Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 48
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Between Helmholtz's first reply and Bertrand'ssecond assault, he suggested to Tait that they should together write a letter to the FrenchAcademy to support Helmholtz. Tait preferred to let Helmholtz 'smash [Bertrand] in hisown way'. Maxwell, who heard of 'Bertrand's refutation' through Thomson, declaredhimself completely confident in the truth of Hehnholtz's theorems and continued exploring their consequences.
In the following years, these theorems became standard knowledgeand found applications to diverse cases of fluid motion, from Thomsen's primitive world-34Bertrand [1 868b].36Helmholtz [1868b].35Saint-Venant [1868b]; Bertrand [1868c].31lbid.; Moigno [J868b]; Bertrand [1868a]. Helmholtz [1 868c].WORLDS OF FLOW1 58fluid to the atmosphere. Reviewing intheorist William Hicks wrote:381 8 8 1 the progress of hydrodynamics, the BritishDuring the last forty years, without doubt, the most important addition to the theoryof fluid motion has been in our knowledge of the properties of that kind of motionwhere the velocities cannot be expressed by means of a potential .
. . Helmholtz firstgave us clear conceptions in his well-known paper [on vortex motion].4.2.5FrictionAs already mentioned, Helmholtz studied vortices in ideal fluids as a step toward includinginternal friction, which implies this kind of motion. We saw in the previous chapter how, inAugust1 859, he rediscovered the Navier-Stokes equation and used it in the analysis of theexperiments he conducted with his student Piotrowski on the damping of the oscillationsof a hollow metallic sphere filled with a viscous fluid.
39In1 869, Helmholtz returned to viscous fluids in a physiological context. Another of hisstudents, Alexis Schklarewsky, had performed experiments on the motion of small suspended particles in capillary tubes, probably with blood circulation in mind. He found thatthe particles moved toward the axis of the tube, in contradiction with a theorem established by Thomson in the case of non-viscous fluids. The anomaly prompted Helmholtz tostudy stationary fluid motion for which the viscous term of the Navier-Stokes equationdominates the nonlinear term. Thus he derived the important theorem according to whichthe real motion is that for which the frictional energy loss is a minimum (for fixedboundary conditions).
Although the result turned out to be irrelevant to Schklarewsky'sexperiments, for which the quadratic terms were non-negligible, the combination ofenergetic and variational principles became an important trait of Helmholtz's theoreticalstyle, as it already was in recent British natural philosophy. The appeal of this method wasthat it determined general properties of physical systems without entering inaccessible ornon-computable structural details.40From an empirical point of view, Helmholtz's most successful discussion of the effectsof viscosity concerned organ pipes. In1 863, he explained the leftover discrepancy betweenhis earlier calculation of the resonance frequency of open pipes and Zamminer's wavelength measurements.
The inclusion of the internal friction term in the equation of motionof the air diminished the propagation velocity in the required proportion. It also explainedwhy the resonance of narrow tubes was not sharp, even though wave reflection at theiropen end was more perfect. Helmholtz further determined the radius of the tube for whichthe sum of viscous and radiation damping was minimal and the resonance was sharpest.This radius is a function of the length of the tube and the mode of resonance.
As theexcitation of a given mode of oscillation by the air from the bellows depends on thesharpness of the corresponding resonance, so does the timbre of the emitted sound. 'Mostsurprisingly', Helmholtz found that his condition of sharpest resonance justified anempirical rule discovered a century earlier by the celebrated organ-maker Andreas Si!ber-38Tait to Helmholtz, 3 Sept. 1 868, HN; Hicks [1882] 63 (quote), pp. 63-8 (detailed review of works on vortexmotion until l 881).39See Chapter 3, p. 144.""Helmholtz [1869].VORTICES159mann, namely, that, for the timbre of a stop to be uniform, the width of the pipes has todecrease with their length, by one-half for the ninth.4 14.3 Vortex sheetsHelmholtz had not yet exhausted the problem of organ pipes.
In a real organ pipe, such asthat shown in Fig. 4.4, the sound is produced by blowing air through the slit cd against theFig. 4.4.Organ pipes. From Helmholtz [1877] p. 149.41Helmholtz [1863b]. Helmholtz only published the results. The full reasoning and calculations are in a long,difficult manuscript ('Die Bewegungsgleichungen der Luft', HN, #582) which Helmholtz intended for publicationin JRAM.WORLDS OF FLOW1 60'lip' ab. Helmholtz wondered by what mechanism this continuous air stream couldproduce an oscillatory motion in the tube. If, he reasoned, the motion of the air in themouth of the tube admits a velocity potential, then this motion should be the same as theflow of an electric current in a uniformly-conducting medium that replaces the air.However, the latter motion is known to be smooth, steady, and progressive, without theoscillations observed in the acoustic case.42Helmholtz knew other cases offluid motions that looked very different from electric flow.As he had seen while expelling smoke with a bellow or by blowing air across a candle flame,when air flows into a wider space through a sharply-delimited opening, it forms a compactjet that gradually spreads into vortices (see Fig.
