Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 64
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7774Rayleigh [1879]. See also Rayleigh [1896] pp. 362-5.75Rayleigh [1880] pp. 474-83.76Ibid. pp. 483-4.17Ibid.pp.484-7.(5.20)WORLDS OF FLOW210The stability equation has the form v" + av = 0, with..,U"U - a"/k(5.21)a = -le- - --- .J lv' l2 dy + J aivf dyMultiplying by the complex conjugate v* of v, and integrating from wall to wall gives=0.J Im(a) lvl2 dy = 0,(5.22)Hence, the imaginary part of the function a must satisfy the conditionorIm(a")2J I U -lvlu/kl2U" dy = 0.(5.23)(5.24)If the sign of U" is constant (and if the perturbation v does not uniformly vanish), then theintegral is nonzero, so that the imaginary part of £T must vanish and the perturbationcannot grow exponentially.
Rayleigh concluded that parallel flow without inflexion of thevelocity profile was stable. As he noted, the criterion is of no help in the jet case for whichU" changes sign. 785.5.2Reynolds's instabilitiesIn this discussion of parallel flow between fixed walls, Rayleigh probably had in mind atwo-dimensional approach to the stability of pipe flow.79 Yet he did not discuss thisapplication, presumably because of the lack of relevant experiments. As we will see inthe next chapter, Osborne Reynolds filled this gap in 1 883 with a thorough study of thetransition between 'direct' and 'sinuous' flow in straight circular pipes. Reynolds had theturbulent eddying in his pipes depend on an excess of the inertial term of the NavierStokes equation over the viscous term. When the flow depends on only one characteristiclength L (the pipe diameter) and on the average velocity V, the ratio between the two termsis governed by the ratio LV I v, where v is the kinematic viscosity J.k/p.
This ratio is nowcalled the Reynolds number. 80Through color-band experiments, Reynolds verified that the critical transitiondepended on this number. He thereby noticed the surprisingly sudden character of thistransition: violent eddying occurred as soon as the critical Reynolds number was reached.Moreover, the flow appeared to be unstable with respect to finite perturbations well beforethe critical number was reached: 8178Rayleigh [1 880] p.
487. Rayleigh also gave (without proot) the criterion in the cylindrical case that 'therotation either continually increases or continually decreases in passing outwards from the axis.'"Rayleigh states this in RSP 3, p. 576.80Reynolds [1883] pp. 54-5. A more detailed account will be given in Chapter 6, pp.
249-52.81 /bid. p. 61. Also, ibidpp. 75-6: 'The fact that the steady motion breaks down suddenly, shows that the fluid isin a state of instability for disturbances of the magnitude which cause it to break down. But the fact that in someINSTABILITY211The critical velocity was very sensitive to disturbance iu the water before entering thetubes . . . This showed that the steady motion was unstable for large disturbanceslong before the critical velocity was reached, a fact which agreed with the full-blownmanner in which the eddies appeared.From casual observations o f conflicting streams of water, Reynolds was aware o f theexistence of another kind of instability for which the transition from direct to sinuousmotion was gradual and independent of the size of the disturbances.
In his memoir of1883, he recounted an elegant experiment in which he had a lighter fluid slide over aheavier one with a variable velocity difference. For a certain critical velocity, the separating surface began to oscillate. The waves then grew with the sliding velocity, until theycurled and broke. 82Reynolds was unaware of relevant theoretical considerations by Helmholtz, Kelvin, andRayleigh. He was therefore 'anxious' to find a theoretical explanation of the two kinds ofinstabilities he had encountered.
He first studied the stability of the solutions of Euler'sequation, with the result that 'flow in one direction was stable, flow in opposite directionsunstable.' As he could only imagine a stabilizing effect of viscosity, the instability of pipeflow puzzled him for a long time. At last, he attempted a similar study in the more difficultcase of the Navier-Stokes equation. He then found that the boundary condition for viscousfluids (vanishing velocity at the walls) implied instability for sufficiently-small values of theviscosity: 'Although the tendency of internal viscosity of the fluid is to render direct orsteady motion stable, yet owing to the boundary condition resulting from the friction at thesolid surface, the motion of the fluid, irrespective of viscosity, would be unstable.'83Reynolds never published his stability calculations. He could conceivably have handledthe inviscid case in a manner similar to Rayleigh's, although the roughness of his statementof the criterion suggests some erring.
