Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 95
Текст из файла (страница 95)
He believed that two-dimensional vortex production at the front of thewing would account for observed drags, and ignored the effect of finite span, even thoughhe had read Lanchester's book and approved the concept of trailing vortices.1 1 57.4 .3 Prandtl's theoryIn the years 1910-1918, Prandtl and his collaborators combined Lanchester's intuition ofthe motion around a three-dimensional aerofoil with the mathematical precision of twodimensional theories. His fullest publications on this topic appeared in 1918/19, with adelay due to wartime secrecy.
As Prandtl himself noted, the organization of these papersdoes not reflect the historical course of his thoughts. This course may, however, be inferredfrom Prandtl's few historical remarks, from earlier fragmentary publications, and from thelogic of the subject.
1 16One ofPrandtl's earliest contributions must have been the explanation of the process bywhich circulation is produced around a streamlined two-dimensional aerofoil. WhileKutta said nothing on this process, Lanchester's falling-plate reasoning could not passfor a proper hydrodynamic demonstration. Yet for anyone versed in fluid mechanics,circulation was only admissible if its genesis could be reconciled with the theorems byLagrange and Kelvin that seemed to forbid it.
Due to his familiarity with Hehnholtz'svortex sheets, Prandtl easily solved the paradox as follows.11 7The irrotational, non-circulatory flow around the aerofoil involves infinite velocity atthe rear edge. In order to keep the velocity finite, a vortex sheet must be generated at thebeginning of the motion, as shown in Fig. 7.32.
This process is perfectly compatible withKelvin's theorem, which only forbids vorticity for fluid particles that have never been incontact with a wall. Neither does the theorem forbid a change in the velocity circulationaround the body. On the contrary, when applied to a curve enclosing both the body andthe emerging vortex sheet (the dotted line in Fig. 7.32), this theorem implies that thevelocity circulation around the body should increase by an amount equal to the totalvorticity of the vortex sheet. After a brief time, this circulation reaches the value for whichFig. 7 .32.Transient pattern of the flowaround a wing, with vortex production atthe trailing edge. From Prandtl [192la]p.
464.1 1 5Joukowski [1916] pp. 184-5, [1910] p. 282 (approving Lanchester).116Cf. Prandtl [1918] p. 322.1 1 7Prandtl's systematic use of Helmholtz's and Kelvin's vortex theorems in wing theory presupposes that thecompressions of the air are negligible, which is true for widely subsonic flight.DRAG AND LIFT317the Kutta condition o f smooth flow i s satisfied, the vortex sheet production ceases, and theresulting vortex flows away.118For a wing of finite span, Prandtl reasoned, circulation must exist at least around thecentral sections of the wing in order to make lift possible.
Such circulation, however,cannot exist without permanent vortex production. This is a consequence of the theoremaccording to which the variation of the circulation around a loop during a continuousdeformation or a displacement of this loop is equal to the number of vortex filaments cutby the loop. 1 19 Consider a loop that embraces a section of the wing, and move it towardone of the tips of the wing. As the circulation necessarily vanishes at the tip (since the loopshrinks to a point), it must cross vortex filaments on its way. Hence vorticity is necessarilyproduced near the tips. More generally, vorticity must be produced whenever the circulation varies between successive sections of the wing.
According to one of Helmholtz'stheorems, the generated vorticity must follow the fluid motion. Therefore, a trailing vortexsheet is formed behind the wing, with an intensity depending on the rate of variation of the0circulation along the span of the wing (see Fig. 7 . 33).12Prandtl once said that he had reached this picture while p=ling over Lanchester'strailing vortex.121 There are significant differences, however. Whereas Lanchester reasoned in an intuitive, qualitative manner based on the 'field of force' of a falling plate,Prandtl applied Helmholtz's vortex theorems to derive a precise quantitative connectionbetween the circulation around the wing and the trailing vortex. Prandtl had the vortexfilaments follow the main flow, whereas Lanchester erroneously gave them a sidewaysinclination.
