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TRANSCRIPT
TA7 W34
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no. H-74-8 c.3
TECHNICAL REPORT H-74-8
UNSTEADY FLOW COMPUTATIONS ON THE
OHIO-CUMBERLAND-TENNESSEE-MISSISSIPPI
RIVER SYSTEM by
Billy 1-1. Johnson
~ydraulics laboratory
U. S. Army Engineer Waterways Experiment Station
- P. 0. Box 631, Vicksburg, Miss. 39180
September 1974
Final Report
Approved For Public Release; Distribution Unlimited
Prepared for U. S. Army Engineer Division, Ohio River
P. 0. Box 1159, Cincinnati, Ohio 45201
------------------------------~------------------~-.
( 'r~31..\ H- q4-9
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Technical Report H-74- 8 4 . TITLE (and Subfltle) s. TYPE O F REPORT & PERI OD COVERED
UNSTEADY FLOW COMPUTATIONS ON THE OHIO- CUMBERLAND- Final TENNESSEE- MISSISSIPPI RIVER SYSTEM report 6 PERFORMI N G ORG. REPORT NUMBER
7 . AUTH O R(s) 8 CONTR ACT OR GRANT NUMBER(s)
Billy H. Johnson
9 . PERFORMING ORGANIZATION NAME AND ADDRE SS 10 . PROGRAM ELEMENT. PROJECT, TASK AREA & WORK U N IT NUMBERS
u. s . Army Engineer Waterways Experiment Station Hydraulics Laboratory P . 0 . Box 631 , Vicksburg, Miss . 39180
11. CONTROLLIN G OFFICE NAME AND ADDRESS 12 REPORT DATE
u. s . Army Engineer Division, Ohio River (ORDED- W September 1974 P . 0 . Box 1159 13. N UMBER OF PAGES
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u. s. llh w [ I lll£ER I:ISTPI~T. LOUISVIUE P. 0. B X ~9. HT!l. li~RARY LOlli ~ILLE, H .. ~C. Y ' 201
19 KEY WORDS (Continue on re,·ers~ stde if n<• essarv nnd tJenllfv b\' block number)
Barkley Reservoir Mississippi River Cumberland River Ohio River Flood control Tennessee River Kentucky Reservoir Unsteady flow Mathematical models
' 20. ABSTR"CT (Continue on rever&<' side II necessary and iden ltly by b lock number)
The U. s . Army Engineer Division , Ohio River , directs the operation of Barkley and Kentucky Reservoirs on the Cumberland and Tennessee Rivers , respectively, during periods of flooding on the lower Ohio and lower Mississippi Rivers . Flood control regulation by these reservoirs is met by controlling , to some degree, the Ohio River stage at Cairo , Illinois . A mathematical model, SOCHMJ , capable of accurately predicting Ohio River stages as a result of reservoir
(Continued)
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Unclassified SECURI TY CLASSIFICATION OF THIS PAGE(When Data Entered)
20 . ABSTRACT (continued) .
operations at Barkley and Kentucky Reservoirs has been developed . SOCHMJ provides the capability of modeling a system containing an unlimited number of junctions. The physical limits of the Ohio- Cumberland- Tennessee- Mississippi system modeled herein are Golconda, Illinois, on the Ohio River; Barkley Dam on the Cumberland River ; Kentucky Dam on the Tennessee River ; Cape Girardeau on the upper Mississippi River; and Caruthersville on the lower Mississippi River. Three applications of SOCHMJ were made in the study. These were (a) an application using 1950 flood data, (b) an application using 1973 flood data, and (c) an application using data from a 3- day period in February 1974 .
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PREFACE
The work described herein and the preparation of this report were
conducted during the period May 1973 to April 1974 for the U. S . Army
Engineer Division, Ohio River (ORD) , by ~he U. S . Army Engineer
Waterways Experiment Station (WES) under the general supervision of
Messrs. H. B. Simmons, Chief of the Hydraulics Laboratory, and M. B.
Boyd, Chief of the Mathematical Hydraulics Division (MHD).
Dr. B. H. Johnson, MHD, conducted the study and prepared the re
port . Mr. P . K. Senter of the Automatic Data Processing Center aided in
the programming of the computer model, and Mr . Ron Yates of ORD aided in
the data collection.
Directors of WES during the conduct of this study and the prepa
ration and publication of this report were BG E. D. Peixotto, CE, and
COL G. H. Hilt, CE. Technical Director was Mr . F . R. Brown .
1
CONTENTS
PREFACE . . . . . . • • • • • • • • • • • • • • • • • • • • • • • •
CONVERSION FACTORS , U. S . CUSTOMARY TO METRIC (SI) UNITS OF M.EAS'UREM.ENT . . . . • . . . . . . . • . . . . . . • . . . . . . .
PART I : INTRODUCTION . . • • • • • • • • • • • • • • • • • • • • •
Purpose and Scope . . . . . . . . . . . . • • • • • • • • • •
Background . . . . . . . . . . . . . . . • • • • • • • • • •
PART II : MULTIJUNCTION UNSTEADY FLOW MODEL (SOCHMJ) • • • • • • •
, Modification of SOCHJ . . . . . . . . . . . . . . . . . . . . Input Data Required by SOCHMJ . . . . . . . . . . . . . . . . SOCHMJ Computation Cycle . . . . . . . . . . . . . . . . . . Output Provided by SOCHMJ . . . . . . . . . . . . . . . . . .
PART III : PRESENTATION OF RESULTS • • • • • • • • • • • • • • • •
Calibration of the Model . . . . . . . . Application to 1973 Flood . . . . . . . . Application Using Hourly Discharge Data .
