NUMERICAL MODELING OF BUOYANT PLUMES IN A
TURBULENT, STRATIFIED ATMOSPHERE
by
Ralph Ge
Bennett
and
Michael W. Golay
Energy Laboratory Report
MIT-EL 79-002
January, 1979
I
NUMERICAL MODELING OF BUOYANT PLUMES IN A
TURBULENT, STRATIFIED ATMOSPHERE
by
Ralph G. Bennett
and
Michael W. Golay
Energy Laboratory
and
Department of Nuclear Engineering
MassachuSetts Institute of Technology
Cambridge, Massachusetts
02139
MIT-EL 79-002
January, 1979
Sponsored by
The Consolidated Edison Company of New York, Inc.
Northeast Utilities Service Corporation
MIT-EL
79-002
ERRATA
,tt
Page
Line
64
5
speed
78
9
atmospheres
84
3
library
116
17
122
8
0.50 OF
126
3
analysis
11
19
uncertainty
The name of Pasquill is in
common usage today for
plume modeling.
The
bivariate Gaussian plume
description was originated
by O. G. Sutton
Soco A.
135,
(Proc. Roy.
pp. 143-165).
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,
bent-over plumes in realistic atmospheres is constructed.
To do this, 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.
A demonstration of the ability of the model to reproduce the solutions to problems that are known is undertaken. Comparisons to buoyant line-thermal laboratory
experiments show that the model calculates the dynamics of
the fluid motions to an acceptable accuracy. Comparisons
to atmospheric plume rise and dispersion experiments show
that the model can simulate individual plumes more accurately than existing correlations because it calculates the
effect of the atmospheric turbulence and stratification
from first-principles. The comparisons also show that
improvements to the model are likely to be made by more
accurately describing the anisotropic nature of atmospheric
turbulence, and the production of turbulence by the sources
of buoyancy.
TABLE OF CONTENTS
1.
2.
ACKNOWLEDGEMENTS
6
Problem Description and Solution
8
1.1
Introduction
8
1.2
Background and Problem Description
9
2.2
9
1.2.1
Historical Background
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
16
Literature Review
2.1
3.
page
Numerical Plume Models
16
2.1.1
Three-Dimensional Models
17
2.1.2
Two-Dimensional Models
19
2.1.3
Experimental Studies
20
Numerical Planetary Boundary Layer Modeling
Hydrodynamic Model Development
21
23
3.1
Introduction
23
3.2
Model Equation Sets
23
3.2.1
3.2.2
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
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.
Equilibrium Cloud Microphysics
Model
49
Latent Heat Source Term
50
Model Solution Methodology
53
3.3.1
The VARR-II Fluid Mechanics Algorithm
53
3.3.3.2
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
3.3.3.7
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
Results
90
5.1
90
Introduction
-5-
page
5.2
5.3
5.4
6.
Comparisons to Analytical Models
93
5.2.1
General Nature of the Solutions
93
5.2.2
Turbulent, Buoyant Line-Vortex Results
108
5.2.3
Brunt-Vaisala Period of a Turbulent,
Buoyant Parcel
112
Comparisons to Field Studies
115
5.3.1
Pasquill Dispersion and Briggs Plume Rise
in Neutral Atmospheres
115
5.3.2
LAPPES S0 2 Dispersion Studies
120
Results of Model Extension
133
5.4.1
Fumigation Episode
133
5.4.2
Plumes with Change of Phase
134
Conc lusions and Recommendations
137
6.1
Model Validation
137
6.2
Recommendations
139
6.2.1
Calculational Scheme to Include Wind Shear
139
6.2.2
Calculational Scheme for Time-Dependent
Releases
142
6.2.3
Cloud Microphysics Model
143
NOMENCLATURE
146
LIST OF FIGURES AND TABLES
151
REFERENCES
154
APPENDIX -- Computer Code Listing
160
-6ACKNOWLEDGEMENTS
This research was conducted under the sponsorship of
Consolidated Edison and Northeast Utilities Service Corp.
The authors gratefully acknowledge their support.
-7-
Blank Page
d
-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
turbulence.
Typically the models are successful at analyz-
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
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.
In
general, the models seek to integrate a closed set of Reynoldsaveraged 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
Stuhmiller 2 , 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 model 3 and
the entrainment model of Richards4 for idealized cases in
which these models apply, and for several cases from the
LAPPES 5 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
are available as computer programs.
Several recent reviews
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 modelingl 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.ll Only integrations of the pollutant
transport equation are needed in Lewellen's model because
of the adoption of a complete PBL solution.
model 1
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. 1522 A mass-consistent threedimensional 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, wet and dry deposition, and surface
terrain have been added.
-19-
2.1.2
Two-Dimensional Models
Two two-dimensional numerical plume models have been
Henninger's model23 solves contin-
found in the literature.
uity, 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
than that of the present work.
to the model in this work
Taft's model24 is much closer
, 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 model2 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 model27 of rain cloud dev-
elopment also assumes a constant eddy viscosity, while
Arnason's model2 8 ignores eddy transports altogether.
Liu's
model 2 9 employs a stratification of atmospheric turbulence
While the treatment
into two constant eddy viscosity layers.
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
rork is validated
against is the Large Power Plant Effluent Study (LAPPES).
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 SO 2 concentration cross sections
are of the most interest in this work.
The Chalk Point
Cooling Tower Project (CPCTP)3 1 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 Tsang33 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
boundary layer has been reported by Deardorff3 4 36 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
Planetarv boundary laver field studies.
To alv the model to a single source of pollutant, a
sinale 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
,
, T, and
i. are
physically measurable values of the fluctuating pressure,
density, temperature, and velocity, respectively:
Continuity Eq:
at+
=
(3.1)
Momentum Eq:
at
3a
i. u) ax.
aa
ij
X - 0gi
ax.x1
+
(3.2)
(3.2)
11-i
Energy Eq:
at
gt+ax
axT)+
(-j+T)
~~
j
=
~
a
+9
Cx.
xj]
T
1
j
cp
t
(3
3)
-25-
Equation of State:
p
= pRT
These equations have property values
depend upon temperature in general.
(3.4)
, 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
the value
of a primitive variabl
its value in
an adiabatic
- atmosphere
(function of
height only
+
a departure
from the
state at
est
(3.5)
or, in terms of the notation in this work
P _. P
+ p
(3.6)
P
+ P
(3.7)
T -
T
+ T
(3.8)
-
0
+ Ui
(3.9)
.i
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 u i to zero.
The continuity and energy equations be-
come trivial under this substitution.
The momentum equation
becomes the hydrostatic equation:
dPo
dz
Pg
=
(3.10)
The equation of state is simply
po = P Ro
(3.11)
The First Law of Thermodynamics for an adiabatic process is
dQ =
= cpdTo - dPo /p
(3.12)
Dividing by a displacement dz gives
dp
dz
dT
=o
Pocp dz°
(3.13)
and substitution of Eq. 3.13 into Eq. 3.10 gives rd, the lapse
rate of the dry adiabatic atmosphere:
dT
T
5
rd
d
9.76
_C/km
dz
(3.14)
cp
To this point the solution of the adiabatic atmosphere
has been presented.
Substituting the reference state decom-
position, Eq. 3.6-Eq. 3.9 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
au.
+
o
axc
a2T-
Ed_ k
0~ = - .
+
Co P- a
POaX
=
(3.15)
au2
aX
(3.16)
Momentum Equation:
au.
P Uj
o at +. + pou
x.
=
-a x.i
Pgi
+
/
Energy Equation:
2
P
aT +
-
oUj
aT
aXjI3
Rd
T
(3.17)
7
Equation of State:
P =
P0
T + P
To
p0
+
pT
(3.18)'
P T
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/p0 -
-T/T
(3.19)
-28-
which is the familiar Boussinesq equation of state.
This
equation allows the buoyancy term (-pgi/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:
a
(fi
11
a.
<
«X.
(3.20)
andll 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.:
a u.
3= 0
(3.21)
ax.
Momentum Eq:
2
au.
1 + u
au.
a
+ U
ao
ap
1
jar
xPo
x.
p
-
11
o
g.
9(3.22)
+
I
au.
1
;x
Energy Eq:
T+u
"
-
~' j~x]
=+ P -aT o
~~O
~~x:
(3.23)
-29-
Define the potential temperature,
, as
T 10p
(3.24)
Differentiating with respect to height finds that the adiabatic atmosphere has a lapse rate of potential temperature of
zero,
de
0
dzdz
(3.25)
or that the potential temperature is a constant in an adiabatic atmosphere.
Errors introduced by evaluating density
with 8 instead of T are assumed to be small (this is investigated in Sec. 3.3.4).
Neglecting the perturbation p with
respect to p 0 in Eq. 3.24, and approximating p
as p(0o ) in
Eq. 3.22, the use of 8 instead of T in the primitive equations (Eq. 3.21-Eq. 3.23) gives
Continuity Eq:
au.
3
(3.26)
Momentum Eq:
9u.
t
u.
+
1
1
9p
P(6 o)
x.
'
z
1
j D.
p(6)-p()
P()
p)
gi
9 2u.
+ v
(3.27)
3
Energy Eq:
a6+
at
u
a X.
vPr
x.
(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,
38
The transients are
lead to strong transient fluid motions.
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 + P'
(3.30)
e
+ 8'
(3.31)
P + P'
(3.32)
-8
P
U.iu.+ U.
1 )
J
(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 j/
x.
= 0
(3.34)
x
0
(3.35)
au'
j
Momentum Equations:
aui
a 1u
-
at +
j
1
Xe
p
ax.
J3
P(e)-P(6 )
p(0 0o)
+
a 2u.
a
1
3x
(u] u'.)
j
au'
at
-1
p(6O)
+
a u!
1
jaX
ap'
ax 1i
,au.
3i X
au!
ax
ax.
3
P(e')-p(e 0 )
P(e)
o
(3.36:)
1
9i
+
.2u
---i
a x2
I
(
J
)
=
(3. 37)
-32-
Energy Equations:
e
U
ae
j aX
ae'I
at
e
-1a2
VPr ax
j axj ja~~~ ~x.
aza
j
J
t+
`at
-
--
=
a
i U
J ~U
i
i
.
Jx
)
j
ael
U
a
-.
-
.
3
a
a
xIj
(3.38)
J
-1 a 2 e,
(u!G'J = VPr
Iax
ax.;
(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!e
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:
au.
a u.
D uu
XXkJ
= U-1lu
Dti;j
a xk
uj3U
-a
I
3Xk (ulu 3u)
-l
PF
1
P0
a
ax
(p'u3)
aU!
pI
1
a
Po aXj
P
1+__
'
J
0
au
i
ax.1/
3
production
terms
turbulent
transport
term
pressure
diffusion
terms
tendency
toward
isotropy
term
-33-
+
buoyant
production
terms
+ g iu-)
1
u-3
+ (gi
(i
2
a (uuT)
+
V
molecular
diffusion
terms
1.
a xk
-2va 1
dissipation
term
3
axk
Xk
(3.40)
D
D
Dt
(e,)
1
'
= -uu'.,
a
ax.
j
j
J
a (!
ax.
a
u
a e
=
U
1 3
, ')
(p
oTO
(P
FO )
~
ae''
p
xi
Po
-- gi 8e8'
1
a xD
U.
ax.
J
production
terms
turbulent
transport
term
pressure
diffusion
term
tendency
toward
isotropy
term
buoyant
production
term
molecular
diffusion
term
-34-
a e'
-au!
-
dissipation
term
1
2v
j
j
(3.41)
A discussion of the individual terms noted in Eq. 3.40
and Eq. 3.41 can be found elsewhere.ll
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
Dt = 2
Dt+
v
ay
+
=
TT,
(aw
)aw
)
2
2
(()av
+y z)
(z
+
yaz
ay
a
z)
cay
atz
-4q2
a z(3.42)
(43q )
-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
Xj
z
=Z3W
I -M.~
w
CROSSWIND
Xz
y
to v
ZZ
DOWNWIND
x,
x
-36-
The off-diagonal terms of the Reynolds stress tensor are related to a scalar eddy viscosity, a,
and
aW
3-a xj
+
u /,
xi/
has the following transport equation:
2
Da
where u!u! =
_
Dt =
+r
3/av 2
q \ta- "a +
ay3
a
aw2
13v
z +a
+
+
aw
z)
2
a
z(g,)
(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'T which is specified along with the three
other turbulence constants a, r,
and r1l.
With this turbulence
closure, the continuity, momentum, and energy equations become,
in a two-dimensional flow
3
at +
Tt
+
+w
3y
(v )
+
(VW)
P(0)
(vw)+
32 (w2)=
a3zt
-e3w
P(e 0
=
)
(3.44)
+
y
3y +3y
a z = aa
zw
a
w-z
+
a
(
)
(3.45
gz
-(0)
ay
g) ) +az (aU--) (
P(
(o)
)
(3.46)
-37-
at
+a; v)+
a zz (ew)=
''d~ay
+
az (Ta-z
)
(3.47)
With an internal energy variable, I, defined as Ic
tions 3.42-3.47 are solved by the VARR-II code.
, equaP
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, x
The pollutant is
is added to the set of Eqs. 3.42-3.47.
assumed to be a neutrally buoyant, passive species, although
The
it may be contained in a buoyant stream of effluent.
assumption that the species is neutrally buoyant could be relaxed, but the model is felt to be useful in modeling most
The turbulent diffu-
dilute pollutants in its present form.
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 derivativej
of X
[turbulent transport
1 =of
X
Vrate of
I - destruction
.of
(3.48)
X
-38-
which is represented here as
a x +3
a
+
a
(i) X
x)
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
X
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)
x
and energy Ei).
A fraction F(i)
x
of the energy is deposited within the plume, yielding an
energy release rate of
-39-
BTU-atoms
- radioactive
= 4.151x1010
,4
xx
PWmolX
MeV-ibm -mole N
/X
¢'
)
x
i
(3.50)
where WmolX is the molecular weight of X in
bm/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.
Moist Equations
3.2.2
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,
-vap,and the amount of cloud liquid
water is described by the liquid density moisture variable, Pliq'
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:
=
dry
+
vap +
(3.51)
liq
In this work the contribution to the density of the typically
small amount of cloud liquid water is ignored, (there is usually 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.
perfect gas law to
Dropping the Pliq term and applying the
dry and Pvap yields:
P
RdT
d
RvT
v
va
(Pdry vap)
RdT
v
dv
(3.52)
RT
dv
-41-
where p is the total pressure, mvap
and mdry are molecular
weights and the virtual temperature, T v , is
I+
T
v
drYPvap/mP
vap.dry
T
Pvap/
1+
(3.53)
dry
It is very important to note in Eq. 3.52 that the virtual temperature is a fictitious temperature that is used in the dry
gas equation of state to give the density of moist air.
