r/EPA
United States
Environmental Protection
Agency
EPA-600/7-82-03 7a
May 1982
Research and
Development
VERIFICATDNAND
THERMAL POLLUTION MODEL
Volume L Verificati
TRANSFER OF
on of
Three -dimensional Free-surf ace Model
Prepared for
Office of Water and Waste Management
EPA REGDNS 1-10
Prepared lj>y
Industrial Environmental Research
Laboratory
Research Triangle Park NC 27711
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RESEARCH REPORTING SERIES
Research reports of the Office of Research and Development, U.S. Environmental
Protection Agency, have been grouped into nine series. These nine broad cate-
gories were established to facilitate further development and application of en-
vironmental technology. Elimination of traditional grouping was consciously
planned to foster technology transfer and a maximum interface in related fields.
The nine series are:
1. Environmental Health Effects Research
2. Environmental Protection Technology
3. Ecological Research
4. Environmental Monitoring
5. Socioeconomic Environmental Studies
6. Scientific and Technical Assessment Reports (STAR)
7. Interagency Energy-Environment Research and Development
8. "Special" Reports
9. Miscellaneous Reports
This report has been assigned to the INTERAGENCY ENERGY-ENVIRONMENT
RESEARCH AND DEVELOPMENT series. Reports in this series result from the
effort funded under the 17-agency Federal Energy/Environment Research and
Development Program. These studies relate to EPA's mission to protect the public
health and welfare from adverse effects of pollutants associated with energy sys-
tems. The goal of the Program is to assure the rapid development of domestic
energy supplies in an environmentally-compatible manner by providing the nec-
essary environmental data and control technology. Investigations include analy-
ses of the transport of energy-related pollutants and their health and ecological
effects; assessments of, and development of, control technologies for energy
systems; and integrated assessments of a wide range of energy-related environ-
mental issues.
EPA REVIEW NOTICE
This report has been reviewed by the participating Federal Agencies, and approved
for publication. Approval does not signify that the contents necessarily reflect
the views and policies of the Government, nor does mention of trade names or
commercial products constitute endorsement or recommendation for use.
This document is available to the public through the National Technical Informa-
tion Service, Springfield, Virginia 22161.
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EPA-600/7-82-037a
May 1982
VERIFICATION AND TRANSFER
OF THERMAL POLLUTION MODEL
VOLUME I: VERIFICATION OF THREE-DIMENSIONAL
FREE-SURFACE MODEL
By
Samual S. Lee, Subrata Sengupta,
S. Y. Tuann and C. R. Lee
Department of Mechanical Engineering
University of Miami
Coral Gables, Florida 33124
NASA Contract No. NAS 10-9410
NASA Project Manager: Roy A. Bland
National Aeronautics and Space Administration
Kennedy Space Center
Kennedy Space Center, Florida 32899
EPA Interagency Agreement No. 78-DX-0166
EPA Project Officer: Theodore G. Brna
Industrial Environmental Research Laboratory
Office of Environmental Engineering and Technology
Research Triangle Park, North Carolina 27711
Prepared for:
U. S. Environmental Protection Agency
Office of Research and Development
Washington, D. C. 20460
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PREFACE
This report is one in a series on the University of Miami's thermal
pollution models. Much of the background, formulations, solutions tech-
niques and applications of these models has been summarized in a three-
volume report by Lee and Sengupta (1978). These mathematical models
were developed by the Thermal Pollution Group at the University of Miami
and were funded by the National Aeronautics and Space Administration.
The primary aim was to have a package of mathematical models which has
general application in predicting the thermal distribution of once-through,
power plant heated discharge to the aquatic ecosystem, The joint effort
was planned so that the calibration and verification of these models de-
pend on simultaneous remote sensing and ground-truth data acquisition
support. The concept is the development of adequately calibrated and
verified models for direct prediction of thermal dispersion by the user
communities. The intended user communities include the utility companies
and the regulatory agencies at the federal and state level.
The purpose of the present effort is to further verify these models
at widely different sites using minimal calibration, and then to provide
the program code and user's manual to the Environmental Protection
Agency (EPA) for future users.
Two sites chosen were Anclote Anchorage on the west coast of
Florida and Lake Keowee in South Carolina. The free-surface model was
applied to Anclote Anchorage and the rigid-lid to Lake Keowee. Two
data acquisition trips, one in the summer and the other in the winter,
were carried out at each site. The acquisition was a collaborated effort
jointly by the UM, EPA, NASA-KSC (Kennedy Space Center) and the
corresponding utility company personnels.
The two-year project consisted of two phases. During Phase I,
the individual model was modified and calibrated to fit the corresponding
site; then the model was verified against the remote sensing isotherms
and in-situ measured velocity and temperature data under both summer
and winter conditions. In general, the computed isotherms were compar-
able with isotherms based on remote sensing. In Phase II, source pro-
grams of both models were documented and transferred to the EPA, and
the user's guides were prepared to familiarize the potential users. The
results of the two-year effort are summarized in a set of three final re-
ports.
ii
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ABSTRACT
To assess the environmental impact of waste heat disposal by power
plant operation into natural water bodies, mathematical models are essen-
tial, especially for predictive studies. The Thermal Pollution Group at
the University of Miami has developed a package of models for this pur-
pose. A joint effort with EPA, NASA, Duke Power Company and Florida
Power Company was conducted to verify these models with remote sensed
IR data and in-situ measurements.
The free-surface model, presented in this volume, is for tidal
estuaries and coastal regions where ambient tidal forces play an important
role in the dispersal of heated water. The model is time dependent, three
dimensional and can handle irregular bottom topography. The vertical
stretching coordinate is adopted for better treatment of kinematic condition
at the water surface. The results include surface elevation, velocity and
temperature.
The mode! has been verified at the Anclote Anchorage site of Florida
Power Company. Two data bases at four tidal stages for winter and summer
conditions were used to verify the model. Differences between measured and
predicted temperatures are on an average of less than 1°C.
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CONTENTS
Preface "
Abstract l»
Figures v
Tables !x
Symbols * *
Acknowledgments Xl
1. 1 ntroduction 1
Background 1
University of Miami models 2
Description of Anclote Anchorage *
2. Conclusions *>
3. Recommendations 8
H. Mathematical Formulation and Model Description 9
General background on free-surface (tidal) models . 9
Governing equations and boundary conditions ...... 12
Uncoupled system 17
Computational grid 18
Finite difference equations 20
Solution procedures 25
Stability 27
5. Application to Anclote Anchorage 29
I ntroduction 29
Choice of domain and grid system 30
Summary of data 30
Calculation of input 32
Results 38
References • 43
iv
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FIGURES
Number
1 Anclote Anchorage location in the state of Florida 52
2 Map of Anclote Anchorage 53
3 Definition sketch of o-coordinate 54
4 Grid arrangement in the horizontal projection 55
5 Four cells in a vertical column with velocities shown at
definition point and scalar variables at the center of cell .. 56
6 Notations and variables used in calculations 57
58
7 Grid work for the Anclote Anchorage
8 Location of stations for in-situ measurement, June 1978 ... 59
9 Velocity from in-situ measurement at 1710-1903,
June 19, 1978 60
10 Velocity from in-situ measurement at 0648-0812,
June 20, 1978 61
11 Velocity from in-situ measurement at 1125-1245,
June 20, 1978 62
12 Velocity from in-situ measurement at 1450-1605,
June 20, 1978 63
13 Daytime flight lines on June 19 and 20, 1978 64
14 Surface temperature in deg C from in-situ measurement
at 1710-1903, June 19, 1978 65
15 Surface temperature in deg C from in-situ measurement
at 0648-0812, June 20, 1978 66
16 Surface temperature in deg C from in-situ measurement
at 1125-1245, June 20, 1978 67
v
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FIGURES
Number Page
17 Surface temperature in deg C from in-situ measurement
at 1450-1605, June 20, 1978 68
18 Location of stations for in-situ measurement, January 1979 69
19 Velocity from in-situ measurement at 1020-1340, January
30, 1979 70
20 Velocity from in-situ measurement at 1440-1800, January
30, 1979 71
21 Velocity from in-situ measurement at 1430-1640, February
1, 1979 72
22 Surface temperature in deg C from in-situ measurement
at 1020-1340, January 30, 1979 73
23 Surface temperature in deg C from in-situ measurement
at 1440-1640, January 30, 1979 74
24 Surface temperature in deg C from in-situ measurement
at 1430-1640, February 1, 1979 75
25 Surface temperature from in-situ measurement at 1020-
1340, January 30, 1979 76
*
26 Surface temperature from in-situ measurement at 1440-
1800, January 30, 1979 77
27 Surface temperature from in-situ measurement at 1430-
1640, February 1, 1979 78
28 Semidiurnal tide for June 19-20, 1978 at south end of
AncJote Key 79
29 Semidiurnal tide for January 30-Fefaruary 1, 1979 at
south end of Anclote Key 80
30 Surface velocity by modeling at 1030, June 20, 1978 81
31 UW velocity by modeling at 1030, June 20, 1978 82
32 VW velocity by modeling at 1030, June 20, 1978 83
33 Temperature from IR at 1130, June 20, 1978 84
VI
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FIGURES
Number
34 Surface temperature by modeling at 1030, June 20, 1978 .. 85
35 Surface Velocity by modeling at 1430, June 20, 1978 86
36 UW velocity by modeling at 1430, June 20, 1978 87
37 VW velocity by modeling at 1430, June 20, 1978 88
38 Temperature from IR at 1500, June 20, 1978 89
39 Surface Temperature by modeling at 1430, June 20, 1978 .. 90
40 Surface velocity by modeling at 1730, June 20, 1978 91
41 UW velocity by modeling at 1730, June 20, 1978 92
42 VW velocity by modeling at 1730, June 20, 1978 93
43 Temperature from IR at 1730, June 20, 1978 94
44 Surface temperature by modeling at 1730, June 20, 1978 .. 95
45 Surface velocity by modeling at 2030, June 20, 1978 96
46 UW velocity by modeling at 2030, June 20, 1978 97
47 VW velocity by modeling at 2030, June 20, 1978 98
48 Temperature from IR at 2000, June 20, 1978 99
49 Surface temperature by modeling at 2030, June 20, 1978 .. 100
50 Surface velocity by modeling at 1100, January 30, 1979 ... 101
51 UW velocity by modeling at 1100, January 30, 1979 ... 102
52 VW velocity by modeling at 1100, January 30, 1979 103
53 Temperature from IR at 1130, January 30, 1979 104
54 Surface temperature by modeling at 1100, January 30, 1979 105
55 Surface velocity by modeling at 1600, January 30, 1979 .. 106
56 UW velocity by modeling at 1600, January 30, 1979 107
vii
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FIGURES
Number Page
••••••. ••••M*...!..
57 VW velocity by modeling at 1600, January 30, 1979 108
58 Temperature from IR at 1700, January 30, 1979 109
59 Surface temperature by modeling at 1600, January 30, 1979 110
60 Temperature from IR at 1600, February 1, 1979 111
vtl!
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TABLES
Number Page
1 Climatic Data for Summer Run at Ancfote Anchorage 46
2 Climatic Data for Winter Run at Anclote Anchorage 49
IX
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SYMBOLS
Ht
B
H
K
K
L
P
t
Vertical_eddy viscosity,
cm2 sec
HorlzontaJ eddy djffusivity,
cm2 sec
Vertical_eddy diffusivity,
cm2 sec ,
Coriolis factor, sec
Relative humidity in fraction
of unit
Acceleration of gravity,
cm sec
Local water depth with re-
spect to mean sea level, cm
Total water depth, cm
Node index in the direction
of the x-axis
Node index in the direction
of the y-axis
Node index in the direction
of the z-axis
Surface heat exchange co-
efficient, BTU ft"Z day
deg
Reference length, cm
Pressure, dynes cm
Time, sec
ave
u
T Water temperature, deg C
T., Air temperature, deg F
Average of air and dew-
point temperatures, deg F
Dewpoint temperature,
deg F
Equilibrium temperature,
deg F
Ambient surface tempera-
ture, deg F
Component of water velo- ,
city along x-axis, cm sec
Wind speed, mph
Component of water velo- ,
city along y-axis, cm sec
Component of water velo-«
city along z-axis, cm sec
n Displacement of the free
surface with respect to the
mean water level, cm _ -
p Water density, gm cm
fl Nondimensional vertical
fluid velocity
cr Nondimensional vertical
coordinate
w
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ACKNOWLEDGMENTS
This work was supported by a contract from the National Aeronau-
tics and Space Administration (NASA-KSC) and the Environmental Pro-
tection Agency (EPA-RTP).
The authors express their sincere gratitude for the technical and
managerial support of Mr. Roy A. Bland, the NASA-KSC project manager
of this contract, and the NASA-KSC remote sensing group. Special
thanks are also due to Dr. Theodore G. Brna, the EPA-RTP project
manager, for his guidance and support of the experiments, and to Mr.
Albert W. Morneault from Florida Power Company (FPC), Tarpon Springs,
and his data collection group for data acquisition. The support of Mr.
Charles H. Kaplan of EPA was extremely helpful in the planning and re-
viewing of this project.
XI
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SECTION 1
INTRODUCTION
BACKGROUND
The problem of disposing of the waste heat produced as a result of
generation of electrical energy, whether by fossil-fuel or by nuclear-fuel,
is a dominant consideration in making power production compatible with
ecological concerns. For every unit of energy converted to electricity,
approximately two units are rejected in the form of waste heat. The ul-
timate heat sink for the heat removed by the condenser-cooling-water
system is the earth's atmosphere. The cooling water taken from natural
or man-made water bodies is circulated once through the condenser, and
the heated water is discharged back to the same bodies, which usually
are lakes, rivers, estuaries, or coastal waters. Eventually, the heat is
transferred to the atmosphere through evaporation, radiation and conduc-
tion over relatively large areas at the air-water interface.
The use of natural water bodies as an intermediate means for dispo-
sal of waste heat must take into account the effect upon the environment
of the circulation and temperature rises produced in the receiving water.
The rate of oxygen consumption by aquatic species increases with rising
water temperature; however, the ability of water to hold dissolved oxygen
decreases with rising water temperature. There are possibilities of im-
pairment of biological functions of fish and of breaking important links
in the food chain. The lethal effects of thermal pollution are sometimes
obvious; the sublethal effects on hydrobiological systems and waste assi-
milation capacities are not easy to foresee unless interactive hydrothermal,
chemical and biological studies are conducted in an integrated fashion.
Accurate understanding of hydrothermal behavior of the receiving
water bodies is an important factor in a power plant system for the fol-
lowing reasons:
I. To provide a priori information about the nature and extent of ther-
mal impact on the aquatic life forms.
2. To analyse the circulation pattern of the receiving water body so
that recirculation between intake and outlet and consequent decrease
in cooling efficiency can be minimized.
3. To assess the thermal impact on the aquatic life forms existing in the
receiving ecosystem so that post-operational remedies can be done to
1
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reduce the hazards.
It is therefore apparent that not only environmental but planning
and designing interests also are at stake.
The above-mentioned objectives can only be met by having large
data sets over the entire discharge flowfield under various extreme hydro-
logical and meteorological conditions. Measurement for temperature and
velocity made over the affected domain could be used to develop maps for
velocity and temperature distributions. These in-situ measurements can
serve for diagnostic and monitoring purposes under limited circumstances;
however, they are not relevant for predictive objective. The physical
modeling is useful for hydrodynamicai behavior studies when properly
verified; it is generally expensive and time-consuming. Frequently,
physical dimensions necessitate distortion in the model, making exact
dynamic similitude impossible. Mathematical modeling is tractable and
predictive; it has become a powerful means in simulating complex environ-
mental flows.
In order to establish the rationale of model formulation, the physical
mechanisms underlying the heat dispersion from a heated discharge need
to be outlined. The following mechanisms govern the heat dispersal.
I. Entrainment of ambient fluid into the thermal discharge.
2. Buoyant spreading of discharged heated effluent.
3. Diffusion by ambient turbulence.
4. Interaction with ambient currents.
5. Heat loss to the atmosphere through air-water interface.
The first four mechanisms redistribute heat and momentum in the re-
ceiving water body. The last mechanism eventually transfers heat to the
atmosphere. It has been customary, therefore, to make assumptions and
approximations which enable the model solvable. For example, the ambient
turbulence is considered by assuming the eddy viscosity and diffusivity
dependent on the mean velocity field. There assumptions necessitate
careful calibration of models to assure reliability. Large data bases are
needed for proper calibration, especially for three-dimensional, time-de-
pendent models. Remote sensing is the only available method of obtaining
large synoptic data bases. Sengupta et al. (1975) has discussed the need
for remotely sensed data for adequate development of time-dependent,
hydrothermal models.
