Optimal perturbations of non-parallel wakes and their stabilizing effect on the global instability

Abstract

We compute the spatial optimal energy amplification of steady inflow perturbations in a non-parallel wake and analyse their stabilizing action on the global mode instability. The optimal inflow perturbations, which are assumed spanwise periodic and varicose, consist in streamwise vortices that induce the downstream spatial transient growth of streamwise streaks. The maximum energy amplification of the streaks increases with the spanwise wavelength of the perturbations, in accordance with previous results obtained for the temporal energy growth supported by parallel wakes. A family of increasingly streaky wakes is obtained by forcing optimal inflow perturbations of increasing amplitude and then solving the nonlinear Navier-Stokes equations. We show that the linear global instability of the wake can be completely suppressed by forcing optimal perturbations of sufficiently large amplitude. The attenuation and suppression of self-sustained oscillations in the wake by optimal 3D perturbations is confirmed by fully nonlinear numerical simulations. We also show that the amplitude of optimal spanwise periodic (3D) perturbations of the basic flow required to stabilize the global instability is much smaller than the one required by spanwise uniform (2D) perturbations despite the fact that the first order sensitivity of the global eigenvalue to basic flow modifications is zero for 3D spanwise periodic modifications and non-zero for 2D modifications. We therefore conclude that first-order sensitivity analyses can be misleading if used far from the instability threshold, where higher order terms are the most relevant.