Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 50

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Rayleigh thus offered a new escapefrom d'Alembert's paradox. 55Helmholtz believed his short paper on discontinuous fluid motion to be 'of greatimportance,' no doubt because it filled some of the gap between the fundamental equa­tions of hydrodynamics and the fluid motions observed in nature. He regarded thesurfaces of discontinuity and their spiral unrolling as basic features of flow around solidobstacles. His contemporaries were divided on this issue. As we will see in the next chapter,Stokes shared Helmholtz's view that discontinuous flows correctly schematized the be­havior ofnearly-inviscid fluids.

William Thomson, despite his friendship with Helmholtz,believed that they contradicted basic dynamical principles. 564.3.3Pipe blowingHelmholtz's most immediate concern was the blowing of organ pipes (see Fig. 4.4).According to the three first editions of the Tonempfindungen, the air stream from themouth cd of the pipe produces a hissing noise while breaking on its lip ab, and the Fouriercomponents of this noise near the resonance frequency of the tube excite the vibrations ofthe air column. In his hydrodynamic paper of 1 868, Helmholtz mentioned that the trueexplanation of the blowing of organ pipes should instead be based on discontinuous airmotion. The details are found in the fourth edition (1 877) of the Tonempfindungen. 57The mouth of the pipe, Helmholtz explained, produces an air blade that would hit thelip ab if no additional motion intervened.

Now suppose that the air in the tube is alreadyoscillating, with alternating compression and expansion. Owing to this motion, air streamsback and forth perpendicular to the blade and forces it alternately in and out of the tube55Kirchhoff [1869]; Rayleigh [1 876a] (resistance), [1 876b] (vena contracta).

Works on discontinuous fluidmotion are reviewed in Hicks [1 882] pp. 68-71 and Lamb [1895] pp. 100-l l . For a recent assessment of Rayleigh'ssolution to the blade problem, cf. Anderson [1997] pp. 100-6.56Helmholtz to Du Bois-Reymond, 20 Apr. 1868 ('Grundgedanken . . . von grosser Tragweite'), in Kirstenet al. [1986].

See Chapter 5, pp. 197-207.57Helmholtz [1 863a] p. 150; Helmholtz [1 877] pp. 1 54-7, 629-3 1 .166WORLDS OF FLOW(since the corresponding vortex sheets must follow the motion of the air). Owing to theinstability of vortex sheets, the air of the blade 'mixes with the oscillating air of the pipe'.The air from the bellows is thus fed into the tube during the compression phase, whereasthis air avoids the tube during the expansion phase.

For a strong supply of air from thebellows, the blade moves suddenly from one side of the lip to the other because of the highintensity of the deflecting stream. Consequently, the driving force is a crenellated functionof time. If the tube is narrow enough not to suppress higher harmonics, the resultingvibrations of the air column are a saw-shaped function of time, as is the case in bowed­string instruments. This is how Helmholtz justified the namesGamba, Violoncell,andViolonbass,Geigenprincipal, Viola dithat German organ-builders had given to the stops58made of strongly-blown, narrow tubes.We have now gone full circle through Helmholtz's hydrodynamic studies.

Schematic­ally, the insufficiencies of his theory of organ pipes led him to investigate internal frictionand vortex motion, the latter study being a preliminary of the former. One simple caseof vortex motion brought him to discontinuous motions, which turned out to be relevantto the blowing of organ pipes.

Here we find the thematic interconnectivity, and theoscillation between the theoretical and the practical that characterized much of Helm­holtz's work.4.4 Foehn, cyclones, and storms4.4.1Outdoor thermodynamicsOrgan pipes are a simple, small-scale, and man-made device that can be manipulated in thelaboratory and subjected to physico-mathematical analysis. The emergence of modemphysics largely depended on the focus on systems of this kind; so did the later progress ofhydrodynamics.

Yet the study of complex, large-scale, natural systems never came to ahalt. It constituted what Wolfgang Goethe called the 'morphological sciences', includingbotany, comparative anatomy, geology, and meteorology. In these sciences, the applica­tion of general physical laws was rarely attempted and the approach was mostly descrip­59tive and non-mathematical.Toward the mid-nineteenth century, the methodological gap between small- and large­scale physics became smaller, mainly thanks to the efforts of British natural philosophers.One important novelty was the introduction of mimetic experimentation, the imitation inthe laboratory of natural phenomena such as clouds, rain, and thunder. Another was thenew thermodynamics, which controlled the global evolution of complex macro-systemsindependently of their detailed constitution.

Meteorology nevertheless retained its descrip­tive, empirical character. Attempts to subject it to general physical principles were rare.58He!rnholtz [1877] pp. 155-6, 629-30. On p. 630n Helmholtz referred to several anticipations of his notion ofblattformiger Luftstrom-by Heinrich Schneebeli (Luftlamme/le), by Herrnann Smith (air reed), and by W.

Sonrek(Anb/asestrom). Schneebeli [1874] demonstrated the air blade experimentally, by means of movable lips, smokedair, and silk paper, and theoretically justified it through Helmholtz's discontinuous surfaces. Helmholtz did notexplain how the motion in the tube was started. Schneebeli and Smith did so in two different manners (cf. Ellis[1885] pp. 396-7). Ellis mentions that the famous French organ-builder .Aristide Cavaille Coli had presented thenotion of anche fibre aerienne to the French Academy of Science in 1840.59Cf.

