J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 88
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The quantityand(11.1.6))(i.e.q=in thehave already beenEE= withSgS+ gS fthe slope of the river bed andSf,,the friction slope, given byManning's formulaHere we assume the channel to be rectangular with breadth B.The numerical data for the problem of a flood in a model of theOhio River are as follows. For the slope S a value of 0.5 ft/mi waschosen, and B is given the value 1000 ft. For y a value of 2500 wastaken (in foot-sec units), corresponding to a value of Manning's2constant n (in the formula y(1.49/n) ) of 0.03. The special problem considered was then the following: At time t = 0, a steady flowof depth 20 ft is assumed.
At the "headwaters" of the river, corresponding to x = 0, we impose a linear increase of depth with time whichbrings the level to 40 ft in 4 hours. For subsequent times the level of40 ft at x =is maintained. The initial velocity of the water corauniform flow of depth y Q = 20 ft is calculated fromtorespondingSf=Sto bei>=2.38mph;the^propagation speed of small disturbances corresponding to thedepth of 20 ft isMATHEMATICAL HYDRAULICSCThe problem then==17.3489mph.to determine the solution of(11.6.1) for v(x, t),times ttheriverx0.
Figures 11.6.1alongand 11.6.2 present the result of the computation in the form ofstageand discharge curves plotted as functions of distance along the riverat various times.is^c(x, t) for all later^In order to indicate how the solution was calculated it is convenient to refer to diagrams in the (x, t) plane givenby Figs. 11.6.3and 11.6.4. According to the basic theory, we know that for xfao+c o)t=19-7*, called regionO^in Fig. 11.6.3, the solutionisgivenLegendt- time10--<;|;:%minstage-distance along Ohio8O60100Fig.
11.6.1. Stage profiles for a flood in the=mmiles^^^sTJG4Ofloodfeetyx^-A%3;e$ A^^20hours after start of-xI2OOhio River=Cc(x, t)(since thev(x t)CQ19.7 mph).forerunner of the disturbance travels at the speed wExperiments were made with various interval sizes and finiteby the unchangedinitial data,9,+=difference schemes in order to try to determine the most efficient wayto calculate the progress of the flood.proceed to describe thevarious schemes tried and the regions in which they were used on theWebasis of Figs. 11.6.3RegionI,and^x^11.6.4.19.7J,^ ^t.4.Quite small intervals ofWATER WAVES490300.Legend260.t *timeQdischargex 3distance along Ohio220.hours after start of floodin1000inc.fs.inmilesta>40206080100140120Fig.
11.6.2. Discharge records for a flood in the160Ohio River1.25Fig. 11.6.3.Ax =lRegions in which various computational methods were triedmile and At= .048 hours were required owing to the sudden= 0, = 0. The finite difference formulas givenincrease of depth at xabove in equationsIn Region II,t(11.5.8),^x^(11.5.9)19.7*,.4were used.^t<^ .7,withAx=1 mile,MATHEMATICAL HYDRAULICS491t2.0-1.5 J1.0JLegend1 sx10Fig.
11.6.4.Net points usedatime in hoursdistancein20in the finite differencemiles30schemes.024 hr, the "staggered" scheme was used. The formulas for thisscheme have been given above in equations (11.5.14). In order tocalculate 0(0, 2), the velocity at the upstream boundary of the river,the formula associated with the backward characteristic, namelyequation (11.5.10), has to be used twice in succession: for the trianglesand(cf. Figs.
11.5.3 and 11.6.5). The values C B and V Barc simply determined by linear interpolation from the values at thepoints F and G.AtFBMMRPWATER WAVES492RegionIII,^ x ^ 5,.7^ t^ 1.25, with Ax =The same procedure was usedRegion IV, 5At=.048 hr.^x^The values19.7J,at theas in.7Region^ ^*1mile, z^= .024 hr.II.1.25,withAx=2miles,boundary between Regions III and.G.HxFig.
11.6.5.Net point arrangement usedat boundary in "staggered" schemeIV were obtained by linear interpolation from the neighboring values."Other quantities were computed by the staggered" scheme as inRegions II and III.^ x ^ Ut, 1.25 ^ t <, 10, Ax = 5 miles, J* = .17 hr.Region V,U represents a variable speed which marks the downstream end ofwhat might be called the observable disturbance (U10 mph).That is, by using an expansion scheme (see the appendix to thischapter) we obtain the solution in&Region VI, defined by Ut ^ as ^ 19.72, back of the forerunnerof the disturbance, in which the flow is essentially undisturbedfor all practical purposes. The expansion valid near the front of thewave and referred to above was used to calculate the various quantitiesRegion VI, and a staggered scheme was used to compute the valuesin Region V.A number of conclusions reached on the basis of the experiencegained from these calculations of a flood in a model of the Ohio Rivercan be summarized as follows:5 feet per hour is extreme, and(a) The rate of rise of the floodsuch a case exaggerates the way in which errors in the finitedifference methods are propagated.
For example, slight inaccuracies at the head, x ~ 0, were found to develop upon increasinginAx interval. In spite of the exceptionally high rateof rise of the flood, the fluctuations created by using finite difference methods were damped out rather strongly (in about810 time steps).
