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# Circulation and point vortices

Consider a contour that encloses a Rankine Vortex tube (Figure (6.5)).

The circulation is defined as

 (6.18)

by Stoke's Theorem. ( is bounded by ).
 (6.19)

Thus the circulation does not depend upon the details of .
(We have a similar result in electromagnetism:

The flow velocity depends only upon the circulation, . If and then (by (6.19)) = const., and the flow velocity will be unchanged.

The limit of and , such that const. results in a point vortex (PV), and is illustrated in Figure (6.6).

What is the flow around a PV? Take to be a circle of radius , then (6.18) becomes

 (6.20)

The flow associated with a PV of circulation consists of circular streamlines (see Figure (6.7)).

Thus one PV just sits there with flow circulating around it (if there is no uniform flow at .) If there is a uniform flow as we can find the solution by adding on to everything - this is equivalent to a frame transformation. The result is that the PV appears to move with a steady velocity .

Subsections

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Andrew Wright
2002-09-16