J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 81
Текст из файла (страница 81)
(The theory developed here could be extended to casesin which the shore reflects all of the incoming energy, it might benoted.) In principle, the calculation of the deflection curve of thestructure, and hence also of the bending stresses in it, as given by thetheory is straightforward, but it is very tedious; consequently onlythe reflection coefficients have been calculated.CHAPTER11Mathematical HydraulicsIn this chapter the problems to be treated are, from the mathemapoint of view, much like the problems of the preceding chapter,but the emphasis is on problems of rather concrete practical signifiticalcance.
Aside from this, the essential difference is that external forcesother than gravity, such as friction, for example, play a major role inthe phenomena. Problems of various types concerning flows andwave motionsin open channels form the contents of the chapter.
Thebasic differential equations suitable for dealing with such flows underrather general circumstances are first derived. This is followed by astudy of steady motions in uniform channels, and of progressing wavesof uniform shape, including roll waves in inclined channels. Floodwaves in rivers are next taken up, including a discussion of numericalmethods appropriate in such cases; the results of such calculationsa simplified model of the Ohio River and for ajunction with the Mississippi are given.
This discussionfollows rather closely the two reports made to the Corps of Engineerstoy a floodmodel ofwaveinitsStoker [S.23] and by Isaacson, Stoker, andTrocsch [1.4], These methods of dealing with flood waves have beenapplied, with good results, to a 400-mile stretch of the Ohio as itactually is for the case of the big flood of 1945, and also to a floodof the U.S.Army bythrough the junction of the Ohio and the Mississippi; these resultswill be discussed toward the end of this chapter.There is an extensive literature devoted to the subject of flow inopen channels.
We mention here only a few items more or less directlyconnected with the material of this chapter: the famous Essai ofBoussinesq [B.17], the books of Bakhmeteff [B.3] and Rouse [R.10,11] (in particular, the article by Gilcrest in [R.ll]), the Enzyklopadiearticle of Forchheimer [F.6] and the booklet by Thomas [T.2].451WATER WAVES45211.1. Differential equations of flow inopen channelsIt has already been stated that the basic mathematical theory tobe used in this chapter does not differ essentially from the theoryderived in the preceding chapter.
However, there are additionalcomplications due to the existence of significant forces beside gravity,and we wishto permit the occurrence of variable cross-sections in thechannels. Consequently the theory is derived here again, and a somewhat different notation from that used in previous chapters is employed both for the sake of convenience and also to conform somewhatwith notations used in the engineering literature.The theory is one-dimensional, i.e. the actual flow in the channel isassumed to be well approximated by a flow with uniform velocity overeach cross-section, and the free surface is taken to be a level line ineach cross-section. The channel is assumed also to be straight enoughso that its course can be thought of as developed into a straight linewithout causing serious errors in the flow. The flow velocity is denotedthe depth of the stream (commonly called the stage in theengineering literature) by y, and these quantities are functions of thebyv,Fig.
11.1.1. River cross-sectiondistance xdownandFig. 11.1.1). Theof the free surface of the stream,the stream and of the timevertical coordinates of thebottom andprofileas measured from the horizontal axis x9t(cf.are denotedbyz(x)andMATHEMATICAL HYDRAULICS453=+with z positive downward, h positive upward; thus yhz.slope of the bed is therefore counted positive in the positive^-direction, i.e. downward. The breadth of the free surface at anysection of the stream is denoted by B.h(x,t),TheThe differential equations governing the flow are expressions of thelaws of conservation of mass and momentum. In deriving them thefollowing assumptions, in addition to those mentioned above, aremade *: 1) the pressure in the water obeys the hydrostatic pressurelaw, 2 ) the slope of the bed of the river is small, 3) the effects of frictionand turbulence can be accounted for through the introduction of aresistance force depending on the square of the velocity va certain way to be specified, on the depth y.andalso, inWe first derive the equation of continuity from the fact that themass gAAx included in a layer of water of density p, thickness Ax,and cross-section area A, changes in its flow along the stream onlythrough a possible inflow along the banks of the stream, say at therate qq per unit length along the river.
The total flow out of the element of volume A Ax is given by the net contributions Q(Av) xAx fromthe flow through the vertical faces plus the contribution @Bh t Ax dueto the rise of the free surface, with B the width of the channel; sinceBh t represents the area change A it follows that the sum [(Av) x+tAt}Ax equals the volume influx qAx over thesides of the channel,with q the influx per unit length of channel.
