Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 92
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Accordingto Bernoulli's law, this difference implies a pressure difference and a transverse deviationof the ball. Magnus tested his explanation with the device of Fig. 7.21. 9 1In a memoir of 1877 'On the irregular flight of a tennis ball', Rayleigh recalled Magnus'sreasoning and noted that the most general irrotational solution of Euler's equation for thetwo-dimensional flow around a cylinder with constant asymptotic velocity had the form( �)l{l = a l -r sin B + .B ln r,(7.30)90Kelvin to Baden-Powell, 8 Dec. 1896, in Gibbs-Smith [1960] p. 35; Rayleigh [1891], [1 883c], [1900] p.
462.91 Magnus [1 853] pp. 5-7.304WORLDS OF FLOWFig. 7.19.Lift and drag components of fluid resistance for separated flow around an inclined plate.Fig. 7.20.Flow around a rotating sphere-��-�according to Magnus [1853]. The straightarrows represent the fluid velocity.Fig. 7 .21.--Magnus's apparatus for demonstrating the pressure difference between the two sides of a rotatingcylinder subjected to the draft from the ventilator F.
The light, horizontally movable blades a and b serve todetect the increase and decrease in pressure when the fan is turned on. From Magnus [1 853] plate.where 1/J is the stream function, r is the distance from the axis of the cylinder, (} is the anglearound this axis, and a is the radius of the cylinder.
The first part of this formula, alreadyknown to Stokes, by itself satisfies the boundary conditions if a is equal to the asymptoticvelocity of the flow. The second part represents a circulation of the fluid around the axis,with a velocity {3/r at the distance r from the axis. Integrating the pressure over the surfaceof the cylinder, Rayleigh found a resultant force perpendicular to the asymptotic velocity,with the intensity 2'1Ta[3.92At the very best, Rayleigh hoped this consideration to be relevant to the Magnus effect,abstraction being made of the circulation-inducing process and of the 'unwillingness of the92Rayleigh [1 877b].
A missing factor of 2 has been corrected.DRAG AND LIFT305stream-lines to close in at the stern of an obstacle'. He did not dream of any application tothe problem of flight. That viscosity could possibly induce a fluid circulation around anon-rotating, flying object was hard to conceive. For a perfect liquid, Kelvin's circulationtheorem seemed to prohibit the genesis of any circulation.
In his treatise of 1 895, Lambreproduced Rayleigh's solution of the cylinder problem as an interesting example of fluidmotion that has circulation despite being everywhere irrotational. No more than Rayleigh3did he perceive any connection with the problem of flight. 9In summary, Rayleigh, Lamb, and Kelvin knew too much fluid mechanics to imaginethat circulation around wings was the main cause oflift. The two men who independentlyhit upon this idea lacked training in theoretical physics.
One of them was an engineer, andthe other was a young mathematician.7 .4.1Lanchester 's theoryFrederick Lanchester was an automobile engineer and industrialist with a passion foraeronautics. In 1892, he imagined a singular theory of what he called an 'aerofoil', that is,the organ of sustentation of airplanes and birds. As he had 'very little acquaintance withclassical hydrodynamics', he reasoned by direct application of the laws of mechanics tothe particles of the fluid.
In the first, Newtonian approximation, the fluid particles hit theaerofoil independently of each other, which leads to a resistance proportional to thesquared sine of the inclination. 94In reality, Lanchester went on, the mutual interaction of the fluid particles implies thatthe layers of air adjacent to the foil react on the neighboring layers, so that a stratum of airof considerable thickness is affected (see Fig. 7.22). Then the flux of deviated particles is nolonger proportional to the sine of the inclination (as it was in Newton's reasoning) but tothe width of the stratum or 'sweep', and the resistance becomes proportional to the sine ofthe inclination, in conformance with small-angle measurements. Lanchester estimated thiswidth from Langley's experiments with superposed planes, which showed that the sustaining power of the planes added up only when the vertical distance between them exceeded acertain value.
Substituting this value into the revised resistance formula, he obtained aboutFig. 7 .22.Provisional, constant-sweep picture ofthe flow past an inclined plate. From Lanchester [1907] p. 227.93Rayleigh [1 877b] p. 346; Lamb [1895] pp. 87-90.94Lanchester [1 894], [1907] p. 143, [1 926] p. 593. On Lanchester's biography, cf. Fletcher [1996]. On hisaerodynamics, cf.
