Linear two-dimensional stability of a Lamb-Oseen dipole as an aircraft wake model

The dynamics of perturbed aircraft wake models is investigated in the two-dimensional limit by means of a linear stability analysis. The base flow, computed by a direct numerical simulation of the incompressible Navier-Stokes equations, is a viscous counter-rotating vortex dipole obtained from an initial condition which is either a superposition of two Lamb-Oseen vortices or a vorticity sheet with elliptical vorticity distribution. The former, referred to as the Lamb-Oseen dipole (LOD), is a model for the far field of the wake and gives rise to a family of quasisteady dipoles parametrized by their aspect ratio only. The later approaches the near-field wake of a wing during the rolling phase which eventually converges towards the LOD model at later times. First, a modal stability analysis of the LOD under the assumption of a frozen base flow is performed for aspect ratios a/b ranging from 0.05 to 0.36 at various Reynolds numbers. Several families of unstable antisymmetric and symmetric modes are observed. The maximal growth rates are reached at low Reynolds numbers. The results are consistent with those obtained by Brion et al. [Phys. Fluids 26, 054103 (2014)] for the higher aspect ratio inviscid Lamb-Chaplygin dipole (LCD). However, the a posteriori verification of the validity of the frozen base flow assumption shows that, except for a few modes occurring at the highest aspect ratios and large Reynolds numbers, these two-dimensional instabilities do not survive the base flow unsteadiness due to viscous diffusion. They are thus not likely to develop in the flow. Second, we focus on the transient dynamics of the dipoles by looking for the optimal perturbations through a nonmodal stability analysis based on a direct-adjoint approach. The observed energy gains are substantial and indicate the potential of transient mechanisms. In the short time dynamics, the optimal perturbation consists of intertwined vorticity layers located within each vortex core and leading to a m = 2 deformation Kelvin wave excited by the Orr mechanism. For moderate to large horizon times, the optimal perturbation takes the form of vorticity layers localized outside the vortex core which eventually give rise to the two-dimensional unstable mode unveiled by the modal analysis through a combination of Orr mechanism and velocity induction. The robustness of these modes is examined by considering the initial stage of the development of aircraft wakes. The optimal perturbations developing on an elliptic and a double-elliptic vorticity sheet present a similar structure and rely on the same mechanisms as the ones observed for the LOD but come with lower energy gains. It is concluded that the rolling-up of the vorticity sheet in the near-field of the wake does not influence significantly the linear development of these two-dimensional perturbations.