coefficients
1/2
FR
Froude Number = U /
0
0
g D)
0
2
FRL
local Froude Number = U ,/[
g
acceleration of gravity
I
inertial force
j
unit vector in vertical direction
L
length of jet-plume element
P
pressure
R
velocity ratio = U/U
S
salinity
g b]
centerline salinity
S
discharge salinity
S
ambient salinity
c
LS
0
s
s
C
0
-s
-s
S
distance along jet axis
T
plume temperature
ambient temperature
U
t
time
U
jet velocity vector
a
ambient velocity vector
discharge velocity
towing or current velocity
U
jet velocity in x-direction
ambient velocity in x-direction
v
plume element velocity in vertical direction
tip velocity due to collapse
tip velocity due to entrainment
V
contribution to tip velocity due to entrainment
3
plume half-width
plume element velocity in z-direction
w
a
ambient velocity in z-direction
horizontal downstream distance from discharge port
vertical distance from discharge port
x, y, z
x_, y, z
rectangular coordinates fixed on discharge port
rectangular coordinates fixed on plume element
Greek Symbols
a
a1
a2
a3, a4
entrainment coefficient
entrainment coefficients for convective descent
entrainment coefficients for collapsing plume
coefficient of thermal expansion
angles ambient current at s makes with x and z
axes, respectively
density gradient
coefficient for density gradient difference inside
and outside of collapsina element; also angle in a
vertical plane between s and the resultant ambient
current
coefficient of salt concentration expansion
density
reference density
ambient density
discharge density
p
0
53
2'
p
0
Co
direction angles of the jet trajectory
angle between the surface projection of the collapsing element centerline and the x-axis
AN EXPERINENTAL/ANALYTICAL INVESTIGATION OF NEGATIVELY
BUOYANT JETS DISCHARGED VERTICALLY UPWARD
INTO A CROSSFLOW CURRENT
I.
INTRODUCTION
With the growing environmental awareness, industry practices are
coming under close scrutiny to ensure that industrial processes and
operations are conducted so as to minimize their impact on the surOffshore oil operations are a case in point.
rounding environment.
In the process of extracting oil from beneath the ocean floor, brine
mixes with the oil.
Serving no useful purpose, the brine is separated
from the oil and discharged into the ocean.
Since the salinity of
the brine is typically 2-5 times that of sea water, the potential for
environmental disruption is significant.
In the interest of obtaining more information on the disposal of
produced water, the Offshore Operators Committee and Exxon Production
Research Company have developed a computer model to simulate and predict the fate of produced water when discharged into offshore receiving water.
The model has the capability to predict the behavior of
produced water discharged into a variety of ambient conditions.
This
model will aid in the evaluation of the environmental impacts of of f-
shore oil operations and will help establish the relative merits of
alternative site selection.
A critical phase in mathematical modeling is the model validation
and calibration phase.
In this phase the model is checked for valid-
ity and calibrated to ensure that the model agrees with actual field
conditions.
The development of such computer models involves the use
2
of many assumptions and empirical coefficients.
As a result, the
assumptions must be checked and verified, and the empirical coeff i-
cients within the model must be adjusted (or "tuned") before the model
can be used as a predictive tool.
The purpose of this investigation was to conduct laboratory simulations of negatively buoyant jets discharged vertically upward into
a crossflow current to obtain information useful in the calibration
and validation of the model.
Specifically, this study included devel-
oping the experimental model for laboratory simulation; conducting
measurements to obtain data for the computer model validation and
calibration; and reviewing and briefly describing the analytical basis
of the produced water computer model.
The investigation was directed
mainly towards the jet and collapse phases of the plume, which included the interaction of the plume with the ocean bottom.
The results of this investigation are presented in two parts.
The first describes the analytical basis of the computer model and a
procedure for tuning the coefficients within the model.
part, concerned with
The second
he experimental modeling involved in the labor-
atory simulation, outlines the experimental procedure and presents
the results to be used in the calibration and validation of the computer model.
3
II.
ANALYTICAL MODELING
This chapter is devoted to the description of the mathematical/
computer model for predicting the fate of produced water disposal.
First, a brief background is given of pertinent plume models, followed
by a description of the basic aspects of the produced water computer
model.
Then the development of the mathematical model is presented,
including a discussion of the solution procedure used in the model.
Lastly, the method of tuning the model using the experimental results
of this investigation is outlined.
INTRODUCTION
A number of models for predicting the fate of marine discharges
have been developed over the past several years.
Early efforts were
directed towards the development of stand alone models for passive
diffusion of pollutants in the deep ocean, and for the behavior of
buoyant plumes from thermal discharges (see Davis and Shirazi1).
Koh
and Chang2 assembled some of these models and added a treatment for
particulate solids contained in the discharge.
The resulting model
assumed the ambient fluid to have a steady current uniform in the
horizontal but varying in direction and speed with depth, and to
be density stratified with an arbitrary gradient uniform in the
horizontal.
Brandsma and Divoky3 built upon the Koh-Chang model and
incorporated a passive diffusion model by Fischer
to develop a gener-
alized model for ambient current variations in three dimensions and
in time, variable water depths, and up to 12 different solids.
A
1.
paper by Brandsma, et. al.5 describes a new model, developed from the
Brandsma and Divoky model and tempered by field measurements made at
drilling sites in the Gulfs of Mexico and Alaska, and on the eastern
The produced water model is this new
coast of the United States.
model with the solids handling feature removed.
In the produced water model, the discharge of produced water
into the ocean is taken to originate as a jet from a submerged pipe
at an arbitrary orientation.
The ocean is assumed to be stratified
with an arbitrary velocity distribution.
The behavior of the material
after release is divided into three distinct phases:
1) convective
descent, during which the material tends to behave as a plume under
the influence of gravity; 2) dynamic collapse, occuring when the
descending plume either impacts the bottom or arrives at the level of
neutral buoyancy, at which time the descent is retarded and horizontal spreading dominates; and 3) passive diffusion, beginning when the
transport and spreading of the plume is determined more by ambient
currents and turbulence than by any dynamic character of its own.
This behavior is illustrated in Figure 1.
The simplified conceptualization just presented is modeled mathematically.
The development of the model is presented below.
This
investigation deals exclusively with the short and intermediate term
regions of the discharge.
Consequently, the model development con-
centrates on the two dynamic phases of plume behavior.
The develop-
ment for the long-term passive diffusion will not be presented here.
Most of the material presented is taken from References 3 and 5.
PLUME IMPACTS
BOTTOM
CONVECTIVE
DIFFUSIVE SPREADING
GREATER THAN
DYNAMIC SPREADING
DYNAMIC COLLAPSE
P H AS E
PASS I V E
DIFFUSION
/
-
Figure 1.
Idealized jet discharge described by mathematical model.
Cross sections are shown at three stages of plume development.
Ui
6
MODEL DEVELOPMENT
n this section the mathematical formulation of the produced
water model is presented.
The flow phenomena of the discharge to be
modeled are described as follows:
the flow near the discharge port
is that of a rising (or sinking) jet in a crosscurrent.
The jet
entrains ambient fluid and momentum, and experiences a drag force
from the ambient fluid due to pressure differences between the upstream and downstream faces of the jet.
As a result, the jet grows
in size, bends in the direction of the ambient current, and is diluted due to the entrainment of ambient fluid.
As the jet extends
further downstream, its centerline velocity approaches that of the
ambient fluid, the influence of the ambient density gradient becomes
dominant, and the jet begins to behave more like a plume (convective
descent phase).
Upon reaching a point of neutral buoyancy or the
ocean bottom, the plume will spread out horizontally and collapse
vertically (dynamic collapse phase).
After spreading out into a
relatively thin layer, further transport and spreading are determined by ambient currents and turbulence (passive diffusion).
The
model describing all but the passive diffusion portion of this
sequence of events is developed below.
Convective Descent
The equations describing a jet in a stratified ambient with an
arbitrary velocity distribution are those for conservation of mass,
momentum, energy, and salt.
