J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 58
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It is the exact counterpart of the case discussed at theend of the preceding section in which the piston was withdrawn fromthe water at a uniform speed.To obtain the conditions at a discontinuity we consider a regiona Q (t)made up of the water lying between two vertical planes xand x = a>i(t) with a^ > a and such that these planes contain alwaysthe same particles. Such an assumption can be made, we recall fromChapter 2, since in our theory the particles which are in a verticalplane at any instant always remain in a vertical plane.
Hence thehorizontal particle velocity component u is the same throughout anyWe nowsuppose that there is a finite discontinuity inthe surface elevation rj at a point x(t) within the column of watera 1 (^), as indicated in Fig. 10.6.2.a Q (t) and xbetween xThe laws of conservation of mass and of momentum as applied toour column of water yield the relationsvertical plane.===d(10.6.1)anddnnftftl(10.6.2)whenfiWA Q^JaS (r,+h)udx= po p,dy-\pi Pl dy\J-n(t)the formulap=J-hy) for the pressure in the watergQ(r)isThe second relation states that the change in momentum of thewater column is equal to the difference of the resultant forces over theused.end sections of the column.Theintegrals in these relations=(x, t)Jawhichhave the form/h(has a discontinuity at xtherelationintegral yieldsin\p(x, t)dld f*dx(t)=dyidx(10.6.3)-dxg(t).Differentiation of thisLONG WAVES IN SHALLOW WATER317Thequantities U Q and u are the velocities a (t) and a^t) at the endsof the column, f is the velocity of thediscontinuity, and y;(_, t) andy(+, t) mean that the limit values of y to the left and to the right ofxf respectively are to be taken.wish to consider the limit casein which the length of the column tends to zero in such away thatthe discontinuity remains inside the column.
When we do so the=Weintegralon the right-hand side of (10.6.3) tends to zero and we obtaindllim(10.6.4)inwhich v l and V Q are the= y^ - Worelative velocities givenby(10.6.5)andand y refer to the limit values of \p to the right and to the leftif?!of the discontinuity, respectively. The important quantities V and vQlare obviously the flow velocities relative to themoving discontinuity.use of (10.6.4) and (10.6.5) for the limit cases which(10.6.1) and (10.6.2) we obtain the following conditionsUpon makingfromarisegfo(10.6.6)+ hfa -+ h)v -Q(rhandeOh+AK^-efoo+A^(io.6.7)weand p (which areintroduce, as in section 10.1, the quantitiesthe analogues of the density and pressure in gas dynamics) by theIfrelations(of.(10.1.3)and(10.6.8)(10.1.4))=Q+ h)e (i,andp(10.6.9)we obtain=^ (r/+A)in place of (10.6.6)andIgi,(10.6.7) the discontinuity conditionsQM =(10.6.10)=iQ OV O ,ande&iVi(10.6.11)Thelasttwo(?(Wo=relations are identical inPoPi-form with the mechanical con-wave in gas dynamics when the latter are expressedterms of velocity, density and pressure changes.ditions for a shockin~WATER WAVES318Henceforth we shall often refer to a discontinuity satisfying (10.6.10) and (10.6.11) as a shock wave or simply as a shock even thoughsuch an occurrence is better known in fluid mechanics as a bore, orif it is stationary as a hydraulic jump.SinceuUQVQvlfrom(10.6.5)itiseasily seen that theshock conditions (10.6.10) and (10.6.11) can be written in the form(10.6.12)m(^IinwhichToVQ )= po - _plfrepresents the mass flux across the shock front.motion on both sides of the shock five quantities arethe particle velocities U Q u v the elevations 77 and rj l (or,fix theneeded;whatm-i.e.,the same, the "pressures" p or the "densities" g as given by(10.6.8) and (10.6.9) on both sides of the shock), and the velocity |of the shock.
Evidently the relative velocities V Q and v l would thenbe determined. Since the five quantities satisfy the two relationsis(10.6.12) we see that in general only three of the five quantities couldbe prescribed arbitrarily. Since the equations to be satisfied are notlinear it is not a priori clear whether solutions can be found for twoof the quantities when any other three are arbitrarily prescribed orwant to investigate thiswhether such solutions would be unique.inanumberofquestionimportant special cases.WeBefore doing so, however, it is important to consider the energybalance across a shock.
The fact is, as we shall see shortly, that thelaw of conservation of energy does not hold across a shock, but ratherthe particles crossing* the shock must either lose or gain in energy.Since we do not wish to postulate the existence of energy sources atthe shock front capable of increasing the energy of the water particlesas they pass through it, we assume from now on that the waterparticles do not gain energy upon crossing a shock front.
