Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 80
Текст из файла (страница 80)
It contained some basic ideas for a statistical approach toturbulence, namely, the mean-mean-motion, the turbulent stress, and the cascading ofenergy from one scale to another. Unknown to Reynolds, Saint-Venant had alreadyintroduced such notions. However, the purposes and manners of reasoning were different.Saint-Venant aimed at a fundamental foundation of hydraulics and offered new effectivestrategies for solving hydraulic problems. Reynolds had a more philosophical ambition:he wanted to reform hydrodynamics by specifying the conditions for turbulent flow andsubjecting this kind of flow to methods similar to those of Maxwell's kinetic theoryof gases.
1 28Although Reynolds's contribution to our understanding of turbulence is the most memorable, it was not the only nineteenth-century achievement on this arduous subject. Forthe investigators honored in this chapter, it was clear that many of the flows occurring innature and in hydraulic systems had the unpredictable, multi-scale whirling character thatwe now call turbulent. Saint-Venant and Poncelet ascribed an essential role to thisproperty in large-scale transfer phenomena.
Saint-Venant, Thomson, Boussinesq, andReynolds inaugurated statistical approaches to turbulent flow, based on effective viscosityand kinetic--molecular analogy. Hagen and Reynolds provided fine descriptions of thetransition from laminar to turbulent flow. In this context, Reynolds introduced importantdimensional considerations and the dimensionless number that relates viscous flows atdifferent scales.This early history of turbulence offers a rich sample of the strategies that physicists maydeploy when confronted with the complex, highly-irregular behavior of a dynamicalsystem. Most primitively, they may completely ignore the complexity and reason on anersatz system that obeys simple mechanical laws.
This is what early backwater theoristssuch as Belanger did with a certain amount of success. In the more refined strategyinaugurated by Saint-Venant, they may try to average out the irregularities, to seek generalrelations between the averages based on symmetry considerations, conservation laws, anddimensional analysis, and to determine the leftover parameters empirically. At a still moreadvanced stage, they may investigate the stochastic processes that relate the macroscopicparameters to the microscopic dynamics.
This is what Reynolds began to do by analogywith the kinetic theory of gases.These achievements guided later theories of turbulence, for instance Ludwig Prandtl's inthe 1 920s and Geoffrey Taylor's in the 1930s. As the latter theories, especially Prandtl's,127Reynolds [1 895] pp. 557-77.128Reynolds was not only interested in the philosophical aspects of hydrodynamics. In 1886, he gave aninfluential theory oflubrication based on the Navier-Stokes equation.
Although this theory involves laminar flowonly, he mentioned the existence of 'molar viscosity' in the turbulent case ([1886] pp. 236-8).TURBULENCE263were the first to provide reliable guidance for hydraulic and aeronautic engineers, andbecause their authors founded important schools of pure and applied fluid mechanics,they are often regarded as the true starting-point of the scientific study of turbulence.
Inthe next chapter, we will see how much twentieth-century success in this field and in therelated resistance problem capitalized on insights from the previous century. 1291 29For a historically sensitive review of the kinetic and statistical approaches to turbulence, cf. Farge andGuyon [1 999]. On Taylor, cf. Battimelli [1990].7DRAG AND LIFTWhen the complete mathematical problem looks hopeless, it is recommended toenquire what happens when one essential parameter of the problem reaches the1limit zero. (Ludwig Prandtl, 1 948)The problem of the forces that a fluid exerts on a solid body challenged hydrodynamicssince d'Alembert's foundational Essai of 1752.
It only found a quantitative, wide-rangingsolution in the first third of the twentieth century, after a few partial successes in thenineteenth century and earlier. The first section of this chapter is devoted to these olderattempts, including Newton's molecular-impact theory, Rayleigh's dead-water theory,and eddy-resistance theories by Poncelet and Saint-Venant. None of these theories trulysucceeded in making quantitative predictions, and they all lacked a solid conceptual basis.Newton's theory artificially neglected the mutual action of fluid molecules, Rayleigh'simplied an absurdly large wake, and Saint-Venant's required some observational input.Yet they all contained important elements of the modern understanding of fluid resistance.Newton understood how a similitude argument constrained inertial resistance to bequadratic.
