Low-order representations of the canonical wind turbine array boundary layer via double proper orthogonal decomposition

Abstract

Wind turbine wakes are investigated in order to characterize the development of energetic turbulence structures. Experimental data from stereo particle image velocimetry render the full Reynolds stress tensor accessible in planes parallel to the swept area of the scale model turbine rotor. Proper orthogonal decomposition (POD) is applied to decompose and analyze structures in the wake. The modes resulting from the decomposition demonstrate that structures grow and develop along the streamwise direction. A second iteration of the snapshot POD, otherwise called double proper orthogonal decomposition (DPOD), is applied to modes of common rank from the span of measurement locations yielding an ordered set of projections. The DPOD describes the sub-modal organization in terms of largest common projection and a series of correction modes with coefficients that are functions of the streamwise coordinate. Sub-structures of POD modes that persist through the wake have a dominant projection that accounts for the character of individual POD modes. Distribution of eigenvalues associated with DPOD modes indicates that near-wake turbulence behavior is superimposed on overall wake structure. High order POD modes do not reveal any common projections in the measurement sets of the wake and associated eigenvalues are nearly equal. The eigenvalues from the DPOD indicate that the structure of the wind turbine wake can be described with a small subset of the original mode basis. The truncated basis of sub-modal structures represents a total reduction to 0.015% of the degrees of freedom of the wind turbine wake. Low-order representations of the Reynolds stress tensor are made for the wake using the most dominant DPOD modes. The stress tensor is corrected to account for energy excluded from the truncated basis. A tensor of constant coefficients is defined to rescale the low-order representation of the stresses to match the original statistics. That a constant correction for each term in the Reynolds stress tensor is sufficient to correct the wake suggests that high order modes account for a nearly constant, isotropic turbulence kinetic energy. The following method reconstructs diagonal elements of the stress tensor to a root-mean-square error within 15% and shear terms to within 3%.