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STABILITY AND MIXING OF A VERTICAL
ROUND BUOYANT JET IN SHALLOW WATER
by
Joseph H. Lee, Gerhard H. Jirka,
and Donald R. F. Harleman
Energy Laboratory
Report No. MIT-EL 74-014
November 1974
400.e
STABILITY AND MIXING OF A VER:IfCAI
ROUND BUOYANT JET IN SHALLOW WAi-R
by
Joseph H. Lee
Gerhard H. Jirka
and
Donald R. F. Harleman
ENERGY LABORATORY
in association with
RALPH M. PARSONS LABORATORY'
FOR
WATER RESOURCES AND HYDRODYNAMICS,
Department of Civil Engineering
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Sponsored by
New England Electric System and Northeast Utilities Service Company
under the
M.I.T. Energy LaboratoryElectric Power Program
Energy Laboratory Report No. MIT-EL 74-014
R. M. Parsons Technical Report No. 195
November
1974
.eW,
2
ABSTRACT
Discharging heated water through submerged vertical round ports
located at the bottom of a receiving water body is a currently used
method of waste heat disposal.
The prediction of the temperature
reduction in the near field of the buoyant jet is a problem of
environmental concern.
The mechanics of a vertical axisymmetric buoyant jet in shallow
water is theoretically and experimentally investigated.
Four flow
regimes with distinct hydrodynamic properties are discerned in the
vicinity of the jet:
the buoyant jet region, the surface impingement
region, the internal hydraulic jump, and the stratified counterflow
region.
An analytical framework is formulated for each region.
The
coupling of the solutions of the four regions yields a prediction of
the near field stability as well as the temperature reduction of the
buoyant discharge.
It is found that the near field of the buoyant jet is stable only
for a range of jet densimetric Froude numbers and submergences.
A theor-
etical solution is given for the stability criterion and the dilution of
an unstable buoyant jet.
A series of experiments were conducted to verify the theory.
experimental results are compared to the theoretical predictions.
agreement is obtained.
The
Good
3
TABLE OF CONTENTS
Page
ABSTRACT
2
ACKNOWLEDGEMENTS
5
AND BACKGROUND
I.
INTRODUCTION
II.
THEORETICAL FRAMEWORK
2.1
2.2
6
12
The Buoyant Jet Region
14
2.1.2
Statement of the problem
14
2.1.2
General characteristics of the axisymmetric jet
16
2.1.3
General analytical treatment
17
2.1.4
Governing equations
19
2.1.5
The entrainment principle
21
2.1.6
Mathematical formulation
25
2.1.7
Mathematical solution
26
The Surface Impingement Region
29
2.2.1
Analysis of the control volume
31
2.2.2
Limiting cases
33
2.3
Radial Stratified Flow
35
2.4
The Radial Internal Hydraulic Jump
40
2.5
Stratified Counterflow Region
52
2.5.1
52
2.5.2
The momentum equation for axisymmetric stratified
flow
Behavior of the counterflow system
2.5.3
Critical flow in a two-layered system
60
2.5.4
Behavior of flow at large distances
65
2.6
Smmary
2.6.1
of Theoretical Framework
Definition of the near field dilution
55
67
67
4
TABLE OF CONTENTS (Continued)
Page
2.6.2
Stable near field dilution
67
2.6.3
Unstable near field dilution
70
2.6.4
Equal counterflow
70
2.6.5
Nonequal counterflow
72
III. EXPERIMENTAL INVESTIGATION
IV.
V.
74
3.1
The Experimental Setup
74
3.2
Experimental Procedure
81
3.3
Experimental Program
85
3.4
Experimental Observation
85
COMPARISON OF THEORY AND EXPERIMENTAL RESULTS
98
4.1
Near Field Stability
98
4.2
Near Field Dilution
98
CONCLUSION
105
REFERENCES
107
LIST OF FIGURES AND TABLES
108
LIST OF SYMBOLS
110
APPENDIX
113
5
ACKNOWLEDGEMENTS
Funds for the publication of this study were made available from
the Waste Heat Management Research Program of the M.I.T. Energy Laboratory.
The material in this study was submitted by Joseph H. Lee,
Research Assistant, to the Department of Civil Engineering in partial
fulfillment
of the requirements
for the degree
of Master
of Science.
Technical and administrative supervision was provided for by Professor
Donald R.F. Harleman, Professor of Civil Engineering and Director of the
R.M. Parsons Laboratory for Water Resources and Hydrodynamics, M.I.T.,
and Dr. Gerhard H. Jirka, Research Engineer, Energy Laboratory, M.I.T.
Several discussions with Professor Ole S. Madsen have been very
helpful.
Mr. Roy C. Milley, machinist, and Mr. Harry Garabedian, graduate
student, assisted in the construction of the experimental setup.
The
manuscript was typed by Ms. Susan Wiseman.
Computational work was carried out at the Civil Engineering Mechanical Engineering Joint Computer Facility at M.I.T.
6
I.
Introduction and Background
With the increasing demand in electric power in the U.S., waste heat
disposal has become a problem of important environmental concern.
Steam
electric power plants, both fossil-fueled and nuclear-fueled, require a
continuous cooling water flow to remove the waste heat from the steam condenser.
Two modes of cooling water operation are possible: In a once-
through system, the cooling water is circulated through the power plant
only once and then discharged as heated water into an adjacent receiving
waterbody.
In closed-loop systems, the cooling water is continuously re-
circulated, the heat being rejected directly to the atmosphere before the
water returns to the plant.
Due to the low efficiencies of existing power plants (determined by
the thermodynamics of the steam cycle), enormous quantities of waste heat
are discharged.
In a nuclear fueled power plant, for every kilowatt of
electrical energy produced, an equivalent of two kilowatts of energy is rejected to the environment in the form of heat.
The artificial heat addition into a natural water body has a definite
impact on the local ecological balance.
Consequently decision makers as
well as ecologists are very concerned with the thermal effects in the natural waterway induced by various methods of condenser cooling water discharges.
Common methods of waste-heat disposal for present once-through cooling water systems can be classified into two categories:
a) Surface Discharge schemes: The condenser cooling water is discharged
through a canal or a number of pipes located at the water surface into the
neighboring waterway.
This method of discharge usually results in a larger
__
7
surface
area with elevated water temperatures, but has the advantage that
the heated water forms a stably stratified surface layer and the effect on
the bottom of the receiving water is reduced.
Pilgrim Nuclear Power Sta-
tion in Plymouth, Mass. is one such example.
b) Submerged discharge schemes: Either single port or multiport submerged
discharges are in common application.
Single port discharges involve a
single (or dual) outlet located at the bottom of the receiving water and
discharging either vertically or horizontally.
Two examples of large
existing vertical single port outfalls on the Pacific coast are:
1. San Onofore nuclear plant.
The cooling water flow from approximately
450 MW generation is 3.2 x 106 ft3 /hr. discharged through a 14-ft. diameter pipe 2600 ft. offshore, about 15 ft. below sea surface.
2. Redondo beach fossil fueled plant with 1612 MW capacity.
One of the
two offshore outfall systems consists of a single 14-ft. diameter pipe
discharge 300 ft. offshore, about 16 ft. below water surface.
Recent innovations propose the use of multiport diffusers as an efficient
way of heat disposal.
This consists of a long pipe with the condenser
flow discharged through many openings spaced along the pipe.
The high
velocity jet discharges induce intense turbulent mixing with the ambient
water, thus achieving rapid temperature reduction of the heated discharge
within a relatively small area.
The ultimate heat sink is the earth's atmosphere.
The entire temp-
erature distribution in the nearby waterway induced by the flow and heat
input of the condenser cooling water is governed by the interaction of a
variety of complicated physical processes and boundary conditions: turbulent mixing of the discharge with the ambient water, the hydrodynamic
8
conditions in the receiving water, conduction, convection, evaporation and
radiation to the atmosphere.
For once-through cooling water systems the
receiving water can be broadly classified into two regions with respect to
the thermal effects of waste heat input: Far from the discharge the temperature pattern is dependent on ambient processes and consequently is not
under the direct control of the engineer: the wind speed, the prevailing
direction and magnitude of the currents, the ambient temperature, humidity
and other meteorological conditions that govern the heat transfer between
the water surface and the atmosphere.
Near the discharge the temperature
distribution is sensitive to the mode of discharge (surface discharge or
submerged discharge) as well as the design characteristics (orientation,
spacing, and number of discharge ports, diameter of port opening, size and
The task facing the engineer is to produce the best
geometry of channel).
design with respect to specified thermal discharge criteria.
The quantity
of interest is often an average dilution defined by the ratio of the temperature rise across the condenser to the temperature rise above ambient
near the discharge.
This serves as a general indicator of the effective-
ness of temperature reduction achieved by the discharge design.
Other con-
siderations include the time of travel of organisms entrained in the discharge, whether the near field is stratified- or fully mixed, the area of
a certain surface isotherm.
The discharge of heated water through a vertical round port located
at the bottom of the receiving water is a currently used method of waste
heat disposal.
The temperature distribution induced by such a method of
discharge entails an understanding of the hydrodynamics of the physical
situation.
The heated discharge entrains surrounding water by virtue of
9
its momentum and its buoyant acceleration as it rises to the water surface,
with a corresponding dilution of the discharge flow.
The mechanics of a
round buoyant jet in an infinite ambient field has been investigated by
many investigators.
However, in many practical situations, these vertical
outfalls are situated in shallow water ( a physical parameter that measures the 'degree of shallowness' is the ratio of the water depth to the
port diameter).
Near field dilution is usually computed by extending the
buoyant jet solution in an infinite field in some arbitrary way.
An at-
tempt at a more refined treatment has been Trent and Welty's (1973) work
on numerical modelling of turbulent jet flows.
These studies, however,
have neglected the important question of hydrodynamic stability of the
near field.
The boundary conditions chosen always dictate a stable near
field, i.e., the heated water always form a stratified flow away and the
jet discharge is always entraining ambient cooling water.
The stability of the near field for a two dimensional slot, buoyant
jet was investigated by Jirka and Harleman.
It has been found that the
densimetric Froude number, the submergence and the angle of discharge of
the jet are the governing parameters that determine the stability of the
near field.
In an unstable near field, it is not possible to distinguish
an upper layer in the flow away zone, and there is continuous heat reentrainment into the jet.
The dilution is hence decreased considerably as
compared to that obtained in a stable near field
(fig. 1-1).
The objective of this thesis is to extend the physical and analytical notions of the two dimensional case to the simplest three dimensional
case - an axisymmetric vertical buoyant (round) jet in stagnant shallow
water.
With the exception of two experimentally determined coefficients,
10
a theoretical solution is derived to determine the near field dilution and
establish the criterion of the stability of the near field.
If a stable
near field exists, the near field dilution is dependent solely on the near
field parameters (jet densimetric Froude numbers, submergence).
In the
case of an unstable near field, the dilution is dependent on both the near
field parameters as well as the far field boundary condition.
A series of experiments were conducted to verify the theory.
11
D
E
COOLING
WATER INFLOW
DISCHARGE
a) STABLE NEAR FIELD
DISCHARGE
b) UNSTABLE NEAR FIELD
FIGURE (1-1)
ILLUSTRATION
STABILITY
OF NEAR FIELD
12
II.
Theoretical Framework
Both the experiments done for a two-dimensional buoyant jet (Jirka
and Harleman, 1973) and the experiments carried out in this study for an
axisymmetric vertical buoyant jet in shallow water (ch. 3) suggest strongly the classification of the near field into several distinct flow regimes; (Fig. 2-1)
A) Buoyant Jet Region:
Before the bouyant jet rises to
the surface of the water, its behavior is postulated to be the same as that
of a buoyant jet in an infinite field.
B) Surface Impingement Region: this
refers to the surface hump formed by the jet impingement on the free surC) The
face, followed by horizontal spreading of the jet discharge.
Internal Hydraulic Jump: An abrupt transition from the high velocity flow
in the surface impingement region to a lower velocity flow away occurs
some distance away from the jet axis, with a thickening and a corresponding decrease of velocity of the upper layer.
D) Stratified Counter-Flow
Region: the flow that occurs after the internal jump is described by a
stratified two-layered slowly varying flow.
Fig. 2-1 illustrated the flow details for the case of a stable near
field condition.
In the case of an unstable near field continuous re-
entrainment of already mixed water into the jet region occurs.
Hence a
large vertical eddy (of toroidal shape in the axisymmetric case) comprises
the near field region.
Outside this region exists a stratified counter-
flow system as in the stable case.
The classification of the problem into distinct flow regimes with
appropriate assumptions renders the description of the flow field amenable
to analysis.
