EPA 660/2-73-029
December 1973
Environmental Protection Technology Series
Mathematical Model For Barged
Ocean Disposal Of Wastes
33
\,
UJ
CD
Office of Research and Development
U.S. Environmental Protection Agency
Washington, D.C. ZG460
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RESEARCH REPORTING SERIES
Research reports of the Office of Research and
Monitoring, Environmental Protection Agency, have
been grouped into five series. These five broad
categories were established to facilitate further
development and application of environmental
technology. Elimination of traditional grouping
was consciously planned to foster technology
transfer and a maximum interface in related
fields. The five series are:
1. Environmental Health Effects Research
2. Environmental Protection Technology
3. Ecological Research
4, Environmental Monitoring
5. Socioeconomic Environmental Studies
This report has been assigned to the ENVIRONMENTAL
PROTECTION TECHNOLOGY series. This series
describes research performed to develop and
demonstrate instrumentation, equipment and
methodology to repair or prevent environmental
degradation from point and non-point sources of
pollution. This work provides the new or improved
technology required for the control and treatment
of pollution sources to meet environmental quality
standards.
EPA REVIEW NOTICE
This report has been reviewed by the Office of Research and
Development, EPA, and approved for publication. Approval does
not signify that the contents necessarily reflect the views
and policies of the Environmental Protection Agency, nor does
mention of trade names or commercial products constitute
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THE GUNTER LIBRARY
GULF COAST RESEARCH LABORATORY
.OCEAN SPRINGS, MISSISSIPPI
MATHEMATICAL MODEL FOR
BARGED OCEAN DISPOSAL OF WASTES
by
Robert C. Y. Koh and Y. C. Chang
Grant No. 16070FBY
Program Element 1BA025
Project Officer
Walter F. Rittall, Civil Engineer
Pacific Northwest Environmental Research Laboratory
National Environmental Research Center
Corvallis, Oregon 97330
Prepared for
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
Washington, D.C. 20460
For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 - Price $4.85
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ABSTRACT
Theoretical and experimental studies were performed on the
dispersion and settling of barge disposed wastes in the ocean. A
computer program based on the mathematical model has also been
written. Comparison of predictions with experiments, both in this
study and from previous investigations were found to be good. Ex-
ample solutions based on the model for prototype situations are also
presented.
The waste is assumed to consist of two phases, i) a. solid phase
characterized by constituents with various densitites and fall velo-
cities, and ii) a liquid phase. The methods of disposal considered
include i) discharge from a bottom opening hopper barge, ii) pumped
discharge through a nozzle under a moving barge-and iii) discharge
into the barge wake. The effects of ambient horizontal currents, den-
sity stratification, variation of diffusion coefficients are incorporated
in the model.
Three phases of dispersion are envisioned: i) a convective phase,
ii) a collapse phase and iii) a long term diffusion phase. Transition
between phases are accomplished automatically in the numerical
model. In addition, the collapse phase may a) be replaced by or b)
include a bottom, spreading phase. Under certain circumstances, the
collapse phase is bypassed.
Every attempt has been made to minimize the amount of input re-
quired in the use of the numerical model. The integration steps and
grid sizes are all automatically chosen by the model. Both detailed
printout and graphic output are incorporated. The solution may also
be terminated at the end of any of the three phases of dispersion.
ii
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TABLE OF CONTENTS
Pag<
1. INTRODUCTION 1
2. CONCLUSIONS AND RECOMMENDATIONS 3
3. DEVELOPMENT OF THE MATHEMATICAL MODEL 6
3.1 Properties of Waste Material 6
3.2 Ambient Conditions 9
3,2.1 Ocean. Density Structure 9
3.2.2 Ocean Curreats and Turbulence 11
3.2.3 Horizontal Diffusion Coefficient -- K (or K ) 14
x z
3.2.4 Vertical Diffusion Coefficient 18
3.3 Barge Operation 1 - Simple Over-Board Dumping 22
3.3.1 Convective Descent 24
3.3.2 Dynamic Collapse 31
3.3.3 Bottom Encounter 39
3.4 Barge Operation 2: Jet Discharge 45
3.4.1 Jet Convection 47
3.4.2 Dynamic Collapse 52
3.4.3 Bottom Encounter 60
3.5 Barge Operation 3 - Discharge Into Barge Wake 66
3.5.1 Initial Mixing in the Near Wake of a Barge 68
3.5.2 Convective Descent 74
3,5.3 Subsequent Motions 76
3.6 Long Term Diffusion 77
3.6.1 Formulation of the Theoretical Model 80
3.6.1.1 Flow Configuration and Basic Equation 80
3.6.1.2 Method of Mpments 85
3.6.2 Diffusion Coefficients 88
3.6.3 Limitations of the Theoretical Model 91
iii
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3.7 Numerical Procedure and Computer Program 93
3.7.1 Numerical Procedure 93
3.7.2 Computer Program 195
3.7.2.1 Empirical Coefficients 105
3.7.2.2 Transition between Different Phases 108
and Input of Solid Particles to Long
Term Diffusion
4. EXPERIMENTAL INVESTIGATION 115
4.1 Objective and Scope of Experimental lavestigations 115
4.2 Apparatus and Procedure 115
4.2.1 Ambient Condition Ufa
4.2.2 Discharge Material 115
4,2.3 Procedure and Data Reduction 117
4.3 Results and Discussions 119
4.3.1 Instantaneous Release of a Three-Dimensional 119
Slug
4.3.2 Continuous Discharge from a Horizontal 143
Travelling Vertical Jet
4.3.3 Instantaneous Release of Two-Dimensional 152
Puff
5. COMPARISON OF EXPERIMENTS WITH THE 153
MATHEMATICAL MODEL
5.1 Barge Operation 1 153
5.2 Barge Operation 2 155
5.3 Discussion 158
6. EXAMPLE SOLUTIONS 159
6. 1 Interpretation of the Results of Computations from 169
the Mathematical Model for Long Term Diffusion
REFERENCES 173
APPENDIX A
APPENDIX B
iv
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LIST OF FIGURES
Figure No. Title Page
3- 2« 1 Ocean Density Structure 10
3- 2. 2 Horizontal Diffusion Coefficient as
a function of Horizontal Scale
(from Orlob 1959) 15
3- 2. 3 Variation of Transport Rates as
Functions of Richardson Number
R., (u1 :horizontal velocity
fluctuation, t1 : temperature fluctuation;
from Webster, 1964) 17
3. 3. 1 Schematic Definition Sketch 23
3.3.2 Definition sketch 35
3.4. 1 Definition sketch 48
3.4.Z Definition sketch 54
3. 5. 1 Definition sketch (After
Naudascher 1968) 69
3. 5. 2 Radial variation of mean-velocity
difference in the wake of a disk 71
3. 5. 3 Radial variation of mean-velocity
difference in the wake of a slender
spheroid 72
3. 5. 4 Axial variation of effective width for
various jet and wake flows 73
3.6. 1 Coordinate System and Ambient
Conditions for long term diffusion
model 81
3. 6. 2 Schematic for boundary conditions
for solids settling to the bed 84
3. 7. 1 Grid System for Long Term Diffusion 94
3. 7. 2 Definition Sketch 104
3. 7. 3 Flow Chart for the Computer Program 106
3.7.4 Definition Sketch 112
4. 3. 1. 1 Comparison of experiment with 133
4.3. 1. 2 numerical model 134
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Figure No. Title Page
4. 3. 1. 3a Ambient density stratification 135
4. 3. 1. 3b-d Comparison of experiment with
numerical model 136-138
4. 3. 1. 3e Waste concentrations from
numerical model 139
4, 3. i. 4_ Comparison of experiment with
4.3.1.6 numerical model 140-142
4. 3. 2. 1- Comparison of experiment with
4.3.2.8 numerical model 144-151
5. 1 Comparison of present theory
with Fan's experiments 157
vi
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LIST OF TABLES
Table No. Title Page
3< -1 Great Lakes Dredging Spoil
Characteristics 8
3« 2. 1 Summary of Values of Vertical
Diffusion Coefficient K in the Ocean 19
3- 2. 2 Summary of Formulas on Correlation
of Vertical Diffusion Coefficient K
with Richardson's Number R. (or y
Density Gradient e) X 21
3« 6 Use of Moments to Describe
Dispersion 89
4- la Summary of Experimental Parameters 120
4. lb Summary of Experimental Parameters 122
4. lc Summary of Experimental Parameters 124
4. 2a Density Stratification for Runs
M-S, S-S and N-S 125
4. 2b Density Stratification for Runs
N.-S and S.-'S 128
J J
4. 2c Density Stratification for Runs
M2-S and S2-S 130
6. 1 Summary of Parameters of Runs Made
for Barge Operation 1 160
6. 2 Summary of Parameters of Runs Made
for Barge Operation 2 162
6. 3 Summary of Parameters of Runs Made
for Barge Operation 3 163
6. 4 Summary of Solid Waste Characteristics
used in Simulation Runs 165
vii
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LIST OF SYMBOLS
drag
cfictn
CM
f
'l.Frictn
For simplicity, symbols of secondary importance which appear only
briefly in the text are omitted from the following list:
^ dissipation parameter based on variance
AT dissipation parameter based on size
g buoyancy
drag coefficients
drag coefficient
friction coefficient
apparent mass coefficient
k, 1^ moment of concentration distribution
concentration of ijth solid
drag force
entrainment function
buoyancy force
frictional force
friction coefficients
depth
inertial force
vor ticity
horizontal diffusion coefficient
vertical diffusion coefficient
vertical diffusion coefficient specified
horizontal diffusion coefficient
length scale
momentum
p.. volume of ijth solid
R. Richardson number
S.. volume of ijth solid settled out of convective
^ element
~t} velocity vector
U ambient velocity vector
V mass
W, surface concentration of floating solid
Wo bottom concentration of settled solid
E
F
F
F
H
I
K
K
K
Kyl,Ky2jKy3
L
"M
viii
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minor axis of collapsing element
major axis of collapsing element
concentration
° gravitational acceleration
distance along jet or wake axis
*• time
u velocity component in x-direction
ua ambient velocity in x-direction
Uao ambient velocity in x-direction at surface
v velocity component in y-direction
vl tip velocity due to collaps'e
V2 tip velocity due to entrainment
w velocity component in z-direction
wa ambient velocity in z-direction
wao maximum ambient velocity in z-direction
ws settling velocity of solid particles under
consideration
wsij settling velocity of ijth solid
x horizontal coordinate
Y vertical coordinate
^kl, ^kZ, yk3, y^4 y-positions where Ky changes values
Yw,yejyu y-positions where ua Wa specified
ix
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eatrainment coefficients
a? CX3 <*4 used in convective and collapse phases
a a absorbancy coefficients for solids settling
' to the bed or free surface
.y y re-entrainment coefficients for solids settling
1» to the bed or free surface
p.. settling coefficient
ft 0_ 9 angles s makes with x, y, z axes
1 j L. f 3
,. c angles ambient current makes with x, z axes
Af = Pa(o) - Pa
A
p /„) ambient density at y
0 . density of i1-^1 solid
c density gradient
2 x-variance of diffusing pool
x
a 2 z-variance of diffusing pool
z
covariance of diffusing pool
si
density
xz
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ACKNOWLEDGMENTS
The support of the project by the Water Quality Office, Environ-
mental Protection Agency and the helpful discussions provided by
Dr. Donald Baumgartner and Mr, Walter Rittal, the Grant Project
Officer, is acknowledged with sincere thanks.
xi
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SECTION I
INTRODUCTION
One method of disposal of concentrated wastes such as dredge spoil,
sewage sludge and industrial wastes is to barge it out to sea and
dump it overboard. Several distinct schemes for the discharge of
the waste cargo from the barge into the sea can be envisioned. These
include: a) dumping the waste load from a bottom opening hopper
barge, b) discharging it by pumping the waste through nozzles at
the bottom of the barge and c) discharging it into the barge wake.
A more detailed review of current practice and typical waste char-
acteristics can be found in Clark, et al (197D.
As a first step in the overall evaluation of the environmental impact
of this practise, it is necessary that the mechanics of dispersion of
the discharged waste be analysed. The analysis must include the de-
termination of the concentration of the waste material in suspension,
in solution and the distribution of disposed solids, either floating at
the surface or settled on the ocean floor.
It is the primary purpose of the present study reported herein to
develop a mathematical model for the analysis of dispersion of
the disposed waste in the ocean environment. The model
is detailed in Section III of this report. A computer program based
on this model is also included as Appendix A in this report. Based
on this model, the fate of the waste can be determined given the waste
characteristics, the ocean environmental conditions and the method
of disposal.
In addition, a series of preliminary laboratory experiments have been
performed and reported in Section IV. The primary purposes of
these experiments are the establishment of the adequacy of the mathema-
tical model and the verification of the important assumptions made.
Comparison of the experimental results with the predictions based on
the mathematical model is presented in Section V.
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A sequence of example cases have been analysed based on the model
and the results are presented in Section VI and Appendix B.
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SECTION II
CONCLUSIONS AND RECOMMENDATIONS
In this study, a mathematical model has been developed to predict the
physical fate of wastes discharged from barges. A computer program
has also been written based on the mathematical model. In addition,
a set of preliminary laboratory experiments were performed to verify
and supplement the mathematical model. Based on the study, the
following conclusions and recommendations are made.
1) The general behavior of the discharged waste, with or without
solid constituents of the type used in the experiments, behaves similarly.
2) The comparison between the experimental results and the theo-
retical predictions is good.
3) The mathematical model requires the knowledge of various empir-
ical coefficients. A set of suggested values are built into the computer
program. However, as better knowledge becomes available through
experiments, these may be modified. It is recommended that experiments
be performed in. the future to better define these coefficients.
4) With the built in coefficients, the model is believed adequate in
predicting the physical dispersion and settling characteristics of typical
solid wastes over the long run when a number of discharges are made
even though the detailed description for each single discharge may not
be as accurately predicted until the coefficients are better defined.
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5) A set of representative runs has been made using the numerical
model and presented in Section VI. However, due to the large number
of parameters, in no way can these runs be considered a parametric
study. It is recommended that a parametric study be performed to
obtain a complete description of the dependence of the result on the
parameters .
6) While the basic mathematical model for long term diffusion allows
the ambient conditions such as the diffusion coefficients, and currents
to be functions of time, they are treated as only functions of the vertical
coordinate in the computer program. It is recommended therefore that
the program be modified to allow them to be functions of time so that
the effects of temporal variations, such as encountered in tidal currents,
can be obtained.
7) The model as developed assumes that the waste will undergo a
phase of descent and that the gross density of the waste is larger than
the ambient. While in practice this should always be the situation lest
the waste would undesirably spread on the surface, .the model can and
should be modified to include this possibility.
8) In the model, after the descent phase, the waste is directed by
the program to either a) collapse if the local ambient is stratified or
if the bottom is encountered, or b) go directly to the long term diffusion
phase; Due to the settling of solid particles, the waste pool may reach
the end of the descent phase in a uniform ambient with or without first
passing through a density gradient. Depending on the latter condition,
going directly to long term diffusion may or may not be valid. The
user of the model must exercise some judgement as to whether or not
to slightly modify the input density stratification to ensure a collapse
phase.
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9) Based on the results presented in Section VI, the method of.
discharge in Barge Operation 1 results in the most likelihood of the
waste cloud reaching the bottom even if there exists a fairly strong
pycnocline. In. Barge Operations Z and 3, a relatively weak pycnocline
is usually sufficient to arrest the vertical descent of the waste plume.
This phenomenon should be further explored in a detailed parametric
study since it has practical implications.
10) One of the main drawbacks of the method of moments used in this
study for the long term diffusion phase is the fact that the ambient con-
ditions cannot be functions of horizontal position. While strictly speaking,
there is no way to overcome this without abandoning the advantages of the
method of moments, physically, it may be argued that as long as the
characteristic horizontal scale of the variation of ambient conditions is
large compared with the size of the diffusing waste pool, horizontal
variation of the ambient condition can be allowed to provide at least a
first approximation. It is recommended that this capability be incor-
porated in the model.
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SECTION III
DEVELOPMENT OF THE MATHEMATICAL MODEL
In this section, a mathematical model will be developed for the pre-
diction of the dispersion and settling of waste material discharged from
barges into the ocean environment. Only the physical aspects of the
phenomenon will be analysed since the chemical, biological and eco-
logical aspects are beyond the scope of the present investigation.
In this investigation, three different modes of operation for the dis-
posal of waste from barges are considered, namely 1) simple over
board dumping, 2) discharge through a nozzle under the barge bottom
and 3) discharge into the barge wake. In spite of the apparently-
different methods of disposal, the waste material is ultimately either
settled to the bottom or mixed into the ambient. The dispersion pro-
cess in each of the three modes can be roughly divided into three
stages. In the early stage, the phenomenon is dominated by the ef-
fects of the initial momentum and buoyancy of the discharge. In the
final stage the waste material is essentially dynamically passive and
subject to diffusion .due to the effects of ambient turbulence. In be-
tween, there is a transition phase between the two stages. In the fol-
lowing, the properties of the waste material and the ambient conditions
will first be discussed, then mathematical models will be developed
for each barge operation and for each stage of the mixing process,
finally a numerical solution to the whole problem and a computer pro-
gram will be presented.
3 . 1 Properties of Waste Material
According to Clark et al. (1971), the most important waste materials
presently being disposed by barges are dredge spoil, sewage sludge and
industrial wastes. The dredge spoil can be organic or inorganic solids
with or without contamination from fertilisers, chemicals, pesticides
as well as a variety of industrial wastes. Among the ecological effects
of the discharge of dredge spoil found are i) a temporary reduction in
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fish abundance and dissolved oxygen, ii) an increase in turbidity, iii)
an increase in the level of pesticide concentration in fish and iv) an
increase in nutrients in the water. For the sewage sludge, the in-
herent public health hazard and the potential for build up of organic
solids on the ocean bottom are most significant. Industrial wastes
vary widely in their properties, depending on their particular origin.
In addition, there are other waste materials which may be discharged
from barges such as radioactive wastes, garbage and refuse.
Dredge spoil constitutes by far the largest percentage of waste material
dumped from barges. For example, in 1968, it contributes to 80%
of all such discharges. The characteristics of dredge spoil for the
Great Lakes region is summarized on Table 3-1. It is observed that
most of the solid particles are in the silt (4U < d< 63 M. ) and sand
(63 JJ.
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00
TABLE 3. 1
GREAT LAKES DREDGING SPOIL CHARACTERISTICS*
Location
Buffalo
Calumet
Cleveland
Green Bay
Indiana
Rouge River
Sodus Bay
Toledo
aj Based on
Percent
Solids
37. 0
40.7
44. 9
43. 0
35.2
43.7
53. 1
Average
Density
gm/ml
1. 27
1. 33
1.36
1.37
1.23
1.28
1. 51
39.0 1.30
30 minute settling
Settling
Velocity
ft/hr a/
0.068
0. 144
0.201
0. 103
0. 150
0.290
0. 506
0.023
Average Percentag
Gravel Sand
d>2 mm 63u
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3 . 2 Ambient Conditions
Knowledge of pertinent aspects of the ocean environment is ess.ential for
predicting the fate of waste material discharged from a barge into the
ocean environment. These include the structures of ocean density,
ocean current and ocean turbulence.
3.2.1 Ocean Density Structure
The density of sea water depends upon temperature, salinity and pres-
sure. In the ocean, the potential density is often inhomogeneous due
to variations in temperature and salinity. A typical ocean density
profile is shown in Figure 3.2.1. In general, the density of sea water
increases with depth. The ocean ijs then said to be stably density
s tratified.
The surface layer in the ocean is characterized by relatively low sa-
linity (35°/oo) and high temperature (20°C). The density is fairly
uniform within this so-called "mixed layer" because of wiad induced
agitation and vertical circulation induced by evaporation and temperature
reversals across the free surface. The thickness of the mixed layer
varies depending on local wind and weather conditions but is typically
of the order of a few hundred feet.
The ocean depths are characterized by higher salinity (37°/oo) and
lower temperature (2°C). It is relatively stagnant and much less dis-
turbed than the mixed layer. In this deep layer, a moderate or weak
density gradient is ususally maintained. According to Defant (1961),
the density gradient e[ = - Up^/dy)] ia the deep layer is roughly
in the range of 10" to 10" /m.
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WIND
EVAPORATION
'HEAT
THERMOCLINE
DEEP LAYER
OCEAN FLOOR
V
To
SALINITY
PROFILE
TEMPERATURE
PROFILE
DENSITY
PROFILE
Figure 3.2.1
Ocean Density Structure
10
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Measurements of ocean currents are made difficult by instrumentation
limitations. Data on the ocean currents are mostly limited to the sur-
face and in the surface mixed layer. Current velocities here often
reach 1 fps or more. Information on the current velocity in the thermo-
cline or the deep layer is scare. Recently, velocity measurements
carried out in the deep layers showed the existence of velocities of
the order of 0. 1 fps.
The distinction between ocean current and turbulence is often not clear
until the mode of averaging is specified. In other words, the length
scales used in the averaging process must be chosen based upon the
scale of the interested phenomenon. Turbulence in the ocean may be
visualized as the coexistence of eddies of various sizes. In the pres-
sent study eddies much greater in size than that of the diffusing pool
should be regarded as currents. For a much larger pool, these ed-
dies should be considered as turbulence.
The detailed characteristics of turbulence in the ocean is almost en-
tirely lacking. The difficulty in instrumentation for measuring tur-
bulence in the ocean is tremendous. Progress is beginning to be made
in direct measurement of ocean turbulence,
In analysing ocean diffusion problems, the turbulent transport is often
represented by semi-empirical diffusion coefficients defined as follows:
., i (— i
K = - —- +D .
,,
K = - ?-±- + D (3.Z.I)
12
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K - - + D
where K and K are the horizontal diffusion coefficients K is
x z y
the vertical diffusion coefficient - u'C1, - v'C1, - w'C' are tur-
bulent transport .quantities in (x, y, z) directions, C is the mean
concentration of the transported material, D , D , D are the
c x y z
molecular diffusion coefficients.
Equation (3.2.1) defines the diffusion coefficients in such a way that
they represent the combined effects of turbulent and molecular trans-
port. Generally, molecular transport is negligible in comparison with
J:he turbulent counterpart. However, in some instances, for example,
when vertical turbulence is completely suppressed by a strong density
stratification, the vertical transport may entirely be due to molecular
diffusion; i.e., K = D .
Y Y
The nature of these diffusion coefficients will now be disucssed. Note
that the values of these coefficients were not usually determined by
measuring turbulent transport quantities as given by Equation (3.2.1)
but from observations on some gross diffusion characteristics such as
the rate of spread of dye patches.
The vertical diffusion coefficient K is usually several orders of mag-
y
nitude smaller than the horizontal diffusion coefficients because of the
presence of stable density stratification and boundaries imposed by the
free surface and the bottom of the ocean. The nature of the vertical
diffusion coefficient is different from its horizontal counterpart and thus
they are discussed separately.
13
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3.Z.3. Horizontal Diffusion Coefficient -- K (or K )
x z[
Most investigators concentrated on the investigation of horizontal tur-
bulent diffusion in the ocean. Dye or other tracer objects were-'re-
leased and the size of the dispersing patch was observed as it grew
with time. Sometimes existing materials in the ocean were used as
tracers. Horizontal diffusion coefficients were then derived based upon
diffusion models disregarding both curr'ent shear and vertical transport.
Extensive information in this respect is available in the literature.
[e.g., Stommel (1949), Richardson and Stommel (1948), Olson and
Ichiye (1959), Munk, Ewing and Revelle (1949), Gunnerson ( I960), Ichiye
(1962), Okubo (1962), Bowden (1962), Joseph and Sendner (1962),
Foxworthy, Tibby and Barsom (1 966 ), and Snyder (1967)].
The values of horizontal diffusion coefficients obtained in the ocean
2 82
ranged from 5x10 to 4 x 10 cm /sec. Most values were obtained
at the ocean surface. The values of K (or K ) were found to increase
x z
with the size of the diffusing patch L as shown in Figure 3.2.2. Al-
though data scatter is significant, the general trend follows a 4/3
power law: i.e.,
KX = ALL4/3 lO^ft^ L< 108 ft (3.2.2)
where AT is a constant called dis sipation parameter (in fps system:
2/3, L 2/3,
ft /sec and in cgs system: cm /sec).
The 4/3 power relation has some theoretical batis according to
Kolmogoroff's similarity hypotheses [Batchelor (1950)]. However,
Equation (3.2.2) is strictly empirical since its range extends well
beyond the limiting scale of the Kolmogoroff inertial subrange. The
value of A varied from 5 x 10" to 1.5 x 10 ft /sec.
14
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o
0)
10"
10'
10'
1 io5
10
UJ
o
u.
u.
LU
8
io
U.
t
Q
t 10
O
N
I i i i r
O OR LOB, 5 = 0.00327, l" MESH
• OR LOB, 5 = 0.00055, T'MESH
O SVERDRUP
C PROUDMAN
O MUNK
A PEARSON
A HANZAWA
Kx= 0.00015 L
Kx=0.005 L
A STOMMEL
• "MOON , ET AL
\- 3 HARLEMAN
a VON ARX
X GUNNERSON
+ HIDAKA
T PARKER-1961-^
. SUMMARY
) BY PEARSON
NRDL
DEEP LAYER
EXPERIMENT
1968
^\f V
W /
'7 '
LIMITS OF
/ /*
DRIFT CARDS
RADIOACTIVITY
/ ' / IN BIKINI LAGOON
' /M— I MILE OUTFALL FIELD
/—CURRENT-CROSS PAIR
DATA, OLSON
AND 1C HI YE _
STREAMS
G
O \ R
GUNNERSON, < M
I960 / C
D
I
OJ I 10 10 10 10 10 10 10 10
HORIZONTAL SCALE OF DIFFUSION PHENOMENON,L-ft.
Figure 3.2.2 Horizontal Diffusion Coefficient
as a function of Horizontal Scale
(from Orlob 1959)
15
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It is expected that as the depth increases and approaches the thermocline,
the horizontal turbulent transport might decrease. Thus one might expect
AT to decrease with increasing density gradient. Munk, et. al. (1949)
J_i
found that in Bikini Lagoon at 50 rn depth, the value of horizontal diffusion
coefficient was only one-third of that observed near the surface. However,
in this case, an error might have been introduced since horizontal exchange
was calculated separately from the vertical exchange. Snyder (1967)
found that at 9 ft depth the value of AT drops to one quarter of the value
2/3
at the free surface (0. 00017 vs 0. 00065 ft /sec). However, the tracer
objects used in the 9 ft depth test appear to respond somewhat slowly to
the turbulence which might have caused such fast drop-off. Nevertheless
at present there is no definitive determination of the variation of hori-
zontal diffusion coefficient as a function of ambient density structure.
In general, the horizontal transport is expected to be less affected by
the density stratification than the vertical transport. Based upon
Webster's (1964) measurement in a uniform shear flow with constant
temperature gradient generated in a laboratory wind tunnel, the hori-
zontal transport (-u't1), as shown in Figure 3. 2. 3a, was found to be
unaffected by the increase of the Richardson number, R., defined as:
(3.2. 3)
>r
\ dy
where e is the density gradient
du
is the velocity gradient
dy
On the other hand, he found that the vertical transport (-v't1) decayed
rapidly with R. as shown in Figure3. 2. 3b. Therefore, at present, a
constant value of AT (independent of depth) may be used as a first approxi-
j_j
mation. It must be reiterated that in the past determinations of these
horizontal diffusion .coefficients, the effects of vertical exchange
16
-------
0.8
2
O
H 0.6
<
_J
UJ
£ 0.4
LL.
0
o
." 0.2
"^ o
1 1 1 . 1
O ^j
— 8 o ° - - "^
no o _ -
o _ - o
o o _ — °-o •" o —
~"°"°" o f 0°
— ° —
0 °
00
0
— —
1 1 1 1
0 0.2 0.4 O.G 0.8
Ri
a) HORIZONTAL TRANSPORT
0.8
O
UJ
ir
cr 0.4
o
o
'- 0.2
1 1 1 1 I
o
o
0 o
O jp
o°!0 8 0
R^-PO
o
1,1,
1 1
—
—
_
O —0
, 1
0 0.2 0.4 0.6 0.8
Ri
b) VERTICAL TRANSPORT
Figure 3.2.3 Variation of Transport Rates as Functions
of Richardson Number R. , (u1 horizontal
velocity fluctuation, v ' : vertical velocity
fluctuation, t1 : temperature fluctuation;
from Webster. 1964)
17
-------
and current shears were neglected. Thus the values of K _ obtained
include implicitly such effects.
x
It is interesting to note that if an experiment is performed at the sea
surface with floating tracer objects, the objects stay at the surface and
are not subject to either vertical exchange or current shear effects.
For these experiments, effects of shear and vertical exchange were
automatically excluded. However, results from such experiments were
not much different from the other sets of data and is in fact within the
range of scattering. Therefore, it is very Likely that values derived in
previous experiments are close to the cases with zero shear effects.
3. 2. 4 Vertical Diffusion Coefficient
The vertical diffusion coefficient in the ocean is generally much smaller
than the horizontal diffusion coefficient because of the suppression of
vertical transport by stable density stratification. Values of vertical
diffusion coefficients have been obtained by numerous investigators under
various ocean conditions. These are summarized in Table 3.2. 1. These
-22 2
values ranged from 10 to 300 cm /sec. (Note that 1 cm /sec = 1. 1 x
- 3 2
10 ft /sec. ) They are much smaller than horizontal diffusion coeffi-
cients as discussed previously. These K values were obtained by using
various methods including observations of salinity and temperature changes
or diffusion of tracer substances in the ocean. There is no apparent uni-
versal law or values for K . Generally, the vertical diffusion coefficient
has its maximum at the surface and decreases as the depth increases.
This is as expected since the surface layer is subjected to wind agitation
and convective instability. That is, in fact, the reason it remains rela-
tively well mixed. As pointed out by Koczy (I960), the vertical diffusion
coefficient decreases as it approaches the thermocline because of the
increase of density stratification. The value of K may decrease close
-3 2 y
to its molecular value (for heat: 1. 5 x 10 cm /sec and for NaCl: 1. 3 x
18
-------
TABLE 3.2.1
SUMMARY OF VALUES
OF VERTICAL DIFFUSION COEFFICIENT K IN THE OCEAN
y
Note: Molecular diffusivity for heat: 1. 5 x 10" cm /sec (at 20°C, 1 atm)
salt: 1. 3 x 10"5 cm2/sec (at 20°C, 1 atm)
Current or oceanic
region
Philippine Trench
Algerian Coast
Mediterranean
California Current
Caspian Sea
Barents Sea
Bay of Biscay
Equatorial Atlantic
Ocean
Randesfjord
Schultz Grund
Kuroshio
Kuroshio
Southern Atlantic
Ocean
Arctic Ocean
Carribean Sea
South Atlantic Ocean
South Atlantic Ocean
West Atlantic Trough
(50°S to 10°N)
North Atlantic
Indian Ocean
Pacific Ocean
Tidal Channel
(Mersey estuary
and Irish Sea)
Near Cape Kennedy,
Florida
Bikini Lagoon
Coast of Denmark
California Coast
Depth of
layer (m)
5000-9788
0- 20
0- 28
0- 200
0- 100
0- 100
0- 50
0- 15
0- 25
0- 200
0- 400
400-1400
200- 500
500- 700
3000-Bottom
Near Bottom
.Near Bottom
Near Bottom
0- 20
(bottom)
Surface Layer
0- 50
(bottom)
4
Vertical Diffusion
Coefficient K
(cm2/sec) Y
Z. 0-3. 2
35-40
42
30-40
1-3
4-14
2-16
320
0. 1-0.4
0. 04-0. 74
30-80
7-90
5-10
20-50
2. 8
4
4
7-50
4-30
2-40
19 (in August)
1. 3 (in Summer
260
0. 05-1
0. 1-10
15-180
(at wind force
3-4)
Reference
Schmidt, 1917 >
Schmidt, 1917
Schmidt, 1917
McEwen, 1919
Stockman, 1936
Subov, 1938
Fjeldstad, 1933
Defant, 1932
Jacobsen, 1913
Jacobsen, 1913
Sverdrup-Staff, 1942
Suda, 1936
Defant, 1936
Sverdrup, 1933
Seiwell, 1938
Defant, 1936
Wattenberg, 1935 -'
Wust, 1955
Koczy, 1956
Bowden, 1965
(with R. from
0. 1 to * 2. 0)
Carter and Okubo,
• 1965
Munk, Ewing and
Revelle, 1949
Harremoes, 1967
Foxworthy, Tibby
and Barsom, 1966
1 ^
Stommel and
Woodcock, 1951 ***
* As given by Defant, 1961
** As given by Bowden, 1962
*** As given by Harremoes, 1967
**«* As given by Wiegel, 1964
19
-------
10 cm /sec for sea water at 20°C and 1 atm) if vertical turbulence
collapses completely. The value of K may increase somewhat from
the thermocline to the deep layer as indicated by Koczy (I960). Varia-
tion of K in the ocean remains to be investigated more extensively.
Present day methods of determination are based upon variation of the
vertical spreading of a certain substance. In the region of small vertical
diffusion coefficient, a long observation period is required to achieve a
moderate accuracy in determining K . In the future, direct measurement
of the vertical transport rate (-v'C1) may become feasible as instrumen-
tation improves.
The vertical transport decreases as R. increases. For R.= 0,
i.e. , no density gradient, the transport rate is at its maximum. The
dependence of K as a function of R. is a basic problem in fluid
mechanics which should be fully investigated. When such a relation is
known, it can be used to estimate K in the ocean if p and du/dy
are known.
A summary of results of various investigators is as shown in Table
3.2.2. These relationships all show correct trend but are all dif-
ferent in quantitative prediction. This remains to be clarified in the
future. Among these eight proposed equations, all have right asympto-
tic behavior for R.=0, i.e., tr-° neutral case, except Kolesnikov et.
al. (1961) and Harremoes (1967). The equations proposed by Kolesnikov
et, al. , and Harremoes are based upon the density gradient ( £ ) only. The
equations porposed by Holzman (1943) and Yamamoto (1959) show that
there is a limiting value of R , or a critical Richardson number, at
i
which K vanishes. The values of the constant 8 in. these equations
have to be determined before these equations can be applies. A major
difficulty is the lack of accurate relevant data on K .
y
For further discussion on the diffusion coefficients, the reader is
referred to Koh and Fan (1969).
20
-------
TABLE 3.2.2
SUMMARY OF FORMULAS
ON CORRELATION OF VERTICAL DIFFUSION COEFFICIENT K
WITH RICHARDSON'S NUMBER R. (OR DENSITY GRADIENT e) '
Note: K 0 : K at R. = 0, i.e., the neutral case 6 : proportionality constant varies
from case to case
Rossby and Montgomery
(1935)*
K
Rossby and Montgomery
(1935)*
K
Holzman (1943)*
K
Yamamoto (1959)*
K
Mamayev (1958)*
K = K Q
y y°
Munk and Anderson
(1948)**
K = K
yO { ° i'
= 3.33 based upon data by Jacobsen (1913)
and Taylor (1931)
Harremoes (1968)
v , ,n-3 -2/3 2,
K = 5 x 10 xe cm /sec
y -i
note: g in m ; approximate experimental
-9 -5-1
range 5x10
-------
3. 3 Barge Operation 1 - Simple Over-Board Dumping
Perhaps the simplest way of discharging waste material into the ocean
from a barge is a simple over-board dumping. After its release from
the barge, the cloud of waste material will descend by virtue of its
momentum and buoyancy. As the cloud moves downward it will push the
ambient fluid around it, experience drag from the flow field, while
entraining some of the surrounding fluid. During this process the solid
particles inside the cloud tend to settle out of it. If the ambient is
density stratified, the cloud could reach a neutrally buoyant position
and oscillate about it. At this point the vertical motion is much reduced
and the effects of ambient density stratification become dominant with
the result that the cloud tends to collapse vertically and spread out hori-
zontally seeking a hydrostatic equilibrium with the ambient fluid.
If the waste material consists of fluid only, then it can be treated as a
buoyant element (sometimes termed a thermal by meteorologists).
Buoyant thermals have been studied in a still water tank by Scorer (1957).
Both Scorer (1957) and Woodward (19&9) reported on the flow field in and
around a thermal. Their description can be summarized schematically
for the barge operation 1 as shown in Figure 3.3.1. The buoyant element
is seen to have the shape of a hemisphere reminiscent of a vortex ring.
The rise of thermals in a stratified still environment has also been inves-
tigated by Morton, Taylor and Turner (1956) and buoyant vortex rings
have been investigated by Turner (1957 and i960). Other related articles
on this subject include Richards (1961, 1965), Turner (1963), Hall (1962)
and Turner (1966). More recently Koh and Fan (1968) applied the thermal
model to the prediction of the radioactive debris distribution subsequent
to a deep underwater nuclear explosion. Their model included both den-
sity stratification and shear current. In the same report, Koh and Fan
(1968) further analyzed the dynamic collapse of a neutrally buoyant cloud
in a quiescent stratified ambient assuming no entrainment.
22
-------
U(t)
y
Figure 3.3.1 Schematic Defiaition Sketch
23
-------
la this study, Koh and Fan's approach is further extended to apply to
a waste cloud with (or without) solid particles in it. In the dynamic
collapse phase, their model is extended to include entrainment and
shear current.
3.3.1 Convective Descent
After being released from the barge, the waste material mass will be
assumed to retain its identity as an entity as it convects. Settling of
solid particles is allowed as the situation occurs. A mean linear size
and mean velocity will be defined for the element as b(t) and U(t) where
t is the time. Let p(fc) be the mean density of the element and p (y)
the ambient density. Let U (y) be the ambient current, assumed hori-
a
zontal and function only of vertical coordinate y (See Figure 3.3.1).
The characteristics of the element are assumed similar at all stages of
its motion, and to retain the shape of a hemisphere. The various solid
particles inside will be assumed to have densities p .(i = l,2,...,n).
S L
Because of size distribution, the solids are further characterized by
their fall velocities w . . , (i = l,2,...,n;j = l,2,...,m). For each
J
specific density p . and fall velocity w .. the concentration is desig-
nated C
Then the equations governing the motion are:
Conservation of Mass:
f - EP. -
Conservation of Momentum:
SSSijPs.U (3.3.2)
24
-------
Conservation of Buoyancy:
dt
.-.. .
f . . ii si
1 J
(3.3.3)
Conservation of Vorticity:
dK
dt
= -AG
(3.3.4)
Conservation of Solid Particles:
TT--
dt ij
"
ij
(3.3.5)
where
V
2 3
p jnb
total mass in the cloud
E = 2nb
-4 ->
U - U
a
entrainment in volume
S.. = nb |w ..
C ..(1 - B.
settled solids in volume
M
"nb U
inomentuni
25
-------
2 3
= JTT b g(p - pa)
buoyancy force
2, I:
x
- u
(u - u )
drag force in x-direction
D
drag force in y-direction
D
(w - w )
ct 3.
drag force in z-direction
B = f nb3(p (o) - p)
j a
buoyancy
26
-------
A = Cb2
pa(o)
coefficient for vorticity dissipation
dP
a
dy
ambienb density gradient
ambient density difference between that at the free
surface and that at the position of the centroid
of the cloud
density difference between that at free surface and
the solid particles
P.. = |nb3C .. (3.3.6)
U 3 sij
solid volume in the cloud
27
-------
'In the above equations, a is the enbrainmenfc coefficient, |3 . . is the settling
~t
coefficient, j is a unit vector in the vertical direction, C is the drag
coefficient, C is the apparent mass coefficient, p (o) is the density at
m a
the free surface, K is the vorticity, and C is a vorticity dissipation
coefficient (equals to 3, according to Turner, [I960]).
In the above formulation, the entrainment is assumed to be proportional to
the relative velocity between the element and the surrounding fluid and to
the area of the hemispherical front. The entrainment coefficient a should
in fact depend on the properties of the cloud and the ambient fluid and the
turbulence structure inside and outside the convective element, a has been
experimentally determined to be approximately 0.25 by Scorer (1957) and
Richards (1961). Generally, it is found to be constant for each single
experiment, but tends to vary significantly among experiments. In one
rare experiment, Richards (1961) even observed a t° change abruptly
from 0.15 up to 0.62 during convection. In studying the motion of a vortex
ring, Turner (I960) found that
a = - S_ , (3.3.7)
by assuming similarity where C,was found to be 0. 16. When waste mate
rial is dumped from a barge, vorticity can be generated from the initial
momentum and buoyancy. However, as the cloud descends the vorticity
becomes zero, Turner's assumption cannot hold. Since a is expected to
approach a found in turbulent thermals, it seems reasonable to postu-
late that the dependence of a on B and K might be of the form
a =a0(tanhf ^ ) 1 (3.3.8)
28
-------
The only justification for Equation (3.3.8) is that it tends to the correct
limits. Thus when K is large, a approaches Turner's assumption,
while when K is small a approaches Q. • However this expression
has not been verified by experiment and is presented here only as a
conjecture .
The mechanism of settling from a convecting cloud is another complicated
subject that has not been thoroughly investigated. As observed in the
related research of buoyant thermals, the convective cloud is often a weak
vortex, and the turbulence inside the cloud is related to the descending
velocity of the cloud. From dimensional analysis, it may be argued that
f (q.., w .., v, b, Q ., C .., C ) = 0
^ij' sij' ' ' Hsi sij' s
where q. . is mass rate settling out of the cloud and C is .£, .<£, C • • ,
the total concentration. Thus
WsijPsi
Equation (3 . 3 . 9) shows that the dimensionless mass rate of settling
is a. function of the ratio of the descending velocity and the fall
velocity of the solid particle, the concentration of each group and the
total concentration. At high concentrations, hindered settling could
occur. Now equation (3 . 3 . 9) can be further simplified to the following
form
q.. = w .. nb2 p . C .. (1 - «..) (3.3.10)
1J S1J S1
where (3. . is defined to be a settling coefficient which depends on
c i
C .. and C. Although the functional form is not available at present,
29
-------
B. . is expected to be between 0 and 1, the extreme representing the two
cases of settling freely or no settling. Discussion of several situations
will help in the understanding of j3... If v is zero and the concentration
J
is low, the solid particles would settle freely and'hence R.. is 0. For
J
particles with positive fall velocity, and for v>w .., it can be visualized
that even if a solid particle does settle out of the cloud, it will be over-
taken by the cloud, i.e., no settling will occur and 8 • • - 1 • However,
when the cloud is moving opposite to the direction of settling, i.e. , float-
ing particles in a descending cloud 'or sinking particles in a rising cloud,
the criteria for 8- • are less obvious. If v is relatively large, the solids
are envisioned to be trapped in the cloud by turbulence, and if v is small
settling will occur. The actual functional form of 8 • • must be determined
by future experimental investigation. In the present model, 8-- is assumed
if I v/w .. I <; 1
1 sij i
o if l^siji^1
where 8 is a constant which is assumed known.
The drag coefficient, C_^ , is in general a function of Reynolds number,
hence it depends on b and lU - U I . In .the convective phase C-.- is chosen
to be 0.5. C^_, the apparent .mass coefficient should be in the range
to be of the form
In Equations (3. 3. 4) the decay of vorticity is assumed to depend on the
ambient stratification only. In fact, the vorticity decay is likely to be much
more complicated and the formulation is therefore subject to change when
better knowledge is acquired.
30
-------
3.3.2 Dynamic Collapse
In undergoing the convecfcive descent phase, the waste material cloud
usually gains a significant amount of mass and momentum through
entrainment, particularly if the ambient is moving. As the cloud comes
close to a neutrally buoyant position, it may have a horizontal velo-
city close to that of the ambient. At the same time, the concentration
of the waste material would be greatly reduced while the vorticity,
having been dissipated through the action of ambient stratification and
turbulence would become insignificant. At this point the further motion
and deformation of the cloud is still governed mostly by its momentum,
buoyancy and ambient density stratification. The momentum tends to
overshoot the convective element beyond the neutrally buoyant position,
while the buoyancy force tends to bring the convective element back to
the neutrally buoyant position. The combination of these two forces
tends to make the cloud oscillate vertically. While the gross vertical
motion of che cloud is largely suppressed, the cloud tends to collapse
vertically and spread out horizontally seeking a hydrostatic equilibrium
with the ambient fluid due to the density stratification in the latter. The
conservation equations used in the convective descent phase still hold.
However, as the cloud is collapsing more dimensions are needed to des-
cribe the phenomenon fully. If the cross section of the cloud is assumed
to retain an ellipsoidal shape, it could be characterized by its major and
minor axes b and a.
Neglecting vorticity, the conservation equations read:
Conservation of Mass
dV _ -.-^ ...._.„
•dt ' Epa-SijPsi (3.3.11)
31
-------
Conservation of Momentum
d.M
dt '
F j - D + Ep U -• EES.. p . U
J a a . . ij r si
i J J
(3.3. 12)
Conservation of Buoyancy
dB
dt
EAr - XIS..A .
f . . ij si
i J J
(3.3. 13)
Conservation of Particles
dP. .
11
dt
(3.3. 14)
where V
4 ,
p -T-TT ab
total mass inside the cloud
2iri b
a2b
x ^
c dt ,
entrainment in volume
S..
b w . .
settled solid in volume
M
i-nomentum
F
4 2
TTab g(p -
buoyancy force
32
-------
D
U - U
(u - u J
drag force in x-direction
D
y
2PaCD
U - U
drag force in y-direction
D
•=• P C nab
2 a D0
U - U
(w - w )
drag force in z-direction
B
p (o)
a
- P
buoyancy
p (o) - p
3. 3.
ambie.nt density difference between that at the free
surface and that at the position of the centroid of
the cloud
si
P (o) - p .
a si
density difference between the ambient at the free
surface and solid particles
P..
U
3
..
sij
solid volume in the cloud
[3.3.15)
33
-------
Most of the symbols are as defined before. Q, is introduced in the
formulation to take into account the entrainment due to the collapse.
The above equations cannot be solved unless pertinent information on
the cloud is further introduced as follows.
In studying the collapse of the cloud, it will first be assumed that the
cloud always retains an ellipsoidal shape characterized by its major
and minor axes b and a. Choosing coordinate axes with origin fixed
on the centroid of the cloud, the shape of the cloud may be represented
bv
,2 ,2
(3.3.16)
where a and b are functions of time. In this formulation, the cloud is
assumed to be symmetrical. It should be noted that in practice a sym-
metrical cloud can only obtain if there is no relative velocity between
the cloud and the ambient fluid. Following the convective descent, the
velocity difference between the cloud and the ambient is expected to be
very small, and its influence on the shape may be assumed insignificant.
The ambient density distribution is
p (y) = (p + Ap)( 1 -e(y)y') (3.3.17)
a \ /
where p is the average density inside the cloud, Ap is the density dif-
ference between the cloud and the ambient at y1 = 0. (Note that Ap can
be positive or negative.) e(y) is the density gradient at y1 = 0. Note
that p, Apande(y) are continuously changing as the cloud moves in an
arbitrary density stratified ambient. The cloud and ambient condition
are shown in Figures 3.3.2a and 3.3.2b.
34
-------
-i-\-;
I?
(a) Configuration of the cloud (b) Ambient conditions (c) Slice segment
Figure 3.3.2 Definition sketch
-------
As the vertical motion of the cloud slows down, the entrained ambient
fluid might also make the density inside stratified. It is therefore fur-
ther assumed that the density distribution inside the cloud is
P*(y', r', t) = p/l -^ e(y)y'j (3.3.18)
where o < y< 1 is a distribution constant to be determined from actual
density distribution inside the region. If the density distribution inside
the region is the same as the environment, i.e., y = 1> there will be no
spreading. Otherwise, v < 1 > an<^ the cloud will collapse to seek a hydro-
static equilibrium.
a
The quantity Y — represents a much simplified approximation to the
3,
ratio of the density gradients inside and outside the cloud. This |hould
be further examine
possibly modified.
be further examined experimentally in the future and the term y —
ct
Since it is assumed that the cloud is always symmetric, the collapse
of the cloud can be determined by studying a slice of a segment as shown
in Figure 3.3.2c. Taking this slice as a free body it is seen that there
are two categories of forces on the slice: a) the driving force arising
from the difference in the density structure, and b) the resistive forces
consisting of i) the local inertia force of the segment, ii) the form drag
of the collapsing segment and iii) the skin friction of the collapsing region.
In computing the driving force, one is referred to Figure 3.3.2. The
pressures at B and B1, C and C1 should be equal respectively. Assum-
ing hydrostatic pressures, the pressure distribution in the ambient can
be calculated. The pressure inside depends on Ap» the acceleration of
the convective element and the parameter y. In the present study, Ap is
always very small and the acceleration effects are generally averaged out
as the cloud oscillates about its neutrally buoyant position, thus the
36
-------
pressure distribution inside the cloud is approximately hydrostatic. Then
the horizontal pressure force acting on the slice is
T5 d8 (3.3.19)
where d0 is the angle of the segment at y1 =0.
In computing the resistive forces, it is assumed that che horizontal
velocities of elements inside the slice are related to the
distance r1 from the centroid of the cloud, and the velocity of hori-
zontal deformation is characterized by the velocity of the centroid of
the slice segment which is linearly related to the major axis of the
ellipsoid b.
In collapsing, the horizontal spread and entrainment occur simultaneously,
and the velocity of the centroid of the slice can be divided into two parts;
one due to entrainment and the other due to collapse. Entrainment adds
mass to the slice segment and it may entrain momentum into the cloud
if the ambient is moving; however, the entrained momentum is already
assumed to contribute to the momentum of the cloud as a whole as shown
in Equation (3 . 3. 12)and the cloud retains a symmetric shape. So when
entrainment is considered for the slice, although the centroid of the
slice is moving, there is no dynamics involved in this velocity. Hence,
we will designate two velocities v, and v? representing the velocity of
the segment tip for collapse and entrainment respectively and
db
dt " Vl V2 (3.3.20)
Then the local horizontal inertia force is
v, }dQ (3.3.21)
37
-------
The form and skin friction drags are
Dd
2
F:, "= C. . . p - v, de (3.3.Z3)
f fnctn^a Za 1 ^
where C, and'C. . are numerical coefficients similar to that of
drag frictn
the drag coefficient for a wedge and the kinematic viscosity of the fluid.
The equation of motion in the horizontal direction for the slice is
~ FD " Dd " Ff (3.3.24)
In Equation (3. 3. 20), the tip velocity is related to the velocity induced by
entrainment v-> . However, in Equation (3.3. 11), only the gross mass
entrained into the cloud is considered. In the dynamic collapse phase,
as the cloud collapses horizontally, it can be visualized that the entrained
mass is mostly added to the tip of the cloud, hence equation (3.3. 11) can
be rewritten to specify how entrainment adds to the growth of b. That is
EP _ Is...p .
a i j ij Msi
gj J (3.3.2b)
.P --Ti ab
Equation (3 . 3 . 25) states that in each step of entrainment the contribution
of the entrainment to the growth of b is approximated by holding p and
a constant.
Equations (3. 3. 11--25) constitute a set of equations readily solvable.
for the pertinent-parameters a,, b, U, P and C; . . given a set of initial
S1J
conditions that ca".-n be obtained-from the solution of the convective
descent phase.
38
-------
3. 3. 3 Bottom Encounter
If the density stratification is not strong enough the waste cloud is
ultimately going to hit the bottom and spread out at the bed while the
settling of the solid particles continues. In this Section, a mathematical
model will be developed for this possibility. The model used will be
essentially an extension of the model for dynamic collapse.
When the cloud hits the ocean bottom, it is assumed that it keeps the
shape of half an ellipsoid as shown in the upper half of Figure 3. 3. 2a.
The equation for its shape is
'2 r'2
— +— = 1 (3.3.26)
Again the cloud is assumed to remain symmetric although velocity dif-
ference between the cloud, the bed and the ambient is allowed. The
ambient density distribution is
?Jy) = (p+Ap)(l- e(y)y') (3.3.27)
3,
where e(y) is the density gradient at the centroid of the half ellipsoid.
The density inside the cloud is assumed to be
/ Ya \
p'V.r'.t) = ,- 1 - —-- e(y)y') (3.3.28)
\ a /
The situation is very close to that of dynamic collapse; the vertical
motion is suppressed by the bottom, and the cloud is mainly undergoing
only horizontal spreading. The equations used in dynamic collapse can
essentially be used again. However, two more forces, the reaction force
and friction force at the bed, must be incorporated into the equations.
The equations are
39
-------
Conservation of Mass
~ = EC - EE S..c . (3.3.29)
dt -a . . 11 si
i J J
Conservation of Momentum
~»
= Ff-S+E; ft- -r.^rs..: .tf-F^ (3.3.30)
J • a a . .. ij • si F
Conservation of Buoyancy
dB _ „ A q " n 3 31)
^T^ -JCrf--i_;_o...... ij.j.ji;
dt f . . ij s]t
1 J
Conservation of Particles
dP..
•~3- = - Sr (3.3. 32)
Dynamic Equation for a Slice of a Half Ellipsoid
I = F.-Dj-F.-FJ: (3.3.33)
Q a f DI
The Tip Velocity of the Segment
db , /-> i i * \
-rr = v,+v (3. 3. 34)
at i c
Contribution of Ei>tra|nment to tne Growth of b
(3.3.35)
E : - i T S. . ;• :
a . i"s
4,0
-------
2
where V - p-r
E
tobal mass in Lhe cloud
Vb2-a2
*• ,,
c dt
entrainment in volume
S.. - nb2 |w .. 1C ..(1 - 5 ..)
13 ' sij sij ij
settled particles in volume
-» 2 2~*
M = C, , -— nab U
M 3
momentum
buoyancy force
D = 4- P CT- nab
x 4 ^a D ' a '
drag force in x-direction
D - 0
y
drag force in y-direction
41
-------
D = -p C nab
z 4 a D,
U-U
(w- w
drag force in z-direction
2 2
u + w
V
resultant horizontal velocity
F_ = F, Frictn uAy
r x D
friction force in x-direction
+ rrb2 Z I |w ..I; . C ..(l-g..
1 sij ' • si sij pij
reaction force at bed
FbFrictnw/'
friction force in z-direction
2 2
B = -^nab ( p (o) - p)
J 3.
buoyancy
p (o) - p
cL d.
ambient density difference between that at the free
surface and that at the position of the cloud centroid
/, . = P (o) - P .
si a si
density difference between the ambient at the free
surface and the solid particles
42
-------
sij
solid volume in the cloud
inertial force of a slice segment of a quadrant
ellipsoidal cloud
driving force of the slice segment
D^ = C, P ¥" KKde
d drag a 4 ' 1 ' 1
form drag of the slice segment
2
F, = Cc . . p — v.d8
f frictnKa2a 1
skin friction of the slice segment
Fbf = FbFFde/2- (3-3.36)
friction force at the bed
In addition to the Equations (3.3.29-35), one more equation is needed for
closure. That is
av
v = 0.75-T-1 (3.3.37)
b
where v is the vertical velocity of the centroid of a half ellipsoid.
43
-------
Equation (3.3.37) is obtained by assuming zero entrainment, and the
vertical deformation is readily obtained from horizontal deformation by
continuity. Note that in computing the friction forces at the bed, reac-
tion force times the friction coefficients F, and F is used.
1 rictn
Now Equations (3.3.29-37) are readily solvable given initial conditions
that can be either obtained from the end of convective descent or from
dynamic collapse. The solution and the links between these three phases
will be deferred to Section 3.7.
44
-------
3 . 4 Barge Operation Z: Jet Discharge
A large number of barges discharge the waste material through nozzles
at the bottom of the brrge, either by gravity or by pumping, while the
barge is cruising at a certain speed. Near the nozzle, the flow phenom-
enon is that of a sinking jet in a cross current. The jet entrains ambient
fluid and momentum while also experiencing a drag force from the ambi-
ent due to the pressure difference between the upstream and downstream
faces of the jet. As a result, the jet grows in size and bends over in the
direction of the ambient current. The waste material is diluted through
entrainment of the ambient fluid and solid particles settle out of the jet
as the situation allows. As the jet goes further downstream, it becomes
less active and the influence from the ambient density gradient becomes
dominant, and the jet fluid will spread out horizontally seeking a neu-
trally buoyant position.
The mixing phenomenon in buoyant jets and plumes has been studied by
numerous investigators. Morton, Taylor and Turner (1956) applied an
integral method to the problem of a buoyant plume discharged from a
point source into a linearly stratified ambient fluid. Brooks and Koh
(1965) analyzed the two dimensional buoyant jet problem with applica-
tion to the design of submerged ocean outfall diffusers. Fan (1967)
examined the case of a buoyant jet discharging at an arbitrary angle
into a linearly density stratified quiescent ambient. Fan also analyzed
the case of a buoyant jet in a uniform cross current. In the Latter case,
the ambient is not density stratified. In treating the problem of a buoyant
jet in a uniform cross current, Fan assumed an entrainment mechanism
based upon the vector difference between the characteristic jet velocity
and the ambient velocity and the existence of a gross drag term for the
unbalanced pressure field on the sides of the jet flow. Fan found that the
drag coefficient needed to be varied from 0. 1 up to 1.7 in fitting the predic-
tions from his mathematical model to experimental data. Abraham (1970)
45
-------
examined the same problem of a buoyant jet in a uniform cross current
employing a similar approach except that the entrainment was assumed
to consist of two parts: one due to the momentum jet and the other due
to the buoyant plume. Abraham was successful in fitting all of Fan's
data with the mathematical model with a single value for the drag coef-
ficient .
Singamsetti (1966) investigated the diffusion of sediment in a submerged
jet. The jet containing sand was injecting vertically downward into a
quiescent ambient. Singamsetti found that the concentration distribution
in the jet is approximately normal and the mass diffusion is slightly
faster than the diffusion of momentum.
In this investigation, Abraham and Fan's approach will be extended to
study a sinking jet containing sediments in a density stratified non-
uniform two dimensional cross stream for the flow pattern near the
nozzle. As the jet goes further downstream, since mass and momentum
of the ambient fluid are continuously entrained into the jet, it will move
more or less at the velocity of the ambient, and the ambient density
gradient effects will become dominant so that horizontal spreading
similar to that treated in Section 3.3.2 for Barge Operation 1 will
occur. Collapse of a two dimensional wake in a density stratified still
ambient has been studied experimentally by Wu (1965) and analytically
by Koh (1967) assuming no entrainment. In the analysis of dynamic
collapse of the jet detailed in the following, Koh's basic approach
extended to incorporate entrainment and cross current will be used.
46
-------
3.4.1 Jet Convection
In this section, equations for a sinking jet in a stratified cross current
will be formulated. Figure 3.4. 1 shows a round jet discharging at a
velocity into a cross current. It is assumed that the jet cross section
remains circular. Top-hat velocity, density and concentration of
waste material distribution are also assumed. Then the pertinent
parameters for the jet are b, U, p and C .. which are the radius,
velocity, density and concentration of waste material respectively.
The ambient density stratification is designated p (y) and the cross
ct
current U (y).
3.
As shown in Figure 3.4. 1, the jet can flow in any direction depending
on its initial momentum and ambient current. In the figure, s is in
the direction of the jet trajectory; 9 . , Q~ and 0 ., are the angles between
s and the x, y and z axes; 6, and ft,, are the directions of the resultant
ambient current at position s with respect to the x and z axes respec-
tively; and Y i-s the angle between s and the resultant ambient current.
Physically, the flow is very similar to that in a momentum jet close to
the nozzle. However, as the jet bends over towards the direction of the
ambient current, the rise or fall of the plume is more like a two dimen-
sional thermal. In this study, it is assumed', following Abraham (1970),
that the entrainment mechanism depends on the local mean flow and con-
sists of two parts as follows.
E = 2TTba.(U-U cosy) (3.4.1)
m la
E^ = 2TTba7U siny (3.4.2)
T £ a
Equation (3.4. 1) states that the entrainment due to a momentum jet is
proportional to the perimeter of the jet and the velocity difference
47
-------
(a) Jet configuration
(b) Ambient density profile (c) Ambient velocity and drag forces
Figure 3,4.1 Definition sketch
48
-------
between the jet and the ambient in the direction of the jet travel.
Equation (3.4.2) states that the entrainment due to a two-dimensional
thermal is proportional to the perimeter of the jet and the velocity of
the thermal. Note that Equation (3.4.2) is formulated by visualizing
that the plume is essentially moving with the ambient velocity with a
rising or descending velocity U siny . In order to adequately take
3.
into account these two types of entrainment mechanisms, it is further
assumed that the entrainment E is given by
E = E +E sin8-> (3.4.3)
lil. -L C*
Thus, the total entrainment is equal to the sum of the entrainment by
momentum jet and the entrainment by a two-dimensional thermal modi-
fied by sinQ?. sin9? is arbitrarily chosen to diminish the thermal
type of entrainment when the jet is close to vertical.
