Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 73
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Effective viscosityparameters remain a practically important approach to turbulent flow both in hydraulics59and in atmospheric physics.For physicists, the most important aspect of Boussinesq's essay was the nonlineartheory of waves discussed in Chapter2. However, this theory was confined to waves oncalm water, for which the effects of internal friction are negligible. In general, Boussinesq'sresults depended little on his specific treatment of turbulence and internal friction. Hisequation for slowly-varying flow had exactly the same form as Coriolis's old equation,even though Coriolis's proof turned out to be unacceptable.
For the curvature-dependentterms, Boussinesq mainly used an approximation in which the Jack of uniformity of thevelocity in a given fluid section was irrelevant. Consequently, Boussinesq's readers couldaccept and rederive his final equations for the shape of the water surface without payingattention to his innovative treatment of turbulence.606.4 The turbulent ether6.4.1 A hydraulic anomalyFrench hydraulics attracted little attention from Britons, who preferred more empiricalmethods. The Glasgow engineer James Thomson was in this respect atypical. In1878, hisreading of Boileau, Darcy, and Bazin prompted him to reflect on an anomaly emphasizedby Captain Boileau, namely that the maximum velocity in a fluid section lay somewhatbelow the water surface, at variance with the 'laminar theory' based on mutual frictionbetween successive fluid layers.
In this context, Boileau introduced the 'oblique motions'58Saint-Venant [1 873] p. xxi; [1880] p. 16.59Graeff [1882], vol. 2, p. 130; Forchheimer [1914] (re-edited in 1924 and 1930); Flamant [1891]; Kozeny [1920]p. 3 1 .60In his Hydrodynamics (Lamb [1 895], [1932] for the sixth edition), Horace Lamb only referred to Boussinesqfor his derivation of the profile of solitary waves (Boussinesq [187la]) and for his calculations of Iaminar flow in·pipes of various sections (Boussinesq [1 868]).240WORLDS OF FLOWthat met Saint-Venant's approval in the earlier cited letter.
James Thomson similarlyassumed transverse, 'commingling' motion of the water:61The laminar theory constitutes a very good representation of the viscid mode ofmotion; but it offers a very fallacious view of the motion in the flow of water inordinary cases in which the inertia of the various parts of the fluid is not subordinatedto the restraints of viscosity .
. . [In these cases], indefinite increase of velocity of thewater situated in the interior of the current is prevented by continual transverse flowsthereto, and commingling therewith, of portions of water already retarded throughtheir having been lately in close proximity to the resisting channel face.To explain Boileau's maximum-velocity anomaly, Thomson assumed that 'deadened'water from the bottom of the channel reached the fluid surface and slowed the laminarmotion there. Gravel, mud, and weeds at the bottom, when present, were a possible cause o fthis effect. However, Thomson believed it also occurred for a smooth bed.
After consultation with his brother William, he assumed a thin layer of dead water at the bottom, onwhich the next layer slid with a finite velocity. This motion, being unstable according toHehnholtz and William Thomson (Helmholtz-Kelvin instability), led to turbulent transverse fluxes. 626.4.2 Turbulent rigidityThese reflections prompted William Thomson to investigate the distribution of the average velocity in the open channel.
He was 'surprised to discover the seeming possibility of alaw of propagation as of distortional waves in an elastic solid.' In a paper of1 887,heapplied a mixture of line, surface, and volume averages, as well as Fourier analysis, to theplane, laminar disturbance of homogeneous, isotropic, turbulent flow. The following is ageneralization of his arguments to an arbitrary disturbance, with the benefit of modemtensor notation. 63Thomson conceived the flow as a superposition of a large-scale regular ('laminar')componentu and a small-scale turbulent component v assumed to be homogeneous andisotropic, in a sense that will become clear.
The total flow obeys Euler's equation(6.20)in tensor notation and for unit density. The partial flows obey the incompressibilityequations81u; = 0and8;v; = 0.Taking the average of Euler's equation over a volumethat is small at the laminar scale and large at the turbulent scale, we obtain8,u; = -8;1> - 8j(u;Uj) - 8j(V;Vj).The last term corresponds to a stress system(6.21)v;vj (the Reynolds stress).6461J.