In the following sections the properties of the flow and
temperature for each region will be analysed.
The coupling of the
13
NEAR FIELD
FAR
FAR
FIELD
FIELD
HEAT LOSS
HEAT LOSS
t
.
.
.4
_
7
//
ft
.I
.
>
®
.4-
t
/////////
//7
TURBULENT
ENTRAINMENT
4fT
/////
////,,
////////
I BUOYANT JET REGION
2 SURFACE IMPINGEMENT REGION
3 INTERNAL JUMP REGION
4 STRATIFIED
COUNTER FLOW REGION
FIGURE (2-1)
FLOW STRUCTURE IN THE PLANE OF SYMMETRY
OF A VERTICAL
WATER
AXI-SYMMETRIC
JET IN SHALLOW
14
analyses of the four regions yields the prediction of the near field
dilution.
Since the near field is of small areal extent, the heat loss from
A scaling argument
the surface is excluded from the subsequent analysis.
demonstrates this assumption is well-justified under typical thermal discharge conditions (Appendix F).
The flow is assumed turbulent for all the analytical treatment in
the following sections.
No generality is lost by considering the specific
case of a hydrothermal jet.
The words 'water' and 'density deficiency' can
be replaced by 'fluid' and 'concentration' without altering the method of
analysis.
2.1
The Buoyant Jet Region
2.1.1
Statement of the problem
Fig. 2-2 shows an axi-symmetric buoyant jet of fluid issuing from
a source of finite diameter vertically upwards into a denser homogeneous
ambient fluid (of infinite lateral extent) at rest.
The physical varia-
bles of interest are the velocity and density of the jet at any particular
position (z,r) in a cylindrical co-ordinate system.
15
I
1
An axi-symmetric jet discharging vertically
Fig. 2-2.
u
(z,r)
p(z,r)
b(z)
g
:
vertical
:
density
velocity
at (z,r)
of fluid at
(z,r)
: width of jet
:
acceleration due to gravity, acting in
direction
-z
Uo
:
exit jet velocity
O
:
initial jet fluid density
D
Pa
nozzle diameter
:
density
of ambient
fluid
16
2.1.2
General Characteristics of the Axi-symmetric jet
The general characteristics of the buoyant jet (or forced plume) in
a fluid of unlimited vertical extent are well established by extensive reIn any given physical situation (convection induced by fires,
search.
plumes rising from smoke stacks, sewage disposal from a submerged outfall),
the fundamental physical variable is the density of the issuing fluid (be
it due to a temperature difference or embodied pollutant), and the characteristic dimensionless parameter that governs the mechanics of the buoyant
uo
jet is the exit densimetric Froude number as defined by F
=
APD
p
where
Apo =
fluid.
a -
o : initial density difference between jet and ambient
This parameter describes the ratio of the sum of all forces per
Apo
2
unit mass,
-
, to the buoyancy force per unit mass
D
When Fo
+
g--
of the fluid.
P
, inertia dominates, and the buoyant jet behaves like a pure
momentum jet.
is small, buoyancy dominates, and a
Conversely, when Fo
plume-like convective motion arises.
In the intermediate case when Fo
Near
has a finite value, both inertia and buoyancy effects are important.
the source the initial momentum dominates and the discharge behaves like
a pure jet.
Far from the source buoyancy predominates and all buoyant
jets behave like plumes.
Near the source of a pure momentum jet, the sharp discontinuity in
velocity between the jet and the ambient fluid creates a region of high
shear.
Such a region is highly unstable; eddies accompanied by turbulent
mixing result, with the effect that ambient fluid is entrained into the
jet, increasing the mass flux of the jet.
The width of the jet, and hence
the dilution of the fluid increases in the direction of the discharge.
The momentum flux is conserved.
17
For a pure plume, the discharge with no initial momentum is continuously accelerated by the buoyancy force.
A certain distance away from the
source, the plume will have acquired enough momentum to entrain the ambient
fluid; the basic turbulent mixing process that ensues after this point is
The
then similar to the momentum jet.
uo yancy flux is preserved in this
case, whereas the mass flux and the momentum flux increases in the direction of the discharge.
2.1.3
General Analytical Treatment
The structure of a submerged buoyant and non-buoyant jet has been
determined from a number of experimental investigations(e.g., Albertson,
Rouse and Yih, Morton):
1.
Near the source, where turbulent diffusion of the momentum has not
penetrated to the center of the jet, the velocity profile consists of a
top hat portion, and a bell-shaped tail approximating the drop in velocity
due to the entrainment of the ambient fluid (Fig. 2-3).
2.
A certain distance away from the discharge, where the central core of
constant exit velocity ceases to exist, the velocity profiles are of bellshaped form.
Gaussian profiles can usually be well-fitted to the experimental
results.
It can also be observed in experiments that the profiles of density
deficiency, defined as
well.
Ap(z,r) = Pa - p(z,r), are of bell-shaped form as
The rate of spreading, however, is larger, indicating that heat or
concentration of a pollutant diffuses faster than momentum.
18
z
r2
uz (z,o)
)= Uz(z,O)t
b(z)
N OF
LlSHED
FLOW
ON OF FLOW
BLISHMENT
r
FIGURE(2-3)
SCHEMATIZED STRUCTURE OF AN
AXISYMMETRIC BUOYANT JET
19
2.1.4.
Governing equations:
A steady state formulation of the problem is presented in this
section:
Continuity: Invoking Boussinesq's(constant mass but variable weight);
r ar
r u
a u(z,r) =
+
Integrating across the jet, we have:
dd
uz(z,r)2rrdr
=
- 2irur(z,r)
o
(2.1.1)
QQe
where
Q
= entrainment flux
The change in the volume flux of the jet is due to the entrainment
of ambient water.
Newton's 2nd Law of Motion:
Navier Stokes equation in the z-direction:
pr[U
where T rz
, T
zz
au, _
au
Duap
r
z
aT zz
a~rT
r_z
-rpgr + r [
a
ar
are turbulent shear terms.
Assuming
zz
rz , i.e., lateral variation of the turbulent shear
az
ar
is much greater than the longitudinal variation and integrating across the
jet (invoking the Boussinesq assumption), we have
a
p a [urz r
Ko
o
o
= -_ az
u
rr)
+ 1 a
U 2r
uz auDr
dr + --f o u z rdr]
2a z
[
0 rp dr] -
fro pgrdr
+
rz r
0
0
-
f0 TIo
rz dr
20
The boundary conditions are:
u z (z,O)
=
rz dr = 0
O0
0
=
trz(Z,W)
=0
; ZFinternal
as no work is done at zero velocity gradient.
Assuming hydrostatic pressure distribution, we obtain
d
u2 2r
f
z 1
dz o
dr
=
Pa
o
g2rr
P
(2.1.2)
dr
(2.1.2)
The change in the momentum flux of the jet is due to that added by
buoyancy.
Heat Conservation:
ar
++
[rpuZT]
z[rT]
= 0
-
= Ta, it can be shown that
Noting that T(z,)
d
where
[rpurT]
r
[
o puz(T-Ta) 2r
T(r,z)
Ta
dr]
0
=
: temperature at (z,r)
: ambient temperature
Alternatively the above heat conservation equation can be formulated
as an equation of conservation of density deficiency by noting that
p
Pa
and using the equation of state in linearized form
T - Ta
dz
dz [
=
8(P -
P)
for small
Oo (Pa - P) u z 2wr dr ]
= 0
AT
;
8 constant
(2.1.3)
21
2.1.5.
The Entrainment Principle
Experiments have shown that the bell-shaped distributions for both
velocity and density deficiency can be approximated by Gaussian functions:
Letting
_r202
u z(,r)
=
u (z,o) e
-r2/x2b2
and
Pa-p(z,r)
where
2
=
[Pa-p(z,o)] e
is the turbulent Schmidt number, a measure for the relative
diffusifities of momentum and heat (or mass).
Morton, Taylor et al (1956) assumed that the entrainment flux is
related to the centerline velocity uc and 'width'
b
of the jet via a
proportional constant:
Qe
a
=
27cabu
= entrainment coefficient.
Substituting the special forms of the velocity and density deficiency
profiles into eq. 2.1.1 - 2.1.3
and carrying out the integrations, the
following set of equations is obtained for the region of established flow:
d (ub)
dz
2
d
2abu
(2.1.4)
c
b2
d (ucb)
dz
=
c
=
gX2b2A
2
(ucb2 Ap)
(2.1.5)
p
(2.1.6)
=O
The problem can be solved numerically, taking care to transfer the
conditions at the source to the beginning of the region of established
flow.
Nevertheless, there are two principal drawbacks.
in a later section, the entrainment coefficient
a
As will be shown
is some function of
the local densimetric Froude number of the jet.
22
This is indicated by
For an axisymmetric jet it varies from 0.085 for a
experimental data:
plume to 0.057 for a pure jet.
Thus the assumption that
a
is a constant
In the mathematical solution employed in this study,
is not a good one.
a better assumption is used to replace eq. 2.1.4.
For buoyant jets in deep water, the region of interest (water depth)
is large compared with the length of the zone of flow establishment Ze .
Neglecting buoyancy in the region of flow establishment, the constancy of
momentum flux yields the relationship between conditions at the source and
those at the end of the region of flow establishment.
However, in many
practical cases of interest (e.g., continental shelf), the submergence
The length of the region of flow establishment can
H/D is less than 50.
constitute a significant portion of the total water depth, and cannot be
In the theoretical solution of the
conveniently left out in the analysis.
study,
ze is derived as a function of the exit densimetric Froude number.
Special Cases:
Valuable information can be derived from eq. 2.1.4 - 2.1.6 by considering the limiting cases of a pure momentum jet and a plume a)
Momentum jet:
Setting
Fo + X
Ap = 0 in eq. 2.1.4 - 2.1.6
it can be shown that
db,
db
dz
-
2a
2
(2.1.7)
2
(2.1.8)
0o
(az
23
Hence in a momentum jet the width increases linearly with z, and
the jet angle is related to the entrainment coefficient.
Consequently
the Reynolds number defined with respect to the centerline velocity and
the width of the jet is a constant.
b) Pure plume:
F
0
In this sub-section it will be proved that at large distances from
the source, the local densimetric Froude number of all plumes approaches
an asymptotic constant value.
The approach employed here is similar to
that by Jirka and Harleman (1973) for the two-dimensional plume.
The local densimetric Froude number is defined as
Uc
F
=
/g P b
The change in the densimetric Froude number can be written as
dF
dF
=
F2
{
u du
-
dz
(Apb)
}
(2.1.9)
u
It can be derived from eq. 2.1.4 - 2.1.6
u_ u2bdbdz
gx2b22
dzudz
and
2 du
b2 u dz
that
b2
+
(2.1.10)
2 db
2u2 ,b.
dz
=
2
2abu
(2.1.11)
Subtracting eq. 2.11 from eq. 2.10 and back substituting, we have
db
2a-
2/F22.1.13)
dz
gx2b2Udz
'dz -
_ u 2 b(2a-X2/F2)
b2
(2.1.14)
24
du
u d
Substituting the expression for
and dz(Apb) into eq. (2.1.9), we
obtain
Thus if
dF
2a
dz
bF
5X 2
2
Fo
4
_
5
F2
4a
the plume will be initially accelerated to in-
crease the local densimetric Froude number; conversely, if F2 >
o
the plume will be decelerated:
5x2
4a
from the source of buoyancy.
Froude number of
F =
5X 2
4a
in both cases an asymptotic densimetric
=
4.30
is approached at large distances
In the region where the asymptotic densimetric Froude number is
approached:
db
dz
dz
6
65
(2.1.15)
5/3
Ap
=
const x z
u
=
const x z
(2.1.16)
_ 2/3
That the jet angle is approximately constant (or more correctly,
varies slowly with F) is easily shown by substituting the values of a ,
for the plume and the jet in eq. 2.1.7 and 2.1.15
Jet:
a = 0.057
Plume:
a = 0.085
db
db
0.114
d
0.104
dz
dz
It can be seen there is only a difference of less than 10% between
the jet angle for the two limiting cases.
In the mathematical formulation of the Buoyant Jet Region Solution
presented in the following section, a constant jet angle assumption is
used to replace eq. 2.1.4.
Besides being a more accurate description of
25
the physical situation, this has the further advantage that an analytical
solution is rendered possible.
2.1.6.
Mathematical Formulation
In this section the assumptions employed to solve the problem of the
bouyant jet region in shallow water will be stated:
db
a) dz =
= constant independent of the local densimetric Froude number
i.e., the spread of the standard deviation of the cross-sectional profiles is linear with z.
