EPA-650/4-74-006
FEBRUARY 1974
Environmental Monitoring Series
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EPA-650/4-74-006
A METHODOLOGY
FOR TREATING
LARGE LOCALIZED EMISSIONS
OF REACTIVE POLLUTANTS
by
A.J. Fabrick, P.I. Nakayama, anclE.J. Fredricksen
JRB Associates, Inc.
1200 Prospect Street
La Jolla, California 92037
Contract No. 68-02-1238
Program Element No. 1AA009
EPA Project Officer: Herbert Viebrock
Meteorology Laboratory
National Environmental Research Center
Research Triangle Park, N.C. 27711
Prepared for
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
WASHINGTON, D.C. 20460
February 1974
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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 develop a numerical model
to accurately calculate the trajectories and concentrations of reactive
pollutants emitted from localized sources. The numerical model
employs the solution of the full three-dimensional Navier-Stokes equa-
tions along with the solution of the species density equation by summing
over Lagrangian mass points. The mass points are transported by the
mean wind field and moved with a random walk technique that simulates
the turbulent diffusion. The effect of chemical reactions are modeled
by reweighting the Lagrangian mass points. The equations are solved
for regions of space where terrain features or buildings may cause
strong distortions in the flow field.
The numerical techniques are verified by comparisons with analytic
formulas, including the boundary layer above a plate and the Gaussian
plume. The method was used to calculate photochemically reacting
plumes using a simplified inorganic photochemical reaction model.
It is concluded that the numerical model shows promise of being able to
assess the impact on air quality of point source emission over complex
terrain. Furthermore with the incorporation of a valid photochemical
reaction kinetics module, the model developed during this study could
be used to construct a parametric study of various plumes in order to
quantify the errors introduced by regional models in averaging point
sources over large areas.
in
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CONTENTS
Page
Abstract „ iii
List of Figures v
Acknowledgments vi
Sections
I Conclusions 1
n Recommendations 3
III Introduction 5
IV Solution of the Hydrodynamic Equations 7
V Verification of Numerical Solutions 17
VI Solution of the Density Equation Using Particles 23
VII Verification of the Particle Method 27
VIH References 37
DC Appendices 39
IV
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FIGURES
No. Page
1. Location of variables in computational mesh. 9
2. Flow diagram for hydrodynamics routine. " 14
3. Grid used for laminar boundary layer calculation 19
4. Initial velocity field for laminar boundary layer
calculation. 19
5. Velocity vectors after 150 computational cycles. 19
6. Horizontal velocity as a function of height for three
locations along a flat plate. 20
7. Computational grid for plate over notch calculation. 21
8. Initial velocity field for flow over notch calculation. 21
9. Velocity vectors after 21 computational cycles. 21
10. Initial velocity field for V notch problem. 22
11. Velocity vectors after 16 computational cycles. 22
12. Particle locations during a VARMINT simulation of a
Gaussian Plume with and without subcycle staggering. 28
13. Comparison of VARMINT calculation and analytic
solution for a point source. • 29
14. Comparison of VARMINT calculation and analytic
solution for a point source with exponential decay. 31
15. Schematic diagram for Eulerian-Lagrangian code. 32
16. Calculation of NO9 plume emitted into clear ambient
air. • * • 34
17. Calculation of an equilibrium plume emitted
into equilibrium ambient air. 35
18. Nitrogen oxide source emitted into ^ppm ozone
background. 36
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ACKNOWLEDGMENTS
The authors would like to express their appreciation to C. W. Hirt
and J. L. Cook for their pioneering work on the techniques employed
under this contract and for their valuable technical assistance during
their study.
This work was supported, in part, by the Environmental Protection
Agency under contract (EPA-68-02-1238).
VI
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SECTION I
CONCLUSIONS
The transport and dispersion of reactive emissions from point sources
has been modeled and incorporated into a numerical solution technique
that was developed under this program. The technique shows promise
of being able to assess the impact on air quality from point source
emissions of reactive pollutants over complex terrain; a capability
that has heretofore not been available.
With the incorporation of an accurate photochemical reaction kinetics
module, the techniques outlined in this report can be used to estimate
the errors introduced by smearing point source emissions over large
areas in regional photochemical air quality simulation models.
. 1
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SECTION n
RECOMMENDATIONS
As will be seen in the text, this program has demonstrated the
feasibility of a numerical simulation of reactive plumes over
irregular terrain. With the addition of an appropriate photo-
chemical reaction kinetics module, the techniques outlined in this
report could be used to estimate the errors introduced by smearing
point source emissions over large areas in current regional photo-
chemical air quality simulation models.
In order to utilize the methods developed for the calculation of the
impact of localized sources in complex terrain several additional
features must be added. Atmospheric density stratification and
the dynamics of buoyant plumes must be incorporated. Turbulence
models that specify eddy viscosity as a function of time and space
should be incorporated. To allow greater generality in specifying
terrain features arbitrary boundaries treatment should be
developed to complement the existing obstacle blocks and half
blocks already encoded.
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SECTION IE
INTRODUCTION
The importance of the role played by localized emissions on air
quality in a large air basin is evident. A source inventory of the Los
Angeles Air Basin, for example, shows that about 20% of the oxides
of nitrogen are emitted from point as compared to area sources (mainly
attributed to the automobile). As automobile emission standards are
made more stringent and as additional fossil fuel power plants are built
to meet energy demands the percentage of air pollutants emitted from
point sources will increase. Because the pollutants of interest are
reactive and their reaction rates are a function of concentration,
smearing point source emissions over a large area or prescribing
an apriori analytical distribution of initial concentration may result
in erroneous estimates of the effect of these emissions on air quality.
In order to assess the magnitude of the errors resulting from the
approximate treatment of point sources employed in current photo-
chemical simulation models a methodology must be developed to
correctly relate point source emissions of reactive pollutants to
ambient air quality.
The methods developed under this contract solve two major problems
associated with reacting plumes. These problems are the specification
of the wind and turbulence fields over complex terrain and the solution
of the species density equations, including chemistry. The solutions
to 'these problems are respectively, the direct numerical calculation
of the three dimensional wind field and a particle-random walk tech-
nique to solve the species density equation. In the former, the full
Navier-Stokes equations with appropriate boundary conditions are used,
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and while the turbulence is not calculated directly its effect on the
mean flow and the dispersal of pollutants can be prescribed as a func-
tion of space and time by assuming the turbulence to be in a state of
equilibrium where the creation of turbulent energy by velocity gra-
dients and decay by molecular viscosity are in balance. In the latter,
the density equation, including chemistry is solved by representing
pollutant species as discrete particles that move within an Eulerian
grid. This technique eliminates the numerical smoothing of diffusion
of species concentration inherent in finite difference solutions of the
density equation. The turbulent diffusion of pollutants is accomplished
by a random walk technique imposed on individual particles and is
independent of the Eulerian grid. The standard deviation of the random
walk is scaled to the local turbulence intensity. Changes in concentra-
tion due to chemical reactions are accomplished by adjusting the
masses of the individual particles. The techniques discussed above
have been incorporated into a computer code, VARMINT. The discus-
sion of this code is divided into four sections. The first two sections
discuss the solutions of the Navier-Stokes equations and the verifica-
tion of the numerical methods employed. The third section deals with
the methods employed to solve the species density equation. The
fourth section presents the verification of the particle method used to
solve the species density equation and preliminary investigation of
reactive plumes using a simplified photochemical reaction mechanism.
