Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 81
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Mariotte andHuygens, who belonged to this commission, were unaware of the vena contracta (the contraction of the fluidvein near to the opening on the vase) first described in the second edition ofNewton's Principia. Cf. Blay [1992] pp.339-42.WORLDS OF FLOW2663lengths and velocities is also a possible motion if all forces acting in the system aresimultaneously rescaled as the inverse of a length times the square of a velocity.Newton next assumed that the particles of the fluid were individually deflected by thesolid surface, and identified the force acting on this surface with the destroyed momentum(see Fig.
7.1). For normal impact, the resulting force on the surface element dS is pv2dS.When the incoming flow makes an angle IJ with the surface element, the particles' flux andthe destroyed momentum of each particles are both multiplied by sin IJ. The resulting forceis normal to the surface and has the intensity �dS sin2 1J. This implies the existence of aprow shape for which the resistance is a minimum for a given maximal breadth, a resultNewton believed to be relevant to navigation. Newton next integrated the differentialpressure over the front half of a sphere and compared the result with his own experimentson metal spheres dropped from the dome of Saint Paul's Cathedral in London.
As theresistance turned out to be smaller than expected, he offered another theory based onefflux through a partially-obstructed opening. Assuming the 'cataract' flow of Fig. 7.2, heequated the resistance to the weight of the static pyramid of fluid above the obstacle.4This odd theory was soon forgotten, except for the dubious implication that the largesttransverse section of the body determines the fluid resistance. The discrete-impact theorysurvived for about two centuries in engineering quarters, for it gave the correct dependenceFig. 7 . 1 .The Newtoruan deflection of a fluid particle against an inclined surface element.��::....,..·.·.·.·.··""""'"--t; . ...f.r,�...�\l\:E0�CFig. 7 .2.!�..
.. .. ............. . .. .s�.E;/i, \:� _,:·/�I! i ll� �' 'j•JIf•:i�··..��....,:ff:lI.Hl Bin---:!D:Newton's cataract (AEPH and BFQH) for the flow through an opening (EF) partially obstructed bythe disc PQ. From Newton [1713]: book 2, sect. 7, prop. 36.'Newton [1 687], [1713], book 2, sect. 7, prop. 32 (sintilitude theorem), 33 (general form of resistance).4Ibid. prop. 34 (implicit sin2 0 law), 35 (resistance of globe), 36-9 (cataracts), 40 (experiments). Cf. SaintVenant [1 887b] pp.
15-19, G. Smith [1998].DRAG AND LIFT267on density and velocity in the simplest manner. Yet careful resistance measurementsdisproved it several times in the second half of the eighteenth century. In the 1760s,Jean-Charles de Borda found that the resistance more nearly depended on the sine ofthe inclination of the surface elements of the body. In the 1 770s, Charles Bossut includedtwo other factors, namely, the dimensions of the towing tank, and the formation of wavesin the case of partially-immersed bodies.
In the 1 880s, Pierre Du Buat discovered that theform of the rear of the body largely controlled the resistance. He also found that a'negative pressure' (a pressure inferior to that of the undisturbed fluid) occurred in thisregion and contributed to the resistance. In the 1 890s, the British Colonel Mark Beaufoyidentified fluid friction along the walls of the body as an important contribution to theresistance.
Through precise experiments with plates towed edgewise, he showed that thisfrictional effect depended on the power 1 .8 of the velocity. In conformance with Borda'sremark, Samuel Vince proved that the resistance offered by a slightly-inclined plate variedas the sine of the inclination, rather than the Newtonian squared sine.5This impressive rise of empirical knowledge went along with a distrust of higherhydrodynamic theory. In 1 768, d' Alembert admitted that he did not know how to avoidthe vanishing resistance predicted by his theory. Two years earlier, Borda deduced theabsence of resistance in a very general manner based on the principle of live forces: heshowed that no work is needed to pull a body uniformly through a still fluid, because thelive force of the fluid motion around the body remains globally unchanged in this process.Even earlier, in 1 745, Euler had used momentum balance along tubes of flow to deduce theabsence of resistance.
In an attempt of 1 760 to escape from this conclusion, he cut off therear part of the tubes, thus effectively returning to the Newtonian theory of resistance andgiving up recourse to his own hydrodynamic equations.6In summary, at the beginning of the nineteenth century the best-founded hydrodynamictheories, namely those of d'Alembert and Euler, led to a vanishing fluid resistance.Newton's old theory accounted for a finite resistance proportional to density and squaredvelocity, but failed in any other respect.7. 1 .2 Discontinuity surfacesA first way to avoid d' Alembert's paradox was to introduce viscosity, as Navier first did in1 822. Twenty years later, Stokes successfully obtained the viscous damping of pendulumoscillations and the linear resistance formula for slowly-moving, small spheres such as thedroplets of clouds. He knew, however, that the most common kinds of fluid resistanceeluded this theory, since their dependence on velocity was quadratic instead oflinear.
