EPA-R2-73-162
MARCH 1973 Lnvironmental Protection Technology
Numerical Thermal Plume
Model for Vertical Outfalls
in Shallow Water
Office of Research and Monitoring
U.S. Environmental Protection Agency
Washington, D.C. 20460
-------
RESEARCH REPORTING SERIES
Research reports of the Office of Research and
Monitoring, Environmental Protection Agency, have
been grouped into five series. These five broad
categories were established to facilitate further
development and application of environmental
technology. Elimination of traditional grouping
was consciously planned to foster technology
transfer and a maximum interface in related
fields. The five series are:
1. Environmental Health Effects Research
2. Environmental Protection Technology
3. Ecological Research
4. Environmental Monitoring
5. socioeconomic Environmental Studies
This report has been assigned to the ENVIRONMENTAL
PROTECTION TECHNOLOGY series. This series
describes research performed to develop and
demonstrate instrumentation, equipment and
methodology to repair or prevent environmental
degradation from point and non-point sources of
pollution. This work provides the new or improved
technology required for the control and treatment
of pollution sources to meet environmental quality
standards.
-------
EPA-R2-73-162V
March 1973
NUMERICAL THERMAL PLUME MODEL
FOR VERTICAL OUTFALLS IN SHALLOW WATER
By
Donald S. Trent
James R. Welty
Project 16130 DGM
Project Officer
Mostafa A. Shirazi
Environmental Protection Agency
National Environmental Research Center
Corvallis, Oregon 97330
Prepared for
OFFICE OF RESEARCH AND MONITORING
U.S. ENVIRONMENTAL PROTECTION AGENCY
WASHINGTON, D.C. 20460
For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402
Price $4.80 domestic postpaid or $4.26 OPO Bookstore
-------
EPA Review Notice
This report has been reviewed by the Environmental Protection Agency
and approved for publication. Approval does not signify that the
contents necessarily reflect the views and policies of the Environ-
mental Protection Agency, nor does mention of trade names or commercial
products constitute endorsement or recommendation for use.
ii
-------
ABSTRACT
A theoretical study of the heat and momentum transfer resulting
from a flow of power plant condenser effluent discharged vertically to
shallow, quiescent coastal receiving water is presented. The complete
partial differential equations governing steady, incompressible, tur-
bulent flow driven by both initial momentum and buoyancy are solved
using finite-difference techniques to obtain temperature and velocity
distributions in the near field of the thermal discharge.
Turbulent quantities were treated through the use of Reynolds
stresses with further simplification utilizing the concept of eddy
diffusivities computed by Prandtl's mixing length theory. A Richardson
number correlation was used to account for the effects of density
gradients on the computed diffusivities.
Results were obtained for over 100 cases, 66 of which are reported,
using the computer program presented in this manuscript. These results
ranged from cases of pure buoyancy to pure momentum and for receiv-
ing "water depths from 1 to 80 discharge diameters deep. Various com-
puted gross aspects of the flow were compared to published data and
found to be in excellent agreement. Data for shallow water plumes
and the ensuing lateral spread are not readily available; however, one
computed surface temperature distribution was compared to proprietary
data and found also to be in reasonable agreement.
This report was submitted in fulfillment of Grant No. 16130-DGM
between the Environmental Protection Agency and the Department of
Mechanical Engineering, Oregon State University.
ill
-------
TABLE OF CONTENTS
Page
CHAPTER 1. INTRODUCTION 1
1.1 Objectives 3
1.2 Summary 4
CHAPTER 2. DISCUSSION OF THERMAL PLUMES AND PROBLEM
DESCRIPTION 7
2.1 The Nature of Thermal Plumes in Marine
Surroundings 7
2.1.1 Discharge Magnitude 8
2.1.2 Outfall Configuration 10
2.1.3 Hydrodynamic Regimes 12
2.1.4 Oceanographic Effects 15
2.1.4.1 Density Stratification 17
2.1.4.2 Effect of Currents 18
2.1.4.3 Ocean Turbulence 21
2.1.4.4 Air-Sea Interactions 22
2.2 Plume Analysis State-of-the-Art 23
2.2.1 Submerged Outfalls 25
2.2.2 Horizontal Shoreline Outfalls 27
2.3 Work Description 28
CHAPTER 3. TRANSPORT EQUATIONS - GENERAL THEORY 30
3.1 Coordinate System 30
3.2 Conservation Laws 31
3.2.1 Continuity 33
3.2.2 The Equations of Motion for Turbulent Flow 34
3.3 The Boussinesq Approximation 36
-------
3.4 The Pressure Equation 39
3.5 r Transport 40
3.5.1 Transport of Heat, Salinity and Buoyancy 40
3.6 The Equation of State for Sea Water 45
3.7 Vorticity Transport 47
3.8 Non-dimensional Form of the Equations of Motion 50
3.9 Further Comments on the Concept of "Eddy Viscosity" 53
3.10 Two-Dimensional Forms of the Transport Equations
in Rectangular and Axisymmetric Coordinates 56
3.10.1 Two-Dimensional Transport Equations in
Rectangular Geometry 58
3.10.2 Two-Dimensional Transport Equations in
Axisymmetric Coordinates 62
CHAPTER 4. PLUME THEORY - SIMILARITY SOLUTIONS 65
4.1 General Description 65
4.2 Simplified Equations for a Vertical Plume 67
4.3 Radial Velocity and Temperature Profiles 71
4.3.1 Zone of Established Flow 71
4.3.2 Zone of Flow Establishment 74
4.4 Zone of Flow Establishment 76
4.5 Governing Differential Equations 77
4.5.1 Initial Condition 80
4.5.2 Evaluation of Terms Involving K and x 81
4.5.3 Homogeneous Receiving Water 82
4.6 Lateral Velocity, u 83
CHAPTER 5. FINITE DIFFERENCE MODELS 84
5.1 Physical System for the Vertical Round Port 85
VI
-------
5.2 Governing Differential Equations 87
5.3 Vorticity Equations 90
5.4 Dimensionless Forms 92
5.5 Coordinate Transformation 94
5.6 Finite Difference Grid System 96
5.7 Difference Equations TOO
5.7.1 Stream Function and Velocity 100
5.7.2 Transport Equations 103
5.7.3 Summary of Required Difference Equations 108
5.8 Boundary Conditions 109
5.9 Rectangular Coordinates 132
5.9.1 Governing Differential Equations 132
5.9.2 Rectangular Difference Equations 135
5.9.3 Rectangular Boundary Conditions 138
CHAPTER 6. CODE DESCRIPTION AND ORGANIZATION 141
6.1 Computational Procedure
6.2 Executive Program and Subroutine Description 143
6.3 Flow Charts 151
CHAPTER 7. CODE VERIFICATION AND NUMERICAL EXPERIMENTS 163
7.1 Deep Water Plumes 164
7.1.1 The Momentum Jet 171
7-1.1.1 Centerline Velocity and Concentra-
tion for Momentum Jets 172
7.1.1.2 Spread of the Momentum Jet 189
7.1.1.3 Radial Distribution of Vertical
Velocity, Concentrations and
Vorticity for the Momentum Jet 192
vii
-------
7.1.1.4 Distribution of Radial Velocity
for the Momentum Jet 201
7.1.1.5 Typical Contours and Three-
Dimensional Plots for a Momentum
Jet 207
7.1.2 Two Cases of Pure Buoyancy 207
7.1.2.1 Center!ine Velocity and Temperature 207
7.1.2.2 Spread of the Pure Buoyant Plume 218
7.1.2.3 Radial Distribution of Vertical
Velocity, Temperature and Vorticity
for Pure Buoyancy 220
7.1.2.4 Radial Velocity and Entrainment
for Pure Buoyancy 228
7.1.3 Mixed Flow - Forced Plumes 237
7.1.3.1 Center!ine Velocity and Temperature
for Forced Plumes 237
7-1.3.2 Rate of Spread and Entrainment 251
7.2 Transport Coefficients 253
7.2.1 The Radial Transport Coefficient, er 266
7.2.2 The Vertical Transport Coefficient, EZ 284
7.3 Numerical Stability and Convergence 298
7.3.1 Numerical Stability 299
7.3.2 Convergence 304
CHAPTER 8. NUMERICAL EXPERIMENTS FOR SHALLOW WATER CASES 319
8.1 Modeling the Vertical Eddy Diffisivity Multiplier,
FZ 319
8.2 Results for Homogeneous Receiving Water 10 Port
Diameters Deep 325
8.3 Results for Homogeneous Receiving Water 5 Port
Diameters Deep 340
8.4 Results for Two Different Methods of Computing FZ 346
viii
-------
8.5 Numerical Experiments Involving Ambient
Stratification 355
8.6 Discharge at Very Shallow Depth 376
8.7 Comparison with Field Data 382
CHAPTER 9. CONCLUSIONS 384
BIBLIOGRAPHY 389
APPENDIX A - CONVECTIVE TRANSPORT DIFFERENCE APPROXIMATION 400
APPENDIX B - FINITE DIFFERENCES FOR IRREGULAR NODE SPACING 415
APPENDIX C - COORDINATE TRANSFORMATION 419
APPENDIX D - SOME RELATIONSHIPS BETWEEN TIME DEPENDENT
AND STEADY STATE NUMERICAL METHODS IN HEAT
TRANSFER AND FLUID FLOW 423
APPENDIX E - LISTING OF SYMJET COMPUTER PROGRAM - 1108 VERSION 434
ix
-------
LIST OF TABLES
Table Title . Page
2.1 Summary of Work Pertinent to Ocean Outfall Plume
Analysis 24
3.1 Differential Equations Required for Velocity-Pressure
and Vector Potential-Vorticity Methods in Two and
Three Dimensions 57
4.1 Values of Terms Involving K and x 81
7.1 Summary of Momentum Jet Verification Cases
(FQ + -) 166
7.2 Summary of Pure Buoyant Plume Verification Cases
(FQ = 0) 168
7.3 Summary of Mixed Flow Verification Cases 169
7.4 Comparison of the Spreading Constant Reported by
Various Investigators 191
7.5 Correlation of the Vertical Diffusion Coefficient
e with the Local Richardson Number, RI 287
7.6 Values of Vertical Eddy Viscosities in the Sea 290
7.7 Convergence Behavior, 40x33 Grid 308
7.8 Convergence Behavior, 31x34 Grid 312
7.9 Convergence Behavior, 26x25 Grid 314
-------
LIST OF ILLUSTRATIONS
Figure Page
2.1 Condenser coolant flow rate as a function of temper-
ature rise and plant electric generating capacity
(fossil fired plant) 9
2.2 Condenser coolant flow rate as a function of tem-
perature rise and plant electric generating
capacity (nuclear plant) 10
2.3 Vertical thermal plume in deep water, illustrating
possible flow regimes 13
2.4 Vertical thermal plume in shallow water, illus-
trating continual transition of the flow field 16
2.5 Possible configuration of a vertical buoyant plume
in stratified receiving water 19
2.6 Possible configuration of a buoyant plume in
stratified receiving water with cross-current,
u on
00 ^(J
3.1 Rectangular coordinate system 30
3.2 Relationship between the buoyancy parameter, A,
and density disparity, A? 45
4.1 Zone of flow establishment for plumes with large
and small densimetric Froude numbers, F 66
4.2 Coordinate system for axisymmetric vertical plume 68
4.3 Typical velocity profile in the zone of flow
establishment for a momentum jet 75
5.1 Physical system for axisymmetric vertical plume
where the bottom boundary is some distance z, t 0
above the outfall port 86
5.2 Physical system for shallow water, axisymmetric,
vertical plume 88
5.3 Computational grid for difference equations 97
XI
-------
Figure
5.4 Typical finite difference cell illustrating indices
for v, n, r, U and V 99
5.5 Typical sea surface boundary and interior cells 115
5.6 Typical centerline boundary and interior cells 118
5.7 Typical inflow boundary and interior cell (deep
water only) 120
5.8 Typical inflow boundary and interior cell (shallow
water case only) 122
5.9 Typical vertical port side boundary and interior
cell (shallow water case only) 125
5.10 Typical bottom boundary and boundary cell 127
5.11 Typical inflow-outflow boundary and interior
cells 129
5.12 Physical system for line plume issuing to flowing
receiving water 133
7.1 Computational grid for the stream function, 4*,
illustrating the effect of the sinh (c) trans-
formation (A? = .14690, AZ = 1.0) 170
1.2 General features of momentum jet centerline
velocity (based on Albertson's data) 173
7.3 Comparison of experimental data and similarity
solution with computed results for a momentum
jet. Centerline velocity and concentration for
case 2 175
7.4 Comparison of experimental data and similarity
solution with computed results for a momentum
jet. Centerline velocity and concentration for
case 4 177
7.5 Computed centerline velocity and concentration
for momentum jet, case 5 178
xii
-------
Figure Page
7.6 Comparison of experimental data and similarity
solution with computed results for a momentum jet.
Centerline velocity and concentration for case 6 179
7.7 Computed centerline velocity and concentration for
momentum jet, case 7 181
7.8 Computed centerline velocity and concentration for
momentum jet, case 8
7.9 Computed centerline velocity and concentration
distribution for momentum jet, case 9 184
7-10 Centerline velocity distributions for cases 4, 7,
and 9, normalized to V = 1.0 186
o
7.11 Centerline velocity and concentration distribution
for case 10 (includes effect of large vertical eddy
diffusivity) 188
7.12 Shape preserving of velocity profiles computed for
an inviscid, rotational fluid (reference case 2) 189
7.13 Computed rate of spread of the momentum jet half-
radius, r1 ,„ 190
7.14 Radial distribution of normalized vertical velocity
for case 2 194
7.15 Normalized radial distribution of axial velocity,
momentum jet case 4 195
7.16 Normalized radial distribution of axial velocity
case 4 196
7.17 Radial distribution of axial velocity at various
elevations case 4 197
7.18 Normalized distribution of axial velocity case 6 198
7.19 Normalized radial concentration distribution,
type 1 boundary condition case 2 199
xiii
-------
Figure
7.20 Normalized radial concentration distribution,
type 2 boundary condition, case 4 200
7.21 Radial vorticity distribution for momentum jet
type 2 boundary condition, case 4 202
7.22 Radial vorticity distribution for momentum jet
at Z = 15. A comparison between type 1 and 2
boundary conditions, and the Gaussian distribution 203
7.23 Normalized radial velocity distribution for
momentum jet 204
7.24 Vertical distribution of stream function at R^,
case 6 206
7.25 Streamlines for case 6 -- momentum jet 208
7.26 Isopycnals for case 6 -- momentum jet 209
7.27 Vorticity level lines for case 6 -- momentum jet 210
7.28 3D illustration of stream function -- psi,
case number 6 211
7.29 3D illustration of stream function -- psi,
case number 6 212
7.30 3D illustration of buoyancy distribution - A,,
case number 6 213
7.31 3D illustration of fluid vorticity - omeqa,
case number 6 214
7.32 Computed centerline velocity and temperature excess
for case 13. Pure buoyancy, F = 0
7.33 Computed centerline velocity and temperature excess
for case 14. Pure buoyancy, FQ = 0
7.34 Computed rate of spread of half-radius, r,/?/D.
Pure buoyancy, case 14 (D = 2 r ) '
219
XIV
-------
Figure Page
7.35 Normalized distribution of computed axial velocity.
Pure buoyancy, case 13 222
7.36 Normalized radial distribution of axial velocity.
Pure buoyancy, case 13 223
7.37 Radial distribution of axial velocity in pure
buoyancy, case 13 224
7.38 Normalized radial distribution of axial velocity.
Stronger source, pure buoyancy, case 14 225
7.39 Normalized distribution of computed radial tempera-
ture excess. Pure buoyancy, case 14 226
7.40 Radial distribution of vorticity. Pure buoyancy,
case 14 ' 227
7.41 Normalized radial velocity distributions for pure
buoyant plume, case 14 229
7.42 Vertical distribution of stream function at R^.
Pure buoyancy, case 14 °° 230
7.43 Streamlines for case 14, pure buoyancy 231
7.44 Isotherms for case 14, pure buoyancy, AT/AT 232
7.45 Vorticity level lines for case 14, pure buoyancy 233
7.46 3D illustrations of stream function - psi, case
number 14 234
7.47 3D illustration of temperature field - AT, case
number 14 235
7.48 3D illustration of fluid vorticity - omega.
Case 14, pure buoyancy 236
7.49 Centerline velocity and buoyancy for cases 15, 16
and 17 ' 240
7.50 Centerline buoyancy distribution for cases 17 and 18 242
-------
Figure Page
7.51 Centerline velocity and buoyancy for cases
17, 19 and 20 243
7.52 3D illustration of vorticity - omega. Case 17,
buoyant plume with running calculation of half
radius 245
7.53 3D illustration of vorticity - omega, case
number 21 246
7.54 Centerline velocity and buoyancy for cases
22 and 23 247
7.55 Comparison between computed results and simi-
larity solution for F = 1.0 249
7.56 Comparison between computed centerline distribu-
tions of velocity and buoyancy for f = 1000
and F •* » 250
o
7.57 Comparison of half-radius spread for various
densimetric Froude numbers 252
7.58 Effect of the eddy Prandtl number on half-radius
spread 252
7.59 Entrainment trends in mixed flows 254
7.60 Streamlines for case 22 - mixed flow, FO = 46 255
7.61 Isotherms for case 22 - mixed flow, FO = 46 256
7.62 Vorticity level lines for case 22 - mixed flow,
FO = 46 257
7.63 3D illustration of stream function - psi.
Case 22 - deep water buoyant jet 258
7.64 3D illustration of temperature field - T.
Case 22 - deep water buoyant jet 259
7.65 3D illustration of fluid vorticity - omega.
Case 22 - deep water buoyant jet ' 260
xvi
-------
Figure Page
7.66 Regional specification for turbulent eddy
coefficient modeling 267
7.67 Computed values of FR for a momentum jet 270
7.68 Computed radial eddy diffusion factors, FR
for deep water plumes at various densi-
metric Froude numbers 271
7.69 Comparison of computation using constant and
variable radial eddy transport coefficients 273
7.70 Concentration distribution in the zone of
flow establishment 275
7.71 Computed potential core and half radius
(FQ = 46) 277
7.72 Centerline velocity and temperature distribu-
tion for 44 diameter deep outfall 279
7.73 Computed centerline velocity and temperature
excess. Cases for 10 diameter deep water 281
7.74 3D illustration of temperature field 283
7.75 Dependence of e on sea state 288
7.76 Correlation of e with density gradient 290
7.77 Observation of flow patterns past the end of
a cylinder 300
7.78 Computed flow patterns past the end of a
cylinder 300
7.79 Convergence behavior, 40 x 33 grid 309
7.80 Convergence history of V and r at selected
cells, momentum jet, 40 x 33 grid 311
7.81 Convergence history of U, V and A, at
selected cells 31 x 34 grid ' 313
7.82 Iteration history for one cell of case 2 316
xvii
-------
Figure
8.1 Computed center!ine velocity and temperature
excess for intermediate depth, cases 48
through 51 (10 diameters deep) 326
8.2 Surface distribution of radial velocity, cases
48 through 51 (see Table 8.1) 327
8.3 Distributions of radial velocity case 50 329
8.4 Maximum radial velocity profiles^ cases 48
through 51 33°
8.5 Radial velocity profiles at r/D=7.32, cases 48
through 51 331
8.6 Surface temperature excess distribution* cases 48
through 51 (see Table 8.1) 332
8.7 Vertical temperature excess distributions for
various radial positions, cases 48 and 50 333
8.8 Streamlines for case 48 - buoyant discharge,
FO = 100 333
8.9 Isotherms for case 48 - buoyant discharge,
FO = 100 333
8.10 Vorticity level lines for case 48 - buoyant
discharge, FO = 100 333
8.11 3D illustration of stream function -- PSI. Case
No. 48, intermediate water outfall, surface 10
diameters above port, FO = 100 334
8.12 3D illustration of temperature field -- AT.
Case No. 48, intermediate water outfall, surface 10
diameters above port, FO = 100 334
8.13 3D illustration of fluid vorticity - OMEGA,
Case no. 48, intermediate water outfall, surface
10 diameters above port, FO = 100 334
8.14 Streamlines for case 49 - buoyant discharge,
FO = 25 334
XVlll
-------
Figure Page
8.15 Isotherms for case 49 - buoyant discharge,
FO = 25 335
8.16 Vorticity level lines for case 49 - buoyant
discharge, FO = 25 335
8.17 3D illustration of stream function -- PSI.
Case No. 49, intermediate water outfall, surface
10 diameters above port, FO = 25 335
8.18 3D illustration of stream function — PSI.
Case No. 49, intermediate water outfall, surface
10 diameters above port, FO = 25 335
8.19 3D illustration of temperature field -- AT.
Case No. 49, intermediate water outfall, surface 10
diameters above port, FO = 25 336
8.20 3D illustration of fluid vorticity - OMEGA.
Case No. 49, intermediate water outfall, surface
10 diameters above port, FO = 25 336
8.21 Streamlines for case 50 - buoyant discharge, FO = 5 336
8.22 Isotherms for case 50 - buoyant discharge, FO = 5 336
8.23 Vorticity level lines for case 50 - buoyant
discharge, FO = 5 337
8.24 3D illustration of stream function -- PSI.
Case No. 50, intermediate water outfall, surface
10 diameters above port, FO = 5 337
8.25 3D illustration of temperature field -- AT.
Case No. 50, intermediate water outfall,
surface 10 diameters above port, FO = 5 337
8.26 3D illustration of fluid vorticity - OMEGA.
Case No. 50, intermediate water outfall, surface
10 diameters above port, FO = 5 337
XIX
-------
Figure 'Page
8.27 Streamlines for case 51 - buoyant discharge, FO = 1 338
8.28 Isotherms for case 51 - buoyant discharge, FO * 1 338
8.29 Vorticity level lines for case 51 - buoyant
discharge, FO = 1 338
8.30 3D illustration of stream function — PSI. Case
No. 51, intermediate water outfall, surface 10
diameters above port, FO = 1 338
8.31 3D illustration of temperature field — AT.
Case No. 51, intermediate water outfall,
surface 10 diameters above port, FO = 1 339
8.32 3D illustration of fluid vorticity - OMEGA.
Case No. 51, intermediate water outfall, surface
10 diameters above port, TO = 1 339
8.33 Computed centerline dimensionless velocity and
temperature excess for shallow water, cases 52
through 55 (5 diameters deep) 341
8.34 Vertical distribution of radial velocity at
various radial positions, case 52 342
8.35 Vertical distribution of temperature excess at
various radial positions, case 52 342
8.36 Streamlines for case 52 - buoyant discharge, FO = 1 344
8.37 Isotherms for case 52 - buoyant discharge, FO = 1 344
8.38 Vorticity level lines for case 52 - buoyant
discharge, FO = 1 344
8.39 3D illustration of stream function -- PSI.
Case No. 55, very shallow water outfall, surface 5
diameters above port, FO = 1 344
8.40 3D illustration of temperature field — AT.
Case No. 55, very shallow water outfall, surface
5 diameters above port, FO = 1 345
XX
-------
Figure Page
8.41 3D illustration of fluid vorticity - OMEGA.
Case No. 55, very shallow water outfall,
surface 5 diameters above port, FO = 1 345
8.42 Computed radial velocity at surface,
cases 58 and 59 348
8.43 Vertical distribution of radial velocity, I),
cases 58 and 59 349
8.44 Vertical distribution of radial velocity, U,
cases 58 and 59 349
8.45 Streamlines for an axisymmetric, vertical plume,
confined by a free surface, case 58 350
8.46 Streamlines for an axisymmetric, vertical plume,
confined by a free surface, case 59 351
8.47 Surface temperature excess, AT , cases 57, 58 and 59 352
8.48 Vertical temperature excess distribution, cases
58 and 59 ' 353
8.49 Vertical temperature excess distribution,
cases 58 and 59 353
8.50 Isotherms for an axisymmetric, vertical plume,
confined by a free surface, case 59 354
8.51 Ambient temperature profiles for cases 60
through 65 356
8.52 Vertical distribution of radial velocity, case 60 357
8.53 Vertical temperature excess distribution, case 60 357
8.54 Streamlines for an axisymmetric, vertical plume,
confined by a free surface, case 60 - intermediate
depth, homogeneous ambient, Mamayev, BETA = .4 358
8.55 Isotherms for an axisymmetric, vertical plume,
confined by a free surface, case 60 - intermediate
depth, homogeneous ambient, Mamayev, BETA = .4 358
XXI
-------
Figure Pa9e
8.56 Vorticity contours axisymmetric, vertical plume,
confined by a free surface, case 60 - intermediate
depth, homogeneous ambient, Mamayev, BETA = .4 358
8.57 3D illustration of stream function — PSI, case
no. 60 358
8.58 3D illustration of temperature field -- AT,
case No. 60 359
8.59 3D illustration of fluid vorticity - OMEGA,
case No. 60 359
8.60 Vertical distribution of radial velocity, case 61 359
8.61 Vertical excess temperature distribution, case 61 359
8.62 Streamlines for an axisymmetric, vertical plume,
confined by a free surface, case 61 - intermediate
depth, with 2 degree thermocline, Mamayev 360
8.63 Isotherms for an axisymmetric, vertical plume,
confined by a free surface, case 61 - intermediate
depth, with 2 degree thermocline, Mamayev 360
8.64 Vorticity contours axisymmetric, vertical plume,
confined by a free surface, case 61 - intermediate
depth, with 2 degree thermocline, Mamayev 360
8.65 Vertical Distribution of Radial Velocity. Case 61 362
8.66 Vertical Temperature Excess Distribution. Case 61 363
8.67 Surface Temperature Excess, AT<~ for Cases 60 and 63 364
8.68 Streamlines for an axisymmetric, vertical plume,
confined by a free surface. Case 63 - inter-
mediate depth with 4 degree thermocline. Mamayev 365
8.69 Isotherms for an axisymmetric, vertical plume,
confined by a free surface. Case 63 - inter-
mediate depth with 4 degree thermocline, Mamayev 365
8.70 Vorticity contours axisymmetric, vertical plume,
confined by a free surface. Case 63 - inter-
mediate depth with 4 degree thermocline, Mamayev 365
xxii
-------
Figure Page
8.71 3D illustration of stream function — PSI.
Case 63 - intermediate depth with 4 degree
thermocline, Mamayev 365
8.72 3D illustration of temperature field -- T.
Case 63 - intermediate depth with 4 degree
thermocline, Mamayev 366
8.73 3D illustration of fluid vorticity - OMEGA.
Case 63 - intermediate depth, with 4 degree
thermocline, Mamayev 366
8.74 Vertical distribution of radial velocity. Case 64 368
8.75 Vertical, excess temperature distribution. Case 64 368
8.76 Streamlines for an axisymmetric, vertical plume,
confined by a free surface. Case 64 - inter-
mediate depth, with 5 degree thermocline, Mamayev 369
8.77 Isotherms for an axisymmetric, vertical plume,
confined by a free surface. Case 64 - inter-
mediate depth, with 5 degree thermocline, Mamayev 369
8.78 Vorticity contours axisymmetric, vertical plume,
confined by a free surface. Case 64 - inter-
mediate depth, with 5 degree thermocline, Mamayev 369
8.79 3D illustration of viscous stream function -
Case 64 - intermediate depth, with 5 degree
thermocline, Mamayev 369
8.80 3D illustration of temperature field -- AT
Case 64 - intermediate depth, with 5 degree
tnermocline, Mamayev 370
8.81 3D illustration of vorticity -- OMEGA
Case 64 - intermediate depth, with 5 degree
thermocline, Mamayev 370
8.82 Vertical distribution of radial velocity. Case 65 371
8.83 Vertical, excess temperature distribution. Case 65 372
8.84 Streamlines for an axisymmetric, vertical plume,
confined by a free surface, case 65 - intermediate
depth, with 5 degree thermocline, Mamayev 373
XXlll
-------
Figure Pa9e
8.85 Vorticity contours axisymmetric, vertical plume,
confined by a free surface, case 65 - inter-
mediate depth, with 5 degree thermoeline, Mamayev 373
8.86 Isotherms for an axisymmetric, vertical plume,
confined by a free surface, case 65 - inter-
mediate depth, with 5 degree thermoeline, Mamayev 373
8.87 Streamlines for an axisymmetric, vertical plume,
confined by a free surface, case 65 - inter-
mediate depth, continued iteration. 373
8.88 Isotherms for an axisymmetric, vertical plume,
confined by a free surface, case 65 - inter-
mediate depth, continued iteration. 374
8.89 Vorticity contours axisymmetric, vertical plume,
confined by a free surface, case 65 - inter-
mediate depth, continued iteration. 374
8.90 Surface radial velocity, case 66 378
8.91 Surface temperature excess, case 66 378
8.92 Vertical distribution of radial velocity at
various radial positions, case 66 379
8.93 Vertical distribution of temperature excess at
various radial positions, case 66 379
8.94 Streamlines for case 66 (1.0 dia deep) FO = .111 380
8.95 Isotherms for case 66 (1.0 dia deep) FO = .111 380
8.96 Vorticity for case 66 (1.0 dia deep) FO = .111 380
8.97 3D illustration of fluid vorticity - OMEGA
Case 66 380
8.98 3D illustration of stream function -- PSI,
Case 66 381
8.99 3D illustration of stream function -- PSI,
Case 66 381
8.100 3D illustration of temperature field -- T.
Case 66 381
XXIV
-------
Figure Page
8.101 3D illustration of temperature field — AT.
Case 66 381
8.102 Comparison of computed surface temperature
with field data 383
A-l Finite difference grid system 404
A-2 Values of X and 4. for a preferred
difference scheme 408
A-3 Convective r flux for an infinitesimal
axisymmetric volume element 410
A-4 Axisymmetric finite difference cell, p, with
the four immediate neighbor cells 412
B-l Irregular spaced grid 415
B-2 Grid layout for vertical differences 417
C-l Ratio of actual to computed node spacing 422
D-l Finite difference cell 428
XXV
-------
NOMENCLATURE
Variables which are not listed in this nomenclature are defined
at the appropriate location within the manuscript. A few variable
names have been duplicated; however, the definitions listed below hold
throughout the text with duplications defined at the point of use.
Dimensions are given in the force-length-time system (F-L-T).
A Constant used in Chapter 7.
p
a Thermal diffusivity, L /T
B Buoyancy parameter defined in Chapter 3.
b Slot width (slot plume), L
C-, ,C2 Constants
c Concentration
D Diameter of outfall port, L
ei, e Unit vector (i = 1,2,3)
E Momentum parameter for plume similarity
solution.
f Coriolis constant, 1/T
FR Radial eddy momentum diffusivity multiplier
FZ Vertical eddy momentum diffusivity multiplier
p
g Gravitational constant, L/T
k Kinetic energy of turbulent motion, FL
K Entrainment parameter
£ Characteristic length, L
L Liebmann acceleration constant
M Total momentum, FT
XXVI
-------
n Unit surface normal vector
P Pressure, F/L2
P° Deviatoric pressure (defined in Chapter 3), F/L
2
P Fluctuating pressure, F/L
2
Q Plume entrainment rate, L/T
r Radial coordinate, L
2
R.. Reynolds stress tensor, F/L
' J
S Salinity, ppt
t Time, T
T Temperature, °C
u Velocity, L/T
u' Fluctuating velocity, L/T
ft, Irrotational velocity vector, L/T
IL Solenoidal velocity vector, L/T
v Vertical velocity, L/T
x horizontal coordinate, L
x- General rectangular coordinate, L
Y Space coordinate, L
z Vertical coordinate, L
XXV11
-------
Greek
a Entrainment constant
B Constant used in vertical eddy diffusivity model
6 Convergence criterion
2
e Eddy diffusivity for momentum, L /T
2
£. Eddy diffusivity for matter, L /T
eu Eddy diffusivity for heat, L2/T
n
e Eddy diffusivity for density, L /T
p
c, Variable defined in Chapter 3.
0 Longitude
K Thermal conductivity, F/LT°C
Dynamic viscosity, FT/L2
Kinematic viscosity, L /T
n 3.14159
P Density, FT2/L4
p1 Fluctuating density, FT2/L4
T Shear stress
4 Latitude
•
4> Source or sink term in r-transport equation, i/|_
2
$ Scalar potential , L /T
2
* Stream function, L /T
£ Vector potential, L2/T
oj vorticity, 1/T
fi* Earth rotation velocity, 1/T
xkviii
-------
Standard Dimensionless Parameters
1 2
EU Euler number, AP/-^ pv!"
"o
fn » FO Densimetric Froude number,
O O ~D
Pr Prandtl number, V/K
PR Eddy Prandtl number, e/e,,
SO,A Eddy Schmidt number, e/e
wfl
Re Reynolds number,
wo
RE Eddy Reynolds number,—-
RI
Local Richardson number, —9- (dP/dz)
po
RI1 Gross Richardson number, - ^— ^
po (AU/AZ)^
for the thermal layer.
Dimensionless Parameters Defined in this Manuscript
C Concentration, c/cn _
°/V Z\3
E* Momentum parameter,
/K
P* Pressure, P°Pr/APQ
R Radial coordinate, r/rQ
E*1/3
R* Density parameter,
xxix
/K (1+x)
-------
t* Time, tvQ/D
U Radial velocity, u /VQ
V Vertical velocity, UZ/VQ
X Space coordinate, x/x
Z Vertical coordinate, z/rQ
Z Vertical coordinate, z/D
r Conservative constituent parameter
Pr-p
A-, Buoyancy parameter, ( )
pr"po
oo .
A? Density disparity parameter, ( )
pr"po
Vs
Salinity parameter, (7—^
r b
Eddy diffusivity for momentum, e/e0
T -T
Temperature parameter, (=—=F-)
o" r
Radial coordinate, sinh"' (R)
Stratification parameter, poo(Z)/p
2
Stream function, y/r v
Vorticity,
-------
Subscripts
The following subscript definitions hold unless otherwise
defined in the text.
b Refers to slot jet width
c Refers to center, or core
e Value at end of zone of flow establishment
E Elliptic partial differential equation
H Refers to heat
i Tensor index
j Tensor index, also computational grid index in
the horizontal (radial) direction
k Tensor index, also computational grid index in
the vertical direction
m Value at jet centerline
max Maximum value
p Computational grid index
port Refers to conditions at outfall port
q Computational grid index
r Refers to radial direction, or reference condition
for scalar quantities
s Refers to condition at surface
T Refers to turbulent conditions, or transport
equation
x Refers to x (horizontal) direction
z Refers to 2 (vertical) direction
XXXI
-------
Greek Subscripts
Y Refers to a conservative constituent
A Refers to the buoyancy parameter
p Refers to density
Y Refers to the stream function
n Refers to vorticity
Other Subscripts
o Refers to conditions of or at the outfall
Refers to conditions far removed from the outfall
1/2 Refers to the half-width
Mathematical Notations
D_ Substantial derivative
Dt
v^ Laplacian operator
->-
v Gradient operator, del
A Finite-difference operator
£ Summation except where otherwise specified
6..: Kronecker delta function
* J
e... Permulation tensor
1 JK
Log Natural logarithm
|| Absolute value
Hat, unit vector
- Overbar, time or space averaging
sinh, cosh, Hyperbolic functions
tanh, coth
xxxii
-------
A NUMERICAL MODEL FOR PREDICTING ENERGY DISPERSION
IN THERMAL PLUMES ISSUING FROM LARGE, VERTICAL OUTFALLS
IN SHALLOW COASTAL WATER
CHAPTER 1
INTRODUCTION
The growing demand for electric power in the United States has
set the stage for an additional environmental concern; the enormous
quantities of waste heat discharged to our natural waterways by
existing and planned large thermal power plants. The concern, of
course, is the impact of the waste heat on the resident ecosystem.
The answer to the underlying question, "are thermal effects a detri-
ment to the environment?", is largely a matter of philosophy since
certain species of the flora and fauna are apt to thrive under the
altered conditions whereas others would doubtless perish.
The central issue is, however, that these large quantities of
discharged waste heat will in fact alter the environment and certain
changes in the ecosystem will occur. Just what changes will take
place and the nature of the shift in the ecosystem are open to numer-
ous questions. Preservation of species, the impact on the overall
food chain, and the encroachment of undesirable species are certainly
compromising aspects. These questions and many others of equal im-
portance are certainly not unattended, but the interaction of the eco-
system with the environment and the complexity of ecodynamics as
influenced by artificial shifts in the environment presents an ana-
lytical and empirical task to arrive at reliable predictive methods of
monumental proportions.
-------
Although the ultimate concern of so-called "thermal pollution"
lies in the ecological impact, it is necessary as a first step to
assess the receiving water temperature changes. Prediction of the
temperature distribution in natural waters is in itself a formidable
task owing to the complexity of such natural phenomena as hydro-
dynamics, dispersion, and atmospheric interaction (transport process-
es). To date, no analytical or empirical tool has been devised to
predict thermal distributions with any degree of confidence for gen-
eral situations. The state-of-the-art has been developed along the
lines of applying the most appropriate simplified analytical or empir-
ical model to an immediate situation. Unfortunately, some situations
are complicated to the extent that simplified methods are a hopeless
exercise and can lead to a valueless or grossly overrestrictive
assessment.
Such complexities lead to methods involving more elaborate
numerical models or physical scale modeling. In this work, we take
the former approach, that of numerical modeling.
As is pointed out in Chapter 2, previous analytical plume
modeling efforts have dealt primarily in two areas which are:
« The initial mixing zone where, in certain cases, simil-
arity solutions apply, and
« The far field where heat transfer is governed by turbulent
diffusion and atmospheric interchange.
-------
The past research has largely neglected an area cf prime impor-
tance, that being the near field of large, vertical outfalls in shal-
low coastal waters. This neglect is in part due to the complexity of
the flow region in question and the fact that it is a new problem.
The near thermal field for such outfalls is, nevertheless, a very
important aspect of plume analysis, and is in need of analytical
attention.
1.1 Objectives
The primary objective of the work contained in this thesis is
the investigation and application of finite-difference methods in ana-
lyzing the dispersion of thermal effluents issuing from large single
port vertical outfalls in shallow coastal receiving water. Such sys-
tems are typical of several existing and/or planned thermal power
plant reject-heat discharge systems. This analysis, constitutes
research needed for future thermal discharge management. Since we are
interested primarily in the hydrodynamics and energy transport for a
shallow water, vertically confined plume, simplified analytical
methods cannot be applied with confidence. Physical modeling holds
some promise as an alternative to numerical modeling, at least in the
near field and in the absence of stratification. Since the numerical
modeling devised in this study was a considerable effort in itself,
physical modeling was not attempted. Verification of the numerical
techniques was rather carried out by testing the computer program for
several cases that could be checked with data published in the
literature.
-------
The secondary objective of the work was to develop a computer
program for analytical study of the above mentioned outfall systems
which would also include use of similarly solutions where applicable,
along with the more elaborate numerical techniques.
1.2 Summary
In the initial scoping of the vertical plume problem it was
planned to investigate both the transient and steady state operation
of the outfall system. Initially, several transient cases were run
which were academically quite interesting but it was soon ascertained
that the application of steady flow techniques was more efficient in
obtaining the desired results—the quasi-steady flow distributions.
Consequently, the transient techniques were abandoned. In general,
the scope of the study encompasses nearly all of the real quasi-
steady flow complication expected in actual situations which conform
to axisymmetric assumptions. The most notable complication is that of
plume induced turbulence.
One exception to the modeling of observed phenomena was the
surface boil; the surface was assumed flat and free-slip in all
instances. This assumption averted the problem of modeling a dis-
torted surface which is thought to be of small importance to the
overall plume characteristics. Other complications accounted for
include the possible existence of a potential core, ambient strati-
fication, and non-homogeneous, anisotropic turbulence in both the
vertical rise and lateral plume spreading. Flows for the entire
-------
•range of densimetric Froude numbers were investigated, including
cases of pure natural convection.
The solution method deemed most practical for purposes of this
study was the stream function-vorticity, finite-difference approach,
in axisymmetric coordinates. The transport equations were used in
their conservative forms and special upstream differencing techniques
were employed for the convective terms.
The finite-difference computation technique verification study
was carried out for three deep water flow categories:
• pure momentum jets,
• pure buoyant plumes, and
• forced plumes where both initial momentum and buoyancy
play important roles.
Results from this portion of the study were compared to data
reported in the literature or valid similarity solutions. These com-
parisons involved:
• centerline distributions of velocity and buoyancy (or
temperature),
• spread of the half-radius,
• radial distributions of vertical velocity and buoyancy
(or temperature),
• radial velocities,
• entrainment trends, and
* eddy diffusivities.
-------
The effects of several different computational aspects were
included which involved effects of the:
& boundary conditions and their computation,
® various models for eddy diffusivities,
« Prandtl (or Schmidt) number effects,
e Richardson number modification of vertical diffusivities,
« potential core,
• ambient turbulence,
» vertical turbulence within the plume, and
e various factors involving numerical stability and con-
vergence.
The general results of this portion of the study showed excel-
lent agreement with experimental data where the eddy diffusivities
are well modeled. Plume generated turbulence was modeled using
Prandtl's mixing length hypothesis in all cases.
In Chapter 8 the plume model is extended to shallow water cases.
Verification is not presented since there are no readily available
appropriate or reliable data.* Here we rely on the extensive veri-
fication study of Chapter 7 mentioned above.
Verification of the surface temperature distribution was obtained
for one case. The data is proprietary, hence no details of operating
conditions are disclosed.
-------
CHAPTER 2
DISCUSSION OF THERMAL PLUMES AND PROBLEM DESCRIPTION
The dynamical behavior of heated water issuing to the marine
environment from an ocean outfall is influenced by a number of variables
which fall into two general categories. The first of these categories
encompasses engineered variables such as outfall design, effluent temp-
erature, etc; and, the second, those variables which we cannot control,
such as the oceanographic and meteorlogical parameters. In this chapter,
we shall illustrate and discuss how ambient and engineered variables
influence the gross behavior of a thermal plume, briefly discuss the
analytical "state-of-the-art," and qualitatively describe the problem
undertaken in this research.
2.1 The Nature of Thermal Plumes in Marine Surroundings
In the following discussion the terms jet flow and plume flow
will be used, and to avoid confusion it is appropriate to outline the
meaning of each at this time. A convective flow in a free environment
caused solely by buoyancy is commonly called a simple plume. In this
case, the general pattern of motion is caused by a density disparity
between the flow and the surrounding environment. Such instances are
atmospheric thermal and the smoke plumes generated by field fires.
A jet, on the other hand, is characterized by source flow inertia where
the flow may not involve a density difference.
The flow which is of primary concern in this discussion is a com-
bination of the above where both initial momentum and buoyancy have
significant influence on the flow behavior. Such a flow might be termed
7
-------
a forced plume. However, in this work the flow field will be called a
thermal plume or plume. Reference will be made to jet flow from time
to time, which will imply that conditions near the outfall, where
initial momentum dominates the dynamic behavior, is the subject of dis-
cussion or that the effluent is neutrally buoyant.
A temperature difference is not the only factor which must be con-
sidered as a buoyancy source in a thermal plume. Differential salt con-
centration is certainly a factor. Salinity differences must be consid-
ered if the power plant condenser coolant is drawn from an estuary and
rejected off-coast,in which case, the effluent would most likely be less
saline than the receiving water and contribute to the overall buoyant
force.
2.1.1 Discharge Magnitude
The volumetric flow rate required by a thermal power station
depends on plant size, steam cycle thermodynamic efficiency, and coolant
temperature rise. Typical installations range from 1000 to 2000 Mwe and
operate at a coolant temperature rise between 15 and 20 °F. Plant
efficiency depends largely on whether the heat source is nuclear or
fossil. The steam cycle thermodynamic efficiency for a typical fossil
fired plant will be in the neighborhood of 42% for optimum conditions,
whereas typical efficiency for a modern nuclear plant operating under
similar conditions is about 32%. Hence, the nuclear plant will reject
about 50% more heat than a fossil fired plant having the same net
electrical output.
The condenser coolant volumetric flow rate required by power sta-
8
-------
tions in the 1000 to 2000 Mwe range is impressive by any standards,
regardless of whether the plant is nuclear or fossil fired. Figures 2.1
and 2.2 illustrate this fact. It is possible that in the future a
particular site will consist of a number of individual units. Thus
the cooling load on a certain ocean locale may result from the produc-
tion of perhaps 10 Gw .
o
o
o
o
o;
30
20
10
(ELECTRIC)
(ASSUMED EFFICIENCY, 42%)
2345 10
TEMPERATURE RISE, AT (°F)
20 30
Figure 2.1 Condenser Coolant Flow Rate as a Function
of Temperature Rise and Plant Electric
Generating Capacity (Fossil Fired Plant)
-------
GO
U_
O
CD
O
o
o:
30
20
10 -
5
4
3 -
2 -
(ASSUMED EFFICIENCY, 32%)
2345 10 20 30
TEMPERATURE RISE, AT (°F)
Figure 2.2 Condenser Coolant Flow Rate as a Function
of Temperature Rise and Plant Electric
Generating Capacity (Nuclear Plant)
2.1.2 Outfall Configuration
Condenser coolant may be rejected to the ocean either at the
shoreline or offshore through a submerged outfall.
The shoreline discharge may be either by canal or conduit.
Examples of such existing systems are the following fossil fired plants
owned by Pacific Gas and Electric [113].
10
-------
1) Contra Costa, 1298 Mwg, rejecting heat to the San Francisco
Bay Delta.
2) Pittsburgh, 1340 Mwe, rejecting heat to the San Francisco Bay
Delta.
3) Morro Bay, 1030 Mwe, rejecting heat to the Pacific Ocean.
Numerous other examples might be cited since the shoreline outfall
system has widespread use.
Submerged, offshore outfalls may be designed in two general
fashions:
1) a single port (dual in some cases) outlet situated either
vertical or horizontal, or
2) a diffuser section at the end of the pipeline consisting of
numerous ports. The diffuser is typical of municipal waste
outfalls.
Some examples of Targe vertical port outfalls are:
1) Moss Landing fossil fired plant. Reject heat from 1500 Mwg
generation, discharged about 800 feet offshore. Dual ports.
2) San Gnofre nuclear plant. Reject heat from approximately
450 Mw generation, discharged through a 14-foot diameter
pipe 2600 feet offshore, about 15 feet below sea surface.
3) Redcmdo Beach fossil fired plant. Reject heat from 1612 Mwg
generation. Two offshore outfall systems: a) two 10-foot
diameter pipes discharging vertically about 2100 feet offshore;
and b) a single 14-foot diameter pipe discharging vertically
300 feet off, about 16 feet beneath water surface.
11
-------
4) El Segundo fossil fired plant. Reject heat from 1020 Mwe
generation. Two offshore outfall systems: a) two 10-foot
diameter pipes, discharging 2100 feet offshore, vertically,
20 feet beneath ocean surface; and b) two 12-foot diameter
pipes, discharging 2070 feet offshore, vertically, 20 feet
beneath ocean surface.
To this author's knowledge, no large power plant uses diffusers for off-
shore ocean discharge at present, although such a system is proposed for
the Shoreham plant [95], discharging to Long Island Sound.
2.1.3 Hydrodynamic Regimes
Experimental observations of forced plumes issuing from submerged
ports have revealed the existence of four distinct flow regimes, as
follows (Figure 2.3):
• Zone of flow establishment (jet flow)
• Zone of established flow (mixed flow)
• Transition from established to drift flow, and
• Zone of drift flow.
The zone of flow establishment is in effect a transition zone from pipe
flow to an established forced plume. Consider fluid issuing from an
outfall port of diameter D (Figure 2.3), to the surrounding ocean, with
a turbulent velocity profile. For the sake of analysis, this profile
is usually assumed uniform with velocity v . Immediately the velocity
begins to deteriorate at the flow boundary as a result of turbulent
mixing with the surrounding ocean water. This region of mixing spreads
both inward toward the center of the plume and outward into the sur-
12
-------
SEA SURFACE
VELOCITY
PROFILE
NOMINAL PLUME
BOUNDARY
(v > v )
^ me — o'
POTENTIAL CORE
OUTFALL
PORT
Figure 2.3 Vertical Thermal Plume in Deep Water,
Illustrating Possible Flow Regimes
13
-------
roundings. Within a short distance, ze, from the outfall port, the
interchange of momentum due to mixing has spread to the center of the
plume. At this point, it is generally assumed that the plume vertical
velocity profile is fully developed, or established.
In the zone of established flow, velocity profiles are approxi-
mately similar at all axial locations and the driving force may be
either initial momentum, buoyancy, or both (mixed flow). As distance
from the outfall increases, the effective width of the plume and the
amount of plume flow increases as a result of lateral mixing or turbu-
lent diffusion (commonly called entrainment). Momentum of the plume at
successive cross-sections is changing according to the density differ-
ence between the plume and surroundings. Maximum velocity, v , of the
plume will decrease, except if the buoyancy is large compared to initial
momentum, in which case the maximum velocity may increase momentarily
near the outfall.
The transition from established flow to drift flow is caused by
the plume encountering the ocean surface or by the plume attaining a
neutrally buoyant condition in a density stratified sea. Here velocity
profiles change drastically with essentially all mean vertical motion
vanishing. The motion at the transition zone termination may be dom-
inated by prevailing ocean currents.
In the zone of drift flow, prevailing ocean currents will generally
dominate the plume motion, although a lateral density flow will persist
if the plume is situated on the ocean surface with buoyancy. Lateral
mixing is dominated by ocean turbulence, whereas vertical mixing depends
on both the plume and environment driving forces.
14
-------
Under certain conditions, all of the above hydrodynamic regimes
will not prevail. For instance, in the case of a large diameter port
issuing in shallow water, the zone of established flow will most likely
be absent. This situation is usually termed a "confined plume"
(Figure 1.4) and the hydrodynamics are characterized by a continuous
transition from pipe flow to drift flow.
An example of a typical confined plume is the thermal effluent of
the Southern California Edison power plant located at San Onofre,
California, discharging approximately 15 feet beneath the sea surface.
The port is vertical and 14 feet in diameter. Based on experiments by
Albertson, et al. [4] concerning neutrally buoyant jets, this depth is
less than the length for flow establishment.
For shoreline outfalls, the same flow regimes exist. However,
the zone of established flow may be less distinct depending on the
relative magnitudes of initial momentum and buoyancy (initial densi-
metric Froude number). This zone will assert itself if buoyancy is
small or initial momentum is large. In the case of small initial
momentum and moderate or large buoyancy, the initial mixing zone will
be a continuous transition from the outfall to drift flow without
established flow in the sense of similar velocity profiles.
2.1.4 Oceanographic Effects
The nature of the surrounding ocean can have a dramatic effect on
the behavior of a thermal plume. Probably the most influential of these
oceanographic variables are the following:
15
-------
SEA SURFACE
POSSIBLE
VELOCITY
PROFILE
NOMINAL PLUME
BOUNDARY
POSSIBLE POTENTIAL CORE
OUTFALL PORT
EXTENT OF TRANSITION
REGION
Figure 2.4 Vertical Thermal Plume in Shallow Water, Illustrating
Continual Transition of the Flow Field
-------
• density stratification,
• currents, and
• turbulence.
2.1.4.1 Density Stratification
In all discussions concerning ambient density stratification,
stable stratification is implied. One effect of stratification is
stabilization of the ambient flow field insofar as vertical convection
and mixing are concerned. However, the discussion in this chapter will
be confined to the direct effect of limitation of height of rise for
plumes issuing from submerged outfalls.
The maximum height that the thermal plane will attain (and whether
the plume will reach the surface or not) depends largely on the ambient
density structure. Obviously, this discussion does not apply to con-
fined plumes, but to cases where the outfall port size is small com-
pared to the ocean depth, as for example, diffuser ports. Both theory
and experiment have shown that the plume will always reach the surface
if the ocean is homogeneous with respect to density.
The ocean, however, is rarely homogeneous, except perhaps in very
shallow coastal waters where good vertical mixing occurs. The reason
that a thermal plume may not penetrate to the ocean surface in a density
stratified environment is that the plume entrains the heaviest water
nearest the outfall. This water causes dilution to some extent and is
carried upward with the plume. As the plume ascends, the density dif-
ference between the plume and surroundings steadily decreases because
the flow is being diluted and cooled through entrainment, and because
17
-------
the density of the surroundings is decreasing upwards.
If the density stratification has sufficient magnitude (among
other considerations which will be discussed later), the plume will
eventually reach a level of neutral buoyancy some distance below the
water surface. At this point the flow continues upward only by virtue
of the vertical momentum it possesses at that point. As the plume con-
tinues upward, it continues to entrain liquid that is now less dense
than the plume flow; hence, the flow is negatively buoyant. Eventually,
all upward vertical momentum is lost and, since the plume liquid is
denser than the surroundings at that depth, the pollutants will cascade
downward around the upward flow.
Small oscillations in the vertical motion will follow and when
these oscillations vanish the plume is said to be "trapped" (Figure 2.5),
At the trap level all mean motion is horizontal since the flow is
neutrally buoyant (assuming that environmental isosteric surfaces are
horizontal).
2.1.4.2 Effect of Currents
Currents have a dramatic effect on plume behavior in nearly all
flow regimes. The types of currents that might have influence are tidal
currents, longshore currents, upwelling, wind driven surface currents,
and persistent currents that are peculiar to a certain locale.
The zone of flow establishment is essentially unaffected by cross
currents; but, in the zone of established flow (deep water), a cross
current will cause the plume to be "bent-over" (Figure 2.6). The most
significant effect of this bending is a decrease in the height of rise,
18
-------
also, the dynamics within the plume are changed.
When the plume is bent over, two distinct counter rotating vor-
tices are formed (Figure 2.6). These vortices are quite apparent in
atmospheric smoke plumes discharging into a cross wind; the same
phenomenon occurs in the ocean.
In the drift flow regime, the plume flow is carried along with
the ocean current nearly as though it were the ambient water. Thus,
ocean currents play a dominant hydrodynamic role on the eventual fate of
the pollutant. Upwelling causes a persistent offshore surface current,
SEA SURFACE
^ESTABLISHED
VELOCITY
PROFILE
OUTFALL PORT
BOTTOM
Figure 2.5 Possible Configuration of a Vertical Buoyant
Plume in Stratified Receiving Water
19
-------
SEA SURFACE
ro
o
(AMBIENT .
CURRENT)
VELOCITY PROFILE
SECTION A-A
COUNTER ROTATING
VORTICES
Figure 2.6 Possible Configuration of a Buoyant Plume in Stratified
Receiving Water with Cross-Current, u
-------
thus, a surface plume could be carried out to sea. Wind driven surface
currents and tidal currents can cause the pollutant to be carried on-
shore or out to sea, and longshore currents can cause the pollutant to
be distributed along the shoreline.
2.1.4.3 Ocean Turbulence
The origin of oceanic turbulence is not fully understood,
although in the surface zone it is probably caused mostly from winde
generated wave action. As such, the turbulence is neither homogenous
nor isotropic, and only the gross behavior can be described.
Ocean turbulence has some effect on all regimes of plume flow.
Turbulence scales that are on the same size or larger than the plume
cross-section will have an effect similar to a crosscurrent, and all
scales should have some influence on the plume entrainment rate
(although it is thought that the influence is small in all zones except
the drift regime, since turbulence generated by the plume dominates
the ocean turbulence). In the zone of drift flow scales of motion
larger than the flow field result in action similar to oceanic currents,
and the pollutant field simply flows along with the turbulent motion.
Smaller scales of motion add to the eddy diffusion of the pollutant;
thus, as the pollutant field spreads, larger and larger scales of eddy
mixing come into play.
Another factor complicating oceanic turbulence is that it is
highly anisotropic, at least in the larger scales of motion. Since
most oceanic waters are density stratified to some degree (except per-
haps in shallow water), vertical mixing is suppressed to a great
21
-------
extent. Thus, a pollution field diluted by eddy diffusion will spread
much more rapidly in the lateral direction than in the vertical.
