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NAVAL POSTGRADUATE SCHOOL

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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.

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Entrainment Modelling of Buoyant Momentum Jetsin Water

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Engineer's Thesis;December 1981

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7. AUTHOR, t)

David Stuart Hilder

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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

OS i——

1

a 1 < ro <C -a1 -O. 1

E 3 fO <a 1 1

3 as a. c** **—-*

ImJ 11 II

-Qa II 11 CD CD

CM «-—*» *^7*%

^—

^

^—** CO e.*—

N

,—-» CM r— CM 1— O •r—

II CD CD x> + .a + u COCO C CM CM CM CM

•1— << r< << r<*"—

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Ume ' U

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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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109

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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