Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 76
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86As Reynolds came to realize, this simple and still popular explanation of the radiometerleads to a serious paradox. Consider two infinite parallel plates immersed in a gas, withdifferent temperatures of their facing sides and equal temperature of their external sides. IfReynolds's simple theory held, the forces acting on the two plates should be different, sincethe temperature asymmetries are different for the two plates.
However, the equality ofaction and reaction implies that these two forces should be equal and opposite. In order toescape from this paradox, Reynolds introduced a finite extension of the plates. With themilitary analogy of two batteries of guns, he explained how the oblique shots near thecorners of the plates disturbed the balance of forces, because for them the recoil of a gunon one plate was not necessarily compensated by the impact of a bullet on the other.87For a consideration ofthis sort to work in the radiometer case, the mean free path ofthemolecules has to be of the same order as the breadth of the vanes.
This explains why aradiometer only works for a very small pressure of the residual gas. Reynolds furthersurmised that a radiometer with very tiny vanes would work at ordinary pressures. Suchan experiment being practically impossible, he turned to the 'inverse phenomenon', that is,the gas motion induced by the temperature gradient of the vanes.
This suggested a thermalcounterpart to the 'transpiration' of a gas through a porous plug that Thomas Grahamhad investigated half a century earlier. Reynolds justified 'thermal transpiration' theoretically through an extension of Maxwell's kinetic theory of gases that included stresses inthermally heterogeneous gases, and experimentally by extending Graham's experiments totemperature gradients.88Graham had noted that the law of transpiration became different for very small pores,but fell short of a theoretical conclusion. In contrast, Reynolds argued that the change oflaw occurred when the size of the pores became of the order of the mean free path.
In thekinetic molecular conception of a gas, transpiration should only depend on the ratiobetween these two quantities. Reynolds therefore expected the transpiration curves to behomothetic whenever the product of the density of the gas and the diameter of the poreswas the same. This he verified by means of a logarithmic plot of his data. He believed that86Reynolds [1874b], [1876]. On early theories of the radiometer, cf.
Everitt [1974] pp. 224-5.87Reynolds [1 879] pp. 304-5 (paradox), 306-9 (gun batteries).88Ibid. p. 261.TURBULENCE249he had thus reached an 'absolute experimental demonstration that gas possesses a heterogeneous structure. ' 896.5.5 Guessing the criterionIn his works on transpiration and kinetic theory, Reynolds wrote as a Cambridge-trainednatural philosopher. He even enjoyed criticism from his hero James Clerk Maxwell, whohad his own theory of stresses in rarefied gases. Yet the engineer was always alive inReynolds.
While philosophizing on the distinction between capillary and ballistic flow, heremembered the existence of a similar distinction between capillary and hydraulic flow, orbetween 'direct' and 'sinuous' motion in his terms. He called for a similar dimensionalanalysis in this case:90As there is no such thing as absolute space and time recognised in mechanicalphilosophy, to suppose that the character of motion of fluids in any way dependedon absolute size or absolute velocity, would be to suppose such motion without thepale of the laws of motion.
If then fluids in their motion are subject to these laws,what appears to be the dependence of the character of the motion on the absolute sizeof the tube, and on the absolute velocity of the immersed body, must in reality be adependence on the size of the tube as compared with the size of some other object,and on the velocity of the body as compared with some 'other velocity'.In the case of pipe flow, the relevant theory was hydrodynamics based on the NavierStokes equation (with uniform viscosity). By the 1 870s, the validity of this equation forcapillary and small-scale motion was established, and its failure for larger-scale motionwas blamed on unknown or uncontrollable circumstances of the motion rather than on theform of the equation itself. In his Royal Institution lecture on vortex motion, Reynoldscriticized the deductive approach to hydrodynamics, but not its fundamental equation.
Ashe remembered in 1 883, 'the equations of [fluid] motion had been subjected to such closescrutiny, particularly by Professor Stokes, that there was small chance of discoveringanything new or faulty in them.' Moreover, Reynolds knew that in the case of gasesMaxwell had provided a kinetic-theoretical derivation of the Navier-Stokes equation.91Reynolds noted that, after the elimination of pressure from this equation (by taking itscurl), the time derivative of the vorticity w = V x v had two terms:8w11- = V x (v x w) + -Llw.8tp(6.32)If the motion depends on a single velocity parameter U and on a single linear parameter L,then the first term has the 'factor' U2 IL2 and the second CP-1p)(UIL3).