PART IV : CONCLUSIONS AND RECOMMENDATIONS . • •
• • • • • • • • • •
• • • • • • • • • •
• • • • • • • • • •
• • • • • • • • • •
BIBLIOGRAPHY
TABLES 1- 2
PLATES 1- 11
• • • • • • • • • • • • • • • • • • • • • • • • • • •
APPENDIX A: MATHEM.ATICAL ASPECTS OF SOCHMJ . . . . . . . • • • • •
Page
1
3
4
4 5
10
10 11 12 13
14 14 16 17
18
20
Al
Governing Differential Equations and Assumptions . . . . . . Al Difference Equations . . . . . . . . . . . . . . . . . . . . A3
APPENDIX B:
APPENDIX C:
LIST OF INPUT REQUIRED BY SOCHMJ .
NOTATION . . . . . . . . . . . . .
2
• • • • • •
• • • • • •
• • • •
• • • •
'
Bl
Cl
CONVERSION FACTORS, U. S . CUSTOMARY TO METRIC (SI) UNITS OF :MEASUREMENT
U. S . customary units of measurement used in this report can be con
verted to metric (SI) units as follows :
Multiply By To Obtain
feet 0.3048 meters
miles (U. s. statute) 1 .609344 kilometers
cubic feet per second 0.02831685 cubic meters per second
3
UNSTEADY FLOW COMPUTATIONS ON THE OHIO-CUMBERLAND
TENNESSEE-MISSISSIPPI RIVER SYSTEM
PART I: INTRODUCTION
Purpose and Scope
1. The U. S. Army Engineer Division, Ohio River (ORD), directs
the operation of Barkley and Kentucky Reservoirs on the Cumberland and
Tennessee Rivers, respectively, during periods of flooding on the lower
Ohio and lower Mississippi Rivers. The primary objectives of flood con
trol regulation by these reservoirs are to:
a. Safeguard the Mississippi River levee system.
b. Reduce the frequency of use of the Birds Point-New Madrid floodway.
c. Reduce the frequency and magnitude of flooding of lands along the lower Ohio and Mississippi Rivers that are unprotected by levees.
These objectives are met by controlling, to some degree, the Ohio River
stage at Cairo, Illinois. It is obvious that a mathematical model capa
ble of accurately predicting Ohio River stages as a result of reservoir
operations at Barkley and Kentucky would be very useful. With such a
model, the most efficient reservoir operation plan could be determined
for a given flow condition on the Ohio and Mississippi Rivers. The
development and subsequent verification of such a mathematical model to
provide ORD with such a capability were the objectives of the project
described herein.
2. The development of a mathematical model capable of providing
ORD with this capability involves the calculation of unsteady flows in
a system composed of portions of the Ohio, Cumberland, Tennessee, and
Mississippi Rivers. The equations which govern such flows are state
ments of the conservation of mass and momentum of the flow field and
may be written as:
Continuity: ah 1 a(AV) -+ at B ax
4
_g_ = 0 B
(1)
where
Momentum :
a/at - rate of change with respect to time*
h - water- surface elevation above mean sea level
B - effective width of water surface
a/ax - rate of change with respect to distance
A - cross- sect i onal flow area
V - mean flow velocity
q - lateral inflow per unit distance along channel and per unit time
g - acceleration due to gravity
n- Manning ' s resistance coefficient
R - hydraulic radius
(2)
These equations are often referred to as the equations of St . Venant .
A brief discussion of their properties and the assumptions underlying
their derivation are presented in Appendix A. In the past , the momen
tum equation has been s i mplifi ed by omitting some of the terms so that
solutions could be more easily obtained . However, the approach taken
here is to solve the complete set of equations as given above .
Background
3. This study is a direct extension of an earlier project autho
rized by the Mississippi Basin Model Board (MBMB) . At its twenty
seventh meeting on 19 May 1970, MBMB approved a study to develop com
puter programs for unsteady flow computations along reaches of the
Mississippi River and its larger tributaries . At the thirty- second
meeting of the Mississippi Basin Model (MBM) working committee on
7 January 1971, in a joint effort with ORD , the area of responsibility
* For convenience, symbols and unusual abbreviations are listed and defined in the Notation (Appendix C) .
5
of the U. S . Army Engineer Waterways Experiment Station (WES) was de
termined to be the Lower Ohio River from Louisville, Kentucky, through
the junction with the Mississippi River . Figure l is a location map
illustrating the region modeled .
4. In the MBMB study, two mathematical models developed by
Tennessee Valley Authority (TVA) for the calculation of unsteady flows
in open channels were employed . These models (called SOCH and SOCHJ)
are described in detail in the final report of the MBMB study as well
as elsewhere .*, ** Both models use an explicit finite difference scheme
proposed by Stokert for the numerical solution of the St . Venant equa
tions. A brief discussion of the solution technique is presented in
Appendix A. A numerical solution is required since analytical solutions
of the St . Venant equations do not exist . It should be noted that
Stoker ' s original work, in addition to the work described herein, was
also funded by ORD . The first TVA model (SOCH) does not allow for any
branching in the system . Therefore, when SOCH is employed, the flow
from tributaries must be treated as lateral inflow into the main
channel . The second model (SOCHJ) allows a system which contains one
junction to be modeled. One type of data required by these models ln
cludes boundary conditions , which must be prescribed as a function of
time and may consist of either water - surface elevations or discharges
or a rating curve . In addition to the boundary conditions, the other
major set of data which must be input consists of tables containing
*
**
t
B. H. Johnson and P . K. Senter , "Flood Routing Pr ocedure for the Lower Ohio River, " Miscellaneous Paper H- 73- 3 , Jun 1973 , U. S . Army Engineer Waterways Experiment Station , CE , Vicksburg, Miss . M. J . Garrison , J .-P. P . Granjo , and T. J . Price , "Unsteady Flow
Simulation in Rivers and Reservoirs--Applicat i ons and Limitations, " Journal , Hydraulics Division , American Society of Civil Engineers , Vol 95 , No . HY5 , Sep 1969 , pp 1559-1576; presented at ASCE Hydraulics Division Specialty Conference at Cambridge , Mass ., 21- 23 Aug 1968.
