Open-loop control of noise amplification in a separated boundary layer flow

Abstract

1Linear optimal gains are computed for the subcritical two-dimensional separated boundary-layer flow past a bump. Very large optimal gain values are found, making it possible for small-amplitude noise to be strongly amplified and to destabilize the flow. The optimal forcing is located close to the summit of the bump, while the optimal response is the largest in the shear layer. The largest amplification occurs at frequencies corresponding to eigenvalues which first become unstable at higher Reynolds number. Non-linear direct numerical simulations show that a low level of noise is indeed sufficient to trigger random flow unsteadiness, characterized here by large-scale vortex shedding.Next, a variational technique is used to compute efficiently the sensitivity of optimal gains to steady control (through source of momentum in the flow, or blowing/suction at the wall). A systematic analysis at several frequencies identifies the bump summit

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

“…As pointed out by , the flow response should therefore be dominated by the optimal response. Boujo et al (2013) observed this behaviour in a direct numerical simulation of a different geometry, with a predominance of the optimal response at optimal frequency. As discussed later in section 5, the sensitivity of the stochastic gain is also dominated by the sensitivity of the optimal harmonic gain at the optimal frequency.…”

“…All calculations are performed using methods described in Boujo et al (2013). The finite element software FreeFem++ is used to generate a two-dimensional triangulation of the domain Ω and, based on P2 and P1 Taylor-Hood elements for velocity and presure respectively, to build all the discrete operators involved in calculations of base flow, eigenvalue and sensitivity, from their corresponding continuous expression in variational form.…”

“…The dependence of the maximal optimal gain on Reynolds number is exponential beyond some value of Re, as illustrated in Figure 3(b), with log(max ω G in,1 ) and log(max ω G vol,1 ) scaling like 0.66 × 10 −2 Re. The exponent for volume forcing was ≃ 0.6 × 10 −2 Re in the flow past a wall-mounted bump with Re based on the bump height (Boujo et al 2013). for a stenotic flow, and 1.18 × 10 −2 for the Γ = 0.5 backward-facing step with Re based on the centreline velocity), while in parallel flows the maximal transient growth only increases like Re 2 (Schmid & Henningson 2001).…”

“…They applied their formula to parallel and non-parallel flat-plate boundary layers and discussed the sensitivity of G(ω) for Tollmien-Schlichting and lift-up instability mechanisms. Boujo et al (2013) used this method to identify sensitive regions in the separated flow past a wall-mounted bump, and designed a simple open-loop wall control able to delay noise-induced subcritical transition. They observed, however, that sensitivity to volume control was dependent on frequency, indicating that a given control could reduce G(ω) at some frequencies but increase it at others.…”

A self-consistent formulation for the sensitivity analysis of finite-amplitude vortex shedding in the cylinder wake

“…Such an approach (herein referred to as the base flow approach) offers an attractive alternative to bottleneck "trial and error" procedures, as it allows exhaustive coverage of large parameter spaces at very low computational costs. It has sparked renewed interest in the last decade, through a substantial body of work focusing on steady and unsteady effects modeling the presence of the control cylinder (Giannetti & Luchini 2007;Marquet et al 2008a), and is now routinely applied to a variety of flow configurations (Meliga et al 2010;Pralits et al 2010;Alizard et al 2010) and control targets, including optimal transient growth and optimal harmonic response (Brandt et al 2011;Boujo et al 2013), recirculation length or aerodynamic forces (Meliga et al 2014).…”