Contents14 Vorticity 114.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 Vorticity and Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2.1 Vorticity Transport Equation . . . . . . . . . . . . . . . . . . . . . . 514.2.2 Barotropic, Inviscid Flows: Vortex Lines Frozen Into Fluid . . . . . . 614.2.3 Tornados . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.2.4 Kelvin’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.2.5 Diffusion of Vortex Lines . . . . . . . . . . . . . . . . . . . . . . . . . 914.2.6 Sources of Vorticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.3 Low-Reynolds-Number Flow – Stokes Flow and Sedimentation . . . . . . . . 1414.3.1 Motivation: Climate Change . . . . . . . . . . . . . . . . . . . . . . . 1414.3.2 Stokes Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1514.3.3 Sedimentation Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1914.4 High-Reynolds-Number Flow – Laminar Boundary Layers . .......... 2014.4.1 Similarity Solution for Velocity Field . . . . . . . . . . . . . . . . . . 2214.4.2 Vorticity Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2514.4.3 Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2614.5 Nearly Rigidly Rotating Flows — Earth’s Atmosphere and Oceans . . . . . . 2814.5.1 Equations of Fluid Dynamics in a Rotating Reference Frame . . . . . 2814.5.2 Geostrophic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3114.5.3 Taylor-Proudman Theorem . . . . . . . . . . . . . . . . . . . . . . . 3214.5.4 Ekman Boundary Layers ......................... 3214.6T2 Instabilities of Shear Flows — Billow Clouds, Excitation of Water Wavesby Wind, T urbulence in the Stratosphere . . . . . . . . . . . . . . . . . . . . 3814.6.1T2 Discontinuous Flow: Kelvin-Helmholtz Instability ........ 3814.6.2T2 Discontinuous Flow with Gravity . . . . . . . . . . . . . . . . . 4314.6.3 T2 Smoothly Stratified Flows: Rayleigh and Richardson Criteria forInstability. . ................................ 440Chapter 14VorticityVersion 1114.1.K, 01 February 2012. — Changes made after version 1 was distributed toYanb ei Chen’s Ph136 class. Subsequent chapters have cross references to verison 1, not thisversion.Please send comments, suggestions, and errata via email to [email protected] or on paper toKip Thorne, 350-17 Caltech, Pasadena CA 91125Box 14.1Reader’s Guide• This chapter relies heavily on Chap. 12, Fundamentals of Fluid Dynamics.• Chapters 14–18 (fluid mechanics and magnetohydrodynamics) are extensions ofthis chapter; to understand them, this chapter must be mastered.• Portions of Part V, Plasma Physics (especially Chap. 20 on the “two-fluid formal-ism”), rely on this chapter.14.1 OverviewIn the last chapter, we introduced an important quantity called vorticity which is the subjectof the present chapter. Although the most mathematically simple flows are “potential flows”,with velocity of the form v = ∇ψ for some ψ so the vorticity ω = ∇ × v vanishes, themajority of naturally occuring flows are vortical. We shall find that studying vorticity allowsus to develop an intuitive understanding of how flows evolve. Furthermore, computing thevorticity can provide an important step along the path to determining the full velocity fieldof a flow.We all think we know how to recognise a vortex. The most hackneyed example is waterdisappearing down a drainhole in a bathtub or shower. Here what happens is that theangular velocity around the drain increases inward, because the angular momentum per unit12mass is conserved when the water slowly circulates radially in addition to rotating. Thisangular momentum conservation means that the product of the circular velocity vφand theradius " is independent of radius, which, in turn, implies that ∇ ×v " 0. So this is a vortexwithout m uch vorticity! (except, as we shall see, a delta-function spike of vorticity right atthe drainhole’s center); see Fig. 14.1 and Ex. 14.14. Vorticity is a precise physical quantitydefined by ω = ∇ ×v,notanyvaguelycirculatorymotion.In Sec. 14.2, we introduce two tools for analyzing and utilizing vorticity: vortex lines,and circulation. Vorticity is a vector field, and therefore has integral curves obtained bysolving dx/dλ = ω for some parameter λ.Theseintegralcurvesarethevortex lines;theyare a nalogous to magnetic field lines. The flux of vorticity!ω ·dΣ across a closed surface isequal to the integral of the v elocity field!v ·dx around the surface’s boundary (by Stokes’theorem). We call this the circulation;itisanalogoustomagnetic-fieldflux. Infact,theanalogy with magnetic fields turns out to be extremely useful. Vorticity, like a magnetic field,automatically has vanishing divergence, whichmeansthatthevortexlinesarecontinuous,just like magnetic field lines. Vorticity, again like a magnetic field, is an axial vector andthus can be written as the curl of a polar vector potential, the velocity v .1Vorticity has theinteresting property that it evolves in a perfect fluid in such a manner that the flow carriesthe vortex lines along with it; we say that the vortex lines are “frozen into the fluid”. Whenviscous stresses make the fluid imperfect, then the vortex lines diffuse through the movingfluid with a diffusion coefficient that is equal to the kinematic viscosity ν.In Sec. 14.3, we study a classical problem that illustrates both the action and t he prop-agation of vorticity: the creeping flow of a low Reynolds number fluid around a sphere.(Low-Reynolds-number flow arises when the magnitude of the viscous stress in the equationof motion is much larger than the magnitude of the inertial acceleration.) The solution tothis problem finds contemp orary application incomputingthesedimentationratesofsootparticles in the atmosphere.In Sec. 14.4, we turn to high Reynolds nu mber flows, in which the viscous stress isquantitatively weak over most of the fluid. Here, the action of vorticity …
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