Thomson [1 878] pp. 114, 1 1 7, 120 (quote); Boileau [1 846]; Saint-Venant to Boileau, 29 Mar. 1 846 (citedearlier on p. 229). J. Thomsen's paper probably inaugurated the hydrodynamic meaning of 'laminar'. Boileau,unlike Thomson, made viscosity responsible for the oblique movements, see footnote 27.62J. Thomson [1878] pp. 121, 124.63W. Thomson [1887e] p. 314.64/bid. See later on p. 261 for the Reynolds stress. Thomson did not reason in terms of stresses. The lastequation is a generalization of his equation (34).TURBULENCEWhen there is no Iaminar motion (umotion implies that241= 0), the homogeneity and isotropy of the turbulent(6.22)where a represents the constant intensity of the turbulence.
Now suppose a laminarmotion to begin. Subtracting eqn (6.21) from eqn (6.20) and multiplying by Vj, we obtain(6.23)The large-scale average of the second of the resulting terms is(6.24)That of the third term vanishes after symmetrizing with respect to the indices i and j:(6.25)The averages of the fourth and fifth terms vanish because they contain odd-degreeproducts of the components of the initial turbulent velocity, which is isotropic. Themost delicate part is the evaluation of the first term.The incompressibility conditions together with eqns (6.20)-(6.22) yieldI:!..(P - P) = -28;UjOjV[.Consequently, the first term of eqn(6.26)(6.23) is-vjB;(P - P) = 2vj8;1:!..-! (OkUfOIVk).(6.27)Since the spatial variation of u is much slower than that of v, this may be rewritten as-vjB;(P P) = 28kUI(vj/:!,.- ! O;OfVk).(6.28)Owing to the isotropy of the original turbulent flow, averaging leads to 6521-1"O;O!Vk = 3 a-8jUi - 3 a2O;Uj.-vjo;(P - P) = 2ukU!VjA?(6.29)Altogether, we have28,(v·vJ·) = - - a2 (o·u·J + o·u·)J3II1•(6.30)Strictly speaking, this expression only holds at the beginning of the Iaminar motion u.However, when this motion is small and periodic, eqn (6.30) remains valid at any time ifthe induced anisotropy of the turbulent motion stays small.
Combining this equation withthe average Euler equation (6.21) yields, to first order in u,65According to Thomson ([l887e] p. 316), isotropy and incompressibility lead to the relationsVxll.-1 &�vx = Vxll.-1 iij;vx = Vxll.-18}vx = a2 /3, Vxll.-1&x&yvy = (l/2)vxll. 1 &x(&yvy + &,v,) == -(lj2)vxll.-1&_;vx= -a2 /6, to similar relations for similar terms, and to the vanishing of other kinds of terms.242WORLDS OF FLOW�zotu-2 2·r�llu = o,(6.31)which means that transverse waves propagate in the turbulent liquid with a velocityproportional to the average velocity of the turbulent motion.
666.4.3The vortex spongeIn Thomson's eyes, this result meant much more than a hydrodynamic curiosity. Heannounced it as 'something seemingly towards a solution (many times tried for withinthe last twenty years) of the problem to construct, by giving vortex motion to an incompressible fluid, a medium which shall transmit waves of laminar motion as the luminiferous ether transmits waves of light.' In 1 847, Thomson had written to Stokes:I perceived a fine instance of elasticity in an incompressible liquid, in a very simpleobservation made at Paris, on a cup of thick 'chocolat au lait'.
Wben I made theliquid revolve in the cup, by stirring it, and then took out the spoon, the twistingmotion (in eddies, and in the general variation of angular vel[ocity] on acc[ount] ofthe action of the spoon overcoming the inertia of the liquid, and the fric[tion] at thesides) in becoming effaced, always gave rise to several oscillations so that before theliquid began to move as a rigid body, it performed oscillations like an elastic(incompressible) solid.In the 1850s, after Thomson developed the mechanical theory of heat, he became convinced that every physical phenomenon could be reduced to pure motion.
In particular, hebelieved that the rigidity of solid bodies or of the ether derived from the centrifugal inertiaof internal, rotary motions.67In this state of mind, Thomson could turn his brother's hydraulic problem into a hopefor ether theory. His formulation was nevertheless cautious in that he only claimed'something seemingly towards a solution' of the ether problem. He feared that thecondition of persistent randomness of the turbulent flow could be 'vitiated by a rearrangement of vortices'. At the end of his paper, he showed that a symmetrical distribution ofvortex rings satisfied the condition, but only if (as he now doubted) the vortex rings werethemselves stable.
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