Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 69
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The surface of the water inequilibrium, Vaulthier reasoned, is horizontal. Then the immediate cause of the flow in achannel must be the slope of the surface. In the uniform case, the descent -d� of the water(x.jpgS)(av + bv2) ds. For a frictionless/2g) of the velocity head, by analogy withsurface is given empirically by the Prony formulafluid it should be equal to the variation d(iffrictionless fall on an inclined plane. Vaulthier simply added these two contributions to getthe backwater equation. His main service to the subject was not this dubious proof, but hismany applications of the backwater equation at a time when French engineers busiedthemselves with the improvement of rivers.2018Belanger [1841/42] pp.
94--6. Belanger obtained the correct expression for thejump by equating the resultantof the pressures acting on the sides of the fluid portion to the variation of its momentum.1 9Poncelet [1 836] pp. 66n; Navier [1 838] p. 190. Cf. Saint-Venant [1887c] p. 158.
Belanger's procedure of 1828can also lead to the Poncelet form of the backwater equation if the s-axis is made parallel to the water surface."'Vaulthier [1 836] pp. 241-313. Cf. Saint-Venant [1887c] pp. 157-8.TURBULENCE227Upon reading Vaulthier, Coriolis offered his own derivation of the backwaterformula, which long remained the standard one.
'The question of the figure of backwaters', Coriolis declared, 'is the most important question that theoretical hydraulicspresents to the engineer.' He meant that, whereas the known laws of efflux and uniformflow were essentially empirical, the backwater problem admitted a theoretical solution.He nonetheless disagreed with Belanger's treatment, which he misread as an incorrectapplication of the theorem of live forces, and with Vaulthier's, which ignored the principles of mechanics. Even Poncelet's treatment fell short of satisfying Coriolis, for itmaintained Belanger's provisional assumption of uniform velocity in a given section ofthe flow.21Avoiding the latter restriction (but still neglecting the curvature of the lines offlow), Coriolis expressed the variation dT of live force of a slice of fluid in the time dtand the work WG of the gravity force and the work Wp of the pressure force in the sametime asJ J pv � dS,WG = dt J J pvgz dS,Wp = -dt J J v[Po + pg([ - z)] dSdT = dt 11-/111(6.10),where 11 denotes the difference between the values that the expression following it takes onthe two sides of the slice, z denotes the height of a point of the fluid section, and theintegrals are performed over this section.
For the work of frictional forces, Coriolis simplyassumed an external friction with an effective sliding velocity equal to the average velocity.The resulting backwater formula isI = - d? = 0._ Fu + i_ds s pg ds(a uz)2g,(6. 1 1)where U denotes the average velocity of the fluid through a given section, and(6.12)This equation only differs from Vaulthier's by the introduction of the coefficient a, whichCoriolis determined from Du Buat's old measurements of the velocity profile.226.
1.5 Rivers and torrentsThere is more to say about the critical depth he. In 1851, Saint-Venant noted that the formof the backwater curves below and above the critical depth corresponded to tumultuous2 1 Coriolis (1 836] p. 314. Cf. Saint-Venant [1 887c] pp. 1 58-67, Rouse and !nee [1957] pp. 150-1.22Coriolis [1 836] p. 318.228WORLDS OF FLOWand tranquil flows, respectively. The subcritical case defmed'torrents, the various parts ofwhich seem to flow independently of each other and whose acquired velocity allows themto flow over small obstacles.' The supracritical case defined'rivers or quiet streams whosesuccessive slices press on each other and move along together, so that they can only getover obstacles by means of the weight of the elevated water and so that every elevation inone part is felt in the upstream direction to a finite distance. '23h >=:J he for the possibility of a jump1870, this means that the velocity ofFor a rectangular canal of small slope, the conditionis equivalent toU >=:J Vifi..
As Saint-Venant noted inthe water is the same as the propagation velocity of a small swell as given by Lagrange. In ahydrostatic canal closed by two distant gates, with a rise of the water level obtained byconstantly feeding water at one of the gates, the higher level propagates as a step along thecanal with the Lagrangian celerity Vifi., as indicated in Fig.6.4.A small hydraulic jumpcan be obtained by superposing with this motion a constant flow at the velocity -Vifi.. Aswe saw in Chapter2, Saint-Venant used this remarkable connection between jumps andwaves to confirm his distinction between rivers and torrents.
In a stream slower than Vifi.,the swells created by an obstacle must propagate in the upstream direction, so that wateraccumulates before it can pass the obstacle. This is the case of a river. In a stream fasterthan Vifi., the water can pass obstacles without previous accumulation. This is the case of atorrent. 24With their sophisticated analysis of backwaters, hydraulic jumps, and critical depths,French hydraulicians could pride themselves on having transcended the more empiricalapproach of their predecessors. Yet they did not base their theories on the fundamentalhydrodynamics of Euler and d' Alembert, which famously failed in most concrete problems.
Following a via media between pure empiricism and fundamental deduction, theydeveloped effective theories that exploited the principles of mechanics but required someexperimental input and various theoretical idealizations.The theorists of pipes and open channels obviously required empirical knowledge of theretarding action of the walls, and also of the transverse velocity profile in Coriolis's case.In a first idealization, they assumed the retarding effect of the walls to be transmitteduniformly to the inner filaments of the fluid by an unknown mechanism roughly independent of the varied character of the flow. Without knowledge of this mechanism, thevelocity profile could not be derived.
Most authors assumed approximate uniformity ofFig. 6.4.The progression (thin arrow) of a swell produced by feeding additional water (thick arrow) at onegate of a hydrostatic canal.23Saint-Venant [1851a] p. 319.24Saint-Venant [1870] pp. 1 86-95. See Chapter 2, p. 82.TURBULENCE229the fluid velocity within a given section.
Although Coriolis avoided this restriction, he stillneglected the curvature of the fluid filaments and the resulting centrifugal force.There was yet another simplification, so obvious that no one cared to mention it. Thedeductions of formulas for backwaters and hydraulic jumps all rested on the assumptionthat the only relevant motion was the average, macroscopic motion measured by standardgauging methods. Their authors must have been aware of the temporal and spatialirregularities constantly encountered in hydraulic experiments. However, they did notsuspect that these irregularities could affect the average flow in pipes or channels ofslowly-varying slope and section.6.2 Saint-Venant on tumultnous waters6.2. 1 TumultuousflowNot all French engineer-mathematicians of this period confined themselves to a semiempirical approach to hydraulics.
An early exception was Navier, who in 1 822 derived adifferential equation of fluid motion based on a simple molecular assumption and intended to describe the behavior of real, viscous fluids. As we saw in Chapter 2, Navierquickly realized that his equation only applied to 'linear motions' (we would say laminar)and not to 'the more complex motions' occurring in typical hydraulic systems. We also sawthat in 1834 his former student and admirer Saint-Venant had begun to develop an equallyfundamental approach. 25Saint-Venant clearly distinguished two scales, namely, a larger scale at which theaverage velocity varies smoothly in space and time, and a smaller scale at whichthe motion can be highly irregular. In his derivation of the Navier-Stokes equation, heused volume elements that included 'the case when partial irregularities of the fluid motionforce us to take faces of a certain extension so as to have regularly varying averages.' Theeffective viscosity parameter 8 defmed at this scale depended on the irregular motions atthe smaller scale.
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