Low-dimensional representations and anisotropy of model rotor versus porous disk wind turbine arrays

An experimental study of the wake of a model horizontal axis wind turbine with a three-bladed rotor and a turbine with a stationary disk is done within a wind turbine array in order to compare the structure of these wakes. Measurements of the flow field surrounding the center turbine in the fourth row of a $4\times 3$ array are made with stereo particle image velocimetry. Rotational effects of the blade are evident in the cross-stream mean velocity component in the rotor case and are absent in the disk case. The second and third invariants of the Reynolds stress anisotropy tensor have larger ranges in the wake of the rotor case than in the disk case, with the disk case displaying higher levels of anisotropy trailing the top tip, a key location relating to kinetic energy entrainment. Application of the proper orthogonal decomposition indicates a greater emphasis on intermediate scales in the near wake of the rotor case in comparison to the disk case, with such differences being mitigated in the far wake. The eigenfunction of the lowest rank mode, which contains the highest turbulence kinetic energy, displays coherence in both the near and far wakes in the rotor case while the disk wake lacks such apparent organization. Based on the discrepancies in the structure and scales of the wake of the rotor versus that of the disk, careful judgment is advised in order to apply stationary disk parametrizations in modeling applications.

Figure 1

The Lumley triangle with annotations showing significant regions of interest.

Figure 2

Experimental setup and turbine models: (a) side view of wind tunnel test section, (b) model rotor, (c) porous disk.

Figure 3

Normalized time-averaged statistics of the rotor and disk cases for (a) the normalized streamwise velocity, $U/U_{\mathrm{hub}}$, (b) the normalized spanwise mean velocity, $W/U_{\mathrm{hub}}$, (c) the normalized in-plane Reynolds shear stress, $\overline{u^{\prime }{v}^{\prime }}/U_{\mathrm{hub}}^2$, (d) the normalized out-of-plane Reynolds shear stress, $\overline{u^{\prime }{w}^{\prime }}/U_{\mathrm{hub}}^2$, and (e) the normalized turbulence kinetic energy, $k/U_{\mathrm{hub}}^2$. In each subfigure, the top row of two panels represents the rotor and the bottom row of two panels depicts the same information for the disk case.

Figure 4

Invariants of the normalized Reynolds stress anisotropy tensor and the anisotropy factor calculated from the time-averaged Reynolds stresses. (a) Second invariant $\eta$, (b) third invariant $\xi$, and (c) aniosotropy factor $F$. In all subfigures, the rotor case is shown in the top row of panels while the disk case is represented in the bottom row of panels.

Figure 5

Anisotropy invariant maps computed from the time-averaged Reynolds stress anisotropy tensor for $0.6\le x/D\le 5.6$ with points shaded by wall-normal distance. (a) All points measured for rotor $0.6\le x/D\le 5.6$, (b)–(e) profiles for the rotor case, (f) all points measured for the disk $0.6\le x/D\le 5.6$, and (g)–(j) profiles for the disk case.

Figure 6

Eigenvalues for the snapshot POD in the near wake $\left(0.6\le x/D\le 2.0\right)$ and far wake $\left(4.2\le x/D\le 5.6\right)$ for the rotor as well as disk cases. (a) Cumulative sum of eigenvalues for all 3000 modes normalized by the sum of all eigenvalues and (b) detailed view of cumulative sum of eigenvalues for 50 modes normalized by the sum of all eigenvalues.

Figure 7

Vectorial components ${\varphi }_u,{\varphi }_v$, and ${\varphi }_w$ of selected POD modes. (a) POD mode 1, (b) POD mode 2, (c) POD mode 13, and (d) POD mode 100. Four contour maps illustrate each vectorial component, with the top row of panels in each component representing the rotor case and the bottom row of each component depicting the disk case.

Figure 8

Turbulence kinetic energy from each component of the fluctuating velocity, ${\gamma }_{ij}^{\left(n\right)}$, where the component $i,j=1$, 2, or 3 for each mode $n$. (a) Rotor near wake measurement plane, (b) rotor far wake measurement plane, (c) disk near wake measurement plane, and (d) disk far wake measurement plane. The near wake measurement plane is $0.6\le x/D\le 2.0$ and the far wake measurement plane is $4.2\le x/D\le 5.6$, which are the same planes provided for the POD spatial eigenfunctions.

Figure 9

Representative Lumley triangles from POD reconstructions of the normalized Reynolds stress anisotropy tensor using modes 1 to $S$, where $S$ is increased from 2 to 6, 12, 50, 200, and finally the full basis of 3000. The POD was applied over a plane in the far wake $\left(4.2\le x/D\le 5.6\right)$ of the rotor case, and invariants for each reconstruction were selected at $x/D=5.00$.

Figure 10

Spatially integrated anisotropy factor, ${\langle F\rangle |}_S$, computed from POD reconstructions using modes 1 to $S$, where $S$ varies from 1 to the full basis of 3000. Spatial integration was done on the POD of a plane in the near wake $\left(0.6\le x/D\le 2.0\right)$ and the far wake $\left(4.2\le x/D\le 5.6\right)$ of both the rotor and disk cases. The corresponding POD eigenvalues and spatial eigenfunctions for these cases are discussed in Sec. 4c.