Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 78
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He gave a precise characterizationof the transition to turbulence for pipe flow with respect to eddying and the discharge law,he defmed the two relevant critical points and their relation to the Reynolds number of the1 04Saint-Venant [185la]; also Woltman [1791-1799], and Hagen [1854].WORLDS OF FLOW256.. f.<.Fig. 6. 1 1 .....-- -tP "r..e01- .Reynolds's log-log plot of head loss versus fluid velocity (Reyno!ds [1883] p. 94).flow, he introduced the distinction between two kinds of hydrodynamic instability, and hegave the now canonical form of the turbulent discharge law in a smooth pipe. However,some of the discoveries usually credited to him were not his.
Hagen already knew ofthe correlation between eddying and the discharge law, as well as the suddenness of thetransition to turbulent flow. Saint-Venant, Hagen, and earlier German hydraulicians hadused discharge laws with fractional powers. 105Reynolds did not provide the dimensional analysis for which he is most famous. He didnot discuss the scaling invariance properties of the Navier-Stokes equation, even thoughhis stability criterion and his discharge laws reflect this invariance.
When Stokes readReynolds's theoretical derivation of the criterion (article6 of the 1883 memoir), he saw init an argmnent of dynamical similarity. Reynolds mildly protested: 'I had no intentionwhatever of laying down the conditions of dynamical similarity, although I now see thatArt.6 not only bears this construction but really fails to express what I meant.' What hetruly meant was a comparison of the magnitude of viscous and inertial forces. 1 06Stokes had noted the similarity property of the Navier-Stokes equation in the context ofhis pendulum studies. There he found that the equation was invariant under a change ofspace scale(L), time scale (T), viscosity (J.L),and density(p)if the ratio(L2/T)/( J.L/p)1 05For blasting criticism of Reynolds's priority claims and concomitant praise of Saint�Venant and Boussinesq, see Knibbs [1897].1 06Reynolds to Stokes, 25 Apr. 1883, in Stokes [1907], vol.
I, p. 232.257TURBULENCEremained the same. This implies the similarity offlows related by the same change. From thisproperty, Stokes inferred that the mass correction of the pendulum, already known todepend on the density of the fluid around it, also had to depend on the pendulum's period.
1 07Stokes expected the similarity to hold only in the case of regular motions for which theNavier-Stokes equation effectively determined the flow. This explains his assessment ofReynolds's contribution: 1 08Professor Reynolds has shown that the same conditions of similarity hold good, as tothe average effect, even when the motion is of the eddying kind; and moreover that ifin one system the motion is on the border between steady and eddying, on anothersystem it will also be on the border, provided the system satisfies the above conditionsof dynamical as well as geometrical similarity.
This is a matter of great practicalimportance, because the resistance to the flow of water in channels and conduitsusually depends mainly on the formation of eddies; and though we cannot determinemathematically the actual resistance, yet the application of the above propositionleads to a formula for the flow, in which there is a most material reduction in thenumber of constants for the determination of which we are obliged to have recourseto experiment.Through this influential reading of Reynolds's contribution, Reynolds found himselfcredited for strategies that he did not use.
Helmholtz was the first physicist, in1 860, toapply the similarity condition of the Navier-Stokes equation to pipe flow. He showed thatD of the pipe's diameter, v of the fluid'sAP I AL of the pressure gradient, would also be linear for the values nD, vln,and APIt.Ln3 of these quantities. In 1 873, the Prussian Ministry asked him to investigatea flow known to be 'linear' for certain valuesvelocity, andthe problem of the steering of balloons.
Since the aerial motion around the rudder was toocomplex to be calculated, he decided to lay out the conditions for the rational use of smallscale experiments. As in Stokes's reinterpretation of Reynolds's memoir, Helmholtzassumed the similarity condition to extend to turbulent motion. 1 09The use of the similarity condition to restrict the form of the resistance law was LordRayleigh's invention. Reynolds discovered the restricted form empirically, due to thelogarithmic plotting of his data.
Rayleigh reasoned by dimensional analysis. If the densityp, the kinematic viscosity v, the velocity U, and the diameter D entirely determine the flow,then dimensional homogeneity requires that the resistance APIAL (pressure fall per unitlength) should have the form (p U2 1D)!f!(DUiv) or (pv2ID3)<f>(DUiv). This is to becompared with Reynolds's relation (6.33), which implies the form (gpvl I�)f(D Ulv)for the resistance (since APIAL = ipg). The occurrence of the acceleration of gravity inthe latter formula confirms Reynolds's neglect of dimensional analysis. In contrast,Rayleigh had been using dimensional analysis since the beginning of his career. He nevermissed an opportunity to denounce its neglect and to emphasize the severe constraint it1 07Stokes [1850a] p.
17. Stokes's analysis did not include the (v · \7)v term.108Stokes's statement in presenting the Royal Medal to Reynolds. in Stokes [1907] vol. 1, pp. 233-4.109Helmho1tz and Piotrowski [1860] p. 173; He1mholtz [1873]. In the same period William Froude applied adifferent similitude argument to naval construction, see later on pp. 278-9. Bertrand [1 848] had discussedmechanical similarity in general.WORLDS OF FLOW258placed on the laws of fluid resistance.
As we will see in the next chapter, dimensional andsimilitude arguments gradually pervaded fluid mechanicsY 06.5.1 1Calor bands, magic cubes, and military disciplineThe calor-band experiments on the two manners of fluid motion lent themselves to brilliantpublic shows. Reynolds did not miss this opportunity. In 1 884, at the Royal Institution, heclaimed a partial verification of his earlier prediction that 'the method of coloured bandswould reveal clues to those mysteries of fluid motion that had baffled philosophy.' Besidesthe transition from direct to sinuous flow in straight pipes, he showed instabilities occurringin jets and in diverging pipes.
John Tyndall had already displayed the sensitivity of smokejets to sound in the same room. Helmholtz had enunciated the theoretical instability of theimplied surfaces ofdiscontinuity. Reynolds showed the transitory formation ofa vortex ringwhen the flow came on gradually, the stabilizing effect of viscosity for slow motion, and theexistence of a critical velocity for which the jet became unstable. 1 1 1Reynolds also offered a few spectacular analogies for the effects o f hidden motion. Atthe beginning of his lectures, he showed the oscillations of two visually identical cubessuspended on springs. One cube obeyed the laws of mechanics, while the other apparentlyviolated them. Reynolds then revealed to his audience that a gyrostat had been mountedinside the second cube.
In a humorous allusion to Newton's apple, he argued that the lawsof mechanics would have been as impenetrable as the laws of fluid motion if apples hadhidden internal motion. 1 1 2To illustrate the dependence of the manner of fluid motion on viscosity, size, andvelocity, Reynolds resorted to an analogy with a disciplined troop.
If the disciplineis respected, then the motion of the troop should be regular. Yet the march may turninto a whirling, struggling mob under an external disturbance. This instability clearlyincreases with the velocity of the march, with the size of the troop, and with the difficultyof the maneuver ordered; it decreases if the discipline is reinforced. In a real fluid,discipline corresponds to viscosity, and the difficulty of the maneuver to the boundaryconditions.113In 1 893, Reynolds gave his last calor-band show at the Royal Institution. This time hedistinguished between two kinds of internal fluid motion. Firstly, wave-like motions arepossible (in an incompressible fluid) when the fluid has a free surface.
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