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NAVAL POSTGRADUATE SCHOOL
Monterey, California
THESIS
ENTRAPMENT MODELLING OFBUOYANT MOMENTUM JETS IN WATER
by
David Stuart Hilder
December 1981
Co-Advisors: B.
M.
GebhartKelleher
Approved for public release; distribution unlimited.
Prepared for:
Naval Sea Systems CommandWashington, D.C.
NAVAL POSTGRADUATE SCHOOLMonterey, California
Rear Admiral J. J. Ekelund David A. SchradySuperintendent Acting Provost
This thesis is prepared in conjunction with research supported in part
by Naval Sea Systems Command under work request N0002481WR10497.
Reproduction of all or part of this report is authorized.
Released as a
Technical Report by:
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4. T!Ti_E mnd Subitum)
Entrainment Modelling of Buoyant Momentum Jetsin Water
5. TYPE OF «EPO*T 4 PERIOO COVERED
Engineer's Thesis;December 1981
S. PERFORMING ORG. REPORT NUMBER
7. AUTHOR, t)
David Stuart Hilder
In conjunction with B. Gebhart and M. Kelleher
• CONTRACT OR GRANT NuMBERf)
N0002481WR10497
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Naval Postgraduate SchoolMonterey, California 93940
10. PROGRAM ELEMENT. PROJECT TASKAREA * WORK UNIT NUMBERS
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IS. SUPPLEMENTARY NOTES
IS. KEY WOROS (Contlnuo an rarer** •<*• II nocaaaarr ana laantlty ar aloe* numbat)
Buoyant Momentum JetsEntrainmentPlumesJet ModellingThermal Discharge
20. ABSTRACT (Conilnua an ravmaa tldo // nacaaaary *n« I fatuity »r black rrumbar)
The general characteristics of buoyant momentum jets in water are des-cribed. Previous analytical modelling techniques utilizing the entrainmentconcept for prediction of trajectory and residual physical properties arediscussed, and an overview of the existing experimental data base is given.The limitations of previous analytical modelling techniques are enumerated,generally resulting from incomplete or inadequate equations of state. Anexisting comprehensive equation of state for pure and saline water is proposed
DDFORM
1 JAN 73 1473 EDITION OF I MOV IS IS OBSOLTTT.S/N 102-0 1*- 440 1
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UNCLASSIFIED"»»«• n*«« Bmlm—4
#20 - ABSTRACT - (CONTINUED)
for use in entrainment modelling. An original computerized procedure,based on appropriate conservation equations, is used to predict trajectoryand physical properties of various buoyant momentum jets. Comparisonis made with previous analytical and experimental results for the casesof quiescent, flowing, and stratified ambients. Finally, the comprehen-sive equation of state, coupled with the present computational procedure,is used to describe a complexly stratified ambient and the behavior ofa buoyant momentum jet discharged into it.
DD Tor^ 1473?_ UNCLASSIFIED1 Jan 73
S/N 0102-114-6601 iJtcj»iT* st-iMifieATiOM o' '*•» »»o*r«»«« a«r« !«(•.•«»
Approved for public release; distribution unlimited,
ENTRAPMENT MODELLING
OF
BUOYANT MOMENTUM JETS IN WATER
by
David Stuart HilderLieutenant, United States Navy
B.S., University of New Mexico, 1975
Submitted in Partial Fulfillment of the
Requirements for the Degrees of
MASTER OF SCIENCE IN MECHANICAL ENGINEERING
and
MECHANICAL ENGINEER
from the
NAVAL POSTGRADUATE SCHOOLDecember 1 981
- yf
ABSTRACT
The general characteristics of buoyant momentum jets in water are
described. Previous analytical modelling techniques utilizing the
entrainment concept for prediction of trajectory and residual physical
properties are discussed, and an overview of the existing experimental
data base is given. The limitations of previous analytical modelling
techniques are enumerated, generally resulting from incomplete or inade-
quate equations of state. An existing comprehensive equation of state
for pure and saline water is proposed for use in entrainment modelling.
An original computerized procedure, based on appropriate conservation
equations, is used to predict trajectory and physical properties of vari-
ous buoyant momentum jets. Comparison is made with previous analytical
and experimental results for the cases of quiescent, flowing, and strati-
fied ambients. Finally, the comprehensive equation of state, coupled
with the present computational procedure, is used to describe a com-
plexly stratified ambient and the behavior of a buoyant momentum jet
discharged into it.
TABLE OF CONTENTS
I. INTRODUCTION — - - 8
II. BASIC CHARACTERISTICS OF ROUND DISCHARGE SYSTEMS 11
III. LITERATURE REVIEW — - 18
A. REVIEW OF PAST MODELLING 18
B. REVIEW OF EXPERIMENTAL STUDIES 29
IV. EQUATIONS OF STATE AND AMBIENT STRATIFICATION MODELLING — 34
V. PRESENT METHOD —
-
41
VI
.
RESULTS 51
A. QUIESCENT AMBIENT -- 51
B. FLOWING AMBIENT 72
C. STRATIFIED AMBIENT 94
VII. EXTRAPOLATION OF PRESENT METHODS TO PARAMETERSOF INTEREST 111
VIII. RECOMMENDATIONS 113
APPENDIX A — — 114
APPENDIX B - 119
APPENDIX C — 124
LIST OF REFERENCES 128
INITIAL DISTRIBUTION LIST 130
NOMENCLATURE
B - Characteristic jet width
b - Dimensionless jet width, B/D
c - Concentration
D - Jet discharge diameter
E - Volumetric entrainment
p a"p 1/2F - Densimetric Froude number, Un/(gD( ))
u p
2p a"
p mF, - Local densimetric Froude number, U /gB( )L m 3
p
«
g - gravity
Q - jet mass flow rate
R - Ambient flow ratio, II /ILa u
r - Radial jet coordinate
S - Streamwise coordinate of jet velocity
s - Dimensionless streamwise jet coordinate, S/D; salinity
t - Temperature
U - Streamwise jet velocity
U* - U - U cos 9, relative local velocity
u - Dimensionless streamwise jet velocity, U/Uq
X - Horizontal Cartesian coordinate
Z - Vertical Cartesian coordinate
GREEK SYMBOLS
a - Entrainment constant
3 - Volumetric coefficient of thermal expansion
y - Volumetric coefficient of concentration expansion
A( ) - ( )1
- ( )j
8 - Local angle of inclination from horizontal
x - Relative spreading ratio
p - Density
cj)- Azimuthal jet angle
SUBSCRIPTS
a - ambient
e - at beginning of zone of established flow
m - at jet center! ine
- at jet discharge
I. INTRODUCTION
The cooling water discharge from a power plant into a large body of
water, the thermally loaded condenser discharge from the condenser of a
moving ship or submarine, and the high temperature gas issuing from a
stack or gas turbine exhaust are all examples of buoyant momentum jets.
The trajectory and behavior of such jets after discharge is influenced
by factors such as initial jet velocity and buoyancy, ambient motion and
stratification, and mixing rate. However, questions such as whether
or not the jet will rise to a certain level, what the jet velocity and
temperature will be at any point along its trajectory, or what effect
ambient stratification will have on behavior, all require an involved
quantitative analysis. Significant effort has been expended in the past
few decades in attempting to understand the mechanics of buoyant jet
mixing and trajectory, with the ultimate objective of developing accurate
models to predict trajectory and decay.
Certainly the need for such predictive models has grown. Contem-
porary nuclear and fossil fueled power plants have thermal efficiencies
on the order of 30-40%. The significant waste heat from these facilities
takes the form of a thermally loaded discharge into either the atmosphere
or a body of water. Sewage is often discharged as treated effluent into
rivers, lakes, and oceans. The proper evaluation of the ecological
impact of such discharges requires that their physical behavior be pre-
dictable. More stringent environmental regulations and heightened public
awareness place a premium on the accuracy of such prediction.
The need to predict momentum jet behavior is not limited to environ-
mental issues. Rapid advancement of the ability to detect small
8
temperature variations, concentration differentials, and turbulence
anomalies may make it increasingly easy to detect various military craft
and vehicles by virtue of propulsion system thermal discharges, wake
turbulence, and wake concentration variations. The implications for
weapons systems and platforms which rely on stealth for effectiveness
are enormous.
Given the wide range of applications in which an analysis of fluid jet
behavior might be used, it becomes obvious that the range of possible jet
or ambient characteristics that may be of interest is equally wide.
Initial jet geometry, discharge parameters, degree of thermal loading,
and turbulence characteristics, as well as ambient flow conditions, turbu-
lence, and stratification, can be combined into an almost infinite number
of scenarios.
This investigation will be restricted to the case of a single, fully
turbulent, circular water jet discharged into a surrounding water ambient.
The case will be further restricted by limiting jet trajectory to two
dimensions--in other words, ambient flow, if present, will be parallel
to the horizontal component of jet velocity. Since jet encounter with
an abrupt ambient discontinuity, such as a water-air interface, will not
be addressed, the ambient will be considered to be infinite.
Among the variables which will be considered are:
(1) buoyancy effects, arising from density differentials between the
jet and the surrounding ambient. These density differentials
may arise from temperature and/or concentration variations.
(2) ambient density stratification, arising from vertical non-
uniformity of temperature and/or concentration in the ambient.
(3) ambient flow conditions, in which the magnitude and orientation
of the ambient flow velocity relative to the jet are varied.
(4) initial jet discharge characteristics, such as momentum
orientation.
10
II. BASIC CHARACTERISTICS OF ROUND DISCHARGE SYSTEMS
The terms "jet", "momentum jet", "forced plume" and "plume" are often
used to qualitatively describe certain characteristics of a discharge
system as it progresses through an ambient medium. It is generally under-
stood that "jet", "momentum jet", and "forced plume" refer to that region
where the momentum of the initial discharge is still sufficient to influ-
ence jet behavior. "Plume" refers to a discharge in which the discharge
momentum is either negligible to begin with, or small relative to the
eventual total momentum produced by buoyancy. It is with this under-
standing that these terms will be used.
The jet/ambient system may be classified according to a number of
characteristics:
(1
)
Jet buoyancy
(a) buoyant (positively or negatively)
(b) neutrally buoyant
(2) Initial jet orientation
(a) horizontal (perpendicular to gravity field)
(b) inclined
(3) Ambient stratification
(a) unstratified
(b) linearly stratified
(c) non-linearly stratified
(4) Ambient motion
(a) quiescent
(b) flowing
11
Regardless of the classification of the jet or ambient, a jet passes
through several flow regimes as it progresses along its trajectory.
