Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 68
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Combined with the equilibrium condition (6.1), this form gives the formulanamed after Antoine Chezy, who proposed it first in an unpublished report on a canalplanned to bring the waters of the River Yvette into Paris. In 1 804, the director of theEcole des Ponts et Chaussees, Gaspard de Prony, inferred from Couplet's, Bossut's, andDu Buat's retardation measurements and from Charles Coulomb's understanding ofsurface friction the perennial form7(6.2)As well as his treatment of uniform permanent flow, Du Buat gave semi-empiricalformulas for weirs and backwaters. One of his main concerns was the improvement ofthe navigability of rivers, then usually achieved by a series of weirs that elevated the waterlevel.
The weir formula gives the height of the water above a weir as a function of the river'sdischarge and the weir's width and height. Upstream from the weir, the water surface has acurved shape that would asymptotically reach the natural level of the river if no other weirinterfered. This is the 'backwater' phenomenon which Du Buat improperly called remou.The navigability of a naturally shallow river is improved by weirs placed so that the depth ofthe backwater of the nth weir at the foot of the (n 1 )th weir exceeds the minimal depthrequired for navigation. A lock on the side of each weir permits the passage of the boats.
8-6.1.2Be/anger's backwater theoryDu Buat contented himself with a circular-arc approximation of the backwater curve,arguing that the knowledge of the relevant differential equation would be of no practicalhelp. Some forty years later, Jean-Baptiste Belanger, an engineer with the Ponts etChaussees, judged differently. Like many former polytechnicians, Belanger had faith inthe practical usefulness of higher mathematics. While working on canals and adjacentrivers, he sought a theory of non-uniform flow that would permit more rational designs.The Royal Academy of Metz had recently advertized a prize for 'determining the curvethat running water forms upstream from a weir'.9The only known open-channel formulas concerned uniform permanent flow, for whichthe section of the channel is uniform and the slope of the water surface is the same as theslope of the bottom.
In his new theory, B61anger admitted a slow variation of the section ofthe channel and slight differences between the surface and bottom slopes. For simplicity,he assumed that the velocity (vector) within a section of the stream was nearly uniform,6Du Buat [1786] vol.8Du Buat [1786] vol.I,I,p. xvii.1Ibid. p. 62.
Cf. Rouse and Ince [1957] pp.141-3.chap. 4, pp. 205-17.9Belanger [1 828] p. iii. Cf. Saint-Venant [1887c] pp. 1 54-7, Rouse and !nee [1957] pp. 148-9. On the prize, cf.Poncelet [1845] p. SI On.TURBULENCE223although he knew from Du Buat that the velocity increased with the distance from thebottom. This assumption agrees with a variable fluid section as long as the departure fromuniformity remains small. 1 0Following Belanger, take the s-axis parallel to the common velocity of the approximately-parallel water filaments, and the x-axis normal to this axis (see Fig.
6.1 ). Denote byy the angle that the s-axis makes with the horizontal. A given particle of the fluidexperiences three forces, namely, its weight, the pressure gradient, and a frictional force,which Belanger took to be the same at every point of a fluid section. 1 1 Its velocity v varieswith the section S, as follows from the constancy of the discharge Q = vS (the volume ofwater crossing a section of the channel per unit time).
Newton's second law, projected ontothe s- and x-axes then gives(6.3a)-aPox- pgcos y = 0.(6.3b)With the origin of the x-axis at the bottom of the section, integration of eqn (6.3b) givesP= (h - x)pgcos y + Po,(6.4)where h is the depth measured in the direction of the x-axis and Po is the atmosphericpressure. 1 2So far, the slope of the s-axis could have any value between the slope of the surface andthat of the bottom.
Belanger ultimately placed the s-axis at the bottom (running throughXsFig. 6. 1 .The geometrical parameters for slowly-varying flow in an open channel.10Belanger [1828] p. 5.1 1 This assumption is clearly valid in the uniform case, since the frictional force is then balanced by thegravitational force, which is a constant.1 2Belanger [1828] p. 8.WORLDS OF FLOW224the lowest point of each section). For any particle of a given fluid section, the equation ofmotion (6.3a) then givesdhX1 dv(6.5)-- cos y + sm y - -Fv = - - .g dtpSgds.Denoting by i the slope sin y of the bottom, Belanger obtained his backwater equation:i ds - vT=i2 dh _ L (av + W) ds + Jt dS = o,gS3pSg(6 .
6)the last term of which comes from the identities dvfdt = d(Q/S)/dt = -(Q/S2)dS/dtand dS/dt = (dS/ds)v = (dS/ds)Q/S. This equation completely determines the backwater curve if the variation of the fluid section S with the depth h and the distance s isknown.13Belanger provided a stepwise integration of this equation in the simple case of thehorizontal aqueduct which had been built recently to bring the waters of the RiverOurcq into Paris.