2.1.4.4 Air-Sea Interactions
Wind and heat transfer are the major air-sea interface phenomena
which may significantly affect thermal plume dynamics. Wind stress at
the sea surface causes two local effects which have previously been men-
tioned: wind driven surface current, and turbulence. And, on a larger
scale, wind is responsible for coastal upwelling. We will only point
out these wind stress effects here and refer the interested reader to
such references as Neumann and Pierson [63] or Wada [107] for addi-
tional details and references.
Heat transfer at the interface is carried on by atmospheric con-
vection, radiation, and evaporation. Evaporation is probably the most
significant of these modes and is materially affected by the surface
temperature and conditions in the atmospheric boundary layer such as
temperature, humidity and turbulence. Again, wind plays an important
role here through promotion of atmospheric turbulence and convective
currents. Radiation heat transfer depends on the sea surface tempera-
ture and albedo, atmospheric conditions such as turbidity, and position
of the sun.
The effect of surface heat transfer is more complicated than
merely heating or cooling of the plume. For instance, if heat is lost
at the surface, convective downcurrents of cooler water may occur,
tending to homogenize the plume vertically. If heat is gained at the
surface, the plume will become more stable and suppress vertical mixing.
22
-------
Atmospheric heat transfer will affect the plume dynamics predomin-
ately in the drift flow regime when the plume is situated at the surface.
The area exposed to the atmosphere in the surface transition (zone 3)
is small on a comparative basis and will likely be unaffected by sur-
face heat transfer.
2.2 Plume Analysis State-of-the-Art
There has been a great deal of theoretical and experimental work
carried out in the past 20 years or so dealing with the dynamics of
buoyant plumes. Most of this work has dealt directly with either atmos-
pheric smoke plumes or ocean plumes caused by submerged offshore indus-
trial and municipal waste outfall systems: (cf. Baumgartner and Trent
[12]). Much lesser and more recent efforts have treated horizontal shore-
line discharges (cf. Stolzenbach and Harleman [94]). More basic studies
concerned with turbulent transport quantities in jet flow have also
received much attention.
In this section we will briefly outline the state-of-the-art and
past studies dealing with plume calculations. Table 2.1 summarizes a
good share of the work related to plume investigations both theoretical
and experimental. This table is by no means all inclusive and the
particular categories may not be completely descriptive of the work
accomplished in the cited references. However, it does serve to illus-
trate where research emphasis has been placed on problems which are
related both directly and indirectly to thermal outfall analysis.
A brief discussion of Table 2.1 will be given separately for
submerged and horizontal shoreline outfalls.
23
-------
TABLE 2.1. SUMMARY OF WORK PERTINENT TO OCEAN OUTFALL PLUME ANALYSIS
Principal
Investigator
Albertson
Albertson
Balnes
H1nze
Schmidt
Rouse
Priestley
Priestley
Morton
Morton
Aoranam
Abraham
Fan
Fan
Keffer
Cederwall
Brooks
Tomlch
Zeller
Jen
Tamaf
Hayashl
Sharp
Frankel
Saml
1 Stolzenbach
Hart
Bosanquet
Hoult
Wada
Wada
Manabe
Okubo
Okubo
Leenderste
Ramsey
Fay
Murota
Masch
Fox
Murgal
Hirst
Schmidt
Hirst
Scorer
Morton
Csanady
Anwar
Abraham
Turner
Rawn
Harremoes
Tulln
Baumgartner
Ref.
4
4
8
41
85
31
73
74
60
58
1
j
26
27
50
17
16
99
112
48
96
38
88.89
30
83
94
37
14 •
45
106
108
56
64
65
53
75
29
62
57
28
61
44
86
43
87
59
22
5
2
102
76
36
101
11
Application
Sub. Jets
Sub. Jets
Sub. Jets
Sub. Jets
Thermals
Thermals
Plumes
Plumes
Thermals
Plumes
Waste outfall
Jets
Waste Outfall
Waste Outfall
Plume
Waste Outfall
Waste Outfall
Jets
Thermal Outfall
Thermal Outfall
Thermal Outfnll
Thermal Outfall
Thermal Outfall
Waste Outfall
Jets
Thermal Outfall
Waste Outfall
Waste Outfall
Plumes
Thermal Outfall
Thermal Outfall
Thermal Outfall
Dispersion
Thermals
Tidal Hydraulics
Heated Jet
Plumes
Jets
Tidal Hydraulics
Plumes
Thermals
Plumes
Smoke Plumes
Plumes
Plumes
Plumes
Plumes
Waste Outfall
Waste Outfall
Waste Outfall
Waste Outfall
Waste Outfall
Waste Outfall
Waste Outfall
Geometry
c
3
S
4, iS«
u *-*"5
•o a o a:
C (/> J=
§t/i o
- -V
°^ 'to *n ~~*
Q C C >
£S8S
tjrt.=
X L
X L
X L
X L
L
L
X L
X L
L
X L
XX L
L
XX L
X L
X L
XX L
X L
X L
X
X
X
X
X L
X I
X
X
X L
X L
X L
X
X
X
NA
L
NA
X
X L
X I
NA
X L
L
X X L
X L
XX L
X L
X L
X L
XX L
X L
X L
X L
X L
X L
X L
Type of
Flow
^ r- f—
IO-Q-
C r— t-
c C c t_>
01 4> C I/I
o *0 0* t/i
X X
X X
X X
X X
X X
X X
XXX
X X
X X
X X
A A A
X X
XXX
X XX
X XX
XXX
X X
X X
X X
X X
X X
X X
X X
X X
X X
X XX
X X X
XXX
X XX
X X
X X
X
X
X X
X
X X
X XX
X X
X
XXX
X X X
X X X X
XXX
X X X X
X X X
XXX
X X X
X X
X X
X X
X X
X X X X
X X
nncipai
Zone
Inves-
1 gated
r- ^
C C C C
o o o o
Mt-JMM
X
X
X
X
X
X
X
X X
X
X
A A
X
X
X
X
X
X
X
X
X
X
A
X
X X
X X
XXX
X
X
X X
X X
X
X
X
X
X
XXX
X X
XXX
X
X
X
X
X
X
X
X
X
X
X
XXX
X
X X
X
+J
«
o
^
u
o
C «J
I £
i- X
£•=
X C
UJ <
X X
X X
X
X
X X
X X
X X
X X
X
X
A A
X
X X
X X
X X
X X
X
X X
X X
X X
X X
X X
X
X
X
X X
X
X X
X
X
X
X
X
X
X X
X
X
X
X X
X X
X
X
X
X
X
X
X
X X
X
X X
X
X
X X
Solutlor
let hods'
I/I
rt
s.
is
c
o
c
g
o5
° 1.
• 4-1 O
t— ia ^.
O 3 -0
(/i a* c
UJ ro
4J C 1—
T -2 3
f2 -3 T
"ifi O ul
X
X
X
X
X
X
X
X
A
X
X
X
X
X
X X
X
X
v
X
X
X
X X
X
X X
X
X
X
X
X
X
X
X X
X X
X
X
X
X
X
X
X
X
X
-D
1n1te
Iff.
c
o
^
o
c
u.
gi
Is
S x
£
(0 •«-
II
X
X
X
X
X
X
*The L/D ratio applies only to submerged outfalls: S, I/O <5; I, 5
-------
2.2.1 Submerged Outfalls
For submerged outfalls the depth of discharge dictates the method
of analysis. Deep water cases are substantially simpler to analyze
than the shallow water counterparts (at least in the absence of cross
currents) which is a result of the applicability of similarity solutions.
Similarity analysis has expedited the theoretical analysis in this zone
and resulted in mathematical models that are sufficiently accurate for
engineering calculations.
Zone 1 has received substantial attention but is of minor impor-
tance in deep water analysis because it is a relatively short-distance
effect (approximately six port diameters or less). Most of the work
involving this zone has been carried out in the absence of buoyant forces.
Abraham [1] presents a mathematical model for cases where buoyant forces
have a significant affect on the zone length. Recently, Hirst [43]
has presented a more thorough analysis.
There has been essentially no theoretical work done for zone 3 of
the deep water plume (i.e., near the surface or in the region of the
maximum height of rise). It is generally assumed that the similarity
solutions of zone 2 hold in zone 3; but, this is a very poor assumption.
Frankel and Gumming [30] have shown through experiment that this is the
case. Sharp [88,89] has experimentally investigated the surface spread
of a hot water plume, and Murota and Muraoki [62] have investigated the
effect of a free surface on plume hydrodynamics.
Very little theoretical work has been done on deep water plumes
in the presence of a crosscurrent. This lack of effort is undoubtedly
25
-------
a result of the solution difficulty since similarity principles are not
strictly valid for this case. However, Fan [26] has treated the cross-
flow problem for a vertical plume using similarity assumptions and
obtained reasonable results. There are serious theoretical questions
concerning the use of similarity profiles in the presence of a cross-
current. Hirst [44] presents analysis for crosscurrents which includes
a stratified ambient medium. Various experimental studies coupled with
dimensional analysis have been carried out for the crossflow problem,
but as yet no generally proven computational model has been published
which relates details of the plume dynamics.
Deep water plume analysis is particularly applicable to waste out-
falls having small ports, common to diffuser systems. Typical submerged
thermal outfalls such as those off the Southern California coast cited
by Zeller and Rulifson [113] utilize very large, vertical single ports.
The amount of receiving water between the port and sea surface may be
on the order of 1-3 port diameters. No published theoretical studies
have treated plumes with such L/D ratios. In this case zone 2 does not
exist and there is no delineation between zones 1 and 3. All that may
be said is that the flow undergoes a transition from pipe flow to drift
flow.
The following general conclusions are made concerning submerged
outfall state-of-the-art computational models.
1. Acceptable computational models are available for deep water
plumes except for;
• Zone 3, the surface or maximum-height-of-rise transition
zone, and
26
-------
• plumes issuing in crosscurrent (existing models to be
proven).
2. There is no acceptable computation model or technique available
for shallow water plumes such as those typical of large thermal
power plant outfalls.
2.2.2 Horizontal Shoreline Outfalls
Horizontal shoreline discharge is also utilized by a number of
thermal power plants throughout the United States. Table 2.1 illustrates
that there has been only modest effort made to analyze this problem.
From a mathematical modeling standpoint the horizontal surface discharge
of a thermal plume is extremely complex since the phenomena involved
are inherently three-dimensional (the same is true for horizontal sub-
merged ports in shallow water, and the case of a crosscurrent in deep
water).
In spite of the three-dimensional aspects of the shoreline plume,
various solutions have been formed using similarity princples (e.g.,
Zeller [112], Jen, et al. [48], Hayashi, et al. [38], Tamai, et al.[96]
and Stolzenbach, et al. [94]. Except for the work of Stolzenbach,
none of these methods are, in this author's opinion, acceptable for
engineering computations. Before a completely acceptable model is con-
structed for general application, three-dimension flow characteristics
will need to be accounted for in some manner along with crosscurrent
effects.
27
-------
2.3 Work Description
The previous section delineates several areas of outfall analysis
which need attention. As a practical matter it is not feasible to
incorporate all of these areas into a general mathematical model which
would apply to all outfall configurations and oceanographic conditions.
The scope of this manuscript is limited to vertical plumes.
We are primarily interested in large single port vertical thermal
outfalls issuing in shallow water (Figure 2.4). Typical existing
configurations are those located at Moss Landing, San Onofre and
Redondo Beach, cited earlier. However, the ultimate objective of the
work is to provide a complete program which mathematically models the
temperature and velocity distribution in a vertical thermal plume, from
outfall port to the drift flow regime (zone 4), regardless of ocean
depth. The transition region, as defined here, refers to any part of
the flow field for shallow water plumes. This region is the portion of
the program which must be treated by finite-difference techniques and
constitutes the principle effort of this work.
In addition we also set down the difference equations appropriate
for a line plume, but do not include these in the modeling program.
In summary, the work covered by this manuscript deals with the
problem of mathematically modeling velocity and temperature distribu-
tions in the locale of vertical thermal outfalls. The techniques for
analysis are as follows:
• Shallow water plumes : finite-differences
28
-------
• Deep water plumes
1. Zone 1 : existing empirical
2. Zone 2 : similarity solution
3. Zone 3 : finite-differences
The primary task described in this manuscript is the finite-dif-
ference application to the confined plume and computation of the entire
flow field dynamics for zones I, II, and III. The circulation of the
ambient is also included. Although there have been various related
studies, none have dealt with the numerical solution of a confined,
vertical plume and radial surface spread. Tomich [99] numerically
modeled the compressible free jet problem, Ma and Ong [55] investigated
an impulsively started momentum jet, but paid little attention to the
more complicated features of the dynamics. Recently, Pai and Hsieh [68]
have carried out numerical work with laminar jets.
29
-------
CHAPTER 3
TRANSPORT EQUATIONS - GENERAL THEORY
In this chapter the fundamental laws and equations which govern
marine hydrodynamics and energy transport are set down. We begin by con-
sidering the fundamental equations for laminar, incompressible flow and
modify these equations so they are appropriate for marine considerations.
These equations are written in various forms which are appropriate
for later discussion concerning theory review, similarity solutions, and
numerical considerations.
3.1 Coordinate System
The governing differential equations are given in Cartesian
tensoral form with coordinates x- (Figure 3.1). For analysis of the
local sea, the geopotential surface is assumed to be flat.
(EARTH ROTATION)
VERTICAL:
EARTH
Figure 3.1
Rectangular
Coordinate
System
30
-------
3.2 Conservation Laws
The differential equations governing the heat and momentum trans-
port of a thermal plume in the oceanic environment may be derived from
the following physical laws:
• Continuity (conservation of mass)
• Newton's Second Law (conservation of momentum), and
• The first law of thermodynamics (conservation of energy)
In addition, an appropriate equation of state is needed to relate sea
water density in terms of local temperature and salinity.
i
Detailed derivation of the primitive conservation equations will
not be discussed here but may be found in such texts dealing with fluid
dynamics (cf. Bird, Stewart and Lightfoot [13], Welty, Wicks and
Wilson [115], Hinze [40]). A few modifications of the standard form of
the conservation equations must be made so that they apply in general
to a thermal plume in the sea. These modifications are chiefly con-
cerned with turbulent approximations, incorporation of coriolis effects,
and the Boussinesq approximation concerning small density variations.
Additional detail concerning these approximations may be found in
standard references dealing with marine hydrodynamics (cf. Hill [39],
Phillips [70]) and the general subject of turbulence (e.g. Hinze [40]).
The primitive equations appropriate for our analysis are pre-
i
sented in Cartesian tensoral form as follows:
^insteinian notation is used where repeated indices imply summation
over all three index values (i = 1,2,3).
31
-------
Continuity:
The operator D/Dt in the above equation is the substantial deriva-
tive and has the usual meaning:
D_ _ 3_
Dt 3t
where t is time and u. is velocity along the j coordinate. In Equa-
J
tion (3.1) the quantity p is density.
Momentum:
Du • sp 3Tii
+ <3-2'
where n* is the component of planetary angular velocity along the j
J
coordinate, P is pressure, g is the local gravitational constant and T-.
' J
is the fluid molecular stress tensor. The symbol e- .. is the usual
cartesian permutation tensor which takes values of zero if any two of
the three subscript are identical, +1 for even permutations and -1 for
odd permutations. The symbol 6.. is the Kronecker delta which is equal
' J
to 1 when i = j, and otherwise 0. Coriolis effects are incorporated
*
into the momentum equation by the term e- .,2pu^n- and, according to
the specified coordinate system, (Figure 3.1) gravitational forces act
only along the x7 direction; hence, 6.. = 6.,.
•j i j i j
In any fluid dynamic system, variations of density may cause
fluid motion due to the action of gravity. In the ocean, these density
variations may be caused by temperature differences and variation of
local salt content, or concentrations of other materials whether in
32
-------
solution or not. Hence, in lieu of the heat transport equation we will
consider at this point a transport equation for a general scalar
quantity, r, where r may be heat, salinity or other dilute transferable
constituents. The r transport equation is:
Dr _ 9 / 3T \ . : /., ,x
~Dt - a*7 (KY^7J ' (3'3)
Constituent sources, sinks and dissipative mechanisms are incorporated
in the term $ and the symbol < is the molecular diffusion coefficient
for the r quantity.
3.2.1 Continuity
In the ocean, and especially in the case of the thermal plume,
-»
the density field, p, varies with both space and time,
P = p(xist). (3.4)
However, essentially all density variation is caused by distributions
of heat content, salinity, etc., as opposed to compressibility effects
(i.e. high speed compressible effects). The local density anomally is
very small compared to the local value of density, and the conservation
of mass (Equation 3.1) may be approximated with sufficient accuracy by
the volume continuity equation
9U.
l - 0. (3.5)
We point out here that although f^- = 0 may be an acceptable approxi-
mation with regard to mass conservation, this quantity cannot be
33
-------
ignored in the momentum equation (see Section 3.3), and is precisely
the coupling between momentum transport and r transport.
3.2.2 The Equations of Motion for Turbulent Flow
Within the framework of assumptions concerning continuous fluid
properties, constant gravitational force, and negligible earth curva-
ture, the momentum transport equations (3.2) are valid regardless of
the nature of the flow or fluid. The usual additional assumptions in
hydrodynamics are that the fluid is Newtonian, incompressible and that
Stokes viscosity relationships are a valid description of the fluid
stress rate-of-strain (cf. Welty et al.). Thus, the stress terms
(Equation 3.2), T.., may be replaced by
' J
3U,
J
where y is dynamic viscosity.
For the purpose of treating turbulent flow, it is assumed that
the velocity components, ui , pressure, P, and density, p, are composed of
mean or average parts and superimposed random fluctuating parts
(cf. Hinze [40]). Symbolically,
ui ="i +">
P = P"+ P', and
p = p + p',
where the everbar represents mean of values and the prime, random
values. These definitions are substituted into the equations of
motion and the result is time averaged term-by-term over a sufficiently
long period of time to obtain
34
-------
Du . a
or —. u'iu + eijk«juk = - 37: -'96i3 + ^r- (3.7)
I
which is seen to be identical in the mean motion with Equation (3.2)
except for the appearance of the term
A new quantity is now defined:
which is called the Reynolds stress. Finally the complete equations
of motion in the rotating Earth reference frame are written as
for the mean flow. Here the overbars denoting average quantities have
been omitted since mean, or average, quantities are implied. The tur-
bulent stress terms may be related to mean flow quantities through the
Prandtl mixing length theorem (cf. Neumann and Pierson [63]) to obtain
Terms envolving fluctuations of pressure and density have been ignored,
35
-------
R.. = -pu'.u'. = PC.:; 37- • (3.10)
1J 1 J j
Hence, using Equations (3.6) and (3.10) in (3.9) yields
(3.11)
i
/Du. \ ap s - • 3Ui-
. I L + ?p 0*11 = - —-—
PI n+- T teT i lx"iul/ / av
where e.. is the eddy diffusion coefficient for momentum, a second order
' J
tensor, and v is kinematic viscosity.
In the case of a thermal plume, e. -»v except where velocity
' J
gradients are small and the flow has strong stratification. We will
assume that e.. includes molecular viscous effects and write the momen-
' J
turn equations as
Du,
DT
3.3 The Boussinesq Approximation
In this work, four quantities of density are defined as follows:
• P = p(x.,t), the density at a point in the thermal plume.
• P = P00(x3), the density distribution which would exist in
the local sea in the absence of the plume.
« pr = Constant, a reference density for the receiving water.
« pQ = Constant, the density of the effluent issuing from
_ the outfall port. _
1 The summation convention for repeated tensoral indices does not apply
to underscored indices in this text.
36
-------
The density distribution of the reference ocean, P^xOj is assumed to
be independent of time and vary with x, alone.
Buoyant forces on a fluid element are established by the density
difference
Ap = p - p^ . (3.13)
So that,
P = PTC + Ap. (3.14)
According to the Boussinesq approximation, (cf. Phillips [70])
when density variations, Ap, are small,(i.e. |Ap/p|«l) these varia-
tions may be ignored as they influence inertial and viscous terms in
the equations of motion, but must be accounted for in the gravitational
term. In view of Equation (3.14), the equations of motion may be
written
Du-
Now, let P° be the pressure difference between a point in the plume
and outside the plume located on the same geopotential surface, so that
. L. . g
-------
Here, we have assumed that the pressure distribution in the reference
ocean is hydrostatic. Hence, Equation (3.15) may be reduced'to:
Du.
JL e..Zl) (3.17)
3xj \
Equation (3.17) is the so-called "advective" form of the equations of
motion. This name has become popular among oceanographers and meterol'
ogists and is so called because the convective terms are expressed in
the form u.au./ax..
J ' J
The convective terms may be written in slightly different form
by noting that
au .u. au • au .
ax.
J
= u.
J ax
u . — d-
1 ax.
J
However, by Equation (3.5)
3u.
j ax..
J
= 0,
so that for an incompressible flow
J ax.
J
Thus, Equation (3.17) may also be expressed as
at
9PC
au.
'i3
(3.18)
38
-------
which is called the "conservative" form of the equations of motion.
3.4 The Pressure Equation
Equation (3.17), or (3.18) contains four unknown quantities;
u-pUpjUoj and P°. Since only three scalar equations are involved, an
additional relationship is required.
An equation for pressure, P°, may be derived by taking the diver-
gence of Equation (3.17). This operation yields:
__
9t
Q 3(p«,-p)
3X.3X.
LI,
J 9X:
9 f 9 9U1 1
ax. [ax. eij 3x.J
= 0.
311.
(3.19)
By continuity
ax.
0,
so that Equation (3.19) is reduced to
92P°
9X.9X-
9U-J
vJ
9X.
J
ax. / \ 9x.
1 ' x J
j 9x.
where B is the buoyancy parameter, defined as
ax,
(3.20)
(3.21)
(3.22)
39
-------
For the case where coriolis forces are neglected and quantities involv-
ing derivatives of eddy viscosity are small compared to other terms,
the presssure equation is
(3'23)
3.5 r Transport
The r transport Equation (3.3) may be modified for turbulent flow
by considering the transported quantity, r, to be composed of a mean
part, r, and a fluctuating part, r', or
r = f + r' .
Then in a manner analogous to the method applied to the equations of
motion, the turbulent r transport equation becomes
or _ s / sr\ , •
Dt ' 8x7 1 £Yj 377 1 $ ' (3.24)
Where e • is the eddy diffusion coefficient and is assumed to include
i J
molecular effects.
3.5.1 Transport of Heat, Salinity and Buoyancy
Letting r = T, in Equation (3.24), where T is temperature, the
heat transport equation is
- 8 9T
Hi 8y
HJ X
(3.25)
40
-------
In this case * corresponds to heat sources and sinks and/or viscous
dissipation. Since none of these effects are significant in an ocean
•
plume, * is neglected. The quantity eH- is the turbulent heat diffusion
coefficient and is assumed to include molecular effects. For salt trans-
port, we let r = S, when S is salinity; hence,
DS _3/3S\ / •} ne\
Dt - W. (tSjJT}' (3'26)
J J
Salinity is a conservative property, thus i is omitted. The quantity
e^j is the combined molecular and turbulent mass diffusion coefficient.
The equations for heat and salinity transport are coupled to the
Equations of motion (3.17) or (3.18) through the buoyancy term
(pep-p)/Pr. For that matter, any r constituent, which when transported
in the system of interest causes density variations to occur, is
coupled in the same fashion. Thus, it is not the absolute value of
temperature, salinity, etc., which is important to the system dynamics,
but resulting density variations in a lateral plane caused by the trans-
port of these quantities. For this reason it is necessary only to deal
with the transport of buoyancy 1n analyzing the dynamical behavior of
the system. However, we must solve the equation for heat or salinity
transport (in a system where differences of salinity and temperature
are the causes of density variations) in order to establish the magni-
tude of temperature and salt content, and to treat certain boundary
conditions. Once the density and temperature (or salinity) distribution
is known, salinity (or temperature) may be calculated from an equation
of state for sea water.
41
-------
A "density transport" equation may be derived by combining Equa-
tions (3.25) and (3.26) [assuming that an equation of state, p = p(S,T)
holds] after the independent variables, T and S, have been changed to p
Hence,
fi. = _L_ (. to- ) + . ' - (3 27)
fit ax. U. ax, ' £P pax I lax.) {6^J}
P. ax
where
(3.28)
A buoyancy parameter may be defined as
Pr " p
pr ' po
and the appropriate transport equation for AI is
(3'29)
nt aV ^t^-i 1w^ ' t -il av I I 3^ /
Ut dX- pj dX- Pj\c'x-i/ \ i '
A second parameter A2 which incorporates p^ may be defined as
A2 = A! - ( r _ °° ) = — ' (3.30)
The transport of A is described by
42
-------
u
-- __
Dt 3 3X3 ax. pj 3X. ' " 3X3 p3 3X
J J 3
where p* = pM/(pr - PQ)
If density is a linear function of both temperature and salinity, that
p - PO = _a (T - TQ) + b(S-So),
then ^ = constant and H/3x. = o.
J
Equations (3.27), (3.29) and (3.21) become
3A1
, and
respectively.
43
_ _ — c . - , an /o oo\
Dt = ax. VEPJ ax. ; (3.33)
J J
-------
The quantity c, is seen to be a correction term which accounts for
nonlinearities in the equation of state, p = p (S,T). As it turns out,
sea water density does vary approximately linearly with salinity
(See Section 3.6) so that p = f (T) for. constant S.
In the remainder of this manuscript, A-| and A2 will be referred
to as
pr - p
- — ! - ' buoyancy parameter,
pr - po
- P
• A2 = - = AJ - AI , density disparity parameter.
-
Pr - PO
The motivation for defining two buoyancy quantities is that it is more
convenient to use A-, in the numerical analysis (Chapter 5), whereas Ap
is convenient for similarity analysis. For consideration of salinity
transport, a third buoyancy term is defined as
S - S
Sr ID'
where Sr and SQ are the reference and outfall effluent salinities,
respectively.
Figure 3.2 illustrates the relationship between the quantities
and A-| and A2 at elevation X3 = constant.
44
-------
CO
UJ
UJ
Qi
X2
Figure 3.2 Relationship Between the Buoyancy Parameter, A-, and
Density Disparity, Ap
3.6 The Equation of State for Sea Water
The density of sea water is a function of pressure, temperature
and salinity, in the absence of other pollutants. Hence, the equation
of state has the form
P = p(P,S,T). (3.35)
Since we are dealing only with rather shallow water on an oceanographic
scale, pressure effects are negligible; therefore,
P = P(S,T). (3.36)
45
-------
If other contaminants, having concentration, C, are present, then
P = p(S,T,C) . (3.37)
In this work, we will deal only with Equation (3.36).
Since density variations are small in the sea, oceanographers
deal with a modified density called sigma-t, defined as
ot = (p-1) x 1000,
which has cgs units and is a measure of the deviation in density from
1.0 gm/ml . The equation of state in general use by oceanographers may
be found in U.S. Navy Hydrographic Office publication number 615 [103]
(or in Hill [39]) and has the form:
°t = Et + (ao + J324) [1 - At + Bt (00
where
It •
- (T - 3.98) „ T + 283
503.370 T + 67.26
At = 10" T(4.7867 - .098185T + .0010843J2)
_6
Bt = 10~~T(18.030 - .8164T + .01667T2)
= -.093 + .8149S - .000482S2 + .0000068S2,
46
-------
In the above equations,! is in degrees Celsius, and salinity in
parts per thousand. The quantity a is the density of sea water, in
sigma-t units at zero pressure and temperature, a is usually expressed
in terms of chlorine content instead of salinity, S, but for purposes
here, salinity will suffice.
3.7 Vorticity Transport - An Alternate Approach
In dealing with geophysical fluid dynamic problems it is frequently
difficult, if not impossible, to set realistic boundary conditions
required for the solution of Equation (3.21). Pressure, and consequently
associated boundary conditions, may be eliminated entirely from consid-
eration by introducing the quantity, vorticity.
A brief summary of the general theory will be presented here for
a homogeneous, isotropic turbulent flow field (i.e., e^- = e = constant)
* V
in three dimension. Additional information concerning vorticity trans-
port may be found in Batchelor [10].
As demonstrated by Batchelor, a conservative fluid velocity field
may be defined by vector addition of an irrotational contribution, Uj
and a solenoidal contribution us, or
u = Uj + us. (3.39)
The solenoidal part satisfies
v • us = 0
whereas the irrotational part satisfies
v x U = 0.
47
-------
In addition the irrotatlonal part of the velocity field, Uj, may be
described in terms of a scalar potential » so that
UT = V*
and the solenoidal part in terms of a vector potential, $, or
jj = v x ^ .
Hence, the total velocity field is described by the vector and scalar
potential as
u • 7* + 7 x $. (3.40)
Vorticity, u>, is defined as
U) = V X U.
Taking the curl of Equation (3.40) and use of the above expression for
vorticity, ytelds
u • v x (7x$). (3.41)
However, by vector identity
v x (vxf) = 7(7-$) - v2? ,
which for an incompressible flow gives
v * " -w (3.42)
since 7«^ = 0.
Equation (3.42) is a Poisson type partial differential equation
relating the vector potential to the distribution of vorticity in the
flow field.
48
-------
The divergence of Equation (3.40) gives
2 -»•
7 4 • V- U
In view of the incompressibility condition,
7 • u =0,
and satisfies LaPlace-'s equation
72$ = 0. (3.43)
Hence, the velocity field may be established through solution of
Equations (3.42), (3.43) and (3.40).
Hirasaki and Heliums [42] have shown that Equation (3.43) is
extremely useful for the purpose of prescribing inflow-outflow boundary
conditions in a three dimensional velocity field. In fact, they have
demonstrated that the flux boundary condition may be prescribed
entirely by the scalar potential, * (velocity potential), or Uj.
Hence, one is permitted to set tangential components of $ = 0 and the
normal derivative of $ = 0 at all boundaries. The utility of this
theory lies in the fact that vector potential boundary condition may
be intractable without consideration of the scalar potential, *. One
exception is the case of flow in a closed system where the boundary
conditions on $ remain as described above and since there is no
boundary mass flux,7$ = 0 everywhere (cf. Aziz [7]).
An equation for vorticity transport may be derived by taking the
curl of the Equations of motions (3.17) (after setting e.. = e). This
' J
operation yields
->• A i,
49
-------
where e3 is a unit vector in the vertical direction.
The vorticity transportation equation was simplified appreciably
by assuming a homogeneous, isotropic turbulence field. If the turbu-
lence field were not treated as such, numerous terms involving the
gradient of &•• would appear. These terms will be investigated in
' \j
Section 3.10, which covers two-dimensional flow fields. The two
dimensional counterpart to Equation (3.44) is
§| = V x Be3 + eV2u , (3.45)
where one coordinate is vertical (x.,) and the other lies in the
lateral plane.
3.8 Non-dimensional Form of the Equations of Motion
A non-dimensional formulation of the equations of motion permits
the investigation of the magnitude of the various forces exerted on a
fluid element in terms of similarity parameters. The importance of
the various parameters may then be analyzed on an order-of-magnitude
basis and the results used to justify simplification of the governing
equations under certain flow conditions. To this end, we define the
following dimensionless variables:
ui = W
P* = P°Pr/AP0,
** *
t* = tv0/D,
Xi = xi/D>
ij = eij/eo-
(3.46)
50
-------
In the above,
v - reference velocity (for the thermal plume we will use the
effluent velocity at the outfall port),
AP - reference dynamic pressure (may be taken as 1/2 p v ^)
f - characteristic coriolis parameter
D - characteristic length (may be taken as the outfall port
diameter)
e - characteristic eddy diffusion coefficient for momentum
(may be set to Cv D, where C is a constant).
Substituting the set (3.46) into Equation (3.18) yields,
3Ui 8UjU' /fo°\ ** / APo ^ 3P*
The dimensionless groups in Equation (3.47) are:
vn2
YJ- = Ro, Rossby number (ratio of inertial forces to coriolis
o
forces),
2
Prv
P = Eu, Euler nipber (ratio of inertial forces to pressure
forces),
vn2 .
— F , densimetric Froude number (ratio of inertial forces
\ n
)gD
/
. , to internal buoyant forces),
51
-------
VD
= Re-,-, turbulent Reynolds number (ratio of inertial
eo
forces to turbulent shear forces).
In terms of the above similarity parameters Equation (3.47) becomes
J
T_ 3P* + l_fi. + J L_(e*
Eu 3X-; Fn "i Rej 3X-; \ la~---i
i o i j — j
(3.48)
Equation (3.48) represents a gross non-dimensionalization.
Ideally, we should treat each component of momentum separately and use
length scales which correspond to the particular coordinates. However,
for purposes here the form of Equation (3.48) is sufficient.
At middle latitudes, the characteristic coriolis parameter, f , is
approximately equal to 10, and vQ/D has magnitude on the order of 1
for a large outfall part. Hence, the Rossby number for the thermal
plume is on the order of 10,000. Where smaller ports are considered
vQ/D may be from 10 to 100, giving Rossby numbers from 105 to 10 .
The densimetric Froude number, F , for a large thermal outfall will be
on the order of 10-50 and the reference Reynolds number ReT will be of
the same order. Also, we cannot neglect pressure effects. All other
terms are on the order of 1 except eddy coefficients in some portions
of the flow field. Hence, it follows that for a thermal plume and the
scales of motion to be considered here, the coriolis term is suffic-
iently small to neglect by virtue of the apparaent size of the Rossby
52
-------
number. In consideration to follow we will deal with the equations of
motion in the general form of
aUj + 8UjUl = ] ap* ,
+ Re? a!j WSTjf* (3.49)
and dimensional variations of the same.
3.9 Further Comments on the Concept of "Eddy Viscosity"
In Section 3.22, we introduced velocity fluctuation, u., as a
means of describing turbulent flow. Without the coriolis term,
Equation (3.9) is known as Reynolds' equation, after Osborne Reynolds
[78] who first expressed the turbulent equations of motion in this
fashion. Reynolds' equation for the mean flow differs from the laminar
flow counterpart only by the Reynolds stress terms, R.^
1 J
The Reynolds equation represents a vast simplification (at least
outwardly) of extremely complex flow conditions. However, the task
still remains in relating the turbulent or "apparent" stresses to mean
flow quantities.
Boussinesq (cf. Hinze [40]) was evidently the first to use the
concept of "apparent" viscosity, in his studies of two-dimensional flow.
He assumed that turbulent stress, T could be expressed in a manner
analogous to molecular viscous stress or
i = -p u1 v" = e -3— • (3.50)
53
-------
In the above, e is the "apparent" or eddy viscosity, u1 and v' are x
and y components of the velocity fluctuation, respectively, and u is
the mean velocity in the x direction.
Prandtl [72] introduced the concept of "mixing lengths" to
describe the turbulent exchange coefficient. This idea was motivated
by the mean free path concept of molecular motion and has turned out
to be a fruitful hypothesis in spite of obvious physical questions.
The idea of mixing length theory is that a small parcel of fluid
containing any transferrable property is transported, unchanged, by
velocity fluctuation from one position, a distance £ to a new position
where it is absorbed in the flow field. The distance a is the mixing
length.
Let u-|(x-| ,x2,x3) be the mean velocity at the origin of the
exchanged fluid parcel, and u-)(x1 + a^ ,x2 + £2»X3 + ^3) be tne mean
velocity at the absorbed position. Then the velocity fluctuation is
(cf. Neumann and Pierson).
3U-, 3U-. 3U-,
oXn t 0X0 O dXn
du,
Then u = -£
Ul(2) = - £2 a3cj (3.51)
au1
UK3) = - £3 9x7
54
-------
I
Here, the fluctuating velocity ui /.» is shown as a second order ten-
sor where the subscript j indicates the particular turbulent component
i
of u-,. Hence, in a somewhat nebulous fashion:
— 9U-
ui> = eii air • <3-52)
J U aXj
Mixing length theory is rather unsatisfying because of the physical
basis; nevertheless, it does accomplish the purpose of relating mean
flow behavior to the Reynolds stresses. Actually, the concept of an
eddy viscosity requires a fourth order tensor quantity (Hinze, Pond
[71]) to satisfy theoretical treatment of the Reynolds stresses.
Such a quantity would be completely unmanageable from a practical
standpoint. Even the second order tensor e.. is difficult, if not
impossible, to calculate from measurable quantities such as frictional
forces and velocity gradients.
Hot wire and laser techniques offer a method for direct mea-
surement of the fluctuating velocities and hence correlation of the
Reynolds stresses through statistics. However, statistical theory has
not yet provided a means for evaluating e-. in practical engineering
' J
calculations.
As a result of our lack of understanding and inability to cal-r
culate or measure E.., further assumptions must be made. In the ocean
' J
we must deal with at least two values of eddy viscosity, a lateral
value and a vertical one. Gross measurements have shown that these
two values are vastly different. Fofonoff (cf. Hill) suggests using
a form from Saint-Guily which gives
55
-------
Rij = - ' - ej^r + ei a*?' (3-53)
J '
where e- is the lateral eddy viscosity for i,j^3 and the vertical
J
for i,j=3.
For the work presented in this thesis, we will use three
components given by e..
J
3.10 Two-Dimensional Forms of the Transport Equations in Rectangular
and Axisymetric Coordinates
In the previous sections of this chapter, the appropriate differ-
ential equations for solving the thermal plume problem in three-space
were layed out. Ideally, we would prefer to solve the plume problem in
this manner since the nature of the flow is distinctly three-dimensional
However, computational requirements necessary to obtain proper resolu-
tion of desired quantities in three dimensions are prohibitive from a
practical standpoint in view of available computer hardware and
economics.
Two-dimensional considerations which demand significantly less
computation time and computer capacity, are appropriate in cases where
flow symmetry is approximately realized. Such cases are the vertical
plume and line thermal investigated in this thesis. Hopefully, compu-
tation economics will permit practical, three-dimensional engineering
calculations in the near future, thus avoiding certain restrictions
inherent with two-dimensional approximations. Table 3.1 gives a sum-
mary of general requirements for two- and three-dimensional forms of
the velocity-pressure and Vorticity-Vector potential equations.
56
-------
TABLE 3.1. DIFFERENTIAL EQUATIONS REQUIRED FOR VELOCITY-
PRESSURE AND VECTOR POTENTIAL-VORTICITY
METHODS IN TWO AND THREE DIMENSIONS
Velocity-Pressure
Equation Set
Vector Potential-
Vorticity
Equation Set
ul
U2
U3
0
P
r
ul
(Oo
(i)-5
fl
\l/
m
$
Parabolic
Parabolic
Parabolic
Elliptic
Parabolic
(1 or more)
Parabolic
Parabolic
Parabolic
Elliptic
Elliptic
Elliptic
Elliptic
3-Dim. 2-Dim. 3-Dim. 2-Dim.
i=l,2,3 1-1,2 i=l,2,3 1=1,2
X X
X X
X
X X
XX XX
X
X
X X
X
X
X X
(X)*
Total of Required
Equations (minimum)
7(8)
*Used only in the case of open boundaries.
57
-------
3.10.1 Two-Dimensional Transport Equations in Rectangular Geometry
The two-dimensional rectangular coordinate system which will be con-
sidered in this study is defined as a plane normal to the geopotential
surface (Figure 3.1). The two coordinates are defined as x and z, where
x is in the xl5x2 plane, with no particular orientation, and z is aligned
with the vertical x3 axis. Corresponding velocity components u and v are
in the x and z directions, respectively.
Velocity-Pressure Equations:
The velocity-pressure equations are as follows.
Continuity:
9u , 9v n
9x 9Y = U (3.54)
Momentum transport:
x-direction,
DU _ 9P° , 9 f 9ul . 3 f 9u]
Dt " "ax 9X leX3xJ 3Z~ [ez 3?J » (3.55)
Dy_ _ 9P° + g + 3_ L ivj + 3_ f 3v_j
Dt 9z 9X ( X 3xJ 9Z [ Z 9zl' (3.56)
In the above momentum transport equations, ex is the lateral eddy diffu-
sivity coefficient and ez is the corresponding vertical value. The
substantial derivative is given in two dimensions as
T» = "- + V^-.
Dt 9X 9Z
Constituent transport:
P^T1 ^ll ^iT1! Cif 'NT1!
L/l O I Oi I i^l Ol !
Dif = "9x"(eYx^j + 9! [EYZ ilj • (3.57)
58
-------
Equations for the transport of specific constituents such as A,, A2, S,
etc. will be developed where appropriate.
The appropriate pressure equation may be obtained from Equation
(3.20) by letting i=2,3 and j=2,3. Hence
V2po= (lii)2
V K ;
IT
where v2 = + •
9X2 3Z2
If turbulent contributions are neglected,
{<\iio l^\i Ci 11 si w i ? I ^ R
/ o U \ ^ o/^*\/^^\ _i / i v -i- ^ ^*
Recall that by Equation (3.16),
P° = P/Pr - 9 J2°Pco/Pr dz'
The most notable work in obtaining numerical solution to the
laminar form of the velocity-pressure equations given above was per-
formed at the Los Alamos Scientific Laboratory by Welch and colleagues
(cf. the "MAC Method" [109]). Based on these pioneering efforts at
LASL, numerous other investigations have employed MAC techniques to
viscous flow problems [6, 23, 46]. Pagnani [67] applied the MAC
59
-------
techniques successfully to natural circulation in an enclosed cell.
Stream Function - Vorticity Equations:
An expression for the stream function in (x-z) coordinates may be
obtained by considering only the x3 component of the vector Equation
(3.43), or
/* = - u, (3.60)
where: v (stream function) = f- and w = cn\
Dt = - 97 + £V w (3'64>
where again we let to = u>3.
However, in general we must consider the two anisotropic, nonhomo-
geneous components ex and ez. In this case, numerous terms involving
derivatives of ex and e,, appear. The vorticity equations are derived
for this case by cross differentiating Equations (3.54) and (3.56), ther
subtracting the latter result from the former to obtain
60
-------
Dw_ !§. + 92t»
Dt " 3X GX 3X'
3e.
+
!!>L i!y. + I!*, alii IfiL i!y. l!z.i!v
82 '9x2 3Z *3Z2 " 8X '3X2 " 3X '3Z2
3 f9ex 3U _, 3eZ 3U
3Z [3X 3X 3Z 3Z
3 f9ex 3V ^ 9ez 3V
3X [3X 9X 3Z 3ZJ (3.65)
If the structure of the turbulent field is homogeneous, and isotropic,
Equation (3.65) simplifies to
= . + e
Dt 8x X
Stream function-vorticity transport solution methods have been
employed for a number of years by oceanographers in computing such geo-
physical phenomena as western boundary currents (e.g. the Kuro Shio and
the Gulf Stream, cf; Neumann and Pierson). But these techniques have
become popular in engineering application only in the past few years, a
result due in part to the recognition that these methods are extremely
well adapted to problems involving natural convection. Solution to the
laminar form of the stream function-vorticity equations given above
have been carried out by a number of researchers [7, 31, 82, 100, 104,
106, 108, 111]. The most notable work being carried out on the turbulent
form of the equations is at the Imperial College by Spa!ding and
61
-------
colleagues [69, 82, 90, 91, 92, 93].
3.10.2 Two-Dimensional Transport Equations in Axisymmetric Coordinates
Again referring to Figure 3.1, the axisymmetric coordinate system
is oriented such that the radial coordinate, r, may be considered a
rotating line in the x,, x2 plane. The vertical coordinate, z, is again
aligned with the x3 direction, normal to a geopotential surface.
Velocity-Pressure Equations:
The velocity-pressure equations are as follows:
Continuity:
1 9urr av
FIT + If ' °' <
where u is the radial velocity.
Momentum transport:
r - direction,
.
Dt 3r 3r V r 9r ) 9z ^ z 2zj ' (3.68)
The substantial derivative in axisymmetric coordinates is:
2_ = JL+u 3_+ v 1- (3.69)
Dt at r ar az
z-direction:
Dv 8P° , n , 1 9 L av] . a f av'i fo
+ B + -TTT re,- 7— + — e, —- , ^.
Dt 9z ' " ' r 3r lcr 9r ' 3Z -z 3z
62
-------
In the above equations, er is the radial eddy diffusivity coefficient
for momentum.
Constituent transport:
Dr _ 1 8 (re arl , 3 f
Dt - ?37lr%r arj + al [EYZ
where evr is the radial coefficient for turbulent r diffusivity.
1 i
The pressure equation may be derived by differentiating Equations
(3.68) and (3.70), then adding these two results to Equation (3.68).
Hence,
n p p
u] faul [3V| + 2 au . avl (3<72)
j (3rJ [9zJ 3z 3rJ v 7
where the operator
V2 = ll. + ll_ + iL. . (3.73)
3r2 r 3r 9z2
Vorticity in (r-z) coordinates is given as
.
3z 3r
Also we define a stream function $ according to
u = _ 1 M. (3.75)
r r 3z
and
(3.76)
63
-------
Substitution of Equations (3.75) and (3.76) into Equation (3.74)
yields
£i_ l|t + l!i = . ru (3.77)
for the stream function *. Note that Equation (3.77) is not the usual
Laplacian for (r-z) coordinates (e.g. Equation 3.73).
The vorticity transport equation is derived by cross-differ-
entiating Equations (3.68) and (3.70) and then subtracting the latter
result from the former. This operation leads to
3u^ + u^" + 3vai _ 3B
3t 3r 3z 3r
3Z2
3z Lr 3r 3r 3z 3z J 3z 9r Lr 3r
r + 3v_ ;
32^ dr Ldr 9r 9Z 3Z
°-lL . _r_ 3_ T3v
3z 372 3r L9r
. I!r .fiiy. +1 IV.T , 3ez . 32v
3r 'Lar2 r 3rJ
3r aZ2 (3.78)
If the turbulent structure of the flow field is homogeneous, and isotro
pic, derivatives of tr and ez vanish and the vorticity transport
equation becomes
3B
3t 3r 3z 3r r 3r
r 3r I -* 9z2 •
(3.79)
64
-------
CHAPTER 4
PLUME THEORY - SIMILARITY SOLUTIONS
As an integral part of the thermal plume dispersion program, this
chapter is concerned with flow regimes 1 and 2, which may adequately
be described by empirical correlations and similarity solutions.
4.1 General Description
The zone of "flow establishment" (Figure 2.3) is a region of
transition from essentially a pipe flow at the outfall orifice to a
fully developed velocity profile some distance downstream. This
situation occurs only in deep water, and when velocity profiles
become fully developed, the flow field is said to be "established."
This zone is characterized by velocity profiles which are very similar
in shape at each axial location.
The zone of flow establishment is a region of intense turbulent
mixing between the plume flow and surrounding water. The mixing
process which starts at the periphery of the outfall port spreads
inward toward the center of the plume and outward into the surround-
ings. Eventually mixing will spread to the plume centerline where
the centerline velocity will begin rapid diminution. Upstream from
this point, flow in an approximate conical section is relatively
unaffected by the mixing process. This zone is called the "potential
core" and is characterized by relatively flat velocity profiles at all
axial locations.
Figure 4.1 illustrates a precise change from one flow regime to
the next. In reality, however, before the velocity field becomes
65
-------
VELOCITY
vg TEMPERATURE
v > v
m o
B. SMALL F
A. LARGE F
Figure 4.1. Zone of flow establishment for plumes with large and small densimetric Froude numbers, FO
-------
fully established in the sense of similar velocity profiles, the
centerline velocity will begin to deteriorate giving a transition
zone between the two regimes. This transition is apparent from the
data of Albertson et al. [4]. Although Murota and Muraoki [62] have
proposed a correlation for this zone, according to Hinze [40] this
distance is relatively short and is generally excluded from analysis.
In the case of a momentum jet (neutrally buoyant flow, or
FQ -»• «>) velocity in the potential core is that of the issuing jet and
analysis is based on the assumption that momentum is conserved at
each axial cross-section. However, in the case of buoyant plumes,
momentum is generated by the density disparity and velocity will
actually increase in the potential core (as indicated in Figure 4.2B).
As mentioned previously, the zone of established flow is
typified by velocity profiles which have nearly the same shape at all
axial locations. For this reason similarity analysis has played an
important role in analysis of this flow regime. Numerous experimental
and analytical studies have been carried out for both the momentum jet
and buoyant plume in the absence of restraining boundaries.
In this manuscript, the work of Albertson and Abraham [1] is
used for modeling Zone 1, and Abraham's work for the established flow
regime is extended for the analysis of Zone 2.
4.2 Simplified Equations for a Vertical Plume
Governing equations for a vertical plume issuing from a round
port are more convenient to derive in axisymmetric coordinates. Thus,
with reference to Figure 2.3 and the coordinate system given in
67
-------
NOMINAL
PLUME
BOUNDARY
Figure 4.2. Coordinate system for axisymmetric vertical pi
ume.
68
-------
Figure 4.2, the following assumptions are posed:
• steady flow
• flow is axisymmetric
• coriolis effects are neglected
3D0
. flow field is assumed hydrostatic throughout: if— = 0
a Z
. density difference between the plume and surroundings is
assumed small compared to the density at any point in the
flow field: IP^-PI « p
• plume is fully turbulent
• eddy transport of momentum and heat is only effective in
the lateral direction (normal to jet axis)
• molecular heat conduction and viscosity are ignored.
With the above simplifications and assumptions it is possible to
disregard a number of terms in cylindrical governing Equations (3.69)
through (3.73) and arrive at the following equation set:
Continuity:
3r
Momentum:
Employing "order of magnitude" analysis common to boundary layer
theory (e.g. Schlichting [84])and incorporating previous assumptions,
we see a need for the z-direction momentum Equation (3.72) only. This
69
-------
equation reduces to
where trz is the turbulent shear stress.
Energy transport may be accounted for by the appropriate
axtsymmetric form of the density transport Equation (3.34) or
(4.3)
9A 9A 9p*
u
r ar
For salinity we use Equation (3.71), with r = A,
\ .
/
9 fe 9A3 I (4.4)
r 3r 3z r 3r \ Sr 3r
with the buoyancy parameter, A,, defined as'
- S
A, -^
3 Sr - SQ • (4.5)
Using the continuity relationship Equation (4.1), Equations (4.2),
(4.3) and (4.4) may be rearranged to yield the following:
1 -i rr ) - v _ _ 1 _L J E !_21» (4.7)
r 9r ^ ur 2' 3z r 3r 1 Hr 3r
70
-------
fi <"3> + F If K 43' - F If | ESr *T \ , (4.8)
respectively.
4.3 Radial Velocity and Temperature Profiles
A large amount of experimental work has been carried out in the
past concerning radial velocity and temperature profiles for free
jets. Earlier work was concerned primarily with momentum jets.
Schmidt [85] in 1941 was evidently the first to consider the mechanics
of convective plumes, such as convective currents over fires, etc.
Schmidt's work was reported in the German literature, and apparently
because of the war, went unnoticed until Rouse et al. [81] carried out
similar work in the early 1950's. Since then a number of researchers
[8, 26, 41, 77, 83] have investigated velocity profiles and associated
transport coefficients for both momentum jets and buoyant plumes.
4.3.1 Zone of Established Flow
The experimental studies have established that velocity and
temperature profiles are approximately similar at all axial locations
in the zone of established flow for all vertical plumes in a stagnant,
free environment. Also, profiles are nearly Gaussian and may be
adequately described by the normal distribution curve:
1 (L\2
v(r,z) = vme~ 2 V (4.9)
for velocity, and
71
-------
.
e(r,z) = eme" 2 V (4.10)
for the temperature distribution. In the above equations the sub-
script m refers to condition at the plume center!ine, A is the eddy
Prandtl number, and a is the standard deviation.
The standard deviation has been found to relate to the vertical
coordinate, z, by
2
2 K
a2 =i | (4.11)
where K is an experimental entrainment parameter. Hence,
K^2
v(r,z) = vme'K (z> (4.12)
and
K x n2
e(r.z) = eme-K x V. (4.13)
It is important to remember that these profiles have no theoretical
basis and are merely the result of curve fitting.
The values K and x must be determined by measurement and have
been found to depend on the extent of buoyancy. For instance, in
the case of a simple plume (pure buoyancy, FQ=0) Schmidt found that
K = 48
x = 1.2 .
The data of Rouse yields
K = 96
and
x = .74
72
-------
for a buoyant point source. Abraham in his analysis of a simple
plume used values
K = 92
and
A = .74.
For the momentum jet case, (neutral buoyancy) Albertson found
K = 77-
Abraham used
A = .80
for this case.
Baines [8] observed in his investigations that the initial
Reynolds number affected the results. He found the following best
fit for his experimental results:
r N
v(r,z) = vme'K (I} (4.14)
where K = 43.3 and N = 1.82 for Rfi = 2.1xl04, and K = 64.4 and N = 1.84
for Rfi = 7x1O4.
Where values for K and A are needed in the present work, the
following are used:
simple plume (pure buoyancy, or FQ^0),
K = 92
A = .74, and
momentum jet (neutral buoyancy, or FQ -> °°),
K = 77
73
-------
A = .80
4,3,2 Zone of Flow Establishment
Figure 4.3 illustrates a typical velocity distribution in
this zone. Albertson estimated this distribution for a momentum jet
by assuming a flat profile across the potential core and a Gaussian
distribution for the mixing zone. Albertson derived an integral
expression for momentum flux across a lateral plane in this zone by
integrating Equation (4.6) with p = p, for r = 0 to r -»• », or
/CO
v!
M „ o = (4J5)
M ./ A
0 V ft
00
The quantity M is total momentum flux crossing a plane normal to the
mean flow and A is cross-sectional area. Thus, Equation (4.15) states
2
that momentum is conserved with M =v A the momentum source strength.
By letting C, = a/z, the momentum flux relationship above yields
^e _ 1 (4.16)
D ~ 2CTJ~
where C-j is an experimental constant. By approximating the potential
core diameter, D , according to
!!c - I 1 (4.17)
D
the mean velocity distribution in this region takes the form
74
-------
-------
V
Equation (4.18) above will be used in the following work when a
velocity distribution near the outfall port is required. Note that
this equation is not correct for buoyant plumes since density dif-
ferences have been ignored. However, very near the outfall port (say
one port diameter downstream), inertial effects are assumed to domi-
nate the flow behavior regardless of the degree of buoyancy. Evalua-
tion of the empirical constant C, and the length z are dealt with in
the next section.
4.4 Zone of Flow Establishment
For a plume issuing from a small diameter port in deep water
the length for flow establishment, z , has relatively small influence
on conditions far downstream except as it enters in the established
flow solutions as a boundary condition. On the other hand, for large
outfall ports, the theoretical zone may extend over a good portion
of the flow field, or even to the ocean surface. In this section, we
will discuss methods for evaluating z in deep water for both the
neutrally buoyant and buoyant cases.
Many experiments have been carried out by various investigators
in an effort to establish the length of the potential core for turbulent
round jets issuing into stagnant fluids. Good reviews of this work
are given by Hinze [40] and by Gaunter, Livingwood, and Haycak [32].
Gaunter et al. in their review, state that values for z /D vary from
76
-------
about 4.7 to 7.7. For Instance, Albertson et al. found that z = 6.2
for their work. Baines reports that jet Reynolds number had
considerable effect on z /D for his experiments. In fact for Re =
1.4xl04, ze/D = 5 and for ReQ = 105, ze/D = 7.
Where buoyancy affects the potential core length, Abraham bases
z on the concentration distribution. Hence, z for concentration is
given by
where x and K take values .8 and 77, respectively. The limiting value
of z /D in Equation (4.19) for F -> °° is approximately 5.6. The value
of z /D for concentration profile establishment is about 10% less than
the value of 6.2 for velocity profiles found by Albertson.
4.5 Governing Differential Equations
To derive the equations governing the dynamics of a vertical
plume in Zone 3 we integrate Equations (4. 6), (4. 7) and (4.8), in a
lateral plane, from r = 0 to r -> °°. Thus, the following expressions
apply as indicated,
Vertical momentum transport:
•/
(4-20)
77
-------
Density disparity transport:
*
9p
vA,rdr - —^- J vrdr = 0 (4.21)
o Z 3Z o
Salinity or concentration transport:
_
dz / vA,rdr = 0 . (4.22)
•r 0 »J
Equation (4.20) may be written in terms of A_ by rearranging the gravi
tational contribution to yield,
fi*> r.
