USERS' GUIDE FOR
NUMERICAL MODELING OF BUOYANT PLUMES IN A
TURBULENT, STRATIFIED ATMOSPHERE
by
Ralph G. Bennett
and
Michael W. Golay
Energy Laboratory Report
MIT-EL 79-004
February, 1979
_I·_IIIII_____I____·
_-LI_----L__II-_LIP*I-···-I^IIIIIIIYI
aI.
H
I------ ----------------I
1
_
_
USERS' GUIDE FOR
NUMERICAL MODELING OF BUOYANT PLUMES IN A
TURBULENT, STRATIFIED ATMOSPHERE
by
Ralph G.
Bennett
and
Michael W.
February,
Golay
1979
MIT-EL 79-004
Energy Laboratory
and
Department of Nuclear Engineering
Massachusetts Institute of Technology
Cambridge, Massachusetts
02139
Sponsored by
The Consolidated Edison Company of New York, Inc.
Northeast Utilities Service Corporation
e ---·----I
LW·I--·m(l3l-I----L-r-_·*_·r_
I21------_--L--.-I··_______--^_
NOTICE
The VARR-II User's Guide mentioned in this report is
currently available in three volumes from:
U.S. Department of Energy
Technical Information Center
P.O. Box 62
Oak Ridge, Tennessee 37830
The report numbers and current
(1979) prices are:
CRBRP-ARD-0106
Vol. I
CRBRP-ARD-0106
Vol. II
CRBRP-ARD-0106
Vol. III .
.
.
.
....
$6.75
$4.50
.
.
$6.00
As Limited Distribution Applied Technology Reports, these
volumes are available to U.S. Citizens, and "any further
distribution by any holder of the documents or of the data
therein to third parties representing foreign interests,
foreign governments, foreign companies, and foreign subsidiaries
or foreign divisions of U.S. companies should be coordinated
with the Director, Division of Reactor Development and Demonstration, U.S. Energy Research and Development Administration."
USERS' GUIDE FOR
NUMERICAL MODELING OF BUOYANT PLUMES IN A
TURBULENT, STRATIFIED ATMOSPHERE
by
Ralph G. Bennett
and
Michael W.
Golay
ABSTRACT
A widely applicable computational model of buoyant, bentTo do this,
over plumes in realistic atmospheres is constructed.
the two-dimensional, time-dependent fluid mechanics equations are
numerically integrated, while a number of important physical approximations serve to keep the approach at a tractable level.
A
three-dimensional picture of a steady state plume is constructed
from a sequence of time-dependent, two-dimensional plume cross
sections--each cross section of the sequence is spaced progressively further downwind as it is advected for a progressively
longer time by the prevailing wind.
The dynamics of the plume
simulations are quite general.
The buoyancy sources in the plume
include the sensible heat in the plume, the latent heat absorbed
or released in plume moisture processes, and the heating of the
plume by a radioactive pollutant in the plume.
The atmospheric
state in the simulations is also quite general.
Atmospheric variables are allowed to be functions of height, and the ambient
atmospheric turbulence (also a function of height) is included
in the simulations.
_I
II
_I
I_
D
_CI
I
__
TABLE OF CONTENTS
1.
2.
ACKNOWLEDGEMENTS
6
PREFACE
7
Problem Description and Solution
8
1.1
Introduction
8
1.2
Background and Problem Description
9
1.2.1
Historical Background
9
1.2.2
Characteristics of Bent-Over Buoyant Plumes 10
1.2.3
Overview of Plume Models
11
1.2.4
Scope of Work
14
Literature Review
16
2.1
Numerical Plume Models
16
2.1.1
Three-Dimensional Models
17
2.1.2
Two-Dimensional Models
19
2.1.3
Experimental Studies
20
2.2
3.
page
Numerical Planetary Boundary Layer Modeling
21
Hydrodynamic Model Development
23
3.1
Introduction
23
3.2
Model Equation Sets
23
3.2.1
Dry Equations
23
3.2.1.1
Reference State Decomposition
24
3.2.1.2
Reynolds Decomposition and Closure 30
3.2.1.3
Pollutant Transport Equation
37
3.2.1.4
Radioactive Decay Heating
38
3.2.2
Moist Equations
39
3.2.2.1
Reference State Decomposition
40
3.2.2.2
Reynolds Decomposition and Closure 46
.~--
-4-
page
3.2.2.3
3.2.2.4
3.3
4.
5.
·_
_I
Equilibrium Cloud Microphysics
Model
49
Latent Heat Source Term
50
Model Solution Methodology
53
3.3.1
The VARR-II Fluid Mechanics Algorithm
53
3.3.2
Orientation of the Computer Mesh
54
3.3.3
Downwind Advection of the Mesh
58
3.3.4
Property Data
60
3.3.5
Input Profiles and Boundary Conditions
64
3.3.5.1
Input Profiles
64
3.3.5.2
Boundary Conditions
64
3.3.5.3
Mesh Initialization
67
3.3.6
Mesh Coarsening Capability
69
3.3.7
Statistics Package
72
Description of Atmospheric Turbulence
74
4.1
Introduction
74
4.2
Atmospheric Profiles of Wind, Temperature, and
Humidity
74
4.3
turbulence in the Planetary Boundary Layer
77
4.3.1
Introduction
77
4.3.2
Layers in the PBL and Important Processes
78
4.3.3
Prescription of Eddy Viscosity
81
4.3.4
Prescription of Turbulence Kinetic Energy
86
Card Input and Sample Problems
90
5.1
Overall Program Support
90
5.2
Card Input Decks
92
CI
_
____
-5page
5.3
6.
5.2.1
The Dry, Buoyant Line-Thermal
92
5.2.2
LAPPES
92
5.2.3
Moist, Buoyant Line-Thermal
Plume of 20 October 1967
98
Sample Problem Results
101
5.3.1
The Dry, Buoyant Line-Thermal
101
5.3.2
LAPPES 5 Plume of 20 October 1967
101
Recommendations for Model Extensions
109
6.1
Calculational Scheme to Inluce Wind Shear Effects 109
6.2
Calculational Scheme for Time-Dependent Release
or Weather
112
Cloud Microphysics Model
113
6.3
NOMENCLATURE
116
LIST OF FIGURES AND TABLES
121
REFERENCES
123
APPENDIX A.
Definitions of Important Variables
Added to VARR-II for Simulating Plume
Behavior
128
APPENDIX B.
Card Formats for Input Variables
130
APPENDIX C.
Statement Number Cross-References
136
APPENDIX D.
Computer Code Listing
152
_____
-6ACKNOWLEDGEMENTS
This research was conducted under the sponsorship of
Consolidated Edison and Northeast Utilities Service Corp.
The authors gratefully acknowledge their support.
,---------
-
-7-
PREFACE
The VARR-II40 computer code has been modified for the
modeling of buoyant plumes in turbulent, stratified atmospheres.
A description of the model and the results and conclusions
that have been obtained to date were released in a previous
report
(MIT-EL 79-002).
The purpose of the present report is
to serve as a users' guide for the model.
The first four
chapters of MIT-EL 79-002 have been included in this report
to introduce and describe the model;
the detailed card input
lists and sample problems are found in Chapter 5, and the
Appendices.
Chapter 6 introduces themodel extensions that were
considered in the original work.
-8-
1.
PROBLEM DESCRIPTION AND SOLUTION
1.1
Introduction
With the rapidly increasing burden of air pollution over
recent decades,
the engineer's ability to analyze the behavior
of an ever-widening assortment of effluents has not kept up
with the importance of the consequences of the releases.
The
reason for this is that the "predictive" models of plume behavior
that are currently available universally suffer from a lack of
extendability.
That is, they need to observe the behavior of
an ensemble of the releases that they wish to model before they
can form an accurate picture of the release.
The models are
useful only to the extent that an appropriate ensemble of plumes
can be created for study, either as full-scale atmospheric
releases, or as scaled-down laboratory experiments.
Inasmuch
as the important turbulent and thermal characteristics of the
atmosphere cannot be simulated in the laboratory, and since
an ensemble of plumes with catastrophic consequences
(e.g.,
radioactive plumes from nuclear reactor accidents) may be
impractical to produce, plume modeling has needed to take a
more universal approach.
The purpose of this
work
is to construct a widely
applicable model of plume behavior in realistic atmospheres.
To do this,
a
"first principles" approach is adopted.
A
-9-
numerical integration of the fluid mechanics equation is
undertaken, while a number of important physical approximations to the problem serve to keep the approach at a
tractable level.
The advantage of the model presented
here is the ability to tackle problems outside of the scope
of existing models without greatly increasing the resources
spent on the analysis.
1.2
1.2.1
Background and Problem Description
Historical Background
Man has produced and observed bent-over buoyant plumes
since the discovery of fire.
However, the bent-over plume
did not have any great impact on society until the advent of
large industrial sources near population centers during the
industrial revolution.
The number of large industrial sources
has increased steadily with the industrialization of many
countries.
In the recent past, the variety of releases from
large industrial sources has increased greatly, and now includes the potentially more harmful effluents from chemical
refining and combustion processes, nuclear power plants, and
large cooling towers.
Also, the steady growth of population
centers almost always dictates that these new sources will
be located in at least moderately populated areas.
Historically, the ability to analyze the effects of
large releases and hence to develop technologies for their
-10-
mitigation has not kept pace with the consequences of the releases.
To date, the advances have been quite modest:
early
observations during the industrial revolution suggested the use
of tall stacks for lessening the effects of large releases.
The strong influence of the synoptic-scale weather on releases
(first investigated in order to increase the effectiveness of
chemical warfare agents) has largely motivated the Pasquilltype correlations of plume behavior.
The hope of simply re-
ducing the consequences by reducing the amount of effluents
has stimulated an abundance of filtering, scrubbing, and
effluent control technologies.
However, increasingly important
releases are certain to occur.
A brief review here of the
existing approaches to plume modeling can indicate the most
promising avenue for study.
1.2.2
Characteristics of Bent-Over Buoyant Plumes
The character of the bent-over buoyant plume is central
to all of the available plume models.
When an effluent stream
with a given upward momentum and initial buoyancy is released
from a stack into a windy atmosphere, the plume is deflected
downwind.
This occurs partly because of pressure forces that
develop around the plume, and partly because the plume entrains
the ambient air, which mixes a lot of downwind momentum into
the plume.
The deflection quickly causes the plume to bend
over (usually within about one stack height) and then to be
- ----
-----------------
-11-
carried downstream.
The buoyancy of the plume is converted
into kinetic energy, and the plume rises under this action
for a considerable distance downwind.
About 20 years ago it
was notedl that the motion of the plume in cross section during
this rise was essentially that of a two-dimensional turbulent
vortex pair.
Initially the vortex pair rises and grows with-
out being too dependent upon atmospheric turbulence (although
atmospheric stratification is always important).
After the
kinetic energy of the cross-sectional motions has essentially
died out, the plume continues to disperse solely by atmospheric motions.
It will be found in the review of plume
models that only the detailed numerical plume models provide
a method that can easily bridge between the regimes where
plume turbulence dominates and where atmospheric turbulence
dominates.
1.2.3
Overview of Plume Models
With regard to the detailed three-dimensional nature of
plume motions, existing models of plume behavior are found
to possess a wide variety of sophistication.
The Pasquill-
type models, the entrainment models, and the numerical models
are considered here.
The Pasquill-type models develop a highly idealized
picture of the fluid motions in and around the plume.
Pollu-
tants in the plume cross section are assumed to fit Gaussian
-12-
distributions of height and width.
In essence, the model
parameters (standard deviations of the Gaussian distributions)
are simply an ad-hoc replica of the experimental results;
as such, the models are unable to predict in cases for which
experiments have not been performed.
The wealth of non-
passive effluents and the rich variations in the meteorological state of the atmosphere serve to guarantee that cases
outside of the Pasquill-type models will always exist.
The entrainment models develop a much less idealized,
and much more physical picture of the fluid motions in the
plume.
In general, the models make use of the very elegant
non-dimensional formulations and similarity relationships
that are central to the theory of homogeneous isotropic
Typically the models are successful at analyz-
turbulence.
ing the initial plume behavior, where the self-generated
plume turbulence dominates over the atmospheric turbulence.
The entrainment models are generally able to analyze plumes
only in fairly simple atmospheres when analytical solutions
are sought.
But this is not the primary limitation of
entrainment models, since in some cases their solutions
are found on computers.
The limitations of the entrainment
models are the condition that the plume self-generated
turbulence is dominant over the atmospheric turbulence,
(which eventually breaks down for all plumes, commonly at
_
I·I
I___·__I I__
_ _C __I
I___
-13-
downwind distances for which the solution is still needed)
and the basic entrainment velocity assumption, which cannot
be obtained from fundamental constants and scales in a
straightforward way.
Numerical plume models are capable of developing the
most detailed picture of the fluid motions in the plume.
general,
In
the models seek to integrate a closed set of Reynolds-
averaged fluid mechanics equations, either in two or three
dimensions.
Turbulence leads to a fundamental closure
problem in writing this set of equations, so that each model
will have a collection of closure assumptions that together
form a turbulence model, aside from other assumptions that
are made concerning the plume behavior.
Numerical plume
models are becoming capable of analyzing the most detailed
cases, yet they are often limited by the large computing
costs.
Aside from the computer costs, the tasks of initial-
izing and validating the problem with fully two- or threedimensional data can also quickly become intractable.
Until
computer costs are reduced greatly, the most useful numerical
plume models will likely have to be two-dimensional.
The
greatest benefit that comes from such models is the wider
range of application of the models, and the ease of extending
them to new cases.
-14-
1.2.4
Scope of the Work
This work constructs a three-dimensional solution of a
steady state plume from a sequence of time-dependent twodimensional plume cross sections; each plume cross section
of the sequence being spaced progressively further downwind
as it is advected for a progressively longer time by the
prevailing wind.
The two-dimensional cross sections are
simulated with a time-dependent turbulent fluid mechanics
code which integrates the time-averaged equations of continuity, momentum, energy, moisture, and pollutant.
The
behavior of an individual plume is modeled in this way until
the height or radius of the plume reaches several hundred
meters, which roughly corresponds to the plume cross section
being tens of kilometers downwind of the source.
The dynamics of the plume simulations are quite general.
The buoyancy sources in the plume encompass the sensible heat
in the plume, the latent heat absorbed or released in plume
moisture processes, and the radioactive decay heating of the
plume by a radioactive pollutant species in the plume.
Buoyancy from chemical reactions could be easily included.
The atmospheric state in the simulations accepts atmospheric
wind, temperature, water vapor, liquid water, background
pollutant, turbulent eddy viscosity, and turbulent kinetic
energy as functions of height.
The turbulence is treated
-15-
with the sophisticated second-order closure model of
2
Stuhmiller , which allows the turbulent recirculation and
entrainment of the plume cross section to be treated in a
very natural way.
The model is validated against the Pasquill model3 and
the entrainment model of Richards 4 for idealized cases in
which these models apply, and for several cases from the
LAPPES5 field data for actual large power plant stacks.
Simulations are obtained for cases outside of the Pasquill
and entrainment models, and while no specific field data
for these cases exists, the behavior of the simulation
agrees with the physical changes imposed on the problems.
-16-
2.
LITERATURE REVIEW
The literature review in this work undertakes a broad
survey of plume modeling.
In the first section, existing
numerical plume models are discussed, along with the experimental data base that is available for the validation of
these detailed plume models.
The first section also includes
the research that has been done on computational and
experimental modeling of two-dimensional line vortex pairs.
It is important to include them since the results of such
work are very easily interpreted in the context of air
pollution problems.
In the second section, existing numerical
models of the planetary boundary layer are discussed.
Again,
these models are very easily extended to air pollution
problems
(with the inclusion of a pollutant transport equation
and pollutant source),
so it is important to include them in
the review.
2.1
Numerical Plume Models
A large number of plume models -have been developed that
Several recent reviews
are available as computer programs.
6-8
have reported dozens of such models, and it is important to
make a distinction regarding them.
A majority of the models
employ the Gaussian plume assumption;
as such, the computer
-17-
is simply being used to look up and present the standard
handbook calculations, with minor modifications in some
cases.
These are not"numerical.plume models" in the sense
that the primitive equations are not being integrated to
show the plume development, although computers are being
used.
Such models are not considered further here.
The
remaining models in the reviews are truly numerical plume
models, and they will be considered next, along with several
models that were reported elsewhere.
2.1.1
Three-Dimensional Models
The most sophisticated numerical plume models to date
have not yet attempted a second-order turbulence closure to
the fully three-dimensional flow field for non-passive pollutants.
Some of these features are found in each of the models
discussed here, but not all of them.
The notes of Rao 9 and
Nappo1 0 discuss the desirable features of three-dimensional
numerical plume models, and provide a good introduction to
future work that may be undertaken.
Donaldson's modeling 1 1 has concentrated on a second-order
turbulence closure for a three-dimensional planetary boundary
layer simulation with a passive pollutant.
Because the pol-
lutant is passive, and hence does not affect the flow field
or its turbulence, the turbulence closure only addresses PBL
-18-
turbulence, and is independent of the behavior of buoyant
plumes.
This is in contrast to the method in this work,
where the second-order closure is "tuned" to the development
of turbulent buoyant plumes, and is largely independent of
PBL turbulence development.
Lewellen's modeling
begins
with a second-order closure to the passive pollutant transport
equation, and then adopts the PBL flow field and turbulence
from Donaldson's model.
Only integrations of the pollutant
transport equation are needed in Lewellen's model because
of the adoption of a complete PBL solution.
model
13
Patankar's
of a deflected turbulent jet in three-dimensions also
uses a second-order closure model, but does not allow for
buoyancy and stratification, although it does allow for nonisotropic turbulent transports in the vertical and horizontal
directions.
A fundamentally different approach to three-
dimensional modeling is found in the Atmospheric Release Advisory Capability (ARAC) system.
15-22
A mass-consistent three-
dimensional wind field is interpolated from a small set of
local tower wind measurements and used to predict the advection of a passive pollutant.
Turbulent diffusion is modeled
with a zero equation model, although many other important
features such as rainout, wt and dry deposition, and surface
terrain have been added.
-19-
2.1.2
Two-Dimensional Models
Two two-dimensional numerical plume models have been
23
found in the literature. Henninger's model
solves continuity, momentum, energy, and moisture with a less-sophisticated
zero-equation turbulence closure, and with a more sophisticated
treatment of moisture.
For plumes in a wind, the model
chooses the mesh alignment shown in Fig. 3.3.2.1b of
Sec. 3.3.2, which is felt to be a less satisfactory choice
24
than that of the present work. Taft's model
is much closer
to the model in this work
, since it adopts the same mesh
alignment (see Fig. 3.3.2.1c in Sec. 3.3.2).
The principal
differences are that Taft's model employs a one-equation turbulence model, uses a more complex moisture model, and does
not make any attempt to describe ambient atmospheric
turbulence.
A number of two-dimensional numerical buoyant thermal
models have evolved in the literature of meteorology, usually
in support of efforts to parameterize the growth of rain
clouds.
The models have not been applied to air pollution
directly, but could be easily converted.
Lilly's model 2 5
seeks a self-preserving solution for the (dry) buoyant line
thermal, and as such, would only be applicable for the early
plume behavior when plume self-turbulence is dominating.
Johnson's model 2 6 is used to study fog clearing on runways
-20-
with helicopter downwash; while the moisture equations are
more complex than that in this work, the eddy viscosity is
assumed to be constant.
Ogura's model 2 7 of rain cloud dev-
elopment also assumes a constant eddy viscosity, while
Arnason's model
model
9
ignores eddy transports altogether.
Liu's
employs a stratification of atmospheric turbulence
into two constant eddy viscosity layers.
