EPA-R4-73-029
June 1973
Environmental Monitoring Series
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EPA-R4-73-029
AIR POLLUTION
TRANSPORT
IN
STREET CANYONS
by
R.S. Hotchkiss andF.H. Harlow
University of California
Los Alamos Scientific Laboratory
Los Alamos, New Mexico 87544
Interagency Agreement No. EPA-IAG-0122(D)
Program Element No. 1A1009
EPA Project Officer: William H. Snyder
Meteorology Laboratory
National Environmental Research Center
Research Triangle Park, North Carolina 27711
Prepared for
OFFICE OF RESEARCH AND MONITORING
U.S. ENVIRONMENTAL PROTECTION AGENCY
WASHINGTON, B.C. 20460
June 1973
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This report has been reviewed by the Environmental Protection Agency and
approved for publication. Approval does not signify that the contents
necessarily reflect the views and policies of the Agency, nor does
mention of trade names or commercial products constitute endorsement
or recommendation for use.
11
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ABSTRACT
This report presents the results of a study to determine the appli-
cability of numerically modeling the transport of pollution in street
canyons. The numerical model employs the solutions of the fully non-
linear, three-dimensional Navier-Stokes equations along with a
transport equation for pollutants, for regions of space in which
obstacles or buildings cause strong distortions in the flow fields.
An analytic formulation of a two-dimensional street canyon is pre-
sented to illustrate the linear theory and the associated principles
of flow in a notch. These results are then compared to the numeri-
cally obtained non-linear solutions to determine the regions of validi-
ty for linear theory.
The numerical technique is also used to model three-dimensional flows
for which some experimental data have been obtained. This includes
calculating the distribution of pollutants in the Broadway Street
Canyon in downtown St. Louis, Missouri.
Finally, the numerical method is used to calculate pollutant distri-
butions in a non-specific street canyon; that is, a street canyon in
which the geometry and other important nondimensional flow parameters
give rise to solutions that are applicable, in a general sense, to a
variety of street canyons.
This report was submitted in fulfillment of Contract Number
EPA-IAG-0122(D), by the University of California, Los Alamos Scien-
tific Laboratory, under the partial sponsorship of the Environmental
Protection Agency. Work was completed as of March, 1973.
ii
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CONTENTS
Page
Abstract ii
List of Figures iv
List of Tables viii
Acknowledgments ix
Sections
I Conclusions 1
II Recommendations 2
III Introduction 3
IV Analytical Derivation 6
V Comparison with the Model of Johnson, et.al. 14
VI The Numerical Approach 15
VII Two-Dimensional Results 19
VIII Three-Dimensional Results 41
IX A Generalized Street Canyon 60
X Computer Requirements 77
XI References -,0
/o
XII Appendix
iii
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FIGURES
No. Page
1. Steady velocity field established in a two di-
mensional square street canyon. 20
2. Experimental flow field observed by Wang, Chang and
Lin for a Reynold's number of 2.1 x 10^. 21
3. Steady pressure distribution isobars for the flow
depicted in Fig. 1. 22
4. Steady particle distribution resulting from a line
source in the bottom of the canyon. 24
5. Isopleths of Fig. 4 showing concentration distri-
bution within the canyon. 25
6. Steady particle distribution resulting from a
point source located at the bottom center of the
canyon. 26
7. Isopleths of Fig. 6 showing concentration distri-
butions in the canyon. 27
8. Isopleths predicted by the analytic model for
concentration distribution in the canyon. 28
9. Steady particle distribution resulting from a line
source in the bottom of the canyon. Results re-
flect a smaller diffusivity. 33
10. Isopleths of Fig. 9 showing concentration distri-
bution within the canyon. 34
11. Steady particle distribution obtained from modeling
the experiment of Wang, Chang and Lin. 36
12. Isopleths of numerically calculated concentrations
in the plane of the source of Fig. 11. 37
13. Isopleths drawn through the data reported by Wang,
Chang and Lin. 38
14. Isopleths of numerically calculated concentrations
for a plane slightly off the plane of the source in
Fig. 11. 39
iv
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FIGURES (continued)
No. Pagt
15. Computing mesh for Broadway Street Canyon -
St. Louis, Missouri. 42
16. Perspective view of velocity vectors in a plane
perpendicular to the z axis at a distance of 15 feet
from the origin. Location of the coordinate system
is shown. 43
17. Perspective view of velocity vectors in a plane
perpendicular to the z axis at a distance of 65 feet
from the origin. 44
18. Perspective view of velocity vectors in a plane
perpendicular to the z axis at a distance of 115 feet
from the origin. 45
19. Plane view of velocity vectors in a plane perpen-
dicular to the y axis at a distance of 55 feet from
the origin. 46
20. Plane view of velocity vectors in a plane perpen-
dicular to the y axis at a distance of 105 feet from
the origin. 46
21. Plane view of velocity vectors in a plane perpen-
dicular to the y axis at a distance of 155 feet from
the origin. The intersection of this plane with the
buildings is shown. 47
22. Plane view of velocity vectors in a plane perpen-
dicular to the x axis at a distance of 175 feet from
the origin. 48
23. Steady particulate distribution in Broadway Street
Canyon resulting from real sources on Broadway, Locust
and Olive Streets. 49
24. Isopleths in a plane perpendicular to the x axis
at a distance of 155 feet from the origin. This plot
corresponds to the measuring station of Ludwig and
Dabberdt. 50
25. Distribution of CO concentration in Broadway Street
Canyon measured by Ludwig and Dabberdt. 51
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FIGURES (continued)
No. Page
26. Isopleths in a plane perpendicular to the z axis at
a distance of 15 feet from the origin. 54
27. Isopleths in a plane perpendicular to the z axis at
a distance of 65 feet from the origin. 55
28. Isopleths in a plane perpendicular to the y axis at
a distance of 55 feet from the origin. 56
29. Isopleths in a plane perpendicular to the y axis at
a distance of 105 feet from the origin. 57
30. Isopleths in a plane perpendicular to the x axis at
a distance of 95 feet from the origin. 58
31. Isopleths in a plane perpendicular to the x axis at
a distance of 205 feet from the origin. 59
32. Generalized street canyon configuration in per-
spective with velocity vectors shown in a plane
perpendicular to the z axis at a distance of 5.5
units from the origin. Location of the coordinate
system is shown. 62
33. Plane views of velocity vectors in a plane perpen-
dicular to the y axis at a distance of 9.5 units
from the origin. 63
34. Steady particle distribution in the generalized
street canyon, as viewed from above, resulting from
a line source perpendicular to the incoming flow. 64
35. Same particle distribution as that shown in Fig. 34,
only viewed from the top of the buildings at mid-
block. 65
36. Isopleths in a plane perpendicular to the z axis at
a distance of 1.5 units from the origin. (Source
perpendicular to inflow). 66
37. Isopleths in a plane perpendicular to the z axis at
a distance of 5.5 units from the origin. (Source
perpendicular). . • 67
vi
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FIGURES (continued)
No. Page
38. Isopleths in a plane perpendicular to the z axis at
a distance of 10.5 units from the origin. (Source
perpendicular). 68
39. Isopleths in a plane perpendicular to the y axis at
a distance of 1.5 units from the origin. (Source
perpendicular). 69
40. Isopleths in a plane perpendicular to the y axis at
a distance of 13.5 units from the origin. (Source
perpendicular). 70
41. Isopleths in a plane perpendicular to the x axis at
a distance of 8.5 units from the origin. (Source
perpendicular). 71
42. Isopleths in a plane perpendicular to the x axis at
a distance of 16.5 units from the origin. (Source
perpendicular). 72
43. Steady particle distribution in the generalized street
canyon as viewed from above, resulting from a line
source parallel to the incoming flow. 73
44. Isopleths in a plane perpendicular to the z axis at
a distance of 1.5 units from the origin. (Source
perpendicular). 74
45. Isopleths in a plane perpendicular to the y axis at
a distance of 13.5 units from the origin, (source
parallel). 75
46. Isopleths in a plane perpendicular to the x axis at
a distance of 8.5 units from the origin. (Source
parallel). 76
vii
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TABLES
No. Page
1. Numerically obtained concentration values associ-
ated with isopleths of Fig. 5. 30
2. Analytically predicted concentration values associ-
ated with isopleths of Fig. 4. 31
viii
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ACKNOWLEDGMENTS
The authors would like to express their appreciation to X. D. Butler
for his valuable technical assistance during this study.
This work was performed, in part, under the auspices of the United
States Atomic Energy Commission and supported, in part, by the
Environmental Protection Agency under contract (EPA-IAG-0122(D)).
ix
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SECTION I
CONCLUSIONS
The transport of air and pollutants in street canyons can be ef-
fectively modeled with numerical techniques that employ solutions
to the full Navier-Stokes equations and pollutant transport
equations in regions where obstacles occur. With a constant
eddy diffusivity turbulence model, we find that the results of
calculating flows in street canyons compare well with experiment.
This is most probably due to the fact that the additional turbu-
lence produced in street canyons by automobiles and by warm
rising gases (plumes), coupled with the vortical flow structure,
causes turbulence to be more nearly uniformly distributed through-
out the canyon than occurs, for example, for flows in a plain
notch. That flow in a notch is extremely dependent on turbulence
intensity distribution is verified by comparing calculational re-
sults with experimental results.
The applicability of numerical calculations to generalized street
canyon configurations is valuable in the understanding of the compli-
cated flow fields that exist in various geometries of city street
canyons. Such techniques can be used as an aid in the development
of more sophisticated analytic models.
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SECTION II
RECOMMENDATIONS
As seen from the text, it is evident that numerical calculations
are very useful in modeling flows in street canyons. However,
it has also been shown that the modeling of turbulence by a
constant eddy viscosity causes inaccuracies in the calculated
pollutant dispersal in pure notch flow. The distributed effects
of a turbulent viscosity in such situations therefore, must be
included in the numerical technique. These effects can be in-
cluded if either a set of mass or momentum diffusivities are
available from experiments, or a full turbulence model is in-
cluded in the numerical method from which a distribution of
turbulence energies can be obtained.
In addition to the above recommendation, further refinements
could be made to the numerical technique that would allow
greater flexibilities and accuracies in computing the structure
of a wide variety of flows. Such refinements include a variable
mesh that would allow for the inclusion of arbitrarily shaped
obstacles, finer resolution of specific regions of interest and
along with a set of turbulence transport equations, better reso-
lution of the spatially distributed mass diffusivities.
