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On the ‘δ-equations’ for vortex sheet evolution

Published online by Cambridge University Press:  26 April 2006

James W. Rottman  and
Peter K. Stansby
James W. Rottman
Affiliation:
Universities Space Research Association. NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA
Peter K. Stansby
Affiliation:
Department of Engineering, Simon Building, The University, Manchester M13 9PL, UK
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Abstract

We use a set of equations, sometimes referred to as the ‘δ-equations’, to approximate the two-dimensional inviscid motion of an initially circular vortex sheet released from rest in a cross-flow. We present numerical solutions of these equations for the case with δ2 = 0 (for which the equations are exact) and for δ2 > 0. For small values of the smoothing parameter δ a spectral filter must be used to eliminate spurious instabilities due to round-off error. Two singularities appear simultaneously in the vortex sheet when δ2 = 0 at a critical time tc After tc the solutions do not converge as the computational mesh is refined. With δ2 > 0, converged solutions were found for all values of δ2 when t < tc, and for all but the two smallest values of δ2 used when t > tc. Our results show that when δ2 > 0 the vortex sheet deforms into two doubly branched spirals some time after tc The limiting solution as δ→0 clearly exists and equals the δ = 0 solution when t < tc. For t > tc, the limiting solution appears to exist if only the converged solutions are used, but it is unclear what relation this limiting solution has to any δ2 = 0 solution for these times.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 247 , February 1993 , pp. 527 - 549
Copyright
© 1993 Cambridge University Press

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