Sensitivity of stochastic perturbations in a noise amplifier flow

It is a well-known fact that small-amplitude perturbations can be amplified in globally stable flows due to non-normal mechanisms. This leads in the time domain to the transient growth of some initial disturbances, and in the frequency domain to the amplification of some harmonic disturbances. It is of interest to look for the most amplified of these disturbances: the optimal perturbations. We consider here random perturbations which are being continuously advected in the laminar flow past a backward-facing step. The response of the flow to this sustained stochastic input can be analysed in the frequency domain in terms of amplification at each individual frequency, and the problem therefore boils down to combining appropriately harmonic optimal and sub-optimal perturbations at all frequencies. In this work we use a variational technique to compute the sensitivity of noise amplification to steady control, either passive (e.g. a small cylinder inserted in the flow) or active (e.g. wall blowing or suction). Sensitivity maps allow the identification of regions where control is most effective in reducing amplification. Our work extends existing sensitivity methods in two ways: (i) from time-harmonic to time-stochastic, (ii) from perturbations distributed in space in the whole flow to perturbations localized at the inlet. We therefore deal with a more realistic representation of incoming noise. We also simplify the process of control design by integrating all harmonic sensitivity maps for each individual frequency into one single stochastic sensitivity map. In the backward-facing step flow, harmonic disturbances are strongly amplified around one preferred frequency. The amplification of stochastic noise can be reduced by inserting a small control cylinder in regions of large streamwise velocity, particularly at streamwise locations between the step corner and the upper reattachment point. Alternatively, boundary control is most effective when using wall suction upstream of the step corner; this results in a shorter separation bubble on the lower (resp. upper) wall when using boundary actuation at the lower (resp. upper) wall, as confirmed by applying sensitivity analysis to recirculation length in this flow. We find that sensitivity maps for stochastic noise are largely similar to those for the optimal harmonic perturbation at the most amplified frequency. This suggests that the design of steady control in strong noise amplifiers (globally stable but convectively unstable flows) can be conducted restricting one's attention to the most dangerous perturbation at the most dangerous frequency.