UNIVERSITY OF MIAMJ MODELS
Critical evaluation of mathematical models used for thermal pollution
analysis has been made by Dunn et al. (1975). They compared the per-
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formance of various models in predicting a standard data base. A gene-
ra! conclusion that can be made from their analysis is that, though some
models may perform well under certain conditions, a generalized three-
dimensional mode! which accounts for wind, current, tide, bottom topo-
graphy and diverse meteorological conditions is yet to be developed.
One of the first three-dimensional models was by Waldrop and Farmer
(1973, 1974 a, b). This model was essentially a free-surface formula-
tion. One of the first three-dimensional models which adequately accounts
for bottom topography and comprehensive meteorological conditions was a
rigid-lid model developed by Sengupta and Lick (1974 a, 1976). They
used a vertical stretching to convert the variable depth, a concept cus-
tomarily adopted by numerical weather forecasting peoples.
The thermal pollution research team at the University of Miami has
for the past several years been developing a package of three-dimensional
mathematical models which could have general application to problems of
power plant heated discharge to the aquatic ecosystem. The primary
motivation behind the effort was to develop a series of models with mini-
ma! restrictive assumptions, enabling applications to diverse basin confi-
gurations and to various driving forces of ambient flow. The effort is
closely integrated with simultaneous remote sensing and ground-truth
data acquisition support. Our aim is to develop adequately calibrated
and verified models for production purpose; that is, for direct applica-
tion by the user communities. The user communities are the power in-
dustries and the regulatory agencies like the Environmental Protection
Agency and the Nuclear Regulatory Commission.
For the time-dependent, free-surface mode! which is the main concern
of this volume, the governing equations are the incompressible Navier-
Stokes equations, conservation of mass, energy and an equation of state.
The conventional hydrostatic and Boussinesq approximations are made.
The mean velocity field closure is assumed in defining eddy viscosity
coefficient. The boundary conditions at water-land interfaces are no-
slip, no-normal velocity and adiabatic conditions. At the air-water inter-
face, wind stress and heat transfer coefficients are specified. At open-
to-sea boundaries, conditions are specified for surface elevation and tem-
perature. Otherwise, normal derivatives of temperature and velocity are
equal to zero. Initial condition is assumed to be equilibrium; that is
"cool start." Conditions at intake and outlet are completely specified in
space and time.
The features of the UM's free-surface model can be summarized as
follows:
1. Gravity, earth's rotation, nonlinear inertia and bottom friction terms
are included in the hydromechanic part.
2. The driving forces include wind, tide, river outflow and power plant
intake and discharge.
3. Convection, diffusion and heat transport at water surface are
3
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included in the thermodynamic part.
H. The a system of coordinate, devised by a meteorologist for avoiding
difficulties at the free surface, Js incorporated.
5. It predicts spatial and temporal variation of surface elevation due
to tide surge and wind set-up.
6. It has means for graphically representing velocity and temperature
fields.
This free-surface model is general enough to be applied with
minor modifications to a large variety of sites, such as lakes, rivers,
estuaries, tidal inlets and coastlines. This model has been successfully
applied to Lake Okeechobee for hydrodynamic study, Biscayne Bay for
dispersion study and Hutchinson Island for hydrotherma! study. This
model can simulate the receiving water body in the far-field and de-
tailed features of thermal plumes and mixing zones in the near-field.
DESCRIPTION OF ANCLOTE ANCHORAGE
The UM's thermal pollution team has been studying the Anclote
Anchorage on the west central coast of Florida near the town of Tarpon
Springs (Figure 1) since the summer of 1978. The Anclote power plant,
operated by the Florida Power Corporation, has two 515 MW, oil-fired
electrical generating units. The once-through cooling water for the
two units is to be drawn from the Anclote River by six pumps deliver-
ing a total of 930,000 gpm. After a temperature rise of 6.1°C, this
water is diluted with a flow of 1,060,000 gpm at ambient temperature to
reduce the temperature rise to 2. 8°C at the outlet to Anclote Anchorage.
This mixing is done in a 1250 meter-long man-made channel leading to
an outlet.
The Anclote Anchorage (Figure 2) consists of shallow coastal water
separated from the Gulf of Mexico by a series of barrier island parallel
to the coast line. The Anchorage has a relatively unrestricted exchange
of water with the Gulf through natural channels to the north and to the
south of the Anclote Keys. Depth ranges from 0. 3 to 3. 6 m, with a
mean of 1.8 m. Shallow regions of less than 0. 6 m comprise approximately
5 km in length and 6 km in average width. Currents in the Anchorage
are tidal- and wind-driven, with the tide entering from the south stronger
than that from the north.
Prior to power plant construction, the Marine Science Institute of
the University of South Florida (USF) was contracted by the Florida
Power Corporation (FPC) to investigate the possible environmental im-
pact of the plant operation. The USF's Anclote Environmental Project
beginning in 1970 was comprehensive in nature. Much of their efforts
was to obtain a detailed picture of the Anclote environment prior to
alternation of the environment by power plant construction and operation.
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Valuable background data were collected in a variety of areas, including
general physical characteristics of the Anclote River and Anchorage,
suspended sediments, turbidity, temperature, salinity, water quality,
seagrasses, benthic invertebrates and fishes. All information was ap-
plied to preconstruction plant design in order to minimize environmental
impact. This application resulted in major changes in intake and dis-
charge channel dredging, outfall design and thermal dishcarge character-
istics. The same team also monitored the undergoing changes of the
environment during and after the dredging operations for the intake and
discharge channels.
Unit I of the Anclote Power Plant commenced operation during the
fall of 1974, after which a post-operation ecological monitoring program
for evaluating the nature and degree of thermal impact was carried out
and maintained by the USF. At the time of this study, the planned
Unit 2 was still pending permission.
Since the nature of thermal impact at Anclote Anchorage is not
clear, it is decided that a joint effort by UM, NASA-KSC, EPA and FPC
to use a three-dimensional mode! with support of remotely sensed data
and in-situ data for calibration and verification may be appropriate to
study this thermal impact at Anclote. It is for this purpose that the
existing free-surface model was adapted to the Anclote site, as we have
done in its application to Biscayne Bay. As the original program was
developed for well-mixed shallow coastal waters, except for some modi-
fication to accommodate the tidally influenced Anclote River flow, the
program was ready for calibration. We shall account for this application
in a later section.
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SECTION 2
CONCLUSIONS
A numerical simulation of the hydrotherma! characteristics of a well-
mixed, shallow, coastal water body at Anclote Anchorage under the effect
of waste heat disposal by a power plant in the summer and winter situa-
tions is presented. The model takes into consideration the effects of
Coriolis force, wind, tide, bottom topography, power plant intake and
discharge, river outflow and surface heat transfer. Results obtained
with the model have been verified with in-situ measurements and 1R data.
Reasonable agreement has been obtained. From the experiences and re-
sults of these simulation runs, the conclusions may be summarized as
follows:
1. Inputs for the description of open boundaries, discharge, wind and
heat transfer must be found experimentally from field data for ac-
curate hydrotherma I predictions.
2. The shape of the thermal plume was dominated by the stage of the
tidal cycles, as clearly exhibited in the plots of IR and calculated
isotherms.
3. Tide plays a main role in the Anchorage as a driving force; its in-
fluence from the south is stronger than that from the north. A
dividing line is observed. On the northern side of this line, the
water flows north, while on the south side it flows south. At this
ridge line, the water has minimal transverse motion. The location
of this line varies with time as the tide from both ends is not in phase.
4. It is important to impose correct boundary conditions, especially the
correct tidal functions on the south and north boundaries, for ob-
taining a good prediction of the thermal plume.
5. Wind does appear to be an important external force affecting surface
currents. It is found that the surface currents are close to the
wind direction when wind speed is in excess of 15 mph.
6. An estimation of the Rossby Number revealed that the nonlinear
inertia terms can be safely neglected for Anclote site. This is be-
cause the ratio of inertia force to the Coriolis force is small compared
with unity.
7. The recirculation of cooling water should be prevented by proper
design of intake and discharge location, since the heated water
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recirculating back to the intake will cause a reduction in the effi-
ciency of the power plant.
Numerically the model behaves very well for both summer and winter
simulation runs. The model is able to include very shallow water depths.
A problem associated with the analysis is computing the response of the
thermal plume to very strong winds such as hurricane, since the formula
used in this study to estimate the vertical eddy viscosity coefficient does
not involve wind speed. A brief investigation was made regarding the
effects of hurricane-force winds. It was found that currents become un-
realistically large in such cases unless the vertical eddy viscosity coeffi-
cient was increased with wind speed. This general problem of computing
the response of shallow coastal water to very strong winds requires addi-
tional research.
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SECTION 3
RECOMMENDATIONS
One of the difficulties encountered in model verification with the
remote sensing temperature field was the lack of measured current data
of comparable accuracy. We believe that, at the verification state, the
accuracy of velocity calculation needs to be assessed in order to clear
ambiguities about the limitations of comparison of calculated surface iso-
thermal maps with the IR imagery mosaics.
In the initial phase of mixing, the plume shape is governed by the
volume of discharge, the geometry of outlet and the initial temperature
difference between the discharge and receiving waters. Consequently,
for predicting the details of the near-field thermal plume a finer grid is
needed near the outlet. Thus, a combination of grid structures would be
desired for the computation with a course grid for far-field and a fine
grid for the near-field. This, however, would result in higher computa-
tional costs.
For any non-reactive, dissolved, chemical constituent, the governing
transport equation is completely analogous to the temperature equation
of the present model. Thus, without much endeavor, the model can be
extended to include the dispersal of these constituents.
For situations of deep and stratified water, the bouyancy effects
are important factors in shaping the plume. Therefore, a coupled system
of momentum and energy equations should be considered as the basis of
formulation.
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SECTION H
MATHEMATICAL FORMULATION AND MODEL DESCRIPTION
GENERAL BACKGROUND ON FREE-SURFACE (TIDAL) MODELS
Recent concern about the ecological future of our estuaries and
coastal waters has generated a need for practical and reliable methods of
predicting the environmental impact of the widespread use of natural sea-
shores as industrial zones. The tendency of setting up once-through,
seawater-cooled, fossil- or nuclear-fueled power generation plants is in-
creasingly strong, due to the demand for electrical power at competitive
prices and prohibitive costs of cooling towers and man-made cooling lakes.
Thus, it is of special importance that mathematical models be developed
for simulating and predicting thermal pollution. The estuaries, tidal
inlets and coastal waters which serve as the receiving body for waste
heat are usually of complex configuration and topography. The flow is
driven by tide, wind, run off and buoyancy force. Thus, a complete
hydrothermal model for coastal waters must, in addition to solving the
three-dimensional equations of mass, momentum and energy, include the
salinity equation to determine closely the local density. A model of this
type is still far beyond feasibility. Therefore, assumptions and approxi-
mations must be made to render the system closed and tractable.
Hindwood and Wallis (1975) have compiled a bibliography of 141.
papers concerning computer models for tidal hydrodynamics. In general,
the output from the hydrodynamic/hydraulic model is recorded on magnetic
tape. This tape is then entered into the thermal dispersion model. The
dispersion model operates through successive solution of the finite differ-
ence equations for the change of temperature (with time and space) due
to diffusion (eddy mixing), advection (velocity transport), and heat
flux through the water surface. Therefore, it is appropriate to say that
the hydrodynamic/hydraulic model serves as the backbone to the thermal
pollution model of the well-mixed water. Commonly, the hydrodynamic
models are two-dimensional and based on the vertically-integrated equations
of motion and continuity for an incompressible fluid. Thus, the vertical
structure of the circulation was not considered. This procedure turned
out to be sufficient for the investigation of tidal processes.
Of the two-dimensional models, we should briefly mention the Leen-
dertse (1967) model and Reid and Bodine (1969) model for their comprehen-
siveness and popularity. In Leendertse's model, the unknowns are verti-
cally-averaged velocities and water level; the bottom friction is in terms
of Chezy coefficient which is to be calibrated. Reid and Bodine used
-------
vertically integrated transports and water level as unknowns and Darcy
friction coefficient. Both models used the space-staggered system which
gives the simplest scheme consistent with the control volume approach to
deriving difference equations. However, in the Leendertse model, the
finite difference equations are expressed in an alternating direction im-
plicit (ADI) form, with two successive time-level operations being executed
during each time cycle; while in the Reid and Bodine model a time-split
explicit (leap-frog) method is used for marching forward, the time step
is limited by numerical stability requirement (cf. Platzman, 1958), less
than the value min(Ax, Ay)/ 2gD ), in which D is the maximum
depth. max max
In the context of two-dimensional tidaUy-driven flows, we should
include the finite element method models which were developed in the last
few years. This is a method combining finite element for spacial discre-
tization and finite difference for temporal discretization. The variables
in discrete element is approximated by simple polynomials whose coeffi-
cients are expressed in terms of nodal values of the variables and their
derivatives. Correctly formulated, the physical conservation principles
are satisfied and, in theory, the element shape and interpolation function
are quite free as long as certain compactibility conditions are satisfied.
The freedom of using irregular grid to fit complicated geometry is the
primary advantage of this method.
At the present stage, FEM (finite element method) models for tidal
hydrodynamics are all two-dimensional. Wang's CAFE (Circulation Analy-
sis by Finite Element) model and Brebbia's shallow water model (in his
Finite Element Hydrodynamic Problem Orientated Language (FEHPOL)
package) are both productive. CAFE (Wang, 1978) has a linear triangu-
lar element for all variables, vertically-integrated transports and water
level, and has split-time method for time integration, Patridge and
Brebbia (1976) use a six-node, quadratic triangular element for all vari-
ables; in this case, they are vertically-averaged velocities and free-
surface elevation. The 4th order Runge-Kutta (explicit) and Trapezoidal
Rule (implicit) methods are used for time integration. In general, the
flexibility of the grid layout, the consistency of FEM formulation and the
easiness in taking into account the spatially variable properties are the
advantages. However, the time integration scheme is very much problem-
oriented, and more fruitful research could lead to a better time integra-
tion method. It is worthwhile to point out that, in the FDE (finite
difference method) for two-dimensional hydrodynamics, the explicit me-
thod had advantage over the implicit method in terms of computer time.
In the classical approach of two-dimensional computations of tides
and storm surges in a shallow sea, it is assumed that the velocities are
uniform over the vertical; however, the vertically uniform velocities are
of little evidence in the case of those propagation and transport processes
which are essentially vertically structured. The application of a three-
dimensional model requires extensive computation on a large number of
grid points if relatively good resolution is needed. This precludes the
10
-------
use of the implicit method which requires that matrix solutions of a row
or column of variables be found. In fact, Leendertse et ai. (1973) deve-
loped a three-dimensional model which really uses the explicit leap-frog
method instead of the implicit methods such as the ADI method Leendertse
used in his two-dimensional model.
Leendertse's three-dimensional model is a vertically-integrated, multi-
layered model, in which the usual assumptions of hydrostatic pressure,
imcompressibility and small density variation are made. The interfaces
are assumed as fixed horizontal planes, while the bottom layer has its
lower face fit to bottom topography and the top layer has its upper face
representing the free surface. The top layer has a time-variable height.
The other layers may be intersected by the bottom and have a height
which is dependent on the bathymetry. The description of the finite-
difference equations from the differential equations is accomplished in
two steps: First, the equations for the layer are derived by vertically
integrating the variables over the layer thickness, and subsequently,
finite difference approximations for the layer equation are developed.
Thus, the vertically-integrated momentum, heat, and salt balance equa-
tions can be presented for each layer. From the continuity equation,
the time derivative of the free surface and the vertical velocity at the
interfaces can be derived. The horizontal pressure gradient in each
layer is approximated from hydrostatic equation with layer-averaged
density, which is tied to salt and heat through the equation of state.
