Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 93
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100Most relevant to his thinking were Helmholtz's vortices and discontinuity surfaces, aswell as Rayleigh's tennis-ball problem. Lanchester now understood that the flow he hadimagined around an aerofoil belonged to the same category as Rayleigh's irrotationallycirculating flow: the same compression of the lines of flow above the flying object andrarefaction below occur in both problems. Lanchester now made circulation the essence oflift. From Helmholtz's law for the velocity induced by a linear vortex, he inferred thedownward precession of the two trailing vortices of the foil. He further suggested thatthese vortices should be replaced by a Helmholtz vortex sheet extending behind the wholebreadth of the foil, as the air skirting the upper surface of the aerofoil reaches its rear edgewith a transverse velocity directed toward the axis of flight, and the air skirting the lowersurface reaches the near edge with an opposite velocity (see Fig.
7.26). Wrongly assumingthat the circulation around every transverse section of the foil caused a deviation of thevortex filaments away from the axis, and taking into account the mutual twisting and theviscous diffusion of these filaments, he obtained the emblematic picture in Fig. 7.27. 101Lanchester published these considerations together with his earlier intuitive theory inhis Aerodynamics, constituting thefirst volume of a complete work on aerialflight of 1907.The book got fair reviews in the British press, and won Lanchester an appointment to theBritish Advisory Committee for Aeronautics.
The president of this committee, LordRayleigh, endorsed Lanchester's boundary-layer consideration, as was mentioned earlier.Despite these welcoming signs, Lanchester's ambition to provide guidance for aeroplanebuilders was largely frustrated. When the Wright brothers' machine was first flown inEurope at Le Mans in 1908, Lanchester found Wilbur Wright very ill-disposed towardtheory. The pioneering constructor dryly commented that the most talkative birdFig. 7.26.The vortex sheet induced by the lateral skirting of the air on the upper and lower surfaces of theaerofoil, seen from behind. From Lanchester [1907] p. 176.1 00Lanchester [1894]; Prandtl [1927b] p.
753. Cf. Lanchester [1907] p. 142.1 0 1Lanchester [1907] pp. 1 62-78. Lanchester confused the circulation around the foil with a real layer ofvorticity around it. He does not seem to have understood the connection between the circulation around the foiland the vorticity of the trailing vortices.309DRAG AND LIFT(a)�--�� -- - --- ---- ---- ----- ·(b)Fig. 7.27.The trailing vortex of a flying wing according to Lanchester [1907] pp. 177-8.(the parrot) was also a poor flier.
To a letter from Lanchester in the following year, hebriefly replied: 102In glancing over [your paper] I note such differences in matters of information,'theory, and even ideals, as to make it quite out of the question to reach commonground by more talk, as I think it will save me much time if I follow my usual plan,and let the truth make itself apparent in actual practice.Although the British aeronautical establishment was more open to theory than theWright brothers, it seems to have ignored Lanchester's aerofoil theory until Prandtl's1 02Cf.
Lanchester [1926] p . 588, Ackroyd [1992].310WORLDS O F FLOWrelated theory became known after the war. The Germans were the most receptive toLanchester's ideas. Soon after the publication of Aerodynamics, Prandtl's prominent colleague Car! Runge contacted Lanchester to propose a translation. He welcomed him toGottingen in September1908, and he arranged conversations with Prandtl, who was thenbusy completing the Gottingen wind channel. Although Prandtl later claimed (in Lanchester's presence) to have reached the main ideas of his wing theory before reading Lanchester,he also admitted that he and his collaborators 'were able to draw many useful ideas' fromAerodynamics.Karman, who had witnessed the Gottingen encounter, suggested thatPrandtl had borrowed more from the English engineer than he was conscious of.
1037.4.2 Kutta's and Joukowski's theoriesIn1 902, Wilhelm Kutta,a mathematics student in Munich with an interest in Lilienthal'sgliding experiments, devoted his dissertation to the flow around the simplest idealizationof Lilienthal's cambered wings, namely a circular arc. His method consisted in applying aconformal transformation z =c:!>(C) to the incompressible flow around a circular cylinder,represented in the complex plane of the variable z = x + iy.
As Rayleigh had shown, themost general irrotational solution to the latter problem with an asymptotic, horizontalvelocityU is given by.q; + u/J =where q; is the velocity potential,U( a2)z+-;-r "I27T1n z,(7.31)1/J is the stream function, a is the radius of the disc, andr is the cyclic period of the potential (the circulationf v dr).
As a mathematician, Kutta·had no objection against the circulatory component of this solution.104Kutta applied to this flow an intricate, double-step conformal transformation thatturned the circular boundary of the cylinder into a circular arc with chord parallel to theasymptotic flow. The velocity at the tips of the arc, he found out, was only finite if thecirculation r had the specific value 27Th U, where h is the maximum height of the arc.Under this condition, the flow has the shape shown in Fig.7 .28.
