Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 94
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He alsosketched a strange wing theory based on an analogy with the fall of a Bristol card (a rigid,rectangular paper strip). The fall of the card from a horizontal position usually implies a1 08Cf. Grigorian [1965], [1976], Strizhevskii [1957].109Joukowski [1890].I thank Yury Kolomensky, who helped me read this paper.313DRAG AND LIFTrotation around its axis o f symmetry, as well as a deviation i n the direction o f thehorizontal velocity component of the lower edge of the strip.
By the adequate loading ofthe strip and the folding of its tail, Mouillard believed he could check the rotation and yetpreserve the deviation from a vertical fall. In his opinion, birds flew according to thismechanism. His theoretical explanation of the rotation involved the dubious principle thatthe center of gravity of any falling body should be displaced by an amount proportional tothe velocity of fall. He remained silent on the cause of the horizontal deviation. 1 10Although Joukowski ignored Mouillard's speculations, he credited him with the description of the 'interesting phenomena' accompanying the fall of a Bristol card.
He alsomentioned Wladimir Koppen's model of an aeroplane with motorized rotating wings,based on the principle that rotation prevents the fall of bodies. Joukowski justified thisprinciple by analogy with the Magnus effect, according to which a rotating projectile issubjected to a deviating force proportional to its rotation.
As he was unaware of Rayleigh's tennis-ball paper, he explained this deviation by a general theorem of his ownY 1If an irrotational, two-dimensional flow with asymptotic velocity [U] surrounds aclosed curve [made of lines of current] on which the circulation of the velocity is [Il,the resultant of hydrostatic pressure on this curve is perpendicular to the velocity [U]and has the value prU.
The direction of this force is obtained through a right-anglerotation of the vector U in the sense of negative circulation.In his demonstration Joukowski imitated Poncelet's and Saint-Venant's recourse tomomentum balance in their theories of resistance, with which he had .become familiarduring a formative stay in Paris. Around the closed curve made oflines of current he drewa circle of large radius (see Fig. 7 .30), and required that the pressures acting on the fluidcontained between the closed curve and the circle should balance the momentum increaseof this fluid:-L -TPn ds = p fv(v · n ds),(7.36)where L represents the resistance (action of the fluid on the body) per unit length, and theintegrals are taken over the circular trace of the fictitious cylinder (n being the unit normalvector). Using Bemoulli's law and retaining only first-order terms in w v, this givesL= p T[(U·w)n - w(U n)] ds·= p TU= -Ux (n x w) ds.(7.37)UAt large distances from the body, w is the velocity of a pure circulation in a directionperpendicular to n.
Therefore, the vector L is directed downward when is directed to the1 1 0Mouillard [1881] pp. 210-17. Unknown to Mouillard, in 1 854 James Clerk Maxwell had explained therotation by the greater resistance of the air to the motion of the lower edge of the plane, and the deviation byperiodic modulations of the net resistance and the fall velocity.!I,1 1 1Joukowski [1906a] p. 52; Koppen [1901] (falling card experiments); Moedebeck [1904] p. 179 (on Koppen'smodel).
Joukowski tried to verify his theorem by measuring the force acting on a rotating blade in the wind tunnelof the Aerodynamic Institute of Koutchino. The director of this pioneering institute, Dimitri Riabouchinski[1909], criticized this procedure as well as Maxwel!'s old theory of the falling paper strip (Maxwell [ 1854]), whichdoes not imply any transverse force when flow relative to a rotating blade is kept constant.,: IWORLDS O F FLOW314Fig.7.30.Joukowski's flow past a blade (shaded rectangle) rotating around the axis Oz (perpendicular toO.xy). The closed curve ABO, made of two converging lines of current, separates the zone ofwhirling motionfrom a zone of larninar, irrotational flow.