In Leendertse's model (Leendertse and Liu, 1975), the balance of
momentum fluxes between the local and convective accelerations, pressure
gradient, Coriolis force, lateral diffusion, and Snterfacial stresses is
accounted for within each layer; likewise, the balance of heat fluxes be-
tween the time rate, convection, lateral diffusion and cross-layer diffu-
sion is for each layer; so is the salt balance. The same lateral diffusion
coefficient is used for both heat and salt. Even postulations are intro-
duced to express the vertical momentum, salt and heat exchange under
vertically stable or unstable stratifications; these layer-averaged equa-
tions clearly point out the foremost problem of his model. There are
many parameters to be determined and an enormous amount of supporting
field data is needed.
Leendertse et al. (1973) discuss the numerical finite difference
solution scheme in some detail. The explicit leap-frog method is used to
avoid the difficulties encountered by ADI. The spatial grid structure is
cell-like with u, v and w components at the center of the corresponding
normal faces and pressure (p), density (p), salinity (S) and temperature
(T) at the center of the cell. For programming reasons, the bottom must
be approximated in steps of layer thickness, causing some numerical pro-
blems at the jumps. But the most troublesome one is the specification of
boundary conditions for the seaward boundary of the model, since field
measurements do not provide enough data to describe that boundary
completely. Numerical experiments involving the simulations of hydrody-
namic behavior of both Chesapeake Bay and San Francisco Bay are given
11
-------
with a graphical representation of velocity, salinity and temperature results.
A specific three-dimensional model has been developed by Sundermann
(1974) and applied to the North Sea. In the model, the usual assumption
of a hydrostatic pressure and the Boussinesq approximation for the turbu-
lent Reynolds stresses are made. However, no horizontal turbulent ex-
change of momentum is retained. To fit the vertical velocity profiles of
a wind-generated surface current and a compensating countercurrent near
the bottom, as observed in nature, it was found necessary to include a
bottom boundary layer that can be modeled by having a vertical eddy
viscosity dependent on the depth. The spatial discretization is carried
out by means of a cubic grid net, while the explicit method of Crank-
Nicolson has been used for approximating the vertical diffusion term.
The a system of coordinates, devised by N. A. Phillips (1957), was
used for numerical forecasting. Its advantages are that the kinematic
boundary conditions at free-surface and bottom are made simple and a
vertical stretching is used to avoid the difficulty of using regular grid
net for irregular bathymetry. Briefly, the (x, y, a, t) system is used
to replace (x, y, z, t), and the free surface and bottom are trans-
formed into a = 0 and a = -1 surfaces. The modified vertical velocity,
Q = dcr/dt, is zero at both these surfaces, while the actual vertical
velocity is w = dn/dt at the free surface and w = 0 at the bottom.
Freeman et al. (1972) introduced u-transformation into their three-
dimensional free-surface model for wind-generated circulation in a closed
region. Sengupta and Lick (1974) incorporated ^-transformation in their
rigid-lid model for wind-driven circulation in lakes. However, the trans-
formation was used only for the advantage of vertical stretching so that
a constant vertical grid size, A a, can be used throughout the domain.
This rigid-lid approximation effectively eliminates surface gravity wave
by imposing a zero-actual vertical velocity at the water surface. An
additional equation, a Poisson equation for surface pressure which con-
tains the rigid-lid condition, can be derived. Therefore, at each time
step the Poisson equation has to be solved by iterative method. However,
for large problems, particularly when normal derivative boundary condi-
tions are used, this can be a time consuming part of the calculation pro-
cedure. As mentioned, this model is not for tidally-driven coastal flow.
The present free-surface model is similar to that of Freeman et al.
in using o-coordinate system. However, the horizontal turbulent exchange
of momentum terms is neglected while the vertical turbulent exchange term
is approximated by using the Dufort-Frankel differencing to ease the con-
straint on numerical stability due to neglect of the horizontal diffusion
terms. This model was developed by Carter (1977) for his study on wind-
driven flow in Lake Okeechobee. Later, Sengupta et al. (1978) applied
it to tidal flow in Biscayne Bay.
GOVERNING EQUATIONS AND BOUNDARY CONDITIONS
Flow of water in estuaries and coastal areas is predominantly hori-
12
-------
zontai. Vertical velocities do occur and are important, as they character-
ize the vertical circulation. However, the vertical acceleration is extremely
small in comparison with the gravitational acceleration, so that the vertical
acceleration is neglected and the vertical equation of motion is then re-
placed by the hydrostatic assumption. The effect of density variations
on the inertial and diffusion terms in the horizontal equations of motion
is neglected. Density variation is retained in the lateral pressure gradi-
ent terms; that is, the effect of bouyancy is accounted for by allowing
density variations in the horizontal pressure gradients which influence
the fluid motion through the horizontal momentum equations. Consequently,
the continuity equation replaces the mass conservation equation. In the
estuaries, the density is influenced by the salinity and temperature; in
this model, the density will be taken dependent only on the temperature.
For turbulent closure, the Boussinesq approximation for the turbu-
lent Reynolds stresses is made, and lateral dispersion of momentum is
further assumed unimportant when compared to the rest of the terms in
the horizontal equations of motion. The vertical eddy viscosity, the
vertical and horizontal eddy diffusivity of temperature are assumed to be
constants, although the horizontal eddy diffusivity has orders of magni-
tude larger than the vertical eddy diffusivity, being due to the much
larger horizontal characteristic length, L, in comparison with the vertical
characteristic length, H.
With these conditions, we can write the set of governing equations
expressing the conservation of mass, mementum, and energy in incompres-
sible flow:
9t ' Tx ' T^ ' ~3z~ iy ' p 3x" "V3^ = ° (1)
3v , 3uv , 3vv , 3vw , r.. . 1 3p „ 32v _ n
" U (2)
-H + oe = 0
3z p§ u (3)
3x 3y 3z (4)
3T L 3uT L 3vT t 3wT „ ,32T , 32T, „ 32T „
P = P(T) (6)
13
-------
where x, y, z = Cartesian coordinates positive eastward, northward and
upward, respectively
u, v, w = Respective components of velocity in x, y, z direction
t = Time
p = Pressure
T
f
B,
B.
= Density
= Temperature
= Coriolis parameter
= Gravitational acceleration
- Vertical eddy viscosity
= Horizontal eddy diffusivity
- Vertical eddy diffusivity
The first three equations represent the equations of motion. Equa-
tion (I) is the equation of continuity, and Equation (5) represents the
energy equation. The equation of state. Equation (6), expresses the re-
lation between the density and the temperature. For a more complete
representation of an estuarine ecosystem, an equation expressing the
balance of the salts dissolved in the water should be included together
with balance equations of dissolved substances that are important to the
analysis. All these equations have the same form as Equation (5). If
salinity is included then the density should be related to both salinity
and temperature.
Boundary conditions for the above equations must be specified:
u = v = w = 0 at z = -h(x, y) (7)
\
p = 0
3u
92
_* av Ty
A ' 82 = pAT,
3T
K
S at z = n(x, y, t)
aif
(Te - V
y~ - w = 0
(8a)
(8b)
(8c)
(8d)
-------
where n = Free-surface elevation above mean water level
T , T = Surface wind stresses
x y
C = Specific heat at constant pressure
K =: Surface heat transfer coefficient
T = Equilibrium temperature
T = Surface water temperature
Also, boundary conditions at closed and open boundaries must be known.
One major difficulty in the treatment of the free-surface mode! is
the fulfillment of kinematic condition at the free surface. The approach
used in the model is to follow a vertical stretching transformation sug-
gested by Phillips (1957) and used successively in lake circulation studies
by Freeman et al. (1972). Using this o-transformation, the free surface
becomes a fixed, flat surface and the variable depth bottom becomes a
fixed, flat bottom of constant depth. Thus, this method allows easy
adaptation to various bottom topographies, and a constant vertical grid
size can be used throughout the domain.
With respect to the mean water level (MWL), z is n(x, y, t) at the
free surface and -h(x, y) at the bottom. The a-transformation of the
vertical coordinate for the free-surface model is obtained by introducing
_ z - TI(X. y, t) z - n ..
a ~ h(x, y) + n(x, y, t) H(x, y, t) (9)
where H = h + n is the total depth measured from local water surface.
Figure 2 shows the (x, y, a) coordinate system. Note that the value
of o decreases monotonically from zero at the free surface to minus unity
at the bottom. The modified vertical velocity, ft, is clearly zero at both
free surface and bottom. From Equation (9), the actual vertical velocity,
w, is related to fl by
w = (h + n) n + (0 + 1)|3 + ff(u|| + vf|) (10)
Differential transformation relationships are required to convert
equations in (x, y, z) to those in (x, y a). The first derivatives can
be written as
3F _ 3F if n— + -fLlhlE
(} = (} " H1 3x 3x 3a
15
-------
l 3H . 3TK3F
r JF, _ 1 3F
I3zjx,y,t H 30 (lie)
where F Is any dependent variable and H = h + n is the total depth and
is independent of a. The continuity equation can be written as
By integrating Equation (12) from the bottom to the surface and observing
the fact that fi vanishes at either face, we obtain the first useful form of
the continuity equation.
The horizontal momentum equations are
3Hvu ^u + (1 + } |u
3t 3x 3y " 3a- 3a
The energy equation can be written as
B
The coupling of momentum and energy equations may be retained via
the density which is assumed to be a function of temperature only. An
empirical formula
16
-------
p = 1.029431 - 0.000020T - 0.000005T2 (
may be used for sea water of salinity 38 parts per thousand. The hydro
static equation can be integrated to obtain a diagnostic equation for p.
The continuity equation, Equation (12), can also be integrated from
the free surface to a to yield the modified vertical velocity n at a plane.
Thus, a second useful form of Equation (12) yields
- £ In - Ira fMH + 3Hv
~ H 3t H\ ( 3x- 3y
-'0
(19)
The actual vertical velocity w is recovered through relationship Equation
(10). In solving parabolic-type partial-differential Equations (14)-(16)
and differential Equation (13), we choose the initial conditions to be zero
elevation from MWL and zero velocities. Given the initial temperature
field, the boundary conditions are u = v = w = 0 on all solid surfaces,
and at a = 0, the wind stresses (t • , T ) are exerted; i.e.
A A
2y3_u _y §z _
pH 3cr ~ Tx' pH 3a Ty at ° ~ U (20a,b)
The adiabatic conditions are assumed on all solid surfaces, and at a = 0,
the heat flux is set proportional to the difference between surface tempera-
ture T and equilibrium temperature T ; i.e.
PC B «T ,,
...P v |£ = K (T - T ) at a = 0 f2;n
H 3cr s e s t.^1-'
The specification of velocities and temperatures at the open boundar-
ies where the tide enters the basin is more difficult. In the present study,
the tides outside the basin are given, and the difference of surface eleva-
tions across the open boundaries is used to determine the normal velocities
there. The temperature at the open boundaries is set equal to ambient
temperature. The velocities and temperature at the discharge is specified
according to plant operation.
UNCOUPLED SYSTEM
The foregoing system may be uncoupled so that the hydrodynamic
model for tidal-driven flow is separated from the thermal dispersion model.
The former system consists of Equations (13), (14), (15) and (19), while
the latter contains Equation (16) only. This decoupling amounts to an
17
-------
assumption that the density variations due to the temperature rises
caused by waste heat dispersion are negligibly small, so that the water
is of constant density and the energy equation can be uncoupled from
the continuity and momentum equations. The lateral pressure gradient
is thus directly related to the gradient of the free surface.
In the present problem of shallow water flow strongly influenced by
tide, the thermal dispersion is mainly caused by velocity transport and
the buoyancy effect is negligible with comparison to the tidal effect.
Numerically, this decoupling has two obvious advantages. Firstly, the
time step of the coupled model is the same as that of the hydrodynamic
model. This time step is limited by the criterion for computational
stability by the explicit method; namely, At must satisfy the condition
cAt/As« 1, where As represents one of the space intervals in the three-
dimensional grid and c denotes the maximum characteristic speed. In
the free-surface model, c is identified with the wave speed and As with
horizontal spacing; thus,
At < A*, Ay)
"• *. . '
max
However, the stability analysis of linearized energy equation alone yields
the thermal time step to be
1 Ax ' Ay * 2Bh(Ax^ H
where u and v are the maximum particle velocities in x and y direction
respectively. For the present problem, these time steps are 50 sec and
100 sec respectively; therefore, in using At =15 sec, we can store the
hydrodynamic results every 20 steps to match At— = 300 sec used in the
detached calculation of temperature field. Secondly, the decoupling
allows one to try for flow solution before it is used for temperature cal-
culations, and the appropriate flow solution may be used for many tem-
perature solutions of various initial and boundary conditions of tempera-
ture.
The effect of stratification Is known to arrest the thermal dispersion.
For a shallow basin having an insignificant river discharge in comparison
with tidal flow, which is the case of concern, the basin is nearly well-
mixed; the density variation is small. Thus, uncoupling is physically
sound and numerically beneficial.
COMPUTATIONAL GRID
To represent the equations in finite difference form, a horizontal
staggered computing grid system is used. Its plan version, shown in
18
-------
Figure 4, indicates the arrangement of field variables in the x, y plane.
Figure 5 shows a vertical fluid column subdivided into four layers; each
has a constant nondimensional thickness, A a = 0.25. The u, v and w
velocities are shown at their definition point respectively. Having divided
the region into cells by a series of grid points which are spaced at dis-
tances of Ax, Ay, and A a, the time variable is differenced into increment
of At such that t = nAt, where n denotes the current time step.
The water depth, h. ., is specified at full-grid point where both
horizontal indices, i and f/ are integers; however, since the total depth,
H - h + n, is needed at half-grid point, where both indices i and j have
half-integer values, (i + $, j + i), the specified water depth at full-grid
points is averaged. The following notations are used to indicate water
depth at the u-, v- and n-points.
(23)
A notation for n at full-grid point is needed for calculating the
nonlinear inertia terms of the momentum equations:
_ ., n , n , n j.ni
Ei,j = Hni,i + Vl.j + Vl.i-1 * Yj-l3 (24)
where n indicates the present-time surface elevation above the mean
water level. With Equations (22) to (24) and the calculated water level,
n.., at present time step, the present total water depths at u-, v- and
n-points are respectively given by the following expressions.
(25)
(26)
H. . = hn
. . - .
i.J i.J (27)
In the momentum equations, values of u and v are required at half-grid
points where values of these variables are not defined and thus, not
stored. In these cases, values are obtained by linear interpolation be-
tween the values stored for that variable at the two neighboring points.
19
-------
Therefore, the following average notations are adopted.
It is clear that notations (28) and (29) give velocities at half-grid point
while notations (30) and (31) give velocities at full-grid point. No
superscript n is needed for these notations. These and other variables
are shown in Figure 6, where u-, v- and n-points are marked.
FINITE DIFFERENCE EQUATIONS
In the equations presented here, the space subscripts are i, j, k
and are all integers, unless otherwise noted. The superscript n refers
to the time level; n is present time, at which the predictive equations,
(13) through (16), are used to advance the fields of u, v, n and T to
new time level n + 1. In this study, first derivatives with respect to
time are always represented by central time difference, i.e., for any
variable F,
- F?:1,
V3t'i,j,k 2At v^" ' (32)
where 0(At2) refers to the order of truncation error associated with this
differencing. The central time scheme is thus of second order accuracy,
since the neglected terms are terms multiplied by (At2) or a higher order
of At. This indicates that the finite difference approximation can be
made to approach the differential by taking sufficiently small At.
The second derivatives with respect to a in Equations (14) to (16)
are written in the DuFort-Frankel format, i.e.
n F F -_ pn-l pn
- i.i.k-1 1.1.k i.i.k l,j
3a2i,j,k (A a)2 (33)
This expression for second order crderivatives when used with the central
20
-------
time difference is known to avoid the inherent instability caused by using
the common form of second order difference; i.e.
The continuity equation, (13), is the predictive equation for surface
elevation n/ and thus applies at n-point. For convenience, we introduce,
at level k of (i,j) -column, a variable
Fn =
k Ax Ay
where the superscript n on the right-hand-side variables has been dropped.
Then the finite difference form of the continuity equation can be written as
n+1 n-1
Using Simpson's rule, the above equation gives
= "U1 + (2At)T CF1 * 4F2 ^ 2F3 + 4F4 + F5] (34)
It is to be noted that Fg = 0 since both u and v vanish at the solid bottom.