Hence Reynolds89Ibid. p. 259.90Reynolds [1883] p. 53. The present reconstruction agrees with Reynold's statement, ibid. p. 54: 'It is alwaysdifficult to trace the dependence of one idea on another. But it may be noticed that no idea of dimensionalproperties, as indicated by the dependence of the character of motion on the size of the tube and the velocity of thefluid, occurred to me until after the completion of my investigation of the law of transpiration of gases, in whichwas established the dependence of the law of transpiration on the relation between the size of the channel and themean range of the gaseous molecules.'91Reynolds [1 883] pp.
54-5. On Stokes's opinion, cf., e.g., Stokes [1 850b].250WORLDS OF FLOWconcluded that 'the birth of eddies depend[ed] on some definite value of [LUpjp.].' Thisdimensionless number is what Arnold Sommerfeld later called the Reynolds number.926.5.6Testing the criterionReynolds tested his theory with two kinds of experiment. In the first, performed in 1 8 80,he injected ink at the center of the conical entrance of a glass tube in a tank of still,isothermal water.
He controlled the water flow by a valve at the lower end of the tube.A preliminary small-scale trial failed to show the transition because the maximal velocityof the flow was too small. Reynolds therefore ordered the larger apparatus of Fig. 6.6.
Fora slow flow, the injected ink formed a steady band along the axis of the tube (see Fig.6.7(a)). At a certain value of the velocity, and at some distance from the entrance, 'thecolour band appeared to expand and mix with the water so as to fill the remainder of thepipe with a coloured cloud' (see Fig. 6.7(b)).
By moving the eye so as to follow the motionof the water, or under a flash of light, the cloud appeared to be made of two or three wavesfollowed by distinct eddies (see Fig. 6.7(c)).93To test his criterion for the transition, Reynolds varied the diameter of the tube and thetemperature of the water. The latter amounted to a variation of viscosity, calculablethrough a formula by Poiseuille. The experiments proved to be difficult, because thetransition occurred suddenly and the slightest disturbance of the water entering the tubelowered the critical velocity. Reynolds had to wait several hours to reach sufficientequilibrium before each run.
Instead of heating up the water, he cooled it with ice so asto minimize convection currents. He thus managed to establish that the critical velocitywas proportional to the diameter of the tube and inversely proportional to the viscosity.This confirmed the theoretical criterion, with a Reynolds number of 6415 and the radius ofthe tube as the characteristic length.946.5.7The two kinds of instabilityThe simple reasoning behind this criterion suggested that the instability of the fluid did notdepend on the size of the disturbances, since it made the relative size of the two terms ofeqn (6.32) depend only on the breadth and the velocity of the flow. The observations onpipe flow sharply contradicted this expectation. The water appeared to be in an unstablestate with respect to finite disturbances well before the critical point was reached. Reynolds verified this condition by artificially inducing a finite disturbance with a wire placedin the tube (see Fig.
6.7(d)). He also noted that, for narrow tubes and velocities slightlyabove the critical point, the eddying occurred in a series of distinct 'flashes' (see Fig.6.7(e)). As Reynolds put it,the critical velocity was very sensitive to disturbance in the water before entering thetubes . . . This showed that the steady motion was unstable for large disturbances92Ibid. p. 55; Reynolds to Stokes, 25 Apr. 1883, in Stokes [1907], vol.93Reynolds [1883] pp.
59-61 , 68-77, 72 (quote).I,pp. 232-3; Sommerfeld [1908] p. 599.94Reynolds [1883] pp. 44, 73-5. Later investigators found much higher critical numbers. According to Drazinand Reid ([1981] p. 216), the Poiseuille flow in a cylindrical pipe is probably stable (no rigorous proof exists), butthere is an instability owing to the boundary layer that forms at the entrance of the pipe.TURBULENCEFig. 6.6.251Reynolds' apparatus for studying the turbulent transition of the flow of water in a tube (fromReynolds [1883] p.
71). Water from the tank enters the horizontal glass tube through the conical funnel. Thevalve with the long handle on the right controls the flux, whose value is inferred from the lowering of thefloater. Ink from the flask is injected continuously in the middle of the entrance of the tube.long before the critical velocity was reached, a fact which agreed with the full-blownmanner in which the eddies appeared.Indeed, the latter observation indicated a kind of snowball effect, a higher instabilityinduced by the disturbances that appeared at the critical velocity. 95Reynolds found this result the more surprising because other kinds of flow agreed withthe expected behavior, namely, the transition from direct to sinuous motion independentof the size of disturbances, and the gradual divergence from direct motion above thecritical velocity.
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