J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 61
Текст из файла (страница 61)
10.8.1. Breaking of adamWATER WAVES384dam is initially free of water. In the present section we considerthe more general problem which arises when it is assumed that thereis water of constant depth on the downstream as well as theupstreamof thedam. Or, as the situation could also be described: a horizontal tank of constant cross section extending to infinity in bothside of the=directions has a thin partition at the section x0. For xhhastheandwaterfor xthe depth h v with Adepth<><hthel9asindicated in Fig. 10.8.1. The water is assumed to be at rest on bothsides of the dam initially. At the time tthe dam is suddenly des-=troyed, and our problem is to determine the subsequent motion ofthe water for all x and t.The special case h =the cavitation case was treated, as wehave already mentioned, at the end of section 10.5.
We found therethat the discontinuity for x =and twas instantly wiped outand that the surface of that portion of the water in motion took theform of a parabola tangent to the #-axis (i.e. to the bottom) at thepoint x=2\/gh l t=2o 1 <, inwhicht isthe time and c x thex=-c,tFree surface at(I)(3)t* t(2)Fig. 10.8.2.
Breaking of adamwaveLONG WAVES IN SHALLOW WATER335speed in water of depth h v If A is different from zero we might therefore reasonably expect (on the basis of the discussion at the beginningof section 10.6) that a shock wave would develop sooner or later onthe downstream side, since the water pushing down from above actssomewhat like a piston being pushed downstream with an acceleration.
In fact, since the water at xseems likely to acquire instan-=taneously a velocity different from zero it is plausible that a shockwould be created instantly on the downstream side. The simplestassumption to make would be that the shock then moves downstreamwith constant velocity(cf. Fig. 10.8.2). If this were so, the state ofthe water immediately behind the shock (i.e. on the upstream sideof it) would be constant for all time, since the velocity u 2 and depthA 2 on the upstream side of the shock would have the constant valuesdetermined from the shock relations for the fixed values UQ =andhh for the velocity and depth on the downstream side and theassumed constant value | for the shock velocity.
However, it is clearthat the constant state behind the shock could not extend indefinitelyupstream since u 2 ^ while the velocity of the water far upstream iszero. Since we undoubtedly are dealing with a depression wave behind the shock it seems plausible to expect that the constant statebehind the shock changes eventually at some point upstream into awave of the type discussed in section 10.5. In Fig.indicate in an #, 2-plane a motion which seems plausible as asolution of our problem.
In the following we shall show that such acentered simple10.8.2wemotion can be determinedina manner compatible with our theoryfor every value of the ratio A //i rAs indicated in Fig. 10.8.2, we consider four different regions in thefluid at any time t/the zone (0) is the zone of quiet downstream=:whichis terminated on the upstream side by the shock wave, or bore;the zone (2) is a zone of constant state in which the water, however, isnot at rest; the zone (3) is a centered simple wave which connects theconstant state (2) with the constant state (1 ) of the undisturbed waterupstream.
We proceed to show that such a motion exists and to determine it explicitly for all values of the ratio h^h^ between zero and one.For this purpose it is convenient to write the shock conditions forthe passage from the state (0) to the state (2) in the form(10.8.1)(10.8.2)-!(*-{)=<$+*;);- f) = - cfecl(u2which are the same conditions as (10.6.18) and (10.6.19; withWATER WAVES386=by c\gh i9 i.e.
by the square of the wave propagation speedwater of depth h t By eliminating c\ from (10.8.1 ) by use of (10.8.2)and then solving the resulting quadratic for u 2 one readily obtainsreplacedin.u 2 /c(10.8.3)=#cc--1+(Vlf and(The plus sign before the radical was taken in order that u 2Weobservealsohavethesamethatshouldonly positivesign.fvalues of | and u 2 are in question throughout our entire discussionsince the side of (0) is the front side of the shock and the positive^-direction is taken to the right.) It is also useful to eliminate u 2from (10.8.3) by using (10.8.2); the result is easily put into the form-=(10.8.4)Thecorelations (10.8.3)(iand(VT+8tfM* -1)}*.(10.8.4) yield the velocityu 2 and the wavebehind the shock as functions of f and the wave speed c inspeedthe undisturbed water on the downstream side of the dam.
We proceed to connect the state (2) by a centered simple wave (cf. the discussion in section 10.5) with the state (1). In the present case thestraight characteristics in the zone (3) (cf. Fig. 10.8.2) are those withc (rather than the slope u + c as in section 10.5);the slope uhence the straight characteristics which delimit the zone (3) are thec 2 )t on the right. Alonglines x =(u 2cj on the left and xeach of the curved characteristics in zone (3) one of these is indicatedschematically by a dotted curve in Fig. 10. 8.
2 the quantity u + 2cis a constant; it follows therefore that on the one handc2sinceu+u(10.8.5)=0,2c=2c xwhile on the other handthroughout the zone(3).Theu 2 /c(10.8.7)must therefore+ 2c =u(10.8.6)hold.+u2+2c 2relation2c 2 /c-2^/CoOur statement that a motionin Fig. 10.8.2 exists for every value of theamounts to the same thing, the ratio cj/cjjof the typedepth ratioish^h^shownwhator,equivalent to the state-ment that the relation(10.8.7) furnishes through (10.8.3) and (10.8.4)an equation for f/c which has a real positive root for every value ofC^CQ larger than one. This is actually the case.
