by
A. M. Teeter and D. J. Baumgartner
Marine and Freshwater Ecology Branch
Corvallis Environmental Research Laboratory
Corvallis, Oregon 97330
CERL PUBLICATION 043
November, 1978
CORVALLIS ENVIRONMENTAL RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
CORVALLIS, OREGON 97330
Page
.
Abbreviations, Symbols, etc.
Acknowledgments
1.
• •
Introduction
General Requirements
Mixing Zone Concepts .....
Ocean Discharges .
Buoyant Plume Models
• •
.
.
..
... . ..
..
• .....
•
.........
..............
Summary and Conclusions
General. . .. . . . . . . . ..... . ......... .
Simplified Results for Plumes in Stratified Ambients . . . .
Jets in a Stagnant Environment.
Jets in a Flowing Environment
Merging or Slot Jets in a Stagnant Environment
Merging or Slot Jets in a Flowing Environment
Non-Uniform Stratification
B.
Recommendations
Methods
Appropriate Conditions
Mixing Zone Specification
4.
PLUME Model
Theoretical Development ..
..
Model Description. . . ...... . . .
Example Input, Output and Model Listing.
5.
OUTPLM Model
Theoretical Development and Model Description.
Example Input, Output and Model Listing.
DKHPLM Model
Theoretical Development. .
Model Description.
. . . .
Example Input, Output and Model Listing.
. . . ...
LIST OF ABBREVIATIONS, SYMBOLS, ETC
C
Radius or characteristic half-width of the plume.Concentration by volume, = 1/Sa.
Diameter of discharge port.
E
-
Entrainment.
Froude .number, = U./(g D ©o/p0)1
Fn
/2
Gravity.
Volume rate of discharge.
-
Volume rate of discharge per unit length of diffuser.
r
-
Radial distance from the plume's centerline.
s
-
Distance along plume axis.
Sa
Flux averaged dilution or the total plume efflux divided by the volume
of the discharge.
Length of the zone of flow establishment.
T
Temperature.
T'
Fluctuating component of T.
U
Ambient current speed.
U.
-
Initial velocity of the jet.
u-
Fluctuating component of V.
V
Velocity at some point in the plume in the s direction.
Velocity normal to the plume's axis.
Fluctuating component of v.
Horizontal distance in the direction of a jet discharge and
currents or the horizontal distance parallel to a diffuser.
y
-
Horizontal distance 90° to x.
z
-
Vertical distance from the discharge depth.
Z
Depth of water.
-
Normalized density disparity, = (p. -
p)/(p. - pd).
pd.
A. -
Initial density disparity of the waste, = Po -
Oh -
Maximum height of rise or height of rise to the trapping level.
p
-
Density of the plume at some point.
-
Ambient density at the level of the discharge.
-
Density of the discharge.
Pd
p ,Ambient density at some level.
dp/dz - Absolute value of the ambient density gradient.
1 m3 = 35.31 ft 3 = 264 gallons.
ACKNOWLEDGMENTS
The assistance and cooperation of Drs. Lorin Davis, Larry Winiarski, and
Safa Shirazi, and Mr. Walter Frick of the Ecosystem Modeling and Assessment
Branch of the U.S. Environmental Protection Agency's Corvallis Environmental
Research Laboratory are gratefully acknowledged. A similar acknowledgment is
due Bill McDougal of the Marine and Freshwater Branch.
SECTION 1
INTRODUCTION
Initial dilution is the rapid turbulent mixing between wastewater discharged at depth in the ocean and seawater. It takes place close to the point
of release, resulting from physical interactions between these components.
The understanding of this process can be used to predict the initial dilution
attained under given conditions. Such information enables comparison of
wastewater constituent concentrations, subsequent to initial dilution, to
water quality standards and thus can assist in evaluating the discharge's
effect on the marine environment.
The purpose of this report is to present procedures by which applicants
and reviewers can calculate initial dilution and describe the zone of initial
dilution surrounding the discharge site. These calculations would be required
under regulations proposed by the U.S. Environmental Protection Agency on
April 25, 1978 (1) implementing section 301(h) of the Clean Water Act (2),
draft regulation implementing section 403(c), and provisions of the State of
California's Ocean Plan (3).
MIXING ZONE CONCEPTS
When wastewater is injected into the ocean, it has buoyancy and momentum.
This energy is used in an initial mixing or dilution process. The wastewater
discharge forms a plume which rises and entrains ambient water and is thus
diluted. The mixing process slows markedly when plumes reach a position of
equal buoyancy and momentum with the ambient (no relative energy) or the
surface. Then the dilution process becomes more dependent on ambient oceanic
processes. Thus, the mixing zone is thus an area where rapid mixing takes
place between waste and ambient waters. It can be thought of either as a
concentration isopleth or as the zone where initial mixing takes place. It is
not a treatment for wastes but a transition region (4). By insuring that
water quality standards are not violated outside the mixing zone, impacts on
the marine environment from the waste discharge should be controlled. Of
course, the dilution of some pollutants may have little effect on their impacts. In general, however, biological uptake, concentration, and deposition,
and physical processes, like aggregation and sedimentation, are concentration
dependent.
OCEAN DISCHARGES
Ocean discharges of municipal wastes have a wide variety of physical
characteristics which can affect initial dilution. Although single port
outfalls are still used, many discharges now are from multi-port designs to
optimize the dilution and vertical rise attained by their plumes. The numbers
of ports and their orientations can form complex patterns reflecting a great
deal of design effort. The discharged material is almost uniform, with physical characteristics similar to those of fresh water. The marine environment
varies greatly along the miles of our country's shoreline. Physical conditions at the discharge site also have a great effect on initial dilution. The
ambient receiving waters are of a higher density than the waste discharge and
are typically vertically stratified. Currents are usually present but may
have periodicity caused by tidal and other forces.
Discharges initially have momentum and buoyancy as they are injected into
the environment. Densimetric Froude numbers (Fn = U/(g D A /po)
1 /2
) describe
the ratio between these two forces. The higher the initial Froude number, the
more closely the resulting plume resembles a momentum or forced jet. The
smaller the number, the more the plume resembles a purely buoyant element.
Because the density difference between the waste and the ambient varies only
slightly and jet velocity is bounded by practical considerations, Froude
numbers for ocean discharges generally vary between 0 and about 30. The ends
of this range are characteristic of single-port and multi-port diffusers,
respectively. In this range of Froude numbers, buoyancy is likely to dominate
the initial mixing process, making volume rate of discharge, currents, ambient
stratification, and possibly water depth, important. Especially in a current,
buoyancy dominates since the plume's momentum becomes insignificant compared
to the momentum entrained from the ambient.
Coastal discharge sites vary in depth up to about 75 meters (currently).
The water columns at these sites are typically stratified vertically by differences in temperature and/or salinity. This leads to density stratifica
tion. The amount or degree of stratification varies widely in time and space.
Proximity of large sources of fresh water, surface heating, advection, and
wind mixing all can influence density stratification. Salt wedges can form
near the mouth of estuaries with large fresh-water flows. Advection and
nonlinear interactions between wind-induced surface turbulence and buoyancy
gradients also can produce pycnoclines, areas of strong density gradients or
discontinuities. Strong pycnoclines can arrest the vertical motion of buoyant
plumes, ending the initial mixing process. Winds of sufficient duration can
erode the density gradient by the turbulence they produce, leaving the ambient
well mixed.
Currents are very complex in coastal areas. These are areas of high
energy received from a number of processes. Currents at discharge sites have
tidal components which can be strong. Oceanic currents impinge on the coast
and intensify on the western edges of the oceans (the Florida Current for
instance). Local forces such as wind shear and waves also generate currents.
Instantaneous current values (time scale of minutes), rather than long-term
(time scale of hours, days, etc.) net currents, are important in the initial
dilution process.
BUOYANT PLUME MODELS
Ocean outfalls can be modeled by properly scaled hydraulic models, or by
mathematical models. The methods described here include only the latter,
although in cases of complex geometry or special conditions, physical or
hydraulic modeling may be appropriate. The characteristics and behavior of
plumes and jets in other aquatic settings, the atmosphere, as well as for
ocean outfalls, have been studied by means of mathematical models. Many
excellent reviews have been published (5, 6, 7).
Early work in this field was concerned with purely buoyant convection (no
initial momentum). Rouse et al. (8) studied the buoyant plume above a continuous heat source in a uniform, stagnant environment. Equations of mass continuity, momentum, and energy were integrated by assuming Gaussian approximations for the lateral distributions of velocity and temperature across the
plume and with experimentally determined spreading constants. The results
were used to describe the distribution of velocity and temperature as functions of mass flow and height of rise. Priestly and Ball (9) studied thermal
plumes issuing into a stratified, as well as uniform, environment. They used
essentially the same equations as Rouse et al. for mass, momentum, and energy,
but their lateral Gaussian profiles of temperature and velocity assumed a
lateral length scale.
Morton et al. (10) studied buoyant plume convection. They introduced an
entrainment function based on local characteristic plume velocity and length
scale to replace the spreading ratio. This approach has been adapted by many
later investigators (5). Later, Morton (11) assumed other lateral profiles of
velocity and temperature to account for the difference between eddy transport
of heat and momentum. Another coefficient was introduced and estimated from
experimental data.
Abraham (12) investigated neutrally-buoyant forced plumes, purely-buoyant
plumes, and plumes with combinations of both properties. Equations of mass,
momentum, and energy were integrated using lateral Gaussian profiles recognizing the difference in transfer rates between momentum and heat (or concentration). Entrainment was considered by integrating the continuity of mass
equation with appropriate spreading coefficients. Abraham's results included
analytic expressions for plume centerline velocity and concentration as func
tions of initial plume characteristics, height of rise, and entrainment coefficients for momentum and heat. The limiting cases of neutrally-buoyant
forced jets and purely-buoyant plumes were considered.
Fan (13) reported on plumes issuing at arbitrary angles into a stagnant,
stratified environment. Velocity and buoyancy profiles were used similar to
those used by Morton (11) and included a lateral characteristic length.
Solutions were found by simultaneously solving six equations (continuity of
mass, vertical momentum, horizontal momentum, conservation of buoyancy, vertical geometry, and horizontal geometry).
Fan (13) also investigated buoyant plumes in a flowing, uniform-density
ambient. A theoretical solution was given as a function of plume drag forces
and an entrainment function. The entrainment function was such that drag and
entrainment coefficients varied with plume conditions, a disadvantage in
applications beyond the range of experimental validation (14). Abraham (14)
introduced an entrainment function based on subdomains influenced by initial
momentum and buoyancy. With entrainment coefficients defined for these two
subdomains and a single coefficient describing the drag force, the theoretical
solution was in good agreement with the experimental results of Fan.
Plumes from multi-port discharges have been studied less extensively.
Cederwall (15) studied the flow regimes of line sources for discharges in
confined and un-confined environments. Roberts (16) studied the flow regimes
of line sources in unstratified and unconfined environments. Sotil (17) posed
a mathematical model for slot jets or continuous line sources in stagnant,
stratified ambients. Koh and Fan (18) posed a mathematical model of multiport diffusers by interfacing single jet and slot jet solutions at a transition point. A good review of this subject is given by Davis (7).
Three models are described in this paper. The PLUME model by Baumgartner
al. (19) simulates a solitary plume in a stagnant environment of arbitrary
stratification. OUTPLM by Winiarski and Frick (20) models a single plume in a
stagnant or flowing environment of arbitrary stratification. DKHPLM by Davis
(21) describes single plumes in flowing, stratified ambients which are allowed
to merge with identical adjacent plumes to simulate closely spaced, multipleport diffusers. The results of these models have been used to develop analytic solutions to describe the initial dilution of ocean discharges in terms of
the principal quantities involved. These three simulation models and the
analytic solutions resulting from them will be presented along with sugges
tions and example applications.
SECTION 2
SUMMARY AND CONCLUSIONS
GENERAL
The plume attributes of interest here are the dilution attained at the
end of convective ascent through ambient waters, and the plume's position.
For our purposes, the poorest or lowest dilutions are important. Dilution is
defined as the total volume of a parcel or sample divided by the volume of
waste contained within it. This is equivalent to the inverse of the volume
fraction (or concentration) of waste that a parcel or sample contains.
The behavior of buoyant plumes can be mathematically modeled by properly
considering mass, momentum, and energy. Some form of entrainment function
must be assumed and fitted to experimental data. Other assumptions generally
made are that all flows are steady and incompressible, pressure is hydrostatic
throughout, the plume is fully turbulent and axisymmetric, and that turbulent
diffusion dominates and is significant only in the radial direction. Distri
butions of velocity and concentration may also be assumed. The physics of
this problem can be solved in various ways. Systems of differential equations
can be integrated across the plume, reducing the equations to one independent
variable, length along the axis of the plume. Difference equations can be
used with Lagrangian coordinants. Simultaneous equations or three-dimensional
numerical schemes also have been used.
Simulation models of buoyant plumes are systems with internal variables
(mass, momentum, and energy), external variables (discharge characteristics,
ambient vertical density, and currents) and internal mechanisms (entrainment).
These models have behavior which is affected by principal quantities. Computer model studies can be used to describe plume behavior of interest in
terms of effecting principal quantities. For stagnant conditions, the principal quantities are initial density difference between the waste and ambient
(to), ambient stratification (dp/dz), and flow rate or flow rate per unit
length for multi-port diffusers (Q or q). Current speed (U), if considered,
is also a principal quantity. Froude number is not important to the dilution
attained in the neighborhood considered (0 < Fn < 30), especially in currents.
The Froude number and orientation of the discharge affect the plume's trajectory in stagnant conditions but have only a small effect in a flowing
ambient. The spacing of multiple ports, for a given discharge rate per diffuser length, does not significantly effect the dilution attained by merging
plumes in a flowing ambient. In stagnant conditions, port spacing has a
moderate (perhaps 20 percent) effect on initial dilution.
Closed-form solutions for the behaviors or attributes of interest (dilution at maximum height of rise and height of maximum rise) have been found in
semi-theoretical, state-transition modeling. These solutions have been mapped
or tuned to the behavior of the computer models. Plume behavior can, therefore, be solved analytically in terms of the principal quantities. These
solutions also allow the importance of the principal quantities to be seen
directly. The computer models are still useful for detailed analysis or
special conditions. In many cases, however, the analytic models are quite
adequate to calculate dilution and maximum height of rise.
The discharge configuations which will be considered include single
ports, and multiple ports spaced such that they interact at some point during
convective ascent. These configuations are termed, respectively, plumes, and
merging plumes. Some other restrictions and assumptions must be made. The
discharge site is assumed to be at least ten port diameters deep and unconfined. Flow in the ambient is assumed to be uniform. Each of the three
models used in this study is suited to certain combinations of conditions.
Recommendations are given in Table 1.
TABLE 1. RECOMMENDED MODEL APPLICATION.
Discharge
Configuration
Single Plumes
Ambient
Density
Ambient
Currents
Stratified Stagnant
H
II
II
Flowing
II
H
Uniform
H
Stagnant
Flowing
Merging Plumes Stratified Stagnant
U
11
H
Flowing
11
II
Surfacing
Plumes?
Analytic
Models
Yes
No
Yes
No
Yes
Yes
X
X
X
X
X
X
Yes
No
Yes
No
X
X
X
X
Computer Models
PLUME OUTPLM DKHPLM
X
X
X
X
X
X
X
X
SIMPLIFIED RESULTS FOR PLUMES IN A STRATIFIED AMBIENT
Analytic solutions are given in this section for dilution and height of
rise of plumes and merging plumes. For stagnant conditions, the height of
rise is taken to the trapping level. The trapping level is where the plume
first loses its positive buoyancy. Above this level the plume is expected to
"pool" and spread horizontally. For flowing conditions, height of rise is
taken to the centerline of the plume where it first becomes horizontal.
Dilutions predictions are made for these locations.
The results described cover ambient conditions including vertical density
stratification, both stagnant and flowing, and the previously described discharge configurations. Vertical density stratification is normally found in
the marine environment and, since it adversely_ affects dilution should be
considered. A summary of the analytic solutions covering the conditions and
configurations listed in Table 1 will be given. Following these, sections on
non-uniform stratification and single plumes in uniform-density ambient are
presented. SI units (kilograms, meters, seconds) are used.
Plumes in a Stagnant Environment
The dilution for a single port discharge into a stably-stratified and
quiescent ambient is given by:
Sa = 0.46 Q
-1/4
A0
3/4
(dp/dz) -5/8(
1 )
The height of the trapping level can be determined by:
Ah = 4.6 (Q 40) 1/4 (dp/dz) -3/8(2)
where Sa is the flux averaged.dilution at the trapping level.
The dilution of the plume is most affected by initial density difference
(6,o) and ambient density gradient (dp/dz), and secondarily, by the volume rate
of discharge (Q). Flux of buoyancy and ambient density gradient both affect
height of rise. The height of rise (Ah) given by equation (2) is a maximum
value for the trapping level. This expression does not consider the angle of
discharge from the vertical or the Froude number which can reduce the actual
trapping level height. If, in application, it is found that the height
of
rise given by equation (2) exceeds the water depth (Z) at the discharge site,
dilution can be calculated by:
Sa = 0.10 Z Q-
1/2 A01/2
(dp/dz) -1/4(3)
This relationship assumes linear dilution with height
of rise which is not
strictly observed in stagnant conditions. As the water depth becomes smaller
than Ah, equation (3) predicts dilution which becomes less than those predicted by the PLUME program.
should be used.
If greater accuracy is needed the PLUME program
Equation (1) is in good agreement with model results for jets with Froude
numbers of up to about 50. For jets with Froude numbers above this value the
PLUME program should be used.
Example
-A discharge has a maximum, flow rate of 0.10 m 3/sec and a density of
1000 kg/m3 . The ambient at the discharge site has a density at the surface of
1020 kg/m3 and, at 30 m depth, a density of 1025 kg/m 3 during periods of
maximum stratification. From equations (1) and (2):
Sa = 0.46 (0.10)
-1/4
3/4
-5/8
(25) (.166)
= 28
Ah = 4.6 ((0.10)(25)) 1/4
.166) -3/8
= 11.3 m
If the depth over the discharge were only 10 m, then from equation (3):
Sa = 0.10 (10.) (0.10)
-1/2
(25)
1/2
(.166)-1/4
= 25
Plumes in a Flowing Environment
The dilution and height of rise of a single plume discharged into a
stably-stratified, flowing ambient are:
Sa = 3.0 (U/Q) 1/3 (A0/(dp/dz)) 2/3(4)
Ah = 2.33 (A. Q/U(dp/dz)) 1/3(5)
The dilution of the plume is most strongly dependent on the ratio of initial
density difference to ambient density gradient. Neither the dilution nor the
height of rise are sensitive to discharge angle or typical Froude numbers.
The maximum height of rise is for the centerline so that the plume actually
extends one radius up from this level. The plume radius can be estimated by:
B = (SaQ/nU/
1/2.
If Ah + B exceeds the depth of water at the discharge site,
the computed dilution can be corrected by:
Sa (corrected) = Sa Z/(Ah + B)
(6)
where B is the plume's radius or half width. By substitution into equation
(6), the dilution of a plume which reaches the surface is:
Sa = 0.91 Z (U/Q) 2/3 (A0/(dp/dz)) 1/3(7)
Example-In the previous example, A. = 25 kg/m 3 , Q = 0.10 m 3 /sec, dp/dz =
0.1667 kg/m 3 /m. If the lowest 10 percentile of current speed observations is
0.10 m/sec, what is the dilution and height of rise?
1/3
2/3
Sa = 3(0.10/0.10) (25/.1667)
= 85
Ah = 2.33((25)(0.10)/(0.10)(.1667))1/3
= 12.4m
The radius can be estimated as: B = (56(0.10)/n(0.10))
1/2
= 5.2 m
If the water at the discharge site is only 10 m deep, dilution as corrected by equation (6) is:
Sa (corrected) = 85(10)/(12.4 + 5.2
= 48
Alternately, dilution can be re-computed using equation (7):
Sa = 0.91 (10) (0.10/0.10)
2/3
(25/.1667)1/3
Merging Plumes in a Stagnant Environment
Brooks (22) gives expressions for the dilution and height of rise of slot
plumes in stagnant, stratified ambients. Slightly re-arranged, re-dimen-
sioned, and tuned to DKHPLM for consistency with the other analyses presented
here, dilution can be predicted by:
Sa = 0.32 q-1/3 Ao
2/3
(dp/dz)
-1/2
(8)
The height of rise of the merging plume is:
Ah = 5.85 (q
©0)1/3
-1/2
(dp/dz)(9)
The height of rise is predicted from equation (9) is the maximum trapping
level height.
Widely spaced plumes which merge after a significant rise will attain a
higher dilution (perhaps 20 percent) than predicted by equation (8).
If the height of rise of the plumes exceeds the depth of water, dilution
can be calculated by:
Sa = 0.055 Z q
-2/3
Do
t/3
(10)
This dilution will be lower than predicted by simulation model DKHPLM because
of the assumed linearity of dilution with height of rise and because the
trapping level is assumed to be the maximum. If more accuracy is needed,
DKHPLM should be used.
Example
-A diffuser discharges 0.02 m3 /sec/m in 50 m of water. If the efflu
ent has a density of 1000 kg/m 3 and the ambient density gradient is 0.05
kg/m3 /m, the dilution by equation (8) will be:
Sa = 0.32(0.02) -1/3 (25) 2/3 0.05)1/2
= 45
The maximum height of rise to the trapping level, from equation (9), will b
Ah = 5.95 ((0.02)(25)) 1/3 (0.05)1/2
= 20.8 m
If the water depth were 10 m, the dilution, from equation (10), would be:
Sa = 0.055 (10) (0.02) -2/3 (25)1/3
= 22
Merging Plumes in Flowing Environment
For multi-port diffusers where plumes interact or for slot plumes, the
Principal quantities which affect dilution and height of rise of the plume are
initial density difference (AO, discharge rate per unit length of diffuser
(q), current speed (U) and ambient density gradient (dp/dz). The dilution
from these discharges can be found by:
Sa = 2.75 (UAo/q(dP/dz))1/2
(11)
and the height of rise by:
Ah = 1.25 (6.0 q/U(dp/dz)) 1/2(12)
Like single plumes, the dilution and height of rise of multiple plumes in a
current are not sensitive to Froude number and discharge orientation. Apparently the port spacing or level of interaction also have little effect on the
values ultimately attained by the plume.
If currents are not perpendicular to the diffuser, the discharge rate (q)
can be adjusted to consider the length of the diffuser projected by the current. Below a relative angle of about 45 degrees, prediction
on stagnant conditions.
The rate of dilution is normally about linear with height of rise so that
equation (6) can be used to correct the dilution if the height of rise plus
the plume half-width exceeds the water depth at the site. The half width of
the plume can be estimated by B = Sa q/2U. The dilution of merging jets
reaching the surface can also be calculated by:
Sa = 1.0 Z U/q
(13)
These analyses, as well as DKHPLM, assume that the diffuser length is very
long compared to the height of rise. Otherwise edge effects complicate the
problem.
