Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 87
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7.9) in the domain x > O,y = 0. The corresponding Eulerian flow is strictlyuniform, so that the pressure gradient vanishes. Prandtl asserted without proof that thevelocity component u was a function of yj ..;X only. Presumably, he guessed that, in the"'Prandtl [1905] pp. 575-6. Cf. Ackroyd et al. [2001] Chap. 9.51 Ibid.
pp. 576-577. Even though Grenzschicht occurs only once in this paper, it is the term that Prandtl laterpreferred.52A similar reasoning is found in Blasius [1908] pp. 2-3. Prandtl is not likely to have used dimensionlessvariables, for these only became popular in later years.DRAG AND LIFT2851\JFig. 7.9.Flow along a flat plate. Here v isthe asymptotic velocity,Ithe length ofthe plate, and is 5 is the thickness of theboundary layer at the end of the plate.From Prandtl l931b: 90.absence of a characteristic length (such as the length of the plate), the parallel-velocityprofiles at different points of the plate only differed by the y-scale. Formally, this meansthat, for any constant a, there are two other constants {3 and 'Y for which the substitutionu,v, x,y -t u, -yv, ax, {3y leaves the boundary-layer equations and the boundary conditions(zero velocity on the blade, u equal to U far from the plate) invariant. This is indeed thecase if a = {32 and {3-y = 1 .
As the solution should be unique for given boundary cmi.ditions, we have u(x,y) = u(ax,y.;a) for any values of x, y, and a. The choice a = l jx leadsto u(x,y) = u(l ,yjyX), in conformance with Prandtl's assertion.Prandtl then solved the resulting ordinary differential equation numerically. Integratingthe stress J.L au;ay on both sides of the blade from the edge to the length l, he reached theresistance formula53R = 1.1byfJ.LplU3,(7.9)where b is the breadth of the blade.
He thereby assumed that the boundary layer of a finitelength blade was approximately the same as the x < l part of the boundary layer of aninfmite plate.Prandtl next proceeded to 'the most important result with regard to application', that is,the separation (Ablosung) of the fluid current from the wall in the presence of an antagonistic pressure gradient. Such a gradient typically occurs at the rear of a bluff-shapedbody, where the lines of the Eulerian flow spread out, the sliding velocity diminishes, andthe pressure therefore increases along the wall (through Bernoulli's law). Owing to viscousdamping, Prandtl reasoned, the fluid in the transition layer may reach a point at which itdoes not have enough kinetic energy to surmount the pressure gradient, in which case itshoots off the wall.
Prandtl drew the evolution of the velocity profile in such cases, andargued that separation occurred at the point 8uj8y = 0, beyond which an absurd backward flow would occur if separation did not prevent it (see Fig. 7.10). He assumed theseparation to result in a vortex sheet a la Helmholtz:A layer of fluid that has been set into rotation through wall friction thus pushes itselfinto the free fluid and there plays the same role as Helmholtz's separation layers(Trennungschichten) in effecting a complete reconfiguration of the motion.53Prandtl [1905] p.
578. A similar reasoning is found in Blasius [1908] p. 5.286WORLDS OF FLOWFig. 7.1 0.The evolution of a boundarylayer in an antagonistic pressure gradient. From Prandtl [1905] p. 578.Prandlt summed up:54The treatment of a given flow process divides itselfin two mutually interacting parts:on one hand we have the free fluid that can be treated as friction-free according toHelmholtz's laws of vortex motion, on the other hand we have the transition layerson the solid boundaries, the motion of which is ruled by the free fluid, but which inreturn give to the latter its characteristic imprint.Prandtl then showed theoretical drawings of the formation of a separation surface at theedge of a plate and behind a cylinder (see Fig.
7.1 1) . He emphasized the instability of thesesurfaces, with the characteristic spiral unrolling identified by Helmholtz. Lastly, he described the apparatus he had used to verify (or reach?) his insights, a waterwheel-drivenwater current with suspended metal dust (see Fig. 7.12), and gave the pictures of theobserved motions behind the edge of a blade and behind a cylinder (see Fig. 7.
13). Inthe latter case, he showed that the separation process could be prevented by pumpingoff the fluid of the boundary layer though a slit on the wall of the cylinder. 55Comparing this communication with earlier notions of a boundary layer and a separation surface by Stokes, Thomson, Rankine, Froude, Boussinesq, and Levi-Civita, twospecificities stand out. Firstly, Prandtl was able to mathematically derive the velocityprofile within a laminar boundary layer and the resulting contribution to the resistance,whereas his predecessors (with the exception of Boussinesq) only had qualitative knowledge of the layer.
