Vortex breakdown in variable-density gaseous swirling jets

Abstract

Theoretical predictions and numerical simulations are used to determine the transition to bubble and conical vortex breakdown in low-Mach-number laminar axisymmetric variable-density swirling jets. A critical value of the swirl number $S$ for the onset of the bubble ( $S^*_B$ ) and the cone ( $S^*_C$ ) is determined as the jet-to-ambient density ratio $\varLambda$ is varied, with the temperature dependence of the gas density and viscosity appropriate to that of air. The criterion of failure of the slender quasi-cylindrical approximation predicts $S^*_B$ that decreases with increasing values of $\varLambda$ for a jet in solid-body rotation emerging sharply into a quiescent atmosphere. In addition, a new criterion for the onset of conical breakdown is derived from divergence of the initial value of the radial spreading rate of the jet occurring at $S^*_C$ , found to be independent of $\varLambda$ , in an asymptotic analysis for small distances from the inlet plane. To maintain stable flow in the unsteady numerical simulations, an effective Reynolds number $Re_{eff}$ , defined employing the geometric mean of the viscosity in the jet and ambient atmosphere, is fixed at $Re_{eff}=200$ for all $\varLambda$ . Similar to the theoretical predictions, numerical calculations of $S^*_B$ decrease monotonically as $\varLambda$ is increased. The critical swirl numbers for the cone, $S^*_C$ , are found to depend strongly on viscous effects; for $\varLambda =1/5$ , the low jet Reynolds number (51) at $Re_{eff}=200$ delays the transition to the cone, while for $\varLambda =5$ at $Re_{eff}=200$ , the large increase in kinematic viscosity in the external fluid produces a similar trend, significantly increasing $S^*_C$ .