Gen-
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:
the value of
a primitive
variable
its value in a uniformly moist adiabatic
atmosphere (function
of height only)
=
a departure
from the
state at
rest
(3.54)
Or, in the notation of this work:
P
PO + p
(3.55)
P
PO + p
(3.56)
Tv - T vo + Tv
(357)
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:
(3.59)
/RdTv
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:
ii-0
_Pog
(3.60)
The equation of state is simply
(3.61)
PO = poRdTvo
The first Law of Thermodynamics for an unsaturated adiabatic
process in this atmosphere is
dQ = 0 = cpmoist dT
dp/p
(3.62)
°
as that
Approximating the heat capacity for a moist gas, c pm oist,
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
vo =
= 9.76 0 C/km
dz
CP
(3.63)
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 temperature.
The equations for the perturbations are identical to
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.
a u.
a3
x0
axj
(3.64)
Momentum Eq.
at
+u u
au
=
x.
P
Po
E
++ T
vo
gi
i +
Po
a
9
(3.65)
phase
(3.66)
x
Energy Eq.
-laT
a2T
aT
at+Uj a
=
vPr
1
P
Dp-
(
Dt
The latent heat release term is considered in Sec. 3.2.2.4.
Define the virtual potential temperature,
ev,
as
-44-
8vv - Tvv
Rd/Cmoist
p
1000
0
p
(3.67)
Again assume that cmoist is essentially equal to c
P
.
Differ-
P
entiating with respect to height finds that the moist unsaturated adiabatic atmosphere has a lapse of virtual potential
temperature that vanishes:
de
(3.68)
= O
vo
dz
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
ev
instead of Tv in the primitive
(Eq. 3.64-Eq. 366) gives
Continuity Eq:
a u.
=
(3.69)
0
Momentum Eq:
3 ui
at
+
+ Uj
3 i
-1
3
-- F
ax
-x
p
p(evo)
vo ax i
P(ev))-p( )0
-
P(eOvo
o
g+v
9
ui
ax2
(3.70)
Energy Eq:
ev +u
+Uj
t
-1
x
Jx
vPr
2 v
2
xj
L
pc
p
yDPvap
phase
Dt
(371)
-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
in the primitive equations.
This formulation is common (al-
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
rate of change
of vapor or
liquid
Diffusion of
vapor or
liquid
Gain or loss of
vapor or liquid
due to phase
changes
+
(3.72)
or, in the notation of this work:
vap
at
Pli
j
3
+ u.
ap
ax
lila
a
p
1
vSc
1
a2 P
xa
2
liq .a.
(DP
Dt/
q
-
phase
(3.73)
D
D
where the gain or loss of vapor due to phase changes,
(Dvap/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!
vap,
term is discussed in Sec. 3.2.2.3.
Dt /phase
-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
8
p + p'
(3.75)
8 + 8'
v
(3.76)
-
p+ p+
Uj 4o
Pvap
Pliq
'
p'
(3.77)
+ u
(3.78)
vap + Pvap
(3.79)
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:
au.
ax.
]
= 0
(3.81)
-47-
Momentum Eq:
au
Uj X.
at ~t~~~~~
3 u.
-1
P' vo
P e )-p(e
ap
a xi
3 2 ui
gi1. + va- X2
)
Vo
-
(U! U )
(3.82)
3x. i 1
Energy Eq:
D
v
t
e
1_W
= vPr
+
j3
a
3x -
2
x 3.
(u!e )-
3
jv
L
Dt
p(Ov ) c
VP
)
phase
(3.83)
and the transport equations for moisture, Eq. 3.73 and Eq. 3.74
yield
Vapor Eq:
3a
at
-1
va
vap
1
=
'vap
+ vScvap
+ Uj =X.
3rvap
(
a 27vap
2
ax
-
+ D vap
x(p' o u ) DDt
) phase
x.va
(3.84)
aX (
(3.84)
Liquid Eq:
!E iq
at
1 iq
+uj
P'liq
3Cx
X
1
=
(
lisL -3vap)
(Piiquj)a
x
phase
Dt
(3.85)
Rather than providing the full equations for the correlated
fluctuations uu,
u!,
i;
u
·
u J vap'
and u'
7,
jPliq'
the turbulence
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:
av
+
ay
aw
=
az
(3.86)
0
Momentum Eqs:
av
a.
-1
+ a (.
3y
p (vo) ay
Dv
Dt
-1'
Dw
Dt
peO Fvo
Pcevo
p'vo"'
+
y
3az
gz+vo) a
(f
(a
)
D
(3.87)
w)+
.+ a-f
a
aw
Energy Eq:
a (c e
a
Dt (p 6V)
-5-
(YTa
a y
(Ce v)
)+ a z
aY
(YT c
L
z)
Dt
P (V)
phase
(3.88)
Vapor Eq:
D
Dt
ay
vap
a
a PvaP)
a
-
(TvCya
+ a
(vC
_vap
I P
yapz) +(
Dt
phase
(3.89)
Liquid Eq:
D
-
a ( y ayiiq)
Dt Pliq = a(YLa
+
a (Y
z
L(Io
Pliq)
z
(
Eddy Viscosity Eq:
Da
2
D5t
q
/
2
a y/
+1(
+W)
2
+ (aw))
q
Dt
phase
(3.9 0)
-49-
+ r((
a a()
aas))- rl(
a aa)+(ak
a a(a)
(3.91)
Turbulence Kinetic Energy Eq:
Dq = 2a
)+
+
1 av +
(I
w)+
as a + aZ a
O ay
az
4qa
sa
z(3.92)
Pollutant Eq:
DXI a
Dt
ay
o -2Y
)( ++
Y)
+Xa
(
z
)
-
X
(3.93)
i=l
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, Pvap'
in the presence of liquid water would be e sat (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: 3 9
log0e
The
(
=
2937.4
1~~~~
T
4.9283 log1OT + 23.5518
0 sat
10(3.94)
) 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
M
w
.9'-4
vapor
en by
nula
.ncluding
line
9-1
.1-4
5
subcoole,
vap,
6.11
liquid-
a p
nr
it
Mc
M5
VAPOR
__
M
273
TEMPERATURE,
Fig
3.2.3.1
T
(°K)
Phase Diagram for Hater Substance
-52transport vapor and liquid using
updated velocities
liquid evaporates
completely
Di vap
\Dt aps
e
Dt Phase
vapor condenses
Dt
Dt/=
Phase
Rsat - pvap
fsat - Pvap
Dt
Fig. 3.2.2.3.2
Dt
eDt
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
Latent Heat Release
lb BTU] = - P( L
DPvap
Dt
phase
(3.95)
where the latent heat of vaporization, L, is assumed to be a
constant, 1075 BTU/lb
m
.
The
vaphase
is found in the logic
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 code40 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.
I
-
-
-54-
The VARR-II fluid mechanics algorithm is the Simplified
Marker and Cell (SMAC) method.
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 knownthe 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
three-dimensional plume in Fig. 3.3.2.1a.
The plume has bent
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. 3.3.2.1b, or (2) 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
t
x = /
(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. 3.3.2.1a 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
5 km
X
Fig. 3.3.2.1c
Mesh Alignment Appropriate for a Point Source Release
7
I1
b
symmetry line
1 km
--Omy
-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
wind
t=t7
=
(tY
-1 9
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:
x =
at
u(z)
bu
g
ffX
X(y,z)dydz
(3.96)
(y, z)dydz
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 propfor T is considered
or
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 e or
The calculation
y substituted into the formula for
also appropriate because
or
v
is
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 o )
p(To)
0
I_
P(8e)-(e
P(Evo)
vo
o
(3.97)
·
-62-
0'
ao
CO
00
.J
r.
Ln
o
o
m
CO
o
U)
4J
'14
m
r"l
0
O
o
o
N
H
o
0
Ln
I
rl
a
kD
I
I
*H
O
.1i
r.
U)
l
X
X
X
H
rH
LO
0
,4
,4
I
0
C)
.1
v
r
44
U,
!
I
>1
o
I
X
.-
a)
0
o
o
X
o
x
o
m
o
I0x
O
I
E-1
X
CS
('
-I
.0
04
O
U
rH
U)
U)
4-i
4)
,
Un
,
-W
44
44
L
H
N"
.
Eq
m
z:
0
O
o
E
-i
NE
.I
E
P-~
_
SQ
r
E-4
c,
.rl
>1
11
I
*r-I
.,
0)
k
-I I 1
4
.r4
0
0
·rl
Q.
H
N
4.,
I
k4 t1
a)
-
>1
r >
o 4O
4. >
>1 *dq
-
g )u n
O
O
4 O U0
U)'~O
(1
.
M
~
U3
4
u
Lr
~04
)
0
4
04
,
-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 e or ev 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
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-
Me;sh Initialization and Boundary Conditions
3.3.5
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 hydro-
static 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 th.e 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.
of the computer mesh are numbered in Fig. 3.3.5.1.
The walls
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
Units
OF
water vapor density
lbm/ft3
cloud liquid water density
lbm /ft 3
eddy viscosity
turbulence kinetic energy
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
I
(crosswind)
EARTH
Fig.
3.3.5.1
all Numbering Scheme
-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
--
----i--il
I
-68-
HZ
H
Z
P:
P;
m
H
H-
r
-
m
U-
P
P
;
I
I
',
0
,-4
CN
4K1
H
H
r-q
(a
C)
U]
C/
r%
P
p)
H
40
r-I
U
r-i
r(0
O
0
M4
Et
11
4w
O
II
0
--
U
>
-H >
.l .)
1
'H
U
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o
r
r
-I
O
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U
>
0
H
H
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o
a)
m
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0
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44
U
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4k
) -
,)
H
m
a)
'H
>1
tD
54
4*
3
En
54
D
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w (1)
00U
0
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.>1
-I
s
'H
40
-H
*1
U)
C
a
14I
0
-I
H
-
>
0)
>1
4
V
a
U)
z
I
(1
O
4
0U
'~
U
>4J
U)
4J
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5-4
o
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k
04
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I
ci)
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Z
I
I
I
I
9P'K
I
I
-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.
-lil
II
I--i-I
---------1--
1111-_
_ _
In
-
-70-
__I
F
-
DZ
D ~E
I
I
I~~
I
~~~~~~~
(a)
Step One - Four Cell Averaging
I
new cell initialization area
2xDZ
old cells transformed
in Step One
(b)
i
I
Step Two - Initialization of New Cells
Figure 3.3.6.1
Mesh Coarsening Procedure Steps
-71-
a first step (a) the entire mesh is swept over, four cells
at a time.
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 th.e 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
2
5
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
Units
Quantity
jl
I_
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
I·
1
^II
111
IIJLIIPI___LII___1·1111111-·---·1-^
-74-
4.
DESCRIPTION OF ATMOSPHERIC TURBULENCE
4.1
Introduction
This chapter describes in detail how atmospheric turbu-
lence is represented in the model.
The description begins
with the knowledge (e.g., from a set of measurements) of the
common atmospheric variables as functions of height:
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.
The relation of these two variables to the
other profiles, 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 approxi44
45
or a thermal
an Ekman spiral,
mate a logarithmic profile,
wind relation,46 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
-77-
a number of approximations
(dry hydrostatic, moist hydro-
static, 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
4.3.1
Turbulence in the Planetary Boundary Layer
Introduction
The planetary boundary layer
CPBLL 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 ca-lled "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 ob-
viously affect the amount of solar heating at the ground, so
that these effects are strongly coupled).
_
__ __
____
The layer nearest
I___
-80V
g
laminar (geostrophic) flow
(very small turbulence kinetic energy)
clouds
buoyancy - dissipation
(small turbulence kinetic energy)
shear + buoyancy - dissipation
(large turbulence kinetic energy)
EART
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.
-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.
---
I
It was mentioned in Sec. 3.3.2
I
- --
-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. 58
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.
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)l alibrary(Z)
(4.1)
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
64
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)
0
U)
a
U)
a
Uw
a
·
z
>1
>1
>1
:>.,c
O
O
a)
a)
U)
a)
U)
a)
tn
a
O
tn
En
a)
a
ao
la-Q
rC
(a
43
¢
r.
a)
U)
.4
C
0
U4
0
a)
I
..i
-
H
r--t Cd
m
c
.4-i
tn
.r-i
r--
U
a
C
a
0
C'
U
a)
>1
..Q H
4
CI1
.ri
C
a
0
4
4
O
U)
: 0
.r
o
U)
0
0
lz
Q-4
a
,4
,l
O
a)
3
N
tO
a
Ss
P4
cn
a)
04
E!
u
z O
3
a)
.,fl
01
a)
4,
.4
to
.a
-,.4
t.t: H
m
"C
S
o-
C
rd
U
to
·U
3
0-.
43
0
O
EH
43 O
N~
d
C
O
o
a
.
Z~~~~a
c
O
0
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 6 1 should be used.
If the wind speed
and direction profiles have not been measured, the prescriptions of Bornstein63 or O'Brien 2 should be used.
The pres-
cription of Nieuwstadt 6 4 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 for mesh 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,
, and TKE, q, are functions
of height, z, only; with the resulting expression being
-88-
s
at
4q2
+
r az c- aa)
z
(4.21
For a properly time-independent TKE, there must be a balance
of dissipation and diffusion in Eq. 4.2; or
g
=
ra (v )
(4.31
Performing a scale analysis of the terms, noting that
r/4,
10
and that the depth of the planetary boundary layer is taken
equal to Ldy , one obtains the result
(4.4)
q 10
eddy
For typical values in the atmosphere, c-100 ft2/sec and
L
eddy
'w103ft,giving the value
q
10 -3
sec -
(4.5)
Note that for a highly idealized picture of turbulence,65 with
eddies of a single size, Leddy
and velocity, Ueddy
q~ueddy '
q-u~2
~(4.6)
(4.71
Ueddy eddy
and therefore
Ueddy
a
Leddy
1 ]
-sec-
(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-10-3secallow q to have a constant value.
a should roughly
The fact that this choice
of q agrees with the eddy turnover time of roughly the most
diffusive atmospheric eddies 6 6
(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(yz,t)> qibrary()
= 10-3sec -1 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.
RESULTS
5.1
Introduction
The discussion of the results of the computer plume sim-
ulations is very naturally divided into two sections corresponding to the two regimes of plume behavior outlined in
Sec. 1.2.2.
