It is routinely and often implicitly assumed in load calculations on wind turbines that the statistics of the turbulence do not change as the flow is approaching the rotor plane. As it is well known that the rotor affects the mean flow in front of the rotor it cannot be ruled out that the turbulence is also affected. In this paper we investigate this assumption experimentally with lidar measurements and large eddy simulation and compare the results with a simple model. We focus on low-frequency wind speed fluctuations.

use vortex particle methods to calculate the effect of the turbine rotor on the incoming turbulence. They calculate the turbulent spectra at several center-line positions upstream of a Nordtank 500 kW wind turbine assuming a fixed thrust coefficient. They conclude that the presence of the rotor does not affect the turbulence spectrum significantly. However, at higher frequencies above 0.1 Hz, they observe a slight decrease in the power spectral density when the presence of the rotor is taken into account, implying marginally lower loads. They see no changes at lower frequencies. emphasize that further investigations are necessary to conclude whether the effects of the stagnation on the turbulence are systematic or not see alsowhich is an expanded version of Branlard's thesis.

measure the turbulent inflow towards a Vestas V27 wind turbine using three synchronized continuous wave, scanning Doppler lidars. They clearly see the stagnation in front of the rotor. The standard deviation of the along-wind velocity component σu decreases slightly close to the rotor plane and they hypothesize that this is linked to the reduced mean velocity, which also affects the low-frequency fluctuations. They do not support this suggestion with spectral analysis and they also point out that the amount of data is limited. An additional complication is that the Doppler lidars average the turbulent flow field in ways that depend on the direction from the lidars to the measurement volumes and their distances.

The change in the turbulence spectrum due to the stagnation in front of the rotor is investigated theoretically using rapid distortion theory by . In a first step, he assumes that the turbulent scales are much smaller than the size of the rotor. He also assumes that the mean flow around the rotor is described by the model of , a linearized actuator disk model, and that the approaching turbulence is isotropic and described by the von Kármán spectrum. With these assumptions, he derives that σu2/σu∞2 increases with the induction factor a of the rotor, reaching a value of σu2/σu∞2≈1.34 at the induction factor of maximal energy extraction a=1/3. Here, σu∞2 is the undisturbed, upstream variance of the longitudinal wind speed component u and σu2 is the local variance. He analytically derives that the amplification of turbulence is not equally distributed on frequencies but rather concentrated at lower frequencies, leaving the inertial subrange almost unchanged. He also derives that the integral length scale of u in the y or z direction (i.e., perpendicular to the mean flow), which indicates how correlated fluctuations are across the rotor, increases as the flow approaches the rotor. The increase is a little less than the stretching by the mean flow in these perpendicular directions. extends the theory to the more realistic case in which the integral length scale of the turbulence is not much smaller than the rotor by concentrating on the flow along the symmetry line of the rotor. The amplification of σu2 is less for the small length scale case than cases with turbulence length scales of the order of or larger than the rotor. The amplification of σu2/σu∞2 decreases from 24 to 7 % for a=1/3 as 2Lu/D increases from 1 to 10, where Lu is the undisturbed integral length scale of u in the flow direction and D the rotor diameter. The variance slowly and asymptotically approaches its upstream value as Lu/D→∞.

perform wind tunnel model studies of the flow upstream of a rotor. They find very little change in σu approaching the rotor, much less than expected from the small-scale rapid distortion limit discussed above. They suggest that the stagnation of the flow almost cancels out the amplification implied by rapid distortion theory.

In Sect.  we briefly discuss how quasi-steady fluctuations in the wind translate into fluctuations in the induction zone and we emphasize the effect of change in the induction with wind speed. That is followed by a discussion of a numerical experiment on turbulence in the induction zone in Sect. . Then we analyze a field experiment measuring low-frequency variations in the induction zone with a pulsed Doppler lidar (Sect. ). Finally, results are presented and discussed in Sects.  and .

Low-frequency fluctuations are the focus of this paper and are discussed first. Then we summarize the results of , which should be valid for all frequencies but have a particularly simple solution for high frequencies. Finally, we discuss the limitation of rapid distortion theory for our particular application.

The experiment took place at a 13-wind-turbine farm in northern Denmark in generally flat terrain. A five-beam pulsed prototype lidar from Avent was mounted on the nacelle of a Siemens 2.3MW wind turbine with a hub height of 81.8 and D=92.6 m. Only the central beam of the Avent lidar looking horizontally upstream of the turbine was used in this investigation. The lidar measured the line-of-sight velocity at 10 range gates centered at 49, 72, 95, 109, 121, 142, 165, 188, 235, and 281 m upstream of the rotor at a sampling frequency of 0.2 Hz. All details about the experiment may be found in .

The line-of-sight velocity in the range gate centered around 235 m from the lidar and the wind speed from a WindSensor cup anemometer at the same distance and at hub height is compared to ensure the viability of the lidar. We find a slope deviating 1 % from 1 and a correlation coefficient of 0.98. The scatter is larger than other similar comparisons see, for example,Fig. 4. Due to the yawing of the turbine, the measurements are rarely collocated. An additional difference between the measurements is that the cup measures the “wind way” , while the lidar measures the component of the wind vector in the direction the wind turbine is pointing.

