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

Abstract

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. In view of 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 for 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 Γ = 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 Γ 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.Key words: Authors should not enter keywords on the manuscript, as these must be chosen by the author during the online submission process and will then be added during the typesetting process (see http://journals.cambridge.org/data/relatedlink/jfm-keywords.pdf for the full list)

Saturation of the response to stochastic forcing in two-dimensional backward-facing step flow: A self-consistent approximation

“…1 for the backward-facing step at the chosen Re = 500, as studied in the literature [12,15,18,27]. For an expansion ratio = 0.5, the flow is globally stable at this Reynolds number, presenting mainly a 2D response to white noise [12] and thus supporting the choice of a 2D analysis.…”

“…While the focus of the study is on the response to stochastic forcing, we describe first the harmonic response to facilitate understanding and to predict in which frequency range larger amplifications are more likely to be observed [4,5,[17][18][19]. For a harmonic forcing f (y,t) = f 1 (y)e iωt + c.c.…”

“…2(b). The structure of the response displays an apparent characteristic wave number that increases with the frequency, and an envelope that migrates downstream for larger gains [16,18].…”

“…The optimal forcing and corresponding response are described by the singular vector of the resolvent operator R(ω) = (iω + L ) −1 [2,4]. Optimal forcing/response structures have been assessed in plane Couette [13] as well as in spatially developing open flows [8,10,11,14] and particularly in the backward-facing step [15][16][17][18][19]. A slightly different approach is undertaken by Garnaud, Lesshaftt, Schmid, and Huerre [14] where, in an attempt to describe more precisely the actual physics involved in the strong noise amplification exhibited in turbulent jets, they apply the optimal gain analysis on a model mean flow instead of the stable steady solution of the Navier-Stokes equations (NSEs) as in the previously mentioned studies.…”

“…Additionally, the response to inlet white noise forcing in the backward-facing step shows that the exact stochastic response from the direct numerical simulation (DNS) is well characterized by the two-dimensional (2D) optimal perturbances [12]. More recently, Boujo and Gallaire [18] have studied the sensitivity of the stochastic response to passive forcing devices, with control applications in mind.…”