FEDERAL WATER QUALITY ADMINISTRATION
NORTHWEST REGIONAL OFFICE
•3 ^^^^^ »
OCEAN
OUTFALL
DESIGN
CLEAN
Part One Literature Review and
Theoretical Development
April 1970
-------
FEDERAL WATER QUALITY ADMINISTRATION
NORTHWEST REGION, PORTLAND, OREGON
James L. Agee, Regional Director
PACIFIC NORTHWEST WATER LABORATORY
CORVALLIS, OREGON
A. F. Bartsch, Director
BIOLOGICAL EFFECTS NATIONAL EUTROPHICATION
Gerald R. Bouck RESEARCH
A. F. Bartsch
CONSOLIDATED LABORATORY
SERVICES NATIONAL THERMAL
Daniel F. Krawczyk POLLUTION RESEARCH
MANPOWER AND TRAINING Fl H' Ra1nwater
Lyman J. Nielson
WASTE TREATMENT RESEARCH
NATIONAL COASTAL Paper and Allied Products
POLLUTION RESEARCH Food Waste Research
D. J. Baumgartner James R. Boydston
NATIONAL COASTAL POLLUTION
RESEARCH PROGRAM
D. J. Baumgartner, Chief
R. J. Call away
M. H. Feldman
G. R. Ditsworth
W. A. DeBen
L. C. Bentsen
D. S. Trent
D. L. Cutchin
E. M. Gruchalla
L. G. Hermes
-------
OCEAN OUTFALL DESIGN
PART I
Literature Review and Theoretical Development
by
D. J. Baumgartner, D. S. Trent
United States Department of the Interior
Federal Water Quality Administration, Northwest Region
Pacific Northwest Water Laboratory
200 Southwest Thirty-fifth Street
Corvallis, Oregon 97330
April 1970
-------
CONTENTS
Chapter Page
"I. INTRODUCTION 1
II. POLLUTANTS 2
Modeling Pollution Fields Through Mathematics ... 2
III. THE NATURE OF POLLUTION PLUMES IN OCEANIC ENVIRON-
MENTS 4
The Regimes of Flow 4
Environmental Effects 7
Density Stratification . 7
Environmental Turbulence 10
Air-sea Interactions 11
Outfall Geometry 11
IV. THE FUNDAMENTAL EQUATIONS GOVERNING THE MOTION AND
DILUTION OF A FORCED PLUME 14
The Differential Equations for a Forced Thermal
Plume 14
Rectangular Coordinates 21
Cylindrical Coordinates 23
Integration of the Governing Differential Equations. 25
Simplified Hydrodynamic Equation 26
Simplified Equations for a Forced Vertical Plume 28
V. INITIAL DILUTION OF THE POLLUTION FIELD — DEVELOP-
MENT OF WORKING MODELS 33
Zone of Flow Establishment 34
The Zone of Established Flow 38
Particular Cases for Buoyant Plumes 39
Vertical Plume - Stratified and Non-stratified
media 40
Inclined and Horizontal Plumes — Stagnant
Environment 58
-------
Vertical Plumes — Crossflow .......... 72
Negatively Buoyant Plumes ............. ^°
Horizontal Surface Discharge .......... 81
VI. ZONE OF DRIFT FLOW ................. w
Submerged or Neutrally Buoyant Drift Flow ..... 86
Comments on the Eddy Transport Coefficient^. ... 89
Point Source Models ............ • • • • 91
Models Based on a Variable Diffusion Coefficient . . 9-5
Line Source Models ................. 96
Surface Drift of a Buoyant Pollutant ........ 100
VII. NUMERICAL TECHNIQUES ................ HI
Numerical Models ............... ... 113
Coastal Hydraulics - Leendertse's Model
Dispersion
...
BIBLIOGRAPHY . . ..................... l19
NOMENCLATURE ........................ 125
-------
LIST OF FIGURES
Figure Page
1 Regions of flow for fluid issuing into a density-
stratified medium with current 5
2 Vertical plume in a non-stratified stagnant medium . . 8
3 Vertical plume in a stratified stagnant medium .... 9
4 Inclined plume in a stratified stagnant medium .... 9
5 Vertical plume issuing into homogeneous crossflow . . 10
6 Interacting plumes from a ported outfall line with
horizontal discharge, in a density-stratified
environment . 13
7 Rectangular coordinate system 15
8 Cylindrical coordinate system 23
9 Flow establishment 35
10 Relative velocity and concentration profiles 36
11 Velocity profile assumed by Morton 47
12 Virtual source 49
13 Comparison of Morton's model to representative
observation 51
14 Horizontal plume issuing into stagnant, density
stratified medium 59
15 Plume issuing at an angle into a stagnant medium ... 62
16 Maximum height of rise versus downstream distance
for horizontal plume in linearly stratified
environment 70
17 Maximum height of rise for a vertical plume in a
linearly-stratified environment 71
-------
18 Vortices associated with a bent-over plume 73
19 Schematic representation of smoke plume separating
into (a) two or (b) three streams due to wind
effects 73
20 Vertical plume issuing into uniform cross-current . . 75
21 Typical flow patterns for plumes with negative
buoyancy 79
22 Initial and steady penetration heights of a vertical
heavy plume in homogeneous medium 80
23 Two types of drift flow resulting from submerged
horizontal discharge of buoyant fluid in a density
stratified ocean with current 85
24 Brooks' line source model (for ported outfall line). 96
25 Interpretation of the Larsen and Sorensen Model
for pure gravitational spread 102
26 Planar view of flow issuing to a semi-infinite
medium 109
-------
LIST OF TABLES
Table Page
1 Radial Diffusion Models from Okubo (37) 95
2 Line Source Solutions 99
-------
INTRODUCTION
The growth of population and industry in the coastal regions
of the country, along with increased use of coastal waters for
disposal of wastes produced inland, presents the potential of new
or increased pollution in a valuable national resource. In order
to assure that marine water quality is maintained and enhanced for
present and future water uses, each source of waste input must be
evaluated for its impact on the environment. Treated wastes must
be discharged in such a manner and location that the resulting con-
centrations and contact times do not violate the water quality
requirements specified for the desired water uses. The disposal
process can be designed within these constraints through a mix of
at least four techniques:
• degree and type of waste treatment;
« site selection;
. outfall or barge distribution of wastes;
. barrier construction.
This report presents a review of the literature pertinent to
the theoretical development of present ocean outfall design tech-
nology as it applies to waste discharges in general. Part II,
entitled "Recommended Procedures for Design and Siting," is in
preparation and will incorporate references and data collection
methods for specific pollutants.
Barge discharge of wastes is becoming more important because
of the increasing production of concentrated wastes, and the
increased concern for continued incineration and ground disposal
of wastes presently practiced inland. This technology will be
explored in a future report.
-------
POLLUTANTS
Material pollutants are those commonly associated with munici-
pal and industrial wastes: biological species, chemicals, sludge,
etc. Another material pollutant, which has caused no problem in
the past but which promises some cause for concern in the future,
is the highly saline reject water from desalinization plants.
A non-material pollutant new to coastal waters is appearing on
the scene, threatening to present problems in the near future
which might surpass those of municipal and industrial wastes. This
pollutant is heat - reject heat from very large thermal power sta-
tions using coastal water for condenser coolant. In either case
we need to be able to predict pollutant concentrations in the local
sea, as a function of both location and time.
Modeling Pollution Fields Through Mathematics
The problem of mathematically modeling pollutant dispersion in
coastal waters involves solving the basic equations of marine hydro-
dynamics, and heat (or mass) transfer. Although the fundamental
equations which describe the differential motion and diffusion are
reasonably well understood, solution of these equations is analyti-
cally intractable without gross simplification of the oceanic motion,
pollutant flow, and boundary conditions. In addition, the terms
associated with pollutant decay and interaction are frequently not
well described.
-------
Analytical research, then, has been a process of fashioning
simplifications in order to derive tractable mathematical models.
Along with the analytical research, laboratory and field experi-
ments have yielded additional information which developed reliable
semi-empirical models.
An approach to the problem of mathematical modeling which has
apparently not yet been fully utilized is the use of numerical tech-
niques in conjunction with high-speed digital computation. The
numerical approach involves approximating the governing differential
equation with finite quantities and carrying out the required numeri-
cal computations on a digital computer. The advantage of this
procedure is that many complications that must be ignored in the semi-
analytic models can be incorporated in the numerical model. Regardless
of which approach is taken, certain data must be obtained in the field.
Generally, mean currents, some estimation of turbulent transport
coefficients, density stratification, and pollutant interaction data
are needed.
Subsequent discussion in this report will be restricted to only
those characteristics of a pollutant which influence the general
description of its fate, such as density. The considerations of
decay, interaction, sedimentation, and air-sea exchange, etc.,
which might be unique for any specific pollutant, are not included.
-------
THE NATURE OF POLLUTION PLUMES IN OCEANIC ENVIRONMENTS
Before investigating theoretical aspects and the various methods
of analyzing the dispersion of pollutants emitted from ocean outfall
systems, the gross, behavior of the pollution flow field will be
reviewed to give a general idea of the nature of the flow.
Fluid which issues continuously from an outfall pipe on the
ocean floor passes through several flow regimes as it disperses
into the surrounding ocean. It may contain pollutants in the form
of either heat or matter; and, when ejected Into the surrounding
environment, it may be buoyant or not. However, unless specifically
stated otherwise, discussion will be concerned with a positively
buoyant fluid. It is common to refer to free convection flow in
the environment as a plume, for in this case the general pattern
of motion is caused by virtue of a density disparity between the
fluid and the surroundings. A jet is characterized by initial momen-
tum of the fluid. If the injected flow has a density disparity and
momentum, then the flow assumes some character of both types of
flow. This mixed flow could be called either a buoyant jet or a
forced plume. Since the overall pollution dispersion in the sea
resembles a plume more than a jet, the term forced plume or plume
will be used here.
The Regimes of Flow
Experimental investigations of forced plumes have revealed the
existence of three distinct flow regimes, as follows (see Figure 1):
-------
SEA SURFACE
Current, U
Established Flow
•FLOW ESTABLISHMENT
potential core (not mixed) with surroundings
port
//// iff / f / f ' ' / / ' it/ / / / ( / / / / i //if/
Figure 1. Regions of flow for fluid issuing into a density-stratified
medium with current.
-------
• zone of flow establishment,
• zone of established flow, and
• zone of drift flow.
The zone of flow establishment is associated with the transi-
tion from pipe flow to an established forced plume flow. The flow
from an outfall orifice or pipe enters the ocean with essentially
a uniform velocity profile, which begins to deteriorate at the flow
boundary as a result of turbulent shear. The region of mixing
spreads both inward toward the center of the jet and outward into
the medium. Within a short distance from the orifice the inter-
change of momentum will have spread to the center of the jet. At
this point, it is generally assumed that the velocity profile is
fully established.
In the zone of established flow, the driving force for the plume
may be either momentum or buoyancy, or both. As distance is increased
from the orifice, the width of the plume increases as a result of
lateral mixing or turbulent diffusion (commonly called entrainment).
Eventually the pollutant will reach a position where momentum
no longer affects the flow and buoyancy may play only a minor
role. In this region, called 'the zone of drift flow' here, the
spread of the pollutant is dominated by the environmental turbulence,
although a lateral density flow may also be present if the pollu-
tion field is situated atthe surface with residual buoyancy.
-------
Environmental Effects
The environment Into which the fluid 1s discharged can Influence
the nature of the plume flow, due to
• density stratification,
• environmental velocity (oceanic currents),
• environmental turbulence,
• air-sea interactions.
Density Stratification
Both theory and experiment have shown that a buoyant plume
will reach the surface if the ambient fluid is homogeneous with
respect to density.
However, the ocean is rarely homogeneous in this respect, except
perhaps in shallow coastal waters where wave action causes good
vertical mixing to occur. Where the ocean is density stratified, it
is possible that the plume will not penetrate to the surface, at
least in the zone of established flow. The reason for this
phenomena is as follows.
The plume entrains the heaviest environmental fluid near the
orifice, decreasing the density disparity between the plume
and surroundings. As the plume ascends the decrease continues
as the density of the surroundings decreases. The plume will
eventually reach a point of neutral buoyancy, which may be below
the surface. At this point the flow continues upward only because
of the vertical momentum it possesses. As the flow continues upward,
-------
8
it continues to entrain liquid, but the plume is heavier than the
surroundings; hence, the flow is negatively buoyant. Eventually,
all upward vertical momentum is lost and, since the plume liquid is
heavier than the surroundings, the pollutant will cascade downward
around the upward flow until it reaches equilibrium with the sur-
roundings.
The general shapes of plumes in non-stratified and stratified
stagnant liquids are shown below.
velocity profile
port
Figure 2. Vertical plume in a non-stratified stagnant medium.
-------
SEA SURFACE, P,
Velocity profile
p < p , p
-------
SEA SURFACE, pg
Plume
Absolute velocity profile
po
Figure 5. Vertical jet issuing into homogeneous cross flow.
Environmental Turbulence
The origin of oceanic turbulence is not fully understood, although
it is probably caused mostly from wind-generated wave action. As
such, the turbulence is neither homogeneous nor isotropic, and
only the gross behavior can be described.
Turbulence scales that are on the same size or larger than the
plume cross-section will have an effect similar to a cross-current,
and all scales should have some influence on the plume entrainment
rate (although it is thought that the influence in the zone of estab-
lished flow is small (Cederwall, 11)). Turbulence has a more
dominating effect in the zone of drift flow. Scales of motion larger
than the pollutant field result in transport action similar to
oceanic currents, while smaller scales of motion add to the eddy
diffusion of the pollutant; thus, as the pollutant field spreads,
larger and larger scales of eddy mixing come into play.
-------
11
Since most oceanic waters are density stratified (except
perhaps near the surf zone], vertical turbulence is suppressed to
a great extent. Thus, a pollution field may disperse much more
rapidly in the lateral direction than in the vertical direction.
Air-sea Interactions
Although wind stress can indirectly affect the motion and
dilution of a submerged pollution field through perturbing the mean
oceanic motion and by causing turbulence, the most dramatic inter-
actions with the atmosphere occur when the pollution field develops
on the surface. The two major interactions are wind stress and
transfer of heat and mass.
Wind-driven surface currents may cause the surface drifting
pollutant to stagnate along the coast or cause it to be carried out
to sea. Transfer at the sea surface is of primary importance in the
case of thermal pollution. Thermal energy interchange at the sea
surface will occur through the mechanisms of convection, evapora-
tion, and radiation. Volatilization and oxidation of oil and other
pollutants are also important considerations.
Outfall Geometry
Outfall geometry also affects the nature of the plume. Greater
dilution can be achieved with outfall structures employing a series
of horizontal rather than vertical ports or orifices. In this case,
each plume will follow a curved path until it reaches either the
maximum height of rise or the sea surface. Port spacing is important
-------
because of the possibility of plume interaction and the resulting
decrease in dilution efficiency. Figure 6 illustrates a possible
flow configuration from a ported outfall with a slight downstream
drift of the density-stratified receiving water. Unless otherwise
indicated, discussions here will be confined to round orifices.