Figure 2 shows a round jet discharging
with a flow rate, Q, into a crosscurrent.
It is assumed that the
7
(a) Jet coafiguration
I
(b) Arnbien density profile
Figure 2.
(c) Ambient velocity and drag furcea
Coordinate system and definition
sketch for a round jet.
8
jet cross section remains circular and that velocity and density
distributions may be approximated by "top hat" profiles.
The jet
properties are then described by its radius, b, velocity, U, and
density, p.
The ambient density and current are designated by
(y,t) and U
(x,y,z,t).
As shown in Figure 2, the coordinate axes are fixed on the discharge port, with s the direction of the jet trajectory; @, O, and
63 are its direction angles;
and
2
are the directions the resul-
tant ambient current at position s makes with respect to the x and z
axes, respectively; and y is the angle in a vertical plane between
s and the resultant ambient current.
Near the discharge port, the flow is very similar to that in a
momentum jet.
In momentum jet theory, variation in the above quan-
tities occur only with distance along the jet axis, s, so rates of
change of the conserved quantities are written as derivatives with
respect to s.
The rate of change of mass flux along the jet axis equals the
rate of ambient fluid mass entrainment per unit jet length, so the
conservation of mass equation is
d
2
(rb pu) = EPa
(1)
where E is the entrainment rate.
The rate of change of momentum flux along the jet axis is equal
to the buoyancy force per unit length plus the rate of ambient fluid
momentum entrainment per unit length minus the drag force.
servation of momentum equation is expressed as
The con-
9
-
ds (rrb2pLi)
=
Fj + E
aa
-
D
(2)
where j is the vertical unit vector.
The drag term in the momentum equation accounts for the action
of the ambient current in bending the jet. The drag force, accounting for the action of the unbalanced pressure field around the jet
caused by the ambient current seen by the jet, is proportional to the
square of the velocity component of the oncoming ambient fluid normal
to the jet axis. Referring to Figure 2c, the magnitude of the drag
force is
FD
CDPa b(iUaI Sin y)
(3)
where CD is the drag coefficient.
The rate of change of energy flux along the jet axis equals the
energy entrained from the ambient fluid,
ds
[irb2
tJ(T - T )} = Tb
dT
(4)
The rate of change of salt flux along the jet axis is equal to
the amount of salt entrained from the ambient fluid,
dC
ds
= irn
(5)
The rate of change of relative buoyancy flux along the jet axis
is equal to the rate of ambient fluid relative buoyancy entrainment,
d
-ds[rrb 2
U(pa (0) - pH = E(p (o) - p
a
a
(6)
10
An equation of state is required to calculate the local density.
p(T, C, F) in a Taylor series about a reference density,
Expanding p
p, and neglecting higher order terms, yields
p=p[l-(T-T)-A(C-c)]
0
0
(7)
where
p0
T
T,P
and
= 0 (incompressible fluid).
T,C
Equation (7) can be rewritten as
p - p
-
(T
- T) + X(c
- C)
(8)
Solving (4) and (5) provides the information required in (8) to calculate the local density.
Following Abraham6, the entrainment function is assumed to de-
pend upon the local mean flow andis the sum of contributions due to
momentum jet entrainment and a two-dimensional thermal type of entrainxnent.
Momentum jet entrainment is proportional to the perimeter
of the jet and the velocity difference between the jet and the velocity
component of the ambient fluid in the direction of jet travel,
= 2Trb ci.1(UI -
UI cos y)
(9)
Entrainment from a two-dimensional thermal is proportional to
the perimeter and velocity of the thermal.
Visualizing the thermal
11
plume moving horizontally with the ambient fluid but with a vertical
velocity U
sin y, the resulting equation is
E = 2Trb c
where
ITJ
2a
and
2
sin y
(10)
are entrainment coefficients.
The total entrainment
is then
+ Et sin 02
ET = E
where sin 82 is arbitrarily introduced as a convenient way to turn on
the thermal type of entrainment as the jet bends over to the horizontal.
The momentum entrainment depends upon the local Froude Number,
2
FR
L
where p
p-p
= U /{
g b]
is the ambient density, p is the plume density, b is the
plume radius, g is the acceleration of gravity, and p
is a reference
density, in this case the ambient density at the water surface.
expression for
l
An
as a function of the local Froude Number is given
by Wu and Koh8:
= 0.0806
for FRL
19.1
(12a)
= 0.1160
for FRL < 19.1
(12b)
Several additional equations are needed to complete the modeling
of the convective descent phase.
Momentum flux
M =
These are:
7rb2
pU
(13)
Buoyancy force
per unit length:
F = Trb
g(p - p
Buoyancy flux
B =
U(P(0) -
2
:
irb2
a
The locus of the jet centerline is found by integrating the
directional cosine differential equations:
dx
ds
- = cos e
ds
= cos 0
2
dz
- = cos 8
ds
3
(l6c)
The trigonometric relation,
cos
2
0
+ cos
2
0
± cos
2
(17)
03 = 1
is also required.
Given a set of initial conditions:
02(0)1 and
03(0)1
Equations (1)
-
U(o), b(o), p(o), e.,(o),
(17) can be integrated using a
numerical integration scheme, resulting in the description of the jet
behavior in the convective descent phase.
When the jet encounters
the sea bottom or a depth where the jet density equals the ambient
density, the calculation is switched to the dynamic collapse phase.
Dynamic Col1ase in the Water Column
If the jet encounters a level of neutral buoyancy, its momentum
will tend to make it overshoot beyond the neutral point causing a
buoyant force to push it back to the neutral position.
The combined
13
action of these forces will cause the plume to undergo a decaying
vertical oscillation.
As the vertical motion of the plume is being
suppressed, the plume tends to collapse vertically and spread out
horizontally, seeking hydrostatic equilibrium within the stratified
ambient fluid.
The jet plume here is expected to behave more like
a two-dimensional thermal than a momentum jet.
The cross section of the two-dimensional thermal is assumed to
b
elliptical, as shown in Figure 3a.
If coordinate axes originate
from the centroid of the thermal, the cross sectional outline of the
thermal is described by
x
y
a
2
b
(18)
2
where a and b, the semi-minor and semi-major axes, respectively,
vary
with time.
The conservation equations for the dynamic collapse in the water
column are formulated considering a portion of the thermal of length
L.
For this case, the conservation equations, written per unit
length L, are:
Mass:
ds
Salt:
Buoyancy:
(19)
a
Fi-D+Eo a
Momentum:
Energy:
(pabTrU) = Ep
d
ds
(20)
a
dT
[rabU(T - TCo H = rabU
d
--- ['rrabtj(C - C
as
dt
U
= E(p
a
(o)
ds
(21)
dC
H = TrabUds -
- p
a
(22)
(23)
14
(a)
Figure 3.
(b)
Geometry of the collapsing'jet plume.
15
ifl
these equations,
Momentum of
the element:
-
M
piTabU
Buoyancy force:
F
lTabg(p
Buoyancy:
B = irab(p
(24)
a
(25)
-
(o)
- p)
(26)
Drag force:
x-direction:
D
x
=
2
C
D3
2a sin()p
a
lu-ua
I
(u-u
a
(27)
2
y-direction:
D
y
c
p
frica /(a2+b
a
cos 0
=-C
2bp a lu-u a lv
2
D4
(28)
-C
-
2
z-direction:
D
fric p a /(a2+b2)2 IUUaJ cos 07
COS ()p
= 2 CD34a
IUUI
(W_Wa)
(29)
-
c
2
p
r / 2
2
frica v(a+b )
Lu-u
a
cos 0
is the
is the drag coefficient for a spheroid wedge, CD
where CD
3
4
3
drag coefficient for a circular plate, C. is a skin friction coefficient,
is the angle between the surface projection of the element
centerline and the x-axis, and 01
and 03 are the angles between
the element centerline and the x, y, and z axes, respectively.
