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WES | Articles | Volume 3, issue 1

Wind Energ. Sci., 3, 293-300, 2018

https://doi.org/10.5194/wes-3-293-2018

© Author(s) 2018. This work is distributed under

the Creative Commons Attribution 4.0 License.

https://doi.org/10.5194/wes-3-293-2018

© Author(s) 2018. This work is distributed under

the Creative Commons Attribution 4.0 License.

Special issue: Wind Energy Science Conference 2017

**Research articles**
24 May 2018

**Research articles** | 24 May 2018

Turbulence approaching a rotor

- DTU Wind Energy, Technical University of Denmark, Roskilde, Denmark

Abstract

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For load calculations on wind turbines it is usually assumed that the turbulence approaching the rotor does not change its statistics as it goes through the induction zone. We investigate this assumption using a nacelle-mounted forward-looking pulsed lidar that measures low-frequency wind fluctuations simultaneously at distances between 0.5 and 3 rotor diameters upstream. The measurements show that below rated wind speed the low-frequency wind variance is reduced by up to 10 % at 0.5 rotor diameters upstream and above rated enhanced by up to 20 %. A quasi-steady model that takes into account the change in thrust coefficient with wind speed explains these variations partly. Large eddy simulations of turbulence approaching an actuator disk model of a rotor support the finding that the slope of the thrust curve influences the low-frequency fluctuations.

How to cite

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How to cite.

Mann, J., Peña, A., Troldborg, N., and Andersen, S. J.: How does turbulence change approaching a rotor?, Wind Energ. Sci., 3, 293-300, https://doi.org/10.5194/wes-3-293-2018, 2018.

1 Introduction

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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.

Branlard et al. (2016) 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. Branlard et al. (2016) emphasize that further investigations are necessary to conclude whether the effects of the stagnation on the turbulence are systematic or not (Branlard, 2017).

Simley et al. (2016) 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 *σ*_{u} 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
Graham (2017). 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
Conway (1995), 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
${\mathit{\sigma}}_{u}^{\mathrm{2}}/{\mathit{\sigma}}_{u\mathrm{\infty}}^{\mathrm{2}}$ increases with the induction factor *a* of
the rotor, reaching a value of ${\mathit{\sigma}}_{u}^{\mathrm{2}}/{\mathit{\sigma}}_{u\mathrm{\infty}}^{\mathrm{2}}\approx \mathrm{1.34}$
at the induction factor of maximal energy extraction $a=\mathrm{1}/\mathrm{3}$. Here,
${\mathit{\sigma}}_{u\mathrm{\infty}}^{\mathrm{2}}$ is the undisturbed, upstream variance of the
longitudinal wind speed component *u* and ${\mathit{\sigma}}_{u}^{\mathrm{2}}$ 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.
Graham (2017) 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 ${\mathit{\sigma}}_{u}^{\mathrm{2}}$ 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 ${\mathit{\sigma}}_{u}^{\mathrm{2}}/{\mathit{\sigma}}_{u\mathrm{\infty}}^{\mathrm{2}}$ decreases from 24 to
7 % for $a=\mathrm{1}/\mathrm{3}$ as 2*L*_{u}∕*D* increases from 1 to 10, where *L*_{u} 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 ${L}_{u}/D\to \mathrm{\infty}$.

Farr and Hancock (2014) perform wind tunnel model studies of the flow upstream
of a rotor. They find very little change in *σ*_{u} 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. 2 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. 3. Then we analyze a field experiment measuring low-frequency variations in the induction zone with a pulsed Doppler lidar (Sect. 4). Finally, results are presented and discussed in Sects. 5 and 6.

2 Theory

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Low-frequency fluctuations are the focus of this paper and are discussed first. Then we summarize the results of Graham (2017), 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.

Low-frequency or quasi-steady fluctuations are defined as variations in the
wind speed *U* that are so slow that the rotor and the upstream flow have
sufficient time to adjust to all the changes such that they appear as if the
wind was steadily blowing at that wind speed. If $D=\mathrm{2}R=\mathrm{100}$ m and the
induction zone extends 3*D* upstream then the low-frequency limit would be
around *f*≈0.03 Hz for a free mean wind speed *U*_{∞} of 10
m s^{−1}.