They are shown in Figure (2-1), and are:
(1) The zone of flow establishment. In this region, flow character-
istics are dominated by discharge conditions. Velocity and scalar
quantity profiles (temperature, salinity, etc.) undergo transition
from their initial discharge shapes through the action of a turbu-
lent shear layer formed on the jet periphery. As mixing with the
ambient progresses, the turbulent shear layer grows inward and
the extent of the core of undisturbed profiles becomes smaller. The
zone of flow establishment ends at the point where turbulent mixing
reaches the jet centerline. The jet behavior in this region is
strongly influenced by initial momentum and discharge conditions,
and is only slightly influenced by the ambient.
(2) The zone of established flow. This region begins when turbulent
mixing reaches the jet centerline. The motion of the jet and its
physical characteristics are governed by its momentum (initial and
acquired), buoyancy, as well as ambient stratification and flow
conditions. Initial discharge conditions play a progressively
smaller role, and and the transition progresses from jet-like to
plume-like behavior.
(3) The far field. In this region the jet's initial momentum has
negligible effect, and the jet may be convected by ambient flow.
The jet fluid may be further diffused by ambient turbulence, and
the distinction of the jet as a separate entity gradually disappears.
The first two flow regimes constitute the near field, and will be the
concern of this investigation.
12
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Experimental work, begun by Albertson et al . [3] and continued and
expanded by many others, has shown that within the zone of established
flow, mean velocity profiles are nearly Gaussian:
U = Um
exp |-r2/B
2l (2-1)
where U is the mean centerline velocity, r is the radial jet coor-m
dinate, and B is a characteristic measure of jet width, or that radial
distance at which U is equal to (1/e) of its mean centerline value.
Profiles of jet scalar quantities, such as temperature and concen-
tration, have also been found to be Gaussian in the zone of established
flow by investigators such as Fan [7], Hoult et al . [16], and others.
The profiles may be expressed as:
At = Atmexp|-rVx B^| (2-2)[-r
2/x
2B2
]
ac = AcmexpI-rVA B
I(2-3)
where At = (t-t ), At = (t -t ), ac = (c-c ), ac = (c -c ), and x isa m ma a m ma
the relative spreading ratio between velocity and density constituent
2scalar properties, x is defined as the inverse of the turbulent Schmidt
Number. Figure (2-2) illustrates these profiles within the jet.
A coordinate system to describe the trajectory and physical dimen-
sions of a jet system is shown in Figure (2-3). The X coordinate is
perpendicular to the gravity field and parallel to the flow (if any) of
the ambient fluid. The Z coordinate is vertical and opposite to the
gravity vector. The streamwise coordinate S defines the direction of
mean centerline jet velocity at any point along its trajectory. The
14
_ ZONE OF *
FLOW
ESTABLISHMENT
« ZONE OF
ESTABLISHED
FLOW
U = U m exp j -r 2 /B 2[-
Figure 2-2. Development of Gaussian velocity profilesin a momentum jet after discharge
15
*> X
Figure 2-3. Coordinate system for physical dimensions
and trajectory of a jet system
16
local angle between S and X, or the inclination of the jet from the
horizontal, is e. <p and r are polar coordinates defining the jet cross
section, normal to S. Herein, any ambient medium motion is assumed to
be horizontal
.
An important quantitative measure of relative momentum and buoyancy
is the densimetric Froude number, F, given by
F = 9 (2-4)
(go^) 1 ' 2
p
The contribution of momentum is reflected in the numerator by the dis-
charge velocity, IL. The buoyancy effect is included in the denominator
by the density differential term. Thus, the value of the densimetric
Froude number ranges from near zero for plumes to infinity for pure, non-
buoyant momentum jets. Hereafter, the term "Froude number" will be
used to mean the densimetric form of Eqn. (2-4).
17
III. LITERATURE REVIEW
A. REVIEW OF PAST MODELLING
Several kinds of predictive models have been developed for the
circular buoyant momentum jet. Although specific calculations consider
different circumstances in origin of buoyancy, stratified/uniform ambi-
ents, quiescent/coflowing ambients, etc., they all may be classified by
basic method:
(1) Algebraic models, based on either empirical data or simplification
of differential model. These most typically predict only trajectory
and jet width. Some, such as the model of Shirazi, McQuivey and
Keefer [26], also predict velocity, concentration, and temperature
residuals. Data-based algebraic models tend to become unreliable
when the basic conditions upon which they were based, such as
general temperature and salinity range of the jet and ambient, are
significantly changed.
(2) Differential models, based on the relevant conservation equations
(mass, momentum, energy, and scalar species). This modelling tech-
nique allows prediction of jet trajectory and width, as well as
velocity, temperature, and concentration decay downstream in the
jet. Stratification and motion of the ambient may also be
accommodated.
Because of their limited scope, algebraic models will not be treated
here. Certainly such models have a place in predictive use when the
jet/ambient system involved is simple, and only information such as
trajectory is required. However, the vast majority of effort in recent
18
years has involved the differential approach to jet modelling. In a
majority of these differential models the entrainment mixing concept is
invoked, rather than approaches utilizing mixing length hypotheses, k-e
models, or eddy diffusivity.
Morton et al . [20] were the first to use the entrainment concept to
develop a buoyant jet model, as previously suggested by Taylor [27] . The
concept supposes that the downstream induction of ambient fluid into the
jet is proportional to the local jet centerline velocity, U , and a
characteristic jet width, B. Thus,
E <X 2ttII Bm
where E represents volumetric rate of entrainment, or ambient inflow,
into the jet, and is defined by
dQ
dS"Jr" - t
where Q is the total mass flow in the jet at any downstream location, s
Defining the constant of proportionality, the entrainment constant or
coefficient as a, the rate of entrainment can be written as:
2TCUmB
Solutions of the governing equations for differential modelling have
been based on the following assumptions for round jets.
(1) The jet flow is steady.
19
(2) The jet flow is fully turbulent. Molecular diffusion can be
neglected in comparison with turbulent transport.
(3) Streamwise turbulent transport is a negligible downstream trans-
port mode, compared with streamwise convective transport.
(4) Variation of fluid density throughout the flow field is small
compared to a chosen reference density. Density variations are
included only in buoyancy terms, the Boussinesq approximation.
(5) Other fluid properties, such as viscosity, are constant over the
range of interest.
(6) Pressure is hydrostatic throughtout the flow field.
(7) The jet remains axisymmetric throughout the near field. Velocity,
temperature, density, and salinity profiles have no circumferen-
tial dependence.
The governing equations in the forms used in differential modelling
are presented in Table (T3-1).
With the exception of Hoult et al . [16], all studies cited in the
following discussion have assumed velocity, temperature, salinity, and
density profiles are Gaussian. This assumption, therefore, limits the
applicability of such models to the zone of established flow.
Hoult et al . circumvented the problem of having the model applicable
only to the zone of established flow by assuming uniform ("top hat")
profiles rather than a Gaussian distribution. This assumption was applied
in the entire near field. As a result, the reduced form of the conserva-
tion equations for this model differ from the reduced form used by others.
Since Hoult 's modelling technique is valid for both the zone of flow
establishment as well as the zone of established flow, initial conditions
20
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also differ. The end result of this approach is that values of p, t, c,
and U ascribed to the jet at various points along the path are mean values
for the entire jet cross section. This is a more limiting case than
models using Gaussian profiles, where maximum values of jet properties
result, and the entire cross section profile may be deduced from the
appropriate Gaussian distribution.
Abraham [2] initially used the vertical and horizontal momentum equa-
tions, as well as the energy equation, to model jets discharged to a
quiescent ambient. The continuity equation was not included. The
solution required a pre-specification of the variation of B as a func-
tion of S. Most other models have included the continuity equation in lieu
of pre-specifying the B variation.
The solution to the seven equations in Table (T3-1) yields values of
jet centerline velocity, U , and temperature and concentration differ-
ences At and ac , as well as jet width D(s) and trajectory as functions
of S. The solution of the equations, of course, also requires that the
entrainment function E be specified. Herein lie the principal differ-
ences between entrainment models. The models fall into two general cate-
gories: those for a quiescent ambient, and those for a flowing ambient.
1 . Quiescent Ambient Media
Albertson et al . [3] and others have verified through measurements
that for non-buoyant momentum jets, F = » , the appropriate value of
within the zone of established flow is 0.057. There seems to be little
disagreement with this value, based on numerous comparisons of differen-
tial modelling and experimental data.
Abraham [2] suggested, also on the basis of experimental evidence,
that for relatively buoyant flows (small F) , a = 0.085. This is in good
agreement with the suggestion of List and Imberger [18] of a = 0.082
for pure buoyant plumes (F = 0). Fan [9] also suggested a = 0.082 for
all flows except pure momentum jets. Fan also recommended, on the basis
of his experiments, a = 0.057 for the pure momentum jet.
In application, however, discharges are seldom either pure jets
or plumes. Typically they are in some stage of transition away from
jet behavior toward plume behavior. Morton et al . [20] proposed to model
this transition by:
a2
a = 0.057 + p^-hL
where a~ is an empirically determined coefficient, and F. is a local
Froude number, based on the local density difference. The same general
form was derived by Fox [10] for a vertically discharged buoyant jet.
Hirst [14] maintained that for a discharge into a quiescent
ambient, the entrainment function should depend on:
(1) local mean flow conditions in the jet, i.e., U and B.
(2) local buoyancy within the jet, as indicated by local Froude
number, and
(3) jet orientation, 8Q
.
The following form was proposed:
a = 0.057 + %^ sin(e)
This is the general form suggested by Morton and Fox, with the constants
defined by fitting the function to known discharge and endpoint conditions
of jet flow.
23
Another entrainment function for initially horizontal buoyant
momentum jets is the jet-plume extrema fit proposed by Riester et al
.
[231:
a = T(0. 057 cos e)2
+ (0.082 sin e)^]1/2
From data on a buoyant jet discharged vertically downward into
a quiescent ambient, Davis et al . [6] proposed:
a -. 0.057 + MM
A tabular summary of entrainment functions for discharges into quiescent
ambient media is given in Table (T3-2).