Another form of continuity equation, (19), can be used to calculate
the modified vertical velocity n at each level of the column. That is
... ,- . n+1 n-1
o - "fk"1^Ag n, .- - n. •
i i k -- H -^ _ 1>J
lf]lJS Hi, 2At
(35)
where the trapezoidal formula has been used and H. . is from Equation (27);
i.e. ''*
H. . =H* = h?. + n?,
i,] i,] i.] i>]
21
-------
In the computation, always the present time ft only is needed; therefore,
no superscript n is required for ft and, thus, one array is needed to
store J2. On the other hand, three arrays are necessary to store ri at
the past (n-1), present (n) and advance (n+1) stages.
The actual vertical velocity itself, w, is not needed in the calcula-
tion. However, when it is desired, it can be obtained from Equation (10),
which is rewritten here in finite difference form.
wi i k = Hi
l.J.'K l
n+1 n-1
2At
2Ax
IJ' 2Ay
n+1 n-1
2At i.j.k 2Ax
2Ay (36)
The predictive equations for velocities are given by Equation (1H]
and Equation (15); in finite difference form, these equations are
Ui,j ui,j,k " Ux,1 ui,j.k _ - vi,j,k + vi-l,j.k + vi-
2At ~ i,j 4
. n-+l n-1
" a " U " "
Uj~ (A a)2 " Xi,j,k
(37)
7n+1vn+1 - V11'1 n~1 U 4- U + +
Ad—Llik LJ—iJLiik_ = -fv> . ^J^ —klli-k i+l,j-l,k i+1'3'k
2At
v i.i.k-l Vi,3,k vi,j,k
Ay V. . (Aa)2 "i.j.k
i.J
(38)
22
-------
where, unless otherwise noted, ail the superscripts are n and thus are
dropped for convenience. The nonlinear inertia terms X. . . and Y. . .
i,J,rvI,J,lN
are
Hi.iuf.i.kuf,j,k
~
Ax
Ay
n-KL n+1 n-1 n-1
"1,1 * Vl,1 " \,] " Vl,i
4At
(39a)
i,j,k " Ay
. (hm.i ^Vi.i^^i.i.kVi.i.k- (hi.i +Ei,i)ui7-n.i.kvf+i.j,k
+ Ax
n+1 n-1 n-1
"1,1-1" "l.j " ni.i-l
4At
(39b)
Unless otherwise noted, ail the superscripts are n and thus dropped,
since the inertia terms are calculated at present time level n. In Equa-
tions (39a,b), r\n and n are presently obtained variables; since at each
time level n and B are calculated first by Equations (34) and (35) respec-
tively, u and v are followed by Equations (37) and (38).
When the calculations for velocities are to be performed using-the
predicative equations, (37) and (38), the variables u. . . and v. . .
which appeared respectively on the right-hand-side of ^Equations1'(37) and
(38) should be rearranged, so that a form for velocities similar to Equa-
tion (34) appears. In this form, all quantities on the right-hand-side are
obtained based on specified h, calculated u and v at present and previous
time levels, newly computed Q and n at three time levels. All the variables
23
-------
and notations that appeared in Equations (37) through (39) have been
defined and are shown in Figure 4. However, in Figure 4, the common
superscript n and subscript k have been omitted for brevity. Figure 4
also shows clearly the horizontal extent of variables involved in the cal-
culation of n, u and v in the cross-hatched region, namely n. ./ u. . .
and v. . . (for k = 1 to 4). I/J I/J'K
i/J/ K
Similar to the treatment of the momentum equations, the energy
equation, Equation (16), can be written In finite difference form as
KLprH-1 Hn-lTn-l
i.j i,j,k " i,j 1,1. k _
2At
T11 9Tn 4. Tn Tn 9T11 -I- T11
.1. k " i,j,k * i-l,j.k , ,]+l,k ^Ai.1.k' i.j-l.k,
n _ Tn+l _ Tn-l Tn
1.
(A
-------
for clearity. Equation (40), similar^tp Equations (37) and (38), can
lead to a predictive equation for T. . . .
In the model, since the density is considered as a constant, the
equation of state, Equations (17), is of no use, and the system is un-
coupled. With that, the temperature field can be solved separately pro-
vided that all results from the hydrodynamic model are available. That
is, the spatial and temporal variation of temperature is solved after the
spatial and temporal variation of velocity fields is known. Alternatively,
Equation (40) can predict Tn ' after obtaining n , 0, un and v" f
at each time cycle. The former involves two separated programs while
the latter is a coupled program only. In coding the model above, both
possibilities are taken into account by using flag statements (see Users
Manual for Free-Surface Model, 1980).
SOLUTION PROCEDURES
Clearly, Equations (37) and (38) are for u and v at interior points;
that is, they are not on the boundaries. Figure 4 implies why these
formulae are not for the boundary points. Therefore, Figure 2 indicates
that the normal velocity along the boundaries must be either specified or
calculated by some other means. For solid lateral boundaries parallel to
x- or y-axis, the normal velocities are specified to be zero all the time.
At the river mouth and discharge outlet, the normal components are given
by the known flowrate and the average water depth. At the seaward
boundary, the imposing tide is specified along a parallel line at half-grid
size away from the boundary. The difference of water elevations at
half-grid points across the open boundary is used to calculate the normal
velocity there.
At the bottom, the no-slip condition requires that u = v = 0 at all
times; hence, there is no need for doing Equations (37) and (38) at K = 5.
At the free surface or k = 1, the wind-produced shear stresses are pre-
scribed as a function of space and time, and can be written as
These conditions demand a modification of predictive formula for un+1 and
v at k = 1. That is, the DuFort-Frankel format for second derivative,
Equation (33), is replaced by (indices i and j are omitted and values of
k are indicated)
, .,
K=l
25
-------
and a similar one for v-momentum equation. In this scheme, the free-
surface,wind drag has been incorporated to drive the current through
the u and v at the free surface.
The boundary conditions on vertical velocity w are the kinetic con-
dition
w = ^ at a = 0
and the rigid bottom condition
w - 0 at a = -1
These conditions expressed in terms of modified vertical velocity fl are
fl - 0 atc = 0ork = 1
Q = 0 at a - -1 or k = 5
As stated earlier, these conditions have been incorporated into the first
integral form of the continuity equation; namely, the predictive equation
for surface elevation. Equation (13) or Equation (14) with F = 0. Conse-
quently, the finite difference formular for Q, Equation (35), is for k = 2,
3 and 4, while Qk=s1 and 8k=5 are set to zero at all times.
The boundary conditions of energy equation are adiabatic on solid
boundaries and known temperature or zero normal derivative of tempera-
ture on open boundaries. It can be seen that if one neglects the lateral
thermal diffusion terms, which are small in comparison with convection
terms anyway, then no adiabatic condition is needed on the solid lateral
wall; this is due to zero normal velocities on the solid lateral wall. How-
ever, condition is still needed on the open boundaries. At the bottom,
the heat flux is zero; while at the surface, the heat flux is proportional
to (T - T ) .These conditions are required when Equation (40) is used
to calculate T at k = 1 and k = 4 respectively.
As in most other hydrodynamic models for transient problems, the
computation routine works step-by-step in time. This means that the
computation proceeds through a sequence of time steps, each advancing
the entire flow configuration through a small, but finite, increment of
time, At. The results of the present and the last steps act as a basis
for the calculation to proceed to the next one, whereby the initial con-
ditions can develop, within the limitations imposed by the boundary con-
ditions, into the subsequent flow configurations. That is, provided that
the values of dependent variables are known initially, the values at sub-
sequent times are obtained by using the explicit scheme. The leap-frog
finite difference formulae. Equations (34), (36), (37) and (40), predict
surface elevation, n, and two horizontal velocities, u and v, and water
temperature, T, at time level n+1. The vertical velocity w does not
appear in these equations and thus, is left out until needed for picturing
26
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the flow configuration. Each time cycle itself contains the following recur-
sive processes.
1. The countinuity equation in the form of Equation (34) is used for n
through the mass conservation of the fluid column at (i,j). At the
same time, the modified vertical velocity fi is also calculated at
k = 2, 3 and 4, since it is also based on the continuity equation but
in a different form; namely, Equation (35).
2. The nonlinear inertia terms that appear in the horizontal momentum
equations, (37) and (38), are considered as driving terms; that is,
they are calculated from formulae (39a,b) by using known results
at the present time (n). This calculation usually takes a significant
portion of the total computing time; thus, it is advised to drop this
calculation whenever justified. For general tidal flows, the inertia
effect is negligibly small in comparison with the Coriolis force. There-
fore, if the Rossby Number, which is the ratio of inertial force to
Coriolis force, is very close to zero, the calculation of X and Y terms
can be skipped entirely.
3. The horizontal momentum equatipns, Equations (37) and (38), are
used to calculate u and vn . Here, the specified boundary con-
ditions, such as prescribed normal velocities, surface wind stresses
and specified tide, come into play.
4. The nonlinear terms, R, in the energy equation, Equation (40), is
calculated by formula Equation (41). Unlike the nonlinear terms,
X and Y of the momentum equations, R is to be included and calcu-
lated under normal circumstances.
5. Temperature at advance step, Tn , is calculated by Equation (40),
and the adiabatic condition, known discharge temperature, given
ambient temperature and surface heat transfer rate play a part in
determining the temperature field.
STABILITY
The leap-frog (explicit) method has a limit on the size of the time
step. Exceeding this limit makes the computation unstable. According to
Platzman (1963), the maximum time step for an inviscid linear system, i.e.
the system with viscous and nonlinear inertia effects neglected, the maxi-
mum allowable time step is
.. r,f 2
Atmax = "I5 +
where H - maximum depth in the problem. If the Coriolis effect is
also neglected, then
27
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At
max
The leap-frog method with the DuFort-Frankel scheme for vertical
diffusion terms has been adopted in the present model. As pointed out
by Forsythe and Wasow (I960), this format is unconditionally stable for
a pure diffusion model, a system in which the viscous diffusion is the
only mechanism responsible for transport. Although, the present hydro-
thermal model is considerably more complex than a pure diffusion model,
the use of the DuFort-Frankel format is an important consideration for
a shallow water system because the vertical diffusion criterion tends to
become relatively more restrictive as the vertical dimension becomes
smaller.
28
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SECTION 5
APPLICATIONS TO ANCLOTE ANCHORAGE
INTRODUCTION
The present model has been successfully applied in thermal disper-
sion studies at the Anclote Anchorage site. The site, located on the
west central coast of Florida near the town of Tarpon Springs, is a shal-
low channel between the mainland and Anclote Key which separates the
channel from the Gulf of Mexico, as shown in Figure 1. Anclote Anchorage
is of interest since it is the receiving water body for the Anclote Power
Plant cooling water discharge. The channel is relatively shallow with
depths ranging from 0.3 to 3.6 m. Shallow regions of less than 0.6 meter
comprise about 35% of the Anclote Anchorage area, which is approximately
5 km in length and 6 km in average width. The principal driving mecha-
nism for current circulation is tidal flux at the north and south entrances
of the channel. The tide is predominantly semidiurnal with a mean range
of two feet. Earlier measurements of temperature and salinity indicated
the currents flow in and out through both entrances. However, the water
exchange appears to be stronger in the south than in the north, or the
currents generally flow in the north direction during flood tide and flow
in the south direction during ebb tide.
The Anclote Power Plant operated by the Florida Power Corporation
has two 515 MW, oil-fired, electrical generating units. Cooling water is
drawn from the Anclote River through a man-made channel. The six
pumps delivering a total of 1,990,000 gpm (125.6 m3/sec) are designed to
raise the water temperature of 2. 8°C above the ambient water temperature.
The heated water is discharged back into the Anclote basin through the
discharge channel with dredged submarine extension. The designed
total flowrate is approximately 53 times the long-term average flowrate
of the Anclote River. At present, only Unit 1 is operative, while Unit
2 is awaiting permission. The present flowrate, therefore, is 995,000
gpm (62.7 m3/sec).
The model as applied to the Anclote Anchorage shows its capacity
for considering the effects of geometry and bathymetry, spatio-temporal
variation of the free surface, various boundary conditions, including tides
of different phase and range, surface heat transfer based on equilibrium
temperature concept, and changing meteorological conditions. In addition,
turbulence has been considered by using the eddy transport approxima-
tions.
29
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CHOICE OF DOMAIN AND GRID SYSTEM
In this study, the Anclote Anchorage is designed to contain the area
of water bounded by the Anclote Keys and sand barriers on the west,
mainland on the east, and two imaginary E-W lines drawn from the northern
tip of the barriers and the south end of Anclote Key to the mainland.
These two imaginary boundaries are considered to be so far that the
thermal plume will not reach them even at low tida! stage. Its location
and schematization are shown in Figure 7. This water is open to tides
from the Gulf of Mexico at both ends. The Anclote River, where the
intake of cooling water is located, is also included since the recirculation
of discharge flow is of concern to the power company.
The adopted horizontal grid layout and its index are also shown in
Figure 7. The grid work Is allowed to orient away from north-south,
east-west system, but in general, the y-axis of the grid system aligns
with south-north. Thus, the subscript i increases eastwardly while j
increases northwardly. The z-axis is chosen upward from mean water
surface while the subscript k increases downward from the water surface.
The selection of the grid size is governed by several constraints.
If the grid size is too large, the approximation of the channel in the
system will be inaccurate, and at certain size, the computation will become
meaningless. However, decreasing the grid size will lead to a considerable
increase in the computer time since the computations must then be made
on more points. In addition, the time step will decrease because the
dispersive properties of the computational method are related to the ratio
of the time step to the spatial grid size. After some numerical experiments,
the model on a 16x 14 x 5 grid with grid size Ax = Ay = 416. 75 m and
A a = 0. 25 is considered a good compromise between the resolution desired
for the region near the discharge and the limitation of the computer
(UNIVAC 1108) at the University of Miami. Also, care has been taken
to have the intake and outlet at grid points. The velocities at these
grid points are specified such that the flowrate and direction can be
easily represented.
SUMMARY OF DATA
Field measurements of current velocities and water temperature have
been made for model calibration as well as verification purposes for the
hydrothermal prediction. Two field trips were carried out in the summer
and winter respectively. The procedures and data results will be discussed
briefly below.
June 1978 Data Acquisition
On June 19 and 20, a team was sent out to the field for measurement
of current velocity and temperature. In coordination, the infrared (IR)
scanner data was obtained by flights over the channel. The current
measurement and temperature readings were obtained at 9 points; 4
30
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points were located at the north end, four others at the south end, and
one in the middle of the channel, as shown in Figure 8. At each point,
the measurements were done at different depths, namely the water surface
and some depths below the water surface.
The in-situ measurements were carried out by personnel on three
boats. The Barnes PRT-5 Radiometer was used to measure the surface
temperature, and a thermister (#9) with floating mechanism was used to
measure the temperature of the water surface. Temperature profiles are
measured by lowering another thermister (#4) into the water in increments
of 2 to 3 feet. Current velocities are measured with a Bendix Model No.
665 w/readout current meter which reads the magnitude and direction of
the current. Figures 9 through 12 show the current velocities at 4 tidal
stages, which give a picture of the current patterns in the channel,
expecially at the boundaries. One may notice that the current is stronger
at the south boundary than at the north boundary. Both boundaries can
exchange water with the Gulf of Mexico.
The flights were made at 609.6 meter or 2000 feet altitude. The
black body of 1R scanning window was set at different ranges for each
flight set. For the first flight, the black body range was 74° to 98°F,
the second was 75° to 99°F, the third was 78° to 102°F and the fourth
was 80° to 104°F. All have a full black body range of 24°F; it provides
a satisfactory resolution of 4°F for each of the six colors between white
and black. A finer resolution was also obtained which reduced the color
band to 0.74°C (1.33°F) temperature spread. At 2000 feet altitude, the
scanning width is 941.8 meters or 3090 feet. With this scanning width,
the whole stretch of the channel was covered by ten east-west flights
as labeled in Figure 13.
The in-situ measurement of water surface temperature at the time
when the airborne IR data was undertaken provides a calibration of IR
temperature. Figures 14-17 show the measured temperature at their
points at four tidal stages of this mission. Since the channel is quite
shallow and only the surface temperature is of primary concern in this
study, only the surface temperature is presented.