A /
rdr ' 9 " 'rdr (4-23)
Integration of Equations (4.21) through (4.23) may be completed by
utilizing profiles given by Equations (4.12) and(4.13) (4.13) for A,
and A2- Hence, the resulting expressions are
Vertical momentum:
3 3
K
Density disparity:
,-V2 ^ (4.25)
K dz
78
-------
Salinity or concentration:
K(A + 1) 4"
Cast in dimension! ess form, the above equations become
where,
Z = z/D
V = v /v
m nr o
(v)
(4.26)
»*•
dR*
dT
A _ T_ (4.29)
1m —4 £J/3Z
E* = -_ni_' (4-30)
E*l/3
R* = _ A2mZ , and (4.31)
VK (1+A)
Alo= K
79
-------
4.5.1 Initial Condition
The solutions of Equations (4.27) and (4.28) are begun at
1=1 , or in the beginning of the established flow regime. Abraham's
relationship (4.19) may be used to evaluate this distance for the
entire range of densimetric Froude numbers, F . Once Zg is known,
the initial values E*fi and R*e may be established. We assume that
ambient stratification may be neglected over Z then
A
2
dp *
oo
HZ"
= 1, and
= c .
"e
Hence, by Equations (4.30) and (4.31)
R * -
Ke " 4- • (4.32)
The initial value of E* may be found by considering Equation (4.30).
For large initial Froude numbers (F -*• °°) V •> 1 , so that
0 IDS
(4.33)
However, for low FQ, Vme is typically larger than 1 and unknown. To
avoid estimation of Vme, we use Equations (4.29) and (4.30) with
A2me = lj to obtain
80
-------
Ee* •
e
vaiiA) (4.34)
For large F , Equation (4.34) reduces to
Ee* 'T ; (4'35)
in which case Zg = 5.6. This result agrees with Equation (4.19).
4.5.2 Evaluation of Terms Involving K and A
Listed in Table 4.1 below are limiting values of K and A as
suggested by Abraham along with limiting and mean values of terms
involving K and A.
TABLE 4.1 VALUES OF TERMS INVOLVING K AND X
Term
K
A
1+A
Momentum
Jet
(F -*• col
ir0 + »;
77
.80
.256
Simple
Plume
(FQ - 0)
92
.74
.245
Mean
Value
1/4
Max.
Error
J%L
2A%
.114 .104 .109 4.8%
.253 .239 .245 2.9%
v/K(A+l)
Using convenient values for the above terms, the governing equations
and initial conditions are:
dE* _ 3^_ Z R*. (4.36)
o
81
-------
dR*
.nzE*1/3
~dT
(4.37)
(4.38)
im
Z E*
1/3
and the initial conditions are:
64
F * = —o
Ee ,
Re* = 1/4 .
4.5.3 Homogeneous Receiving Water
(4.39)
(4.40)
For the case of homogeneous receiving water the above equations
may be solved analytically since dp^/dZ = 0. Therefore, from Equa-
tion (4.37)
R* = 1/4,
and Equation (4.36) becomes
dE* 32
dT~ " T6F '
o
Equation (4.42) may be integrated immediately to yield
(4.41)
(4.42)
'
(5*
2 21
-ze]
} •
(4.43)
Centerline concentration is then given by Equation (4.41) as
i i r\
73
im
_/Z3
•[64
64
2 -, 2,
- Ze J
(4.44)
82
-------
Apparently, the stratified case must be solved numerically.
4.6 Lateral Velocity, u
Once the plume centerline velocity, v, has been calculated,
and the lateral distribution of axial velocity has been established,
it is a simple matter to calculate u from the continuity equation,
r 8r 9r * (4.45)
Since v(r,z) is known, Equation (4.45) may be written as
[urr] = rf(v) (4.46)
or
ur = - f f(v) xdx. (4.47)
J(\
83
-------
CHAPTER 5
FINITE-DIFFERENCE MODELS
The finite-difference models developed in this chapter are applic-
able to the following two situations:
. Vertical round ports issuing into quiescent receiving
water, and
• Line plumes which may include ambient current effects.
From a practical standpoint, the vertical round port in shallow water
is of foremost importance because this configuration is typical of pres-
ent and planned installations. The line thermal model would find appli-
cation in analyses of the plume which develops over a diffuser line once
the individual round plumes have interferred with one another.
The numerical models are formed in two dimensions for steady flow
conditions. In the case of a vertical round plume, a two-dimensional
model will not accommodate any ambient cross flow which would destroy
the plume symmetry. Hence, the solution is strictly valid only during
slack tide conditions in the absence of prevailing local currents.
However, cross currents, tidal or otherwise, have little effect on the
initial mixing (near-port locale) of plume flow from large outfalls in
shallow water. The reason for this is that the effluent momentum
dominates the ambient flow. At the San Onofre outfall, data show that
isotherms in the near vicinity of the outfall are reasonably concentric
even in the presence of tidal currents [24]. In view of available data
it appears that a two-dimensional axisymmetric model for the vertical
round plume should give adequate results for the initial mixing region,
84
-------
in spite of ambient cross flow.
The line thermal model may accommodate ambient flow perpendicular
to the plume since in this case the phenomenon remains two-dimensional.
End affects are, of course, ignored in this case.
Difference models are based on the vorticity-stream function equa-
tion described in Chapter 3. Where the finite-difference solution is
started some distance above the outfall port, boundary conditions are
obtained from available data or similarity solutions as described in
Chapter 4. As indicated by Table 3.1, the minimum number of equations
required is three. We will also consider salinity transport so that
four partial differential equations are required, these being one
Poisson type equation for the stream function and a total of three
transport equations for vorticity and two r constituents.
5.1 Physical System for the Vertical Round Port
The physical system of primary concern is a large, single port,
submerged vertical thermal outfall issuing to stagnant receiving
water. Figure 5.1 illustrates this system in axisymmetric coordinates
(r, z). Later, conditions for a line plume will be discussed in an
appropriate cartesian coordinate system. The receiving water has
depth, L, and is assumed stratified with density pro(z). Flow enters
the system along the bottom boundary (z = zb) with some known velocity
and temperature distribution. In all cases to be analyzed the inflow
will occur only over a small portion of this boundary, which extends
from the plume centerline to a point rb, the nominal plume boundary.
For the shallow water cases, rfc = RQ the outfall port radius
85
-------
SEA SURFACE
RECEIVING WATER P°°
z INFLOW
BOUNDARY
O
CO
O
I
NOMINAL PLUME BOUNDARY
Figure 5.1 Physical System for Axisymmetric Vertical Plume
Where the Bottom Boundary is Some Distance
zb t 0 Above the Outfall Port
86
-------
(Figure 5.2). It is assumed that no flow crosses that portion of the
bottom boundary extending from r. to r^.
The plume centerline and ocean surface form no-flow boundaries,
or a reference streamline. A free-slip condition is assumed at the
ocean surface, but this surface is not allowed to distort vertically.
The flow boundary condition at r = r^ is free except that streamlines
are assumed to have constant slope. Flow will both enter and exit
over portions of this boundary; the exact distribution is a part of
the numerical computation. The mean velocity might be assumed wholly
horizontal since r^ is a large distance compared to r, , and since
density stratification will impede vertical flow. This assumption
would lead to level streamlines.
For shallow water geometry, (Figure 5.2) the ocean bottom is
assumed flat and z. = 0. The port side and ocean floor are assumed
no-slip boundaries.
5.2 Governing Differential Equations
For incompressible, turbulent flow in axi symmetric coordinates,
the differential equations describing continuity, linear momentum
and buoyancy transport were given in Section 3.10.2 and are reiterated
below.
Continuity:
1 1ML + 9v
r ar 3z
(5.1)
87
-------
PLUME CENTERLINE
= L
INFLOW
BOUNDARY
I I
t 4
SEA SURFACE
RECEIVING WATER
Pro (z)
OUTFALL PORT
OCEAN FLOOR (z b = 0)
o:
<
o
o
'CO
U- O
•o
i
UJ
on
Figure 5.2 Physical System for Shallow Water, Axisymmetric,
Vertical Plume
-------
where u and v are radial and vertical velocity components, respectively
(note that u is used instead of u as in Section 3.10.2).
Momentum transport:
r = direction,
ft'-ir *
z = direction,
In Equations (5.2) and (5.3) above, derivatives of tf and ez have
been ignored.
Buoyancy transport:
In lieu of the energy equation, the transport equation for AI is
considered,
DA, e 3A, 32A,
L - Pr £_ t~ L\ j. c L (e; A}
W ~ r 3r (r 3r ' £Pz 3Z2 ' { '
where again derivatives of the eddy buoyancy diffusivities, Epr and t^
have been ignored. The buoyancy parameter, A-J , as defined in chapter 3
is
A = r
pr ' po
89
-------
5.3 Vorticity Equations
For the problem at hand, it is more convenient to deal with
vorticity transport rather than linear momentum transport. In dealing
with vorticity, we need not be concerned about pressure and need to
consider one less partial differential equation. The appropriate
vorti city-stream function equations were given in Section 3.10.2 and
as a matter of convenience are listed below.
Stream function, Y:
2 2
3 * 1 3* , 9 * _ (5 z\
~ — ~ ~ ru P-3;
3r
Vorticity,
_ 3Vu) _ 3lB + 3_ / J_ 3gjr_x + 3 (u (3.79)
E e
_
at 3r 3z 3r r 3r r 3r
+
ez 2
where vorticity is defined as
3U 3V
Once having solved for the stream function distribution (Equation 5.5)
the velocity field is found by the relationships,
ll. 1C C\
(5.6)
and
v = - ^
r 3r
In the remainder of this work we will consider only steady flow. Hence,
the vorticity transport Equation (3.81) has the form
90
-------
r, / \ O
^p°°~ p' , 9/1 9^r\ , 3 o
a? — + Er a? ( FaF-} +£z ~
0 o Z
where B has been replaced by the definition Equation (3.22). Steady
flow transport of the buoyancy parameter A-, is given by
(r ; e
r 3r 3z r 3r v 3r ' pz 2 /5 gx
The convective terms in Equation (5.9) are in "conservative" form
which was obtained from Equation (5.4) through the use of the contin-
uity Equation (5.1).
In summary, the equations to be solved for the axisymmetric
plume dispersion are (5.4), (5.8) and (5.9) along with (5.5) and (5.6).
Equation (3.76) will be considered to evaluate vorticity boundary con-
ditions. To account for salinity transport (if applicable) a second
Equation (5.9) will be solved with A-J defined as a salinity parameter,
A3,where
3 Sr - SQ '
Temperature distributions may be calculated from the Equation of State
(3.38) once A, and A~ have been established. Hereafter only the con-
• 0
servative form of the transport equations will be considered. Although
the pressure distribution is not considered in this work, it could be
calculated through Equation (3.74).
91
-------
5.4 Dimensionless Forms
To cast the governing equations in dimensionless form consider
the following dimensionless variables:
R = r/r0,
Z - z/rQ, X
U - u/v0,
V = v/v0.
and, the dimensionless parameters:
r v
RE = , (radial, turbulent Reynolds number)
r v
RE = , (vertical, turbulent Reynolds number)
er
PR = •;— , (radial, turbulent Prandtl number)
r Kr
PR, = rr- , (vertical, turbulent Prandtl number)
2 Kz
FQ = — — - - (densimetric Froude number).
2 r g
Note that a second dimensionless vertical distance is used in this
manuscript defined as Z = z/D and should not be confused with Z.
92
-------
In the above definitions r is the outfall port radius and v
is the effluent velocity issuing from the port.
With these variables, the system of governing equations is written
as
stream function:
2 , 2
3 f 1 W , 3 y _ Do /c lnx
— 9 " D" W — 2" = - Rfi. (5.10)
8R2 R 9K 3Z
vorticity:
Note that in Equation (5.8) the Boussinesq term may be rewritten as
(Pr"p) +
PO 3r PO 9r pQ 9r
since p is a function of Z alone. Hence,
-f (un) + rf- (Vn) = - J p1
dr\ d Z. L.\ ^ t\
2 \
fi,l3fifi\,
" ~ ~
8R R Z 3Z
buoyancy parameter:
2 \ 2
3 Ai , 8Ai \ i 3 AI
11111 -7- (5.12)
REfPRz I 9R? R 3R \ REZPR2
93
-------
along with
and
u = -ff!i • (5J3)
t\ oL
"'IT Iff ' (5'14)
5.5 Coordinate Transformation
When solving partial differential equations numerically, it is
desirable to have fine grid space resolution where large derivatives
of the dependent variables are expected. In the present problem, a
fine grid spacing is needed in the radial direction near the outfall
port and plume centerline. At large distances from the centerline,
large grid spacing may be used since radial changes in the dependent
variables are expected to be small. To this end, a non-linear trans-
formation is employed on the radial coordinate, of the form
R = sinh 5 . (5.15)
This transformation has the desirable properties:
R = 5, AR = A5 for small R,
and
R = 75- e?, AR * |5- e5 for large R.
In terms of transformed coordinates, the governing equations are:
stream function:
sech
- (tanh £ +coth f) ~
94
~ = - sinh £ n . (5.16)
-------
vorticity:
sech
se
RE
sech'
2 tanh
Ml _ o coth2
Re.
(5.17)
buoyancy parameter:
|sech_U.l_
\sinh 5J K
sinh n UT
sech
3Z
REr PRr
sech . 3r
tanh £ ' 3?
3Z
2
(5.18)
Transformed expressions for velocity are given by:
U =
_^_
sinh g 3Z
(5.19)
V = sec.h € . 91
sinh 9
(5.20)
95
-------
Finite-difference calculations will be based on even increments of the
transformation coordinate, 5.
In the vertical direction, fine resolution is needed in the
region where the plume spreads laterally. In all thermal plume cases
of interest, this region is in the vicinity of the receiving water sur-
face. However, for other pollutants, such as municipal and industrial
wastes, lateral spread may take place below the surface and pollutant
concentration information is needed in the vicinity of this plane.
Since methods presented here are also applicable to these pollutant
plumes, a fine grid arrangement near the surface is not specified as a
general case. Rather, the vertical grid spacing will be treated as
node-wise variable and exact specification left to the discretion of
the computer program user.
5.6 Finite-Difference Grid System
The finite-difference grid layout consists of two grid systems.
One grid is usedto calculate the stream function, y, which provides
information to compute velocity components, U and V. This system
coincides with the physical boundaries and is illustrated by the wider
lines on Figure 5.3. The stream function is calculated at the interior
intersection points designated by the solid round symbols. Solid box
symbols represent boundary points.
Velocities are not calculated at these same points. The U com-
ponents are computed at vertical midpoints which are designated by open
circle symbols; whereas, the V components are computed at horizontal
midpoints U coordinate) and designated by open box symbols. In this
96
-------
COMPUTATIONAL POINT LEGEND
• -V; X-A, ft; O-U; D-V
BOUNDARY VALUE LEGEND:
• -V; O-A, ft
/ / f/f/ttf/ff/
Figure 5.3 Computational Grid for Difference Equations
97
-------
manner, the stream function grid layout defines a system of cells with
the stream function, f, computed at each corner point (or set by
boundary conditions, as the case may be) and velocities defined at the
center of the cell face (see Figure 5.4).
The second grid system is used to calculate vorticity, ft, and
buoyancy parameter, A,, (also A3) and is illustrated in Figures 5.3
and 5.4 by the narrow lines. This layout completely overlaps the
grid (and physical system) with interior intersection points centered
in the cells defined by,the f grid system. These interior grid points
are indicated by crosses with boundary values at cross-and-box points.
The reason this staggered grid system is used is for computa-
tional convenience in treating boundary conditions and to permit con-
vective transport terms to be evaluated at cell faces.
In Figure 5.3, the Y grid system is sized by NJ and NK grid
points in the £ (or R) direction and vertical direction, respectively.
The ft, A-| system has size NJ + 1 and NK + 1 in the respective direc-
tions. Points on the f grid are indicated by j, k, whereas points on
the n, A-| grid are indicated by p, q. In this figure, Z. defines the
bottom boundary of the stream function grid (physical boundary) and Zh
the top (sea surface). Vertical spacing for the system is defined by
AZ^ and may be variable. Grid spacing along the ? coordinate is even,
designated by A£. System boundary points for the n, A-, grid are
located at Zfa - ^ AZ2 for the bottom \ * \ AZNK-1 at the top> "I A?
on the left and ^ + ^ A£ at the right boundary, where ^ is the
assumed right hand physical boundary. Figure 5.4 also illustrates
98
-------
q+1
COMPUTATIONAL POINT LEGEND:
• - V x - r, Q
O-U D-V
j-l
j
P+l
<>
/
-c
\
)
-c
\
)
V L J
^ V.
,
} \
AZk
, ,
1 '
^ r
r
\ J ' [
P [
)J •*•! " N
)
1 [
\ N
^ w *
f A s
S
i \f
\ J,K
J 1
C C
3 f
^ f
r
i j'k c
5 L
\ J' N
J /
J.k-1
' L
^ ^
V '
X
k
3-
/
V
3-
^_
V
k-1
q-1
1-1
R.
j-l
R
P
R.
Figure 5.4 Typical Finite Difference Cell
Illustrating Indices for
-------
indices, computed quantities, cell size and radial distances for a
typical interior cell.
5.7 Difference Equations
Standard difference representation is used wherever possible in
this work. Central differences are used for both first and second
partials except for convective terms where a special donor-cell method
is used. Techniques for uneven spacing are used for the vertical
differences.
5.7.1 Stream Function and Velocity
Consider the stream function grid system illustrated in
Figures 5.3 and 5.4. The finite difference representation of Equa-
tion (5.16) based on central differences for both first and second
partials is as follows:
9
t_
sech 5- •,
V |
? r
sech" £_•
+
4.
J r
/
j
2 r
sech C-
j „
2
AC
1 1 +^nl>
i tanr
\
?, + coth cj • r
j j/ ^
tanh £. + coth r,l . ^k
J ^J/ 2
w
j+l,k
(5.21)
100
-------
In the above difference equation, the quantity fi- . is the average
J j K
value of a at point (j,k), hence the overbar. This average value must
be used since n does not lie on the ¥ computational grid points.
p »q
Vorticity is averaged for the four cells neighboring point (j,k) as
follows:
where
and
Velocity is calculated in first-order manner as
and
sech Cn
V = —=—;—2- (y. ,-y. , ,) (5.24)
j.k sinh £_AE v j.k j-1,k'
Thus far, we have discussed differencing the governing equations
only in transformed radial coordinate, ?. To permit more versatile
computation we also include provision for calculation directly in
(R.Z) coordinates. This is easily done by collapsing the hyperbolic
functions so that
101
-------
sinh 5 + £ = R, (also set tanh 5=0)
and
cosh £ -»• 1
giving
A? = AR.
Hence, for linear radial coordinates Equations (5.21), (5.22),
(5.23) and (5.24) collapse to
i-l,k
AZk(AZk+l +AZk) ^^
+ JAR n, . (5.25)
J , N
Velocity:
R-component
Z-component
V
J.K l]^2 "
102
-------
In Equations (5.25) through (5.27) sinh ?. is replaced by JAR
J
and sinh ? by (j - -~) AR.
5.7.2 Transport Equations
Except for the convective transport terms, central differences
are used to approximate all derivatives in the transport Equations
(5.17) and (5.18). Special consideration is given the convective
terms which involves basing numerical approximations on transport
integral techniques (see Appendix A).
Referring to the p,q grid system illustrated in Figures 5.3 and
5.4 the difference representation of the steady flow vorticity trans-
port Equation (5.17) is written as (after collecting terms)
coth
2
REzAZk
1
[AZk+AZk+l
, 1
AZk+AZk-l_
|Sech ?p
RErA£2
sech
v. , I + v. , +
p.q
sech E
P
2AE
I
|U. , , + U. , ,
1 1 -1 V 1-1 K
J 1 » l» J ' J ^
sech 5p
9
^ A?sech Si
1 2 tanh £
p-l.q
Equation (5.28) continued on next page.
103
-------
sech
sech
AZk+AZk-l
'p.q-1
2AZ,.
sech
IE .
A1p+l,q"Alp-l,q
Wl (5.28)
The turbulent Reynolds numbers, RE and RE , in the above difference
equation are point variables of the form RE (p,q) and RE (p,q).
Derivatives of these quantities are neglected in the above equations
but are accounted for in the computations.
Equation (5.28) may be collapsed to radial coordinates in the
same fashion as illustrated in Section 5.7.1. Hence, in non-
transformed radial coordinates the vorticity transport difference
equation, after collecting terms, is (note that numerically p = j-1/2)
104
-------
I
[2AZk j.k-l j,k-l REzAZk
Z (|Vj,kl ' Vj,k) +REZ" (AZ n
p,q+l
(Vl.q " Vl.q) ' (5-29)
The convectlve terms are formed In a manner such that vortldty con-
vected out of cell (p,q) has the value n_ _ and vortldty flowing Into
P«H
the same cell is convected in with the value of the cell where it
originated, regardless of the directional sense of fluid motion. This
character of convective transport is essential in properly conserving
105
-------
the transported quantity and in avoiding certain computational
difficulties.
The difference formulation of the buoyancy equation after col
lecting terms is written as
|PRzREzAZk
1
1
Azk+l+Azk
2 sech^ r sinh r . sech
IP A J
PRrRErZ
sinh £. i • sech £
+ J"' E.
O A >• " • " I ^
2A? sinh
2AZ,
.
'p.q
sinh5, , • sech. £
'"^nhtp
sech
sech
1 -
2 tanh
sinh £. • sinh. 5
2 tanh
'P J
'p+l,q
PR2REzAZk
AZk-l+AZk
'p.q-1
I2AZ,
PRzREzAZk
AZk+l+AZk
(5.30)
106
-------
In linear radial coordinates, Equation (5.30) reduces to
PRrRErAR'
PRzREzAZklAWAZk ' AZk-l+AZk
j-l
p.q
2pAR
r r
-i»q
'p+i.q
PRzREzAZk
I
'p.q-l
A!
p»q+i .
(5.31)
The A., transport difference equation corresponding to Equations
(5.30) and (5.31) are obtained simply by replacing A-, with A3 and noting
that the eddy Schmidt number, SC, should be used in the case of material
transport, instead of the eddy Prandtl number, PR. Materials other than
salt may be treated in a similar fashion.
107
-------
5.7.3 Summary of required difference equations
The difference equations to be solved are:
Transformed coordinates (?, Z),
f - Equation (5.21)
a - Equation (5.28)
A-j - Equation (5.30)
A3 - Equation (5.30)
U - Equation (5.23)
V - Equation (5.24)
Linear Coordinates (R, Z)
V - Equation (5.25)
n - Equation (5.29)
A-| - Equation (5.31)
A- - Equation (5.31)
U - Equation (5.26)
V - Equation (5.27)
5.7.4 Vertical Grid Space Restrictions
Although the vertical grid spacing is variable, there are three
locations where an exception is expedient for the treatment of bound-
ary conditions (Section 5.8). These exceptions are as follows:
1. At the grid system bottom boundary AZ~ = AZ-,
2. At the sea surface
where Zh = ]T AZK
NK
]T
k=2
108
-------
3. At the level of the plume inflow boundary
AZ
Kp
where KP is the grid boundary location.
These exceptions place no serious limitation on vertical grid spacing
and are incorporated only to expedite computer bookkeeping in treating
the various boundary conditions.
5.8 Boundary Conditions
Attention is now focused on evaluation of boundary conditions
necessary to carry out solution of the equation sets summarized in
Section 5.7.3.
Referring to Figures 5.1 and 5.2, the sea surface (Z = Z, ) is
considered a free-slip boundary which is vertically rigid. A specified
flow enters the bottom inflow boundary where R <_ R . Depending on the
water depth, this boundary may constitute the outfall port orifice
(shallow water case, see Figure 5.2) or an arbitrary lateral plane
through the plume (deep water case, see Figure 5.1) at elevation
Z = Z, . In the former case, the port geometry must be considered along
with the ocean floor. The radial velocity distribution, VQ, depends
on R and the port side and ocean floor are no-slip surfaces. In the
latter instance, the velocity distribution is obtained either directly
from data (hydraulic model or prototype) or calculated by the similarity
techniques described in Chapter 4. Outside the plume nominal boundary
(Figure 5.1) the bottom boundary is assumed slip-free.
109
-------
Surface heat transfer is neglected in this study since the sea
surface area is relatively small and surface heat exchange will have
very little effect on the overall temperature distribution. Boundary
condition sets a and b given below refer to the physical systems
shown in Figures 5.1 (deep water) and 5.2 (shallow water) respectively.
To eliminate confusion, the boundary conditions are stated in terms
of R (in lieu of the transformed coordinate, ?)•
1. Sea Surface ( OfR^, Z = Zh)
a. 41 = Constant = ^ ,
n = 0 (5.32)
3A,
-T— = 0 (adiabatic condition)
3A,
b. Same as above.
2. Plume Centerline
a. R = 0, Zb <_ Z <_ Zh
v = Constant = Y,
3A,
XT - °
b. R = 0, ZQ £ Z £ Zh
Same as above.
110
(5.33)
-------
3. Inflow Boundary
a. Z = Zb, 0 <_ R <_ Rfa
/-R
= *1 +J V(R,Zb)RdR (5.34)
= Alb
b. Z = Z . 0 < R < Rrt
o o
* = f1 +7 V(R,ZQ)RdR (5.35)
o
_ _ 9U. 9V_
" " 8Z " 3R
4. Port Side (R = RQ, Zb = 0 <. Z <. ZQ)
a. Not applicable
VR
Where the reference velocity, V = 1
= . V (no siip) (5.36)
m
-------
3A,
- -
3A,
5. Bottom Boundary
a. Z = Zb, Rb < R < Ro
f "b
¥ = V] + J V(R,Zb)RdR = *2 (5.37)
ft = 0
3A,
3Z
3A
3- =0
b. Z = Zb = 0, RQ <. R <
n - |y (no slip) (5.38)
3A,
3Z
6. Inflow-Outflow Boundary (R = RM, Zfc <^ Z <_ Zj
The distance to the inflow-outflow (or free-flow) boundary, R^,
must be chosen in advance and this distance must be large enough such
112
-------
that boundary conditions listed below prevail approximately.
a- ||=0; or 4 =0
8R 8R
Meaning that streamlines are level, or the streamlines do not change
slope, respectively.
« • S - & (5.39)
P - P^,
A-, = (Ambient condition),
S - S
°v« °00
A3 = S—^~T~ (Ambient condition)
r o
The conditions on A, and A., are valid so long as convection
dominates the transport at the boundary and upstream differencing is
used.
Now consider the difference form of these equations. Again,
refer to Figure 5.3 and note that boundary values for the (j,k) grid
(vgrid) fall on the boundary of the physical system; whereas, on the
(p,q) grid (grid for n, A, and A3) the boundary cells are fictitious
in that they fall outside of the physical system. These cells are for
the purpose of obtaining specific conditions at the real boundary.
Again conditions a and b refer to cases given in Figures 5.1 and 5.2,
respectively. The difference forms are given in terms of the trans-
formed variable, 5, for computer application.
113
-------
1. Sea Surface (k = NK, q = Nq + 1)
a. Deep Water (Refer to Figure 5.5)
Velocity:
Uj,NK+l =Uj,NK (^ee Slip) (5.40.1)
Vj,NK=0 f5'40-2>
Stream Function:
Vj NK = 1 (Arbitrary) (5.40.3)
Vorticity:
Let fi be the vorticity at point (p, Nk). By the free
slip velocity condition above and the fact V. ... = 0,
J »INK
Hence, H is the nodewise average value at Zh> or
so that
(5.40.4)
114
-------
•C
1
U.M,NK.
s
boundary
!.i-l .NK r
interior
V
f 1
r
r S
cell
j V.i.NK ,
cell
r
p,Nq
f
, VJ,NK-1 t
\ \
q - Nq+1
I^.i.NK k = NK
(sea surface)
u .
^ ' q = No
V .
Figure 5.5 Typical Seo Surface Boundary and Interior Cells
[r indicate any of the cell centered
quantities n, A-, and A-,.]
115
-------
Buoyancy:
Since the adiabatic condition prohibits heat transport
across the surface,
1 =0,
Hence,
1 . .
AL7.7 ^Alp,Nq+l " Alp,NqJ " ' (5.40.5)
NK
or
Alp,Nq+1 = Alp,Nq.
Salinity:
Likewise,
- '3p,Nq •
b. Shallow Water
Same as deep water case above.
116
-------
2. Plume Center! ine (R = 0)
a. Deep Water Case (Refer to Figure 5.6)
Velocity:
U, . = IL . (velocity gradient vanishes) (5.41.1)
Vljk=0. (5.41.2)
Stream Function:
Y, , = 1 (Must be consistent with condition l.a). (5.41.3)
Vorticity:
From the conditions on velocity given above, the centerline
vorticity, ^_ - 0, or averaging across the centerline,
V 1/2 ( "l,q + "2,q) =0"
Hence,
nl.q = - n2,q- (5.41.4)
Buoyancy:
At the centerline, the buoyancy gradient must vanish.
Hence,
77
-------
R = 0, £ = 0
.,k
boundary cell
Y^
2 k
*
interior cell
2,k-l
k
.K
k-1
P = 1
j = 1
p = 2
j = 2
Figure 5.6 Typical Centerline Boundary and Interior Cells
[r indicates any of the cell centered
quantities, n, A-, and A,.]
118
-------
Salinity:
Since the same conditions hold for salinity and buoyancy
transport at the centerline,
A3 = A3 (5.41.6)
Jl,k J2,k
b. Shallow Water Case
Same as deep water case above.
3. Plume Inflow Boundary
a. Deep Water Case (Z = Zb, 0 ^ R •_ Rb;
(Refer to Figure 5.7)
Velocity:
\ AZ^ (5.42.1)
Calculated by methods in Chapter 4.
Vj., -vu.zb).
Data function, or calculated by methods in
Chapter 4.
Stream Function:
sinh -cosh C . (5.42.3)
P=2
119
-------
)t-x^
interior cell
',1-1.1 r^LL
boundary cell
JM,1
k = 2
q = 2
k = 1
q = 1
j-l
Figure 5.7 Typical Inflow Boundary and Interior Cell
(deep water case only).
[rp,q indicates any of the cell centered
quantities n, AI and A.,.]
120
-------
Vorticity:
Vorticity at the inflow boundary, n b is calculated from,
Hence,
"n i = - °n 9 + 7T- [U4 o-U, i + U. ,
p,l p. 2 AZ J,2 j,l .1-1,
Buoyanpy:
AIP,I = Al^J'1/2^5» Zb'1/2 AZ1)]- (5.42.5)
Data, function, or calculated by methods in Chapter 4.
Salinity:
A, « Ao[(M/2)A£, Z.-1/2 AZ,)] (5.42.6)
3p,l 3 b '
Data, function, or calculated by methods in Chapter 4.
b. Shallow Water Case (Z=ZQ, OfRfRQ; Refer to Figure 5.8)
Velocity;
Uj)Kp = 0 (5.43.1)
Vj,KP = Vo = Constanti or» vj,kp = V(C.Zb). (5.43.2)
Stream Function:
1 + A5Z V(n,KP) sinh ?_ cosh £_ (5.43.3)
n=2 P P
121
-------
Uj-l,KP+2
f_2_^
tLr
. 4fj-l,KP+2 lVj,KP+2
interi
y»
i — T I^P+T
K 1
r
or cell
LVJ,KP+1
Jp,QP+l
, 4/.i.KP+2
UJ,KP+2
V.
,UJ,KP+1
Uj-l,KP
interior cell for A-|&A.J
[boundary cell for
)
J-l.KP
U
'j.KP
boundary cell
|FP.QP
UJ,KP
k = KP+2
q = QP+2
q = QP+1
k = KP
(port orifice)
= QP
Figure 5.8 Typical Inflow Boundary and Interior Cell
(shallow water case only).
trp,q indicates cell centered quantities
ft, A-| and A,.]
122
-------
Vorticity:
Since in one case, V. •, = V_ is assumed constant over the port
J > ' o
radius, we choose to evaluate vorticity at QP+1 , instead of at
the port orifice. HD QD+, will then become the boundary value.
Convenience is the primary reason for doing this, because to
remain consistent with V = constant, n. np is impossible to
0 J ,ljr
define correctly at the port edge. This procedure is also help-
ful in using power law profiles for V(s,Zu) (see Chapter 7).
Vorticity at a point (p,k) is given by
«p,k ' 1 «Wl+flp.q> *
Hence,
°p.q = • <
at q = QP + 1,
[U j,KP+2+U j-l,Kp+2"U
" A?cosh5p ^Vj+l,KP+rVj-l,KP+lJ
(5.43.4)
Buoyancy:
A-, = Constant = A, (5.43.5)
'p.QP ]o
Salinity:
A- = Constant = A- (5.43.6)
123
-------
4. Port Vertical Side
a. Deep Water Case - Not applicable
b. Shallow water case (R = RQ, 0 <_ Z <_ ZQ; Refer to
Figure 5.9)
Velocity:
UNp>k=0 (5.44.1)
VNP,k = " VNP+l,k (N°-s1iP condition). (5.44.2)
Stream Function:
. NP
f = 1 + A? V(n,KP.) sinh £ cosh 5. (5.44.3)
n=2
P=2
1 2
Although the exact value of 4* , = 1 + j R V , the
lir j K 0 0
difference approximation will lead to a slight deviation.
Vorticity:
Hence,
"MP.q = - "MP+l,q ' ACcosh {VNP+1 ,k+VNP+l .k-
Buoyancy:
A, = A, (Adiabatic condition) (5.44.5)
'MP.q 'MP+l,q
124
-------
Port side, R=RQ, 5=
-
boundary cell
-
_[
.rMP,q (
1
,VNP,k-l .
V V
NP,k lvnPH,k
r^
interior cell
,UNP'k .
LfNP.k-1 r
FMP+I.Q e
f L
SfNP+Lik-1.
V+l.k
[WLik_
JNP+TJJ^I
k-l
p=MP j=NP
j=NP+l
Figure 5.9 Typical Vertical Port Side Boundary and Interior Cell
(shallow water case only).
Tr indicates cell centered quantities n,
1 p»q
A-J and A.J.]
125
-------
Salinity^
5. Bottom Boundary
a. Deep Water Case (R. <_ R <_ R^, Z = I^\ Refer to Figure 5.10)
Velocity:
U. i = U. j, (free-slip condition) (5.45.1J
J 5 * tj 9 *~
V. , =0 (level stream line condition) (5.45.2)
J > '
Stream Function:
NB
*. , = 1+A5V V(n,l)sinh Sn tosh E . (5.45.3)
J » i £-> P P
n=2
p=2
where NB is the number of inflow cells to the nominal plume
boundary.
Vorticity:
Qn i = ~ nn 9 (free-slip condition) (5.45.4)
[J 9 I (J 9 C.
Buoyancy:
A, = A, (5.45.5)
'p,l 'p,2
Salinity:
(5,45.6)
A = A
126
-------
11*2.
k = 2
•P.2
interior cell
J-1.1
boundary cell
UJ,1
q = 2
k = 1
(boundary)
q = 1
Figure 5.10 Typical Bottom Boundary and Boundary Cell
[r indicates cell centered quantities
n, A, and A,.]
127
-------
b. Shallow Water Case (RQ <. R i R^, Z = Zfa = 0;
Refer to Figure 5.10)
Velocity:
U. , = - U. 9 (No-slip condition) (5.46.1)
J >i J >^
V, , = 0 (5.46.2)
J »i
Stream Function:
y = i + 1 R 2v (5.46.3)
Vorticity:
n = " n 2 + AT" ^U> 2 + U'-l 2^' (No's11P Condition) (5.46.4)
Buoyancy:
A, = A, (Adiabatic condition) (5.46.5)
Vl 'p.2
Salinity:
A, = A, (5.46.6)
dp,l P»2
6. Inflow-Outflow Boundary
a. Deep Water Case (R = R^ , Zfa <_ Z <_ Zh; Refer to Figure 5; 11)
j = NO, p = Np
Velocity:
U k. t i = "• ---••..-..-- . ,.^.— [OJ _ m \ /r>lTT\
NJ.k Sinhf,N,AZ, v NJ.k NJ,k-r (5.47.1)
128
-------
.R=R , g = NJAC
J-l,k
w V 1 w
.NJ-l.k !vNJ.k l¥NJ,k J
i *
interior
^NJ-l.k-l p
1
FNP,q
cell
,VNJ,k-l .
'
UNJ,k
,VNJ+l,k
_rNP+l ,q
T
boundary cell
/NJ.k-l
T 1 1
VNJ+l,k-l
f
j=NJ-l p=NP
j=NJ
k-1
p=NP+l j=NJ+1
Figure 5.11 Typical Inflow-Outflow Boundary
and Interior Cells
["iv „ indicates cell centered
L P»q
quantities n, A, and A^.J
129
-------
VNJ+l,k = VNJ,k (5-47
Condition 5.47.2 results follow from the stream function
condition given below.
Stream Function:
*NJ,k = *NJ-l,k (Level Stream 11nes) (5.47.3)
Vk = 2*NJ-l,k - *NJ-2,k (N° Chan9e of
Vorticity:
av
Since |jj- =0 (Equation 5.47.3)
the vorticity, nNp+] k is given by
0 - o."
"NP+l,q 9Z
NJ,k
Note that (3U/8Z)NJ k has been replaced by a central difference
form using even spacing of AZ. For the more general case of
uneven AZ, , refer to Appendix B •
Buoyancy:
pr ~pn
A-, = (-—^ ) (5.47.6)
NP+l,q pr po Ambient
Salinity:
sr - s
A3 = ^ S—:S ^ (5.47.7)
NP+l,q ^r ^o Ambient
130
-------
6. Shallow Water Case
Same as above.
A number of assumptions and restrictions are involved with the
above boundary values. For instance, the sea surface is restricted
to remain flat, although visual observation indicates that a slight
"boil" will occur at the plume center!ine, At the bottom boundary
of the deep water case, beyond Rb, it is assumed that there is neither
a vertical component of mean velocity nor any change in the horizontal
velocity profile. Additionally it is assumed that neither A, (nor r)
is diffused across this boundary.
Transported quantities are assumed constant at R = RM. Within
the framework of the difference scheme, this is a perfectly valid
assumption if convective terms, acting normal to this boundary,
dominate the diffusion terms. Any quantity convected into the system
is assumed to have the ambient value. Stream lines are assumed flat
or having constant slope at this point and recirculation of flow out
of the system is prohibited.
Many of the above assumptions are a result of ignorance with
regard to processes outside the chosen system boundary. Since there
is no way of regulating these processes, assumptions based on physical
insight are the only viable alternative. Fortunately, the more
nebulous assumptions occur at points far removed (at R = Rro and
bottom boundary) from the region of prime interest. And there is some
recourse, in that numerical experiments are possible which give in-
sight to the importance and effect of these assumptions. Results
131
-------
given in Chapter 7 reveal that the boundary specifications at R^
have little influence on the numerical solution as long as R^ is a
reasonable distance from the port (approximately two plume diameters).
5.9 Rectangular Coordinates
The previous sections have dealt exclusively with (R,Z)
coordinates or transformed, (£,Z) coordinates. In this section we
treat the governing differential and difference equations in rectang-
ular (X,Z) coordinates.
Detailed derivation of these forms are omitted; only the
results are presented. In contrast to previous considerations neither
transformed coordinates or unequal grid spacing will be considered.
The physical problem which we wish to analyze is a two-
dimensional line plume that forms over a multiport diffuser line.
This condition is approximately realized once the flows from a series
of single round ports spread and interfere with one another parallel
to the diffuser line. In dealing with the single round port, we were
restricted to stagnant environment because any cross-current would
destroy the problem symmetry and require a three-dimensional analysis.
In the line plume case we may consider environmental velocity com-
ponents which fall in the (X,Z) plane. Figure 5.12 illustrates the
physical system for the line plume considered here.
5.9.1 Governing Differential Equations
Differential equations for the X,Z coordinate system comparable
to Equations (5.10), (5.11), (5.12), (5.13), and (5.14) given in
132
-------
SEA SURFACE
RECEIVING
WATER
PLUflE IN
z=z,
NOMINAL
PLUME
BOUNDARY
o
3
O
Figure 5.12 Physical System for Line Plume Issuing
to Flowing Receiving Water
133
-------
Section 5.4 are:
Stream Function:
-------
n = <"/(v0/b0) ,
where VQ and bQ are reference plume velocity and width, respectively.
Dimensionless parameters are:
£x
F -
defined as in Section 5.4, keeping in mind that the radial direction
is simply the "X" direction in this section.
5.9.2 Rectangular Difference Equations
Rectangular difference equations are formulated on a grid iden-
tical to that illustrated in Figures 5.3 and 5.4 with the corresponding
change from the £ coordinate to the X coordinate.
Here we consider only a regular grid, which has spacing AX and
AZ. Difference equations are given below.
135
-------
Stream Function:
+ v.
AZ
+_L\AZ
AX AZ 7
"j.k
AZ
(5.55)
Vorticity, n. k, is the average value for the four surrounding cells
(see Figure 5.4) and given as
Velocity is calculated by
U
j,k
j,k
and vorticity by
REVAX RE AZ'
p»q
l
.q+0
(5.56)
(5.57)
(5.58)
136
-------
2F7X
The buoyancy parameter, A], is calculated by,
RE PR AX2 RE PR AZ2 J Vq
(IU. ,
\ J'
u.
REPR AX
X X
i k) + — ] — z- 1
,K; R^PR^ J
Ai
'
137
-------
The salinity or A3 transport equation is given exactly by
Equation (5.60) with PR replaced by SC, the eddy Schmidt number.
5.9.3 Rectangular Boundary Conditions
Boundary conditions for the rectangular problem are substantially
the same as in the axisymmetric problem. Notable differences are pro-
vision for crossflow and lack of problem symmetry.
Referring to Figure 5.12 boundary conditions are as follows:
1. Sea surface (OfXfX^, Z=Zh)
y = constant = y,
n = 0 (free slip condition)
8A-j
-ry = 0 (adiabatic condition)
3A,
2. Inflow boundary (X=0, Z^
dz
r
* = fi - I
Zh
3U
8 •
3Z
D ~ 0
r °°
pr- po
.
. _ r
43 '
138
-------
3. Bottom boundary (O^UXC - ^ Xb, Z=Zb)
f Zb
* = v-i - / U (Z) dZ = constant
\ "
n = 0 (free slip condition)
3A-,
1 = 0
3Z
4. Plume inflow boundary (Xc - ^ X^X^ + ^ X., Z=Zb)
Assume that V, A-, and A3 are known from data or empirical
relationships.
X
VbdX
Xc ' 7 Xb
n = iy.. M.
5. Bottom boundary (X + i X.< X
-------
3A,
3Ao
6. Inflow-outflow boundary (X=Xoo, Z,
-------
CHAPTER 6
CODE DESCRIPTION AND ORGANIZATION
The computer program described herein obtains the solution of
the transformed difference Equations (5.21), (5.23), (5.24), (5.28),
and (5.30) for the quantities f, U, V, n, and A-| , (or r), respectively.
Through input option one may also obtain these solutions in ordinary
radial coordinates (see summary Section 5.7.3). A program which
obtains the solutions through the use of the density disparity param-
eter A2 (as opposed to AI) has been used but is not presented in this
manuscript.
The program consists of 20 subroutines and/or functions which in
part are managed by an executive routine called "SYMJET". Initially,
the code was set up for the Oregon State University CDC 3300 time
sharing system. This system, although extremely handy for program
development, is too small in terms of available core and too slow for
economically treating large problems. The code version presented
here is adapted to the Computer Science Corporation Univac 1108
located in Richland, Washington. This version of the code has also
been successfully executed on the Control Data Corporation 6600
located in Palo Alto, California, and on the CDC 6400 system at the
Battelle Memorial Institute in Columbus, Ohio.
6.1 Computational Procedure
The primary task at hand involves the simultaneous solution of
one elliptic partial differential equation for the stream function, y,
141
-------
(Equation 5.21) and two parabolic transport equations for the
vorticity, n, and the buoyancy, A-J , [Equations (5.28) and (5.30),
respectively]. Equations for U and V (5.23 and 5.24, respectively)
may be considered as auxiliary, but are, nevertheless, essential and
need to be solved along with (5.21), (5.28), and (5.30) during itera-
tion. In the case of neutral buoyancy, only Equations (5.21) and
(5.28) need to be solved simultaneously.
The iterative procedure is built about the equations for Y, n,
and A-,. The technique used in the Gauss-Seidel method for all quanti-
ties defined by second order partial differential equations. Liebmann
acceleration is employed with the alternatives of both under and over
relaxation. Assuming all boundary conditions are set and pertinent
variables are initialized, the procedure is as follows:
1. Compute A, _ and r using Equation (5.30) based on
1K»H PJ^
previously calculated values of U. .,V. . and appropriate
J »K J ,K
transport coefficients.
2. Compute n using Equation (5.28) and the previously
V »H
computed values of U. . ,V. ,., A, and appropriate trans-
j »* j»K- i p »q
port coefficients.
3. Update necessary boundary values for Apr, and n.
4. Use the newly computed values of n to compute the stream
function distribution from Equation (5.21). One or more
iterations may be required to arrive at a satisfactory
solution for v. Compute a new velocity field V. . and U. ,.
J »K J ,K
from the newly calculated f distribution.
142
-------
5. If the eddy transport terms are not constant, compute multi-
pliers FR and FZ from new velocity field (for definition of
the FR and FZ multiplier, see Chapter 7).
6. Repeat Steps 1 through 5 until a preset convergence criter-
ion is satisfied or a specific number of iterations has been
completed.
6.2 Executive Program and Subroutine Description
As mentioned previously, the computer code consists of an execu-
tive routine called "SYMJET" and 20 subroutines and/or functions. The
following discussion relates the primary duties served by each of
these routines.
SYMJET
Executive routine
1. Reads case header and integer case set-up information.
2. Reads alphanumeric data for line printer output array
option, plot tape options, isoline interpolation options,
and program control.
3. Calls subroutines for data input, problem set-up and
initialization, and problem execution. The subroutines
called are (in the calling sequence):
• INPUT
. READY
. PLABAK
• STREAM (for inviscid flow solution)
. SSCOMP
143
-------
. INTERP
4. Performs other miscellaneous tasks such as clock initial-
ization, tape rewind, presetting variables, etc.
SUBROUTINE INPUT
General data input routine
1. Reads restart tape if required.
2. Reads remaining input data from cards.
3. Converts portions of input data to appropriate quantities
and units (e.g., temperature data to density data).
Subroutine is called once during execution.
SUBROUTINE READY
Problem set-up routine
1. Sets all computed constants.
2. Sets constant boundary conditions.
3. Presets turbulence multipliers.
4. Option to call SUBROUTINE SIMJET.
5. Option to call SUBROUTINE GAUSS.
Subroutine called once during execution.
SUBROUTINE PLABAK
General information and debug output
1. Writes to line printer various computed and input supplied
variables and the operation modes of current case.
2. Writes to line printer constant arrays used in the difference
equation computations.
Subroutine is called once or not at all at the user's option.
144
-------
SUBROUTINE STREAM (IT, NSKIP)
Solves for stream function, Y
1. Computes the viscous or inviscid stream function (Equation
5.21) by Gauss-Siedel iteration. When called, this sub-
routine iterates on y (PSI) "IT" times.
2. Upon completion of "IT" iterations the velocity components
U. ,, and V. b are computed by the auxiliary Equations (5.23)
j »*• J »K
and (5.24).
For an inviscid flow computation (the inviscid flow solution
may be called for the purpose of initializing the viscous flow
computation if desired) STREAM is called and returns control to
the executive routine. When STREAM is called from SSCOMP,
which computes the viscous flow field, control is then returned
to SSCOMP. Subroutine STREAM constitutes what is referred to
in this manuscript as the "inner iteration loop" (subroutine
SSCOMP constitutes the "outer iteration loop") and is called at
least once for each "outer iteration".
SUBROUTINE SSCOMP
Computes steady flow solution of all transport equation
1. Solves transport equations for
. A1
• r and
• ft,
using Gauss-Siedel iteration with Liebmann acceleration
(deceleration).
145
-------
2. Updates boundary values of L, r, and :<-.
3. Computes convergence rate information and the cell
indices having the slowest convergence.
4. Calls subroutine STREAM to compute velocity field.
5. Calls subroutine EDDY to compute eddy transport multipliers
as required.
6. Writes out monitor node values.
7. Calls subroutine OUTPUT for either interim or final array
output.
8. Generates plot data tape.
9. Computes surface area above Tamb in 1 °C increments.
10. Performs a Gamma constituent balance error, (rin~rout)/r- »
for the overall system and then returns control to the
executive routine.
This subroutine is referred to as the "outer iteration loop" and
is called but once during a case execution. The code spends
the majority of the execution time in this routine.
SUBROUTINE EDDY (M)
Computes eddy transport multiplier FR and FZ
1. Computes potential core.
2. Computes plume half radius, RI .^ and nominal plume boundary,
R405» at each vertical grid point.
3. Computes FR from mixing length theory.
FR=Vmax ' Rl/2
146
-------
4. Computes FZ based on mixing length theory and incorporates
Richardson number modification (computes point Richardson
number, RI, and calls function RCHMOD for modifier).
If eddy multipliers are computed based on the velocity distri-
bution, this subroutine is called once during each "outer
iteration". Either FR, FZ or both may be computed selectively.
(Parameter M in the call list specifies the option). Details
of the particular eddy transport models used and regions of
applicability are discussed in Chapter 7. Also, this sub-
routine may be bypassed a set number of iterations for computa-
tion stability purposes (discussed in Chapter 7).
SUBROUTINE OUTPUT (MODE)
Primary line printer output call routine
1. The primary purpose of this routine is to call selectively
the output array writer subroutine, AROUT, based on the
alpha input read in through the executive routine. The
arrays and array header Holleriths are aligned in the call
list of AROUT. This subroutine may be called selectively
for array writing through the input Fortran variable NOUT.
That is, every time that the "outer iteration" number is
divided by NOUT and yields a whole number, the array
writing routine is called. The parameter, MODE, is an out-
put option.
2. The secondary purpose of subroutine OUTPUT is to write out
selectively the convergence rate information computed in
147
-------
subroutine SSCOMP, that is, maximum changes in *, A and n
and the nodal location of these changes, during successive
iterations. The iteration numbers selected for output are
specified by the input Fortran variable NTTY, in the exact
manner that NOUT is used in 1. above.
SUBROUTINE AROUT (list)
General array writer
This subroutine is used to write out all computed arrays speci-
fied for printing. The appropriate array, header and grid
coordinates are aligned in the call list at subroutine OUTPUT.
Miscellaneous computations are also performed here as necessary.
For instance, if normalized arrays are desired, these are
normalized in AROUT and if temperature arrays are required the
buoyancy parameter (A-|) array is converted to a temperature
array through successive calls to function TEMP.
SUBROUTINE INTERP
Calling routine for isoline interpolation
The only job performed by this subroutine is selectively setting
up arrays to be interpolated by the general interpolator
routine, IS06EN. Selection is made through input of the Fortran
alpha array TERP during execution of the executive routine. The
particular array, header and other appropriate data are aligned
in the call list of ISOGEN. This subroutine is optionally
called through the executive routine following execution of
SSCOMP.
148
-------
SUBROUTINE ISOGEN (list)
General isoline interpolator
The function of ISOGEN is to interpolate a given array, aligned
in memory through the subroutine call list, for isolines whose
values are selected at input and specified by the Fortran
array, ISOLN. For a specific array (say the stream function
array) the coordinates of an isoline (streamline) are quad-
ratically interpolated and coordinates printed. Contouring may
be accomplished by hand plotting the results. Automated
plotting of the computed points would be quite difficult since
i
the points are not ordered.
SUBROUTINE GAUSS (N)
Optionally computes Gaussian distributions for inflow
This subroutine computes Gaussian boundary distributions for V,
A,, and r in either the zone of flow establishment or the zone
of established flow. The particular option is determined by the
parameter, N. These computations are based on the Albertson
et al. [4] data and theoretical results given by Abraham [1].
The routine is called once from subroutine READY.
SUBROUTINE SIMJET (list)
This routine computes the center!ine distributions of V, Ag and
r from the similarity solutions of a vertical plume given in
Chapter 4. For the homogeneous problem, V is calculated from
Equation (4.43) and A2 from (4.4). In the case of stratification
Automated contouring is accomplished using a special contouring
routine.
149
-------
these quantities are computed from Equations (4.36), (4.37),
and (4.38) using the fourth order Runge-Kutta technique.
Results from this routine may be used for inflow boundary
information in the more elaborate finite-difference method for
the confined plume. Calling is through subroutine READY and
is performed at most once.
FUNCTION SIGMAT (SAL, T, N)
Given the salinity, SAL, and temperature, T, this function com-
putes Sigma-t (at, see Section 3.6) based on algebraic equations
given in the U.S. Navy Hydrographic publication number 615 [103]
or as given in Hill [39].
FUNCTION TEMP (SALT, SIGMA)
Given the salinity, SALT, and the density in Sigma-t units,
SIGMA, this function solves the equations referenced above for
the temperature in degrees centigrade by the Newton-Raphson
method. The function SIGMAT (SAL, T, N) is repeatedly called
during the iteration process.
FUNCTION SANK (X, N)
Hyperbolic sine coordinate transformation function which yields
Sinh (X) for N = 1 and X for N = 0 (linear radial coordinates,
no transformation).
FUNCTION CASH (X, N)
Hyperbolic cosine transformation function which yields COSH (X)
for N = 1, and 1.0 for N = 0.
150
-------
FUNCTION RCHMOD (N, RICH)
Computes Richardson number (RICH) modification of the vertical
eddy viscosity coefficient by one of five different models
(option given by N). These models are given in Chapter 7
(cf. Table 7.5).
6.3 Flow Charts
Detailed flow charts of all subroutines in the SYMJET computer
code would require an extensive amount of space. For this reason
only the main subroutines and the executive program will be illus-
trated. The charting of these will also be somewhat abbreviated.
A partial bibliography of the computer variables may be found in the
program listing (Appendix E).
151
-------
SYMJET FLOW CHART
(Executive Routine)
Read alpha case header
Read integer set-up data
/ Read alpha TLIST option
| Set write option arrays
AOUT, PLOT, TERP, AND CONT
Set auxiliary Indices
Initialize arrays and constants
Compute monitoring arrays
Generate array
Instruction vectors from
TLIST Options: NRITE(J
N3DPT(J), fSOPT(J) CONTR
152
-------
CALL INPUT: Reads
main data file from
cards and optionally
Initializes arrays from tape
CALL READY: Completes
Initializations, computes
constants, and sets
fixed boundary values.
Positions output tape.
CALL PLABAK:
Write out computed
and supplied constants,
and debug arrays.
CALL STREAM: Compute
1nv1sc1d flow solution
153
-------
CALL SSCOMP : Compute
solution to transport
and auxiliary equations
CALL INTER? : Compute
contour coordinates
REWIND output
tape
154
-------
SUBROUTINE INPUT FLOW CHART
READ LUN 7
UNO. n, a,, U, V, y and r
Data from previous computation
for Initialization or
continued iteration.
READ data card
DATA. JI, KI, NI
GO TO (N,, N2—N12), NI+1
N. : Replace appropriate
variables with DATA
155
-------
SUBROUTINE READY FLOW CHART
Set up various computed
Constants: e.g., SC(J,L), SZ(K,L)
Preset variables: e.g., FZ(J,K)
FR(O.K)
Set up coordinate systems: Z(K),
ZC(K), X(J), XR(J), R(J), RC(J)
Set inflow boundary
velocity according to
"INMODE"
156
-------
Compute and set all
fixed or Initialized
boundary condition not
treated above
157
-------
SUBROUTINE STREAM FLOW CHART
SET STREAM FUNCTION
Inflow-outflow boundary
condition. * (NJ.K)
Compute vortlclty, n
at cell corners (OMEGA 3)
Compute f (J.K
Equation (5.27
Accelerate (or decelerate)
Solution, v (J.K)
Compute U(J,K) and V(J.K)
from * (J,K), Eqs. (5.23) and (5.24)
158
-------
SUBROUTINE SSCOMP FLOW CHART
Compute
Compute
i
^ (p.q).
r (P.q).
