Absolute instability of hot round jets discharging from tubes

The spatiotemporal, inviscid linear instability of hot gas jets emerging from a round tube of radius $a$ is studied for jet Reynolds numbers $Re \gg 1$. The analysis focuses on the influence of the injector length $l_t$ on the stability characteristics of the resulting jet, whose base velocity profile at the exit is computed in terms of the dimensionless tube length $L_t=l_t/(Re \, a)$ by integrating the boundary-layer equations along the injector. Both axisymmetric modes ($m=0$) and first azimuthal modes ($m=1$) of instability are investigated for values of the jet-to-ambient density ratio $S=\rho_j/\rho_{\infty}<1$. For short tubes $L_t \ll 1$ the jet becomes absolutely unstable for critical density ratios $S_c \simeq (0.66,0.35)$ for $m=(0,1)$, in agreement with previous results of uniform velocity jets. For increasing $L_t$ both modes are seen to exhibit absolutely unstable regions for all values of $L_t$ and small enough values of the density ratio. For $m=1$ we find a critical density ratio which increases monotonically with $L_t$, reaching its maximum value $S_c \simeq 0.5$ as the exit velocity approaches the parabolic profile for $L_t \gg 1$. In the case $m=0$ the critical density ratio achieves a maximum value $S_c\simeq 0.9$ for $L_t \simeq 0.04$ and then decreases to approach $S_c=0.7$ for $L_t \gg 1$. The absolute growth rates in this limiting case are however extremely small, in agreement with the fact that the parabolic velocity profile is neutrally stable to axisymmetric disturbances.