Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 66
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This layer includes a periodicarray of vortices, as shown in Fig.55.14.The peripheral velocity of the vortices is of orderv / V, where v0 is the maximum velocity of the particles of the plate, and Vis the celerity ofthe two progressive waves of which the standing wave motion of the plate is a superposition. As Rayleigh emphasized, this vortical velocity does not depend on the value of theviscosity v: 'We cannot, therefore, avoid considering this motion by supposing the coefficient of viscosity to be very small, the maintenance of the vortices becoming easier in thesame proportion as the forces tending to produce the vortical motion diminish.'98Rayleigh anticipated a similar singularity of the zero-viscosity limit in the case of planeparallel flow. This view agreed with Reynolds's assertion that intense shear near the wallscaused the instability observed in pipe-flow experiments.
As we will see in Chapter7, in1 921 Ludwig Prandtl described a destabilizing mechanism for plane Poiseuille flow. In1 829 and 1 847, his disciple Waiter Tollmien proved the correctness of this intuition. In1 824, Heisenberg independently derived the instability of plane Poiseuille flow, through amethod of approximation whose validity could only be established much later by ChiaChiao Lin and others. For circular pipes, the flow is probably stable at any Reynoldsnumber, although a complete proof is still lacking.
The latter problem is mathematicallysimilar to plane Couette flow, for which a rigorous proof of stability is now available.Nineteenth-century experts on fluid mechanics did not possess the mathematical techniques that have proven necessary even in the simplest problems of viscous-flow stability.Yet they could anticipate various causes of instability, such as finite disturbances, intenseshear in boundary layers, and irregularity of walls.995.5.6Reynolds's energetic approachReynolds offered a last nineteenth-century approach to parallel-flow instability in a memoir of 1 894.
His reasoning was based on an equation he derived for the variation in time ofthe energy of the eddying motion. He thereby assumed the existence of a macroscopicaveraging scale for which the mean motion no longer involved turbulent eddying. Underthis assumption, the energy of the eddying motion is borrowed inertially from the energy ofthe mean motion and damped by viscous forces. As a stability criterion, Reynolds requiredthe dominance of the damping term of his eddying-energy equation over the inertial term"Rayleigh [1883a] p.
246.99For modern knowledge regarding the stability of parallel flow, cf. Drazin and Reid [1981] pp. 212-13 (planeConette flow), 221 (plane Poiseuille flow), 219 (Poiseuille flow in a circular pipe); also Lin [1966] pp. 1 1-14.INSTABILITY217for any choice of the eddying motion. By laborious calculations he estimated the corresponding Reynolds number in the case of flow between two fixed parallel plates. 100Reynolds's method can at best yield a value of the Reynolds number below which themotion must be stable.
It does not allow one to determine the Reynolds number fromwhich certain perturbations (not necessarily of the random eddying kind) will grow. Thegeneral idea of studying the evolution of the energy of a perturbation of the laminarmotion has nevertheless seduced later students of hydrodynamic instability, includingHendrik Lorentz, William Orr, Theodor von Karman, and Ludwig Prandtl. In somecases, as the Prandtl-Tollmien boundary-layer instability, it provides some physicalunderstanding of the mechanism of instability. 101Thomsen's, Rayleigh's, and Reynolds's mathematical studies of parallel flow show howimpenetrable the caprices of fluid motion could be to the elite of nineteenth-centurymathematical physics.
Where stability was hoped for, for instance in Kelvin's vortexrings, it turned out to be highly improbable. Where instability was observed, for instancein Reynolds's pipes, it turned out to be very hard to prove. The first failure threatened theBritish hope of basing the entirety of physics on the perfect liquid. The second stood in theway of concrete applications of fluid dynamics to hydraulic or aerodynamic processes. Yetthe few mathematical successes obtained in simple, idealized cases, together with inspiredguesses on general fluid behavior, opened a few paths of the modern theory of hydrodynamic instability.In general, the nineteenth-century concern with hydrodynamic stability or instability ledto well-defined, clearly-stated questions on the stability of the solutions of the fundamental hydrodynamic equations (Euler's and Navier's).