Secondly, Prandtl saw that the separation process and the departurepoint of discontinuity surfaces depended on how the velocity profile of the layer evolvedalong the walls, whereas previous advocates of flow separation and discontinuity surfacesignored viscosity and the role of viscous stress in determining the separation point. 567.3 .2Prandtl's heuristicsAccording to Prandtl, a first key to his success in this and other problems was his ability todevelop an intuitive, visual understanding of the phenomena before trying to set them intoequations:54Prandtl [1905] pp.
578-9.55Ibid. pp. 580-4. Prandtl later ([1927b] pp. 768-9) gave aerodynamic illustrations of the prevention ofseparation.56ln their correspondence ofDecember 1898 (see earlier on p. 269), Kelvin and Stokes regarded small viscosity asbeing responsible for a high-shear instability in the boundary layer. However, they did not relate the separation pointwith the velocity prof!le in this layer. According to Prandtl, the position of the separation point does not depend onthe value of the viscosity since the condition fJujfJy = 0 does not.
Yet the separation mechanism requires a finitevalue of the viscosity (see Chapter 5, pp. 214-5 for Rayleigh's discovery of a similar occurrence in 1883).287DRAG AND LIFTFig. 7.1 1 .Initial stages of the discontinuous fluid motion (a) past an edge, and (b) behind a cylinder. FromPrandtl [1905] p.
579-80.Fig. 7 . 12.Prandtl's apparatus forstudying the flow past a solid obstacle (at c). The water is set intomotion by the paddle-wheel. Thefour sifters at b homogenize theflow after the sharp turn at a.From Prandtl [1905] p. 581.Herr Heisenberg has . . . alleged that I had the ability to see without calculation whatsolutions the equations have. In reality I do not have this ability, but I strive to formthe most penetrating intuition [Anschauung] I can of the things that make the basis ofthe problem, and I try to understand the processes.
The equations come only later,when I think I have understood the matter.The sort of intuition he had in mind was acquired by 'special training', in the mannerexemplified in his Digest [Abriss] ofthe science offlow. Instead of deriving the fundamentalhydrodynamic equations and then discussing their consequences for a given hydrodynamicWORLDS OF FLOW288510Fig. 7 . 1 3.1112Pictures of the initial flow past the edges of flat and curved plates (2-6); past a cylinder, without(8-9) and with (1 1-12) suction through a slit.
From Prandtl [1905] plate.system, he directly applied Newton's laws of motion to slices of the tubes of flow of thesystem, thus combining geometrical representation and dynamical understanding ofthe flow. Intuition was the experience gained by working out series of concrete examplesin this manner. As Prandtl remembered, 'in the examples of mechanics, I gradually gotused to "see" the forces and accelerations in the equations and sketches or to "feel" themby muscular sense.' When he learned the Navier-Stokes equation, he studied examples ofviscous flow in order to appreciate the relative importance of each term and thus 'topenetrate the mode of action of this equation'.57There was, however, a more specific key to Prandtl's invention of boundary-layertheory:When the complete mathematical problem looks hopeless, it is recommended toenquire what happens when one essential parameter of the problem reaches the limit57Prandtl [1948] pp.
1604-5.289DRAG AND LIFTzero. It is assumed that the problem is strictly soluble when this parameter is set to zerofrom the start and that for very small values ofthe parameter a simplified approximatesolution is possible. Then it must still be checked whether the limiting process and thedirect way lead to the same solution. Let the boundary conditions be chosen so that theanswer is positive.
The old saying 'Natura nonfacit saltus' decides the physical soundness of the solution: in nature the parameter is arbitrarily small, but it never vanishes.Consequently, the first way [the limiting process] is the physically correct one!From this, we may infer that Prandtl conceived the boundary layer and the separationprocess by requiring that the zero-viscosity limit of the viscous flow should resemble theperfect-fluid flow.
The finite fluid slide on a rigid wall in the latter case suggests a thin layerof intense shear in the former. Also, Helmholtz's recourse to discontinuity surfaces(altered boundary conditions) suggests separation in slightly-viscous fluids. Reciprocally,the working out of boundary-layer dynamics informs the genesis of separation surfaces. 587.3.3 Lanchester and RayleighIn 1907, the British automobile engineer and flight enthusiast Frederick Lanchesterpublished his Aerodynamics, including a description of boundary layer and separationthat was clearly independent of Prandtl's. Lanchester gave much importance to Helmholtz's surfaces of discontinuity, to the point of defining a streamlined body as a body forwhich motion through a fluid does not give rise to a surface of discontinuity. For nonstreamlined bodies, he ascribed most of the resistance to low pressure in the dead-waterregion within the surface of discontinuity.
Around any body within a stream of a viscousfluid, he argued, there must be a layer of dead water adhering to the surface of the body. Ifthe viscosity is small, then this layer is extremely thin near the cutwater, but grows in thesternward direction owing to internal friction. Along a curved surface, this dead watertends to move toward the places of lower pressure. For instance, in the case of a sphere thedead water tends to accumulate near the equator (the axis being parallel to the flow).
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