Inviscid Centre-modes and Wall-modes in the Stability Theory of Swirling Poiseuille Flow

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Abstract
The properties of the stability equation are examined for this swirling flow at Rossby number ε with particular reference to the dependence of the complex wave number ω on the real azimuthal and axial wave-numbers n and α. Two broad classes of eigenfunction are identified: (a) centre-modes in which the critical layer is very close to the axis of the pipe and the precise position of the pipe-wall plays a minor rôle in controlling the properties of the eigenfunctions; (b) wall-modes in which the critical layer is a significant distance from the axis and the boundary condition at the pipe-wall is of importance to the structure of the eigenfunction. Generally, these modes are distinct, but in certain limiting situations, one of the distinguishing features may be lost. The modes can arise only if 0 ≤ λ ≤ 1 where λ = (εn−α)/εn. A precise theory of the modes is developed for λ « 1 and 1−λ « 1, while an approximate theory is presented for intermediate values of λ. Good agreement is obtained with earlier analytic studies, when ε « 1 and when n » 1, and with numerical studies of the neutral mode at n = 1. Further evidence is provided that the range of α over which the modes are unstable is greater for centre-modes than for wall-modes.

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