In the presence of a cross current, a force arises due to the unbalanced
pressure field at the upstream and downstream faces of the jet. A gross
drag force will be introduced and assumed to be perpendicular to the
trajectory of the jet. The force is
p (U siny)
F = C-J - 2b (3.4.4)
where C is the drag coefficient. The drag force components in x, y
shown to be
-cosy cos 9, + cos&
*
and z directions can be shown to be
p - - -
Dx siny D
-cos'r cos 87
p - - £. p
Dy siny r D
Dz siny D
49
.,
F (3.4.4a)
-------
The settling of solid particles from a jet is a most complicated phenom-
enon. The s<>lid particles in the jet tend to settle out by gravity, however,
they are also kept in the main stream by the turbulence in the jet. In the
formulation of the settling, a settling coefficient a., will be introduced
just as in Section 3.3.1. Then the term for settling of the solid becomes
S.. = 2b w .. I C ..(1 - 8..) (3.4.5)
ij sij i sij ij
Based on the mechanisms of entrainment, drag and settling represented
by Equations (3.4. 1-5), the conservation equations become
Conservation of Mass
_ F TVqo
^— — J-/ P - Jir .*» O . . H . / o . / \
ds a . . ij si (3.4.6)
Conservation of Momentum
si-D (3.4.7)
Conservation of Buoyancy
f- .. . ,, . Q»
ds f . . ij si (3.4.8)
Conservation of Particles
dP..
ds " ""ij (3.4.9)
11 -S..
where
V
flux of mas s
50
-------
"* 2 ~*
M = rrb p UU
flux of momentum
JT b g ( P - pa)
buoyancy force per unit length
if = Pa(o)-p
d. d.
density difference between free surface and
position s
B = (p (o)- p)nb2 U
3,
flux of buoyancy
density difference between ambient free surface
and solid particle
rrb2UC ..
f lux of solid particles (3.4.10)
Together with the geornetric relationship
cos fl +cos 82+cos6=l (3.4.11)
Equations (3.4.6-9) and Equation (3.4. 11) constitute seven-simultaneous
ordinary differential equations for seven unknowns.
51
-------
The equations can be solved with the initial conditions
U(o) = UQ, b(o) = b0, p(o)=p0, Cg..(o) = (
8l(o)=8lo' e2(o)=82o' 83(o)=93o (3.4.12)
The trajectory of the jet center line can be obtained through the follow-
ing equations
dx Q
-T— = COS8,
ds 1
= COS87
ds 2
dz _ Q
ds ~ C°S 3 (3.4. 13)
Note that the formulation above is slightly different from that of Fan and
Abraham. In particular, top-hat distributions are assumed for density
deficiency, velocity and solid concentrations. However, since the inte-
gral approach integrates the distribution over the jet cross section,
there is no essential difference in the resulting equations except for
some modifications on the coefficients which are normally obtained
through experiments.
3.4.2 Dynamic Collapse
When the jet plume is far downstream from the nozzle, it no longer
behaves like a jet. Similar to the dynamic collapse phase of Barge
Operation 1, the jet-plume tends to collapse vertically arid spread out
horizontally seeking a hydrostatic equilibrium in the ambient density
52
-------
gradient. In this section, a mathematical formulation is derived
accounting for both the convection and the collapse of the waste mate-
rial plume similar to that in Section 3.3.2.
Far downstream from the nozzle, the jet is expected to be moving
approximately with the same velocity as the ambient, and the plume
would be more like a two-dimensional thermal rather than a jet. In
analogy with the arguments presented in Section 3.3.2, the cross
section of the two-dimensional thermal will be assumed to have the
shape of an ellipse.
,2 ,2
•2=- + •%- = 1 (3.4. 14)
a* b^
where x1 and y' are coordinates with the origin fixed on the centroid
of the thermal, a and b are minor and major axes of the ellipse and
are functions of time. Again, the ambient density distribution is
pa(y) = (p+ApMl - e(y)y') (3.3.15)
and the density distribution inside the cloud is
Y a
p*(y',x',t) = p'l - -^-eWy' i (3.3.16)
The configuration of the two-dimensional thermal is shown in Figure
3.4.2.
By considering a two-dimensional thermal with a length L, the conserva-
tion equations are:
Conservation of Mass
— = E S2
dt ~ pa " 1 i
53
-------
2 ,2
UQ
y
I
(d)
Figure 3.4.2 Definition sketch
-------
Conservation of Momentum
->
.-) Tyf _». -> -» .. „ _»
~- = F. - D + E o U - *? S p.U (3.4.18)
dt j pa a i j ij Hsi
Conservation of Buoyancy
^ = EAf - *4 S..A - (3.4.19)
dt i j ij si
Conservation of Particles
dP..
—rfL = - S.. (3.4.20)
dt 11
where
V = p rr abL
total mass in the buoyant element
V 2 ,, 2 .» ^
E = 2n ^a :"b L(a.,|u- U , ,-.
dt J cl TC
total entrainment in volume
S.. = 2bJL
w ..1C ..(1 -8..)
sij| sij ij
settled solids in volume
M = CMprrabLU
momentum
TTabL(p - p )g
cL
buoyancy force
55
-------
D = ic^ 2a L sin cop I U - U I (u - u )
x 2 D a i a ' a
drag force in x-direction
D = yCp, 2bLP U- U I v
y 2 D a a'
drag force in y-direction
. -* -»
D^ = T^n ^aJLcoscop |U- U& (w - w )
3 a
drag force in x-direction
B = nab L(P (o) - p)
3,
buoyancy
ambient density difference between that at the free
surface and that at the position of the centroid of
the buoyant element
. = p (o) -p .
si a si
density difference between the ambient at the free
surface and the solid particles
P.. = nabLC .. (3.4. 21)
ij S1J
solid volume in the buoyant element
56
-------
In the above equations, cc3 and a4 are entrapment coefficients for
convection and collapse respectively. CDS is drag coefficient for a
two-dimensional streamlined wedge, CD4 is drag coefficient for a
two-dimensional plate, and cp is the angle between L and thex-axis.
The equation for the dynamic collapse of a quadrant of the elliptical
cylinder with length L is (in analogy with Section 3.3.2)
1 = F-D-F 0.4.22)
where
d /ab
inertia
driving force
P a
DD = '
form drag
Ff = CLv (3.4.23)
skin friction
The tip velocity of the segment is given by
|p = vj + v3 (3.4.24)
57
-------
In the Equations (3. 4.22 ) and (3. 4. 24) v, is the tip velocity due to
collapse and v? is the combination of the tip velocity due to dynamic
collapse and that due to the stretching of L.
D d 1—I / Q ^ *3 £ ^
V2 = vl - L^F (3.4.25)
v_ is the contribution to tip velocity from entrainment by instantan-
eously holding p, a and L constant.
In the present formulation, at the end of jet convection, the two-
dimensional thermal model immediately takes over the convection of
the waste material plume. At that moment, the horizontal velocity of
the element may be different from that of the ambient velocity. As the
buoyant element is moving downstream, it will be slowed down or
speeded up by entrainment of ambient momentum and by the drag force
applied on it. However, the supply of waste material is continuous from
the jet, hence, the two-dimensional element L should be able to be
either stretched or squeezed in order that the trajectory of one convec-
fcive element be capable of representing the steady picture of a continu-
ous plume. In the estimate of the stretching of L, it is assumed that
= constant (3.4.27)
The contribution from the stretching of L to the tip motion is obtained
by assuming that the minor axis a is kept constant, and that no entrain-
ment occurs at that moment.
58
-------
The trajectory of the two-dimensional buoyant element is furnished by
dx
dF = U
dt
|p = w (3.4.28)
The initial conditions for dynamic collapse are from the information at
the end of jet convection.
59
-------
3.4.3 Bottom Encounter
Just as in the case of Barge Operation 1, the waste material plume can
reach the ocean bottom and spread out there if the ambient density
stratification is not strong enough to arrest the vertical descent of the
plume somewhere above the bottom. A mathematical model \vhich is
an extension of the model presented in Section 3. 4. 2 will be developed in
this section for this special situation.
At bottom encounter, the cross section of the plume is assumed to have
a half elliptical shape as shown in the upper half of Figure (3. 4. 2a)
,2 ,2
£_. + 2L_ = i (3.4.29)
L, ,t,
a b
The ambient density distribution is
p (y) = (p+ AP)(1- -c(y)y') (3.4.30)
Si
The density distribution inside the plume is
>•- Ya
p'V.x'.t) = p(l-—- e(y)y') (3.4.31)
cL
The ocean bed is assumed to be horizontal. The plume is allowed to move
as an entity with respect to the ambient fluid and the ocean bed, while it
collapses vertically and spreads out horizontally. By following es-
sentially the same arguments as presented in Sections 3. 3. 2, 3. 3. 3 and
3.4.2, the equations become
Conservation of Mass
dV
3- =Ep-L;;S..p. (3.4.32)
dt Ka . . ij Ksi
1 J J
60
-------
Conservation of Momentum
Conservation of Buoyancy
- ESS.. A (3.4.34
f . .. !j 81
Conservation of Particles
dP..
where
V = — prrabL
total mass in the buoyant element
db ,
entrainment in volume
S.. = 2bL |w .. |C ..(1-R..
ij ' sij ' sijv Hij'
settled solid in volume
| CMpnabLtf
momentum
-nab L( p- p )g
LJ 3,
buoyancy force
61
-------
Fx b rictnv
bottom friction force in x-direction
F_ = - F, = -F + ~c.. pnabLv) + £ ES..VP .
Fy b dt 2 M K . . 11 si
y J
reaction force at the bed
F_ = F, F . , (w-w, )
Fz b nctn b
bottom friction force in z-direction
Dx
resultant velocity difference between the plume element
and the bed
drag force in x-direction
D = -cn.2bLp |- j v
y 2 D4 Ka ' a1
drag force in y-direction
drag force in z-direction
B = j TrabiXp , (o)- p)
u 3.
buoyancy
62
-------
M = Pa(o)-pa
ambient density difference beyween that at the free
surface and that at the position of the buoyant element
Asi = pa(o) " psi
O i Q. S i
density difference between the ambient at the free
surface and the solid particles
P.. = 4-nabLC ..
13 2" sij
solid volume in the buoyant element (3. 4. 36)
In the above equations, u, and w, are velocities of the bed with respect
to a coordinate fixed on the moving barge.
The equation for the dynamic collapse of a quadrant of the elliptical
cylinder is
I = F -D - F - F (3.4.27)
where
I -A^L v )
inertial force
driving force
63
-------
form drag
F, = C, . -Lv.,
f fric a 2
skin friction
F = F F F
bf b rictn 1
friction at the bed (3. 4. 38)
The tip velocity of the segment is
f = v1+v3 (3.4.39)
In the above equations, v, is the tip velocity due to collapse, v- is
tip velocity due to the combination of collapse and the stretching of
element length, and v, is the tip velocity due to entrainment (see Section
(3.4.2)). And the vertical velocity of the centroid is obtained by
continuity assuming no entrainment and settling at an instant.
The trajectory is again furnished by
dx
dF
= v (3.2.41)
dz
64
-------
The initial conditions for this phase can be obtained from jet convection
or dynamic collapse.
65
-------
3 . 5 Barge Operation 3 - Discharge into Barge Wake
In the disposal of highly harmful waste material, it may be desirable to
discharge it into the wake of a moving barge to achieve high initial mixing,
After the initial mixing, the waste material plume is again expected to
undergo a convective descent phase due to the density difference between
the mixed waste material and the ambient fluid, and a dynamic collapse
phase when the plume reaches a neutrally buoyant position.
The initial mixing process in the wake of a moving barge is essentially
a mass transfer process in separated flow. The flow field in the wake
of a moving barge is complicated by flow separation from the stern, wave
generation, turbulence developed in the shear zone, and back flow fol-
lowing the separation. The phenomena of mass and heat transfer in
separated flow has been investigated in the literature; for example,
Richardson (1963) studied experimentally the heat and mass transfer in
the wake of a cylinder. He found that shear and mass transfer in a
separated region is proportional to the two thirds power of Reynolds
number. Ruckenstein (1970) and Spalding (196?) developed some theo-
retical relationships between the mass transfer coefficient and physical
parameters of the flow for two simple flow conditions: constant shear
and linear shear. There is some confirmation between the studies by
Richardson and Ruckenstein. Hanson and Richardson (1968) further
investigated the flow field in the wake of a cylinder in great detail in
an attempt to find a relationship between the flow field and heat or mass
transfer. Elzy et al (1968) studied the heat transfer for a porous cylin-
der in a cross flow where the local heat transfer coefficient for different
injection rate and turbulence intensity were obtained. Mass transfer in
separated flow depends heavily on the pattern of the flow field. For the
flow field in the wake, in spite of its being a classical problem, theo-
retical development has been mainly limited to the far field of the wake
66
-------
as presented by Swain (1929). Carmody (1964) and Chevray (1968)
made extensive experimental and analytical studies on the near wake
of a disk and an ellipsoid respectively. The results were found to
depend on the form of the body. Kuo and Baldwin (1969) examined the
formation of wakes behind elliptical plates •• Strong dependence of the
far field on the results of the near field was found. • Naudascher (1967,
1968) in a -study of the general characteristics of jet and wake flows,
developed a general self-preservation hypothesis. The lateral distri-
bution of various mean-flow and turbulence characteristics obtained
from measurements in axisymmetric jets and wakes were shown to
confirm his concept of self-preservation.
Ketchum and Ford (1952) have analyzed the discharge of fluid waste
material into the wake of a moving barge where mixing was observed
to be instantaneous in the vertical direction; The transverse dispersion
coefficients were calculated from prototype experimental data. Abraham
et al (1970) investigated the mixing of acid into a propeller stream of a
moving ship for the case where the mixed stream density is very close to
the ambient density so that the mixed stream stays'-clos'e to the free sur-
face. The dilution in the wake was measured..
In the -following, an initial mixing phase will.be-formulated based on the
assumption that the material is perfectly mixed into the main stream as
calculated by Naudascher',s- semi-empirical analysis. After the initial
mixing a two dimensional thermal approach will be-'iis-ed for the convec-
tive de.scent phase. For the subsequent dynamic collapse phase, the
analysis presented in Sections 3 . 4. 2 and 3'. 4 .3' wild be' us'ed.
67
-------
3.5.1 Initial Mixing la the Near Wake of a Barge
The mixing phenomenon in the near wake of a barge is so complicated
that a complete analytical solution to the problem is impossible. Here
a simple gross approach will be used. This is desirable from a prac-
tical point of view since at least some insight into the significance of
disposal into the barge wake can be obtained from the present analysis .
It is first assumed that the flow rate of the waste material is compara-
tively small so that it is completely mixed into the wake by the turbu-
lence without altering the wake flow pattern. Secondly, the effect of
waves is disregarded so that the flow pattern can be approximated from
Naudascher's analysis of jet and wake flows.
In generalizing the distribution and development of mean-flow and tur-
bulence characteristics in jet and wake flows, Naudascher reduced the
flow pattern in the near field of disk and ellipsoid wakes (Figure 3.5.1)
to a similarity form by introducing a length scale 4 which is a func-
tion of measured velocity distributions . The velocity distributions are
approximated by
as shown in Figures 3.5.2 and 3.5.3. In Equation (3.5.1), U , is the
local velocity deficit, U , is the maximum velocity deficit and
n =
where y is the radial distance. The length scale &Q was found to vary
for different flow conditions as shown in Figure (3. 5.4). It can be ap-
proximated by
68
-------
vO
B
VT
-u
•£
Q
oi
(O)
(X)
Figure 3.5.1 Definition sketch (After Naudascher 1968)
-------
(3.5.2)
where C, is a constant for each group of data and approximately varies
from 0. 15 up to 1. 0 for the two extreme cases of a blunt disk and a
streamlined body of revolution. R is a characteristic length of the body
which can be chosen as the geometric mean of the barge depth and half
width in the present analysis.
By assuming that similarity exists after a distance x = C_R downstream
from the barge stern, the velocity distribution at that position becomes
(3. 5. la)
By referring to Figures 3. 5. 2 to 3.5.4, if we take T)= 2.0 as the
nominal boundary of the wake, then the radius of the wake is
Y = 2C1C*/3R (3.5.3)
and the mean velocity is
U = —^- I U,(l-e" Tl2)2n-ndT1 = 0. 75 U, (3.5.4)
4 n k d d
at x = C_R where similarity of flow pattern begins.
If the waste material is assumed to mix uniformly intovthe wake within
the nominal boundary, then the concentration can be estimated by
Q C
d ° (3.5.5)
0.5nY2U
where Q, is the discharge rate and C.. is the initial concentration of the
d 6 ijo
solid particles. Note that the formulation in Equations (3. 5. 3, 3. 5. 5)
70
-------
•o
ID
0
2DO
Figure 3.5.2 Radial variation of mean-velocity difference in the wake of a disk.
(After Naudascher 1968) (Data by Carmody for UR/v~35,000;
m = -0.57.)
-------
tM
0
0
0.25
0.50
0.75
1.00
1.25
1.50
2.00
Figure 3.5.3 Radial variation of mean-velocity difference in the wake of a
slender spheroid. (After Naudascher 1968) (Data by Chevray
for UR/v~ 1,375,000; m = -0.03.)
-------
20
10
8
DC.
\
o
,-k
tr
r 1 I i i i i
tQ/R IM/R
o CD Chevray (1967), wake of slender spheroid
O e Carmody( 1963), wake of disk
-O c Uj/Uo=3.72l Curtet &Ricou(l964)
o- 3 2.03J round jet
9 2.00 Ortega(1968), round jet
SLOPE I'-3
o
1.0
0.8
0.6
0.4cr
0.2
e
O
o
-o
o-
€
-O
SLOPE 1:3
I i
8 10
20
40
60 80 100
200
x/R
Figure 3.5.4 Axial variation of effective width for various
jet and wake flows. (After Naudascher (1968))
73
-------
is quite general and by properly choosing C, and C_ the flow pattern
of any barge wake can be approximated. The judgment as to the exact
values of C, and C? should be based on further experimental information.
3. 5. 2 Convective Descent
In the initial mixing phase, it is tentatively assumed that because of
the strong turbulent mixing in the near wake, the buoyancy effect is
of secondary importance. However, after the initial mixing, as the
turbulence subsides the buoyancy will make the half cylinder waste
material plume descend vertically to seek a neutrally buoyant position
while it is convected downstream by the ambient current.
In this convective descent phase, it is assumed that the plume retains
a half cylindrical shape so that a two-dimensional thermal approach
can be used. Then by considering a two-dimensional thermal with a
length L, the conservation equations become
Conservation of Mass
—• = Ep -2SS..p . (3.5.6)
Q.U cL • * 11 SI
i j J
Conservation of Momentum
T-D + E0 ft - ZES..p .ft (3.5.7)
dM
dt
Conservation of Buoyancy
.. .
dt . ij si
Conservation of Particles
dP..
Af - ££ S.. A . (3.5.8)
- -S.. (3.5.9)
74
-------
where
V
total mass in the buoyant element
lav + aJ (u -u
2
= nbL |av + a,( lu -u ) + (w-w )
cL 3,
total entrainment in volume
S.. = 2bL |w JC ..(1-p..)
settled solids in volume
momentum
F = jnb2L(p-pa)g
buoyancy force
Dx
drag force in x-direction
D = cr^ 2bL
y 2 D
drag force in y-direction
D = ic bLcoscpp \tf-ti j(w-w )
z £ LJ -, a a a
drag force in z-direction
B = ynb2L(p (o) - p)
£> 3.
buoyancy
75
-------
Af
ambient density difference between that at the free
surface and that at the position of the centroid of the
buoyant element
A . = p (o) - p .
si Ma psi
density difference between the ambient at the free
surface and the solid particles
P.. = ^rrb2LC ..
ij 2 sij
solid volume in the buoyant element (3. 5. 10)
In the above equations, a and a, are the entrainment coefficients for a two-
dimensional thermal and a momentum jet respectively, C_ and Cn
Ul U2
are the drag coefficients for a two-dimensional wedge and a two-
dimensional cylinder respectively, cpis the angle L makes with the
x-axis.
The element length and velocity relationship is assumed to be such that
L
J
\u
= constant (3. 5. 11)
2 2
w
The contribution from stretching of L to the growth of b is obtained by
assuming no entrainment or settling occurring at that moment. The
Equations (3. 5. 6-11) are readily solvable given initial conditions from
the initial mixing phase.
3. 5. 3 Subsequent Motions
Following the convective descent phase, the subsequent motions and
dispersion of the waste plume is obtained by using the same model
developed for the corresponding phases for jets (see Section 3.4).
76
-------
3, 6 Long Term Diffusion
In the long run, regardless of the exact method of disposal of the waste
material from the barge, after the initial phases of convection and col-
lapse the waste plume becomes dynamically passive subject only to
turbulent diffusion, advection and settling of the solid particles.
Ocean diffusion has been studied by investigators from various disciplines
including geophysicists, oceanographers 'and engineers. Previous
studies of marine diffusion have not only determined quantitatively ocean
transport characteristics as discussed in Section 3.2, but also established
different methods of analyses. In general, most of the analysis starts
Trom the concept of the conservation of material expressed as
u|£ + v|£ -f w|£
ox oy da
(3.6. 1)
where c is the concentration of the diffusing substance, (u, v, w) are local
instantaneous velocities in (x, y, z) directions, (D ,D ,D ) are molecular
x y 2
diffusion coefficients in (x, y, z) directions. In turbulent flow, Equation
(3. 6. 1) may be further simplified by resolving the instantaneous con-
centrations and velocities into the sum of time averaged and fluctuating
components.
-
c = c + c', u= u + u'
where the overbar indicates the averaged value which is the result of
averaging over a sufficiently long period to permit convergence of the
averages of the primed quantities to zero. Making the appropriate
substitution in Equation (3. 6. 1), and averaging the resulting equation
over a sufficiently long period, introducing the definitions (3.2. 1)
77
-------
-, -W
K = !__ + D , K = + D , K = _«_ + D
x ^— xy ^ — y z ^ — z
oc y dc y dc
ox oy dz
(3.2. 1)
and dropping the overbars for simplicity gives
|£ + u|£ +v|£ +w|£ -_ B /K I^V-r
ot ox oy oz ox ^ xdxy oy
(3.6.2)
Most of the early investigations were confined to the study of horizontal
diffusion only, employing simplified versions''of Equation (3.6.2); current
shear and vertical exchange were ignored, i. e. , in Equation (3. 6. 2)
K , v and w are taken to be zero. Horizontal diffusion has been in-
y
vestigated by stommel (1949), Joseph and Sender (1962), Okubo (1962),
Carter and Okubo (1965), Snyder (1967) and others. These studies
yielded information on the horizontal transport characteristics of the
ocean, as was summarized in Section 3. 2. Perhaps the most significant
result from these studies is the empirical 4/3 power law, (Equation
(3.2.2)) for the horizontal diffusion coefficient.
A model neglecting the effects of vertical shear current and vertical dif-
fusion is inadequate in describing the whole dispersion phenomenon in
the ocean. The shear current tends to stretch out the distribution and
shift the centroids of material distributions at different levels. Vertical
diffusion tends to redistribute the material in the vertical direction.
These effects result in greater dilution of the diffusing substance than
a model which only considers horizontal diffusion.
The effect of current shear in ocean (or lake) diffusion problems haye
been studied by Okubo (1968), Okubo and Carter (1966), Bowden (1965),
and Csanady (1963, 1966). These analyses were mostly aimed at the
determination of apparent longitudinal diffusion coefficient. Recently,
Koh and Fan (1968, 1969), studied the diffusion of the radioactive debris
78
-------
distribution subsequent to a deep underwater nuclear explosion. In this
investigation, shear current and vertical diffusion were incorporated.
For the horizontal diffusion, the empirical 4/3 power law were used.
Jobson and Sayre (1970) examined the vertical transfer of fluid and
sediment particles in open channel flow. From experimental data of
the vertical distribution of the dispersant at sections downstream from
a continuous source across the width of a channel, they evaluated the
vertical transfer coefficient as function of depth. It was found that for
fine sediment (w = 0.035 fps) the vertical transfer coefficient distribu-
tion is close to the momentum transfer coefficient obtained by assuming
a logarithmic vertical velocity distribution. For coarse sediment
(w = 0. 2 fps), more diviation between the two coefficient distributions
s
were observed. However, from the results of numerical simulation of
the vertical transfer phenomenon, Jobson and Sayre concluded that the
predicted concentration profiles were not very sensitive to the distribu-
tion of the vertical mass transfer coefficient.
Sayre (1968) investigated the longitundinal dispersion of silt in open
channel flow. Vertical momentum transfer coefficient was used for
the mass transfer coefficient. Comparison of the results with exper-
imental data was found to be good.
The approach used by Bowden (1965), Okubo (1968), Sayre (1968) was the
so called method of moments. The method begins by defining the moments
of the concentration distributions across horizontal planes. Equation
(3. 5. 2) is then transformed into a set of equations for these moments
with t and y as the independent variables. The equations of moments
are considerably simpler to solve, and the important characteristics
of the diffusing substance can be determined from only the first few
moments. The problem then is solved with no sacrifice of rigor but at
the expense of some loss of detail. With its simplicity and capability
of incorporating fairly realistic ambient conditions, the method of mo-
ments will be adopted in this investigation and presented in detail in
the following sections.
79
-------
3. 6. 1 Formulation of the Theoretical Model
For solving the long term diffusion problem of waste material in the
ocean environment, the method of moments will be used. Coordinate
system, ambient conditions and basic equation will be given in Section
3. 6. 1. 1, Derivation of equations will be presented in Section 3. 6. 1. 2.
Numerical solution of the equations and the computer program will be
deferred to Section 3. 7.
3. 6. 1. 1 Flow Configuration and Basic Equation
The coordinate system is chosen to be as shown in Figure 3. 6. 1. x, z
are horizontal coordinates, y is the vertical coordinate pointing down-
ward from the free surface. The ambient current in the ocean are
designated as (u , o, w ) in (x, y, z) directions. Note that the mean v-
a a
component is neglected here, considering the effect of the stable
density stratification in the ocean. The mean u and w components
a a
are taken to be independent of the x and z coordinates but may be functions
of y and t. These are all assumed to be known functions. By con-
sidering a specific group of the waste material particles settling with
fall velocity w while being convected and diffused by the ambient cur-
s
rent and turbulence, the conservation of material equation is similar
to Equation (3. 6. 2) except for one more term involving the fall velocity
as follows.
ac , ac , ac a /,_ acV 8 (v ac\, a (v ac\ a / J
•vr— + u -r— +w -^— = 3— |K -r— I + -—IK —- I -t- -—|K — I- T—-fw C
ot 3idx a.dz oxV xdxy dy y yBy/ 9z I z Bz / ayl s ,
(3.6.3)
where C is the time averaged concentration of the waste material,
u and w are ambient currents and w is the fall velocity. Note that
cl 3, S
Equation (3. 6. 3) is satisfied by both the liquid and the solid part of the
waste matter depending on the value assigned to the fall velocity w .
S
80
-------
oo
y=0
WASTE
MATERIAL
(a) Coordinate System
0
Yk2
H
0
K.
K -profile
y
'U
H
'oo
H
u -profile w -profile
Figure 3.6. 1 Coordinate System and Ambient Conditions for long term diffusion model
-------
Corresponding to Equation (3. 6. 3), boundary conditions are needed for
each of the waste material treated as follows.
i) For fluid w = 0
s
y = 0,H ~ = 0 (3.6.4)
oy
Equation (3. 6.4) indicates that there is no diffusion of material through
the boundaries, and the boundaries act as reflecting barriers to the
diffusion material.
ii) For sinking particles w = positive
S
y = 0, K - w C = 0 (3. 6. 5a)
' y dy s
y = H, K- (l-0,)wC-YW = 0 (3. 6. 5b)
oW
- - owC + YW = 0 i3.6.5c)
iii) Floating particles; w = negative
S
y = 0, K - (l-a1)wgC + Y1W1 = 0 (3. 6. 6a)
aw ; aw aw
at
u ,-3-- + w . -~ - + a,w C +v,W.
al ax al bz Is '11
-. aw \ , / aw .
= -1/K -— 1 )+ -2- K —JL (3.6.6b)
ax \ xw ax
82
-------
y=H, K ~ - w C = 0 (3.6. 6c)
7 ydy s
In the above equations, ais a bed absorbency coefficient representing
the probability that a particle of sediment coming into contact with the
bed is deposited, W represents the amount of sediment stored per unit
area of bed surface (or free surface), and yis an entrainment rate
coefficient, defined in such a way that yAt is the probability that a
typical particle resting on the bed is entrained during a short time
interval of duration At. yW represents an average rate of entrainment.
The subscripts 1 and 2 signifies the values at the free surface and at
bed respectively.
As shown in Equations (3. 6. 5, 3. 6. 6), the boundary conditions for solid
particles are fairly complicated. The settling of the sinking solid
particles will be further discussed below. At the free surface the
transport by fall velocity is balanced by diffusion as shown in Equation
(3. 6. 5a). At the bed, a fictitious double layered bed is introduced;
the first boundary is the boundary of the fluid and the second is the
boundary where the solid particles can be stored. At the first boundary
there are three mechanisms for transport, namely convection by fall
velocity, diffusion by turbulence and entrainment by ambient flow. These
are schematically presented in Figure 3. 6. 2 . As shown in Figure
5.6.2, CL, w C represents the rate of deposition down to the second
fw S -v «
boundary, (l-a->)w C is the rate of retention in the fluid, K -r — is the
s y ay
rate of diffusion and Y -^ ? =LS ^e rate of entrainment of particles from
the second boundary back into the fluid. Equation (3. 6. 5b) summarizes
these mechanisms and indicates that the combined actions of diffusion,
entrainment and settling contributes no net transport across the boundary.
On the second boundary, whatever is settled there will increase the
storage of waste material which in turn again is subjected to detrain-
ment. Equation (3. 5. 5c) just expresses the conservation of waste
material while allowing the waste material to be stored,
83
-------
y = 0
wsc
K
dc
VT7
' ' ' X"X" ^ /" /
\
Q2wsc
I(l-a2)wsc
Ai^ dc
pvay
r
rzw2
^^^^^
Figure 3.6.2 Schematic for boundary conditions for solids
settling to the bed
84
-------
and exchanged with those in the fluid. For the floating particles the
same argument applies. However, after the solids have settled to the
surface, it is also subjected to horizontal diffusion by turbulence and
advection by ambient currents at the free surface and therefore these terms
are added a.s shown in Equation (3. 6. 6b).
3. 6. 1. 2 Method of Moments
Equation (3. 6. 3) with any one of the corresponding boundary conditions
for fluid, sinking particles orfloating particles can be solved given the
initial conditions of the concentration distribution in the fluid and the
deposition of the waste material on the bed or free surface. However,
Equation (3. 6. 3) can have analytical solutions only in simple cases, and
with its four independent variables x,y, z and t, a numerical solution of
Equation (3. 6. 3) will require considerable storage and computer time.
Also, as noted in Section 3. 2, the values of the diffusion coefficients
are often determined based upon the gross characteristics of the dispersant.
It is believed to be more consistant in the analysis to ignore the detailed
distributions but solve for only the gross characteristics of the dis-
persant as functions of time and depth.
Applying the technique of the moment method, as given previously by
Saffman (1962), Smith (1965), Tydesley and Wallington (1965) Bowden
(1965). and Sayre (1968), the moments of the concentration C(x, y,z,t)
and waste material deposition W(x,y,z,t) across horizontal planes are
defined as follows:
/*°° r m k 0
C =11 x z Cdxdz (3.6.7)
k, A J J_
85
-------
"xVwdxdz (3.6.9)
These express respectively the k, 4'th moment of the horizontal distribution.
Thus, for example taking k = 4 = 0, Equations (3. 6. 7-9) represents
respectively: (1) The amount of dispersant contained in a lamina of unit
thickness at a distance y below the free surface; (2) all the waste material
in suspension; (3) the amount of dispersant on the bed or free surface.
The sum of Equations (3. 6. 8) and (3. 6. 9) M, +W , is the k, !L 'th
K, Hi rC, X,
moment of the horizontal distribution of all the waste material.
k JL
Multiplication of each term in Equation (3. 6. 3) by x z followed by
integration over the horizontal plane yields the transformed equation,
rr—C, - ku C, . ..-^w C, ,
dt k, SL a k- 1, H a k, 4- 1
Ek
dC
The boundary conditions transform to
i) Fluid w = 0
s
y = 0,H = 0 (3.6. 11)
ii) For sinking particles w = positive
S
(3.6. 12a)
,
86
-------
ac
(1-a)wc-Yw = ° (3-6-12b)
dW
at ' * - cuw C, t + v_W_, = 0 (3. 6. 12c)
ot 2 s k, -i, '2 2k, £
iii) For floating particles; w = negative
S
= 0 (3.6. I3a)
r-W,, - ku .W.. . -jgw .W,. . , + 0^ C.
3t Ik, A al lk-l,£ al Ik, Sb-1 1 s k,
+ Y W.. = k(k-l)K C, . .
1 1 Ik, JL xw k-2, S,
SC
y = H, K ——— - w C, , = 0 (3. 6. 13c)
y . y 3y s k, I
The initial distribution of deposited waste material and that in suspension
can be transformed in the same manner.
It should be noted that the independent variables x and z have been eliminated
in the moments. The solutions to the new set of equations are correspondingly
87
-------
simpler. The penalty for this is that the solutions give only the
moments of the horizontal distribution of waste material rather than
the distribution itself. Still, the behavior of the moments provides a
sufficient basis for the description of the dispersion process. In
principle, any desired degree of detail can be achieved by solving the
equations for sufficiently high values of k and £,. In practice, the first
three moments are usually all that are required.
Table 3. 6 shows how the O'th, first and second moments can be used to
calculate statistical parameters which are useful for describing various
aspects of the dispersion process. In the formulas the subscripts s
and w denote respectively the suspended and deposited components of
the waste material, and T denote the sum of both. The argument (y, t)
indicates the value of the function at y and t. The argument (t) in con-
junction with the subscript s indicates a depth integral for all the suspended
material.
3. 6. 2 Diffusion Coefficients
From Equation (3. 6. 10), it is seen that the diffusion coefficients K ,
X.
K and K are important to the dispersion process. Following the dis-
cussion of Section 3. 2, the relationship between the horizontal diffu-
sion coefficient and cloud size can be defined. However, the interrela-
tionship between the vertical diffusion coefficient and the ambient con-
dition is less well known. In this study the vertical diffusion coefficient
will be assumed to be as given in Figure 3. 6. Ib; the K value is
minimum at the thermocline, and maximum in the mixed layer.
For the horizontal diffusion coefficient, the 4/3 - power law in the ocean.
as discussed in Section 3. 2. 3 will be used. The relation is given by
Equation (3. 2. 2)
88
-------
TABLE 3.6
USE OF MOMENTS TO DESCRIBE DISPERSION
Moment
0
1
2
Desired Statistical Parameter
Volume under concentration
curve
Mean displacement
Variance
Formula
V8(y,t)
vs(t)
vT(t)
xs(y,t)
*s(y,t)
Y(t)
Xw(t)
2
axs
2
azs
2
a
xzs
2
a
y
2
axw
2
zw
2
^xzw
= co.o
= MQ(t)
= M0(t)+W00(t)
= Total waste material
= ci.o/co o.
= C0 1/C0 Q
= M, / M«
1 0
- w /w
W1.0' 0 0
- r / r x 2
~ C2 0' 0 Os
- r / r 7 2
0 2' 0 0~ s
= ci i/co o:xszs
= M-/ Mn - Y2
2 0
- W / W _ x
* * O O ' ** A A
20 00 w
- W / W - x 2
~ W0 21 W0 0 w
= W, ,/ Wn n- x z
11 00 w w
89
-------
K ,K - ATL4/3 (3.2.2)
x z J-'
where A is a constant called dissipation parameter, and L is the size
L
of the diffusing patch.
The size of a pool of waste material L is defined to be 4 times the
standard deviation a of the distribution, i.e. L = 4 cr. Equation (3. 2. 2)
can be written also in terms of a, i. e.
.
K ,K = Aa (3. 2.2a)
x z
where A = 6. 34 A is a dissipation parameter defined based upon
J_j
the value of standard deviation or L/4.
Since a and a in general are not equal, and a may not be zero,
X Z XZ
there are several possibilities in adopting the horizontal diffusion law
(3. 2. 2) for use in the present theoretical model. One way is to
define K and K separately by their corresponding a and CT values,
X Z X. Z
i. e.
4/3 4/3
K = Aa and K = Aa (3. 2. 2b)
XX Z Z
Another method is to define the horizontal diffusion coefficients based
upon the geometric mean of the standard deviations along the principal
axes, . e.
Kx
Equation (3. 2. 2b) is not entirely logical because the K and K values
X Z
depend on the choice of the coordinate system. Solutions will change
with a change of coordinate directions. Equation (3.2.2c) gives identical
K and K values but the length scale chosen is invariant under coordinate
A. £*
transformations. Moreover, in the field determinations of the values
of AL or A, the length scale was often taken to be proportional to the
90
-------
square root of the patch area which directly corresponds to the quantity
.22 2 ,1/4
(a a - a )
X Z XZ
The value of A in Equation (5. 2. 2c) based upon a choice A, = 0. 00015
2/3 2/3
feet /second is A = 0. 001 feet /second. In the model the relation
(3. 2. 2c) is chosen to represent the horizontal diffusion law for the
waste material in suspension, in solution or on the free surface.
Note that the theoretical model can handle more complicated diffusion
laws. e. g. when A is a function of y, the depth instead of being a
constant.
3. 6. 3 Limitations of the Theoretical Model
The theorectical model formulated in the previous section can handle
cases where the ambient velocities, u and w , and the diffusion
a a
coefficients, K , K and K , are functions of both time t and depth y.
x y z }
As described in Section 3.2. 3, these environmental quantities in the
ocean are, in fact, generally functions oft and y. Therefore the model
is thought to be sufficiently general to describe adequately the long-
term diffusion of waste material in the ocean. However, the model is
subject to the limitation that these environmental characteristics are
independent of the x, z coordinates or invariant over the horizontal
planes. This limitation is minor especially in the open ocean. In the
practical case, if the environmental condition is slowly varying in the
horizontal planes, or the horizontal variation across the pool of waste
material is small, the average condition within the pool of waste
material can be used at each instant as a good approximation.
Also note that in the formulation, the waste material is assumed to be
in an open sea bounded only by horizontal bottom from below. However,
if the ambient is bounded horizontally, such as an inland lake or a bay,
the ass '.imptions should be modified. If there is ambient current, since
the velocity at the boundary has to be zero, by continuity the vertical
91
-------
velocity cannot be zero everywhere. In such situations, care must be
taken in the application of the present model. If the area of the
bounded ambient is large, and the discharge site is far from the boundary
present assumptions can still be valid. If the bounded ambient is
small horizontally, then the convective descent and dynamic collapse
phases are of most interest anyway.
-------
3. 7 JMumerJcaJ^ Procedure and Computer Program
In the previous sections, sets of equations for different stages of mixing
for different barge operations were formulated. These sets of equa-
tions do not yield closed form analytical solution?. Therefore, they will
be solved numerically on a digital computer. In the following sections,
the numerical procedures for the solution of these equations will be
presented.
3.7.1 Numerical Procedure
In Sections 3.3, 3.4, and 3. 5, the formulations presented for convective
descent and dynamic collapse phases consisted of systems of ordinary
non-linear differential equations which are readily solvable given a
set of initial conditions. A standard fourth order Runge Kutta method
is employed in the solution of these equations.
For long term diffusion, in solving the hyperbolic partial differential
equations, Equation (3.6. 10), the Crank-Nicolson method with a relaxa-
tion factor of 0.5 is used and this will be briefly presented in the
following.
A grid system shown in Figure 3.7.1 is chosen for finite difference
approximation. In the diagram, i signifies the vertical grid points,
j indicates the time steps and &y(i) is the distance between y. and y-,i •
For the unsteady term, a forward difference is used.
3Ck t Ck *(i'J+1) - Ck *(i'j)
K> JL _ K,I k, & (3.7.1)
dt At
93
-------
At
Jt
4
,
1
>.i
•
1 .
0 I 2345
Ay, Ay, Ay2
n n+l
Ayn Ayn
Figure 3.7.1 Grid System for Long Term Diffusion
-------
The diffusion term is approximated by a central difference averaged
over times j and j+1.
dy \ dy
Ay(i][Ay(i)
Ay(i-l)]
(i.
Ay(i)[Ay(i) + Ay(i-l)]
(i,j) - C(i-l.j)]
Ay(i-l)]
(3.7.2)
For the advective term involving the fall velocity, a backward difference
is taken for a positive fall velocity and a forward difference for a
negative fall velocity. The finite difference is also averaged over times
j and j + 1.
C^ .(i+k, ;il) - C^ , * = 0. 5w
dy s
k.i.j+i-k.Jt-2..i+i
[kjAyU) +k2Ay(i-D]
C. ,(i+k. .) - C, .(i-k- .)
•H 0 5w k- * 1»3 k' * _2'-T
s [kjAy(i) +k2Ay(i-l)]
(3.7.3)
where
and k.
1 >0
if w^
0
<0
95
-------
At the boundaries, the diffusion term, is approximated by a central
difference using fictitious grid points beyond the real boundaries. Thus
f
at y=
sck A ck J1^1'^ - ck ji(0>V
n j TT KI * K.I Jfr K-j * /o -7 .1 \
and at y=H dy - - TfcfW - (3.7.4)
Substituting Eqs. (3. 7. 1-4) into Eq. (3. 6. 10) and the boundary conditions
yields a system of simultaneous difference equations as follows.
w >0, particles in suspension:
s
K (1) 0. 5(k2-k1)^>
2 + (k+k)Ay(l) ^T) (kkK (l)
•H
Air/H V 1 7 ' ' V ' " » ' ^I'^T/^ V
'-•y^j-/ ^^. i/cy
C (1) 0. 5(k -k.)we
y j_ ^ J- s
0.
Ayd-)2 + (ki+k2)Ay(lS7Ck^(2lj)J
96
"I
JCk,
k(k-l)Kx(l, j)(Ck_2i ^l.j, -f-
Ay(l)2
-------
i=2 to n-1
0.5w k_
s 2
i) + Ay(i-l)]
K (i)Ay(i-l) + K (i-l)Ay(i)
1 A y . . y
At Ay(i;
K a;
DAy(i) 0. 5w (KO-KJ -,
O M 1 I
, S |/~* li i _L T \
"7—T"—TTT "" i 7 7^\—i—. t,—rr JO, „(!> 1T J. j
Ay(i-l)] k, Ay(x) + k7Ay(i-l) k, Ax
i t- j
r %UJ
[Ay(i)[Ay(i) + Ay(i-
0. 5w- k
.
j.
1)] kjAyd) +k2Ay(i-l)_
K (i-1)
y
(i. J)
0. 5w k
s 2
Ay(i-D]
K (i)Ay(i-l) + K (i-l)Ay(i)
•DAy(i) 0. 5(k0-k.)wc \
. • -. ? 1 s .. \
Ay(i-l)] kjAy(i) + k Ay(i-l) I
/ K (i) 0.5k w \
/ y 1 s \r ,. , .
I Ay(i)[Ay(i) + Ay(i-l)] " k Ay(i) + k Ay(i-1)1 k,/' i>3
\ i £ i
(3.7.6)
i=n
0.5(k -k )
r K (n-1) 0.5(k,-k.)w (l-a,)w k (l-a,)w2 "I
_ _L + y. + 2 1 s ^ s + i ,—§ C (n i
[At , n2 (k +k ) Ay (n-1) ' Ay (n-1) (k,+k )K (n- 1)J k,/ '-1
Liy ^ii"" if J. C* *t< y
H-l
97
-------
= -0.5(kua(n)(Ck_ljA(n,j)
+k(k-l)Kx(n,j)(Ck_2>je(n,j)
(n, j)(C (n,j) + C (n, j
-
K (n-1)
r /is. (n-
M -
L\Ay(n-
Ay(n-l)'
0. 5(fc0-k.)w
^ i i
K (n-1) 0. 5(k -kjjw
Ay(n-i;
(l-o2)wg kj(l
"STKTir " (k1+k2)K
2
r
s
k w
(3. 7. 7)
For particles deposited on the bed:
W.
•f 0.
(3. 7. 8)
°kwsCk,je(h>j) ' Y2W2k,je(j))
w <0 particles floating on the free surface:
S
98
-------
w
(3.7. 9)
w <0 particles in suspension:
AT
,
K (1) 0.5(k_-kJw (1-oJw (l-ou)k
V ,
-------
K(l) 0.
I At A /i\2 (k,+k_)Ay(l) Ay(l)
\ Ay(l) V127V ' v '
K (1) 0.5(k0-k,)w
+ ' "
Ay(l)2
i=2 to n-1 same as Eq. (3. 7. 6)
i=n
K (n-1) 0.
= -0.
K.(n-l)
«K (i
__X_
Ay(n-
0.5(k -k )w \ / K
Z 1 S
Ay (n-1)2
100
K (n-1) 0.5(k-,-k1)w w k,w
_X + 21 S . ^B + 1 8 JC ( j + 1)
X (lr -T-lr \ Airfn I 1 i\\r (-n I 1 (lr -r-lr J K fr> 111 ir 6 **
A / i\'-' IKi'*^^/ uy IH— J. J tav 1*1*- J. J lix-.~iv— iJ.\. IH—J./ / Kjji
Ay (n-1) 1 £ ' ' IZy
-------
2
0.5(k_-k,)w w k.w
21s, s Is
" (kj+k2) Ay(n-l) + Ay(n-l) " (kj+k^K (n-1) k, 4>
(3.7. 11)
Note that the solution for the fluid part of the waste can be obtained by
using either equations (3. 7. 5-7) or Equations (3. 7. 9-11) only by setting
w to zero.
s
The implicit schemes in Eqs. (3. 7. 5-7) and Eqs. (3.7. 9- 11) are in
the form of
= di
a.u. . + b.u. -}• c.u. , . = d., i = 2, 3, .... n-1 (3. 7. 12)
a u , + b u = d
n n-1 n n n
or
[A][U] = [D] (3.7. 13)
where [A] is a tridiagonal matrix. The system of equations is solved
by the Thomas algorithm as presented by Ames (1965)
u = dn'
n
vu = d/ - c. 'u , i=n-l, n-2, . . . , 1 (3. 7. 14)
where
ci = ci/br di = di/bi
. + 1
d. , ,- a. , ,d.'
' - l+ i 1+i 1
.,-, - -r - r
1+1 bi+rai+ici
-10 i it n ic\
1= 1, 2, . . . , n- 1 (3.7.15)
101
-------
The above numerical procedure for the solution of the equations for long
term diffusion follows the Standard Crank Nicolson scheme. The only
point which should be emphasized is the difference scheme employed for
the term involving the fall velocity w . As mentioned before, different
s
difference approximations are used for the term involving the fall
velocity w based on the sign of w . This is important as will be shown
in the following analysis. In the following, subscripts k and & will be
dropped for simplicity.
For a simple derivation of the effects of the term
6C
w
s
on the transport phenomenon, terms involving ambient currents and
diffusion are deleted from Equation (3. 6. 10). Equation (3. 6. 10) then
becomes
L£ (3<6> 10 }
y
The solution to Equation (3. 6. 10 a) is simply
C = f(y - w t) (3.7. 1)
&
which indicates that given an initial distribution
C(y,0) = f(y) (3.7.2)
the functional form of f(y) will be preserved at any instant t subjected only
to a linear transformation w t. Now let us try to solve Equation (3. 6. 10a)
by a finite difference approximation, using a forward difference for both
the independent variables y and t. It is obtained
Ct) (3.6. 10b)
Equation (3. 6. lOb) is incapable of correctly predicting the concentration at
time t+At from a known distribution at time t. For instance, for a
102
-------
concentration distribution as shown in Figure (3. 7.2 ), Equation (3. 6. lOb)
will predict at t+At a negative concentration at position A and a zero
concentration at position B which is completely wrong according to the
analytical solution in Equation (3. 7. 1). If a central difference is used
for
ac
97 '
a similar result may be found depending on the relation "between the
concentration distribution and grid point position. If a backward dif-
ference is used for the term
dy
Equation (3. 6. 10a) becomes
C(y,t+At) = C(y,t)-wJ|(Cy>t-Cy_Ayit) (3. 6. lOc)
Critical points such as A and B in Figure (3. 7.2) can again be tested.
The results show that at t+At, the concentration at A will be zero, and at
B the concentration will increase which is exactly what it should be.
Following the same argument a forward difference should be used for the
ac
W -r
s dy
term in the finite difference approximation for solving Equation (3. 6. 10a)
if w <0.
O
103
-------
A--
B •
> t+At
Figure 3.7.2 Definition Sketch
104
-------
3.7.2 Computer Program
Using the numerical procedures in Section 3.7.1, a computer program
has been developed for the prediction of the mixing characteristics in the
various phases of dispersion when waste is discharged from a barge into
the ocean. A flow chart of the program is shown in Figure 3.7.3. The
input sequences and detailed explanations of the input and output symbols
are presented in Appendix A. Attention should be called to the many
options of printing and graphing in the outputs, to whether or not built in
suggested coefficients are to be used, and to the options of terminating
the computation at the end of different phases of dispersion. In the fol-
lowing, some of the details of the program are discussed.
3.7.2.1 Empirical Coefficients
For each of the three barge operations, values of a package of empirical
coefficients introduced in the formulations are needed. A set of coeffi-
cients are built into the program based on either available knowledge or
educated guess. These are subject to change when better estimates are
available through experiments. The user of the program has the option
of inputting his own estimates of the coefficients.
In Barge Operation 1, the entrainment coefficient ALPHAO is set to be
0.235 which is the mean of the best fit values from numerical simulation
on the experimental data in density stratified ambient experiments in
this study. It ranges from 0.21 up to 0.265 as will be discussed in
Section V. The coefficient is believed to be accurate to at most two
digits. BETA is set to 0. which is believed true when the solid concen-
trations are relatively low. CM is set to 1. CD, CD3 and CD4 are
estimated to be 0.5, 0.1 and 1.0, from drag coefficient diagrams for
sphere, spheroidal wedge and circular plate respectively. For the coef-
ficients GAMA, CDRAG, CFRIC, ALPHAC, FRICTN and Fl, esti-
mates are based only on educated guesses. More extensive experiments
are needed to provide these coefficients.
105
-------
READ CONTROL PARAMETERS
AND
AMBIENT CONDITIONS
I
BARGE OPERATION
1 \
PUFF
CONVECTION
1
DYNAMIC
COLLAPSE
BARGE OI
Z
BOTTOM
TEST METHOD OF
DISPOSAL
' BARGE OPERATION
3EI
ENCOUNTER
NATION ' l)f 3
JET
CONVECTION
WAKE-PLUME
CONVECTION
\ 1
J
DYNAMIC BOTTOM
COLLAPSE ENCOUNTER
I
UPDATE INPUT
DATA FOR LONG
TERM DIFFUSION
LONG TERM DIFFUSION
UPDATE INPUT
DATA FOR LONG
TERM DIFFUSION
END
Figure 3. 7. 3 Flow Chart for the Computer Program
106
-------
In Barge Operation 2, in the jet convection phase the entrainment coeffi-
cients for a momentum jet and a two-dimensional thermal, ALPHA1 and
ALPHA2, are set to 0.0806 and 0.3536 respectively. The values are
from Abraham (1970) after correction by a numerical factor arising from
the different similarity distributions for density deficiency, velocity, etc.
used in Abraham's and the present formulations.
The settling coefficient BETA is set to ?ero. The drag coefficient CD
is set to be 1. 3 as obtained from the drag diagram for a two-dimensional
cylinder. The above coefficients are considered quite good since numer-
ical simulations using those coefficients fit the experimental data well.
In the dynamic collapse phase, the entrainment coefficient ALPHA3 is
again set to 0.3536. The approximation should be good if the shape of
the elliptical cylinder is not too far off from a circular cylinder. The
drag coefficients CDS and CD4 for a two-dimensional wedge and two-
dimensional plate are estimated to be 0. 2 and 2. 0 from the drag coeffi-
cient diagram. There is again no experimental information on GAMA,
CDRAG, CFRIC, FRICTN and Fl. The estimated values are accord-
ingly subject to change when better knowledge is available.
In Barge Operation 3, Cl and C2 are important in determining the
initial mixing and are estimated to be 0.6 and 4 respectively. Since
there is no field data, these coefficients are set to the above values in the
numerical simulation. Their choice can be made with confidence only
when adequate experimental data is available. For the convective descent
phase, the entrainment coefficient ALPHA is set to be 0.3536. The drag
coefficients CD1 and CD2 are again estimated to be 1.3 and 0.5 from
drag coefficient diagrams for a two-dimensional cylinder and a two-
dimensional wedge. For the dynamic collapse phase Barge Operation 2
and Barge Operation 3 use the same coefficient.
For the long term diffusion phase, if there is no ambient current near the
free surface and the bed, the boundary absorbency coefficients ALFA1
and ALFA2 should be 1 and the re-entrainment coefficients GAMA1 and
107
-------
GAMA2 should be 0. However, if there is a current near the boundaries,
the coefficients may take on other values. The dissipation parameter
AJLAMDA is set to 0. 001 which is the value for the ocean (see Section
3.2.2). It may be smaller for inland lakes.
3.7.2.2 Transition between Different Phases and Input of Solid
Particles to Long Term Diffusion
During the mixing processes beginning with the discharge of waste mate-
rial and ending with the end of long term diffusion, there is no clear
demarcation between the different stages of convection, collapse and
long term diffusion. In the computer program, the first stage of convec-
tion is either terminated by bottom encounter or when the cloud reaches
the first neutrally buoyant position for Barge Operations 1 and 3, and the
first horizontal position of the jet trajectory for Barge Operation 2. The
dynamic collapse phase for all three barge operations stops when the
estimated horizontal spreading due to diffusion is larger than that due to
dynamic collapse. In this second stage of computation, the computation
can switch from dynamic collapse to bottom encounter if the cloud or
plume hits the bottom or from bottom encounter back to dynamic collapse
if the reaction force at the bed is less than or equal to zero.
During the convection and collapse phases, the mechanism of entrainment
is important in the mixing phenomenon. In the dynamic collapse and
bottom encounter phases, the entrainment due to both convection and
collapse are included in the formulation. However, it is believed that
the entrainment is dominated by that due to convection and collapse at
the beginning and end of dynamic collapse respectively. In the program,
it is assumed that the entrainment is solely due to collapse after either
the cloud reaches the bottom or the cloud passes the second neutrally
buoyant position. Also after the waste pool starts to collapse, the cloud
or plume is more like a disk or a two-dimensional plate and the added
mass coefficient in the vertical direction must be modified. It is postu-
lated herein that CM in the vertical direction is increased by the ratio
108
-------
of b/a. This is equivalent to taking the hydrodynamic mass in the ver-
3 2
tical direction to be proportional to b in the case of a disk and to b L
in the case of a plate. While the numerical value of Cx, is uncertain,
M
this functional behavior is expected to be correct.
For a reasonable presentation of the characteristics of the waste cloud
or plume in these two stages, a maximum of 600 computation steps is
allowed; 100 to ZOO steps for the convection stage and 100 to 400 points
for the collapse or bottom stage. DINCR1 and DINCR2 are controls of
integration steps for the first and second stages respectively. The com-
puter proceeds during the first stage with step size of DINCR1 times a
program estimated step size (based on density gradient and/or depth)
until either a normal termination (see previous paragraph) or the compu-
tation exceeds 600 steps. At that moment the computer prints DINCR1,
the step size, the number of computation points and checks whether the
computation points are satisfactory. If it is not, DINCR 1 is modified
and the computation is started over. The computer enters the second
stage only if both the physical and the computation points criteria are
fulfilled. In analogy to the first stage, the computation of the second
stage is completed b,y repeated trials of modifying the step size to ful-
fill both the physical and computation points criteria. For each stage a
maximum of five trials is allowed. If the computation is not satisfactory
within five trials, the computer prints a diagnosis and exits. Then by
the trial history of DINCR 1 or DINCR2, the user should be able to pick
a best trial value for DINCR 1 or DINCR2 and start the program over.
Situations requiring more than five trials have not been observed in any
normal run.
At the end of the dynamic collapse phase, the waste material is con-
sidered to be dynamically passive and long term diffusion takes over.
However, for those solid particles which settled out of the waste cloud
or plume in the first and second stages, long term diffusion must start
whenever they are out of the main cloud. Hence the long term diffusion
109
-------
computation in the program can be further divided into two phases: before
and after the end of the dynamic collapse phase. Before the end of dynamic
collapse, there are inputs of solid particles into the long term diffusion
mode] while the diffusion proceeds. After the end of collapse only diffu-
sion and settling are involved.
For Barge Operation 1, if there are solid particles settled out of the main
cloud between two time steps, the settled particles are considered to form
a circular slab of the same radius as the cloud right beneath or above the
cloud depending on the sign of the fall velocity. The thickness can be
evaluated by the fall velocity and positions of the cloud. While it is pos-
sible to input each slab individually into the long term diffusion model and
let the model perform the calculations for diffusion and settling between
successive inputs it is much more efficient in terms of computation time
to wait until several or possibly many slabs have settled out. A charac-
teristic time interval given by the settling velocity and the vertical grid
spacing in the diffusion model is chosen and all slabs which settle out of
the cloud during each such period are inputted into the Long term diffusion
model simultaneously. However each slab is updated to the time of input
by allowing it to settle, move with the ambient currents and diffuse accord-
ing to a simple diffusion law. During each such time interval, the solid
particles already in the long term diffusion model undergo several time
steps of diffusion. In this manner, the solid particles settled out of the
cloud are updated toward the end of dynamic collapse. At that moment
all the solid particles which still remain in the cloud are thrown into the
long term diffusion model all at once. As for the fluid part of the waste
material, it is thrown into the long term diffusion model only at the end of
dynamic collapse. For input of a circular slab of solid particles into the
long term diffusion model, the following formulas are used
110
-------
CQO = Cnb2
C01 = zC00
C20 = C0o(x2+ib2)
r r ( 2 ± l T\
C02 = C00\z +4b2/
Cll = xzC00 (3.7.2.1)
where C is the concentration, b is the radius, x and z are the
centroid of the slab.
For Barge Operations 2 and 3, because of the limited discharge time,
attention should be called to the front and tail of the plume in inputting
solid particles to the long term diffusion model. In Figure 3.7.4, curve
ABC shows the front of the plume in space-time, point B is where solid
particles start to settle and point C is when long term diffusion takes
over for the front of the plume. Curve A'B'C1 shows the tail of the
plume. Starting from T,, there should be solid particles to be put into
the long term diffusion model. The solid particle slabs are thrown into
the long term diffusion model at intervals of At. It is therefore seen that at
T, + At, there are slabs only below positions i, iH-1 and i+2 to be
taken care of. As time moves from T, + At, to T . + At, + A t^, the
number of slabs increases as shown in the figure. When time is larger
than T-,, solid particles inside the main plume have also to be put into
the long term diffusion. And when time is larger than T,, there no
longer are any slabs under position j as shown in the figure. When
time is larger than T , all the solid particles are in the long term
diffusion model.
Ill
-------
Til
Figure 3. 7.4 Definition Sketch
112
-------
For inputting the solid particle slab into the long term diffusion model,
rectangular horizontal slabs just above or beneath the plume are used
according to the plume sizes, positions, solid particle fall velocity and
time interval At. For computing the moments, the following formulas
are used
C00 = C2bL
C -
So ~ oo
coi - zCoo
, .
00X + T2COS Y+— sin Y
XZ +snY COSY ~
(3.7.2.2)
where b is the half width of the plume, L, is the length of the plume
segment, x and z are the horizontal centroid and Y is the angle
between L, and the x-axis.
After all the waste material is in the long term diffusion model (at the
end of dynamic collapse for Barge Operation 1 and at the time when the
tail of the plume comes to the end of dynamic collapse for Barge Opera-
tions 2 and 3), the model will continue to perform the dispersion calcu-
lations until terminated either when the time exceeds a read-in time
limit or by the fact that a substantial amount of the solid particles have
settled to the boundaries or that the fluid waste is well mixed vertically
into the water column.
In the long term diffusion part of the program, special consideration has
been given to the change of grid size and computation time step according
113
-------
to the characteristics of the diffusing pool such as the vertical spread
of the plume, the fall velocity of the solid particles and the vertical
diffusion coefficient. The change is automatically done in the program.
The vertical grid lines are chosen to be close together within and
near the waste material and farther apart where there is little waste.
Thus, for example, if at the beginning of long term diffusion, all the
waste is within a thickness of 5 feet in an ocean of depth 100 feet, the
grid lines would be packed close together within and around the cloud.
