Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 72
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. . We do not yet havereasonable notions about the internal motions of fluids and the mutual actions oftheir molecules.Boussinesq cut the Gordian knot by guessing the form of the effective viscosity andverifying observable consequences of the guess. 47According to Boussinesq's intuition, whirling originates in the macroscopic fluid slideon an unavoidably irregular wall and then propagates through the rest of the fluid, withan intensity depending globally on the breadth of the flow and locally on the distancefrom the wall. The effective viscosity then results from the additional momentumexchange that local agitation implies between successive fluid filaments.
In the case ofa wide rectangular channel, Boussinesq assumed an effective viscosity proportional to thesliding velocity vo on the walls and to the depthh of the water.In the case of a circularchannel, he took it to be proportional to the sliding velocity on the walls, to the radiusof the channel, and to the ratio rradius.48IRRof the distance from the axis of the channel to itsBoussinesq substituted these values into the Navier-Stokes equation, which he carefully rederived for large-scale, secular motions, in a manner reminiscent of SaintVenant's reasoning of1843.For the boundary conditions, Boussinesq assumed a frictionproportional to the square of the velocity slip on the bottom and a vanishing stresscomponent on the surface. He then derived and discussed relations between surface slopeand discharge in the following cases of increasing difficulty: permanent and uniform,permanent and slowly varying, permanent and quickly varying, and non-permanent.
Theconsideration of the last two cases was new, except for some anticipations by SaintVenant. The consideration of the first two cases led to important corrections to Belanger's and Coriolis's theories.49For a better control of approximations, Boussinesq generally started from the NavierStokes equation in an adequate coordinate system and integrated over a fluid section. Thefollowing is a sketch of simpler but less rigorous deductions based on a momentum46Boussinesq [1 868].47Boussinesq [1 870], [1871b], [1872c]; Saint-Venant [1872] p.
774; Darcy and Bazin [1 865a] p. 30.48Boussinesq [1877] pp. 45-51 .49Cf. Saint-Venant [1 873].236WORLDS OF FLOWbalance for fluid elements or slices. Although Boussinesq knew of such deductions, SaintVenant was the first to publish them in full in a suggestive memoir of 1 887. 506.3.3 Corio/is revisitedIn the permanent, uniform case, the main novelty of Boussinesq's approach was thedetermination of the velocity profile. For a wide rectangular channel, the viscosity e is aconstant which is proportional to the depth and to the sliding velocity at the bottom.Balancing the weight of a truncated fluid filament with the stresses acting on its lower andupper sides (the pressures on the right and left sections cancel each other), Boussinesqfound that(6.1 3)where x is the distance of the fluid filament from the surface and i is the slope.
This impliesa quadratic velocity profile, in conformance with Bazin's (still partial) measurements (seeFig. 6.5).51Denote by v0 the velocity on the bottom, and h the distance between the bottom andthe surface. The expression e = pgAhvo for the viscosity, the formula F = pgBifo for thefriction on the bottom, the second-order equation (6.13), and the boundary conditionsv'(O) = 0 and, -ev'(h) = F lead to the formula( y}-)vB1.= +:;; 1 2A - /;2(6.14)Hence the average velocity U is proportional to the velocity vo, which is itself related to theslope i by the formula pghi = F = pgBifo which expresses the balance between the weightof a normal slice of fluid and the external friction F. In conclusion, the Cht\zy formula (6.1)still holds, despite the transverse variation of the velocity.
520XFig. 6.5.Boussinesq's velocity profile (horizontal arrows) for permanent flow in a wide rectangular channel.50Saint-Venant [1 887c] pp. 168-76. See also the simplified deductions for the case of a broad, rectangularchannel in Boussinesq [1897].5 1 Boussinesq [1 877] p. 72.52/bid. p. 73.TURBULENCE237In the permanent, slowly-varying case, consider a slice ds of fluid perpendicular to thes-axis, which is chosen to be parallel to the surface. The s-component of the weight of thisslice is pgSI ds, where S is the fluid section and I is the slope of the surface. The pressureson both sides mutually cancel (up to a higher-order term), since their transverse variationis hydrostatic.
The frictional force is -ds JF0 dx, where F0 denotes the wall friction perunit surface area for the shift velocity v and the integration runs over the wetted perimeterx (along which v may vary). The momentum variation per unit time is -d JJ v (pvdS),where the integration is performed over the section S. Denoting by TJ the number suchthat(6.15)and neglecting the variation of this number with s, the dynamical equilibrium of the slicereduces to(6.16)This equation differs from Coriolis's equation (6.
1 1) in two ways. Firstly, the coefficientI + TJ differs from Coriolis's a. Secondly, the velocity v in the frictional force differs fromthe average velocity U. 53The extent of the latter difference can be derived from the ,velocity profile. By areasoning similar to that used in the uniform case, for a wide rectangular channel thisprofile satisfies the equation(6.17)where v denotes the acceleration of a fluid particle that results from the variation of thefluid section. This acceleration is given by -ifh-1 dh/ds in the approximation for whichthe slope of the fluid filaments varies linearly between the bottom and the surface. Since itis not uniform in a given section of the fluid, a distortion of the velocity profile results.Boussinesq determined this distortion approximately by replacing the acceleration withthe value it would have if the velocity profile were still parabolic.
