Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 89
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In 1 9 1 1, Prandtl's brilliant, Hungarianborn student Theodore von Karman understood that the succession of vortices producedby the instability of the separation surface could only be stable if the vortices werearranged according to the double-alternating row of Fig. 7.14, where the distanceh between the two rows is a definite fraction (0.283) of the spacing l between two successive vortices. As Rayleigh later saw, this periodic shedding of vortices explains the'Aeolian harp' heard by sailors when strong wind blows past the shrouds of a mast.Unfortunately, this is about the ouly case where something simple can be said about aturbulent wake.6865Cf.
Rotta [1990] p. 9 (Klein's comment, reported by Sommerfeld), Hanle [1982] chap. 3 (Klein's project),chap. 4 (Prandtl's call).66For histories of boundary-layer theory, cf. Tani [1977], Dryden [1955]. For a critical assessment of separationprediction, cf. Batchelor [1967] pp. 325-9.67B!asius [1908].
Cf. Tani [1977], Ackroyd et al. [2001] chap. 1 1.68Karman [1911]; Rayleigh [1915a]. In 1908, Henri Benard had published a careful experimental study of whatis now known as the 'K:irman vortex street'. Cf. K:irman [1954] pp. 67-72.DRAG AND LIFTFig. 7 . 14.293Lines of flow for Kinnin's vortex street, (a) theoretical, and (b) experimental. From Kirmin[19l l], and Prandtl [1931b] p.
133.7.3.5Turbulent layersIn 1 9 1 3, while trying to verify the U312 resistance law for a plate, Blasius found that thislaminar law ceased to be valid for a critical Reynolds number Uljv of about 450000,beyond which turbulence occurred in the boundary layer and the resistance becameproportional to uL864, in conformance with Beaufoy's and Froude's earlier measurements.
The following year, Prandtl used the turbulent boundary layer to explain a strangeanomaly in experiments performed by Gustave Eiffel on spheres suspended in a windtunnel. Against any received theory, Eiffel found a sudden diminution of the resistance ofhis spheres beyond a certain critical velocity. Prandtl suspected that at that point thelaminar boundary layer became unstable before the (laminar) separation point, and thatthe resulting eddies 'washed away the thin wedge of quiet air behind this point', thusretarding the separation of the flow. He succeeded in visualizing this effect with smoke inthe Gottingen wind tunnel, but found a higher critical velocity than that measured in Paris.To explain this last anomaly, he noted that Eiffel's flow-homogenizing device causedturbulence of the incoming air and thus induced an earlier transition of the boundarylayer from laminar to turbulent.
Lastly, he confirmed this view by showing that aturbulence-inducing wire attached around a parallel of the sphere similarly retarded theseparation of the flow (see Fig. 7.15).69As Prandtl immediately saw, in the case of airships and airplanes, the boundary layeralways becomes turbulent before the laminar separation point. Consequently, the trueseparation point is very close to the rear end of the flying body, the global flow is nearlypotential except in a narrow wake, and most of the resistance is frictional (unless there isalso drag-related, induced resistance).
Prandtl liked to emphasize the paradoxical role ofturbulence in this felicitous cancellation of eddy resistance: 'It is precisely these turbulentflows of low resistance around bodies that can be so closely represented by the theory of aperfect fluid.' At the break of World War I, Prandtl worried that the boundary layer mightnot be turbulent in some model experiments, which would jeopardize predictions of fullscale resistance. Fortunately, he found this was generally not the case for the elongatedbodies that imitated zeppelins or airfoils.7069B!asius [1913] pp. 25-7; Eiffel [1912]; Prandtl [1914] p. 600.70Prandtl [1914] pp.
605-8, [1927] p. 773.WORLDS OF FLOW294{b)Fig. 7 .15.Separated flow around a sphere: (a) with laminar boundary layer, and (b) with turbulence inducedin the boundary layer through the wire a From Prandtl [1914] p. 605; [1926] p. 720.7.3.6 InstabilitiesDespite the high technical importance of turbulent boundary layers, Prandtl held backtheir theoretical study.
His priority of the 1910s was wing theory, for which it wassufficient to know that separation only occurred at the rear edge of the wing and thatthe flow was laminar and potential everywhere except in the boundary layer. As thefrictional resistance of the wing could be evaluated from the measured flat-surface friction,its theory could be postponed. Prandtl began his theoretical investigation of turbulence in1921, with the onset of turbulence in Poiseuille flow and in boundary layers.
7 1At that time, the received wisdom was that Poiseuille flow was always stable under aninfmitesimal perturbation, but unstable with regard to finite perturbations. Prandtl's.ownexperiments on the critical transition of open-channel flow contradicted this view, as theyshowed that growing wave-like oscillations next to the walls preceded the transition toturbulence.