In the region of established flow this assump-
tion is equivalent to that of a linear jet.
b) In the region of flow establishment, a linear spread is assumed for
the development of the central core region (Fig. 2-3).
uz(z,r)
=
uo
=
u-(r-b)
r < b
2
/b2
r > b
X = spreading
coefficient
=
Ap(z,r)
AP0
2
= AP e-(rb') /2b2
r < b
r>b
The assumptions in the region of flow establishment are good only
when the exit densimetric Froude number is greater than the asymptotic
In laboratory practice laminar effects will come
value of the plume.
into play near the nozzle for extremely low densimetric Froude numbers,
and jets with small F
(Ungate, 1974).
may possess a different turbulent structure
In such cases the above stated assumptions will break
down and there is no accurate analysis possible to determine the length
26
of the region of flow establishment.
2.1.7
Mathematical Solution
An analytical solution is given in this section for the region of
established flow and the region of flow establishment.
tions are the same as used by Abraham (1963).
The basic assump-
The analytical treatment,
however, is different in two respects:
1.
The assumptions that lead to the evaluation of the length of the zone
of flow establishment is explicitly stated.
In his evaluation of ze
Abraham evaluated the buoyancy flux using Albertson's result that assumes
a constant momentum flux.
The buoyancy flux is correctly evaluated in
present solution.
2.
Two boundary conditions are invoked to couple the solution of the
region of established flow with that of the region of flow establishment:
The resulting differential equations are then explicitly solved subject to
the boundary conditions rather than using an integral approach as employed
by Abraham.
Region of established flow
In the region of established flow assumption (a) can be used along
with eq. 2.1.5 - 2.1.6 to yield an analytical solution.
1
By employing a change of variables
m3
= uc
and solving the
transformed equations, the following solution can be obtained:
u(z)
c
-
=
1
{uz
u Ap z
Ap(z)
+
ee
z
e
2
gueJZ
3X
3 3
(z
3gX2ue
{3z3
=
UC(z = ze)
Ape
=
AP0
1/3
2
2
27
(2.1.17)
-e)
z
+
2
2
(Z -Ze)
ee
ue
e
e
a
2
e e
z
where
2
Ap
2
-1/3
(2.1.18)
2pa
a
e
c
by definition
Hence uc, Ap in the region of established flow are reduced to a
It is evident that eq. 2.1.17 and eq. 2.1.18
function of z and ze.
exhibit the expected behavior of a buoyant jet.
_1
uc
Z
For z sufficiently large,
5
z T
Ap - z 3
Ze' Uc
and
Z
; this agrees with the behavior of a plume.
For
, resembling the motion of a momentum jet.
Assuming that the velocity profile is Gaussian at z = ze (density
deficiency), heat conservation gives
Heat flux at z = z (density)
this gives
2
Apu2wrdr =Ap -D2u°
=
D2Uo( l+X2)
(2.1.17a )
e e422
Also, it can be shown that
uee
z
uoD
where
[
Me
Mo
1
22
2
2
]
Me : momentum flux at z = ze(density)
M
O
: initial momentum flux
(2.1. 18a)
28
Substituting eq. 2.1.17aand eq.2.1.18a into eq. 2.1.17- 2.1.18 yields
u
D
Uo
z
(
1M
3/2
e
3(1+X2 )
+
2M
{ ( z ) 2_(
8E2F
O
D
Ap
=( +) D)([(
Apo
(2) 2
Apo
4X:
8c F 2
0
2e M
Z
0
)
} ]
1/3
(2.1.19)
D
2 ) { ()2
+ 3(+X
2
2M2
D
e
1
D
(2.1.20)
I
Determination of the Length of Flow Establishment
Referring to Fig. 2-3 for the region of flow establishment:
b' =
By similarity
(1-
Z
z/ze)
e
The momentum flux at z = z
Ze
M
= M
e
f
+
o
co (Pa -p )g2 wrdr dz
o
By invoking assumption (b) the bouyancy contribution to Me can be evaluated
as
z
fo0e
Hence
Apg 2rdr
where
2p
=
.2
3
+
e
{(
326
+
-D-_2
}
Ze
Me/Mo can be expressed as
Me
M
Ze
gX232z3
f0
4
=
_
2
0
c
[12
12
+ 4
i
A£
12
12
c = ze (density)/D
At
z
e
Apo
=
1
2
+
c3-
X2£2
c
3
]
(2.1.21)
29
eq. 2.1.20 then gives
1
M
1+
I
)
( 2E2 M
=
2
1
2 4
1
2
(2.1.22)
(2.1.22)
c
Equating the expressions for Me/Mo derived from eq. 2.1.21 and eq. 2.1.22
we have
4
F2
{c
_/
12
2+
12c
c 2
+
X2C2
c
3
} =
1+%2
2 2 2
2 z4X2
0
(2.1.23)
c
Eq. 2.1.23 describes c as a function of the exit densimetric Froude
number.
In the limiting case of a momentum jet F
=
+
1
This value is similar to that given by Albertson et al (1950).
Given F
(e and X are approximately constants) equation 2.1.23 can
Fig. (2-4) shows the value of c as a function of
be solved numerically.
F0 for X = 1.14 and
= 0.109 (these are respectively intermediate values
for the jet-plume range:plume
Splume = 0.104).
= 1.12,
jet
=
1.16,
jet = 0.114,
It can be seen c increases rapidly from zero for
Fo = 0.0 to an asymptotic value of 5.74 for F
beyond 25.0.
of interest for buoyant jet applications is 4.3 < F
<
X
The region
where
4.3 is
the asymptotic value for the densimetric Froude number of the pure plume.
2.2
The Surface Impingement Region
When the buoyant jet impinges on the free surface, the surface
pressure, documented as a surface hump, causes horizontal spreading of
the heated discharge.
Intense turbulent mixing occurs in this region
30
O
-Iw
a)
r')
4°
I
ma
w-
0)
d
0u
0
LAJ
so
CJ
N
U
N
(
o-0
I
Cfl4
u
I
O
9
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
(D U') 10n
I
t3
L.
C:
t,
-
-
LL
>- -j 0.
l
U.ZZ
U,)
a
I
lo
L1
Nn
-
lio
C
31
and a detailed analysis of the exact flow and temperature distribution
within this region is deemed impractical.
Instead a control volume ap-
proach is taken to couple the flow conditions just before and after impingement.
A definition sketch is given in fig. 2-5.
The heated flow enters as
a jet through section i and leaves the control volume at section I.
is assumed to be fully established in section i.
Let R I be the radial
position at which the free surface returns to level
.
R I is related to
the standard deviation of the incoming jet flow by R I = aobi , and
evaluated from experiments.
Flow
ao is
Let uI and hI be the velocity and depth of
the upper layer and uniform distributions over the thickness h I are
assumed.
2.2.1
Analysis of the Control Volume
Continuity:
Neglecting entrainment in the surface impingement region and invoking the Boussinesq assumption, one obtains
bi
i
= 2 a
h I uI
Heat Conservation:
Assuming the linearized equation of state and equating the inflow
and outflow heat fluxes:
B Pa
0
u Ap2wrdr
=
PT
-
u 2rdr
section
section
i
i
wbiPi(2aohIIU)TI
32
"
Ir
I
FA
1
1e
* -
SECTION I
)N
i
z
I
r
I--Rr
FIGURE (2-5) THE SURFACE IMPINGEMENT REGION
33
Invoking continuity, we have
X2
Conservation of energy:
In a conservative buoyant force field an energy potential Apgz can
be defined.
Assuming an energy loss of the form Kx(Kinetic
energy flux lin)
where KL is a head loss coefficient, conservation of energy then gives
u2i
(1 - KL) 6g
2
UI
+
2g
=
hI
g
Recapitulating the complete set of equations for the Surface Impingement Region:
biu 1
=
APi
=
P u
2- (1-A)
2g
2hiuIa 0
(2.2.1)
API
U2
Pa
2g
Va2g
(2.2.2)
Ap
I h
1-2 I
(2.2.3)
Eq. 2.2.1 - 2.2.3 can be solved iteratively with eq. 2.1.19 - 2.1.20
to find hI and the densimetric Froude numbers of the upper and lower
layers at the end of zone 2 (Fig. 2-1).
2.2.2
Limiting Cases
Insight can be gained by considering the two limiting cases of a
momentum jet and a plume:
34
A.
/
3
(2.2.4)
4(1-KL)a2O
Ih
into eq. (2.2.4)
dz=
Substituting
in eq. 2.1.1-2.1.3 gives
Setting Ap = 0
Momentum Jet:
the following equation is
obtained.
hI
1
H
2 /(-KL),2
3
3
For a momentum jet a = 0.057
-=0.114
for different values of KL and ao, we have
Evaluating hI
H
0.09
0.0994
0.113
ao =1
0.05
0.06
0.07
Lo = 1.73
hI
H
KL = 0.4
KL = 0.2
KL =
ao = 1.73 corresponds to a R I where the vertical velocity is 5% of
the centerline velocity.
B.
Assuming the asymptotic value of the local densimetric Froude
Plume:
number is reached before impinging the free surface
2
Fi
From eq. 2.1.15
512
-
4a
bi
i
= 6
5
hI )
(H
I
Substituting b i into eq. 2.2.1 - 2.2.3, we obtain
2
1
Fi
{
(1-KL)
3
bi
2
1
I ( ~
4a2
)~~~
4or
2
1+ X
1
(bi/hI)
35
For a plume X = 1.12,
a = 0.082
By iteration, we get
KL
KL = 0.2
=O
KL
= 0.4
hI
0.081
0.091
0.107
a
= 1
H
0.048
0.053
0.06
a
= 1.73
The above analysis demonstrates the weak sensitivity of hI/H to the
range of densimetric Froude numbers.
This approximately constant value
can serve as a useful starting point in the numerical solution of
eq. 2.2.1 - 2.2.3 and 2.1.19 - 2.1.20.
Lower bounds for the densimetric Froude numbers of the respective
layers after surface impingement are given for the case of the plume as
F1 = 4.12, F2 = 0.21 where subscript 1 refers to the upper layer and 2
the lower layer in the impingement zone.
In the theoretical solution K L = 0.2 is assumed for a 90° bend and
a wide range of curvature (Jirka and Harleman, 1973).
As it is experi-
h
mentally observed that
I
H
analysis.
0.1,
a
= 1 is assumed in the subsequent
0
Thus, the outer radius, section I, of the surface impingment
region is assumed to be equal to the radius of the jet at section i.
2.3
Radial Stratified Flow
In this section the basic equations that govern the flow of a
stratified two-layered system are derived and presented.
A slowly-
varying flow situation with a distinct interface is schematized as shown
in fig. (2-6).
For a two layer system with low densimetric Froude numbers
there is very weak turbulent entrainment from the lower layer into the
36
FREE SURFACE
I
1
-
m
p -p
z
I
-I
i
p
r
h2
///,/j///
..
r
BOTTOM /
FIGURE (2-6)
NTERFAC E
///BOTTOM////
RADIAL STRATIFIED FLOW
IN A TWO-LAYERED SYSTEM
37
upper layer (Ellison and Turner, 1959).
can hence be regarded as constants.
The densities of the two layers
The Navier Stokes equations are aver-
aged in the vertical direction, and the resulting equations are further
developed for the internal hydraulic jump as well as the stratified counterflow in later sections.
The steady state Navier-Stokes eq. in the radial direction in a cylindrical co-ordinate system (z,r) is
au
p( u
au
+
pu
z )=
zr
+
a
ar
p
az
q
=
(u,w) : velocity vector at (z,r)
p
=
pressure
=
turbulent shear stress
T
zr
The kinematic and dynamic boundary conditions are:
A) Kinematic Boundary Condition:
Surface
w
s
=
u a(hl+2
s
ar
Interface
ah2
Wi =
Bottom
i ar
wb = 0
(no bottom slope)
B) Dynamic Boundary Condition:
Surface
P
= 0
PE
p
Ts
s
=
(free surface)
z
au
j
T
rz
l
Interface
i
Pi
Bottom
a
Trz
au
Tb
where
H
azb
fz
rz
b
Ts
= surface shear
Ti
=
interfacial shear
Tb = bottom shear
(2.3.1)
38
Defining the average velocities of the two layers as:
Upper layer =
1l
Lower layer =
where
=
2
1
1
hl+h2
h=
h
h2
q2
1
h2
h2
h2
o
udz
udz
are flow per unit width of the respective layers
ql' q2
ql
=
Q2
2rw
q2
h
2r
Q1' Q2
are constants
= upper layer depth
h 2 = lower layer depth
Assuming hydrostatic pressure distribution; we have
upper layer =
p=
lower layer =
p= p1g h + p2 g (h2 - z)
plg (h1 + h2 - z)
P1 = density of upper layer
P 2 = density of lower layer
By continuity:
a8u
aw )
u ( Dr + r + aa
= -
therefore
au 2
3r
-u
au
ar +
Dr
auw
9z
aw
u a
au 2
auw
u2
ar
az
r
Integrating eq. 2.3.1 in the z-direction over the upper layer
39
hlh
2
2
au
h2 ar
dz +
h
f 1+
hh h
1
2 auw
h2
u dz
z
h2
I+
hl+h
u2
2
-d =
rr
h2
Pa
2
h2
th
ah
ar
r
2
au~~ ~~ara
+ (
u
hl+h2
z aZ
h
2
2
a
hl+h2
a(h1 +h 2 )
2
u dz
- u
U
2 a (hl+h2)
ar
+
-u.w.
