Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 90
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Exploiting a private suggestion by Prandtl, Karman used dimensional74Prandtl [192Ib] pp. 692-3; Tollmien [1929]. Cf. Prandtl [1930] p. 791. Prandtl borrowed the expression for theenergy transfer between macro.flow and micro·perturbation from Reynolds [1 894] (see Chapter 6, p. 261).Heisenberg gave another proof of this instability in 1 923.75Karman [1921]. The expression ausbebildete Turbulenz already appeared in Prandtl [1913] p. 1 19.DRAG AND LIFT297considerations to derive the velocity profile from Blasius's law. If the (large-scale) velocityu of the water does not depend on the radius of the tube,distance y from the wall, the wall stressthe quantities� and vjyTo,(To)(n+IJI2(Y)"then it can only depend on thethe density p, and the kinematic viscosity v. Asboth have the dimension of velocity, the only possiblemonomial expression for the fluid's velocity ispu=A -(7.
17)- ,1J. where A is a dimensionless constant. Accordingly, the wall stress and the loss of head varyas the power2/(n + 1)of the average velocity in the pipe's section. Compatibility withBlasius's law then implies that n =1/ 7.Having found satisfactory agreement with measured pipe velocity profiles, Karman assumed the similar formif =7m ll(7.18)for the velocity profile of a turbulent boundary layer, withTo (!!...)=P714A(!::.)o1 14(7.19).Substituting these two expressions into the momentum equation(7. 1 6), he found that theU is a constant)boundary-layer thickness o of a flat plate (for which the impressed velocityvaried as the power 4I 5 of the distance x from the cutwater, and that the corresponding5 of the impressed velocity U. Accordingly, the growth ofresistance varied as the power 9Ia turbulent boundary layer is faster than that of a laminar one, and the resistance hasnearly the same form as found by Beaufoy and Froude.
Karman found even betteragreement with more recent experimental data.767.3.8The mixing lengthNo matter how successful it was, Karman's approach remained semi-empirical, for itborrowed the law of pipe retardation from experiment. Prandtl wanted a more fundamental theory of developed turbulence from which velocity profiles and the form of theresistance law would result without experimental input. In1925, he announced significantprogress toward this deductive goal.
Starting from 'Boussinesq's formula'T=edudy(7.20)for the shear stress in a turbulent flow with the transverse, large-scale velocity gradientdu/dy, he followed up Saint-Venant's idea that the effective viscosity e resulted frommomentum transfer through velocity fluctuation. He had just read a popular book bythe Viennese meteorologist Wilhelm Schmidt, who subsumed the transport of momentum,76Blasius [191 3]; Karman [1921]. Unlike Boussinesq and French hydraulicians, Karman did not assume a finitevelocity at the walls.
The y111 law nonetheless gives a very rapid increase of the velocity near the walls.298WORLDS OF FLOWheat, and electricity in the atmosphere under the unified concept of 'turbulent exchange'(Austausch) and 'mixing' a la Reynolds.77Knowing that e/p has the dimension of length multiplied by velocity, Prandtl sought anintuitive representation of the relevant length and velocity. For this purpose, he imaginedthat, owing to the turbulent fluctuation, balls of fluid were constantly carried over adistance of the order 1 from one layer of the fluid to another, with a transverse velocityIdentifying the resulting momentum exchange,w.wpldufdy, with the Boussinesq stress, heobtained e = pwl. He then made the transverse velocity w result from the collision of twoballs of fluid with different u, which gives the estimate w "" lldu/dyl .
The resultingexpression for the turbulent stress isT\ l2 du du= p1dy dy '(7.21)This only improves on Boussinesq's formula if the length 1 is a simpler function of the flowthan the coefficient e. Prandtl showed that this was indeed the case for the 'free turbulence'occurring in the boundary layer of an air jet. With Tollmien's help, he proved that thesimple Ansatz 1=Cx, where x is the distance the jet has traveled from the nozzle and C is adimensionless constant, matched observations of the layer. The case ofpipes did not workso well, since Blasius's law then required that !varies as the power 6 I 7 of the distance fromthe wall-not so simple a Jaw.