To illustrate the two regimes, typical values
of effluent temperature, velocity, and pollutant are shown in
Fig. 5.1.1 for several stations downwind of a large combustion
source.
At the stack exit, the plume rushes upward at
20m/sec, is about 100 0 C above the ambient air temperature,
and has an SO 2 concentration of about 100 000 pphm (parts per
hundred million).
At the second station the plume has become
diluted about 200 times.
Without a detailed picture of its
cross section, it may be stated generally that its average
temperature excess is now only about 0.5°C and its turbulent
velocity fluctuations (disregarding those induced by its buoyancy) are about 10 cm/sec--and these are just about on the
level of observed atmospheric fluctuations.
However, the
plume SO2 concentration is still many times higher than the
background SO 2 level, so that the plume is recognizable by
its SO 2 concentration field, but not by its temperature or
velocity fields.
Throughout this first regime the plume
-91x
0
0
En
4
c
..4
.C
"4
C)
H
C
E
0
-C
0
Ua
4
C
Oan
0
rd
.9-4
U
in
.r4
H
0
f
O
0
0oo
In
0
m
0-
4:J
H
0
0
Oz
C)
E-4
4
O
H
W
O
0
H
..4
U)
C
:>
g
C ·
0
0
P4
Ln
E4
C)
0W
w
rM
ri
U
0-
0
o
°
CU
2
r
.
,-
I
_
_
0
0
C)U
O
l)
U
U
0 x
H
E-
4
4J
4-,o
C)
4U-4
C 4
V ,'
0C
*a
0
n
o
¢>
~
-N
o
00
_C_)
z:
C
Cm
-92-
velocity and temperature fluctuations have been stronger
than the atmospheric fluctuations, so that in a large part
the plume motions have been dominated by the plume properties.
Throughout the second regime the atmospheric motions are responsible for the plume dilution to the point where the pollutant becomes indistinguishable from the background level,
and the plume disappears.
The selection of these regimes is very natural in the
discussion of the results.
The results that are applicable
in the plume dominated stage will address the question of how
adequately the dynamics of a buoyant, deflected plume are
simulated.
Such results are found in Sec. 5.2.
The results
that are applicable in the atmospheric dominated stage will
address the question of how adequately the atmospheric turbulence is being simulated.
Such results are found in Sec. 5.3.
The results in Sec. 5.4 are essentially model extensions that
are applicable in the atmospheric dominated regime for
Sec. 5.4.1, and the plume dominated regime for Sec. 5.4.2,
but which do not have a body of experimental results to be
compared with.
The plume simulations that are presented in this chapter
have been included for several different reasons.
The general
simulation in Sec. 5.2.1 is included to acquaint the reader
with the general features of the buoyant line-vortex.
The
-93-
detailed comparison of simulations like the one in Sec. 5.2.1
with experimental results is made in Sec. 5.2.2.
The effect
of thermal stratification on the buoyant line vortex is then
developed in the simulation of Sec. 5.2.3.
The simulations
for comparison with actual field studies again considers the
neutral atmosphere case in Sec. 5.3.1.
The effect of thermal
stratification is then developed in the simulation of
Sec. 5.3.2, where a large stack plume in a low-level inversion
is studied.
Simulations that demonstrate the model extensions
are found in Sec. 5.4.
5.2
5.2.1
Comparisons to Analytical Models
General Nature of the Solutions
The general nature of all the computer solutions in this
work is discussed in this section.
All of the simulations are
performed on a 20 cell by 20 cell mesh, although the cell
height and width vary between different simulations.
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,6
7
where it always chooses the smallest step size from a choice of
-94-
a diffusion condition,
TSTEP
.. T
DT =
max
( ;Dy7t
1
5 .1)
Dz 2)
a Courant condition,
DT =
DT
TSTEP min(Dx,Dz )
max v,w)
5.2)
C5.2)
or a simple rate of change condition,
DT
.2 max (v,w)
(max(v,w) - max(vold,wold) + 10
where TSTEP usually has a value of 0.01.
(5.3)
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.
This allows the codeto
conserve energy in the computer 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 advec-
tion terms is found to give the best answers in the simulations--less than full donor cell differencing produces noticeable nonlinear instabilities in the flow.
A simple plume development is found in Figures 5.2.1.1
to 5.2.1.6.
Figure 5.2.1.1 interprets the mesh cell quant-
ities found in the following figures.
In Fig. 5.2.1.2, the
-95-
ordered-pair notation for locating cells
e.g., 3rd cell from left, 2nd from bottom
w velocity, ft/sec
f^
cell
temperature,
OF
0. 0
XXXXX
XXXXX
( 3, 2)
" 49.999 F
ZLL
x
X
X
Y
X
X
DZ X
0.1
TNU= 1.000E+00 X
X
CHI= 1.OOOE-05 X
X
VAP= 1.OOOE-07 X
X
LIQ= 1.OOOE-07 X
X
1
X TKE
1.0OOOE-03 X
v velocity,
ft/sec
_-xxxxxxxxxxxxxxxxxxx
k-
DY-|
eddy viscosity in ft2/sec
turbulence kinetic energy in ft2/sec2
CHI is pollutant concentration in lbm/ft 3
VAP is water vapor density in lbm II2 0/ft 3
LIQ is liquid water density in lbm H20/ft3
TNU is
TKE is
Fig. 5.2.1.1 Key to Cellwise Quantities for Figs.
5.2.1.2, 5.2.1.3, 5.2.1.4, 5.2.1.6, and 5.3.2.1.
-96XKKK
0.0
XKXXXxxK
0.0
KKXXKKXx
X'
A
X
K
4 2, 9)
49.999 F
0.0
X
X
( 3, 81
49.999 F
X
X
0.0
(
4 8
49.q99 F
0.0
XXKXXXXK
X
X
X
XKXXXXXXx
X
XXXXX
X
I 5 8)
49.9)9 F
X
X
X
X
X
X
X
I 6,
-"49.99
X
X
X
8)
F
-.
1
O.0.
0.0
I
0.0
I
0.0
0
I
1.COOE0a X
TNU= t.COOE+OO X r
TNU
X TNU, 1.000E+0 X TNUs 1.OOOE00 X TNU' l.OOOE.o00
s
X tKE I.OOOE-03 X TKE 1.O000E-03 X TKE- I.COOE-03 X TKE= l.00OE-03 X. TKE= I.0OOE-03. X
1.00E-05 X
X CHIs 1.000E-05 X CHIs 1.000E-05 X CHI" 1.013OE-05 X CHIs 1.OOE-05 X CH-I
1, COOE-7O X
X VAP, l.03OE-07 X VYAP- I.COOE-07 X VAP- 1.00E-07 X VAP-4.OO0-07 X VYAP
X L1Q, l.OOCE-37 X LEO
1.0006-07 K LXG. 1.00OE-07 X LIO0
1.COOE-07 X LIQ- 1.OO00-07 X
0.0
XXXxx.
XXXXX
0.0
XXXXXXX
0.0
XXXXXXXXX
0.0
XXtX..XXXX
0.0
XXXXXXXXX
X
X
X
X
X
X
X
1 2, 7)
X
1 3, 7)
X
I 4, 7)
X
I 5, 7)
X
I 6, 7)
X'
X
X
49.999 F
X
49.999 F
X
49.991 F
X
49.999 F
X
49,.999 F
X
X
X
X
x
X
0
I
0.0
1
0.0
1
0.0
1
0.0
I
O.C
=
X TUO 1.OOE+00 X TNUu 1.00CE+00 X TNU 1.000E*00 x ThNU
1.0000 X ThU- 1.COOEOC X
X TKEw 1.000C-03 X TKE a 1.00E-03 X TKE
.OO000E-03 X TKE 1.000OE-03 X TKE- I1.nOE-03 X
X CHI, 1.000E-05 X CHI
1.0OOE-05 X CH- I.COOE-05 X CHI 1.000E-05 X CHII l.COOE-O5 X
X VAP- .000EX VAP
.COEX
AP07
X YAP
AP
X.COOE-07
.COOEVYAP.COOE-C X
XX
I LQ10 t.OOOE-07
0
X L
1OCOE-07
LIQ= 1.006E-0
EX LIC 1.000E-0
X
E
X LIQ
X
.OOEX
XXXX
0.0
XXXXXXXXX
0.0
XXXXXXXXX
0.0
xXXXXXXXX
0.0
XXXXX0XXXX
0.0
XX
X
X
X
X
X
X
I S 6)
X
I 6, 6
X
1 2 6
X
1 3, 6)
X
I 4* 6)
X
X
.X
49.999 F
X
49.999 F
X
49.999 F
X
4q.999 F
X
49.999 F
X
X
X
X
X
X
X
0
1
0.0
1
0.0
1
0.0
I
0.0
1
O.C
X TNUJ 1.000E+OO0 X TNUs 1.000E+00 X TNU- 1[.E00
X THU= L.CCOE+00 X TNA
U l.00OE00 X
X TKEx 1.00E-03 X TKES 1.0005-03 X TKE 1000OOE-03 X TKE 1i.GOOE-03 X TKE- 1.000E-03
X CHII 1.COCE-35 X CHi
1.C00E-05 X CHII 1.030E-05 X CHI'
1.000E-05 X Cl!
1.OOCF-C5 X
X YAP.OOOE-07 X VYAPs 1.COOE-07 X VPw 1.00OE-07 X VAP 1.0006E-07 X YAP
.00OOOE-C? X
X LIQ t.COCE-07 X LIO- 1.COOE-07 X LIQ 1.OOO-07 X LIQ
L.000E-07 X LIQ0
1.000E-C1 X
XXXXX
0.0
XXXrXXXXX
C.C
XXXXXXX
0.0
XXXxxXXxx
0.0
XXXXAXXXX
0.0
XXA;X
X
X
X
X
X
X
X
( 5, S5
X
! 6, 51
X
.(.'5)X
S
X
( 3, 51
X
(2,
X
49.999 F
X
49.999 F
X
49.999 F
X
49.999 F
X
. 49.999 F
X
X
X
X
X
X
'
X
a
1
0.0
I
0.0
1
0.0
1
C.O
1
0.0
X TNU- 1.OOCE+00 X TNU- 1.CGOE00 X TUs I.000E+00 X TNU. 1.000E+OC X TNUM l.OOCE*CC X
X TKE- I.COCE-03 X tKE" 1COGE-03 X TKE.0OOOE-03 X TKE 1.000E-03 X TKCs 1t000E-03 X
X ;H I 1.000EC05 X CHI- 1.000E-05 X CHI- [.00OE-05 X CHIt 1.000E-05 X CHI l.0OOE-CS
'X
X VAP- 1.000E-07 X
YAP- 1.00-07 X VYAP- 1.000E-07 X VAP- L.OOOE-07 X VAPs 1.00OE-C7 X
X LIQ- L.OOCE-07 X L1- [.CCOE-07 X LO= 1.000E-07 x LIC= 1.OOOE-07 X LICs 1.000E-07 X
XXXXX
0.0
XXXXXXXXX
0.0
XXXXXXXXX
0.0
XXXXXXXXX
0.0
iXXXXXXx
0.0 -XXXXX:
X
X
X
X
X
X
X
X
I 4i 41
X
( 5, 4)
X
6, 41
X
( 2 41
X
t 3, 4
X
61.088 F
X
49.999 F
X
49.999 F
X
49.999 F
X
49.999 F
X
X
.
X
X
X
X
o
1
0.0
1
0.0
01 0
1
C.O
.C
1
X TNUw 1.OOOE02 X TNU- I.COOE+00 X TNUI
l.000E+00 X TNU. l000E'00 X TNU. I.OGOECC X
X TKE- I.OCCE+02 X TKE- I.COOE-03X TKE- 1.000E-03 X TKE s 1.000E-03 X TKE l.O00CE-C3 X
X CHI- 1.00E00 X CHIs 1.CCOc-35 X CHI 1.0O0EE-05 X CHI' 1.COOE-05 X CHI- I.COOE-05 X
X VAP- 1.OOQE-07 X VAP- 1.000E-07 X vAP. I.OOOE-07 X VAP.00OOE-07 X VAP- 1.000E-07 X
X LIQ- 1.OOE-07 X LIQ- L.000E-OT X LO- 1.QOOE-07 X LQ = 1.COOE-07 X LOQ 1.000E-07 X
XXXXX
0.0
XKXXXK
0.)
X. 'KXXXXXXXX 0.0
XXXXXXXX
0.0
XXXXXXXXX
0.0
XXXXX
X
X
X
X
X
X
X
X
I 4, 3)
X
{ 5
3
X
I 6w 31
X
( 2 3
X
( 3, 31
X
49.999 F
X
49.999 F
X
49.999 F
X
49.999 F
X
49.S99 F
X
X
X
X
X
X
X
0
1
0.0
1
0.0
0.0
1
0.0
1
0.0
X TNU- l.00E*00 X TNU 1o000f00
.
X TNU- 1.0O0E+00 X TNU= 1.C00E+00 X TNU= 1.OOE+00 X
X TKE= 1.00GE-03 X TKEs l.C00E-03 .
TKE- I.COO0-03 X rKEw l.COOE-o3 X TKE= I.CCE-C3
X CHI 100E-05
O
X CHI= 1.000E-05 X CHI I 1.OOCC-05 X CHI= 1.C30E-05 X CHI= 1.0OOE-G
X
X VAPw 1.000E-07 X VAP- 1.00OE-07 X VAP t.COOOE-07 x VAP = 1.COOE-07 X VAP= I.COOE-07 X
I.OCOE-07 X LIC= 1.COOE-07 X
X LIO L.000E-07 X LIQO 1.000E-0? X LI0' 1.003E-Q7 XKL1Q
XXXXX
0.0
XXxXXXXXX
O.0
XXXXXXXXX
C.0
XXXXX
0.0
XXXXXXXX
0.0
XXXXXXX
X
X
( 2, 2)
49.999 F
X
X
X
X
X
(
3 2
49.999 F
I
0
X
X
X
X
1
0.0
TNUs l.000E00 X
TKE- l.OnOC-03 X
CHI- 1.00CE-5 X
.OOrF.-7 X
V.AP-
5.2.1.2
X
I 4, 2)
49.999 F
0.0
TNU.
TKECHI*
VAP.
IOIxEXXXXXX
X
I 5, 2)
4',.999 F
X
X
X
1
0.0
Trlu. 1.00OEO00 X
TKE. l-OOC-7'3 XK
CHII L.000-aS X
VAPs l.oonnr-o7 x
X
LQ= tl.0'-)7
X L IQ= I.0CE-o
X
.XXXXE
0.0
XrKxxxXXX
Fig.