Rayleigh's theory foreshadowed the separation process now admitted fornon-streamlined flow. Saint-Venant correctly described the eddy resistance resultingfrom the instability of separated flow.Section7.2 is devoted to ship resistance. The development of steam navigation in theVictorian empire motivated the efforts of a few learned engineers to reflect on the optimalshape of ship hulls. John Scott Russell saw how to minimize wave resistance. WilliamRankine clearly distinguished skin friction, large-eddy resistance, and wave resistance.Lastly, and most importantly, William Froude expressed the conditions for a rational useof models and developed the relevant experimental techniques.
In his analysis of skinfriction, he finely described what Prandtl later called a turbulent boundary layer. His andRankine's insights into the mechanisms ofhigh-Reynolds-number resistance neverthelessremained qualitative. The means were still lacking to turn them into efficient computational schemes.In Section7.3, we will step into the twentieth century and follow the successful development of Ludwig Prandtl's boundary-layer theory of fluid resistance at high Reynoldsnumbers. Contrary to a well-spread myth, this theory was not suddenly born out of aPrandtl paper of1904. Prandtl benefited from the rich conceptual resources of earlierhydrodynamics, and many years elapsed before the aims expressed in this paper were trulyreached. The boundary-layer concept only became practically useful around1 830, afterPrandtl and Theodore von Karman understood the role of turbulence in these layers.1Prandtl [1 948] p.
1 606.DRAG AND LIFT265Building on Boussinesq's and Reynolds' intuition of eddy viscosity, they discovered thelogarithmic velocity profile on which the modern understanding of hydraulic pipe retardation and turbulent boundary-layer theory is based. One important aspect of this evolution was the pervasive use of similitude and dimensional arguments that Stokes andRayleigh pioneered in the previous century.Section7 .4, the final section of this Chapter, will take us into the air with the firstsuccessful theories of a peculiar, useful form of fluid resistance, namely aerodynamic lift.Whereas the leaders of late-nineteenth-century fluid mechanics adhered to concepts offluid resistance that made it very difficult for birds and planes to fly, a British automobileengineer, Frederick Lanchester, and a young German mathematician, Wilhelm Kutta,independently arrived at the circulatory flow that occurs around any properly workingwing.
In Russia, Nikolai Joukowski clarified the relation between circulation and lift, andgreatly generalized Kutta's two-dimensional theory. Amidst the German war effort,Prandtl capitalized on these analytical considerations and on Lanchester's intuitions todevelop modern wing theory in three dimensions.
Together with boundary-layer theory,this achievement crowned nineteenth-century efforts at reconciling theoretical and realflows. These efforts, recounted in the previous chapters, indeed brought many of theconcepts that permitted Prandtl's breakthroughs: vortex structures, discontinuity surfaces,viscous stress, parallel-flow instability, eddy viscosity, conformal transformations, andsimilitude.7.1 Tentative theories7.
1 . 1Early views on fluid resistanceThe earliest quantitative studies of fluid resistance occurred in the seventeenth century,when Edme Mariotte and Christiaan Huygens investigated the impact of a fluid jet on aplate, with waterwheels and windmills in mind. They found the impulsion on the plate tobe proportional to the density of the fluid and to the squared velocity of the jet. Huygensexplained this result by analogy with the pressure that the water exerts on the bottom of�pifS for theS is the section of the jet.the vessel from which the jet issues. In modern terms, he obtained the formimpulsion, wherep is the fluid's density, v is its velocity,andCloser to our point, Mariotte believed the resistance encountered by a solid body movingthrough a fluid to result from the impact of fluid veins acting like the jets on which heexperimented.
2In hisPrincipia,Isaac Newton obtained a similar formula while trying to show thatDescartes' matter-filled space led to an absurd resistance to planetary motion. He firstused a similitude theorem to show that the resistance was necessarily proportional to thefluid density, to the square of the linear dimensions of the body, and to the square of itsvelocity. According to this influential theorem (anticipated by Mariotte), for any possiblemotion of a mechanical system, the similar motion obtained by uniformly rescaling all2Cf. Saint-Venant [1 887b] pp. 12-15, Rouse and !nee [1957] pp. 65-7. These researches originated in anAcademic commission of 1668 mainly devoted to the verification of Torricelli's law of efflux.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