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SECTION III
INTRODUCTION
The transport of pollution in the air can be significantly af-
fected by the presence of terrain sculpture of buildings. The
dispersal properties of a noxious plume depend quite strongly
on the position of the source relative to surrounding structures,
for example, a source in a valley surrounded by hills. Likewise
the distribution of carbon monoxide from automobile exhaust de-
pends in complicated fashion on the modifying effects that ad-
jacent buildings have on both the mean wind pattern and the turbu-
lence .
The primary source of information on these effects is derived from
experimental observations, both in the field and from laboratory
models. To correlate these observations requires a combination of
both theoretical and empirical reasoning. The value of such rules
of correlation is maximal if they can be applied to a wide range
of circumstances, but at the same time are simple and convenient
to use. Confidence in their wide applicability requires extensive
testing with as much data as possible, while simplicity and con-
venience implies formulation in terms of short analytical or tabu-
lar expressions.
The purpose of this study is to develop correlation expressions
for a particular class of circumstances, namely the dispersal and/
or buildup of pollution concentration in street canyons (i.e., the
gaps carved by streets in high density metropolitan areas) mainly
as a result of automobile exhaust and smoke or fumes from the
buildings. To this end, our investigation has consisted of three
parts, as follows:
1. As a basis of insuring realism, we have gathered together the
available field data that are essential to the accomplishment of
discriminating comparison. Especially valuable for this purpose
-------
have been the measurements conducted in St. Louis.
2. Because the amount of field data is severely restricted, how-
ever, we have also developed and utilized computer generated data
derived from solutions of the full Navier-Stokes equations in
three dimensional configurations, coupled with the full transport
equations for pollution dispersal in a turbulent atmosphere. Confi-
dence in these results has been inspired by their favorable compari-
sons with the available field data, and use of them has been made
both for the investigation of circumstances not covered by those
data, and as a basis for testing predictions from the correlation
formulas.
3. We have used a combination of analysis and empiricism for the
formulation of analytical expressions and rules by which to describe
and/or predict the pollution levels to be expected in various street-
canyon configurations. The goal has been to achieve a compromise be-
tween analytical elegance and completeness on the one hand, versus
simplicity and convenience on the other. The latter attributes,
however, have not been allowed to override the requirements of ap-
plicability and accuracy.
These three topics are presented in reverse order. We first discuss
the derivation of an equation for the pollution concentration in a
simple notch. The result is quite similar to an empirical formula
~2
proposed by Johnson, et. al. The present derivation serves,, however,
to show more clearly the basis for their formula, modifies the form
to increase accuracy and applicability, and enables a more critical
appraisal of the limitations that can be expected from such models.
In addition, it is shown how the analysis might be extended to other
configurations, by a combination of analytical and heuristic reason-
ing .
The second topic is a discussion of the numerical method employed
for the complete three-dimensional solutions. The goal has been to
provide data for circumstances not easily amenable to the collection
-------
of field data, the results being used both as an end in themselves
(to illustrate the nature of pollution dispersal in various commonly-
occurring street-canyon configurations), and for comparison purposes
(with both the field data and the correlation formulas). This dis-
cussion then blends into the third topic in which the results and
comparisons are brought together. Finally we discuss the conclusions
from the study so far, and indicate the directions that appear ap-
propriate for extension.
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SECTION IV
ANALYTICAL DERIVATION
This analysis is addressed to the problem of determining the distri-
bution of pollutant in a simple notch, this being the idealization
of a street canyon with automobile exhaust emitted at the bottom and
a cross wind passing over the top. While a complete analysis re-
quires numerical solution of the appropriate equations, the approxi-
mate derivations presented here are useful in showing how the princi-
pal features can be parameterized, as well as the manner by which
previous semi-empirical formulas can be extended.
The specific configuration is described as follows.
The origin of coordinate is located at the upper-left
corner of the notch, with x positive to the right and
y positive upwards into the overlying external flow.
The velocity components in these two directions are
u and v, respectively. The width of the notch is W,
and its depth is D (the bottom lying at y = - D). We
define k 5 ir/W. The walls of the notch allow free
slip of the fluid. Across the bottom there is a pre-
scribed flux of pollutant, S, resulting in a concen-
tration field,
-------
.
3y 1
in which v.. is the (constant) coefficient of kinematic eddy viscosi-
ty. Eliminating the pressure and defining
_ 3u 9u
u = 3y " 3^ '
we get
7 0) = 0 .
A suitable solution is
a) =
-------
As described below, we use this approximate form for the benefit
of simplicity, rather than the full solution, which we have
found to produce an extremely complicated concentration formula
without, however, the virtue of significantly greater accuracy.
It may be observed that this solution gives a non-uniform
(sinusoidal) horizontal velocity profile across the top of the
notch, of which the maximum has been equated to the speed of the
external flow.
To solve for the pollutant concentration within the notch, we
utilized the combined convection-diffusion equation
in which v is a (constant) coefficient of eddy diffusivity, closely
related in value to the kinematic eddy viscosity coefficient, V- .
Our procedure is to derive the solution as a power series in u :
* = *b + *o + Uo *1 + "" ' (6)
in which <(>, is the background concentration level carried by the
external flow. Within the notch, the boundary conditions on <|> are
b at y = 0, (7)
=0 at x = 0 and x = W, (8)
oX
v = S at y = - D. (9)
dy
-------
This last describes a constant, uniform flux of pollutant across
the entire bottom. For reasons to be discussed below, however,
the uniformity of flux will be required only in the zero-order
part of the solution. Equation (8), which forbids a flux of
pollutant into the walls of the notch, is easily satisfied to
first order.
Substitution of Eqs. (1), (2) and (6) into Eq. (5) leads to
2 2
Z Sy
for which the appropriate solution is
To the next order in u ,
o
i
3x2 3 2 ~ 2
y v uo
For this, a particular solution can be found of the form
^ = f (y) cos kx . (12)
With
(i -
-------
the equation for £ becomes
2
d f ,2 , ky
—2 ~ k F = Ry (e 7 - Be
dy
for which the following solution is appropriate
f - ^ [eky (1 - ky) - Be-^ (1 + ky)l
L\T L J
4k
Accordingly, the solution for <|> becomes at this stage
„ Ru y cos kx
_
4k
,
ky
Although this solution satisfied the boundary conditions in Eqs.
(7) and (8), it departs from the requirement of Eq. (9), in that
u D cos kx
) = S '
/y - - D
This could be remedied by adding an appropriate part of the re-
duced (homogeneous) solution, but the added complexity does not seem
necessary for the purpose at hand. Indeed, the non-uniform flux is
no greater an error than several others that have been introduced
for the sake of tractability. Equations (1) and (2) could, for
example, be replaced with the exact solution for the velocity pro-
file that results from the stated vorticity solution. While this,
in itself, is not severely more complicated, the resulting distri-
bution is described by a vastly more lengthy expression, not war-
ranted by the circumstances at hand. In addition, the non-uniformi-
ty of flux described by Eq. (15) is not inconsistent with the
10
-------
experimental observations of a strong low-level non-uniformity in
concentration from surface winds at the bottom of the notch, and
accordingly we continue with the derivation from Eq. (14), with-
out requiring further refinements.
The next step in the analysis is to determine <|L , described previ-
b
ously as the background concentration level carried by the external
flow. From the viewpoint of the .notch, , means the concentration
lying just above. This is composed of two parts, the ambient con-
centration carried to the region from far upstream and the spilled-
out concentration coming from the notch itself. The first of these
contributes an additive level of concentration that is easily super-
imposed onto the final solution. The second, which is assumed to be
constant across the level of the notch top, is to be determined by
the following derivations.
Neglecting diffusion in the x direction above the notch, we write,
analogous to Eq. (5).
36 326
uo af = v 72 '
3y
and look for a solution subject to the conditions
6=0 at x = 0
6 = 0 at y = °°
* = 6 at y = 0, for x > 0
D
dx = SW at y = 0 .
The analysis is easily accomplished, proves to be over determined, and
11
-------
therefore allows us to derive the result
in which has been added to describe the far upstream ambient
a
concentration in the wind approaching the notch region.
With all of this, Eq. (14) can be written in the useful form
(TT \
-W_\
vu J
cos kx
vu v I 4kv (1 - B)
- 6e"ky (1 + ky)|> . (17)
•I}-
For comparison with the street-canyon models of other authors, however,
it is convenient to identify some of the factors in this equation in
terms of a somewhat different set of parameters.
The simplest theories of turbulence eddy diffusivity show that
T] - q » as)
in which L is the integral scale of the turbulence, q is the turbu-
lence energy per unit mass, and a factor of order unity has been
omitted. The magnitude of q, in turn, can be related to the wind
speed, u , and the stirring speed, (from vehicle motion), u , by
O S
q = Yl Uo + Y2 Us '
The factor, y., describes the relative stability of the external
atmosphere; for a very stable atmosphere, YI « 0> while for a very
12
-------
unstable atmosphere, Y-I can be as large as 0.1, perhaps even larger.
In contrast, YO is independent of the atmospheric stability pro-
perties, and should in principle depend on the distance away from
the vehicle path. Having ignored the spatial dependence of v, how-
ever, we must postulate a constant, effective value for Y2» which
will require empirical determination. Note, incidentally, that the
presence of u in Eq. (19) is crucial, both to avoid a singularity
s
as u ->• 0 as well as to represent the observed contrast between
idling traffic and rapidly moving traffic. The magnitude of L will
be given further consideration below.
The modified concentration distribution equation thus becomes
. o
r
2 ^ 2\% , / 2 ^ 2
U0 + Y2 Us J L Yl Uo + Y2 Us
COS ~ ' ^ (1 - ky) - 0e -' (1 + ky)
4kL (1 - 3) Y, + Y,
2 ° u ' (20)
1}
13
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SECTION V
COMPARISON WITH THE MODEL OF JOHNSON, ET. AL
Johnson and associates have proposed a street canyon model that bears
some similarity to Eq. (20), and has been proven to agree moderately
well with experimental data. For the leeward side of the building,
they write, in the present nomenclature,
Ǥ - (2i)
W
in which u =0.5 m/sec, d = 2 m, and a is a constant. On the
windward side,
ctS
* = *a + (u + u ) W ' (22>
o c
The result is a discontinuity in concentration at x = W/2. Apart
from this discrepancy, there is agreement with Eq. (20) in the de-
pendence upon S, and qualitatively in the dependence on u . The
nature of that agreement suggests that we follow their prescription
for the turbulence scale, and choose L proportional to W. Actually,
the scale of turbulence produced by the vehicles should be signifi-
cantly smaller than W, whereas that carried by the external flow
depends upon meteorological conditions and may be somewhat larger
than W. As in the case of Y?» the factor, e, in the relation L = eW,
will probably require empirical determination.
