THE EFFECT OF SWIRL ON JETS AND WAKES - LINEAR INSTABILITY OF THE RANKINE VORTEX WITH AXIAL-FLOW

The effect of swirl on jets and wakes is investigated by analyzing the inviscid spatiotemporal instability of the Rankine vortex with superi mposed plug flow axial velocity profile. The linear dispersion relatio n 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 numbe r In, there exists a single unstable Kelvin-Helmholtz mode and an infi nite number of neutrally stable inertial waveguide modes. Swirl decrea ses the temporal growth rate of the axisymmetric Kelvin-Helmholtz mode (m = 0), which nonetheless remains unstable for all axial wave number s. For helical modes (m not equal 0), small amounts of swirl lead to t he widespread occurrence of direct resonances between the unstable Kel vin-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 crit ical value, all bubbles merge to give rise to unstable wave numbers th roughout, but the growth rate envelope decreases continuously with inc reasing swirl. In the high wave number range, negative helical mode gr owth rates are enhanced for small swirl and decrease continuously for large swirl, while positive helical mode growth rates monotonically de crease with increasing swirl. For a given positive swirl, negative mod es are more unstable than their positive counterparts, although their growth rates may not necessarily be larger than in the nonrotating cas e. The absolute/convective nature of the instability in swirling jets and wakes is determined in an a - S control parameter plane by numeric al implementation of the Briggs-Bers criterion. In the absence of swir l, 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 inc reased, 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 co unterflow 1 + a > 0.091. For S > 0.47, AI occurs even for coflowing wa kes (a < -1) and, as S increases, the transitional mode to AI successi vely becomes m = -1, -2, -3, etc. This instability analysis is found t o 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 R o = S-1 similar to 0.62, which very favorably compares with available experimental and numerical threshold values for vortex breakdown onset . (C) 1998 American Institute of Physics.