Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 86
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Whereas he did not comment on the relation between exponent and roughness, hegave the following discussion of the unexpectedly slow decrease of the friction a few feetbehind the cutwater.44Assuming an approximately linear transverse variation of the velocity in the currentinduced by the plank's motion on each face, denoting by H the thickness of this current atthe end of the plank, U the velocity of the plank, and p the density of water, Froudeestimated the momentum flux (per unit breadth) in the wake to be pHU2/3. Equating thisvalue to the measured resistance in Rankine's manner, he derived values of H thatmatched observations and that increased with the length of the plank.
This growth ofthe favoring current suggested a rapid decrease of the friction with the distance from thecutwater, in contradiction with the measurements. As a solution to this paradox, Froudeimagined that violent eddying in the boundary current fed undisturbed fluid particles fromthe outer margin of the current to the surface of the plank.4543Froude [1 874a] p. 253 (quote), [1869a] pp. 212-13 (computation).
The modern reader may recognizePrandtl's assumption for the stress within a turbulent boundary layer in the case of a constant mixing lengthVa/P (in reality, the mixing length grows linearly with the distance from the wall), as well as Rayleigh's idea([1911]) of connecting temporal and spatial growths of a boundary layer. See later on pp. 290-1, 297-9.44Froude [1872], [1874a].45Froude [1874a] p. 253.
In conformance with Froude's view, Karmin's theory of 1921 yields a turbulentboundary layer growing as the power 4 I 5 of the distance from the cutwater, and a wall stress decreasing as thepower -1/5 of this distance. See later on pp. 296-7.282WORLDS OF FLOWIn the same period, Froude performed full-scale experiments on HMS Greyhound andcompared the results with measurements done on a model of this ship.
This time Froudeno longer assumed a quadratic form of the skin friction. Instead, he computed the skinresistance by extrapolation from his plank measurements. Then he subtracted this resistance from the measured total resistance, and applied his scaling rules to the remainingresistance. An impressive match between the model data and the full-scale data resulted.Froude concluded:46The experiments with the shlp, when compared with those tried with her model,substantially verify the law of comparison which has been propounded by me asgoverning the relation between the resistances of shlps and their models.
This justifiesthe reliance I have placed on the method of investigating the effects of variation ofform by trials with varied models-a method which, if trustworthy, is equallyserviceable for testing abstract formulae, or for feeling the way towards perfectionby a strictly inductive process.Froude's main service to naval engineering was indeed the development of the rationaluse of models. As he showed, the proper exploitation of model data required the knowledge of the scaling laws for non-frictional resistance, and some understanding of themechanism of skin friction.
He modestly admitted to having borrowed most of histheoretical ideas from colleagues with higher mathematical skills (Rankine, Thomson,and Stokes): 'I am but insisting on views which the highest mathematicians ofthe day haveestablished irrefutably; and rriy work has been to appreciate and adapt these views whenpresented to me.' Froude nevertheless grasped aspects of fluid motion that had eluded hispredecessors. He understood that the variation of friction along a ship hull depended onan internal fluid-stress mechanism acting within a growing boundary layer of draggedfluid. Moreover, he foresaw the role of destructive wave interference in lowering the waveresistance of some ship shapes, such as the Swan of 1 867, and he described important wavephenomena, such as group velocity and echelon waves, thus stimulating mathematicalstudies by Stokes, Rayleigh, and Kelvin.47A last service of Froude was his simple, pedagogical explanation of the principles of shipresistance for lay audiences.
Unconsciously imitating Euler, he derived d'Alembert's paradox through momentum balance along tubes of flow, thus condemning the fallacy of 'headresistance' that had so long impeded the progress of naval architecture. The true causes ofship resistance, he went on, were skin friction, wave emission, and large-eddy production.About each kind ofresistance he had a simple wisdom to offer. Skin resistance is about thesame on a ship hull as on a flat surface, wave resistance only counts at velocities for whichthe wavelength is comparable to the ship's dimensions, and eddy resistance essentiallydepends on the tendency of stream lines to separate from a blunt stem and thus to form adead-water, eddying region: 'Blunt tails rather than blunt nose cause eddies.'4846Froude [1874b] p.
59.47Froude [1877a] p. 213, [1 877b] (on ship waves). See Chapter 2, pp. 85-6.48Froude [1875], [1877a] p. 205. Rayleigh ([1918] p. 553) claims to have obtained from Froude the idea (usuallyattributed to Prandtl) that separation is due to 'the loss ofvelocity near the walls in consequence of fluid friction,which is such that the fluid in question is unable to penetrate into what should be the region of higher pressure.'I have not been able to locate any statement of this sort in Froude's writings.DRAG AND LIFT283In summary, the development of steam-powered navigation prompted scientific studiesof fluid resistance by RusseU, Rankine, and Froude, spanning from the mid-1830s to the1870s. These three investigators recognized, with increasing accuracy, the importance ofwave resistance for partiaUy-immersed bodies moving at sufficient speed.