-------
s s s / / s
SIDE VIEW
/ f / f
/ f f f T f / f f r f I / • / f
END VIEW
Figure 6. Interacting plumes from a ported outfall line
with horizontal discharge, in a density-stratified
environment.
-------
14
THE FUNDAMENTAL EQUATIONS GOVERNING THE MOTION AND DILUTION
OF A FORCED PLUME
Prior to assessment of various analytical approaches for
evaluating plume dynamic behavior, the fundamental laws and equa-
tions which govern the flow will be introduced. The reason for
this introduction is to clarify the physical grounds upon which
plume mathematical models are based.
There are two basic approaches one may follow in deriving plume
models - one is based on a differential analysis, and the other on
an integral analysis. The differential approach involves solving
the general partial differential equations of motion and heat (or
mass) diffusion to arrive at velocity and temperature (or concentra-
tion) distributions. In the case of an integral analysis, specific
shape of the profiles must be assumed, based on intuition or experi-
ment. Since the integral analysis involves several simplifications
and assumptions about the flow field, analytic discussions of them
will be deferred until special cases are considered.
The Differential Equations for a Forced Thermal Plume
To be specific, the discussion here will be based on the con-
sideration of a thermal plume where heat is the pollutant. Where
desirable, analogous equations will be written and discussions
presented for the case where matter is the pollutant.
The differential equations for a thermal plume are derived
from consideration of the following physical laws:
-------
15
• Continuity (conservation of mass,
• Newton's Second Law (conservation of momentum),
• First Law of Thermodynamics (conservation of energy), and
• An Equation of Volumetric Expansion.
Detailed derivation of the equations of motion and energy will
not be given here, since these equations can be found in nearly
usable form in texts dealing with fluid mechanics or convective heat
transfer (e.g. Ref. 7). A few modifications must be made for appli-
cation to a plume in an oceanic environment. The coordinate system
for the following discussion is shown in Figure 7.
(Earth rotation)
Vertical: x.,, z
EARTH
Figure 7. Rectangular coordinate system.
-------
16
The differential equations for incompressible, density strati-
fied laminar flow written in Cartesian tensoral form are as
follows:
Continuity:
3p .n. 8P +D 3Ui _ 0
•5T+UT5*7 P3x7" °'
In the ocean, density varies with both time and space as indicated
by the above equation. However, changes caused by temperature or
mass concentration variations are usually quite small and may be
neglected in consideration of mass conservation. Thus, the continuity
equation may be approximated with acceptable accuracy by
5-.
It will be shown later, however, that these variations cannot be
neglected in the momentum equation and are the point of concern in
the energy equation.
Momentum :
3U.
i . ,, 1
+ U.
at wj ax.
J
3P . . »'U1
oX
•
I
Energy:
_ o ^
- + $ + q . * (3)
Volumetric Expansion:
1 9v _ 1 3p (4)
a"'
* The molecular fluid properties jj and k are assumed isotropic
in this work.
-------
17
In the above equations the Einsteinian notation is used (a
repeated index in a term implies, summation over all three indexes--
1, 2, and 3).
The symbol 6-3 is the ICronecker delta and is equal to 1 when
i = 3, and otherwise zero. The term $ which appears in the energy
equation is a viscous dissipation term and may be neglected. For
simplicity, changes in p are assumed a function of T alone in
equation (4).
Consider the fluid density expressed as p = p^Cx ) + Ap,
where p (x ) is the reference density.*
Equation 2 may be written as:
32U.
at
ax
- (p^+ApJgfi^ + u
J J
Let p be a constant reference density, specifically p =
Division by PQ, noting that (PQ = p) yields
3U.
9P
"*j
92U.
This manipulation was done to separate out the buoyancy effect
The important idea to note here is that Ap has significance in
establishing the above buoyancy term in the momentum equation;
(5)
* Hereafter, the reference density p CxJ will be written simply as
*** 3
p with dependence on coordinate xg always implied.
-------
18
however, ignoring small variations of p will not otherwise affect
the outcome significantly Cthe so-called "Boussinesq approxima-
tion"). The symbol v indicates kinematic viscosity and is defined
as v = y/p.
Equations (3) and (5) must now be modified for application to
the oceanic environment. First of all, the equations of motion
must be modified to account for the earth's rotation (the Coriolis
effect). If a is the earth's rotation vector, equation (5) becomes
3U_. 3U_. i ^n IP.--P r 3 U..
po
-1 g6.3 + v
where e.. - is the permutation tensor which takes values of zero if
any two of the three subscripts i, j and k are identical, +1 for
even permutations, and -1 for odd permutations of the subscripts.
Next the equations of motion and the energy equation must be
modified to account for oceanic turbulence. To do this the depen-
dent variables are considered as composed of a mean or average
part and a superimposed random fluctuating part; for instance,
U1 = U. + u.,
where U.. is the mean velocity and u. is the fluctuation part. These
definitions are substituted into the equations of motion, and the
result is time averaged term by term over a sufficiently long
period of time.
The resulting equations of motion for turbulent flow are
then
-------
3U.
31L
3t
1 3P
p 9x .
Uj c
'X • oX
j j
. Po
tu • u • j T tc . ., !£ .
1 i iitn
• J i j i^ j
i3 Bx.9x.
j J
(6)
which is seen to be identical in the mean motion with equation
(5) except for the appearance of the term
A new quantity is now defined
19
which is called the Reynolds stress. Finally the complete equations
of motion in the rotating earth reference frame are written as
9U.
8U.
-l
3R..
(7)
for the mean flow. Here the overbars denoting average quantities
have been omitted. Another way in which the turbulent stress terms
are handled is through the Prandtl mixing length theory (e.g. Ref. 36)
In this case
-------
20
so that the equations of motion become:
)U. 3U. T 3P
1 + It 1 + ?n 0 II - ' H
)t j 3x. ijk j k pn 3x.
J 01
P "P
K°° -j
PO
nf + 3
96i3 3xj
O U •
1 -. •> 1 ^ 1 ^^^^M.
•
(8)
where again variations in p are considered small and e.. is the
' J
eddy diffusion coefficient for momentum, a second order tensor.
The turbulent form of the energy equation may be derived in
a similar fashion. For constant p and C the energy equation for
laminar flow is written as
+ U .
3t uj 3x.
3T_
3x,
PC.
where a is the molecular thermal diffusivity,
-------
21
where C is the pollutant concentration, e^. is the eddy diffusivity
of mass, V is the molecular diffusion coefficient, and rh is the
production or loss of the species mass per unit volume. Equation
(10) is based on the diffusion of a single species whose concentra-
tion is C(x , x , x , t). These equations,(1), (4), (8), and (9)
1 2 3
(or (10)), then form a complete set of equations from which a plume
may be analyzed for dynamic behavior. Unfortunately, the equations
are extremely complex and evidently cannot be solved analytically
for the general case.
Rectangular Coordinates
For analysis of the local sea, the geopotential surface may be
assumed flat. Rectangular coordinates are defined in Figure 7.
The Coriolis term in equation (8) may be expanded, as:
= 2
so that:
2eijknJuk •
- U Cos e.
x y
0 nCos
u v
Cos - V
W
(11)
(12)
where e , e , and e are unit vectors for the east, north, and
x y z
vertical directions, respectively. Then in component form the
governing equations become:
-------
22
Continuity:
9£ 9V_
9x 9y
s-«-
(13)
x-Momentum:
2fi(W CoscD - V Sincfr) = - -
IF
y-Momentum:
£. 00
Dt
~yy
"yz' 9z
(15)
z-Momentum:
DW _
Dt "
3_ f
ay
2 J2U Cosc|) =
[V+P ) ^-1 H
lv EzyJ 3y
1 3P +
-pQ 3z
P.-P
Po
• *!x
I + 1 IW
\ v •» v * Civ
iA OA
h 3_ f/^e \ 3H1 (
9z (V^£zz) 9z ' (
(16)
* The operator,
stan.tial derivative.
» is known as the sub~
-------
23
Energy:
Dt =
ay
(17)
Cylindrical Coordinates
Figure 8 below illustrates the cylindrical coordinate
system.
(Earth rotation)
z (vertical)
East
EARTH
Figure 8. Cylindrical coordinate system.
-------
In this system, the z-direction is perpendicular to the geo-
potential surface,> 1s the latitude, r 1s a rotating vector which
lies 1n the flat geopotentlal plane at an angle 8 measured from
east, and fi 1s again the earth's rotational velocity.
The corloHs components 1n cylindrical coordinates are given
by:
ncossine ncos^Cose nsin
u
W
08)
The notations ew, eQ and 3, designate unit vectors 1n the r, 0 and
r o z
z directions, respectively.
Thus, the governing equations for an incompressible flow may
be written 1n cylindrical coordinates as follows:
Continuity:
r 3r
r 36
. 0.
(19)
Motion (In terms of fluid stress
r-direction:
DU
Q
+ nW CosCose -
1 3P 1
» «*^
P,
3rT
3T
rr .I >e _ ^ee
(20)
* The equations of motion 1n terms of stress T are more appropriate
for development to be considered later. y
JJA D 9j.lL34.634.y3L
** In cylindrical coordinates, Qt*st + ^r3rr~9e3z '
-------
25
6-dtrection:
DU UfU
Dt r
1 3P L
" P0r 3e " PC
z-direction:
Sin(j>-nHCos<|>Stne
1 9Tee + BT6A
r 36 6z
(21)
1 3_ f ,
r 3r lrTzrJ
Energy:
PI.
Dt '
3i
zz
3z
(22)
1 3
r 3r
Ca+EHr>r If
1 3
r2 36
3T
V " ^Lf Q / 'iO
no oo
(23)
Integration of the Governing Differential Equations
The differential equations governing the motion and dilution
of a thermal plume were set down in equations (1), (8), and (9).
The problem of predicting pollutant dispersion, therefore, lies in
gaining the solution to these equations, once having established
such requirements as boundary conditions, eddy transport coefficients,
etc. However, these equatfons are evidently impossible to solve
analytically while maintaining all terms of the equations required
to describe the motion and dilution of the pollution plume from its
source to its ultimate fate.
-------
26
Simplified Hydrod.ynamjc Equation
Only a very few analytical solutions to the equations of
motTon (8) are known and these have come by accepting gross simpli-
fications of the physical phenomena. For instance, gross motion
in the deep oceans as influenced by Coriolis effects Cearth rota-
tion), sea surface slope, density anomalies, etc., fall into a
class of motion called "friction! ess flow," implying the fractional
terms in the equations of motion have been neglected. In other
simplified flows, frictional terms are included under specific
conditions. Also, vertical momentum is usually ignored. Examples
of frictionless flow are as noted:
• Inertia! Currents -
^ = 2flVSin4> (24)
{j£ = -2^Sin4>. (25)
The term 2ftWCos<|> has been neglected in equation (24), since it
is small compared to 2VftSin. Equations (24) and (25) are used to
approximate the motion of a water mass during an acceleration where
horizontal pressure forces are comparatively small.
• Geostrophic Currents -
For a water mass in steady motion, Coriolis forces are balanced
by lateral pressure forces, so that:
lf£ (26)
(27)
-------
27
Gradient Currents:
If inertia! terms are important for a water mass in steady
motion, such as a gyre, then
(28)
Some examples of frictional flow are:
Non-accelerated Drift Currents:
Where Cortolis forces are balanced by frictional forces in a
homogeneous ocean,
2UnSin =£2 0 (30)
- 2VOST n4> - T (31 )
Solution of these equations gives the so-called "Ekman Spiral"
(see Ref. 36) for the motion of the water mass. This type of flow
has notable implication for wind-driven drift flow.
Density Flow :
For the case where frictional forces are balanced essentially
by a horizontal pressure gradient,
IE. - JL f ^M} + JL f jy.}
3x ~ 3x lex 3x1 3y ley By I (32)
3y Sx [^x axj 3y
-------
28
The above examples of frictional and frictionless flows are
examples where analytic solutions are possible, but may be diffi-
cult to obtain because of complicated boundary conditions.
To completely describe the pollution plume, however, neither
inertia! terms or eddy diffusion terms can be neglected over the
entire range of flow, but certainly some kind of simplification
must be made if the problem is to be solved at all. For this
reason it has been the course of both theoretical and experimental
investigations to consider each flow regime as a separate problem,
within which separate simplifications may be warranted.
Simplified Equations for a Forced Vertical Plume
The zone of established flow has received much analytical and
experimental attention. In this regime both inertial and viscous
forces play an important role in determining the nature of the
flow. The simplest case is the axisymmetric vertical plume. Since
most of the plume theories center around assumptions fundamental
to the solution of the vertical flow problem, the equations for
this particular case will be derived from the general equations
of motion.
The general assumptions associated with the vertical plume are
as follows:
• Steady flow: 9U./3t = 0
• Flow is axisymmetric: UQ = 0
• Coriolis effect is neglected: ZeU = 0
-------
29
3P
• Flow field Is assumed hydrostatic throughout: -^ = -prog
• Density difference between the plume and surroundings is
assumed small compared to the density at any point in the flow
field: |poo-p|«p
• Plume is fully turbulent
- Eddy transport of momentum and heat is only effective in
the lateral direction (normal to jet axis)
• Molecular heat conduction and viscosity ignored.
With the above assumptions and simplifications it is possible
to disregard a number of terms in the governing differential equa-
tions. Also, for the axi symmetric round jet it is more convenient
to consider the general equations in cylindrical coordinates. In
simplified form the governing equations become:
Continuity:
3W _ 0 (34)
3r 3z
Momentum:
Schlicting (1960, ref. 45a) employed an "order of magnitude"
analysis incorporating the previous assumptions, which can be used
to show that equation (22) reduces to:
--J-^-CrT,). (35)
,. 3W .,, BW
W 31 + Ur 9f
Energy:
In order to deal with specific quantities, a thermal plume
will be considered. However, the end result of this derivation will
-------
30
be valid for pollutant plumes in general.
The energy equation (23) reduces to
_ P r - u +
r 3r eHrr 3r ur 3r
21
32 '
(36)
It is sometimes more convenient to consider the energy equation in
terms of a temperature disparity,
G = T-TQ,
where TQ is a reference temperature, specifically the temperature
of the surrounding medium at the jet orifice. The energy equation
is then written
1 3
I "u 4- U "^
V 3r 32 '
(37)
Equations (35), (36), and (37) may be written in slightly simpler
terms by considering the continuity relationship (34). Since
3 _ (\J2\ = \j 3" j. u 3"
3Z Vw ' W 3z W 32 '
3r
= rU - + W
rur 3r + W 3r
then equation (35) is written as
3W2 w 3W
32 32
'PCO-P'
PO
g_
,
1
V
1_ 3_
r 3r
3 ,
• 3r '
r 3r
(38)
-------
31
By multiplying equation (34) by W and substituting the result
into equation (38), equation (35) is simplified to
"' *
ar
^0 J
The same procedure may be applied to the energy equation (36) when
the continuity equation is multiplied by T. In this case, equation
(36) is simplified to:
F IF
or in terms of a temperature disparity 0,
rlrh/lr] ' F!T <"V8> +
By using the definition of volumetric expansivity (equation 4) a
relationship between p and T (or ©) may be developed. By defini
tion
' R - 1 3P
6 ' - p 3T '
and to a first order approximation
(4)
AT = -
so that
(43)
Then equation (43) may be written as
-------
32
1JL
r 3r
£Hrr
(44)
with p B = constant.
