Helicity effects on inviscid instability in Batchelor vortices

Abstract

In this paper we investigate the instability properties of Batchelor vortices with a large swirl number and a fixed axial velocity deficit. In particular, it elucidates the effect of the helicity profile on the instability of the vortices as swirling wakes. In a linear stability analysis, a negative helicity profile destabilised a vortex with a large swirl number; the name given to this instability is ‘helicity instability’. Note that helicity instability is qualified for the case of axial flow with wake. In contrast, a conventional Batchelor vortex was stable at swirl numbers above a value of circulation, which is determined by the axial velocity deficit. The instability was related to a parameter proportional to the square of the inverse azimuthal vorticity thickness. Decreasing this helicity-profile parameter increased the growth property of the vortex. Such unstable features (helicity effects) were also studied in direct numerical simulations of vortices subjected to small random disturbances at Mach numbers 2.5 and 5.0. The instability based on the vorticity thickness originally grew at the outer edge of the vortex, whereas the instability waves in a conventional Batchelor vortex originate inside the vortex core. The simulation results support the results of the linear stability analysis on the helicity profile when the parameter is small. Because of the helicity instability, the nonlinear developments yielded a large fluctuation field with many small scales and high radial spreading rates. Even at the Mach number of 5.0, negative helicity exerted a much greater destabilisation effect than a zero entropy gradient. Therefore, the investigated novel effect established a reasonably powerful instability in compressible fluids, which is favourable for supersonic mixing.