From Joukowski [1906a].right and the circulation is oriented trigonometrically, and the intensity2Kutta-Joukowski formula (7 .35). 1 1L agrees with theAlthough Joukowski knew his theorem to apply to the case in which the closed line ofcurrent is the frontier of an immersed solid, he only applied it to a rotating blade immersedin a uniform stream, assuming that the flow was smooth and irrotational outside a pearshaped zone (ABD in Fig. 7.30) delimited by converging lines of current. In another paperof 1906, he introduced the notion of 'bound vortices', that is, a series of virtual or realvortex lines contained within a closed curve and able to represent the circulatory part ofthe flow outside this curve.
He enunciated theorems relating the force and angularmomentum resulting from the pressure on this curve to the strength of the vortex linesand the fluid velocity on these lines. He applied these notions to the rotating blade and tothe vortex pair behind a plate immersed perpendicularly in a uniform stream. He did notconsider the case of an airfoil or wing, in which he may not yet have understood thatcirculation-flow occurred. 1 1 3l12Joukowski[1906a], [1906b].Cf. Ackroydet al. [2001]pp.88-106.In the first paper, Joukowski uses thebalance of angular momentum around an arbitrary axis instead of the momentum balance.
Had Joukowskifollowed Saint-Venant closer, he would have used a fictitious surface of large rectangular section. In this case themomentum variation of the enclosed fluid vanishes, and the resistance is simply given by the pressure difference onthe two horizontal walls of the fictitious cylinder.113Joukowski[1 906b].In1909,Joukowski obtained the two-dimensional flow around a curved plate as anextension of Kirchhoff's flat-plate solutionwith surfaces ofdiscontinuity: cf. Chaplygin [1911], who gives Kuttafull credit for conceiving the possibility of smooth, circulatory flow around a cambered foil.
I thauk my friendGuenaddi Sezonov who helped me with the Russian (before I became aware of Anatoly Ruban's translation inAckroydet al. [2001] pp. 88-104).DRAG AND LIFT315In 1910, Kutta extended his calculation of the lift of an arc of a circle to the case of aninclined flow. In the same year, Joukowski's brilliant disciple Sergei Alekseevich Chaplygin rediscovered Kutta's solution for the smooth flow around a circular arc, using themuch simpler conformal transformation? = z - ia +A2-.- ,z - ta-(7.38)where a and A are real constants. Kutta and Chaplygin both noted that, in the inclinedcase, an infinite velocity could only be avoided at one extremity of the arc, say the rear one.In order to avoid the remaining infinite velocity and vortex-sheet formation, Kuttarounded the front edge through a complicated numerical procedure, and Chaplygingrafted a disc onto it. Joukowski then found that a horizontal shift of the origin in thez-plane magically thickened the arc-shaped foil, leaving only one sharp edge at the rear.The transformation(7.39)now called the Joukowski transformation, turns a circle of radius A centered at the originof the z-plane into a segment of length 4A in the ?-plane.
A horizontal (real) shift of theorigin of the circle turns the segment into a fish shape. A vertical (purely imaginary) shiftof the origin turns it into a circular arc. Both shifts combined lead to a cambered fish shape(see Fig. 7.31). 1 14++·H ·FEt+f-H-WH-H -e -Fig. 7.3 1 .Joukowski's theoretical wing profiles. From Joukowski [1916] p . 105.1 14Kutta [1910); Chaplygin [1 9 1 0); Joukowski [1910), [1916) chap.
6. Cf. Ackroyd et al. (2001), chaps 12-14.For a modem account, cf. Batchelor [1967) pp. 445-9. According to Chaplygin ([1945] p. 5), his paper was alreadyin press when Joukowski told him about Kutta's earlier work. Chaplygin ([1 9 1 1 ) pp. 17-18) briefly mentioned thenecessity of tip vortices in the case of finite span, but gave them an erroneous mustache shape.316WORLDS OF FLOWThe two-dimensional approach to wing theory culminated with these Russian fmdings.Joukowski's suggestion for obtaining a finite drag within this theory was, however,misconceived.
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