Thus, equations (34), (39), and (44) are one form of the set
of differential equations which must be solved to establish the
flow behavior of a vertical plume.
-------
33
INITIAL DILUTION OF THE POLLUTION FIELD -
DEVELOPMENT OF WORKING MODELS
In the previous section the fundamental equations governing
the motion and dilution of a marine pollution field were developed.
It is the purpose of this section to illustrate how analytical and
experimental research has proceeded, using these equations, in the
development of working mathematical models which describe the fate
of a plume from its formation at the orifice to its equilibrium
level in the ambient field. But it is not the purpose here to
provide a detailed literature review of this subject. Excellent
reviews of this material may be found in Hinze (21) for momentum
jets and in Baumgartner (6) or Cederwall (11) for buoyant jets.
Research on jet flow has shown that the zone of initial dilu-
tion consists of three flow regimes, which are:
a potential core region, followed by
a transitional flow, leading to
the zone of established flow.
In the first two zones, it has been illustrated analytically
that similarity solutions* are not possible. On the other hand,
once the flow becomes fully established, similarity approximations
* The premise on which similarity solutions for jet flow are based
is that velocity profiles are geometrically similar at all axial
locations, differing only by a magnification factor; that is,
U(r,z) = f[z'F(r)].
For a plume, U(r,z) = U(z)f(r/b(z)).
-------
34
are valid. The transitional zone is usually very short and is
generally ignored in model analysis (reference 21) - as will be
done here. In the literature, the flow regime between the orifice
and the point where the flow becomes established is called "the zone
of flow establishment," and the distance required for flow estab-
lishment is denoted by Z .
e
Only the mainstream of research on this subject will be presented
here - with major emphasis on the zone of established flow, since
it is much more important. The discussion on established flow is
presented in some detail so that a clear understanding of underlying
principles and assumptions is at hand.
Zone of Flow Establishment
As far as the overall plume dispersion is concerned, the zone
of flow establishment is of less interest (because it is a compara-
tively short distance) except as it enters into established flow
solutions for integration limits or boundary conditions. Thus, in
this light, it is important to know the length for flow establish-
ment, Z .
It is usual to assume that the polluted fluid enters the local
environment from an orifice as a plug flow (velocity is uniform
over the diameter). As the flow penetrates the surrounding fluid,
there is a zone of intense shear where eddy mixing causes the flow
of pollutant at the edge of the plume to decelerate and the adjacent
surrounding fluid to be accelerated. This zone of turbulent mixing
-------
35
(Figure 9) spreads inward toward the center of the plume and outward
into the surrounds as distance from the orifice is increased.
mixing
zone
Fully
Established
Flow
Figure 9. Flow establishment.
The region where the plume flow is undisturbed by the lateral
eddy diffusion is usually called the "potential core." Once the
lateral spread of eddy diffusion has penetrated to the centerline,
the flow is considered to be fully established. Distinction must
be made between the development of the velocity profile and the
temperature or concentration profile. It has been established
experimentally that heat and mass diffuse more rapidly than momen-
tum, in free turbulence. That is to say, that the eddy Prandtl and
Schmidt numbers are smaller than 1.0 in free turbulence. The net
result of this difference is that temperature and concentration
profiles develop more rapidly than do velocity profiles.
-------
36
velocity profile
concentration profile
Figure 10. Relative velocity and concentration profiles.
Many experiments have been carried out by various investigators
in an effort to establish the length and shape of the potential
core for round jets issuing into stagnant fluids. A good review
of this work with references is given by Hinze (21). According to
Hinze, there is general agreement in most results. The shape of
the potential core is evidently conical, and, according to measure-
ments of Hinze and Van der Hegge Zijnen (22), 2Q = 60 to 8D for
the velocity profile. The work of Albertson, Dai, Jensen and Rouse
(4), which seems to be the most frequently quoted, showed that
I - 6.2D for the neutrally-buoyant momentum jet.
The results above are associated with neutrally-buoyant jets.
If the issuing fluid has significant upward buoyant force which
causes changes of momentum similar in magnitude to the initial
-------
37
momentum, the zone of flow establishment takes on different charac
ter. For instance, the potential core velocity increases so that
at Zg, Um>Uj, where U is the plume center! ine velocity. Abraham
(1) presents a method for calculating the length for concentration
profile establishment in terms of the initial densimetric Froude
number Fr,, as follows:
u
1,42 c 3+c2_ u a o. (45)
t~r, 22 o
J
where y is the eddy Schmidt or Prandtl number and K is an entrain-
ment constant. The densimetric Froude number is given as
Frj • V
Dg, (46)
where U, is the orifice velocity, p density of surrounding fluid at
the level of the jet, g acceleration of gravity, and p, density of
the issuing fluid. According to Abraham, the constants should take
values: u = .80 and K = 77. Then, once Fr, is established, the
j
quantity C can be found which relates Z and D as
2 °
C2 = Ze/D.
For Fr,-**>, C approaches about 5.6, which is about 10 percent less
J 2
than the 6.2 found by Albertson et al. for the velocity flow
establishment length of neutrally-buoyant jets. The difference here
arises from the difference between the rate of spread of matter (or
heat) and momentum, for if both are equal, y= 1, and for Frj-*=°,
C ->€.2.
-------
38
The Zone of Established Flow
This particular flow regime has evidently received the majority
of theoretical and experimental attention to this time - and, from
a practical standpoint, with good reason. This zone is extremely
important, because it establishes the initial character of the pol-
lutant dispersion. For instance, with a proper diffuser design
(possible only with a reasonably accurate forced plume model) the
pollution field can be made to spread at the surface or at a submerged
level in a density stratified field, depending on which is the most
propitious for the specified water quality situation. For a waste
heat discharge, surface spreading will accomplish a greater rate of
temperature reduction. Submerged fields of other wastes reduce un-
sightliness and also may reduce beach hazards associated with onshore
surface currents.
Theoretical analysis of neutrally buoyant jet flow dates back
to Tollmien's (51) work in 1926, but evidently Schmidt (47) in 1941
was the first to consider the mechanics of convective plumes, or
plumes with no initial momentum, such as convective currents over
fires, etc., from both the theoretical and experimental views.
Schmidt's work was reported in the German literature and, apparently
because of the war, went unnoticed until Rouse et al. (45) had carried
out similar work in the early 1950's. Since that time a number of
researchers have investigated the established flow regime of a
forced plume, both theoretically and/or experimentally. From these
studies various mathematical models have been proposed and applied
-------
39
to both oceanic and atmospheric problems. Although the models
differ in some ways, the theoretical approaches are somewhat simi-
lar, and there is reasonably consistent agreement in results.
Parti cular Cases for Buoyant PIumes
A set of simplified governing equations for a vertical plume
issuing from a round port were set down in Chapter III. Even though a
vertical plume is not to be expected from an outfall system issuing
to receiving waters which are in lateral motion, or for outfalls
issuing the pollutant other than vertically, the ideas involved in
the vertical plume analysis are basic to more complicated flow
problems. For a buoyant plume, researchers have constructed
mathematical models for the particular cases which follow (in
order of increasing complexity):
1) flow issuing vertically into a stagnant, homogeneous medium;
2) flow issuing vertically into a stagnant, density-stratified
medium;
3) flow issuing inclined to the vertical for the above two
cases; and
4) flow issuing vertically into a homogeneous medium with a
uniform cross-current.
In addition, some work has been carried out for the above cases with
a negatively buoyant pollutant, which will be discussed briefly at
the end of this section. No models have been developed specifically
for an inclined forced plume in a density-stratified medium with a
cross-current.
-------
40
Vertical Plume - Stratified and Non-stratified Media
The analysis of plume dynamics by Schmidt (47) and Rouse
et al. (45) dealt with pure buoyancy or thermal convection.
Later research has generalized the flow models to include all
three classifications of flow.
Rouse, Yih and Humphreys (45). Rouse et al. reported analytical
and experimental results for the dynamic and thermodynamic behavior
of a convective plume in 1952. In this study, Rouse considered
turbulent convective flow above continuous line and point sources
of heat rising in a stagnant, homogeneous medium.
To analyze the dynamic and thermodynamic behavior of the
axisymmetric plume, Rouse considered the equations (34), (39), and
(44). (For the line source, the rectangular counterparts of these
equations were used.) A momentum integral equation was derived by
integrating equation (39) from r=0 to r-*», which is given by:
pW2rdr = - ^rdr. (46)
In equation (46), the quantity Ay is the buoyant force and sym-
bolically given as:
If the energy equation (44) is integrated over the same range of
r, the result is
r°°
WAyrdr =0. (47)
-------
41
As a third equation, Rouse uses a statement of the conserva-
tion of mechanical energy, which is derived by multiplying the
momentum equation (39) by W and integrating with respect to r to
obtain
d W3
dT PF-
rdr = -
WAyrdr -
T
o
3W
rz 3r
rdr. (48)
Now equations (46), (47) and (48) present three equations with
which to solve for the three unknowns W, Ay, and T. However, these
equations cannot be completely solved unless the functional forms
of the unknowns are given beforehand. Rouse assumed functions
for these quantities in accordance with ideas of dynamic similarity,
In this way the following relationships were established:
1) lateral spread, cr^z
2) W (center!ine) ^z"1/3
m
3) AYm (center!ine) %z~ ' .
Finally, it was established through experimentation that axial
velocity and temperature were well represented by
,, V3 -1/3
W(0,z) = Wm = 4.7 (^) z (49)
and AT(0,z) - ATm = -ll[p(-w)*]1/3z-5/3, (50)
where w represents the mass flow rate given by
w = -gH/CT. (51)
-------
42
The subscript m relates to plume center! ine condition.
In equation (51), H is the heat input and C is specific heat.
The experimental portion of this study also revealed two other
important findings ~ first, that velocity and temperature profiles
were essentially similar at all elevations; and second, that these
profiles could be closely approximated by the Gaussian distribution
curve; that is,
H(r.z) = W^xpC-K^rVz2] (52)
and AT(r,z) = ATmexp[-K2r2/z2] (53)
where Kx and K2 are experimentally determined constants found to
be 96 and 71, respectively, for the point source.
Priestley & Ball (41). In 1955 Priestley and Ball reported
results for studies on thermal plumes rising vertically in a
stratified atmosphere. Assumptions are those that give rise to
equations (34), (39), and (41). Equation (39) is multiplied through
by W, and the term (p^-p)^ is replaced by (T-Tj/T0 (perfect gas
law, p=P/RT). Also, in equation (41), 6=T-T , instead of T-T .
oo (J
The above-mentioned set of equations are then integrated from
r=0 to r-x» to yield the following:
r T-T«
pW2rdr = pg
To
rdr. (54)
-------
43
d
dz
.00
£• pW3rdr =
n '
pgw
ft
fT-T ]
oo
0
rdr -
i
(55)
"o
d
ay
pW(T-Tjrdr -
pWr
dz.
(56)
These equations are identical to those used by Rouse, except
for nomenclature and the right-hand side of equation (56). This
term represents the effect of density stratification which Rouse
does not consider.
Priestley then argues that, based on the findings of Rouse,
velocity and temperature profiles are Gaussian (including the case
of density stratification) and to a first approximation these
profiles (heat and momentum) are identical; that is,
«- = ^- = exp[-r«/2Rc]
/57\
ij TT =*HL-' -/ '-rv^J VJ' I
\ Alm
where the subscript again designates conditions at the plume axis
and R is a characteristic lateral scale for the plume. It is
also assumed that
1
2
pv
(58)
Applying equations (57) and (58) to the equations (54), (55),
and (56) yields the following set of differential equations:
-------
44
d
Hz
d
dT
2RC2
in
m
m <»
g -
dT
oo
3z~
m
(59)
(60)
(61)
From these equations Priestley established that the constant
c, (a profile constant or spreading constant) may be determined by
dR
c
which results by combining equations (59) and (60).
For neutral conditions in the atmosphere (dT^/dz =0), the
general solutions are:
3Ag
B
2V-22
1/3
-1/3
and T -T =
GO c 2Z2
_3Ag_ . B
2Toc- z
(62)
(63)
where A and B are constants of integration and may be determined
from boundary conditions.
For the case where dT^/dz ^ 0, apparently no analytical solu-
tion exists. However, it is a relatively simple task to carry out
numerical solutions with a digital computer.
-------
45
Priestley and Ball reported experimental results for tempera-
tures measured between 5 and 70 cm above the heat source for the
neutral (non-stratified) case. These results did not compare
favorably with equation (63) (calculated results were about 75 per-
cent too high); but the measured results did have the same general
shaped curve as equation (63). In other words,
-1/3
(64)
m oo^
A
c 2z2
1
3Ag
[2V.2z
B
z3
Measurements transverse to the axis of flow revealed that the tem-
perature profiles were indeed nearly Gaussian.
Morton, Taylor, and Turner (34). Morton, Taylor and Turner
also studied the problem of gravitational convection from a continu-
ous point source. Although they employed the same fundamental
principles as previous investigators, their development has a
slightly different approach, which has been adopted by several
other investigators. One difference between this development and
that of Priestley and Ball (36) is that here an unknown entrainment
velocity must be contended with, whereas Priestley and Ball dealt
with an unknown spreading coefficient.
At this point two assumptions are made:
1) The entrainment velocity is proportional to the vertical
plume velocity; and
2) that the plume has some defined boundary b; that is,
R=b, where b is not the actual radius of the plume, but some
characteristic radius.
-------
46
Morton considers two different velocity profiles (see Figure
11} as noted:
1] the so-called "top hat," or
W(r,z) - W, r
-------
47
Velocity
profile: W(r,z) = W
m
Velocity profile
tr\:
W(r,z) = We"M?'
"Top hat" profile Gaussian profile
Figure 11. Velocity profile assumed by Morton.
It is assumed that a is constant with respect to changes of
Wm or z, which may not be the actual case. Also, a must be
determined from experimental or observational data.
For the case of a non-stratified medium p =p and dpoo/dz=0,
so that from (67)
m-
= constant,
(68)
and for this case the solutions are:
b = - az ,
Wm =
1/3 -1/3
z
(69)
(70)
-------
48
-1/3 -5/3
Z
(71)
Thus, the simple "top hat" profile yields the same general
flow behavior as found by previous investigations.