Prandtl related the variation of the circulation and the production of vorticityto the variation of the wing's section along its span, whereas Lanchester reasoned on aconstant section.Lastly and most importantly, Prandtl was able to apply his picture of wing flow to aquantitative determination of lift and drag, whereas Lanchester's considerations remainedmostly qualitative. In a first approximation, Prandtl reasoned, the lift is the sum of the liftsgiven by the Kutta-Joukowski theorem applied to the successive sections of the wing (as ifthey belonged to infinite cylinders), and there is no drag.
In a second approximation, thevelocity field of the trailing vortex must be taken into account. In the vicinity of a givensection of the wing, this induced flow is approximately uniform and in the downwardvertical direction (see Fig. 7 .3 4). Therefore, the net flow impressed on this section has adownward inclination, and the corresponding reaction, being rotated by the same angle,now has a finite drag component and a slightly diminished lift component.122Prandtl had this general picture by1912. The mathematical implementation did notgo as smoothly as he had hoped. The simplest conceivable case is that of constant1 18Prandtl [1 9 1 3] pp.
1 1 8-19, [1918] pp. 325-8, [1921a] pp. 463-4.1 19This theorem results from the divergenceless character of the vorticity: the flux of the vorticity across thesurface swept by the loop must be equal to the variation of its flux across a surface bounded by the loop, which byStokes's theorem is equal to the circulation around the loop.120Prandtl [1913] p. 1 12, [1918] pp. 324-5, [1921a] pp. 465-6.121Prandtl [1 948] p.
1607n.122Prandtl [19 1 8] pp. 337-8, [!921a] p. 477. On Lanchester's few quantitative attempts, cf. Ackroyd [1992].WORLDS OF FLOW318arctg rv/V.zFig. 7. 3 3.A trailing vortex sheet with vorticityFig. 7 . 34 .The Inclination of the resistanceprofile dr/dx corresponding to the (elliptic) cirowing to the vertical induced velocity w superculation profile r around a flying wing. Fromposed to the unperturbed, horizontal air flowPrandtl [19 1 8] p.
337.V. From Prandlt [1918] p. 337.circulation r along the span of the wing, for which the trailing vortex has the horseshoeshape of Fig. 7.35. The corresponding value of the velocity w(x) of the induced flow at theabcissa x along the span of the wing is given by the Biot and Savart law asw(x) =rA...-n(-- --)1a-x+1a+x,(7.40)if 2a is the span of the wing. According to the reasoning outlined above, the resultingdrag isJ+aD=p-afwdx.(7.41)DRAG AND LIFTFig. 7.35.3 19The horseshoe vortex behind a flying wing.
From Prandtl [192la] p. 466.This integral diverges logarithmically. Prandtl was thus compelled to use a variablecirculation f(x). By the above-mentioned theorem, a vortex filament of intensity f'(x)dxtrails behind the element dx of the wing's span. The resulting induced velocity isx) = __!._w(4rrJ+af'(s)dfs-x(7.42)The divergence of this integral for g = x is easily avoided by taking its principal value inCauchy's sense. Prandtl tried a number of simple expressions for the circulation profilef(x), but kept obtaining an infinite result for the drag integral (7.41).
For a while, he putthis difficulty on hold, and considered the non-divergent effect of the induced velocity onother wings in the same aeroplane. In the case of a biplane or a single wing interacting withits mirror image through a solid wall, his collaborator Albert Betz published considerations of this kind and their wind-tunnel confirmation in 1912/ 14. 1 23As last, in November 1 91 3, Betz and Prandtl found that the elliptic profiief(x) =yielded the constant induced velocitythe liftwroy�1 - di(7.43)= fo/a. For a horizontal velocity V of the wing,J+aL = pVf dx(7.44)takes the value(7.45)and the drag isD = �L = .!lL.VaV(7.46)Remembering that the circulation fo is proportional to the velocity V (owing to Kutta'ssmooth-flow condition), the lift is proportional to the squared velocity of the wing and to123Prandtl [1913] p.
376; Betz [1912], [1914]. Cf. Anderson [1997] p. 285.320WORLDS OF FLOWits span. The ratio of drag to lift is independent of the wing's velocity, and diminishes with2the span, as was to be expected. 1 4In sum, Prandtl and Betz accidentally discovered the elliptically-loaded wing in anattempt to avoid drag-integral divergence.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
