  A method is developed to estimate the properties of a global hydrodynamic
instability in turbulent flows from measurement data of the limit-cycle
oscillations. For this purpose, the flow dynamics are separated in
deterministic contributions representing the global mode and a stochastic
contribution representing the intrinsic turbulent forcing. Stochastic models
are developed that account for the interaction between the two and that allow
determining the dynamic properties of the flow from stationary data. The
deterministic contributions are modelled by an amplitude equation, which
describes the oscillatory dynamics of the instability, and in a second approach
by a mean-field model, which additionally captures the interaction between the
instability and the mean-flow corrections. The stochastic contributions are
considered as coloured noise forcing, representing the spectral characteristics
of the stochastic turbulent perturbations. The methodology is applied to a
turbulent swirling jet with a dominant global mode. PIV measurements are
conducted to ensure that the mode is the most dominant coherent structure and
further pressure measurements provide long time series for the model
calibration. The supercritical Hopf bifurcation is identified from the linear
growthrate of the global mode and the excellent agreement between measured and
estimated statistics suggest that the model captures the relevant dynamics.
This work demonstrates that the sole observation of limit-cycle oscillations is
not sufficient to determine the stability of turbulent flows since the
stochastic perturbations obscure the actual bifurcation point. However, the
proposed separation of deterministic and stochastic contributions in the
dynamical model allows identifying the flow sate from stationary measurements.
