Inviscid instability of the Batchelor vortex: Absolute-convective transition and spatial branches

The main objective of the study is to examine the spatio-temporal instabili ty properties of the Batchelor q-vortex, as a function of swirl ratio q and external axial flow parameter a. The inviscid dispersion relation between complex axial wave number and frequency is determined by numerical integrat ion of the Howard-Gupta ordinary differential equation. The absolute-convec tive nature of the instability is then ascertained by application of the Br iggs-Bers zero-group-velocity criterion. A moderate amount of swirl is foun d to promote the onset of absolute instability. In the case of wakes, trans ition from convective to absolute instability always takes place via the he lical mode of azimurhal wave number m = -1. For sufficiently large swirl, c o-flowing wakes become absolutely unstable. In the case of jets, transition from absolute to convective instability occurs through various helical mod es, the transitional azimuthal wave number m being negative but sensitive t o increasing swirl. For sufficiently large swirl, weakly co-flowing jets be come absolutely unstable. These results are in good qualitative and quantit ative agreement with those obtained by Delbende el al.(1) through a direct numerical simulation of the linear response, Finally, the spatial (complex axial wave number, real frequency) instability characteristics are illustra ted for the case of zero-external flow swirling jets. (C) 1999 American Ins titute of Physics. [S1070-6631(99)04306-8].