Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 79
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In this case, 'thecolour bands, however much they may be distorted, cannot be relatively displaced,twisted, or curled up, and in this case motion in water once set up continues almostwithout resistance.' In the other class of internal motion, the colored band becomesthoroughly mixed with the rest of the water. As in cooking, wool spinning, and metalrolling, the mixing process involves 'folding, piling, and wrapping, by which the attenu-1 10Rayleigh [1892]. See also Rayleigh [1904], [1909] (plate resistance), [1915b] (many examples).
Cf. Rott[1992], Roche [1998] pp. 208-9. Although similarity arguments (Helmholtz) and dimensional analysis (Rayleigh)are not quite the same, they are intimately related to each other: in one case the scale of the actual system ischanged, in the other the scale of units is changed.111Reynolds [1 884] pp. 154 (quote), 1 58-9 Gets).mIbid.p. 154.mIbid.pp.
155-6.TURBULENCE259ated layers are brought together.' Such mixing occurs when waves break at the free fluidsurface, or when the fluid slides over a solid surface with sufficient velocity. It explains theease with which stirring brings uniformity in a liquid, compared to the slowness ofmolecular diffusion. 1 14Lastly, Reynolds showed the precise mechanism by which the motion of a spoon or vaneachieved mixing, namely, the creation of a spiraling vortex sheet behind the vane, togetherwith wave motion outside the sheet.
The theory of the teacup was not the sole ambition ofthis displayY5We can hope to interpret the parallel of the vortex wrapped up in the wave, as appliedto the wind of heaven, and the grand phenomenon of the clouds, as well as thosethings which directly concern us, such as the resistance of ships.6.5.12The philosophy of scalesReynolds intended his experiments, especially the color-band shows, to suggest newtheoretical insights. Conversely, his experimental work depended on theoretical analysis.As we saw, his original analysis of pipe discharge depended on his theoretical understanding of the instabilities that led to turbulent motion. In his memoir of1883, he mentionedthat he had spent many months studying the infinitesimal perturbation of stationarysolutions to the Euler and the Navier-Stokes equations.
Yet he never published hiscalculations, perhaps because two giants of British hydrodynamics, Lord Kelvin andLord Rayleigh, controlled the field.1 1 6Reynolds's idea that the dimensionless numberUDjv determined the character of thefluid motion also derived from a theoretical consideration, namely, comparing the inertialand viscous terms in the vorticity equation.
In1894,Reynolds published a purely theoretical memoir on this criterion. Instead of the vorticity equation, he now considered theenergy distribution at various scales. His inspiration came from an analogy with thekinetic theory of gases and also from Stokes's analysis of dissipation in a viscous fluid.In1 850, after being converted to Joule's concept of heat as a kind of motion, Stokesexplained that the internal work of viscous stresses implied a continual conversion of theobservable fluid motion into the microscopic motion called heat.
1 1 7Stokes derived the rate of this dissipation from the Navier-Stokes equation. I n areformulation inspired by Maxwell's identification of stress with momentum flux, Reynolds wrote this equation as(6.35)with(6.36)" 'Ibid. p. 534."4Reynolds [1893] pp.
529, 53 1."'Reynolds [1883] p. 63. Cf. earlier pp. 253-4 Chapter 5, p. 2 1 ! .11 7Reynolds [1895); Stokes [1850b] pp. 67-70.260WORLDS OF FLOWThe part Pii of this tensor gives the viscous stresses and corresponds to molecular momentum transfer. The rest corresponds to molar momentum transfer. 1 18The convective derivative of the macroscopic kinetic energy density ( l/2)pv2 follows:(6.37)After integration over an internal portion of the fluid, the first term yields the work of thepressures acting on the portion's surface.
For an incompressible fluid, the second termshould therefore represent the 'vis viva consumed by internal friction' and 'converted intoheat'. The expression(6.38)of the stress system implies that this term is always negative, in conformance with itsdissipative character. 1 19As Reynolds correctly pointed out, this derivation assumes the possibility of discriminating between small-scale motion to be identified with heat, and larger-scale motion to bedescribed by the Navier-Stokes equation. Since the distinction involves some arbitrariness, Stokes's interpretation of the dissipative term in his energy balance is not entirelycompelling.
