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Extension of the Prandtl–Batchelor theorem to three-dimensional flows slowly varying in one direction

Published online by Cambridge University Press:  17 June 2010

M. SANDOVAL  and
S. CHERNYSHENKO
M. SANDOVAL*
Affiliation:
Department of Aeronautics, Imperial College, London SW7 2AZ, UK
S. CHERNYSHENKO
Affiliation:
Department of Aeronautics, Imperial College, London SW7 2AZ, UK
*
†Email address for correspondence: m.sandoval07@imperial.ac.uk
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Abstract

According to the Prandtl–Batchelor theorem for a steady two-dimensional flow with closed streamlines in the inviscid limit the vorticity becomes constant in the region of closed streamlines. This is not true for three-dimensional flows. However, if the variation of the flow field along one direction is slow then it is possible to expand the solution in terms of a small parameter characterizing the rate of variation of the flow field in that direction. Then in the leading-order approximation the projections of the streamlines onto planes perpendicular to that direction can be closed. Under these circumstances the extension of the Prandtl–Batchelor theorem is obtained. The resulting equations turned out to be a three-dimensional analogue of the equations of the quasi-cylindrical approximation.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 654 , 10 July 2010 , pp. 351 - 361
Copyright
Copyright © Cambridge University Press 2010

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References

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