Oneof Stokes's early suggestions for solving this difficulty was that a dead-water regioncircumscribed by surfaces of discontinuity occurred in the wake of bodies traveling throughan inviscid fluid. Such discontinuities were indeed compatible with Euler's equations.7In 1 868, Helmholtz independently introduced surfaces of discontinuity (Trennungsfliiche) in order to explain jet formation within a fluid. He believed that such surfaces5Cf. Saint-Venant [1 887b] pp.
27, 37-9, Rouse and !nee [1957] pp. 128-9, 133-4.6Cf. Saint-Venant [1 887b] pp. 9-1 1 , 21-37, Truesdell [1954] p. XL.7See Chapter 3.268WORLDS OF FLOWwere formed whenever the pressure of the hypothetical irrotational flow became negative,typically near a sharp edge of a solid wall. He interpreted them as infinitely-thin vortexsheets and used his vortex theorems of 1 858 to show their tendency to spirally unroll underan infinitesimal perturbation.
In geometrically simple cases of two-dimensional flow, heintroduced the velocity potential cp and the stream function !f; in the manner of d'Alembert,Lagrange, and Stokes, and managed to determine the form of the discontinuity surfaces byseeking a holomorphic function cp + i!f; that satisfied the required boundary conditions inthe plane of the complex variable x + iy. 8Kirchhoff and Rayleigh soon applied this technique to the motion around a flat plate (asegment in two dimensions, see Fig. 7.3) to derive the resistance formulaR=1r sin 1J2.
8 pv S,4 + 1T S!ll(7.1)where p is the fluid density, v is the constant fluid velocity far from the plate, S is thesurface of the plate, and IJ is its inclination. Whereas the normal direction of the resistance,and its dependence on density and velocity agree with Newton's theory, the angulardependence does not and fits experiments much better, as Rayleigh judged on the basisof Vince's old data. Helmholtz agreed that the formation of surfaces of discontinuity wasthe main source of resistance in any large-scale (high-Reynolds-number) motion, althoughthe instability of these surfaces cast doubt on any quantitative use of them.
9The recourse to surfaces of discontinuity and dead water was a controversial issue.William Thomson strictly rejected them, even though he was the British physicist closest toHelmholtz. According to Thomson, surfaces of discontinuity could never be formed in aperfect liquid, because Lagrange's and other theorems forbade the creation of vorticity;and the discontinuous state of motion, with its infinitely-long dead-water wake, waspatently absurd. In his view, the true cause of any apparent departure from potentialflow was viscosity or cavitation.
In the viscous case, the intense shear of the flow past anedge induced the production of a series of vortices that roughly imitated Helmholtz'syFig. 7.3.Flow around an inclined plate (thick line) according to Kirchhoff [1869] p. 425. The two lines ofcurrent from the edges ofthe plate delimit the dead-water region.8Helmholtz [l 868b].
See Chapter 4, pp. 163-5.9Kirchhoff [l869]; Rayleigh [l 876b]; Helmholtz [1873]. See Chapter 4, pp. 1 64-5.DRAG AND LIFT269vortex sheets. In his playful correspondence with Thomson on this matter, Stokes valiantlydefended the discontinuity surfaces, arguing that none of Helmholtz's theorems forbadethe growth of surfaces of discontinuity from a germ on the wall surface, and that the zeroviscosity limit of viscous flow led to surfaces of discontinuity whenever the lines of theEulerian flow diverged too much (typically, behind a body with a bluff rear).
1 0In 1 898, Stokes's focus on the zero-viscosity limit of viscous flow prompted Kelvin toreflect on the nature of the flow around a globe. He imagined that the motion of the globewas impulsively started from rest and then kept uniform. In the first instant, he reasoned,the induced motion is very similar to the potential flow ruled by Euler's equation, with afinite sliding velocity on the walls. The corresponding infinitesimal layer ofvorticity is thensubjected to three effects, namely, viscous diffusion, convection through the impressedflow, and shear instability.
The competition between the first two effects leads to deviationsfrom the potential flow confined within an adjacent fluid layer that grows downstream:If the velocity is sufficiently great, the motion of the fluid at small distances from itssurface all round will always be very nearly the same as if the fluid were inviscid, andthe difference will be smaller near the front part than near the rear of the globe.Shear instability within the boundary layer, Thomson went on, induces a trail of turbulentmotion behind the globe. Stokes agreed with this scenario, except that, in his view, thetrail commenced with a surface of sudden slip whose instability led to the observedturbulence.
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