While the treatment
of turbulence in these models is verysimple,
it should be
emphasized that these models are focussed on precipitation
modeling, and they are likely to be helpful in the improvement
of the moisture model in this work.
A recent review of pre-
cipitation modeling is found in Cotton. 3 0
2.1.3
Experimental Studies
The field study that the model in this
ork is validated
against is the Large Power Plant Effluent Study (LAPPES).5
Complete field data for stack plumes from three mine-mouth
coal-fired plants are found in the four volumes of the study:
wind, temperature, and humidity profiles, plant operating
characteristics, and plume S0 2 concentration cross sections
are of the most interest in this work.
Cooling Tower Project
(CPCTP)
31
The Chalk Point
is also of interest to this
work since it provides cooling tower plume cross sections, but
plant operating data 3 2
was not available during this work.
-21-
The experimental laboratory studies that this work is
validated against are the papers of Tsang
33
and Richards.
4
The experiments study the behavior of two-dimensional line
thermals released in a water tank.
The ambient receiving
fluid in the tank is both laminar and unstratified, and the
thermals are fully turbulent.
2.2
Numerical Planetary Boundary Layer Modeling
A three-dimensional numerical model of the planetary
34-36
boundary layer has been reported by Deardorff
that
could easily be adapted to local air pollution studies,
although the expense is likely to be prohibitive.
The model
solves the complete set of primitive equations (with an
eighteen-equation model of turbulence) in a box that ranges
5 km on a side and 2 km deep.
The numerical experiments to
date have compared very favorably with several well-documented
planetary boundary laver field studies.
To applv the model to a sinqle source of pollutant, a
sinqle mesh cell could be initialized with sources of momentum, heat, moisture, pollutant, and turbulence.
To accommodate
this, a pollutant transport equation would have to be added,
and an additional three-equation model of turbulent pollutant
fluxes would need to be developed.
Time-dependent or steady-
state releases could be modeled in great detail in this way.
-22-
However, the model currently requires 15 seconds of CPU on
a CDC-7600 to simulate 1 second of flow in the atmosphere.
Also, the specification of boundary conditions on a threedimensional mesh with accurate time-dependent micrometeorological data would require a very elaborate reporting network.
Nonetheless, the model represents a more sophisticated and
potentially more accurate approach than the model in this
work.
-23-
3.
HYDRODYNAMIC MODEL DEVELOPMENT
3.1
Introduction
In order to model buoyant plumes in the atmosphere, the
equation set contained in the VARR-II computer code is reinterpreted and expanded.
A reinterpretation of the hydrody-
namic variables is necessary in order to satisfactorily account for the compressible nature of an atmosphere that is at
rest.
The euations are expanded in Sec. 3.2.1 to include
the transport of a pollutant and radioactive decay heating
by the pollutant, and in Sec. 3.2.2, where the transport of
water vapor, cloud liquid water, and the energy released or
absorbed during the phase changes of water substance
considered.
are
Since so many fundamental changes are made here
in reinterpreting the VARR-II equation set, this discussion
of the model development undertakes a derivation of the equations; for completeness it reiterates the important assumptions
ontained in the VARR-II code which were developed out-
side of this-work.
3.2
3.2.1
Hydrodynamic Model Equation Sets
Equations for Dry Atmospheres
The equations for a dry atmosphere are derived in this
-24-
section.
When the potential temperature is simply reinter-
preted as the virtual potential temperature, these equations
are applicable to moist plumes in moist atmospheres if none
of the moisture undergoes a change of phase, and if the turbulent diffusion coefficients of heat and moisture are equal.
A further discussion of virtual potential temperature is found
in Sec. 3.2.2.1.
3.2.1.1
Reference State Decomposition
As a starting point for the model development, consider
the three-dimensional compressible fluid mechanics equations,
,
where the six primitive variables
,
,
and u. are
physically measurable values of the fluctuating pressure,
density, temperature, and velocity, respectively:
Continuity Eq:
IL + -k-~
ax.
'
(3.1)
0
=
Momentum Eq:
aa~1(
xJtOj("p~i1
at(~i)+
ax.
p9i
2)
2
+11 ax 1-
(3.2)
Energy Eq:
a
Ot(~)'+
x
atax
(ijT)
J
=c
c
k
+
ax.
J~~~~
J
.adB
ax.
+
T
Tj
]R
J
1
C
c
at
(3.3)
.
-25-
Equation of State:
=
(3.4)
RdT
These equations have property values
depend upon temperature in general.
, cp, and k, which may
The energy equation has
neglected the kinetic energy in the fluid motions, and the
equation of state is that for an ideal dry gas.
The variations of temperature, pressure, and density in
a static atmosphere are usually "subtracted out" of these
equations in meteorological analyses by a reference state
decomposition.
That is, equations of motion for perturba-
tions about an adiabatic atmosphere are sought by decomposing
the primitive variables as
a departure
its value in
from the
an adiabatic
state at
+
=,atmosphere
rest
(function of
height only)
(3.5)
the value
of a primitive variabl
or, in terms of the notation in this work
P
PO
+ p
(3.6)
P
P
+ p
(3.7)
T -
T
+ T
(3.8)
+ u.1
(3.9)
O
.i
~1
o
The state of the dry, adiabatic atmosphere is found by
-26-
making the substitutions Eq. 3.6-Eq.
3.9 into Eq. 3.1-Eq. 3.4,
and setting the time derivatives and the perturbations p, p,
T, and ui to zero.
The continuity and energy equations be-
come trivial under this substitution.
The momentum equation
becomes the hydrostatic equation:
dp o
d-Z
-
Pog
(3.10)
The equation of state is simply
Po = PoRdTo
(3.11)
The First Law of Thermodynamics for an adiabatic process is
dQ =
= c dT
- dPo/P o
(3.12)
Dividing by a displacement dz gives
dpo
dz
dT
dz PoCB
p
dz
d7
(3.13)
and substitution of Eq. 3.13 into Eq. 3.10 gives rd, the lapse
rate of the dry adiabatic atmosphere:
dT
rd
o
dz
a
=
9
.7 6
C/km
(3.14)
cp
To this point the solution of the adiabatic atmosphere
has been presented.
position, Eq. 3.6-Eq.
Substituting the reference state decom3.9 into the equations of motion,
-27-
Eq. 3.1-Eq. 3.4, and using the results of the adiabatic atmosphere, Eq. 3.10 and Eq. 3.14, gives the equations of motion
for the perturbations:
Continuity Equation
u. a
au.
-- J=
ax.
+
+
p 01 ax.o
ka
_4
2T
!d
a
co, kp0 ~T
"o
0°
(3.15)
Momentum Equation:
au.
2
au.
ojax.P
Pgi +
-at
u.
Xa
(3.16)
Energy Equation:
aT
Po -t
+ pu
pu
Rd
aT
Rd
aT
aT
(3.17)
Equation of State:
po
To
So p T + pT
p0
P 0o
T
(3.18)
The fluid perturbations will generally be assumed to
be incompressible in the Boussinesq sense.
That is, changes
in fluid density are assumed to be produced only by temperature changes, and not by pressure fluctuations.
Neglecting
the pressure fluctations in the equation of state, and noting
that generally pT<<poTo , the equation of state becomes
P/po
-To
(3.19)
-28-
which is the familiar Boussinesq equation of state.
This
equation allows the buoyancy term (-pgi/p o ) in the momentum
equation (Eq. 3.16) to be similarly approximated.
The con-
tinuity equation (Eq. 3.15) becomes that of an incompressible
fluid, assuming that the fluid motions do not rapidly mix
37
deep layers of the fluid,
e.g., comparing length scales of
velocity and density:
Iu«
(
)-1<<
(3.20)
11
and 1 1 that the heat conduction term in Eq. 3.1 is a small
contribution to the divergence.
Making these approximations,
the equations for the perturbations may be written as
Continuity Eq.:
au.
I
x.
= 0
(3.21)
Momentum Eq:
2
ui
at
+
U.
Ui
a 1 ap
-
jaxjx.
_
1
~
P a xi
P
-
~- g
+u
g
io
+
__
x
1
(3.22)
Energy Eq:
2
aT +
at
aT
ax.
-1
P r
1o
-
P2
3TTo
a0
x.
(3.23)
-29-
Define the potential temperature, 0, as
6
T
3.24)
d/cp
1000mb)
Differentiating with respect to height finds that the adiabatic atmosphere has a lapse rate of potential temperature of
zero,
dO
dz
0_= 0
(3.25)
or that the potential temperature is a constant in an adiaErrors introduced by evaluating density
batic atmosphere.
with 8 instead of T are assumed to be small (this is investNeglecting the perturbation p with
igated in Sec. 3.3.4).
in Eq. 3.24, and approximating p
respect to p
as p(0o ) in
instead of T in the primitive equa-
Eq. 3.22, the use of
(Eq. 3.21-Eq. 3.23) gives
tions
Continuity Eq:
au.
a
-
(3.26)
0
Momentum Eq:
3u.
Ii + U
u.
1
p
j
aj
Po
P( )-p()
a p- P(ax)
g
+
32U.
x
2
(3.27)
.
Energy Eq:
a e + Uja
-=\Pt
J
j
a 0
~x.
J
(3.28)
-30-
The utility of the potential temperature formulation
is that strong variations of pressure and density with height
in the hydrostatic approximation of Eq. 3.10 are no longer
present in the primitive equations.
Initialization errors to
the hydrostatic state, if included in the primitive equations,
lead to strong transient fluid motions.38
The transients are
neatly avoided by this formulation.
To this point the fully three-dimensional fluid mechanics equations have been decomposed into an adiabatic reference state, and a flow field of perturbations about this
state.
A number of approximations have simplified the equa-
tions for the perturbations to those of a Boussinesq incompressible flow.
The equations need to be ensemble-averaged and
a turbulence closure formulated, and then the set must be
finite-differenced for computer solution.
3.2.1.2
Reynolds Decomposition and Closure
To model the effects of turbulence on the mean flow,
each primitive variable in the equation set is decomposed
into its time-averaged and fluctuating parts as
(the value of the
perturbation of
a primitive
variable
=
its ensembleaveraged
value
+
any fluctuations about
its ensembleaverage value
'3.29)
-31-
which is represented here by the decompositions
p p +
'
(3.30)
08
- + 8'
(3.31)
p -p + p'
(3.32)
u.+u.+ u.'
(3.33)
Under this transformation, by selectively ensemble-averaging
and subtracting the equations, and by making use of the continuity equation, the primitive equations become
Continuity Equations
a
uj/x.=
UJA
au
/
I AX
=
0
(3.34)
0
(3.35)
Momentum E:quations:
a u.
i +
at
au.
1
j
j
I
_
p(a0) a x
+ v
at
-1
-p(
, au.
a ui
+ u
+
u
.
·
aj +
a
o )a
i
p ( 0 ) ax.
P p()')-P ()
p(0 )
0
o
4
i
-
-\
P O /
I
+
a 2ax a
I
au. I
-
a
1
D X.
I
au iI
! 3ax
(e )
u')
(Vu
1
((
_ aX.
x.j
gi + va. 2ui
2
ax
I
(3.36:)
)
=
(3.37)
-32-
Energy Equations:
=
+ U
a- -
xj
vPr
(-
j33
at
(3.38)
j
+
UU
j x
0 )
Uj ax
Uj ax.
x
(u!e') = VPr
x.
x
(3.39
The set of ensemble-averaged equations (i.e., Eq. 3.34,
Eq. 3.36 and Eq. 3.38) suffer from the well-known closure
problem due to the generation of the uu' and u'
terms by
the non-linear advection terms in Eq. 3.27 and Eq. 3.28.
Equations 3.37 and 3.39 may be manipulated to produce transport
equations for these two new variables:
a u.
au.
= - u'u'a
i k xXk
Dt(Dtu!)
1 j
k
a xXk
production
terms
(u!uu)
z
turbulent
transport
term
j
a3
3 Xk
-1
+
a
(p'u)
po
1
o0
x.1 j
pia
o x. j
-
u! +
1
a
i
_(')
pressure
diffusion
terms
tendency
toward
isotropy
term
-33-
+
buoyant
production
terms
')
0 (gi uO' + gj
2
a
uu)
a
k
a u'
3u
-2v -
molecular
diffusion
terms
ax
xk
dissipation
term
I
k
(3.40)
D
D
ai
a30
(uiO') = -
-'
3x.
i
D--
-
]]
(u'uO')
a
3x. iJ
J
a
-- P~
0
+
+
u e' ax.1
]
Vo
a
e
xxi
gi 0'0'
00
2
(u!'
+V
X2
turbulent
transport
term
pressure
diffusion
term
-..tX,
,
1
production
terms
J
tendency
toward
isotropy
term
buoyant
production
term
molecular
diffusion
term
-34-
-2v
dissipation
1
j
j
term
(3.41)
A discussion of the individual terms noted in Eq. 3.40
and Eq. 3.41 can be found elsewhere. 11
These equations were
closed by Stuhmiller2 and the results are listed here for
completeness.
In Eq. 3.40, the tendency toward isotropy term
is neglected, because the turbulence is assumed to be homogeneous, and the molecular diffusion term is neglected because
the flow is expected to be highly turbulent.
The buoyant pro-
duction term is also neglected, mainly in order to see how well
the turbulence model can do without it, since it was neglected
in Stuhmiller's turbulence model.
It is found that the
incorporation of this term would probably aid the model in
reproducing the buoyant line-thermal results (see Sec. 5.2.2).
By further making the assumption that the average flow is twodimensional in the y-z axes of Fig. 3.1, the following closure
is made for the trace of Eq. 3.40, which is the turbulence
kinetic energy, q, q
uu,
2
V)
=
Dt
+ay
+
1a
V
2 + -z
+
2
Y
P(
ycY ay/
+(aW
+ra-
)
_
-
F az a(3.42)
4aq2G-1
4az c
-35-
Figure 3.1
Flow Field Orientation
The flow field of Eqs. 3.42-3.47 is time-dependent and
two-dimensional in the y-z axes. The relationship of the timedependence to the (downwind) x-axis is discussed in Sec. 3.3.3.
VERTICAL
x'-3
l
Z
,
CROSSWIND
X
y
-z',Z V
U
=-ZC
DOWNWIND
-36-
The off-diagonal terms of the Reynolds stress tensor are related to a scalar eddy viscosity, a, where u!u
1
=
+ 3u
IJ
a3 xj 3xi/
and T has the following transport equation:
Da
=
Dt
+
(av
2
q
2y
3a
/a
q
ay
aw2
+
-
+
qa3a
y
+(aW)
Y
2-q
~
% z -
lz
a a _q
a
3ay
a
z
az
(3.43)
Finally, the turbulent fluxes of heat in Eq. 3.41 are related
to the turbulent momentum fluxes through a reciprocal turbulent
Prandtl number,
,T'which
is specified along with the three
other turbulence constants a, r, and rl.
With this turbulence
closure, the continuity, momentum, and energy equations become,
in a two-dimensional flow
3v
3w
3 + Lw =0
ay
az
at+ay(V)a3
aw +
3t
3
ay
(vw)+
a )+
vw
a
at
2t
p( )
ayl (00) a
+
-0
(w2)=
P
=
60
p
z
(3.44)
+ a+
(
-(
awljpO)
)
a
aw
a.3
a a z
ay+a(a
ayY ) ++(
(3.45
:)-p(+O)
gy
gz
)
( ( 0) (6, )gz
(a)- P(3
(3.46)
-37-
at
+
atev)+
aya
(Ow)=ay(
ay)+
y-azTaz
-(T(a)
(3'473
(3.47)
With an internal energy variable, I, defined as I-Cp , equations 3.42-3.47 are solved by the VARR-II code.
Additional
pollutant and moisture transport equations are discussed in
the next two sections, and possible modifications to these
equations are discussed in section 6.2.
3.2.1.2
Pollutant Species Transport Equation
A transport equation for a pollutant species density,
is added to the set of Eqs. 3.42-3.47.
X
The pollutant is
assumed to be a neutrally buoyant, passive species, although
it may be contained in a buoyant stream of effluent.
The
assumption that the species is neutrally buoyant could be relaxed, but the model is felt to be useful in modeling most
dilute pollutants in its present form.
The turbulent diffu-
sion of the pollutant is related to the eddy viscosity of
momentum by a reciprocal turbulent Schmidt number, 7X.
The
transport equation may be written down as
[substantial derivative]
of X
j =
urbulent transporti
of X
-
-
1
rate of
destruction
(3of
48)
(3.48)
-38-
which is represented here as
a
w
+
t
z()
X
2y
(3.49)
i=l
in the notation of Fig. 3.1.
The destruction of X is assumed to be by radioactive
decay into any of N decay channels, so that the rate of destruction of
is the product of X and the sum of its radio(i)
active decay constants
AX
,
in Eq. 3.49.
This formulation
makes no account of sources of the pollutant species through
decay of radioactive precursors.
It also ignores chemical
reactions which could alter the pollutant concentration.
How-
ever, the extension of the model to include these effects is
straightforward.
3.2.1.4
Radioactive Decay Heating
The thermal energy released by radioactive decay of the
pollutant is added to the specific internal energy of the fluid.
Pollutants may decay by any one of N different decay channels
with decay constant
(i)
and
energy E(i)..
andXE.
A
fraction F(i)
A fraction
Fx
of the energy is deposited within the plume, yielding an
energy release rate of
-39-
10
(CP ae)
radioactive
4.151x10
PW olx /X
BTU-atoms
X
MeV-lb -moleE,,dii)A)
i=
(3.50)
where WmolX is the molecular weight of X in lbm/lbm-mole.
Daughter radiations have been ignored in this formulation, but
could be included with their own transport equation.
Similarly,
alterations of the energy balance caused by chemical reactions
has not been treated in this work, but would be easy to address
in extensions of this work.
3.2.2
Moist Equations
The inclusion of moisture is considered in this section
with the purpose of pointing out the assumptions that allow
the equations to be formulated with the concept of virtual
potential temperature, in addition to two other moisture variables.
The assumptions that are made in this section are
important--the moisture model is not meant to be perfectly
general; it is expected to do poorly when these assumptions are
not valid.
-40-
3.2.2.1
Reference State Decomposition
Atmospheric moisture is assumed to be in either the liquid
or vapor phases.
The amount of vapor is described by the vapor
density moisture variable, Pvap, and the amount of cloud liquid
water is described by the liquid density moisture variable,
liq'
Transport equations for these two.variables are written that
take note of the turbulent transports of vapor and liquid, and
the processes of evaporation and condensation that cause the
interchange of vapor and liquid.
First, however, the effect of
moisture on the buoyancy of a parcel of air is developed and
applied to the description of a hydrostatic reference state.
The density of a parcel of moist air is the sum of the
dry air, vapor, and liquid densities:
P
=
yap +
dry
liq
(3.51)
In this work the contribution to the density of the typically
small amount of cloud liquid water is ignored,
(there is usu-
ally no liquid water present in the simulations, and when it
is present, it is typically less than 1% of the mass of the
fluid), so that the concept of virtual potential temperature
can be explored.
Dropping the
perfect gas law to
dry +
RdT
d
dry and
vap=_=
Rv T
liq term and applying the
vap yields:
dryvap)
R dv
T
(3.52)
_
RdT
dv v
-41-
where p is the total pressure, mvap and mdry are molecular
weights and the virtual temperature, T v , is
p
mdryPvap/m
.. y...
vap .dry
1+
T=
(3.53)
vap/
v
dry
It is very important to note in Eq. 3.52 that the virtual tem-perature is a fictitious temperature that is used in the dry
Gen-
gas equation of state to give the density of moist air.
erally the virtual temperature is no more than a few degrees
higher than the thermodynamic temperature for typical atmospheric conditions.