J . J. Stoker , "Numerical Solution of Flood Prediction and River Regulat ion Problems ; Report I , Derivation of Basic Theory and Formul ation of Numerical Methods of Attack , " Report No . 200 , Oct 1953 , New York University , Institute of Mathemat ical Sciences , New York , N. Y; prepared for U. S . Army Engineer Division , Ohio River , under Contract DA- 33- 17- eng- 223 .
6
Louisvi lie
GREEN RIVER INFLOW
WABASH RIVER INFLOW
SALINE RIVER INFLOW
CUMBERLAND RIVER INFLOW
Thebes
HW 166
Figure l. Location map for MBMB study
geometric information . Such a table , consisting of top width, flow
area, and (hydraulic radius) 2/ 3 , all as functions of elevation , must be
input at each point of the finite difference net . Additional data such
as control parameters and initial values of elevations and discharges
must also be input .
5. For the I~MB study , geometric data were obtained for the Ohio
River from Louisville to Golconda (see Figure 1) from the MBM . However,
the data used for the remainder of the region had been collected many
years earlier for large , unequal reaches . As noted by Johnson and
Senter , * these reaches had to be broken up and then recombined into
equal reaches since the models require all reaches, i.e., ~x , on a
branch to be equal . With the above geometric data, the models were
calibrated using MBM results from the 1950 flood . After the calibra
tion phase , the models were then applied to the 1945 flood. The re
sults of these applications are presented by Johnson and Senter . *
6. At the conclusion of the above study, it was decided that in
order to provide ORD with a model capable of being used to predict
stages accurately at Cairo for given release schedules at Barkley and
Kentucky Reservoirs , the Cumberland and Tennessee Rivers should be
treated as dynamic branches of the system . As illustrated in Figure 1 ,
these tributaries were treated merely as lateral inflows into the Ohio
River in the MBMB study . In addition , it was also considered necessary
to obtain more accurate geometr ic data from the MBM . The physical
limits of the mathematical modeling effort to provide ORD with a model
to aid in planning releases at Barkley and Kentucky Reservoirs are shown
in Figure 2 .
* Johnson and Senter , op . cit .
8
Grand Chain Cape Girardeau IS 0
50 . ~ Thebes Beechndge o
c.. 15 ~ 30 ~
/"'$) ~ss ~· Price Landing
New Madrid ~ -(I\ (f)
Golconda
Paducah
Metropolis
Lateral Inflow
Wickliffe
Hickman
LEGEND
KENTUCKY DAM
e GAGING STATIONS • DISCHARGE STATIONS ® MILES FROM OHIO-
MISS JUNCTION
HW- 166 100
Ca ruthersvi lie
Figure 2 . Location map for application of SOCHMJ
9
PART II : MULTIJUNCTION UNSTEADY FLOW MODEL (SOCHMJ)
Modification of SOCHJ
7. As previously noted, the physical limits of the mathematical
modeling effort reported herein are shown in Figure 2 . This system is
seen to be composed of three junctions and seven branches. These are:
Junction
1
2
3
Branch
1
2
3
4
5
6
7
Location
Ohio-Cumberland Rivers
Ohio- Tennessee Rivers
Ohio-Mississippi Rivers
Location
From Golconda on the Ohio River to junction 1
From Barkley Dam on the Cumberland River to junction 1
The Ohio River between the Cumberland and Tennessee Rivers
From Kentucky Dam on the Tennessee River to junction 2
The Ohio River between the Tennessee and Mississippi Rivers
From Cape Girardeau on the Upper Mississippi River to junction 3
From the Ohio- Mississippi junction to Caruthersville
As discussed earlier , the digital computer program SOCHJ developed by
TVA can be applied only to a system containing one junction. Therefore ,
a major modification of SOCHJ was required in order to model the Ohio
Cumberland- Tennessee- Mississippi system composed of the three junctions
and seven branches listed above . The modified model (called SOCHMJ) can
be applied to a system composed of any number of junctions and branches ,
including a system containing no junctions . One other major difference
exists between the original SOCHJ model developed by TVA and SOCHMJ . In
SOCHMJ , Manning ' s n is allowed to vary with elevation as well as with
10
distance along the channels , whereas the original SOCHJ allows only for
variation with distance along the channel .
Input Data Required by SOCHMJ
8. Data required for the operation of SOCHMJ are read from cards .
The first data card contains basic information such as the total number
of net points in the system , the total number of junctions and branches,
and the time step employed in the computations . The second group of
data contains information about each branch which consists of the num
bers of the first and last net points of each branch ; the outer boundary
of the branch, whether upstream or downstream; the type of boundary con
ditions prescribed at the various boundaries; and the size of the spa
tial step to be employed for each branch . The third data group contains
information about the junctions in the system being modeled . The pro
gram is instructed as to the numbers of the branches associated with
each junction and which of these branches are upstream or downstream of
the junction . The fourth major data group consists of the tables of
geometric data. A table of top width, flow area , (hydraulic radius) 213 ,
and Manning ' s n , all as functions of elevation, must be input at each
net point in the system. A discussion of how the physical model geo
metric data are converted to these tables is presented by Johnson and
Senter . * The fifth data group specifies initial values of the elevation
and discharge at all grid points on the first two time lines . These are
required in order to initiate the computations . A brief discussion of
the initial conditions required for a solution of the equations of St .
Venant is presented in Appendix A. The final major data group required
by SOCHMJ consists of the time- dependent boundary conditions which must
be prescribed at each open boundary . At such a boundary , either eleva
tions , discharges , or a rating curve may be prescribed as the boundary
condition . It should be realized from this statement that SOCHMJ can be
applied only to subcritical flows. This is discussed further in
* Johnson and Senter, op . cit .
