J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 74
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10.12.1. HydraulicWejumphave already encountered an interesting example of a flowis in part supercritical, in part subcritical, i.e. the case of ahydraulic jump in which the character of the flow changes on passagethrough the discontinuity. Figure 10.12.1 is a photograph, takenfrom the paper of Preiswerk, of such a hydraulic jump. Figure 10.12.2,also taken from the paper of Preiswerk, shows a more complicatedcase in which hydraulic jumps occur at oblique angles to the directionof the flow.
The picture shows a flow through a sluice in a dam, withconditions (i.e. depth differences above and below the dam) such thatsupercritical flow develops in the sluice, and changes in level take placeso abruptly that they might well be treated as discontinuities (as wasdone in earlier sections in the treatment of bores). The two discontinuities at the sides of the sluice (marked 1 and 2 in the figure) areturned toward each other and eventually intersect to form a stillwhichWATER WAVES408higher one (marked 1+2). Such oblique discontinuities can betreated mathematically; the details can be found in the works citedabove.Another interesting problem of the category considered hereisthe1+2Fig.
10.12.2. Hydraulicjumps at oblique angles to thedirection of the flowproblem of supercritical flow around a bend in a stream. This type ofproblem is relevant not only for flows in water, but also for certainflows in the atmosphere (for which see Freeman [F.9]). It is possiblein these cases to have flows of the type which are mathematically ofthe kind called simple waves in earlier sections. This means that oneof the families of characteristicsis a set of straight lines along each ofalsoareconstant. Even the notion of a cen(henceh)tered simple wave can be realized in these cases. Suppose that thewhich wandwflow comes with constant supercritical velocity along a straight wallLONG WAVES IN SHALLOW WATER409(cf.
Fig. 10.12.3) until a smooth bend begins at point A. The straightcharacteristics are denoted by C + in the figure; they form a set ofFig. 10.12.3. Supercritical flow around aparallel lines in the region of constant flow,along the C+ characteristic through the pointsmooth bendwhich then terminatesA, where the bend be-gins; beyond that characteristic a variable regime begins. The straightcharacteristics themselves are called Mach lines; they have physicaland would be visible to the eye: the Mach lines are lineswhichinfinitesimal disturbances of a supercritical flow arealongpropagated; in an actual flow they would be made visible because ofthe existence of small irregularities on the surface of the wall ofthe bend.
If the bend contracts into a sharp corner, the straightcharacteristics, or Mach lines, which lie in the region in which theflow is variable, all emanate from the corner, as indicated in Fig.10.12.4; the flow as a whole consists of two different uniform flowssignificanceFig. 10.12.4. Supercritical flowaround a sharp cornerconnected through a centered simple wave. If the bend in the streamis concave toward the flow, rather than convex as in the precedingtwo cases, the circumstances are quite different, since the Mach lineswould now converge, rather than diverge, in certain portions of theflow, as indicated in Fig.
10.12.5. Overlapping of the characteristicswould mean mathematically that the depth and velocity would bemulti- valued at some points in the flow; this being physically impos-WATER WAVES410sible it is tobe expected that something new happens and, in fact,jump is to be expected. If the bendthe development of a hydraulicFig. 10.12.5.Machlines fora supercritical flow around a concave benda sharp angle, as in Fig. 10.12.6, the configuration consisting of twouniform flows parallel to the walls of the bend and connected by anoblique hydraulic jump is mathematically possible, and it occurs inispractice.Having considered flows delimited on one side only by a wall, it isnatural to consider next flows between two walls as in a sluice orchannel of variable breadth.
(Such flows are analogous to two-dimen-Fig. 10.12.6. Oblique hydraulicjumpsional steady flows through nozzles in gas dynamics.) The possibilitieshere are very numerous, and most of them lead to cases not describ-able solely in terms of simple waves.They are of considerable importance in practice. For example, v. Karman [K.2] was led to the studyof particular flows of this type because of their occurrence in bends inthe concrete spillways designed to carry the flows of the Los AngelesLONG WAVES IN SHALLOW WATER411river basin through the city of Los Angeles; the seasonal rainfall is soheavy and the terrain so steep that supercritical flows are the rulerather than the exception during the rainy season. Experiments forsluices of special form were carried out by Preiswerk; Fig. 10.12.7,b)gemessenFig. 10.12.7.beiha =31,1mmLaval nozzle a) Machlines b)contour lines of the water surfaceexample, shows the result of an experiment in a particular case.figure shows the Mach lines, the lower figure shows thecontour lines of the water surface as given by the theory as well as byexperiment; as one sees, the agreement is quite good.Finally, we discuss briefly some applications of interest because oftheir connection with aerodynamics.