As the diffusion and settling proceeds however, the grid automatically
changes to accommodate. The time step of integration is also auto-
matically controlled based on the characteristics of the waste pool and
the diffusion coefficient. However a provision for change by the user
is incorporated.
It should be remarked that the computer program developed herein must
not be regarded as the ultimate achievable in the solution of the basic
problem. Modifications and improvements are possible which can greatly
enhance its utility. It should further be pointed out that the program is
not designed to be able to handle all situations and any user must first
fully understand its limitations. For example, it is assumed in the
program that the waste will go through a stage of convective descent.
Thus, the gross density of the waste discharged must be greater than the
ambient density and the initial downward velocity of the waste should be
larger than 0. Moreover, when the convective descent phase ends,
either at the first neutrally buoyant position for Barge Operations 1 and 3
or a horizontal position in Barge Operation 2, the local ambient may not
be stratified. In that event the program will bypass the collapse phase
and go directly to the long term diffusion phase. This is not necessarily
always valid because the pool may return upward into a density gradient
and collapse resulting in different predictions. In the latter case the
program user may force the collapse simply by recalculating the case
after first introducing a very slight density gradient at the proper
location.
114
-------
THE GUNTER LIBRARY
GULF COAST RESEARCH LABORATORY
.OCEAN SPRINGS, MISSISSIPPI
SECTION IV
EXPERIMENTAL INVESTIGATION
4. 1 Objective and Scope of Experimental Investigations
The mathematical model developed in this study is intended to predict
the physical behavior of the waste material discharged from a barge into
the ocean. To test the applicability of the theory, experiments are needed
to confirm the hypotheses made and to obtain the coefficients introduced
in the formulation. For a complete experimental check, a large number
of experiments would have to be performed in a complex ambient with
cross currents and density stratifications similar to that of the prototype.
However, in the present study, only a set of simplified but representative
preliminary experiments were performed in a stagnant laboratory tank with
or without density stratifications. Results of the experiments were used
to evaluate mainly the entrainment coefficient and test the applicability of
the mathematical model up to the onset of the long term diffusion phase.
115
-------
4- 2 Apparatus and Procedure
4.2.1 Ambient Condition
Experiments were performed in a tank 9 inches wide, 15 inches deep
and 40 inches long. For uniform density ambient runs, the tank was
filled with cold water. For runs with density stratification, the tank was
first filled to half depth with cold water, then hot water was gently intro-
duced on top of the cold water through a cup residing at the water surface,
with holes on the sidewalls near its bottom. The position of the cup was
adjusted as the water surface rose. Thus the lighter hot water spread out
horizontally on the heavier cold water. When the water level reached the
top of the tank, there were two layers of water with an interface of fairly
strong temperature gradient. To destroy the interface and obtain a nearly
linear gradient, a piece of plastic cloth about 4 inches high and 9 inches
wide was put vertically with its middle at the interface and towed slowly
horizontally by two rods attached to the ends of the plastic cloth. After
about 10 minutes, the turbulence generated by the towing of the plastic
cloth died away and a nearly linear stratification was obtained. The
temperature distribution was measured and the density stratification was
calculated from the temperature distribution.
4.2.2 Discharge Material
Four different discharge materials were used in the experiments, namely:
a) mud, b) digested sewage sludge c) dredge spoil and d) salt solution.
Mud was obtained by mixing a quantity of earth with water. The digested
sludge was provided by EPA and LA County Sanitation District. The
coarser particles in the mud and larger sticks in the sewage sludge were
filtered off by a screen of about 1/32 inch mesh size. The dredge spoil,
obtained at San Francisco Bay, was provided by the Corps of Engineers
at San Francisco. The particles in the mud were in the range of clay
to silt, and those in the dredge spoil were small sands. The mud and
dredge spoil are non-cohesive in nature. The digested sewage sludge
116
-------
consisted of very fine particles up to about 1/32 inch in size and it
seemed tobe slightly sticky at high concentration. Salt solution is ob-
tained by dissolving table salt into water.
4.2.3 Procedure and Data Reduction
Three different modes of discharge were used including i) an instan-
taneous three-dimensional slug, ii) a continuous discharge from a hori-
zontally travelling vertical jet, and iii) an instantaneous two-dimensional
line. The first mode simulating Barge Operation 1 was achieved by in-
jecting the waste material through a modified syringe, cut-off at the
shoulder and covered by a screen of about 1/32 inch mesh size. The
special feature of the syringe was that the screen could hold the waste
material in the syringe by surface tension when the syringe was held in
the vertical position in air. For injection, the syringe was lowered and
when the front of the syringe was just below the water surface, the waste
material was injected downward by the plunger. The second mode of
discharge simulating Barge Operation 2 was achieved by discharging the
waste material continuously through a vertical glass tube fixed on a
carriage which was towed horizontally along the center line of the tank
by a system consistingof a cable, pulleys, and a variable speed drill
motor. The glass tube is connected by a Tygon tube to a constant head
Mariotte tank which provides constant discharge. In the third mode, a
half-cylindrical trough filled with the waste spanning the width of the
tank was inverted to simulate a two-dimensional puff.
For the experiments, the aforementioned materials were made to various
densities by mixing with water. The density was measured by weighing a
knownvolume of was te material or by a hydrometer . After the waste
material was injected into the water, the position and size was recorded
by time lapse photography. The timing of the pictures taken was indica-
ted by either a clock placed in front of the tank or a wax ball of known
fall velocity set free in the tank before the injection. The intervals be-
tween successive pictures were variable depending on the time scale of
the dynamics. The shortest interval achieved was about 0.3 second.
117
-------
For the position and size of the waste material after injection, the time-
lapse pictures on the 35 mm film were projected frame by frame by an
enlarger. The pictures were adjusted to be one-third of life size. A
sheet of graph paper was lined up accurately on each frame using the
markers on the tank as a guide. The outlines of the cloud were then
traced.
For injection modes 1 and 3, the distance of the centroids of the cloud
below the injection point and its horizontal and vertical dimensions were
measured off from the traced picture. For each position, the corres-
ponding time starting from the injection was obtained from the clock or
the position of the wax ball, and the position-time plot was extrapolated
to the point of injection. The initial velocity for each run was also esti-
mated from the slope of the position-time plot. For injection mode 2,
the traveling velocity of the jet was measured from the horizontal posi-
tion vs. time plot of the glass tube, and the trajectory and size of the
plume was obtained by taking the average of several traces.
118
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4 . 3 Results and Discussions
Seventy experiments were performed in this investigation. The experi-
mental parameters and density stratification profiles are presented in
Tables 4. 1 and 4.2. The run number designations listed in the tables are
as follows: The first letter in the designation signifies the nature of the
waste material: M for mud, S for sludge, D for dredge spoil, and N
for salt water. The subscript on the first letter signifies the mode of
discharge: blank for three-dimensional instantaneous, 2 for two-dimen-
sional instantaneous and j for jet. The second entry in the designation
signifies the density stratification in the receiving water: 0 for uniform
and S for stratified. The last entry is just the numerical sequence
within each group of experiments.
4.3.1 Instantaneous Release of a Three-Dimensional Slug
The pertinent parameters on those runs involving an instantaneous re-
lease of a three-dimensional slug, i.e. , simulation of Barge Operation 1,
are summarized in Table 4. la. Those experiments in Table 4. la can
be subdivided into two groups a) with uniform ambient and b) with strati-
fied ambient. The primary purposes of the runs in the first category are
to obtain an estimate of the entrainment coefficient a and to attempt a
correlation of the entrainment coefficient with other measurable quan-
tities in the experiments. The experiments in the second category are
used to verify the mathematical model.
In the formulation of the mathematical model for Barge Operation 1, it
was assumed that the cloud is always in a hemispherical shape. It was
noted in the experiments that the initial momentum and buoyancy some-
times generated a vortex inside the cloud, and the shape of the cloud was
strongly affected by the strength of the vorticity. The increase of vor-
ticity changed the shape of the cloud from spherical to hemispherical
119
-------
TABLE 4. la
Summary of Experimental Parameters
Run No. V(cc) p (gm/cc) a
M-0- 1
M-0- 2
M-0- 3
M-0- 4
M-0- 5
M-0- 6
M-0- 7
M-0- 8
M-0- 9
M-0-10
S-0- 1
S-0- 2
S-0- 3
S-0-- 4
.S-0- 5
S-0- 6
S-0- 7
S-0- 8
S-0- 9
S-0-10
S-0- 11
S--0-12
S-0-13
D-0- 1
D-0- 2
D-0- 3
D-0- 4
.86
.86
.86
.86
.86
.86
.372
.372
.372
.74
.74
3.66
3.66
5.. 16
5.16
8.55
3.66
0.86
3.66
NA
t -» r\
1.053
1.053
1.143
1.143
1.217
1.217
1.053
1.053
1.053
1.217
1.077
1.139
1.11
1.11
1.11
1.139
1.074
1.108
1.13
1.016
1.13
1.108
1.108
1.997
1.894
1.997
1.997
,29
.39
.35
.25
.25
.26
.29
.40
.27
.27
.34
.45
.25
.40
.20
.19
.22
.19
.18
.44
.23
.19
.16
.25
.26
.23
.30
120
-------
(Table 4. la continued)
&un No.
M-S-1
M-S-2
M-S-3
M-S-4
S-S-1
S-S-2
S-S-3
S-S-4
S-S-5
S-S-6
S-S-7
N-S-1
N-S-2
N-S-3
N-S-4
N-S-5
V(cc)
3.66
3.66
3.66
3.66
3.66
3.66
3.66
3.66
3.66
3.66
3.66
3.66
3.66
3.66
3.66
3.66
p (gm/cc)
1.432
1.202
1.290
1.290
1.128
1.128
1.1233
1.0655
1.0655
1.0655
1.1233
1.080
1.080
1.179
1.179
1.179
u (fps)
0.294
0.190
0.236
0.318
0.350
0.292
0.244
0.143
0.152
0.242
0.156
.210
.181
.240
.250
.3
a
.265
.265
.22
.23
.235
.25
.22
.255
.255
.25
.26
.21
.25
.22
.225
.230
V = volume of waste discharged
p = overall density of waste discharged
tt = observed entrainment coefficient
u = velocity of waste at discharge point
121
-------
TABLE 4. Ib
Summary of Experimental Parameters
Run No.
Nj-0- 1
Nj-0- 2
Nj-0- 3
Nj-0- 4
Nj-0- 5
Sj-0- 1
Sj-0- 2
Sj-0- 3
Sj-0- 4
Sj-0- 5
Sj-0- 6
Nj-S- 1
Nj-S- 2
Nj-S- 3
Nj-S- 4
Nj-S- 5
Nj-S- 6
Nj-S- 7
Nj-S- 8
Nj-S- 9
Nj-S-10
Nj-S-11
D. (in)
,0725
,0725
,0725
0725
0725
0725
0725
0725
125
125
125
0725
0725
0725
0725
0725
0725
0725
0725
0725
0725
0725
U. (ft/sec)
3.86
3.86
3.86
3.86
3.86
2.05
2.05
2.05
.593
.593
.593
3.7
3.7
3.7
3.7
3.2
3.7
3.2
3.7
3.2
3.2
3.2
U.(ft/sec) k=U./
D J
.442
.432
.238
.275
.216
.142
.146
,146
.145
.146
.140
.150
.464
.276
.135
.141
1.03
.141
.341
.258
..45
.3
8.73
8.95
16.20
14.00
17.85
14.42
14.05
14.05
4.09
4.06
4.23
24.64
8.00
13.40
27.40
22.70
3.60
22.70
10.85
12.40
7.12
10.65
Ub p(gm/cc) F
1.1455
1.1455
1.1455
1.1455
1.1455
1.030
1.030
1.030
1.030
1.030
1.030
1.134
1.134
1,134
1.134
1.163
1.163
1.163
1.163
1,163
1.163
1.163
18.40
18.40
18.40
18.40
18.40
690
690
690
3.5
3.5
3.5
19.00
18.00
18.00
18.00
9.00
12.00
9.10
12.00
8.70
8.70
8.85
122
-------
(Table 4. Ib continued)
Run No.
Sj-S-
SJ-S-"
Sj-S-
Sj-S-
Sj-S-
1
2
3
4
5
D. (in)
.0725
.0725
.125
.125
.125
Uj(ft/sec)
2.05
2.05
.593
.593
.593
Ub(ft/sec)
.147
.146
.153
.142
.138
17.
17.
3.
4.
4.
k
00
12
87
18
30
p(gm/cc) F
1.
1.
1.
1.
1.
03
03
03
03
0
0
0
0
030
540
540
2.
3.
3.
6
3
3
D. = ID of discharge tube
U. = velocity of discharge
U, = horizontal velocity of discharge tube
p = density of waste
7 - _
= U7/g - - - D.
J B p 1
123
-------
TABLE 4. Ic
Summary of Experimental Parameters
Run No. p(gm'/cc) u (ft/sec) a
.45
.34
M.,-0-1
s2-o-i
M2-S-1
M2-S-2
s2-s-i
S2-S-2
S2-S-3
1.171
1.138
1.138
1.060
1.060
1.1233
,0983
.0865
.095
.126
.0716
p = density of waste
u = velocity of waste at discharge point
a = observed entrainment coefficient
124
-------
Table 4. 2 a. Density Stratification for Runs M-S, S-S and N-S
Run
y
0. 0
0. 083
0. 166
0. 250
0. 333
0.416
0. 500
0. 583
0. 666
0. 750
0. 833
0. 917
1. 000
1. 083
1. 168
M-S-1-2
P
0. 9911
0. 9911
0. 9911
0. 9911
0. 9911
0. 9911
0. 9913
0. 9922
0. 9938
0. 9949
0. 9958
0. 9965
0. 9971
0. 9975
0. 9978
M-S-3
P
0. 99125
0. 99125
0. 99125
0. 99128
0. 99142
0. 99176
0. 99278
0. 99395
0. 99505
0. 99588
0. 99650
0. 99695
0. 99725
0. 99755
0. 99776
M-S-4
P
0. 99182
0. 99183
0. 99185
o. 99190
0. 99200
0. 99225
0. 99280
0. 99395
0. 99505
0. 99588
0. 99650
0. 99695
0. 99725
0. 99755
0. 99776
125
-------
(Table 4.2a continued)
S-S-1-2
S-S-3
S-S-4-5
S-S-6
S-S-7
0. 9959
0. 9959
0. 9959
0. 9959
0. 9959
0. 9960
0. 9961
0. 9963
0. 9964
0. 9966
0. 9967
0. 9969
0. 9970
0. 9972
0. 9976
0. 99388
0. 99389
0. 99390
0. 99391
0. 99398
0. 99440
0. 99498
0. 99537
0. 99572
0. 99600
0. 99625
0. 99645
0. 99660
0. 99669
0. 99672
0. 99422
0. 99422
0. 99422
0. 99422
0. 99430
0. 99450
0. 99498
0. 99537
0. 99572
0. 99600
0. 99625
0. 99645
0. 99660
0. 99669
0. 99672
0. 99280
0. 99280
0. 99280
0. 99281
0. 99285
0. 99297
0. 99330
0. 99386
0. 99455
0. 99530
0. 99595
0. 99651
0. 99695
0. 99724
0. 99745
0. 99225
0. 99228
0. 99233
0. 99245
0.99265
0. 99295
0. 99335
0. 99380
0. 99435
0. 99490
0. 99548
0. 99607
0. 99652
0. 99684
0. 99715
126
-------
(Table 4.2a continued)
N-S-1
N-S-2
N-S-3-4-5
0. 99315
0. 99315
0. 99315
0. 99315
0. 99316
0. 99321
0. 99330
0. 99345
0. 99368
0. 99406
0. 99442
0. 99480
0. 99519
0. 99550
0. 99580
0. 99285
0. 99285
0. 99285
0. 99287
0. 99289
0. 99300
0. 99317
0. 99344
0. 99368
0. 99406
0. 99442
0. 99480
0. 99519
0. 99550
0. 99580
0. 99253
0. 99253
0. 99253
0. 99255
0. 99262
0. 99281
0. 99305
0.99342
0. 99368
0. 99406
0. 99442
0. 99480
0. 99519
0. 99550
0. 99580
127
-------
Table 4. 2b. Density Stratification for Runs N.-S and S.-S
J J
tv
oo
N.-S-l
J
0. 99205
0. 99205
0. 99205
0. 99205
0. 99205
0. 99235
0. 99280
0. 99343
0. 99435
0. 99557
0. 99645
0. 99713
0. 99765
N.-S-2-3-4
J
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
98960
98986
99025
99080
99155
99242
99350
99435
99510
99572
99630
99680
99730
N.-S-5-6
J
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
99500
99500
99503
99510
99521
99532
99548
99563
99581
99608
99621
99643
99668
N.-S-7
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
J
99550
99552
99553
99555
99560
99562
99569
99575
99581
99591
99600
99610
99625
N,-S-8
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
J
99550
99552
99553
99555
99560
99562
99569
99575
99581
99591
99600
99610
99615
N.-S-9-10
J
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
99255
99260
99272
99295
99332
99372
99425
99480
99540
99608
99678
99748
99823
N.-S-ll
J
0. 99333
0. 99333
0. 99340
0. 99360
0. 99390
0. 99430
0. 99476
0. 99530
0. 99590
0. 99652
0. 99720
0. 99790
0. 99865
-------
(Table 4.2b continued)
S.-S-l
J
0. 98900
0. 98900
0. 98900
0. 98912
0. 98985
0. 99U2
0. 99335
0. 99488
0. 99612
0. 99710
0. 99772
0. 99808
0. 99925
S.-S-2
J
0. 99020
0. 99020
0. 99021
0. 99025
0. 99065
0. 99165
0. 99335
0. 99488
0. 99612
0. 99710
0. 99772
0. 99808
0. 99925
S.-S-3
J
0. 98938
0. 98938
0. 98940
0. 98960
0. 99022
0. 99121
0. 99250
0. 99390
0. 99520
0. 99635
0. 99720
0. 99795
0. 99820
S-S-4-5
0. 99065
0. 99065
0. 99065
0. 99072
0. 99105
0. 99175
0. 99280
0. 99395
0. 99505
0. 99617
0. 99700
0. 99765
0. 99850
129
-------
Table 4.2c Density Stratification for Runs M -S and S -S
J_> L*
M2-S-1
0. 99438
0. 99439
0. 99439
0. 99440
0. 99441
0. 99448
0. 99460
0. 99475
0. 99500
0. 99530
0. 99565
0. 99615
0. 99670
0. 99710
0. 99744
M -S-2
S--S-1
L-t
0. 99163
0. 99163
0. 99165
0. 99165
0. 99165
0. 99167
0. 99168
0. 99173
o. 99190
0. 992ZO
0. 99258
0. 99315
0. 99383
0. 99455
0.99542
S2-S-2
0. 99550
0. 99551
0. 99552
0. 99552
0. 99554
0. 99558
0. 99564
0. 99573
0. 99582
0. 99592
0. 99602
0. 99613
0. 99627
0. 99640
0. 99655
S2-S-3
0. 99291
0. 99293
0. 99294
0. 99297
0. 99300
0. 99312
0. 99337
0. 99380
0. 99435
0. 99490
0. 99548
0. 99607
0. 99652
0. 99684
0. 99715
130
-------
and finally to a convective vortex ring. It was also noted in the experi-
ments that the shape of the cloud could depend on the injection mode. A
single well formed cloud was not always produced; every once in a while,
a cloud consisted of a better shaped front with some residue behind it,
similar to the dirty thermal found by previous investigators. Although
there exist anomalies in the experiments, such as dirty thermals, in
general the cloud was of a shape which can be best described by a hemi-
sphere. Most significant of all, the experiments showed no significant
and consistent .differences among different waste materials, with or
without solid particles settling out of the cloud.
In the integral technique used in the formulation of Barge Operation 1,
one of the most important items requiring experimental determination is
the entrainment coefficient. From the conservation of mass equation, by
assuming no settling, small density difference between the cloud and the
ambient and that the entrainment is only through the front part of the
hemispherical shaped cloud, the entrainment coefficient can be easily
shown to be equal to the tangent of half of the angle the cloud spreads,
i.e., a = —. Strictly speaking, — can be taken as the entrainment coef-
ficient only if the assumptions are fulfilled. From the general observa-
tions made in the experiments, it is believed that this estimate for the
entrainment coefficient is the best available. The estimates of entrain-
ment coefficient from experimental data are shown in Table 4. la. It varies
from 0.16 up to 0.45, with most of its values around 0.24.
From simple dimensional analysis, it can be shown that the entrainment
rate depends on the buoyancy and vorticity. However, no consistent
definite correlation has been found in the present investigation. It is
speculated that the entrainment rate is related to the structure and the
development of the vorticity inside the cloud, a detail which is not
131
-------
included in the present integral approach. As will be shown later, the
entrainment rate is important to the accuracy of the prediction of the
waste material's final position in a stratified ambient. The entrainment
rate and its relation to the buoyancy, vorticity and other parameters
should be further investigated.
Since in most cases the ambient density in the ocean is stratified due to
temperature and salinity variations, the adequacy of the mathematical
model depends on its verification by the experiments in a stratified
ambient. Experiments with instantaneous release of a three-dimensional
slug in a stratified ambient are summarized in Table 4. la. Selected
detailed results are shown in Figures 4.3.1,1-6, consisting of graphs of
i) centroid locations vs. time, ii) ambient stratification, iii) vertical
size vs. time and iv) horizontal size vs. time. Comparison with numer-
ical predictions are also shown.
In the experiments for runs with uniform ambient condition, the cloud always
reached the bottom and spread out horizontally there. For runs with a
stratified ambient, the cloud may or may not reach the bottom, depending
on the strength of the stratification. If the cloud did not reach the bottom,
it was observed to first descend, then halt, return upward and oscillate
very slowly. In general the cloud had initiated its horizontal collapse
before it came to the first halt. From the general observation of the
experiments, the transition from convective descent to dynamic collapse
was found to be a continuous process; the shape of the cloud changed con-
tinuously from approximately hemispherical to approximately ellipsoidal
near the neutrally buoyant position. There was no clear demarcation to
separate the convective descent and dynamic collapse phases. However,
the experiments clearly showed the general characteristics such as
entrainment, convective descent and collapse as postulated in the for-
mulation .
132
-------
OJ
OJ
THEORY
EXPERIMENT
5,0
t, TIME , ft.
Figure 4.3.1.1 Comparison of experiment with numerical model (Run N-S- 1 )
-------
THEORY
o EXPERIMENT
10.0
t, TIME ,ft.
Figure 4.3,1,2 Comparison of experiment with numerical model (Run N-S-2)
15,0
-------
o.o
0.2
0.4
LU
Q
DC
O
O
0.6
0.8
ce
s
^ i.o
1.2
1,4
0.990 0.992
RUN S-S-5
0.994 0-996 0.998
pt gm/cc.
Figure 4.3.1.3a Ambient density stratification
for Run S-S-5
1.000
135
-------
0.4
0.3
LJ
M
UJ
J-
cr
UJ
0,1 -
o.o
RUN S-S-5
0.0
5,0
THEORY
EXPERIMENT
I OX)
15.0
t, TIME , sec.
Figure 4. 3. 1 . 3b Comparison of experiment with numerical model (Run S-S-5)
-------
0.4
- 0.3
LJ
N
0)
o
N
o:
o
0.2
O.i
0.0
RUN S-S-5
0.0
5.0
THEORY
EXPERIMENT
10.0
ISO
t, TIME, sec.
Figure4.3.1.3c Comparison of experiment with numerical model (Run S-S-5)
-------
u*
00
THEORY
o EXPERIMENT
100
t, TIME , sec.
Figure 4.3.1.3d Comparison of experiment with numerical model (Run S-S-5)
ISO
-------
UJ
0,8
RUN S-S-5
SOLID PARTICLE
CONCENTRATION
FLUID CONCENTRATION
t, TIME, sec.
Figure 4. 3. 1. 3e Waste concentrations from numerical model (Run S-S-5)
-------
THEORY
EXPERIMENT
5.0
10,0
t, TIME , sec.
Figure 4.3.1.4 Comparison of experiment with numerical model (Run S-S-6)
15.0
-------
- 0.8 -
ui
5
6
DC
o
o
o
o
I-
a:
UJ
THEORY
o EXPERIMENT
10.0
t, TIME ,sec.
Figure 4.3.1.5 Comparison of experiment with numerical model (Run M-S-1)
15,0
-------
THEORY
EXPERIMENT
5.0
10,0
150
t, TIME ,sec.
Figure 4.3. 1.6 Comparison of experiment with numerical model (Run M-S-2)
-------
It is interesting to note that dirty thermals were never found in the runs
with stratified ambient. This could be explained in terms of the stability
in the flow field. In the uniform density runs, the density of the cloud is
always larger than the ambient, hence it is always unstable, and a devia-
tion of the cloud shape caused by the initial injection tends to be magnified
by the unstable condition. However, in a stratified ambient, when the
cloud penetrates the neutrally buoyant position, the stability condition
changes from that of unstable to stable, and any deviation in shape tends
to be restored.
4.3.2 Continuous Discharge From a Horizontal Travelling Vertical Jet
Tables 4. Ib and 2b summarize the experiments simulating Barge Operation
2 involving a jet type discharge. Selected detailed experimental results
are shown in Figures 4.3.2.1-8 where the predictions based on the mathe-
matical model are also shown. It is readily observed that the comparison
between the model and the experiments is very good.
Two different materials were used in the experiments, namely salt water
and sewage sludge. Experiments with a salt water jet in a uniform density
were used to test the adequacy of the mathematical model in the simplest
situation. Salt water jet in a stratified ambient was used to delineate the
influence of the density stratification on the behavior of the jet. The experi-
ments using sewage sludge were employed to verify the compatibility of the
model for handling both the ambient density stratification and the settling
of solid particles .
In the experiments performed the velocity ratio of the jet to the cross
current ranges from 3.6 up to 24.64, while the densimetric Froude number
varies from 2.6 up to 1840.
The experiments simulating Barge Operations 1 and 2 will be further dis-
cussed and used to verify the mathematical models in Section 5.
143
-------
0.0
UJ
CC
o
o
o
o
H
tr
UJ
0.5
1.0
I
PREDICTED JET
TRAJECTORY
TRACE OF JET
BOUNDARY
0.0
0.5 I JO
x , HORIZONTAL COORDINATE ,ft
1.5
Figure 4.3.2.1 Comparison of experiment with numerical model (Run N.-O- 1)
-------
0.0
Ul
UJ
o
ac
o
O r\ 5
o u
o
H
cc
UJ
.0
0.0
RUN Nj-0-2
PREDICTED JET
TRAJECTORY
TRACE OF JET
BOUNDARY
0.5 I,O
x, HORIZONTAL COORDINATE , ft
1,5
Figure 4.3.2.2 Comparison of experiment with numerical model (Run N.-O-2)
-------
LU
h-
O
or
O
o--
O
I-
o:
LU
PREDICTED JET
TRAJECTORY
TRACE OF JET
BOUNDARY
05 1.0
x , HORIZONTAL COORDINATE , ft
Figure 4.3.2.3 Comparison of experiment with nvimericai model (Run S.-O-l)
-------
0.0
111
o
cc
o
o
o
o
H-
QC
LiJ
0.5
0.0
RUN Sj-0-2
-o
PREDICTED JET
TRAJECTORY
TRACE OF JET
BOUNDARY
I
0.5 1,0
x, HORIZONTAL COORDINATE ,ft
Figure 4.3.2.4 Comparison of experiment with numerical model (RunS.-O-2)
1,5
-------
ao
00
LJ
h-
a
tr
o
o
o
o
h-
ce
UJ
0.5
RUN Nj-S-l
PREDICTED JET TRAJECTORY
TRACE OF JET BOUNDARY
0.0
Figure 4. 3. 2. 5
0.5 1,0
x , HORIZONTAL COORDINATE , ft
Comparison of experiment with numerical model-(Run N.-S-l)
1.5
-------
0.0
vO
UJ
Q
a:
o
o
o
<
o
h-
-------
LU
o
a:
o
o
o
o
h-
(T
LU
RUN Sj-S-4
$S SOLIDS SETTLING
OF JET ^
PREDICTED JET
TRAJECTORY
TRACE OF JET
BOUNDARY
0.5 1,0
x , HORIZONTAL COORDINATE , ft
Figure 4.3.2.7 Comparisoa of exper-iment with numerical model (Run S.-S-4)
-------
0.0
llj
a
tr
o
o
o
o
ft
UJ
0.5
0.0
PREDICTED JET
TRAJECTORY
TRACE OF JET
BOUNDARY
1
0.5 1.0
x, HORIZONTAL COORDINATE, ft
1.5
Figure 4.3.2.8 Comparison of experiment with numerical model (Run S.-S-5)
-------
4,, 3. 3 Instantaneous Release of Two-Dimensional Puff
Tables 4. lc and 2c summarize the pertinent parameters and density
stratifications for experiments involving the instantaneous release of a
two-dimensional puff. The entrainment coefficient evaluated from the
experiments in a uniform ambient was found to be close to 0. 3536 as
found by Richards (1965) who used salt water in a uniform ambient.
Again these runs also showed the general characteristics of entrainment,
convective descent, and collapse of the two-dimensional puff as
postulated in the mathematical model.
152
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SECTION V
COMPARISON OF EXPERIMENTS WITH THE
MATHEMATICAL MODEL
In this section the predictions based on the mathematical model developed
in Section III will be compared with the results of the experiments reported
in Section IV. In the following, comparison will be made for Barge Opera-
tions 1 and 2. The three experiments performed for the two-dimensional
thermal were exploratory in nature and no comparison will be made herein.
5 . 1 Barge Operation 1
In the comparison, the initial velocity, volume, gross density of the dis-
charge and density stratifications obtained directly from experimental data
are used in the numerical simulation. The fall velocity of the particles was
estimated by putting a small amount of very dilute waste material in the tank
and timing the time needed for the particles to pass a fixed distance. For
the mud and sewage sludge the fall velocities are found to be less than 0. 005
fps and 0. 02 fps respectively. The density and concentration of the solids
are estimated. For the various coefficients needed for the numerical simu-
lation, the Tetra Tech suggested values are used except for the entrainment
coefficient which is allowed to vary for different experiments and the best
fit value is chosen. The best fit entrainment coefficients for the stratified
ambient runs are presented in Table 4. la and are found to be between 0.21
and 0.265.
All the experiments with stratified ambient listed in Table 4. la have been
compared with the mathematical predictions. However, to avoid excessive
repitition only six selected runs are presented in Figures 4. 3. 1. 1-6. All
other runs compare at least as well as those selected. The positions of
the centroid of the waste cloud are plotted vs. time and presented for all
the runs selected. Further information such as ambient density stratifica-
tion, gross size and variation of concentration inside the cloud are shown
only for RUN S-S-5.
153
-------
la Figures 4.3.1.1-6, the waste clouds are seen to descend and then
oscillate about a neutrally buoyant position. The comparisons between
the experiments and the theory are found to be fairly good. Note that
in spite of some uncertainty in the values of some of the coefficients, it
seems that the positions of the cloud can be predicted with good accuracy
by only varying the entrainment coefficient between 0.21 and 0.265 in
the present investigation.
Figure 4.3.1.3a shows a typical ambient density stratification used in the
Experiments .
In Figures 4. 3 . 1. 3b and 4. 3 . 1. 3c, the gross size of a sewage sludge cloud
is compared with the numerical simulation. The comparison for a , the
vertical size, is good. However, the experimentally obtained horizontal
size b is larger than the theoretical prediction beyond a certain time.
Since the experiments were performed in a 9 inch wide tank, the discrep-
ancy in the prediction of b is believed to be from the influence of the
limited width. The presence of the walls-forced the cloud to spread more
in the other directions which led to the apparent excess expansion observed
in the experiment. Note that salt water was used in the experiments for
RUNS N-S-1 and N-S-2. No significant difference was observed between
these runs and those with solid particles in the cloud except for the phenom-
enon of settling.
Figure 4.3.1.3e shows the fluid and solid concentrations in the cloud as pre-
dicted by the mathematical model for RUN S-S-5. The concentrations are
seen to decline very rapidly in the early stage of convection due to entrain-
ment. This rapid decline of the solid concentration suggests that the
phenomenon of hindered settling would not be of major concern in the
selection of the settling coefficient B.. as discussed in Section 3. 2. 1.
No experimental data is obtained on the concentrations. However, since
the size of the cloud is directly related to the concentration during convec-
tion when settling was minor, the favorable comparison on the position and
size between prediction and experiment indicates that the model is adequate.
154
-------
From general observations of both the numerical simulation and the
experimental data, it is found that the initial gross density and velocity
of the waste cloud play important roles in the convection phase. For a
waste cloud with high density, the initial velocity is of less importance.
However, if the gross density is nearly the same as the ambient density,
then the influence of the initial velocity dominates.
In the numerical simulation for comparison with the experiments the
density and concentration of the solid particles were estimated. However,
it was found that changes of the inputs for solid density and concentration
over a factor of two did not change the results significantly. The reason
is that the fall velocities are very small for the mud and digested sewage
sludge used in the experiments . The mechanics is affected much more
by the dilution and entrainment than by settling in the whole period of con-
vection and collapse. Hence the gross density is made more important
rather than the solid density and concentration. However, regarding the
final phase of long term diffusion of the waste material, accurate informa-
tion on the concentration would be needed to predict the concentration at
the end of dynamic collapse for input to the long term diffusion model.
5 . 2 Barge Operation 2
In the comparison between predictions and experiments for Barge Opera-
tion 2, the jet discharge characteristics and ambient conditions used in
the predictive model are those measured in the experiments. The empir-
ical coefficients used are those built into the program. The concentration
and density of the solid particles in the discharge were estimated based
on gross measurements. For comparison, the centerline trajectories of
the jet obtained by numerical simulation are plotted against experimental
data of the jet boundaries. Although all runs were compared, only eight
representative runs are selected and presented in Figures 4.3.2.1 to
4.3.2.8 to avoid excessive repetition.
155
-------
For showing the influence of the ambient density and solid particles on the
gross behavior of the jet, the data will be presented in the order of N.-O,
S.-O, N.-S and S.-S successively, The designations are for salt water jet
J J J
(N.) and sewage sludge jet (S.) in a homogeneous density (0) or density
stratified (S) cross current respectively. The results of the first group
are shown in Figures 4.3.2.1-2. The jet is seen to bend over in the direc-
tion of the cross current and it tends to reach the bottom. Figures
4.3.2.3-4 show that the solid particles inside the sewage sludge jet do
not change the jet behavior significantly. The effects of density stratifica-
tion on the salt water jet are shown in Figures 4.3.2.5-6. Figure 4.3.2.5
shows that the salt water jet is trapped in a neutrally buoyant position; the
jet first overshoots the neutrally buoyant position; however, it is forced
back by the density stratification. Figure 4.3.2.6 does not show the
influence of the density stratification since the jet has not reached the posi-
tion of strong density gradient yet. Figures 4.3.2.7-8 show clearly that
the main core of the jet is trapped in a neutrally buoyant position while
particles are settling out of the jet as shown in the figures.
For another check, two runs were made to compare with Fan's (1967)
experimental data and theoretical predictions. The comparisons are
found to be very good as shown in Figure 5. 1.
From, the comparisons of numerical simulation to the experimental data it
is found that the jets with or without solid particles behave closely to what
is assumed in the formulations. In the experiments, the development of
horse-shoe profiles as found by other investigators were observed; how-
ever, the assumption of top-hat profiles appears adequate to provide useful
predictions. Hayashi (1971) reported that a bent-over plume splits sideways
into two concentrated regions with a clear space between them when the
upward injected plume comes close to the free surface. This phenomenon
was not observed when the jet plume was trapped in a neutrally buoyant
position in a density stratified ambient in the present investigation. The
156
-------
200
I I 1 ! 1 1 1 I
FANS THEORY
TRACE OF JET BOUNDARY
(0.5lem ORIFICE, PHOTO N0.763O.!}
X PRESENT THEORY
• EXPERIMENT
100
a)
100
X
•
FAN'S THEORY
TRACE OF JET BOUNDARY
{PHOTO NO. 7628.10)
PRESENT THEORY
EXPERIMENT
f I I |._ 1
50
b)
F=20
Figure 5. I Comparison of present theory with Fan's
experiments (from Fan (1967))
157
-------
free surface can be considered as a position with an infinite density
gradient and perhaps if the density stratification is sufficiently strong in
the ambient the phenomenon of splitting a plume into two separated regions
may occur.
5.3 Discussion
From the above comparisons, it can be concluded that the mathematical
model developed is able to provide good estimates on the behavior of wastes
discharged from barges. Perhaps the most important verification which
can be inferred from these experiments is that the general behavior of the
buoyant (or sinking) elements and jets are not materially affected if at all
by the presence of solid particles of the type encountered in typical wastes
(those with relatively small settling velocities).
158
-------
SECTION VI
EXAMPLE SOLUTIONS
The computer program based on bhe mathematical model developed in
Section III has been used to obtain the dispersion characteristics in a
variety of cases. These will be presented in this section.
Before the computer program can be used, it is first necessary to specify
a) the ambient conditions, including the currents, diffusion coefficients,
and density stratification, b) the waste characteristics, including the
density, size, and solid characteristics, c) the discharge method and other
control parameters, and d) the various numerical coefficients. Examina-
tion of the list of inputs presented in Appendix A reveals that a great
number of parameters are needed for the complete specification of each
case. A parametric study based on this model, although highly desirable,
would entail a very large number of cases and is beyond the scope of the
present investigation. While a parametric study should be performed in
the future, in the following, only representative cases are presented to
a) test the model and program and b) delineate the effects of some of the
more important parameters.
Tables 6.1, 6.2, and 6. 3 summarize the parameters used in the various
example runs for Barge Operation 1, 2, and 3 respectively.
The computations for most of the cases are terminated at the conclusion
of the dynamic collapse phase. A few representative cases are selected
from the tables for which the computations are allowed to proceed further
into the long term diffusion phase.
The run designations used for identification consist of three entries, e.g.
D-S-6. The first letter signifies the barge operation (D for bottom open-
ing hopper dump, J for jet discharge, and W for discharge into the barge
159
-------
TABLE 6. 1
SUMMARY OF PARAMETERS OF RUNS MADE
FOR BARGE OPERATION 1
RUN NO. p_ . Remarks
D-L-1
D-L-2
D-L-3
D-L-4
D-L-5
D-JL-6
D-L-7
D-L-8
D-JL-9
D-L-10*
D-L-11
D-S-1
D-S-2
D-S-3*
D-S-4
D-S-5
D-S-6
D-S-7
D-S-8
D-S-9
D-S-10
D-S-11
D-S-12
D-S-13
D-S-14
a
1.
1.
1 .
1 .
1.
l.
l .
l.
1 .
l .
1 .
1 .
1 .
1 .
1 .
1.
1 .
1.
1 .
1.
1.
1 .
1 .
l .
1 .
030
029
028
02?
026
025
024
0235
024 UAO = 1
024 UAO = 1, WAO = 1
030 U(l) = l, W(l) = l
028
027
026
025
024
0235
024 UAO = 1
024 UAO = 1, WAO = 1
026 V(l)=0
026 V(l) = 5
026 V(l)=5, H=65
026 RB = 10
024 FI=10
024 FI=1
160
-------
Table 6. 1 (continued)
RUN NO. p Remarks
3. 1
D-S-15 1.024 FRICTN = 0.5
D-S-16 1.024 FRICTN=0.
D-S-17 1.026 CFRIC = 0.01
D-S-18 1.026 ALPHAC = 0.1
D-S-19 1.026 ALPHAC = 0.
D-LL-1* 1.023 V(l) = 0.6 Collapse phase bypassed
D-F-1 1.0231 V(l) = 0.6
UAO=1, Y(F) = 0, 20,40, 100, ROA(I)=1. 02, 1.02, 1.03,
1 . 035, YK1 = 18, YK2-22, YK3-38, YK4=42, YU = 20,
20, YE=40
D-FS-2* Same as D-FS-1 except GAMA1 = 0.02
D-2S-1 UAO=1, YU=30, YW=40, YE=50, ROA(I)=1 . 023, 1 . 023,
1. 024, 1. 0245
Calculations carried to long term diffusion phase
161
-------
TABLE 6.2
SUMMARY OF PARAMETERS OF RUNS MADE
FOR BARGE OPERATION 2
RUN NO. P . Remarks
J-L-1
J-L-2
J-L-3
J-L-4
J-L-5
J-S-1
J-S- ;.
J-S.-3
J-S-4
J-5-5
J-S-6V
J-S-7
J-S-8
J-S-9
J-S-10
J-S- 11
J-S-12
OL
1.
1.
1.
1.
1.
1.
1.
1 .
1.
1 .
1.
1 .
1.
1.
1.
1.
1.
JL
024
0235
0231
024
024
024
0235
0231
0230
024
024
0235
0235
0235
0235
0235
02311
UAO = 1
UAO = 1, WAO = 1
Collapse phase bypassed
UAO = 1
UAO=WAO = 1
BC(1) = 1
U(l) = 10
CY(1)=20
UB = 5
THETA2(1)=45
J-F-1 1.024 SAI=225
J-F-2 1.024 SAI=225, UAO=1,
J-2S-1 1.024 UAO=1,YU=30, YW=40, YE=50, UB = 5
J-4S-1 1.024 UAO = 1, WAO = 1
* Calculations carried to long term diffusion phase
162
-------
TABLE 6.3
SUMMARY OF PARAMETERS OF RUNS MADE
FOR. BARGE OPERATION 3
RUN NO. P
Remarks
W-L-1
W-L-2
W-L-3
W-L-4
W-L-5
1
1
1
1
1
a J.
.0232
.0230
.0232
.0232
.0232
BB = 10
QD = 100
UB=5
W-S-1 1.0232
W-S-2* 1.0230
W-4S-1 1.0232
UB=5
Calculations carried to long term diffusion phase
163
-------
wake). The second letter signifies the solid whose characteristics are
listed in Table 6.4. The last number is simply a numerical sequence
within each group.
i) Ambient density stratification.
The ambient density ROA(I) is specified by four points at depths Y(I)
taken to be 0. , 40. , 60. , and 100 feet. The density at the surface and
40 foot depth is 1.023 gm/cc. The density p , at 60 and 100 feet varies
from 1 . OZ3 to 1.030 for different runs. The value of p , is indicated in
the tables. For some runs, a density gradient is inserted between 60 and
100 feet. In such cases, ROA(I) is provided in the tables. A few runs
also use different values for Y(I). These are also indicated in the tables.
ii) Ambient currents, (see Figure 3.6.1)
The ambient currents are specified by the parameters YU, YE, YW,
UAO,WAO. These are taken to be 50., 50., 100., 100., 0., 0., respec-
tively unless specified otherwise in the remarks column.
iii) Vertical diffusion coefficients, (see Figure 3.6.1)
The vertical diffusion coefficient profile is specified by the parameters
YK1, YK2, YK3, YK4, AKY1, AKY2, AKY3. These are taken to be
35., 45., 55., 65., 0.05, 0.005, 0.01 respectively unless specified
otherwise in the remarks column.
iv) Waste characteristics
The waste characteristics are specified by its gross density ROO, and
the solids characteristics. The characteristics for the various solids are
summarized in Table 6.4.
164
-------
TABLE 6.4
SUMMARY OF SOLID WASTE CHARACTERISTICS
USED IN SIMULATION RUNS
esignation ROC CS
(gms /cc)
S
L
LL
F
FS
2S
4S
1
1
1
1
1
1
1
. 12
.3
.
.05
. 10
. 13
.2
0.
0,
0.
0.
0.
0.
0.
0.
0.
0.
0.
15
2
2
2
2
1
1
2
15
005
ROAS
(gms/c
1.
2.
2.
--
1.
0.
2.
1
1
2
2
8
5
5
-
0
9
5
WS
c) (ft/sec
0.
0.
0.
0.
-0.
-0.
0.
-0.
-0.
0.
0.
005
05
5
01
05
05
1
001
15
0015
165
-------
v) Discharge characteristics
In Barge Operation 1, the discharge characteristics are characterized
by RB the radius of the waste load, and U(l), V(l), W(l), the velocity
components of the waste at the point of discharge. These are taken to be
5, 0, 1, 0. respectively unless specified otherwise in the remarks column.
In Barge Operation 2, the discharge characteristics are characterized by
the jet radius BC(1), discharge angle THETA2(1) with respect to the
vertical, discharge velocity U(l), and discharge depth CY(1). These
are taken to be 0. 5, 0. , 5. , and 10. respectively unless specified other-
wise in the remarks column.
In Barge Operation 3, the discharge characteristics are characterized by
the discharge rate QD which is taken to be 10. In addition the barge
width BB and barge depth DD are taken to be 25 and 10 respectively
unless specified otherwise .
Moreover, in Barge Operations 2 and 3, the quantities UB, the barge
velocity, SAI, the barge direction, and TIME the discharge time are
needed. These are taken to be 7 . 5, 180°, and 500 respectively unless
specified otherwise.
In all runs, built in coefficients are used except for runs D-S-13 through
19 where the deviations from the built in coefficients are noted in Table
6.1.
The graphic output from the calculations for the runs are grouped in the
order given in the tables and presented in APPENDIX B. Each
graph is identified by the run number. The case data are collectively pre-
sented at the beginning of the appendix before the graphs for ease of access.
166
-------
The computer runs presented in the tables can be divided into three
groups: 1) runs using standard inputs for solid L, 2) runs using standard
inputs for solid S and 3} runs using non-standard inputs and/or other
solids. Within groups 1 and 2, the only quantity which varies between runs
is the ambient density stratification parameter p , . By comparison among
these runs, the effect of ambient density stratification can be observed.
By comparing runs in group 1 with the corresponding runs in group 2, the
effect of solid characteristics can be seen. For the solid (L) with larger
fall velocity, the waste pool sometimes reaches the bottom and then leaves
the bottom when sufficient solids settle out. The effect of some of the
other input parameters such as those relating to ambient currents and
waste discharge characteristics can be seen by comparing ru"ns in group
3 with corresponding runs in the first two groups. An effort has also
been made to investigate the effect of the numerical coefficients. Only
Barge Operation 1 is used for this purpose. Runs D-S-13 through D-S 19
are the same as run D-S-3 or D-S-5 except for one of the coefficients.
The net effect of these coefficients are found to be as expected, e.g.
increasing the friction coefficient Fl hastens the transition from collapse
to long term diffusion while decreasing the rate of spreading due to
collapse. It is believed that while the effect may be significant for each
single discharge, the integrated effect for many discharges such as in the
prototype would not be of major influence to the predictions.
An item of some interest is the reentrainment coefficients GAMA1 and
GAMA2. Two runs D-FS- 1 (GAMA 1 = 0) and D-FS-2 (GAMA2=0. 02 ) are
made to investigate the effect of this coefficient. It is found that the rate
of material accumulation on the surface is decreased for D-FS-2. At
the same time, the values of x, z, a , a at the surface and those
.X Z
very near the surface tend to be more nearly equal in D-FS-2. Both
phenomena are expected physically.
167
-------
In summary, the model predictions are all reasonable physically. When
the coefficients are fully defined through experiments, both in the labor-
atory and the field, it is believed that the model will be a valuable tool
in the assessment of the physical impact of barge discharge of solid
wastes. Even in the present state, before the accurate determination of
some of the coefficients, the model is believed capable of predicting the
overall effects over the long run even though for each single discharge,
the details may not be as accurately predicted.
It should be pointed out that in the graphs, wherever several different
quantities are plotted versus the same independent variable, the scale of
the ordinate for the various quantities are different. They are automati-
cally set in such a way as to result in graphs which span a reasonable
scale. For comparison purposes, attention is called to the headings
where the maximum and minimum values of each plotted variable are
noted. It should further be mentioned that the last point plotted corres-
ponding to the largest value of the independent variable is a fictitious
point used in setting the scale and should therefore be disregarded.
The coordinate system used for Barge Operation 1 is always an inertial
system with the origin attached to the earth. For Barge Operations 2
and 3, the origin of the coordinate system is attached to the barge which
is moving. However, after the time when the input to the long term dif-
fusion model is completed, the motion of the coordinate system is
arrested.
168
-------
Interpretation of the Results of Computations from the
Mathematical Model for Long Term Diffusioa
The mathematical model developed for evaluating the long term
diffusion phase of the phenomenon is based on the method of moments.
Instead of obtaining the detailed concentration of the waste (both fluid
and solid phases) in the environment, only their first few moments
are obtained as functions of time and of the vertical coordinate. This
greatly reduces the computational labor and yet retains the most
significant aspects of the solution. The primary result of the cal-
culations in each specific case consists , then, of C..(y, t),(i,j=0,l,2),
th ^
the ij moment of the concentration defined previously in Section 3.6.2.
Their relations to the physically more meaningful quantities of cen-
troid locations x . z , and variances c a , o~ have also been
o o x ' z xz
discussed in Section 3.6.2. In the following, an interpretation will be
given on the physical significance of each of the quantities.
a) C
oo
The quantity C is simply the total amount of waste material at
depth y and time t within a horizontal slab of unit thickness.
b) x , z
o o
The quantities x , and ZQ» are x and z coordinates of the cen-
troid of the waste at depth y and time t.
, 2 2
c) a , a , a
x ' z ' xz
These are the variances of the distribution at depth y and time t
(see below for further discussion).
169
-------
d) C
max
This is the peak concentration at depth y and time t based on the
assumption that the distribution is Guassian.
All the above quantities are self-explanatory. Their physical signi-
ficance are also clear. However, in the event a ^ 0, the relation
between the variances o , a. , a and the physical shape of the
waste pool is somewhat obscure. An attempt will now be made to
clarify this aspect.
A Guassian distributed waste pool with parameters
r x z 2
C ' O' Z0' av ' °7 ' av7
can be written
C
oo
, / 2 2 2
2nVa._ cr - a,_
2 2
~ax az
22 2
Ma a -a )
x z
X Z XZ
X - X
X
z - z
2a
xz
2 2
°xaz
x - x
By a rotation and a translation of the coordinates from x, z to x, ,
z,, it is possible to remove the terms containing x , z and xz in
the exponent. The new axes x, z, are then the principal axes and
2 2
the corresponding a , Q , then correspond to the variances in the
xl Zl
principal axes whose physical interpretation is more obvious. It can be
demonstrated that the origin of the principal axes are at x , *. with
respect to the old axis and the clockwise angle Q which the old axes
needs to be rotated is such that
170
-------
2a
tan 29 = -> XZ i
ax " az
It can further be shown that the variances
Xl
2 2
» CT
in the principal axes are
_ 2 2 //2 2\2
x + CTz ACTx + az j
2 ' V ' 4 ' '
a 2+a 2 '(a 2_ a 2f
V "7 / \ V T /
- .X - Z */ VJS z / 4-
2
xz
2
x,
_ 2
az _ _ _
Zl ~ 2 4 xz
from which it can be seen that
22 22 2
a a = a a - a
Xi Z, X Z XZ
With the help of these equations, the physical interpretation of the
variances can be readily made.
When solid waste such as dredge spoil is discharged into the ocean,
an item of interest is the build up of the discharged material on the
ocean floor. In the mathematical model developed in this investigation
based on the method of moments, the following quantities are obtained
as functions of time.
W Total deposited solid
X x-coordinate of centroid of deposited solid
171
-------
Z z-coordinate of centroid of deposited solid
a x-variance of deposited solid
0 z-variance of deposited solid
Z
a covariance of deposited solid
xz r
The significance of the variances have already been discussed.
Thus, a set of principal axes can be found and the variances in these
axes obtained. In the present situation involving deposited solids,
an additional item of interest is the maximum height h of the mound
built up on the ocean floor. This can be shown to be
h =
where p is the porosity in the deposited solids. In the above ex-
pression, aGuassian distribution is assumed. If the distribution were
assumed to be more nearly uniform over the range
t xl l
-------
REFERENCES
Abraham, G., "The Flow of Round Buoyant Jets Issuing Vertically into
Ambient Fluid Flowing in a Horizontal Direction," Proc. of
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Abraham, G., van Dam, G.C., Eysink, W.D., Miiller, K. and Sydow,
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Ames, W.F., "Nonlinear Partial Differential Equations in Engineering, "
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Batchelor, G.K., "The Application of the Similarity Theory of Turbu-
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Bowden, K.F., "Horizontal Mixing in the Sea Due to a Shearing Current,"
J. of Fluid Mechanics, Vol. 21, pp. 83-95, 1965.
Bowden, K.F., "Turbulence, " Chap. VI of "The Sea" Vol. I edited by
M.N. Hill, Interscience Publishers, New York, 1962.
Brooks, N.H. and Koh, R.C.Y., "Discharge of Sewage Effluent from a
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1965.
Chevray, R. , "The Turbulent Wake of a Body of Revolution, " J. of
Basic Engineering, June 1968, pp. 275-284.
Clark, B.D. , Rittal, W.F., Baumgartner, D.J. and Byram, K.V.,
"The Barged Disposal of Wastes, A Review of Current Practice
and Methods of Evaluation, " Environmental Protection Agency,
Water Quality Office, Northwest Region, Pacific Northwest
Water Laboratory, Corvallis, 1971.
173
-------
Csanady, G.T., "Turbulent Diffusion in Lake Huron, " J. of Fluid
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Csanady, G.T., "Accelerated Diffusion in the Skewed Shear Flow of
Lake Currents," J. of Geophy. Res., Vol. 71, No. 2,
pp. 411-420, Jan., 1966.
Defant, A., "Physical Oceanography, " Vol. 1, The MacMillan Co. ,
N.Y., 1961.
Elder, J. W. , "The Dispersion of Marked Fluid in Turbulent Shear
Flow," J. of Fluid Mechanics, Vol. 5, pp. 544-560, 1959.
Elzy, E. and Wicks, C.E., "Transpirational Heat Transfer from a
Cylinder in Cross Flow Including the Effects of Turbulent
Intensity, " Chemical Engineering Progress Symposium,
Series 82, Vol. 64, pp. 150-165, 1968.
Fan, L.N., "Turbulent Buoyant Jets into Stratified or Flowing Ambient
Fluids," W.M. Keck Laboratory of Hydraulics and Water
Resources Division of Engineering and Applied Science,
California Institute of Technology, Pasadena, California,
Report No. KH-R-15, June 1967.
Fischer, H. B. , "Longitudinal Dispersion in Laboratory and Natural
Streams, " W.M. Keck Lab. of Hyd. and Water Res. , Tech.
Rep. KH-R-12, June, 1966, California Institute of Technology.
Foxworthy, J.E., Tibbey, R.B., and Barsom, G.M., "Dispersion of
a Surface Waste Field in the Sea, " J. of Water Poll. Control
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Gunnerson, C. G. , Discussion on "Eddy Diffusion in Homogeneous
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Hall, W.S. , "The Rise of an Isolated Thermal in Wind Shear, " Quar-
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Hanson, H.B. and Richardson, P.D., "The Near-Wake of a Circular
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1968.
Harremoes, P. , "Diffuser Design for Discharge to a Stratified Water, "
The Danish Isotope Center, Copenhagen, Denmark, 1967.
Hayashi, T. , "Turbulent Buoyant Jets of Effluent Discharged Vertically
Upwards from an Orifice in a Cross-Current in the Ocean, "
Proc. , Hydraulic Research and Its Impact on the Environment;
Vol. 1, 14th Congress of the Intn. Assoc. for Hyd. Res. ,
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174
-------
Ichiye, T., "Studies of Turbulent Diffusion of Dye Patches in the
Ocean," J. of Geophy. Res., Vol. 67, No. 8, pp. 3213-3216,
July, 1962.
Jobson, H.E. andSayre, W.W., "Vertical Transfer in Open Channel
Flow," J. Hydr. Div. Proc. ASCE, March 1970.
Joseph, J. and Sender, H. , "On the Spectrum of the Mean Diffusion
Velocities in the Ocean," J. of Geophy. Res., Vol. 67,
No. 8, pp. 3201-3205, July, 1962.
Ketchum, B. H. and Ford, W. L. , "Rate of Dispersion in the Wake of a
Barge at Sea, " Trans. AGU, Vol. 33, No. 5, pp. 680-684,
Oct. 1952.
Koczy, F.F. , "The Distribution of Elements in the Sea, " "Disposal
of Radioactive Wastes, " Vol. II, Proc. of Scient. Conf. on
Disposal of Radio. Wastes, Monaco, Nov. 1959, Pub. by
IAEA, Vienna, pp. 191-197, I960.
Koh, R.C.Y. et al, "A Survey of Radiometric Detection, " Tech. Rept
(Classified SECRET) National Engineering Science Company,
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the Radioactive Debris Distribution Subsequent to a Deep
Underwater Nuclear Explosion", Tetra Tech Inc. Tech. Rept
TC-154, 1969.
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Kuo, Y.H. and Baldwin, L.V., "The Formation of Elliptical Wakes,"
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Proc. Roy. Soc. A, Vol. 234, pp. 1-23, 1956.
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Lagoon," Trans. Amer. Geophy. Union, Vol. 30, No. 1,
pp. 59-66, Feb. 1949.
175
-------
Naudascher, E. , "On a General Similarity Analysis for Turbulent Jet
and Wake Flows," IIHR Report No. 106, Iowa Institute of
Hydraulic Research, The University of Iowa, Iowa City, Iowa,
Dec. 1967.
Naudascher, E. , "On the Distribution and Development of Mean-Flow
and Turbulence Characteristics in Jet and Wake Flows, " IIHR
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University of Iowa, Iowa City, Iowa, Aug. 1958.
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Diffusion from an Instantaneous Source," Int. J. of Oceanology
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1961.
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176
-------
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177
-------
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178
-------
APPENDIX A
COMPUTER PROGRAM
The computer program is described and listed in this appendix. It should
be noted that the program listed was tested on a CDC 6600. Modifications
may be necessary if it were to be adapted to other systems. It should
further be cautioned that since the choice of such parameters as integration
step size and grid points are built into the program, the efficiency of the
processing is a function of the data for the particular case. Under certain
circumstances, the program may not be able to choose a satisfactory step
size to complete the calculations. The choices of step sizes are made
assuming that
1) the waste load will descend into the ocean
2) the convective phase will terminate (either by reaching
bottom or by reaching neutral buoyancy condition) in a
reasonable amount of time.
3) the collapse phase can occur (either by reaching bottom
or by being in a density gradient) and will terminate in
a reasonable amount of time.
Several tries may be necessary in cases when the discharge and ambient
conditions deviate from those envisioned in typical coastal waters.
Section 3. 7 should be consulted for more details.
Al
-------
READING SEQUENCE IN BARGE OPERATION 1
Symbol
METHOD, IGCN, IGCL, IGLT, IPCN,
IPCL, IPLT, NSCALE, NX, NY, IBUG
(TRIGER(I), 1=1,4)
*
N
(Y(I), 1 = 1, N)
(ROA(I), 1=1, N)
YK1, YK2, YK3, YK4, AKY1.AKY2, AKY3
YU, YW. YE, H, UAO, WAO
TSTOP
**
ISIZE , KEY1, KEY2, KEY3
DINCR1, DINCR2 +
ALPHAO, BETA, CM, CD +
GAMA, CDRAG, CFRIC, CDS, CD4, ALPHAC
FRICTN, Fl +
ALFA1, ALFA2, GAMA1.GAMA2, ALAMDA +
RB, ROO.U(l), V(l), W(l)
K, L++
(ROAS(I), 1=1, K)
((CS(I, J), J=l, L), 1=1, K)
((WS(I, J), J=l, L), 1=1, K)
Format
1615
8F10. 5
110
8G10.4
8G10.4
7G10.4
6G10.4
G10.4
41 10
2G10.4
4G10.4
8G10.4
5G10.4
5G10.4
2110
4G10.4
8G10.4
8G10.4
Subroutine
DISPSN
AMBIENT
DUMP
* N< 30
** Option of 51, 101 or 151
+ Skip card if KEY 1 = 1
++ K< 4, L< 2
A2
-------
EXPLANATION OF THE INPUT SYMBOLS
In Program
METHOD
IGCN
IGCL
IGLT
IPCN
IPCL
IPLT
In Text Remarks
Method of barge operation must be 1
for barge operation 1
IGCN = 0 No graph plotted for convective phase
IGCN = 1 One graph plotted for convective phase
IGCN=2 Extra graphs of concentrations plotted for
convecfcive phase
IGCL=0 No graph plotted for collapse phase
IGCL=1 Graph plotted for collapse phase
IGLT = 0 No graph plotted for long term diffusion phase
IGLT = 1 Graphs of CQQ, Size and settled solids
characteristics plotted
IGLT = 2 Extra graphs of centroid locations plotted
IGLT > 2 All graphs plotted for long term diffusion
IPCN = 0 No detailed printed output for convective phase
IPCN = 1 Detailed printed output included
IPCL=0 No detailed printed output for collapse phase
IPCL=1 Detailed printed output included
IPLT = 0 No detailed printed output for long term
diffusion phase
IPLT> 0 Approximate number of pages of detailed output
desired before input to long term diffusion
completed
A3
-------
In Program In Text
Remarks
NSCALE
Scale for plotting
NX
Size of graph along independent variable
NY
Size of graph along dependent variable
IBUG
Number of integration steps between inputs to
long term diffusion
TRIGER(I)
N
Triggers for graphing in long term diffusion
(graphs are at times when the peak value of
Cnn in the y direction first falls below
TRIGER(I) times the maximum of CQO ever
achieved after input to long term diffusion
completed).
