lecture8 - 2.20 Marine Hydrodynamics Spring 2005 Lecture 8...

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Lecture 8 - Marine Hydrodynamics Lecture 8 In Lecture 8, paragraph 3.3 we discuss some properties of vortex structures. In paragraph 3.4 we deduce the Bernoulli equation for ideal, steady flow. r u 3.3 Properties of Vortex Structures 3.3.1 Vortex Structures r r A vortex line is a line everywhere tangent to ±ω . vortex line ω 2 Ω 2 1 Ω 1 1 r u 2 A vortex tube (filament) is a bundle of vortex lines. vortex tube r u r vortex lines 1 2.20 - Marine Hydrodynamics, Spring 2005 2.20
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A vortex ring is a closed vortex tube. A sketch and two pictures of the production of vortex rings from orifices are shown in Figures 1, 2, and 3 below. (Figures 2,3: Van Dyke, An Album of Fluid Motion 1982 p.66, 71) side view v u U v u Γ v u ω v cross section v u v v v u U Figure 1: Sketch of vortex ring production 2
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±±± ±± ²³´µ 3.3.2 No Net Flux of Vorticity Through a Closed Surface Calculus identity, for any vector ±v : ∇· ( ∇× )=0 ² ³´ µ ±ω =0 = ± ω ± · n ˆ dS V Divergence S vorticity flux Theorem i.e. The net vorticity flux through a closed surface is zero . (a) No net vorticity flux through a vortex tube: (Vorticity Flux) in = (Vorticity Flux) out ( · n ˆ) in δ A in =( ω ± · n out δ A out 0 ˆ = ω n v v n ˆ ) out v n ˆ ) in (b) Vorticity cannot stop anywhere in the fluid. It either traverses the fluid begin- ning or ending on a boundary or closes on itself (vortex ring). ω r r ω 3
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± ²² 3.3.3 Conservation of Vorticity Flux 0=Γ 3 = ±v · d±x = ± ndS =0 ω · ˆ C 3 S 3 C 1 C 2 1 2 C 3 1 ˆ n 2 ˆ n ± Γ 1 = · = ω ± · n ˆ 1 dS = ω ± · n
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.20 taught by Professor Dickk.p.yue during the Spring '05 term at MIT.

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lecture8 - 2.20 Marine Hydrodynamics Spring 2005 Lecture 8...

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