The resulting differencebetween Boussinesq's and Coriolis's frictional terms has the same form as the inertial term,so that eqn (6. 1 6) can be rewritten as54(6.18)This equation has exactly the same form as Coriolis's, but with a new value for thenumerical coefficient of the gradient of the velocity head. As Boussinesq emphasized, thissimilarity of form should not be mistaken for a justification of Coriolis's method. On the53Ibid.
p. 66,54Boussinesq [1877] pp. 92, 1 1 2.WORLDS OF FLOW238contrary, the new derivation makes it clear that Coriolis erred in equating the work ofpressure, gravity, and frictional forces to the increase of large-scale kinetic energy; whenever the fluid section changes, part of the energy of the large-scale motion goes to smallerscale, eddying motion. Coriolis also underestimated the error he committed in assumingthe same expression for the work of frictional forces in the uniform and varying cases. 55Boussinesq then approached the more delicate problem of quickly-varying permanentmotion.
In this case, a non-negligible centrifugal force acts on the fluid filaments. To firstorder in the filaments' curvature, and for a wide rectangular channel, the resultingequation is( ) [I3 (2g )d u2xFu1 = - + (1 + 71 + /3) - - - h2Spgds 2g]1 u2 d2id3 u2- +--- .ds32 gh ds2- -(6.19)Boussinesq deduced the profile of a hydraulic jump and confirmed Bazin's distinctionbetween two sorts ofjumps with or without long-range oscillations. He also refined SaintVenant's distinction between torrents and rivers, now introducing an intermediate category of 'moderate torrents' for which jumps can occur, but only long and wavy ones.Lastly, Boussinesq obtained equations for non-permanent flow, and used them to discusswaves, river tides, tidal bores (mascarets), and floods.
566.3.4 Praise and neglectBoussinesq's theory of open channels appeared in several papers between 1870 and 1872,and formed, together with an extension to pipe flow, the substance of a long essaysubmitted in 1872 to the Academie des Sciences and published five years later in itsMemoires des savants etrangers together with Saint-Venant's laudatory report. SaintVenant had himself introduced an equation for non-permanent flow in 1 871 and hadintegrated it in a simple case of fluvial tide, though only for negligible curvature.
Also, hehad obtained the equation for rapidly-varying permanent flow as early as 185 1. After hisdeath, Boussinesq discovered and published the relevant manuscript as a homage to hismentor's modesty. Probably not to discourage his beloved disciple, Saint-Venant hadhidden his priority on this important aspect of the theory of open channels. 57In his report on Boussinesq's essay, Saint-Venant emphasized its recourse to subtlemethods of approximation and its agreement with Darcy's and Bazin's data:These numerous results of a high analysis, founded on a detailed discussion and onjudicious comparisons of quantities of various orders of smallness, sometimes to bekept, sometimes to be neglected or abstracted, and their constant conformity with theresults obtained by the most careful experimenters and observers, appear mostremarkable to me.55Boussinesq [1877] pp.
1 12-13.56Ibid. pp. 193, 196-217, 242-529. There is also a fourth part which includes notes on efflux, weirs, turns, andsome effects of capillarity.57Boussinesq [1 870], [1 871b], [1 872c], [1 877]; Boussinesq's introduction to Saint-Venant [1887b] pp. 5-6(hidden priority); Saint-Venant [1 873], [1871a], [1851b].TURBULENCE239Elsewhere, Saint-Venant praised 'the man who possessed intuition no less than highcalculus, who knew how to invent new integrals for the needs of thisintimate mechanicsborn in our century from our France, and pertaining to the real things of the terrestrial58world that we inhabit.'The reception of Boussinesq's theory was not as warm as Saint-Venant's commentswould suggest. Most hydraulicians could not follow its analytical sophistication andregarded the practical circumstances of hydraulics as too complex for precise quantitativeanalysis. For example, in his authoritative treatise of1882, the French hydraulician MichelGraeff condemned the application of backwater theory to the navigability of riversbecause of the complex and variable shape of real beds.
After praising Saint-Venant'sand Boussinesq's analytical skills, he put forward his own empirical methods. British andAmerican hydraulicians ignored the higher French theories. Some German-languagehydraulicians picked up on Boussinesq's theory. In his major treatise of1914, the AustrianPhilipp Forchheimer gave detailed accounts of this and earlier French theories, whichSaint-Venant's disciple Alfred Flamant had made more accessible in his clear, pedagogicaltreatise of1891.In1920,Josef Kozeny improved Boussinesq's theory of pipe flow byintroducing a new form for the effective viscosity in circular pipes.
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