For the case of a stratified medium (dp^/dzjK)), Morton uses
the Gaussian profiles to obtain the following set of governing
equations:
d
BY
d_
dz
d
BY
= 2b2g
(72)
' 2b2Wm
dP0
dT
Note that a factor 2 appears on the right-hand side of the last two
equations, the only difference between this set and (65), (66), and
(67). These equations were rewritten with normalized variables and
solved numerically. For a linearly-stratified medium, the dimension-
less z is found to achieve a maximum value of 2.8. This is related
to the actual maximum height of rise, through use of a scale factor
proportional to the fourth root of the initial buoyancy flux and in-
versely proportional to the 3/8 power of the uniform gradient, i.e.
1/4
zmax ^ ~~f \3/
AZ
-------
49
Since the equations were developed for a point source, the
height zmau is measured from a virtual source (see Figure 12} which
max
lies some distance below the real source (which has finite diameter)
The virtual source for their experiments was found to be
zs = 7*8 •
where z is the distance the virtual source lies below the real
source and D is the real source diameter.
From the same experiments it was also found that
a = .093,
the proportionality constant between the vertical velocity at the
plume centerline and the lateral entrainment velocity.
Plume
Real source
-* o*-z_Virtual source
Figure 12. Virtual source,
-------
50
Morton (33). Morton exffended this work in two papers published
in 1959. In this work he assumed that the characteristic lengths
for the velocity and temperature (or buoyancy) profiles are different
to account for the difference between the eddy transport of heat and
momentum. He also extended the 1956 work to include a flux of mass
and momentum from the source, whereas the 1956 work considered only
a case of pure buoyancy such as a "thermal."
To include the effect of a different rate of spread between heat
and momentum, he assumes the characteristic length for the velocity
profile to be b as before, but the characteristic length of heat
(or buoyancy) to be \b. Under these assumptions, the governing
equations:
2abw
m
(74)
d_
dz
d
BT
d
Hz
The only change from the set of equations, (65). (66), and (67)» is
the appearance of X2 in the second equation of set (74).
Using the experimental results from Rouse et al. for a point
source, Morton ascertained that X=1.16, and from his previous
experimental results, a=.Q82, if Gaussian profiles are assumed. In
the case of "top hat" profiles, a is multiplied by SZ to obtain
a=.116 andA= T1*116 = 1.08.
-------
51
The assumptions in Morton's model that entrainment occurs
through the rise of the plume to zm^ and that the similarity of
HI a X
profiles is likewise maintained implies that no fluid leaves the
plume until zmav, at which point it is not accounted for.
HlttA
Schematically, this implication is superimposed on the observed
pattern in Figure 13.
Morton's Model
Observed
Figure 13. Comparison of Morton's model to representative
observation.
The consequence of this is that the model underestimates the
penetration of the plume (zmax) if one accepts Morton's entrain-
ment coefficient, or one must find a lower value for the entrainment
coefficient to match the heights. In any event, there is no infor-
mation provided for the thickness or location of the spreading
-------
52
field, which is felt to be more important than zmax. Morton et al.
(34), however, do point out that the plume should begin to spread
horizontally at the level where Ap is first diminished to zero.
Abraham (1). Abraham sought to overcome part of this objection
by allowing entrainment to become negative above the equilibrium
level, thus accounting in an artificial manner for transfer of mass
\
to the horizontal layer.
Abraham considers the motion and dilution of forced vertical
plumes in both stratified and homogeneous surroundings. He also
generalizes the solution to include the three types of flow men-
tioned earlier: neutrally buoyant, pure buoyancy, and mixed flow.
To derive the equations considered by Abraham for the axisym-
metric round jet, one may begin with equations (34), (39), and (44)
Following integration, one obtains the following set of integral
equations for a homogeneous environment:
d
BT
2TtWrdr = -2?
0
.D
d
dT
d
dz
bp2TrW2rdr =
(
rRbUr(Rb)
.R
b 27r(poo-p)grdr
'Rh
D 2TrW(pQ-p)rdr = 0.
0
(75)
-------
53
Abraham also assumes that the velocity and buoyancy profiles
are Gaussian, but differ in the dispersion scale to account for
differences of the lateral eddy transport of momentum and matter
(or heat). These assumed distributions have the form:
W(r,z) = W exp[-K(^-) ] (76)
C(r,z) = C_ exp[-Ky(^-) ] (77)
where K and u are experimentally determined entrainment coefficients
for momentum and matter, respectively, and Cm is concentration
defined as
c • • . (78)
m P P
In the above equation, p, is the effluent density as it leaves the
orifice. Note that the above definition of Cm is not a valid
definition of concentration for pollutants in stratified ambient
media. In the general case the numerator must contain p^ instead
of PO.
For the purpose of integration, Abraham permits Rb-*», and the
equation set (75) is written as:
d_
dz
d
dT
2TiWrdr = Q1 (79)
2irpW2rdr =
0
2ir(prt-p)grdr (80)
-------
54
d_
dz
2TrW(p0-p)rdr = 0.
(81)
In the continuity equation (79) above, Abraham does not assume a
relationship between VI and Up as did Morton et al. (34), but simply
insists that continuity is satisfied by some unknown inflow, Q',
to the plume. Also in the second equation Abraham uses PJ instead
of p and in the second and third equations POO has been replaced by
D since p =p for a nonstratified medium.
^o o °°
If equation (80) is multiplied by dz and the result integrated
from z to z (z is the point where the flow becomes fully estab-
lished - see discussion on zone of flow establishment), the result
is:
2-rrp
W2rdr =
,00
dz
2ir(pQ-p)rdr,
(82)
where M is the jet momentum at z = ze or the left-hand side of the
equation evaluated at z = zg. The following expression for the
plume center!ine momentum as a function of z is obtained when (76),
(77), and (78) are inserted into (82) and integrated:
(po'pJ} C z2dz.
(83)
When equation (81) is divided by (PQ-PJ), (76) and (77) are
used to replace W and the quantity (p0-p)/(pQ-PjK and the result
integrated from zfi to z:
-------
55
IT
KU+y) m1 m'
(84)
is obtained, where C, and W, are plume conditions at the orifice
j
-------
56
Hence, the dilution of a vertical plume
s (r>z) = 1 (88)
is found from equations (86) and (87). As mentioned previously
in the discussion on the zone of flow establishment, the quantity
z is found by the equation:
3 2.
1.42
Fr, D
8
= 0, (89)
where the values of k and u are 77 and .80, respectively, for equa-
tion (89).
Abraham also presents solutions for the limiting cases where
either inert!al forces dominate the flow behavior (FRj-*~) or where
buoyant forces dominate (FR..-K)). For the uniform medium the follow-
ing limiting solutions are available:
. Inertial dominated flow: Fr,-*»
u
H. v V2 n
" = (|) £ . (90)
J
in which case K=77 and y=.80.
-------
57
Buoyancy dominated flow:
m
W.
D
z
3 1 (l+uw
8 FrT ( u ^
-------
58
where
and
Wm,n-l z
n
K + K
m,n-l
K = 1 -
i
3 1
i-l
K -1-1
n-1
_ (Wm.n-l)'
m "
pn"pm,n-l
i PJ J
P-Pn
9zn-l
"m
,n-Tpn
1/3
(93)
-1/3
(94)
Inclined and Horizontal Plumes - Stagnant Environment
In more recent outfall construction it has been the practice to
orient the diffuser ports so that the waste material is issued
horizontally into the receiving water (Figure 14). Both theory
and practice have shown that the horizontal plume is more efficient
in diluting the pollutant with respect to the final height the
pollution field achieves. The analyses offered thus far for the
horizontal plume differ little from vertical plume analysis, except
that a horizontal component of momentum is considered.
-------
59
Abraham (1). Abraham also presents an analysis for horizontal
round plumes issuing into stagnant, nonstratified media.
SEA SURFACE, pr
ps < po
port
velocity profile
Figure 14. Horizontal plume issuing into stagnant, density
stratified medium.
For the horizontal plume, the natural coordinates are s and r
(Figure 14), where s is the distance along the plume axis and r
always lies in a plane normal to this direction. Although the
velocity and buoyancy profiles are certainly not Gaussian, they
are assumed as such to simplify the analysis. These profiles are
represented by:
U(r,s) = Umexp[-K(f)
and C(r,s) = Cmexp[-Ku(£) ].
(95)
(96)
-------
60
The z-x coordinates for the flow are given by:
rs/D
Z_ _
D "
_
D
0
s/D
0
Cosed(^),
(97)
(98)
Equation (so) can be modified for this coordinate system
simply by replacing z with s to obtain:
(99)
However, in the momentum equation both z-directional and
x-directional momentum must be considered. For the z-direction,
the equation analogous to (82) is
Sine
a
rS r°°
ds
se '•
2ir(pA-p)rdr,
(100)
and for the x-direction (horizontal):
Cose pjU2rdr = Mj,
•* o
(101)
where M, is the momentum at the orifice.
j
Equations (lOO) and (lOl) a^e integrated using profiles
(95) and (95) and added vectorially to obtain
1/2
m
(102)
-------
61
With use of equation (99), equation (102) may be reduced
to the differential form
U,
m s
U,
111 1 1
I, D K2" 4
1/2
<103>
Equation (103) is then used to solve for Um/Uj as a function
of s/D. Equation (99) is used to evaluate Cm/Cj, equation (101)
to find 6 and eventually (97) and (98) to find z/D and x/D. To
evaluate the entrainment coefficients, Abraham uses the following
relationships:
K = -304(1) + 228(|) + 77 (104)
3 Q 2
y = .96(1) - .72(1) + .80. (105)
Obviously, the solution for the horizontal plume problem must
be carried out numerically. Also, there is some question concerning
the basis of equations (104) and (105) (Fan (14)).
Fan (14). Work carried out by Fan and reported in 1967 represents
valuable extensions to established jet and plume theory. Part of
this work is concerned with plumes issuing horizontally (and at
arbitrary angles above the horizontal) into a stagnant, linearly
density-stratified medium. Fan's work is based on the Morton et al.
(34) theoretical approach, along with the Albertson et al. (4)
experimental work to define the length for flow establishment.
-------
62
In Figure 15 the z',x' coordinates are referred to the actual
source, and the z,x coordinates are referred to the point where the
flow becomes fully established with respect to the velocity pro-
file, or 6.2D from the z'.x' origin at an angle Q.
Figure 15. Plume issuing at an angle into a stagnant medium.
Equations are again needed for continuity, z and x momentum,
buoyancy, and geometry. These expressions are written in terms
of coordinates along the jet axis, s, and a normal to this direc-
tion, r. Velocity and buoyancy profiles are assumed Gaussian in
the region of established flow:
-------
63
U(r,s) = Um(s) exp[-r2/b2]
O-P = Cp(B-Pm(s)]exp[-r2/(x2b2)],
where b is a characteristic width.
The governing equations are as follows:
Continuity: - (b2Um) = 2bUma (106)
z-momentum: - (ni sine) = gX2b2-- (107)
o
d ,b2!lv,2 ^ /
x-momentum: - ( m Cose) = 0 (108)
Buoyancy: _ b2[p = I±Xi 2 (109)
«-^
Jm" L^-^jy X2 w um ds
z-coordinate: ^| = Sine (110)
x-coordinate: $r = Cose (111)
Concentration: CmUmb2 = Const. = UjCQbo2 (112)
To obtain the solutions in terms of U_, b, 6, p -pm, x, and
m °° m
z, the above six differential equations must be solved simultaneously.
The initial conditions are Um(0)=U,, b(0)=b (b =D//2~), p(0)=p,,
MI j oo m j
e(0)=6 , and x=z=0 at s=0. Fan normalized these equations in much
the same fashion as did Morton et al. (34) and worked with dimen-
sionless parameters. Solutions were carried out numerically by
digital computer (see reference 14a).
Cederwall (11). Based on equations developed by Bosanquet et
al. (8) for plumes issuing into fluids of other density at various
-------
64
inclined angles, Cederwall proposed the following set of equations
for horizontal plumes in homogeneous receiving waters:
c 7/16
*" 1/2
m = r^= -54 Fr,
m C_ J
T72
J
; z/D<.5 Fr (113)
Cl l/2f z 15/3 1/2
S = 7^- = .54 Fr,1'^ .38 z * „ + .66 ; z/D>.5 Fr,
01 Cm J DFr,1/2 J °
J (114)
Hu 10Fr1/2
Um J
v j '
These equations are for conditions along the jet axis, with
the zone of flow establishment being neglected. Cederwall comments
that Abraham's model can be written in similar form if the zone
of flow establishment is neglected.
For the case where the receiving water is stratified, Ceder-
wall suggests a step-by-step numerical approach using the above
set of equations. For application, Reference 11 should be con-
sulted.
Trent, Baumgartner and Byram (52). Theoretical work on inclined
jets issuing into arbitrarily stratified media is also being carried
on at Corvallis by Trent et al . This work follows more along the
work of Abraham than that of Morton et al . It is felt that the
introduction of the characteristic length, b, used by Morton (and
later by Fan) is an unneeded complication.
Refer to Figure 15 and consider the velocity and density dis-
parity profiles to be given by:
-------
where K and u are possible functions of s. Let
R =
65
.s) - Um(s)exp[-K(r/s)2]
(116)
Poo-p(r,s) P00-Pm(s)
= - exp[-Ky(r/s)2]
D ~Pi P ~P i
0 u 0 u
(117)
The governing equations become, after simplification:
Axial velocity:
dE 3E L._dK - 3g_
ds s " 2K ds " y
Density disparity:
PJ
RSin e
1/3
dR . 2R R
ds s"
(118)
(119)
Angle:
2/3 c 2
9 - co*"1 r r Coseo
(120)
Geometry:
x = x + CosedC
(121)
z = z.
Sineds
(122)
-------
66
Continuity of pollutant:
r -
in
E
,1/3
e
2
(123)
where C is any convenient measure of centerline concentration. It
should be pointed out here that in the case of a hot water plume
having the same salinity as the ambient, A is the measure of pol-
lutant concentration and equation (123) is unnecessary.
In order to evaluate equations (118) and (119), it is necessary
to evaluate
1 dK
2KHF
and
These quantities are functions of the as-yet-undefined flow field.
And, even if the flow dynamics were known, just how K and y vary
is open to question. We do have, however, Abraham's limiting
values for small and large Frj (see page 57). In this work
we assume that K and y change slowly along s, at least over
the range of s where a similarity solution is valid. Then terms
involving derivatives of K and y are small compared to other terms
of the equations and as a first approximation may be ignored.*
* Once the approximate flow dynamics have been established, one
could, at least in principal, use this information to estimate
derivatives of K and y and thus improve the solution.
-------
67
Equations (118) and dig) become:
dJL
ds
dR
II.
s
2R
s
RSine
(124)
(125)
The quantity R could be eliminated from (124) by using (125) .which
would yield a second order equation in E. However, the equations will
eventually be solved numerically, so that two first-order equations
turn out to be the more convenient form.