The subscripts x and trefer, of course, to partial derivatives with respect to these variables.Tfye equation of continuity therefore has the form(Av) x(11.1.1)+A =It should be observed that Athings a given function of y andtq.A(y(x,t) 9x)isin the nature ofx, although y(x, t) is an unknownfunction to be determined; in addition, qq(x, t) depends in generalon both x and / in a way that is supposed given. In the importantof constant breadth B, so thatspecial case of a rectangular channelA=By, the equation of continuity takes the form(11.1.2)v xy+ vy x + y =tq/B.The equation of motion is next derived for the same slice of massm = qAAx by equating the rate of change of momentum d(mv)jdt* Theseassumptions are not the minimum number necessary: for example,on the verticalassumption 1 ) has as a consequence the independence of the velocitycoordinate if that were true at any one instant (cf.
the remarks on this pointin Ch. 2 and Ch. 10).WATER WAVES454to the net force on the element.Wewrite the equation of motion forthe horizontal direction:(AvAx) =Q(11.1.8)HAxF Ax cosf9?dt+QgA Ax sinq>.HIn this equationrepresents the unbalanced horizontal pressureforce at the surface of the element. The angle 9? is the slope angle of thebed of the channel, reckoned positive downward. The quantity f re-Fpresents the friction force along the sides and bottom of the channel,and the term QgAAx sin <p represents the effect of gravity in accelerating the slice down-hill as manifested through the normal reaction ofwas assumed small we may replace sin (p bydz/dx and cos 99 by 1. In the frictional resistance termthe stream bed. Sincethe slopeweS9?setis an empirical formula called Manning's formula.
The resistancethus proportional to the square of the velocity and is opposite to itsdirection; in addition, the friction is inversely proportional to theThisis4/3-power of the hydraulic radius R, defined as the ratio of the crosssection area A to the wetted perimeter (thus RByj(B -\- 2y) for achannelandRforwidearectangularveryrectangular channel),yand inversely proportional to y, a roughness coefficient.We calculate next the momentum change Qd(AvAx)/dt. In doing so,we observe that the symbol d/dt must be interpreted as the particle==Chapter 1.1 and equation (1.1.8)) d/dt + vd/dx sinceNewton's law must be applied in following a given mass particle alongits path x = x(t).
However, the law of continuity (11.1.1 ) derived aboveis clearly equivalent to writing d(AAx)ldtqAx, with d/dt again in-derivative(cf.terpreted as the particle derivative. Sincedtit(AvAx)=vdt(AAx)+AAxdtfollows thatdt(AvAx)AAx(vv x +vt)-}-qvAx.HAxof the pressure forces over theFinally, the net contributionsurface of the slice is calculated as follows: The total pressure over avvertical face of the slabisgiven bythe hydrostatic pressure law(cf.fQg[y(x,t)]b(x, f )Fig.
11.1.1); while thedfromcomponentMATHEMATICAL HYDRAULICS455in the ^-direction of the totalpressure over the part of the slice incontact with the banks of the river is given bytry"~Ax, we have for HAx the following equation:S]b x (x f ) d6S[y\9Ij(11.1.5)/omyfWO-t-f= --[1@gyxb(%, %)d!;~I~ogAy x.JoIn this calculation the integrals involving b x cancel out, andused the fact that y x is independent of fAdding all of the various contributions we havewe have.vt(11.1.6)+w +x-v=Sg-S,g-gy xscLupon defining whatcalled the friction slopeS f bythe formulaS,(11-1.7)withisFf definedbyshould perhaps be mentioned that thehand side of (11.1.6) arises because of the(11.1.4). Itterm qv/A on the lefttacit assumption that flows enter the main stream from tributariesor by flow over the banks at zero velocity in the direction of the mainstream; if such flows were assumed to enter with the velocity of themain stream, the term would not be present it is, in any case, aterm which is quite small.
If we introduce A = A(y(x, t), x) in(11.1.1) the result(11.1.8)The twoisA yy xv + A xv + Av x + A yydifferential equations (11.1.6)tandq.(11.1.8),which serveto determine the two unknown functions, the depth y(x, t) and thevelocity v(x, t), are the basic equations for the study of flood waves inand flows in open channels generally. For any given river orchannel it is thus necessary to have data available for determiningthe cross-section area A and the quantities y and R in the resistanceterm Ff as functions of x and j/, and of the slope S of its bed as afunction of x in order to have the coefficients in the differential equations (11.1.6) and (11.1.8) defined.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
