Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 82
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1 1These private considerations did not foster any quantitative estimate of resistance. InBritain, the Rayleigh-Kirchhoff dead-water theory remained the only published, quantitative estimate of resistance in a fluid of small viscosity. Before the end of the century,Samuel Langley's and William Henry Dines's resistance measurements showed that thistheory failed by at least a factor of three in the case of a blade moving parallel to itself.Helmholtz's surfaces of discontinuity nonetheless enjoyed some popularity among physicists and mathematicians.
Horace Lamb and Alfred Basset devoted to them long sectionsof their widely-used treatises. In Germany they graced the lectures of the influentialMunich professor August Foppl (published in 1899), who even declared: 12The consideration of separation surfaces represents the first and the most importantstep toward a theory that better accounts for the facts . . .
Helmholtz's doctrine offluidjets is therefore to be regarded as a remarkable progress of hydrodynamics, eventhough it leaves much to be desired with regard to physical exactness.In 190 1 , the Italian mathematician Tullio Levi-Civita made discontinuity surfacesresponsible for the fluid resistance of bodies of any shape (see Fig. 7.4). Like Stokes, hetraced the failure of early ideal-fluid theories of resistance to an unwarranted assumption:The [usual] analytical formulation of the problem [of the flow of an ideal fluid arounda solid body] introduces some elements, seemingly innocuous but much more remote10See Chapter 5, pp.
197-207.1 1Thomson to Stokes, 27 Dec. 1898, ST; Stokes to Thomson, 30 Dec. 1898, ST. See Chapter 5, pp. 204-6. Themodem reader may recognize a boundary layer with separation.12Langley [1891]; Dines [1891]; Foppl [1899] p. 396.WORLDS OF FLOW270AFig. 7 .4.The discontinuity surface (8) and dead-water region (B) for the flow past a bluffbody moving at thevelocity v through a perfect liquid. From Levi-Civita [1907] p. 131.from reality than is the character of the perfect fluid. Such is, in my opinion, thehypothesis of the continuity of the movement of the fluid in the entire space aroundthe body.Again like Stokes, Levi-Civita assumed the formation of discontinuity surfaces even forround bodies.
He then showed that a dead-water wake generally implied a resistanceproportional to the fluid density and to the squared velocity of the body. However, he didnot address the question of the location of the curve s from which the discontinuity surfacedeparted. Nor did he know, in190 1 , how to solve the system of equations that determinethe shape of the discontinuity surface for a given curves.He accomplished this difficulttask in a larger memoir of 1907 through a broad extension of Helmholtz's complex-planemethodYIn summary, at the turn of the century, discontinuity surfaces remained the mainanalytical approach to the resistance problem for a slightly-viscous fluid.
Yet they hadwell-identified shortcomings, namely: they led to utterly instable and physically impossiblemotions, they gave smaller resistances than in reality, and they were essentially indeterminate in the case of smoothly-shaped bodies. For a Kelvin, these defects were fatal. For aRayleigh, a Foppl, or a Levi-Civita, discontinuity surfaces marked a significant steptoward a successful theory of resistance.7 .
1 .3Turbulent wakes and shear layersAnother way to solve d'Alembert's paradox was to assume some instability of the laminarflow of a slightly-viscous fluid that prompted turbulent eddying in the rear of the body.Stokes first suggested this option in1 843. Poncelet and Saint-Venant made it the basis of1 846, Saint-Venant placed the fixed body within thecurrent of a cylindrical pipe (see Fig. 7.5) which was so wide as to leave the resistancequantitative resistance estimates. In13Levi-Civita [1901] p.
130, [1 907].DRAG AND LIFTFig.2717.5. Flow around a truncated body (BDD) placedwithin a cylindrical pipe. From Saint-Venant [1887b]p. 89.unchanged. The momentum which the incompressible fluid conveys to the body in a unittime is equal to the difference P0S - PtS between the pressures on the faces of a column offluid extending far before and after the body, because the momentum of the fluid columnremains unchanged. The work (PoS - P1 S) vo of these pressures in a unit time is equal thelive force of the 'non-translatory motions' generated in the fluid.
Hence the resistance isgiven by this live force divided by the original velocity of the fluid. In the non-translatorymotion, Saint-Venant included both the small-scale motions that are a direct consequenceof viscous stress and the 'tumultuous', whirling motions observed at the rear of bluffbodies. 1 4Saint-Venant then improved on a method invented by Poncelet to estimate the magnitude of the resistance, and based on the assumption that the wall pressure behind aseparation point does not differ much from the value that Bernoulli's law gives it in themost contracted section of the flow.