2. Flowing Ambient Media
Hirst [15] proposed an entrainment function applicable to 3-
dimensional buoyant jet flow, in which the horizontal component of initial
jet velocity was not necessarily parallel to ambient flow. Eliminating
such terms, the form for two dimensional buoyant jets becomes:
a2
E = (a, + r- sin 9)b[|U -U cos 8 1+ a,U sin 9]
I r I ma o a
where the term
U -U cos am a
represents the relative velocity of the jet with respect to the ambient,
in the direction of jet flow. It is a pure "coflow" term. The term
24
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acuens-<T3
SZuco
+J r—c (Ocu a•^ •r—
jd •M= S_rt3 cu
• A >4-> 4-)
« c C Acu cu 0J -MP= -r- u CZ3 J3 I/) • »« CD— ^ cu -U> -r-
Q. (T3 •r— c a3 CD -I-
(-> +J cr •r- <+-
c c XI M-fO cu 03 E cu
>> u 03 OO co o (_>
3 CU +J +JJ3 •!- c -a
3 "O cu cu
>> er s_ CJ c03 CO -r-
CL fO "S CD EE cr •r- S_•r- O ^ 3 CDCO +-> O CT-M
T3 CUO CO 03 "O-M CU >>
i— i
—
O >scu en r— -t-J 1
—
r— C 03 r—_Q ra U >5 0303 •r— r— Uu cn-t-> r— t—•r- C s_ 03 S-r^ "r— OJ +-> T-a. >> > C CLQl s- O E<C fa -o M LU
> OJ •r- *CD i_-—
s
'—»4J S_ O r^CM <T3
SZx: —
-
T3 U "OCU co CD • •>
• > CD •r— CD CD+-> i- TD ^ ECU 03 03 3<-n^z +-> JZZ r—
o CD CJ cxE co -r-> CO3 -i- •i- -u>
+-> -a 4-> -a cc C 03cu -u> ra 4-> >iE CU >> (U oo •<-> o •'-> 3E 3 J2
+J ^ +->
cu c C CDr— 03 to 03 i
—
Q. >, >, a-E O s_ O E•r- 3 a 3 r-CO _Q C|- _Q CO
O O +J o o-U +J •r— -M 4->
CU CD CD CD
i3J2 <Oi2Xl03 03 U 03 0JO O •!- U <J•r— 'r— S— *r— t—
cx a. a. a. a.aas aa<=t <: lu <c =c
i— CO <^- CO CO
25
a, U sin aJ a
represents the contribution to the total value of entrainment of pure
cross flow, or ambient motion normal to the jet axis, as the jet turns
due to buoyancy.
Hirst specified values of a-, and a2
so that the flowing ambient
entrainment function reduced to the quiescent ambient function if U = 0.a
He specified a value of 9.0 for a.,, based on a best fit to available data
This entrainment function thus became:
E = (0.057 + V^sine)b[|U -U cose| + 9. OIL sin e]r
,
ma a
Ginsberg and Ades [12] performed a least squares analysis on a
large set of laboratory trajectory data. Using Hirst's entrainment model
with a-| and a2
as specified, they found that a large variation in the
value of a3
was necessary to fit predicted results with the data. They
constructed a correlation for a3
as a function of F and the coflow ratio:
a3
- 25.810[F- 19464
R' 35155
] - 10.825
where the coflow ratio, R, is defined as:
U
u
Other entrainment functions for coflowing ambients have appeared
using variations of the governing differential equations. Schatzmann [25]
proposed an entrainment function similar in form to that of Hirst, but
26
for use in a set of governing equations in which the Boussinesq approxi-
mation was not invoked. Fan [9] proposed an entrainment function for
coflowing ambients which included a drag term as well as the standard
proportionality of entrainment with centerline velocity and jet width.
He found that the value of the drag coefficient used, as well as the
entrainment constant (a = 0.082 in this case) had to be readjusted to
make the prediction conform with data with each change in discharge or
ambient conditions. The entrainment functions of Hirst, Ginsberg and
Ades, and Schatzmann are collected in Table (T3-3).
The progression of differential modelling, once the entrainment
function is specified, is similar in most cases. First, the governing
equations are integrated over the jet cross section so that they appear
in differential form. They are then non-dimensional ized with respect to
chosen reference variables. Initial conditions are specified. The re-
sulting equations are then numerically integrated over the desired range
of the streamwise pathlength, S. However, different downstream trajec-
tory and decay may be calculated for a given jet due to differences in
any of the following:
(1) Entrainment function chosen;
(2) Initial conditions specified for the beginning of the zone of
established flow;
(3) Equation of state specified for the density of the fluid;
(4) Computational technique.
Examples of predicted jet trajectories and phsyical properties
are presented in Figures (6-1) to (6-53). Comparison between models for
identical or closely similar discharge and ambient conditions are pre-
sented in Figures (6-11) to (6-15). Discussion of these predictions and
comparisons will be made later.
27
CO+->
cCU
c
3CUCO
o
coco•r™+->
uc2
CCU
03i-
c
on
<u
_203
CD
CO
ou
I
03
II
*
MCO)
Ecto
• • s.CO -t->
-t-> CC LU0)•r- U-Q •«-
E S-< +->
OJCD EC 3
2 OO >•+- cuO 4J<_> cu
-C Q.4-> E•<- o3 c_>
so«»-
COtoos_o
-M +->
c: ccu cu•r~ •^J3 -Ogs E03 03
CD enc c
•r~ •!-•
s so on— r™4- 4-
03 03
o O-t-> 4->
to COCU CUf— r—CD CDc cfO 03
COen CD4->c c c
• r- •r- cu
>> >>T-_ s- u03 03 i-> > 4-
4-4-> +-> CU03 03 O
U-o -oCU cu -aCD en cu1- S- £=03 03 -i-
JZ J= Eu o s-CO CO CUr— •r- -t->
"O -o cuT3
+-> 4->
O) CU >,-rn •"—5i
—
+-> -i-> 03c c u03 03 -i-
>> >> s-
O O -r-
3 3 GL-2 -Q E
• CUO -o o+-> OI 4-> «
+-> r—
i
CU 4-> CU CDi—
•
•r— r^ ^-—
'
ja E J303 O 03 COu U 03•^ co •p—
r— E r- CUQ. E Q- EQ. cu a. 03< +j <C OO
GO CT> O
28
B. REVIEW OF EXPERIMENTAL STUDIES
By far the most comprehensive and often cited set of data for buoyant
water jets resulted from the work of Fan [7], The experiments concerned
two classed of buoyant jets:
(1) inclined jets discharged into a stagnant environment with linear
density stratification.
(2) buoyant jets discharged into a uniform cross stream, en
= 90°,
with no ambient stratification.
The experiments for the flows of group (1) above were conducted in
a 2.26m * 1.07m tank with a depth of 0.61m. The tank was stratified
with successive 3 cm to 5 cm layers of aqueous salt solutions. Tank
temperature remained constant within a 2°C range over the duration of the
experiments.
Nozzle diameters varied between 0.223 and 0.762 cm. Flow into the
nozzle was provided from an unregulated head tank, which provided a
discharge rate estimated by Fan to be constant within 3%. Measurements
in each experimental run were limited to jet trajectory and half width,
observed photographically by use of a tracer dye premixed into the dis-
charged fluid.
Fan described one of the dilemmas of conducting experiments on jets
of small physical scale:
For complete experimental check on theory, it is necessary todetermine values of the jet velocity and density. Practicallyhowever, laboratory experiments on density stratified flows areusually limited in scale and do not allow the time requiredin measuring time fluctuating quantities. On the other hand,the jet trajectories and half widths can be determined con-veniently by photographic means. These two quantities areinterrelated with other jet characteristics. Thus the com-parison of the observed and calculated values of these twoquantities is believed to be indicative of the applicabilityof the theoretical solutions. [8]
The experimental discharge Froude number ranged from 10 to 60, with the
exception of 3 runs with a non-buoyant momentum jet (F = »).
In Fan's second group of experiments, with a flowing, unstratified
pure water ambient, the saline jets were actually negatively buoyant.
Conductivity measurements were taken by variable position probes at a
number of downstream stations. The locus of stations of maximum concen-
tration was defined as the jet centerline. The jet width was defined
from concentration readings taken radially outward from the centerline.
These experiments were conducted in a 40m flume, 1.1m wide, with a
water depth of .51m. Flow was induced by inclining the flume. A region
in the core of the flume flow with the least shear effects from wall
boundaries was selected to introduce the jet via a nozzle. Variation of
ambient flow in this region was estimated at +6% to -9%.
Experimental runs were made for a Froude number range of 10 to 80,
ambient flow of R = 0.0625 to 0.25, and a discharge diameter of 0.5 cm
to 0.762 cm.
In an attempt to ascertain the effect of ambient turbulence and shear
introduced by the restricted cross-sectional dimensions of the flume,
a limited set of concentration measurements were made in which the fluid
in the flume was stagnant and the jet discharge towed through the ambient
by a carriage mounted over the flume. In this case, conductivity probes
were fixed to the carriage and moved to different relative positions in
successive runs.
The experiments of Fan have been used by numerous modellers, including
Fan himself, as a basis for analytical/experimental comparison. The
experiments also served to validate earlier hypotheses and observations
regarding the Gaussian distribution of concentration in the zone of
established flow.
30
Riester, Bajura, and Schwartz [23] studied horizontally discharged
buoyant fresh and salt water jets. The ambient was quiescent and un-
stratified. A 6.2 x 1.1 x 0.8 m tank was used in conjunction with a 0.87
cm diameter discharge nozzle. Jet trajectory and width were recorded
photographically by a tracer dye in the jet fluid. Temperature distri-
butions were measured by a rake of thermocouples. Jet centerline was
determined through the measured temperature distributions.
A novel aspect of these experiments was the wide range of ambient
temperatures utilied (4.5°C-43.0°C) , and the use of both salt and fresh
water jets. Some possible implications of the results will be discussed
in a later section.
Davis, Shirazi, and Slegel [6] measured the behavior of single and
multiple port salt water discharges directed vertically downward into a
fresh water ambient. A flowing ambient was simulated by mounting the
discharge nozzle on a moving carriage mounted over a 17.1m tank. Water
depth in the tank was 0.91m. Concentration profiles were measured by
conductivity probes, and velocity measurements were made by hot film
anemometry. Several runs of these experiments made with the carriage
stationary led to the proposal of the entrainment function:
_ n nc-7 0.083
a = 0.057 +
r 3
These same experiments verified the Gaussian nature of velocity profiles
in the zone of established flow.
Shirazi, McQuivey, and Keefer [26] studied buoyant jets in flowing
turbulent ambients. An inclined 120 ft. flume was used to produce
31
ambient flow. Turbulence was introduced by imposing a layer of varying
sized rock in the flume bed. Temperature and salinity concentration
were measured using conductivity probes. Turbulence was monitored by
hot film anemometers.
As a result of these experiments, a set of algebraic correlations
were made, expressing temperature and concentration residuals, jet width,
and trajectory as functions of downstream distance, F, and R. It was
determined that as the level of turbulence increases, the rate of decay
of centerline temperature and concentration with respect to the stream-
wise coordinate increases. Not surprisingly, it was also found that
correlation of the data became more difficult as turbulence level
increased.