14
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SECTION VI
THE NUMERICAL APPROACH
The street canyon problem involves very complex phenomena. For a
complete understanding, much more experimental data will have to
be obtained and analyzed. These data will need to be extensive in
content, not only measuring pollution distributions, but velocity,
temperature and turbulence intensity distributions as well, with
significant spatial resolution to account for the complex inter-
actions that occur between buildings and fluid. Even with this
large amount of data, the fact still remains that no two street
canyons are exactly alike, thus implying that the categorization
of street canyon results will require more than a single extensive
experiment. Not only are data such as these difficult to obtain,
they are also very expensive.
The use of three dimensional computer programs to model such flows
can greatly alleviate the need for many of the costly experiments
and reinforce those experiments that are needed, by indicating the
basic structures of these flows and their related implications to
pollution dispersal. Data generated in this way not only familiar-
ize the experimentalist with flow patterns for which he must be
concerned, but allow him to concentrate his efforts on those
portions of the street canyon for which there is interest, with
cognizance of the complications of the nearby flow. Data generated
in this way also give the analyst information with which to compare
and extend his analytic models of these complicated phenomena at
significantly reduced costs.
A numerical technique that can be used to perform such calculations
at these, allows the detailed consideration of flows of incompressi-
ble, buoyant fluids in and around three-dimensional obstacles and
their associated effects on pollutant dispersal. The equations used
15
-------
to calculate the motions of such fluids are the continuity equation,
"^T = ° ' (23)
the Navier-Stokes equation with buoyancy,
- + - + v <24>
and the heat equation,
in which
i,j = 1, 2 or 3
t = time
u. = the component of velocity in the x. direction
g. = the acceleration of gravity in the x direction
p = the pressure per unit constant density
v = a kinematic molecular or eddy viscosity, here assumed
constant, but easily generalized to include space-time
variations
T = temperature
a = thermal diffusivity
T -a reference temperature
o
B = coefficient of volumetric expansion
and the summation convention of repeated indices is implied.
16
-------
The method by which numerical solutions for these equations are
3
obtained is described by Hirt and Cook (a preprint of which is
included in this report as an appendix) and is an extension of
4
the Marker-and-Cell method originated by Harlow and Welch.
This method approximates the above equations with finite differ-
ences by dividing the calculational region spanning the problem,
into Eulerian cells. After initial and boundary conditions are
specified, the dependent variables, recorded at specific locations
on each cell, are advanced through time in short time steps thus
providing the time dependent solution.
Obstacles, which may occur anywhere in the mesh, are specialized
cells that never contain fluid and which impose internal boundary
conditions on the fluid. Since obstacles are constructed by de-
noting any desired combination of cells as these specialized cells,
this technique can be used to study a wide variety of complex flow
problems.
In this technique, pollutants are represented as discrete parti-
cles , each particle being spherical in shape and having the mass
of a prescribed amount of pollutant. Particle motions are in-
fluenced by such forces as gravity, Stokes drag and a diffusive
force that statistically represents the drag force exerted on a
particle by the turbulent eddy spectrum. The movement of parti-
cles is governed by the particulate transport equations,
du
P •*• /•*••*• \ . •*•
—^ =g + a (u - u ) + a u,.c,
dt m p m diff
with
u = particle velocity
P
u = the fluid velocity evaluated at the position of the
particle
17
-------
g = the acceleration of gravity
u,.,;- = a random diffusion velocity
a = 4.5 o = the coefficient of Stokes drag divided
m / th
mm by the mass of the m particle species.
p = fluid density
p = density of the m particle species
r = radius of the m particle species
A species. of particles is defined by the species' size and density.
It is evident that particle transport is coupled to fluid transport
through the fluid velocity u in Eqs. (24) and (26). The manner in
which u,.,.,. is calculated and the details of the particulate trans-
dltl 5 •>
port scheme are described by Hotchkiss and Hirt. Basically, u,.,.,:
is chosen to have randomly generated components that are Gaussian
distributed in each coordinate direction.
18
-------
SECTION VII
TWO-DIMENSIONAL RESULTS
Various problems have been studied numerically, not only to com-
pare numerical results with experimental results and demonstrate
credibility in the numerical technique, but to generate information
with which analytic model results can be compared.
Shown below are the results from a series of calculations on a two-
dimensional street canyon problem. Fig. 1 shows the steady, state
flow velocity distribution resulting from a uniform inflow of unit
magnitude velocity from the left and existing the mesh on the right.
Even though the picture is shown in perspective, the problem is two-
dimensional. The velocity vectors shown indicate the magnitude and
direction of the flow in the notch. Comparison of this figure with
the observed results of Wang, Chang and Lin, Fig. 2, shows excellent
correlation of flow structure. Note, for example, the location of
the vortex center in the two figures. Also comparison of Fig. 1 with
7 8
the experimental results of Reiman and Sabersky and Mills shows
excellent agreement. The Reynolds number for the flow of Fig. 1 was
4 9
100 while that of Fig. 2 was 2.1x10 , however, Jacobs and Button have
shown that there is no appreciable difference in flow structure, at
least calculationally, for the two cases. The Reynolds number re-
ported by Reiman and Sabersky is 143.2, while that of Mills is 10 .
The calculation depicted in Fig. 1 was performed with uniform inflow
and free slip walls. Thus any secondary vortices that appear in
experimental results do not appear here, although the overall flow
structure is very similar. The steady pressure distribution for
this flow is shown in Fig. 3. The contour values for the isobars
are in non-dimensional units measured relative to an ambient value
of zero.
With this flow field established, particles are fluxed into the
system from various sources, with varying amounts of diffusion and
19
-------
INF L
1 length unit
time unit
Fig. 1. Steady velocity field established in a two-
dimensional square street canyon.
-20-
-------
Fig. 2. Experimental flow field observed by Wang
Chang, and Lin for a Reynold's number of 2.1 x 10^
Reproduced with the permission of Prof. P. C. Chang,
-21-
-------
Fig. 3. Steady pressure distribution isobars for
the flow depicted in Fig. 1. Pressures are in non-
dimensional units relative to an ambient pressure
of zero.
-22-
-------
allowed to move to a statistically steady distribution. For example,
Fig. 4 shows the steady distribution of pollutants input from a line
source at a rate of 40 particles per unit time into the velocity field
established in Fig. 1. Each particle has a non-dimensional mass of 1
mass unit. The line source, is across the bottom of the street canyon
corresponding to a uniform area source for the street of which this is
a cross section. This resulting distribution is more specifically ana-
lyzed in Fig. 5 which shows the isopleths of Fig. 4 within the canyon.
These lines of constant concentration, measured in units of mass per
unit volume of space, are labeled in the figure and show the effect of
the notch vortex on the pollutants' distribution. The circulating
vortex convects a large amount of pollution to the lee side of the
canyon causing higher concentrations there than on the side with a
downwash wind in front.
It must be noted here that all of the isopleth plots presented in this
paper reflect the averaging of instantaneous particle distributions
over relatively long periods of time at steady state. The results in
Fig. 5, for example, were averaged over 50 time units. This procedure
is performed since instantaneous distributions are dependent upon time
varying random fluctuations and therefore never come to a true steady
state. The contour plots in this paper are produced directly with
the aid of the CDC 7600 computer and a Stromberg-Carlson 4020 micro-
film printer. Since contour lines are scaled and drawn automatically
by placing ten contour lines between the maximum and minimum contour
values to be plotted, the contour lines usually have fractional numbers
associated with them.
Figure 6 shows the particle distribution from a similar calculation
to those shown in Figs. 4 and 5, except here a point source is
positioned at the bottom center of the canyon. As before, 40 mass
units per unit time are input into the steady velocity field shown in
Fig. 1. The isopleths in Fig. 7 show contours of the number of mass
23
-------
•••:• .• . *. .•. .»•*•••• • •••... A
••;••• •/•'•:•"•'.• / •: .;•'•<:• •••..'.• ...".'".'Vf
• -.v .-..« ^ •• ••..• •. .....-.•:••. • .-;• '• '• •• ' '
»•/•. '•« '• •'•':•'•• •>•/• •'• •' ' •- :-/-^:•.' •:"..:
^.-vviv:^&!-.^^-/f;v
.:v>-.;:-::^<--^--::^.v^;-:v;v.v.-;-;
.;r VvV-v.;;->?>4 -':;:.~4 • ...-?.. .»•.-.-..•.; .•
::^:^^/fy^-^:^v-y>:.::
-/•/;*.•• .Vi. ;•':; ':' ;.?>::•"•*•"'•' ^'* -*:
•/••.V'-'./-- ;:\"fv" Vyc/ v-:\/. :'''•
Fig. 4. Steady particle distribution resulting from
a line source in the bottom of the canyon.
-24-
-------
INFLOW
14.24
27.43
Fig. 5. Isopleths of Fig. 4 showing concentration
distribution within the canyon. Units are particles
per cell volume.
-25-
-------
Fig. 6. Steady particle distribution resulting from
a point source located at the bottom center of the
canyon.
-26-
-------
Fig. 7. Isopleths of Fig. 6 showing concentration
distributions in the canyon in units of particles
per cell volume.
-27-
-------
INFLOW
13.09 ••
Fig. 8. Isopleths predicted by the analytic model
for concentration distribution in the canyon. Units
are the same as those in Figs. 5 and 7.
-28-
-------
units per unit volume that result at statistically steady state
from this type of source. (Here, we are looking at the effects
of a single lane of traffic in the center of the canyon.) Com-
parison of the results in Figs. 7 and 5 show similarities between
the two source types. Comparison of the two particle distributions
reveals that the contour lines from the lower middle portion of the
canyon (« 36 mass units per unit volume) on up to the top of the
canyon have not only very similar shapes but very nearly equal
magnitudes as well. The similarities of these two results are not
too surprising since they both were performed with an enhanced parti-
cle diffusivity A = 1.0 as compared to the fluid diffusivity of
v = 0.1. These calculations do, however, present the solution to a
well posed mathematical problem with which the analytic results as
shown in Fig. 8 can be compared. These isopleths exhibit the so-
lution of Eq. (20) with S = 4.0, W = D = 10.0, u = 1.0,
X = Ly^u2 + y0u2 = 1.0 and 4> = 0.0 .
v 1 o 2 s a
The agreement between analytic results and the numerical results
of Fig. 5 are extremely good on the left side of the canyon with
increasing divergence in agreement on the right side. The reason
for the disagreement on the right side, however, arises from the
boundary condition given by Eq. (7) which prohibits the concentration
generated within the cavity from being convected away at the top in
the analytic model. A detailed comparison of concentration distri-
bution for the two cases is enabled with the uses of Tables 1 and 2,
which shown concentrations in mass units per unit volume throughout
the canyon. The numerical results are concentrations that have been
averaged over a long period of time after a statistically steady
particle distribution has been obtained.