X
X
x
( 6, Z)
49.999 F
X
X
X
X
1
0.0
l.OOaF+t0 X
I.COOE-03 X
1
0.0
TP!j= 1.3iCSO
tKE= 1.CZOt:-O3 x
X
X
X
1
0.0
CIr= 1.03r-O
X
[.Onfr-nO7 X
VAP= I.CXO'E-O
X
1.OGOE+CO0 X
Tn
rX
TrK.= 1.C0C-03 X
CHI = I.C00-C5 X
V.M=
0.0 O:!-A7
1.;
X
.I
1.OCOF-0
0.0
X
X
XXXXXXXXX
.:).:E-
O.n
XOXXXXXXt
1. COCt -C
x
0.0
xxxxx
Initialized Plume Cross Section at 0 sec.
DY = 100 ft,
DZ = 200 ft.
Disregard moisture values.
-97-
entire 400 mesh cells are initialized at rest and at 50°F,
except for one warm cell(2,4)centered 500 ft/high (and on the
left boundary) which is at rest and at 68 F.
is 200 ft high and 100 ft wide.
Each mesh cell
The ambient turbulence is
set uniformly to 1 ft /sec in eddy viscosity and 10
ft2/sec 2
in turbulence kinetic energy, which are essentially laminar
values compared to the values that develop inside the plume
cell.
After 20 seconds of development (see Fig. 5.2.1.3)
a
vortex circulation has formed in the vicinity of the warm
fluid, and mixing has brought the warmest fluid cell (,4)
from 68 F to 61.24°F.
The strongest updraft
(6.57 ft/sec)
occurs in the warmest cell, and the downdrafts tend to be
weaker, since they are spread over a larger area.
After 80 seconds of development (.see Fig. 5.2.1.4) the
plume has risen 379 ft.
Considerable mixing has reduced the
warmest cell temperature to 52.9°F from 680 F, and the updraft
has now increased to 10.8 ft/sec.
Again, the vortex circula-
tion is very easy to identify and it occupies a progressively
larger area as the plume cross section grows.
The turbulence
kinetic energy field at this point in time is illustrated in
Fig. 5.2.1.5, where the maximum TKE occurs in the updraft
region and is about 1600 times that of the ambient field.
After 200 seconds of development (see Fig. 5.2.1.6),
the
-98XXXXX0.215
YXXXXNXXC.107 xxxXxxx
0.142 XXXXXXXXX
0.073 xxxxxxxxx
0.041 XXXX)
X
X
X
X
I e6, )
X
X
{ 2 8)
X
( 3, 8)
X
4 8)
X
5S, 8)
X
49.922 F
X
X
50.137 F
X
X
49.942 F
X
50, .019 F
49.922 F
X
X
X
X
0
1
0.073
1
0.120
1
0.148
1
C.121
0. 126
X TNU 1 .005E+00 X TNU- 1.C07E+30 X ThU- 1.005E+00 X TNU- 1.033E+OC X TNU- 1.00E+CC X
X TKE- 1.039E-03 X TKE= 1.014E-03 X TKE- 1.009E-03 X TKE= 1.006E-03 X TKE- 1.OGIE-03 X
X CHI5 1.084E-05 X CHI- 1.00£E-O5 X CHI 1.030OE-05 X CHI- 9.S99E-06 X CHl- 9.99E-06 X
X VAP= 2.000E-07 X VAP- 2.000E-37 X VAP= 2.031E-07 X VAP-.2.COOE-07 X VAP- 2. CJE-07 X
X LQ10 0.0
X LIO= 0.C
X Lt0= 0.0
X
X LIQ- 0.0
x L10- 0.0
XXXXX
0.348 XXXXXXXXX
0.232 XXXXXXXXX
0.209 XXXXXXXX
0.005 XXXXXXX
0.029 XrxX
X
X
X
X
X
X
X
1 2, 7)
X
I 3, 7)
I 5 7
X
I 6, 71
X
t 4, 7)
X
F
X
X
49.957 F
X
49.777 F
X
50.t09 F
X
49. .964 F
X
50 .089
X
X
X
X
X
X
O
1
0.209
1
0.294
1
0.277
1
C.254
1
0.218
=
X TNU 1.000E00 X TNU= I.C39E+00 X TNU- 1.024E+00 X TNU= 1.010E+CC X TNU= 1.004t+CC X
X TKE= 7.111r-03 X TKE- 1.239E-03 X. TKE= 1.050QE-03 X TKE= 1.019E-C3 X TKE= 1.0)37-C3 X
X CHI- 2.144E-04 X CHI= 1.398E-05 X CHIt
1.004E-05 X CHI- 9.996E-06 X CH= 1.000E-05 X
m=
2.030E-C7
X VAP= 2.000E-07 X VAP- 2.000E-07 X VAP= 2.030E-07 X VAPX 1.999E-07 X VA
X LO1 0.0
X LIC- 0.0
X L IQ 0.0
X
X LIQ1 0.0
X LIQa C.O
0.380 XXXXXXXXX
0.150 XXXXXXXXX -0.045 XXXXXXXXX
-0.028 XXXXX
XXXXX
3.791 XXXXXXXX
X
X
X
X
X
4 6
X
1 5, 6)
X
X
I 2, 6)
X
6 6)
X
3, 6)
X
50.172 F
X
49. .985 F
X
50.012 F
X
5C.040 F
X
50.068 F
X
X
X
X
X
X
1
0.756
1
0.543
1
C.430
1
0.276
1
0.8375
=
X TNU 2.255E+00 X TNU= 1.018OlE+00 X TNU 1.047E+00 X TNU- 1.028E+CC X TNUt 1.027E+CC X
X TKEw= 3.841E-0 X TKE= 1.945E-02 X TKE- 1.626E-03 X TKE- 1.061E-03 X fKE- 1.052E-03 X
X CHI
1.611E-02 X CHI
5.960E-04 X CHI- 2.124E-05 X CHI- 1.012E-05 X CHI- L.C00E-05
X VAP- 2.O00E-07 X VAP- 2.COOE-07 X VAP= 2.COOE-07 X VAP- 2.COE-07 X 'IAP= 2.000E-07
X LIQ= 0.0
X LI0.0
X LIQ= 0.0
x LIQ- 0.0
X LIQ- 0.0
X
.XXXXX
2.530
XXXXAAXX
0.153 XXXXXXX
-. 260 XXXXXXXXX -0.266 XXXXXXXXX -0.345 XXXXX
X
X
X
X
X
X
X
1 2, 5)
X
1 3, 5)
X
I 4, 5)
X
1 5, 5)
X
6 5
X
X
54.088 F
X
50.324 F
X
50.061
F
X
50.234 F
X
49.888 F
X
X
X
X
X
x
0
1
2.009
1
1.302
1
0.711
I
C.392
1
0.180
X TMUu 1.840E+01 X TNU- 3.753_+00 X TNU= 1.141E+00 X TNU= 1.084E+CC x TNI- 1.0TeE+CC X
X TKE* 4.772E+00 x TKE- 7.725E-IL X TKEw 5.368E-02 X TKE= 2.?22E-C3 x TKE- 1.176E-C3 X
X CHI- 2.405E-01 X CHI= 3.218E-02 X CHI= 1.717E-03 X CHIII 4.576.-05 X CHI- 1.032E-05 X
X VAPw 2.000E-07 X VAP- 1.999E-07 X VAP- 2.00CE-07 X VAP- 2.COIE-07 X VAP- 2.COGE-07 X
X LIQ- 0.0
X LQ= 0.0
X LI0- 0.0
X Ll0= 0.0
X LIQ- C.0
X
:XXXXX
6.566XXXXXXXXXX
-1.302
xxxXXXXX -1,462 XXXXXXXXX
-Q.889 XXxxxxxxx -. 738 xxxX
X
X
X-'
K
X-
-
-
-
X-
X
( 2 4
X
, 4)
4
X
4)
X
S 4
X
( 6, 4)
X
X
61.240 F
X
50.768 F
X
49.888 F
X
49.908 F
X
49.770 F
X
X
X
X
X
X
X
0
1
-0.536
1
-0.461
1
-0.319
1
-0.260
1
-0.157
X TNU* 4.347E+01 X TNUm 2.428E+01 X TNU- 4.466E00 X TNU- 1.035E+00 X TNU- 1.071E+00 X
X TKE- 1.122E+01 X TKE- 3.101E+00 X TKE- 9.614E-02 X TKE= 1.592E-03 X TKE- 1.146E-03 X
X CHIt 6.256E-01 X CHI- 6.555E-32 X CHI- 1.012E-03 X CHI- 1.180E-05 X CHI- 9.999E-G6 X
X VAP- 2.COCE-07 X VAP= 1.S99E-07 x VAP= 2.000E-07 X VAP- 2.OOO2-07 X VAP= 1.999E-C7 X
X LTI- 0.0
X LO= 0.0
0.0
X L
X LTO= 0.0
X LTC= 0.0
X
-0.582 rNrXX
-1.117 XXXXXXXXX -0.775 XXXXXXXXX
-1.077 XX'XXXXXX
XXXXX
5.434 XXXXXXXXX
X
X
X
X
X
I
X
I 2, 3)
X
( 3, 3)
X
( 4, 3
X
5, 3)
X
1 6, 31
X
X
49.735 F
X
50.207 F
X
49.860 F
X
49.943 F
X
50.116 F
X
X
X
X
X
X
x
0
1
-2.248
1
-1.472
1
-0.924
1
-0.626
1
-0.426
X T
3. 3E+U
3.36E00
TNU 3.074E+30 X TNU- 1.050E+OO X TNU- t.151E+CC X TNU- 1.073E+CC X
X TYrE 8.033E-01 X tKE- 2.349E-31 X TKE- 5.556E-33 X TKE- 1.33E-C X TKE- 1.14eE-C3 X
X CHI- 1.297E-02 x CHI- 3.665E-03 X CHI- 4.321E-n05
CH[= 1.C04E-05 X CHI= 1.000E-05 X
X VAP- 1.99)E-07 X VAP- 2.001E-07 X VYAP
1.999E-07 X VAP= 2.CCOE-37 X VAP- 2.000-07 X
X LIQ- 0.0
x LIQ- 0.0
X LIQ0 0.0
X L1it
0.0
X LIQ 0.0
X
XXXXX
1.012 XXXXXXXXX
0.435 XXXXXXXXX
-0.102 XXXXXXXXX -0.217 xxxxxxxxx
-0.154 XXXXX
X
X
X
X
! 2, 2)
49.659 F
X
X
X
X
( 3 2
40.999 F
X
X
X
X
I 4, 2)
49.957 F
X
-0.492
1
-0.710
TI{Jr 1.0CRF+00 X TlU- 1.CCOE*0 X TNIJTKE- 4.085£-)
X TKE- 1.iSlFE-)3 X TKE
X
CHI
CH{I
1. 10-O5
X
s 2
4S.915 F
X
1
0
X
X
2.746f-05 x
X
X
Cl1=
X
X
X
( 6, 2)
49.964 F
X
X
X
X
1
-0.618
I
-C.583
I
-0.50
1.147f+00
X T :U= l.06PE+CC X TNL-= 1.03E+CC X
t.323F-Ol X TKE
.t31. -3
X TrKF; 1.OSC-C3
I.0(,':-)5
X
CMl= q9.96E-06
X
CFl=
.99E:-C6
X
X YaPr I .99E-0
X VAP- 2.COOC-07 X V6Pz ?.000-01 X VAP= 2.OOOE-C7 X VAP= 2.C00O-C7 x
x L1Q 0.0
x LIO= C.
x LI)= 0.,)
x LIO 0J.O
X LI,
0o.0
x
xxxxxxxx
0.3
xxxxxxxxx
0.0
XXIX:
xxxxx
0.0
xxxvxgxxx
C.C
xxxxxxxxx
0.0
Fig. 5.2.1.3
Plume Cross Section at 20 sec.
DY = 100 ft, DZ = 200 ft.
Disregard moisture values.
-99-0.170 IX.IXx
0.168 XXXXXXX
xxxXXxxXX
I.C4C XXXXXXX
0.609 xxxxXXXXX
X
X
X
X
X
x
X
6., 83
I 5, 8)
X
2
8)
X
3 8
X
( 4, 8
X
X
49.936 F
X
5C. 137
F
X
X
X
49.999 F
50.809 F
50.082 F
X
X
X
X
X
X
X
1
0.843
1.585
1
1.118
I
1
1.592
1
1.860
0
X - TNi I- 5.087E*00 X TNU. 2.169E.00 X TNJU 1.658C+00 X TNUs 2.01ZE+CC x TNUI 1.370E+CC X
:4.702E-01
X
TKE.
1.135E-01
x
IKI
E2.026E-02
X
TKE6.234E-C3
X
TKE2.041E-C3
X TKI
s
1.078E-02 X CHI. 1.448f-03 X CHI- 1.383E-C4 x CHIs 1.682E-CS X
X CHI !- 4.895E-02 X CHI
'2.00CE-07
X
VAP2.000E-07 X VAIP 2.000E-07 x VAP- 2.000E-07 X VAP: 2.000E-C7 X
X YV
X L[I '.0.0
X LIQ- 0.0
X LIQO 0.0
X
X LIQU 0.0
XI LIC1- 0.0
-0.752 XXXXIXXXX -0.7114 XXXX
XXXXX
5.963 XXXXXXXKX
1.573 XXXXXxXXX
0.074 XXXXXXXX
X
X
X
X
X
X
X
X
! 6 7
I S 7)
X
1 2 7
X
t 3 7
X
1 41 7)
X
X
X
49.999 F
50.165 F
X
49.888- F
X
50.927 F
X
X
52.397 F
X
X...
