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lecture8

# 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 ω 1 Ω 1 1 r u 2 A vortex tube (filament) is a bundle of vortex lines. vortex tube ω 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 ω v ω v v u U Figure 1: Sketch of vortex ring production 2
3.3.2 No Net Flux of Vorticity Through a Closed Surface Calculus identity, for any vector v : · ( ∇ × v ) = 0 ω · ω = 0 ∇ · ω = ω · n ˆ dS = 0 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 ( ω · n ˆ) in δ A in = ( ω · n ˆ) out δ A out 0 ˆ = ω n v ( ω v n ˆ ) out ( ω 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).

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