That he could derive a boundary-layer instability inthe viscous case seems highly implausible, considering the subtlety of the later considerations of that sort by Prandtl, Heisenberg, and Tollmien.845.5.3Thomson 's proofs of stability in viscous casesIn his presidential address to the British Association meeting of 1 884, Rayleigh praisedReynolds's contribution to the study of the transition between laminar and turbulent flow.His view of the future of the subject was singularly optimistic: 'In spite of the difficultieswhich beset both the theoretical and the experimental treatment, we may hope to attainbefore long to a better understanding of a subject which is certainly second to none inscientific as well as practical interest.' It is likely that he and Stokes were responsible forthe subject of the Adams prize for 1889: 'On the criterion of the stability and instability ofcondition it will break down for a large disturbance, while it is stable for a smaller disturbance, shows that there isa certain residual stability, so long as the disturbances do not exceed a given amount .
. . It was a matter of surpriseto me to see the sudden force with which the eddies sprang into existence, showing a highly unstable condition tohave existed at the time the steady motion broke down.-This at once suggested the idea that the condition mightbe one of instability for disturbances of a certain magnitude, and stable for small disturbances.'82Jbid.
pp. 61-2.83/bid. pp. 62-3.840n the later considerations, cf. Drazin and Reid [1981] chap. 4, and Chapter 7, pp. 294-6.212WORLDS OF FLOWthe motion of a viscous fluid'. After a reference to Reynolds's work, the announcement ofthe prize read: 85It is required either to determine generally the mathematical criterion of stability, orto find from theory the value [of the critical Reynolds number] in some simple case orcases. For instance, the case might be taken of steady motion in two dimensionsbetween two fixed planes, or that of a simple shear between two planes, one at restand one in motion.The only theorist to claim success in solving these two cases was no beginner in need ofthe £170 prize; it was Sir William Thomson.
86 In the second case (plane Couette flow), 87the simpler one because of its constant vorticity, Thomson provided a fairly explicitprocedure for deriving the evolution of an arbitrary small perturbation of the flow.From the Navier-Stokes equation and the incompressibility condition, he first obtainedthe linearized equation(�+u.!._-v!:>.) !:>.vexatv,=0•(5.25)which only contains the second component,of the velocity perturbation of the basicflow U = f3y (the y-axis being perpendicular to the plates, and the x-axis being parallel tothe motion ofthe moving plate). As he astutely noted, this equation and those for the othercomponents u and w can be solved explicitly for any initial value of the perturbed velocitywhich is compatible with the incompressibility condition, if only the real boundarycondition (vanishing relative velocity on the plates) is replaced with the sole condition ofvanishing normal velocity at the plates for t = 0.
This may be called the relaxed solution.Thomson next used Fourier analysis to find the 'forced solution' of the linearizedequations for which the velocity perturbation on the plates was a given function of time,vanishing for negative time and opposite to the velocity of the relaxed solution on theplates for positive time. As Thomson believed the latter solution to vanish identically fornegative time, he regarded the sum of the relaxed and the forced solutions to be therequested solution of the real initial-value problem. The relaxed solution is easily seen todecrease exponentially in time. This implies the same behavior for the forced and thecomplete solutions.
Thomson concluded that the simple shear flow of the prize questionwas stable.In the other case of the Adams prize (plane Poiseuille flow), Thomson could no longerobtain the relaxed solution. Instead, he directly applied Fourier analysis to the real initial"Rayleigh [1 884b] p. 344; G. Taylor, G. H. Darwin, G.
G. Stokes, and Lord Rayleigh (examiners), 'TheAdams Prize, Cambridge University', PM 24 (1887), pp. 142-3.86Thomson [1 887c]. According to The Cambridge review 9 (1 889), p. 156, the prize was not adjudged in defaultof candidates.87The Couette flow is the steady viscous flow between two concentric parallel cylinders, one of which isrotating at a constant speed.
Following a suggestion by Max Margules, in 1890 Maurice Couette measured theviscosity of various fluids from the torque exerted on a cylinder immersed in the fluid contained in a rotating,coaxial cylinder (Couette [1 890a]). This method permitted a better control of pressure (in the gas case), betterprecision, and a wider range of velocity gradients than Coulomb's and Maxwell's earlier methods (thus permittinga more extensive confirmation of the Navier-Stokes equation). Couette [1890b] described the instability of thisflow beyond a critical velocity of the rotating cylinder.213INSTABILITYvalue problem. He seems to have believed that both the boundary condition and the initialcondition could be satisfied by superposing Fourier components of the formf(y)ei(a-r+kx+=l , where the frequencies u, k, and m are real numbers.
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