X
X
X
1
1.345
1
0. 966
1
2.327
1
1.898
0
1
1.790
.560E+CC
X TNU 1. 364E01 X TNU- 6.0942E00 X TNU- 2.564E*00 X TNUs 1.746E4CC X TNUX TKE, 1.42CE*00 X TKE- 5.864E-01 X .TKE- 1.550E-01 X TKE- 3.042E-02 X TKE- 5.69E-C3 :
CHI- 2.2u9E-C4 X
X CHIS 1.422E-01 X CHI& 5.942E-02 X CHI 1.527E-02 X CHI- 2.390E-03
X VAP- 2.000E-07 X VAP- 2.COOE-07 X VAPz 2.000E-07 X VAP- 1.991E-07 X VAP- 2.000E-C? X
X
X LIQa 0.0
X
X LIQz
Q 0.0
X L QO 0.0
X LtQ
00
0.0
X LIQ 0.0
-. 471 XXXXX
-0.778 XXXXXXXXX -1.852 XXXXXXXXX
9.S35 XXXXXXXXX
2.644 XXXXXXXX
XXXXX
X
X
X
X
X
X
X
1 6, 6)
X
( 5, 6)
X
1 4, 6)
3 6)
2 61
X
X
50.006 F
X
5C.137 F
g
X
X
5C.733 F
X
51.870 F
X
52.903 F
X
X
X
X
X
X
0.313
C.562
1
0.740
1
1.127
1
1
0
1
0.651
X TNU. 1. 802E+01 X TNU 1.130E+01 X TNU= 4.335E+00 X TNU 1..775+CC X tNU- 1.98CE4OC X
X TEN 1.815E+00 X TXE- 1.200E+00 X TKEs 3.780E-01
X TKE- 7.838E-02 X TKE- 1.715E-C2 X
X CHI. 1.763E-01 X CHI- 1.146E-01 X CHI= 4.295E-02 X CHIs 8.16TE-03 X CHI- 9.445E-04 X
X VAPO 1.999E-07 X VAP. 2.000-07 X V.Ps 2.C00E-07 X VAP- 2.000E-07 X VAP- 2.COOE-07 X
X
X LIO- 0.0
X LIO- 0.0
X LWQ- 0.0
X LIQz 0.0
X LIQ 0.0
XXIXXX
-1.998
XXXXX 10.633 XXXXXXXXX
3.637 XXXXXXXXX -1.545 XXXXXXXXX -2.236 XXXXXXXXX
'
X
X
X
X
X
X
( e, 5)
X
X
!
5, 5)
4 5)
X
1 3, 5)
X
2 5
X
X
49.943 F
X
X
5C.096 F
50.324 .F
X
X
51.607 F
X
52.119 F
X
X
X
X
X
1
-0.833
1
-C.988
1
-1.372
1
-1.622
1
-1.157
0
TNU t.922E+CC X
X TNU- 1.397E+0C
TNU 283298E+00 X
X TNU- 1.677E+01 X TNU. 1.1982.01
TKE- 8.061E-03 X
X TKE. 1.643E*00 X TKE 1.242E+00 X TKEz 2.058E-01 X TKE- 3.337E-02
1.046E--01 X CHI- 2.520E-02 X CHI- 2.410E-03 X CHI- 1.479E-C4 X
X CHIN 1.300E-01 X CHI
VAP- 2.00OE-C7 X
VAP- 2.000E-07
X VAP- 2.00CE-07 X VAP. 1.999E-07 X VAPs 2.000E-07
X
_:
. x
x.
J-..0[s-.O
'..
... X I:t
.... X..LZG- .0.0
X. LIQ-O.O
IxXxxxxx - ---r. 665 -XXXI.
.- 'tr..O6l XXXxxxXX --*t. 48
XXXXX -' 8';533-XXXkXXC---*.-676 xxXXXXx'
X
X
X
X
X
X
X
( 6 4
X
I 5, 4)
X
1 4, 4)
X
X
( 3, 4)
( 2, 41
X
X
X
49.811 F
49.902 F
X
X
50.061 F
X
50.394 F
50.858 F
X
:XXXXX
X
X
2.792
X
..
.
X
X
X
1
-1.2 69
1
-1.733
1
-2.173
1
-2.552
1
-1.958
O
TNUs 1.141E+O0 X NhUs 4.348E00 X TNU- 2.004E+00 X TNU, 2.577E+CC X
X TNU. 1.359E+*01
X
TKE8.188E-03
X
TKE1.617E-02
X TKE 1.2b7E+00 X TK E- 8.776E-01 X TKE= 1.592E-01 X
X CHlm 5.792E-02 X CH I 3.061E-02 X CHI- 8.101E-03 X CHI- 5.101E-04 X CHI- 2.975E-CS X
YAP- 2.0002-07 X VAP- 1.999E-07 X
X V AP 2.000E-07 X VAPAP 2.CCCE-07 X YAP. 2.000E-07
X
X LIQ' 0.0
X LIQ. 0.0
X LIOJ 0.0
.
X L1 0. 0.0
X LIQ- 0.
4.668 XXXXXXXXX
1.477 XXXXXXXXX -0.350 XXXXXXXXX -0.614 XXXXXXXXX -0.739 XXXXX
'XXXXX
X
X
X
X
X
X
X
X
I6 3
1 5, 3)
X
X
I 4, 3)
X
1 39 3)
X
I 2, 31
X
X
50.103 F
49.936 F
X
X
49.714 F
49.735 F
F
X
X
49 .978
X
X
x
X
X
X
-1.166
-1.SO0
1
1
1
-1.970
I
-1.76S
0
1
-1.689
3.1925+00 X ThUs 2.453E+00 X TNUw 2.1t50E0C X TNU' 2.20CE+CC X
X TNU- 5.732E+00 X Th
1.307E-01 X TIE- 4.032E-02 X TXE- 7.152E-03 X TKE- 5.154E-C3 X
X TXE 4.311E-01 X TI
2.758E-03 X CHI- 9.150E-04 X CHI= 5.188LE-5 X CHI- t.13E-05 X
X CHI- 6.642E-03 X C
YAP' 2.000E-07 X VAP- 2.COCE-C7 X
1.599E-07 X VAP. 1.qq9E-07
X YAP. 2.000E-07 X VA
X
X LIQ- 0.0
X LICs 0.0
0.0
X LIQ= 0.0
X LIQ. 0.0
X LI
XXXXX
1.202 XIXXXXXXX
0.886 XXXXXXXXX
0.151
XXXXXXXXX
-0.036 XXXXX.
0.01 XXXXXXXXX
X
X
X
X
1 2, 2)
X
3, 2)
x
I 4 21
X
( 5 2)
X
I 6, 2)
X
X
49.583 F
X
.061
F
X
50 .089
F
X
49.995 F
X
49.874 F
X
X
X
x
X
X
0
1
-0.619
L
-1.082
1
-1.137
l
-1. 113
1
-1.078
X TNU- 2.11SE,00 X TNU s 2.726E+00 x TNU. 3.23CE+00 X TNU-U 2.270ECC ( TrNL 1.936E2G X
X TKE 2.4?2E-02 X TKE 1.;24E-02 x TKE% 1.210E-02 X TXF= 5.171E-C3 X TKE- 3.723E-C3 x
X CHI. 7.8dlE-05 X CHI- 2.39LE-05 X CHI- 1.320C-05 X CHI= 1.031E-05 X C1iI
1.001E-05 X
X VAPt 1t.98E-07 X VAP9 2.COOE-Q0 X YAP- 2.000o-07 x VAP- 2. 00E-C X VAP- 1.99'9E-C7 X
X L Ir3 0.0
'X L:U' 0.0
X Llq= 0.0
x LIQ= 0.0
x LIC= 0.0
X
XXXX
0.0
xxxxxxIxx
0.0
IxxxxtxEX
0.0o
XXXXXXx
0.0
JlXXXX'XXX
0.0
xIxxxX
Fig. 5.2.1.4
Plume Cross Section at 80 sec.
DY = 100 ft, DZ = 200
ft.
Disregard moisture values.
-100-
z
0.005
0.1
1.0
1.5
Y
Fig. 5.2.1.5 Turbulence Kinetic Energy Profile at
80 sec. Contours of TKE in ft 2 /sec 2 .
The maximum
TKE is 1.82 ft2 /sec2 , and the minimum TE is 0.001
ft 2 /sec2 (throughout the far field).
-101XXXX
4. 66h
XXXZXtZZ
X
X
{ 2,12)
X
50.588
X
f
X
X
0
X
X
1.270
1
TaU= 8.396.3-00
X
-3,*%-%%TWIK.XX 0.917 XXXXXXXA.X
9.397 XXXXXXXX
0.014 XXXX
Y
X
X
X
( 3,12)
T.
4,1'2)
X
( 5,12)
x
( 6,12)
X
5..047 r
50.255
%
49.950 7 'X
.
49.929 ?
F
X
K
1
2.036
T'US 6.390+03J
X
X TE=r
X CHI=
I VAP.
X LIQ=
XXXXX
X
X
1
2.231
1
2. 37
T'1 3-SJ
'.089E40
X T~IN1 2 .9q4'03 X TNU
'E 1.147E-01 X rKE= 3.51JE-02 X TXE
Tl Es;
1
1.739
2.371E+00 X
1.140O-02
5.049E-01
X TKEs 3.143E-01
3.858E-02 X CHL- 1.840E-02
CHI Et= 6.147E-03 X
X CHIs
CHI- 1.435r-03
1.435-03 T
X CHI*
CHIs 2.628R-04
2.628X-04 X
3.857E-02 X VAP = 1.834E-02 X VAT?a
VAP- 1.425E-03
X
YAP=
2.52SE-0
,a 6.137E-03 T
x
TAP·
1.425E-03
YAP' 2.528E-04 T
S
7.714E-02 X LIQ 3.678_-02 X LI'av
2? 1.223E-32 X LIQ 2.850_-;)3 X LI;) 5.056Z-04 X
7.1b5 XXXXXXXX
3.oo1 XXIIXAIIXX . 11.249
29 XXXXXXXXX
-- '.O.,'CF,7
XXXXx. -.
0Fi7 XXxxvxxrr
XXXXXXX -0.%Rfi
-0.586 xxxxv
XXXX
%
X
i
S
X
( 2,11)
X
( 3,11)
X
( 4,11)
X
( 5,1)
F
6,11)
S
x
50.990
X
50.699 F
X
50.352
x
50. 14;F
:
50.012 F
I
x
X
X
X
X
x
0
1
1.191
1
2.153
1
2.442
1
2.207
1
1.781
· TiUS 1.321E+O01
TNUl 1.106101 X T1U0 6.546.+00 X TNU 4.055E00 X T'u 2.917E+00
I TKs 8.242E-01 X 'KE: 7.557E-01 X 'rr
3.537E-01 X 'KE 1.313-Cl X TKIE 4.229E-02
X CHIl 6.5191-02 X CHl- 4.620E-02 X CRII 2.2921-02 X CH= 8.101_-03 X CHI 2.071E-03 S
I VAP 6.518E-02 X TIP= 4.6202-02 X VTAr 2.291E-02 X
AP- 8.091E-03 X VAP: 2.0C61-03 X
I LUQ- 1.304E-01 X LIZQ 9.239-02 S LXIQ 4.582E-02 X LIQ= 1.6183-02 % LQ= 4.123E-03 X
XxxIX
9.549 XXXIXXXXX
5.520 XXXXXXX
1.d25 XXXXYXXXX -. 401 XXXXXXXS -1.429 XXYXX.
X
X
X
I
0
S
(2,10)
51.031 F
K
X
X
( 3,13)
50.962
X
F
x
x
X
( 4,10)
50.560 F
X
X
( 5,10)
5s.186 r
x
x
I
r
%
( 6,10)
49.999 TF
. K
1
0.616
1
1.358
1
1.676
1
1.520
1
1.199
1 TUw 1.373.+01
11NU 1.4495+01 X TNU= 9.508E+00 X TNU 5.302E+00
TIU- 3.155+,00
I TIE. 8.007E-01 X
IKE- 1.077 '+3) X TKE- 6.129E-01 X TE 2.499E-01 X TIEE 8.170E-02 X
I CUl
6.9821-02 X CrIZ 6.276E-32 X CE1I 3.956tE-02 X CHI= 1.769E-02 X CHII 5.453M-03 X
s
X TAPs 6.982Z-02 X VA,
6.2752-02 X YAP: 3.9555-02 I VAP' 1.768E-02 r
VAP 5.443E-03 X
I KLXQ: 1.3963-01 X LIQw 1.255E-01 X L1;= 7.91CE-02 X L'2 3.536E-02 I LIXQ 1.089E-02 X
XXXII
10.783 XXXXXXXX
7.0039 XXXXXXX
2.458 XXXXXXXX
-0.720 XIXXXXXIX -2.077 XXXXX
I
X
X
X
t
X
K
( 2, 9)
I
(3,
) ..
X
44( 9)
X
5, )
I
{
6, 9)
X
K
50.837 Ft
50.934 r
x
50.657 -F
X
50.269
Fr
50.006
r
x
K
X
X
X
K1
0
-0,308
1
0.230
1
0.396
1
0.339
1
0.298
TUO- 1.2433+01
TINU 1. 430E*01
TN= 1.331E+01
TNU= 5.267E+00 X TIJ 2.875E*CO X
I TlK6.9412-01 X TK1.077E03 X TKE- 6.959E-01 X T2.578E-01
TE= 7.728E-02 r
S CI1I 6.017E-02 K CHII 6.223E-02 X CIT= 4.516E-02 X CII= 2.060E-02 X CHRI
6.351E-a3 X
I
APs 6.016£-02 X VAPw 6.222E-02 X YA#P 4.515E-02 X
AP- 2.059E-32 X VAP= 6.341E-03 X
S-tIQ: 1.-203-01 X LIQ 1.245-51-1 K LOIQ 9.:30E-02.X .Lt. a .117;-02
LQ. 1.ti2f8E-Q2 X
xftf -"fb;.76' XXXIXXJX'i ' 7.474.--XXXXftt
2i795 ZtKStXzX
-'.R23 .%XXtItXXt1
-2, 149 1tXXX
X
;
X
X
X
I
( 2, 8)
X
( 3, 8)
x
( 4, 8)
X
t 5, 8)
;
( 6, 8)
X
K
50.636 F
X
50.754 F
X
50.539 F
X
50.179 P
X
50.026 F
X
S
''S
X
O
1
-0.501
1
-0.384
1
-0.849
1
-0.39
1
-0.602
IrTUII= 1.112*.01
TI
1.2+01
154+01 X TNIw 8.778100 X TNHU= 4.489E+00 k TINO 2.716E+00
TKE: 1.973l-01 I TXI 6.111B-02
X TiE 6.3041-01 X TIKE 8.2111-01 X TEE'' 5.6741-01
C
4.694E-02 X CUIs 5.021E-02 X CHI= 4.157E-02 X CHI 1.996E-02 X CHI- 6.007'-03 X
X
AP- 4.6932-02 X TAPw 5.0201-02 X VAP=t .156E-02 X VAP- 1.995-02
TAP: 5.997E-03 X
V
K LIQ 9.3d63-02 X LIQ- 1.004-C
X LI2: 8.310E-02 X L3:= 3.991.-02 X LIQw 1.199E-02 X
XXSIX
5.763 XXXXXXXXX
6.718 XXXXXXXXX
2.861
%.089E*00
XXXXXXX -0.613 XXXXXXXXX -1.882 XXXX.
x
X
X
x
X
x
K
( 2 7)
7)
(4, 7)
X
(5
7
( 6, 7)
K
__( 3,
X
50. 08
X
50.498
F
X
50.442
F
X
50. 165
F
.