Example
-A discharge of 12.5 m 3/sec (N 300 MGD) is made in 50 m of water.
The ambient density during maximum stratification varies from 1024 kg/m 3 at
the surface to 1025 kg/m 3 at the bottom. The diffuser is 1250 m long so that
the discharge rate per unit length of diffuser (q) is 0.01 m 3/sec/m. Density
of the effluent is 1000 kg/m 3 . If the lowest 10 percentile of current speed
occurrences is 0.05 m/sec, the dilution will be about:
Sa = 2.75((0.05)(25)/(0.01)(0.02))1/2
= 217 And the expected height of rise will be:
Ah = 1.25((25)(0.01)/(0.05)(0.02))1/2
= 20 m
If the discharge site were only 20 m deep, equation (13) predicts the dilution
to be:
Non-Uniform Stratification
The preceding analyses are for uniform, or monotonical, density gradients. Often density gradients deduced from field measurements are not strictly
monotonical but can be approximated as such. For cases where the vertical
density distribution is interrupted by a sharp discontinuity or pycnocline,
other approximations are necessary. Brooks (22) gives one method. The height
of rise can first be estimated by using the density gradient of the lower
layer. If this height of rise extends to the upper layer, the density gradient is taken as the average gradient to that point. It may be necessary to
re-estimate maximum height of rise if the first adjustment resulted in a large
change in the average density gradient. The density gradient can then be
re-estimated. By this method, an average density gradient over the expected
height of rise of the plume can be estimated and dilution calculated. If the
density structure of the ambient is very complex, or if more accuracy is
needed, then PLUME or OUTPLM models should be used.
Uniform Density Ambient
Conditions may exist at a discharge site which mix the ambient to a
uniform state. Normally these are not the conditions of interest since ambient stratification is more critical to the initial dilution process. The
analytic solutions for single plumes in stratified ambients, because of their
form, approach infinity as the ambient density gradient approaches zero.
Thus, they may over-estimate dilution when stratification is slight. Different analytic solutions are necessary for single plumes in uniform ambients.
The analytic solutions for merging plumes in stagnant and flowing conditions,
equations (10) and (13), are applicable to uniform density ambients.
Analytic solutions will be given here for cases of single plumes in
stagnant and flowing ambients with little or no stratification. For a single
plume in a stagnant, uniform ambient:
Sa = 0.074 Z
513 Q-2/3
For a single plume in a flowing, uniform ambient, dilution can be calculated
Equation (15) should be applied when:
(dp/dz)/A0 < 0.194 Z -3 U -2 Q
(16)
No criteria is presented for the application of equation (14). This is
because of the uncertainty in predicting dilutions in stagnant, stratified
ambients when the height of rise is limited and much less than the trapping
level height. In very slight stratification, this is the case. However, in
either the flowing or the stagnant case, results from equation (14) or (15)
should be compared with equation (3) or (6) when stratification is slight.
The lower result is correct since stratification should reduce the dilution of
single plumes.
SECTION 3
RECOMMENDATIONS
METHODS
The buoyant-plume phase of waste dispersion can be described by a variety
of published methods. However, to evaluate of water quality impacts of municipal ocean discharges for permitting and other purposes, the methods presented
here are well suited and recommended for consistency. The analytic emthods or
computer models can be used as the situation warrants or requires. When
especially high resolution is necessary, the computer models should be used.
For unusual situations or conditions, other methods will be considered.
Although these methods are considered reliable, they should be continually evaluated evaluated to theoretical developments and measurements, especially quality field measurements.
APPROPRIATE CONDITIONS
The dilution achieved during the initial mixing process is dependent on
ambient and discharge conditions and is, therefore, highly variable. In
evaluating a discharge's effect on water quality, the appropriate conditions
to consider are those which result in the lowest dilution and those which
occur at times when the environment is most sensitive. For example, minimum
dilution will be predicted using a combination of maximum vertical density
stratification, maximum waste flow rate, and minimum currents for a particular
site. Other situations may be more critical to water quality depending on the
ambient water quality, and applicable water quality standards.
This approach relies on the availability of quality measurements or data
with which to estimate absolute worst case conditions, and is therefore not
without problems. There are inherent biases to some, oceanographic measurements. For example, current measuring instruments have finite thresholds. It
therefore becomes difficult to distinguish low values (which may be as high as
0.05 m/sec) from zeroes with these data. In estimating environmental conditions, a more reliable estimation can be made at the lowest 10 percentile of
occurrences. Data on ambient density structure are not collected by remote
means so that the frequency of observations is generally low. To increase the
reliability of worst-case estimates, data should be evaluated not only from
the discharge site but from nearby coastal areas of similar environmental
setting. Given the sensitivity of dilution to environmental conditions and
the ranges of, these conditions, the evaluation of data and selection of model
input conditions are critical to the proper evaluation of water quality impacts from municipal ocean discharges.
The characteristics of the discharge needed for evaluation are:
a)
port sizes, spacings, and orientations,
b)
flow rates through ports,
c)
wastewater density, temperature and apparent salinity at the point
of discharge.
Apparent salinity refers to the major inorganic ions of the wastewater
which contribute to its density. It has been found that a salinity, based on
conductivity ratios for seawater, gives an acceptable value. Alternately, an
apparent salinity which corresponds to a measured density can be used. Ap
oparent salinities of municipal effluent generally range from 1 to 5 /oo._
In unusual cases, the ports may resemble thin (< 1/10 D), sharp-edged
orifices. In these cases, the port diameter may be taken as 0.62 times the
actual diameter of the orifice and the discharge velocity adjusted accordingly. This will increase the Froude number of the discharge. Where the
orifice is rounded or thick, as is generally the case, no correction should be
made.
As mentioned, the principal quantities to consider in dilution prediction
are volume rate of discharge (Q) or volume, rate of discharge per unit diffuser
length (q), ambient stratification (dp/dz), initial density difference between
waste and ambient (AO, and ambient currents (U). These quantities should be
considered at:
a)
times of seasonal maximum waste outflow,
b)
times of seasonal maximum and minimum stratification,
c)
times of low ambient water quality,
d)
times of low net circulation or flushing, and
e)
times of exceptional biological activity.
The quantities selected to represent these periods should reflect conditions
of worst (or worst 10 percent) occurrence during these periods; highest volume
rate of discharge, extreme ambient stratification, lowest initial density
difference, and lowest current speed.
MIXING ZONE SPECIFICATION
The mixing zone is a zone of rapid mixing resulting from the buoyancy and
momentum of the waste material. Conceptually, it terminates when plumes reach
the end of convective ascent (buoyancy equilibrium without vertical velocity
or surface encounter). Currents greatly affect the horizontal extent of
convective ascent. Defining a zone in which convective ascent always occurs
is impractical and unnecessary. Ambient conditions could not be specified
with enough certainty to include the maximum distances which might occur and a
great deal of analysis might be necessary to find the combination of conditions which fosters the maximum horizontal extent. The predicted dilutions
used in evaluating water quality impacts are for conditions which will result
in small mixing zones. If the mixing zone were specified using these conditions, lower dilution than predicted might occur at the mixing zone boundary
at times when the plume trajectory becomes more horizontal. However, the
mixing zone should always bound the lowest or most sensitive dilutions and
thus define the area of possible water quality degradation and waste contact
with organisms.
The mixing zone should be defined in a practical sense as the area always
bounding the minimum or most sensitive dilution. It can be specified as the
radial distance equal to the height of rise of the minimum, or most sensitive,
dilutions predicted, measured horizontally from the outfall diffuser or port.
Its extent, therefore, will be intermediate to the conceptual definition and
the conceptual definition applied to the minimum or most sensitive dilution
prediction. This distance will often equal the depth of water at the discharge site.
SECTION 4
"PLUME" MODEL
THEORETICAL DEVELOPMENT
The computer model PLUME (5, 19) considers a buoyant plume issuing at an
arbitrary angle into a stagnant, stratified environment. Two zones of plume
behavior are considered. Profiles of velocity and concentration are considered to be uniform or "top hat" at the port orifice. The region required to
develop fully established profiles is the zone of flow establishment. Beyond
this point, similarity profiles are assumed for velocity and concentration.
This allows the equations for mass continuity, momentum, and density disparity
to be integrated across the plume, reducing them to one independent variable,
the length along the axis of the plume.
Zone of Flow Establishment
The length of flow establishment (s e ) has been found to depend on the
initial Froude number (12). As initial Froude numbers approach infinity, s e =
5.6 D. At low Froude numbers, this distance can be much shorter. The centerline value of concentration is assumed equal to the initial discharge condi
tions. The inclination of the plume's axis at s e is determined by linear
interpolation using 6, s e , and Fn. The centerline velocity at s e depends on
the Froude number and, at low Froude numbers, may be greater than the initial
velocity. To avoid estimation, the velocity at s (V ) is determined by s
e me
e
and empirical spreading coefficients.
momentum,
3 3 3/2
d/ds (Vm s /k ) = 3 DUPO-P )/P V m A
m m
pk
3/2
sine
(23)
density disparity,
d/ds (Vm
2
AM s
/k(p + 1)) = (1/(Po-Pd)) V m s /k-dp./ds
(24)
conservation of pollutant,
V C0/4
VmC ms2 /k(p + 1) =
(25)
The equations can be cast dimensionlessly as:
momentum,
04.10/2.s
) ^ R* sine
dE*/ds* = 3/Fn -((1
(26)
density disparity
dR*/ds* = s*/k
1/2
E*1 3
(dp/ds*)
(27)
conservation of pollutant,
Cm
= (k
1/2
1 3 *
(p + 1))/(4E* / s )
where s* = s/D, V* = V m/Vo, E* = (TM' s * /k 1/2 ) 3 ,
E*
1/3
Om S*/k
1/2
(28)
= V* A
m m
*2/k(.1 4. 1) =
(p + 1), and P.* P./(Po-Pd)
The initial conditions for the zone of established flow are obtained by evaluating E* and R* at s* = s eIt is assumed that ambient stratification is
negligible in the zone of flow established so A m = Cm and Re * = 1/4. Since
C
me
= C0 E * = (k 1/2 (p + 1)/4 s *) 3 . The values of p and k are mean values
e
of the limiting cases of purely-buoyant and purely-momentum jets found by
Abraham (12).
The angle of the plume (0) can be evaluated by considering conservation
of horizontal momentum,
d/ds j VV2cost) r d
which combined with equation (1J) yields
= cos
1
ft
((E */E*)2/3
(30)
MODEL DESCRIPTION
The zone of flow established is first solved depending on the Froude
number:
s* = 2.8 Fn2
.3
Fn
<
2 < Fn < 3.1
s* = 0.113 Fn 2 + 4.
s* = (5.6 Fn
2
)/(Fn
4
2
+ 18)1/2
Fn > 3
e
Then other initial conditions, 0 , E *, and R *, are solved for, Equations
e
e
7
(26) and (2T) are then solved at steps along the plume's axis by Runge-Kutta
A
approximation. E* is used to calculate the inclination 0 and concentration C.
Then the position of the plume's axis can be incremented by dx/ds = cos0 and
dz/ds = sine, where x and z are horizontal and vertical coordinates, respectively.
The centerline concentration is inverted to describe the centerline
dilution. To convert the centerline dilution to a flux averaged dilution, the
distribution of concentration must be weighted by, distribution of velocity.
With the assumed distributions, the flux average dilution can be found by:
Sa = 1.77 (1/C m ) or = 1.77 S m(31)
where Smis the centerline dilution from PLUME. Calculation of dilution is
terminated when the density of the plume equals the density of the ambient.
This is normally somewhat below the maximum height of rise but is where sim
ilarity ends and above which the plume will spread and become passive, possibly interfering with further dilution.
Computation of time and plume diameter have been added to the original
program. Those changes are set off by asterisks in the program listing given
here. The centerline velocity, averaged over each step length, is divided
into the step length to obtain time. The plume diameter is found by D =
0.308 s.
EXAMPLE INPUT, OUTPUT, AND MODEL LISTING
Input
The example problem is that of a single port with a diameter of 0.25 m
discharging 0.1 m3 /sec of effluent with a density of 1000 kg/m 3 . The discharge is horizontal. The discharge site varies in density from 1020 kg/m 3 at
the surface to 1025 kg/m3 at the discharge depth of 30 m.
A schematic of the computer-punch input cards for this problem follows.
Fitill t 1 iti
1_,_,±.1_,■_,J,L1
H.I
1
;
1.•. '
, +
--'11
I
IM
,,,,,
1-.'
1,4-1t
.."
...""rr...""" • A:.
_ ____ __
ij
1
11
II
I.N!
QI
IC)1
•
1 I
I
C•s:
111 ii
..laill,,
NI
111111
1
Itti.
r".
•
4
..._..
,t-'
•••
4
.,
ctsuaraturratwzwressw=a—macreva.sarsl-mressaesionaaarrsorarsadwansrenrantaraamaraccamcosx,raterneaneressaa.5.4.•gas_warvase.m.:0
usacsaurtex=amtamor
fEPN
1TTG
S
L
1 R_______
DEPTH
DIA
s
--=mrmactustscomegariarr.....nwqmicoomaseemericra....anscrwscserNzArnocc...nue
RHOJ
RFD
I
I I
1 1 1
I
1-
METERS
NPTS
HANGLE'
DEPTH.
•RHOJ
RFD
Q
FN
PS
-
,
.....r....„,........
1
LI
0
1
G,
%
1 1
i' ril ...!
l I ."-."'"11—''.
1 I4.H....J.iI
2..1
0!
I
Units, 1=MKS, blank=:FPS
Number of ambient density points to be specified,
Port angle measured from the horizontal
Depth of port
3
Density of effluent, gm/cm
Blank
Total volume rate of discharge
Number of ports
Print out interval
•
I'
1 u-",0
'
[,1 IL
--,, I
•
I
1 'TT-Tr-71"""r-7.i 4 !i : I 4 7 4
.1 1I
t
1 1...!
1 :. LI f f'
' ' • '''
L.L.1.11.
1 °8 1.
-,..-z-,.....:.,x1=
j I
i_fi
.smslarlikt---tme."=,..t.a.-7...:4
..1,41S=1:1=i••••35••••■11011•01,11•31•7001...1.17.==.41
•
-aZia.W.:
•
DP(r)
-
RHO(I)
0.0
1.0200
SAL
---..........1
Card 3
Card
1111111111111
mull
i;
t-
a1 i 17-17,
a. I
:,. .
i
Li--
0
•
1.....1
L4.....J
I-N
it11
)
4
...........4
7-s.----47
17-1177 !
1
4.4
W
1..1
It)
..11,-=>Azi:13.:=716.,....5.7:=WYCZW=LIC...03:ste911===.- •`
DP(I) - Depth measured from the surface
Density (gm/cm 3 ), or temperature (°C) if Sal specified
- If density not specified in . RHO(I) then salinity, o/oo.
* SAL
I
--10
7.1--'
IZI
--• - • •
-,
■ ■ • -- -
-
!
i
I
I
1
,
1 !
:=4'
(-...'r
TX.=2,1l5.A.,:4:-W.:=-7----,' a:
Output
The first part of the model output repeats the input and provides other
computed initial conditions. The Froude number given is the square of the
definition used here. The discharge velocity and characteristics of the zone
of flow establishment are also given in this section.
The next section of the output traces the plume and its characteristics
along its trajectory. T is a time based on the centerline velocity. S is the
distance along the centerline. X and Z are distances measured horizontally in
the direction of the discharge and vertically from the water's surface to the
center of the plume, respectively. Elevation is the vertical distance to the
centerline of the plume from the level of the discharge. D is the diameter of
the plume and DILN is the centerline dilution. The dilution and height of
rise to the trapping level are also given. As mentioned, equation (31) is
used to convert this to a flux averaged dilution. In this case, Sa = 1.77 Sm
= 27.3. The example output and program listing follow.
S3 PLUME VERSION 2.3 9/12/77
********A BUOYANT PLUME IN A DENSITY STRATIFIED MEDIA********
PLUME TEST CASE
CASE NO. 1 INITIAL CONDITIONS
UNITS: MKS
PORT ANGLE . • • OOOOOOO
O
•
FROUDE NUMRER
LENGTH FOR FLOW ESTABLISHMENT • . •
INTEGRATION STEP LENGTH
PRINTOUT INTERVAL
X0
ZO • • •
• • • • • • . •
• • •
DISCHARGE DENSITY
PORT DEPTH
FLOWRATE • • • OOOOO • OOOOO
NUMBER OF PORTS • • • • • • • • • •
DISCHARGE . VELOCITY
PORT DIAMETER
• •
• • • • • • •
DENSITY STRATIFICATION DEPTH RHO
-67.7
1.40
.076
3.00
1.39
29.84
1.00000
30.00
1.00000E-01
1
2.04
2.50000E-01
30.00
T
S
X
Z
D
ELEV
THETA
2.35
4.29
4.06
28.85
1.32
1.15
36.2
6.43
7.25
5.99
26.62
2.23
3.38
59.1
11.51
17.64
10.22
13.18
7.27
8.30
9.41
.23.95
21.17
18.42
3.15
4.06
25.61
16.14
4.97
33.91
18.12
10.74
--17.07
5.58
LAST LINE ABOVE IS FOR MAXIMUM HEIGHT OF RISE.
TRAPPING LEVEL IS
21.5 WITH DILUTION OF
HEIGHT OF RISE= 28.3 PERCENT OF DEPTH
6.05
8.83
11.58
12.93
68.1
70.3
63.1
DILN
3.3354
7.0703
11.6926
00001
00002
00003
00004
00005
00006 110
.00007
PROGRAM PLUME
COMMON G9FK9FM9COSTH.SINTH9COSTHE,DS9C19C29E139FLAG9GRAV
DIMENSION ZD(50),DG(50),RHO(50),DP(50),CMT(S)
INTEGER NDC9TRAPPD9FLAG9CHGDEN9METERS
WRITE(69110)
FORMAT("053 PLUME VERSION 2.3 9/12/77")
NOCASE=0
00008
CUE:WT=1./3.
00009
TWOTHD=2./3.
00010 20
READ(5,222)(CMT(I),I=1,5)
00011 222
FORMAT(5A8)
00012
IF(EOF(5)) CALL EXIT
00013
READ(5,73) NDC9METERS,NPTS,ANGLE•DIA9DEPTH,RHOJ9RFD909FN9PS
00014
IF(EOF(5)) GOTO 74
00015 73
FORMAT(211913,F5.0.7F10.0)
00016
FD=A8S(RFD)
00017
DO 102 I=1,NPTS
00018
READ(5.75) DP(1)9RHO(I),SAL
00019
IF(EOF(5)) GOTO 999
00020
IF(SAL.E0.0.) GOTO 102
00021
RHO(I)=1.+.001*SIGMAT(SAL9RHO(I))
00022 102
CONTINUE
.00023
CALL ASORT(DP,RHO,SAL,NPTS)
00024
999 IF(NPTS.NE.1) G0T0.76
00025
NPTS=2
00026
DP(2)=DEPTH
00027
RHO(2)=RHO(1)
00028 76
NOCASE=NOCASE+1
00029 75
FORMAT(8F10.0)
00030
GRAV=32.172
00031
IF(METERS) GRAV=9.80665
00032
DO 55 I=1,NPTS
00033
IF(DP(I).GE.DEPTH) GOTO 56
00034 55
CONTINUE
00035
WRITE(6959) NOCASE
00036 59
FORMAT( "-NO DENSITY INFORMATION FOR JET LEVEL. EXECUTION FOR",
00037
0 " CASE NO."9129" DELETED.")
00038
GOTO 20
00039 56
NP=I
00040
NM=I-1
00041
RHO8=(DEPTH-.DP(NM))*(RHO(NP)-RHO(NM))/(DP(NP).-DP(NM))+RHO(NM)
00042
DISP=RHOJRHO8
00043
DO 54 I=1,NM
00044
J=NP-I
00045
ZD(I)=(DEPTH-DP(J))/DIA
00046 54
DG(I)=(RHO(J+1)•RHO(J))*DIA/(DISP*(DP(J+1)-DP(J)))
00047
IF(NDC) GOTO 58
00048
U0=0/(FN*.7853982*DIA*DIA)
00049
RFD=UO*U0*RHOJ/(DISP*DIA*GRAV)
00050
FD=ABS(RFD)
00051 58
IF(FO.LE.4.01.0R.FD.GE.9.99) GOTO 61
,
00052
S=.113*FD+4.
00053
GOTO 62
---m--'00054 61
IF(FD.LE.4.01) S=2.8*FD**CU8RT
00055
IF(FD.GT.10.) S=5.6*FD/SORT(FO*F04.18.)
00056 62
TEMP=ATAN(1.416667*S/FD)
00057
TMETAO=.0111*ANGLE*(1.5708-TEMP)+TEMP
00058
COSTHE=COS(THETA0)
00059
SINTHE=SIN(THETA0)
00060
OSI=DEPTH/(177.*DIA)
00061
IF(DG(1).E0.0.) GOTO 77
00062
DGTEMP=.01/DG(1)
00063
IF(DGTEMP.LE.0.) DGTEMP=-DGTEMP
00064
00665
00066
00067
00068
00069
00070
00071
00072
00073
00074
00075
00076
00077
00078
00T79
00080
,00081
00082
00083
00084
0 0 085
00086
00087
00088
- 00089
00090
`00091
00092
00093
00094
00095
00096
00097
00098
00099
00100
00101
00102
00103
00104
00105
00106
'00107
00108
00109
00110
00111
00112
00113
00114
00115
00116
m•—••••••••••‘,-,* 00117
00118
00119
00120
00121
00122
00123
-00124
00125
00126
00127
00128
00129
DSI=.12*1.6**(ALOG10
NPO=IFIX(PS/(051*DIA
N=0
Z=S*SINTHE
X=S*COSTHE
E=(4./S)**3
R=.25
C1=E**TWOTHD
C2=.75/RFD
IPTS=1
G=DG(IPTS)
ZLIM=ZD(IPTS)
CHGDEN=0
FLAG=0
TRAPPD=0
XP=X*DIA
ZP=DEPTHZ*DIA
OSIP=DSI*DIA
SP=S*DIA
C ************************************
T=2.*SP/(U0*(1.+336.67*(S*S)))
C *************************************************
C
C WRITE INITIAL CONDITIONS
C
WRITE(6,23) (CMT(I),I=1.5),NOCASE
FORMAT("0********A BUOYANT PLUME IN A DENSITY STRATIFIED
23
*IIMEDIA********"/"0".5A8/
*"-+CASE NO. ",12," INITIAL CONDITIONS ....")
IF (NDC) WRITE(6.104)
UNITS: NON DIMENSIONAL")
FORMAT("0
104
IF(.NOT.NOC.AND.METERS) WRITE(6,105)
UNITS: MKS")
FORMAT("O
105
IF(.NOT.NDC.AND..NOT.METERS) WRITE(6.106)
UNITS:FPS")
FORMAT("O
106
WRITE(6,108)ANGLE.RFD,SP.DSIP.PS.XP,ZP,RHOJ,DERTH
FORMAT(
108
• ..F7.1/
PORT ANGLE
*40H
..F7.1/
... •
.
FROUDE NUMBER .