The auxiliary equations needed are those for entrainment of
ambient fluid and for the collapse of the plume.
Entrainment is
assumed to be the sum of contributions due to convection of the element through the ambient fluid and to the collapse of the element.
16
Each type occurs over the surface area of the element.
The total
entrainment is given by
3a' + a4
ET = 2r(22)
where
and
(30)
a4 are the entrainment coefficients for convection and
collapse, respectively, and
is the tip velocity of the collapsing
plume.
The mechanism that drives the plume collapse is the density difference between the inside and outside of the plume.
It is assumed
that because of turbulent mixing the density gradient inside the
plume is less than that outside. This difference is assumed to be
ya
constant at
, where 'y is a coefficient, and a is the radius of
a
0
the plume at the end of convective descent.
If it is assumed that
the plume is resting at the level of neutral buoyancy, that the am-
bient density at this level is p, and
is the normalized ambient
density gradient,
1
p
(31)
y
then the ambient density seen from the plume centroid is
= p(1 - cy')
(32)
and the density inside the plume is
' a
p
= p0(l -
cy')
(33)
Brandsma and Divoky3 evaluate the force driving the collapse of
the plume as
17
'a
gL(l -) aa
1
3
a
3
The forces resisting the collapse are form drag,
4 Cg
=
a
Lav2v2
and skin friction
Lv
frica
2
These forces all act at the centroid of the quadrant of the element,
and their resultant equals the inertia of the quadrant,
I=FD -F f -D D
The horizontal inertia of the quadrant is the time rate of change of
the product of its mass and the horizontal velocity of its centroid
I =
d
ab
(-i---- L p v1)
(3'3)
The tip velocity of the quadrant is
db
= V1 + V3
In these equations, v1 is the quadrant tip velocity due to collapse, and
is the combination of the tip velocity due to collapse
and due to stretching of the element (obtained by assuming that the
minor axis a is kept constant and that no entrainment occurs at that
moment), expressed as
V2 = V1 -
b dL
:
18
As the buoyant element velocity approaches the ambient velocity,
it will either increase or decrease due to the entrainment of ambient
momentum and the drag forces applied to the element.
material is supplied
continuously
Since the
from the jet, the element should be
capable of stretching or squeezing so that the trajectory of one
element can represent the steady picture of a continuous plume.
The
stretching or squeezing of the element is assumed to be represented
by
L
/22
= constant
(41)
(u +w
V
In Equation (39), v3 is the contribution to tip velocity due to
entrainment.
Its magnitude is obtained by solving for v3 =
in
Equation (19) while instantaneously holding p and a constant, which
yields
E
V
(42)
3
rap
The trajectory of the two-dimensional buoyant element is determined from:
dx
dt
U
V
dz
= w
at
Equations (18) through (43) can be integrated using a numerical
integration scheme, using the jet characteristics at the end of the
convective descent phase as initial conditions.
The development of
19
the plume collapse is computed until spreading of the plume due to
collapse is less than that due to diffusion.
to handle passive diffusion is used.
At this point a routine
If the collapsing plume en-
counters the bottom, calculation is switched to a different set of
equations designed to handle collapse on the bottom.
Dynamic Collapse on the Bottom
If the plume does not encounter a level of neutral buoyancy it
will eventually hit the bottom.
This section presents a variation of
the model for collapse in the water column to apply to a plume collapsing on the bottom.
It is assumed that the shape of the plume
cross section is changed to, and maintained as, a half-ellipsoid, as
shown in the upper half of Figure 3a.
Equation (18) describing the
cross section still applies.
Velocity differences between the plume element, the bottom, and
the anthient fluid are allowed.
The bottom is assumed horizontal in
the region of plume collapse.
Except for minor modifications due to
different geometry and to account for the reaction and friction
forces on the bottom, the conservation equations are very similar to
that for collapse in the water column:
Mass:
Momentum:
Energy:
Salt:
(- p'rrabu)
--- (- pTrab[J)
d
=
(44)
a
= Fj - D ±
1
- [- rabU(T - T)J =
d
ds
1
- [- iabU(C - C )] =
-
1
2
Ecu - Ff
dT
rrabu a-
(45)
(46)
dC
TrabUds -
(47)
20
d
Buoyancy:
1
[
rab(o
(o)
-
E(p(o)
-
(48)
The dynamic collapse of a quadrant of the semi-elliptical element
is expressed as
I = F
(49)
- Ff - Fbf
-
where the inertial force is
aD
I=
L p v1)
(50)
Two forces drive the collapse of the plume on the bottom:
the
force as developed for collapse in the water column, and the force due
to the difference in mean density between the plume and ambient fluid.
The combined collapse driving force is given by:
ia
3
gL(l -
F'D =
_a)
2
Bp
+
- (p -
the form drag is
=
2
C
drag
p LaIv v
a
2
2
the skin friction is
F
f
C
Lv
frica
2
the friction force on the bottom is
F
bf
=Fb F rictn
F
1
As with the collapse in the water column,
db
= vi + V3
g L
21
b dL
V2V1----
L
/22
Y
(56)
= constant
(57)
(u +w
EPa
1
prra
Entrainment is given by
/(a2b2)
db
(c
ET
2
c4d)
a
3
Momentum:
M
Buoyancy force:
F =
Buoyancy:
B =
=
4
1
pabU
(60)
6)
7rabg(p-o)
iab(p (o)-p)
2
(62)
a
The trajectory of the plume may be determined by
dx
U
(63a)
dy
(63b)
dz
(63 c)
dt
Equations (44) through (63) can be integrated using a numerical
integration scheme, using the jet characteristics prior to entering
the bottom collapse phase as initial conditions.
the plume collapse on the bottom is computed
until
The development of
horizontal spread-
ing due to collapse is less than that due to diffusion.
At this
point, calculation is switched to the passive diffusion phase.
22
Solution Technique
Once the ambient conditions, topography of the ocean bottom, and
discharge conditions have been defined, the model can be used.
The
discharge is described by the volumetric flow rate and density of the
discharge material, the initial jet radius, and the depth and vertical angle of the discharge port.
The convective descent phase of the jet is computed by integrating Equations (1) through (17) using a standard fourth-order RungeKutta integration routine.
This routine simultaneously integrates
the equations of motion over small increments of arc length along
the jet axis, resulting in a complete history of the convective
descent.
If the jet encounters neutral buoyancy, its cross sectional
shape is instantly transformed from a circle to an ellipse.
Equa-
tions (18) through (43) are then integrated by the same routine as
that for the convective descent.
Since many of the equations in this
phase contain time derivatives, the model uses derivative routines
before integration for converting time derivatives to derivatives
with respect to an element of plume length, ds.
Initial conditions
are provided from the final step of the convective descent.
The
history of the plume collapse is computed until either the plume
hits the bottom or the spreading of the plume due to collapse is less
than that due to diffusion.
If the descending jet or collapsing plume encounters the bottom,
its cross sectional width is immediately transformed to a half
ellipse.
Equations (44) through (63) are then integrated using the
23
same routine as described above for collapse in the water column.
History of the collapsing plume on the bottom is calculated until
passive diffusion dominates.
There are a number of numerical toefficients used in the model.
Many are the same as those used by Koh and Chang1-.
Table 1 lists the
coefficients, and default numerical values, which may be modified
during program execution.
is obtained from Equation (12), and 02
is obtained from Abraham6, after Brandsma and Divoky3 corrected for
the difference in the similarity distributions used in Abraham's and
the present formulations.
The value of 03
the entrainment coeff 1-
cient for a convecting thermal, is also obtained from Abraham6.
Values for 041 the coefficient for entrainment due to collapse, and
-y,
the coefficient used to simulate the effect of density gradient
differences in causing plume collapse, are suggested by Koh and Chang;
these particular default values are based on educated guesses.
The
default values for the drag coefficients were obtained, considering
the range of Reynolds numbers expected, from diagrams presented by
Hoerner7 for solid shapes in fluids.
applicable to this work.