+ 1r hlh
Usws
1 1
arh
2
a2
1
T -T.
arlTh
a
r
P
S
a
+
hl
0a
Carrying out a similar integration for the lower layer and invoking the
kinematic and dynamic boundary conditions at the points of discontinuity,
the following equations of motion for the two layers are obtained by neglecting surface shear ( Ts= 0).
Upper layer:
Q1 2
1
Dh
Lo
1
P1
r
(2=
lay1
a
3(hl+h 2)
Ti
h
+
(2.3.2)
a
Lower layer:
Q2
(2-~rr)
[-
1 h 22
ar +
2
2
1
rh2
2
ah1
-
aI
a
1
r
h2
+ p2
Dr
(Ti-Tb)
I
h2
a
(2.3.3)
2.4
The Radial Internal Hydraulic Jump
40
The internal hydraulic jump in a two-dimensional two-layered system
has previously been treated by Yih (1955), Jirka and Harleman (1973).
For
an axi-symmetric jet in shallow water, a two-layered counterflow system
consisting of a heated flow away in the upper layer and an ambient inflow
induced by jet entrainment in the lower layer is set up in the near field.
If a stable near field exists, an internal jump is always observed.
The
transition is accompanied by an energy loss and possibly turbulent entrainment at the interface.
An approximate analysis is presented in this sec-
tion to solve for the conjugate jump heights of the respective layers.
These represent two possible dynamic states for the same given momentum
flux.
A simplified asymptotic solution is also derived as a special appli-
cation to submerged discharge problems.
As a first approximation, a momentum analysis of the two layers is
carried out by neglecting shear stresses.
Because of the expanding cross-
sections of a radial system, this assumption may introduce a substantial
error in the computation of the exact conjugate jump height.
It will be
seen that this simplified analysis still gives valuable insight into the
stability of the near field.
With the above-stated assumptions, the vertically averaged equations
of motion for the radial stratified flow of an axisymmetric two-layered
system eq. 2.3.2 - 2.3.3 become
41
HEATED
FLOW AWAY
/
I
I
t -h
-
1
i
AMBIENT INFLOW
INDUCED BY
-
-
''1
h2
JET ENTRAINMENT
.h'2
or
///////.
- -
----
-
----------------//////1
//////
,,,,,,
,
r2
FIGURE (2-7)
THE RADIAL INTERNAL JUMP
. m
.
42
Q1
1 dhl
[ 2 dr
2
1
+
h
1
Q27rr
(2wfr)
[
+
dr
(2.4.1)
a
1
dh2
2
P 1 d (hl+h2 )
-1
1
rh2
[P
=
pa
2
1
dh
+
P2
2
dr
dh
-d2
(2.4.2)
dr
where Q1 , Q2 are flows in the respective layers.
Noting that
1
dhl
1
1
dr
[ --
rh]
dr
1
E.24bcmh
1
Eq. 2.4.1 becomes on simplification
Q1 2
rh1
P1
2')r
dr
Pa
Integrating from r
h(1 =
hl+h2
r
dr
from
rto
r2 assming
2
to r2
1
assuming r
2
and an average head
in the interval, we have
2h
[Ph1
Q
d(hl+h2)
=
-,rh]
g
(rl+r
)2
(h 1+h -hh )2
(2.4.3)
Pa
where h
, h
are the conjugate jump heights of the respective
layers
A similar integration for the lower layer then gives
Q
2)
2ir
2
1
1
[ rh
2
rlh2
r2h]
'2h2
=
g
(h 2 +h2)
2
2
(rl+r
2
2
2
)
P
{
(hi-hl)+
a
a
(h2-h2)}(2.4.4)
43
Defining free surface Froude numbers as:
Ql
2
,2
-(2nrlh
gh1
Q2
,2
2
(2Tn h2
2
gh 2
Equation 2.4.3-2.4.4 can then be reduced to:
Upper Layer:
rp, ,
*22
1h
21
'
p1
1
r1 +r2
a
2
(2.4.5)
2
Lower Layer:
1
*2 2
2r2
1
rrlh 2
r2h
(h2 +h2)
]
2
(rl+r2 ) P 1
p
2
P
22[-h-hl+
lh2 '2ia
P]
(h
p
a
2 -h
2
)
hi (2.4.6)
Eq. 2.4.5 and 2.4.6 constitute an approximate momentum analysis of
an internal hydraulic jump in a general two-layered system.
Most sub-
merged discharge designs, however, are characterised by small density differences and negligible free surface Froude numbers, but finite densimetric Froude numbers.
An asymptotic solution can be obtained as follows:
Rearranging eq. 2.4.5 and eq. 2.4.6 we have
21
r2 h
1
r
F1 [1[1 r 2
h
2[
2h
;
rlh 2
r2h
r2
*2
2
A[
[1+
r2hl
r 1
F
F;2 [1-
ph
1
4
[2h-
r
h
2
p rlh[1
h;
h
h;
h2 h
l
2] [1
+
r2
(1- hhi
h' +
h'
2
(1-
h
+
] [1+P 12 h h
27-)1
h2
On further algebraic manipulation we obtain
hl+h(2.4.
h2[1h----1
2+h (4F12[ -1 +
h2
r22
[- AP
p
l
h2] -
hi]
"1]
rlhl
["+r2
t"+h I
rl
h2
44
2
4F~
[+rhhI
+ (1-1
4*'2
2
h
h2
rh
1
2
h2
r2h2
[ 1+
] [+ IF
- Ti 1
rlh 2
2
hh2 hg
A)
(1 -
h
2 1
1
(1-
+1
(2.4.8)
P
p
It can be derived from eq. (2.4.7)
that
h"-hl
A h2
hj-h1
hi-hl
-
1
r 2h
*2
4F1
where
r-
+
iL1
A
r 2h;
rlhI
[1 +
rlhl
(2.4.9)
]
h
h2
2 ][1
ri
-
h1
(1 -
A)
p
Two alternative expressions can be derived from eq. (2.4.8)
B hI
-h-h
(1 -
- (1.)
p
a)
(2.4.10)
and
hl-h
h2-h2
h{-h
(hi-h 2 ) (1 - AP)
(2.4.11)
(1- AP)B h I
-(h'-h )
45
*2
4F2
2
where
r2h2
rlh2
r2 h
[- 1+
h2
r h2 ]
r1h
h1
h!
2
r2
+ 22I]
[ 1 + r]--[1
1
2
Subtracting eq. 2.4.9 from eq. 2.4.10
we get
*2 r2h;
h
1 [ rh
h2
*2 2h
APrl
p
r1
1]
2
h2
2
h2
h1
'-h
(1+
(1+
r1 h1
4F2
h2
r2 hh'
[i
[_1-ApP ]
h1 -hl
h1
(h;-hl)
rh2
(1+
r
rlh2
-
h2
(2.4.12)
Ap
+
p
Subtracting eq. 2.4.9 from eq. 2.4.11 we obtain an independent eq.
*2
4F 1
[1
r h'
p
r1 h1
r2h;
rlh 1
r2
h'
h
=
1
-
(1+-)(1+
) (1-h~1)
r1 h1
(2.4.13)
r2h2
2
r1h2
r1
_rh2 (
4F*2
2
In the limit when
Ap+ 0,
F1 , F2
r2 h
(1-2
rh
12
+
0
h
h2
1+ (h2(1h2
(
2
Li'
2h
2
rlh2
rlh2 (+rri
2h
h2)
h2
(1-
h2
2)
2
46
*2
and
F2
finite
1
p
P
*2
F2
2
2
Ap
P
Eq. 2.4.12 and 2.4.13 reduce to
4F 1
r 2h
rl]
+ hi ] [
hl
r2 hi
[ r-l
- 1=
hi
1
[ 1+
4F2
r11
[1
(1+
h1
1 h1
h1
(1-
hI
2
1-2
rh 2 hi
r2
r 2h2
(1+ -)(1+
r
rlh 2
h )4F
(1-
2
1
hi
hi
h2
h1
r2h
r 2h 2
2lh1
rlh
1 2
11
h+2
r2
)(1
(2.4.14)
1h
1)
-4.15)
(2(2.4.15)
-
2(
4F 2 (1-
r2h2
r1 h2 )
rl'h2
r2h;
16 F2 F2 [
1
2
1-
r2hI
2
rh 2
rlh1
These 2 equations describe an asymptotic solution to the radial internal jump problem.
Given the densimetric Froude numbers of the respec-
tive layers, a numerical solution can be determined by relating the jump
length (r2 -r1 ) to the jump height (h{-hl).
The radial free surface hydraulic jump has been studied by
Sadler et al (1963).
The momentum equation assuming a finite jump length
for this case is
T
r·y 1
2ifgry =
2
r2Y2
2
+
o
2rgr2 y 2
This can be gotten from eq. 2.4.4 by setting Pl=0
P2 = Pa
Q2 =Q
47
In the free surface case an experimentally determined coefficient of 4 is
found for the ratio of the jump length to the jump height.
The theoretical
investigations of a radial two-layered system in the next section, however,
shows a drastic difference in its behavior as compared to the free surface
counterpart.
No attempt is hence made in using this coefficient.
Valuable insight can be obtained by treating the case of negligible
jump length, i.e., r2 = rl.
A main concern of this study is to determine
the criterion of the near field stability, that is, the locus of (Fo, H/D)
that characterises a stable-unstable near field transition.
In view of the
exclusion of shear stress in the momentum equations and the unknown relationship between jump length and jump height, it is judged that the solution of the radial internal jump problem in the context of a negligible
jump length should furnish adequate information concerning the existence
of a jump.
By setting r2 = r1 in eqs. 2.4.14-2.4.15 we obtain
2 1
1
h
1
2
(1+
2F
)
h hi
1
-2F
hI
(1
(h1
h
]
1
h2
[ h
2
(2.4.16)
h
22
hI
2
2
2
<+
h2
1
h'
1
hh'
1 12 1
[h(l)
h1
'
2F2
2
h2
]=
4 F
F2
2
1
(2.4.17)
The above equations are the same solution obtained for a two dimensional internal jump by Jirka and Harleman (1973).
Combining the two
eqs. we get
h'
2
+F
2
2
4
h
F
+
4+1)F2
h'
h 1
F2
(2.4.18)
2
18
48
From eq. (2.4.17), we have
'i h'
h'
2
=
1
h
-1)
11(-l + 1)
1
~2
h
h
2
- 2 F2
(2.4.19)
1+
~-2
1
F2 hl (1+h
2
II
I1
h2 h'
Substituting
(--
the value of
2
+ 1)
in eq. 2.4.18 into eq. 2.4.19 the
h2 h2
following relationship is obtained
h'
1
h2
or
h
+
-
h'
h
h1
h2
(2.4.20)
= h2 + h
Under such limiting conditions the total water depth remains unchanged.
Substituting the value of h
in terms of h
into eq. 2.4.17 we have the
single asymptotic form:
hl
Ihi
1
32
-2
1
h h
2
4 ] [hj-(h + 1) - 2 F ]=
h' h;
h-(hf + 1)2F
2 (2.4.21)
which has been given by Jirka and Harleman (1973).
In the limiting case of a critical section h
=h
eq. 2.4.21 reduces
to
2
1
F
2
2
+ F2
(2.4.22)
1
Eq. 2.4.22 can be viewed as a defining statement of a critical section in
a two-layered system.
For some combinations of F
solution.
F2 ,
jh , eq. 2.4.21 does not yield a
h2
This indicates a hydrodynamically unstable situation: even the
longest waves at the interface amplify in magnitude; the excess kinetic
energy is dissipated by turbulent diffusion over the near field region,
leading to heat re-entrainment into the jet.
49
The implicitform of eq. 2.4.22 is plotted for a typical stable case
and a typical unstable case (Fig. 2-8).
In the case of a stable near
field, two roots are always detected, the root with the larger value being
disregarded by energy considerations.
