Downstream development of the turbulence intensity

Figure 5.3.: Centerline evolution of the to the inflow velocityu₀=7.5 m/s normalized mean velocity downstream turbine 1 (purple *), turbine 2 mid (turquoise x) and turbine 2 side (green +). TheY-coordinate is with respect to the turbine center.

(c) (b) (a)

Figure 5.4.: Evolution of the to the maximum turbulence intensity normalized turbulence intensity downstream turbine 1 (a), turbine 2 mid (b) and turbine 2 side (c).

Figure 5.5.: Comparison of the to the turbulence intensity peak on the centerline normalized turbulence intensity downstream turbine 1 (purple *), turbine 2 mid (turquoise x) and turbine 2 side (green +). TheY-coordinate is with respect to the turbine center.

Closer to the turbine, different span-wise positions show a strong variation of the turbulence intensity evolution. An enhanced turbulence intensity at the rotor blade tip is for example visible up to approximately 4D in the near wake. Downstream turbine 2, a profile evolves already aroundX/D≈2. The influence of the different inflow situations is seen in the variation of the profile development. Downstream turbine 2 side, the profile shows close to the rotor a stronger curvature, as at theY/D=−0.72 position, the sensor is exposed to the still laminar inflow. As the wake expands, the turbulence intensity increases downstream.

The centerline evolution of the normalized turbulence intensity shown in figure 5.6 enables a qualitative comparison with results from the preceding chapter and a quantitative comparison of the three scenarios investigated in this chapter. The curves develop similarly to the curves presented in the preceding chapter, and the four previously introduced wake regions can be identified. A drop under the lee of the nacelle characterizes the near wake and is followed by the transition region where the turbulence builds up. Finally, after the local maximum, the decay region starts. As the wake evolution is sensitive to the inflow turbulence in this setup, the regions are not fixed to certain downstream positions, but vary. For turbine 1, the measured maximum is located atX/D=6.24, for turbine 2 mid atX/D=1.59 and for turbine 2 side at X/D=2.10. The precise positions can be found in table E.2 in appendix E. At the respective last measurement position, the normalized turbulence intensity has decayed to T I_end ≈0.50·T I_CLpeak downstream turbine 1, T I_end ≈0.22·T I_CLpeak downstream turbine 2

Figure 5.6.: Centerline evolution of the to the peak turbulence intensity normalized turbu-lence intensity downstream turbine 1 (purple *), turbine 2 mid (turquoise x) and turbine 2 side (green +). TheY-coordinate is with respect to the turbine center.

mid and T I_end ≈0.33·T I_CLpeak downstream turbine 2 side. To further investigate the decay of the normalized turbulence intensity, the centerline evolution of the normalized turbulence intensity is plotted with logarithmic axes in figure 5.7, and the decay region is fitted according to T I/T I_CLpeak ∝α·(X/D)^β. Again, the fitting range is split into two parts, as indicated by the dashed vertical lines. The fit regions and the exponentβ can be found in table 5.2.β1<β2for all scenarios indicates that in the beginning, the turbulence decays faster. Whileβ1andβ2are quite similar for turbine 1,β2decreases by approximately 25% downstream turbine 2. This is equivalent to a less fast decay downstream turbine 2.

As already discussed in chapter 4, this change of the turbulence decay takes up results obtained downstream regular grids where the decay process of turbulence also changes [Hinze (1975)].

To relate the change of the decay exponent to the variance, the evolution ofσ²is plotted in figure 5.8. The downstream positions where the decay exponent changes from β1 to β2 are marked with dashed lines for the respective wake. Close to the rotor, the variance decreases in

Table 5.2.: Investigation of turbulence intensity decay: fit parameterβ with fit error χ²for fit regions 1 and 2 downstream the disc and the model turbine for all inflow conditions.

β1 χ²(β₁) X/D β2 χ²(β₂) X/D turbine 1 -1.16 0.08 6.59 - 7.79 -1.03 0.02 7.97 - 12.62 turbine 2 mid -1.27 0.04 1.76 - 3.14 -0.76 0.02 3.14 - 8.66 turbine 2 side -1.04 0.05 2.45 - 4.00 -0.79 0.01 4.00 - 8.66

Figure 5.7.: Centerline evolution of the to the peak turbulence intensity normalized turbu-lence intensity downstream turbine 1 (purple *), turbine 2 mid (turquoise x) and turbine 2 side (green +) with fits within the decay region. TheY-coordinate is with respect to the turbine center.

all cases. After an increase, the variance starts to slowly decay after a local maximum. While downstream turbine 1, this decay starts aroundX/D≈8.31, it already starts atX/D≈2.10 in case of turbine 2 mid andX/D≈3.14 in case of turbine 2 side. Therefore, it can be concluded that the change of the decay exponent happens when the variance decreases.

Overall, the turbulence intensity evolves as expected from the experiments evaluated in the preceding chapter and also (qualitatively) as compared to e.g. Kermaniet al.(2013). The positive influence of the inflow turbulence on the accelerated wake recovery could be shown.

Here, the recovery does not only include the mean velocity upswing but also the turbulence intensity decay. The accelerated turbulence decay in case of turbulent inflow downstream turbine 2 was assigned to the extended fit region ofβ₁downstream turbine 2 as compared to turbine 1: Asβ₁>β₂, the decay of the turbulence intensity is thus accelerated. In combination with the above-mentioned downstream positions ofT I_peak, the approximate downstream borders of the four wake regions can be identified for the three scenarios. The precise position of local

Table 5.3.: Identification of wake regions

X/D near wake X/D transition region X/D decay region X/D far wake turbine 1 0.55 - 4.00 4.00 - 6.24 6.59 - 7.97 7.97 - 12.52 turbine 2 mid 0.55 - 0.90 0.90 - 1.76 1.76 - 3.14 3.14-8.66 turbine 2 side 0.55 - 1.07 1.07 - 2.45 2.45 - 4.00 4.00-8.66

Figure 5.8.: Centerline evolution of the varianceσ²downstream turbine 1 (purple *), turbine 2 mid (turquoise x) and turbine 2 side (green +). TheY-coordinate is with respect to the turbine center.

extrema and changes in the evolution can be found for different quantities in table 5.3. Thus, the existence of the before identified wake regions is confirmed.