However, in the absence of more relevant
data, these values are used.
coefficients (C
drag
,
C.
rric
As such, they are not strictly
,
The default values for the remaining
F
ricrn
,
and F
1
)
were presented by Koh
and Chang based on educated guesses, and as such are subject to revision.
All default values of the coefficients are used if the user
cannot, or chooses not to, supply his or her own values.
of a procedure to adjust (or
An outline
'tun&') some of these coefficients
based on the experimental results of this investigation is presented
below.
24
Table 1.
COEFFICIENT
Coefficients used in the present model and
their default numerical values.
DEFAULT NUMERICAL
VALUE
DESCRIPTION
Entrainment coefficient for a
momentum jet
a2
CD
Entrainment coefficient for a
two-dimensional thermal
0.3536
Entrainment coefficient for a
convecting thermal
0.3536
Entrainment coefficient for a
collapsing plume
0.001
Drag coefficient for a twodimensional cylinder
1.3
Drag coefficient for a two-dimensional wedge
CD
I
C
drag
C.
fric
rictn
F,
Equation (12)
-
0.2
Drag coefficient for a twodimensional plate normal to flow
2.0
Coefficient to simulate denity
gradient differences causing collapse
0.25
Drag coefficient for an elliptical
cylinder edge
1.0
Skin friction coefficient
0.01
Eottom friction coefficient
0.01
Modification factor for bottom
friction
0.1
25
TUNING THE MODEL
The intent of this investigation was to provide information to
validate and calibrate the produced water computer model.
The assuxnp-
tions used in the model must be verified, and the coefficients within
the model must be tuned before the model can be used as a predictive
tool.
The procedure for tuning the model by adjusting the coeff i-
cients is outlined below.
A prominent feature of the plume behavior in this investigation
was the collapse of the plume on the bottom.
As a result, the major-
ity of the experimental data corresponds to this phase of plume development and will tune mainly-those coefficients in the bottom collapse
routine of the model
for CDf CD
1
4
mended for use.
2'
(
,
4
C
C
,
D3
frc
,
F
rlctn
)
.
The default values
and c3 are fairly well established and are recoin-
Due to the lack of more relevant data, default values
are also recommended for y, F
and C
,
1
drag
.
The following procedure
is recommended for tuning the model:
Run the model at conditions similar to the experimental
conditions investigated using default values for all
coefficients.
Check for agreement on dilution and plume width between
the model output and the experimental results.
Adjust
to make the dilution output from the model
agree with the experimental results.
Adjust individually C
D3
,
C
fric
,
and F
rictn
to make the
plume width output from the model agree with the
experimental data.
26
5)
Iterate steps 3 and 4 until good agreement is obtained
between the model output and experimental results.
27
EXPERIMENTAL MODELING
III.
This chapter details the experimental modeling involved in the
laboratory simulation of negatively buoyant jets discharged into a
crossflow current.
First, the parameters involved in the modeling
are introduced, then the apparatus and data acquisition system are
Following this are details of the data treatment process
described.
Finally, the results of the laboratory
and uncertainty analysis.
simulation are presented and discussed.
MODELING PARAMETERS
In experimental modeling, certain conditions of similarity must
be observed to ensure that the model test data are applicable to the
TW9 kinds of conditions must be satisfied:
prototype.
1) geometric
similarity of the physical boundary, and 2) dynamic similarity of the
flow fields.
analysis.
Relations for similitude are obtained from a dimensional
Such an analysis yields the following independent vari-
ables necessary for dynamic similarity:
1) the densirnetric Froude
1/2
Number, FR =
g 0)
,
which is the ratio of inertial to
0
buoyant forces; and 2) the velocity ratio,
P. = U/U, which is the
ratio of current to discharge velocities.
Since the discharge is
turbulent, Reynolds number effects are negligible.
The discharge in
the experimental model must, therefore, also be turbulent.
The dependent variables resulting from the dimensional analysis
include:
1) the excess salinity ratio, IS /IS
C
0
= (S
C
- S_)/(S
0
- S ),
which is the ratio of local excess salinity to the excess salinity at
discharge; and 2) the dimensionless plume half-width, W/D.
28
The independent parameters were varied in this study as follows:
FR = 0.5, 1.0, 1.5
R = 0.5, 1.0, 1.5
All combinations of these variables were considered.
Salinity measurements were made along a vertical traverse of the
plume centerplane at X/D values of 2.5, 5, 10, 20, and 30.
A schema-
tic showing the plume coordinates and line of traverse is given in
Figure 4.
For most conditions, the plume collapsed rapidly on the
bottom and spread out in a thin layer.
So for many runs, the traverse
would yield a measurement at only one Y/D value very close to the
bottom.
Consequently, most measurements were taken along the bottom
at Y/D '\
0.1.
A detailed discussion of the apparatus and measuring
techniques is given in the next section.
A partially randomized experimental plan was developed to reduce
trend errors associated with natural effects (changing barometric
pressure, temperature, etc.), human activities (increasing skill or
boredom), and mechanical effects (sticky instruments, hysteresis).
The plan consisted of selecting at random the sequencing of sampling
events.
The Froude Number, FR, was first randomly selected from the
possible values, then the velocity ratio, R.
Following this was the
random selection of the downstream distance, x/D, at which measure-
ments were to be taken.
This procedure was followed until all com-
binations of independent variables were considered.
Data collection
yielded plume excess salinity ratio and plume half-width.
LINE OF TRAVERSE
PROBE
CENTER PLANE OF PLUN&_-
PLANE OF TRAVERSE
EDGE OF CHANNEL
Figure 4.
Plane of traverse of the probe.
30
APPARATUS AND DATA ACQUISITION
The experiments were conducted in Graf Hall on the Oregon State
University campus.
In each run, a jet of heavy salt water was dis-
charged vertically upward, from a false bottom, into a towing channel.
Information desired from the experiments included dilution, as expressed by the excess salinity ratio, and plume half-width.
Salinity
measurements were taken of the discharge solution, ambient fluid, and
along the vertical centerplane of the plume at several stations down-
stream from the discharge port to evaluate the excess salinity ratio.
Plume width measurements were made using photographs of dyed jets.
The experimental apparatus, electronic instrumentation, and sequence
of events termed a run are described in detail below.
A schematic of
the apparatus is given in Figure 5.
The apparatus consisted of a salt water discharge into a 12.2 m
x .6 m x .9 m towing channel.
Two carriages containing the discharge
and conductivity measuring systems were mounted on rails along the
top of the channel and connected to a motor driven tow cable.
The
discharge system, supported by the first carriage, included a salt
water reservoir, a main discharge valve, a flow control valve, 1.3 cm
PVC pipe for the manifold, and 2.5 cm PVC pipe for both the reservoir
outlet piping to the manifold and the discharge port.
A false bottom
was used to minimize the disturbance from the discharge piping along
the channel bottom and to allow the discharge port to be flush with
the bottom.
31
POTENTIOMETER GEARED
TO VERTICAL HEIGHT PROSE
SALT WATER RESERVOIR
DC DRIVE MOTOR
MAIN DISCHARGE VALVE
FLOW CONTROL VALVE
FALSE BOTTOM
CHANNEL
BOTTOM
BALL-SCREW
DRIVE
POTENTIOMETER SIGNAL
PROBE SIGNAL
000
O 'I
p'i 1101
r
0000000
I
I
-
I
I
000
CARRIER
LIFIER
LIGHT-SENSITIVE PRINT
Figure 5.
Schematic of the experimental apparatus
and electronic instrumentation.
32
The sealed salt water reservoir was kept at constant head by
bubbling air in as the water level dropped.
voir is shown in Figure 6.
The constant head reser-
As water is discharged from the reservoir,
air pressure pushes the water from the bubbling tubes until air
issues from the tubes into the reservoir.
In this manner, the level
of ambient air pressure is kept constant at the level of the bottom
of the bubbling tubes.
Baffles were included in construction of the
reservoir td damp out waves that would form when the reservoir was
towed.