January 1979 Data Acquisition
Field measurements by boat and by IR scanning were schedualed
on January 30 and 31. On previous experience, the current measurement
and temperature reading were carried at different points from the pre-
vious mission. Figure 18 shows the location of all points where the data
were collected by three boats. The route for boat #1 was along points
Ji' .§' 1' 1 and Jj' and for boat #2/ the route was along points 8, 7, jj,
W, and JI MeaTiwhile, boat #3 was working on the region near" the coast
between outlet and intake. The water temperature and current velocity
were measured by the same instruments used in the previous data acqui-
sition. The measurements were taken at the surface and successive
depths of 3 ft (1 m) intervals. The flights were coordinated at the same
31
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time as for IR scanning, while the boats were collecting the data. These
ten flight lines, shown in Figure 13, were made at 609.6 meters or 2000
feet altitude. The black body of IR scanning range was set at 44-80°F
so each of the six color bands between white and black would represent
a 6°F interval.
On January 30, the weather permitted both morning and afternoon
data collection. However, on January 31, the conditions was so bad that
the mission had to be postponed to February 1 afternoon. Figures 1 9
through 21 show the current velocities at three tidal stages; namely,
flood tide, ebb tide and high tide respectively. Figures 22 through 24
show the surface temperature distribution at corresponding stages.- To
further show the measured temperature fields, isotherms were interpolated
from these in-situ data and presented in Figure 25 through 27 correspond-
ing to each tidal stage respectively.
In these two data acquisitions, although a plan of synchronized
measurements of current at tide changes was carried out in order to
provide current data at slack, flood and ebb, the effort was not so
successful due to technical difficulties in obtaining reading and other un-
expected circumstances. Therefore, the current data could only be of
use as reference.
CALCULATION OF INPUTS
The important input parameters and some specification of boundary
conditions, such as intake and discharge velocities, discharge temperature,
tidal condition, river flow, surface heat transfer, and wind stress, will
be presented in this section.
1. Time step, DT
In order to determine the time step, DT, the stability criterion
has to be followed:
Dx _ 41760
About 1/3 of this value is reasonably safe to use.
Here, we use DT = 15 sec.
2. Vertical eddy viscosity, A
The vertical eddy viscosity 5s estimated by means of the formula
where H is the local depth and C is an empirical constant. This
type of formula for horizontal turbulent diffusion was originally
32
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suggested by Richardson (1926). Since that time, the 4/3 power has
been substantiated by a considerable amount of empirical evidence and
theoretical analysis. The so-called constant C may vary with the in-
tensity of the turbulence and is not well established. In this study,
H is used as maximum depth and C is 0.002, so that we have
Av = .002x(360) = 6 cm2 /sec
For shallow, well-mixed tidal water, about three times the calculated
value was found suitable. Here, we use A = 20 cm2 /sec.
3. Horizontal eddy diffusivity, B.
The horizontal eddy diffusivity is calculated by the same formula
as mentioned above. However, the maximum depth is replaced by the
maximum length of the domain, which is 6 km in this study. So we
have
Bh = .002x(600,000)It/3 = 100,000 cm2 /sec
4. Vertical eddy diffusivity, B
In this study, the turbulent Prandtl Number is assumed as 1.
Thus, the vertical eddy diffusivity is equal to the vertical eddy vis-
cosity or B = A =20 cm2 /sec.
5. Surface heat transfer coefficient, K
The procedures for K calculation are as follows:
a. T . = T - (14.55 + 0.11VT )(1-f) - [(2.5 + 0. 007T )(1-f)]3
Q o a 3
where Tj = dewpoint temperature, °F
T = air temperature, °F
3
f = relative humidity in fraction of unit
b. 6 = 0.255 - 0.0085Tave + 0. 000204Tave2
where Tgve = fl"s + Td) /2, and B is an intermediate step
T = ambient surface temperature, °F
c. f(u) = 70 + 0. 7u2
where u = wind speed, mph
d. KS = 15.7 + (p + 0. 26)f(u)
where KS is the surface heat transfer coefficient in BTU/ft2 day.
33
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The values of T , f, u, T needed for this calculation are read
from the climato&gical data3.
6. Wind drag coefficient, C ,
The wind stresses on the free surface are introduced into the
mode! as
v 3ui _ v 3 v i _
pH 80'0=0 = Tx " pH 3~0'a=0 ~ Ty
The subscripts x and y indicate the shear stress acting in the x and
y direction respectively. The relation of these stresses to the wind
speed at a certain height is very difficult to determine theoretically
and its value is usually based on semi-empirical formulae. The well
known form of the relationship between shear stress T and wind
speed U (usually measured at a height of ten meters] is
* = PaCdU2
where p is the air density and C , is a dimensionless drag coefficient.
In this study, the drag coefficient formulae obtained by Wu (1969) is
adopted, and its value was given by
C . = 0.00125 U2 for U < 1 m/sec
d
= 0.0005/CT for 1 < U < 15 m/sec
= 0.0026 for U > 15 m/sec
7. Intake and discharge velocities
The intake and discharge velocities are calculated according to
the discharge flow rate from power plant data, the grid size and the
average depth at the intake and discharge outlet. The procedures
are shown as follows:
a. Flowrate = 955,000 gpm (from power plant physical data)
= 62.8 m3/sec
* 62. 8 x 10 cm3/sec
b. Both intake and discharge channels are at 45° from N,
therefore,
31.4 x 10 cm3/sec is crossing the Ax and Ay at the point of
intake and discharge.
c. The average depth at intake and discharge outlet is approxi-
mately 41 or 122 cm, and the width is Ax = Ay = 41760 cm;
so the cross-sectional area is 41760 x 122 cm2.
34
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d. The average velocities:
U = V
ave ave
31.4 x 10
41760 x 122
e. The velocity profiles are assumed as shown.
1
2
= 6.163 cm/sec
7.0 cm/sec
Intake: V3(14, 4, k) = 7 - 3 x
U3(15, 3, k) = V3(14, 4, k)
Discharge: V3(14, 8, k) = 7 - 3 x
k 3
4
5
f. To allow for channel storage during tide change we assume the
intake and discharge velocities to be sinusoidal, i.e.
(EST - 7.625)]
for k = 1, 2, 3, 4
y^ (EST - 7.5)]
U3(15, 8, k) = -V3(14, 8, k) for k = 1, 2, 3, 4
where 7. 625 and 7. 5 are taken to be the values of the phase shift
which takes into account the time to travel from the south end of
Anclote Key to the concerned point.
8. Tidal condition on June 19, 1978
Simulated diurnal tide is shown in Figure 28, where
a. Period = 12.5 hr
b. Stage = short term average sea level - MSL = 48 cm
c. Amplitude = £ short term average tide range = 65 cm
d. Time shift = 7.125 hr
i.e. at 7.125 am, June 19, 1978, the tide at the south end of
Anclote Key was zero.
e. W - E lapse = 0.014 hr/DX
Wave propagation speed C =J 2gh -J 2x980x360 = 850 cm /sec
(H = 360 cm is the maximum depth of the Anchorage.)
35
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The time needed to travel one grid distance is
iff?- SO sec- 0.0,4 hr
We use 0.014 hr per DX for phase shift in W - E direction and
the imposing tide at the south entrance is
n =48 + 65 sin[~I- (EST - 7.125 - 0.014(1 - 1)]
S 1 £•• 5
1 = grid no. in W - E direction.
f. S - N lapse = 0.15 hr
Distance from south entrance to north entrance is about 543000 cm.
Time for wave to travel this distance is 54300° =0.18 hr.
We take 0.15 hr as phase difference betweln°the south and the
north boundaries; there, the imposing tide at the north entrance
is
n =48+65 sin!—— (EST - 7.125 - 0.15 - 0.014(1 - 1)]
n 1 it o
9. Tidal condition on January 30, 1979
Simulated diurnal tide is shown in Figure 29. The calculation
procedures are of the same as summer tidal conditions.
a. Period = 12.0 hr
b. Stage = 36. 6 cm
c. Amplitude = i short-term average tide range = 42. 7 cm
d. Time shift = 10 hr
i.e. at 10 am, January 30, 1979, the tide at the south end of
Anclote Key was zero.
e. W - E lapse = 0.015 hr/DX
This value is slightly higher than the summer case since the
maximum water depth in the winter is less than the maximum water
depth in the summer. The imposing tide at the south entrance is
ns = 36. 6 + 42.7 sin[j| {EST - 10 - 0.015(1 - 1)]
f. S - N lapse = 0.2 hr/DX
This is the time for wave to travel from south entrance to north
entrance. So the imposing tide at the north entrance is
TI = 36.6 + 42.7 sin[|| (EST - 10 - 0.2 - 0.015(1 - 1)]
n 12
36
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10. Anclote River flowrate and temperature
a. The distance traveled from South Anclote Key to Tarpon Springs
is 20 DX. We estimate a time lapse of 0.5 hr to account for the
retardation due to buffering effect of river storage and Anclote
River's natural outflow.
b. The average current is estimated to be 20 cm/sec,
therefore, we take
U3(16, 1, k) = 20 cosly— (EST - 7.625)]
V3(15, 1, k) =-20 coslj^L (EST - 7.625)]
for k = 1, 2, 3, 4
c. The surface elevation at Tarpon Springs is to be calculated.
d. To be in accordance with given velocities at Tarpon Springs,
the temperature there is also assigned and its value has a
24 hr period instead of 12.5 hr. This temperature on June
19, 1978 is
T3(15, 1, k) =26.9 + 0.5 sin[|| (EST - 12)]
while on January 30, 1979 the temperature was assumed as
T3(15, 1, k) =11.9+0.5 sin[|| (EST - 12)]
where the 12 hr shift is to make the peak temperature occur
at 1800. Thus, the water in and out at Tarpon Springs has a
temperature ranging from 26. 4 (before dawn) to 27. 4 (late after-
noon) in the summer and ranging from 11.4 (before dawn) to
12.4 (late afternoon) in the winter.
11. Discharge temperature
On June 19-20, 1978, the recorded discharge temperature at
daytime was in the range of 29.3 - 30. 3°C, while on January 30-
February 1, 1979, the temperature range was 16. 4 - 15,2°C. To
account for the further drop of discharge temperature due to cooler
ambient temperature at nighttime, we assumed a sinusoidal variation
of discharge temperature with diurnal period.
a. Discharge temperature is estimated for June 1978
T3(14, 8, k) =29.4+0.4 sin[|| (EST - 12)]
Therefore, the highest discharge temperature of 30. 3°C happens
at 6 pm and the lowest 29. 3°C at 6 am.
37
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b. Discharge temperature is estimated for January 1979
T3(14, 8, k) = 15. 8 + 0.6 sin[~ (EST - 12)]
The highest discharge temperature is 16. 4°C at 6 pm and the
lowest discharge temperature is 15. 2°C at 6 am.
12. The Gulf temperature
The Gulf water outside the Anclote Anchorage as well as the
atmosphere is sink to the heat disposal from the power plant;
therefore, the boundary conditions on temperature at the north and
south entrance are not considered as adiabatic as in normal case of
far-field thermal pollution problem. Instead, we specify the outside-
Anchorage ambient temperatures. Again, they are 24 hr periodic
and their values should be in accordance with the measured tempera-
ture in the same neighborhood. Here in compliance with measured
data, we use
Tab ~ 27*° + °'2 sintl (EST ~
for the summer simulation during June 19-20 I978 and use
Tab = 11*8 + °'4 sin[15 (EST ~ 12)]
for the winter simulation during January 30-February 1, 1979.
RESULTS
Due to the fact that the archival data for Anclote site is inadequate,
the following inputs are used for the computer runs: data from NOAA
tide table, solar radiation, wind and the power plant operating conditions.
Tables 1 and 2 show the operating conditions of Anclote Power Plant
during June 19-20, 1978 and January 30-February 1, 1979 respectively.
The isotherms obtained from the IR data from each flight are interpolated
by hand from mosaic digicolor films and then plotted by the computer.
These isotherm plots are presented in this section for easy comparison
with the predicted isotherms. The average deviation of the calculated
temperatures from IR temperatures is also indicated. This deviation is
calculated by simply averaging the temperature differences between the
measured and calculated temperatures at each point of the domain. The
average deviation is given by
S2 = S[TB(i, j, D - TlR(i, j)]2/Z (i, j)
i'j i,j
where TB is the calculated temperature, T1R is the IR temperature and
I(i, j) is the number of surface half-grid points in the domain.
U
The T1R and TB isotherms are then compared to assess the accuracy
of the model in predicting the dispersion of the waste heat. It is recog-
38
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nized that the principal factors affecting the flow pattern and the shape
of the thermal plume are the tide driving through the north and south
boundaries, wind effects on the water surface, bottom topography, heat
transfer through the air-water interface, and the intensity of eddy vis-
cosity and diffusivity. The effects of each of these factors on the flow
pattern have been discussed in detai! by Lee et al. (I978a, b). These
effects are important in understanding the numerical behavior of the model,
and play a leading role in proving the capabilities of the model. In this
report, however, only the results of the verification runs are presented.
The results of the summer and winter verifications are discussed in the
next two sections.
Summer Results
The figures from the summer simulation show the hydrothermal dis-
persion of waste heat under the conditions of June 19-20, 1978, as de-
scribed in the previous section. The simulation run started at 0400 EST,
June 18, with initial conditions of zero velocity field, equilibrium water
level and constant ambient temperature (so-called "cool start"). The tide
and heated discharge are then imposed. After 20 hours of "warm up,"
the thermal plume can be seen to develop. Experience indicates that the
initial condition is not important, as its effects die out in the first one
or two hours. However, it is convenient to begin a simulation through a
"cool start."
Figure 30 shows the surface flow pattern at 1030 EST, June 20,
which corresponds to the high tide at south Anclote Key. The current
velocity is relatively small and the mainstream flows in the north direc-
tion. A recirculation can be seen to occur as a result of the flow of
water from the Anchorage into the Anclote River. The net effect is the
flow of part of the discharge into the river entrance where the intake
structure is located.
Figure 31 shows the resultant velocity at high tide. The result of
the velocity components, u and w, is plotted on the vertical cross sections
at J = 4, 8 and 12. It is believed that the currents at these sections are
most affected by the plan-form configuration, bathymetry and other fea-
tures such as the river mouth, discharge and tidal boundary etc. The
u-w velocity profiles show clearly the effects of bottom friction in retard-
ing the flow. Figure 32 shows the v-w resultant velocity on the vertical
cross sections at I = 4, 8 and 12. In these two figures, the vertical
velocity component w has been exaggerated to make the vertical circula-
tion of the current detect!ble. One may notice that, since the velocity
vector at the top level (a = 0} is plotted right on the water surface,
joining the tails of these velocity vectors would show the free-surface
profile.
Figure 33 shows the isotherms plotted from the IR data. The com-
puter-plotted points were obtained through visual interpolation from the
mosaic IR image, as discussed earlier. Due to the difficulties of visual
39
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interpolations and large grid size, the computer-plotted isotherms un-
avoidably differ somehow from the real isotherms in the digicolor film.
This figure shows that the heated water is recirculated back to the intake
through the Anclote River. This will cause an increase in the input
temperature of the condenser cooling water, thus reducing the efficiency
of the power plant.
Figure 32 shows the computed isotherms at the same tidal stage.
Comparing this figure with Figure 33, the good agreement of these
temperature fields can be easily seen. The average deviation of the
simulated temperature from 1R temperature is only 0. 359°C.
Figures 30 through 34 form the first set of results of the
summer simulation. This set of plotted results, which includes the
surface velocity, u-w velocity, v-w velocity, IR temperature and the
calculated temperature, coincides with the high tide at south Anclote
Keys. Figures 35 through 39 is the second set of results of the same
simulation run, at 1430 EST, June 20, or at a time corresponding to
maximum ebb tide at south Anclote Keys. Therefore, this set shows
the current and temperature fields of the Anchorage four hours after
the first set of results was taken. It is also seen that the Anclote
River empties into the Anclote basin while the basin drains northward
and southward. There are strong ebb flows occurring at both the north
and south end of the Anclote Keys barriers. The dividing line lies along
the j = 10 grid line. On the northern side of this line, the water flows
north, while on the southern side, it flows south. At this ridge line,
the water has minimal transverse motion. The location of this Tine varies
with time as the tides from both ends are not in phase. In reality, most
of the flow in the Anclote Anchorage comes primarily from the south end;
therefore, the ridge line is observed to exist at a region close to the
northern end of the Anchorage. Also, this line is observed to move
southward during ebb tide and northward during flood tide. The flow
pattern corresponds with the observed flowfield at a similar tidal stage.
