Second-order perturbation of global modes and implications for spanwise wavy actuation

Sensitivity analysis has successfully located the most efficient regions in which to apply passive control in many globally unstable flows. As is shown here and in previous studies, the standard sensitivity analysis, which is linear (first order) with respect to the actuation amplitude, predicts that steady spanwise wavy alternating actuation/modification has no effect on the stability of planar flows, because the eigenvalue change integrates to zero in the spanwise direction. In experiments, however, spanwise wavy modification has been shown to stabilize the flow behind a cylinder quite efficiently. In this paper, we generalize sensitivity analysis by examining the eigenvalue drift (including stabilization/destabilization) up to second order in the perturbation, and show how the second-order eigenvalue changes can be computed numerically by overlapping the adjoint eigenfunction with the first-order global eigenmode correction, shown here for the first time. We confirm the prediction against a direct computation, showing that the eigenvalue drift due to a spanwise wavy base flow modification is of second order. Further analysis reveals that the second-order change in the eigenvalue arises through a resonance of the original (2-D) eigenmode with other unperturbed eigenmodes that have the same spanwise wavelength as the base flow modification. The eigenvalue drift due to each mode interaction is inversely proportional to the distance between the eigenvalues of the modes (which is similar to resonance), but also depends on mutual overlap of direct and adjoint eigenfunctions (which is similar to pseudoresonance). By this argument, and by calculating the most sensitive regions identified by our analysis, we explain why an in-phase actuation/modification is better than an out-of-phase actuation for control of wake flows by spanwise wavy suction and blowing. We also explain why wavelengths several times longer than the wake thickness are more efficient than short wavelengths.