In Fig. 10.8.3 we haveplotted curves for u 2 /cQ9 2c 2 /cQ and u 2 /cQ + 2c 2 /c Q as functions of f /cOnce the curves of Fig. 10.8.3 have been obtained, our problem can,.LONG WAVES IN SHALLOW WATER337be considered solved in principle: From the given value ofwe can determine /c from the graph (or, by solving (10.8.7)).The values of u 2 /c Q and c 2 /c are then also determined, either from thec il c ograph or by use of (10.8.3) and (10.8.4). The constant state in thezone (2) would therefore be known. In zone (3) the motion can nowbe determined exactly as in section 10.5; we would have along thestraight characteristics in this zone the relationsdxx= U -c = 2c=-1-3c =,^-c1from which= j2ciC2(10.8.8)l-l,and"(10.8.9)Thus the water surface in the zone (3) is curved in the form of aparabola in all cases.* At the junctions with both zones (1) and (2)the parabola does not have a horizontal tangent, so that the slope ofthe water surface is discontinuous at these points.Someinteresting conclusions can be drawn from (10.8.8) and(10.8.9).
By comparison with Fig. 10.8.2 we observe that the J-axis,i.e. the line x0, is a characteristic belonging to the zone (3) pro-vided that u 2=^c2onsince the terminal characteristic of the zone (3)the J-axis or to the right of it in this case. If thison the rightliesconditionsatisfiediswe observe fromare then valid on the /-axisat x=(10.8.8)that c anduand(10.8.9)whichare both independent ofwhich means that the depth of the water anditstu0,velocityarc both independent of t at this point, i.e. at the original location ofthe dam, and hence that the volume of water crossing the originaldam site per unit of time (and unit of width) dQ/dt uh is independ-=not a steady motion.ent of time although the motion as a wholeIn fact, hf c x for all time t at this point.
In\h l and uis=addition, u andas long as u 2 ^and thusalso dQ/dt, are not only independent of tbut also independent of the undisturbed depth Aon the lower side of the dam if A x is held fixed. Of course, it is clear thatAO/A! must be kept under a certain value (which from section 10.5evidently must be less than 4/9) or the condition u 2 I> c 2 could not bec,c2,* Relations(10.8.8) and (10.8.9) are exactly the same as (10.5.8) and (10.5.7)except for a change of sign which arises from a different choice of the positiveaj-direction.WATER WAVES888V'o7,234567IIIIIIFig.
10.8.3. Graphical solution forIn fact, the critical value of the ratio /^//^ at which u 2 = c 2can be determined easily by equating the right hand sides of (10.8.3)and (10.8.4) and determining the value of |/c for this case, afterfulfilled.whichc 2 /c= Vh /h2the critical caseh2= |Aj_able toiseitherknown from (10.8.4).from the known factin this case, orcompute thefromcritical(10.8.8) withvalue ofcj/cj=ASince c 2thatx1 /A.A~ fqinwe still havewe thus arenumericalcal-culation yields for the critical value of the ratio h^h^ the value 7.225,or for AQ/A! the value .1384. Thus if the water depth on the lower sideof thedamislessthan 13.8 percent of the depth above thedamthedischarge rate on breaking the dam will be independent of the originaldepth on the lower side as well as independent of the time.
However,AQ/AJ exceeds the critical value .1384, the depth, velocity, and discharge rate will depend on h ; but they continue to be independent ofifLONG WAVES IN SHALLOW WATER330= in the x> J-plane is under the latter circumstances contained in the zone (2), which is one of constant state.The above results, which at first perhaps seem strange, can bemade understandable rather easily from the physical point of view,as follows.
If the zone (3) includes the /-axis (i.e. if A /A t is below thecritical value) we may apply (10.8.8) and (10.8.9) for x =to obtainthe time since the line x=at this point cufcj. In other words, the flow velocity at thedam site is in this case just equal to the wave propagation speedthere. For x0, i.e. downstream from the dam, we observe from>(10.8.8) and (10.8.9) that u is greater than c. Since c is the speed atwhich the front of a disturbance propagates relative to the movingwater we see that changes in conditions below the dam can have noeffect on the flow above the dam since the flow velocity at all pointsbelow the dam is greater than the wave propagation speed at thesepoints and hence disturbances can not travel upstream.
However,once AQ//^ is taken higher than the critical value, the flow velocityat the dam will be less than the wave propagation speed at this point,as one can readily prove, and we could no longer expect the flow atthat point to be independent of the initial depth assumed on thedownstream side.The discharge rate dQ/dthu per unit width at the dam, i.e. atx = 0, is plotted in Fig. 10.8.4 as a function of the depth A In accordance with our discussion above we observe that dQ/dt remains con.0.30.20.10.20.40.60.8Fig. 10.8.4. Discharge rate at the=1.0dam.296h l c l until h^h^ reaches the criticalstant at the value dQ/dtvalue .138, after which it decreases steadily to the value zero whenh/ir i.e. when the initial depth of the water below the dam is the=sameas that above thedam.WATER WAVES340Another feature ofinterest in the present problem is the height ofthe quantity h 2A as a function of the original depthwe know that there is no bore and the waterratio h^jh v When Asurface (as we found in section 10.5) appears as in Fig.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