Kutta then integrated thefluid pressure (as given by Bernoulli's law) to obtain the lift(7.32)Comparing this theoretical result with Lilienthal's measurements, Kutta fourid a25%excess that could plausibly be explained by vortex formation and a finite span.105Through a consideration of energy, Kutta also related this lift to the cyclic period r ofthe potential. The work done by the lift during a (virtual) vertical displacement 8y of the103Prandtl [1927b] pp.
753 (quote), 776 (Lanchester remembering Gottingen); Karman [1967] pp. 50-3. Runge,who had an English mother, was the interpreter. His and his wife Aimee's translation of Aerodynamics appeared in1909.1 04Kutta [1902a]. Sebastian Finsterwalder, a mathematics professor and ballooning expert at the TechnischeHochschule in Munich, suggested the topic of Kutta's Habilitationsschrift (cf. Kutta [1910] p.
4). As was wellknown. the compressibility of the air can be neglected in any resistance problem for which the velocity of the airremains small compared to the celerity of sound waves.108Kutta [1902a], [1902b]. Cf. Ackroyd et al. [2001] pp. 70-6.311DRAG AND LIFT...----t" _- - - - - c - - - - - - - - --.-_ _ _ _ __ __- - - - - - - - - -+-"":' _ _ _---- -"':- -�- - -Fig. 7.28.Kutta's flow around a circular arc. From Kutta [1902b] p . 133.arc, he reasoned, should be equal to the energy produced by the annihilation of ahorizontal fluid slice of the same breadth at a large positive ordinate[ +_Looof another slice at the symmetric ordinateL=Y�oo-- Y.
This+oo� pv1(x, Y) dx +Using the asymptotic approximation_2v-(x,Y) � U2 +ru27TY and the creationgives, for the lift L,_[� pv1(x, -Yx2 +]Y) dx .y2 ,(7.33)(7.34)and lightheartedly assuming a mutual cancellation of infinite terms, Kutta found thatL = pr u,(7.35)in conformance with the result (7.32) of direct pressure integration.106In summary, Kutta's mathematics led to a flow around a thin curved foil that strikinglyresembled the one Lanchester predicted. Instrumental to his derivation was the conditionthat the velocity of the flow should remain everywhere finite, which is now called the Kuttacondition. The remarkably simple formulaL = pr U is now called the Kutta-Joukowskir with the circulation of the airtheorem.
However, Kutta did not explicitly identifyaround the foil. Nor did he refer to Rayleigh's tennis-ball problem as the origin of formula(7.31) for the irrotational flow around a circular cylinder. In the semi-popular summarypublished in the Illustrirte aeronautische Mittheilungen, he did not give the general relationbetween lift and circulation. Instead, he argued that, in order to prevent the formation ofvortex sheets at the extremities of the arc foil, the velocity of the air had to be tangential.This implies a higher velocity above and a lower velocity below the foil, and a liftingpressure difference by Bernoulli's law.1071 06Kutta [1910] pp.
19-20. In this article Kutta described the reasoning as belonging to his Habilitationschrift[1902a], which I have not been able to find. Joukowski ([1910] p. 282) accepted this claim.1 07Kutta [1902b]. Kutta ([1910] p. 3) credited Lanchester for the concept of wing circulation.312WORLDS OF FLOWFig. 7.29.A tentative flying device by Joukowski. The twisted rubber band C induces the rotation of thepaddle-wheels A and B. From Joukowski [1890] p. 350.Unlike Lanchester and Kutta, the Russian physicist Nikolai Joukowski was a highlyprofessional physicist of international repute, and head of the mechanics department atMoscow University.
Much of his early work was in theoretical hydrodynamics, with anemphasis on potential flow and complex-variable methods. He published his first significant paper on aerodynamic lift in 1906, after several years of interest in the problem offlight. From then on, he played a leading role in developing the aeronautical industry in hiscountry. His main contribution of 1906 was a rigorous and general derivation of thetheorem that relates circulation and lift for the two-dimensional flow around a solidcylinder.108In an address of 1 890 on the theory of flight, Joukowski argued that paddle propulsionwas only possible if the fluid motion implied discontinuity surfaces,.
viscous stress, orwhirling motion. In the last case, he imagined and constructed the device shown inFig. 7.29, in which each of the rotating paddle-wheels is subjected to the upward currentinduced by the rotation of the other. Although the device turned out to be too heavy to fly,Joukowski found that the rotation of the wheels diminished its apparent weight. There isno hint, in this communication, that whirling motion may also occur around static wings,nor that it may imply a transverse, lifting force when the whirl progresses horizontally.Worth noting, however, is the general idea of exploiting vortex motion for the sake ofartificial flight. 109Before 1906, Joukowski had read Louis Pierre Mouillard's L'empire de !'air, a book of1881 familiar to several pioneers of aeronautics. By careful observation of bird flight,Mouillard hoped to help in the successful design of gliders and 'aeroplanes'.
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