Number of points where ambient density is
specified
Vertical position from free surface where
ambient density is specified (ft)
ROA(D
Ambient density at Y(I), (gram/cc)
YK1, YK2,
YK3, YK4
yk3'
AKY1, AKY2, k ,, k 0
yl' y2
AKY3 k „
Positions where the vertical diffusion coef-
ficient changes (ft)
Vertical diffusion coefficients: near to the
free surface, at thermocline and below the
thermocline (ft /sec)
YU, YW, YE y , v
' 7 r
H
H
Vertical positions where ambient velocity is
specified (ft)
Total depth (ft)
A4
-------
In Program la Text
Remarks
UAO, WAO
u , w
a' a
Maximum horizontal velocities in x and
z directions (ft/sec)
TSTOP
ISIZE
Physical time limit in second in the long
term diffusion computation. Time started
from the commencement of dumping.
Vertical grid point size in the long term dif-
fusion computation, 3 options: 51, 101, and
151
KEY1
KEY1 = 1 Use Tetra Tech suggested coefficients
KEY1=2 Use read in coefficients
KEY2
KEY2 = 1 The computation stops at the end of
convective descent phase
KEY2 = 2 The computation stops at the end of
dynamic collapse
KEY2=3 The computation stops at the end of
long term diffusion
KEYS
KEY3 = 0 Long term diffusion computation also
performed for the fluid part
KEY3 = 1 No long term diffusion performed for
the fluid part
DINCR 1
Factor in obtaining suitable trial time step in
the convective descent phase
DINCR2
Factor in obtaining suitable trial time step in
the dynamic collapse phase
A5
-------
la Program In Text
Remarks
ALPHAG
ALPHAO
BETA
CM
CD
a
a
C
m
C
D
Entrainment coefficient for collapsing
Entrainment coefficient for a thermal
Settling coefficient
Added mass coefficient
Drag coefficient
GAMA
CDRAG
CFRIC
CDS
CD4
FRICTN
Fl
ALFA1
ALFA2
Y
C
drag
Cfric
C
C
D3
D4
•p
rictn
a
Gradient factor in the cloud
Form drag coefficient for the quadrant of
a collapsing ellipsoid
Skin friction coefficient for the quadrant of
a collapsing ellipsoid
Drag coefficient for an ellipsoidal wedge
Drag coefficient for a plate
Friction coefficient between the cloud and
ocean bottom
Modification factor used in computing the
resistance of the friction force to the col-
Ipse of a quadrant of an ellipsoid
Absorbency coefficient of the bed to the
solid particles
Absorbency coefficient of the free surface
to the solid particles
A6
-------
In Program In Text
R emarks
GAMA1
Entrainmenfc coefficient (for resuspension)
at the bed
GAMA2
Entrainment coefficient (for resuspension)
at the free surface
ALAMDA
Dissipation factor used in computing the
horizontal diffusion coefficient
RB
Initial radius of the waste material (ft)
ROO
Density of the waste material (gram/cc)
Initial velocity in x direction (ft/sec)
Initial velocity in y direction (ft/sec)
w
Initial velocity in z direction (ft/sec)
K
K
Number of different solid density
L
ROAS(I)
CS(I, J)
Number of different solid fall velocities for
each solid density
Solid density (gm/cc)
Concentration of solid particles with a
specific, density ROAS(I ) and fall velocity
WS(I, J), (ft3 /ft3)
WS(I, J)
W
Fall velocity of the solid particles with
density ROAS(I) (ft/sec)
A7
-------
EXPLANATION OF OUTPUT SYMBOLS
In Convective Descent and Dynamic Collapse Phase
In Program In Text
Remarks
TIME
Physical time from conmencement of
dumping (sec)
X, Y, Z
x, y, z Position of the centroid of the cloud in
x, y, and z direction respectively (ft)
U, V, W
u, v, w Velocity of the cloud in x, y, and z direction
respectively (ft/sec)
DEN-DIF
RADIUS
VORT.
P - P.
K
Density difference between that of the cloud
and the ambient (gram/cc)
Radius of the hemispherical cloud (ft)
Vorticity of the cloud (sec" )
ALPHA
FLUID CONC.
a
KL
Entrainment coefficient
j C .. Concentration of the fluid part of waste
J -j -3
material (ft /ft )
SOLID VOL.
CONCENTRATION
C
s
A a
Solid volume of waste material with a specific
fall velocity (ft3)
Concentration of the solid particle with a specific
fall velocity
Minor axis of the waste material in dynamic
collapse phase
Major axis of the waste material in dynamic
collapse phase
A8
-------
In Long Term Diffusion
In Program.
TIME
MEAN Y
FALL VEL.
TSS
In Text
Y
X, Z
SIGMAX,
SIGMA Z,
SIGMXZ
CONG.
COO
w
V
xz
oo
Physical time from the commencement of
barge operation
The centroid of waste material in y direction
Fall velocity of the solid particles
Total suspended solid
Total boundary material
Position of grid point
Centroid of waste material on a horizontal
plane at Y
Variances of the waste material on a
horizontal plane at Y
Concentration assuming the waste material
is normally distributed
f f
J^coJ _o
C dxdz at Y
For Solid on the Bed or Free Surface
X, Z
SIGMAX,
SIGMAZ,
SIGMXZ
SIZE
DEPOSITED SOLID
Centroid of deposited waste material
Variances of the deposited waste material
Geometric mean of 0 ancj 0 along principle axes
x z
Solid deposited either on bed or free surface
A9
-------
READING SEQUENCE IN BARGE OPERATION 2
Symbol
Format
Subroutine
METHOD, IGCN, IGCL, IGLT, IPCN,
IPCL, IPLT, NSCALE, NX, NY, IBUG
(TRIGER(I), 1=1,4)
1615
8F10. 5
DISPSN
N I 10
(Y(I), 1=1, N) 8G10.4
(ROA(I), 1=1,N) 8G10.4
YK1, YK2, YK3, YK4, AKY2, AKY3 7G10.4
YU, YW, YE, H, UAO, WAO 6G10.4
TSTOP G10.4
ISIZE**, KEY1, KEY2, KEY3 4110
DINCR1, DINCR2* 2G10.4
ALPHA1, ALPHA2, BETA, CD+ 4G10.4
GAMA, CDRAG, CFRIC, CD3, CD4, ALPHA3, 8G10.4
ALPHA4, FRICTN, Fl, CM +
ALFA1, ALFA2, GAMA1, GAMA2, ALAMDA+ 5G10.4
BC(1),ROO, THETA2(1), U(l), CY(1) 5G10.4
K, L++ 2110
(ROAS(I), 1=1, K) 4G10.4
((CS(I, J), J = l, JL), 1 = 1, K) 8G10.4
((WS(I, J), J=l, L), 1=1, K) 8G10.4
UB, SAI, TIME 3G10.4
AMBIENT
JET
N< 30
Option of 51, 101, or 151
Skip card if KEY 1 = 1
K< 4, L< 2
A10
-------
EXPLANATION OF THE INPUT SYMBOLS
In Program In Text
METHOD
Remarks
Method of barge operation must be 2 for
barge operation 2
IGCN
IGCN = 0 No graph plotted for convective phase
IGCN= 1 One graph plotted for convective phase
IGCN=2 Extra graphs of concentrations plotted for
convective phase
IGCL
IGCL=0 No graph plotted for collapse phase
IGCL=1 Graph plotted for collapse phase
IGLT
IPCN
IGLT=0 No graph plotted for long term diffusion phase
IGLT=1 Graphs of COQ, Size and settled solids
characteristics plotted
IGLT=2 Extra graphs of centroid locations plotted
IGLT>2 All graphs plotted for long term diffusion
IPCN=0 No detailed printed output for convective phase
IPCN= 1 Detailed printed output included
IPCL
IPCL=0 No detailed printed output for collapse phase
IPCL=1 Detailed printed output included
IPLT
IPLT=0 No detailed printed output for long term
diffusion phase
IPLT >0 Approximate number of pages of detailed output
desired before input to long term diffusion
completed
Al 1
-------
In Program In Text
Remarks
NSCALE
Scale for plotting
NX
NY
Size of graph along independent variable
Size of graph along dependent variable
IBUG
Number of integration steps between inputs to
long term diffusion
TRIGER(I)
Triggers for graphing in long term diffusion
(graphs are at times when the peak value of
CQQ in the y direction first falls below
TRIGER(I) times the maximum of C ever
00
achieved after input to long term diffusion
completed).
N
Number of points where ambient density is
specified
Vertical position from free surface where
ambient density is specified (ft)
ROA(I)
Ambient density at Y(I), (gram/cc)
' k2
YK1, YK2,
YK3, YK4
AKYl, AKY2, k j, k 2
AKY3 k ,
Positions where the vertical diffusion coef-
ficient changes (ft)
Vertical diffusion coefficients: near to the
free surface, at thermocline and below the
thermocline (ft /sec)
YU YW YE V , v v Vertical positions where ambient velocity is
' u Jw' 7
H
H
specified (ft)
Total depth (ft)
A12
-------
In Program In Text
Remarks
UAO, WAO u , w
' a' a
Maximum horizontal velocities in x and
z directions (ft/sec)
TSTOP
Physical time limit in second in the long
term diffusion computation. Time started
from the commencement of discharging
through the jet.
ISIZE
Vertical grid point size in the long term dif-
•fusion computation, 3 options: 51, 101, and
151
KEY1
KEY1 = 1 Use Tetra Tech suggested coefficients
KEY1=2 Use read in coefficients
KEY2
KEY2 = 1 'The computation stops at the end of
jet convection
KEY2=2 The computation stops at the end of
dynamic collapse
KEY2 = 3 The computation stops at the end of
long term diffusion
KEY3
DINCR1
DINCR2
KEY3 = 0 Long term diffusion computation also
performed for the fluid part
KEY3 = 1 No long term diffusion performed for
the fluid part
Trial value in obtaining distance step DS =
DINCR1 x BC(1) in the jet convection phase
where BC = jet size
Trial value in obtaining distance step DS =
DINCR2 x BC(INDEX) in the dynamic col-
lapse phase, where INDEX is the step at the
end of jet convection
A13
-------
In Program In Text
Remarks
ALPHA1
ALPHA2
BETA
a-
Entrainment coefficient for a momeatum jet
Entraiament coefficient for 2-D thermal
Settling coefficient
CD
GAMA
CDRAG
CFRIC
CDS
CD4
FRICTN
Drag coefficient for a cylinder
Gradient factor in the cloud
Form drag coefficient for the quadrant of
a collapsing elliptical cylinder
Skin drag coefficient for the quadrant of
a collapsing elliptical cylinder
Drag coefficient for an elliptical wedge
Drag coefficient for a two-dimensional plate
Friction coefficient between the cloud and
ocean bottom
Fl
CM
ALFA1
Modification factor used in computing the
the resistance of the friction force to the
quadrant of an elliptical cylinder
Added mass coefficient
Absorbency coefficient of the bed to the solid
oarticles
ALFA2
GAMA1
Absorbency coefficient of the free surface to
the solid particles
Entrainment coefficient (for re suspension) at
the bed
A14
-------
GAMA2
A LAM DA
BC(1) b
ROO p
THETA2(1) 9.
i.
U(D u
CY(1) y
K
L
ROAS(I)
CS(I, J)
WS(I, J)
LIB
SAI
TIME
Entrainment coefficient (for re suspension)
at the surface
Dissipation factor used in computing the
horizontal diffusion coefficient
Radius of nozzle (ft)
Density of the waste material (gram/cc)
Angle of the nozzle (degree)
Velocity of the waste material at the nozzle (ft/sec)
Vertical position of the nozzle (ft)
Number of different solid density
Number of different solid fall velocities
for each solid density
Solid density (gm/cc)
Solid concentration of the solid particles
with density ROAS(I) and fall velocity
WS(I, J) (ft3/ft3)
Fall velocity of the solid particles (ft/sec)
Barge velocity (ft/sec)
Barge moving direction (limited to 180 - 270 )
Time of continuous discharge (sec)
A15
-------
EXPLANATION OF OUTPUT SYMBOLS
la Jet Convection And Dynamic Collapse Phase
In Program In Text
Remarks
TIME
Physical time taken for the front of jet to
move from the nozzle to position S (sec)
Distance along the axis of the jet (ft)
X, Y, Z
x, y, z
Position of the center of the jet at position S (ft)
U
u
Velocity of the jet at S (ft/sec)
RADIUS
Radius of the jet (ft)
DEN-DIF
P ' P:
Density difference between that of the jet and
the ambient (gram/cc)
FLUID CONG.
Concentration of the fluid part waste material
FLUX OF SOLID
Flux of solid particles through a cross section
urr b2Cg.. at S (ft3/ft3)
J
CONCENTRATION
C ..
Concentration of solid particles with a specific
fall velocity (ft3/ft3)
The minor axis of the two dimensional thermal
in the dynamic collapse phase (ft)
B
The major axis of the two dimensional thermal
with length DS (ft)
A16
-------
In Program
In Texb
Remarks
SOLID VOL.
Solid volume in the two-dimensional thermal
with length DS (ft3)
In Long Term Diffusion
See Explanation of Output Symbols In Barge Operation 1
A17
-------
READING SEQUENCE IN BARGE OPERATION 3
Symbol Format
METHOD, IGCN, IGCL, IGLT, IPCN, 1615
IPCL, IPLT, NSCALE, NX, NY, IBUG
(TRIGER(I), 1=1,4) 8F10.5
N* 110
(Y(I), 1 = 1, N) 8G10.4
(ROA(I), 1=1, N) 8G10.4
YK1, YK2, YK3, YK4, AKYl, AKY2, AKY3 7G10.4
YU, YW, YE, H, UAO, WAO 6G10.4
TSTOP G10.4
ISIZE**, KEY1, KEY2, KEYS 4110
Cl, C2+ 2G10.4
DINCR1, DINCR2+ 2G10.4
ALPHA, BETA, GDI, CD2, CM , ALPHA 1 + 8G10.4
GAMA, CDRAG, CFRIC, CD3, CD4, ALPHAS, 8G10.4
ALPHA4, FRICTN, Fl +
ALFA1, ALFA2, GAMA1, GAMA2, ALAMDA+ 5G10.4
ROO, QD, BB, DD 4G10.4
K, L++ 2110
(ROAS(I), 1=1, K) 4G10.4
((CS(I.J), J=l, L), I=1,K) 8G10.4
((WS(I, J), J=l, L), 1=1, K) 8G10.4
UB, SAI, TIME 3G10.4
Subroutine
DISPSN
I
AMBIENT
WAKE
* N< 30
** Option of 51, 101, 151
+ Skip card if KEY 1 = 1
++ K< 4, L< 2
A18
-------
EXPLANATION OF THE INPUT SYMBOLS
In Program In Text
Remarks
METHOD Method of barge operation must be 3
for barge operation 3
IGCN IGCN=0 No graph plotted for convective phase
IGCN=1 One graph plotted for convective phase
IGCN=2 Extra graphs of concentrations plotted for
convective phase
IGCL
IGCL=0 No graph plotted for collapse phase
IGCL=1 Graph plotted for collapse phase
IGLT
IPCN
IPCL
IPLT
IGLT=0 No graph plotted for long term diffusion phase
IGLT=1 Graphs of COQ, Size and settled solids
characteristics plotted
1GLT=E Extra graphs of centroid locations plotted
IGLT>2 All graphs plotted for long term diffusion
IPCN=0 No detailed printed output for convective phase
IPCN= 1 Detailed printed output included
IPCL=0 No detailed printed output for collapse phase
IPCL^l Detailed printed output included
IPLT=0 No detailed printed output for long term
diffusion phase
IPLT>0 Approximation number of pages of detailed output
desired before input to long term diffusion
completed
A19
-------
In Program In Text
Remarks
NSCALE
Scale for plotting
NX
Size of graph along independent variable
NY
Size of graph along dependent variable
IBUG
Number of integration steps between inputs to
long term diffusion
TRIGER(I)
Triggers for graphing in long term diffusion
(graphs are at times when the peak value of
CQO in the y direction first falls below
TRIGER(I) times the maximum of C.,,. ever
achieved after input to long term diffusion
completed}.
N
Number of points where ambient density is
specified
Yd)
Vertical position from free surface where
ambient density is specified (ft)
ROA(I)
Ambient density at Y(l),, (gram/cc]
YK1, YK2,
YK3, YK4
Positions where the vertical diffusion coef-
ficient changes (ft)
AKY1, AKY2, k p k 2
AKY3 k ,
Vertical diffusion coefficients: near to the
free surface, at thermocline and below the
2
thermocline (ft /sec)
YU, YW, YE, y , y^, yg Vertical positions where ambient velocity is
H
H
specified (ft)
Total depth (ft)
A20
-------
In. Program
UAO, WAO
TSTOP
ISIZE
KEY1
In Test
u . w
a' a
Remarks
Maximum horizontal velocities in x and
z directions (ft/sec)
Physical time limit in second in the long
term diffusion computation. Time started
from the commencement of discharging into
the wake
Vertical grid point size in the long term dif-
fusion computation, 3 options: 51, 101, and 151
KEY1=1 Use Tetra Tech suggested coefficients
KEY1 -2 Use read in coefficients
KEY2
KEY2=1 The computation stops at the end of
convective decend
KEY2=2 The computation stops at the end of
dynamic collapse
KEY2=3 The computation stops at the end of
long term diffusion
KEY3
Cl, C2
DINCR1
V C2
KEY3=0 Long term diffusion computation also
performed for the fluid part
KEY3=1 No long term diffusion performed for
the fluid part
Barge shape factors, used in computing plume
size BC(1) = 2 (Cj x GZ 1/3) x (BB x DD) 1/Z
where BC = plume size, BB = barge width,
DD = barge draft
Trial value in obtaining distance step DS =
DINCR1 x BC (1) in the convective decend phase
A21
-------
In Program In Text Remarks
DINCR2 Trial value in obtaining distance step DS =
DINCR2 x BC (INDEX) in the dynamic collapse
phase, where INDEX is the step at the end of
convective decend
ALPHA
BETA
GDI
CD2
CM
ALPHA1
GAMA
CDRAG
CFRIC
CD3
CD4
FRICTN
a
P
C
Dl
'D2
m
Y
C
drag
fric
C
C
D3
D4
rictn
Entrainment coefficient for 2 - D thermal
Settling coefficient
Drag coefficient for a half cylinder wedge
Drag coefficient for a cylinder
Added mass coefficient
Entrainment coefficient for a jet
Gradient factor in the cloud
Form drag coefficient for the quadrant of
a collapsing elliptical cylinder
Skin drag coefficient for the quadrant of
a collapsing elliptical cylinder
Drag coefficient for an elliptical wedge
Drag coefficient for a two-dimensional plate
Friction coefficient between the cloud and
ocean bottom
Fl
CM
C
m
Modification factor used in computing the
the resistance of the friction force to
quadrant of an elliptical cylinder
Added mass coefficient
A22
-------
In Program In Text
Remarks
ALFA1
al
Absorbency coefficient of the bed to the solid
particles
ALFA 2
a-
Absorbency coefficient of the free surface to
the solid particles
GAMA1
Yl
Entrainment coefficient at the bed
GAMA2
Entrainment coefficient at the bed
ALAMDA
Dissipation factor used in computing the
horizontal diffusion coefficient
ROO
Density of the waste material (gram/cc)
QD
Discharge rate of the waste rnaterial(ft /sec)
BB
B
Barge width (ft)
DD
D
Barge depth (ft)
K
K
Number of different solid density
L
Number of different solid fall velocities for
each solid density
ROAS(I)
Solid density
CS(I, J)
C
Solid concentration of the solid particles with
density ROAS(I) and fall velocity WS(I, J) (ft3/ft3)
WS(I, J)
W
Fall velocity of the solid particles
UB
Barge velocity (ft/sec)
A23
-------
In Program In Text Remarks
SAI Barge moving direction (limited to 180 ^270 )
TIME Time of continuous discharge (sec)
A24
-------
EXPLA-NATION OF OUTPUT SYMBOLS
In Convective Descent And Dynarnic Collapse
In Program In Text
Remarks
TIME
x.y.z
x,y, z
U
RADIUS b
DEN-DIF p-p
FLUID CQNC.
SOLID VOL.
CONCENTRATION
C ..
Physical time taken for a two-dimensional
thermal to move to positions (sec)
Distance along the path of a two-dimensional
thermal (ft)
Position of the two-dimensional thermal at
S (ft)
Velocity of the two-dimensional thermal at
S (ft/sec)
Radius of a half cylinder thermal (ft)
Density difference between that of the two-
dimensional thermal and the ambient (gram/cc)
Concentration of the fluid part waste material
Solid volume in the two-dimensional thermal
with lengthDS (ft)
Concentration of solid particles with a specific
fall velocity (ft3/ft3)
The minor axis of the two-dimensional thermal in the
in the dynamic collapse phase (ft)
A25
-------
Program In Text Remarks
The major axis of the two-dimensional
thermal in the dynamic collapse phase (ft)
In the Long Term Diffusion Phase
See Explanation of Output Symbols in Barge Operation 1
A26
-------
PROGRAM DISPSN ( .
C PROGRAM FOR PREDICTING THF FATF OF WASTfc MATERIAL OISCHARGtD
C FROM A i'ARGfc INTO THE OCEAN ENVIRONMENT
r.
C
F.XTFRNAL DfcfclVEl ,DERIVE?,DtRIVF3,DERIvE MDTRN, INDFX, IfiEl), ILF.AVE
,IJK,FBFO,AMO,OLDT,METHOD»TSTOP
JPCL,IPtT,NSCALfc
INDE ,THIGEH(«),CMAXMAX,TOT^AX,KKK
C
DATA G,PAI/32.2,3,H1S9/
C
0 CONTINUE
URXsO,
WHZ*0,
RFAO(5,10) METHDO,I6CM»16CL»IGLT,IPCN,IPCL»IPLT,NSCALE,NX,NY,1BUC
IKMETHOO.LT.l) CALL EXIT
20 FORMAT C8F10.5)
TlMt«)sO,
OAXMAXsO,
IF(I5UM,E'G,0) CALL EXIT
CALL AMHIFNT
10 FOWf.AT(lfcI5)
C NO, OF BARGE OPF.RATIOM
C 1 3 DIMENSIONAL AXI -SYMHKTPIC INSTANTANEOUS RfLEASt OF
c THF WASTF. LOAD FROM A SARGF
C 2 TIME LIMITED CONTINUOUS DISCHARGE CJF THE WASTE MATERIAL THROUGH
c A MOZZLF UNDER THt BOTTDH OF A MOVING BARGE
C 3 TIMf LIMITED CONTINUOUS DISCHARGE. OF THt wASTt MATERIAL INTO
c THE WAKE CT A MOVING BARGF
A27
-------
GO TO (1,2,3) METHOD
1 CALL DUMP
IFdJK ,r,t. 5) GO TO «0
IFCKEY2 ,EQ, 1) GO TO 100
IFdPl.UWG.EQ.l) GO TO 800
DO 88P TIIsl,NMN
IF(CYdSTE.P).GE,YdII).ANO,CYdSTEP),LE,YdII+l)) DENGRAs
* ROAUll + n-RQAdll)
888 CONTINUE
1F(OENGRA,GT,1,E»10) GO TO 800
IF(xEY2.fcQ,2) GO TO 100
00 2001 lAMsi,N\CC
2001 CMAX(IAM)*SS(IAM,ISTtP)
YYsCY(ISTtP)
AO=0,5*BC(ISTEP)
IF(YY-AO,LT,0,) YY=AO
666 FflRMATClHlfiOX, 38HGR ADIEKiTsO GO DIRECTLY TO LT DIFFUSION)
CALL DIFU8W1
GO TO 100
800 CONTINUE
CALL COLAPSJ
IP(IJK ,6t , 5) GO TO ^0
IF(KfcY2 ,EQ. ^) GO TO 100
11 CALL D1FUSN1
GO TO 100
2 CALL JET
IFdJK ,GE, 5) GO TG «0
IFCKEY2 .FQ, 1) GO TO 100
.F0,l) GO TO 700
00 777 IIIa
IF
-------
3 CALL wAKfc
IFCIJK ,GE, 5) GO TO UO
IFCKEY2 ,EQ. 1) GO TO 100
IF(NOTf ,EQ,1) GO. TO 900
DO 999 III»i
IF(CY(ISTFP).GE.YC!I!).ANO,CY(ISTEP).LE.YCIII+1)) DENGRA
* ROA(in + l)-RUA(!II)
999 CONTINUE
IF(t)ENGRA.GT.l.E»10) GD TO 900
IF(KEY2,E.Q,2) GO TO 100
M
DO
2003
CMAX(K'MCC+1)=USEJET(ISTEP)
IF(WS(1,1).FO.O,)
INDFXelSTFP
YYsCY(ISTEP)
AO«sft.5*8C(ISTEP)
IF(YY-AO,LT.O,) YYaAO
CALL PIFUSN2
GC TO 100
- 900 CClMIWUE
CALL COLAPS2
IFCIJK ,6E. 5) GD TO «0
IF (KtY2 .EO. ?) GU TO 100
CALL. DIFUSN2
GO TO 100
«0 WPITE(6,1000i)DT
JOOOt frOR/M4Tnx,28H***THE S7FP
, 34H LONG TFRM DIFFUSION* IN TROUBLE***, 3H *
100 CONTINUE
GO TO U
END
vv
A29
-------
/Af»CO/ V( 30)»Rl)AC 30) , YK1 ,YK?,YK3, YK4,AKY1,AKY2,
YIJ,YW,YE,H,UAO,WAO,N,CKYU5n»UA(15n,iNA(l$l)
/TRASFER/ KK, JJ.TT, ISIZt,NOTF, *0, I Y , w(j T »N , I K'PE. X , I BED , T L fc A VE
C
C
RE An(5,1 )N
1 FORMATdlO)
C N-Mfl, OF DATA PTS, ON DENSITY
2
C Y(^) V.IJST Bfc. L'OUAL TU THp: TUTAL DFPTH
C Y-DEPTH
C ROA-AMbjfNiT ntNSITY
WPITF(6,3)
3 FORMATUH1,10X, IHHAMBIE.'vT CUNDTTIONS/1 OX, IttHDSPTH IN FT.
, ^SHDFNSITY IN GRA>' PEW cc, ,^IMKY IN SOFT PER src.
, 14HVEH. .
on 10 jsirK:
IFC.JJ ,GE. fOJJ = w
«RITfc(6,«)(Y(I),I=J,JJ)
a FORMAT(10X,SHDFPTH,9X,8E12,^)
5 FORl*ATCtOX,liHA*8-DF.NSITY,
10
6
DO 20 1=1
20
C YK-THF VERTICAL POSITIO^S WHE«E KY CHANGES VALUES
C AKY-KY VALUE
fcRITE{6,7)YKl,YK2,YK3,YK4,AKYl,AKY2,AKYl
7
C YO-THfc POSITinN UA STARTS TO DECREASE
c YW-THF POSITION »IA HAS ITS HAX,VALUF
c YE-THE POSITION A^B-VFI, GO TO
C W-TOTAL DEPTH
C UAO WAO»
8 FORMAT (10X,3HYU ,E10,a,3HYW ,E10,U,3HYE ,E10,U,2HH ,E10.U/10X,
,3HUA ,E10,a,3H^A , E10.il/)
lYsN
RETURN
A30
-------
C
r
SUBROUTINE DUMP
3-OIMfcNSIONAL AXI
LOAD FROM BARGE
-SYMMETRIC INSTANTANEOUS RELEASE OF THE
EXTERNAL TFR I VE1, DERI VE2, DERIVE 3, DERI VE«,DEKIVf 5, DJRIVE6, DERI VET
COMMON /R/DINCR1,01NCR2,KEY1,KF. Y2,KEY3
/A6CD/ Y( 30),KOA< 30 ) , YK 1 , YK2 , YK 3 , YK,
,SS(8,600),5AVE(600),DUMY?(6,600),FC(600),BC(600),WS«l,?),
, VF ,
, 0(8,lbl),YJ Clbl),Y2(lSl) ,PY
, CCMAX(15n,VAR(lbl)
COMMON /cON55T/G,PAi,AuPHAof ALPHA, AL.PH AC, ALPHA i,ALPNA2,
ALPHA3,ALPHA4,GAMA,CDRAG,CFRIC,BE TA,CM,CD,CDl,CD2,FRICTNi,Fl
COMMON-1 /TORS/ DT.DT1 , DT2, CDT , NGH ID, MGRID1 , ^(iRID2« NGRIO3 ,
COMMOM /TRASFER/ KK , J J, TT , IS I ZE , NOTF. , AO , I Y , MOT RM, I KiDfTX , I BED , RE AVE
, , IJK,FBF. 0, AMD, OLD T,*FTHOO,T STOP
IJKsO
KVsl
KV CONST
IPF.OSO
ILF.AVE=0
ABOUT VORTICITY GENERATION
c
c
c
c
c
c
c
2 FORMAT( //10X,17H8ARGE OPERATION 1,
, //10X,37w«ASTE MATpRIAL DUMPfcD
, 15H INTO THt OCEAN,
, //10X,50HTHE SHAPE OF Tut CLOUD IS
RF. ADtb, DTSTOP
,/)
iMSTANTAMt'CUSLY,
ASSUMED TO BE HfrM]SPHFRE>
KEY 1=2
FORMAT(/I OX,5HTSTOP,t1?,4,UHSFC,
TIMf. LIMIT IN THE COMPUTATION
READCb, 6)ISm,KEYl,KEY2,KEY3
USE TETRA TECH SUGGESTED COEFFICIENTS
USF READ IN COEFFICIENTS
COMPUTATION STOPS AT THE EMQ OF CONVECTIVfc, DESCENT
COMPUTATION STOPS AT THE, END OF OYN'AMK COLLAPSE
COMPUTATION STOPS AT THE END OF LUN'G TERM DIFFUSION
LONG TERM DIFFUSION FOR FLUID PART ALSO
MO LONG TERM DIFFUSION FOR FLUID PART
KEY3eO
KFY3*1
WRITE.C6,9007a)ISIZF,KEYt,KFY2,KFY3
9007U FORMAT(lOX,lbHGHIO POINT SIZt»I^»5H
IF (KEY1 .EG, 2) GO TO 500
KfYi,I5»5H KEY2,I5,SH Kt.Y?,IS)
90070
FOHMATf/10X/37HUSE
DINCHlsl.
TETRA TECH SUGGESTED COEFFICIENTS/)
ALPHAQs.235
BFTA=0,
CHs.5
A31
-------
CDRAG=1 .
CF»JC*,01
C03s,l
ALPHAC=,001
FHICT'^s.Ol
F 1 s . I
AlFAlSl,
ALF A2si.
AL i
r,(.: TO soi
500 kRTTE(fe»90071)
90071 FnR"AT(/lCX»24HUSF- HEAD IN COEFFICIENTS/)
RFAPC5,
1 FliRMAT(8G10,4)
C ALPH.*0«»F'NTRAJNN£-'NT COFh,
C HFTA-SFTTlIMG COEF,
C CHADDED MASS COEF,
C CD-DRAG COEF,
REAO(5, UGAMA,CORAG,CFRIC,C03,C04,AIPHAC ,FRICTN,FJ
READC5, !)ALFA1,A|.FA2,GAMA1,GA*A2,AUMDA
bOl
90073 FUR*AT(10X,feHDlNCKl.F10,4,7H DINCR2»FI 0,«)
WR1TE(6,4)ALPHAO,BETA,CM,CD
a FORMAT(10X,6H*LPHAO,FiO,a,5H Pt" T A , F 1 0 , 4 , 3H C.M,Fin,a,3H CD,F10,4
WRITE(6,90072)GAMA,CDRAG,CFRIC,CD3,CD«.ALPHAC ,FR1CTN/F1
90072 KORMATdOX, 4HG AM A t Fb, 2 , 6H COR AfJ, Fb' ,2, 6H CFRIC, F'Sti^H CD3,F-15,,
, 4H C04,F5,2,7H ALPHAC,F10.4 ,/9X, ,7H FRICTM,F10,4,
, 3H F1,F10,4)
90068 FORMAT ( 1 0 X , 5HALF A 1 , f- 10,4, 6H Al.F A? , F 1 0 , 1 , 6H GAM A 1 , F 1 0 ,
, 6H GAMA2,Fl0.a,/9X,7H AL AHO A , F 1 0 ,
C INITIAL CONDITIONS
c RB-PADJUS
C ROD -DENSITY CF WASTt HAThRlAL
c u,v,w,-i< INITIAL VEL,
RPO»ROO*1,V4
READC5,fe)K,L
C K.NO, f)F DIFFFREMT SOLID DENSITIES
C L-NO, OF DIFFERENT FALL VEL,FOR A SPECIFIC DENSITY
WRITF(fa,7)K,L
10 FORMAT(10X,16HOENSITY OF SOLID, 8X,4E1?.,4)
^RnF(fe,ll)
11 FORMAT(10X,13HCUNCENT«ATION)
00 101 jsl,l
A32
-------
101 WP!TE(6,16MCS(l,J),Isl,K)
16 FORMAmux,Ufcl2.«)
WRITE(6,12)
12 FORMAT (10X,22HFALL VELOCITY OF SOLID)
DO 102 J«1,L
102 WRITF.tfe, 16) (wS(I,J) ,Isl,K)
DO 200 1*1 ,K
200 ROAS(I)=R{.iAS(I)*l .94
ROAAsRnA(l)
Cle(Rno«RrjA(l})/ROA(l)
El=CROA(IY)-ROA(l))/(H*ROACl))
F SV(1)/SQRTCG*C1*RB)
F.E1=E1*RB/C1
IFCROAUY) ,EQ, ROAC1))3
-------
C VOLUME r.1F SOLID
C VF-VOLUME OF THE FLUID PART OF WASTE MATEP-IAL
T(l)sO,
C MF.-NO, OF TOTAL EQS
IVsl
C IY-INOEX OF DfcPTH WHERF DENSITY SPECIFIED
ISTEPsl
C ISTEP»TIMF STF.P
C NC IS FQN, NO, FOR SOLIDS
C NU1RL si IKiDlCATE NEUTRAL POSITION REACHED
IPLUNG«O
C IPLUNG*INDICATQR OF HITTING BOTTOM
111 CX(ISTfcP)=FC13
V(ISTEP5aFC6)*CMHASS
ENTRCn£lSTFP)*ALPHA
IF CF(9),LF, 0) E(<>)30,
WHEN VORTICITY GOES TO ZERO, IT IS SET TO ZFRO
VDRT(ISTEP)«f.(9)
BC(IS1EP)B£VDLUMF*3,/C2,*PAI))**,35333^
BC-RADIUS OF CLOUD
AA(ISTFP)8.5*BCC1STEP)
KKsl
OD 120 Isl,K
DO 120 jsl,L
SS C KK, I STfeP)»E(<> + KK)/ VOLUME
120 KKsKK+1
SS SOLID CONCENTRATION IN VOLUME RATIO
FCCISTEP)*VF-/VOLUMF
DEMDIF(lSTEP)»CROO"ROAA)*,5i5«5
IF(£CY(ISTEP)+3,*8C(ISTEP)/8t) ,GE, H)
IFdPLUNG ,EQ, 1) GO TO 1000
IF(NUTRL ,EQ, 1)GQ TO 1000
IFCISTEP ,GE, fcOO) GO TO 1000
C&LL «UNGSCDERIVE1»NE)
ISTFPsISTEP+1
TClSTFP)sT(ISTEP-n*DT
GO TO 111
1000 WRITEU,22)IJK,DTrIPLUN.G,NUTRL>ISTEP
22 FQPMATC/5X,i5,Ei2tU,3I5)
IFCISTEP ,LT. 100 ,OR. ISTEP ,GT, 200)210,220
210 OT*OT*ISTf.P*OINCR/lt>0,
IFUJK.EQ, 5) GO TO 220
GO TO 25
220 CONTINUE
IF(IPCN,EQ,0) GO TO 8001
WRITEC6,13)
13 FORMATC/7 8X ,i)HTIME,5x , 1HX,7X , 1HY ,7X, I HZ, bX, IHUr 6X, 1HV,SX, 1HW, feX ,
, 7HOEN-DIF,3X,6HRAOIUS,1X,5HVQRT,,2X,5HALPHA,2X,
, 12HFLUID CONC, ,
A34
-------
, 10HSiJlID*vrJL.»2X,
OH ?«0 Jjjsl ,ISTrP,NG«ID
DO 23C1 KKsl,NC
230 ACONC(K*)=2,*PAI*BCCJJJ ) * *3* SS ( K K , J J J ) /3 ,
WP1TF(6,1U)T(JJJ),CX(JJJ>,CY(JJJ),CZ(JJJ),U(JJJ),VCJJJ'I,
, «(.JJJ),OFNOlF(JJ.n,f5C(,M,n , VO«T(JJ,T) , ENTRCO C ,J JJ ) , FC ( JJ J ) ,
, ACOMC( 1) rSSCl ,JJJ)
la FUKHATC aX,«Fe,£,^6.2,F7,3,F6.?,t 1 2 . 4 , 2F7 , 2 , F7 , a , 3E. 1 2 , « >
IF(MC ,f:Q, U fin TO 240
KK*2,K'C
NC(KK),SS(KK, JJJ)
IS FCIHMATUOOX,2tl3.«)
2ao CONTINUE
flOOl IF(IGCM.tO.O) GO TO 6002
ISTEPlsISTtP+1
T(ISTEP1)B2,*T(ISTEP)-T(1STEP«J)
CX(ISTEPl)=2,*CX(lSTF:P).CX{I8TFP«n
CZCISTtPJ5=2
*CZUSTtP)-CZ(ISTFP*n
CY(ISTF.Pl)sO
RC(ISTFP1)*0
Ff.(!STtPl)=0,
CALL DRAW(T,T,T,T,CY,BC,CX,CZ , ISTt-.Pl , 1, MSC AL t, U )
IF(IGCN.FO,1) GO TO 8002
DO 8004 1=1,NC
on sooa jsi,isTfc'p
800« AUX(J,I)sSSCI,J)
DO R005 .' = 1,NC
800S AUX(ISTFP1»J)»0.
IF(N'CP1,GT.4)
CALL PRA*(T,T»T,T,FCf AUX(l,i)f Ai)X(l,2), AUX(1,3),ISTEP1, lfr»N3CALt.»
*NCPt 3
IF(NC.LT,4) 50 TO 800?
N C M 1 * N C
IFfNC^l.GT.a) GO TO 8006
CALL DRAW(T,T»T,T,AUXCl,i),AllX(l,2),AUX(l,3),AUX(l,4),I8TFPl»10,
GO TO 8002
CALL DRA»(T»T,T,T»AUX(1, 1) » AUX ( 1 , 2 ) , AUX { 1 , 3) , AUX ( 1 , U ) , ISTtP 1 , 10,
CALL ORAW(T,T»T,T,AUX(l,5),AUX(1,6)fAUX(l,n,AUX(l,8),ISTEPl,ll,
*NSCALfc"^CM5)
8002 CONTINUE
RTlsDT
DO 300 KK=1,NC
KKKsNC-KK+i
300 E
INOfcXsISTEP
IFdPLUNG .FO. 1) GO TO 310
E(10)sO.
AO=.5*8CCI5TtP)
RETURN
310 E(10)3ROO*PAI*E
A0= SCCISTFP)
RETURN
A35
-------
SUBROUTINE. JE.T
TIME LIMITED CONTINUOUS DISCHARGE THROUGH A NOZZLE UNDER A
MOV I MR BARGf.
EXTERNAL DERIVE 1, OF. RIVF2, OF & I Vf3,l>F. RIVE «, PERI Vfb, DERIVE*., nfRIVF7
COMMON/ AUXX/AUX(hOO, 8)
Ci'KMON/GP/IGCN,IGCL»IGLT,IPCN,!PCL,IPl.T,NSCALE
/B/DINCRi,OlN'CR2,KEYl,KEY2,KEY3
/APCD/ Y( 30),ROA( 30 ) , YK 1 , YK2, YK3, YK4 , AK Y 1 . AK Y? , AK Y3 ,
N CX(600),CY(bOO),THETA2(600),UC600),
SZtTA(600),CZFTA(600),AA CfcOO ) » DFNDIF (600) , DL(600),
V2,CMAX(9),YY
/COMPUT/Ft(7.
COMMON /CONST/G,PA1/ Al.PHAO* ALPHA, ALPHAC, ALPHA 1 , ALPHA2,
ALPHA3,ALPHAq,GAMA,CORAG,CFHIC,MtTA,CM,CO,COi,C02
/TORS/ DS,OTt,r)TZ,CDT,NGRIO,NGRl01,NGRI02,NGR!D3,NCD
COMMON /TRASFEiP/ KK , .J J, TT , ISI Zt , NOTE, AO , I Y , NHT«N , J.NOE.X , I BED, ILE AVE
IJKSO
IHEDau
ILEAVEsO
wRITECb,90051)
90051 FOR^ATC //I OX, 1 7H8 ARf.E OPERATION 2/10X,
, 47HDISCHARGE THROUGH A N'OZZLE UNDER A MOVING
REAOC5,9000nTSTOP
kRITF(b,90069)TSTOP
90069 FORMAT C/10x,5HTSTOP,E12,<4,UHSFC,/)
C TIME LIMIT IN THE COMPUTATION
U C A Pi ? CT Qrtftft!3^TyT7t.° If t V t tf t V 2 Lf (r V 1
n(_fll_/VJ^^>'v^JC / Awi/.L^'^w.T \ ^ ^ i. TC^^p TJ
C KtYlsl USE TETRA TECH SUGGESTED COEFFICIENTS
C KEY1=2 USF. READ IN COEFFICIENTS
C KEY2rl COMPUTATION STOPS AT THF END OF JET PHASE
C KEY2=2 COMPUTATION STOPS AT THE END OP DYNAMIC COLLAPSE
C KEY2=3 COMPUTATION STOPS AT THfc END OF LONG TERM DIFFUSION
C KEY3sO LONG TF.RM DIFFUSION FOR FLUID PART ALSO
C KFY3sl NO LONG TERM DIFFUSION FOR FLUID PART
fR!TE(6,9007«)I3lZe,Km,KF.Y2,KEY3
9007a FORMATClOX,1SHGRID POlMT SIZE,IS,'iH KbYl,I5,5H KEY2,I5,5H KEY3,I5)
GO TO (1,2)KFY1
1 WHlTECfe,900705
90070 FORHAT(/10X,37HUSt TETRA TECH SUGGESTED COEFFICIENTS/)
DINCRlsl,
DINCR2sl,
ALPHilsO.0806
Al.PHA2s.3536
BPTAsfl,
CDsl,3
CDRAG=1 ,
A36
-------
ALPHA3=,3536
A I PH Ail = ,001
FRICTNs.Ol
ALFAlsl.
GAMA2«0,
ALAMOAr.OOl
GO 70 3
? KHITE(6, 90071)
90071 FORMATUOX, 2UHUSE READ IN COEFFICIENTS)
RFAD(5,90001)niNCR1.0INCR2
«FAO(5',90001)ALPHA1,ALPHA2,BFTA,CD
FORMAT(8G10,5)
ALPHA1-FNTRAJNMENT COEF FOR JtT
AL.PHA2-FNTRAINMENT COEF FOR 2-0 THF.RMAt
BETA-SETTLING COEF.
CD-DRAG COEF,
RF AD (5, 90 001) GAM A, CDRAG,CFR 1C , C03,C04, ALPHAS, ALPHA4I, FRICTN, Fi , CM
RF.AD(5,90001) ALFAt,ALFA2,GAMAl,GA'-IA2,ALAMDA
3 CONTlNUe
wWITt(6, 90073)01 NCR1. 01 NCR2
90073 FUHMAT(10X,6Ht)lNCHl/F10ttt,7H DINCR2 » K 1 0 ,4)
90001
C
C
C
C
900S2 FORMATC10X,6HALPHAt ,HO,«,8H ALPH A2 , F 1 0 , U , 6H BtTA,F10.a,
. 3H CO»F10,a)
WRITE(6,90072)GAMA,CDRAG,CFRIC.CD3,CD«, AI.PHA3, ALPHA4,FRICTN,F1,CM
90072 FORMAT C 9X,bH GAHA,F5.2»6H CDR AG,F5 .2, 6H CFRIC, F5,3,ttH CD3,F5,2,
, /9X,7H ALP*AU,F10,4,7H PRICTM,F10.'
, 3H F1,F10,4,3H CM,F10e(4)
WRlTF(6,900K)
RtAO(5,90001 )((WSU,J),J*l.L)f 1 = 1 ?K)
00 15 I=l,K
wRITE(&,90055)ROAS(I)
90055 FORMATUOX, 16HDENS1TY np SOL 10 , 8X , F 1 2 ,«
hRTTE(6,90056)(CS(I,J ),Jsl,L>
90056
90057 FORMAT (10X,22hKALL VELOCITY OF SOLI D, 2X , 8E1 ?. . <
RF.AO(5,90001)U8fSAI,TIMfc
C SAI IS LIMITED TO 180-270 DFGREES
A37
-------
90058 FDRMAT(lOX,14HRARGt VELOCI T Y ,H 0 ,4 , 18H ANGlb *ITH X- AX I S , F 1 0 , a ,
, flH DEGREES,
-------
CFsCF-CS(I,J)
C FLUID CONCENTRATION AT THE 8
30 KK3KK+1
DINCRsOINCRl
DO 31 JjBifNE
31 SAVF(JJ)sEMJ)
wRITE(6>,197J)
1973 FO«MATflHl,10X,tSHJET CONVECTION /)
25 r>0 26 JJBI,N,E
26 E(JJ)»SAVF(JJ)
DSsOINCR*8C(l)
OSAVEa20e*DS
IJKaIJK+1
T(l)aO.
S(i)aO,
IY=1
C IY-INDEX OF DEPTH WHERE DENSITY SPECIFIED
ISTEPsl
C ISTEP-COMPUTATION STEP
NOTF - INDICATE HITTING BOTTOM
NUTRL - FIRST HORIZONTAL POSITION RFACHED
100 CX(ISTEP)«E.C13
CY(ISTEP)s£C2)
CZ(ISTFP)s£(3)
BC(I8TEP)eSRRT(CE(«)+E(8)>/(ROA(i)*J(131EP)*PAn)
AA(IST£P)e8C(ISTEP)
FUJXeu(ISTEP)*PAJ*8C
-------
!F(IST£P .(it, 600) GO TO 200
IFCNOTE ,EQ. 13 GO TO 200
103 CALL RUW6S(DERIVF3,Mn
DSsDSM ,1
IF(DS.GT.DSAVE) DSsDSAVE
OT= DS/UCISUP)
ISTEPelSTfrP+i
TUSTEP)sTCISTEP-l)*DT
SCZ(JJJ)»U(JJJ),
, BC(JJJ),RFNDIf:(JJJ)fFC(JJJ),ACONC(l),S3(i»JJJ)
9006U FORHAT(8X,5F9t2,2F8.3,«E12,
-------
hfi (101 KKrr'J,?
'JO 1 fc ('IK ) nf- (KK 1 *UU
on iio? KK = I,M;
•'402 F ( 1 ( i +• K K ) = I. ( 1 0 -t K' •< ) * U 1 1
v^"Mf isTf:p)*r.f-S(THL'iA;'CiSTr;p) )
F(9)r.ur. ( i sim
V r - C. f- *• H C ( 1 ) * * c? * ' I ( i) / ( .••. C ( 1. S T L P ) * * f! * U ( KS T t P ) )
FlUJD CHNCr.' TK4THK' AT 1 ;n- fMi) i)i .H. T
II- C'-UTt ,rri, ) )i;n TO 'JlO
F.(10)=0,
AO = Mf:( ISTPP)
RF Ti;i = (^Aj*.ciS
f-'NO
A41
-------
SUBROUTINE
WASTE MATERIAL DISCHARGED INTO THt BARGE
EXTERNAL DERIVF..l,DERIVE2,QERIVFi,DtRlVE4,DERIVE5,D£RIV£6,DERIVE7
COMMON /B/DINCfn,DlNC*2,KEYi,KEY2,KEY3
COMMON /A6CO/ Y( 30),«OA{ 30) , YKl , YK2, YK3, YK« , AKY1 , AKY2, AKY3,
, YU,YW,YE,H,UAO,WAO,N,CKYn51),UA(15i),WA(15n
COMMOM CX(600),CY(600),TH£TA2(600),U(600),
, SZETA(600),CZ£TA(foOO),AA (60 0 ) , DENDI F ( 600 ) , DLC600),
, CZ(600),SS(8,600),SAVEC600),DUMY2C5,600),FCC600),T«»00),
, BC(600)»«SC4,2),
, ROAS(a),S(600),K,L.IPLUNG,NUTRU,UB,SAI,TIKE,I3TEP,Vr,
, Ul3XfW8Z, V2»CMAX(9) ,YY
COMMON /COMPUT/£E(7,l'3l),E(20},ACONC(123),C3(^,2),
CCMAX(1S1),VAR(151)
COMMON /CONST/G,PAI»ALPHAO, ALPHA, ALPHAC, ALPHA 1 , ALPH A3,
, AlPHA3,AlPHA4,GAMA,CORAG,CFRICfBETA,CM,CD,CDl,CD2,FRICTN,Pl
CQMNGN /TORS/ DS,OTl,DT2«CDT,NGRID,MGRIDl,HGRID2fNGRlD3,NCD
COMMON /TRASFER/ KK, JJ,TT, I8IZE,NOTE» AO, IY,NOTRN, INDEX, 1BED»1LEAVE
COMMQN/A UXX/AUXC 600 ,8)
COMMON/GP/IGCN,IGCL/IGLT,IPCN,IPCL,IPLT,NSCALF
IJKsO
INDfcXsO
If3EOsO
ILEAVEaO
WRITECfe, 90051)
FORMATC / 1 OX, 1 7HB ARGE OPERATION 3 //IOX,
, 62HWASTE MATERIAL IS DISCHARGED CONTINUOUSLY INTO THE BARGE WAKE,
,/10X,56HAKTF« THE INITIAL MIXING, THE WASTE MATERIAL IS ASSUMED
, /10X,«iHTQ IJE. IN A FORM OF HALF CYLINDER THERMAL, /)
READ(5,90001)TSTOP
WRITE(6,90069)TSTOP
FORMAT(/10X,SHTSTOP,E12,a,4HSEC./)
TIME LIMIT IN THE COMPUTATION
RFAD(5,90002)ISIZE,KEY1,KEY2,KEY3
KEYIsi USE TET9A TECH SUGGESTED COEFFICIFMTS
KEY1=2 USfc READ IN COEFFICIENTS
COMPUTATION STUPS AT THE END OF CONNECTIVE Dt'SCFNT
COMPUTATION STOPS AT THE FNQ OF DYNAMIC COLLAPSE
COMPUTATION STOPS AT THE END Of- LONG TFRM DIFFUSION
LONG TERM DIFFUSION FOR FLUID PART ALSO
NCI LONG TERM DIFFUSION FOR FLUID PART
WRITE(6,9007tt)ISlZF,KFiYi,KtY2,KEY3
9007a Fn«MAT(10X,15HGRID POINT SIZE, IS, 5H KtYi,Ib»5H KEY2,I5,5H KEY3,I5)
GO TO (1,2)KEY1
WRITEU, 90070)
FORMAT(/10X,37HUSfi TETRA TE.CH SUGGESTED COEFFICIENTS/)
Cls, 6
90069
C
C
C
C
C
C
C
C
90070
KfcY2si
KEY2a2
KF.Y2 = 3
KEY3=0
KEY3el
DINCRU.2b
ALPHAsO.3536
ALPHA130.0806
BFTAcO,
CMsl.
A42
-------
GAMAS.2S
CDRAG=1.
CKWICs.O
CD3=,2
ALPHAUx.OOl
FKK.TNs.Ol
Fts.l
GAMAl 80 .
GAMA2=0.
Al AMOAs.OOl
GU TO 3
2 HRITEC6, 90071 )
90071 FCRMATdOX, 2«HUSE «EAO IN COEFFICIENTS)
READ(5,90001)C1,C2
RE-AD(5,90001)DINCR1,OINCR2
REAnC5,90001)ALPHA, tip TA,C.D1,CD2, CM, ALPHA!
90001 FORMAT C8G10. iO
C ALPHA -FNTRAINMfcNT CDEf- FOR 2.r> THERMAL
C BETA-SETTLING COtF,
C C01 -DRAG COEF. (IF A HALF SPHERE "EOGt.
C C02-DMAG COP.F, FOR CYLlMOtR
C CM • ADDFD MASS CUEF,
Rt.A[){S,90001)GAMAf CO«AG,CFHlC,CD3,COa,ALPHA3, ALPHA«,FRJCTN,F1
RE'AD(5»90001)ALFA1,ALFA2, GAMAl,
3 CONTINUE
9007.3 FDRMAT(inx,6HOlNCRl ,F10.
-------
90055 FQRMATdOX, 16HDFNSITY OF SOLIO,flX,F12,4)
WRITEC6,90056)(CS(I,J ),Jsl,L)
90056 FORMATC10X,13HCONCENTRAT10N,11X,8E1?,4)
WRITE(6,90057)(WSU,J),Jsl,L)
90057 FQRMATUOX,22HFALL VELOCITY OF SOLIO,?X, 8E12,4)
15 CONTINUF
REAO(5,90001)U8,SAI,TIM£
C SAI IS LIMITED TO 180^270 DEGREES
WRITE(6,90058)UB,SAI,TIME
90058 FORMATC/10X,14HBARGF VELOCITY,F10,4,18H ANGLE w!
20
, 8H OEGRttS,4H FOR, 6.10,4, 5N SEC,)
DO 20 Isl,K
ROAS(I)=RQAS(I)*1,94
X*AXISrF10 ,4,
SAHaSAI
S A IBS A I*P A I*, 005555556
UBXaUB*COS(SAI)
180,)WBZaO,
270,)UBXaO,
IFCSAII ,feO.
IFtSAII ,EQ,
UlsUAO«U8X
U2*U1
U3««UBX
W3S-KBZ
WPITEC6,90059)
90059 FORMAT(/10X,45HAMBIENT CURRENT FOR A MOVING COORDINATE FIXED,
, 13H ON THE BARGE)
rtRITfc C6,90060)
FORMAT (HX,4X»i HO, 6X ,2HYU,6X, 2HY* , 6X/
WRITE(6,90061)U1,U2,U3,U4
FORMAT(9X,3H UA,2F8,3,8X,2F8,3)
WRITEC6,90062)Wl,W2fW3,M
FORMAT(9X,3H HA,F8,3,8X,3FR.3)
90060
90061
, 6X , 1HH)
90062
C----
BPsSQRT(B8*DD*,5)
VB-AMBIfeNT VfcLOClTY I M THE BARGE DIRECTION1
C3«l ,/(Cl*C2**. 333333)
FtUXs,5*PAI*BCCl)**2*VB*,75
CF*1,
DO 21 !el,K
DH 21 J«1,L
CFsCF-CS(I»J)
CnNCfcMTRATIUN OF FLUID
21 eS(I,J)sCS(I,J)*QO/FLUX
CF*CF*QO/FLUX
RDUsROU
: INCREMENT IN OS - DINCR*BCU)
WPITF(6,1973)
1973 FORHATflHl,10X,18HCONVECTIVE DESCENT,/)
25 ROOs(Rf3U*OD+CFLUX-QD)*ROA(l))/FLUX
A 44
-------
( 1 )
1
DS«DIN.'CR*flC(l)
E(n«0.
E(2)=«,*BC(n/(S.*PAI)
E(3)«0.
E«J)"Rnn*,5*PAI*RC(l)**2*DS
E(5)*F*»CM*COS(SAI-PAl)*t750*VB
Et6)«o,
E(7)=F;acM*SIN(SAl*PAI)*,7SO*V6
IF(SAT! ,EQ, 180,)F(7)sO.
IFCSAT: .to. 27o,5F(S)so,
DO 30 Iel,K
DO 50 J«1,L
E(8+KK)e,5*PAl*BC(t)**2*CS(I,J)*OS
30 KKsKK+1
Vf--«CF*f5*PAI*BC(l)**2*(3S
VOLUME OF FLUID
Ttl)sO,
S(l)nO,
C NONQ, OF TOTAL FQS
c IY-INDEX OF DEPTH WHERE DENSITY SPECIFIED
ISTEPsl
C I5TEP-COMPUTATIC3N STEP
C NC IS EGJN, *C1, FOR SOLIDS
NUTRLsO
C NUTHL -MUTRAL BUOYANT POSITION HtACHtP
c NOTE -INDICATOR OF HITTING BOTTOM
c....... ..—.................
100 CX(ISTEP)sEU)
CYCISTEP)=E(2)
CZ(1STEP)=E(3)
IFCISTEP ,EQ. 1)111,112
111 DL
-------
Ui'=l,/UU
SZETAdSTEP)sV3*UU
C2ETAdSTFP)sVl*UU
SIN AND COS OF ZETA
ROAAsROAdY)+(E(2)-YdY))*(ROAdY+n»ROAdY))/{YdYM)-YdY))
1=1, K
J = 1,L
110
KKsl
DO 110
00 110
SSCKK
KKsKK+1
FCdSTFPJsVF/VOlUMF.
IF((CYdSTEP)+BC(ISTEP)*,
-------
If- (NCCC,GT,4) *
CALL DWAW(CX,CX,CX,CX,AUX(l,n,AUX(l,2),AOXU,3),AUXn,<4),ISTePl,
* JO,NSCAL.E,NCCC)
!FCNC.LE.
-------
r.
c
SUBROUTJNF rOLAPSl
EXTERNAL OERIVEl,DERIVE2»DFHlVt3,OF.RIVEa,DERlVE5,DERIVE6,DERIVE7
/8/DINC«l,DINCR2,KEYi,KEY2,KEY3
/ABC0X YC 30),<*QA( 30),YK1,YK2,YKJ, YM,AKY1,AKY2,4KY3,
YU.YW,YF. ,H,UAO,WAO,M,CKY(151},UA(15i),WA(151)
CX(600),CY(f>00), AA (600) ,U(600),
, VC600), W(AOO),VORH600)/DEMDIFf600),ENTHCn(A0n),
,CZ(600),SS{6,600),SAVF(600),DUMY2(6,600),FC(600),BC(600),WS(U,2),
, R0*8(«)f T(600),K,l..IPlUNGfNUTKL,UB»SAI,nME,lsn.P, VF,
ALPHACtALPHAl ,A|.PHA2,
AIPHA4,
/cnHPUT/EE(7ilbl)fE(20),ACONC(l23)^CSU,2),
0(8/151), Yl(iSl),Y2Cl51)»OYn501,F(8),FwC8),Wrt
CCMAXU51),VAR(lbl)
/TORS/ OT/OTl»OT2»COT,ivGRIO,MGRI01»N6RI02»NGHI03,MCD
COMMON' /TPASRtR/ i^K,JJ/TT/l SIZE, NOTE, AO , 1Y,NOTRN, INDEX, I BED, I LEAVE
,IJK,FbFD,AMO,DlDT,MEtHriD,TSTOP
IJKsO
,fcOt
3 Els(ROA(IY*n«HOA(IY))/(HUA(l)*(Y(IY+l)"YUY)))
INDEX) **3*,8y*EG*i 000, )**,«2857
DT2B.OOl*(Bl/BCClNDtX))**'5/EG*tl
DTSOT2
GO Td a
1 Fn«MAT(lHi,////10X,l«HCOLlAPSE PHASE)
MC»K*L
NC IS EON. NQ, FOR SOLIDS
00 100
100 SAVE (KK)sE(KK)
2S DO 101 KK*i,Nr
101 E(KK)rSAVE(XK)
IJKsIJK+1
ISTtPsINDEX
NUT«L=0
NUTRL=3 DIFFUSION TAKES OVER OYMAMJC COLLAPSE
IFdSTEP ,EO, IBEO
IPLUNG-.INOICATCR OF HITTING BOTTOM
IFCISTEP ,ER, I8FD )GO TD IbOO
IFdSTEP .60, INDFX) GO TO 1??