11
Appendix A. A detailed listing of the individual data cards required by
SOCHMJ is presented in Appendix B.
SOCHMJ Computation Cycle
9. As previously noted , SOCHMJ utilizes an explicit finite dif
ference scheme for the numerical solution of the St . Venant equations .
Wi th such an explicit scheme , the solution on a particular time line can
be directly determined at each net point since it involves only known
values on the two previous time lines plus boundary values which are spec
ified . In other words, the solution marches forward in time from one
time line to another . The computations performed by the program from
one time line to the next are referred to as the computation cycle .
10 . When SOCHMJ is ready for operation on a particular time line,
the computation cycle is initiated by checking to see if new values of
lateral inflows at those points in the system where they occur (along ~
with the number of time steps at which they later apply) are to be read
in . If not , linear interpolation is used between the previous two values
input to yield values at this time line . The program next checks each
open boundary (beginning with branch 1 , then branch 2 , and so on) to see
if new boundary values of elevations or discharges are to be read in .
As with the lateral inflows , linear interpolation is used to determine
boundary values at intermediate time lines . The computation cycle then
proceeds with the solution of the difference equations presented in
Appendix A for the water- surface elevations, cross- sectional velocities,
and corresponding discharges at the intermediate points of each branch .
At this point in the computation cycle, the value of the dependent
variable not prescribed as a boundary condition is also computed at each
open boundary of the system . The last computation of the cycle involves
the computation of the dependent variables at each junction . These
calculations are made through an iterative process which ensures that
the elevation at a junction point is the same for all branches forming
the junction . After redefining variables so that values on only three
12
time lines need to be stored, the computation cycle ends.and is ready to
begin again at the next time step.
Output Provided by SOCHMJ
11 . Output can be obtained from SOCHMJ only in printed form . The
user specifies through input data those net points at which output is
desired as well as the type of output . Output in the form of only
elevations, velocities , and discharges at the net points can be re
quested, or the user can request that geometric data such as flow area,
top width, (hydraulic radius) 2/ 3 , and Manning ' s n also be printed .
Output is provided after a certain interval of time steps . The value of
this interval is variable and is specified in the input data .
13
PART III : PRESENTATION OF RESULTS
12 . The application of SOCHMJ to the Ohio- Cumberland- Tennessee
Mississippi River system shown in Figure 2 consisted of two phases . In
the first phase , the model was calibrated using recorded MBM values of
elevations and discharges for the 1950 flood . The second phase then
consisted of an application of the model to the 1973 flood using re
corded field data obtained from ORD . In addition , an application us1ng
both hourly and average daily values of discharges at Barkley and
Kentucky Dams for a 3- day period in February 1974 was made .
Calibration of the Model
13 . As previously discussed , the geometri c tables consisting of
top width , flow area , (hydraulic depth) 213 , and Manning ' s n , all as
functions of elevation , which are required as input data at each net
point of the system, were derived from basic storage data obtained from
the MBM . Johnson and Senter* indicate the manner in which the basic
storage data , which consisted of storage volume and an average top width
versus elevation for each small reach, were converted to the geometric
tables . It should be noted that hydraulic depth rather than hydraulic
r adius was employed .
14 . The boundary conditions input at the upper model limits shown
1n Figure 2 were obtained from the MBM for the 1950 flood . The boundary
conditions specified are as follows :
*
a . Elevations were specified at Golconda on the Ohio River .
b . Discharges were specified at Barkley Dam on the Cumberland River .
c . Discharges were specified at Kentucky Dam on the Tennessee Ri ver .
d . Elevations were specified at Cape Girardeau on the Upper Mississippi River .
Johnson and Senter , op cit.
14
e . A rating curve in the form of a table of elevations versus discharges was specified at Caruthersville on the Lower Mississippi River.
The boundary elevation and discharge hydrographs are presented in
Plate 1 . Table 1, which was obtained from data provided by the U. s . Army Corps of Engineers, Memphis District, presents elevations and
corresponding discharges used as the boundary condition at Caruthers
ville . It should be noted that the looped nature commonly observed in
actual rating curves is not allowed in specifying a boundary condition
for SOCHMJ.
15 . With the spatial steps shown in Table 2 employed for the
seven branches of the system, a common time step of 300 sec was found to
be sufficient to yield stable computations . The stability criterion
which must be satisfied by the time and spatial steps is presented in
Appendix A.
16 . The model was calibrated by comparing calculated and recorded
values of elevation and discharge from the MBM at several points in the
system. The points at which the comparisons were made are indicated in
Figure 2 . The calibration or "matching" of elevations was accomplished
by varying the values for Manning ' s n at net points in the neighbor
hood of the check point . Increasing n at upstream points decreases
the elevation whereas increases at downstream stations increase the
elevations . Discharges were brought into agreement at Metropolis on the
Ohio River, Thebes on the Upper Mississippi River , and at Wickliffe on
the Lower Mississippi River by changing the n values at all net points
on a particular branch by the same amount. A good point to keep in mind
when calibrating such a mathematical model is that elevations are
strongly dependent upon the gradient of Manning ' s n , whereas discharges
are dependent upon the magnitude of n itself . As previously noted,
SOCHMJ allows for the variation of n with elevation as well as with
distance along the channel. However, this capability was not nearly so
essential as in the earlier applications of SOCH and SOCHJ in the MBMB
study, due primarily to the more accurate geometric data used in SOCHMJ .
In keeping with the actual operation of the MBM when the 1950 flood was
15
reproduced, lateral inflows were input at three points. These are indi
cated in Figure 2 and are seen to lie in the neighborhood of Paducah on
the Tennessee River, Grand Chain on the Ohio River, and Beechridge on
the Upper Mississippi River. The inflow hydrograph at each of these
points was the same and is presented in Plate 1 along with the boundary
hydrographs.