Following the development in Sec. 3.2.1.1, the variations
of virtual temperature, pressure, and density of a static atmosphere are "subtracted out" by making a reference state
decomposition:
its value in a uni-
the value of
a primitive
formly moist adiabatic+
=
atmosphere (function
of height only)
variable
I
(3.54)
Or, in the notation of this work:
P
PO + p
(3.55)
Po +
(3.56)
T v + T vo + T v
(3.57)
+ u.1
(3.58)
-
+
u.
1
-42-
the only difference here to the reference state decomposition
of Eqs. 3.6-3.9 is in the use of the (fictitious) virtual
temperature in order to allow the use of an equation of state
that is analogous to Eq. 3.4:
p =
P/RdTv
(3.59)
Substituting Eqs. 3.55-3.58 into the primitive equation set
(Eqs. 3.1-3.3 and Eq. 3.59), and setting the time derivatives
and perturbations to zero yields the state of the moist adiabatic atmosphere.
The continuity and energy equations are
trivial (as before), and the momentum equation becomes the
moist hydrostatic equation:
dpo
=
z
Pog
(3.60)
The equation of state is simply
pP0
=o
PRT
PoRdTvo
(3.61)
The first Law of Thermodynamics for an unsaturated adiabatic
process in this atmosphere is
dQ = 0 = cmoist dT
p
vo
- dpo/p
o
o
°
(3.62)
Approximating the heat capacity for a moist gas, cmoist, as that
P
of a dry gas, cp dividing by dz and substituting Eq. 3.62 into
3.60 yields an approximate lapse rate for a moist, unsaturated
atmosphere which is the same as that for a dry adiabatic
atmosphere:
-43-
-dT
g = 9.76 0 C/km
vo
dz
(3.63)
C
To this point the resting state of a moist adiabatic
atmosphere has been presented.
The neglect of the effect of
the liquid water on the total density has allowed the treatment of moisture to duplicate the dry atmosphere equations
after the transformation of temperature to virtual temperaThe equations for the perturbations are identical to
ture.
those of the dry atmosphere developed in Sec. 3.2.1.1, except
that temperature is replaced by virtual temperature, and a
latent heat release term is included:
Continuity Eq.
au.
I
x.
o
=
(3.64)
Momentum Eq.
3u.
. 1
at
au.
U 1
T
_
jax. j
P
ax.
1
T
i
vo
+
o
P
o
a 2u.
(3.65)
ax 2(3.65)
Energy Eq.
aT
3T
at+U
l
= vPr 1
2T
vap
J
Dp
phase
(3.66)
p
The latent heat release term is considered in Sec. 3.2.2.4.
Define the virtual potential temperature,
v,
as
-44-
v
R
T= (100)
mo1000ist
/cip
d
(3.67)
moist
is essentially equal to c . DifferAgain assume that cp
p
p
entiating with respect to height finds that the moist unsaturated adiabatic atmosphere has a lapse of virtual potential
temperature that vanishes:
de
vo
dz vo
(3.68)
O
The result here is that the virtual potential temperature is
a constant in the reference state.
Neglecting the perturbation pressure, p, with respect
to p
in Eq. 3.56, the use of
equations
(Eq. 3.64-Eq.
v instead of T
in the primitive
366) gives
Continuity Eq:
a u.
DXI.= 0
(3.69)
3x.
Momentum Eq:
aui
t+ u.
9t
a Ui
a
p(ev )-p(
-1
p(v)
Dx.
3
PO
DX.
1vo
e vo
gi x+
32
2
X
(3.70)
Energy Eq:
evv
t
1
u
j
I
v
2
-
X~ DPvp
D
L
pc
~~~
Dvap
Dt
phase
(3.71)
-45-
The result here is the same as in Sec. 3.2.1.1:
the
strong variation of pressure with height is no longer present
This formulation is common (al-
in the primitive equations.
though in slightly different forms) among papers in meteorology.
Transport equations may be written down for the water
vapor and liquid water densities according to the conservation
scheme:
Eulerian time
Diffusion of
rate of change _ vapor or
+
liquid
of vapor or
liquid
or, in the notation of this work:(372)
Gain or loss of
vapor or liquid
due to phase
changes
or, in the notation of this work:
vapp
Pvaavap
at
a P
at
u
].
-Y~2ap
= VSy -1
x.
C vap
]
a Plq_
+ U 3-lliq
3x.
J
a
2P
ax
+ _apDPvap)
2
xi
_VS
a 2l
liq
3
] ~x]~~~~~~~
/ phase
(3.73)
DPvap)
Dt / phase
(3.74)
Dt
X)
where the gain or loss of vapor due to phase changes,
(D/vap/Dt)phase, identically shows up as a loss or gain of
liquid, and Schmidt numbers that describe the molecular diffusion of vapor and liquid are introduced, respectively.
The
terminal fall velocities of the liquid water droplets are ignored.
The(DPvap)
term is discussed in Sec. 3.2.2.3.
phase
Dt
-46-
Note that any constant background (ambient atmospheric) value
of Pvap and Pliq trivially satisfied these equations, so that
no new information would be brought into the specification of
the reference state by decomposing the variables in these transport equations.
That is, Pvap and Pliq do not have a reference
state "subtracted away" from them, unlike the other primitive
variables
3.2.2.2
,
v, and p.
Reynolds Decomposition and Closure
A Reynolds decomposition of the primitive equations is
made as in Sec. 3.2.1.2.
Each primitive variable in the equ-
ation set is decomposed into its ensemble-averaged and fluctuating parts:
p
P + p'
(3.75)
e + 8 + 8'
(3.76)
P
P'
(3.77)
+ u
(3.78)
v
p+
uj +0
Pvap
Pvap + Pvap
(3.79)
Pliq +
Pliq + Piiq
(3.80)
By selectively ensemble-averaging and subtracting the equations,
and by making use of the continuity equation, the primitive
equations yield the following relationships:
Continuity Eq:
u.
Dx3=X
x.
j
(3.81)
-47-
Momentum Eq:
3u.i
a t
u.i
j Xu
-ix
IPvo
+
)
'V
a
3E
1
3 2u.
(.vl-P6)
p
a3
a xiI3x.
-1
I'_vo
gi + Va-
a]X
p(e vo )
(u!U!)
(3.82)
Energy Eq:
aeVv
t
+ u.
v
a.
-
vPr
-1 a
(D
3X x (Uj
) - p(- v ) c
j1.v
laX2
3xi
L
,a
2
vap
Dt /phase
P
j
(3.83)
and the transport equations for moisture, Eq. 3.73 and Eq. 3.74
yield
Vapor Eq:
a
3at Pvap
+ u.
3 /°vap
j
=
x.
j
-
--1
UScvap
2- a2
aPvap
3(P
_
a
u)
x
a
+
x
phase
Dt
(3.8)
(3.84)
Liquid Eq:
aT
at
liq
iq
+ uj
1i
- Plq
=liq
3i
1
-
J
-2
219=
ja x.13x
ax
x.
Pi s'U ;
iq
i
phase
-Dt
J
(3.85)
Rather than providing the full equations for the correlated
and develapoped
uP' q' the
ujO',
fluctuations uiu
j,is
in turbulence
Sec 3.2.1.iq
from that
closure
extended
simply u'vap
closure is simply extended from that developed in Sec. 3.2.1.2.
The closed set of equations in two-dimensions is,
in the
-48-
notation of Fig. 3.1
Continuity Eq:
aw
az
av
ay
0
(3.86)
Momentum Eqs:
Dv
-1
Dt
p(8vo)
+
(
y
a ) +
y
3
a
(a
a
V)
(3.87)
Dt
p(o
Vo
-p
p ()
-1
Dw
)
(evo)
aya
9z +
P evo
aw
a
aw
aY
(°a)
Energy Eq:
a
Dt (Cpav)
ce
(YTa
))
(cPa
(C v )
a z )-
a
) +
a y
az (YT
L
P(
v )
fDPvap)
phase
Dt
(3.88)
Vapor Eq:
-
a
Dt Pvap
y
D
a
(Yv
v
(y
a az (v
ay)+
a
Papz( Dtap)
phase
(3.89)
Liquid Eq:
ig )
a (YLa
y
3
y 'La
D
Dt Piq
a
+ z(
L
li)
L CYa
Eddy Viscosity Eq:
D
_
Dt -q
2
/
2av2
2
l2 3 V
a y)
z
Dt
aphase
phase
Dt
(3.90)
-49-
a (I
a
r1C a 3 q
-a)+(z
a ff-fz -z
iF a-y-1:4 y (Y 3 --"~ z
a 3y
+ra(( q~~~~
(3.91)
Turbulence Kinetic Energy Eq:
1
(•a
av +w+ 1 (2-V
Dq = 2 (()
Dt
r
+
+
a-q
~C--y y
a
z
) +w2(+--.) 2)- - 4qC
(3.92)
ca
Pollutant Eq:
N
Dt
ay (
ay)
+
(-XE a
)
(0
X(
(3.93)
where reciprocal turbulent Prandtl and Schmidt numbers have
been introduced, and are assumed to be constants.
3.2.2.3
Equilibrium Cloud Microphysics Model
The cloud microphysics model simply assumes that water
vapor and liquid are always in equilibrium.
Further, the
surface tension of the liquid droplets is ignored.
That is,
phase equilibrium over a flat surface of water is assumed to
exist.
A phase diagram that illustrates this equilibrium is
-50-
sketched in Fig. 3.2.2.3.1.
The liquid-vapor equilibrium
curve above 2730K is the locus of points that the saturation
vapor pressure, esat (T), may take.
The vapor density,
vap '
in the presence of liquid water would be esat(T)/RvapT.
If
there is no liquid available to evaporate, then the vapor density may be less than this saturation value.
Below 273°K the
subcooled liquid-vapor equilibrium (dashed line) is obeyed.
ice formation is allowed.
No
The entire liquid-vapor equilibrium
curve is given by Magnus' formula:
gl0esat
2937.4 - 4.9283 logl 0 T + 23.5518
(3.94)
The (Dt
/ phase term of Eq. 3.89 and Eq. 3.90 is
simply adjusted to make the liquid and vapor coexist.
of the moisture model is illustrated in Fig. 3.2.2.3.2.
The logic
Liquid
and vapor are advected and diffused in an initial calculation
for each computer cell.
This generally results in a non-equi-
librium moisture state in the cell, so the cell is allowed to
evaporate or condense water in order to restore the equilibrium.
The amount of evaporation or condensation in each cell is noted
in order to provide the latent heat release term in the energy
equation.
3.2.2.4
Latent Heat Source Term
The latent heat source term is calculated in each cell
-51-
II
In
k
.1-
Hi
r-4
rH
·
vapor
en by
nula
4
subcoole,
including
vap
6.11
line
r
N
nt
VAi UK
l
I
273
TEMPERATURE, T
Fig. 3.2.2.3.1
( K)
Phase Diagram for Water Substance
-52-
transport vapor and liquid using
updated velocities
liquid evaporates
completely
vapor condenses
(DfDap)
Dpva
s
Dt ~hase
Dpt
ar
Dt 7phase
fsat - pvap
fsat -
vap
Dt
Fig.
3.2.2.3.2
Dt
Dt
Logic Diagram for the Equilibrium Moisture
Calculation in a Single Cell during a Single
Timestep
-53-
at every step depending on whether evaporation or condensation
takes place.
The latent heat release term is calculated as
lb
Latent Heat Release
BTU
) I=
sec
L
Dpvap
(By (Dt
Dt
phase
phase
(3.95)
where the latent heat of vaporization, L, is assumed to be a
constant, 1075 BTU/lbm.
In
The(
yap
Dt
is found in the logic
a
phase
diagram of Fig. 3.2.2.3.2.
3.3
3.3.1
Model Solution Methodology
The VARR-II Fluid Mechanics Algorithm
The VARR-II computer code 4 0 is the starting point for
the model development methodology in this work.
In its ori-
ginal form, the VARR-II code solves the two-dimensional timedependent turbulent fluid mechanics equations of continuity,
momentum, and energy for a Boussinesq fluid.
(The Boussinesq
approximation to the momentum equation is considered in
Sec. 3.2.1.1.)
Two closure variables, the eddy viscosity, a,
and the turbulence kinetic energy, q, are also calculated from
their own transport equations.
The original VARR-II computer
code is quite flexible in the choice of boundary conditions,
allowing no-slip, free-slip, continuative inflow/outflow, or
prescribed inflow/outflow boundaries.
-54-
The VARR-II fluid mechanics algorithm is the Simplified
Marker and Cell (SMAC) method.
41
The computer mesh for this
method is Eulerian in either Cartesian or cylindrical geometry,
and the primitive variables are solved directly, with no transformation to vorticity-stream function variables.
The algorithm
divides naturally into two sections during each time step:
In
the first section the velocity field is updated using the previous velocity and pressure fields with mixed central and donorcell differencing42 of the equations.
These velocities gener-
ally do not satisfy the continuity equation, so in a second
section a pressure iteration adjusts these velocities until
they satisfy continuity.
Once the divergence-free updated
velocity field is known,the energy and turbulence transport
equations are updated, completing the calculational cycle
of the time step.
The basic SMAC fluid mechanics algorithm has not been
modified in this work.
Pollutant and moisture transport equ-
ations have been added to the equation set, and they are updated in the same manner as the energy and turbulence variables, using the divergence-free updated velocity field.
The stability of the method for problems of an atmospheric
scale is considered in Sec. 5.2.1.
3.3.2
Orientation of the Computer Mesh
The optimal orientation of the two-dimensional computer
-55-
solution mesh is discussed here.
Consider the representative
The plume has bent
three-dimensional plume in Fig. 3.3.2.1a.
over in the imposed (one-dimensional) wind field, and the
plume boundaries monotonically expand as the plume proceeds
downwind.
The most natural possibilities of orienting a two-
dimensional solution mesh on this flow are:
(1) to align the
mesh parallel to the wind and through the center of the plume,
as in Fig. 3.3.2.lb, or (21 to align the mesh perpendicular
to the flow, as in Fig. 3.3.2.1c.
The advantages of the "crosswind" alignment of
Fig. 3.3.2.1c over the "downwind" alignment of Fig. 3.3.2.1b are
immediately apparent.
In the crosswind alignment a three-
dimensional simulation results since in the downwind Lagrangian
translation of the computational mesh the time variable becomes
a surrogate for the downwind position x, where
x
/
U(z(t)dt.
The downwind alignment
is appropriate only for cases of line-source plumes--in which
internal recirculation and entrainment will be of secondary
importance to buoyant plume rise and atmospheric turbulent entrainment.
Further, the crosswind alignment can take advantage
of the centerline symmetry of the turbulent vortex pair to reduce the total mesh area by a factor of two, while the downwind
-56-
z
X
Y
Fig. 3.3.2. la Bent-Over Buoyant Plume
with Ambient Thermal Stratification.
-57-
Fig. 3.3.2.1b
Mesh Alignment Appropriate for a Line Source Release
1 km
z
I
5 km
X
Fig. 3.3.2.1c
Mesh Alignment Appropriate for a Point Source Release
'7
I
1,--
symmetry line
1 km
-58-
alignment scheme needs an extraordinarily long x-axis to
model the same plume.
Overall, the crosswind alignment scheme
is about five times smaller than the downwind scheme.
The vel-
ocity field in the crosswind alignment is that of a two-dimensional turbulent vortex, which typically exhibits strong shearing
and entrainment of fluid.
The velocity field in the downwind
alignment is that of a two-dimensional turbulent deflected jet,
which over most of the flow field exhibits a much smaller amount
of shearing and entrainment.
Clearly, the crosswind alignment
scheme is expected to simulate the more important features of
the flow.
The singular disadvantage of the crosswind alignment
scheme is that it cannot explicitly calculate the shearproduced turbulence of the mean wind field, since the mean
wind has no component in the y-z plane.
The resolution of
this problem is discussed in Sec. 4.3.3.
3.3.3
Downwind Advection of the Mesh
From the discussion in Sec. 3.3.2, the computer solution
mesh is aligned perpendicular to the wind.
The time evolution
of the flow field of the plume cross section is drawn in
Fig. 3.3.3.1.
The choice of an appropriate downwind advection
velocity of the computer mesh is needed in order to reconstruct
-59-
t 3 )t
2
>tl
z
X2 = -
wind
) t2-1
-
t=t
-Y
Fig. 3.3.3.1 Reconstruction of the Three-dimensional
Plume. Wind vectors as a function of height are shown.
-60-
the full steady state plume, i.e., the time of the computer
simulation must be related to a downwind distance.
The choice
is difficult because the wind profile dictates that fluid elements at different heights will advect downwind at different
rates.
A simple approximation is that the advection velocity
should be equal to the "pollutant averaged" wind speed:
Ax
u(z)
X (y,z)dydz
(3.96)
f X (y,z)dyez
The finite difference form of Eq. 3.96 is written in Fig. 3.3.3.1.
The calculation of this quantity is performed in the "statistics
package" of Sec. 3.3.7.
A further refinement of the solution
scheme is discussed in Sec. 6.2.1.
In practice, for plumes that are released from tall
stacks, the amount of wind shear that the plume encounters is
ordinarily moderate and does not greatly alter the plume
behavior.
3.3.4
Property Data
The original VARR-II computer code allows for quadratic
fitting of air property data versus temperature.
In view of
the fact that potential temperature is substituted for
.
.
. .
-61-
temperature in moist simulations, the scheme of fitting property data to temperature must be examined.
The air property
data to be fitted includes density, specific internal energy,
dynamic viscosity, thermal conductivity, and heat capacity at
constant pressure.
The coefficients of the quadratic fits for
dry air data43 are listed in Table 3.3.4.1, along with the
quadratic form that they are used in.
erty value of the substitution of
The effect on the prop-
or
for T is considered
next.
The use of T or T v in the perfect gas law yields, by
definition, the correct density of a dry or moist parcel of
air, respectively.
A quadratic fit of the perfect gas law
over a small temperature range of interest would yield essentially exact results for the density as well.
of densities with
The calculation
or 8y substituted into the formula for T is
also appropriate because
or
v
vary from T by very little
compared to the absolute temperature.
Recall that
or
is used in the problem formulation to eliminate the compressible nature of the hydrostatic atmosphere.
The relevant density
variations in the momentum equation are the relative density
variations, and the criteria for the use of, say
v for T
is that
p.(T)-p(T)
p(T )
. P0..) -p (.vo
P( vo)
(3.97)
-62-
oI
0rt
u
r-
o
m
tn
00
0
ogl
00
0
N
o
I-r
m
Iz
o
Io
Io
o
L
0
I
0
ao
X
0(a] ~Q
~4
0
4
0
r-
x
r-I
4-)
U)
rl
H-
r4
o
0
.-
I
U
U)
a)
-H
I
o
0
s-i
(a
x
-P
0t
0
I
o
o
0
E-,
x
rlx
x
x
-
PI
0
0
0
r4
0*
U)
~j
Cl
P
4J
4) £
44
N
r:;
H
a)
5
'0
I
E
Q)
N
EU
m
H
-H
mt
H
E-4
II
.>1
4-)
I
>1
1
a
)
04
0
U)
0f
-HiI
4
.,,
Hd
P
cd
>1
4-)
U -H
ul
r
U) -4
>1
4-)
I-,-
0
(a
U
O
( do
En
O.
tQ
U)
a
4-JU)
a
s
< a)
*HU )
C
rTJ
a
H
?2
;
U
(N
Cl
r
Ln
H
0
rU
>iU) CE
2
-0
4
O
(d
(1) -H 0
u
0
O
U
a)
0
0-
a4
10
H
-63-
with a similar condition for
in dry simulations.
This rela-
tion holds with about four percent accuracy for the most
extreme cases encountered in this work.
The specific internal energy is originally fitted versus
T.
Again, the fact that
or 8v is close to T compared to the
absolute temperature allows them to be interchanged without
significant error.
