Processing time-series of randomly forced self-oscillators: The example of beer bottle whistling

We present a model-based, output-only parameter identification method for self-sustained oscillators forced by dynamic noise, which we illustrate experimentally with a simple aeroacoustic setup: a turbulent jet impinging a beer bottle and producing a distinct whistling tone in a finite range of jet angles and jet velocities. Given a low-order model of the system, the identification is based on analyzing stationary time series of a single key observable: the acoustic pressure fluctuations inside the bottle. Noting that this observable exhibits all the characteristics of weakly non-linear self-oscillations, we choose as a minimal model the classic Van der Pol (VdP) oscillator: a linear acoustic oscillator (the bottle) subject to stochastic forcing and non-linear deterministic forcing (the turbulent jet). Although very simple, the VdP oscillator driven by random noise proves to be a sufficient phenomenological description of the aeroacoustic limit cycle for the purpose of model-based linear growth rate identification. We derive the associated stochastic amplitude equation, which allows us to describe, on both sides of the bifurcation, the stochastic fluctuations of the acoustic amplitude resulting from a competition between (i) linear growth rate induced by the coherent unsteady vortex force, (ii) random forcing induced by the turbulence and (iii) non-linear saturation of the coherent flapping motion of the jet. Finally, we use the associated adjoint Fokker-Planck equation to reveal both the deterministic potential well and the stochastic forcing intensity that govern these random fluctuations. Additional experiments with an external control validate the identification results. We also observe that different operating conditions can yield similar statistics but different dynamics, showing that the identification method can disentangle deterministic and stochastic effects in systems forced stochastically. This output-only parameter identification approach can be used in a wide range of phenomena exhibiting stationary self-sustained oscillations in the presence of random forcing.