Finite rotating and translating vortex sheets

We consider the rotating and translating equilibria of open finite vortex sheets with endpoints in two-dimensional potential flows. New results are obtained concerning the stability of these equilibrium configurations which complement analogous results known for unbounded, periodic and circular vort...

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Bibliographic Details
Published in:Journal of fluid mechanics 2021-09, Vol.923, Article A23
Main Authors: Protas, Bartosz, Llewellyn Smith, Stefan G., Sakajo, Takashi
Format: Article
Language:English
Subjects:
Aspect ratio
Computational fluid dynamics
Dimensional stability
Equilibrium
Fields
Fluid mechanics
Fluids
Growth rate
JFM Papers
Perturbation
Perturbations
Potential flow
Rotation
Stability
Stability analysis
Two dimensional flow
Velocity
Vortex sheets
Vortices
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Summary:We consider the rotating and translating equilibria of open finite vortex sheets with endpoints in two-dimensional potential flows. New results are obtained concerning the stability of these equilibrium configurations which complement analogous results known for unbounded, periodic and circular vortex sheets. First, we show that the rotating and translating equilibria of finite vortex sheets are linearly unstable. However, while in the first case unstable perturbations grow exponentially fast in time, the growth of such perturbations in the second case is algebraic. In both cases the growth rates are increasing functions of the wavenumbers of the perturbations. Remarkably, these stability results are obtained entirely with analytical computations. Second, we obtain and analyse equations describing the time evolution of a straight vortex sheet in linear external fields. Third, it is demonstrated that the results concerning the linear stability analysis of the rotating sheet are consistent with the infinite aspect ratio limit of the stability results known for Kirchhoff's ellipse (Love, Proc. Lond. Math. Soc., vol. s1–25, 1893, pp. 18–43; Mitchell & Rossi, Phys. Fluids, vol. 20, 2008, p. 054103) and that the solutions we obtained accounting for the presence of external fields are also consistent with the infinite aspect ratio limits of the analogous solutions known for vortex patches.
ISSN:0022-1120
1469-7645
DOI:10.1017/jfm.2021.572