Numerical experience have shown
that solving eq. 2.4.16 - 2.4.17 always gives the correct conjugate jump
height.
50
z
z
o
z0
0I0
0
0,w
Z
:)-
0
co
w
II
c
I
w
!
I
I
w 0 OC
I
-
I
I
51
-
(
c
z
0
-J
0
V)
0
0
>-ULL
OD
to
4%
oz
1AW
w
crj
(D
-a
w (D
Z
m
OD z
Co
v~
ao
(0
I
1
O' 0
I
I
I
O0 0 0000
1
I
I
.I .
- _.
I
1
I
I
I
I
I
I
I
_a _
00
I
CJ
I
-
I
(
I
52
2.5
Stratified Counterflow Region
When an unstable near field is present, there is heat re-entrainment
of the jet, and a critical section is established near the discharge (at
the critical section there is a sharp change in the interface)(Fig. 2-9).
The subsequent fluid motion is described by a stratified counter-flow
system.
In the following sections the basic mechanics of the flow is dis-
cussed with respect to a far field condition similar to that in the experimental set up of this study (no imposed physical boundaries; ambient fluid
at rest).
In the prototype heat loss effects may govern the far field
boundary condition.
The fundamental behavior of the governing equations
are presented and contrasted with the two-dimensional counterpart.
Finally
the predictions of the near field dilution for unstable jets are given.
2.5.1
The Momentum Equation for Axi-symmetric Stratified Flow
Noting that P 2 =
P
= Pa
a
AP > 0, eq. 2.3.1-2.3.2 can be
-
simplified to give:
dh
*2
F1
1
[jh-
*2 dh2
F2 [
+
+
dr
h1
1
r
h2
2
d(h +h)
dr
dr
(1 -
d(h1 +h2 )
dr
Under the limiting conditions Ap
(TT
-
-
+
0 , F1 , F2
P dr
*2
(2.5.1)
gh
p dh
d2
r
T
+pgh
(2.5.2)
pgh2
*2
+
0.
It was shown
in a previous section that the total water depth is a constant
d(hl+h2 )
dr~+h
dr
=
0
(2.5.3)
53
CRITICAL
SECTION
HEATED
)
I
FLOW AWAY
AMBIENT
INFLOW
,2
4
I
7/7777
·
4
.
§'7~i~LL~/Y7/77
-
-
-
-
-
-
-
-
-
-
-
-
"/1////171/
rc
FIGURE (2-9)
STRATIFIED COUNTER FLOW SYSTEM
IN AN UNSTABLE NEAR FIELD
-
-
-
54
Substituting eq. (2.5.3) in eq. 2.5.1-2.5.2 and rearranging, one obtains
the following expression governing the radial variation of the interface
dh2
*2
2
F2r ]
d2 [o _ F*
_
1
dr
F
Remembering
=
2 A
=
2 h2
F
*2 h1
r
-
F
2
- F1
2
(Ti-T)
Ti
ib
+ pgh
-
Pgh2
1
, we get
/
1
2 h2
dh2
F2 r
dr
2
F
1
h1
F T. F p+
+
r+
pgh 1
(ri-Tb)
b(2.5.4)
pgh2
Ap
2
2
1
- F2
1 -F
=
At the critical section, the sharp change in the interface can be
dh2
described mathematically by
dr + o , giving again the critical condition
F
2
1
+ F
2
2
= 1.
The interfacial and bottom shear are related to the velocities in
the two layers in the usual quadratic friction relationships:
Til
'f
2
(Ul-u2)2
-8
(
'8
-
{Ti
8
2
Q2
Q,
Pfi
i
Q8
1F2rQ2
1
(2rr)2
h1
2
Prefixing the known directions of our counterflow system, one obtains
i
Pfi
Q1
=8-
2 r-
)
8
Similarly
o=
Pfo
8
2
Q2 h1
1
2
Q2 2
2wrr
2-Q2 Ql
2
Q
>
>
Q
1
h2
Substituting the expressions for Ti and T o into eq. (2.5.4) we obtain
the radial variation of the interface in the stratified counterflow
system:
2h
LL
2 h
"1
L
1 r
8
F 2 2 2 1 +i
dh2
dr
F2r
2
I~
1
h2
2(1+
1
~~lL-~ n
-1 +
Qh2
o2
8
2
(2.5.5)
1 - F1
1
2 F2
2
2
55
2.5.2
Behavior of the counterflow system
The entire physical situation can be described by eq. 2.5.5 subject
to a far field boundary condition which will be discussed later.
2
2
1
2
critical section r = r , the critical flow condition F 1 + F2
c
At the
= 1 has to
be satisfied.
In the sequel, the essential features of eq. 2.5.5 are discussed by
considering the special case of equal counterflow
represents the case of high dilutions.
Q1 = Q2
which indeed
For this case the problem can be
shown to be dependent on a single dimensionless parameter.
2
Defining F
Q
(2irr H)
c
as
g
2
constant densimetric Froude number based
the total water depth
H
the problem can be cast in dimensionless form:
(l-H
) 2
Rc 2
(I- H2 ) + o2
F2
dH
dH 2
2
dFH
H2
1- F2
1 -F
s.t. at the critical section R
R
(-H
H2
2
Rc2
R c2
dR
8
F2
F
3
R
H3
c
H2 satisfies
2
1
1
H (1-H2 )3
2
where H2 = h2 /H
R = r/H
3
H
H2
R
= rc/H
The radius rc is to be determined from experimental results.
For a given FH , H 2 (r = rc) can be found by solving the critical
flow condition.
One H2c is known, eq. 2.5.6 can then be solved numerically
56
as an initial-value problem, using numerical methods, such as a fourth
order Runga-Kutta
scheme .
Since the derivative
dH2
dR
is infinite at the
starting point, the first few points of the interface is found by inverting
the derivative and solving the inverse problem with
dH
; after marching
a few steps out, the formal derivative can be used again.
The change in the interface for different values of FH and different
friction coefficients is illustrated in fig. (2-10).
Two remarks can be
inferred:
1) For small R and in particular, R - 0(1), which is experimentally obc
servedthe
c
inclusion of frictional effects has a negligible effect on the
shape of the interface.
In such cases the radial inertial effects pre-
dominate, and a frictionless flow situation can be adequately assumed.
2) The interface always approaches an asymptotic value horizontally.
value increases as FH increases.
The
In the limit as FH approaches 0.25, the
interface attains a maximum asymptotic value of 0.5 in the far field.
These behavior can be readily explained by studying eq. 2.5.5 in
detail.
For R c ~
0(1) and H 2 finite the numerator of the derivative
approaches zero as r +
.
Hence H2 attains a constant value for large r.
Insight into the mechanics of the flow can be gained by contrasting
eq. 2.5.5 with a radial free surface inward flow and a two-dimensional
stratified counter-flow system.
Sadler et al (1963) have derived the
free surface curve for frictionless radial inward flow to be
dy
dR
F2
1 - F2
Y
R
where F = free surface Froude number
y = water depth
(2.5.7)
57
a,
al
w
U
Z
z
r
I-
0L.
L
O
-i
vJ
0
In
0
d
(D
0
I)
0
dNo
0
2:
0
N
0
-0
O~~~(
58
This can be attained from the more general eq. 2.5.5 by setting At=
O
f. = f
= 0 and
= 1
P
F 1 = 0.
It can be seen from eq. 2.5.7 that for subcritical flow (F < 1) the
water depth is always increasing with
dy=
dR
0
is asymptotically approached
in the far field.
Jirka (1973) treated the two-dimensional counterpart of the present
problem.
For r
+
f
namely
dh2
=8
eq. 2.5.5 reduced to this two-dimensional case,
o
2
2
fi
8
dx
2
1 (12
1 h
2 H
Qrh2 ) h 2
5.8)
2
where Qr = Q1/Q2
Again eq. 2.5.8 can be obtained directly from eq. 2.5.5 by neglecting the radial components.
In fact, equation 2.5.5 can be made to exhibit
a two-dimensional behavior by artificially setting Rc very large, thus
destroying the radial dependence of the equation.
As illustrated in
fig. (2-11), in these cases a second critical section is always found by
marching out the solution.
The interfacial height at this second critical
section is approximately conjugate to the starting point.
The physical
implication is that in subcritical flow roughness effects always tend to
raise the interface; however, because of the physical constraint imposed
by the free surface, a critical section has to be formed some distance
from the starting point.
In a radial stratified counterflow system with Rc - 0(1), however,
the radial expansion allows one more degree of freedom; this stabilizes
the flow and a second critical section is not formed near the starting
point.
For the range of interest, 0.4 < R
< 2 strong self-similarity is
59
0Z
w
U
t-
0
._!
(.3
zZw
0
0
z
0 C
O_
Od
F-
zo
0<
z
0
W
-
o Ir--
d
C.)
z
3
a LL
tn,.
a-::)
0
00C
03e
I
w
cD
Q
(.
00
Ll-
Ccj
0
('4
O
-C
3:
,_0
60
found in the behavior of eq. 2.5.5.
All the information can be summarized
by plotting h 2 against r/rc (Fig. 2-12).
In the general case of non-equal counterflow the problem can be
2
Q2
2
shown to be dependent on F 2 H and Qf where F2 H = (2rH)
The shape of the interface as a function of Q
is illustrated
rQ 2
in fig.
2.5.3
(2-13).
Critical Flow in a two layered system:
Since the critical flow condition is vital to the understanding of
many stratified problems, and is very much related
to the prediction of
dilution in this study, a short discussion is deemed appropriate.
In open channel flow, as well as in two-layered systems, a critical
section is often formed by an imposed control such as at a free overfall,
sudden expansion from confinement into infinite space, etc.
It has an
implication on flow geometry, namely - a sharp change in the interface
position.
For the case of equal counterflow, the same governing condition
can be derived from an independent energy principle (Appendix D).
With
respect to submerged buoyant discharges and other stratified flow problems
the critical condition has the further implication of limiting the exchange flow.
Consider the general case of a counterflow system:
At the critical section:
2
1
F 1 +F
or
2
F2H [
Qr
(-H
)3
2
+
1
H2
]
2
2
=
=
1
1
(2.5.9)
61
0
z0
0I--:
0-
-
9
*Ib
031
2::
O
C z
Z'HZ
N
tr
O
to:
cN
II
0II
o.L.,
II
L,
LLU
zS0 0
d
.L2:
I
a_,0
CQ
C"L~~~~~L.,I
CK
LI.
I
'1
L
d
I
k
L
I
II
;I-
d
I
rl)
d
C_
d
I
d
62
;
LO
In
' O
11-
VI
00
U-
O
L
cr
w
LL LL
zo
Qo
CI-
<o
L.
IO%%
<0
tO
cM
L)3
Cd
o
tO)
CN
d
-
-
6
63
Q2
where
2
2r H
F2
Qr
Q1/Q2
Fig. 2-14 shows the variation of H2c as a function of F2H for different
values of Qr
exchange
f'low
For a given ratio of flows in the two layers, Q a maximum
Q1 + Q 2
corresponds to a maximum FH.
By rewriting eq.
eq. 2.5.9 as
2H
H2 H3
3
(2.5.10)
QrH2 + (1-H )3
dF2
and setting the derivative
H
dH 2
H2
to zero we have
1
(2.5.11)
l+Qr
Substituting eq. 2.5.11 into eq. 2.5.10 we obtain
2
=
2H
-
+1
(2.5.12)
[1+ Q1 /2 ]
r
Hence for a given Qr we can compute the value of F2H that will give the
maximum exchange flow.
In the special case of an equal counterflow Q r=l
F2H = 0.25 is the limiting condition when a maximum exchange
flow is created.
In a two-dimensional two-layered system friction effects tend to
oppose a.condition of maximum exchange flow.
The radial expansion
of the
flow in the three dimensional case (in the absence of physical boundaries),
however, enhance the formation of such a condition at the critical section.
64
OK
O
0O
I~
0
)
o,
lL
o
H
0O
zO
I.,..
O
z
I
Ll
O
t.
L
()
I
-C
IT
w
o
laJ
rs
-J
I-
0
.(
0
00
CJ
(~Q
I
C-
0 To
65
2.5.4
Behavior of Flow at large distances
Fig. 2-10, 2-12, shows that at 'large distances' (r
1OH) from the
jet discharge, an asymptotic behavior of the interface is approached.
In
the absence of any physical boundaries and ambient currents in the far
field, flow is postulated at minimum energy dissipation.
