Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 83
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Consequently,the retarded layer reaches the thickness R for a distance of the order x = UR2 jv, wbichmeans that the thickness grows with x as .jvxj U.Z1radius of the tube,To summarize these French contributions, Saint-Venant's semi-empirical approach tohydraulic questions led to a well-defined strategy to take into account the turbulentcharacter of the .fluid motion in the resistance problem and in similar problems of retardation. Most essential were the recourse to momentum and energy balance in astutelychosen spatial domains, and the concept of effective stress depending on eddy viscosity.Saint-Venant and Boussinesq thus made sense of a large number of hydraulic measurements.
Their theories nonetheless lacked predictive power, for they involved adjustablefunctions giving the distribution of the turbulent eddying in the fluid. This objection doesnot apply to laminar flows, in which case Boussinesq obtained the beginning of the motionby purely deductive means. He did not extend these insights to the resistance problem, forhe lived in a world of rivers and canals rather than ships or airplanes.7.2 Ship resistanceUntil the 1 830s, the form of ship hulls was usually decided according to conservative andempirical principles.
Naval architects distrusted theory-for good reason, as we mayretrospectively judge. Contemporary hydraulics and hydrodynamics yielded an abouteven share of correct and incorrect ideas. True were the mostly quadratic dependence ofresistance on velocity, Bossut's wave contribution, and Beaufoy's skin friction. Wrongwere the concept of a bow resistance resulting from the impact of repelled water, theproportionality of the resistance with the mid-ship section, and the notion of a solid ofleast resistance. The latter ideas were dangerously stamped with Newton's authority.22Pierre Bouguer enshrined them in his widely-used Traite du navire ( 1 746).For most kinds of commercial ships, the resistance of water was only a minor consideration among others that determined the preferred form of the hull.
The required amount of21Boussinesq [1890], (1891]. The modem reader recognizes Prandtl's law for the growth of a laminar boundarylayer.22Cf. Wright (1983] chap. 2.274WORLDS OF FLOWwood, the weight and volume of the intended cargo, and the stability at sea were mostimportant. A better understanding of ship resistance only began to matter with the development of steam-powered, high-speed navigation in the first decades of the nineteenthcentury. One of the most important duties of the British Association for the Advancementof Science, founded in 1 8 3 1 , was precisely to favor the scientific study of navigation.
A seriesZ3of expert committees were formed to study ship resistance, stability, and propulsion.Scott Russell steered a couple of these committees in the 1 8 30s, and thus promoted hisideas on the contribution of wave formation to ship resistance. The hollow lines of thebow, and the proportions he gave to the rest of the hull were meant to minimize waveformation.
Although they improved on more conservative designs, they rested on a fragiletheoretical basis. Russell understood little of the principles of mechanics, and reasonedmo�tly through intuition, analogy, and empirical induction. Where we would see energywasted through the constant emission of periodic waves, he instead saw a conflict betweenZthe 'bow wave' (surge of water) and the progression of the ship. 47 .2.1Rankine's friction layer and stream linesThe first British theorist of ship resistance who knew enough fluid mechanics was theGlasgow engineering professor William Macquorn Rankine.
Educated at the UniversityofEdinburgh, experienced in railways and hydraulic engineering, and a major contributorto the new mechanical theory of heat, Rankine best embodied a rising engineering sciencethat profited from the fundamental theories of physics. His first considerations of shipresistance derived from his friendship with the Scottish shipbuilder James Napier who, in1 857, asked him for advice about the engine power necessary to propel a ship of givenshape and size. Apparently, Napier did not trust the 'Admiralty formula' that had so farbeen used for this purpose:(7.4)where II is the power,25S is themid-ship section,V is the velocity, and C is an empiricalconstant.Rankine communicated his own formula privately to Napier, and, 'for the sake ofrecord', as an anagram in the August 1 858 issue of the Philosophical Magazine. He proudly26announced:In the course oflast year there were communicated to me in confidence the results of agreat body of experimentation on the engine power required to propel steam-slrips ofvarious sizes and figures at various speeds.
From these results I deduced a generalformula for the resistance of ships having such figures as usually occur in steamers,wlrich on the 23'd of December, 1857, I communicated to the owner of the experimental data; and he has since applied it to practice with complete success.Five years later, Rankine revealed his secret theory to the learned public. For shipsdesigned according to Russell's wave principle, Rankine reasoned, the main cause ofresistance had to be 'skin friction', that is, the force exerted tangentially by the water sliding23Cf.
Wright [1983] Chap. 3.25Cf. Wright [1983] pp. 89, 106-19.24See Chapter 2, p. 51.26Rankine [1858] p. 238.DRAG AND LIFT275on the hull. In approximate conformance with Beaufoy's old measurements and by analogywith pipe retardation, Rankine took this force to be proportional to the fluid density and tothe square of the sliding velocity.
Unlike Beaufoy, however, he did not assume this velocityto be constant along the hull. Instead, as 'the only assumption', he propounded 'that theagitation in the water caused by the friction on the ship's bottom extends only to a layer ofwater which is very thin as compared with the dimensions of the ship.' Beyond that layer, heassumed a smooth motion obeying Euler's equation. In the case ofRussell's trochoidallines,Rankine determined the corresponding flow in Gerstner's manner and thus obtained thesliding velocity v as a function of the curvilinear abscissa s along the hull.Z7Rankine then equated the propelling engine power to the work done by the frictionalforces:(7.
5),wherejis a friction coefficient borrowed from Julius Weisbach's pipe-retardation formula,p is the density of water,breadth, and£1G is the mean girth of the ship,V is its velocity,B is its greatestis the length between the bow and the stem. The parenthesis or 'augmented length' factor contains the effect of the curvature of the hull. Rankine further obtainedthe resistance as the ratio PI V. The direct summation of the longitudinal component ofthefrictional force leads to a smaller result. Rankine attributed this discrepancy to a reactionof the hull on the water that slightly deformed the lines of flow and thus lowered the8pressure on the stem.
2In•••1 864, Rankine clearly distinguished three contributions to ship resistance:a blunt stem leading to large eddies;a front surge leading to surface waves;frictional eddies.For a fair-shaped ship at moderate velocity, this last cause was the only important one.Rankine described the relevant process in a manner probably reminiscent of Darcy: 29The resistance due to frictional eddies .
. . is a combination of the direct and indirecteffects of the adhesion between the skin of the ship and the particles of water whichglide over it; which adhesion, together with the stiffness [viscosity] of the water,occasions the production of a vast number of small whirls or eddies in the layer ofwater immediately adjoining the ship's surface. The velocity with which the particlesof water whirl in these eddies bears some fixed proportion to that with which theseparticles glide over the ship's surface: hence the actual energy of the whirling motionimpressed on a given mass of water at the expense of the propelling power of the ship,being proportional to the square of the velocity of the whirling motion, is proportional to the square of the velocity of gliding.27Rankine [1862] p. 23.
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