Eq.
Eq.
(5
(5
.30)
.30)
Accelerate (or decelerate)
(p.q) and r (p,q)
Set boundary values for
next Iteration on AI, r and n
Reset iteration
limits;
initialize eddy
factors, FR & FZ
159
-------
• Compute n (p,q) by
Equation (5.28)
• Compute maximum change 1n
n (p,q) and value
of p and q for location
• Accelerate or decelerate
solution n (p.q)
Compute updated boundary
conditions for n (p,q)
(
Write out monitor
node Information
CALL EDDY
Computes eddy
transport multipliers
FZ(J.K) and FR(J.K)
160
-------
Write to LUN 8 (MAG. TAPE)
ITNO, n, A, U. V. v, r
CALL OUTPUT (1)
Calls array writer
CALL OUTPUT (2)
Call Intermediate output
161
-------
Create plot files
LUN 8 according
to N30PT(J)
Compute surfacs
Isotherms 1n
Increment of 1 °C
Perform GAMA
Sum convergence
check
Print GAMA Sum error
162
-------
CHAPTER 7
CODE VERIFICATION AND NUMERICAL EXPERIMENTS
In this chapter we are concerned with verification of the
numerical model. Ultimately, the program is to be used in describing
the plume resulting from large vertical thermal outfalls in shallow
water, and, as previously mentioned, published field data concerning
velocity and temperature distributions along with other pertinent data
needed for evaluation or verification are essentially non-existent for
these cases. Even laboratory data from hydraulic models are scant and
steady flow experiments to model quasi-steady oceanic conditions with
stratification are essentially impossible.
Verification of the numerical techniques will be carried out by
using the code described in Chapter 6 to simulate various problems
which have been well studied, both experimentally and analytically,
and for which much information has been published in the literature.
One such problem which the code can easily handle is the deep water
momentum jet. In this case much knowledge has been compiled concern-
ing velocity distributions, concentrations, and turbulence parameters.
The computer code can easily handle interacting buoyancy for the same
geometry. Although there is a lesser amount of experimental data pub-
lished in the open literature for buoyancy cases, especially on turbu-
lent parameters, there is enough information for meaningful comparisons
with the numerical model.
163
-------
Once the computer program is verified using this published infor-
mation, the program can be applied with confidence to conditions of
more interest and practical value, such as shallow water and stratified
ambient cases. Having checked the program against experimental results
for simple cases, we know at least that the numerical procedures are
working correctly, although auxiliary models (e,g., turbulence) may not
be entirely correct.
Also presented in this chapter are some of the code operating
experiences, turbulence modeling, solution convergence and stability,
and discussion of some of the more troublesome boundary conditions.
7.1 Deep Water Plumes
By deep water plumes we are implying that the effluent is dis-
charging into a semi-infinite water body, although as a practical
matter computational boundaries must be finite. For program verifica-
tion, we use the following deep water flow categories:
• Momentum Jet - the fluid motion is induced entirely by the
effluent initial momentum. Buoyancy is also calculated but
is decoupled from the momentum equation and may be used as
a measure of concentration. This case is indicated by
FQ + ».
• Pure Buoyant Plume - in this instance there is no effluent and,
consequently, no initial momentum. The driving force is pure
buoyancy caused by a source of heat located in the position
of the outfall port. An arbitrary reference velocity is used
164
-------
along with a length scale that corresponds to a port radius.
This case is indicated by F =0.
• Mixed Flow - both initial momentum and buoyancy have varying
degrees of importance. In this case 0 < FQ < <*>.
Various cases of the above categories have been checked against avail-
able experimental data and similarity solutions. These cases are
itemized in Tables 7J, 7.2, and 7.3.
Four different effluent velocity profiles and concentrations (or
temperature) have been used in this work which are:
• Type 1 : Gaussian profiles, established at 4.5 diameters
from the port exit,
• Types 2, 3 : Power law velocity profile at the port exit with
a constant radial concentration (or temperature)
distribution, and
• Type 4 : Constant radial distribution of all quantities
at the port exit.
Equations for these profiles are given in Table 7.1.
Figure 7.1 illustrates a typical grid system in R-Z coordinates.
Note the effect of the hyperbolic sine transformation in stretching
the cell widths as the distance R is increased. The computation grid
(C,Z-coordinates) has uniform radial cell widths as illustrated in
Figure 5.3.
165
-------
TABLE 7.1. SUMMARY OF MOMENTUM JET VERIFICATION CASES (Fc
PR,, = .80, PR, = .80
Case
No.
1
2
3
4
5
6
7
8
9
10
11
12
Grid
Size
26 x 40
35 x 40
26 x 40
40 x 33
40 x 33
40 x 33
40 x 33
40 x 33
40 x 33
40 x 33
30 x 26
30 x 26
^
.2
.1
.2
.12591
.12591
.12591
.12591
.12591
.12591
.12591
.2
.2
AZ
2
2
2
2
2
2
2
2
2
2
2
2
Z
(Surface)
43.5
43.5
43.5
64
64
64
64
64
64
64
43.5
43.5
R
00
74.2
14.96
74.2
67.85
67.85
67.85 '
67.85
67.85
67.85
67.85
74.2
74.2
Boundary
Type
1
1
1
2
N=7
3
N=7
N=10
2
N=10
3
N=7
2
N=6.6
3
N=10
1
1
er
Type2
3
1
1
3
3
3
3
4
3
3
Inviscid
Test
Creeping
Test
ez
.0001
.0001
.0001
.0001
.0001
.0001
.0001
.0001
.0001
e =e
z r
-------
Inlet velocity profile type:
. V(R,Z) = V(0,Z)
2. .v(Rf0) = Ai-JLLj2N+ll
2N2
3. V(R,0) = (1-R)1/N
-^ 4. V(R,0) = Vrt = Constant
CT> 0
2
Radial eddy viscosity calculation type:
1. er = .0295 rQv0 = Constant
2. e = .0256 r,/?v : Prior specification of r1 ,2 from Gaussian distribution of velocity,
v calculated iteratively.
m
3. er = .0256 r1/2vm: Iterative calculation of both r]/2 and vm.
4. er = .0263 r1/2vm: Same as Type 3.
-------
cr>
CO
TABLE 7.2. SUMMARY OF PURE BUOYANT PLUME VERIFICATION CASES (FQ = O)
PRf = .714, PRZ = .714
Heat
Case Grid Z R er Source
No. Size Ag AZ (Surface) °° Type Condition
13 40 x 33 .12591 2 64 67.85 3 1
14 40 x 33 .12591 2 64 67.85 3 2
i
Reference densimetric Froude number is not zero but based on a reference velocity since there
is no inflow at the source.
2See Table 7.1.
3
Heat Source Type:
1. Weak Source: Simulated heated plate maintained at AT = 25 °C.
Heat transferred to fluid by conduction alone over range 0
-------
TABLE 7.3. SUMMARY OF MIXED FLOW VERIFICATION CASES
VO
Case
No.
15
16
17
18
19
20
21
22
23
24
25
26
27
i
See
Grid
Size
26 x 40
26 x 40
26 x 40
26 x 40
26 x 40
26 x 40
26 x 40
40 x 33
40 x 33
^40 x 33
40 x 33
40 x 33
40 x 33
Table 7.1.
M.
.2
.2
.2
.2
.2
.2
.2
.12591
.12591
.12591
.12591
.12591
.12591
AZ
2
2
2
4
2
2
2
2
2
2
2
2
2
#e = 1.0, Gaussian distribution
Z
(Surface)
43.5
43.5
43.5
82.5
43.5
43.5
43.5
64
64
64
64
64
64
R
00
74.2
74.2
74.2
74.2
74.2
74.2
74.2
67.85
67.85
67.85
67.85
67.85
67.85
e" A ^Zs
p Boundary
o Type
52
52
52
52
35
106
52
45.5
45.5
1
1000
45.5
45.5
rZ)
1
1
1
1
1
1
1
2
N=7
2
N=7
2
N=7
2
N=7
2
N=10
2
N=IO
Type
1
2
3
3
3
3
3
3
3
3
3
3
3
ezQ
.0001
.0001
.0001
.0001
.0001
.0001
#See
Below
.0001
.0001
.0001
.0001
.0001
.0001
PRr
.714
.714
.714
.714
.714
.714
.714
.714
.80
.80
.80
.714
.714
PRZ
.714
.714
.714
.714
.714
.714
.714
.714
.80
.80
.80
.714
.714
-------
£t-
22-
?fi
18-
Ifi-
M
» Id
g"~
i—
co 12
S "'
D-
_l
S in
x IU
<
oo
m «
uj o-
o
h— 1
^ 6
£ 6
i—i
o
4_
2 _
0-
ca -
23 -
91 .
1Q .
17.
1 c;
13 -
It
K
11 -
Q .
7 .
5 -
3 -
1 .
1
0
10
2
1
5
4
e
20
E
\
i
10
12
14
2i
16
18
1
20
DIMEiMSIONLESS RADIAL POSITION, R
Figure 7.1. Computational Grid for The Stream Function, v,
Illustrating the Effect of the Sinh U)
Transformation (A? = .14690, AZ = 1.0)
170
-------
7.1.1 The Momentum Jet
A vast amount of information has been gathered concerning the
dynamic behavior of momentum jets dating back to Tollmien's [98] work
of 1926. Hence, there is sufficient data reported in the literature
to check all of the gross aspects of the jet structure computed. In
verifying the computational technique with the published data we use
the following jet characteristics:
• Centerline velocity and concentration,
• Radial distribution of axial velocity and concentration,
• Rate of jet spread, and
• Radial velocity.
Although there is a vast amount of published data available for veri-
fication, the primary data used is from Albertson, et al. [4], Baines
[8], Abraham [1] and information obtained from several researcher's
published in Chapter 24 of Schlichting's text "Boundary Layer Theory"
[84]. Additional information is obtained from reviews by Gauntner
et al. [32] and Chapter 6 of Hinze's text "Turbulence" [40].
Some of the relevant restrictions in this section are:
• Vertical turbulence is negligible; one case is run to
verify this fact.
, The computational grid system has an impermeable upper
boundary. Hence, velocity profiles begin to "feel" the
boundary some distance before it is reached.
Aside from the quantitative verification mentioned above, illustrations
of streamlines, concentration, and vorticity contours, and three-
171
-------
dimensional plots of the same information are provided for additional
qualitative assessment. Table 7.1 summarizes the momentum jet cases
run.
7.1.1.1 Centerline Velocity and Concentration for Momentum Jets
A similarity solution for vertical plumas was given in Chapter 4
as
E* = ^ + Jp- (Z2 - Z2) (4.43)
Z3 3ZFQ e
In the case of a momentum jet F -»• » so that,
E*= ^ • (7.1)
Je
Then by definition
^•f (7,,
/FT Ze
where again Vm is the centerline velocity, K is related to the plume
entrainment (see Table 4.1), numerically equal to 77, and Z is the
potential core length based on concentration (cf. Abraham [1]).
By Equation (7.2)
(7-3)
According to Abraham I ^ 5.6; hence,
\ - 6-2/2 (7.4)
which is also the result obtained by Albertson.
172
-------
Equation (7.4) implies that a plot of the dimensionless center-
line velocity, Vm, versus axial distance in port diameters Z has slope
of -1 when plotted to Log-Log scale, and has an intercept of 6.2 on the
^-coordinate when Vm-= 1. Experiments carried out by Albertson are
probably the most frequently quoted data bearing out Equation (7.4).
Various other researchers have carried out similar experiments (e.g.,
Baines, Tollmien and Reichardt [77]). Although there seems to be
general agreement that V ^ Z~ , there is some disagreement on the
potential core length (hence, the constant of proportionality), or the
Log-Log plotted intercept value mentioned above. A review of a portion
of this work is given by Gauntner. It is noteworthy to point out here
that the potential core length (see Figure 4.1) is assumed to be the
centerline velocity plot intercept (Vm=l, Z=6.2; see Figure 7.2),
AXIAL DISTANCE, t
Figure 7.2. General Features of Momentum Jet Centerline
Velocity (Based on Albertson's data)
173
-------
although the actual potential core length may be somewhat smaller.
For instance, Albertson measured an actual length of approximately 4.5
whereas their similarity solution is based on 6.2. The reason for
using the value 6.2 is that it is more representative of downstream
data than 4.5. As a matter of fact, similarity solutions are not valid
out to approximately 10 to 12 diameters. In Figure 7.2 , the distance
Z = 4.5 is the approximate distance where deterioration of centerline
velocity is first apparent.
Figure 7.3 illustrates centerline velocity, V , and concentration,
Cm, comparisons for
• Similarity theory
• Experiment, and
• The present computational technique.
The similarity theory concentration distribution along the centerline
is
Cm = 5.6/Z (7.5)
as given by Abraham.
Figure 7.3 indicates remarkable agreement between the computed
and measured centerline velocity distribution. Concentrations agree
with the similarity curve almost identically past t ^ 20. These results
are based on the Type 1 boundary conditions (Section 7.1). Computa-
tional runs 1 and 3 also use the Type 1 boundary condition, for differ-
ent water depths and node spacing; although these cases are not plotted,
centerline distributions nearly identical to those depicted in Figure
7.3 were obtained. The only deviation found between experimental and
174
-------
U
en
1.0
.5
.4
.3
.2
0-1
-
I
i
VELC
• EXF
• CO
°IK
A
)CIT
=»ERI
MPL
/IILA
-
Y
MENTAL RESULTS ALBERTSON ET AL. [4]
JTED
R SOLUTION V- ^p
. v CONCENTRATION
>
' i
^
\
\
4
i\
V
1
\
^
o CO
SI
AE
MPUTED
MILARITY SOLUTION c^
JRAHAM [1] C - *%
\ TYPE 1 BOUNDARY CONDITION
•xW v(R A *>) •=. c*~TT (.%/*)
Vx
\
V
i
23*5 10 20 3
I
1
-
-
-
0 4-0 SO 1C
AXIAL DISTANCE, 2
Figure 7.3. Comparison of Experimental Data and Similarity Solution with Computed
Results for a Momentum Jet. Center!ine Velocity and Concentration for Case 2.
-------
computed centerline velocity in these cases is that a very slight dif-
ference in slope was noted, whereby the computed slope was very
slightly less steep than -1.
Similar results for Case 4, which uses the Type 2 boundary condi-
tion, are given in Figure 7.4. Note that the 1/7 power velocity pro-
file gives a centerline value of 1.22 for an average jet exit velocity
of VQ = 1. These centerline velocity results are somewhat higher than
Albertson's data, but agree well with the data obtained by Baines for
an initial Reynolds number of 7 x 10 . Baines contends that there is
a Reynolds number effect on the potential core length and offers data
which apparently substantiates his assertion. According to Gauntner,
this facet of jet theory is apparently still unresolved.
The computed data for this case reveals the relationships:
Vm * 7/Z. (7.6)
and
Cm2i5.1/Z. (7.7)
Again, the computed velocity distribution is very slightly less steep
than a slope of -1.
Figures 7.5 and 7.6 illustrate centerline velocity and concentra-
tion distributions for Cases 5 and 6. Both of these cases again use a
Type 2 boundary condition with the inflow velocity distributions given
by
V(R,0) = (1-R)1/N. (7.8)
Case 5 uses N equal to 7 whereas N in Case 6 is equal to 10.
176
-------
1
1*0 1
E
U
i= :>
It"''
LU t J .•?
U 0
z: _i
0 U
u >
0 (
VELOCITY
• EXPERIMENTAL DATA FROM BAINES [8] REYNOLDS NO.-7*I04
• COMPUTED
•
»• i
-
-
-
-
i
<
1
<
i
(
i
v
\'
[k
I
\
v
V
<
\
i '
ss
i X
\
i
s
i
CONCENTRATION
\
•
N.,
'
o COMPUTED
r _ 5.1
m Z
\ ™
KV(.
VVI
\ N
\
\
i
PE 2 BOUNDARY CONDITION
R,0)» 1.22(1 -R)'/r
\
*X ^>s
^V i
V
i
\
\
,
^
^o
i
\
? f
\
-
•~
-
2345 10 20 30 40 50 100
AXIAL DISTANCE,
Figure 7.4. Comparison of Experimental Data and Similarity Solution with Computed Results
for a Momentum Jet. Centerline Velocity and Concentration for Case 4.
-------
1 .0
o
o
o
.5
.3
.2
. 1
VELOCITY:
• COMPUTED
Vn = 5.8/2
CONCENTRATION:
o COMPUTED
c = 5.1
TYPE 2 BOUNDARY CONDITION
V(R,0) = (1-R)1/7
5 10
AXIAL DISTANCE, 2
Figure 7.5. Computed Centerline Velocity and Concentration
for Momentum Jet, Case 5
50
-------
vo
£
U
Z
o
l-
z
(J
u
o
U >
1.0
.2
O.I
VELOCITY
• EXPERIMENTAL RESULTS ALBER^
•COMPUTED
— SIMILAR SOLUTION Vm a ^f
-
-
-
-
<
1
__..
i
4
i
\
N
,
i
\
V
\
i
\
\
\ I
\
V
(
\
K
CON(
o rn
V 1 YPt
^%
\
rSON ET AL.
:ENTRATION
[4]
MPUTED
'••¥
E 2 BOUNDARY CONDITION
t
\
1
1^
-
-
-
2345 10 20 30 40 50 10
AXIAL DISTANCE, 2
Figure 7.6. Comparison of Experimental Data and Similarity Solution with Computed Results
for a Momentum Jet. Centerline Velocity and Concentration for Case 6.
-------
According to Schlichting, these profiles correspond to pipe Reynolds
numbers of 1.1 x 105 and 3.2 x 106, respectively. The computing tech-
nique shows a marked difference between the asymtotic centerline veloc-
ities for these two cases, that is, for large 2,
Case 5: Vm ^ 5.8/2 (7.9)
Case 6: Vm ^ 6.2/2 (7.10)
Although the slope is still approximately -1 and the asymtotic concen-
tration for both cases is given by,
Cm^5.1/Z. (7.11)
Note that
V(0,0) = 1
which results in an average inflow velocity less than unity.
From these results it is tempting to conclude that since the
inflow velocity profile has an effect on the -1 slope intercept,
a Reynolds number effect on the potential core length is demonstrated.
However, it is felt that the lack of finite difference resolution and
shortcomings in modeling turbulence in the zone of flow establishment,
are sufficient to shadow such a conclusion. Comparing Case 4 where,
V(R,0) = 1.22 (1-R)1/7 (7.12a)
and Case 7 (Figure 7.7) where
V(R,0) = 1.155 (1-R)1/10 (7.12b)
reveals asymtotic velocity profiles,
Vm-7/Z (7.13)
and concentration
Cl 5.1/2. (7.14)
180
-------
I I I
1 .0
oo
.5
.1
VELOCITY:
• COMPUTED
Vm = 7/2
CONCENTRATION:
o COMPUTED
Cm = 5.1/2
TYPE 2 BOUNDARY CONDITION
V(R,0) - 1 .ISSO-R)1/™
1
Figure 7.7.
5 10
AXIAL DISTANCE, 2-
Computed Center-line Velocity and Concentration
for Momentum Jet, Case 7
100
-------
It is important to note that the jet exit average velocity in both
Cases 5 and 6 is unity whereas it is 1.22 in Case 4 and 1.155 in
Case 7.
In all computer runs cited thus far, the radial eddy viscosity
has been computed from Prandtl mixing length theory. This particular
aspect of the work is discussed in more detail in Section 7.2.
Essentially, the eddy viscosity is calculated by
er = c vmax rl/2> (7J5)
where vmav is the centerline velocity, r,/0 is the jet half radius and
ITiaX I / L.
c is a constant having the value .0256 for an axisymmetric momentum jet
(cf. Schlichting [84], p. 699); all cases thus far use c = .0256.
Case 8 (see Figure 7.8) uses c = .0263 (picked quite arbitrarily and
as a fraction is 1/38 = l/REr) and is to be compared to Case 5,
Figure 7.5. The net effect of this change is a slight shift in the
velocity slope toward -1 (difficult to see slope shift from compared
figures, but numerical results bear out the change). Although the
higher value of c appears to yield a velocity slope nearer -1, the
value c = .0256 is used for all following computations in this
manuscript.
Case 9 (Figure 5.9) represents an additional case using Type 2
boundary conditions with a velocity profile at the jet exit given by
1
V(R,0) = 1.24 (1-R)6'6 . (7.16)
Note that all cases (4, 7 and 9) use the boundary velocity profile
182
-------
T
1 .0
CO
oo
.5
E
. 1
VELOCITY:
• COMPUTED
- Vm = 5.7/Z
CONCENTRATION :
o COMPUTED
TYPE 2 BOUNDARY CONDITION
V(R,0) = (1-R)1/7
j i
10
AXIAL DISTANCE,
TOO
Figure 7.8. Computed Centerline Velocity and Concentration
for Momentum Jet, Case 8
-------
1 .0.
oo
-p.
.5
.1
VELOCITY:
• COMPUTED
CONCENTRATION:
o COMPUTED
-- Cm = 5.1/Z
TYPE 2 BOUNDARY CONDITION
V(R,0) = 1 .24(1-R)
5 10
AXIAL DISTANCE, Z-
Figure 7.9. Computed Centerline Velocity and Concentration
Distribution for Momentum Jet, Case 9
i i i
100
-------
V(R.O) = N+12N+1) (1_R)N (7.17)
2fT
for the jet. In all of these cases the asymptotic centerline velocity
profiles are essentially identical and represented quite accurately by
Vm^7/2 (7.18)
and concentration given by
(7.19)
Figure 7.10 illustrates these cases where the distribution is
normalized by dividing each value by the corresponding value of V(0,0).
The net result of this operation is that the solution collapses to the
cases using corresponding values of N and where V(R,0) is set by
Equation (7.8). Although this result was certainly expected, it serves
to illustrate that the computer program is functioning correctly in
this sense and to bear out again the velocity profile effect on the
asymptotic centerline velocity distribution (Figure 7.10). Computa-
tionally, this condition is apparently caused by the differences of the
jet exit vorticity distribution.
Vertical eddy diffusion, which should be of minor importance in
the jet mainstream, has also been ignored in cases cited to this point.
By ignored, it is meant that the value has been set to compare with
molecular viscosity which is perhaps three orders of magnitude smaller
than the jet induced eddy viscosity. The primary reason for vertical
diffusion being set to a very small value in these verification studies
185
-------
1 .0
oo
CT>
.5
CASE 9
I I
1 0
AXIAL DISTANCE, Z-
100
Figure 7.10. Centerline Velocity Distributions for Cases 4,
7, and 9, Normalized to VQ = 1.0
-------
is so that vertical entrainment near the surface where the jet is
spreading laterally will be minimized.
With a large value of vertical diffusion, in nonstratified
media, streamlines outside the jet would be distorted upward because of
the vertical entrainment in the lateral spread and would not be a
realistic representation of deep water conditions.
In Case 10 (Figure 7.11) the vertical eddy viscosity has been
accounted for by setting
= er
Figure 7.11 is to be compared with Figure 7.5 (Case 6). Case 10
shows a slight increase of centerline velocity over Case 6 which is an
effect to be expected if vertical diffusion has any importance, since
the shape-preserving vorticity will be transported downstream at a
slightly higher rate.
As further discussion of the above statement, Case 11 has been
run where the fluid was considered as inviscid, although rotational.
The numerical fluid reacted in a manner such that the jet exit velocity
profile was completely shape preserved until the surface effects were
encountered (see Figure 7.12). Considering the opposite extreme of
a hypothetical fluid where vertical diffusion completely dominates
radial transport, the same shape preserving nature would exist. Case
11 also served to illustrate the computational stability of the differ-
encing technique used for cases where Rer = Rez -*• ».
187
-------
1 .0
I t
oo
oo
.5
.3
.2
.1
VELOCITY:
• COMPUTED
CONCENTRATION:
o COMPUTED
__ c =5. 1/2-
m i /in
TYPE 2 BOUNDARY CONDITION V(R ,0) = (1 -R) '' 'U
I .... I
5 10
AXIAL DISTANCE, Z-
I '' r
50
Figure 7.11. Center-line Velocity and Concentration Distribution for Case 10
(Includes effect of large vertical eddy diffusivity.)
-------
linn/
VELOCITY
PROFILE
OUTFALL
PORT
Figure 7.12.
Shape Preserving of Velocity Profiles
Computed for an Inviscid, Rotational
Fluid (Ref. Case 2)
7.1.1.2 Spread of the Momentum Jet
The rate of spread of the half radius, ry2 is illustrated in
Figure 7.13-A and compared to measurements in Figure 7.13-B. The com-
puted rate of spread is given by
where
rl/2 = C1Z
C1 = .0875.
(7.20)
For the several momentum jet computations carried out, the above
equation holds. Table 7.4 compares some of the reported values of (^.
189
-------
o
^.
JP
10
SLOPE = .0875
I
20 30 40
AXIAL POSITION,?
50
60
JP
S-
• MEASUREMENTS BY TAYLOR ET AL. [97\
—REICHARDT [77],
o COMPUTED
8
12
16
20
24
B
28
Figure 7.13,
AXIAL POSITION, =t
Computed Rate of Spread of the Momentum
Jet Half-Radius, r,/2
32
190
-------
TABLE 7.4. COMPARISON OF THE SPREADING CONSTANT REPORTED BY
VARIOUS INVESTIGATORS
Investigator Comment
Albertson et al . [4]
Baines [8] Reynolds Number
7 x ID4
Baines [8] Reynolds Number
2.1 x 104
Reichardt [77]
Taylor et al. [97]
Corrsin and Uberoi [20]*
Keagy and Weller [49]*
Present numerical computation
Cl
.095
^ .085
^ .095
.0848
.0854
.0814
.0888
.0875
*Based on momentum measurements.
As Table 7.4 indicates, there is no universal agreement of the
value for C, among the cited investigators. These discrepancies are
possibly due to measurement methods and/or flow condition dependence.
Again, Baines offered data which tends to confirm the role of the
latter. Hence, the computed value of .0875 seems to be a realistic
value in view of reported measurement, but cannot be compared as an
absolute because of experimental discrepancies. Variations in the
half-radius may also be observed from Figure 7.12.
191
-------
7.1.1.3 Radial Distribution of Vertical Velocity, Concentrations
and Vorticity for the Momentum Jet
The radial distribution of vertical velocity for a momentum jet
is essentially Gaussian. For instance the data obtained by Albertson
is adequate expressed by
V = Vm e I , (7.21)
where
K = 77.
Likewise, concentration distributions are adequately given by
-AK(i)
C = Cm e Z (7.22)
where X is the eddy Schmidt number and equal to .8. The coefficient
K will vary from experiment to experiment similar to the variation in
data measured to establish the length of the potential core. As given
in Chapter 4, Baines found
-64.4(4) '
V = Vm e L (7.23)
for a Reynolds number of 7 x 10 and
R J-82
-43.3(f)
V = Vm e L (7.24)
for a Reynolds number of 2.1 x 104. Gortler [34] found K = 100. For
a summary of additional experimental data on the value of K one may
refer to Abraham [1],
192
-------
One should bear in mind that the use of the Gaussian distribution
has no theoretical basis, but is a result of curve fitting. Figures
7.14 through 7.18 all illustrate the vertical velocity profiles plotted
against different coordinates. Figure 7.14 illustrates the distribu-
tion of computed velocity for comparison with the data of Albertson
for Case 2 which uses the Type 1 boundary condition. Figure 7.15
relates this same type of information for Case 4 compared to the data
of Reichardt (cf. Schlichting). Figures 7.14 and 7.15, along with
Figure 7.16 (Case 6) provide a comparison with the Gaussian distribu-
tion. Computed information shows excellent agreement with the data and
essentially the same deviation from the Gaussian curve. Unfortunately,
correct numerical modeling at the jet boundary is practically unob-
tainable because of numerical smearing and inability to correctly model
turbulence at the jet boundary. These facets account for deviations
at the boundary and the fact that the computed velocity does not
attain zero at a finite radius.
Figures 7.16 and 7.18 also bear out the similarity of the com-
puted velocity profiles whereby the computed velocity at elevation
I = 10 shows the only appreciable deviation from complete similarity.
Baines' data is also illustrated in Figure 7.15. The various other
momentum jet case runs showed, upon spot check, that these curves are
typical of all cases run with similar assumptions.
Typical computed concentration profiles are shown in Figures 7.19
and 7.20. Again, as in the case of velocity, striking similarity is
evidenced in the radial distributions at all elevations. One noticable
fact is the deviation from a Gaussian distribution is more pronounced
193
-------
COMPUTED AT ELEVATIONS
•2=5.5
• =t = 10.5
= 15.5
= 20.5
GAUSSIAN, „
°D , ••
04
.08
12
. 16
.20
.24
R/Z
Figure 7.14. Radial Distribution of Normalized Vertical
Velocity for Case 2
-------
10
en
1.0
o
_J
LLJ
.6 —
.4 —
.2 —
EXPERIMENTAL DATA OF REICHARDT REPORTED
IN SCHLICHTING [84]
COMPUTED BY PRESENT TECHNIQUE
• 2 = 20
o f = 30
D 2 = 40
CURVE FIT OF COMPUTED POINTS
GAUSSIAN, i
1 .0
RADIAL POSITION, R/R.
1 .5
2.0
Figure 7.15.
Normalized Radial Distribution of Axial Velocity,
Momentum Jet Case 4
-------
CTi
1.0
.9
.7
.6
.4
.3
.2
.1
0
COMPUTED AT ELEVATIONS:
o 2 = 10
• 2 = 20
A I = 30
A 2 = 40
-I- EQUATION 7.24
x EQUATION 7.23
BAINES [b]
I
x
• fc
02 .04 .06 .08 .10 .12 .14 .16 .18 .20 .22
R/Z
Figure 7.16. Normalized Radial Distribution of
Axial Velocity Case 4
-------
(£>
COMPUTED AT ELEVATIONS
o 2 = 4
• 2 = 10
2 = 20
RADIAL POSITION, R
Figure 7.17. Radial Distribution of Axial Velocity
at Various Elevations Case 4
-------
10
00
o
o
1.0
.9
.8
.7
.5
.4
.3
.2
.1
0 *
COMPUTED AT ELEVATIONS:
o =t = 10
2 = 20
A* O
02 .04 .06 .08 .10 .12 .14 .16 18 20 22 24
R/Z
Figure 7.18. Normalized Distribution of Axial
Velocity Case 6
-------
. 4
COMPUTED AT ELEVATION:
o 2=5
• Z = 10
A Z = 20
T E = 30
A 2 = 3!5
V Z = 25
GAUSSIAN:
_
e
-68.1
m
0 .02 .04 .06 .08
.16
n 17
K, L
24 .26 .28 .30 .32
Figure 7.19. Normalized Rad'al Concentration Distribution,
Type 1 Boundary Condition Case 2
-------
ro
o
o
o
a:
o
o
1.0
.9
.8
,£.7
.6
.5
.4
.3
.2
.1
0
COMPUTED AT ELEVATIONS:
o 2=11
• Z = 21
A Z = 31
0 .02 .04 .06 .08 .10 .12 .14 .16 .18 .20 .22 .24
R/Z
Figure 7.20. Normalized Radial Concentration Distribution,
Type 2 Boundary Condition Case 4
-------
for these profiles. As in the case of velocity, concentration is
smeared to some extent across the jet boundary.
Figure 7.21 illustrates the vorticity profiles at several loca-
tions and Figure 7.22 compares the computed vorticity to the Gaussian
vorticity at elevations Z = 11, 31 and 41. Note that the computed
vorticity maxima occur nearer the jet centerline than similar maxima
for the Gaussian velocity profile. This fact is also revealed by the
experimental velocity data presented in the literature (cf. Figure 7-15).
7.1.1.4 Distribution of Radial Velocity for the Momentum Jet
A typical normalized distributional of radial velocity is illus-
trated in Figure 7.23 (Case 6). The solid line represents the Albertson
et al. theory and the dashed line represents an approximate envelope of
their experimental data. Albertson was unable to resolve clearly the
difference between the theory and his data. Misinterpretation of the
collected data may have been the cause of such a large discrepancy for
it hardly seems logical that his theory (based largely on empirical
results) could be so far in error. The radial velocities computed in
this study show good agreement with Albertson's empirical model, at
least over the range of positive velocities. Again, Albertson's data
shows gross disagreement with computed and experimental results for the
distributions of vertical velocity. The effect of this discrepancy
should be revealed most clearly along the jet centerline which is not
apparent from results (cf. Figure 7.6).
Figure 7.23 also reveals the similarity of radial velocity. It
is difficult to compare computed entrapment rates with the result
201
-------
,20
.15
>-
I—
o
a:
o
.05
COMPUTED AT ELEVATIONS:
• ? = 11
o Z = 21
o 2 = 31
6810
RADIAL POSITION, R
12
16
Figure 7.21.
Radial Vorticity Distribution for Momentum Jet
Type 2 Boundary Condition Case 4
202
-------
.15
• CASE 4
o CASE 1
— GAUSSIAN:
. 144 Vm R
10 —
,05
01234567
RADIAL DISTANCE, R
Figure 7.22. Radial Vorticity Distribution for Momentum Jet at
Z = 15. A Comparison Between Type 1 and 2 Boundary
Conditions, and the Gaussian Distribution.
203
-------
COMPUTED AT ELEVATIONS
* 2=11
° Z=21
• Z = 31
o Z=41
R/Z
.3
f *
.2
I APPROXIMATE BAND
OF DATA REPORTED
BYALBERTSON
-0.1
0.1
0.2
0.3
0.4
UZ
Figure 7.23. Normalized Radial Velocity Distribution
for Momentum Jet
204
-------
given in the literature because we have typically assumed a jet nozzle
extending into the fluid, whereas reported data is usually for wall
flush jets. In Case 6, this distance is four port diameters. Typical
experimental data may be correlated by
where C-| is an empirical constant, Q is the total vertical flow at
elevation Z, and QQ is the jet flow. Albertson gives C1 as .32.
Equation (7.25) indicates a constant entrainment rate for momen-
tum jets, or
§=C,Q0. (7.26)
Figure 7.24 is a plot of the computed stream function vertical distri-
bution at the inflow-outflow boundary (i.e., ^(R^.ZjJfor Case 6. By
definition the differential stream function along this vertical plume
is a measure of the entrained flow; that is,
A4- = - URAZ. (7.27)
The total flow through the plane Z = 4 is given by
4) -
-------
The straight line fit of the computations illustrated in
Figure 7.24 is
, Z) = .233 (Z-Zport) + 1.75. (7.29)
Then, based on the intercept with 2 = 4,
= -33 ("* • (7-30>
Hence, using QQ as the total of the jet effluent plus entrainment
from below the port gives a lateral entrainment rate comparable to the
reported work where the fluid issues from a wall-flush jet.
12
10
8
r+J
2 6
—*
^
4
j I
I
10 20 30 40
AXIAL DISTANCE,
50
60
Figure 7.24. Vertical Distribution of Stream
Function V , Case 6
206
-------
7.1.1.5 Typical Contours and Three-Dimensional Plots for a Momentum
Jet
Additional information may be obtained by inspecting the level
lines and distribution surfaces of the stream function, concentration
and vorticity. The centerline and surface streamlines are set at
¥ = 1.0. This information is illustrated in Figures 7.25 through 7.31.
The three dimensional plots (Figures 7.28 through 7.31) have been arbi-
trarily scaled to fit a prescribed data box and are valuable for quali-
tative reasons alone.
7.1.2 Two Cases of Pure Buoyancy
To check the computer program and computational techniques where
buoyancy is the sole driving force, t.wo cases were run where the out-
fall port or jet was replaced by a heat source (see Table 7.2). In the
case of pure buoyancy, we are checking the same general features of the
plume as in the case of the momentum jet. However, there is much less
information published. Here we check the computed
• Centerline velocity and temperature,
. Radial distribution of axial velocity and temperature, and
• Rate of plume spread
for a very weak and intermediate strength buoyant source. Both cases
are well within the validity of the Boussinesq approximation. Solution
restrictions are the same as those pointed out in Section 7.1.1.
7.1.2.1 Centerline Velocity and Temperature
For a purely buoyant source (and also for effluent cases where
F0 = 0) it has been established by Rouse et al. [8]] and Schmidt [85]
207
-------
ro
o
c»
S-OD
1O.OO I5-OQ
RRDIRL D1RECTIQN. R/D
ZO.QO
FIGURE 7.25
STRERMLINES F9R CRSE 6 -- MOMENTUM JET
-------
ro
O
5-oo
10.DO 15-DO
RRDIRL DIRECTION. R/D
ZD.OQ
FIGURE 7.26. ISQFYCNRL5 FQR CRSE 6 -- MOrlENTUM JET
-------
5-DO
10.00 15-DO
RRDIRL DIRECTION. R/D
za.aa
FIGURE 7.27. VQRTICITY LEVEL LINES FOR CRSE 6 -- MOMENTUM JET
-------
ro
FIGURE 7.28
ILLU5TRRTI0N 9F STRERM FUNCTIQN -- P3I. CflSE NO. 6
-------
ro
r>o
FIGURE 7.29. 3D ILLUSTRRTIQN 0F STRERM FUNCTIQN -- PSI. CASE NO. 6
CRSE - DEEP WRTER MOMENTUM JET - VfR.OJ = (1-R)••(1f\ 0 )
-------
IN3
CO
FIGURE 7.30. 30 ILLU5TRRTION 0F BUOYANCY DISTRIBUTION -
CRSE NQ.6
-------
ro
FIGURE 7.31. 3D ILLUSTRRTI0N 0F FLUID VQRTICTY - QMEGR. CRSE NQ.6
-------
that
V'V? / "7 O 1 \
m — (7.31)
and
ATm - Z" (7.32)
In the case of an effluent with little initial momentum and strong
buoyancy, Abraham [1] gives
Vm = 4.4(FQZ)- (7.33)
ATm = 9'5 Fo Z" (7-34)
based on Rouse's data.
Figure 7.32 illustrates the center! ine velocity and temperature
for Case 13. In this case, the source is very weak and gives a maxi-
mum fluid temperature rise of only .95 °C. The maximum velocity is a
little above .09 ft/sec occurring at an elevation of about seven
source diameters above the source. The flow apparently does not
become established until an elevation of 15 to 20 diameters has been
reached. Above this approximate region the computed center! ine
velocity shows decay very closely approximating the -1/3 law given by
Equation 7.31. Velocities computed above Z = 50 (surface at Z = 64)
show influence of the free surface.
Temperature decay, on the other hand, begins to follow Equation
s
(7.32) at approximately Z = 10 and computed values are extremely close
to a - 5/3 slope. However, there is no apparent surface effect on
temperature, whereas Case 14 (Figure 7.33) reveals noticeable change
215
-------
1.0
.5
.1
.05
.01
SLOPE = -5/3
SLOPE = -1/3
CENTERLINE VALUES OF:
•VELOCITY, Vm
0 TEMPERATURE EXCESS, AT
I I . I ... I I ,
m
5 10
AXIAL DISTANCE,
50
Figure 7.32 Computed Center!ine Velocity and Temperature
Excess for Case 13. Pure Buoyancy, F = 0.
216
-------
1.0
.5
.1
.05
.01
0
T T
T 1 ' | I I I I.
*»*•
SLOPE = -1/3
v
\
\ SLOPE = -5/3
\
CENTERLINE VALUES
•VELOCITY, V
m
\
TEMPERATURE EXCESS, AT
m
\
. . i .... I
5 10
AXIAL DISTANCE,-Z
50 100
Figure 7.33. Computed Centerline Velocity and Temperature
Excess for Case 14. Pure Buoyancy, FQ = 0.
217
-------
in slope near the surface. It is felt that continued iteration would
have shown somewhat larger deviation from the -5/3 slope near the sur-
face in both of these cases.
Figure 7.33 (Case 14) illustrates similar results for a situation
where the fluid directly in contact with the heat source was maintained
at a 25 °C temperature rise. Under these conditions the maximum
velocity was about 0.8 ft/sec occurring at approximately 8 diameters
above the source. The shape of the center!ine velocity distribution is
very nearly the same as in Case 13 and achieves the -1/3 slope at
approximately 20 diameters above the source. The temperature distribu-
tion, however, shows some differences in that the -5/3 decay is not
attained until about 20 diameters and, as mentioned previously, there
is demonstrated a marked surface effect. Results for both of these
cases could be improved somewhat by continued iteration in the vicinity
of the surface. Convergence was slow in this region for both runs, but
temperature changes indicated an increased surface effect. Another
aspect is that vertical turbulence has been essentially neglected, a
poor assumption in the surface effects region. A realistic approxima-
tion of vertical turbulence here would also tend to increase the sur-
face temperature.
7-1.2.2 Spread of the Pure Buoyant Plume
The rate of spread of the half radius, r, ,„, for pure buoyancy is
demonstrated in Figure 7.34 for Case 13. Case 14 was found to be
essentially identical to Case 13. Based on Rouse's data, Abraham ascer-
tained that the half radius is approximated by
218
-------
o
<
00
O
= 20 -
x
<
0
Figure 7.34.
'1/2
D
Computed Rate of Spread of Half-Radius,
Pure Buoyancy, Case 14 (D=2rQ)
1/2
D
L69
K
Z = .0866 Z
(7.35)
where K = 92.
The data obtained by Rouse revealed K = 96, at least for the
selected curve fit. Abraham's theory and experiments yield K = 92,
and according to him, no major discrepancy in results is obtained in
either case. Figure 7.34 reveals a computed spread of approximately
219
-------
.092 I. Not only is this rate of spread different from the rate based
on a Gaussian profile, but the rate is greater than in the case of
pure momentum (r1/2/D ± was computed). Gaussian profiles show the
opposite to be true. The reason for these discrepancies has not been
completely resolved.
Barring difficulties with the computer code, which has been
checked, the discrepancy may be caused by incorrect modeling of the
turbulence in the presence of buoyancy. It is also possible that the
data obtained from flame sources in air may be significantly influenced
by effects not accountable through the Boussinesq approximations. That
is, the Boussinesq approximation would not be valid for modeling plumes
over diffusion flame plumes because of the large density variations
compared to the reference density, even though temperature will decay
quite rapidly. In both Cases 13 and 14, the density variations may
influence the rate of spread and explain the present discrepancy.
Additional data for a low Froude number flow case is presented in
Section 7.1.3.2.
7.1.2.3 Radial Distribution of Vertical Velocity, Temperature and
Vorticity for Pure Buoyancy
The data obtained by Rouse and Schmidt demonstrate that the
normal distribution curve again fits the buoyant plume radial profiles
quite well.
In this case, data obtained by Rouse gives
220
-------
^2
VI \
"" »"T~'
V = V P / •? i/- ^
vm (7.36)
where K = 96, and
AT = Tme (7.37)
where
A = .74.
However, the Gaussian curves used for comparisons here will be based on
Abraham's value of K = 92 which yields xK = 68.1. As in the case of
the momentum jet, these distributions have no theoretical basis, but
are a result of curve fitting.
Radial distributions for Case 13 arc illustrated in Figures 7.35,
7.36 and 7.37, for various elevations. Computed results show excellent
similarity at all elevations except near the source (Figures 7.35 and
7.36).
Figure 7.37 shows the velocity profiles for Case 13 as computed.
Figure 7.38 again shows excellent similarity at all elevations except
near the source for Case 14.
A normalized temperature profile is illustrated in Figure 7.39
and vorticity at various elevations is plotted in Figure 7.40. One
notable feature revealed by Figure 7.39 is that the temperature distri-
bution is in considerably closer agreement with the Gaussian curve in
the case of a momentum jet (cf. Figure 7J
221
-------
ro
IN}
ro
O
o
1.2
1.0
.8
.6
.4
.2
0
-A.
0
COMPUTED AT ELEVATIONS:
• 2=20
*2=30 GAUSSIAN:
R 2
A -2 = 40 V -92(f)
V"=e Z ;
m
ABRAHAM [1]
5 1.0 1.5
RADIAL POSITION, R/R1/2
2.0
Figure 7.35. Normalized Distribution of Computed Axial Velocity.
Pure Buoyancy, Case 13
-------
r\>
ro
CO
1 .0
E .8 ~
.6 —
-------
ro
ro
CJ
CD
.10
08
06
.04
02
—
_
••°°o
_ • o
• 0
LODD8°D8
AA A A A A A A 8
_ Q ft A
A
• DA
~ * o a
• _ a
__ • o
a
• o
~ • °
o
•
HEAT SOURCE •
•
COMPUTED AT ELEVATIONS:
• ? = 4
o Z = 10
a 2 - 20
A I = 30
A
A
A
D A
a A
D
o a
0 D
0
o
o
• I • 1
1 • • • 1
—
^_
_
««•
—
, — _
A
a —
o
4
R
8
Figure 7.37. Radial Distribution of Axial Velocity in Pure Buoyancy, Case 13
-------
ro
IN3
en
o
o
1 .0
.8 -
.6 -
4 -
X
•=c
I I I T i \ I I I I I I
COMPUTED AT ELEVATIONS:
t = 10
Z = 20
Z = 30
* = 40
GAUSSIAN:
y = v e-92(R/Z)2
m
ABRAHAM [1]
I I
• ma
J I
o-
J I
.04 .08 .12 .16 .20
RADIAL POSITION, R/Z
.24
.28
Figure 7.38. Normalized Radial Distribution of Axial Velocity.
Stronger Source, Pure Buoyancy, Case 14.
-------
ro
ro
CTi
X* w
! AT =e L '
\S \° ABRAHAM [1]
, 1 , I>H , 1 , "
.12 .16 .20 .24 .2
R/Z
Figure 7.39. Normalized Distribution of Computed Radial Temperature Excess.
Pure Buoyancy, Case 14.
-------
.20
.15
G
>-"
o
£• in
o •"
>
.05
0 i
0
1 ' 1 ' 1 '
*\
(?>
o
~ o —
o COMPUTED AT ELEVATIONS:
o
o 2 =11
0 -, 2 01
0 * Z = 21
* 2 =31
o
0
0
^»_
w®
0 • • _
• «
• o
9
O A •
X*°*
• / »
-• a o *
° » A
-)• A
J A 0 • A
o • A
• A ° ^ A
AA o • A
? , , , 1° «. o' ,' • 1
4 8 12 1(
R
Figure 7.40.
Radial Distribution of Vorticity.
Pure Buoyancy, Case 14.
227
-------
7.1.2.4 Radial Velocity and Entrainment for Pure Buoyancy
The normalized distribution of radial velocity for Case 14 is
given in Figure 7.41. As opposed to momentum jet results (Figure 7.23),
similarity of the radial flow is not apparent using the coordinates
R/Z and UZ. Also note that, compared to the corresponding momentum
jet data, the magnitude of negative radial flow is somewhat larger,
indicating an increased radial entrainment rats. Although it has not
been plotted, the radial flow below about six source diameters is
negative over the entire flow field.
From similarity theory it has been established (cf. Abraham)
that
§ = W- (7-3S>
By Equation (7.31)
2/3
—*• = C^
Then integrating Equation (7.39) yields
Q = C3 Z5/3. (7.40)
The values C-j, C2 and C3 are appropriate constants; magnitudes are
unimportant since we are interested only in how Q varies with Z.
Figure 7.42 illustrates the value of ¥(Roo,Z) as a function of Z, and
since i^R^.Z) is directly proportional to the entrainment Q, this plot
reveals the variation of Q with Z for pure buoyancy. The computed data
in Figure 7.42 is obviously represented by a functional relationship
more complicated than Equation (7.40). At lower elevations (Z^ 6 to 15)
228
-------
.6
.5
Ivl
•^
ct
.4
.3
.2
T 1
COMPUTED AT ELEVATION:
-.8
-.6
-.4
-.2
.2
U2
Figure 7.41. Normalized Radial Velocity Distributions
for Pure Buoyant Plume, Case 14
229
-------
TOO
r+j
50
10
I I
SLOPE = 5/3
SLOPE = 1
I I I I
10
50
AXIAL DISTANCE, Z
100
Figure 7.42. Vertical Distribution of Stream Function
at R . Pure Buoyancy, Case 14
Q -\, Z. (7.41)
Thus, in this range the plume entrains ambient fluid proportional to a
momentum jet. The 5/3 slope is never indicated by the data, but
Figure 7.42 shows that the entrainment data would apparently approach
a 5/3 slope asymtotically for sufficient depth.
Figures 7.43 through 7.45 illustrate streamlines, isotherms and
vorticity level lines for Case 14. As in all cases reported, the cen-
terline value of the stream function is 1.0. Three-dimensional illus-
trations of the same information is displayed in Figures 7.46 - 7.48.
230
-------
ro
CO
in.an IS.QQ
RRD1RL DIRECTION. R/D
ZO.DD
Figure 7.43. Streamlines for Case 14, Pure Buoyancy
-------
ro
co
rvi
.005
Isotherms not labeled (from heat source)
No. AT/AT0
.75
, 50
.25
.20
.15
5-aa
la.aa iS-aa
RflDIBL DIRECTIQN. R/D
zo.oa
Figure 7.44. Isotherms for Case 14, Pure Buoyancy, AT/ATQ
-------
r-o
CA>
co
PJ
JS-00
RRDIRL DIRECTION. R/0
za.oa
Figure 7.45. Vorticity Level Lines for Case 14, Pure Buoyancy
-------
ro
CJ
Figure 7.46. 3D Illustration of Stream Function - PSI, Case No. 14.
-------
ro
CO
en
Figure 7.47. 3D Illustration of Tenrperature Field - AT, Case No. 14.
-------
Figure 7.48. 3D Illustration of Fluid Vorticity - Omega, Case 14, Pure Buoyancy.
Temperature Difference of Source Maintained at 25 oc.
-------
7.1.3 Mixed Flow - Forced PI
umes
In Sections 7-1.1 and 7.1.2 we have checked in some detail the
computed flow characteristics at both ends of the dynamic spectrum--
pure momentum and pure buoyant flows. This section deals with flows
having dynamic characteristics of both which are appropriately classi-
fied as "forced plumes' as coined by Morton [58]. Cases used to com-
pare with similarity solutions and experimental data are summarized in
Table 7.3. To this end, a variety of effluent boundary conditions have
been investigated.
The cases here are too numerous to treat each in full detail so
that only the general characteristics of
« Centerline velocity and temperature, and
• Rate of spread and entrainment
will be illustrated, along with selected contour and three-dimensional
plots. The similarity solution discussed in Chapter 4 will be used for
comparison.
7.1.3.1 Centerline Velocity and Temperature for Forced Plumes
In Chapter 4 the following similarity solution was given for
vertical forced plumes:
(7'42)
e
and
I- 1/3
(7.43)
237
-------
the variable
V_Z
*
E = -— , (7.44)
~
and Z is based on Equation (4.19).
The above equations, except for (7.44), do not reveal variations
in the values of K and \. These values and their effect on the govern-
ing equations have been discussed in Chapter 4 and are summarized in
Table 4.1. The largest error in velocity is seen to be introduced by
1//K (4.8% deviation from the mean value) but is absorbed in E*.
Equations (7.42) and (7.43) reveal the use of simple fractions
which simplify the equations and are very close to the mean values
given in Table 4.1. Since these variations are small, and in view of
experimental data scatter, it does not seem justified to use more
complicated relationships for K and x as did Abraham, at least for the
vertical plume. At any rate, the subject equations yield results that
are in good agreement with Abraham's computations and yield excellent
agreement with Fan's [27] data concerning the maximum height of rise
where stratification is of concern (cf. Baumgartner and Trent [12] )•
Thus Equations (7.42) and (7.43) will be used to compare with the finite
difference results.
Cases 15, 16 and 17 compare the effect of three different methods
of computing the radial component of eddy momentum diffusiviity, e .
In all cases er is computed from
- -0256 VmaxRl/2 ' -0256
238
-------
however, different methods for computing FR are used. A detailed dis-
cussion for this computation is given in Section 7.2.
Case 15:
FR = constant = 1.178
Case 16:
FR = .180 VmZ.
where Vm is the currently ...aiculated value of center!ine
velocity at elevation Z.
Case 17:
FR = VmRl/2
with running calculation of both V and R-i/2>
all other conditions for these cases remain fixed.
Figure 7.49 illustrates the centerline velocity, V , and buoyancy,
A, , for these three cases. The significant feature of results shown
in this figure is that using a constant value for er (Case 15) gives
results with appreciable error in buoyancy (or temperature). The use
of a pre-calculated half-radius (mixing length) based on a Gaussian
velocity distribution gives somewhat better results (Case 16). The
similarity solution is found to give quite accurate results for I > 15-
20 and Case 17 shows buoyancy results in excellent agreement with the
similarity solution, although the velocity distribution shows a sizable
difference. The large discrepancies in both velocity and buoyancy at
lower elevations (Z ^ 10) are expected since similarity solutions are
not valid in this range.
239
-------
1 .0
ro
-P.
o
.5
.4
.3
.2
.1
CASE 15 16
'1m
m
A
A
17
a
SIMILARITY SOLUTION
I
... I
5 10
AXIAL DISTANCE, I
50
100
Figure 7.49. Centerline Velocity and Buoyancy for Cases 15, 16 and 17
-------
These three cases also represent progressively more difficult
computational problems owing to the non-linearity of the eddy
diffusivity.
Case 15, where a constant value of e is used, caused no compu-
tational difficulties and is of course the fastest with regard to com-
puter time. This problem is quite similar to the laminar flow plume
problem, but er can be several orders of magnitude larger than the
counterpart molecular momentum diffusivity. Case 17, where V and
RI12 are computed iteratively is the most difficult and requires the
most computer time. The computational difficulty stems from the fact
that velocity profiles at the initiation of the FR computation cannot
be too far in error or a numerical instability will result. In addi-
tion, the convergence rate is slowed by continuous updating of FR.
Returning to the discussion of momentum jets (Section 7.1.1),
only Cases 2 and 3 used e = constant, all other cases used FR calcu-
lated as in Case 17. However, in the case of a momentum jet, FR is
indeed constant so that any of the three methods for computing er
should yield essentially identical results (see Section 7.2). Only in
the case where buoyancy is present will variations in FR become appar-
ent, and for this reason, demonstration of results was deferred to
cases dealing with mixed flow.
Case 18 is identical to Case 17, except the vertical grid spacing
has been doubled giving an overall depth of 82.5 port diameters.
Figure 7.50 illustrates centerline buoyancy for the case compared with
the results of Case 17 along with the similarity solution. Slightly
higher values for buoyancy were calculated in Case 18 compared to
241
-------
Case 17, an effect of doubling the vertical grid spacing,
o:
<
D_
o
rj
CO
1.0
.5
.1
.05
01
• CASE 18
• CASE 17
— SIMILARITY
SOLUTION
10
AXIAL DISTANCE, Z
100
Figure 7.50. Centerline Buoyancy Distribution
for Cases 17 and 18
Figure 7.51 illustrates the centerline velocity and buoyancy
distributions for Cases 17, 19 and 20 where the densimetric Froude
numbers are 52, 35 and 106, respectively. All other variables are
fixed for these cases. Case 21 is identical to Case 17 except the
vertical eddy momentum diffusivity, EZ, was assumed to have the form
ez = ezoe
-A2(ZS-Z)2
(7.46)
242
-------
1.0
10
20 30
AXIAL DISTANCE, 2
Figure 7.51. Centerline Velocity and Buoyancy for Cases 17, 19 and 20
243
-------
where A is a constant, Z is the surface elevation and e is a
o
reference eddy diffusivity. The objective of this case was to illus-
trate the effect, on the plume flow, of substantial eddy diffusion
confined near the surface. Although the exact values of e and A are
zo
of little importance to this end, they have values: e = 1 and A = .2.
zo
The only significant effects caused by this treatment of e are in the
radial spread and vertical diffusion of vorticity and radial velocity
at the surface. In the case of negligible vertical momentum diffusion,
vorticity tends to accummulate in the surface nodes and the mass tends
to spread frictionlessly within these surface nodes at high velocities.
