Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 71
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During the construction of a new water supply system for the·fountains of Dijon, Darcy had found faults in Prony's old retardation formula. Hisappointment as head of the municipal water service of Paris gave him the opportunity toperform experiments on pipes of various diameters and wall structure. These were by far themost extensive and sophisticated measurements of this kind since Bossut's and Du Buat's.Unlike his predecessors, Darcy used graphical plots as well as the method of!east squares tofind the best-fitting resistance laws.
His most important finding was that retardationdepended on the roughness of the walls. Du Buat had excluded such an effect, because hebelieved that an adhering layer of fluid prepared a smooth surface for the gliding fluid.35Darcy further demonstrated that the friction per unit surface depended on the diameterD of the tube according to an a + f3 /D law, in conformance with the natural expectationthat the effect of roughness should be more important for small pipes than for large pipes.Although he did not offer any precise theory, he suggested that the roughness of the wallplayed an essential role in determining the nature ofthe flow. He rejected the usual parallelfilament picture, because asperities on the wall implied 'gyratory motion by moleculargroupings' and a concomitant loss of live force for the progressive motion.
Far from thewalls, or for a smooth wall, he believed that undulations or oscillations of eventual fluidfilaments would also trigger gyratory motion at the surface of mutual contact.3 6In this view, as in Saint-Venant's, the velocity profile had special theoretical interest.Darcy measured it with unprecedented accuracy thanks to an improved Pitot tube, andfound a semi-cubic profile for a circular section.37 For this profile, the dynamical equilibrium of successive cylindrical layers of the fluid may be obtained by making the internalfriction proportional to the square of the velocity gradient.
Darcy favored this oddtheoretical choice 'against the opinion of several eminent hydraulicians' . Among theeminences, he mentioned Saint-Venant for his notion of a Newtonian internal friction(proportional to the velocity gradient) with a coefficient depending on ruptures andwhirling motions. 38After completing his work on pipes, Darcy turned to open channels. He plannedexperiments on the canal of Burgundy and its reaches, with generous funding by the35Darcy [1856], [1857]; Du Buat (1786] vol.
1 , p. 41: 'Considering how water itself prepares the surface on whichit flows, we see that the difference of the matters of which the wall may be composed cannot have a truly sensibleeffect on the resistance.' Cf. Rouse and !nee (1 957] pp. 1 69-173, Brown, Garbrecht, and Hager [2003].36Darcy [1857] pp. vi, 10, 188-93, 202-1 9. That Prony had failed to detect this dependence in the dataaccumulated by Couplet, Bossut, and Du Buat did not worry Darcy, because the larger pipes of Couplet'sexperiments were older and therefore rougher than the smaller pipes of Bossut and Du Buat.37Darcy [1 857] p.
128. The original Pitot tube, invented by Henri de Pitot in 1732 (see Rouse and !nee [1957]pp. 1 1 5-16) was made of two parallel glass tubes, one being straight, and the other bent through a right angle at itslower end; the water-level difference in the two tubes after vertical immersion in a stream gives the pressure at theirlower end.38Darcy [1857] pp. 1 81-2. Bazin's assumption for the internal shear stress is equivalent to a constant mixinglength in Prandtl's later theory of turbulent flow (see Chapter 7, pp. 297-9). It does not really contradict Saint·Venant's theory, for the effective viscosity e may depend on the velocity gradient.
Saint-Venant ((1869] p. 585)welcomed Drazin's suggestion, but approved Maurice Levy's rejection of generalizations that contradicted theform e(81v1 + 81v1) of the stress system.TURBULENCE233Ministry of Public Works. After his sudden death in 1 8 5 8, his gifted disciple Henri Bazincompleted the measurements and published them in 1 8 6 5 . As in the pipe case, Darcy andBazin found that the roughness of the walls played an important role. They noted thepronounced irregularities of motion, 'the very sudden jolts' and 'ruptures' that AndreBaumgarten (from the Ponts et Chaussees) had long ago observed on the River Garonnewith Reinhardt Woltman's velocity-measuring mill.
They determined the velocity profileand found it to be quadratic in the case of a wide rectangular section, and cubic in the caseof a semicircular section. 39Bazin also studied varied flow, thus partly confirming previous backwater and jumptheories, but also pointing to discrepancies and new phenomena. For example, he described the undulations of the jumps occurring in small-sloped channels, and he found theheight of the jumps to be generally smaller than the change in velocity head, owing to'losses by tumultuous motion'. Lastly, Bazin provided much data on non-permanent flow,including solitary waves and tidal bores, for which no satisfactory theory yet existed.
Forabout half a century, Darcy's and Bazin's measurements remained the most reliablehydraulic data in France and abroad. Their wealth of new regularities and phenomenadefied theory, Bazin thought: 'Maybe such a delicate part of science must long remain in40the realm of experiment.'6.3 Boussinesq on open channels6 . 3 .1Terrestrial physicsAmong those undeterred by this pessimistic forecast was Saint-Venant's protege JosephBoussinesq. Born in Herault and from a family of small farmers, Boussinesq became ahigh-schoolsurveillant and teacher.In this position he found time to take analysis andmechanics at the University of Montpellier and to study works of higher analysis byhimself.