49.971
F
X
x
X
X
X
4
I
.865
I
-1.592
1
-1.882
I
-1.710
1
-1.444
1.
I TUJ 9. 805E+00 X
tV1 8.357E+03 X T'YU 53.846&*00
X T:llJU 3.547£'3o X Tiqil' 2.6R5E+OC
S.612E-01 X T$;E 5.3230-31 X Trt.: 3.141E-01 X TE: 1.2211-01 X ?KSl 4.248E-32 X
X- CUts 3. 38 1E-02 X ClIt 3.36iE-C2 I CHIs 2.736E-02
1.'410E-02 x Cit.
3.847E-03 X
1.223E-32 X CiP.
TAP: 3. 31-02
I
X
VAP' 3.367!-C2 X YAPa 2.735E-02
YAP. 1.'09tE-02 X YAP: 3.8171-03
I
LIQ. 6.7bOE-02 X LIQ' 6.7322-02 X ,I=Q
5.1471-02 X LIn- 2.814-02 X LIQ= 7.6751-03
8. 017 IXXx YXXXXX
.XXX
5.265 XX
XTX'.:-KK
O.2_51
JTX
-1.341
XXXXX
X
X
X
X
X
X
x
( 3, 6)
S
4.
)
x
( 2, 6)
( 5,
)
X
( 6, 6)
K
.2.9
r
X
50.151
r
x
50.433
F
X
49.'564 F
K
49.9'0
?
X
0
X
1
-0.
t
X
-1.128
CTNII8. 57.Etn10 X
4.8043J1-01 X
2. 207K-0
X
· .',,I.JoX
X IL;L/
XW ;XE--
Fig.
2
A
A
=
1
X
-1.917
T2NU . 132F*C1
tE
1.23lE--1
CllIs 1.78h!.-02
VAP
1--t.7:r*':;
tI.-'-.
L
1
X TJI1
%. .K=
CX lil
-2.37.1
n%,.0n
4.
1. 41heE-O1
. 15% .-. 2 ( I
VA"=· :. .- e
! ?.
i .;
5.2.1.6
t
-2. 183
I
-1. 4
T;l= 3.tR1EfnO X TNI= 3.128v*13 K
TK I?- 6f.r.6-O'
J TKF= 3.1.11I2
X
!. :( .l' t
. t ' XX iA. X
s!C
,711 F--ll X Cll=
1.h3-:-31 v
.7'
-e
r.
X
Va
hn'1
X
If
L
-. .'"-~'
!.1x&_'.X
'~. 1 X/$!'
XX
-)
.. :"' ;X(X
Plume Cross Section at 200 sec.
DY = 100 ft,
DZ = 200 ft.
Disregard moisture values.
147E-012. X
231 LIQX
T' (IrK
E- 2.R50%-,13
23.
1.9Q!6!:&00
15I 3 E-02X
2.1137
XX LI;)x
TNEI-x 5.056Z-04
TXEx
2.371!+03
1
1.140'3-02
1.739
1
X
-102-
plume has risen 1003 feet.
The warmest cell in the plume is
barely 1°F warmer than the surroundings, and the updraft
velocity has remained constant.
The plume cross section has
grown considerably, and the maximum TKE has dropped by a factor of two from its value at 80 seconds.
The effect of adding an initial internal circulation to
the plume cross section is developed in Figures 5.2.1.7 and
5.2.1.8.
In Fig. 5.2.1.7, the initial circulation is shown.
The mesh cells are not 50 ft by 50 ft, although the same size
plume is initialized as in the earlier discussion.
The
uniform 3 ft/sec circulation pattern is simply a rough guess
at the actual circulation.
Simulations to 40 seconds with and
without the circulation are found in Fig. 5.2.1.8.
The
presence of an initial circulation makes only a small difference between the runs, as seen in the selected velocity and
temperature values.
The initialization of all of the subse-
quent simulations with no initial circulation is presumed to
introduce little error into the results, i.e., the dynamics
are strongly affected by the buoyancy, and not by the initial
circulation.
The effect of ambient atmospheric turbulence is developed
in Figures 5.2.1.9 and 5.2.1.10.
The initialization is the
same as in Fig. 5.2.1.7 without the initial circulation, but
with 50 ft square cells.
In Figure 5.2.1.9, the ambient
-103-
symmetry line
+4
+
+
+
+
+
+
+
+
+
++
+
++
+
++
++
Plum ie:
4
-i
+
i
I
+
ia_
+
+
4.
+
+
+
+
+
+
+
+
--
4
+
+
Ambient Field:
at rest
50 0F everywhere
+
+
+
+ +
300 ft
ft
300
+
++
+ +
+
+T+
++
+
200 ft on a side, total
18 0 F above ambient
internal circulation as shown
(all arrows are 3 ft/sec)
+
+
I
-
+
50 ft x 50 ft cells
400 cells total
+
+
+
+
I
Fig. 5.2.1.7
Initialized Plume Cross Section
with Internal Circulation.
+
-104-
No initial circulation
3.67 ft/sec
3 ft/sec initial circulation
4.14 ft/sec
57.
c
ft/sec
Fig. 5.2.1.8
4.81 ft/sec
Plurme comparison at 40 sec of simulation
showing the small effect of an initial circulation on
the plume development.
-105-
computer mesh boundary
2/sec]
I
!
!
.-
6.0
armbient
%
475 ft
1
Fig. 5.2.1.9 Plumne development in a moderately
turbulent atmosphere. Mean flow field velocity
vectors are shown.
15
15
15
15
15
15
15
-106-
.-o1
L
A
_n
-1
l
"/secl
I
at 40 sec
I
10 2
10-2
I
f I
/
1
le
warmest cell is
475 f
1.94 F above
*_~ ambient
0.
jI
r
15
Fig. 5.2.1.10 Plume developmlent in a very turbulent
atmosphere.
Mean flow field velocity vectors are shown.
II
I
-107-
eddy viscosity is maintained at a uniform
15 ft 2 /sec.
The
resultant plume cross section at 40 seconds has developed
the usual circulation, and its warmest cell is found to be
6.05°F above ambient.
In strong contrast to this is the plume
cross section of Fig. 5.2.1.10 which has a much stronger
ambient turbulence field below 850 ft, and a much weaker
turbulence field above 850 ft.
The fictitious eddy viscosity
profile quoted in Fig. 5.2.1.10 reflects a very turbulent
boundary layer whose depth is about 850 ft.
The resultant
plume cross section at 40 sec is markedly different.
The
strong atmospheric dispersion has resulted in a much more
diffuse plume whose maximum cell temperature is about onethird that of the previous run, although the plume rise is
quite similar.
This agrees with the notion that plume rise
is dominated by the thermal stratification of the atmosphere
(which is neutral in both cases here), and to a much lesser
extent by other factors.
Plume dispersion, which is very
different in the two cases here, is affected strongly by the
turbulent state of the atmosphere
(which in turn is strongly
affected by the thermal stratification of the atmosphere,
among other factors).
The effects of using continuative outflow
versus
free-slip solid walls for the top and right boundaries was
studied.
The alternative assumptions produce little difference
-108-
between runs.
The solid free-slip walls give more satisfac-
tory results, although they are somewhat unrealistic physically, as are the continuative walls.
Further refinement of
the boundary conditions is expected to have little influence
on the solutions.
5.2.2
Turbulent, Buoyant Line-Vortex Results
The results of the plume simulations in the plume domi-
nated regime (see Fig. 5.1.1) are discussed here.
To obtain
these results, the ambient turbulence level should be less
than one-tenth of the plume turbulence, so that the ambient
turbulence will have only a small effect on the results.
Plume
simulations are compared to the experimental results of
Richards
and Tsang.
Tsang's results are generally more
accurate since his experimental technique is more sophisticated, but Richards was first to set down the basic similarity arguments.
Similarity and dimensional analyses by Richards and Tsang
have revealed the formula for the plume top height, Z, versus
the plume radius, R, and the formula for Z versus time, T.
The concept of a virtual origin of Z and T simplifies the
results in their analyses.
(T,,Z)
Briefly, the virtual origin
is the limit where the plume radius vanishes, much
as if the plume had emanated from a single point at time T.
-109-
This is shown in Fig. 5.2.2.1, where the two formulas are
quoted.
Two universal constants, N and C, are found in
the formulas.
Tsang found that N = 3.0 and C = 1.9 provided
a very good fit to dense salt water line thermals released in
a tank of still, fresh water.
The flow inside the line ther-
mals is turbulent.
Tsang's results are simulated with the computer and
presented in Fig. 5.2.2.2.
Essentially, the virtual origin
(T*, Z,) is free to be chosen to provide the best agreement
between experimental and calculational results.
center height (not top height)
The plume
is to be compared--the formula
quoted in the figure is readily derived from the formulas in
Fig. 5.2.2.1.
The calculated values are represented by the
points, and the experimental results
(with an optimal T
and Z)
are represented by the solid line.
Since ambient atmospheric turbulence is not important, the
comparison serves to test the turbulence model by making sure
that it can reproduce the self-similar plume development.
The results are acceptably accurate through several hundred
seconds of development.
The calculated plume is found to
rise a little too fast, so that a more "diffusive" turbulence
model would be more accurate.
a,
The VARR turbulence constants,
, and Fl, were varied in an effort to accomplish this.
The dissipation constant, a, was decreased tenfold to allow
-110-
Z-Z.= NR ,
c[ Ao AP 9 "/2(T-T)
(Z- Z~)
't)7,3 / 2
P1
circle of equivalent area
R
oundary
Z (T-T*)
Z *)
)wn)
Fig. 5.2.2.1
Geometry for plume analyses.
-111 -
o
o
m~~~
4J
4t
9Q:
3x
H
a
4)i
o
0E
o
o
aw
N
t r-4 o
0
H
u
a
E>
04
E-4
8
r,
rz-
or
M
0-
0
o
s
o
~ ~ ~
oa~~
U20 *
CO~
'U
U
U
O
tnt
o
a
HX
0)
C4
P3
*H X
0~~
o
o
o
o
o
o
U
(4
o
o
o
)
Z
o
o
U
' HDIH HaN3D alnqd
o
N
-112-
the turbulence to persist with less dissipation.
lent transport constants, r and r,1
to enhance turbulent diffusion.
The turbu-
were increased tenfold
Alone or in any combination
these variations produced little more than a 20 ft decrease
of plume rise at 200 sec.
Thus, these line-thermal results
are largely independent of the model constants.
The only
term not associated with these constants (see Eq. 3.43) is
the production term.
It is suggested here that the production
term is probably too small because it neglects buoyant production in favor of mechanical production alone.
This hypo-
thesis was not tested further in this work, however.
5.2.3
Brunt-Vaisala Period of a Turbulent, Buoyant Parcel
As a test of the hydrodynamic model, the Brunt-Vaisala
period of a buoyant parcel in a stably stratified atmosphere
is calculated.
Briefly, the Brunt-Vaisala period is the per-
iod of the oscillation of a parcel of fluid that is perturbed
from its equilibrium level in a stably stratified fluid.
A
consideration of the restoring force on the parcel yields
the formula6 9
Brunt-Vaisala period =
2[r
if az
sec]
s
(5.4)
(
-113-
For typical atmospheric values of T and d/dz, the period is
hundreds of seconds.
Computer simulation to hundreds of
seconds is too costly, so the stratification, d/dz, is increased to 0.1 F/ft--about a twentyfold increase over typical
atmospheric values--which decreases the predicted period to
81 sec and allows much more inexpensive simulations.
The entire computer mesh, 103 ft by 103 ft, was initialized to this stable stratification, and a warm parcel was
placed at an elevation of 300 ft.
The results of two differ-
ent runs are shown in Fig. 5.2.3.1.
In a first run, the par-
cel had a small buoyancy parameter:
F
U-
gQh
-4.6
=
s CpTsU
-=cTU
x 10
3
3
2
ft3/sec
which resulted in the lower curve.
The curve exhibits a Brunt-
Vaisala period of 92 seconds, and a fair amount of "jitteriness"--which is not surprising since the total parcel motion
is much less than one cell spacing, so that the motion is not
very well resolved on the mesh.
In a second run, the parcel
had a larger buoyancy parameter:
=
4
.6 x 104 ft3/sec
which resulted in the upper curve.
The curve exhibits a
Brunt-Vaisala period of 102 seconds, and a much smoother
motion since several mesh cells have been traversed, and thus
-114I
I
I
I
440
__
,,- parcel rising in a neutral
I41g*O
a txnosphere
I
I
420
I
.****4
0
4
0a
I.
0
I.
400 -
e
0a
S
00
.00·
O
00
.... e.0
.
0
eOOO0**
0
380
I.
t
I
360 .0
0
I
I
102 sec period
I
44
E4
j
I
340 -
/
H
92 sec period
I
320 -
.06 0*00.0@
.00
.
6.6
10
O.
4
OO*.OOO~t80
*........
O
.
.
I',0.
"@** 60-9*0
o6a
.600
0I
5
_
300
0
L,*
0
_ __
__
I
L
50
I
100
200
TIME (seconds)
Fig. 5.2.3.1 Demonstration of the 13runt-Vaisala period.
Parameters for these runs are discussed in the text.
-115-
the motion is better resolved on the mesh in this run.
Overall the agreement between calculated and observed
values is good, considering that the classic Brunt-Vaisala
problem allows no turbulent mixing, while the simulations in
this work allow it.
Generally, the action of turbulent mixing
is to rapidly diffuse the temperature field and to slow the
period of oscillation.
For comparison to these results, the parcel motion in a
neutral atmosphere for the stronger (F/U = 4.6 x 104 ft3 /sec 2 )
run has been included in Fig.
5.2.3.1.
The stratification
thus has a very strong effect on the motion.
5.3
5.3.1
Comparisons to Field Studies
Pasquill Dispersion and Briggs Plume Rise in
Neutral Atmospheres
A comparison of plume simulation and experiments both in
the plume and atmospheric dominated regimes
(see Fig. 5.1.1)
are discussed here for neutral atmospheres.
To obtain these
results, the ambient atmospheric turbulence is estimated from
the discussion in Chapter Four.
