Sensitivity and open-loop control of stochastic response in a noise amplifier flow: the backward-facing step

The two-dimensional backward-facing step flow is a canonical example of noise amplifier flow: global linear stability analysis predicts that it is stable, but perturbations can undergo large amplification in space and time as a result of non-normal effects. This amplification potential is best captured by optimal transient growth analysis, optimal harmonic forcing, or the response to sustained noise. With a view to reducing disturbance amplification in these globally stable open flows, a variational technique is proposed to evaluate the sensitivity of stochastic amplification to steady control. Existing sensitivity methods are extended in two ways to achieve a realistic representation of incoming noise: (i) perturbations are time-stochastic rather than time-harmonic, (ii) perturbations are localised at the inlet rather than distributed in space. This allows the identification of regions where small-amplitude control is the most effective, without actually computing any controlled flows. In particular, passive control by means of a small cylinder and active control by means of wall blowing/suction are analysed for Reynolds number $Re=500$ and step-to-outlet expansion ratio $\Gamma=0.5$. Sensitivity maps for noise amplification appear largely similar to sensitivity maps for optimal harmonic amplification at the most amplified frequency. This is observed at other values of $Re$ and $\Gamma$ too, and suggests that the design of steady control in this noise amplifier flow can be simplified by focusing on the most dangerous perturbation at the most dangerous frequency.