Bi-global Optimal Perturbation of a Laminar Separation Bubble using Harmonic Resolvent Analysis

The bi-global optimal perturbations of an adverse-pressure-gradient-induced separated unsteady flow over a flat plate are investigated. Using linear stability and resolvent analyses of the stationary and time-varying base flows of the laminar separation bubble (LSB), we systematically examine the roles of the time-invariant and unsteady components in amplifying perturbations. The incorporation of the unsteady base into the resolvent analysis framework provides additional insight into the downstream evolution of coherent perturbations in the flow, a feature lacking in the traditional method. The LSB amplifies a wide range of frequencies, in addition to the primary vortex-shedding frequency, as found from classical resolvent analysis using the spanwise-time-averaged flow as the time-invariant base state. The harmonic resolvent analysis is performed by linearizing the Navier-Stokes equation about a spanwise-averaged time-varying base with the same periodicity as the dominant vortex-shedding frequency. Both studies, using time-invariant and time-varying base flows, reveal 3D perturbation amplification at low frequencies relative to the vortex shedding frequency. The modal structures show that the unsteady base flow amplifies the perturbation near and after the reattachment of the separation bubble, compared to the time-invariant base flow, which exhibits perturbation amplification only within the separation zone.