Ill CXUST£P)sF(iJ
CY(ISTEP)sFC2)
CZ(!5TEP)sE(3)
A48
-------
CMMASS*! ,
) «3,* VOLUME/ C«,*PAI*E(9)**i>)
RADIUS
BC(ISTEP)sfcC9)
BC-MAJQR RAIUS OF CLOUD
UClSTt:P}=fc(5)*CMMASS
VCISTFP)=E- (b)*AA(ISTEP)*CKMASS/RC(ISTEP)
RnAA«ROA{IY)*(t<2)-YClV))*(HpA(IY*l)»ROA(lY))/miY+l)»Y(lY))
DD 120 1*1, K
OP 120 J=1,L
SSCKK, IS TFP)«E(10+KK) /VOLUME
120 KKSKK+1
SS SAVE SOLID CONCENTRATION FOR DIFFUSION
FC(ISTEP)=VF/V0LUME
IF UPIUNG .EG, 4) GO TO 5
IF(CCYClSTtP)+3t*2.*AA(ISTEP}/8,} .GE, H) IPLUNG«2
b AKXaa.*ALAMDA*(.5*E ( 9 ))**. 333333
AKX - CHANGE DF 8 BY DIFFUSION
BBOTs(8C(ISTEP)-RC(ISTEP«l))/DT
IF(CY(ISTEP)«AA(ISTEP).LE.O.) GO TO 420
IFCISTEP .UP, iNDEX+b) GO TD 121
IF(AKX .GE, BBDT)NUT»La3
1?1 IF(\'UT«L .feO. 3) GO TO £000
IF (IPLUK'G ,EQ, 23 GO TG 1000
IKISTEP ,GF, 599) GO TO 2000
122 CALL RUNGS(DERIVE2,NE)
T(ISTEP)BT(ISTEP-U*OT
GO TO 111
1000 IPFD=ISTEP
DRTsE(10}*lfc,/(pAI*AA(ISTEP)*BC(!STFP)**?*ROU)
£(10)sRnO*PAI*AA(ISTEP)*BC(ISTF.P)**2
» *(DBT+,6666fefe7*8CClSTEP3*V(ISTEP)/AA(ISTEP))/ 8,
AO=AO*2,
1500 CALL BOTTOM1
IFCISTFP ,GF, 600) GO TO 2050
IFCNUTRL ,EO, 3) GO TO 2000
TFtlPLUNG .EG, «)1600,1500
1600
OBT»EC10)* «./'(PAI*AA(!STf P ) *BC ( 1STEP 3
E(10)=RnO*PAI*AA(ISTEP)*SC(ISTEP)**2*OOT/16,
AOsAO*,5
ILEAVF.sISTFP
GO TO 122
ISTEPsISTfP-1
2000 WRITE<6,22)IJK,DT,IPL.UNG,MUTRL»ISTEP
22 Fn*?MAT(lX,l5,E12,
-------
13 FORM Alt/ 3X,UHTIME,8X, lHX,9X,lHY,9X,lHZ,6X,lHU,6X,lHVf 5X,1HW,
, 6X,7HDEN«OIF,SX,1HA,9X,1HB,5X,11HFLUID CONC,, Sx,
, 10HSaLlOVOL.»2X,13HCONCFNTRATIUN)
NCR IDBCISTEP- INDEX) /iOO
IFCNGRID ,LT. I)
DO «30 JJjBl*DEX
00 442 KK*1,NC
ACOK.'C(KK)sa,*PAI*AA(JJJ)*BC(JJJ)**2*SS(KK, JJJ)/3,
WRITE(6,U)T(JJJ),CX(JJJ),CY{JJJ),CZ(JJJJ,U(JJJ),V(JJJ),
W(JJJ),DENr)IF(JJJ)» AACJJJ),BCCJJJ)»FC(JJJ),ACONC(1),
ssa.jjj)
3F12.4)
IFCNC .EG. 1) GO TO «JO
DO «50 KK82»NC
WRITE.C6,15)ACONC(KK),SS(KK,JJJ)
IS FQRMATU01X,2tl2,T,T,T,AA,BC,CX,CY » ISTEP1 ,2» NSCALE,4)
CALL ORAW(CX,T,T»T,CZ,T,T,T»ISTEPt»15fN8CALE»t)
8002 COMTINUE
DT2*DT
DO iao KK«I,NC
CMAX(KK)=SS(KK,ISTF.P)
140 CONTINUE
CMAX(NC+l)sFC(ISTEP)
FOK LCJNG TERM DIFFUSION U8F
IFChSCl,!} ,EO, 0,)C«AX(1)«CMAX<2)
AOsAA(ISTtP)
IF(CY(ISTEP)»AA(ISTF,P),LE.O.) YY*AA(I8TfP)
RETURN
END
A50
-------
SUHHOUT JN-fc Ct'LAPS2
CnHMON/GP/IGCK',IGCU»IC.LT,IPCN,IPCLpIPLT,MSCALE
COMMON /A/ tP(20)
COMHQN /B/OlNCRl,OlNCH2,KEYl,Kf.Y2,KEY.S
COMMON /Af00),
, SZETA(600),CZHA(600), A A (600 ) , OF.NO IF (60 0 ) , CL<600),
V2,CMAX(9),YY
/TUPS/ DS,Oti fOT^^CDT^NQHIDfNG
/CnNST/G,PAI/ ALP^AO, ALPHA, ALPHA Cr ALPHA 1, ALPHAS,
/COMPUT/FF(7,lbl) ,E(205 » ACOKCU23) ,CSC«,2) ,
,tbn,Yl(15n»Y2C15n»DY(lbO),F(fl),FW(8),taW(9),
CCMAX(151 ) »VAP(151)
/TkASFfH/ KK,JJ,TT, I S I ZF , NOTE , A 0 , I Y , MCJTfrM , I NDfc X , I BtO •
FHtDi AfO,OLOT,METHnp,TSTOP
C
C
OSS=DL(ISTfcP)
IFC*FTHOD .1-0, 2) AA(I3TF.P)«BC(1STFP)
IJKsO
•?
90051 FORMAT(1H1,///10X,27HCOLLAPSF PHASE OF THE
INDFXsISTfcP
N'P. OF TOTAL. tQS
DO 101
101 SAVE CKKJstMKK)
25 00 102 KK«i,Mf
102 E'(KK)«SAVE(KK)
IJKsIJK+1
ISTEP=INDEX
UUUsUOLD
V2 = V5
DO 103 KKs«,8
103 F(KK)sf (KK)*OS
00 104 KKslO,Nf
100 E(KK)»F(»
-------
C NOTRN -INDICATE ENTRAPMENT FITHEK BY CONVECTION OR BY COLLAPSF
NUTRlsO'
C WJTRL -INDICATE DIFFUSION TAKING OVER
C IPLLINOINDKATOR OF HITTIMG BOTTOM
IFCISTEP ,EQ, IBED)IPLUMGsl
IFCK.OTE ,FO, 1) GU TO 500
IFC1STEP .EQ. INOEX)G(j TO 121
CY(ISTEP)sE(2)
CZ(ISTEP)*E(3)
i./ccM*t'(a))
UUs SURT(V1**2+V3**2)
OL(ISTFP)=UU*DL(ISTEP»1)/UUU
DLDTs(DL(ISTEP)»DL(ISTEP«l))*UUU/OL(ISTEPi-l)
AREAaVfiLUMF/OLClSTEP)
BC(ISTEP)sE(<»)
BC-MAJOR RAIUS OF CLOUD
AA(ISTEP)»AREA/(PAI*E(9J)
V2SF(6)*CMMASS*AA(TSTEP)/BC(1STFP)
U(ISTEP)sSORT(Vl**2*V2**2+Vi**2)
THETA2(IRTEP)=,5*PAI
DS=U(ISTFP)*DL(ISTEP)/UU
AKXs2,29*ALAMDA* E ( 9 ) **t^33333
BBDTs(BC(ISTEP)-BCCISTEP»l))*UUU/OL(ISTEP-l)
IFCISTFP.IF.IM3FX+5J GO TO 111
IFCAKX ,GE. BBOT)NUTRL»3
IF(CY(ISTFP)-AA(ISTEP).LE,0.) on TO 220
111 CONiTIWUF
UUU*UU
UUsl./UU
SZETA(ISTEP)sV3*UU
CZETAf ISTFP)sVl*IJU
DO 120 IS!,K
DO 120 J=1,L
SS(KK ,ISTEP)sE(10+KK)/VOLUME
120 KKsKK+1
SS SAVE SOLIDS FOR LONG TPRw DIFFUSION;, CO^C, IN VOL. RATIO
FCf ISTEP)SVF/VOLU«E
IFtlPLUN-G ,EQ, «) GO TO 122
IF(CCY(!STFP)+AA(ISTEP)*,B5) .GF, H)
122 IFCNUTRL .EG. 3) GO TO 200
IF (IPLUNG .EG. 2) GO TO 300
IFUSTEP ,GE, 599) GO TO 200
121 CALL RUNGS(DERtVE«»NE)
DT« DS/UdSTFP)
ISTEP=1STEP-H
TCISTEP)sT(ISTEP-l)*DT
S(I5TEP)=S(ISTFP»1)+DS
GO TO 110
300 CONTINUE
OBTsE(10)*3t/(AA(ISTEP)*8C(ISTFP)*ROO*DL(ISTEP))
A52
-------
E(10)BRDO*AA(ISTfcP)*BCdSTEP)*((5BT
, +.375*PAI*BC(ISTEP)*V2 /A A(ISTfcP))/1,5*CL(ISTEP)
IBED*ISTEP
500 CALL BOTTOM2
IFCISTEP ,GE. 600) GO TO 2020
IF(MJTRL tEO, 3) GO TO 200
IFdPLUNG ,FQ, 4)ifeO,SOO
160 E(6)«CM*E«O*V2
ROnsE(a)*ROA(l)/(E(tt)+E(8))
PBT*E(10)*1,5/(AA(ISTEP)*BC(ISTEP)*ROO*OL(ISTEP))
F(10)eROO*AA(ISTEP)*BC(ISTEP)*OBT/S,*DLdSTEP)
AOsAO*,5
!LEAVF=ISTEP
GO TO 121
2020 ISTFPBlSTfcP-1
200 WRITE(6,90056)IJK,[)lNCRf
90056 FORMAT(1X,IS,2F1?,4,3I5)
IF(IJK ,EQ. S) GO TO 220
IFmSTFP-I^eX) ,LT, 100 .OR, (ISTtp.IMDEX) ,GT. «00) 210,220
210 DINCH«OINCR*(ISTEP»IMDEX
GO TD 2^
220 COMTINJUE
IF(IPCL.F-Q.O) GO TO 8001
WRITEC6, 90053)
90053 FURMATdBXMHT/TX^lHS^X
,,7HOEKi-DIF,2X,llHFLUID CONC,,13H SOLID VOL, »2X , 13HCOMCENT H AT ION)
NGRIDe(ISTFP-!NOF.X)/100
IFC^GRID ,LE, 03 NGRIDSI
DO 240 JJJsINOFX, ISTEP, MGRID
DO 230 KKel/NC
230 ACQNC(KK)8 PAI*8CCJJJ)*AA(JJJ)*SSCKK,JJJ)*DL(JJJ)
WRTTE(6,9005a)T(JJJ).SfJJJ),CX(JJJ),CV(JJJ),CZ(JJJ)rU(JJJ),
, AA(JJJ),BC(JJJ),DE.MOIF(JJJ)»FC(JJJ}»ACONC(1)»SS(J»JJJ)
IF(NC §EQ. 1) GO TO 2ilO
00 201 KKe2,NC
201 WRITE C6,900bS) ACONC(KK),SSCKK,JJJ)
90055 FO»PAT(101X,2fc'12,^)
2«0 COK.TINUF
6001 IFCIGCL.FQ.O) GO TO 8002
ISTEPlsISTEP*!
CZ ( I STEPl)s2,*CZ(I8Tfc'P)-CZ( ISTfcP- 1)
CX(ISTEPl)s2t*CX(I3TKP)"CX(ISTfcP-l)
AA(ISTEPl)sO,
CYdSTEPDeO.
CALL ORAW(CX,CX,CX,CX,AA,8C,CZ,CY , ISTEPI,«»^SCALE»'
8002 CONTINUE
DO 260 KKel,NC
CMAX(KK)SSS(KK,ISTEP)
260 Cf.'N'TINUE
CHAX(NC+l)sFCCISTEP)
!F(wSd,l) ,EQ, 0,)CMAX(l)sCMAX(2)
YY=F(2)
IF(CY(ISTrP)-AA(!3TEP),LE,0.) YY«AA(ISTEP)
AOsAA(ISTEP)
DT2«DS/U(ISTFP)
RFTURM
(-MD
A53
-------
FXTFRMAL
Y( SO},HfUC 30 ) , YK J , Y«2 , VK 3 , YK tt , AK Y i , AK Y2 , AK Y3 ,
CX(bOO) ,CY(feOO) , AA (600) ,IJ(600),
V(bOO), n(600),VORH600),OtNniF«>00),COLAPV(faOO),
,C?(600),SS(8,600),SAVf: (600),nUMY2(6,600),FC(600),BC(600),wS(U,2),
, RnASU),T(feOO),K,l,IPUJNG,Num.liH,SAl, TlMf.iSTEP. VF,
/TPHS/ DT , DT 1 , DT? , CDT , ^GRI 0 , NG« ID1 , N(;P ID?, NGRI 05 ,
/CnMP
n(8,lbl)
/TRASFERX KK, JJ, TT, IS I2t, MOTE , AO , 1 Y, NJOTR*, INDEX , l8ED,IlEAVfc
(8))/RDA(l
IFCISTEP ,fcO, I8EO)12a,125
FREDsO.
GO TO 126
CX(I5TEP)af (1)
CY(1STEP)=E.(2)
CZ(15TEP)sE(3)
NUSTEP)=fc fS
VCISTEP)s.75*16.*F(10)/(PAl*BC(ISTtP)**3*P-nn)
«ClSTF.P)=t (7
KKsl
DM 120 I=i,k
DC 120 JsJ,L
IF (ABS(VnSTEP)) ,U. ABS ( *S (I, J) ) ) 121,122
121 RFTAAsi.
GO TO 123
122 BET*A=fitTA
FBED*F8FD-PAl*BC(ISTfcP)**?*ABS(wS(I,J))*ROAS(I) *SS(KK,ISTEP)
, *(1.-BF.TAA)*V(ISTEP)
120 KKsKK+i
FC(ISTtP)sVF/VCLUHE
RfUABRriA(IY)*(F(2)»Y(IY) ) * (ROA CI Y* 1) -KG A (I Y) ) / ( Y ( I Y+ 1 )-Y(IY))
FRED = FBFD+1,33333*PA I*A A(I STEP)*BC(1STEP)**2*(RQO-ROAA)*G
, -C«*CKC«)*V(ISTEP}-E(fc)*V(ISTEP»i))/DT
AKXsjJ,*AL AMD A*(,b*E (9))**,333333
C AKX » CHANGE OF R BY DIFFUSION
BPOTs(BC(ISTEP)-BC(ISTEP-l)5/DT
IFfAKX ,CE. BBOT)
126 E(6)=F{U)
C STOKF OLD MASS IN
IKCFBF.O ,LT. 0,5
IF(MUTRL .EG, 3) GO TO 200
IFUPLUMG ,ER. a) ILF.AVEsISTFP
SEQ, A) GO TO 200
A54
-------
CALL RUMGS{nFRIVE6,NiE)
JSTfPsjSTFP+1
T(ISTEP)sT(ISTEP"l)+DT
200 CONTINUE
RETURN
A55
-------
SUBROUTINE BOTTO*2
EXTERNAL DERlVFl,DERlVF?,DERIVt3,DERlVfcil,DEHlVFS,DJfR!Vfc6,DERlVE7
COMMON /ABCO/ Y( 30),RDA( 30 ) , YK I r YK2, YM, YKU , AKY 1, AK Y2, AK Y 3,
, YU,YW,YEfH,UAO,WAO,N,CKY(15l),UA(151),WA(lbl)
COMMON CX(600J,CY(600),THETA2(600),UC600),
, SZETA(600),CZETA(600),AA (600),DFND1F(600), DLCbOO),
, C2(fcOO),SS(8,600),SAVE(600),DUMY2(5,600)fFC(feOO),T(feOO),
ROAS(«)»S(600),K/L»IPLUNG,NUTRL»UBlSAI,TlME»ISTEPfVF,
UBX,WBZ, V2»CMAX(9),YY
COMMON /Ct)NST/G»PAI,ALPHAO»ALPHA,ALPHAC,ALPHAl,ALPHA2,
ALPHAS/ At PHA«,GAt*A, COR A6»CFKIC» BETA, CM, CD»CD1,C 02, FR!CTKi»Ft
,ALFA1,ALFA2,GAMA1,GAMA2,ALAMDA,CD3,CO<»
CUMMOK' /TORS/ OS,OTl,OT2,CD
-------
SS SAVE SOLIDS FUR LO^G TFRw DIFFUSION. CONC, IN VOL, RATIO
FC(ISTEP)=VF/VOLUME
FBEDsFBf O+PAI *AA(ISTF.P3*8C(1STF.P)*(RUO-RQAA)*G*DL(ISTFP)
)*uuu/oi. USTFP-U
1KAKX .GT, BBDT)NUTRLS3
126 f (fa)*t«J)
va = v2
STORE 010 MASS IN E(6)
IF(FBfcD ,LT, 0,) IPLUNGsil
IF CMUTRL .tO, 3) GO TO 200
,tQ, a) ILEAVE*ISTEP
,EO. 4) GO TO 200
CALL RUfcGS(DFRIVE7»NE)
OTs DS/UC1STEP)
ISTEP*ISTFP*1
T(ISTFP)aT(lSTEP»l)*DT
S(ISTFP)sStISTtP«l)+OS
200 CONTINUE
RFTURN
END
A57
-------
SUBROUTINE RUNGS(DERJVE,NF>
COMMON /A/ FPC20)
COMMON /COMPUT/tFC7,lSn,E(20),Wl(20),W2(2Q),w3
-------
DFP-IVUCt)
N f
CUMHI1M /A/ EP(20)
/ARCH/ Y( 40),«OA( 30),Y*UYK2,YK'4,YM,AKY1,AKY2,AKY3,
CX(e>00),CY{600) , AA C 600 ) , U ( 600 ) ,
V(600). i*(600) ,VO«T(feOO) ,DF.ND!F(600) f
CZ(feOO)
O ,T(60
-------
EP(«)=ENTRV*ROAA
EP(fl)*FN!7RV*(K[}AU)"RnAA)
EP (<»)*• 3, *B**2*f5*CE/RQACl)
KKst
DD 150 IBJ,K
00 150 J*lfL
IP(&BSWS-ABS(VV)) 151 ,151,152
C IF FAIL VEL. IS SMALLER THAN THE COVECWE VEL, NO SETTLING OCCURS
151 BFTAAH.
GO TO 153
152 BFTAAsBETA
153 SETLVBPAI*B**2*ABS(WS(I,J))*(1 ,-RtTAA)*E ' (9 + KK) /VGLUMF
I) 3
I) 3*UU
I) )*VV
EP(7)«EP(7)"SfcTLV*(ROAS(I)
EP(e}=E
EP(9+KK
150 KKSKK4-1
vv
A60
-------
DFklvfc2(F)
t (20)
/A/ EP(J?0)
/ARCP/ Y{ 30),«IU( 30 ) , YK 1 , YK2, YK 3 , YK a , t h Y 1 , AK Y?, AK Y3 ,
CX(600),CY(feOO), AA ( 600 ) , U (60 0 ) ,
V(fcOO),
a) rT(t>00)
iJRX,wB
COMMON /CONST/G,PAI, A LPHAO. ALPHA, ALPH AC, ALPHA 1 , ALPNA2,
KK , JJ, TT, ISIZE ,NOTe , AO , I Y ,NDTRw, INDEX, IBtP, ILf- A VE
C
c
IF (F(?) ,Gt .0,) GG Tf 88B
FHR^ATl U7H Y LT 0 -- CHAK'GF INPUT DATA TO FMSUNE
CALL EXIT
100 IF(t(2) .LE, Y(IY*m GP TO 101
IYsIY+1
GO TO 100
101 IF Ct (2)-Y(IY))102,200,200
102 IYsIY-1
GO TO 100
Kr.AAsROACIY)+(E(?)»Y(lY))*(ROA(IY+l)-fOA(IY))/(Y(IY+l)^Y(IY))
IF(ROO .Gh, PUAA)
IFfMOTR^1 ,FO, 1) 10/20
10 ALPHAIsO
ALPHAsALPHAC
?0
UA«0,
!F{fc(2) ,LE. YU)
IF(F(2) ,GT. YU ,AN-D, E(2) ,LT, YE) GO TO 110
GO TO 120
110 LIAS (YE»F(2))*UAO/(YE-Ytl)
120 WAso,
IF(t(2) .LE. Yw) WA«sE(2)*WAO/YW
1F(E(2) ,GT, Yh ,AMO, §(2) ,LT, YE) GO TO 130
GO TO 140
130 WAs(YE-
140 B=f(9)
CMMASSSsCMMASS*B/A
UUBEtSi/CMMASS
VV*t (6)/CMMASSS
PHlsSQRT((UU-UA)**2*VV**2*(W|«/»HA)**2)
EP(
-------
GO TO 3
2 APEA2sA**2*B/DUMY
AREA2sAREA2*ALOG((DUMY+A)/A)
C EMRV-EK-TfrAIMMENT IN VOLUME
C MAIN COMPUTATIONS
f PCi)aUU
EPf2)=VV
EP(3)aww
DRAG= PAI*ROAA*PHI*,5
FP(U)aENiTRV*ROAA
EP(5)sEHTRV*ROAA*UA-ORAG*A*B*(IJU-UA)*f:D3
EPC8)sEMTPV*CROA(13-ROAA)
EPdOJs PAI*(l,«GAMA*AO/A)*CE'*G*A**X*B/lfe,
»CDRAG*ROAA*A*8*FP(
-------
SUBROUTINE DERIVES CE)
DIMENSION EC20)
COMMON /A/ EP(HO)
COMMON XABCH/ Y( *0),ROA(
CX(600),CY(600),THETA2(600),U(600>,
t SZ£TA(fcOO) ,CZ6TA(tiOO),AA (600 ) ,DENDIK CfeOO) , DL(feOO),
» CZ(600),SSC8, 600), DUMY2 (8 , 600 ) , BC (600) , WSC4 , 2 ) ,
» ROAS(tt) ,S(600),K,L,IPLUNG,NUTRL,UB,SAI,TIME,ISTEP,VF,
» UBX,«8Z, V2,CMAX(9),YY
COMMON /CONST/G,PAI, ALPHA 0, ALPHA, A tPH AC; ALPHA 1, At PHA2,
COMMON /TORS/ DS,OTl,DT?,CDT,MGRID,NGRI01,NGRJD2fMGRID3,NCD
COMMON /TRASFER/ KK, JJ, TT, I SIZfc , NOTE, AO , I Y , NQTRN, INDEX, I BED, REAVE
, • UK, FBED,APO,DLDT, METHOD, TSTOP
c
c
IF(F(2),Gt ,0.) GO TO 888
CALL EXIT
999 FORMAT( fl?H Y LT 0 -- CHANGE INPUT DATA TO ENSURE DESCENT )
888 CONTINUF
100 IF(F(2) ,LE. YCIY*!)) GH TO 101
IYsIY+1
GO TO 100
101 IF(F(2)-YCIY))102,200,200
102 IYalY-1
GO TO 100
200 ROAAsROACIY)+(E (2)"Y ( I Y ) )* (ROA ( I Y+i )»RQA ( I Y) ) / ( Y ( I Y* 1 )«Y ( IY) )
BHsSQRT((F.(«)+F(B))/(ROA(l)*UU*PAI))
FLUX«PAI*BB**2*UU
UASO.
IFCFC2) ,LE. YU) UAsUAO
IF(E(2) ,GT. YU .AND, E(2) ,LT. YE) GO TO 110
GO TO 120
110 DAB (YE-E(2))*UAO/tYF.-YU)
120 WA=0,
IF(E(2) ,LE. Yi«) WA*E(2)*WAO/YH
IF(E(2) iGT. YW .AND, E(2) .LT, YE) GO TO 130
GO TO lan
130 wAB(YF«E(2)
UO CONTINUE
UA=UA»UHX
OEN1*S(3RT(UA**2*WA**2)
CQSD3KWA/DEN1
OENOef («)*UU
D!RCOSisEC5)/OENO
OIRCOS2*EC6)/DENO
COSGAMA«C0301*OIRCOS1*-COSD3*OIRC083
AGAMAc ACDS(COSGAMA)
FD=CO*ROAA*fUA**2+^A**2)*SIN(AGAMA)**2*BB
OENO»8IN(AGAMA)
FDXc(cOSDlfCOSGAMA*OlRCOSl)*FD/DENO
A63
-------
FDY*«eQSGAMA*[JIRCOS2*FD/DENO
FDZB/FLUX
RUUsROAS(I)*UU
FP(5>sEP(5)-SI:TL*RUU*OIRCOSl
EP(fe)sFP(6)»8ETL*RUU*OIRCOS2
EP(7)*EP(7>»SETL*RUU*nlRC083
FP(8)sEP(8)»SETL * (RDA ( 1 J-ROAS ( I)
EPCB+KK Js.SETL
150 KKSKK+1
RETURN
A64
-------
SUBROUTINE DERlVfcU(E)
DIMENSION t(20)
C
c
/A/ E.PC20)
/APCD/ Y( iO),ROA< 30 ) , V K 1 , YK2 , YK3 , YM , AK Y 1 , AK Y2 , AK Y3 ,
CX(600).CY(600),THETA2C600),U(600),'
, SZFTA(feOO) ,CZETA(fcOO),AB (600 ) , DENDIF (600 ) , DLC600),
, CZ(6003,SS(8, 600),DUNYa(8,fcOO),BC(600),WSC
-------
10 ENTRsALPHA3*SF*PHl*OL(ISTEP)
GO TO 30
20 £NTRsALPHA«*SF*tPC
-------
SUBROUTINE T--ERIVE5 CE)
DIMENSION EC20)
COMMON /A/ FPC20)
COMMON /ABCF/ Y( 'iOfHUAC 30 ) , YK 1 , YK2 , YK 3 , YK U , A K Y 1 , AK Y2 , AK Y 3 ,
CX(600),CY(600),THFTA2(60Cn,UUOO),
, SZFTA(600),CZETA(fcOO), A A (6005 , OENOIF (600) , DL
. CZ(bOOD ,SSC8, 600),DU^Yi>(e,feOO) ,BC(feOO) ,hS(«,2
, ROAS(u),S(600},K,L,IPLUNG,NUm »UB/SAI, T IMF ,ISTt.p, VF,
, UBX|WB£,RDAF- ,C«AX(9) , YY
/CCNST/G, PA!, AI.PHAO, ALPHA, ALPHAC,ALPHA1,AI.PHA2,
COMMfik /TRASFFR/ KK , J J, T T , IS I ZE , NOTE , AO , I Y , K.OTRK , IK'OF X , I BED, I l.fc A VE
, , TJK,F BFO,Af'0,DLDT, METHOD fTSTOP
C
C
IF(E(2).GF ,0,) GO in H88
CALL EXIT
999 KORMATC ^?H Y LT o -- CHANGE INPUT DATA TO ENSURE DESCENT )
888 COMTINUE
100 IF(F(2) .LE, Y(IY + D) GO TO 101
IYBjY+1
GO TO 100
101 IFCE(2)-YCIY))102,200,200
102 IY»IY«1
GO TO 100
200 RnAA«RO*(IY)*(fc(2)"Y(IY))*(ROACIY+l)«ROA(IY))/(YCIY*l)*Y(lY))
CF«CRPA(IY*l)-ROA(IY))/(Y(IY*n»Y(IYj)
UAnO,
IF(F(25 .LE, YU) UAsUAO
IF(E(2) ,GT. YU .AND, F(2) .LT, YF.) GO TO 110
GO Tf 120
110 UAs (YF-E(2))*UAO/(YE»YU)
120 HA«0,
IF(E(2) ,LE, Yw) WAst(2)*WAO/YW
IF(E(2) ,GT, Yin ,AMO, E(2) ,LT. YE) GC' TO 130
GO TO 140
130 WAs(YF-E(2)
HO CONTINUE
UAsl'A-UBX
AREAevnLUHE/DLdSTtP)
IFCROO ,LT, RCAA
BRsSORT(2,*AREA/PAl)
CMMASSaS ,/(CM*t(«))
V2BE(fe)*CHMASS
PHI*SOPT((VUUA)**2*V2**2+CV3-1«(A)**2}
PCY«SORT( (Vl»UA)**2+ (V3«1*A)**23
ENTR*PAI*BB*DL(I3TEP)*(ALPHA*V2tRCY*Al.*»HAl)
c ENTR -FNTRAIKMENT IN VOI.UHE
A67
PX*V3*UUU
PZ=VI*UUU
-------
DRAGJ=,b*CDl*ROAA*PHl
FP(1)*V1
FP(?)cV2
EP(5)sEMTR *ROAA*UA»DRAGt*RB*PX*(vr-UA)*OL(TST!P)
tP(6)aG*(ROf)»HCAA)*PAI*B8**a*.5*OL(ISTKP)»ORAG2*BB*va*2,*DL(I3TtP)
F.P(7)*ENTR *ROAA*WA«0«AG1*BB*PZ*(V3»WA)*DUISTFP)
UU*lt/UU
KKxl
DO 150 islfK
00 150 J*l,L
AHSWSsABS(WS(I/J))
IFCABSWS«ABS(V2 ) ) Ib I , 15 t f 152
IF FALL VEL. IS SMALLER THAM THt CONVFCTIVE VEL, NO SFTTLING OCCURS
151 BETAAsi.
GO TO 153
152 BETAA*BETA
i53 SFTL s2.*BB*ABSWS*OUISTEP)MU-BETAA)*f (
EPCa)s(EP(«)«SfTL*ROAS(I)>
F.P(5)«(EP(5)«5ETU*ROAStI)*Vl)
FP(fe)s(FP(6)«SETL*RCAS(I)*V2)
EP(7)*(EP(7)"SETL*WOAS(I)*V3)
FP(8)e(fP(8}*SETL * ( ROA ( i) -ROAS ( I ) ) )
tP( 8 + KK)s«Sf.TU*UU
IbO KKSKK4-J
DO 160 KK*1,8
160 EP(KK)Bf P(KK)*UU
RETURN
A68
-------
DIMENSION E(20)
COMMON /A/ EPC20)
COMMON XARCOX Y( 30),RQA( 30 ) t YK 1 , YK2 , YK3, V«l , AK Y 1 , AK Y2 , AK YS ,
CX(600),CY(600),AA ( 600 ) / IJ C 600 ) ,
, V(fOO), «(600)»VOHT(600)»l)F NDU' (600 ) » FMTRCOC600 } ,
ROAS(a),T(600).K,L ,IPLUNGfNUTRt , HB» S* I , T IHE, ISTFP » VF ,
UBX,WBZ,(?nAf-,C^AX(9) , YY
COMMON /CONST/ G^PAI.ALPHAO, A LPHA.ALPHAC, ALPHA i,
/TRASFtH/ KK , J J, T T , I S 1 ZF , NOTE , A 0 , I Y, NOTRfc, INPFX, T BED, REAVE
, , UK , FBtD, AMD, DIDT, METHOD,! STOP
IF(t (2),Gfc.O,) GO TO BRP
CALL fcXlT
Ff)R^AT( ^7H Y IT 0 »- CHA^Gt I^PUT 0/»TA TO ENSURE DfcSCfcNT )
888 CUMlMJt
100 IF(t(2) .LE. Y(IY + 1)5 G(i TO 101
GO TO 100
101 IF (E(?)-Y(IY))102,200,200
102 IYsiY.1
GO TO 100
200 ROAAsRQA(IY) + (F(2)»Y(IY))*(f
-------
2
AREA2»AREA2*ALOG{ (DUMY*A)/A)
3 EMTPVB PAl*(AREAl + AREA2)*
C ENTRV-»EMRAlNHfcMT IN VOLUME
C MAIN COMPUTATIONS
EPU)=UU
EP(2)«VV
PAI*PCAA*PHl*,5
EP(8)sENTRV*(ROA(l)-RUAA)
E'P(10)s PAI*(l.»GAMA*AO/A)*CE*G*A**1i*B/16,
, «CDRAG*ROAA*A*B*EP(9)* ABS (EPO) ) /« ,
»CFRIC*RDAA*B**2*EP(9)/(2.*A)
-Fl* PBFD*FRICTN/C2,*PA1)
OVaEKTHV*HOAA
DO 150 jsl,K
DO 150 J«1»L
APSWSaABS(W3(l,J3)
!F(ABSWS»A8SCVV))15if i51f 15?
C IF FALL VFL, IS SMALLER THAM THE CONVECTlVt VI- L, NU SETTLING OCCURS
151 BETAAsl.
GP TO 153
152 BFTAAsRF.TA
153 SfcTLV*PAl*B**2*ABS(wS(I/ J) ) * (1 ,»3ETAA) *h ( 1 0 + KK)/ VOLUME.
EP(4)eEP(tt)«SFTLV*(ROAS(I) )
FP(5)sfp(5)-StTLV*(ROAS(I) )*UU
EP(7)sF.P(7)-SETLV*CROAS(I)
EP(l
-------
DIMENSION EC20)
COMMON /A/ EP(20)
COMMON /ABCO/ YC 30),MOA( 30 ) , YK 1 , YK2 , Y*3 , YK« , AK Y i , AK Y2 , AK Y3 ,
, YU,YW,YE,HfUAO,wAO,N,CKY(l51),UAC15l)»WAn5l)
COMMON CX(600),CY(600),THETA2(feOO),U(600),
, SZFTA(600),CZtTA(600),AB (600 ) , flENDIF (600 ) • 01_(600},
, CZC600;»SS(8, 600) »DUKY;>C8,600),BC(600),»S(4,
, ROAS(«),S(600), K,L,IPLUNG,MUTRLiU6»SAI,TIME,ISTEP,VFf
COMMON /cnNST/C/PAI,ALPHAO,ALPHA,ALPHAC»4LPHAl » ALPHA2,
COMMON /TRASFtR/ KK , J J , TT , I SIZfr., NOTE , AO , I Y , NOTRN, I MOEX , I BED/ ILE A VE
, i UK, F RF D » AMD, DLDT, METHOD,! STOP
IF(E(2).GE.O.) GO TO 888
WHITE (6,999)
CALL FXIT
999 FORMAH a?H Y LT 0 -- CHANGE INPUT DATA TO ENSURE DESCENT )
888 CONTINUF
100 IPCFC2) .LE. YCIY41J) GO TO 101
IYsIY+1
GO TO 100
101 IF(E(2)"Y(IY))102»200,200
102 IY*IY»1
GO TO 100
200 ROAA=«OA(IY)*(F(2)-Y(IY})*(ROACIY-H)«RUA(1Y))/(Y(1Y+1)«YCIY))
UAsO,
IF(F(R) ,Lt, YU)
IF(E(2) ,GT, YU .AND, EC?) .LT, YE) 50 TO 110
GO TO 120
110 UA= (YF"EC2n*UAO/(YE-YU)
120 fcAsOg
IF(F(2) .LE. YW) WAtEC2)*WAO/Yw
IF(E(2) ,GT, YW .AND, E(2) ,LT, YE) GO TO 130
GO TO HO
130 VJAs(YE»E(2)
1«0 CONTINUE
VOLUMFB(t(«)+E(8))/ROA(l)
ROOeE(tt)/VOLUMfc-
AREAsVOLUMf/DLdSTEP)
AAa2,*ARf.A/(PAI*BB)
CMMASSBl./(CM*E(«))
V3=E(7)*CMMASS
EP(9)=E(10)*3,/(ROO*UU*AA*BB*DLUSTEP))
EP(9)*EP(9)»BB*DLDT/DL(ISTEP)
ENTRs.5*SF*DLtISTEP)*C ALPHA«*EP(9)*UU)
C F.NTW »eNTRAINMENT IN VOLUMF
c MAIK COMPUTATIONS
UUU=1./SORT(V1**2*V3**2)
A71
-------
PXeV3*UUU
PZ«V1*UUU
DRAGls,5*CD3*ROAA*PHl
UiKVUUBX
U3*V3+WBZ
FP(1)=V1
FP(2)*V2
EP(3)»V3
*ROAA
*RCAA*UA«OR*61*AA*PX*(V1»UAJ*2.*DL(I8TEP)
+FRICTN*FBED*U1*PH
*ROAA*(NA«ORAG1*AA*P2*(V3«WA)*2,*DL(ISTEP)
+FRICTN*F8tD*U3*PM
EP ( I 0)=. 1666667 *CE*(l.»GAMA*AO/AA)*AA**3*G*DL(ISTtP)/UU
»CDRAG*.b*ROAA*AA*UU*EP(9)*A8S(EP(9) 3 *OLCISTEP)
«CFRIC*BB*tP(9)*DL(ISTEP)/AA
«F1 *FRICTN*FBED
UUsl ,/UU
KKeJ
00 150 1=1, K
DO 150 J*1,L
1F(ABSWS-ABS(V2 ))151,151»1S2
IF FALL VEL. IS SMALLER THAN THE COMVECTIVE VEL, NO SETTLING OCCURS
151 BETAAsl.
GO TO 153
152 BFTAAsRETA
153 SFTL »?,*B8*ABS«VS*DL(ISTEPJ*{Jf»BETAA)*F(10*KK)/VOLUME
EPU)s(£PU)«SeTl*ROAS(I>)
FPt7) = (EP(7)»SET|.*ROAS(l)*V3)
EPCe)"(EP(8)»SETL * (ROA( 1 )«ROA5 C I) ) )
150 KKaKK+1
FP(9)sEP(9)+nV*UU/(PAI*AA*ROO*DLUSTEP))*2t
nn uo KXBii8
160 EP(KK)*fPCKK)*UU
RfTURM
END
vv
A12
-------
SUBROUTINE DIFUSN1
COMMON /B/DINCRl,OINCR2,KEYl,KfcY2,KEY3
COMMON /C/XO.ZO
COMMON /MATRIX/ A(lbl),B(151),C(151),CP(151)
COMMON /ABCD/ Y( SO),RQA( 30) , YK1 , YK2, YK3, YKa , AK Yl , AKY2, AKY3,
YU,YK,YE,H,UAO,WAO,N,CKY(15i),UA(151),HA(151)
COMMON /COMPUT/EE<8,151)»
E(e,lbl),Yl(151),Y2(151),DY(i50).FCfl),FW(8),WW(9),
CCMAXC151),VAR(151)
COMMON /TORS/ OT,DT1,DT2,CDT,NGRIO,NGRID1,NGRID2,NGRID3,NCD
COMMON /CONST /G, PA I, ALPHAO, ALPHA rALPH AC »ALPHA1, ALPHAS,
ALPHA3,ALPHA
-------
110 t(I,J)sO,
C Sf-T VALUES IN E AND F TO 0,
IFfKK ,EO. k'CPl ) GO TO 250
DO 120 !«l»J2
OY(I)sH/J2
120 Cf.lMTINUE
YKUsO,
Y2(l)«0.
OH 130 1*2, Jl
YlCIJsYl (I-l)*DY(I»l)
Y2(I)«Y?U-n+DY(l-l)
130 CONTIMJf
C SET UP A G»10 OF EQUAL SIZE
CALL IfcABCD
CALL SUMAPCO
DO 131 JJM»IST£P
131 SVOI.(JJ)«1,333333*PA!*AACJJ)*BC(JJ)**2*SS(KK,JJ)
DH 132 JJs?»lSTEP
132
DO 133 JJ*2,ISTEP
IFfSVOLUJ) fGT, ORIGIN) GO TO
133 CONTINUE
JJsISTEP
TTsT(JJ)
T2GRIDB H/(J2*ABS(Ki.'(K)<)n
DTOaTgGPID
TPLOTBT(ISTfP)
IF(TPIOT/T2GK!D,GT,100,) T2GRIDaO ,Ot*TPl.OT
NCAKE«(TPLOT»TT)/T2GRIO
IF(IPLT.EO.O) GP TO 12347
IF(NOUTPT,LT,n
GO TO 12348
123a7 NOUTPTsS
CONTIMUE
IFCISTART .fcQ, ISTtP) GO TD !<»$
13
-------
194 CALL SOLUTN
ZAMQsO,
Df! 997 1*1,J2
997 ZAMOeZAMO+(E(l,mE(lf I«-l))*DYU)*0,5
If-UEWTOT.LF.O.) 60 TO 991
RTOTsOLDTOT/ZtwTOT
DO 99fe 1*1, Jl
996 EU,I)sECl,I)*RTOT
FU)eF(l)*RTOT
991 CONTINUE
191 TT*T(ISTOP)
DO 190 JJ*ISTART,ISTOP
IFCSVOLCJJ) tGT, QRIGIKOGO TO 142
SVOL(JJ)sO.
GO TO 190
142 IF(WWCKK)) 170, 170, Hi
UPDATING FOR SOLID WITH POSITIVE FALL VELOCITY
143 IFCJJ ,LE, INDEX) 144,145
IN THE CONVtCTIVF. PHASE
144 THlCKCJJjsCYUJ-l) +WW (KK ) * £ T < J J )»T ( J J" 1) )
, »CYCJJ)
PUSITNCJJ)sCY(JJ)+,fc25*BC(JJ)+,5*THlCKCjJ)
GO TO 146
145 IF(JJ ,GT, IRFO .AMD, JJ ,U. HEAVE) 151,158
H' THE DYNAMIC COULAPSf PHASE
158 THICK (JJ)*CY(JJ-»1)+AACJJ»1)*WW(KK)*(T(JJ)«T(JJ»1))*CY(JJ)«AA{JJ)
POSITNCJJ)*CY{JjHAACJ
F(1)BF(1)+RASVOL
F(2)sF(2)+RASVOL *CX(JJ)
F(5)*F(3)+RA8VDL *CZ(JJ)
F(4)BF(4)+RASVOL * ( CX C J J ) **2+ , 2S*BC ( J J ) **2 )
Ft5)sF(5)+RASVOL * (CZ ( JJ)**2+ ,2b*BC ( JJ)**2)
F(6)«F(6)'«-RASVOL *CX ( JJ )*CZ ( JJ)
SVQL(JJ)=(1 ,»RATIO)*SVOL(JJ)
THICK (JJ)*(1,-RAT 10) *THIC«(JJ)
PDSITN(JJ)«H»,501*THICK(JJ)
DISTa,50l*THlCK(JJ)
400 IF(JJ ,feO. ISTOP) 401,402
401 XCCJJ)sCX(JJ)
ZC(JJ)sCZCJJ)
CBCJJ)sBC(JJ)
SVOL(JJ)*SVCL(JJ)/(PAI*CB(JJ)**2*THICK(JJ))
GO TO 190
402 CONTINUE.
DO 147 J»1,J1
IF(Y2(J) ,GE, POSITM(JJ)) 148,147
147 CONTIMUE
148 RATIO s(P(.iSITN(JJ)-Y2(J-l))/DY(J"l)
SUMKYlsSCKYCJ-n*CSCKY(J)-SCKY(J-l))*RATIO
SUMUA1BSUA{J«1)+CSUACJ)-SUA(J«1))*RATIO
IF(WW(KK)*(T(ISTOP)-T(JJ)) ,GE, OIST) 152,155
152 THICK(JJ)sSQRT(THICK(JJ)**2*2«,*(SCKY(jn»-8UMKYl)*WlNV)
A75
-------
XC(Jkn=C.X( J.I) + (3UA( J1 1-S
lJM^Al)*
,06*ALA
iGf, OIST+,5*THICK(JJJ)
THF rtnun HITS
isi xcuj)sc:x(jj)
ZC(JJ)sCZ(JJ5
CBCJJ)SHC(JJ)
1U9 F(1)*F(1)+SVOL(JJ)
K2)=FC2)+SVUL(JJ)*XC(JJ)
FC3)=F m*SVOL(JJ)*ZC(JJ)
F()*F(5)+SVOL(JJ)*(ZC(JJ)**2+,2b*C8(JJ)**2)
F(fe33F(fc)+SVnL(JJ)*XC(JJ)*ZC(JJ)
SVDL(JJ)=0.
THOSt PEACHF.D THE BtO AKF OFPOSITED
GP TO 1QO
KATlO=fcMTBFDXTHICK(JJ)
F(1)SF(1)+RASVCL
Ff2)sF(2)+RASVOL *XC(JJ)
*ZC(JJ)
*(XC(JJ)**2+,2S*CB(JJ)**2)
F(S)=F(5)+RASVOL *(ZC(JJ)**2+,25*f.B(JJ)**2)
P(6)sF(6)+RASVQL *XC(JJ)*ZC(JJ)
SVDL(JJ)sSVOL(JJ)/(PAI*CB(JJ)**2*THlCK(JJ))
THKK(JJ)s(l.-PATin)*THICK(JJ)
POSITN(JJ)SH-.501*THICK(JJ)
SAVE THOSE NOT REACHED BOTTOM FOR LONG TERM DIf-FUSlON
GO TO 190
153 POSlTK'(JJ)sPf3SITN(JJ)+WW(KK)*(T(ISTOP)»TCJJ))
DO 155 KSJ,JI
IF(Y2(K) ,GE, P03ITNCJJ))
155 CONTINUE
156 RATin
ZC(JJ)aCZ(JJ)*(SL'M»*BC(JJ)«(CY(JJ"l)-,375*BC(JJ«n
, *w«(KK)*(T(JJ)«T(JJ-l)})
GO TO 176
175 1F(JJ ,GT, IBFO .AND. JJ ,LE. U.EAVF.) 159,lfcO
c THE CLOUD HITS BOTTOM
159 THICK (JJ)*CY(JJ)-lf 25* AA(JJ).(CY(JJ»l)t.lt25*AA(JJ»l)
PnSITN(JJ)aCY(JJ)-1.25*AA(JJ)-,S*THICK(JJ}
GO TO 176
160 THICK (JJ)sCY(JJ)»AA(JJ)*tCY(JJ.[)-AA(JJ»t)-).Wtv(KK)*CTf JJ)-T(JJ-J)))
POSITN(jJ)aCY(JJD«AA(.JJ)-.S*THICK(Jj)
-------
176 DlSTnPOSITN(JJ)
IMDISU.5*THICK(JJ) ,LE, 0.) 60 TO 171
IFCDIST .ST. ,5*7HICK(JJ>) GO TO 410
EMTSURFtt ,5*THICK(JJ)*DIST
RATIOENTSURP /THICK(JJ3
RASVQLsRATIU*SVOLCJJ)
F(2)*F(2)+RASVCL *CX(JJ)
F(3)BF(3J+RASVOL *CZ{JJ)
F(U)EF(a)+RASVOL *
F(fc)sF(fe)*RASVOL *CX(JJ)*CZ(JJ)
SVCllCjJ)s(l,.f»ATIO)*SVOU(JJ)
TH1CK(JJ)«(1.-RA7!0)*THICK(JJ)
POSI7N(JJ)B ,501*7HICK(JJ)
OISTs,501*7HlCK(JJ)
«10 IF(JJ ,F.Q. ISTDP) «11,«12
ail Xf (JJ)aCX(JJ)
ZC(JJ)»CZ(JJ)
CB(JJ)*BC(JJ)
SVOLCJJ5*SVOL(JJ)/(PAI*CB(JJ)**2*TH!CK(JJ})
GO 70 190
«12 CON7IMUE
DO 177 J«1»J1
IKY2(J) .GF. POSITN(JJ)} 178,177
177 COMTI^UF
178 RATIO s{PDSITN(JJ)-Y2(J-l))/DY(J*n
SUMUAlsSUA(J«l)+(3UA(J)»SUA(J.l))*RATIO
SUMWAl»SWA(J«l)+(SWA(J)-5WAt J«1))*RATIO
IF(-WW(KK)*(T(ISTOP)«T(JJ)) ,RE, DIST) 182,183
182 THICK (JJ)*SO»T( THICK (JJ}**2»2flt*SUMKYl*^lMV)
XC(JJ)aCX(JJ)- SUMUA1 *WINV
ZC(JJ3=CZ(JJ)« SUHwAl *«TMV
C8 ( JJ) a f8C(JJ)**. 666666-1. 06* ALAMf>A*DIST*l*INV)** 1,5
IF(-W»/(KK)*(T(ISTOP)-T(JJ)) ,GE. DIST* ."SuTHlCK ( JJ) ) 179, 18»
171 XCCJJ)SCX(JJ)
ZC(JJ)»CZ(JJ)
CB(JJ)sBCCJJ)
179 C^^TI^^UE
F Cl)sF(l)*3VOL(JJ)
f C?)aF(2)+SVQL(JJ)*XC(JJ)
F (3)«F(3)+SVOL(JJ)*ZC(JJ)
F (iOeF(«)+SVOL(JJ)*(XC(JJ)**2+.25*CBCJJ)**2)
F (5)*F(5)+SVDL(JJ)*(ZCCJJ)**2+,?5*CB(JJ)**2)
F (fc)sF(6)*SVOL(JJ)*XC(JJ)*ZC(JJ)
SVOL(JJ)=0,
TwOSF REACHED 7HE SURFACE ARE CONSIDERED TO STAY THERF.
GO TO 190
lea ENTSURFe-Wh(KK)*(T(ISTaP)-T(JJ)3»DIST+,b*THlCK(JJ)
R^TIQ sEMTSURF/THICKfJJ)
RASVOL «RATIO*SVOL(JJ)
F (l)ep(l) + RASVOt.
F (?)«tFf2)+«ASVOL *XC(JJ3
F (3)8F(3)+RASVnt *ZC(JJ)
F (aj«F(a)+RASVt)L *(XC(JJ)**2+.25*CB{JJ)**2)
F (5)sF (5)+RASVOL * (ZC ( JJ) **2*,25*CB(.U ) **2)
F (6)sF(6) + RASVfJL *XC ( J J) *ZC ( J J)
THOSE REACHFD THE SURFACF IS CONSIDERED TO STAY THFRE"
SVOl. (JJ)sSVOL(JJ)/(PAl*C8(JJ)**2*THlCK(JJ)J
A77
-------
THICK(JJ)=(1.»RATIO)*THICK(JJ)
POSITN(JJ)=.50!*7HICK(JJ)
SAVF THOSF NOT REACHED SURFACE FOR t.ns,G TERM DIFFUSION
GO TCI 190
183 PPSITN(JJ)sPOSITM(JJ)+WW(KK)*(T(lSTOP3»T(JJ))
DO 165 K = 1,J
IFCY2CK) ,GE. POSIWJJ)) 186,185
185 CONTINUE
186 RATIOe(POSITN(JJ)"Y?(K«l ))/DYCK-.l)
SUMWA?sSWA(K»l)*(SVU(K)«»SWA(K"i))*RATIl)
THICK (JJ)sSSRT( THICK (JJ)**2+2tt,*(SUMKY2»SUMKYl}*W!NV)
IF(»WMKK)*(T(ISTOP)»T(JJ)) .GE, OIST* ,5*THICK( JJ) ) 184,187
187 SVDL(JJ)sSVCL(JJ)/(PAI*C6(JJ>**2*THICK(JJ))
GO TO 190
190 CONTINUE
00 192 JJslSTART,ISTOP
1F(SVOL(JJ3 ,LE, 0.) GO TO 192
CALL INDATAI
192 CONTINUE
IFCISTGP.EQ.ISTEP) GO TO 12346
KJI*IJK/NOUTPT
XIJKzIJK
XXXsXIJK/K'OUTPT
IFfABS(XXX-KJI),LT.l,f»10) 1 2346,1 2.545
12346 CALL OUTPT
IF(IPLT.eo.O,AND.NOHOW,EQ.1) GO TU 1000
12345 CONTINUE
CALL SHIFT
CALL GRIOSA
CALL IMABCD
CALL SUMARCD
CALL INSOL
IFCISTOP .EO, ISTEP) 195,196
GO TO 139
195 CONTINUE
COTsOTO/IBUG
lF(kh(KKn 257,258 ,257
257 CALL CU^VKT2
CALL ADD
GO TO 259
258 IF(KEY3 .EG. J) GO TO 1001
CALL COK)VRT2
OH 256 J=1,J1
OH 256 1=1,6
259 TTeT(ISTfcP)
CALL OUTPT
CALL INABCD
2 FORMAT(lHJ,loX,y?H**lMP(iT TO LONG TERM DIFFUSION COMPLETED** ,/,10
IF(IPLT.FO.OtANiO,NOMOR,EO,i) GH TO 1QOO
A78
-------
NOTE*0
255 CALL GRIOSB
CALL INABCD
CALL HATXCO
CALL INSOI.
CALL SHIFT
OLDTOTOAMO + FU)
DO 300 !Jsi,NGR!D
CALL SOI.UTN
TTsTT+DT
300 CONTIKiUt
ZAMQsO,
DO 999 Isl,J2
999 ZAMOs2AMQ>(Ml»i
ZEWTOTsZAMO + FU)
IF(ZfWTOT.LE,Of) GO TO 992
KTOT*OlDTOT/ZfUTOT
00 998 1*1,Jl
998 Etl»I)*E(l»I)*»TnT
F(U»F(t>*RTOT
992 COWTIK'UF
CALL OUTPT
!F(MOMOR,EO,1) GO TO 1000
IF(TT ,GE, TSTOP) GO TO 1000
IF (WW(KKmiO,3?0 >310
310 IF(AMO,LT.ORIGIN) 60 TO 1000
GO TO 255
3?0 VMAXsOe
VMlNslOOOOOO,
DO 340 1*1,Jl
IF(E(1,I) ,LE. 1,OOOE-»1S) GO TO 340
COKC*E(l»n/VAR(I)
IF (VMAX ,LT, CONC) VMAXaCONC
IF< VMJKi ,GT, CONC) VHIK8CONC
3«0 CONTINUE
RATIOsVMlN'/VHAX
IFCKATIO .GE. 0,9b) GO TO 1000
GO TO 255
1000 CONTINUE
IFIMSHsl
CALL OUTPT
DO 1003 Isl,IST£P
CX(I)sCXCI)+XO
1003 CZUJsCZCn
1001 CONTINUE
RETURN
END
vv
A79
-------
SUBROUTINE
COMMOhJ/SHtAKE./AVtDTU
XB/DlNC«l,OI'lM:P<15n
/AHCD/ Y( 30)>WCA( 30 ) , YK 1 , Y*2? Vk 3, YK4 , AK Y 1 , AK Y2 , AKY3,
COMMON /COMPUT/FP (8,
CCMAXC151),VAR£iSi)
/THRS/ DT,OTJ,bT2/COT,WGRIOiNGKJ01,NGKI02,KGKID3,AMCVT5
/CONST/6, PA I, ALPHAO,ALPH A, ALPH AC, ALPHA i
CX(600),CY(600)»THETA2(600)fU(600),
, SZtTA(600),CZfcTA(600)f AA (600), DTT(fcOO), 01(600),
, CZ(bOO),SS(8f 600),
f SVOL(600),XC(600)f ZC(600),POSITN(600), T C600) , THICK (600 5 ,
, SUA(aoO),S*A(200),SCKY(200),CON
-------
TCTSTtP)=TUSTE.PH.5*DL(ISTtP)/U(ISTfcP>
CX(IS7FP)sCXUSTEP)*.S*(CX
-------
IFCINDEX.EQ.ISTFP) GO TO U14
DO U12 JJsIlvDEX,ISTFP
SVOL(JJ)«PAI*/tA(JJ)*BC(JJ)*DLCjJJ*SS(KK,JJ)
412 CONTINUE
DO «13 JJ*INDEX,ISTEPNi
IF(SVOLCJJ) ilE, QRIGIN)SVOL(JJ)»O
CONTINUE
«l« CONTINUE
SVni.CISTEP)«0,
DO U20 JJ«1,ISTF.P
IF(SVOICJJ) ,GT. 0,)
420 CONTINUE
JJsISTEP
TT«T(JJ3
ISTARTsJJ
T1*T(JJ)
T?ET(I3TEP)
T3BTIHE+T1
TUBTIME+T2
IF(JJ ,FO. ISTF.P) GO TO 250
IF(TPinT/TGRID,&T,100,} T6RIO»TPLQT*,0 J
TSTARTeTCJJ)
IF(IPLT,EO,0) GO TO
NDUTPT*MCAKE/JPLT
GO TO 123«8
123U8 CONTIMUF
IF (NDUTPT ,LE, 0)NOUTPTBl
fcOO CONTINUE
IJKsIJK+1
,GT.
,tT. T3) GO TO U2fe
DO i)28 JJsISTARTiISTFP
IF(TEND ,LF. TlME+T(JJ))a29,«28
U28 CONTINUE
GO TO U26
00 a2« JJsIBEGIN,I3TfP
IF (TFND ,LF, T C J J) ) «25,A2«
CONTINUE
425 TFNOeTCJJ)
CONTINUE
DTsTEK'D-TSTART
NNssIBUG*DT/DTO
IF(NN ,LE. O
DT*DT/NN
IFdJK ,£0, J)CO TO 473
CAUL MATXCO
DO U7Q 11*1, NN
DO
9871 OAMOBOAMO+(E(l»I)+FCl,l4-l))*OY(n*,5
A82
-------
OTOTsOAMO+FfU
IFCIJK ,FO. 1) 471,47?
472 CALL SOIUTM
DO 9872 1=1, J2
9872 ZAMPsZAMD+Ct CUimtl, ! + !))* DY(J)*, 5
ZTOT=ZAMfHF(l)
IFCZTOT ,LE, 0,)GO TO 9873
RTtlTsOTUT/ZTOT
DO 9874 1=1, Jl
9870 EU»n*F
-------
IF (Ufc(KK) .LT. 0.) GO in 500
IF(MFTHfiO ,fO, 3) GO TO
IFfJJ .If. IfcDEX)«at,/.|C(.lJ)sCONC(JJ)/(2t*BC(JJ)*DL(JJ)*THICK(JJ))
(iO TO U30
,J = CX{JJ)-(OT4»0,ri*DT3)*UPX
CZZ.fJJsCZ(J.l)-(OTa-0,5*DT3)*WBZ
F(1 )*F(1)+RASVOL
F(2)=F(2)+RASVOL*CXXJJJ
F(3)sF(3)+RASVOL*CZZJJJ
F(a)sF(^J)*HASVCL*(CXXvlJJ** 2+ (Dl(JJ)*CZtTA(JJ))**2*. 083333
. +CBC(JJ)*SZtTACjJ))**2*,333333)
F{5)rF(5)*RASVQL*(CZZJJJ**<'+(DL(JJ)*SZETACJJ))**2*,083333
, +(BC(JJ)*CZFTA(JJ))**2*. 333333)
F(fe)sr(6)+»ASVDt*(CXX,UJ*C7?JJJ+S2ETA(JJ)*CZETA(JJ)*
, (PL (JJ)**2*,oem3-BC(JJ)**2*. 333333))
CONC(JJ)=0,
GO TH 430
CaMC(JJ)=CDWC(JJ)/(2,*BC(JJ)*OL(JJ)*THItK(JJ) )
GO TO «30
SOO IK^ETHHO ,tO, 3) GO TO 5a5
IF(JJ ,LT, INCF.X)S«1 »545
501 POSITMf JJ)=CY(JJ)»SIMaHF.TA2(JjM*flC(JJ)+*n(KK)*DTa + ,5*TMICKf JJ)
GO TP 549
IFCJJ ,LT. U'DEX) 5«6,5«7
PPSITM(JJ)=CV(J.n-.37b*PC(JJ)4wlN(KK)*DT*t+f5*THICK(JJ)
GU TO 5U9
IFC'JJ ,PT. IBtO .AND, JJ .Lf. I LKAVt )5bl
GO TO 509
A 84
-------
PPSlTfcCjJ)*CY(JJ)»l,3*AA(JJ)+Kw(KK)*DT4+tS*THICK{JJ)
Cr.NTlNUF
CONC(JJ)sSVOL(JJ3*DT3
DISTBPOSITN(JJ)
iMflST *.5*THICK(JJ3 8L£. 0,) GO TO 550
IFCDIST ,GE» ,5*THICK(JJ)) GO TO 560
EMTSURFs,5*THlCK(JJ)-DIST
RATIDaFNTSURF/lHICK(JJ)
RASVOLSRATIO*CONC(JJ)
CXXJJJsCX(J,J)-(in4«O.S*D73* RATIO) *UBX
CZZJJjBCZ(JJ)»(OTa«0(!5*DT3*RATIC»**BZ
F(2)*F(?)+RA8VOL*CXXJJJ
F(3)*F(3)-»-RASvGL*CZZJJJ
FCa)cF(iJ)*RAS VOL *(CXXJJJ**?+(DL(JJ)*CZETA(JJ))**2*. 083333
, +(Rf(JJ)*SZfcTA(JJ))**2*,333333)
FC 53 sF(S)+P AS VOU*(CZZJJJ**2+(DL(JJ)*SZtTACJJ))**2*. 083333
» +(BC(JJ)*CZETACJJ)3**2*f333333)
F(63eF(fr)+RASVCU*(CXXJJJ*CZ7JJJ*SZETA( JJ}*CZF7
. C[)L(JJ)**2*.083333-BCCJJ)** 2*. 33333:5))
COKiC(JJ)s(J .-RATIQ)*CO^C(,!J)
THlCK(JJ)s(i,-HATIO)*THICK(JJ)
PHSITK'CJJ)* ,501*THICK(JJ)
CONC(JJ)scr.lNC(JJ)/(2,*BC(JJ)*DL(Jxl)*THlCK(JJ)3
RO TO «30
550 RASVOL*CONC(JJ)
CXXJJ,!sCX(JJ)-(OTa»0,5*DT3)*UBX
F(3)»F(3)*RASVnL*CZZJ>lJ
F{a)sKii)+RASVPL*(CXXjJJ**2+(DLCJJ)*CZETA(JJ)3**2*.083333
, +(BC(.fJ3*SZETA(JJ)3**2*,333333)
F(5)eF(5)+RASVOl*l(JJ)*SZF.TACJJ))**2*,083333
, +CflC(JJ)*CZFTA(JJ) )**?*. 3333333
F(fa3e^(6)t«ASVOL*(CXXJJJ*CZ2JJJ+SZtTA(JJ3*CZETA(JJ3*
, (DL(JJ)**2*,083333"BC(JJ3**2*,333333)3
GO TO
560 CnN
i)30 CONTINUE
DO 465 JJ«ISTART,IEMD
IF(CONC(JJ3 ,LE, 0,3 T,U TO C65
CALL INDATA2
U65 Cd^TI^UE
CALL GRJDSA
CALL INAbCD
CALL INSOI.
