Short-wave cooperative instabilities in representative aircraft vortices

This paper considers the short-wave cooperative instabilities in a family of vortices representative of aircraft wakes. These vortices are characterized by two core scales, an internal core scale a₁ and an external core scale a₂, and their azimuthal velocity follows a power law V(r)∼r^-α and the intermediate zone (a₁<r<a₂). The results are compared to the cases of the Rankine (constant) and Lamb-Oseen (Gaussian) vortices, and to the elliptic instability theory. The unstable wavelengths and the structure of the unstable modes are characterized as functions of the base flow parameters α and a₂/a₁. For 0.5≤α≤1, the wavelength of the instability is of the order of the internal scale a₁ and the unstable modes only affect the internal core. In this case the growth rate of the instability is in accordance with the predictions of the elliptical instability theory and is a growing function of the parameter a₂/a₁ . For 0≤α<0.4, the wavelength of the instability is of the order of the external scale a₂ and the unstable modes extend into the intermediate zone. In this case the growth rate of the instability differs from the predictions of the elliptical instability theory and is independent upon the parameter a₂/a₁- Interestingly, a sharp transition between these two regimes occurs for 0.4<α<0.5, in a range of parameters corresponding to experimentally measured trailing wakes. In this range, the bands of wave numbers affected by the instability are particularly large and may coalesce into a broadband spectrum.