Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 91
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Hence the average velocity must have the form(7.26)where k and K are two universal numerical constants. The pressure gradient, which isbalanced by the integral of the wall stress ro over the perimeter of the pipe, is thusproportional to the square of the velocity, in conformance with the usual assumptionmade by hydraulicians.If the walls are smooth, then the velocity formula (7 .24) can only hold at a sufficientlylarge distance from them.
Near the walls the flow is controlled by the viscosity v. Fordimensional reasons, the thickness of this viscous sublayer must be of the order vvPJTand the velocity at the border of this layer must be of the order v:rJP. Assuming thatformula (7.24) begins to apply at this border, Kan:min required that(7.27)where a is a numerical constant.
Hence the velocity profile must have the form(7.28)where f3 and y are two nmnerical constants. The resulting average velocity has the form(7.29)where k and K' are two nmnerical constants. Karman found excellent agreement with thelatest pipe-retardation data provided by the Gottingen experimentalist Johann Nikuradse.Since this epoch-making paper, the problem of pipe retardation is reduced to the empiricaldetermination of the two numerical constants k and K' . As Karman later remembered, hissubsequent communication at the third international congress of applied mechanics inStockhohn signaled his victory in a tacit competition with his mentor:85I came to realize that ever since I had come to Aachen my old professor and I were in akind of world competition.
The competition was gentlemanly, of course. But it wasfirst-class rivalry nonetheless, a kind of Olympic Games, between Prandtl and me, and84Karman ([1930] p. 61) noted this property, but did not exploit it in the rest of his calculations. The followingreasoning is found in Karman [1932] p. 409.85Karman [1930] pp. 69-72, [1967] p. 135 (quote).Cf. Battimelli [1984] pp. 86-92.DRAG AND LIFT301beyond that between Gottingen and Aachen. The 'playing field' was the Congress ofApplied Mechanics.
Our 'ball' was the search for a universal law of turbulence.In the report of his Tokyo lectures, presumably written after he saw Karman's paper,Prandtl admitted the logarithmic velocity profile (7.22) that he had originally rejectedbecause of its divergence on the wall. At a sufficient distance from the wall, he nowreasoned, the mixing length must have the form Cy because, in the absence of viscosity,y is the only relevant length. Hence the profile must be logarithmic. Next to the wall theflow is laminar and the mixing length must be v/Pfi.
In 1 833, Prandtl added that, in thecase of the rough wall, the natural choice I = a + Cy immediately leads to Karman'svelocity profile (7 .25). In the case of a smooth wall, he directly replaced the roughnesswith the length v/Pfi and thus obtained a formula similar to Karman's profile (7.28).86Commenting on Karman's 'very much noticed paper', Prandtl noted that his andKarman's approach coincided only for a constant stress r.87 For a variable r, bothapproaches become more arbitrary: Prandtl's does not take into account another characteristic length of the problem, which is r/(drjdy), while Karman's overlooks derivatives ofu of order higher than two. Fortunately, most applications only require knowledge of thevelocity profile in regions of approximately constant r. Prandtl's approach is then recommended, since it is the simpler one. Prandtl attributed this simplicity to his focus on themixing length as the main parameter of turbulent momentum transport.
Yet he could alsohave reasoned directly in terms of Boussinesq's eddy viscosity e. The only expression ofthis parameter that can be built from y, p, and r is Kyy'/Yi', where K is a dimensionlessconstant. Then the relation r = e dujdy leads to dujdy = (1/Ky)/Pfi, from whichPrandtl derived the logarithmic profile.In subsequent years, Karman's and Prandtl's derivations of the velocity profile of aturbulent boundary layer were improved in various manners. It was understood that theassumption of an overlap region between the turbulent layer and the laminar sublayersufficed to establish the logarithmic form of the velocity profile, and more precise estimates of the numerical constants were given.
From a practical point of view, the discoveryof the logarithmic profile of turbulent boundary layers marked the successful completionof Prandtl's program for determining fluid resistance at high Reynolds numbers. Since18 14, it was clear that the resistance of well-designed airships, airfoils, and ship hulls, aswell as hydraulic pipe retardation, depended on the formation of turbulent boundarylayers. By 1930, the relevant wall stress could be computed directly from the logarithmicvelocity profile in the hydraulic case, and indirectly via Karman's momentum equation inthe nautical and aeronautical cases.
