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

Some nonlinear dynamical systems are metastable: subject to a weak noise, they randomly switch back and forth between two configurations, after long and unpredictable times. However, because the mean transition time to change from one attractor state to the other is considerable, this situation will usually not be easily detected by simply running the numerical models. Instead, rare event algorithms based on the large-deviation theory are generally used.

Alternatively, to compute the flow statistics at a cheaper numerical cost, we propose to apply a multiple-scale weakly nonlinear expansion technique to the Navier-Stokes equations. Specifically, these latter are forced by a weak, narrow-band noise, acting on the same slow time scale as the amplitude of the dominant mode shortly after a bifurcation point. In the case of a steady symmetry-breaking bifurcation, we rigorously derive a stochastically forced Stuart-Landau equation for the slowly varying amplitude premultiplying the dominant symmetry-breaking mode.

The validity of this reduced order model is tested on the flow past a sudden expansion. Given the nature of the amplitude equation, the noisy dynamics derive from a potential and the probability density function of the solution is then easily determined by solving the Fokker-Planck equation. At a very low numerical cost, the statistics thus obtained accurately reproduce those of long-time direct numerical simulations of the forced Navier-Stokes equations.