For a particular wind turbine, the mean wind speed on a line extending
upstream from the center of the rotor depends on the ambient wind speed
*U*_{∞} and the distance from the rotor normalized by the rotor radius
$\mathit{\xi}=x/R$ and is given by

$$\begin{array}{}\text{(1)}& f(\mathit{\xi},a,{U}_{\mathrm{\infty}})\equiv {\displaystyle \frac{U}{{U}_{\mathrm{\infty}}}}=\mathrm{1}-a\left(\mathrm{1}+{\displaystyle \frac{\mathit{\xi}}{\sqrt{\mathrm{1}+{\mathit{\xi}}^{\mathrm{2}}}}}\right).\end{array}$$

A slow fluctuation in the ambient wind speed *U*_{∞} will produce slow
variations in the wind speed in the induction zone *U*(*x*). The power spectral
density at low frequencies is therefore amplified as

$$\begin{array}{}\text{(2)}& {\displaystyle \frac{S\left(x\right)}{{S}_{\mathrm{\infty}}}}={\left({\displaystyle \frac{\partial U}{\partial {U}_{\mathrm{\infty}}}}\right)}^{\mathrm{2}},\end{array}$$

where *S*(*x*) is the power spectral density (so the amplitude squared) at low
frequencies at the position *x* and *S*_{∞} is the upstream, undisturbed
spectrum. The partial derivative can be expanded as follows:

$$\begin{array}{}\text{(3)}& {\displaystyle \frac{\partial U}{\partial {U}_{\mathrm{\infty}}}}={\displaystyle \frac{\partial f}{\partial {U}_{\mathrm{\infty}}}}{U}_{\mathrm{\infty}}+f=f-\left(\mathrm{1}+{\displaystyle \frac{\mathit{\xi}}{\sqrt{\mathrm{1}+{\mathit{\xi}}^{\mathrm{2}}}}}\right){\displaystyle \frac{\partial a}{\partial {U}_{\mathrm{\infty}}}}{U}_{\mathrm{\infty}}.\end{array}$$

Typically, *a* does not change for ambient wind speeds below rated wind
speed, so the second term is negligible. The spectral amplification in
Eq. (2) will then be proportional to the square of
relative slowdown, which is of the order of but less than unity. Above
rated, $\partial a/\partial {U}_{\mathrm{\infty}}$ will become negative and a positive
amplification should be seen. A similar quasi-steady model for how low-frequency fluctuations of turbulence are modified by topography is presented
by Mann (2000).

Rapid distortion theory (RDT) for smaller turbulent scales corresponding to more rapid fluctuations is investigated by Batchelor and Proudman (1954) and Townsend (1976). Townsend calculates the response of initially isotropic turbulence to a contraction (or expansion) of the mean flow, which to some extent is what is happening in front of a rotor. The theory is used in Graham (2017) to produce amplifications of the velocity variance and the low-frequency part of the velocity spectrum shown in Fig. 1.

The theory assumes that the vorticity lines are advected by the mean flow and
that the approaching turbulence is isotropic as described in the Introduction
and by Conway (1995). The theory implies that the
amplification is strongest at the lowest frequencies and almost absent at the
highest frequencies. Their results in the limit of turbulent scales much
smaller than the rotor diameter are shown in Fig. 1. In a
wind tunnel contraction, the *u* component is diminished, also relative to
the other components. In contrast, the *u* component is enhanced in the
diverging flow in front of a rotor.

Now we analyze the applicability of RDT by estimating the relevant spatial and
temporal scales. The term “rapid” in RDT means that the turbulent eddies
should not be able to interact during the time it takes for the distortion of
the flow to take place. The eddies should not “rotate” several times during
the distortion phase; in other words, their lifetime *τ* should be longer
than the distortion time 𝒯_{D}.

$$\begin{array}{}\text{(4)}& {\mathcal{T}}_{D}\ll \mathit{\tau}\end{array}$$

Another requirement for traditional RDT (Batchelor and Proudman, 1954; Townsend, 1976)
but not for the work by Graham (2017) is that the shear or strain
deforming the eddies is constant over the extent of the eddy. The strain
changes appreciably in the wind turbine inflow zone over a length of the
order of the size of the turbine, like its diameter *D*, so

$$\begin{array}{}\text{(5)}& D\ll {k}^{-\mathrm{1}},\end{array}$$

where we use the inverse wavenumber to characterize the size of the eddy. We
now estimate the terms in Eqs. (4) and (5) to get a
sense of the applicable frequency or wavenumber ranges of traditional RDT. We
estimate the distortion timescale as ${\mathcal{T}}_{D}\approx D/U$, where *U*
is the mean wind speed at hub height. In the work by Mann (1994) the
simplest model for the eddy lifetime in the neutral atmospheric
surface layer was

$$\begin{array}{}\text{(6)}& \mathit{\tau}=\mathrm{\Gamma}{\left({\displaystyle \frac{\mathrm{d}U}{\mathrm{d}z}}\right)}^{-\mathrm{1}}{\left(kL\right)}^{-\mathrm{2}/\mathrm{3}},\end{array}$$