Having ensured the quality of the measurements we calculate the 10 min average of the u component of the wind and fit Eq. () to the measurements. That gives the value of a as a function of U∞, which we assume is equal to the velocity measured at the furthest range gate. The induction factors from the undisturbed sector see are shown in Fig. b together with a smooth curve through the points, which is later used to compute ∂a/∂U∞. For wind speeds below rated the induction factor reaches levels above 0.4, which is higher than the expectation of approximately 0.3. The reason for this is that the quasi-steady model assumes a uniform load distribution and therefore tends to underestimate the induced velocity of real rotors, which have a nonuniform loading, as shown by . For a given CT, this bias causes an overestimation of the induction factor when the model is fitted to measurements of the upstream velocity. The bias is particularly dominant for the Siemens 2.3 MW because it has a high local loading below rated wind speed; see . Figure a shows values of the measured u(x)/U∞ averaged in intervals of U∞ with fits of Eq. () superimposed. It can be seen that the intervals below the rated wind speed 6<U∞<8 and 8<U∞<10 m s-1 almost coincide.

We now calculate the power spectrum of the velocity at each range gate in all 2 m s-1 intervals of U∞. These are based on 10 min time series so the lowest frequency investigated is f=1/600Hz=0.00167 Hz. Two examples are shown in Fig.  for U∞ where a is constant with U∞ and for a velocity where a rapidly decreases as a function of U∞. For low frequencies, the power spectra for the low U∞ coincide for most ranges except for those closest to the rotor when they are slightly but significantly lower. Conversely, for the higher U∞, the spectra close to the rotor are significantly higher than the upstream spectra. The experiment has the great advantage that the measurements at all range gates are done simultaneously with the same instrument, making the detection of small differences possible.

The experimental results are summarized in Fig. . Here we calculate the low-frequency spectral amplification as measured by the lidar Slow(x)/Slow,∞, where Slow≡∫1/600Hz4/600HzS(f)df; i.e., we add the four lowest-frequency bins of the 10 min average spectra. We calculate the low-frequency fluctuations using different upper frequency limits. The results vary but the trend remains. On the y axis, we plot the expected amplification according to the quasi-steady model in Eq. () for which the induction factor and its slope are derived from the solid curve in Fig. b. The cloud of points corresponding to each U∞ bin represents results from nine range gates (the tenth is used for normalization assuming it is far enough away to represent the ambient flow). The trend that the low-frequency fluctuations are reduced below rated and amplified above is captured but the exact magnitude is not.

We now turn to the analysis of the LES simulations. Since the turbulence is not completely homogeneous in the stream-wise direction, we determine the effect of the rotor on the fluctuations at a position x by comparing the two simulations with and without the rotor at that position. In Fig.  we show the relative changes of turbulence divided into low and high frequencies. At low frequencies the fluctuations below rated wind speed are reduced while they are amplified above rated. The high-frequency fluctuations, or what the LES can resolve of them, change very little.

In Fig.  we summarize the results and compare them with the quasi-steady model. The theoretical prediction is based on the thrust curve shown in Fig. . We put a fifth-order spline through the points to be able to do derivatives and then we use the relation a=121-1-CT to get the induction factor . We are then able to use Eq. () to predict the change in low-frequency fluctuations. We do that for two distances from the rotor, x=0 and x=-R. Again the model has the trends right, but the magnitude, especially below rated, is exaggerated.

Since the theory by predicts an increase in low-frequency u fluctuations near the rotor, a combination of the two models could potentially improve the results.

The often used assumption that the statistics of turbulence approaching a wind turbine rotor are unaltered relative to its upstream values is investigated in this paper. Since the mean wind speed is reduced in the induction zone one cannot rule out the possibility that the turbulence is also affected.

A nacelle-mounted forward-looking pulsed lidar is used to measure low-frequency wind fluctuations upstream of a wind turbine rotor situated in flat, homogeneous terrain. It measures wind speeds simultaneously at 10 ranges between 0.5 and 3 rotor diameters upstream sampling at 0.2 Hz. The integral of the velocity spectrum up to a frequency of 1/150 Hz is reduced by up to 10 % at 0.5 rotor diameters upstream and above rated enhanced by up to 20 %. The changes disappear rapidly further upstream.

A quasi-steady model that uses the CT curve partly predicts the variation, but overestimates the changes. The model differs from a recent development in rapid distortion theory that is also applicable to low-frequency fluctuations .

An implementation of an actuator disk model in a large eddy simulation is used to investigate the changes in detail. The simulation is not completely homogeneous in the along-wind direction so the changes in turbulence statistics are found by comparing otherwise identical simulation runs with and without the rotor at corresponding positions. The simulations support the finding that the slope of the thrust curve influences the low-frequency fluctuations, but the simple quasi-steady model overestimates the changes. The exact consequences for loads are not investigated in this work.