Contents13 Vorticity 113.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Vorticity and Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2.1 Vorticity Transpo r t . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2.2 Tornados . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.2.3 Kelvin’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.2.4 Diffusion of Vortex Lines . . . . . . . . . . . . . . . . . . . . . . . . . 813.2.5 Sources of Vorticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . 913.3 Low Reynolds Number Flow – Stokes Flow, Sedimentation and Nuclear Winter 1113.3.1 Stokes Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1213.3.2 Sedimentation Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1613.4 High Reynolds Number Flow – Laminar Boundary Layers . . . . . . . . . . 1713.4.1 Vorticity Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2213.4.2 Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2313.5 Kelvin-Helmholtz Instability — Excitation of Ocean Waves by Wind . . . . 2413.5.1 Temporal growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2713.5.2 Spatial Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2813.5.3 Relationship between temporal and spatial growth; Excitation of oceanwaves by wind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2813.5.4 Physical Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . 2913.5.5 Rayleigh and Richardson Stability Criteria. . . . . . . . . . . . . . . . 290Chapter 13VorticityVersion 0613.2.K, 2 February 2007. Changes from 0613.1.K: some trivial typos fixed; equa-tions boxed; two new Boxes added: Reader’s Guide and Important Concepts; BibliographicNote added.Please send comments, suggestions, and errata via email to [email protected] l tech.ed u or on paperto Kip Thorne, 130-33 Caltech, Pasadena CA 91125Box 13.1Reader’s Guide• This chapter relies heavily on Chap. 12 , Fundamentals of Fluid Dynam i c s.• Chapters 14–18 (fluid mechanics and mag netohydrodynamics) are extensions ofthis chapter; to understand them, this chapter must be mastered.• Portions of Part V, Plasma Physics (especially Chap. 2 0 on the “two-fluid formal-ism”), rely o n this chapter.13.1 Ove rviewIn 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. Here what happens is that water at largedistances has a small angular velocity about the drain, which increases as the water flows12towards the drain in order to conserve angular momentum. This in turn means that theproduct of t he circular velocity vφand the radius r is independent of radius, which, in turn,implies that ∇ × v ∼ 0. So this is a vortex without much vorticity! (except, as we shallsee, a delta-function spike o f vorticity right at the drainhole’s center). Vorticity is a precisephysical quantity defined by ω = ∇ × v, not any vaguely circulatory motion.1In Sec. 13.2 we shall introduce several tools for analyzing and utilizing vorticity: Vorticityis a vector field and therefore has integral curves obtained by solving dx/dλ = ω for someparameter λ. These are called vortex lines and they are quite analogous to magnetic fieldlines. (We shall also introduce an integral quantity called the circulation analogous to themagnetic flux and show how this is also helpful for understanding flows.) In fact, the analogywith magnetic fields turns out to be extremely useful. Vorticity, like a mag netic field, hasvanishing divergence which means that the vortex lines are continuous, just like magneticfield lines. Vorticity, again like a magnetic field, is an axial vector and thus can be writtenas the curl of a polar vector potential, the velocity v. Vorticity has the interesting propertythat it evolves in a perfect fluid in such a manner that the flow carries the vortex lines alongwith it. Furthermore, when viscous stresses are important, vortex lines diffuse through themoving fluid with a diffusion coefficient that is equal to the kinematic viscosity.In Sec. 13.3 we study a classical problem that illustrates both the a ction and the propa-gation of vorticity: the creeping flow of a low Reynolds number fluid around a sphere. (LowReynolds number flow arises when the magnitude of the viscous stress in the equation ofmotion exceeds the magnitude of the inertial acceleration.) The solution to this problemfinds contemporary application in computing the sedimentation rates of soot particles in theatmosphere.In Sec. 13.4, we turn to high Reynolds number flows, in which the viscous stress isquantitatively weak over most of the fluid. Here, the action of vorticity can be concentr atedin relatively thin boundary layers in which the vorticity, created at the wall, diffuses awayinto the main body of the flow. Boundary layers arise because in real fluids, intermolecularattraction requires that the component of the fluid velocity parallel to the boundary ( notjust the normal component) …
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