Ackroyd [1992], [1996], Ackroyd et al. [2001] pp. 57-69. The theory of 1892 is given inLanchester [1907] pp. 143-62 (the manuscripts are lost).WORLDS OF FLOW306half of the measured drag. Consequently, the flow of Fig. 7.22 could not accuratelyrepresent reality. 95The reason for this discrepancy, Lanchester surmised, was the dubious assumption thatthe air encountered by the front edge of the moving foil was at rest. In reality, air must flowfrom the region below the foil to the region above it in order to prevent the accumulationand rarefaction of fluid in these two regions. To make this clear, Lanchester decomposedthe motion of an approximately-planar foil into a component parallel to the plane and anormal component.
Then the resistance problem is the same as in the case of a falling platesubjected to a simultaneous (faster) horizontal motion. The fall of the plate, Lanchesterreasoned, induces a fluid motion of the sort represented in Fig. 7.23, with an upwardcurrent or 'vortex fringe' in front of the plate. In its forward motion the plate interceptsthis upward current, and thus experiences a stronger lift than it would by the soleproduction of a downward current. 96To refine his reasoning, Lanchester gave the plate infinite span and loaded it with asmall weight.
He assimilated the effect of the weight with the creation of an accelerationfield of the form given in Fig. 7 .24. The horizontal air flow with respect to the plate bringsnew fluid particles into this acceleration field. Their trajectory has the shape indicated inFig. 7.23.Vortex fringe for a plate falling through the air with the velocity w. From Lanchester (1907] p. 145.Fig. 7 .24.Acceleration field around a falling plate.
From Lanchester (1907] p. 176.95Lanchester (1907] (1892) pp. 144-5.96Ibid. pp. 145-46.307DRAG AND LIFTFig. 7.25.Fig.Horizontal flow modified by the acceleration field of Fig. 7.24. From Lanchester [1 907] p.159.7.25. Owing to the left-right symmetry of the acceleration field, the particles leave itwith their original velocity and their original height. Therefore, the only effect of the smallloading of the plate is to produce 'a supporting wave' traveling together with it. No work isneeded to preserve the horizontality of the motion. In modern words, Lanchester imagineda state of fluid motion such that the lift exactly compensates the load of the plane, without7any induced drag.
9This state of motion is only possible if the plate is given a small curvature, so that theundulating trajectories of the fluid particles do not cross the foil. Using this principle,Lanchester drew the proflle marked by the thick line of Fig.7 .25. The curvature increaseswith the load of the plate. An aerofoil can thus produce lift with vanishing inclination, aslong as this foil is curved. Lanchester regarded the observed shape of bird wings as avindication of this theory.
Lilienthal and Langley also used cambered wings in theirgliders, and the former had given precise experimental proof of their superiority. 98Lastly, Lanchester considered the more difficult case of an aerofoil with finite span.Owing to the lateral spread of the field lines in this case, the ascending field that actsaround the edges of the plane is weaker than the descending field that acts underneath andabove the plane. Consequently, the fluid particles that travel through these two fieldsemerge with a downward velocity; there is a downward current in the wake of the foil,compensated for by two upward currents caused by the ascending field that acts alonebeyond the tips of the foil.
Since the accelerating field, seen from behind the foil, has aform similar to that drawn in Fig.7.24, it must induce a whirling motion of the fluid,essentially two vortices starting from the tips of the aerofoil. The continual production ofthese vortices and the formation of the downward and upward currents spend energy, sothat an induced drag necessarily accompanies the lift of a finite aerofoil.99Lanchester expounded these ideas at the annual meeting of the Birmingham NaturalHistory and Philosophical Society on1 9 June 1894.
In 1 897, the Physical Society of91Ibid. pp. 149-56. Lanchester's acceleration field has the same geometry as the velocity field of a moving plate(indeed, the motion can be regarded as being impulsively started from rest).98Ibid. pp. 1 58-60. Lilienthal attributed the superiority of cambered wings to the reduced production of eddies(on his resistance measurements, cf. Anderson [1997] pp. 138-59).99Lanchester [1907] pp. 1 56-8.308WORLDS OF FLOWLondon rejected a fuller account. As Prandtl later put it, 'Lanchester's treatment isdifficult to follow, since it makes a very great demand on the reader's intuitive perceptions.' Only a reader who would have known the results to be essentially correct wouldhave bothered penetrating the car maker's odd reasoning. Lanchester must have becomeaware of this communication problem, since he immersed himself in Lamb's Treatise andsought more academically acceptable justifications of his intuition of the flow around anaerofoil.
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