Pryputniewicz and Bowley [22] conducted measurements of a buoyant
jet discharged vertically upward into a uniform quiescent ambient.
Temperature profiles were measured by a rake of thermistors in the flow
field. Of interest in these experiments was the presentation of data
which showed temperature residuals as measured near the terminal point
of rise at the surface. Froude numbers of 1 to 50 and nozzle diameters
of 0.425 and 0.55 inches were used.
A summary of the range of parameters studied in various experiments
is presented in Table (T3-4).
32
ins-
<U
EtoJ_03Q_
c01
s-
OlQ.
<+-
o
s-
03
3CO
i
co
<u
.a03
Lf5 ID O r^CM CM I*-* • .—i •
cu 1 LQ .
• OO l
1 i—i i—
i
O l
«—t CO
O l
1
r_ E E E E +-> -M03 <+- 14-
• O 0) CD r—
t
r—i: 00 «—
i
IQ 4-> C O •- LO LO • cr> CM LOET i. rtj q: • • o •
a) c^ o o o>
I— CU E Q. X X Xto
CUB
T3 -r- c_> o c u X X XS- -r-
3 4->
13 03
>0Jr— O O >- -M >, X X
3" -r- "O +> .C X X X X X XOl 3s: c .
\— S_ 03 T-3 X X X X X X
+->
c U. r- O S 'r C O) X X XOl -
•1— to +J S- ns -P X XE s> sr «/) +j s- «o +j X X X X
lo o , ,
1 1 r—1 CMu_ o o o o . LO LO CO 1—
1
LO<—
«
lo <-H CO CO 1 • CO1—
1
• CMLO
1
1—
1
E E
>— CU
o a E E E c1 u u 1 1 •—
03 +-> C\J CM 1
•f- 4-> CU co lo i lo r^ r—
1
CO CM LO LO•P 0) E cnj r^ r-» CO CM CO LO CO CM LO
i—• QCM • LO • • r-t •=3- CO «3" •
• o • O o • CO • • Oo O I—
1
O ^H O
S-
CU4-> 1— 1 MC 03 03 (JO) r— •1—
E +-> 03 -M 2 >>"f <U CU cu cuS_ -t-> •r— r—CU S_ CU • 1— C 2Q. a> Nl 4-> OX +-) c/> 03 3 COLU m •!-" <~ Q.
E c CU > •<— >>-aft3 03 •r— 03 x: s- cU_ u_ DC Q 00 Q_ 03
33
IV. EQUATIONS OF STATE AND AMBIENT STRATIFICATION MODELLING
The determination of the discharge Froude number and the buoyancy
force requires the evaluation of the temperature, t, and concentration,
c, effects on density. Often the density, p(t,c,p) is a sufficiently
linear function of both t and c over the range of temperature and concen-
tration difference between the jet and the ambient. Then density differ-
ences may be accurately estimated in terms of the two volumetric
coefficients of expansion,
w r c,p Mr t,p
where p is some reference value of p, say p Q= p(t ,c ,pQ
). Then
Ap = -Bp At and Ap = -yp Ac
for the separate t and c effects on density. The density at some t and
c, in terms of the initial jet density Pq , is written as
P (t,c,p )- p(t
Q,c ,p ) = -D(t ,c ,p )[B(t-t )- Y (c-c )]
or
pq
[1 - s(t-t ) - y(c-c )]
In particular, the initial density difference between the local
ambient and the initial jet, which appears in the Froude number, is
24
p af)"p nJ2_i = B (t -t
a) Y( c -c
a)
In an unstratified ambient medium, the reduced, non-dimensional forms
of equations (3-3,4 and 5) will contain the following three non-dimensional
terms:
p -p t -t c -c/ a m x / m a >, / ma x
pa0
p La0
c ca0
With the values of \ in Equations (2-2 and 3) the same, profiles of
t and c are the same and
/_m___a__\ / m a \
^t -t ' ^c -c 'L So c ca0
at all points along a trajectory at which (tQ-t Q ) f and (Cq-c q) t 0,
Thus, for constant values of 3 and y in an unstratified ambient,
p -p t -t c -c
(-^-) = (^L-L.) = (J0-J-)pa0
p z La0
c ca0
and defined values of 3 and y are not required for integration, computa-
tion, and solution, beyond initial definition of discharge Froude number.
A similar situation exists for some special cases of linearly strati-
fied ambients. If p = p(t), and density stratification is defined solely
by a temperature stratification parameter
3t
Then the definition of 6 beyond the initial computation of discharge
Froude number is not required. For such flows,
p -p t -t£ajL, =
(
ma}'
pa0~
p V^O
pertains throughout. In addition, if p f p(c), an assumption that might
be made in a jet with a tracer dye, the concentration and vertical momen-
tum equations are not coupled. Concentration computations can proceed
independently, even to the point of specifying a concentration stratifi-
cation if desired.
Parallel reasoning holds true for the case of p = p(c), in which case
a density dependence on concentration stratification parameter would
couple the concentration and vertical momentum equation. Presumably
temperature would be uniform throughout the system in such a case.
These simple, degenerate cases of the overall modelling problem are
important because they represent conditions under which measurements
are often taken. Modellers then specify these conditions to compare
data with their analytical models.
The underlying assumptions and limitations of such formulation are:
(1) With no equation of state incorporated in the computational
process, only unstratified ambients may be acommodated if
p - p(t,c), and then only if s and y are assumed constant.
(2) If no equation of state is included in the model, density
stratification in the ambient may be specified as either
3t
1 aO~V
36
or
3Ca
= p(c)
or
8pa
/. n i. (directly)
Ambients with more than one density-constituent gradient cannot be
accommodated, since temperature, concentration, and density residuals
would be independent of each other and profiles could be dissimilar.
The more general case of an ambient medium with temperature and
concentration stratification requires an equation of state for solution
of the governing equations. Such a temperature/concentration/density
relation may be used in one of two ways:
(1) Internally in the calculational scheme, with ambient temperature
and concentration gradients specified. The local density differ-
ences are computed downstream from jet temperature and concentration
decays calculated from the energy and concentration equations.
(2) External to the actual integration calculations, by using speci-
fied ambient stratification to calculate the ambient density
variation a priori. The resulting ambient density gradient is then
used in an equation of density excess or deficiency of the form:
. 2tt °° do 2n x
W^I I U(p -Pm )rdrd4}= --jA/ / Urdrd*}
ai '0 a a5 '0
37
assuming the same Gaussian profile for density as for two of
its constituent properties, temperature and concentration.
This extra equation effectively uncouples the temperature and
concentration equations from the vertical momentum equation.
Method (1) above was used by Hirst [151 . The second method above was
used by Fan [9] . Other models specify the method of use of an equation
of state, only as noted below.
Hirst specified values of 3 and y» assumed constant throughout the
downstream integration of the governing equations. The equation of
state was:
-f_ = 3(t-t ) + y(c-c n )V MV- "aO
In reality, of course, density of water is a complex and, under some
conditions, highly non-linear, function of three variables—temperature,
species concentration, and pressure. Many modelling situations might
arise in which the range of these variables clearly argue against the use
of constant values of y and 3, as well as the omission of the pressure
dependence of density. Table (T4-1) shows some of the variation of 3
encountered over small temperature ranges, for both saline water at 35 °/O0
(typical of sea water) and fresh water. The pressure in both cases is
one atmosphere.
In their model, Riester, Bajura, and Schwartz [231 attempted to
account for the effect of variable 3 by representing it in the form
l
a= a-, + a
2t + a
3t (4-1)
38
<uO)c<ns_
0)s-3+Jtos-0)Q.ECD+->
-oCU+J s-
•i— O)
E -t->
•^ (O"— 2S- <u0) c> 'r—
o r—<T3
CQ (/)
<+- "Oo c
<TJ
co 0)•^ s--l-> 3fO Q.•1—
s_ S_
<T3 O> 4-
.
r—
•=3"
1—
0)r—-QIT3
2:I—
00go
as:
0£o
Gv-l-U
— 0^
CTt LOC\J O
CO • •
*d- <sO
1—
t
cr> 101—
1
U3r^» • •
00 >^t-
.—
1
CTl C\J
O C\J
<o • •
C\J roI—
1
r-^ COCT> 10
un • •
O 1—
1
i-H
en ID00 O
vT • •
**"* CTt OUJcs: r^ r-»=> 10 l*»
\— ro • •
<c CO f-H
onUJ
1
Q_ CT> 1—
1
2: •d" mUJ CM • •
1— r~^ m1
r^.
1—
1
C\J
«-H
O O
Or—
1
3s*
-»
>- IDh- m <
1—
1
z1—
<
_l<x.
OO
39
where the coefficients a-,, a2
, and a3were evaluated for the temperature
range pertinent to a given calculation. The discharge Froude number was
then "adjusted" using the relation
F" (9D8
a(t -t
a ))l/Z
They summarized the need for an effective equation of state as follows:
...if the initial temperature difference between the jet and
the ambient is greater than 3°C, the effect of the temperature-density relationship of water on the trajectory must be con-
sidered. This result is of particular significance whenlaboratory models are operated at high temperature differencesto simulate prototype flow characterized by low temperaturedifferences the effects of temperature on the trajectorycan be minimized by using the (adjusted) Froude number ...
to define flow conditions. .. .Additional work is required to
determine the correlation application to jet flows whichexperience reversing buoyancy for cases near 4°C.
Salt water jets cannot simulate the characteristics offresh water jets operating over large temperature differencessince the salt models do not duplicate the temperature-density characteristics of the fresh water system. [24]
In view of this last observation, it is noted that, although the proposed
model attempted to compensate for the variable temperature effects on
density, no dependence on salinity was included except for that which
may have been implicit in the choice of the three coefficients in
Equation (4-1).
The ideal equation of state for buoyant jet modelling would compre-
hensively represent density as a function of temperature, species concen-
tration, and pressure. Such an equation should be applicable in all
temperature, concentration, and pressure ranges of interest. Also, for
most modelling of fresh/salt water systems, the concentration variable
should be salinity. An equation of state which seems to fulfill these
requirements will be suggested in the following section.
40
V. PRESENT METHOD
The foregoing modelling system assumptions regarding the flow field
also apply to all the calculations done in this work. Generally, they
assume a steady, axisymmetric jet with negligible molecular transport,
negligible streamwise turbulent transport, and small curvature effects,
operating in a hydrostatic pressure field under the Boussinesq approximation
The governing equations in the zone of established flow follow.
