Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 74
Текст из файла (страница 74)
'I am thus driven to admit', he concluded, 'that the most favourableverdict I can ask for the propagation of laminar waves through turbulently movinginviscid liquid is the Scottish verdict of not proven.' 68The Irish ether theorist George Francis FitzGerald embraced Thomson's deductionwithout the worries about vitiating effects. Two years earlier, while reviewing variousmechanical theories ofthe ether, he had introduced the 'vortex-sponge theory' of the etherthat became his foremost philosophical project:It seems certain that the only way in which a perfect liquid can become everywhereendowed with properties analogous to rigidity is by being everywhere in motion.
The66Equations (6.30) and (6.31) generalize the equations (50) and (51) ofThomson [1887e].61/bid. p. 308; Thomson to Stokes, 20 Oct. [1 847], in Wilson [1990]. Cf. Smith and Wise [1989] chap. 12. Therelevance of Thomsen's casual observation is questionable as hot chocolate is hardly a perfect liquid.68Thomson [1 887e] pp. 317, 320. On the instability of vortex motion, see Chapter 5, pp. 191-5.TURBULENCE243most general supposition of this kind would be, that it was what Sir William Thomsonhas called a vortex sponge, i.e.
everywhere endowed with vortex motion, but with thismotion so mixed up as to have within any sensible volume an equal amount of vortexmotion in all directions. There are many ways in which this supposition seems to be inaccordance with what we know of the properties of the ether.Thomson arrived at the vortex sponge in the 1 870s while considering the degradation of acylindrical vortex, in connection with his speculations on vortex atoms. With little proofand strong faith, FitzGerald made this residue of Thomson's matter theory the primitivematerial of the universe.69Impressed by Thomson's deduction of laminar motion in a turbulent fluid, FitzGeraldtried to interpret it in electromagnetic terms.
This was part of the Maxwellian endeavor tofind a mechanical medium whose equations of motion would correspond to Maxwell'sequations. FitzGerald had already done so on the basis of lames MacCullagh's rotationally-elastic medium. The turbulent liquid, or vortex sponge, was a more appealing candidate, for it derived elastic behavior from pure motion. In a first attempt published in 1889,FitzGerald identified the Reynolds stress system v;vj with Maxwell's electromagneticstress system (E;Ei + H;llj for the off-diagonal elements), and the fluid velocity with theelectromagnetic momentum flux E x H (the vectors E and H denote the electric andmagnetic field vectors, respectively).
This works in the case of plane disturbances, theonly ones considered by Thomson and FitzGerald. 70At the close of the century, FitzGerald still believed in the vortex sponge as the ultimatebasis for a theory of ether and matter. He reasserted his conviction that Thomson'svitiating rearrangement was improbable. He upheld the electromagnetic interpretation,though in a different guise; he now compared the Iinearized, plane-disturbance counterparts of eqns (6.21) and (6.30) directly with Maxwell's equations, so that the large-scalevelocity u corresponded to the electric field vector and the Reynolds stress v;vj to themagnetic field.
7 1FitzGerald's analogies between the electromagnetic ether and a turbulent liquid fail forarbitrary, non-plane disturbances. Whether a better analogy of the same kind can befound is an open question. From a historical point of view, it matters only that theturbulent perfect liquid nourished the hopes of a few British ether theorists, in spite ofor because of its inexhaustible complexity.6.5 Reynolds's criterionGirard's discharge experiments with narrow tubes in the 1 8 10s showed that, for a givenhead, the character of pipe flow depended on the diameter of the pipe.
For small diameters, the flow was 'linear', that is, divisible into straight or slightly-curved filaments;69FitzGerald [1885] p. 1 54. Cf. Hunt [1991] pp. 96-107. On the origin of the vortex sponge, see Chapter 5,pp. 194-5.70FitzGerald [1889]. With the extension provided above, FitzGerald's analogy is easily seen to fail in thegeneral case.71 See FitzGerald [!899a], and [1 899b] for a more picturesque expression of the same analogy in terms ofspiraling vortex filaments.WORLDS OF FLOW244the efflux was proportional to the head and it depended strongly on temperature.
For thelarge diameters encountered in hydraulics, the flow was 'complicated', that is, the fluidparticles followed tortuous, variable paths; the efflux was roughly proportional to thesquare root of the head. Girard, Navier, Poiseuille, Darcy, and Bazin all knew of thisdifference. Saint-Venant and Boussinesq first took it into account in theories of hydraulicflow.