A program users guide and listing are provided in appendices.
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SECTION IV
SOLUTION OF THE HYDRODYNAMIC EQUATIONS
The VARMINT code solves the standard Eulerian incompressible
equations using a variation of the Marker and Cell Technique which is
o •*
similar to that employed by Hirt et. al. •' A pressure iteration is
utilized to insure the incompressibility condition. The governing
equations are
bU.
-1=0 . (2)
ax.
J
•In Cartesian coordinates
. 5 6u 6 bu
CT" +~CT
.. _ _
at bx by "" bz "ax 6xCT&x" b~yCTby 5lCT6^x
2
Y_ 4. $Xw - 5P 6 &v 5 &v a bv (3)
by +' bz ~ ~bT +blEor^ + 5ya^y +^CTS¥ + gy
2
bw t bwu bwv ; _ _ ^+__ _ __ __
bt bx by bz ~ " bz bx a bx + by a 6 y + bz a bz gz
bu + bv + bw = 0 (4)
by by bz
Here P is the ratio of pressure to constant density, a is a momentum
eddy diffusivity, and g , g , g are prescribed body accelerations
x y z
that arise from gravity, buoyency etc. The region in which computa-
tions are to be performed is divided into a set of small rectangular
cells having edge lengths &x, A7> Az> which need not be the same for
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all cells. With respect to this set of computational cells, velocity
components are located at cell faces and pressure values are at cell
centers as shown in 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.
A time-dependent solution is obtained by advancing the flow field varia-
bles through a sequence of time steps of duration At. The advancement
through one time step is calculated in two stages. The first stage or
tilde phase consists of a simple explicit calculation, in which the velocity
components (u, v, w), are advanced using the previous state of the flow
to calculate the accelerations caused by convection, diffusion, body
forces, and pressure gradients. This explicit time advancement does
not necessarily lead to a velocity field with zero divergence, i. e., to
one that insures mass conservation. We can, however, insure incom-
pressibility by adjusting the pressure in each cell in such a way that
there is no net mass flow in or out of any cell. Adjusting the pressure
in one cell will affect neighboring cells so that this adjustment must be
performed iteratively until all cells have achieved a mass change not
exceeding some small prescribed amount. The explicit finite differ-
ence equations for the tilde phase are given in Appendix A. These
equations allow the user to specify doner cell or cell centered con-
vective flux approximations or any combination of the two methods.
8
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x
Figure 1. Location of variables in computational mesh.
- 9
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BOUNDARY CONDITIONS
Several boundary condition options are available in VARMINT. These
include no slip and free slip rigid walls, continuative output walls, and
V
periodic boundaries. For simplicity VARMINT assumes all physical
boundaries coincide with cell boundaries. Internal obstacles consisting
of cells or half cells (diagonal) are also available with no slip and free
slip boundary conditions.
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. If the boundary is
periodic the value must be chosen equal to the corresponding velocity
one wavelength away. For outflow boundaries there is no unique pre-
scription, but the general idea is to choose boundary conditions that
have the least upstream influence. It has been found that for this pur-
pose 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 iteration.
The tangential velocities in cells immediately outside the fluid region
are set equal to the adjacent velocities inside the fluid when it is
desired that the boundary represent a free-slip wall, 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 vanishing shear or vanishing velocity
10
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at a rigid wall. If the boundary is periodic these external velocities
are set equal to their counterparts one wavelength away. At an outflow
boundary, they are set equal to the adjacent velocities inside the fluid.
f
The boundary conditions are set after the tilde velocities are calculated
and after each pass through the pressure iteration.
The boundary conditions on internal obstacles are the same as for the
actual mesh boundaries except that before each pass through the pres-
sure iteration all velocities inside the obstacles are set to zero. This
method allows for the correct stress boundary conditions during the
tilde calculation and for no divergence boundary conditions during the
pressure iteration.
PRESSURE ITERATION
Since tilde equations do not necessarily result in a velocity field that
satisfies Eq. (4), some adjustment of the tilde velocities must be
made to insure mass conservation.
The iterative process used to modify the tilde velocities is discussed
below. In each cell (i, j, k) the value of the velocity divergence
D is calculated as:
V = SET (Vijk-Vijk'
-------
The pressure is then changed by an amount
6P = - 0D
where
1 2At
(6)
or
1*t \JJ^ • * 1 -4
_ _ _1_ i]k _ _1_
* ~*o bPijk.=" ^o
2At
Ax (Ax +AX. T) Ax (Ax+Ax. )
1 1 i-t-J. Jl JL 1—A
2At
j (Ay^+Ay. x)
2 At
2At
2 At
j (Ay.+Ay.^) AZj, (Azk+Azk+1)
o
2 At *
AxrAxi^i
Ax.+Ax. '
Ayj
(6a)
Here, BQ is a relaxation factor, where overrelaxation and under-
relaxation correspond to 8Q greater than or less than unity. For
stability in the iteration, £o must remain less than 2.
12
-------
Once AP has been calculated it is necessary to add it to the total
pressure P^ and to adjust the velocities:
, 2At
2At
2At ^ ,p
V,_..it. -» v«.iu •*" V AW>+Ay.—) OP
3 3+l
2At
M
This process is repeated successively in all cells until no cell has a
magnitude of D greater than e . Where e is a function of the magnitude
of the velocities calculated.
When the iteration has converged, the adjusted velocities satisfy the
mass conservation condition (4), and this completes the necessary cal-
culations for advancing the flow field through one cycle in time.
A flow diagram of the hydrodynamics routine of VARMINT is given in
Figure 2.