It is possible to control these inaccuraciesthe size of theMATHEMATICAL HYDRAULICS493sizes. The process by which thesmall errors of the finite difference scheme are caused to die outmay be described as follows: A value of v which is too largesimply by using small intervalproduces a correspondingly larger friction force which slows downthe motion and produces at a later time a smaller velocity. Thelower velocity in a similar way then operates through the resistanceto create a larger velocity and the process repeats in an oscillatoryfashion with a steady decrease in the amplitude of variation.The(b)accuracy of our computation (as a function of the intervalsize) was checked by repeating the calculation for two differentinterval sizes over the same region in space and time.(c)Alinearized theory of wave propagation, obtained by assuminga small perturbation about the uniform flow with 20 ft depth, iseasily obtained, and the problem was solved using such a theory.However, it does not give an accurate description of the solutionof our problem.
It was found that the stage was predicted too lowby the linear theory by as much as 2 feet after only 2 hours avery large error.(d) It would be convenient to be in possession of a safe estimate forthe maximum value of the particle velocity, in order to select anappropriate safe value for the time interval At, since we musthave At 5g Ax/(v -)- c) in order to make sure that the finite difference scheme converges. The calculations in our special caseindicate that this may not be easy to obtain in a theoretical way,since thereedsitsmaximum0, for example, greatly exvelocity at xasin Fig.
11.6.6. In aindicatedvalue,asymptoticv(0,t)mph5.4-velocity for40ft steadyflow24t4Fig. 11.6.6.Waterhoursvelocity obtained at "head" of rivercomputation for an actuallikely to result, since cishowever, no real difficultyin general much larger than v andriver,determined by the depth alone.isisWATER WAVES494(e)As was already indicated above, the curves of constant stageturn out to have slopes which are closer to 5 mph (the speed withwhich a steady progressing flow, 40 ft upstream and 20 ft downstream, moves) than they are to the 19.7 mph speed of propagation of small disturbances.
This is shown by Fig. 11.6.7.region ofthourspracticallyundisturbedflow10.5.Smph-slope50Fig. 11.6.7. Curves of constant stageI97mph-slope100comparison withsteady progressing flow velocityThe regionxmilesfirst characteristicandof practically undisturbed flow (determined by an19.7J, for whichexpansion about the "first" characteristic xsee the appendix to this chapter) is shown above.
In an actual=we would of course expect the local runoff discharges and thenon-uniform flow conditions to eliminate largely the region ofpractically undisturbed flow. For this reason it is not feasibleto use analytic expansion schemes as a means of avoidingriver,computational labor.turn next to our model of the junction of the Ohio and Mississippi Rivers and the problem of what happens when a flood wavecomes down the Ohio and passes through the junction.* The physicaldata chosen are the same as were used above in sec.
11.2 in discussingthe problem of a steady flow at a junction.We suppose the upstream side of the Mississippi to be identicalwith the Ohio River i.e. that it has a rectangular cross-section1000 ft wide, a slope of .5 ft/mile, and that Manning's constant n hasthe value .03. The downstream Mississippi is also taken to be rectangular, but twice as wide, i.e. 2000 ft in width, Manning's constant isagain assumed to have the value .03, but the slope of this branch isgiven the value .49 ft/mile. This modification of the slope was madeWe* Theanalogous problem in gas dynamics would be concerned with the propagation of a wave at the junction of two pipes containing a compressible gas.MATHEMATICAL HYDRAULICSin order tomakeform flow of 20possibleaninitial solution495corresponding to a uni-depth in all three branches.
(Such a change isinorderto overcome the decrease in wetted perimeternecessaryftwhich occurs on going downstream through the junction.) Figure11.6.8 shows a schematic plan of the junction. The concrete problemto be solved is formulated as follows. A flood is initiated in the OhioL3JDownstreamMississippiFig. 11.6.8.Schematic plan of junctionat a point 50 miles above the junction by prescribing a rise in depth ofthe stream at that point from 20 ft to 40 ft in 4 hours in otherwords, the same initial and boundary conditions were assumed as forthe case of the flood in the Ohio treated in detail above.
After about2.5 hours the forerunner, or front, of thewavein theOhio caused bythe disturbance 50 miles upstream reaches the junction; up to thisinstant nothing will have happened to disturb the Mississippi, andthe numerical calculationsmade abovefor theOhio remain validduring the first 2.5 hours.
Once the disturbance created in the Ohioreaches the junction, it will cause disturbances which travel bothupstream and downstream in the Mississippi, and of course also areflected wave will start backward up the Ohio. The finite differencecalculations therefore were begun in all three branches from themoment that the junction was reached by the forerunner of the Ohioflood, and the solution was calculated for a period of 10 hours.Weproceed to describe the method of determining the numericalc (3) represent the velocity v and thesolution. Let r (1) c (1) i> (2) c (2)(8)propagation speed c for the Ohio, upstream Mississippi, and down,,,,,WATER WAVES496stream Mississippi, respectively. A "staggered" scheme was used with.17 hr as indicated in Fig. 11.6.9.intervals Ax = 5 miles and AtThe junction point is denoted by x = 0, the region of the Ohio andxLXA.KOhiox.R.FXB.GDownstream Mississippi [31[IIUpstream Mississippi-P*MC2JJunctionFig.
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