The effect of swirl on jets and wakes: Linear instability of the Rankine vortex with axial flow

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The effect of swirl on jets and wakes is investigated by analyzing the inviscid spatiotemporal instability of the Rankine vortex with superimposed plug flow axial velocity profile. The linear dispersion relation is derived analytically as a function of two nondimensional control parameters: the swirl ratio S and the external axial flow parameter a (a > –0.5 for jets, a < –0.5 for wakes). For each azimuthal wave number m, there exists a single unstable Kelvin–Helmholtz mode and an infinite number of neutrally stable inertial waveguide modes. Swirl decreases the temporal growth rate of the axisymmetric Kelvin–Helmholtz mode (m = 0), which nonetheless remains unstable for all axial wave numbers. For helical modes (m0), small amounts of swirl lead to the widespread occurrence of direct resonances between the unstable Kelvin–Helmholtz mode and the inertial waveguide modes. Such interactions generate, in the low wave number range, neutrally stable wave number bands separated by bubbles of instability. As S increases above a critical value, all bubbles merge to give rise to unstable wave numbers throughout, but the growth rate envelope decreases continuously with increasing swirl. In the high wave number range, negative helical mode growth rates are enhanced for small swirl and decrease continuously for large swirl, while positive helical mode growth rates monotonically decrease with increasing swirl. For a given positive swirl, negative modes are more unstable than their positive counterparts, although their growth rates may not necessarily be larger than in the nonrotating case. The absolute/convective nature of the instability in swirling jets and wakes is determined in an a – S control parameter plane by numerical implementation of the Briggs–Bers criterion. In the absence of swirl, jets (a > –0.5) become absolutely unstable (AI) to the axisymmetric mode m = 0 only for a sufficiently large external axial counterflow a < –0.15. AI is found to be significantly enhanced by swirl: for S > 1.61, AI occurs, even for coflowing jets (a > 0). As S is gradually increased, the transitional mode to AI successively becomes m = 0, –1, –2, –3, etc. In the absence of swirl, wakes (a < –0.5) become absolutely unstable to the bending modes m = ±1 only for sufficiently large counterflow 1 + a > 0.091. For S > 0.47, AI occurs even for coflowing wakes (a < –1) and, as S increases, the transitional mode to AI successively becomes m = –1, –2, –3, etc. This instability analysis is found to provide a preliminary estimate of the critical Rossby number for the onset of vortex breakdown: for zero external axial flow jets (a = 0), absolute/convective transition first takes place at a Rossby number RoS^–1 ~ 0.62, which very favorably compares with available experimental and numerical threshold values for vortex breakdown onset. ©1998 American Institute of Physics.