The rate of energy dissipation, or work done against dissipative
forces, can be expressed as:
fi
Ediss
l3
3
P8 (u1 + u2 )
fo 3
+ P
u
2
Q1
2rh
1
Q2
2
1
2wrh 2
Assuming h1 + h 2 = constant,
dEdiss
dh2
= 0
gives
f
[Qrh2 + h]
For the case of equal counterflow
hi 4
2
2
fi
4
hl
f.
h2 4
2
Qrh2
Q
-
h]
= 1
=O
(2.5.13)
this reduces to
hl 4
(0j-))i
=
(2.5.14)
Since the interfacial shear is always about 4 times that of bottom
shear
gives
(fi
h1
0.5 f
h2
H
H
h2 /H to fi/fo.
(2u)3
8u3 ), a limiting
approximation
of f/f
+
o
Fig. 2-15 illustrates the weak sensitivity of
66
w
z
H
Z
_0
ILLii o
-
o
z
I°
ZZ
0Z
C-
w
O~~D~
C\i
0
I
O
I
co
I
I
r(D
t
C--
I
I_
C
:
(D
LL.
67
In the prototype far field (r >> H) the boundary condition may be
In view of the small areal extent within
determined by heat loss effects.
which asymptotic behavior of the interface is established, the boundary
condition presented in this section is judged to be independent of heat
loss in the far field.
2.6
Summary of Theoretical Framework
The coupling of the theory outlined for the four regions to give the
near field dilution is described in the following sections.
2.6.1
Definition of the Near Field Dilution
The near field dilution S is defined volumetrically as the ratio of
the flow away in the upper layer to the initial jet discharge flow,
S =
Q/Qo
.
In the absence of heat losses, heat conservation implies this
definition is equivalent to S =
A
, where
AT
2.6.2
AT
= temperature rise above ambient in the near field
AT
= discharge temperature rise above ambient
Stable Near Field Dilution
For a given (Fo, H/D) the velocity and the upper layer thickness in
the surface impingement region can be obtained by solving eq. 2.2.1-2.2.3
in conjunction with eq. 2.1.19-2.1.20.
By visual observation, confirmed
by temperature data, it is found that the internal jump occurs at
rj = 0.57 H from the jet axis.
F1 , F2 and hl/h2 can then be computed and
used as input to the internal jump equations.
Existence of a conjugate
68
height implies a stable near field.
It can be inferred from the two-dimensional buoyant jet experiments
done by Jirka and Harleman (1973) that the ratio of the jump length to
jump height is approximately 4.
It is expected that this number is smaller
for a three dimensional buoyant jet.
Unfortunately, the arrangement of the
temperature probes in the near field is not dense enough to resolve the
shape of the jump interface from temperature data.
A zero jump length is
assumed as a first approximation in the theoretical solution.
This is
chosen in light of the stability analysis, with the main purpose of evaluating the near field stability rather than the exact shape of the internal
jump region.
For submergence (H/D) less than the length of the zone of flow establishment, the theory outlined in sec. 2.1 is not directly applicable.
A
simplified analysis based on the assumption of a momentum jet is substituted as an approximation in this range (Appendix C).
If a stable near field exists, the dilution is given by the solution
of the surface impingement region.
A different theory for the prediction
of near-field dilution is posed in the next section for the case of an
unstable near field.
The prediction of the near field stability is shown in fig. 2-16
along with the near field dilutions.
For H/D > 6.0 the stability transi-
tion can be described by the criterion
F
(2.6.1)
= 4.4 H/D
In view of the assumptions
embodied
in the analytical
framework,
the
stability criterion should be interpreted as a narrow band rather than a
69
REGION
100
OF
H/D
10
I
I
I
I
I
I I!
I
I
I
I II
I
100
10
F0 -Fosg aUoD
p
FIGURE (2-16)
GENERAL THEORETICAL SOLUTION OF
NEAR FIELD DILUTION
I
I
I
I
IIIl
70
single line delimiting the stable region on the graph from the unstable
The 'transition' from a point in the stable region to one in the
region.
unstable region is continuous in nature, as exemplified by the weak instability (submerged jump) observed (Ch. 3).
The same statements apply to
the two-dimensional case (Jirka and Harleman, 1973).
For low submergences (H/D < 5), the stability criterion is determined
This is due to the fact a different
by a line with a different slope.
model is assumed for the zone of flow establishment.
Fig. 2-17 illustrates the sensitivity of the stability criterion to
the assumed location of the internal jump at r.
As this is well estab-
lished from experimental data, this sensitivity should not have an important effect of the overall prediction.
2.6.3
Unstable Near Field Dilution
Based on the theoretical discussions presented in sec. 2.5, two
assumptions are made:
1) the radial variation of the interface can be described by a
frictionless flow situation.
2) at large distances from the jet, bottom shear is negligible
compared with interfacial shear.
2.6.4
Equal Counterflow
For the case of high dilutions, an equal counterflow system can be
= 0.5; given the behavior
assumed: the far field boundary condition is
H
of the interface,
a limiting
condition
of FH = 0.25 has to be established
71
,)rj
100
fr RilH
-
Ri 0.57
R1i
0.!
j 0.60
H/D
IO
I
I
I
I i I
I
10
II
I
I
IilI
I
I
100
Fo
FIGURE (2-17)
SENSITIVITY OF STABILITY TRANSITION
TO THE LOCATION OF THE JUMP TOE
I
I
I
1111
72
at the critical section r
large distances.
in order to match the boundary condition at
This has the physical implication that a maximum exchange
By definition
flow is generated in the counterflow system.
F2
64R (H/D)5
2
3
APgH
F 2 = - Q)
H
2
P
R
c
=r /H
c
c
The solution for high dilutions is given by the limiting condition
2
F
H
2
(0.25)
= 1/16.
4 R2 (H/D)5
1/3
C
S
i.e.
R
2.6.5
c
=
[
(2.6.2)
]
F2
0
is the second experimentally determined coefficient.
Non-equal Counterflow
For low dilutions the equal counterflow approximation is not valid
and the general case of non-equal counterflow has to be considered.
The formal approach is to assume a starting value for the dilution,
solve the initial value problem defined by eq. 2.5.5 iteratively until the
asymptotic value of h 2 in the far field matches with that obtained by
solving the far field boundary condition.
involved is deemed not necessary.
The large numerical efforts
Instead a concept derived in the equal
counterflow case is postulated to carry over to the non-equal counterflow
case: a condition of maximum exchange flow has to be created.
By definition
F2
2H
2S(S-1)2 F2
=
2
64R
c
(H/D)5
(2.6.3)
73
Combining eq. 2.6.3 with eq. 2.5.12 and noting that Q
=
S
the
ddilution
nfor
unstable
erically.
buoyant
jets
can
be
solved
near field dilution for unstable buoyant jets can be solved numerically.
74
III.
Experimental Investigation
A series of experiments were conducted to test the behavior of the
axisymmetric buoyant jet in stagnant ambient water.
In an experimental
basin of limited extent boundary effects will influence the stratified
flow pattern in the far field.
In order to minimize these effects a plane
of symmetry was assumed at one basin wall and a half jet in lieu of the
This has the additional advantage of being able
full round jet was used.
to visually observe the physical phenomenon through the water and the plane
of symmetry.
3.1.
The Experimental Setup
The experiments were carried out in a 37' x 18' x 1' hydraulic model
basin.
Fig. 3-1 illustrates the general experimental setup.
To ensure
good heat insulation, the bottom of the model basin was covered with 1"
thick styrofoam material.
A plastic liner was laid on top of the insula-
tion material to prevent any possible leakage of water.
layer of 1" thick styrofoam and 1
An additional
" thick concrete blocks formed a false
floor.
Near one wall of the basin a partition was constructed along the
whole length of the model.
the partition.
This created a 16' x 34' area on one side of
In order to visualize the flow pattern of the jet,the cen-
ter portion of the partition was constructed of two 6' x 10" plexiglass
pieces (
1" thick).
The rest of the partition was made from 14" high
plywood sheets and styrofoam material, both of which were braced and weighted by concrete blocks.
axi-symmetric jet.
The partition formed a plane of symmetry of the
75
14"HIGH PLYWOOD
rs·
-WilI
mnumLtnIAlKm
C
I
I
.ToA
u
-Uurrui
Ull
MATTING IN FRONT
4 SCREWJACKS
FIGURE (3-1a)
PLAN VIEW OF EXPERIMENTAL SET-UP
76
THERMISTOR PROBES
MOUNTED AT THE
SAME LEVEL
TYGON TUBING
TO HEAD TANK,
DISCHARGE
TO
)
BYPASS AND
FLOWMETER
|WOODEN
SUPPORTING
STEEL ANGLES
DISCHARGE"'"
PARTITION
/
PLEXI-GLASS
FLOW
ADJUSTABLE
PATO
INJECTON.
I 5/8CONCRETE BLOCKS
BASIN
I
SWALLRO FOA
''/
A////A
// ./ /
Z/ /
STYROFOAM; ,
HALF-JET
INSULATION
OPENING
FIGURE ( 3-
b)
I
I
/
/
77
+temperature
scanner
Fig. 3-1(c) Experimental Set-up
78
An existing circulating water system capable of generating currents
This ensured a
across the model was used to mix the water in the basin.
uniform ambient water temperature before the experiment starts.
Two
4" x 14.5' diffuser pipe manifolds were installed in two 1.3' wide channels
at either end of the basin.
The two pipe manifolds were connected by 3"
PVC piping to a flow meter system.
(25 HP, 500 GPM).
Flow is generated by a large pump
The lateral uniformity of the crossflow was improved by
horsehair matting and vertical slotted weirs at the basin ends.
The flow injection device for the half-jet is a rectangular plexiFlow enters
glass box composed of two parts, as illustrated in fig. 3-2.
the box at one end and exits upwards through a semi-circular hole.
Fig.
The other part consisted of a
3-2a illustrates the core part of the box.
glass plate of the same thickness as the upper face of the central core
Different pieces of semi-
with a semi-circular hole cut in fig. 3-2b.
circular plexiglass with the desired semi-circular opening (0.25", 0.5",
1") cut at the center can then be fitted onto the glass plate.
A half-jet
of a desired diameter is formed by fitting the appropriate glass plate onto the core part and sealed with construction sealant.
A
" x 6" slot is
The injection device was
cut off the center portion of the partition.
then sealed onto the plexiglass wall by fitting it inside the slot and
aligning the dividing line E-E of the box with the inner edge of the plexiglass wall.
The device was then installed in place such that the upper
face of the box is level with the floor.
obtained by changing the plexiglass piece.
Jets of different diameters are
To avoid flow separation the
exit section of the half jet was rounded off smoothly.
Hot water obtained from a heat exchanger flows to a discharging
79
e
-6
GLASS
IC
E
COPPER FITTING
PLEXI-GLASS
WALL
COPPER FITTING
a TYGON TUBING
(
gm~\
xX
+
L
N
: - DISCHARGE
3
-
I
r-
FLOW
I
4
SECTION D-D'
SECTION C-C'
FIGURE (3-2) THE FLOW INJECTION
DEVICE
80
Fig. 3-2(c) The Flow Injection Device
81
piping system that consists of a
bypass and a connection to the flow in-
jection device via a flexible tygon tubing and copper fittings.
Depending
on the amount of flow needed, two types of flowmeters were used to monitor
the flow.
used.
For flows higher than 0.5 GPM, a calibrated Brooks rotameter is
A different type of rotameter (Brooks, Model 1560) was used to mon-
itor flows below 0.5 GPM accurately.
Forty-four Yellow Springs therimistor probes (Series 701, Time Constant = 9.0 sec., accuracy 0.30 F) for temperature measurement were set up
and mounted at the same horizontal level on a wooden platform supported on
four screw jacks.
The probes were identically mounted on four different
radial lines, as shown in fig. 3-3.
Six additional probes were used to
monitor the discharge and ambient water temperature at fixed positions.
Temperature readings were recorded by an electronic scanner and printed on
paper to the nearest 0.01 F.
By turning the screw jacks manually, the
elevation of the wooden platform can be adjusted.
Thus through the move-
ment of the wooden platform vertical temperature profiles can be taken.
Temperature data was punched on cards and processed by a data reduction computer programme that prints out the experimental run parameters
and the temperature excess along the radial lines for different vertical
positions (see Appendix E).
3.2.
Experimental Procedure
Before the start of each run the circulating water system was oper-
ated to mix the water in the basin.
Hot water was allowed to flow through
the bypass at the desired rate until a steady desired hot water temperature was attained.
The depth of the water in the basin was measured by
82
I
SECTION
Iv|
|
V^
Y
aN
aV
%
1-1
PROBE LAYOUT ALONG A RADIAL
V
Y
V
r%
%
%
u
V
1
LINE
-
1
row
13'
.L
FIGURE (3-3)
SCHEMATIC OF TEMPERATURE
SET-UP
A
P.4
PROBE
83
taking readings with a point gauge.