The governing equation may be non-dimensional ized by defining
the following parameters:
E* - E/EJ • W
s* = s/D, x* = x/D, z* = z/D
R* • R/RJ -
* =
Hence, the equation set in dimensionless form becomes
dE* , 3E*
yrr.
R*Sin0
(126)
l£- ' C127>
along with appropriate dimensionless forms for equations (120). (121 )»
(122), and (123) if needed. Initial conditions are:
-------
68
at s* = s *: E* = E * = 1
e e
R* = Re* = 1
c *= r * = i
Sn Le ' '
Here s * is 5.6, based upon concentration profile establishment
for Fr,>100.
J
From equations (126) and (127), the obvious simplifications
are:
• Vertical buoyant plume: e = Const = 90°
• Homogeneous medium: dpro*/ds* = 0.
These equations also reveal two similarity parameters (aside from
6):
U
Fr, =
PJ
Dg
* d
or
(z/D)
For linear stratification:
* HP
Ap
K
= -6
where G is the stratification parameter, and Apj = PO~PJ
-------
69
Numerous calculations have been carried out based on various
values of G and Fr . The dimensionless height of z*max and cor-
responding downstream distance x* are plotted in Figure 16 for
max r
parametized values Fr. and G, where 6 =0°. Figure 17 illustrates
z*max for a vertica^ Jet as a ^unction of Fr, for various values
of G. These results are in excellent agreement with Fan's experi-
mental results, and may be used as a guide for ocean outfall diffuser
designs.
The information in Figure 17 may be approximated by the equa-
tion:
n 1/8 -3/8
z*max = rFrJ G •
_3
For values of G>2xlO this equation begins to lose accuracy, as
illustrated by the dashed line in Figure 17.
-------
70
200
fM
TOO
4x10
-4
10
-3
2x10
-3
G=10
-2
100
200
max
Figure 16. Maximum height of rise versus downstream distance for horizontal
plume in linearly stratified environment.
-------
71
s 100
I ' I
— Numerical solution to equations(126)&(127)
max
n 1/8 -3/8
11 Fr, G
Solutions coincide for G<10
500
Fr.
1000
2000 3000
Figure 17. Maximum height of rise for a vertical plume in a linearly
stratified environment.
-------
72
Vertical Plumes - Crossflow
The previous discussions were concerned with plumes in either
stratified or nonstratified stagnant receiving waters. However,
the ocean is seldom stagnant and cross-currents can have a
dramatic effect on the dilution and height of rise of a buoyant
piume.
For analysis of the bent-over plume, simplifications and
assumptions associated with the vertical plume analysis are main-
tained. One must realize that in approaching the problem from
this viewpoint, gross approximations must be made. For instance,
the assumption of an axisymmetric Gaussian velocity profile along
the plume axis is certainly not a characteristic of the bent-over
plume. In fact, a cross-section of the plume normal to the axis
of flow will reveal a variety of forms depending on the magnitude
of the current and the geometry and dynamic character of the plume,
Under most conditions the situation may be similar to Figure 18,
a situation which has been observed experimentally by a number of
investigators. The counter-rotating motion shown in section A-A
of Figure 18 has also been calculated numerically by Lilly (28)
for a line thermal. Casual observation of smoke issuing from a
chimney into a cross-wind has shown that when wind and plume
conditions are adjusted in some as-yet-undefined proportions, the
plume may disintegrate into two or even three distinguishable
streams (Figure 19).
-------
SECTION A-A
73
Figure 18. Vortices associated with a bent-over plume.
Figure 19. Schematic representation of smoke plume separating into (a) two or (b) three
streams due to wind effects.
-------
74
The amount of analytical work carried out on bent-over plumes
is meager and in some cases quite flimsy - probably owing to the
difficulty of the analysis. Priestley (42) was evidently the
first to propose a working model for atmospheric plumes. Schmidt
(46) has also done some work on smoke plumes, among a few others.
However, Fan (14) presents what is considered here the most systematic
approach to the problem of buoyant plumes in ocean currents. For
this reason only Fan's work will be discussed.
Fan considered the problem of a vertical round port discharging
buoyant pollutant to a homogeneous ocean with a uniform (top to
bottom) cross-current (Figure 20).
The absolute velocity of the plume along the s-direction is
assumed to be axisymmetric, and composed of two parts: 1) the
component of the cross-current in the direction of s; 2) the
relative velocity which is assumed to be Gaussian with respect to
r and similar with respect to s. Thus the velocity along s with
respect to a fixed reference frame is
U*(s,r) = U^Cose + Umexp[-r2/b2], (131)
where U is the relative velocity along the jet axis. Density
deficiency is given by
P.rpCr.s) - [pw-pm]exp[-r2/b23. (132)
-------
75
Uoo -
(cross-current)
Figure 20. Vertical jet issuing into uniform cross-current.
-------
76
The superscript * refers to the absolute velocity in the plume.
Again, the governing equations must satisfy the conditions of
continuity, momentum, and energy.
Continuity. Continuity is assumed to be satisfied
by:
d_
r 'A
where the quantity under the absolute value sign is the relative
velocity between jet and environment. By integrating the component
of the environmental velocity in the direction of r from 0 to /Z b
(the nominal radius of the jet is assumed to be /Z b) and inte-
grating the Gaussian part of the profile from O-H», the continuity
relationship becomes:
3s I U*dA '
[b2(2UooCose+Um}] = 2ab(Uro2Sin2e+Um2) . (133)
Momentum. Momentum components for both the z and x direction
must be considered. But unlike the case for a uniform environment,
some compensation must be made for drag on the plume, according to
Fan. This drag force is assumed to have the form:
where Cd is a coefficient of drag which must be found experi-
mentally. Then for x-directional momentum,
V2
2obUjUco2Sin2e-HJm2) + -- UjbSiVe, (135)
-------
and for z-directional momentum,
2~
U 2b Sin26Cos6,
oo
(136)
Density deficiency. The relationship for density deficiency
for a non-stratified medium is
U*(pTO-P)dA= 0.
(137)
Substituting the expressions (131) and (132) for U* and p, and
integrating from 0-*», yields
;!-[b2(2U Cose+U)(p -Pm)] = 0. (138)
ds L °° m 'oo Knr J
The geometric relationship between s-6 coordinates and x-z
coordinates are given as
=Sin9,
and
(139)
(140)
Initial conditions are referred to the origin of the s-6 coor-
dinate system and have the following values: Um(0)=Uj, b(0)
, pm(0)=p,, 6(0)=9 . z=x=0 at s=0. Equation (138) can
=b.
n
be integrated immediately to yield:
77
b2(2UooCose+Um)(poo-pm) = constant,
(141)
-------
78
but the five remaining differential equations ((133), (135), (136),
(137), and (138)) must be solved simultaneously. Since the ambient
medium is assumed non-stratified, concentration may be calculated
using equation (141) with (p^-pj replaced by Cm. Fan carried out
these computations numerically for various cases after normaliza-
tion of equations and non-dimensionalizing parameters. One other
point that should be mentioned here is that 9 is no longer 90° from
the horizontal where established flow begins.
To calculate eQ at S=SQ, Fan uses the empirical relationship
60 = 90-110K- (142)
where K1 is a function of s/D (see reference 14, page 94).
A very brief and incomplete summary of Fan's work has been
presented here. Reference 14 should be consulted for the detailed
theory and results.
Negatively Buoyant Plumes
Plumes that have negative buoyancy result when heavier fluid
is discharged into lighter receiving water such as water with
heavy salt concentration rejected from a desalinization plant.
The dynamics are essentially the same as in the case of a buoyant
plume, except the sign of the buoyancy term is reversed. However,
in the case of the vertical plume the heavier fluid will cascade,
causing interaction between the initially formed plume and the
cascade flow. For cases where the fluid issues from an inclined
port, and in the same general direction as the ambient current,
-------
79
this interaction will not take place. Figure 21 illustrates
typical negative buoyancy flows.
plume flow
cascade
flow
Vertical Discharge
Inclined Discharge
Figure 21. Typical flow patterns for plumes with negative
buoyancy.
Analysis similar to the positively buoyant plume should hold
for the inclined discharge if the sign of the buoyancy term is
changed in the governing differential equation. However, with
the vertical discharge there is some question whether an adequate
similarity solution can be expected because of the interaction
between flows. It has been the practice thus far to assume that
there is no mixing between the two flows. In this way solutions
for the positive buoyancy plume can be fashioned into solutions
for the heavy plume.
Cederwall (11) gives the following comparisons of theoretical
ceiling heights for heavy fluids discharged vertically into homo-
geneous environments:
-------
80
zmax * 'M ffj'2°
Abraham
Priestly & Ball
Morton
(143)
V2r
The work by Trent et al. yields: zmax = 1.80 Frj ' D. The
length for flow establishment has been ignored in the above rela-
tionships.
Turner (54) carried out analysis and experiments of heavy jets
and plumes and found that the initial penetration of the fluid is
somewhat higher than the steady-state penetration height (Figure 22)
max
D
initial transient height
steady state
\-
r/D
Figure 22. Initial and steady penetration heights of a vertical
heavy plume in homogeneous medium.
-------
81
Turner's experimental results (for comparison with (143))
yield
z =1.74 Fr.1/2D (steady state)
max u
and zma = 2.48 Fr,1/2D (initial transient). (144)
max o
The initial transient penetrates higher apparently because
there is no cascading flow to interact with, as is the case of
steady state. Turner's theoretical penetration height for steady
state is:
z = 2.9Fr,1/3D. (145)
max J
Horizontal Surface Discharge
There may be benefit in discharging certain pollutants at the
sea surface. For instance, thermal interchange with the atmosphere
would hasten the dissipation of heat and lower the overall heat-up
of the local sea. Also, shoreline discharge is economically
attractive because the cost of constructing an expensive marine
discharge line is eliminated.
Very little analysis has been carried out for this type of
discharge. The dynamics of the pollution field are again charac-
terized by a zone of initial dilution and a zone of drift flow.
However, the zone of initial dilution is influenced primarily by
the discharge momentum. If the pollutant is lighter than the
receiving water, a gravitational spread must be considered along
with the turbulent diffusion.
-------
82
In some cases, such as low velocity canal discharge of power
plant reject heat, Initial momentum of the coolant may play only
a minor role on the overall pollution field dynamic behavior. In
such instances, the pollutant field dynamics must be treated by
methods discussed in chapters V and VI.
Jen et al. (24) have carried out analytical and experimental
investigations for horizontal surface jets having initial densi-
metric Froude numbers in the range of 324 to 32,400. The results
of this study showed that the surface temperature distribution
could be approximated by
1 _.. l/*,Vx-| (146)
•m
for x/D<100. In the above equation, Tro is the receiving water
temperature, T is the jet center!ine temperature, x is the coor-
dinate along the jet axis, and y is normal to the jet axis.
For values of x/D<100 the center!ine temperature is expressed
by
^=7.0$, (147)
lj~'oo X
where T, is the effluent temperature.
The above equations were developed in a manner similar to the
methods used by Abraham. Also, these equations are only valid
where initial momentum is dominating the flow dynamics (large Fr,).
It is doubtful that the analysis presented by Jen et a!. could
be applied to practical cases with a great deal of confidence,
-------
83
because the range of Froude numbers covered by the investigation
does not correspond to most practical situations. For instance,
the densimetric Froude number for canal discharge of hot water
may be on the order of 10 or less.
Zeller (61) considered the surface jet from a viewpoint similar
to Fan's (14) analysis of the buoyant plume. However, unlike the
work of Oen et al., Zeller incorporates the effects of cross-currents
and wind stress.
Hayashi and Shuto (20) also studied the problem of hot water
spreading over colder receiving water. These authors considered
the case of a very low initial densimetric Froude number (Frj-1)
and ignored the non-linear terms in the equations of motion. Hence,
this work is not defined in the category of jet flow, but is dis-
cussed in Chapter V under "Surface Drift of a Buoyant Pollutant."
-------
84
ZONE OF DRIFT FLOW
The zone of drift flow is characterized by the pollutant field
drifting with the oceanic currents nearly as if it were a part of
the oceanic velocity field. In the drift flow regime the pollutant
field continues to dilute and disperse into the oceanic environment
primarily by two possible mechanisms. The first of these mechanisms
is turbulent transport or eddy mixing; the second is a density
disparity between the pollutant flow field and the environment.
The first mechanism, turbulent mixing, will always be present in
the ocean, but spreading by virtue of a density disparity may not
be. Lateral spread of a pollution field under the action of gravity
is nearly entirely a surface phenomenon* and is characterized by the
polluted stream from an outfall reaching the ocean surface without
sufficient dilution so that it remains on the surface, with posi-
tive buoyancy (or on the bottom with negative buoyancy). As a
result of the density disparity, the action of gravity causes the
lighter pollution to spread laterally over the heavier receiving
water. At the same time lateral eddy diffusion is also acting.
Depending on the intensity of vertical turbulence, vertical dispersion
may not be significant. Figure 23 illustrates the different types of
drift flow schematically for a submerged horizontal discharge of
buoyant fluid in a density-stratified receiving water.
* Cederwall (11) points out that a submerged lateral density spread
will occur under certain conditions and may, in fact, become an
important consideration in some locales.
-------
85
A) Submerged Drift Flow
Initial
Dilution
Drift
X ' / ' I**/} I 7 T~l
B) Surface Drift Flow
Figure 23. Two types of drift flow resulting from submerged
horizontal discharge of buoyant fluid in a density
stratified ocean with current.
-------
86
Submerged or Neutrally Buoyant Drift Flow
Neutrally buoyant flow is by far the simplest type of motion
in the drift flow regime because gravitational forces need not be
considered.
Consider the momentum equation (8):
Ji
3U.
it uj 3xT "ijk"juk
j
Pco-P
_PO
njr + 3 - T
90iq ^ ay £TT ay
1 W OA • 1 J O A *
J J
(8)
Since we are dealing with a neutrally buoyant pollutant field,
and equation (8) may be written as
3D.
3U.
3P_.
oX j
3x.
J
3Ui
i^r.
(148)
Now, if the outfall discharge momentum does not perturb the
oceanic motion to any appreciable extent, solution of the above
equation is independent of the pollutant field.
The pollutant dispersion is governed by the mass diffusion
equation (10),
3t
9C _ 3
3x. " 3X
J
.
J
3X,
m
(149)
-------
87
where C is concentration of the pollutant.
The quantity rfi represents a source or sink that is used to
account for BOD decay, growth of bacteria, sedimentation, etc.
For oceanic flow V«£-> so that
_ --
3t j 8x. 8X,
J J
_
Mj 3X.
— J
-Hi, (150)
where eu. is a function of the flow field and is not a fluid
Mj
property. To solve equation (150) for C, the quantities Um and
eM. must be known.
Mj
The velocity components, U., may be calculated using equation
J
(148), or they may be determined by oceanographic measurements. The
eddy diffusion coefficients, EM., are determined by measurement and
may be constant or assumed to have some functional form (to be dis-
cussed later).