13
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Input Hydrodynamics Data
I
CO
i— i
O
O
1
o
•r-<
13
t
O
U
^' jLiiut: jfiia.se .Dounuaiy ^onaiuons
Calculate u, v, w
1
-i_>
c^
CD
O
CD
O
O
O
•g
c
03
.2
f-l
CD
-M
•r-l
ki Mo T^i ^0^^011(^0 T^rmnrliT'T <~'n~
^^ J.iv/ JL/l VC i ^C11\^C X./UllIlLlctx y v^OIiUXLiUIlo
1
Form 6P the Pressure Difference
i
Update Velocities
I
Check for Convergence
6P/P
-------
NUMERICAL STABILITY
The basic restriction on the size of the time step, At, is that the fluid
must not be permitted to flow across more than one computation cell
in one time step, that is
A, ^ . Ax Ay Az /ox
At < mm , , - (8)
This is both a numerical accuracy condition (because the tilde equa-
tions assume exchanges between adjacent cells only) and a numerical
stability condition.
The VARMINT code may be used either with donor cell or with centered
difference for the convection term. When centered difference is used,
the equations will be unstable unless the kinematic viscosity, a, is approx-
imately
^ At r 2 2 2-, ,ON
or > max [u , v , w ] (9)
When donor cell is used this restriction may be relaxed.
Finally, when low Reynolds number flows are simulated the time step
may be restricted by the condition
At < 1 , *
v 0/ 11T
I—2" +—2 +—2
\Ax Ay Az
In analogy with the interpretation of equation (8) this restriction can
be described as limiting the distance over which momentum diffuses
during one time step to less than one cell width.
15
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16
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SECTION V
VERIFICATION OF THE NUMERICAL SOLUTION
Several qualitative checks were made in order to determine if the
methodology incorporated into VARMINT was encoded correctly.
As a check on the variable zoning feature of the code as well as on the
diffusion, advection and boundary condition sections, a laminar
boundary layer building over a flat plate was calculated.
The grid used for this problem is shown in Figure 3. Initial conditions
for the problem shown in Figure 4 is unit velocity over the plate.
Figure 5 shows the flow after 150 cycles when the boundary layer has
been established. An examination of the results of this problem is
shown in Figure 6. The u velocity is plotted as a function of height
for three locations at the front, middle and rear of the plate. Note the
linear gradient in velocity at the plate and the increase in boundary
layer thickness with increasing distance from the front of the plate.
To the extent that comparison with the analytic solution is possible
these results show good agreement.
The check of the internal obstacles was accomplished by calculating
the flow over a flat plate with a notch. The first problem shown in
Figures 7-9 is a notch formed by rectangular obstacles. The flow is
from left to right with unit velocity over the plate and zero velocity in
the notch. The flow at the left boundary is maintained at unity through-
out the calculation. Figure 7 shows the computational grid with the
plate and notch indicated. Figures 8 and 9 show the resultant velocity
vectors at cycles 1 and 21 respectively. Note that by cycle 21 the
rotational flow inside the notch has been established. Notice also that
17
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the velocity vectors at the left are tilted upward, indicating a small
upward component of velocity. This is the start of the boundary layer
development.
7
Diagonal obstacles were also incorporated into the code and checked.
Figures 10 and 11 show the results of the calculation of a flat plate
with a V notch. The initial conditions were similar to the rectangular
notch problem with unit velocity above the plate and zero velocity in
the notch. Although too small to see in Figure 11, a rotational flow
field has been established at the bottom of the notch.
' 18
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I I
Figure 3. Grid used for laminar boundary layer calculation.
Figure 4. Initial velocity field for laminar boundary layer calculation.
Figure 5. Velocity vectors after 150 computational cycles.
19
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•s
rt
14
12
10
8
6
4
2
= 47.0
.4 .6 .8 1.0 1.2 1.4
u Arbitrary Units
Figure 6. Horizontal velocity as a function of height for
three locations along a flat plate.
20
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1
1
1
;j t .
i 2
•
: ^
:
J
f
,
....
Figure 7. Computational grid for plate over notch calculation.
Figure 8. Initial velocity field for flow over notch calculation.
Figure 9. Velocity vectors after twenty-one cycles of computation.
21
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Figure 10. Initial velocity field for V notch problem.
Figure 11. Velocity vectors after 16 computational cycles.
22
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SECTION VI
SOLUTION OF THE DENSITY EQUATION USING PARTICLES
The equation describing turbulent atmospheric diffusion including
pollutant sources and chemical reactions,
be b b , be a ,, /...,\
r-r = -r—-u. c +-r— k., -5;— + S +R (11)
bt bx. i bx. ]k bx,
1 J K
equates the time rate of change of concentration bc/bt, to the advective
rate of change of -(b/bx.) u. c, due to the mean velocity field plus the
gradient of the turbulent flux with the position dependent eddy diffusivity
(k-•), (b/bx.) k., (bc/bx,), plus source of concentration, S, plus the
lj ] JK K
change in concentration by chemical reactions, R.
The VARMINT code uses Lagrangian point particles to represent
pollutant mass, so that Eulerian species concentration is obtained by
summing the masses of the appropriate species in any volume of interest.
Changes in particle position are calculated to simulate pollutant mass
transport due to both advection and diffusion as specified by the turbulent
atmospheric diffusion equation. In addition, changes in particle mass
are calculated to simulate changes in pollutant concentration due to
chemical reactions.
4
This technique was developed by Sklarew et al and modified by Hotchkiss
g
and Hirt. In the limit of a large number of particles, equation (11) is
solved without the numerical errors introduced by Eulerian finite
difference techniques. To simulate advection and diffusion the particles
are moved an increment dx. in the i direction where,
dx. = u.dt +dw. (12)
This increment is composed of two parts. The fluid velocity times the
time step simulates the advection, while dw. is a random displacement
which simulates turbulent diffusion.
23
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The random displacement dw. is a variable whose average value is zero and
1 i
whose variance is equal to (2k^ dt) . This method developed by
5
Hotchkiss and Hirt approximates the process of eddy diffusion as a
random walk scaled to the local eddy diffusivity. This technique
cannot directly model the effect of the off diagonal terms in the eddy
diffusion tensor k.., however if these terms are significant (i.e.
comparable in magnitude to the diagonal terms) we note that a
symmetric tensor can be brought into diagonal form by an orthogonal
transformation. Therefore a rotation can always be defined such
that the eddy diffusion tensor is diagonal. Thus the random movement
of the particles can be done in the rotated frame.
Localized sources are modeled by emitting new particles of the appro-
priate mass and number densities each cycle. For a source strength
Q(gm/sec), a time step of At (seconds) and a particle mass m, the
number of particles added each cycle is Q»n/m. The particle mass
m must be chosen so that the total number of particles are large
enough to accurately solve equation (11) yet minimize computer run
time and storage requirements. For each localized source about 10
to 100 particles emitted per cycle seems to satisfy these criteria.