When the temperature scanner indicated a uniform ambient temperature,
the bypass was turned off and the jet discharge was initialized.
Shortly
after the experiment started, dye was injected to observe the flow pattern.
The first scan of the surface temperatures was started when the dye front
had gone past a substantial area.
After two or three surface scans had
been taken, the wooden platform was then lowered to record vertical temperature profiles.
The experiment was stopped shortly after the dye cloud
had reached the basin boundaries.
majority of runs.
This took about 20 minutes for the
Since the response time of the thermistor probes is
9 sec., 15 sec. was allowed to elapse after each adjustment of the platform
before starting the scan.
To ensure that some kind of quasi-steady state situation was reached
in the experiment, a surface scan was always taken at the end of the experiment.
In all the runs the temperature recordings of the last surface
scan in the near field were very close to those of the first few initial
scans.
As a confirming check, a particular run was carried out for as
long as an hour.
Fig. 3-4 illustrates that the near field temperature re-
duction remains fairly stable with time.
As no
suction device had been installed to withdraw the basin water,
the water depth was increasing during the course of the experiment.
Due to
the large size of the basin, the maximum and average relative deviation in
water depth was only 0.04 and 0.01 respectively for the range of water
depths and flow rates used in the set of experiments performed.
84
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85
3.3.
Experimental Program
Experiments were conducted for a sufficiently wide range of densi-
metric Froude numbers and submergence in order to cover the stable-unstable
transition region.
Runs were made in the highly unstable region to obtain
experimental comparison with the theoretical prediction of near field dilution for unstable jets.
The summary of run parameters and observed near field dilution for
the experiments performed in this study is presented in Table 3-1.
The
near field dilution corresponds to the temperature recordings of the thermistor probes at the nearest radial position (2" from the jet axis).
3.4.
Experimental Observation
Dye injections were used to visually observe the flow pattern of the
jets.
However, due to the oblique angle of observation it was difficult
to obtain good quality photographs of the cross-sectional flow profile
through the water and the plexiglass partition.
A turbulent jet is always observed for the range of the Reynold's
numbers tested.
The erratic, eddying motion of the fluid particles ac-
company the linear spread of the jet.
As the jet impinges on the free
surface, a surface boil is observed, which fluctuates in intensity, creating a disturbance that generates easily observable circular wave fronts on
the free surface.
As outlined in Ch. 2, the stability of the near field depends on two
dimensionless parameters, the submergence of the jet H/D and the discharge
densimetric Froude number F
.
For high submergence and low Froude num-
bers, a weak surface boil is observed, followed by a jet like horizontal
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89
spreading, usually accompanied by a weak jump.
As the submergence is de-
creased and (or) the Froude number is increased while still maintaining
near field conditions, the near field structure is even more clearly observed, namely a thin upper layer of approximately 1/10 of the total water
depth in the surface impingement region is found.
This thin layer spreads
out horizontally with no apparent change in thickness, and an internal
jump is always observed at a radial distance of approximately 0.6 H.
As H/D is decreased further and (or) Fo is increased, a weak instability is observed in the near field.
This is characterised by a thicken-
ing of the upper layer in the near field, followed by an internal jump possessing a conjugate depth that touches the bottom (submerged jump).
re-entrainment of the upper layer water is observed.
Weak
The region of insta-
bility extends some distance off the jet axis, and a critical section is
observed at the end of the field of instability.
For sufficiently high Fo and (or) low H/D an instantaneously unstable
near field is observed.
Intense re-entrainment occurs and the linear
spread of the jet is no longer visible.
The region of instability is con-
centrated near the jet axis, with the establishment of a critical section
at some distance from the jet.
The intense instability creates a strong
counterflow system which results in a critical section close to the jet.
The observations are schematized in Fig. 3-5.
Fig. 3-6 shows temperature transects for a typical case of each of
the three cases mentioned above.
The normalized temperature rise
AT
AT
plotted beside the location of each thermistor probe.
Radial symmetry of the dye pattern was not obtained in all runs.
For runs with an unstable near field, reasonable symmetry was observed.
is
90
szJ
ELD
CRITICAL SECTION
MERGED
P
CRITICAL SECTION
TABLE
R FIELD
FIGURE (3-5)
OBSERVED
NEAR FIELD STABILITY
91
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94
For runs with a stable near field, protruded fronts near the partition and
normal to it are frequently observed, with a slight dent in a narrow portion of the circumference, as illustrated in fig. 3-7.
Possible explana-
tions for this phenomenon are: the presence of the basin boundary has the
effect of creating a recirculation into the near field, causing the observed dent, fig. 3-8.
Constrained by the model geometry, the exit section
of the injection device (0.5" long) is not large compared with the jet
diameter.
The exit flow may have a stronger component in the forward
direction (
= 90°), again creating a weak recirculation into the near
field for a narrow portion of the circumference.
In every case the temperature rise of the four radial lines for different vertical positions delimits very distinctly the three cases of a
stable, weakly stable (submerged jump) or an unstable near field.
The effect of the partition wall, which was located at one jet symmetry plane, can be assumed as negligible for the submergence tested
(max. 32).
This is based on a consideration of the wall jet data on
centerline velocity by Rajaratnam (1974), which can be compared to the
For
free jet solution by Albertson et al. (1950), as shown in fig. 3-9.
the range of submergence tested, the deviation due to additional wall
shear can be neglected.
95
Fig. 3-7:
Observed Indentation:
Slight Asymmetry of The Dye Front
96
MODEL
/
///
/ ////
/ ////
BOUNDARY
/ ///
/
/
//
//
/ /////
i?
FIGURE (3-8)
WEAK ENTRAINMENT INDUCED
BY MODEL BOUNDARY
//
/
97
to0
0
>0.
x'-CENTERLINE
DISTANCE
A - AREA OF NOZZLE
Umo-CENTER LINE VELOCITY
0.
v
Vn- NOZZLE
0
VELOCITY
10 2C)030 40 50 60 70 80 90 100
XI
FIGURE (3-9)
COMPARISON OF WALL JET DATA WITH
ALBERTSON'S FREE JET DATA:DECAY OF
MAXIMUM VELOCITY ALONG CENTER PLANE
98
IV.
Comparison of Theory and Experimental Results
The experimental observation of the near field dilution in relation
to the theoretical prediction outlined in Ch. 2 is discussed in the sequel.
The results of the theoretical solution are then compared with experimental
data and empirical coefficients are evaluated.
4.1.
Near Field Stability
The prediction of the near field stability as discussed in sec. 2.6.
is compared with experimental data in fig. (4-1).
It can be seen that the
stability is well-predicted by the theory.
4.2.
Near Field Dilution
The theoretical predictions for the near field dilutions are evalua-
ted for the exact densimetric Froude numbers and submergences of the experimental runs.
The results are compared with the observed near field
dilutions in Table 4-1.
Stable Jets:
In general reasonable agreement is obtained.
tions are always higher than predicted.
Observed dilu-
This may be ascribed to additional
entrainment in the surface impingement region and the weak re-entrainment
on the surface caused by the slight asymmetry observed.
Unstable Jets: Using experimental results of runs with an unstable near
field and near field dilution greater than 3.0, an average value of
Rc = 0.47 is obtained by fitting the data with eq. 2.6.2.
Theoretical
predictions computed with this value of Rc are compared to the observed
dilutions.
99
-
-
EXPERIMENTAL
DATA
a:
STABLE RUNS
a:
SUBMERGED
0:
UNSTABLE
JUMP
RUNS
RUNS
THEORETICAL
100
_
0
HiD
o
o 0 00
0
Al
a
IO
0
0
A
L6e
00
0
0
o
O
00
0
I
I
I
I
I I
0
I il
0
I
I
I0
I
I
I
I I I
I
I
100
Fo
FIGURE (4-1)
NEAR FIELD STABILITY OF AN AXISYMMETRICJET IN SHALLOW WATER
I
I I
iiil
100
Although the coefficient Rc is derived from experimental results
with dilutions greater then 3.0, very good agreement is obtained with data
characterised by dilutions less than 3.0.
This confirms the validity of
the postulated structure of the theory for the stratified counterflow system in an unstable near field.
Although the theory requires two experimentally determined coefficients: namely the location of the jump R
for a stable near field and the
length of the mixing region for an unstable near field R,
the near field
dilution predictions as well as the experimental data demonstrate a consistent trend which could be understood in terms of our physical notions
of buoyant jets in shallow water.
As a turbulent jet was always observed for the range of Reynold's
number tested and frictional effects are shown to be unimportant at large
distances from the jet (sec. 2.5, stratified counterflow region), the findings of this study can be extended to prototype conditions.
The experimental data is compared with the general theoretical predictions in fig. 4-2.
101
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105
V.
Conclusion
The mechanics of a vertical axisymmetric jet in stagnant water is in-
vestigated both theoretically and experimentally.
Four flow regimes with
distinct hydrodynamic properties are discerned in the near field of the
jet: the Buoyant Jet region, the Surface Impingement region, the Internal
Hydraulic Jump region, the Stratified Counterflow region.
the flow in each region are formulated analytically.
The mechanics of
Insight is gained by
examining in detail the mathematical behavior of the theoretical framework.
The solutions of the four regions are coupled to give a prediction of the
near field stability and the near field dilution as a function of the jet
characteristics.
To verify this theory, a series of experiments were
carried out with a half-jet.
It is found that the near field stability is dependent on the densimetric Froude number and the submergence of the jet.
The criterion that governs
tions of the two, an instability is detected.
transition
the stable-unstable
is found
to be
For certain combina-
F
0
= 4.4 H/D
for H/D > 6.
In the case of a stable near field, the dilution is governed only by the
jet characteristics.
When an unstable near field exists, there is heat
re-entrainment from the stratified flow away, and the dilution is correspondingly lessened.
In this case the dilution is governed by the far field
boundary condition in addition to the jet characteristics.
The basic
mechanics of the flow for an axisymmetric buoyant jet can be understood
in terms of the theory developed in this study.
The theory is solved on a generic basis and the general results presented.
The characteristics of the four flow regimes and the phenomenon
of instability are experimentally confirmed.
The observed near field
dilution are compared with the theoretical predictions.
106
Good agreement is
obtained.
Recommendations for future research include: investigation of the
behavior of buoyant jets in an ambient crossflow, the effect of the angle
of discharge on the near field stability, and testing the theory in this
study against experiments carried out with a full round jet.
107
References:
1.
Abraham, F., "Jet Diffusion in Stagnant Ambient Fluid", Delft Hyd.
Lab., Publ. No. 29 (1963)
2.
Albertson, M. L., Dai, Y. N., Jensen, R. A. and Rouse, H., "Diffusion
of Submerged Jets", Trans. ASCE, 115 (1950)
3.
Brooks, N. H., "Dispersion in Hydrologic and Coastal Environments",
Keck Lab. of Hydraulics and Water Resources, Report No. KH-R-29,
California Institute of Technology
4.
Ellison, T. H. and Turner, J. S., "Turbulent Entrainment in Stratified
Flows", J. F. M., Vol. 6, Pt. 3, Oct. (1959)
5.
Jirka, G. and Harleman, D. R. F., "The Mechanics of Submerged Multiport Diffusers for Buoyant Discharges in Shallow Water", R. M. Parsons
Lab. of Water Resources and Hydrodynamics, Report No. 169 (1973)
6.
Morton, B. R., G. I. Taylor and J. S. Turner, "Turbulent Gravitational
Convection from Maintained and Instantaneous Sources", Proc. Roy. Soc.
of London, Series A, 1956, No. 1196, Jan., pp. 1-23.
7.
Rajaratnam, N. and Pani, B., "Three Dimensional Turbulent Wall Jets",
ASCE, Vol. 100, No. HY1, Jan. 1974.
8.
Rouse, H., C. S. Yih and H. W. Humphreys, "Gravitational Convection
from a Boundary Source", Tellus 4, 1952, No. 3, Aug., pp. 201-210.
9.
Sadler, C. D. and Higgins, M., "Radial Free Surface Flow", M.S. Thesis,
Dept. of Civil Engineering, M.I.T., 1963.
10.
Trent, D. and Welty, J., "Numerical Thermal Plume Model for Vertical
Outfalls in Shallow Water", EPA-R2-73-162, Environmental Protection
Technology Series, March 1973.
11.
Ungate, C., "Temperature Reduction in a Submerged Vertical Jet in the
Laminar-Turbulent Transition", S.M. Thesis, Dept. of Civil Engineering,
M.I.T., Aug. 1974.
12.
