J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 90
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What we now have are resistanceandcross-section areas that represent averages over anygiven reach. However, the reaches are too long to serve as intervalsfor the method of finite differenceswhich is basic for the numericalcoefficientsintegration of the differential equations. Rather, an interval between*Each suchintervalregulation problems.iscalled a reachby those who workpractically with riverMATHEMATICAL HYDRAULICS501net points (in the staggered scheme described in the preceding section)of 10 miles was taken in order to obtain a sufficiently accurate approx-imation to the exact solution of the problem.Atime interval ofCincinnatiHuntmgtonFig. 11.7.1.
Reaches in the Ohio9 minutes was used. Actually, calculations were first made using a5-mile interval along the river, but it was found on doubling the interval to 10 miles that no appreciable loss in accuracy resulted.To begin with, flood predictions for the 1945 flood were made, starting at a time when the river was low and the flow was practically asteady flow. Calculations were first made for a 36 hour period duringwhich the flood was rising; as stated earlier, these were made usingthe measured inflows from tributaries, and the estimated run-offin the main valley. Upon comparison with the actual records, it wasfound that the predicted flood stages were systematically higher thanthe observed flood stages, and that the discrepancy increased steadilywith increase in the time. It seemed reasonable to suppose that theerror was probably due to an error in the resistance coefficient.
Consequently a series of calculations was made on the UNIVAC in whichthis coefficient was varied in different ways; from these results, corrected coefficients were estimated for each one of the reaches. ActuallyWATER WAVES502this was done rather roughly, with no attempt to make correctionsthat would require a modification in the shape of these curves in theirdependence on the stage. The new coefficients, thus corrected on thebasis of 36-hour predictions (and thus for flood stages far under themaximum), were then used to make predictions for various 6-dayperiods, as well as some 16-day periods, with quite good results, onthe whole.might be said at this point that making such a correction of theon the basis of a comparison with an actual floodcorresponds exactly to what is done in making model studies.
There,Itresistance oceff icientalways necessary to make a number of verification runs after theis built in order to compare the observed floods in the modelwith actual floods. In doing so, the first run is normally made withoutmaking any effort to have the resistance correct in fact, the roughit ismodelness of the concrete of the model furnishes the only resistance at thestart.
Of course it is then observed that the flood stages arc too lowbecause the water runs off too fast. Brass knobs are then screwedinto the bed of the model, and wire screen is placed at some parts ofthe model, until it is found that the flood stages given by the modelagree with the observations. This is, in effect, what was done inmaking numerical calculations. In other words, the resistance cannotbe scaled properly in a model, but must be taken care of in an empirical way. The model is thus not a true model, but, as was stated earlier,it is rather a calculating machine of the class called analogue computers.
It is, however, a very expensive calculating machine which can,in addition, solve only one very restricted problem. A model of twofair sized rivers, for example, consisting of two branches perhaps 200miles in length upwards from their junction, together with a shortportion below the junctions, could cost more than a UNIVAC.It has already been stated that average cross-section areas for theindividual reaches were used in making the numerical computations,while in the model the cross-sections arc obtained from the contourmaps. In operating numerically it is possible to change the local crosssection areas without any difficulty, and this might be necessary atcertain places along the river.Some idea of the results of the calculations for the 1945 flood in theOhiogiven by Fig.
11.7.2. The graph shows the river stage at Poas a function of the time. At the other stations the results wereismeroyon the whole more accurate. The graph marked "computation withoriginal data", and which covers a 36 hour period, was computed onMATHEMATICAL HYDRAULICS503the basis of the resistance coefficients as estimated from the basicflow data for theriver.As onesees, these coefficients resulted intoo high stages, and corrections tomuchthem were made along theriverComputed hydrogroph(resistance adjusted)5541212Feb 28Feb 27Fig.1 1.7.2.Comparison of calculated with observed stages at Pomeroy1945 flood in the Ohio Riverresults of this computation. Afterwards, floodmade for periods up to 16 days without furtheron the basis of thepredictions worefor theThebesMetropolisMISSISSIPPI32m.HickmonFig. 11.7.3.The junction of the Ohio and theMississippiWATER WAVES504The graph indicates results for a 6 daywhichthefloodwas rising.