The concentration of a species is determined by summing the mass of
the particles inside a prescribed fluid volume and then dividing by that
volume. In the current.version of VARMINT the fluid volumes chosen
correspond to the cells used in the hydro calculation although this is
not required. For computations involving a single species, volume
weighting the mass of a particle in the neighborhood of a point where
24
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the concentration is required can eliminate discontinuous jumps in the
concentration as a mass particle enters and leaves a fluid volume,
however this procedure leads to additional diffusion-like errors when
chemistry involving multiple species is present.
Once the concentration of all the relevant species in a fluid element
are known, the chemical reactions over a time period At can be
solved and new concentrations computed. Any method of solution of
the chemical rate equations is allowed. The VARMINT code treats
the chemistry as a "black box" into which is entered At and the
species concentrations, and from which the concentrations at time
plus At is returned. Each particle mass is then adjusted so that
n+1 n cn+1
m = m •
cn
where mn+ is the particle mass at time t plus At, mn is the particle
mass at time t.
Multiple species can be handled in two ways. Each specie can be
treated as individual particles or each particle can have multiple
masses representing each specie associated with its location. Both
methods seem to give the same accuracy while the second method
saves both computer time and storage by reducing the total number
of particles required.
Boundary conditions for the particles are handled in a manner consistent
with the boundary conditions in the hydrodynamics section of VARMINT.
25
-------
Rigid walls and obstacles are treated by reflecting the particles,
continuitive boundaries are treated by allowing the particle to leave
the computational mesh and periodic boundaries are treated by
allowing the particle to leave the computation mesh and return at
the other end of the mesh.
26
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SECTION VH
VERIFICATION OF THE PARTICLE ROUTINE
The initial validation of the particle section of VARMINT was done by
comparing the VARMINT calculation of a point source in a uniform
wind field with the analytic solution of an inert Gaussian plume. The
inert plume calculations was made to assess the advection, diffusion,
concentration and boundary condition sections of the code. Initially,
difficulties were experienced because without diffusion in the downwind
direction all the particles emitted during a particular time step were
constrained to have the same downwind coordinate, causing them to
"line up." Emitting the particles throughout the time step instead of
in a single puff corrected this difficulty. This effect is illustrated in
Figure 12. The calculation illustrated on the top of Figure 12 is com-
pared with the Gaussian formula in Figure 13. The concentrations are
o
in general units of mass per length . The solid lines are the analytic
results computed from the Gaussian formula. The plotted points repre-
sent the average value of the computed concentrations while the bars
associated with each point represent the standard deviation of concentra-
tion values about this mean value. Both ground level concentration
and concentrations at the level at which the plume was emitted are
presented for both the down wind and the cross wind direction.
Additionally the variation in concentration with height is also plotted.
In all plots the mean values of the VARMINT calculation agrees with
the Gaussian formula.
To test the method for adjusting particle masses due to changes by
chemical reactions the simple rate equation dC/dt = -XC was chosen
-------
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to allow comparison with an analytic solution. The results of this test
run is shown in Figure 14. Again the units of concentration are general.
These two comparisons demonstrate that the methods incorporated into
VARMINT can reproduce the analytical formulas and' could simulate
more realistic situations with wind shear and variable diffusivity,
terrain and time varying source strengths.
As a further examination of the particle mass change logic and as a
preliminary introduction to reactive plumes a chemistry model which
solved the inorganic photochemical reactions
(hv) + NO2 -» NO + O
O + (02) + (M) -> 03 +(M)
O, + NO -» N09 + (OJ
O £1 ft
was incorporated into VARMINT and several problems were rum Since
the Gaussian formulas cannot take into account nonlinear reactions, a
simple code was written that solved the finite difference form of the
diffusion equation in the y - z plane while advancing along the x-axis
in a Lagrangian sense. The method is illustrated in Figure 15.
A problem of interest is that of a nitrogen-dioxide source emitted into
clear ambient air. This example was computed by both codes and
compared in Figure 16. The initial source strength was 90ppm NO0,
ti
and most of the NO0 remained intact with only a small amount becoming
CJ.
NO and OQ. The circles and dots represent the mean concentrations
«J
predicted by VARMINT, while the solid lines are the results calculated
by the Eulerian-Lagrangian code. The bars through the dots and
circles for the downwind center line concentration are the variance of
30
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31
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32
-------
the concentration predicted by VARMINT. The bars have been left off
of the other graphs to make them more legible.
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t
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SECTION vm
REFERENCES
1. "Air Pollution in California - Annual Report 1970," Resources
Agency — Air Resources Board, State of California (January 1969).
2. Hirt, C. W., and Cook, S. L., Calculating Three-Dimensional
Flows Around Structures and Over Rough Terrain, Los Alamos
'Scientific Laboratory Report, LA-DC-13289 (November 1971).
3. Hirt, C. W. and Cook, S. L., Calculating Three-Dimensional
Flows around Structures and over Rough Terrain, Journal of
Computational Physics, Vol. 10, No. 2,(October 1972).
4. Sklarew, R. C., Fabrick, A. J. and Prager, S. E., A Particle-
in-Cell Method for Numerical Solution of the Atmospheric Diffu-
sion Equation and Applications to Air Pollution Problems, Systems,
Science and Software Report 3SR-844, (November 1971).
5. Hotchkiss, R. S. and Hirt, C. W., Proceeding Summer Computer
Simulation Conference, San Diego, California (June 1972).
37
-------
(This Page Intentionally Left Blank)
-------
SECTION DC
APPENDICES
Page
A. Users Guide for VARMINT 41
B. VARMINT Listing 73
39
-------
(This Page Intentionally Left Blank)
40
-------
A. USERS GUIDE FOR THE VARMINT CODE
The VARMINT code has been written for and run on the CDC 7600,
using FORTRAN IV. The main program (VRC) is the driver program.
There are eight subroutines, seven of which are standard, with the
eighth (KEM) being an offshoot of the particle movement section and
not accessing information by means of COMMON storage. This
chemistry subroutine will differ for types of species and kinds of
reactions desired. The subroutines are:
VRS - setup subroutines for calculational arrays,
FILMCO - sets film constants,
FLMCAL - calls IGSB plotting routines,
VRM - main calculations,
VRPS - particle movement,
KEM - chemistry,
VRIO - output section for printing and plotting,
VRTAP - tape handling.
The logic flow for these is shown in Figure 1.