78The following year, Prandtl gave a somewhat different interpretation ofthe length 1, theone most commonly known today. He now reasoned by analogy with the notion of themean free path in the kinetic theory of gases and on the basis of Reynolds's stress formulaTwhereii and v represent theAccording to the fluid-ball picture,= piiv,follows. Prandtl now called1 theturbulent fluctuations of the velocity components.ii "" v "" ±l du/dy, and the stress formula (7.21)(Mischungsweg), in conformance with'mixing length'Reynolds's and Schmidt's emphasis on mixing. 79Prandtl returned to the mixing length in his Tokyo lectures of Octoberalong a smooth wall, he then noted, the simplest possible1929. For flowAnsatz is l = Ky, where y is thedistance from the wall and K is a numerical constant (l must vanish at the wall, since thereis no room for fluctuation there).
Within the boundary layer the stressindependent of y, so that eqn(7.21) leads to1u =K/P- (lny + C).pTis nearly(7.22)Prandtl rejected this option, because it implied the absurd uhe went on, dimensional homogeneity requires the form l= -oo for y = 0 . In general,= ycp(R.), with R. = (yj11)...[i7P.77Prandtl [1925] p. 716 (Boussinesg), [1927al (Schmidt); Schmidt [1 925]. Foppl ([1909] vol. 4, pp. 364-5)already emphasized Mischbewegung, Platzwechsel, and Austausch.
Prandtl ([1913] p. 120) used Mischbewegungand briefly described Boussinesq's and Reynolds's approaches.78Prandtl [1925]. Cf. Battimelli [1984] pp. 83-6. Darcy ([1 857], see Chapter 6, p. 234), a few other Frenchengineers, and Froude ([J869b], see earlier on p. 281) had used a similar stress formula (with an uninterpretedconstant instead of pi2) in analogy with the quadratic form of waJI friction.79Prandtl [1926], [1 927a].299DRAG AND LIFTAs he had already shown in1825,the choice=AK; 1/7leads toBlasius's law for pipe retardation. Implicitly, Prandtl confined hiscp(R,)length to simple algebraic expressions meant to apply to the0variable. 87.3.9The logarithmic profileKarman got rid of this prejudice in an important memoir of1930.u ex y l/7 and toAnsiitze for the mixingwhole range of the yInstead of speculatingon the form of the mixing length, he assumed that turbulent fluctuations at differentlocations of a fluid only differed in their temporal and spatial scales.
He also implicitlyassumed that these fluctuations were entirely determined by the first and second derivatives of the macroscopic velocity function u(y) in plane-parallel or circular-cylindricalflows. These two assumptions together imply that the Reynolds stresson the characteristic length Lr can only depend= u ju', the characteristic time T = 1/rl, and the density p.In order to be homogenous to a pressure, it must then have the formwherer=kpL2kpu'4---:rr- = u't2 ,(7.23)k is a numerical constant. For a constant r, this equation leads to the logarithmicprofile8 1(7.24)If the wall is rough, then eddy viscosity is dominant even next to the wall, and thisformula applies to arbitrary small values of the variable y. Furthermore, the characteristiclength L at the bottom must be of the order of the size a of the asperities of the wall.
Takinginto account the vanishing of the velocity at the wall, this gives(7.25)This formula holds as long as the stressr can be regarded as constant. In a circular pipe,obtained the counterparts of formulas(7.24)this stress grows linearly with the distance from the axis, as required by the balancebetween the pressures and stresses acting on the surface of a volume element. 82 Karmanderive the retardation Jaw. 83yand(7.25)in this case, and used them toIt may be noted, however, that r remains approximately constant as long as the distancefrom the wall does not exceed a small fraction, say1 o-2,of the radiushof the tube.80Prandtl [1930] (translation of notes taken by a Japanese auditor).81Karman [1930] pp.
58-65. Saint-Venant ([1887b] pp. 133-4) had used a logarithmic profile, with finite slideson the walls, for the flow between two coaxial circular cylinders, one of which moves along the axis (with stress asthe inverse of the distance from the axis, and a constant effective viscosity).82For a fluid disc of radius r and thickness dx, this balance requires 21rrrdx .,.,.,.,_ ( - dP/dx) dx.83Ibid. pp. 65-8, 74.=300WORLDS OF FLOWDenoting the wall stress by r0 , it may also be noted that for dimensional reasons the ratiouj � must be a universal function of the ratio y I h provided that y is much larger thanthe size a of the asperities.84 Consequently, the average value of the former ratio over thecross-section of the pipe differs from its value for yjh = I0-2 by a universal constant(provided that ajh << I0-2).
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