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

Abstract

Selective noise amplifiers are characterized by large linear amplification to external perturbations in a particular frequency range despite their global linear stability. Applying a stochastic forcing with increasing amplitude, the response undergoes a strong nonlinear saturation when compared to the linear estimation. Building upon our previous work, we introduce a predictive model that describes this nonlinear dynamics, and we apply it to a canonical example of selective noise amplifiers: the backward-facing step flow. Rewriting conveniently the stochastic forcing and response in the frequency domain, the model consists in a mean flow equation coupled to the linear response to forcing at each frequency. This coupling is attained by the Reynolds stress, which is constructed by the integral in frequency of the independent responses. We generalize the model for a response to a white noise forcing δ-correlated in space and time restricting the flow dynamics to its most energetic patterns calculated from the optimal harmonic forcing and response of the flow. The model estimates accurately the response saturation when compared to direct numerical simulations, and it correctly approximates the structure of the response and the mean flow modification. It also shows that the response undergoes a selective process governed by the nonlinear gain, which promotes a response structure with an approximately single frequency and wavelength in the whole domain. These results suggest that the mean flow modification by the Reynolds stress is the key nonlinearity in the saturation process of the response to white noise.

Frequency prediction from exact or self-consistent mean flows

“…The resolvent and generalizations of it have been used in Refs. [17][18][19][20][21][22][23][24][25][26][27] to approximate the optimal forcing or the energy spectrum of complex and even turbulent flows.…”

“…Mantič-Lugo and Gallaire [21] have carried out a study of the optimal forcing response in the backward facing step, comparing fully nonlinear results [retaining in (15b) the nonlinear terms g of Eq. (11)] with linear results from the resolvent Eq.…”

“…The SCM has also been used to treat acoustic emissions in the compressible wake of a cylinder [15] and the two-dimensional shear driven cavity [16]. Other reduced-order models in which sets of modes or interactions are omitted have been proposed and implemented for many other hydrodynamic phenomena, notably in aeronautics and fluid mechanics [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. and in geophysics and astrophysics [35][36][37][38][39][40].…”

Spectral energetics of a quasilinear approximation in uniform shear turbulence

“…The modern approaches share similar ideas with these early ones, but they take more flexible and delicate approaches for modelling of the self-interaction term of the second group (e.g. stochastic forcing, eddy viscosity, etc); for example, stochastic structural stability theory (S3T) (Farrell & Ioannou 2007, 2012), direct statistical simulation (DSS) (Marston, Conover & Tobias 2008; Tobias & Marston 2013), self-consistent approximations (Mantič-Lugo, Arratia & Gallaire 2014; Mantič-Lugo & Gallaire 2016), restricted nonlinear model (RNL) (Thomas et al. 2014, 2015; Farrell, Gayme & Ioannou 2017), generalised quasilinear approximations (GQL) (Marston, Chini & Tobias 2016; Tobias & Marston 2017) and minimal quasilinear approximation augmented with eddy viscosity (Hwang & Eckhardt 2020).…”

Frequency selection in a gravitationally stretched capillary jet in the jetting regime

“…12) for a jet defined by Oh in = 0.3, Bo in = 0.1 and We in = 1.75 for (a) the optimal forcing frequency ω opt and (b) the breakup length l c at different inverse forcing amplitudes 1/ . with the scheme presented in Section 2 by exciting the jet at the nozzle by a white noise ξ(t) defined in the time interval [0 T ] and formulated in a similar way as in Mantič-Lugo & Gallaire (2016). The white noise signal ξ(t) is characterised by a constant power spectral density (PSD) S ξξ (ω) = |ξ(ω)| 2 whereξ(ω) is the Fourier transform of ξ(t) and has an infinite power P defined as, (5.35) where σ is the variance.…”