Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 96
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They soon realized that this circulation profilewas the one for which the drag was a minimum for a given lift. Lagrange's variationalmethod leads to a simple demonstration of this fact (not Prandtl's and Betz's original one).Denoting byAthe Lagrange parameter for the constraint of constant lift, the minimumdrag corresponds to8D - A8L = pThe relationJ+aIPw + (w- A V) 8f] dx = 0.(7.47)(7.42) between w and r further leads toJ+a-ar 8 w dx =J+a-aw ar dx,(7.48)by analogy with the symmetry of mutual inductance coefficients.125 Hence the vanishing ofthe integral in eqn.(7.47) for an arbitrary variation ar requires w to be a constant.
Thisonly happens for the elliptic circulation profile.As Prandtl noted, the corresponding pattern of the induced flow in a vertical planecontaining the wing is that of a horizontal plate suddenly set into motion with thedownward velocity w, for this is the only irrotational flow that satisfies the boundaryconditions (see Fig.7.23). Amazingly, this flow is exactly the one on which Lanchesterbased his elementary, intuitive reasoning! 1 26With this treatment of the elliptically-loaded lifting line, Prandtl had in hand the basicelements of his wing theory. During the war, Betz and another outstanding collaborator,Max Munk, helped Prandtl solve the following problems.(i) Determine the form and size of the sections of the wing that produce a givencirculationf(x).(ii) Determine the circulation f(x) and the corresponding drag and lift for a given shapeof the wing.The first problem is easily solved by noting that, according to the Kutta-Joukowskitheory, the circulation around a given section of the wing has the formf(x)=(ae' + {3) Vl(x)(7.49)to first order in the effective angle of attack e', if l(x) denotes the chord of the section at theabscissax.To first order in the induced velocityw(x),the effective angle of attack differs12'Prandtl [1918] p.
342, [192la] pp. 478-80. Hints to these results are in Betz [1914].125Munk obtained this relation in 1918, cf. Prandtl [192la]126Prandtl [1921a] pp. 464-5.p.489.321DRAG AND LIFTfrom the real angle of attack e by the amount -wjV. Taking into account the relation(7.42) between r and w, Prandtl obtained the equationr(x) + l(x) ..:!_47TJ+a-ar'(s)(s - x)-1 ds =(o:e + f3) Vl(x).(7.50)Problem (i), that is, the determination of the chord l(x) for a given circulation r(x), onlyinvolves a simple integration.
In contrast, the inverse problem (ii) involves an integrodifferential equation that required the full skills of Betz and Munk. 1 27By the end of the war, Prandtl and his collaborators could legitimately claim a mathematical, quantitative solution of the wing problem. The only leftover task was to justifythe various approximations that Prandtl had introduced at various steps of the reasoning.For this purpose, Prandtl started his memoir of 1918 with the exact, general equations ofthe problem.
For a given wing at a given inclination, a first equation gives the velocity fieldas a function of the asymptotic velocity V, the trailing vortex sheet, and fictitious boundvortices that replace the boundary conditions on the wing. Reciprocally, the trailing vortexsheet depends on the velocity field through two conditions, namely, that the vortexfilaments must be lines of flow for the velocity field, and their intensity must be given bythe gradient of the circulation along the span of the wing. In principle, this mutualcoupling should determine both the vortex sheet and the velocity field, and a threedimensional generalization of the Kutta-Joukowski theorem then gives the force actingon the wingY8In practice, various approximations must be made. Treating the circulation and theinduced velocities as small quantities, Prandtl argued that, in a first (linear) approximation, the vortex sheet was parallel to the unperturbed flow V and the correspondingvelocity field simply added to the unperturbed flow in the force formula.
He also arguedthat, in the calculation of this first-order induced velocity field, the aerofoil could bereplaced by a line of vorticity r(x)-hence the name 'lifting-line theory' now given to hiswing theory. Lastly, he argued that along most of the span the motion could be regarded asbeing approximately two-dimensional, which makes the circulation a function of the angleof attack and the sectional form only.1 29Although Prandtl's justifications for these assumptions lacked rigor, experiments performed during the war in the Giittingen wind tunnel vindicated them. Post-war British andAmerican experiments further confirmed Prandtl's theory.
