J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 86
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For example, the depth y (correspondingto the stage of the river) or the velocity v (which together with thecross-section area A fixes the rate of discharge) might be given interms of the time.x>Initial conditionsdownstream fromthis point(i.e.might also be prescribed. Suppose, for example, that theoftheriver is prescribed at x -= 0, i.e. that j/(0, t) is known, andstagethat the calculation had already progressed so far as to yield values offor0)net points along a certain line parallel to the #-axis andthecontainingpoints L, M, JB, as indicated in Fig.
11.5.3. It is clearthat the determination of the values of v and c at point P can be ob-vandc attained from their values at L, M, R by using (11.5.8) and (11.5.9),above discussion of the initial value problem, and similarlyas in theMATHEMATICAL HYDRAULICS479P v P2 etc. However, the value of v at Q must be detera different manner; for this purpose we use the equation(11.5.7) with the subscript Q replacing P, L replacing Af, andis alsoreplacing R. Since V M C M V R C R are supposed known, andfor pointsmined,inM,,CQ,'t-PMFig. 11.5.3. Satisfyingboundary conditionsknownsince the values of y arc proscribed on the /-axis, it follows thatequation (11.5.7) contains VQ as the only unknown; in fact it is givenby the equationv Q ^v L +At\(11.5.10)The reason(c Lfor using (11.5.7) instead of (11.5.0)is,of course, that thebackward characteristic, andpoints M and Q are associated with thehence (11.5.2) should be used to approximate the ^-derivatives at..Ldifference(where the differential equations are replaced bythe same general procedure could beequations).
It is quite clear thatused to calculate CQ if the values of v had been assumed given alongthe /-axis. If, on the other hand, we had a boundary condition on theof our domain instead of on the left, as above, we could make usepointrightof (11.5.6) for the forward characteristic as a basis for obtaining theformula for advancing the solution along the /-axis.The above discussion would seem to imply that under all circumstances only one boundary condition could be imposed that is, thateither v or c could be prescribed at a fixed point on the river, but notsince prescribing one of these quantities leads to a unique determination of the other.
This is, indeed, true in any ordinary river,bothbut not necessarily in allin the above discussion,from any point of thecases. In fact,i.e.we made atacitassumptionthat of the two characteristics issuing/-axis only theforward characteristic goes intoWATER WAVES480the region xv+and vc>-to the right of the J-axis, and this in turn implies thatc, which fix the slopes of the characteristics, are op-posite in sign.
The physical interpretation of this is that the value ofv (which is positive here) must be less than c^gy, i.e. that theflow must be what is called tranquil, or subcritical.* Otherwise, as=wevseeandcfrom Fig.11.5.4,we should expectat points close toto determine the values ofand to the right of theJ-axis,say atK9byMFig. 11.5.4.Acase of super-critical flowboth v and c along the segment LQ, its domain ofdependence. The scheme outlined above would therefore have to bemodified in a proper way under such circumstances.
One sees, however, how useful the theory based on the characteristics can be eventhough no direct use of it is made in the numerical calculations (asidefrom decisions regarding the maximum permissible size of the ^-interutilizing values forval).The procedure sketched out above, whileit isrecommendedfor use.PMFig.
11.5.5.Astaggered netIn gas dynamics the flow in an analogous case would be called subsonic.MATHEMATICAL HYDRAULICS481when a boundarycondition is to be satisfied, is not always the bestone to use for advancing the solution to such points as P, Pl9 P 2in Fig. 11.5.3. For such "interior points" a staggered rectangular net,as indicated in Fig. 11.5.5, and a difference equation scheme based onthe original differential equations (11.4.6) may be preferable (cf.Keller and Lax [K.4] for a discussion of this scheme). The equations(11.4.6) were,(11.5.11)2vccv2c t=...0.MThe values V M and C Mat the mid-point(which is, however, not aare defined by the averages:net point) of the segmentLR(11.5.12)which the derivatives atafterway byM are approximatedina quite naturalthe difference quotients''AxAx(11.5.13)uMLMAtUponsubstitution of these quantities into (11.5.11), evaluation of theand subsequent solution of theat pointc, u, andtwo equationsMEcoefficientsfor v pandcp,9the resultis7Ax(11.5.14)Ax(11.5.8) and (11.5.
