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Vortex sheet roll-up revisited

Published online by Cambridge University Press:  18 September 2018

A. C. DeVoria Open the ORCID record for A. C. DeVoria [Opens in a new window]  and
K. Mohseni Open the ORCID record for K. Mohseni [Opens in a new window]
A. C. DeVoria
Affiliation:
Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611, USA
K. Mohseni*
Affiliation:
Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611, USA Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL 32611, USA
*
†Email address for correspondence: [email protected]
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Abstract

The classical problem of roll-up of a two-dimensional free inviscid vortex sheet is reconsidered. The singular governing equation brings with it considerable difficulty in terms of actual calculation of the sheet dynamics. Here, the sheet is discretized into segments that maintain it as a continuous object with curvature. A model for the self-induced velocity of a finite segment is derived based on the physical consideration that the velocity remain bounded. This allows direct integration through the singularity of the Birkhoff–Rott equation. The self-induced velocity of the segments represents the explicit inclusion of stretching of the sheet and thus vorticity transport. The method is applied to two benchmark cases. The first is a finite vortex sheet with an elliptical circulation distribution. It is found that the self-induced velocity is most relevant in regions where the curvature and the sheet strength or its gradient are large. The second is the Kelvin–Helmholtz instability of an infinite vortex sheet. The critical time at which the sheet forms a singularity in curvature is accurately predicted. As observed by others, the vortex sheet strength forms a finite-valued cusp at this time. Here, it is shown that the cusp value rapidly increases after the critical time and is the impetus that initiates the roll-up process.

JFM classification

Vortex Flows: Vortex dynamics Vortex Flows: Vortex Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 855 , 25 November 2018 , pp. 299 - 321
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Copyright
© 2018 Cambridge University Press 

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