Most answers to these questions weretentative, controversial, or plainly wrong. The subject that Rayleigh judged 'second tonone in scientific as well as practical interest' remained utterly confused. Apart from theHelmholtz-Kelvin instability and Rayleigh's inflection theorem, the theoretical yield wasrather modest.
There was Stokes's vague, unproved instability of divergent flows, Thornson's unproved instability of vortex rings, the hanging question of the formation ofdiscontinuity surfaces, and two illusory proofs of stability for simple cases of parallelviscous flow. 102The situation could be compared to number theory, which is reputed for the contrastbetween the simple statements of some of its problems and the enormous difficulty of theirsolution. The parallel becomes even closer if we consider that some nineteenth-centuryproblems of hydrodynamic stability, for example the stability of viscous flow in circularpipes or the stability of viscous flow past obstacles, are yet to be solved, and that the fewavailable answers to such questions were obtained at the price of considerable mathematical efforts. This long persistence of basic questions of fluid mechanics is the more strikingbecause in physics questions tend to change faster than their answers.In number theory, failed demonstrations of famous conjectures sometimes broughtforth novel styles of reasoning, interesting side problems, and even new branches of1 00Reynolds [1 895].
See Chapter 6,'mean-mean-motion'.10 1 Cf. Lin [1 966] pp. 59-63.pp.259-62. What I here call 'mean motion' corresponds to Reynolds's102Rayleigh [1 884b] p. 344.218WORLDS OF FLOWmathematics. Something similar happened in the history of hydrodynamic stability,though to a less spectacular extent. Stokes's and Helrnholtz's surfaces of discontinuitywere used to solve the old problem of thevena contracta and to determine the shape ofliquid jets.
They also permitted Rayleigh's solution(1 876) of d'Alembert's paradox, and(1904). Rayleigh'sinspired some aspects of Ludwig Prandtl's boundary-layer theoryformulation of the stability problem in terms of the real or imaginary character of thefrequency of characteristic perturbation modes is the origin of the modem method ofnormal modes. 1 03As a last important example of fruitful groping, Stokes, Thomson, and Rayleigh allemphasized that the zero-viscosity limit of viscous-fluid behavior could be singular. Stokesregarded this singularity as a symptom of the instability of inviscid, divergent flows;Thomson regarded it as an indication that the formation of unstable states of parallelmotion required finite viscosity; Rayleigh regarded it as a clue to why some states ofparallel motion were stable for zero viscosity and unstable for a small, finite viscosity.Rayleigh even anticipated the modem concept of boundary-layer instability: 104But the impression upon my mind is that the motions calculated above for anabsolutely inviscid liquid may be found inapplicable to a viscid liquid of vanishingviscosity, and that a more complete treatment might even yet indicate instability,perhaps of a local character, in the immediate neighbourhood of the walls, 'when theviscosity is very small.'In the absence of a mathematical proof, such utterances are of dubious value.
Rayleighhimself warned that 'speculations on such a subject in advance of definite arguments arenot worth much.' Many years later, Garrett Birkhoff reflected that speculations wereespecially fragile on systems like fluids that have infinitely many degrees of freedom. Yet,by imagining odd, singular behaviors, the pioneers of hydrodynamic instability avoidedthe temptation to discard the foundation of the field, the Navier-Stokes equation; andthey sometimes indicated fertile directions of research. 1 05Early struggles with hydrodynamic stability are not only interesting for the clues theygive on the later development of this topic; they also reveal fine stylistic differences amongleaders of nineteenth-century physics. Due to the lack of rigorous mathematical solutionsfor the outstanding problems of fluid dynamics, these physicists had to rely on subtle,individual combinations of intuition, past experience or experiment, and improvisedmathematics.
They ascribed different roles to idealizations such as inviscidity, rigidwalls, or infinitely-sharp edges. For instance, Helmholtz and Stokes believed that theperfect liquid provided a correct intuition o f low-viscosity liquid behavior, if only discontinuity surfaces were admitted.
Thomson denied that, and reserved the perfect liquid(without discontinuity) for his sub-dynamics of the universe. As the means to excluderigorously one of these two views were lacking, the protagonists preserved their colorfulidentities.103Kirchhoff [1869]; Rayleigh [1876b]; Prandtl [1905]. Cf.
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