4.5), whereas electricity would flow in everydirection. He did not regard internal friction as a plausible explanation for this phenomenon, for he believed it could only smooth out the flow. Instead, he turned his attention to thetangential discontinuities of motion that he knew to be possible in a non-viscous fluid.434.3.1Discontinuous motionHydrodynamicists had generally assumed the continuity of velocity, which the differentialcharacter of Euler's equations seemed to require.
The only exception was Stokes, in briefsuggestions for solving paradoxes of fluid motion (see Chapter 5). Helmholtz arrived at hisconcept of discontinuity surfaces independently, through his study of vortex motion. AsFig. 4.5.Smokejets, with (a) spontaneous instability, and (b) sound-triggered instability. From Becker andMassaro [1968]; plate 1.
Courtesy of Henry Becker.42Helmholtz [1868dj p. 147.43Ibid. pp. 146-50.VORTICES161we saw earlier, a thin vortex sheet implies a discontinuity of the parallel component of thevelocity. Conversely, any possible discontinuity is reducible to a vortex sheet. Helmholtzused this representation to derive some basic properties of discontinuity surfaces.44According to one of Helmholtz's theorems on vortex motion, the vortex filaments of thesheet must move together with the fluid particles. Let, for instance, the fluid move with thevelocity v on one side of a plane and be at rest on the other side. Within the sheet the fluidmoves on average with the velocity v/2. Therefore the sheet must move paraUel to itselfwith the velocity vj2. The vortex-sheet picture also gives an immediate intuition of theeffects of viscosity on a discontinuity surface.
By internal friction, the rotating particles ofeach vortex filament gradually set into rotation the neighboring particles. Consequently,the sheet grows into a row of finite-size whirls. 45Most importantly, surfaces of discontinuity are highly unstable. Helmholtz referred toexperiments by his friend John Tyndall that showed the astonishing sensitivity of a jet ofsmoky air to sound waves. 'Theory', Helmholtz went on, 'allows us to recognize thatwherever an irregularity is formed on the surface of an otherwise stationary jet, thisirrregularity must lead to a progressive spiral unrolling of the corresponding portion ofthe surface, which portion, moreover, slides along the jet.' This peculiar instability ofdiscontinuity surfaces was essential to Helmholtz's uses of them. Yet Helmholtz neverexplained how it resulted from theory.
The behavior of discontinuity surfaces under smallperturbations is a difficult problem which is still the object of mathematical research. Ascan be judged from manuscript sources, Helmholtz probably used qualitative reasoning ofthe following kind.46Consider a plane surface of discontinuity, with opposite flows on each side. In this case,the velocity field is completely determined by the corresponding vortex sheet. Let a smallirrotational velocity perturbation cause a protrusion of the surface. The distribution ofvorticity on the deformed surface is assumed to be approximately uniform.
Then thecurvature of the vortex sheet implies a drift of vorticity along it, at a rate proportionalto the algebraic value of the curvature. Indeed, at a given point of the vortex sheet thevelocity induced by the neighboring vortex filaments is the vector sum of the velocitiesinduced by symmetric pairs ofneighboring filaments, and each pair contributes a tangential velocity as shown in Fig. 4.6(a). Consequently, vorticity grows around the inflectionpoint on the right-hand side of the protrusion, and diminishes around the inflection pointon the left-hand side of the protrusion (see Fig.
4.6(b) ). The excess of vorticity on theright-hand slope induces a clockwise, rotating motion of the tip of the protrusion (seeFig. 4.6(c)). The upward component of this motion implies instability. The rightward44Discontinuous motions are not possible in the (stationary) electric case because the current is irrotational in auniform conductor. Helmholtz may already have been working on discontinuous fluid motion in 1 862, for at thattime he asked Thomson about the potential near a 'Kante', which was relevant to Helmholtz's argument for theformation of discontintuities (Thomson to Helmholtz, 29 Nov. 1 862, HN).45Helmholtz [1868d] pp. 1 5 1-2.46Ibid. pp.
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