He was mainly self-taught, which makes his writings sometimes difficult topenetrate. Saint-Venant noticed him in 1 867, upon reading his memoir on anisotropyinduced by compression. The two men had a common interest in applied mechanics andalso in religion. Their correspondence covers both topics with equal prolixity. SaintVenant knew that a religious provincial without higher academic training had little chanceto gain recognition.
With his usual generosity, he supported Boussinesq so efficiently as towin him a chair at the University of Lille in 1 873, the chair of experimental and physical4mechanics at the Sorbonne in 1 885, and election to the Academy of Sciences in 1 886. 1Boussinesq defined 'the aim of his life' as 'the study of mathematics as they came alive inCreation, or, if you prefer, the study of the traces left in nature by the geometer whoorganized her when he produced her.' He shared Saint-Venant's dislike for the abstractions of rational mechanics, and worked hard to develop a 'physical mechanics' based on amore realistic, molecular conception of matter.
According to Saint-Venant, the latter39Darcy and Bazin [1 865a] pp. 23-5; Baumgarten [1 847]. Thesevelocity profiles, together with the finite slip atthe walls, can now be seen as approximations of the logarithmic profiles given by turbulent boundary-layer theory,cf. Prandtl [1 933] p. 833n.40Darcy and Bazin [1865a] pp. 30-7, [1865b].4 1 Cf. Le Toumeur [1954], Picard [1933], Douysset [undated], Blaquii:re [1931].WORLDS OF FLOW234mechanics, developed in the 1 820s and 1830s by Poisson, Navier, Coriolis, Poncelet,Belanger, and himself, later met considerable skepticism from the powerful JosephBertrand and other academicians, who pursued 'an absolute and immediate rigor thatprohibited any application of analysis to phenomena, even in celestial mechanics.'42Saint-Venant presented Boussinesq as the savior of the true, physical, 'terrestrial', or'intimate' mechanics:43The sterility that seemed to strike [this mechanics] some twenty years ago and theresulting skepticism in some excellent minds only derived from the manner offormulating questions; one tried to solve the general problem ofeach science, dealingwith it in all generality and all inextricable complication, whereas what needed to bedone was to seek a simplifying cause to permit simple, approximate laws or give ahandle for successive approximations, without which celestial mechanics itself couldnot have been built.Laplace and Poisson excepted, the promoters of physical mechanics had all been engineerstrained at the Ecole Polytechnique and one of the Ecoles d'application.
Boussinesq wasnot. Saint-Venant nevertheless found him to possess a strong sense of the concrete:44Very well versed in analysis . . . his positive spirit yet never rests in abstractions; eventhough he never went through any engineering school, he has passion for the real, theconcrete as presented by our terrestrial world, which is certainly more difficult toknow than the planetary world. The general law of the latter is already known,whereas the study of the other requires that a law be extracted for each subject,and a constant effort to discern the relative magnitudes and orders of approximation.6.3.2 Eaux courantesBoussinesq's biographers report that his interest in hydraulics started early, with thecontemplation of whirls and waves during walks along the River Herault with his highschool teacher.
Some of his early works, published in the Journal de mathematiquespures etappliquees, bore on internal fluid friction. They display the combination of analyticalpower and concrete sense praised by Saint-Venant. For example, Boussinesq solved theproblem of the Poisenille flow in pipes of elliptical, rectangular, and triangular sections,and applied this kind of flow to the phenomena of infiltration and transpiration (inGraham's sense). Before William Thomson, who is usually credited for this discovery,he derived the helicoidal nature of the flow in a curved tube or channel, and used it toexplain the evolution of meanders in rivers.45In these studies Boussinesq confined himself to larninar flow in small sections. Henonetheless gave the most striking illustration of the failure of this assumption in hydraulicpipes and channels: regular flow at ordinary temperature in a semicircular channel wouldyield, for a diameter of one meter and a slope of l o-4, the absurdly high velocity of 187 m/ s42Boussinesq to Saint-Venant, quoted in Picard [1933] p.
14; Boussinesq to Saint-Venant, 21 Apr. 1876,Bibliotheque de I'Institnt.43Saint-Venant [1876].44Saint-Venant [1880] p. 23.45Cf. Douysset [undated] (Herault), Boussinesq [1883].TURBULENCE235for the central fluid filament. In harmony with his mentor's view, Boussinesq concludedthat in this case irregular transverse motions induced a much higher effective velocity thanoccurs in regular flow. Aware of Darcy's and Bazin's relevant measurements, he began athorough investigation of open-channel flow that took the whirling agitation of the waterinto account.46Boussinesq borrowed from Saint-Venant the fundamental idea of a large-scale, effectiveviscosity that depended on the 'intensity of the whirling agitation'. Saint- Venant judgedthe distribution and effect of this agitation to be 'a hopeless enigma'. Similarly, Bazinconcluded his hydraulic research with the pessimistic words:The question grows more complex and obscurejust when new, numerous, and preciseexperiments would be expected to throw a brighter light .
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