Second order sensitivity, Ginzburg-Landau equation and the stabilizing mechanism of streaks on 2D absolute and global instabilities
Résumé
Global and local absolute instabilities of 2D wakes are known to be stabilized by spanwise periodic modulations of the wake velocity [4, 3, 1, 2] but the nature of this stabilizing mechanism is not yet completely understood. Assuming that the leading effect of the spanwise modulations of the wake streamwise velocity profile is associated to the spanwise modulation of the advection velocity of unstable waves, we investigate the effect of spanwise periodic modulations of the wave advection velocity in a generalization of the linear complex Ginzburg-Landau equation, which approximates the spatio-temporal evolution of unstable waves. We show that the enforced spanwise modulations of the wave advection velocity have a stabilizing effect on the global instability of the non-parallel complex Ginzburg-Landau equation. The growth rate of the unstable global mode is shown to decrease quadratically with the amplitude of the modulations and the global instability is suppressed for large enough modulation amplitudes. It is then shown that a second order structural sensitivity analysis provides accurate predictions of the variation of the growth rate of the unstable global mode with respect to the amplitude of the advection velocity modulations. This second order analysis is then applied to compute the influence of these modulations on the absolute instability growth rate in the parallel case. It is shown that the leading order effect of the advection velocity modulations is to alter the wave diffusivity in the local dispersion relation. This altered diffusivity is associated to a decrease of the absolute growth rate. A straightforward local stability analysis of the global mode stabilization obtained in the non-parallel case shows that at criticality the spanwise advection velocity modulations have reduced the pocket of local absolute instability to the same level that it would have had by reducing the global bifurcation parameter µ max to its critical value µ c. This confirms that the suppression of the global instability is induced by the reduction of the local absolute instability. These results, which are in complete agreement with what is found in non-parallel wakes [1, 2], show that the simple spanwise modulation of the advection velocity plays the key role in the stabilization mechanism. As no vorticity is defined in the Ginzburg-Landau equation, this key stabilizing mechanism is simpler and of more general nature than explanations based on the vortex dynamics of wake vortices in the presence of 3D modulations. This stabilizing effect is also more general in the sense that it can probably be applied to other physical systems described by the complex Ginzburg-Landau equation. Spanwise modulations of the advection velocity may e.g. play a role in the analysis of the solar dynamo mechanism [6] or of the cyclogenesis in the atmosphere [5], where the absolute and global instabilities play an important role.