GF .Til) GO TO 123U6
KJIsUK/NOUTPT
XIJKSIJK
XXXSXJJK/KOUTPT
IF(ABS(XXX«KJI)ftT.l.E.-lO) 12 3« b, 1
123a6 CALL OUTPT
IFdPLT.BQ.O.AK'O.NQMCR.EQ.l) GO TU 1000
CGNTIMUF
CALL SHIFT
IF(TEMO ,GE, T«) GO TO 480
A85
-------
IFHSTART ,GT, T3) U75,600
UTS DO U76 JjslSTART»lSTEP
IF(TIM£ +TCJJ) ,GE, TSTART)
476 CONTIMIF
477 ISTARTsJJ
GO TO 600
?50 COWTINUF
1FCYY ,Lt. YKl)YKKsAKYl
IFCYY ,GT, YKJ .AMD, YY ,LT, YK? ) YKKs ,5* C AK Y 1 + AK Y2 }
IF(YY ,GF.t YK2 .AND. YY ,LE.» YK3)YKKsAKY2
IFCYY ,KT, YK3 .AND. YY ,LT, YK4 ) YKKs,5* C AK Y2+ AK Y3)
JFfYY ,GE. YKU) YKKSAKY3
IF(NGRID1 ,tt. 0)251,252
2S1 AHCVT3e(T««.T2)/DT2
CALl
DO 750
750 Yl (
OH 760 1*1, b
DCl 760 jsl,Jl
760 eCI.JJsFEtlf J)
TT = TU
CALL OUTPT
1F(IPLT.EO.O()AND,NOHUR.EQ,1) GO TO 1000
CALL INABCD
GO TO HBO
252 AMCVT3=DT/DT2
CALL CONVRT3
DH 770 1=1, Jl
770 Yi(I)eY2U)
DP 780 1=1 »6
DO 780 jsj,jl
780 F. CJ,J)stT(!,J)
CALL OUTPT
IFCIPLT.tG.O.AND.NOMOR.EO.l) GO TO 1000
IFC^GPIOUNF, 1) GO TO 253
AKf.VT3s(TIMF«DT)/OT2
CALL
CALL MATXCG
CALL SOL.UTM
CALL SHIFT
CALL CO^VRT3
CALL ADD
CALL OUTPT
IFCIPl T.EO.O.ANO.NOHOR.f 0,1) GO TO 1000
GO TO 080
253 CONTINUE
CALL GH10SA
CALL INABCD
CALL INSOL
CALL MATXCO
CALL SHIFT
AMCVT3=DT/DT2
00 254 JJ=2,NGRID1
A86
-------
CALL S0|. MTV
CALl
CALL
TT=TT+DT
I F (.1J , f Q , 1 0 * ( J J / 1 0 ) )
CALL OUTPT
If- (l^LT ,F ft, 0 , ANin, N^MHS 0 E Q 11 ) QH yn 1000
25a CONTINUE
DT=AMCVT3*DT£
CALL VATXC.O
CALL SOLUTN
CALL
CALL
CALL CHITPT
IF ( JPl T.ECO.O , ANO.NO^O*'.FG, 1 ) GO TO 1000
tt 8 0 C 0 K" T I NI L) F
"RITE(6,?) TT
1H1, 10X,i)2H**IMPUT TO LOMT, TER^ DIFFUSION CO^PLF. Tf D** ///10
K'OTF OF COMPUTING SPRf-.ADO ftf.FORE UO^G TFWM DIFFUSION
CDT=DTO/IBUR
NGRIPSNGRIP3
CALL
CALL
CALL HATXCD
CHI TMSOt
CALL SHIFT
DO 300 IJsi,N)GRID
CALL SOLUTE
TTsTT+DT
COMTINUF
ZAMf.lsO,
DO 9877 Ir|,j2
9877 ZAMpBZAMO+(F.(l fIJ
ZTOTsZAMO + FM)
IPfZTHT ,LE. 0,)G
RTOTsOTUT/ZTDT
DO 9879 lsj,jl
9879 F(i,nsfc(l,i
9fl78 CDNTIN'UF
CAUL OUTPT
lF(MfjMQR,fcO,l) GO TO 1000
IF(TT ,Gh, TSTOP) GO TO 1000
IF (Wi*-'(KK))310,320 .310
310 IF(AMO ,Lt, ORIGIN) GO TO 1000
GO TO 25b
320 V^AXsO,
VMJN*1000000,
DO 3UO I=1,J1
IF(E(1,I) ,LF, 1.0006*15) GO TO
CONCsEd »I)/VAR( J)
IF (VM&X ,t.T, COMC)
:N ,GT. CONC)
A87
-------
IfCRATin ,GE, 0,95) GO TO 1000
GO TO 255
teat CONTINUE
CALL DUTPT
00 1003 I*i,ISTEP
CX(I}«CX(I)*XO
CZ(I>»CZ(IHZO
RETURN
fNP
wv
A88
-------
V-ATXCO
c
C COMPUTE THE FLEMfNTS OF A TR I -D I AGHN AL K
C
COMMON XAPCD/ YC 30),KOA( 30 5 , YK1 , YK2, YKJ, YKil , AK Y 1 , AK Y2, AKY3,
, YU,Yh.YtfH,UAO,WAO,N,CKY(15i),UAU51),WA(l51)
COMMON /COMPUT/EE(8,151),
. fc(8,lbl),YlClbn,Y2(lM),DY(150),FC8),FW(fi),WtK(9),
MAX
COMMON /MATRIX/ AUM), 8(151), CU51),CPH51)
COMMON /CONST/G,PAI,ALPHAO, ALPHA, ALPHAC,ALPHA1, ALPHA2,
, ALPHA3,ALPHA«,GAMA,CDRAG,CFRIC»BETA,CM,CD,C01,CD2,F'RICTM,F1
, ,ALFAl,ALFA2,GAMAl,GAMA2,ALAMOA,CD3,COa
COMMON /TORS/ OT,DTl,OT2»CDT,NGRH),NGRIf)l,NGRID2rMGR10.1J,NCO
COMMON /TRASFtR/ KK,JJ,TT,ISIZF,NOTE,AO,IY,MOTRN,lNDex,IBED,ILEAV6
, , IJK,FBtD,AMO,OL^T,MrTHOD, TSTHP
C
C
J1=ISIZE
J2=Jl«l
DODDTel ,/DT
IF(wh(KK))
'300 K 1 s l
K
-------
DY1=1./DY(I)
DYinrl,/(DY(I-l
A(!)= CKYU»1)*DYI1*DYI1I
+K2*rtKDY
eCI)=-(rDDDT+(CKY(I)*OY(I-l HCKYCI»n*DY(n)*DY!i*DYI*OY!lI
A(Jl)sCKY(J?)/DY(J2)**?
+,5*(KP-K1)*WW(KK)/((KUK2J*DY(J2))
B(Jl)s.(ODDDT+ CKY(J2)/DY(J2)**2
, •(l,»ALFA?)*W*.tKK)/DY(J?)
,
200
C(Jl)sO,
DH 220 Jel,J2
DFNOB8CJ+l)-A(J+i)*CP(J)
220 rP(J+l)«C(J*t)/OFNO
HETURM
FMD
vv
A90
-------
c
C SOLUTION BY THOMAS A
C
MNDN DH(lbl) , n(iSl) , SOLdbt )
XABCU/ Y( 30),RPA( 30 ) , YK 1 , YK 2, YK 5 , YK n , A K Y 1 , AK Y^ , AK Y
E(fl,lbl),YlClbl)»Y2(15n»DY(150),K«),F»((i3),wW(9),
CCMAXC1«51 ),VARC151 )
CX(600),CY(600) ,AA ( 60 0 ) , U ( 60 0 ) ,
V(600), ALOW(600) ,VMBT(6001 ,DENOIK600),6NTRC(*(«>005i
CZ(600),SS(8, 600),
SVOL(600),XC(600J ,ZC(600),POSITN(fcOO),CHt/SOfl),THlCK(f>00) ,
BC(600) ,wS(a
RfJASf«) , T
UBXfWB2fWDAF,C^AXCc')f YY
/TGHS/ DT,DT!,nT2,COT,^GRIO,WGPID! , NRRI D?^-{,«1D1, N.CO
/MATRIX/ ACISD ,R(isn »cdbi) ,CP(ibi )
/CONS T/G,PAI,ALPHAO, AL^HA, AUPHAC,ALPH At, ALP HA?,
ALPHAI, ALPHAS, GAMA, CORA G,CFRIC:,BFTA,CW,CD,CPI, en?, f-ff c TN,FI
/TRASFtR/ KK , J J, T T , ISI ZF , NOTE , A 0 , 1 Y , NiOTRN , I K-DF X , TBfeO, ILE
T STOP
BUXsUBX
IF(TT .LT. TPLPT) GO TO «00«
BUX=0,
400U
J 1 = I S I Z (
J2SJ1-!
DDDDTsl ,/DT
hswu ( KK )
DO 993 Jsl,Jl
F.(7,J)sEC»»J)
993 F(8f J)sF(b.J)
F (7)»F(a)
F(8)*Ff.5)
DO 99ft 1*1,8
VAR(n=F(T3
00 998 J«=1,J1
998 EE(I,J)at(I,J)
IF(w> 995,996,997
995 Ki=l
K2 = 0
GO TO 999
996 Kl=l
GO TO 994
997 KlsO
K2sl
SULUTION rnw POSITIVE FALL VEL,
994 DO 1000 1=1,8
GO TO (1,2,3,4,5,6,7,9),!
1 DP 101 J*1,J1
101 DCJ)=0.
A91
-------
G(- TO 200
00 102 J=1 ,J1
D(J)s-UACJ)*(Mi,J)+EE(l,J))*0,5
GO TO 200
3 DO 103 J=1,J1
103 D(J)s.wA(J)*(E(l,JHFt*O.S
F-if:(ns-H^2*(F- (n+VAW(l))*,b
GO TO 200
4 DO 104 J=1,J1
lOtt D(J)=-(UA(J)*(E(2, J)+FF(2, J))fDISPC(ALAMDA, J)*(E(1 ,J
F*(I)s-RUX*(F(2)+VARC2))
GO TO 200
5 DO lOb J*1,J1
105 0(J)s.(WA(J)*(EC3-,J)*E.E(3,jmi>ISPCCALAf'DA,J)ME(l,J
Fw(l)=-BWZ*CF (3)+VAP(3)3
GO TO 200
fc CD 106 J=1,J1
106 DCJ)*-(UA(J)*(fc(3»J)+fcF.(3,J))+wA(J)Mt(2,J)+tf-(2,J)))*0,S
F^(I)=-(8UX*(F(3)»VA»(3))+Bi«Z*(F{2}+VAH(?)))*,s
GP TO 200
7 DO 107 jsl.Jl
107 0(J)=-(U»fJ)*(E(2,J)+t.K2,J))+OISPCC(ALAMOA,J)*(t(l»J)+bE(l,jn)
Fw{I)e.PUX*(F(2)+VAR(?))
GO TO, 200
8 DO 108 J=l,Jl
108 D(J)s-(WA(J)*(E(3»J)+tt(3»J) )+DISPCC(ALAMDA,J)*(t ( 1, J ) +h fc ( 1 t J ) ) )
200 F(I) sFd)* (F*(mA|_FAJ>*W*e (J,JlJ-GAMAi>*F(I))*DT
,«»(DDDDT-CKYC1)/DY(1)**?-(K2»K1 )*,•}**/( (K 1 +K?) *DV U ))
, .K/DY(J)-K2*W**2/((Kl+K2)*CKY(l)))*t(T»n
00 210 J=2,J2
DYJsl,/OY(J)
DYJ1J=1 ./(DYCJ)+OY(J-1))
210 D(J)=D(,!)-( CKYCJ)*DYJ*DYJ1J
>"CDDODT -
,
.-( CKY(J-1)*DYJ1*OYJKT
, 4K?*^DY
D(Jl)BD(Jn-(CKY(J2)/nY(J2)**2
, +.5*(K?-Kl)*^/C(K
-CKY(J2)/I)Y(J25**2
*(l,»Al FA2)*K/DY(J2)
-KJ*(1 ,»ALFA2)*w**2/((Kl+K23*CKY(J2) ))*F (I, J1)
-(1./OY(J2)-K1*K-/((K1+K2)*CKY(J?)))*GAK.A2*(F(I)+VAPCI))
DO 220 J=1,J2
DENO=B(J*1 )»A(J+1)*CP(J)
220 r>p(j + l)s(0(J+l)-A(J+l
SnL(Jl)=OP(Jl)
230 SPL(JP)=DPCJP)"CP(JP)*Snu(JP+l
on 300 j=ifji
300 fc U,J)=SOL (J)
A92
-------
GO TO 1001
SnU'TJOK FOk KFGATIvE FALL VFL.
Of 10000 1 = 1,6
GO T(i f 10,r>0,.SO,UO,SO,hP,70,80) , 1
10 F*(I)=0,
Dn 1010 JslrJl
1010 r>(j)=o,
GO in 20 on
20
bO
nt 1020 J=i,Ji
1020 DtJ)=-UA(J)ME
GP TO ?noo
30 Fw(I)rwAtl)*(F
r»f 1030 Jsi.Ji
1030 DCJ5s-w*(J)*(F
GO TL' 2000
DC 10«0 Jel,Jl
[)(J)s-(UA(J)*(
GO TO 2000
Fw{!) = l.*(WA(l
DO J050 J=1,J1
GO Tfl 2000
60 FMI)s(L'Ad )*(
DO 1060 J=1,J1
1060 D(j)s*(uA(j)*(
GO TO 2000
70 FMJ) = aiA(l)*(
DP 1070 jsl,Jl
1070 &Cj)=-(uA(J)*(
GO TO ?000
80 Fw(n = l,*
00 1080 J
1060 D(J)B»i f*
/ )
2000 Ftl) sp.-(I
(1,J))*O.S
J)
2)})*0,5
-Cl.-ALFAl)*K2*u**2/CKY(l))*t(I»l)
no 2100 J=2,J2
DYJsJ ,/DY(J)
DYJ1J=1,/(DY(J)+OYCJM))
2100 DfJ]so(J)-( CKYCJ)*DYJ*DYJ1J
-K1*WDY )*F(I^J+1)
'CDDODT - (CKY(J-n*OY(J) + CKY(J)*DY(J-l) ) * D Y J * D YJ 1 *D Y J U
CKY(J-1)*DYJ1*OYJ1J
*E(I»J2)
,5*(«2»Kl )*w/((Kl+K2)*DY( J2) )
rKi*w**2/((Kl+K2)*CKY(J2)))*E(I»Jl)
A93
-------
00 2200 Jsl,J2
DENOsB(J+l)-A(J+l)*CP(J)
2200 DP(J+l)=CD(J+l)-A(J+l)*nP(J))/OFMO
DO 2300 J»1»J2
2300 SOL(JP)=DP(JP)-CP(JP)*SOL(JP+1)
00 5000 J=1,J1
3000 t(I/J)=SOL(J)
10000 CDNTlKiUE
1001 CUMIMUF
00 1002 JaliJi
E(4,J>*.5*(F«l,.J>*E<7,J)>
1002 E(5» J)s.b*(F{b
RETURN
END
vv
A94
-------
SUBROUTINE GRIDSA
C
C COHPDTt THE GRID SIZF ACCORDING TO THE SPRMDIK.G OF THt CLOUD
*..
COMMON /COMPUT/tfc(8,151),
, E(fl,151),Yld51),Y2d$n»DYd50),F(tt),Fw(a),|«w{9>,
t CCMAXdSl) ,VARdSl )
CPMKON /TORS/ DT,DT1,OT2,CDT,NGRIO,NGRICM,NGRID2,MGRI03,AMCVT3
COMMON /CONST/G,PAI,ALPHAO,ALPHA, A.LPHAC,ALPHAI, ALPHA2,
, ALPHAS, AIPHA4,GAMA,CDRAG,CFRIC»BF.TA, CM, CO, CDJ»CD2,FRICTN,F1
, »ALFA1,ALFA2,(;AMA1,GAMA2,AL.AMOA,CD3,CD«
COMMON /ABCD/ Y( 30),ROA( 30} , YKl , YK2 , YK3, YKU , AKY 1 » AKY? , AK Y3 ,
» YUfYWfYF.Hrl.UO/WAOfN^CKYdSnfUAdSn^WAdBl)
/TPASFFR/ KK , J J, TT , I SI ZE , NOTE , AO , 1 Y , NOTPi^, TNDEX , I 8FD, ILF- AV?
J3=U*J2/10
AMO=0.
A M 1 s 0 ,
AM2=0,
DO 170 J=
170
IFCAMO .EG, O.DGO TO 230
IF(YL ,LE. 0,)YL=0,
YHsYBAR+SPRFAD
IF(YH .RE. H)YHSH
SPREADS, 5*{YH-YL)
YfiARsYL + SPRF.AD
IK^W(KK) ,LT, 0,)GU TD 5bO
!P(H -(YHAR + 8PRe'AO + HW(KK)*NGRID*DT) ,LT. 0,)SOO,600
500 DHs(H.(YBAR«SPRtAD))/J3
GO TO 650
550 IF(YBA»-SPREAD*wwCKK) *WGW!0*DT ,LT, 0,3560,600
560 DHs(YBAR+SPREAD)/J3
GO TO 650
600 DHs(2.*SPf>FAD+ABS(M(KK))*NGRlO*PT)/J3
650 ODDHSI./OH
IF (WW(KK) ,LT,0.) GO TO 300
N2=(H.(Y8AR + SPRtAD +h^ ( KK ) *MGR I 0
IF (M2 ,LT, 0) M2=0
Nl=(YBAR"SPRfcAD )*DOOH
I F { N 1 . 1. E , 0) NisQ
NNSN1+M2
IF (MM ,LT, J«) GO TO 210
MJsKl* J4/KN
IF (M2 ,FO, 0) GO TG 171
AO+WW ( KK ) *NGR I D*DT )-M2*OH ) *2 , / (M2*DH*
171 IF(M1 ,FO, 0) GO TO 400
AA sCYPAK-SPRtAD -M 1 *OH D *2 , / ( M 1 *DH* ( M 1 + 1 ) )
GO TO «00
A95
-------
300 N/l = (YBAR-SPRt AD * WM KK ) *MGR ID*OT ) *DDDH
IF (M ,U, 0} NisO
N2s(M..(Y8AR + SPRf AD ))*DDDh
IF(N2 ,Lfc. 0) N2*0
IF f*JM ,LT, J4) i;u TO 210
MlSNli* Jtt/NN
M2=j««Mi
If (Ml ,EQ? 0) GO TO 301
AA s(YBAR»SPREAD + *« (KK)*N(fRI D*»T«Ml *DH) *H,/ (MI *(>H* (MI * 1 ) )
301 IF(M? ,FO. 0) GO TO «00
BB s(H»( YBAR+SPRFAO )«M2*DH) *2 ./ («2*OH*
IF (Ml .EG.O) GO TO
DO 180 1*1^1
IJs^l-I+1
DY(1)=(IJ*AA+1)*OH
ieo
181 CONTINUE
DO 190 I»1»J3
190 CONTIMUf
IF (M2 ,EO, 03 GO TO 2
00
-------
SljRRCJUTINfc GRIDSB
COMMON /CPMPUT/tF(8,15n ,
KB,lbl),YlClbl},Y2(lSl),DYU50),F(8),FWCB>,lM
CC.MAX(151),VARUb'l)
COMMON /TOWS/ DT.DT1 , OT? , CDT, KTU
GO Td 26?
A97
-------
200 DTs,004*SPRfc.AD**2/VMAX
26? CONTINUF
IF(Wi-l(KK3 ,LT, 0,3GO TO
IF(H .(YBA» + SPREAD+wW(KK)*MGRir>*DT) ,LT, 0.3510,600
510 DHs(H.(YBAR-SPRFAD3)/J3
GO TU 650
550 lP(YB4»-SPREAD + -«/w(KK} *NGRID*DT ,LT, 0,3560,600
560 DHs(YBAR+SPREAD)/J3
GO TO 650
600 DHS(2.*SPREAD*ABSCW*(KK))*NGRID*DT)/J3
650 DDDhsi,/DH
IF (WW(KK) .LT.O.) GO TO 300
N2e(H»(YBAR+SPREAD +WW (KK 3*NGRID *OT))*ODDH
IF (N2 ,LT, 0) *2*0
M*(YBAR"SPREAO }*ODOH
IF (Ml ,i.E. 0) MSQ
M M s b 1 + M ^
IF (MM .IT. JO) (JO TO 310
M J
IF (M? ,f.Q. 0) GO TO 261
OH2B(H«(Ye/iR + SPRfc:AD +Ww (KK) *MGRID *DT )-H2*DH) *2,/
261 IF (MI ,EQ, 0) GO TO «00
OHle(YBAR-SPRFAD -Ml *OH) *2 , / ( M 1 *DH* (MU 1) )
GP TO 400
300 Nls(YBAP«SPRfcAD 4-WW ( KK ) *NGR ID*OT ) *ODOH
IF (Ml .IE, 0) Nl«0
IKN2 ,LE.
NNsNl+Nt2
IF (NN ,LT, Jfl) GO TU 310
IF (Ml ,EfJ, 03 GO TO 301
DHls(YBAR-3PRtAO +WW ( KK)*N6RID*DT«M1 *PH)*2 t
301 IF( M2 ,EQ. 0) GO TO aoo
AO 3 -H2*DH 3 *2 , / ( M2*OH* (
IF (Ml ,EQ,0) GO TO
DO MO I*l» Ml
DO
V2(Ml+i+I)=Y2(Hlf I3+OH
DY(M1+I)=DH
420 CONTINUE
IF (M2 .EQ, 0) GO TO 500
DO U30 I=l/ H2
)s(I *OH2+1)*DH
U30
GO TO 500
310 DHSH/JP
Y?(l)aO.
00 320 isl.jg
Y?(I+l)sY?(l3*DH
OY(I)sDH
A98
-------
•500
CONTINUE-
CONTINUE
RETURN
f MO
A99
-------
C
C iNTERpru ATP DATA u> SOLIDS FROM ro.-vEt.TtvF DESCENT PHASF OR
c DYNAMIC COLLAPSE RHASf- rn PR in vt SUITABLE. FOR LOG Tt»^ UT
c COMPUTATION
C
/c/xo,zo
F. (8,lM},Yl(l5n,Y?
CC.MAX(151)»VAR(t«51 )
/IPWS/ DT,01 1 ,nT2,C[>T,NGWlO/N6Hlr>l,NGHID2»NGKID.J,
- /C OK; ST/GfPAl,ALPHAO, ALPHA, ALPHAC,ALPHA1,&LPHA2,
CK(bOO),CY(600) , AA
, K(600).VQWT(600) ,O
C/(600),SSC», 600),
5 VOL (600) ,XC(600),ZC(hOO),MOSlTN(600),CH(feOO),7HlCKt600).
SUA(?OOJ,SwA(200),SCKYC200) , OUMY^ ( 60 0 ) ,
«) ,f(600) ,K,L,]PLUNfi,NUTRL.Ut3,SAI,TIMF,ISTFP, VF
UBX,wBZ,RQ4F,CMA)
-------
'Ai>
E(l,J-l)=t(l
XCCsXC(JJ)
ZCC=ZC(JJ)
E(2,J-l)=t(2
£.(4,>!)«£.(«> J"1)+I*«A1*(XCC **2K8CJJ5**2*.25)
t"(5, J-l )=F (5,,
f:'«J.J-i)sf(fefJ«'l)+RWAi*XCC *ZCC
E(t,J)sfc(l,J
E(3.J)st(3
fc: C«,J)st(a
E(5,J)sFCb,J)+RRA2*(ZCC
E (6, J)=E.(fcf J) + RRA2*XCC *ZCr
IF CMOPF .to, o ) GU TO l&o
PSNl=Yi(J)
150
IfaO
END
A101
-------
SUBROUTINE' INDATA2
c
C U'TFRPOLATF DATA ON SOLIDS f-KO1" PLUMfc, CR JfT TO GRID Yl FOR LUNG
C TEP* DIFFUSION
C
XC/XO,ZO
/COMPUT/FF(8,1M) ,
/Tf)HS/ DT»OTl
/CONST/G, PAI,AI_PHAO, HPHA, ALP n AC, ALPHAI, ALPHA?,
»ALFA1
OMKON CX(600),CY(600),THETA?(600),U(600),
SZfcTA{hOn),C2tTAC600),AA (600)» OTTC600), r>L
r.Z(600),SS(8, 600),
SVni(600),xe(600),ZC(hOO),POSITN(600),
SUA(200),SWA(2do)fSCKY(200),COMC (600),
RnAS(«),S(600),K,L,IPLUNGfiviUTRL»UB,SAI,TlME,ISTFP, VF ,
UPX,wB£,RUAF,CMAX(9),YY
/TPASFER/ KK, JJ.TT, ISIZF, MOTE, A O,IY,NOTRN,IN-REX,IBED,I LEAVE
D,AKO,nLOT
AKE/AVfDTU
J?sJl-l
PSNl=pnsjTN(JJ)-THICK(JJ)*.5
1FCPSN2 ,GT, YKJD)
00 100 1=1, Ji
IF (PSNJ-YKD) 110, 10n, 100
100 CfiNTIMlF
I=J1
110 no iso ,i = i, Ji
IF (PSM2-YlfJ)) 120,120,130
130
GO TO
120 MORF=0
IMJ .fcfJ. 2 .OR. J ,EQ. JI) 1U1,1^2
IF(J ,EQ, ?) l«3,iay
DtMl= CY1 (J)»CFNTtR)*( nY(J-D)
, +(CtNTER»Yl(J-l))*(DY(J-l)*nY(J))
GO in las
DFA'()= (Y1(J)»CFNTEK)*(OY(J-2)*OY(J-1))
, +(CFVTFK-YUJ«1))*(DY(J-1) )
GO TO 1US
DFNO= (Yl (J)«CENiTER)*(OY(J»2)+OY(J-l))
J-1 HDY(J) )
"Yl (J-l))/ntNO
AREAs2,*(3C(JJ)*!)L(JJ)
KP = CC.'NCt JJ)*AREA*(POS2-PSM1)
CJP4J =RR*A1
-------
CZZsCZ(J«J)»AVFDT4 *0
EM , J»1)*EU, J-O+RRA1
F(2, J-1)=E<2, J»l)+RRAl*CXX
FC3,J-1)3E(3,J-1>+RRA1*CZZ
EC6
Dl ( JJ)*CZET A (JJ))**2*, 083333
Dt. (JJ)*3ZET A (JJ) )**2* .083333
+(BCCJJ)*CZETA(JJ))**2*,333333)
J^1)=F(6, J»1)+RRA1*(CXX *CZZ +SZET A ( JJ)*CZETA( J J)
*(OL(JJ)**?*I083333-.fiC(JJD**2*,333333))
E(l,J)sE(l,J)+RRA2
f.(2,J)sF(2,J)+RRA2*CXX
fc (3» J)«E !C3,J)+RRA2*CZZ
F(«, J)sE (a,J)+RRA2*(CXX **2+ (DL ( JJ)*CZfcTA ( JJ) )**2*,083333
+(BC(JJ)*SZETA(JJ))**2*,333333)
f (5,J)«E(b,J)*RRA2*(CZZ **2+ (DL ( JJ)*SZET A (JJ))**2*. 063333
+(BC(JJ)*CZETA(JJ))**H*,333333)
F.C6, J)sE«b» J)+RRA2*(CXX *CZZ +SZET A ( J J) *CZET A ( J J) *
IF (MORE. ,EO,
PSNI=Y1 (J)
150 CONTT.NUF
160 CONTINUE
RETURN
END
0 ) GO TO
A103
-------
Nif- CGMRT2
c
C COK'VLRT DATA FROM DYNAMIC COLLAPSE TO DATA FUR LONG TER* DIFFUSION!
C
XC/XO,ZO
/cnMPUT/EF(8,lbl),
F.(8,151)fYl(lbl),Y2(151),OY(l50)»F(B),Fi«(8)jWK(9),
CC*AXU51)fVAR(l51}
COMMON /TORS/ OT»DTl,OT2,CDT,NGRID,NGPIni,Wf;RID2,NGR]:DS,NCO
COMMON /cnNST/G,PAI,ALPHAO, ALPHA, ALPH AC, ALPHA 1.ALPHA2,
ALPHAS, ALPHAS, G A MA, CDRAG/CF«IC»8tTA, CM, CO,CD1»C 02, FKICTN,F1
,ALFAl,ALFA2,GAMAi,GAMA2,ALAMDA,C05,COtt
OM CX(600),CYC600),4A ( 600 ) , U (600 ) ,
V(fcOO), w(600),VOHT(6005 ,PE!M01F(600),EMTRCO(600),
CZ<600),SSC8, 600),
SVnL(600),XC(600) , ZC(600),PUSITN(600),CB(fcOO) ,THICK(600),
8UA<200),SWA(200),SCKY(200),OUMY2(600),
Rf3AS(U)»T(600),K,L,IPLUNG,NUT«l.,UB,SAI,TIHE,ISTEP, VF
UBX,wBZ,ROAF,CHAX(9) fYY
/APCD/ Y( 50),ROA( 30J,YKl,YK2,YK3,YKa,AKYl,AKY?,AKY5,
COMMON) /TRASFfcR/ KK , J J, TT , ISI ZE, MTTE, AO , I Y , NUTRKi, INDFX, I BF.O, I LEA VE
, , UK, FBfcD,AKO,OLOT, METHOD,? STOP
C
C
OsAO
IF(YY*0 ,GE, H) YY=H"D
Ji=!S!ZF
J2SJ1-1
J3s2*J2/in
D IN'DICATFS THF MINOR AXIS np THE. CLOUD
I F ( N 1 , L F . 0 ) N 1 = 0
K^rPN2/nH
IF(K;2 ,LF. 0)N2sn
N>OSK'I +KI?
IKK'Ni ,LE. J5) GO TO 500
IF (MI ,e.o. 03 GQ Tn 90
AAs(PHl-Mi *[)H)*2./(Ml*
IF(M? ,EO. 0) GU TO <» 1
91 Y2(l)sO,
IK MI ,FQ. 0 5 GO TO ill
00 JOO T = l^'l
IJ="1-I+1
OY(I)=( AA*IJ+1)*OH
Y2(I*1 )=YP t D + DYCI )
100 CONTINUE
111 CPNTTMl'F
DO 200 1=1 ,ja
DY(
A104
-------
IF (M2 ,CO, 0) GO TO 301
OH 300 jsl,M2
= U *Hrt+l)*DH
300 CONTINUE
GO TO 301
500 OHSH/J2
Y2(l)=0.
DO SOI I=?,J1
501
301 C.nNCsCMAX (KK )
C2Z=CZ(ISTEP)
on aoo isi,ji
IF(ABS(Y2(I)-YY) ,GF, 0) GO TO «10
AREA sPAI*BC (1ST FP)**2*(1,-((YY-Y2(I))/0)**2)
HSQSARE.A/PAI
ffc (i ,I)=CDNC*ARF.A
EE(2.I)=EF(lrI)*CXX
EF.(«,I)=tE (1,1) *(CXX
EE(5,IJ=FF.(1,I) *(CZZ
EF.(fc,I)at:E(l,I)*CXX *CZZ
RO TO UOO
on a^o *HCY£i,6
EF (KKCY,I)=0,
CHNTIMUE
COMTIV'UF
KfcTURM
A105
-------
CON'VWT'i
c
C CONVERT DATA FROM FNJD Of- PLUME TO OATA HI* LO*G TERM
C
COMMON) /C/XO,ZO
COMMON /C.nMPiJT/Efc(8,lb'l)»
CCMAXUbl
/TORS/ DT.nTl,OT2,CDT,Njr,RIO,N;Gf<'IDl,'JGR!tV,NGPir>3>AMCVT3
/CON!ST/t;,PAl,AlPHAO, ALPHA, ALP HA C,AIPHA1,AIPHA2,
, , ALFA1 , AI..FAi?,GAMAl ,KAf'A;?, Al A MO A , T D J5 , CPU
COMMON; CXChOO ) ,CY (600) . THETA2 (600) , U(600 )»
, SZt"TA(600) fCZE'TA(600),A8 (600)» OTT(600)» OL
, C7(600),SSC8, 600),
, SVPL (600),XC(600),ZC(600),PUS1TN(600), T (600) , THICK { 600 ),
, SUAf?00),SWA(iJOO)»SCKY(200),XXXX (600),
, RC(600),'*S(«,2),
a) ,5(600) ,K,L,!PLUNG,NLITRL,UB,SAI,TIMf, ISTpP, VF ,
! /A8CD/ Y( 30),HOA( 30 ) , YK 1 , YK2, YK3, YK« , AKY 1 , AKY? , AK Y i ,
COMMON /T RASPER/ KK,JJ,TT,ISIZF.,NflTE,AO, IY,MGTRM,IMDEX,I6ECi,ILEAVF
, , ux, r Bf- o, A MO, OLOT, METHOD,? STOP
c
c
n=Ao
IF(VY+D .PF. H)
J 1 = I S I Z fc
D INDICATES Thf MINOR AXIS OF THE CLOUD
2sH-( YY+0)
NlcPNl /OH
I f ( N 1 . L E . 0 ) M a 0
,LE.
IFfNNJ .I.E. Jb) GO TO 500
Ml S J5*N1 /MM
M2S.I5-M1
IF (MI ,tQ, 0) GO TO 90
AAs(PM-Ml*r)H)*i»./(Ml*l)H*(Mm
90 IF(Mi> tfQ, 0) GO TO 91
91 Y2(l)sO.
1F( Ml ,tO. 0 ) GO TO ill
no 100 1=1, MI
IJsMl-T+1
nY(I)s(AA*TJ+l )*DH
YJ?(H-l)sY2(I)+DY(l)
100 CONTINUE
HI Cn^TI^Uf
DO 200 lsl,J«
DYU + M )aOH
A106
-------
200 CONTINUE
Tf-' (M2 ,EQ. 0} on TO 401
nr. 300 isi ,M2
300 CONTINUE
GO TO 301
bOO DHSH/J?
Y2(l)sO.
DO 501 1=2, Jl
501
301
OLLsA«CVT3*OL(ISTEf)
CXXscx(ISTEP)t,5*(OLL-DI. < ISTF.P) ) *CZtT A (
CZZrCZ(ISTE^) + ,cJ*(DLL -DL ( I STFP) ) *SZF. 1 A(ISTKP)
DO aOO Jsl , Jl
IF (AHSCY2C D-YY) .Gfc, 0) GO TO 410
. (I3TFP)*nL(ISTtP)*cc
F»CCCsRCCISTEP)*CC
rF.(2,I)sF.F.(l,I)*CXX
FE(5»I)stE(l»I)*CZZ
EF(ttfnsfcr(l,l) *(CXX **2+(DLU *C?ET A ( I STfc P) ) **2* . 083333
EE(5.I)*Ft(l»I) *(CZZ **2+(DLL *SZfcTA{ ISTF.P) )**2*. OB3333
*(RCCC *rZETA(ISTFP))**2*,333J33)
F.E(6,I)stt(lf I)*(CXX *CZZ +SZETA(I5TEP)*CZETA(1STEP)
, *(HLU **2*,083333-BCCr **2* , 333333) )
GO TO 400
00 ^20 KRCYsl.6
thCKRCY, I)sO.
CONTINUfc
400 CONTINUE
HET
FND
vv
A107
-------
MF SUMA8CD
COMMON XABCD/ V( iO)»«OA( 30) ,YKl , YKR, YK3, YKU, AKY1, AKY2, AKY3,
, CCM&Xdbl )
COMMON CX(feOO) ,CY(6005 , AA (600 ) , U C600 ) »
, V(hOO), W(ftOO),VOKT(600),OfcNDIKC600) , F NTRCO C 600 ) ,
, CZC600),8S(8, 6005>
, SVOL(feOO) , XC (600 ), ZC (600 ), POSITN (600 ),C8( 600), THICK (600),
, SUA(aOO)»SwA(200),SCKY(200) ,DUMY2(600),
RC(hOO),wSC«,2),
VF,
COMMON /T«ASFf«/ KK,JJ,TTfI3IZt,NaTE,AO,IY,N()ThM,IMDKX,IRCDfILtAVt
SUA(J )30.
SWA(l)sO,
SCKYd)sO.
00 100 Is2,Jl
8CKY(T)sSCKY(l-n*(CKYCI»l)+CKY(l))*DY(I.l)*fb
100 CONT1NUF
RFTtJKN
END
A108
-------
stfc
INTERPOLATE: THE
Y(
T 1 DNS TH Y? f-Win
30) , YKI , Y*2, YKS, YM , AKYI , AT(600)/K,L»
Cn>»MnM
|. ,HR>SAI,
YY
KK,JJ,TT,IS!Zt , MUTK ,60, I Y ,
T,MFTHf
» T PLOT
Ej ISTfP. VF,
I NOf.X , IBtO,lLFAVE
JUISJ7F
J2=J1"1
on 170 isi.ji
UA(I)sO.
IF (Y2(I) .IF-. YU) UA(I)=UAO
IF fY2(T) .GT, VI) .AND, Y?(I) .LT, Yh) U A ( I ) = ( YF.- Y? C I) ) *UAO/
IF fY?(I) .I.E.
IF (Y?(I) ,KT.
wA(I)s
,AND, Y2(I)
Yi>(T)*'-AO/Y*'
L^. Yt) - A ( I ) a C Yf- - Y? ( I ) ) ** A 0 / ( YF
IF fY2(T) .Lfc. YKl) CKY(I)=AKY1
IF CY2CI) ,GT. YK1 .AND, Y2(I) .LT,
, *(YK2-Y?(I) )/(YK2"YKl)
IF (Y?(I) ,GF. YK? .AND, Y2(I) .l.K.
IF (Y?(I) ,t;T, YK^ .AND, Y2(I) .UT,
, *(Y2(I)»YK5)/(YK«-YKj)
I? (Y2CJ) ,r,E. YK4) CKY(I)cAKY^
170 CONTINUE
DO 180 1 = 1,, 12
1 ft 0 CKYC I) = {CKY(D+CKY(I+1))*,5
IF C^f THnn.tt-1.!) HETUKM
IF(TT,Gf .TPLOT) WE.TUHN
DO 190 lst,Jl
UACI)=llA(I)-!.IHX
CKYfl) * AK Y2+ ( AK Y 1 . AK Y2 )
CKY(I)=AKY?
CKY(I)=AKY?+CAKY%»AKY?)
A109
-------
SlWRfHITINfc
c
c INTFHPOLATE DATA ON YI TO Y?
c
COMMON /COMPIIT/EFC8, 151),
, Efa,l51),Yl(lSl)fY2{151),OY(lb
, CCMAX(151),VAR(151)
COMMON' /TRASFt.H/ KK, J J , T T , I SI ZE , MOTfc , AO , J Y , NOTRN , I NDEX , I BEI5 , HE A VE
, , IJK,FBED,AMO,DLDT,METHOD,TSTOP
C
C
JlsISIZF
J2=ISJZt-»l
DO 90 Ksl,6
DO 90 Jsl,Jl
90 EE(K,J)sO,
DO 100 jal,Jl
IF(PM2 ,LE. YKJ)) GO TO 110
100 CONTINUF
no on i?o Ksi,f>
120 VAR(K)=E (K, J-1) + (F.CK,.J)«E(K,J»1 ))*
, (PN2-YKJ-i))/(Yl(JJ-Yi(J-t))
IF(J ,F.O, 2) GO TO ISO
on i«o K»i,6
DO 130 jsl,jP?
130 Et(K,n=Et(K,l
GO TO 151
IbO DO 152 Ksl.
152 FE{K,l)s
JP1*J
151 CONTIMJF
DO 3in Is2,
PM=PN2
DO 160 js
IF(PN2 .LF, Yl(JI) GO TO 170
160 COMTINUF
170 DO 180 K=l,6
180 VAR(K + 6)at(K,J-n-Kfc(KfJ)-t(
IF(JP1 .fc'O, J) GO TO 190
IF (JPJ + 1 ,e.Q, J1 GO TO 200
JP2=J-2
DO 210 Ksl,6
210 ttCK,I)=fcF(K,I)+
DO 220 Ksi.b
DO 230 jsjPl ,JP2
230 E6CK,I)=f.F(K,I)*
220 f.t. (.<, I)=2,*(f-f «, I) + ,5«(F fK, JP2+1 )+VAK(6 + K))*(PN2- YKJP2+1)
,
JP1=JP2+?
GO TO 300
190 DO 191 Ksj,6
191 fc.F.(K,l)
,
JPlsjf'l
G" TL! 300
A110
-------
200 DO 201 Kcj ,6
201 EE(K,I)s2.
300 CONTINUE
DO 310 Ke 1,6
510 VAR(K)svAR(K+6)
PN13PN2
JP2sj?
IFCJPl .EQ, Jl) GO TO 320
DO 330 K=l,6
DO 340
DC 3bO
350 EE(K,j
340 ef(K,.Jl)B2t*EE(K,Jl)/ DYCJ1-1)
GCI TO «00
320 DO 360 K*l,6
360 EE(K,Jl)s (VAH(K)+ECK,jm*(Yl(Jl)«PNl)
400 CONTINitjf-
00 500 1=1, Jl
Yl (I)sY2(I)
00 500 K=l,6
SOO CONTINUE
RtTURN
Alll
-------
TIMf" ADD
C
C Anr> OfcTA ON Y2 TO OAT& ON Yi
C
CHMMOM CX(600),CY(60(n,THETA2C600), UC600),
, SZFYA(bOO) »C/!ETA(6005 , AB C600), OTT(600), 01(600),
, C2C600) ,SS(8« 600) ,
, SVOL(600),CC(6,200),PC1SITN(600) , T ( 600 3 , THICK ( 600 3 t
, SUA(200),SwA(200),SCKY(200),XXXX (600),
,3(6005 ,K,l, IPLUMG,MJTRI»UB,SA1, TIME , ISTtP,
U BX,^ HZ, «O
M /CflMPUT/EF (8, 1 5 1 ) ,
RX
J1-1SIZF
J2 = ISI7.e-i
DO 90 ,?al , J2
90 OY(.J)=YtCJ*l)«Yl(J)
DO 95 !«!,&
DG Q5 Jsi , Jt
95 CCU, J)=0«,
00 300 J=!?J1
IF(PV? 0Lt. . Y2(J)) fil.l TO 11Q
100 C CM T I ,v u t.
110 DO 120 «=!,&
1 ? 0 V A w C K ) = h fc C *< , J - i ) * C F. F C K , J > - P. E ( K , J * I) ) *
CP^2-Y2(J-1))/(Y2(J)-Y2(J»1))
U"(j ,F.O, a) e;o TO iso
J P 2 - J »• 2
DO i'-JO >n 130 K = i , 6
r /(V2(j)-Y2(J-n)
IFCJPJ .EO. J) GO TO 190
IF CJPl-i-i , hQ, J) GO TO 200
JP2=J-2
Oil (> 1 0 K = 1 , 6
A112
-------
230 CCfK,I)sCC(K,
220 CCCK, n=2,*(CC(K,l) + ,5*(tFCK, JP2+l)+VAR(6+K))*(PN2» Y2(JP2+1)))
,
JPl=JP2+2
60 TIT 300
190 DO 191 K=l,6
JPlsJPi
GO TO 300
200 00 201 K=i,6
201 CC(K,I)e2,
, f ,5*(F.f
JPlejPUl
300 CONTIMUfc
r>n 310 Ksj
310
1FCJP1 ,fC4, Jl) RO TO 3?0
DO 330 K=l,6
330 CC(K,Jl)sCC(K,Jl)+.S*(VAR(K)+FE(K,JPl))*(Y2(JPl)»PNl)
DO 340 K=l,6
DO 350
350 CC(K/J
3«0 CC(K,Jl)s2t*cc(Kf Jt>/ DY(Jl-i)
GO TO ilOO
320 00 360 K=l,6
360 CC(K,Jl)s (VARCK) + F:t(K.Jl))*(Y2CJl)»PNl) /DYCJ2)
aoo CONTINUE
on
DM
E(IrJ)sf CI» J)+CC(1> J)
All 3
-------
SHIF r
l(151),Xl(15l>f Zl(ll>nfS!GX(lSl),SI(;Z(151),SIGXZCl!>l>
COMMON /C/XO,ZO
COMMON CX(600),CY(600),TH6TA2(600),U(600),
, SZf-TA(fcOO),CZF.TA(600),AA (600), OTT(feOO), DLC600),
, CZ(600),SS(8, 600),
, SVni.C600),XC(600),7C(600),POSITN(600), T(600),THICK(600),
, SUA(200),SWA(i>00)fSCKYC200),C:ONC (600),
»OAS(U), 3(600), K,U,IPUUNG,^UTRLfUB,SAI, TIME, ISTFP, VF,
URX,»/BZ,KO
COMMON /COMpuT/te(8,lbl ),
CCMAX(151),VAR(151)
/THASFtH/ KK> J J , TT , ISIZK , NHTt , AO , I Y , NQTRN, INDEX , IBtOr ILfc AVE
, , UK, FBED, AMD, PLOT, MET HOD, 1ST OP
JcO
00 5 1=1, Jl
IF(E(1,I) ,FO.
IF(J .EO. Jl)6,7
6
7
00 1 I«1,J1
IF(E(1,I) ,FO. 0.) GO TO 2
E1CI)«1./E(1»I)
XHr>«F(2»I)*fi(I)
Z1(I)*K3»IJ*E1(I)
SlGX(i)sFta,I)*El(I)-XHI)**2
8lGZ(l)sEf5»I)*El(I}"Zl(I)**2
SIGXZ(I)=E(6,I)*tl(I)-Xl(I)*ZKI)
GO TH 1
2 Xl(I)sO.
zi m=o.
SIGX(I)cO.
S!GZ(I)sO,
SIGXZ(I)sO,
1 CONTINUE
IF(KCl) ,F.Q, Of)RO TO <)
F1«1./F(1)
XX1«F(2) *F1
ZZ1»F(3) *F1
SIGXX=F («)*Fi-XXl**2
SlfiZZaF (5)*F1-ZZ1**2
SIGXXZZ«F(6)*FI»XX1*ZZ1
GH TO 8
9 XXlsQ,
ZZlsO,
SIGXXsO.
SIGZZsO.
SIGXXZZaO.
8 COMTIMJE
AM2BO.
DO 3 1*1, J2
E(1,I)+E C1,H-1))*DY(I)*,5
A114
-------
3
XOi=AMl/AMO
00 a !sj,Jl
E(a,I)sE(l,I)*(Xl (I)-XOi)
€ (3,I)sF(i»I)*(ZlU)»Z01)
E(6fI)*F. (l»I)*C(Xl(I)*XOl)*(Zl(I)»ZOi)*SIGXZ(l))
u COM IN/ME
F(2)=F(1)*(XX1»X01)
F («) sABS (F (!)*(( XXI -XO 1 ) **2 + S IGXX) )
F(5)eA8S(F(l)*((ZZl-Z01)**?+SrGZZ))
Ff6)sF(J)*((XXl«X01)*(771-Z01)*SIGXXZZJ
Z08ZO+Z01
DO 11 Iel,ISTtP
CX(!)*CX(J)«X01
11 CZ(I)«CZ(I)-Z01
f NO
w
A115
-------
DISPC(AL
/COMPUTE!- (a, isn >
, t.(fl,lbl),YlUbl),Y2(151),nY<150)fKfl)/F*C P. ),'««(<»),
, CCM4XC1S1) ,VA»(lbi)
c
C FUNCTIOS' FOR Cf>PiJTlMU THF DI SPf- WS TfJM CPtfFICIEM KX,K
C
IFfFLd.J) .I.e. 1.00E-1S) GH TO 1
D1SPC SAL /SMr-A*(ARS(4»S(FF(^,J)*FF.( I » J)-fcT(2, J)**2>*
, •CEF(6,J)*tt(l.JJ-EE(?»J)*KC3,jn**2)/FE(l,J)**
-------
J)
1*51 ),
IF( E(1,J) ,Lt. . l.OOE-lb) GO TO 1
D!SPeCsALAMOA*UBS(AB!3( E(£J»J)* fc
, ABSC fc(5»J)* F(1»J)» F(
, •( FCfe,J)* E(i^J)- F.
, **0, 3333333
RF.UIRKi
1 PlSPCCaO,
v v
All?
-------
/C'iMPU1/Ft(B,m),
C
t FU^CTIOK' HOK COMPUTING THt DISPFRS10N C'lfcFF ICIE.MT KX,KZ FOR 501,10
C OKi SURFACh
C
iFrvAR(i) ,I.E, i,oof«is5 c;u in i
A«3S(ABStVAP(a)* VAK(1)»
A118
** 0 ,
RF. TURN;
1 DISPwsO.
-------
FUNCTION
COMMON /CnMPUT/FU8.151),
n8,15l),Yl(151),Y2(151),DV(150),K8),FWC8),V..«l(<»,
CCMAX(i51),VAR(15i)
IK Ml) .IE. 1 .OOE-15) Gtl TO 1
OISPWwsALAMDA*(AHS(ABS( F(4)* F(l)« F(2)**2)*
, ABS( F(5)* F(l)- F(3)**2)
-( F(fe)* F(l)- F(?)* f(3))**2)/ F(l)**a)**0,33333J5
RETURN
KFTUHKi
A119
-------
NF TUTPT
c
C OUTPUT SUbRfUTlNF
C
DIMENSION! Sr.U52,
-------
I F f. H I* ( K K ) ) 110,120 ,110
110 IF (F(l) ,U , ELIMIT ) 111,112
111 X=0.
Z*0,
sic;x=o,
SIGZsO.
SIGXZ=0,
GO TO 120
112 Fl = l ,/Ffl)
X=F(2) *(• 1
Z=F (3)*F1
SIGX = A6S(K1)*F1 -X**2)
SIGZeAHS(F(b)*Fl -Z**2 )
SJGXZsF (6)*F1 -X*Z
x=x+xo
ZsZ+ZO
120 AMO=0,
A M 1 = 0 ,
nn I3o i*i,, IP
F(J,I)+F. (1,I + 1))*OY(I)*,S
130 CONTINIJF
IF (IFIMJSH ,F.Q, 1)GO TO 7007
IF(IPLT.F.C,0) GO TO 8001
nn 7002 isi,ji
IF (F ( 1 , 1 ) .GT , CDOMX ) CQOMXsfc 1 1 , 1 )
7002 CONTINUE
IFCAKO.FQ.O.) GO TO 7001
IF(TT.LE.TPLOT) GO TO 7001
IF(A5S(fCOOMX»(Jir>MX)/OlC>MX),LT.O,l .AND, (GAMA 1 *GAMA2) ,CE , 1 ,
* GO TO 8001
7001
7007
WKlTF-(6,n
1 FORM AT (t HI )
IF(AMO ,EQ, Of)131,l32
132 YRARsAMl/AMQ
GO TO 133
131 YBA»BO.
133 CONTINUE
IF(HI»(KK)) iaO,150,160
1«0 WKITF(6,2)TT,YBAW,WOAS(IP),WW(KK),AMO
2 FORMAT(10X,«HTlMt,F12.fl»1HSEC,f2X, 6HMtAN Y, F9,3,3HFT,, 2X,
, 13HSOLIO DENSITY, F5. 3, 5HG«/CC,2X,
, 9HFALL VEL,,F12,«,feHFT/SEC,«H TSS, E 1 2, a ,5HCUFT, )
WRITF(6,3)
3 FORMAT(/25X,?OHSOLIO ON THE SURFACE/)
U FORMAT(3bX,lHX,9X,lH2,7X,6HSIGMAX,<»X,t.HSIGMAZ,6X,7HSIGMAXZ,5X,
, IUHFLOATING SOLID)
WRITE(6,20)X,Z,SIGX,SIGZ,SIGXZ,F(1)
20 FORMAT(2ex,6E12.U/)
IFCAMO ,EO. 0,)GO TO 200
^RITE(6,5)
5 FOHMAT{/2bX,19HSOLID IN SUSPENSION/)
WRITF(6,10)
10 FORMAT(25X,1HY,9X,1HX,9X,1HZ,7X,6HSIGMAX,6X,6HSIGMAZ,6X,7HSIGMAXZ,
, UX,5HCGNC.,7X,3HCOO)
DO 1«1 I31,J1,JS
A121
-------
1«1 HRITE(6,21)Yim,XX(I),ZZU),SlGMAXm,SIGMAZ(l),SIGMAXZCn,
, CCMAX(I),E(i, I)
21 FORMAT(20X,F6tl,7E12,«)
GO TO 200
150 WRITE(6,7)TT,YBAP,A«0
7 FnRMAT(25X,UHTm,E12.4,4HSEC.,?X, 6HPEAN Y, F9,3,3HFT,,
, 12H TOTAL HASTF.,E12t4,5HC(5FT,)
WRITE(6,B)
« FOPMATf/25X,28HFLUID PART OF WASTE MATERIAL/)
WRITEC6,6)
6 FORMAT(25X,1HY,9X,1HX,9X,1HZ,7X,6HSIGHAX,6X,6HSIGMAZ,6X,7HSIGMAXZ,
, «X,10H CONC, ,5X,3HCOO)
00 151 I=l»JlfJ8
151 HRITE(6f2l)Yl(I),XX(I),ZZ(J),5IGMAX(n»SI6MAZ(I),3I6HAXZ(I)p
, CCMAXCI)»E(1,I)
GO TO 200
160 WRlTE(6»2)TT»YBAR,ROAS(IP),Kh{KK),AMO
IFCAMQ ,EQ. 0,)60 TO 16?
WR!TE(6,5)
WRITF(6,10)
DO 161 I»1,J1,JS
161 WRITfc(6,21)Yl(I),XX(I),ZZ(IJ»3I6MAX(I)»8I6MAZ(I),SIGMAXZ(I)»
, CCMAX(I),E(1.I)
162 h»ITE.(6,9)
9 FnRM&T(/25X, IfcHSOLin ON THE BED/)
w«ITE(6,l 1)
, 15HDEPOSITEO SOLID)
HRITE(6»20)X,Z,SJ6X,SI6Z»816XZ,Ftl)
200 cnNTIKiUF
R001 IF(IGLT.EQ.O) GO TO 8002
IF(WW(KK)) 8000,800,8000
8000 IFCKKK.fQ.i99) GO TO 800
TP(KKK)sTT
FKC(KKK)sFd)
FKX(KKK)BX
FKZ(KKK)*Z
FKSX(KKK)sSlGX
FKSZ(KKK)cSIGZ
FK8XZ(KKK)8SIGXZ
FSIZ(KKK)B»1,
TFMPORs SIGX*SIGZ»SIGXZ**2
IF(TEMPDR.GE,Ot) FSIZ (KKK)sSQRT (SORT (TEMPOR) ) *2 ,
800 CON-TINUF
IF(TT,LT.TPUOT) GO TO 8002
COGMAXsO,
DO 8003 1*1, Jl
I F ( E ( 1 , 1 ) . GT . CfiOM AX ) COOM AX»E (1,1)
8003 CONTINUE
IF(AMO,FQ,0.) GO TO 800«
IFCCOQMAX.GT.CMAXMAX) CMAXMAX*COQMAX
IF(AMO.GT.TOTKAX) TOTMAX«AMO
CnORsCDOMAX/CMAXMAX
TOTRsAMO/TOTMAX
IFCCOOR.GT.TRIGERdNOF. ) , AND . TOTH ,GT .TRIGfR ( INOE" )) GO TO 8000
lF(TIM(i|).GT.O,5) GO TO 800«
A122
DO 8005 I«1,J1
8G(1,INDE
-------
Sf (I,I*f>F )=F ( 1 , I)
sxn, IMDF ) = xxm
SZCI.IMU )=ZZ(I)
I,I*'OF )=SIGMAZCI)
SXZ(I,TMM 5=SIGMAX7(I)
SlZCIf IMnt)=-l.
Tt-MPOs S
IFCTf^pr: .RF.O.) STZCI»
8005 CPNTlMiie
IF UNOE ,to,a) GO TO «ona
IMDF =
sooa CONTIHUF
FO.O) GH TO 800?
IF twiw(KK) ,(-0.0,) Gil TO 601 1
KKKSKKK+1
TP(KKK)s2,*TP(KKKM)-TP(KKK-2)
FKC(KKK)sO,
FKX f KKK )sn ,
FKZ(KKK)=0,
FKSX (KKK )eO ,
FKSZ(KKK)sO.
FKSXZ(KKK)sO,
FSIZ(KKK)=0,
8011 IFCTMOF. .fcO.l) GO TO 801^
JJJ=J1*1
Df R007 ! = 1,INDF.
SG CJJJ ,1 )s2.*Yl (Jl )-Yl (Jl-1)
SC (JJJ »I )sO.
SX (JJJ ,1 )sQ,
SZ (JJJ ,1 )sO,
SSX(JJJ ,1 )*0,
SSZCJJJ ,1 )=0,
SXZ(JJJ ,1 )sO.
SIZUJJ tl )=0,
8007
IF(TjM(«).GT,(b)
CALL nRAW(SG(lM)»SG(l,2),SG(l,3)»S6(l»«)»3C(l>l)»SC(l,2)»SC(l»i)f
8012 IFCWirf(KK) .F.Q.O.) GO TO 8010
CALL DRA"(TP,TP,TP»TP,FKCfFKX,FKZ,FSIZ,KKK,12,KSCALEf4)
CALL f>RAI*(TP,TP»TP»TP.FKSX,FKSZfFKSXZ,FKC,KKK,ri,NSCALt,3)
8010 CONTINUE
IF (INDt.ER.i) GO TO 800h
CALL DRAW(SG(l»l)»SG(l,2),S6(l,-J),SG(l»a).SIZ(l,l),SIZ(l,2)»SlZ(l,
iF(IGLT.fcO.l) GO TO 8006
CALL DRA»<(SGCl,l),SG(l,;?),SG(l,3),SG(l,U),SX(l,n,SX(l,2),SX(l,3),
CALL nRArt(SG(l»l)fSG(l,2),S6{l,i),SC(lffl},8Z(lf 1)»8Z(1,2),8Z(1»J)»
1F(IGLT.EQ.2) GO TO 8006
CALL DRAMSGU,nrSGCl,2),SG(1,3),Sr,Cl,4),SSX(l,n,SSX(l,2),SSX(l,
CALL nRAW(SG(i,n,sG(i,2),sG(i,3),sG(i,ii),ssz(i,n/ssz(i,2),ssz(i,
3)»SSZC1»«),JJJ,7,NSCALE,INDF)
CAUL D«AW(S6(l/l)fS6(l,2),S6{J,3),SCCl»a),SXZf 1,1)»SXZ(1,2),8XZ(1»
A123
-------
8006 CONTINUE
KKKsO
CMAXMAXsO,
8002 CONTINUE
IFU'ISHsO
RETURN
END
vv
A124
-------
Nfe DRAW (Xl,X2,X3,X«,Yl,Y2,Y3,YO,K.,IG,K/SCAtE,NCURV)
N Xl(l),X2(l),X3(l),X«(n,Yl(l),Y2d),Y3(l),YU(n,X(800),
*Y(800),YY(800),SYM(u),SIM(lh)»P(2a.OO)
COMMON/GRAPH/TJM(/O
DATA SIV1HY,1HB, 1HC,1HS,1HA,1H1, 1H2, )H3,1HU,1HS, iHfc, 1H7, 1H8,1HT,
* IHX,!HZ/
IF(K»CU«V.LT.l) RETURN!