Continuity of mass equates the downstream change in total mass of
the jet to the mass of fluid entrained. The entrainment concept results
in the following:
Lij j pUrdrd*} = p(2^aUB)uo
The first Boussinesq approximation, concerning density level, yields:
35-i/ / Urdrd<j)} = 2™UmB (5-1)
Horizontal momentum is conserved in a hydrostatic pressure field.
Thus, the change in horizontal momentum within the jet is equal to the
horizontal momentum of the fluid entrained:
2 ^
L{j j P U2coserdrd^} = P UEaa a v
Again, invoking the Boussinesq approximation:
41
tr{j j U2
coserdrdc})} = UE (5-2)dO a v
The change in the vertical momentum of the jet system is the result
of the action of the buoyancy force over the extent of the jet:
2fT o° p 2tT °°
%A( I pUsinerdr} =/ / ( p .- )grdrd<j> (5-3)
0:3 a
In a stratified ambient, the ambient temperature, as well as the
temperature of the jet, may be variable along the trajectory. The energy
equation relates the change in energy of the system, as expressed by a
temperature excess or deficiency relative to the ambient, to the rate
at which the ambient temperature is changing.
O A 4. p
Lij C nP U(t-t Jrdrdd,} = - gJ./ / p c Urdrd*
a* P p
Assuming constant specific heat and the Boussinesq approximation,
2tt °° dt 2tt °°
2~{f ( U(t-t)rdrd?} = - ^A ( ( Urdrdcfc (5-4)dS i
QJ
Qa dS J
QJ
Q
Similarly, the change in concentration excess relative to the
ambient is related to the rate at which the ambient reference concentra-
tion is changing:
j 2tt °° dc 2tt <=°
uf{| ( oU(c-c,)rdrd<j>} = - J- / / pllrdrdcj)db
'o 'oa db
Again, invoking the Boussinesq approximation:
42
L{j ! U(c-c.)rdrd0}- = - -gJ- / / Urdrd<j> (5-5)ai
Geometrically, the incremental trajectory of the jet may be
described by:
dX = dS cos 9, (5-6)
dZ = dS sin 8 (5-7)
It will be assumed that the Gaussian velocity, temperature, and
concentration profiles cited previously are valid for the zone of flow
establ ishment.
Appendix A contains the development of the governing equations from
their stated form above to their reduced form. In Appendix B the equa-
tions are non-dimensional ized. The dimensionless equations appear in
Table (T5-1). The forms of the equations for degenerate cases are
presented as follows:
(1) Flowing unstratified ambient—Table (T5-2);
(2) Quiescent, stratified ambient--Table (T5-3);
(3) Quiescent, unstratified ambient—Table (T5-4).
The above equations apply only in the zone of established flow,
which begins at S , see Figures F-2-1 and F-2-2. Estimates of the end
of the region of flow establishment have been made. The procedures
followed here in calculations are based on the experimental work of
Abraham [2] and the analytical development by Hirst [13]. They are also
consistent with assumptions made in references [14], [15], and [12].
The initial conditions are:
43
CDC•r—
cS~
01>o +>C3 e
a>o» •r—
-C -O.4-> E
«=C»4-
O "CO)
E •r-s- 4-o •r—Ll_ 4->
rQr— S-(T3 -M
•r— CO-l->
C A
O) ens- ca> •r*i+- 514- O•^ r~Q Lu.
r— rOfOe s-
o o•r-» 4-COc CO0) cE o•r- •^-o M
1 rac 3o cr
I
LO
<o*
CDC•r-
CO1—1 OSo COc fO•r-(/) +acCO CM
fO CD -Q CMco CM Lu (*•» !-»*
+ O ^C CM CM(J CM| -Q -Q^_DC • E _ Eo 1
^—
»
3 3CO e O --j wo 3 E «Li (T3
o — E a +j <" u co
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1
a 1 < ro <C -a1 -O. 1
E 3 fO <a 1 1
3 as a. c** **—-*
ImJ 11 II
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CM «-—*» *^7*%
^—
^
^—** CO e.*—
N
,—-» CM r— CM 1— O •r—
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•1— << r< << r<*"—
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47
Ume ' U
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(c -c) n r
2
m a u c\c
The value of S , relative to the point of discharge, is
Se/D = 6.2 F > 40
Se/D = 3.9 + 0.057F
25 < F < 40
Se/D = 2.075 + 0.425F
21 < F < 5
S /D = 0. < F < 1e —
All that remains to be specified to integrate the equations down-
stream along the S coordinate are the entrainment function to be used,
the ambient stratification condition, and the density equation of state.
The calculations used an IBM 360 and IBM 3033 computing system.
The method of integration is a trapezoidal rule expression of the
governing equations with adjustable step size. Two versions of the
48
general method were used here. The first is used to evaluate flows in
unstratified ambients, or in ambients characterized by linear gradients
in temperature, concentration, or density itself. No explicit equation
of state is necessary, only 3 and y. The second version, with the same
computational technique, uses an accurate density equation.
Since the objective is to develop a predictive model applicable to
large scale discharges in ocean water, the density equation used in
version two is that of Gebhart and Mollendorf [11]. This relation
correlates the temperature dependence of density as an expansion around
the density extremum temperature at any given level of pressure and
salinity. The resulting expression is fitted to a comprehensive set
of experimental data over temperature, salinity and pressure maximums
of t = 20°C, s = 40%, and p = 1000 bars absolute. Density is in kg/m .
This equation is given in Appendix C.
For each of these calculations, the density equation was first used
to establish the density field in the ambient. This requires input of
at least one reference temperature and salinity level and appropriate
gradients, or specification of the complete temperature and salinity
fields. The water surface pressure is assumed to be one atmosphere. The
relation is first used at the surface, and density is calculated pro-
gressively downward, using hydrostatically integrated values of pressure
along the way.
Once the ambient density field is established, the relation is then
used to calculate the density terms in the vertical momentum equation at
each calculational interval, using the jet temperature and concentration
values predicted by their respective equations. Ambient density and
pressure are assumed to be linear between the points calculated for the
ambient.
49
The relation is also used to calculate the initial density differ-
ence between the jet and the ambient at the point of discharge, given
temperature and salinity level, or to calculate the initial temperature
of the jet, given Froude number and salinity.
The advantages of using an accurate density relation include:
(1) The full density dependence on temperature, salinity and pressure
is taken into account in all calculations.
(2) The actual ambient density field of either a hypothetical or
real circumstance is accurately established, utilizing appro-
priate temperature and salinity input.
(3) The relation is readily applicable to computerized analysis, and
is computationally compact.
50
VI. RESULTS
Comparison between the jet trajectories and properties calculated
here and earlier calculations, using the same entrainment function, has
been generally good. This is in spite of differing computing techniques
used in different studies. The resulting trajectories will be compared
with earlier calculations.
A. UNSTRATIFIED, QUIESCENT AMBIENT
This jet/ ambient system represents by far the most frequently modelled
case. Comparison here is for horizontally discharged buoyant jets.
Figures (6-1) and (6-2) compare the trajectories predicted by these
calculations, for F = 10 and 20, with the calculations of Abraham [31,
both using a = 0.085. Correspondence between the two is consistent, with
the present method predicting slightly less horizontal penetration of
the ambient than Abraham's model at all Froude numbers compared.
Figures (6-3) and (6-4) show the uniformly close correspondence between
these calculations and those of Hirst [14], [15], both using Hirst's en-
trainment function, entry 3 in Table T3-2, for F = 4,6,8 and 10.
Figures (6-5) through (6-8) again compare present calculations with
those of Hirst, but with both using the entrainment functions a = 0.057
and a = 0.082. Data of Fan is included for comparison where available.
The results, combined with those in Figures (6-3) and (6-4) indicate that
within the Froude number range compared, correspondence of present calcu-
lations with Hirst's model is yery good regardless of the choice of
entrainment function.
51
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Comparison of calculations with those of Fan [9], using a = 0.082,
is shown in Figure (6-9). In this case, the present method predicts
greater horizontal penetration of the jet and, therefore, somewhat less
vertical penetration at most values of F.
A comparison of the present calculations, using no equation of state,
with the calculations of Riester et al . [23] , both using
2 2 1/2a = [(0.057 cose) + (0.082 sine) ] , showed reasonably good agreement
for high temperature ambients, where the 3 correction factor of Riester
et al . resulted in little adjustment of the Froude number. Progressively
poorer comparison was evident with their prediction for lower temperature
ambients, where the Froude number adjustment became larger. This is shown
in Figure (6-10)
.
Figures (6-11) and (6-12) compare the calculated jet trajectories at
F = 1,2,4,6,8,10,50,100,150, and 200 for the five quiescent ambient
entrainment functions. The ambient is unstratified and no equation of
state is used. The models which relate to plume and non-buoyant jet
flows understandably begin to deviate significantly from the norm of the
trajectories when F is not in their intended range of use. This is
particularly true for the buoyant plume entrainment function, a = 0.082
(model 2). It consistently predicts lower entrainment and higher trajectory
at all but the lowest Froude numbers. Albertson's momentum jet function,
a = 0.057, (model 1) consistently predicts the highest entrainment and,
therefore flattest, trajectory at all values of F. The "end-point"
correlation of Riester et al . , model 5, is also seen to predict higher
entrainment than the norm for all values of F.
The generally accepted values of entrainment functions for the extrema
of Froude numDers are a = 0.082 for F = 0, and a = 0.057 for F = ». The
50
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64
two models which most closely reflect the Froude number/entrainment
relation over the range of F considered here are the function of Hirst
(model 3) and the empirically derived function of Davis et al . (model 4).
Hirst's function actually predicts the most buoyant, least entraining
behavior for the lowest Froude numbers (<_ 10).
All models predict strikingly similar behavior in the decay of
centerline velocity, temperature, and concentration. The decay of jet
properties along the pathlength S are not strongly dependent on F or
entrainment function. Figure (6-13) shows the superposition of center-
line velocity decay, U/IL, for all jets depicted in Figure (6-11) (F = 10,
50,100,150,200). The trend is seen to be an extremely rapid initial
velocity decrease immediately after discharge. For jets with F >_ 50,
U/IL = 0.1 around 50 diameters along the trajectory. Further downstream,
the residual velocities decrease less rapidly. Figure (6-13) also shows
that higher velocities persist downstream at smaller values of F, i.e.,
for more buoyant or less vigorous jets.