It should be pointed out that the analytic model, as previously
formulated, gives valid results only when the coefficient,
29
-------
TABLE 1
U =
14.3
20.1
26.9
32.5
36.8
41.4
45.5
48.9
53.0
58.2
14.2
20.6
27.0
32.2
36.8
41.2
45.9
50.3
53.0
57.9
15.4
20.9
26.6
31.8
35.8
38.7
44.0
48.0
51.5
56.4
16.3
21.2
25.8
31.0
35.0
38.2
42.9
46.0
51.5
56.0
17.2
22.2
26.0
29.4
33.8
36.2
40.5
46.1
50.5
53.2
18.4
22.2
26.6
28.9
32.6
35.4
41.4
45.0
49.0
52.4
19.5
22.7
25.5
29.4
31.5
34.2
39.6
43.8
46.3
49.5
20.1
23.4
25.4
28.3
31.5
34.6
37.9
42.6
45.4
50.2
21.4
24.6
25.9
29.0
31.0
33.8
37.4
41.0
45.2
48.2
23.9
24.6
26.6
28.4
31.5
32.5
37.8
41.0
45.2
49.1
Numerically obtained concentration values associated with
isopleths of Fig. 5. S = 4, A = 1.
30
-------
TABLE 2
U =
16.2
23.0
29.1
34.4
39.0
43.2
47.0
50.8
54.7
58.9
16.1
22.6
28.4
33.6
38.2
42.3
46.2
50.0
53.9
58.1
15.8
21.8
27.2
32.2
36.6
40.8
44.7
48.5
52.4
56.5
15.4
20.6
25.6
30.2
34.5
38.6
42.5
46.4
50.3
54.4
14.9
19.3
23.7
27.9
32.0
36.0
40.0
43.9
41.9
52.0
14.4
18.0
21.6
25.4
29.3
33.3
37.3
41.4
45.4
49.3
13.9
16.7
19.7
23.1
26.8
30.7
34.8
38.9
43.0
46.9
13.5
15.5
18.1
21.1
24.7
28.5
32.6
36.8
40.9
44.8
13.2
14.7
16.9
19.7
23.1
27.0
31.1
35.3
39.4
43.2
13.1
14.3
16.2
18.7
22.3
26.1
30.3
34.5
38.6
42.4
Analytically predicted concentration values associated
with isopleths of Fig. 4. S = 4, A = 1.
31
-------
u /4kA(l - g), is less than unity (by virtue of the solution's ex-
pansion in u ). In other words, the analytic model can only be
applied with confidence to problems in which the horizontal vari-
ation of concentration, as predicted by the model, does not differ
by more than about a factor of 2 from one side to the other. Since
the analytic model was derived from first principles and only has a
limited range of applicability, the implication can be made that
the analogous Johnson model cannot be applied to generalized circum-
cumstances without taking cautions to ensure proper normalizations.
A street canyon with smaller diffusivity, modeled in two dimensions,
is depicted in Fig. 9. In this calculation, particles are input
uniformly along the bottom of the canyon at a rate of 40 particles
(i.e., mass units) per time unit. The particle diffusivity and the
kinematic eddy viscosity of the fluid are equal: A = v = .1. The
steady flow field used for particle movement is again that shown
in Fig. 1. The concentrations produced by this calculation are
considerably greater than those presented in the previous results.
The overall concentration distribution is also somewhat different
as shown in Fig. 10. The effects of the vortical flow structure
on pollution distribution are now extremely evident. This purely
two-dimensional case should be able to be compared with two-
dimensional experimental results, however, purely two-dimensional
experiments in which concentrations are measured are not easily
accomplished. The water tunnel experiment performed by Wang, Chang
and Lin involved three-dimensional phenomena as far as the concen-
tration measurements were concerned. Since a dye was injected at a
single point beneath the surface of the water, diffusion proceeded
in all three directions, thus reflecting much lower concentrations
in the plane of the source than a truly two-dimensional experiment
would have produced. Additional complications to the flow field can
be expected in their experiment, as a vertical flow structure, pro-
duced by an Eckman layer on the floor of the water tunnel, was most
32
-------
X^]^:^ff^V£f''$-i<-^'J-f:t-:' '..-.-i
yy*»»^;•/ v* • ••• ••..••••*•> > *xj-\.JL-
^^Sw^-^^SS^^T^^^^^^^'^^T^tTTT^'^.^T^^iy.'^TA'*^^*1"^^.^^"*^?
Fig. 9. Steady particle distribution resulting from
a line source in the bottom of the canyon. Results
reflect a smaller diffusivity.
-33-
-------
Fig. 10. Isopleths of Fig. 9 showing concentration
distribution within the canyon in units of particles
per cell volume.
-34-
-------
probably present thus distributing concentration by a vertical
convective process.
Since the two-dimensional results previously reported cannot be
compared with the experimental data obtained by Wang, Chang and
Lin, a three-dimensional version of the problem was modeled as
shown in Fig. 11. Here, a steady particulate distribution result-
ing form a point source, located midway between the front and back
of the mesh (which is seven cells deep), is shown. Diffusion in
all three directions is allowed, however, the mean-flow field is
actually two-dimensional because all walls are assumed to be free
slip and the acceleration of gravity is assumed to be zero, hence,
convection only occurs in two dimensions. A comparison of Fig. 12,
which contains the numerically calculated contour lines of concen-
tration (in ppb) in the plane of the source, with Fig. 13, which
contains contour lines for the experimentally obtained concen-
trations at various points throughout the canyon, shows the areas
of agreement and areas of disagreement between calculation and ex-
periment. It is evident that the calculated concentrations in the
plane of the source are considerably higher than the experimentally
obtained values. However, it is interesting to note Fig. 14, which
contains contour lines of concentration for a plane parallel to but
slightly off the plane of the source. Here the agreement is much
better.
It is clear from the calculation, that all of the physics of the
flow is not being properly modeled. For example, a calculation of
this type confirms the fact that a constant eddy diffusivity scheme
has certain limitations and that the selection of a proper variation
for X is a critical step in the calculation. Clearly, a full set of
turbulence equations that would automatically couple the turbulent
shear stresses with particulate motions would alleviate this problem
considerably.
35
-------
!•*•'•' •'•*••''- .'*•'•'.
--/I
Fig. 11. Steady particle distribution obtained from
modeling the experiment of Wang, Chang and Lin.
-36-
-------
INFLOW
»
u = 13.8 cm/sec
E
u
*•
CVJ
id
15.24 crrr
Fig. 12. Isopleths of numerically calculated concen-
trations (in ppb) in the plane of the source of
Fig. 11.
-37-
-------
INFLOW
u=!3.8
cm /sec
Fig. 13. Isopleths drawn through the data reported
by Wang, Chang and Lin (in ppb) (results approximate)
-38-
-------
INFLOW
u = 13.8 cm/sec
E
u
-------
4
Since the source strength in the experiment was 3.8 x 10 ppb, one
would initially expect to find extremely large concentrations in
the vicinity of the source. Consequently, since these high concen-
trations are not found, it becomes evident that the pollutants re-
main near the wall for some distance away from the source before
they become dissipated throughout the rest of the fluid. The
measuring technique of the experimentalists did not allow them to
see these high concentration areas because their measurements were
taken fairly far from the wall. However, calculationally the parti-
cles near the wall are still considered to contribute to the concen-
tration in the layer of cells next to the wall thus explaining the
difference between the large calculated concentrations and the re-
latively small experimentally obtained concentrations near the walls.
The remainder of the field, however, does show a relatively good
comparison between calculations and experiment; it is evident that a
distributed mass diffusivity would allow even better correlation.
40
-------
SECTION VIII
THREE-DIMENSIONAL RESULTS
A fully three-dimensional investigation of the Broadway Street
canyon in St. Louis, Missouri was made in order to compare numeri-
cal predictions with the experimental results obtained by Ludwig
and Dabberdt. The case chosen for comparison involved an incoming
wind (at approximately 2 m/sec) perpendicular to Broadway over the
top of the First National Bank and down Locust and Olive Streets.
The computing mesh was selected to span the space between the center
of Locust Street and the center of Olive Street as indicated in
Fig. 15. The lateral span of the mesh only encompasses portions of
the buildings (approximately forty feet) on both sides of Broadway
and the height of the mesh is the equivalent of 16 stories. Actually,
using the coordinate system specified in Fig. 16, the mesh extends
300 feet (30 cells) in the x direction, 160 feet (16 cells) in the
y direction and 160 feet (16 cells) in the z direction.
The incoming wind, flowing from left to right in Fig. 16, causes
an extremely complex velocity field to exist in the canyon at steady
state as shown in Fig. 16 through 22. These figures show various
views of velocity vectors in planes perpendicular to the direction
indicated at a distance (in feet) from the origin of the coordinate
system. Mental superposition of the various figures indicates the
overall flow field structure.
Traffic exhaust emissions are assumed to originate uniformly at
ground level on each street. The equivalent number of particles in-
put per unit time for Broadway was 66.7 while Locust and Olive were
assumed to have forty percent of Broadway's input rate. Particles
were uniformly distributed at height z = 0 as they modeled continuous
area sources. They were allowed to be convected by the steady ve-
locity field and randomly diffused until they reached a statistically
41
-------
FEDERAL RESERVE
BANK
(S)
:LOW * j
i
FIRST NATIOl
BANK
PERIMENTAL S
FLOW
COMP
®
IAL
TA. -
(
0
®
d
u1
•v
1*
Y
LOCUST
| BOATMANS BLDG
INFLOW
OLIVE
Fig. 15. Computing mesh for Broadway Street Canyon
St. Louis, Missouri. Circled numbers represent
building heights in stories.
-42-
-------
Fig. 16. Perspective view of velocity vectors in a
plane perpendicular to the z axis at a distance of
15 feet from the origin. Location of the coordinate
system is shown.
