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

摘要

The two-dimensional backward-facing step flow is a canonical example of noiseamplifier flow: global linear stability analysis predicts that it is stable,but perturbations can undergo large amplification in space and time as a resultof non-normal effects. This amplification potential is best captured by optimaltransient growth analysis, optimal harmonic forcing, or the response tosustained noise. In view of reducing disturbance amplification in theseglobally stable open flows, a variational technique is proposed to evaluate thesensitivity of stochastic amplification to steady control. Existing sensitivitymethods are extended in two ways to achieve a realistic representation ofincoming noise: (i) perturbations are time-stochastic rather thantime-harmonic, (ii) perturbations are localised at the inlet rather thandistributed in space. This allows for the identification of regions wheresmall-amplitude control is the most effective, without actually computing anycontrolled flows. In particular, passive control by means of a small cylinderand active control by means of wall blowing/suction are analysed for Reynoldsnumber $Re=500$ and step-to-outlet expansion ratio $\Gamma=0.5$. Sensitivitymaps for noise amplification appear largely similar to sensitivity maps foroptimal harmonic amplification at the most amplified frequency. This isobserved at other values of $Re$ and $\Gamma$ too, and suggests that the designof steady control in this noise amplifier flow can be simplified by focusing onthe most dangerous perturbation at the most dangerous frequency.