where Γ is a constant of the order of 3, *L* is a length scale
representative of the energy containing eddies, and d*U*∕d*z*
the shear at hub height. The model implies that small eddies (large *k*) have
a shorter lifetime than larger eddies, so it is in the limit of small eddies
that RDT has a limitation. The same eddy lifetime model was use to extend
the model by Mann (1994) to a semi-Lagrangian model
(de Maré and Mann, 2016). For simplicity, we assume a logarithmic wind profile
$U\left(z\right)={u}_{\ast}/\mathit{\kappa}\mathrm{log}(z/{z}_{\mathrm{0}})$ (Wyngaard, 2010) so
$\mathrm{d}U/\mathrm{d}z={u}_{\ast}/\left(\mathit{\kappa}z\right)$ and that the diameter of the
rotor is close to the hub height above the ground *D*≈*z*. Then, by
isolating *k* in Eq. (4) we get

$$\begin{array}{}\text{(7)}& k\ll {\displaystyle \frac{{\left(\mathrm{\Gamma}\mathrm{log}(z/{z}_{\mathrm{0}})\right)}^{\mathrm{3}/\mathrm{2}}}{L}}\approx \mathrm{5}{\mathrm{m}}^{-\mathrm{1}},\end{array}$$

where we assume a hub height of *z*=80 m, a roughness length of *z*_{0}=0.05 m, and use IEC (2005) to get Γ=3.9 and
*L*=33.6 m. This inequality states that only eddies smaller in scale than a
meter or so will be short-lived enough not to feel the entire distortion in
the induction zone. Equation (5) implies *k*≫0.013 m^{−1},
so looking at Fig. 4 we conclude that traditional RDT should
be valid for frequencies higher than the peak of the spectrum (most energy
containing eddies) and all the way up to the highest frequencies measured.
Having estimated the range of applicability of traditional RDT we now turn
our attention to an extension of RDT to larger scales.

The novelty of Graham (2017) is that he succeeds in calculating the
velocity spectrum and variance without assuming that the length scale of the
longitudinal turbulence *L*_{u∞} is much smaller than the rotor. Graham
develops the theory of Hunt (1973) further and cleverly exploits the
axisymmetry of the mean flow to make the calculations feasible. As a function
of *L*_{u∞}∕*R*, the low-frequency part of the velocity spectrum decays
slowly to the ambient value after a small initial increase. This is in
contrast to Eqs. (2) and (3), which predict a
reduction of the velocity variance equal to *f*^{2} for below rated where
$\partial a/\partial {U}_{\mathrm{\infty}}=\mathrm{0}$. The cause of this difference is that
Graham (2017) does not take into account the interaction of the
vorticity lines with the actuator disk. The velocity field that a vorticity
line induces will be nonzero at the rotor plane, particularly for turbulence
scales larger than the rotor, and the actuator disk will reduce the
fluctuation caused by this vorticity line near the rotor. The same is assumed
in Lee (1989) on the axisymmetric expansion of turbulence because in that
work there is no rotor to interact with after the expansion.

3 Numerical techniques

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The wind turbine rotor is modeled as an actuator disk (AD) using the implementation proposed by Réthoré et al. (2014). Meyer Forsting et al. (2017) use the same model to simulate the induction zone of a 500 kW turbine and validated their predictions with lidar measurements.

The thrust force per unit area applied on the disk is assumed uniform and given by

$$\begin{array}{}\text{(8)}& {\displaystyle \frac{\mathrm{d}{F}_{\mathrm{T}}}{\mathrm{d}A}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\mathit{\rho}{C}_{\mathrm{T}}\left({U}_{\mathrm{\infty}}\right){U}_{\mathrm{\infty}}^{\mathrm{2}},\end{array}$$

where *ρ* is the density of air and *C*_{T}(*U*_{∞}) is the thrust
coefficient as a function of the free-stream velocity *U*_{∞}. The
free-stream velocity is the velocity that would be at the disk location if
the disk was not present. This velocity is not known a priori in an unsteady
turbulent setting and therefore it is convenient to express the loading of
the rotor in terms of the velocity averaged over the rotor disk,
*U*_{disk}. For this reason we define a modified thrust coefficient,
${C}_{\mathrm{T}}^{*}\left({U}_{\text{disk}}\right)$, as a function of the disk-averaged
velocity:

$$\begin{array}{}\text{(9)}& {C}_{\mathrm{T}}^{*}\left({U}_{\text{disk}}\right)={C}_{\mathrm{T}}\left({U}_{\mathrm{\infty}}\right){\displaystyle \frac{{U}_{\mathrm{\infty}}^{\mathrm{2}}}{{U}_{\text{disk}}^{\mathrm{2}}}}\end{array}$$

such that Eq. (8) becomes

$$\begin{array}{}\text{(10)}& {\displaystyle \frac{\mathrm{d}{F}_{\mathrm{T}}}{\mathrm{d}A}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\mathit{\rho}{C}_{\mathrm{T}}^{*}\left({U}_{\text{disk}}\right){U}_{\text{disk}}^{\mathrm{2}}.\end{array}$$

