A note on the stability of steady inviscid helical gas flows

Eckhoff, Knut S.; Storesletten, L.

doi:10.1017/s0022112078002669

Cited by 45 publications

(24 citation statements)

References 6 publications

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“…Recently, Hattori & Fukumoto (2012) performed a local stability analysis to study the effects of axial flow on the so-called curvature instability of a helical vortex tube for which the angular velocity is constant up to the first order of a small parameter; elliptic instability was not considered in this study. Eckhoff & Storesletten (1978) employed the local approach to investigate the effects of an axial flow on an axisymmetric vortex in a compressible flow, and in a follow-up paper, Eckhoff (1984) showed that the results from the local approach agreed with the asymptotic analysis of Leibovich & Stewartson (1983) in the limit of incompressibility. Le Duc & Leblanc (1999) and Leblanc & Le Duc (2005) have further extended the study of axisymmetric vortices with an axial flow to establish the connection between the local approach and the high-wavenumber asymptotic limits of the normal mode approach.…”

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confidence: 79%

Effects of an axial flow on the centrifugal, elliptic and hyperbolic instabilities in Stuart vortices

et al. 2014

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Linear stability of the Stuart vortices in the presence of an axial flow is studied. The local stability equations derived by Lifschitz & Hameiri (Phys. Fluids A, vol. 3 (11), 1991, pp. 2644-2651 are rewritten for a three-component (3C) two-dimensional (2D) base flow represented by a 2D streamfunction and an axial velocity that is a function of the streamfunction. We show that the local perturbations that describe an eigenmode of the flow should have wavevectors that are periodic upon their evolution around helical flow trajectories that are themselves periodic once projected on a plane perpendicular to the axial direction. Integrating the amplitude equations around periodic trajectories for wavevectors that are also periodic, it is found that the elliptic and hyperbolic instabilities, which are present without the axial velocity, disappear beyond a threshold value for the axial velocity strength. Furthermore, a threshold axial velocity strength, above which a new centrifugal instability branch is present, is identified. A heuristic criterion, which reduces to the Leibovich & Stewartson criterion in the limit of an axisymmetric vortex, for centrifugal instability in a non-axisymmetric vortex with an axial flow is then proposed. The new criterion, upon comparison with the numerical solutions of the local stability equations, is shown to describe the onset of centrifugal instability (and the corresponding growth rate) very accurately.

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confidence: 79%

Effects of an axial flow on the centrifugal, elliptic and hyperbolic instabilities in Stuart vortices

et al. 2014

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“…Kurzweg (1969) was one of the earliest to consider non-axisymmetric disturbances, and derived a sufficient condition for stability for a smooth vortex with radial density variation. Necessary conditions for the stability of the same flow were given by Eckhoff & Storesletten (1978, 1980. To the best of our knowledge, there is no necessary and sufficient condition for instability of a vortex subject to radial density stratification.…”

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Stability of a vortex in radial density stratification: role of wave interactions

Dixit

Govindarajan

2011

J. Fluid Mech.

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We study the stability of a vortex in an axisymmetric density distribution. It is shown that a light-cored vortex can be unstable in spite of the 'stable stratification' of density. Using a model flow consisting of step jumps in vorticity and density, we show that a wave interaction mediated by shear is the mechanism for the instability. The requirement is for the density gradient to be placed outside the vortex core but within the critical radius of the Kelvin mode. Conversely, a heavy-cored vortex, found in other studies to be unstable in the centrifugal Rayleigh-Taylor sense, is stabilized when the density jump is placed in this region. Asymptotic solutions at small Atwood number At show growth rates scaling as At 1/3 close to the critical radius, and At 1/2 further away. By considering a family of vorticity and density profiles of progressively increasing smoothness, going from a step to a Gaussian, it is shown that sharp gradients are necessary for the instability of the light-cored vortex, consistent with recent work which found Gaussian profiles to be stable. For sharp gradients, it is argued that wave interaction can be supported due to the presence of quasi-modes. Probably for the first time, a quasi-mode which decays exponentially is shown to interact with a neutral wave to give exponential growth in the combined case. We finally study the nonlinear stages using viscous direct numerical simulations. The initial exponential instability of light-cored vortices is arrested due to a restoring centrifugal buoyancy force, leading to stable non-axisymmetric structures, such as a tripolar state for an azimuthal wavenumber of 2. The study is restricted to two dimensions, and neglects gravity.

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“…As pointed out by Eckhoff (1984), this condition is a particular case of a more general condition derived by Eckhoff & Storesletten (1978) for swirling compressible flows with arbitrary radial distribution of density (see also Sipp et al 2005;Leblanc & Le Duc 2005;Di Pierro & Abid 2010, 2012. When the condition (1.2) is satisfied, the Leibovich-Stewartson asymptotics predict that the maximum growth rate is reached in the limit |m| → ∞ with a dimensionless axial wavenumber such that |k/m| O(1) in agreement with numerical stability results for the Batchelor trailing line vortex (Lessen, Singh & Paillet 1974;Duck & Foster 1980;Mayer & Powell 1992;Delbende & Rossi 2005).…”

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confidence: 90%

A unified criterion for the centrifugal instabilities of vortices and swirling jets

Billant

Gallaire

2013

J. Fluid Mech.

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Swirling jets and vortices can both be unstable to the centrifugal instability but with a different wavenumber selection: the most unstable perturbations for swirling jets in inviscid fluids have an infinite azimuthal wavenumber, whereas, for vortices, they are axisymmetric but with an infinite axial wavenumber. Accordingly, sufficient condition for instability in inviscid fluids have been derived asymptotically in the limits of large azimuthal wavenumber m for swirling jets (Leibovich and Stewartson, J. Fluid Mech., vol. 126, 1983, pp. 335-356) and large dimensionless axial wavenumber k for vortices (Billant and Gallaire, J. Fluid Mech., vol. 542, 2005, pp. 365-379). In this paper, we derive a unified criterion valid whatever the magnitude of the axial flow by performing an asymptotic analysis for large total wavenumber √ k 2 + m 2 . The new criterion recovers the criterion of Billant and Gallaire when the axial flow is small and the Leibovich and Stewartson criterion when the axial flow is finite and its profile sufficiently different from the angular velocity profile. When the latter condition is not satisfied, it is shown that the accuracy of the Leibovich and Stewartson asymptotics is strongly reduced. The unified criterion is validated by comparisons with numerical stability analyses of various classes of swirling jet profiles. In the case of the Batchelor vortex, it provides accurate predictions over a wider range of axial wavenumbers than the Leibovich-Stewartson criterion. The criterion shows also that a whole range of azimuthal wavenumbers are destabilized as soon as a small axial velocity component is present in a centrifugally unstable vortex.

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A note on the stability of steady inviscid helical gas flows

Abstract: A necessary condition for linear stability of steady inviscid helical gas flows is found by the generalized progressing-wave expansion method. The criterion obtained is compared with the known Richardson number criteria giving sufficient conditions for stability.

Cited by 45 publications

References 6 publications

Effects of an axial flow on the centrifugal, elliptic and hyperbolic instabilities in Stuart vortices

Effects of an axial flow on the centrifugal, elliptic and hyperbolic instabilities in Stuart vortices

Stability of a vortex in radial density stratification: role of wave interactions

A unified criterion for the centrifugal instabilities of vortices and swirling jets

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