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Vortex cusps

Published online by Cambridge University Press:  11 November 2019

Volker W. Elling Open the ORCID record for Volker W. Elling [Opens in a new window]
Volker W. Elling*
Affiliation:
Department of Mathematics, Academia Sinica, Taipei 10617, Taiwan
*
†Email address for correspondence: [email protected]
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Abstract

We consider pairs of self-similar two-dimensional vortex sheets forming cusps, equivalently single sheets merging into slip condition walls, as in classical Mach reflection at wedges. We derive from the Birkhoff–Rott equation a reduced model yielding formulas for cusp exponents and other quantities as functions of the similarity exponent and strain coefficient. Comparison to numerics shows that the piecewise quadratic and higher approximation of vortex sheets agree with each other and with the model. In contrast, piecewise linear schemes produce spurious results and violate conservation of mass, a problem that may have been undetected in prior work for other vortical flows. We find that vortex cusps only exist if the similarity exponent is sufficiently large and if the circulation on the sheet is counterclockwise (for a sheet above the wall with cusp opening to the right), unless a sufficiently positive strain coefficient compensates. Whenever a cusp cannot exist a spiral-ended jet forms instead; we find many jets are so narrow that they appear as false cusps.

JFM classification

Vortex Flows: Contour dynamics Interfacial Flows (free surface): Contact lines Wakes/Jets: Jets
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 882 , 10 January 2020 , A17
Copyright
© 2019 Cambridge University Press 

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