Instability and transient growth for two trailing-vortex pairs

Abstract

The stability of two vortex pairs is analysed as a model for the vortex system generated by an aircraft in flaps-down configuration. The co-rotating vortices on the starboard and port sides tumble about one another as they propagate downward. This results in a time-periodic basic state for the stability analysis. The dynamics and instability of the trailing vortices are modelled using thin vortex filaments. Stability equations are derived by matching the induced velocities from Biot–Savart integrals with kinematic equations obtained by temporal differentiation of the vortex position vectors. The stability equations are solved analytically as an eigenvalue problem, using Floquet theory, and numerically as an initial value problem. The instabilities are periodic along the axes of the vortices with wavelengths that are large compared to the size of the vortex cores. The results show symmetric instabilities that are linked to the long-wavelength Crow instability. In addition, new symmetric and antisymmetric instabilities are observed at shorter wavelengths. These instabilities have growth rates 60–100% greater than the Crow instability. The system of two vortex pairs also exhibits transient growth which can lead to growth factors of 10 or 15 in one-fifth of the time required for the same growth due to instability.