Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 67
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Drazin and Reid [1981] pp. 10-1 1 .1 04Rayleigh [1892] p . 577.105Ibid. p . 576; Birkhoff [l 950].6TURBULENCEIf the velocities [of water in rivers] remained constant in each point ofthe traversed space, the surface of the liquid would look like a plate of iceand the herbs growing at the bottom would be equally motionless.
Far fromthat, the stream presents incessant agitation and tumultuous, disordered movements, so that the velocities change in an abrupt and most diverse manner fromone point to another and from one instant to the next. As noted by Leonardo daVinci, Venturi, and especially Ponce!et, one can perceive eddies, large and small,with a vertical mobile axis.
One can also see, at the surface, bouillons, or eddieswith a nearly horizontal axis, that constantly surge from the bottom and thusform genuine ruptures, with the intertwining and mixing motions that M.Boileau observed in his experiments. 1 (Adbemar Barn& de Saint-Venant, 1 872)Hard to gain though it may be, any understanding of hydrodynamic instabilities is of anegative kind. Namely, it only tells us when and why the rigorous solutions of thehydrodynamic equations under given boundary conditions fail to represent naturalflows. It does not tell us much on the sort of motion into which the unstable system settlesafter perturbation.
From common observations, everyone knows the great complexity ofthis motion. The capricious eddying of water behind obstacles, or the hesitating, convoluted rise of smoke from a fire have indeed inspired poets with metaphors for theunpredictability of human life.William Thomson began using the term 'turbulent' in the 1 880s to characterize suchirregular motions, as opposed to the 'laminar' flows in which successive fluid layers glidesmoothly over each other.
Much earlier, in 1 822, Navier opposed 'linear' to 'nonlinear' flow,and from the 1 830s Saint-Venant opposed 'tumultuous' to 'regular' flow. The unpredictability captured in this terminology has long deterred the theorists of fluid motion. Yet, theintellectual mastery of some aspects of turbulent flow has proved possible. 2Turbulence studies began in the nineteenth century with what French engineers calledeaux courantes, or open-channel flow. Pipe flow and fluid resistance took second priority,perhaps because turbulence is less visible in pipes and more heterogeneous around anobstacle, but also because in those years French engineers were busy building new canalsand improving the navigability of rivers.
In 1 822, just after proposing his equations forviscous-fluid motion, Navier recognized its impotence for describing the 'nonlinear' flowsencountered in hydraulics. In the 1 830s and 1 840s, Saint-Venant suggested that the same1 Saint-Venant [1872] p. 6502Cf. Thomson [1887e], [1 894]. Navier's and Saint-Venant's contributions are discussed later on pp. 229-3 1 .The opposition between turbulent and laminar flow i s used here as roughly as i t was in the nineteenth century, withno consideration of intermediate, oscillatory forms of motion.220WORLDS OF FLOWequation could be applied to the large-scale average of a tumultuous flow if the viscosityparameter was made to depend on the circumstances of the flow. In the 1 870s, his discipleJoseph Boussinesq implemented this approach in a monumental Theorie des eauxcourantes.These early quantitative and statistical theories of turbulent flow are described inSections 6.2 and 6.3.
Section 6.1 does not deal with turbulence per se, but with anteriorstudies of open channels in the 1 820s and 1 830s, mainly the problem of backwaters thatlargely motivated Saint-Venant's and Boussinesq's work. The authors of these studies didnot calculate from the fundamental equations of hydrodynamics. Instead, they developeda semi-empirical approach that combined a parallel-slice idealization of the flow, mechanical principles, and some experimental input for wall friction. They ignored the turbulent character of the motion.
In contrast, Saint-Venant argued that insights into the natureof turbulence would permit more fundamental solutions of hydraulic problems.The mathematical theory of open-channel flow was a mostly French topic, usuallyavoided by British engineers. There was a significant exception, the brother James ofWilliam Thomson, who kept up with literature on this subject and agreed with his Frenchcounterparts that turbulence played a significant role in determining the flow pattern.While helping James explain an anomaly of the velocity profile, in 1887 William Thomsondiscovered that the turbulent fluid had effective rigidity and could thus propagate largescale transverse vibrations.
For a short, exhilarating time, he believed to have found thekey to the perfect-fluid theory of the luminiferous ether. George Francis FitzGerald, whosimilarly dreamt of a 'vortex-sponge' theory of the ether, extended Thomsen's speculationwith much enthusiasm. These theories are described in Section 6.4.Thomsen's and FitzGerald's ether theories, as for Saint-Venant's and Boussinesq'shydraulics, only involved developed turbulence. They did not require an understandingof the transition from laminar to turbulent flow.