None of them,however, could theoreticaUy predict the amount of this resistance. Froude remedied thisweakness by a rational use of model measurements. Rankine and Froude recognized thatthe motion of water around a fair-shaped ship huU was mostly governed by the corresponding irrotational solution of Euler's equation, except for a layer of fluid adjacent tothe hull, in which complex eddying motion occurred.
Froude understood that the behaviorof this layer and the resulting skin friction depended on internal friction within the layer.As he did not have the means to develop a quantitative theory of this behavior, he againrelied on smaU-scale experiments, and extrapolated the results to large-scale skin friction.This research better achieved its aim, namely the prediction of ship resistance, than theresistance theories discussed above and based on the concepts of discontinuity surfacesand eddy viscosity.
The key to this empirical efficiency was not the elaboration of aquantitative, deductive theory. It was a qualitative understanding of the implied physicalprocesses along with the rational exploitation of smaU-scale experiments.7.3 Boundary layers7.3. 1Prandtl's Heidelberg paperAfter completing his engineering studies at the Technische Hochschule in Munich, LudwigPrandtl obtained a doctorate in 1 898 under Ludwig F6ppl on the lateral instability ofbeams in bending.
From F6ppl he learnt a kind of engineering science that relied on highermathematical skills, fundamental physical theory, and multifarious approximation strategies. He went on to work in the Maschinenfabrik Augsburg-Niirnberg, where he wasasked to improve a suction device for the removal of shavings. While working on thisproject, he realized that the pressure rise expected in a sharply-divergent tube failed tooccur because the lines of flow tended to separate from the walls-as Daniel Bernoulli hadlong ago noted in a similar hydraulic case. Prandtl later remembered this observation tohave started the chain of reasoning that led him to the boundary-layer approach toresistance in slightly-viscous fluids.49In the foUowing years, Prandtl developed his resistance theory and tested it with a watertank of his own making, while teaching mechanics at the Technische Hochschulein Hannover.
At the third international congress of mathematics held in Heidelberg in1 904, he had ten minutes to announce results that inspired much fruitful research insubsequent years. In the short, dense report published in 1905, he began with the 'unpleasant properties' of the Navier-Stokes equation(7.6)Solutions were known, he noted, for the simpler equations obtained by omitting either thenonlinear term p(v v)v or the viscous term iJ-AV.
No non-trivial solution had yet been·49Prandtl [1 948]. For biographical data, cf. Lienhard [1975], Rotta [1990].WORLDS OF FLOW284found for the complete equation. For a slightly-viscous fluid such as water or air, a naturalcourse was to omit the viscous term. Alas, the resulting solutions to resistance problemsdiffered widely from the observed behavior. A different strategy was needed.
50Prandtl assumed that the viscous term JJ-I1V could be neglected everywhere except in a'boundary layer' (Grenzschicht) or 'transition layer' (Obergangschicht) of fluid near thesolid walls on which the fluid adheres. This layer remains thin only if the path of fluidparticles along the walls is not too long. Without proof, Prandtl further asserted that, if theviscosity JJ- is an infinitesimal of second order, then the width of the transition layer and thenormal velocity within the layer are of first order, and the normal pressure gradient andthe curvature of the lines of flow are negligible.
He presumably reasoned as follows. 51For a two-dimensional flow, denote by 8 the thickness of the transition layer, u theparallel velocity, v the normal velocity, x the curvilinear abscissa along the wall, and y anormal curvilinear coordinate. As long as the curvature radius of the surface is largecompared to the thickness of the layer, the differential equations of the motion within thelayer have the same form as if x and y were Cartesian coordinates. As the velocity withinthe layer varies much faster in the normal than in the parallel direction, B2ujax? isnegligible compared to &uj8y2, and the Navier-Stokes equation for u readsBuBuBu1 BPB 2u- + u- + v- = - - - + v- .BtBx Byp BxBy2(7.7)The continuity equation readsau + av0.ax By =(7.8)In the zero-viscosity limit, and at a given fraction y/8 of the transition layer, the termsuBujBx, - (ljp)BPjBx, and BujBx in these equations must remain finite; the termv&uj8y2 is of the order of vj82, and the term Bv/By is of the order ofv/8.
Consequently,v is of the same order as 8, which is of the same order as .fii, and all of the terms of eqn(7.7) are ofthe same order. The Navier-Stokes equation for v further implies that BPjBy isnegligible, because all other terms are of the same order as 8. Therefore, the term-(1/p)BPjBx in eqn (7.7) may be regarded as a known function of x only that can beobtained by solving the Eulerian flow problem along the given solid body. Prandtlobtained the velocity profile of the boundary layer through the numerical, stepwiseintegration of eqns (7.7) and (7.8). 52The simplest case is that of a uniform flow of velocity U encountering a parallel, infiniteblade (see Fig.
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