Evidently, the calculatedperiod duringand observed stages agree very well.correction of these coefficients.300296292-288StageJan15182421atHickman3027304-Observed stages300Computed hydrographStageJan151821Fig. 11.7.4. Calculated2427at Cairo30and observed stages at Cairo and HickmanIn Fig. 11.7.3 a diagrammatic sketch of the junction of the Ohioand the Mississippi is shown indicating the portions of these riverswhich entered into the calculation of a flood coming down the Ohioand passing through the junction.
The flood in question was that ofMATHEMATICAL HYDRAULICS5051947. It was assumed that the stages at Metropolis in the Ohio (about40 miles above Cairo) and at Thebes in the upper Mississippi (alsoabout 40 miles above Cairo) were given as a function of the time. AtHickman in the lower Mississippi (about 40 miles below Cairo) thestage-discharge relation at this point, as known from observations,as a boundary condition. The results of a calculation for awas used16 day period are shown in Fig. 11.7.4, which gives the stages atCairo, and at the terminating point in the lower Mississippi, i.e. atHickman. As one sees, the accuracy of the prediction is very high,the error never exceeding 0.6 foot. It might be mentioned that aprediction for 6 days requires about one hour of calculating timeon the UNIVAC, so that the calculating time for the 16 day periodwas under 8 hours, which seems reasonable.
This problem of routing a flood through a junction is, as has been mentioned before,one which has not been dealt with successfully by the engineeringmethods used for flood routing in long rivers.*Appendix to Chapter 11Expansion in the neighborhood of thefirstcharacteristicbeen mentioned already that whereas the forerunner of adisturbance initiated at a certain point in a river at a moment whenthe flow is uniform travels downstream with the speed v + \/gy, theIt hasmain part of the flood wave travels more slowly (cf. Deymte [D.9]),depending strongly on the resistance of the river bed.
An investigationof the motion near the head of the wave, i.e. near the first characteris<?tic (cf. the first part of sec. 11.6) with the equation x)<,(V Qshows immediately why the main part of the disturbance will in=+behind the forerunners of the wave.The motion is investigated in this Appendix by means of an expansion in terms of a parameter that has been devised by G. Whithamand A. Troesch and carried out to terms of the two first orders for themodel of the Ohio River, and to the lowest order in the much moregeneralfall* Added inproof: In the meantime, calculations have been completed ( see [I.4a] )for the case of floods through the Kentucky Reservoir at the mouth of theTennessee River. The calculated and observed stages differed only by inches fora flood period of three weeks over the 186 miles of the resevoir.WATER WAVES506complicated case of the junction problem. The results obtained makepossible to improve the accuracy of the solution near the firstcharacteristic which separates the region of undisturbed flow fromthat of the flood wave.
It turns out that the finite difference schemeitwhich are tooyields river depthslarge, as indicatedbyFig. 11. A.I.computed byv profilefinite differenceswovefront'IRegion ofundisturbed'uniform flowFig. 11. A.I. Error introduced by finite difference scheme in neighborhood offirst characteristic of a rapidly rising flood waveIn order to expand the solution in the neighborhood of the wavewe introduce new coordinates f and r as follows:front,=such that the-axisx and r(i.e.r==(v+cx)t0) coincides with the first characteristic.Near the front of the wave T will be small, and the expansion will becarried out by developing v and c in powers of r. The basic system ofequationsisrestated for convenience:%cc xcv xUpon+v + vv x - gS + gS =+ 2vc x + 2c = 0.cT )0,tsubstitution of the(11.A.1)ftnewvariables |+ v(vt-v ++ c(0-0 +)(vT)2(rand r we find+ c )v r - gS + gS, + c Q )c = 0.0,rwhere the friction slope Sf for a rectangular channel of widthB is givenby4/3MATHEMATICAL HYDRAULICS507We expand v and c as power series in r with coefficients that are functions ofas follows:v=<>we are inThis expansion is to be used for ronly, since for rthe undisturbed region and all the functions ^(1), u 2 (),., c x (f ),c intotheforseriesvandinsertIfwec 2 (),vanish identically...T, we getThe equationsv l = 2Cj Theequations (ll.A.l) and collect terms of the same order inordinary differentialresultingequations for^(f ),from the terms of zero orderc x (f ),....in r yieldfirst order terms become, after thus eliminating v l..1dc=*>o(11.A.2)Byadding these two equations and removing thewefind the differential equation for ^(1)1Although the solution ofcommonfactor 4,is:21this differential equation for-^(f )0.iseasilyobtained, the result expressed in general terms is complicated, and itthe case of the model of the Ohio Riverispreferable to give it only for.wove frontthe front of aFig.
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