Absolute instability of light jets emerging from circular injector tubes

Abstract

We present a numerical study of the spatiotemporal, inviscid linear instability of light jets emerging from round tubes for values of the Reynolds number, Rej=ρjQ∕(πaμj)⪢1, where Q is the volumetric flow rate, ρj and μj are, respectively, the jet density and viscosity, and a is the injection tube radius. The analysis focuses on the influence of the injector length lt on the stability characteristics of the resulting jet, whose base velocity profile at the exit is computed in terms of the dimensionless tube length Lt=lt∕(Reja) by integrating the boundary-layer equations along the injector. Both axisymmetric (m=0) and helical (∣m∣=1) modes of instability are investigated for different values of the jet-to-ambient density ratio S=ρj∕ρ∞<1. For short tubes Lt⪡1 the base velocity profile at the tube exit is uniform except in a thin surrounding boundary layer. Correspondingly, the stability analysis reproduces previous results of uniform velocity jets, according to which the jet becomes absolutely unstable to axisymmetric modes for a critical density ratio Sc≃0.66 and to helical modes for Sc≃0.35. For tubes of increasing length the analysis reveals that both modes exhibit absolutely unstable regions for all values of Lt and small enough values of the density ratio. In the case of the helical mode, we find that Sc increases monotonically with Lt, reaching its maximum value Sc≃0.5 as the exit velocity approaches the Poiseuille profile for Lt⪢1. Concerning the axisymmetric mode, its associated value of Sc achieves a maximum value Sc≃0.9 for Lt≃0.04 and then decreases to approach Sc≃0.7 for Lt⪢1. The absolute growth rates in this limiting case of near-Poiseuille jet profiles are, however, extremely small for m=0, in agreement with the fact that axisymmetric disturbances of a jet with parabolic profile are neutrally stable. As a result, for S<0.5 the absolute growth rate of the helical mode becomes larger than that of the axisymmetric mode for sufficiently large values of Lt, suggesting that the helical mode may prevail in the instability development of very light jets issuing from long injectors.