A job will normally operate with external hardware for storage as well
as I/O. Besides the card reader and line printer, it can require large
core memory (LARGE statement), microfiche (TAPE9 for data,
TAPE10 for plotting), and magnetic tapes (TAPE7 for tape dump writes,
TAPES for tape dump reads). Small core memory houses the common
blocks:
VRCOM contains the array A equivalenced to the flow
field parameters, plotting parameters, and cell
volumes;
CONCEN contains chemical concentration related arrays;
AMIX contains a temporary array;
41
-------
VRC - CONTROL PROGRAM
Dimension Arrays; Starts and
Terminates Problem; Calls VRS
and VRM. Initiates Graphics
VRS - SET-UP SUBROUTINE
Reads Input Data; Initializes
Constants, Index Quantities,
Cell Flags for Obstacles; Calls
FILMCO to Establish Film
Constants
FILMCO - FILM CONSTANTS
Sets Film Constants; Computes
Film Plane Coordinates and
Tests for Those Out of Range;
Writes Film Labels and Draws
Frames
FLMCAL - CALLS PLOTTING ROUTINES
Calls Berkeley IGS Subroutines
VRM - MAIN CALCULATIONS
Boundary Conditions for Mesh
and Obstacles; No-slip Boundary
Conditions, Tilde Equations;
Pressure Iterations; Updates Time
and Cycle; Checks on Output Time
for Data Print, Plots, And/Or Tape
Dumps; Calls Particle Subroutine
VRPS - PARTICLE MOVEMENT
Injects Particles into the Mesh;
Initializes Particle Location and
Mass; Initializes Background;
Particle Chemistry; Moves and Edits
Particles; Printer and Tape Output
of Concentrations
VRIO - OUTPUT SECTION
Printer Output of Cell Quantities;
Plots Mesh Zones, Velocity Vectors,
and Contours on Film
VRTAP - TAPE HANDLING
Output Mesh Quantities to Write Tape;
Input (Restart) Mesh Quantities From
Read Tape; Change any Input Quantities
Figure 1. VARMINT flow chart.
42
-------
FL contains masking variables;
FLS contains masking array;
VRCON contains VARMINT constants;
MODSET contains IGS film constants; »
SAFLM contains VARMINT generated film constants;
ZLAST contains a dummy variable ZLAS which must
always be the last variable in the last common
block. The buffer length for tape dumps is
calculated from the number of words between
and including A(l) and ZLAS.
VARMINT INPUT DEFINITIONS
Rather than presetting all its constants within the code, VARMINT
reads most of the non-calculated constants from cards. For each
problem, the first two cards are read from VRC:
IBR - number of computational cells in x direction,
if negative ~> stop, if 0 ^ restart;
JBR - number of computational cells in y direction;
KBR ' - number of computational cells in z direction,
if IBR = 0 -> KBR = tape dump number for restart;
PRFLM - film flag determines whether or not graphics are
requested
0 H. no film
1 -> yes
LABEL: problem title.
VRS reads in the next cards:
DXC
DYC
DZC
GX
GY
GZ
size of cell in x
size of cell in y
size of cell in z
body force in x
body force in y
direction
direction
direction
direction
direction
- body force in z direction
43
-------
ALX
ALY
ALZ
KWR
KWL
KWF
KWA
KWT
KWB
BO
CEP
DT
TPLT
TPRT
TFIN
TWTD
NPRT
NGOP
NOVP
LPR
DROTJ
donor or cell centered in x direction
donor or cell centered in y direction
donor or cell centered in z direction
0 -» cell centered
1 -» donor
boundary condition flag for right wall
boundary condition flag for left wall
boundary condition flag for front wall
boundary condition flag for aft wall
boundary condition flag for top wall
boundary condition flag for bottom wall
1 -» rigid
2 -» continuative
3 -» periodic
relaxation factor
convergence criterion for pressure iteration
- time step
cell size
(min)
4 velocity/
(max)
time increment between plots
time increment between long prints
time when to finish
time when to dump on tape
initial print options
0 -» no first print
1 -» print cycle 0
2 -» print cycle 0, 1 and at time TPRT
number of graphics options (5 are available)
number of variables to be plotted by the contours
option
long print option
1 -» no long print
2 -» long print on microfiche
3 -» long print on paper
scaling factor for velocity vector plots
44
-------
NWPC - number of words per cell equivalenced to A..,
NWPP - number of words per particle 1J
- particle coordinates and mass
NTPE - particle type
1 -> marker particles ,
2 -» turbulent particles
GPX - body force on particles, x direction
GPY - body force on particles, y direction
GPZ - body force on particles, z direction
NPMX - maximum particles allowed
UI - initial u velocity
VI - initial v velocity
WI - initial w velocity
NU - diffusion constant
ISFLG - slip flag for boundary conditions
0 -» free slip
1 -* no slip
NRC - number of obstacle blocks
IL - left-most boundary cell of block
IR - - right-most boundary cell of block
JA - aft-most boundary cell of block
JF - front-most boundary cell of block
KB - bottom-most boundary cell of block
KT - top-most boundary cell of block
OB - type of obstacle
0 -» cube
1-12 -» diagonal blocks (see Figure 2)
IF NGOP^O,
IGOP. - graphics options
1 1 =» particles
2 -=» zones
3 -* velocity vectors
4 -» contours
5 -* finish (this must be an included option)
45
-------
For Cube: code = 0
\
type A
m
type B
ra-
in-
type C
type D
For Diagonals:
code
type
-t-axis
m-axis
1
A
K
I
2
C
K
I
3
A
K
J
4
C
K
J
5
D
K
I
6
B
K
I
7
D
K
J
8
B
K
J
9
A
J
I
10
D
J
I
11
C
J
I
12
B
J
I
I: horizontal
J: fore-aft
K: vertical
Figure 2. Obstacle axis code.
46
-------
IF NOVP^O,
VNTPj
NCVTPi
IPMM
IPMm
JPPM
JPMm
KPMM
KPMm
IPPP
JPPP
KPPP
EPROJ
- contour options
1 -» pressure
2 -» u velocity
3 -» w velocity
4 -» v velocity
- number of contours per contour plot
- number of cuts in I for J * K plane
(vectors and contours)
- locations of cuts (i. e., I cell no.)
- number of cuts in J for J * K plane
- locations of cuts (i. e., J cell no.)
