08%20Vorticity%20Dynamics%20EGM%206812%20F11

08%20Vorticity%20Dynamics%20EGM%206812%20F11 - EGM 6812,...

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EGM 6812, F11, University of Florida, M. Sheplak 1/18 Section 8, Vorticity Dynamics 8 Vorticity Dynamics 8.1 Introduction to vorticity/circulation Vorticity: although not a primary variable in fluid mechanics, it is important for interpretation of various fluid flows 8.1.1 Kinematic property k i ijk j V u x    8.1.2 Various Physical Interpretations is equal to twice the angular velocity of a point particle   ang. vel. 1 2 r dV dr    Recall Stokes’ Theorem Figure 1 Stokes Theorem (Panton pg 58) Definitions: C a closed curve n is a unit vector perpendicular to dS dS is an infinitesimal area t unit vector tangent to C V  t dl n dS V C
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EGM 6812, F11, University of Florida, M. Sheplak 2/18 Section 8, Vorticity Dynamics dl is an infinitesimal line element on C dr tdl   Note: dS cannot contain any singularities. Stokes Theorem:   CS V dr V dS   vs. nV in same direction (+), opposing (-) Counter clockwise along C with n pointing up is (+) Circulation:   C S S V dS     A ndA    So the around the curve C is equal to the flux of through the surface ii d n dA Vorticity is the circulation per unit area for a surface perpendicular to the vorticity vector. 8.2 Helmholtz’s Vortex Theorems 8.2.1 Vortex Lines Vector that is everywhere tangent to (analogous to streamlines). Note that that the vorticity field is solenoidal, dA
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EGM 6812, F11, University of Florida, M. Sheplak 3/18 Section 8, Vorticity Dynamics   2 antisymmetric symmetric 0 Proof: 0 kk ijk ijk i j i j uu V x x x x          “rate of expansion” of the Vorticity = 0 Math and Physical Implications Curves tangent to a solenoidal vector can’t end in a region within which the vector field is defined. Therefore, vortex lines must form “closed loops”, initiate and terminate at a boundary, or extend to infinity.   zero net flux 0 0 dV ndA    Therefore, the net circulation over the entire area is zero. 8.2.2 Vortex Tubes It is bounded by all vortex lines that pass through points on a closed surface around which there is a specified and constant circulation   .
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EGM 6812, F11, University of Florida, M. Sheplak 4/18 Section 8, Vorticity Dynamics Because vortex tubes are comprised of vortex lines, they are subject to the above implications. Vortex tubes cannot end within the fluid. They must either form a closed loop, extend to , or intersect a wall at a place where 0 . Consider the following: A “portion” of a vortex tube with end caps 12 , AA 1 A 2 A Vortex lines Via Gauss 0!
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This note was uploaded on 01/17/2012 for the course EGM 6812 taught by Professor Renweimei during the Fall '09 term at University of Florida.

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08%20Vorticity%20Dynamics%20EGM%206812%20F11 - EGM 6812,...

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