Contents14 Turbulence 114.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 The Transition to Turbulence - Flow Past a Cylinder . . . . . . . . . . . . . 414.3 Semi-Quantitative Analysis of Turbulence . . . . . . . . . . . . . . . . . . . 914.3.1 Weak Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.3.2 Turbulent Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.3.3 Relationship to Vort icity . . . . . . . . . . . . . . . . . . . . . . . . . 1214.3.4 Kolmogorov Spectrum for Homogeneous and Isotropic Turbulence . . 1314.4 Turbulent Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . 2014.4.1 Profile of a Turbulent Boundary Layer . . . . . . . . . . . . . . . . . 2014.4.2 Instability of a Laminar Boundary Layer . . . . . . . . . . . . . . . . 2314.4.3 The flight of a ball. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2414.5 The Route to Turbulence: Onset of Chaos . . . . . . . . . . . . . . . . . . . 2714.5.1 Couette Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2714.5.2 Feigenbaum Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . 290Chapter 14TurbulenceVersion 0814.1.K, 4 February 2009Please send comm ents, suggestions, and errata via email to kip @tapir.caltech.edu, or onpaper to Kip Thorne, 130-33 Caltech, Pasadena CA 91125Box 14.1Reader’s Guide• This chapter relies heavily on Chaps. 12 and 13.• The remaining chapters on fluid mechanics and magnetohydrodynamics (Chaps.15–18) do not rely significantly on this chapter, nor do any of the remaining chaptersin this book.14.1 Ove rviewIn Sec. 12.7.6, we derived the Poiseuille formula for the flow of a viscous fluid down a pipeby assuming that the flow is laminar, i.e. that it has a velocity parallel to the pipe wall. Weshowed how balancing the stress across a cylindrical surface led to a parabolic velocity profileand a rate of flow proportional to the fourth power of the pipe diameter, d. We also definedthe Reynolds number; for pipe flow it is Rd≡ ¯vd/ν, where ¯v is the mean speed in the pipe.Now, it turns out experimentally that the flow only remains laminar up to a critical Reynoldsnumber that has a value in the range ∼ 103− 105depending upo n the smoothness of thepipe’s entrance and roughness o f its walls. If the pressure gradient is increased further (andthence the mean speed ¯v and Reynolds number Rdare increased), then the velocity field inthe pipe becomes irregular bo t h temp orally and spatially, a condition known as turbulence .Turbulence is common in high R eynolds number flows. Much of our experience of flu-ids involves air or water for which the kinematic viscosities are ∼ 10−5and 10−6m2s−1respectively. For a typical everyday flow with a characteristic speed of v ∼ 10 m s−1and acharacteristic length of d ∼ 1m, the Reynolds number is huge: Rd= vd/ν ∼ 106−107. It is12therefore 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 Rdincreases using a particularly simple example, the flow of a fluid pa st a circular cylinderoriented perpendicular to the line of sight. We shall see how the flow pattern is dictated bythe Reynolds numb er and how the velocity changes from steady creeping flow at low Rdtofully-developed turbulence at high Rd.What is turbulence? Fluid dynamicists can certainly recognize it but they have a hardtime defining it precisely,1and an even harder time describing it quantitatively.At first glance, a quantitative description a ppears 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,so there will be coupling between different modes (akin to wave-wave coupling between lightmodes in a nonlinear crystal, discussed in Chap. 9). Analyze that coupling perturbatively.The resulting weak-turbulence theory (some of which we sketch in Sec. 14.3 and Ex. 14.3)is useful when the turbulence is not too strong. In this theory, among other things, oneaverages the spectral energy density over many realizations of a stationary turbulent flowto obtain a mean spectral energy density for the fluid’s motions. Then, if t his energy den-sity extends over several octaves of wavelength, scaling arguments can be invoked to inferthe shape of the energy spectrum. This produces the famous Kolmogorov spectrum for tur-bulence. This spectrum has been verified experimentally under many different conditions.(Another weak-turbulence theory which is developed along similar lines is the quasi-lineartheory of nonlinear plasma interactions, which we shall develop in Chap. 22.)Most turbulent flows come under the heading of fully developed or strong turbulence andcannot be well described in this weak-turbulence manner. Part of the problem is that the(v · ∇)v term in the Navier-Stokes equation is a strong nonlinearity, not a weak couplingbetween linear modes. As a consequence, eddies of size ℓ persist for typically no more thanone turnover timescale ∼ ℓ/v before they are broken up, and so do not behave like weaklycoupled normal modes. Another, related, problem is that it is not a good approximationto assume that t he phases o f the modes are ra ndom, either spatially or temporally. If welook at a snapshot of a turbulent flow, we frequently observe large, well-defined coherentstructures like eddies and jets, which suggests that the flow is more organized than a purelyrandom superposition of modes, just as the light reflected from the surface of a paintingdiffers from that emitted by a black body. Moreover, if we monitor the time variation ofsome fluid variable, such as one component of the velocity a t a given point in the flow,we can recognize intermittency – the irregular starting and ceasing of strong turbulence.Again, this is such a marked effect that there is more than a random-mode superposition atwork, reminiscent of the distinction between noise and music (at least some music). Strongturbulence is therefore not just a problem in perturbation theory; and alternative, semi-quantitative approaches must be devised.In the …
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