- number of cuts in K for I * J plane
- location of cuts (i. e., K cell no.)
particle plot projection for I
particle plot projection for J
particle plot projection for K
0 -> not this plane
1 <=* this plane
perspective view plotting flag
(not used at present so = 0)
0 -» no perspective
1 =» use perspective
plane
plane
plane
•47
-------
The particle subroutine reads in the rest:
!(if IPL = 0, skip particle calculation)
side of particle source block cell number
IPL - left
IPR - right
JPA - aft
JPF - front
KPT - top
KPB - bottom
XPCEN - x coordinate for particle point source location
YPCEN - y coordinate for particle point source location
ZPCEN - z coordinate for particle point source location
(if XPCEN = 0, particle will be placed at
random inside source block above)
NOPI - number of particles to input
PDX - size of cell in x direction for particle meshj
PD Y - size of cell in y direction for particle mesh | (not used
PDZ - size of cell in z direction for particle mesh) at present
TWO? - time when to input particles
UPI - particle velocity in x direction
VPI - particle velocity in y direction
WPI - particle velocity in z direction
SIGMA - diffusivity
VARMINT is constructed for 3-dimensional simulation. For efficiency
however, all arrays are singly dimensioned variables. For each time
cycle, progression through the mesh is ordered by columns (K), next
by rows (I), and finally by planes (J). The number of words per cell
is essential for correct machine locations of the flow field variables
operated on in each cell, for all positioning is done with indexing
counters.
' 48
-------
VRC is the control program of VARMINT. In VRC the problem date
and time are called and printed before the input and output files are
declared. The main array dimensions are read in along with the film
option flag for plotting. The error flag is initialized,5 and if the film
plot option is on, the graphics are initiated.
An array dimension determines the next logic sequence. If T is
negative, the problem terminates; if T equals zero, restart procedures
are instigated; if I is greater than zero, the value is assumed to be
the size of the real mesh in the I, or horizontal direction, and the
problem is initialized.
When the problem is started by this last method, the problem label is
the second data card read in. VRC then calls VRS to set up the problem.
After VRS returns control, VRC calls VRM, which is the main calcula-
tional subroutine. If the problem were to be restarted, using the
second method described above, VRC would call VRM immediately,
bypassing VRS because the values therein would have been stored on
magnetic tape which was generated during a previous run.
Following the calls to VRM and VRS the program checks the status of
the error flag. If this had been tripped within one of the subroutines,
graphics is concluded (when the plot flag is on), and the run is termi-
nated with a printed diagnostic. Otherwise, we return to the beginning
of the program and anticipate another problem. Hence, to end this run,
the next data card must contain the negative L This negative T sends
us to the section which concludes the graphics. We then exit, and the
run terminates.
49
-------
VRS sets up a specific problem for us. It reads the following from
data cards:
• The cell size constants for three dimensions,
• The prescribed body accelerations for 3 dimensions,
• The constants determining amount by which flux is
calculated by donor or cell centered method for
3 dimensions,
• The boundary condition flags for the 6 walls:
1 -* rigid
2 -» continuative
3 -» periodic,
• The relaxation factor for pressure iteration,
• The convergence criterion for pressure iterations,
• The time criteria,
• The diffusion constant,
• The obstacle no-slip flag, and
• The print options.
Then these input variables are printed.
VRS initializes constants and calculates cell index quantities. It'checks
for programming errors such as array dimensions exceeded. Whenever
errors are found, the error flag is tripped and control returns to the
calling program.
Next, VRS reads the obstacle limit cards and defines the obstacles in
the mesh. Obstacle surface flags are set for no-slip boundary conditions.
It reads and writes both graph options and contour plot values which are
needed by VRIO. Graphic quantities are read in subsequently. Fluid
flow arrays are initialized to zero. Velocities and cell sizes are
initialized to prescribe input values. This section allows for variable
zoning. Mesh coordinates are initialized. Cell volumes are calculated
with allowances with respect to the obstacles. Plotting constants are
set next with a call to FILMCO. Following this call, control is
returned to VRC.
50
-------
VRM is the main calculational subroutine. This is where the flow
field is established and from which I/O and particle input are controlled.
First, VRM initializes PMAX for the pressure iteration and defines
loop constants to be used in cell prints from VRIO. This section is
never returned to during the course of this problem.
The calculational sequence for every time step begins at statement
number 1000. Machine time for the cycle is noted and the pressure
iteration counter is set to zero. A time dependent flag determines
whether or not the hydro will be skipped; this flag allows passage on
initial cycle. If the flag is tripped, we bypass the wind field calcula-
tions and decide whether or not to call the particle movement sub-
routine VRPS. This subroutine is only called when the maximum
number of particles allowed in the system is greater than zero.
When the hydro is not to be bypassed, I must next be interpreted.
If I equals zero, we jump to the final section of VRM where VRTAP,
the tape I/O subroutine, is called. Hence, we are restarting a pre-
viously run problem. We read the dump tape until we arrive at the
time T from which we wish to proceed. The information from the
tape dump supplies all the parameters normally initialized in the
set up subroutine VRS as well as the state of the flow field at time T.
This data has been stored in the labeled common blocks VRCOM,
CONCEN, AMIX, FL, FLS, VRCON, MODSET, SAFLM and ZLAST
and are accessed by all subroutines and VRC. We return from
VRTAP to statement number 1000 and continue with I now containing
the integer number of computational cells in the x(or i) direction.
51
-------
When I is greater than zero, it is assumed that we also have the
number of computational cells in the y direction (J) and the z
direction (K). First, the boundary conditions are set for the right
and left walls, top and bottom walls, and then the fore and aft walls
of the mesh. For each of the above, assigned GO TO statements
route the user to the rigid, continuative, or periodic boundary con-
ditions. No slip boundary conditions can also be calculated for
these when the proper flag is set. Next, the free slip obstacle
boundary conditions are set, conditions depending on the obstacle
type. Following this, the no-slip boundary conditions for obstacles
are calculated, but only if the flag is set and the pressure iteration
is not in progress.
Once boundary conditions have been set, we are free to calculate the
tilde velocities. First, face No. 1 (j = 1 plane) and the left column
•
and bottom row of velocity components for computational cells are
transferred to storage arrays. Cell quantity indices are calculated
along with diffusion. The u-, v-, and w-tilde fluxes for each cell
are then calculated. Where
52
-------
2 At
"U
53
-------
V.,
2
54
-------
2 At
At
55
-------
These equations combine donor cell and cell-centered difference
schemes; a = l giving full donor cell, a= 0 yielding centered differencing,
a fraction in between promoting a combination thereof. Quantities
needed at positions other than when they are defined'are calculated as
simple averages. The donor cell method is suggested for most cases
because of the greater stability in the calculations without an additional
viscous term. The diffusivity constant, a , is an input quantity read
from data cards in VRS or from the dump tape in VRTAP.
The tilde velocities must now be modified to insure conservation of
mass. This is done in the pressure iteration section after boundary
conditions have again been set. The velocity divergence D is
calculated, and then the pressure is changed by an amount
6P = -0D
where
x TV I-
2At 1 2At
—1+
1-1 bDiJk _ 1 [ 1
*~VPiJk U^V^i
2AtJ^ 2At 1 2At
Ay. (Ay. + Ay. ,) y. (Ay.