Figure 36 shows a strong u-velocity component at the river mouth,
while Figure 37 shows a strong v-ve!ocity component near the south
boundary. Figures 38 and 39 are two corresponding isotherm plots at
this tidal stage. It can be seen that the thermal plume moves according
to the stage of the tidal cycles, as would be expected. The 27. 50°C
isotherm from the simulated results covers a larger area than that of the
IR plot. Generally speaking, the simulated results are in good agreement
with the IR data.
Figures 40 through 45 is the third set of results of the simulation
run, but at 1730 EST, June 20, which corresponds to low tide at the
south Anclote Keys. The current field is seen to be generally quite small
except at Anclote River region. The river is still undergoing its outflow-
ing process and continues to empty its tidal storage into the basin, while
the open sea flows into the Anchorage through both open boundaries.
It has to be mentioned here that the lower part of the 27. 50°C isotherm
from the IR data is not very visible in the IR digicolor film. As a result,
40
-------
the contour shown in Figure 43 is estimated by extrapolation from the
IR data. The calculated isotherm at this temperature, as shown in
Figure 44, has a peculiar shape at the same region. This may be due
to the fact that this region, north of the dredged ship channel leading
to Tarpon Springs, is relatively shallow and becomes especially so during
low tidal stage. The sum effect is that the heat transport, due to con-
vection, becomes small. This shallow region can be easily seen from
Figure 42 along the I = 8 cross section at a location about J = 4, 5 and
6. If this isotherm is ignored, then the remaining isotherms, shown in
Figures 43 and 44, have the same sort of tongue-shape profiles, show-
ing some degrees of good agreement.
Figures 45 through 49 is the last set of results of the summer
simulation. The time is 2030 EST at the same day and corresponds to
flood tide at the south end of the Anchorage. The incoming tide from the
south end drives northward into Anclote basin. The dividing fine dis-
cussed earlier, does not appear, as the northbound tidal current changes
course into the northeast direction. Both Figures 48 and 49 show that
the thermal plume has been pushed back, and the isotherms become more
compact. These features could be explained by considering that the
changing tide retards the convective transport of the thermal plume.
The 27. 50 and 28. 25°C isotherms in Figure 49 are seen to be pushed
toward the northeast direction rather than being pushed toward the east
coast, as shown by the corresponding IR isotherms. Figure 48. This
could be because the north and south tidal conditions are no longer con-
sistent with the real tidal conditions after such a long time of simulation
(64.5 hours). At this stage, it seems that the northern tide could in-
duce a stronger current to push the plume toward the southeast direction.
If correct tidal data had been provided, the results might have improved.
Winter Results
The winter simulation run started at 2200 EST, January 29, 1979.
The meteorological data and input conditions are as discussed in the pre-
vious section. Here, we present only two sets of results corresponding
to the successive flood tidal and maximum ebb tidal stages at the Anchorage
on January 30, 1979. These tidal stages were recorded during the first
and second data acquisition missions which took place on January 30, 1979,
at about 1100 EST and 1600 EST respectively. It should be pointed out
that, due to sudden change in weather condition, the second and third
missions were about 48 hours apart. In between these missions, the sea
had become very rough under very stormy conditions. The effect of the
stormy weather can be clearly seen by comparing the two IR temperature
fields, Figures 58 and 60; the former was taken at about 1600 EST,
January 30, while the latter was taken at about 1600 EST, Feburary 1.
There is therefore a 48 hour time lag. Figure 58 shows the thermal
plume at ebb tide of a typical winter day. This is a great contrast to
Figure 60 which shows the plume during high tide but after the in-
fluence of the stormy weather. It is to^ be noted that the isotherms
shown in Figure 60 are not of the same values as those shown in Figure
58. In fact, there was a 4°C drop in temperature, and a 2. 3°C drop in
41
-------
the discharged water temperature. Particularly, the wind condition had
gone through a very substantial change during this 48 hour time gap.
This wind speed, shown in Figure 60, was still 50% stronger than that
shown in Figure 58. Therefore, it was no surprise to find that the
thermal plume, shown in Figure 60, was so compact and,shrunken.
The windy condition and the relatively low ambient temperature were
the apparent causes of this much dwarfed plume. Observing the com-
pactness of the neighboring isotherms and judging the resolution power
of the present grid system, one is tempted to suspect the ability of
modeling under this kind of extremely severe meteorological condition.
Experience shows that it is difficult to obtain such a compact thermal
plume through simulations of such adverse natural conditions. Consider-
ing the roughness of the grid system, this is actually not a surprise.
Therefore, for the winter simulation only two sets of results, namely,
1100 EST and 1600 EST on January 30, are presented.
Figures 50 through 54 is the first set of results of 1100 EST,
January 30, 1979, which corresponds to the flood tidal stage at south
Anclote Key. Figure 50 shows the surface-flow pattern from calculated
results. The incoming tide from the south entrance drives directly into
the Anclote River. There is a dividing line across the Anchorage at
this tidal stage. This ridge line is located along the J = 9 grid line,
at which the incoming tides from both north and south meet. This can
also be seen from the in-situ measurement, as shown in Figure 19. The
discharge at the outlet is pushed head-on by this flow. As a result,
the thermal plume is squeezed by this flow. The isotherm plots in
Figures 53 and 54 show this effect very clearly. Although Figures 53
and 54 show similar tendencies, the calculated isotherms seem not to be
in good agreement with the IR isotherms; particularly so for the 15. 40°C
isotherms. However, the calculated plume areas are close to those ob-
tained from the IR data. Both temperature fields show the dispersion
of the thermal plume along the shore. The recirculation of heated water
is clearly indicated in both figures, showing that a recirculation actually
occurs at this tidal stage under the meteorological influence.
Figures 55 through 59 show the second set of results of the winter
simulation runs. This set of plotted results shows the current and
temperature fields at 1600 EST, January 30, 1979, during ebb tidal con-
ditions at the Anchorage. Figure 55 shows a strong current driving from
northeast to southwest along the channel. No ridge line is observable,
and the discharged water is not entrained by this main current; in fact,
most of the discharged water flows directly through the river mouth into
the intake of the power plant. Thus, serious recirculation happens at
this time; this is also shown in Figure 58 of the IR isotherms and in
Figure 59 of the calculated isotherms. The calculated current pattern
shown in Figure 55 is in good agreement with that of the in-situ mea-
surements shown in Figure 20. A similar agreement can be seen between
the calculated thermal plume shown in Figure 59 with that of the IR data
shown in Figure 58. It can also be seen that all the calculated isotherms
cover a slightly larger area than the corresponding IR isotherms; this
difference is shown to be insignificant by a low deviation of 0.65°C above
the IR measured temperature.
42
-------
REFERENCES
Carter, C. V. The Hydrothermal Characteristics of Shallow Lakes. Ph.D.
Dissertation, University of Miami, Florida, 1977.
Dunn, W. E., Policastro, A. J. and R. A. Paddock. Surface Thermal
Plumes: Evaluation of Mathematical Models for the Near and Com-
plete Field. Part One and Two, Energy and Environmental Systems
Division, Great Lakes Project, Argonne National Laboratory, May
1975.
Forsythe, G. E. and W. R. Wasow. Finite-Difference Method for Partial
Differential Equations. J. Wiley and Sons, 1960.
Freeman, N. G., Hale, A. M. and M. B. Danard. A Modified Sigma
Equations Approach to the Numerical Modeling of Great Lakes
Hydrodynamics. J. Geophysical Research, 77(6), 1050-1060, 1972.
Hinewood, J. B. and I. G. Wallis. Classification of Models of Tidal
Waters. J. Hyd. Div., ASCE, Vol. 101, No. HY10, 1975.
Leendertse, J. J. Aspects of a Computational Mode! for Long-Period
Water-Wave Propagation. RM-5924-Pr, The Rand Corp., 1967.
Leendertse, J. J., Alexander, R. C. and S. K. Liu. A Three-Dimen-
sional Model for Estuaries and Coastal Seas: Volume I, Principles
of Computation. R-1417-OWRR, The Rand Corp., December 1973.
Leendertse, J. J. and S. K. Liu. Modeling of Three-Dimensional Flows
in Estuaries. Proc. 2nd Annual Symposium of the Waterways, Har-
bors and Coastal Engineering Division of ASCE, September 1975 or
Symposium on Modeling Techniques - 75.
Lee, S. S. and S. Sengupta. Three-Dimensional Thermal Pollution Models.
NASA CR-154624, 1978(a).
Volume 1 - Review of Mathematical Formulations
Volume II - Rigid-Lid Models
Volume III - Free-Surface Models
Lee, S. S. and S. Sengupta. Demonstration of Three-Dimensional Ther-
mal Pollution Models at Anclote Rivers, Florida, and Lake Keowee,
South Carolina. NASA Project, Mid-Term Report, Department of
Mechanical Engineering, University of Miami, Florida, December
1978(b).
43
-------
REFERENCES
Lee, S. S., Sengupta, S., Tuann, S. Y., and C. R. Lee. User's
Manual for Three-Dimensional Free-Surface Model, NAS 10-9410.
Final Report, August 1980.
Patridge, P. W. and C. A. Brebbia. Quadratic Finite Elements in
Shallow Water Problems. Proc. ASCE, Vol. 102, No. HY9, Septem-
ber, 1976.
\
Phillips, N. A. A Coordinate System Having Some Special Advantages
for Numerical Forecasting. J. Meteorology. 14:184-185, 1957.
Platzman, G. W. A Numerical Computation of the Surge of 26 June 1954
on Lake Michigan. Geophysics. 6:407-438, 1958.
Platzman, G. W. The Dynamic Prediction of Wind Tides on Lake Erie.
American Meteoroligical Society, Vol. 4, No. 26, September 1963.
Reid, R, 0. and B. R. Bodine. Numerical Model for Storm Surges in
Galveston Bay. Proc. ASCE, Vol. 94, No. WW1, February 1968.
Richardson, L. F. Atmospheric Diffusion Shown on A Distance-
Neighbor Graph. Proc. Roy. Soc., 110:709, 1926.
Sengupta, S. and W. J. Lick. A Numerical Model for Wind-Driven
Circulation and Temperature Fields in Lakes and Ponds. FTAS/TR-
74-98, Case Western Reserve University, 1974(a).
Sengupta, S., Veziroglu, T. N., Lee, S. S. and N. L. Weinberg. The
Application of Remote Sensing to Detecting Thermal Pollution. NAS
10-8470, Mid-Term Report, May 1975.
Sengupta, S. A Three-Dimensional Model for Closed Basins. ASME Paper
76-WA-HT21, 1976.
Sengupta, S., Lee, S. S. and H. P. Miller. Three-Dimensional Numerical
Investigations of Tide and Wind-Induced Transport Processes in
Biscayne Bay. SEA Grant Technical Bulletin No. 39, University of
Miami, 1978.
Sundermann, J. A Three-DimensSona! Model of a Homogeneous Estuary.
Proc. 14th Coastal Engineering Conference, Vol. Ill, June 1974.
Waldrop, W. and R. C. Farmer. Three-Dimensional Flow and Sediment
Transport at River Mouths. Coastal Studies Institute, Louisiana
State University, Baton Rouge, Technical Report No. 150, Septem-
ber 1973.
44
-------
REFERENCES
Waldrop, W. and R. C. Farmer. T hree-Dimensional Computation of
Buoyant Plumes. J. Geophysical Research, Vol. 79, No. 9, 1974(a).
Waldrop, W. and R. C. Farmer. Thermal Plumes for industrial Cooling
Water. Proceedings of the 1971 Heat Transfer and Fluid Mechanical
Institute, Davis, L. R. and R. E. Wilson (ed.), Stanford, California,
Stanford University Press, June 1974(b).
Wang, J. D. Real-Time Flow in Unstratified Shallow Water. Proc. ASCE,
Vol. 104, No. WW1, February 1978.
Wu, J. Wind Stress and Surface Roughness at Air-Sea Interface. J.
Geophysical Research. 74(2) : 444-455, 1969.
45
-------
TABLE 1 CLIMATIC DATA FOR SUMMER RUN AT ANCLOTE ANCHORAGE
Date
6/18
•
6/19
EST
04
05
06
07
08
09
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
01
02
03
(1)
Tair(°C)
23.3
23.3
23.3
23.3
25.0
26.7
28.3
30.0
31.1
32.2
31.6
31.1
30.6
30.0
29.4
28.8
28.3
27.2
26.1
25.0
23.9
22.2
21.7
21.1
(2)
Humidity
.90
.90
.90
.90
.84
.77
.70
.65
.60
.56
.57
.58
.59
.61
.64
.67
.70
.73
.77
.75
.74
.73
.76
.80
(3)
Wind
Speed
357.6
268.2
312.9
312.9
312.9
447.0
536.4
536.4
536.4
536.4
491.7
536.4
581.1
581.1
625.8
581.1
447.0
402.3
312.9
402.3
312.9
223.5
223.5
312.9
(4)
Wind
Direction
50..
50.
50.
50.
50.
70.
80.
80.
70.
90.
80.
80.
90.
80.
80.
80.
80.
80.
80.
SO.
80.
80.
60.
40.
. (5)
Solar
Radiation
0.0
0.0
0.0
0.05
0.40
0.75
1.05
1.40
1.60
1.70
1.60
1.50
1.30
1.10
0.'70
0.30
0.05
0.0
0.0
0.0
0.0
0.0
0.0
0.0
(6)
T
surf
26.4
26.4
26.5
26.5
26.7
26.8
26.9
27.0
27.0
27.1
27.1
27.2
27.2
27.2
27.1
27.0
26.9
26.9
26.8
26.8
26.8
26.8
26.8
26.8
-------
TABLE 1 (Continued)
Date
6/19
EST
04
05
06
07
08
09
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
(1)
Tair(°C)
21.1
21.7
22.2
22.8
23.9
25.0
26.7
27.8
29.4
30.6
30.0
29.4
28.8
28.3
27.2
26.1
25.6
25.0
25.0
25.0
24.4
(2)
Humidity
.84
.82
.80
.79
.73
.67
.60
.56
.53
.50
.53
.57
.61
.65
.69
.74
.76
.78
.79
.79
.79
(3)
Wind
Speed
312.9
312.9
312.9
312.9
357.6
402.3
447.0
491.7
536.4
447.0
312.9
268.2
447.0
536.4
581.1
581.1
357.6
402.3
357.6
312.9
357.6
(4)
Wind
Direction
50.
50.
70.
70.
70.
80.
80.
90.
80.
80.
90.
80.
350.
350.
360.
350.
60.
120.
110.
110.
110.
(5)
Solar
Radiation
0.0
0.0
0.0
0.10
0.25
0.40
0.60
1.25
1.25
0.60
0.80
1.20
0.70
0.50
0.15
0.15
0.05
0.0
0.0
0.0
0.0
(6)
Tsurf
26.8
26.9
26.9
27.0
27.0
27.0
27.0
27.0
27.0
27.0
26.9
26.9
26.9
26.9
26.9
26. 8
26.8
26.7
26.7
26.6
26.5
-------
TABLE 1 (Continued!
Date
6/20
-
EST
01
02
03
04
05
06
07
03
09
10
11
12
13
14
15
16
17
18
19
20
(1)
Tair(°C)
24.4
23.9
24.4
25.0
25.0
25.0
25.0
26.1
27.2
28.8
28.9
28.4
30.0
30.6
30.6
30.0
30.0
29.4
28.8
28.0
(2)
Humidity
.79
.80
.81
.82
.82
.82
.82
.78
.74
.70
.66
.62
.59
.58
.57
.57
.56
.55
.53
.50
(3)
Wind
Speed
312.9
268.2
357.6
357.6
402.3
357.6
357.6
357.6
491.7
581.1
447.0
268.2
357.6
402.3
402.3
268.2
402.3
312.9
223.5
223.5
(4)
Wind
Direction
100.
100.
110..
110.
110.
110.
90. ,
90.
110.
110.
110.
110.
90.
90.
110.
100.
100.
110.
90.
90.
