Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 84
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Rankine identified the flow in a horizontal plane with the vertical section of a Gerstnerwave (see Chapter 2, pp. 73-5). This kind of flow is not irrotational, unlike those later favored by Rankine.28Rankine [1862] p. 24, [1863] p. 137, and [1864] p. 323 for the indirect effect of viscosity.29Rankine [I 864] p. 322. For Darcy's and Du Buat's similar ideas, see Chapter 6, pp. 224, 234.WORLDS OF FLOW276In 1 871, Rankine refined his description:It is well known through observation: that the friction between a ship and the wateracts by producing a great number of very small eddies in a thin layer of water close tothe skin of the vessel, and also an advancing motion in that layer of water; that thisfrictional layer (as it may be called) is of insensible thickness at the cutwater, andgradually increases in thickness towards the stern, by communication of the combined whirling and progressive motion to successive streams of particles; and that,finaJly the various elementary streams of which the frictional layer is composed,uniting at the stern of the ship, form the wake-that is, a steady or nearly steadycurrent, full of small eddies, which follow the ship, but at a speed relatively to stillwater which is less than the speed of the ship.From this picture, Rankine derived the equality of the resistance with the momentum fluxin the wake.
If V is the velocity of the ship, A is the area of a section of the wake, and Uis its average velocity, Rankine reasoned, then the mass of water fed into the wake ispA( V - U) per unit time, and its momentum is pA(V - U)U. Rankine chose U = V/2,which gives the smallest wake section, 4R/pV2, for a given resistance R.30As the dominance of skin resistance depended on the fairness of the ship's shape,Rankine wondered whether Russell's wave lines were the only ones that prevented waveformation.
In order to answer this question, he considered simple, two-dimensional,potential flows obtained by superposing the flows defined by two opposite foci (a sourceand a sink) and a uniform flow directed along the lines joining the two foci. Figure 7.7represents the lines of flow that asymptotically merge with the uniform flow.
Rankinecalled them 'oogenous neolds', for they correspond to the potential flow around a solidc•c•c•==enc\ .XLFig. 7. 7.0Rankine's bifocal lines of flow. A is one of the foci (the other being its mirrorimage through OY), LBis the limiting oval, and PQ is the 'lissenoid' or line of minimal vertical disturbance. The lines AC areconstruction lines. From Rankine [1865] plate."'Rankine [1871] pp.
300-1.DRAG AND LIFT277limited by the central oval. Yet he did not regard the oval as a plausible ship shape, sincethe resulting flow implies an abrupt vertical disturbance (through Bemoulli's law, the localdepression of the free surface varies as the square of the fluid velocity). Rather, he selectedthe stream lines for which the velocity differs the least from that of the neighboring lines.As these 'lissenoids' unfortunately have parallel asymptotes, Rankine cut them off at thepoint of slowest gliding and completed them with a plausible, edge-shaped stem and bow.The resulting shapes resembled Scott Russell's wave linesYWhen Rankine published these considerations, in 1 864, he seems to have believed thatlaminar flow was only possible around special, simple solids such as those given by thebifocal method.
In reality, Laplace's equation for the velocity potential admits a solutionthat meets the boundary conditions for a body of any shape. Being close to WilliamThomson, to whom potential theory had no secret, Rankine could not remain long inerror. In a note of 1 870 he explained:Although every surface is a possible stream-line, the surface of a ship is not evenapproximately an actual stream-line surface unless it is such that she does not dragalong with her a mass of eddies of such volume and shape as to cause the actual tracksof the particles of water to differ materially in form from those which would bedescribed in the absence of eddies.Being now aware of William Froude's water-bird proflles (to be discussed shortly),Rankine added two more foci to his previous scheme and obtained the 'cynoid' lines. Asthe number of foci was in principle unlimited, there seemed to be no limit to the variety ofimaginable ship shapes.
While gaining generality, Rankine's method lost predictivepower.327.2.2Froude's modelsEven though Rankine's contributions marked a significant progress in the understandingof ship resistance, they turned out to be of little value in the computation of resistance, orso it appears from the report of the British Association Committee on 'Resistance ofwater' that Russell, Napier, Rankine, and Froude directed from 1 863 to 1 866. This failureprobably motivated William Froude's experiments of 1 865-1867 on models.
