Pseudo spectral analysis of the energy entrainment in a scaled down wind farm

Particle Image Velocimetry from the centerline of a 3 × 5 scaled down array of wind turbines in a wind tunnel was analyzed to gain further understanding of how turbulent transport brings mean kinetic energy (MKE) into the array from the neutrally stable Atmospheric Boundary Layer (ABL) above. Vertical fluxes of MKE due to the Reynolds stresses were computed i.e.−〈U〉〈u′v′〉 A modal expansion for 〈U〉〈u′v′〉 was constructed based on the Proper Orthogonal Decomposition (POD). By determining each mode's fractional contribution to the total entrainment it was shown that a small number of modes (the first 25) account for 75% of the entrainment. The remaining 25% is achieved asymptotically as the remainder of the modes are included in the representation. Based on this behavior the labels “idiosyncratic” and “asymptotic” were applied to the different mode types. A characteristic wavelength for each mode was defined as the length of a mode's longest positive contribution to the energy entrainment. By this definition it was shown that idiosyncratic and asymptotic modes are characterized by wavelengths greater than and less than D (rotor diameter) respectively so that large percentages of the energy brought into the wind farm are done at scales greater than D. Physical reasoning indicates the idiosyncratic modes are associated with larger scale coherent motions whereas the asymptotic modes are associated with small scale turbulent fluctuations. The analysis was repeated for PIV data without turbines. It was shown that the idiosyncratic modes represent the scales which are affected by the presence of the turbines. This further established that the idiosyncratic modes were connected with the larger scales of turbulent motion.