Anisotropic character of low-order turbulent flow descriptions through the proper orthogonal decomposition

Abstract

Proper orthogonal decomposition (POD) is applied to distinct data sets in order to characterize the propagation of error arising from basis truncation in the description of turbulence. Experimental data from stereo particle image velocimetry measurements in a wind turbine array and direct numerical simulation data from a fully developed channel flow are used to illustrate dependence of the anisotropy tensor invariants as a function of POD modes used in low-order descriptions. In all cases, ensembles of snapshots illuminate a variety of anisotropic states of turbulence. In the near wake of a model wind turbine, the turbulence field reflects the periodic interaction between the incoming flow and rotor blade. The far wake of the wind turbine is more homogenous, confirmed by the increased magnitude of the anisotropy factor. By contrast, the channel flow exhibits many anisotropic states of turbulence. In the inner layer of the wall-bounded region, one observes onecomponent turbulence at the wall; immediately above, the turbulence is dominated by two components, with the outer layer showing fully three-dimensional turbulence, conforming to theory for wall-bounded turbulence. The complexity of flow descriptions resulting from truncated POD bases can be greatly mitigated by severe basis truncations. However, the current work demonstrates that such simplification necessarily exaggerates the anisotropy of the modeled flow and, in extreme cases, can lead to the loss of three-dimensionality. Application of simple corrections to the low-order descriptions of the Reynolds stress tensor significantly reduces the residual root-mean-square error. Similar error reduction is seen in the anisotropy tensor invariants. Corrections of this form reintroduce three-dimensionality to severe truncations of POD bases. A threshold for truncating the POD basis based on the equivalent anisotropy factor for each measurement set required many more modes than a threshold based on energy. The mode requirement to reach the anisotropy threshold after correction is reduced by a full order of magnitude for all example data sets, ensuring that economical low-dimensional models account for the isotropic quality of the turbulence field.

Stochastic Wake Modelling Based on POD Analysis

“…As Hamilton et al [39,47], we find that the spatial dependence of the local SKE u (y, z) 2 t of the wake flow can already be captured by a few modes (Figure 8), in our case approx. four modes.…”

“…This indicates that most relevant large-scale effects are captured by 6 modes while the small-scale effects are captured by the homogeneous turbulent field, which is moving with the wake structure. Similarly, Hamilton et al [47] found that lower order POD modes are responsible for the anisotropy in the wake while the higher order modes mostly contribute to higher-order and small-scale turbulence. There is however a fundamental difference in our approach since we add the homogeneous field only to the instantaneous wake deficit and not to the ambient field.…”

“…Another dynamic approach, also followed in this work, is to analyse and model wind turbine wakes using modal decompositions [21,[35][36][37][38][39][40][41][42][43][44][45][46][47] which describe the velocity field as a linear superposition of spatial modes with time-dependent weighting coefficients. It has been shown that a few spatial modes can already capture important features of the wake flow [35][36][37][38][39]41,43,47].…”

“…It has been shown that a few spatial modes can already capture important features of the wake flow [35][36][37][38][39]41,43,47]. In the case of a large eddy simulation of an infinite row of turbines, Andersen et al [35] found that a few modes stemming from the proper orthogonal decomposition (POD) already yield a good description of the velocity field on large spatial scales.…”

A generalized framework for reduced‐order modeling of a wind turbine wake

“…Truncating the basis of POD modes before reconstructing velocity snapshots as in Equation results in a filtered description of the turbulence. Low‐rank modes are taken to be the most energetic and anisotropic structures in the flow field; intermediate and high‐rank POD modes account for turbulence kinetic energy that is more homogeneously distributed and more isotropic, detailed in Hamilton et al The point at which the POD basis is truncated is typically determined by choosing a desired level of the TKE according to the eigenvalues, λ .…”