At any rate, the problem of calculating C from equation (150) is
considerably simplified since the velocity field is assumed to be
known beforehand. This situation contrasts the case of a buoyant
surface drift where the momentum equation (8) must be considered
simultaneously with equation (150) because of the interaction
between pollutant field concentration and momentum (i.e. density
disparity, p^-p, causing a gravity spread of the field).
Now if the coordinate system is assumed to move along with the
ocean current, then the convective terms in equation (150) vanish and
+ (151)
+
_ _
3t 3x . Mj 3x.
j •— J
-------
If the ocean has a steady uniform velocity field, U., and
J
the pollutant has been issuing from a continuous source long enough
for steady-state conditions to prevail at downstream points, then
at
Hence, equation (150) becomes
(152)
M
3C_ 3
-
_
'Mj 3X.
~ J
m .
(153)
For the case of homogeneous, isotropic turbulence &.. is constant
with respect to spatial coordinates so that from equation (151),
(154)
3X.3X.
J J
or, from equation (153),
y 3C_ = 32C
J Xj Xj
m •
(155)
Finally, if the substance is conserved, m=0, then
32C
3t
_
3x.3x.
w w
(156)
and
32C
_
3X-
J
_
3X-3X.
J J
(157)
-------
89
Comments on the Eddy Transport Coefficient,
In order to solve the above diffusion equations analytically,
the eddy diffusion coefficient, e«-» must have some functional form.
Obviously, the simplest case is that of a homogeneous isotropic
turbulence field where EM^EM* a constant. However, it may not be
reasonable to assume the ocean turbulence is either homogeneous
or isotropic. For example, wind stress at the sea surface causes
ocean turbulence which diffuses with decreasing intensity to lower
depths. Also, density stratification depresses the component of
vertical mixing so that lateral turbulent transport is much greater
than the vertical transport.
Another complicating factor is that as the plume grows in size,
larger and larger scales of turbulence take part in the diffusion
process. In the ocean, turbulence of all scales can exist, from
the isotropic microscales to the highly anisotropic macroscales
which are limited in size by solid boundaries and the sea surface.
Various authors have attempted to account for this phenomena by
assuming that the diffusion coefficient is a function of the plume
size and/or is a function of diffusion time.
In the open sea (see Reference 36) measurements have shown
that the lateral eddy diffusion coefficient is proportional to
4/3
L , where L is the scale size (the width of a dye plume, for
instance). However, in the open sea very large scales of turbulence
are possible, whereas in a coastal region the scales would be
limited in size by the coastline and bottom effects.
-------
90
Orlob (38) carried out various eddy diffusion studies in
open channel flow. From experiment he found that lateral dis-
persion patterns would not grow indefinitely according to the
4/3 power law, but would become parabolic in shape as governed
by the largest sized eddies participating in the diffusion process.
These findings indicate that when the pollution field is smaller
than the largest eddies, the eddy diffusion coefficient is propor-
4/3
tional to L , but when the plume grows to a size larger than the
largest eddies, the eddy diffusion coefficient is constant (actually
there is a transition zone separating these two effects).
Cederwall (11) argues that in shallow coastal waters the growth
of turbulent eddies is limited, and at a distance not too far from
the source, the plume will grow to a size where the maximum possible
sized eddies are affecting the diffusion. Thereafter the eddy
diffusion coefficient is essentially constant - at least with regard
to maximum scale sizes. For this reason Cederwall maintains that
it is more reasonable to base diffusion models on a constant diffusion
coefficient. As an example, Harremoes (18) has found from radioactive
tracer studies off Ha'lsingborg, Sweden, that the coefficient of
lateral eddy diffusion was essentially constant (eM lateral - 0.5
meters 2/sec) in these waters. He also found that the vertical eddy
diffusion could be approximated by
« 5x1 0~ meters2/sec,
-------
91
for the same region. Foxworthy et al. (15) also questions the
validity of using the 4/3 power law in coastal waters, based on
measurements off Southern California which indicated considerable
disagreement with calculations based on the 4/3 power law.
Ozmidov (39) has proposed a function, f(L/H), which corrects
the 4/3 power law for shallow coastal water. Using this correction
factor Ozmidov writes the coefficient of eddy diffusion as
eM(L,L/H) = kL4/3f(L/H),
where k is a proportionality constant. However, this proposal has
apparently not received general acceptance. A good summary of
both lateral and vertical eddy transport coefficient models is
given by Koh and Fan (25).
The eddy diffusion coefficients are fundamental to the solu-
tion of the mass diffusion equation. At present there is no
general agreement as to the functional form of these coefficients
for applications in coastal water. Obviously, a good deal of
research is needed concerning large-scale turbulence in shallow
coastal waters.
Point Source Models
A number of solutions to the diffusion equation are possible
based on boundary effects and the assumed form of the diffusion
coefficient. A few of these proposed diffusion models are pre-
sented here for a conservative pollutant (m = 0).
-------
92
Fickian diffusion models* are based on a constant value of
eddy diffusion which are not necessarily isotropic.
For the case of an instantaneous release of an amount, M,
of pollutant in a uniform current having velocity components
Uoo» v«>» and Woo in the x»y> and z directions, respectively, the
concentration at a point is given by
C(x,y,z,t) =
»3/2,
k1/2*3/2
exp -
(x-U t)2
4eMxt
(y-vj:)2 (z-w^t)2
—, __ + _. .—
(158)
For a fixed source, a continuous volumetric discharge, q, of
pollutant into a uniform current flowing in the x-direction only,
results in
C(x,y,z) =
^VW
exp -
eM x
My
In the above equation, turbulent transport in the x-direction
is ignored; specifically, ex/Uoox«l is assumed.
To account for the sea surface boundary
* This work has been adapted from Reference 11.
-------
93
C(x,y,z) =
^VW""*
exp -
z2U
(160)
where z=0 may be taken as the sea surface (the plane of reflection)
Models Based on a Variable Diffusion Coefficient
For a conservative pollutant, the simple radial flow diffusion
equation takes the form
9C_ _ 1 9
8t r 9
Mr
(161)
Specifically, this equation is used to describe the diffusion of
the instantaneous release of an amount M of pollutant. The pollu-
tant is assumed to spread only in a radial plane and any solid
boundary effects are ignored. Different models may be derived from
equation (161) by assuming different functional forms for eM .
Okubo (37) has generalized these various solutions by assuming
that the eddy transport coefficient eM may be a function of space
and time, defined functionally by
m
(162)
Then equation (161) may be written as
. _
9t ~ r 9r
0
(163)
where eM and m are constants and f(t) is a function of time related
to the concentration distribution.
-------
94
The general solution to (163) is:
exp -
(2-m)M
for 00
C(r,t) = 0, for r<0, t>0
f C(r,t)2irrdr = M, for t>0
J
Table I summarizes the models derived from equation (163)
(or (164)) proposed by various authors.
Brooks (reference 60) has presented solutions based on the
assumption that the coefficient of eddy diffusion is a function
of the pollution field width, L, alone. The results of this
work are:
C (x) =-^ - (J-r1, e=a,L,
m 2Sw7 U°°
- -3/2 4/3
-------
95
TABLE 1
RADIAL DIFFUSION MODELS FROM OKUBO (37)
Experimental Proposed
f(t) m Parameter by:
C(r,t)= — latoi- 1 0 eMo "Fickian"
M exp[- jjr]
C(r,t)= - Ei 1 1 prDiffusion Joseph &
2?rp2t2 Velocity Sender
M
C(r,t)= - 1 4/3 a: Energy Ozmidov
6ira3t3 Dissipation
M exp[-m-]
C(r,t)= - ^— t 0 wrDiffusion Pritchard &
iro)2t2 Velocity Okubo
t ^
C(r,t)= 572— t 2/3 a:Energy Okubo
(3/4h ' a3t3 Dissipation
2
CXp[ 33]
C(r,t)= - — t2 0 BrEnergy Obukhov
Tt$3t3 Dissipation
-------
96
where a, and a2 are determined by experiment and L 1s the width
of the pollution field.
Line Source Models
Brooks (10) considered the dispersion of matter from an ocean
outfall line with pollutants issuing from a number of ports along
the line. For analytical purposes, he considered the outfall
system a steady line source issuing horizontally in a uniform
stream U^. By neglecting vertical and longitudinal eddy transport,
Brooks writes equation (153) as
3C _ 9
3C,
II ov< _ o / ow
00 8x ~ 3y ^ M By-
Figure 24 illustrates the plan view of the idealized physical
system.
(165)
r
b
Hultiport
Outfall
Line — ^
"I
"~T c
~7 (y=°)-2.
i
^j
~1
.
Pollutant field
C(x,y)
Figure 24. Brooks' line source model (for ported outfall line).
-------
97
Brooks assumes that for biological and/or solid pollutants
the die-off rate and/or sedimentation are proportional to the con-
centration, so that
m = kC.
This assumption has a net effect on the final solution of
multiplication of C by exp(-kx/Uoo). Since this is the only effect,
m can be ignored in solving the diffusion equation, keeping in mind
that the solution must be multiplied by this exponential factor.
Brooks also maintains that equation (165) may be solved considering
^ as a function of x alone. Thus, equation (165) may be written as
3C' _ fM
3x " U
oo
where C1 is defined by C=C' exp(-kx/Uj.
Now let EW(X) = £M0f(x) where f(o) = 1, and define x' such that
dx1 = f(x)dx.
The quantity EMO is the eddy diffusion coefficient at x = o.
Then equation (166) becomes:
3C' EMo 32C'
oX U oV
oo •*
The solution for (167) is
C
b/2
o
C'(x,y) = exp
2/ireMot' -b/2
-(y-y1)
4e4.lt'
dy1 (168)
-------
98
where C is the concentration of the plume as it enters the zone
of drift flow, and t'= x'/U .
00
Brooks defines the field width as
L = 2 v^T a
where a, the standard deviation, is defined by
a2=^V fy2C(x,y)dy. (169)
n »
0 -co
Brooks evaluates the solution for three different assumptions
of e». These cases are listed in Table II. In Table II, g is
defined by:
Yudelson (60) discusses the previous diffusion equation solu-
tions in somewhat more detail than given here, and he also compares
theory with results from several experimental investigations (pri-
marily dye-patch experiments). These comparisons will not be
discussed in detail here except to point out that according to
Yudelson, Brooks' solution using the 4/3 power law seems to
yield reasonably accurate results. He also notes that Brooks'
solutions have been used in designing a number of ocean outfalls
in Southern California which are performing satisfactorily.
-------
TABLE 2
LINE SOURCE SOLUTIONS
e(x)
%
L
Mo F
L 4/3
L
b
(1 + 2 8 X)
(1 + B x)
u p b'
2 x 3/2
0+f 0£)
x1
X
h v 2
. rn + p x) n
23 LU p b; IJ
b 2 x 3
(L 0 D
Cmax(x)
C erf •
0
CQ erf -
C erf '
0
f ^
Is
,ovn( ^•^^
v| 43x/b cxP^Kt^
*• \
J3/2 / i,.\
'Cxp^-Ki;
(1+3 |-)2-l J
3/2
J (1+2/3 3 |)J-1J
t = x/U
ID
-------
100
Surface Drift of a Buoyant Pollutant
A buoyant pollutant not only disperses by the action of
turbulent mixing, as in the case of neutrally buoyant drift, but
also spreads because of the density disparity between the plume and
surrounding water. This means that the velocity terms in equation
(9) (or (150)} are not independent of the plume temperature (or con-
centration) and cannot be established beforehand through oceano-
graphic measurement or by independent solution of the equations of
motion. This complication leads to the necessity of solving the
equations of motion and heat diffusion (or mass diffusion) simul-
taneously.
Consequently, the solution to the buoyant drift problem is
considerably more complicated than the neutrally buoyant drift
case. In fact, it is worthwhile investigating ways to determine
if for a specific case the gravity spread is negligible compared
to the turbulent diffusion. If this can be established, then
methods used for neutrally buoyant drift can be employed.
However, for the general case, gravitational effects which
cause the plume to spread laterally cannot be ignored and we are
left with a difficult problem of solving equations (8) and (9)
(or (150)) simultaneously.
Density profiles may show essentially a discontinuity between
the pollution field and the receiving water, giving rise to a two-
layer flow, or the profiles may be smooth, with the density becoming
-------
101
a minimum at the surface. Whether or not the layering effect
prevails depends on the density and miscibility of the pollutant
and the vertical eddy mixing of the receiving water. In either
case gravity will tend to spread the lighter material laterally
over the heavier receiving water, and air-sea interactions may
also have a significant effect on the dispersion of the pollutant.
Apparently, there has been very little analytical work done
for this type of flow, probably owing to the complexity of the mathe-
matics. Cederwall (11) presents a calculational method from Lar-
sen and Sorensen (26, not seen) for predicting the spread of
layered flow under the action of gravity only, in a current with
velocity U^ (see Figure 25). The procedure is based on the main
assumption that the two layers remain perfectly identified with
no frictional effects between layers, which implies: 1) no
vertical mixing, and 2) a homogeneous pollution field. Also, only
lateral spreading is considered (normal to current flow). Thus,
the velocity of advance of the pollution front is given entirely
by conditions at the front. By continuity of the pollutant
qo = 2Uoob(x)h(x), (170)
where qQ designates volumetric flow of the pollutant from the
source, U^ is the current velocity, b(x) is the half-width of the
field a distance x downstream from where the pollutant appears at
the surface, and h(x) is the average cross-sectional depth of the
-------
102
U^, Current
P
Lateral spread, Velocity = U,.
A
2b(x), width of field
Pollutant field density=P
Ap = PS-P
Pollutant first appears
at sea surface, x=0,
j_J '°
Lateral spread
SECTION A-A
h(x), average depth of
pollution field
Idealized cross-section
SECTION A-A
Possible actual shape
of field cross-section
Surrounding
ocean, p_
Figure 25. Interpretation of the Larsen and Sorensen Model
for pure gravitational spread.
-------
103
pollution field. By conservation of mechanical energy, the velocity
of spread of the pollution front, U, is given by
Uf =
2gh(x)
(171)
where a3 is a constant on the order of 1, and Ap is the density
disparity between layers. Because
db
dx'
b(x) =
3/2 +
2/3
(172)
where2b is the initial width of the pollution field. As an
approximation,for b «b(x),
b(x)«
2/3
(173)
Thus, equation (173) may be used to estimate the pollution
field width,2b, at downstream points, x. However, this equation
was developed from a very crude model and considers only gravita-
tional effects. For these reasons, this model must be considered
only a very rough approximation of the actual dispersion process.
Akira Wada and colleagues in a series of Japanese papers
(55, 56, 29), have taken a more rigorous and realistic approach
in analyzing the recirculation of condenser cooling water for
nuclear power stations located on certain bays in Japan. The
approach taken by Wada is to consider the fundamental equations
of momentum and energy,
-------
104
au,
j- ax.,
PO 3x.