*40H0
.14'8.2/
LENGTH FOR FLOW ESTABLISHMENT •
*40H
. • . .,F9.3/
INTEGRATION STEP LENGTH
*40H
.,F8.2/
PRINTOUT INTERVAL ......
*40H
......... ,F8.2/
XO • . • • • . •
*40H
ZO • • • • • • .......... • tF8•2/
*40H
,F11.5/
DISCHARGE DENSITY
*40H
*40H PORT DEPTH ....... • ..F8.2)
IF(.NOT.NDC) WRITE(6.60)O.FN.UO,DIA
FORMAT(
FLOWRATE
• . • •
*40H
NUMBER OF PORTS •
*40H
.
•
•
DISCHARGE VELOCITY
*40H
*40H PORT DIAMETER . .... •
WRITE(6.24) ((0P(1),RHO(I)).1=1.NPTS)
RHO"//
FORMAT("-.. DENSITY STRATIFICATION DEPTH
*10(24X.F6.2,F11.5/))
77
DS=DSI
WRITE(6.109)
C *** ....... ***************************************
FORMAT(1H-...6Xe"T",9X."5".9X,"X".9X."Z".9X."0".7X0ELEV".
109
*6X."THETA",4X,"DILN")
C *** ...... ***** ********** * ***** *******************
GOTO 16
DS=DSI
45
DELX=COSTH*DS
00130
DELZ=SINTH*DS
00131
00132
DELE=FK
DELT=FM
00133
SPOS2=S+DS*.5
00134
00135
DO 10 1=1.2
CALL SDERIV(SPDS2.E•.5*FK9R+.5*FM)
00136
DELX=DELX+2.*COSTH*DS
- 00137
DELZ=DELZ+2.*SINTH*DS
00138
DELE=DELE+2.*FK
00139
00140
DELT=DELT+2.*FM
CONTINUE
00141
10
CALL SDERIV(S+DS,E+FK.R.FM)
00142
00143
ZLAST=Z
ZINCR=(DELZ+SINTH*DS)/6.
00144
Z=Z+Z1NCR
00145
IF(CHGDEN) GOTO 41
00146
IF(Z.GT.ZLIM) GOTO 40
00147
X=X+(DELX+COSTH*DS)/6.
00148 43
*****************************
00149 C
00150
EOLD=E
*****************************
00151
C
E=E*(DELE•FK)/6.
00152
R=R+(OELT+FM)/6.
00153
*************************************************
00154 C
DT=.218*DIA*DS/(U0*(EOLD**(1./3.),S+E**(1./3.)/(S+05)))
00155
T=T•DT
00156
*************************************************
00157 C
S=S+DS
00158
IF(E.LE.0.) FLAG=1
00159
C
00160
C THIS STOPPING CRITERIA IS BASED ON VELOCITY GOING TO ZERO
00161
00162 C
IF(FLAG) GOTO 13
00163
CALL SDERIV(S,E,R)
00164
16
IF(TRAPPD) GOTO 14
00165
IF(R.GT.0.) GOTO 15
00166
RAT=R/(R0.4i)
00167
E13TRP=E13+(E13—•E130)
00168
ZTRAP=DEPTH(Z+ZINCR*RAT)*DIA
00169
SMTRAP=.245*(5.05*RAT)*E13TRP
00170
TRAPPD=1
00171
00172
GOTO 14
RO=R
15
00173
00174
E130=E13
00175
14
N=N+1
00176
IF(N — (N/NP0)*NPO.NE.0) GOTO 11
XP=X*DIA
00177
13
ELEV=Z*DIA
00178
ZP=DEPTH—ELEV
00179
SP=S*DIA
00180
*************************************************
00181
fl=.308*SP
00182
**** ***** ****************************************
00183
DILN=.245*S*E13
00184
THETA=ACOSF(COSTH)*57.2958
00185
IF(.NOT.TRAPPD) GOTO 72
00186
**** ***** **** ***** *******************************
00187
WRITE(6,101) T,SP.XP.ZP.B.ELEV,THETA
00188
****** ******* ************041#4******43,041**41.0*******
00189
GOTO 71
00190
C ************* ***** ********************** ***** ****
00191
WRITE(6.101) I.SPIoXP.ZP98.ELEVeTHETA,PDILN
-00192
72
FORMAT(6110.2.F10.1.F10.4)
00193
101
00194 'C *4741p*******0********OAM.** ****** ******* e*******
71
IF(.NOT.FLAG) GOTO 11
00195
-00196
POEPTH=100.-(ZTRAP/DEPTH*100.)
00197
WRITE(6.65)
00198 65
FORMAT("-LAST LINE ABOVE IS FOR MAXIMUM HEIGHT OF RISE.")
00199 67
FORMAT("ATRAPPING LEVEL IS",F10.1.
00200
* " WITH DILUTION OF"lF10.4)
00201
IF(TRAPPD) WRITE(6,67) ZTRAP,SMTRAP
00202
IF(.NOT.TRAPPD) WRITE(6,68)
00203
IF(TRAPPD)WRITE(6.69) PDEPTH
00204 69
FORMAT(" HEIGHT OF RISE=".F5.1," PERCENT OF DEPTH")
00205 68
FORMAT("ATRAPPING LEVEL NOT REACHED")
00206
GOTO 20
00207 C
00208 C FIND NEXT STRATIFICATION AND RECOMPUTE LAST STEP IF
00209 C NECESSARY
00210 C
00211 40
DS=DS*(ZLIM-ZLAST)/(Z-ZLAST)
00212
CALL SDERIV(S,E,R)
00213
CHGDEN=1
00214
Z=ZLAST
00215
GOTO 45
00216 41
CHGDEN=0
00217
IPTS=IPTS•1
00218
IF(IPTS.GT.NM) GOT() 42
00219
G=DG(IPTS)
00220
ZtIM=ZD(IPTS)
00221
GOTO 43
00222 42
WRITE(6,44)
00223 44
FORMAT("OPLUME HITS SURFACE"/
00224
*" HEIGHT OF RISE=100.0 PERCENT OF DEPTH")
00225
FLAG=1
00226
GOTO 43
00227 74
CALL EXIT
00228
END
00229
SUBROUTINE SDERIV(S,E,R)
00230
COMMON G.FK,FM,COSTH.SINTH.COSTHE.DS.C14,C2.E13,FLAG.GRAV
00231
INTEGER FLAG
00232
F13=0.
00233
IF(E.LE.0.) GOTO 3
00234
E13=E**(1./3.)
00235
COSTH=COSTHE*C1/(E13*E13)
00236
IF(COSTH.LT.1.) GOTO 1
00237 C
00238 C THIS STOPPING CRITERIA IS
00239 C AGAIN
00240 C
00241
FLAG=1
00242
SINTH=0.
00243
COSTH=1.
00244
GOTO 3
00245
SINTH=SORT(1.-COSTH*COSTH)
00246 3
FK=C2*S*R*SINTH*DS
00247
FM=.109*S*EI3*G*DS
00248
RETURN
00249
END
00250
FUNCTION SIGMAT(SAL,T)
00251
SIG0=(((6.8E-6*SAL)-4.82E-4)*SAL•.8149)*SAL....093
00252
8=1.E6*T*((.01667*T:.8164)*T*18.03)
06253
A=.001*T*((.0010843*T-.09818)*T.4.7867)
00254
SUMT=(T-3.98)*(7-3.98)*(T•283.)/(503.57*(T+67.26))
00255
5IGMAT=(SIG04..13241*(1.-A.B*ISIGO-.1324))-SUMT
00256
RETURN
00257
END
00258
-SUBROUTINE ASORT(OP.RHO,SAL.NPTS)-00259
DIMENSION DP(1).RHO(1),SAL(1)
00260
NESTEO=NPTS
00261
L=NESTED-1
00262
00263
00264
00265
00266
00267
00268
00269
00270
00271
00272
00273
00274
00275 10
00276
00277
00 10 M=1,1_
NESTED=NESTED-1
DO 10 I=1,NESTE0
IF(DP(I).LE.DP(I.1)) GOTO 10
DUMMY = OP(I)
DP(I)=DP(I•1)
DP(I•1)=DUMMY
DUMMY=RHO(I)
RHO(I)=RHO(I•1)
RHO(I•1)=DUMMY
DUMMY=SAL(I)
SAL(I)=SAL(I.1)
SAL(I+1)=DUMMY
CONTINUE
RETURN
END
In the computer program, the entrained mass (DM) is added to the element's mass (PM) to become the new mass (SUM). The new temperature and salinity of the element are the averages of the old values and the entrained
ambient values (TA and SA) weighted by their relative masses. The new density
(DEN), and thus buoyancy, creates a vertical acceleration on the plume segment. Since the element is considered to be one of a train, each following
the preceding element, drag is assumed to be negligible and the vertical
velocity is incremented (by DV). The horizontal velocity is found by conservation of horizontal momentum with the current-induced entrainment (EINS)
having the velocity of the current. The segment length (H) is changed in
proportion to the total velocity (VEL) to conserve mass and pollutant. The
radius (B) is changed to correspond to the new mass and density. Dilution
(DIL) is calculated by comparing the initial volume to that of the element.
The program terminates execution when the vertical velocity reaches zero, the
surface is reached, or length scales or execution step limits are reached.
EXAMPLE INPUT, OUTPUT, AND MODEL LISTING
Input
Consider, for example, a horizontal, single port discharge into a flowing, stratified ambient. The maximum discharge velocity is 2.0 misec. The
discharge port radius is 0.125 m. The temperature and salinity of the waste
are 15.0°C and 1.09 °/oo, respectively. The ambient at the surface is 15.0°C
and 27.2 °/oo. At 30 m, the level of the discharge, the ambient is 15.0°C and
33.71 0/m. Currents are assumed to be 0.10 m/sec. The input punch cards for
this case follow.
1171.4e=CM74.101.1.
Wfr...71=4,1,1171=i====
.
IletteEMILittbtlleanC61.
.
,. , -
- ....
,
viesaan.acacal.
B )1
UW
/
0.0
0.10
15.00
1.09
.125
1.0
F10.3
F10.3
F10.3
F10.3
F10.3
F10.3
I
1
' 1-11-7-1--7-1-1-1171,7-77-7
mil
L,_11_L_L.__L.
ii ^-1
V .UWT S H E A -
,G _---
1.1.A..1
L. !
1 I
1‘,..
sz,......,..,,
sut-W-.",-*-' .....-Zrzams.c.
F10.3
F10.3...............-..............t
r"...777.7717"'
. L.....L.
/
1
i
.1.:
10)
f
1
.
I1.. - armacl-,..a. ak.Lcsal.-1
-,- ez...=o=r.--:.....=t-r..xu,..uzw=:rr==-y.d
'
1 . ,
1e- i
•
,- i-r',-...essar-w.,
.125
0.10
..1.,‘,.., 1.C.*.,,.
Vertical velocity of the plume, m/sec.
Current speed, m/sec.
Temperature of the plume, °C.
Salinity of the plume, o/oo.
Thickness of the plume element (=8), m.
Impingement entrainment coefficient (recommended = 1.0)
Aspiration entrainment coefficient (recommended = 0.1)
Plume radious, in.
Orea:Cv-Ictrastal:Z=msztax="...con
se,-xemar=tcusrcsaasesccs,
1-
L.':
; ,...
,
SECTION 5
"OUTPLM" MODEL
THEORETICAL DEVELOPMENT AND MODEL DESCRIPTION
The computer model OUTPLM (18, 23) considers a plume element or puff. By
following this element, the characteristics of the entire plume are described.
The original model has been adapted for ocean discharges. Density, velocity,
temperature, and salinity are assumed to be average properties of the element.
The quantities of mass, momentum, and energy are conserved. Two explicit
functions for entrainment are used. An equation of state is used to calculate
the density of the ambient and the plume element at each time step.
Initially, the plume segment has a horizontal velocity component (computer variable U) and a vertical component (V). It has temperature (T) and
salinity (S) and is injected into a uniformly flowing (UW) ambient. The
segment has a radius (B) and thickness (H) and is tracked in two dimensions,
depth (Z) and horizontal distance (X).
Entrainment brings ambient mass (plus momentum, temperature, and salinity) into the plume element. Entrainment is assumed to consist of two mechanisms. One mechanism (EINS) is due to the impingement of current on the plume.
That mass which crosses the exposed area of the plume element projected by
currents is entrained. The other mechanism (ZWEI) is aspiration entrainment
which captures 0.1 times the product of the external area of the plume element
and its shear velocity. The assumption that the total entrainment is the
larger of these two mechanisms gives good agreement with data.
Zone of Established Flow
The following assumptions are made:
a)
flow is steady and incompressible,
b)
the plume is fully turbulent,
c)
turbulent diffusion is significant only in the radial direction,
d)
pressure is hydrostatic throughout,
e)
plume flow is axisymmetric.
It is also assumed that the distributions of velocity and concentration are
Gaussian across the plume, so that:
V/V
C/C
•
m
m
= exp (-k (r/s) )
2
= exp (-kp(r/s) )
(18)
where V and C are the velocity and concentration, respectively, at some radial
distance, r, and the subscript m refers to the centerline. The distance along
the plume's trajectory is s, and k and p are the empirical coefficients.
The integrated governing equations are:
conservation of mass,
d/ds fc° V r dr = -r u
0
(19)
conservation of momentum,
d/ds fc° V 2 r dr = sine f (p . - p)g r/p d dr
0
0
(20)
conservation of density disparity,
d/ds f V A r dr = fc° V r dr • dpolds
(Po-Pd)
conservation of pollutants,
d/ds fVCrd
0
Substituting in the velocity profiles (17) and (18), and simplifying
yields:
(21)
Output
Model output and program listing follow. The line above the header
echoes the first two input cards. Ambient stratification is then given with
input depth, salinity, temperature, and computed sigma-t densities (density,
kg/m 3 , -- 1000). Other discharge conditions are given. K is a velocity
ratio, = U/UW. Q is the volume rate of discharge. The first data line in the
listing are the initial conditions. The horizontal distance inthe direction
of the jet discharge is the X dimension. Z is depth measured from the water's
surface. B is the radius, THICK is the plume element thickness along the
centerline. MASS is the mass of the plume element being considered. EINS and
ZWEI are the previously mentioned entrainments. DENDIF is the plume's density
difference with the ambient. DILUTION is an average volumetric dilution, the
inverse of CONC. TEMPDIF is the temperature difference between the plume
element and the ambient at that level.
.,.__.
■•■5441.
_.
,,
•
U
XSCALE YSCAL:
NPTS FTERB
2.0
500
501
2
1000
5
E10.3
F5.0
F5,1
15
15
15
1111111
EI
TR
___-....._ _ _
.,.....1.L.,
1...1_5_,
_LL.,..__LLIIIN
...
1.1._ 1;.[...,__,.. j
I I( I 1 11 .- 1111 1_,...,.,
II F.,•,.
....=if_i
, . it3 -
17_11_
U
Horizontal Velocity of the plume.
.XSCALE
Number of diameters in the X direction at which run will be cut-off.
YSCALE - Number of diameters in the Z direction at which run will be cut-off.
NPTS
- Number of points at which ambient is specified.
ITERB -',Number of steps allowed.
IR
-'j Interval of printout.
1_
_
DP(I)
SA(I)
TA(I)
0.0
27.20
15.00
Card.
-
30.0
33.71
15.00
Card 4
F10.3
F10.3
F10.3
all
11110111.11
Killi
.2
II
i
.--1.7"-,J,.t. . k__
DP(I) Depth where ambient is specified, m.
SA(I) - Salinity, o/o0.
TA(I) - Temperature, °C.
_ _ _ ___ .___ __ _
i
3
••■■■••■■■••.............clow.....g1
1111111
.. al ,..
g
1 1
IZI
, Ur I TirT1
J
I
,-'4
.100
15.000
1.090
.125
1.000
.100
.125
2.000 500 500
2 1000
25
4.00
OUTFALL BUOYANT (JET) PLUME IN FLOWING, STRATIFIED AMBIENT
0S3 VERSION 2.2 5-31-78
AMBIENT STRATIFICATION- DEPTH,M SALIN TEMP,C SIGMAT
0 27.20 15.00 20.01
30.00 33.71 15.00 25.00
00 9.82E-02 1.00E-01
X
OE 00
4.43E-03
1.22E-01
2.68E-01
4.45E-01
6.58E-01
9.16E-01
1.23F 00
1.60E 00
?.04E 00
'2.56E 00
3.I3E 00
9.78E
1.05E
I.13E
1.20E
1.28E
I.37E
I.48E
00
01
01
01
01
01
01
3.0nE
3.00E
3.00F
3.0oF
3.00E
3.00F
3.00F
2.99F
2.99F
2.97F
2.96F
2.93F
2.1IF
2.02F
1.94F
1.88E
1.83F
1.80E
1.7AF
(LINE . 1) AND MODEL OUTPUT
Z
:
B
THICK
,
MASS
EINS
01 1.25E-01 1.25E-01 6.14E 00
OE 00
01 1.25E-01 1.24E-01 6.18E 00
OE 00
01 1.48E-01 1.05E-01 7.30E 00 3.08E-03
01 1.75E-01 8.85E-02 8.68E 00 4.50E-03
01 2.08E-01 7.46E-02 1.03E 01 . 6.66E-03
01 2.47E-01 6.29E-02 1.23E 01 1.00E-02
01 2.93E-01 5.31E-02 1.46E 01 1.53E-02
01 3.46E-01 4.51E-02 1.74E 01 2.39E-02
01 4.08E-01 3.86E-02 2.06E 01 3.76E-02
01 4.77E-01 3.35E-02 2.45E 01 5.90E-02
01 5.53E-01 2.97E-02 2.92E 01 9.05E-02
01 6.32E-01 2.71E-02 3.47E 01 1.34E-01
01 7.12E-01 2.53E-02 4.13E 01 1.88E-01
01 7.95E-01 2.42E-02 4.91E 01 2.55E-01
01 8.83E-01 2.33E-02 5.84E 01 3.35E-01
01 9.79E-01 2.25E-02 6.94E 01 4.31E-01
01 1.09E 00 2.17E-02 8.26E 01 5.51E-01
01 1.21E 00 2.08E-02 9.82E 01 6.78E-01
01 1.36E 00 1.96E-02 1.17E 02 8.06E-01
01 1.54E 00 1.81E-02 1.391 02 9.59E-01
01 1.76E 00 1.65E-02 1.65E 02 1.14E 00
01 2.03E 00 1.48E-02 1.96E 02 1.36E 00
01 2.36E 00 1.30E-02 2.331 02 1.61E 00
01 2.76E 00 1.13E-02 2.781 02 1.92E 00
01 3.231 00 9.77E-03 3.30E 02 2.281 00
01 3.77E 00 8.57E-03 3.93E 02. 2.71E On
01 4.34E 00 7.69E-03 4.67E 02 3.23E 00
01 4.71E 00 7.29E-03 5.25E 02 3.63E 00
ZWEI
1.91E 01
1.91E 01
4.97E-02
5.91E-02
7.03E-02
8.36E-02
9.94E-02
1.18E-01
1.41E-01
1.67E-01
1.99E-01
2.38E-01
2.83E-01
3.37E-01
4.01E-01
4.77E-01
5.67E01
6.46E-01
6.93E-01
7.24E-01
7.31E-01
6.98E-01
6.18E-01
4.94E-01
3.60E-01
2.58E-01
2.13E-01
4.67E-01
DILUTION DENDIFF
1.00E 00 2.50E 01
1.01E 00 2.48E 01
1.18E 00 2.10E 01
1.40E 00 1.77E 01
1.66E 00 1.48E 01
1.98E 00 1.25E 01
2.34E 00 1.05E 01
2.78E 00 8.83E 00
3.31E 00 7.42E 00
3.93E 00 6.22E 00
4.66E 00 5.21E 00
5.54E 00 4.34E 00
6.59E 00 3.59E 00
7.83E 00 2.94E 00
9.31E 00 2.38E 00
1.11E 01 1.88E 00
1.32E 01 1.45E 00
1.56E 01 1.08E 00
1.86E 01 7.51E-01
2.21E 01 4.74E-01
2.63E 01 2.42E-01
34131 015.16E-02
3.72E 01-9.62E-02
4.42E 01-2.00E-01
5.26E 01-2.63E-01
6.251 01-2.91E-01
7.44E 01-2.98E-01
7.44E 01-2.92E-01
HOR VEL VER VEL TOT VEL TEmPDIF
2.00E 00 OE 00 2.00E 00 OE 00
1.99E 00 5.43E-04 1.99E 00-2.33E-10
1.68E 00 1.37E-02 I.68E 00 2.33E-10
1.42E 00 2.80E-02 I.42E 00 OE 00
1.19E 00 4.33E-02 1.19E 00 1.40E-09
1.00E 006.02E-02 1.0IE 00 1.16E-09
8.46E-01 7.92E-02 8.50E-01 4.66E-10
7.15E-01 1.01E-01 7.22E-01 6.98E-10
6.05E-01 1.26E-01 6.18E-01 6.98E-10
5.13E-01 1.54E-01 5.36E-01 9.31E-10
4.38E-01 1.84E-01 4.75E-01 4.66E-10
3.76E-01 2.14E-01 4.33E-01 4.66E-10
3.26E-01 2.40E-01 4.05E-01 9.31E-10
2.86E-01 2.61E-01 3.87E-01 9.31E-10
2.53E-01 2.74E-01 3.73E-01 1.40E-09
2.26E-01 2.80E-01 3,60E-01 4.66E-10
2.05E-01 2.80E-01 3.47E-01 6.98E-10
1.88E-01 2.74E-01 3.32E-01 6.98E-10
1.74E-01 2.60E-01 3.13E-01 6.98E-10
1.63E-01 2.41E-01 2.90E-01 1.16E-09
1.53E-01 2.16E-01 2.64E-01 9.31E-10
1.44E-01 1.88E-01 2.37E-01 2.33E-10
1.37E-01 1.56E-01 2.08E-01'2.33E710
1.31E-01 1.24E-01 1.80E-01-6.98E-10
1.26E-01 9.20E-02 1.56E-01-1.40E-09
1.22E-01-.1.24E-02 1.37E-01 4.66E-10
1.19E-01 3.28E-02 1.23E-01 9.31E-10
1.17E-01-2.50E-03 1.17E-01
OE 00
uuout
00002
00003
00004
00005
00006
00007
00008
00009
00010
00011
00012
0C413
00014
00015
00016
00017
00018
00019
00020
00021
00022
00023
00024
00025
00026
00027
00028
00029
00030
00031
00032
00033
00034
00035
00036
00037
00038
00039
00040
00041
00042
00043
00044
00045
00046
00047
00048
00049
00050
00051
00052
00053
00054
00055
00056
00057
00058
00059
00060
00061
00062
00063
PRObHAM OUIPLa
DIMENSION oP(30),s4(30),TA(30).DENp(30)
DATA (G=9.8).(TWO=2•),(PI=3•1416),(ZERU=0.)
