Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 63
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The modem reader may wonder which of the protagonists wasright, and whether they anticipated later insights into low-viscosity fluid behavior. Here isa brief answer to these ahistorical questions.Consider first the formation of discontinuity surfaces. As Stokes correctly argued, noneof the theorems invoked by Thomson prohibits the formation of such surfaces, even in the66Stokes to Thomson, 30 Dec.
1898, ST. The modern reader may recognize Prandtl's separation process for theboundary layer.67Stokes to Thomson, 5 Jan. 1 899, 19-20 Dec. 1900 (quote), ST.INSTABILITY207absence of a sharp edge. These theorems presuppose the continuity of the motion. Forexample, the demonstration of Lagrange's theorem requires the finiteness of the term(w \7)v in the vorticity equation and therefore the continuity of the velocity.
6 8 If the flowis continuous at a given time, then it remains so at subsequent times. If, however, a tinysurface of discontinuity is grafted onto the wall, then Helmholtz's theorems and theelectromagnetic analogy imply that it should grow at a rate given by the velocity discontinuity at its origin, with a spiral unrolling of its extremity. 69In order that the discontinuity be finite, the fluid should be stagnant at one side of theorigin of the discontinuity surface, and move continuously on the other side.
Consequently, the surface must depart tangentially from the wall (in the case of an edge, it is,at any time, tangent to one side of the edge). As far as Marcel Brillouin and Felix Kleincould see, there is nothing in Euler's equations that contradicts this growth process.Neither is there anything in this equation that restricts the points from which an embryonic surface would grow (at least in the two-dimensional case). In summary, in an Eulerianfluid surfaces of discontinuity can be formed as Stokes wished, but their departure point ismore arbitrary than experiments on real fluids would suggest.
70Another important issue of the Stokes-Thomson debate is the connection betweeninviscid and viscous behavior. According to Ludwig Prandtl's later views, at high Reynolds numbers the flow of a real fluid along a solid obstacle is irrotational beyond a thinboundary layer of intense shear.
Unless the solid is specially streamlined, this layerseparates from the body at some point (line) of its profile. The resulting flow resemblesthe surfaces of discontinuity imagined by Stokes for the Eulerian fluid. However, theseparation point can only be determined through the Navier-Stokes equation (eventhough it does not depend on the value of the viscosity parameter!). Hence Stokes wasright to expect a resemblance between the low-viscosity limit of real flows and discontinuous Eulerian flow; but Thomson was also right to lend viscosity a decisive role in formingthe thin vortex layers that imitate discontinuity surfaces.71·68This is emphasized in Stokes [1 849a] pp.
106-13.69Jacques Hadamard ([1903] pp. 355-61) gave a proof that surfaces of discontinuity could not be formed in aperfect fluid as long as cavitation is excluded. This proof, however, does not exclude the growth of a pre-existing, tinysurface of discontinuity. Marcel Brillouin [1911] made this point, described the growth process, and extended theconformal methods of Helmholtz, Kirchhoff, and Levi-Civita to curved obstacles devoid of angular points. FelixK1ein [1910] described the evolution of a surface of discontinuity formed by immersing an infinitely-thin blade(concretely, a rudder) perpendicularly to the liquid surface, pulling it at uniform speed in the direction of its normal,and suddenly withdrawing it.
He resolved the apparent contradiction between Helmholtz's vorticity theoremsand the formation of discontinuity processes as follows: 'Clearly the source of [the contradiction] is that wehave now admitted the confluence of two originally separated fluid masses, whereas the usual foundation of thetheorem presupposes that fluid particles that once belonged to the surface of the fluid must indefinitely belong tothis surface.'70Brillouin [191 1]; Klein [1910]. According to Brillouin, in the two-dimensional case the departure point of asteady surface of discontinuity must be beyond a certain point of the surface of the body.71 Prandtl [1905].