13
3Xj
(8)
with molecular dlffusivity, v, ignored, and
n
at j 3x.
J
3x.
J
3T
axT
J (
(9)
o p
subject to certain assumptions, and carry out the solutions numer-
ically. These assumptions are:
• steady flow;
. inertial terms
3U
U.
are small compared to turbulent
transport and pressure terms;
. Coriolis forces are neglected;
. the motion is a two-dimensional lateral flow restricted to
the region above the thermocline.
Equations (8) and (9) are then written in x,y,z coordinates as:
x-direction momentum,
_ _
3x x 3x ay
y-direction momentum,
IP. l_ fe 3V) + l_ /e
ay ax ^x ax; ay v y
(174)
(175)
-------
105
Energy,
U|l+V9l=9_f 31)+3_fr 3T^
3* 9y 3x UHx 3x; 3y UHy 3y'
+ 3_ (, 9T, Q0
9z l£Hz 9F + —^ > (176)
pupn
where QQ is net surface heat transfer,*Cp is specific heat, and
h is the surface layer thickness. The quantity Q the net heat
exchange at the water surface, is composed of the net radiant
exchange, evaporation, and convection. Note that in contrast to the
single layer flow assumption for the equations of motion, heat
transfer is treated in three dimensions.
By assuming the eddy diffusivities of momentum, e and e
to be constant, and by cross-differentiation, equations (174)
and (175) become:
92P
9x9y
32
= e _ H. e 3
e + £
(177)
x 9x 9X2 y 37 ayT • (178)
Now a stream function, ^, may be defined such that
Us|y (179)
(180)
Strictly speaking, the term Q0/pCDh should not be included in
equation (176) since this term is a boundary condition and
should be treated as such.
-------
106
which automatically satisfy the continuity equation,
-ii+3V = 0. (181)
3x 3y
By subtracting (178) from (177), the terms involving pressure
are eliminated. By replacing U and V with equations (179) and
(180) in this result, an equation for horizontal motion in terms
of the stream function may be written as
= 0. (182)
The quantity 6 in equation (182) is the ratio ey/ex-
Wada solved equation (182) for the stream function, q>, using
finite difference techniques. Velocity components U and V were
established using equations (179) and (180). Once the velocity
components were determined, solution to equation (176) was found
numerically, thus establishing the temperature distribution for
the particular bay.
The approach taken seems to be a reasonable one. However, the
hydraulics may be oversimplified by the single layer assumption.
Also, it is not clear that all inertial terms in the equations of
motion can be neglected - especially in the vicinity of the hot water
source.
Hayashi and Shuto (20) also investigated the case of hot water
spreading over a colder stagnant receiving water. Their approach
was based on vertically integrated equations of motion and energy
-------
107
conservation, and similarity approximations concerning vertical
velocity and temperature profiles. Except for the inclusion of
vertical variations, these authors make the same flow field assump-
tions as did Wada et al, The governing equations become:
x-momentum,
Pdz = 0; (183)
y-momentum,
rd
Pdz = 0; (184)
J J J f J J -Tf
z-momentum,
pgdz.
Here z is vertically downward, £ is the ordinate at the water
surface, d is the water depth, and e, is the lateral eddy viscosity
which is assumed to be the same for both the x and y directions.
The vertically integrated continuity equation is
Udz +|~ f Vdz = E'v'uW , (185)
where E' is the entrainment parameter (entrainment from lower depths,
z direction).
The energy equation used by Hayashi and Shuto is written as
= Ff-(Ts-Tj. (186)
In this equation T^ and T$ are the temperatures of the receiving
water and hot water surface, respectively, and hg is an overall
-------
108
heat transfer coefficient at the air-sea interface. Note that
turbulent heat transport is not accounted for.
According to Hayashi and Shu to, at low values of Frj (the
length term in the densimetric Froude number is based on the
thickness of the thermal layer), the entrainment parameter is
essentially zero, so that continuity is given by
Itfwz t fyjjdz = 0. (187)
Now a stream function may be defined such that
(188)
and a'hY = -|£, (189)
where a1 is a parameter related to the vertical velocity profile
and h is the thickness of the layer of motion.
It is now possible to arrive at equation (182) with <5=l(e /e =1),
x y
so that the stream function is given by
V> = 0 (190)
where v" is the bihartnonic operator. For a simple geometry the
above equation may be solved analytically. Hayashi and Shuto
assumed the flow was issuing to a semi-infinite medium, as shown
in Figure 26. For this geometry,
* = T2T [(y+lHan-\^)-(y-l)tan-^)]. (191)
From this equation, velocities U and V may be obtained from the
-------
N
N
\
N
N
\\\N>
Boundary
Flow
qj
Receiving water
1
B/2
- B
Figure 26. Planar view of flow issuing to a semi-infinite medium,
o
10
-------
no
definition of the stream function and applied 1n the energy equation
to arrive at the temperature distribution,
h. ...
T=T +(T,-T )exp
ao * J
-------
Ill
NUMERICAL TECHNIQUES
Advances in computer technology during the past few years have
opened a new frontier in solving partial differential equations
such as those describing the motion and diffusion of pollutants in
coastal waters. Before the age of high-speed computers, numerical
techniques which had been set down in theory some years previous,
were impractical to carry out because of the monumental amount of
calculation involved. But today such techniques as finite differ-
ences, finite elements, and the Monte Carlo method are indeed prac-
tical and are being applied to a wide variety of practical engineering
problems. In some cases these methods are so successful that they
are used to save engineering time in spite of known analytic solu-
tions.
The most attractive feature of numerical solutions is that many
complexities associated with real physical phenomena do not have
to be ignored as in the case of analytical solutions. Complex
boundary geometry and boundary conditions can be accounted for and
non-linear effects may be incorporated in the analysis. On the
other hand, numerical solutions are not without drawbacks. Such
problems as computation time and numerical stability in some cases
negate the attractiveness of the numerical approach. Nevertheless,
the numerical solution is a practical alternative to the oversimpli-
fied analytical solution.
The most complete treatment of numerical modeling would include
the following considerations:
-------
112
time dependence,
computation of velocity and diffusion in three-space,
interaction between the flow field and pollutant field,
variable eddy diffusivity coefficients, and
arbitrary boundary conditions such as irregular solid
boundaries, heat transfer and wind at the sea surface, and variable
currents and water levels at the flow boundaries.
At present, there is no computer program available which in-
cludes all of the above complications. Typically, computations
are carried out in two space dimensions, usually the horizontal
plane. However, diffusion computations are in some cases extended
to three dimensions. For instance, Wada (55 ) computed the velocity
field for certain bays in Japan, using the two-dimensional hydro-
dynamics equations, but considered heat transfer in three dimensions.
If one is to consider interactions between the flow field and
density distribution, which may be very important during initial
dilution, three space dimensions must be considered unless the
region of interest has symmetry to the extent that one space variable
may be realistically eliminated. In coastal situations, symmetry
is usually lacking because of irregular currents and boundaries.
Wada (56) considered a horizontal shoreline discharge of hot water
which included buoyancy interaction. However, the analysis was
carried out in two dimensions, vertical and parallel to the dis-
charge. Since no variation was considered parallel to the shoreline,
-------
113
the numerical model was not realistic in this respect. On the
other hand, the analysis demonstrated significant differences
in the flow and temperature fields between the cases of interac-
tion and no interaction.
For the drift flow regime lateral density gradients caused by
a contaminant are usually quite small. Thus, it is typical to
neglect the interaction between buoyancy and flow and thus analyze
the hydraulic problem separate from the pollutant diffusion.
Velocity distributions calculated from hydraulic analysis or
estimated from field measurements are then used as known input
to compute convective terms in the transport equation.
Numerical Models
A considerable amount of work has appeared in the literature
in recent years dealing with numerical methods for solving the
equations of motion and diffusion ((5), (17), (49), (50),(53), and
(57)). However, most of this work deals with specific cases and
is not directly suited for coastal pollution problems. At this
time there are only a few programs published and available for
coastal hydraulic computation, and there is no program generally
available as a complete package for predicting both coastal
hydraulics and dispersion.
The remainder of this section will be confined to a brief
discussion of published computer programs which under certain
conditions are suitable for predicting coastal hydraulics and/or
dispersion in the drift flow regime.
-------
114
Coastal Hydraulics - Leendertse's Model
The numerical model developed by Leendertse (27) is based on
a finite difference representation of the vertically integrated
equations of motion and continuity. The model is specifically
designed for tidal computations along a coast having arbitrary
geometry and bottom topography. An operational computer program
is listed in reference (27) along with input instructions.
Dronkers (13) gives a review of the numerical model.
Numerical modeling is based on the following partial differ-
ential equations:
Motion.
1/2
= F(x) (193)
9 -- ' U
= y
where U and V are the vertically averaged velocity components u and
v defined by
U = nir I udz
-h
V =
-------
115
P' and F' are forcing functions from wind stress and barometric
pressure fluctuations, Ch is the Chezy coefficient, and t, is the
fluctuating water height measured from a reference. This reference
plane is a distance H from the ocean bottom.
Continuity. The vertically integrated continuity equation
leads to the following expression for the fluctuating water level
3t 3x 3y '
The method of solution is what Leendertse terms a multi -opera-
tional technique. This technique is very closely associated with
the alternating direction method for two-dimensional partial differ-
ential equations first discussed by Peaceman and Rachford (40a)
concerning the heat equation. Details of the solution technique
will not be discussed here; however, the interested reader may con-
sult Leendertse ( 27 ) or Dronkers (13).
Leendertse has carried out detailed studies of the method
stability and has ascertained that the model is unconditionally
numerically stable. However, solution accuracy deteriorates as the
time step becomes large.
Input to the computation program includes system geometry-flow
boundary water levels as a function of time, and Chezy coefficients.
Output includes water level and vertically integrated velocity at
each grid point in the system as a function of time.
-------
116
Although the vertically integrated model may not yield the
velocity field detail that is required for studying localized
dispersion, it does provide flow information such that gross
transport and dispersion may be estimated.
Dispersion
Three available computer codes are briefly discussed
here:
1) NRDL codes (23) based on the mathematical solution of
the Carter-Okubo (IQa) model;
2) Tetra-Tech code(25) used for studying long-term transport;
3) New York University,(30), based on the Monte Carlo method.
In each of the above codes the velocity field must be estab-
lished independently, and each has limited application.
Naval Radiological Defense Laboratory Codes. The Naval
Radiological Defense Laboratory (NRDL) has developed computer
codes for estimating radioactivity levels which result from instan
taneous and continuous point releases in the ocean. These models
are based on the analytical solution of the Carter-Okubo shear
f1ow model (1Oa):
<196>
The terms fl1 and fi1 • are velocity gradients in the y and z direc-
tion and U is the mean current. These terms are assumed constant
o
-------
117
for a given solution to the above transport equation.
The computer programs calculate the dimensions and volume,
and amount of contaminant, in a dispersing patch caused by a
point or continuous release. Program input includes the empiri-
cal constants U , n1 , n'z, e x, e and e z- This program is
not applicable where finite boundaries, such as the water surface
and coastline, affect dispersion.
Tetra-Tech Code. Koh and Fan (25) have developed a computer
program for estimating long-term transport for a pool of radio-
active debris submerged in the ocean, a problem similar to the one
investigated by NRDL, described above. However, Koh and Fan,
instead of using the analytical solution of the Carter-Okubo
transport equation as was done in the NRDL programs, solve the
basic diffusion equation,
3C , „ 3C , ,, 3C _ 3 / 3C v
at + u W+ v ay - a^( MX W}
3C ) + i_ (e,, 3), (197)
3y; 3z {\*\z Sz;'
by the "method of moments." The above equation does not include
vertical convective transport of the pollutant.
Details of the method of moments are not discussed here (e.g.,
see reference 48), but it is a technique of lumping quantities in
a horizontal plane so that the concentration moments are a function
-------
118
of z and t alone. This method then greatly reduces the amount of
time necessary for numerical solutions to the transport equation.
The computer program application is limited to a submerged
pool with no coastline effects. Horizontal velocities are assumed
constant in a lateral plane, but may vary with depth. Velocities
must be supplied to the program as input. Also, no transport across
the water surface is allowed.
New York University Code. Mehr (30) has developed a dispersion
code based on the Monte Carlo method using random numbers. At this
time full detail concerning the program and solution technique has
not been published, but may be obtained from Mehr.
According to Mehr, the program will predict contaminant distri-
butions for a two-dimensional horizontal flow system having laterally
irregular solid boundaries. As in the case of previously discussed
dispersion codes, the velocity field must be supplied by the program
user. The general idea of the method is that a large number of
contaminant particles are tagged. Beginning with an initial distri-
bution of these tagged particles, they are convected through the
flow field according to the velocity distribution, and diffused
according to random number components. Solid boundaries reflect
these tagged particles and cross ing over flow boundaries causes the
particles to be lost from the system. The program will accommodate
time dependent particle sources; hence, an outfall may be simulated.
-------
119
BIBLIOGRAPHY
1. Abraham, G., "Jet Diffusion in Stagnant Ambient Fluid." Delft
Hydraulics Publication No. 29, 1963.
2. Abraham, G., "Entrainment Principle and its Restriction to
Solve Problems of Jets." Delft Hydraulics Publication
No. 55, 1965.
3. Abraham, G., "Jets with Negative Buoyancy in Homogeneous Fluid."
Delft Hydraulics Publication No. 56, 1967.
4. Albertson, M. L., Y. B. Dai, R. A. Jensen, and H. Rouse,
"Diffusion of Submerged Jets." Transactions American
Society of Civil Engineers, Volume 115, Number 150,
pp. 639-697.
5. Aziz, K. and J. D. Heliums, "Numerical Solution of the Three-
Dimensional Equations of Motion for Laminar Natural
Convection." The Physics of Fluids, Volume 10, Number 2,
1967, pp. 314-324.
6. Baumgartner, Donald J., "Vertical Jet Diffusion in Non-linear
Density Stratified Fluids." Oregon State University,
Corvallis, Ph.D. Thesis, 1966, 166 p.
7. Bird, R. B., W. E. Stewart, and E. N. Lightfoot, Transport
Phenomena. John Wiley and Son, Inc., New York City,
New York, 1960, 780 p.
8. Bosanquet, C. H., G. Horn, and M. W. Thring, "The Effect of
Density Differences on the Path of Jets." Proceedings
Royal Society of London, A263(1961), pp. 340-352.
9. Brooks, Norman H., "Limiting Height of Rise of a Buoyant Plume
in a Stratified Environment." W. M. Keck Laboratory of
Hydraulics and Water Resources, Division of Engineering
and Applied Science, Pasadena, California, Technical
Memorandum No. 63-6, 1963, 4 p.
10. Brooks, Norman H., "Diffusion of Sewage Effluent in an Ocean
Current." Proceedings of the First Conference on Water
Disposal in the Marijie Environment, Pergamon Press, Inc.,
Long Island City, New York, 1959, pp. 246-267.
lOa. Carter, H. H. and Okubo, A., "A Study of the Physical Processes
of Movement and Dispersion in the Cape Kennedy Area."