Close inspection of Figure (6-13) reveals an apparent anomaly. For
F = 10, the Hirst model predicts the lowest residual velocity along most
of the trajectory. The highest residual velocity is predicted for the
buoyant plume entrainment function a = 0.082. Yet Figure (6-11) for
F = 10 shows that these two entrainment functions predict the highest,
most buoyant trajectory, with the Hirst function representing the extreme.
The behavior of the Hirst function, on one hand displaying buoyant charac-
teristics indicative of low levels of entrainment, and on the other
hand displaying high velocity decay indicative of high entrainment,
seems to contradict the consistent behavior of the buoyant plume function.
65
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66
This behavior may be explained by close examination of Figure (6-14),
an enlargement of the F = 10 results on (6-11). In the first 20 diameters
of trajectory, the Hirst model is seen to be lower because of more rapid
entrainment in comparison to the buoyant plume model. It then curves
rapidly upward, displaying the most buoyant trajectory farther downstream.
The velocity level apparently decreased early due to rapid entrainment.
However, the jet still retained enough buoyancy to cause the sharp upturn
characteristic of the later trajectory. These results are indicative of
the importance of the early level of entrainment in determining properties
much further downstream.
An even more uniform behavior among the group of 25 calculations con-
sidered in the illustration of velocity decay is shown in a superposition
of temperature difference decay curves in Figure (6-15). Jets in this
F range of 10 to 200 have temperature residuals of less than 0.2 at 20
diameters downstream and less than 0.1 at 50 diameters. Both of these
decay curves are in close agreement with existing data. Note that in
this formulation, concentration residuals decay in the same way.
An example of the radial or diameter growth of a jet, using entrain-
ment model 2, is shown in Figures (6-16) and (6-17). These figures
illustrate that the physical extent of a jet generally increases more
rapidly with increasing F.
In summary, the various past entrainment models for jets in a quies-
cent ambient medium predict reasonably similar behavior. Most notable
exceptions to this conclusion are the results using the buoyant plume
entrainment function, model 2, at higher F. This is well outside the
range of intended use. Further, all of the models are in reasonably
close agreement with meager existing data for small diameter jets.
57
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Twenty years of "entrainment function generation" in the literature not
withstanding, it may be stated that in a quiescent ambient, predicted
jet behavior is not strongly dependent on the particular entrainment
model used.
B. FLOWING UNSTRATIFIED AMBIENT
The two entrainment functions proposed for a buoyant momentum jet in
a flowing ambient are models 8 and 9. It will be seen that they do not
result in agreement between predicted results characteristic of the
quiescent ambient models.
Figure (6-18) shows the trajectory predicted by the two entrainment
functions for a vertical jet in a horizontal cross flow. The conditions
are R = 0.25 and F = 2.83, in an unstratified ambient. This set of values
of R and F is especially significant because it is one of the few cases
in which the models are known to be in close agreement. For all F less
than 2.83, model 9 predicts a higher, more buoyant trajectory than model
8. Conversely, at F > 2.83, model 9 generally predicts lower trajectory
and somewhat more rapid decay of velocity, temperature, and concentration.
For F = 2.83, model 9 predicts lower trajectory for all R < 0.25, and vice
versa. Figure (6-19) shows the effect of lessening the ambient flow
rate to R = 0.125 for a jet of F = 2.83 with all other conditions the
same as Figure (6-18)
.
Figures (6-20), (6-21) and (6-22) illustrate that the disparity between
the trajectory predictions of the flowing ambient entrainment models is
not limited to the case of discharge nromal to a cross flow. In these
cases, the discharge angle is 45°. The same behavior is evident—reason-
ably good agreement for F = 2.83 and R = 0.25 as in Fig. 6-20. However
72
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model 8 predicts a higher trajectory for R > 0.25, Fig. (6-21), and
model 9 a higher trajectory for R < 0.25, Fig. (6-22).
As F and R are increased, the upward penetration of the jet decreases.
This is due to more rapid entrainment and lower initial buoyancy of the
high Froude number jets, as well as the stronger horizontal "sweeping"
effect of the more rapidly flowing ambients. In most cases, however,
the increase in these effects do not lessen the disparity between the
relative magnitudes of rise predicted by the two entrainment models.
Figure (6-23) depicts another case of relatively good agreement at F = 10,
R = 0.125, 9Q
= 90°, compared with the data of Fan at these values. How-
ever, Figure (6-24) shows that model 8 diverges from the other two indi-
cated trajectories when R is increased to 0.25 at the same value of F.
Figure (6-25) shows the same disagreement between models 8 and 9 at 6q = 45°
For higher Froude numbers, the disparity persists, as shown in Figures
(6-26) to (6-31).
The results for initial pure "coflow", in which the jet is discharged
horizontally parallel to the ambient flow, eQ
= 0, are shown in Figures
(6-32) to (6-37). The conditions are F = 20 and 40 for R = 0.125, 0.25,
0.4 and 0.8. The same characteristic disagreement is evident. At small
values of R, the disagreement between the models is 15 to 40%. At larger
R values, the disagreement approaches 100%. The two models are inconsistent
throughout. These high R values are beyond the data used in the fit by
Ginsberg and Ades in model 9.
In summary, the disparity between the predicted jet trajectory using
the two flowing ambient entrainment functions is generally most apparent
in flows with significant vertical components in their trajectory. Obvi-
ously, the models are very sensitive to how the cross flow entrainment
73
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93
effect is accounted for. These low Froude number, low R flows often
result in calculated differences of vertical rise of many diameters in
magnitude. This disagreement is found at all discharge angles. At
higher values of F and R, the absolute difference in predicted vertical
rise becomes small, but relative to the total rise of the jet, the dif-
ference may be many times greater than for lower values of F and R.
C. STRATIFIED AMBIENTS
A stable stratified ambient, in which density increases with increasing
depth, has the general effect of restricting the vertical motion of an
initially buoyant jet. This is because of the compound effect of strati-
fication and of the jet losing buoyancy due to entrainment of ambient
fluid. This occurs regardless of the direction of buoyancy, ambient
stratification, or of the magnitude of the stratification.
Figure (6-38) illustrates the effect of increasingly strong stable
stratification, due to an ambient temperature gradient. The calculation
is for an upwardly buoyant jet of F = 50, discharged horizontally into a
quiescent ambient. For the least stratification, A, the jet has followed
a trajectory similar to the unstratified case, replotted from Figure 6-11.
There is a yery slight vertical restriction of the trajectory. In case
B, near the end of the computed trajectory, a noticeable change in trend,
or re-bending of the trajectory, is evident. The point of inflection in
this curve is yery significant. Calculations show that this is the verti-
cal level at which the jet has become neutrally buoyant, due to entrainment
on one hand and the decreasing density of the surrounding fluid on the
other. Further upward rise beyond this point is due solely to the momentum
then existing in the jet. This momentum is gradually decreased by the
downward force of negative buoyancy.
94
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Case (C) shows the eventual course of events for a jet which has
become vertically "trapped", due to increased ambient stratification.
The trajectory has undergone complete re-bending, to horizontal flow
with no buoyancy. Standard practice in entrainment modelling, which
will be followed here, is to halt computation at the point of maximum
rise. Beyond this point, all but the most vigorous jets will have
acquired "far field" characteristics. Assumptions of axi symmetric jet
shape, tenuous at best in a stratified ambient, certainly cannot be made
beyond this point, and initial momentum and buoyancy have become largely
irrelevant to future behavior in most cases.
Figures (6-39), (6-40), and (6-41) illustrate the effect of the
same degrees of stratification on jets at a higher discharge Froude number,
F = 100, 150 and 200. The general behavior characteristics are similar
to those previously noted, with ambient stratification further restricting
the vertical rise already limited by low initial buoyancy.
The intuitive reasoning that the trajectory of initially more buoyant
jets is less affected by stratification is supported by these calculations.
Figures (6-42), (6-43), and (6-44) compare the trajectories of various
Froude number discharges in an environment of three different stratifica-
tion levels. The lowest F jets are seen to be negligibly affected over
this range of stratification, while the higher F jets exhibit varying
degrees of trajectory flattening, vertical trapping, and terminal rise.
Given strong enough stratification in the ambient, of course, any initially
buoyant jet will eventually experience negative buoyancy and terminate its
rise at some elevation in a sufficiently extensive ambient. No elevation
overshoot is seen.
96
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102
The above circumstances demonstrated are typical of past jet modelling
in a stratified ambient. Stratification, as indicated in the earlier
discussion of equations of state, is usually represented in terms of a
single, constant temperature, salinity, or density gradient. Rarely,
as in Reference [151, temperature and salinity gradients may be treated
together.
The capability of a comprehensive equation of state to more realis-
tically describe a particular water ambient and the behavior of a jet
within it is illustrated in Figures (6-45) to (6*50). Figure (6-45)
characterizes the temperature, salinity, and density stratification of
an area of the Northern Pacific Ocean to a depth of 500 m. June tempera-
ture and salinity data for a point 53° 04' N, 175° 35' W were taken from
Reference [4], and approximated by a series of 5 temperature and 4 salinity
gradients. The Gebhart-Mollendorf relation was then used to establish
the corresponding density field shown, which includes the contribution
of hydrostatic pressure.
Figure (6-46) shows the predicted trajectories of 5 buoyant momentum
jets, for F = 10,20,30,40 and 50, discharged horizontally in this density
field at a depth of 200 m. In each case, the initial jet diameter is 1 m.,
while F ranges from 10 to 50. Initial salinity difference between the
jet and ambient was assumed to be zero. Buoyancy was due to elevated
temperature, however the effect of the salinity stratification gradient
is included in the calculation of jet density and buoyancy along the
trajectory.
All five of these jets exhibit the progression from positive to nega-
tive buoyancy, as well as a level of maximum upward penetration. All dis-
play a rather smooth curvature and re-curvature, due to the relatively
103
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ENTRAINMENT
MODEL
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constant density gradient encountered in the vertical range of penetra-
tion. Again, no overshoot occurs.
Figure (6-47) illustrates another characteristic oceanic temperature-
salinity-density field, in this case from the tropical Atlantic at a point
02° 03' S, 39° 20' W. The data for this construction was taken in Febru-
ary and is from Reference [21]. The trajectory of jets F = 10,20,30,40
and 50 discharged into this density field at a depth of 200 m. is shown
in Figure (6-48). The vertical penetration of these jets in this case
is significantly less than calculated for the Northern Pacific density
profile, due to the much stronger density gradient in the tropical ocean.
An October temperature-salinity-density field for the Arctic Ocean
at 81° 28' N, 8° 05' E is shown in Figure (6-49). Data is from Reference
[17]. The strong salinity discontinuity near the surface is due to melt
from the adjacent Arctic ice pack. The behavior of jets discharged into
this ambient, Figure (6-53), is especially interesting compared to Figures
(6-49) and (6-51), because of the varying recurvature rates encountered.
