Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 85
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265-75. On the broader history of similitude and models, cf. Wright [1983]chap. 8, [1992].DRAG AND LIFT279not negligible. With this restriction, Reech concluded that 'Newton's theorem of similitudewould always be the best and often the unique foundation ofmany practical applications [ofmechanics]', and contrasted this power of similitude arguments with the meager yield ofhigher theories based on Euler's or Navier's 'special equations'.In order to verify his (or Reech's) scaling rules, in 1 867 Froude built a series ofmodels atdifferent scales (3, 6, and 1 2 feet) for two shapes, namely, a wave-line shape he calledRaven, and a water-bird shape he called Swan (see Fig.
7.8). The results confirmed hisexpectations, although modern analysis of his data has shown enormous errors (up to50%!), probably due to a flawed dynamometer. He also concluded that the odd water-birdshape was superior to the wave-line shape at high velocities. This finding justified the needfor further model experiments in which a large variety of unusual shapes could be tested.Froude soon planned the construction of a towing tank that would permit sufficientprecision in such experiments.The project required important funds, which Froude secured from the Admiralty. Thechief constructor of the Navy, Edward Reed, approved Froude's exploratory approach toship form, in part because for iron-clad warships the high cost of iron excluded the slenderforms recommended by Russell. Civil naval engineers were far less enthusiastic.
In aBritish Association report of 1 869, the Principal of the Royal School of Naval Architecture and Marine Engineers, Charles Merrifield, pointed to Reech's similitude conditionsand held ignorance of these conditions responsible for the past failures of the modelsapproach. Yet his general distrust of theory prompted him to recommend a new series offull-scale experiments in the name of the BA committee.373839Fig. 7.8.Froude's Raven and Swan models. From Froude [1 957] p. 132 (photos), [1869b] (half water lines).37Reech [1852] p.
274."cr.Wright [1983] pp. 1 31-6.39Merrifield [1869] pp. 24-5. Merrifield had read Eugime Flachat's treatise on navigation, which reproducedwritings by Sim6on Bourgois and Stanislas Dupuy de L6me, including discussions ofReech's similitude conditions(Fiachat [1866] vol. I , pp. 1 65n-167n, 214n). Bourgois noted that the frictional resistance measured by Beaufoy didnot meet these conditions, so that their application to the total resistance could only be approximate (ibid. p. 167n).280WORLDS OF FLOWFroude, who belonged to this committee, defended his own model approach in a longappendix to the report.
He did the same at the Institution of Naval Architects, whereRussell cited his own past failure to exploit model data and ironically questioned thefuture of this approach:You wiJI have on the small scale a series of beautiful, interesting little experiments,which I am sure will afford Mr. Froude infinite pleasure in making them, as they didto me, and will afford you infinite pleasure in the hearing ofthem; but which are quiteremote from any practical results upon the large scale.Froude's defense brought forward the similitude conditions, the need to explore unusualshapes, the practical impossibility of predicting wave resistance, the agreement of his viewswith Rankine's earlier theories, and the success of his preliminary experiments on Ravenand Swan.
He conceded difficulties with small-scale towing, especially in the achievementof uniform speed, but felt able to surmount them. Merrifield, whose own full-scale towingproject had just been rejected by the Admiralty, rejoiced magnanimously over the supportgiven to 'a man of proven ability' (Froude).40Froude built a 25-foot long, 33-foot wide, 1 0-foot deep tank in Chelton Cross, near hishome town of Torquay.
In his first experiments in this tank, reported in 1872, he towed aplate edgewise through the water, with the skin friction of ship hulls in mind. LikeRankine, he believed that the resistance of any fair-shaped ship was mainly due to skinfriction, and therefore computable if the laws of this sort of resistance were known.However, he doubted the correctness of Rankine's and others' assumptions about theselaws. His suspicion derived from his involvement in a water-main problem in Torquayaround 1 869.
After a few tests, he had determined that the deplored loss of head was notdue to obstructions, but to the roughness of the oxidized internal surface of the pipe.Scraping solved the problem.41While pondering on the effect of roughness-which he wrongly believed to be unknownto hydraulicians-Froude came to question Beaufoy's and Rankine's assumption that thefriction on a plate moving edgewise was uniform along it. In a memoir of 1 869, he2explained why it should not be so:4It is certain that the anterior portions of the surface, in rubbing against the particleswhich it passes, and experiencing resistance from them,equivalent force in the direction of the motion, andmust impress on them anmust impart to them somevelocity in that direction.
Thus, though it may be in some sense asserted that theanterior portions of the plane rub against the contiguous particles with the entirevelocity of the plane, since these particles are undisturbed, this cannot be trulyasserted of the posterior portions of the plane, since the particles against whichthese rub have already received a velocity conformable to that of the plane; and a'state of motion' will be thus produced in the contiguous particles involving awidening body of fluid, and with increasing velocity imparted to it, as we recedefoot by foot sternward along the plane; forming in fact a 'current', created and left40Froude [1869b]; Merrifield [1870] pp. 82 (Russell's comment), 87-90 (Froude's defense), 80-1 (Merrifield).41See R.
Froude [1869].42W. Froude [1869a] p. 212. Darcy had earlier emphasized the role of roughness, see Chapter 6, p. 232.Froude's description anticipates three features of modern boundary-layer theory, namely, the growth of the layer,the gradual decrease of wall stress along the wall, and the momentum balance.DRAG AND LIFT281behind, by the transit of the plane, such that if we could integrate the volume ofcurrent created in each unit of time, and the exact velocity possessed by each of itsparticles, the aggregate momentum must be precisely that which is due to thefrictional resistance of the entire plane acting during that unit of time. Obviouslythe sternward portions of the plane moving forward in such a favouring current, mustexperience a less intense frictional resistance than the anterior portions.With Rankine, Froude shared the idea of a growing layer of dragged fluid and therelation between wake momentum and resistance.
Unlike Rankine, he did not regard thefriction on the walls as being determined by the sliding velocity of the potential flow alongthe surface of the body. Instead, he made this friction depend on the normal gradient of thelongitudinal fluid velocity. Whereas Rankine's sliding velocity is a constant along a plane,Froude's wall stress decreases along the plate owing to internal fluid friction. As Froudelater wrote, 'it is the motion of the surface relative to contiguous particles, and not relativeto distant ones, that governs the resistance.' Based on this idea, he indicated a way tocompute the spreading of the motion from an infmite plate suddenly set in uniform motionin its own plane: he assumed the frictional force to be a(8u/ay)2 between consecutive layersof the fluid, and balanced the inertial force of each layer (of thickness 71) with the difference27Ja(8uj8y)(82 uj8y2) between the frictional forces on its two faces.
Froude next suggestedthat the solution would also apply to the case of a finite plane penetrating a still fluid with aconstant velocity. However, he was reaching the limits of his mathematics.43In his plank-towing experiments of 1 872 and 1 874, Froude verified that the resistancewas not proportional to the length of the plank, and that it depended on the roughness ofthe surface, with a velocity exponent ranging between 1 .83 for the smoothest surface(varnished) and 2.0 for the roughest one (sand-coated) in the case of the longest plank(50 feet).
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