Streamwise Evolution of Statistical Events in a Model Wind-Turbine Array

Abstract

Hot-wire anemometry data, obtained from a wind-tunnel experiment containing a $3 \times 3$ model wind-turbine array, are used to conditionally average the Reynolds stresses. Nine profiles at the centreline behind the array are analyzed to characterize the turbulent velocity statistics of the wake flow. Quadrant analysis yields statistical events occurring in the wake of the wind farm where quadrants 2 and 4 produce ejections and sweeps, respectively. The scaled difference between these two events is expressed via the $\Delta R_{0}$ parameter and is based on the $\Delta S_{0}$ quantity as introduced by M. R. Raupach (J Fluid Mech 108:363–382, 1981). $\Delta R_{0}$ attains a maximum value at hub height and changes sign near the top of the rotor. The ratio of quadrant events of upward momentum flux to those of the downward flux, known as the exuberance, is examined and reveals the effect of root vortices persisting to eight rotor diameters downstream. These events are then associated with the triple correlation term present in the turbulent kinetic energy equation of the fluctuations where it is found that ejections play the dual role of entraining mean kinetic energy while convecting turbulent kinetic energy out of the turbine canopy. The development of these various quantities possesses significance in closure models, and is assessed in light of wake remediation, energy transport and power fluctuations, where it is found that the maximum fluctuation is about $30\%$ of the mean power produced.