9\ these equationsThe criterion for convergence reones.theearlierare simpler thanmains the same as before, i.e. that P should lie within a triangle formedAs weseeon comparison withby the segmentends.LRand the twocharacteristics issuingfromitsWATER WAVES48211.6. Flood prediction in rivers. Floods inandmodels of the Ohio Riverjunction with the Mississippi RiveritsThe theory developedin the preceding sectionscan be used to makepredictions of floods in rivers on the basis of the observed, or estimated, flow into the river from its tributaries and from the local run-off,together with the state of the river at some initial instant. Hydraulicsengineers have developed a procedure, called flood-routing, to accomplish the same purpose. The flood-routing procedure can be deducedas an approximation in some sense to the solution of the basic differential equations for flow in open channelsGilcrest in the book by Rouse [R.ll] ), but itdifferential equations.(cf. the article by B.
R.makes no direct use of theHowever, the flood-routing procedurein ques-tion seems not to give entirely satisfactory results in cases other thanthat of determining the progress of a flood down a long riverforofwhattheatasuchofasthatexample,problemhappensjunction,the Ohio and Mississippi Rivers, or the problem of calculating thetransient effects resulting from regulation at a dam, such as theKentucky dam at the mouth of the Tennessee River, seem to be difficult to treat by methods that are modifications of the more or lessstandard flood-routing procedures. Even for a long river like theOhio, the usual procedure fails occasionally to yield the observed riverstages at some places.
On the other hand, the basic differential equations for flow inopen channels are in principle applicable in all casesand can be used to solve the problems once the appropriate data describing the physical characteristics of the river and the appropriateinitial and boundary conditions are known.The idea of using thedifferential equations directly as a means ofof flow in open channels is not at all new. In fact,treating problemsgoes back to Massau [M.5] as long ago as 1889. Since then the ideahas been taken up by many others (mostly in ignorance of the workitof Massau)Thomasfor example,[T.2],and Stokerby Preiswerk [P.16], von Karman [K.2],[S.19].
Thomas, in particular, attacked thehis noteworthy and pioneering paper andflood-routing problem inoutlined a numerical procedure foritssolution based on the idea of us-ing the method of finite differences. However, his method is very laborious to apply and would also not necessarily furnish a goodapproximation to the desired solution even if a large number ofdivisions of the river into sections were to be taken. In general, theamount of numerical work to be done in a direct integration of theMATHEMATICAL HYDRAULICS483differential equations looked too formidable for practical purposesuntil rather recently.During and since the late war new developments have taken placewhich make the idea of tackling flood prediction and other similarproblems by numerical solution of the relevant differential equationsquite tempting. There have been, in fact, developments in two different directions, both motivated by the desire to solve difficult problemsin compressible gas dynamics: 1) development of appropriate numerical procedures for the most part methods using finite differencesfor solving the differential equations, and 2) development of com-puting machines of widely varying characteristics suitable for carrying out the numerical calculations.
As we have seen, the differentialequations for flood control problems are of the same type as those forcompressible gas dynamics, and consequently the experience and calculating equipment developed for solving problems in gas dynamicscan be used, or suitably modified, for solving flood control problems.In carrying out such a study of an actual river it is necessary tomakeuse of a considerable bulk of observational data cross-sectionsand slopes of the channels, measurements of river depths and discharges as functions of time and distance down the river, drainageareas, observed flowsfromtributaries, etc.in order to obtain theinformation necessary to fix the coefficients of the differential equations and to fix the initial and boundary conditions. This is a taskwith many complexities.
For the purposes of this book it is morereasonable to carry out numerical solutions for problems which aresection has as itssimplified versions of actual problems. The presentpurpose the presentation of the solutions in a few such special cases,together with an analysis of their bearing on the concrete problemscase, the generalmethodsforan actual riverfor actual rivers.
Inanywould be the samethere would simply be greater numerical compli-cations.simplified models chosen correspond in a rough general wayOhio River and (b) to the Ohio and(a) to two types of flow for theof constant slope, withMississippi Rivers at their junction. Riversand with conuniformabreadth,cross-sectionshavingrectangularThestant roughness coefficients are assumed. In this way differentialresult.
The values of these quanequations with constant coefficientstities are, however, taken to correspond in order of magnitude withthose for the actual rivers. In the model of the Ohio, for example, theto be 0.5 ft/mile, the quantity nslope of the channel was assumedWATER WAVES484(the roughness coefficient in Manning's formula) was given the value0.03, and the breadth of the river was taken as 1000 feet. It is assumedthat a steady uniform flow with a depth of 20 ft existed at the initialthe depth of the water was increasedinstant t0, and that for t=>=from 20 ft to 40 ft within 4at a uniform rate at the point xhours and was then held fixed at the latter value. (These depths arethe same as for the problem of a steady progressing wave treated insec. 11.2 above.) The problem is to determine the flow downstream,i.e.
the depth y and the flow velocity v as functions of x (for x0)>andThe methods used to obtain the solution of this problem of a flood ina model of the Ohio River, together with a discussion of the results,will be given in detail later on in this section.
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