Optimal streaks in the circular cylinder wake and suppression of the global instability

The steady, spanwise-periodic, symmetric (varicose) optimal blowing and suction that maximizes energy amplification in the circular cylinder wake is computed at Reynolds numbers ranging from 50 to 100. It is found that the cylinder wake can sustain large energy amplifications that are associated with the generation by the optimal blowing and suction of streamwise vortices near the cylinder, which then induce the transient spatial growth of high-energy streamwise streaks further downstream. The most amplified perturbations have spanwise wavelengths ranging from five to seven times the cylinder diameter at the Reynolds numbers considered, with the corresponding optimal streaks reaching their maximum amplitude in the near wake, inside the pocket of absolute instability which sustains the global instability. The optimal blowing and suction is shown to stabilize the global linear instability. The most stabilizing spanwise wavelengths are in good agreement with previous findings. The amplitude of optimal blowing and suction required to suppress the global instability decreases when the Reynolds number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Re}$ is increased from 75 to 100. This trend reveals the key role played by the non-normal amplification of the streaks in the stabilization process, which is able to overcome the increase of the uncontrolled global growth rate with $\mathit{Re}$. Finally, it is shown that the global instability can be suppressed with control amplitudes smaller than those required by 2-D (spanwise-uniform) control. This result is not what would be expected from first-order sensitivity analyses, which predict a zero sensitivity of the global instability to spanwise-periodic control and, in general, a non-zero sensitivity to spanwise-uniform control.