17. Plates 2- 4 are plots of calculated versus MBM elevations at
various points in the system. They represent the final results of the
calibration phase of·SOCHMJ . With the exception of parts of the Price
Landing and Beechridge hydrographs on the Upper Mississippi, the calcu
lated and MBM values are seen to lie within 1 . 0 to 1 . 5 ft* of each
other . The difficulty on the Upper Mississippi was probably due to the
geometric representation of that portion of the river composed of the
large "S" loop which covers approximately the first 30 miles upstream
of the Ohio- Mississippi junction. Plate 5 consists of similar plots
of ~ischarges on the Ohio and Upper and Lower Mississippi Rivers .
Application to 1973 Flood
18 . After SOCHMJ was calibrated using MBM results for the 1950
flood, the model was applied to field data for the 1973 flood . In this
application there was no further variation of Manning ' s n . As in the
calibration phase, elevations at Golconda and Cape Girardeau and dis
charges at Barkley and Kentucky Dams were input as upstream boundary
conditions. These hydrographs are presented in Plate 6. At Caruthers-.
ville on the Lower Mississippi, the same rating curve used in the cali-
bration phase, i.e ., Table 1, was employed as the downstream boundary
condition .
19 . Plates 7 and 8 illustrate the comparison of calculated and
field values of elevations at the points where data were available .
Plate 9 is a similar comparison of discharges at Hickman on the Lower
* A table of factors for converting U. S . customary units of measurement to metric (SI) units is presented on page 3.
16
Mississippi River. From an inspection of these plots, it is obvious
that the calculated results from the model were in very good agreement
with the recorded field values. This agreement 1s as good (if not bet
ter in some places) as the results from the calibration phase . The major
exception to this occurs as one gets closer to the lower boundary on the
Mississippi (consider the plot at New Madrid, Plate 8). This exception
is probably due to the influence of the approximate rating curve speci
fied at the lower boundary. However , it should be noted that this in
fluence does not appear to extend far enough upstream to affect the
results at Cairo, which is the major point of interest.
Application Using Hourly Discharge Data
20 . This application was performed to illustrate the use of hourly
discharge data at Barkley and Kentucky Dams . As in the two prev1ous
applications, elevations at Golconda and Cape Girardeau , discharges at
Barkley and Kentucky Dams, and rating curve values at Caruthersville
(Table 1) were prescribed as boundary conditions . These boundary hydro
graphs are presented in Plate 10. Note that the hydrographs that result
from us i ng average daily val ues as wel l as hourly discharges at Barkley
and Kentucky Dams are presented. As in the application to the 1973
flood , there was no variation of Manning ' s n 1n this application.
21 . Plate 11 illustrates the compari son of computed and field
values of elevations at Paducah and Cairo. Calculated elevat i ons re
sulting from first prescribing hourly discharge data at Barkley and
Kentucky Dams and then as a result of applying average daily values are
both presented . From these plots it is obvious that much better results
are obtained at Paducah , which is about 22 miles do1mstream of Kentucky
Dam, when hourly discharges are prescribed. This of course was to be
expected . Applications which ORD will make in planning release sched
ules at Barkley and Kentucky Dams for given flow conditions on the Ohio
and Mississippi Ri vers at Golconda and Cape Gi rardeau, respectively,
will probably employ hourly discharge data at Barkley and Kentucky Dams .
17
PART IV : CONCLUSIONS AND RECOMMENDATIONS
22 . In this study , numerical solutions of the unsteady flow
equations applied to a system composed of portions of the Ohio, Cumber
land , Tennessee, and Mississippi Rivers have been obtained. The model
limits were Golconda on the Ohio, Barkley Dam on the Cumberland,
Kentucky Dam on the Tennessee, Cape Girardeau on the Upper Mississippi ,
and Caruthersville on the Lower Mississippi. The mathematical model was
first calibrated by application to the 1950 flood on the MBM . This was
accomplished by varying Manning ' s n with distance along the rivers , as
well as with elevation at particular points , until recorded and computed
values of elevations and discharges compared favorably . The calibrated
model was then applied to field data from the 1973 flood to determine
its predictive capability . An additional application of the model was
performed for three days i n February 1974 to illustrate the use of
hou~ly discharge data at Barkley and Kentucky Dams .
23 . A comparison of elevation plots at the same location for both
the 1950 and 1973 floods shows that the comparison between computed and
recorded values in the 1973 application is even better than the compari
son in the calibrati on phase at most locations . It should be remembered
that the calibration of the model was undertaken with MBM results whereas
the 1973 application was made using field data . From the one discharge
plot available for the 1973 flood, it is obvious that excellent agree
ment of discharges was also realized. The conclusion can be reached
then that the mathematical model can indeed be used in a predictive
sense . It can also be concluded from the results presented that the
influence of an approximate rating curve employed as a downstream
boundary condition does not extent far enough upstream to affect the
results at Cairo appreciably . As a final note , a continuity check,
which compared the sum of the integrated areas under the discharge
hydrographs at Golconda , Barkley Dam , Kentucky Dam , Cape Girardeau, and
the lateral inflows for the 1950 flood with the area under the discharge
hydrograph at Caruthersille , revealed a difference of less than one
percent .
18
24. Based upon the results of this study, it is concluded that
SOCHMJ provides ORD with a potentially useful model to aid in the plan
ning of reservoir releases from Barkley and Kentucky Dams during periods
of flood control on the lower Ohio and Mississippi Rivers. The model
will also be utilized in developing stage and discharge forecasts for
the Corps' need at other pertinent locations on the lower Ohio and along
the lower Cumberland River. As previously noted, the output from
SOCHMJ can be obtained only in printed form. Graphical output would
result in tremendous savings of time and would make SOCHMJ a much better
operational model. Therefore, it is recommended that the inclusion of
this capability be investigated in the near future. Another area which
should be investigated concerns the downstream boundary condition.