-43-
-------
Fig. 17. Perspective view of velocity vectors in a
plane perpendicular to the z axis at a distance of
65 feet from the origin.
-44-
-------
Fig. 18. Perspective view of velocity vectors in a
plane perpendicular to the z axis at a distance of
115 feet from the origin.
-45-
-------
* * v \ \ \ » »
* x ^\\ \ I '
Fig. 19. Plane view of velocity vectors in a plane
perpendicular to the y axis at a distance of 55 feet
from the origin.
Fig. 20. Plane view of velocity vectors in a plane
perpendicular to the y axis at a distance of 105
feet from the origin.
-46-
-------
• • *
• • .
• • •
• • «
' • %
' • t
1 • 1
' • 1
• • »
. . .
* • •
I • •
1 • •
1 • •
I • •
1 • •
/ " *
i • !
% - •
Fig. 21. Plane view of velocity vectors in a plane
perpendicular to the y axis at a distance of 155
feet from the origin. The intersection of this
plane with the buildings is shown.
-47-
-------
Fig. 22. Plane view of velocity vectors in a plane
perpendicular to the x axis at a distance of 175
feet from the origin.
-48-
-------
Fig. 23. Steady particular distribution in Broadway
Street Canyon resulting from real sources on Broadway,
Locust and Olive Streets.
-49-
-------
Fig. 24. Isopleths in a plane perpendicular to the
x axis at a distance of 155 feet from the origin. This
plot corresponds to the measuring station of Ludwig and
Dabberdt. (Units in ppm of CO).
-50-
-------
I -3 m/sec
<
CD
-I
<
Z
o
CO
or
o
z
o
-J
CD
o
CO
Fig. 25. Distribution of CO concentration in Broadway
Street Canyon (in ppm) measured by Ludwig and Dabberdt.
-51-
-------
steady distribution. This final distribution is pictured in Fig. 23.
Isopleths. of this distribution in the same plane as that measured by
Ludwig and Dabberdt are shown in Fig. 24 in the units of parts per
million.
In order to determine the proper units for these numerical results,
it is necessary to know two of the parameters involved in the ex-
periment. These may include, for example, a source strength in
equivalent units of mass per unit area and time and a background
concentration, a value of concentration at any point within the
street canyon and a background concentration or any other pair of
similar values. Since a background concentration does nothing more
than elevate the entire concentration field by that same amount, a
numerical calculation may proceed without a background concentration,
and the results elevated accordingly. The calculation previously
depicted proceeded in this way.
In order to scale our results to the units reported by Ludwig and
Dabberdt, an experimental value of concentration (in ppm) from the
CO-detector at the lower corner of Broadway and the First National
Bank was chosen equal to our calculated concentration (in particles
per cell volume) at the same point after all concentrations had
been elevated by their experimentally determined background concen-
tration. Thus we were able to determine the mass of pollutant that
was being represented by each particle. This approach was taken
since it was not possible to determine the flux of pollutants at the
street level pertinent to those results presented in Fig. 21 of their
referenced paper. A comparison of Fig. 24 with their results as
presented in Fig. 25 with the wind between 1-3 m/sec shows excellent
correlation.
As a matter of fact, the above approach proves to be an advantage
if one elected to use a numerical technique of this type to monitor
CO levels in downtown street canyons. Time dependent CO-detectors
52
-------
located at two points within the street canyon could be used with
the numerical technique to monitor the levels of CO at every point
within the street canyon over long periods of time.
After having determined the mass of pollutant per particle in this
way, we can calculate the average value of the source flux as being
-3 2
approximately 4.3 x 10 gm-CO/m sec (based on properties of air
at standard temperature and pressure) in the Broadway street canyon
during the data taking phase of this experiment.
The concentration plots in Figs. 26 and 31 show the distribution
of pollution in planes, again perpendicular to the direction indi-
cated and at distances from the coordinate system, as previously
discussed. The presence of buildings on the sides of the streets
shows how pollutants are allowed to build up in the canyon. How-
ever, at the intersections of Locust and Olive, it is evident that
the concentrations are greatly reduced as they are swept out of the
computing region.
53
-------
Fig. 26. Isopleths in a plane perpendicular to the z
axis at a distance of 15 feet from the origin (ppm) .
-54-
-------
Fig. 27. Isopleths in a plane perpendicular to the z
axis at a distance of 65 feet from the origin
-55-
-------
7.8'
Fig. 28. Isopleths in a plane perpendicular to the y
axis at a distance of 55 feet from the origin (ppm).
-56-
-------
9.IO>
10.20 9.70 7.90
Fig. 29. Isopleths In a plane perpendicular to the y
axis at a distance of 105 feet from the origin (ppm).
-57-
-------
Fig. 30. Isopleths in a plane perpendicular to the x
axis at a distance of 95 feet from the origin (ppm).
-58-
-------
Fig. 31. Isopleths in a plane perpendicular to the x axis
at a distance of 205 feet from the origin (ppm).
-59-
-------
SECTION IX
A GENERALIZED STREET CANYON
As previously mentioned, the uniqueness of each street canyon geo-
metry complicates analysis of flow patterns and pollution transport
since each canyon must be analyzed separately. However, an alter-
nate approach to such analyses is to generalize street canyons in
categories defined by height to width ratios, incoming wind veloci-
ties, source strengths, etc., and then calculate the flow and pol-
lution dispersal patterns for such a class of problems. For example,
geometries such as that depicted in Fig. 32 can pertain to a large
class of problems for which an effective canyon height to width ratio
is unity and for which the building heights on both sides of the
street are nearly equal. Since half of the canyon block is being
modeled, the front and back planes in Fig. 32 are chosen to be planes
of symmetry, implying that mirror images of the geometry viewed ex-
ist on the opposite side of those planes. The incoming wind is
chosen to be perpendicular to the main canyon, not only for simplicity
in the calculational setup but for witnessing the "worst case" type
of problem. The steady velocity distribution resulting from the
uniform inflow of unit magnitude is presented in Figs. 32 and 33 in
a similar manner to the previous results. The calculational region
is 18 cells in the x direction, 18 cells in the y direction and 15
cells in the z direction with each cell having unit length in each
coordinate dimension. Thus with the appropriate selection of length
and time scales, the calculation can be made to apply to any desired
situation involving this geometry.
A line source of pollutants, placed along the bottom center of the
longitudinal street, influxes 360 particles per unit time uniformly
along the length of the street in the velocity field previously pre-
sented. At statistically steady state, the particle distribution in
the canyon as viewed looking downwards and from mid-block is shown in
60
-------
Figs. 34 and 35. Concentration plots, in units of particles per
unit cell volume, are given in Figs. 36 and 42 for various planes
within the canyon. Again these contour lines can be scaled to any
appropriate units by assigning a mass of pollutant to each particle,
and dividing this number by the true length scale cubed.
Another line source, parallel to the incoming wind, is placed along
the bottom of the plane of symmetry at the street intersection and
emits pollutants into the steady velocity field. The steady distri-
bution from this source and the associated concentration contour
plots are shown in Figs. 43 through 46. The mass diffusivity for
particle motion in each of these cases was chosen to be A = .25
while the turbulent eddy viscosity of the fluid was v = .1, however,
as previously mentioned, the proper choice of X is somewhat uncertain.
After having scaled each of these results to the appropriate units,
a superposition of the two produces the distribution of concentration
in the canyon for the cases in which traffic density is different
along each of the two streets. Although a single line source in each
street may not be the appropriate type of source for most common
studies, the technique is very flexible to allow area, volume or even
point sources to be considered along with the previously calculated
steady velocity field, in a fairly efficient manner.
61
-------
Fig. 32. Generalized street canyon configuration in
perspective with velocity vectors shown in a plane perpen-
dicular to the z axis at a distance of 5.5 units from the
origin. Location of the coordinate system is shown.
-62-
-------
INFLOW
» \ X X
Fig. 33. Plane views of velocity vectors in a plane perpen-
dicular to the y axis at a distance of 9.5 units from the
origin.
-63-
-------
Fig. 34. Steady particle distribution in the generalized
street canyon, as viewed from above, resulting from a line
source perpendicular to the incoming flow.
-64-
-------
Fig. 35. Same particle distribution as that shown in Fig.
34, only viewed from the top of the buildings at mid-block.
-65-
-------
Fig. 36. Isopleths in a plane perpendicular to the z axis
at a distance of 1.5 units from the origin. (Source perpen-
dicular to inflow). Concentrations in particles per cell
volume.
-66-
-------
Fig. 37. Isopleths in a plane perpendicular to the z axis
at a distance of 5.5 units from the origin. (Source perpen-
dicular) .
-67-
-------
INFl
OW
Fig. 38. Isopleths in a plane perpendicular to the z axis
at a distance of 10.5 units from the origin. (Source perpen-
dicular) .
-68-
-------
INFLOW
20.0
24.0
\X32.0
X36.0
Fig. 39. Isopleths in a plane perpendicular to the y axis
at a distance of 1.5 units from the origin. (Source perpen-
dicular) .
-69-
-------
38.'0 48'. 8 4 43.4
Fig. 40. Isopleths in a plane perpendicular to the y axis
at a distance of 13.5 units from the origin. (Source perpen-
dicular) .
-70-
-------
55.9 49.7 43.5
Fig. 41. Isopleths in a plane perpendicular to the x axis
at a distance of 8.5 units from the origin. (Source perpen-
dicular) .
-71-
-------
0.0
Fig. 42. Isopleths in a plane perpendicular to the x axis
at a distance of 16.5 units from the origin. (Source perpen-
dicular) .
-72-
-------
Fig. 43. Steady particle distribution in the generalized
street canyon as viewed from above, resulting from a line
source parallel to the incoming flow.
-73-
-------
INFLOW
0.08
t
Fig. 44. Isopleths In a plane perpendicular to the z axis
at a distance of 1.5 units from the origin. (Source perpen-
dicular) .
-74-
-------
Fig. 45. Isopleths in a plane perpendicular to the y axis
at a distance of 13.5 units from the origin. (Source perpen-
dicular) .
-75-
-------
0.0
Fig. 46. Isopleths In a plane perpendicular to the x axis
at a distance of 8.5 units from the origin. (Source perpen-
dicular) .
-76-
-------
SECTION X
COMPUTER REQUIREMENTS
The calculations presented in this paper were all performed on a
CDC 7600 computer with a 65,536 60-bit word small core memory and
a 512,000 60-bit word large core memory. The computer program used
to perform these calculations (entitled S-TRES) is designed to use
almost all of both memories in order to reach its maximum resolution
which is a mesh the equivalent of 27 cells cubed, and 30,000 parti-
cles.
