Prediction of vortex precession in the draft tube of a model hydro turbine using mean field stability theory and stochastic modelling

In this work we employ mean field stability theory (MFST) to predict the onset of the precessing vortex core (PVC) in the draft tube of Francis turbines. MFST is based on the linear stability analysis of the mean field of turbulent flows. Recent work shows that MFST very accurately predicts the formation of coherent structures in turbulent shear flows, such as the PVC. MFST may further reveal the flow regions that are most susceptible to flow actuation to suppress the PVC, which is of great practical relevance. In this work, MFST is accompanied by a data-driven approach to predict the linear growth rate of the PVC based on pointwise wall pressure measurements. The method is based on statistical evaluation of the probability density function of the PVC amplitude at limit cycle. It makes use of the intense noise induced by the background turbulence, which is expected to be a major driver of hydrodynamic instabilities. The empirical and analytic results are compared to phase-locked LDV measurements conducted inside the draft tube at various operating conditions, to assess the quantitative accuracy of the approach. The methodologies outlined in this work will be of relevance for future design of hydro turbines to run stable over a wide range of operating conditions.