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On vortex-sheet evolution beyond singularity formation

Published online by Cambridge University Press:  30 November 2023

D.I. Pullin Open the ORCID record for D.I. Pullin [Opens in a new window]  and
N. Shen Open the ORCID record for N. Shen [Opens in a new window]
D.I. Pullin*
Affiliation:
Graduate Aerospace Laboratories, California Institute of Technology, CA 91125, USA
N. Shen
Affiliation:
The Fluid Dynamics of Disease Transmission Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
†Email address for correspondence: [email protected]
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Abstract

We consider the evolution of a spatially periodic, perturbed vortex sheet for small times after the formation of a curvature singularity at time $t=t_c$ as demonstrated by Moore (Proc. R. Soc. Lond. A, vol. 365, issue 1720, 1979, pp. 105–119). The Moore analysis is extended to provide the small-amplitude, full-sheet structure at $t=t_c$ for a general single-mode initial condition in terms of polylogarithmic functions, from which its asymptotic form near the singular point is determined. This defines an intermediate evolution problem for which the leading-order, and most singular, approximation is solved as a Taylor-series expansion in $\tau = t-t_c$, where coefficients are calculated by repeated differentiation of the defining Birkhoff–Rott (BR) equation. The first few terms are in good agreement with numerical calculation based on the full-sheet solution. The series is summed, providing an analytic continuation which shows sheet rupture at circulation $\varGamma =0^+$, $\tau >0^+$, but with non-physical features owing to the absence of end-tip sheet roll up. This is corrected by constructing an inner solution with $\varGamma < \tau$, as a perturbed similarity form with small parameter $\tau ^{1/2}$. Numerical solutions of both the inner, nonlinear zeroth-order and first-order linear BR equations are obtained whose outer limits match the intermediate solution. The composite solution shows sheet tearing at $\tau =0^+$ into two separate, rolled up algebraic spirals near the central singular point. Branch separation distance scales as $\tau$ with a non-local, $\tau ^{3/2}$ correction. Properties of the intermediate and inner solutions are discussed.

JFM classification

Vortex Flows: Vortex dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 976 , 10 December 2023 , A17
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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