Optimal streaks in the wake of a blunt-based axisymmetric bluff-body and control of vortex shedding
Résumé
We are interested in the control of global instabilities developing in the wake of a blunt-based axisymmetric bluff-body with an ellipsoidal nose by means of the optimal forcing of large-amplitude steady streaks in the wake, thus extending to three-dimensional wakes the approach used in [1, 2] to control global instabilities in two-dimensional wakes. Optimal perturbations are forced by means of a radial velocity of azimuthal wavenumber m applied on the lateral skin of the body. The optimal distributions of such a forcing, maximizing the spatial energy growth of steady perturbations in the wake, are computed as in [1, 2] with a scalable adjoint-free subspace expansion method which is found to quickly converge to the optimal solutions. Optimal perturbations with m = 0 lead to the amplification of streamwise streaks in the wake (see panels a and b in Figure 1) and are obtained with zero mass flux, while m = 0 perturbations correspond to a non-zero mass flux. We find that m = 1 optimal perturbations forced with finite amplitude have a stabilizing effect on the large-scale unsteady vortex shedding in the wake and that m = 0 optimal perturbations are stabilizing only for negative amplitudes (suction). The forcing of m ≥ 2 modes can significantly reduce the amplitude of the unsteady lift force exerted on the body (see panel c in Figure 1). When combined with low levels of base bleed these perturbations can completely suppress the unsteadiness in the wake with reduced levels of mass injection in the flow. (c) Figure 1. Cross-stream (y − z) view of the velocity perturbations forced by the m = 2 optimal blowing and suction at the bluff-body stern (panel a) and at the position of maximum streak amplitude (panel b). The scales used to plot the cross-stream v-w velocity components (streamwise vortices, arrows) and the streamwise u component (streamwise streaks, contour-lines) are the same in both panels. The circular cross-section of the bluff-body base is also reported as a (red) circle for reference. In panel c we show the temporal history of the lift coefficient at Re = 500 associated to the uncontrolled flow (Aw = 0) and to increasing amplitudes Aw of the m = 2 optimal blowing and suction.