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Unformatted text preview: Chapter 14 Turbulence Version 0214.2, 5 February 2003 Please send comments, suggestions, and errata via email to email@example.com and also to firstname.lastname@example.org, or on paper to Kip Thorne, 130-33 Caltech, Pasadena CA 91125 14.1 Overview In Chap. 12, we derived the Poiseuille formula for the flow of a viscous fluid down a pipe by assuming that the flow is laminar, i.e. that it has a velocity parallel to the pipe wall. We showed how balancing the stress across a cylindrical surface led to a parabolic velocity profile and a rate of flow proportional to the fourth power of the pipe diameter, d . We also defined the Reynolds number for pipe flow, R vd/ , where v is the mean speed in the pipe. Now, it turns out experimentally that the flow only remains laminar up to a critical Reynolds number that has a value in the range 10 3- 10 5 depending upon the experimental conditions. If the pressure gradient is increased further, then the velocity field in the pipe becomes irregular both temporally and spatially, a condition we call turbulence . Turbulence is common in high Reynolds number flows. Much of our experience of flu- ids involves air or water for which the kinematic viscosities are 10- 5 and 10- 6 m 2 s- 1 respectively. For a typical everyday flow with a characteristic speed of v 10m s- 1 and a characteristic length of d 1m, the Reynolds number is huge: R = vd/ 10 6- 10 7 . It is therefore not surprising that we see turbulent flows all around us. Smoke in a smokestack, a cumulus cloud and the wake of a ship are three examples. In Sec. 14.2 we shall illustrate the phenomenology of the transition to turbulence as R increases using a particularly simple example, the flow of a fluid past a circular cylinder oriented perpendicular to the line of sight. We shall see how the flow pattern is dictated by the Reynolds number and how the velocity changes from steady creeping flow at low R to fully-developed turbulence as R is increased. But what is turbulence? Fluid dynamicists can certainly recognize it but they have a hard time defining it precisely, and an even harder time describing it quantitatively. At first glance, a quantitative description appears straightforward. Decompose the ve- locity field into Fourier components just like the electromagnetic field when analysing elec- tromagnetic radiation. Then recognize that the equations of fluid dynamics are nonlinear, 1 2 so there will be coupling between different modes (akin to that between the phonons dis- cussed in Sec. 11.6). Analyze that coupling perturbatively. The resulting weak-turbulence theory (some of which we sketch in Sec. 14.3) is useful when the turbulence is not too strong....
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