The presence of significant vertical eddy transport diffuses the
vorticity and velocity further downward into the ambient fluid.
Figure 7.52 shows a vorticity ridge near the surface for an essentially
frictionless flow (Case 17), whereas Figure 7.53 (Case 21) illustrates
considerable mitigation of this ridge through vertical diffusion.
Cases 22 and 23 differ from the preceding mixed flow computation
in that a Type 2 boundary condition is used with N equal to 7 (refer to
Equation 7.17). Various other differences are noted from Table 7.3
(e.g. the Froude number and finite-difference grid). These two cases
are identical to one another except the eddy Prandtl number in Case 22
is .714 whereas in Case 23, .80 is used. These computations were per-
formed primarily to determine the effect of the Prandtl number on the
rate of spread (Section 7.1.3.2). However, an appreciable effect is
also noted on the centerline buoyancy distribution (Figure 7.54),
whereas little difference was found in centerline velocity for the two
244
-------
ro
-P.
en
FIGURE 7.52. 3D ILLUSTRflTION 0F V8RTICITY OMEGfl
CASE 17 BUGYflNT PLUME WITH RUNNING CflLCULflTION OF HRLF RROI'JS
-------
ro
-pa
en
FIGURE 7.53. 30 ILLUSTRRTION QF VORTICITY --- GMEGfl - CflSE NO. 21
-------
1 .0
ro
.1
.04
g
CASE 22, PR = .714
BUOYANCY:
D CASE 23, PRr = .800
• SIMILARITY SOLUTION
VELOCITY:
• CASE 22
• CASE 23
— SIMILARITY SOLUTION
, , 1
Figure 7.54.
10
AXIAL DISTANCE, Z
Center-line Velocity and Buoyancy for Cases 22 and 23
100
-------
cases. Since the power law effluent velocity profile is used, with N
equal to 7, the maximum centerline velocity is approximately 1.2.
This boundary condition indicates much better agreement with the simi-
larity solution for downstream velocity than was obtained using the
Gaussian profile (Type 1 boundary condition) in preceding mixed flow
cases.
Figure 7.55 shows centerline distributions for Case 24 which is
identical to Case 23 except the densimetric Froude number is 1.0 as
opposed to 46. Unfortunately, the eddy Prandtl number for this case
was not reset to .714 (.8 was used). This error was not discovered
until the contents of the restart tape were destroyed; hence, for
economic reasons the case was not rerun (cases for very low Fn are
slow in converging). However, the slope of the buoyancy curve is
essentially identical to the similarity solution and, borrowing the
trends of Cases 22 and 23, the buoyancy curves would nearly coincide if
PRr equal to .714 had been used. Also, from Figure 7.54 we would
expect no appreciable change in the velocity distribution of Figure
7.55.
Figure 7.55 illustrates that for low F , the velocity initially
increases due to the large relative buoyancy, reaching a maximum at
about 5 diameters downstream. The velocity distribution then tends to
a - 1/3 slope as in the case of pure buoyancy. Likewise, the buoyancy
distribution tends to a - 5/3 slope as in purely buoyant plumes.
Figure 7.56 illustrates centerline distributions for F = 1000
(Case 25) compared to computed results for a momentum jet (Case 4).
248
-------
10
1.0
.1
.001
SIMILARITY SOLUTION
COMPUTED (PRr • .80)
VELOCITY. V
5 10
AXIAL DISTANCE. Z
50
100
Figure 7.55. Comparison Between Computed Results and Similarity
Solution for FQ = 1.0
249
-------
ro
en
1 .0
.5 -
. 3
.2
. 1
10
AXIAL DISTANCE, I
50
100
Figure 7.56. Comparison Between Computed Center!ine Distributions
of Velocity and Buoyancy for FQ = 1000 and FQ -> ~
-------
7.1.3.2 Rate of Spread and Entrainment
Results from momentum jet computation revealed that the jet half
radius spreads according to
r]/2 = .0875 z. (7.47)
and pure buoyant plume calculations yielded
r]/2 ^ .092 z. (7.48)
Although these results showed a reverse trend from experimental
observation, absolute values are not in large disagreement with experi-
ment. Figure 7.57 illustrates the rate of half-radius spread for
FQ = 0, 1, 46 and ». The effect of different eddy Prandtl numbers for
FQ = 46 is revealed by Figure 7.58 (Cases 22 and 23). As pointed out
earlier, the case for FQ = 1 was inadvertently run using PRr = .8 and
Figure 7.57 shows that this case has the same spread rate as the case
where FQ -»• °°. Thus, the fact that one case is dominated by initial
inertia and the other by buoyancy seemed to have no effect on the
half-radius spread rate. This being a fact of the computational tech-
nique then explains why the plume has a larger computed spread rate
where PRr = .714 as opposed to PRr = .8. It is expected that had
PR = .714 been used in the FQ = 1 computation, the half-radius curve
would have coincided with the curve for F = 0. Case 22 where
F = 46 shows that the half-radius begins to spread as a momentum
jet (I <_ 10-12), passes through a transition and then spreads at the
same rate as a purely buoyant plume, at far downstream points. Case
23 begins to spread as a momentum jet, then passes through a
251
-------
en
ro
rtj
eC
»—i
X
60
50
40
30
20
10
~\ I
(CASE 22) -^
60
50
40
<" 30
i—i
Q
x 20
-------
transition to a wider spread, and at far downstream points, again
spreads like a momentum jet (but wider).
Figure 7.59 shows the variation of ^(R^.Z), a measure of entrain-
ment, with elevation. Again we cannot expect good correspondence with
wall-flush jets at lower elevation since for the cases illustrated the
outfall port has finite height above the bottom. At higher elevations
we note that for FQ = 1000 a slope of 1 is attained which is appropriate
for momentum jets. The case for F = 1 has obtained a slope of
approximately 1.4 and is increasing. Had the solution been carried to
higher elevations, the experimental value of 5/3 would perhaps be
attained. For F = 46 we find intermediate values of \HR »2) with the
o °°
slope tending toward that of the case for F = 1. Again the slope is
increasing and would perhaps attain the value of 5/3 as in pure
buoyancy, at increased axial distance.
Figures 7.60 through 7.62 illustrate streamlines, isotherms and
level lines of vorticity, respectively, for Case 22. Figures 7.63
through 7.65 show this same information in three-dimensional plots.
7.2 Transport Coefficients
In obtaining the results presented thus far, we have made use of
certain transport coefficient models which describe the required compo-
nents of radial and vertical turbulent diffusion. This thesis, in the
main, is not a study of modeling these coefficients but through neces-
sity one must utilize reasonable methods for modeling these quantities
if reliable results are to be obtained. For the momentum jet issuing
to a semi-infinite medium, the important transport coefficient models
253
-------
TOO
O
O
c
3
U_
E
fO
0)
-M
OO
50 -
10 —
5 -
AXIAL DISTANCE,
Figure 7.59. Entrainment Trends in Mixed Flows
254
-------
IV)
on
cn
10-00 15.00
RRDIRL DIRECTION, R/0
20-00
FIGURE 7.60. STREAMLINES FOR CRSE 22 - MIXED FL.0W. FO = 46
-------
ro
01
O1
ISOTHERMS NOT LABELED:
NO. T, °C
30
28
26
25
24
°o
10.00
RRDIRL DIRECTION. R/D
FIGURE 7.61
15.00
ISOTHERMS FOR CRSE 22 - MIXED FLOW, FO = 46
20.00
-------
I\J
in
LEVEL LINES NOT LABELED:
NO. ft
.5
.25
.15
s.t
10.00
RflDIRL DIRECTIQN. R/D
15-00
20.00
FIGURE 7.62. VORTICITY LEVEL LINES FOR CflSE 22 - MIXED FLOW, FO - 46
-------
en
CO
\
FIGURE 7.63. 3D ILLUSTRRTI ON QF STRERM FUNCTION -- PSI
CRSE 22 - DEEP URTER BUOYRNT JET
-------
r>o
en
FIGURE 7.64- 3D IL.LUSTRRT I ON QF TEMPERRTURE FIELD -- T
CRSE 22 - DEEP WRTER BUQYRNT JET
-------
ro
en
o
R --
FIGURE 7.65- 3D ILLUSTRRTI0N QF FL.UID VQRTICTY - QMECfl
CRSE 22 - DEEP KRTER BUQYRNT JET
-------
turn out to be trivial since they are constant. However, where
buoyancy plays a role and the buoyant surface spread in stratified
media is of concern these models are quite complicated and in certain
instances (surface spread) the theoretical and experimental efforts
are sadly lacking.
In this work it is necessary to model the momentum diffusion
coefficients for the radial and vertical directions, er and e , along
with the corresponding Prandtl (or Schmidt) numbers, PR and PR .
Turbulence contributions may be considered to fall into the following
two categories:
1. that generated by the effluent stream, and
2. the ambient contribution which has origin from
• wind stress and wave action,
• shear flow at solid boundaries, and
• contributions depending on the local history
and/or convection across system boundaries.
In general, the effluent generated turbulence will dominate the
ambient contribution within the plume except in the surface zone
where plume velocities may be low and wind and wave action under a
high sea state dominate the effluent induced effects. However, in
the circulating portion of the flow field, ambient contributions will
dominate.
The turbulence models used in the present work are based on
Prandtl's second hypothesis which is appropriately modified to include
the influence of stratification. Experience has found that Prandtl's
261
-------
hypothesis may be applied with good results where mean velocity grad-
ients have reasonable magnitude and a mixing length may be easily
defined, but breaks down entirely, at least computationally, where
velocity gradients are very small, or confused, and the mixing length
has dubious interpretation (e.g., the circulating flow). Prandtl's
hypothesis, as stated by Schlichting [84], is
8V . / \ 3V
T = pe — - pC-, b(V -V • ) —
where T is fluid stress, C, is an empirical constant, and b is the
width of the mixing zone. The eddy diffusivity for momentum er is
then
er = ClV <7-
where £ is the mixing length of an axisymmetric plume and assumed to
be the width of the half-radius in established flow. An equivalent
relationship may be written for e , the vertical component, in the
zone of surface spread. In the mainstream of the plume, the usual
case is that-only one or the other of the transport coefficients will
have a significant effect on the flow dynamics. For instance, in the
vertical rise, er is of utmost importance ,whereas e may be neglected
as a practical matter. However, EZ is included in the computations,
and may in fact be important near the surface where vertical velocity
may be small. In the lateral spread, the opposite is true where e
has relatively small influence. The value v_, in the zone of plume
max r
rise is easily defined as the centerline velocity, u in the lateral
max
spread will occur at the surface for a buoyant flow in homogeneous
262
-------
surroundings. In both cases the maximum velocity has sufficient magni
tude compared to velocities outside the plume so that v »v . and
r max mm
vmax
umax>>umin- Hence>
and
where £ is an as yet undefined vertical mixing length in the vertical
direction. Note, that Equation (7.51) includes no compensation for
stratified flow.
Equations (7.50) and (7.51) are adequate for modeling the tur-
bulence inside the plume and are relatively convenient to use, but
only because we have prior knowledge of the plume geometry. Outside
the plume, in the region of flow induced circulation, these expres-
sions are useless because we have no adequate criterion for mixing
lengths and, in fact, velocity gradients may have nothing to do with
the primary contribution to the field of turbulence. Fortunately, for
the problem at hand, turbulence in the circulating field is of nominal
importance, and except for the fact that some degree of viscosity in
this region helps to speed the numerical computation, we could assume
the fluid as inviscid.
It is recognized that Prandtl's second hypothesis has limited
application in the numerical computation of circulating and recircu-
lating fluid flow. Prandtl recognized the shortcomings of this
hypothesis in that it could be applied with confidence only to rea-
sonably simple, steady-state flows. Various other investigators also
263
-------
recognized that a more fundamental approach needed to be employed.
Such an approach needed to consider such various aspects as
• convection,
• diffusion,
, creation, and
• dissipation
of the turbulence which could be related in some fashion or another to
mean flow quantities. Earlier models were based on the transport of
turbulent energy. However, these models still depended on the defini-
tion of a mixing length to relate the dissipation or decay. Chou [18,
19] sought to overcome this difficulty by introducing a second trans-
port equation for decay scale. Rotta [79,80] developed these ideas
even further and set down the transport equations for the complete
Reynolds stress tensor.
Based on the pioneering work of Rotta, Spalding [92] and his
colleagues at the Imperial College in London, have had considerable
success in applying these ideas to generalized numerical computation
in recirculating flow fields. Spa!ding's model for computing turbu-
lence quantities involves transport equations (cf. Reference [93])for
• k, the kinetic energy of turbulent motion,
. W, which may be considered as the average value of the
fluctuations of the fluid vorticity, and
• g, the average value of the square of the fluctuating
component of the mass fraction of injected fluid.
264
-------
Spalding defines a length scale as:
* = (k/W)1/2, (7.52)
hence,
e = C3pk1/2£. (7.53)
Thus, in addition to equations for the stream function, vorticity,
buoyancy, and/or other required constituents, transport equations of
the following types are also required:
.. 3k. , 3k ]_ 9_ /e 3kx
pu 3X pv 3r " r ar ^k 3r;
I/? II 2 k3/2
- C4 Pk1/2£ (^) - C5 pf • (7.54)
where the C's are constants defined by Spalding. Similar equations
are required for W and g. As testimony to these and similar methods
the reader is referred to the following work carried out at the
Imperial College: Patankar and Spalding [69], Gosman, Pun, Runchal,
Spalding and Wolfshtein [35], Bradshaw and Ferriss [15], and Spalding
[89].
Although solving additional transport equation for turbulence
quantities, such as Equation (7.54), appears to be a considerable
effort in itself, this approach offers a realistic and negotiable
compromise to otherwise unapproachable problems in turbulent flow.
These, or similar methods have not been employed in the present study,
but only because the flow field offered enough a priori knowledge to
justify and permit the use of simpler mixing length models.
265
-------
7.2.1 The Radial Transport Coefficient, er
To model the plume and circulating flow fields, the radial
component of eddy diffusion must be modeled throughout the fluid
system. To this end, four flow regions are defined which are illus-
trated in Figure 7.66.
These regions are defined as follows:
Region I: Zone of.established plume flow
Region II: Zone of flow establishment
Region III: Circulating ambient
Region IV: Lateral surface spread
Each of these regions has special characteristics and must receive
special attention.
Region I
Equation (7.50) relates the radial component of momentum trans-
port as
er - ClV vmax <7-55)
For Region I (established flow), C-j = .0256, *r = r1 /2, the plume
half-radius, and vmax as the centerline velocity. Tomich [99] used a
similar relationship for his analysis of a compressible free jet. In
dimensionless form
r^T = '°256 Rl/2 Vmax (7-56)
where
266
-------
ro
PLUME BOUNDARY
VERTICAL CUT-OFF
RADIAL CUT-OFF
FLOW ESTABLISHMENT
OUTFALL PORT
R
Figure 7.66.
Regional Specification for Turbulent
Eddy Coefficient Modeling
-------
and
u = Vmax
max ~ v
As a reference value for er we set
e = .0256 (7.57)
ro
^ere R1/2- RQ - 1. 0 and Vmax-V0- 1.0.
So that
e
1
r vn RE
oo r
0
Hence, the reference radial Reynolds number is
REr = 39, (7.58)
ro
the value used in all computations except Case 8.
To obtain the point Reynolds number RE/.. .<> (the indices on REr
will be omitted hereafter with the point value always implied), we
define
er = er0FRjk
where, er may be viewed as the point value of eddy diffusivity with
subscripts omitted. With the definition Equation (7.59)
and
FRjk = Rl/2 Vmax«
REr = REr /FRjk . (7.61)
For the momentum jet, in the zone of established flow (cf. Sec-
tion 7.1), it has been established that
Vmax =12'4/Z (7-62)
268
-------
and at the half radius
V
\T— = -5 = e (7.63)
max
or
• (7-64)
Using the value K = 77 from Abraham [1],
FRjk = 1.176, (7.65)
for a momentum jet (subscripts on FR will be omitted hereafter, with
the point value implied).
Equation (7.65) represents an empirical value for FR. Two
numerical experiments were carried out for the momentum jet, one case
where FR = 1.176 was held constant and the other where I-K was computed
according to Equation (7.60). The centerline velocity distributions
for both cases were found to be essentially identical. Figure 7.67
illustrates the result of iteratively computing FR = R1/oVm,v. In
I / i. in a A
both of these cases a Gaussian profile at Z = 4.5 was used for the
inflow boundary condition (Type 1 boundary condition - Region II does
not enter into compution). Figure 7.68 illustrates FR for cases having
varying degrees of buoyancy using the Type 2 inflow boundary condition
(power law velocity profile). Note that in this instance FR ^ 1 for
the momentum jet (F -»• ») and is owed to a slightly higher centerline
velocity, the ratio approximately equal to the value of Equation (7.65),
269
-------
ro
--4
1 .2
1 .1
cc
1 .0
0.9
EQUATION (7.65)
I
I
I
10 20 30
AXIAL DISTANCE, I
40
Figure 7.67. Computed Values of FR for a Momentum Jet
-------
ro
o
I—
CJ
O
LlJ
=c
O
10 -
8 -
6 -
4 -
2 -,
EQUATION (7.67) (F = 1 )
10
20 30
AXIAL DISTANCE,
40
50
60
Figure 7.68. Computed Radial Eddy Diffusion Factors, FR for Deep Water
Plumes at Various Densimetric Froude Numbers
-------
For the cases dominated by buoyancy, Equation (7.33) gives
or
Vmax = 4-4(2)(F0Z)' (7.66)
The radial velocity distribution is again given by Equation (7.63) with
K = 92.
In this instance
FR = (2)1/34.4(FZ)-1/3 V Z'
FR =
FR =
based on
Fo
Z = Z/r0.
Equation (7.67) is also plotted on Figure 7.68 for comparison of
the empirical approximation and computed value of FR for F =1.
Aside from merely illustrating how the radial eddy transport
coefficient er varies as a function of the degree of buoyancy,
Figure 7.69 also reveals that the use of a constant transport coeffic-
ient is untenable in buoyant plume flow computations and can lead to
order-of-magnitude errors. A numerical experiment was carried out to
272
-------
1.0
.5
I r
Fo = 52
FR = 1 .176 1
FR = V R.
max !
10
50
AXIAL DISTANCE, I
100
Figure 7.69. Comparison of Computation Using Constant and
Variable Radial Eddy Transport Coefficients
ascertain these differences for F = 52 using the Type 1 boundary
condition. Figure 7.69 illustrates that large errors will occur in
both the centerline velocity and buoyancy distribution if er = con-
stant is used. The curves corresponding to Cases 15 (FR = constant)
and 16 (FR = VmR1/9). Also refer to Figure 7.49.
max
In the early development of the computer program, how to effect
the variable transport computation iteratively was unknown and such
273
-------
attempts led to numerical instability. Although this problem was sur-
mounted later (see Section 7.3 for stability problems related to var-
iable e and ez), as an interim step, the mixing length R was com~
puted prior to computation from similarity assumptions for the given
Froude number. With R, .^ fixed, FR was computed from Equation (7.60),
and eliminated this source of numerical instability. However, the
solution was only nominally more accurate than using FR = constant.
Hence, this method was adjudged inadequate and, as mentioned earlier,
later abandoned.
Region II
The zone of flow establishment is characterized by turbulence
regimes (see Figure 7.70), 1) the potential core, a roughly conical
region, where mixing is dictated by the convected pipe flow turbulence,
and 2) the zone of intense mixing lying outside the potential core,
spreading into the ambient, and created by the shear between the
effluent and the ambient fluids. A mixing length, «, , may be philos-
ophically defined as being proportional to the width of the shear
region. However, the geometry is difficult to define and the criterion
as to the width of the mixing zone is quite arbitrary. Also the length
of the mixing zone, Ze, that is, the point where the zone of intense
mixing reaches the plume centerline, is also quite arbitrary and cer-
tainly is not defined by a sharp point as Figure 7.70 indicates.
Tomich [99] bypassed the mixing length problem in the region by
setting er = .2 times the value in the established flow regime. For a
274
-------
momentum jet, this value was found to yield downstream results in good
agreement with experimental results.
MIXING
ZONE
POTENTIAL
CORE
Figure 7.70. Concentration Distribution in the Zone
of Flow Establishment
In this study, we have not followed Tomich's method since we
deal with cases of high relative buoyancy (low densimetric Froude
numbers).
To set up a turbulence model for this regime we need:
1) a mixing length, and
2) a definition of the region of application:
• radial region
• vertical extent.
275
-------
To compute a mixing length, a reasonable criterion is
*c = rl/2 - rc
where r is the radius of the potential core and r1 ,2 is again the
half-radius. The transport coefficient is then defined by,
erc - .0256 (r1/2 - rc) v^. (7.68)
Physically, e would apply over the region r. - r , where r. is the
mixing zone outer boundary.
The next problem then is to define r based on some relevant
mean flow quantity. In the present work, the concentration profile
was used for such a criterion. Velocity could not be used because of
power law boundary profiles and because buoyancy tends to distort the
velocity profile. The criterion was set as
rc - r>95 (7.69)
or r extended to the point where the concentration was decreased to
95% of the centerline value. The outer boundary was set as
rc = r.05 (7-7°)
or where the concentration had decreased to 5% of the centerline value.
The length of the potential core was computed from the criterion
zc = z.90 (7-71)
where the concentration at the plume centerline is reduced to 90% of
the initial value. The numerical model does not account for
276
-------
derivatives resulting from variations in z , and where convection
terms dominate transport, this deletion is valid. However, in the flow
establishment region, this treatment can lead to large constituent
discrepancies if the gradients of er are not accounted for. For this
reason and other computational difficulties, e has been assumed
radially constant at a given elevation, laterally to the plume cut-off.
Based on Equation (7.68) along with criterion Equations (7.69)
and (7.71) a typical computed potential core and half-radius is illus-
trated in Figure 7.71 for FQ = 46. This method for computing the
transport coefficient was felt to
be unsatisfactory in that the
computations were slowed down
compared to preset specification,
and definition of the potential
core appears to have questionable
accuracy. However, one fact was
established as a result of these
experiments in that R, ,~ ^ 1.0
for all cases run. The method
finally used was to define the
.5
1 .0
R
Figure 7.71. Computed Potential
Core and Half-
Radius FQ = 46
length, Z , based on a criterion
similar to Equation (7.71), use a
straight line fit between the
Refer to Equations (3.73) and (3.80) and note the computer program
does not contain any viscous terms envolving derivatives of er. It
may be shown that such terms are small except in Region II.
277
-------
points (0,Z ) and (1.0, Z ) to define the potential core, and set
R1/2 = 1.0 up to Zg (see dashed lines on Figure 7.71). This procedure
was found to be satisfactory and added speed to the computation.
The remaining problem, in computing quantities within Region II,
is that the computer model treats e constant across a lateral plane,
where, in fact, there is considerable variation. Treating e constant
in this fashion is to overestimate the diffusion coefficient within the
core since the value used is typical of the turbulent mixing region.
The net result of this procedure is to effectively reduce the computed
core length which can result in downstream errors. One way to bring
the computed core length more in line with experimental results con-
cerning the core lenth is to reduce the value of e .
One such model, which is based solely on numerical experiment is
given by,
erc - .0256 (r1/2-rc)(r1/2-r0)/r1/2 vmax (7.74)
which is the same as Equation (7.68) except for the multiplication fac-
tor (ri/2~rc)/ri/2' This factor nas tne effect of reducing the eddy
diffusion, given by Equation (7.68), near the outfall and has decreas-
ing importance as the end of the potential core is approached. This
model for radial eddy diffusivity gives good results over the entire
range of Froude numbers for deep water plumes (see Figure 7.72) and is
the preferred method of computing e . All cases discussed earlier are
based on Equation (7.68) where applicable. The case for F = 1 illus-
trated in Figure 7.72 may be compared with Figure 7.55 (Case 24).
Cases displayed in Figure 7.72 were computed on a 26x25 grid.
278
-------
2.0
1.0
.50
.10
CASE F Vm
—o— —m
.05
44
45
46
47
1
5 a
25 *
100 v
V(R,0) - 1.15
SIMILARITY SOLUTION
FOR AT/AT
m0
1
.01
10
AXIAL DISTANCE, I
50
Figure 7.72.
Center!ine Velocity and Temperature Distribution
for 44 Diameter Deep Outfall
100
279
-------
For the purpose of comparing results using the two different
methods for computing c in the core, refer to Figure 7.73, which
illustrates the centerline temperature distributions for intermediate
water depth cases. A summary of these four cases may be found in
Chapter 8, Table 8.1 listed as Cases 48, 49, 50 and 51 for Froude
numbers 100, 25, 5 and 1, respectively. Note that Equation (7.68)
(Figure 7.73) yields much more rapid deterioration of the centerline
temperature than Equation (7.72) (Figure 8.1, Chapter 8).
Results from Figure 8.1 may also be compared to Figure 7.72.
Note that for the deep water cases at low Froude number, the centerline
temperature distribution again decays more rapidly near the source
than for corresponding cases at intermediate depths. This discrepancy
is caused by lack of axial finite difference resolution in the core
region of the deep water results.
Region III
In the region outside the plume, the value of er is set to a
reference constant that is descriptive of the ambient conditions.
Reasonable variations of this value have been found to have little
effect on the circulation patterns or on the plume computed quantities.
In fact, several early runs were made letting e in the ambient take
the same value as computed within the plume. Only slight differences
were noted in the plume size when the value of the ambient was set to
1% of the plume interior value.
Most calculations and the present version of the computer program
use a "cut-off" point (see Figure 7.66) for e at a point just outside
280
-------
1.0
0.1
* *"**—•-•
EDDY COEFFICIENTS IN CORE
COMPUTED BY EQUATION (7 . 70)
i v i (
i
1
10
AXIAL DISTANCE, Z
Figure 7.73. Computed Center!ine Velocity and Temperature
Excess. Cases for 10 Diameter Deep Water.
281
-------
the plume where radial convective effects dominate radial transport
(again radial derivatives in e are neglected). The first attempt to
establish a radial cut-off was based on r (concentration) dropping to
5% of the center!ine value. This seemed to be a reasonable criterion
but proved to be computationally unacceptable because oscillation of
the cut-off point position between nodes, near the plume boundary,
dramatically slowed convergence and grossly added to the computation
time. The convergence problem was eliminated by extending the cut-off
point two nodes beyond the r = 5% criterion, but resulted in a
"ragged" plume edge, the raggedness being unrelated to flow physics
(Figure 7.74). The next step was to preset an envelope in which the
plume would always exist and e could be held constant at a particular
elevation. This envelope extends two to five nodes beyond the r = 5%
criterion but is computationally very attractive because convergence
is significantly speeded with no real loss of accuracy.
Region IV
In the lateral surface spread, the plume boundary is defined by
the presence of the circulating or reverse flow field. For a vertical
cut-off point, the boundary is extended two nodes below this region of
negative radial velocity. The value of er is set to the value com-
puted within the vertical rise region and being held radial constant.
For all cases run, the convective effects are reasonably large in this
region; hence, e is of minor importance.
282
-------
ro
CO
CO
FIGURE 7JI4 . 3D ILLUSTRflTION OF TEMPERRTURE FIELD--A T
-------
7.2.2 The Vertical Transport Coefficient, e.
Referring to Figure 7.66, the unique regions of vertical eddy
diffusion computation are identical to those of the radial component.
However, it is generally true that for the present model only one of
the coefficients, e and e , will be of major importance in a given
region. For instance, in Regions I and II, e was found to play a
major role in computing the plume dynamics, whereas, for all intents
and purposes, e may be ignored. This statement is proved by numerical
experiment (Case 10, Figure 7.11) where e was set to er in the mixing
zone. Only minor differences were noted between Case 10, and Case 6
where e was set to a constant value of .001. From our knowledge of
jet induced turbulence we expect that point-wise, e and e, should be
i b
nearly the same in Regions I and II. (cf. Hinze [40]). Some differ-
ences may be noted near the surface where larger vertical mixing
scales are suppressed.
The fact that vertical mixing is of little importance in Regions
I and II may be ascertained on theoretical grounds by comparing the
order of magnitude of the various vertical transport terms in the
Equations of motion (3.67). Although the details are not presented
here, one finds that vertical convection dominates vertical diffusions
in these regions, an expected result, except near the surface where the
two transport mechanisms may play equally important roles.
Hence, we may dispatch concern for EZ in Regions I and II remote
from the surface, without further investigation. However, numerical
experiments have shown that e is very important in Region IV and there
284
-------
is, nevertheless, incentive for extending the vertical cut-off to the
plume centerline to overlap that portion of Region II.
Region III
The vertical transport coefficient associated with Region III is
that of the ambient sea, and as such, e depends on water depth,
currents, sea state and ambient stratification. Extensive work has
been carried out by the Oceanographic community to determine e as
influenced by the above mentioned variables. Summaries and discussions
of this work may be found in work by Koh and Fan [52], and Wada [107].
The presence of vertical stratification can dramatically impede
vertical mixing, whereas shear force tends to enhance this mixing.
Hence, the vertical mixing coefficient must depend on, in some fashion,
the relative importance of the stabilizing effect of stratification
and the destabilizing forces of shear flow. The local Richardson
number, RI, relates the relative importance of these forces through
the ratio
RT = stabilizing forces
destabilizing forces '
a dp.
RI = . £_dz . (7.73)
In terms of the dimensionless quantities defined in this manuscript,
285
-------
RI = 9r • r • \7 •
If RI < 0, the flow is obviously unstable. Various researchers have
proposed methods for computing t using a Richardson number correla-
tion. The most notable of the efforts are summarized in Table 7.5.
Note that in this discussion we are speaking of a general vertical
eddy transport coefficient with no distinction between the transport
of material, heat or momentum. Since any correlation for general
application is at best a rough approximation, we are assuming that the
vertical Prandtl (or Schmidt) number is unity.
The various correlations given in Table 7.5 are essentially
Richardson number modifications of the neutral diffusion coefficient
ez . Thus, the first task lies in determining e for a neutral am-
0 zo
bient (RI=0). Kent and Pritchard [51] give one such correlation for
the wave induced component, for the James river estuary, as
d H
ez = .Old (1 - {4 f e (7.75)
o
where
d = distance from the surface,
L = depth of the water body,
H = wave height,
T = wave period, and
SL = wave length.
For a well mixed surface layer only, Golubeva [33] and Isayeva
286
-------
TABLE 7.5. CORRELATION OF THE VERTICAL DIFFUSION COEFFICIENT
ez WITH THE LOCAL RICHARDSON NUMBER, RI
(extracted from Koh and Fan [52])
Note: e = e at RI = 0, i.e., the neutral case, 3: proportionality
o
constant; varies from case to case.
Rossby and Montgomery
(1935)*
Rossby and Montgomery
(1935)*
Holzman (1943)*
Yamamoto (1959)*
Mamayev (1958)*
Munk and Anderson
(1948)**
ez =
ez =
ez =
(1+6 RI)
(1 + 3 RI)
-1
-2
0
(1 - 3 RI)
(1-3
''•V
-3 RI
«'4
nr 1
ez =
(1 + 3 RI)"3/2
*As given by Okubo (1962)
**As given by Bowden (1962)
3 = 3.33 based upon data by
Jacobsen (1913) and
Taylor (1931)
287
-------
and Isayev [47] give
H
T
(7.76)
Figure 7.75 (extracted from Reference [52]) illustrates the
relationship between e and the local sea state.
400
01
300 -
200 -
100 -
Figure 7.75. Dependence of
EZ on Sea
State
For the case of tidal currents, Wada [107] gives
K2US /d" z
L/L log
(7.77)
288
-------
where K is the Karman constant, Ug is the surface current, L is the
scale of the bottom roughness. Where both components, tidal currents
and wind waves, are acting, Wada gives
K2(d+l_J z2 /L IL ftH -2Trd/£
e = \ +^le (7.78)
zo \- Y^d + LQ Z '
Various measured values of pe are given in Table 7.6 (extracted from
Reference [107]).
In the absence of ambient currents Harremoes [36] gives
•3 / A i X-2/3
3
cm/sec
where z is in meters. This correlation was obtained off the coast of
Denmark. Koh and Fan have obtained the relationship
{7'80>
dz
where again z is in meters. Data used in obtaining this result is dis-
played in Figure 7.76.
Any estimate of e or e in the ambient sea has questionable
o
accuracy. At best, these correlations, and measurements for that
matter, are accurate only for the observed conditions, conditions
which may change drastically with time and location. Aside from this
complication, just how the researcher deduced the transport coefficient
value from physical measurements may shadow the validity of results
289
-------
ro
io
o
o
0)
to
LU
I—I
O
LU
O
•a:
o
10'
10
I I I I I I I I I I I I 1 I I I 1
A IIARREHOS (1967)
ov KOLESNIKOV (1961)
• JACOBSEN (DEFANT, 1961)
a FOXWORTHY (1968) PATCH
• FOXWORTHY (1968) PLIU1E
FOXWORTHY (1968) POINT SOURCE
1 _
0.1 —
0.01
10
• • x \
X a N
\ -N
I ill I II
-7
10
-6
,-5
,-4
10 10 ^ 10 J
(-1/p) dp/dz, DENSITY GRADIENT, m"1
Figure 7.76. Correlation of ez with Density Gradient
10
-2
10
-1
-------
TABLE 7.6. VALUES OF VERTICAL EDDY VISCOSITIES IN THE SEA
ro
Current or Sea
Region
All oceans
North Siberian Shelf
North Siberian Shelf
North Siberian Shelf
Schultz Grund
Caspian Sea
North Sea
Danish Waters
Kuroshio
Japan Sea
off San Diego
a) W = wind velocity in m/sec
b) z = distance from sea bottom in meters
c) Very great stability
Layer
Surface
0 to 60m
0 to 60m
0 to 22m
0 to 15m
0 to 100m
0 to 31m
0 to 15m
0 to 200m
0 to 200m
Near the
sea bottom
pt in y/ v*m/ acv.
a)pe_ = 1.02W3
(W 6m/sec)
= 4.3W2
(W 6m/ sec)
75-260
10-400
b) Z+Q i 3/4
385(~227p)
1.9-3.8
0-224
75-1720
c)1.9-3.8
d)680-7500
150-1460
e) 93-(z+0.02)
pe Derived From
Thickness of upper
homogeneous layer
(wind currents)
Tidal currents
Tidal currents
Wind currents
Reference
Thorade, 1914
Eckman, 1905
Sverdrup, 1926
Fjeldstad, 1936
Fjeldstad, 1929
Jacobson, 1913
Stochman, 1936
Strong tidal currents Thorade, 1928
All currents Jacobson, 1928
All currents Suda, 1936
All currents Suda, 1936
Tidal currents Revelle & Fleming
d) Very strong currents
e) z = distance from sea bottom in meters
-------
and the application to numerical modeling. Generally, these coeffi-
cients are deduced from concentration measurements and back-calculated
through an analytical diffusion equation. Hence, the values are valid
only for the diffusion equation used to calculate them in the first
place. Just how appropriate these values are as they enter into more
elaborate numerical computation is open to question in this author's
opinion. It is felt that the determination of ambient diffusion coef-
ficients is an area that needs extensive research.
In the present computer model for Region III diffusion coeffi-
cients, various of the models discussed above were tried. Very little
difference was noted in the Region III circulation patterns in any case.
Influence on the plume was noted only when the value of t was unreal-
istically large, in which case the flow dynamics took on the charac-
teristics of a creeping flow. For this reason e was set to a value
-4 -3 2
on the order of 10 to 10 ft /sec in Region III for succeeding
computation, a value corresponding to moderate stratification, low sea
state, and low ambient current.
Region IV
In modeling the plume lateral spread, the vertical turbulence
component is of utmost importance. As the plume encounters the surface
and begins the radial surface spread, plume induced turbulence dominates
the mixing phenomena. At increased radial distance, the induced turbu-
lence decays and is suppressed by stratification. Generation of tur-
bulent energy by virtue of the lateral shear flow is also declining
because of smaller velocity gradients. At some larger radial distance
292
-------
the field of turbulence will be dominated by ambient effects such as
sea state.
We have just discussed the ambient contribution to e and indi-
cated rough methods for such calculation. The plume induced turbulence
in the zone of initial spread (or the transition zone) is the important
feature of Region IV. Unfortunately there is very little data avail-
able in the literature which is directly applicable to the problem of
turbulence modeling in this zone.
From a theoretical point of view, we assume that Prandtl's second
hypothesis holds, or that
ez =Cl*2umax <7'81)
o
for the neutrally buoyant case. We also expect that a Richardson
number modification of Equation (7i81) would suffice for the case of a
spreading thermal layer, of the form
ez = Cl*z "max
For the neutrally buoyant situation we may gain some insight as
to how the produce £, u_,v behaves by assuming the flow can be approxi
Z nloiX
mated by a radial jet similarity solution. The appropriate similarity
equations for a radial jet following the methods devised by Morton, et
al. [60] for a vertical jet, are
Continuity:
3F < V> - <"um (7'83)
293
-------
Radial momentum:
3? (um rt) = °' (7'84)
In the above equation a "top-hat" velocity profile has been
assumed, where um is the mean radial velocity, T is the characteristic
thickness of the jet and a is the usual entrainment constant. The use
of the top-hat velocity profile is entirely satisfactory for purposes
here, since we are only interested in the relative behavior of T and
u , which is insensitive to the similarity profile used.
Solving these equations, one finds
u r = constant (7.85)
and
p
UITT = constant (7.86)
Hence,
(umr)(umT) = constant,
and
umT = constant. (7.87)
Equation (7.87) reveals that if the velocity field is approxi-
mately similar then the eddy coefficient ez must be constant in view of
Prandtl's second hypothesis (a result identical to the axisymmetric
jet). Hence
e.. = C-,4, u _ = Ci-Constant (7 88)
Z- \ 2. max ' \/ .00;
The remaining problem lies in evaluation of C, and (a u
In the present work, C] is assumed to take the value .0256 as in
max
294
-------
the case of the axisymmetric flow region.
The quantity &,um_v was treated by four different methods during
Z iTlaX
numerical experiments as listed below.
Method 1 :
Compute the value of a from the local velocity profile based, on
the distance from the level of maximum lateral velocity to the level
where lateral velocity is 1/2 the maximum value. That is,
lz = zl/2'
This method is identical to that used to compute FR, but in the instance
of lateral flow was found to be unsatisfactory because of numerical
instability. All attempts to compute FZ where
FZo = Zl/2Umax' <7'89)
iteratively from local information were found to be unstable and the
method was abandoned.
Method 2:
Use a constant value of Z- based on the value of R-| ,^ at the
point of lateral spread. This method proved to yield diffusivities
which were too large.
Method 3:
Use a constant value of FZ = Z, ,2 Umax where Z-j ,2 and Umax for
the entire system are computed in the vertical plane where the maximum
lateral velocity occurs. This method, based on the insight given by
the similarity solution, also yielded diffusivities which were
295
-------
too large. This method was applied only to cases having buoyancy;
hence, the failure may have been due to an inappropriate Richardson
number modification of FZ .
Method 4:
Use the method given immediately above, except scale the result
by the local ratio (Umax): (Umax)system. As in the two methods
immediately above, this calculation proved to be numerically stable
under all conditions once a reasonably realistic lateral velocity dis-
tribution was established. But, unlike the above methods, local dif-
fusivities are computed which give more realistic velocity fields.
Hence,
REZO = REz(ref)/FZo (7-9«>
and
FZo = Zl/2 Umax
where Z,y2 is calculated at the system maximum lateral velocity and
U is the local maximum lateral velocity. The subscript o again
indicates the condition of neutral buoyancy.
To account for local stratification, the local Richardson number
model due to Mamayev (cf. Reference [108]) was employed,
o (7.91)
where RI is again the local Richardson number as defined by Equation
(7.76) and $ is an empirical constant. Wada [108] used Equation (7.91)
in his study of planar thermal outfalls discharging horizontally, but
used a constant value of ezo.
296
-------
Although there is no data known to the author relating point
eddy diffusivities to the point Richardson number in turbulent jets,
data has been obtained which relates the entrainment of such flow to
the overall Richardson number (cf. Ellison and Turner [25]). Stolzen-
bach and Harleman [94] have illustrated that the data of Ellison and
Turner may be adequately represented by the form,
(7.92)
where a and a are the entrainment coefficients for buoyant and
o
neutral spreading surface flows, respectively, and RI' is the gross
Richardson number. Stolzenbach also illustrates the relationship
between eddy viscosity and entrainment as
ez _ az
EZn "Zo
0 0
Thus, based upon the data of Ellison and Turner, and the func-
tional relationship, Equation (7.92), derived from this data, the
Manayev Equation (7.91) is apparently a credible method for modifying
point-wise neutral eddy diffusion coefficients for application in
laterally spreading buoyant plumes. In the computer program, we use
the form,
e • e_ + t e-3RI (7.93)
ambient o
The computer program is also set up to use the various other
models given in Table 7.5. These models have not been used owing
297
-------
primarily to lack of appropriate information concerning the empirical
constant B.
The value of B (for Equation 7.93) used by Wada [108] was .8 for
momentum diffusivity and .4 for heat diffusivity based on ambient
conditions. According to work done by Stolzenbach this value should
be appreciably higher for plume flow. Computations using various
values of B for the present work are illustrated in Chapter 8.
In the present work, another form of e has been used, primarily
for starting solutions where Equation (7.93) results in numerical
instability. This form is given by the equation
= s7 e (7.94)
z zref
where d is depth or distance from the surface. The result is a
Gaussian depth decay of eddy momentum diffusivity from a surface
reference value. Equation (7.94) is used in computation merely as a
computational aid and is abandoned in favor of Equation (7.93) once
reasonable velocity and temperature profiles are established, or a
numerically stable situation is attained.
7.3 Numerical Stability and Convergence
During the course of this investigation various experiments were
performed dealing with solution stability and convergence. For each
case run, at least five node points were monitored for convergence
rates of U, V and r. Additionally, the program computes the maximum
change of ty, ft, and AI throughout the system at selected iterations,
298
-------
and an overall r balance error is computed at the end of each run.
Liebmann relaxation factors were employed to each of the equations for
if;, n, A,, and r to either accelerate or decelerate solutions.
7.3.1 Numerical Stability
To define what is meant by numerical stability in this manu-
script, we take the opposite view—that of numerical instability. The
reasoning for this view is that it is entirely possible that the
system of buoyant fluid may have physical instabilities which are not
divergent. The solution which we are trying to attain may, in fact,
be physically unsteady, and may never be attained by steady flow
methods. Since the Gauss-Seidel method with under/over-relaxation is
not unlike certain transient methods (see Appendix E), continued
iteration may reveal a cyclic behavior of the computations. This
situation cannot be termed a numerical instability. It merely illus-
trates the inability of steady flow techniques to simulate transitory
flow physics.
To demonstrate this idea, the computer program was set to a
different task, that of predicting the flow field past the end of a
cylinder contained in a larger pipe. From experiments we know that,
at low Reynolds numbers, streamlines past the end simply are distorted
toward the centerline, much as the case of irrotational flow (Figure
7.77-A). At much higher Reynolds numbers, vortex shedding from the
end of the cylinder will occur and the flow field is termed unsteady
although patterns may be repeated in time or in a cyclic fashion
(Figure 7.77-B).
299
-------
PIPE WALL
//////////;//A////////////////////////////////////////////////////////////////////////
CYLINDER
A. LOW REYNOLDS NUMBER (CREEPING FLOW)
REGION OF
VORTEX SHEDDING
B. HIGH REYNOLDS NUMBER
Figure 7.77. Observation of Flow Patterns
Past the End of a Cylinder
A. LOW REYNOLDS NUMBER (CREEPING FLOW)
HIGH REYNOLDS NUMBER
Figure 7.78. Computed Flow Patterns Past
the End of a Cylinder
300
-------
We expect that the steady flow computer program would converge
to a steady solution at low Reynolds number, and, in fact, this was
the result as illustrated in Figure 7.78-A. At a high Reynolds number,
however, a converged solution could not be attained. Computed quanti-
ties demonstrated quite the same behavior that one would expect from a
transient solution to this problem. Generally, as computation pro-
ceeded, a recirculation pattern formed behind the cylinder, grew by
elongation, and collapsed to nearly circular form, and elongated again
(see Figure 7.78-B). This process occurred repeatedly as computation
continued. Although it is impossible to quantify the physics from
these results, it is reassuring to know that the numerical technique
will reveal the presence of a physical instability, or unsteady flow,
and not converge to an erroneous steady solution.
Thus, it is entirely possible to have non-converging (although
not diverging) solutions that are not associated with numerical
instability. Hence, we define numerical instability as that situation
which upon repeated iteration leads to increasingly divergent and
physically ridiculous results.
As a general observation, involving perhaps a hundred or more
computer runs of various duration, the numerical techniques used were
found to be unconditionally numerically stable provided that:
. All Liebmann acceleration factors were less than unity,
• All eddy diffusion coefficients were constant, or the
velocity field at the beginning of computation has at least
reasonable similarity to the final solution.
301
-------
These observations are a result of flows having Reynolds numbers
from 0 to infinity and a variety of other testing conditions. Based
on these numerous experiments, difficulties encountered by other
authors, the accuracy outlined in Section 7-1, and comments made by
Spalding [91], the present difference formulations and grid system used
is extremely attractive.
At an early date in this investigation it was discovered that
solutions invariably became unstable if the acceleration factor, Lj,
for the n and A-I transport equations was greater than unity. However,
the value Lr = 1.6 was used for the stream function, y, elliptic equa-
tion without difficulty. Later, it was discovered that under some
flow condition, the value of Lp also needed to be less than unity to
avoid instability. For cases involving constant eddy coefficients,
only the transport equations needed to be decelerated. After these
initial investigations, the general rule used was to decelerate all
equations or set L£ and LJ < 1.0. The general form of the decelerated
solutions is
rV (7'95)
where the subscript p indicates the nodal point in question, n is the
n iteration and r is the result of the n+1 unaccelerated Gauss-
Seidel iteration.
A value
(7.96)
302
-------
was found to be satisfactory for nearly all cases. In a few instances
of very shallow water and non-linear e and e , values as low as
LT = .80 were used. In all cases, the acceleration factor is applied
as soon asr1 is computed at a node.
No attempt of a theoretical analysis of stability will be pre-
sented here since the presence of non-linear eddy coefficients negate
meaningful analysis and the case of constant diffusion coefficients
has been presented by various authors, [7,111], at least for time
dependent problems. Some insight to stability of steady state computa-
tions is given in Appendix A. Further insight into this question may
be gained by the analysis given in Appendix D which compares the Gauss-
Seidel iteration technique to an appropriate (similar) transient
solution.
It was, perhaps, propitious that a superior grid system was
devised at the outset of this study (see Figures 5.3 and 5.4). In a
recent publication, Spalding [91] points out that making vorticity
adjustments in a cluster of five adjacent points and the stream func-
tion at the central point has a striking effect on divergence removal
for reasons unknown. Unlike the grid system to which Spalding refers
where vorticity and the stream function are computed at the same space
points, the present grid system is staggered. The vorticity values
which interact as a source for the stream function elliptic equation is
averaged from the four adjacent neighbor points, which is closely akin
to the method referred to by Spalding and may be responsible in part
for the seemingly inherent stability of the present method.
303
-------
Another aspect of the present computational technique is that
linear gradients are always used for flux terms whether on the boundary
or in the interior, by the use of fictitious boundary cells. Spalding
again points out that higher order methods for treating boundary con-
ditions may in fact lead to less accurate results due to violation of
reciprocity and conservation principles at boundaries. In the present
method, through the use of the correct conservative difference equa-
tions and fictitious boundary cells, quantities are identically con-
served. This feature may also contribute to the success of the tech-
nique in avoiding instabilities propagated from system boundaries.
7.3.2 Convergence
The question of solution convergence has been partially answered
in the preceding section. It is obvious that solutions which are
numerically unstable will not converge. On the other hand, it is also
possible that a solution which is numerically stable will not converge
as demonstrated in the example of flow past the end of a cylinder at
high Reynolds number (Section 7.3.1).
The condition for convergence used in this work is defined by
fn+l _ -n
P " P
f
P
n+l
(7.97)
where 6f is the convergence criterion for the quantity f. The sub-
script p again indicates the nodal point in question and n is the nth
iteration. The condition for 6f approaching zero is not a sufficient
304
-------
condition to guarantee solution accuracy, however, since the numerical
procedure may in fact converge to an erroneous solution. The method
used in this work to decrease the probability of erroneous solutions
was to check the continuity of matter by evaluating net flux of
matter at the system boundaries and selected interior planes. This
check is subsequently referred to as the r-balance error (6r),r
referring to a conservative constituent. This quantity is effectively
given as a surface integral for the system in the form of
6F =
r f r(tf-n) dS + / (-Verr)-n
dS
ST ST
Sin Sin
dS + J (-verr)-n) dS
f, 1
100% (7.98)
where ST represents a vertical plane in the flow field and S is a
1 in
radial plane at the inflow boundary extending to RQ. Equation (7.98)
gives 6r as a percent error of the system inflow.
Typical results showed the r-balance error to be on the order
of 1%.
General observation of the various numerical experiments illus-
trated the following:
• The convergence rate decreased significantly with increased
grid size.
• The stream function distribution converged with respect to
6 more rapidly than vorticity and buoyancy (or r).
305
-------
Vorticity was the slowest to converge and also the most
erratic.
Convergence of all quantities near the outfall was much
more rapid than in the far field. Thus, sizable errors in
the far field did not influence the validity of solutions
near the outfall.
The relative magnitude of buoyant fortes compared to inertial
forces played a significant role in the rate of convergence.
Highly buoyant effluents (low F ) converged much slower than
pure inertial flows.
Runs made with constant eddy diffusivities converged much
more rapidly than those runs using variable coefficients.
One inner iteration (stream function elliptic equation) was
sufficient. Increasing the number of inner iterations
served to aggravate the convergence rate.
Deeply stratified cases (as opposed to surface layer strati-
fication) significantly aggravated the convergence rate.
This item is discussed further in Chapter 8.
Neglecting derivaties of EZ in the transport equations led
to r-balance errors on the order of 10-20% where variable e
was employed.
Beginning a solution from an irrotational flow solution as
opposed to zero velocity everywhere, appeared to have no
particular advantage, and in some instances tested, actually
slowed convergence.
306
-------
Most computer runs were initialized from a restart tape gener-
ated by a previous case. There was considerable economy in this
action since a solution would need to be started from a zero initial
velocity distribution (or irrotational distribution) only when the
grid layout was changed. Unfortunately, from another aspect however,
not many solutions beginning at iteration number one and ending at
convergence are available for comparison. To illustrate the conver-
gence behavior, some of the computational aspects will be compared for
identical grid layouts. 'This information is displayed in Tables 7.7,
7.8 and 7.9 for grid layouts of (JxK) 40x33, 31x34, and 26x25,
respectively.
The four cases cited in Table 7.7 constitute the worst lot as
far as convergence lethargy is concerned. The starting run was the
momentum jet case which took 800 iteration cycles to converge properly.
All succeeding cases used the momentum jet solution as initializing
information. Of these succeeding cases, the run for F = 1 (buoyancy
dominated) was the most reluctant to converge. Convergence lethargy
in this lot is laid chiefly to grid size although there is some
suspicion that cell aspect ratio and position of the inflow-outflow
boundary also have some effect. Figure 7.79 shows the convergence
history of Y, A-J and a for the 40x30 grid layout. This illustrates
the behavior of & for these variables where again the momentum jet
max
is used as a starting solution (first 800 iterations) for the succeed-
ing runs FQ = 46 and FQ = 1. As noted previously, 6max is the maxi-
mum relative change in the entire system and does not always occur
307
-------
TABLE 7.7 CONVERGENCE BEHAVIOR, 40x33 GRID
CO
o
OD
A 5
A Z
Fo
Start Variable
e and £7, Iterations
Total Iterations
Incremental Iterations
r - Balance Error
6 Stream Function
at Node
'max Vorticity
at Node
&m*v Buoyancy Parameter
fflclX
at Node
Starting
Case
.12591
2.0
00
75
800
800
- .1281
1.1 95x1 O"4
37,10
7.221xlO'3
32,6
3.210xlO~3
37,2
Succeeding Cases
A
.12591
2.0
46*
801
1100
300
.734
6.571xlO"5
20,37
3.191xlO"3
26,5
7. 380x1 O"4
2,33
B
.12591
2.0
1.0*
801
1400
600
- .4641
2.92xlO~5
14,32
1.466xlO~3
37,12
3. 345x1 O"4
2,33
C
.12591
2.0
1000*
801
1200
400
- .9133
6.989xlO"5
37,9
9.422xlO"3
32,6
1. 508x1 O"3
37,2
indicates variable changed in restart case.
-------
10
X
*t
-1
-2
-3
-4
-5
MOMENTUM JET
STARTING SOLUTION
500
1000
1500
ITERATIONS
Figure 7.79. Convergence Behavior, 40x33 Grid
309
-------
at the same cell. Figure 7.80 illustrates the convergence history of
the starting solution for V at nodes (2,20) and (2,30) and r at node
(10,33).
Table 7.8 illustrates similar data for a 31x34 grid layout.
Convergence in this lot is rather slow also. Note that the values for
6 _w are considerably larger in this lot than in the lot given in
max
Table 7.7, although the r-balance error is about the same. The explan-
ation is that &„ gives a relative change, and these changes are
max
occurring where the absolute value of the quantity is very small. For
instance, the maximum relative change of vorticity in the starting
case is .1595, whereas the value of vorticity at this point is
-9.76x10 ( the maximum value in the flow field is 2.944). Figure
7.81 shows convergence history of selected data.
Table 7.9 illustrates the convergence characteristics for the
26x25 grid. Note that each case is not finely converged with respect
to 6may, whereas the r-balance error is less than 1% in all cases.
Thus, this table illustrates that the system may be reasonably well
converged with regard to absolute quantities although relative changes
in part of the system may be comparatively large. Also, only 150
iterations were required to obtain each solution using the starting
run initialization.
For this case, the primary concern was plume centerline condi-
tions. Changes of velocity and temperature were occurring only in
the fourth and fifth significant figures along the centerline, indi-
cating that computation time may be saved by using a regional
310
-------
.4
.3
.2
START MIXING LENGTH CALC
V(2,10)
V(2,30V
r(10,35)
500 1000
ITERATIONS
Figure 7.80. Convergence History of V and r at Selected Cells,
Momentum Jet, 40x33 Grid
-------
TABLE 7.8 CONVERGENCE BEHAVIOR, 31x34 GRID
co
ro
AS
AZ
Fo
Start Variable
e and £7 Iterations
Total Iterations
Incremental Iterations
r -Balance Error
Stream Function
II id A
at Node
W Vorticity
at Node
6 , Buoyancy Parameter
at Node
Starting
Case
.14690
.2
51
150
600
600
- .4381
1.350 x 10~3
28,25
1.595 x 10~]
23,24
2.918 x 10"2
20,5
Succeeding Cases
A
.14690
.175*
105*
601
900
300
.2778
1.115 x 10"3
26,12
1.455 x 10"1
23,23
4.126 x 10"2
30,13
*Indicate changed variable
-------
oo
CO
Ui?
1 0
on
n_c\
OA
O.^i
A
•
a
•
0
rf
e
0
o
*
*
* (
a
« a
a
X
o
(
ff°
^
f
Jt
«
/•
a
i •
*
;
*
i
o <
X i
1 •
* '
• 1
' * :
• i
O i
•M :
' U I
1 • '
c
^
2
L
n
ii
'X
^
3^-
D^
•K
"•*
> i
\
)
./
j
W
/
/
a
•
, « J
1
L X .