Through the use of Manning's equation, it is possible in such unsteady
flow computations to generate a rating curve as the downstream boundary
condition, as the computations march forward in time. Some thought
should be given to the possibility of modifying SOCHMJ to allow for
this.
25. A further effort has been funded by ORD to extend this model
from Golconda to Louisville utilizing the Wabash River from Mt. Carmel,
Illinois, to the mouth; the Green River from Livermore, Kentucky, to the
mouth; and the Salt River from Shepherdsville, Kentucky, to the mouth as
dynamic branches. The effort will also embrace an algorithm to clrcum
vent the discontinuity in the flow regime caused by the navigation dams
in the Ohio River. The model could then be applied in a single run to
the total flow regime of the Ohio River from Louisville through Cairo.
19
BIBLIOGRAPHY
Isaacson, E., Stoker, J. J., and Troesch, A., "Numerical Solution of Flood Prediction and River Regulation Problems; Report II, Numerical Solution of Flood Problems in Simplified Models of the Ohio River and the Junction of the Ohio and Mississippi Rivers," Report No. 205, Jan 1954, New York University, Institute of Mathematical Sciences, New York, N. Y.; prepared for U. S. Army Engineer Division, Ohio River, under Contract DA-33-017-eng-223.
20
Table 1
Rating Curve Employed at Caruthersville*
Elevation ft msl
255.0 257.0 259 . 0 261.0 263.0 265 . 0 267.0 269.0 271.0 273.0 275 . 0 277 . 0 279.0 281 . 0 283.0 285.0
Discharge cfs
200,000 320,000 440,000 575,000 700,000 820,000 940,000
l,050,0DO 1,170 , 000 1,290,000 1,410,000 1,550,000 1,710,000 1,920,000 2,240,000 2,900,000
* Obtained from data provided by the U. S. Army Corps of Engineers , Memphis District .
Table 2
Spatial Steps Employed in the Mathematical Model
No. of ~x ' s
Branch ~x, miles per Branch
1 4.956 4 2 4.597 6 3 3.319 4 4 3.592 6 5 4.560 10 6 5.237 10 7 4.992 24
APPENDIX A: MATHEMATICAL ASPECTS OF SOCHMJ
Governing Differential Equations and Assumptions
1. Free- surface waves are fluctuations in the surface level of a
fluid . If the ratio of the depth of the mean water level to the wave
length is < l/20 , the waves are classified as shallow- water waves or
long waves. In addition , free- surface waves can also be grouped as
waves of oscillation and waves of translation . As a translatory wave
advances, the water over which it passes is also moved forward in the
direction of motion . Translatory waves are thus related to unsteady
flows. An example of a long translatory wave in a natural channel is a
flood wave .
2 . The process of propagation of long waves in open channels is
described by the St. Venant equations. These equations represent the
conservation of mass and momentum of the fluid.
3. The assumptions that are made in the derivation of the St .
Venant equations are as follows :
a . The flow i s assumed to be one- dimensional , i . e. , the flow in the channel can be approximated with uniform velocity over each cross section and the free surface is taken to be a horizontal line across the section .
b . The pressure is assumed to be hydrostatic , i.e ., the vertical acceleration is neglected and the dens i ty of the fluid is assumed to be homogeneous .
c . The effects of boundary friction and turbulence can be accounted for through the introduction of a resistance force which is described by the empirical Manning friction factor equation . With the additional assumption that the bottom slope of the channel is small , the governing equa-
·tions become :
Continuity :
Momentum:
~ + l a(AV) at B ax
Al
~ = 0 B
(1 bis)
(2 bis)
where
a/at - rate of change with respect to time
h - water- surface elevation above mean sea level
t - time
B - effective width of water surface
a/ax - rate of change with respect to distance
A - cross- sectional flow area
V - average flow velocity
x - distance along channel
q - lateral inflow per unit distance along channel per unit time
g - acceleration due to gravity
n- Manning ' s resistance coefficient
R - hydraulic radius
4. Since aA/ax can be expressed as a function of ah/ax , the ,
previous equations constitute a system of two nonlinear , first order ,
fir~t degree, partial differential equations with two independent varia
bles x and t and two dependent variables h and V . Upon casting
the equations into the matrix form
[U]t + [A][U]x - [B]
where
[] - matrix, either square or column
U - column matrix with h and V as elements
it can easily be seen that the two eigenvalues of [A] are both real
and [A] is capable of being diagonalized. Therefore , the above system
of equations is classified as a hyperbolic system.* Thus, so long as
the flow remains subcritical , i . e ., only one of the eigenvalues of [A]
is positive, only one of the two dependent variables , or a relation be
tween them, must be specified as a function of time at the two
* P . D. Lax, Partial Differential Equations , New York University , Institute of Mathematical Sciences , New York , N. Y., 1950 .
A2
boundaries defining the reach of open channel of interest. Therefore,
at both the upstream and downstream ends, either the stage or the veloc
ity (usually the discharge since the discharge Q is given by mean
velocity times cross-sectional flow area) or a rating curve, i.e., h
versus Q , must be specified. In addition, initial values of h
and Q must be specified as functions of x . A steady-flow profile,
or a flat pool-zero flow profile, can be prescribed for the initial
conditions. In addition, a transient profile from previous computations
can be used. The specification of initial conditions is flexible due to
the characteristic of hyperbolic equations that the solution becomes
independent of initial conditions after a sufficient length of time.
5. A different form of the basic equations is used in the compu
tation of boundary values of the unspecified dependent variable Q when
h is input as the boundary condition. Multiplying Equation 2 by
+(AB/g)1 / 2 and adding to Equation 1 yields
a(AV) + B ah _ ax at + t)~ 0 (Al)
where the sign is used at the left or upstream boundary and the +
sign is used at the right or downstream boundary.
