Some Remarks on Free Boundaries of Recirculating Euler Flows with Constant Vorticity

In this investigation, we revisit the question of linear stability analysis of two-dimensional steady Euler flows characterized by the presence of compact regions with constant vorticity embedded in a potential flow. We give a complete derivation of the linearized perturbation equation which, recognizing that the underlying equilibrium problem is of a free-boundary type, is carried out systematically using methods of shape-differential calculus. Particular attention is given to the proper linearization of contour integrals describing vortex induction. The thus obtained perturbation equation is validated by analytically deducing from it stability analyses of the circular vortex, originally due to Kelvin, and of the elliptic vortex, originally due to Love, as special cases. We also propose and validate a spectrally accurate numerical approach to the solution of the stability problem for vortices of general shape in which all singular integrals are evaluated analytically.

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Eroding dipoles and vorticity growth for Euler flows in : axisymmetric flow without swirl

Journal of Fluid Mechanics ◽

10.1017/jfm.2016.573 ◽

2016 ◽

Vol 805 ◽

pp. 1-30 ◽

Cited By ~ 1

Author(s):

Stephen Childress ◽

Andrew D. Gilbert ◽

Paul Valiant

Keyword(s):

Mirror Symmetry ◽

Axisymmetric Flow ◽

Vortical Structures ◽

Piecewise Constant ◽

Constant Vorticity ◽

Vortex Dipole ◽

Euler’S Equations ◽

Euler Flows ◽

Euler's Equations ◽

Structurally Stable

A review of analyses based upon anti-parallel vortex structures suggests that structurally stable dipoles with eroding circulation may offer a path to the study of vorticity growth in solutions of Euler’s equations in $\mathbb{R}^{3}$. We examine here the possible formation of such a structure in axisymmetric flow without swirl, leading to maximal growth of vorticity as $t^{4/3}$. Our study suggests that the optimizing flow giving the $t^{4/3}$ growth mimics an exact solution of Euler’s equations representing an eroding toroidal vortex dipole which locally conserves kinetic energy. The dipole cross-section is a perturbation of the classical Sadovskii dipole having piecewise constant vorticity, which breaks the symmetry of closed streamlines. The structure of this perturbed Sadovskii dipole is analysed asymptotically at large times, and its predicted properties are verified numerically. We also show numerically that if mirror symmetry of the dipole is not imposed but axial symmetry maintained, an instability leads to breakup into smaller vortical structures.

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