1
m ,
NODE
(2,11) U
. (2,21) V
/"p 31) v
l^->01/ »
^2 34) A
(£.tj"+j z\|
(2 1,34) A,
40 SO 120 160 200 240 280 320 360 400
ITERATION NO.
Figure 7.81. Convergence History of U, V and
at Selected Cells 31x34 Grid
-------
TABLE 7.9 CONVERGENCE! BEHAVIOR, 26x25 GRID
A?
AZ
Fo
Start Variables and e Iterations
Total Iterations
Incremental Iterations
r -Balance Error
<5m,v» Stream Function
Hi a X
at Node
W Vortldty
at Node
6 , Buoyancy Parmater
at Node
Starting
Case
.14690
.50
1.0
100
400
400
.8035
1. 644x1 O"4
(24,9)
1. 776x1 O"3
(20,14)
1. 976x1 O"3
(24,6)
Succeeding Cases
A
.14690
.50
5.0*
400
550
150
.3700
l.OllxlO"3
(24,10)
9.350"2
(20,20)
4. 683x1 O"2
(21,20)
B
.12591*
.50
25.0*
600
600
150
- .9936
2.135xlO"3
(24,12)
5. 345x1 O"2
(22,20)
4.850xlO"2
(24,3)
C
.14690*
.50
100*
800
750
150
- .3203
2t210xlO"3
(24,11)
5.435X10"1
(22,17)
3.440xlO"2
(24,5)
GO
-f=>
indicates changed variable.
-------
convergence criterion. In the computer program one has some control
over this criterion by applying the convergence check only out to a
set radius.
As a final illustration of numerical convergence behavior,
Figure 7.82 shows the iteration history of V and A, at one cell for
Case 2 (see Table 7.1). The significance of this plot is that the
velocity initialization is the irrotational flow solution (for the
other cases cited, U and V are zero everywhere except the inflow
boundary). Note that velocity V shows considerable oscillation.
The theoretical development of difference equations in this
manuscript is based on Equations (5.8) for vorticity, Equations (5.9)
for the transport of buoyancy and equations similar to (5.9) for the
transport of materials. These equations make no allowance for contri-
butions, or more accurately, corrections, issuing from variable eddy dif-
fusivities. In the instance of Equation (5.9), these'corrections may be
made rather straightforwardly by adding the terms
9AlV3evr
9r
However, in Equation (5.8) the appropriate correction terms add con
siderable complication as noted by comparing Equations (3.80) and
(3.81). Fortunately not all of the terms involving derivatives of
and e need to be incorporated into the numerical model either
because they are zero in accordance with assumptions concerning the
eddy coefficient model, or transported quantities are minute where
315
-------
CO
<4
3,
2.
C
£*»
k
• *
'•
:
-
•
• i
y
• V
> 1C
!*•
• •
* * * *
* • • **
• • t
• * ,
• v
\j *
X> 2(
t~'-- ,—^ ^,
J
X
K
)0 3
^^>^_J^t . ^^r-
00 4(
^V
A,
DO 5
00 6C
ITERATION CYCLES
Figure 7.82. Iteration History for One Cell of Case 2
-------
the variations occur. For instance, we may neglect all terms involv-
ing 3er/3r since er is constant where diffusion is important., and
convective terms dominate the transport where step changes in e
occur. Likewise, other order-of-magnitude approximations may be made.
Having eliminated these second order factors one is left with the cor-
rection terms for vorticity of:
1 „ 3FZ 3 U , 3U 3 FZ
RF *• a7 2 a? ?
Kt7 dL a7 9Z s7
L Of- OL
where FZ is again the vertical eddy diffusion multiplier (cf. Section
7.2).
Similar approximation for Equation (5.9) yields the correction
term,
1 3FZ . 9A1
REzPRz 3Z 3Z
with a similar correction for r transport.
The importance of these terms was ascertained by the system r-
balance. Without the corrections, the r-balance error ran as high as
20%. For the same conditions, addition of the correction terms
reduced the error to less than 1%.
Before closing the subject of convergences the author wishes to
note that in all cases run where the transport equations were decel-
erated and turbulence modeling did not lead to numerical instability,
the stream function convergence was extremely well behaved. This
behavior was obtained by iterating only once on the * elliptic equa-
tion; additional iterations were noted to aggravate the convergence of
317
-------
¥ as well as the transported quantities n, A, and r. It is entirely
possible that the system would have converged equally well if even
fewer ¥ iterations were performed, that is, iteration on v only once
for every two, three or perhaps five outer iterations. This facet
was not investigated in the present study, but such experimentation
could yield fruitful results in terms of computer time.
318
-------
CHAPTER 8
NUMERICAL EXPERIMENTS FOR SHALLOW WATER CASES
Material presented in this chapter deals with application of the
numerical techniques discussed earlier in shallow water situations.
All computer runs presented here are for cases where the assumed water
depth is ten or less port diameters above the outfall discharge. The
techniques used are identical to those applied for the verification
studies presented in the previous chapter.
Unlike cases in Chapter 7, however, applicable data are not
available except for one case where surface temperature field data are
available. Hence, we rely substantially on the verification study
as an indicator of the validity of the computational techniques.
Table 8.1 summarizes the cases to be discussed and illustrated in this
chapter; those listed are only a portion of the total shallow water
computer runs made during the course of the present research. None-
theless, these cases are typical and space limitations preclude further
illustrations.
8.1 Modeling the Vertical Eddy Diffusivity Multiplier, FZ
In the region of the lateral surface spread of shallow water
plumes, modeling the vertical component of the pointwise eddy diffus-
ivity plays an important role in determining the flow behavior. Con-
siderable effort was devoted to this subject in Section 7.2.2; the
computational methods used to obtain results presented in this chapter
will be briefly reviewed.
319
-------
CO
r^o
o
Case
No.
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
Grid
Size
26x25
26x25
26x25
26x25
26x25
26x25
26x25
26x25
31x34
31x34
31x34
31x34
30x26
30x26
30x26
30x26
30x26
30x26
29x20
A£
.14690
.14690
.14690
.14690
.12591
.12591
.12591
.12591
.14690
.14690
.14690
.14690
.14690
.14690
.14690
.14690
.14690
.14690
.12591
AZ
.5
.5
.5
.5
.25
.25
.25
.25
.20
.20
.175
.175
.40
.40
.40
.40
.40
.40
.15
TABLE 8.1
Depth R
2 oo
10
10
10
10
5
5
5
5
5.6
5.6
4.97
4.97
8.4
8.4
8.4
8.4
8.4
8.4
1.42
19.66
19.66
19.66
19.66
11.62
11.62
11.62
11.62
41.00
41.00
41.00
41.00
35.40
35.40
35.40
35.40
35.40
35.40
16.97
SUMMARY OF
F D
o ft
100
25
5
1
100
25
5
1
51
51
105
105
45
45
45
45
45
45
.111
10
10
10
10
10
10
10
10
14
14
16
16
10
10
10
10
10
10
21
SHALLOW WATER CASES
vo Boundary o
ft/ sec Type* °C
10.15
5.075
2.270
1.015
10.15
5.075
2.270
1.015
7.25
7.25
11.10
11.10
7.00
7.00
7.00
7.00
7.00
7.00
.574
2,N=10
2,N=10
2,N=10
2,N=10
2,N=10
2,N=10
2,N=10
2,N=10
2,N=7
2,N=7
2,N=7
2,N=7
4
4
4
4
4
4
2,N=10
18.12
18.12
18.12
18.12
18.12
18.12
18.12
18.12
8.33
8.33
8.33
8.33
11.10
11.10
11.10
11.10
11.10
11.10
13.80
Gtr
(amb)
27.98
27.98
27.98
27.98
27.98
27.98
27.98
27.98
18.01
18.01
18.01
18.01
27.78
24.78
24.78
24.78
24.78
24.78
25.57
Stratifi-
cation**
None
None
None
None
None
None
None
None
None
None
None
None
None
2 °C
3 oc
4 °C
5 °C
5 oc
None
* See Table 7.1
** Stratification extends somewhat deeper than in Case 63.
-------
The general form of the vertical diffusion coefficient is
e= e, + e (8.1)
ambient plume
In the region of plume flow the ambient contribution will be insigni-
ficant; hence,
EZ^EZ . (8.2)
plume
The plume generated turbulence is a function of both mean flow char-
acter and thermal character.
Recall from Section 7.2.2,
e
o
(8.3)
where ez is the vertical diffusion coefficient for neutrally buoyant
o
conditions and f is a function of the point Richardson number, RI.
Likewise, the vertical component multiplier, FZ, may be expressed as
FZ • FZQ f(RI), (8.4)
where FZ is the neutral buoyancy multiplier. The model for FZQ may
be expressed as (See Section 7.2.2)
• (8-5)
where Z1/2 is the radial plume half-depth, and the radial velocity
difference,
max" mi n " max '
since Um1n - 0 . Then
Fzo ' Zl/2 Umax' (8'6>
321
-------
If we followed the same method used for computing the radial
multiplier,
FR = Rl/2 (8'7)
Zi, /0 and V would be computed iteratively and pointwise to establish
i / c. max
FZQ (Method 1, Section 7.2.2). However, all attempts to calculate
FZ based on local values of I-, /9 and IJ led to numerical instability.
0 B / u ffloX
Exactly why this condition persisted, especially in view of excellent
success with Equation (8.7), was never ascertained. After several
numerical experiments and correctional efforts without success, it
was decided to stabilize the computation by restricting the computed
value of the plume half-depth, Zy2» since this value seemed to ex-
hibit the most unstable character in previous experiments. This
decision led to Methods 2S 3 and 4, described in Section 7.2.2.
Method 2 used Z. . based on R-, .^ computed at the elevation of
lateral flow. Z, ,„ was then held constant for that iteration but the
local value of Umax was used. This method led to eddy diffusion coef-
ficients which were quite large and a correspondingly unrealistic
flow field; hence, the method was quickly abandoned.
Method 3 computed a constant value of FZQ to be applied every-
where in the lateral plume spread. The value of FZQ in this method
was set by computing Z at the radial position corresponding to the
maximum radial velocity; the value of Umax at this point along with
Zy2 was used 1>n Equation (8.6). Results from this method are pre-
sented in Section 8.4. Again, this method yielded diffusivities
of excessive magnitude.
322
-------
However, experimentation with this method was carried out, in
every case, in conjunction with thermal flows (as opposed to neutrally
buoyant conditions). It is possible that the Richardson number
modifier, f(RI), was inaccurate.
Finally, the most realistic results were obtained by Method 4
which in principle uses the technique of computing Z,/2 of Method 3,
but bases U ,v on the local value. This method was found to be always
iTIaX
stable once the general, but approximate, flow patterns were
established.
Table 7.5 summarizes several models for f(RI); however, the
Mamayev correlation has been employed exclusively in this work which
has the form,
f(RI) = e"8RI, (8.8)
where 6 is an empirical constant.
The value of 3 to be used presents an additional uncertainty in
computing FZ. Wada [108] used the value @ = .8. However, Stolzen-
bach [94], based on the data of Ellison and Turner [25], suggests the
value 6 y 5.0 when using the gross Richardson number. We have used
values ranging from .4 to 2.0 in this work (Table 8.2).
Table 8.2 below summarizes the computation of FZ for results
presented in this chapter.
323
-------
TABLE 8.2 SUMMARY OF FZ COMPUTATION
Case Method
48-55 4 1.0
56 3 .4
57 3 .8
58 3 1.0
59 4 1.0
60-65 4 .8
66 4 2.0
Actual computation of FZ proceeds as follows: (sequence of
operations for one outer iteration).
Based on the computed values of Us V and A-| for the present
iteration:
• Compute the array of local Richardson numbers,
,„ 2
J,K
9 Scan the U array to establish the maximum value of U and the
corresponding index, J.
® Compute the plume half -depth, Z. . at index J.
® Use this value of Z, ._ to compute
where Umax takes on different values at each radial grid
point.
Compute
FZ(jsK) = FZJJ.K) e' eRI(J'K)
Use the above value of FZ in computing transported quantities
for the next iteration.
324
-------
8.2 Results for Homogeneous Receiving Water 10 Port Diameters Deep
Results for plumes issuing in homogeneous receiving water at a
depth of ten port diameters are reported as Cases 48 through 51. For
these cases, the initial temperature excess is 18.12 °C and the value
of sigma-t for both the effluent and reference ambient is 27.98.
Each case represents a different densimetric Froude number as indi-
cated in Table 8.1. Changes in the Froude number were effected by
varying the effluent velocity. All initial velocity profiles are
assumed to be turbulent and follow a profile given by Equation (7.17)
with the exponent equal to 1/10. The port diameter is held constant
at 10 feet and the lateral spread is computed out to about 10 port
diameters.
Centerline distributions of velocity and temperature excess are
illustrated in Figure 8.1 for all four cases. Note that the plume
accelerates for low Froude numbers (FQ = 1,5), but for Froude numbers
of 25 and above, very little acceleration is noted even though tur-
bulent mixing (as a function of distance from the port) is decreased
(temperature excess curves). Velocity of the lateral surface spread
is illustrated in Figure 8.2 for these same cases. Maximum velocity
in each case occurs at radial distance between 1.5 and 2.0 diameters.
In the highly buoyant Case 51, the maximum lateral velocity is nearly
as great as the initial velocity. Note that these results are norm-
alized to the average effluent velocity; hence, for FQ = 1 the maximum
lateral velocity is about 1 fps, whereas for FQ = 1, the corresponding
velocity is about 3.5 fps. Vertical profiles of lateral velocity for
325
-------
3.0
1 .0
0.5 -
0. 15
AXIAL DISTANCE, Z
Figure 8.1. Computed Center!ine Velocity and Temperature
Excess for Intermediate Depth, Cases 48
Through 51 (10 diameters deep)
326
-------
co
ro
o
<:
t—I
Q
<:
o:
GO
o
*—t
CO
1.0 U
.8 \-
.6 U
.4 U
.2
3456
RADIAL DISTANCE, r/D
Figure 8.2. Surface Distribution of Radial Velocity,
Cases 48 Through 51 (see Table 8.1)
-------
Case 50 are illustrated in Figure 8.3.
Comparison of the radial velocity profiles are illustrated by
Figures 8.4 and 8.5. Figure 8.4 is for a radial position of r/D=1.9,
which corresponds approximately to the position of maximum velocity
in all four cases. This figure also illustrates that radial entrain-
ment occurs below the depth of about 10.5 (1.5 diameters from the
surface) for these cases. At 7.32 diameters (Figure 8.5) the spread-
ing surface layer is slightly thinner.
The small cross-hatched rectangle shown in Figure 8.5 illustrates
the variation of the spreading depth for these cases. Greater penetra?
tion is noted at Froude number 100; FQ = 1 shows the least penetration.
The distributions of temperature excess at the surface
(ATS/ATQ) are illustrated by Figure 8.6. Vertical profiles of excess
temperature (iTs/ATQ) for Cases 48 and 50 are shown in Figure 8.7
(A and B). Note that the temperature profiles penetrate slightly
deeper than the velocity profiles and indicate some minor recircula-
tion of the heated water takes place.
A complete set of contour plots and three-dimensional illustra-
tions for the stream lines, temperature and vorticity for Cases 48
through 51 are given in Figures 8.8 through 8.32.
328
-------
RS
DIA
i— «
rss
i— «
o
IA
oo
rw
i '
§
o
«—»
SCALE: U =.5
SURFACE
''V Y V\
~~i—r T—I
0
RADIAL
VELOCITY
PROFILES
REG I ON OF
NEGATIVEU
•DISCHARGE PORT
ro - D/2
l
468
RADIAL POSITION, r/D
FREE BOUNDARY
r
10
Figure 8.3. Distributions of Radial Velocity
Case 50
329
-------
12
11
10
c/l
ee.
H-•
Q
rw 8
7
o
a.
>* c
< 6
ELEVATION
• OF
PORT
RADIAL POSITION rr/D'v,!. 9
l I I i I t
-0.1 0 0.5
DIMENSIONLESS-RADIAL VELOCITY, U
1.0
Figure 8.4. Maximum Radial Velocity Profiles, Cases 48 Through 51
330
-------
12
11 —
£10
UJ
H4
9 -
8 —
CO
o
Q_
7
X
6 -
5 —
4 -
-0.3 0 0.5
DIMENSIONLESS RADIAL VELOCITY, U
Figure 8.5. Radial Velocity Profiles at r/D=7.32,
Cases 48 Through 51
1.0
331
-------
CO
CO
ro
3456
RADIAL DISTANCE, r/D
Figure 8.6. Surface Temperature Excess Distribution
Cases 48 Through 51 (See Table 8.1)
-------
CO
CO
CO
.1
4T/4T
FIGURE 8.7 . VERTICAL TEHPERATURE EXCESS DISTRIBUTIONS
FOR VARIOUS RADIAL POSITIONS. CASES 48 > SO.
RflOlRL OIRECTIBN. R/d
FIGURE 8.8- STRERHLINES FBR COSE 48 - 8U8TBNT DISCHARGE. FO = 100
RBDIBL DIRECT1BH. R/D
FIOURE 8.9. 1S3THERHS FBR CPSE 48 - BUBrnNT DISCHARGE. FO = 100
rte
RBDIBL OIRECTIBN. R/l)
FieORE 8.10. VBRTIC1TT LEVEL LIMES FBR CBSE 48 - BUBYHNT DISCHflRGE. FO =
-------
OJ
GO
FIBURE8.il. 3D ILLUSTRRTI8N 8F STRERH FUNCTiaK — PSl . CBSE MB. 48
INTERtlEOIRTE HRTER BUTFHLL. 3URFRCE 10 DIHHETERS HBBVE P8RT. F0=100
FIGURE 8.12. 30 ILLUSTRHT18N BF rEHPERBTURE FIELD —4T. CBSE NB. 48
INTERHE01HTE HRTER 8UTFHLL. 8URFBCE 10 DIBHETERS BBBVE PBRT. FD=100
FIOURE 8.13. 30 1LLU3IRRTI8N ttf FLUID VORTICITT - BflEOB. CR8C NB.4*
INTCRHEOIRTE MBTE* BUTFRLL. 8URFBCE 10 D1R«TER8 M8VE r«KT. F0=|00
te rte rte rte~
RROIBL OIRECTI8N. R/0
FIOURE 8.H. STKBI1LIIIES FBR CBSE 49 - BUBTflNT OISCHRROE. FO = 25
-------
co
CO
171
FIGURE 8.15. ISBTHERflS FOR CRSE 49 - BUBYRNT OISCHRRGE. FO = 25
RRD1RL DIRECTION. R/D
FIGURE 8.16. V6RTICITY LEVEL LINES F6R CflSE 49 - 8UOf»NT OISCHHRGE. FO = 25
FIGURE 8.17. 30 ILLUSTRRTIBN BF STREBM FUNCTIBN — PS1 . CASE H8. 49
INTERHEDIRTE NRIER BUTFRLL. SURFHCE 10 DIMETERS BBBVE PBRT. FO
FIOURE 8.11. 3D ILLUSTRBTIUM OF STRERd FUNCTIBN -- PSI. CRSE NO. 49
INTERdEOIflTE URTER OUTFRLL. SUPFflCK 10 OIRnETERS RBBVE PORT. FO =25
-------
co
oo
en
FIGURE 8.19. 3D ILLUSTRBT18N BF TEHPERHTURE FIELD —»T. CRSE NB.49
INTERtlEOlfiTE MRTER BUTFRLL. SURFACE 10 DIMETERS RBBVE PORT. FO =Z5
FIGURE B.20. 3D ILLUSTRRT1BN BF FLUID VORTICITY - BHEGB. CRSE MB. 49
IHTERHED1BTE MRTER BUTFBLL. SURFHCE 10 DIRflETERS HBBVE PBRT. FO =ZS
g
RH01RL OIRECT1BN. R/D
FIGURE 8.21. STRERHLINES FOR CRSE SO- BU8VRNT D1SCHRRGE. FO - S
RRDIRL DIRECT IBM. R/D
FIGURE B.22. ISOTHERMS FOR CRSE 50- BUBVRNT OI-SCHRRGE. FO = 5
-------
CO
Co
RfiOIRL DIRECTION. R/0
LEVEL LINES FOR CASE SO- BUflVflNT OISCMHRGE. FO = 5
FIGURE s.24. 30 ILLUSTRATION BF STRERH FUNCTION -- PSI- CASE NQ.SO
IffTEKDlATC HATCH OUTFAU. SUtFACE 10 DIJWTERS WOVE PORT. FO • 5
FIGURE I.ZS. 30 ILUfSTWTIOd OF TDTCMTUK FIELD - AT. CASC K. 50
INTEIKOIATC WTU OUTFAU. SUtfACE 10 OMC7EH MOV KIT. FO - S
_ _
lOUffC •.». 30 ILLUSTRATION 0F FLU 10 m
e HRTCR OUTFRLL.
- 0HEGfl.
t 10 oin
CASE Hfl. 50
rCRS ABflve rasr, FO = s
-------
CO
co
oo
RfiDIflL DIRECTION. R/D
:e.27. sTRERniiNES FOR CRSE 51 - BUOYRNT OISCHHRGE. FO = i
RROIRL DIRECTION. R/D
FIGURE 8.28. ISOTHERfIS FOR CHSE 51 - BUOYRNT OISCHflRGE. FO = 1
TTfeo j.tw
RflOIflL DIRECTION, R/D
FIGURE 8-M. VORTICITY LEVEL LINES FOR CRSE SI - BUeTRNT DISCHRRGE. FO = 1
•"IOURE 8.30. 30 ILLUSTRRTIBN 8F STRERH FUNCTION -- P3I . CRSE NO. 51
INTERHEDIRTE HBTER 8UTFSLL. SURFRCE 10 OIBHETERS RB8VE P8RT, FO = 1
-------
FIGURE 8.31. 30 ILLUSTRRTI0N 3F TEMPERRTURE FIELD --4T. CRSE NO. 51
1NTERMEDIRTE HRTER 0UTFRLL. SURFRCE 10 OIRMETERS RS0VE PQRT. FO = 1
FIGURE 8.32. 30 ILLUSTRRTI0N 0F FLUID VORTICITY 0MEGR. CRSE NO. 51
INTERMEOIRTE WRTER 0UTFRLL. SURFRCE 10 DIRMETERS RB0VE PORT, FO - 1
339
-------
8.3 Results for Homogeneous Receiving Water 5 Port Diameters Deep
Results for outfalls issuing to receiving water 5 port diameters
deep are given as Cases 52 through 55 for Froude numbers of 100, 25,
5 and 1, respectively (See Table 8.1). All boundary conditions and
parameters for these cases correspond to those- of similar Froude
numbers for the 10 diameter deep cases given in Section 8.2. Actual
water depth here is 6 diameters with the outfall port rising one diam-
eter above the bottom.
Center!ine distributions of velocity and temperature excess are
shown in Figure 8.33 for Cases 52 through 55. As was illustrated by
Case 50 and 51, the plume also accelerates for Cases 54 and 55 as a
result of dominant buoyant forces. For Froude numbers of 25 and
above the centerline velocity remain essentially constant until sur-
face effects are encountered. On comparing Figure 8.1 with 8.33, one
notes that at 5 diameters the temperature excess given in Figure 8.33
is slightly higher than for corresponding cases given in Figure 8.1.
The decreased dilution is a result of the surface proximity.
The vertical distribution of radial velocity, U, is illustrated
by Figure 8.34 for Case 52. The lateral spread is seen to be quite
thin (approximately .8 D) at least out to 4 diameters. Figure 8.35
shows that temperature effects somewhat deeper (approximately 1.2 D)
and some recirculation of heated water is indicated. At r/D = 1.0,
the temperature distribution lies within the rising portion of the
plume above 2 ^ 2.5 (1.5 above the port) and is not to be interpreted
as penetration of the lateral spread.
340
-------
co
-p.
1 .0
5 -
.5 1.0
Axial Distance, Z (Diameters)
Figure 8.33. Computed Centerline Dimensionless Velocity and Temperature Excess for Shallow
Water Cases 52 Through 55 (5 Diameters Deep)
-------
CO
-p»
ro
r/0
o 1 .08
a 2.10
A 3.97
I
l
I
-0.4 0 0.4 0.8 1.2
Dimension!ess Radial Velocity, U
Figure 8.34. Vertical Distribution of Radial Velocity at Various
Radial Positions, Case 52
^ 4
AMBIENT,
r/D
o 1 .00
D 2.24
* 4.23
TEMPERATURE, T, C
10
Figure 8.35. Vertical Distribution of Temperature Excess at Various
Radial Positions, Case 52
-------
Contour plots and 3-dimensional illustrations of the stream
function, temperature and vorticity are given in Figures 8.36
through 8.41.
343
-------
CO
-p.
RBOlflL DIRECTION. R/0
FIGURE B.36. STREflHLINES F9R CflSE 52 - BUBYflNT DISCHARGE. FO = 1
RBOIHL DIRECTION. R/0
8.37. ISOTHERMS F8R CBSE 52 - BUBYRNT DISCHHRGE. FO =
RROIHL DIRECT IBM. R/0
FIGURE 8.38. V8RTICITY LEVEL LINES F8R CRSE 62 - BU6TRNT DISCHARGE. FO = 1
FIGURE B.JS. 30 ILLUSTRBT1BN OF STRERH FUNCTIBN — fSl. CHSE HO. 55
VERT SHHLLBH HBTER BUTFBLL. SURFBCE S OIBHETERS BBBVE PORT. F0= 1
-------
FIGURE 6.40. 30 ILLUSTRflTION OF TEMPERRTURE FIELD --4T. CflSE NO. 55
VERY SHRLLOH HRTER OUTFflLL. SURFRCE 5 DIRHETERS RBOVE PORT. F0= 1
FIGURES.Hi. 3D ILLUSTRHTION OF FLUID VORTICITY OflEGH. CfiSE NQ. 55
VERY SHHL1.3W WHTER 3UTFHI.L. SURFRCE S OIRMETERS HBQVE PORT. F0= 1
345
-------
8.4 Results for Two Different Methods of Computing FZ
Cases 56 through 59 are results illustrating the effects of
using Methods 3 and 4, and different values of the constant 3, for
computing the vertical eddy diffusivity multiplier, FZ (refer to
Tables 8.1 and 8.2). Cases 56 and 57 are for receiving water 5.6
diameters deep, using Method 3 to compute FZ with Froude number,
F =51. Case 58 has FQ = 105, with 4.97 diameter deep water using
Method 3. Case 59 is the same as Case 58 except Method 4 is used to
compute FZ.
Cases 56 and 57 were run to observe the effect of changing
6 = .4 to 3 = .8, respectively. Comparative results are not shown,
but this change of 3 did not alter the computed velocity and tempera-
ture profile a great deal.
It was observed, however, that computation of FZ by Method 3
resulted in excessive vertical diffusivities. Case 58 also employed
Method 3 and exhibited excessive vertical diffusivities (in this
Case 6 = 1.0). As pointed out in Section 8.2, Stolzenbach suggests
the value of 3 = 5.0 based on the gross Richardson number; however,
values using 3 > 1.0 were not tried in these cases. Using the larger
value of 3 could have a major effect on the velocity and thermal dis-
tributions computed by the present techniques using Method 3. The
use of large 3 would significantly reduce vertical mixing in the
thermal boundary region, but allow substantial vertical exchange
within the spreading plume where thermal gradients are expected to be
small.
346
-------
Figure 8.42 shows the comparison of surface spread velocity
between Cases 58 and 59. The difference here is not of major impor-
tance, but Figures 8.43 and 8.44 illustrate a significant difference
in vertical entrainment. Significant differences between stream-
line patterns is revealed by comparing Figures 8.45 and 8.46. The
contours shown in Figure 8.45 (Case 58) are more indicative of
creeping flow in the spreading portion of the plume than a high
Reynolds number flow (Case 59, Figure 8.46).
The distribution of surface temperature excess is shown in
Figure 8.47 for Cases 57, 58 and 59. Case 57 shows lower temperature
at the centerline as a result of the port being in deeper water.
Case 58 may be compared to Case 59 and exhibits a lower surface temp-
erature (also, refer to Figures 8.48 and 8.49). This result is due
to the larger values of vertical mixing employed in the computation
of Case 58. Isotherms for Case 59 are illustrated by Figure 8.50.
347
-------
CO
-p>
CO
5 10
RADIAL POSITION, r /D
15
Figure 8.42. Computed Radial Velocity at Surface, Cases 58 and 59
-------
CO
?.84
-0.04
CASE 58
RADIAL POSITION:r/D=9.83
I 1 I I
0.05 0.1 0.15
DIHENSIONLESS RADIAL VELOCITY, U
0.2
Figure 8.43. Ve-tical Distribution of Radial Velocity. U.
Cases 58 and 59.
0.24
RADIAL POSITION:r/D=13.19
I I
0 0.05 0.1 0.15
DIMENSIONLESS RADIAL VELOCITY, U
Figure 8.44. Vertical Distribution of Radial Velocity,
Case 58 and 59
0.20
U.
-------
CO
en
O
10.00
RROIRL DIRECTION. R/D
FIGURE 8.15. STRERMLINES FOR RN RXI SYMMETRIC. VERTICPL PLUME. CONFINED BY R FREE SURFRCc.
CASE 58.
15.00
-------
oo
en
s-bo
IQ.OO
RRD1RL DIRECTION. R/0
1S.OD
FIGURE 8. 46. STRERMLINES FOR RN RXISYMMETRIC , VERTICRL PLUME. CONFINED BY R FREE SURFRCE .
CASE 59.
-------
CO
U1
ro
5 -
4 -
3 -
2
*- ••••
o.
LU 1
(J I
«t
u_
o:
3
01
Figure 8.47.
RADIAL POSITION r/D
Surface Temperature Excess, ATS> Cases 57, 58, and 59
-------
CO
en
CO
RADIAL POSITION:r/D=9.13
2.5
RADIAL POSITION:r/DH4.2
0.25
Figure 8.48. Vertical Temperature Excess Distribution.
Cases 58 and 59
Figure 8.49. Vertical Temperature Excess Distribution.
Cases 58 and 59
-------
co
en
10-00
RRDIIU DIRECTION. R/D
FIGURE 8.50. ISOTHERMS F0R RN RX[SYMMETRIC. VERTICRL FLUME. CONFINED BY R FREE SURFRCE.
CASE 59.
-------
8.5 Numerical Experiments Involving Ambient Stratification
Results involving the effects of stratification are given by
Cases 60 through 65. Case 60 is a base case to be used for comparison
and is for a homogeneous ambient. The remaining cases have different
degrees of ambient stratification. In all cases the ambient (also,
effluent) salinity is constant at 35 ppts hence the ambient density
structure is a function of the temperature distribution alone. In
this section, all results use Method 4 to compute F and 6 = 1.0.
Unlike all previous cases presented in this chapter, the effluent
velocity profile is assumed flat.
Figure 8.51 illustrates the assumed ambient density structure
for the six cases.
Results for the base Case 60 are illustrated by Figures 8.52
through 8.59. One significant feature of the Case 60 results concern
velocity distribution and may be noted in Figures 8.52 and 8.54.
Figure 8.52 illustrates that radial velocity profiles for the spreading
plume continue to penetrate deeper into the ambient with increasing
radial distance from the outfall. For this case, temperature differ-
ences are small between the plume flow and ambient as illustrated by
Figure 8.53. The upward-distorted streamlines illustrated in Figure
8.54 indicate that there is significant upward entrainment into the
plume lateral spread.
The influence of a 2 °C ambient thermocline situated as shown
by Figure 8.51 is illustrated by Figures 8.60 through 8.64. Comparison
of Figures 8.62 and 8.54 shows that the presence of the thermocline
355
-------
ELEVATION OF PORT
20
21 22 23 24 25 26
TEMPERATURE, °C
Figure 8.51. Ambient Temperature Profiles
for Cases 60 Through 65
356
-------
CO
en
10
9.6 I—
9.2 —
£• 8.8 —
CK
&
I ••«--
O
* 8.0 I—
2 7.6
7.2
6.8
6.4
6.0
5.6
5.2
4.8
-0.1
0 0.1 0.2
MHEMSIONLESS RADIAL VELOCITY, U
0.3
Figure 3.52. Vertical Distribution Of. Radial Velocity. Case 60
Figure 8.53. Vertical Temperature Excess Distribution. Case 60
-------
CO
en
00
SH01RL DIRECTION.
FIGURE a.54. STREAMLINES FOR AN AXISTMHETRIC, VERTICAL PLUME. CONFINED BY A FREE
SURFACE CASE 60 - INTERMEDIATE DEPTH, HOMOGENEOUS AMBIENT, HAMATEV
22 tl.5 21 20.75 T - 20.5°C
RflDIRL DIRECTION. R/O ""™
FIGURE «,55. ISOTHERMS FOR All AI1STMMETRIC, VERTICAL PLUME, CORFINEO BY A FREE
SURFACE CASE 60 - INTERMEDIATE DEPTH, HOMOGENEOUS AMBIENT, HAMATEV
VOKTICITY CONTOURS AlISYMNETRIC, VERTICAL FLUNC. CONFINED IT I FtEI
SURFACE CASE 60 - INTERMEDIATE DEPTH. HOMKENCOUS AHIIENT. MMTEV
FIMIRE 1.97. IB ILLUSTRATION OF STKM FMCTION - PSI. USE NO. CO
-------
CO
(Jl
VO
8.58. 3D ILLUSTRflTIW Of TCnfeMTUftC FIELD —JT. CMC W. M
riOUK 8.5*. SO ILLUSTRDTI^M OF FLUID TOTIC1TY - MlCOfl. USE M. 60
THERHOCLIHE - 2°
r/0
o!J.26
a 9.13
• (.80
• 5.07
T... - 20°C
1.0
it. °C
-------
CO
CT>
O
r«R m miSTiwCTRic. »EKTIC«. rune, cwfinto IT R mcc jwrncc
tEmCOIKU OtPTH. KITH I KOttl THCMKl.INC. MMTC>
FtCURCLG* . VQftllCi" (.OHTOURr- M1S rnntl RIC. VCRT1CAL ft -rtC. CaNTINCO 6r ft rlfEC SURFRCE
CBScVl- illlCRBCOIRTE DEPTH. MITH 2 D€«t£E TME«nBtLl«E. MH1BYE*
-------
causes significant flattening of the streamlines, or reduced vertical
entrainment by the spreading plume. This reduction of vertical entrap-
ment is caused by suppression of vertical mixing by the presence of the
thermocline. In this case the plume flow spreads above the thermo-
cline, Also, the plume destroys the thermocline in the discharge
locale but the "convecting in" of the ambient density structure has a
significant effect beginning at distances approximately 5 diameters
out. Note the diverging of isotherms in Figure 8.63 and the tendency
for the isotherms to attain the ambient condition.
Increasing the magnitude of the thermocline results in further
reducing the vertical entrainment and stream line flattening as
illustrated by the results of Case 63 (Figures 8.65 through 8.71,
respectively). In this case the vertical location of the thermocline
is the same as in Case 61, but the magnitude of the thermocline is
4 °C instead of 2 °C.
The effects of a thermocline on the temperature structure are
most clearly revealed by Figures 8.66 and 8.67. Also note that out
to about 5 diameters the ambient density structure is again completely
destroyed by the plume flow. This feature coupled with the upwelling
of cooler water from beneath the thermocline results in a phenomenon
whereby there is a thermal peak above the outfall, but this peak
rapidly deteriorates radially to a temperature which is cooler than
the surface (see Figure 8.67). Unlike the base Case 60 where
vertical entrainment cools the plume, vertical entrainment warms the
361
-------
1
f
c2
z
Q
n
;y
0
1
p™
00
O
Q.
<
X
0
9.
9.
8.
8.
8
7
7
6
6
6
5
5
4
4
4
3
?
6
2
8
4
^p
1 1 p 71
£> J* ^^
Ai^^"^
sg/
- I//
••on
o- I//
*h "I'
1
on
.2- II
.8
.4
.0
.6
.2
.8
.4
.0
.6
?
CD
tl
— u
a
\
D
1
?
—
_
—
—
—
|
i
—
—
—
—
—
r/D ~
o 13.19
a 9.83
• 7.32 ~"
• 5.45 _
—
1 1 1
-0. 1
0 0.1 0.2
DIMENSIONLESS RADIAL VELOCITY, U
0.3
0.4
Figure 8.65. Vertical Distribution of Radial Velocity.
Case 63
362
-------
I I I I I I I I I I
2.0
AT, °C
Figure 8.66.
Vertical Temperature Excess Distribution.
Case 63
2.8
363
-------
to
o
o
5 -
4 -
3 -
2 -
T
BASED ON 20 °C AMBIENT TEMPERATURE
(SEE FIGURE 8.51 FOR VERTICAL STRUCTURE, CASE 63)
CASE 63
10
RADIAL POSITION, r/D
AMBIENT SURFACE
CASE 63 (ATC =4°C]
15
20
Figure 8.67. Surface Temperature Excess, AT<- for Cases 60 and 63
-------
CO
CT>
RADIAL DIRECT1BN. R/DIA
lfH. OIRCCTIBN. R/OIA
riniRF •« 3T«FBBLIHt3 FM (» BIISrnnET«IC. VCRTICHL FLUHE. Off I"™ •I-iJ"" SU«FBCE FIGimE U. ISOTncuns rgi B« DKISTKnCTKIC. »E«TIC«. TLunt. C»«fl«D er d r«EE SUKFHCE
FIGURE !•»• 3crfJE"n^1_"YM;[';1,,E|,1BT£ OErTH. """ • OEOIEE THEBnOCLIHE. MmBYEV Co5t B- l«Tt«BtolBI[ OErlH. «ITH « OECUtE IHEHnKLIMt. mtMrcv
voRTicnr CONTOURS Hxisi'finETRic. VERTICBI PLUHE. cowfiNeo BT n FREE SURFHCE
CBSL O- INTCRnEOIftTC DEPTH. HITH « DEGREE THEHMOCLLNE. HRtlHYEV
FICURPS-71- 30 ILLUSTRflTI9N 6F STREflH FUNCTieN -- PS1-
CflSE 63 - INTERPIEOfflTE DEPTH. HITH 4 DEGREE THERH0CLINE. nfitlHYEV
-------
lateral spreading flow since the cooler water is now on the surface
in the region of radial spread. This is, of course, a thermally
unstable situation, but the configuration is maintained by the flow
dynamic forces. This phenomenon is not uncommon and has been observed
on several occasions by Eliason [24] through areal infrared photog-
raphy. We would expect, however, that once dynamic forces are
mitigated to the point where buoyant forces (if they still persist)
dominate, local upwelling within the lateral spread would occur. Our
steady flow computer program cannot reveal these local time dependent
effects, but they are indicated by numerical cycling and reluctance to
converge. Since the case converged without difficulty, we conclude
that the flow field is dynamically stable, at least for the parameters
used.
Figures 8.69 and 8.72 again show the thermal effects of "con-
vecting in" or recirculating the ambient thermal structure and the
tendency of the thermal distribution to attain the ambient structure.
Figures 8.74 through 8.81 show results for Case 64 where the
thermocline is 5 °C, although the thermal gradient is identical to
Case 63 (see Figure 8.51). Comparison with appropriate results of
Case 63 shows little influence from this change.
In Case 65 the shape of the thermocline was assumed to be the
same as in Case 64 except situated at a somewhat greater depth
(Figure 8.51). Figures 8.82 through 8.90 illustrate results for this
case. For the problem posed, computation could not be carried out
to achieve a steady flow converged solution. Instead numerical
366
-------
FIGURE 8-72- 30 ILLUSTRRTI8N 3F TEMPERflTURE FIELD --4T.
CflSE 63- INTERMEDIRTE DEPTH. H!TH 4 DEC-REE THERM3CLINE. MRflRYCV
FIGURE 8.73. 30 ILLUSTRPiTI 3N 3F FLUID VQRT1CTY 3HEGH.
CHSE 63 INTERMt'DIfiTE DEPTH. WITH 4 DEGREE THERMOCLINE
367
-------
co
o
00
10.0
9.2 -
8.4
7.6 -
6.8 _
6.0 -
5.2 -
4.4 -
10.0
-0.05 0 O.I 0.2 0.3
DIMErtSIONLESS RADIAL VELOCITY, U
Figure 8.74. Vertical Distribution of Radial Velocity.
Case 64
4.4 -
1.0
2.0
2.8
AT, °C
Figure 8.75. Vertical, Excess Temperature Distribution.
Case 64.
-------
co
en
<£>
RRDIRL DIRECTION. R/o
FIGURE 1.76. STRERHUNES FBR AN RXISTfnlETRIC. VERTICAL PLUHE. CONFINED BT R FREE SURFACE
CBSE M - IHTERnEOIRTE DEPTH. HITH 5 DECREE THERIWCLI HE. IUUIRTEV
RRO"IBL DIRECTION. R/tr
FIGURE 1.77. ISBTHERHS FDR B« RXI3THIETR1C. VERTICRt. PtUHC. CONFIICO »T R FREE SUKFRCE
CRSE H - IHIERItEOIBIt DEPTH. HITH 5 DEOHEE THERMtl-IIC. MM)rE>
FIGURE 1.19. 30 ILLUSTmilM or VI3CMJS SIKEBn FWCTIW
CRSE M - IHTERNCDIRTE DEPTH. HITH 5 DCCiCC Tl
HCmmCLIIC. MMTEV
-------
FIGURE I.SO. 30 ILLUSTRATION BF TEMPERATURE FIELD—AT .
CASE H - INTERMEDIATE DEPTH. WITH S DECREE THERXOCLINE. NfmRTEV
FIGURE ».«!•
3D ILLUSTRATION OF VORTICITT — ONEGA
CASE M - INTERNED IBTE DEPTH. MITH 5 DEGREE THERMOCLINE. HAHAYEV
370
-------
IU.O
9.2
CO
UJ
Ejj 8.4
0
rw 7.6
ft
0
s 6-8
D.
_J
*~r*
< 6.0
5.2
4.4
3.6
-
—
- //
/*/*/y
• •00
I///
-1
• A
- \\
- \\
- \
: "
i
J^^^J "
.^D^^
^Jx
TAMB = 25
i
'
THERMOCLINE = 5
r/D
n 1 "\ 1 Q
w 1 o . I y ^~~~
i
a 9.83
• 7.32 T = 20
. 3.45 AMB
1 1 1 1
°C
°C
Op
-0.05 0 0.1 0.2 0.3 0.4
DIMENSIONLESS RADIAL VELOCITY, U
Figure 8.82. Vertical Distribution of Radial Velocity.
Case 65
371
-------
GO
~«4
IN5
10.0
9.2 -
to
ae.
at Q A
i—* O . •*
o
«t
»—I
X
7.6 -
6.8 -
6.0
5.2 -
4.4 ~
2.0
3.0
AT, °C
4.0
Figure 8.83.
Vertical, Excess Temperature Distribution.
Case 65
-------
CO
^J
CO
"RHDIRL DIRECTION. R/Q
RHDIRL 0
IRECTION. R/D
FIGURE S-flU - STRERI1LINES FOR RN RXISYHHETR1C. VERTICAL PLUME. CONFINED BY R FREE 9URFRCE
CRSE ffi • IHTERflEDlflTE DEPTH. WITH 5 DEGREE THERHOCLINE. HRHRYEV
FIGURE 8,85 . VORTICITY CONTOURS RXISYRRETRIC • VERTICHL PLUHE
CR3ES - INTERHEDIRTE DEPTH. HITIi 5 DECREE THE
. CONFINED1 BY R FREE SURFRCE
ISOTHERMS F3R RN RXiSYnflETRIC. VERTICAL PLUME. CONFINED BY R FREE SURFRCE
CRSE ffi - INTERMEOIRTE DEPTH. HITH 5 DEGREE THERHOCL1NE. MfmRYEV
'RflOIflL DIRECTION. R/D
FIGURE B.W . STRERHLINES FOR RN RxisrnMETRic. VCRTICRL PLUHC. CONFIWEO er R FREE SURFRCE
CHSE 65- iNTERttEOIRTE DEPTH. CONTINUED 1TERRTION.
-------
RBD°IAL DIRECTION. R/D
FIGURE 8.88. ISOTHERMS FOR FIN AXI3YMNETR1C. VERTICAL PLUME. CONFINED BY ft FREE SURFACE
CASE 65 - INTERMEDIftTE DEPTH. CONTINUED ITERATION.
RBDIAL DIRECTION. R/0
FIGURE 8.89. VORTICITY CONTOURS RX1SYMMETRIC. VERTICAL PLUME. CONFINED BT A FREE SURFACE
CASE 65- INTERMEDIATE DEPTH. CONTINUED ITERATION.
374
-------
cycling occurred. Results after 1000 iteration cycles are shown by
Figures 8.84 through 8.86. Figures 8.87 through 8.89 reveal results
after 300 additional iterations.
Although, the case as posed may not conform to a physically real
situation (in particular, the ambient density structure), a thermal
instability is suspected which may be either real or perhaps incited
by numerical perturbations. Inspecting Figure 8.86 illustrates a
large region of cooler water above the thermocline. Continued itera-
tion showed that the ambient isotherms, within the circulating ambient,
begin to fluctuate vertically out to about 7 diameters. Further itera-
tion resulted in the development of two recirculating regions: one
above the thermocline and the other below (see Figure 8.87). That is,
some of the plume flow attempts to spread beneath the thermocline. If
the iterative computation is continued, streamline patterns closely
resembling those shown in Figure 8.84 will redevelop (single recircu-
lating region).
The investigation of Case 65 was carried out through approximately
three cycles of the flow changing from one recirculating region, to
two regions and back to one region again. These computations showed
neither the tendency for the solution to converge or diverge numeric-
ally. It is difficult to derive much incite from steady flow computa-
tions possessing such behavior except that a thermal instability is
either present or close at hand. A transient computation of the same
flow conditions would doubtless reveal similar oscillations during the
initial transient, caused by the pulsed plume flow starting condition.
However, we would expect the oscillations to damp out with time except
375
-------
if a true thermal instability were present. For our steady flow
computation, real conditions may be near to those for a real thermal
instability and the nature of the steady flow numerical techniques
may be perturbing the solution to a point which prevents a converging
result.
-------
computed velocity occurs about 0.7 diameter from the plume centerline,
which is about the edge of the "boil" for a real outfall of these pro-
portions. Figure 8.91 indicates that the plume has undergone only
slight cooling on reaching the surface (^ 3/4 °C), but cools very
rapidly out to about 2 diameters and decreases to about 2 °C above
ambient at 8 diameters.
The radial velocity profiles at selected locations are shown in
Figure 8.92 which shows that the plume along with entrained flow,
spreads in a fairly shallow sheet at the surface, penetration being
less than 0.4 diameter. Temperature profiles (Figure 8.94) penetrate
slightly deeper. In fact, the computation shows that plume thermal
effects penetrate into the negative flow region, hence there is some
indication of plume heat recirculation.
Streamlines, isotherms and level lines of vorticity are illus-
trated in Figures 8.94, 8.95 and 8.96, respectively. The inward
bending of the streamlines (Figure 8.94) above the discharge port
indicates considerable acceleration of the effluent. Maximum vorticity
for this case occurs near the surface and near the point of maximum
lateral spread. This region of high vorticity is also the region where
one would expect the edge of the surface boil to occur in a real flow.
Three-dimensional surfaces are plotted in Figures 8.97 through 8.101
for the stream function, temperature excess and vorticity.
377
-------
3.0
2.5 .
2.0 _
1.5 -
1.0
23456
RADIAL DISTANCE, r/D
Figure 8.90. Surface Radial Velocity.
Case 66
23456
RADIAL DISTANCE, r/D
Figure 8.91. Surface Temperature Excess.
Case 66.
378
-------
1.425
1.20 _
RADIAL POSITION
r/D
2.10
3.97
4.81
7.48
.30 _
.5 1.0
RADIAL VELOCITY, U
1.5
2.0
1.425
Figure 8.92. Verttcal Distribution of Radial Velocity ik Various Radial Positions.
case 06
fc
o
Q.
.90
.60
.30
SURFACE
\J
RADIAL POSITION
r/D
• 2.24
• 4.23
D 5.45
O 7.02
TEMPERATURE EXCESS. AT UC
Figure 8.93. Vertical Distribution of Temperature Excess at Various Radial Positions.
Case 66
379
-------
co
8
FIGURE 3.34. STREflnUNES FOR CflSE 66 I 1 .T ulP DLtf i FO ; .III
Tte • rte-
RBOIHL OIREETIBN. R/0
F1BURE 8.95. IS6IHERHS F8R CflSE 66 (I.0 OIR DEEP) FO r .111
RflOlHL OIRECTI8N. R/D
FIGURE 8.96. YeRTIClTY F8R COSE 66 11 .0 DIB DEEP) FO =: .111
FIGURE 8.97. 3D ILLUSIRfll 18N aF FLUID VBRTICTr - SIIEGfl. CR3E N8. 66
-------
OJ
00
FIGURE 8,38. 30 ILLUSTRRTI8N BF STREfln FUNCTION — PSI. CR3E N8.
FIGURE S.99. 30 ILLUSTRHTI8N 6F STRERn FUHCTI8N -- PS1. CRSE N8. 66
FIGURE 8.100. 3D 1LLUSTRRT1BN BF TEBPERRTURE FIELD -- T. CHSE HO. 66
8.101. 3D IU.U3TRRTIBN BF TEHrERRTURE FIELD —»T. CR3E MB. 66
-------
8.7 Comparison with. Field Data
At a late-date in this study* the author was able to obtain
reliable field data for one shallow water application. This data,
obtained for a customer by Battelle-Northwest, is proprietary and
details cannot be disclosed. However, the discharge depth is less
than one port diameter and the densimetric Froude number is on the
order of 2.5.
Figure 8.102 shows a comparison between the computed results and
the field measurements. As can be seen, there is reasonably good
agreement between data and computation. The computer program predicts
surface temperatures which are about 50% high out to about 10 diameters.
Temperatures equal to the effluent temperature are predicted at the
surface directly over the outfall, whereas the field data indicates
an average of about 70% of this value. This discrepancy illustrates
that an improved turbulence model is needed for the transition region
and perhaps a better representation of the cascading caused by the boil
formation. Nonetheless, this result is very encouraging because the
computation was performed before the infra-red field data were reduced
to temperature information, indicating that at least for very shallow
water cases the computer code is a useful predictive device which
requires little use of empirical constants.
This result is only one check point and additional field or
laboratory data are certainly needed for further verification. Such
information could also be used for improvement of the eddy diffusivity
model—which is sorely needed.
382
-------
oo
CO
JU
,28
26
24
22
20
oo
oo
uj 1;
16
14
12
10
8
6
4
2
,
A FIELD DATA
• MODEL DATA
MAXIMUM OBSERVED SURFACE TEMPERATURE
A
A
i i i
i i i
i i r
1 i ! i 1 i
100 200 300 400
RADIAL DISTANCE FROM DISCHARGE, FEET
Figure 8.102. Comparison of Computed Surface Temperature with Field Data
500
-------
CHAPTER 9
CONCLUSIONS
The work contained in this manuscript represents an extensive
numerical study of axisymmetric plume flow. Various computational
details dealing with practical applications have been investigated
along with an extensive verification study comparing numerical
results with available published data.
The objective of developing a computer code for general use for
vertical plume rise in shallow water and the ensuing lateral spread
was riot entirely realized. The code developed is more of a research
tool than a design tool. The primary reason for this result was the
difficulty in modeling turbulent diffusivities. Such models are well
established for the vertical rise, but relatively little is known
about vertical diffusivities in the lateral spread. Hence, for this
and other investigative reasons the computer code suffered through
various changes and adaptions during the study; the code listed in
Appendix E is one of these later versions.
The more significant conclusions from this study are as
follows:
, The steady flow vorticity-stream function technique
along with the use of a coupled buoyancy transport equa-
tion is an effective and accurate method for computing
buoyant plume hydrodynamics up to our ability to model
turbulent transport coefficients.
* The iterative use of Prandtl mixing length theory
(Prandtl's second hypothesis) is entirely satisfactory
384
-------
for computing radial eddy transport coefficients in
the plume-rise regime. In addition
- the computations predicted er to be essentially
constant for a pure initial inertia! flow which is
also demonstrated by published experimental data).
- depending on the extent of buoyancy, the computations
predicted er to vary a great deal with axial positiop,
and that using a constant value of e in a buoyant
flow can lead to large errors in the computed plume
velocity and temperature distributions.
The iterative use of Prandtl mixing length theory for the
vertical eddy transport coefficient was used in this work
but was found not to be entirely satisfactory for the plume
lateral surface spread. That is, limitations had to be
imposed on the maximum size of the computed mixing length
to prohibit numerical instability resulting from an unstable
mixing length computation. Vertical eddy diffusion was found
to have little effect on computed quantities within the plume
vertical rise.
Mixing length theory was found to be entirely unsatisfactory
/
for the circulating (ambient) flow field.
Solution convergence was slowed dramatically by:
- Iterative computation of eddy transport coefficient
(as opposed to constant values),
- flow coupled with buoyancy transport (as opposed to a
385
-------
pure inertial flow),
- multiple iteration on the stream function elliptic equa-
tion between each iteration of the transport equations.
In addition to the third point mentioned immediately above, in
every case tried one psi inner iteration (stream function) per out-
ter iteration (vorticity and buoyancy transport) was found to
be satisfactory for convergence. It is strongly suspected that
once the approach to convergence for the stream function has
become smooth more than one outer iteration per inner itera-
tion would not significantly affect the convergence rate.
This action would, however, result in decreased computation time.
The numerical techniques were found to be stable for every case
tried except for the following two instances:
- over relaxation of the transport equations,
- use of iteratively computed eddy transport coefficients
, before reasonable.velocity profiles were obtained by
using constant coefficients.
It was found that over-relaxation of the vorticity equation
always led to a numerical instability for the cases tried.
This problem was rectified by using LT = .999. In no case
using constant transport coefficients and LT <_ .999, was an
instability noted.
The stream function elliptic equation could be over-relaxed
in some cases (deep water cases) using LF = 1.6.
386
-------
However, in the shallow water cases (Z$ <_ 5) numerical insta-
bilities were noted using LE = 1.6. Subsequently, L < 1 was
used with general success.
• Based on results shown in Figure 8.102, it is concluded that
the computational methods presented herein can be a very accur-
ate mechanism for computing the surface temperature distribu-
tion in the near field of a large, vertical, shallow water
coastal thermal outfall. Hence, the primary objective of this
study is successfully accomplished.
The result shown in Figure 8.102 is very encouraging since
the computed surface temperature distribution was found to be
in excellent agreement with field measurements and the fact
that this agreement was obtained without prior knowledge of
the field results. However, this is the only case where
computation was compared to field data and other situations
may reveal discrepancy. Obviously, complete validity of the
model can only be ascertained by further comparison with field
measurement.
From the results of this study it is generally concluded that the
numerical techniques used are a viable and practical method for comput-
ing thermal dispersion in confined steady-flow plumes up to our ability
to model the plume-generated turbulence. The numerical approach is
extremely attractive from the viewpoint that important complexities
can be incorporated in the analysis which cannot be accommodated with
387
-------
similarity techniques.Hence, the numerical model, which may be cali-
brated with field data, will yield reliable computed information and
permit a more competent thermal analysis. However, this study has
shown that there is indeed a great need for research in turbulence
modeling and the application of these models in numerical computation.
388
-------
BIBLIOGRAPHY
1. Abraham, G. Jet diffusion in stagnant ambient fluid. Delft,
Delft Hydraulics Laboratory, 1963. 183 p. (Publication No.
29* series 1, group 14, section 14.42)
2. Abraham, G. Horizontal jets in stagnant fluid of others density.
Proceedings of the American Society of Civil Engineers, Journal
of the Hydraulics Division 86:1-13. 1960.
3. Abraham, G. Jets with negative buoyancy in homogenous fluid.
Journal of Hydraulic Research 4:235-248. 1967.