For F = 30, the jets exhibit a smooth curvature and recurvature. The
F = 10 and F = 20 jets, however, show a much sharper recurvature because
of the encounter with the steeper density gradient above the 100 m. level.
These examples illustrate the advantage of using a full equation of
state in entrainment modelling. The dependence of the model on assumed
values of 3 and y is eliminated, as is the requirement to express ambient
stratification in terms of constant temperature, salinity, or density
gradients. The accuracy of the ambient density characterization is limited
only by the availability and spacing of the field data. Ultimately, the
use of a full equation of state can reduce one of the primary uncertain-
ties in entrainment model ling--that of evaluating the temperature-
salinity-pressure-density relationship.
110
VII. EXTRAPOLATION OF PRESENT METHODS TO PARAMETERS OF INTEREST
In reviewing the progress to date in entrainment modelling of buoy-
ant momentum jets, some rather striking matters become apparent. The
first of these is the very small physical scale of the experimental studies
which has been made to underlie subsequent modelling schemes. Jet dis-
charge diameters were mostly about 1 cm, the largest being 3.9 cm. The entire
jet trajectories were usually only a few meters in length. Great lengths
arose in some of the experiments done with flowing ambients, but even
these studies used similarly small diameter discharges. These limitations,
of course, arose, at least in part, through the innate limitation in size
and economy which applies to experimental work.
Still, the data base from these small scale discharges are often used
to develop models for real world jets, such as power plant discharges and
sewer outfalls. These are commonly hundreds to thousands of times larger.
Such upscaling of the implications of the experiments implies a faith in
the similarity properties of the densi metric Froude number. This may or
may not be justified. Further, such a large scale difference between
supporting data and real system projection, in the case of an underwater
momentum jet, does not necessarily imply similar scaling in turbulence
within the jet and in the immediately surrounding ambient. Fan [7] found
that the effect of ambient turbulence on jet behavior was profound. It
is not unreasonable to expect that the scale of turbulence within the
jet, or the relative scale between ambient turbulence and jet dimensions,
may not have equally important effects.
The second striking aspect of experimental and analytical work is the
relatively low densimetric Froude number range used. Admittedly, more
111
buoyant, lower Froude number jets are more "interesting" in terms of
possible trajectories, and may be of more importance in many environmen-
tal modelling situations. Froude numbers in the range of 10 to 40 are
common in the literature. Values as high as 100 are rare. There are a
number of "real" cases, however, in which entrainment modelling might be
used, where Froude numbers might be in this higher range, yery basic
calculations dealing with unclassified physical and propulsive properties
of submarines, for instance, indicate that the propeller outflow, with
all condenser efflux entrained and mixed within, might represent a Froude
number in the range of 50 to 200. Ship or submarine condenser efflux by
itself might be expected to have a range inclusive of all Froude numbers
to about 200, depending on discharge size, flow rate, and thermal loading.
A third limitation in the present state of entrainment modelling is
the inconsistent predictive quality of the collection of available flowing
ambient entrainment functions in their supposed range of applicability.
A fourth shortcoming is the range of R in which data is available to guide
modelling, ^ery little experimental data exists for R greater than 0.25.
Almost no data exist for an initially co-flowing ambient. The data is not
sufficient to definitively select either a general entrainment function
or even one valid over a restricted range of F and R.
Given the limitations of the current background data to support any
modelling technique, a high level of confidence in predicting large scale
jet trajectories, involving conditions of ambient flow, turbulence, or
high Froude number discharges is unwarranted. There are simply too many
unknowns to proceed confidently, limited by contemporary methods and
information.
Toward the objective of developing large scale entrainment modelling,
the following suggestions are made.
112
VIII. RECOMMENDATIONS
If entrainment modelling is to be made applicable to large scale
discharges involving relatively high ambient flow velocities, a more
comprehensive data base than now exists needs to be established. Such
a data base would include:
(1) higher Froude number flows, up to a range of 200.
(2) larger scale discharges, with particular attention to turbulence
scale and its effects.
(3) higher R coflows.
Future analytical models, drawing from such an enhanced data base,
need to specifically address:
(1) determination of a dependable entrainment function for submerged
jets discharged to a flowing ambient.
(2) determination of methods to include turbulence effects in the
model
.
(3) inclusion of an accurate and comprehensive equation of state,
such as the one proposed here, in the model.
113
APPENDIX A
A. ASSUMED GAUSSIAN DISTRIBUTIONS
U Um
exP[--^] (2-1)
T-T 2
r-4- = exp[-^2] (2-2)
ma x B
C-C 2
r4- = expt-^-] (2-3)V La *V
P a "P _ r2
p aDm a
2B2
B. DEVELOPMENT OF THE GOVERNING EQUATIONS
1 . Continuity
jL/ / Urdrd^} = 2™UmB (5-1)uo
^U2tt/ Urdr} = 2iTaUmB
Substituting Equation (2-1)
Ioo 2
i-{/ U exp[^V]rdr} = aUBdS L J
Qm
r2 m
114
db m 2 m
iiUB 2} = 2aUB (A-l)
aS m m
For the case of a coflowing ambient, the entrainment rate is
expressed as a function of jet center! ine velocity and jet width which
is more complex than the simple linear relationship of the quiescent
ambient case. For coflowing ambients,
. 2tt °°
%{( / Urdrd<<>} = 2irf(a,U ,B,R)^ m
By development identical to the quiescent ambient case
^UmB2
} = 2f(a,Um,B,R) (A-2)
2. Conservation of Energy
. 2tt =» dt 2tt °°
%{j ( U(t-tjrdrdd)} = -^J / Urdrdct} (5-4)05
o 'o
Integrating and substituting Equations (2-1) and (2-2),
U D AD
dt 2
"df*2 * /
Q
Umexp[^]rdr}
115
dVVVV f] expt^^lrdr)
dt, -r2^ U
m / exp[^]rdr}
. ,2D 2 dt R 2
d5 m m a «/,2+ -j\ dS m 2
3. Conservation of Concentration or Other Scalar Species
2lT °o
^/ / U(c-c-)rdrdd>}^ a
dc 2tt °°
~i|{/ / Urdrd*} (5-5)a;>
By development identical to conservation of energy in (2) above, using
Equation (2-3),
d a2R2 dS R
2
dVyv ca^^y]} = -tKV (a" 4)
4. Conservation of Horizontal Momentum
For the case of the quiescent ambient,
2tt
4k/ / U2coserdrddi} = (5-2)
^
d 7jkiZ^i IT coserdr} =
Substituting Equation (2-1),
116
^2ttU 2cos9 / exp[ :=^-]rdr}
Q5 FT1
g g^
d_{2 , u
2cose[^]} . o
^{U 2B2cose} =
d5 m(A-5)
For the case of the flowing ambient.
H 2tt °°?
£r[/ / lTcoserdrd<t>} = 2Trf(a,U ,B,R)U.as m <
3^(2* f U coserdr}dS
27Tf(a,Um,B,R)U
a
Substituting Equation (2-1),
d 2 2r^l£cose / expt^-jrdr} f(a,U
m,B,R)U
a
^U 2C0S6[1V> f(a,U
m,B,R)U
a
^coseB 2} 4f(a,U
m,B,R)U
a(4-6)
5. Conservation of Vertical Momentum
2tt
^-{/ / pU sinerdrdcj)} =
2tt
(p -p)grdrd<j> (5-3)a
117
oo oo n — o
^2tt / U2sinerdr} = 2tt / (——)grdr
Substituting Equations (2-1) and (2-2),
^ Umsine f
Q
exp[^-]rdr} = ^ g f exp[^]rdr
oV Umsin4V }
V p m ,A2B2
^
^{U 2sineB
2}
dS m^(2gx 2
B) (A-7)
6. Horizontal Trajectory
dX = dX cos e
dSA = COS 8 (A-8)
7. Vertical Trajectory
dZ = dS sin 9
dSZ = sin (A-9)
118
APPENDIX B
A. DIMENSIONLESS VARIABLES
The following dimensionless quantities will be used in the non-
dimensional ization of the governing equations:
_U_ ^a!
