J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 89
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11.6.9. Junction net pointscheme^the upstream Mississippi are represented by x0, while the downstream Mississippi is described for x S> 0. The time t2.5 hrs, as=explained above, corresponds to the instant that the forerunner ofthe flood reaches the junction.The values of the quantities v andas follows:Assume that the valuesc at the junction were determinedof v and c have been obtained at allnet points for times preceding that of the boundary net point P, whichrepresents a point at the junction. We use at this point the relations=\/gy and the water level is the same in the three branches atthe junction. In addition, we havesince cwhat flows into the junction from the upstream side of the Mississippi and from the Ohio must flow out of the junction into the downstream branch of the Mississippi.
If the values of v and c were knownsinceMat the pointin Fig. 11.6.9 in the respective branches of the rivers,could find the values atfrom equation (11.5.6) for the Ohio andwePthe upstream side of the Mississippi, and equation (11.5.7) for theMATHEMATICAL HYDRAULICSdownstreamWeside of the Mississippi.497rewrite the equations forconvenience, as follows:CP(l)=C+="<V(3)>P(2)IJ=P(2)(2 ^P(3)with=Cy (l)(since=y (2)=j/ (3) ),nifi9\(11.6.2),+-----,--{...\>./,.(Of(J)+PMU))l+VL(,)\\----Ijandl)----M(3)"A/(3)"""27IThe above system,,--^jof six linear equations determines uniquely thez;c (3) atin terms of their values at the{3)values u (1) c (1) u (2) c (2),AAxAt(,P,preceding points L, M and R. The equations can be solvedThe values of the relevant quantities at M are determined in the sameexplicitly.way fromF and B.
The values at A and Bthe neighboring pointsbetweeninterpolationthe preceding values at A,arc determined byrespectively (sec Fig. 11.6.9). Of course, it is nemotions in each of the branches away from thethetotreatcessarymethods as were described for the problem ofthesamejunction by(K,F) and (F, G)the Ohio treated above, and thisisfeasibleonce the values of v andchave been obtained at the junction.Theresults of the calculations areshownin Fig. 11.6.10,whichfurnishes the river profiles, i.e.
the depths as functions of the locationin each of the three branches, for times t(K 2.5, 4, and 10 hours=after the beginning of the flood 50 miles up the Ohio. The curves fortoo are those for the steady flow which was calculated above in=Fig. 11.2.3).
The calculations indicate that the unsteadyflow does tend to the steady flow as the time increases. Another nosec*.11.2(cf.the backwater effect in the upper branch of theMississippi. For example, the stage is increased by about 2 feet at apoint in the Mississippi 20 miles above the junction and 7.5 hours afterticeable effectisthe flood wave from the Ohiofirstreaches the junction.WATER WAVES498It might be mentioned that the forerunners of the flood in all threebranches were computed by using the expansion scheme which isexplained in the appendix to this chapter.xy==distonceinmilesmeasured from junctionstage measured in feett= time in hours after startof flood40'10'////ii i i i ii1 1 iii iiiii1 1iiFig.
11.6.10 River profiles for the junction11.7.Numerical prediction of an actual flood in the Ohio, and at itsjunction with the Mississippi. Comparison of the predicted withthe observed floodswave problems inabove and applied to simplified models of the Ohioand its junction with the Mississippi have been used to predict theprogress of a flood in the Ohio as it actually is, and likewise to predictthe progress of a flood coming from the Ohio and passing through thejunction with the Mississippi. The data for the flood in the Ohio weretaken for the case of the big flood of 1945, and predictions were madeThe methodsfor numerical analysis of floodrivers developednumerically for periods up to sixteen days for the 400-mile longstretch of the Ohio extending from Wheeling, West Virginia, toCincinnati, Ohio.
For the flood through the junction, the data forthe 1947 flood were used, and predictions were made in all threebranches for distances of roughly 40 miles from the junction alongMATHEMATICAL HYDRAULICS499each branch. In each case the state of the river, or river system, wastaken from the observed flood at a certain time t = 0; for subsequenttimes the inflows from tributaries and the local run-off in the mainriver valley were taken from the actual records, and then the differential equations were integrated numerically with the use of theUNIVAC digital computer in order to obtain the river stages and discharges at future times.
The flood predictions made in thisthen compared with the actual records of the flood.way wereA comparison of observed with calculated flood stages will be givencan be said in general that there is no doubt thatthis method of dealing with flood waves in rivers is entirely feasiblesince it gives accurate results without the necessity for unduly largeamounts of expensive computing time on a machine such as theUNIVAC. For example, a prediction for six days in the 400-milestretch of the Ohio requires less than three hours of machine time.This amount of calculating time which is anyway not unreasonablycould almost certainly be materially reduced by modifyinglargeappropriately the basic methods; so far.
no attention has been givento this aspect of the problem, since it was thought most importantfirst of all to find out whether the basic idea of predicting floods bylater on; however,itintegrating the complete differential equations is sound. The fact thatsuch problems can be solved successfully in this way is, of course, amatter of considerable practical importance from various points ofview.
For example, this method of dealing with flood problems inrivers is far less expensive than it is to build models of a long river ora river system, and it appears to be accurate. Actually, the twomethods empirically by a model, or by calculation from the theory- -are in thepresent case basically similar, since the models are reallyhuge and expensive calculating machines of the type called analoguecomputers, and the processes used in both methods are at bottom thesame, even in details. An amplification of these remarks will be madelater on.would require an inordinate amount of space in this book to dealmethods used to convert the empirical data for ariver into a form suitable for computations of the type under discussionhere, and with the details of coding for the calculating machine; forItin detail with thethis, reference is made to a report [1.4].
Instead, only a brief outlineof the procedures used will be given here.In the first place, it is necessary to have records of past floods withstages up to the maximum of any to be predicted. It would be idealWATER WAVES500to have records of flood stages and discharges (or, what comes to thesame thing, of average velocities over a cross-section) at points closelyspaced along the river at ten mile intervals, say. Unfortunately,measurements of this kind are available only at much wider interof the order of 50 to 80 miles or more even in the Ohiovals *which the data are more extensive than for most rivers inforRiver,the United States. From such records, it is possible to obtain the coefficient of the all-important resistancetermin the differential equa-tion expressing the law of conservation of momentum.
This coefficientdepends on both the location of the point along the river and thestage. The other essential quantity, the cross-section area, also as afunction of location along the river and of stage, could in principlebe determined from contour maps of the river valley; this is, in fact,the method used in building models, and it could have been used insetting up the problem for numerical calculation in the manner underhad been done, the results obtained wouldmorebeenhaveaccurate; however, such a procedure isprobablyextremely laborious and time consuming, and since the other equallyimportant empirical element, i.e. the resistance coefficient, is knownonly as an average over each of the reaches (this applies equally tothe models of a river), it seems reasonable to make use of an averagecross-section area over each reach also.
Such an average cross-sectionarea was obtained by analyzing data from past floods in such a wayas to determine the water storage volumes in each reach, and from theman average cross-section area as a function of the river stages wasdiscussion here. If thatcalculated.
In thisway the coefficients of the differential equations areobtained as numerically tabulated functions of x and y. (It mightperhaps be reasonable to remark at this point that the carrying outof this program is a fairly heavy task, which requires close cooperationwith the engineers who are familiar with the data and who understandalso what is needed in order to operate with the differential equations).In Fig. 11.7.1 a diagrammatic sketch of the Ohio River betweenWheeling and Cincinnati is shown, together with the reaches andobservation stations at their ends.
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