Yih, C. S. and Guha, C. R., "Hydraulic Jump in a Fluid System of Two
Layers", Tellus, VII (1955).
LIST OF FIGURES AND TABLES
108
Page
Fi.
1-1
Illustration of near field stability
11
2-1
Flow structure in the plane of symmetry of a vertical
axisymmetric jet in shallow water
13
2-2
An axisymmetric jet discharging vertically
15
2-3
Schematized structure of an axisymmetric buoyant jet
18
2-4
Length of zone of flow establishment as a function of
the exit densimetric Froude number
30
2-5
The surface impingement region
32
2-6
Radial stratified flow in a two-layered system
36
2-7
The radial internal jump
41
2-8
Behavior of asymptotic solution
50
2-9
Stratified counterflow system in an unstable near field
53
2-10
Radial variation of the interface
57
2-11
Illustration of two dimensional component feature of
axisymmetric radial stratified flow
59
2-12
Interface profile as a function of radial distance normalised with respect to location of critical section
61
2-13
Radial variation of interface for non-equal counterflow
62
2-14
Critical depth H2c as a function of F2 H, Ir'
64
2-15
Variation
2-16
General theoretical solution of near field dilution
69
2-17
Sensitivity of stability transition to the location of the
jump toe
71
3-1
Experimental set-up
75
3-2
The flow injection device
79
3-3
Schematic of temperature probe set-up
82
3-4
Variation of near field temperature rise with time
84
of the far field
as a function
interface
of fi/f
66
109
Fig.
Page
3-5
Observed near field stability
90
3-6
Temperature transect for the near field
91
3-7
Observed indentation of dye front
95
3-8
Weak entrainment induced by model boundary
96
3-9
Comparison of wall jet data with Albertson's free jet
data: decay of maximum velocity along centerplane
97
4-1
Near field stability of an axisymmetric jet in shallow water
99
4-2
Near field dilution as a function of F , H/D; vertical axisymmetric jet in stagnant water
101
TABLES
3-1
Smmary
of run parameters and experimental results
86
4-1
Summary of run parameters and experimental results
102
110
LIST OF SYMBOLS
Subscripts:
1,2
upper, lower layers in stratified flow
c
critical section in stratified flow
e
end of region of flow establishment
s,i,b
surface, interface, bottom boundary conditions
j
internal jump section in stratified flow
i
inflow section of impingement zone
I
outflow section of impingement zone
a
ambient variables
o
discharge variables
z
vertical direction
r
radial direction
--
-
-
--
b
jet width
b'
width of mixing region in zone of flow establishment
c
dimensionless length of zone of flow establishment
c
P
specific heat
D
jet diameter
F
layer densimetric Froude number
FH
densimetric Froude number based on total water depth
F
free surface Froude number
f.
interfacial stress coefficient
fb
bottom stress coefficient
g
acceleration due to gravity
H
total water depth
LIST OF SYMBOLS (Continued)
h
layer depth in stratified flow
h'
conjugate jump height
hI
thickness of jet impingement layer
KL
head loss coefficient for impingement
M
momentum flux
p
pressure
Q
layer flow in 2 layer system
Qe
entrainment flux
Qr
flow ratio in 2 layered system
q
flow per unit width
qH
heat flux
rc
location of critical section for unstable jet
rj
location of internal jump
r
radial co-ordinate
R
Reynolds number
RI
radial position of outflow section of surface
impingement region
rlr2
toe and end of internal jump
S
dilution
T
temperature
T
equilibrium temperature
e
(u,w)
velocities in axisymmetric cylindrical co-ordinate
system
"u9u2
averaged layer velocities for stratified 2-layered
system
uc
jet centerline velocity
111
112
LIST OF SYMBOLS (Continued)
u.
jet velocity at inflow section of impingement zone
y
water depth for free surface flow
z
vertical coordinate
entrainment coefficient
jet spreading angle
X
jet spreading ratio between mass and momentum
AT
temperature rise above ambient
ATo
discharge temperature rise
Ap
density deficiency
p
density
rT
shear stress
coefficient of thermal expansion
Appendix A
113
Stable Near Field Solution:
The solution for the average dilution in a stable near field can be
obtained by solving eq. 2.2.1-2.2.3 in conjunction with eq. 2.1.19-2.1.20.
The following set of non-dimensionalized algebraic equations are arrived
at.
These two equations are solved numerically by the Newton Ralphson
method.
u
=L[-l+
2
1
czu
H*z
u 2 (1-KL)
X2
+
482 C
i
u-
zz=
3
{z2 H*2
o
c2}
]
3
(l-z)
+
2H (4e2 )z2F 2u
=
/H =
1
'IO
1
3 (l+X2)I
2
2
I CM
i
z
z(l-z)ao
6
where
3
1l+A2 11
{1
/D
c = z /D
e
Having solved for u, z the densimetric Froude numbers of the upper
and lower layer can be computed and used as input for obtaining the conjugate jump height.
Assuming that the internal jump occurs atr=R.H
from the jet axis,
and experimental observation indicates there is practically no change in
the thickness of the upper layer prior to the jump.
The densimetric
Froude numbers of the respective layers can then be related to the jet
characteristics:
2
L=
u2
_
A2p
|
=
J
F2
2
=(Si2S
(1-z)3
z
F 21
c
(_2)
( ; R.2
z
(1-z)3
H*
114
Appendix B
The Internal Hydraulic Jump
The conjugate jump height of the radial internal jump can be solved
numerically by the Newton
r2 - r1
Raphson method.
T constant
T(h 1 - h 1)
=
By assuming
such that
h
R* =
r2
- 1)
hi
Th 1
a =
where
(6
l+a
rl
I
rl
By rearranging eq. (2.4.14) and (2.4.15), we obtain the following set of
two algebraic equations.
4F
F1 (X1
F 2 (X1
Y1 )
Y)
1
-R*x]
I
R* (+R*)x (-
=
)
hi
X1 -'
Yl
=
h2
h2
-Ry 1 1
= 0
- R*(+R*)(l+xl)(ll)x
0
-
where
2 2h
R (I+R )yl(l+Yl)(1l-x)
4F2 yl(l+X)
9'
4
-1-
4F2 h1 (R*x
)
2
Having evaluated the partialderivatives of F(x
the two equations can be solved by iteration.
1,
Y1 ) and F 2 (Xl, y1 )
- N '-, u,,
.C C C 0 0
,,C. 0c
cC 0
a 0
tc[~Cr- cc CO
'-- () 0" C N e 4 L(' O C 0'
0 cr - N
N t
C 0 D)
O,
C
G
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.
,
0
122
123
Appendix
C
For H/D smaller than approximately 6.0 the theory outlined in ch. 2
and the previous appendix does not strictly hold as the flow is not fully
established when the jet reaches the free surface.
By assuming momentum
dominates in such cases, a simplified analysis is done to derive an average
dilution in the near field.
From Albertson et al (1950),
z
Q/Qo = 1.0 + 0.083
where
Q is the total
+
0.0128
z2
flow of the jet.
Assuming the depth of the upper layer = BH , the dilution in the near
field is given by
S = 1.0 + 0.083(1-8)H
+
0.0128 (1-) 2H*2
Assuming that the jump occurs at rj=RjH from the jet axis, the densimetric Froude numbers of the respective layers prior to the internal
jump can be related to the jet characteristics and experimental coefficients:
2
1
1
S3F2
1
_
1
_U
g
h
64 R2
2
u2
p
p
H*5
J
2g hl
S-1 2 ()3
F2
2
In the numerical solution
B
is assumed to be 0.1.
124
Appendix D
Energy Approach to critical flow in a two-layered counterflow system:
It is well known that for open channel flows, the critical flow
condition can be interpreted as that which minimizes the specific energy
for a given flow.
The following is an extension of the same principle to
a two-layered counterflow system.
-
-
_
-
-
#
) il
ZA
10
I
P-A
k
.I
< .
r
R,
Z
!
D
v
The two dimensional case is treated here.
However, the analysis is
also applicable to axi-symmetric flows.
Kinematic Boundary Condition:
free surface:
d(hl+ h 2)
=
w
dx
interface:
dh 2
wi
Ui
dx
A small fluid particle of mass
energy:
E6V =
1
2
2
{pgz + 212 (u
u+ + w)}
)
pV
6V
g
possesses potential and kinetic
125
First Law of Thermodynamics:
= (
DE
=
DE
aE
at
Dt
(Work)
ax
ax
azt6a
+ ap ) 6V
az
Assuming steady state and neglecting friction losses: we have
a (E+p)w =
az
ax (E+p)u +
Integrating this vertically and applying Leibnitz rule:
d
fhl+h
dx o
2
{(E+p)u} dz
-
(E+p)
u
d(h
+ h2 )
+ (E+p) w
= 0
dx
Invoking the 'kinematicboundary condition at the free surface,
d fhl+h2{(E+)u}dz
w <<
Assuming
fh2 {pgz + 1 p(u2 +w 2) + (p-Ap)gh1 + pg(h 2 -z)Judz
=
h2 (E+p)udz
0
=
u
and
-3
u
u
2
=
we have
2
Pu2 h + {pg(hl+h
2) For
the
system:
we have
q
2 counterflow
Apg
} h2 n
= h[q2
For the counterflow system: we have q2 = -
121
therefore
3
fh2 (E+p) udz
o
1
q2
2 P 2 h2
h3
q2
-
pg(h +h2 ) - Apghl} h 2 ( -2 )
h2
Similarly,
hl+h 2
hl+h2
2
(E+p)u
2
dz
u{-A)gz
=
+ PU+(-A)g(h+hz)}d
3
½p(--)
=
h
+
(P-ap)g(hl+h
2) (h-) h1
1
h1
ql>
Total energy at any
x
0
can be defined as:
3
q2
E(x) = -
{pg(h I+h2 ) - Apghl } q2
P22 ql3
+ p()
h1
E
For extremum,
( p-Ap)g(hl+h2)
+
aE
=
g q 2 + (p-Ap)
-(p-Ap)
0
ah-=
2
ah1
ql
WIe have
3
Pql1
gq
g(
3
1 -
hi
=
0
(1)
3
- pg q2
+
(p-Ap) g ql
+
Pq 2
=
3
2
(1) - (2):
-pq33
h
3
pq33
Apg q2
-3
=
h2
ql=
q2
gives
F 2 + F 22
1
2
=
1 .
0
(2)
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129
Appendix F
Heat loss effects in the Near Field:
In this section it is shown that heat loss effects are insignificant
in the near field of the bouyant jet.
Neglecting molecular transport process, heat conservation implies:
u3aTr + w3aT
=
az
where
T'u'
-w
r
u 1 w'
T'w'
z
(F.1)
= velocity fluctuations
= temperature fluctuation
T'
Integrating eq. F.1 vertically for the upper layer, using Leibnitz
rule and invoking kinematic boundary conditions at the free surface and
the interface (assuming no free surface slope) it can be shown that
u1 -1d=
TqHs-qHi
d
T1
dr
pch
where
=
surface heat flux
=
interfacial heat flux
T1
=
average temperature of upper
layer
c
P
=
specific heat of water
qHs
H
Putting the heat fluxes in the form:
qHs
i
=
qHi
=
-
k(T 1 -T
kz(T 1
-T
e)
2)
130
where
k
=
surface heat loss coefficient
k
=
interfacial heat loss coefficient
=
average temperature of lower layer
=
Equilibrium air temperature
z
T2
T
e
Doing a scalling and replacing T 1 with the temperature excess above
ambient AT1 , we halve
*
* dAT1
u1
PcHu
-[
dr
ap
o
*
]
h
(T-AT)
(AT-AT2 )
k R
(T 1-AT)
kR
-
[
1
]
u
c
.2)
(F
h
o
ap
where
1
u
AT
0
o
U
l/U
r
=
r/H
hI
=
h 1 /H
AT1
°
=
AT/AT
:
characteristic
:
characteristic temperature excess above ambient
of upper layer
iupper layer velocity
San Onofore Power plant, as an example of a submerged discharge
Using the theory
3.2 x 106 cf/hr.
design, has a condenser flow rate of
outlined in this study, the upper layer velocity can be estimated to be
approximately
0.2 ft./sec.
at
r
= 10.
The values
of the heat loss
coefficients are given by:
k
=
150 BTU/°F-ft.2-day
k
=
10- 4 ft2/sec
=
62.5 BTU/ft3
z
PCp
Jirka and Harleman
(1973)
]
131
Substituting these numbers into eq. F.2 the dimensionless parameters
in brackets can be shown to be 0.03 for the surface heat loss and 0.0001
for the interfacial mixing.
Hence heat loss effects are not important in the region of interest
(r < lOH) treated in this study.
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