J J J J J
.56
-------
and j3 is the relaxation factor. The velocities are then adjusted by
adding the 8P at the cell faces to the corresponding velocity component.
This process is repeated successively in all cells until no cell has a
magnitude of D greater than 6 , where € is a function of the magni-
tude of the velocities calculated.
When the pressures converge within 1000 iterations, the time counter
is updated by dt and VRM calls for cell prints in VRIO. These will
not necessarily be printed every cycle but rather at a predetermined
time interval TPRT.
The boundary conditions are reset. If particles are to be allowed in
the field, the particle movement subroutine VRPS is called. At
appropriate time intervals, the particles will move within the wind
field. This calling section is the first section of operation for the
time cycle when the hydro is bypassed. When the hydro has been
skipped, the timer is not updated, so the time is updated here under
these circumstances.
-------
The subroutine FILMCO contains entry points under which various
film constants are computed or tested. Also, the film frame is
appropriately labeled. A call to FILMCO checks that film option is
on, sets mesh length reciprocals, sets film coordinate starting
points, and initializes do loop constants. Entry at FPLAN computes
film plane coordinates and constants for the I * K, IXJ or JxK
plane. Entry at FCTEST performs a test to determine whether film
coordinates are out of range and resets those which are. Entry at
FLMID labels SAI, date, time and cycle and draws the frame.
The subroutine FLMCAL contains the calls to the Berkeley SC 4060
graphic routines. Entry FLMADV" records the display and advances
the film. Entry FLMFIN terminates the graphic output. Entry
FLMINIT initializes the graphics package. Entry FLMLAB labels
the film frame. Entry FLMLIN draws noncontiguous line segments.
Entry FLMPNT plots points. Entry FLMOPT computes graphics
options such as redefining the plot point, character size, line width
and density, and point size.
58
-------
The particle subroutine VRPS is responsible for the injection of
particles into the mesh; for initializing particle location and mass,
for initializing background, for particle chemistry (through calls to
KEM), for moving and editing the particles, and for printer and tape
output of particle information.
VRPS is called by the main calculations subroutine VRM when the
maximum number of particles allowed in the mesh (NPMX, an input
parameter) is greater than zero. The first pass through VRPS invokes
the read statements which initialize particle input locations and
specify point or area assignments, time when to input particles,
particle velocities in x, y, and z directions, the number of particles
to input per pass, and the diffusivity. Background and concentration
constants are set, as well as the probability curves used in the turbu-
lent dispersion section. If the time to input particles is greater than
the time of the initial call to VRPS, control is returned to VRM.
At the appropriate time interval, particles are emitted by one of two
methods: assignments to a point or an area. When the input variable
specifying the x-coordinate of the point location (XPCEN) is less than
or equal to zero, the area source is calculated from a given left, aft,
and bottom local boundary and a fraction of the corresponding
upper bounds, the fraction being determined by a random number
generator. When XPCEN is greater than zero, a point source loca-
tion is designated using the coordinates given in the input variables
(XPCEN, YPCEN, ZPCEN). The particle locations for both cases
are stored in the dimensional variables XPC, YPC, and ZPC. Up
59
-------
to ten species, or chemicals, are allotted to each particle location.
Each has an initial mass which is assigned to the variable PM, PM
increasing by the number of species. The number of particles
emitted each time is the same for each species and has been assigned
by the constant NOPI.
The first time that particles are emitted, masses and locations are
assigned, and then control is returned to VRM. Thereafter, entries
to VRPS bypass the initializing sequence and preassign the logic flow
for the marker particles and the turbulent dispersion equations. The
concentration array is zeroed out, if that option is on, and then begins
the major loop for the calculation of particle transport.
The location of each particle composite (or block), a composite con-
taining N species, is arrived at by taking the integer part of the
location variable. This tells us which cell the block is in. To find out
how far into the cell the block has progressed, we use the decimal
part of the same variable. Using the velocity components of the flow
field for this and adjacent cells, we interpolate the u, v, and w fluid
velocities for simple marker particle advection in the following manner:
define DC1 = DC + sgn (XW - . 5)*1. 0 , XW1 = XW - . 5
IY1 = IY + sgn (YL - . 5)*1. 0 , YLl = YL - . 5
' IZ1 =IZ +sgn (ZH - .5)*1.0 , ZHl = ZH - . 5
60
-------
where DC, IY, and IZ are the cell numbers in the x, y, and z
directions, respectively, and XW, YL, and ZH are the correspond-
ing fractions describing the distance the particle has moved within
y
the cell.
II, Jl, and Kl are the resulting cell numbers described by IXI,
IYI, and IZI, respectively.
The flux velocity components of the particle are:
Up =XW{(1-YL1)
+ (YL1)
(YL1)
+ -
.^.^)] } ;
Wp= ZH|(1-XW1)
(XW1)
(1-ZH){(1-XW1)
+ (XW1)
+
Vp = YL
J(l-ZHl)
(XH1)
+ -
+ -
(ZHi)
Second order particle advection can be calculated in like manner.
61
-------
Another option available is turbulent dispersion. The displacement
is added to the fluid velocity component and is defined as
6X = y4adt W sgn (l-2£) where £ is chosen randomly and
satisfies the conditions
T f
" Jo
W 2
"Z dz
O is the local diffusion coefficient;
dt is the time step;
W 1 - wt\
*£±± — * ) ( t - R ) such that 20k-values are
" k
W
are found for W = — — k , k=l , ... , #k and V? is an
upper limit such that particles will seldom shift more than one cell.
R, are solutions for
2 /.k 2
RT, = JW J e dz , R, < q < R^i
k v Q k k+i
The particle is then moved and given its new location as
U dt
XW =XW , . +
new old
V dt
YW =YW . , +
new old
W dt
ZW =ZW . . +
new old
62
-------
after a particle has been reflected off an obstacle or the ground if its
new displacement would have put it there. When a new displacement
places the particle outside of the mesh, its new location is not
retained. It is considered as outside the area of interest.
New concentrations must now be calculated for each cell. The
particle masses for each species are summed for the cell into which
the particle block has entered. This process continues until all
particles have been moved.
We then loop through all species and all mesh cells to calculate the
molecular concentrations for each cell from particle mass, cell
volume, molecular weight, ppm conversion factor, weight of air, and
background. At the appropriate time interval, these concentrations
are printed or written on tape.
Next, the cells alone are looped over and the chemistry subroutine
KEM is called to prescribe the reactions of the N species and
deliver a new concentration for each. Then, each particle is re-
weighted as a function of the ratio of old concentration to new
concentration.
If the time is less than the next time when to input particles, then the
time, cycle number, particle masses and locations, and the number
of particles in the system are printed and control is returned to VRM.