The reservoir was filled with salt water of the proper salin-
ity and sealed with a rubber stopper prior to each run.
A second carriage supported the conductivity measuring system,
responsible for measuring the electrical conductivity in the field of
the jet.
A conductivity probe (Figure 7) connected to a Tektronix
Carrier amplifier monitored the electrical conductivity at given
values of X/D and Y/D.
Calibration of the probe prior to its use
provided information to obtain salinity values from the conductivity
measurements.
The probe was mounted on a rod that traversed veri-
cally through the plume.
The mechanism to move the probe vertically
employed a double-ball-screw drive powered by a remotely controlled
D.C. motor.
A potentiometer geared to this mechanism monitored the
vertical position of the probe.
The probe was positioned laterally
on the rod such that it followed the vertical centerplane of the jet,
as shown in Figure 4.
The conductivity signal from the Carrier ampli-
fier arid the potentiometric position signal were recorded using a
Honeywell Visicorder (see Figure 5)
FILLING STOPPER
SEALED TANK
BUBBLING TUBES
n
0
C
0
0
BAFFLES
MAIN DISCHARGE VALVE
ATMOSPHERIC PRESSURE LEVEL
AMBIENT WATER LEVEL
Figure 6.
Schematic of the constant head salt water reservoir.
34
ELECTRICAL LEAD
ThNGS I LN LEADS
GROUNDING
CTR0DE
PLATINUM
ELECTRODE
4
Figure 7.
3nm
H-
Enlarged view of conductivity probe.
35
As mentioned previously, the rapid collapse and spreading of the
plume into a thin layer on the bottom required many measurements close
to the bottom.
Because the grounding electrode protruded from the end
of the probe, measurements attempted directly on the bottom grounded
the probe.
To avoid this, the probe was positioned at the lowest pos-
sible height, about Y/D "-' 0.1.
The discharge flow rate was adjusted using the flow control
valve.
Flow measurements were made by observing the reservoir level
drop over a given length in time.
Temperature measurements of the
salt water and ambient were made with a mercury-in--glass thermometer.
The fluids were kept within about 100 of each other during the experiments.
For FR = 1.5, the nominal discharge velocity was 9.1 cm/sec and
the nominal difference between discharge salinity and ambient salinity
was about
220/00 (parts per thousand), depending on ambient salinity.
At FR = 1.0, the nominal U
difference of 47°/oo.
was 8.9 cm/sec, with a nominal salinity
For FR = 0.5, U
salinity difference was 150°/oo.
= 8.2 cm/sec and the nominal
Ambient salinity varied from
00/00
to a high of about 8°/oo at times during the FR = 0.5 cases.
The conductivity probe was calibrated using standard salt solutions of differing salinities.
A salinometer from the OStJ School of
Oceanography was used to standardize the salt solutions.
The uncer-
tainty in the salinity measurements using the salinometer was about
± 0.2%.
Calibration of the probe was obtained by immersing the probe
into these standard solutions and recording the corresponding signal
with the Honeywell Visicorder.
Since the calibration drifted slightly
36
on occasion, the calibration curve was checked and adjusted periodically.
Figure 8 shows a typical calibration curve for the probe.
Dilution Measurements
In order to have reasonable confidence in the dilution results,
duplicate runs were needed for each FR and R condition at each X/D
station downstream from the port.
Typically three runs were made at
each station and condition; however, when two points showed favorable
replication, a third run was not made.
Due to the multitude of runs,
exact duplication of conditions was impossible.
The resulting stan-
dard deviation was about 6% for the Froude Nunther and about 6.5% for
the velocity ratio.
The X/D and Y/D values were reasonably exact.
The sequence of events which constitute a run for measuring dilution is described as follows:
Calibrate the conductivity probe and the potentiometer
connected to the vertical height control.
Prepare and align the traversing mechanism for the
particular downstream distance, X/D and vertical
height, Y/D (usually starting from the bottom).
Fill the reservoir with salt water of desired salinity.
Adjust the flow control valve to attain the desired
discharge velocity.
Measure the discharge salinity and temperature and
ambient salinity and temperature.
Adjust towing speed to desired value.
Open main discharge valve to begin the discharge.
37
2
4
6
8
10
12
14
PROBE OUTPUT(DIVISIQNS)
Figure 8.
Tyoical calibration curve for the
Conductivity probe.
38
Initiate tow.
After towing speed is reached, begin
traversing the jet with the probe.
At the conclusion of the tow, record the distance of
the tow and the time associated with that towing distance to measure the towing speed.
Shut the main discharge valve off.
Record the reservoir
level drop and the associated time for the drop to measure the volumetric flow rate.
Use continuity to cal-
culate the average discharge velocity.
Plume Width Measurements
Photographs of dyed jets using a 35 nmi camera provided plume
width measurements.
A 4 cm square grid was drawn on the false bottom
to provide a reference for the measurements.
Runs were made similar
to that described above, except salinity measurements were not needed.
Instead, top view color transparencies were taken of the dyed plume
for each FR and R condition investigated.
These photos were then
used to plot plume width, W/D, as a function of downstream distance,
X/D, for each condition.
Another study was conducted to determine the character of the
boundary layer along the upper half of the false bottom.
Velocity
measurements within the boundary layer were made using a Thermal
Systems, Inc. hot wire anemometer.
During a tow, the sensor traversed
the boundary layer vertically to provide velocity measurements at
various heights for a given X,D station.
These measurements were
plotted to give the velocity distribution within the boundary layer.
39
In general, the experimental apparatus operated as desired and
had acceptable error.
A couple of problems arose in the course of
experimentation, including:
1) variations in towing speed at slow
speeds, which required great care to ensure that measurements were
taken only over the portion of the channel with uniform towing speed;
and 2) fouling of the conductivity probe when using high saline discharges, which required periodical cleaning of the probe with methyl
alcohol.
At the end of the experimentation, an effort was made to measure
conductivity directly on the bottom by modifying the apparatus
A ulifterfi plate, illustrated in Figure 9, was attached to
slightly.
the false bottom to allow for bottom measurements.
Measurements were
taken at the end of the lifter and represent actual plume bottom conductivities.
ments.
Time did not allow for replication of these measure-
Rather, the information resulting from this effort was used
to verify that bottom salinities were significantly higher than at
Y7D " 0.1.
DATA TREATMENT
A typical Visicorder plot of the conductivity and position signals is shown in Figure 10.
For each probe position, the conductivity
signal was examined and by visual scrutiny the mean (time averaged)
value of the signal was determined.
This value and the value ascribed
to the position signal were referenced back to their respective calibration curves to obtain a salinity value at the associated probe
height.
A flowchart outlining the data treatment process is shown
in Figure 11.
40
LIFTER PLATE
CHANNEL
CONDUCT IV ITY
BOTTOM
PROBE
FALSE BOTTOM
mm
/ / / // /
Figure 9.
/ / /
/ /
/
Section view showing lifter plate used for
measuring conductivity on the false bottom.
41
C(N)UCTIVfly PROBE
SI(NAL
INCREASING
SALINITY PND
PROBE HEIGHT
POTENTI C(9EJER
PUS I TI'SI Ei"JAL
Figure 10.
Typical Visicorder print of conductIvity
and position signals.
42
DATA TREATMENT
Read Instrument
Output
Record Values and
Other Importart Information
Enter Data
on Data File
Normalize Data and
Compute Dimensionless
Parameters by Computer
Statistically Analyze
Eliminate Outlying
Data Points
Data.
Plot Data on Graphs,
Draw Curve Through
Ilean of Data Groups
Run Regresslon
Analysis on Data
Figure 11.
Flowchart showing data treatment process.
43
These values and other important information were recorded on
The data for all runs was assembled and entered on data
data sheets.
files in the computer.
A computer program then normalized and re-
duced the data to the forms iS /iS
c
0
,
FR, R, X/D, Y/D.
This output
was statistically analyzed by computing the mean and standard deviation for each Froude Number and velocity ratio condition investigated.