In 1839, the German hydraulicianGotthilf Hagen discovered the sudden character of this transition in the case of pipeflow. His original purpose was to provide engineers with more exact retardation formulas,so he did not dwell on this curious phenomenon. In contrast, Reynolds's hydrodynamicinvestigations of the 1 880s, described in Section 6.5, the flnal section of this chapter,concerned this transition and its 'criterion'.Problems of navigation, rather than hydraulics, motivated Reynolds's interest in turbulence. While reflecting on propellers, wakes, and sea waves, he surmised that mosthydrodynamic paradoxes and anomalies resulted from our ignorance of invisible vortexmotion. William Thomson and James Clerk Maxwell had already made vortices in apervasive, ideal fluid responsible for the magnetic properties of the ether and for thestability ofmatter.
Reynolds made their continual production the main cause of resistanceand retardation in real fluids. To reveal the secrets of fluid motion, he only needed a fewdrops of ink.A more surprising source of Reynolds's reflections on turbulent flow was the kinetictheory of gases. Following an investigation ofWilliam Crookes's radiometer and ThomasGraham's transpiration phenomena, Reynolds argued that the nature of the flow of adilute gas depended on the 'dimensional properties of matter', specifically on the ratiobetween the dimensions of the flow (vane size or tube diameter) and the mean free path.Similarly, he expected the nature of the flow of a denser fluid to depend on the dimensionalTURBULENCE221properties of the Navier-Stokes equation.
This led him to the idea of a transition controlled by the Reynolds number, to the experimental verification and sharpening of thisidea, and to his later kinetic-statistical theory of the turbulent transition.6.1 Hydraulic phenomenology6. 1 . 1Hydraulics versus hydrodynamicsAs the rational hydrodynamics of d'Alembert and Euler proved inept at practical problems of hydraulics and navigation, empirical or semi-empirical methods began to thrive.When, in the 1870s, the minister Turgot consulted d'Alembert, the Marquis de Condorcet,and the abbot Charles Bossut about the project of an underground canal in Picardie, theyperformed towing experiments that showed, among other things, that the resistanceincreased with the narrowness of the canal.
Bossut taught empirical hydrodynamics atthe Ecole Royale du Genie de Meziere. The Ministry of War funded his numerousexperiments on retardation in pipe and channel flow. The second edition of the resultingtreatise, published in 1 786/87, long remained a reference for hydraulic engineers.3Bossut praised the 'very profound and very generous method' of his friend d'Alembertas well as the 'scope and generality' of Euler's contribution. However, he did not try toapply these theories in the real world:These great geometers seem to have exhausted the resources that can be drawn fromanalysis to determine the motion of fluids: their formulas are so complex, by thenature of things, that we may only regard them as geometrical truths, and not assymbols fit to paint the sensible image of the actual and physical motion of a fluid.Bossut measured the loss of head in pipes and channels of various breadths, for which heprovided a wealth of numerical tables and the inference that the loss was roughly proportional to the square of the average velocity.4The other French master of late-eighteenth-century hydraulics, Pierre Du Buat, agreedwith Bossut that urgent hydraulic problems could only be solved by the experimentalmethod.
Yet he had the more theoretical ambition of providing general formulas for pipeand channel flow, as well as a detailed discussion of the course of rivers, which was hismain interest. He applied Newton's second law to the bulk motion of water, guessed theform of retarding forces by molecular intuition, and inferred relevant parameters fromabundant measurements. 5Du Buat formulated the 'key to hydraulics' as the balance between the accelerating force(due to pressure gradient or gravity) of a fluid slice and its friction on the walls.
Foruniform, permanent flow in an open channel, this leads to the equation (in Prony's laternotation)pgSi = x Fu,3Cf. Dugas [1950] pp. 300-3, Rouse and !nee [1957] pp. 126-8, Redondi [1997].4Bossut [1786/87] p. XV.5Du Buat [1786]. Cf. Dugas [1950] pp. 303-5, Rouse and !nee [1957] pp. 129-34.(6.1)222WORLDS OF FLOWwhere p is the density of water, g is the acceleration of gravity, S is the normal fluid section,i is the slope of the bottom (the sine of the angle that it makes with a horizontal plane), x isthe wetted perimeter of the channel, Fu is the retarding force per unit length, and U is theaverage velocity.6Du Buat's intuition of fluid tenacity and 'molecular gearing' yielded an intricateexpression for the retarding force Fu, which reduces to the quadratic form bU2 in mostpractical cases.
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