The purely empirical methodsof early aeronautics gradually made room for refined theoretical considerations. Inparticular, Prandtl and his group computed the effect that the walls of the tunnel had onthe vortex trail of the wings, and subtracted it from raw model data in order to improvefull-scale predictions.
After some hesitation on the British side, by the mid-1920s this'Prandtl correction' became a routine procedure in any wind-tunnel experiment.1301 27Prandt1 [1918] pp. 339-40. [192Ja] pp. 484-7.128Prandtl [1918] pp. 329-35.129Ibid. pp. 335-9. Prandtl ([1918] pp. 336) used the words tragender Faden and tragende Linie.1 30Cf. Anderson [1997] pp.
292-4. On the Prandtl correction, cf. Hashimoto [2000] pp. 231-5.322WORLDS OF FLOWNo matter how much it owes to Lanchester's intuitive mechanics, to Kutta's conformaltransformations, or to Joukowski's interest in bird flight, the Gottingen wing theory maybe seen as a splendid application of Helmholtz's theory of vortex motion, includingdiscontinuity surfaces and conformal methods. As in the case of boundary-layer theory,Prandtl astutely combined and extended nineteenth-century concepts through intuitivepictures related to asymptotic approximations.
Under the stimulus of the rising field ofaeronautics and with the strong support of Gottingen institutions, his group put an end tothe engineers' legitimate distrust of the theoretical predictions of fluid mechanics.8CONCLUSIONHydrodynamics evolved considerably in the course of its application to various phenomena. So did all major theories of mathematical physics. The myths that make Newton thesole creator of mechanics, Cauchy the father of the theory of elasticity, Clausius thefounder of thermodynamics, and Maxwell the unique inventor of modem electrodynamicsdo not resist historical analysis. These theories have changed so much since the firstformulations of their fundamental principles and equations that a modem physicist whoreads the mythical founders can barely recognize a kinship with present theories.
Thisestrangement is not limited to notations and styles of presentation, but runs very deeplyinto the conceptual structure of the theory.In many cases, these structural changes have occurred during attempts to apply thetheory to a specific class of phenomena. For example, William Rowan Hamilton's attemptto apply mechanics to light rays led to the Hamiltonian formulation of mechanics;Charles-Eugene Delaunay's application of the same theory to the motion of the Moonyielded a new perturbation theory based on action and angle variables; Saint-Venant'sapplication of the theory of elasticity to the flexion and torsion of prisms produced thesemi-inverse method of approximation; the application of thermodynamics to mixturesand chemical reactions led to the concept of thermodynamic potential; Hendrik Lorentz'sapplication of Maxwell's electrodynamics to certain optical phenomena led him to separate ether and matter; the application of quantum mechanics to solid-state physics engendered the theory of bands; and its application to field-mediated interactions promptedRichard Feynman's path-integral formulation.In this small sample, four kinds of theory change are involved.
In an order of increasingmagnitude, they imply new methods of resolution or approximation (Saint-Venant, Delaunay), new derived concepts (thermodynamic potentials, bands), a reformulation of thefoundations (Hamilton, Feynman), and the replacement of a basic principle (Lorentz).Although such innovations are most frequent during the early applications of a theory,they may occur many years later. They affect the very life of the theory, that is, the class ofproblems to which it is believed to be relevant, the communities that use it, theway it is taught,its conceptual hierarchy, the attached paradigms, and its relationships to other theories.Such wide-ranging feedback effects of application are rarely acknowledged.
Mostcommonly, applications are regarded as 'runs' of a theory, for utilitarian purposes or fortransmitting implicit knowledge to students. According to Thomas Kuhn, applicationscontribute to the smooth, gradual expansion, and consolidation of normal science. Significant conceptual change can only result from the accumulation of major anomalies, inwhich case a global revolution occurs and a new paradigm emerges. 1 The above-cited1 See, e.g., Kuhn [1961].324WORLDS OF FLOWexamples of application-induced change fit neither the smooth paradigmatic phase nor therevolutionary one.
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