INTEGER NPTS
•••'
5 FORMAT(/,9F10.3,2F5.0,315/"
2/" OUTFALL BUOYANT (JET) PLUME IN FLOWING, STRATIFIED AMBIENT"/
- '•
3/" 0S3 VERSION 2.2 5-31-78"/
1/" AMBIENT STRATIFICATION- DEPTH,M SALIN TEMP,C SIGMAT"//)
6 FORMAT(8F10.3,/:,1F10.3,2F5.0,315)
9 FORMAT(3F10.3)
14 FORMAT (26X,4F6.2/)
READ (10,6) V,UW,T,S,H,E,A,B,U,XSCALE,ZSCALE,NPTS,ITERB,IR
WRITE(11,5) V.UW,T,S,H.E.A,B,U,XSCALE,ZSCALE,NPTS,ITER8,IR
•
DO 8 I=1,NPTS
READ (10.9) DP(I),SA(I),TA(I)
DENP(I)=SIGMAT(SA(I).TA(I))
8 CONTINUE
WRITE (11,14) ((oP(I)s54(I),TA(1),DENP(I)),I=1,NPTs)
nr=i•
=ZERO
X=DD=B5AVE=ZWEI=DH=DP
7=0P(NPTS)
nEL=T-TA(NPTS)
AK=9999.
DTL=1.
RA=B
80=B*TWO
VI=VEL=SORT(NON+U*U)
DFNA=1000.+SIGMAT(SA(NPTS).TA(NPTS))
DEN=1000•4.5IGMAT(S,T)
RHO=DEN
DENDIFF=DENA-DEN
PM=PM0=PI*B*8*H*DEN
0=PM/DEN*VEL/H
TF(UW.NE.ZERO) AK=VEL/UW
FR=VEL/SORT(DENDIFF/DEN*TW0*B4G)
lo,
7 FORMAT(/"
CURRENT"/1X4E9.2//
0
K
FROUDE
1"
"/"
X
MODEL INPUT (LINE 1) AND MODEL OUTPUT
2"
ZWEI DILUTION OENDIFF
EINS
MASS
THICK
8
3 Z
4 HOR VEL VER VEL TOT VEL TEMPDIF")
WRITE(11,7) AKtFR,O,UW
UO=U
RATIOX=RATIOZ=1.
DO 99 J= 1,ITER8
no 88 I=1.NPTS
IF (DP(I).GE.Z) GO
CONTINUE
10 FORMAT("NO DENSITY
WRITE(11.10)
GO TO 999
888 LL=I
-,-.LU=I-1
TT=TA(LL)+(DP(LL)-Z)*(TA(LU)-TA(LL))/(DP(LL)-0P(LU))
SS=SA(LL)-(DP(LL)-Z)*(SA(LL)-SA(LU))/(DP(LL)-DP(LU))
DENA=1000•+SIGMAT(SS.TT)
FTN5=E*OENA*UW*DT*B*H*(TWO*V/VEL+PI/D0*(U*08/VEL+B/TWO*(U/VEL
1-UO/V1)))
...L.ZWEI=TWO*PI*DENA*H*B*ABS(VEL-UW*U/VEL)*A*DT
nM=EINS
TF(ZWEI.GT.EINS) DM=ZWEI
GAMMA=DM/.0069555/PM
nT=DT/GAMMA
FINS=EINS/GAMMA
00064
00065
00066
00067
00068
00069
00070
00071
00072
00073
00074
00075
00076
00077
00078
00079
00080
00081
00082
00083
00084
00085
00086
00087
00088
00089
00090
00091
00092
00093
00094
00095
00096
00091
00098
00099
00100
00101
00102
00103
00104
00105
00106
00107
00108
00109
00110
00111
00112
00113
00114
00115
OM=DM/GAMMA
13 FORMAT (1)(914E9.2)
TF(J.E0.1)WRITE(11.13)X,Z,R,H,PM.EINS.ZWEI
1.0ILIDENDIEFoUfV,VEL,DEL
SUM=PM+DM
UO=U
U=(PM*U+EINS*UW)/SUM
T=(PM*T+DM*TT)/SUM
s=(PM*S+DM*SS)/SUM
OFL=T–TT
OFN=1000.•SIGMAT(S,T)
DENDIFF=DENA–DEN
V=PM*V/SUM+OENDIFF*G/DEN*DT
V1=VEL
PM=SUM
VFL=SORT(U*U•V*V)
OZ=V*DT
DX=U*DT
IF(DZ.LE.ZERO)G0 TO 999
00=VEL*07
RSAVE=B
R=SORT(PM/(DEN*PI*H))
OR=(B-8SAVE)
H=VEL/VI*H
X=X+OX
Z=Z–OZ
TF(Z.LE.0.)GO TO 999
IF (J.E0.1)G0 TO 98
IF (J/IR – (J-1)/1R.NE.1)G0 TO 99
IF (RATIOZ.GT.ZSCALE.OP.RATIOX.GT.XSCALE)G0 TO 999
OiL=(RHO*PM)/(PM0*DEN)
98 RATIOX=X/80
PATIOZ=Z/80
OIL=(RHO*PM)/(PMO*DEN)
WPITE(11,13)X,Z,B,H,PM,EINS,ZWEI,DIL,DENDIFF
I.U4V9VEL,DEL
99 CONTINUE
999 CONTINUE
wRITE(11+13)X,Z,H,H,PM.EINS,ZWEI+DIL,DENDIFF
1,U,V+VEL,DEL
15 FORMAT(/"NUMBER OF STEPS="I5/)
wRITE(11.15)J
CALL EXIT
END
FUNCTION SIGMAT(SAL,T)
SIG0=(((6.8E-6*SAL)-4.82E-4)*SAL+.8149)*SAL–.093
C=1.E-6*7*((.01667*T–.8164)*T+18.03)
0=.001*T*((.0010843*T–.09818)*T+4.7867)
SUMT=(T-3.98)*(T-3.98)*(7+-283.)/(503.57*(T+67.26))
SIGMAT=(SIG0+.1324)*(1.–D+C*(5IGO–.1324))–SUMT
RFTURN
END
SECTION 6
"DKHPLM MODEL
THEORETICAL DEVELOPMENT
The computer model DKHPLM (7, 21, 24) is an approach to the problem of
merging plumes which considers the detailed dynamics of the process instead of
simplifying the problem to a idealized slot plume or to a combination of plume
and slot plume. DKHPLM considers three zones of plume behavior; zones of flow
establishment, established flow, and merging. The first two zones are based
on the analyses of Hirst (25, 26) for a plume in a stratified, flowing environment. In the zone of merging, neighboring plumes are superimposed. This
allows a smooth transition as single plumes begin to compete for dilution
water then gradually merge with their neighbors. Equations for the conservation of mass, pollutant, energy, and momentum are developed for all three
zones. Entrainment is an explicit function dependent on the local Froude
number, plume spacing, excess velocity, and ambient velocity. Similar lateral
profiles, a 3/2 power approximation of a Gaussian, are assumed, for velocity,
concentration, and temperature. These profiles are superimposed in the merging zone. A complete theoretical development of this model is beyond the
scope of this paper but can be found in (21, 24, 25, and 26). The following
is a brief summary.
Zone of Flow Establishment
All quantities are assumed uniformly distributed in the plume at the
point .of discharge. In the zone of flow establishment, these uniform profiles
change to similar profiles as the boundary layer diffuses inward to the centerline of the jet. The rate at which the profiles of velocity, concentration
and temperature develop may vary. The integrated forms of the governing
equations are:
conservation of mass,
d/ds fe° V r dr = -lim r 00 ((r v) =
0
(32)
conservation of energy,
d/ds
V (T-T, ) r dr = - d T./ds f V r dr - lim r -
r v"T")
(33)
- 00 ((r v' C')
(34)
conservation of pollutant,
d/ds fc° V (C-C. ) r dr = -d Colds fw V r dr - lim
conservation of momentum in the s equation,
d/ds fw V2 r dr = U E sine l cos0 2 + fw g (p. - p)/p d ) r dr sin02
- lim r +
(r u' v')
(35)
where 0 1 is the horizontal angle between the centerline and the x axis and 02
is the angle between the centerline and the horizontal. Two additional integral equations have been developed from equation (35) to describe momentum in
two additional plume coordinants. These "natural" coordinants of the plume,
described in (25), are converted to conventional three-dimensional Cartesian
coordinates for model output. Implicit in the derivation of these equations
are the assumptions that:
a)
flow is steady in the mean,
b)
flow is fully turbulent,
c)
fluid is incompressible and density variations are included only in
the buoyancy terms,
d)
all other fluid properties are constant,
e)
no frictional heating,
f)
pressure variations are purely hydrostatic,
changes in density are small enough to be approximated by a linear
equation of state,
h)
flow within the jet is axisymmetric,
i)
flow within the jet can be approximated as boundary layer flow,
j)
the ambient is infinite in extent.
Several of the assumptions are compensated for in the solution. The zone of
flow establishment uses a special entrainment function (see equation (128) of
(24)) which is a function of local Froude number, velocities, port diameter,
spacing, and thickness of the developed flow region.
Zone of Established Flow
The equations of mass, energy, pollutant, and momentum (equations 32 35), and the additional momentum equations mentioned above are also solved in
the zone of established flow, but are cast into a slightly different form.
The governing equations are written in a cylindrical coordinant system where 0
is the circumferential angle around the plume and cross section and the independent variables are r and s. These are evaluated using the assumed 3/2
power approximation to Gaussian lateral profiles. The angle 0 is the angle
between the centerline projected to the xy plane and the x axis, 0 2 is the
angle between the centerline and the xy plane. These angles relate the two
coordinate systems. Another entrainment function is used in the zone of
established flow which is a function of the local Froude number, velocities,
plume diameter, and spacing.
Zone of Merging
When adjacent plumes begin to overlap, the discharge is no longer considered axisymmetric. The distributions of plume properties are superimposed.
Another entrainment function is used which also considers the variable entrainment surface during merging. A drag term is also introduced to account
for the additional bending of the plumes after merging.
MODEL DESCRIPTION
An equation of state for sea water has been added to the original program
(subroutine SIGMAT) replacing the linear equation of state. In the zone of
flow establishment, the system of six governing equations are solved simultaneously (subroutine SIMQ) and stepped forward in space by a Hamming's modified predictor-corrector method (subroutine HPCG). This procedure continues
until velocity, temperature and concentration become fully developed. Subroutine OUTP1 contains the results which are stored as initial conditions for
the zone of established flow.
In the zone of established flow, similar profiles and the integral method
allow solution of the six governing equations for the six unknowns initially
by Runge-Kutta integral approximation and then by the Hamming's modified
predictor-corrector method. At the point where the plumes overlap, the assumed similarity no longer applies. The merging plumes have axes of symmetry
along the discharge line and normal to it. Only one quadrant of a plume,
taken to a midpoint of the overlap area, is evaluated. The profiles are
superimposed in this region using integral similarity coefficients. Above the
point where the plume and the ambient have equal density, results are obtained
by extrapolation.
The program itself contains many comments and explanations which serve as
EXAMPLE INPUT, OUTPUT, AND MODEL LISTING
Input
As an example case, consider a diffuser 1250 m long with discharges 12.5
m3 /sec. The ports are 0.178 m in diameter, oriented vertically, and space 5 m
apart. The ambient currents are assumed to be 0.05 m/sec and normal to the
discharge line. (Note that the angle of currents to the diffuser should be
within about 20° of normal.) The velocity through the discharge ports is 2.0
m/sec. The effluent is 15.01°C and 1.09 0/00. (Note that the effluent should
have some finite temperature difference with the ambient.) The ambient at the
level of discharge is 15.0°C and 33.71 0/00. The ambient temperature gradient
is negligible but must be finite. The ambient salinity gradient is -0.026
o
/oo per m. Depth of the discharge is 50 m. Computer punch card format for
this input follows.
_.
........„,.........
,
1
J3
HEADER INFORMATION (N11)
1
DKHPLM EXAMPLE RUN
14
16A4
..._...dk.,..„L.
-
.
.......--,
*
ti
_
„,
11
0
WM
II
Lai
0
;t
G
r
..........g. rn c...) =,
X 0.,
ru -s =
g
0_ p2
a
fro
4
y
.....1 .....1
! m
=
-o
a
(-I-
1
;
1
...
17=rial:C=116,M922500,.
i
.2¢1.51OCIVISOILVOMMI=911=1.1203M.S.."30.Z20=X. "^" r" ' ^ , —.4---ar...ac.-4=4.,,A
g
ar antzsavramsaav-
GILL,=1:17111tIMICCII2121=
.MISOMMEHMILVSLCOMMTVI
,
#-;
UA
TO
CO
TAO
CAO
TIZ
2.0
0.05
15,01
1.09
15.00
33.71
.00001
-.026001
F10.6
F10.6
F10.61
;
F10.6
IMUIIII
1
1
•••■•
UO
F10.6
,
VIWJE.ITAXIMWOU2C=WOM======5
11151=1,0■=1.7. -Jr•1:1,241GINEtt,
-,3 1
i
[
•-
F10.6
III
1
F10.6
F10,6
'
CIZ
NM • 411MCMANOINTMINIMIMM./IINMIIMMII.*
_.7_,..,....„1
T
_.111.j_
1I 1 1
lu'o,_..._____
_...–,7-77
,, ___ :=1`...__
11
1 L!
,...a--...—p------7—
..,1
i:
t.-1..;
_}.70,.L..,_!.,,,....,,,
,,,
UO - Initial jet velocity, m/sec.
UA. - Current 'speed, m/sec.
TO - Initial jet temperature, °C.
CO - Initial jet salinity, o/oo.
TAO- Ambient temperature at the level of discharge, °C.
CAO- Ambient salinity at the level of discahrge, °C.
TIZ- Temperature gradient, °C/m,
CIZ- Salinity gradient, o/oo/m.
•
•,I=TXMO.WIC111,1;
,
_
DO
THT
TH2
SF
SPACE
0.175
90.0
90.0
500.0
5.0
OA
F10.6
F10.6
F10.6
F10.6
F10.6
F10.6
1 r" 1
,
,ZWer72,4X.7-W.NAWIEuszvu,.....
-
-----_---
i-T--T-1
,z
.„,
. . . ,,._.1= ==.. . .C. . p _,
-• Port-diameter, m.
- Horizontal discharge angle' (DEG) relative to the current (70<THl<110).90
- Angle of diSdharge from the horizontal, DEG.
- Number of diameters along centerline at cutoff.
- Port spacing, m.
- Depth, m.
9
.
VfialffMti=aff3aSZEGII2d=3.1Wwle:=21
.
--,--------4
ci
i .:
, 1 L=.22
,
,
normal to current.
Output
Example output and program listing follow. The initial input conditions
are printed along with dimensionless conditions. The first section of data
listing concerns the zone of flow establishment. The columns list: length
along the plume axis (S), horizontal distance parallel with discharge line
(X), horizontal distance normal to the discharge line (Y), vertical distance
from the level discharge (Z), the horizontal angle from the plume's axis to
the discharge line (TH1), the angle the plume's axis makes from the horizontal
(TH2), the radius (B), potential core widths for velocity, temperature and
concentration (RU, RT, and RC), normalized centerline disparities of velocity,
temperature, and concentration with the ambient (DUCL, DTCL, and DCCL), the
ambient density normalized by the density of the discharge (PI), and nondimensional time (TIME). In the zone of established flow, width and average
dilution (Q/Q 0 ) are also given. Time is given in seconds. The centerline
dilution before zone of mergin is:
Sm= 0.52 Q/Q 0(36)
after complete merging (which never actually occurs):
S = 0.70 Q/Qo
(37)
based on the assumed distributions of concentration and velocity.
If stagnant conditions are specified, the dilution should be taken at the
point of buoyant equilibrium with the ambient. The location is announced in
the model output.
SOLUTION TO MULTIPLE BUOYANT DISCHARGE PROBLEM WITH AMBIENT CURRENTS AND VERTICAL GRADIENTS
DKHPLM EXAMPLE CASE
I.D. =
DISCHARGE VELOCITY = 2.00-M/S
* TEMP. = 15.01-DEG C
DIAMETER =
.18-M
** SPACING = 5.00-M
** DEPTH =
AMBIENT CONDITIONS AT DISCHARGE ELEVATION. VELOCITY =
**
AMBIENT STRATIFICATION GRADIENTS ** TEMPERATURE
AMBIENT CONDITIONS AT DISCHARGE - NONDIMENSIONAL
TEMP= .99933 SALINITY=30.92661 DENSITY= 1.02501
** SALINITY = 1.09-PPT
50.00-M
.05-M/S ** TEMP. =
15.00-DEG C ** SALINITY =
33.71 PPT
.000010-DEG C/M
** SALINITY ** -0.026000-PPT/M
TEMPGR= .000000 SALGR=-0.002123 DENGR=-0.0000018
PORT SPACING L/D. = 28.09
FRounE
NO =
9.58
VELOCITY RATIO =
.025
STRATIFICATION NO = 14096.6
OF FLOW ESTABLISHMENT -- ALL LENGTHS ARE IN METERS
Z
RT
TH1
RU
TH2
B
0
0
0
0
90.00
.09
.00.09
.00
90.00
i
.18
.00
.00
.1A
90.00
.27
.00
..00
.27
90.00
.36
.00
..00
.36
90.00
.44
.00
..00
.44
90.00
.53
.00
:.00
.53
90.00
.62
.00
.00
.62
90.00
.71
-0.00
'.01
.71
90.00
.80
-0.00 1
.01
.80
90.00
STARTING LENGTH. T =
.844
-0.00 '
.01
.84
.84
90.00
STARTING LENGTH. VELOCITY =
.866
.87
-.0.00 .
.01
.87
90.00
-0.00
.00
.00
.00
.00
.00
DTCL
DCCL
PI
TIME
0
.018
.038
.059
.082
.107
.133
. 162
.193
.226
.089
1.000
1.000
1.000
.083
1.000
1.000
1.000.
.077
1.000
1.000
1.000
007770
.0773
..0070
.5
1.000
.070
1.000
1..000
.062
.062
1.000
::::
1.000
1.000
.053
1.000
1.000
1.010
.043
.043
.999
1.010
1.000
:00;
::::
.033
1.000
.999
1.000
.021
1.000
99
.9
.999
=
.007
.007
.999
.999
.999
1.02500
1.02500
1.02500
1.02500
1.02500
1.02499
0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
88.62
.243
.004
.999
.999
.999
1.02499
.4
88.57
.252 -0.000
.999
.930
.930
1.02499
9.7
.089
.083
.089
.083
WIDTH
DUCL
DTCL
DCCL
PI
90.00
90.00
90.00
90.00
90.00
90.00
88.57
84.59
80.81
77.58
74.86
72.50
.51
1.24
2.03
2.82
3.59
4.35
.997
.429
.284
.223
.190
.169
.930
.354
.196
.126
.088
.065
.930
.354
.196
.127
.089
.066
1.02499
1.02498
1.02497
1.02495
1.02494
1.02492
.87
5.03
90.00
1.14
5.69
90.00
6.34
1.43
90.00
1.74.
6.9R
90.00
7.63 . 90.00
2.081
2.77 :- 8.88 '90.00
3.49
-90.00
10.11
69.97
66.86
64.78
63.27
62.08
60.32
58.89
5.09
5.71
6.27
6.81
7.35
8.44
9.64
.154
.144
.138
.132
.126
.113
.099
.050
.040
.033
.027
.022
.015
.010
.051
.041
.034
.028
.024
.017
.011
1.02491
1.02490
1.02488
1.02487
1.02486
1.02483
1.02481
. .01
.05
_.14
.28
.45
.65
.87
1.58
2.2A
2.98
3.67
4.36
DUCL
90.00
89.91
89.81
. 89.70
89.57
89.43
89.28
89.11
88.92
88.72
ZONE OF ESTABLISHED FLOW
Z
TH1
TH2
.87
1.58
2.29
300
.
3.71
4.43
RC
TIME
.43
1 1.00 .
2.03
3.43
5.12
7.04
9.14
11.40
13.76
16.20
18.81
24.21
30.19
1.02501
1.02501
1.02501
1.0d501
1.
0/00
2.07
5.48
9.99
15.67
22.48
30.37
39.29
47.81
55.90
63.73
71.60
86.36
100.48
I.D. =
OKHPLm EXAMPLE CASE
Z
X
TH1
TH2
4.24
11.32
90.00
57.55
5.04
90.00
12.55
56.06
54.44
5.85
13.72
90.00
6.70
90.00
14.86
52.58
7.58
15.98
90.00
50.38
S REACHED ITS EQUILIBRIUM HEIGHT
8.10
16.59
48.94
90.00
.00
8.22
16.72
90.00
48.60
.00
8.34
16.86
90.00
48.25
8.46
16.99
.00
90.00
47.89
.00
8.58 ! 17.12
'90.00
47.52
8.70
.00
17.25
'90.00
47.15
.00
8.82
17.38
90.00
46.76
.00
8.94
17.51
90.00
46.36.
9.06
17.64
.00
90.00
45.95
9.19 , 17.77
.00
90.00
45.53
.00
9.31
17.89
90.00
45.09
.00
9.44
18.02
90.00
44.65
.00
9.57
18.14
90.00
44.19
.00
9.70' 18.27
90.00
43.72
9.82
.00
18.39
90.00
43.23
9.95
18.51
90.00
.00
42.73
.00
10.09 1 18.63
90.00
42.21
.00
10.22
18.75
90.00
41.68
.00 110.35 ! 18.87
90.00
41.13
.00 '10.49 ! 18.98
90.00
40.57
.00
10.69 , 19.16
90.00
39.68
.00
10.97
19.38
90.00
38.44
.00
11.25
19.60
90.00
37.12
.00
11.53
90.00
19.81
35.70
.00
11.83 1- 20.01
90.00
34.19
12.12
20.21
.00
90.00
32.58
.00
12.43
20.40
90.00
30.86
12.73
.00
20.57
90.00
29.02
.00
13.05
20.74
90.00
'27.05
.00
13.37 ! 20.90
90.00
24.96.