Because of the analogy of theshallow water theory with compressible gas dynamics, it is of coursepossible to interpret experiments with flows in shallow water in termsforThe upperof the analogous flows in gases. Sinceit ismuch cheaper andsimplerWATER WAVES412to obtain supercritical flows experimentally in water than it is toobtain supersonic flows in gases, it follows that "water table" experiments (as they are sometimes called) may have considerable import-ance for those whose principle interestisinaerodynamics. Thereisaconsiderable literature devoted to this subject; we mention, for example, papers by Crossley [C.12], Einstein and Baird [E.5], HarlemanFig.
10.12.8.Photogram of hydraulic-jumpintersection[H.8], Laitone [L.I], Bruman [B.19]. Figure 10.12.8 is a photograph,taken from the paper by Crossley, showing the interaction of twohydraulic jumps; this is a case essentially the same as that shown byFig. 10.12.2. The ripples with short wave lengths constitute an effectdue to surface tension, and the discontinuities are smoothed out so thata hydraulic jump does not really occur; the changes in depth are quiteabrupt, however.
Another important case that has been studied bymeans of the hydraulic analogy is, as a matter of course, the flowLONG WAVES IN SHALLOW WATERpattern which resultsairfoil) isimmersedin413when arigid body (simulating a projectile or ana stream. Figure 10.12.9 shows a photograph ofFig. 10.12.9.Shock waveFig. 10.12.10.in front ofFlow pattern of aa projectileprojectilesuch a flow (taken from the paper by Laitone). The shock wave infront of the projectile is well shown. Figure 10.12.10 is another photo-graph made by Preiswerk; here, Machlines are clearly visible.WATER WAVES41410.13.
Linear shallow water theory. Tides. Seiches. Oscillations inharbors. Floating breakwatersUptonowin this chapterwe have considered problemsofwavein water sufficiently shallow to permit of an approximation interms of what we call the shallow water theory. This theory is non-motionand consequently presents difficulties which areoften quite formidable. By making the assumption that the waveamplitudes in the motions under study are small in addition to thelinear in character,assumption that the water is shallow, it is possible to obtain a theorywhich is linear and thus attackable by many known methodsand which is also applicable with good approximation in a varietyof interesting physical situations. We begin by deriving the linearshallow water theory under conditions sufficiently general to permitus to discuss the cases indicated in the heading of this section.
(Abrief mention of the linear shallow water theory was made in Chapter2 and in Chapter 10.1.)The linear shallow water theory could of course be derived byappropriate linearizations of the nonlinear shallow water theory. It is,however, more convenient and perhaps also interesting from thestandpoint of methodto proceedbylinearizing first the basic general1, and afterwards making the approximations arising from the assumption that the water is shallow.In other words, we shall begin with the exact linear theory of Chapter2.1, and proceed to derive the linear shallow water theory from it. Oneof the advantages of this procedure is that the error terms involvedin the shallow water approximation can be exhibited explicitly.We suppose the water to fill a region lying above a fixed surfacetheory as developed in ChapterFig.
10.13.1. Linear shallow water theorybottom) y =the motion of which(theh(z, z) 9 and beneath a surface yY(x 9 z\ t),for the time being supposed known (cf. Fig.isLONG WAVES IN SHALLOW WATER41510.13.1). The t/-axis is taken vertically upward, and the x, s-planehorizontal. The upper surface of the water given by yY(x, z;=will consist partly of the free surface (toist)be determined, for example,by the condition that the pressure vanishes there) and partly of theimmersed bodies; it is, however, not necessary to specifymore about this surface for the present than that it should representsurfaces ofalways a small displacement fro?n a rest position of equilibrium of thecombined system consisting of water and immersed bodies.We recapitulate the equations of the exact linear theory as derivedin Chapter 2.1. The velocity components are determined as the derivatives of the velocity potential 0(x, y.
z; t) which satisfies the Laplaceequationxx+(10.13.1)yv+ zz==Qspace filled by the water. It is legitimate to assume that allboundary conditions at the upper surface of the water are to be satisin thefied at the equilibrium position; this positionissupposed given byy ^fj(x,z).(10.13.2)to the fact that fj could also berj pointsastheofthe water in the importantinterpretedaverage position(The bar over the quantityspecial case inwhich the motiona simple harmonic motion in thetaken in the undisturbed position of the freeisThe x, z-planc isand this in turn means that fj in (10.13.2) has the value zerothere.
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