The specific internal energy is accurate
to about 4 percent under this substitution.
The values of dynamic viscosity and thermal conductivity
are important only if the flow becomes laminar.
None of the
simulations in this work are expected to encounter regions of
laminar flow, so the fitted values of molecular viscosity and
thermal conductivity are unimportant.
The specific heat varies slowly with temperature, and the
substitution-of e or
ev
for T results in only a 0.02 percent
error for typical cases.
The necessary property data for equilibrium conditions of
water vapor and cloud liquid water are included in Secs. 3.2.2.3
and 3.2.2.4.
The inclusion of water in the simulations is
assumed to have a negligible effect on the property data of
the air-water mixture, except for the density, which is corrected through the use of the virtual temperature.
-64-
3.3.5
Mesh Initialization and Bo'undary Conditions
3.3.5.1
Input Profiles
Seven vertical profiles are required for a simulation.
Five of the profiles serve to specify the boundary conditions
on the computer mesh, one profile (the mean wind wpeed) is
needed by the statistics package, and one profile (the hydrostatic pressure) is needed by the equilibrium moisture thermodynamics model.
3.3.5.1.
The required profiles are listed in Table
Each vertical profile consists of a set of values
that are representative of the cell-centered temperature, ' wind
speed, etc.
The number of values is obviously equal to the
number of fluid cells in the z-direction.
The extension of
the model to time-dependent vertical profiles is considered
in Sec. 6.2.2.
3.3.5.2
Boundary Conditions
Boundary conditions must be specified for each of eight
variables on the four walls of the computer mesh.
The walls
of the computer mesh are numbered in Fig. 3.3.5.1.
Wall #1
is in the plume centerline with the real computer simulation
to its left.
wall.
For this purpose, wall #1 is a free-slip solid
Wall #4 always represents the earth, and is specified
to be a no-slip wall.
The earth is assumed to be a perfect
-65-
Table 3.3.5.1
Required Input Profiles
Atmospheric Profile
virtual potential temperature
water vapor density
cloud liquid water density
eddy viscosity
turbulence kinetic energy
Units
OF
3
lb/ft
lb /ft3
3m
lbm/ft
ft2 /sec
ft2/sec2
mean wind speedA
ft/sec
hydrostatic pressureB
millibars
A.
The mean wind speed is required by the statistics package
of Sec. 3.3.7.
B.
The hydrostatic pressure is required by the equilibrium
moisture thermodynamics model of Sec. 3.2.2.3.
-66-
Z (Vertical)
symmetry line
EARTH
Fig. 3.3.5.1
Wall Numbering Scheme
.
(crosswlnc)
-67-
reflector of pollutant and humidity in this work.
This
assumption could be easily modified to account for deposition
of pollutant, sources of humidity, etc., for any case of
specific interest.
Walls #2 and.#3 are chosen to be sufficient-
ly far away from the plume so that negligible error is introduced
in making them solid and free-slip.
In practice, the plumes
rise toward wall #3 and begin to deflect when their 10% boundary intersects the wall.
This serves as a rough criterion on
when to stop the computer simulation.
A summary of the boundary conditions is found in Table
3.5.5.2.
The solid-wall, no-slip and free-slip conditions are
found in the specification of the two velocity components, v
and w.
The reflective conditions are due to the "perfect ref-
lecting walls" assumption; they are foregone at wall #2 for the
five variables that are known as functions of height.
3.3.5.3
Mesh Initialization
The entire computer mesh in Fig. 3.3.5.2 is first initialized with the known atmospheric profiles of virtual potential
temperature, eddy viscosity, turbulence kinetic energy, water
vapor density, and cloud liquid water content.
The entire
mesh is initialized with. a single background value of pollutant,
and the velocity field is initialized to be at rest.
The plume
cells in the figure are then initialized by volume-averaging
-68-
I'll
rd
:R:
x
0
r
II
A
Ha
l-4
H,
Cd4
H
HCd
0
-I
U)
U)
r4
*
tn
*
0
-
uu
m
II
r--
(a
V
0
>
-H
O
>
)
4.
Cd
>1
>1-·1.
*H
U)
U
U1
4>1
~~k
a~
U > a
O)
-.
U
o
>
*H
O
U
H
4
0-I
),O
H
*w
k
, Hd,.-I Cd
Cd
,
>
Cd
-rd
P ·
H
a)
d
0
H
*H
-rH
.C
4·
~).,)-4
H
Od
H
oO
tr7
>
4-
0
Cd
3
r
4
U)
>1
..
U
H
II
11
H
,1
0
0
>)
>
#
U
H
4-1
O
o
r-4
(N
)
.)a
@
·
,.l
Ci
O
>
4.)
0
04
Q
5-I
4P
Cd
U
U zH
,-i
pI
I0
V
U)
-H-I
rH
Z:
)
4I
)
4t
5I - U
I
r4
k C
-69-
the plume sources of energy, pollutant, and moisture over those
cells, using mean wind speed at that height to define the depth
of the cells swept out in one second.
The initial eddy visco-
sity and turbulence kinetic energy in the plume cells are set
to about 100 times that of the surrounding atmosphere--in practice, the plume turbulence values very quickly relax into values
that are consistent with the flow field.
No initial volume-
averaged momentum is given to the plume cells.
Instead of this,
an effective stack height increment due to momentum is added
to the actual stack height in specifying the location of the
center of the plume cells.
3.3.6
Mesh Coarsening Capability
Model programming has been undertaken to allow the mesh
spacing to be doubled periodically during the simulations,
while keeping the same number of fluid cells on the whole computer mesh.
The motivation for this is the desire to keep the
growing plume cross section away from the unphysical (solid
wall) top and right mesh boundaries.
When the simulation is
"coarsened," the mesh spacing doubles, which reduces the plume
cross section by a factor of four.
The calculation is restarted,
and the simulation proceeds on a mesh that has four times the
area of the old mesh, but the same number of fluid cells.
The coarsening procedure is outlined in Fig. 3.3.6.1.
In
-70-
-~"
~~~~~--
O
t
I
(a)
Step One - Four Cell Averaging
(b)
Step Two - Initialization of New Cells
Figure 3.3.6.1
Mesh Coarsening Procedure Steps
l
DZ
DZ.
-71-
a first step
at a time.
(a) the entire mesh is swept over, four cells
Note that the number of cells vertically or hor-
izontally must be even in order to do this.
The fluid vari-
ables in these four cells are averaged in the following way:
the cell specific internal energy, momenta, and turbulence
kinetic energy are mass-averaged over the four cells, since
these variables are defined on a per unit mass of air basis.
The cell pollutant, eddy viscosity, and moisture variables are
simply averaged over the four cells, since these variables are
not defined on a per unit mass of air basis.
The cell pressure
is set to zero, which conforms with the usual starting guess
procedures in running VARR-II.
The average cell made up from
these four cells is now stored in its proper place on the larger
mesh, which is half of the distance to the origin vertically
and horizontally.
When the entire mesh has been swept, four
cells at a time, the old mesh has now been relocated in the
lower left corner, and is one-fourth of its old size.
In a second step (b) the remaining three-quarters of the
mesh needs to be initialized.
row in ascending order,
This "new" area is swept row-by-
The velocity field is assumed to be
initially at rest, and the pressure field is initially set to
zero.
The remaining atmospheric state variables are all spec-
ified from a master library of profiles.
When the "new" area
has been initialized, the calculation is restarted with the
-72-
vertical and horizontal mesh spacings doubled.
The computer mesh may be coarsened up to five times during
a simulation--this would result in a final mesh that is
5
5
2 x 2 = 1024 times as large as the original mesh. The five
times are user specified, and need not take place at regular
intervals.
3.3.7
Plume Statistics Package
At regular intervals specified by the user, the program
calls on a statistics package to calculate a number of important plume statistics without printing out the data of the
entire computer mesh.
The quantities that are reported by the
statistics package are listed in Table 3.3.7.1.
The average
plume advection velocity is the feature discussed in Sec. 3.3.3,
and is defined in Eq. 3.96.
------
-73-
Table 3.3.7.1
Data Reported by the Plume Statistics Package
Quantity
Units
Time of Simulation
sec
Total Number of Problem Iterations
(none)
Current Number of Pressure Iterations
(none)
Current Time Step Size
sec
Center Height of Pollutant Field
ft
Total Specific Internal Energy on Mesh
BTU
Average Downwind Advection Velocity
ft/sec
Plume Downwind Distance
ft
-74-
4.
DESCRIPTION OF ATMOSPHERIC TURBULENCE
4.1
Introduction
This
chapter describes in detail how atmospheric turbu-
lence is represented in the model.
with the knowledge
(e.g.,
The description begins
from a set of measurements)
common atmospheric variables as functions of height:
of the
the set
includes the wind speed and direction, virtual potential temperature, water vapor density,
and cloud liquid water density.
The important processes that are responsible for the characteristic shapes of these profiles are outlined, and the concept
of layers in the atmosphere arises naturally in the explanation
of the interdependencies of the profiles.
With a working know-
ledge of the dominant phenomena in the atmospheric layers,
the problem of prescribing the atmospheric turbulence is undertaken.
For the model in this work,
the atmospheric turbulence
is specified with profiles of eddy viscosity and turbulence
kinetic energy.
other profiles,
The relation of these two variables to the
and their inclusion into the model occupies most
of this chapter.
4.2
Atmospheric Profiles of Wind, Temperature, and Humidity
The vertical atmospheric profiles considered in this work
-75-
are assumed to have been measured with some appropriate meteorological instruments over a flat terrain.
For instance, a
tower with a series of instruments at various heights would
produce essentially pointwise values of the variables, which
could then be linearly interpolated between the measurement
heights to produce the full profiles.
It is assumed that the
measurements were time-averaged for at least 20 minutes so that
there is very little time-dependence in the profiles.
Alterna-
tively, a radiosonde (balloon) ascent is commonly used for
measuring vertical profiles, although the measurement averaging
times are not long enough to completely average over the larger
atmospheric eddies.
The measured atmospheric wind profiles have several common
features.
First, the atmospheric wind vanishes at the ground.
This is in accord with the no-slip velocity boundary condition
of real fluids.
Second, the time-averaged (i.e., averaged over
about 20 minutes') vertical velocity is very small at any height.
This is because the very low frequency (of the order of 1 per
day) vertical velocities are due to the 'synoptic scale subsiding
or lifting motions associated with fronts; these velocities are
usually only about 10 cm/sec.
Because the average vertical
velocities are small, the wind at any height is assumed to be
parallel to the ground.
Generally, the wind speed increases
with height and commonly exhibits some turning with height--
-76-
especially in the first several hundred feet of elevation,
where pressure gradient, Coriolis, and frictional forces are
all important.
The fact that the wind vector may very roughly approximate a logarithmic profile,
wind relation,
46
44
an
Ekman spiral,
45
or a thermal
is only of minor interest here since the
actual wind profile determines the behavior of an individual
plume. In this work, the turning of the wind with height is not
represented in the hydrodynamic simulations, although the prospect of including it is considered among the extensions of the
model outlined in Sec. 6.2.
Also, the difficulty of defining
an average wind direction when there are only light, variable
winds at a station dictates that the computer simulations are
not expected to be accurate for winds of less than about 5 knots.
The temperature and humidity profiles directly provide the
information about the local stability of vertical atmospheric
and plume motions.
No approximations to the temperature or
humidity profiles are needed to incorporate them into the simulations.
The temperature and humidity profiles are used to
evaluate the virtual potential temperature profile:
Note that
in defining equations for virtual potential temperature
(Eq. 3.53 and Eq. 3.67) the temperature, humidity, and pressure
are required at any height.
To this end the pressure profile
could have been measured by itself, or calculated with any of
Il______sly____ll__rlll_-III_--L^X
WT---
-·-··Pl···---fC-11·_-l.l1l____s_
___1__1·_
-77-
a number of approximations (dry hydrostatic, moist hydrostatic, various interpolations between points, etc.)
Whatever
assumptions are made, the pressure profile consistent with
these assumptions must be input to the simulation where it is
used to recalculate the correct temperature from the virtual
potential temperature and humidity for the equilibrium moisture
thermodynamics model.
4.3
Turbulence in the Planetary Boundary Layer
. . ..
4.3.1
. . ..
.. . .
. . ..
Introduction
The planetary boundary layer (.PBLI is a boundary layer'
in a rotating, stratified, multi-component fluid whose moisture
component can undergo changes of phase.
Further, the boundary
conditions on fluxes of momentum, sensible and latent heats,
and radiant energy can vary greatly over large and small distances
(i.e.,
distances that are large or small in comparison
to the depth of the boundary layer), and are typically strongly
coupled to the flow.
Although. a number of excellent reviews
have been written 4 7 - 5 7 at many levels of detail, the basic
notions of turbulence in the planetary boundary layer are developed here with the aim of pointing out the limitations of the
description of the PBL turbulence embodied in the computer
simulations.
-78-
4.3.2
Layers in the' PBL and Important Processes
Without much loss in generality, it is assumed in this
work that all of the energy in turbulent atmospheric motions
ultimately comes from the sun.
Although it is possible to
conceive of special situations where this is not quite true
(for example, the turbulence near a busy expressway, much of
which is caused by mechanical stirring and buoyant exhausts),
the atmopsheres' which are encountered in this work are free
of man-made turbulence, except for the buoyant plumes themselves!
For the purposes' of illustration, the solar energy
which produces atmospheric turublence may be divided into two
streams:
(1) that part of the solar energy that produces the
large synoptic-scale pressure patterns on the earth, which in
turn drives the wind and produces turbulence in regions of the
atmosphere of sufficiently large wind shear, and (2) that part
of the solar energy that produces the local thermal stratification of the atmopshere, which. in turn produces turbulence in
regions of sufficiently unstable stratification.
The turbul-
ence that is produced by the first stream is called "mechanically produced turbulence," and that produced by the second
stream is called "buoyancy produced turbulence."
The thermal
stratification that is produced by the second stream is usually
formulated in terms- of virtual potential temperature, so that
--
·----
--
-79moisture and latent heat effects are naturally included in the
"buoyancy produced turbulence."
There are two mechanisms that
destroy atmospheric turbulence:
(1) viscous dissipation, which
is always at work in a turbulent flow, and (2) buoyant destruction, which is present in regions of stable thermal stratification.
From the preceding discussion it is expected that in a
region in steady state the mechanisms of turbulence production
and destruction will be balanced, and that the turbulence
kinetic energy will maintain a value that is commensurate with
the destruction rate.
Very commonly in micrometeorological
studies, the regions that these processes are studied in are
simplified to layers, so that the description of atmospheric
turbulence becomes' one-dimensional--the single dimension is then
height.
The situation is illustrated in Fig. 4.3.2.1.
In the
uppermost layer of laminar flow, the strong geostrophic winds
usually have very small wind shears with height, and are usually associated with stably stratified air, so that there is
little or no turbulence.
The next layer down usually is a
region of buoyancy produced turbulence with only small wind
shear--the buoyancy is typically from solar heating at the
ground and latent heat release in cloud formation (clouds obviously affect the amount of solar heating at the ground, so
that these effects are strongly coupled).
The layer nearest
-80V
laminar (geostrophic) flow
(very small turbulence kinetic energy)
clouds
buoyancy
dissipation
(small turbulence kinetic energy)
shear + buoyancy - dissipation
(large turbulence kinetic energy)
EARTH
Fig. 4.3.2.1 The Concept of Layers in the Planetary
boundary Layer. Atmospheric turbulence is assumed to be
variable in one-dimension only in this figure. The
turbulence is steady-state.
11
11
1
11
1
1
1
__
-81-
to the ground typically exhibits a lot of wind shear due to
the no-slip condition at the ground, so that mechanically
produced turbulence is present in addition to buoyant production, and turbulence kinetic energy is usually a maximum somewhere in this layer.
The particular illustration of atmospheric layers in
Fig. 4.3.2.1 is certainly not unique.
Many investigators
have coined names for layers to illustrate different refinements
on the processes in the PBL.
Such terms as the surface layer,
Ekman layer, subcloud layer, cloud layer, inner layer, outer
layer, tower layer, convection layer, inversion layer, superadiabatic layer, and viscous sublayer are common, but they do
not represent anything more sophisticated than treating the
atmosphere as one-dimensional.
The prospect of treating the atmospheric state as two-dimensional--now including its downwind development as well as its
profile with height--is considered in Sec. 5.4.1 in conjunction
with the modeling of a fumigation episode.
4.3.3
Prescription of the Eddy Viscosity
The prescription of the eddy viscosity in the two-dimen-
sional mesh of the crosswind alignment scheme of Fig. 3.3.'2.1c
is considered in this section.
It was mentioned in Sec. 3.3.2
-82-
that the absence of any mean wind component (by definition)
in the crosswind direction means that, away from the plume,
and as far as the computer simulation is concerned, there
is no explicit mechanical production of turbulence in the
atmosphere.
In fact, what takes place in the atmosphere is
that the turbulence kinetic energy component, u' , and the
Reynolds stress, u'w', of the downwind x-z plane are feeding
into the crosswind y-z plane turbulence kinetic energy component, v' , and Reynolds stress, v'w', through the return to
isotropy term in Eq. 3.40.
For this work, the assumption is
made that the return to isotropy term is very strong, so that
the turbulence is isotropic.
Experiments on atmospheric return
to isotropy indicate that this assumption is reasonably good.5 8
It is seen in the discussion of the results in Chapter Five
that this is probably the most limiting assumption in the work
with regard to being able to model real atmospheres.
The eddy
viscosity as a function of height in the downwind x-z plane
is estimated from a number of prescriptions for eddy viscosity
that are correlated from mean wind and temperature profiles,
then the eddy viscosity in the crosswind y-z plane is assumed
to be the same as in the x-z plane under the assumption of
isotropy.
The incorporation of an ambient eddy viscosity profile
on the simulation mesh finds two problems.
_I_________________
_______
___
First, any
-83-
arbitrary eddy viscosity imposed on the mesh cells at the
start of the simulation will, in the absence of sufficient
mechanical and buoyant production, rapidly decay down to the
molecular kinematic viscosity.
Second, the turbulence field
inside the plume must be allowed to develop on its own.
The
method of incorporating the ambient eddy viscosity profile in
light of these problems is as follows:
to start the simulation,
the cells outside of the initial plume cells are initialized
with the eddy viscosity profile, depending on their height in
the mesh.
After each time step, each cell on the mesh is tested
to see if it has fallen below the prescribed eddy viscosity
profile at its height.
If it has, its eddy viscosity is simply
reset to the ambient value.
If it has not fallen below the
ambient value, presumably because either the plume-induced
turbulence or the turbulently diffused turbulence from neighboring cells is dominating, then the cell eddy viscosity value
is left alone.
In this way, the far field always maintains
the ambient atmospheric turbulence values, and the plume
turbulence, if greater than the ambient turbulence, is left
to develop on its own.
Overall, this method has the effect
of adding a non-uniform source term to the eddy viscosity
equation--the term always adjusts itself to yield the original
eddy viscosity in the far field, and to "turn itself off" if
the plume turbulence is dominating.
Mathematically, the
-84-
inequality
a(y,z,t)b
(4.1)
library(Z)
has been added to the equation set, where
libary(Z) is the
prescribed eddy viscosity profile as a function of height.
Before discussing the available prescriptions of eddy
viscosity, it should be noted that the potentially most accurate method of prescribing the eddy viscosity for an individual
release would be to actually measure it in the field--perhaps
simply by estimating it from bivane wind fluctuation data.
The effort in this work to arrive at workable prescriptions
from the micrometeorological literature is motivated by the
total absence of these measurements in. existing plume field
data.
The particular prescriptions that are recommended here
are used only because they offer a simple way to estimate the
eddy viscosity profile.