The amount of computer time used in achieving steady state on a
particular problem depends on such factors as the number of cells
used in spanning the geometry, the size of the time step used, the
size of the particle diffusivity and the amount and frequency of
computer generated output. For example, the Broadway street canyon
problem took approximately 90 min of computer time to reach hydro-
dynamic steady state and an additional 40 minutes to reach particle
steady state; thus implying approximately 2.2 hours of total computer
time to obtain the results presented. These numbers, of course, do
not reflect setup of debugging time. S-TRES executes its function
in an average time of approximately 5 sec for each hydrodynamic
cycle and 3 sec for each particle moving cycle. These last numbers
can vary greatly with the complexity of a problem.
77
-------
SECTION XI
REFERENCES
1. Ludwig, F. L. and Dabberdt, W. F., "Evaluation of the
APRAC-1A Urban Diffusion Model for Carbon Monoxide,"
Stanford Research Institute Final Report (February 1972).
2. Johnson, W. B., et al,, "Field Study for Initial Evaluation
of an Urban Diffusion Model for Carbon Monoxide," Stanford
Research Institute Comprehensive Report (June 1971).
3. Hirt, C. W. and Cook, J. L., "Calculating Three-Dimensional
Flows Around Structures and Over Rough Terrain," J. Comp.
Phys. 10, 2 (1972).
4. Harlow, F. H. and Welch, J. E., Phys. Fluids .8, 2182 (1965),
2, 842 (1966); Welch, J. E., et al., "The MAC Method," Los
Alamos Scientific Laboratory report LA-3425 (November 1965);
Amsden, A. A. and Harlow, F. H., "The SMAC Method," Los
Alamos Scientific Laboratory report LA-3470 (May 1970).
5. Hotchkiss, R. S. and Hirt, C. W., "Particulate Transport in
Highly Distorted Three-Dimensional Flow Fields," Proceedings
of the Summer Simulation Conference, San Diego, California,
June 14-16, 1972.
6. Wang, P. N., Chang, P. C. and Lin, A., "Circulation and
Diffusion of the Separated Flow in a Rectangular Trough,"
University of Utah Scientific Report for the Period 1 May 1970
to 30 April 1972, prepared for the Environmental Protection
Agency under Grant AP 01126, May 1972.
7. Reiman, T. C. and Sabersky, R. H., "Laminar Flow Over
Rectangular Cavities," Int. J. of Heat and Mass Transfer, 11
(1968).
8. Mills, R. D., "On the Closed Motion of a Fluid in a Square
Cavity," J. of the Royal Aeronautical Society, 69, (1965).
9. Jacobs, H. R. and Button, S. B., "A Numerical Analysis of
Steady Separated Flows in Rectangular Cavities With and Without
Mass Additions," University of Utah Scientific Report for the
Period 1 May 1970 to 30 April 1972, prepared for the Environmental
Protection Agency under Grant AL 01126, May 1972.
10. Daly, B. J. and Harlow, F. H., "Transport Equations in
Turbulence," Phys. Fluids, 13, 2634 (1970).
78
-------
SECTION XII
APPENDIX
A PREPRINT OF
"Calculating Three-Dimensional Flows Around Structures And
Over Rough Terrain," by C. W. Hirt and J. L. Cook, published
in Journal of Computational Physics, JLO, 2, 1972.
79
-------
LA- DC - 13289
CALCULATING THREE-DIMENSIONAL FLOWS AROUND STRUCTURES
AND OVER ROUGH TERRAIN
by
C. W.Hirt and J.L. Cook
PREPRINT FROM
scientific laboratory
of the University of California
LOS ALAMOS, NEW MEXICO 87544
UNITED STATES
ATOMIC ENERGY COMMISSION
CONTRACT W-74O5-ENG. 36
-------
CALCULATING THREE-DIMENSIONAL FLOWS AROUND STRUCTURES
AND OVER ROUGH TERRAIN*
C. W. Hirt and J. L. Cook
University of California
Los Alamos Scientific Laboratory
Los Alamos, New Mexico 87544
November 12, 1971
Copies Submitted: 3
Manuscript Pages: 28
Figures: 11
Tables: 0
*This work was performed under the auspices of the United States Atomic
Energy Commission.
-------
-2-
Running Head: Three Dimensional Flows
C. W. Hirt
University of California
Los Alamos Scientific Laboratory
Los Alamos, New Mexico 87544
-------
-3-
ABSTRACT
A computing technique for low speed fluid dynamics has been developed
for the calculation of three dimensional flows in the vicinity of one or
more block type structures. The full time-dependent Navier-Stokes equations
are solved with a finite difference scheme based on the Marker-and-Cell meth-
od. Effects of thermal buoyancy are included in a Boussinesq approximation.
Marker particles that convect with the flow can be used to generate streak-
lines for flow visualization, or they can diffuse while convecting to rep-
resent the dispersion by turbulence of particulate matter. The vast amount
of data resulting from these calculations has been rendered more intelligible
by perspective view and stero view plots of selected velocity and marker par-
ticle distributions.
-------
-4-
I. INTRODUCTION
Finite difference solutions have been obtained for many complicated
fluid flow problems, but until recently, there have been relatively
few of these involved with three-dimensional transient flows. The three
dimensional calculations that have been reported have been restricted in
scope, having been developed for the solution of specific problems, for
F21
example, the structure of the planetary boundary layer, for Benard con-
vection, and for flow between two concentric cylinders. In this paper
a method is described for calculating transient three-dimensional flows about
large obstacles and over irregular boundaries. The technique is based on a
simple variant of the Marker-and-Cell method for the solution of the in-
compressible Navier-Stokes equations. Thermal buoyancy effects are included
in a Boussinesq approximation, and a technique developed by Sklarew is
used to represent the convection and diffusion of particulate matter. Only
confined flow calculations are reported here. Extensions to three-dimensional,
free surface, flows over and around obstacles will be reported elsewhere.
A major problem with three-dimensional calculations is the limited numer-
ical resolution that may be obtained with the fast access memories of even the
largest computers. Of course, additional external storage devices may be em-
ployed but these usually require much larger amounts of computer time. The
program used for the examples in this paper is limited to a maximum of 3375
computational cells when run on a CDC 7600 computer with a 64,000 word fast
core memory. This is not large, since 3375 cells is equivalent to a cubical
-------
-5-
mesh with only 15 cells on an. edge. Nevertheless, sample calculations show
that meaningful and interesting calculations can be performed even with this
limited resolution.
Another problem associated with three-dimensional calculations is how
to reduce the vast amounts of computed data into easily assimilated forms.
Displays of velocities, contours, and other kinds of data taken from two-
dimensional slices through a three-dimensional mesh are not always suffi-
cient to form a clear picture of the complete flow pattern. To reconstruct
a composite three-dimensional mental picture from a collection of two dimen-
sional slices is not an easy task. An alternative and more efficient means
of displaying data is described in this paper. The technique is based on a
hidden-line perspective view plot routine designed especially for finite
rgi
difference calculations. A perspective picture of, for example, velocity
vectors associated with a given two dimensional plane of calculational cells
shows not only the three-dimensional variations of the vectors, but also
their orientation with respect to all nearby obstacles. An even better dis-
play method consists of making two perspective views from slightly different
observation points. When correctly done the result can be combined into a
stereoscopic view, which is the ideal way to see the structure of three-
dimensional flows.
Examples of these various display methods are described in more detail
in the text. In Section II a description of the basic fluid dynamic com-
puting technique is presented together with some of its properties. Section
III contains descriptions of the buoyancy and particulate transport models.
-------
-6-
II. THE BASIC TECHNIQUE
A. Finite Difference Approximations
The Marker-and-Cell technique for the calculation of incompressible
fluid flow is an Eulerian finite difference approximation to the Navier-
Stokes equations,
2 222
9u , 9u 9uv 9uw _ 9p , /9 u . 9 u 9u.
"97 + "9T~+9y~ + "9T~~~9x + 8x~l"v (7T + 7~2 + TT
3 9x 3y 3z
2 222
3w , 3wu , 9wv . 3w 3p . . , /3_w , 3 w . 3
H+97-+3r+3^=-9t+sz + v (7T + 7T + 7
7 3x 9y 9z
and the mass equation
9u.9v.3w.. .„.
IT" 157 = 0 » (2)
where p is the ratio of pressure to constant density, g , g , g are pre-
scribed body accelerations and V is the coefficient of kinematic viscosity.
In addition to solving directly for the velocity components and pressures,
the Marker-and-Cell method also uses marker particles that are convected
-------
—7—
about by the fluid to record the locations of free surfaces. In this paper
only confined flows are considered so that marker particles are not required
for this purpose, but they are used to represent distributions of particulate
matter as described in Section III-B.
The region in which computations are to be performed is divided into a
set of small rectangular cells having edge lengths fix, <5y, and fiz. With
respect to this set of computational cells, velocity components are located
at cell faces and pressure values are at cell centers, see Fig. 1. Cells
are labeled with an index (i,j,k), which denotes the cell number as counted
from the origin in the x, y, and z directions respectively. Also p , is
i, J »K
the pressure at the center of cell (i,j,k), while U..T. . . is the x-direction
1"t^»J »K
velocity at the center of the face between cells (i,j,k) and (i+l,j,k), and
so on.
A time dependent solution is obtained by advancing the flow field var-
iables through a sequence of short time steps of duration fit. The advance-
ment for one time step is calculated in two stages. First the velocity
components are all advanced using the previous state of the flow to calcu-
late the accelerations caused by convection, viscous stresses, body forces,
pressure gradients, etc. In other words, stage one consists of a simple
explicit calculation. However, this explicit time advancement does not
necessarily lead to a velocity field with zero divergence, that is, to one
that conserves mass. Thus, in stage two, adjustments must be made to insure
mass conservation. This is done by adjusting the pressure in each cell in
such a way that there is no net mass flow in or out of the cell. A change
in one cell will affect neighboring cells so that this pressure adjustment
-------
-8-
must be performed iteratively until all cells have simultaneously achieved
a zero mass change.
In the original Marker-and-Cell method the pressures in stage two were
obtained from the solution of a Poisson equation. A related technique de-
F31
veloped by Chorin involved a simultaneous iteration on pressures and
rq]
velocity components. Viecelli has shown that the two methods as applied
to the Marker-and-Cell method are equivalent. In this paper we have chosen
the second procedure and simultaneously iterate both pressures and veloci-
ties. This choice simplifies the applications of boundary conditions as
discussed in Section II-C.