(5)
Solar
Radiation
0.0
0.0
0.0
0.0
0.0
0.0
0.05
0.20
0.60
0.30
0.40
0.40
0.60
0. 50
0.55
0.40
0.30
0.20
0.15
0.05
(6)
Tsurf
26.4
26.5
26.6
26.6
26.7
26.7
26.7
26.8
26.8
27.0
27.0
27.0
27.0
27.1
27.0
27.1
27.2
27.1
27.0
27.0
-------
TABLE 2 CLIMATIC DATA FOR WISTEB, RUN AT ANCLOTE ANCHORAGE
Date
1/29
1/30
EST
22
23
24
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
(1)
Tair(°c>
8.8
8.8
8.8
8.3
7.7
7.2
6.7
6.1
5.5
5.0
6.7
8.3
10.0
11.6
13.8
15.5
17.2
18.3
19.4
18.8
18.3
17.2
16.1
(2)
Humidity
.98
.93
.89
.86
.87
.88
.89
.89
.89
.89
.75
.63
.54
.50
..47
.43
.39
.35
.31
.38
.45
.52
.59
(3)
Wind
Speed
350
350
350
350
350
325
300
280
270
250
280
310
350
375
400
400
400
375
350
310
280
250
230
(4)
Wind
Direction
320
320
320
330
340
350
360
10
20
40
60
80
100
140
180
220
220
220
210
200
170
140
190
(5)
Solar
Radiation
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.05
0.25
0.45
0.75
0.95
1.15
1.30
1.15
1.00
0.80
0.40
0.10
0.0
(6)
T
surf
11.8
11.8
11.8
11.7
11.7
11.7
11.7
11.7
11.7
11.7
11.7
11.8
11.9
12.0
12.1
12.2
12.4
12.2
12.1
12.1
12.0
12.0
11.9
-------
TABLE 2 (Continued)
Date
1/31
1/31
EST
21
22
23
24
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
(1)
Tair(
15.5
15.0
15.0
15.0
14.4
14.4
13.8
13.3
12.7
12.2
11.6
12.7
14.4
16.1
17.2
16.6
16.1
15.0
13.8
12.7
11.6
10.5
9.4
(2)
..Humidity
.66
.72
.70
.68
.66
.66
.65
.65
.77
.89
1.00
.97
.95
.93
.86
.79
.72
.74
.76
.77
.77
.77
.77
(3)
Wind
Speed
210
200
210
230
250
250
270
290
320
360
400
420
500
600
700
800
800
850
900
900
700
600
600
(4)
Wind
Direction
250
310
340
10
40
50
60
70
90
110
140
180
220
250
280
310
330
330
330
330
330
330
330
(5)
Solar
Radiation
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.05
0.15
0.25
0.35
0.45
0.55
0.50
0.50
0.40
0.30
0.15
0.05
(6)
T
surf
11.9
11.8
11.8
11.8
11.7
11.7
11.7
11.7
11.7
11.7
11.8
11.8
11.8
11.9
12.0
12.1
12.2
12.1
11.9
11.7
11.5
11.3
11.0
50
-------
TABLE 2 (Continued)
Date
2/1
EST
20
21
22
23
24
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
(1)
Tair(°c)
8.8
8.3
8.3
7.7
7.7
7.2
6.7
6.1
5.5
4.4
3.8
3.3
3.8
5.0
6.1
7.2
8.3
9.4
10.0
10.5
11.1
10.0
8.3
(2)
Humidity
.79
.81
.83
.80
.77
.74
.74
.73
.73
.73
.73
.73
.68
.58
.51
.45
.39
.33
.30
.32
.34
.40
.46
(3)
Wind
Speed
600
600
550
500
500
500
550
550
550
500
500
500
550
600
650
650
650
600
600
600
600
600
600
(4)
Wind
Direction
330
330
340
340
340
340
340
350
350
350
360
360
360
360
360
350
350
350
340
340
330
330
330
(5)
Solar
Radiation
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0,0
0.0
0.0
0.05
0,15
0.25
0.40
0.55
0.70
0.85
0.80
0.70
0.55
0.40
(6)
T
surf
10.7
10.4
10.0
10.0
10.0
10.0
10.0
9.9
9.9
9.8
9.8
9.7
9.8
9.9
10.0
10.1
10.2
10.2
10.1
10.1
10.0
10.0
10.0
51
-------
Jacksonville
Daytona Beach
Tallahassee
Gainesville
Anclote
Anchorage
Anclote R.
Tarpon Springs
Tampa
Anclote
Key
Lake
Okeechobe
Ft.
Myers Ft
Lauderdale
Melbourne
GULF OF MEXICO
.JWest
Palm
Beach
*) Miami
Figure 1. Anclote Anchorage location in the state of Florida
52
-------
U7
U)
Q>
km
rage
-------
_ z-n(x,y,t)
h(x,y) + n(x,y ,
h(x,y)
Fiaure 3. Definition sketch of a-coordinate
54
-------
All Point Have
Same i
i=
All Points
Have Same j
Figure 4. Grid arranqement in the horizontal projection
• (full-grid) depth (h) point; — u velocity point;
i v velocity point; + (half-grid) w,ft,p,T,p and n point
55
-------
v(i,j,5)=0
,J, 5)=0
Fiqure 5. Four cells in a vertical column with velocities
shown at definition point, and scalar variables
at the center of cell f»l
56
-------
(i)
Half-grid
point
Full-grid point
. Fiqure 6. Notations (left) and variables (right) used in calculation
of.T (#), n (-»), u (>*) and v f^)" within fi,i)-block
(cross-hatched)
57
-------
Ul
00
Figure 7. Grid work for the Anclote Anchorage
-------
Ul
to
Figure 8. Location of stations for in-situ measurement, June 1978
-------
en
o
Discharge Rates 994,000gpm
Wind Velocity: 12.5 mph
(155°)
Figure 9. Velocity from in-sttu measurement at 1710-1903, June 19, 1978
-------
4 56 7 8 9 10 II
Discharge — •> l m
0 10 20cm/sec
Discharge Rate: 994,000 gpra
Wind Velocity: 8.0 mph(90°)
Figure 10. Velocity from in-situ measurement at 06H8-0812, June 20, 1978
-------
en
ro
Discharge Race: 994,000gpm
Wind Velocity: 8.0 mph(110}
Figure 11. Velocity from in-situ measurement at 1125-1245, June 20, 1978
-------
O>
to
Discharge Rate: 994,000 gpm
Wind Velocity: 7.5
Figure 12. Velocity from in-situ measurement at 1150-1605, June 20. 1978
-------
Ol
N
2000 feet altitude
3090 flight line overlap
5 feet resolution
FL 1
FL 2
FL 3
FL 4
FL 5
FL 6
PL 7
FL 8
t
ANCLOTE
ANCHORAGE
1 km
FL 9
FL 10
Figure 13. Daytime flight lines on June 19 and 20, 1978
-------
U1
IDischarge Temp: 30.3°C
Wind Velocity: 12.5 mph(355^
Fiqure 11. Surface temperature in degree C from in-situ measurement at
1710-1903, June 19, 1978
-------
en
01
u\
26.9
26.9
27.J
27.2
26.8
t
27.1
26.9
26.7
Tidal Cycle
'Intake
j Discharge Temp: 29.3°C
V-S~| wltld Velocity: 8.0 mph(90°)
10
Fiaure 15. Surface temperature in degree C from in-situ measurement at
0648-0812, June 20, 1978
-------
13 -,
II
3 1
27.3
27.0
27.3
27.7
27.1
27.6
27. •
26.9
9 tO II ,2 ~~I3~ ,-Jl4 IS
jDischarge Temp: 29.6°C
Wind Velocity: 8.0 nmh(110°
Figure 16. Surface temperature in degree C from in-situ measurement at
1125-1215, June 20, 1978
-------
01
CO
[Discharge
| Discharge Temp: 29.6 C
Hind Velocity: 7.5 mph
(100°)
34 * l> 7 89 10 11
Figure 17. Surface temperature in degree C from in-situ measurement at
11(50-1605, June 20, 1978
-------
1 K
5 »
en
10
lx
14
H
5x
IK
13
6x
NA-5
n
12
9«
17
Ix
9 10 I) 12 13
Figure 18. Location of stations for in-situ measurement, January 1979
-------
Discharge: 994,000 gpro
Wind Velocity: 4.0 mph(320°)
Figure 19. Velocity from in-situ measurement at 1020-1340, January 30, 1979
-------
Discharge: 994,000 gpm
Wind Velocity: 6.0 mph
Fiqure 20. Velocity from in-situ measurement at U40-1800, January 30, 1979
-------
Discharge > 2 •
0 10 20cm/sec
J. . Tidal Cycle
Discharge: 994,000 gpm
Velocity: 14.0 mph
6 7 S 9 10 II 12 13 r->\* IS
Figure 21. Velocity from in-situ measurement at 1430-1640, February 1, 1979
-------
12.1
12.1
12.3
12.6
12.3
11.8
12.0
13.6
14.0*
14.7
I
13.1
11.9
12.2
13.8
r_
11.8
11.8
12.0
12.0
11.9
12.2
11.9
^Discharge
4-N--I
ikm
! Tidal Cycle
I Discharge Temp: 15.3°C
v j Wind Velocity: 4.0 roph
N.I (320°C)
56 789 1O
Figure 22. Surface temperature in degree C from in-sltu measurement at
1020-1310, January 30, 1979
-------
7 t 9 10 II |2 13 j~>\4 IS
Figure 23. Surface temperature in degree C from in-situ measurement at
1 WO-1800, January 30, 1979
-------
en
Figure 21. Surface temperature in degree C from in-situ measurement at
1130-1610, February 1, 1979
-------
Figure 25. Surface temperature from in-situ measurement at
1020-1310, January 30, 1979
-------
Figure 26. Surface temperature from in-situ measurement at
1 MO-1800, January 30, 1979
-------
00
Fi«;jure 27. Surface temperature from In-situ measurement at
U30-16UO, February 1, 1979
-------
-Tide at south end of Anclote Key
2 TT
-Simulated tide for calculation n= 48 + 65 x sin l~ (H>))
-Average Level
Figure 28. Semidiurnal tide for June 19-20, 1978 at south end of Anclote Key
-------
CO
o
Tide at south end of Anclote Key
simulated tide for calculation n=36.6+42.7xsin[~- ^t-^
Figure 29. Semidiurnal tide for January 30-February 1, 1980 at south end of Anclote" Key
-------
TIHECJUNE 2Q.197S)>
HINO SPEEO(CH/SEC)«
MIND OIRECTiaN(OEG/N)»
BIR TEMPERBTUREtDEO-Oi
OISCHflRGE TEHPtOEG-Oi
OI3CH FLOHRflTECCUM/SEC) i
LENGTH SCflLEClCMs X CHJi
VELOCITY SCflLEtCM/SEC)>
10. S
450.0
110.
28.8
29. S
62.7
41013.
52.49
k 4
I 4
% h
. 4
' '
. t
1
• * t f f » , . .
• »/*«»..,
t >>«,,,,.
>»»«...* ^
>'<«*»..—.
'»•«».» 4
'^<««444
\
\
» «•
HIGH TIDE
Figure 30. Surface velocity, Anclote Anchorage by modeiina
81
-------
TIHEtJUNE 20.1978)1
HIND SPEEDtCM/SEC)i
MIND DIRECTIONOEG/N)i
HIR TEHPERRTUREtOEG-CJi
OISCHflRGE TEHP(OEG-CJ>
OI3CH FLOHRBTEtCUM/SEC)i
LENGTH SCRLEtlCflr X CM J»
VELOCITY SCflLErCM/SECJ«
10.S
450.0
UQ.
28.8
29.5
62.7
41019.
52.43
J= 8
J= -4
HIGH TIDE
Figure 31. UW velocity, Anclote Anchorage by modeling
82
-------
TIHEUUNE 2Q.1978)>
WIND SPEEOlCH/SECJi
MIND DIRECTIONOEG/N)i
flIR TEHPERflTUREtOEG-C)i
OISCHflRCE TEHPCOEO-CJi
OI3CH FLQHRflTE(CU»/3ECJi
LENGTH SCRLEtlCtts X CfM i
VELOCITY SCRLECCM/SEC)«
10.5
450.0
110.
Z8.S
29.5
62.7
41019.
52.49
1= 12
1= 8
1= 4
HIGH TIDE
Figure 32. VW velocity, Anclote Anchorage by modeling
83
-------
TIHEtJUNE 20.1978J*
HIND SPEEOtCM/SEC)i
HIND OIRECTIONCOEO/N)«
RIR TEMPERRTUREtDEC-CJi
DISCHARGE TEf1P(D£0-C)i
OISCH FLQMRflTEfCUM/SEC)»
LEMOTH SCRLE(1CM= X Cf1)»
VELOCITY SCflLEfCM/SECJi
11.5
420-0
110.
29.1
29.6
6Z.7
41019,
52.49
HIGH TIDE
Figure 33. Temperature from IR
84
-------
TIJ1EUUNE 20.1978)1
HINO SPEED!CM/SECJ>
HIND OIRECTI8N(OEG/N)t
BIR TEWERflTURElOEG-CJi
DISCHflRQE TEHP(OEG-C)»
DI3CH FL0HRaTEtCUH/SEC)i
LENGTH SCRLEUCfls X CHli
VELOCITY SCHLEtCM/SEC)»
10. S
450.0
110.
28.8
29.5
62.7
41019.
52.49
OEVIHTI9N FR8M IR TEMP«
HIGH TIDE
0.359
Figure 34. Surface temperature, Anclote Anchorage by modeling
85
-------
TIME(JUNE 20.1978)1
HIND SPEED(CM/SEC)i
HIND OIRECTI8N(DEG/N)»
flIR TEHPERF1TURE£ DEO-C )i
DISCHARGE TEHP(OEG-C)i
OISCH FLQHRBTECCU«/3EC)i
LENGTH 3CflLE(lCM= X Ctt)>
VELOCITY SCHLEtCn/SECJi
14.5
400.0
110.
30.6
30.0
62.7
41019.
52.49
*
-
» - . / ^^^.----.N
•»«/X^^--*-^
' » J ///---**•
• M / / '
' / / / / / " -" • "~ N
4 / / / / ' ^ ^ " ' N
* * *
\ i, ',',', ','.:::
. . ,
» > %
» ^^
-
*
\ \
N \
/ '
EBB TIDE
Figure 35. Surface velocity, Anclote Anchorage by modelinq
86
-------
TIHECJUNE 20.1978Ji
MIND SPEECHCtl/SEC) i.
HIND DIRECTIBWOEG/NJi
HIR TEJIPERflTURECOEOrOi
OI3CHHROE TEf1P{OEG-CJ«
D13CH FLOHRflTECCUM/SEC)»
LENGTH SCflUEUCtls-^CfllJ
VELQCITY SCflLE(CM/SEC J a
14.5
400.0
110.
30.6
30.0
62.7
41019.
52.49
Ja 12
Js 6
J= 4
EBS TIDE
Fiqure 36. UW velocity, Anclote Anchorage by modeling
87
-------
TIMECJUNE 20.1978Jt
HIND SPEEOCCM/SEC)i
MIND OlRECTIQN(OEG/N)i
HIR TEflPERflTURECOEG-CJi
OISCHRRGE TEHPCOEG-CJ*
OI3CH FL3HRflTECUm/3EC)>
CENCTH SCflLEncn= X C«JJ
VELOCITY SCflLECCM/SEC)»
14.5
400.0
110.
30.6
30.0
62.7
41019.
52.49
1= 12
1= 8
1= 4
EBB TIDE
Figure 37. VVV velocity, Anclote Anchorage by modeling
88
-------
TIHEUUNE 20.1978J>
HIND SPEED(Cfl/SECn
HINO OIRECTI3N{OEG/N)i
RIR TEflPERflTUREtOEG-CJi
OISCHflRGE TEflPfOEO-CJi
OISCH FLQHRRTEICUH/SECJi
LENGTH SCflLEriCHs X CflJi
VEL3CITY SCflLECCM/SEC)i
15.0
400.0
110.
30.6
29.S
62.7
41019.
52.49
EBB TIDE
Figure 38. Temperature from !R
89
-------
7IMEUUNE 20,1978)1
HIND SPEEOtCtl/SECJi
HIND OIRECTIONCOEG/HJi
RIR TEflPERflTURECQEG-CJj
OISCHRRGE TEHP{DEO-C)i
OISCH FLQHRRTE(CUM/SEC)»
LENGTH SCflLEdCtls X Ct1)»
VEtaCITTf SCflLEtCII/SEOi
14. S
400.0
no.
