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Numerical stability analysis of a vortex ring with swirl

Published online by Cambridge University Press:  04 September 2019

Yuji Hattori Open the ORCID record for Yuji Hattori [Opens in a new window] ,
Francisco J. Blanco-Rodríguez  and
Stéphane Le Dizès Open the ORCID record for Stéphane Le Dizès [Opens in a new window]
Yuji Hattori*
Affiliation:
Institute of Fluid Science, Tohoku University, Sendai 980-8577, Japan
Francisco J. Blanco-Rodríguez
Affiliation:
Aix Marseille Université, CNRS, Centrale Marseille, IRPHE, UMR 7342, 13384 Marseille, France Área de Mecánica de fluidos, Departamento de Ingeniería Aeroespacial y Mecánica de fluidos, Universidad de Sevilla, Avda de los Descubrimientos, s/n, 41092 Sevilla, Spain
Stéphane Le Dizès
Affiliation:
Aix Marseille Université, CNRS, Centrale Marseille, IRPHE, UMR 7342, 13384 Marseille, France
*
†Email address for correspondence: hattori@fmail.ifs.tohoku.ac.jp
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Abstract

The linear instability of a vortex ring with swirl with Gaussian distributions of azimuthal vorticity and velocity in its core is studied by direct numerical simulation. The numerical study is carried out in two steps: first, an axisymmetric simulation of the Navier–Stokes equations is performed to obtain the quasi-steady state that forms a base flow; then, the equations are linearized around this base flow and integrated for a sufficiently long time to obtain the characteristics of the most unstable mode. It is shown that the vortex rings are subjected to curvature instability as predicted analytically by Blanco-Rodríguez & Le Dizès (J. Fluid Mech., vol. 814, 2017, pp. 397–415). Both the structure and the growth rate of the unstable modes obtained numerically are in good agreement with the analytical results. However, a small overestimation (e.g. 22 % for a curvature instability mode) by the theory of the numerical growth rate is found for some instability modes. This is most likely due to evaluation of the critical layer damping which is performed for the waves on axisymmetric line vortices in the analysis. The actual position of the critical layer is affected by deformation of the core due to the curvature effect; as a result, the damping rate changes since it is sensitive to the position of the critical layer. Competition between the curvature and elliptic instabilities is also investigated. Without swirl, only the elliptic instability is observed in agreement with previous numerical and experimental results. In the presence of swirl, sharp bands of both curvature and elliptic instabilities are obtained for $\unicode[STIX]{x1D700}=a/R=0.1$, where $a$ is the vortex core radius and $R$ the ring radius, while the elliptic instability dominates for $\unicode[STIX]{x1D700}=0.18$. New types of instability mode are also obtained: a special curvature mode composed of three waves is observed and spiral modes that do not seem to be related to any wave resonance. The curvature instability is also confirmed by direct numerical simulation of the full Navier–Stokes equations. Weakly nonlinear saturation and subsequent decay of the curvature instability are also observed.

JFM classification

Instability: Parametric instability Vortex Flows: Vortex instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 878 , 10 November 2019 , pp. 5 - 36
Copyright
© 2019 Cambridge University Press 

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