The plume simulations are
started in the plume dominated regime and the simulations are
run out to times where the plume excess temperature is very
small, and the plumes are followed with the pollutant species
concentration.
The atmospheres in this section all have dry
-116-
adiabatic lapse rates of temperature.
A comparison to Briggs' 70 plume rise for neutral atmospheres is made in Fig. 5.3.1.1.
Briggs' work found that the
plume rise and downwind distance, when nondimensionalized with
a length L,
L
-
F
=
gQh
ft 4/
T
3
cpP U 3
ps
3
(5.5)
s
yields a 2/3 power law relation between the plotted values for
a wealth of field data.
To interpret the data from the sim-
ulations, the plume rise is taken as the plume center height
minus the virtual origin height (i.e., the rise from the virtual origin), and the downwind distance is then the product
of the downwind velocity and the elapsed time from the virtual
origin (see Fig. 5.2.2.1).
For the run in Fig. 5.3.1.1, the
distance, L, for a 100OMWt release in a 30 mph wind is 11.3 ft
when calculated with Eq. 5.5.
The agreement is generally
good between calculation and experiment; the errors of estimation of Z, and T, and the undertainty in the ambient turbulence level all contribute to the discrepancy.
Also, the
data point at X/L = 78 is taken from the initialized plume
cross section at time t = 0 which is not a physically accurate
picture of the plume.
The good agreement between calculation
and experiment at this point is felt to be simply a cancellation of opposing errors.
-117-
004
me
0S
mo
-40o
4
.a
4.)
Z4
,-4%4
,-4
.4 X
44 XE-
Ono
0o
0
·04
o
O0rC0
d4)c
C I
-4 U
CoQ
.Z4U
N
*O
UZ
m
0
O
E
o
*
^
-118-
A comparison to Pasquill's plume dispersion in neutral
atmospheres (exactly the Pasquill D class) is made in Fig.
5.3.1.2.
The Pasquill dispersion curves are taken from
Turner's workbook 1, whose values are corrected from older
The calculated plume dispersion
sources of dispersion data.
is taken from plume cross section printouts at four different
times during the simulations.
The calculated plume dispersion
follows the Class D dispersion fairly well, but with a trend
toward overpredicting the dispersion at points closer than
the point 1/2 km downwind.
This overprediction is again
related to the finite plume size at time t = 0 in the initialization scheme.
This error affects the earlier solution
greatly, but has a decreasing effect on the solution at longer
times.
The error bars in the figure represent the error
associated with increasing or decreasing the plume cross sectional area by one mesh cell.
This gives a rough notion of
the errors expected when the mesh cells are interpreted as
Note
being either entirely inside or outside of the plume.
that these one-cell error bars decrease as the total number of
cells in the plume increases with downwind distance.
The
trend to underpredict the dispersion at large distances
reflects probable errors in the estimation of the ambient
turbulence.
Also note that since the turbulence is assumed
to be isotropic, the calculated horizontal and vertical
-119-
10
M
W
rn
4)
43
4)
0
I
b
1
z
H
r4
C12
M
H
in
E-1
z
0o
N
H
0:
o
100
10 000
1000
DISTANCE DOWNWIND
(meters)
Turner's7 lHorizontal Dispersion versus
Fig. 5.3.1.2
Distance Downwind Compared to a Computer Simulation
of a 1000 MWt Release in a 30 mph Wind.
-120-
dispersion will not differ greatly; therefore, Pasquill dispersion cases that have significantly different a 's, and
y
ai's (e.g., extreme stability) will be difficult for this
z
model to duplicate.
LAPPES SO 2 Dispersion Studies
5.3.2
A number of comparisons to a well-studied plume from
the LAPPES field experiments are made in this section.
The
plume emanating from stack No. 1 of the Keystone coal-fired
generating station at about 8 a.m. on October 20, 1967 is
modeled with a computer simulation.
Information about the
ambient weather and plant operating characteristics are provided in the LAPPES study.
Experimental helicopter SO2
plume cross sections and S02 bubbler data are available for
comparison.
The computer simulation is initialized in Fig. 5.3.2.1
The stack is releasing heat at
in the following way:
6
28.6 x 10 cal/sec. Half of this is to be arbitrarily put
into three mesh cells that are 164 ft (50 m) high and 492 ft
(150 m) wide.
The other half of the heat resides in the
mirror image of these cells.
Using a Briggs plume rise cor-
relation for the rise induced by the initial momentum of the
effluent (20 m/sec exit velocity) yields a rise of about
100 ft.
The three cells are then to be centered at the
121..
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Fig. 5.3.2.1 Initialized Plume Cross Section for the
Keystone No. 1 Stack on 20 October 1967. DY = 150 m,
DZ = 50 m.
-122-
stack height (800 ft) plus the momentum rise height (100 ft),
or 900 ft.
If the cells are all 164 ft high, then the center
of cell (2,7) is at 902 ft--thus cells (2,6),
(2,7), and
(2,8), are to be initialized with half of the heat release.
Checking the prevailing winds for cells (2,6),
2,7),
and
(2,8) finds that they sweep out 7.3 million cubic feet in one
second.
Releasing 14.3 x 106 cal into 7.3 x 106 ft3
gives
a temperature rise of 0.47°F, and this is added to the ambient
air temperature in these cells in Fig. 5.3.2.1, which shows
the computer initialization in the vicinity of the plume
cells.
This whole initialization process is admittedly
crude, particularly in the treatment of momentum, but it gives
very satisfactory answers.
The experimental results for this plume are found in
Figs. 5.3.2.2 and 5.3.2.3.
The former figure shows the pre-
vailing wind speed, direction and potential temperature.
The
weather was clear on that morning, and a sizeable low-level
inversion had formed during the night to about 250 m depth.
The flow above 250 m was essentially neutrally stratified and
flowed from the west.
The turning of the wind with height
is ignored in the computer simulation, but the wind speed
and potential temperature values are input directly onto the
computer mesh.
The ambient humidity was fairly low due to
a wide subsidence inversion over most of the computer mesh,
-123-
i
i
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to
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-124-
.,4
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,lHi
oli
.,-I
-125-
so humidity is neglected in this simulation.
Turbulence
values are calculated for a neutral atmosphere from Blackadar,
S9
and the turbulence in the inversion layer is supp-
ressed by a factor of 100 in the absence of any better information about the turbulence in the stably stratified region.
The eddy viscosity is of the order of 100 ft2/sec in the
neutral region, and about 1 ft2/sec in the stable region.
The helicopter plume cross section at 4.8 km downwind is
drawn in Fig. 5.3.2.3 as the jagged outline.
The outline con-
nects the measured SO2 horizontal traverses, and represents
essentially a 1 percent boundary of SO 2.
A mass balance of
SO 2 in the plume cross section finds only 55 percent of the
SO 2 that was emitted at the stack.
It is suggested that much
of the remaining 45 percent of the SO 2 could be found below
200 m, since the helicopters flew no lower than this
safety) yet were still finding SO 2 at this level.
(for
The comp-
uter simulation at 600 sec is superposed on the experimental
plume outline.
Again, the computer trace represents about
a 1 percent boundary of SO 2.
of SO 2,
Except for the two large "wings"
the agreement is fairly good.
The "wings" are likely
produced by low frequency horizontal turbulent eddies generated
in the region of the turning of the wind--but since the turbulence on the computer is assumed to be
isotropic, this can-
not be corrected in these runs in any simple way.
-126-
To put the computer and experimental results into perspective, a handbook calculation of the plume S
taken here.
2
is under-
The trouble with the anlaysis of the plume of
October 20 is to decide whether the stable or neutral conditions have the greatest effect on the behavior, since only
one stratification can be used in handbook estimates.
The
plume rise and dispersion in an F class (stable inversion)
is presented in Fig. 5.3.2.4.
Note that the plume rise fits
the data well, but the dispersion is too small
sion is the 3a, or 1 percent level value).
and dispersion in a D class
Fig. 5.3.2.5.
(the disper-
The plume rise
(neutral layer) is presented in
Note that the plume rise is too large, while
the dispersion is fairly close to the actual, but is also not
able to reproduce the "wings" of S
2.
It is seen that either
single choice of stability does not agree as well as the
computer simulation (which was able to follow the plume
through both regions of stability).
This generality in the
computer simulations appears to be a major source of improvement over the handbook estimates.
The comparison of experimental and computational
plume cross sections is continued in Fig. 5.3.2.6 for the
plume at 10.0 km.
The helicopter results have the jagged
outline in Fig. 5.3.2.6.
A mass balance of SO 2 yields only
29 percent of the emitted S2, which brings the experimental
-12 7-
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-128-
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-129-
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-130-
results into question.
The computer simulation is super-
posed, but not much emphasis should be placed on the comparison since the helicopter results appear to be inaccurate
on the basis of the low mass balance.
Two other tests of the model with the October 20 plume
are found in Figs. 5.3.2.7 and 5.3.2.8.
Four S02 bubblers
were placed on a small hill at 65 m elevation above the stack
base, at 6.5 km downwind.
The bubbler 1/2 hour averages
were all averaged together to yield a 12 pphm ground-level
SO2 concentration.
The central region of the simulated
plume cross section is copied in Fig. 5.3.2.7.
Each S02
concentration represents the value in a single computer mesh
cell.
The dashed line is drawn through the plume at the 65 m
elevation, where the 12 pphm experimental value compares very
well with the predicted values inside the plume
The entire simulation was carried to 12 km downwind.
The maximum ground level SO 2 concentration is plotted at 2 km
intervals in Fig. 5.3.2.8.
yet reached a maximum.
Even at 12 km the S02 has not
However, the maximum S02 in the plume
(calculated at 12 km) is 33 pphm, while the ground level is
already 17 pphm--so the maximum calculated value will have to
be between 17 and 33 pphm.
A handbook estimate of the maxi-
mum ground level concentration (taken from Eq. 3.146 of
Slade) 2 yields 47 pphm at 17 km downwind in neutral D class
-131-
-F c
plume centerline
II
I
I
I
75
48
84
48
16
88
45
14
60
36
13
22
22
9
m
65 m
11
m
8
m
15
m
_m
9
_
_
_
9
_
7
_
_
7
m
m
7
7
---
EARTH
Fig. 5.3.2.7 Comparison of the computer simulation of
the 20 Oct 67 Keystone #1 plume cross section with the
half-hour average SO2 concentration at 65 m. The values
in the boxes are the mesh cell
[SO 2 ] in pphm predicted
by the code at 6.5 km downwind. An average [SO 2] of
12 pphm was recorded in four SO2 bubblers at 65 m
elevation and 6.5 km downwind. The dashed line at 65 m
represents the bubbler elevation on the computer mesh.
-132-
4.I
.J
u
O 4
0 gu
1,
H
0
.4k
0
4
O
W
4J
r4
U
U
'UN~~U
C 40
0
0
O
e4
0
0
O
5
0X
H
0
q
£
0
.rq
Ez
0
Q
(%
4.u
.
Ua
0
E)
10
E
f....
'U
:r
0d
*e
*e
0
,_
h u
044-
c
to
UZ
~
-133-
stability (which is roughly an upper bound for this value) and
22 pphm at 90 km downwind in stable E-class stability (which
is roughly a lower bound for this value).
The calculated
value must therefore fall somewhere between these handbook
estimates which lends credence to the results, even though
the actual maximum ground level S0 2 value was not calculated.
5.4
5.4.1
Results of Model Extension
Fumigation Episode
A computer simulation that approximates a fumigation
episode is presented in this section.
No particular episode
is intended to be represented by this run, but several of the
general features of a fumigation episode near a shoreline
site in the Great Lakes area are included.
73
These episodes
are commonly characterized by a wind off of a large cool
lake on a sunny spring day.
The air that has traveled over
the lake has developed a deep stable layer because of sensible heat exchange with the cool lake water.
As this deep
stable layer streams inland, the strong solar heating at
the gound causes a deepening unstable thermal boundary layer
to develop.
This layer is characterized by strong mixing as
vigorous turbulent thermal convection sets in.
Plumes re-
leased in the stable air exhibit small plume rise and
dispersion until they encounter the growing boundary layer
-134-
from below.
Quite rapidly they have their pollutants spread
to the ground, in a sense "fumigating" a relatively small
area with high pollutant concentration.
A rough calculation was performed to demonstrate this.
The plume in a stable inversion from Sec. 5.3.2 was released
over a deepening turbulent layer coming up from the ground.
The situation is shown in Fig. 5.4.1.1.
At 0, 20, 50, 100,
and 200 seconds the turbulent layer (eddy viscosity =
1000 ft2/sec) is deepened by an increment of 100 meters--this
is represented by the staircase in Fig. 5.4.1.
The plume
that is released at 0 km is about half engulfed in the turbulent layer at 1 km, and almost entirely engulfed at 2 km.
The strong turbulent mixing has produced ground level concentrations a factor of 4 and 8 times higher at these stations than the results of Sec. 5.3.2.
This agrees qualita-
tively with actual fumigation episode results, and serves to
demonstrate the ability of the model to extend into these
important cases.
A more refined calculation of the boundary
layer, and actual weather and plume data from a shoreline
site would be needed to more carefully test this type of
simulation.
5.4.2
Plumes with Change of Phase
Model validation has not been carried out for plumes
-135-
0
4.J
x
o
I
II
!
I
0
a
.O
k
.,4
.,4
4J U)
,U
0
0
4-
.
I
I
0
O
,n CU
. )
UO)
La
oO
r,
.q
o0
.,4
aC3
Ln
w
U
-136-
with change of phase.
The available Chalk Point data were
incomplete during this work and could not be used.
The bal-
loon data from several mechanical draft towers at the Commanche plant74 suffers from a lot of scatter from a variety
of sources, and data were only obtained very close to the
towers, which casts doubt on the ability of the simulations
to handle this case.
From preliminary work with saturated parcels of air on
the computer mesh, the model is found to suffer from oscillations that are due to the explicit nature of the moisture
equilibrium calculation.
The oscillations can be brought
under control by reducing the timestep size, but a more fundamental solution to this problem is recommended in Sec. 6.2.3.
-137-
6.
CONCLUSIONS AND RECOMMENDATIONS
6.1
Model Validation
A widely applicable calculational model of buoyant
bent over plumes has been developed.
The advantage of the
model is its ability to treat problems outside the scope of
existing plume models without greatly increasing the resources required for the analyses.
The acceptance of the
model, however, must begin with a demonstration of its
ability to reproduce the solutions to problems that are known
to be solvable.
