Stability of a four-vortex aircraft wake model

The stability of an aircraft wake model composed of an external vortex pair (modeling the wing tip vortices) and an internal vortex pair rotating in the opposite direction (modeling the vortices generated by the fuselage and the horizontal tail) in a stationary configuration is investigated with the vortex filament stability method used by Crow [AIAA J. 8, 2172 (1970)] and Crouch [J. Fluid Mech. 350, 311 (1997)]. It is shown that this configuration is unstable with respect to two-dimensional and three-dimensional disturbances. For long wavelength three-dimensional symmetric perturbations, the rapid growth observed in the numerical simulations of Rennich and Lele [J. Air. 36, 398 (1999)] is found. Moreover, the analysis allows one to show that without an excitation of the long-wave mode, the system will naturally develop short wavelength instabilities localized within the inner vortices which do not affect the outer vortices. Inspection of the initial value problem shows that the long-wave modes can be efficiently initiated by the introduction of perturbations on the internal vortices.