This countrygentleman worked as a railway engineer until 1 845 and retired at the early age of thirtyfive to look after his ailing father. He had an elementary knowledge of mathematics, but avery good understanding of the laws of mechanics. In 1 856, his former employer and ChiefEngineer of the Great Eastern, Isambard Brunei, asked him for help in the study of waveinduced ship rolling.
Froude's outstanding contributions to this subject won him thefavors of the British Association, the ear of the Admiralty, and a membership of theInstitution of Naval Architects. Worth noting are his consideration of skin friction as oneof the damping factors of the rolling motion, and his use of similitude conditions to exploitrolling measurements carried out on a model of the Great Eastem.333 1Rankine [1865].32Rankine [1871] p.
267n (note dated Dec. 1870).33For biographical information, cf. Abell [1933], Brown [1992]. For a penetrating analysis of his works, cf.Wright [1983] chaps 6, 7.WORLDS OF FLOW278In the absence of a priori means to determine the most advantageous ship shapes,Froude consulted experiments. As full-scale trials excluded any radically innovativeshape, he built models of small dimensions and towed them in Dartmouth harbor. Unlikeprevious ship-model experimenters, Froude understood that the rational use of models toderive the behavior of full-scale ships required adequate scaling rules.
On the basis of theexpression)gA./217"for the celerity of a deep-water wave of length A, he argued thatsimilar wave patterns for models at different scales required a towing velocity proportionalto the square of the dimensions of the model. Assuming that the total resistance varied asthe square of the velocity and the square of a 'ruling dimension', he further expected this4resistance to vary as the cube of the dimensions of the model. 3Such were the first scaling rules explained by Froude in an unpublished report to theAdmiralty of April l 868.
In an improved report of December 1868, he recognized that thewave component of the resistance did not generally vary as the squared velocity, butnonetheless varied as the cube ofthe dimensions of the ship. The advocated reason for thissimple law was that the height of the waves as well as their length and breadth varied as thelinear dimensions of the ship, so that their energy varied as the cube of these dimensions.As for the skin resistance, Froude believed that pipe-retardation measurements sufficientlyproved its quadratic form.
In his opinion, Beaufoy's 1 . 8 exponent probably resulted from5experimental errors-a view that Froude revised a couple of years later. 3Froude did not explicitly introduce the 'Froude number' of modern navigation theory.Nor did he reason from fundamental principles or equations. Newton had briefly done soin the section of thePrincipia devoted to fluid resistance,and Joseph Bertrand had giventhe general similitude conditions of rational mechanics in 1 848. Most relevantly, thedirector of the Ecole d'Application du Genie Maritime in Lorient, Ferdinand Reech,derived the similitude conditions for ship models in 1 844 and included them in his lectureson mechanics. Although Froude seems to have been unaware of Reech's reasoning, itsgenerality and rigor deserve a few lines.
36In any mechanical system, Reech reasoned, the equations of equilibrium are unchangedunder global change of the length scale, as long as this change affects all forces in the sameproportion. Taking d'Alembert's principle into account, the equations of motion areunchanged through a change of the length and velocity scales, if this change affects allforces, including the inertial forces, in the same proportion. Denote by a and {3 the factors bywhich the lengths and velocities are respectively multiplied.
Then inertial and gravitational2forces are multiplied by a2[3 and a3 , respectively. For a system in which the acting forces are2inertial and gravitational, the equations of motion will be invariant if a2{3 = a3 , or a = {32 ;so velocities must vary as the square root of a length in order that both kind of forces vary asthe cube of a length.
As Reech further noted, atmospheric pressure and viscosity forcesbehave differently, so that the rule no longer applies in problems for which these forces are34Unpublished report sent to Edward Reed, discussed in Wright [1983] p. 210. Rankine ([I 862] p. 28) hadearlier expressed the condition that 'the velocities of the model and of the ship should be proportional to the squareroots of their linear dimensions' in order that the wave effects should be comparable.35Froude [1868].36Reech [1844] p. 166, [1852] pp.
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