This will ineffect furnish us with an inequality which in conjunction with thetwo shockrelations (10.6.12) leads in all of our cases to unique solutions of the physical problems.turn, then, to a consideration ofthe energy balance across a shock, which we can easily do by followingWe* It isimportant to observe that the water particles always do cross a shockfront: the quantityin (10.6.12), the mass flux through the shock front, isVQdifferent from zero if there is an actual discontinuity since otherwise v l0,m==UQ,and p=p==in other words the motion isand hence @ = ^continuous.
There is thus no analogue in our theory of what is called a contactdiscontinuity in gas dynamics in which velocity and pressure are continuous,but the density and temperature may be discontinuous.tijlLONG WAVES IN SHALLOW WATER319the same procedure that was used to derive the shock relations (10.6.10) and (10.6.11 ). For the rate of change dE/dt of the energyin theEwater column of Fig. 10.6.2 we have, as one can readily verify:dEd<*o)(10.6.1,3)PiJand_hl-h*this in turn yields in the limit(10.6.5),(10.6.8),(10.6.9),whena-+ a vthrough use ofand the hydrostatic pressure law, therelationdE~~(10.6.14)which energy is created or destroyed at the shock front.If we multiply (10.6.11) by | on both sides and then subtract from(10.6.14) the result is an equation which can be written after somemanipulation and use of (10.6.5) in the formfor the rate atdE-=(10.6.15)dtm {%(v*vl)f2(~pi/eip/Q)}mis the mass flux through the shock front defined inInthis way we express dE/dt entirely in terms of the(10.6.12).relative velocities V Q and v l and the change in depth.
By eliminatinginvlwhichandv through use of v 1=m/g x and=vM/{J Oand replacing p l andterms of Q I and Q O we can express dE/dt in terms of Q O and g^;pthe result is readily found to be expressible in the simple formindE(10.6.16)mg-7dtWe sec therefore that energy isQ3(PApi), _*Qi6Q.not conserved unless g==Q 19i.e.unless~~the motion is continuous. Since QQQip(^o^i) ^ follows from(10.6.16) that the rate of change of the energy of the particles crossingthe shock is proportional to the cube of the difference in the depth ofthe water on the two sides of the shock, or as we could also put it inis considered to be a small quantity: the rate of changecase rj Qr/ lof energy is of third order in the "jump" of elevation of the watersurface.The statement that the law of conservation of energy does not holdin the case of a bore in water must be taken cum grano salis.
What wemeanisof course that the energy balance can not be maintainedWATER WAVES320through the sole action of the mechanical forces postulated in theabove theory. The results of our theory of the bore and the hydraulicjump are therefore to be interpreted as an idealization of the actualoccurrences in which the losses in mechanical energy are accountedfor through the production of heat due to turbulence at the front ofthe shock (cf. the photograph of the bore in the Tsien-Tang rivershown in Fig. 10.6). 8). In compressible gas dynamics the theory usedFig.
10.6.3.Borein theTsien-Tang Riverallows for the conversion of mechanical energy into heat so that thelaw of conservation of energy holds across a shock in that theory.The analogueof the loss in mechanical energy across a shock in waterthe increase in entropy across a shock in gas dynamics; furthermore,both of these discontinuous changes are of third order in the differenisces of "density"Weon the twosides of the shock.havetacitly chosen as the positive direction of the #-axis, andhence of all velocities, the direction from the side toward the side 1m is assumed to beFig.
10.6.2). Suppose now that the mass fluxpositive; it follows from (10.6.12) and the fact that(cf.and Q I are posij5tive that VQ and v l are also positive and hence that the water particlescross the shock front in the direction from the sidetoward the side 1.Our condition that the water particles can not gain in energy on crossing the shock then requires, as we see at once from (10.6.16) since w,LONG WAVES IN SHALLOW WATER321<anc^ ^i are a ^ positive, that g>{>og la In other words, our energycondition requires that the particles always move across the shock froma region of lower total depth to one of higher total depth.* Since the massis not zero unless the flow is continuous, and hence there is nofluxg>mpossible to define uniquely the two sides of the shock by theusefulconvention: the front and back sides of the shock arefollowingdistinguished by the fact that the mass flux passes through the shockshock,it isfrom front to back, or, as one could also put it, the water crosses theshock from the front side toward the back side.
Our conclusion basedon the assumed loss of energy across the shock can be interpreted interms of this convention as follows: the water level is always lower ononthe front side of the shock thanthe back side.For the further discussion of the shockrelationsit isimportant toobserve thatenergyloss,all of them, including the relation (10.6.16) for thecan be written in such a way as to involve only the velocitiesand v lof the water particles relative to the shock front and not the absolute velocities U Q and u r It follows that we may always assume one ofVQthe three velocities u u l9 to be zero if we wish, with no essential lossof generality, because the laws of mechanics are in any case invariantwith respect to axes moving with constant velocity, and adding the,%and does not affect the values of v and i^.same constant to M OLet us assume then that u Q0, i.e.
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