Acoustic instability of a circular vortex with a smoothed vorticity profile in the subsonic and supersonic case

It is known that disturbances of a localized vortex can have two specific mechanisms of interaction with the surrounding flow. The first is associated with energy loss by the flow, which leads to instability when the energy of the vortex disturbances is negative. The second is the Miles mechanism of interaction between vortex core oscillations and disturbances in the vicinity of the critical layer (the streamline along which the phase velocity of the disturbances coincides with the velocity of the mean flow). This is accompanied by an energy flux from this vicinity, leading to damping of the oscillations in the case of negative energy (and, conversely, to Miles instability when the energy of the core disturbances is positive). A flow in which both of these mechanisms are realized simultaneously is considered for the first time. For this purpose, disturbances of circular vortices with negative energy are considered, for which both acoustic instability and Miles damping are realized. It is shown that in the case of weak compressibility, the Miles mechanism can completely suppress acoustic instability. However, in the case of stronger energy loss due to acoustic radiation, acoustic instability will dominate. The influence of various parameters on these effects is analytically studied, and a quantitative criterion for the acoustic instability of a vortex with a smoothed vorticity profile is established. The effect of acoustic instability is considered for high velocities in the vortex core, including supersonic flow. Flow velocity enhances the acoustic instability increment due to more efficient sound radiation, which enables instability of vortices with stronger smoothness. This effect demonstrates that the behavior of vortex structures in high-speed jets can differ fundamentally from the case characterized by a low Mach number and, due to acoustic instability, intensity vortex oscillations, which in the subsonic case are characterized by strong damping. It is also shown that in an incompressible flow with a vortex confined by impedance walls, an alternative energy loss mechanism to the acoustic one is realized. In this case, Miles damping can also be overcome. Moreover, unlike the mechanism realized by outgoing acoustic waves, energy loss by the vortex due to absorption by the cylinder walls can be significantly more effective, leading to an expansion of the instability region to flows with smoother vorticity profiles.

Rankin vortex, eigen-oscillations, acoustic instability, Landau damping, supersonic vortex