30.S
30.0
6Z.7
41019.
52.49
so
OEVIflTION FRQfl IR TEHP t
ES8 TIDE
0.361
Figure 39. Surface temperature, Anclote Anchorage by modeling
90
-------
TIt1E(JUNE 20.1978)1
HIND SPEED CCt1/3ECJ»
HIND OIRECTIQNCOEG/N)!
RIR TEHPERflTUREtOEG-CJi
OI3CHRRCE TEHPCOEG-CJi
DI3CH FL8HRRTEtCUH/3EC)i
LENGTH SCRLE(1C«= X CM) i
VELOCITY SCRLEt CM/SEC )i
17.5
310. Q
110.
29.4
30.2
62.7 ;
41019.
52.49
.. » *
»»»»
\
\
N\
L8H TIDE
Figure 40. Surface velocity, Anclote Anchorage by Modeling
91
-------
TIMEfJUNE 2Q.1978}»
HIND SPEEDtCn/3EC)i
HIND OIRECTIQNdJEG/NJi
fl!R TEMPERRTURECQEG-Cli
DISCHARGE TEMP(DEG-CJt
OI3CH FLQHRRTEtCUH/SECli
LENGTH SCRLEUCris X CM It
VELQCITY SCflLEtCM/SEC)»
17.5
310.0
no.
23.4
30. Z
SZ.7
41019.
52.49
J= 1Z
J= 8
Js 4
LGH TIDE
Figure
UW velocity, Anclote Anchorage by modeling
92
-------
TIMEUUNE 20.1978)1
HIND SPEED!CM/SEC)«
HIND OIRECTIQN(DEG/N3i
RIR TEMPERflTURE(DEC-C)i
OISCHfiRGE TEMPJDEO-CJi
OISCH FL3HRaTE(CUH/3ECJ>
LENGTH SCflLE(lCh= X Cf1)i
VELOCITY SCHLEICH/SEC)«
17.5
310.0
110.
29.4
30.2
S2.7
41019.
52.49
1= 12
1= 8
1= 4
LOU TIDE
Figure 42. VW velocity, Anclote Anchorage by modeling
93
-------
TIME!JUNE 20.19783.
HIND SPEEDCCn/SEOi
WIND DIRECTIQN(DE&/NJ>
HIR T£f1PERRTUREtOEC-C)j
OlSCHflRGE TEMPtOEO-CJi
DI3CH FL9HRHTEtrUH/3EC)»
LENGTH SCRLEUCtts X CHJ»
VELOCITY SCflLE(Cf1/SEC3>
17.S
310- 0
110.
29.4
30.2
62.7
41019.
52.49
L9H TIDE
Figure 43. Temperature from IR
-------
TIflEtJUNE 20.1378)1
HINO SPEEOCCM/SEC)*
HIND OIRECTIQNCOEG/NJi
HIR TEMPERRTUREtOEG-C)»
OISCHflROE TEMP(DEG-C)i
OI3CH FL8HRflTE(C'Jtt/SECJ>
LENGTH SCRLEI1CH= X CH)>
VELOCITY SCflLECCJ1/SEC)«
17.5
310.0
110.
29.4
30.2
62.7
41019.
52.49
so
OEVIRTIQN FROM IR TEMP*
LOW TIDE
0.538
Figure 44. Surface temperature, Anclote Anchorage by modeling
95
-------
.
»
-
TIHECJUNE ZQ,1978)i 20.5
HIND SPEED (Cfl/SECJi 225. 0
HIND QIRECTI9N(DEG/M)» 90-
RIR TEHPERRTUREtOEO-C)i 28.0
OI3CHRROE TEf1PtOEG-C)i 30.1
OI3CH FL8HRRTEfCUH/3EC)J 62.7
LENGTH SCRLEdCtls X CHJ« 41019.
VELQCITY SCRLEt CM/SEC J» 52.49
, , % ,, .. — ^ * * *
•• « i t t s
f f , .
lilt//,*..
/* t
////,,..
lit//,*...
* * 4
* • *
» * .
» *
b
i
. \
\_- ,
\
N \
FLQO TIDE
Flqure 45. Surface velocity, Anclote Anchorage by modeling
96
-------
TIflECJUNE 20.1978)1
HINO SPEEOtCM/SEC)«
HIND OIRECTiamOEG/NJi
flIR TEMPERRTURE(OEO-C)«
OI3CHRRSE TEMPI OEG-Oi
OI3CH FL8HRFITEtCUf1/3EC)»
LENGTH SCRLEtlCMs X CM) i
VELQCITY SCHLEtCM/SEOi
20.5
225.0
90.
28.0
30.1
62.7
41019.
52.49
= a
J= 4
FLQD TIDE
Figure 46. UW velocity, Anclote Anchorage by modeling
97
-------
TIHECJUNE 20.1978)1
HIND SPEEDCCn/SEC)i
HIND OIRECTIQN(OE&/N)i
RIR TEllPERRTURECDEO-CJi
OISCHflROE TEMP(OEG-C)J
OISCH FLQHRRTEtCUM/3EC)i
LENGTH SCRLEUCf1= X CflJi
VELOCITY SCRLECC«/SEC)«
20.5
225.0
30.
28.0
30.1
62.7
41019.
52.49
1= 12
Is 8
la 4
FLOO TIDE
Figure 47. VW velocity, Anclote Anchorage by modeling
98
-------
TIHECJUNE 20.1978)»
WIND SPEEOCCtt/SECJi
HIND OIRECTIQNCDEG/N)!
RIR TEMPERflTUREIDEC-CJi
OISCHflRGE T£f1P(DEG-C)i
OISCH FLQHRflTEfCUM/SECJ»
LENGTH scm.Eticri= x cnj«
VELOCITY SCflLECCM/SEC)i
20.0
225.0
90.
28.0
29.3
62.7
41019.
52.49
FLQD TIDE
Figure 48. Temperature from IR
99
-------
T1HEUUNE 20.1978]i
HIND 3PEEOCCM/SEC) i
HINO DIRECTION!OEG/N)i
flIR TEHPERRTUREfDEG-CJi
OISCHflRGE TEMP(OEO-C)»
OI3CH FL3HRRTEtCU«/3EC)i
LENGTH SCRLEllCns X CMJi
VELOCITY SCflLECCM/SECli
20. S
225.Q
90.
28.0
30.1
62.7
41019.
52.49
OEVIflTION FRQM IR TEMP>
FLQO TIDE
0.361
Figure 49. Surface temperature, Anclote Anchorage by modeling
TOO
-------
TIMEfJBNURRY 30.1970) :
HIMO SPEED(CN/SEOi
NINO OIRECTlQNfCEC/NJ:
flIR TEMPERHTUREf DEC-CM
01SCHRRCE TEflPCDEG-C)!
DISCH FLQHRflTEfCUM/SECJ»
LENGTH 5CRLEriCn= X CMJ>
VELOCITY SCflLEfCH/SECJ»
11.0
4QQ.Q
320.
12.5
15.6
62.7
41013.
52.49
\ \ \
\ \ \
« \ \
! /
« / s
f / /
/ / /
/ / /
\
\
\
FLQO TIDE
Figure 50. Surface velocity, Anclote Anchorage by modeling
101
-------
TIflEf JflNUflRY 30.1979)>
HIND SPEEOrCM/SEC)>
HIND 01RECnON(OEG/N)»
HIR TE«PERflTUREfOEO-C)«
OlSCHflRGE 7EflPCDEG-C)>
01SCH FLOHRRTEfCUM/SECJ«
LENGTH SCRLEflCff= X CH)»
VELOCITY SCRLEfCn/SECJ«
11.0
400 .0
320.
12.5
15- S
G2.7
41013.
52.49
J= 12
J= 8
J= 4
FLQO TIOE
Figure 51. UW velocity, Anclote Anchorage by modelina
102
-------
T'lEtJflNUBRY 30.1979)>
HIND SPEEOfCM/SECJi
HIND DIRECTIQNfOEO/N).
BIR TEflPERfiTUREfOEO-CJs
OISCHflROE TEHP(OEG-CJ«
OISCH FLOHRfUEf CUM/SEC J«
LENGTH SCBLEflCNr X CHJ»
VELOCITY SCRLEfCM/SECJ«
11.0
4QQ.O
320-
12-5
15.6
62.7
41Q13,
52.49
1= 12
1= 8
1= 4
FLOD TIDE
Ficure 52. VW velocity, Anclote Anchoraqe by modeling
103
-------
Tlft£{JRNUflRY 30.1979}!
HiNC SPEEO(Cn/SEC)J
HIND OIRECT1QN(OEG/NJJ
flIR TEHPERflTUREfOEG-C)!
QlSCHflRCE TEflPtOEG-CJ:
OiSCH FLQHRRTEfCUM/SECH
LENGTH SCRLEHCHr X CM J s
VELOCITY SCfiLEfCfl/SECJs
It .5
400.0
320.
12.7
IS.5
62.7
41019.
52.49
PLOD TIDE
Figure 53. Temperature from 1R
104
-------
11HECJBNURRY 30.1379)>
H1NO SPEEDfCM/SEC)»
H1NO DIRECT1QN(OEO/N)»
R1R TEf1PERRTURE(CEC-C)»
01SCHHRCE TEMPCDEO-C)>
OISCH FLQHRflTE(CUM/SEC)i
LENGTH SCflLEftCf1= X CM)
VELOCITY SCRLEfCM/SEC).
11.0
400.0
320.
12.5
15-6
G2.7
41019.
52.49
OEYIRT1QN FROM IR TEMP*
FLQO TIDE
0.742
Figure 54. Surface temperature, Anclote Anchorage by modeling
105
-------
TlftEfJflNUflRY 30.1979)1
HIND SPEED fCft/3EC)»
HINO DIREC71QNtDEO/NJ>
ftlR TErtP£RRtUREfOEO-C)«
01SCHRRGE TEf1PCOEG-C}«
01SCH FLQHRRTEfCUM/SECJ»
LENGTH SCRlEflCfls X CM)»
VELOCITY 5CfllEfCft/SECJt
16.0
400.0
360.
14.0
1G.3
52.7
41019-
52.49
/ / /
1 It//'-' '
Figure 55. Surface velocity, Anclote Anchorage by modelina
106
-------
T1HEURNURRY 30.19791*
HIND SPEEOfCM/SECJ»
WIND DlRECTIQNfOEO/NJ»
R1R TEHPERflTUREfOEC-CJ>
DI5CHRROE T£HP(OEO-CJi
OISCH FLQHRRTEfCUM/SECJ»
LENGTH SCRLEflCNr X CMJ»
VELOCITY SCRLEfCH/SECJ»
16.0
400.0
360.
14.0
16.3
62.7
41019.
52.49
Ja 12
J= 4
EBB TIDE
Figure 56. UW velocity, Anclote Anchorage by modeling
107
-------
nflEfJflNURRf 30,1379)>
HIND SPEEDfCfl/SECJi
H1NO DIRECTIQN(OEQ/N)»
flIR 7EflPERfnUREfOEC-C)»
DISCHARGE TEMP(OEO-C)>
01SCH FLQ«RflTEfCU«/SECJ«
LENOTH SCflLEflCf1= X CM)i
VELOCITY 5Cm.EfCH/SECJ«
16.0
400.0
360.
14.0
16.3
62.7
41013.
52.49
1= 12
r 4
EBB 710E
Figure 57. VW velocity, Anclote Anchorage by modelinq
108
-------
TIMEURNURRY 30.1979) i
HIND SPEEOfCM/SEC )>
HIND DIRECTIQNfOEO/NJ:
RIR TEflPERRTUREfOEG-C):
01SCHRRGE TEf1PtDED-CJ»
DISCH FLOHRRTEfCUM/SECJs
LENGTH SCRLEflCM= X CMJ:
VELOCITY SCRLEfCfl/SECJs
17.0
400.0
360.
18.0
16.3
62.7
41019.
52.43
£83 TIDE
Figure 58. Temperature from 1R
109
-------
MIND SPEEQfC.fi/SECJi
HIND OIRECTION(QEO/N)«
HIR TEf1PERfnUREfDEC-C)J
OISCHBROE TEMPfOEO-Oi
01SCH FLQHRflTEfCUf1/SECJ«
LENGTH SCflLEflCMs X CH)«
VELOCITY SCflLEfCM/SEC It
16.0
400.0
360.
14.0
IG.3
82.7
41013.
52.43
OEVlflUQN PROM IR
£88 TIDE
0.350
Figure 59. Surface temperature, Anclote Anchorage by modeling
110
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TlflEClST FEB. .1979Ji
HINO SPEEOtCtl/SECU
WIND OIRECTI9NCOEG/N}:
flIR TEflPERHTURE(OEG-C)«
DISCHflRGE TEHPtOEG-CJi
DISCH FLQHRRTECCUM/SEC) :
UENGTH SCfllEClCMr X CM) :
VELOCITY 5CRLEICM/SEC )J
16.0
600.
330.0
11-1
U.O
62.7
41019.
52.49
HIGH TIDE
Figure 60. Temperature from !R
111
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TECHNICAL REPORT DATA
(Please read Inunctions on the reverie before completing)
1. REPORT NO. 2.
EPA-600/7-82-037a
4.T.TLEANDSUBT.TLE Verification and Transfer of
Thermal Pollution Model; Volume I. Verifica-
tion of Three-dimensional Free-surface Model
7.AUTHOR(s> s.S.Lee, S.Sengupta, S.Y.Tuann, and
C.R.Lee
9. PERFORMING ORGANIZATION NAME AND ADDRESS
The University of Miami
Department of Mechanical Engineering
P.O. Box 248294
Coral Gables, Florida 33124
12. SPONSORING AGENCY NAME AND ADDRESS
EPA, Office of Research and Development
Industrial Environmental Research Laboratory
Research Triangle Park, NC 27711
3. RECIPIENT'S ACCESSION- NO.
5 REPORT DATE
May 1982
6. PERFORMING ORGANIZATION CODE
8. PERFORMING ORGANIZATION REPORT C
10. PROGRAM ELEMENT NO.
11. CONTRACT/GRANT NO.
EPA IAG-78-DX-0166*
13. TYPE OF REPORT AND PERIOD COVERS
Final: 3/78-9/80
14. SPONSORING AGENCY CODE
EPA/600/13
15. SUPPLEMENTARY NOTES IERL-RTP project officer is Theodore G.Brna, Mail Drop
61, 919/541-2683. (*) IAG with NASA, Kennedy Space Center, FL 32899,
subcontracted to U. of Miami under NASA Contract NAS 10-9410.
e.ABSTRACT The six-volume report: describes the theory of a tnree-aimen-
sional (3-D) mathematical thermal discharge model and a related one-
dimensional (1-D) model, includes model verification at two sites, and
provides a separate user's manual for each model. The 3-D model has tw
forms: free surface and rigid lid. The former, verified at Anclote An-
chorage (FL), allows a free air/water interface and is suited for sign
fleant surface wave heights compared to mean water depth; e.g., estu-
aries and coastal regions. The latter, verified at Lake Keowee (SC), i
suited for small surface wave heights compared to depth (e.g., natural
or man-made inland lakes) because surface elevation has been removed a
a parameter. These models allow computation of time-dependent velocity
and temperature fields for given initial conditions and time-varying
boundary conditions. The free-surface model also provides surface
height variations with time. The 1-D model is considerably more econo-
mical to run but does not provide the detailed prediction of thermal
plume behavior of the 3-D models. The 1-D model assumes horizontal
homogeneity, but includes area-change and several surface-mechanism
effects.
17. KEY WORDS AND DOCUMENT ANALYSIS
a. DESCRIPTORS
Pollution
Thermal Diffusivity
Mathematical Models
Estuaries
Lakes
Plumes
13. DISTRIBUTION STATEMENT
Release to Public
b. IDENTIFIERS/OPEN ENDED TERMS
Pollution Control
Stationary Sources
19. SECURITY CLASS (This Report)
Unclassified
20. SECURITY CLASS (This page)
Unclassified
c. COSATI Field/Group
13B
20M
12A
08H,08J
21B
21. NO. OF PAGES
123
22. PRICE
EPA Form 2220-1 (9-73)
112
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