Three-Dimensionnal Instabilities and Transient Growth of Trailing Vortices in Homogeneous and Stratified Flows

Abstract

An aircraft wake is made of counter-rotating vortices and is known to be affected by a long (Crow) and a short (elliptic) wavelength instabilities. Numerical investigations on the threedimensionnal instabilities and transient growth of such dipole are performed. By means of a three-dimensionnal linear stability analysis, we retrieve the instability bands corresponding to the Crow and elliptic modes but we also observe less unstable oscillatory modes with very broad peaks. The transient growth of perturbations on this dipole, investigated by computing the optimal linear perturbations with a direct-adjoint technique, demonstrates the crucial role of the region of maximal strain at short time and of the hyperbolic point at intermediate time . Investigations on the three-dimensionnal dynamics of trailing vortices in stratified fluids are performed. The elliptic instability is almost unaffected by weak and moderate stratifications. Trailing vortices behind aircrafts consist of a horizontal pair of counter-rotating vortices propagating downwards. Depending on atmospheric conditions, such dipole can persist over a long time or be rapidely destroyed. If the vortex pair remains coherent, it can be hazardous to following aircrafts, especially during take-off and landing thus limiting the frequency between airplanes at airports. Studies of the dynamics of a pair of counter-rotating vortices in unstratified flows have shown that this vortex pair is unstable with respect to three-dimensionnal perturbations. [Crow (1970)] has discovered a long-wavelength instability, symmetric with respect to the plane separating the two vortices. The existence of a short-wavelength elliptic instability has been revealed by [Tsai & Widnall (1976)], [Moore & Saffman (1975)] and numerous articles ever since for both symmetric and antisymmetric modes. This instability, due to the elliptic deformation of the core of the vortices, is a resonant interaction between the strain and Kelvin waves of azimuthal wavenumbers m = 1 and m = 1 when both waves have the same frequency ! and are particularly intense for ! = 0. However in many atmospheric situations, as such dipoles propagate downwards, they evolve under the influence of the stable stratification of the atmosphere and the three-dimensionnal dynamics of this vortex pair in stratified flow has yet received much less attention. Direct numerical simulations of [Nomura et al. (2006)] on the short-wavalength instability of a counter-rotating vortex pair in presence of stable stratification have suggested that the instability mechanism corresponds, despite the stratification, to the elliptic instability as in homogeneous media. The instability appears earlier than in the unstratified case, owing to the decrease due to the stratification of the separation distance between the vortices as they propagate downwards, decrease that induces larger ellipticity of the vortices and then enhances the instability. In this paper, we perform a three-dimensionnal linear stability analysis of a Lamb-Oseen vortex pair in unstratified fluid in section 2. The transient growth of perturbations on this vortex pair, investigated by computing the optimal perturbations with the direct-adjoint technique introduced by [Corbett & Bottaro (2000)], is presented in section 3. In the stratified case, the two-dimensional flow is unsteady and the optimal perturbations are computed at several times, with a direct-adjoint technique similar to the one used in the steady case and which takes into account the evolution of the flow. The results of this study are presented in section 4.