The specific finite difference expressions used for the steps described
above may assume many forms. Those that follow are essentially direct ex-
tensions of the original Marker-and-Cell method. The stage one, explicit,
advancement of velocities, resulting in quantities labeled by superscript
tildes, is
fe (pi,J,k -
~
oz
-------
-9-
6t {
(vi,j+3/2,k ~
V
6x£
-------
-10-
Quantities needed at positions other than where they are defined are calcu
lated as simple averages, e.g., u± . fc = y (u±+% -j k + ui-*- i k^ ' and the
2
square of a quantity, e.g., u at (i,j,k) is the square of the average,
2 27
(u. , ) , rather than the average of the squares, u4JJ, . . and u
The computations indicated in (3) are made for all (i,j,k), and rep-
resent a straightforward explicit finite difference approximation to (1).
Although centered differences have been used in approximating the convec-
tion terms, the resulting equations will be stable provided sufficient
viscosity is applied. This is similar to the MAC method, and is more
fully described in Section II-E.
B. Pressure Iteration
Equations (3) do not necessarily result in a velocity field that
satisfies (2), so that some adjustment of the tilde velocities must be made
to insure mass conservation. An iterative process is used for this purpose,
in which the cell pressures are modified to make the velocity divergence
vanish. In each cell (i,j,k) the value of the velocity divergence, D, is
calculated as
-------
-11-
If the magnitude of D is less than some prescribed small value, e, the
flow is locally incompressible and no change in the cell velocity is
necessary. However, if the magnitude of D is larger than e then, the
pressure is changed by
<5p = - 3D , (5)
where 8 is given by
26t (-^ + -~
6xZ 6y
The constant 8 is a relaxation factor, where over-relaxation and under-
o '
relaxation correspond to 8 greater than or less than unity respectively.
For iteration stability it is necessary to keep 8 < 2. A value of
i
8 « 1.7 is commonly used, but this is occasionally too large when there
are strong flow distortions. The value of 8 giving the most rapid convergence
can, in general, only be determined by experimentation.
Once 6p has been calculated for a cell (i,j,k) it is necessary to add it
to the pressure p. . . , and to adjust the velocity components on the sides of
^-»j »K-
cell (i,j,k) according to:
-------
-12-
-------
-13-
calculations for advancing the flow field through one cycle in time.
If, in addition, it is desired to permit the transport of heat or
pollution concentrations these field quantities must also be advanced
one time step before beginning the next fluid dynamic cycle. Likewise,
discrete marker particles used to define particulate distributions, or
for flow visualization purposes, must be moved before starting the next
cycle.
C. Boundary Conditions
The five principle kinds of boundary conditions to be considered are:
rigid free-slip walls, rigid no-slip walls, inflow and outflow boundaries,
and periodic boundaries. For simplicity it will be assumed that all
physical boundaries coincide with cell boundaries. The inclusion of more
general boundary configurations is a difficult problem, but a good start
[9]
in this direction for two-dimensional flows has been made by Viecelli.
The prescription of boundary conditions consists of a choice for both
the normal and tangential velocities at the boundary. The normal velocity
is easy to prescribe when the boundary coincides with a cell edge, since
it is the normal velocity that is stored for each cell face. For a rigid
boundary this velocity is set to zero, while for an input boundary it is
assigned the desired input value. If the boundary is periodic the value
must be chosen equal to the corresponding velocity one wavelength away.
-------
-14-
For outflow boundaries, however, there is no unique prescription, but the
general idea is to chose boundary conditions that have the least upstream
influence. It has been found that for this purpose a useful prescription
consists of setting the normal tilde velocity on the outflow boundary equal
to the corresponding tilde velocity immediately upstream, and then letting
the velocity on the boundary relax as it wishes during the pressure itera-
tion. This appears to keep the flow going smoothly out of the boundary in
the examples tested.
Tangential velocities are needed in cells immediately outside the
fluid region in order to specify the appropriate viscous stress at the
boundary. These velocities are set equal to the adjacent velocities inside
the fluid when it is desired that the boundary represent a free-slip wall
(plane of symmetry), and they are set equal to the negative of the adjacent
fluid velocities when the boundary is to be no-slip. In other words, the
external velocities tangent to a boundary are chosen to give either van-
ishing shear or vanishing velocity at a rigid wall. A more complete dis-
cussion of these alternatives and the conditions under which each should
be used is contained in reference (11). If the boundary is periodic then
these external velocities are set equal to their counterparts one wave-
length away, and at an inflow boundary they are prescribed to give the
desired input flow. At an outflow boundary they are set equal to the
adjacent velocities inside the fluid, which encourages a smooth transition
through the outflow boundary.
-------
-15-
To aid in the identification of various kinds of boundaries a flagging
scheme is employed in the computer program, which assigns to each cell a
number that identifies it as an obstacle cell, inflow cell, outflow cell,
etc. In this way it is easy to arrange a distribution of obstacles in a
mesh, and to have various combinations of inflow and outflow boundaries.
Several examples illustrating different combinations of boundary con-
ditions are shown in Figs. 2-6. In Fig. 2 a horizontal layer of velocity
vectors is shown in perspective for steady flow around a simple rectangular
structure. A uniform flow is entering the computing region (large rectan-
gular box) through the left face and is leaving through the right face.
Each vector (short line segment) is drawn from the corner of a computing
cell with a direction and magnitude representing the average velocity about
that corner.
A recirculation in the wake region is clearly evident in the figures.
It consists of a pair of counter rotating eddies that are small near the
top of the structure, but large near its base. The x-y components of the
same set of velocity vectors have been plotted in Fig. 3. Here the double
eddy structure is more clearly seen, but no indication of the distribution
of z-component velocity is available in this kind of plot. Velocity vectors
for a similar calculation, but involving a more complicated obstacle, are
shown in Figs. 4-5. The three-dimensionality of the velocities is most
clearly seen in Fig. 5.
-------
-16-
In the previous examples the inflow is normal to the front face of
the obstacle, but by making two adjacent sides of the mesh inflow bound-
aries and the opposite two sides outflow boundaries, the incident flow
can be adjusted to any angle. Fig. 6, for example, shows the results of
a calculation with the flow passing through the mesh from left to right,
and oriented 45 to the large faces of the two obstacles.
D. Computer Requirements
In the previously described calculations the total number of computa-
tional cells used was 3344, requiring an average calculation time of 1-2
seconds per time cycle on a CDC 7600 computer. With this number of cells
the computer program absorbed nearly all the storage available in a 64,000
word fast core memory. Fortunately, even with this limited resolution
there are many interesting calculations that can be performed.
The problem of what to do when more resolution is needed, however,
is an interesting one that deserves further comment. Clearly, the simplest
approach is to use auxiliary memory units. Although more computer time is
needed when operating with this kind of storage, because of the longer
time needed to retrieve data, the calculation time for the examples illus-
trating this paper could easily be increased by an order of magnitude with-
out becoming too unreasonable. An order of magnitude increase is roughly
equivalent to doubling the finite difference resolution, since that re-
quires a factor of eight increase in the number of cells and a somewhat
larger increase in calculation time.
-------
-17-
Nevertheless, it is easy to think of three-dimensional problems in
which still larger increases in resolution are required, and aside from
relying on the development of larger and faster computers, it is clear
that more effort must be devoted to improving both computer programming
and numerical approximation methods.
E. Numerical Stability
No additional stability conditions are introduced in the Marker-and-
Cell method when it is used for three-dimensional computations, but the
stability conditions previously reported ' ' must be appropriately
modified.
The basic restriction on the size of the time step, 6t, is that fluid
must not be permitted to flow across more than one computational cell in
one time step, that is
6z
6t
This is clearly a numerical accuracy condition, because the convective flux
approximations used in the tilde calculations (3) assume exchanges between
adjacent cells only. This condition must also be satisfied for numerical
stability, as can be verified by linearizing the difference equations and
[12]
performing a Fourier analysis on them.
-------
-18-
The linear analysis also reveals that the equations will always be
unstable unless the kinematic viscosity, V, is large enough; a good
approximation is
V > ^max [u2, v2, w2] . (9)
This condition follows easily from a heuristic stability analysis,
which shows that V should also satisfy the following approximate in-
equality ,
M >» r1 x 2l9u| 1 ~ 2|3v| 1 ,-.
V > max [- <$x |j^|, -j 6y 11. "J fiz
The last two conditions imply a lower bound on the kinematic viscosity,
which imposes an upper limit on the flow Reynolds number. This Reynolds
number restriction is not unique to the Marker-and-Cell method, but is a
necessary feature of all finite difference methods. The reasons for this
can be shown in many ways . One way is to argue as follows : Truncation
errors are unavoidable in finite difference approximations, and even though
they do not always lead to instabilities that require restrictions like (9)
or (10), they do influence the accuracy of a calculation. For purposes of
accuracy, if the effects of V are not to be obscured by truncation errors
it is necessary that
v > a Ax Au , (11)
where a is some numerical factor of order unity, Ax is a typical cell
-------
-19-
dimension, and Au is a typical velocity change across a cell. This rela-
tion is based on the observation that a difference approximation of order
(p + 1) will have truncation error terms that modify V by a contribution
like,
In a finite difference approximation this quantity will be approximated by
oAxAu, which defines the value of a. For the order of magnitude estimate
wanted here, a can be replaced by unity. Thus, (11) is simply the state-
ment that V must be larger than these errors for an accurate calculation.
Now if a typical dimension in a flow, L, is resolved by N finite differ-
ence cells, L = NAx, and if a typical velocity U is NAu, then (11) also
states that the flow Reynolds number, R = ——, must be less than N . In
other words, the condition
R < N2 (12)
is a necessary restriction for accurate finite difference calculations.
It may be noted that a few finite difference approximations, for ex-
ample, those using the so called donor cell approximation, have even
larger truncation errors that lead to the more restrictive condition,
R < N (12a)
-------
-20-
Condition (12) is a rough estimate for the maximum allowable Reynolds
number obtainable with any finite difference approximation. It is primarily
an accuracy condition, but it often happens, as in the present case, that
it is a condition for stability as well.
