Stochastic modelling of a noise driven global instability in a turbulent swirling jet.

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.