In the simple case of the truncated body of Fig. 7 .5,the pressure at the rear is thus made equal to the pressure Pt in the section w 1 • SaintVenant further included a stress acting tangentially to the walls of the body, mostly due toeddy viscosity in the case of a turbulent incoming flow. In conformance with the standardtreatment of pipe and channel retardation, he assumed a large-scale sliding velocity of the1fluid along the walls, and made the friction proportional to this velocity squared. 5In the case of a plate parallel to the flow, for which wall friction is clearly the only causeof resistance, Saint-Venant compared Beaufoy's and Du Buat's measurements with thethen-accepted friction coefficient in cylindrical pipes.
The result indicated that the slidingvelocity along the plate had to be smaller than the velocity of the incoming flow. SaintVenant explained this difference as a retarding effect of eddy viscosity for the large-scale16flow beyond the slide on the walls (see Fig. 7. 6).Although Saint-Venant's disciple Boussinesq did not address the resistance problemper se, he abundantly developed Saint-Venant's idea of eddy viscosity in pipe and openchannel flow, with velocity profiles that had finite slide on the walls and parabolic (for14Saint-Venant [1 846b]. See Chapter 3, pp.
134-5.15/bid. pp. 28, 72-8, 120-1; Saint-Venant [1 887b] pp. 56-192.16/bid. (from a MS of 1847) pp. 1 16-49. The modern reader may recognize here a turbulent boundary layer,although Saint-Venant neglected any variation of this layer along the wall (in conformance with his assumption ofa quadratic dependence on velocity).272WORLDS OF FLOWFig. 7.6.Velocity profile (FGHI, with the velocitiesDF, KG, LH, EI) for the flow between a body(DBD') and a cylindric wall (MM'NN'). FromSaint-Venant [1887b] p.
132.M'lE'rectangular channels) or cubic (for circular pipes) increase from the wallsY In 1 880,Boussinesq discussed the more academic question of the role played by viscosity andadherence to the walls at the beginning of a laminar flow. He did this in reaction to JacquesBresse's erroneous extension to viscous fluids ofLagrange's theorem, according to which avelocity potential exists for any fluid motion started from rest. 1 8With Boussinesq, consider the simple case of a constant, uniform, and horizontalaccelerating force pk applied at time zero and onward to the entire mass of a viscousfluid resting over the horizontal plane z = 0.
The resulting flow is obviously parallel to theplane, and its velocity u vanishes on the plane at any time. The Navier-Stokes equation fora kinematic viscosity v gives(7.2)in which Boussinesq inunediately recogrlized Fourier's equation for the diffusion of heat.The relevant solution is(7.3)with a = z/2-/Vi. Consequently, the retarding effect of the wall is only sensible in a layerwhose thickness is comparable to yVi.1 9In a second note, Boussinesq insisted that wall stress played an essential role indetermining the nature of the motion in any hydraulic problem and that his simplecalculation revealed the general mechanism through which rotational motion began inany flow:2017These profiles agreed with Darcy's and Bazin's measurements. They can now be seen as approximations tothe logarithmic profiles given by turbulent boundary-layer theory (cf.
Prandtl [1933] p. 833n).1 8Boussinesq [1880a].19 Stokes had already treated a similar problem in his pendulum memoir of 1 850 (see later on pp. 290-1).20Boussinesq [1880b] p. 967.DRAG AND LIFT273The retarding influence of a wall wiiJ first only be sensible in the vicinity of this wall.Hence some time will elapse before the similar influences of the other walls reach thisregion, and it will therefore be permitted to evaluate the velocity variation at thebeginning of motion as if . . .
the wall under consideration had infinite breadth andthe fluid mass had infinite thickness . . . Hence, [my previous calculation] most simplyexpresses what happens at the beginning of any flow, and demonstrates the generalmechanism, abstractedfrom accessory complications.Ten years later, Boussinesq examined a similar question in an attempt to correct forentrance effects in some of Poiseuille's experiments, namely: how does the velocity profileof a viscous fluid entering a capillary tube evolve toward the uniform, parabolic profile?Assuming a rectangular profile at the entrance of the tube, he showed that an annular layerof retarded fluid grew from the walls until it reached the central part of the tube.
Throughan approximate solution of the Navier-Stokes equation, he found that the departure fromis thethe steady profile varied as e-I6vxfUR' , where x is the distance from the entrance,RU is the average velocity, and v is the kinematic viscosity.
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