Finally, when very low Reynolds number flows are to be simulated the
time step is additionally restricted by the condition
V
-------
-21-
III. AUXILIARY FEATURES
A. Thermal Buoyancy
A heat equation may be simultaneously solved with the fluid equations
in order to simulate the effects of thermal buoyancy that are important for
many meteorological applications. The differential equation governing con-
vection and diffusion of temperature, T, is
~+ V«Tu - V-(AVT)
where X may be chosen to represent both turbulent and molecular diffusion
processes. The finite difference expression used to approximate (14) assumes
that T. . , is located at the center of cell (i,j,k),
- 2Ti,j,k
6y
2
-------
-22-
A constant diffusion coefficient has been assumed for simplicity, but this
can be easily changed. The notation (Tu) ^, . . means that the flux be-
l'fc2)J , K.
tween cell (i,j,k) and (i+l,j,k) is to be evaluated by the donor cell
rule,[13J that is,
Donor cell fluxes are used here to insure numerical stability and to avoid
negative temperatures upstream from a local hot spot.
The most common boundary condition on the temperature is that of zero
flux, which corresponds to a nonconducting wall or a plane of symmetry.
Heat sources can be added in a variety of ways. Either selected portions of
the boundaries can be given prescribed temperatures, or prescribed energy
fluxes, of energy can be deposited directly into selected regions of the
fluid.
The effects of temperature variation are assumed to influence the fluid
motions through a Boussinesq approximation, which consists of the addition
of buoyancy terms to the right sides of the tilde equations (3). For ex-
ample, the following term is added to the w-tilde equation,
-------
-23-
The constant T is an initial reference temperature and $ is the coefficient
of thermal expansion. This term requires a temperature at the boundary be-
tween two cells, which is equal to the average of the two cell temperatures.
An additional numerical stability condition is needed when equation
(15) is used. This condition, which is analagous to (13), is
6x 6y
~)
(16)
The temperature equation can also be used to represent the transport
of particulate matter when temperature effects are not of interest, in
which case T is interpreted as the particulate concentration. For example,
Fig. (7) shows a particulate distribution calculated in this way faith $
equal to zero)- The air flow is incident at 45 to the buildings, as shown
in Fig. (6). There is a constant source of particulate matter being in-
serted at the center of the base of the large obstacle on the side furthest
from view. The particulate concentration is shown in Fig. 7 as a distri-
bution of particles. This was made by plotting in each cell a number of
particles proportional to the cell concentration, T, and with positions dis-
tributed randomly within the cell.
B. Marker Particles
The above technique for particulate transport is not very refined and
does not work well for problems having sharply defined regions of particulate
-------
matter. A better technique has been devised by R. C. Sklarew. He keeps
track of individual particles and uses a clever trick to move them so that
their distribution represents a solution of (14). The trick is to rewrite
this equation as
(17)
Now it is evident that if particles are moved (convected) with the effective
velocity
u - -| VT (18)
they will approximate a solution of (17). Another way to say this is that the
total flux of T resulting from convection and diffusion is equivalent to a
pure convection with the velocity (18). The concentration, T, in a cell is
then proportional to the number of marker particles in the cell. The diffu-
sion coefficient can vary arbitrarily in space and time, and the method is
stable provided no particle moves more than one cell width in one time step.
Figure 8 shows an application of the Sklarew method to the flow of a
slowly dispersing plume passing over the top of a rectangular structure. The
flow is the same as that shown in Figs. (2-3). Particles are seen trapped
and recirculated in the wake region.
-------
-25-
A similar calculation is shown in Fig. 9 for a more complex building and
with particulates emitted from a vent centrally located on top of the principle
structure. The flow field for this problem is identical to that of Figs (4-5).
The numerical prescription used for moving particles is based on a
straightforward extension of the technique used in the original Marker-and-
Cell method. Each particle is moved with a velocity obtained from a
linear interpolation among the eight nearest cell velocities. The same inter-
polation is used whether the particles are to move with the fluid or with the
effective velocity (18).
The only difficult problem in moving particles is to account for the
presence of various boundary conditions. In the examples shown here, this
has been accomplished by suitably adjusting the velocity interpolation factors
when particles are near a boundary.
C. Data Display Techniques
Most of the figures have displayed data in the form of perspective views.
These views give a much better picture of the three-dimensional flow fields
than could be obtained from sets of purely two-dimensional plots. In addi-
tion to the velocity vectors and particle distributions shown, it can be
useful to plot perspective views of contour lines, streak lines, and, in
general, anything having a three-dimensional distribution.
T81
The perspective plots used here1 have been designed especially for
three dimensional finite difference calculations. They are so efficient
-------
-26-
that movies of transient flow phenomena can be made at little additional
expense to a calculation. Movies can also be made with the observation
point continually changing position, to give an even better feel for the
three-dimensionality of a problem.
Stereo pictures of velocity vectors and particle distributions have
proven themselves to be extremely useful, but unfortunately they are not
easily presented in journal articles. The usual procedure is to print,
side by side, two perspective views made from slightly shifted observa-
tion points, as in Fig. lOa. The left view is the correct perspective
for the left eye and the right view is correct for the right eye. To see
in stereo it is necessary to hold the figure approximately 18 inches in
front of the eyes and to let the eyes move apart so that the combined eye
images merge together at some distance beyond the page. Unfortunately,
many persons cannot keep their eyes in focus while forcing them to move
apart (walleyed). On the other hand, a large fraction of these people can
keep them focused when they are moved together (crossed). Thus, in Fig.
10b the left and right images shown in Fig. lOa have been reversed. This
figure will appear in stereo when the eyes are crossed to bring the images
together at a point in front of the page. Admittedly it takes some practice
to get a stereo view in either case, but the results are generally worth the
effort.
-------
-27-
IV. ACKNOWLEDGMENT
The authors would like to express their appreciation to Robert Hotchkiss
who has supplied many valuable additions to the program, and who performed
the calculations illustrated in Figs. (8-10).
-------
-28-
REFERENCES
1. F. H. Harlow, "Numerical Methods for Fluid Dynamics — An Annotated
Bibliography," Los Alamos Scientific Laboratory Report LA-4281 (1969).
2. J. W. Deardorff, Geophys. Fluid Dynamics _!, 377 (1970); J. Fluid Mech.
41, 453 (1970).
3. A. J. Chorin, AEG Research and Development Report, NYO-1480-61 (1966).
4. G. P. Williams, Jour. Fluid Mech. 37_, 727 (1969).
5. F. H. Harlow and J. E. Welch, Phys. Fluids JJ, 2182 (1965); J. E. Welch,
F. H. Harlow, J. P. Shannon, and B. J. Daly, Los Alamos Scientific
Laboratory Report, LA-3425 (1966).
6. R. C. Sklarew, Paper presented at 63rd Annual Meeting Air Pollution
Control Assoc., St. Louis, Missouri, June (1970).
7. B. D. Nichols and C. W. Hirt, Manuscript in preparation.
8. C. W. Hirt and J. L. Cook, Manuscript in preparation.
9. J. A. Viecelli, Jour. Comp. Phys. _8, 119 (1971).
10. C. W. Hirt, Jour. Comp. Phys. .2, 339 (1968).
11. B. D. Nichols and C. W. Hirt, To be published in Jour. Comp. Phys.
12. B. J. Daly and W. E. Pracht, Phys. of Fluids 11., 15 (1968).
13. R. A. Gentry, R. E. Martin, and B. J. Daly, Jour. Comp. Phys. _!, 87
(1966).
-------
LIST OF SYMBOLS
V
6
e
->•
>
<
1-1
a
A
X
Nu
Delta
Beta
Arrow
Greater than
Less than
Absolute value
Alpha
Cap. delta
Lambda
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FIGURE CAPTIONS
1. Location of velocity components on a typical Eulerian cell (i,j,k).
2. Perspective view of velocity field about a single building.
3. Projection of velocity vectors seen in Fig. 2 on a z = constant plane.
4. Perspective view of velocity field near the bottom of a complicated
structure.
5. Perspective view of velocity field near the top of a complicated structure.
6. Perspective view of velocity field in vicinity of two buildings. Incident
flow is oriented 45 with respect to large faces of the buildings.
7. Perspective view of particulate distribution in flow field shown in Fig. 6.
8. The dispersal of a narrow plume passing over a single building. Recircula-
tion in wake region is clearly evident.
9. The dispersal of pollutant from a flush vent on the top of a complex build-
ing structure.
10. The two perspective views in (A) appear in stero when viewed "walleyed",
while those in (B) appear in stero when viewed "crosseyed".
-------
-------
I • • • »
, • ' '
-------
r r r 11 \
\
-------
'
-------
it
-------
B
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BIBLIOGRAPHIC DATA
SHEET
1. Report No.
EPA-R4-73-029
3. Recipient's Accession No.
4. Title and Subtitle
Air Pollution Transport in Street Canyons
5. Report Date
June 1973
6.
7. Author(s)
R. S. Hotchkiss and F. H. Harlow
8. Performing Organization Rept.
No.
9. Performing Organization Name and Address
University of California
Los Alamos Scientific Laboratory
Los Alamos, New Mexico 87544
10. Project/Task/Work Unit No.
11. Contract/Gram No.
EPA-IAG-0122 (D)
12. Sponsoring Organization Name and Address
EPA, Meteorology Laboratory
National Environmental Research Center
Research Triangle Park, North Carolina 27711
13. Type of Report & Period
Covered
Final Report
14.
15. Supplementary Notes
16. Abstracts
This project was conducted to demonstrate the applicability of numerically modeling
the transport of pollution in street canyons. The numerical model employs the solutions
of the fully nonlinear, three-dimensional Navier-Stokes equations along with a transport
equation for pollutants, for regions of space in which obstacles or buildings cause
strong distortions in the flow fields.
The numerical technique is used to model three-dimensional flows for which some ex-
perimental data have been obtained. This includes calculating the distribution of pol-
lutants in the Broadway Street Canyon in downtown St. Louis, Missouri. Also, the nu-
merical method is used to calculate pollutant distributions in a non-specific street
canyon; that is, a street canyon in which the geometry and other important non-
dimensional flow parameters give rise to solutions that are applicable, in a general
sense, to a variety of street canyons.
17. Key Words and Document Analysis. 17a. Descriptors
17b. Identifiers/Open-Ended Terms
17c. COSATI Fie Id/Group
18. Availability Statement
19..Security Class (This
Report)
UNCLASSIFIED
20. Security Class (This
Page
UNCLASSIFIED
21- No. of Pagen
113
22. Price
FORM NTIS-35 IREV. 3-721
USCOMM-DC I4952-P72
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Guidelines to Format Standards for Scientific and Technical Reports Prepared by or for' the Federal Government,
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FORM NTIS-35 IREV. 3-721 USCOMM-DC 149S2-P72
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