Mertz [1965]pp. 200-26.VORTICES1 67As late as 1 890, the American meteorologist Cleveland Abbe still deplored this state ofaffairs: 'Hitherto, the professional meteorologist has too frequently been only an observer,a statistician, an empiricist-rather than a mechanician, mathematician and physicist.' 60Helmholtz loved mountaineering and boating, and was a keen observer of naturalphenomena. From Thomson's yacht, he watched and measured the waves on the sea. Inthe Alps, he scrutinized cloud and storm formations and admired the Mer de glace: 'Atruly magnificent show, this motion, so slow, so constant, and so powerful and irresistible.'Like his British friends Thomson and Tyndall, in these beautiful, sometimes strangephenomena he saw an opportunity to demonstrate the powerful generality of physicallaws.

The Italian physicist Pietro Blasema, who often accompanied Helmholtz on hishikes, recalled: 61He loved to climb mountains and glaciers and to enjoy the wonderful views thatnature generously offers from their heights. He was a strong and confident climber,for whom four to six hours climbing was nothing . . . It was very interesting to walka glacier with him. His eyes were everywhere, and he immediately turned anyremarkable phenomenon or formation that ice could offer into an object of investi­gation.In 1 865, Helmholtz lectured on 'Ice and glaciers', elaborating on James Forbes's andJohn Tyndall's studies.

At the beginning of his talk, he discussed the temperature gradientof the atmosphere, which determines the existence and height of the snow line. This ledhim to a brief thermodynamic explanation of a peculiar meteorological phenomenon, thefoehn. In the first step of this explanation, warm, humid air from the Mediterranean Seaexpands adiabatically while rising over the Alps and thus cools down. Owing to theprecipitations occurring around the summits, the air becomes warmer and drier. Then itflows down the northern side of the Alps, and the resulting adiabatic compression makes iteven warmer and drier.

At that stage it is experienced as the foehn wind. 62This explanation of the foehn was original at the time Helmholtz proposed it. Sinceabout 1 850, Swiss meteorologists believed in a Saharan origin of the foehn. More recently,the leading German meteorologist, Heinrich Dove, had proposed an equatorial origin ofthis wind, against the Swiss evidence for its dryness.

The controversy between Dove andthe Swiss lasted even after the publication of Helmholtz's theory, which was apparentlyignored. Only after Julius Hann independently proposed and powerfully defended thesame theory did meteorologists change their mind. 63Even though the foehn is a minor meteorological phenomenon, the Helmholtz-Hannexplanation of it has broader historical significance, as one of the first successful applica­tions of thermodynamics to the atmosphere.

Although the meteorological importance of60Abbe [1 890] p. 77. On mimetic experimentation, cf. Galison [1997] pp. 80-1, Schaffer [1995]. On the state ofmeteorology, cf. Garber pp. [ 1976] pp. 52-3, Kutzbach [1979] pp. 1-3, a book on which I heavily rely. Other usefulsources are Khrgian [1970], Schneider-Carius [1955], Brush and Landsberg [1985].61Helmholtz [1865] p. 1 1 1; Blaserna, quoted in Koenigsberger [1 902] vol.

2, p. 66.62Hehnholtz [1865] p. 97.OnHelmholtz's glacier theory, cf. Darrigo! [ 1 998] pp. 31-3.63Cf. Kutzbach [1979] pp. 58-62. James Espy had proposed a theory of the foehn similar to Helmholtz'saround 1 840, without success, cf. Khrgian [1970] pp. 1 65-6.WORLDS OF FLOW168adiabatic convection had been known since the 1840s, the process was then analyzed interms of the caloric theory of heat. The first applications of the new thermodynamics toadiabatic convection in the atmosphere occurred in the mid-1860s.

64In 1 862, William Thomson first applied thermodynamics to the rate of temperaturedecline in the atmosphere, and showed that the adiabatic convection of saturated airwould result in a smaller rate than for dry air, in conformance with observations. In1 864, the German mathematician Theodor Reye independently published a thoroughanalysis of the role of the adiabatic expansion of saturated air in the formation ofascending currents, to which we will return shortly. Originally, these considerationsattracted even less attention than the Helmholtz-Hann theory of the foehn.

They becamestandard meteorological knowledge after Reye included them in his influential DieWirbelstiirme, Tornados und Wettersiiulen, published in 1872.654.4.2General circulationHelmholtz returned to meteorology in 1875, in a popular lecture on 'cyclones and storms'.The incentive was probably Reye's book, which he admired for its insights into the role ofadiabatic processes. He may also have been struck by the magnificent pictures of atmos­pheric vortices that Reye provided (see Fig.

4.8). 66At the beginning of his lecture, Helmholtz briefly discussed the difficulty of applying thegeneral laws of physics to atmospheric phenomena. The beholder of a cloudy sky, henoted, could not help feeling that 'the rebellious and absolutely unscientific demon ofchance' was at work. Yet physicists did not doubt that meteorological phenomena obeyedthe laws of hydrodynamics and thermodynamics. The true difficulty, Hehnholtzexplained, was that the very nature of the system forbade detailed predictions:The only natural phenomena that we can pre-calculate and understand in all theirobservable details, are those for which small errors in the input of the calculationbring only small errors in the final result.

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