Chesapeake Bay Institute, The Johns Hopkins University,
Baltimore, Maryland, Report No. NYO-2973, March 1965.
-------
120
11. Cederwall, K., "Hydraulics of Marine Waste Water Disposal."
Hydraulics Division, Chalmers Institute of Technology,
Goteborg, Sweden, Publication No. 42, January 1968.
12. Csanady, G. T., "The Buoyant Motion within a Hot Gas Plume
in a Horizontal Wind." Journal of Fluid Mechanics,
Volume 22, Part 2, 1965, pp. 225-239.
13. Dronkers, J. J., "Tidal Computations for Rivers, Coastal Areas,
and Seas." Proceedings of the American Society of Civil
Engineers, Journal of the Hydraulics Division, Volume 93,
1969, pp. 29-77.
14. Fan, Loh-Nien, "Turbulent Buoyant Jets into Stratified or
Flowing Ambient Fluids." W. M. Keck Laboratory of Hydraulics
and Water Resources, California Institute of Technology,
Pasadena, California, Publication No. KH-R-15, June 1967.
14a. Fan, Loh-Nien and Norman H. Brooks, "Numerical Solution of
Turbulent Buoyant Jet Problems." W. M. Keck Laboratory
of Hydraulics and Water Resources, California Institute
of Technology, Pasadena, California, Publication No.
KH-R-18, January 1969.
15. Foxworthy, J. E., R. B. Tibby, and G. M. Borsom, "Dispersion
of a Surface Waste Field in the Sea." Paper presented
at the April 1965 meeting of the California Water Pollution
Control Association, San Diego, California.
16. Frankel, Richard J. and James D. Cumming, "Turbulent Mixing
Phenomena of Ocean Outfalls." Proceedings of the American
Society of Civil Engineers, Journal of the Sanitary Engineer-
ing Division, Volume 91, 1965, pp. 33-59.
17. Fromm, Jacob E., Finite Difference Methods of Solution of Non-
linear Flow Processes with Application to the Benard
Problem.Los Alamos, New Mexico, LA-3522, 1967, 135 p.
18. Harremoes, Paul, "Diffuser Design for Discharge to a Stratified
Water." Preprint, Water Research, 1965.
19. Hart, William E., "Jet Discharge into a Fluid with a Density
Gradient." Proceedings of the American Society of Civil
Engineers, Journal of the Hydraulics Division, Volume 87,
1961, pp. 171-200.
20. Hayashi, Taizo and Nobuo Shuto, "Diffusion of Warm Water Jets
Discharged Horizontally at the Water Surface." Proceedings
of the Twelfth Congress of the International Association
of Hydraulics Research, Fort Collins. Colorado, 1967.
-------
121
21. Hinze, J. 0., Turbulence. McGraw-Hill Book Company, New York
City, New York, 1959, 586 p.
22. Hinze, J. 0. and B. G. Hegge Zijnen, v.d., "Transfer of Heat
and Matter in the Turbulent Mixing Zone of an Axially
Symmetrical Jet." Applied Scientific Research, Section
A, No. 1, 1949, pp. 435-461.
23. James, L. R., S. Z. Mikhail, and V. E. Schrock, "A Parametric
Study of Three-dimensional Ocean Diffusion." Environmental
Science Associates, Burlingame, California, NRDL-TR-69-50,
July 1969.
24. Jen, Yuan, Robert T. Wiegel, and Ismail Mobarek, "Surface
Discharge of Horizontal Warm-water Jet." Proceedings of
the American Society of Civil Engineers, Journal of the
Power Division. Volume P02, April 1966, pp. 1-30.
25. Koh, R. C. Y. and Loh-Nien Fan, "Prediction of the Radioactive
Debris Distribution Subsequent to a Deep Water Nuclear
Explosion." Tetra-Tech, Inc., Pasadena, California, Final
Report NRDL-TRC-68-60, 1968.
26. Larsen, I. and T. Sorensen, Qualitative Description of Gravita-
tional Spread, Mathematical Formulation and Basic Assump-
tions.NEWGOHOMD, Copenhagen, Denmark, 1967.
27. Leendertse, Jan J., "Aspects of a Computational Model for Long-
period Water-wave Propagation." The Rand Corporation,
Santa Monica, California, RM-5294-PR, 1967.
28. Lilly, Douglas K., "Numerical Solutions for the Shape-preserving
Two-dimensional Thermal Convection Element." Journal of
Atmospheric Sciences. Volume 21, 1964, pp. 83-91T
29. Manabe, Kyohei, Yoshio Watanabe, and Akira Wada, "Study on
Recirculation of Cooling Water of Tsuruga Nuclear Power
Station Sited on Urayoke Bay." Coastal Engineering in
Japan. Volume 9, 1966.
30. Mehr, Emanual, New York University, Letter Report to Federal
Water Pollution Control Administration, October 24, 1969.
31. Miyakoda, Kihuro, "Contribution to the Numerical Weather Predic-
tion-Computation with Finite Differences." Japanese Journal
of Geophysics, Volume 3, No. 1, 1962.
-------
122
32. Morton, B. R., "On a Momentum Mass Flux Diagram for Turbulent
Jets, Plumes, and Wakes." Journal of Fluid Mechanics,
Volume 10, 1961, pp. 101-llf:
33. Morton, B. R., "Forced Plumes." Journal of Fluid Mechanics,
Volume 5, 1959, pp. 151-163.
34. Morton, B. R., G. I. Taylor, and J. S. Turner, "Turbulent
Gravitational Convection from Maintained and Instantaneous
Sources." Proceedings of the Royal Society of London,
Volume A234, 1956, pp. 1-23.
35. Murgai, M. P. and H. W. Emmons, "Natural Convection Above
Fires." Journal of Fluid Mechanics, Volume 8, 1960,
pp. 611-623"
36. Neumann, Gehard and Willard J. Pierson, Jr., Principles of
Physical Oceanography. Prentice Hall, Inc., Englewood
Cliffs, New Jersey, 1966, 545 p.
37. Okubo, A., "Horizontal Diffusion from an Instantaneous Point-
source Due to Oceanic Turbulence." Chesapeake Bay
Institute, Johns Hopkins University, Baltimore, Mary-
land, Technical Report 32, Reference 62-27, 1962, 22 p.
38. Orlob, G. T., "Eddy Diffusion in Homogeneous Turbulence."
Transactions of the American Society of Civil Engineers.
Volume 126, Part I, 1961, pp. 397-438.
39. Ozmidov, R. V., "On the Dependence of the Coefficient of Hori-
zontal Diffusion in Wind Currents upon the Relative Depth
of a Water Basin." IZV. Geophysical Series, 1958,
pp. 1242-1246.
40. Phillips, 0. M., Dynamics of the Upper Ocean. Cambridge Press,
Cambridge, Massachusetts, 1967, 320 p.
40a. Peaceman, D. W., and H. H. Rachford, "The Numerical Solution
of Parabolic and Elliptic Differential Equations," Journal
of the Society of Industrial Applied Mathematics, Volume
3, No. 1, March, 1955.
41. Priestley, C. H. B. and F. K. Ball, "Continuous Convection from
an Isolated Source of Heat." Quarterly Journal of the
Royal Meteorological Society. Volume 81, 1955, pp. 144-157.
-------
123
42. Priestley, C. H. B., "A Working Theory of the Bent-over Plume
of Hot Gas." Quarterly Journal of the Royal Meteorological
Society, Volume 82, 1956, pp. 165-176.
43. Rawn, A. M., F. R. Bowerman, and Norman H. Brooks, "Diffusers
for Disposal of Sewage in Sea Water." Proceedings of the
American Society of Civil Engineers, Journal of the
Sanitary Engineering Division, Volume 86, 1960, pp. 65-105.
44. Ricou, F. P. and D. B. Spalding, "Measurements of Entrapment
by Axisymmetrical Turbulent Jets." Journal of Fluid
Mechanics, Volume 11, 1961, pp. 21-32.
45. Rouse, H., C. S. Yih, and H. W. Humphreys, "Gravitational
Convection from a Boundary Source." Tell us, Volume 4,
1952, pp. 201-210.
45a. Schlichting, Hermann, Boundary Layer Theory, tr. by J. Kestin.
McGraw Hill Company, New York City, New York, 4th Edition,
1960, 647 p.
46. Schmidt, F. H., "On the Rise of Hot Plumes in the Atmosphere."
International Journal of Air and Water Pollution. Volume
9, 1964, pp. 175-198.
47. Schmidt, W., "Turbulente Ausbreitung eines Stromes erhitzter."
Luft. Zeitschr. Angew. Math, und Mech., Volume 21, 1941,
pp. 265-278; pp. 351-363.
48. Smith, F. B., "The Role of Wind Shear in Horizontal Diffusion
of Ambient Particles." Quarterly Journal of the Royal
Meteorological Society, Volume 91, 1965, pp. 318-329.
49. Terry, Michael D., "A Numerical Study of Viscous, Incompressible
Fluid Flow Problems." Oregon State University, Corvallis,
Oregon, MS Thesis, 1968, 95 p.
50. Tomich, John F. and Eric Weger, "Some New Results on Momentum
and Heat Transfer in Compressible Turbulent Free Jets."
American Institute of Chemical Engineers Journal, Volume
13, No. 5, 1967, pp. 948-954.
51. Tollmien, W., "Strahlverbreiterung." Zeitschr. Angew. Math.
und Mech.. 1926, pp. 468-478.
52. Trent, D. S., D. J. Baumgartner and K. V. Byram, "Forced Plumes
in Stratified, Quiescent Media." To be published.
-------
124
53. Trent, D. S. and J. R. Welty, "Numerical Methods for Solving
Problems in Transient Heat Conduction." Oregon State
University Experiment Station Bulletin. To be published.
54. Turner, J. S., "Jets and Plumes with Negative or Reversing
Buoyancy." Journal of Fluid Mechanics, Volume 26, Part
4, 1966, pp. 779-792.
55. Wada, Akira, "Study on Recirculation of Cooling Water of Power
Stations Sited on a Bay." Coastal Engineering in Japan.
Volume 10, 1967.
56. Wada, Akira, "Numerical Analysis of Distribution of Flow and
Thermal Diffusion Caused by Outfall of Cooling Water."
Central Research Institute of Electric Power Industry,
Abiko-cho, Japan, Technical Report C:67004, 1969.
57. Welch, J. Eddie, Francis H. Harlow, John P. Shannon, and Bart
J. Daly, The MAC Method. Los Alamos, New Mexico, LA-3425,
1966, 1461T
58. Yih, Chia-Shun, Dynamics of Nonhomogeneous Fluids. MacMillan
Company, New York City, New York, 1965, 306 p.
59. Yuan, S. W., Foundations of Fluid Mechanics. Prentice Hall, Inc.,
Englewood Cliffs, New Jersey, 1967, 608 p.
60. Yudelson, J. M., "A Survey of Ocean Diffusion Studies and
Data." W. M. Keck Laboratory of Hydraulics and Water
Resources, California Institute of Technology, Pasadena,
California, Technical Memorandum No. 67-2, September 1967.
61. Zeller, R. W., "Cooling Water Discharge into Lake Monona."
University of Wisconsin, Madison, Wisconsin, Ph.D. Thesis,
1967, 287 p.
-------
125
NOMENCLATURE
The following notations are used unless otherwise specified
in the text.
a Molecular thermal diffusivity
b Characteristic plume radius
b(x) Half-width of pollution field
C Concentration of pollutant
C . Coefficient of drag
d
C. Chezy coefficient
C Concentration of pollutant at centerline
m
C Specific heat
D Diameter of discharge port
p Molecular mass diffusivity
E Cube of centerline velocity (Um3)
e.., Permutation tensor (third order)
e Unit vector, radial
r
e Unit vector, east
x
e Unit vector, north
e Unit vector, vertical
e Unit vector, angular
9
erf Error function
-------
126
f Function
f Coriolis parameter
P ' Forcing function
F, Drag force
Fr Densimetric Froude number
Fr, Densimetric Froude number at discharge port
\j
g Gravitational constant
G Linear stratification parameter
h Average thickness of pollution field
h Heat transfer coefficient
H Water depth
k Proportionality constant
K Entrainment parameter for momentum
L Lateral scale for drifting plume
1 Width of pollution field
m Mass species loss or production per unit volume
M Instantaneous source strength
M Momentum of plume at beginning of established flow
M. Momentum at orifice
P Pressure
q Volumetric heat generation
q Volumetric flow rate of pollutant
Q Volume flow rate
Q Surface heat transfer
^o
Q1 Volumetric entrainment
-------
127
r Radial coordinate
R Product (UmAm)
Rb Radius of plume boundary
RC Characteristic radial scale for plume
R.. Reynolds stress tensor
• w
s Axis for curved plume
S Dilution (1/C)
S1 Source strength
Sg Length to established flow, curved plume
Sm Centerline dilution
t Time
t' Length of time for averaging turbulent quantities
T Temperature
T Reference temperature (T^o))
Tm Temperature at plume centerline
T^z) Temperature of environment
u. Fluctuating velocity along i— coordinate (description
of turbulent flow)
U Velocity (x-direction)
I). Velocity along i— coordinate
Um Centerline velocity (horizontal and inclined plumes)
Ur Velocity (radial direction)
U^ Environmental velocity, x-component
Ufl Velocity (angular)
-------
128
UQ,Uj Orifice velocity
v Volume
V Velocity (y-direction)
V^ Environmental velocity, y-component
W Velocity (z-direction)
W, Orifice velocity
u
Wm Vertical velocity at center!ine
Wro Environmental velocity, z-component
x Space coordinate
x. i— space coordinate
y Space coordinate
z Space coordinate, vertical
z ,x Length for flow establishment
e e 3
z . Maximum height of rise at plume centerline
rnax
Greek
a Entrainment coefficient
6 Temperature coefficient
A Density disparity parameter
Ay Buoyant force
6 Ratio, e /e
* j
5.. Kronecker delta function (second order tensor)
• J
£ Water level
e.. Eddy momentum diffusivity tensor (second order)
' *J
eu. Eddy thermal diffusivity tensor (first order)
nj
-------
129
e. Eddy momentum diffusivity
J
0 Temperature disparity
6 Angular coordinate
K Thermal conductivity
A. Entrainment coefficient for heat or matter (Morton, Fan)
u Entrainment parameter for heat or matter
v Molecular kinematic viscosity
p Density
p. Density of pollutant at orifice
u
p Plume center!ine density
p Density of environment at orifice
Ko J
p Density of environment at surface
poo(z) Density of environment
TT Pi, 3.141
T Fluid stress tensor (second order)
$ Viscous dissipation term in energy equation
$ Latitude
4>(x.) Function
¥(x.) Function
ty Stream function
n1 Velocity gradient
------- |