Part Load Vortex Rope as a Global Unstable Mode

Renewable energy sources (RES) have reached 23.7% of the worldwide electrical generation production in 2015. The hydraulic energy contribution amounts to 16.6% and comes mainly form large-scale hydropower plants, where Francis turbines represents 60% of the generating units. However, the future massive development of RES will require more advanced grid regulation strategies that may be achieved by increasing the operation flexibility of the Francis generating units. Part load operating condition of these turbines is hindered by pressure fluctuations in the draft tube of the machine. A precessing helical vortex rope develops in this condition, which imperils the mechanical structure and limits the operation flexibility of these turbines. A thorough description of the physical mechanism leading to the vortex rope is a prerequisite to develop relevant flow control strategies. This work, based on a linear global stability analysis of the time-averaged flow field, including a turbulent eddy viscosity, interprets the vortex rope as a global unstable eigenmode. In close resemblance to spiral vortex breakdown, a single-helix disturbance develops around the time-averaged flow field and growths in time to finally form the vortex rope. The frequency and the structure of this unstable linear disturbance are found in good agreement with respect to the three-dimensional (3D) numerical flow simulations.