Otherwise, more particles, determined by the constant NOPI, are
added to the system at the predetermined source location and mass is
assigned to each one. Then the printing described above occurs and
control is returned to VRM.
63
-------
The current subroutine KEM solves the inorganic loop of the photo-
chemical air pollution reactions
kl
1. (hi/) + N02 4 NO + O
k2
2. O + (09) + (M) 3 0« + (M) (1)
£| O
k3
3. O3 + NO * NO2 + (02)
where h^ represents the ultraviolet photons of the solar flux, the third
body, M , is the total concentration of all species present and the k's
are the rate coefficients. The species in parens are assumed to remain
constant and their values are incorporated into the rate constants. All
concentrations are in parts per million (ppm). The values for k are
estimated by Eschenneder to be
k. = 0. 4 min
k2 = 2. 64 x 106 min"1
kg = 30-40 ppm"1 min"1
This set of reactions can be solved analytically as discussed below.
Noting that the value of k,, is much greater than either k. or k« , we
can assume that the second reaction is in pseudo-equilibrium when com-
pared to the other two reactions. Further, at equilibrium, we see that
the ratio of (X to 0 is k? or that atomic oxygen, O , is about one
millionth of the concentration of ozone, O« .
64
-------
Assuming that all oxygen is either CL or
-------
becomes
dN09
-^ = - k N09 + k.(NOY - NOJ(TOX - NOJ . (5)
Qt 1 & O J\. £t £t
Defining y = NO,, we can express Eq. (5) in the standard quadratic form
^JL- = ay2 + by + c (6)
where
b = - kx - kg (TOX + NOX)
c = kg • NOX • TOX
If the rate of change of y (NOJ is zero, the chemical reactions
Ct
are in equilibrium and we can solve for the equilibrium value of y .
_ - b ± //fa - 4ac
ye ~ 2a
Expanding the coefficients we find that the physically correct solution is
- 4ac
If the time rate of change of y is not equal to zero we must solve the
integral equation
yQ ay + by + c
66
-------
The solution for the integral
r
j
ay + by + c
r
is given as
2 "1 -2a+K b2 > 4ac
=r 5
xx it 99
(2ay + 1>) < b - 4ac
2 ctnh-i
rt 99
(2ay + b) > b - 4ac
where
Solving equation (8) we get
R = (b2 - 4ac)* .
for R ^ 0 , where f is either tanh or ctnh depending on the
2 2
value of (2ay + b) - b - 4ac, and C is a constant of integration.
We can solve for C by noting at t = 0, y = y so
9 .1 2avn + b
C =|f 1(— £ - ) . (11)
Solving Eq. (10) for y gives
y = (-b + R-f (- (t-C)R/2))/2a (12)
Note as t -» » , f -> 1 , and y-^y as t-^0 y = -b R-f(cR/2)/2a and by
Eq. (11) y = yQ .
67
-------
Subroutine VRIO handles the printing and graphics output. The first
section prints cell quantities and the second handles graphic output.
Each can be executed without entering the domain of the other.
The first section encountered is the line printer section. After the
subroutine is entered, if long print is not desired, if the hydro has
been skipped, or if printing is not desired at this time of entry, then
the printing section is skipped. Otherwise, VRIO sets up printer
counters and flags and writes the time, cycle number, the number of
iterations needed to converge during the pressure iteration, and the
time increment dt. At cycle zero, the cell coordinates and cell sizes
are displayed. Then and thereafter, the cell numbers and their
respective pressures and velocity components are printed.
The graphic section is entered whether or not long print is required.
Only the input flag for plotting being off can return control to the
calling routine at this intermediate step. Otherwise, control is
returned at the-completion of the requested plots.
The graphic section can be separated into several parts, each part
dealing with one of the available graphic options. After the options
are initialized, the option counter is incremented, sending VRIO to
either particle, mesh zone, velocity, vector, and/or contour plotting.
The first part plots particle coordinates, initializes constants
necessary in plotting the particles, and then sets constants for either
the I x K, the I * J, or the J * K plane. It is possible to obtain a view
from each of these planes, but each view must be a separate plot.
The mesh zone plot follows the same general set up for plane constants
as the particle plot. It then draws line segments to display the mesh
68
-------
from any of the three planes. Zones, however, will only be plotted
once; the first time VRIO is called for plotting purposes. The velocity
vectors and the contours are viewed from a specific plane, or cut,
•r
within the mesh. This is unique in that, while particles and zones are
viewed from only the front, top, or right of the mesh, looking across
all of the cells, the cut limits the view to a 2-D case. The user
designates his I x K, I * J, or J * K viewing plane and will receive
plots of only the quantity within that plane.
The velocity vector plot follows the same pattern of setting up con-
stants for vectors and film plane coordinates as the earlier parts. It
uses the velocity vector components appropriate for the particular
view plane in computing the coordinates between which it will draw a
line segment proportional to the magnitude and in the direction of the
velocity, neglecting .the velocity component normal to the plane/ It
then plots the vector.
The contouring plot again initializes general constants to plot the
contours on film. The number of contour lines is optional. It then
sets loop constants for the I*K, I*J, or J x K planes and encodes
the title and labels the film. It transfers the desired variable to a
contour storage array and computes the maximum and minimum
values of this variable. The contour loop flags each contour cell,
sets coordinates for the desired plane, and checks the contour
variable value in each cell with respect to neighboring cells. When
contour demarcation values are within this neighborhood, the vertex
position might need a shift. If so, it calculates volume constants,
contour cell volume, and the shifted vertex position. It then plots
the contour on film.
69
-------
After each plot, VRIO can go to various exits. It will either set up
for a new plane and plot the same quantity from a different view point,
or it will return to the beginning of the plotting section to increment
the plotting counter and, by means of a computed GO TO, go to another
plotting part. The last plotting option must always be included (now
equal to 5), because this directs logic flow to return to the calling
routine.
70
-------
VRTAP is called for magnetic tape manipulations. Upon entry, I is
interogated. If it is not zero, VRTAP proceeds to write mag tape
records of the buffer length, the common blocks starting from VRCOM
down to ZLAST, and the particle array stored in large core memory.
It prints the tape dump number, the program time, and the cycle
number for the user's reference.
When I equals zero, the subroutine assumes a restart request. It
rewinds the dump tape and reads records until it arrives at the tape
dump number requested by the initial K. It then prints the time,
cycle, and tape dump number. After this point, the users may insert
any desired changes to common stored information, such as time when
to finish. It then calls FILMCO and VRIO before returning to the
calling routine.
71
-------
(This Page Intentionally Left Blank)
72
-------
B. VARMINT LISTING
73
-------
(This Page Intentionally Left Blank)
74
-------
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