Using a method known as the Chauvenet's Criterion9, data with significant deviation from the investigated conditions were eliminated
from the data base.
The normalized data was then plotted on graphs, and curves were
drawn through the mean of each data group.
An example of the plots
and data points for excess salinity ratio as a function of X/D and
Y/D are shown in Figures 12 and 13.
Some of the data points have
been shifted off the true X/D position in order to clarify the plots.
Appendix A contains all the curves obtained in this experimental inLike the others, the curves are drawn through the mean
vestigation.
of the data groups, and are restricted to the X/D range investigated.
A table of all data contributing to the curves in Appendix A is given
in Appendix B.
In addition to having plots with curves drawn through the mean
of data groups, a regression analysis was performed to offer an unbiased examination of the collected data.
Employment of the Statisti-
cal interactive Programming System (SIPS) available at Oregon State
University's Computer Center provided a least-squares regression
curve fit.
The model proposed to SIPS was a logarithmic transforma-
tion of the function
FR = 1,L4L
R = 1.02
A Y/D0
0 Y/D0.1
0
(I)
(j U
F
0
0
0
10
20
3)
1()
FOUZItffAL DtSTN'cE - X/D
Figure 12.
Example of typical excess salinity raho data, as a function of
X/D, arid its' represenLajve curve,
0,6
0.5
FR = 1.01
R = 1.118
XJD = 5
0.2
0
0. ]
0
'
0
1
0
0.2
0.Lj
0,6
0.8
1.0
VERTICAL If IGIIT - y/lJ
Figure 131.
Examplo of typical excess salinity ratio data, as a function of
VD, and its' representative curve,
46
1s
C
= exp(a) (X/D)
b
d
c
(R)
(FR)
0
The transformed equation
LS
ln(-) - a + bln(X/D)
cln(R) + dln(FR)
provided the unknown coefficients (a,b,c,d) in linear form as required
for a least-squares linear regression analysis.
UNCERTAINTY ANALYS IS
An uncertainty analysis of the data was performed to obtain a
general indication of the quality of the data.
The standard approach
to uncertainty analysis requires knowledge of the error in the rnea-
sured quantities, and the mathematical relationship between the measured quantities and the desired result, before the propogation of
errors to the final result can be assessed9.
Since the process of
making salinity measurements was involved (it required making stan-
dard salt solutions, diluting a portion of each solution for the
salinometer measurements, calibrating the conductivity probe using
the standard solutions, making a conductivity measurement, and relating instrument output back to the calibration curve to arrive at a
salinity value), an estimate of the error in the salinity measurement
would at best be an educated guess.
As such, the standard approach
to uncertainty analysis was abandoned.
Instead, 90% confidence inter-
vals were estimated for the data groups using a method for small data
samples outlined by Ang and Tang10.
Figure 14 is a repeat of Figure
12 with the 90% confidence interval included to indicate the quality
of the data.
FR =
1,LI11
R = L02
0
(I)
0
U
V)
Y/r=0
YiD0.1
90% ConfIdence Interval
10
20
FKJRIZCTAL MSTNCE
Figure 14.
XiB
A repeab of Figure 12 with 90% confidence inberval included to
indicate the quality of Lhe experimental data.
48
Three factors explain the spread of the data illustrated by the
confidence interval:
1) the small number of data points within each
data group; 2) usual measurement errors or uncertainties; and 3) the
sensitivity of excess salinity ratio to vertical height, as shown in
Figure 13.
Small variations in probe height around Y/D = 0.1 yield
significant deviations of excess salinity ratio from that at Y/D = 0.1.
Since it was virtually impossible to align the probe exactly at
Y/D
0.1 for each run, the vertical sensitivity added to the data
spread.
The data obtained in this investigation, however, was suffi-
cient to define the functional relations between the variables and to
recognize established trends.
The results of the experimental inves-
tigation are discussed in detail below.
RESULTS
The effects of FR and R on dilution are best demonstrated by the
nf
plots
C
versus X/D and Y/D for the various combinations of FR
0
and R, as given in Appendix A. The individual effects of Froude Number and velocity ratio on dilution are described as follows:
1)
Within the range of values investigated, the effect
of velocity ratio, F, on dilution is minor, as demonstrated in Figure 15.
All measurements were made
within the boundary layer above the false bottom
(see Figures 29-31)
.
Since the magnitude of the
velocity within the boundary layer at Y/D
0.1
(where most measurements were made) was much lower
than the freestrearn current, the effects of the
.25
FR = :1.01
.20
20
X./D
0
A
Y/D=O
0
V/DuO,1
C,,
U
C.,)
.15
10
MS
0
0
00
0.-
0.5
0
1,0
1.5
WLOCI1Y RATIO - R
Fiqure 15.
Excess salinity ratlo as a function of velocity ratio.
50
ambient current were reduced.
As a result, the net
effect of the velocity ratio on dilution was found
to be small.
The most critical parameter affecting dilution is the
2)
Froude Number, FR.
It was found that dilution in-
creases with decreasing FR, as shown in Figure 16.
The dense fluid associated with low FR numbers
spreads very rapidly on the bottom following collapse
of the plume.
This rapid spreading is thought to
increase entrainment and thereby increase dilution.
The results of the photo study of dyed jets, providing plume
width information, are presented as plots bf plume half-width versus
X/D, for the various FR and P. conditions investigated, in Figures
17-19.
The results indicate that:
1) plume width increases with
decreasing velocity ratio; and 2) plume width increases with decreas-
ing Froude Number, supporting the explanation of greater spreading
with lower FR.
The regression analysis on the data yieldsthe following regression model:
Ls
- (0.363) (x/D)"02 (R)024 (FR)277,
0
valid for Y/D
0.1.
An examination of the exponents in the regression model supports the
results presented earlier.
The Froude Number, with the largest ex-
ponent, has the greatest effect on dilution.
On the other hand, the
velocity ratio has the smallest exponent and hence has the least
effect on dilution.
R
1.00
X/D = 5
X/D10 - X/D=30
V/B 0.1
.20
/
0,5
1,0
1.5
FROUDE NUMBER - FR
Figure 16.
Excess salinity ratio as a function of Froude Number.
2.0
10
/
/
/
/
/
I/I
FR=0.5
1/
/
-12
0
R=0.5
0- -0
R1.0
o----0
R=1,5
0
U
Li
8
HORIZONTAL DISTANCE - X/D
Figure 17.
Plunie half-width as a function of X/D and R for FP = Ob.
12
10
/
I/
7,,
1/'
/
IL
/
1
2
/
,'
7'
FR1:0
0
0
R10
G--
0
0
11
8
12
20
16
HORIZONTAL DISTANCE - X/D
Figure lB.
Plume haif-width as a function of X/D and B. for FR
1 .0.
FR=L5
0
0
R=O,5
0- -0
R=L0
°
R=1,5
°
8
12
HORIZONTAL DiSTANCE
16
20
X/D
Figure 19. Plume half-width as a function of X/D and R for FR = 1.5.
2L
55
The regression analysis results are shown graphically, and corn-
pared to the data average curves, in Figures 20-28 for the selected
values of FR and R.
The graphs show reasonable agreement in shape
and trend between the proposed model and the data average curves
for most of the conditions investigated.
The relatively low value for
the coefficient of determination (R2 = 0.53) suggests that:
1)
im-
provements could possibly be made in the proposed model to better fit
the data, and/or 2) the spread of the data may affect the regression
analysis to the extent that a good fit cannot be obtained.
The velocity distributions within the boundary layer over the
false bottom for each of the R values investigated are presented in
Figures 29, 30, and 31.
The graphs indicate that virtually all salin-
ity measurements were made within the boundary layer.
Due to the
complexity of ambient currents along the ocean bottom, no attempt was
made to replicate the ocean bottom conditions in this investigation.
Instead, the study provides information on the actual velocity distribution within the boundary layer over the false bottom associated
with the data obtained.
