J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 87
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Before doing so, a fewgeneral remarks and observations about them should be made at thispoint. In the first place, it was found possible to carry out the solutionnumerically by hand computation over a considerable range of distances and times (values at 900 net points in the #, 2-plane were determined by finite differences), and this in itself shows that theproblems are well within the capacity of modern calculating equipment. It might be added that the special case chosen for a flood in theOhio was one in which the rate of rise at the starting point upstreamwas extremely high (5 feet per hour, in comparison with the rate oft.during the flood of 1945 one of the biggest ever recorded in thewhich was never larger than 0.7 feet per hour at Wheeling,West Virginia), so that a rather severe test of the finite differenceriseOhiomethod was made in view of the rapid changes of the basic quantitiesin space and time. The decisive point in estimating the magnitude ofthe computational work in using finite differences is the number ofnet points needed; for a river such as the Ohio it is indicated that aninterval Ax of the order of 10 miles along the river and an interval Atof the order of 0.3 hours in time in a rectangular net in the x, J-planewill yield results that are sufficiently accurate.
(Of course, a problemfor the Ohio in its actual state involves empirical coefficients in thedifferential equations and other empirical data, which must be codedfor calculating machines, but this would have no great effect on theseestimates forAx and might under extremeflood conditions reduce Atby a factor of 1/2.)As we know fromsec.
11.3above, therea case in which an exactknown, i.e. the case of aissolution of the differential equations issteady progressing wave with two different depths at great distancesMATHEMATICAL HYDRAULICS485upstream and downstream. The exact solution obtained in sec. 11.3wave of depth 20 ft far downstream and 40 ft farwastakenas furnishing the initial conditions at tforupstreama wave motion in the river. With the initial conditions prescribed inthis way the finite difference method was used to determine the mofor the case of ation at later times; of course the calculation,if accurate, should furnish a wave profile and velocity distribution which is the same attimeas at the initial instant=except that all quantities are displaced downstream a distance Ut, with U the speed of the steadyprogressing wave.
In this way an opportunity arises to compare thettapproximate solution with an exact solution. In the present case thephase velocity U is approximately 5 mph. Interval sizes of Ax = 5miles in a "staggered" finite difference scheme (cf. equations (11.5.14))with At = .08 hr were taken and a numerical solution was workedout.Wereport the results here. After 12 hours, the calculated valuesy agreed to within .5 per cent with the exact values.for the stageThe discharge and thevelocity deviatedbylessthan.8per centfrom the exact values.Oneof the valuable insights gained from working out the solutionof the flood problem in a model of the Ohio was an insight into thefor example, byrelation between the methods used by engineersthe engineers of the Ohio River Division of the Corps of Engineers infor predicting flood stages, and the methods explainedCincinnatiuse of the basic differential equations.
At first sightwhichmakehere,the two methods seem to have very little in common, though both, in'the last analysis, must be based on the laws of conservation of massand momentum; indeed, in one important respect they even seem tobe somewhat contradictory. The methods used in engineering prac-(which make no direct use of our differential equations) tacitlyassume that a flood wave in a long river such as the Ohio propagatesonly in the downstream direction, while the basic theory of the difticeferential equations we use tells us that a disturbance at any point ina river flowing at subcritical speed (the normal case in general andalways the case for such a river as the Ohio) will propagate as a wavetraveling upstream as well as downstream.
Not only that, the speedof propagation of small disturbances relative to the flowing stream, asdefined by the differential equations, is \/gy for small disturbancesand this is a good deal larger (by a factor of about 4 in our model ofthe Ohio) than the propagation speed used by the engineers for theirflood wave traveling downstream.
There is, however, no real dis-WATER WAVES486crepancy. The method used by the engineers can be interpreted as amethod which yields solutions of the differential equations, with cer-good approximations (though not underall circumstances, it seems) to the actual solutions in some cases,among them that of flood waves in a river such as the Ohio.