This demonstration has proceeded along two
lines in this work--problems in which the plume properties
dominate the flow, and problems in which the atmospheric
turbulent mixing dominates the flow.
Some overlap between
these simple regimes occurs, but overall this organization
serves to highlight the causes of the particular successes
and discrepancies in the model validation work.
The results of Sec. 5.2 deal mainly with the plume dominated motions.
Generally it is found that very good agree-
ment with laboratory experiments is obtained.
In particular,
the buoyant line-vortex motions and the Brunt-Vaisala period
of a buoyant cylinder of fluid are studied, and they compare
very favorably with the predictions.
However, the
-138-
unsuccessful attempts to "tune" the turbulence model coefficients point out the rather limiting assumptions contained in
the present turbulence model, particularly with regard to
buoyant production of TKE.
been noted 75
To the model's credit, it has
that wall-free turbulent flows are the most
difficult to "tune," and that the model does a credible job
in its current form.
The results of Sec. 5.3 deal mainly with the atmospheric
dominated motions.
There is a large amount of overlap into
the plume-dominated motions in the Briggs plume rise and
Pasquill dispersion results, but these cases both represent
experiments that were actually performed in the atmosphere,
and they exhibit a fair amount of scatter in their data because of this.
Again, the agreement between calculation and
experiment is good.
The results of the LAPPES individual
plume study provides the best indication of where the calculational model is expected to benefit the modelers of plumes.
In the limited number of calculations contained in that section, it is found that the calculational model agrees with
the experimental results more closely than the current handbook estimates simply because it has made a more fundamental
calculation, taking into account the actual micrometeorological profiles.
Furthermore, the model provides a relatively
accurate starting point for the detailed description of other
-139-
important processes in a plume:
chemical reactions, visi-
bility, radiation dose rates, etc.
6.2
6.2.1
Recommendations
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 (though it can be in steady state), but a
very restrictive assumption may afford a useful recasting of
the two-dimensional problem.
next.
This assumption is discussed
-140-
-L
R
bm/sec
-
X(z) - R/u(z)
1 ft2/
u (z)
I
r
-
R lbm/sec -- mm
1 ft 2
u (z
o)
z
z
X(z) > X(Z)
for
EARTH
Fig. 6.2.1.1
Dilution of a Steady Re leas e
of Pollutant.
-141-
Consider the advection and turbulent diffusion of a
pollutant in three dimensions (the results extend directly
to momentum, sensible heat, etc.):
ax + u ax + v
at
ay
+
aXz = Y
CPX
7
(6.1)
X
Assuming that the system is in steady state, we have
u a
ax
+ v
ay
+ w a
Z = Xa
x
(6.2)
In the presence of a steady uniform wind field, u,
the first
term is commonly interpreted as the time-rate-of-change for
an observer moving with the wind, and is written as t
'
0
where uot o = 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)
where u
ax-_X u(z)
u(z )
(uU(z)
o ax
u(z0 )
x
at
0
has been arbitrarily chosen to be u(zo .
(6.3)
This
interpretation allows the equation to be formulated as
-142-
a-
+
u (z
u(z o
)
ay
+
u (.Zol
(z
2
x )
(6.4)
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 corresNote that it also causes parcels that
pondingly diluted.
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.
6.2.2
Calculational Scheme for Time-Dependent Release or
Weather
The simulation of "mildly"
made with the model.
time-dependent plumes can be
Essentially, the governing assumption
here is that the prevailing weather or effluent properties
-143-
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
to 100 sec, so a second simula-
tion is made with properties QCto + 100) to 200 sec.
the plume properties changed at time t
Again
+ 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.
6.2.3
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
-144-
I
simulation with plume properties at time t
I
I
simulation with plume properties at time t+100
I
I
__
I
I
I
II
simulation with plunae properties at time to+ 2 0 0
I
8I
mill
-
III II
a
I
100
200
Time
Fig. 6.2.2.1
I
I
I
00
I
I
_
300
(sec)
Simulation of a Time-Dependent Plume in
Steady-State Weather. Scheme is discussed in the text.
-145-
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.
-146-
NOMENCLATURE
A
0
2
initial line-vortex area (ft
)
C
experimental constant, 1.9
Cp
heat capacity at constant pressure (BTU/lbm °R)
c m oist
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/sec 2 )
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)
-147time average pressure perturbation
p
(mb)
P
p
fluctuating pressure perturbation (mb)
Pr
Prandtl number
q,q(y,z,t)
turbulence kinetic energy per unit lb
qlibrary
(Z) prescribed turbulence kinetic
energy profile
(ft2 /sec 2 )
Q
heat (BTU)
QH
heat emitted at stack exit (BTU/sec)
R(T)
plume radius as a function of time (ft)
Rdry
Rd
Rvap
Rv
gas constant for dry air
(ft3 mb/lbm°R)
gas constant for water vapor
(ft3mb/lb °R)
Sliq
Schmidt number for liquid water
Sc
Schmidt number for water vapor
vap
t,T
time (seconds)
t
x/u O
o
)
2
(ft2/sec
(sec)
T*
time coordinate of virtual origin (sec)
T
physically measurable temperature (R)
T
temperature perturbation about an adiabatic
reference state (R)
T
temperature in a quiet adiabatic atmosphere
T
temperature of stack effluent (R)
s
virtual temperature
Tv
T
virtual temperature in a quiet adiabatic
atmosphere (OR)
vo
u
U
(R)
downwind velocity (ft/sec)
windspeed, constant with height (ft/sec)
U
Ueddy
turbulent velocity scale in an eddy (ft/sec)
ui,u
j
velocity (ft/sec)
Ui,u
j
velocity (ft/sec)
(R)
-148ui.Uj
time average velocity (ft/sec)
ui',uj'
fluctuating velocity (ft/sec)
Reynolds stress tensor (ft2/sec2
)
correlation of fluctuating velocity and temperature
Ui
,'
'uk,
triple correlation of fluctuating velocity (ft3/sec3 )
v
crosswind velocity (ft/sec)
Vg
geostrophic wind (ft/sec)
w
vertical velocity (ft/sec)
W ol
molecular weight of the pollutant species
(lbm/lbm-mole)
X
downwind distance (ft)
X i jX j
cartesian coordinate (ft)
y
crosswind distance (ft)
z
height (ft)
*,
height coordinate of virtual origin (ft)
-149-
a, a1
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)
Eh
eddy diffusivity of heat
Em
eddy diffusivity of momentum (ft2 /sec)
(ft2/sec)
eddy diffusivity of pollutant
e
potential temperature
(ft2/sec)
(R)
potential temperature perturbation about an
adiabatic reference state (R)
8e
potential temperature in a quiet adiabatic atmosphere (R)
time average potential temperature
(R)
(R)
8'
fluctuating potential temperature
8v
virtual potential temperature
8vo
virtual potential temperature in a quiet adiabatic
atmosphere (R)
8v
time average virtual potential temperature
ev
fluctuating virtual potential temperature
(i)
X x
decay constant for i t h radioactive decay channel
from pollutant species X (sec -1 )
(R)
(OR)
(R)
-150-
1j
dynamic viscosity (lbm/sec. ft)
V
kinematic viscosity (ft2/sec)
Pdry
Pliq
density of dry air
(lbm/ft 3)
liquid water density (lbm/ft 3)
Pliq
time average liquid water density (bm/ft
P liq
fluctuating
PS
density of stack effluent (lbm/ft3)
Psat
saturation water vapor density (lbm/ft 3)
Pvap
Pvap
yap
a, a (y,z,t)
library(z)
X
3)
liquid water density (lbm/ft3)
water vapor density (lbm/ft3)
time average water vapor density (lbm/ft 3)
fluctuating water vapor density (lbm/ft 3)
eddy viscosity (same as
m)
(ft2/sec)
prescribed eddy viscosity profile
pollutant density (lbm/ft 3)
(ft2 /sec)
-151LIST 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
Figure 3.3.2.la
Bent-Over Buoyant Plume with Ambient Thermal Stratification
56
Figure 3.3.2.lb
Mesh Alignment Appropriate for
a Line Source Release
57
Figure 3.3.2.1c
Mesh Alignment Appropriate for
a Point Source Release
57
Figure 33.3.3.1
Reconstruction of the Three-dimensional Plume. Wind vectors
as a function of height are
shown.
59
Table 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
Figure 4.3.2.1
The Concept of Layers in the
80
Table 4.3.3.1
Planetary Boundary Layer.
Comparison of Eddy Viscosity Prescriptions
85
Figure 5..11
Figure 5.2.1.1
Plume Regimes
Key to Cellwise Quantities for Figs.
5.2.1.2, 5.2.1.3, 5.2.1.4, 5.2.1.6,
91
95
and 5.3.2.1.
Figure 5.2.1.2
Initialized Plume Cross Section at
0 sec.
96
Figure 5.2.1.3
Plume Cross Section at 20 Sec.
98
-152page
99
Figure 5.2.1.4
Plume Cross Section at 80 sec.
Figure 5.2.1.5
Turbulence Kinetic Energy Profile
at 80 sec.
100
Figure 5.2.1.6
Plume Cross Section at 200 sec.
101
Figure 5.2.1.7
Initialized Plume Cross Section
with Internal Circulation
103
Figure 5.2.1.8
Plume Comparison at 40 sec of
simulation showing the small
effect of an initial circulation
on the plume development.
104
Figure 5.2.1.9
Plume development in a moderately
turbulent atmosphere
105
Figure 5.2.1.10
Plume development in a very turbulent atmosphere
106
Figure 5.2.2.1
Geometry for plume analyses
110
Figure 5.2.2.2
Comparison of Computational and
Experimental Results for Plume
Rise versus Elapsed Time for
Buoyant Line-Thermals
111
Figure 5.2.3.1
Demonstration of the BruntVaisala period
114
Figure 5.3.1.1
Briggs' 70 Comparison of Plume
Rise in Neutral Atmospheres
Compared with a Computer Simulation of a 1000 MWt Release in a
30 mph Wind.
117
Figure 5.3.1.2
Turner's 7 1 Horizontal Dispersion
versus Distance Downwind Compared to a Computer Simulation
of a 1000 MWt Release in a 30
mph Wind.
119
Figure 5.3.2.1
Initialized Plume Cross Section
for the Keystone No. 1 Stack on
20 October 1967.
121
Figure 5.3.2.2
Weather for 20 October 1967 at
the Keystone Plant
123
-153page
*
_ 1_1_1
Figure 5.3.2.3
Comparison of Computational and
Helicopter Results at 4.8 km
Downwind
124
Figure 5.3.2.4
Comparison of Handbook Calculations and Helicopter Results at
4.8 km Downwind
127
Figure 5.3.2.5
Comparison of Handbook Calculations and Helicopter Results at
4.8 km Downwind.
128
Figure 5.3.2.6
Comparison of computational and
experimental results at 10.0 km
downwind for the 20 October '67
Keystone #1 plume.
129
Figure 5.3.2.7
Comparison of the computer simulation of the 20 Oct 67 Keystone
#1 plume cross section with the
half-hour average SO 2 concentration at 65 m.
131
Figure 5.3.2.8
Maximum ground-level SO concentrations predicted in te computer simulation of the 20 Oct 67
Keystone #1 plume
132
Figure 5.4.1.1
Fumigation Episode at a Shoreline Site
135
Figure 6.2.1.1
Dilution of a Steady Release of
Pollutant
140
Figure 6.2.2.1
Simulation of a Time-Dependent
Plume in Steady-State Weather
144
____)__1
_I_
II_
_ __
-154-
REFERENCES
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.
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F. Pasquill, Atmospheric Diffusion, Second Ed.,
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4.
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11.
C. DuP. Donaldson, "Construction of a Dynamic Model of
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W. Lewellen, and M. Teske, "Second-Order Closure
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K. Rao, et. al,, "Mass Diffusion from a Point Source
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R. Lange, "ADPIC: A 3-D Computer Code for the Study
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M. Dickerson t et. al., "Concept for an Atmospheric
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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
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Validation Against Regional Tracer Studies,"
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22.
C. Sherman, "Mass-Consistent Model for Wind Fields
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23.
R. Henninger, "A Two-Dimensional Dynamic Model for
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24.
J. Taft, "Numerical Model for the Investigation of
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___
I
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----
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Ibid., Chapter 18.7.
46.
Ibid., Chapter 12.
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H. Tennekes, "The Atmospheric Boundary Layer," Physics
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C. Priestly, Turbulent Transfer in the Lower Atmosphere,
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R. Scorer, Natural Aerodynamics, New York, Pergamon
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A. Monin, "The Atmospheric Boundary Layer," in
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.-
51.
J. Businger, "The Atmospheric Boundary Layer," V. Derr,
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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.
--------c-
II
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54.
G. Csanady, Turbulent Diffusion in the Environment,
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55.
J. Lumley, and H. Panofsky, The Structure of Atmospheric
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56.
A. Monin, and A. Yaglom, Statistical Fluid Mechanics,
Vol. 1, Cambridge, MA, MIT Press, 1971.
57.
A. Monin, and A. Yaglom, Statistical Fluid Mechanics,
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58.
Ibid., Vol. 1, p 280.
59.
A. Blackadar, "The Vertical Distribution of Wind and
Turbulent Exchange in a Neutral Atmosphere," Jour.
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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.
63.
R. Bornstein, "The Two-Dimensional URBMET Urban
Boundary Layer Model," J. Appl. 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.
Ibid., p. 37.
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.
-159-
69.
J. Holton, An Introduction to Dynamic Meteorology,
New York, Academic Press, 1978, pp 167-169.
70.
G. Briggs, "A Plume Rise Model Compared with Observations,"
Jour. Air Poll. Con. Assoc., Vol. 15, No. 9, pp 433-438,
Sep. 1965.
71.
D. Turner, Workbook of Atmospheric Dispersion Estimates,
Cincinnati, Ohio, Us. Dept. HEW, 1969.
72.
D. Slade, ed., Meteoroloqgy and Atomic Enerqv--1968,
Oak Ridge, Tenn., USAEC, TID-24190, July, 1968.
73.
R. Meroney and J. Cermak, "Modeling of Atmospheric
Transport and Fumigation at Shoreline Sites," Bound.
Lay. Met., 9, pp 69-90, 1975.
74.
R. West, "Field Investigation of Cooling Tower and
Cooling Pond Plumes," EPA-600/7-78-059 t Corvallis,
Oregon, April, 1978.
75.
J. Lumley, and B. Khajeh-nouri, "Computational Modeling of Turbulent Transport," in Advances in Geophysics, 18A, New York, Academic Press, 1974,
pp 169-192.
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