Since turbulence is a second order effect,
the difference in boundary layer between the experimental model and
actual field conditions should have a negligible effect on the
results.
0.125
I
I
I
I
FR = 0,50
R = 0.50
0.10
fr0
0-
0
V/i)
0,1
DATA ATRNES
REGRESSIIflf4I(L
.075
;.ty50
10
20
FURIZ1ffAL D1STM
Figure 20,
30
LIt)
- X/D
Comparison of regression model with data averaqes curve
FR
0,5, R = 0.5.
for the condit:iong;
0.125
FR =
R = 0,99
0.10
Y/DO.1
-o
DATA AVERAGES
o
REGRESSI
tfla
C)
.050
.cY25
20
3)
-j
Lb
HOMZCt'LTPL MSTi4E - XJD
Fiqure 21.
Comparison of regression model with data averages curve
for the conditions:
FR = 0.5, R
0.99.
FR = 0.50
R = 1.1i7
Y/D = 0.1
WTA AVERAGES
RE3RESSI
0
10
fI[L
20
HORIZ(1iThL MSTN4E - )VD
Figure 22.
Comparison of regression model with data averages curve
for the conditions:
FR
0.5, R = 1.47.
20
uoKI21Ta
Figure 23.
iSTP4CE - XJD
cornpar°
regressiOfl model with data av6raq
for the cofld°
FR = 1.01, R = 0.48.
curve
FR = 1.0].
R = 1.01
v/f)
-.
0.1
ETA AVERPLES
RERESSJt
0
2
10
20
30
tkRJ711'lTAL DISTP14E
Fiqure 24.
JEL
Ito
X/[)
Comparison of reqression model with data averages curve
for the conditions:
FR
1.01, R = 1.01.
FR = 1,01
R = LLI8
Yin
o,i
LVTA AEMfS
0
1ESSI1J tUIEL
0,2
0.1
-
0
0
10
20
33
I10(UZ1SffAL D)STP1KE - XJD
Figure 2.
Comparison of regression model with data averages curve
for the conditions: FR
1.01, R
1.48.
0.6
FR = 1MLI
R=0.52
0.5
-
V/B
-
0,1
IYTA ARAGES
REGRESSILli WEL
-
cD
0.3
0,2
0.1
-
if)
0
20
30
L10
HORIZ(Nf/L MST!E - XJD
Figure 26.
Comparison of regression model with data averages curve
for the conditions
FR = 1,44, R
0.52.
50
0.6
FR = 1.LILt
0.5
R = 1.02
Y/D
A- -A
0
0
0.1
DATA AVERA(iS
REGFSSIct1 MJDEL
0.3
0.2
0,1
0
L
I
10
20
IERIZTL D!STJN
Figure 27.
I
LiO
- XJD
Comparison of regression model with data averages curve
for the conditions: FR
1.44, R = 1.02.
0.6
FR = 1.'14
R = 1.49
0.5
A0
0
Yin
0.1
DATA AVER(TS
IGRESSII fvTIL
(1
0,3
0.2
0.1
50
VERTICAL HEIGHT - V/I)
F'igure 29.
Velocity distribution in the boundary layer above
the false bottom for R = 0.5.
1.0
0,8
8
-4
0.4
R = 1.00
-0-0,2
0,25
0.50
0.75
a
X1DJ5
o
X/D=30
1.00
1.25
VERTICAL hEIGHT - V/I)
Figure 30.
Velocity distribution in the boundary layer above
the false bottom for R = 1.0.
1.50
0.8
0,6
0.L1
0.2
\iERTICAL lElGif - Y/B
Figure 31.
Velocity distribution in the boundary layer above
the false bottom for P = 1.5.
68
IV.
CONCLUSIONS
The purpose of this investigation was to conduct laboratory
simulations of negatively buoyant jets discharged vertically upward
into a crossflow current to obtain information useful in the validation and calibration of the produced water computer model.
The re-
suits of the experimental modeling effort may be summarized as
follows:
The Froude Number has the greatest effect on dilution.
Both dilution and plume width increase with decreasing
Froude Number.
Within the range of values investigated, the velocity
ratio has a minor effect on dilution.
Plume width
decreases with increasing velocity ratio.
In the analytical portion of this study, the produced water computer model was reviewed, the development of the mathematical model
was presented, and the solution technique used in the model was discussed.
Then a procedure was outlined to tune the coefficients with-
in the model using the experimental results of this
investigation.
Using the experimental results presented here, the near and
intermediate portions of the model can be validated as representing
actual field situations and calibrated for use as a predictive tool
to determine the fate of produced water disposal.
69
V.
REFERENCES
Davis, L. R. and Shirazi, M. A., A Review of Thermal Plume Modeling, Keynote Address, Proceedings of the 6th International Heat
Transfer Conference, Toronto, Canada, August 1978.
Koh, R. C. Y. and Chang, Y. C., Mathematical Model for Barged
Ocean Disposal of Wastes, U.S. E.P.A. Report EPA-660/2-73-029,
December 1973.
Brandsma, M. G. and Divoky, D. J., Development of Models for Prediction of Short-Term Fate of Dredged Material Discharged in the
Marine Environment, U.S. Army Engineer Waterways Experiment
Station Report D-76-5, May 1976.
Fischer, H. B., A Method for Predicting Pollutant Transport in
Tidal Waters, U.C. Berkeley Water Resources Center, Report 132,
March 1970.
Brandsma, M. C., Davis, L. R., Avers, R. C. and Sauer, T. C., A
Computer Model to Predict the Short-Term Fate of Drilling Discharges in the Marine Environment, Proceedings of the Symposium:
Research on Environmental Fate and Effects of Drilling Fluids
and Cuttings, Lake Buena Vista, Florida, January 1980.
Abraham, C., The Flow of Round Buoyant Jets Issuing Vertically
Into Ambient Fluid Flowing in a Horizontal Direction, Proceedings: Fifth International Conference on Water Pollution Research,
San Francisco, Paper 111-15, July 1970.
Hoerner, S. F., Fluid Dynamic Drag, Published by the Author,
Brick Town, New Jersey, 1965.
Wu, F. H. Y. and Koh, R. C. Y., Mathematical Model for Multiple
Cooling Tower Plumes, U. S. E. F. A. Report EPA-6007-78-102,
1978.
Schenck, H., Theories of Engineering Experimentation, 3rd Ed.,
McGraw-Hill, New York, 1979.
Ang, A.H-S. and Tang. W. H., Probability Concepts in Engineering
Planning and Design, John Wiley, New York, 1975.
APPENDICES
70
APPENDIX 1\
What follows are graphs of the experimental data obtained in
this investigation.
The graphs include centerline excess salinity
ratio plotted against downstream distance, X/D, and vertical distance,
Y/D, for conditions of FR = 0.5, 1.0, 1.5 and R
0.5, 1.0, 1.5.
curves presented are drawn through the mean of the data groups.
The
For
clarity, some of the data points have been shifted off the true X/D
position.
0 25
FR = 0.50
R = 0.50
Y/D
0.1
0.2
(40
U
C,)
':3
0.15
I.-
;
Oil
0105
0
0
10
.1
20
30
HORIZONTAL DISTANCE - X/D
Fiqure ii-l.
Plume centerline excess salinity ratio versus X/D, for
FR= 0.50,R = 0.50, and Y/D
0.1.
025
0.50
R = 0.99
0.1
YID
FR
0.2
0
U)
U
U)
0
::
0.05
o
0
0
p
I0
0
0
00
20
30
HORIZONTAL DISTANCE - X/O
Figure A-2.
Plume centerline excess salinity ratio versus X/D, for
FR = 0.50, F = 0.99, and Y/D 0.1.
0.25
FR
= 0.50
R
1.147
Y/D0.1
0.2
0
(4
U
(4
cJ
c.
0.15
0.1
0.05
0
A
0
0
I
10
20
30
HORIZONTAL DISTANCE - X/D
Figure A-3.