For a viscous fluid, separation is not an instability issue. However, it is so in the ideal fluidcase according to Stokes.208WORLDS OF FLOW5.5 Parallel flow5.5.1 From dancingflames to the inflection theoremIn the course of his acoustic studies, the London professor John Tyndall heard about thesensitivity of flames to sound that his American colleague John Le Conte had observed ata gas-lit musical party. The flames from 'fish-tail' gas burners danced gracefully as themusicians played a Beethoven trio, so that 'a deaf man might have seen the harmony'. In1867, Tyndall displayed this strange phenomenon at the Royal Institution, as well as asimilar effect with smoke jets, and published an account in the Philosophical magazine.When subjected to various sounds, the jet shortened to form a stem with a thick bushyhead (see Fig.
5.12). The length of the stem depended on the pitch, and high-pitch noteswere ineffective. Tyndall made this instability the true cause of the dancing of flames, buthe did not propose any theoretical explanation. 72Tyndall's work attracted Lord Rayleigh's attention. This country gentleman had anuncommon disposition for physics, both mathematical and experimental.
Coached byEdward Routh and inspired by Stokes's lectures at Cambridge, he emerged as seniorwrangler and Smith's Prizeman in 1 866. Until his appointment as Cavendish Professor onMaxwell's death (1879), his main research interests were in optics and acoustics. Hiselegant and masterful Theory of sound, first published in 1 877, became one of the fundamental treatises of British physics, and remains an important reference to this day.73Rayleigh, the theorist of sound, was naturally interested in Tyndall's observations aswell as in Felix Savart's and Joseph Plateau's earlier experiments on the sound-triggeredinstability of water jets.
In the latter case, the determining factor is the capillarity of thewater surface, which favors a varicose shape of the jet and its subsequent disintegrationinto detached masses whose aggregate surface is less than that of the original cylinder. In1879, Rayleigh determined the condition for the growth of an infinitesimal sinusoidalperturbation of the jet surface, as Thomson had done in the case of wind over water.
Healso gave a theory of smoke-jet instability, in even closer analogy to Thomson's wavetheory. The relevant instability is that of a cylindrical surface of discontinuity for the air'sFig. 5. 12.irvSmoke jets subjected to sounds of various pitch. From Tyndall (1 867] p. 385.72Le Conte (1858] p. 235; Tyndall (1867].73Cf. Lindsay [1976].209INSTABILITYmotion. Neglecting capillarity, Rayleigh showed that, on a jet of velocity V, a sinusoidalperturbation with the spatial period A grew as e Vt/!t .
74This result contradicted Tyndall's observation that short sound waves were ineffective.Rayleigh traced the discrepancy to the viscosity of the air. In the case of two-dimensionalparallel motion, the Navier-Stokes equation implies that the vorticity w evolves accordingto the equationowat!!:.= 6.w(5.17)P(the convective terms vanish), so that vorticity is 'conducted' through the fluid accordingto the same laws as heat. Consequently, any vortex sheet or discontinuity surface evolvesinto a layer of vorticity of finite thickness.
Rayleigh then examined the stability of a finitelayer of uniform vorticity. Switching off viscosity, he found that the layer became stablewhen its thickness somewhat exceeded the wavelength of the perturbation. This resultmade it likely that viscosity, by smoothing out the velocity discontinuity, should stabilize ajet for high-pitched sounds. 75After thus resolving the discrepancy between fluid mechanics and Tyndall's experiments,Rayleigh proceeded to the theoretically similar problem of two-dimensional parallel flowbetween fixed walls. He first studied the stability of successive finite layers of uniformvorticity with perturbed separating surfaces, using Helmholtz's analogy between vorticityand electric current.
The result suggested that, for a continuous variation of the vorticity w,stability would depend on the constancy of the sign of the variation dw/ dy between the twowalls. In other words, the curvature d2 Ujdy2 of the velocity profile could not change sign. 76Rayleigh then offered the more direct approach to the stability problem that has nowbecome standard.
Let Ox denote an axis parallel to the flow, Oy the perpendicular axis,U(y) the original velocity, and u(x, y) and v(x, y) the components of a small velocityperturbation . The vorticity equation givesowowow(U )v = 0,fii + + u OX + oy(5.18)with(5.19)Retaining only first-order terms in u and v, assuming that u and v vary as ei(kx-m) , andeliminating u by means of the continuity equation 8u/8x 8vj8y = 0, Rayleigh reached+the stability equation(U_?:_) (82v - k!v) - d2 U v = 0.k8y2dy2He derived his stability criterion in the following ingenious manner.
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