4. Albertson, M. L., Y. B. Dai, R. A. Jensen, and H. Rouse.
Diffusion of submerged jets. Transactions of the American
Society of Civil Engineers 115:639-664. 1950.
5. Amsden, A. A. and F. H. Harlow. The SMAC method - a numerical
technique for calculating incompressible fluid flows.
Los Alamos, Los Alamos Scientific Laboratory. 1970. 85 P-
(Document No. LA-4370)
6. Anwar, H. 0. Behavior of buoyant jet in calm fluid. Proceedings
of the American Society of Civil Engineers, Journal of the
Hydraulics Division 97(4):1289-1303. 1969.
7. Aziz, K. A numerical solution of cellular convection. Doctoral
dissertation. Houston, Rice University, 1966, 116 numb.
leaves. (Microfilm)
8. Baines, Douglas W. Discussion of submerged jets by Albertson
et al. Transactions of the American Society of Civil Engineers
115:677-684. 1950.
9. Barakat, H. Transient natural convection flows in closed
containers. Doctoral dissertation. Ann Arbor, University of
Michigan, 1965. 220 numb, leaves.
10. Batchelor, G. K. An introduction to fluid dynamics. London,
Cambridge University Press, 1967. 615 p.
11. Baumgartner, D. J. Vertical jet diffusion in non-linear density
stratified fluid. Doctor dissertation. Corvallis, Oregon State
University, 1966. 166 numb, leaves.
389
-------
12. Baumgartner, D. J. and D. S. Trent. Ocean outfall design,
part one, literature review and theoretical development.
Corvallis, 1970. 129 p. (Federal Water Quality Administra-
tion, Environmental Protection Agency)
13. Bird, R. G., W. E. Stewart, and E. N. Lightfoot. Transport
phenomena. New York, John Wiley and Son, 1960. 780 p.
14. Bosanqet, C. H., G. Horn and M. W. Thring. The effect of
density differences on the path of jets. Proceedings of the
Royal Society of London, Ser. A, 263:340-352. 1961.
15. Bradshaw, P. and D. H. Ferriss. Application of a general
method of calculating turbulent shear layers. London, Imperial
College, 1970. 17 p. (Department of Aeronautics. Report
No. 70-06)
16. Brooks, N. H. Conceptual design of submarine outfalls - I.
Pasadena, California Institute of Technology, 1969. 19 p.
17. Cederwall, K. Hydraulics of marine water waste disposal.
Goteborg Hydraulics Division, Chalmers Institute of Technology,
1968. 273 p. (Publication No. 42)
18. Chou, P. Y. On velocity correlations and solutions of equations
of turbulent fluctuations. Quarterly Journal of Applied
Mathematics 3(l):38-54. 1945.
19. Chou, P. Y. Pressure flow of turbulent fluid between two
infinite parallel planes. Quarterly Journal of Applied
Mathematics 3(3):185-197. 1945.
20. Corrsin, S. and M. S. Uberoi. Further experiments on the flow
and heat transfer in a heated turbulent jet. Washington, D. C.
1947. 61 p.
21. Crowley, W. P. Numerical advection experiments. Monthly
Weather Review 96(1):1-11. 1968.
22. Csanady, G. T. The buoyant motion within a hot gas plume in a
horizontal wind. Journal of Fluid Mechanics 22(2):225-239.
1965.
390
-------
23. Donovan, L. F. Numerical solution of the unsteady Navier-Stokes
equations and application to flow in a rectangular cavity with
a moving wall. Washington, D. C. 1971. 56 p. (NASA TN D-6312)
24. Eliason, J. R. Battelle-Northwest. Personal communication.
Richland, Washington. 1972.
25. Ellison, T. H. and J. S. Turner. Turbulent entrainment in
stratified flows. Journal of Fluid Mechanics 6(3):423-448.
1959.
26. Fan, Loh-Nien. Turbulent buoyant jets into stratified or flowing
ambient fluids. Pasadena, California Institute of Technology,
1967. 196 p. (W. M. Keck Laboratory of Hydraulics and Water
Resources. Report No. KH-R-15)
27. Fan, Loh-Nien and Norman H. Brooks. Numerical solution of
turbulent buoyant jet problems. Pasadena, California Institute
of Technology, 1969. 94 p. (W. M. Keck Laboratory Hydraulics
and Water Resources. Report No. KH-R-18)
28. Fay, J. A., M. Escudier and D. P. Holt. A correlation of field
observations of plume rise. Journal of the Air Pollution
Control Association 20(6):391-397. 1960.
29. Fox, D. G. Forced plumes in a stratified fluid. Journal of
Geophysical Research 75(33):6818-6835. 1970.
30. Frankel, R. J. and J. D. Cummings. Turbulent mixing phenomena
of ocean outfalls. Proceedings of the American Society of
Civil Engineers. Journal of the Sanitary Engineering Division
91(2):33-59. 1965.
31. Fromm, J. E. Numerical methods for computing non-linear, time
dependent, buoyant circulation of air in rooms. IBM Journal
of Research and Development 15(3):186-196. 1971.
32. Gaunter, J. W., J. N. B. Livingood and P. Hrycak. Survey of
Literature on flow characteristics of a single turbulent jet
impinging on a flat plate. Washington, D. C. 1970. 42 p.
(NASA TN D-5652)
391
-------
33. Golubeva, V. Nf The formation of the temperature field in a
stratified sea, TR. by F. Goodspeed. Bulletin of Academy of
Science of the USSR Geophysical Series 5:4670-4671. 1964.
34. Gortler, H. Berechung von aufgaben der.freinen turbulenz auf
grund eine$ neuen naherungsansatzes. zeitschrift fur Angewandte
Mathematik Und Mechanik 22:257-267. 1936.
35. Gosman, A. D. et al. Heat and mass transfer in recirculating
flows. London, Academic Press, 1969. 338 p.
36. Harremoes, P. Diffuser design for discharge to a stratified
water. Copenhagen, Danish Isotope Center, 1967. 18 p.
37. Hart, W. E. Jet discharge into a fluid with a density gradient.
Proceedings of the American Society of Civil Engineers, Journal
of the Hydraulics Division 87:171-200. 1961.
38. Hayahsi, T. and N. Shuto. Diffusion of warm water jets
discharged horizontally at the water surface. In: Proceedings
of the Twelfth Congress of the International Association of
Hydraulic Research, Fort Collins, Colorado, 1967. p. 47-59.
39. Hill, M. N. (Editor). The sea. Volume 1. New York Interscience
publishers, 1962.
40. Hinze, J. 0. Turbulence. New York, McGraw-Hill, 1959. 586 p.
41. Hinze, J. 0. and B. G. Van Der Hegge Zijnen. Transfer of heat
and matter in the turbulent mixing zone of an axially symmet-
rical jet. Applied Scientific Research (A) 1:435-461. 1949.
42. Hirasaki, G. J. and J. D. Heliums. Boundary conditions on the
vector and scalar potentials in viscous three-dimensional
hydrodynamics. Quarterly of Applied Mathematics 28(2):293-296.
1970.
43. Hirst, E. A. Analysis of buoyant jets within the zone of flow
establishment. Oak Ridge, Oak Ridge National Laboratory, 1971,
41 p. (Report No. ORNL-TM-3470)
392
-------
44. Hirst, E. A. Analysis of round turbulent, buoyant jets dis-
charged to flowing stratified ambients. Oak Ridge, Oak Ridge
National Laboratory, 1971. 36 p. (Report No. ORNL-4685)
45. Hoult, D. P., J. A. Fay and L. J. Forney. A theory of plume
rise compared with field observations. Journal of the Air
Pollution Control Association 19(8):585-590. 1969.
46. Hwang, J. D. On numerical solution of the general Navier-Stokes
equations for two-layered stratified flows. Doctoral disser-
tation. Corvallis, Oregon State University, 1968. 134 numb.
leaves. (Microfilm)
47. Isayeva, T. and I. L. Isayev. Determination of vertical eddy
diffusion in the upper layer of the black sea by a direct
method. TR. by Scripta Technica, Inc. Soviet Oceanography
Trans, of the Marine Hydrophysical Institute, Academy of Science
of the USSR, 1963 series 2:22-24. 1963.
48. Jen, Y., R. L. Wiegel and I. Mobarek. Surface discharge of
horizontal warm water jet. Proceedings of the American Society
of Civil Engineers, Journal of the Power Division 92(2):i-3U.
1966.
49. Keagy, W. R. and A. E. Weller. A study of freely expanding
inhomogeneous jets. In: Proceedings of the Heat Transfer and
Fluid Mechanics Institute, University of California, Berkeley,
June 22-24, 1949. New York, American Society of Mechanical
Engineers, p. 89-98. 1949.
50. Keffer, J. F. and W. D. Baines. The round turbulent jet in a
crosswind. Journal of Fluid Mechanics 15(4):481-497. 1963.
51. Kent, R. E. and D. W. Pritchard. A test of mixing length
theories in a coastal plain estuary. Journal of Marine
Research 18(l):62-72. 1959.
52. Koh, R. C. Y., and L. N. Fan. Mathematical models for the pre-
diction of temperature distributions resulting from the dis-
charge of heated water into large bodies of heated water.
Washington, D. C. 1970. 219 p. (FWQA Publication No, 16130
DW010/70, Tetra-Tech Inc.).
393
-------
53. Larkin, B. K. Some stable explicit difference approximations
to the diffusion equation. Math. Computation 18(86):196-202.
1964.
54. Leenderste, J. J. Aspects of a computational model for long
period water-wave propagation. Santa Monica, The Rand Corpor-
ation. 165 p. 1967. (Document No. RM-5294-PR)
55. Ma, A. S. C. and K. S. Ong. Impulsively started incompressible
jet. London, Imperial College, 1970. 28 p. (Department of
Mechanical Engineering. Report No. EF/TN/A/28)
56. Manabe, K., Y. Watanabe and A. Wada. Study on recirculation
of cooling water of Tsuruga nuclear power station sited on
Urazoko Bay. Civil Engineering in Japan 9:156-171. 1966.
57. Masch, F. D. et al. A numerical model for the simulation of
tidal hydrodynamics in shallow irregular estuaries. Austin,
The University of Texas, 1969. 123 p. (Office of Water
Resources Research, D. I. Document No. PB 184834)
58. Morton, B. R. Forced plumes. Journal of Fluid Mechanics
5:151-197. 1959.
59. Morton, B. R. The ascent of turbulent forced plumes in a calm
atmosphere. International Journal of Air Pollution 1:184-197.
1959.
60. Morton, B. R., Sir Geoffrey Taylor, and J. S. Turner. Turbulent
gravitational convection from maintained and instantaneous
sources. Proceedings of the Royal Society of London, Ser. A,
234:1-23. 1956.
61. Muragi, M. P. and H. W. Emmons. Natural convection above fires.
Journal of Fluid Mechanics 8(4):611-624. 1960.
62. Murota, A. and K. Muraoka. Turbulent diffusion of the vertically
upward jet. In: Proceedings of the Twelfth Congress of the
International Association of Hydraulic Research, Fort Collins,
Colorado, 1967. p. 60-70.
63. Neuman, G. and W. J. Pierson Jr. Principles of physical ocean-
ography. Engelwood Cliffs, Printice Hall, 1966. 545 p.
394
-------
64. Okubo, A. Horizontal diffusion from an instantaneous point
source due to oceanic turbulence. Baltimore, Johns Hopkins
University, 1962. 22 p. (Chesapeake Bay Institute, Tech.
Report No. 32)
65. Okubo, A. Fourth report on the rising plume problem in the sea.
Baltimore, Johns Hopkins University, 1968. 19 p. (Chesapeake
Bay Institute, Reference 68-3)
66. Oregon State University. Department of Civil Engineering.
Tidal flats in esturine water quality analysis. Corvallis, 1969.
10 numb, leaves. (Progress Report. Grant No. WP-01385-01 of the
Federal Water Quality Administration)
67. Pagnani, B. R. An explicit finite-difference solution for
natural convection in air in rectangular enclosures. Doctoral
dissertation. Corvallis, Oregon State University, 1968.
176 numb, leaves. (Microfilm)
68. Pai, S. I. and T. Y. Hsieh. Numerical solution of laminar jet
mixing with and without free stream. College Park, University
of Maryland, 1969. 32 p. (The Institute for Fluid Dynamics and
Applied Mathematics. Technical Note BN-627)
69.
70.
Patankar, S. V. and D. B. Spalding. A finite-difference
cedure for solving the equations of the two-dimensional
boundary layer. International Journal of Heat and Mass
Transfer 10(10):1389-1412. 1967.
pro-
Phillips, 0. M. The dynamics of the upper ocean.
Cambridge University Press, 1966. 261 p.
London,
71. Pond, S. Oregon State University. Personal communication.
Corvallis, Oregon. 1969.
72. Prandtl, L. Uber die ausgebildete turbulenz. Zeitschrift Fur
Angewandte Mathematik Und Mechanik 5:136-139. 1925.
73. Pristly, C. H. B. A working theory of the bent-over plume of
hot gas. Quarterly Journal of the Royal Meteorological
Society 81:144-157. 1955.
395
-------
74. Priestly, C. H. B. and K. F. Ball. Continuous convection from
an isolated source of heat. Quarterly Journal of the Royal
Meteorological Society 81:144-157. 1955.
75. Ramsey, J. W. and R. J. Goldstein. Interaction of a heated jet
with a deflecting stream. Transactions of the America! Society
of Mechanical Engineers, Journal of Heat Transfer, Series C,
93(4):365-372. 1971.
76. Rawn, A. M., F. R. Bowerman and N. H. Brooks. Diffusers for
disposal of sewage in sea water. Proceedings of the American
Society of Civil Engineers, Journal of the Hydraulic Division
86:65-105. 1960.
77. Reichardt, H. Impuls - und warmeaustauch in freier turbulenz.
Zeitschrist Fur Angewandte Mathematik Und Mechanik 24:268-272.
1944.
78. Reynolds, 0. On the dynamical theory of incompressible viscous
fluids 'and on the determination of the criterion. Philosophical
Transactions of the Royal Society, Series A 186:123-164. 1894.
79. Rotta, J. C. Statistische theorie nichthomogener turbulenz.
Z. Fur Physik 129:547-572. 1951.
80. Rotta, J. C. Statistische theorie nichthomogener turbulenz.
Z. Fur Physik 131:51-77. 1951.
81. Rouse, H., C. S. Yih and H. W. Humphreys. Gravitational
convection from a boundary source. Tellus 4:201-210. 1952.
82. Runchal, A. K. and M. Wolfshtein. Numerical integration
procedure for the steady state Navier-Stokes equations.
Journal of Mechanical Engineering Science 11(5):445-453. 1969.
83. Sami, Sedat. Space correlation in round turbulent jet. Proceed-
ings of the American Society of Civil Engineers, Journal of the
Hydraulics Division 97(3):907-917.
84. Schlichting, H. Boundary layer theory, TR. by J. Kerstin.
6th Ed. New York, McGraw-Hill, 647 p. 1968.
396
-------
85. Schmidt, Wilhelm. Turbulente ausbreiting eines stromes
erhitzter lift. Zeitschrift Fur Angewandte Mathematik Und
Mechanik 21:265-278, 351-363. 1941.
86. Schmidt, F. H. On the rise of hot plumes in the atmosphere.
International Journal of Air and Water Pollution 9:175-198.
1964.
87. Scorer, R. S. The behavior of chimney plumes. International
Journal of Air Pollution 1:198-220. 1959.
88. Sharp, J. E. Spread of buoyant jets at the free surface.
Proceedings of the American Society of Civil Engineers, Journal
of the Hydraulics Division 95(3):811-825. 1967.
89. Sharp, J. J. Unsteady spread of buoyant surface discharge.
Proceeding of the American Society of Civil Engineers, Journal
of the Hydraulics Division 97(9):1471-1491. 1971.
90. Spalding, D. B. Concentration fluctuations in a round turbulent
free jet. Chemical Engineering Science 26:95-107. 1971.
91. Spalding, D. B. Numerical methods in fluid mechanics and heat
transfer. London, Imperial College, 1970. 37 p. (Department
of Mechanical Engineering. Report No. EF/TN/A/34)
92. Spalding, D. B. Heat and mass transfer in recirculating flows.
London, Imperial College, 1968. 28 p. (Department of
Mechanical Engineering. Report No. EF/TN/A/8)
93. Spalding, D. B. Mathematical models of turbulent flames.
London, Imperial College, 1969. 24 p. (Department of Mechan-
ical Engineering. Report No. BL/TN/B/22)
94. Stolzenbach, K. D. and D. F. Harleman. An analytical and
experimental investigation of surface discharges of heated
water. Cambridge, Massachusetts Institute of Technology, 1971.
212 p. (Ralph M. Parsons Laboratory for Water Resources and
Hydrodynamics. Report No. 135)
95. Stone and Webster Corp. Circulating water discharge system,
Shoreham nuclear power station unit 1. Hicksville, Long
Island, Long Island Lighting Comp. February 22, 1971.
397
-------
96. Tamai, N., R. L. Wiegel and G. F- Tornberg. Horizontal surface
discharge of warm water jets. Proceeding of the American
Society of Civil Engineers, Journal of the Hydraulic Division
95(2)253-276. 1969.
97. Taylor, J. F., H. L. Grimmett and E. wV Comings. Isothermal
free jets of air mixing with air. Chemical Engineering
Progress 47:175-180. 1951.
98. Tollmien, W. Strahlverbreiterung. Zeitschrift Fur Angewandte
Mathematik Und Mechanik 6:468-478. 1926.
99. Tomich, J. Heat and momentum transfer from compressible
turbulent jets of hot air impinging normally on a surface.
Doctoral dissertation. St. Louis, Washington University,
1967. 216 numb, leaves. (Microfilm)
100. Torrance, K. E. and J. A. Rockett. Numerical study of natural
convection in an enclosure with localized heating from below
creeping flow to the onset of laminar instability. Journal
of Fluid Mechanics 36(l):33-54. 1969.
101. Tulin, M. P. and J. Shwartz. Hydrodynamic aspects of waste
discharge. Los Angeles, AIAA 3rd Fluid and Plasma Dynamics
Conference, June, 1970. (AIAA Paper No. 70-755)
102. Turner, J. S. Jets and plumes of negative or reversing
buoyancy. Journal of Fluid Mechanics 26(4):779-792. 1966.
103. U. S. Navy Hydrographic Office. Tables for sea water density.
Washington, D. C. 1952. 265 p. (H. 0. Pub. No. 615)
104. Van Sant, J. H. Laminar free convection of heat generating
fluids in horizontal pipes. Livermore, Lawrence Radiation
Laboratory, 1969. 15 p. (Report No. UCRL-71569)
105. Van Sant, J. H. Lawrence Radiation Laboratory. Personal
communication. Corvallis, Oregon. 1969.
106. Wada, Akira. Numerical analysis of distribution flow and
thermal diffusion caused by outfall of cooling water.
Tokyo, Central Research Institute of Electric Power Industry,
1969. 23 p. (Technical Report:C:67004)
398
-------
107. Wada, Akira. Effects of winds on a two layered bay. Japan
Society of Civil Engineers 9:137-156. 1966.
108. Wada, Akira. Study on recirculation of cooling water of power
stations sited on a bay. Japan Society of Civil Engineers
10:143-170. 1967.
109. Welch, J. E., F. H. Harlow, J. P. Shannon and B. J. Daly.
The MAC method - a computing technique for solving viscous,
incompressible, transient fluid-flow problems involving free
surfaces. Los Alamos, Los Alamos Scientific Laboratory, 1965.
146 p. (Document No. LA-3425)
110. Welty, J. R., C. E. Wicks and R. E. Wilson. Fundamentals of
momentum, heat and mass transfer. New York, John Wiley and
Sons, 1969. 697 p.
111. Wilkes, J. 0. and S. W. Churchill. The finite-difference com-
putation of natural convection in a rectangular enclosure.
American Institute of Chemical Engineers Journal 12(1):161-166.
1966.
112. Zeller, R. W. Cooling water discharge to Lake Monona.
Doctoral dissertation. Madison, University of Wisconsin, 1967.
287 numb, leaves.
113. Zeller, R. W. and R. L. Rulifson. A survey of coastal power
plants. Portland, Oregon, 1969. 27 p. (Federal Water Quality
Administration Report)
399
-------
APPENDIX A
CONVECTIVE TRANSPORT DIFFERENCE APPROXIMATION
Differencing the convective terms is the most troublesome
aspect of solving transport equations numerically. The mathematical
principles for treating these quantities are available, but one must
exercise extreme caution when applying these principles or grossly
inaccurate solutions will result if not numerical instabilities.
When forming difference equations for convective transport, prime
consideration must be given to the directional nature of these terms.
A number of papers have been written and studies made con-
cerning numerical convection experiments. Perhaps one of the best
studies on higher order methods has been carried out by Crowley [21 J.
Crowley carried out numerical experiments using a number of difference
techniques in solving the "color equation" due to R. Lelevier,
Here r is a scalar quantity transported with the flow in a manner
such the total derivative is zero along an instantaneous streamline.
Crowley refers to Equation (A-l) as the advective form of the r trans
port equation. An alternative way to write Equation (A-l) is
at 3x ay sx 3y -
which Crowley refers to as the "conservative" form of the transport
equation. By continuity,
400
-------
. -
3X 9y ~ U>
However, in numerical approximation,
3U 3V .
97 + 3j <<: ]
but never zero. For this reason, the right hand side of Equation
(A-2) is sometimes included with the analysis in an attempt to reduce
accumulating numerical error.
As a point of criticism, in view of transport physics, it is
correct to write
and
3r + 3(ur) + 3(vr;
3t 3x 3y
instead of Equations (A-l) and (A-2), respectively.
In the paper cited, Crowley carried out various numerical
experiments with first, second and fourth order approximations for
Equations (A-l) and (A-2), and the one-dimensional counterpart of
these equations. For the one-dimensional tests, he concluded that a
second order process using the "conservative" Equation (A-2) was the
most accurate. In two dimensions he found that fourth order methods
were the most accurate but could not ascertain which equation gave
the best results. However, he does recommend that the conservative
equation be used.
Reference [66] reports results of numerical experiments con-
cerning the one-dimensional transport equation,
401
-------
Unlike Crowley's work, this work was concerned with the directional
nature of u and the proper method for differencing 9F/3X (forward,
backward or central) to minimize numerical error and achieve stable
computation.
For these experiments u was assumed positive and steady, with
the corresponding explicit difference equation written as:
n+1 n _ uAt n n , n n
r ' r ~
1 i " IT Lv'-»xM1i-rV "6x ui i+l'J (A-5)
where the superscript n refers to the nth time step. The parameter
£ varies from 0 to 1. The following difference techniques are
obtained from Equation (A-5) for the corresponding values of £ :
/\
£ = 0 backwards or upstream method
X = .25 so-called "quarter point" method
i = .5 central method
A
f$x = 1. 0 forward or downstream method
The results of these numerical experiments are compared with the
analytical results for various time steps and total elapsed time, and
found that the upstream difference (backward to the direction of flow)
gave the superior results.
Note, that in all but the upstream method, downstream quantities,
to some extent, are used to establish upstream results. In the case
of pure convection these formulations are physically incorrect.
402
-------
Lelevier (cf. [21]) was evidently the first to introduce the
upstream differencing technique. Crowley reports that a great deal
of numerical damping results with this method, applied to the
"advective" equation, over long integration periods. Nevertheless,
the upstream method (also called, unidirectional or one-sided deriva-
tive), has been used extensively in solving transport equations.
For instance, Van Sant [104] used the "advective" form to solve the
vorticity transport equation. Torrance and Rockett [100] solved the
"conservative" form of the vorticity equation in this fashion, and
Runchal and Wolfshtein [84] used upstream differencing to solve for
steady flow vorticity transport in "advective" form. Van Sant [105]
stated that he was unable to obtain a solution to the steady flow
vorticity equation using central differences.
One trouble with using any method except the upstream method
is that truncation and numerical round off can cause serious errors
and even destroy the solution through numerical instability. Higher
order methods (central difference, for instance) in spite of their
purported higher degree of accuracy may be inferior if the direction
nature of the flow is not considered. Runchal and Wolfshtein present
some clarification of this subject. We will pursue the matter here
by formulating convective difference schemes using one-sided and
central techniques.
Consider the incompressible steady flow transport equations, with
constant eddy coefficients for a conservative scalar quantity r in
(x,y) coordinates:
403
-------
sr L „ sr 1
2 2
) r . 9 r
(A-6)
dA dy IHrj II
where NR is the Reynolds number and
., _ momentum diffusivity
r r diffusivity
The finite difference grid system (Figure A-l) has constant and equal
spacing in the x and y directions.
I
j-l
k+l
— k
j+l
Figure A-l: Finite-Difference Grid System
404
-------
Suppose we now apply a general difference scheme to the convec-
tive terms of Equation (A-6) which, for the time being, disregards the
directional sense of the velocity components u and v. Then,
VP
F
where the constant subscript has been suppressed and point (j,k) is
replaced by p for convenience. In the above equation, X and X are
x y
factors corresponding to difference schemes in the x and y directions.
These quantities (X, and X ) take values of 0, 1/2 and 1 for backward,
x y
central, and forward differences, respectively. The quantity F is
equal to NDN_h. Solving for r yields
K 1 p
= UP 1-rj-r*xrj+i + VP
(A-8)
Case 1. Central difference scheme, |JX and X = 2
Equation (A-8) reduces to
405
-------
V
(A-9)
If F is very small, implying a very small Reynolds number
(creeping flow) or a very small grid spacing, h, Equation (A-9) will
usually converge. However, for large F,
rp a 'j-r + 'k-rVi (A-io)
Hence, small errors in the differences are magnified by a large coef
ficient, F, which will eventually destroy the computation through
instability. For this reason the central difference scheme is not
desirable for either transient or steady state application for inter
mediate and large values of F.
Case 2. Forward difference scheme, X and $ = 1-
x y
Equation (A-8) reduces to
• (A-ll)
Equation (A-ll) poses additional complications because of the presence
of the negative sign in the coefficient multiplying r . For positive
u and v and
406
-------
Equation (A-ll) is unmanageable. For large values of F, the differ-
ence scheme becomes
If either u or v is positive, this equation is physically incorrect
because we would be basing upstream computation on downstream informa
tion. On the other hand, if both u and v are negative, then
r r
Equation (A-ll) becomes
if +lJrp-f <|up|rj+1* |»p|rw)
which may be shown to be computationally stable for all values of F
and is a preferred scheme. This equation is also physically correct
since upstream quantities are used for downstream computation.
Case 3. Backward difference scheme, X and (( = 0.
x y
Equation (A-8) reduces to
If velocities u and v are both positive we have a computationally
stable scheme which is posed physically correct. However if either
velocity component is negative, we have the same type of situation
discussed in Case 2 where the scheme may be unstable and is not posed
407
-------
correctly with regard to transport physics.
Clearly, it is necessary to have a computationally stable and
correctly posed difference scheme for all values of F. It is impos-
sible to meet this criterion in a general flow system without cogniz-
ance of velocity directional sense and magnitude at each and every
boundary and computation point in the difference network. A sound
scheme may be obtained by choosing <( and <( according to the sign of
x y
the velocity components. We disregard / and X = 1/2 because of
A y
instability at large F.
Vp
Figure A-2 Values of <( and / for a Preferred Difference Scheme
A Jf
Figure A-2 summarizes the upstream difference method. Since the
velocity sign must be checked at each point in order to decide which
value of rfx and rf is to be used, an alternate method is formed which
is well adapted to computer application. Consider Equation (A-6),
specifically the term
up[(l-iJx)(rp-r..1)-(5x(rp-r.+1)].
408
-------
Let
i r UDF^ UD ^s p°s''tive
p ^x ? ' pi p' I 0,if u is negative '
i. p
,if u is positive
,if u is negative '
hence,
II
UP 3X
l»pl ' uP>(VrM»
which always gives the correct difference regardless of the sign of u .
The upstream difference technique applied to Equation (A-8) yields
1 l
+ 7TIU ~U)r. + 7T
2 ' p' p J^l 2
+ ? d vpi • Vvi+ r'
Solving for r yields
rP
!{(lu
4+ F (|u| + Jv|
p
409
-------
Upstream Differencing for Conservative Forms
Previous discussion of upstream differencing has dealt entirely
with convective differences in the "advective" form, u.ar/3x..
J J
However, this form is a result of mathematical manipulation of
the correct "conservative" form, a(u.r)/ax.. The conservative form
J J
is a direct result of a r balance in terms of infinitesimal quantities
and is the correct method for proper conservation of a transported
quantity in numerical analysis.
Consider the convective balance of r in r,z coordinates
(Figure A-3).
r(v-n)dA
'1
fr(v-n
\»
)dA
(r.z)
I
7 4- AZ
z + 2-
•/
r(v-n)dA
z -
Az
j: Ar I r(\r-n)dA Ar
r - o~~ « r + -s—
Figure A-3 Convective r Flux for an Infinitesimal
Axisymmetric Volume Element
410
-------
The steady flow convective balance equation for volume element
p is given by
I r(v-n)dA = J r(v-n)dA + J r(v-n)dA
A A A
MT rt] A2
(A-17)
J r(v-n)dA + I
+ / r(vn)dA + I r(v-n)dA = 0.
A3 A4
In Equation (A-17) and Figure A-3, A-j, Ap, etc., are element
S*.
areas corresponding to side 1, 2, etc., and n is a unit normal vector,
with outward, the positive sense and inward, negative. Like direc-
tional sense is used for the boundary velocity vector v.
Now refer to the grid system shown in Figure A-4. This grid has
constant Ar and A.Z, and velocities u and v are specified at the cell
face, whereas r is cell centered at point p (also see Figure A-2).
In setting up the difference scheme based on Equation (A-17) we want
to:
1) convect into the cell, p, the value of r at the upstream
neighbor, and
2) convect out of cell p, the value of r at p.
Hence, the value of r to be used in Equation (A-17) is given by
r =
r , for |v-n| = v-n
.value at upstream neighbor for |v-n| ^ v-n. (A-18)
411
-------
j-i
* j ^ 1
'.-.-1
r .
"2
k + 1
r * *
P
O___ . .__ _
k-l
r . . , ».
Jtl I
r i+i- • A z
I
Figure A-4 Axisymmetric Finite-Difference Cell,
p, with the Four Immediate Neighbor
Cells
412
-------
Unlike typical difference schemes, Equation (A-17) provides
flexibility of convecting into or out of any cell face. For the ele-
ment, Equation (A-17) may be written as
(v°n)
)AZF (v-n)
Ar r(v^n)
+i:r Ar r(v-n| = 0
z—
AZ
AZ
Dividing by volume (2-irrArAz) yields
r(v-n)
rAr
rAr
r(v
-i
/\
•n)
r
Z- —IF
=— 4- —
(v
A,
•n)
**!
AZ
AZ
= 0-
(A-19)
In accordance with Equation (A-18) and Figure A-4, Equation (A-19) may
be expressed as
r Ar
P
2 rj
-------
4
- r
k+l
AZ ;
<=?>
1_ a(rur) +
r ar
(A-20)
The above form is used throughout in this thesis for convective dif-
ferences. Vorticity transport has a slightly different form in the
convective terms,
3r
which amounts to deletion of rj_-|/2» rj+i/2 and rn 1n the first two
terms of Equation (A-20).
414
-------
APPENDIX B
FINITE-DIFFERENCES FOR IRREGULAR NODE SPACING
A.I General
Consider the irregular grid shown in Figure B-l below.
Bh
,1,
*
AX . . fc
•*- nA-,_l »
I
i
- AX- f
* 1 *]
,1
. AX n »
^ 1+1 m
\
Figure B-l. Irregular Spaced Grid
i
The width of node i is designated AXi and the nodal points are all
cell centered. Finite-difference approximations for the first and
second derivatives at node i are developed as follows.
Let,
h = \ (AX.^ + AX.)
and
6h = \ (AX. +
Then a Taylor series expansion of a function f about point i is
given by the equations:
415
-------
fi*r fi
i
ehf
.3.3 „, 4h4
f-+tf f, +6h
(B-l)
and,
TW
(B-2)
Now, divide Equation (B-l) by 6 and add the result to Equation (B-2)
to obtain the difference approximation for the second derivative of f:
*^ + _^
2f.
1 + (3-1) Oh + Oh . (B-3)
h'(3+l)
Fbr 3=1, Equation (B-3) reduces to the familiar central difference
form:
fi+l+fi-T2fi 2
1+1 I ' 2. + oh^
(B-4)
A finite-difference approximation for the first derivative of f at
point i may be found by subtracting Equation (B-2) from (B-l), up to
and including terms involving f". Hence,
fi+rfi-i
9f
3X
Again with 6 = 1 the familiar central difference form results:
(B-5)
9l
-1
'
~2T
(B-6)
416
-------
Equation (B-5) is a first order approximation of |y- . A second order
method may be developed by reducing the coefficients of f." to 1 in
Equations (B-l) and (B-2). Equation (B-2) is then subtracted from
(B-l) to obtain:
1 _. 9_ _ _ «_1_ 9
(B-7)
Equation (B-7) collapses to (B-6) for 6 = 1.
A.2 Computer Application
For computer application, irregular spaced first and second
derivatives difference forms are needed for both points (j,k) and (p,q)
in the vertical direction (Figure B-2).
t
A;
i
AZ
i
AZ
I
1
i
k
k-1
X
X - — —
x
- J.K+I
P,q+l
P.q
_ i k-1
P,q-l
Figure B-2. Grid Layout for Vertical Differences
417
-------
The following forms are used for differencing a general quantity, F(the
subscripts p and j have been suppressed).
Point (j.k)
First derivative of F:
3F
AZk Fkfl AZk+1Fk-l
Second derivative of F:
Fk+rFk
3Z'
AZ, -/
!w4Zk
(B-8)
Fk"Fk-l
(B-9)
Point (p,q)
First derivative of F:
q+l
If.
3Z
Second derivative of F:
q-l
- D
[k-l AZk+l
AZ|<+AZk_1
(B-10)
q 7 AZk^AZk+l+AZk) 7
(B-ll)
418
-------
APPENDIX C
COORDINATE TRANSFORMATION
The required partial differential equations are given in
Chapter 5 by Equations (5.10) through (5.14) and are restated here
for reference.
Stream Function:
4 - 1 U + 4 - - Rn. (5.10)
9R R 9R 3Z
Vorticity:
w;
Buoyancy Parameter:
= ._ . + ^ + _! J. , (5.12)
along with
U = - 113L , (5-13)
and
1 aw
V"lf W ' (5-14)
419
-------
These same expressions are given in transformed coordinates by
Equations (5.16) through (5.20), respectively. The transformation
to £ coordinates by setting
R = sinh £ (C-l)
has the desirable properties mentioned in Section 5.5. Details of
the transformation are given in the following discussion.
Consider a quantity F and first and second derivatives of this
quantity in R coordinates. The general transformation of these deriva-
tives to 5 coordinates is derived as follows:
dF _ dF d? _ -
Then
or
d2F d£ _ d£ d£ _ d__ r df_ d£-, d£
~ ~ dR " d? ' dR " dc L de ' dRJ ' dR
d? , d
dR ; d?
Now,
d_ /d^x dH^ dH d_R
dc ^dR; " dc ~ dR * d?
= dj*
Hence,
420
-------
From Equation (C-l),
41 = 1 (C-4)
dR cosh £
and
Then,
f (C-6)
and
?_ ?_
e If.). (c-7)
Substitution of Equations (C-l),(C-6) and (C-7) into Equations (5.10)
through (5.14) yields the transformed set (5.16) through (5.20).
One discomforting feature of non-linear transformations
is that small errors are introduced in calculating areas and dis-
tances in the transformed coordinates. For instance the distance AR
in real coordinates is given by
ARA = sinh (5 + AC) - sinh (5) .
In the difference computation,
ARC = cosh (5 + T^- ) A5 .
Taking the ratio of these two expression yields, after manipulation of
identities:
A . Actual spacing = 2_ sinh { ^i } (C-8)
Computed spacing AC v 7- ; .
421
-------
As Figure C-l indicates, AC should be kept as small as possible.
1 .05
1 .04 -
1 .03 -
o
1 .02 -
1 .01 -
I I I I I I I I I
.2 .4 .6 .8 1.0
Figure C-l. Ratio of Actual to Computed
Node Spacing
422
-------
APPENDIX D
SOME RELATIONSHIPS BETWEEN TIME
DEPENDENT AND STEADY STATE NUMERICAL
METHODS IN HEAT TRANSFER AND FLUID FLOW
The general transport equation for a conservative quantity,
T, is written in tensor form as:
9U T
where the summation convention does not extend over the underscored
indices and source and sink terms are negligible. The symbols in
the above equations are:
t = time
x. = jth spatial coordinate
J
U. = jth velocity component
J
a. = diffusion coefficient along the jth coordinate
J
For simplicity in this discussion, we will ignore the convective
terms, consider a as a constant, and write Equation (D-l) as
\ J J
For steady flow,
ax. ax.
J J
(D-3)
The usual technique for solving the above equation is either by
Gauss-Seidel or Gauss iteration, where the former is much faster
423
-------
than the latter and, consequently, the most popular technique. In
both cases successive over-relaxation (SOR, extrapolated Liebmann
method) is employed.
It is the task here to illustrate that certain methods for solving
Equations (D-2) and (D-3) above are identical up to the Liebmann
extrapolation factor, L, in the steady state technique ahd the time
scale factor, a, in certain time dependent methods.
D.I Correspondence Between the Classical Explicit and Gauss Methods
The classical explicit and most common method for solving Equa-
tion (D-2) is given in difference form for an evenly spaced grid as
follows:
T 3k1 -T Jk ' « . (D-4)
where a = .
AX
The superscript n denotes the nth time step. One may rearrange
Equation (D-4) to give
• «
-------
An algorithm for Gauss iteration of Equation (D-3) may be written as
Tjk' ' l TJ*k + <'-L> Tjk, (D-7)
where s denotes the sth iteration and L is again the Liebmann
extrapolation (or SOR) factor. We note that Equations (D-6) and
(D-7) are identical insofar as
L = 4 a. (D_8)
In Equation (D-7), L is greater than 1, but must be less than 2
to prevent solution divergence; that is, for over-relaxation
1 <_ L 1 2.
Hence, as a maximum value
4 aAt 0 aAt
— ~- »
AX AX
which is exactly the explicit method stability criterion.
D.2 Correspondence Between ADEP Transient Methods and the Gauss-
Seidel Technique
Alternating direction explicit procedures (ADEP) are relative
newcomers to the field of applied numerical analysis. The prototype
ADEP was conceived by the Russian mathematician, Saul 'ev, in 1957.
Since then other methods have been presented such as those proposed
by Larkin [53] and Barakat [9]. These methods, which have been
demonstrated to have good accuracy and incredible stability, have
basic algorithms identical to the Gauss-Seidel method with SOR.
425
-------
A. Saul 'ev Method
The Saul 'ev method consists of alternate directional sweep-
ing of the grid system. A forward sweep is written as
n+1 Tn /Tn+l . Tn+l Tn Tn 2 Tn - 2 Tn+1 }.
Tjk - Tjk = ° {Tj-U + Tjk-i + Tj+l k + Tjk+l * Tjk 'jk >•
(D-9)
Note that there is equal weighting on the n and n+1 time levels.
Rearranging Equation (D-9) into the context of Gauss-Seidel iteration
with SOR yields
*
(1 + 2 .) 1$ . 4 . T™ + (1-2 .) T^
4T
Tn+l _ 4 a 1 T*n+l |l-2 gj ,n
Hence, Tjk - ^p^j Tjk + \^-^J T..k
(D-10)
Comparing Equation (D-10) to the Gauss-Seidel algorithm,
(1-0 Tj^ (D-ll)
again shows equivalence insofar as
1= -4a
or
L =
426
-------
Now Lim —r -> 2;
hence, the upper limit of the Liebmann extrapolation constant is
satisfied from the standpoint of stability irregardless of the
size of the time step, At. Fora = .5,
Tn+l T*n+1
'jk " jk
which is identical to the Gauss-Si edel method without SOR.
For the Saul 'ev method, the next time level computation involves a
similar backward sweep.
B. Larkin's ADEP
Larkin's ADEP is actually one of several methods discussed by
Larkin in the cited reference. The method here is very similar in
the mechanics to the prototype Saul 'ev ADEP, except that the for-
ward and backward sweeps are averaged to form a time level.
Larkin's methods yield the same relationship between L and a given
in Equation (D-12).
D.3 Further Comparisons Between Larkin's ADEP And The Gauss-Seidel
Iterative Technique
Consider the two-dimensional form of Equation (D-1),
/ ? 9 \
3T . 3UT , SVT a/3T + lfl\ . (D-13)
-- a — ~ —
Based on upstream differencing of the convective terms, the forward
sweep ADEP finite-difference equation would be,
427
-------
Tn+l
Tjk
- U.
Tn+l
jk
jk
Tn+l
Jk
Tn+l
Tjk
- Vjk
jk+1 >
jk
{Tn+l Tn Tn Tn+l Tn+ Tn Tn Tn
'j-1k + 'j+1k " 'jk " 'jk ^ 'jk-1 + 'jk+1 " 'jk " 'j
^7~~^ 1?
(D-14)
Figure D-l illustrates a finite-difference cell and the relative
locations of the quantities T, U, and V.
-D
VJk
x O
Tjk
ujk
Figure D-l. Finite-Difference Cell
428
-------
Note that in Equation (D-14) if velocity is negative T.. is evalu-
' J
uated at n, whereas for positive velocity T.. is evaluated at n+1.
' J
The backward sweep would use the opposite sense. Also, this
convention is not a necessity and other time level evaluation schemes
may be used as long as they are computationally explicit.
n+1
Solving for T'k yields
At
At
'V
4. u
v 4. 4.
' Vjk-l «?"
aAtn+l
(D-15)
429
-------
For a short hand notation let:
D.. = a<-V+-
J ' A AY2
Then,
At T
n+1
jk-1
'jk+1
(D-16)
430
-------
The Gauss-Seidel scheme yields
+ -^'M-lk
' (D-17)
Substituting Equation (D-17) into (D-16) yields,
(Cjk + Djk) ^} ^ - (Cj
Yields
(C.k + 2D )At
L =
431
jk jk jk j
(D-18)
or in terms of iterations s,
s+, ,
Comparing to
= L
(D-20)
-------
Thus,
Lim CJk + 2Djk _ fo + 2Djk
A t-**0 I Z - C. i + D. i
uv • 4. r -u n ilc TK
TT" ~ ".Six ~ u-ib J1^ J *
At j f> j i>
The condition
2V
leads to some restrictions on the over-relaxation factor t.
For the case where convection effects are very small, characteristic of
a creeping flow,
2D.,,
-, - & - < 6 (D-22)
k+DJk "
The question is what values of 6 are possible in Equation (D-22).
For At -H», 6 = 2 and for At •*•• 0,6 = 0.
" 0 l L i 2.
For very high Reynolds number flow, viscous effects become relatively
small and
for At -*», 6 -»• 1 and for
At = 0, 6 = 0; hence,
0 < L < 1
432
-------
This preceding analysis indicates that it is impossible to accelerate
the Gauss-Seidel technique for flows where viscous effects are
negligible. In the general case there will be regions in the flow
field where the local Reynolds number will be such that D.. ^ 0.
If the condition 0 < L < 1 is violated, then an instability will propa-
gate from this local point.
433
-------
APPENDIX E
LISTING C-F SYMJET COMPUTER CODE - UNlvAc HO* VERSION
FOLLOWING ARE THF PROGRAM PARAMETERS AND DIMENSIONS.
CO
4s.
c*
c*
CGMLST* FCOPY
PARAMETER
COMMON
1
5
COMMON
COMMON
COMMON
i
2
REAL
INTEGER
LOGICAL
END
GAM(LJrLK),SC(LJf 15 > • R 'UGRAD(LjfLK) 'PlCH(LjtLK> »RB(LK) »
R051LK) »TEMdO) » XR (LJ) »ZC (LK ) »ISOLM(5»30) »N3DPT(5) »
OATE(2) »TjM(2) »TLABEL
DPSAX » DOMAX » DDMAX » DZc rEZ • S IGTS » VMB » DMB » S I GTB » ZRP »
G » OMB • SALR r SAuJ » DS/iLT»RMIN»tXR»EXS» EXT » BETA »AK1»
SIGTJ» TINT » START* RATIO rGAWEND»ERATTO»VMBl»PLX
MJ»HJJ»NH,NL»NOUT»NPT»IN»OHT»IPMAX»KASE»KT»INMODE»
ITMAX»ITf!0»NB»NTTY»NK.»NKK»NCR»NPT»MMAD»NX»DZT5»TO»TR»
I TEMP » NED » NEDDY » ^JOTE^•P t JpORT » KPORT r I TNOO »
ISOLN
CUT
CONTRL
ApDIM* FCOPY
PARAMETER
END
C*
C
-------
C UN1VAC 1108-VERSION
C* THIS VERSION oF THE SYMJET RRoGnAM HAS BEEN CHECKED-OUT FOR THE
c* FOLLOWING OPTIONS:
C*
C* 1) AOUTr OPTION (OUTPUT ARRAYS)
c* PSIP»'PSIV» »LELT».OMEG'
C* UFAC»'VFAC»»RICH»
C* 2) PLOT* OPTION(CREATES PLOT ARRAYS FOR SUBSEOUENT
C* CONTOURING AND 3-D PLOTS)
C* ALL OPTIONS WORK
C* 3) TERP» OPTION (CALCULATrS UNORDERED CONTOUR VALUES)
C* ALL OPTIONS WORKED CORRECTLY ON PREVIOUS VERSION'
C* BUT HAVE NOT BEEN CHECKED FOR THIS VERSION.
C* **> COMT» OPTION IPROGRAM CONTROL)
C* BUOY*»TRAN»»TEMP».MONT»»TAPE»»SAVEr»INVS»»TUR8»r
C* CENO»'CENI»
C*
C*
C* INMODE = *
C*
C**** NOTE ****
C*
C* THIS CODF VtPSIO!; HAS BEEi^ DEBUGGED FOR OPTION INMODE = * ONLY
C* -HICH TREATS SHALLOW WATER PLUMES WITH PQ«FR LAW INFLOW VELOCITY
C* PROFILE. <-,QME CHANGES HAVE BEEN MADE ON THL LATERAL DIFFUSIVITY
C* MODEL SINCE RUNNING OF CASE 66. HENCF RESULTS WILL NOT CHECK
C* PRECISELY.
C*
C* SYSTEMS ROUTINES USED bY CODE.
C*
C* TOY (F)
C* OOY (F)
C* ETIME (F>
-------
CO
c*
L*
C*
C*
C*
C*
C*
c*
c*
c*
c*
c*
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
WHEf-'
THEr
THE POLL
THE OPT
********
CELT
RSEA»
Dl^LNSlONLEbS VELOCITY AT POINT (X»Z)»
RADIAL EDDY MULTIPLICATION FACTORS-
WHEN NEDOY=0 »FR(K)=1.178
rtHEN NEDDYri .FR(K)=R.5*V^AX»WITH R
WHEN NEDt)Y=2 r FR ( K ) =R ,5*VMAX » WITH R
EXECUTION
VERTICAL EDDY MULTIPLICATION FACTORS
RADIAL DISTANCE TO OnTER CELL SIDE J'K
RADIAL DISTANCE TO CENTER CELL 0»K
DEfjsiTY STRATIFICATION OF AMRIENT(SIGMA UNITS)
TRANSFORMED RADIAL DIMENSION TO riGDAL POTNT
DENSITY DIFFERENCE BETWEEN PLUME AT PORT AND REF.AMBIENT
CENTEPLINF VALUE OF CELTA AT Z=Z3
X-COMPoNENT
Z-COMPONENT
,5
.5
SPECIFIED
CALC DURING
OF PROGRAM
-------
CO
c
C
C
C
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
DSIGT
DZ
DX
DX2
DZ
DZ2
DIAC
EZ
FO
G
6MB
R.5
RER
REZ
RRP
RO
SALJ
SALR
DSALT
SIGTJ
SIGTR
SIGTB
TLABEL
TLIST(
VERTICAL
VERTICAL
WIDTH OF
DX*OX
VID1H OF
DZ*DZ
DIAMETER
VERTICAL
DENSITY CHANGE OVFR DZ IF CONSTANT(SIGMA UNITS
NODE THICKNESS IF CcNSTANT»DELTAZ/DTA
NODEr X-DIRECTICN
NODE* Z-DlRrCTION
OF OUTFALL PORT
EDDY TRANSPORT COEFFI IENT
DENSIMETFIC FROUDE NyMBE" AT OUTFALL PORT
LINEAR STRATIFICATION PARAMETER SIMILARITY SOLUTION
CENTERLINE GAMMA (GA^M/GAMO) AT 2 = ZB
RADIAL DISTANCE TO HALF VELOCITY (MIXING LENGTH APpRQX)
RADIAL REFERENCE TL'RPULEMT REYNOLDS NUMBER
VERTICAL PEFEKENCE TijRBULENT REYNOLDS NUMBER
RADIAL REFERENCE pRApDTL MUMpEP
RADIUS OF OUTFALL POPT
SALINITY OF PLUME AT OUTFALL PORT
SALINITY OF REFERENCE AMPlENT (ASSUMED CONTSTANT WITH Z
SALR-SALJ
DENSITY Op PLUME AT OUTFALL(SIGMA UNITS)
DENSITY OF REFERENCE AMBIENT(SIGMA UNITS)
DEUS. OF REF. AMBIENT AT Z - Zp (SIGMA UNITS)
(J)ALPHANUMERIC CASE HEADfR ARRAY
J) ALPHANUMERIC DATA INPUT FOP CERTAIN CONTROLS AS FOLLOWS:
SET UP ARRAY WRITER WITH OLIST(I)
OPTIONS.
INTEFPOLATE ARRAYS GIVEN BY ELIST(I)
FINDS ISOLINES OF VALUE ISOLN(K»N) FOR
ARRAY MATCHING ELlST
-------
CO
oo
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
TLIST = CONTr
SET ijP PPOGRAM LOGICAL CONTROL FROM
DIRECT(I) DATA.
VO
VMB
ZR
ZRP
IPMAX
INMODE
ITMAX
ITAPE
ITtMP
KASE
KT
NEDDY
INMODE=2
1NMODE=3
INMODE=i*
NCR
NJ
CENTERLI^E VELOCITY THERMAL pLUME AT SYSTEM IN-BOUNDARY
CEN'TERLINE VELOCITY (VM/VO) AT Z=ZB
ELEVATION JO GRID ROTTOM PHYSICAL BOUNDARY »Z/DI A
VERTICAL REFERENCE Pr-ANQTL NUMBER
MAXIMUM NUMBER OF ITERATIONS FOR psi ITERATION
INFLOW BOUNDARY INPUT DATA MnDE +
INMODE=0 INPUT FROM DATA
GAUSSlAN-FlOW ESTABLISHMENT ZONE
GAUSSIAN-ESTABLISHMENT
INPUT CALCULATED FROM SIMILARITY SOLUTION
INFLOW DATA AT PORT ORlFICF
TOTAL NUM8ER OF ITERATIONS
SIGNAL FOR CONTINUED ITERATION OF OLD CASE*
ITAPE=0 » NEW CASE
ITAPE=1 » CONTINUE ITERATIONS OF OLD CASE
SIGNAL FOR DENSITY OR TEMPERATURE INPUT
ITEMP r 0» SIGMA-T INPUT
ITEMP = 1» TEMPERATURE IN^UT
CASE NUMBER
SIGNAL FOR TRANSFORM OF LINEAR RADIAL COORDINATES+
KT = 0» LINEAR RADIAL COORDINATES
KT r 1» TRANSFORMED ACCORDING TO R=SANH(X)
SIGNAL FOR TYPE OF RADIAL EDDY TRANSPORT COEFF CALCULAT
NEDDY=Q • ER = CONSTANT
NEDDY=1 ' ER = FO*R.5*VMAX'PRIOP SPECIFICATION OF R.
NEDDY=2 » ER = EO*R.5*VMAX»RUNNING CALCULATION OF R.
NUNBER OF ITERATIONS PERFORMED AT ER=Eo*i.i78 BEFORE
RUNNING MIXING LENGTH CALCULATIONS
USED WHEN NEDDY = 2
RAnlAL CONVERGEN E RANGE
NUMBER OF MODES» RADIAL DIRECTION
-------
-p.
CO
c
C
C
C
C
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
NJJ
NK
NKK
NL
NH
NOX(J)
NOUT
NMAD
NJ-1
NUf-'bEP OF NODES' VERTICAL DIRECTION
NK-i
NJ+1
NK-H
NUVBEPING
NUMBER OF
FOR OUTPUT HEADING* SET IN MAIN PROGRAM
ITERATIONS FOR LINE PRINTER OUTPUT
SIGNAL TO CALL RICHARDSON VODIFIER ROUTINE
NMAD = Of DONOT CALL
NMAD = 1-b S£E SUBROUTINE RCHMOD
NTTY NUMBER OF ITERATIONS FOR CALCULATION MONITORING OUTPUT
uPT =1 » CALL PLABAK
NPI NUMbER OF ITERATIONS ON STREAM FUNCTION IN MAIN COMP
WRITE(J) SIGNAL TO CALL OUTPU- OF SPECIFIC DATA
NX MAXIMUM VALUE OF INDFX j FOR PLOTTING
OLIST(J) CHARACTER DATA INPUT SIGNAL OUTPUT ARRAYS DESIRED
OPLIST(J) MATCHES IM«EDDFD DATA DLIST(J)
TO SET VALUE OF NRlTf(J)
POTENTIAL FLOW STREAM FUNCT
VISCOUS FLOW STREM FUNCT,
DENSITY DISPARITY
VORTIClTY
VERTICAL VELOCITY
RADIAL VELOCITY
GAMMA CONSTITUENT
TEMPERATURES
NORMALIZED DENS. DISP.
NORMALIZED VERT. VELOCITY
NORMALIZED TEMPERATURE
RADIAL EODY FACTORS
VERTICAL EDDY FACTORS
RICHARDSON NUMBERS
ISENT
DIRECT(I) LOGICAL CHARACTER DATA FOR PROGRAM CONTROL
REAL) IN UNDER TLIST OPTION CONT».
OLIST
OLIST
OLIST
OLIST
OLIST
OLIST
OLIST
OLIST
OLIST
OLIST
OLIST
OLIST
OLIST
OLIST
OLIST
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
1)
2)
3)
4)
5)
6)
7)
R)
9)
10
11
12
13
14
15
-
—
—
~
—
~
—
—
—
) =
) =
) =
) =
) =
) =
PSIP
PSIV
DELT
OMEG
VELV
VELR
GAVIA
TEMP
NOEL
NVEL
NTEM
PFAC
VFAC
RICH
f
t
t
f
f
»
t
>
t
i
r
»
r
r
BLANK
slRITF.
viRITF
'/RITE
-RITE
VRITE
WRITE
^RITE
WRITE
'.-,'RITF
WRITF
•JRIT^
'.--'RITE
AiRITE
WRITE
«T PR
-------
-PS-
CD
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
DIRECT (D = 6UOY» :
BLANK
DIRECT(2> = UNCP»:
l,TPECT(3) = GRAD' :
bLANK
uIREcTu) = TRAN» :
bLANK
DlPECT(5) = TEMP»:
BLANK
= MOMT»:
DIRECT(7> =
BLANK
NPCH»:
BLANK
Bt'OYANCY COUPLED FLOW.
CONTRLU) = .TRUE.
MOMENTUM FLOW ONLY* NO BUOYANCY
CONTRLd) = .FALSE.
NO BUOYANT INTERACTION1* BUT BOTH
TEMPERATURE AND SALINITY OR
CONCENTRATION APE COMPUTED.
CONTKH2) = .TRUE.
AMBIENT STRATIFICATION
CONTRL<3> = .TRUE.
IF HOMOGENEOUS AMBIENT
CONTRL<3) = .FALSE.
TRANSFORM RADIAL COORDINATE
ACCORDING TO R = SINH(XI>
CONTRLU) = .TRUE.
