Determination of Wind Turbine Near-Wake Length Based on Stability Analysis

Abstract

A numerical study on the wake behind a wind turbine is carried outfocusing on determining the length of the near-wake based on the instability onset ofthe trailing tip vortices shed from the turbine blades. The numerical model is based onlarge-eddy simulations (LES) of the Navier-Stokes equations using the actuator line(ACL) method. The wake is perturbed by applying stochastic or harmonic excitations inthe neighborhood of the tips of the blades. The flow field is then analyzed to obtain thestability properties of the tip vortices in the wake of the wind turbine. As a mainoutcome of the study it is found that the amplification of specific waves (travelingstructures) along the tip vortex spirals is responsible for triggering the instabilityleading to wake breakdown. The presence of unstable modes in the wake is related tothe mutual inductance (vortex pairing) instability where there is an out-of-phasedisplacement of successive helix turns. Furthermore, using the non-dimensional growthrate, it is found that the pairing instability has a universal growth rate equal to π/2.Using this relationship, and the assumption that breakdown to turbulence occurs once avortex has experienced sufficient growth, we provide an analytical relationship betweenthe turbulence intensity and the stable wake length. The analysis leads to a simpleexpression for determining the length of the near wake. This expression shows that thenear wake length is inversely proportional to thrust, tip speed ratio and the logarithmicof the turbulence intensity