J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 82
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Three of these quantities arepurely geometrical in character and could in principle be determinedriversWATER WAVES456from an accurate contourmapof the river valley, but the determina-tion of the roughness coefficient y of course requires measurements ofactual flows for its determination.11.2. Steady flows.Ajunction problemWe define a steady flow in the usual fashion to be one for which thevelocity v and depth y are independent of the time, that is, v t y t 0.In this section channels of constant rectangular cross-section andconstant slope will be considered for the most part. It follows from theequation of continuity (cf.
(11.1.2)):Vt+ vy x + yv x =0,that for steady flow(11.2.1)(vy) xwhen no flow=whence vyD (Dainto the channel fromitsconstant),sides occurs(11.1.2)). Similarly, the equation of motion(cf.(i.e.q=in(11.1.6))^ + vv x + gy x + g(S, -S) =yields(11.2.2)It followsvv x+ gy, + g(St -S) =from equationvQ.(11.2.1) thatD=_A vandxy= --D-y2yx,so that equation (11.2.2) becomesHere the hydraulic radius is given by R = y/(l + 2y/B) because thechannel is assumed to be rectangular in cross-section.For a channel with given physical parameters such as cross-section,resistance coefficient, etc. the steady flows would provide what arecalled backwater curves. In general, one could in principle alwaysfind steady solutions y = y(x) and v = v(x) for a non-uniform channel.
The explicit determination of the stage y and discharge rate BDas functions of x would be possible by numerical integration of thepair of first order ordinary differential equations arising from (11.1.6)and (11.1.8) when time derivatives are assumed to vanish.MATHEMATICAL HYDRAULICSWe note that equation (11.2.3) has the simple solution y =for457constanty satisfyingThis means that we can find a flow of uniform depth andvelocityhaving a constant discharge rate BD (B is, as in the preceding section,the width of the channel). Conversely, by fixing the depth y we canfind the discharge from (11.2.4) appropriate to thecorrespondinguniform flow.
Physically this means that the flow velocity is chosenso that the resistance due to turbulence and friction and the effect ofgravity down the slope of the stream just balance each other. We remark that if (11.2.4) is satisfied at any point where the coefficientD 2 ly z of y x in (11.2.3) does not vanish, then y = constant is thegonly solution of (11.2.3) because of the fact that the solution is thenuniquely determined by giving the value of y at any point x. We noteD==corresponds to vVliJ/> *- e to a fl w at criticalspeed (a term to be discussed in the next section), since Dvy.Furthermore, the differential equation (11.2.3) can be integrated tothat gyield/y3when xa?-=x as a function of(11.2.5)We2y:=foryyQ.makeuse of (11.2.5) in order to study a problemat the junction of two rivers each having aaflowinvolvingsteadyrectangular channel.
Later on, the same problem will be treated butfor an unsteady motion resulting from a flood wave traveling downone of the branches, and such that the steady flow to be treated hereisexpected to result as a limit state after a long time. The numericalproceed todata arc chosen here for the problem in such a way as to correspondroughly with the actual data for the junction of the Ohio River withthe Mississippi River. Thus the Ohio is assumed to have a rectangularchannel 1000 feet in width and a constant slope of .5 feet/mile.
InManning's formula for the resistance the constant y is assumed given2by y =. (1.49/n) in terms of Manning's roughness coefficient n, andn is given the value 0.03. The upstream branch of the Mississippi wastaken the same in all respects as the Ohio, but the downstream branchis assumed to have twice the breadth, i.e. 2000 feet, and its slope toWATER WAVES458have aslightly smaller value, i.e.
0.49 feet/mile instead of 0.5 feet/mile.these values of the parameters, a flow having the same uniformdepth of 20 feet in all three branches is possible the choice of theWithLowerMississippiFig. 11.2.1. Junction ofOhio and Mississippi Riversvalue 0.49 feet/mile for the slope of the downstream branch of theMississippi River was in fact made in order to ensure this. Later onwe intend to calculate the progress of a flood which originates at amoment when the flow is such a uniform flow of depth 20 feet. Theflood wave will be supposed to initiate at a point 50 miles up the Ohiofrom the junction and to be such that the Ohio rises rapidly at thatpoint from the initial depth of 20 feet to a depth of 40 feet in 4 hours.A wave then moves down the Ohio to the junction and creates waveswhich travel both upstream and downstream in the Mississippi as wellas a reflected wave which travels back up the Ohio.
After a long timewe would expect a steady state to develop in which the depth at thepoint 50 miles up the Ohio is 40 feet, while the depth far upstream inthe Mississippi would be the original value, i.e. 20 feet (since we wouldnot expect a retardation of the flow far upstream because of an inflow atthe junction). Downstream in the Mississippi we expect a change in theflow extending to infinity. It is the steady flow with these latter characteristicsWethatwe wish to calculate in the present section. See Fig. 11.2.1)remark(firstofallthat the stream velocities inallof the threebranches will always be subcritical in fact, they are of the order ofa few miles per hour while the critical velocities \/gt/ are of the orderof 15 to 25 miles per hour.