.00
21.04
90.00
22.73
.00
21.17
20.36
90.00
.00
21.29
17.85
90.00
.00
21.39
90.'00
15.21
.00
21.48
90.00
12.45
.00
.00
.00
.00
19.00
19.18
19.36
19.53
19.71
19.89
20.07
20.25
20.42
20.60
20.78
20.96
21.14
21.31
21.49
21.67
21.85
22.03
22.20
22.47
22.83
23.18
23.54
23.89
24.25
24.61
24.96
25.32
25.67
26.03
26.39
26.74
27.10
27.45
WIDTH
11.18
12.63
14.13
15.73
17.43
STRATIFIED
18.45
18.68
18.91
19.14
19.38
19.62
19.86
20.11
20.35
20.60
20.85
21.10
21.36
21.62
21.87
22.14
22.40
22.67
22.93
23.20
23.60
24.15
24.70
25.27
25.84
26.41
26.98
27.55
28.12
28.67
29.21
29.72
30.20
30.64
31.04
DUCL
DCCL
OTCL
.087
.006
.084
.004
.080
.002
.076
-0.000
.071
-0.002
ENVIRONMENT
.068
-0.003
.067
-0.003
.066
-0.003
.065
-0.004
.065
-0.004
.064
-0.004
.063
-0.004
.062
-0.004
.062
-0.005
.061
-0.005
.060
-0.005
.059
-0.005
.058
-0.005
.058
-0.005
.057
-0.006
.056
-0.006
.055
-0.006
.054
-0.006
.053
-0.006
.052
-0.007
-0.007
.051
.049
-0.007
-0.007
.047
.045
-0.008
.044
-0.008
-0.008
.042
.040
-0.009
.038
-0.009
.036
-0.009
.034
-0.010
.032
-0.010
.U30
-0.010
.029
-0.010
.027
-0.011
-0.011
.026
PI
TIME
WU 0
.008
.005
.004
.002
.000
1.02479
1.02476
1.02474
1.02471
1.02469
36.87
44.28
51.69
59.38
67.41
114.06
127.61
140.44
152.93
165.00
-0.000
-0.001
-0.001
-0.001
-0.001
-0.001
-0.001
-0.002
-0.002
-0.002
-0.002
-0.002
-0.002
-0.003
-0.003
-0.003
-0.003
-0.003
-0.003
-0.003
-0.004
-0.004
-0.004
-0.004
-0.005
-0.005
-0.005
-0.006
-0.006
-0.006
-0.006
-0.006
-0.007
-0.007
-0.007
1.02468
1.0246a
1.02467
1.02467
1.02467
1.02467
1.02466
1.02466
1.02466
1.02466
1.02465
1.02465
1.02465
1.02465
1.02464
1.02464
1.02464
1.02464
1.02463
1.02463
1.02463
1.02462
1.02462
1.02462
1.02461
1.02461
1.02460
1.82460
1.02460
1.02459
1.02459
1.02459
1.02459
1.02458
1.02458
72.11
73.17
74.24
75.32
76.40
77.50
78.60
79.72
80.84
81.97
83.11
84.26
85.42
86.59
87.77
88.96
90.17
91.38
92.6193.85
95.73
98.29
190.91
103.58
106.32
109.13
112.01
114.96
118.00
121.11
124.30
127.58 ,
130.95
134.39
137.91
171.58
173.02
174.45
175.88
177.29
178.69
180.08
181.46
182.84
184.20
185.55
186.89
188.21
189.53
190.83
192.12
193.39
194.65
195.90
197.13
198.95
201.32
203.62
205.85
208.00
210.06
212.02
213.89
215.66
217.30
218.83
220.22
221.47
222.58
223.52
PI
TIME
I.D.
TH2
WIDTH
DUCL
.00
15.40
27.81
'21.54
90.00
.025
9.57
31.36
28.34 15.93
90.00
21.61
' 5.09
.00
31.71
.023
28.88
.00. 16.46
21.64 '90.00
.50
31.88
.023
29.06 !
.00
16.64, 21.64
.023
90.00
-1.03
31.91
PLUME HAS REACHED ITS MAXIMUM HEIGHT - STRATIFIED ENVIRONMENT
MAXIMUM PLUME HEIGHT. EXTRAPOLATED -!
NO OF INTEGRATION STEPS= 898
LINEAR =
NO OF HALVINGS= 11
DTCL
-0.011
-0.011
-0.011
-0.011
44.52
PARABOLIC =
ABSERR=
.00100
DCCL
-0.007
-0.007
-0.007
-0.007
1.02458
1.02458
1.02458
1.02458
141.49
146.96
-152.51
154.36
0/0
224.31
225.16
225.63
225.72
43.41
PRMT(5)=
1.000 FINAL SPACE =
4.000
00001
PROGRAM DKHPLM
00002 C THIS PROGRAM CALCULATES THE CHARACTERISTICS OF A LINE OF EQUALLY
00003 C SPACED BUOYANT DISCHARGES , INTO A FLOWING STRATIFIED AMBIENT
00004 C IT IS ASSUMED THAT BUOYANCY IS DUE TO TEMPERATURE AND SALINITY
00005 C THE METHOD OF SOL*N INVOLVES .7 ORDINARY DIFFERENTIAL EQUATIONS
00006 C WHICH ARE 1. CONSERVATION OF MASS; 2. CONSERVATIONS OF ENERGY;
00007 C 3. CONSERVATION OF , CONCENTRATION (SALINITY); 4. DENSITY DEFICIENCY
00008 C (DERIVED FROM 2)3); 5,6,7. MOMENTUM EONS IN THE Z (AXIAL),
00009 C K (VERTICAL) ) J(HORIZONTAL AND PARALLEL TO AMBIENT CURRENT) DIRECTION
00010 C EQN 4 IS USED. ONLY IN ALGEBRAIC FORM.
00011 C THE SIX
EQUATIONS ARE WRITTEN FOR A CONTROL VOLUME WHICH IS FINITE
00012 C IN A DIRECTION PERPENDICULAR TO THE JET AND INFINITESIMAL IN THE
00013 C DIRECTION OF THE JET AXIS.
00014 C
00015 C
00616 C
****** INPUT VARIABLES *******
00017 C
00018 C
00019 C INPUT CONSISTS OF THREE CARDS FOR EACH CASE RUN - SETS OF THREE
00020 C CARDS CAN BE STACKED FOR MULTIPLE RUNS
00021 C
00022 C
*** CARO NO. 1 ***
00023 C
00024 C 1. THE FIRST INPUT IS AN INDICATOR AS TO THE NUMBER OF RUNS
00025 C
REMAINING AFTER PRESENT RUN.
ZERO OR BLANK INDICATES LAST
00026 C
RUN OR A SINGLE RUN.
FORMAT
(I4)
00027 C 2. THE REST OF THIS CARD CONTAINS ANY INFORMATION THE USER WISHES
00028 C
TO HAVE PRINTED OUT AT THE TOP OF EACH PAGE.
00029 C
00030 C
*** CARD NO. 2 ***
FORMAT (8E10.0)
00031 C
00032 C 1. DISCHARGE VELOCITY (M/S)
00033 C 2. AMBIENT VELOCITY - (ASSUMED UNIFORM) (M/S)
00034 C 3. DISCHARGE TEMPERATURE (DEG C)
00035 C 4. DISCHARGE SALINITY (PPT)
00036 C
THIS VALUE CANNOT BF ZERO. FOR FRESH WATER SET TO LOW VALUE.
00037 C
0.0001 IS SUGGESTED IF A BETTER VALUE IS NOT KNOWN.
00038 C 5. AMBIENT TEMPERATURE AT DISCHARGE LEVEL (DEG C)
00039 C 6. AMBIENT SALINITY AT DISCHARGE LEVEL (PPT)
00040 C 7. LINEAR AMBIENT TEMPERATURE GRADIENT (DEG C/M)
00041 C 8. LINEAR AMBIENT SALINITY GRADIENT (PPT/M)
00042 C
00043 C
*** CARD NO. 3 ***
FORMAT
00044 C
00045 C 1. DISCHARGE DIAMETER (M)
00046 C 2. HORIZONTAL DISCHARGE ANGLE (DEG) RELATIVE TO THE AMBIENT CURRENT
00047 C
90 DEG IS FOR CURRENT NORMAL TO LINE OF DISCHARGE PORTS
00048 C
MODEL IS NOT GOOD FOR LARGE VARIATIONS FROM THIS DIRECTION
00049 C 3. DISCHARGE ANGLE (DEG) RELATIVE TO THE HORIZONTAL.
00050 C 90 DEG IS VERTICAL - ZERO IS HORIZONTAL DOWN STREAM
00051 C 4. DISTANCE ALONG PLUME CENTERLINE SIMULATION IS TO STOP
00052 C 5. DISTANCE BETWEEN PORTS (M)
00053 C
USE A LARGE VALUE TO SIMULATE SINGLE PORT DISCHARGES
00054 C 6. DEPTH OF DISCHARGE _(M)
00055 C
00056 C
00057
EXTERNAL DERIV,OUTP
-00058
-REAL
II1,1I2,II3,II4
00059
DIMENSION PRMT(5), AUX(16,6), F(6), FP(6) .N11(18)
00060
COMMON /AMH/ TIZ,CIZ,CIO,TIO,PIO,GA,AL,BE+FR,R1,FRL
00061
COMMON /OUT/ L,THI,TH2,H,Z•SNEW,DS.R,K,N11,J3,J2,J5,JJ,SPACE,SIGmA
00062
X.DO,CD,A7•XXX
00063
COMMON /PLTE/ J6+.17,J8,J9,J10,X1M,Y1M,X2M,Y2M,Y3M
00064
00065
00066
00067
00068
00069
00070
00071
00072
00073
00074
00075
00076
00077
00078
00079
00080
00081
00082
00083
00084
00085
00086
00087
00088
00089
00090
00091
00092
00093
00094
00095
00096
00097
00098
00099
00100
00101
00102
00103
00104
00105
00106
00107
00108
00109
00110
00111
00112
00113
00114
00115
00116
00117
00118
00119
00120
00121
00122
00123
00124
00125
00126
00127
00128
00129
COMMON /ZF9/ SET,XE,YE,ZEOBE9DCE,DTE,LCNT.RUE,RTE,RCE.T1E9T2E
COMMON /STR95/ II,M,R3
COMMON /CONS/ II1,112.II3,I14,III
COMMON/WGM/MERGCK•ROVO
MFRGCK=0
XXX = 0.16
A7 = 11.5
TI1 = .1285714
112 = .0667582
113 = 0.45*1.07034
114 = 0.31558*1.094
READ (5,102) J3,N11
TF(EOF(5).NE.0) GO TO 221
J5=0
•
'
RE=1.4253E—'4
G A=-7.6689E'.-4
C0=2.
os=4.
TTI=1
JJ = 1
J1=0
J2=1
J6=0
II=1
M = 16
C NI1 CONTAINS ALPHAMERIC INFO WHICH IS PRINTED IN OUTPUT. J3 IS ID NO.
102 FORMAT (I4,16A4,2A2)
220 PRINT 200, N11,J3
200 FORMAT (1H1."
SOLUTION TO MULTIPLE BUOYANT DISCHARGE",
I" PROBLEM WITH AMBIENT CURRENTS AND VERTICAL GRADIENTS",//,
2 20X91804," I.D.
".I49//)
42 CONTINUE
37 READ(5,105)U0.UA,TO.009TAO.CAO.JIZ,CIZ
105 FORMAT (8E10.6)
READ(59100)009TH1,TH2,SF,SPACE,H
PRINT 20009U09TO,C0900.SPACE,H
PRINT 20019UA,TA0,CA0
PRINT 2002,TIZoCIZ
FORMAT (1H ,"DISCHARGE VELOCITY = ",F5.2,"—M/S
2000
TEMP. = "9E5.2
X,"—DEG C
** SALINITY = "9E5.29"—PPT",/91H ," DIAMETER = "9E5.2,
X"--M
** SPACING = ".F5.2," — M
** DEPTH = ".F7.20—M")
FORMAT (1H ,"AMBIENT CONDITIONS AT DISCHARGE ELEVATION, VELOCITY
2001
X= ".F7.2,"—M/S ** TEMP. = ",F7.2."—DEG C ** SALINITY = "9E7.20
XPPT")
2002
FORMAT (1H ,"AMBIENT STRATIFICATION GRADIENTS ** TEMPERATURE
** SALINITY ** le,F10.6,"—PPT/M"//)
X ** "9E10.80 — DEG C/M
H = H/DO*2.
CALL SIGMAT(PIO.TO,C0)
CALL SIGMAT(PAO,TAO,CAO)
ROV0=0.5*0O/U0
R=UA/U0
BE=BE*TO/PIO
GA=GA*CO/P10
PID=ABS(PAO—P10)
FR=SORT(PID/PIO*00*9.8)
FR=UO/FR
TIZ=TIZ*00/(2.*TO)
CIZ=CIZ*00/(2.*C0)
TIO=TAO/TO
CIO=CAO/C0 SF = SF/DO*2.
SPACE = SPACE/DO
100 FORMAT (8F10.6)
9876 RI = R
RE = 1.0 E•6
PIO = 1.0 •BE*(1.0,—TIO) •GA*(1.0—CIO))
00130
PTZ = — BE*TIZ — GA*CYZ
00131
PRINT 207, TIO,CIO,PIO,TIZ,CIZ,PIZ
00132
207 FORMAT (1H ," AMBIENT CONDITIONS AT DISCHARGE
_
00133
1" NONDIMENSIONAL",/, " TEMP=",F8.5.2X,"SALINITY=",F8.5,2X,
00134
Z "DENSITY=",F8.5,3X."TEMPGR=",F9.6,2X,"SALGR=".F9.6,2X,
00135
3 "DENGR=",F10.7./)
00136
40 PRINT 202, SPACE
00137
202 FORMAT (18 ,"
PORT SPACING L/D = ",F62)
00138
IF (ABS(PIZ) — 1.0E-10) 90,90,91
00139
90 T = 1.0 E+8
00140
GO TO 92
00141
91 T = (P/0-1.0)/(—PIZ)
00142
92 PRINT 206, FR,R,T
00143
206 FORMAT (1H0," FROUDE NO- =" • ,F9.2,4X,"VELOCITY RATIO =",
00144
1 F7.3,4X,"STRATIFICATION NO =-",F7.1,/)
00145
FR = 2.*FR*FR
00146
SPACE = 2.*SPACE
00147
nsil = DS
00148
TTT1 = TH1
00149
TTT2 = TH2
00150
R3 = R*SIN(0.01745329*TH1)*COS(0.01745329*T82)
00251
IF (R3-1.0) 60,61.60
00152
61 P3 = 0.0
00153
60 CALL ZFE
00154 C ZFF TAKES CARE OF COMPUTING IC AT END OF ZFE.
00155
DS = DS11
00156
THI = TTT1*0.01745329
00157
TH2 = TTT2*0.01745329
00158
TIE = TIE*0.01745329
00159
T2E = T2E*0.01745329
00160 C CONVERT TH FROM DEGREES TO RADIANS
00161 C TH ANGLE AT S=0, TE = ANGLE AT SE
00162
SI = SIN(T1E)
00163
C2 = COS(T2E)
00164
RS12 = R*S1*C2
00165
TF (RE — 1.0E+4) 20,21,21
00166
20 PRINT 214, RE
00167
214 FORMAT (1H ,"
FLOW MAY NOT HE FULLY TURBULENT, RE = ",F8.0s/)
00168
21 PRMT(1) = SET
00169
PRMT(2) = SF
00170
PRMT(3) = DS
00171
PRMT(4)= 0.001
00172
36 L = —I
N=6
00173
00174 C L=-1 USED TO INITIALIZE OUTP.
00175
K=
0
•
00176 C K COUNTS NO OF INTEGRATIONS,
00177
DO 2 I=1,6
2
00178
FP(I) = 0.1666667
00179
SIGMA = BBE
00180
RBE2 = BBE*BBE
00181
-F(1) = BBE2*(II1+(.5—II1)*RS12).
00182
F(2)=DTE*BBE2*(II2+(I11—II2)*RS12)
00183
F(3) = DCE*RBE2*(II2+(111—II2)*RS12)
00184
F(4)=88E2*II2*(1.—RS12)**2+2.*(1.+-RS12)*RS12*BBE2*III
00185
1+RS12*RS12*PBE2*.5
00186
F(5) = TIE
0.0187
F(6)- = T2E
00188 C SET _INITIAL CONDITIONS AT ENO OF POTENTIAL CORE.
00189
PRINT 208
.
00190
208 FORMAT (1H0,20X,"ZONF OF ESTABLISHED FLOW",/
00191
1
IH .5X."S", 8X,"X",7X,"Y",7X,"Z",6X."TH1",5X,"TH2",6X,
_00192
1"WIDTH",5X,"DUCL",5X,"DTCL",5X0DCCL"t5XOPI"t7XOTIME",5X,"0/110"
00193
X,/)
00194
CALL OUTP (PRMT(1),F.FP.IHLF,N.PRMT)
00195
CALL HPCG(PRMT.F,FP.N,IHLF.DERIV,OUTP,AUX)
-
00196
00197
00198
00199
00200
00201
00202
00203
00204
00205
00206
00207
00208
00209
00210
00211
00212
00213
00214
00215
00216
00217
00218
00219
00220
00221
00222
00223
00224
00225
00226
00227
00228
00229
00230
00231
00232
00233
00234
00235
00236
00237
00238
00239
00240
00241
00242
00243
00244
00245
00246
00247
00248
00249
00250
00251
00254
00255
00256
00257
00258
00259
00260
00261
IF (IHLF — 11) 10.11911
CALL EXTR (S,X,Y,Z,B.DU.DT,5,00)
11
10 PRINT 212. K.IHLF,PRMT(4),PRMT(5).PRMT(3)
212 FORMAT (1H ../."
NO OF INTEGRATION STEPS=".I4.3X."NO OF HALVINGS"
11"=".I3.3Xt"ABSERR=".F9.5.3X."PRMT(5)=".F9.3." FINAL SPACE ="
2 ,F9.3)
GO TO 1
221 STOP
END
SUBROUTINE SIGMAT(RHO.T.S)
SIG0=(((6.8E-6*S)-4.82E-4)*S+.8149)*S—.093
C=1.E-6*T*((.01667*T—.8164)*T+18.03)
0=.001*T*(1.0010843*T—.09818)*1+4.78671:
SUMT=(T-3.98)**2*(7+283.)/(503.57*(7+67.26))
RHO =(SIG0+.1324)*(1.—D+C*(SIGO—.1324))—SUMT
RHO =RHO/1000.+1.
RETURN
END
SUBROUTINE OERIV (S,F,FP)
DI M ENSION F(6),FP(6), NI1(18)
COMMON /OUT/ L.TH19TH2sHsZ,50L1).D5994K.N114J39J2,J5,JJ,SPACEtSIGMA
X.DO.0O347,XXX
S1=SIN(F(5))
S2=SIN(F(6))
.
C1=COS(F(5))
C2=COS(F(6))
71 = Z+ 52*(5 — SOLD)
(Z1.CI,TI.PI.TIZ.CIZsPIZ.VTI.VCI.VUI.VV1)
CALL AMBIEN
C AMBIENT COMPUTES GRADIENTS OF T.0 ) TURBULENCE QUANTTIES IN FR STREAM.
CALL PRFLE (F.F.8.E.8)
C PRFLF COMPUTES INTEGRAL OUANITITIES WITHIN THE JET
C F8 IS THE INTEGRAL OF DENSITY DIFFERENCE. AND B THE WIDTH OF TE JET
FP(1) = E
FP(2)= — F(1)*TIZ*52 + VII
FP(3) =—F(1)*CIZ*S2 + VCI
FP(4) = E*R*S1*C2 +FR*52 + VU!
F9 = F(4) -• 0.25*E*E • VVI
IF (4BS(C2) — 0.01) 1.1. 2
F(5) = F(6)
1
FP(5) = 0.0
GO TO 3
2 FDY=0.
11NX=—(C2**2*C1*S1)
UNY=S2**2+C2**2*C1**?
UNZ=—(S2*C2*S1)
UN=SORT(UNX**2+UNY**2+UNZ**2)
CDP = CD*SRACE/B
IF -(SPACE/B.GE.2.) COP = CD*B/SPACE
FD = CDP*B*R*R*UN/6.2832
YY=ABS(Cl*C2)
IF(YY.GT.0.)FDY=F0*(52**2+C2**2*C1**2-51**2*S2**2)/(F9*C1*C2)
FP(5) = (E*R*C1)/(C2*F9)+FDY
FP(6) = (F8*C2
— (E*R+FD)*S1*S2)/F9
3
K=K+I
RETURN
END
SUBROUTINE XINT(I.A.FIINT.F2INT,F3INT)
DIMENSION XA(25).XI1(25).X12(25)
DATA (XA=1.0E-10,1..1.1.1.2.1.3.1.4.1.5.1.6.1.7.1.8.1.9. XP.09100.),(XI1=.4314..4314..42459.42599..42389.42219.4209,
X.4201..4196..4194..4193..4193).(X12=.436..3905..3576.
X.3348..3201..3114..3068..3047..304..3038..30377..30377)
I IF(A — XA(I)) 2.3.4
2 I = I-1
GO TO 1
3 FlINT=XII(I)
00262
00263
00264
00265
00266
00267
00268
00269
00270
00271
00272
00273
00274
00275
00276
00277
00278
00279
00-2'80
00281
00282
00283
00284
00285
00286
00287
00288
00289
00290
00291
00292
00293
00294
00295
. 00296
00297
00298
00299
00300
00301
00302
00303
00304
00305
00306
00307
00308
00309
00310
00311
00312
00313
00314
00315
_ 00316
00317
00318
00319
00320
00321
00322
00323
00324
_00325
00326
00327
F2INT=XI2(I)
GO TO 8
4 IF(A-XA(I+1))5,6,7
6 FlINT=XI1(I+1)
F2INT=XI2(I+1)
1=1.1
C
C
C
C
C
C
GO TO 8
7 1=1+1
GO TO 4
S rlINT=XI1(1)+(XI1(1+1)-XI1(I))/(XA(1+1)-XA(I))*(A-XA(1))
F2INT=X12(I)+0(I2(I+1)-XI2(I))/(XA(I+1)-)0(I))*(A-XA(1))
A F3INT= A*SORT-(1.-(A/2.)**2)/2..+ASINE(A/2.)
RETURN
END
SUBROUTINE AMBIENT (Z+CI+TI,PI,TIZ,CIZ,PIZ,VTI,VCI,VUI,VVI)
AMBIENT COMPUTES H THE'VALUES OF PARAMETERS IN THE FREE STREAM.
COMMON /AMB/ TIZO,CIZO,C10,7101PIO.GA,AL,BE,FR,R1+FRL
TIZ=TIZO
CIZ=CIZO
PIZ= -8E*TIZ - GA*CI7
GRADIENTS OF T,C,P
CI'
CIZ*Z • CIO
TI = TIZ*Z + TIC)
PI = PIZ*Z • PIO
FREE-STREAM VALUES OF C,T.P
VTI = 0.0
VCI = 0.0
VUI = 0.0
VVI = 0.0
FRET-STREAM VALUES OF TURBULENCE LEVELS
RETURN
END
SUBROUTINE PRFLE (F.FR,E,B)
REAL
111,112,113,114
DIMENSION F(6), N11(18)
COMMON /CONS/ 111,112,113,114,111
COMMON /A.M8/ TIZ,CIZ,CIO,TIO+PIO,GA,AL,AE,FR,R1,FRL
COMMON /OUT/ L,TH1,TH2,H,Z,SNEW,DS,R,K,N11,J3,J2;J5,JJIISPACE+SIGMA
X.00.CD,A7,XXX
COMMON /WGM/MERGCK.ROVO
I =III
pI=3.14159
IF (F(4)) 12,12,11
11 512=SIN(F(5))*COS(E(6))
012=R*512
**** SINGLE PLUME ****
A = SPACE/SIGMA
TF(4-2.)501,500+500
501 TF(A-.95)503,503,502
500 C1=II1
C2=U12/2.
C3=II2
C4=2.*U12*111
C5=U12*U12/2.