IF (KJSCALE.nT.60) NSCALE=feO
IN B N/NSCALE
IFdM.LT.J) IN=1
,JsO
DO 1 ISJ,M,IIM
J = J + 1
X(J)sXld)
1 Y(J)»Y1(I)
J = J+1
X( J)cXi (N)
YCJ)sYl(N)
NNeJ
IF (NCURV.f-.Q.l) GO TO 5
DO 2 Isl,M,JM
X(J)«X2(I)
Y(J)«Y2(I)
Y(J)*Y2(K')
IF(MCURV,FG,2) GO TO 5
DP 3 JSI,M,IM
J = J+l
X(J)=X3(I)
3 YCJ)sY3(I)
lF(MCURV,tQ,3) GO TU
00 H I = l,Ni,IN.'
J = J41
X(J)sXfld)
Y(J)sY«(N)
-
Gf.i TU (10, 20, 10, 20, 30, 30, 30, 30. 30. SO, hO, 70, 80, 90, 2000, 1000, 3000)
* ,TG
10 DO 11 I «1, a
11 SYMd)BSlM(I)
IFdG.EQ.J) SYM(3)=SIM(15)
CA|.L NOR M ( Y , Y Y , N N , 1 , , 0 f , A M X Y , A M N Y )
NNlsNN* 1
CALL MOPM(Y(NN.'l ),YY(MNil),WNJ,0,«,0,,AMXH,AMN8)
MKil SNNI +^^^l
IMIG.FQ.3) CALL NORM ( Y { NN'l ) , YY (K
-------
* , AhNY , AMKR, A*N.'C , AMNS
IF(IG. 1:0,3) wKITt (*»» MO) X ( I ) . X f NN ) , A V X Y , A •" X B , A "X I'. , Af'-X S
* , 4 * N Y , 4 « N B > A * N C , A M M S
CALL SPLOT(YY,X,P,J,NY,NX,l,NN,i,a,SYK)
110 FPRMATdwl ,/////»21X,JflHOATA POP GRAPH IMMEDIATELY FOLLOWING
*///, 10X,«6HlNDFPEMOfM VARIABLE IS TlKfT nvt'R RANGE »2X
1 2G12, *>,///, 30X, 19HDfPFMf?FN.!T VAHIABLFS , // , 1 0* , 6HSYMSOL , 1 JX
3 3(?XfG12.5)»/, lOX^UHMlN PLOTTfO , JX, G 1 2, 5, 3 f 2X, Gl ^, S) , / , 1 OX,
* 7HKFMA»KS»flX, 1 1HVFP-T, DIST, ,UX/9H HAOIUS
a MM WOP, DIST. ,3x,lln HOW, njST, )
130 FO»MATfJH3 ,/////, 21X,38^DATA FOR G«APH IMULDI ATfL Y FOLU-mNG
*///, 1 OX,«6HIMDF.PFND(:NT VARIABI.F IS HOP..OIST, IWF» RAN'GL »?X,
1 2Gl2.5,///,30X,19HnEPF.NDE'"T VARIABLES , // , 1 OX, 6HSYMHOL , 1 3X f
2 )HY,l«X,lHt!,13X, lHC,r3X,lHZ,/,10X,llnt«AX PLOTTfr. ,3X,G12,S,
3 3(2X,G12,5) »/» IDXfllHMiM PlOTTtO , 3X,G12 fb » 3 (i>X, Gl ?„•?),/, 1 nx,
* 7HRt.MARKSr«X, 1 IHVtRT. DIST, ,«X,9H RADIUS
« 11WFLUID CONC, ,?X,11M HOP, OIST, )
20 SYM(1 )*SIM
IF(IG, fcQ.il)
SYMJUjsSIfd)
CALL MJRM ( Yf YY,NM, J , , 0 , , AMXA, AMNA )
N H 1 c N ^i + t
CALL NORM (Y(NN1 ) ,YY(K-N1) t tvH t ,5,0, , AMXf, AMNB)
NN1 SNN1 +NN'
CALL NMRM (Y(K-N1) ,YY(MM1) , MM, ,ft,0.» AMXC,
CALL MHRM (YCMN1 ), YY(NN1 ) , NK> 1 , , 0 , , A KX S , A MNS )
IF(IG,f Q.2)' WRITE (6,120) X ( 1 ) , X (MM) , AMX A , A«XH, AhXC , AMXS
* ,AMNA,AMK'B»AMNC»AMNS
IF(IG.FO.a) t^RJTP.(6»HO) X ( 1 ) , X (NN) , AMX A , AMX8, AKXC , AMXS
* , AMNA, AMNR, AMNC, AMNS
CALL SPL.OT(YY,X,P,J,NY,NX, l,NK!,l,a,SYK)
120 FnRMATdm, /////, 21X»38HOATA FDR GRAPH IM^FDIATUY FOLLOWING ,
*///,10X,
-------
Gf TO (41,42»43,44,AHXX,AMNX)
CALL wORMfY(NM) , YY(NMl) , NN),0,5 , 0 . , AMXZ ,
M NJ 1 s \j H] + N K
V(NM ),YY(NN1) ,N'N,0.6 , 0 .
CALL SPLOT(YY,X,P,J,NY,K'X,1,K:N,1, NCURV, SYM)
701 FORMATdHl, /////, 2X, 7«HPLC)T OF TOT BDRY MAT (T) CENTROID LOCATION
*S (X,Z) AND SIZE (S) VS TJKf. ,// »2X, 12HT1MF RAKifiE ,4X,
1 2G20.8// ,2X,16HMAXIMUM T,X,Z,S ,aG20 ,8»/»2x, 16HMINIMIJM T,X,Z,S
3 , 4G20.fl,7(/))
R E TURN!
80 CONTINUE
A127
-------
SYM(1
SYM(2)«SIMU6)
CALL NORMCY,YY,NM,l,,o,
NK'lsNKi+1
CALL NOPM(Y(NM),YY(N*l) »K'N,0.5 ,0,, A^XX,
CALL N00M(Y(MM),YY(NN1},N*,0,75,0.,AMXZ,
*RITE(6,80i)
CALL SPLOT(YY,X,P,J,*Y,*x,l,NN,l,MCURv,SYM)
801 FORMATUH1, /////, 2X,70HPLOT OF BOUNDARY MATERIAL SIGMA* AND 2 (X,2
*) AND SIGMAXZ (S) VS, TIME. ,// ,2X, 12HTIMf RANGF= .flX,2G20 .8, /
1 ,2X, J6HMAXIMUH X,Z,S , 3G20 . 8 , / , 2X , 1 fcHM I MMUM X,Z,S ,
2 3G20,B,7C/))
RFTURM
90 DO 91 Id,
-------
SUBROUTINE SPLOT(8,A,P,H,I,M,JSTR,MREP,NUM8,NSYM,SYM) SPLOT
DIMENSION AU),B(1),SYM(1) ,P(1),0(20),H(10)
DATA 0/20*5H-»--I/
DATA STR/lHs/
DATA END/IH//
DATA BL6/6H /
DATA EYE/1HI/
DATA BINK/1H /
LQBL6-1
NNUM8N*
MUMlBNUMB-1
JZA»0
ZPOsQ.
DO 1916 J«1,ML
1916 P(J)»8L6
FMSM-1
AMXsACl )
DO 100 J*2,N
100 IF(A(J).LT.AMN) AMN=A(J)
DO 105 J«2,NNUM
105 IF(BCJ) .I.T.BMN) 8MN*B(J)
DAa(AMX»AMN)/FM
KRITEC6.2000) AMX,AMN,DA
2000 FG»HAT(/////f 1X,26HMAX,MIN,INC, OF JND.VAR,
2001 FO«MAT(//,1X,26HMAX,MIN,INC, OF DEP. VAR, ,/,IX,6G20.«)
2002 FORMATC36(/))
TESTAaAMX*AMM
TFSTBaBMX*BMN
IF(TRSTA) 1971,1972»19T2
1971 JZAs-AMM/DA
1972 IFCTESTB) 1973, 1 975, 1975
1973 Ifls-6MN/DB
LIA8-L6
DO 197« J«l,H
LIA»LIA+L6
197a CALL PFIX(P,IB,LIA,EYE)
1975 CONTINUE
L10*LOLD/20*1
DB?OB20,*D8
HBM=BMN-D820
DO 2020 Jsl.LlO
2020 H(J)*HBH + .J*DB20
WRITE(6»2021) (H(J)r Jsl»HO
2021 FORMATC 3X,6G20.5)
WRITE(6,1200) (8(J)»Jel»20)
1200 FORMAT(16X,20A5')
A129
-------
vv
00 200 JsJPQ,NPQ
DO 2f>0 KsJKNP, JNUC
CALL Pf-'IX(P,IB,L,IA,SYM(ISY"))
200 CONTINUE
DO 300 Jsl,M
JUi=(J/10)*10
JHI»JLO+LO
WHITE (6,1000) (P(K),KsJLO,JHl)
IF(J.EO,J2A) WRITF(6^i550) Z»0, {Q(X) ,K«1 ,20)
IF(JO,NF.J) GO TO
( J-i )
300 CONTINUE
1000
(H(J),J*l»L10)
UMN
tNO
A130
-------
SUBROUTINE RANGt. CA,N,AMX, AMN, JMX,
DIMENSION A(i)
AMX=A(1)
AMNsA(l)
PO 100
IFtA(J).LT.AHX) GOTO 50
JHXsJ
AMXcACJ)
50 CHN/TINUE
IKA(J).6TtAMIw) GOTO 100
JHKrj
AMMSA ( J)
100 CONTINUE
RETURN
A131
-------
SURRPUTINF NORM(A,B»NfCl»C2,AMX, AMN)
f.ALL
CC=C1-C2
Xs(AHX»AMN)/CC
IMX,EO.O,)XS1,
Y = (Cl*AMKi-C2*AHX)/CC
Zsl./x
DO 100 J=1»N
B(J)=(A(J)-Y3*Z
100 CONTINUE
RET
FND
A132
-------
I^E PF IX(P,IB,1 IA.SYI*)
DIFFUSION BUFC6),P(1)
IB6SIP/6
LIBsLIA+IBfi+1
OFCOpfc (6,1000,P(LIB)) BU^
1000
PUF (IRES)sSYV-
FNr.OOf(6,1000,P(Llfl)) PUF
PFTUPM
A133
-------
APPENDIX B
GRAPHIC OUTPUTS
(See Tables 6. 1 through 6. 4 and Section 6. 4
for explanation)
The results presented in this appendix may not be exactly reproduced by
running the program listed in Appendix A due to i) a difference in computer
system, and ii) some of the runs in Appendix B may have been obtained
with an earlier version of the program.
Bl
-------
RUN D-L-l
AMBIENT CPNDITTPNS
DEPTH TN, FT. DENSITY IM GRAM PFR CC, KY IN SOFT PFR SF.C. Vtl.. IN FPS.
npPTH n. O.OOOOF+OI 6.oooo.F+oi I.OOOOF. + O?
AMR-nfcNSTTY 1.0210E+00 1.0230F+00 1.0300F. + 00 1.0300E+00
YK 3.5000F+01 4.5oooE+oi 5.5000F+01 ft
KY b.OOOOF-O^ b.OOOOF-03 l.COOOF-0?
Yll b.OOOOf+OIYW b.OOOOF+01YF 1.0000E+02H 1.0000F+02
UA 0, WA 0.
RAWGF OPFWATIUN 1
WASTF ^ATFPIAL PLIMPEO INSI AMTANEPUSLY INTO THE
THF SHAPF f:F THF CLO'IO IS ASSHMFD TO RF HFM
TSTHP i.nooor + o'iSEC.
r-RTD PtJTM SIZF 51 KFY1 1 KFY? 2 KEY3 0
USF TFTRA Tfc'CH SUGHFSTFO CdFFF TCIEMS
DP:CR1 1.0000 DTMCK? J.OOOO
iLPHAO .2^50 BETA 0.0000 CM l.OQOO CO .5000
T,AMA .25 r.DRAr, i.oo CFRIC ,010 CD? .10 CD« i.oo AI.PHAC .0010
FRICTN .0100 Ft .1000
ALFAI i.oooo ALFA? i.oooo GAHAI o.oooo GAMA2 o.oooo
ALAMDA .0010
PB 5.0000F+00 WOP 1.^00OF+OOU 0. V 1.0000E+00^ 0.
K 1 L 1
DENSITY (JF SPLID 2.5000F + 00
COMCFNTRATTON
2.0000E-01
FALL VPl.OCITY PF SOLID
5.0000F-OP
B2
-------
RUN D-L-2
AMBIENT CONDITIONS
DEPTH TW FT. DENSITY IN GRAM PER CC. KY IM SQFT PEP SFC. VFL. I to FPS,
DEPTH 0. a.OOOOE+01 6.0000F+01 1 . OOOOF. + 02
AMB-DFNSITY 1.0P30F+00 1.0230F+00 1.0290F+00 1.0290E+00
YK 3.5000E+01 a.SOOQF. + Ol 5.5000E+01 6.5000Ft01
KY 5.0000E-02 5.0000E-03 l.OOOOE-02
YU 5.0000F+01YW S.OOOOF+01YE 1.OOOOE+0?H l.OOOOE+0?
UA 0. WA 0.
BARGF. TiPFRATION 1
WASTE MATERIAL DUMPED INSTANTANEOUSLY INTO THE UCEAN
THF S"HAPF OF THE CLOUD IS ASSUMED TH BF HEMISPHERE
TSTHP 1.0000E+03SEC.
GRID POINT SIZE 51 KEY! 1 KEY? 2 KFY3 0
USE TFTRA TECH SUGGESTED COEFFICIENTS
DINCR1 1.0000 DINCR? 1.0000
ALPHAO .2350 BETA 0.0000 CM 1.0000 CD .SOOO
GAMA .25 CDRAG 1.00 CFRIC .010 CD3 .10 CD4 1.00 ALPHAC .0010
FRTCTN .0100 F1 .1000
ALFA1 1.0000 ALFA2 1.0000 GAfAl 0.0000 GAMA2 0.0000
ALAMDA .0010
RR 5.0000F+00 RHO 1.3000F+001I 0. V l.OOOOE+OOW 0.
K 1 L 1
DENSITY OF SOLID 2.5000E+00
CONCENTRATION
2.0000E-01
FALL VELnCITY OF SOLID
S.OOOOE-02
B3
-------
RUN D-L-3
AMPTFf-'T rr-NOITTDNS
HFPTH I*1 FT. OFNSTTY TN
PFPTH n.
CC. KY TM SOFT PFR SFC. VF.|_
1 .OOOOF + O?
FPS,
KY
YU 5.nnn^F + oi YUI s.
ii* n . WA o .
s.onooF-o? 5.nnooF-o3 I.OOOOF-O?
YF 1 . noOf)F-»-o?H i.nonoF + o?
WA^TF "ATFPT4I. HIIMPFO T^PTAMTAMFnilSLV INTO THF
TMF CHAPF PF THF ci nun T? ASSMMFO TO RF HFMTSPHFQF
TSTOP 1 .ooftOF+nuRFc.
) KFY? ?
r,RTH PPT^T CJ7F 51 KFY]
URF T^TPA TFi-H SUGGF. tTFn
r^T>"C
-------
RUN D-L-4
FT. nFN^TTY T N' n
n.
M PFR CC. KY I M SOFT PFP SFC. VFl
y.ooooF+A} ^.nonoF + oi i.oonoF+o?
T *' FPS,
vw
WA 0.
a . "? n n p F + 0 1
nPFRATin\i 1
'-'iTcPT4| nUMPFH T MS T A K'T 4 MFOI |C (_ Y T
SHAPF PF TMF C I OHH TS AP-SII^FH TO Rf
TSTPP 1 . nnAnp + n/JSPr .
^.PT^P^'T^•T<5T7F ^IKFYI IKFY?
IIRF T^'TPi TFrH SI'^OFSTFn rnFFFTrTF''1TS
i.nooo OTMrpp t
P-AWA ,?s rn^Ar; i.on rFPTC .oin
FPTTTM . o 1 00 F 1 ,inon
AIFAI i.nnnn JLFA? i . o o o o
/, i A M n A .nn\r>
to
DR
t I
HP pnirn
!• TTHN
FAI i VFI/TTTY n^ ''HI r
n n.
?. 0001F-0 1
THF
i.no Ai.pw,»r
v t.ooooF+on w n.
B5
-------
RUN D-L-5
TN FT. DFNSTTY
DFPTH 0.
GRAM PFR CC. KY TM SOFT PF« SET. VFL. IN FPS,
U. 000^ + 01 ft.nOOOF + 01 l.OOOnF+0?
YK
KY
YU s.
1 1 A 0 .
3.5nnoF+oi
1.00POF-0?
YF
VI ft 0 .
WASTF WATFPT4I. OUMPFH T MS T i NT i NFOI I PL Y TNTH THF
THF ^HA°F OF IMF cinnn TS ASsiiMFD Tn RF
TSTOP i .oonoF+
f.RTO °nTMT ST7F
i)SF T^TD* Tjr(-H
njwr.Pi i . n ft o n r
«i PHAO
KFY1 1 KFY? ?
AI.FAI
: rn«Ap; i.oo
.m no F1
i.nnno A
pnn i
K 1 I 1
OF sni rn
=-A! I VFl.nryjy
CM t.onoo rn
.010 cn^ .in cnu j.on AIPNAC
o.onoo
v i.oonnF+nn
,ooin
n n.
p.snooF + r.n
B6
-------
RUN D-L-6
AMBIENT CONDITIONS
PFPTH IKJ FT. DENSITY IN GRAM PER re. KY IN SOFT PER SEC. VEL. IN FPS
D(rpTN 0. a.OOOOE+01 6.0000F4-01 l.OOOOE + O?
AMB-DE^STTY 1 . 02^0E+00 1.0230F+00 1.0250F+00 1.02SOE+00
i(.SOOOE^01 5.5000F*01
s.ocone-n? ?.ooooE-o3 I.OOOOE-O?
YU S.OOOOF + OIY") S.OOOOFfOtYF l.nnonE + 0?H i.OOOOE+0?
It A 0. WA fl.
1.
WASTE HATF.PIAL DUMPED INSTANT ANFOUSLY IMTO THF OCEAN!
THE SHAPF GF THE CLOUD IS ASSUMED TO BE HF.MISPHFRE
TSTOP l.OOOOE+OaSFT.
GRID PDJMT SIZE 51 KEY1 i KEY2 ? Kf.Y3 0
USF TETRA TECH Sl'GGESTFO COEFFICIENTS
OlNCRi 1.0000 DIMCP.2 1.0000
ALPHAD .23SO BETA 0.0000 CM 1.0000 CD .5000
GA^A ,?5 CD^AG 1.00 CFRIC .010 CD3 .10 CD« 1.00 AlPHAC ,0010
FRJCTN .0100 Fl .1000
ALFA! 1.0000 4LFA2 1.0000 GAMA1 0.0000 GAMA2 0.0000
.0010
P8 S.OOOOE+00 POO 1.3000E+00 U 0. V l.OOOOEtOO W 0.
K 11. 1
DENSITY CF SOLIH 2.5000E+00
CPNCf MTP4TICN
2.0000E-01
FALL VFLHCITY OF SOLID
5.0000E-02
B7
-------
AMBIENT
DFPTH INJ FT
DEPTH
RUN D-L-7
DENSITY U1 GRA" PER CC. KY IN SPFT PE" SEC. VFL. IN FPS,
0. 4.0000E+01 6.0000F+01 l.OOOOF+02
i .0230F+00 1 . 0?30K. +0 0 1.0240P+00 1.02UOt>00
3.5noOF-i-ni
VK
YU S.noTOF + 0 t Y/l 5.00POF.
MAO. « A 0 .
fi3 l.OOOOF-02
YF 1.0000E + 02H 1.0000E + 0?
THF
P4RG.-
«.'ASTF MATERIAL OUMPF.D I'JSTANTAMF.DUSLY IM
THF SHAPF OF THF rinup is A.SSUMFD in HF
TSTOP 1 .nonnp + O'jSFr.
f-PIO POINT SIZE SI KF.Y1 1 K.FY? ? KFY3
i;SF TFTR4 TPfh SUGr.FSTFn COFFFIC IEK«TS
1.0000 OTNC1?? 1. 00(10
HKTA o.ouoo CM
l.OQ CFPK .010 CD3
. 1 1 0 0 F 1 .
4LPHAO
r.A^A
F P I C T N
ALFAI
ALAMOA
«n s.ooooFtoo woo i.?ooot>oo ii o.
K ILI
r-FNSTTY OF- SMI. 10 2 . SO 0 0 tf + 0 0
cr^rr- ^TWATTHM
P.OOOOE-01
FALL VFIJ'CTTY f)F S-'lt. 10
j.onoo en
10 CM 1.00 AI.PpAC
i.ocnoE + oo
.0010
o.
B8
-------
RUN D-L-8
AM«IE>T CONOT T I HNS
DEPT^ JN FT. DENSITY T * G3AM PER TC. KY I Ki SOFT PFR SEC. VFL.
"FPTH n. u.OOOOE+01 6.0000F+01 l.OOOOF. + O?
AMp-nF.v?ITY 1.0P30F+00 i.0230F+00 1.0P35F+00 1 . 0??1^!-: + 0 0
I K! FPS,
KY
YU
3.SOOOF+01 u.500nf+oi S.5000E+01
O? S.onooE-oS l
I.OOOOE + OPH i.noooe
n .
IMF ?HSPF
1ST I.F I . .
Q K I D P P I ^ T ? I 7 F S1KFY1
U?K TETkA TFCH .SUGGESTED
r)l|MPFP TN?T ANTAMFOUPLY JNTO THF. OCFANi
'D IS AS-SUMFr> jn «F HEMISFHBRF
1KEY?
DT^CRl l.nnno OT\T,t?S 1.0000
A I P M 4 0 .23SORF.TA 0. "0 00 CM
r.AMA . pf. r.DRAG 1.00 CFPTC .010 CD?
FRJCTM .0100 Ft .1000
*LFAi l.oooo ii.r'A? 1.0000
4LiMOA .0010
5.0000F + 00 C'
K II.
i ! s T T Y nc 5 n i i n
rON'CFMTRATJ PN
1.?OOOF + 00 IJ 0.
1
p . s o o o E + o o
FALL VFIHCITY ilF S n|. in
P.OOOOE-01
5 , 0 0 0 0 F - 0 2
PKF.Y?
1 . 0 0 n 0 C 0 .SOOO
10 CHa 1.00 4I.PWAC
.0010
o.ooon
o.oooo OAMA?
V 1.0000F + 00 W 0.
B9
-------
RUN D-L-9
A'-'-BIFNT f. 0 H 0 I T 10 N ^
IN FT. DENSITY IN r,PAM PER CC. KY IN ?PFT PER SFC. Vfl.. IN; FPS.
0 l!
YK 3
*Y S.OOOOF-O? •'S.noooE-oi i.nnooF-o?
YU 5.0000E+01YW 5.0000F+01YF i.ononF+o?H l.onnnF+0?
:IA ) .OOOdF + OOWA 0.
|1PK'PATTI1M 1
WASTF MATFfvjiL ^UMPFD J «ST 4 M T ANpOUSL Y TNTD THF f.iCF
THp SHAPF OF THF CLHUO IS ASSUMFD TC"! BE Hf.MISPHF.RE
r-PTD PPlivT SIZE 51 KFY\ ') KEY? ?
USF TFTPA T'FCH SUGGESTFD fllfc. FF I C I F*TS
i.nnoo DINCR? i.nooo
.P350 BF.TA O.OO'OO C^ 1.00^0 C r> .'iOOO
,?S r.HRAG 1.00 fPRTC .010 CD? .10 TDU l.nn 41 PH4C .0010
FRICTN .0100 FI .1000
ALFA1 1,0000 4LFA? 1.0000 GA("A1 0.0000 f, AMA2 0.0000
AL.AMPA .0010
RR S.OOOOt + 00 WOO 1.3000E + OOLI 0. V l.OOOOF + 00 w 0.
K 1 I 1
MENSTTY OF SOI. IP 2.5000F + 00
2.0000E-01
FALL VF-LnrjTY OF SOLID
S.OOOOF-O?
BIO
-------
RUN D-L-10
AM8TEM CONDITIONS
DEPTH IN FT. DENSITY" IN GRAM PEP CC. KY TN SOFT PER SEC. VEl. IN F?S,
DEPTH o. u.onooF+ni fc.ooooE+oi t.oooop+o?
V 1.0?30fc+00 1.0230F+00 1.0PUOE+00 l
3 . 5000 F. -f 01
5. ooo of- o?
YK
KY
Ytl 5.0000F + 01YW 5
Li A l.Ononf + OOWA 1.0000F + 00
5.SOOOE + 01
«.SOOri£ + oi
5.ooooF-o3
l.onOOF-fO?H 1.0000F + 0?
, '5 n 0 0 F. + 0 1
RARGE nppRATION 1
WASTF MATERIAL DUMPED INSTANTAMFHIISI Y INTO THF: OCFAN
THF SHAPF OF THF. CLOUD IS ASSHMFD Tl) RF HEMTSPHFPF
TSTHP 3.hOOO£403SFC.
GRID PPIKT SIZE SI KFY1 1 KFY? 3 KFY3 0
USE TFTRA TF.CH SUGGESTED COEFF I C I FNTS
DINCR1 1.0000 DINCR2 1.0000
AL.PHAO ,?3SO BETA 0.0000 CM"
PAHA .25 TDRAG 1.00 CFRTC .010 CD3
FRICTN , 0 1 0 0 F 1 .1000
&LFA1 1.0000 ALKA2 1.0000
ALAMDA .0010
RB S.OOOOFfOO ROl'r t.30nOF + 00 II
Kill
DFNSITY OF SOLID P
CONCENTRATION
2 . 0 0 0 0 fc - n t
FALL VFLUCTTY OF SOLID
S.OOOOF-n?
1.0000 CD .S'OO"
10 CDa l.On ALPHAC
. 0 0 1 n
0.0000
0.000" GA*A?
V I.OOOOhtOO W In.
Bll
-------
RUN D-L-11
AMBIENT CONDITIONS
DFPTH in f-T. ()tMSITY IN GRAM Pt« CC. KY IN SOFT PFK SEC. VEL. IN FPS,
DEPTH 0. q.OOOOfcfOl 6.0000t+01 l.OOOOt+02
AMB-DhNSJIY 1. 02301+00 1.0230E+00 1.0300E+00 1.0300t+0u
YK .S.50COF-KJ1 «,^iOOOE + 01 b.SOOOfiOl 6.bOOOf.-»01
KY 5,OOUOt-C
-------
RUN D-S-I
AMRTFNT nPNDTTTnNS
HFPTH TN FT. nFNSTTY TM GRAM PFP CC . KY TW SfiFT PFR SF.C. VEL. TN FPS,
nF^TH o. u.ooooF+01 f.ooooF+oi
AMP-HFNSTTY I
Y ! i s.onnnc + ojYW ^.no^OF + ^IYF" 1.00nPE*npH l.
1 1 A n . win.
4?TF K'ATFRTAI.. rvjMOfrpv T M <5 T A W T 4 MFOU^(_ Y INTO THF
nF TMF ri.nl'r T S A^^llMFn TQ RF
T ST7F SI KFY1 \ KFY? ? KFY"? 0
TFTR& TFCU SIlRRF^TFn fOFFF I C T FMT ^
r. T w r P t t . o o o f> n T w r P >
, o.onnorM t.nooo rn
.?c rn^jp '1. .00 CFRTC .^10 cn^ .in rr»u i.oo ALPHAT .noto
. 0 1 (1 0 F 1 .1^00
AIPA1 1.0000 Aj. FA? 1.. 0000 GAMA1 0.0000 RAMA? 0.0000
.0010
OP c;.ooooFfOo pnn 1 . 1 ?noF + oo n o. v I.OOOOF + OO w o.
V 11 1
TTY OF Sn|TH l.ROOOF + nO
FAI.I VFLn(".TTY HF S^l. JH
^ . noooF-o"?
B13
-------
RUN D-S-2
T rrMr)T TTTiTv
TV PT. . ncr.^TTY IN n°AM PF9 TC. KY T N SOFT PF0 SfT . VFI
^ F " T M n. u. oooo r + 01 f, . o P o o F + o 1 I.OOOOF+O?
ys M n - n p Ki c. T T Y i
. TM FPS,
ICY
TF
n.nnn T?
TP
TCTP6
1KFY?
AI PWAn
<~Aw4 .
r " T r T u
AIFA1
AlAMhi
i.nnno OTK:rop l
QFTA o.onnn CM
.oo pFPTr .010 cn?
. o i o o F 1 .1000
1.0000 ALFAP 1.0000 HA^
.00)0
1 I
1.1?OOF + 00 " ft.
1
i.nnpn rr?
10 rnu i.no *IPHAT .0010
O.OOOO r, A M A ? 0.0000
V I.OOOPF + OO w o.
F4| I. VF|. TCTTY OF Snl. TO
1 .SOOOF-0 1
S.OOOOF-O-?
B14
-------
RUN D-S-3
fiMT CCiNDT ri'.JMS
DEPTH IN FT. OFNS11Y IN GHAI PfK CC. KY IN SOFT PtK SEC. VEL. I IN FPS.
DtPTH o. U.OOOOEtOl 6,0000t + 01 1 . 0 0 0 0 1" + 02
AMb-OtK'SITY l.0230r+00 1.0250ttOO 1.0260t+00 l.0?6()k+00
KY
YU 5.00HOF
DA 0.
i.SOOOt+01 U.bOOOfc+01 S.6000K+01 h.bOOOh+01
S. 0000 £-02 b.OOOOE.-0'i l.OOOOt-02
Yw
F 1 . OOOOt" + 02H l.OOOOfc + 02
0.
RAKGF UPFNAlIOiM 1
WASTK MAff-.kiAL DUHPLO INST AMANFOUSLV INTU
THE. SHAPf: UF THK CLC'UD IS ASSUMED T U Bt HEMJSPHtKt
TSTOP 2.1000L + 0.4SF.C.
RHID PliI NT SIZE b'l KEY! 1 KfcY^ 3 HF.Yi 0
USE 1ETKA TF'CN SUGGESTED CdfcF F It I F.N I S
DINCR1 1,0000 DINCK2 1.0000
ALPHAO .^:SbO iiiLTA 0.0000 CM
(JAMA .2b CDRAG 1.00 CFRIC .010 CD3
F R 1 C T N . 0 1 0 0 F 1 .I0i>0
i.oooo ALFA? i.oooo GA^AI
.0010
«b s.oooot+oo nno i.iifoottoo u o.
K i L i
DENSITY OF SUI..ID l.SOOOt + 00
CUNCf-.lvTkAT 1UM
1 .bOOOE-01
FALL VtLUCITY OF SOLID
S.OOOOE-03
1,0000 CD .5000
10 CO'J 1.00 AI.PHAC
.0010
o.oooo
ti.ooco GAMAS
v i.oooot + oow o.
B15
-------
RUN D-S-4
AMBIENT CONDITIONS
DEPTH IN FT. DENSITY IN GPAM PER CC. KY IN SOFT PFR SEC. VFL. IN FPS,
DEPTH 0. y.OOOOE+01 6.0000E+ni l.OOOOE+02
A.MR-DENSITY i.OSSOE+OO 1.0250E+00 1.0250E+00 1.0?50E+00
YK
KY
3.5000E+01 U.5000E+01 5 .5000E + 0 1 6.5000E+01
5.000PE-02 5.0000F.-03 l.OOOOE-0?
YD 5.0000E+01YW 5.0000E+01YF 1.0000Et02H l.OOOOE+02
U4 0. WA 0.
RAPGE
KASTF MATERIAL PUMPEH I NST AMT ANFOUSL Y INTO THR OCEAN
THF SHAPE OF THF CLOUD IS ASSUMED TO RE "EMTSPHFRif
TSTOP J.OnOOFfO'JSEC.
PHID POI*'T SIZE 51 KEY1 1 KEY2 2 KFY3 0
USE TFTBA TECH SUGr,E?TED rnEFFlCIEKTS
t.onon DINCP? i.oooo
Al.PHAO .2^50 BETA n.OOOO CM
,?s COPAG i.oo CFRIC .010 rn?
.0100 Fl .1000
ALFAI i.oooo ALFA? i.oooo GAMA
ALAMOA .0030
KR S.OOOOF+OO RCO 1.1200F+00 U 0.
K 1 L 1
DENSITY OF SOLID 1.8000E+00
1 .SOOOE-01
5.0000E-03
FAI L VFLTC1TY OF SOLID
1.0000 CD
to cna i.
.5000
ALPHAC
.ooto
o.oooo
n.oooo GAMA?
V l.OOOOE+OO W 0.
B16
-------
IN FT.
DEPTH
DF.MSITY !M Gf
o.
1.0230F + 00
RUN D-S-5
M PF« CC. KY IN VSPFT PFR SFC. VEL
U.OOOOF+C1 f-.^POOFfOl l.OOOOE + 02
1.0P30F + 00 !.0?40E + 00 1.0?aoe*00
FPS,
YU s.
3.SOOOF+01 a.5000E+0! 5.5000E+01 6.SOOOE+01
5.0000E-0? S.nOOOF-03 l.OOOOE-0?
YF i
'rt a o .
•s'43TF
TH£
TSTHP 1 .
GPID POIN'T SI?F. SJ KFYi
t'SF TET^A
L ni|MCF!> JNST AMI A\'K.riilSL.Y TMTf! THF PT.F4W
THE CLn:i[) IS ASSUMED TO Pfe HFMJSPHFPF
1 KFY? ?. KFY3
1.0000 D
AlPHiQ .?^508ETA O.OOOnrM 1 . 0 0 0 0 C D .5000
r,iMi .PC; CORAS. 1.00 C F " T C .010 Cr>3 .10 CDil 1.00 aiPH*c .0010
F P I C T M . 0 1 0 0 F I .1000
4LFM 1.0000 4 1. FA? 1.0000 Giwi1 O.OPOO '~AMA2 0.0000
Al A^PA .0010
C;.OOOOF+-OO Ron i.i?'ioF*oo u ^
K i L 1
OF SGI TO 1.8000F + 0
FALL VFLOriTY OF SOLID
5.0000E-03
v I.OOOOF + OO K o.
B17
-------
RUN D-S-6
DENSITY IN G»AM PER CC. KY ]> SOFT PEP SEC. VFl. IN EPS,
0. 4.0000E+01 6.0000F + 01 l.OOOOE+0?
l.o?30F+no 1.0?30E+oo 1
3.5000F+01 ii.SOOOE + 01 5.SOOOE+01 6.5000E+01
5.000nE-0? 5.0000E-03 l.OOOOF-0?
l.OOOOE + 02
0.
MATE"TAL
INSTANTANEOUSLY INTH THE OCEAN
SM4PE OF THF CLOIJH IS ASSUMED TO «F HE»-
-------
HEPTH
r F D T H
FT.
RUN D-S-7
TicviS
DENSITY I* GPA" PFR CC. KY IK- SC.'F'T PFP SEC. VEL. IN FPS,
0. 4. 0000 E +01 ^.OOOOF+ni 1.0000E+02
1.0?30F+00 1.0230F+00 1
KY
YU
5.0000E-0? 5.0nnnF-03 l.OOOOF-02
^IYF l.nooOE + n^H l.OOOOF + 0.?
i ) A l . 0 o 0 0 F + n 0 w A 0 .
TNTH THF. OCFAN
THF PHAPE OF THE CLOUD IS ASSUMED TO BE HFMISPHF.PE
r.PTO PnjijT STZF 51 KEY] i KEY2 ?. KEY3 0
IISF TFTRA TEC* SUG^FSTED CPFPFTCIEMS
AlPHiO
f-A^-A
FRICTN
4LFA1
ALAHOA
1.0000 DIMTR2 J.OOOO
,?351RETA n. 0000 CM
CDPAG 1.00 CFRIC .010 CH3
.0100 Fl .1000
1.0000 ALFA2 1.0000
.0010
RB ^.OOOOE + 00 POO 1.1200E + OOU 0.
K 1 L 1
DENSITY PF SOLID l.«OOOE+00
CONCFNTRATJON
1.5000F-01
FALL VELOCITY OF SOLID
5.0000E-03
1.0000 CT) .5000
10 CQ4 1.00 ALPHAC
0.0000
.0010
0.0000
l.OOOOF + OO
'0.
B19
-------
RUN D-S-8
AMBIENT CONDITIONS
DFPTH JM FT. DENSITY JN GPAM PER CC. KY IN SOFT Pp.R SEC. VEL . IN FPS,
DEPTH 0. 4.0000F+01 6.0000F+01 1.0000E+0?
TTY 1.0??OE + 00 1 . 0?30{- +00 l.OPUOF+00 1.0?«OE + 00
YK 3.5000E+01 4.5000F. + 01 5. 5000 F. + 01 6.5000E + 05
KY 5.0100F-02 5.00006-03 l.OOOOE-0?
Yll 5.flinOF+OtYW 5.0000F+01 YE l.OOOOE+OPH l.OOOOE+02
UA l.noor.FfOOWA l.nnonF + 00
F MATERIAL DUMPFD INS T A MT ANEOUSL Y INTO THF. DCEAN
THE SHAPF (IF THF CLOUD IS ASSUMED TO <3F. HEMISPHERE
TSTOP 1 .OOOOF-t-OUSEC.
G&ID POIMT SI7E SI KEY1 1 KEY? ? KEY^ 0
USE TETRA TFCH SUGRFSTFD COEFFICIENTS
C'lNCKl 1.0000 DIN'CR? 1.0000
AI.PHAO ,?350 RPTA 0.0000 CM J.OOOO CD .5000
GAMA .?S CORAG 1.00 tFRTC .010 CD3 .10 CD4 1.00 ALPHAC .0010
FPICTN . 0100 F1 .1000
AI.FA1 1.0000 ALFA? 1.0000 GAMAJ 0.0000 GAMA? 0.0000
AL.AMHA .0010
CP S.OOOOF+00 ROD 1.1200F+00 U 0. V 1.0000E+00 K 0.
K 1 L 1
HFK'SITY rip SOLID 1.8000E + 00
CONCFNTRATIflM
1 .5000E-01
FALL VFLDCITY OF SULID
B20
-------
AMHIFNT
DEPTH IN FT.
f.'EPTH
AMR-hENiSTTY
RUN D-S-9
DENSITY T.M GRA" pfR c C . KY TS SOFT PFP S(-C
n. a.oooot+oi 6.0000F+01 l
00 1 . 0?30F- +00 1.0260F+00 I
VFI . TM FPS.
YU s.
MAO.
S.ononp-n?
i.ooooF
\fl A 0 .
WASTF WATH.KIAL DUMPF.D TMSTAMTANFOUSLY TMTO IMF
THF SHAPF PF TMF ClHIin IS ASSU^F.n TIT HI-
rstnp i.onooE+oasFc.
ORIO POINT SIZE SI KFY1 1 KFY?
USF TETPA TFCH SUGGFSTFD CHfFF TC IF^'TS
1,0000 DINCR? i.nooo
,?i,bo PF;TA o.oooo rn
GAMA .?S CDHAG 1.00 CFPIC .010 COT
. 0 1 0 0 F 1
1.0000 ALFA?
.0010
FRTCTN
AI..FA1
ALiMDA
Rnn i.i
K 1 I 1
nf- SOLID
CflMCFNTRATIGN
KAIL VFL"CTTY OF SOL.IO
1000
1.0000
ti o.
i,«ooor + oo
1 .SOOOF.-01
I
1.0000 en
10 Cnu t.OO A| PI-AC
0,0000 GAMA?
o.
.0010
0.0000
« o.
B21
-------
RUN D-S-iO
AMBIENT CONDITIONS
DEPTH TM FT. DENSITY IN GRAM PER CC. KY IN SQFT PER SEC. VtL. IN FPS.
DEPTH 0. U.OOOOF. + 01 6.0000F+01 1.0000F. + 02
AMR- DENSITY 1.0230F+00 1.0?30F+00 1.0260F+00 1.0260F+00
YK
KY
3.SOOOF. + 01 4.5000F.+ 01 5.5000F+01
5.0000E-0? 5.00QOE-03 l.OOOOE-0?
YU S.OOOOF+OtYW S.OOOOF+OlYfc 1.00(10E + 02H 1.0000E. + 0?
UA 0. WA 0.
RARGF
WASTE MATFRIAL PUMPED TMSTANTAMFHUSLY INTO THF OCFAN
THF SMAPf fF THE Cl ODD IS ASSUMED TO Btf HF^ISPHfcPf
TSTOP 1 .onoOF + OISEC.
RRTO Pr.IKiT SIZE SI KFY1 1 KEY? 2 KFY3 0
USE TFTRA TECH SURGESTFD COEFFICIENTS
DIMCP.1 1.0000 DINCR? 1.0000
ALPHAO .2350 RFTA 0.0000 CM
GAMA ,2S CDWAt; 1.00 CFRIC .010 CD3
FPICTN . 0 1 0 0 F 1. .1000
AL.FA1 1.0000 ALFA2 1.0000 r, A *» A
4| AMDA ,0010
PH s.oonoF + 00
K 1 L
DENSITY np SOLID
tOKCFNTRATTON
1.1200F + 00 U 0.
1
l.BOOOf+00
FALL VELOCITY OF Si.H.TD
! .SOOOF-01
S.OOOOt-03
1.0000 CO .5000
10 C0<4 1.00 AIPHAC
0.0000 GAMA2
.0010
0.0000
o ^ 0.
B22
-------
RUN p-s-ii
AMBIENT CONDITIONS
DEPTH IN FT. DENSITY IN GRAM PER CC. KY IN SOU PER SfC. vfL. IN FPS.
DEPTH 0. /J.OOOOe+Ol fc.OOQOE+01 6.5000F+01
AMB-DfcNSlTY 1.0230F+00 1.0230F+00 1.0260F+00 1.0P60E+00
YK 3.SOOOE't-01 a.5000f + 01 b.bOOOE + 01 b
KY b.OOOOE-02 5.0000fc-03 l.OOOOE-02
YU 5.0000E+01YH b.OOOOEtOlYt o.SOOOh+Olh 6.bOOOE+01
U A 0 , W A 0 .
BARGF DPtRATIUN 1
wASTt MATERIAL DUMPED I MS] A NT ANhCuJSl Y INTO THE OCtAK
THE SHAPE OF THE CLOUD IS ASSUMED TO BE rifc'M I SI'HERF
TSTQP 1 .OOOOE + 02SEC.
GHIO POINT SIZF SI KEY I 1 KEY2 £ KEY3 0
USt 7FTHA TECH SUGGESTF.O CUt.FF 1C ItM T S
DINCH1 1.0000 OINCK2 l.OOOu
ALPHAO .2450 BETA O.OOOU CM 1.0000 CL> .bOOO
KAMA .25 CD«AG 1.00 CFRIC .'.HO CDS .10 C\)U l.UO Al PrAC .OHIO
F R I C T N . 0 1 0 0 F 1 .1000
ALFAI i.oooo ALFA? i.oooo GAMAI o.oooo HAMA^ n
ALAMQA .0010
H6 5.0000E + 00 ROH 1.1200F.+OD u 0, V b . iKi'Mfc-nxi ••• 0 .
K 1 L 1
DENSITY l!F SOLID 1.8000E + 00
CONCFNTRATION
1 . 5 0 0 0 fc - 0 1
FALL VH f)CI TY OF SiJLlD
b.OdOOE-lii
B23
-------
RUN D-S-12
IN FT. OffNSTTY IN r;WA* PfJR CC. *Y TN SOFT PtR SFT. Vfcl. T
PFPTH o. 4.ooooF+oi 6,onnofc+oi i.ooon
AMf-r>h>isTTY j .op.^oE' + no i.op^or + fio i.n?hOF + nn i.o^^or
YK
KY
HA o.
?.5nnoF+oi /4.SOOOF + 01 s
S.ooooF-n? s.oonofr-03 l.nnonF-o?
1.0000F+OP
o ,
WASTF MATFRIAI DIJMPFn T H.STAN' T A NFPIJSI. Y 7 M 7 f) IMF OTF4N
THF .SHAPF PF THF CLOUD IS ASSHMF.O TO BF HEMJSPHFHF
TSTQP ! .OOOOKtOaSFC.
RWTD PRINT SIZF SI KfY1 i KEY? 2 KKY^ 0
IISF TETR/s TFCH SHRRfSTFO CHFFF If. TF^'TS
4LPHAO
i.nnno OTNTR? i.oono
.2^50 HETA 0,0000 CM l.nono CO .5000
CDRAG 1.00 Cf-RIT .010 fUi? ,10 CDa 1.00 ALPHAC
F « T C T N .0100F] ,1000
AIFAI i.oooo ALFA? i.nooo
ALAMDA .0010
RH l.OOOOF + 01 HHI] 1.1200F + 00 If 0.
K It. 1
DFN'STTY OF SOLID l.«OOOF + 00
1 .SOOOF-01
'j.OOOOF-03
FAIL VFl.HCTTY fjF SOL TO
.00(0
o.oooo
o.oooo I;AMA?
V l.OOOOF + 00 w 0,
B24
-------
RUN D-S-13
AMBTFNT CPWrnTiriNS
DFPTH IN FT, DFNSITY IN GRAM PF.R CC. KY IN SOFT PfcS SFT. VFI_. IN FPS,
DF.PTH 0. 4.0000F+01 6.0POOF+01 l
YK
KY
Yll 5.1
UA 0.
YW S
WA 0
). .1
1 .OOOOF+02
BARGF PPFRATinw J
WASTF WATFPTAL OHMPFH U'st ANTAMpniisi. v TNTO
THF SMAPF OF THF CLnilD IS ASSUMFn TO RF
TSTOP 1 .nnonFtoysec.
PPTI.) POINT ST7F SI KFY1 ?. KFY?
MSF READ TW COFFFICIFNTS
1.0000 DTNTR? 1.000ft
ALPHAO .2350 RFTA 0.0000 T
GAMA .2S CDRAG 1.00 CFRIC .010
FP, TTTM .01 00 F 1 10.0000
ALFA1 1 . 0 0 ft 0 A L F A 2 1.0000 KAMA
At.AMDA .0010
PR S.OOOOF + 00 R'lO 1.1200F
K 1 I 1
DENSITY np sni.in I.ROOOF + OO
t .
FAl L VFLOCTTY OF
OCFAN
i.oooo r. r> .5000
, i o cna l.oo AI.PHAT
0.0000 f, A M A ? 0.0000
V 1.OOOOFtOO W 0.
B25
-------
RUN D-S-14
AMRTRMT CONDTTinMS
IN FT. DFMSTTY IN RRAM PfP fC. KY IN SOFT PFR SEC. Vfl . TH FPS.
o. 'i .OOOOF + O i fc.ooooF+o) 1 .ooniu'
VK ^
KY s.noonF-o? s.onnoF-oj
YII 5.ononF+o i YW s.nooor+oi YF i.nonnF+o?H i.ooooK+n?
IIA n . WA a.
RARGF OPFRATTHN )
WASTF HATFRTAL DPMppp TNST AMT ANFOlISI Y T^'TH TMF
THF SHAPF HF THF ci.nun is ASSU^FD TH HE HEHISPHFRF.
TSTOP i .nonnF+nusFc.
nPJD PHTNT SI7F SI KFY1 ? KF-'Y? ? Kf-Y'* 0
IISF "FAO TN c
1.0000 DTNCP? 1.0000
AtPHAO .PSSO PFTA 0.0000 CM 1.0000 CH .5000
T,AMA ,?5 CORAG 1.00 rFPTf .oio cn^ .10 cna i.no AIPHAC .onto
FRTCTN .0100 F1 t.0^00
ALFA1 1.0000 AI.FA? 1.0^00 RAMA1 0.0000 RAMA? 0.0000
.0010
Rfi 9.0000F400 PHfl 1.1200F+00 II 0. V 1.0000F + 00 w ri .
K 1 L 1
HF snirn j
1 .SOOOE-01
FALL VFLHCTTY OF SOI TD
s.noooF-oi
B26
-------
RUN D-S-15
AMBIENT CONDITIONS
DF.PTH IN FT. DENSITY IN GRAM PER CC. KY IN SOFT PER SEC. VEI . IN FPS,
HFPTH 0. a.OOOOF-fO! 6.0000E+01 l.OOOOE+n?
AMB.DENSITY 1 .0 ?30E+00 1.0?30F+00 1.0?aOE+00 1.0?
-------
RUN D-S-16
DEPTH TK FT. DENSITY TN GRAM PFW ft'.. KY J K' SOFT PF.R SFf. VFL. TN
o. u
YK 3.snooE+oi a.soooE+nt s
KY S.nooOF-O? S.ooonE-03 i.naooF-o?
vti S.oooflF+oi YW s.nooor+niYF i.ooonE*o?H j
1)40. l»' A 0 ,
RARGF
WASTF KATFRTAI DIIMPFD I MS T ANT AMFOUSl Y TN'TO THF
THF SHAPF Op THE CLnilD TS ASStJMFH TO 8F HF.MTSPHFRF
TRTOP \ .OOnnF + OdSFC.
HRIO POINJT SI7f «51 KFYt ? KFY? ? *FY3 0
USF RfeAD TM COFFFICIFWTS
i.ooon DTNCR?
ALPHAO .2^50 RFTA 0.0000 CM 1.0000 CH .SOQO
RAMA .?S TDRAG 1.00 CFRTC ,OJO Cn? .10 C^/J 1.00 Al PMAC .0010
FRTCTN 0.0000 F1 .1000
ALFA1 1.0000 ALFA? 1.0000 RAHA1 0.0000 r,AMA? 0.0000
ALAMDA .0010
PR S.OOOOF400 ROO !.1?OOF + 00 U 0. V 1.0000F. + 00 W 0.
K 1 L 1
DFNSTTY OF SOLID l.BOOOE+00
mNCFMTRATTON
t .
FAt | VFLHCTTY OF SOLID
B28
-------
RUN D-S-17
AMBIF.MT CHNntTinNS
DFPTH TM FT. DENSITY TN GRAM PFP CC. KY T» SOFT PER SFC. VF-.l. . IN FPS,
HFPTH o. u.onooF+01 *.OOOOF+OI I.OOOOF+O?
i.o?60F+oo
YK 3
KY s
YU 5.0000F4-01 YW s,
UA o. ^A n.
S.SnonF + 0
5.ooooF-o^
YF
P4RGF
WASTF MATF»IAL D'
THF SHAPF PF THE ci.nnn is
TSTHP 1 .ftftOOF + OUSFC.
RR70 POINT SIZE S! KF-Yt
USF RFAD TM fOFFFICIFNTS
twin TUF
TO MF H
? KFY?
? KFY^
DINCRi 1.0000 OT^TR? 1.0000
AIPHAO ,?350 BFTA o.oooo CM i.nnnn r:n
RAMA ,?S TDRAG 1.00 CFRTC .100 CD'S .10 C^/i 1.00 AIPHAC .onto
FRTCTN .OtOO Ft .1 000
ALFA1 t.oooo ALFA? i.oooo GAMAI o.ooon GAMA? o.oooo
ALAMPA .0010
PR s.oonof+oo Rnn i.iponFtoo n o. v I.OOOOF+OO w o.
K 1 L 1
nFMSTTY OF SOt TD 1.POOOF+00
1 .SOOnE-01
S.OOOOF-03
FAI.l VFLHCTTY OF SOLIO
B29
-------
nFPTH TN FT.
HFPTH
RUN
DENSITY TM KRAM Pf-R PC. KY IN SOFT PFR St'C . VFI
o. a.oooftFffM fc.oonoF+fti i.oonnp + n?
I.O?IOE+OO i.o?30Ffon 1 .
IN EPS
KY
YU s.
DA o.
WA n,
U .
B.'SOOOF. + Ot fe.SOOOF +
t.onnnF-n?
t ,nonnF+o?H i.noooE-»-n?
RARGF OPFRATTHM 1
WASfff MATFRrAl. OI/MPEO rMST/SMTAWEOUSI.Y INTO THf
THF SHAPF OF THE rt nun is ASSUMED TO RF HC^
TSTOP i .OOOOF + O^JSFC.
RRTD POINT SI7F SI KFYt ? KFY? ? KEY3
USE P-EAD IM COFFFirTFNTS
DINCR1 1.0000 DTNTR? t.OOOO
ALPHAO .?"?50 RFTA 0.0000 CM
GAMA ,?S CDRAG 1.00 CFHTC .010 CD3
FRTCTN .0100 Fl .lOftn
AI.FA1 1,0000 ALFA? .1.0000 GAMA1
ALAMOA .0010
RB S.OOOOE+00 RQn 1.1POOF+00 U 0.
K 1 I 1
DENSITY OF SOLID l.flOOOEtOO
rnNCFNTRATTON
1 .SOOOF-01
FAIL VFLPriTY OF SOLID
1.0000 CD .SOOO
10 COO 1.00 AlPHAC
.1000
0.0000
0.0000 RAMA?
V t.OOOOF+00 W 0
B30
-------
RUN D-S-19
AMRTFMT CONDITIONS
DFPTH TN FT. OENSTTY TN GRAM PFR CC. KV TN SOFT PFR SFC . VFl . IN FPS.
OEPTH o. ^.OOOOF+OI 6.ooooE«-oi i.ooooE+02
1.0210F+00 1.0P30F+00 1.0260F+00 1.0260F. + 00
YK s.snooF+oi a.5onnE+ot s.soooFtoi 6.5nooF+oi
KY S.OnOOE-02 5.0000E-03 l.OOQOF-0?
YU S.OOOOEtCHYW S.OOOOF + 01YF l.f»OOOE + 02H l.OOOOE + 0?
IIA 0. WA 0.
HAPRF nPFHATTON 1
WASTE ^ATFPTAl nUMPFD I WST 4 MT ANFOUSt Y J WTO THF
THE SHAPF OF THF CLOUD IS ASSUMED TO HF
TSTHP i.ooncF+ft«SFr.
T.RTD POINT SIZE 51 KFY1 2 KEY? ? KFY^
USF «FAO IN
HTNCR1 1.0000 DINCR? 1.0000
ALPHAO ,23so RFTA o.oooo CM i.oooo rn ,«sooo
r,AMA ,?S f.DRAG 1.00 CFPTT .010 CDl ,tO Cny t.OO At PHAC 0.0000
FPTCTN .0100 Fl .1000
AI.FA1 1.0000 4LFA? f.OOOO GAMA1 0.0000 GAMA? 0.0000
. 00 J 0
RR ^.OOOOFtOO Rnn 1.1200F+00 H 0. V l.OOOOF+00 W 0.
K II. 1
DFMSITY np sni.in I.ROOOF*OO
t .5000E-01
FALL VF| nrTTY flF SHL.m
S.OOOOF-0?
B31
-------
RUND-LL-1
CHNO J f JiiN.S
H IN FT. DH-lSUY IN GkArt PI- K 1C. KY II* SUFI PfcR SPC. Vf.L. IN FPS
•). a.OOOOltOl b.OOOOt + 01 i.OOOOE+02
I Y l
VK 3
KY s.oooot-o^ 5.oooot-o3 i.ooooe-ca
YIJ b.OOOOf+OlYw b.DOOOt + OlYK 1.0000F + Oi?H l.
L'A 0. MO.
WASTi- MAlF^IAL UUMPtO INSTANIANHUUSLY IiMlLi Tut Utfc'AN
Ihfc" SHAPF OH TliF CLdUU IS A.SSUMhD TU lit
t,Hlu t'UIhl bl/t SIKfYI 1KF.Y? 3 Kf. Yi 1
HSK Tt'THA TFCH SUGUt'STFfj CUFFT 1C 1 1 N I S
IUNCK1 1.0000 OlNt;«2 1.0000
AL^HAO .^^50 bh.TA 0.0000 CM 1.0000 CD .5000
GAMA .25 CDKAii 1.00 CFNIC .010 CDS .10 CD'* 1,00 ALPHAC .0010
FHTCTN .0100 Kl .1000
ALFA1 1.0000 ALFA? 1.0000 T,AMA1 0.0000 GAMA2 0.0000
ALAMDA .0010
Rb 5.0000h+00 KUU l.OSOOEtOO II 0 . V 6.0000E-01 rt 0.
K 1 L 1
Dt'NSlTY IjF SOLID 2 .50 00f>00
CUMC6N1RATION
2.0000L-01
FALL VELOCITY OF SOLID
B32
-------
RUN D-F-1
AMBItNT CUMHTIUNS
DEPTH IN FT. OFNSITY IN GRAM PER CC. KY IN SOFT PER SEC. VEL. IN FPS,
DEPTH 0. 4.0000Et01 6.0000E+01 1.0000E.+ 02
AMH-OENSiTr I .ci^OE + OO 1.0230E+00 1.0231EtOO 1 . 0. 5 K F. Y 3 0
usr U-TKA IFCH s^G^FSitu ccFFt-ICIE^TS
Dlr.'O 1 l.OfUjd OIMI'.»2 1.0000
Al PHAD .^ibO riEIA 0.0000 CM 1.0000 CO .5000
GA.-IA ,,fS C.I)HA(, I.no fF^lt .010 CO3 . 1 «'> C'i« 1.00 ALPHAC .0010
FU'ICT:! .0100 Fl ,1000
ALF/'l J.OooO ftl.FAf.' l.uoOO bAHAl O.CUOO f.A^A^ 0.0000
A I Ai-IH A .OHIO
'•'« S.fiOr. Of >0'J f^1111 1 • '''Si/01 +00 U n. V 6.0000fc-01 « 0.
K ! L 1
DtUSlT/ i/F S'H I'.1 1 ,0230k. tOO
CUNfK'. l ^A i If1;,
o.
FA1.L VKI ' :i. J T Y uF S"l ID
0.
B33
-------
RUN D-FS-l
AMBIENT CONDITIONS
DEPTH IN FT, DENSITY IN GRAM PER CC. KY IN SOFT PER SEC, VEL. IN FPS.
DEPTH 0. 2.0000E+01 4.0000E+01 l.OOOOE+02
AM8-DENSITY 1.0200E + 00 1.0200E+00 1.0300E+00 1.0350E400
YK I.BOOOEfOl 2,2000Et01 3.BOOOE+01 4.2000E+01
KY 5.0000E-02 5,0000t-03 l.OOOOE-02
YU 2.0000E + OIYW 2. OOOOE-t-0 1 YE ^.OOOOE-fOlH l.OOOOE + 02
0.
BARGE GPfcHATIQN 1
KASTE MATERIAL DUMPED INSTANTANEOUSLY INTO THE OCEAN
THF SHAPE OF THE CLOUD IS ASSUMED TO BE HEMISPHERE
TSTOP 3.6000E+03SEC,
GRID POINT SIZE SI KEY1 1 KEY? 3 KEY3 1
USE TETRA TECH SUGGESTED COEFFICIENTS
DINCKi 1.0000 DINCR2 1.0000
ALPHAO ,?350 BETA o.oooo CM i.oooo CD ,5000
GAMA .25 CDRAG 1,00 CFH1C ,010 CD3 ,10 CD4 1,00 ALPHAC .0010
FRICTN ,0100 Fl ,1000
ALFA1 1.0000 ALFA2 1.0000 GAMA1 0.0000 GAMA2 0.0000
ALAMDA .0010
RB 5.0000E+00 ROO l.lOOOfctOO U 0, V l.OOOOE+00 W 0.
K 1 L 1
DENSITY OF SOLID l.OOOOE+00
CONCENTRATION
2.0000E-01
FALL VtLUCITY OK SOLID
-l.OOOOE-02
B34
-------
RUN D-FS-2
AMBIfcNT CONDITIONS
DEPTH IN FT. DENSITY IN GRAM PER cc. KY IN SOFT PER SEC. VEL, IN FPS.
DEPTH 0. 2.0000E+01 4.0000E+01 l.OOOOE+02
AMB-DENSITY 1.0200E+00 1.0200EtOO 1.0300E+00 1.03bOh+00
YK l.flOOOE+01 2.2000E+01 3.8000£tOi 4.2000E+01
KY S.OOOOE-02 5,0000t-03 l.OOOOfc-02
YU 2.0000E+OIYH 2.0000E+01YE 4.0000fc+01H l.OOOOE+02
LIA l.OOOOE + OOWA 0.