Yet none of these investigators examined the transition between the two kinds offlow. Their lack of interest in this question is not surprising: they believed the transitionto depend on accidental circumstances (entrance effect, irregularities of the walls, etc.),and they probably expected it to be gradual, in analogy with Coulomb's study offluid friction. 726.5.
1 Hagen 's transitionIn 1 839, the German hydraulician Gotthilf Hagen accidentally observed the transitionwhile experimenting with small pipes. In order to enhance the effect of wall friction, heused hydraulically unrealistic diameters, of the order of a millimeter. While varying thehead of water h, he observed a sudden change in the efflux for velocities larger than acertain (small) fraction of Vlift. He also noted73an essential change of appearances themselves when this limit was crossed .
. . whichwas very clearly marked in all series of observations: when I let the water flow freelyin the air, for smaller pressure heads the jet had a permanent shape, and near the pipeit looked like a solid piece of glass; but as soon as the velocity, by stronger pressure,exceeded the given limit, the jet started to fluctuate and the efflux was no longeruniform and occurred by pulses.As we saw in Chapter 3, he carefully established that below the turbulence threshold the lossofhead per unit length ofthe tube was proportional to Q/R 4, where Q denotes the efflux andR is the radius of the tube (Hagen-Poiseuille law). In the turbulent case, he found the loss ofhead to be much larger and roughly proportional to the square of the discharge, inconformance with previous hydraulic knowledge.
However, he focused on the regular case,in his view the only one suited to precision measurement. For turbulent flow he believed that'the water [in the tube] lacked the tension [Spannung] necessary for the transmission ofpressure', so that the conditions of motion were inherently underdeterruined.74Some fifteen years later, Hagen carefully studied the effect of temperature on the flow inhis small pipes.
He corroborated his earlier observation that the loss of head dependedstrongly on temperature in the non-turbulent case, whereas it did not in the turbulent case.He also observed that an increase in temperature could induce a transition from nonturbulent to turbulent flow. In his interpretation, the temperature increase implies adiminution of the internal friction of the fluid, which in turn causes the tension of thefluid to vanish at some point and fluctuations to occur. Then part of the pressure head is72Girard [1 8 1 6].
See Chapter 3, pp. 1 04-6.73Hagen [1839] p. 424. Cf. Schiller [1933] pp. 83-4, Rouse and !nee [1957] pp. 1 57-61 . Hagen's criterion for thetransition agrees with Reynolds's, because below it, in the larninar mode, the viscous force is comparable to thepressure head.74Hagen [1839] p. 442.TURBULENCE245lost in the production of 'internal motions, which are induced by the smallest irregularitiesof the walls, or perhaps during the entrance into the tube.'75Hagen insisted on the existence of these internal motions besides the motion measuredby hydraulicians: 76Being reckoned from the efflux, the velocity [of the water] is only measured along theaxis of the pipe and does not take into account internal motions and eddies. Hence itdoes not represent the total motion of the water; it only represents the part of themotion that corresponds to the progression of the whole mass.
Special observations,which I performed with glass tubes, show the two kinds of motion very clearly. As Ilet saw dust enter these tubes along with the water, I observed that for small pressuresthe saw dust propagated only in the direction of the tube, whereas for strongpressures it shot from one side to another and often assumed an eddying motion.Non-German physicists and hydraulicians long remained unaware of Hagen's remarkable observations. Poiseuille is usually credited for the laminar discharge law, and Reynolds for the discovery of the suddenness of the transition to turbulent flow and for itscriterion. The latter attribution does not misrepresent history as much as the foi:mer,because Hagen's research focused on the laws of capillary flow, not on the turbulenttransition.
The study of this transition remained wide open.6.5.2 An eccentricphilosopher-engineerBorn in Belfast in 1842, Osborne Reynoldsentered Queen's College at Cambridge University after completing an apprenticeship in mechanical engineering. He graduatedseventh wrangler in 1867. The following year he obtained the new chair of engineeringat Owens College, Manchester. Reynolds belonged to a new kind of British engineeringprofessors and consultants, well versed in higher mathematics and familiar with recentadvances in fundamental physics.
Throughout his life he maintained a double interest inpractical and philosophical questions.77Reynolds's teaching and research styles seemed highly idiosyncratic to his students andcolleagues. His scientific papers were written in an unusually informal, concrete, andseemingly naive language. At first reading they often seem obscure, but tend to makemore sense after careful study. They rely on astute analogies with previously knownphenomena rather than deductive reasoning. Even though some of these analogies laterproved superficial or misleading, in most cases Reynolds gained valuable insight from them.As he did not bother scanning older literature on his subject, he often duplicated previouslyknown results.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
