oEPA
United States
Environmental Protection
Agency
Environmental Sciences Research ..„
Laboratory
Research Triangle Park NC 27711"
Research and Development
EPA-600/S4-81-008 Apr. 1981
Project Summary
Potential Flow Model for
Gaussian Plume Interaction
With Simple Terrain Features
A. Bass, D. G. Strimaitis, and B. A. Egan
The theory of turbulent plumes
embedded within potential flow fields
is discussed for flows modified by
special complex terrain situations.
Both two- and three-dimensional
isolated terrain obstacles are consid-
ered. Concentration estimates are
evaluated using a Gaussian solution to
the appropriate diffusion equation;
dispersion coefficients are modified to
account for terrain-induced kinematic
constraints, and plume centerline
trajectory is obtained from a stream
line of the potential flow. Specific
limitations to the theory and its appli-
cability are reviewed.
A computer algorithm is developed
and documented to perform these
calculations. Dispersion estimates
and ground-level concentrations are
given for a variety of meteorological
situations. Parameters of the problem
include obstacle height, effective
source height, distance between source
and obstacle, crosswind aspect ratio
of the obstacle, and atmospheric
stability. The potential flow theory.
originally applicable to neutral flows.
is extended by an empirical approxi-
mation to slightly stable flows. Addi-
tionally, an interpolation scheme is
proposed for objects of arbitrary cross-
wind aspect ratio between the limiting
cases of a hemisphere and a half-
circular cylinder. Model computations
are compared to laboratory experi-
mental results and to field measure-
ments.
This Project Summary was devel-
oped by EPA's Environmental Sciences
Research Laboratory. Research Tri-
angle Park, NC, to announce key find-
ings of the research project that is fully
documented in a separate report of the
same title (see Project Report ordering
information at back).
Introduction
This study has been motivated by the
requirement for a treatment of plume
dispersion in complex terrain that is
practical for regulatory use, yet retains
much of the essential physics of the
problem. For regulatory purposes, the
modeling approach should emphasize
cases with potential for high ground-
level concentrations. These include:
• stable conditions and low wind
speeds, under which direct plume
impact or blocking by nearby terrain
obstacles may occur, and
• neutral or slightly stable conditions
and moderate or high wind speeds,
under which the plume centerline
trajectory passes over and close to
the terrain surface.
This report addresses the development
of modeling methods for neutral and
slightly stable conditions. The general
approach employed follows the theory
of turbulent plumes embedded within
potential flow fields. The theory is
applied to the calculation of ground-
level concentrations using a Gaussian
form of solution to the diffusion equa-
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tion. Streamfunctions proper to potential
flow over a cylinder (aspect ratio = «>)
and to potential flow over a sphere
(aspect ratio = 1) form cornerstones of
the model. These are extended to de-
scribe flows over terrain features of
intermediate crosswind aspect ratio by
a weighting of the two limiting stream-
functions. This weighting scheme was
derived in part using results from wind
tunnel experiments for flows over ob-
stacles of intermediate aspect ratio.
In addition, although the model is
strictly applicable only to neutral flows,
an empirical approximation scheme is
included to define streamline lowering
caused by increased stratification. The
empirical basis for this portion of the
model is derived from stratified tow-
tank experiments.
Other restrictions in the use of the
model have not been addressed. These
limitations to the model arise from the
neglect of boundary layer phenomena
such as flow separation, unsteady wake
effects, time-dependent effects of sta-
bility (e.g., lee wave generation), and
surface heating effects. In addition, the
theory is applicable in a strict mathe-
matical sense only for thin plumes.
The report discusses the rationale for
selecting particular modeling approaches,
provides full technical documentation
for the algorithms developed, and pre-
sents, for a number of specific test
situations, the results of comparisons
between model calculations and labora-
tory and field observations.
Approach
In complex terrain, moderate or high
wind speeds with neutral or slightly
stable stratifications often result in high
pollutant concentrations because, as
the plume is transported over terrain
features, it is forced to pass close to the
terrain surface. Physical mechanisms
relevant to these conditions include
terrain-induced alteration of the plume
centerline trajectory and kinematic
constraints on horizontal and vertical
dispersion.
The modeling approach used here
applied potential flow theory to a Gaussian
point source model. The model incor-
porates a theory for turbulent plumes
embedded within potential flow fields
based on solutions to the diffusion
equations describing flow fields over
two-dimensional and three-dimensional
axisymmetric terrain obstacles. Quali-
tatively, these solutions are of Gaussian
form, with crosswind and vertical dis-
persion coefficients evaluated as line
integrals of the velocity field along the
plume centerline trajectory.
Evaluating the terrain-influenced
dispersion coefficients for this model
requires specifying the crosswind and
the vertical diffusivities. To compare
model calculations with analogous flat
terrain situations, an approximation
scheme was implemented, using the
PGT dispersion coefficients as a cali-
brating scale. Qualitatively, the diffu-
sivity at a given distance from the
source along the plume centerline
streamline is taken as that for the same
transit time in flat terrain. The conse-
quence of this assumption is that model
calculations of dispersion coefficients
reduce to flat terrain values in the limits
of large downwind distances or small
obstacles.
To account for the effects of stability,
an approximation consistent with labor-
atory observations is adopted to lower
the height of the neutral streamline
within two obstacle heights of the
obstacle center by an amount deter-
mined by the height of the obstacle and
the Froude number.
A second approximation is derived to
account for an obstacle with arbitrary
crosswind aspect ratio. In this case,
two- and three-dimensional streamlines
are weighted to provide an intermediate
plume centerline trajectory.