With Oskar Tietjens's help, Prandtl examined the stability of non-viscous,71 Prandtl [192lb].295DRAG AND LIFTparallel, two-dimensional flow under an infinitesimal perturbation. In agreement withRayleigh's theorem of 1 880, he found that an inflection of the velocity profile led toinstability. As Prandtl had known since 1904, the evolution of a boundary layer alongthe wall leads to an inflected profile beyond a certain point (see Fig.
7.10). At thatpoint, the boundary layer should become unstable, as long as viscous damping does notprevent the growth of perturbations. 72Prandtl and Tietjens next took into account viscosity, which leads to the much moredifficult problem of Kelvin and Orr (in two dimensions). In order to simplify the calculation, they replaced the continuously-curved profile with the broken profiles of Fig. 7 .16.They found instability even for convex profiles, and for any value ofthe Reynolds number.As this was more instability than Prandtl wanted-pipe flow has to be stable for asufficiently high viscosity-Prandtl surmised that the broken-profile idealization was notpermitted.
In the real, continuously-curved case, he knew from Rayleigh that, at points ofthe velocity profile for which the celerity of the sine-wave perturbation equals the flowvelocity, a special kind of motion occurs, namely the Kelvin 'cat-eye' pattern of Fig. 7.17.Prandtl suspected a connection with the wave-like behavior he had observed as a preludeto turbulence in channel flow.73In the absence of viscosity, the cat-eye motion is stationary. The most evident· effect ofviscosity is a damping of the whirling motion in the eyes of the pattern.
Prandtl speculatedthat the viscous stress also induced a phase difference between the u- and v-componentsof the oscillations of the fluid particles, in which case the energy I I (d U/ dy)puvdxdy thatthe unperturbed motion U conveys to the oscillatory perturbation may have a positivevalue which exceeds the viscous damping. Another student of Prandtl, Waiter Tollmien,confirmed this intuition in 1929, thus providing one of the first proofs o( the instability ofmm®'??;mmm®mmmm'l?.//____,.---;.Fig. 7 . 1 6.�hJJ..fi.J?.&d,L&%�12Ibid. pp. 688-9.73Ibid.
pp. 689-93.Broken-line velocity profiles. From Prandtl[1921b] p. 691.Fig. 7.17.Kelvin's cat-eye flow pattern. FromThomson (!880c] p. 187.WORLDS OF FLOW296plane Poiseuille flow. Kelvin also believed the cat-eyes to cause instability, no doubtbecause he intuitively connected whirling motion with turbulence. He did not realize,however, the essential role that viscosity played in permitting the growth of the whirls. 747.3.
7Developed turbulencePrandtl's next concern was the 'developed turbulence' (ausgebildete Turbulenz) that occurswell after the critical point of instability has been reached. On this question he foundhimself in competition with Karman, who now held the chair of mechanics and aerodynamics in the Aachen Technische Hochschule. In 1921, Karman propounded a semiempirical derivation of the basic properties of a turbulent boundary layer. Borrowingfrom Boussinesq's Eau courantes, he assumed that the average fluid motion obeyed anequation of the same form as the Navier-Stokes equation, but only if the ordinaryviscosity was replaced by an eddy viscosity (Turbulenzfaktor) depending on the momentum convection caused by turbulent fluctuation. This implied that Prandtl's boundarylayer equation also held for the average motion, but only if the effective viscosity replacedthe molecular viscosity.75As efforts to solve this equation had been largely frustrated, even in the simpler laminarcase, Karman replaced it with the momentum equation obtained by integrating Prandtl'sequation over the thickness 8 of the boundary layer (taking into account the continuityequation):J8J8J8aaaaP+ - prldy - U- pudy = -8 - - ro .8t pudy 8xaxax00(7.16)0On the left-hand side of this equation, the first term represents the acceleration of a normalthin slice of the layer multiplied by its mass, the second term represents the difference ofthe momentum fluxes across the two sides of the slice, and the third term represents themomentum of the fluid that enters the tip of the slice with the asymptotic velocity U.
Onthe right-hand side, the first term represents the impressed pressure difference on the twosides of the slice, and the second term represents the wall friction ( -JL dufdy[y=O in thelaminar case). If a reasonable Ansatz is made on the form of the velocity profile in theboundary layer (giving u/U as a function of y/8), then the above equation becomes adifferential equation for the unknown function 8(x,t), or just 8(x) in the steady case. Thismathematical problem is much easier than Prandtl's original problem.Karman drew his Ansiitze for the velocity profile and wall stress of a turbulent boundarylayer from pipe-retardation data. In 1913, through careful experiments performed at theVersuchsanstalt fiir Wasserbau und Schiffbau and through compilation of older smoothpipe data, Blasius had found the loss of head to vary as the power 7/4 of the sectionaverage velocity.
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