A number of prescriptions for the eddy viscosity in the
outer boundary layer of the atmosphere as a function of height
have been reviewed. 5 9 - 6 4 A summary of the various prescriptions
is presented in Table 4.3.3.1, where they are separated into
two major groups--those that require wind speed and direction
profiles, and those that do not.
Those which do not require
wind profiles as input are easier to use because the wind profiles need not be measured (e.g., with instrumented towers or
-85-
a)
Q
Eo
a
0
1= 3(
U)
a
U
a
3
U
w
U)
w
wU
U)
a)
U)
a(
U)
a)
Cd
4.
U)
a
-V
U)
-I
4
0
m0
H
rl-i
C.)
3
O
O
a
a)
0
a
Q
-i
1
>1
C
U
a)
.H Ha
u
rl
4
-'
H
04
d
rr
3)
U)
U)
U)
U)
n
U
U)
N
>1
a)
>1
O
.,
u
1-1
4.)
-H
s-I K
.1i
.,3-i
rO
-
a 4-)
4"
o
0
4) a)
.I
0
04
a
InI
P-4
1 0
04
o
rd
UW
0
4.
ra
a)
Nl
.I,
o)
3
U)
-H
a)
5-I
-,
0
u
o
X4
o
r-I
C?)
-,
U
r
;:
U)
a)
-)
r-I
i
5-
0m
'0m
00
,-
,.
-
a)
fo
Z
(n
(d
H-4
5-'C
z
4J
a
a)
k4
04
ms
0
a,
.10
U)
0-i
0
m
-86-
balloons).
However, they are not expected to be as accurate,
since the wind profile has taken an ideal shape.
All of the
models in Table 4.3.3.1 are searched for applicability to
neutral, stable, and unstable atmospheres.
It is recommended that if the wind speed and direction
profiles have been measured, the prescriptions of Blackadar,5 9' 6 0
and Yamamoto and Shimanuki 61 should be used. If the wind speed
and direction profiles have not been measured, the prescriptions of Bornstein6
or O'Brien
cription of Nieuwstadt6
should be used.
The pres-
requires a substantial numerical
analysis of the profiles and has not been tested.
Any of these prescriptions must be used with caution since
all of them are only capable of providing an estimate to the
eddy viscosity.
The greatest difficulty in using these pre-
scriptions is that they typically require values for quantities
that were not measured, such as the heat flux at the ground,
the roughness height, geostrophic velocity, etc.
4.3.4
Prescription of the Turbulence Kinetic Energy
The prescription of the turbulence kinetic energy (TKE)
in the two-dimensional mesh of the crosswind alignment scheme
of Fig. 3.3.2.1c is considered in this section.
The turbulence
kinetic energy suffers from exactly the same problem as the
eddy viscosity in Sec. 4.3.3.:
in the absence of explicit
-87-
buoyant and mechanical production of turbulence on the twodimensional mesh., the turbulence kinetic energy would gradually
decay away entirely.
To satisfactorily avoid this problem,
the concept of the turbulent "return to isotropy" is again
invoked to allow the turbulent kinetic energy produced by the
mean flow shearing and buoyancy to be fed into the crosswind
motions.
A turbulence kinetic energy profile is needed, so
that it may maintain the turbulence formesh cells that lack
the sufficient turbulence production in exactly the same way
that an. eddy viscosity profile maintains the eddy viscosity
for the mesh.
Ideally, the TKE profile should be measured or deduced
from other profiles for an actual atmosphere.
In fact, how-
ever, prescriptions for the turbulence kinetic energy from
mean wind and temperature profiles are not generally available
in the literature.
The actual prescription of the turbulence
kinetic energy profile in this work has had to come from the
following, very approximate analysis of the transport
equations.
Consider the TKE transport equation in a region away from
the plume.
The vertical and horizontal velocities, v and w,
are zero, and the eddy viscosity, a , and TKE, q, are functions
of height, z, only; with the resulting expression being
-88-
aq=
q
4at
a
+
-
Pr
z
(a
(o
(4.2)
For a properly time-independent TKE, there must be a balance
of dissipation and diffusion in Eq. 4.2; or
a
q'
= r az (o
araz )
(4.3)
Performing a scale analysis of the terms, noting that F/4~
10
and that the depth of the planetary boundary layer is taken
equal to Leddy
, one obtains the result
.2
q
10
a
Leddy
(4.4)
For typical values in the atmosphere, Oc100 ft2/sec and
Leddy
edd103ft, giving the value
i10
- 3
sec -
.
4.5)
Note that for a highly idealized picture of turbulence,65 with
eddies of a single size, Leddy, and velocity, Ueddy,
(4.6)
q-ueddy ,
~
Uededdy eddy
(4.7)
and therefore
q
Q
Ueddy
Leddy
[sec-l
.]
(4.8)
-89-
This states that q/a is simply the inverse of the eddy turnover time.
The scale analysis (Eq. 4.5) of the q transport
equation shows that the choice q-10- 3sec
allow q to have a constant value.
ao should roughly
The fact that this choice
of q agrees with the eddy turnover time of roughly the most
diffusive atmospheric eddies66 (103 seconds, or about 15
minutes) lends support to the idea that a and q have been
chosen consistently in this scheme.
The crude specification of qlibrary (z) has been found to
be satisfactory in this work primarily because the turbulence
kinetic energy only indirectly influences the eddy viscosity,
so that errors in estimating TKE are tolerated much more than
the errors in estimating the eddy viscosity.
The preceding
analysis, since it is a scale analysis, only provides a
very approximate estimate of the turbulence kinetic energy
profile.
Mathematically, the inequality
q(y,z,t)> qlibrary()
= 10
sec
library
(Z)
(4.9)
has been added to the equation set, where qlibrary (z) is the
prescribed turbulence kinetic energy profile as a function
of height.
-90-
5.
CARD INPUT AND SAMPLE PROBLEMS
The VARR-II40 computer code has been modified for the
modeling of buoyant plumes in turbulent, stratified atmospheres.
This chapter is intended to illustrate the use of the modified
code for users that are already familiar with the VARR-II code.
Users that are not familiar with VARR-II should consult the
VARR-II Users' Guide.
All users should note that a previous
68
alteration
of the VARR-II code from CDC to IBM machines has
removed the extensive film plotting routines (Sec. III.C in the
VARR-II Users' Guide), and has also removed the packing of the
cell flag index (Sec. III.F).
The current modifications have not
updated the program tape dump and restart capability (Sec. III.D),
although this could be re-established with a minimal amount of
programming.
The remainder of this chapter will discuss the alterations
to the input data cards found in Sec. III.B of the VARR-II Users'
Guide 40, and then several numerical examples are presented.
5.1
Overall Program Support
The modified program has a moderately larger storage require-
ment than the previous code, but the program support devices
are the same.
The code now stores 25 variables for each fluid
cell (real or imaginary), where the previous code only stored
19.
In the current version, the total program storage for a
20 x 20 (real) cell mesh is about 175K bytes of CPU.
reads input on device 5 in card format
No. 94 in Appendix D
The program
(when IVDI=5 in Statement
and writes output on device 6 (when IVD0=6
-91-
in Statement No. 95 in Appendix D).
Device 6 should be a
line printer since nearly the full 120 characters per line
are used.
Of course, disc, mag tape, or microfiche could be
used instead of a line printer.
Running time in CPU is, of course, dependent on the
time step size in the problems.
The time step size is
selected by the program at each time step, and is usually from
one-tenth of a second to several seconds.
The selection of a
time step size is performed by the code.67 where it always
chooses the smallest step size from a choice of a diffusion
condition,
TSTEP
DT =
(5.1)
TSE
max(a)
(
Dy2
+
1 )
2
DZ
a Courant condition,
DT = TSTEP
min (DY,DZ)
max (v,w)
(5.2)
I
or a simple rate of change condition,
.2 max (v,w)
(max(vw) (vdOldwold)
(max(v,w) - max (voldwold
+
)
+
0
-6
_max
(5.3)
where TSTEP in the VARR-II Users' Guide has a value of 0.25.
As a practical matter, it is found that the time steps have to
be reduced beyond these conditions by about a factor of 25 for
the mesh cell sizes encountered in this work.
reduction is accomplished by giving TSTEP
This 25-fold
a value of 0.01
-92-
(note that Eq. 5.3 is virtually never the most limiting step
size).
This allows the code to conserve energy in the com-
puter mesh cells within an acceptable tolerance.
The non-
conservation of energy arises from the first order accuracy
of the differencing scheme for the advection terms.
Full
donor cell differencing of the advection terms is found to
give the best answers in the simulations--less than full
donor cell differencing produces noticeable nonlinear instabilities in the flow.
Running times on the IBM370/168 are
usually about 10 to 15 minutes per thousand time steps.
5.2
Card Input Decks
The generation of input decks is considered in detail in
the Appendices, where the additional input variables are
defined and the detailed card formats are listed.
Many of
the input variables are discussed in Section 3.3 and in
Chapter 4.
To illustrate several different problems, three
input decks are listed in the following sections..
5.2.1
The Dry, Buoyant Line-Thermal
Card input for the dry, buoyant line-thermal is listed
in Table 5.2.1.
Briefly, the problem simulates the rise of
an initially quiet, warm parcel of air in a neutrally stratified
dry atmosphere.
Results from this problem are given as a
sample problem in Sec. 5.3.1.
5.2.2
LAPPES 5 Plume of 20 October 1967
Card input for the essentially dry, buoyant plume from the
Keystone #1 combustion stack is listed in Table 5.2.2.
-93,-I
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-98-
Briefly,
the problem simulates the behavior of the plume
emanating from an 800 ft tall
Time.
stack at about 7 A.M.
The plume and ambient weather were documented
LAPPES study 5
(Volume 2).
in the
Results from this problem are pre-
sented as a sample problem in Sec. 5.3.2.
This input deck is
included to illustrate the input of a pollutant
SO 2
Eastern
(here
in gm/ft 3 ), and to illustrate the input of available wind
speed and temperature profiles, and the inclusion of ambient
atmospheric turbulence profiles.
the neutral layer
(above 350 m elevation) is calculated from
Blackadar's prescription.59
stable layer
The ambient turbulence in
The ambient turbulence in the
(below 350 m elevation)
is simply Blackadar's
prescription59 suppressed by a factor of 100 to make a
rough account of the suppression of the turbulence by the
stable stratification.
Comparisons between the numerical
simulations and the LAPPES experiments are found in
MIT-EL 79-002.
5.2.3
Moist, Buoyant Line-Thermal
Card input for the moist, buoyant line-thermal is listed
in Table 5.2.3.
Briefly, the problem simulates the rise of
a very nearly saturated warm parcel of air in a cool environment.
As the parcel begins to rise, its temperature falls
rapidly and moisture begins to condense within a few seconds.
This input deck is included to illustrate the input of moisture
-99-
1'% m =s un \o r
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-101-
variables.
Temperatures must now be interpreted as virtual
potential temperatures since there is an appreciable amount
of moisture in the cells.
The absolute pressure profile is
not a real profile, but simply is an estimated profile.
In
practice, the unstable nature of the explicit differencing
of the moisture model imposes a very small time step on the
problem (about 100 times smaller than usual).
Results from
this calculation are not given as a sample problem because of
this restriction.
5.3
Sample Problem Results
The interpretation of the sample problem output listing
is illustrated in Fig. 5.3.1.
Each box in the figures that
follow Fig. 5.3.1 represents an individual fluid cell.
A map
of the entire mesh cell storage is produced by the VRPRT
subroutine.
Only a small region of the entire problem is
found in each of the figures--the figures are roughly centered
on the regions of strongest flow.
5.3.1
The Dry, Buoyant Line-Thermal
Results from the dry, buoyant line-thermal are presented
in Figures 5.3.2 to 5.3.5.
regarded in the figures.
The moisture values may be disA further discussion of the results
is found in MIT-EL 79-002.
5.3.2
LAPPES5 Plume of 20 October 19'67
Results from the tall stack of the Keystone plant are
presented in Fig. 5.3.6.
Only the mesh cells at the beginning
-102-
of the simulation are shown.
The plume several kilometers
downwind extends over many cells and is discussed further in
MIT-EL 79-002.
-103-
ordered-pair notation for locating cells
e.g., 3rd cell from left, 2nd from bottom
w velocity, ft/sec
cell
temperature
OF
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( 3 , 2)
49.9' )9
X
X
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X
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X LIQ= 14.OOOE-07 X
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|
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v velocity,
ft/sec
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TNU is eddy viscosity in ft2 /sec
TKE is turbulence kinetic energy in ft2/sec 2
CHI is pollutant concentration in lbm/ft 3
VAP is water vapor density in lbm Ii 0/ft 3
2
3
LIQ is liquid water density in lbm H 2 0/ft3
Fig. 5.3.1.
Key to Cellwise Quantities for Figs.
5.3.2, 5.3.3, 5.3.4, 5.3.5, and 5.3.6.
-104-
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X CHI- 1.000 E+00 X CHI[
1.CCOE-35 X CHIl 1.030E-05 X CH I- 1.OOE-05 X CHI- 1.CGOE-0 X
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X TKE- 1.00GE-03 X TKE" 1.COOE-03
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X CHI= 1.000E-05 X CHI
1.000OE-05 X CHI
1.00C5-05 X CHIs I. COOE-C OSX CHI- I. 00E-GC X
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X TKC1 .IOC
CO.-x
.ona0C-o) X TKE- 1.0005-73
I.OCOF-0s
C£I.- 1.OC3E-0 X C11 K' I.COOr-C5 X
1.00E-95 X CHI
X CHI
1.000-ni X CHI
X
VAP
1.GOOnr-n
X
YAP.
l.C.0-07
X
VAi
1.00F-07
'A
YAPIP
t.C3OE-7
X
X VAP1-'
:.CJ;C-O1
I
IU-.C9C-C7 X
X L IO l!.0CE-7 X LIQO 1.030-37 X LIOQ 1.00r-'7 X LC
0.0
YX XX
CXXXIX
0.0
XXXXXXXX
0.0
XAXXXXXX
0.0
XXXXX
0.0
XWXXXXXXX
.XX X
0.0
X
.
Fig.
5.3.2.
Initialized Plume Cross Section at 0 sec.
DY = 100 ft, DZ = 200 ft.
Disregard moisture values.
-105XXXXX
0.215 XXXXXXXXX
C.107 XXXXXXXXX
0.1'2 XXXXXXXXx
0.073 XXXXXxxxX
0.041 XXX))
X
X
X
X
X
X
X
2
)
X
I 3, 8)
X
4 8)
X
I 5,
)
X
I , 8)
X
X
49.9q2 F
X
5.19
F
X
50.137 F
X
49.922 F
X
49.922 F
X
X
X
X
X
X
X
O
1
0.073
1
0.1-20
I
0.148
I
C.121
1
0.1;26
X TNU= 1.005E+00 X TNU- I.CO7E*3 )0 X ThU- 1.005E+C X TNU- 1.033E+OC X TNU- I.OOIE*CC
X TKC 1.039E-03 X TKE- 1. 014E-0 ?3 X TKE = 1.009E-03 X TKE= 1.006E-03 X TKE- I.OGIE-03 X
X CHI- 1.084E-05 X CHI= 1.001E-C DS X CHI- 1.030E-05 X CHI- 9.99E-06 X CHI- 9.99')E-06 .X
X VAP- 2.030E-07 X VAP. 2.000E-3 T3 X VAP- 2.091E-07 X VAP=.2.COOE-07 X VAP- 2. C3E-07 X
X LIQ= 0.0
X LIQ= 0.0
X LI.Q- 0.0
X LTQ= O.0
X LTO= 0.0
X
0.029 XXX
XXXXX
0.348 XXXXXXXXX
0.232 XX
KXXXXXXX
0.209 XXXXXXXXX
0.005 XXXxXXXX
X
X
X
X
2 X
I 6, 7
X
( 5 7
X
( 3, 7I
X
( 4, 71
X
X
X
I 2, 71
49.957 F
X
X
49. .964
49.777 F
X
50.109 F
X
F
50 .089 F
X
X
X
ZC
'X
X
0
I
C.254
1
0.209
1
0.294
1
0.277
t
.0.218
X TNU= 1.03CEOO X TNU= I.C39E.00 X TNU= 1.024E+00 X TNU= I.OIOE*CC X TrNU · 1.004c+CC X
X TKE. 7.111t-03 X TKE- 1.239E-03 X. TKE= 1.050E-03 X TKE= 1.019E-C3 X TKE== 1.037E-C3
X CHI 2.144E-04 X CHIt 1.398E-05 X CHI- 1.004£-05 X CHI 9.996£-06 X CHI- Z1.000E-05 X
X VAP. 2.QOOE-07 X VAP- 2.000E-07 X VAP
2.030E-07 X VAP= 1.999E-07 X VAPI = 2.030-C?7 x
X LIC 0,0 .
S LIC- 0.0
X
X LT0.0
X LIQ- C.0
X L 0 = 0.0
KXXXX
0.380 XXXXXXXXX
0.150 XXXXXXXXX --0.045 XXXXXXXXX -0.028 XXXXX
3.791 XXXXXXXXX
X
X
X
X
X'
X
X
I 4. 6)
I 2.
I 3, 6)
X
S. 61
X
II 6.
31
x
6
X
)
3
6)
6 61
X
49.985 F
X
5C.O40 F
X
50.172 F
X
50.012 F
X
5S1.068 F
X
X
X
X
X
X
X
0
1
0.875
1
0.756
1
0.543
1
C.430
1
0.276
X TNU= 2.255E+00 X TNU: I.018E+00 X TNU= 1.047E+00 X TNU= 1.028E+CC X TNU- 1.027E+CC X
X TKEw 3.841E-01 X TKE- 1.945E-02 X TKE = 1.626E-03 X TKE- 1.061E-03 X TKE- 1.OS2E-03 X
X CHI- 1.611E-02 X CHI= 5.960E-04 X CHI- 2.124E-05 X CHI- 1.C12E-05 X CHI- 1.C0OE-05 X
X VAP. 2.CO0E-07 X VAP- 2.COOE-07 X VAP= 2.COOE-07 X VAP- 2.COOE-07 X 'AP- 2.000E-07 X
X LIQ= 0.0
X LIQ- 0.0
X LIQ= 0.0
X
X LIQ= 0.0
X LIQ= 0.0
;XXXX
2.530 XXXXAAAXX
0.153 XXXXXXXXX -0.260 XXXXXXXXX -0.266 XXXXXXXXX -0.345 XXXXX
X
X
X
X
X
X
X
X
1 6, 5)
X
I 5, 5)
X
4 5
X
3 5)
X
! 2, 5
X
49.ee8 F
X
X
50.324 F
X
50.061 F
X
50.234 F
X
54.088 F
X
X
X
X
X
1
0.180
1
1.302
1
0.711
1
C.392
O
2.009
X THNU 1.840EOL X TNU- 3.753E+00 X TNU= 1.141E+00 X TNU- 1.04E+CC X T7Ntss 1.07E+CC X
1.176E-C3 X
X TKEw 4.772E+.0 X TKE- 7.725E-31 X TKEw 5.368E-02 X TKEO 2.722E-C3 X TKE
X CHI- 2.405E-01 X CHI= 3.218E-02 X CHI= 1.717E-03 X CHI= 4.576E-05 X CHI= 1.032E-05 X
X VAP= 2.000E-0T X VAPs 1.999E-07 X VAP. 2.000E-07 X VAP= 2.COIE-07 X YAP= 2.COCE-07 X
X
LQtQ
0.0
X L10= 0.0
X LIQ= 0.0
X LIQ= C.0
X
-. _889
*(XXXXXXX.X -.738 'xxxx
:XXXXX
6.566' XXXXXXXXX -1.302 XXXXXXX. -1,462 XXXXXXXX.