Results
Model computations indicate that
maximum concentrations vary signifi-
cantly with obstacle size, effective stack
height, and relative distance between
the stack and the obstacle. Comparisons
of model predictions with available
observations test the model performance
for a limited number of possible com-
binations of these and other factors.
Table 1 summarizes the range of model
parameters involved in the comparisons.
Note that A is the crosswind aspect ratio
of the obstacle, Fr is the Froude number
characterizing the importance of densi
stratification in defining the flow, X«/
is the distance between the stack ar
the obstacle normalized by the obstacl
height, and H,/a is the effective stac
height normalized by the obstacle heigh
The "smooth tunnel" comparisor
and the "tow tank" comparisons te
the model under conditions that min
mize the influence of processes ni
contained in the model. They show th.
the model is able to predict the observe
maximum surface concentration with
a factor of two (generally overpredictinc
depending on the interpretation of th
observed plume properties in the al
sence of the obstacle.
The "rough tunnel" comparisor
include the effects of a strong bounda
layer on flow over obstacles with
triangular cross section. Model predii
tions of maximum surface concentn
tions are again within a factor of tv\
(overprediction) for obstacles of aspe
ratio 1. However, as the crosswin
aspect ratio increases, the concentn
tion predictions fall below the observ:
tions. The data indicate that the plurr
size is significantly enhanced upstreai
and over hills with the larger aspei
ratios. The deformations included in th
potential flow field approximation ai
only partly responsible. A better unde
standing of plume dispersion in a d<
forming boundary layer flow is likely 1
be needed to describe these experiment
more accurately.
Plume interactions with two terrai
features, a ridge and an isolated moum
near the Widows Creek Steam Electr
Power Plant in Alabama have bee
modeled. Meteorological conditior
used in the model correspond to seve
hours selected from nine months i
hourly S02 and meteorological dal
collected by the TVA Air Quality Brand
The hours selected for comparison wei
derived by matching hours of hig
measured SO2 concentrations on th
two terrain features with neutral-tc
stable atmospheric conditions. In add
Table 1. Range of Model Parameters Evaluated in Laboratory and Field Tests of th
Potential Flow Model
Comparison Study
Fr
Smooth Tunnel
Tow Tank
Rough Tunnel
Widows Creek
1
1
1,2,3,°°
4
999
0.97
999
1.3-1.7
0.4-1.0
3.7
3.7
3.7
32
12
0.4,0.
0.4
0.5.1.
>1
>1
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Table 2. Comparison of Observed Concentrations (/jg/m3) at Widows Creek and Predicted Concentrations Based on the Potential
Flow Model with Buoyant Plume Enhancement
Julian
Day
3
40
190
230
4
166
222
Multilayer
Plume Height
(m)
422
402
320
370
301
373
392
Complex
Observed
Concentration Stability C
Ridge Impact
393 982
550 729
576 —
603 —
Mound Impact
1,179 —
367 —
353 —
Terrain Model Predictions
Stability D
270
157
4,599
12,038
1,529
99
2,123
Stability E
—
—
3,354
7,711
531
16
246
tion, only those hours with nearly
coincident vertical temperature and
velocity profiles were considered. Of the
seven hours selected, four are associated
with impacts on the ridge, and three are
associated with impacts on the isolated
mound.
A comparison of observed and calcu-
lated concentrations for the two most
appropriate dispersion parameter
classes are given in Table 2. Most of the
observations fall between the model
predictions for the two dispersion
parameter classes.
Uncertainties in the meteorological
conditions at plume height, and in the
emissions from the facility, cloud the
comparison of model predictions with
observations. Given the range of data,
reasonable combinations of assumptions
can produce good correspondence be-
tween the concentrations in six of the
seven cases. This does not, however,
constitute an adequate evaluation of the
model because of the uncertainties
underlying these assumptions. The
three greatest uncertainties lie in the
specification of the dispersion param-
eters, the final plume height, and the
actual emission rate.
Conclusions
These preliminary assessments sug-
gest that with verification and refine-
ment, the approach may be applicable to
the following situations:
• isolated, single terrain obstacles of
arbitrary height, of cross-section
approximately circular in a plane
parallel to the wind direction, and
of arbitrary aspect ratio in the
crosswind direction; and
• neutral to slightly stable strat-
ifications.
A number of limitations arise mainly
from physical effects that are not
described by the potential flow model.
The model should not be applied to the
following situations:
• stable flow cases in which the
plume may directly impact the hill;
• dispersion cases dominated by
surface boundary layer effects;
• unstable cases (e.g., strongly
convective situations) for which
potential theory is unsuitable; and
• cases dominated by wake effects.
The range of suitable applications of
the model is also limited by the
theoretical approximations made, and
by the limited configurations studied
experimentally. These limitations
include:
• the "thin plume" approximation
(
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A. Bass, D. G. Strimaitis, and B. A. Egan are with Environmental Research and
Technology, Inc.. Concord, MA 01742.
J. Clarke is the EPA Project Officer (see below).
The complete report, entitled "Potential Flow Model for Gaussian Plume Inter-
action withSimple Terrain Features," (Order No. PB81-171 837; Cost: $17.00,
subject to change) will be available only from:
National Technical Information Service
5285 Port Royal Road
Springfield, VA 22161
Telephone: 703-487-4650
The EPA Project Officer can be contacted at:
Environmental Sciences Research Laboratory
U.S. Environmental Protection Agency
Research Triangle Park, NC 27711
United States
Environmental Protection
Agency
Center for Environmental Research
Information
Cincinnati OH 45268
Postage and
Fees Paid
Environmental
Protection
Agency
EPA 335
Official Business
Penalty for Private Use $300
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