Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 77
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Reynolds knew this from casual observations of crossing streams of water95Reynolds [1883] pp. 61 (quote), 76-7.WORLDS OF FLOW252(a)(b)(c)(d)Fig. 6. 7.=--[:::[:::3'>�7� .... �Reynolds' drawings of kinds of flow in a tube, as indicated by the ink jet method (from Reynolds[1883] pp. 59-60, 76-77): (a) 'direct' flow, (b) 'sinuous' flow, (c) the same observed with a flash of light(d) the disturbance of direct flow by a wire, and (e) intermittent 'flashes' of eddying.(see Fig.
6.8). He studied the corresponding type of instability with a clever device in whichhe made two fluids of different density slide over each other with a gradually increasingvelocity (see Fig. 6.9). In this 'very pretty experiment', small waves appeared beyond awell-defined critical velocity and then grew until they curled and broke, 'the one fluidwinding itself into the other in regular eddies'.
In modem terms, what he observed was asimple spectrum of perturbations followed by developed turbulence.966.5.8 Private calculationsHaving empirically established the existence of two different kinds of instability, Reynoldswas 'anxious to obtain a fuller explanation . . . from the equations of motion.' He firststudied the stability of steady solutions ofEuler's equation for frictionless fluids and foundthat parallel flow in one direction was stable, while parallel flow in opposite directions wasunstable. Since Reynolds assumed that viscosity could have only a stabilizing effect, hecould not explain the observed instability of pipe flow.
After a long period of p=lement, atthe end of 1882 he attempted a similar study in the more difficult case of the Navier-Stokesequation. He then found that the boundary condition for viscous fluids (vanishing velocityat the walls) implied instability for sufficiently small values of the viscosity.
The transitionbetween stability and instability still depended on the value of the Reynolds number.9796Ibid. pp. 56, 61 (pretty), 62 (eddies).97/bid. pp. 62-3. See Chapter 5, p. 215. The boundary-layer instability of the plane Poiseuille flow wassuggested by Prandtl in 1821, and proved by Waiter Tollmien in 1 829 (cf. Drazin and Reid [1981] p. 216, thisbook, Chapter 7, pp. 294-6).
It seems doubtful that Reynolds anticipated this difficult analysis.TURBULENCE253Fig. 6.8. A case of unstable parallel flow (Reynolds [1883] p. 56).Fig. 6.9. Reynolds's device for studying the instability of the sliding of two fluids over one another (Reynolds[1883] pp. 61-2). (a) In the original configuration the higher-density colored fluid and the lower-density fluidrest horizontally in two superposed layers.
(b) A double, conflicting flow is obtained by inclining the tube.Above a certain velocity, waves appear on the separating surface.Reynolds never published the relevant calculations. In his memoir of1883 he only gave theresults, together with a few empirical confirmations. In the course of experiments performedin1876 to study the calming effect of oil on wind waves, he had incidentally observed anothereffect of the wind, namely, the formation of eddies beneath the oiled water surface.
In thelight ofhis new theory, he argued that the stiffness of the oil film introduced a new boundarycondition on the water surface and thus destabilized the parallel flow beneath it.98As for the lowering of the critical velocity under finite disturbances in the case of pipeflow, Reynolds explained that 'as long as the motion was steady, the instability dependedupon the boundary action alone, but once eddies were introduced, the stability would bebroken down.' He thereby meant that the introduction of an eddy changed the distributionof velocity and thus induced an instability of the frictionless kind (inflection in the velocityprofile). The latter instability could overcome a much higher viscous damping than theboundary-based instability.996.5.9 Pipe dischargeAs a corollary, there should be a value of the Reynolds number below which instabilitywith respect to finite disturbances disappears.
Reynolds inferred the existence of a secondcritical velocity of pipe flow, 'which would be the velocity at which previously existingeddies would die out, and the motion become steady as the water proceeded along98Reynolds [1883] pp. 58-9, 63.99Reynolds [1883] p. 63.254WORLDS OF FLOWthe tube.' The method of colored bands could not test this conjecture, since the diffusion ofthe ink in turbulent water is irreversible.
Reynolds therefore appealed to the law ofdischarge, the study of which he had so far avoided because of its greater experimentaldifficulty.100For temperature stability and to assure a wide range of pressure, Reynolds usedthe water from the Manchester main. He measured the fall of pressure within pipes ofvarious diameters using a differential pressure gauge, and the corresponding dischargeby means of a special weir gauge (see Fig.6.1 0).Since the pressure from the main couldvary during a run, Reynolds's assistant kept it constant with an additional valve andmanometer. The pipes were fed through a T-shaped connection that caused considerabledisturbance of the entering water. Reynolds placed the differential pressure gauge awayfrom this connection, in order to give time for the regularization of the flow in thesubcritical case.
He conducted the experiments in the workshop of Owens College,'which offered considerable facilities owing to arrangements for supplying and measuringthe water used in experimental turbines', and with the help of a Mr Forster, a skillful andclever technician.101DU jAs Reynolds expected the character of the flow to be the same for equal values of thep f.L, he used the 'method of logarithmic homologues' that had served him sonumberUiwell in his transpiration studies. He plotted the logarithm of the pressure slope (the fall ofpressure head per unit length) versus the logarithm of the velocity(see Fig.6.1 1). The1 015.curves had a well-defined critical point, corresponding to a Reynolds number ofThey were composed of two straight lines, save for a small curved portion around the(Djv),D3i =!(DU)·(D3jv3)critical point.
They could be very accurately superposed through a shift of the ordinates by2and of the abscissas by Inwhere v is the ratio f.L/p . 1 0InAccordingly, the relation between pressure slope and velocity has the general form(6.33)vv2Up to the critical point, the function f is linear, in conformance with Poiseuille's law.D3iv2 (DU)"v ,A little beyond the critical point, Reynolds's discharge law takes the formex:with a =(6.34)1.753. Reynolds confined his measurements to pipes of relatively small sections.(6.34) still held, but with anFor larger pipes, he relied on Darcy's raw data. The lawexponent depending slightly on the roughness of the pipe's walls.103Unknown to Reynolds, German and French hydraulicians had already suggestedaUfractional-power discharge laws.
For example, inover Prony's+ b U21 00Reynolds [1883] pp. 64.1 851Saint-Venant favored such a lawlaw to permit the use of logarithms in backwater calculations.101/bid. pp. 78-SS (quote on p. 79).102Ibid. pp. 93-4. The expressions for the abscissa and ordinate shifts result from the validity of the Poiseuillelaw below the critical point.1 03Reynolds [1883] pp. 94-7 (formula), 98-105 (on Darcy).TURBULENCEFig. 6.1 0.255Reynolds's apparatus for studying the loss of charge in a tube as a function of the discharge rate(Reynolds [1883] p. 79).
Water from the main is fed at a constant rate into the lower horizontal tube, towhich a differential manometer (the vertical U-shaped tube) is connected. The discharge rate is measuredwith the cylindrical weir gauge on the right-hand side.Reynolds's motivation, as well as the special form he gave to the discharge law, werenonetheless new. They reflected the dimensional properties of his stability criterion.1046.5. 1 0The Reynolds legendIn retrospect, Reynolds's achievement seems enormous.
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