Difference Equations
6. The previous system of equations cannot be solved analytically,
and must be solved by numerical methods such as finite differences. When
using finite differences to find numerical solutions, values of the un
known variables are obtained at a discrete set of points called net points.
The simplest representation uses a rectangular net of lines superimposed
on the x,t plane such that one family of lines is parallel to the x
axis and the other is parallel to the t axis. In general, the lines
are assumed equispaced with x interval ~x and t interval ~t ·
7. Both implicit and explicit finite difference schemes are em
ployed in solving differential equations. If an implicit representation
A3
is employed, one is confronted with the problem of solving a system of
algebraic equations at each time level. Usually, iterative techniques
must be employed . In contrast, in explicit schemes values of the de
pendent variables at each time level depend only upon values at previous
time levels . Thus, explicit schemes are very desirable due to the ease
with which computations are made. The major reason for choosing im
plicit finite difference schemes over explicit ones is due to stability
considerations . In general, implicit schemes are much more stable than
explicit ones . However, an explicit- centered difference scheme proposed
by Stoker* has been found to be sufficiently stable and convergent for
the numerical solution of the unsteady flow equations if the following
relation is satisfied:
(v gn2 Jvl ----..:......,-.:.,....- l'l t 2.21 R4/ 3
where ~t = the time increment and l'lx = the distance increment.
, 8. Stoker's scheme utilizes a staggered rectangular net as il
lustrated in Figure Al. With such a staggered net, values of h and V
are computed only at every other point on a particular time line, with
the points at which computations are made alternating from one time line
to the next. Considering the appropriate circled area of Figure Al
interior points, the spatial derivatives are replaced by a centered
difference scheme, i.e . , at the interior point labeled P
where in this and the following equations:
* J. J . Stoker, "Numerical Solution of Flood Prediction and River Regulation Problems; Report I, Derivation of Basic Theory and Formu-lation of Numerical Methods of Attack," Report No . 200, Oct 1953, New York University, Institute of Mathematical Sciences, New York, N.Y.; prepared for U.S . Army Engineer Division, Ohio River, under Contract DA- 33- 17- eng- 223 .
A4
T
LB
" ""' w R 81' p 4 ' /" "'\ '
3 I' < ,... Ll'" p • M
"\ \, \,
"' 0~p //~ w/
/ / \ / L' M' ' -,
[7 ' [\ ~~--p •
J L I 1 "\ -< !"' '-./ '
M L R L\T M' L' /"' ...,M 0 \ ./ •
0 1 2 3 4 5 6 7 8 ~
Figure Al . Centered difference computation net
1 ,1 ' ,M,M' , R,W -(subscripts)
p P' -' (subscript s)
points in the x,t plane where discharge, mean velocity, and water-surface elevation are known
points in the velocity , and calculated
x,t, plane where discharge, mean water-surface elevation are to be
whereas time derivatives are replaced by
( oh) ..__ hP - ~ ot P - 2llt
Therefore, the finite difference forms of Equations 1 and 2 used for
computations at interior points are :
X
llt 1 h = h --
p M llx BM (A2)
where 1R(subscript) - distance along the channel from point 1 to
point R of the x,t plane, and
At VR + v1 At V u ( V V ) :;-X ( hR - h_ ) - M - llx 2 R - 1 - g u -~
-
A5
-2lltgn~RvMivMI
2.21 ~13 (A3)
Also,
(A4)
9. The difference representation of the equations to be applied
at the boundaries is different . At the left boundary, if Q . lS pre-
scribed , the finite difference representation of Equation l used for
the computation of ~ is
l 2 (hL ' + ~~ )
2 ~t
(A5)
If h is prescribed as the left boundary condition, Stoker ' s scheme
utilizes a finite difference representation of Equation Al with the
negative slgn . The difference scheme is such that one first computes
V on time lines that do not contain boundary points . For example, the
computation of V at point P ' (VP , ) is made using the following dif
ference form of Equation Al . ,
D.t - q_L ' M'
- 0 (A6)
Next, the computation of V at the boundary point P is obtained from
A6
Equation A6 with the primes dropped and 2!1x replaced by ~x .
10. At the right boundary, with Q prescribed, h is computed
from Equation A5 with ~x replaced by -~x . If h is prescribed as
the right boundary condition, V is computed from the difference rep
resentation of Equation Al with the positive sign. The resulting equa
tion for computation of VP on nonboundary time lines is
+ B ' M ~ · - ~'
!1t
~I - 0 (A7)
Then the computation of V at the boundary point P lS obtained from
the above equation with the prlmes dropped and 26x replaced by 6x •
11. If a rating curve in the form of a table of h values
versus Q values is specified as a boundary condition, h is computed
from an equation such as Equation A5. Then Q is determined from the
elevation- discharge table . This value is then used to recompute h .
This process continues until the computations for h have sufficiently
converged.
A7
APPENDIX B: LIST OF INPUT REQUIRED BY SOCHMJ
Input is submitted on cards 1n the following order and format.
l. TITLE (A80)
2 . NSTA, NBRCHS, JUNCS, IS5, NSTEP, MNTH, KDAY, KYEAR, I.MTE, TIME, DT (9I5, 5X, 2Fl0 .3)
NSTA - Sum of total number of grid points on each branch . Junction point counts on each branch
NBRCHS - Total number of branches
JUNCS
IS5=l
=0
NSTEP
MNTH
KDAY
KYEAR
lMTE
TIME
DT
- Total number of junctions
-Elevation, disc~irge, velocity, area, width, (hy-draulic radjus) 3, Manning ' s n are printed
-Elevation, discharge, and velocity are printed
-Number of entries in geometric tables ( <9)
- Starting month
- Starting day
- Year
- Print interval
- Starting time
- Time step in seconds
3 . NBRCH, IBRNCH (I,l) , IBRNCH (I , 2), IDIR (I), NEQ (I), TMILE (I), DX (I)
(515, 5X, 2Fl0 . 0 )
(One card for each branch)
NBRCH = Number of this branch
IBRNCH (I,l) = Number of first grid point in the Ith branch
IBRNCH (I,2) = Number of last grid point in the Ith branch
If NBRCHS = 2, set IBRNCH (l,l) = l and IBRNCH (1,2) = NSTA and IBRNCH (2,1) = IBRNCH (2,2) = NSTA .
