Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 70
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Thus it could vary from one point of the fluid to another, and from onekind of flow to another. 26Among the irregularities of motion on which the variable 8 depended, Saint-Venantincluded the undulations of molecular paths he had described in his memoir of 1834 asbeing caused by the sliding of successive fluid layers over each other.
In a letter to PierreBoileau of March 1846, he evoked the further possibility that 'the internal frictioncoefficient may vary with the general dimensions of the current and with the freedomthat oblique motions and eddies thus have to develop and to disseminate live force, as youhave very well said. mLeonardo da Vinci and Daniel Bemoulli had long ago noted the whirling motionsinduced by the sudden enlargement of a pipe or by obstacles. As we saw in Chapter 3, in1799 the Italian hydraulician Giovanni Battista Venturi pleaded for a more realistic25Navier [1823c] pp.
389-440.26Saint-Venant [1843c] p. 1242n.27Saint-Venant to Boileau, 29 Mar. 1846, Fond Saint-Venant, Ecole Polytechnique. Boileau ([1 846] p. 215) hadwritten: 'The viscosity of liquids seems to play [in the retardation of the upper fluid layers] a more important andmore complex role than has been admitted by geometers, by giving birth to molecular motions oblique to thestream and by disseminating the live force of the fluid filaments . . . in a manner related to their mutual friction.'WORLDS OF FLOW230conception of fluid motion in which 'the lateral communication of motion' and theresultant eddies played an essential role.
According to Venturi, 'the eddies of the waterin rivers are produced by motion, communicated from the more rapid parts of the streamto the lateral parts, which are less rapidly moved.' They contribute to the retardation of theflow:28One of the principal and most frequent causes of retardation in a river is produced bythe eddies incessantly formed in the dilations of the bed, the cavities of the bottom,the inequalities of the banks, the bends or windings of its course, the criss-crossingcurrents, and the streams that intersect with different velocities.In a study of pressure losses in the pipes of steam engines published in 1 838, SaintVenant emphasized the role of 'extraordinary friction, usually called loss ofliveforce, anddetermined by the eddying of fluids especially at points where the section of the flowsuddenly increases.' The following year, the military engineer Jean-Victor Poncelet published the second edition of his celebrated course 'for the artists and workers' of Metz, inwhich he gave much importance to the whirls observed during the sudden alteration of aflow.
These motions, 'much more complicated than one usually thought', involved pulsations, intermittence, and the conversion of large-scale motions to smaller-scale ones,perhaps thus cascading to the molecular level:29Careful observation of the facts justifies the belief that independently of the gyratorymotions shared by a whole portion of the fluid mass, there are also secondary or lessapparent motions that involve smaller groups of molecules and develop in theintervals between the former motions .
. . . We may further assume without muchrisk that the motions of rotation or oscillation thus impressed on individual molecules or on the smallest groups of molecules are, in addition to adhesion andcohesion . . . one of the most important causes of the loss of motion in fluids andespecially of the resistance that their stream lines experience when gliding on eachother or on the surface of solids.Poncelet thus provided the mechanism through which Joule and others later interpretedthe dissipation of macroscopic motion into heat.
He even explained Brownian motion as aconsequence of the ensuing molecular agitation, instead of the vitality of organic particlesimagined by naturalists. Less speculatively, he regarded the formation of eddies as 'one ofthe means that nature uses to extinguish, or rather to dissimulate the live force in thesudden changes of motion of fluids, as the vibratory motion themselves are another causeof its dissipation, of its dissemination in solids.' He also believed that smaller-scalewhirling largely contributed to the effective friction between fluid filaments. 30Saint-Venant approved these considerations and brought them to bear on hydraulicproblems.
In 1 846, he examined Borda's old formula for the loss of head during a suddenenlargement of a pipe. 'The molecular gearing [engrenement moleculaire]', he wrote,'creates whirls and other non-translatory motions indicated by D. Bernoulli and byM. Poncelet, which, after being conserved for some time in the fluid, end up being28Venturi [1797] prop. X. Cf. Rouse and Ince [1 957] pp. 1 34-7. See also Chapter 3, pp. 105-8.29Saint-Venant [1838] p. 47; Poncelet [1839] pp. 529-30.30Ibid.TURBULENCE231dissipated under the effect of friction and extraordinary resistance.' He then offered asimple derivation of Borda's formula, based on the balance of live forces in a referencesystem moving at the final velocity of the fluid.3 1In the same year, Saint-Venant also considered the old, difficult problem of fluidresistance.
As we saw in Chapter 3, he related the resistance to the live force of the nontranslatory fluid motions induced by the immersed body. When tumultuous, whirlingmotion occurs at the rear of the body, the resistance largely exceeds the value it would havefor a perfectly smooth flow. 326.2.2 The effective viscosityIn a memoir of 1851 on retardation formulas and backwater tables, Saint-Venant publicized the idea of small-scale tumultuous motions being responsible for the variable 8 thathe had championed since 1 83 4:33Newton's hypothesis, as reproduced by MM. Navier and Poisson, consists in makinginternal friction proportional to the relative velocity of the filaments sliding on oneanother; if it can be approximately applied to the various points of the same fluidsection, every known fact indicates that the proportionality coefficient must increasewith the dimensions of transverse sections; which may be to some extent explained bynoticing that the filaments do not proceed in parallel directions with a regulargradation of velocity, and that the ruptures, the whirls and other complex and obliquemotions that must considerably influence the intensity of friction develop better andfaster in large sections.Saint-Venant found much evidence for this view in experimental studies of openchannel flow.
Boileau's contribution has already been mentioned in Chapter 3. In 1868,the American hydraulicians Andrew Humphreys and Henry Abbot published the resultsof their measurements and observations on the Mississippi River. From these sources,Saint-Venant extracted the contrast between small-scale disorder and large-scale orderthat justified the effective 8 approach: 34Beyond this disorder [in the local fluid motion], as was especially noted by [CaptainBoileau] and as has been observed in larger masses by American engineers [Humphreys and Abbot], a certain order is nevertheless observed; for the same particularities of the velocity of the fluid quickly repeat themselves everywhere, so that themotion, if determined by constant causes, settles up by periodicity [Humphreys andAbbot's 'river pulses'] .
. . and the effective velocities undergo complex but smalloscillations around constant averages relative to each point. These local averagevelocities for fluid translation or transport are 'those measured by floats and otherhydrometric instruments and they determine the flows to be computed.'31 Saint-Venant [1 846a] p. 147. Saint-Venant's reasoning was a simplification of an earlier reasoning byBelanger that combined momentum and live-force balance in the natural reference system. Rather than directlyestimating the live force lost to whirls, Belanger and Saint-Venant assumed the pressure on the walls of theexpanding part of the pipe to vary hydrostatically.32Saint-Venant [1 846a] pp.
28, 72-8, 120-1.33Saint-Venant [185la]34Humphreys and Abbot [1868] pp. 1 65-94; Saint-Venant [1 872] p. 650.p.229.232WORLDS OF FLOW6.2.3 Darcy's and Bazin's measurementsIn 1 857, still another engineer at the Ponts et Chaussees, Henry Darcy, published systematic hydraulic measurements.
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