Part of the small-scale internal motion produced by dissipation might bedifferent from heat. 120Reynolds's lengthy discussion of this point boils down to the simple requirement thatthe average velocity v in the Navier-Stokes equation should vary very slowly at the scale ofthe averaging length and time. Concretely, if v(r) represents the spatial average of thelower-scale velocity in a domain of radius R centered on the point r, then the variation ofthe function v(r) should be very small over the length R. As Reynolds emphasized, thisproperty depends on the nature of the mechanical system.
Reynolds hoped that theelucidation of this 'discriminative cause' would clarify the 'hitherto obscure . . . connection between thermodynamics and the principles of mechanics.'1216.5.1 3Proving the criterionIf the requirement of smooth averages is met at one scale, then it will be for anyneighboring scale for which the variation of averages remains slow.
This defines a certainscale band of consistent averaging. The equation of motion is approximately the same forany scale in this band, and implies the same dissipation rate. The Navier-Stokes equationcorresponds to a first scale band. From hydraulics and from previous observations of1 18Reynolds [1895] p. 544. Of course, Reynolds did not use the tensor notation. FitzGerald ([1889] p.
254)introduced the momentum-transfer form of the Euler equation.1 1 9Stokes [1850b] p. 70; Reynolds [1895] p. 545.1 20Ibid. p. 546.121Ibid. pp. 537--44, 547-50, 560 (quote). Similarly, Burbury's and Boltzmann's contemporary notion ofmolecular chaos allowed for equations of large-scale evolution (the Boltzmann equation) that no longer referredto the low-scale, molecular dynamics.TURBULENCE261turbulent flows, Reynolds assumed a similar band to exist at a much higher scale, of thesame order of magnitude as the 'pulse' and spatial 'period' of turbulence. 122This scale is also the averaging scale that Saint-Venant and Boussinesq had introducedin their theories.
Reynolds, who had not read them, called the corresponding averagemotion the 'mean-mean-motion', for it could be obtained by averaging the 'mean-motion'of the Navier-Stokes equation, which itself comes from averaging the molecular motion.Whereas Saint-Venant determined the equation of mean-mean-motion by a symmetryargument, Reynolds relied on an analogy with the kinetic theory of gases. In1866,Maxwell derived the Navier-Stokes equation for the mean-motion by averaging overmolecular processes.
Similarly, Reynolds obtained his equation of mean-mean-motionby averaging the Navier-Stokes equation.123Specifically, Reynolds divided the mean-motion v into the mean-mean-motion v and the'relative-mean-motion' v'. Averaging the formleads to(6.35) of the Navier-Stokes equation then(6.39)This equation has the same form as the Navier-Stokes equation, save for the additionalstress system-pv;vj. The latter stress corresponds to momentum transfer between consecutive layers of the fluid through the relative-mean-motion, whereas the viscous stress Piicorresponds to momentum transfer through molecular motion.
For large Reynolds numbers, the viscous stres.s is negligible.124Having no way to determine the statistical behavior of the relative-mean-motion,Reynolds could not reach a more determinate form of the equation of mean-mean-motion.He could, however, discuss energy transfers in a manner similar to Stokes. In the counterpart of the earlier balanceitional term-pv;0a;vj,(6.37), the additional stress system-pv;0 implies an addcorresponding to the conversion rate of mean-mean-motionenergy into relative-mean-motion energy.125Another equation can be written for the convective variation of the kinetic energy of therelative-mean-motion:(6.40)withpij = Pii - Pii for the viscous stress of the relative-mean-motion.
Once integrated overa fluid portion, at the borders of which the relative-mean-motion vanishes, this equationmeans that the variation in the energy of the eddying motion in this portion is the sum oftwo terms, namely, a negative one corresponding to the conversion of eddying motion intoheat, and a positive one corresponding to the conversion of mean-mean-motion intoeddying motion.1261 22Reynolds [1895] pp. 538, 551-3.1 25Reynolds [1895] p. 555.123Ibid.
pp. 553-4.126Ibid. p. 556.124Ibid. p. 554.262WORLDS OF FLOWReynolds applied the latter balance to the determination of the Reynolds number belowwhich turbulent flow becomes impossible, namely, the damping term dominates in theabove equation for any relative-mean-motion that is compatible with the Navier-Stokesequation. With much laborious calculation, the best Reynolds could do was to determine alower limit (51 7) of this number for the flow between two fixed parallel plates.127Reynolds's memoir should not be judged only for the 'determination of the criterion' promised in its title.
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