Instantaneous linear stability of plane Poiseuille flow forced by spanwise oscillations

In the present work, the stability of a plane Poiseuille flow forced by spanwise oscillations is studied via the instantaneous linear stability theory (LST). For streamwise Poiseuille flow and a spanwise Stokes layer, the superposition of these two linearly stable flows can lead to transient growth of perturbations. Periodic and aperiodic growth rates over time are found. Mode-crossing is observed in the aperiodic mode. Effects of oscillation amplitude W₀ and frequency Ω are studied. The maximum instantaneous growth rate increases with increasing W₀. At low W₀, the stability of the flow is dominated by the corresponding Poiseuille flow or by the interactions between the two flows. At high W₀, the stability characteristics of the Poiseuille-Stokes flow are very similar to those of the corresponding Stokes layer. It is demonstrated that oscillations have destabilizing effects in short-time intervals of one oscillation period. Oscillations of extremely low or high Ω have much weaker effects than those of medium Ω. The transient growth of the most unstable mode is traced by using direct numerical simulation (DNS). Instantaneous LST can capture the transient growth but fails to predict the accurate growth rate when the amplification ratio is higher than e⁹. The difference between LST and DNS is mainly due to the incorrect production of LST without nonlinear transportation. From the growing phase to the decay phase of the transient mode, the production term shifts from the destabilizing role to the stabilizing role because of the correlation-phase reversal of the perturbation velocities.