X
.:
X:'
.xX.
X
( 6 4"
X
X
I 5, 41
4, 4)
X
X
3, 4)
I 2 4
X
X
X
49.888 F
X
49.908 F
X
49.770 F
X
61.240 F
X
50.768 F
X
X
X
X
X
X
1
-0.319
1
-0.260
1
-0.157
0
1
-0.536
1
-0.461
X TNUw 4.347E+0 X TNU- 2.428E+01 X TNU- 4.466E+00 X TNU 1.035E'00 X TU= 1.071E40 X
X TKE= 1.122EO1 X TKE= 3.101E+00 X TKE- 9.614E-02 X TKE- 1.592E-03 X TKE t1.146E-03 X
X CHI= 6.256E-01 X CHI 6.55SE-32 X CHI- 1.012E-03 X CHI" 1.180E-05 X CHI- 9.999E-06 X
X VAP 2.C0CE-07
P. X YAP. I.99E-07 X VAP- 2.0002-07 X VAP- 2.000E-07 X VAP= 1.999E-C7-X
X
X LIO= 0.0
X LIC= 0.0
X LO
0.0
X LIQ- 0.0
X LTQ- 0.0
XXXXX
5.434 XXXXXXXXX
-1.077 XX4,XXXXXX -1.117 XXXXXXXXX -0.775 XXXXXXX -0.582
XXXX
X
X
-X
X
X
3
X
6 3
X
X
X
I 5 3)
X
1 4, 3)
X
1 3 3)
1 2, 3)
49.943 F
X
50. 116 F
X
X
50.207 F
X
49.860 F
X
49.735 F
X
X
X
X
X
X
1
-1.472
1
-0.924
1
-0.626
1
-0.426
1
-2.248
0
X TNU= t.IS1E*CC X TNU- 1.073E+CC X
X T.U= 3.361Ee00 X TNU- 3.074E+0 K TJU= I.OSOE0
8.0,3E-0 X TKE= 2.349E-01 X TKE- 5.556E-33 X TKE= 1.33E-C3 X TKE= 1.14E-C3 X
X TfrE
X CHI= 1.297£-02 A CHI- 3.665E-03 X CHI 4.321E-05 x CHI= 1.C04E-05 X 'CHI= 1.000-05 X
VAP 2.000E-07 X
X VAP= 1.999E-07 X VAP= 2.OOIE-07 X VAP= 1.999E-07 X VAP. 2 C0OE-x07
X
X LIQ= 0.0
X Lt4 0.0
X LIO= 0.0
X LIQ- 0.0
X L 10 0.0
-0.154 XXXXX
-0.217
XXXXXXXXX
-0.102 XXXXXXXXX
0.435 XXXXXXXX
1.012 xxxxXXXxx
XXXXX
X
X
X
X
3
X
X
( 3 21
X
( 4, 21
X
I 5 2J
1 6, 2
X
X
I 2, 2
X
49.999 F
X
49.957 F
X
A9.915 F
X
49.964 F
X
X
49.659 f
X
X
X
X
X
X
X
1
-0.618
I
-C.583
1
-0.50
1
-0.T10
1
-0.492
0
1.147F.00
X
1T&U
1.CCOE
00
X
TN1=
IU
1.OtPE+CC
X
1.04E+CC X TN
X TfJU- 1 .On+O00 X TSNU"
X rKE- 1. 323F-3
X TKC- 1.13E -03 x TVE= I1.0,'E-C3
X T Ea 4.085--1 X TKE. L.Ri8E-)
.99C-06 X
1.005£r-05 X C I - 9.9V9E-06 X CF '-!I
1. 101o-s5 X C-ll
X CHI- 2.741F-05 X CHI
l.qq9E-07 X VAP
2.COE-07 X VAP== 2.000r-o X VAP. 2.1030-C 7 X VA,P. 2.COOi-CT XX V AP
X LIOt 0.0
X LiO 0.0
X LI G- 0.O
X
X LIO . C.O
X L 10O 0.0
0.. IXXxoxxxx% 0.0
0.0
X XxICx Y
C.C
XXXXXXXEK
3 Xx)
0.0
XIXx3,XXX
XXXXX
Fig. 5.3.3.
Plume Cross Section at 20 sec.
DY = 100 ft, DZ = 200 ft.
Disregard moisture values.
-106:KXXXX
2.792 XXXXXXXXX
1.C4C XXXXXXX~X
0.609 XXXXXXXX
0.168 XrXXXXXX
-0.170 NIx)x
X
X .
X
X
.
X
1 2, 8)
X
X
X
3 8
X
.X
I t, 81
( 4, 81
X
I S 8
50.809 F
X
X
5C. 137 F
x
50.082 F
X
49.999 F
X
49.936 F
X
X
X
X
X
X
0
1
1.592
1
1.860
I
1.55
1
1.118
1
0.843
X- TNU- 5.087E00 X TNU= 2.169E00 X T 'U 1.658t40O X THU. 2.012E+CC X TNUIj 1.3TOE*CC X
X T K- 4.702E-01 X TKE= 1.135E-01 X TK.E, 2.026E-02 X TKE- 6.234E-C3 X TKEa 2.047E-C3 a
X CHI- 4.895E-02 X CHI
1.078E-02 X CtII- 1..48E-03 X CHI- 1.383E-C4 X CHI- L.682E-CS X
X V4AP 2..00CE-07 X VAP= 2.000E-07 X VALP= 2.000E-07 X VAP- 2.000E-C7 X VAP= 2.030E-C7 X
X LIl= 0.0
X L 0- 0.0.
X L IQ= 0.0
X LIQ 0.0
X LIQ 0.0
X
XXXXX
5.963 XXXXXXXXX
1.573 XXXXXXXXX
0.074 XXXKXXXXX -0.752 XXXX)XXXX -0.714 XXXX).
X
X
X
X
X
X
X
I 2, 71
( 3 7)
X ·
I 4, 71
X
X
X
6, 7)
( S 7)
49.999 F
X
X
50.165 F
X
49.888- F
X
X
52.397 F
X
50.927 F
X
X
X
X
X
X ...
1
2.327
1
1.898
l
1.345
1
0.966
1
1.790
X TNIU- 1.364E.01 X TN U- 6.094E+00 X TU- 2.564E+00 X rNUA - 1.746E4CC X TNU= 1.560E+CC X
X To 'En 1.42CE+00
TKlE= 5.864E-01 X .TKE= 1.5502-01 X rKE _ 3.042E-02 X TKE= 5.69CE-C3
5.S42E-02 X CH[I 1.527E-02 X CH " 2.390E-03 X CH[
2.2d9E-C4
X CHII= 1.422E-01 X CH
2.C00'-07 X VAP- 2.000E-07 X 1
VAP== 1.99E-0 X VAP- 2.000E-G7 X
X VA mP= 2.00CE-07 X VA
X LQ0 0.0
X LIQ - 0.0
X LIQo I
0.0
.
X
X LI OIQ 0.0
X LI Q= 0.0
2.644 XXXXXXXX
-0.778 XXXXXXXXX -1.852 XXXXXXXX
- 1.471 XXXX
XXXX) C 9.535 XXXXXXXXX
X
X
X
X
X
X
X
X
4 6)
X
5 6)
t 6, 61
I 3 6)
6
X
£ 4,
1 2, 61
X
51.870 F
X.
5C.733 F
X
50.137 F
g
50.006 F
X
X
52.903 F
X
X
X
X
X
X
X
1
1.127
1
0.740
1
C.562
1
0.313
1
0.651
O
X Th U. 1.802E+01 X TNIU= 1.130E+0 X TNU= 4.335*00 X TNU= 1..775+CC X T NU= 1.98CE+0C X
X TO,E- 1.815E+00 X TK:E= 1.200E+00 X TKE,= 3.780E-01 X TKE= 7.E38E-C2 X TKE- 1.715E-C2
1.763E-01 X CHII 1.146E-01 X CHI 4.295E-02 X CH- 8.167E-03 X CHI= 9.445E-04 X
X CH
X VAP- 2.COOE-07 X
X VA[P. 1.999E-07 X VA lP= 2.000E-07 X WAP= 2.000E-07 X VAP- 2.C00E-07
X LI Q= 0.0
0.0C.
X Ll
X LIO n 0.0
X
*X L IP '0.0
3.637 XXXXXXXXX -1.545 XXXXXXXXX -2.236 XXXXXXXX -1.998 XXXXX
XXXXX 10.833 XXXXXXXXX
X
X
X
X
X
1 3, 5)
X
4 5)
X
5 5
X
I e, 5
X
X
I 2. 5
X
49.943 F
X
X
50.324 .F
X
5C.096 F
X
X
51.607 FX
52.119 F
X
X
X
X
X
X
1
-1.157
1
-1.622
1
-1.372
1
-C.988
1
-0.833
0
1.677E01 X TNU-n 1.198E+01 X TNU= 2.832E+00 X TNU= 1.397E+OC X TTNU 1.922E+CC X
X' TN U=
X TK E 1.643E+00 X TKE 1.242E+00 X TKE- 2.05SE-01 X TKE= 3.337E-G2 X TKE- 8.061E-03 X
X CH I- 1.300E-01 X CHII- I.C46E-01 X CHI" 2.520E-02 X CHI= 2.410E-03 X CHIn 1.479E-C4 X
X VA P- 2.000E-07 X VAP- 1.999E-07 X VAP= 2.000E-07 X VAP. 2.000E-07 X VAP= 2.000E-C7 X
.0
...
_X.U
.
0
L=.O0.0
........
X, 1J.-..0
,
X...
....
X. LI Q- 0.0
)XXXXX
xx'x -' t.4588XX'XxxxX ---t.665 -XxxXx.
X xxX
XX
:.2;67
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XXXXX r 8'533'XXX% XX XXXX
X
X
X
X
X
X
X
1 6, 4)
X
S 4)
X
X
1 4, 4)
1 3, 4
X
X
1 2, 4
50.858 F
X
50.394 F
X
50.061 F
X
49.902 F
X
49.811 F
X
X
X
.X
.X
X
...
.
1
-2.552
1
-2.173
1
-1.733
1
-1.269
0
1
-1. 958
X TNIUM 1.359E.01 X TNU- 1.141E01 X THNU 4.348E+00 X TNU- 2.004E+00 X TNU= 2.577E+CC X
X TKXE- 1 .2b7?E00 X TKE- 8.776E-01 X TKE- 1.592E-01 X TKE- 1.617E-02 X TKEn 8.188E-03
X CH41. 5.792E-02 X 'HI- 3.061E-02 X CHI- 8.101E-03 X CHI 5.101E-04 X CHIn 2.975E-C X
X VAiP- 2.000E-07 X VAP= 2.CCOE-07 X VAP= 2.000E-07 X VAP- 2.000E-07 X VAP. 1.999E-07 X
X LIQ= 0.0
X LO- 0.0
X L1Q 0.0
X LICO 0.0
X
X LI [Q= O.66 ..
1.477 XXXXXXXXx -0.350 XXXXXXXXx -0.614 XXXXXXXXX -0.739 xxxKX.
XXXXX
4.668 XXXX xxXXX
X
X
X' .
X
X
X
- X
X
6 3)1
I 5, 31
4 3
X
X
( 2, 31
X
I 3, 3)
X
X
49.735 F
X
49.714 F
X
49.936 F
X
50.103 F
X
49.978 F
X
X
x
X
X
X
X
1
-1.970
1
-1.765
1
-1.500
I
-1.'166
0
1
.
-1.689
X -TNU- 5.732E+00 X TNU= 3.192E.00 X ThUU 2.453E+00 X TNU- 2.150*+C X TNU- 2.20CE+CC X
X TKE- 4.311E-01 X TKE- 1.307-01 X TKE- 4.032E-02 X TKE- 7.152E-03 X TKE- 5.154E-C3 X
X CHI= 6.642E-03 X CHI= 2.758E-03 X CHI- 9.150E-04 X CHI= 5.188E-05 X. CHI t.13E-05 X
X VAP= 2.000E-07 X VAP= 1.99E-07 X VAP= 1.999E-07 X VAP- 2.000E-07 X VAP 2.COCE-C7 X
X LEO 0.0
X
X LJQz 0.0
X L1C 0.0
X LIQ- 0.0
X LIO= 0.0
XXXX
1.202 XXXXXXXXX
0.886.XXXXXXXX
0.151 XXXXXXXX
-0.036 XXXXX.
O.018 XXXXXXXXX
X
X
x
X
X
1
X
I 20 2)
X
( 3, 2)
X
4 2)
X
( 5, 2)
X
t 6 2
X
X
49.583 F
X
50..061 F
X
50.089 F
X
49.995 F
X
49.814 F
X
X
X
K
X
0
1
-0.619.
I
-I.082
1
-1.137
1
-1.113
I
-1.078
X TNU- 2.115
SE00 X TNU- 2.726E+00 x
Nr;U 3.23E*00 X TNU = 2-270ECC
Tl ;ti- 1.936E4CG X
X TKE- 2.472E-02 X TKt
1.924E-02 X TKEs 1.210S-02 X TKF
5.71E7-C3 X TKE. 3.723E-C3 X
X CHI- 7.8dlE-05 X CHI 2.391E-05 X CHI= 1.32GC-05 X CHtI 1.038E-05 X Cliil- 1.001E-05 X
X VAP= 1.998E-07 X VAP- 2.COOE-07 X VAP- 2.000E-07 X VAP. 2.000E-C X VAIP- 1.939E-C? x
X LOQ= 0.0
X L IO 0.0
'X LOQ. 0.0
X LIQ= 0.
X LI 0: 0.0
X.
0.0
XXXxXXXX
0.0
xxxxxWxx
KXXX
0.0
KxXxxxxKK
0.0
xxxxxKxxx
0.0
AXXX
Fig. 5.3.4.
Plume Cross Section at 80 sec.
DY = 100 ft,
DZ = 200 ft.
Disregard moisture values.
-107XXXXx
.Z.34~'.3PCX'r1~
4.626
XXXX..(
0XX:9XXX
0.917
0.01,
KXXXXx
'.397
X
X
X
XXXX
X
X
. ( 6,12)
,12)
512)
X
(s
1.
(3.12)
X
(212)
K
i
149.929 F
49.950 7
X
5).047 F
Z
X'' 50.2S5 r
50.588 r
X
X
K
X
X'
K
1.739
1
2.037
1
2.231
1
2.'03L
1
1.270
1
0
=
TXli, 2.q&t*_00 X THU 2.3711400 X
TNU %.089£-0 '
X TNU= 83962*00 X TNUu 6.393i403
TKE 1.14701 X TIEK 31.513-02 X TKK!= 1.140-02 X
IE 5.06861-0 X TKEz 3.143 1-01
X
CHI= 6.147E-03 X CHfZI 1.435P.-03 X CHIK 2.628-0 X
X CHI- j.85E-02 X CMI- 1.840E-02 t
6.137.-03 X VA?= 1.4251E03 K TAl- 2.5281-04 X
X VA'r
X VAP= 3.8571-02 X VAP 1.839E-0
5.056-01 X
1.22E-32 ; LIE= 2.CB50.-03 X. L13
X L1Q= 7.71'4-02 X LTQ= 3.67.'--02 X LJ'2
XXXI]r -0.586 XXXXX
TXi '1.2149 XXXKXXX'.:.'O&7
3.601 XXAXXX
7.165 XXAXXXXXX
XXXXX
X
K
K
IX
(6,11)
( 5,11)
(4,11)
3,11)
X
2,11)
.(
x
K
50.012 Y
50.14
X
50.352 T
X
50.699 F
X
50.990 F
x
X
X
X
X
1.794
1
2.207
1
2. 442
1
2.153
1
1.191
1
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X T:U= 4.055E30
1.106Et01 X TU-- 6.546E+03
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3.537F-01 X
X TKE= 8.2142E-01 X 'IKE- 7.557E-01 X 'E=
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CHI- 6.519]-02 X CHI= 4.6201-02 X CHIi 2.292E-02 X CHI 8.1011-03
·
X TAP= 6.518Z-02 X TAP= 4.620-02 X VAI= 2.2911-02 X V.O=? 8.0911-03 X VAIP 2.0611-03 X
X LIQ= 1.61*3-02 K- LI3m 4.123E-03 X
=
I LIQ= 1.304E-01 X LIQ= 9.2393-02 K LIQ 4.5821-02
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5.3.5.
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X V'l.;
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X.'yIXII
Plume Cross Section at 200 sec.
DZ =t 200 ft.
DY = 100 ft,
Disregard moisture values.
LZ-
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x
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Fig. 5.3.6.
I;eystone No.
DZ = 50 rm.
I
CC
(. 71+3CC X
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I
C.C
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A KUIKvf<
Initialized Plume Cross Section for the
1 Stack on 20 October 1967. DY = 150 m,
X
X
AKAI
-109-
6.
RECOMMENDATIONS FOR MODEL EXTENSIONS
6.1
Calculational Scheme to Include Wind Shear Effects
A brief overview of a plausible calculational scheme
that would address one of the important effects of wind shear
on the plume dynamics is discussed here.
The effect is that
of the dilution of the plume properties as the plume rises
into progressively stronger winds.
The process is sketched
in Fig. 6.2.1.1, and is well-known to plume modelers.
A
constant release of pollutant (illustrated in Fig. 6.2.1.1),
momentum, sensible heat, moisture, etc., diluted into air
that moves with a velocity u(Z o ) will have a density proportional to the inverse of the velocity.
A plume property that
is released into a stronger wind, u(Z), will be correspondingly more dilute.
This effect is important in buoyant
plumes when the plume updrafts and downdrafts in the presence
of a wind shear cause parcels of the plume to change their
downwind advection rate. Clearly, the problem is fully
three-dimensional (although it can be in steady state), but a
very restrictive assumption may afford a useful recasting of
the to-dimensional problem.
next.
This assumption is discussed
-110-
-~~~~~
-
R lbmrl/sec
X(z)
1 ft
2
J
R/u(z)
u(z)
2
I
.
m
R libm/sc
IX(
1 ft2/
I
I,
RIz,)
u( z0 )
z
z
X(Z 0)
>
X(z)
for u(z) >
0
u(Z 0 )
r
1
EARTH
Fig. 6.2.1.1
Dilution of a Steady Release
of Pollutant.
__
--
-111-
Consider the advection and turbulent diffusion of a
pollutant in three dimensions (the results extend directly
to momentum, sensible heat, etc.):
aX
at
+ u u+
v
x
X
+ w
X =
z
v
6.1)
x)
Assuming that the system is in steady state, we have
uX
ax
+
ay
+ wX
az
= y
(6.2)
In the presence of a steady uniform wind field, u o,
the first
term is commonly interpreted as the time-rate-of-change for
aX
an observer moving with the wind, and is written as a to,
where ut
= X
This contains the important assumption that
the plume always has the downwind velocity u--implying infinite accelerations at the stack exit, to be sure.
In a
strong wind field the downwind diffusion is commonly neglected
with respect to the downwind advection, so the gradient
operator has only y and z derivatives.
If the wind field -is
allowed to have shears, then the first term may be
represented as
u(z) aX
ax = u(z)
U(z )
where u
u(z)
(6.3)
=
ax(U(Z)
u(z ) a) t
has been arbitrarily chosen to be u(zol.
This
interpretation allows the equation to be formulated as
-112-
uaX O)(VaX
a
= ;(- 0
Crx
2
)4)
(6z
This equation holds the assumption that any parcel of air in
the plume, when advected into a region of stronger wind,
immediately takes on the 'local wind velocity and is correspondingly diluted.
Note that it also causes' parcels that
are decelerated to concentrate their properties, which is
physically unrealistic, but hopefully is not too serious an
error since plume rise and updrafts are almost always stronger
than downdrafts.