C6=II2
C7=U12*II1
GO TO 504
**** MERGING PLUME ***4*
502 CONTINUE
519 FORMAT (/20X,"ZONE OF MERGING
IF(MERUCK.E0.0)PRINT 519
MERGCK=1
CALL XINT(I.A,AII+AI2,AI3)
C2=U12*AI3/PI
C3=2.*114*Al2/PI
00328
C4=4.*II3*AI1*U12/PI
00329
C5=U12*U12*A13/PI
00330
C6=2.*114*Al2/PI
00331
C7=2.*I13*U12*AI1/PI
00332
GO TO 504
00333 C **** MERGED PLUME ****
00334
503 CALL XINT ( I •A,AI 1 tAI2.413)
00335
C1=.45*A13/P1
00336
C2=U12*AI3/PI
00337
C3=.315S8*A13/P1
00338
C4=2.*U12*C1
00339
C5=U12*C2
00340
C6=C3
00341
C7=0.5*C4
00342
504 TE(ABS(U12)—.001)112.112.111
00343
112 RSOR=E(1)*E(1)*C3/(F(4)*C1*C1)
00344
GO TO 12
00345
113 AA=C3*C2*C2/C1/C1—C4*C2/C1tC5
00346
88=C4*F(1)/C1—C3*2.*C2*E(1)/C1/C1—F(4)
00347
CC=C3*F(1)*F(1)/C1/C1
00348
RSOR=-88-5URT(81.3*BB-4.*AA*CC)
00349
8SQR=BSOR/(2.*AA)
00350
12 R=SORT(BSQR)
00351
SIGMA = B
00352
DLUM=E(1)/(RSOR*C1)—C2/C1
00353
UCL = DLUM+U12
00354
DTCL = F(2)/(RSQR*(OLUM*C6+C7))
00355
DCCL = F(3)/(8SUR*(0LUM*C6+C7))
00356
PICL
RE*DTCL + GA*DCCL
00357
IF (FR — 0.98E+6) 1,2,2
00358
1
IF(A.GT.2.)F8=PICL*BSOR*II1/((PIO-1.)*ER)
00359
TF(A.LT.2.)E8=2.*BSOR*I/3*PICL*AI1/(PI*FR*(PI0-1.))
00360
TF(A.LE..95)F8=F8*A13/A11/2.14068
00361 C
COMPUTE F8 — INTEGRAL OF DENSITY DIFFERENCE
00362
FRLI = PICL*B/(FR*(P10-1.0)*UCL*UCL)
00363
GO-TO 3
00364
2
F8 = 0.0
00365
FRLI = 0.0
00366
3
,IF (ABS(FRLI) — 10.0*ABS(1.0/FRL)) 10,13,13
00367
13 FRLI = 1.0/FRL
00368
10 CONTINUE
00369
Al = .05
00370
422 = 0.0
00371
AA = Al • A22*FRLI*SIN(F(6))
00372
CC=A7*R*SORT(1.0-512*S12)
00373
813=8*ABS(UCL—R*S12)
00374
28 IF(A.GE.2.)E=AA*(88*(1.—XXX/A)+CC*8)
00375
TF(A.LT.2.)E=AA*(88*(1.—XXX/2.)*(1.-2./PI*ACOSF(A/2.))
00376
X+ CC*SPACE/2.)
00377
ITI=1
00378 C
COMPUTE E — ENTRAINMENT FUNCTION
00379
RETURN
00380
END
00381
SUBROUTINE OUTP (S.F.FP.IH.N.PRMT)
00382
REAL
1111112.113.114
00383
DIMENSION F(6).FP(6),PRMT(5).50(4),SN(4).N11(18)
00384
COMMON /OUT/ L.TH1.TH2.H,Z.S1.0S.R.K.N11.J3.J2,J5IJJ,SPACE.SIGMA
00385
X..00.CD.A7,XXX
00386
COMMON IAMB/ TIZ.CIZ,CIOtTIO*PIO.GA,AL,BF,FR.R1,FRL
00387
COMMON /PLTE/ J6.J7.J8.J9,J10.X1M.Y1M.X2M9Y2M.Y3M
00388
COMMON /ZF9/ SET.XE+YE.ZE.BBE.00E,DTE.LCNT.RUE.RTE.RCEs T1E+T2E
00389
COMMON /STR95/ 11.M,R3
00390
_COMMON /CONS/ 111.112,113.114+III
00391
COMMON/WGM/MERGCK.ROVO
00392
I=III
00393
P11=3.14159
00394
IF (L) 6,8,7
00395
6 L= 0
00396
L19 = 0
00397
ml =-1
00398
M2 = 0
00399
71 = -DS
00400
SNEW = PRMT(1)
00401
SN(1) = SIN(T1E)
00402
SN(2) = SIN(T2E)
00403
SN(3) = COS(T1E)
00404
SN(4) = COS(T2E)
00405
TIME = SET
00406
X = XE
00407
Y = YE
00408
Z = ZE
00409
UCC = 1.0
sl= PRMT(1)
00410
00411
T3I = TI0
00412
C3I = CIO
00413
IF(CIO-1.0.LT..000001)C3I=0.
00414
IF (110- 1.0) 63,64,63
00415
64 T3I = 0.0
00416
63 GO TO 10
00417
8 L=1
00418
SNEW = DS + SI
00419
SO(1) = SN(1)
00420
SO(2) = SN(2)
00421
SO(3) = SN(3)
SO(4) = SN(4)
00422
7
IF (M2) 18.18,19
00423
00424
19 ns = S-SNEW
00425
TF (S-51 - 1.0) 83.1,1
00426
18 IF (S - SNEW) 20,111
00427
20 TF (A9S(S-SNEW) - 0.001*SNEw) 1,1,83
00428
1 SN(1)=SIN(F(5))
00429
SN(2)=SIN(F(6))
00430
SN(3)=COS(F(5))
00431
SN(4)=COS(F(6))
00432
X= X +0.5*(SN(3)*SN(4) • SO(3)*S0(4))*(5-51)
00433
Y = Y +0.5*(SN(1)*SN(41 +50(1)*S0(4))*(S-S1)
00434
71 = Z
00435
Z=Z+0.5*(SN(2) + S0(2))*(S-51)
00436
no 9 1=1.4
00437
9
SO(I)=SN(I)
00438
10 PIZ = -9E*TIZ - GA*CTZ
00439
TI F TIO + TIZ*Z
00440
CI = CIO + CIZ*Z
00441
PI = PIO + PIZ'Z
00442
TF (F(4)) 21.21,55
00443
55 R14=R*SN(1)*SN(4)
00444
A=SPACE/SIGMA
00445
IF(A-2.)501,500.500
00446
501 IF(A-.95)503,503.502
00447 C **** SINGLE PLUME ****
00448
500 C1=II1
00449
C2=R14/2.
00450
C3=II2
00451
C4= 2. *R14*II1
r5=R14*R14/2.
00452
00453
C6=II2
00454
C7=1I1*R14
00455
GO TO 504
**** MERGING PLUMES
00456
00457
502 CONTINUE
00458
519 FORMAT(/20X, fl ZONE OF
00459
1F(MERGCK.E0.0)PRINT
00460
MERGCK=1
00461
CALL XINT(I,A,A11,AI2,AI3►
00462
C1=2.*I13*AI1/PII
00463
C2 =R14*AI3/PII
00464
C3=2.*I I4*A I2/P I I
00465
C4=4.*II3*AI1*R14/PI I
00466
C5=R14*R14*4 I3/P I I
00467
C6=2.*114*412/PI I
00468
C7=2.*II3*R14*AI1/PIT
00469
GO TO 504
00470 C **** MERGED PLUMES ****
00471
503 CALL XINT(1,A,A11,AI2+AI3►
00472
C1=.45*A13/PII
00473
C2=R14*A13/PII
00474
C3=.31558*A13/PII
00475
C4=2.*R14*C1
00476
C5=R14*C2
00477
C6=C3
00478
C7=0.5*C4
00479
504 iF(ABS(R14) — .001) 12.12.99
00480
12 RSOR=F(1)*F(1)*C3/(F(4)*C1*C1)
00481
GO TO 15
00482
99 AA=C3*C2*C2/C1/C1—C4*C2/C1+C5
00483
RB=C4*F(1)/CI—C3*2.*C2*F(1)/C1/C1—F(4)
00484
CC=C3*F(1)*F(1)/Cl/C1
00485
RSOR = —88—SORT(88*B8-4.*AA*CC)
00486
RSUR = BSOR/(2.*AA)
00487
15 R = SQRT(BSQR)
00488
SIGMA =8
00489
SIGMAO =SIGMA*00
00490
mum = F(1)/(BSOR*C1)—C2/C1
00491
UCL = DLUM • R14
00492
nTa. = F(2)/(BSwF(*(DLUM*C6+C7))
00493
DCCL = F(3)/(BSOR*(DLUM*C6•C7))
00494 C THE ABOVE CARDS DEFINE THE COEFFICIENTS OF THE CONTINUITY,MOMENTUM
00495 C AND ENERGY EQUATION FOR THE SINGLE PLUME, MERGING PLUME AND
' 00496 C MERGED PLUME REGIONS BASED ON PROFILES OF THE FORM (1.—(X/8)**3/2)
00497 C SQUARED
00498
DELU = (UCL — R14)/(1.0 — R3)
00499
DELT = DTCL/(1.0 — T3I)
00500
DE-Lc = DCCL/(1.—C3I)
00501
PCL = PI — RE*OELT*(1.0-731) —GA*DELC*(1.0—CIO)
00502 C DELU= (UCL
UI*S1*C2)/(00 — UI)
DELT = (TCL — TI)/(TO — TI)
00503
F55 = F(5)*57.29578
00504
F66 = F(6)*57.29578
00505 C CONVERT ANGLES BACK TO DEGREES
00506
UMA = 0.5*(UCL • UCC)
00507
TIME = TIME • (S-51)/UMA
00508
IF (FR — 0.98E+6) 110.111,111
00509
110 IF (ABS((PI — PCL)*B) — 1.0E-6*A8S(UCL**2*FR*(PIO-1.0))) 1129113.113
00510
113 FRL = FR*(PIO-1.0)*UCL*UCL/((PI—PCL)*B)
00511
GO TO 112
00512
111 FRL = FR
00513
112 IF (L19) 70.70,71
00514
70 SO0 = S/2.*DO
00515
X0 = X/2.*DO
00516
YO = Y/2.*DO
00517
ZO = Z/2.*DO
00518
R0=2.*F(1)
00519
TIMER=TIME*ROVO
00520
PRINT 104.500,X0,Y0,70,F55.F66.SIGMAO.DELU.DELT,DELC.
00521
*PI,TIMER.R0
00522
104 FORMAT (1H
F9.2,1X.5F8.2, 1X,F8.2,3F9.3,1X,F9.5, 2E9.2)
00523
MI = M1 " a 1
00524
M=M+1
00525
L19 = L19 •
00526
00527
00528
00529
00530
00531
00532
00533
00534
00535
00536
00537
00538
00539
00540
00541
00542
0A543
00544
00545
00546
00547
00548
00549
00550
00551
00552
00553 C
00554 C
00555
00556
00557 C
00558
00559
00560
00561
00562
00563
00564
00565
00566
00567
00568
00569
00570
00571
00572
00573
00574
00575
00576
00577
00578
00579
00580
00581
00582
00583
00584
00585
00586 C
00587 C
-00588
00589
00590
00591
GO TO 72
71 LI9 = 0
72 PICL = PI-PCL
PICLO = PI0-1.0
IF (M2) 40,40,42
40 IF (SIGN(1.0,PICL) - SIGN(1.0,PICLO)) 41,42,41
41 DS = 0.0
MI = 0
M2 = 1
MEXT=1
PRINT 213
213 FORMAT (1H ,10X,"PLUME HAS REACHED ITS EQUILIBRIUM HEIGHT - I',
1 " STRATIFIED ENVIRONMENT")
42 IF (M1
10) 36,32,32
32 MI = 0
OS = 2.0*DS.
36 IF (Z) 3,3,31
31 IF (M2) 90,90,91
91 TF (Z-Z1) 21,90,90
21 PRMT(5) = 1.0
TH = 11
PRINT 212
212 FORMAT (1H ,9X," PLUME HAS REACHED ITS MAXIMUM HEIGHT - "
1 "STRATIFIED ENVIRONMENT",/)
IF (J6) 83,83,81
90 IF (Z-H) 3,3,4
4 PRMT(5) = 1.0
THE CARDS BELOW ARE FOR USE WHEN THE PROGRAM IS USED FOR
DISCHARGE INTO WATER OF DEPTH H. THIS MUST BE READ INTO PROGRAM
PRINT 214
214 FORMAT (1H ,"PLUME HAS REACHED WATER SURFACE",/)
IF WE HAVE REACHED THE WATER SURFACE - STOP.
IF (J6) 83,83.81
3
SNEW = SNEW + OS
SI = S
UCC = UCL
TF (M2) 52,52,51
51
ns
0.0
MI = 0
CALL EXTR (S,X,Y,Z,STGMA,DELU,DELT,MEXT,DO)
MEXT= MEXT+ 1
52 IF (M-40) 2,13,13
13 PRINT 207,N11,J3
207 FORMAT (1H1,1OX, 1844, " I.D. ="014,/)
PRINT 208
208 FORMAT (1H ,5X,"S", 8X,"X",7X,HY",7X,"Z",6X+"TH1",5XOTH2",6X,
1"WIDTH",6XODUCL".5X0DTCL",5X0OCCL",5X0PI",7X4-"TIME",5X0Q/Q
X0"/)
M=0
2 _IF (J6) 83,83,84
84 IF (SNEW-PRMT(2))
81 CONTINUE
PRMT(5) = 1.0
RETURN
80 IF (M2 -1) 83,85,83
85 IF (ABS(Z-Z1) -0.05*PRMT(3)) 81,83,83
83 RETURN
END
SUBROUTINE EXTR (S,X,Y,Z,B,DU,DT,M,00)
DIMENSION S1(5),X1(5),Y1(5),Z1(5),B1(5),DU1(5),DT1(5)
THIS SUBROUTINE STORES VALUES OF THE OUTPUT WHEN THE PLUME HAS PASSED
ITS EQUILIBRIUM HEIGHT IN A STRAT AMBIENT. AT THE END OF THE
RUN, THIS ROUTINE-COMPUTES THE MAX HEIGHT-OF-PLUME BY EXTRAPOL-ATION.
IF (M°3) 1,1,2
2 TF(M-4) 3,3,4
3
no 51=1,2
00592
S1(I) = S1(I*1)
00593
Xl(I) = X1(I+1)
00594
sem) = Y1(1.1)
00595
zI(r) = 11(m)
00596
R1(I) = B1(I+1)
00597
OU1(I) = DU1(I*1)
00598
DT1(I) = DT1(I+1)
00599
5
CONTINUE
00600
M=3
00601
51(M) =
00602
X1(M) = X
00603
Y1(M) =!.Y
00604
Z1(4) = Z
00605
RI(M) = B
00606
nul(m)
DU
00607
011(m) = DT
00608
MR=M
00609
RETURN
00610
4
IF (MR-3)- 202.69202
6 mp 2 = MR-2
00611
00612
MR1=MR-1
00613
7MAX = Z1(MR1) * DUl(MR1)*(Z1(MR1)—Z1(MR))/(DUl(MR)-4)U1(MR1))
00614
7MAX = ZMAX*DO
00615
DZ1 = Z1(MR1) — Z1(MP)
00616
022 = z1(mm2) - Z1(MR)
00617
DUll = oultmR1) - DU1("R)
00618
0u12 = DUI(MR2) - oul(mR)
00619
nEr
DzI*072*(oz2-071)
00620
DErt
1.0/DET
00621
R = DETI*(072*0Z2*OU11 — DZ1*OZ1*DU12)
00622
C = DETI*(-072*DU11
DZ1*DU12)
00623
DISC. = R*8-4.0*DU1(MR)*C
00624
IF (DISC) 206.205.205
00625
206 7M = 0.0
00626
GO TO 207
00627
205 7M = (Z1(MR) — (B*SORT(DISC))/(2.0*C))*00
00628
207 PRINT 102.ZMAX, ZM
00629
102,FORMAT (1H0,"
u,
MAXIMUM PLUME HEIGHT. EXTRAPOLATED
00630
1 "LINEAR = "'F8.2."
PARABOLIC =",F8.2,/)
00631
RETURN
- --00632
202 PRINT 203
00633
203 FORMAT (1H s" ERROR IN HPCG, IHLF =-11",/)
00634
RETURN
00635
END
00636
SUBROUTINE ZFE
00637
C THIS PROGRAM COMPUTES THE SOLUTION TO BUOYANT JET PROBLEMS WITHIN
00638
C THE ZONE OF FLOW ESTABLISHMENT, I.E. UP TO XE.
00639 C THE SOLUTION CALCULATES RU,RT,RCIPH,TH1,TH2 AS FUNCTIONS OF S.
00640
DIMENSION PRmr(s).Aux(16,6),E(6),FP(6),N11(18)
00641
COMMON /AMR/ TIZ.CIZoCIO.TIO.PIO.GA,AL,RE.FR.R1.FRL
00642
COMMON /OUT/ L.rmlo-H2.m,z.sNEw.os,m.K,N11.„13....)2,15,J.J.sPAcE,siomA
00643
X.Do.co,Ar,xxx
00644
COMMON /ZF9/ SET.XE.YE.ZE.BRE,DCE.DTE.LCNT,RUE.RTE.RCE.T1E.T2E
00645
EXTERNAL DERIV1.OUTRI
00646
PIZ = — 13E*TIZ — GA*CIZ
00647
TH1 = 0.01745329*TH1
00648
TH2 = 0.01745329*T02
00649
PRMT(1) = 0.0
00650
PRMT(2) = 100.0
00651
PRMT(3) = 0.01
00652
PRMT(4) = 0.0001
00653
IFAABS(COS(TH2))
00654
TH1 = TH2
00655
N=6
00656
L=0
00657
LCNT =
00658
00659
00660
00661
00662
00663
00664
00665
00666 C
00667
00668
00669
00670
00671
00672
00673
60674
00675
00676
00677
00678
00679
00680
00681
00682
00683 C
00684
00685
00686
00687
00688
00689
00690
00691
00692
00693
4
00695
00696
00697
00698
00699
00700
00701
00702
00703
00704
00705
00706
00707
00708
00709
00710
00711
00712
00713
00714
00715
00716
00717
00718
00719-00720
00721
00722
00723
no 31=1,6
3
FP(I) = 0.16666667
F(1) = 1.0
F(2) = 1.0
F(3) = 1.0
F(4) = 0.0
F(5) = TH1
F(6) = TH2
F(1) = RU. F(2) = RT. F(3) = RC. F(4) =8, F(5) = TH1, F(6) = TH2
IHLF = 0
PRINT 210
210 FORMAT 11H0,20X,"ZONF OF FLOW ESTARLISHMENT
ALL LENGTHS ",
1uARE IN METERS '
",/
1H ,5X,"S",9X,"X",7X,"Y",7X,"Z",6)(0TH1",5X,"TH2",9X
1
1 ."8 ".5X,"RU",6XORT",6XORC",5XODUCL",4X0DTCL",4X,"DCCL",
? 6XOPI",74,"TIME",/)
CALL OUTP1 (PRMT(1),F,FP,IHLF,N.PRMT)
CALL HPCG (PRMT,F,FP,N,THLF,DERIVI,OUTP1,AUX)
IF (L+2)6.21.6
21 F(1) = RUE
F(2) = 1.0 ••• TIO • TIZ*Z
F(3) = 1.0 — CIO — CIZ*Z
F(4) = BRE
F(5) = TlE*0.01745329
F(6) = T2E*0.01745329
FI=RU/RO, F2=(TM — TI)/(TO—TI), F3=(CM — CI)/(CO — CI), F4 =8/RO
TH1 = F(5)
TH2 = F(6)
DO 9 1=1,6
9
FP(I) = 0.1666667
PRMT(1) = SET
PRMT(2) = SET + 100.0
PRMT(3) = 0.01
PRMT(4) = 0.0001
IHLF = 0
CALL OUTP1 (PRMT(1),F,FP,IHLF,N,PRMT)
CALL HPCG (PRMT,F,FP,N,THLF,DERIV1,OUTP1.AUX)
GO TO 95
6
PRINT 231, IHLF, PRMT(5)
231 FORMAT (180," ERROR IN HPCG. NO. OF HALVINGS = 11 913," PRMT5=".F8.2)
95 RETURN
END
SURROUTINE DERIV1 (S,F,FP)
DIMENSION F(6),FP(6),A(4,4),N11(18),A2(3.3),A3(3,3)
COMMON /AM8/ TIZ,CIZoCIO,TIO,PIO,GA,AL.8E,FR,R1,FRL
COMMON /OUT/ L,TH1,TH2,H,Z,SNEW0DS,R,K,N11,J3,J2,J5,JJ,SRACE,SIGMA
X,DO,CD,A7,XXX
COMMON /ZF9/ SET,XE,YE.ZE,8RE,DCE,DTE,LCNT,PUE,RTE,RCE.TIE,T2E
•
VUE=0.
VTE=0.
VCE=0.
VVE=0.
Fl = 1.06
F2 = 34.0
F3 = 6.0
F4 = .4
S1 = SIN(F(5))
S2 = SIN(F(6))
Cl
COS(F(5))
C2 = COS(F(6))
R12 = R*51*C2
--512 =-SI*C2
DT0=1.0-0.10 — TIZ*Z
oc0 =1.0-cIo - CIZ*Z
E = E1*(0.02044.0.0144*F(4))*(1.0+E2*S2/FR) *(A8S(1.0 — R12) +E3*R
1 •SORT(1.0-.512**2))*(1—E4/5PACE)
•
00724
00725
00726
00727
00728
00729
00730
00731
00732
00733
00734
00735
00736
00737
00738
00739
00740
00741
00742
00743
00744
00745
00746
00747
00748
00749
00750
00751
00752
00753
00754
00755
00756
00757
00758
00759
00760
00761
00762
00763
00764
00765
00766
00767
00768
00769
00770
00771
00772
00773
00774
00775
00776
00777
00778
00779
00780
00781
00782
00783
-00784
00785
00786
00787
00788
00789
C E IS THE ENTRAINMENT FUNCTION.
01=.45+.55*R12
011=2.*(.12857+.37143*R12)
02=.315584.13442*R12
03=2.*(.06676+.06181*R12)
04=.31558+.26884*R124.41558*R12*R12
05=2.*(.066764.12362*R124.30962*R12*R12)
06
DTO*BE*(F(2)*F(2)*0.5 +0.12857*F(4)*F(4) +.45*F(2)*F(4))
06=136+ DCO*GA*(F(3)*F(3)*0.5 4. 0.12857*F(4)*F(4) +.45*F(3)*F(4))
0 = 0.5*F(1)*F(i) +F(1)*F(4)*04 •F(4)*F(4)*0.5*05 -0.25*E*E
AAI =0.5*F(1)-*.F(1) +01*F(1)*F(4) +011*0.5*F(4)*F(4)
A(1,1) = F(1)
D1*F(4)
A(1.2) = 0.0
A(1.3) = 0.0
A(1.4) = D1*F(1) + D11*F(4)
C FIRST EQUATION IS CONTINUITY.