In this case, the practical question was the height that the water musthave at the beginning of the aqueduct for a given height at the end. Belanger also gave afew examples of calculations of the backwaters before a weir, with the navigability of riversin mind. 146.1.3 Hydraulic jumpsIn most practical cases, the values of the parameters in eqn (6.6) allow integrationthat does not conflict with the starting assumption of a slow variation of the depth h.However, Belanger noticed the possibility of different behaviors. Consider the case ofa straight canal with a wide rectangular section and with a purely quadratic friction.Denote by q the discharge per unit breadth, ho the depth (bq2/pgi)113 that the flow wouldhave in the uniform case, and he the depth (q2 fg)1 /3 • In these terms, eqn (6.6) takesthe simple formdhh3 - h�tgy.(6.7)h3 - h3eAlthough this equation can be integrated explicitly, the variations of h are more conveniently inferred from the sign that the derivative dh/ds takes according to the relative valuesof h, h0 , and he.
In the frequent case of a swell (h > ho) on a small-sloped bed (ho > he), thecurve h(s) is concave and has an upstream asymptote parallel to the bed and a horizontaldownstream asymptote (see Fig. 6.2(a)). This means that the flow is asymptoticallyuniform in the upstream direction and then swells owing to a downstream cause, whichcould be a weir or the merging into a lake. 1 5ds1 3Belanger [1 828] p. 10.=14/bid.
pp. I I-28.1 5Cf. Bresse [1 860] pp. 218-30, Flamant [1891] pp. 237, 263-4, Forchheimer [1927] pp. 1 81-3.225TURBULENCEFig. 6.2.The backwater curves in the small-sloped case, and their concrete realizations according to Forchheimer [1927] p. 1 8 1 .In the case h < he < ho, which would occur when water is forced through a sluice gateinto a small-sloped channel or when a high-sloped channel turns into a small-sloped one,the depth increases in the downstream direction until it reaches the critical value he forwhich the slope dh / ds becomes infinite (see Fig.
6.2(b)). The part of the curve close to thiscritical point cannot be trusted, for it contradicts the approximation of parallel-slice flow.Belanger surmised that in this case the water level would suddenly increase to a valuehigher than critical, and then again vary smoothly according to eqn (6.7). He identified thisbehavior with the 'hydraulic jumps' that the Italian hydraulician Giorgio Bidone hadstudied in the 1 820s.
16In order to determine the height of the jump, Belanger appealed to the theorem of liveforces which his colleague Claude-Louis Navier and his friend Gaspard Coriolis had beenapplying to the theory of machines. In Coriolis's statement of this principle, the variationof live force of a mechanical system during a given time must be equal to the work of theforces acting on the system during this time.
Belanger considered a portion of fluiddelimited by two planes perpendicular to the bottom and situated before and after thejump. Denote by !; and !;' the surface heights (measured from a fixed horizontal plane) inthese planes, v and v' the corresponding fluid velocities, and za and z'a the heights of theirgravity centers (see Fig. 6.3). During a time dt, the live force of the portion of fluid variesby (I j2)p(VZ - v2)Qdt. The work of the pressures acting on the sections during the sametime is pg(!; za !:' + z'a)Qdt, because the pressures vary hydrostatically in the twosections.
The corresponding work of gravity is pg(za - z'a)Qdt. Neglecting the work offrictional forces, the theorem of live forces gives--if v12!:' - !: = - - - .2g 2g(6.8)In the parlance of hydraulicians, the jump equals the decrease of the velocity head. 171 6Belanger [1828] pp. 29-31; Bidone [1820]. Cf. Rouse and !nee [1957] pp. 143-4, 149.1 7Belanger [1 828] p.
32.WORLDS OF FLOW226zFig. 6.3.6. 1 .4Geometrical parameters for a hydraulic jump.Backwater energeticsAs Belanger later realized, this reasoning errs by ignoring the loss of live force that theabrupt change of motion implies. 1 8 For a gradually-varying flow, the balance of live forceseems to apply to an infinitesimal change d� of surface height over the infinitesimaldistanceds.Taking into account the workthe equationd� +-(x./S)FvQdt of frictional� Fv ds +Sgd(�)=forces, this gives0,(6.9)which is a simpler form of Belanger's backwater equation, because of the relationsd�= -i ds + dh ../f=i2 and v = QjS(see Fig.6.1). Poncelet and Navier obtained thebackwater equation in this manner, the former in the same year as Belanger. 1 9In1 836, Pierre Vaulthier, who only knew Belanger's proof, obtained the Poncelet formof the backwater equation by suspiciously simple reasoning.
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