The important feature that this scheme hopes
to address is the dilution Cusually by about 10 to 50 percent)
of plume buoyancy, momentum, and moisture, which affect the
plume dynamics.
The procedure could be extended to every
transport equation in the equation set--only the effect on
the divergence condition in the fluid mechanics algorithm
has not been studied.
Its satisfaction would still be.re-
quired as a constraint on the solution.
6.2
Calculational Scheme for Time-Dependent Release or
Weather
The simulation of "mildly" time-dependent plumes can be
made with the model.
Essentially, the governing assumption
here is that the prevailing weather or effluent properties
-113-
will advect downwind, and never affect the flow that precedes
or follows it.
The situation is developed in Fig. 6.2.2.1,
where a stack is assumed to have a set of exit properties,
,
that are piecewise-constant in time over periods of 100 sec.
To reconstruct the behavior, an initial simulation with the
properties at time to,
(to) is made to 300 sec.
properties changed at time t
+
The plume
100 sec, so a second simula-
tion is made with properties Q(to + 100) to 200 sec.
Again
the plume properties changed at time to + 200 sec, so a third
simulation is made with properties Q(to + 200) to 100 sec.
The actual plume is then "cut and pasted" from the pertinent
data in the simulations as shown at the bottom of the figure.
The calculation is somewhat wasteful, since 600 sec of simulation produces only 300 sec of results--but the scheme surely
saves time and storage over a fully three-dimensional calculation.
Eventually, for sufficiently "strong" time-dependence
the scheme becomes too laborious with respect to a threedimensional calculation.
6.3
Cloud Microphysics Model
The limited success of the equilibrium moisture thermo-
dynamics model is due to its explicit differencing.
In short,
the model is ignorant of the latent heat released in a current
timestep, and it adjusts the equilibrium conditions without
-114-
I
sirlulation with plume properties at tiae to
I
I_
~ simulation
.
O
with plume properties at t
1~~~~~ie
simulation with plumne properties at tiae tn +100
~~~~~~~0
,
I.
__
F%~~~~~~~~~~~
I
I
I]
I
I
s imulation with plume properties at time t +200
0
I
I
I
I
I
J
I
]1
[[
_
I-
0
[I
I
Fig. 6.2.2.1
Steady-State
_
~~~~~~~~~~~~~~~~.
I
200
T ime
l [
____
-rI
I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
100
Il
laws.
300
(sec)
Simulation of a Tire-Dependent Plunme in
ileather.
Scheme is discussed in the text.
-115-
this knowledge.
The resultant oscillations in the equilib-
rium conditions are not surprising, nor is the ability to
control them with very small timesteps.
If the calculation
was made implicit--essentially iterating on the coupled latent
heat release equation and Magnus' formula for liquid-vapor
equilibrium--the timesteps could be relaxed back to their
original size.
The prospect of incorporating a non-equilibrium moisture
model is not investigated in this work.
The limitations.of
the equilibrium model have not been sufficiently explored
to justify the change at this point.
-116-
NOMENCLATURE
Ao
2
initial line-vortex area (ft
C
experimental constant, 1.9
Cp
heat capacity at constant pressure (BTU/lbm °R)
cmoist
P
heat capacity of moist air at constant pressure
(BTU/lb °R)
m
DT
timestep size (sec)
DY
cell width (ft)
DZ
cell height (ft)
esat (T)
saturation vapor pressure of water (mb)
E (i)
X
energy of the i t h decay channel from pollutant
species X (MeV)
F(i)
X
fractional energy deposition for i t h radioactive
decay channel from pollutant species X
g
acceleration due to gravity (ft/sec2)
gi
vector acceleration due to gravity (ft/sec2)
I
specific internal energy (BTU/lbm)
k
thermal conductivity (BTU/ft-sec-°R)
L
buoyancy parameter (ft)
Leddy
eddy length scale (ft)
Lva p
latent heat of vaporization of water (BTU/lbm )
N
experimental constant, 3.0
p
physically measurable pressure (millibars)
P
pressure perturbation about an adiabatic
reference state (mb)
pO
pressure in a quiet adiabatic atmosphere (mb)
)
-117-
time average pressure perturbation (mb)
p
P
p
fluctuating pressure perturbation (mb)
Pr
Prandtl number
q,q(y,z,t)
turbulence kinetic energy per unit lbm
qlibrary
(ft2/sec2 )
(z) prescribed turbulence kinetic energy profile
(ft2 /sec 2 )
heat (BTU)
Q
heat emitted at stack exit
QH
(BTU/sec)
plume radius as a function of time (ft)
R(T)
dry
Rd
gas constant for dry air (ft3 mb/lbm°R)
Rvap, Rv
gas constant for water vapor (ft3mb/lbm°R)
SCliq
Schmidt number for liquid water
Sc
vap
t,T
Schmidt number for water vapor
t
x/u O (sec)
0
time (seconds)
T*
time coordinate of virtual origin (sec)
T
physically measurable temperature (R)
T
temperature perturbation about an adiabatic
reference state (R)
temperature in a quiet adiabatic atmosphere (R)
T
Ts
temperature of stack effluent (R)
virtual temperature (R)
V
Tv
T
virtual temperature in a quiet adiabatic
atmosphere (R)
vo
downwind velocity (ft/sec)
U
U0
windspeed, constant with height (ft/sec)
U
Ueddy
Ui,u
turbulent velocity scale in an eddy (ft/sec)
velocity (ft/sec)
j
Uit Uj
velocity (ft/sec)
-118-
UjiUj
time average velocity (ft/sec)
u. ,uj.
fluctuating velocity (ft/sec)
,
U
U.
I
Reynolds stress tensor
U 0
(ft2/sec2
)
correlation of fluctuating velocity and temperature
U.
3
u'
u.' uk'
1
U.,
uk
triple correlation of fluctuating velocity (ft3/sec3)
'
v
crosswind velocity (ft/sec)
Vg
geostrophic wind (ft/sec)
w
vertical velocity (ft/sec)
W
molX
molecular weight of the pollutant species
(lbm/lb -mole)
X
downwind distance (ft)
X i Ix j
cartesian coordinate (ft)
y
crosswind distance (ft)
z
height (ft)
*,
height coordinate of virtual origin (ft)
-119-
a, a
turbulence constants
YL
reciprocal turbulent Schmidt number for liquid
water
YT
reciprocal turbulent Prandtl number for heat
YV
reciprocal turbulent Schmidt number for water
vapor
YX
reciprocal turbulent Schmidt number for pollutant
r
turbulence constant
r1
turbulence constant
rd
dry adiabatic lapse rate
(R/ft)
Ch
eddy diffusivity of heat
(ft2/sec)
m
eddy diffusivity of momentum (ft2/sec)
£m
eddy diffusivity of pollutantum
(ft2/sec)
6
potential temperature
6
potential temperature perturbation about an
adiabatic reference state (R)
0
potential temperature in a quiet adiabatic atmosphere (R)
(R)
8
time average potential temperature
0'
fluctuating potential temperature (OR)
ev
virtual potential temperature
0
virtual potential temperature in a quiet adiabatic
atmosphere (OR)
6v
v
time average virtual potential temperature (OR)
a8v
fluctuating virtual potential temperature
(i)
X x
decay constant for it h radioactive decay channel
from pollutant species X (sec -1 )
(OR)
(R)
(OR)
-120-
ft)
(lbm/sec
11
dynamic viscosity
V
kinematic viscosity (ft2/sec)
Pdry
Pliq
Pliq
P liq
PS
Psat
Pvap
Pvap
I
density of dry air
(lbm/ft 3)
liquid water density (lbm/ft 3)
3)
time average liquid water density
(bm/ft
fluctuating
(lbm/ft 3)
liquid water density
density of stack effluent
(lbm/ft 3 )
saturation water vapor density
(lbm/ft 3)
water vapor density (lbm/ft 3 )
time average water vapor density
(bm/ft
P vap
fluctuating water vapor density (bm/ft
a, a(y,z, t)
eddy viscosity (same as
alibrary (z)
prescribed eddy viscosity profile
X
pollutant density
m)
(lbm/ft 3 )
3)
3)
(ft2/sec)
(ft2 /sec)
-121-
LIST OF FIGURES AND TABLES
page
Figure 3.1
Flow Field Orientation
35
Figure 3.2.2.3.1
Phase Diagram for Water Substance
51
Figure 3.2.2.3.2
Logic Diagram for the Equilibrium
Moisture Calculation in a Single
Cell during a Single Timestep
52
Bent-Over Buoyant Plume with
Ambient Thermal Stratification
56
Mesh Alignment Appropriate for
a Line Source Release
57
Mesh Alignment Appropriate for
a Point Source Release
57
Reconstruction of the Three-dimensional Plume.
Wind vectors as a
function of height are shown.
59
Figure 3.3.4.1
Property Values of Air
62
Table 3.3.5.1
Required Input Profiles
65
Figure 3.3.5.1
Wall Numbering Scheme
66
Table 3.3.5.2
Boundary Conditions
68
Figure 3.3.6.1
Mesh Coarsening Procedure Steps
70
Table 3.3.7.1
Data Reported by the Plume Statistics Package
73
The Concept of Layers in the
Planetary Boundary Layer
80
Figure 3.3.2.1a
Figure 3.3.2.1b
Figure 3.3.2.1c
Figure 3.3.3.1
Figure 4.3.2.1
85
Table 4.3.3.1
Comparison of Eddy Viscosity
Prescriptions
Table 5.2.1
Card Input for the Dry,
Buoyant Line-Thermal
93
Card Input for the LAPPES 5
Plume of 20 October 1967
95
Card Input for the Moist,
Buoyant Line-Thermal
99
Table 5.2.2
Table 5.2.3
-122-
page
Figure 5.3.1
Key to Cellwise Quantities for
Figs. 5.3.2, 5.3.3, 5.3.4, 5.3.5,
and 5.3.6.
103
Initialized Plume Cross Section
at 0 sec.
104
Figure 5.3.3
Plume Cross Section at 20 sec.
105
Figure 5.3.4
Plume Cross Section at 80 sec.
106
Figure 5.3.5
Plume Cross Section at 200 sec.
107
Figure 5.3.6
Initialized Plume Cross Section
for the Keystone No. 1 Stack on
20 October 1967.
108
Dilution of a Steady Release of
Pollutant
110
Simulation of a Time-Dependent
Plume in Steady-State Weather
114
Profile Generation for Problems
with Mesh Coarsening.
134
Profile Generation for a Particular
Problem.
135
Figure 5.3.2
Figure 6.2.1.1
Figure 6.2.2.1
Figure B.1
Figure B.2
-12 3-
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1.
R. Scorer, Natural Aerodynamics, New York, Pergamon
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2.
J. Stuhmiller, "Development and Validation of a TwoVariable Turbulence Model," SAI-74-509-LJ, 1974.
3.
F. Pasquill, Atmospheric Diffusion, Second Ed.,
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4.
J. Richards, "Experiments on the Motions of Isolated
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5.
F. Schiermeier, Large Power Plant Effluent Study, Vol.
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R. Sklarew, and J. Wilson, "Air Quality Models Required Data Characterization," Palo Alto, CA, EPRI
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M. Winton, "Computer Codes for Analyzing Nuclear
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K. Rao, "Numerical Simulation of Turbulent Flows-A Review," ARATDL internal note, April, 1976.
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C.
Nappo,
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11.
C. DuP. Donaldsont "Construction of a Dynamic Model of
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12.
W. Lewellen, and M. Teske, "Second-Order Closure
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-124-
13.
S. Patankar, et. al., "Prediction of the Three-Dimensional Velocity Field of a Deflected Turbulent Jet,"
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14.
K. Rao, et. al., "Mass Diffusion from a Point Source
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15.
M. Dickerson, and R. Orphan, "Atmospheric Release
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16.
R. Lange, "ADPIC: A 3-D Computer Code for the Study
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17.
M. Dickerson, et. al., "Concept for an Atmospheric
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18.
J. Knox, "Numerical Modeling of the Transport, Diffusion,
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19.
J. Knox, "Atmospheric Release Advisory Capability:
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20.
R. Lange, and J. Knox, "Adaptation of a 3-D Atmospheric
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21.
R. Lange, "ADPIC: A 3-D Transport-Diffusion Model for
the Dispersal of Atmospheric Pollutants and its
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22.
C. Sherman, "Mass-Consistent Model for Wind Fields
Over Complex Terrain," UCRL-76171, May 1975.
23.
R. Henninger, "A Two-Dimensional Dynamic Model for
Cooling Tower Plumes," Trans. ANS, Vol. 17,
1973, pp 65-66.
24.
J. Taft, "Numerical Model for the Investigation of
Moist Buoyant Cooling-Tower Plumes," in S. Hanna and
J. Pell, coords., Cooling Tower Environment-1974 r
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USERDA, Conf-740302, April, 1975.
25.
D. Lilly, "Numerical Solutions for the Shape-Preserving
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-125-
26.
D. Johnson, et. al., "A Numerical Study of Fog Clearing
by Helicopter Downwash," Jour. Appl. Met., 14,
pp 1284-1292, 1975.
27.
Y. Ogura, "The Evolution of a Moist Convective Element
in a Shallow, Conditionally Unstable Atmosphere: A
Numerical Calculation," Jour. Atmos. Sci., 20,
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28.
G. Arnason, et. al., "A Numerical Experiment in Dry
and Moist Convection Including the Rain Stage,"
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29.
J. Liu, and H. Orville, "Numerical Modeling of Precipitation and Cloud Shadow Effects on Mountain-Induced
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30.
W. Cotton, "Theoretical Cumulus Dynamics," Rev. Geophys.
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31.
Chalk Point Cooling Tower Project, Vol. 1-3, PPSP-CPCTP-16,
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Auust, 1977.
32.
Chalk Point Cooling Tower Project, PPSP-CPCTP-11 and PPSPCPC'P-12, Applied Physics Lab., Johns Hopkins Univ.,
Laurel,
D, 1978.
33.
G. Tsang, "Laboratory Study of Line Thermals," Atmos.
Environ., 5, pp 445-471, 1971.
34.
J. Deardorff, "Numerical Investigation of Neutral and
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29, pp 91-115, 1972.
35.
J. Deardorff, "Three-Dimensional Numerical Modeling of
the Planetary Boundary Layer," in D. Haugen, ed.,
Workshop on Micrometeorology, Ephrata, PA, Science
Press, AMS, 1973.
36.
J. Deardorff, "Three-Dimensional Numerical Study of
the Height and Mean Structure of a Heated Planetary
Boundary Layer," Boun. Lay. Met., 7, pp 81-106, 1974.
37.
E. Spiegel, and G. Veronis, "On the Boussinesq
Approximation for a Compressible Fluid," Astrophys.
Jour., 131, pp 442-447, 1960.
38.
L. Cloutnman, C. iHirt, and N. Romero, "SOLA-ICE: A uhierical
Algorithmi for Transient Compressible Fluid Flows," UC-34,
July, 1976.
-126-
39.
J. Iribarne, and W. Godson, Atmospheric Thermodynamics,
Holland, Reidel Publ. Co., 1973.
40.
VARR-II--A Computer Program for Calculating TimeDependent Turbulent Fluid Flows with Slight Densi
Variation, Vols 1,2,3, Madison, PA, Westinghouse
Adv. React. Div., May 1975.
41.
A. Amsden, and F. Harlow, "The SMAC Method: A Numerical
Technique for Calculating Incompressible Fluid Flows,"
LA-4370, May 1970.
42.
R
43.
E. Eckert, and R. Drake, Heat and Mass Transfer, New
York, McGraw-Hill Book Co., 1959.
44.
S. Hess, Introduction to Theoretical Meteorology,
New York, Holt, Rinehart, and Winston, 1959, Chap 18.6.
45.
Ibid., Chapter 18.7.
46.
Ibid., Chapter 12.
47.
H. Tennekes, "The Atmospheric Boundary Layer," Physics
Today, Jan. 1974, pp 52-63.
48.
C. Priestly, Turbulent Transfer in the Lower AtmosPhere,
Chicago, U. of Chicago Press, 1959.
49.
R. Scorer, Natural Aerodynamics, New York, Pergamon
Press, 1958.
50.
A. Monin, "The Atmospheric Boundary Layer," in
M. Van Dyke, ed., Ann. Rev. of Fluid Mechanics,
Vol. 2, pp 225-250 1970..
51.
J. Businger, "The Atmospheric Boundary Layer," V. Derr,
ed., Remote Sensing of the Troposphere, Washington,
D.C., U.S.Dept. of Commerce, NOAA, Govt. Printing
Office, Aug. 1972.
52.
H. Panofsky, "The Boundary Layer Above 30 M," Bound.
Lay Met., 4, pp 251-264, 1973.
53.
H. Panofsky, "The Atmospheric Boundary Layer Below
150 M," in Annual Review of Fluid Mechanics,
pp 147-177, 1974.
Gentry, et. al., "An Eulerian Differencing Method
for Unsteady Compressible Flow Problems," Jour. of
Comp. Phys., Vol. 1, 1966, pp 87-118.
-127-
54.
G. Csanady, Turbulent Diffusion in the Environment,
Boston, MA, Reidel Publ., 1973.
55.
J. Lumley, and H. Panofsky, The Structure of Atmospheric
Turbulence, New York, John Wiley & Sons, 1964.
56.
A. Monin, and A. Yaglom, Statistical Fluid Mechanics,
VolA.1 Cambridge, MA, MIT Press, 1971.
57.
A. Monin, and A. Yaglom, Statistical Fluid Mechanics,
Vol. 2, Cambridge, MA, MIT Press, 1975.
58.
Ibid., Vol. 1, p 280.
59.
A. Blackadar, "The Vertical Distribution of Wind and
Turbulent Exchange in a Neutral Atmosphere," Jour.
Geophys. Res., 67, pp 3095-3102, 1962.
60.
A. Blackadar, and J. Ching, "Wind Distribution in a
Steady State Planetary Boundary Layer of the Atmosphere with Upward Heat Flux," AF(604)-6641 Dept.
of Meteor., Penn. State Univ., pp 23-48, 1965.
61.
G. Yamamoto, and A. Shimanuki, "Turbulent Transfer in
Diabatic Conditions," J. Meteor. Soc. Japan, Ser. 2,
44, pp 301-307, 1966.
62.
J. O'Brien, "A Note on the Vertical Structure of the
Eddy Exchange Coefficient in the Planetary Boundary
Layer," J. Atmos. Sci., 27, pp 1213-1215, Nov. 1970.
63.
R. Bornstein, "The Two-Dimensional URBMET Urban
Boundary Layer Model," J. App . Met., 14, pp 14591477, Dec. 1975.
64.
F. Nieuwstadt, "The Computation of the Friction
Velocity, u, and the Temperature Scale, T
from
Temperature and Wind Velocity Profiles by LeastSquare Methods," Bound. Lay. Met., 14, pp 235-246, 1978.
65.
H. Tennekes, and J. Lumley, A First Course in Turbulence,
Cambridge, MA, MIT Press, 1972.
66.
A. Monin, Weather Forecasting as a Problem in Physics,
Cambridge, MA, -MIT Press, 1972.
67.
VARR-II Users' Guide, Op. Cit., p. 88.
68.
Y. Chen, "Coolant Mixing in the LMFBR Outlet Plenum,"
MIT PhD Thesis, Nuclear Engineering Dept., May 1977.
