Contents15 Turbulence 115.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.2 The Transition to Turbulence - Flow Past a Cylinder . . . . . . . . . . . . . 415.3 Empirical Description of Turbulence . . . . . . . . . . . . . . . . . . . . . . . 1115.3.1 The Role of Vorticity in Turbulence . . . . . . . . . . . . . . . . . . . 1215.4 Semi-Quantitative Analysis of Turbulence . . . . . . . . . . . . . . . . . . . 1415.4.1 Weak Turbulence Formalism . . . . . . . . . . . . . . . . . . . . . . . 1415.4.2 Turbulent Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1715.4.3 Turbulent Wakes and Jets; Entrainment, and the Coanda Effect . . . 1815.4.4 Kolmogorov Spectrum for Homogeneous and Isotropic Turbulence . . 2315.5 Turbulent Boundary Layers ........................... 2815.5.1 Profile of a Turbulent Boundary Layer ................. 2815.5.2 Coanda Effect and Separation in a Turbulent Boundary Layer .... 3015.5.3 Instability of a Laminar Boundary Layer ................ 3215.5.4 The Flight of a Ball. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3315.6 The Route to Turbulence: Onset of Chaos . . . . . . . . . . . . . . . . . . . 3515.6.1 Couette Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3515.6.2 Feigenbaum Sequence and Onset of Turbulence in Convection . . . . 370Chapter 15TurbulenceVersion 1115.1.K, 6 February 2012Please send comments, suggestions, and errata via email to [email protected], or on paper toKip Thorne, 350-17 Caltech, Pasadena CA 91125Box 15.1Reader’s Guide• This chapter relies heavily on Chaps. 13 and 14.• The remaining chapters on fluid mechanics and magnetohydrodynamics (Chaps.16–18) do not rely significantly on this chapter, nor do any of the remaining chaptersin this book.15.1 OverviewIn Sec. 13.7.6, 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. W eshowed ho w balancing the stress across a cylindrical surface led to a parabolic velocity profileand a rate of flow propo rtional to the fourth power of the pipe diameter, d.Wealsodefinedthe Reynolds number; for pipe flow it is Red≡ ¯vd/ν ,where¯v is the mean speed in the pipe.Now, it turns out experimentally that the pipe flow only remains laminar up to a criticalReynolds number that has a value in the range ∼ 103−105depending on the smoothness ofthe pipe’s entrance and roughness of its walls. Ifthepressuregradientisincreasedfurther(and thence the mean speed ¯v and Reynolds number Redare increased), then the velocityfield in the pipe becomes irregular both temporally and spatially, a condition known asturbulence.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 ∼ 10−5and 10−6m2s−1respectively. For a typical everyday flow with a characteristic speed of v ∼ 10 m s−1and a12characteristic length of d ∼ 1m, the Reynolds number is huge: Red= vd/ν ∼ 106− 107.Itis therefore not surprising that we see turbulent flows all around us. Smoke in a smokestack,acumuluscloudandthewakeofashiparethreeexamples.In Sec. 15.2 w e shall illustrate the phenomenology of the transition to turbulence as Redincreases using a particularly simple example, the flow of a fluid past a circular cylinderoriented perpendicular to the line of sight. We shall see how the flow pattern is dictated bythe Reynolds number and how the velocity changes from steady creeping flow at low Redtofully-developed turbulence at high Red.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. So typicallyfor a definition they rely on empirical, qualitative descriptions of its physical properties (Sec.15.3). Closely related to this description is thecrucialroleofvorticityindrivingturbulentenergy from large scales to small (Sec. 15.3.1).At first glance, a quantitative description of turbulence appears straightforward. De-compose the velocity field into Fourier components just like the electromagnetic field whenanalyzing electromagnetic radiation. Then recognize that the equations of fluid dynamicsare nonlinear, so there will be coupling between different modes (akin to wave-wave cou-pling between optical modes in a nonlinear crystal, discussed in Chap. 10). Analyze thatcoupling perturbatively. The resulting weak-turbulence formalism (some of which we sketchin Secs. 15.4.1 and 15.4.2, and Ex. 15.4) is useful when the turbulence is not too strong.2How ever, most turbulent flows come under the heading of fully developed or strong tur-bulence,andcannotbewelldescribedintheweak-turbulencemanner. Partoftheproblemis that the (v ·∇)v term in the Navier-Stokes equation is a strong nonlinearity, not a weakcoupling between linear modes. As a consequence, eddies of size " persist for typically nomore than one turnover timescale ∼ "/v before they are broken up, and so do not behavelike weakly coupled normal modes.In the absence of a decent quantitative theory of strong turbulence, fluid dynamicistssometimes simply push the weak-turbulence formalism into the strong-turbulence regime,and use it there to gain qualitative or semi-quantitative insights (e.g. Fig. 15.7 and associ-ated discussion in the text). A simple alternativ e (which we will explore in Sec. 15.4.3 inthe context of wakes and jets, and in Sec. 15.5 for turbulent boundary layers) is intuitive,qualitative and semiquantitative approaches to the physical description of turbulence. Weemphasize, the adjective physical, because our goal is not just to produce empirical descrip-tions of the consequences of turbulence, but rather to comprehend the underlying physicalcharacter of turbulence, going beyond purely empirical rules on one hand and uninstructivemathematical expansions on the other. Thismeansthatthereadermustbepreparedtosettle for order-of-magnitude scaling relations based on comparing the relative magnitudesof individual terms in the governing fluid dynamical equations. At first, this will seem quiteunsatisfactory. However, much contemporary physics has to proceed with this methodology.It is simply too hard, in turbulence and some other phenomena, to discover elegant mathe-matical counterparts to the Kepler problem or the solution of the Schr¨odinger equation for1The
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