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Unformatted text preview: vector integraldisplay Fluids Lecture 6 Notes 1. 3-D Vortex Filaments 2. Lifting-Line Theory Reading: Anderson 5.1 3-D Vortex Filaments General 3-D vortex A 2-D vortex, which we have examined previously, can be considered as a 3-D vortex which is straight and extending to . Its velocity field is V = V r = 0 V z = 0 (2-D vortex) 2 r In contrast, a general 3-D vortex can take any arbitrary shape. However, it is subject to the Helmholtz Vortex Theorems : 1) The strength of the vortex is constant all along its length 2) The vortex cannot end inside the fluid. It must either a) extend to , or b) end at a solid boundary, or c) form a closed loop. Proofs of these theorems are beyond scope here. However, they are easy to apply in flow modeling situations. 2-D Vortex 3-D Vortices The velocity field of a vortex of general shape is given by the Biot-Savart Law . d vector + vector r V ( x, y, z ) = (general 3-D vortex) 4 | vector r | 3 The integration is performed along the entire length of the vortex, with vector r extending from the point of integration to the field point x, y, z . The arc length element d vector points along the filament, in the direction of positive by right hand rule....
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This note was uploaded on 01/28/2012 for the course AERO 16.01 taught by Professor Markdrela during the Fall '05 term at MIT.
- Fall '05