APPENDIX A
Definitions of Important Variables Added to the VARR-II Code
for Simulating Plume Behavior
-128-
Definitions of Important Variables Added to VARR-II for
Simulating Plume
Behavior
Meaning
Variable
Units
BKGND
background pollutant concentration
lbm/ft 3
CHI (IK)
pollutant concentration, X
lbm/ft3
CHI0 (IK)
pollutant concentration at the
previous timestep
lbm/ft 3
pollutant concentration input
variable
lbm/ft 3
DWNDS
downwind distance
ft
EFRAC (J)
fraction of the energy of the Jth
radiation deposited in the plume,
CHII
F(j)
x
ELAM (J)
energy of the jth radioactive decay
channel, E(j)
x
MeV
GAML
reciprocal turbulent Schmidt number
for cloud liquid water, yL
GAMV
reciprocal turbulent Schmidt number
for water vapor, yV
GAMX
reciprocal turbulent Schmidt number
for pollutant, OyX
LIQ (IK)
cloud liquid water density, Pliq
lbm/ft 3
LIQ0 (IK)
cloud liquid water density at the
previous timestep
3
lbm/ft
LIQI
cloud liquid water density
variable
input
lbm/ft 3
NCHAN
number of decay channels to be input
NPR0F
number of input profiles to be input
RLAM (J)
decay constant of the jth decay
channel, X(i)
NCHAN
x
sec
RLAMB
net decay constant,l
sec
SER
specific energy release rate of the
radioactive pollutant
MeV/sec
total internal energy on the mesh
(assumed 1 ft deep)
BTU
SMSIE
jl
(j)
x
-1
-1
-129-
Variable
TRSTRT(J)
Meaning
Units
time for program coarsening
(J=l,
2,
...
5)
VELCHI
pollutant-averaged velocity
sec
ft/sec
VAP (IK)
water vapor density, Pvap
lbm/ft 3
VAP0 (IK)
water vapor density, at the previous
timestep
lbm/ft 3
VAPI
water vapor density input variable
lbm/ft 3
WM0LX
molecular weight of the pollutant
lbm/lbm-mo le
WWSP (J)
wind speed profile variable
ft/sec
WZAP (J)
absolute pressure profile variable
mb
WZLQ (J)
liauid water profile variable
lbm/ft 3
WZSIE (J)
specific internal energy profile
variable
BTU/lbm
turbulence kinetic energy profile
variable
ft 2 /sec
WZTS (J)
eddy viscosity profile variable
ft 2 /sec
WZVP (J)
water vapor profile variable
lbm/ft 3
YPLUME
plume center height
ft
WZTQ (J)
APPENDIX B
Card Formats for Input Variables
-130-
Card Formats for Input Variables
Input Variables/Card
Card No.
Format
1*
IBR, KBR, IPRFM
(2(5X,I5),7X,I2)
2
LABEL
(10A8)
3*
DT, TPRT, TPLT, TWTD, TFIN,
ITAPW, NPRT, IDIAG, LPR, IBS,
IDG, KDG
(5F8.3,
DX, DZ, GX, G Z, ALX, ALZ, CYL,
B o , EPS, VMIN
(10F8.3)
4*
512,
213)
5
KWR, KWL, KWT, KWB, FSLIP,
ALP, GAM, ALP O, GAM 1 , NU, TQJET,
(412, 8F8.3)
TSJET
6
AW, BW, CW, WEPS, KDERBC, UBRI,
UBRI, UBLI, WBTI, WBBI
(4F8.3,I2,4F8.3)
7
W0BI, UBI, CSUBP0
(3F8.3)
8*
TGAM, T,
NRESEX
9
TI, TSTEP, MAT,
(4F8.3,
212)
AI, BI, CI, AR, BR, CR, AMU,
BMU, CMU
(10F8.3)
10
AK, BK, CK, ACP, BCP, CCP
(10F8.3)
11*
NFLOW, NT1, NT2, NT3, NT4,
NT5, NTAU
(7X, I3, 5(5X,I3),
7X, I3)
12*
TYMF(I),
13*
TYMYT1 (I), TIN (I)
(8F8.3)
14*
TYMT2 (I), T2N (I)
(8F8.3)
15*
TAU (I)
(8F8.3)
16*
I,
COFA, CFB, CFC
(3X,I3,2X,3F8.3)
17*
I, C0FA, C0FB,COFC
(3X,I3,2X, 3F8.3)
18*
I, CFA, CFB,C0FC
O3X, I3, 2X, 3F8. 3)
F'N(I)
(8F8.3)
-131-
Card No.
Input Variables/Card
Format
19*
I, CFA, CFB, CFC
(3X,I3,2X,3F8.3)
20*
I, CFA, CFB, C0FC
(3X,I3,2X,3F8.3)
21*
I, CFA, CFB, C0FC
(3X,I3,2X,3F8.3)
22*
I, K, RXC, RZC
(2 (3X,I3), 2 (5X,F8.3)
23*
NL, NR, NB, NT, ICELTYP
(415,I2)
24*
SIEI, TQI, TSI, UI, WI, CHII,
VAPI, LIQI
(8F8.3)
25
NG0P, NOVP, DR0U, XDIV, YDIV
(2(7X,I2),3(6X,F8.3))
26
IGP (I)
(6X,6I2)
27
VNTP(I), NCVTYP(I)
(4 (5X, I2,1X, 1I3))
28
IBR, KBR, IPRFM
(2(5X,I5),7X,I2)
29*
GAMX, NCHAN, WM0LX, GAMV,
GAML, BKGND, DWNDS
(F8.3,I8,5F8.3)
30*
RLAM, ELAM, EFRAC
(3F8.3)
31*
NPROF, TRSTRT(1), TRSTRT(2),
TRSTRT(3), TRSTRT(4), TRSTRT(5)
(I8, 5F8.3)
WZSIE, WZTQ, WZTS, WZVP, WZLQ,
WZAP, WWSP
(7F8.3)
IBR, KBR, IPRFM
(2(5X,I5), 7X,I2)
32*
33*
See Special Notes to Input
Cards No. 25-28 in the original VARR-II input have been
omitted entirely in this version.
-132-
Special Notes to Input
Card No.
1
Do not attempt to "restart" the program (i.e.,
storing or retrieving the program storage on
or from magnetic tape or disc) unless
additional programming has been performed that
would "restart" the variables added to VARR-II.
Set IPRFM=l to allow the statistics package
(instead of the film generated plots) to be
printed out.
3
TPLT is now "time when to print statistics"
(sic).
Always set TWTD > TFIN, and/or make
device 8 equal to DUMMY. Set ITAPW = 8.
In IDIAG, grind time and CPU time have been
omitted.
Set LPR = 3, and IBS = 0.
4
Set ALX = ALZ = 1.0,
8
TI is ignored by the program.
a value of about 10-2.
and CYL = 0.0
TSTEP should have
11-15
All of the values on these cards are unimportant
for problems that have no inflow/outflow and no
obstacle cells.
However, the code cannot
skip these cards, so dummy information must
be supplied.
16-22
Set I =
23-24
Because of the solid-wall boundary conditions,
only the real fluid cells on the mesh need to
Cards of type 23 and 24 can
be initialized.
be repeated as many times as desired when initializing; the sequence is terminated when NL = 0
Note that CHII, VAPI, and LIQI have
on Card 23.
been added to Card 24.
25-28
These cards have been dropped from the input
list.
0.
29
See Appendix A
for definitions.
30
This card is repeated NCHAN times with the
See
values corresponding to J=1,...NCHAN.
Appendix A for definitions.
31
See Appendix A
for definitions.
-133-
Card No.
32
This card is repeated NPROF times with the
values corresponding to J=l, ...
NPROF. For
a problem that is 20 real cells high, the code
needs 20 values for each variable if there are
no mesh coarsenings.
If there is one mesh
coarsening carried out, the code needs 30
values (consisting of the first 20 plus 10
more to initialize the new cell area).
If there
are two mesh coarsenings carried out, the code
needs 40 values, etc. The cell center height
at which a profile needs to be defined can be
generated by considering Figs B.1 and B.2.
See
Appendix A for definitions.
33
Set IBR = -1
to terminate the problem.
-134WZ
(J) to be
defined at a height:
J
4KBR
2+
1
4KBR
2
KBR
cells, each with height 4DZ
2
KBR(2DZ) + 6DZ
+ 1
3KBR
2
3KBR
2
JI
KBR(4DZ) - 2DZ
3KBR
2
3KBR
KBR(4DZ) + 4DZ
KBR(2DZ) + 2DZ
KBR(2DZ) - DZ
- 1
KBR(2DZ) - 3DZ
KBR
2R cells, each with height 2DZ
KBR + 2
KBR(DZ) + 3DZ
KBR + 1
KBR(DZ) + DZ
KBR
KBR(DZ) - DZ/2
KBR - 1
.
KBR cells, each with height DZ
3
5DZ/2
2
3DZ/2
1
DZ/2
7_
T
/ EARTH
__
/
I/
r
-
Figure B.1 Profile generation for problems with mesh coarsening. KBR real cells
are placed on the computer mesh at a time. The cell height is DZ initially.
The cell center height at which each of the seven profiles must be specified is
developed in the figure.
-135-
WZ __.J) to be
found at a height:
J
42
4200 f
41
-
1
10 cells, each 400 ft. high
i
__________________________________________
3900
40
10 cells, each 200 ft. high
32
31
2100
30
1950
10 cells, each 100 ft. high
22
1150
21
1050
___________________________________________
20
975
20 cells, each 50 ft. high
125
3
75
i
25 ft
12
I
e
0
EARTH
y
7I
-
EARTH
Fig. B.2. Profile generation for a particular problem.
The problem is 20 cells high, each of which is iO feet
tall initially.
APPENDIX C
Statement Number Cross-References
-136-
Statement Number Cross-References
The statement number cross-references are intended
to aid the user in understanding the purposeof the altered
or added FORTRAN statements.
Any statement that was
altered or added is labeled in columns 73-80 with a "card
entry" that begins with the initials RGB or MWG.
The first cross-reference consists of 12 effects or
problems that are solved by the cards that are listed
below the problem statement.
For example, a constant heat
flux from the top wall of an obstacle cell (Effect No. 12)
is permitted when a single statement (No. 1688) is altered.
Statements that were removed from the code are not listed.
A second cross-reference allows any altered or added
statement to be identified with the problem that it helps
to solve.
For example, statement #1688 is identified with
effect #12--and looking up effect #12 in the first crossreference shows that no other cards were involved in the
solution to that problem.
-137-
EFFECTS OR PROBLEM(S)
1.
EFFECT NO. 1
CONSIDERED
Non-executable statements for COMMON, DIMENSION,
REAL,
INTEGER, or EQUIVALENCE CARDS.
(Two executable statements
(91,92) are included here
since they are related to program control.)
SUBROUTINE (S)
MAIN
CARD ENTRIES
PROGRAM STATEMENT NUMBERS
RGB MN60A,B
32,36,53,55,56,86,91,92
II
RGB MNO 1A
35,85
I!
RGB VM62A
50,52
RGB VM55A, B
51,54
RGB MN60OA, B
125,129,146,148,149,179
i,
IDLE
If
It
VRPRT
II
RGB MN01A
128,178
RGB VM62A
143,145
RGB VM55A,B
144,147
RGB MN60OA,B
392,398,415,417,418,448
RGB MNO1A
397,447
RGB VM62A
412,414
It
RGB VM55,A,B
413,416
VSET
RGB MN60,A, B
512,516,533,535,536,566
I!
RGB MN01A
515,565
IN
RGB VM62A
530,532
II
RGB VM55,A,B
531,534
-138-
EFFECTS OR PROBLEM(S)
non-executable
SUBROUTINE (S)
MESHMK
f
VM
CONSIDERED
EFFECT NO. 1
(continued)
CARD ENTRIES
RGB MN60A,B
PROGRAM STATEMENT NUMBERS
685,689,706,708,709,739
RGB MN01A
688,738
RGB VM62A
703,705
RGB VM55A,B
704,707
RGB MN60A,B
1071,1077,1094,1096,1097,1127
I!
RGB MN01A
1076,1126
I,
RGB VM62A
1091,1093
I,
RGB VM55A,B
1092,1095
-139-
EFFECTS OR PROBLEM(S) CONSIDERED
1.
EFFECT NO. 2
Boundary conditions for crosswind runs for the variables
U, W, SIE, TQ, TS.
2.
Set
SUBROUTINE(S)
l1
ibrary (Z)
CARD ENTRIE
and q > qlibrary(Z)
PROGRAM STATEMENT NUMBERS
VM
RGB VM5OA
VM
RGB VM51A
1301,1305,1307,1309,1467,
RGB VM02A
1489,2199,2200
1302,1369-71, 1379
VM
1466,1847
-140-
EFFECTS OR PROBLEM(S)
1.
EFFECT NO. 3
CONSIDERED
Boundary conditions for crosswind runs for the variable,
CHI.
2.
Pollutant transport equation.
SUBROUTINE (S)
VM
l·-·------·Plsl··li·V-lll-----
CARD ENTRIES
PROGRAM STATEMENT NUMBERS
RGB VM52A,B,C
1310-12,1475-77,1959,1980,
1988,2047-52,2277-78,2358,2366,
2469
---L-L-IYLI
-
LII__-_
1I_III_11I_____
-141-
EFFECTS OR PROBLEM(S)
1.
EFFECT NO. 4
CONSIDERED
Cell initialization from input data.
SUBROUTINE (S)
MESHMK
I
CARD ENTRIES
PROGRAM STATEMENT NUMBERS
RGB MK01A,02A
946,952,960
RGB MK60A
930-31,953-54,961-62,1027,
1049-50
-14 2-
EFFECTS OR PROBLEM(S)
1.
EFFECT NO. 5
CONSIDERED
Atmospheric profile read-in and printout for variables,
WZSIE
(internal energy)
WZTQ
(turbulence kinetic energy)
WZTS
(eddy viscosity)
WZVP
(water vapor)
WZLQ
(cloud liquid water)
WZAP
(pressure)
WWSP
(wind speed)
CARD ENTRIES
PROGRAM STATEMENT NUMBERS
VM
RGB VM62A
1153-58,1162,1169
VM
RGB VM55A
1151-52, 1159-60,
1165-67, 1172-74, 1179-81
VM
RGB VM70A
1161, 1168,1175
SUBROUTINE (S)
-143-
EFFECTS OR PROBLEM(S)
1.
EFFECT NO. 6
CONSIDERED
Moisture model in subroutine VM
moisture is done elsewhere),
(initialization of cell
including input, boundary
conditions, and transport equations.
CARD ENTRIES
PROGRAM STATEMENT NUMBERS
VM
RGB VM60OA,B
1072,1133-37,1163-64,1170-71,
1177-78,1313-16,1478-81,
1960-61, 1981-82,1989-90,
2053-64,2359-60,2367-68,2470-71
VM
RGB VM62A
2225-26,2230-55
VM
RGB VM61A
2201-24
SUBROUTINE (S)
-144-
EFFECTS OR PROBLEM (S)
1.
CONSIDERED
EFFECT NO. 7
Mesh coarsening capability.
CARD ENTRIES
PROGRAM STATEMENT NUMBERS
IDLE
RGB ID55A
202-209, 214-327
IDLE
RGB ID70A
210-213
VM
RGB VM55A,B
1132,2553-56,2564-68
VM
RGB ID55A,B
2557-63
SUBROUTINE (S)
-145-
EFFECTS OR PROBLEM(S) CONSIDERED
1.
EFFECT NO. 8
Cell by cell output of variables by the printing subroutine.
SUBROUTINE (S)
VRPRT
H
CARD ENTRIES
RGB M060A
RGB M01A
PROGRAM STATEMENT NUMBERS
394
451-507*
The entire section was altered, although not all of the
cards have card entries.
-146-
EFFECTS OR PROBLEM (S)
1.
Plume
CONSIDERED
EFFECT NO. 9
statistics package.
SUBROUTINE (S )
CARD ENTRIES
PROGRAM STATEMENT NUMBERS
VM
RGB VM54A,B
2518-21, 2529-30,2545-47
VM
RGB VM70A
2514-17,2522-28,2531-33
VM
RGB VM55B
2548-50
-147-
EFFECTS OR PROBLEM(S) CONSIDERED
1.
EFFECT NO. 10
Radioactive decay heating.
SUBROUTINE (S)
VM
CARD ENTRIES
PROGRAM STATEMENT NUMBERS
RGB VM56A
1073,1138-50,2227-29,2273-74,
2276
-148-
EFFECTS OR PROBLEM(S)
1.
Turbulence model errors.
2.
SIEX error.
SUBROUTINE (S)
VM
MWG 04/78
If
A,
_
_
,
CARD ENTRIES
RGB VM000
It
.__
CONSIDERED
RGB VMO1A
.
.
EFFECT NO. 11
PROGRAM STATEMENT NUMBERS
2193-94
2195-96
1361,1416 J1526,1591,1816,
1877
-149-
EFFECT NO. 12
EFFECTS OR PROBLEM(S) CONSIDERED
1.
Constant heat flux from obstacle top wall.
SUBROUTINE (S)
VM
CARD ENTRIES
RGB VM03A
PROGRAM STATEMENT NUMBERS
1688
-150-
_____IIXIII__IIII__l.^l_.-_--
Statement
Number
Effect
Number
32
1
946
4
35
1
9-954
4
36
1
960-962
4
50-56
1
1027
4
85-86
1
1049-50
4
91-92
1
1071
1
125
1
1072
6
128-129
1
1073
10
143-149
1
1076-77
1
178-179
1
1091-97
1
202-209
7
1126-27
1
210-213
7
1132
7
214-327
7
1133-37
6
392
1
1138-50
10
394
8
1151-62
5
397-3 98
1
1163-64
6
412-418
1
1165-69
5
447-448
1
1170-71
6
451-507
8
1172-76
5
512
1
1177-78
6
515- 516
1
1179-81
5
530-536
1
1301-02
2
56-566
1
1305
2
685
1
1307
2
688-6 89
1
1309
2
73- 709
1
1310-12
3
738-739
1
1313-16
6
930-9 31
4
1361
11
Statement
Number
Effect
Number
-151-
Statement
Number
Effect
Number
Statement
Number
Effect
Number
1369-71
2
2276
10
1379
2
2277-88
3
1416
11
2358
3
1466-67
2
2359-60
6
1475-77
3
2366
3
1478-81
6
2367-68
6
1489
2
2469
3
1526
11
2470-71
6
1591
11
2514-33
9
1688
12
2545-50
9
1816
11
2553-68
7
1847
2
1877
11
1959
3
1960-61
6
1980
3
1981-82
6
1988
3
1989-90
6
2047-52
3
2053-64
6
2193-96
11
2199-00
2
2201-26
6
2227-29
10
2230-55
6
2273-74
10
APPENDIX D
Computer Code Listing
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ARE APRAY NAMES AND Nl,N2,... ARE INTEG ER VALUES OR
EXPRESSIONS GIVIN3 THE ARRAY SIZES.
I.E. - CALL ERASE(C,26*31,N.7*31.E,254)
START 0
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BALR
12, 0
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SR
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L
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4,4 (2,1)
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7,0 (4)
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LTR
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TEST FOR LAST ARGUMENT IN LIST
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RETN
A
2,=FP8'
B
El
PICK UP NEXT ARGUMENT PAIR
RETN
RETURN (14,12),T
END
INTEGER BUFL,C,CF1 ,CFB,CFC,CFI,CFL,CFR,CFS,CFT,CQF,ERF,TD, V NTP,
1
VTP
REAL NU,LI Q, LI QO,LII, LOUT
DIMENSION C(1),CQ(1) ,QCON (1 ) P(1),RX(1)RZ (1),TQ(1) ,TS(1),U(1),
1
W( 1),ER(1),FX3(3 2),FFY3(102),PBTI ( 2),UO (1),WO(1),TQO(1),
2
TSO (1), SIE(1) ,SIE (1),CH(1)CHIO(1)
A,VAP (t),VAPO (1),LIQ(1) ,LIQO( 1)
RGBMN6 OA
RG BMNO 1A
RGBMN6 OA
PAGE
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