Prediction of the hub vortex instability in a wind turbine wake: stability analysis with eddy-viscosity models calibrated on wind tunnel data

The evolution and downwind recovery of wind turbine wakes are crucial factors for the evaluation of waketo-wake interactions within a wind farm and, in turn, for the prediction of wind power harvesting. Wind turbine wakes are affected by the design of wind turbine blades and the produced aerodynamic loads, by the characteristics of the incoming wind, such as turbulence, shear, thermal stability, and by the topography of the site. Moreover, it is observed that dynamics and instabilities of vorticity structures present in a wind turbine wake can affect remarkably the downstream evolution of the whole wake. Indeed, the flow past a wind turbine is characterized by the presence of two main large-scale vorticity structures: the helicoidal tip vortices, which detach from the tip of each turbine blade, and the hub vortex, which is a streamwise-oriented vorticity structure located approximately at the wake center. The dynamics of these vortices and their mutual interactions determine their downstream extent. Consequently, these wake vorticity structures can modify significantly the flow entrainment from the external flow field into the wake, thus the recovery of the wind turbine wake. Instabilitymodes of the helicoidal tip vortices have been deeply characterized by several authors in the past; however, dynamics of the hub vortex are not clearly identified yet. With this spirit, wind tunnel experiments were performed for the wake produced by a three-bladed wind turbine immersed in uniform flow. It was observed that the hub vortex is characterized by oscillations with frequencies lower than that connected to the rotational velocity of the rotor, which previous works have ascribed to wake meandering. This phenomenon consists in transversal oscillations of the wind turbine wake, which might be excited by the vortex shedding from the rotor disc acting as a bluff body. In this work, temporal and spatial linear stability analyses of a wind turbine wake are performed on a base flow obtained with time-averaged wind tunnel velocity measurements. This study shows that the low-frequency spectral component detected experimentally matches the most amplified frequency of the counter-winding single-helix mode, and the hub vortex instability has been spatially reconstructed, as shown in figure 1. Then, simultaneous hot-wire measurements confirm the presence of a helicoidal unstable mode of the hub vortex, with a streamwise wavenumber roughly equal to that predicted from the linear stability analysis1. More recently, stability analysis has been performed taking into account the Reynolds stresses, which are modeled via eddy-viscosity models calibrated on the wind tunnel data. This new formulation leads to a univocal detection of the hub vortex instability and the prediction of the related instability frequency.