It follows that the quantity gD 2 /t/3 in theintegrand of the basic formula (11.2.5) for the river profiles (i.e. thecurve of the free surface) is always positive. The integrand I(y) inMATHEMATICAL HYDRAULICS459that formula has the general form indicated byFig. 11.2.2 in the caseof flows at subcritical velocities.
The verticalasymptote correspondsto the value of y for which a steady flow of constantdepth existsFig. 11.2.2.The integrandiu thewaveprofileformulasince the square bracket (the denominator in the intevanishes for this value. It follows that x can become positive(cf.
(11.2.4)),grand)y only if y takes on somewhere this value;but in that case we have seen that the whole flow is then one withconstant depth everywhere. Consequently the downstream side of theMississippi carries a flow of constant speed and depth, though thevalues of these quantities are not known in advance. However, in theupstream branch of the Mississippi the flow need not be constant, andof course we do not expect it to be constant in the Ohio: in thesebranches x must be taken to be decreasing on going upstream andinfinite for finite values ofconsequently the negative portion of I(y) indicated in Fig.
11.2.2at the junction.comes into use since we may, and do, set xFor the sake of convenience we use subscripts 1, 2, and 3 to refer toall quantities in the Ohio, the upstream branch of the Mississippi, and=the downstream branch of the Mississippi respectively. The conditionsto be satisfied at the junction are chosen to be(11.2.6)* = y, = y = y,(11.2.7)D,1+D2WATER WAVES460Thefirst condition simply requires the water level to have the samevalue yj (which is, however, not known in advance) in all three branches, while the second states, upon taking account of the first condition,that the combined discharge of the two tributaries makes up the totalThe quantitydischarge in the main stream.isupper Mississippi,is supposed knownByknownsince the flow fara uniform flow of depth 20 feet.branch of theit isi.e.Z) 2 , the discharge in theupstream in this branchusing (11.2.7) in (11.2.4) as applied to the lowerMississippi (inwhich the flowisknownto be constant)we haveNext, we write equation (11.2.5) for the 50-mile stretch of the Ohiowhich ends at the point where the depth in that branch was prescribedto be 40 feet (and which was the point of initiation of a flood wave);the resultis50-(11.2.9)inwhichit isindicated thatI(y,D l ,B l )dy\D and B(as well as all otherparameters)are to be evaluated for the Ohio; the quantity y has the value 40/5280in miles.
Equations (11.2.8) and (11.2.9) are two equations containing-j/ ;and D!asunknowns, sinceZ) 2 isDknown. They were solved by aniterative process, i.e. by taking foran estimated value, determiningla value for y j from (11.2.9), reinserting this value in (11.2.8) to deter-mine a new valuefor Z) 1 etc.,Theresults obtainedby such acalcula-tion are as follows:y\=vlThe=2/24.83 miles/hour,=2/3v2==Vt= 81-2feet1.53 miles/hour,profiles of the river surfacecanv3~3.18 miles/hour.now be computed from(11.2.5);the results are given in Fig. 11.2.3.The solution of the mathematical problem has the features wewould expect in the physical problem.
The flow velocity and stage areincreased at the junction, even quite noticeably, by the influx fromthe Ohio. Upstream in the Mississippi the stage decreases ratherrapidly on going away from the junction, and very little backwatereffect is noticeable at distances greater than 50 miles from the junction.This illustrates a fact of general importance,i.e.that backwaterMATHEMATICAL HYDRAULICSeffects in long rivers arisingfrom even461fairly large discharges of tri-main stream do not persist very far upstream,but such an influx has an influence on the flow far downstream.For unsteady motions this general observation also holds, and is infact one of the basic assumptions used by hydraulicsengineers inbutaries into they feet40OhioDownstreamMississippiUpstreamMississippi4--50,Fig.
11.2.3.Junction50100milesSteady flow profile in a model of the Ohio and Mississippi Riverscomputing the passage of flood waves down riversflood routing by them). Later on, in sec. 6 of thisdeal with the unsteady motion described above inOhio-Mississippi system, and we will see that the(a process calledchapter, we shallour model of theunsteady motionapproaches the steady motion found here as the time increases.11.3. Progressingwaves offixed shape. RollwavesIn addition to the uniform steady flows treated above there alsoexist a variety of possible flows in uniform channels in the form ofwithoutprogressing waves moving downstream at constant speedchange in shape. Such waves arc expressed mathematically by depthsy(x, t) and velocities v(x t) in the form9(11.8.1)y(x,t)=y(xUt)<v(x, t)=v(xUt),U=const.WATEE WAVES462The constantUof course the propagation speed of theisviewed from a fixed coordinate system;ifwaveasviewed from a coordinatesystem moving downstream with constant velocity U the wave profilewould appear fixed, and the flow would appear to be a steady flowrelative to theriablebymoving system.Itisconvenient to introduce the va-settingC(11.3.2)=so that y and v are functions ofx-Utonly.
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