  We present a model-based, output-only parameter identification method for
self-sustained oscillators forced by dynamic noise, which we illustrate
experimentally with an aeroacoustic setup: a turbulent jet impinging a beer
bottle and producing a whistling tone in a finite range of jet angles and
velocities. Given a low-order model of the system, the identification analyzes
stationary time series of a single observable: acoustic pressure fluctuations
in the bottle. As this observable exhibits the characteristics of weakly
non-linear self-oscillations, we choose as a minimal model the Van der Pol
(VdP) oscillator: a linear acoustic oscillator (bottle) subject to stochastic
forcing and non-linear deterministic forcing (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 model-based
linear growth rate identification. The associated stochastic amplitude equation
allows us to describe the stochastic fluctuations of the acoustic amplitude
resulting from a competition between (i) linear growth rate from the coherent
unsteady vortex force, (ii) random forcing from the turbulence, (iii)
non-linear saturation of the coherent flapping of the jet. Finally, the
associated adjoint Fokker-Planck equation reveals the deterministic potential
well and the stochastic forcing intensity governing these random fluctuations.
Additional experiments with an external control validate the identification
results. We 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 identification approach can be used in a wide
range of phenomena exhibiting stationary self-sustained oscillations in the
presence of random forcing.