A(2.1) = 0.0
A(2.2) = DTO*F(2) + 02*RTO*F(4)
A(2,3) = 0.0
A(2,4) = DT0*(02*F(2) + 03*F(4))
C SECOND EQUATION IS ENERGY.
A(3.1) = 0.0
A(3.2) = 0.0
A(3,3) = DC0*(F(3) + D7*F(4))
A(3,4) = DC0*(02*F(3) + D3*F(4))
C THIRD EQUATON IS CONCENTRATION.
A(4,1) = F(1) + 04*F(4)
A(4.2) = 0.0
A(4,3) = 0.0
A(4.4) = D4*F(1) + DS*F(4)
C FOURTH EQUATION IS MOMENTUM
FP(1) = E
TIZZ = TIZ
FP(2) = TIZZ*S2*(-AAT+0.5*F(2)**2
F(2)*F(4)*D2 +0.5*F(4)**2*D3)
FP(2)=FP(2)+VTE
FP(3) = CIZ*S2*(-AAI +0.5*F(3)**2 +F(3)*F(4)*D2°+0.5*F(4)**2*D3)
FP(3) = FP(3)+VCE
jF (L-2) 3.4,3
4
A(2,2) = 0.5*03*F(4)**2
A(2,4) = D3*F(2)*F(4)
A(3.3) = A(2,2)
A(3.4) = 03*F(3)*F(4)
FP(2)=-TIZZ*S2*AAI•VTE*0TE
FP(3) = -CIZ*S2*AAI+VCE*DCF
06 = (F(2)*BE 4 F(3)*GA)*0.12857*F(4)**2
3
IF (FR - 0.98E4. 6) 11.12,12
11 FR = D6/(FR*(P10-1.0))
GO TO 13
12 F8 = 0.0
13 FP(4) = F8*S2 • E*R12+VUE
J=4
IF(DCO.E0.0..AND.OTO.E0.0.)G0 TO 10
IF(070.EQ.0.)G0 TO 6
IF(DCO.EQ.O.)GO TO 20
CALL SIMQ(A.FP,J,K)
GO TO 7
C SIMO SOLVES THE SYSTEM OF J LINEAR EQUATIONS FOR THE FP. A IS THE
C COEFFICIENT MATRIX IN EON A.FP = FP. THE INPUT FP ARE DESTROEY0
C BY SIMQ AND REPLACED BY THE SOLUTION VECTOR, FP.
•
C
C INSERTIONS FOR DCO AND/OR DTO =0 PER SHIRAZI 10/6/71
C
10 J=2
02(1.1)=A(1.1)
A2(1.2)=A(1.4)
A2(2,1)=A(4.1)
00790
A2(2,2)=A(4.4)
00791
FP(2)=FP(4)
00792
CALL SIMQ(A2.FP.J,K)
00793
GO TO 7
00794
20 J=3
00795
A3(191)=A(1.1)
00796
A3(1.2)=0.
00797
A3(1.3)=A(1.4)
00798
A3(2,1)=0.
00799
A3(2.2)=A(2.2)
00800
A3(2,3)=A(2,4)
00801
A3(3.1)=A(4,1)
00802
A3(3.2)=0.
00803
A3(3.3)=41494/
00804
FP(3)=FP(4)
00805
CALL SIMO(A3.FP.J.K)
00806
GO TO 7
00807
6 J=3
U808
A3(1,1)=A(1.-.1)
00809
A3(1.2)=0.
00810
A3(1,3)=A(1.4)
00811
A3(2.1)=0.
00812
A3(2.2)=A(3.3)
00813
A3(2.3)=A(3.4)
00814
A3(3,1)=A(4,1)
00815
A3(3.2)=0.
00816
A3(3.3)=A(4.4)
00817
FP(2)=FP(3)
00818
FP(3)=FP(4)
00819
CALL SIMO(A3.FP1J,K)
00820
7 IF (K) 1.291
00821
2 IF(J-3)131.132,136
00822
131 FP(4)=FP(2)
00823
FP(3)=0.
00824
FP(2)=0.
00825
GO TO 136
00826
132 IF(DCO.NE.0.)G0 TO 134
00827
FP(4)=FP(3)
00828
FP(3)=0.
00829
GO TO 136
0
134 FP(4)=FP(3)
00831
FP(3)=FP(2)
00832
FP(2)=0.
00833
136 ONX=-(C2**2*C1*S1)
00834
UNY=S2**2+C2**2*C1**2
00835
UNZ=-(52*C2*S1)
00836
UN=SORT(UNX**2•UNY**2+UNZ**2)
00837
CRP = CD/SPACE
00838
FD = CDP*R*R*UN/6.2832
00839
FOY = 0.
00840
YY=ABS(C1*C2)
00841 IF(YY.GT.0.)FDY=FD*(52**2.+C2**2*C1**2-51**2*S2**2)/(0
00842
FP(5) = E*R*C1/(0*C2)+FDY
00843
FP(6) = (F8*C2 - (E*P+FD)*S1*S2)/Q
00844
RETURN
00845
1
WRITE(6.100)
00846
100 FORMAT (1H0,"SINGULAR MATRIX-EXECUTION TERMINATED AT THIS POINT")
00847
STOP
.00848
END
00849
SUBROUTINE OUTP1 (S.F.FP,IH.N,PRMT)
00850
COMMON /AMB/ TIZ,CIZ,CIO,TIO.PIO.GA,AL.RE/FR.R1.FRL
00851
COMMON /OUT/ L.TH1.TH2.H.Z,SNEw,DS.R.K.N11,J3,J2.J5.JJ.SPACE.SIGMA
- 00852
X.DO,CD.A7,XXX
00853
COMMON /ZF9/ SET,XE,YE.ZE.BBE.DCE.OTE.LCNT.RUE.RTE.RCE.T1EIT2E._
00854
COMMON /STR95/ II.M.R3
00855
COMMON/WGM/MERGCK,ROVO
00856
DIMENSION Fo6),0(6).PRmT(5),so(4),sN(4),N11(18)
00857
IF(L) 191+2
00858
1
SN(1) = SIN(TH1)
00859
SN(2) = SIN(TH2)
00860
SN(3) = CO5(TH1)
00861
SN(4) = COS(TH2)
00862
TJ = PRMT(1)
00863
DS = PRMT(3)
00864
RUN = 0.0
00865
RTN=0.0
00866
RCN=0.0
00867
RN=1.0
00868
SSN=0.0
00869
T1N=TH1*57.29578
00870
T2N=TH2*57.29578
00871
XN = 0.0
00872
YN = 0.0
00873
7N = 0.0
00874
X=0.0
00875
v=0.0
00876
7=0.0
00877
JP = 100
00878
IF (L) 30+31 +30
00879
31 L = 1
00880
GO TO 6
00881
30 L = 2
00882
X = XE
00883
Y = YE
00884
7 = ZE
00885
GO TO 6
00886
2
IF (ABS(S-TJ )-0.01) 3+3+4
00887
3
SN(1) = SIN(F(5))
00888
SN(2) = SIN(F(6))
00889
SN(3) = COS(F(5))
00890
SN(4) = COS(F(6))
00891
X = X + 0.5*(5N(3)*SN(4) + SO(3)*SO(4))*DS
00892
V = Y + 0.5*(SN(1)*SN(4) + SO(1)*S0(4))*DS
00893
7 = Z + 0.5*(SN(2) + SO(2))*DS
00894
6
no 5 I=1+4
00895
5
SO(I) = SN(I)
00896
TIME = S
00897
PIZ = -BE*TIZ - GA*CIZ
00898
CI = CIO +CIZ*7
00899
TI = TIO+TIZ*Z
00900
PI = PIO
PIZ*Z
00901
F55 = F(5)*57.29578
00902
F66 = F(6)*57.29578
00903
TF (JP-100) 50+51+51
00904
51 DUU = (1.0-R*5N(1)*SN(4))/(1.0-R3)
00905
DELT = (1.0-TI)/(1.0-TIO)
00906
nELc=o.
00907
DFNOM=1.-CIO
00908
IF(DENOM.NE.0.)DELC=(1.0-CI)/DENOM
00909 C F(2) = RT OR (TM-71)/TO. E(3) SAME FOR C00910 C DUU = (U0-U12)/(U0 - U120), DELT = (TM-TI)/(70-TIO)
00911
IF (L-2) 22+23922
00912
23 DELT = F(2)/(1.0-TIO)
00913
DELC=0.
00914
DENOM=1.0-CIO
00915
IF(DENOM.NE.0.)DELC=F(3)/DFNOM
00916
100 FORMAT (1H .F9.2+1)(.5F8.2+2X+2F8.3,16X43F8.3+2X+ F9.5,F9.1)
00917
22 SOO = S/2.*DO
00918
X00 = X/2.*DO
00919
Y00 = Y/2.*DO
00920
700 = Z/2.*DO
00921
RRO = F(4)/2.*DO
00922
• RU00 = F(1)/2.*DO
00923
DT00 = F(2)/2.*DO
00924
ncoo = F(3)/2.*DO
TIMER=TIME*ROVO
00925
00926
IF (L-2)1001.1002,1001
1002 PRINT 100,500,X00,Y00,200,F55,F66,BBO+R000,DUU,DELT,
00927
IOELC,PI,TIMER
00928
GO TO 1004
00929
1001 PRINT 1003,5000(00,Y00.Z00,F55*E66•1180,RU00,0T00,0C00-,DUU,
00930
00931
IDELTIPDELC,PI,TIME
1004 PPLOT=F(4)*2.*SORT(2.)
00932
1003 FORMAT (1H ,F9.2,1X.5F8".2.2X.7F8.3.2X,F9.5.E9.1)
00933
WRITE(7,1000)StY,X,Z,BPLOT,F(1),F(2),E13),DUU.DELT,DELC,
00934 C
00935 C
*TIMER
00936 C1000 FORMAT(5F8.3,6F5.3,E9.3)
U937
M = M + 1
IF (M-40) 43,44,44
00938
44 M = 0
00939
00940
PRINT 207,J3
00941
207 FORMAT (11-41,60X,"I.D. = ",I4./1
00942
43 JP = 0
00943
50 JP = JP + 1
00944
LCNT = LCNT+1
00945
IF (LCNT - 4000) 41,41,42
00946
42 PRINT 206
00947
LCNT = 0
206 FORMAT (1H1)
00948
00949
41 RUO=RUN
RTO=RTN
00950
RCO=RCN
00951
80=BN
00952
00953
SS=SSN
T10=T1N
00954
1.20=T2N
00955
X0 = XN
00956
00957
YO = YN
00958
ZO = ZN
00959.
RUN=F(1)
00960
RTN=F(2)
00961
RCN=F(3)
PN=F(4)
00962
00963
SSN=S
00964
T1N=F55
.00965
T2N=F66
00966
XN = X
00967
YN = Y
00968
7N = Z
00969
IF (L-2) 20,21,20
20
IF (F(2)) 14,14,13
00970
14 SET = SSN •+RTN*(SSN-SS)/(RTN-RTO)
00971
RUE = RUO + (SET-SS)*(RUO-+RUN)/(SS-SSN)
00972
RTE = RTO • (SET-SS)*(RTO-RTN)/(SS-SSN)
00973
00974
RCE = RCO • (SETSS)*(RC0+-RCN)/(SS-SSN)
PRE = BO • (SET-SS)*(80-BN)/(SSSSN)
00975
00976
TIE = 710 + (SET-SS)*(T10-T1N)/(SS-SSN)
72E = 120 + (SET-SS)*(T20-T2N)/(SS-SSN)
.00977
00978
00979
XF = XO • (SET-SS)*(X0-XN)/(SS-SSN)
00980
YE -= YO • (SET-SS)*(YO-YN)/(SS-SSN)
_00981
ZE = ZO + (SET-S5)*(70-ZN)/(SS-SSN)
00982
- SOO = SET/2.*DO
--PRINT 200, SOO
00983
200 FORMAT (1H ."
00984
STARTING
00985
L = -2
00986
PRMT(5) = 1.0
00987
RETURN
rb
00988
00989
00990
00991
00992
00993
00994
00995
00996
00997
00998
00999
01000
01001
01002
01003
01004
01005
01006
01007
01008
01009
01010
01011
01012
01013
01014
01015
01016
01017
01018
01019
01020
01021
01022
01023
01024
01025 C
01026 C
01027
01028 C
01029 C
01030 C
-01031
01032
01033
01034
01035
01036
01037
01038
01039
01040 C
01041 C
01042 C
01043
01044
01045
01046
7
01048 C.
01049 C
01050 C
01051
13 TJ = TJ + PRMT(3)
RETURN
40 TJ = TJ + PRMT(3)
RETURN
21
IF (F(1)) 10,10,40
10 PRMT(5) = 1.0
SFT = SSN - RUN*(S5N-SS)/(RUN-RUO)
RUE = RUO
(SET-SS)*(RUO-RUN)/(SS-SSN)
RAE = BO + (SET-SS)*(RO-RN)/(S5-SSN)
OTE = RTO + ('SET-SS)*(RTO-PTN)/(SS-55N)
ncE = RCO + (SET-S5)*(RCO-RCN)/(5S-S5N)
TIE = 1. 10 + (SET-55)*(TI0-T1N)/(SS-SSN)
T2E = T20 • (SET-S5)*(1-20-T2N)/(SS-SSN)
XE = X0-* (SET-55)*(X0-XN)/(SS-S5N)
YE = Y0 + (SET-SS)*(YO-YN)/(SS-SSN)
7E = ZO
(SETSS)*(70-7N)/(55-55N)
OUT = (1.0-R*SIN(0.01745*T1E)*COS(0.01745*T2E))/(1.0-R3)
OTT = OTE/(1.0-TI0)
DCT=0.
DENOM=1.0-CIO
IF(DENOM.NE.0.)DCT=DCE/DENOM
PI = PIO + PIZ*ZE
SOO = SET/2.*DO
PRINT 205,500
205 FORMAT (1H ."
STARTING LENGTH, VELOCITY =", F9.3)
X00 = XE/2.*DO
YO0 = YE/2.*DO
700 = ZE/2.*DO
RRO = 88E/2.*DO
RU00 = RUE/2.*DU
nroo = OTE/2.
ncoo = DCE/2.
PRINT 100,500,X00,Y00,Z00,11.1E,T2E,RBO,RU00,DUT,DTT,
1DCT,PI,TIME
4
RETURN
END
SUBROUTINE SIMO(A,B,N,KS)
DIMENSION A(1),8(1)
FORWARD SOLUTION
TOL=0.0
KS=0
JJ=-N
no 65 J=1,N
JY=J+1
JJ=JJ+N+1
RIGA=0
IT=JJ-J
00 30 I=J,N
SEARCH FOR MAXIMUM COEFFICIENT IN COLUMN
IJ=IT*I
IF(ABS(BIGA)-ABS(A(IJ)))
20 RIGA=A(IJ)
TMAX=I
30 CONTINUE
TEST FOR PIVOT LESS THAN TOLERANCE (SINGULAR MATRIX)
TF(A8S(BIGA)-TOL) 35.35.40
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
500
SIMQ
SIMQ
SIMQ
SIMO
01054 C
01055 C
INTERCHANGE ROWS IF NECESSARY
01056 C
01057
40 T1=J+N*(J-2)
01058
TT=IMAX-J
01059
no 50 K=J,N
01060
T1=I1+N
01061
T2=I1+IT
01062
SAVE=A(11)
01063
A(I1)=A(I2)
01064
A(I2)=SAVE
01065 C
01066 C
DIVIDE EQUATION BY LEADING COEFFICIENT
01067 C
01068
50 A(I1)=A(11)/BIGA
01669
SAVE=B(IMAX)
01070
R(ImAX)=9(J)
01071
8(J)=SAVE/BIGA
01072 C
01073 C
ELIMINATE NEXT VARIABLE
01074 C
01075
TF(J-N) 55.70,55
01076 .55 TO5=N*(J-1)
01077
DO 65 IX=JY,N
01078
TXJ=IQS+IX
01079
TT=J-IX
01080
no 60 JX=JY.N
01081
TXJX=N*(JX-1)+IX
01082
JJX=IXJX+IT
01083
60 A(IXJX)=A(IXJX)-(A(IXJ)*A(JJX))
01084
65 8(IX)=B(IX)-(H(J)*A(TXJ))
01085 C
01086 C
BACK SOLUTION
01087 C
01088
70 NY=N-1
01089
TT=N*N
01090
no 80 J=1,NY
01091
IA=IT-J
01092
TB=N-J
01093
TC=N
01094
no 80 K=1,J
01095
B(IB)=B(IB)-A(IA)*B(IC)
01096
TA=IA-N
01097
80 IC=IC-1
01098
RETURN
01099
END
01100 C
01101 C
01102
SUBROUTINE,HPCG(PRMT.Y,DERY.NDIM.IHLF.FCT.OUTP,AUX)
01103 C
SUBROUTINE HPCG
01104 C
01105 C
PURPOSE
01106 C
TO SOLVE A SYSTEM OF FIRST ORDER ORDINARY GENERAL
01107 C
DIFFERENTIAL EQUATIONS WITH GIVEN INITIAL VALUES.
01108 C
01109 C
USAGE
01110 C
CALL HPCG (PRmT.Y,DERY - ,NDIM,IHLF,FCT,OUTP.AUX)
01111
CPARAMETERS FCT AND OUTP REQUIRE AN EXTERNAL STATEMENT. 01112 C
01113 C
DESCRIPTION OF PARAMETERS
- 01114 C
-PRMT
- AN INPUT AND OUTPUT - VECTOR - WITH -DIMENSION GREATER
01115 C
OR EQUAL TO 5. wHICH SPECIFIES THE PARAMETERS OF
01116 C
THE INTERVAL AND OF ACCURACY AND WHICH SERVES FOR
01117 C
COMMUNICATION BETWEEN OUTPUT SUHROUTINE (FURNISHED
01118 C
BY THE USER) AND SUBROUTINE HPCG. EXCEPT PRMT(5)
01119 C
THE COMPONENTS ARE NOT DESTROYED BY SUBROUTINE
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ .
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMO
Sim()
Sim()
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMO
SIMQ
SIMO
SIMO •
SIMQ
SIMQ
SIMQ
SIMQ
SIMQ
SIMO
5180
SIMO
SIMO
SIMQ
SIMQ
SIMQ
SIMQ
-SIMQ
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
01120
01121
01122
01123
01124
01125
01126
01127
01128
01129
01130
01131
01132
01133
01134
01135
01136
01137
01138
01139
01140
01141
01142
01143
01144
01145
01146
01147
01148
01149
01150
01151
01152
01153
01154
01155
01156
01157
01158
01159
01160
01161
01162
01163
01164
01165
01166
01167
01168
01169
01170
01171
01172
01173
01174
01175
01176
01177
01178
01179
01180
01181
01182
01183
01184
01185
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
HPCG
HPCG AND THEY ARE
HPCG
PRMT(1)- LOWER BOUND OF THE INTERVAL (INPUT),
HPCG
PRMT(2)- UPPER ROUND OF THE INTERVAL (INPUT),
HPCG
PRMT(3)- INITIAL INCREMENT OF THE INDEPENDENT VARIABLE
HPCG
(INPUT).
HPCG
PRMT(4)- UPPER ERROR BOUND (INPUT). IF ABSOLUTE ERROR IS
HPCG
GREATER THAN PRMT(4), INCREMENT GETS HALVED.
HPCG
IF INCREMENT IS LESS THAN PRMT(3) AND ABSOLUTE
ERROR LESS THAN PRMT(4)/50, INCREMENT GETS DOUBLED.HPCG
HPCG
'THE USER MAY CHANGE PRMT(4) BY MEANS OF HIS
HPCG
OUTPUT SUBROUTINE.
HPCG
PRMT(5)- NO INPUT PARAMETER. SUBROUTINE HPCG INITIALIZES
HPCG
PRMT(5)=0. IF THE USER WANTS TO TERMINATE
HPCG
SUBROUTINE HPCG AT ANY OUTPUT POINT. HE HAS TO
CHANGE PPMT(5) TO NON-ZERO HY MEANS OF SUBROUTINE HPCG
HPCG
OUTP. FURTHER COMPONENTS OF VECTOR PRMT ARE
FEASIBLE IF ITS DIMENSION IS DEFINED GREATER 'HPCG
HPCG
THAN 5. HOWEVER SUBROUTINE HPCG DOES NOT REQUIRE
HPCG
AND CHANGE THEM. NEVERTHELESS THEY MAY BE USEFUL
HPCG
FOR HANDING RESULT VALUES TO THE MAIN PROGRAM
HPCG
(CALLING HPCG) WHICH ARE OBTAINED BY SPECIAL
MANIPULATIONS wITH OUTPUT DATA IN SUBROUTINE OUTP. HPCG
HPCG
Y
- INPUT VECTOR OF INITIAL VALUES. (DESTROYED)
HPCG
LATERON Y IS THE RESULTING VECTOR OF DEPENDENT
HPCG
VARIABLES COMPUTED AT INTERMEDIATE POINTS X.
HPCG
- INPUT VECTOR OF ERROR WEIGHTS. (DESTROYED)
DERY
HPCG
THE SUM OF ITS COMPONENTS MUST BE EQUAL TO 1.
HPCG
LATERON DERY IS THE VECTOR OF DERIVATIVES. WHICH
HPCG
BELONG TO FUNCTION VALUES Y AT A POINT X.
HPC
HPCG
- AN INPUT VALUE, WHICH SPECIFIES THE NUMBER OF
NDIM
HPCG
EQUATIONS IN THE SYSTEM.
EQU
;HPCG
- AN OUTPUT VALUE, WHICH SPECIFIES THE NUMBER OF
IHLF
BISECTIONS OF THE INITIAL INCREMENT. IF IHLF GETS HPCG
HPCG
GREATER THAN 10, SUBROUTINE HPCG RETURNS WITH
HPCG
ERROR MESSAGE IHLF=11 INTO MAIN PROGRAM.
HPCG
ERROR MESSAGE IHLF=12 OR IHLF=13 APPEARS IN CASE
PRMT(3)=0 OR IN CASE SIGN(PRMT(3)).NE.SIGN(PRMT(2)-HPCG
HPCG
PRMT(1)) RESPECTIVELY.
HPCG
FCT
- THE NAME OF AN EXTERNAL SUBROUTINE USED. IT
HPCG
COMPUTES THE RIGHT HAND SIDES DERY OF THE SYSTEM
HPCG
TO GIVEN VALUES OF X AND Y. ITS PARAMETER LIST
HPCG
MUST RE X,Y,DERY. THE SUBROUTINE SHOULO NOT
HPCG
DESTROY X AND Y.
HPCG
- THE NAME OF AN EXTERNAL OUTPUT SUBROUTINE USED.
OUTP
ITS PARAMETER LIST MUST HE X,Y,DERY,IHLF.NDIM.PRMT.HPCG
HPCG
NONE OF THESE PARAMETERS (EXCEPT, IF NECESSARY,
HPCG
PRMT(4),PRMT(5),...) SHOULD BE CHANGED BY
SUBROUTINE OUTP. IF PRMT(5) IS CHANGED TO NON-ZERO,HPCG
HPCG
SUBROUTINE HPCG IS TERMINATED.