BARGE OPERATION 1
HASTE MATERIAL DUMPED INSTANTANEOUSLY INTO THE OCEAN
THE SHAPE OF THE CLOUD is ASSUMED TO BE HEMISPHERE
TSTOP 2.0000E+03SfcC.
GHID POINT SIZE bl KEY1 2 KEY2 3 KfcYJ 1
USF READ IN COEFFICIENTS
DINCR1 1.0000 DINCR2 1,0000
ALPHAO .21^0 BETA o.oooo CM i.oooo CD .5000
GAMA .25 CDRAG 1,00 CFKIC .010 CD3 ,10 C[>4 1,00 ALPHAC .0010
FWICTN .0100 Fl ,1000
ALFA1 1,0000 ALFA2 1,0000 GAMA1 .0200 GAMA2 0.0000
ALAMDA ,0010
R6 5,OOOOE*00 ROU 1.1000E+00 U 0. V l.OOOOE+00 W 0.
K 1 L 1
DENSITY OF SOLID l.OOOOE+00
CONCENTRATION
2.0000E-01
FALL VELOCITY OF SOLID
-l.OOOOE-02
B35
-------
RUN D-ZS-1
AMBTFt"T CONDITIONS
HFPTH IN FT. DENSITY IN f,PAM PF.R PC. KY TM SQFT PRR SRC. VEl . IN FPS.
o. 4.ooooE+oi 6.ooonF>oi I.OOOOE+O?
t.O?.30F4-00 1
5.5000F+01 fr.SOOQE-t-01
KY 5.0nnOF-n? S.OOOOF-03 l.OOOnF-02
vii 3.ooooF4.niYw a.nnooF + ni YF. S.IOOOE + OIM i.ooooF + o?
114 t . OOf>nf*OOwA 0.
WfiSTF ^iTFPUL niJMPF.n TMSTiNTAMFOuSL V I^TO IMF
THF SHAPE OF THF rL^IlD Tc ASSUMED TO BF
TSTOP 1 .OOOOF + O'iSF.C.
aRin PrilMT ST?F S1 KFY1 ? KfY?
IJSF PEAH TN C
i.oonn DIMCP? i. oooo
.(^so BFT4 o.onno r.«
G4'-'A ,?S rrjHAU 1.00 CFPTC .OtO CD3 .10 CHU 1.00 41.PHAC .0050
F (-• T C T M . M n o F i .1000
AI.FA1 1.00^0 ALF4? 1.0000
.0010
RP s.oonoF + no won i.t?ooF + oo u o. v I.OOOOF + OO ** o.
K ? I 1
PENSITY CF SHI.JO
-------
AM8IFMT
nfPTH TM FT.
DEPTH
SUN J-L-I
DENSITY JM RRAM PF.R CC. KY IN SOFT PFR SFC. VFL.
0. U.OOOOF+OJ 6.0000F+ni l.OOOOEfO?
i.o?30E+oo i.o?aoF+oo 1.0240E+00
FPS,
VK
KY
YU S.
i.l A 0 .
3.5000F+01
5.5000E+01 6.5000F. + 0
5.0000F-03 l.OOOOE-02
l.OOOOF + 0??
npffR«TION 2
ne TKPnucM 4 NOZZLF UMOFP A
TSTllP 1 .OOOOF-fOUSKC.
POJMT SIZF =>! KFYJ ). KFY?
TFTRA TFO SUfir.FPTFP COFFF T r 1 EM
'•>r»iCRl
RARGF
? KFY3
A |_ F A 1
4 I. 4 M r* A
l.onno OI
..1806 ALPHA?
l.iiO CFPJC
I . n n n o & L F A ?
. 0 0 t 1
1.0000
HFTA O.nooo C r> t.3000
CD3 .20 CH« ?.on ALPH43 .3536
.0100 Ft .1000 CM 1.0000
1,0000 R A c. A i 0.0000
.TFT RADIUS
VFLOTJTY
K
1
I.
.sooo DENSITY 1,3000 ANGI.F OF JFT
AT 1f>,0000 FT.
1
o.oonn
PALL VP.I. TCTTY op s^i. in ^
PAPGH VFIOC7TY 7.5000 ANT-LE WITH X-AXIS ISO.nOOP OFGRFF.S F'lRS . OQIOF + n? SF.C,
i'F SOI 70
0
7.SOOC1
- n . o n p i
Yl..:
FHQ t MI;. VI MR C(inRDINATF FIXfD ON THF
Y* YF »
7.5^00 7.5000
o.nnno -o.oooo - o . o n o o
B37
-------
RUN J-L-2
'T C(in TIT TOMS
IN FT. PF'iSnY I K: G&AM PJ-R CC. KY IN SOFT PF.P SEC. VEI . TN FPS.
E ° T -< a. y. nooo F + 01 *> . 0 n 0 0 F. + n l t.ooooF+02
i ,i
1 .
I .
YK
"Y
YH s.
i ' 4 n .
«.500nE+ol S.^noiF+n i
S.^nonp-oS 1 . '• oo^F.-n?
F: 1.0000F + n2H l.onooF + o?
A K:077LF. UMREW A
TSTOP 1 .OO^-I^ + O'JSFC.
r,Rin onT\'T si?F 51 KFYI i KEY? ?
USf TFT-iA TFCH SUGQKSTFO C'lFFFTCIFN'TS
nr^'C^i i.oooo oi NCR? i.o^oo
ALPHA) ' .0806 ALPHA? .^S^6 RFT\
G/SMA .?S CO«AG 1.00 CFRIC .010 CD? .20
ALf^AU .0010 FP1C1N .0100 Fl
4LFA1 1.0000 ALF4^ 1.0000 GA'-'A]
.0010
FALL VELOCITY OF S^H. in S.OOOOE-0?
CIARGF VFLOCITY y.sooo AK>GLF. WITH X-AXIS iso.oooo OFGRKES FnR5.noooE + o? SEC,
AMHTFUT riiffpNT F0« A MOVING COQROINATF FIXfO ON THE
0 Ylf YW YE w
7. SO on 7.S010 7.5001 7.SOOO
O.OOOn -0.0000 -0.0000
0.0000 Cr: 1.3000
2.00 ALPHA?
. 1 0 n 0 C^ 1.0000
0.0000 GAMA? 0.0000
JFT RADIUS .SOOO OR'-i?ITY 1.3000 ANGLF. flF .TRT 0.00^0
VFI.OCTTY 5.0000 4T 10.0000 FT.
K 1 L 1
nfrSITY OF Sfllln ?.5000F.+ C'0
B38
-------
RUN J-L-3
AM8IENT CONDITIONS
DEPTH IN FT, DENSITY IN GRAM PtR CC, KY IN SOFT PER SEC, VEL, IN FPS,
DEPTH 0. 4,OOOOE+01 6.0000E+01 l.OOOOE+02
AMB-DENSITY 1.0230E+00 1.0250E+00 1.0231E+00 1.0231E+00
YK 3,5000F.tOi 4.5000E+01 5.5000E+01 6.5000E+01
KY 5.0000t-02 5.0000E-03 l.OOOOt-02
YU 5.0000E+01YW b,OOOOE+01YE 1.0000E+02H l.OOOOE+02
UA 0. WA 0.
BARGF OPERATION 2
01SCHARGF THROUGH A NOZZLE UNDER A MOVING BARGE
TSTUP l.OOOOE + OySP.C.
GRID POINT SIZE 51 KtYl 1 KEY2 2 KEY3 0
USE TETRA TECH SUGGESTED COEFFICIENTS
DINCR1 1.0000 DINCR2 1.0000
ALPHA1 ,0806 ALPHA2 .3536 BETA 0,0000 CD 1.3000
GAMA ,2b CORAG 1,00 CFHIC .010 CD5 ,20 CD« ?,00 ALPHA3 ,3b36
ALPHAS .0010 FRICTN .0100 Fl .1000 CM 1.0000
ALFA1 1.0000 ALFA2 1.0000 GAMAJ 0,0000 GAHA2 0.0000
ALAMDA .0010
JET RADIUS .5000 DENSITY 1.3000 ANGLE OF JET 0.0000
VELOCITY s.oooo AT 10.0000 FT.
K 1 L 1
DENSITY OH SOLID 2.5000E+00
CONCENTRATION 2.0000E-01
FALL VELOCITY OF SOLID 5.0000E-02
BARGE VELOCITY 7,5000 ANGLE WITH X-AXIS 180,0000 DEGREES fDH5,OOOOEt02 SEC.
AMBIENT CURRENT FOR A MOVING COORDINATE FIXED ON THE HARGt
0 YU YW YE H
7.5000 7,5000 7.5000 7,5000
-0.0000 0,0000 -0,0000 -0,0000
B39
-------
DFPTH IN' FT.
C-EPTH
RUN J-L-4
OFMSTTY IN RRAM PER CC. KY IM SOFT P£R SEC. VEl.. Jw FPS,
o. 'j.ooooF. + oi 6.ooooF+nt i.ooooF+02
1.0230F+00 1.0230F+PO t
VK
3.500PF. + 01 /I.5000F + OI 5.50PPK+01 6.5000E+01
5.oocPE-o2 s.ooooE-03 l.ooooE-c?
YD s.ooooE+oiYy S.ooooF+o i YE
i.t A 1 . n n o ft c + 0 'H. 4 n .
l.poooE + n?
P THPllur-H a t:n?Zl.E UNDER 4 MrvtMG
fi L P H A a
A L c A 1
pnir.'T SfZE M KFY] i KEY? 2 KPY3
TFCH SUGGESTED C HFFF 1C IEMS
1.0000 njMrP? 1.0000
.OP.rift ALPHA? .3S36 HFTA O.OOin CO 1.3000
CPPftG 1.oo TFPTC .010 CD3 .20 COU ?.00 ALPHAS .3536
. (i 0 1 n F c J f T v . 0 1 0 0 p 1 . 1 P 0 0 C '•' 1.0000
1,0000 i\ L F fi ? 1.0000 GA"A1 0.0000 RA-A2 0.0000
.0010
.'FT HADIUS .sooo HF^STTY i.soon ANGLE np JET
vnnrrTY s.ooofi AT in. oono FT.
1
K 11.
PFMPTTY rr RPLIO
?.SoooF*no
vrirrjTY
VFLOTITY
FM ri.iRRF.vT
0 YD
P. SO O1"' ?
- f) . 0 6 0 f-
o.oooo
si a in S.OOOOF-O?
7.^000 A">'GLF WITH X-A^IS 180,00nn DEGREES FPS)5 . 0 0 noF + 0 2 SEC,
'ltJ A MOVTMT, Cnn^nTMATF F I V F P> PU THE
Y'-f YE H
7.SOOO 7.SOO"
0 . Q o 0 0 -0.0000 -0.0000
B40
-------
RUN J-L-5
ip>. T rrt>:ri T T T^Mt;
TM FT. np^PTTY TM T.PAM opo r f . XV T »> S'""'PT OF" ?Ff. . VFl. Tkl
<•> F r- T P n . /4 . A r. ft n F + n 1 A.npprF*l1 l.ftftOOF. + ft?
"c.TTv t . ipinF^n 1 . /v Ti F + 1 p. t ,r>?urF +no I . HP/IPF* on
.n 1 a . s nnoF + n 1
Hi j , o f> n « F + n o t: A i.
°APf:r nppr.- 1. y { n»i p
T|JOr,i,r.u ^ V077| F
T?THP i^
r.STP PPTVT ST7F SI KFY1 1 KFY? j> K F Y "?
M.cc 7FTPA TFr^J Slirr.p P
-LP wTTH X-AVTC jpo.p-ftnn r.FrroFFS FOP*;. rr.ooF-t-n? ?FC.
p T Y F ^ Ofi TWF
B41
-------
RUN J-S-1
AMRIfcNT rr-J'ITT Ilj'.S
OFPT>< IN FT. HKMSfTY TN fiPiM P£P CC. *Y 1 1> SOFT PFP SEC. VFI. . TM FPS.
o. U.OOOOF+OI h.^nooF+ni I.OOOOF+O?
YK
•^Y S.OOOdF-0? S.QiOOE-O^
A M r p T(."^nijGr( A ST771 F I'Nirif.rf 4 KnvIMG i*
: pi ] > T ST 7f si KPYI 1 Ki-Y2 P KKY3 0
TFIPA TFC^ 5ur,r-FSTPn CnFFFIC TEMS
Al,PHA1 .0^06 ALPHA? ."5S"!,6 SFTA 0.0000 TO 1 . T, fi fl 0
.0010 ForrTf» .OlrtO Fl .1000 C'' 1.0000
AI^A) 1,0^00 AI.FA? i.nooo GA^AI o.ooon GA^A,? o.onoo
4 L !•• * r1 A . n o j n
.'FT RADIUS .SOOO HF^SITY 1.1205 AMGLF. '"'F JF.T p.onno
VFlHTTTY S.O^PO AT 10.^000 FT.
' 11. 1
f-F f-r-Lir.- l.«QOOt>oO
FAI.L VFI/TTTY nc SOLIO S.noooe-03
Pr.F VFIPCITY 7.5ono ANGLE *TTU X-AXIS i K.I, noon HF.GPFFS FORs.ooooF + ns SF.C,
T rf'PFN'T FO^ i >-:OVIN!G COMKO I N A TE FTXFO ill., THF PAPGF.
fj Y|i Yw YF H
7.^000 7.^000 7 . S n 0 0 7. SO 00
-0.00 Of- O.OnOO -n.1000 -0.0000
B42
-------
DEPTH JM FT.
n-PTH
KUN j-s-2
T Tn»c;
OFMSITY IN GPAM PF.tf CC. KY I N SnFT Pf1? SFf. VEl. Il>: FPS,
0. '4.0000F + 01 6.0000E+01 l.OOOOE+02
'Y
YU 5.
"40.
ni a.SOOOE + ni 5.
S.OTIOF.-O? S.O^OnE-^S t.OOfiriK-02
l.OOOOE + 0?
i Y'V 5 . no 0 »P + n ] YF l.OOOOF.
A' 4 0 .
1 .
r, R T n P n I v T S T 7 E
4 MPVJNG PAPT,F
1KEYS
?KFY3
i.nnoo O!\T
r,A-'A .?s CD^AP i.oo
ALPHA/I .0010 Fwtc
ALFAI 1.0000 ALFA2
.0010
-J^1 RADIUS .soon
VELOCITY ^.oooo AT
K 1 l_ t
OF^SITY PF SOI. TO
VFI^CTTY
VFLHCITv
i.oono
.JS56 «FTA n.nooo CD \.
C .OM cn^ .20 cou ?,on ALPHAS
.0100 PI ,iooc c>- i.oooo
l.oooo GA^AI 0.0^01 RAMA^ o.
nh'-'ST.TY . 1.t?no
10.0100 FT.
l.«OOOF + 00
OF JFT
A",PTFMT
0
-o.o
YU
7.500ft A'T^LE «'ITH X-AVTS 16(1.00^0 PFC-PFFS
F(i-" A M-nviMU COOPOTNATE F1XFO OH THF P
Y"f YE. W
^ . 5 0 o o 7.S010
o.oooo -o.f>ooo -o.oooo
. 0 0 0 OF + 02 SEC.
B43
-------
RUN J-S-3
AM (4 IE. NT CtMHTI'iNS
ffepTh IN FT. IJK-JSITY JK OAM PER cc. KY TC. SUM PE» SEC. VEL. IM FPS.
DEPTH 0. 4.000'JttOl 6.0000E+01 i.OOOOE+02
AMH-TEMSJIY 1.0230E+00 1.0230F+00 1.02ME. + 00 1.0231E+00
YK 3.SOOOE+01 y.SOOOE+01 S.5000E+01 6.5000E*01
KY lj.OOOOF-0? b.OOOOE-OJ l.OOOOt-02
YU 5.UOUUE + 01 YH S.OOOut + OlYF l.OOOOttfliJH I.on00tt02
LI A 0 . K A 0 .
BAkfaK I PEKil Il'li-l r'
iHUfiljGH A MJ^LF UNOEW A MOVING
1ST UP 1.0000KtC4SFC.
GhlU HdlM SI2R: S) KfYl 1 Kfct2 .5
ust ILIWA TECH SUGGESTED rdrFur.itf'is
I>1NCH1 1,0000 UJNCH? I.OOUO
ALPHAl .uHOb ALPHA2 .3S36 HtU 0.0000 CD 1.3000
fiAMA ,'eLb. COriAG l.'*0 C^hlt: .010 CU3 .^0
ALPnA A .0010
Jti f-At'U.-.s .boon IJF^SITY i.i?oo ANGLE I.IF- JET o.oooo
VFLtJCHY b.0<>00 A I lO.OuOO KT.
K 11. 1
HfKSirr C'f SfJL]0 1.HOOOF. + 00
CDNCf-Ml^Al HJN l.bOUOE-01
FALL V'Hi'C.11Y nF Sill ID b'.OOOOF-OS
VFl.OCllY V.bOuO Ai-,r,i.r. v-JTh x-AXJS 160.0000 Dtlil-FFS t- L'^S . OOOOF •» 0
-------
RUN j-s-4
AMBIENT CONDITIONS
DEPTH IN FT. DENSITY IN GRAM PER CC. KY JN SOFT PER SfcC. VfcL. IN FPS.
DEPTH 0. U.OOOOE+01 6.0000E+01 l.OOOOE+02
AMB-DENSITY 1.0230E>00 1.0230E+00 1.0230F+00 1.0230E+00
YK 3.SOOOF+01 4.5000E+01 5.5000E+01 6.5000E»01
KY 5.0000E-02 5.0000E-03 l.OOOOE-02
.YU 5.0000E+01YW b,OOOOE + 01 YE 1.0000E+02H l.OOOOE+02
UA 0. WA 0.
BARGE UPERATIUN 2
DISCHARGE THROUGH A NOZZLE UNDER A MOVING BARGE
TSTOP 1.0000E+04SEC.
GRID POINT SIZE 51 KEU I KEY2 2 KEY3 0
USE TETRA TECH SUGGESTED COEFFICIENTS
DINCR1 1,0000 DINCR2 1.0000
ALPHA1 .0806 ALPHA2 .3536 BETA 0.0000 CO 1.3000
GAMA ,2b CDRAG 1.00 CFRIC ,010 CD3 .20 CD4 2.00 ALPHA3 .3536
ALPHAS .0010 FRICTN .0100 Fl .1000 CM 1.0000
ALFAI i.oooo ALFA2 i.oooo GAMAI 0,0000 GAMA2 o.oooo
ALAHDA .0010
JET RADIUS ,5000 DENSITY 1,1200 ANGLE OF JET 0.0000
VELOCITY b.OOOO AT 10,0000 FT.
K I L 1
DENSITY OF SOLID 1 ,8000E + 00
CONCENTRATION 1.5000E-01
I-ALL VELOCITY OF SOLID S.OOOOE-OS
BARGE VELOCITY 7,5000 ANGLE WITH X-AXIS 180,0000 DEGREES FURS,OOOOE+02 SEC.
AMBIENT CURRENT FOR A MOVING COORDINATE FIXED ON THE BARGE
0 YU YW YE H
7,5000 7.5000 7,5000 7.5000
-0.0000 0,0000 -0,0000 -0.0000
B45
-------
RUN j-s-s
TNj FT. DFWSITY TK! GRAM PER C. C . KY IN SOF T PFP SEC. VEL. I *•' FPS,
0. U.OOOOF+OI 6.oonoE+oi i.onooF+os
'"SITY i.n?30F+no i.o?30f+oo i .
YU S.
"'A l .
S.nnnnE-n? 5.nnoof-oJ
i YW
o .
i.oonnp+o?H l.onnnp. + n?
A MOV IMC- °-4PGF.
GRID pniri ?i7F st KFYI i KF.Y?
n$F TFTR4 TFC4 S'JRRESTF.D rnFFFICIE^
i.nooo OTNT.P? 1.0000
^fc"T4 n.oooo rn 1.3000
1.00 CFRir .010 CDS .?0 COU 2.00 ALPHA} .3536
AI.PHAU .nnio F«ICT*I .0100 Ft .jonn C1-1. 1.0000
it.FAJ t.OOni 4 (. c A ? 1.0000 G A * A 1 0.0000 RAMA? 0.0000
& I.. A * fi A .0010
'FT RADIUS .^ooo DENSITY I.IPOO
VEI.nr.ITv S.OOOrt AT 10.0000 FT.
y. 11. 1
r.f'SITY rp SCLIO l.BO'JOF. + OO
OF JFT
n.ooon
FAI.i. VFi.PCITY OF SOLIO ^.OOOOE-03
OAT.F Vp|.nriTv T.sboo A^GLF WITH V-AXIS i^o.ooon OFGRFES Fnos.nnn'oF. + oa
AMRJFMT CU^RF.k'T FH-> A MOVJMG COORDINATE FIVFD HM THE PAf^RF
0 YU YW YE M
A.S^O.I ft.sooo 7.soor> y.sooo
- o . o n o •" o.onno -o.ooo" -o.oooo
B46
-------
RUN j-s-6
AMiTFM (.OMJ t I ili'iS
JN Fl. DFMSUY JN UWAM PHW CC. M I* SUH l PFK Sfcf. . vtl. IN
0. '4.0000F + 01 6.GOOOF+01 1.0UOOF. + 0?
'^S 11 V 1 .
YK
KY
YH S.DOOnr+01 Yrt
U A 1 . 0 C1 0 0 b + U 0 Vi A ! . 0 0 0 0 f + 0 0
S'.bOOOCtOl 6.5000t+OI
5.i)«oot-0.i l
1 .OOI)Ott02H 1.00oOfc>02
OPfrlVAT I(H«I 2
IHRlUJGH A
A MlwTNli
1HTIJP 1 .bOOOH tOaStC.
GrtJO PlilNt SIZE SI KFY1 i KtYc; .5 KFY3
USf TFlkA TtCri SUGGt S I t.O (Ml I- f l(. I (• MS
ALPHAl
,?'
AlPHA«
i.oooo OJNCK?
.OHOf) ALPHAS
CiJKAb i.ou CFK
.0010 K^ICTN
i.ouoo
,3b5h Rb.fA 0 . 0 0 0 0 CO 1.3000
.010 CDi .<^o COa ^.00 ALPHAS
.Oltin (• 1 . lOOu CM l.OODU
ALF'Al 1.0000 ALFA2
ALAMDA .0010
1.0000 GAUAJ
0.0000 (,AMA0
COMCtNT^Ant:^ 1.5000t-01
FALL VF.L{lCITy OF SOLID b.OOOOt-03
BAKUK vhl.nr.ITY 7. bOOt) AM(,LE WITH X-AX]S IbO.OOOO DEGKFfcS F URb . 00 0 OF. + 02 SEC.
AUblFNT CURKF.wr FUK A ML'VINif, COI.lHD I NA T R HXfcO UN FHF BAKGL
0 YU VW YE H
8. SOOO B.bOOO 7.bOOQ 7.SOOO
- 0 . U 0 0 0 1.0000 -0,0000 -0.0000
B47
-------
RUN J-S-7
CONDITIONS
OEPTH IN FT. DENSITY IN GRAM PF.H CC. KY IN SOFT PFR SEC. VEL. IN FPS,
DEPTH 0. U.OOOOF+01 6.0000E+01 t.OOOOE+02
AMB-DENSITY 1.0P30E+00 1.0230F*00 1.0235F+00 1.0335E+00
YK 3.5000E+01 «.5000F+0) 5.5000E+01 6.5000F+01
KY 5.0000E-02 5.0000E-03 l.OOOOE-02
YU 5.0000E+01YW 5.0000F + 01YE l.onooE + 02H l.noOOF + 0?
HA 0, WA 0.
BARGE OPERATION 2
THROUGH A N07ZI F UNDER A MOVING BARGE
TSTOP l
GRID PRINT SIZE 51 KFY1 1 KEY? 2 KEY3 0
USE TETRA TFCH SUGGESTFD COEFFICIENTS
OINCP.1 1.0000 DINCR2 1.0000
ALPHAI .0806 ALPHA? .3536 BFTA o.oooo co
RAMA ,?5 CDRAG 1.00 CFRIC .010 C03 .20 CD/4 2.00 ALPHAS .3S36
ALPHA« .0010 FRICTN .0100 Fl .1000 CM 1,0000
ALFA1 1.0000 ALFA2 1.0000 GAMA1 0.0000 GAMA2 0.0000
ALAHHA ,0010
JET RADIUS 1,0000 DENSITY 1.1200 ANGLE OF JET 0.0000
VELOCITY 5.0000 AT 10.0000 FT.
K 1 L 1
DENSITY PF SOLID 1.8000F+00
CONCENTRATION 1.5000E-01
FALL VELOCITY OF SOLID 5.0000E-03
BARGF VELOCITY 7.5000 ANGLE WITH X-AXIS iso.oooo DFGRFES FrjRs.ooooE+o? SEC.
AMRIENT CURRENT FOR A MOVING COORDINATE FIXED ON THE 8ARGE
0 YU YU YF H
7.5000 7.5000 7.5000 7.SOOO
-0.0000 0.0000 -0,0000 -0.0000
B48
-------
RUN j-s-a
AMBIENT rONOTTiniMS
OFPTM IM FT. OFNSTTY IM GRAM PKP CC. KY IN SOFT Pf» SfC. VFL. TM
OFPTH n. u,oooor+oi h.oooo£+oi I.OOOOF. + O?
AMR-OENSTTY l
YK 3.5000F+01 «.5onnE+ot 5.5onoF+oi
KY s.nnoor-n? S.oonnf-03 i.nnnnf-n?
YD S.OOOOF + OIYW 5 . n none •»• o 1 YF i.nnont + o?H i.onoOF + o?
I.'A 0. WA 0.
FJARGF
PISCH4KGF THROUGH A NOZ7LF UNOER A MOV] MR PAPRE
TSTOP 1 .OOOOE-f
RRTH POINT SIZF 61 KFY] 1 KFY? 2 KF.V5
iiSF TFTHA TECH SUBRFSTFO CHFFF T
1.0000 OINCR2 1.0000
ALPHA1 .0806 ALPHA? .T53A RFTA 0.0000 CO t.3000
GA^A .?S COHAG 1.00 CFRIC .010 CD3 ,?0 Cr>U 2.00 ALPHA3 ,3536
ALPHAa .0010 FPICTN .0100 Fl .1000 CM 1.0000
ALFA1 i.oooo ALF-A? 1.0000 GAMAI o.oooo GAMA? o.oooo
ALAMDA .0010
JET RADIUS .SOOO DENSITY 1.1POO ANGLE OF JFT 0.0000
VELOCITY 10.0000 AT 10.0000 FT.
K 1 L 1
DENSITY PF SOLID l.flOOOF+00
CONCENTRATION l.SOOOF-01
FALI VFLHCTTY Of SOLID S.OOOOF-0?
RAPGF Vfl nr.ITY 7.5000 ANGLF WITH X-AXTS 180.0000 DECPtFS FDRS. OOOOE +0? SFC,
CURRENT FOP & ^(.IVTwy C'JDRfJlNATF HXFO ON THF
0 YD YW YF H
7.5000 7.SOOO 7.5000 7.5000
-0.0000 0.0000 -0.0000 -0.0000
B49
-------
RUN j-s-9
AMBIFNT CPNOITinNS
DEPTH IN FT. DENSITY IN GRAM PER CC. KY IM SQFT PER SFC . VEL. IN FPS.
OF.PTH 0. «.OOOOF>01 6.0000F + 01 l.OOOOF+02
AMR.pF.NSI TV ! ,n?30£->00 1.02TOF+00 1.023SF+00 l
YK
KY
YU 5,
UA 0.
5.soooE+ot
S.OOOOF-n? "5.0000F-03 l.OOOOF-02
i Ye. i.nonoe+o?H j.noooE+o?
WA 0
PAPGK PPF. RATION ?
THRHIIGH A NCZZLF.
A MOVING
TSTHP 1 ,0000£4-0«SFr.
GRID PPIKiT STZt 51 KEY1
USF TFTRA Tfc'CH SUG^ESTFD
1 KEY?
? KFYT
ALPHA1
i.onno
.0806 ALPHAS
i.oooo
RFTA
0.0000 CH
GA"A ,?5 CO«AG 1.00 CFRIC ,010 Cf>3 ,?0 COO 2.00 ALPHAS
ALPHA4 .0010 FRICTN .0100 Ft ,1000 CM 1.0000
ALFAI i.oooo ALFA? i.oooo GA^AI o.oooo RAMA? o.oooo
Al AMOA
.0010
.JFT RADIUS .SOOO DFNSITY 1.1200
veinrnv s.oooo AT 20.0000 FT.
K i L i
nFNSTTY OF SOLID l.flOOOEtOO
rONCFNTRATinN 1.5000F-0!
FAIL VULHCTTY OF SOI ID S.OOOOE-03
HARGF VFLOriTY
7.SOOO AK'GLF.
AMBIFNT CURRENT FOR A MOVTMT, COnPRTNATE
0 YD YK YF H
7. SOOO 7.SOOO 7,5000 7.5000
-0.0000 0.0000 -0.0000 -0.0000
OF .JTT
0.0000
X-AXIS IBO.nflOO 0EKHFFS FHRS . 00 0 OF + 02 SfcC,
ON ?HF. BARGE
B50
-------
SUN j-s-io
AMBIENT CONDITIONS
DEPTH IN FT. DENSITY IN GRAM PER CC. KY IN SQFT PER SFC. VEL. IN FPS.
DEPTH 0. a.OOOOE+01 6.0000E+01 l.OOOOF+0?
AMR-DENSITY 1.0P30E+00 1.0230F+00 1.0235E+00 1.0235E+00
YK 3.5000E+01 4.5000E+01 S.SOOOF+Ol 6.SOOOF+01
KY 5.0000E-02 5.0000F-03 l.OOOOF-02
YU 5.0000E+01YW 5.0000F. + 01 YE 1.0000E+02H l.OOOOE + 0?
IJA o. WA o.
BARGF OPFRATION 2
DISCHARGE THROUGH A Nn?ZlF. UNOFR A MHVING RARGE
TSTOP 1 .OOOOFfOUSEC.
GRTD POINT SIZE SI KFY1 1 K£Y2 ? KFY3
USF TFTRA TFCH SUGGESTED COEFFICIENTS
1.0000 DTNTR2 1.0000
ALPHA t ,0806 ALPHA2 ,353ft BETA 0,0000 CD 1.3000
GAMA ,?S CDRAG 1,00 CFRIC ,OJO CO} ,20 CD4 ?,00 ALPHA3 ,3"536
ALPHAS .0010 FRICTN .0100 M .1000 CM 1.0000
AtFAl 1.0000 AI.FA2 t.OOOO GAMA1 0,0000 GAMA2 0.0000
Al AMDA .0010
JET RADIUS .SOOO HEMSITY 1.1?00 ANRI.F OF JET 0.0000
VEinciTY 5.0000 AT 10.0000 FT.
K 1 L 1
DEMSTTY np SGI.ID i.floooE+oo
rONCFNTRATTON 1.5000E-01
FALL Vtl.rCTTY OF S'lLTD 5.0000E-03
BARGE VELOCITY S.OOdO AMGLF WITH X-AXIS 180,0000 DFGWEES FH»S . ^OOOf+02 SfcC,
AMPJFNT fliRRENT FQP 4 MOVING C'lnRDTNATF FI*Fn ON THF HARGF
0 YD Yw YE H
5.0000 5.0000 S.OOOO 5.0000
-o.oooo n.onoo -O.OOPO -o.oooo
B51
-------
RUN j-s-ii
AMHIfcNT CONDITIONS
DEPTH 1H M. fJF'ojSUY I'M GUAM PfR Cti. Kr IN SUM PKk SFl. vtL. IM F'HS.
DtPTH 0. 4.0COOF+01 6.0000t*-u! 1 . OOOOt + 1)2
AMR-DF.NSJTY l.o230EtOO 1.02.SOF + OQ 1 . 0255F + 00 1.0235MUO
YK 3.500oe+ni U.bOOGb + 'il b.5000k+OI h.SOOOfc + 01
KY b.OOOUt-02 b.OOOOL-Oi l.OOOOt-02
YU S.OOOnt+OlYK S.OOOOK + 01 YF 1.0000L+02H l,OOOOt
U A 0 . w A 0 .
bAKGE uPF RATION ^
DISCHAKGb IHKOUGM A MJ/iJLt' UMDfcH A hUVIN'i liAKlit
mop i .oooob
f,RJU PfilMT SIZt SI KFY1 1 KEY? i5 KEY3
DSf: ItTHA ItCh SUG'ifcSIEU
1,0000 DINCK? 1.0000
ALPhAl .0806 ALPHAS .3536 BETA 0.0000 CO 1.3000
GAMA ,2b (.DKAG l.ou Cf-KIC .010 CD3 .20 GDI 2.00 AI.PHA3 ,3S'i6
ALPHAS .0010 F-KKIN .0100 Fl .1000 CM i.oooo
ALFAl 1.0000 ALKA^ 1.0000 GAMA1 0.0000 GAMAr> 0.0000
ALAMUA .0010
JE1 RADIUS .bOOO DENSITY 1.1200 ANGLT OF JfT aS.OOOO
VELOCITY ^.0000 AI 10.0000 FT.
K 1 L 1
DENSITY OF SOLID I.fl000t>00
CUNCFMRATIGN 1.5000F-01
I-ALL VELOCITY UF SML10 b.OOOOE-Oi
BAKGF. VELOCITY 5.0000 ANGLE WITH X-AXIS iso.oooo OEGKEES FURG.OOOOE+O^ stc,
AMPIENT CURHtfil FOH A MOVllMf, COOKOINATE FIXED UN THE RANGE
0 YU Y* YF H
S.OOOO b.OOOO 5.0000 b.OOOO
•0.0000 0.0000 -0.0000 -0.0000
B52
-------
RUN J-S-12
.T (.bNDl I IIIIMJ
DfPTn 1M KT. DE^SIIY IN KKAM PER CC. KY IN SMM Pfr.K Sfc C . VEL . 1 M FPS.
otPTh o. ^.Doooe to i b.oonor +01 i . oooot. + o^
AMb-|)FNSITY 1.U230E+HO 1.0250E+HO I.0231E+00 l.023l£ti>0
1
Y K i . 5 0 <> 0 K + 0 1 a . S 0 0 0 f. + 0 1 b . S 0 0 0 1 + 0 1 6 . b u 0 0 E + 0 1
KY S.()()OflF.-Oc> 5.000ut-03 l.OOOOt-02
YU b.OOOOE + DlY« S.'-iUliOF + Ol YF 1.0000£ + Oi>H 1 .OOOOL + ii^
L'A 0. WA 0.
BARGF UPFKATIUN ^
UlSChAKtiE THKIlUGH A MUZ2LF UNDER A MDVlMi; BARGE
1S1DH I .OOOOK + 04SLC.
GRID PLUM SIZE bl KhYl 1 Kf.Y? 2 KFY5
USE TfcfRA ThCH SUUGtSitU IHKKI- R I EM -S
1.0000 OINCHi? i.OOOO
AI.PHA1 ,0fl06 ALPHAS .%S36 BF1A 0.0000 CD 1.3000
RAMA ,2S CURAG l.ou CFKIC .010 COi .20 C0« 2.00 AI.PHM ,3S"46
ALPHAS .0010 FWICTN .0100 f- 1 .1000 CM 1.0000
ALFA1 1.0000 ALFA2 1.0000 GAHA1 U.OOOO GAMA? 0.0000
.0010
JEl RADIUS .5000 Ofc'MSITY 1.1POO ANGLE OF Jfc I 0.0000
VELOCITY b.OOOO AT 10.0000 FT.
K 1 L 1
DENSITY OF SOLID 1.8000F+00
CCiNCFNTRATICiN 1.5000E-01
FALL VELOCITY Oh SOLID 5.0000E-03
PARGF VFLnciTY /.SOOO AwULE ^ITH X-AXIS 180.0000 OFGHEtS FOK5 . OOOOE+02 SEC.
NT CURRENT FUR A MLIVJNG COdHDINATE HXFO ON THE BARGE
0 Yll Yh YF h
7. SOOO 7.bOOO 7.5000 7.5000
-0.0000 0.0000 -0.0000 -0.0000
B53
-------
RUN J-F-I
AMBIENT CONDITIONS
DEPTH IN FT. DENSITY IN GRAM PER CC. KY IN SUM PfcH SFC. vtL. IN FPS.
DEPTH 0. «.OOOOE+01 6.000CF. + (J| 1.0000F. + 02
AMB-DENSITY 1.0230E+00 1.02.30E+00 1.0? 1 6 .50 OuEtO 1
KY 5.0000E-02 S.OOOOfc-03 1.0000t-0,f
YU 5.0000t + OlYW b.OOOOEtOl Yt 1 . l'000h + 02n l.OOOOt + 02
UA 0. hA 0.
8ARGF OPERATION 2
DISCHAh'GF iHROiJCH A NUZZLE L»iJL>tK A Ml.iVlM; HAt^Gt
TSTOP 1.0000F+03StC.
GHlO PLUM SIZE Si KEV! 1 KtY2 \ KKY3
USt TfclKA UCH SUGGESTED COLF F 1 C I EN T i>
1.0000 Q1NCK2 1.0000
AUPHAl ,080b ALPHA2 .4S36 BtTA 11.0000 CO l..<0i;0
GAMA .2S CDKAG 1.00 CTKIC .010 C03 .20 COM 2.0(>. ALPnAj
ALPHAS .0010 f-KlCTN .0100 hi .1000 C^ l.OfluO
AL^Al 1.0000 ALFA^ 1.0000 GAMA1 O.OfiOO l,A^/ O.
ALAMI1A .0010
JtT RADIUS .3000 Oht-SIlY 1.1200 ANCLK OH .1(1 O.OOiiD
VELOCITY 5.0000 AT 10.0000 FT.
K 1 L 1
DENSITY Of- SOLID l.OOOOKtOO
CIINCFMRATin.N 0.
FALL VfLOClFY OF SdLlO ".
BARGt VELOCITY /.booo ANGLE ^irn X-AXIS 22'j.oonn ftG'VhKS I 'ird .I.IHMU '+02 si-c.
AMBIFNl CU^KtNT FL1« A HOVlNi; CUdrtDTiMA TF F1XH) (.<'< l^F hAi-T,t
0 YU Yis i*b H
5.3033 S.3033 5.30^ S.3033
b.3033
B54
-------
RUN J-F-2
FM CUMD! T FOMS
Ii! M. uF'-.SHY JN GUAM P£H CC. KY IN SOU PEP SEC. VfcL. IH I-PS.
OfcPTh 0. U.OOOOEtOl b.OOOOF. + OI l.OOOOEtOS
Ar-iK-UKNSn i I . !i£l'i
Vu f).00u0t + 01 1M S. OllOOF. + Ol Yt 1.0000fi02h l.OOOOfc + 02
UA 1 ,00 DOR tO flu, A l.OyliOK
l;PfcKATlON f.
OISCHAkCF: fHKUUfJH a t-OZZLF UNDER A MOVING OARGt
TSUIP 1 . OOOUb
UK ID Hii^.f SJ7E f>l KFY1 1 KEY? i.KtY3 0
uSt TETRA 1ECM SuGGESieO CUEFFICIEMS
DINCH1 1.0000 IJlNC«/e 1.0000
ALPriAJ .OflOo ALPHAS .3S36 BETA 0.0000 CO 1.5000
.25 CUKAG 1.00 CFKIC .010 CD3 .20 CD4 2.00 ALPHA? .3S36
.0010 FrflCTN .0100 f- 1 .1000 CM 1.0000
ALFAl 1.0000 ALKA2 1.0000 GAh'Al 0,0000 GAMA2 0.0000
ALAMO A .0010
JF.T UA01US .SUOO UE-NSITY 1.1200 ANGLE OF JFT 0.0000
VLl'jriTY S.OOOO AT 10,0000 FT.
K 1 L 1
DENSITY OF SUL10 l.OOOOEtOO
CDMCEi^TKATIOni 0.
FALL VFLUCJlY UF SOLID 0.
VFLUCHY y.booo ANGLE MTH X-AXIS 22^.0000 DEGREES FORS.OOOOE+OI SEC,
AMBIENT CUKkENT FOri A MOVING COORDINATE FIXFU ON THE BAHGfc
0 YU Y^ rt' H
6.3033 6.3033 S.3033 b.3034
, b.3033 6.3033 5.3033 jb. 3033
B55
-------
RUN J-2S-1
4MBIFNT CONDITIONS
DEPTH IN FT. DENSITY I * GRAM PF." CC. KY IN SOFT PER SEC. VEL. IN FPS.
DEPTH 0. q.OOOOF. + Ol 6.0000E + 01 l.OOOOE + 0?
AMP-DENSITY 1.P230F+PO 1.0230F+00 1.0240E+OP 1.02aSE+00
YK
KY
3.5000E+01 4.5000E+01 5.5000F+01 6.5000E+01
5.0000F-0? 5.0000F-03 1.0000E-02
YU 3.00POF.+ 01 YW U.nOOOF + O
U A 1.000nF + OOW4 0.
5.0000E + 01H l.OOOOF + 02
P.APGF
?.
THROUGH A V07ZLE
A MOVING BARGF
TSTGP i .
?. KEY?
? KFY3
1
GPTD pfir-'T SI7.F 51 KFY1
i'SF PEAD IN COEFFICIENTS
1.0000 DI^CK? 1.0000
.0806 ALPHA? .3536 BETA 0.0000 CO 1.3000
GAM* .25 CTFMr, 1.00 CFRTC .010 CD3 .20 CD« 2.00 ALPHA3 .3536
Al PHAU .0010 FPTCTN .0010 Fl .1000 CM 1.0000
1.0000 GAMAl .1000 GAMA2 0.0000
ALFA!
1.0000 ALFA2
.0010
JET RADIUS .5000 DENSITY 1.1300 AMGLE OF JET
VELOCITY 7.5000 AT 10.0000 FT.
K ?. I 1
0.0000
DFNSTTY np
FALL VFI.PCITY OF SOLID
PF
FALL VFLfiflTY OF
?.5000F+00
1 .OOOOF-Ot
5.0000E-02
Q.OOOOE-Ol
2.0000E-01
-5.0000E-02
5.0000 ANGLE WITH V-iVTS IfiO.OQOO PEG«EES FQR6.000OE+02 SEC,
APRTFNT CIJPREMT FOP A "fWIMG COOPOINATF FIVFO ON THE RAHGF
0 YU YW YE H
6.0000 6.0000 5.0000 5.0000
-.0000 -."000 -.0000 -.0000
B56
-------
BUN J-4S-1
AMBIENT Cl'NOITKiNS
DEPTH IN FT. DENSITY IK GHAH PFR fC. KY T IV S'JFT Hh.H StC. vFL. 1 ••) FPS.
DEPTH o. '1. OnOOF + 01 6.0000f+(;J l.OGOOE+02
AMR-DENSITY 1.023Ut+00 1.0230t+00 1 . <)2'U)f + 0 0 I.u?«0fc+o0
YK 3.5000E+01 y.SOOOt' + Ol S.SQOutt'il 6.HUOCF + 0
KY S.OOOOi-02 S.OOOOt-03 1.0noOt-0?
YU S.OOOOF+OlYin S.OOOOf-' + OlYt l.«OOOt + 02H l.OOOOF + 0^
UA l.OOOOt+OOKA l.DOOOE+Ou
BARGt OPERATION 2
DISCHARGE THROUGH A t-flJ^ZLt LifJOtH A
TSTUP 1.500oF+OaSEC.
GRID PLUM SIZE bl Kf.Yl 1 KF.Y2 ? Kf.YJ 1
USE TETRA TECH SUGGfcSTtU CflEFF 1C IE* IS
DINCR1 1.GOOD D1NCK2 1.0000
ALPHA! .0606 ALPHAS ..5536 fe 1 * O.OOC'O CO 1.3l>0<>
f.AMA .^S CDRAG 1.00 CFRIC .010 COS ,?0 CDo P.OO ALPHAS
ALPHAS .0010 FK1CIN .0100 (- 1 .1000 t> 1.0000
ALFA1 1.0000 ALKA2 1.0000 (JAMA1 0.0000 GAMA? (1.0000
ALAfDA .0010
JET RADIUS ,r)000 DENSITY 1.2000 Ah'GLF dF JET O.OOO'i
VELOCITY S.OOOO AT 10.0000 FT.
K 21, 2
DENSITY OF SOLID l.OOOOEtOu
CUKiCENTRATION I.OOOOF-ul 2.00f>OE-01
FALL Vtlf.'ClIY UF SuiID -l.QOOOk'-Ol -l.OOOOt-03
DENSITY UF SOL IlJ 2.0U()Ot*00
CQNCfcNTHATIUN l.SOOOE-01 b.OOOOF-02
FALL Vt'LUClTY iJF SOLID 1.S-OOOF.-01 1.500UF.-U3
BARGE VFLOCUY 7.SHOO ANfjI.F nl]t- X-AXIS 160.0000 UEGRtES H'M'3. OOOOfc + 02 SEC
AM8IFNT CURRENT FOR A MUVIMG C'KI«DJNATK FIXED nn IMP.
0 YU Yk YE H
8.5000 8.5000 / . b 0 0 0 7. '^ 0 0 n
-0.0000 1.0000 -0.0000 -0.0000
B57
-------
RUN W-L-1
AMBIENT CONDITIONS
DEPTH IN FT. DENSITY IN GRAM PER CC, KY IN SClFT PER SEC. VEL. IN FPS.
OtPTH 0. 4.0000E+01 6.0000E+01 l.OOOOE+02
AMR- DENSITY 1.0230E+00 1.0230E+00 1.0P32E+00 1 . 0232E + 00
YK 3.5000E+OV «.SOOOE+01 5.5000E+01 6.5000E+01
KY 5.0000E-02 5.0000E-03 l.OOOOE-02
Y(J 5.0000F+0! Yrt 5.0000M01 YE I.OOOOE+02H t.OOOOE + Oi!
UA 0. WA 0.
BARGE OPERATION 3
WASTE MATEKIAL IS UlSCHARGtD CONTINUOUSLY INTO THE BAKGE
AFTER THf INITIAL MIXING, IHE ^ASTE MATERIAL IS ASSUMED
TO BE JN A FORM OF HALF CYLINDER THERMAL.
TSTOP l.OOOOE+OiSEC.
GRID POINT SIZE 51 KEY1 1 KEY2 I KEY3 1
USE TETKA TECH SUGGESTED CUEFFICIENTS
DINCR1 .2SOU OINCR2 1.0000
SHAPE FACTORS el .60 ca i.oo
ALPHA .3536 BETA o.oooo GDI .booo en? 1.3000 CM i.oono
GAMA .25 CDRAG 1.00 CFHJC .010 C03 .20 CD« 2,00 ALPHA3 .3536
ALPHA« .0010 PRICTN .0100 n ,1000 ALPHAI .0806
ALFA] 1.0000 ALFA2 1.0000 GAMA1 0.0000 GAMA2- 0.0000
ALAHOA .0010
DENSITY 1.3000 DISCHARGE RATE 10.0000 BARGE UIIHH 25.00 BARGE OlfPIH 10.00
K 1 L 1
DENSITY OF SULID 2.5000EfOO
CUNCENTRA110N 2.0000E-01
FALL VELOCITY OF SOLID 5.0000E-02
BAKGF. VELOCIir 7.5000 ANGLE "ITH X-AXIS 180.0000 DfcGKt.ES FDR5 . OOOOE + 02 SfcC.
CURRtNT FOR A HCWlNC, COURU1NA1E f-'IXFO (IN THE
0 YU YW Y£ H
UA 7.500 7.SOO 7.5UO 7.500
WA -0.000 0.000 -0.000 -0,000
BS8
-------
t
-------
KUN W-L-3
AMBIENT CONDITIONS
DEPTH IN FT. DENSITY IN GRAM PER CC, KY IN SOFT PER StC. VfcL. IN FPS.
DEPTH 0. 4.00006*01 6.0000E+01 1.0000E*02
AMU-DENSITY 1.0230E+00 1.0230E*00 1.0232E*00 1.0232E+00
YK 3.SOOOE+01 4.5000E+01 S.SOOOfc+01 6.5000E+01
KY S.OOOOf-02 5.0000E-03 l.OOOOE-02
YU 5.0000E+01YW 5.0000E+01YE 1.0000E+02H l.OOOOE+02
UA 0, WA 0.
BARGE OPERATION 1
KASTt MATERIAL IS DISCHARGED CONTINUOUSLY INTO THE BARGE WAKE,
AFTER THE INITIAL MIXING, THE WASTE MATERIAL IS ASSUMED
TO BE IN A FORM OF HALF CYLINDER THERMAL.
TSTfJP 1.0000F+04SEC.
GKIO POINT SIZE SI KEY1 1 KEY2 2 KEYS 0
USE TETRA TECH SUGGESTED COEFFICIENTS
DINCR1 .2500 DINCR2 t.OOOO
BARGE SHAPE FACTORS Cl .60 C2 a.00
ALPHA ,3S36 BETA o.oooo CDI ,5000 002 1,3000 CM i.oooo
GAMA .25 CORAG 1.00 CFKIC ,010 C03 .20 C0« 2.00 ALPHA3 .3546
ALPHA« .0010 FRICTN ,0100 Fl ,1000 ALPHA1 .0806
ALFA1 I.OOOO ALFA2 I.OOOO GAMAI 0.0000 GAMA2 0,0000
ALAMDA .0010
DENSITY 1,3000 DISCHARGE RAIE 10.0000 BARGE WIDTH 10.00 BARGE DEPTH 10,00
K 1 L 1
DENSITY OF SOLID 2.5000EtOO
CONCENTRATION 2.0000E-01
FALL VELOCITY OF SOLID s.ooooe-02
BARGE VELOCITY 7.5000 ANGLE WITH X-AXIS 180.0000 DEGREES FQR5. OOOOE + 02 SEC.
AMBIENT CURRENT FOR A MOVING COORDINATE FIXED ON THE BARGE
0 YU YW YE H
UA 7.500 7.500 7,500 7.500
WA -0,000 0.000 -0.000 -0.000
B60
-------
W-L-4
AMBIENT CONDITIONS
DEPTH IN FT. DhNSITY IN CHAM Pf.K CC. KY IN SOFT PER SEC. VF.L. IN FPS.
DEPTH 0. a.OOOOE»01 6.0000E+01 l.OOOOE+02
AMB-nENSITY 1.0230F. + 00 1.02SOE+00 1.0232EtOO 1.02i2E+00
YX 3.5000E+01 «.5000£*01 5.5000E+01 6,SoOOEf01
KY 5.0000E-02 5.00006-03 l.OOOOE-02
YU 5.0000E+01YW 5.0000E+01YE 1.0000fe+02H l.OOOOE+02
UA 0. WA 0.
BARGE OPFRATION 3
WASTF. MATERIAL is DISCHARGED CONTINUOUSLY INTO THE BARGE NAKE,
AFTER THE INITIAL MIXING, THE WASTE MATERIAL IS ASSUMED
TO BE IN A FORM OF HALF CYLINDER THERMAL.
TSTOP i.OOOOE+04SfcC.
GRID POINT SIZE 51 KEYI 1 KEY2 2 KEY3 0
USE TETRA TECH SUGGESTED COEFFICIENTS
DINCR1 .2500 DINCR2 1.0000
BARGE SHAPE FACTORS Cl ,60 C2 U»00-
ALPHA .3536 BETA 0.0000 CD! ,5000 COS 1,3000 CM 1.0000
GAMA .25 CDRAG 1.00 CFRIC .010 Cl)3 .20 CD4 2.00 ALPHA3 ,3S36
ALPHA4 .0010 FRICTN .0100 Fl ,1000 ALPHA1 .0806
ALFA1 1.0000 ALFA2 1.0000 GAMA1 0.0000 GAMA2 0.0000
ALAMDA .0010
DENSITY 1,3000 DISCHARGE RATE 100.0000 BARGE WIDTH 25,00 BARGf- DEPTH 10.00
K 1 L 1
DENSITY OF SOLID 2.5000E+00
CONCENTRATION 2.0000E-01
FALL VFLDCITY OF SOLID 5,ooooE-o2
BARGE VELOCITY 7.5000 ANGLE WITH X-AXIS 180,0000 DEGREES FCIK5,OOOOE+02 SEC,
AMBIENT CURRENT FOR A MOVING COORDINATE FIXED ON THfc BARGE
0 YU YW YE H
UA 7.500 7.500 7.500 7.500
WA -0.000 0.000 -0.000 -0.000
B61
-------
W-L-5
AMBIENT CONDITIONS
DEPTH IN FT. DENSITY IN GRAM PfR CC. KY IN SOFT PER SEC, VEL. IN FPS,
DEPTH 0. q.OOOOE+01 6.0000E+01 l.OOOOE+02
AMB-OtNSITY 1.0Z30E+00 1.0230E+00 1.0232E+00 1.0232E+00
YK 3.5000£+01 «,5000E+01 S.SOOOEtOl 6.5000E+01
KY 5.0000E-02 5.0000E-03 l.OOOOE-02
YU 5.0000t+01YW 5.0000E+01YE 1.0000E+02H l.OOOOE+02
UA 0. WA 0.
BARGE UPERA1ION 3
HASTE MATERIAL IS DISCHARGED CONTINUOUSLY INTO THE BARGE WAKE,
AFTER THE INITIAL MIXING, THE WASTE MATERIAL IS ASSUMED
TO BE IN A FORM OF HALF CYLINDER THERMAL.
"TSTUP 1.0000E + 03SEC.
GRID POINT SIZE 51 KtYl t KEY2 2 KEY3 I
USE TETRA TECH SUGGESTED COEFFICIENTS
DINCR1 .2500 DINCR2 1,0000
BARGE SHAPE FACTORS Cl ,60 C2 <4.00
ALPHA .3S36 BETA 0.0000 CD1 ,5000 CD2 1.3000 CM 1.0000
GAMA .25 CDRAG 1.00 CFKIC .010 CD3 .20 CD« 2.00 ALPHAS .3536
ALPHAS .0010 FRICTN .0100 FI .1000 ALPHAI .0806
ALFAl 1.0000 ALFA2 1,0000 GAMA1 0.0000 GAMA2 0.0000
ALAMDA .0010
OFNSITY 1.3000 OISCHAHGE RATE 10.0000 HARGE WIDTH 25.00 BARGE DEPTH 10.00
K 1 L 1
DENSITY OF SOLID 2.5000E+00
CONCENTRATION 2.0000E-01
FALL VELOCITY OF SOLID 5.0000E-02
BARGfc VELOCITY 5,0000 ANGLE WITH X-AXIS 180.0000 DEGREES FORS.OOOOE+02 SEC.
AMBIENT CURRENT FOR A MOVING COORDINATE FIXED ON THE BARGE
0 YU YW YE H
UA 5.000 5,000 5,000 5.000
WA -0.000 0,000 -0,000 -0,000
B62
-------
RUN w-s-i
AMBIENT CONDITIONS
DEPTH IN FT. UENSITY IN GRAM PER CC. KY IN SOFT PtR SEC. VFL. IN FPS.
DtPTH 0. '4.0000E + 01 6.0000t + 01 l.OOOOF+02
AMB-DENSJ1Y I.0230E+00 1.0230E+00 1.0232E+00 1.0232E+00
YK JS.bOOOt + 01 1.SOOOE+01 5.5000E+01 6.500Utt01
KY b.OOOOE-02 b.OOOOE-03 1.0000t"-0?
YU 5.0000E+01 Yrt S.OOOOFtOlYF 1 . OOOOtt H2H l.OOOOE+02
U A U . »•! A 0 .
UARGF UPfcHATlON 3
WASTfc MATtftlAl. IS OISCHAHGtP C'JN I 1NUUUSL Y I H TO THF. BARGE HAKf ,
AFTE.N THE INITIAL MIXING, THE WASH MATERIAL IS ASSUMED
Til BE IN A KDKM OF HALF CYLINDER THERMAL.
TSTOP l
GRID PPIdl SIZE SI KEY! 1 KEY2 2 KEYS 1
USt TETKA TFCH SUGGESTED COEFFICIENTS
OlNCftl .HbOO D1NCR2 UOOno
BARGE SHAPE MCTU«S Cl .60 C2 U.UO
ALPHA .J536 BETA O.OOOu CU1 .bOnO CD? 1-.3000 CM 1.0000
GAMA .2b CDRAG 1.00 CFRIC .010 CL)3 .20 Cl><) 2.00 ALPHA3 .ibib
ALPHA4 ,0010 FWK.TN ,0100 H .1000 ALPHA1 .0806
ALf-Ai i.oooo AI.FA? i.oooo GAMAI o.oooo GAMA^ o.oooo
AUAHOA .utiio
DENSITY 1,1200 OI8CHAKGL HAH. 10.0000 UARGt «10!H Hb.OO (UK UK OfPTH 10.00
K 1 L 1
OENSIlY OF SIJI.JD l.flOOOE + 00
CUNCFNTKA110M l.bOOOF-01
FALL VtLOLllY Uh SuLID 5.0000E-03
BAKGF VELI.'CJrY /.50»0 ANGLE WITH X-AXIS 180.0000 UEGKFES ^ OK5 . OOOOE + 02 SFC.
UUHRF.Nl FU>< A hOvi'NG CilltkDlNAlt (-1XF1) ON 1 HK BARGfc
0 YU YW YF H
UA 7.bOO V.SOO 7.bO() 7.500
"A -o.oi<0 0.000 -0.000 -0.000
B63
-------
HUN w-s-2
AMBIENT CONDITIONS
DEPTH IN FT, DENSITY JN GRAM PEW CC. KY IN SOFT PER S€C. VEL. IN FPS.
DEPTH 0. 1.0000M02
AMB-DENSITY 1.0230E+00 1.0230E+00
YK 3.5000E+01 «,5000E+01 5.SOOOE+01 6,5000fc+01
KY 5.0000E-02 5.0000E-03 l.OOOOE-02
YU S.OOOOE+01YW 5.0000E+01YE 1.0000E+02H l.OOOOE+02
UA 0. WA 0.
HARGE OPERATION 3
WASTE MATERIAL IS DISCHARGED CONTINUOUSLY INTU THE BARGE WAKE,
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"U.S. GOVERNMENT PRINTING OFFICEM971 546-315/200 1-3
-------
SELECTED WATER
RESOURCES ABSTRACTS
INPUT TRANSACTION FORM
'. Re,-
w
Mathematical Model for Barged Ocean Disposal of Wastes
Robert C. Y. Koh and Y. C. Chang
Tetra Tech, Inc.
630 North Rosemead Boulevard
Pasadena, California
•'• ,R art Di
6.
#. Performifl Orgxr ution
16070FBY
13 Type. Repo; >nd
Period Covered
Environmental Protection Agency report number,
EPA-660/2-73-029, December 1973.
Theoretical and experimental studies were performed on the dispersion and
settling of barge disposed wastes in the ocean. A computer program based on the mathe-
matical model has also been written. Comparison of predictions with experiments, both
in this study and from previous investigations were found to be good. Example solutions
based on the model for prototype situations are also presented.
The waste is assumed to consist of two phases, 1) a solid phase characterized by
constituents with various densities and fall velocities, and 2) a liquid phase. The
methods of disposal considered include 1) discharge from a bottom opening hopper barge,
2) pumped discharge through a nozzle under a moving barge, and 3) discharge into the
barge wake. The effects of ambient horizontal currents, density stratification, varia-
tion of diffusion coefficients are incorporated in the model.
Three phases of dispersion are envisioned: 1) a convective phase, 2) a collapse
phase, and 3} a long term diffusion phase. Transition between phases 1s accomplished
automatically in the numerical model. In addition, the collapse phase may a) be re-
placed by, or b) include a bottom spreading phase. Under certain circumstances, the
collapse phase is bypassed.
Every attempt has been made to minimize the amount of input required in the use of
the numerical model. The integration steps and grid sizes are all automatically chosen
by the model. Both detailed printout and graphic output are incorporated. The solution
may also be terminated at the end of any of the three phases of dispersion.
Mathematical model, ocean disposal, dredge spoils, sewage sludge, liquid wastes,
barge
•
Waste dilution
dispersion
ocean dumping
•• • •
Unclassified
19.
(Report)
20. Seciuny Class.
(Ptgf)
21. f. . of
Pages
2,2. Price
Send To :
WATER RESOURCES SCIENTIFIC INFORMATION CENTER
US DEPARTMENT OF THE INTERIOR
WASHINGTON. D C. 2O24O
Halter F. Rittall
EPA. PNERL. Coastal Pollution Branch
-------
KD-674
ENVIRONMENTAL PROTECTION AGENCY
Forms and Publications Center
Route 8, Box 116, Hwy. 7O, West
Raleigh, North Carolina 27612
POSTAGE Af
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EPJ
IOCNCY
Official Business
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\22z
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