The *C*_{T} curve used in the present AD simulations is obtained from
steady-state simulations of the Siemens turbine presented by
Troldborg and Meyer Forsting (2017) with a rated power of 2.3 MW at
11.5 m s^{−1}. From their simulations, we also extract the relation
between *U*_{∞} and *U*_{disk} and thereby the ${C}_{\mathrm{T}}^{*}$
curve. Figure 2 shows the variation in *C*_{T} and
${C}_{\mathrm{T}}^{*}$ with respect to *U*_{∞} and *U*_{disk},
respectively. As expected ${C}_{\mathrm{T}}^{*}$ reaches greater levels than
*C*_{T} because *U*_{disk} is lower than *U*_{∞}.

The loading and power of the real Siemens turbine is controlled by regulating
the rotational speed and pitch of the blades. The control essentially depends
on the local flow conditions at the rotor disk. Thus, using *U*_{disk}
to determine the load level at each instant in time is a simple method for
mimicking the behavior of the controller.

The computational domain is Cartesian and has dimensions
$({L}_{x},{L}_{y},{L}_{z})=(\mathrm{40}R,\mathrm{25}R,\mathrm{25}R)$, where *L*_{x}, *L*_{y}, and *L*_{z} are the domain
length, width, and height, respectively, and *R*=46.3 m. The rotor is located
in the center of the domain, i.e., $(x,y,z)=(\mathrm{20}R,\mathrm{12.5}R,\mathrm{12.5}R)$ with its
center
axis aligned with the *x* direction (flow direction). The number of grid
points in each direction of the domain is $({N}_{x},{N}_{y},{N}_{z})=(\mathrm{320},\mathrm{128},\mathrm{128})$. In
the region defined by $\mathrm{8.5}R\le x\le \mathrm{21.5}R$, $\mathrm{11.05}R\le y\le \mathrm{13.95}R$, and
$\mathrm{11.05}R\le z\le \mathrm{13.95}R$, the grid cells are cubic with a side length of
*R*∕27.5. The reason for concentrating cells in this part of the domain is to
better resolve the turbulent fluctuations in the region upstream of the
rotor. Outside of this region, the cells are stretched towards the outer
boundaries.

The boundary conditions are as follows: a fixed uniform velocity
is prescribed at the inlet (*x*=0), bottom (*z*=0), and top (*z*=25*R*)
boundaries. Periodic conditions are applied at the sides (*y*=0 and *y*=25*R*)
and a zero-gradient Neumann condition is applied to the velocity at the
outlet (*z*=40*R*).

The turbulent inflow is generated using the model of Mann (1994). The three parameters governing the Mann spectral tensor model are selected according to the findings of Peña et al. (2017) and represent the best fit to the measured conditions at Nørrekær Enge, which is the site of the lidar turbulence measurements. The output of the Mann simulation algorithm (Mann, 1998) is a spatial box of turbulent fluctuations, which are converted to time domain via Taylor's frozen turbulence hypothesis. The dimensions of the generated box are $({L}_{X},{L}_{Y},{L}_{Z})=(\mathrm{512}R,\mathrm{16}R,\mathrm{16}R)$, with a resolution of $\mathrm{\Delta}=R/\mathrm{8}$.

The turbulent fluctuations are introduced into the computational domain in a
cross section located 8.25*R* upstream of the rotor using the technique
described by Troldborg et al. (2014). Note that only one-quarter of the
full cross-flow extent of the box is introduced in the simulations in order
to avoid any influence of periodicity in the turbulence.