Plume centerline excess salinity ratio versus X/D, for
0.1.
FR = 0.50, R = 1.47, and Y/t)
FR = 1.01
R
A
Y/D
Y/D
0
0i
0
A
10
20
30
HORIZONTAL DISTANCE - X/D
Figure A-4,
Plume centerline excess salinity ratio versus X/D, for
0.48, Y/D = 0, and Y/D 0,1.
FR
LO
0.6
0.5
0.1
0.2
O.'4
0.6
0r8
VERTICAL HEIGHT - Y/D
Figure A--5.
Plume centerline excess salinity.ratio versus Y/D, for
1.01, R = 0.48, and X/D = 2.5, 5 and .10.
F1
1.0
0.6
0
FR = 1.01
0.5
R
A V/U
0
A
V/U
1.01
0
0.1
0,1
A
A
A
0
0
10
20
30
HORIZONTAL DiSTANCE - X/D
Figure A-6.
Plume centerline excess salinity ratio versus X/fl, for
FR = 1.01, R = l0i, Y/D = 0, and Y/D
0.1.
'10
0.6
I
I
I
FR = 1.01
0.5
R=1.01
X/D = 2.5
- X/D = 5
0
- - - X/D = 10
.
0.3
0.2
0.1
0
0.2
0
0.11
0.6
0.8
VERTICAL HEIGHT - Y/D
Figure A-7.
Plume centerline excess salinity ratio versus Y/D, for
FR = 1.01, R = 1.01, and X/D = 2.5, 5 and 10.
1.0
0.6
I
I
I
05
FR = 1.01
R
0
A
C,,
(-I
C.')
.
0 Y/D
nU.
I-
1148
Y/D =0
0.1
£
0.3
a
L)
0,2
8
0.1
A
£
10
0
20
HORIZONTAL DISTANCE
Figure A-8.
'30
X/D
Plume centerline excess salinity ratio versus X/D, for
0.1.
FR = 1.01, R = 1.48, Y/D = 0, and Y/D
LjQ
0.6
0.5
FR = 1.01
R
1.48
X/D = 2.5
- - X/D = 5
0.1
0.2
0.4
0.6
0.8
VERTICAL hEIGHT - Y/D
Fiçure A-9.
Plume centerline excess salinity ratio versus Y/D, for
FR = 1.01, F = 1.48, and X/D = 2 5 and 5.
1.0
0.6
0.5
0,1
0
10
20
HORIZONTAL DISTANCE
Figure A-JO.
30
X/D
Plume centerline excess salinity ratio versus X/D, for
FR = 1.44, R = 0.52, Y/D = 0, and Y/D
0.1.
110
0.5
FR = 1,L14
Ii
0,52
X/D
2.5
___ - X/D= 5
- -X/D= 10
>_J
A
'J.
C/)
0.1
0.2
0.6
0.8
VERTICAL HEIGHT - Y/D
Figure A-il. Plume centerline excess salinity ratio versus Y/D, for
FR = 1.44, R = 0.52, and X/D = 2.5, 5 and 10.
0.6
0.5
FR =
0
C,,
R = .1.02
Y/D = 0
0.1
U
0 Y/D
0,1k
:
0.3
A
0.2
0.1
0
10
20
30
HORIZONTAL DISTANCE - X/D
Figure A-12.
Plume centerline excess salinity ratio versus X/D, for
1.02, Y/D = 0, and Y/D
1.44, R
0.1.
FR
LQ
0.6
FR
0.5
= 1.02
X/D=2.5
0
C,,
C')
3
1.1111
- X/D = 5
= 10
A
us
c
5-
I-
0.3
3
0.2
C-i
0.1
0-0
0.2
0.11
0.6
- I
0.8
VERTICAL HEIGHT - Y/D
Figure A-13.
Plume centerline excess salinity ratio versus Y/D, for
1.02, and X/D = 2.5, 5 and 10.
FR
1.44, P
1,0
0.5
FR
R= 1.'i9
It Y/D
0 Y/D
0
01
0
£
0
óôó
0.1
4
0
10
20
30
hORIZONTAL DISTANCE - X/D
Figure A-i4. Plume centerii!ne excess salinity ratio versus X/I), for
1.44, R = 1.39, Y/D = 0, and Y/D 0.1.
FR
tO
0.6
FR
0.5
R
1.44
1.49
2.5
----X/D=10
-X/D5
X/D
(I)
4
Lii
L)
11
I).
0.1
___J
1-
0.2
0.4
0.6
0.8
VERTICAL HEIGHT - Y/D
Figure A-iS. Plume centerline excess salinity ratio versus Y/D, for
FR = 1.44, P. 1.49, and' X/D = 2.5, 5 arid 10.
1.0
86
APPENDIX B
Appendix B contains a complete table of the experimental data
obtained in this investigation.
The data, normalized by a computer
program, include FR, R, X,'D, Y/D, iS/IS, U, S, S, T, T.
In
this listing
U
FR
= Discharge densimetric Froude Number
/ po-p=
p0
R = U /U
0
gD
= Velocity ratio
X/D = Dimensionless downstream distance from the discharge port,
in port diameters
Y/D = Dimensionless vertical distance from the discharge port,
in port diameters
is/IS
0
0
S -s
SQ -s
- Centerline excess salinity ratio
CO
= Discharge velocity, in cm/s
S
= Discharge salinity, in ppt (°/oo)
Ambient salinity, in ppt (°/oo)
T
T
= Discharge temperature, in
Ambient temperature, in ac
C
87
xiJ
1.L,
,L45
.50
5.3
.50
20.3
0
C
.
.,,
.22
..0
.05
.-.'
.:3
23.0
0.3
:7.3
'.0
23,3
0.3
:7.0
7
,
0.0
.IJ
.33
1.33
.53
2.
1 33
.55
5*0
.J
* :0
3
i.L+2
.52
20.0
.13
.33
I
.52
30.3
.0
I
:
2.5
.J
;:;
.33
e_L
UI J
4
1
-7
.t
C
SU
,
.33
3
6.0
16.0
U7
Ii
.06
.72
22.0
3.0
.5
.72
22.3
0.3
0.0
:.0
0.00
..J
._.,.
:6.3
:s.a
:.:
1U
- I,
.55
:3.3
.13
.55
?I:3
22.3
0.3
16.8
160
.31
.33
23.:
.
_0. -
a 1)
,7
'r
_4J
.j
:6.5
:.o
.
-
4n
*
.13
i...#o
,5
.
88
V/C
1.38
.95
5.3
.13
1.38
.95
Z0.j
.13
55
C
/t.S 0
U
S
0
22.
.31
37
65
2.00
0
0
3.3
15.5
3.3
:6.5
1.G
1.3
1.85
10.3
.13
.12
22.5
0.3
:.5
:5.5
1.39
1.06
2.5
..0
.53
.23
22.5
0.0
16.5
:,0
1.03
30.3
.05
22.0
3.3
b.3
:5.3
:.so i.C3
L.Q3
2:.3
C.3
:5.3
:5.
_,
-tu.0
U.0
-
1.58
s')L
4
.65
.13
.55
0.00
.13
.05
3.03
.5
.:3
'. -
a
r
_..JJ
1.36
I
1.12
51
S
1.51
.
ri
1.03
i.
2O
s.o
)
.55
.07
-
.13
.a7
.57
?.32
10.0
.12
.05
.
.2
' 33
.
.
.13
.3
-,
5.3
1.03
13.0
.
4S_/
:2.3
.7
.5,
.:
33;;
J.
r.
)
'
-
22.2
3.2
6.3
.3?
22
0.3
:o.0
:.3
,79
233
.
:6.5
:6.3
3.2
:5.5
7.J
15
-
.:3
23.0
89
FR
3
/
R
I0
tS C/L\S 0
T
1.5; 23.J
.3
.37
3.3S
22.5
r.3
:6.5
:.o
5.3
.0
.55
.3:
9,2L
22.5
3.3
5.5
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