FOR LINEAR PADIAL
CONTRL(U) = .FALSE-
FLUID STATE INPUT DATA TO BE
GIVEN IN TLRMS OF TEMPERATURE
(DEG. C OR F) AND SALINITY (PPT)
IF TEMP* OPTION USED WITH INPUT
IN DEGRFES C» THEN CENI» OPTION
MUST ALSO BE USED.
CONTRL<5) = .TRUE.
FLUID STATE GIVEN IN TERMS OF
SIGMA-T AND SALINITY.
CONTRL<5) = .FALSE*
MONITOR INFORMATION TO BE PRINTED
AT EACH ITERATION.
CONTRL<6) = .TRUE.
DO NOT MONITOR.
PUNCH RESTART DATA TO CARDS
CONTRL<7) =• .TRUE.
DO NOT PUNCH
-------
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
DIPECT(8> = TAPF»:
BLANK
DlRECT(g) = SAVF»:
BLANK
GlRECT(10)=
BLANK
DIPECT(11)= TURB»:
DIRECT(12)= CENT' :
BLANK
DIRECT(13)=
BLANK
INITIALIZE ARRAYS FROM RESTART
DATA FILL OR TAPE. MUST EQUIP OR
ASSIGN LUN 7.
CONTRL(P) = .TRUE.
00 NOT READ RESTART DATA FILE
SAVE ARRAYS FOR RESTART FILE' OR
PLOT FILE. MUST EQUIP OR ASSIGN
CONTRL19) = .TRUE.
00 NOT SAVE
PERFORM INVISCID FLOW COMPUTATIo
FOR CASF INITIALIZATION
CONTRL<10)= .TRUE.
NO INVISCID COMPUTATION
COMPUTE AMBIENT TURBULENCE AND/O
CONSIDER OERIVATIVES OF THE EDDY
TRANSPORT TFRMS.
CONTRL<11)= -TRUE.
TEMPERATURE INPUT DATA SPECIFIED
IM DEGREES CENTIGRADE.
CHNTRL<12)= .TRUE.
TEMPERATURE INPUT DATA SPECIFIED
IN PEGREES FAHRENHEIT.
TEMPERATURE OUTp-jT RESULTS
SPECIFIFO IM DLiritES CENTIGRADE
CONTRL(13)= .TRUE.
TEMPERATURE OUTPUT RESULTS
SPECIFIED IN DEGREES FAHRENHEIT.
IN
OUT
UNUSEC CONTRL OPTIONS :
CARD READER LOGICAL UNIT
LINE PRINTED LOGICAL UNIT
r »CL15» »CL16» »CL17» ,
-------
DIMENSION OLIbT(l5> .DLIST(IR) .pLlSK*) .ELISTA) »RLIST<5)
DIMENSION OPTIONU) »DATA(l5) .CLISTU7) »DTRECT(17)
INCLUDE COMLST.LlST
DATA < DL1ST(I),1 = 1.15)/bHPSIr..5nPSIV..bHDELT..5HOMEG»»5HVELV,,
1 5HVE.LR . » 5HGAMA » » 5HTE"'P» . SHND^L • r 5HMVEL • ,
1 5HNTEM..^HRFAC..^HVFAC..SHPICH.»5H /
DATA.1=1»5)/5HPbIV..SHnELTr.^HGAMA,.SHTE^P.»5HOMEG»/
DATA
IF(h'PI.EU.U) NPI = 1
IF(IPN'AX.EO.C) IPMAX = 100 '
IFlNCR.rQ.o) NCR = ^J-l
IF(NX.EO.O) NX = NJ-1
IF(KASE.EO.O) STOP
WRITE (OUT. 1001+) TLABEL'DATE.TIM
READ
-------
-p*
.£»
OJ
7
6
9
10
11
12
13
14
15
18
GO TO 15
DO 81 =
OLIbT(l)
60 TO b
DO 10 I
RLISTd)
KPLQT
60 TO 6
DO 12 I
PLlST(l)
GO TO 6
DO 14 I
CLISTd)
GO TO 6
NJJ
KKK
NH
NL
JPOHT
KPOh'T
NB
ITERS
ITNOO
ITUO
DO 18 "-'
NOX(M)
DO 20 J
DO 20 K
PSl(J'K)
OELTt J'K
OVEG
-------
20
30
100
lOb
107
RlCHlJri') = 0.
UGR/\D(J,K) = 0.
L,GKAD( j»K) = 0.
CONTINUE
DO 30 n
K
MOMK + 1)
MOi\ilK+2)
MONIK+3)
1»3
(N-1>*10
2
MK/5
MCN(K+b) = ?
MON-(K+7)
MON(K+8)
MON(K+9)
t.'K
I.JJ/2
liK
fJJ-1
CONTINUE
DO 100 T = 1§15
DO 100 J = 1 »15
IF(oLIST(T) .EG.OLIST(J) ) NRlTE(j) = I
CONTINUE
DO 105 I = 1,5
DO 105 J =
IF(ELIST(T)
!F(tlLIST(I)
EQ
EO
RLIST(J) )
PLIST(J) )
NiDPT(j)
ISOPT(j)
= I
= I
CONT«L < I ) =
CONTINUE
DO 107 I = 1,17
DO 107 J = 1 »17
IF (QlRECT(I) .EQ.CLIST(J) )
CONTINUE
IF(KPLOT.fr&.l.ANC'..NCT.CoNTRL(9) ) GO TO 160
KT =0
iitnP = o
IF(CONTPL(4) ) KT = 1
IF(CONTRL(b) ) 1TEMP = 1
-------
cn
110
120
150
160
1000
1001
1002
1003
CALL INPUT
CALL READY
CALL PL4BAK
CALL ETIMEF (START)
WR1TE
FORMAT (////
1 3bH St4*
2/26H PROGRAM
SET-UP TIME = F5.2' 5H SFC /
tSsSSs^fsS'*;*
1001 FOHMAT(///12A6»5XHA6)
1005 FORHAT(//Q YOU CAN NOT SAVE A PLOT plLE WITHOUT ASSIGNING
*VE FILE TO LUN 8C/W EITHER DELETE PLOT FILE CALL OR EQUIP
*8ffl/Q PUtJ ABORTED - - TRY AGAINQ)
ENU
A SA
LUN
SUBROUTINE
DIMENSION
INCLUDE
INPUT
DATAdO)
COriLST»LlST
-------
10
20
CTl
25
30
IF( .NOT.CCUTRUS) ) GO ro 10
READ < 7) ITNO»OMEG»D^LT»UX »UZ»PS T *
ITNOO = ITNO
REWIND 7
N6 = 0
REIAQ(IN»1000) DATA»JI»KI»N1
(100»20»30»*40'60»60»60»60»70»70»7P»RO)
= DATA(l)
= DATA (2)
= DATA (3)
NI
GO TO
DIA
DX =
DZC =
ZO =
VO =
JPORT =
KPORT =
IF(DATA(8)
IF(DATA(9).F:(5.0.)
IF(DATA(10).EQ.O.)
DATA (5)
OATA(6)+.Q1
DATA(7)+.Q1
= 1.
DMB = i»
RO
IF
DO 25
DZ(K)
60 TO
TBOT
TO
TR
K =
10
,5*DIA
) GO TO
1»NH
DATA<2)
DATA(3)
CONTINUE
DATA<1)
DA1A(2)
DATA<3)
10
QATA(3)
DATA(D
DATA(3>
12)) GO TO 32
3.0) GO TO 32
5./9.*(DATAd)-32.)
5«/9.*
-------
-pi
-vl
60
SIGTJ = PATA(l)
SIOTR = DATA(2)
SIGTB = PATA(3)
OSIGT =: DAT A (1)
SALP = DATA(5)
SALJ = HATA(6)
LXb = DATA<7)
EXK = DATA(S<
IFlEXS.iy.o.) tXS = .999
IFltXR.EQ.O.) tlXR = .999
D5ALT = SALK-SALJ
bELTJ = SlGTR-SlC-TJ
R5tA ERATIO =- .01
iFtf.Z.Er .0. ) EZ = .1
IF(RRP.!=:Q.U. ) RRP = I./.714
IF(ZRP.rQ.O.) ZRP = l./.71<*
CONTINUE
GO TO lo
DO 65 -I = JIfKl
KAT = N-Jl+1
-------
IF(NI*EQ*5>
uz m»i
DtLT(M»l
RSLA(N)
00
65
70
75
60
85
1000
100
!F(Nl.Eu
CONTINUE
iFChl.EO
GC TO 10
JI
KI
DO 75 r;
KAT
MON(N)
CONTINUE
GO TO 10
NN
DO 85 M
NA
ISOLN(KI
CONTINUE
GO TO 10
FORMAT ( 1
RETURN
ENu
.8)
.5)
—
—
—
TI
—
—
—
—
DZ(N) = D
MB = KI
(fJ!-9)*l041
JI+9
JI»KI
N-JI+1
DATA(KAT)+.0001
JI-1
1»10
NN+N
»MA) = DATA(N)
OF5
.0*315)
DATA(KAT)
DATA(KAT)
DATA(KAT)
DATA(KAT)*?.
SUtJROUTTNE RFADY
INCLUDE COMLST»LIST
NOTE^P = 1
1F(KPORT.E(J.O) KPORTz 1
IFdNMOOE.EQ.M-.AND.CONTRLCf) )
RER = 39,
REZ = RO*VO/EZ
2<1) = ZB
ZC(1) = 2(1)-.25*OZU)
= .8pl37^59/(JPORT-i)
-------
ZPORT r o.
DO 5 K = 2rNH
IF(K.LE.KPOPT) ZPORT = ZPQRT-«-.5*DZ (v )
Z = l.b+ZC(K)-ZPOKT
5 CONTINUE
DZTOT = ?(NK)-ZB
IF( iNMOoE.LQ.it) DZTOT = Z(NK)-ZrORT
DZT5 = DZTOT*. 5
IF(DSlGT.Eti.O) GO TO 15
RSEAd) = SIGTR
DO 10 K = 2rNH
RSEA(K) = R5EA(K-D+DSIbT*DZ(Kl/(Z(NK)-ZR)*.5
10 CONTINUE
15 DO 20 K = 1»NH
IF(.NOT.CONTRL<12).ANO.lTEMP.r E.O) PSEA(K) = 5./9.*(RSEA (K)-32« )
IF(.NOT.CONTRL(3) ) RSEA(K) = Slf;TR
IF(CONTRL(3» RSEA(K) = blGMAT < f,ALR,RSLA (K ) , JTEMP)
RStA'K) = RSEA<«)+IOOO.
20 CONTINUE
DXi: = DX*DX
FO = VO*VO/(DELTJ/(SIGTJ+100C.)*2.*PO*32.2)
IF(PELTJ.EQ.O.) FO =0.
DO 50 K = 2rMH
DELT/DELTj
6AM(NL»K)= 0.
iF(.NOT.CClMTRLdl) > GO TO 50
EDI = (RSEA(K)-R5EA(K-1))/;RSEA(1)*RO*DZ(K)*.3048)
IFCED1.EQ.O.) EDl=-l.E-t*
IFC.NOT.COi^TRLl^J) EDI = -1.E-*
ED = -l.E-7/tDl
FZ(1»K) = EO/tZ
FZ(UL»K) = FZ(1»K)
-------
C
C
C
4-0
5(J
55
60
AK1 -
DO 40 J =
FZ(J'K) =
CONTINUE
CONTINUE
IF(KPORT.LE
DO 55 J =
DO 55 K =
FZ(J'K) = .
CONTINUF
.5*uZTOT*SQKT(.689)/(Z(NK>-Zp
prMJ
EXp(-(AKl*Zl>**2)+FZ(l»K)
.1.0R.CONTRL<11> > 60 TO 60
If ML
1'NH
0001
ABS( (SI6TR-SI6TR)/?B)/DELTJ
G = 0.
UZ(NK)
SET-UP FOR Z-DIRECTlON CONSTANTS
70
DO 70 K
SZ(K'D
SZ(K'2)
SZ(K'3)
SZ(K'4)
SZ(K'5)
SZlK '6)
SZ(K»7) =
SZ(K'B) = 2
SZ(K'9) = i
SZ(K'IO) =
SZ(K'll) =
SZ(K»12) =
SZ(K'13) =
SZ(K'l^) =
SZ(K'15) =
SZ(K'l6) =
CONTINUF
2»NK
2./REZ*
)/DZ(K)
-------
c
C SET-UP FOR R-DIRECTION
C
R(
XX
X(
XR
uo
x(
IF
XX
XR
RC
M
1)
i)
(1
J)
(J
}
00 j
• EG.
«
—
z
—
~
—
2)
0.
n.
^ *
0.
?»
X(
x(
5*DX
ML
J-D
1) =
+QX
0«
= xx+ux
(J
(J
J)
)
)
_
~
n
XX
5 A
SA
NH 'KT
X»KT)
)
(K
X
{
(
»KT)
()(X»KT)*CASH(XX»KT)/K
*DV)**P)/RER
J) »KT
CASH(
/(CASH t
*RRP
( J»f
*sci
)/
• *DX)
J]
(J)
>KT)*DX
+CONC)
-CONC >
*
*
)**2)
•C!
. c.
J)>
J)*CASh(X(J)
lF(CONTf?L(ii) ) SC(J»5) = 0.
IF(.NOT.CONTRL(1» 5c(j»b)= 0.
80 COIMTlNUF
RC(1> = -RC(2)
-------
DO 90 J = ] »riL
DO 90 K - 1»NH
IF(RB(K).LT.KCIJ).O^.K.LT.KPOPT) FR(J'K) = EkATlO
90 CONTINUE
iF(UJMOrE.NE.O) GO TO 15U
DO 100 J = 2rNJ
PSi(J'l) = PSI(J-lrl) + uZ(j»l>*RC(j)*CASH(X(vJ) »KT)*DX
100 CONTINUE
150 lF(lNMODE.Ln.3) CALL sIMjET (fCn»ZB»DZ (1 > »G»FO r V«B» VMB1 »r,MB»DMB)
LG.l) CALL GAUSS (1)
E-Q.S.OR.INi^OUE.EQ.S). CALL GAUSS (2)
UZ(1»D = UZ(2»D
IF IINMOOE.NE.^) GO TO 20U
DO 160 J = \fJpOPT
DELT(J»KPOKT)= 1.0
GAM(J'KPOPT) = 1.0
IF(J»EQ,1) GO TO IbO
UZU'KPORT) = (PLX + 1)*(2*PLX + 1)/(2*PLX*PLX)*(1»-RC(J) )**(1./PLX)
PSI(J'KPORT) = PSI
160 CONTINUE
PSIR = PSKJPORTrKPOKT)
UZH'KPORT) = UZ(2,KPORT)
LO 170 K = 1»KPOPT
17U PSI(JPORT»K)= PSIB
DO 180 J r JPORTiNJ
PSIU'I) = PSIB
180 CONTINUE
NB = JPORT
200 CONTINUE
RETURN
END
SUBROUTINE PLABAK
-------
INCLUDE COMLST.LTST
DATA/DF/lHP/CF/lHC/
TU r pF
IF(CONTPL(1?) ) TU = CF
EPR = l./KRP
EPZ = l
toRITE(OUT»lPOl)
M H 1 TE ( Oi iT » 1 0 02 ) N J » NK » OX » RO . VO » FO » TU » TU r TU » TK r SAL J » SALR » S I GT J
i SIGTR»KEK»KEZrPL.XrUZ(2'KPORT)
V.R I TF ( OUT • i (} 07 ) NMAD » KPOKT » JPOR? » NEDDY r EXS ? BETA » AK 1
5 ftRlTE(0'jT»10in) DAT?:»T1M, (NOX(K) »K=lffl) '
DO 1^0 j r l,rjj
v\R!TE(OUT»10l2) J» X < J) »RC ( J) »R ( J) » KC ( J,L) »L=1 »fl)
CONTINUE
WRITE(OUT»101Q) DATF-»TIM» (NOX(K) »K=Q»15)
DO 145 J = ItHj
WRITE (0( IT. 10 12) J»XL=9»15)
COI4TINUE
•A'RITE(OUT»1014) OArErTIM» (NOX(K) »K=1»P)
DO 150 y = 1»MH
150 D»RITE(0«.iTilOl2) K »DZ (K ) »ZC (K ) »Z (K ) » (SZ (K »D »L=1 '8)
WRITE(OUTrl01*i) DATE»TIM» (NOX(K) »K=q»16)
DO 155 K = 1»NH
WRITE (OUT ,1012) K»D2(K) 'ZC(K) »Z(K) • (SZ(K»L) »L=9»16)
155 CONTINUF
WRITE(OUT»lQle») CATE»TIM» (NOX(K) »K= 17»20)
DO 165 K = IrNH
WR ITE( OUT , 1006) K »U? (K) '1C (K ) »Z (K ) » (S7(K»L ) »L=1?»20 )
1 RSEA(K)
165 CONTINUE
WRITECOL.'T»1004)
190 DO 200 J = I»NJ
L = KPORT
-------
VvRITE<0'iT.1005) PSI < J»D »U^ (J»L) *UX (J»L) *DE.LT(J'L) *GAM(J*L)
200 CONTINUF
cn
1001 FOKf>.AT(/// <+OH PARAMETERS FOR THERMAL PLUME CASE 13)
1002 FORMAT (///
1/55H NUMBER OF RADIAL NODFS ( X-DIPLCTION) - - - - -
3/55H NUMRFH OF VERTICAL NODFS (?-DIRECTIOM) - - - - -
U/55H RADIAL NODE THICKNESS (X-DIRFCTl^N) * DX
b/5b(i PLUME OUTFALL PORT RADIUS (X-CoORD)* RO- - - - -
7/55H PLUME OUTFALL PORT VELOCITY *FT/SEC )» - - - - -
£/5bH DENSlMETRiC FROUDE NO* AT OUTFALL PGRT(VO**2)
A/5XQTEMPERATURL OF REFERENCE AMpIF.NT
-------
CLNTE.KLINE VALUt. OF GAyMA-CONSTlULNT ' GMB = F9«3
CLNTFRLINE VALUE OF BUCYANCY PARAMETER »OMB = F9.3)
1004 FOKMAT(//Q RADIAL DlbTRlHUTlONS O/
1/OX65HPSI VERT vtLO RAn VEI 0 OE'LT GAMMA
2 //)
lOOb FORMATltP6(tTl3.3»lX»
1006 FOR!iAT( 13 » F7.2» 2F10 • 3 t iPbEll. * »Fl3. 5 >
1010 FORMAT (1H1»K DATE Q2Ab»ioi Tl^r Q2A6/
i bun COMPUTED CONSTANTS FOR RADIAL DIFFERENCED - -SC(J»D //
S 30h J X(J) R/RO R(J) »8(I6»5X>/)
1012 FORMAT(i3fF7.2r2F10-2rlPoFilO)
1013 FORMAT(/J
1014 FORMAT ( 1H1 tQ DATE Q2Ab»r»i TTMf 02A6/
1 60H COMPUTED CONSTANTS FOP VERTICAL DIFFERENCES - -SZ //
in c. 3UH K LiZ(K) Z/UO 7//
2 30H K UZ(K) Z/DO Z »i*(I6»5Y) '
3 30H DELT(lMLfK)
RETURN
END
SUBROUTINE STREAM (iT
SUBROUTINE CALCULATES THL TWO OlMENSlO^AL STREAM FUNCTION' PSKj
INCLUDE CO^LST^LIST
OM£GA3= 0.
10 DO 120 I = 1,JT
QPMAX = 0.
SET OUT-BOUNDARY STREAM FUNCTION FOP NEXT ITERATION CYCLE
DO 20 K = 2»NKK
PS I ( N J» K ) = ?,. *RS I ( N J-1»K ) -PS I ( N J-2 »K >
20 CONTlNUr
DO 100 J = 2» i\|jJ
-------
Al = SC(J,11)
A2 = SC(J»12)
A3 = 5C(J.13)
00 100 K = ?»MK'
IF(J.LE,JPGRT.Ar'C'.K»LE«KPOrtT)
TO
AH-SZ(K»B)
PSKJ'K)
.O) GO TO 50
= ,5*(OMEG
= OMEGA1+SZ(K»12)*S I1J t K-l) +SZ (K»10) *PSI (J, K+l > -»-OMEGA3
*R(J))/CON
)/PSI(J»K))
tn
CON
PS 10
IFtNSKlp
OML6A1
OMLGA2
**5 CONTINUE:
OM£GA3
50 PSKJ'K)
1
2
DEL = ABS(
IF(J.GT.NCR) GO TO 95
QPMAX = AMAXKDPA'AX'DED
iFtDPf^Ay.Gl .DEL) GO TO 9b
NODE(5) = J
NODE(6) = K
9b PSKJ'K) = PSIO+EXT * (PSI (JrK )-pSIO)
100 CONTINUE
IF(DPMAX-LE..0005) GO TO 130
120 CONTINUE
130 iF(fjSKlp.EG.l) GC TO IbO
ITNC = I
CALCULATE VELOCITY FIELD
15U DO 250 J r 2»NJ
Al = SC(J»14)
DO 250 v= 2»NK
IF(J.LT.JPORT.ANC.K.LT.KPORT) GO TO 250
UX(J'K) = -(PSKJ»K)-PSKJ»K-D )/(R(J)*D7(K)
UZ(J'K) = (PSlf J'K>-PbKJ-l»K)
IFlj-Eu.2) UZd'Kl = UZ(2»K)
-------
UX(J»NH> = UX
IF
-------
en
UMAX = ITNO+ITMAX
N2 = IT^AX
NED = NED + IT^O
IF(NED.GT.ITNO.OR.NEDDY»tO.O) GO TO 15
CALL EUpY(NEOnY)
15 CALL ETIMEF(START)
DO «00 L - M»N2
ITNO = L
DDMAX - 0.
IF( .NOT.CONTRLU) .AND. .N Go TO 200
MASK = 0
IF(J.EGI.JPORT.AND.K.EQ.KPOHT+1) MASK = 1
20 Al = SC(J»7)
A2 = SC(J»8)
A3 = SC(J»9)*FR(J»K-1)
AU = 'SC(0»10)*FR(J»K-1)
A5 = SC(J»6)*Fp(o»K-l)
UPOSl = ABS(UX(J-1»K» * UY(J-1»K)
UPOS2 = ABS(UX(J»K)) * UX(j»K)
UNLGl = ABS(UX(J-1»K)) - UX(j-l»K>
UNE62 = ABStUXU'K) ) - UX
-------
DGAM = GZCON*DFZ*(DPi*GAM(J,K + l)-nMl*GArv-(JfK-l)-DCO* -AM(J
DGKAD(J»K) = DPl*DELT(JfK+D-DMl*DELT(J»K-l)-DCO*bELT(J»K)
CDELT = GZCON*DFZ*DGRAO(J»K)
5C COUTlNur
BO = FZ(J'K)*SZ(K»b)
AJ1 = (Al*UPOSl+A3)
AJ2 =
AK1 =
AK2 = (SZ(K»4)*VNEG2+SZ(K'7)*FZ(J'K»
DJi = AJ1*OELT*OELT
GJt = AK2*GAM(JfK+l)
GAMO = GAi-I(J»K)
DELTO - DELT
GAM (J»K)=
DEL - ABS((DELT(jrK)-DELTO)/DELT(J,K)
IF(J.GT.NCK) GO TO 195
DDMAX r AMAXl (DL)MAX»DEL)
iF(DDMAy.GT.DEL) GO TO 195
NOUEtl) = J
NOUE<2) = K
195 CELT(J»K>= CELTO+EXR*(DELT(J»K>-DELTO)
GAM(J»K) = eANO+EXH*(GAM(J»K)-GAMO)
200 CONTINUE
SET BOUMDAKY VALUES FOR UELTA(J»K)
DO 230 J = 2fNJ
GAM(J»NU)= GAM(J»NK>
Jr'iH)= DELT(J»NK>
-------
IF(J*LT.NP) (30 TO 230
DELT= DELTAS'
GAM(J'l) = GA,y(Jr2)
230 CONTINUE
IF(lNMOHE.NE.t) GO TQ 25Q
DO 2<+0 K. = 2»KPOPT
DELT
GAi*i( JPORTfK) = GAM
UNEti2 = ABS(UX(J'K» - UX(j»K)
VPOSl = ARS(UZ(J'K-D ) * UZ(J»K-1)
VPOS2 - ABS(UZ(J'K)) + U7
-------
UP1 = .5*(UX(
UCO = ,5*(UX(J'K )HIX(J-1.K
DFZ = SZ(K»15)*FZ(J»K-H)-S? ) )*UGRAD(J'K)
293 CONTINUE
BO = FZ(J»K)*Sz(K»l)
01 = 1»/-OM£GO)
300 CONTINUE
CALL STREAM (ITL'PS»1)
C SET CENTEPLINE AND CUT BOUNDARY
DO 310 K = 2»MK
OMEG
-------
VORT2
00
DUZ
DVR
en
ro
SET BOTTOM BOUNDARY VORTICITY
IF(INMOOE.EG.'4) GO TO 35u
SET SLIP BOUNDARY
DO 330 J = UBrNJ
OMLG(J»t>= -OMEG(Jr2)
330 CONTINUE
SET INFLOW BOUNDARY VORTICITY
= l./DZ+UX(J-:
DVR = .5*(UZ(J+1»2)-UZ(J-1»2))/(RC(J+l)-PC= DUZ-DVR
360 CONTINUE
C SET PORT SIDE NO-SLlP BOUNDARY
DOES NOT ENTER IN CALCULATIONS
DO 370
UKP1
UK Ml
UKC
DVR
K = 2»KPOPT
= .5*(UX(JPORT»K+1)+HX(JPORT+1»K+1))
= .5*1UX(JPORT»K-1)+UX(JPOPT+1»K-1))
= ,5*(UX(jPORT»K )+uX(JPORT+1tK ))
= SZ(K»15 >*UKPl-bZ(K'14)*UKM1-SZ(K »16> *UKC
= . 5* (UZ (JPORT+2»K) +UZ (JPORT+2, K-l) -HjZ (JPORT+11 K)
UZ(JPORT + 1»K-l))/(RC(JPORT+2)-RC(JPORT))
370 CONTINUE
SET INFLOW BOUNDARY VORTICITY
-------
LO 380 j = ?>jpORT
DVR = (UZ(J+1»KPQRT)-UZ(J-1»KPORT) )/(RC(J+D-RC
-------
IF(KOD(L»NTTY) .EG.O) CALL. OUTPUT (£)
800 CONTINUE
IF(lNMonE.LO.U) TEMPER(JPORT»KPORT)=TFMPEP(JPORT-1»KPORT)
1F(.NOT.CONTRL(9)) GO TO 880
iivR ITE (8) I Tf ;0»OMEG»DEL.T»UX r(JZ» PC I' GA M
N3UPTS = 0
DO 810 j = i»5
IFtf--}3DPT(J) .EO.O) GO TO 810
N3UF-TS = N3DPTS+1
810 CONTINUC
UZ = NK
IFCr,l3DPTS.£.e.C) GO TO 830
WRITE(8) KASE»DATE»TlM»TLA6EL»N3DPTS'JPoRTrKPORT»NX»NZ
DO 820 J = 1,5
L = N3DPT»(Z (N)»N=1»NZ)»
1 ((PSi (NrM)»N=1»NX)»M=1»NZ)
IF
IF(L«EQ.5) URITE(8) L»(RC(N)»N=1»NX)»(ZC(N)»u=2»NH)
1 ((OMtG(N»M)»N=lrNX)»M=2»NH>
LL = L
IFtL'EQ.l) WRITE(OUTrlOOH) LL
IF(L-E0.2) WRITE(OUT»1005) LL
IF
-------
IF(TEMPrR(2,NK>-TEMPER(NL»NK).LT..1) GO TO BfaO
IF(CONTf>L(l3» WRITE(OUT»1001> OC
IF(.NOT.CONTRLll3)) WRITE(OUT.lOOl) DF
DO 850 j r 1 ,p,o
TDLl_T(J) = TE"!pFp(uL,NK)+J
IF(TOELT(J).LT.TEMPER(2»NK)) GO TO 850
LAbT - J-l
GO TO 855
3bU CONTINUE
855 DO 870 L = l»LAf,T
DO 860 J = ?»-\>j
IF(TEMPER(J»NK).GT.TOELT(L)) GO TO 860
RAD = (RC(J-D + (TDELT(L)-TEMPER(J-1»NK)
1 -TEMPER(J-1»NK))**DZ(K)+GAM(NCR»K)*i jFACE
lF(uFACt.LT.O,)GAMCON=GAMCON+2.*R(NCR)*DZ(K)*GAM(NCp+l»K)*UFACE
GAMP IF ~ GAMC*FR (NCR' K ) *DZ < K > * (GA^ (NCR»K ) -G <\M < NCR-H t K ) ) +GAMDIF
895 CONTINUE
GAMCON r 100.*GAtfCON/GAMlN
-------
6AMDIF = 100.*GAMDlF/GAMIN
GAMSUM = GAMCON+GAMDIF
GAMERR = GAMSUM-lGO.
WR I TE ( OUT » 1 0 1 1 ) J » GASCON » 6 AMD IF r GAMFRR
900 CONTINUE
RETURN-
1000 FORMAT(I6»**Xl5(F6.tt»2X))
1001 FORMAT (//JSOH SURFACE ISOTHERM DATA //
1/15H DEGREES 'Al»3bH AREA IN RADIUS OF
2/60H APOVE AM6 SQ. FF-^T ISOTHERM » FEET
3/>
1002 FORMAT (lln»?(luXFlO*l) )
1003 FORMAT (///bXl5f ® THREE-D PLOT RECORDS WRTTTEM ON TAPEQ/
1 fa) SET PLOT PARAMETERS NJ = Ql3»0 NK = QI3)
1004 FORMAT (.-D STREAM FUNCTION PECOPD Vi/RlTTEN TO TAPE - RECORD NO SI3)
lOOb FORMATtQ BUOYANCY PARAMETER RECORD WRITTEN TO TAPE - RECORD N00I3)
1006 FORMAT (E GAMMA-COMSTlTUENT RECORD WRITTEN TO TAPE - RECORD NO 013)
j.007 FORMAT (Q TEMPERATURE RECORD WplTTEN TO TAPE - RECORD NO 013)
1008 FORMAT (Q VURTICITY RECORD WRITTEN TO TAPE - RFCORD NO 013)
1010 FORMAT (1H1»//Q GAMMA-CONSTITUENT BALANCE ERROR Q /// •
1 loi NET COMVECTIVE h'ET DIFFUSIVE GAMMA BALANCE
2I3/ 0 OUTFLOW » PERCENT OUTFLOW » PrRCENT ERROR » PERCEN
1011
END
SUBROUTINE
INCLUDE COMLST»LlST
DIMENSION FCORE(LK>fRb(LK)»KR
RATIO = REZ/RER
VEDC = .015
GO TO ( I0»i:0»120»20'20»5u0r50p) rM
C* CALCULATE RADIAL EDDY FACTORS USING PRESCRIBED MIXING LENGTH
-------
10 DO 15 K = 2,NK
VMAX = .5*IUZ(2'K)-UZ(2»K-1) )
FR(J'K-D= . 180*(Z(K)-.25*DZ(K) )*VMAX
15 CONTINUE
20 C OUT I NUT
C* CALCULATE HADIAI. EQPY FACTORS BASED ON /i RULING CALC. OF MIXING L
C* BASL LENGTH OF POTENTIAL CORE Or PERCENT GAMvA DECREASE AT CENTERL
iFUNMonL.LT.u) GO TO 40
DO 25 K = K PORT • Nil
IF(6AM(2fK ) .LT.GAMtNQ) GO TO '0
25 CONTINUE
IF = 1.
IFlK-LT.KCORE) GO TO 60
R5(K> = RCCNSO + ^iJZtNbOfKJ-VsO/dJZtNSO^KJ-UZCNISn + lfK) )*
60 ROb(K) - Fi'C(N05) + (UZ(Nu5fK)-\/n5)/(UZ(N05fK}-UZ(MOtS+lfK) )*
-------
1 (PC(N05+D-KC(N05)
70 CONTINUE
IFU"»EQ.l) GO TO 100
DO 90 J = 2'NJ
IFU"«EQ.2) GO TO 75
IF(HB(K) .LT.RCU-D } GO TO 100
75 CONTINUE
FR(J'K) r (P5(K)-KCORE(FR( J'K-i>+FR(J»K) )
90 CONTINUE
100 CONTINUE
IFIK.EQ.1 .OR.M.EQ.2) GO TO 400
C* CALCULATE VERTICAL EDDY FACTORS FZ(J'K) IN SURFACE SPREAD
120 CONTINUE
00 1^0 K = KPOKTfNK
DO 130 j = g,NJ
IF(RC(J-1).GT.RR(K» GO TO 140
FZ(J'K) r RATIO*FH
-------
152 CONTINUE:
155 CONTINUF
FZC = VFOC*ZLE.N*UWAX*REZ*2.
DO 200 J z 2»NJ
DO 150 K = X'NH
UAVE: = .5*(ux(j-irK>+ux(j»K) )
IF(UAV£.LT.O. ) KB(J) = K
150 CONTINUE:
DO 160 K = 2»NK
IF(RC(J-1) .LE.RB(K) .AND.K.GT.KPoRT) GO JO 1S5
FZ(J»K) = FZ(NLrK)
IF(K+2.LT.KtMj) .OR.K.LE.NK/2) Go TO 160
IF(J.NE.NJ) Ff;(J»K-D = FR(2rK-l)
FZ(J'K) = FZ(J»K) + FZC*UX(J»NK)/UM4X
15b lF(f,MAD.E(?.0) GO TO 160
C* MODIFY RY KICHARDSON NUMBEH MODrL (IN LATFRAL PLUME SPRF.AD ONLY)
^ RICHNO z +.b/FO*DGRAD(J»K)/(U<$RAD(J»K)**2)
10 !F(HlCh,jO.LT.O. ) RICHNO = 0
RICH = FZ(JrNK)
200 CONTINUE
UOO IF(?"OD( TTNO»NOuT) ,NE.O>
ftRITE(0!.!T»1001) ITNO
DO 450 K z 2»NK
KN = KK+2-K
Dl = .5*KCCR£.
-------
500 RETURN
inuu f-OKf;AT( IHlX/fo) PLUME COKE F.XTENDS TO 3UKFACE K
I/ fc CORE AbSUwEH TO END AT K = MH FOR PURPOSES OF Er,
^DY CALCI'LATJOM, FR(JtK-l) 0 >
3001 POKVAT(// I? pLiiME LATERAL SECTION GFO^hTRY» ITERATION N0» 0
lIb//R K Z/D
1002 F-CRi AT ( Tlf:»Fo.ii»2X) '
1 HO' AX»NO^E(i) »NOPF('U »n{jvAX»NOnt(I) f NODE (2)
DO 5 J - lf';J
DO 5 K = IrNK
IF(i'Z(J»K) .LE.5. )GO TO 5
iB/RUE (OUT r 1003)
ISTCP = °9999
GO TO IP
5 CONTIMUF
*FSTART)/NOUT
DO 100 J - 1»15
L = Ni'ITEU)
IF (L .fu.6) NOTL'MP =ii
IF(L»F-:U.O) GO TO 10°
IF (MODE.. EO. 1 ) GO TO Q0
IF(L.EG.I) CALL APoUT(L»Zr R»rSl»
1 <+2r|STRLAivi FUNCTION - IKROTAT10NAL FLOW
-------
IF(,'10DL.EQ.O) tO TO 10U
90 IF(L»EQ.2) CALL AROUT(L»L»R.PSl»
j U2HSTREAM FUNCTION - VISCOUS FLOW )
IF(L«E^.3) CALL AROUT(L»ZC»RC»D£LT»
1 42KBUOYANCY PARAMETER " DFLT )
IF(L'EG.U) CALL AROUT(Lr^C»RC»0"EG»
1 42HVORTICITY - CME6 )
iF(L-EQ.b) CALL ARO^T(L»2»RC»UZ.
1 42HVERTICAL VFL-OCITY COMPONENT - U^ )
IF(L«EQ.6) CALL AF^QUT (L»^C» R»Ux>
i 42HRADIAL VELOCITY COMPONENT - ux )
IFCL-EQ.7) CALL AROUr(L»ZC»RCf6AM»
i 42HGAMMA-CONSTITUENT )
IF(.NOT.CO(JTRL(13) ) GO Tu 92
IF(L»E0.8) CALL AROUT(L»ZC»RC»DPLTr
1 42HTEMPERATURE' DEGREES CENTIGRADE )
tO TO 94
92 IF(L'EQ.ti) CALL AROUT (L»^.C »RC »Dr"LT»
1 U2HTEMPEKATURF' DEGREFS
94 CONTINUE
IF(L»EO,9) CALL APO^T (L »ZC » RC »Dr.LT»
1 U2HNORMALIZED BUOYANCY
lF(L«EQ.lnjCALL AROUT(L»^»RCrUZ,
i <42HNORMALIZED VFRTICAL VELOCITY COMPONENT >
IF(L«EQ.11)CALL AROUT(Lt2lC»RC»D|-:LT»
1 U2HNORMALIZED TEMPERATURE DISTRIBUTION )
IF(L*Eu.i?)CALL AROUT(U»ZC»RC»FP»
i 12HRADIAL EDDY •. IXIMG FACTORS )
IF(L»EQ.1?)CALL APOUj (L »ZC f RC »F;?»
i 12HVERTICAL EDDY MIXING FACTORS >
IF(L»EG.14)CALL ARuUT(L»ZC»RC»RfCHr
1 U2HPICHARDSON N
IF(L»EQ.lf))no TO loO
100 CONTINUE
lF(lSTOp.E(i.0999q> STOP
-------
CALL ETIMEF(START)
RETURN
1000 FORMAT(]H1
NO.
MAX
2/
*»7H
3/ 55H
*»7H
<+/ 35H
*»7H (
1001 FORMAT(iHl
I/
) ^
35H SEC
1002 FORMATI/
26H
OF PSI
CHANGE
KESULTS
ITERATIONS-
IN Pbl
FOR ITER.
MAX CHANGE IN OMEG
CHANGE IN DELT
NO. T5/
I7»10XfHNODF
1PE10.3
ElO»3
ElO«3
U5H STREAM FljNCTjON RESULTS FOR ITERATION
30H MAXIMUM RELATIVE ERROR IS 1PF12.3/
23H TIME REQUIRED FO" I3»14H ITERATIONS = F6«2»
///)
1CH ITERATION
I5/
1 10H VERTICAL VELOCITY
2 40H ********* RADIAL VELOCITY *************
3 ^Oh BUOYANCY PARAMETpR ..........
U/10H NUMBER 15(2n (12,lH'12,1H))//)
1003 FORMAT( ///a THIS CASE IS APPARENTLY UNSTABLE* RUN ABORTED 0)
END
10
INCLUDE
DIMENSION
DIMENSION
REAL
DO 10 J =
HCOC-RD(J) =
COWTlNUf
M2 = 0
Nl
N2
COM'LSTrLlST
ARrJAME(LJ»LK>
HCOORD(LJ) »VCOORD(LK)
LABEL
1 r/NL
5*RCOORD(J>
-------
IF(N2.GT«NL)N2=NL
fe/RITE(OUT»1000) DATE*T!MrLABEL»lTNO»TINT
WRITE (OUT* 10-0*»)
WRITE(OUTfiOOl) (NOX(K)»K=N1»N2)
65 WRITE (OUT »1002) (HCOOROdO »K. = Nl»N2>
70 DO 200 K = 1»NH
KN = NH-K*1
IF(N•EQ.fa.OR.N.EQ.ll) 60 TO 160
IF(N«NE.9.AND.N»NE.10) GO TO I5n
AMAX = ARNAME(2»KM)
lF(ARNA;-*E(d»KN> .GT.AMAX) AMAX=ApNAME<3
DO 100 J = N1»N2
loo ANORM(J) = ARNAME:(J'KN)/AMAX
WR1TE(OUT»1003) KN»VcOORD(KM) » = TEMP(SALrSIGT)
IF(.NOT.CONTRL(13) ) TEMPER(J'KN) = 1.8*TE!*PER (JiKN)+32.
165 CONTINUE
1MAX = TEMPER(2»KN)
IF(K:.EQ.ID GO TO i?o
167 WRITE»J=NlfN2»
200 CONTINUE
IFCN2.NE«NL) GO TO 60
1000 FORMAT! 1H1»D DATE Q2Ab»U Tlf/E C
-------
7 ITERATION' DUMBER 0 »I5»
1 25H COMPUTATION SPEED = F6,3»1SH SEC/ITERATION )
1001 FORMAT(/17X3HJ = lC(I8'3X))
1002 FORMAT (l2Xh;RCOORD =I3» 10 (F9»2»2X)/12X(3ZCoOPDQ)
1003 FORMTC bH K - 12, 5H Z = F6.2 »2XlPlOEl 1.3)
1004 FOkr.;AT(;.! COOFDlNATtS GIVEN IN PnPT DlAMETFRSf Z/D OR R/D (3 >
FUUCTIO':
IF(N»EO.
SIGU
b
A
SUMT
KETURM
10 SIGVAT
RETURN
END
GO TO 10
= LE-6*T*(
= .Of'l*T*((
= T
.814^) *SAL-. 093
-SUMT
FUNCTIOii
ERROR
T
SI60
SI60
DO 100
TSQD
TQBC
F
DSUI'.T
TEMP(SALT»SIGMA)
•uEWTON RA^HSON METHOD FOR CALCULATING TEMP.
SALINITY AND REFERENCE DENSITY
I =
.01
20.
-.093+.ei/+9*SALT-.On0482*SALT*bALT
SIGO+6.8EI
] »bO
T*T
= SIGMAT(SALT»Trl)-SlGMA
= (215.74* (T-3. Q8 )**2)/ (503 .570* (T + 67 .26 )**'2>
-------
100
150
DSUMT
DA
DB
DF
Tl
ER
ER
T
IF(£R.LT»ERROR)GO
CONTINUE
TEMP = T
RETURN
= DSUMT-2*(T-3.9e)*(Tf293.)/(503.57n*(T-»-b7.26)
= .001*U'7867-.196?7*T+.P03252C>*TSQD>
= l.E.-6*(lR.03-1.6328*T+.05*TsQn)
= (SIGO+.1324)*(-DA+DR*(SIGO-.132'4) ) +DSUMT
= T-F/DF
= Tl-T
= ABS(ER)
= Tl
TO
en
FUNCTION RCHM~!D(M»RlCHrBETA)
I CHOOSE r-ETA CONSTANT FOR APPROPRIATE MOpEL AT INPUT
GO TO (I0»20»30»^0»50fb0)»M
C* 10 ROSSBY /\ND MONTGOMERY (1935)
10 RChMOD = l./(l.+BETA*RICH>
RETURN
C* 20 ROSSBY ANP MONTGOMERY (1935)
20 RCHXOD = l./(l.+BETA*RICH)**2
RETURN
C* 30 hOLZMAN (1935)
30 RCHMOD = AMAXl(0.»l.-BtTA*RlCn)
RETURN
C* ^0 YAMAMOTr> (1959)
i^O RCHMOD = 5Qt-'T(ANiAXl(0.rl.-RFTA*RICH) )
RETURN
C* 50 MAMAYEV (19^8)
50 RCHMOD - EXP(-RETA*RICH)
RETURN
C* 60 MUNK ANC: ANDERSON
-------
60 RCril^OD = (l.+RETA*PICH)**l,^
RETURN
END
SUdROUTINF ISOGEN(L'R »PSI»ISOLN,L »NJ» NK »LABEL)
INCLUDE ARQlMrLlST
DIMENSION Z(LK>'R(LJ)rPSllLJ»LK)»ISOLN(5»3n)tLABEL(6)
DIMENSION XP(iiOO) »ZP(200) »ROOT(3)
REAL LABEL»IS°LN
INTEGER OUT
OUT = 6
V»RITE
-------
C ********* INTERPOLATION *********
40 M=K-1
C S,$$$S$$$$ QUADRATIC INTERPOLATION ,„,,,„„
C EQUATION FOR INTERPOLATION IS Op FORM Y = AX**2+BX+f>
IF(43,45,45
C ML CORRESPONDS TO I~l
C MM CORRESPONDS TO I
C MM CORRESPONDS TO I+l
C BRANCH TO 43—USE pOlNTS K-2'K-l»AND K F0» THE
C QUADRATIC INTERPOLATION
43 ML=K-2
MM=K-1
MH=K
GO TO 44
C BRANCH TO 45—USE POINTS K-1»K» AND K+l FOR THE
C QUADRATIC INTERPOLATION
45 ML=K-1
MM=K
MH=K+1
44 DENOM=(Z)
BNUM=(PSI(J» MH)-PSI< J,MM))*(Z(MM)**?-Z(ML)**£)-(PSI(J»MM)
1-PSIU»ML))*(Z(MH)**2-Z
AA = ANL'M/QENOM
BB = BNUM/DENOM
D=PSl (J»MM)-AA*Z(MMH*2-BB*Z(r/M)
TERM=SQRT(dP**2-4.*AA*CD-PSIC))
ROOT (i > = t -BB+TL.RM f /(2. *A A >
ROOT<2)=(-fB-TERM)/^2.*AA)
DO 57 1=1,2
IF(MM.EQ.K>60 TO 61
-------
IF(ROOTd) .LT.Z(MH) .AND.ROOT .GT.Z(MM) )GO TO 60
IF I ROOT d) .LT,Z .GT.ZGO T^ 60
IF(ROOT(I).LT.Z(ML)»AND«KOOT(I>.GT.Z= XCOORD>
4=. GO TO 5
» 85 CONTINUE
DO 185 K=KN»NK
j = i
90 IFlpSI (jfK)-PSlC )100»200»300
100 J=J+1
IF(J.GT.NJ)GO TO 185
IF(PSI (J»K)-PSK
300 J=J+1
IF(J«GT.NJ)GO TO 185
IF(PSI (J»K J-PSIC
c ********* INTERPOLATION *********
400 M=J-1
QUADRATIC IMTLRPOLATIOM $$$$$$$$
410 IF( (J-1)-1
<+20 IF((PSIC -PSI<^»K)
C ML CORRESPONDS TO 1-1
C MM CORRESPONDS TO I
C MH CORRESPONDS TO I+l
-------
C BRAIiCH JO 130--USE POINTS J-2,J-1»AND J FOR THE
C QUADRATIC INTERPOLATION
130 ML=J-2
MM=J-1
MH=J
GO TO 1-40
C BRANCH TO 450—USE POINTS J-liJr AND J+l FOft THE
C QUADRATIC INTERPOLATION
150 ML=J-1
11 u DENOM=(o(MM)* t^-R(ML)**2)*(R(MH)-RI MM))-(R(MH)**2-R(MM)
1*(K(MM)-R(ML))
ANUM=
-------
615 XPUOUNT+1) = XCCORD(ROOT
60 TO 90
185 CONTINUE
ouTrj.001) NI»PSIC»KOUNT
01JT f 1002)
DO 500 KK= l»KOUNTfln
KT = ABSlKK-l)
KS = KT+10
IF(KS.GE*KOUNT) KS = KOUNT
'HRITE(OUT»1003) KTr ( XP (Khi) »KR = KK»KS)
wRITE(OUTrlOOU) (ZP
1004 FORMAT(5X10(F3.2»2X> )
1005 FORMAT(//& *****SET PLOT PARAMETERS* NJ=0»I3rQ NK=BI3»0 *****Q)
END
SUBROUTINE
INCLUDE COVLST»LIST
DO 100 J = 1»5
L = ISOPT(J)
-------
IF(L-EG.O) GO TO 100
IF(L-EQ.l) CALL ISO<5FNlZ»XR»Pc;If ISOLN»L.NJ»NK»
i 36HVISCOUS STREAMLINES
IF(L»EG.«n CALL ISOGEN (ZC » X»DrLj» !SnLN»L »MJ»fjK »
1 36HbUOYANcY PARAMETER ISOLlNES*
IF(L.E0.3) CALL ISOGEN(ZC» A»GAM. ISOLN»L,NJ»riK »
1 36HSALIUITY iSOLINESfPARTS PER THOUSAND
IF(L.EQ,<*> CALL isoGEN(zc»XrTEMPER*isoLN'LrNj»NK»
1 36MTEMPERATURE CONTOURS» DF6 CENTIGRADE
100 CONTINUF
RETURN
END
oo
GAUSS(N)
COMLST»LlST
PS8*.5
2.*(ZB-DZ(2>)
SUBROUTINE
INCLUDE
DIMENSION
DELS =
ZR =
ZP =
ZP1 =
PSB(D =
IF(N.NE.I) GO TO 100
RMIND = 1.-ZP/8.
1.-ZR/9.
1.-ZR1/9*
1./9.
l./tt.
2»NJ
1.0
1.
IF(RC(J).GE.RK'IN) UZ(J»1) = EXP (-**0 . 5* (C* (RC < J) "1 • > /ZR) **2)
IFCRCCJV.GL.RMINI) Up, = EXP (-»0 .5* (C+(RC (J)-l . ) /ZR1) **2)
RMIfjl
C
Cl
DO 10 j
UZ(J'l)
UB
-------
iFUiZ(Jrl) .LE..01) UZ(JH)
IF(UB.LE..01) UB = 0.
PSKJ'H = PSI (J-1»1)+UZ(J» 1)*RC+UB*Rt(J)*CASH(X(J)»KT)*DX
10 CONTINUE
DO 50 j = 2»NJ
DEL"T.LE«.01> DtLT(Jrl) = 0.
GAM(J'l) = DELT(J»1>
bO CONTINUE
GO TO 120
100 DO 110 J = 2tNJ
UZ(J'D = EXP(-92.*(RC(J)/ZR)**2)*VMB
UB = EXP<-92»*(RC(J)/ZRl)**2)*VMBl
lF((UZ(jil)/VMB>.LE.-.Ol> UZ(J»1) = 0.
IK (UB/VMB1.LE«»01) UB = 0.
PSI(J'l) = PSI(J-l»D+UZ(J»l)*Rr(J)*CASH(X(j) »KT)*OX
Pb[3(J) = PSB(J-1>*UB*RC(J)*CASH(X(J)»KT)*DX
EXPART = FXP(-68.*(RC(J)/ZP)**2)
IF(EXPAr?T.LE..OD EXPART = 0.
DELT= EXPART*QMR+DELP
GAM(J»1) = EXPART+GMB
11U CONTINUE
120 DO 150 J = 2»NJ
IF-
-------
00
GO
SUBROUTINE
SUBROUTINE
DIMENSION
AK =
A(l) =
A<2) =
A(3) =
AU) =
FRACT r
EX1 =
DELZ =
ZB3 =
ZP2 -
zai =
FlUD LE^GTf
ZO =
DO 10 K =
ZE =
DEL =
iFtoEL.LE.
10 CONTINUE
Ib I
L-B1
EB2
EB3
VMB1
SB
VMB
DMti
SI ff JET ( NCO» ZB > DZr/ » T » ^0 » VP'B » WB1 »GMR » DMB)
OBTAINS SIMILARITY SOLUTION FOP VERTICAL PLUME
A (4) r AF.C*>»AR(**) »D7lC*>»Ml^)
84.
0.
.5
.5
1.
l./o.
1./3.
.5*UZM
2B
ZR-DELZ
i FOK FLOW ESTABLISHMENT
5.57/( (1
ABSUZE-ZO)
0001) GO TO
)) GO TO 30
(i*./ZE)**3+3./32./FO*(ZBl**2-ZF**2>
(4./ZE)**3+3./32./FO*(ZB2**2-ZE**2)
(4./ZE)**3+3./3ii./FO*(ZB3**3-ZE**2)
EB1**EX1*AK**.5/ZB1 '
l./SB
DMti
RETURN
30 DZl(l)
- INTIZE+1O-ZE
-------
00
50
100
150
160
200
DZK2)
DZK3)
DZ1C+)
Nl ( 1 >
Nl(2>
Nl(3>
Niu)
£
R
Z
DO 200 L
DZ
NSTEPS
DO 100 J
DO 50 K
AE(K>
AR
CONTINUE
Z
E
R
CONTINUE
IF(L*NE.3
0Mb
6MB
60 TO 160
IF(L»EQ.l
VMB
IFlL.EQ.2
CONTINUE
CONTINUE
RETURN
END
=
~
—
-
—
—
—
—
—
-
.1
DEi_Z/10»
DZK3)
1
10.*(ZB1-ZE>
5
5
(4./ZE>**3
.25
ZE
li<4
DZKL)
NKD
l»NSTtPS
1»M
DZ*FE ( A ( K ) *L)Z+Z » A < K > * AE ( K-l ) +E » A ( K ) *AR ( K-l ) *R ,FO )
=QZ*FR(A(K)*DZ+Z»A(K)*AE(K-i)+E»A(K)*AR(K-l)+R»T)
—
—
~
)
—
—
)
—
)
Z+DZ
E+FK ACT * < AE < 1 > +2 . * ( Ar < 2 ) + AE ( 3 » ^AE ( 1 > >
R+FRACT*URd)+2.*(AP(2)-»-AP(3) )+AR(<*) >
GO TO 150
1«S.*R/(E**FX1*Z)
l./(.245*Z*£**EXD
GO TO 160
F**£X1*AK**.5/Z
VM81 = V^B
-------
FUNCTION
RETURNS ZERO WHENEVER N IS EVpNtY DIVISIBLE BY M
END
FUNCTION SANH(XrN)
SANh = .5*(EXP(X)-EXP(-X))
IF(N.EQ.O) SANH=X
END
FUNCTION CftSH(X»N)
ro CASH = .5*(EXP(X)-»-EXP(-X))
- IF(N.EQ.O) CASH=1.
END
FUNCTION XCOORU(X)
XCOORD = .5*SINH(X)
RETURN
END
FUNCTION FR(Z»E»R»T)
FR = -.109*E**(1./3.)*T*Z
RETURN
END
-------
FUNCTION FE
-------
SELECTED WATER
RESOURCES ABSTRACTS
INPUT TRANSACTION FORM
Rti-
w
Numerical Thermal Plume Model for Vertical
Outfalls in Shallow Water
5, R
6.
H.
Vp.'totmiD-- Oiga;' f.tioa
Donald S. Trent and James R. Welty
Department of Mechanical Engineering
Oregon State University
Corvallis, Oregon 97331
16130 DGM
12, Span: if lag O
Environmental Protection Agency
Environmental Protection Agency
Report Number,, EPA-R2-73-162, March 1973.
A theoretical study of the heat and momentum transfer resulting from a flow
)f power plant condenser effluent discharged vertically to shallow, quiescent coastal re-
ceiving water is presented. The complete partial differential equations governing steady,
Incompressible, turbulent flow driven by both initial momentum and buoyancy are solved
ising finite-difference techniques to obtain temperature and velocity distributions in the
lear field of the thermal discharge.
Turbulent quantities were treated through the use of Reynolds stresses with
further simplificati"on"uti±izing the concept of eddy diffusivities computed by Prandtl's
aixing length theory. A Richardson number correlation was used to account for the effects
density gradients on the computed diffusivities.
Results were obtained for over 100 cases, 66 of which are reported, using the
computer program presented in this manuscript. These results ranged from cases of pure
moyancy to pure momentum and for receiving water depths from 1 to 80 discharge diameters
ieep. Various computed gross aspects of the flow were compared to published data and
round to be in excellent agreement. Data for shallow water plumes and the ensuing lateral
spread are not readily available; however, one computed surface temperature distribution
?as compared to proprietary data and found also to be in reasonable agreement.
17s..
Waste heat disposal, heated shallow discharge, turbulent buoyant jets,
temperature prediction, thermal pollution.
05G
15, S- urity C *ss.
(Kepon)
21.
of
Send To:
WATER RESOURCES SCIENTIFIC INFORMATION CENTER
U.S. DEPARTMENT OF THE INTERIOR
WASHINGTON, D. C. 2O24O
Author
Oregon State Un-fvers-n-y
------- |