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CALTECH PH 136A - Vorticity

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Chapter 13VorticityVersion 0413.1.K 27 Jan 05Please send comments, suggestions, and errata via email to [email protected] or on paperto Kip Thorne, 130-33 Caltech, Pasadena CA 9112513.1 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. Here what happens is that water at largedistances has a small angular velocity about the drain, which increases as the water flowstowards the drain in order to conserve angular momentum. This in turn means that theproduct of the 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 of 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 analogy1Incidentally, in a bathtub the magnitude of the Coriolis force resulting from the earth’s rotation withangular velocity Ω is a fraction ∼ Ω(r/g)1/2∼ 3 × 10−6of the typical centrifugal force in the vortex (whereg is the acceleration of gravity and r ∼ 2 cm is the radius of the vortex at the point where the water’ssurface achieves an appreciable downward slope). Thus, only under the most controlled of conditions willthe hemisphere in which the drainhole is located influence the direction of rotation.12with magnetic fields turns out to be extremely useful. Vorticity, like a magnetic 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 action 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 concentratedin 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) vanish. It is the vanishing of both components of velocity thatdistinguishes real fluid flow at high Reynolds’ number (i.e. small viscosity) from the solutionsobtained assuming vanishing vorticity. Nevertheless, it is often a good approximation toadopt a solution to the equations of fluid dynamics in which vortex-free fluid slips freely pastthe solid and then match it onto the solid using a boundary-layer solution.All the above issues will be discussed in some detail in this chapter. As a final issue,in Sec. 13.5 we shall consider a simple vortex sheet ignoring viscous stresses. We shall findthat this type of flow is generically unstable. This will provide a good introduction to theprincipal topic of the next chapter, turbulence.13.2 Vorticity and CirculationWe have already defined the vorticity as the curl of the velocity ω = ∇ × v, analogous todefining the magnetic field as the curl of a vector potential.We can illustrate vorticity by considering the three simple 2-dimensional flows shown inFig. 13.1:Fig. 13.1(a) shows uniform rotation with angular velocity Ω = Ωez. The velocity fieldis v = Ω × x, where x is measured from the rotation axis. Taking its curl, we discover thatω = 2Ω.Fig. 13.1(b) shows a flow in which the angular momentum per unit mass j = jezisconstant, i.e. v = j × x/r2(where r = |x|). This is the kind of flow that occurs arounda bathtub vortex, and around a tornado. In this case, the vorticity is ω = (j/2π)δ(x),.i.e. it vanishes everywhere except at the center, x = 0. What is different in this case isthat the fluid rotates differentially and although two neighboring fluid elements, separatedtangentially, rotate about each other with an angular velocity j/r2, when the two elements3yx(a)(c)rvrv(b)Fig. 13.1: Illustration of vorticity in three two-dimensional flows. a) Constant angular velocityΩ. If we measure radius r from the center P, the circular velocity satisfies v = Ωr. This flow hasvorticity ω = 2Ω. b) Constant angular momentum per unit mass j, with v = j/r. This flow haszero vorticity except at its center, ω = (j/2π)δ(x). c) Shear flow in a laminar boundary layer,vx= ωy. In this flow the vorticity is ω = vx/y.are separated radially, their angular velocity is −j/r2. The average of these two angularvelocities vanishes, and the vorticity vanishes.The vanishing vorticity in this case is an illustration of a simple geometrical descriptionof vorticity in any two


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CALTECH PH 136A - Vorticity

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