From an academic, Giittingen-centered activity,boundary-layer theory gradually evolved into a widely-known procedure for determiningfluid resistance in the real world. 88We may now reflect on the reasons for this success. In their major advances on the fluidresistance problem, Prandtl and his disciples relied on the nineteenth-century key conceptsof discontinuity, similitude, instability, and mixing. However, they transcended the original use of these concepts in various manners. Whereas earlier users of Helrnholtz's86Prandtl [1931a], [1933].87Prandtl [1933] p. 827."cr. Tani[1977] pp.
102-3.302WORLDS OF FLOWsurface of discontinuity reasoned in a purely Eulerian context, Prandtl extracted part of thebehavior of these surfaces from local, high-Reynolds-number approximations of theNavier-Stokes equation. Whereas previous similitude arguments by Stokes, Helmholtz,Rayleigh, and Froude were confined to the interpretation of model measurements and tothe dimensional homogeneity of resistance formulas, Prandtl and Karman brought them tobear on the internal processes of a system: they saw that in some circumstances differentparts of the system only differed in scale. Whereas Rayleigh's and Kelvin's theories ofparallel-flow instability had no practical import, Prandtl, Tietjens, and Tollmien showedthat properly completed and applied to boundary layers they bore on crucial mechanismsof fluid resistance and retardation.
Whereas Boussinesq and Reynolds remained unable toquantify the mixing process that they regarded as the essence of turbulence, Karman's andPrandtl's insights into the similitude properties of this process led to accurate laws of piperetardation and turbulent-boundary-layer resistance.Prandtl's extraordinary ability at combining and extending received theoretical concepts within a coherent, productive picture did not completely solve the resistance problem, however. When it comes to separated flow, today's physicist can predict little morethan Saint-Venant did in the mid-nineteenth century.
Prandtl only told us how to avoidseparation, so that the resistance be small and computable through the boundary-layerapproximation. Fortunately, except for parachutes or braking flaps, low resistance is mostfrequently desired in technical applications.7.4 Wing theoryIn the 1 890s, interest in flying contraptions grew tremendously, partly as a consequence ofOtto Lilienthal's invention of the man-carrying glider in 1 889 (see Fig. 7.18). The prospects of building a motor-powered, piloted airplane seemed high in some engineeringquarters.
They materialized in 1903 when Wilbur and Orville Wright flew the first machineof that kind. Theory played almost no part in this spectacular success. Analogies withflying animals, experiments with models, and broad engineering ability were all theinventors needed. Although the most learned of them, Samuel Langley, contributedimportant measurements of lift and drag, refuted the Newtonian sin2 () dependence onthe incidence angle, and even noted that this law would made artificial flight nearlyimpossible, he still refrained from higher hydrodynamic theory.
89The contemporary flight frenzy nonetheless prompted theoretical comments and reflections, ranging from flat rejection to elaborate support. Most negative was Lord Kelvin,who refused an invitation to join the Aeronautical Society of London with the comment:'I have not the smallest molecule of faith in aerial navigation other than ballooning or ofexpectation of good results from any of the trials we hear of.' Lord Rayleigh was far morefavorable. Commenting on Langley's inclined-plane measurements, he noted qualitativeagreement with the formula he had derived in 1 876 on the basis of Helmholtz's discontinuity surfaces; he tentatively ascribed the remaining quantitative disagreement to aviscosity-driven suction at the rear of the plate; and he agreed with Langley that theresults justified optimism for the possibility of mechanical flight.
Rayleigh also appliedenergy and momentum considerations to a global understanding of the conditions of89Cf. Gibbs-Smith [1960]. Anderson [1997].DRAG AND LIFTFig. 7.18.303Otto Lilienthal on a biplane glider in 1895. From Deutsches Museum collection.flight, insisting on 'the vicarious principle' that 'if the bird does not fall, something elsemust fall' (a downward air current). However, he did not attempt any detailed theory ofthe flow around the wings of flying objects. 90Rayleigh presumably believed that his and K.irchhoff's solution of the two-dimensionalinclined-plate problem offered a general explanation of the existence of lift. Indeed,discontinuity surfaces and dead water not only solved d'Alembert's paradox, but theyalso made the resistance perpendicular to the plate, which implies a finite lifting component when the plate is moving horizontally (see Fig.
7.19). Yet Rayleigh knew of a specialcase of fluid resistance in which the reaction was normal to the velocity of the movingbody, that is, a pure lift without drag. In 1853, Gustav Magnus had explained the longknown deviation of spinning bullets by an induced whirlwind. The superposition of thewhirling motion with that resulting from the translational motion of the ball impliesdifferent fluid velocities on the two sides of the ball, as indicated in Fig. 7.20.
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