Noise-induced transitions past the onset of a steady symmetry-breaking bifurcation: The case of the sudden expansion

We consider fluid flows, governed by the Navier-Stokes equations, subject to a steady symmetry-breaking bifurcation and forced by a weak noise acting on a slow timescale. By generalizing the multiple-scale weakly nonlinear expansion technique employed in the literature for the response of the Duffing oscillator, we rigorously derive a stochastically forced Stuart-Landau equation for the dominant symmetry-breaking mode. The probability density function of the solution, and of the escape time from one attractor to the other, are then determined by solving the associated Fokker-Planck equation. The validity of this reduced order model is tested on the flow past a sudden expansion for a given Reynolds number and different noise amplitudes. At a very low numerical cost, the statistics obtained from the amplitude equation accurately reproduce those of long-time direct numerical simulations.

Figure 1

(a) Example of random signals as a function of time. The blue continuous line is a sampling-limited white noise $\xi \left(t\right)$ (multiplied by a factor 10 for better visualization), whereas the blue continuous line with bullets is the slowly varying version $\xi \left(\alpha \epsilon t\right)$. (b) Modulus of the Fourier transforms of the signals shown in (a). The specific values $\epsilon =0.0026, \alpha =1/8$, and a sampling time step $\mathrm{\Delta }t=0.025$ have been selected; this sets ${\omega }_d=125.7$ and ${\omega }_{\mathrm{co}}=0.04$.

Figure 2

Deterministic case $\varphi =0$. (a) Snapshots of the streamwise velocity of the stable steady solution(s) obtained by DNS at $\mathrm{Re}={\mathrm{Re}}_c=79.3$ (top) and at $\mathrm{Re}=100>79.3$ (center and bottom); in the latter case, two equilibrium solutions are found. (b) Measure $M$ that quantifies the asymmetry of the flow, as defined in (25), and computed in the steady regime for both WNL and DNS approaches. The continuous blue line is associated with the stable equilibrium solutions predicted by the deterministic amplitude equation $\pm \overline{M}$ with $\overline{M}=\sqrt{\epsilon }\beta \overline{A}$, whereas the dotted line corresponds to the unstable equilibrium.

Figure 3

Temporal signal of $M$ in a stochastically forced regime, with forcing amplitudes $\varphi \in [10,14,18,22]$. The noise realization $\xi \left(\alpha \epsilon t\right)$ is the same for all the results shown. A $\mathrm{Re}$ number of $\mathrm{Re}=100$ is chosen, implying $\epsilon =0.0026$. A time step $\mathrm{\Delta }t=0.025$ was found sufficient for the convergence of the results, which sets ${\omega }_d=125.7$. We further choose ${\omega }_{\mathrm{co}}=0.04$, which determines $\alpha =1/8$ that we check to be $O\left(1\right)$ indeed. The two deterministic attractors $\pm \overline{M}$ with $\overline{M}=\sqrt{\epsilon }\beta \overline{A}=0.1935$ are highlighted by horizontal black dotted lines. The small crosses highlight a “transition” (see definition in text) of the WNL signal from the neighborhood of one attractor to the neighborhood of the other.

Figure 4

Probability density function of $\left|M\right|$ for the four different forcing amplitudes considered in Fig. 3. The external parameters are also the same as for Fig. 3. For the blue dots “WNL” and the red dashed-dotted line with the square markers “DNS,” 10 simulations, each corresponding to a different noise realization and long of $t=1.1\times {10}^4$ units of times, were performed and post-treated. The magenta line “WNL, $\overline{P}$”, defined in (23), is the steady solution of the Fokker-Planck equation (22). The black dashed line “WNL, ${\overline{P}}^{\text{w}}$”, defined in (24), also corresponds to the steady solution of the Fokker-Planck equation but in a case of “pure” white noise where $\alpha {\omega }_d\to \infty$. The insets show to the same data as the main figures, but as a function of $\left|M\right|$ instead of $\left|M\right|/\overline{M}$. The vertical red dashed line is the deterministic attractor $\overline{M}$ of the DNS whereas the vertical, dotted, blue line is the one from the WNL approach [see Fig. 2].

Figure 5

Probability density function of the escape time $\mathrm{\Delta }T$ between two transition events from the neighborhood of one attractor to the neighborhood of the other. Lighter colors correspond to larger forcing amplitude $\varphi$ where $\varphi \in [10,18,26]$. The parameters are the same as for Fig. 3, specifically $\mathrm{Re}=100$ and ${\omega }_{\mathrm{co}}=0.04$. For the markers “WNL” (dots for $\varphi =10$, squares for $\varphi =18$ and diamonds for $\varphi =26$, 10 direct simulations of (16), each corresponding to a different noise realization and long of $t=1.1\times {10}^6$ units of times, were performed and post-treated. The continuous lines “WNL, F.-P.” (F.-P. for Fokker-Planck), are obtained by marching in time the Fokker-Planck equation (22) with appropriate initial and boundary conditions (see main text). Panels (a) and (b) show the same data, (a) in linear-linear scale and (b) in log-linear scale. The thin black dashed- lines are exponential laws $bexp(-b\mathrm{\Delta }T)$, where $b=b\left(\varphi \right)$ is a fitted parameter.

Figure 6

Mean value and standard deviation of $\mathrm{\Delta }T$. The parameters are the same as for Fig. 3, specifically $\mathrm{Re}=100$ and ${\omega }_{\mathrm{co}}=0.04$. The blue dash-dotted line is obtained as follows: For each forcing amplitude 10 direct simulations of (16), each corresponding to a different noise realization and long of $t=1.1\times {10}^6$ units of times, were performed and post-treated. The continuous magenta line “WNL, F.-P.” (F.-P. for Fokker-Planck), is obtained by marching in time the Fokker-Planck equation (22) with appropriate initial and boundary conditions. In (b) the inset shows the fitted parameter $b$ of the exponential law writing $bexp(-b\mathrm{\Delta }T)$. The mean value and the standard deviation reconstructed from this exponential law, both equal to $1/b$, are shown by means of the blacked dashed line. The red dotted line is the Eyring-Kramers formula as given in (29).

Figure 7

The red continuous line is the (linear) transfer function $\left|\right|R\left(\omega \right)\mathbf{f}\left|\right|$ [norm of the expression (32)] with $\mathbf{f}={\mathbf{q}}^{†}$ of the sudden expansion flow linearized around its symmetric neutral equilibrium at $\mathrm{Re}={\mathrm{Re}}_c$. The blue dashed line is the Lorentzian function (33) only accounting for the response of the virtually neutral mode $\mathbf{q}$, resonant at $\omega =0$. Both curves would collide exactly if $L$ was a normal operator, for ${\mathbf{q}}^{†}$ would then excite only $\mathbf{q}$. Four horizontal black dashed lines are drawn at $\omega \in [4,8,16,32]\times {10}^{-2}$.

Figure 8

Probability density function of $\left|M\right|$, similarly as in Fig. 4 but for a fixed $\varphi =12$; to save some computational time, the 10 simulations, each corresponding to a different noise realization have been shortened to $t=1.1\times {10}^3$ units of times with respect to the computations shown in Fig. 4. The data are shown for four different values of the cut-off frequency ${\omega }_{\mathrm{co}}\in [4,8,16,32]\times {10}^{-2}$. These four specific frequencies are highlighted with vertical black dashed lines in Fig. 7.