ta2"
ta]
)
U At " <VWU AC (C -C ,
)
_ _m a =* a2 al
'
m UQ
ac (c -caQ )
h B ^m |VVD At ' ^aP~
J_ ^m (y ca>
s " D At ^rS^"
X .Ua
x " D~K " U~
_Z_
D
B. NONDIMENSIONALIZATION OF THE GOVERNING EQUATIONS
1 . Continuity
For the case of the quiescent ambient,
TrKUB2
} = 2aUB (A-l)db m m
Substituting,
119
oW^O5^ 2auUnbDm U
-rriu b }as m
2au bm
(B-l)
For the case of the flowing ambient,
ab mO / 2f(a,U
m,B s R) (A-2)
f(a,Um,B,R) (0.057 +^sine)B( |U-U cose
|+a-U sine)
i"| mi a o a
db m2(0.057 +^- sine )B(
|
Um-U cose
|+a-U sine)
Substituting,
as D m m
= 2(0.057 +^-sine)bD (lu U -RILcosej +a,RUnsine)r. ' m m U ' 3 U
d, .2,^Hu b -
ds m2(0.057+^|4ine)b( |u -Rcose |+a.Rsine) (B-2)
l|
III O
2. Conservation of Energy
d2R2
dS m^ m a'L
2 ( A2+1)
J1am
"dS"1
2J (A-3)
Substituting,
120
2 2 2At rtro/
ds Dl
m AtQ
v aO ;
2 (\2+l)
Ata (V t
aO ) b2 °2
= 1 u aur u u
AtQ^TD LU
mu
2 '
a At ,2,2 At,d
{|Jm A_b
}u . a_r
u b2
} (B . 3)ds
1
m AtQ ( x
2+1
\ AtQ
ds m x '
3. Conservation of Concentration or Other Scalar Species
H >2R2 dS R
2
dS m x m a ?(a2+ i\ dS m 2
By development identical to conservation of energy in (2) above,
. ACm ,2. 2 AC adr
u —Dl A_b}
= . —i-j-fu b2
} (B .4)ds
1m ac
q T^VJT acq
ds- mj v y
4. Conservation of Horizontal Momentum
For the case of the quiescent ambient,
^KU2B2cosel = (A-5)
do m
Substituting,
^u 2U2b2D2cose} =
^uVcose} = (B-5)
121
For the case of the flowing ambient,
^U 2coseB
2} = 4f(a,U
m,B,R)U
a(A-6)
f(a,U B,R) = (0.057+^sine)B(|U -U acose|+a,Usine)m r . ni a o a
dP mILcoseB' ;
4U (0.057+^sine)B( IU-U cose |+a.U sine)a i~i in a J a
Subsituting,
^(u 2U2b2D2cose}
4RUn (0.057+-4^sine)bD ( |u IL-RlLcose |+a„RUnsine)U r, ' m U U ' o u
^r{u2b2cose} = 4R(0.057+^sine)b(|u
m-Rcose|+a
3Rsine)
(B-6)
5. Conservation of Vertical Momentum
d_{U
2sineB
2}
a !l^m(2gx
2B) (A_ 7)
The denominator of the density term on the right side of Equation
(A-7) was transposed from the Bousinenesq term on the left side of the
equation. It will be assigned a value equal to the reference discharge
density. Substituting dimensionless terms,
122
<^tf$Vi1ne) - ^(2gx 2b2D2
)
l_f,A2c -i„ fl i
pa"
pmr
p aQ- pmQ A
^^-3-{umb sine} = (
) 5—^ m p p aO" pmO U2
dr
2.2 • .-, / a m \ 2a b /„ 7 \
-|-{u b sine} = ( —
)
—-~
—
B-7ds m p a(T p mO F
2
6. Horizontal Trajectory
~ X = cos e (A-8)
Substituting,
Vrr x D = cos 9
^x = cos 9 (B-8)
7. Vertical Trajectory
^Z = sin 9 (A-9)
Substituting,
irr 1 D = sin 9
d
dsz = sin e (B-9)
123
APPENDIX C
The Gebhart-Mollendorf relation for the density of saline water [11]
is of the form
P (t,s,p) = p m(s,p)[l-a(s,p)|t-t
m(s,p)|
q(s ' p)]
where t is temperature (°C), s is salinity (%o )» P is pressure (bars
absolute), p(s,p) is the density extremum at the given values of s and
P. t (s,p) is the temperature corresponding to the density extremum for
the same s and p values, a(s,p) is a temperature term coefficient, and
q(s,p) is a temperature term exponent. These values are in turn given
by
P m(s,p) = p m
(0,l)[l+f1
(p)+sg1(p)+s
2h
1(p)]
a(s,p) = a(0,l)[l+f2(p)+sg
2(p)+s
2h2(p)]
tm(s,p) = t
m(0,l)[l+f
3(p)+sg
3(p)+s
2h3(p)]
q(s,p) = q(0,l)[l+f4(p)+sg
4(p)+s
2h4(p)]
where
3 13
^i(p)= l fii(p-n
J,
g-jCp) = I g ii (p-DJ
j=i1J
>1
j=o 1J
h.(p) -I
h (p-l) J
1
j=0 1]
124
o (0,1) = 999.972 kg m'3
t (0,1) = 4.029325 °Cm
a(0,l) = 9.297173 x 10' 6 °C' q
q(0,l) = 1.894816
and the values of f.., g.., and h.. are given in tabular form in Table
(TC-1).
The relation is fitted in the range of temperature to 20°C, pressure
to 1000 bars absolute, and salinity to 40 °/ OQ .
Comparison of the density predicted by this relation with the data
of Chen and Mi Hero [51 is shown in Figure (C-l). The overall rms differ-
ence between the relation and this data, for pure and saline water at all
temperatures and pressures considered, is reported to be 9.0 ppm.
125
30
I
Een
Ox
n
<!/-
20
10
10
-20
200 bars
1 bar
100 bars
Figure C-l . Comparison of results of Gebhart and
Mollendorf [11] with data of Chen and
Mi Hero [5]
126
co•r-
ns
<+-
S-
ocO)
I
i-
J=X}0)CD
01
+-)
c
c03
•i—raG7>3
r-O
o </>
cto at0) Qi— i-
(O O> 4-
Io11
-Q
CO o o i—
i
* CM O CM LO CO CM «3-
LU UJ1
LU LU LU1
LUi
LU LU LU1
LU1
LU1
LUcn o VO cn CO CM CO r—4 o cn CO LO<—
i
<?r «d- LO LO LO CO o 53" r^ 1—4 oon VO CM 00 CM '-O "vT 1—
1
LO ^- CO 1—1 i-H
CM co 00 r»» CX> VO i-H r^ CM cn 1—1 o^~ o CO LO r^ V0 I—
1
CO i-H r^ LO r^00 Cn o CO cr> cn "vT cn '-" O LO CO
r~» CM r^ LO CM vo i—
4
CMi
i-H cn1
LO
cn VO r-v r*» o CO 1 cn CM cn cn 1—
1
oi
o o O i—
t
O o o 1—4 o o r—
I
LU UJ LU LU LU LU LU LU LU LU1
LU1
LUCO 00 1—
t
cn LO CO CO r^ CO LO 1^ or^ <3- 00 Cn 00 ^r LO LO CO r*» r-«» r^
CM cn VO 1—
1
cn 1—4 CM O LO r—4 LO o cn»—
4
rv. o 00 l-H CO CM CO CO 00 ^r loo cn CM CO CO LO cn LO o I~- cn VOvo ^3- r» CM o o t~^ «3- CM cn lo i-H
CM1
i—
i
r>- 1—1
1
r—
4
001
CM1
<=3- CM 1—4 r-«
1
•I-
)
LO «=r co «d- CO LO LO LO cn 1^ lo COo o O1
o O o1
o O o o o oUJ UJ UJ LU LU LU LU LU
1
LU LU UJ1
LU00 *a- o 00 VO i—
1
r—4 r»» o CM o LOcn 00 o LO cr> LO CO CO cn i-H LO i—
1
i—
t
cn LO o [-» CO cn r-- cn 1—4 cn LO eno 1
—
o 00 «3- CX> vo CO r^ CM CO COus r-» co i—
t
Cn CM cn CO <d- LO CO i—
i
<T> co «=r <—
t
1—4 l-H •3- cn co CO LO r-^
^r T—
4
LOi
I—
4
1
LO1
I—
1
r-^ CM 1—4
i
CO1
CO1
*r CM CM CO r^ <3- LOo o o o O O oLU UJ LU UJ
1
LU LU1
LUCM LO cn o "3" LO o COLO LO o CO co LO o r^o CM CO LO LO CO CO o COCM CO LO LO CO LO o cncn CM LO CO 1—1 VO o cn<T> LO CM i-H en LO o LO
r~>. <—
4
LO1
CO1
I-H
1
o r^
4-^" ^ ^ 4-T cn cn en -c"T
127
LIST OF REFERENCES
1. Abraham, G. , Transactions ASCE, J. Hydraulics Div., vol. 86, HY6,
1960, pp. 1-13.
2. Abraham, G., Proceedings ASCE, J. Hydraulics Div., vol. 91, HY4,
1965, pp. 139-154.
3. Albertson, M.L., Dai, Y.B. , Jensen, R.A., and Rouse, H., TransactionsASCE, vol. 115, 1950, pp. 639-697.
4. Barstow, D., Gilbert, W., Park, K. , Still, R., and Wyatt, "Hydro-graphic Data from Oregon Waters 1966", Dept. of Oceanography, OregonState Univ., Corvallis, OR, 1966.
5. Chen, C.T., and Millero, F.J., Deep-Sea Research, vol. 23, 1976,
pp. 595-612.
6. Davis, L.R., Shirazi, M.A., and Siege! , D.L., "Measurement of BuoyantJet Entrainment from Single and Multiple Sources", Heat TransferDivision of ASME Paper 77-HT-43, Aug. 1977.
7. Fan, L.H., "Turbulent Buoyant Jets into Stratified and FlowingAmbient Fluids", California Institute of Technology, W.M. KeckLab., Report KH-R-15, 1967.
8. Ibid., pp. 52-53.
9. Fan, L.H., and Brooks, N.H., "Numerical Solutions to TurbulentBuoyant Jet Problems", California Institute of Technology, W.M.
Keck Lab., Report KH-R-18, 1969.
10. Fox, D.G., J. Geophysical Research, vol. 75(33), 1970, pp. 6818-6835.
11. Gebhart, B., and Mollendorf, J.C., Deep-Sea Research, vol. 24, 1977,
pp. 831-848.
12. Ginsberg, T., and Ades, M., Transactions ANS, vol. 21, 1975, pp. 87-88.
13. Hirst, E.A., "Analysis of Buoyant Jets within the Zone of FlowEstablishment", Oak Ridge National Laboratory, Report ORNL-TM-3470,1971.
14. Hirst, E.A., J. Geophysical Research, vol. 76, 1971, pp. 7375-7384.
15. Hirst, E.A., "Analysis of Round, Turbulent, Buoyant Jets Dischargedto Flowing Stratified Ambients", Oak Ridge National Laboratory,Report ORNL-4685, 1971.
128
16. Hoult, F.A., Fay, J. A., and Forney, C.J., J. of Air PollutionControl Association, vol. 19, 1969, pp. 585-590.
17. Johannessen, J. A., and others, "A CTD-Data Report from the NorsexMarginal Ice Zone Program North of Svalbard in September-October1979", University of Bergen, Royal Norwegian Council of Scientificand INdustrial Research, Bergen, Norway, 1980.
18. List, E.J., and Imberger, J., Proceedings ASCE, J. HydraulicsDivision, HY9, 1973, pp. 1461-1474.
19. Madni , I.D., and Pletcher, R.H., Transactions ASME, J. of HeatTransfer, vol. 99, 1977, pp. 99-104.
20. Morton, B.R., Taylor, A.G., and Turner, J.S., J. Royal Society ofLondon, vol. A234, 1956, pp. 2-23.
21. Neumann, G. , and Pierson, W.J., Principles of Physical Oceanography ,
Prentice-Hall Inc. , 1966.
22. Pryputniewicz, R.J., and Bowley, W.W., Transactions ASME, J. ofHeat Transfer, vol. 97, 1975, pp. 274-281.
23. Riester, J.B., Bajura, R.A., and Schwartz, S.H., Transactions ASME,J. of Heat Transfer, vol. 102, 1980, pp. 557-562.
24. Ibid., p. 23.
25. Schatzman, M., "A Mathematical Model for the Prediction of PlumeRise in Stratified Flows", Proceedings of the Penn State Symposiumon Turbulent Shear Flows, 1977.
26. Shirazi, M.A., McQuivey, R.S., and Keefer, T.N., Proceedings ASCE,J. of the Hydraulics Division, vol. 100, HY7, 1974, pp. 919-934.
27. Taylor, G.I., Proceedings Royal Society of London, vol. 201A, 1950,
p. 175.
129
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