IDIR (I) - (+) - Ith branch has upstream outer boundary
- (0) - Ith branch is an interior branch
NEQ (I)
- ( - ) - Ith branch has downstream outer boundary
_ 0 - Elevations are prescribed for boundary conditions of Ith branch
Bl
NEQ - l - Discharges are prescribed for boundary con-ditions of Ith branch
- N - Rating curve is used for boundary conditions of Ith branch and N values of elevation versus discharge must be read in
TMILE (I) - Extreme upstream mileage of Ith branch
DX (I) Spatial th - step used for I branch
4. NJUNC, N, (IJUNC (I, J), J = l, N), (IFLOW (I, J), J- l , N) (l2I5)
(One card for each junction)
NJUNC = Junction number
N = Number of branches coming into this junction
(IJUNC (I, J), J- l , N) - Branch numbers which comprise the Ith junction
(IFLOW (I, J), J- l, N) = (- ) = Jth branch is downstream of Ith junction
- 2 = Side branch coincides with junction
- ( + ) th th = J branch is upstream of the I junction
If NBRCHS - 2, this card is omitted .
5 . (PRMILE (I) , I- l, NSTA) - Print selection control
Input l ' s at those net points for which output is desired
6 . ( ELEV ( K, I ) , AREA ( K, I ) , R2 3 ( K, I ) , WIDTH ( K , I ) , CR (K, I), K = l, NSTEP)
Geometric data: Begin with first grid or net point on first branch and follow through last grid point on last branch
7. (ELTAB (I, J) , QTAB (I, J) , J = l, NEQ (I))
Rating curve for outer boundary of Ith branch if NEQ (I) > l
8. H(IMIN , 2), Q(IMIN, 2), H(IMAX, 2), Q(IMAX, 2)
Elevation and discharge at the first and last grid points on the second starting line
9. H(JJ, 1), Q(JJ, 1), H(JJ + l , 2), Q(JJ + l, 2)
Elevation and discharge at interior grid points of this branch on first and second time lines
B2
(38I2)
(5Fl0 . 0)
(8Fl0 . 0)
(5Fl0.0)
(8Fl0 . 0)
10 . QLAST (I), ELAST (I)
QLAST (I) - Discharge at outer boundary of on third time line
(5Fl0 . 0) Ith branch
ELAST (I) - Elevation of outer boundary of Ith branch on third time line
This card is input only if this branch contains an outer boundary.
Cards 8, 9, and 10 (if 10 is applicable) are input in sequence for each branch in the system. If a branch consists of only one grid point such as a branch coinciding with a junction or a downstream boundary, cards 8, 9, and 10 are not input for that branch .
ll . NRCH , (IRCH (I) , I= l, NRCH) (l2I5)
NRCH = Number of reaches that contain lateral inflow
IRCH (I) = The upstream station numbers of the reaches that contain lateral inflow
12. (XINFL (I), I- l, NRCH)
XINFL (I) - Third line of lateral inflows
This card is omitted if NRCH = 0 .
(8Fl0 . 0)
13. IFCNT, (XINFL (I), I= l, NRCH) (I5, 5X, 7FlO . O, /, (lOX, 7Fl0 . 0)
IFCNT = Number of time steps until the new lateral inflows which are about to be read apply
XINFL (I) = New line of lateral inflows
This card is omitted if NRCH = 0 .
14. ELAST (II) , IECK (II)
ELAST (II) = Elevation specified as the boundary condition for the IIth branch
IECK (II) = Number of time steps before a new value of ELAST (II) is read in
If IDIR (II) = 0 , this is an interior branch and there will be no card . In any case, this card is present only if NEQ (II) = 0 .
15. QLAST (II), IQCK (II)
QLAST (II) = Discharge specified as the boundary condition for the IIth branch
(FlO . O, I5)
(Fl0. 5, I5)
IQCK (II) = Half the number of time steps before a new value of QLAST (II) is read in
B3
If IDIR (II) = 0, this is an interior branch and this card is omitted . In any case, this card cannot be present unless NEQ ( II) = l .
Cards 12, 13, and 14 are repeated . Remember that the check on whether to read in new cards or not is first on l ateral inflows and. then on boundary conditions .
B4
APPENDIX C: NOTATION
A Cross-sectional flaw area, or s~uare coefficient matrix
B Effective width of water surface, or a column matr.ix
g Acceleration due to gravity
h Water - surface elevation above mean sea level
n Manning's resistance coefficient
~
Q
R
t
u
Lateral inflow per unit time
Discharge given by flow area
Hydraulic radius
Time
Column matrix with
V Mean flow velocity
unit
mean
h
x Distance along channel
At Time increment
Ax Distance increment
distance along channel and per
velocity times cross -sectional
and v as elements
a/at Rate of change with respect to time
a/ax Rate of change with respect to distance
[] Matrix, either s~uare or column
Subscripts
L '1 I ,M ,M' 'R 'w Points in the x,t plane where discharge, mean velocity, and water - surface elevation are known
LR Distance along the channel from point 1 to point R of the x, t plane
P,P' Points in the x,t plane where discharge, mean velocity, and water - surface elevation are to be calculated
Cl