Journal of Fluid Mechanics

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Global linear stability of the non-parallel Batchelor vortex

C. J. HEATONa1 c1, J. W. NICHOLSa1 and P. J. SCHMIDa1

a1 Laboratoire d'Hydrodynamique (LadHyX), CNRS–École Polytechnique, 91128 Palaiseau, France

Abstract

Linear stability of the non-parallel Batchelor vortex is studied using global modes. This family of swirling wakes and jets has been extensively studied under the parallel-flow approximation, and in this paper we extend to more realistic non-parallel base flows. Our base flow is obtained as an exact steady solution of the Navier–Stokes equations by direct numerical simulation (with imposed axisymmetry to damp all instabilities). Global stability modes are computed by numerical simulation of the linearized equations, using the implicitly restarted Arnoldi method, and we discuss fully the numerical and convergence issues encountered. Emphasis is placed on exploring the general structure of the global spectrum, and in particular the correspondence between global modes and local absolute modes which is anticipated by weakly non-parallel asymptotic theory. We believe that our computed global modes for a weakly non-parallel vortex are the first to display this correspondence with local absolute modes. Superpositions of global modes are also studied, allowing an investigation of the amplifier dynamics of this unstable flow. For an illustrative case we find global non-modal transient growth via a convective mechanism. Generally amplifier dynamics, via convective growth, are prevalent over short time intervals, and resonator dynamics, via global mode growth, become prevalent at later times.

(Received October 14 2008)

(Revised January 29 2009)

Correspondence:

c1 Email address for correspondence: cjh43@damtp.cam.ac.uk

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