Viscous and inviscid instabilities of a trailing vortex

Abstract

A spectral collocation and matrix eigenvalue method is used to study the linear stability of the trailing line vortex model of Batchelor. For both the inviscid and viscous stability problem, the entire unstable region in the swirl/axial wavenumber parameter space is mapped out for various azimuthal wavenumbers m. In the inviscid case, the non-axisymmetric perturbation with azimuthal wavenumber m = 1 has an unstable region of larger extent than any other, with an unusual two-lobed structure; also, the location and numerical value of the maximum disturbance growth rate previously reported for this case are shown to be incorrect. Exploiting the increasingly localized structure of perturbation eigenfunctions allows accurate results to be obtained up to values of m more than 3 orders of magnitude larger than previously, and the results for the most unstable mode are in excellent agreement with the asymptotic theory of Leibovich & Stewartson. A viscous analysis of these fundamentally inviscid modes reveals that the critical Reynolds number at which instability first occurs increases as O(m2) for m [Gt ] 1, and finds the critical values of swirl and wavenumber, which approach limiting values as m → ∞.In the viscous case, the instabilities for m = 0 and 1 recently reported by Khorrami are found via a simplified numerical approach and the entire unstable region for each of these modes is mapped out over a wide range of Reynolds numbers. The critical Reynolds numbers for these modes are found to be 322.42 and 17.527, respectively, the latter having been unreported previously. The instabilities persist in the limit of large Reynolds number, with corresponding disturbance growth rates decreasing roughly as 1/Re. In addition to the primary mode, a new family of long-wave viscous instabilities is found for the m = 1 case.

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

“…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).…”

“…This allows to compute near neutral modes for which the singularities are close to the real r-axis (Leibovich & Stewartson 1983;Mayer & Powell 1992). Note that δ should be positive in order to avoid the critical points in the correct direction in the complex r-plane.…”

“…and Mayer & Powell (1992) have reported that the flow is inviscidly unstable in the range 0 < q 1.5. Inviscid instabilities are strongest for q = 0.87 with a growth rate being 0.46 (Mayer & Powell 1992).…”

“…and Mayer & Powell (1992) have reported that the flow is inviscidly unstable in the range 0 < q 1.5. Inviscid instabilities are strongest for q = 0.87 with a growth rate being 0.46 (Mayer & Powell 1992). The inviscid stability threshold q 1.5 is close but significantly higher than the value predicted by the Leibovich & Stewartson (1983) criterion: q = √ 2.…”

“…Nevertheless, the maximum growth rate is very small in the range 1.5 q < 2.31 and decreases exponentially with q (Heaton 2007a) explaining why the instability threshold has been reported to be q 1.5 by and Mayer & Powell (1992). There also exist instabilities of a viscous nature Khorrami 1991;Mayer & Powell 1992;Delbende, Chomaz & Huerre 1998;Olendraru & Sellier 2002;Fabre & Jacquin 2004;Le Dizès & Fabre 2007;Fabre & Le Dizès 2008;Le Dizès & Fabre 2010) whose growth rates are maximum for a finite value of the Reynolds number and decay to zero in the inviscid limit. Their typical growth rate is however much smaller than inviscid instabilities.…”