Thetain terms neglected, that areneglect of terms in the differential equations in this approximatetheory is so drastic as to make the theory of characteristics, fromwhich the properties of the solutions of the differential equations werederived here, no longer available. The numerical solution presentedhere of the differential equations for a flood wave in a model of theOhio yields, as we have said, a wave the front of which travels downstream at the speed \/^y; but the amplitude of this forerunner isquite small,* while the portion of the wave with an amplitude in therange of practical interest is found by this method to travel withessentially the same speed as would be determined by the engineers'approximate method.
What seems to happen is the following: smallforerunners of a disturbance travel with the speed \/gy relative to theflowing stream, but the resistance forces act in such a way as to decrease the speed of the main portion of the disturbance far below thevalues given by i/gy, i.e. to a value corresponding closely to the speedof a steady progressing wave that travels unchanged in form.
(Onecould also interpret the engineering method as one based on the assumption that the waves encountered in practice differ but little fromsteady progressing waves). As we shall see a little later, our unsteadyflow tends to the configuration of a steady progressing wave of depth40 ft upstream and 20 ft downstream.This analysis of the relation between the methods discussed herein engineering practice indicated why itthelatterbethatmethods, while they furnish good results inmayfailto mirror the observed occurrences in othermany important cases,and those commonly usedFor example, the problem of what happens at a junction of twomajor streams, and various problems arising in connection with thecases.operation of such adam as the Kentucky Dam inthe Tennessee Riverseem to be cases in which the engineering methods do not furnishaccurate results. These would seem to be eases in which the motionsof interest depart too much from those of steady progressing waves,and cases in which the propagation of waves upstream is as vital as thepropagation downstream.
Thus at a major junction it is clear thatIn an appendix to this chapter an exact statement on this pointismade.MATHEMATICAL HYDRAULICS487considerable effects on the upstream side of a main stream are to beexpected when a large flow from a tributary occurs. In the sameadam in a streamway,any obstruction, or change in cross-section, etc.)causes reflection of waves upstream, and neglect of such reflectionsmight well cause serious errors on some occasions.The above general description of what happens when a flood wavestartsdown(ora long streamit has alengthy frontportion which travels fast, but has a small amplitude, while the mainpart of the disturbance moves much more slowly has an importantbearing on the question of the proper approach to the numerical solution by the method of finite differences.
It is, as we shall seeshortly,in particular, thatnecessary to calculate or else estimate in some way the motion upto the front of the disturbance in order to be in a position to calculateit at the places and times where the disturbances arelarge enough tobe of practical interest.
This means that a large number of net pointsin the finite difference mesh in the #, J-plane lie in regions where thesolution is not of much practical interest. Since the fixing of the solution in these regions costs as much effort as for the regions of greaterinterest, the differential equation method is at a certain disadvantageby comparison with the conventional methodinsuch a case. However,it is possible in simple cases to determine analytically the characterof the front of the wave and thus estimate accurately the places andtimes at which the wave amplitude is so small as to be negligible;these regions can then be regarded as belonging to the regions of thex, f-planc where the flow is undisturbed, with a corresponding reduction in the number of net points at which the solutions must beAmethod which can be used for this purpose has beenderived by G. Whitham and A.
Troesch, and a description of it isgiven in an appendix to this chapter. If a modern high speed digitalcalculated.computer were to be used to carry out the numerical work, however,it would not matter very much whether the extra net points in thefront portion of the wave were to be included or not: many suchmachines have ample capacity to carry out the necessary calculations.We proceed to give a description of the calculations made for ourmodel of the Ohio, including a discussion of various difficulties whichoccurred for the flood wave problem near the front of the disturbance,0,and particularly at the beginning of the wave motion (i.e.
near xwhichcalculationthefeaturesofofthe/0), and an enumeration==must play asimilar role in themore complicated cases presented byby a description ofrivers in their actual state. This will be followedWATER WAVES488the method used and the calculationsa flood coming down the Ohio andmadefor aits effectproblem and its solution giveobservations which will be made later on.The differential equations to be solved areMississippi. This(11.6.1)+ 2vc x + cv x = 0,= Vgy thevelocity, and cproblem simulatingon passing into therise to furthergeneral2c twith v(x9t)thesmall disturbances.The assumptionpropagation speed ofof a uniform cross-section andthe assumption that no flow over the banks occursbasic differential equations (11.1.1)is given byused.
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