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

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 $\delta$-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.

Figure 1

Sketch of the flow configuration of the backward-facing step with two recirculation bubbles at $\text{Re}=500$, one at the top and one at the bottom.

Figure 2

(a) Linear gain $G_B\left(\omega \right)$ around the base flow and (b) contours of the velocity in the $y$-direction of the linear response for different forcing frequency $\text{St}=\omega /2\pi$ for a harmonic forcing of the form of a Poiseuille profile ${\mathit{f}}_1=y(1-y)$. The linear gain of the selected frequencies in (b) is marked as black circles in the curve in (a). The maximal gain is attained at the optimal frequency $\text{St}={\omega }_{\text{opt}}/\left(2\pi \right)=0.075$. Plots for the backward-facing step at $\text{Re}=500$.

Figure 3

(a) Realization of a white noise signal with unit power $P=1$, comparing a signal without filtering ${\omega }_b/2\pi =1/\left(2\delta t\right)=25$ and a filtered one with a band-limiting frequency ${\omega }_b/2\pi =1$. (b) Comparison of the power spectral density for these two signals.

Figure 4

(a) Amplitude of the linear response $R$ and (b) total linear gain $G_{\text{tot}}$ as a function of the band-limiting frequency ${\omega }_b$ for a band-limited white noise with constant PSD $2|{\widehat{\xi }}_b{\left(\omega \right)|}^2=1$ and varying power $P\left({\omega }_b\right)$ (9), where the spatial distribution is in the form of a Poiseuille profile ${\mathit{f}}_1=\sqrt{30}y(1-y)$. The response amplitude and total gain are computed from the integration of the linear gain $G_B\left(\omega \right)$ around the base flow for the backward-facing step at $\text{Re}=500$.

Figure 5

Nonlinear total gain from DNS $G_{\text{tot}}$ and linear total gain $G_{B\text{tot}}$ as a function of the amplitude of the forcing $A$. The figure shows the saturation of the gain and the variation of the response structure. The insets show the perturbation velocity in the $y$-direction and the perturbation energy. $\text{Re}=500$.

Figure 6

(a) Gain and (b) response as a function of the forcing amplitude $A$ of the band limited white noise for the DNS (circles), SC model with $n_f=21$ (triangles), SC model with $n_f=1$ at ${\omega }_{\text{opt}}$ (squares), and the linear estimation (dash dotted line). $\text{Re}=500$.

Figure 7

Total gain of the exact DNS (solid line) and SC model as a function of the number of discrete frequencies ${\omega }_i$ given by $n_f$ for the saturated mean flow with a forcing amplitude $A=0.1$ and $\text{Re}=500$.

Figure 8

(a) Gain distribution function of the frequency and total gain values for the DNS and SC model saturated with a forcing amplitude $A=0.1$. The exact DNS total gain is $G_{\text{tot}}=12$ (thick solid line) and $G_{\text{tot}}=9.7$ for the linear prediction around the DNS mean flow (thin solid line). The SC model integrated in frequency has a total gain $G_{\text{tot}}=12.7$ (squares) for $n_f=21,G_{\text{tot}}=12.5$ for $n_f=9$ (circles), and $G_{\text{tot}}=10.3$ for $n_f=1$ (thick dashed line). (b) PSD as a function of the frequency for different positions $x$ at the centerline, $\left(y=0\right)$, for the DNS fluctuating velocity field for $A=0.1$ and $\text{Re}=500$.

Figure 9

Energy distribution of the response fluctuation for (a) linear around base flow, (b) DNS, (c) SC with $n_f=21$, and (d) SC with single frequency approximation ${\omega }_{\text{opt}}$. Fluctuation velocity in the $x$-direction, $u_x^{\prime }$ for (e) linear around base flow, (f) DNS snapshot, (g) SC with $n_f=21$ constructed with arbitrary ${\varphi }_i$ in Eq. (18), and (h) SC with $n_f=1$, single frequency approximation at ${\omega }_{\text{opt}}$. Forcing amplitude $A=0.1$ and $\text{Re}=500$.

Figure 10

Position in the $x$ coordinate of the maximum of the energy of the perturbation $x_{\text{max}}$ for SC and DNS as a function of the forcing amplitude $A$ for $\text{Re}=500$.

Figure 11

Comparison of the mean flow and base flow for the velocity in the $x$-direction: (a) base flow ${\mathit{U}}_B$ and (b) DNS mean flow $\mathit{U}$. Difference between the mean flow and the base flow $\mathrm{\Delta }\mathit{U}$ defined as mean flow correction in Eq. (13), (c) $\mathrm{\Delta }\mathit{U}$ for the DNS mean flow, and (d) $\mathrm{\Delta }\mathit{U}$ for the SC mean flow for $n_f=21$. Forcing amplitude $A=0.1$ and $\text{Re}=500$.

Figure 12

Position of the mean flow recirculation bubbles (a) bottom and (b) top for the DNS (squares), SC integrated in frequency (circles), and SC with optimal frequency approximation (triangles) as a function of the forcing amplitude $A$. $\text{Re}=500$.

Figure 13

(a) Gain and (b) response as functions of the forcing amplitude $A$ of the band-limited white noise for the DNS (circles), SC model integral in frequency with $n_f=21$ (triangles), SC model with $n_f=1$ at ${\omega }_{\text{opt}}$ (squares), and the linear estimation (dash-dotted line). $\text{Re}=700$.

Figure 14

Position of the recirculation bubbles (a) bottom and (b) top for the DNS (squares), SC model integrated in frequency (circles), and SC model with optimal frequency approximation (triangles) as a function of the forcing amplitude $A$. (c) Position maximum of the energy of the perturbation $x_{E\text{max}}$ for the SC model and DNS as a function of the forcing amplitude $A$. $\text{Re}=700$.