The simulations are carried out using the incompressible Navier–Stokes flow
solver EllipSys3D
(Michelsen, 1992, 1994; Sørensen, 1995). EllipSys3D
solves the finite-volume discretized equations in general curvilinear
coordinates utilizing a collocated grid arrangement. The code uses a modified
Rhie–Chow algorithm (Réthoré and Sørensen, 2012; Troldborg et al., 2015) to
avoid pressure velocity decoupling. The simulations are carried out as
detached eddy simulations (DESs) using the *k*−*ω* SST (shear stress
transport) model by Strelets (2001). The convective terms are
discretized using a hybrid scheme, which switches between the Quadratic
Upstream Interpolation for Convective Kinematics (QUICK) scheme
(Leonard, 1979) in the Reynolds-averaged Navier–Stokes (RANS) regions
and a fourth-order central difference scheme in the large eddy simulation
(LES) regions. The switching is determined through a limiter function given
by Strelets (2001). The coupled momentum and pressure-correction
equations are solved using the Semi-Implicit Method for Pressure Linked
Equations (SIMPLE) algorithm (Patankar and Spalding, 1972). The solution is advanced
in time using a second-order iterative time-stepping method using a time step
of Δ*t*=0.08 s. Simulations are carried out at free-stream velocities
of *U*_{∞}=7, 8, 9, 11, and 13 m s^{−1} in order to
cover operations both below and above rated wind speed. In addition, we also
do a simulation at 11.5 m s^{−1} where the amplification of the
low-frequency fluctuations should peak. Simulations are conducted both with
and without a turbine included in the domain such that a one-to-one map in
both space and time can be made of the influence of the rotor induction zone
on the turbulence. The benefit of this approach is that it is insensitive to
the distortion of the inserted turbulence, which is known to occur when the
fluctuations are not in balance with the flow in which they are inserted.

4 Lidar experiment

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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 Peña et al. (2017).

5 Results

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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 (Sathe et al., 2015). 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” (Kristensen, 1999), 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. (1)
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
(Peña et al., 2017) are shown in Fig. 3b together
with a smooth curve through the points, which is later used to compute
$\partial a/\partial {U}_{\mathrm{\infty}}$. 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
Troldborg and Meyer Forsting (2017). For a given *C*_{T}, 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 Troldborg and Meyer Forsting (2017). Figure 3a shows values
of the measured *u*(*x*)∕*U*_{∞} averaged in intervals of *U*_{∞} with fits
of Eq. (1) superimposed. It can be seen that the
intervals below the rated wind speed $\mathrm{6}<{U}_{\mathrm{\infty}}<\mathrm{8}$ and $\mathrm{8}<{U}_{\mathrm{\infty}}<\mathrm{10}$ 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 $f=\mathrm{1}/\mathrm{600}\phantom{\rule{0.125em}{0ex}}\mathrm{Hz}=\mathrm{0.00167}$ Hz. Two examples are shown in Fig. 4 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. 5. Here we calculate the low-frequency spectral amplification as measured by the lidar ${S}_{\text{low}}\left(x\right)/{S}_{\text{low},\mathrm{\infty}}$, where

$$\begin{array}{}\text{(11)}& {S}_{\text{low}}\equiv \underset{\mathrm{1}/\mathrm{600}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\mathrm{Hz}}{\overset{\mathrm{4}/\mathrm{600}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\mathrm{Hz}}{\int}}S\left(f\right)\mathrm{d}f\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}};\end{array}$$

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. (3) for which the induction factor and its slope are derived from
the solid curve in Fig. 3b. 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. 6 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. 7 we summarize the results and compare them with the quasi-steady model. The theoretical prediction is based on the thrust curve shown in Fig. 2. We put a fifth-order spline through the points to be able to do derivatives and then we use the relation

$$\begin{array}{}\text{(12)}& a={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\left(\mathrm{1}-\sqrt{\mathrm{1}-{C}_{\mathrm{T}}}\right)\end{array}$$

to get the induction factor (Hansen, 2015). We are then able
to use Eq. (3) 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 Graham (2017) predicts an increase in
low-frequency *u* fluctuations near the rotor, a combination of the two
models could potentially improve the results.

6 Conclusions

Back to toptop
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 *C*_{T} 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 (Graham, 2017).

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.

Data availability

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Data availability.

The simulation data are available on request from Niels Troldborg (niet@dtu.dk) and the lidar measurement data from Alfredo Peña (aldi@dtu.dk).

Author contributions

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Author contributions.

Special issue statement

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Special issue statement.

This article is part of the special issue “Wind Energy Science Conference 2017”. It is a result of the Wind Energy Science Conference 2017, Lyngby, Copenhagen, Denmark, 26–29 June 2017.

Acknowledgements

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Acknowledgements.

This work is partly funded by the Unified Turbine Testing (UniTTe) project
funded by the Innovation Fund Denmark (1305-00024B).

Edited by: Jens Nørkær Sørensen

Reviewed by: Mike Graham and one anonymous referee

References

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