HPCG
AUX
- AN AUXILIARY STORAGE ARRAY WITH 16 ROWS AND NDIM
HPCG
COLUMNS.
HPCG
REMARKS
THE PROCEDURE TERMINATES AND RETURNS TO CALLING PROGRAM, IF HPCG
HPCG
(1) MORE THAN 10 BISECTIONS OF THE INITIAL INCREMENT ARE
t=
NECESSARY TO GET SATISFACTORY ACCURACY (ERROR MESSAGE
IHLF=11),
(2) INITIAL INCREMENT IS EQUAL TO 0 OR HAS WRONG SIGN .HPCG
HPCG
(ERROR MESSAGES IHLF=12 OR 1HLF=13),
HPCG
(3) THE WHOLE INTEGRATION INTERVAL IS WORKED THROUGH.
HPCG
(4) SUBROUTINE OUTP HAS CHANGED PRMT(5)-TO NON-ZERO. - HPCG
HPCG
SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED
THE EXTERNAL SUBROUTINES FCT(X,Y,DERY) AND HPCG
OUTP(X,Y,DERY.IHLF,NDIM,PRMT) MUST BE FURNISHED BY THE USER.HPCG
HPCG
01186
01187
01188
01189
01190
01191
01192
01193
01194
01195
01196
01197
01198
01199
01200
01201
01202
01203
01204
01205
01206
01207
-01208
01209
01210
01211
01212
01213
01214
01215
01216
01217
01218
01219
01220
01221
01222
01223
01224
01225
01226
01227
01228
01229
01230
01231
01232
01233
.01234
01235
01236
01237
01238
01239
01240
01241
01242
-"'"
0.1243
01244
-01245
01246
-01247
01248
01249
01250
01251
•
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
1
C
C
2
3
C
C
4
C
C
5
6
7
8
C
C
11
12
METHOD
HPCG
EVALUATION IS DONE BY MEANS OF HAMMINGS MODIFIED PREDICTOR- HPCG
HPCG
CORRECTOR METHOD. IT IS A FOURTH ORDER METHOD, USING 4
PRECEEDING POINTS FOR COMPUTATION OF A NE4 VECTOR Y OF THE HPCG
HPCG
DEPENDENT VARIABLES.
FOURTH ORDER RUNGE-KUTTA METHOD SUGGESTED BY RALSTON IS
HPCG
HPCG
USED FOR ADJUSTMENT OF THE INITIAL INCREMENT AND FOR
COMPUTATION OF STARTING VALUES.
HPCG
SUBROUTINE HPCG AUTOMATICALLY ADJUSTS THE INCREMENT DURING HPCG
HPCG
THE WHOLE COMPUTATION BY HALVING OR DOUBLING.
TO GET FULL FLEXIBILITY IN OUTPUT, AN OUTPUT SUBROUTINE
HPCG
MUST BE - CODED BY THE USER.
HPCG
HPCG
FOR REFERENCE. SEE
HPCG:
(1) RALSTON/WTLF, MATHEMATICAL METHODS FOR DIGITAL
HPCG
COMPUTERS. WILEY. NEW YORK/LONDON, 1960. PP.95-109.
(2) RALSTON, RUNGE-KUTTA METHODS WITH MINIMUM ERROR BOUNDS,HPCG
HPCG
MTAC, VOL.16. ISS.80 (1962), PP.431-437.
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
DIMENSION PRMT(1),Y(1),DERY(1),AUX(16,1)
HPCG
HPCG
HPCG
N=1
HPCG
THLF=0
X=PRMT(1)
HPCG
H=PRMT(3)
HPCG
PRMT(5)=0.
HPCG
HPCG
DO 1 I=1,NDIM
AUX(16,I)=0.
HPCG
AUX(15,I)=DERY(I)
HPCG
HPCG
AUX(1,I)=Y(I)
TF(H*(PRMT(2)-X))3,2,4
HPCG
HPCG
HPCG
ERROR RETUrNS
THLF=12
HPCG
HPCG
GOTO 4
HPCG
IHLF=13
HPCG
HPCG
COMPUTATION OF DERV FOR STARTING VALUES
CALL FCT(X,Y,DERY)
HPCG
HPCG
HPCG
RECORDING OF STARTING VALUES
HPCG
CALL OUTP(X,Y,DERY,IHLF,NDIM,PRMT)
TF(PRMT(5))6,5+6
HPCG
TF(IHLF)7,7,6
HPCG
RETURN
HPCG
HPCG
DO 8 I=1,NDIM
AUX(8,I)=DERY(I)
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
---HPCG
TNCREMENT H IS TESTED BY MEANS OF BISECTION
IHLF=IHLF•1
HPCG
HPCG
X=X-H
HPCG
nO 12 1=1.NDIM--AUX(4,I)=AUX(2.I)
HPCG
HPCG
H=.5*H
01252
01?53
01254
01255 C
01256
01257
01258
01259
01260
01261
01262
01263
01264 C
01265 C
01266
01267
01268
01269
01270
01271
01272 C
01273 C
01274
01275
01276
01277 C
01278 C
01279
01280
01281
01282
01283
01284
01285
01286
01287 C
01288
01289
01290
01291
01292
01293
01294
01295
01296
01297
01298
01299
01300
01301
01302
01303
01304
01305
01306 C
01307
01308
01309
01310
01311
01312
01313
01314
01315
01316
01317
N=1
TSw=2
GOTO 100
13 x=x+H
CALL FCT(X,Y,DERY)
N=2
nO 14 I=l+NDim
AUX(2•I)=Y(T)
14 AUX(9+I)=DERy(I)
TSw=3
GOTO 100
COMPUTATION OF TEST VALUE DELT
15 nFLT=0.
DO 16 I=1,NDIM
16 nELT=DELT+AUX(15+I)*ABS(Y(I)-4UX(4+I))
nFLT=.06666667*DELT
TF(DELT-PRmT(4))19,19.17
17 IF(THLF-10)11.18,18
NO SATISFACTORY ACCURACY AFTER 10 BISECTIONS. ERROR MESSAGE.
18 THLF=11
x=x+H
GOTO 4
THERE IS SATISFACTORY ACCURACY AFTER LESS THAN 11 BISECTIONS.
19 x=X+H
CALL FCT(X9Y,DERY)
nO 20 I=1,NDIM
AUX(3+I)=Y(I)
20 Aux(101I)=DERY(I)
N=3
TSw=4
GOTO 100
21 N=1
x=x+H
CALL FCT(x.Y,DERY)
x=PRmT(1)
no 22 1=1,NOIm
AUX(11.I)=DERy(I)
220Y (I ) =AUX ( 1 e I ) +H* ( •375*AUX ( 89 I ) +.7916667*AUX (9, I )
1-.2083333*AUX(1091)+.04166667*DERY(1))
23 X=X+H
N=N+1
CALL FCTIX,Y,DERY)
CALL OUTP(X.Y,OERY,IHLF,NDIM,PPMT)
TF(PIRMT(5))6,24,6
24 TF(N-4)25,200,200
25 no 26 I=1,NDIN
AUX(NsI)=Y(I)
26 AUx(N+7,I)=DERY(I)
TF(N-3)27,29,200
27 nO 28 I=1,NOIM
OFLT=AUx(9.1)+AUX(9,1)
DELT=OELT+DELT
Y(I)=AUX(1,I)+.3333333*H*(AUX(89I)+DELT+AUX(10,I))
GOTO 23
29 nO 30 I=1,NDIm nFLT=AUx(9.I)+AUx(10.I)
nELT=DELT+DELT+i)ELT
30 Y(I)=AUX(1.1)+.375*H*(AUX(8,I)+DELT+AUX111,I)1
GOTO 23
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
01318 C
HPCG
01319 C
THE FOLLOWING PART . OF SUBROUTINE HPCG COMPUTES BY MEANS OF
HPCG
01320 C
RUNGE-KUTTA METHOD STARTING VALUES FOR THE NOT SELF-STARTING
HPCG
01321 C
PREDICTOR-CORRECTOR METHOD.
HPCG
01322
100 no 101 I=1*NDIM
HPCG
01323
7=H*AUX(N+79I)
HPCG
01324
AUX(5,I)=Z
HPCG
01325
101 Y(I)=AUX(N.I)+.4*Z
HPCG
01326 C
7 IS AN AUXILIARY STORAGE LOCATION
HPCG
01327 C
HPCG
01328
7=X+.4*H ,
HPCG.,
01329
CALL FCTIZIY.DERY)
HPCG
01330
00 102 I=1.NDIM
HPCG
01331
Z=H*DERY(I)
HPCG
01332
AUX(6,I)=Z
HPCG
01333
102 Y(I) = AUX( N 91) + .2969776*AUX(5,1)+.1587596*Z
HPCG
01134 C
HPCG
01335
Z=X+.4557372*H
HPCG
01336
CALL FCT(Z,Y,DERY)
HPCG
01337
no 103 I=1.NDIM
HPCG
01338
7=H*DERY(I)
HPCG
01339
AUX(79I)=Z
HPCG
01340
103 Y(1) =AU X1 N ,D • .2181004*AUX(5,I)-3.050965*AUX(6,1)+3.832865*Z
HPCG
01341 C
HPCG
01342
Z=X+H
HPCG
01343
CALL FCT(Z,Y.DERY)
HPCG
01344
00 104 I=1,NDIM
HPCG
01345
1040Y(I)=AUX(N/1/+.1747603*AUX(591)-.5514807*AUX(6,1/
HPCG
01346
1+1.205536*AUX(7,I)+.1711848*H*DERY(I)
HPCG
01347
GOTO(91,13.15,21),ISW
HPCG
01348 C
HPCG
01349 C
POSSIBLE BREAK-POINT FOR LINKAGE
HPCG
01350 C
HPCG
01351 C
STARTING VALUES ARE COMPUTED.
HPCG
01352 C
NOW START HAMMINGS MODIFIED PREDICTOR-CORRECTOR-METHOD.
HPCG
01353
200 ISTEP=3
HPCG
01354
201 1F(N-8)204,202,204
HPCG
01355 C
HPCG
01356 C
N=8 CAUSES THE ROWS OF AUX TO CHANGE THEIR STORAGE LOCATIONS
HPCG
01357
202 no 203 N=2,7
HPCG
01358
DO 203 I=1,NDIM
HPCG
01359
AUX(N-1.1)=AUX(N,I)
HPCG
01360
203 AUX(N+6,I)=AUX(N+7,I/
HPCG
01361
N=7
HPCG
01362 C HPCG
01363 C
N LESS THAN 8 CAUSES N+1 TO GET N
HPCG
01364
204 N=N+1
HPCG
01365 C
HPCG
01366 C
COMPUTATION OF NEXT VECTOR Y
HPCG
01367
f.)0 205 I=1,NDIM
HPCG
01368
AUX(N-1,I)=Y(I)
HPCG
01369
205 AUX(N+6.D=DERY(I)
HPCG
01370
X=X+H
HPCG --''''','-01371
206 ISTEP=ISTEP+1
HPCG
01372
no 207 I=1,NDIM
„.„
HPCG
01373
OOFLT=AUX(N-4.D+1.333333*H*(AUX(N+6.1)+AUX(N+69I)-AUX(N+5,I)*
HPCG
01374
1AUX(N+4.I)+AUX1N+4,11)
HPCG
01375
Y(1)=DELT-.9256198*AUX(16.1)
HPCG
01376
207 AUX116.1)=DELT
-HPCG
01377 C
_PREDICTOR IS NOW GENERATED IN ROW 16 OF AUX, MODIFIED PREDICTOR -HPCG
01378 C TS GENERATED IN Y. DFLT MEANS AN AUXILIARY STORAGE.
HPCG
01379 C
HPCG
01380
CALL FCT(X.Y,DERY/
HPCG
01381 C
DERIVATIVE OF MODIFIED PREDICTOR IS GENERATED IN DERY
HPCG
01382 C
HPCG
01383
00
208
I=1*NDIM
HPCG :.
01384
ODELT=.125*(9.*AUX(N-1.I)-AUX(N-3.I)+3.*H*(DERY(I)+AUX(N+6.I).
HPCG
01385
1AUX(N.6,I)-AUX(N+5,I)))
HPCG
01386
AUX(16.I)=AUX(16.1)-DELT
HPCG
01387
208 Y(I) = DELT .. .07438017*AUX(16.I)
HPCG
01388
C
HPCG
01389 C
TEST WHETHER M MUST RE HALVED OR DOUBLED
HPCG
01390
DELT=0.
HPCG
01391
DO 209 I=1,NDIM
A
01392
209 DELT=DELT+AUX(15.I)*ARS(AUX(16,I))
=
0 1 393
TF(DELT-PRMT(4))210.222,222
HPCG
01394 C
HPCG
01395 C
H MUST NOT BE-HALVED. THAT MEANS Y(I) ARE GOOD.
HPCG
0 1 396
210 CALL FCT1k.Y.DERY)
HPCG
01397
CALL OUTP(X.Y.DERY.IHLF,NDIM.PRMT)
HPCG
01398
TF(RRMT(5))212,211,212
HPCG
01399
211 IF(IHLF-11)213,212,212
HPCG
01400
212 RETURN
HPCG
01401
213 IF(H*(X-PRMT(2)))214.212.212
01402
214 TF(ABS(x-PHMT(2))-.1*AHSTH))212,215,215
=
01403
215 IF(DELT-.02*PRMT(4))216.216,201
HPCG
01404 C
MPCG
01405 C
01406 C
H COULD BE DOUBLED IF ALL NECESSARY PRECEEDING VALUES ARE
1;117g
01407 C
AVAILABLE
01408
216 IFUHLF)201,201,217
=
01409
217 IF(N-7)2019218,218
HPCG
01410
21H IF(ISTEP-4)201,219,219
HPCG
219 imoo r. ISTEP/2
01411
HPCG
01412
IF(ISTEP-IMOD-IMOD)201.220.201
MPCG
01413
220 H=H•H
HPCG
01414
IHLF=IHLF-1
HPCG
01415MPCG
ISTEP=0
01416
DO 221 I=1.NDIM
HPCG
01417
AUX(N-1.I)=AUX(N-2,I)
HPCG
01418
AUX(N-2.I)=AUX(N-4.I)
HPCG
01419
AUX(N-3.I)=AUX(N-6,I)
HPCG
01420
AUX(N+6.I)=AUX(N+5,I)
HPCG
01421
AUX(N+5,I)=AUX(N+391)
HPCG
01422
AUX(N4.4.1)=AUX(N+1,I)
HPCG
01423
DELT=AUX(N+6.1)+AUX(N+5.I)
HPCG
01424
DELT=DELT+DELT+DELT
HPCG
01425
2210Aux(16,I)=8.962q63*(r(i)-AUX(N-3.I)).3.361111*H*(0ERY(I).OELT
HPCG
01426
1+AUx(N4.4.I)1
HPCG
01427
GOTO 201
HPCG
01428 C
HPCG
01429 C
HPCG
01430 C
H MUST 8E HALVED
HPCG
01431
222 IHLF=IMLF*1
01432
IF(IMLF-10)223,223,210
. hi=
i
01433
223 H=.5*H
HPCG
01434
ISTEP=0
HPCG
01435
- DO 224 I=1.NDIM
01436 0Y(I)=.00390625*(80.4AUX(N-1,I)+135.*A0X(N-2,I)+40.0AUX(N-3.1).
=
01437
1AUX(N4,I))-.1171875*(AUX(N+6.1)-6.*AUX(N+5,I)-AUX(N•4,I))*H
HPCG
01438
OAUX(11-4.1)=.00390625*(12.*AUX(N-1.1)4.135.*AUX(N-2,1)4.
HPCG
01439
1108.*AUX(N--3.I)+AUX(N•4.I))-.0234375*(AUX(N+6.I)+18.*AUX(N+5.1)- HPCG
01440
29.*AUX(N+4,I))*(-1
HPCG
01441
AUX(N-.3.I)=AUX(N-2.I)
HPCG
01442
224 AUX(N+4.1)=AUX(N+5,I)
HPCG
01443
X=X-H
HPCG
01444 DELT=X-(H.H)
HPCG
01445
'CALL FCT(DELT.Y.DERY)
HPCG
01446
DO
225
I=1tNDIM HPCG
01447
AUX(N-2.I)=Y(I)
HPCG
01448
AUX(N+5.I)=0ERY(I)
HPCG
01449
225 Y(I)=AUX(N-4,I)
HPCG
01450
01451
01452
01453
01454
01455
01456
01457
01458
01459
DELT=DELT-(H•H)
CALL FCT(DELT,Y,DERY)
no 226 I=1,NDIM
DELT=AUX(N+9,I)+AUX(4,I)
DELT=DELT*DELT•DELT
OAUX(16,I)=8.962963*(AUX(N-1,I)-Y(I))-3.361111*H*CAUXIN+6,1)•DELT
1•DERY(I))
226 AUX(N+3,I)=DERy(I)
GOTO 206
FND
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
HPCG
REFERENCES
1.
U.S. Environmental Protection Agency. April 25, 1978. Modification of
Secondary Treatment Requirements for Discharges into Marine Waters.
Federal Register, Vol. 43, No. 80, p. 17484-17508. Washington, D.C.
2.
33 USC 1251, et seq. PL 92-500 as amended December 28, 1977.
3.
California State Water Resources Control Board. 1978. Water Quality
Control Plan for Ocean Waters of California. Sacramento, California.
4.
EPA. 1976. Quality Criteria for Water. U.S. Environmental Protection
Agency, Washington, D.C. EPA-440/9-76-023.
5.
Baumgartner, D. J. and D. S. Trent. 1970. Ocean Outfall Design: Part
I. Literature Review and Theoretical Development. U.S. Dept. of Int.,
Federal Water Quality Adm. (NTIS No. PB-203 749).
6.
Briggs, G. A. 1969. Plume Rise. U.S. Atomic Energy Commission, Oak
Ridge Tennessee. (NTIS No. TID-25075).
7.
Davis, L. R. and M. A. Shirazi. 1978. A Review of Thermal Plume Modeling. Keynote address. In: Proceedings of the Sixth Internati9nal Heat
Transfer Conf., ASME, Aug. 6-11, 1978, Toronto, Canada.
8.
Rouse, H., C. S. Yih and H. W. Humphreys. 1952. Gravitational Convection from a Boundary Source. Tellus, Vol. 4, pp. 201-210.
9.
Priestley, C. H. B. and F. K. Ball. 1955. Continuous Convection from an
Isolated Source•of Heat.
Vol., 81, p. 144-157.
Quarterly Journal of the Royal Meteor. Soc.,
10.
Morton, B. R., G.
Taylor and J. S. Turner. 1956. Turbulent Gravi-
tational Convection from Maintained and Instantaneous Sources. Proc. of
the Royal Soc. of London, Vol. A234, pp. 1-23.
11.
Morton, B. R. 1959. Forced Plumes. Journal of Fluid Mechanics, Vol. 5,
pp. 151-163.
12.
Abraham, G. 1963. Jet Diffusion in Stagnant Ambient Fluid. Delft
Hydraulics Publication No. 29. Delft, Netherlands.
13.
Fan, L. H. 1967. Turbulent Buoyant Jets into Stratified and Flowing
Ambient Fluids. Cal. Inst. Tech., Keck Hydraulics Lab., Rep. No. KH-R15. Pasadena, California.
14.
Abraham, G. 1971. The Flow of Round Buoyant Jets Issuing Vertically
into Ambient Fluid Flowing in a Horizontal Direction. Delft Hydraulics
Publication No. 81, Delft, Netherlands.
15.
Cederwall, K. 1971. Buoyant Slot Jets into Stagnant or Flowing Environment. Cal. Inst. of Techn., Keck Lab. Rep. No. KH-R-25. 'Pasadena,
California.
16.
Roberts, P. J. W. 1977. Dispersion of Buoyant Waste Water Discharged
from Outfall Diffusers of Finite Length. Cal. Inst. of Techn., Keck Lab.
Report No. KH-R-35. Pasadena, California.
17.
Sotil, C. A. 1971. Computer Program for Slot Buoyant Jets into Stratified Ambient Environments. Cal. Inst. of Techn., Keck Lab. Tech. Memo
71-2. Pasadena, California.
18.
Koh, R. C. and L. N. Fan. 1970. Mathematical Models for the Prediction
of Temperature Distribution Resulting from the Discharge of Heated Water
in Large Bodies of Water. U.S. Environmental Protection Agency, Water
Poll. Cont. Res. Series Rep. 1613 ODWO/70.
19.
Baumgartner, D. J., D. S. Trent, and K. V. Byram. 1971. User's Guide
and Documentation for Outfall Plume Model. U.S. Environmental Protection
Agency, Pac. N.W. Water Lab. Working Paper No. 80. Corvallis, Oregon.
(NTIS No. PB 204-577/BA).
20.
Winiarski, L. D. and W. E. Frick. 1976. Cooling Tower Plume Model.
U.S. Environmental Protection Agency, Corvallis Environ. Res. Lab. EPA600/3-76-100. Corvallis, Oregon.
21.
Davis, L. R. 1975. Analysis of Multiple Cell Mechanical Draft Cooling
Towers. U.S. Environmental Protection Agency, Corvallis Environ. Res.
Lab. EPA-660/3-75-039. Corvallis, Oregon.
22.
Brooks, N. H. 1973. Dispersion in Hydrologic and Coastal Environments.
U.S. Environmental Protection Agency, Corvallis Environ. Res. Lab. EPA660/3-73-010. Corvallis, Oregon.
23.
Winiarski, L. D. and W. E. Frick. 1978. Methods of improving Plume
Models. Presentation at Cooling Tower Environ.--1978, Univ. of Maryland,
College Park, Maryland.
24.
Kannberg, L. D. and L. R. Davis. 1976. An Experimental/Analytical
Investigation of Deep Submerged Multiple Buoyant Jets. U.S. Environmental Protection Agency, Corvallis Environ. Res. Lab. EPA-600/3-76-001.
Corvallis, Oregon.
25.
Hirst, E. A. 1971. Analysis of Round, Turbulent, Buoyant Jets Discharged into Flowing Stratified Ambients. U.S. Atomic Energy Commission,
Oak Ridge Nat. Lab., Rep. ORNL-4685. Oak Ridge, Tennessee.
26.
Hirst, E. A. 1971. Analysis of Buoyant Jets within the Zone of Flow
Establishment. U.S. Atomic Energy Commission, Oak Ridge Nat. Lab., Rep.
N. ORNL-TM-3470. Oak Ridge, Tennessee.
Prediction of Initial Mixing for
Municipal Ocean Discharges
by
Allen M. Teeter
Internship Report
Submitted to
Marine Resource Management Program
School of Oceanography
Oregon State University
1978
in partial fulfillment of
the requirements for the
degree of
Master of Science
Internship: Marine and Freshwater Ecology Branch
Corvallis Environmental Research Laboratory
U.S. Environmental Protection Agency
Corvallis, Oregon